LMFDB - Embedded newform 354.3.d.a.235.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [354,3,Mod(235,354)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(354, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("354.235");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 354 = 2 \cdot 3 \cdot 59 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	354.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$9.64580135835$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 300 x^{18} - 88 x^{17} + 37952 x^{16} + 27000 x^{15} - 2574056 x^{14} - 3295208 x^{13} + \cdots + 2455573689828 $ x^20 - 300x^18 - 88x^17 + 37952x^16 + 27000x^15 - 2574056x^14 - 3295208x^13 + 98138134x^12 + 195465720x^11 - 1935828112x^10 - 5590976152x^9 + 12661641204x^8 + 55962519336x^7 + 75365256136x^6 + 219349710792x^5 + 570681930321x^4 + 614778274152x^3 + 1559028888924x^2 + 534075251472x + 2455573689828
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{11}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			235.12
Root			$-4.74052 - 1.41421i$ of defining polynomial
Character	$\chi$	$=$	354.235
Dual form			354.3.d.a.235.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -1.73205 q^{3} -2.00000 q^{4} -4.74052 q^{5} -2.44949i q^{6} +12.4076 q^{7} -2.82843i q^{8} +3.00000 q^{9} +O(q^{10})$	\(q+1.41421i q^{2} -1.73205 q^{3} -2.00000 q^{4} -4.74052 q^{5} -2.44949i q^{6} +12.4076 q^{7} -2.82843i q^{8} +3.00000 q^{9} -6.70411i q^{10} +1.77650i q^{11} +3.46410 q^{12} +3.41653i q^{13} +17.5470i q^{14} +8.21082 q^{15} +4.00000 q^{16} -5.78514 q^{17} +4.24264i q^{18} +10.5232 q^{19} +9.48104 q^{20} -21.4906 q^{21} -2.51235 q^{22} +41.0431i q^{23} +4.89898i q^{24} -2.52748 q^{25} -4.83170 q^{26} -5.19615 q^{27} -24.8152 q^{28} -33.6345 q^{29} +11.6119i q^{30} +46.9951i q^{31} +5.65685i q^{32} -3.07699i q^{33} -8.18142i q^{34} -58.8186 q^{35} -6.00000 q^{36} +41.9921i q^{37} +14.8820i q^{38} -5.91760i q^{39} +13.4082i q^{40} +23.8134 q^{41} -30.3923i q^{42} +1.60072i q^{43} -3.55300i q^{44} -14.2216 q^{45} -58.0437 q^{46} +6.39642i q^{47} -6.92820 q^{48} +104.949 q^{49} -3.57439i q^{50} +10.0202 q^{51} -6.83305i q^{52} -2.92135 q^{53} -7.34847i q^{54} -8.42153i q^{55} -35.0940i q^{56} -18.2267 q^{57} -47.5664i q^{58} +(48.0608 - 34.2223i) q^{59} -16.4216 q^{60} +10.8622i q^{61} -66.4611 q^{62} +37.2229 q^{63} -8.00000 q^{64} -16.1961i q^{65} +4.35152 q^{66} -40.9681i q^{67} +11.5703 q^{68} -71.0887i q^{69} -83.1820i q^{70} -30.6899 q^{71} -8.48528i q^{72} +118.342i q^{73} -59.3858 q^{74} +4.37772 q^{75} -21.0464 q^{76} +22.0421i q^{77} +8.36875 q^{78} +62.0124 q^{79} -18.9621 q^{80} +9.00000 q^{81} +33.6773i q^{82} +57.1672i q^{83} +42.9813 q^{84} +27.4246 q^{85} -2.26376 q^{86} +58.2567 q^{87} +5.02470 q^{88} +96.6560i q^{89} -20.1123i q^{90} +42.3910i q^{91} -82.0862i q^{92} -81.3978i q^{93} -9.04590 q^{94} -49.8853 q^{95} -9.79796i q^{96} -73.5083i q^{97} +148.420i q^{98} +5.32950i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 40 q^{4} + 8 q^{7} + 60 q^{9}+O(q^{10}) $ 20 * q - 40 * q^4 + 8 * q^7 + 60 * q^9	$ 20 q - 40 q^{4} + 8 q^{7} + 60 q^{9} - 24 q^{15} + 80 q^{16} + 72 q^{19} + 16 q^{22} + 140 q^{25} + 64 q^{26} - 16 q^{28} + 56 q^{29} - 80 q^{35} - 120 q^{36} - 8 q^{41} + 16 q^{46} + 52 q^{49} + 32 q^{53} - 48 q^{57} + 192 q^{59} + 48 q^{60} - 16 q^{62} + 24 q^{63} - 160 q^{64} + 96 q^{66} - 568 q^{71} - 288 q^{74} - 96 q^{75} - 144 q^{76} + 192 q^{78} + 528 q^{79} + 180 q^{81} + 568 q^{85} - 416 q^{86} - 216 q^{87} - 32 q^{88} - 480 q^{94} - 456 q^{95}+O(q^{100}) $ 20 * q - 40 * q^4 + 8 * q^7 + 60 * q^9 - 24 * q^15 + 80 * q^16 + 72 * q^19 + 16 * q^22 + 140 * q^25 + 64 * q^26 - 16 * q^28 + 56 * q^29 - 80 * q^35 - 120 * q^36 - 8 * q^41 + 16 * q^46 + 52 * q^49 + 32 * q^53 - 48 * q^57 + 192 * q^59 + 48 * q^60 - 16 * q^62 + 24 * q^63 - 160 * q^64 + 96 * q^66 - 568 * q^71 - 288 * q^74 - 96 * q^75 - 144 * q^76 + 192 * q^78 + 528 * q^79 + 180 * q^81 + 568 * q^85 - 416 * q^86 - 216 * q^87 - 32 * q^88 - 480 * q^94 - 456 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/354\mathbb{Z}\right)^\times$.

$n$	$61$	$119$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$			1.41421i			0.707107i
$2$			1.41421i			0.707107i
$3$	−1.73205			−0.577350
$3$	−1.73205			−0.577350
$4$	−2.00000			−0.500000
$5$	−4.74052			−0.948104			−0.474052	−	0.880497i	$-0.657209\pi$
$5$	−4.74052			−0.948104			−0.474052	+	0.880497i	$0.657209\pi$
$6$		−	2.44949i		−	0.408248i
$7$	12.4076			1.77252			0.886259	−	0.463191i	$-0.153296\pi$
$7$	12.4076			1.77252			0.886259	+	0.463191i	$0.153296\pi$
$8$		−	2.82843i		−	0.353553i
$9$	3.00000			0.333333
$10$		−	6.70411i		−	0.670411i
$11$			1.77650i			0.161500i	0.996734	+	0.0807500i	$0.0257315\pi$
$11$			1.77650i			0.161500i	−0.996734	+	0.0807500i	$0.974268\pi$
$12$	3.46410			0.288675
$13$			3.41653i			0.262810i	0.991329	+	0.131405i	$0.0419488\pi$
$13$			3.41653i			0.262810i	−0.991329	+	0.131405i	$0.958051\pi$
$14$			17.5470i			1.25336i
$15$	8.21082			0.547388
$16$	4.00000			0.250000
$17$	−5.78514			−0.340302			−0.170151	−	0.985418i	$-0.554426\pi$
$17$	−5.78514			−0.340302			−0.170151	+	0.985418i	$0.554426\pi$
$18$			4.24264i			0.235702i
$19$	10.5232			0.553852			0.276926	−	0.960891i	$-0.410684\pi$
$19$	10.5232			0.553852			0.276926	+	0.960891i	$0.410684\pi$
$20$	9.48104			0.474052
$21$	−21.4906			−1.02336
$22$	−2.51235			−0.114198
$23$			41.0431i			1.78448i	0.451560	+	0.892241i	$0.350868\pi$
$23$			41.0431i			1.78448i	−0.451560	+	0.892241i	$0.649132\pi$
$24$			4.89898i			0.204124i
$25$	−2.52748			−0.101099
$26$	−4.83170			−0.185835
$27$	−5.19615			−0.192450
$28$	−24.8152			−0.886259
$29$	−33.6345			−1.15981			−0.579905	−	0.814684i	$-0.696911\pi$
$29$	−33.6345			−1.15981			−0.579905	+	0.814684i	$0.696911\pi$
$30$			11.6119i			0.387062i
$31$			46.9951i			1.51597i	0.652272	+	0.757985i	$0.273816\pi$
$31$			46.9951i			1.51597i	−0.652272	+	0.757985i	$0.726184\pi$
$32$			5.65685i			0.176777i
$33$		−	3.07699i		−	0.0932420i
$34$		−	8.18142i		−	0.240630i
$35$	−58.8186			−1.68053
$36$	−6.00000			−0.166667
$37$			41.9921i			1.13492i	0.823400	+	0.567461i	$0.192074\pi$
$37$			41.9921i			1.13492i	−0.823400	+	0.567461i	$0.807926\pi$
$38$			14.8820i			0.391632i
$39$		−	5.91760i		−	0.151733i
$40$			13.4082i			0.335205i
$41$	23.8134			0.580815			0.290408	−	0.956903i	$-0.406209\pi$
$41$	23.8134			0.580815			0.290408	+	0.956903i	$0.406209\pi$
$42$		−	30.3923i		−	0.723627i
$43$			1.60072i			0.0372260i	0.999827	+	0.0186130i	$0.00592504\pi$
$43$			1.60072i			0.0372260i	−0.999827	+	0.0186130i	$0.994075\pi$
$44$		−	3.55300i		−	0.0807500i
$45$	−14.2216			−0.316035
$46$	−58.0437			−1.26182
$47$			6.39642i			0.136094i	0.997682	+	0.0680470i	$0.0216768\pi$
$47$			6.39642i			0.136094i	−0.997682	+	0.0680470i	$0.978323\pi$
$48$	−6.92820			−0.144338
$49$	104.949			2.14182
$50$		−	3.57439i		−	0.0714879i
$51$	10.0202			0.196474
$52$		−	6.83305i		−	0.131405i
$53$	−2.92135			−0.0551198			−0.0275599	−	0.999620i	$-0.508774\pi$
$53$	−2.92135			−0.0551198			−0.0275599	+	0.999620i	$0.508774\pi$
$54$		−	7.34847i		−	0.136083i
$55$		−	8.42153i		−	0.153119i
$56$		−	35.0940i		−	0.626679i
$57$	−18.2267			−0.319766
$58$		−	47.5664i		−	0.820110i
$59$	48.0608	−	34.2223i	0.814589	−	0.580038i
$59$	48.0608	−	34.2223i	0.814589	−	0.580038i
$60$	−16.4216			−0.273694
$61$			10.8622i			0.178069i	0.996029	+	0.0890346i	$0.0283782\pi$
$61$			10.8622i			0.178069i	−0.996029	+	0.0890346i	$0.971622\pi$
$62$	−66.4611			−1.07195
$63$	37.2229			0.590839
$64$	−8.00000			−0.125000
$65$		−	16.1961i		−	0.249171i
$66$	4.35152			0.0659321
$67$		−	40.9681i		−	0.611464i	−0.952118	−	0.305732i	$-0.901099\pi$
$67$		−	40.9681i		−	0.611464i	0.952118	−	0.305732i	$-0.0989011\pi$
$68$	11.5703			0.170151
$69$		−	71.0887i		−	1.03027i
$70$		−	83.1820i		−	1.18831i
$71$	−30.6899			−0.432253			−0.216126	−	0.976365i	$-0.569342\pi$
$71$	−30.6899			−0.432253			−0.216126	+	0.976365i	$0.569342\pi$
$72$		−	8.48528i		−	0.117851i
$73$			118.342i			1.62112i	0.585655	+	0.810561i	$0.300838\pi$
$73$			118.342i			1.62112i	−0.585655	+	0.810561i	$0.699162\pi$
$74$	−59.3858			−0.802511
$75$	4.37772			0.0583696
$76$	−21.0464			−0.276926
$77$			22.0421i			0.286261i
$78$	8.36875			0.107292
$79$	62.0124			0.784968			0.392484	−	0.919759i	$-0.371616\pi$
$79$	62.0124			0.784968			0.392484	+	0.919759i	$0.371616\pi$
$80$	−18.9621			−0.237026
$81$	9.00000			0.111111
$82$			33.6773i			0.410699i
$83$			57.1672i			0.688762i	0.938830	+	0.344381i	$0.111911\pi$
$83$			57.1672i			0.688762i	−0.938830	+	0.344381i	$0.888089\pi$
$84$	42.9813			0.511682
$85$	27.4246			0.322642
$86$	−2.26376			−0.0263227
$87$	58.2567			0.669617
$88$	5.02470			0.0570989
$89$			96.6560i			1.08602i	0.839725	+	0.543011i	$0.182716\pi$
$89$			96.6560i			1.08602i	−0.839725	+	0.543011i	$0.817284\pi$
$90$		−	20.1123i		−	0.223470i
$91$			42.3910i			0.465835i
$92$		−	82.0862i		−	0.892241i
$93$		−	81.3978i		−	0.875246i
$94$	−9.04590			−0.0962330
$95$	−49.8853			−0.525109
$96$		−	9.79796i		−	0.102062i
$97$		−	73.5083i		−	0.757817i	−0.925434	−	0.378909i	$-0.876299\pi$
$97$		−	73.5083i		−	0.757817i	0.925434	−	0.378909i	$-0.123701\pi$
$98$			148.420i			1.51449i
$99$			5.32950i			0.0538333i
$100$	5.05496			0.0505496
$101$			126.711i			1.25456i	0.778794	+	0.627280i	$0.215832\pi$
$101$			126.711i			1.25456i	−0.778794	+	0.627280i	$0.784168\pi$
$102$			14.1706i			0.138928i
$103$		−	140.694i		−	1.36596i	−0.730438	−	0.682979i	$-0.760684\pi$
$103$		−	140.694i		−	1.36596i	0.730438	−	0.682979i	$-0.239316\pi$
$104$	9.66339			0.0929173
$105$	101.877			0.970255
$106$		−	4.13141i		−	0.0389756i
$107$	−76.3548			−0.713597			−0.356798	−	0.934181i	$-0.616132\pi$
$107$	−76.3548			−0.713597			−0.356798	+	0.934181i	$0.616132\pi$
$108$	10.3923			0.0962250
$109$		−	160.557i		−	1.47300i	−0.676439	−	0.736499i	$-0.736478\pi$
$109$		−	160.557i		−	1.47300i	0.676439	−	0.736499i	$-0.263522\pi$
$110$	11.9098			0.108271
$111$		−	72.7325i		−	0.655248i
$112$	49.6305			0.443129
$113$			59.5527i			0.527015i	0.964657	+	0.263508i	$0.0848794\pi$
$113$			59.5527i			0.527015i	−0.964657	+	0.263508i	$0.915121\pi$
$114$		−	25.7764i		−	0.226109i
$115$		−	194.565i		−	1.69187i
$116$	67.2690			0.579905
$117$			10.2496i			0.0876032i
$118$	48.3976	+	67.9682i	0.410149	+	0.576002i
$119$	−71.7798			−0.603192
$120$		−	23.2237i		−	0.193531i
$121$	117.844			0.973918
$122$	−15.3615			−0.125914
$123$	−41.2461			−0.335334
$124$		−	93.9901i		−	0.757985i
$125$	130.495			1.04396
$126$			52.6411i			0.417786i
$127$	−19.7373			−0.155412			−0.0777058	−	0.996976i	$-0.524759\pi$
$127$	−19.7373			−0.155412			−0.0777058	+	0.996976i	$0.524759\pi$
$128$		−	11.3137i		−	0.0883883i
$129$		−	2.77252i		−	0.0214924i
$130$	22.9048			0.176190
$131$			22.2772i			0.170055i	0.996379	+	0.0850276i	$0.0270979\pi$
$131$			22.2772i			0.170055i	−0.996379	+	0.0850276i	$0.972902\pi$
$132$			6.15398i			0.0466210i
$133$	130.568			0.981712
$134$	57.9376			0.432370
$135$	24.6325			0.182463
$136$			16.3628i			0.120315i
$137$	103.894			0.758353			0.379176	−	0.925324i	$-0.376207\pi$
$137$	103.894			0.758353			0.379176	+	0.925324i	$0.376207\pi$
$138$	100.535			0.728512
$139$	−92.2479			−0.663654			−0.331827	−	0.943340i	$-0.607665\pi$
$139$	−92.2479			−0.663654			−0.331827	+	0.943340i	$0.607665\pi$
$140$	117.637			0.840265
$141$		−	11.0789i		−	0.0785739i
$142$		−	43.4021i		−	0.305649i
$143$	−6.06946			−0.0424438
$144$	12.0000			0.0833333
$145$	159.445			1.09962
$146$	−167.361			−1.14631
$147$	−181.777			−1.23658
$148$		−	83.9843i		−	0.567461i
$149$		−	264.023i		−	1.77196i	−0.463719	−	0.885982i	$-0.653485\pi$
$149$		−	264.023i		−	1.77196i	0.463719	−	0.885982i	$-0.346515\pi$
$150$			6.19103i			0.0412736i
$151$			102.361i			0.677886i	0.940807	+	0.338943i	$0.110069\pi$
$151$			102.361i			0.677886i	−0.940807	+	0.338943i	$0.889931\pi$
$152$		−	29.7641i		−	0.195816i
$153$	−17.3554			−0.113434
$154$	−31.1723			−0.202417
$155$		−	222.781i		−	1.43730i
$156$			11.8352i			0.0758666i
$157$			198.566i			1.26475i	0.774662	+	0.632375i	$0.217920\pi$
$157$			198.566i			1.26475i	−0.774662	+	0.632375i	$0.782080\pi$
$158$			87.6988i			0.555056i
$159$	5.05993			0.0318234
$160$		−	26.8164i		−	0.167603i
$161$			509.247i			3.16302i
$162$			12.7279i			0.0785674i
$163$	−143.430			−0.879941			−0.439970	−	0.898012i	$-0.645011\pi$
$163$	−143.430			−0.879941			−0.439970	+	0.898012i	$0.645011\pi$
$164$	−47.6269			−0.290408
$165$			14.5865i			0.0884031i
$166$	−80.8467			−0.487028
$167$	9.22359			0.0552311			0.0276155	−	0.999619i	$-0.491209\pi$
$167$	9.22359			0.0552311			0.0276155	+	0.999619i	$0.491209\pi$
$168$			60.7847i			0.361814i
$169$	157.327			0.930931
$170$			38.7842i			0.228142i
$171$	31.5695			0.184617
$172$		−	3.20143i		−	0.0186130i
$173$			20.9989i			0.121381i	0.998157	+	0.0606903i	$0.0193302\pi$
$173$			20.9989i			0.121381i	−0.998157	+	0.0606903i	$0.980670\pi$
$174$			82.3874i			0.473491i
$175$	−31.3600			−0.179200
$176$			7.10600i			0.0403750i
$177$	−83.2437	+	59.2747i	−0.470303	+	0.334885i
$178$	−136.692			−0.767934
$179$		−	226.280i		−	1.26413i	−0.774914	−	0.632067i	$-0.782207\pi$
$179$		−	226.280i		−	1.26413i	0.774914	−	0.632067i	$-0.217793\pi$
$180$	28.4431			0.158017
$181$	−201.036			−1.11070			−0.555349	−	0.831618i	$-0.687415\pi$
$181$	−201.036			−1.11070			−0.555349	+	0.831618i	$0.687415\pi$
$182$	−59.9499			−0.329395
$183$		−	18.8139i		−	0.102808i
$184$	116.087			0.630910
$185$		−	199.065i		−	1.07602i
$186$	115.114			0.618892
$187$		−	10.2773i		−	0.0549588i
$188$		−	12.7928i		−	0.0680470i
$189$	−64.4719			−0.341121
$190$		−	70.5485i		−	0.371308i
$191$		−	1.70993i		−	0.00895250i	−0.999990	−	0.00447625i	$-0.998575\pi$
$191$		−	1.70993i		−	0.00895250i	0.999990	−	0.00447625i	$-0.00142484\pi$
$192$	13.8564			0.0721688
$193$	49.8788			0.258440			0.129220	−	0.991616i	$-0.458753\pi$
$193$	49.8788			0.258440			0.129220	+	0.991616i	$0.458753\pi$
$194$	103.956			0.535858
$195$			28.0525i			0.143859i
$196$	−209.898			−1.07091
$197$	146.114			0.741694			0.370847	−	0.928694i	$-0.379067\pi$
$197$	146.114			0.741694			0.370847	+	0.928694i	$0.379067\pi$
$198$	−7.53705			−0.0380659
$199$	−372.185			−1.87027			−0.935137	−	0.354286i	$-0.884724\pi$
$199$	−372.185			−1.87027			−0.935137	+	0.354286i	$0.884724\pi$
$200$			7.14879i			0.0357439i
$201$			70.9588i			0.353029i
$202$	−179.196			−0.887108
$203$	−417.324			−2.05578
$204$	−20.0403			−0.0982368
$205$	−112.888			−0.550673
$206$	198.971			0.965878
$207$			123.129i			0.594827i
$208$			13.6661i			0.0657024i
$209$			18.6944i			0.0894470i
$210$			144.075i			0.686074i
$211$		−	155.113i		−	0.735131i	−0.929998	−	0.367566i	$-0.880191\pi$
$211$		−	155.113i		−	0.735131i	0.929998	−	0.367566i	$-0.119809\pi$
$212$	5.84270			0.0275599
$213$	53.1565			0.249561
$214$		−	107.982i		−	0.504589i
$215$		−	7.58823i		−	0.0352941i
$216$			14.6969i			0.0680414i
$217$			583.097i			2.68708i
$218$	227.061			1.04157
$219$		−	204.974i		−	0.935955i
$220$			16.8431i			0.0765594i
$221$		−	19.7651i		−	0.0894347i
$222$	102.859			0.463330
$223$	425.113			1.90634			0.953169	−	0.302439i	$-0.0978010\pi$
$223$	425.113			1.90634			0.953169	+	0.302439i	$0.0978010\pi$
$224$			70.1881i			0.313340i
$225$	−7.58244			−0.0336997
$226$	−84.2202			−0.372656
$227$		−	140.065i		−	0.617028i	−0.951220	−	0.308514i	$-0.900168\pi$
$227$		−	140.065i		−	0.617028i	0.951220	−	0.308514i	$-0.0998317\pi$
$228$	36.4534			0.159883
$229$		−	433.918i		−	1.89484i	−0.319998	−	0.947418i	$-0.603682\pi$
$229$		−	433.918i		−	1.89484i	0.319998	−	0.947418i	$-0.396318\pi$
$230$	275.157			1.19634
$231$		−	38.1781i		−	0.165273i
$232$			95.1328i			0.410055i
$233$		−	271.351i		−	1.16460i	−0.812975	−	0.582298i	$-0.802154\pi$
$233$		−	271.351i		−	1.16460i	0.812975	−	0.582298i	$-0.197846\pi$
$234$	−14.4951			−0.0619448
$235$		−	30.3224i		−	0.129031i
$236$	−96.1215	+	68.4445i	−0.407295	+	0.290019i
$237$	−107.409			−0.453201
$238$		−	101.512i		−	0.426521i
$239$	−213.395			−0.892866			−0.446433	−	0.894817i	$-0.647306\pi$
$239$	−213.395			−0.892866			−0.446433	+	0.894817i	$0.647306\pi$
$240$	32.8433			0.136847
$241$	−53.4060			−0.221602			−0.110801	−	0.993843i	$-0.535342\pi$
$241$	−53.4060			−0.221602			−0.110801	+	0.993843i	$0.535342\pi$
$242$			166.657i			0.688664i
$243$	−15.5885			−0.0641500
$244$		−	21.7244i		−	0.0890346i
$245$	−497.513			−2.03066
$246$		−	58.3308i		−	0.237117i
$247$			35.9527i			0.145558i
$248$	132.922			0.535976
$249$		−	99.0166i		−	0.397657i
$250$			184.547i			0.738189i
$251$	−426.149			−1.69780			−0.848902	−	0.528550i	$-0.822736\pi$
$251$	−426.149			−1.69780			−0.848902	+	0.528550i	$0.822736\pi$
$252$	−74.4457			−0.295420
$253$	−72.9130			−0.288194
$254$		−	27.9127i		−	0.109893i
$255$	−47.5007			−0.186277
$256$	16.0000			0.0625000
$257$	434.787			1.69178			0.845890	−	0.533358i	$-0.179070\pi$
$257$	434.787			1.69178			0.845890	+	0.533358i	$0.179070\pi$
$258$	3.92094			0.0151974
$259$			521.022i			2.01167i
$260$			32.3922i			0.124585i
$261$	−100.904			−0.386604
$262$	−31.5048			−0.120247
$263$	90.5138			0.344159			0.172080	−	0.985083i	$-0.444951\pi$
$263$	90.5138			0.344159			0.172080	+	0.985083i	$0.444951\pi$
$264$	−8.70304			−0.0329660
$265$	13.8487			0.0522593
$266$			184.651i			0.694175i
$267$		−	167.413i		−	0.627015i
$268$			81.9361i			0.305732i
$269$			458.484i			1.70440i	0.523216	+	0.852200i	$0.324732\pi$
$269$			458.484i			1.70440i	−0.523216	+	0.852200i	$0.675268\pi$
$270$			34.8356i			0.129021i
$271$	76.4145			0.281972			0.140986	−	0.990012i	$-0.454973\pi$
$271$	76.4145			0.281972			0.140986	+	0.990012i	$0.454973\pi$
$272$	−23.1406			−0.0850756
$273$		−	73.4233i		−	0.268950i
$274$			146.929i			0.536236i
$275$		−	4.49006i		−	0.0163275i
$276$			142.177i			0.515135i
$277$	−321.427			−1.16039			−0.580193	−	0.814479i	$-0.697023\pi$
$277$	−321.427			−1.16039			−0.580193	+	0.814479i	$0.697023\pi$
$278$		−	130.458i		−	0.469274i
$279$			140.985i			0.505323i
$280$			166.364i			0.594157i
$281$	30.4551			0.108381			0.0541906	−	0.998531i	$-0.482742\pi$
$281$	30.4551			0.108381			0.0541906	+	0.998531i	$0.482742\pi$
$282$	15.6680			0.0555602
$283$			410.231i			1.44958i	0.688971	+	0.724789i	$0.258063\pi$
$283$			410.231i			1.44958i	−0.688971	+	0.724789i	$0.741937\pi$
$284$	61.3799			0.216126
$285$	86.4040			0.303172
$286$		−	8.58351i		−	0.0300123i
$287$	295.468			1.02951
$288$			16.9706i			0.0589256i
$289$	−255.532			−0.884194
$290$			225.489i			0.777550i
$291$			127.320i			0.437526i
$292$		−	236.684i		−	0.810561i
$293$	−196.579			−0.670917			−0.335458	−	0.942055i	$-0.608891\pi$
$293$	−196.579			−0.670917			−0.335458	+	0.942055i	$0.608891\pi$
$294$		−	257.072i		−	0.874393i
$295$	−227.833	+	162.231i	−0.772315	+	0.549937i
$296$	118.772			0.401256
$297$		−	9.23096i		−	0.0310807i
$298$	373.385			1.25297
$299$	−140.225			−0.468979
$300$	−8.75544			−0.0291848
$301$			19.8611i			0.0659837i
$302$	−144.760			−0.479338
$303$		−	219.469i		−	0.724320i
$304$	42.0927			0.138463
$305$		−	51.4925i		−	0.168828i
$306$		−	24.5443i		−	0.0802100i
$307$	362.671			1.18134			0.590669	−	0.806914i	$-0.298864\pi$
$307$	362.671			1.18134			0.590669	+	0.806914i	$0.298864\pi$
$308$		−	44.0843i		−	0.143131i
$309$			243.689i			0.788636i
$310$	315.060			1.01632
$311$	−30.0272			−0.0965506			−0.0482753	−	0.998834i	$-0.515372\pi$
$311$	−30.0272			−0.0965506			−0.0482753	+	0.998834i	$0.515372\pi$
$312$	−16.7375			−0.0536458
$313$		−	145.855i		−	0.465992i	−0.972478	−	0.232996i	$-0.925147\pi$
$313$		−	145.855i		−	0.465992i	0.972478	−	0.232996i	$-0.0748529\pi$
$314$	−280.814			−0.894313
$315$	−176.456			−0.560177
$316$	−124.025			−0.392484
$317$	567.000			1.78864			0.894322	−	0.447424i	$-0.147659\pi$
$317$	567.000			1.78864			0.894322	+	0.447424i	$0.147659\pi$
$318$			7.15582i			0.0225026i
$319$		−	59.7517i		−	0.187309i
$320$	37.9242			0.118513
$321$	132.250			0.411995
$322$	−720.184			−2.23660
$323$	−60.8781			−0.188477
$324$	−18.0000			−0.0555556
$325$		−	8.63520i		−	0.0265698i
$326$		−	202.841i		−	0.622212i
$327$			278.092i			0.850435i
$328$		−	67.3546i		−	0.205349i
$329$			79.3644i			0.241229i
$330$	−20.6285			−0.0625105
$331$	251.778			0.760659			0.380329	−	0.924851i	$-0.375811\pi$
$331$	251.778			0.760659			0.380329	+	0.924851i	$0.375811\pi$
$332$		−	114.334i		−	0.344381i
$333$			125.976i			0.378308i
$334$			13.0441i			0.0390543i
$335$			194.210i			0.579731i
$336$	−85.9625			−0.255841
$337$		−	446.903i		−	1.32612i	−0.748566	−	0.663060i	$-0.769257\pi$
$337$		−	446.903i		−	1.32612i	0.748566	−	0.663060i	$-0.230743\pi$
$338$			222.494i			0.658268i
$339$		−	103.148i		−	0.304272i
$340$	−54.8491			−0.161321
$341$	−83.4867			−0.244829
$342$			44.6461i			0.130544i
$343$	694.194			2.02389
$344$	4.52751			0.0131614
$345$			336.997i			0.976804i
$346$	−29.6969			−0.0858291
$347$		−	619.129i		−	1.78423i	−0.451804	−	0.892117i	$-0.649219\pi$
$347$		−	619.129i		−	1.78423i	0.451804	−	0.892117i	$-0.350781\pi$
$348$	−116.513			−0.334809
$349$			475.410i			1.36221i	0.732187	+	0.681104i	$0.238500\pi$
$349$			475.410i			1.36221i	−0.732187	+	0.681104i	$0.761500\pi$
$350$		−	44.3497i		−	0.126714i
$351$		−	17.7528i		−	0.0505777i
$352$	−10.0494			−0.0285494
$353$		−	584.466i		−	1.65571i	−0.560941	−	0.827856i	$-0.689561\pi$
$353$		−	584.466i		−	1.65571i	0.560941	−	0.827856i	$-0.310439\pi$
$354$	−83.8271	−	117.724i	−0.236800	−	0.332555i
$355$	145.486			0.409820
$356$		−	193.312i		−	0.543011i
$357$	124.326			0.348253
$358$	320.008			0.893878
$359$	458.625			1.27751			0.638753	−	0.769411i	$-0.279450\pi$
$359$	458.625			1.27751			0.638753	+	0.769411i	$0.279450\pi$
$360$			40.2246i			0.111735i
$361$	−250.263			−0.693248
$362$		−	284.308i		−	0.785381i
$363$	−204.112			−0.562292
$364$		−	84.7819i		−	0.232917i
$365$		−	561.002i		−	1.53699i
$366$	26.6069			0.0726964
$367$			320.755i			0.873991i	0.899464	+	0.436996i	$0.143957\pi$
$367$			320.755i			0.873991i	−0.899464	+	0.436996i	$0.856043\pi$
$368$			164.172i			0.446120i
$369$	71.4403			0.193605
$370$	281.520			0.760864
$371$	−36.2470			−0.0977008
$372$			162.796i			0.437623i
$373$	338.130			0.906514			0.453257	−	0.891380i	$-0.350262\pi$
$373$	338.130			0.906514			0.453257	+	0.891380i	$0.350262\pi$
$374$	14.5343			0.0388617
$375$	−226.023			−0.602728
$376$	18.0918			0.0481165
$377$		−	114.913i		−	0.304810i
$378$		−	91.1770i		−	0.241209i
$379$	174.618			0.460735			0.230367	−	0.973104i	$-0.426007\pi$
$379$	174.618			0.460735			0.230367	+	0.973104i	$0.426007\pi$
$380$	99.7707			0.262554
$381$	34.1859			0.0897269
$382$	2.41820			0.00633038
$383$	633.347			1.65365			0.826824	−	0.562461i	$-0.190145\pi$
$383$	633.347			1.65365			0.826824	+	0.562461i	$0.190145\pi$
$384$			19.5959i			0.0510310i
$385$		−	104.491i		−	0.271406i
$386$			70.5393i			0.182744i
$387$			4.80215i			0.0124087i
$388$			147.017i			0.378909i
$389$	4.33499			0.0111439			0.00557196	−	0.999984i	$-0.498226\pi$
$389$	4.33499			0.0111439			0.00557196	+	0.999984i	$0.498226\pi$
$390$	−39.6722			−0.101724
$391$		−	237.440i		−	0.607263i
$392$		−	296.841i		−	0.757247i
$393$		−	38.5853i		−	0.0981814i
$394$			206.636i			0.524457i
$395$	−293.971			−0.744231
$396$		−	10.6590i		−	0.0269167i
$397$			52.4426i			0.132097i	0.997816	+	0.0660486i	$0.0210392\pi$
$397$			52.4426i			0.132097i	−0.997816	+	0.0660486i	$0.978961\pi$
$398$		−	526.348i		−	1.32248i
$399$	−226.150			−0.566791
$400$	−10.1099			−0.0252748
$401$			395.028i			0.985108i	0.870282	+	0.492554i	$0.163937\pi$
$401$			395.028i			0.985108i	−0.870282	+	0.492554i	$0.836063\pi$
$402$	−100.351			−0.249629
$403$	−160.560			−0.398412
$404$		−	253.421i		−	0.627280i
$405$	−42.6647			−0.105345
$406$		−	590.186i		−	1.45366i
$407$	−74.5990			−0.183290
$408$		−	28.3413i		−	0.0694639i
$409$		−	390.371i		−	0.954451i	−0.878781	−	0.477226i	$-0.841642\pi$
$409$		−	390.371i		−	0.954451i	0.878781	−	0.477226i	$-0.158358\pi$
$410$		−	159.648i		−	0.389385i
$411$	−179.950			−0.437835
$412$			281.387i			0.682979i
$413$	596.320	−	424.617i	1.44387	−	1.02813i
$414$	−174.131			−0.420606
$415$		−	271.002i		−	0.653018i
$416$	−19.3268			−0.0464586
$417$	159.778			0.383161
$418$	−26.4379			−0.0632486
$419$		−	11.6755i		−	0.0278651i	−0.999903	−	0.0139325i	$-0.995565\pi$
$419$		−	11.6755i		−	0.0278651i	0.999903	−	0.0139325i	$-0.00443501\pi$
$420$	−203.753			−0.485127
$421$			404.443i			0.960673i	0.877084	+	0.480337i	$0.159486\pi$
$421$			404.443i			0.960673i	−0.877084	+	0.480337i	$0.840514\pi$
$422$	219.362			0.519816
$423$			19.1893i			0.0453647i
$424$			8.26282i			0.0194878i
$425$	14.6218			0.0344043
$426$			75.1747i			0.176466i
$427$			134.774i			0.315631i
$428$	152.710			0.356798
$429$	10.5126			0.0245049
$430$	10.7314			0.0249567
$431$		−	195.172i		−	0.452835i	−0.974030	−	0.226418i	$-0.927299\pi$
$431$		−	195.172i		−	0.452835i	0.974030	−	0.226418i	$-0.0727014\pi$
$432$	−20.7846			−0.0481125
$433$	−327.408			−0.756139			−0.378070	−	0.925777i	$-0.623412\pi$
$433$	−327.408			−0.756139			−0.378070	+	0.925777i	$0.623412\pi$
$434$	−824.624			−1.90005
$435$	−276.167			−0.634867
$436$			321.113i			0.736499i
$437$			431.904i			0.988338i
$438$	289.877			0.661820
$439$	−366.054			−0.833836			−0.416918	−	0.908944i	$-0.636890\pi$
$439$	−366.054			−0.833836			−0.416918	+	0.908944i	$0.636890\pi$
$440$	−23.8197			−0.0541356
$441$	314.847			0.713939
$442$	27.9520			0.0632399
$443$		−	620.656i		−	1.40103i	−0.713638	−	0.700514i	$-0.752954\pi$
$443$		−	620.656i		−	1.40103i	0.713638	−	0.700514i	$-0.247046\pi$
$444$			145.465i			0.327624i
$445$		−	458.200i		−	1.02966i
$446$			601.201i			1.34798i
$447$			457.301i			1.02304i
$448$	−99.2610			−0.221565
$449$	−251.128			−0.559304			−0.279652	−	0.960101i	$-0.590219\pi$
$449$	−251.128			−0.559304			−0.279652	+	0.960101i	$0.590219\pi$
$450$		−	10.7232i		−	0.0238293i
$451$			42.3046i			0.0938017i
$452$		−	119.105i		−	0.263508i
$453$		−	177.294i		−	0.391378i
$454$	198.082			0.436305
$455$		−	200.955i		−	0.441660i
$456$			51.5529i			0.113055i
$457$			524.698i			1.14814i	0.818808	+	0.574068i	$0.194635\pi$
$457$			524.698i			1.14814i	−0.818808	+	0.574068i	$0.805365\pi$
$458$	613.652			1.33985
$459$	30.0605			0.0654912
$460$			389.131i			0.845937i
$461$	455.129			0.987265			0.493632	−	0.869671i	$-0.335669\pi$
$461$	455.129			0.987265			0.493632	+	0.869671i	$0.335669\pi$
$462$	53.9920			0.116866
$463$		−	692.510i		−	1.49570i	−0.663867	−	0.747851i	$-0.731086\pi$
$463$		−	692.510i		−	1.49570i	0.663867	−	0.747851i	$-0.268914\pi$
$464$	−134.538			−0.289953
$465$			385.868i			0.829824i
$466$	383.748			0.823494
$467$		−	418.310i		−	0.895738i	−0.894099	−	0.447869i	$-0.852183\pi$
$467$		−	418.310i		−	0.895738i	0.894099	−	0.447869i	$-0.147817\pi$
$468$		−	20.4992i		−	0.0438016i
$469$		−	508.316i		−	1.08383i
$470$	42.8823			0.0912389
$471$		−	343.926i		−	0.730204i
$472$	−96.7952	−	135.936i	−0.205075	−	0.288001i
$473$	−2.84367			−0.00601199
$474$		−	151.899i		−	0.320462i
$475$	−26.5971			−0.0559939
$476$	143.560			0.301596
$477$	−8.76405			−0.0183733
$478$		−	301.786i		−	0.631352i
$479$	389.623			0.813410			0.406705	−	0.913560i	$-0.366678\pi$
$479$	389.623			0.813410			0.406705	+	0.913560i	$0.366678\pi$
$480$			46.4474i			0.0967654i
$481$	−143.467			−0.298269
$482$		−	75.5275i		−	0.156696i
$483$		−	882.042i		−	1.82617i
$484$	−235.688			−0.486959
$485$			348.467i			0.718489i
$486$		−	22.0454i		−	0.0453609i
$487$	−667.023			−1.36966			−0.684829	−	0.728704i	$-0.740123\pi$
$487$	−667.023			−1.36966			−0.684829	+	0.728704i	$0.740123\pi$
$488$	30.7230			0.0629569
$489$	248.429			0.508034
$490$		−	703.589i		−	1.43590i
$491$	−276.799			−0.563746			−0.281873	−	0.959452i	$-0.590956\pi$
$491$	−276.799			−0.563746			−0.281873	+	0.959452i	$0.590956\pi$
$492$	82.4922			0.167667
$493$	194.580			0.394686
$494$	−50.8448			−0.102925
$495$		−	25.2646i		−	0.0510396i
$496$			187.980i			0.378992i
$497$	−380.789			−0.766175
$498$	140.031			0.281186
$499$	573.362			1.14902			0.574511	−	0.818497i	$-0.305192\pi$
$499$	573.362			1.14902			0.574511	+	0.818497i	$0.305192\pi$
$500$	−260.989			−0.521978
$501$	−15.9757			−0.0318877
$502$		−	602.666i		−	1.20053i
$503$			31.1451i			0.0619186i	0.999521	+	0.0309593i	$0.00985623\pi$
$503$			31.1451i			0.0619186i	−0.999521	+	0.0309593i	$0.990144\pi$
$504$		−	105.282i		−	0.208893i
$505$		−	600.674i		−	1.18945i
$506$		−	103.115i		−	0.203784i
$507$	−272.499			−0.537473
$508$	39.4745			0.0777058
$509$			392.968i			0.772038i	0.922491	+	0.386019i	$0.126150\pi$
$509$			392.968i			0.772038i	−0.922491	+	0.386019i	$0.873850\pi$
$510$		−	67.1762i		−	0.131718i
$511$			1468.34i			2.87347i
$512$			22.6274i			0.0441942i
$513$	−54.6801			−0.106589
$514$			614.882i			1.19627i
$515$			666.961i			1.29507i
$516$			5.54504i			0.0107462i
$517$	−11.3632			−0.0219792
$518$	−736.837			−1.42247
$519$		−	36.3711i		−	0.0700792i
$520$	−45.8095			−0.0880952
$521$	688.699			1.32188			0.660939	−	0.750439i	$-0.270158\pi$
$521$	688.699			1.32188			0.660939	+	0.750439i	$0.270158\pi$
$522$		−	142.699i		−	0.273370i
$523$	511.550			0.978106			0.489053	−	0.872254i	$-0.337342\pi$
$523$	511.550			0.978106			0.489053	+	0.872254i	$0.337342\pi$
$524$		−	44.5545i		−	0.0850276i
$525$	54.3171			0.103461
$526$			128.006i			0.243357i
$527$		−	271.873i		−	0.515888i
$528$		−	12.3080i		−	0.0233105i
$529$	−1155.53			−2.18437
$530$			19.5850i			0.0369529i
$531$	144.182	−	102.667i	0.271530	−	0.193346i
$532$	−261.135			−0.490856
$533$			81.3592i			0.152644i
$534$	236.758			0.443367
$535$	361.962			0.676564
$536$	−115.875			−0.216185
$537$			391.928i			0.729848i
$538$	−648.394			−1.20519
$539$			186.442i			0.345903i
$540$	−49.2649			−0.0912313
$541$			17.5878i			0.0325098i	0.999868	+	0.0162549i	$0.00517432\pi$
$541$			17.5878i			0.0325098i	−0.999868	+	0.0162549i	$0.994826\pi$
$542$			108.066i			0.199384i
$543$	348.205			0.641261
$544$		−	32.7257i		−	0.0601575i
$545$			761.122i			1.39655i
$546$	103.836			0.190176
$547$	−397.445			−0.726590			−0.363295	−	0.931674i	$-0.618348\pi$
$547$	−397.445			−0.726590			−0.363295	+	0.931674i	$0.618348\pi$
$548$	−207.789			−0.379176
$549$			32.5866i			0.0593564i
$550$	6.34991			0.0115453
$551$	−353.942			−0.642363
$552$	−201.069			−0.364256
$553$	769.427			1.39137
$554$		−	454.567i		−	0.820517i
$555$			344.790i			0.621243i
$556$	184.496			0.331827
$557$	427.468			0.767448			0.383724	−	0.923448i	$-0.374641\pi$
$557$	427.468			0.767448			0.383724	+	0.923448i	$0.374641\pi$
$558$	−199.383			−0.357318
$559$	−5.46889			−0.00978334
$560$	−235.274			−0.420133
$561$			17.8008i			0.0317305i
$562$			43.0700i			0.0766370i
$563$			712.655i			1.26582i	0.774227	+	0.632908i	$0.218139\pi$
$563$			712.655i			1.26582i	−0.774227	+	0.632908i	$0.781861\pi$
$564$			22.1578i			0.0392870i
$565$		−	282.311i		−	0.499665i
$566$	−580.154			−1.02501
$567$	111.669			0.196946
$568$			86.8043i			0.152824i
$569$		−	104.521i		−	0.183693i	−0.995773	−	0.0918465i	$-0.970723\pi$
$569$		−	104.521i		−	0.183693i	0.995773	−	0.0918465i	$-0.0292769\pi$
$570$			122.194i			0.214375i
$571$			691.173i			1.21046i	0.796050	+	0.605231i	$0.206919\pi$
$571$			691.173i			1.21046i	−0.796050	+	0.605231i	$0.793081\pi$
$572$	12.1389			0.0212219
$573$			2.96168i			0.00516873i
$574$			417.855i			0.727970i
$575$		−	103.735i		−	0.180410i
$576$	−24.0000			−0.0416667
$577$	133.428			0.231244			0.115622	−	0.993293i	$-0.463114\pi$
$577$	133.428			0.231244			0.115622	+	0.993293i	$0.463114\pi$
$578$		−	361.377i		−	0.625220i
$579$	−86.3927			−0.149210
$580$	−318.890			−0.549811
$581$			709.309i			1.22084i
$582$	−180.058			−0.309378
$583$		−	5.18978i		−	0.00890185i
$584$	334.721			0.573153
$585$		−	48.5883i		−	0.0830570i
$586$		−	278.004i		−	0.474410i
$587$			244.215i			0.416039i	0.978125	+	0.208020i	$0.0667018\pi$
$587$			244.215i			0.416039i	−0.978125	+	0.208020i	$0.933298\pi$
$588$	363.554			0.618289
$589$			494.538i			0.839623i
$590$	−229.430	−	322.204i	−0.388864	−	0.546109i
$591$	−253.077			−0.428217
$592$			167.969i			0.283731i
$593$	474.801			0.800677			0.400338	−	0.916367i	$-0.368893\pi$
$593$	474.801			0.800677			0.400338	+	0.916367i	$0.368893\pi$
$594$	13.0546			0.0219774
$595$	340.273			0.571888
$596$			528.045i			0.885982i
$597$	644.643			1.07980
$598$		−	198.308i		−	0.331618i
$599$	−314.784			−0.525515			−0.262758	−	0.964862i	$-0.584632\pi$
$599$	−314.784			−0.525515			−0.262758	+	0.964862i	$0.584632\pi$
$600$		−	12.3821i		−	0.0206368i
$601$			353.585i			0.588327i	0.955755	+	0.294164i	$0.0950410\pi$
$601$			353.585i			0.588327i	−0.955755	+	0.294164i	$0.904959\pi$
$602$	−28.0878			−0.0466575
$603$		−	122.904i		−	0.203821i
$604$		−	204.722i		−	0.338943i
$605$	−558.642			−0.923375
$606$	310.376			0.512172
$607$	495.558			0.816406			0.408203	−	0.912891i	$-0.366156\pi$
$607$	495.558			0.816406			0.408203	+	0.912891i	$0.366156\pi$
$608$			59.5281i			0.0979081i
$609$	722.827			1.18691
$610$	72.8215			0.119379
$611$	−21.8535			−0.0357668
$612$	34.7108			0.0567170
$613$		−	81.5110i		−	0.132971i	−0.997787	−	0.0664853i	$-0.978821\pi$
$613$		−	81.5110i		−	0.132971i	0.997787	−	0.0664853i	$-0.0211785\pi$
$614$			512.894i			0.835332i
$615$	195.528			0.317931
$616$	62.3446			0.101209
$617$	972.066			1.57547			0.787735	−	0.616014i	$-0.211253\pi$
$617$	972.066			1.57547			0.787735	+	0.616014i	$0.211253\pi$
$618$	−344.628			−0.557650
$619$	−245.762			−0.397031			−0.198516	−	0.980098i	$-0.563612\pi$
$619$	−245.762			−0.397031			−0.198516	+	0.980098i	$0.563612\pi$
$620$			445.562i			0.718648i
$621$		−	213.266i		−	0.343424i
$622$		−	42.4649i		−	0.0682716i
$623$			1199.27i			1.92499i
$624$		−	23.6704i		−	0.0379333i
$625$	−555.425			−0.888680
$626$	206.271			0.329506
$627$		−	32.3797i		−	0.0516423i
$628$		−	397.131i		−	0.632375i
$629$		−	242.930i		−	0.386217i
$630$		−	249.546i		−	0.396105i
$631$	−502.846			−0.796904			−0.398452	−	0.917189i	$-0.630452\pi$
$631$	−502.846			−0.796904			−0.398452	+	0.917189i	$0.630452\pi$
$632$		−	175.398i		−	0.277528i
$633$			268.663i			0.424428i
$634$			801.859i			1.26476i
$635$	93.5649			0.147346
$636$	−10.1199			−0.0159117
$637$			358.561i			0.562890i
$638$	84.5017			0.132448
$639$	−92.0698			−0.144084
$640$			53.6329i			0.0838013i
$641$	−1162.01			−1.81281			−0.906403	−	0.422415i	$-0.861183\pi$
$641$	−1162.01			−1.81281			−0.906403	+	0.422415i	$0.861183\pi$
$642$			187.030i			0.291325i
$643$	−40.2407			−0.0625827			−0.0312913	−	0.999510i	$-0.509962\pi$
$643$	−40.2407			−0.0625827			−0.0312913	+	0.999510i	$0.509962\pi$
$644$		−	1018.49i		−	1.58151i
$645$			13.1432i			0.0203770i
$646$		−	86.0946i		−	0.133273i
$647$	−812.933			−1.25646			−0.628232	−	0.778026i	$-0.716221\pi$
$647$	−812.933			−1.25646			−0.628232	+	0.778026i	$0.716221\pi$
$648$		−	25.4558i		−	0.0392837i
$649$	60.7958	+	85.3799i	0.0936762	+	0.131556i
$650$	12.2120			0.0187877
$651$		−	1009.95i		−	1.55139i
$652$	286.861			0.439970
$653$	534.281			0.818194			0.409097	−	0.912491i	$-0.365844\pi$
$653$	534.281			0.818194			0.409097	+	0.912491i	$0.365844\pi$
$654$	−393.282			−0.601349
$655$		−	105.606i		−	0.161230i
$656$	95.2537			0.145204
$657$			355.026i			0.540374i
$658$	−112.238			−0.170575
$659$		−	496.424i		−	0.753299i	−0.926356	−	0.376650i	$-0.877076\pi$
$659$		−	496.424i		−	0.753299i	0.926356	−	0.376650i	$-0.122924\pi$
$660$		−	29.1730i		−	0.0442016i
$661$	−155.910			−0.235870			−0.117935	−	0.993021i	$-0.537627\pi$
$661$	−155.910			−0.235870			−0.117935	+	0.993021i	$0.537627\pi$
$662$			356.068i			0.537867i
$663$			34.2341i			0.0516352i
$664$	161.693			0.243514
$665$	−618.958			−0.930765
$666$	−178.158			−0.267504
$667$		−	1380.46i		−	2.06966i
$668$	−18.4472			−0.0276155
$669$	−736.318			−1.10062
$670$	−274.654			−0.409932
$671$	−19.2967			−0.0287582
$672$		−	121.569i		−	0.180907i
$673$		−	566.257i		−	0.841393i	−0.907202	−	0.420696i	$-0.861786\pi$
$673$		−	566.257i		−	0.841393i	0.907202	−	0.420696i	$-0.138214\pi$
$674$	632.016			0.937709
$675$	13.1332			0.0194565
$676$	−314.655			−0.465466
$677$	967.786			1.42952			0.714761	−	0.699369i	$-0.246536\pi$
$677$	967.786			1.42952			0.714761	+	0.699369i	$0.246536\pi$
$678$	145.874			0.215153
$679$		−	912.062i		−	1.34324i
$680$		−	77.5684i		−	0.114071i
$681$			242.600i			0.356241i
$682$		−	118.068i		−	0.173120i
$683$			156.240i			0.228755i	0.993437	+	0.114377i	$0.0364873\pi$
$683$			156.240i			0.228755i	−0.993437	+	0.114377i	$0.963513\pi$
$684$	−63.1391			−0.0923086
$685$	−492.513			−0.718997
$686$			981.739i			1.43111i
$687$			751.567i			1.09398i
$688$			6.40287i			0.00930649i
$689$		−	9.98087i		−	0.0144860i
$690$	−476.586			−0.690705
$691$		−	387.135i		−	0.560254i	−0.959963	−	0.280127i	$-0.909623\pi$
$691$		−	387.135i		−	0.560254i	0.959963	−	0.280127i	$-0.0903766\pi$
$692$		−	41.9977i		−	0.0606903i
$693$			66.1264i			0.0954205i
$694$	875.581			1.26164
$695$	437.303			0.629213
$696$		−	164.775i		−	0.236745i
$697$	−137.764			−0.197653
$698$	−672.332			−0.963226
$699$			469.994i			0.672380i
$700$	62.7200			0.0896000
$701$		−	376.867i		−	0.537613i	−0.963194	−	0.268806i	$-0.913371\pi$
$701$		−	376.867i		−	0.537613i	0.963194	−	0.268806i	$-0.0866292\pi$
$702$	25.1062			0.0357639
$703$			441.891i			0.628579i
$704$		−	14.2120i		−	0.0201875i
$705$			52.5199i			0.0744962i
$706$	826.560			1.17076
$707$			1572.18i			2.22373i
$708$	166.487	−	118.549i	0.235152	−	0.167443i
$709$	−728.582			−1.02762			−0.513809	−	0.857904i	$-0.671766\pi$
$709$	−728.582			−1.02762			−0.513809	+	0.857904i	$0.671766\pi$
$710$			205.749i			0.289787i
$711$	186.037			0.261656
$712$	273.384			0.383967
$713$	−1928.82			−2.70522
$714$			175.824i			0.246252i
$715$	28.7724			0.0402411
$716$			452.560i			0.632067i
$717$	369.611			0.515497
$718$			648.594i			0.903334i
$719$			456.694i			0.635180i	0.948228	+	0.317590i	$0.102874\pi$
$719$			456.694i			0.635180i	−0.948228	+	0.317590i	$0.897126\pi$
$720$	−56.8862			−0.0790087
$721$		−	1745.67i		−	2.42118i
$722$		−	353.925i		−	0.490201i
$723$	92.5019			0.127942
$724$	402.072			0.555349
$725$	85.0105			0.117256
$726$		−	288.658i		−	0.397600i
$727$	−669.263			−0.920582			−0.460291	−	0.887768i	$-0.652255\pi$
$727$	−669.263			−0.920582			−0.460291	+	0.887768i	$0.652255\pi$
$728$	119.900			0.164697
$729$	27.0000			0.0370370
$730$	793.376			1.08682
$731$		−	9.26037i		−	0.0126681i
$732$			37.6278i			0.0514041i
$733$	−570.834			−0.778764			−0.389382	−	0.921076i	$-0.627311\pi$
$733$	−570.834			−0.778764			−0.389382	+	0.921076i	$0.627311\pi$
$734$	−453.616			−0.618005
$735$	861.718			1.17240
$736$	−232.175			−0.315455
$737$	72.7798			0.0987514
$738$			101.032i			0.136900i
$739$			1360.08i			1.84043i	0.391409	+	0.920217i	$0.371988\pi$
$739$			1360.08i			1.84043i	−0.391409	+	0.920217i	$0.628012\pi$
$740$			398.129i			0.538012i
$741$		−	62.2720i		−	0.0840377i
$742$		−	51.2610i		−	0.0690849i
$743$	1175.48			1.58207			0.791035	−	0.611771i	$-0.209542\pi$
$743$	1175.48			1.58207			0.791035	+	0.611771i	$0.209542\pi$
$744$	−230.228			−0.309446
$745$			1251.60i			1.68001i
$746$			478.187i			0.641002i
$747$			171.502i			0.229587i
$748$			20.5546i			0.0274794i
$749$	−947.382			−1.26486
$750$		−	319.645i		−	0.426193i
$751$		−	734.037i		−	0.977413i	−0.872448	−	0.488706i	$-0.837469\pi$
$751$		−	734.037i		−	0.977413i	0.872448	−	0.488706i	$-0.162531\pi$
$752$			25.5857i			0.0340235i
$753$	738.112			0.980228
$754$	162.512			0.215533
$755$		−	485.243i		−	0.642706i
$756$	128.944			0.170561
$757$	−946.059			−1.24975			−0.624874	−	0.780726i	$-0.714850\pi$
$757$	−946.059			−1.24975			−0.624874	+	0.780726i	$0.714850\pi$
$758$			246.948i			0.325789i
$759$	126.289			0.166389
$760$			141.097i			0.185654i
$761$	−1135.31			−1.49187			−0.745933	−	0.666021i	$-0.767996\pi$
$761$	−1135.31			−1.49187			−0.745933	+	0.666021i	$0.767996\pi$
$762$			48.3462i			0.0634465i
$763$		−	1992.13i		−	2.61091i
$764$			3.41986i			0.00447625i
$765$	82.2737			0.107547
$766$			895.688i			1.16931i
$767$	116.921	+	164.201i	0.152440	+	0.214082i
$768$	−27.7128			−0.0360844
$769$		−	764.684i		−	0.994387i	−0.867640	−	0.497193i	$-0.834364\pi$
$769$		−	764.684i		−	0.994387i	0.867640	−	0.497193i	$-0.165636\pi$
$770$	147.773			0.191913
$771$	−753.074			−0.976749
$772$	−99.7577			−0.129220
$773$			685.199i			0.886416i	0.896419	+	0.443208i	$0.146160\pi$
$773$			685.199i			0.886416i	−0.896419	+	0.443208i	$0.853840\pi$
$774$	−6.79127			−0.00877424
$775$		−	118.779i		−	0.153263i
$776$	−207.913			−0.267929
$777$		−	902.437i		−	1.16144i
$778$			6.13060i			0.00787994i
$779$	250.593			0.321686
$780$		−	56.1050i		−	0.0719294i
$781$		−	54.5207i		−	0.0698088i
$782$	335.791			0.429400
$783$	174.770			0.223206
$784$	419.796			0.535454
$785$		−	941.305i		−	1.19911i
$786$	54.5679			0.0694248
$787$	−491.981			−0.625135			−0.312568	−	0.949896i	$-0.601189\pi$
$787$	−491.981			−0.625135			−0.312568	+	0.949896i	$0.601189\pi$
$788$	−292.228			−0.370847
$789$	−156.775			−0.198700
$790$		−	415.738i		−	0.526251i
$791$			738.907i			0.934143i
$792$	15.0741			0.0190330
$793$	−37.1110			−0.0467983
$794$	−74.1650			−0.0934068
$795$	−23.9867			−0.0301719
$796$	744.369			0.935137
$797$			1102.47i			1.38328i	0.722244	+	0.691639i	$0.243111\pi$
$797$			1102.47i			1.38328i	−0.722244	+	0.691639i	$0.756889\pi$
$798$		−	319.824i		−	0.400782i
$799$		−	37.0042i		−	0.0463131i
$800$		−	14.2976i		−	0.0178720i
$801$			289.968i			0.362007i
$802$	−558.654			−0.696577
$803$	−210.234			−0.261811
$804$		−	141.918i		−	0.176514i
$805$		−	2414.09i		−	2.99888i
$806$		−	227.066i		−	0.281720i
$807$		−	794.117i		−	0.984036i
$808$	358.392			0.443554
$809$		−	906.426i		−	1.12043i	−0.828348	−	0.560214i	$-0.810719\pi$
$809$		−	906.426i		−	1.12043i	0.828348	−	0.560214i	$-0.189281\pi$
$810$		−	60.3370i		−	0.0744901i
$811$			1164.85i			1.43631i	0.695882	+	0.718157i	$0.255014\pi$
$811$			1164.85i			1.43631i	−0.695882	+	0.718157i	$0.744986\pi$
$812$	834.649			1.02789
$813$	−132.354			−0.162797
$814$		−	105.499i		−	0.129606i
$815$	679.934			0.834275
$816$	40.0806			0.0491184
$817$			16.8446i			0.0206177i
$818$	552.067			0.674899
$819$			127.173i			0.155278i
$820$	225.776			0.275337
$821$		−	1040.13i		−	1.26690i	−0.773783	−	0.633451i	$-0.781638\pi$
$821$		−	1040.13i		−	1.26690i	0.773783	−	0.633451i	$-0.218362\pi$
$822$		−	254.488i		−	0.309596i
$823$			230.161i			0.279660i	0.990175	+	0.139830i	$0.0446557\pi$
$823$			230.161i			0.279660i	−0.990175	+	0.139830i	$0.955344\pi$
$824$	−397.942			−0.482939
$825$			7.77702i			0.00942669i
$826$	600.499	+	843.323i	0.726996	+	1.02097i
$827$	−1521.53			−1.83982			−0.919908	−	0.392134i	$-0.871737\pi$
$827$	−1521.53			−1.83982			−0.919908	+	0.392134i	$0.871737\pi$
$828$		−	246.258i		−	0.297414i
$829$	−189.919			−0.229094			−0.114547	−	0.993418i	$-0.536542\pi$
$829$	−189.919			−0.229094			−0.114547	+	0.993418i	$0.536542\pi$
$830$	383.255			0.461753
$831$	556.728			0.669950
$832$		−	27.3322i		−	0.0328512i
$833$	−607.145			−0.728865
$834$			225.960i			0.270936i
$835$	−43.7246			−0.0523648
$836$		−	37.3889i		−	0.0447235i
$837$		−	244.194i		−	0.291749i
$838$	16.5116			0.0197036
$839$		−	251.046i		−	0.299220i	−0.988745	−	0.149610i	$-0.952198\pi$
$839$		−	251.046i		−	0.299220i	0.988745	−	0.149610i	$-0.0478018\pi$
$840$		−	288.151i		−	0.343037i
$841$	290.281			0.345161
$842$	−571.969			−0.679299
$843$	−52.7498			−0.0625739
$844$			310.225i			0.367566i
$845$	−745.813			−0.882619
$846$	−27.1377			−0.0320777
$847$	1462.16			1.72629
$848$	−11.6854			−0.0137799
$849$		−	710.540i		−	0.836914i
$850$			20.6784i			0.0243275i
$851$	−1723.49			−2.02525
$852$	−106.313			−0.124781
$853$	1323.84			1.55198			0.775992	−	0.630743i	$-0.217250\pi$
$853$	1323.84			1.55198			0.775992	+	0.630743i	$0.217250\pi$
$854$	−190.600			−0.223185
$855$	−149.656			−0.175036
$856$			215.964i			0.252295i
$857$			279.071i			0.325637i	0.986656	+	0.162819i	$0.0520585\pi$
$857$			279.071i			0.325637i	−0.986656	+	0.162819i	$0.947941\pi$
$858$			14.8671i			0.0173276i
$859$		−	991.453i		−	1.15419i	−0.816676	−	0.577097i	$-0.804185\pi$
$859$		−	991.453i		−	1.15419i	0.816676	−	0.577097i	$-0.195815\pi$
$860$			15.1765i			0.0176470i
$861$	−511.766			−0.594385
$862$	276.015			0.320203
$863$			1164.79i			1.34970i	0.737954	+	0.674851i	$0.235792\pi$
$863$			1164.79i			1.34970i	−0.737954	+	0.674851i	$0.764208\pi$
$864$		−	29.3939i		−	0.0340207i
$865$		−	99.5455i		−	0.115081i
$866$		−	463.025i		−	0.534671i
$867$	442.595			0.510490
$868$		−	1166.19i		−	1.34354i
$869$			110.165i			0.126772i
$870$		−	390.559i		−	0.448918i
$871$	139.968			0.160699
$872$	−454.123			−0.520783
$873$		−	220.525i		−	0.252606i
$874$	−610.804			−0.698861
$875$	1619.13			1.85043
$876$			409.948i			0.467977i
$877$	1319.14			1.50415			0.752074	−	0.659079i	$-0.229054\pi$
$877$	1319.14			1.50415			0.752074	+	0.659079i	$0.229054\pi$
$878$		−	517.679i		−	0.589611i
$879$	340.484			0.387354
$880$		−	33.6861i		−	0.0382797i
$881$		−	349.356i		−	0.396544i	−0.980147	−	0.198272i	$-0.936467\pi$
$881$		−	349.356i		−	0.396544i	0.980147	−	0.198272i	$-0.0635330\pi$
$882$			445.261i			0.504831i
$883$	365.342			0.413751			0.206875	−	0.978367i	$-0.433671\pi$
$883$	365.342			0.413751			0.206875	+	0.978367i	$0.433671\pi$
$884$			39.5301i			0.0447174i
$885$	394.618	−	280.993i	0.445896	−	0.317506i
$886$	877.740			0.990677
$887$		−	1111.94i		−	1.25360i	−0.779181	−	0.626799i	$-0.784365\pi$
$887$		−	1111.94i		−	1.25360i	0.779181	−	0.626799i	$-0.215635\pi$
$888$	−205.719			−0.231665
$889$	−244.892			−0.275470
$890$	647.992			0.728081
$891$			15.9885i			0.0179444i
$892$	−850.226			−0.953169
$893$			67.3107i			0.0753759i
$894$	−646.721			−0.723402
$895$			1072.68i			1.19853i
$896$		−	140.376i		−	0.156670i
$897$	242.876			0.270765
$898$		−	355.148i		−	0.395488i
$899$		−	1580.66i		−	1.75824i
$900$	15.1649			0.0168499
$901$	16.9004			0.0187574
$902$	−59.8277			−0.0663278
$903$		−	34.4004i		−	0.0380957i
$904$	168.440			0.186328
$905$	953.016			1.05306
$906$	250.732			0.276746
$907$	−930.561			−1.02598			−0.512988	−	0.858396i	$-0.671462\pi$
$907$	−930.561			−1.02598			−0.512988	+	0.858396i	$0.671462\pi$
$908$			280.131i			0.308514i
$909$			380.132i			0.418187i
$910$	284.193			0.312301
$911$	−144.909			−0.159066			−0.0795331	−	0.996832i	$-0.525343\pi$
$911$	−144.909			−0.159066			−0.0795331	+	0.996832i	$0.525343\pi$
$912$	−72.9067			−0.0799416
$913$	−101.558			−0.111235
$914$	−742.035			−0.811855
$915$			89.1877i			0.0974729i
$916$			867.835i			0.947418i
$917$			276.408i			0.301426i
$918$			42.5119i			0.0463093i
$919$			1480.93i			1.61146i	0.592284	+	0.805729i	$0.298226\pi$
$919$			1480.93i			1.61146i	−0.592284	+	0.805729i	$0.701774\pi$
$920$	−550.314			−0.598168
$921$	−628.164			−0.682046
$922$			643.650i			0.698102i
$923$		−	104.853i		−	0.113600i
$924$			76.3562i			0.0826366i
$925$		−	106.134i		−	0.114740i
$926$	979.357			1.05762
$927$		−	422.081i		−	0.455319i
$928$		−	190.266i		−	0.205028i
$929$			252.022i			0.271283i	0.990758	+	0.135642i	$0.0433096\pi$
$929$			252.022i			0.271283i	−0.990758	+	0.135642i	$0.956690\pi$
$930$	−545.700			−0.586774
$931$	1104.40			1.18625
$932$			542.702i			0.582298i
$933$	52.0087			0.0557435
$934$	591.579			0.633382
$935$			48.7197i			0.0521066i
$936$	28.9902			0.0309724
$937$			431.511i			0.460524i	0.973129	+	0.230262i	$0.0739584\pi$
$937$			431.511i			0.460524i	−0.973129	+	0.230262i	$0.926042\pi$
$938$	718.868			0.766383
$939$			252.629i			0.269041i
$940$			60.6447i			0.0645156i
$941$		−	406.044i		−	0.431503i	−0.976448	−	0.215751i	$-0.930780\pi$
$941$		−	406.044i		−	0.431503i	0.976448	−	0.215751i	$-0.0692201\pi$
$942$	486.385			0.516332
$943$			977.377i			1.03645i
$944$	192.243	−	136.889i	0.203647	−	0.145010i
$945$	305.630			0.323418
$946$		−	4.02156i		−	0.00425112i
$947$	−340.427			−0.359480			−0.179740	−	0.983714i	$-0.557526\pi$
$947$	−340.427			−0.359480			−0.179740	+	0.983714i	$0.557526\pi$
$948$	214.817			0.226601
$949$	−404.318			−0.426046
$950$		−	37.6140i		−	0.0395937i
$951$	−982.073			−1.03267
$952$			203.024i			0.213260i
$953$	1104.08			1.15853			0.579264	−	0.815140i	$-0.303340\pi$
$953$	1104.08			1.15853			0.579264	+	0.815140i	$0.303340\pi$
$954$		−	12.3942i		−	0.0129919i
$955$			8.10595i			0.00848790i
$956$	426.790			0.446433
$957$			103.493i			0.108143i
$958$			551.011i			0.575168i
$959$	1289.08			1.34419
$960$	−65.6866			−0.0684235
$961$	−1247.54			−1.29816
$962$		−	202.893i		−	0.210908i
$963$	−229.065			−0.237866
$964$	106.812			0.110801
$965$	−236.452			−0.245028
$966$	1247.40			1.29130
$967$		−	1615.17i		−	1.67029i	−0.550033	−	0.835143i	$-0.685385\pi$
$967$		−	1615.17i		−	1.67029i	0.550033	−	0.835143i	$-0.314615\pi$
$968$		−	333.313i		−	0.344332i
$969$	105.444			0.108817
$970$	−492.807			−0.508049
$971$	−48.0398			−0.0494745			−0.0247373	−	0.999694i	$-0.507875\pi$
$971$	−48.0398			−0.0494745			−0.0247373	+	0.999694i	$0.507875\pi$
$972$	31.1769			0.0320750
$973$	−1144.58			−1.17634
$974$		−	943.314i		−	0.968495i
$975$			14.9566i			0.0153401i
$976$			43.4489i			0.0445173i
$977$			238.680i			0.244299i	0.992512	+	0.122149i	$0.0389787\pi$
$977$			238.680i			0.244299i	−0.992512	+	0.122149i	$0.961021\pi$
$978$			351.331i			0.359234i
$979$	−171.709			−0.175393
$980$	995.026			1.01533
$981$		−	481.670i		−	0.490999i
$982$		−	391.453i		−	0.398629i
$983$		−	1918.33i		−	1.95150i	−0.218884	−	0.975751i	$-0.570242\pi$
$983$		−	1918.33i		−	1.95150i	0.218884	−	0.975751i	$-0.429758\pi$
$984$			116.662i			0.118558i
$985$	−692.655			−0.703203
$986$			275.178i			0.279085i
$987$		−	137.463i		−	0.139274i
$988$		−	71.9055i		−	0.0727788i
$989$	−65.6983			−0.0664291
$990$	35.7295			0.0360904
$991$			1385.49i			1.39807i	0.715087	+	0.699036i	$0.246387\pi$
$991$			1385.49i			1.39807i	−0.715087	+	0.699036i	$0.753613\pi$
$992$	−265.844			−0.267988
$993$	−436.092			−0.439167
$994$		−	538.517i		−	0.541768i
$995$	1764.35			1.77321
$996$			198.033i			0.198828i
$997$	182.890			0.183441			0.0917203	−	0.995785i	$-0.470763\pi$
$997$	182.890			0.183441			0.0917203	+	0.995785i	$0.470763\pi$
$998$			810.856i			0.812481i
$999$		−	218.198i		−	0.218416i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	354.3.d.a.235.12	yes	20
3.2	odd	2		1062.3.d.f.235.8		20
59.58	odd	2	inner	354.3.d.a.235.2	✓	20
177.176	even	2		1062.3.d.f.235.18		20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
354.3.d.a.235.2	✓	20	59.58	odd	2	inner
354.3.d.a.235.12	yes	20	1.1	even	1	trivial
1062.3.d.f.235.8		20	3.2	odd	2
1062.3.d.f.235.18		20	177.176	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 354 = 2 \cdot 3 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	354.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -1.73205 q^{3} -2.00000 q^{4} -4.74052 q^{5} -2.44949i q^{6} +12.4076 q^{7} -2.82843i q^{8} +3.00000 q^{9} +O(q^{10})\)	\(q+1.41421i q^{2} -1.73205 q^{3} -2.00000 q^{4} -4.74052 q^{5} -2.44949i q^{6} +12.4076 q^{7} -2.82843i q^{8} +3.00000 q^{9} -6.70411i q^{10} +1.77650i q^{11} +3.46410 q^{12} +3.41653i q^{13} +17.5470i q^{14} +8.21082 q^{15} +4.00000 q^{16} -5.78514 q^{17} +4.24264i q^{18} +10.5232 q^{19} +9.48104 q^{20} -21.4906 q^{21} -2.51235 q^{22} +41.0431i q^{23} +4.89898i q^{24} -2.52748 q^{25} -4.83170 q^{26} -5.19615 q^{27} -24.8152 q^{28} -33.6345 q^{29} +11.6119i q^{30} +46.9951i q^{31} +5.65685i q^{32} -3.07699i q^{33} -8.18142i q^{34} -58.8186 q^{35} -6.00000 q^{36} +41.9921i q^{37} +14.8820i q^{38} -5.91760i q^{39} +13.4082i q^{40} +23.8134 q^{41} -30.3923i q^{42} +1.60072i q^{43} -3.55300i q^{44} -14.2216 q^{45} -58.0437 q^{46} +6.39642i q^{47} -6.92820 q^{48} +104.949 q^{49} -3.57439i q^{50} +10.0202 q^{51} -6.83305i q^{52} -2.92135 q^{53} -7.34847i q^{54} -8.42153i q^{55} -35.0940i q^{56} -18.2267 q^{57} -47.5664i q^{58} +(48.0608 - 34.2223i) q^{59} -16.4216 q^{60} +10.8622i q^{61} -66.4611 q^{62} +37.2229 q^{63} -8.00000 q^{64} -16.1961i q^{65} +4.35152 q^{66} -40.9681i q^{67} +11.5703 q^{68} -71.0887i q^{69} -83.1820i q^{70} -30.6899 q^{71} -8.48528i q^{72} +118.342i q^{73} -59.3858 q^{74} +4.37772 q^{75} -21.0464 q^{76} +22.0421i q^{77} +8.36875 q^{78} +62.0124 q^{79} -18.9621 q^{80} +9.00000 q^{81} +33.6773i q^{82} +57.1672i q^{83} +42.9813 q^{84} +27.4246 q^{85} -2.26376 q^{86} +58.2567 q^{87} +5.02470 q^{88} +96.6560i q^{89} -20.1123i q^{90} +42.3910i q^{91} -82.0862i q^{92} -81.3978i q^{93} -9.04590 q^{94} -49.8853 q^{95} -9.79796i q^{96} -73.5083i q^{97} +148.420i q^{98} +5.32950i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 40 q^{4} + 8 q^{7} + 60 q^{9}+O(q^{10}) \) 20 * q - 40 * q^4 + 8 * q^7 + 60 * q^9	\( 20 q - 40 q^{4} + 8 q^{7} + 60 q^{9} - 24 q^{15} + 80 q^{16} + 72 q^{19} + 16 q^{22} + 140 q^{25} + 64 q^{26} - 16 q^{28} + 56 q^{29} - 80 q^{35} - 120 q^{36} - 8 q^{41} + 16 q^{46} + 52 q^{49} + 32 q^{53} - 48 q^{57} + 192 q^{59} + 48 q^{60} - 16 q^{62} + 24 q^{63} - 160 q^{64} + 96 q^{66} - 568 q^{71} - 288 q^{74} - 96 q^{75} - 144 q^{76} + 192 q^{78} + 528 q^{79} + 180 q^{81} + 568 q^{85} - 416 q^{86} - 216 q^{87} - 32 q^{88} - 480 q^{94} - 456 q^{95}+O(q^{100}) \) 20 * q - 40 * q^4 + 8 * q^7 + 60 * q^9 - 24 * q^15 + 80 * q^16 + 72 * q^19 + 16 * q^22 + 140 * q^25 + 64 * q^26 - 16 * q^28 + 56 * q^29 - 80 * q^35 - 120 * q^36 - 8 * q^41 + 16 * q^46 + 52 * q^49 + 32 * q^53 - 48 * q^57 + 192 * q^59 + 48 * q^60 - 16 * q^62 + 24 * q^63 - 160 * q^64 + 96 * q^66 - 568 * q^71 - 288 * q^74 - 96 * q^75 - 144 * q^76 + 192 * q^78 + 528 * q^79 + 180 * q^81 + 568 * q^85 - 416 * q^86 - 216 * q^87 - 32 * q^88 - 480 * q^94 - 456 * q^95

Introduction

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