LMFDB - Embedded newform 387.5.b.c.343.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [387,5,Mod(343,387)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(387, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("387.343");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 387 = 3^{2} \cdot 43 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	387.b (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:	$40.0041757134$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 142x^{10} + 7173x^{8} + 157368x^{6} + 1510016x^{4} + 5098688x^{2} + 90352 $ x^12 + 142x^10 + 7173x^8 + 157368x^6 + 1510016x^4 + 5098688*x^2 + 90352
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 43)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			343.9
Root			$3.65497i$ of defining polynomial
Character	$\chi$	$=$	387.343
Dual form			387.5.b.c.343.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.65497i q^{2} +2.64117 q^{4} -45.6695i q^{5} +34.3337i q^{7} +68.1330i q^{8} +O(q^{10})$	\(q+3.65497i q^{2} +2.64117 q^{4} -45.6695i q^{5} +34.3337i q^{7} +68.1330i q^{8} +166.921 q^{10} +103.099 q^{11} +134.875 q^{13} -125.489 q^{14} -206.765 q^{16} -240.390 q^{17} -100.800i q^{19} -120.621i q^{20} +376.825i q^{22} +475.879 q^{23} -1460.70 q^{25} +492.966i q^{26} +90.6813i q^{28} +159.825i q^{29} +1118.26 q^{31} +334.405i q^{32} -878.620i q^{34} +1568.00 q^{35} -2451.77i q^{37} +368.423 q^{38} +3111.60 q^{40} -1069.81 q^{41} +(1742.01 - 619.833i) q^{43} +272.303 q^{44} +1739.32i q^{46} +2933.56 q^{47} +1222.20 q^{49} -5338.84i q^{50} +356.229 q^{52} +825.685 q^{53} -4708.49i q^{55} -2339.26 q^{56} -584.155 q^{58} +1241.46 q^{59} -6202.15i q^{61} +4087.22i q^{62} -4530.49 q^{64} -6159.69i q^{65} +265.952 q^{67} -634.912 q^{68} +5731.02i q^{70} +3861.16i q^{71} +8274.70i q^{73} +8961.14 q^{74} -266.231i q^{76} +3539.78i q^{77} +4953.26 q^{79} +9442.88i q^{80} -3910.14i q^{82} +13512.1 q^{83} +10978.5i q^{85} +(2265.47 + 6367.01i) q^{86} +7024.45i q^{88} -5359.31i q^{89} +4630.77i q^{91} +1256.88 q^{92} +10722.1i q^{94} -4603.51 q^{95} +3191.32 q^{97} +4467.09i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 92 q^{4}+O(q^{10}) $ 12 * q - 92 * q^4	$ 12 q - 92 q^{4} + 182 q^{10} + 180 q^{11} - 216 q^{13} - 732 q^{14} + 1076 q^{16} - 678 q^{17} - 1566 q^{23} - 174 q^{25} + 5710 q^{31} - 936 q^{35} - 1242 q^{38} - 2618 q^{40} - 4878 q^{41} - 1108 q^{43} + 15168 q^{44} + 5526 q^{47} - 8544 q^{49} + 24084 q^{52} - 1212 q^{53} + 10152 q^{56} - 4666 q^{58} - 14016 q^{59} - 15580 q^{64} - 1088 q^{67} - 15186 q^{68} + 7674 q^{74} + 24302 q^{79} + 7032 q^{83} + 14412 q^{86} - 48354 q^{92} - 606 q^{95} - 5842 q^{97}+O(q^{100}) $ 12 * q - 92 * q^4 + 182 * q^10 + 180 * q^11 - 216 * q^13 - 732 * q^14 + 1076 * q^16 - 678 * q^17 - 1566 * q^23 - 174 * q^25 + 5710 * q^31 - 936 * q^35 - 1242 * q^38 - 2618 * q^40 - 4878 * q^41 - 1108 * q^43 + 15168 * q^44 + 5526 * q^47 - 8544 * q^49 + 24084 * q^52 - 1212 * q^53 + 10152 * q^56 - 4666 * q^58 - 14016 * q^59 - 15580 * q^64 - 1088 * q^67 - 15186 * q^68 + 7674 * q^74 + 24302 * q^79 + 7032 * q^83 + 14412 * q^86 - 48354 * q^92 - 606 * q^95 - 5842 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/387\mathbb{Z}\right)^\times$.

$n$	$46$	$173$
$\chi(n)$	$-1$	$1$

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 $

$2$			3.65497i			0.913743i	0.889533	+	0.456872i	$0.151030\pi$
$2$			3.65497i			0.913743i	−0.889533	+	0.456872i	$0.848970\pi$
$3$		0			0
$3$		0			0
$4$	2.64117			0.165073
$5$		−	45.6695i		−	1.82678i	−0.407085	−	0.913390i	$-0.633455\pi$
$5$		−	45.6695i		−	1.82678i	0.407085	−	0.913390i	$-0.366545\pi$
$6$		0			0
$7$			34.3337i			0.700688i	0.936621	+	0.350344i	$0.113935\pi$
$7$			34.3337i			0.700688i	−0.936621	+	0.350344i	$0.886065\pi$
$8$			68.1330i			1.06458i
$9$		0			0
$10$	166.921			1.66921
$11$	103.099			0.852059			0.426030	−	0.904709i	$-0.359912\pi$
$11$	103.099			0.852059			0.426030	+	0.904709i	$0.359912\pi$
$12$		0			0
$13$	134.875			0.798079			0.399039	−	0.916934i	$-0.369344\pi$
$13$	134.875			0.798079			0.399039	+	0.916934i	$0.369344\pi$
$14$	−125.489			−0.640249
$15$		0			0
$16$	−206.765			−0.807678
$17$	−240.390			−0.831800			−0.415900	−	0.909410i	$-0.636533\pi$
$17$	−240.390			−0.831800			−0.415900	+	0.909410i	$0.636533\pi$
$18$		0			0
$19$		−	100.800i		−	0.279225i	−0.990206	−	0.139613i	$-0.955414\pi$
$19$		−	100.800i		−	0.279225i	0.990206	−	0.139613i	$-0.0445858\pi$
$20$		−	120.621i		−	0.301553i
$21$		0			0
$22$			376.825i			0.778564i
$23$	475.879			0.899582			0.449791	−	0.893134i	$-0.351498\pi$
$23$	475.879			0.899582			0.449791	+	0.893134i	$0.351498\pi$
$24$		0			0
$25$	−1460.70			−2.33713
$26$			492.966i			0.729239i
$27$		0			0
$28$			90.6813i			0.115665i
$29$			159.825i			0.190041i	0.995475	+	0.0950207i	$0.0302917\pi$
$29$			159.825i			0.190041i	−0.995475	+	0.0950207i	$0.969708\pi$
$30$		0			0
$31$	1118.26			1.16364			0.581822	−	0.813316i	$-0.302340\pi$
$31$	1118.26			1.16364			0.581822	+	0.813316i	$0.302340\pi$
$32$			334.405i			0.326568i
$33$		0			0
$34$		−	878.620i		−	0.760051i
$35$	1568.00			1.28000
$36$		0			0
$37$		−	2451.77i		−	1.79092i	−0.445145	−	0.895459i	$-0.646848\pi$
$37$		−	2451.77i		−	1.79092i	0.445145	−	0.895459i	$-0.353152\pi$
$38$	368.423			0.255140
$39$		0			0
$40$	3111.60			1.94475
$41$	−1069.81			−0.636416			−0.318208	−	0.948021i	$-0.603081\pi$
$41$	−1069.81			−0.636416			−0.318208	+	0.948021i	$0.603081\pi$
$42$		0			0
$43$	1742.01	−	619.833i	0.942138	−	0.335226i
$43$	1742.01	−	619.833i	0.942138	−	0.335226i
$44$	272.303			0.140652
$45$		0			0
$46$			1739.32i			0.821987i
$47$	2933.56			1.32800			0.664002	−	0.747731i	$-0.268857\pi$
$47$	2933.56			1.32800			0.664002	+	0.747731i	$0.268857\pi$
$48$		0			0
$49$	1222.20			0.509036
$50$		−	5338.84i		−	2.13553i
$51$		0			0
$52$	356.229			0.131741
$53$	825.685			0.293943			0.146971	−	0.989141i	$-0.453047\pi$
$53$	825.685			0.293943			0.146971	+	0.989141i	$0.453047\pi$
$54$		0			0
$55$		−	4708.49i		−	1.55653i
$56$	−2339.26			−0.745937
$57$		0			0
$58$	−584.155			−0.173649
$59$	1241.46			0.356639			0.178320	−	0.983973i	$-0.442934\pi$
$59$	1241.46			0.356639			0.178320	+	0.983973i	$0.442934\pi$
$60$		0			0
$61$		−	6202.15i		−	1.66680i	−0.552673	−	0.833398i	$-0.686392\pi$
$61$		−	6202.15i		−	1.66680i	0.552673	−	0.833398i	$-0.313608\pi$
$62$			4087.22i			1.06327i
$63$		0			0
$64$	−4530.49			−1.10608
$65$		−	6159.69i		−	1.45791i
$66$		0			0
$67$	265.952			0.0592452			0.0296226	−	0.999561i	$-0.490569\pi$
$67$	265.952			0.0592452			0.0296226	+	0.999561i	$0.490569\pi$
$68$	−634.912			−0.137308
$69$		0			0
$70$			5731.02i			1.16959i
$71$			3861.16i			0.765952i	0.923758	+	0.382976i	$0.125101\pi$
$71$			3861.16i			0.765952i	−0.923758	+	0.382976i	$0.874899\pi$
$72$		0			0
$73$			8274.70i			1.55277i	0.630260	+	0.776384i	$0.282948\pi$
$73$			8274.70i			1.55277i	−0.630260	+	0.776384i	$0.717052\pi$
$74$	8961.14			1.63644
$75$		0			0
$76$		−	266.231i		−	0.0460926i
$77$			3539.78i			0.597028i
$78$		0			0
$79$	4953.26			0.793665			0.396833	−	0.917891i	$-0.370109\pi$
$79$	4953.26			0.793665			0.396833	+	0.917891i	$0.370109\pi$
$80$			9442.88i			1.47545i
$81$		0			0
$82$		−	3910.14i		−	0.581520i
$83$	13512.1			1.96140			0.980700	−	0.195519i	$-0.0626392\pi$
$83$	13512.1			1.96140			0.980700	+	0.195519i	$0.0626392\pi$
$84$		0			0
$85$			10978.5i			1.51952i
$86$	2265.47	+	6367.01i	0.306311	+	0.860872i
$87$		0			0
$88$			7024.45i			0.907084i
$89$		−	5359.31i		−	0.676595i	−0.941039	−	0.338298i	$-0.890149\pi$
$89$		−	5359.31i		−	0.676595i	0.941039	−	0.338298i	$-0.109851\pi$
$90$		0			0
$91$			4630.77i			0.559204i
$92$	1256.88			0.148497
$93$		0			0
$94$			10722.1i			1.21345i
$95$	−4603.51			−0.510084
$96$		0			0
$97$	3191.32			0.339177			0.169588	−	0.985515i	$-0.445756\pi$
$97$	3191.32			0.339177			0.169588	+	0.985515i	$0.445756\pi$
$98$			4467.09i			0.465128i
$99$		0			0
$100$	−3857.97			−0.385797
$101$	−6784.27			−0.665059			−0.332529	−	0.943093i	$-0.607902\pi$
$101$	−6784.27			−0.665059			−0.332529	+	0.943093i	$0.607902\pi$
$102$		0			0
$103$	−650.667			−0.0613316			−0.0306658	−	0.999530i	$-0.509763\pi$
$103$	−650.667			−0.0613316			−0.0306658	+	0.999530i	$0.509763\pi$
$104$			9189.46i			0.849617i
$105$		0			0
$106$			3017.86i			0.268588i
$107$	−15902.5			−1.38899			−0.694495	−	0.719497i	$-0.744372\pi$
$107$	−15902.5			−1.38899			−0.694495	+	0.719497i	$0.744372\pi$
$108$		0			0
$109$	17290.7			1.45533			0.727663	−	0.685935i	$-0.240607\pi$
$109$	17290.7			1.45533			0.727663	+	0.685935i	$0.240607\pi$
$110$	17209.4			1.42226
$111$		0			0
$112$		−	7099.03i		−	0.565930i
$113$		−	15504.7i		−	1.21425i	−0.794607	−	0.607124i	$-0.792323\pi$
$113$		−	15504.7i		−	1.21425i	0.794607	−	0.607124i	$-0.207677\pi$
$114$		0			0
$115$		−	21733.2i		−	1.64334i
$116$			422.125i			0.0313707i
$117$		0			0
$118$			4537.51i			0.325877i
$119$		−	8253.49i		−	0.582832i
$120$		0			0
$121$	−4011.56			−0.273995
$122$	22668.7			1.52302
$123$		0			0
$124$	2953.52			0.192086
$125$			38166.2i			2.44264i
$126$		0			0
$127$	−24303.2			−1.50680			−0.753400	−	0.657562i	$-0.771588\pi$
$127$	−24303.2			−1.50680			−0.753400	+	0.657562i	$0.771588\pi$
$128$		−	11208.3i		−	0.684102i
$129$		0			0
$130$	22513.5			1.33216
$131$		−	6058.45i		−	0.353036i	−0.984297	−	0.176518i	$-0.943517\pi$
$131$		−	6058.45i		−	0.353036i	0.984297	−	0.176518i	$-0.0564834\pi$
$132$		0			0
$133$	3460.85			0.195650
$134$			972.046i			0.0541349i
$135$		0			0
$136$		−	16378.5i		−	0.885516i
$137$		−	20546.4i		−	1.09470i	−0.836905	−	0.547349i	$-0.815637\pi$
$137$		−	20546.4i		−	1.09470i	0.836905	−	0.547349i	$-0.184363\pi$
$138$		0			0
$139$	−7143.71			−0.369738			−0.184869	−	0.982763i	$-0.559186\pi$
$139$	−7143.71			−0.369738			−0.184869	+	0.982763i	$0.559186\pi$
$140$	4141.37			0.211294
$141$		0			0
$142$	−14112.4			−0.699883
$143$	13905.5			0.680010
$144$		0			0
$145$	7299.12			0.347164
$146$	−30243.8			−1.41883
$147$		0			0
$148$		−	6475.54i		−	0.295633i
$149$			29115.5i			1.31145i	0.754999	+	0.655726i	$0.227637\pi$
$149$			29115.5i			1.31145i	−0.754999	+	0.655726i	$0.772363\pi$
$150$		0			0
$151$			11488.6i			0.503864i	0.967745	+	0.251932i	$0.0810659\pi$
$151$			11488.6i			0.503864i	−0.967745	+	0.251932i	$0.918934\pi$
$152$	6867.83			0.297257
$153$		0			0
$154$	−12937.8			−0.545530
$155$		−	51070.5i		−	2.12572i
$156$		0			0
$157$			2810.63i			0.114026i	0.998373	+	0.0570131i	$0.0181577\pi$
$157$			2810.63i			0.114026i	−0.998373	+	0.0570131i	$0.981842\pi$
$158$			18104.0i			0.725206i
$159$		0			0
$160$	15272.1			0.596568
$161$			16338.7i			0.630327i
$162$		0			0
$163$			506.838i			0.0190763i	0.999955	+	0.00953815i	$0.00303613\pi$
$163$			506.838i			0.0190763i	−0.999955	+	0.00953815i	$0.996964\pi$
$164$	−2825.56			−0.105055
$165$		0			0
$166$			49386.3i			1.79222i
$167$	45960.0			1.64796			0.823980	−	0.566619i	$-0.191749\pi$
$167$	45960.0			1.64796			0.823980	+	0.566619i	$0.191749\pi$
$168$		0			0
$169$	−10369.6			−0.363070
$170$	−40126.1			−1.38845
$171$		0			0
$172$	4600.95	−	1637.09i	0.155522	−	0.0553368i
$173$	23016.4			0.769033			0.384516	−	0.923118i	$-0.374368\pi$
$173$	23016.4			0.769033			0.384516	+	0.923118i	$0.374368\pi$
$174$		0			0
$175$		−	50151.4i		−	1.63760i
$176$	−21317.4			−0.688189
$177$		0			0
$178$	19588.1			0.618234
$179$			54490.4i			1.70065i	0.526260	+	0.850324i	$0.323594\pi$
$179$			54490.4i			1.70065i	−0.526260	+	0.850324i	$0.676406\pi$
$180$		0			0
$181$	24427.1			0.745616			0.372808	−	0.927909i	$-0.378395\pi$
$181$	24427.1			0.745616			0.372808	+	0.927909i	$0.378395\pi$
$182$	−16925.3			−0.510969
$183$		0			0
$184$			32423.1i			0.957675i
$185$	−111971.			−3.27161
$186$		0			0
$187$	−24784.0			−0.708743
$188$	7748.04			0.219218
$189$		0			0
$190$		−	16825.7i		−	0.466086i
$191$		−	25639.4i		−	0.702815i	−0.936223	−	0.351408i	$-0.885703\pi$
$191$		−	25639.4i		−	0.702815i	0.936223	−	0.351408i	$-0.114297\pi$
$192$		0			0
$193$	−45243.0			−1.21461			−0.607305	−	0.794469i	$-0.707749\pi$
$193$	−45243.0			−1.21461			−0.607305	+	0.794469i	$0.707749\pi$
$194$			11664.2i			0.309921i
$195$		0			0
$196$	3228.03			0.0840282
$197$	19109.6			0.492401			0.246200	−	0.969219i	$-0.420818\pi$
$197$	19109.6			0.492401			0.246200	+	0.969219i	$0.420818\pi$
$198$		0			0
$199$			41414.6i			1.04580i	0.852395	+	0.522899i	$0.175149\pi$
$199$			41414.6i			1.04580i	−0.852395	+	0.522899i	$0.824851\pi$
$200$		−	99522.2i		−	2.48805i
$201$		0			0
$202$		−	24796.3i		−	0.607693i
$203$	−5487.38			−0.133160
$204$		0			0
$205$			48857.9i			1.16259i
$206$		−	2378.17i		−	0.0560413i
$207$		0			0
$208$	−27887.6			−0.644590
$209$		−	10392.4i		−	0.237917i
$210$		0			0
$211$		−	40579.3i		−	0.911464i	−0.890117	−	0.455732i	$-0.849377\pi$
$211$		−	40579.3i		−	0.911464i	0.890117	−	0.455732i	$-0.150623\pi$
$212$	2180.78			0.0485221
$213$		0			0
$214$		−	58123.4i		−	1.26918i
$215$	−28307.5	−	79556.9i	−0.612384	−	1.72108i
$216$		0			0
$217$			38394.1i			0.815351i
$218$			63197.1i			1.32979i
$219$		0			0
$220$		−	12435.9i		−	0.256941i
$221$	−32422.7			−0.663842
$222$		0			0
$223$		−	76426.0i		−	1.53685i	−0.639940	−	0.768425i	$-0.721041\pi$
$223$		−	76426.0i		−	1.53685i	0.639940	−	0.768425i	$-0.278959\pi$
$224$	−11481.4			−0.228822
$225$		0			0
$226$	56669.4			1.10951
$227$			44850.6i			0.870395i	0.900335	+	0.435197i	$0.143321\pi$
$227$			44850.6i			0.870395i	−0.900335	+	0.435197i	$0.856679\pi$
$228$		0			0
$229$	−68025.3			−1.29718			−0.648589	−	0.761139i	$-0.724640\pi$
$229$	−68025.3			−1.29718			−0.648589	+	0.761139i	$0.724640\pi$
$230$	79434.1			1.50159
$231$		0			0
$232$	−10889.3			−0.202314
$233$		−	61038.6i		−	1.12433i	−0.827026	−	0.562164i	$-0.809969\pi$
$233$		−	61038.6i		−	1.12433i	0.827026	−	0.562164i	$-0.190031\pi$
$234$		0			0
$235$		−	133974.i		−	2.42597i
$236$	3278.91			0.0588716
$237$		0			0
$238$	30166.3			0.532559
$239$	18090.2			0.316699			0.158349	−	0.987383i	$-0.449383\pi$
$239$	18090.2			0.316699			0.158349	+	0.987383i	$0.449383\pi$
$240$		0			0
$241$			63866.5i			1.09961i	0.835293	+	0.549805i	$0.185298\pi$
$241$			63866.5i			1.09961i	−0.835293	+	0.549805i	$0.814702\pi$
$242$		−	14662.1i		−	0.250361i
$243$		0			0
$244$		−	16380.9i		−	0.275144i
$245$		−	55817.1i		−	0.929897i
$246$		0			0
$247$		−	13595.5i		−	0.222844i
$248$			76190.5i			1.23879i
$249$		0			0
$250$	−139497.			−2.23195
$251$	−29813.8			−0.473227			−0.236613	−	0.971604i	$-0.576037\pi$
$251$	−29813.8			−0.473227			−0.236613	+	0.971604i	$0.576037\pi$
$252$		0			0
$253$	49062.7			0.766497
$254$		−	88827.5i		−	1.37683i
$255$		0			0
$256$	−31521.7			−0.480983
$257$			56144.1i			0.850037i	0.905185	+	0.425018i	$0.139732\pi$
$257$			56144.1i			0.850037i	−0.905185	+	0.425018i	$0.860268\pi$
$258$		0			0
$259$	84178.3			1.25487
$260$		−	16268.8i		−	0.240663i
$261$		0			0
$262$	22143.5			0.322584
$263$		−	7995.51i		−	0.115594i	−0.998328	−	0.0577969i	$-0.981592\pi$
$263$		−	7995.51i		−	0.115594i	0.998328	−	0.0577969i	$-0.0184076\pi$
$264$		0			0
$265$		−	37708.6i		−	0.536969i
$266$			12649.3i			0.178774i
$267$		0			0
$268$	702.424			0.00977980
$269$	−6211.47			−0.0858401			−0.0429200	−	0.999079i	$-0.513666\pi$
$269$	−6211.47			−0.0858401			−0.0429200	+	0.999079i	$0.513666\pi$
$270$		0			0
$271$	−15284.9			−0.208125			−0.104063	−	0.994571i	$-0.533184\pi$
$271$	−15284.9			−0.208125			−0.104063	+	0.994571i	$0.533184\pi$
$272$	49704.4			0.671826
$273$		0			0
$274$	75096.4			1.00027
$275$	−150597.			−1.99137
$276$		0			0
$277$			44696.5i			0.582525i	0.956643	+	0.291262i	$0.0940753\pi$
$277$			44696.5i			0.582525i	−0.956643	+	0.291262i	$0.905925\pi$
$278$		−	26110.1i		−	0.337846i
$279$		0			0
$280$			106833.i			1.36266i
$281$	−91345.4			−1.15684			−0.578421	−	0.815739i	$-0.696331\pi$
$281$	−91345.4			−1.15684			−0.578421	+	0.815739i	$0.696331\pi$
$282$		0			0
$283$	25305.7			0.315970			0.157985	−	0.987442i	$-0.449500\pi$
$283$	25305.7			0.315970			0.157985	+	0.987442i	$0.449500\pi$
$284$			10198.0i			0.126438i
$285$		0			0
$286$			50824.4i			0.621355i
$287$		−	36730.7i		−	0.445929i
$288$		0			0
$289$	−25733.6			−0.308109
$290$			26678.1i			0.317219i
$291$		0			0
$292$			21854.9i			0.256320i
$293$	−5876.11			−0.0684470			−0.0342235	−	0.999414i	$-0.510896\pi$
$293$	−5876.11			−0.0684470			−0.0342235	+	0.999414i	$0.510896\pi$
$294$		0			0
$295$		−	56696.9i		−	0.651501i
$296$	167046.			1.90657
$297$		0			0
$298$	−106417.			−1.19833
$299$	64184.3			0.717938
$300$		0			0
$301$	21281.2	+	59809.8i	0.234889	+	0.660145i
$302$	−41990.5			−0.460402
$303$		0			0
$304$			20842.0i			0.225524i
$305$	−283249.			−3.04487
$306$		0			0
$307$	−82720.6			−0.877681			−0.438841	−	0.898565i	$-0.644611\pi$
$307$	−82720.6			−0.877681			−0.438841	+	0.898565i	$0.644611\pi$
$308$			9349.16i			0.0985533i
$309$		0			0
$310$	186661.			1.94236
$311$	−160383.			−1.65820			−0.829102	−	0.559097i	$-0.811148\pi$
$311$	−160383.			−1.65820			−0.829102	+	0.559097i	$0.811148\pi$
$312$		0			0
$313$		−	182768.i		−	1.86557i	−0.360429	−	0.932787i	$-0.617370\pi$
$313$		−	182768.i		−	1.86557i	0.360429	−	0.932787i	$-0.382630\pi$
$314$	−10272.8			−0.104191
$315$		0			0
$316$	13082.4			0.131013
$317$	99340.5			0.988571			0.494286	−	0.869300i	$-0.335430\pi$
$317$	99340.5			0.988571			0.494286	+	0.869300i	$0.335430\pi$
$318$		0			0
$319$			16477.8i			0.161926i
$320$			206905.i			2.02056i
$321$		0			0
$322$	−59717.5			−0.575957
$323$			24231.4i			0.232260i
$324$		0			0
$325$	−197013.			−1.86521
$326$	−1852.48			−0.0174308
$327$		0			0
$328$		−	72889.7i		−	0.677514i
$329$			100720.i			0.930517i
$330$		0			0
$331$			128433.i			1.17225i	0.810221	+	0.586124i	$0.199347\pi$
$331$			128433.i			1.17225i	−0.810221	+	0.586124i	$0.800653\pi$
$332$	35687.7			0.323775
$333$		0			0
$334$			167982.i			1.50581i
$335$		−	12145.9i		−	0.108228i
$336$		0			0
$337$	−78727.5			−0.693213			−0.346606	−	0.938011i	$-0.612666\pi$
$337$	−78727.5			−0.693213			−0.346606	+	0.938011i	$0.612666\pi$
$338$		−	37900.8i		−	0.331753i
$339$		0			0
$340$			28996.1i			0.250831i
$341$	115292.			0.991493
$342$		0			0
$343$			124398.i			1.05736i
$344$	42231.1	+	118689.i	0.356874	+	1.00298i
$345$		0			0
$346$			84124.3i			0.702699i
$347$		−	136310.i		−	1.13205i	−0.824387	−	0.566027i	$-0.808480\pi$
$347$		−	136310.i		−	1.13205i	0.824387	−	0.566027i	$-0.191520\pi$
$348$		0			0
$349$			48465.9i			0.397910i	0.980009	+	0.198955i	$0.0637548\pi$
$349$			48465.9i			0.397910i	−0.980009	+	0.198955i	$0.936245\pi$
$350$	183302.			1.49634
$351$		0			0
$352$			34476.9i			0.278255i
$353$	−25854.2			−0.207483			−0.103741	−	0.994604i	$-0.533081\pi$
$353$	−25854.2			−0.207483			−0.103741	+	0.994604i	$0.533081\pi$
$354$		0			0
$355$	176337.			1.39923
$356$		−	14154.9i		−	0.111688i
$357$		0			0
$358$	−199161.			−1.55395
$359$	125163.			0.971155			0.485577	−	0.874194i	$-0.338609\pi$
$359$	125163.			0.971155			0.485577	+	0.874194i	$0.338609\pi$
$360$		0			0
$361$	120160.			0.922033
$362$			89280.5i			0.681301i
$363$		0			0
$364$			12230.7i			0.0923097i
$365$	377901.			2.83657
$366$		0			0
$367$	112207.			0.833082			0.416541	−	0.909117i	$-0.363242\pi$
$367$	112207.			0.833082			0.416541	+	0.909117i	$0.363242\pi$
$368$	−98395.3			−0.726572
$369$		0			0
$370$		−	409251.i		−	2.98941i
$371$			28348.9i			0.205962i
$372$		0			0
$373$			35635.8i			0.256135i	0.991765	+	0.128068i	$0.0408775\pi$
$373$			35635.8i			0.256135i	−0.991765	+	0.128068i	$0.959123\pi$
$374$		−	90585.0i		−	0.647609i
$375$		0			0
$376$			199872.i			1.41376i
$377$			21556.4i			0.151668i
$378$		0			0
$379$	−205315.			−1.42936			−0.714679	−	0.699452i	$-0.753427\pi$
$379$	−205315.			−1.42936			−0.714679	+	0.699452i	$0.753427\pi$
$380$	−12158.6			−0.0842012
$381$		0			0
$382$	93711.3			0.642193
$383$			164110.i			1.11876i	0.828912	+	0.559380i	$0.188961\pi$
$383$			164110.i			1.11876i	−0.828912	+	0.559380i	$0.811039\pi$
$384$		0			0
$385$	161660.			1.09064
$386$		−	165362.i		−	1.10984i
$387$		0			0
$388$	8428.81			0.0559890
$389$			190984.i			1.26211i	0.775738	+	0.631055i	$0.217378\pi$
$389$			190984.i			1.26211i	−0.775738	+	0.631055i	$0.782622\pi$
$390$		0			0
$391$	−114397.			−0.748272
$392$			83271.8i			0.541908i
$393$		0			0
$394$			69845.0i			0.449928i
$395$		−	226213.i		−	1.44985i
$396$		0			0
$397$	−170575.			−1.08226			−0.541132	−	0.840938i	$-0.682004\pi$
$397$	−170575.			−1.08226			−0.541132	+	0.840938i	$0.682004\pi$
$398$	−151369.			−0.955590
$399$		0			0
$400$	302023.			1.88765
$401$	−52348.9			−0.325551			−0.162775	−	0.986663i	$-0.552045\pi$
$401$	−52348.9			−0.325551			−0.162775	+	0.986663i	$0.552045\pi$
$402$		0			0
$403$	150826.			0.928679
$404$	−17918.4			−0.109783
$405$		0			0
$406$		−	20056.2i		−	0.121674i
$407$		−	252775.i		−	1.52597i
$408$		0			0
$409$			164844.i			0.985430i	0.870191	+	0.492715i	$0.163996\pi$
$409$			164844.i			0.985430i	−0.870191	+	0.492715i	$0.836004\pi$
$410$	−178574.			−1.06231
$411$		0			0
$412$	−1718.52			−0.0101242
$413$			42624.0i			0.249893i
$414$		0			0
$415$		−	617090.i		−	3.58305i
$416$			45103.0i			0.260627i
$417$		0			0
$418$	37984.1			0.217395
$419$			37268.6i			0.212283i	0.994351	+	0.106141i	$0.0338496\pi$
$419$			37268.6i			0.212283i	−0.994351	+	0.106141i	$0.966150\pi$
$420$		0			0
$421$			99646.0i			0.562206i	0.959678	+	0.281103i	$0.0907003\pi$
$421$			99646.0i			0.562206i	−0.959678	+	0.281103i	$0.909300\pi$
$422$	148316.			0.832844
$423$		0			0
$424$			56256.4i			0.312925i
$425$	351139.			1.94402
$426$		0			0
$427$	212943.			1.16790
$428$	−42001.4			−0.229285
$429$		0			0
$430$	290778.	−	103463.i	1.57262	−	0.559562i
$431$	−144175.			−0.776133			−0.388067	−	0.921631i	$-0.626857\pi$
$431$	−144175.			−0.776133			−0.388067	+	0.921631i	$0.626857\pi$
$432$		0			0
$433$			220583.i			1.17651i	0.808675	+	0.588256i	$0.200185\pi$
$433$			220583.i			1.17651i	−0.808675	+	0.588256i	$0.799815\pi$
$434$	−140329.			−0.745022
$435$		0			0
$436$	45667.8			0.240235
$437$		−	47968.8i		−	0.251186i
$438$		0			0
$439$	68794.6			0.356965			0.178482	−	0.983943i	$-0.442881\pi$
$439$	68794.6			0.356965			0.178482	+	0.983943i	$0.442881\pi$
$440$	320803.			1.65704
$441$		0			0
$442$		−	118504.i		−	0.606581i
$443$	−321595.			−1.63871			−0.819355	−	0.573286i	$-0.805668\pi$
$443$	−321595.			−1.63871			−0.819355	+	0.573286i	$0.805668\pi$
$444$		0			0
$445$	−244757.			−1.23599
$446$	279335.			1.40429
$447$		0			0
$448$		−	155549.i		−	0.775015i
$449$			2687.03i			0.0133285i	0.999978	+	0.00666423i	$0.00212131\pi$
$449$			2687.03i			0.0133285i	−0.999978	+	0.00666423i	$0.997879\pi$
$450$		0			0
$451$	−110297.			−0.542264
$452$		−	40950.7i		−	0.200440i
$453$		0			0
$454$	−163928.			−0.795318
$455$	211485.			1.02154
$456$		0			0
$457$		−	222198.i		−	1.06392i	−0.846770	−	0.531959i	$-0.821456\pi$
$457$		−	222198.i		−	1.06392i	0.846770	−	0.531959i	$-0.178544\pi$
$458$		−	248631.i		−	1.18529i
$459$		0			0
$460$		−	57401.0i		−	0.271271i
$461$	63915.9			0.300751			0.150375	−	0.988629i	$-0.451952\pi$
$461$	63915.9			0.300751			0.150375	+	0.988629i	$0.451952\pi$
$462$		0			0
$463$			94323.6i			0.440006i	0.975499	+	0.220003i	$0.0706067\pi$
$463$			94323.6i			0.440006i	−0.975499	+	0.220003i	$0.929393\pi$
$464$		−	33046.2i		−	0.153492i
$465$		0			0
$466$	223094.			1.02735
$467$			288915.i			1.32476i	0.749170	+	0.662378i	$0.230453\pi$
$467$			288915.i			1.32476i	−0.749170	+	0.662378i	$0.769547\pi$
$468$		0			0
$469$			9131.11i			0.0415124i
$470$	489673.			2.21672
$471$		0			0
$472$			84584.4i			0.379670i
$473$	179600.	−	63904.3i	0.802757	−	0.285632i
$474$		0			0
$475$			147240.i			0.652586i
$476$		−	21798.9i		−	0.0962100i
$477$		0			0
$478$			66119.0i			0.289381i
$479$	−328139.			−1.43017			−0.715084	−	0.699039i	$-0.753612\pi$
$479$	−328139.			−1.43017			−0.715084	+	0.699039i	$0.753612\pi$
$480$		0			0
$481$		−	330683.i		−	1.42929i
$482$	−233430.			−1.00476
$483$		0			0
$484$	−10595.2			−0.0452292
$485$		−	145746.i		−	0.619602i
$486$		0			0
$487$	−225179.			−0.949447			−0.474723	−	0.880135i	$-0.657452\pi$
$487$	−225179.			−0.949447			−0.474723	+	0.880135i	$0.657452\pi$
$488$	422571.			1.77443
$489$		0			0
$490$	204010.			0.849687
$491$			119348.i			0.495053i	0.968881	+	0.247526i	$0.0796177\pi$
$491$			119348.i			0.495053i	−0.968881	+	0.247526i	$0.920382\pi$
$492$		0			0
$493$		−	38420.3i		−	0.158076i
$494$	49691.1			0.203622
$495$		0			0
$496$	−231218.			−0.939849
$497$	−132568.			−0.536693
$498$		0			0
$499$		−	184960.i		−	0.742807i	−0.928471	−	0.371404i	$-0.878877\pi$
$499$		−	184960.i		−	0.742807i	0.928471	−	0.371404i	$-0.121123\pi$
$500$			100804.i			0.403214i
$501$		0			0
$502$		−	108968.i		−	0.432408i
$503$			340779.i			1.34691i	0.739230	+	0.673453i	$0.235190\pi$
$503$			340779.i			1.34691i	−0.739230	+	0.673453i	$0.764810\pi$
$504$		0			0
$505$			309834.i			1.21492i
$506$			179323.i			0.700382i
$507$		0			0
$508$	−64188.9			−0.248732
$509$	−35186.6			−0.135813			−0.0679066	−	0.997692i	$-0.521632\pi$
$509$	−35186.6			−0.135813			−0.0679066	+	0.997692i	$0.521632\pi$
$510$		0			0
$511$	−284101.			−1.08801
$512$		−	294544.i		−	1.12360i
$513$		0			0
$514$	−205205.			−0.776715
$515$			29715.7i			0.112039i
$516$		0			0
$517$	302448.			1.13154
$518$			307669.i			1.14663i
$519$		0			0
$520$	419678.			1.55206
$521$			167672.i			0.617712i	0.951109	+	0.308856i	$0.0999460\pi$
$521$			167672.i			0.617712i	−0.951109	+	0.308856i	$0.900054\pi$
$522$		0			0
$523$			388697.i			1.42105i	0.703674	+	0.710523i	$0.251541\pi$
$523$			388697.i			1.42105i	−0.703674	+	0.710523i	$0.748459\pi$
$524$		−	16001.4i		−	0.0582768i
$525$		0			0
$526$	29223.4			0.105623
$527$	−268819.			−0.967918
$528$		0			0
$529$	−53380.2			−0.190752
$530$	137824.			0.490652
$531$		0			0
$532$	9140.71			0.0322966
$533$	−144292.			−0.507910
$534$		0			0
$535$			726262.i			2.53738i
$536$			18120.1i			0.0630711i
$537$		0			0
$538$		−	22702.8i		−	0.0784358i
$539$	126007.			0.433729
$540$		0			0
$541$	258688.			0.883856			0.441928	−	0.897050i	$-0.354295\pi$
$541$	258688.			0.883856			0.441928	+	0.897050i	$0.354295\pi$
$542$		−	55865.9i		−	0.190173i
$543$		0			0
$544$		−	80387.8i		−	0.271639i
$545$		−	789659.i		−	2.65856i
$546$		0			0
$547$	−11333.8			−0.0378791			−0.0189395	−	0.999821i	$-0.506029\pi$
$547$	−11333.8			−0.0378791			−0.0189395	+	0.999821i	$0.506029\pi$
$548$		−	54266.5i		−	0.180705i
$549$		0			0
$550$		−	550430.i		−	1.81960i
$551$	16110.4			0.0530644
$552$		0			0
$553$			170064.i			0.556112i
$554$	−163365.			−0.532278
$555$		0			0
$556$	−18867.8			−0.0610338
$557$	90762.0			0.292546			0.146273	−	0.989244i	$-0.453272\pi$
$557$	90762.0			0.292546			0.146273	+	0.989244i	$0.453272\pi$
$558$		0			0
$559$	234955.	−	83600.2i	0.751900	−	0.267537i
$560$	−324209.			−1.03383
$561$		0			0
$562$		−	333865.i		−	1.05706i
$563$	−198527.			−0.626329			−0.313165	−	0.949699i	$-0.601389\pi$
$563$	−198527.			−0.626329			−0.313165	+	0.949699i	$0.601389\pi$
$564$		0			0
$565$	−708094.			−2.21817
$566$			92491.8i			0.288716i
$567$		0			0
$568$	−263073.			−0.815415
$569$	−204563.			−0.631832			−0.315916	−	0.948787i	$-0.602312\pi$
$569$	−204563.			−0.631832			−0.315916	+	0.948787i	$0.602312\pi$
$570$		0			0
$571$			103248.i			0.316670i	0.987385	+	0.158335i	$0.0506127\pi$
$571$			103248.i			0.316670i	−0.987385	+	0.158335i	$0.949387\pi$
$572$	36726.9			0.112252
$573$		0			0
$574$	134250.			0.407465
$575$	−695119.			−2.10244
$576$		0			0
$577$			214909.i			0.645510i	0.946482	+	0.322755i	$0.104609\pi$
$577$			214909.i			0.645510i	−0.946482	+	0.322755i	$0.895391\pi$
$578$		−	94055.5i		−	0.281533i
$579$		0			0
$580$	19278.2			0.0573075
$581$			463920.i			1.37433i
$582$		0			0
$583$	85127.5			0.250457
$584$	−563780.			−1.65304
$585$		0			0
$586$		−	21477.0i		−	0.0625430i
$587$		−	508043.i		−	1.47443i	−0.675657	−	0.737216i	$-0.736140\pi$
$587$		−	508043.i		−	1.47443i	0.675657	−	0.737216i	$-0.263860\pi$
$588$		0			0
$589$		−	112721.i		−	0.324919i
$590$	207226.			0.595305
$591$		0			0
$592$			506941.i			1.44648i
$593$		−	182995.i		−	0.520390i	−0.965556	−	0.260195i	$-0.916213\pi$
$593$		−	182995.i		−	0.520390i	0.965556	−	0.260195i	$-0.0837869\pi$
$594$		0			0
$595$	−376933.			−1.06471
$596$			76899.1i			0.216486i
$597$		0			0
$598$			234592.i			0.656011i
$599$	173564.			0.483733			0.241867	−	0.970310i	$-0.422240\pi$
$599$	173564.			0.483733			0.241867	+	0.970310i	$0.422240\pi$
$600$		0			0
$601$		−	370900.i		−	1.02685i	−0.858134	−	0.513426i	$-0.828376\pi$
$601$		−	370900.i		−	1.02685i	0.858134	−	0.513426i	$-0.171624\pi$
$602$	−218603.	+	77782.1i	−0.603203	+	0.214628i
$603$		0			0
$604$			30343.4i			0.0831744i
$605$			183206.i			0.500529i
$606$		0			0
$607$			274917.i			0.746147i	0.927802	+	0.373074i	$0.121696\pi$
$607$			274917.i			0.746147i	−0.927802	+	0.373074i	$0.878304\pi$
$608$	33708.2			0.0911861
$609$		0			0
$610$		−	1.03527e6i		−	2.78223i
$611$	395665.			1.05985
$612$		0			0
$613$	358829.			0.954919			0.477459	−	0.878654i	$-0.341558\pi$
$613$	358829.			0.954919			0.477459	+	0.878654i	$0.341558\pi$
$614$		−	302342.i		−	0.801975i
$615$		0			0
$616$	−241176.			−0.635583
$617$	−275204.			−0.722909			−0.361455	−	0.932390i	$-0.617720\pi$
$617$	−275204.			−0.722909			−0.361455	+	0.932390i	$0.617720\pi$
$618$		0			0
$619$	−303567.			−0.792270			−0.396135	−	0.918192i	$-0.629649\pi$
$619$	−303567.			−0.792270			−0.396135	+	0.918192i	$0.629649\pi$
$620$		−	134886.i		−	0.350900i
$621$		0			0
$622$		−	586196.i		−	1.51517i
$623$	184005.			0.474082
$624$		0			0
$625$	830093.			2.12504
$626$	668013.			1.70466
$627$		0			0
$628$			7423.36i			0.0188227i
$629$			589380.i			1.48968i
$630$		0			0
$631$		−	350000.i		−	0.879041i	−0.898233	−	0.439520i	$-0.855148\pi$
$631$		−	350000.i		−	0.879041i	0.898233	−	0.439520i	$-0.144852\pi$
$632$			337481.i			0.844918i
$633$		0			0
$634$			363087.i			0.903300i
$635$			1.10991e6i			2.75259i
$636$		0			0
$637$	164844.			0.406251
$638$	−60225.9			−0.147959
$639$		0			0
$640$	−511879.			−1.24970
$641$		−	404169.i		−	0.983665i	−0.870690	−	0.491832i	$-0.836327\pi$
$641$		−	404169.i		−	0.983665i	0.870690	−	0.491832i	$-0.163673\pi$
$642$		0			0
$643$	488598.			1.18176			0.590881	−	0.806759i	$-0.298780\pi$
$643$	488598.			1.18176			0.590881	+	0.806759i	$0.298780\pi$
$644$			43153.3i			0.104050i
$645$		0			0
$646$	−88565.2			−0.212226
$647$		−	111849.i		−	0.267191i	−0.991036	−	0.133595i	$-0.957348\pi$
$647$		−	111849.i		−	0.267191i	0.991036	−	0.133595i	$-0.0426523\pi$
$648$		0			0
$649$	127994.			0.303878
$650$		−	720077.i		−	1.70433i
$651$		0			0
$652$			1338.65i			0.00314899i
$653$		−	92008.4i		−	0.215775i	−0.994163	−	0.107887i	$-0.965591\pi$
$653$		−	92008.4i		−	0.215775i	0.994163	−	0.107887i	$-0.0344086\pi$
$654$		0			0
$655$	−276686.			−0.644919
$656$	221201.			0.514019
$657$		0			0
$658$	−368129.			−0.850254
$659$	532415.			1.22597			0.612985	−	0.790095i	$-0.289969\pi$
$659$	532415.			1.22597			0.612985	+	0.790095i	$0.289969\pi$
$660$		0			0
$661$	−770144.			−1.76266			−0.881331	−	0.472499i	$-0.843352\pi$
$661$	−770144.			−1.76266			−0.881331	+	0.472499i	$0.843352\pi$
$662$	−469418.			−1.07113
$663$		0			0
$664$			920618.i			2.08806i
$665$		−	158055.i		−	0.357410i
$666$		0			0
$667$			76057.3i			0.170958i
$668$	121388.			0.272034
$669$		0			0
$670$	44392.9			0.0988926
$671$		−	639437.i		−	1.42021i
$672$		0			0
$673$		−	296438.i		−	0.654492i	−0.944939	−	0.327246i	$-0.893879\pi$
$673$		−	296438.i		−	0.654492i	0.944939	−	0.327246i	$-0.106121\pi$
$674$		−	287747.i		−	0.633418i
$675$		0			0
$676$	−27388.0			−0.0599332
$677$		−	255234.i		−	0.556880i	−0.960454	−	0.278440i	$-0.910183\pi$
$677$		−	255234.i		−	0.556880i	0.960454	−	0.278440i	$-0.0898174\pi$
$678$		0			0
$679$			109570.i			0.237657i
$680$	−747998.			−1.61764
$681$		0			0
$682$			421389.i			0.905970i
$683$	−659236.			−1.41319			−0.706594	−	0.707620i	$-0.749769\pi$
$683$	−659236.			−1.41319			−0.706594	+	0.707620i	$0.749769\pi$
$684$		0			0
$685$	−938343.			−1.99977
$686$	−454671.			−0.966159
$687$		0			0
$688$	−360188.	+	128160.i	−0.760944	+	0.270755i
$689$	111365.			0.234589
$690$		0			0
$691$			428774.i			0.897993i	0.893534	+	0.448996i	$0.148218\pi$
$691$			428774.i			0.897993i	−0.893534	+	0.448996i	$0.851782\pi$
$692$	60790.2			0.126947
$693$		0			0
$694$	498208.			1.03441
$695$			326250.i			0.675430i
$696$		0			0
$697$	257173.			0.529370
$698$	−177141.			−0.363588
$699$		0			0
$700$		−	132459.i		−	0.270324i
$701$	−446529.			−0.908686			−0.454343	−	0.890827i	$-0.650126\pi$
$701$	−446529.			−0.908686			−0.454343	+	0.890827i	$0.650126\pi$
$702$		0			0
$703$	−247139.			−0.500070
$704$	−467090.			−0.942443
$705$		0			0
$706$		−	94496.5i		−	0.189586i
$707$		−	232929.i		−	0.465999i
$708$		0			0
$709$	−43598.6			−0.0867321			−0.0433660	−	0.999059i	$-0.513808\pi$
$709$	−43598.6			−0.0867321			−0.0433660	+	0.999059i	$0.513808\pi$
$710$			644509.i			1.27853i
$711$		0			0
$712$	365146.			0.720288
$713$	532157.			1.04679
$714$		0			0
$715$		−	635059.i		−	1.24223i
$716$			143919.i			0.280731i
$717$		0			0
$718$			457469.i			0.887386i
$719$	−46047.8			−0.0890741			−0.0445370	−	0.999008i	$-0.514181\pi$
$719$	−46047.8			−0.0890741			−0.0445370	+	0.999008i	$0.514181\pi$
$720$		0			0
$721$		−	22339.8i		−	0.0429743i
$722$			439183.i			0.842502i
$723$		0			0
$724$	64516.2			0.123081
$725$		−	233457.i		−	0.444151i
$726$		0			0
$727$		−	275601.i		−	0.521450i	−0.965413	−	0.260725i	$-0.916038\pi$
$727$		−	275601.i		−	0.521450i	0.965413	−	0.260725i	$-0.0839615\pi$
$728$	−315508.			−0.595317
$729$		0			0
$730$			1.38122e6i			2.59189i
$731$	−418763.	+	149002.i	−0.783670	+	0.278841i
$732$		0			0
$733$			324631.i			0.604201i	0.953276	+	0.302101i	$0.0976879\pi$
$733$			324631.i			0.604201i	−0.953276	+	0.302101i	$0.902312\pi$
$734$			410114.i			0.761223i
$735$		0			0
$736$			159137.i			0.293775i
$737$	27419.4			0.0504804
$738$		0			0
$739$			359031.i			0.657421i	0.944431	+	0.328710i	$0.106614\pi$
$739$			359031.i			0.657421i	−0.944431	+	0.328710i	$0.893386\pi$
$740$	−295735.			−0.540056
$741$		0			0
$742$	−103614.			−0.188197
$743$		−	164101.i		−	0.297259i	−0.988893	−	0.148629i	$-0.952514\pi$
$743$		−	164101.i		−	0.297259i	0.988893	−	0.148629i	$-0.0474862\pi$
$744$		0			0
$745$	1.32969e6			2.39573
$746$	−130248.			−0.234042
$747$		0			0
$748$	−65458.9			−0.116994
$749$		−	545994.i		−	0.973249i
$750$		0			0
$751$		−	862173.i		−	1.52867i	−0.644818	−	0.764336i	$-0.723067\pi$
$751$		−	862173.i		−	1.52867i	0.644818	−	0.764336i	$-0.276933\pi$
$752$	−606559.			−1.07260
$753$		0			0
$754$	−78788.1			−0.138586
$755$	524679.			0.920449
$756$		0			0
$757$		−	878771.i		−	1.53350i	−0.641946	−	0.766750i	$-0.721873\pi$
$757$		−	878771.i		−	1.53350i	0.641946	−	0.766750i	$-0.278127\pi$
$758$		−	750419.i		−	1.30607i
$759$		0			0
$760$		−	313651.i		−	0.543024i
$761$		−	452736.i		−	0.781763i	−0.920441	−	0.390882i	$-0.872170\pi$
$761$		−	452736.i		−	0.781763i	0.920441	−	0.390882i	$-0.127830\pi$
$762$		0			0
$763$			593655.i			1.01973i
$764$		−	67718.1i		−	0.116016i
$765$		0			0
$766$	−599816.			−1.02226
$767$	167442.			0.284626
$768$		0			0
$769$	379858.			0.642346			0.321173	−	0.947021i	$-0.395923\pi$
$769$	379858.			0.642346			0.321173	+	0.947021i	$0.395923\pi$
$770$			590863.i			0.996564i
$771$		0			0
$772$	−119495.			−0.200500
$773$			466697.i			0.781044i	0.920594	+	0.390522i	$0.127705\pi$
$773$			466697.i			0.781044i	−0.920594	+	0.390522i	$0.872295\pi$
$774$		0			0
$775$	−1.63345e6			−2.71958
$776$			217434.i			0.361080i
$777$		0			0
$778$	−698041.			−1.15325
$779$			107838.i			0.177703i
$780$		0			0
$781$			398083.i			0.652636i
$782$		−	418117.i		−	0.683729i
$783$		0			0
$784$	−252708.			−0.411137
$785$	128360.			0.208301
$786$		0			0
$787$	946342.			1.52791			0.763956	−	0.645268i	$-0.223254\pi$
$787$	946342.			1.52791			0.763956	+	0.645268i	$0.223254\pi$
$788$	50471.7			0.0812822
$789$		0			0
$790$	826803.			1.32479
$791$	532335.			0.850810
$792$		0			0
$793$		−	836517.i		−	1.33024i
$794$		−	623446.i		−	0.988912i
$795$		0			0
$796$			109383.i			0.172633i
$797$	561047.			0.883248			0.441624	−	0.897200i	$-0.354403\pi$
$797$	561047.			0.883248			0.441624	+	0.897200i	$0.354403\pi$
$798$		0			0
$799$	−705199.			−1.10463
$800$		−	488468.i		−	0.763231i
$801$		0			0
$802$		−	191334.i		−	0.297470i
$803$			853115.i			1.32305i
$804$		0			0
$805$	746181.			1.15147
$806$			551264.i			0.848574i
$807$		0			0
$808$		−	462232.i		−	0.708007i
$809$	938377.			1.43377			0.716886	−	0.697190i	$-0.245567\pi$
$809$	938377.			1.43377			0.716886	+	0.697190i	$0.245567\pi$
$810$		0			0
$811$			526766.i			0.800896i	0.916319	+	0.400448i	$0.131146\pi$
$811$			526766.i			0.800896i	−0.916319	+	0.400448i	$0.868854\pi$
$812$	−14493.1			−0.0219811
$813$		0			0
$814$	923886.			1.39434
$815$	23147.1			0.0348482
$816$		0			0
$817$	−62479.4	−	175596.i	−0.0936036	−	0.263069i
$818$	−602499.			−0.900430
$819$		0			0
$820$			129042.i			0.191913i
$821$	1.19525e6			1.77325			0.886626	−	0.462486i	$-0.153043\pi$
$821$	1.19525e6			1.77325			0.886626	+	0.462486i	$0.153043\pi$
$822$		0			0
$823$	230617.			0.340480			0.170240	−	0.985403i	$-0.445546\pi$
$823$	230617.			0.340480			0.170240	+	0.985403i	$0.445546\pi$
$824$		−	44331.9i		−	0.0652923i
$825$		0			0
$826$	−155789.			−0.228338
$827$	353042.			0.516197			0.258099	−	0.966119i	$-0.416904\pi$
$827$	353042.			0.516197			0.258099	+	0.966119i	$0.416904\pi$
$828$		0			0
$829$		−	1.18220e6i		−	1.72021i	−0.510116	−	0.860106i	$-0.670397\pi$
$829$		−	1.18220e6i		−	1.72021i	0.510116	−	0.860106i	$-0.329603\pi$
$830$	2.25545e6			3.27399
$831$		0			0
$832$	−611051.			−0.882736
$833$	−293804.			−0.423416
$834$		0			0
$835$		−	2.09897e6i		−	3.01046i
$836$		−	27448.2i		−	0.0392737i
$837$		0			0
$838$	−136216.			−0.193972
$839$		−	229327.i		−	0.325785i	−0.986644	−	0.162892i	$-0.947918\pi$
$839$		−	229327.i		−	0.325785i	0.986644	−	0.162892i	$-0.0520823\pi$
$840$		0			0
$841$	681737.			0.963884
$842$	−364204.			−0.513712
$843$		0			0
$844$		−	107177.i		−	0.150458i
$845$			473577.i			0.663250i
$846$		0			0
$847$		−	137732.i		−	0.191985i
$848$	−170723.			−0.237411
$849$		0			0
$850$			1.28340e6i			1.77634i
$851$		−	1.16674e6i		−	1.61108i
$852$		0			0
$853$	736313.			1.01196			0.505981	−	0.862545i	$-0.331131\pi$
$853$	736313.			1.01196			0.505981	+	0.862545i	$0.331131\pi$
$854$			778301.i			1.06717i
$855$		0			0
$856$		−	1.08349e6i		−	1.47869i
$857$	−1.18206e6			−1.60945			−0.804726	−	0.593646i	$-0.797688\pi$
$857$	−1.18206e6			−1.60945			−0.804726	+	0.593646i	$0.797688\pi$
$858$		0			0
$859$			1.14109e6i			1.54645i	0.634133	+	0.773224i	$0.281357\pi$
$859$			1.14109e6i			1.54645i	−0.634133	+	0.773224i	$0.718643\pi$
$860$	−74764.9	−	210123.i	−0.101088	−	0.284104i
$861$		0			0
$862$		−	526957.i		−	0.709186i
$863$			81940.4i			0.110021i	0.998486	+	0.0550106i	$0.0175193\pi$
$863$			81940.4i			0.110021i	−0.998486	+	0.0550106i	$0.982481\pi$
$864$		0			0
$865$		−	1.05115e6i		−	1.40485i
$866$	−806224.			−1.07503
$867$		0			0
$868$			101405.i			0.134593i
$869$	510677.			0.676250
$870$		0			0
$871$	35870.3			0.0472823
$872$			1.17807e6i			1.54931i
$873$		0			0
$874$	175325.			0.229520
$875$	−1.31039e6			−1.71153
$876$		0			0
$877$	−347511.			−0.451824			−0.225912	−	0.974148i	$-0.572536\pi$
$877$	−347511.			−0.451824			−0.225912	+	0.974148i	$0.572536\pi$
$878$			251443.i			0.326174i
$879$		0			0
$880$			973553.i			1.25717i
$881$	−150733.			−0.194204			−0.0971018	−	0.995274i	$-0.530957\pi$
$881$	−150733.			−0.194204			−0.0971018	+	0.995274i	$0.530957\pi$
$882$		0			0
$883$	−1.40226e6			−1.79849			−0.899246	−	0.437443i	$-0.855884\pi$
$883$	−1.40226e6			−1.79849			−0.899246	+	0.437443i	$0.855884\pi$
$884$	−85633.9			−0.109583
$885$		0			0
$886$		−	1.17542e6i		−	1.49736i
$887$		−	900825.i		−	1.14497i	−0.819916	−	0.572484i	$-0.805980\pi$
$887$		−	900825.i		−	1.14497i	0.819916	−	0.572484i	$-0.194020\pi$
$888$		0			0
$889$		−	834419.i		−	1.05580i
$890$		−	894581.i		−	1.12938i
$891$		0			0
$892$		−	201854.i		−	0.253693i
$893$		−	295704.i		−	0.370813i
$894$		0			0
$895$	2.48855e6			3.10671
$896$	384824.			0.479343
$897$		0			0
$898$	−9821.02			−0.0121788
$899$			178726.i			0.221140i
$900$		0			0
$901$	−198487.			−0.244502
$902$		−	403133.i		−	0.495490i
$903$		0			0
$904$	1.05638e6			1.29266
$905$		−	1.11557e6i		−	1.36208i
$906$		0			0
$907$	754265.			0.916874			0.458437	−	0.888727i	$-0.348409\pi$
$907$	754265.			0.916874			0.458437	+	0.888727i	$0.348409\pi$
$908$			118458.i			0.143679i
$909$		0			0
$910$			772972.i			0.933429i
$911$		−	834583.i		−	1.00562i	−0.864398	−	0.502809i	$-0.832300\pi$
$911$		−	834583.i		−	1.00562i	0.864398	−	0.502809i	$-0.167700\pi$
$912$		0			0
$913$	1.39308e6			1.67123
$914$	812128.			0.972148
$915$		0			0
$916$	−179666.			−0.214129
$917$	208009.			0.247368
$918$		0			0
$919$	−558842.			−0.661696			−0.330848	−	0.943684i	$-0.607335\pi$
$919$	−558842.			−0.661696			−0.330848	+	0.943684i	$0.607335\pi$
$920$	1.48075e6			1.74946
$921$		0			0
$922$			233611.i			0.274809i
$923$			520776.i			0.611290i
$924$		0			0
$925$			3.58131e6i			4.18560i
$926$	−344750.			−0.402052
$927$		0			0
$928$	−53446.3			−0.0620614
$929$			911473.i			1.05612i	0.849208	+	0.528059i	$0.177080\pi$
$929$			911473.i			1.05612i	−0.849208	+	0.528059i	$0.822920\pi$
$930$		0			0
$931$		−	123198.i		−	0.142136i
$932$		−	161213.i		−	0.185596i
$933$		0			0
$934$	−1.05598e6			−1.21049
$935$			1.13187e6i			1.29472i
$936$		0			0
$937$		−	297178.i		−	0.338484i	−0.985575	−	0.169242i	$-0.945868\pi$
$937$		−	297178.i		−	0.338484i	0.985575	−	0.169242i	$-0.0541319\pi$
$938$	−33374.0			−0.0379317
$939$		0			0
$940$		−	353849.i		−	0.400463i
$941$	44565.6			0.0503292			0.0251646	−	0.999683i	$-0.491989\pi$
$941$	44565.6			0.0503292			0.0251646	+	0.999683i	$0.491989\pi$
$942$		0			0
$943$	−509102.			−0.572508
$944$	−256691.			−0.288049
$945$		0			0
$946$	233568.	+	656433.i	0.260995	+	0.733514i
$947$	1.06777e6			1.19064			0.595319	−	0.803490i	$-0.297026\pi$
$947$	1.06777e6			1.19064			0.595319	+	0.803490i	$0.297026\pi$
$948$		0			0
$949$			1.11605e6i			1.23923i
$950$	−538157.			−0.596296
$951$		0			0
$952$	562335.			0.620470
$953$			1.12239e6i			1.23583i	0.786246	+	0.617913i	$0.212022\pi$
$953$			1.12239e6i			1.23583i	−0.786246	+	0.617913i	$0.787978\pi$
$954$		0			0
$955$	−1.17094e6			−1.28389
$956$	47779.2			0.0522785
$957$		0			0
$958$		−	1.19934e6i		−	1.30681i
$959$	705433.			0.767041
$960$		0			0
$961$	326988.			0.354066
$962$	1.20864e6			1.30601
$963$		0			0
$964$			168682.i			0.181516i
$965$			2.06623e6i			2.21883i
$966$		0			0
$967$	769222.			0.822619			0.411310	−	0.911496i	$-0.365072\pi$
$967$	769222.			0.822619			0.411310	+	0.911496i	$0.365072\pi$
$968$		−	273319.i		−	0.291689i
$969$		0			0
$970$	532697.			0.566157
$971$	−1.12938e6			−1.19785			−0.598926	−	0.800804i	$-0.704406\pi$
$971$	−1.12938e6			−1.19785			−0.598926	+	0.800804i	$0.704406\pi$
$972$		0			0
$973$		−	245270.i		−	0.259071i
$974$		−	823025.i		−	0.867551i
$975$		0			0
$976$			1.28239e6i			1.34623i
$977$	986156.			1.03313			0.516567	−	0.856247i	$-0.327210\pi$
$977$	986156.			1.03313			0.516567	+	0.856247i	$0.327210\pi$
$978$		0			0
$979$		−	552540.i		−	0.576499i
$980$		−	147422.i		−	0.153501i
$981$		0			0
$982$	−436213.			−0.452351
$983$			777551.i			0.804677i	0.915491	+	0.402339i	$0.131803\pi$
$983$			777551.i			0.804677i	−0.915491	+	0.402339i	$0.868197\pi$
$984$		0			0
$985$		−	872725.i		−	0.899508i
$986$	140425.			0.144441
$987$		0			0
$988$		−	35908.0i		−	0.0367856i
$989$	828987.	−	294965.i	0.847530	−	0.301563i
$990$		0			0
$991$			1.84394e6i			1.87759i	0.344478	+	0.938794i	$0.388056\pi$
$991$			1.84394e6i			1.87759i	−0.344478	+	0.938794i	$0.611944\pi$
$992$			373953.i			0.380009i
$993$		0			0
$994$		−	484533.i		−	0.490400i
$995$	1.89139e6			1.91044
$996$		0			0
$997$		−	105165.i		−	0.105799i	−0.998600	−	0.0528996i	$-0.983154\pi$
$997$		−	105165.i		−	0.105799i	0.998600	−	0.0528996i	$-0.0168463\pi$
$998$	676023.			0.678735
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	387.5.b.c.343.9		12
3.2	odd	2		43.5.b.b.42.4	✓	12
12.11	even	2		688.5.b.d.257.6		12
43.42	odd	2	inner	387.5.b.c.343.4		12
129.128	even	2		43.5.b.b.42.9	yes	12
516.515	odd	2		688.5.b.d.257.7		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
43.5.b.b.42.4	✓	12	3.2	odd	2
43.5.b.b.42.9	yes	12	129.128	even	2
387.5.b.c.343.4		12	43.42	odd	2	inner
387.5.b.c.343.9		12	1.1	even	1	trivial
688.5.b.d.257.6		12	12.11	even	2
688.5.b.d.257.7		12	516.515	odd	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:	\( N \)	\(=\)	\( 387 = 3^{2} \cdot 43 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	387.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.65497i q^{2} +2.64117 q^{4} -45.6695i q^{5} +34.3337i q^{7} +68.1330i q^{8} +O(q^{10})\)	\(q+3.65497i q^{2} +2.64117 q^{4} -45.6695i q^{5} +34.3337i q^{7} +68.1330i q^{8} +166.921 q^{10} +103.099 q^{11} +134.875 q^{13} -125.489 q^{14} -206.765 q^{16} -240.390 q^{17} -100.800i q^{19} -120.621i q^{20} +376.825i q^{22} +475.879 q^{23} -1460.70 q^{25} +492.966i q^{26} +90.6813i q^{28} +159.825i q^{29} +1118.26 q^{31} +334.405i q^{32} -878.620i q^{34} +1568.00 q^{35} -2451.77i q^{37} +368.423 q^{38} +3111.60 q^{40} -1069.81 q^{41} +(1742.01 - 619.833i) q^{43} +272.303 q^{44} +1739.32i q^{46} +2933.56 q^{47} +1222.20 q^{49} -5338.84i q^{50} +356.229 q^{52} +825.685 q^{53} -4708.49i q^{55} -2339.26 q^{56} -584.155 q^{58} +1241.46 q^{59} -6202.15i q^{61} +4087.22i q^{62} -4530.49 q^{64} -6159.69i q^{65} +265.952 q^{67} -634.912 q^{68} +5731.02i q^{70} +3861.16i q^{71} +8274.70i q^{73} +8961.14 q^{74} -266.231i q^{76} +3539.78i q^{77} +4953.26 q^{79} +9442.88i q^{80} -3910.14i q^{82} +13512.1 q^{83} +10978.5i q^{85} +(2265.47 + 6367.01i) q^{86} +7024.45i q^{88} -5359.31i q^{89} +4630.77i q^{91} +1256.88 q^{92} +10722.1i q^{94} -4603.51 q^{95} +3191.32 q^{97} +4467.09i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 92 q^{4}+O(q^{10}) \) 12 * q - 92 * q^4	\( 12 q - 92 q^{4} + 182 q^{10} + 180 q^{11} - 216 q^{13} - 732 q^{14} + 1076 q^{16} - 678 q^{17} - 1566 q^{23} - 174 q^{25} + 5710 q^{31} - 936 q^{35} - 1242 q^{38} - 2618 q^{40} - 4878 q^{41} - 1108 q^{43} + 15168 q^{44} + 5526 q^{47} - 8544 q^{49} + 24084 q^{52} - 1212 q^{53} + 10152 q^{56} - 4666 q^{58} - 14016 q^{59} - 15580 q^{64} - 1088 q^{67} - 15186 q^{68} + 7674 q^{74} + 24302 q^{79} + 7032 q^{83} + 14412 q^{86} - 48354 q^{92} - 606 q^{95} - 5842 q^{97}+O(q^{100}) \) 12 * q - 92 * q^4 + 182 * q^10 + 180 * q^11 - 216 * q^13 - 732 * q^14 + 1076 * q^16 - 678 * q^17 - 1566 * q^23 - 174 * q^25 + 5710 * q^31 - 936 * q^35 - 1242 * q^38 - 2618 * q^40 - 4878 * q^41 - 1108 * q^43 + 15168 * q^44 + 5526 * q^47 - 8544 * q^49 + 24084 * q^52 - 1212 * q^53 + 10152 * q^56 - 4666 * q^58 - 14016 * q^59 - 15580 * q^64 - 1088 * q^67 - 15186 * q^68 + 7674 * q^74 + 24302 * q^79 + 7032 * q^83 + 14412 * q^86 - 48354 * q^92 - 606 * q^95 - 5842 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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