LMFDB - Embedded newform 2178.3.c.p.485.8

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2178,3,Mod(485,2178)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2178, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2178.485");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 2178 = 2 \cdot 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	2178.c (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:	$59.3462015777$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 102x^{6} + 3599x^{4} + 51708x^{2} + 249001 $ x^8 + 102x^6 + 3599x^4 + 51708*x^2 + 249001
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 198)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			485.8
Root			$-4.57204i$ of defining polynomial
Character	$\chi$	$=$	2178.485
Dual form			2178.3.c.p.485.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -2.00000 q^{4} +9.68596i q^{5} +8.70191 q^{7} -2.82843i q^{8} +O(q^{10})$	$q+1.41421i q^{2} -2.00000 q^{4} +9.68596i q^{5} +8.70191 q^{7} -2.82843i q^{8} -13.6980 q^{10} +2.23218 q^{13} +12.3064i q^{14} +4.00000 q^{16} -6.99671i q^{17} +6.91224 q^{19} -19.3719i q^{20} +34.1701i q^{23} -68.8178 q^{25} +3.15677i q^{26} -17.4038 q^{28} -48.7585i q^{29} -7.91853 q^{31} +5.65685i q^{32} +9.89484 q^{34} +84.2863i q^{35} -22.1003 q^{37} +9.77538i q^{38} +27.3960 q^{40} +64.4211i q^{41} -60.7684 q^{43} -48.3238 q^{46} +53.8920i q^{47} +26.7232 q^{49} -97.3231i q^{50} -4.46435 q^{52} +37.7268i q^{53} -24.6127i q^{56} +68.9550 q^{58} +46.6859i q^{59} +25.2133 q^{61} -11.1985i q^{62} -8.00000 q^{64} +21.6208i q^{65} -15.8370 q^{67} +13.9934i q^{68} -119.199 q^{70} -29.9277i q^{71} -32.4754 q^{73} -31.2546i q^{74} -13.8245 q^{76} -23.4865 q^{79} +38.7438i q^{80} -91.1052 q^{82} +44.7064i q^{83} +67.7698 q^{85} -85.9395i q^{86} -118.461i q^{89} +19.4242 q^{91} -68.3402i q^{92} -76.2148 q^{94} +66.9516i q^{95} -157.020 q^{97} +37.7923i q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{4} + 8 q^{7}+O(q^{10}) $ 8 * q - 16 * q^4 + 8 * q^7	$ 8 q - 16 q^{4} + 8 q^{7} - 12 q^{10} - 36 q^{13} + 32 q^{16} + 72 q^{19} - 64 q^{25} - 16 q^{28} - 88 q^{31} - 56 q^{34} + 108 q^{37} + 24 q^{40} - 220 q^{43} - 68 q^{46} + 56 q^{49} + 72 q^{52} + 112 q^{58} - 160 q^{61} - 64 q^{64} - 276 q^{67} - 92 q^{70} - 488 q^{73} - 144 q^{76} - 368 q^{79} - 388 q^{82} + 248 q^{85} - 356 q^{91} - 120 q^{94} - 832 q^{97}+O(q^{100}) $ 8 * q - 16 * q^4 + 8 * q^7 - 12 * q^10 - 36 * q^13 + 32 * q^16 + 72 * q^19 - 64 * q^25 - 16 * q^28 - 88 * q^31 - 56 * q^34 + 108 * q^37 + 24 * q^40 - 220 * q^43 - 68 * q^46 + 56 * q^49 + 72 * q^52 + 112 * q^58 - 160 * q^61 - 64 * q^64 - 276 * q^67 - 92 * q^70 - 488 * q^73 - 144 * q^76 - 368 * q^79 - 388 * q^82 + 248 * q^85 - 356 * q^91 - 120 * q^94 - 832 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2178\mathbb{Z}\right)^\times$.

$n$	$1333$	$1937$
$\chi(n)$	$1$	$-1$

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
$3$	0	0
$3$	0	0
$4$	−2.00000	−0.500000
$5$	9.68596i	1.93719i	0.248640	+	0.968596i	$0.420016\pi$
$5$	9.68596i	1.93719i	−0.248640	+	0.968596i	$0.579984\pi$
$6$	0	0
$7$	8.70191	1.24313	0.621565	−	0.783363i	$-0.286497\pi$
$7$	8.70191	1.24313	0.621565	+	0.783363i	$0.286497\pi$
$8$	− 2.82843i	− 0.353553i
$9$	0	0
$10$	−13.6980	−1.36980
$11$	0	0
$11$	0	0
$12$	0	0
$13$	2.23218	0.171706	0.0858529	−	0.996308i	$-0.472638\pi$
$13$	2.23218	0.171706	0.0858529	+	0.996308i	$0.472638\pi$
$14$	12.3064i	0.879025i
$15$	0	0
$16$	4.00000	0.250000
$17$	− 6.99671i	− 0.411571i	−0.978597	−	0.205786i	$-0.934025\pi$
$17$	− 6.99671i	− 0.411571i	0.978597	−	0.205786i	$-0.0659749\pi$
$18$	0	0
$19$	6.91224	0.363802	0.181901	−	0.983317i	$-0.441775\pi$
$19$	6.91224	0.363802	0.181901	+	0.983317i	$0.441775\pi$
$20$	− 19.3719i	− 0.968596i
$21$	0	0
$22$	0	0
$23$	34.1701i	1.48566i	0.669482	+	0.742828i	$0.266516\pi$
$23$	34.1701i	1.48566i	−0.669482	+	0.742828i	$0.733484\pi$
$24$	0	0
$25$	−68.8178	−2.75271
$26$	3.15677i	0.121414i
$27$	0	0
$28$	−17.4038	−0.621565
$29$	− 48.7585i	− 1.68133i	−0.541556	−	0.840664i	$-0.682165\pi$
$29$	− 48.7585i	− 1.68133i	0.541556	−	0.840664i	$-0.317835\pi$
$30$	0	0
$31$	−7.91853	−0.255437	−0.127718	−	0.991810i	$-0.540765\pi$
$31$	−7.91853	−0.255437	−0.127718	+	0.991810i	$0.540765\pi$
$32$	5.65685i	0.176777i
$33$	0	0
$34$	9.89484	0.291025
$35$	84.2863i	2.40818i
$36$	0	0
$37$	−22.1003	−0.597306	−0.298653	−	0.954362i	$-0.596537\pi$
$37$	−22.1003	−0.597306	−0.298653	+	0.954362i	$0.596537\pi$
$38$	9.77538i	0.257247i
$39$	0	0
$40$	27.3960	0.684901
$41$	64.4211i	1.57125i	0.618706	+	0.785623i	$0.287657\pi$
$41$	64.4211i	1.57125i	−0.618706	+	0.785623i	$0.712343\pi$
$42$	0	0
$43$	−60.7684	−1.41322	−0.706609	−	0.707604i	$-0.749776\pi$
$43$	−60.7684	−1.41322	−0.706609	+	0.707604i	$0.749776\pi$
$44$	0	0
$45$	0	0
$46$	−48.3238	−1.05052
$47$	53.8920i	1.14664i	0.819332	+	0.573319i	$0.194344\pi$
$47$	53.8920i	1.14664i	−0.819332	+	0.573319i	$0.805656\pi$
$48$	0	0
$49$	26.7232	0.545371
$50$	− 97.3231i	− 1.94646i
$51$	0	0
$52$	−4.46435	−0.0858529
$53$	37.7268i	0.711827i	0.934519	+	0.355913i	$0.115830\pi$
$53$	37.7268i	0.711827i	−0.934519	+	0.355913i	$0.884170\pi$
$54$	0	0
$55$	0	0
$56$	− 24.6127i	− 0.439513i
$57$	0	0
$58$	68.9550	1.18888
$59$	46.6859i	0.791287i	0.918404	+	0.395643i	$0.129478\pi$
$59$	46.6859i	0.791287i	−0.918404	+	0.395643i	$0.870522\pi$
$60$	0	0
$61$	25.2133	0.413332	0.206666	−	0.978412i	$-0.433739\pi$
$61$	25.2133	0.413332	0.206666	+	0.978412i	$0.433739\pi$
$62$	− 11.1985i	− 0.180621i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	21.6208i	0.332627i
$66$	0	0
$67$	−15.8370	−0.236373	−0.118187	−	0.992991i	$-0.537708\pi$
$67$	−15.8370	−0.236373	−0.118187	+	0.992991i	$0.537708\pi$
$68$	13.9934i	0.205786i
$69$	0	0
$70$	−119.199	−1.70284
$71$	− 29.9277i	− 0.421517i	−0.977538	−	0.210758i	$-0.932407\pi$
$71$	− 29.9277i	− 0.421517i	0.977538	−	0.210758i	$-0.0675933\pi$
$72$	0	0
$73$	−32.4754	−0.444869	−0.222434	−	0.974948i	$-0.571400\pi$
$73$	−32.4754	−0.444869	−0.222434	+	0.974948i	$0.571400\pi$
$74$	− 31.2546i	− 0.422359i
$75$	0	0
$76$	−13.8245	−0.181901
$77$	0	0
$78$	0	0
$79$	−23.4865	−0.297298	−0.148649	−	0.988890i	$-0.547492\pi$
$79$	−23.4865	−0.297298	−0.148649	+	0.988890i	$0.547492\pi$
$80$	38.7438i	0.484298i
$81$	0	0
$82$	−91.1052	−1.11104
$83$	44.7064i	0.538631i	0.963052	+	0.269316i	$0.0867975\pi$
$83$	44.7064i	0.538631i	−0.963052	+	0.269316i	$0.913203\pi$
$84$	0	0
$85$	67.7698	0.797292
$86$	− 85.9395i	− 0.999296i
$87$	0	0
$88$	0	0
$89$	− 118.461i	− 1.33102i	−0.746389	−	0.665510i	$-0.768214\pi$
$89$	− 118.461i	− 1.33102i	0.746389	−	0.665510i	$-0.231786\pi$
$90$	0	0
$91$	19.4242	0.213453
$92$	− 68.3402i	− 0.742828i
$93$	0	0
$94$	−76.2148	−0.810795
$95$	66.9516i	0.704754i
$96$	0	0
$97$	−157.020	−1.61876	−0.809381	−	0.587283i	$-0.800197\pi$
$97$	−157.020	−1.61876	−0.809381	+	0.587283i	$0.800197\pi$
$98$	37.7923i	0.385635i
$99$	0	0
$100$	137.636	1.37636
$101$	106.922i	1.05864i	0.848423	+	0.529319i	$0.177552\pi$
$101$	106.922i	1.05864i	−0.848423	+	0.529319i	$0.822448\pi$
$102$	0	0
$103$	108.474	1.05314	0.526572	−	0.850130i	$-0.323477\pi$
$103$	108.474	1.05314	0.526572	+	0.850130i	$0.323477\pi$
$104$	− 6.31355i	− 0.0607072i
$105$	0	0
$106$	−53.3538	−0.503338
$107$	185.324i	1.73200i	0.500045	+	0.866000i	$0.333317\pi$
$107$	185.324i	1.73200i	−0.500045	+	0.866000i	$0.666683\pi$
$108$	0	0
$109$	−40.2769	−0.369512	−0.184756	−	0.982784i	$-0.559150\pi$
$109$	−40.2769	−0.369512	−0.184756	+	0.982784i	$0.559150\pi$
$110$	0	0
$111$	0	0
$112$	34.8076	0.310782
$113$	− 91.3733i	− 0.808613i	−0.914624	−	0.404307i	$-0.867513\pi$
$113$	− 91.3733i	− 0.808613i	0.914624	−	0.404307i	$-0.132487\pi$
$114$	0	0
$115$	−330.970	−2.87800
$116$	97.5171i	0.840664i
$117$	0	0
$118$	−66.0238	−0.559524
$119$	− 60.8847i	− 0.511636i
$120$	0	0
$121$	0	0
$122$	35.6569i	0.292270i
$123$	0	0
$124$	15.8371	0.127718
$125$	− 424.417i	− 3.39534i
$126$	0	0
$127$	48.9115	0.385130	0.192565	−	0.981284i	$-0.438319\pi$
$127$	48.9115	0.385130	0.192565	+	0.981284i	$0.438319\pi$
$128$	− 11.3137i	− 0.0883883i
$129$	0	0
$130$	−30.5764	−0.235203
$131$	− 124.127i	− 0.947535i	−0.880650	−	0.473768i	$-0.842894\pi$
$131$	− 124.127i	− 0.947535i	0.880650	−	0.473768i	$-0.157106\pi$
$132$	0	0
$133$	60.1496	0.452253
$134$	− 22.3969i	− 0.167141i
$135$	0	0
$136$	−19.7897	−0.145512
$137$	95.6379i	0.698087i	0.937107	+	0.349043i	$0.113493\pi$
$137$	95.6379i	0.698087i	−0.937107	+	0.349043i	$0.886507\pi$
$138$	0	0
$139$	83.7527	0.602538	0.301269	−	0.953539i	$-0.402590\pi$
$139$	83.7527	0.602538	0.301269	+	0.953539i	$0.402590\pi$
$140$	− 168.573i	− 1.20409i
$141$	0	0
$142$	42.3241	0.298057
$143$	0	0
$144$	0	0
$145$	472.273	3.25706
$146$	− 45.9272i	− 0.314570i
$147$	0	0
$148$	44.2007	0.298653
$149$	127.277i	0.854207i	0.904203	+	0.427104i	$0.140466\pi$
$149$	127.277i	0.854207i	−0.904203	+	0.427104i	$0.859534\pi$
$150$	0	0
$151$	7.39913	0.0490008	0.0245004	−	0.999700i	$-0.492200\pi$
$151$	7.39913	0.0490008	0.0245004	+	0.999700i	$0.492200\pi$
$152$	− 19.5508i	− 0.128623i
$153$	0	0
$154$	0	0
$155$	− 76.6986i	− 0.494830i
$156$	0	0
$157$	285.563	1.81887	0.909437	−	0.415842i	$-0.136513\pi$
$157$	285.563	1.81887	0.909437	+	0.415842i	$0.136513\pi$
$158$	− 33.2150i	− 0.210221i
$159$	0	0
$160$	−54.7921	−0.342450
$161$	297.345i	1.84686i
$162$	0	0
$163$	−207.070	−1.27037	−0.635185	−	0.772360i	$-0.719076\pi$
$163$	−207.070	−1.27037	−0.635185	+	0.772360i	$0.719076\pi$
$164$	− 128.842i	− 0.785623i
$165$	0	0
$166$	−63.2244	−0.380870
$167$	− 83.9846i	− 0.502902i	−0.967870	−	0.251451i	$-0.919092\pi$
$167$	− 83.9846i	− 0.502902i	0.967870	−	0.251451i	$-0.0809077\pi$
$168$	0	0
$169$	−164.017	−0.970517
$170$	95.8410i	0.563771i
$171$	0	0
$172$	121.537	0.706609
$173$	102.532i	0.592669i	0.955084	+	0.296335i	$0.0957644\pi$
$173$	102.532i	0.592669i	−0.955084	+	0.296335i	$0.904236\pi$
$174$	0	0
$175$	−598.846	−3.42198
$176$	0	0
$177$	0	0
$178$	167.529	0.941173
$179$	− 116.254i	− 0.649465i	−0.945806	−	0.324732i	$-0.894726\pi$
$179$	− 116.254i	− 0.649465i	0.945806	−	0.324732i	$-0.105274\pi$
$180$	0	0
$181$	127.554	0.704718	0.352359	−	0.935865i	$-0.385380\pi$
$181$	127.554	0.704718	0.352359	+	0.935865i	$0.385380\pi$
$182$	27.4699i	0.150934i
$183$	0	0
$184$	96.6476	0.525259
$185$	− 214.063i	− 1.15710i
$186$	0	0
$187$	0	0
$188$	− 107.784i	− 0.573319i
$189$	0	0
$190$	−94.6839	−0.498336
$191$	− 179.115i	− 0.937773i	−0.883258	−	0.468887i	$-0.844655\pi$
$191$	− 179.115i	− 0.937773i	0.883258	−	0.468887i	$-0.155345\pi$
$192$	0	0
$193$	−163.821	−0.848811	−0.424406	−	0.905472i	$-0.639517\pi$
$193$	−163.821	−0.848811	−0.424406	+	0.905472i	$0.639517\pi$
$194$	− 222.060i	− 1.14464i
$195$	0	0
$196$	−53.4463	−0.272685
$197$	148.625i	0.754440i	0.926124	+	0.377220i	$0.123120\pi$
$197$	148.625i	0.754440i	−0.926124	+	0.377220i	$0.876880\pi$
$198$	0	0
$199$	119.556	0.600784	0.300392	−	0.953816i	$-0.402883\pi$
$199$	119.556	0.600784	0.300392	+	0.953816i	$0.402883\pi$
$200$	194.646i	0.973231i
$201$	0	0
$202$	−151.211	−0.748570
$203$	− 424.292i	− 2.09011i
$204$	0	0
$205$	−623.980	−3.04380
$206$	153.405i	0.744686i
$207$	0	0
$208$	8.92870	0.0429265
$209$	0	0
$210$	0	0
$211$	−154.169	−0.730658	−0.365329	−	0.930878i	$-0.619044\pi$
$211$	−154.169	−0.730658	−0.365329	+	0.930878i	$0.619044\pi$
$212$	− 75.4536i	− 0.355913i
$213$	0	0
$214$	−262.088	−1.22471
$215$	− 588.600i	− 2.73767i
$216$	0	0
$217$	−68.9063	−0.317541
$218$	− 56.9601i	− 0.261285i
$219$	0	0
$220$	0	0
$221$	− 15.6179i	− 0.0706692i
$222$	0	0
$223$	−332.614	−1.49154	−0.745771	−	0.666202i	$-0.767919\pi$
$223$	−332.614	−1.49154	−0.745771	+	0.666202i	$0.767919\pi$
$224$	49.2254i	0.219756i
$225$	0	0
$226$	129.221	0.571776
$227$	− 227.994i	− 1.00438i	−0.864757	−	0.502190i	$-0.832528\pi$
$227$	− 227.994i	− 1.00438i	0.864757	−	0.502190i	$-0.167472\pi$
$228$	0	0
$229$	1.07741	0.00470483	0.00235241	−	0.999997i	$-0.499251\pi$
$229$	1.07741	0.00470483	0.00235241	+	0.999997i	$0.499251\pi$
$230$	− 468.062i	− 2.03505i
$231$	0	0
$232$	−137.910	−0.594440
$233$	240.036i	1.03020i	0.857131	+	0.515098i	$0.172245\pi$
$233$	240.036i	1.03020i	−0.857131	+	0.515098i	$0.827755\pi$
$234$	0	0
$235$	−521.995	−2.22126
$236$	− 93.3718i	− 0.395643i
$237$	0	0
$238$	86.1040	0.361781
$239$	− 202.855i	− 0.848765i	−0.905483	−	0.424383i	$-0.860491\pi$
$239$	− 202.855i	− 0.848765i	0.905483	−	0.424383i	$-0.139509\pi$
$240$	0	0
$241$	162.375	0.673753	0.336877	−	0.941549i	$-0.390629\pi$
$241$	162.375	0.673753	0.336877	+	0.941549i	$0.390629\pi$
$242$	0	0
$243$	0	0
$244$	−50.4265	−0.206666
$245$	258.840i	1.05649i
$246$	0	0
$247$	15.4293	0.0624669
$248$	22.3970i	0.0903105i
$249$	0	0
$250$	600.217	2.40087
$251$	213.139i	0.849157i	0.905391	+	0.424579i	$0.139578\pi$
$251$	213.139i	0.849157i	−0.905391	+	0.424579i	$0.860422\pi$
$252$	0	0
$253$	0	0
$254$	69.1714i	0.272328i
$255$	0	0
$256$	16.0000	0.0625000
$257$	− 125.892i	− 0.489853i	−0.969542	−	0.244926i	$-0.921236\pi$
$257$	− 125.892i	− 0.489853i	0.969542	−	0.244926i	$-0.0787638\pi$
$258$	0	0
$259$	−192.315	−0.742529
$260$	− 43.2415i	− 0.166314i
$261$	0	0
$262$	175.542	0.670009
$263$	− 329.312i	− 1.25214i	−0.779769	−	0.626068i	$-0.784663\pi$
$263$	− 329.312i	− 1.25214i	0.779769	−	0.626068i	$-0.215337\pi$
$264$	0	0
$265$	−365.420	−1.37895
$266$	85.0644i	0.319791i
$267$	0	0
$268$	31.6740	0.118187
$269$	− 72.0601i	− 0.267881i	−0.990989	−	0.133941i	$-0.957237\pi$
$269$	− 72.0601i	− 0.267881i	0.990989	−	0.133941i	$-0.0427632\pi$
$270$	0	0
$271$	226.677	0.836445	0.418222	−	0.908345i	$-0.362653\pi$
$271$	226.677	0.836445	0.418222	+	0.908345i	$0.362653\pi$
$272$	− 27.9868i	− 0.102893i
$273$	0	0
$274$	−135.252	−0.493622
$275$	0	0
$276$	0	0
$277$	83.2686	0.300609	0.150304	−	0.988640i	$-0.451975\pi$
$277$	83.2686	0.300609	0.150304	+	0.988640i	$0.451975\pi$
$278$	118.444i	0.426058i
$279$	0	0
$280$	238.398	0.851420
$281$	− 192.738i	− 0.685899i	−0.939354	−	0.342950i	$-0.888574\pi$
$281$	− 192.738i	− 0.685899i	0.939354	−	0.342950i	$-0.111426\pi$
$282$	0	0
$283$	−488.760	−1.72707	−0.863533	−	0.504292i	$-0.831754\pi$
$283$	−488.760	−1.72707	−0.863533	+	0.504292i	$0.831754\pi$
$284$	59.8554i	0.210758i
$285$	0	0
$286$	0	0
$287$	560.586i	1.95326i
$288$	0	0
$289$	240.046	0.830609
$290$	667.895i	2.30309i
$291$	0	0
$292$	64.9509	0.222434
$293$	25.4233i	0.0867689i	0.999058	+	0.0433845i	$0.0138140\pi$
$293$	25.4233i	0.0867689i	−0.999058	+	0.0433845i	$0.986186\pi$
$294$	0	0
$295$	−452.198	−1.53287
$296$	62.5092i	0.211180i
$297$	0	0
$298$	−179.997	−0.604016
$299$	76.2737i	0.255096i
$300$	0	0
$301$	−528.801	−1.75681
$302$	10.4639i	0.0346488i
$303$	0	0
$304$	27.6489	0.0909505
$305$	244.215i	0.800704i
$306$	0	0
$307$	−166.206	−0.541388	−0.270694	−	0.962665i	$-0.587253\pi$
$307$	−166.206	−0.541388	−0.270694	+	0.962665i	$0.587253\pi$
$308$	0	0
$309$	0	0
$310$	108.468	0.349897
$311$	520.793i	1.67458i	0.546761	+	0.837288i	$0.315860\pi$
$311$	520.793i	1.67458i	−0.546761	+	0.837288i	$0.684140\pi$
$312$	0	0
$313$	−23.1201	−0.0738663	−0.0369331	−	0.999318i	$-0.511759\pi$
$313$	−23.1201	−0.0738663	−0.0369331	+	0.999318i	$0.511759\pi$
$314$	403.847i	1.28614i
$315$	0	0
$316$	46.9731	0.148649
$317$	62.3754i	0.196768i	0.995149	+	0.0983839i	$0.0313673\pi$
$317$	62.3754i	0.196768i	−0.995149	+	0.0983839i	$0.968633\pi$
$318$	0	0
$319$	0	0
$320$	− 77.4877i	− 0.242149i
$321$	0	0
$322$	−420.509	−1.30593
$323$	− 48.3629i	− 0.149730i
$324$	0	0
$325$	−153.613	−0.472657
$326$	− 292.842i	− 0.898288i
$327$	0	0
$328$	182.210	0.555519
$329$	468.963i	1.42542i
$330$	0	0
$331$	356.863	1.07814	0.539068	−	0.842263i	$-0.318777\pi$
$331$	356.863	1.07814	0.539068	+	0.842263i	$0.318777\pi$
$332$	− 89.4128i	− 0.269316i
$333$	0	0
$334$	118.772	0.355605
$335$	− 153.397i	− 0.457901i
$336$	0	0
$337$	367.866	1.09159	0.545795	−	0.837919i	$-0.316228\pi$
$337$	367.866	1.09159	0.545795	+	0.837919i	$0.316228\pi$
$338$	− 231.956i	− 0.686259i
$339$	0	0
$340$	−135.540	−0.398646
$341$	0	0
$342$	0	0
$343$	−193.851	−0.565163
$344$	171.879i	0.499648i
$345$	0	0
$346$	−145.002	−0.419080
$347$	84.6523i	0.243955i	0.992533	+	0.121977i	$0.0389235\pi$
$347$	84.6523i	0.243955i	−0.992533	+	0.121977i	$0.961076\pi$
$348$	0	0
$349$	432.333	1.23878	0.619388	−	0.785085i	$-0.287381\pi$
$349$	432.333	1.23878	0.619388	+	0.785085i	$0.287381\pi$
$350$	− 846.896i	− 2.41970i
$351$	0	0
$352$	0	0
$353$	536.582i	1.52006i	0.649887	+	0.760031i	$0.274816\pi$
$353$	536.582i	1.52006i	−0.649887	+	0.760031i	$0.725184\pi$
$354$	0	0
$355$	289.878	0.816559
$356$	236.921i	0.665510i
$357$	0	0
$358$	164.408	0.459241
$359$	67.0464i	0.186759i	0.995631	+	0.0933793i	$0.0297669\pi$
$359$	67.0464i	0.186759i	−0.995631	+	0.0933793i	$0.970233\pi$
$360$	0	0
$361$	−313.221	−0.867648
$362$	180.389i	0.498311i
$363$	0	0
$364$	−38.8484	−0.106726
$365$	− 314.556i	− 0.861797i
$366$	0	0
$367$	464.098	1.26457	0.632286	−	0.774735i	$-0.282117\pi$
$367$	464.098	1.26457	0.632286	+	0.774735i	$0.282117\pi$
$368$	136.680i	0.371414i
$369$	0	0
$370$	302.731	0.818191
$371$	328.295i	0.884893i
$372$	0	0
$373$	300.514	0.805668	0.402834	−	0.915273i	$-0.368025\pi$
$373$	300.514	0.805668	0.402834	+	0.915273i	$0.368025\pi$
$374$	0	0
$375$	0	0
$376$	152.430	0.405398
$377$	− 108.838i	− 0.288694i
$378$	0	0
$379$	−151.282	−0.399161	−0.199580	−	0.979881i	$-0.563958\pi$
$379$	−151.282	−0.399161	−0.199580	+	0.979881i	$0.563958\pi$
$380$	− 133.903i	− 0.352377i
$381$	0	0
$382$	253.306	0.663106
$383$	− 357.332i	− 0.932983i	−0.884526	−	0.466491i	$-0.845518\pi$
$383$	− 357.332i	− 0.932983i	0.884526	−	0.466491i	$-0.154482\pi$
$384$	0	0
$385$	0	0
$386$	− 231.677i	− 0.600200i
$387$	0	0
$388$	314.040	0.809381
$389$	− 201.046i	− 0.516827i	−0.966034	−	0.258413i	$-0.916800\pi$
$389$	− 201.046i	− 0.516827i	0.966034	−	0.258413i	$-0.0831996\pi$
$390$	0	0
$391$	239.078	0.611453
$392$	− 75.5845i	− 0.192818i
$393$	0	0
$394$	−210.187	−0.533469
$395$	− 227.490i	− 0.575923i
$396$	0	0
$397$	−177.510	−0.447129	−0.223564	−	0.974689i	$-0.571769\pi$
$397$	−177.510	−0.447129	−0.223564	+	0.974689i	$0.571769\pi$
$398$	169.078i	0.424818i
$399$	0	0
$400$	−275.271	−0.688178
$401$	− 562.024i	− 1.40156i	−0.713380	−	0.700778i	$-0.752837\pi$
$401$	− 562.024i	− 1.40156i	0.713380	−	0.700778i	$-0.247163\pi$
$402$	0	0
$403$	−17.6756	−0.0438599
$404$	− 213.845i	− 0.529319i
$405$	0	0
$406$	600.040	1.47793
$407$	0	0
$408$	0	0
$409$	655.446	1.60256	0.801279	−	0.598291i	$-0.204153\pi$
$409$	655.446	1.60256	0.801279	+	0.598291i	$0.204153\pi$
$410$	− 882.441i	− 2.15229i
$411$	0	0
$412$	−216.948	−0.526572
$413$	406.256i	0.983672i
$414$	0	0
$415$	−433.024	−1.04343
$416$	12.6271i	0.0303536i
$417$	0	0
$418$	0	0
$419$	741.961i	1.77079i	0.464840	+	0.885395i	$0.346112\pi$
$419$	741.961i	1.77079i	−0.464840	+	0.885395i	$0.653888\pi$
$420$	0	0
$421$	−316.416	−0.751583	−0.375791	−	0.926704i	$-0.622629\pi$
$421$	−316.416	−0.751583	−0.375791	+	0.926704i	$0.622629\pi$
$422$	− 218.028i	− 0.516653i
$423$	0	0
$424$	106.708	0.251669
$425$	481.498i	1.13294i
$426$	0	0
$427$	219.403	0.513825
$428$	− 370.648i	− 0.866000i
$429$	0	0
$430$	832.406	1.93583
$431$	− 487.300i	− 1.13063i	−0.824876	−	0.565314i	$-0.808755\pi$
$431$	− 487.300i	− 1.13063i	0.824876	−	0.565314i	$-0.191245\pi$
$432$	0	0
$433$	601.307	1.38870	0.694350	−	0.719637i	$-0.255692\pi$
$433$	601.307	1.38870	0.694350	+	0.719637i	$0.255692\pi$
$434$	− 97.4483i	− 0.224535i
$435$	0	0
$436$	80.5537	0.184756
$437$	236.192i	0.540485i
$438$	0	0
$439$	498.971	1.13661	0.568304	−	0.822819i	$-0.307600\pi$
$439$	498.971	1.13661	0.568304	+	0.822819i	$0.307600\pi$
$440$	0	0
$441$	0	0
$442$	22.0870	0.0499706
$443$	− 287.634i	− 0.649288i	−0.945836	−	0.324644i	$-0.894756\pi$
$443$	− 287.634i	− 0.649288i	0.945836	−	0.324644i	$-0.105244\pi$
$444$	0	0
$445$	1147.41	2.57844
$446$	− 470.387i	− 1.05468i
$447$	0	0
$448$	−69.6153	−0.155391
$449$	683.808i	1.52296i	0.648190	+	0.761478i	$0.275526\pi$
$449$	683.808i	1.52296i	−0.648190	+	0.761478i	$0.724474\pi$
$450$	0	0
$451$	0	0
$452$	182.747i	0.404307i
$453$	0	0
$454$	322.433	0.710205
$455$	188.142i	0.413499i
$456$	0	0
$457$	−301.238	−0.659165	−0.329583	−	0.944127i	$-0.606908\pi$
$457$	−301.238	−0.659165	−0.329583	+	0.944127i	$0.606908\pi$
$458$	1.52368i	0.00332682i
$459$	0	0
$460$	661.940	1.43900
$461$	− 210.253i	− 0.456080i	−0.973652	−	0.228040i	$-0.926768\pi$
$461$	− 210.253i	− 0.456080i	0.973652	−	0.228040i	$-0.0732317\pi$
$462$	0	0
$463$	−344.360	−0.743757	−0.371879	−	0.928281i	$-0.621286\pi$
$463$	−344.360	−0.743757	−0.371879	+	0.928281i	$0.621286\pi$
$464$	− 195.034i	− 0.420332i
$465$	0	0
$466$	−339.462	−0.728459
$467$	246.571i	0.527990i	0.964524	+	0.263995i	$0.0850403\pi$
$467$	246.571i	0.527990i	−0.964524	+	0.263995i	$0.914960\pi$
$468$	0	0
$469$	−137.812	−0.293843
$470$	− 738.213i	− 1.57067i
$471$	0	0
$472$	132.048	0.279762
$473$	0	0
$474$	0	0
$475$	−475.685	−1.00144
$476$	121.769i	0.255818i
$477$	0	0
$478$	286.880	0.600168
$479$	339.973i	0.709756i	0.934913	+	0.354878i	$0.115478\pi$
$479$	339.973i	0.709756i	−0.934913	+	0.354878i	$0.884522\pi$
$480$	0	0
$481$	−49.3318	−0.102561
$482$	229.632i	0.476416i
$483$	0	0
$484$	0	0
$485$	− 1520.89i	− 3.13585i
$486$	0	0
$487$	182.012	0.373741	0.186870	−	0.982385i	$-0.440166\pi$
$487$	182.012	0.373741	0.186870	+	0.982385i	$0.440166\pi$
$488$	− 71.3139i	− 0.146135i
$489$	0	0
$490$	−366.054	−0.747050
$491$	946.754i	1.92822i	0.265509	+	0.964108i	$0.414460\pi$
$491$	946.754i	1.92822i	−0.265509	+	0.964108i	$0.585540\pi$
$492$	0	0
$493$	−341.149	−0.691987
$494$	21.8204i	0.0441708i
$495$	0	0
$496$	−31.6741	−0.0638591
$497$	− 260.428i	− 0.524000i
$498$	0	0
$499$	774.386	1.55188	0.775938	−	0.630809i	$-0.217277\pi$
$499$	774.386	1.55188	0.775938	+	0.630809i	$0.217277\pi$
$500$	848.835i	1.69767i
$501$	0	0
$502$	−301.423	−0.600445
$503$	− 430.846i	− 0.856553i	−0.903648	−	0.428276i	$-0.859121\pi$
$503$	− 430.846i	− 0.856553i	0.903648	−	0.428276i	$-0.140879\pi$
$504$	0	0
$505$	−1035.65	−2.05078
$506$	0	0
$507$	0	0
$508$	−97.8231	−0.192565
$509$	237.103i	0.465822i	0.972498	+	0.232911i	$0.0748251\pi$
$509$	237.103i	0.465822i	−0.972498	+	0.232911i	$0.925175\pi$
$510$	0	0
$511$	−282.598	−0.553030
$512$	22.6274i	0.0441942i
$513$	0	0
$514$	178.038	0.346378
$515$	1050.67i	2.04014i
$516$	0	0
$517$	0	0
$518$	− 271.975i	− 0.525047i
$519$	0	0
$520$	61.1528	0.117601
$521$	− 27.8316i	− 0.0534196i	−0.999643	−	0.0267098i	$-0.991497\pi$
$521$	− 27.8316i	− 0.0534196i	0.999643	−	0.0267098i	$-0.00850301\pi$
$522$	0	0
$523$	408.302	0.780693	0.390346	−	0.920668i	$-0.372355\pi$
$523$	408.302	0.780693	0.390346	+	0.920668i	$0.372355\pi$
$524$	248.254i	0.473768i
$525$	0	0
$526$	465.717	0.885393
$527$	55.4037i	0.105130i
$528$	0	0
$529$	−638.595	−1.20717
$530$	− 516.783i	− 0.975061i
$531$	0	0
$532$	−120.299	−0.226126
$533$	143.799i	0.269792i
$534$	0	0
$535$	−1795.04	−3.35521
$536$	44.7939i	0.0835706i
$537$	0	0
$538$	101.908	0.189421
$539$	0	0
$540$	0	0
$541$	303.270	0.560574	0.280287	−	0.959916i	$-0.409570\pi$
$541$	303.270	0.560574	0.280287	+	0.959916i	$0.409570\pi$
$542$	320.569i	0.591456i
$543$	0	0
$544$	39.5794	0.0727562
$545$	− 390.120i	− 0.715817i
$546$	0	0
$547$	646.908	1.18265	0.591324	−	0.806434i	$-0.298606\pi$
$547$	646.908	1.18265	0.591324	+	0.806434i	$0.298606\pi$
$548$	− 191.276i	− 0.349043i
$549$	0	0
$550$	0	0
$551$	− 337.031i	− 0.611671i
$552$	0	0
$553$	−204.378	−0.369580
$554$	117.760i	0.212562i
$555$	0	0
$556$	−167.505	−0.301269
$557$	57.2563i	0.102794i	0.998678	+	0.0513970i	$0.0163674\pi$
$557$	57.2563i	0.102794i	−0.998678	+	0.0513970i	$0.983633\pi$
$558$	0	0
$559$	−135.646	−0.242658
$560$	337.145i	0.602045i
$561$	0	0
$562$	272.572	0.485004
$563$	− 464.393i	− 0.824854i	−0.910991	−	0.412427i	$-0.864681\pi$
$563$	− 464.393i	− 0.824854i	0.910991	−	0.412427i	$-0.135319\pi$
$564$	0	0
$565$	885.038	1.56644
$566$	− 691.211i	− 1.22122i
$567$	0	0
$568$	−84.6483	−0.149029
$569$	788.819i	1.38632i	0.720782	+	0.693162i	$0.243783\pi$
$569$	788.819i	1.38632i	−0.720782	+	0.693162i	$0.756217\pi$
$570$	0	0
$571$	964.425	1.68901	0.844505	−	0.535547i	$-0.179895\pi$
$571$	964.425	1.68901	0.844505	+	0.535547i	$0.179895\pi$
$572$	0	0
$573$	0	0
$574$	−792.788	−1.38116
$575$	− 2351.51i	− 4.08958i
$576$	0	0
$577$	69.5270	0.120497	0.0602487	−	0.998183i	$-0.480811\pi$
$577$	69.5270	0.120497	0.0602487	+	0.998183i	$0.480811\pi$
$578$	339.476i	0.587329i
$579$	0	0
$580$	−944.546	−1.62853
$581$	389.031i	0.669588i
$582$	0	0
$583$	0	0
$584$	91.8544i	0.157285i
$585$	0	0
$586$	−35.9540	−0.0613549
$587$	− 671.636i	− 1.14418i	−0.820189	−	0.572092i	$-0.806132\pi$
$587$	− 671.636i	− 1.14418i	0.820189	−	0.572092i	$-0.193868\pi$
$588$	0	0
$589$	−54.7348	−0.0929283
$590$	− 639.504i	− 1.08391i
$591$	0	0
$592$	−88.4014	−0.149327
$593$	− 175.258i	− 0.295545i	−0.989021	−	0.147772i	$-0.952790\pi$
$593$	− 175.258i	− 0.295545i	0.989021	−	0.147772i	$-0.0472103\pi$
$594$	0	0
$595$	589.727	0.991137
$596$	− 254.554i	− 0.427104i
$597$	0	0
$598$	−107.867	−0.180380
$599$	621.366i	1.03734i	0.854975	+	0.518670i	$0.173573\pi$
$599$	621.366i	1.03734i	−0.854975	+	0.518670i	$0.826427\pi$
$600$	0	0
$601$	13.8039	0.0229682	0.0114841	−	0.999934i	$-0.496344\pi$
$601$	13.8039	0.0229682	0.0114841	+	0.999934i	$0.496344\pi$
$602$	− 747.837i	− 1.24225i
$603$	0	0
$604$	−14.7983	−0.0245004
$605$	0	0
$606$	0	0
$607$	−486.294	−0.801143	−0.400571	−	0.916266i	$-0.631188\pi$
$607$	−486.294	−0.801143	−0.400571	+	0.916266i	$0.631188\pi$
$608$	39.1015i	0.0643117i
$609$	0	0
$610$	−345.372	−0.566183
$611$	120.296i	0.196884i
$612$	0	0
$613$	−276.391	−0.450883	−0.225441	−	0.974257i	$-0.572382\pi$
$613$	−276.391	−0.450883	−0.225441	+	0.974257i	$0.572382\pi$
$614$	− 235.051i	− 0.382819i
$615$	0	0
$616$	0	0
$617$	− 5.53948i	− 0.00897808i	−0.999990	−	0.00448904i	$-0.998571\pi$
$617$	− 5.53948i	− 0.00897808i	0.999990	−	0.00448904i	$-0.00142891\pi$
$618$	0	0
$619$	−222.292	−0.359115	−0.179558	−	0.983747i	$-0.557467\pi$
$619$	−222.292	−0.359115	−0.179558	+	0.983747i	$0.557467\pi$
$620$	153.397i	0.247415i
$621$	0	0
$622$	−736.513	−1.18410
$623$	− 1030.83i	− 1.65463i
$624$	0	0
$625$	2390.44	3.82471
$626$	− 32.6968i	− 0.0522313i
$627$	0	0
$628$	−571.126	−0.909437
$629$	154.630i	0.245834i
$630$	0	0
$631$	531.909	0.842962	0.421481	−	0.906837i	$-0.361510\pi$
$631$	531.909	0.842962	0.421481	+	0.906837i	$0.361510\pi$
$632$	66.4300i	0.105111i
$633$	0	0
$634$	−88.2121	−0.139136
$635$	473.755i	0.746071i
$636$	0	0
$637$	59.6508	0.0936434
$638$	0	0
$639$	0	0
$640$	109.584	0.171225
$641$	835.157i	1.30290i	0.758693	+	0.651448i	$0.225838\pi$
$641$	835.157i	1.30290i	−0.758693	+	0.651448i	$0.774162\pi$
$642$	0	0
$643$	−62.9205	−0.0978547	−0.0489273	−	0.998802i	$-0.515580\pi$
$643$	−62.9205	−0.0978547	−0.0489273	+	0.998802i	$0.515580\pi$
$644$	− 594.690i	− 0.923432i
$645$	0	0
$646$	68.3955	0.105875
$647$	143.240i	0.221391i	0.993854	+	0.110695i	$0.0353078\pi$
$647$	143.240i	0.221391i	−0.993854	+	0.110695i	$0.964692\pi$
$648$	0	0
$649$	0	0
$650$	− 217.242i	− 0.334219i
$651$	0	0
$652$	414.141	0.635185
$653$	760.175i	1.16413i	0.813143	+	0.582063i	$0.197754\pi$
$653$	760.175i	1.16413i	−0.813143	+	0.582063i	$0.802246\pi$
$654$	0	0
$655$	1202.29	1.83556
$656$	257.684i	0.392811i
$657$	0	0
$658$	−663.214	−1.00792
$659$	995.342i	1.51038i	0.655505	+	0.755191i	$0.272456\pi$
$659$	995.342i	1.51038i	−0.655505	+	0.755191i	$0.727544\pi$
$660$	0	0
$661$	−163.048	−0.246669	−0.123334	−	0.992365i	$-0.539359\pi$
$661$	−163.048	−0.246669	−0.123334	+	0.992365i	$0.539359\pi$
$662$	504.680i	0.762357i
$663$	0	0
$664$	126.449	0.190435
$665$	582.607i	0.876100i
$666$	0	0
$667$	1666.08	2.49788
$668$	167.969i	0.251451i
$669$	0	0
$670$	216.936	0.323785
$671$	0	0
$672$	0	0
$673$	−650.691	−0.966852	−0.483426	−	0.875385i	$-0.660608\pi$
$673$	−650.691	−0.966852	−0.483426	+	0.875385i	$0.660608\pi$
$674$	520.241i	0.771871i
$675$	0	0
$676$	328.035	0.485259
$677$	866.842i	1.28042i	0.768201	+	0.640208i	$0.221152\pi$
$677$	866.842i	1.28042i	−0.768201	+	0.640208i	$0.778848\pi$
$678$	0	0
$679$	−1366.37	−2.01233
$680$	− 191.682i	− 0.281885i
$681$	0	0
$682$	0	0
$683$	− 489.451i	− 0.716620i	−0.933603	−	0.358310i	$-0.883353\pi$
$683$	− 489.451i	− 0.716620i	0.933603	−	0.358310i	$-0.116647\pi$
$684$	0	0
$685$	−926.345	−1.35233
$686$	− 274.147i	− 0.399630i
$687$	0	0
$688$	−243.074	−0.353305
$689$	84.2129i	0.122225i
$690$	0	0
$691$	786.806	1.13865	0.569324	−	0.822113i	$-0.307205\pi$
$691$	786.806	1.13865	0.569324	+	0.822113i	$0.307205\pi$
$692$	− 205.064i	− 0.296335i
$693$	0	0
$694$	−119.716	−0.172502
$695$	811.225i	1.16723i
$696$	0	0
$697$	450.736	0.646679
$698$	611.411i	0.875947i
$699$	0	0
$700$	1197.69	1.71099
$701$	− 838.171i	− 1.19568i	−0.801616	−	0.597839i	$-0.796026\pi$
$701$	− 838.171i	− 1.19568i	0.801616	−	0.597839i	$-0.203974\pi$
$702$	0	0
$703$	−152.763	−0.217301
$704$	0	0
$705$	0	0
$706$	−758.841	−1.07485
$707$	930.429i	1.31602i
$708$	0	0
$709$	425.353	0.599934	0.299967	−	0.953950i	$-0.403024\pi$
$709$	425.353	0.599934	0.299967	+	0.953950i	$0.403024\pi$
$710$	409.950i	0.577394i
$711$	0	0
$712$	−335.057	−0.470586
$713$	− 270.577i	− 0.379491i
$714$	0	0
$715$	0	0
$716$	232.508i	0.324732i
$717$	0	0
$718$	−94.8179	−0.132058
$719$	350.067i	0.486880i	0.969916	+	0.243440i	$0.0782759\pi$
$719$	350.067i	0.486880i	−0.969916	+	0.243440i	$0.921724\pi$
$720$	0	0
$721$	943.929	1.30919
$722$	− 442.961i	− 0.613520i
$723$	0	0
$724$	−255.108	−0.352359
$725$	3355.46i	4.62821i
$726$	0	0
$727$	235.723	0.324240	0.162120	−	0.986771i	$-0.448167\pi$
$727$	235.723	0.324240	0.162120	+	0.986771i	$0.448167\pi$
$728$	− 54.9399i	− 0.0754669i
$729$	0	0
$730$	444.849	0.609382
$731$	425.179i	0.581640i
$732$	0	0
$733$	239.458	0.326682	0.163341	−	0.986570i	$-0.447773\pi$
$733$	239.458	0.326682	0.163341	+	0.986570i	$0.447773\pi$
$734$	656.334i	0.894187i
$735$	0	0
$736$	−193.295	−0.262629
$737$	0	0
$738$	0	0
$739$	661.919	0.895696	0.447848	−	0.894110i	$-0.352191\pi$
$739$	661.919	0.895696	0.447848	+	0.894110i	$0.352191\pi$
$740$	428.126i	0.578549i
$741$	0	0
$742$	−464.280	−0.625714
$743$	894.704i	1.20418i	0.798429	+	0.602089i	$0.205665\pi$
$743$	894.704i	1.20418i	−0.798429	+	0.602089i	$0.794335\pi$
$744$	0	0
$745$	−1232.80	−1.65476
$746$	424.991i	0.569693i
$747$	0	0
$748$	0	0
$749$	1612.67i	2.15310i
$750$	0	0
$751$	−372.915	−0.496558	−0.248279	−	0.968689i	$-0.579865\pi$
$751$	−372.915	−0.496558	−0.248279	+	0.968689i	$0.579865\pi$
$752$	215.568i	0.286659i
$753$	0	0
$754$	153.920	0.204137
$755$	71.6676i	0.0949240i
$756$	0	0
$757$	−777.061	−1.02650	−0.513250	−	0.858239i	$-0.671559\pi$
$757$	−777.061	−1.02650	−0.513250	+	0.858239i	$0.671559\pi$
$758$	− 213.945i	− 0.282249i
$759$	0	0
$760$	189.368	0.249168
$761$	1360.15i	1.78732i	0.448742	+	0.893661i	$0.351872\pi$
$761$	1360.15i	1.78732i	−0.448742	+	0.893661i	$0.648128\pi$
$762$	0	0
$763$	−350.485	−0.459352
$764$	358.229i	0.468887i
$765$	0	0
$766$	505.344	0.659718
$767$	104.211i	0.135869i
$768$	0	0
$769$	306.481	0.398545	0.199273	−	0.979944i	$-0.436142\pi$
$769$	306.481	0.398545	0.199273	+	0.979944i	$0.436142\pi$
$770$	0	0
$771$	0	0
$772$	327.641	0.424406
$773$	317.268i	0.410438i	0.978716	+	0.205219i	$0.0657906\pi$
$773$	317.268i	0.410438i	−0.978716	+	0.205219i	$0.934209\pi$
$774$	0	0
$775$	544.936	0.703143
$776$	444.120i	0.572319i
$777$	0	0
$778$	284.321	0.365452
$779$	445.294i	0.571622i
$780$	0	0
$781$	0	0
$782$	338.108i	0.432363i
$783$	0	0
$784$	106.893	0.136343
$785$	2765.95i	3.52351i
$786$	0	0
$787$	613.426	0.779449	0.389724	−	0.920932i	$-0.372570\pi$
$787$	613.426	0.779449	0.389724	+	0.920932i	$0.372570\pi$
$788$	− 297.249i	− 0.377220i
$789$	0	0
$790$	321.719	0.407239
$791$	− 795.122i	− 1.00521i
$792$	0	0
$793$	56.2804	0.0709715
$794$	− 251.037i	− 0.316168i
$795$	0	0
$796$	−239.112	−0.300392
$797$	− 1277.52i	− 1.60291i	−0.598054	−	0.801456i	$-0.704059\pi$
$797$	− 1277.52i	− 1.60291i	0.598054	−	0.801456i	$-0.295941\pi$
$798$	0	0
$799$	377.066	0.471923
$800$	− 389.292i	− 0.486615i
$801$	0	0
$802$	794.821	0.991049
$803$	0	0
$804$	0	0
$805$	−2880.07	−3.57773
$806$	− 24.9970i	− 0.0310137i
$807$	0	0
$808$	302.422	0.374285
$809$	− 380.980i	− 0.470927i	−0.971883	−	0.235464i	$-0.924339\pi$
$809$	− 380.980i	− 0.470927i	0.971883	−	0.235464i	$-0.0756609\pi$
$810$	0	0
$811$	1223.59	1.50875	0.754373	−	0.656447i	$-0.227941\pi$
$811$	1223.59	1.50875	0.754373	+	0.656447i	$0.227941\pi$
$812$	848.585i	1.04505i
$813$	0	0
$814$	0	0
$815$	− 2005.68i	− 2.46095i
$816$	0	0
$817$	−420.045	−0.514131
$818$	926.941i	1.13318i
$819$	0	0
$820$	1247.96	1.52190
$821$	19.5687i	0.0238352i	0.999929	+	0.0119176i	$0.00379358\pi$
$821$	19.5687i	0.0238352i	−0.999929	+	0.0119176i	$0.996206\pi$
$822$	0	0
$823$	−854.145	−1.03784	−0.518922	−	0.854822i	$-0.673666\pi$
$823$	−854.145	−1.03784	−0.518922	+	0.854822i	$0.673666\pi$
$824$	− 306.810i	− 0.372343i
$825$	0	0
$826$	−574.533	−0.695561
$827$	− 1061.84i	− 1.28396i	−0.766721	−	0.641981i	$-0.778113\pi$
$827$	− 1061.84i	− 1.28396i	0.766721	−	0.641981i	$-0.221887\pi$
$828$	0	0
$829$	−21.6173	−0.0260763	−0.0130382	−	0.999915i	$-0.504150\pi$
$829$	−21.6173	−0.0260763	−0.0130382	+	0.999915i	$0.504150\pi$
$830$	− 612.389i	− 0.737818i
$831$	0	0
$832$	−17.8574	−0.0214632
$833$	− 186.974i	− 0.224459i
$834$	0	0
$835$	813.471	0.974217
$836$	0	0
$837$	0	0
$838$	−1049.29	−1.25214
$839$	412.784i	0.491995i	0.969270	+	0.245998i	$0.0791156\pi$
$839$	412.784i	0.491995i	−0.969270	+	0.245998i	$0.920884\pi$
$840$	0	0
$841$	−1536.40	−1.82687
$842$	− 447.480i	− 0.531449i
$843$	0	0
$844$	308.338	0.365329
$845$	− 1588.67i	− 1.88008i
$846$	0	0
$847$	0	0
$848$	150.907i	0.177957i
$849$	0	0
$850$	−680.941	−0.801107
$851$	− 755.171i	− 0.887392i
$852$	0	0
$853$	19.1501	0.0224503	0.0112251	−	0.999937i	$-0.496427\pi$
$853$	19.1501	0.0224503	0.0112251	+	0.999937i	$0.496427\pi$
$854$	310.283i	0.363329i
$855$	0	0
$856$	524.175	0.612354
$857$	554.266i	0.646751i	0.946271	+	0.323375i	$0.104818\pi$
$857$	554.266i	0.646751i	−0.946271	+	0.323375i	$0.895182\pi$
$858$	0	0
$859$	−105.229	−0.122501	−0.0612506	−	0.998122i	$-0.519509\pi$
$859$	−105.229	−0.122501	−0.0612506	+	0.998122i	$0.519509\pi$
$860$	1177.20i	1.36884i
$861$	0	0
$862$	689.147	0.799474
$863$	1536.37i	1.78027i	0.455701	+	0.890133i	$0.349389\pi$
$863$	1536.37i	1.78027i	−0.455701	+	0.890133i	$0.650611\pi$
$864$	0	0
$865$	−993.119	−1.14811
$866$	850.377i	0.981960i
$867$	0	0
$868$	137.813	0.158770
$869$	0	0
$870$	0	0
$871$	−35.3510	−0.0405867
$872$	113.920i	0.130642i
$873$	0	0
$874$	−334.026	−0.382180
$875$	− 3693.24i	− 4.22085i
$876$	0	0
$877$	757.668	0.863932	0.431966	−	0.901890i	$-0.357820\pi$
$877$	757.668	0.863932	0.431966	+	0.901890i	$0.357820\pi$
$878$	705.651i	0.803703i
$879$	0	0
$880$	0	0
$881$	843.725i	0.957690i	0.877899	+	0.478845i	$0.158944\pi$
$881$	843.725i	0.957690i	−0.877899	+	0.478845i	$0.841056\pi$
$882$	0	0
$883$	440.702	0.499096	0.249548	−	0.968362i	$-0.419718\pi$
$883$	440.702	0.499096	0.249548	+	0.968362i	$0.419718\pi$
$884$	31.2358i	0.0353346i
$885$	0	0
$886$	406.777	0.459116
$887$	− 534.128i	− 0.602174i	−0.953597	−	0.301087i	$-0.902651\pi$
$887$	− 534.128i	− 0.602174i	0.953597	−	0.301087i	$-0.0973494\pi$
$888$	0	0
$889$	425.624	0.478767
$890$	1622.68i	1.82323i
$891$	0	0
$892$	665.228	0.745771
$893$	372.514i	0.417149i
$894$	0	0
$895$	1126.03	1.25814
$896$	− 98.4508i	− 0.109878i
$897$	0	0
$898$	−967.050	−1.07689
$899$	386.096i	0.429473i
$900$	0	0
$901$	263.964	0.292967
$902$	0	0
$903$	0	0
$904$	−258.443	−0.285888
$905$	1235.48i	1.36517i
$906$	0	0
$907$	982.540	1.08329	0.541643	−	0.840609i	$-0.317803\pi$
$907$	982.540	1.08329	0.541643	+	0.840609i	$0.317803\pi$
$908$	455.989i	0.502190i
$909$	0	0
$910$	−266.073	−0.292388
$911$	1151.06i	1.26351i	0.775168	+	0.631755i	$0.217665\pi$
$911$	1151.06i	1.26351i	−0.775168	+	0.631755i	$0.782335\pi$
$912$	0	0
$913$	0	0
$914$	− 426.016i	− 0.466100i
$915$	0	0
$916$	−2.15481	−0.00235241
$917$	− 1080.14i	− 1.17791i
$918$	0	0
$919$	−1028.96	−1.11966	−0.559828	−	0.828609i	$-0.689133\pi$
$919$	−1028.96	−1.11966	−0.559828	+	0.828609i	$0.689133\pi$
$920$	936.125i	1.01753i
$921$	0	0
$922$	297.343	0.322497
$923$	− 66.8039i	− 0.0723769i
$924$	0	0
$925$	1520.90	1.64421
$926$	− 486.998i	− 0.525916i
$927$	0	0
$928$	275.820	0.297220
$929$	− 537.319i	− 0.578385i	−0.957271	−	0.289192i	$-0.906613\pi$
$929$	− 537.319i	− 0.578385i	0.957271	−	0.289192i	$-0.0933867\pi$
$930$	0	0
$931$	184.717	0.198407
$932$	− 480.072i	− 0.515098i
$933$	0	0
$934$	−348.705	−0.373345
$935$	0	0
$936$	0	0
$937$	−1327.73	−1.41700	−0.708501	−	0.705710i	$-0.750628\pi$
$937$	−1327.73	−1.41700	−0.708501	+	0.705710i	$0.750628\pi$
$938$	− 194.896i	− 0.207778i
$939$	0	0
$940$	1043.99	1.11063
$941$	1159.21i	1.23189i	0.787790	+	0.615944i	$0.211225\pi$
$941$	1159.21i	1.23189i	−0.787790	+	0.615944i	$0.788775\pi$
$942$	0	0
$943$	−2201.27	−2.33433
$944$	186.744i	0.197822i
$945$	0	0
$946$	0	0
$947$	− 242.837i	− 0.256427i	−0.991747	−	0.128214i	$-0.959076\pi$
$947$	− 242.837i	− 0.256427i	0.991747	−	0.128214i	$-0.0409243\pi$
$948$	0	0
$949$	−72.4909	−0.0763866
$950$	− 672.720i	− 0.708126i
$951$	0	0
$952$	−172.208	−0.180891
$953$	530.435i	0.556595i	0.960495	+	0.278297i	$0.0897700\pi$
$953$	530.435i	0.556595i	−0.960495	+	0.278297i	$0.910230\pi$
$954$	0	0
$955$	1734.90	1.81665
$956$	405.710i	0.424383i
$957$	0	0
$958$	−480.795	−0.501874
$959$	832.232i	0.867812i
$960$	0	0
$961$	−898.297	−0.934752
$962$	− 69.7658i	− 0.0725216i
$963$	0	0
$964$	−324.749	−0.336877
$965$	− 1586.76i	− 1.64431i
$966$	0	0
$967$	275.752	0.285163	0.142581	−	0.989783i	$-0.454460\pi$
$967$	275.752	0.285163	0.142581	+	0.989783i	$0.454460\pi$
$968$	0	0
$969$	0	0
$970$	2150.86	2.21738
$971$	− 1238.62i	− 1.27561i	−0.770197	−	0.637806i	$-0.779842\pi$
$971$	− 1238.62i	− 1.27561i	0.770197	−	0.637806i	$-0.220158\pi$
$972$	0	0
$973$	728.808	0.749032
$974$	257.403i	0.264275i
$975$	0	0
$976$	100.853	0.103333
$977$	320.911i	0.328466i	0.986422	+	0.164233i	$0.0525148\pi$
$977$	320.911i	0.328466i	−0.986422	+	0.164233i	$0.947485\pi$
$978$	0	0
$979$	0	0
$980$	− 517.679i	− 0.528244i
$981$	0	0
$982$	−1338.91	−1.36346
$983$	779.752i	0.793237i	0.917983	+	0.396619i	$0.129816\pi$
$983$	779.752i	0.793237i	−0.917983	+	0.396619i	$0.870184\pi$
$984$	0	0
$985$	−1439.57	−1.46149
$986$	− 482.458i	− 0.489308i
$987$	0	0
$988$	−30.8586	−0.0312335
$989$	− 2076.46i	− 2.09956i
$990$	0	0
$991$	−1400.10	−1.41281	−0.706407	−	0.707806i	$-0.749685\pi$
$991$	−1400.10	−1.41281	−0.706407	+	0.707806i	$0.749685\pi$
$992$	− 44.7940i	− 0.0451552i
$993$	0	0
$994$	368.301	0.370524
$995$	1158.01i	1.16383i
$996$	0	0
$997$	1288.17	1.29205	0.646023	−	0.763318i	$-0.276431\pi$
$997$	1288.17	1.29205	0.646023	+	0.763318i	$0.276431\pi$
$998$	1095.15i	1.09734i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2178.3.c.p.485.8		8
3.2	odd	2	inner	2178.3.c.p.485.1		8
11.3	even	5		198.3.k.a.53.3	yes	16
11.4	even	5		198.3.k.a.71.2	yes	16
11.10	odd	2		2178.3.c.m.485.4		8
33.14	odd	10		198.3.k.a.53.2	✓	16
33.26	odd	10		198.3.k.a.71.3	yes	16
33.32	even	2		2178.3.c.m.485.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
198.3.k.a.53.2	✓	16	33.14	odd	10
198.3.k.a.53.3	yes	16	11.3	even	5
198.3.k.a.71.2	yes	16	11.4	even	5
198.3.k.a.71.3	yes	16	33.26	odd	10
2178.3.c.m.485.4		8	11.10	odd	2
2178.3.c.m.485.5		8	33.32	even	2
2178.3.c.p.485.1		8	3.2	odd	2	inner
2178.3.c.p.485.8		8	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 2178 = 2 \cdot 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	2178.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.00000 q^{4} +9.68596i q^{5} +8.70191 q^{7} -2.82843i q^{8} +O(q^{10})\)	\(q+1.41421i q^{2} -2.00000 q^{4} +9.68596i q^{5} +8.70191 q^{7} -2.82843i q^{8} -13.6980 q^{10} +2.23218 q^{13} +12.3064i q^{14} +4.00000 q^{16} -6.99671i q^{17} +6.91224 q^{19} -19.3719i q^{20} +34.1701i q^{23} -68.8178 q^{25} +3.15677i q^{26} -17.4038 q^{28} -48.7585i q^{29} -7.91853 q^{31} +5.65685i q^{32} +9.89484 q^{34} +84.2863i q^{35} -22.1003 q^{37} +9.77538i q^{38} +27.3960 q^{40} +64.4211i q^{41} -60.7684 q^{43} -48.3238 q^{46} +53.8920i q^{47} +26.7232 q^{49} -97.3231i q^{50} -4.46435 q^{52} +37.7268i q^{53} -24.6127i q^{56} +68.9550 q^{58} +46.6859i q^{59} +25.2133 q^{61} -11.1985i q^{62} -8.00000 q^{64} +21.6208i q^{65} -15.8370 q^{67} +13.9934i q^{68} -119.199 q^{70} -29.9277i q^{71} -32.4754 q^{73} -31.2546i q^{74} -13.8245 q^{76} -23.4865 q^{79} +38.7438i q^{80} -91.1052 q^{82} +44.7064i q^{83} +67.7698 q^{85} -85.9395i q^{86} -118.461i q^{89} +19.4242 q^{91} -68.3402i q^{92} -76.2148 q^{94} +66.9516i q^{95} -157.020 q^{97} +37.7923i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4} + 8 q^{7}+O(q^{10}) \) 8 * q - 16 * q^4 + 8 * q^7	\( 8 q - 16 q^{4} + 8 q^{7} - 12 q^{10} - 36 q^{13} + 32 q^{16} + 72 q^{19} - 64 q^{25} - 16 q^{28} - 88 q^{31} - 56 q^{34} + 108 q^{37} + 24 q^{40} - 220 q^{43} - 68 q^{46} + 56 q^{49} + 72 q^{52} + 112 q^{58} - 160 q^{61} - 64 q^{64} - 276 q^{67} - 92 q^{70} - 488 q^{73} - 144 q^{76} - 368 q^{79} - 388 q^{82} + 248 q^{85} - 356 q^{91} - 120 q^{94} - 832 q^{97}+O(q^{100}) \) 8 * q - 16 * q^4 + 8 * q^7 - 12 * q^10 - 36 * q^13 + 32 * q^16 + 72 * q^19 - 64 * q^25 - 16 * q^28 - 88 * q^31 - 56 * q^34 + 108 * q^37 + 24 * q^40 - 220 * q^43 - 68 * q^46 + 56 * q^49 + 72 * q^52 + 112 * q^58 - 160 * q^61 - 64 * q^64 - 276 * q^67 - 92 * q^70 - 488 * q^73 - 144 * q^76 - 368 * q^79 - 388 * q^82 + 248 * q^85 - 356 * q^91 - 120 * q^94 - 832 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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