LMFDB - Embedded newform 1600.3.g.f.351.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,3,Mod(351,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 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("1600.351");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1600 = 2^{6} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1600.g (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:	$43.5968422976$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{18} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			351.7
Root			$-0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	1600.351
Dual form			1600.3.g.f.351.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.14626 q^{3} -13.7980i q^{7} +0.898979 q^{9} +O(q^{10})$	$q+3.14626 q^{3} -13.7980i q^{7} +0.898979 q^{9} +17.2884 q^{11} -11.3137i q^{13} -19.6969 q^{17} +22.6595 q^{19} -43.4120i q^{21} +21.7980i q^{23} -25.4880 q^{27} -22.6274i q^{29} -2.20204i q^{31} +54.3939 q^{33} -23.3273i q^{37} -35.5959i q^{39} +16.7980 q^{41} -31.1769 q^{43} -36.4041i q^{47} -141.384 q^{49} -61.9718 q^{51} +35.7839i q^{53} +71.2929 q^{57} +74.5889 q^{59} +60.9540i q^{61} -12.4041i q^{63} -18.0525 q^{67} +68.5821i q^{69} -114.788i q^{71} -44.6867 q^{73} -238.545i q^{77} -44.9898i q^{79} -88.2827 q^{81} -139.389 q^{83} -71.1918i q^{87} +99.6765 q^{89} -156.106 q^{91} -6.92820i q^{93} -20.3837 q^{97} +15.5419 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 32 q^{9}+O(q^{10}) $ 8 * q - 32 * q^9	$ 8 q - 32 q^{9} - 40 q^{17} + 200 q^{33} + 56 q^{41} - 504 q^{49} + 296 q^{57} + 152 q^{73} - 40 q^{81} - 104 q^{89} + 464 q^{97}+O(q^{100}) $ 8 * q - 32 * q^9 - 40 * q^17 + 200 * q^33 + 56 * q^41 - 504 * q^49 + 296 * q^57 + 152 * q^73 - 40 * q^81 - 104 * q^89 + 464 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1600\mathbb{Z}\right)^\times$.

$n$	$577$	$901$	$1151$
$\chi(n)$	$1$	$-1$	$-1$

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 $

$2$	0	0
$2$	0	0
$3$	3.14626	1.04875	0.524377	−	0.851486i	$-0.324298\pi$
$3$	3.14626	1.04875	0.524377	+	0.851486i	$0.324298\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 13.7980i	− 1.97114i	−0.169277	−	0.985568i	$-0.554143\pi$
$7$	− 13.7980i	− 1.97114i	0.169277	−	0.985568i	$-0.445857\pi$
$8$	0	0
$9$	0.898979	0.0998866
$10$	0	0
$11$	17.2884	1.57167	0.785836	−	0.618434i	$-0.212233\pi$
$11$	17.2884	1.57167	0.785836	+	0.618434i	$0.212233\pi$
$12$	0	0
$13$	− 11.3137i	− 0.870285i	−0.900362	−	0.435143i	$-0.856698\pi$
$13$	− 11.3137i	− 0.870285i	0.900362	−	0.435143i	$-0.143302\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−19.6969	−1.15864	−0.579322	−	0.815099i	$-0.696683\pi$
$17$	−19.6969	−1.15864	−0.579322	+	0.815099i	$0.696683\pi$
$18$	0	0
$19$	22.6595	1.19261	0.596303	−	0.802759i	$-0.296636\pi$
$19$	22.6595	1.19261	0.596303	+	0.802759i	$0.296636\pi$
$20$	0	0
$21$	− 43.4120i	− 2.06724i
$22$	0	0
$23$	21.7980i	0.947737i	0.880596	+	0.473869i	$0.157143\pi$
$23$	21.7980i	0.947737i	−0.880596	+	0.473869i	$0.842857\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−25.4880	−0.943998
$28$	0	0
$29$	− 22.6274i	− 0.780256i	−0.920761	−	0.390128i	$-0.872431\pi$
$29$	− 22.6274i	− 0.780256i	0.920761	−	0.390128i	$-0.127569\pi$
$30$	0	0
$31$	− 2.20204i	− 0.0710336i	−0.999369	−	0.0355168i	$-0.988692\pi$
$31$	− 2.20204i	− 0.0710336i	0.999369	−	0.0355168i	$-0.0113077\pi$
$32$	0	0
$33$	54.3939	1.64830
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 23.3273i	− 0.630468i	−0.949014	−	0.315234i	$-0.897917\pi$
$37$	− 23.3273i	− 0.630468i	0.949014	−	0.315234i	$-0.102083\pi$
$38$	0	0
$39$	− 35.5959i	− 0.912716i
$40$	0	0
$41$	16.7980	0.409706	0.204853	−	0.978793i	$-0.434328\pi$
$41$	16.7980	0.409706	0.204853	+	0.978793i	$0.434328\pi$
$42$	0	0
$43$	−31.1769	−0.725045	−0.362522	−	0.931975i	$-0.618084\pi$
$43$	−31.1769	−0.725045	−0.362522	+	0.931975i	$0.618084\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	− 36.4041i	− 0.774555i	−0.921963	−	0.387277i	$-0.873416\pi$
$47$	− 36.4041i	− 0.774555i	0.921963	−	0.387277i	$-0.126584\pi$
$48$	0	0
$49$	−141.384	−2.88538
$50$	0	0
$51$	−61.9718	−1.21513
$52$	0	0
$53$	35.7839i	0.675169i	0.941295	+	0.337584i	$0.109610\pi$
$53$	35.7839i	0.675169i	−0.941295	+	0.337584i	$0.890390\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	71.2929	1.25075
$58$	0	0
$59$	74.5889	1.26422	0.632110	−	0.774879i	$-0.282189\pi$
$59$	74.5889	1.26422	0.632110	+	0.774879i	$0.282189\pi$
$60$	0	0
$61$	60.9540i	0.999247i	0.866243	+	0.499623i	$0.166528\pi$
$61$	60.9540i	0.999247i	−0.866243	+	0.499623i	$0.833472\pi$
$62$	0	0
$63$	− 12.4041i	− 0.196890i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−18.0525	−0.269440	−0.134720	−	0.990884i	$-0.543014\pi$
$67$	−18.0525	−0.269440	−0.134720	+	0.990884i	$0.543014\pi$
$68$	0	0
$69$	68.5821i	0.993944i
$70$	0	0
$71$	− 114.788i	− 1.61673i	−0.588682	−	0.808364i	$-0.700353\pi$
$71$	− 114.788i	− 1.61673i	0.588682	−	0.808364i	$-0.299647\pi$
$72$	0	0
$73$	−44.6867	−0.612147	−0.306074	−	0.952008i	$-0.599015\pi$
$73$	−44.6867	−0.612147	−0.306074	+	0.952008i	$0.599015\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 238.545i	− 3.09798i
$78$	0	0
$79$	− 44.9898i	− 0.569491i	−0.958603	−	0.284746i	$-0.908091\pi$
$79$	− 44.9898i	− 0.569491i	0.958603	−	0.284746i	$-0.0919091\pi$
$80$	0	0
$81$	−88.2827	−1.08991
$82$	0	0
$83$	−139.389	−1.67939	−0.839694	−	0.543060i	$-0.817265\pi$
$83$	−139.389	−1.67939	−0.839694	+	0.543060i	$0.817265\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	− 71.1918i	− 0.818297i
$88$	0	0
$89$	99.6765	1.11996	0.559980	−	0.828506i	$-0.310809\pi$
$89$	99.6765	1.11996	0.559980	+	0.828506i	$0.310809\pi$
$90$	0	0
$91$	−156.106	−1.71545
$92$	0	0
$93$	− 6.92820i	− 0.0744968i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−20.3837	−0.210141	−0.105070	−	0.994465i	$-0.533507\pi$
$97$	−20.3837	−0.210141	−0.105070	+	0.994465i	$0.533507\pi$
$98$	0	0
$99$	15.5419	0.156989
$100$	0	0
$101$	− 165.577i	− 1.63938i	−0.572810	−	0.819688i	$-0.694147\pi$
$101$	− 165.577i	− 1.63938i	0.572810	−	0.819688i	$-0.305853\pi$
$102$	0	0
$103$	− 34.7878i	− 0.337745i	−0.985638	−	0.168873i	$-0.945987\pi$
$103$	− 34.7878i	− 0.337745i	0.985638	−	0.168873i	$-0.0540126\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−87.3345	−0.816211	−0.408105	−	0.912935i	$-0.633810\pi$
$107$	−87.3345	−0.816211	−0.408105	+	0.912935i	$0.633810\pi$
$108$	0	0
$109$	193.290i	1.77330i	0.462440	+	0.886650i	$0.346974\pi$
$109$	193.290i	1.77330i	−0.462440	+	0.886650i	$0.653026\pi$
$110$	0	0
$111$	− 73.3939i	− 0.661206i
$112$	0	0
$113$	79.9898	0.707874	0.353937	−	0.935269i	$-0.384843\pi$
$113$	79.9898	0.707874	0.353937	+	0.935269i	$0.384843\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 10.1708i	− 0.0869298i
$118$	0	0
$119$	271.778i	2.28384i
$120$	0	0
$121$	177.889	1.47016
$122$	0	0
$123$	52.8508	0.429681
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 153.394i	− 1.20783i	−0.797050	−	0.603913i	$-0.793607\pi$
$127$	− 153.394i	− 1.20783i	0.797050	−	0.603913i	$-0.206393\pi$
$128$	0	0
$129$	−98.0908	−0.760394
$130$	0	0
$131$	49.1620	0.375282	0.187641	−	0.982238i	$-0.439916\pi$
$131$	49.1620	0.375282	0.187641	+	0.982238i	$0.439916\pi$
$132$	0	0
$133$	− 312.655i	− 2.35079i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	209.788	1.53130	0.765649	−	0.643259i	$-0.222418\pi$
$137$	209.788	1.53130	0.765649	+	0.643259i	$0.222418\pi$
$138$	0	0
$139$	52.0578	0.374517	0.187258	−	0.982311i	$-0.440040\pi$
$139$	52.0578	0.374517	0.187258	+	0.982311i	$0.440040\pi$
$140$	0	0
$141$	− 114.537i	− 0.812318i
$142$	0	0
$143$	− 195.596i	− 1.36780i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−444.830	−3.02606
$148$	0	0
$149$	− 125.594i	− 0.842911i	−0.906849	−	0.421455i	$-0.861519\pi$
$149$	− 125.594i	− 0.842911i	0.906849	−	0.421455i	$-0.138481\pi$
$150$	0	0
$151$	181.353i	1.20101i	0.799620	+	0.600507i	$0.205034\pi$
$151$	181.353i	1.20101i	−0.799620	+	0.600507i	$0.794966\pi$
$152$	0	0
$153$	−17.7071	−0.115733
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 34.8979i	− 0.222279i	−0.993805	−	0.111140i	$-0.964550\pi$
$157$	− 34.8979i	− 0.222279i	0.993805	−	0.111140i	$-0.0354501\pi$
$158$	0	0
$159$	112.586i	0.708086i
$160$	0	0
$161$	300.767	1.86812
$162$	0	0
$163$	220.364	1.35192	0.675962	−	0.736936i	$-0.263728\pi$
$163$	220.364	1.35192	0.675962	+	0.736936i	$0.263728\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	160.363i	0.960259i	0.877198	+	0.480130i	$0.159410\pi$
$167$	160.363i	0.960259i	−0.877198	+	0.480130i	$0.840590\pi$
$168$	0	0
$169$	41.0000	0.242604
$170$	0	0
$171$	20.3704	0.119125
$172$	0	0
$173$	100.424i	0.580483i	0.956953	+	0.290242i	$0.0937357\pi$
$173$	100.424i	0.580483i	−0.956953	+	0.290242i	$0.906264\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	234.677	1.32586
$178$	0	0
$179$	267.304	1.49332	0.746659	−	0.665207i	$-0.231657\pi$
$179$	267.304	1.49332	0.746659	+	0.665207i	$0.231657\pi$
$180$	0	0
$181$	− 257.486i	− 1.42258i	−0.702900	−	0.711288i	$-0.748112\pi$
$181$	− 257.486i	− 1.42258i	0.702900	−	0.711288i	$-0.251888\pi$
$182$	0	0
$183$	191.778i	1.04796i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−340.529	−1.82101
$188$	0	0
$189$	351.682i	1.86075i
$190$	0	0
$191$	303.778i	1.59046i	0.606309	+	0.795229i	$0.292649\pi$
$191$	303.778i	1.59046i	−0.606309	+	0.795229i	$0.707351\pi$
$192$	0	0
$193$	−30.4847	−0.157952	−0.0789759	−	0.996877i	$-0.525165\pi$
$193$	−30.4847	−0.157952	−0.0789759	+	0.996877i	$0.525165\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 75.0673i	− 0.381052i	−0.981682	−	0.190526i	$-0.938981\pi$
$197$	− 75.0673i	− 0.381052i	0.981682	−	0.190526i	$-0.0610194\pi$
$198$	0	0
$199$	134.020i	0.673469i	0.941600	+	0.336735i	$0.109323\pi$
$199$	134.020i	0.673469i	−0.941600	+	0.336735i	$0.890677\pi$
$200$	0	0
$201$	−56.7980	−0.282577
$202$	0	0
$203$	−312.212	−1.53799
$204$	0	0
$205$	0	0
$206$	0	0
$207$	19.5959i	0.0946663i
$208$	0	0
$209$	391.747	1.87439
$210$	0	0
$211$	−274.011	−1.29863	−0.649315	−	0.760520i	$-0.724944\pi$
$211$	−274.011	−1.29863	−0.649315	+	0.760520i	$0.724944\pi$
$212$	0	0
$213$	− 361.153i	− 1.69555i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−30.3837	−0.140017
$218$	0	0
$219$	−140.596	−0.641992
$220$	0	0
$221$	222.845i	1.00835i
$222$	0	0
$223$	− 243.596i	− 1.09236i	−0.837668	−	0.546179i	$-0.816082\pi$
$223$	− 243.596i	− 1.09236i	0.837668	−	0.546179i	$-0.183918\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	257.894	1.13610	0.568049	−	0.822995i	$-0.307699\pi$
$227$	257.894	1.13610	0.568049	+	0.822995i	$0.307699\pi$
$228$	0	0
$229$	210.132i	0.917607i	0.888538	+	0.458803i	$0.151722\pi$
$229$	210.132i	0.917607i	−0.888538	+	0.458803i	$0.848278\pi$
$230$	0	0
$231$	− 750.524i	− 3.24902i
$232$	0	0
$233$	59.6163	0.255864	0.127932	−	0.991783i	$-0.459166\pi$
$233$	59.6163	0.255864	0.127932	+	0.991783i	$0.459166\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 141.550i	− 0.597256i
$238$	0	0
$239$	4.76734i	0.0199470i	0.999950	+	0.00997352i	$0.00317472\pi$
$239$	4.76734i	0.0199470i	−0.999950	+	0.00997352i	$0.996825\pi$
$240$	0	0
$241$	−1.09082	−0.00452620	−0.00226310	−	0.999997i	$-0.500720\pi$
$241$	−1.09082	−0.00452620	−0.00226310	+	0.999997i	$0.500720\pi$
$242$	0	0
$243$	−48.3690	−0.199049
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 256.363i	− 1.03791i
$248$	0	0
$249$	−438.555	−1.76127
$250$	0	0
$251$	−224.528	−0.894533	−0.447266	−	0.894401i	$-0.647602\pi$
$251$	−224.528	−0.894533	−0.447266	+	0.894401i	$0.647602\pi$
$252$	0	0
$253$	376.852i	1.48953i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	224.384	0.873088	0.436544	−	0.899683i	$-0.356202\pi$
$257$	224.384	0.873088	0.436544	+	0.899683i	$0.356202\pi$
$258$	0	0
$259$	−321.869	−1.24274
$260$	0	0
$261$	− 20.3416i	− 0.0779371i
$262$	0	0
$263$	− 38.1612i	− 0.145100i	−0.997365	−	0.0725498i	$-0.976886\pi$
$263$	− 38.1612i	− 0.145100i	0.997365	−	0.0725498i	$-0.0231136\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	313.609	1.17456
$268$	0	0
$269$	− 130.936i	− 0.486751i	−0.969932	−	0.243375i	$-0.921745\pi$
$269$	− 130.936i	− 0.486751i	0.969932	−	0.243375i	$-0.0782547\pi$
$270$	0	0
$271$	− 124.545i	− 0.459575i	−0.973241	−	0.229788i	$-0.926197\pi$
$271$	− 124.545i	− 0.459575i	0.973241	−	0.229788i	$-0.0738032\pi$
$272$	0	0
$273$	−491.151	−1.79909
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 251.444i	− 0.907741i	−0.891068	−	0.453871i	$-0.850043\pi$
$277$	− 251.444i	− 0.907741i	0.891068	−	0.453871i	$-0.149957\pi$
$278$	0	0
$279$	− 1.97959i	− 0.00709530i
$280$	0	0
$281$	−335.535	−1.19407	−0.597037	−	0.802214i	$-0.703655\pi$
$281$	−335.535	−1.19407	−0.597037	+	0.802214i	$0.703655\pi$
$282$	0	0
$283$	74.3995	0.262896	0.131448	−	0.991323i	$-0.458037\pi$
$283$	74.3995	0.262896	0.131448	+	0.991323i	$0.458037\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 231.778i	− 0.807587i
$288$	0	0
$289$	98.9694	0.342455
$290$	0	0
$291$	−64.1324	−0.220386
$292$	0	0
$293$	156.549i	0.534297i	0.963655	+	0.267149i	$0.0860815\pi$
$293$	156.549i	0.534297i	−0.963655	+	0.267149i	$0.913919\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−440.646	−1.48366
$298$	0	0
$299$	246.616	0.824802
$300$	0	0
$301$	430.178i	1.42916i
$302$	0	0
$303$	− 520.949i	− 1.71930i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	126.140	0.410878	0.205439	−	0.978670i	$-0.434138\pi$
$307$	126.140	0.410878	0.205439	+	0.978670i	$0.434138\pi$
$308$	0	0
$309$	− 109.451i	− 0.354212i
$310$	0	0
$311$	381.716i	1.22738i	0.789546	+	0.613692i	$0.210316\pi$
$311$	381.716i	1.22738i	−0.789546	+	0.613692i	$0.789684\pi$
$312$	0	0
$313$	504.302	1.61119	0.805594	−	0.592468i	$-0.201846\pi$
$313$	504.302	1.61119	0.805594	+	0.592468i	$0.201846\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 416.321i	− 1.31332i	−0.754188	−	0.656658i	$-0.771969\pi$
$317$	− 416.321i	− 1.31332i	0.754188	−	0.656658i	$-0.228031\pi$
$318$	0	0
$319$	− 391.192i	− 1.22631i
$320$	0	0
$321$	−274.778	−0.856005
$322$	0	0
$323$	−446.323	−1.38181
$324$	0	0
$325$	0	0
$326$	0	0
$327$	608.141i	1.85976i
$328$	0	0
$329$	−502.302	−1.52675
$330$	0	0
$331$	209.621	0.633297	0.316649	−	0.948543i	$-0.397442\pi$
$331$	209.621	0.633297	0.316649	+	0.948543i	$0.397442\pi$
$332$	0	0
$333$	− 20.9708i	− 0.0629753i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−138.435	−0.410785	−0.205393	−	0.978680i	$-0.565847\pi$
$337$	−138.435	−0.410785	−0.205393	+	0.978680i	$0.565847\pi$
$338$	0	0
$339$	251.669	0.742387
$340$	0	0
$341$	− 38.0698i	− 0.111642i
$342$	0	0
$343$	1274.71i	3.71634i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−185.530	−0.534669	−0.267334	−	0.963604i	$-0.586143\pi$
$347$	−185.530	−0.534669	−0.267334	+	0.963604i	$0.586143\pi$
$348$	0	0
$349$	− 274.772i	− 0.787311i	−0.919258	−	0.393656i	$-0.871210\pi$
$349$	− 274.772i	− 0.787311i	0.919258	−	0.393656i	$-0.128790\pi$
$350$	0	0
$351$	288.363i	0.821548i
$352$	0	0
$353$	242.000	0.685552	0.342776	−	0.939417i	$-0.388633\pi$
$353$	242.000	0.685552	0.342776	+	0.939417i	$0.388633\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	855.084i	2.39519i
$358$	0	0
$359$	− 182.161i	− 0.507413i	−0.967281	−	0.253706i	$-0.918350\pi$
$359$	− 182.161i	− 0.507413i	0.967281	−	0.253706i	$-0.0816497\pi$
$360$	0	0
$361$	152.454	0.422310
$362$	0	0
$363$	559.685	1.54183
$364$	0	0
$365$	0	0
$366$	0	0
$367$	25.1714i	0.0685870i	0.999412	+	0.0342935i	$0.0109181\pi$
$367$	25.1714i	0.0685870i	−0.999412	+	0.0342935i	$0.989082\pi$
$368$	0	0
$369$	15.1010	0.0409242
$370$	0	0
$371$	493.745	1.33085
$372$	0	0
$373$	560.671i	1.50314i	0.659654	+	0.751569i	$0.270703\pi$
$373$	560.671i	1.50314i	−0.659654	+	0.751569i	$0.729297\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−256.000	−0.679045
$378$	0	0
$379$	−29.9088	−0.0789151	−0.0394575	−	0.999221i	$-0.512563\pi$
$379$	−29.9088	−0.0789151	−0.0394575	+	0.999221i	$0.512563\pi$
$380$	0	0
$381$	− 482.618i	− 1.26671i
$382$	0	0
$383$	176.504i	0.460846i	0.973091	+	0.230423i	$0.0740110\pi$
$383$	176.504i	0.460846i	−0.973091	+	0.230423i	$0.925989\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	−28.0274	−0.0724222
$388$	0	0
$389$	− 601.354i	− 1.54590i	−0.634469	−	0.772948i	$-0.718781\pi$
$389$	− 601.354i	− 1.54590i	0.634469	−	0.772948i	$-0.281219\pi$
$390$	0	0
$391$	− 429.353i	− 1.09809i
$392$	0	0
$393$	154.677	0.393579
$394$	0	0
$395$	0	0
$396$	0	0
$397$	768.960i	1.93693i	0.249156	+	0.968463i	$0.419847\pi$
$397$	768.960i	1.93693i	−0.249156	+	0.968463i	$0.580153\pi$
$398$	0	0
$399$	− 983.696i	− 2.46540i
$400$	0	0
$401$	423.191	1.05534	0.527669	−	0.849450i	$-0.323066\pi$
$401$	423.191	1.05534	0.527669	+	0.849450i	$0.323066\pi$
$402$	0	0
$403$	−24.9133	−0.0618195
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 403.292i	− 0.990889i
$408$	0	0
$409$	54.3735	0.132942	0.0664712	−	0.997788i	$-0.478826\pi$
$409$	54.3735	0.132942	0.0664712	+	0.997788i	$0.478826\pi$
$410$	0	0
$411$	660.048	1.60596
$412$	0	0
$413$	− 1029.18i	− 2.49195i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	163.788	0.392776
$418$	0	0
$419$	−434.859	−1.03785	−0.518925	−	0.854820i	$-0.673667\pi$
$419$	−434.859	−1.03785	−0.518925	+	0.854820i	$0.673667\pi$
$420$	0	0
$421$	− 234.531i	− 0.557082i	−0.960424	−	0.278541i	$-0.910149\pi$
$421$	− 234.531i	− 0.557082i	0.960424	−	0.278541i	$-0.0898508\pi$
$422$	0	0
$423$	− 32.7265i	− 0.0773677i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	841.041	1.96965
$428$	0	0
$429$	− 615.396i	− 1.43449i
$430$	0	0
$431$	601.394i	1.39535i	0.716417	+	0.697673i	$0.245781\pi$
$431$	601.394i	1.39535i	−0.716417	+	0.697673i	$0.754219\pi$
$432$	0	0
$433$	200.444	0.462919	0.231459	−	0.972845i	$-0.425650\pi$
$433$	200.444	0.462919	0.231459	+	0.972845i	$0.425650\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	493.931i	1.13028i
$438$	0	0
$439$	725.494i	1.65261i	0.563226	+	0.826303i	$0.309560\pi$
$439$	725.494i	1.65261i	−0.563226	+	0.826303i	$0.690440\pi$
$440$	0	0
$441$	−127.101	−0.288211
$442$	0	0
$443$	147.996	0.334078	0.167039	−	0.985950i	$-0.446579\pi$
$443$	147.996	0.334078	0.167039	+	0.985950i	$0.446579\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 395.151i	− 0.884007i
$448$	0	0
$449$	101.111	0.225192	0.112596	−	0.993641i	$-0.464083\pi$
$449$	101.111	0.225192	0.112596	+	0.993641i	$0.464083\pi$
$450$	0	0
$451$	290.410	0.643924
$452$	0	0
$453$	570.585i	1.25957i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−579.161	−1.26731	−0.633656	−	0.773615i	$-0.718446\pi$
$457$	−579.161	−1.26731	−0.633656	+	0.773615i	$0.718446\pi$
$458$	0	0
$459$	502.035	1.09376
$460$	0	0
$461$	447.347i	0.970385i	0.874407	+	0.485193i	$0.161250\pi$
$461$	447.347i	0.970385i	−0.874407	+	0.485193i	$0.838750\pi$
$462$	0	0
$463$	− 192.445i	− 0.415648i	−0.978166	−	0.207824i	$-0.933362\pi$
$463$	− 192.445i	− 0.415648i	0.978166	−	0.207824i	$-0.0666381\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	638.502	1.36724	0.683621	−	0.729837i	$-0.260404\pi$
$467$	638.502	1.36724	0.683621	+	0.729837i	$0.260404\pi$
$468$	0	0
$469$	249.088i	0.531104i
$470$	0	0
$471$	− 109.798i	− 0.233117i
$472$	0	0
$473$	−538.999	−1.13953
$474$	0	0
$475$	0	0
$476$	0	0
$477$	32.1690i	0.0674403i
$478$	0	0
$479$	− 439.414i	− 0.917358i	−0.888602	−	0.458679i	$-0.848323\pi$
$479$	− 439.414i	− 0.917358i	0.888602	−	0.458679i	$-0.151677\pi$
$480$	0	0
$481$	−263.918	−0.548687
$482$	0	0
$483$	946.294	1.95920
$484$	0	0
$485$	0	0
$486$	0	0
$487$	880.586i	1.80818i	0.427338	+	0.904092i	$0.359452\pi$
$487$	880.586i	1.80818i	−0.427338	+	0.904092i	$0.640548\pi$
$488$	0	0
$489$	693.322	1.41784
$490$	0	0
$491$	196.125	0.399439	0.199720	−	0.979853i	$-0.435997\pi$
$491$	196.125	0.399439	0.199720	+	0.979853i	$0.435997\pi$
$492$	0	0
$493$	445.691i	0.904038i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−1583.84	−3.18679
$498$	0	0
$499$	−3.46410	−0.00694209	−0.00347104	−	0.999994i	$-0.501105\pi$
$499$	−3.46410	−0.00694209	−0.00347104	+	0.999994i	$0.501105\pi$
$500$	0	0
$501$	504.545i	1.00708i
$502$	0	0
$503$	358.829i	0.713377i	0.934223	+	0.356688i	$0.116094\pi$
$503$	358.829i	0.713377i	−0.934223	+	0.356688i	$0.883906\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	128.997	0.254432
$508$	0	0
$509$	5.01469i	0.00985205i	0.999988	+	0.00492602i	$0.00156801\pi$
$509$	5.01469i	0.00985205i	−0.999988	+	0.00492602i	$0.998432\pi$
$510$	0	0
$511$	616.586i	1.20663i
$512$	0	0
$513$	−577.545	−1.12582
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 629.368i	− 1.21735i
$518$	0	0
$519$	315.959i	0.608785i
$520$	0	0
$521$	−763.343	−1.46515	−0.732575	−	0.680687i	$-0.761682\pi$
$521$	−763.343	−1.46515	−0.732575	+	0.680687i	$0.761682\pi$
$522$	0	0
$523$	−730.559	−1.39686	−0.698431	−	0.715677i	$-0.746118\pi$
$523$	−730.559	−1.39686	−0.698431	+	0.715677i	$0.746118\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	43.3735i	0.0823026i
$528$	0	0
$529$	53.8490	0.101794
$530$	0	0
$531$	67.0539	0.126279
$532$	0	0
$533$	− 190.047i	− 0.356561i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	841.009	1.56613
$538$	0	0
$539$	−2444.30	−4.53487
$540$	0	0
$541$	238.988i	0.441752i	0.975302	+	0.220876i	$0.0708916\pi$
$541$	238.988i	0.441752i	−0.975302	+	0.220876i	$0.929108\pi$
$542$	0	0
$543$	− 810.120i	− 1.49193i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	288.988	0.528314	0.264157	−	0.964480i	$-0.414906\pi$
$547$	288.988	0.528314	0.264157	+	0.964480i	$0.414906\pi$
$548$	0	0
$549$	54.7964i	0.0998114i
$550$	0	0
$551$	− 512.727i	− 0.930538i
$552$	0	0
$553$	−620.767	−1.12254
$554$	0	0
$555$	0	0
$556$	0	0
$557$	756.760i	1.35864i	0.733844	+	0.679318i	$0.237724\pi$
$557$	756.760i	1.35864i	−0.733844	+	0.679318i	$0.762276\pi$
$558$	0	0
$559$	352.727i	0.630996i
$560$	0	0
$561$	−1071.39	−1.90979
$562$	0	0
$563$	280.522	0.498262	0.249131	−	0.968470i	$-0.419855\pi$
$563$	280.522	0.498262	0.249131	+	0.968470i	$0.419855\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	1218.12i	2.14836i
$568$	0	0
$569$	704.171	1.23756	0.618780	−	0.785564i	$-0.287627\pi$
$569$	704.171	1.23756	0.618780	+	0.785564i	$0.287627\pi$
$570$	0	0
$571$	−702.628	−1.23052	−0.615261	−	0.788323i	$-0.710949\pi$
$571$	−702.628	−1.23052	−0.615261	+	0.788323i	$0.710949\pi$
$572$	0	0
$573$	955.764i	1.66800i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	573.120	0.993276	0.496638	−	0.867958i	$-0.334568\pi$
$577$	573.120	0.993276	0.496638	+	0.867958i	$0.334568\pi$
$578$	0	0
$579$	−95.9129	−0.165653
$580$	0	0
$581$	1923.29i	3.31030i
$582$	0	0
$583$	618.647i	1.06114i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	160.338	0.273147	0.136574	−	0.990630i	$-0.456391\pi$
$587$	160.338	0.273147	0.136574	+	0.990630i	$0.456391\pi$
$588$	0	0
$589$	− 49.8972i	− 0.0847151i
$590$	0	0
$591$	− 236.182i	− 0.399631i
$592$	0	0
$593$	1047.64	1.76669	0.883343	−	0.468727i	$-0.155287\pi$
$593$	1047.64	1.76669	0.883343	+	0.468727i	$0.155287\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	421.664i	0.706304i
$598$	0	0
$599$	− 113.312i	− 0.189169i	−0.995517	−	0.0945845i	$-0.969848\pi$
$599$	− 113.312i	− 0.189169i	0.995517	−	0.0945845i	$-0.0301523\pi$
$600$	0	0
$601$	945.797	1.57371	0.786853	−	0.617141i	$-0.211709\pi$
$601$	945.797	1.57371	0.786853	+	0.617141i	$0.211709\pi$
$602$	0	0
$603$	−16.2288	−0.0269135
$604$	0	0
$605$	0	0
$606$	0	0
$607$	796.686i	1.31250i	0.754545	+	0.656249i	$0.227858\pi$
$607$	796.686i	1.31250i	−0.754545	+	0.656249i	$0.772142\pi$
$608$	0	0
$609$	−982.302	−1.61298
$610$	0	0
$611$	−411.865	−0.674084
$612$	0	0
$613$	180.204i	0.293971i	0.989139	+	0.146985i	$0.0469570\pi$
$613$	180.204i	0.293971i	−0.989139	+	0.146985i	$0.953043\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	950.604	1.54069	0.770344	−	0.637629i	$-0.220085\pi$
$617$	950.604	1.54069	0.770344	+	0.637629i	$0.220085\pi$
$618$	0	0
$619$	716.042	1.15677	0.578386	−	0.815763i	$-0.303683\pi$
$619$	716.042	1.15677	0.578386	+	0.815763i	$0.303683\pi$
$620$	0	0
$621$	− 555.585i	− 0.894662i
$622$	0	0
$623$	− 1375.33i	− 2.20760i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	1232.54	1.96577
$628$	0	0
$629$	459.477i	0.730487i
$630$	0	0
$631$	281.312i	0.445820i	0.974839	+	0.222910i	$0.0715556\pi$
$631$	281.312i	0.445820i	−0.974839	+	0.222910i	$0.928444\pi$
$632$	0	0
$633$	−862.110	−1.36194
$634$	0	0
$635$	0	0
$636$	0	0
$637$	1599.57i	2.51110i
$638$	0	0
$639$	− 103.192i	− 0.161490i
$640$	0	0
$641$	609.918	0.951511	0.475755	−	0.879578i	$-0.342175\pi$
$641$	609.918	0.951511	0.475755	+	0.879578i	$0.342175\pi$
$642$	0	0
$643$	−85.5751	−0.133087	−0.0665436	−	0.997784i	$-0.521197\pi$
$643$	−85.5751	−0.133087	−0.0665436	+	0.997784i	$0.521197\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 288.363i	− 0.445693i	−0.974854	−	0.222846i	$-0.928465\pi$
$647$	− 288.363i	− 0.445693i	0.974854	−	0.222846i	$-0.0715348\pi$
$648$	0	0
$649$	1289.52	1.98694
$650$	0	0
$651$	−95.5951	−0.146843
$652$	0	0
$653$	− 708.520i	− 1.08502i	−0.840049	−	0.542511i	$-0.817474\pi$
$653$	− 708.520i	− 1.08502i	0.840049	−	0.542511i	$-0.182526\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−40.1725	−0.0611453
$658$	0	0
$659$	−932.376	−1.41483	−0.707417	−	0.706796i	$-0.750140\pi$
$659$	−932.376	−1.41483	−0.707417	+	0.706796i	$0.750140\pi$
$660$	0	0
$661$	714.677i	1.08121i	0.841278	+	0.540603i	$0.181804\pi$
$661$	714.677i	1.08121i	−0.841278	+	0.540603i	$0.818196\pi$
$662$	0	0
$663$	701.131i	1.05751i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	493.232	0.739478
$668$	0	0
$669$	− 766.417i	− 1.14562i
$670$	0	0
$671$	1053.80i	1.57049i
$672$	0	0
$673$	−529.151	−0.786257	−0.393129	−	0.919484i	$-0.628607\pi$
$673$	−529.151	−0.786257	−0.393129	+	0.919484i	$0.628607\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 563.913i	− 0.832959i	−0.909145	−	0.416480i	$-0.863264\pi$
$677$	− 563.913i	− 0.832959i	0.909145	−	0.416480i	$-0.136736\pi$
$678$	0	0
$679$	281.253i	0.414217i
$680$	0	0
$681$	811.403	1.19149
$682$	0	0
$683$	−93.7620	−0.137280	−0.0686398	−	0.997642i	$-0.521866\pi$
$683$	−93.7620	−0.137280	−0.0686398	+	0.997642i	$0.521866\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	661.131i	0.962344i
$688$	0	0
$689$	404.849	0.587589
$690$	0	0
$691$	541.119	0.783095	0.391548	−	0.920158i	$-0.371940\pi$
$691$	541.119	0.783095	0.391548	+	0.920158i	$0.371940\pi$
$692$	0	0
$693$	− 214.447i	− 0.309447i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−330.868	−0.474704
$698$	0	0
$699$	187.569	0.268339
$700$	0	0
$701$	− 279.786i	− 0.399125i	−0.979885	−	0.199562i	$-0.936048\pi$
$701$	− 279.786i	− 0.399125i	0.979885	−	0.199562i	$-0.0639520\pi$
$702$	0	0
$703$	− 528.586i	− 0.751900i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−2284.62	−3.23143
$708$	0	0
$709$	− 423.436i	− 0.597230i	−0.954374	−	0.298615i	$-0.903475\pi$
$709$	− 423.436i	− 0.597230i	0.954374	−	0.298615i	$-0.0965246\pi$
$710$	0	0
$711$	− 40.4449i	− 0.0568845i
$712$	0	0
$713$	48.0000	0.0673212
$714$	0	0
$715$	0	0
$716$	0	0
$717$	14.9993i	0.0209196i
$718$	0	0
$719$	− 923.010i	− 1.28374i	−0.766813	−	0.641871i	$-0.778159\pi$
$719$	− 923.010i	− 1.28374i	0.766813	−	0.641871i	$-0.221841\pi$
$720$	0	0
$721$	−480.000	−0.665742
$722$	0	0
$723$	−3.43199	−0.00474688
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 155.151i	− 0.213413i	−0.994291	−	0.106706i	$-0.965970\pi$
$727$	− 155.151i	− 0.213413i	0.994291	−	0.106706i	$-0.0340305\pi$
$728$	0	0
$729$	642.362	0.881155
$730$	0	0
$731$	614.090	0.840068
$732$	0	0
$733$	46.3978i	0.0632984i	0.999499	+	0.0316492i	$0.0100759\pi$
$733$	46.3978i	0.0632984i	−0.999499	+	0.0316492i	$0.989924\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−312.099	−0.423472
$738$	0	0
$739$	574.678	0.777643	0.388821	−	0.921313i	$-0.372882\pi$
$739$	574.678	0.777643	0.388821	+	0.921313i	$0.372882\pi$
$740$	0	0
$741$	− 806.587i	− 1.08851i
$742$	0	0
$743$	1182.44i	1.59144i	0.605662	+	0.795722i	$0.292908\pi$
$743$	1182.44i	1.59144i	−0.605662	+	0.795722i	$0.707092\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−125.308	−0.167748
$748$	0	0
$749$	1205.04i	1.60886i
$750$	0	0
$751$	1006.30i	1.33995i	0.742384	+	0.669975i	$0.233695\pi$
$751$	1006.30i	1.33995i	−0.742384	+	0.669975i	$0.766305\pi$
$752$	0	0
$753$	−706.423	−0.938145
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 256.530i	− 0.338877i	−0.985541	−	0.169438i	$-0.945805\pi$
$757$	− 256.530i	− 0.338877i	0.985541	−	0.169438i	$-0.0541954\pi$
$758$	0	0
$759$	1185.68i	1.56215i
$760$	0	0
$761$	−733.949	−0.964453	−0.482227	−	0.876046i	$-0.660172\pi$
$761$	−733.949	−0.964453	−0.482227	+	0.876046i	$0.660172\pi$
$762$	0	0
$763$	2667.00	3.49542
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 843.878i	− 1.10023i
$768$	0	0
$769$	−498.081	−0.647699	−0.323850	−	0.946109i	$-0.604977\pi$
$769$	−498.081	−0.647699	−0.323850	+	0.946109i	$0.604977\pi$
$770$	0	0
$771$	705.970	0.915655
$772$	0	0
$773$	− 841.555i	− 1.08869i	−0.838862	−	0.544344i	$-0.816779\pi$
$773$	− 841.555i	− 1.08869i	0.838862	−	0.544344i	$-0.183221\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−1012.69	−1.30333
$778$	0	0
$779$	380.634	0.488618
$780$	0	0
$781$	− 1984.50i	− 2.54097i
$782$	0	0
$783$	576.727i	0.736560i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	408.401	0.518934	0.259467	−	0.965752i	$-0.416453\pi$
$787$	408.401	0.518934	0.259467	+	0.965752i	$0.416453\pi$
$788$	0	0
$789$	− 120.065i	− 0.152174i
$790$	0	0
$791$	− 1103.70i	− 1.39532i
$792$	0	0
$793$	689.616	0.869630
$794$	0	0
$795$	0	0
$796$	0	0
$797$	387.395i	0.486066i	0.970018	+	0.243033i	$0.0781424\pi$
$797$	387.395i	0.486066i	−0.970018	+	0.243033i	$0.921858\pi$
$798$	0	0
$799$	717.049i	0.897433i
$800$	0	0
$801$	89.6072	0.111869
$802$	0	0
$803$	−772.562	−0.962095
$804$	0	0
$805$	0	0
$806$	0	0
$807$	− 411.959i	− 0.510482i
$808$	0	0
$809$	−407.535	−0.503751	−0.251876	−	0.967760i	$-0.581047\pi$
$809$	−407.535	−0.503751	−0.251876	+	0.967760i	$0.581047\pi$
$810$	0	0
$811$	917.402	1.13120	0.565600	−	0.824680i	$-0.308645\pi$
$811$	917.402	1.13120	0.565600	+	0.824680i	$0.308645\pi$
$812$	0	0
$813$	− 391.851i	− 0.481982i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−706.454	−0.864693
$818$	0	0
$819$	−140.336	−0.171351
$820$	0	0
$821$	800.429i	0.974944i	0.873139	+	0.487472i	$0.162081\pi$
$821$	800.429i	0.974944i	−0.873139	+	0.487472i	$0.837919\pi$
$822$	0	0
$823$	− 1004.16i	− 1.22012i	−0.792355	−	0.610060i	$-0.791145\pi$
$823$	− 1004.16i	− 1.22012i	0.792355	−	0.610060i	$-0.208855\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−661.390	−0.799746	−0.399873	−	0.916571i	$-0.630946\pi$
$827$	−661.390	−0.799746	−0.399873	+	0.916571i	$0.630946\pi$
$828$	0	0
$829$	− 133.337i	− 0.160841i	−0.996761	−	0.0804205i	$-0.974374\pi$
$829$	− 133.337i	− 0.160841i	0.996761	−	0.0804205i	$-0.0256263\pi$
$830$	0	0
$831$	− 791.110i	− 0.951998i
$832$	0	0
$833$	2784.83	3.34313
$834$	0	0
$835$	0	0
$836$	0	0
$837$	56.1255i	0.0670556i
$838$	0	0
$839$	134.384i	0.160171i	0.996788	+	0.0800856i	$0.0255194\pi$
$839$	134.384i	0.160171i	−0.996788	+	0.0800856i	$0.974481\pi$
$840$	0	0
$841$	329.000	0.391201
$842$	0	0
$843$	−1055.68	−1.25229
$844$	0	0
$845$	0	0
$846$	0	0
$847$	− 2454.50i	− 2.89788i
$848$	0	0
$849$	234.081	0.275713
$850$	0	0
$851$	508.488	0.597518
$852$	0	0
$853$	1155.10i	1.35416i	0.735911	+	0.677079i	$0.236754\pi$
$853$	1155.10i	1.35416i	−0.735911	+	0.677079i	$0.763246\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	754.678	0.880604	0.440302	−	0.897850i	$-0.354871\pi$
$857$	754.678	0.880604	0.440302	+	0.897850i	$0.354871\pi$
$858$	0	0
$859$	214.906	0.250182	0.125091	−	0.992145i	$-0.460078\pi$
$859$	214.906	0.250182	0.125091	+	0.992145i	$0.460078\pi$
$860$	0	0
$861$	− 729.233i	− 0.846961i
$862$	0	0
$863$	− 872.000i	− 1.01043i	−0.862994	−	0.505214i	$-0.831413\pi$
$863$	− 872.000i	− 1.01043i	0.862994	−	0.505214i	$-0.168587\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	311.384	0.359151
$868$	0	0
$869$	− 777.802i	− 0.895054i
$870$	0	0
$871$	204.241i	0.234490i
$872$	0	0
$873$	−18.3245	−0.0209903
$874$	0	0
$875$	0	0
$876$	0	0
$877$	− 1559.40i	− 1.77811i	−0.457798	−	0.889056i	$-0.651362\pi$
$877$	− 1559.40i	− 1.77811i	0.457798	−	0.889056i	$-0.348638\pi$
$878$	0	0
$879$	492.545i	0.560347i
$880$	0	0
$881$	342.686	0.388974	0.194487	−	0.980905i	$-0.437696\pi$
$881$	342.686	0.388974	0.194487	+	0.980905i	$0.437696\pi$
$882$	0	0
$883$	−179.844	−0.203674	−0.101837	−	0.994801i	$-0.532472\pi$
$883$	−179.844	−0.203674	−0.101837	+	0.994801i	$0.532472\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	1032.93i	1.16452i	0.813004	+	0.582258i	$0.197831\pi$
$887$	1032.93i	1.16452i	−0.813004	+	0.582258i	$0.802169\pi$
$888$	0	0
$889$	−2116.52	−2.38079
$890$	0	0
$891$	−1526.27	−1.71298
$892$	0	0
$893$	− 824.899i	− 0.923739i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	775.918	0.865015
$898$	0	0
$899$	−49.8265	−0.0554244
$900$	0	0
$901$	− 704.834i	− 0.782280i
$902$	0	0
$903$	1353.45i	1.49884i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−76.4317	−0.0842687	−0.0421344	−	0.999112i	$-0.513416\pi$
$907$	−76.4317	−0.0842687	−0.0421344	+	0.999112i	$0.513416\pi$
$908$	0	0
$909$	− 148.850i	− 0.163752i
$910$	0	0
$911$	110.910i	0.121746i	0.998146	+	0.0608728i	$0.0193884\pi$
$911$	110.910i	0.121746i	−0.998146	+	0.0608728i	$0.980612\pi$
$912$	0	0
$913$	−2409.82	−2.63945
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 678.335i	− 0.739733i
$918$	0	0
$919$	− 1493.11i	− 1.62471i	−0.583164	−	0.812355i	$-0.698185\pi$
$919$	− 1493.11i	− 1.62471i	0.583164	−	0.812355i	$-0.301815\pi$
$920$	0	0
$921$	396.868	0.430910
$922$	0	0
$923$	−1298.68	−1.40702
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 31.2735i	− 0.0337362i
$928$	0	0
$929$	−262.163	−0.282199	−0.141100	−	0.989995i	$-0.545064\pi$
$929$	−262.163	−0.282199	−0.141100	+	0.989995i	$0.545064\pi$
$930$	0	0
$931$	−3203.69	−3.44112
$932$	0	0
$933$	1200.98i	1.28722i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0.443872	0.000473716 0	0.000236858	−	1.00000i	$-0.499925\pi$
$937$	0.443872	0.000473716 0	0.000236858		1.00000i	$0.499925\pi$
$938$	0	0
$939$	1586.67	1.68974
$940$	0	0
$941$	581.057i	0.617489i	0.951145	+	0.308744i	$0.0999088\pi$
$941$	581.057i	0.617489i	−0.951145	+	0.308744i	$0.900091\pi$
$942$	0	0
$943$	366.161i	0.388294i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	224.068	0.236609	0.118304	−	0.992977i	$-0.462254\pi$
$947$	224.068	0.236609	0.118304	+	0.992977i	$0.462254\pi$
$948$	0	0
$949$	505.573i	0.532743i
$950$	0	0
$951$	− 1309.86i	− 1.37735i
$952$	0	0
$953$	1132.58	1.18843	0.594216	−	0.804306i	$-0.297462\pi$
$953$	1132.58	1.18843	0.594216	+	0.804306i	$0.297462\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 1230.79i	− 1.28610i
$958$	0	0
$959$	− 2894.64i	− 3.01840i
$960$	0	0
$961$	956.151	0.994954
$962$	0	0
$963$	−78.5120	−0.0815285
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 137.616i	− 0.142313i	−0.997465	−	0.0711563i	$-0.977331\pi$
$967$	− 137.616i	− 0.142313i	0.997465	−	0.0711563i	$-0.0226689\pi$
$968$	0	0
$969$	−1404.25	−1.44918
$970$	0	0
$971$	358.042	0.368735	0.184368	−	0.982857i	$-0.440976\pi$
$971$	358.042	0.368735	0.184368	+	0.982857i	$0.440976\pi$
$972$	0	0
$973$	− 718.292i	− 0.738224i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	498.705	0.510445	0.255223	−	0.966882i	$-0.417851\pi$
$977$	498.705	0.510445	0.255223	+	0.966882i	$0.417851\pi$
$978$	0	0
$979$	1723.25	1.76021
$980$	0	0
$981$	173.764i	0.177129i
$982$	0	0
$983$	− 575.778i	− 0.585735i	−0.956153	−	0.292868i	$-0.905391\pi$
$983$	− 575.778i	− 0.585735i	0.956153	−	0.292868i	$-0.0946095\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−1580.37	−1.60119
$988$	0	0
$989$	− 679.593i	− 0.687152i
$990$	0	0
$991$	− 1354.57i	− 1.36687i	−0.730013	−	0.683434i	$-0.760486\pi$
$991$	− 1354.57i	− 1.36687i	0.730013	−	0.683434i	$-0.239514\pi$
$992$	0	0
$993$	659.524	0.664174
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 1046.86i	− 1.05001i	−0.851100	−	0.525004i	$-0.824064\pi$
$997$	− 1046.86i	− 1.05001i	0.851100	−	0.525004i	$-0.175936\pi$
$998$	0	0
$999$	594.565i	0.595160i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1600.3.g.f.351.7	yes	8
4.3	odd	2	inner	1600.3.g.f.351.2	yes	8
5.2	odd	4		1600.3.e.k.799.2		8
5.3	odd	4		1600.3.e.h.799.8		8
5.4	even	2		1600.3.g.g.351.2	yes	8
8.3	odd	2	inner	1600.3.g.f.351.8	yes	8
8.5	even	2	inner	1600.3.g.f.351.1	✓	8
20.3	even	4		1600.3.e.k.799.1		8
20.7	even	4		1600.3.e.h.799.7		8
20.19	odd	2		1600.3.g.g.351.7	yes	8
40.3	even	4		1600.3.e.k.799.8		8
40.13	odd	4		1600.3.e.h.799.1		8
40.19	odd	2		1600.3.g.g.351.1	yes	8
40.27	even	4		1600.3.e.h.799.2		8
40.29	even	2		1600.3.g.g.351.8	yes	8
40.37	odd	4		1600.3.e.k.799.7		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1600.3.e.h.799.1		8	40.13	odd	4
1600.3.e.h.799.2		8	40.27	even	4
1600.3.e.h.799.7		8	20.7	even	4
1600.3.e.h.799.8		8	5.3	odd	4
1600.3.e.k.799.1		8	20.3	even	4
1600.3.e.k.799.2		8	5.2	odd	4
1600.3.e.k.799.7		8	40.37	odd	4
1600.3.e.k.799.8		8	40.3	even	4
1600.3.g.f.351.1	✓	8	8.5	even	2	inner
1600.3.g.f.351.2	yes	8	4.3	odd	2	inner
1600.3.g.f.351.7	yes	8	1.1	even	1	trivial
1600.3.g.f.351.8	yes	8	8.3	odd	2	inner
1600.3.g.g.351.1	yes	8	40.19	odd	2
1600.3.g.g.351.2	yes	8	5.4	even	2
1600.3.g.g.351.7	yes	8	20.19	odd	2
1600.3.g.g.351.8	yes	8	40.29	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:	\( N \)	\(=\)	\( 1600 = 2^{6} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1600.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.14626 q^{3} -13.7980i q^{7} +0.898979 q^{9} +O(q^{10})\)	\(q+3.14626 q^{3} -13.7980i q^{7} +0.898979 q^{9} +17.2884 q^{11} -11.3137i q^{13} -19.6969 q^{17} +22.6595 q^{19} -43.4120i q^{21} +21.7980i q^{23} -25.4880 q^{27} -22.6274i q^{29} -2.20204i q^{31} +54.3939 q^{33} -23.3273i q^{37} -35.5959i q^{39} +16.7980 q^{41} -31.1769 q^{43} -36.4041i q^{47} -141.384 q^{49} -61.9718 q^{51} +35.7839i q^{53} +71.2929 q^{57} +74.5889 q^{59} +60.9540i q^{61} -12.4041i q^{63} -18.0525 q^{67} +68.5821i q^{69} -114.788i q^{71} -44.6867 q^{73} -238.545i q^{77} -44.9898i q^{79} -88.2827 q^{81} -139.389 q^{83} -71.1918i q^{87} +99.6765 q^{89} -156.106 q^{91} -6.92820i q^{93} -20.3837 q^{97} +15.5419 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 32 q^{9}+O(q^{10}) \) 8 * q - 32 * q^9	\( 8 q - 32 q^{9} - 40 q^{17} + 200 q^{33} + 56 q^{41} - 504 q^{49} + 296 q^{57} + 152 q^{73} - 40 q^{81} - 104 q^{89} + 464 q^{97}+O(q^{100}) \) 8 * q - 32 * q^9 - 40 * q^17 + 200 * q^33 + 56 * q^41 - 504 * q^49 + 296 * q^57 + 152 * q^73 - 40 * q^81 - 104 * q^89 + 464 * q^97

Introduction

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