LMFDB - Embedded newform 1840.3.c.b.1151.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1840,3,Mod(1151,1840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1840, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1840.1151");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1840 = 2^{4} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1840.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:	$50.1363686423$
Analytic rank:	$0$
Dimension:	$56$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1151.7
Character	$\chi$	$=$	1840.1151
Dual form			1840.3.c.b.1151.50

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.41875i q^{3} +2.23607 q^{5} +4.25333i q^{7} -10.5254 q^{9} +O(q^{10})$	$q-4.41875i q^{3} +2.23607 q^{5} +4.25333i q^{7} -10.5254 q^{9} -12.1282i q^{11} -0.465703 q^{13} -9.88063i q^{15} +19.1848 q^{17} -17.0061i q^{19} +18.7944 q^{21} +4.79583i q^{23} +5.00000 q^{25} +6.74018i q^{27} +18.2506 q^{29} +2.24023i q^{31} -53.5916 q^{33} +9.51074i q^{35} +29.9880 q^{37} +2.05783i q^{39} -22.9811 q^{41} -6.12942i q^{43} -23.5354 q^{45} -12.3459i q^{47} +30.9092 q^{49} -84.7728i q^{51} -46.8587 q^{53} -27.1195i q^{55} -75.1457 q^{57} -5.33963i q^{59} -2.01650 q^{61} -44.7678i q^{63} -1.04134 q^{65} -82.3451i q^{67} +21.1916 q^{69} -87.0732i q^{71} -88.4411 q^{73} -22.0938i q^{75} +51.5854 q^{77} -7.99745i q^{79} -64.9451 q^{81} -85.0877i q^{83} +42.8985 q^{85} -80.6449i q^{87} -17.7157 q^{89} -1.98079i q^{91} +9.89904 q^{93} -38.0268i q^{95} -22.7399 q^{97} +127.654i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 56 q - 120 q^{9}+O(q^{10}) $ 56 * q - 120 * q^9	$ 56 q - 120 q^{9} - 56 q^{13} - 96 q^{17} + 104 q^{21} + 280 q^{25} - 76 q^{29} + 240 q^{33} - 88 q^{37} - 76 q^{41} - 356 q^{49} - 88 q^{53} - 256 q^{57} + 376 q^{61} + 120 q^{65} + 192 q^{73} - 168 q^{77} - 392 q^{81} - 60 q^{85} + 368 q^{89} + 216 q^{93} + 264 q^{97}+O(q^{100}) $ 56 * q - 120 * q^9 - 56 * q^13 - 96 * q^17 + 104 * q^21 + 280 * q^25 - 76 * q^29 + 240 * q^33 - 88 * q^37 - 76 * q^41 - 356 * q^49 - 88 * q^53 - 256 * q^57 + 376 * q^61 + 120 * q^65 + 192 * q^73 - 168 * q^77 - 392 * q^81 - 60 * q^85 + 368 * q^89 + 216 * q^93 + 264 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times$.

$n$	$737$	$1151$	$1201$	$1381$
$\chi(n)$	$1$	$-1$	$1$	$1$

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
$3$	− 4.41875i	− 1.47292i	−0.676483	−	0.736458i	$-0.736497\pi$
$3$	− 4.41875i	− 1.47292i	0.676483	−	0.736458i	$-0.263503\pi$
$4$	0	0
$5$	2.23607	0.447214
$5$	2.23607	0.447214
$6$	0	0
$7$	4.25333i	0.607619i	0.952733	+	0.303809i	$0.0982586\pi$
$7$	4.25333i	0.607619i	−0.952733	+	0.303809i	$0.901741\pi$
$8$	0	0
$9$	−10.5254	−1.16948
$10$	0	0
$11$	− 12.1282i	− 1.10257i	−0.834318	−	0.551283i	$-0.814138\pi$
$11$	− 12.1282i	− 1.10257i	0.834318	−	0.551283i	$-0.185862\pi$
$12$	0	0
$13$	−0.465703	−0.0358233	−0.0179117	−	0.999840i	$-0.505702\pi$
$13$	−0.465703	−0.0358233	−0.0179117	+	0.999840i	$0.505702\pi$
$14$	0	0
$15$	− 9.88063i	− 0.658708i
$16$	0	0
$17$	19.1848	1.12852	0.564258	−	0.825598i	$-0.309162\pi$
$17$	19.1848	1.12852	0.564258	+	0.825598i	$0.309162\pi$
$18$	0	0
$19$	− 17.0061i	− 0.895058i	−0.894269	−	0.447529i	$-0.852304\pi$
$19$	− 17.0061i	− 0.895058i	0.894269	−	0.447529i	$-0.147696\pi$
$20$	0	0
$21$	18.7944	0.894972
$22$	0	0
$23$	4.79583i	0.208514i
$23$	4.79583i	0.208514i
$24$	0	0
$25$	5.00000	0.200000
$26$	0	0
$27$	6.74018i	0.249636i
$28$	0	0
$29$	18.2506	0.629332	0.314666	−	0.949203i	$-0.398108\pi$
$29$	18.2506	0.629332	0.314666	+	0.949203i	$0.398108\pi$
$30$	0	0
$31$	2.24023i	0.0722656i	0.999347	+	0.0361328i	$0.0115039\pi$
$31$	2.24023i	0.0722656i	−0.999347	+	0.0361328i	$0.988496\pi$
$32$	0	0
$33$	−53.5916	−1.62399
$34$	0	0
$35$	9.51074i	0.271735i
$36$	0	0
$37$	29.9880	0.810486	0.405243	−	0.914209i	$-0.367187\pi$
$37$	29.9880	0.810486	0.405243	+	0.914209i	$0.367187\pi$
$38$	0	0
$39$	2.05783i	0.0527648i
$40$	0	0
$41$	−22.9811	−0.560515	−0.280257	−	0.959925i	$-0.590420\pi$
$41$	−22.9811	−0.560515	−0.280257	+	0.959925i	$0.590420\pi$
$42$	0	0
$43$	− 6.12942i	− 0.142545i	−0.997457	−	0.0712724i	$-0.977294\pi$
$43$	− 6.12942i	− 0.142545i	0.997457	−	0.0712724i	$-0.0227060\pi$
$44$	0	0
$45$	−23.5354	−0.523009
$46$	0	0
$47$	− 12.3459i	− 0.262678i	−0.991338	−	0.131339i	$-0.958072\pi$
$47$	− 12.3459i	− 0.262678i	0.991338	−	0.131339i	$-0.0419276\pi$
$48$	0	0
$49$	30.9092	0.630800
$50$	0	0
$51$	− 84.7728i	− 1.66221i
$52$	0	0
$53$	−46.8587	−0.884127	−0.442064	−	0.896984i	$-0.645753\pi$
$53$	−46.8587	−0.884127	−0.442064	+	0.896984i	$0.645753\pi$
$54$	0	0
$55$	− 27.1195i	− 0.493083i
$56$	0	0
$57$	−75.1457	−1.31835
$58$	0	0
$59$	− 5.33963i	− 0.0905023i	−0.998976	−	0.0452511i	$-0.985591\pi$
$59$	− 5.33963i	− 0.0905023i	0.998976	−	0.0452511i	$-0.0144088\pi$
$60$	0	0
$61$	−2.01650	−0.0330573	−0.0165287	−	0.999863i	$-0.505261\pi$
$61$	−2.01650	−0.0330573	−0.0165287	+	0.999863i	$0.505261\pi$
$62$	0	0
$63$	− 44.7678i	− 0.710600i
$64$	0	0
$65$	−1.04134	−0.0160207
$66$	0	0
$67$	− 82.3451i	− 1.22903i	−0.788904	−	0.614516i	$-0.789351\pi$
$67$	− 82.3451i	− 1.22903i	0.788904	−	0.614516i	$-0.210649\pi$
$68$	0	0
$69$	21.1916	0.307124
$70$	0	0
$71$	− 87.0732i	− 1.22638i	−0.789934	−	0.613191i	$-0.789885\pi$
$71$	− 87.0732i	− 1.22638i	0.789934	−	0.613191i	$-0.210115\pi$
$72$	0	0
$73$	−88.4411	−1.21152	−0.605761	−	0.795647i	$-0.707131\pi$
$73$	−88.4411	−1.21152	−0.605761	+	0.795647i	$0.707131\pi$
$74$	0	0
$75$	− 22.0938i	− 0.294583i
$76$	0	0
$77$	51.5854	0.669940
$78$	0	0
$79$	− 7.99745i	− 0.101234i	−0.998718	−	0.0506168i	$-0.983881\pi$
$79$	− 7.99745i	− 0.101234i	0.998718	−	0.0506168i	$-0.0161187\pi$
$80$	0	0
$81$	−64.9451	−0.801791
$82$	0	0
$83$	− 85.0877i	− 1.02515i	−0.858641	−	0.512577i	$-0.828691\pi$
$83$	− 85.0877i	− 1.02515i	0.858641	−	0.512577i	$-0.171309\pi$
$84$	0	0
$85$	42.8985	0.504688
$86$	0	0
$87$	− 80.6449i	− 0.926953i
$88$	0	0
$89$	−17.7157	−0.199053	−0.0995263	−	0.995035i	$-0.531733\pi$
$89$	−17.7157	−0.199053	−0.0995263	+	0.995035i	$0.531733\pi$
$90$	0	0
$91$	− 1.98079i	− 0.0217669i
$92$	0	0
$93$	9.89904	0.106441
$94$	0	0
$95$	− 38.0268i	− 0.400282i
$96$	0	0
$97$	−22.7399	−0.234432	−0.117216	−	0.993106i	$-0.537397\pi$
$97$	−22.7399	−0.234432	−0.117216	+	0.993106i	$0.537397\pi$
$98$	0	0
$99$	127.654i	1.28943i
$100$	0	0
$101$	48.3404	0.478617	0.239309	−	0.970944i	$-0.423079\pi$
$101$	48.3404	0.478617	0.239309	+	0.970944i	$0.423079\pi$
$102$	0	0
$103$	82.9340i	0.805185i	0.915379	+	0.402592i	$0.131891\pi$
$103$	82.9340i	0.805185i	−0.915379	+	0.402592i	$0.868109\pi$
$104$	0	0
$105$	42.0256	0.400244
$106$	0	0
$107$	− 103.988i	− 0.971852i	−0.874000	−	0.485926i	$-0.838482\pi$
$107$	− 103.988i	− 0.971852i	0.874000	−	0.485926i	$-0.161518\pi$
$108$	0	0
$109$	60.2498	0.552750	0.276375	−	0.961050i	$-0.410867\pi$
$109$	60.2498	0.552750	0.276375	+	0.961050i	$0.410867\pi$
$110$	0	0
$111$	− 132.509i	− 1.19378i
$112$	0	0
$113$	−47.4128	−0.419582	−0.209791	−	0.977746i	$-0.567278\pi$
$113$	−47.4128	−0.419582	−0.209791	+	0.977746i	$0.567278\pi$
$114$	0	0
$115$	10.7238i	0.0932505i
$116$	0	0
$117$	4.90169	0.0418948
$118$	0	0
$119$	81.5992i	0.685708i
$120$	0	0
$121$	−26.0939	−0.215652
$122$	0	0
$123$	101.548i	0.825592i
$124$	0	0
$125$	11.1803	0.0894427
$126$	0	0
$127$	112.650i	0.887010i	0.896272	+	0.443505i	$0.146265\pi$
$127$	112.650i	0.887010i	−0.896272	+	0.443505i	$0.853735\pi$
$128$	0	0
$129$	−27.0844	−0.209957
$130$	0	0
$131$	− 175.301i	− 1.33817i	−0.743184	−	0.669087i	$-0.766685\pi$
$131$	− 175.301i	− 1.33817i	0.743184	−	0.669087i	$-0.233315\pi$
$132$	0	0
$133$	72.3326	0.543854
$134$	0	0
$135$	15.0715i	0.111641i
$136$	0	0
$137$	−102.474	−0.747984	−0.373992	−	0.927432i	$-0.622011\pi$
$137$	−102.474	−0.747984	−0.373992	+	0.927432i	$0.622011\pi$
$138$	0	0
$139$	75.7793i	0.545175i	0.962131	+	0.272587i	$0.0878794\pi$
$139$	75.7793i	0.545175i	−0.962131	+	0.272587i	$0.912121\pi$
$140$	0	0
$141$	−54.5533	−0.386903
$142$	0	0
$143$	5.64816i	0.0394976i
$144$	0	0
$145$	40.8096	0.281446
$146$	0	0
$147$	− 136.580i	− 0.929115i
$148$	0	0
$149$	−246.548	−1.65469	−0.827343	−	0.561697i	$-0.810149\pi$
$149$	−246.548	−1.65469	−0.827343	+	0.561697i	$0.810149\pi$
$150$	0	0
$151$	105.511i	0.698750i	0.936983	+	0.349375i	$0.113606\pi$
$151$	105.511i	0.698750i	−0.936983	+	0.349375i	$0.886394\pi$
$152$	0	0
$153$	−201.927	−1.31978
$154$	0	0
$155$	5.00932i	0.0323182i
$156$	0	0
$157$	93.6526	0.596513	0.298257	−	0.954486i	$-0.403595\pi$
$157$	93.6526	0.596513	0.298257	+	0.954486i	$0.403595\pi$
$158$	0	0
$159$	207.057i	1.30225i
$160$	0	0
$161$	−20.3983	−0.126697
$162$	0	0
$163$	75.6895i	0.464353i	0.972674	+	0.232176i	$0.0745846\pi$
$163$	75.6895i	0.464353i	−0.972674	+	0.232176i	$0.925415\pi$
$164$	0	0
$165$	−119.835	−0.726270
$166$	0	0
$167$	− 12.9108i	− 0.0773101i	−0.999253	−	0.0386550i	$-0.987693\pi$
$167$	− 12.9108i	− 0.0773101i	0.999253	−	0.0386550i	$-0.0123073\pi$
$168$	0	0
$169$	−168.783	−0.998717
$170$	0	0
$171$	178.995i	1.04676i
$172$	0	0
$173$	29.7898	0.172195	0.0860977	−	0.996287i	$-0.472560\pi$
$173$	29.7898	0.172195	0.0860977	+	0.996287i	$0.472560\pi$
$174$	0	0
$175$	21.2667i	0.121524i
$176$	0	0
$177$	−23.5945	−0.133302
$178$	0	0
$179$	11.0292i	0.0616158i	0.999525	+	0.0308079i	$0.00980801\pi$
$179$	11.0292i	0.0616158i	−0.999525	+	0.0308079i	$0.990192\pi$
$180$	0	0
$181$	−166.068	−0.917505	−0.458753	−	0.888564i	$-0.651704\pi$
$181$	−166.068	−0.917505	−0.458753	+	0.888564i	$0.651704\pi$
$182$	0	0
$183$	8.91040i	0.0486907i
$184$	0	0
$185$	67.0552	0.362460
$186$	0	0
$187$	− 232.677i	− 1.24426i
$188$	0	0
$189$	−28.6682	−0.151684
$190$	0	0
$191$	93.1777i	0.487841i	0.969795	+	0.243921i	$0.0784337\pi$
$191$	93.1777i	0.487841i	−0.969795	+	0.243921i	$0.921566\pi$
$192$	0	0
$193$	−192.316	−0.996456	−0.498228	−	0.867046i	$-0.666016\pi$
$193$	−192.316	−0.996456	−0.498228	+	0.867046i	$0.666016\pi$
$194$	0	0
$195$	4.60144i	0.0235971i
$196$	0	0
$197$	240.236	1.21947	0.609736	−	0.792605i	$-0.291276\pi$
$197$	240.236	1.21947	0.609736	+	0.792605i	$0.291276\pi$
$198$	0	0
$199$	10.3965i	0.0522436i	0.999659	+	0.0261218i	$0.00831577\pi$
$199$	10.3965i	0.0522436i	−0.999659	+	0.0261218i	$0.991684\pi$
$200$	0	0
$201$	−363.863	−1.81026
$202$	0	0
$203$	77.6259i	0.382394i
$204$	0	0
$205$	−51.3873	−0.250670
$206$	0	0
$207$	− 50.4778i	− 0.243854i
$208$	0	0
$209$	−206.254	−0.986860
$210$	0	0
$211$	109.021i	0.516689i	0.966053	+	0.258344i	$0.0831770\pi$
$211$	109.021i	0.516689i	−0.966053	+	0.258344i	$0.916823\pi$
$212$	0	0
$213$	−384.755	−1.80636
$214$	0	0
$215$	− 13.7058i	− 0.0637479i
$216$	0	0
$217$	−9.52846	−0.0439099
$218$	0	0
$219$	390.799i	1.78447i
$220$	0	0
$221$	−8.93442	−0.0404272
$222$	0	0
$223$	− 22.6211i	− 0.101440i	−0.998713	−	0.0507200i	$-0.983848\pi$
$223$	− 22.6211i	− 0.101440i	0.998713	−	0.0507200i	$-0.0161516\pi$
$224$	0	0
$225$	−52.6268	−0.233897
$226$	0	0
$227$	− 241.533i	− 1.06402i	−0.846737	−	0.532011i	$-0.821436\pi$
$227$	− 241.533i	− 1.06402i	0.846737	−	0.532011i	$-0.178564\pi$
$228$	0	0
$229$	−175.038	−0.764356	−0.382178	−	0.924089i	$-0.624826\pi$
$229$	−175.038	−0.764356	−0.382178	+	0.924089i	$0.624826\pi$
$230$	0	0
$231$	− 227.943i	− 0.986766i
$232$	0	0
$233$	178.807	0.767411	0.383705	−	0.923456i	$-0.374648\pi$
$233$	178.807	0.767411	0.383705	+	0.923456i	$0.374648\pi$
$234$	0	0
$235$	− 27.6062i	− 0.117473i
$236$	0	0
$237$	−35.3388	−0.149109
$238$	0	0
$239$	− 370.430i	− 1.54992i	−0.632012	−	0.774959i	$-0.717771\pi$
$239$	− 370.430i	− 1.54992i	0.632012	−	0.774959i	$-0.282229\pi$
$240$	0	0
$241$	−351.286	−1.45762	−0.728809	−	0.684717i	$-0.759926\pi$
$241$	−351.286	−1.45762	−0.728809	+	0.684717i	$0.759926\pi$
$242$	0	0
$243$	347.638i	1.43061i
$244$	0	0
$245$	69.1150	0.282102
$246$	0	0
$247$	7.91980i	0.0320639i
$248$	0	0
$249$	−375.982	−1.50997
$250$	0	0
$251$	371.767i	1.48114i	0.671978	+	0.740572i	$0.265445\pi$
$251$	371.767i	1.48114i	−0.671978	+	0.740572i	$0.734555\pi$
$252$	0	0
$253$	58.1649	0.229901
$254$	0	0
$255$	− 189.558i	− 0.743364i
$256$	0	0
$257$	265.231	1.03203	0.516014	−	0.856580i	$-0.327415\pi$
$257$	265.231	1.03203	0.516014	+	0.856580i	$0.327415\pi$
$258$	0	0
$259$	127.549i	0.492466i
$260$	0	0
$261$	−192.094	−0.735994
$262$	0	0
$263$	55.2452i	0.210058i	0.994469	+	0.105029i	$0.0334935\pi$
$263$	55.2452i	0.210058i	−0.994469	+	0.105029i	$0.966506\pi$
$264$	0	0
$265$	−104.779	−0.395394
$266$	0	0
$267$	78.2812i	0.293188i
$268$	0	0
$269$	41.4668	0.154152	0.0770758	−	0.997025i	$-0.475442\pi$
$269$	41.4668	0.154152	0.0770758	+	0.997025i	$0.475442\pi$
$270$	0	0
$271$	− 482.719i	− 1.78125i	−0.454736	−	0.890626i	$-0.650266\pi$
$271$	− 482.719i	− 1.78125i	0.454736	−	0.890626i	$-0.349734\pi$
$272$	0	0
$273$	−8.75262	−0.0320609
$274$	0	0
$275$	− 60.6411i	− 0.220513i
$276$	0	0
$277$	105.850	0.382128	0.191064	−	0.981578i	$-0.438806\pi$
$277$	105.850	0.382128	0.191064	+	0.981578i	$0.438806\pi$
$278$	0	0
$279$	− 23.5793i	− 0.0845135i
$280$	0	0
$281$	−171.507	−0.610346	−0.305173	−	0.952297i	$-0.598714\pi$
$281$	−171.507	−0.610346	−0.305173	+	0.952297i	$0.598714\pi$
$282$	0	0
$283$	225.382i	0.796402i	0.917298	+	0.398201i	$0.130365\pi$
$283$	225.382i	0.796402i	−0.917298	+	0.398201i	$0.869635\pi$
$284$	0	0
$285$	−168.031	−0.589582
$286$	0	0
$287$	− 97.7462i	− 0.340579i
$288$	0	0
$289$	79.0559	0.273550
$290$	0	0
$291$	100.482i	0.345299i
$292$	0	0
$293$	408.120	1.39290	0.696451	−	0.717605i	$-0.254762\pi$
$293$	408.120	1.39290	0.696451	+	0.717605i	$0.254762\pi$
$294$	0	0
$295$	− 11.9398i	− 0.0404739i
$296$	0	0
$297$	81.7464	0.275241
$298$	0	0
$299$	− 2.23343i	− 0.00746968i
$300$	0	0
$301$	26.0705	0.0866128
$302$	0	0
$303$	− 213.604i	− 0.704964i
$304$	0	0
$305$	−4.50902	−0.0147837
$306$	0	0
$307$	− 455.075i	− 1.48233i	−0.671323	−	0.741165i	$-0.734274\pi$
$307$	− 455.075i	− 1.48233i	0.671323	−	0.741165i	$-0.265726\pi$
$308$	0	0
$309$	366.465	1.18597
$310$	0	0
$311$	− 282.237i	− 0.907515i	−0.891125	−	0.453758i	$-0.850083\pi$
$311$	− 282.237i	− 0.907515i	0.891125	−	0.453758i	$-0.149917\pi$
$312$	0	0
$313$	290.288	0.927438	0.463719	−	0.885982i	$-0.346515\pi$
$313$	290.288	0.927438	0.463719	+	0.885982i	$0.346515\pi$
$314$	0	0
$315$	− 100.104i	− 0.317790i
$316$	0	0
$317$	208.883	0.658936	0.329468	−	0.944167i	$-0.393131\pi$
$317$	208.883	0.658936	0.329468	+	0.944167i	$0.393131\pi$
$318$	0	0
$319$	− 221.348i	− 0.693880i
$320$	0	0
$321$	−459.498	−1.43146
$322$	0	0
$323$	− 326.258i	− 1.01009i
$324$	0	0
$325$	−2.32852	−0.00716467
$326$	0	0
$327$	− 266.229i	− 0.814156i
$328$	0	0
$329$	52.5110	0.159608
$330$	0	0
$331$	367.866i	1.11138i	0.831390	+	0.555690i	$0.187546\pi$
$331$	367.866i	1.11138i	−0.831390	+	0.555690i	$0.812454\pi$
$332$	0	0
$333$	−315.634	−0.947851
$334$	0	0
$335$	− 184.129i	− 0.549640i
$336$	0	0
$337$	372.343	1.10488	0.552438	−	0.833554i	$-0.313698\pi$
$337$	372.343	1.10488	0.552438	+	0.833554i	$0.313698\pi$
$338$	0	0
$339$	209.505i	0.618010i
$340$	0	0
$341$	27.1701	0.0796776
$342$	0	0
$343$	339.880i	0.990904i
$344$	0	0
$345$	47.3858	0.137350
$346$	0	0
$347$	307.406i	0.885897i	0.896547	+	0.442948i	$0.146067\pi$
$347$	307.406i	0.885897i	−0.896547	+	0.442948i	$0.853933\pi$
$348$	0	0
$349$	−190.537	−0.545951	−0.272975	−	0.962021i	$-0.588008\pi$
$349$	−190.537	−0.545951	−0.272975	+	0.962021i	$0.588008\pi$
$350$	0	0
$351$	− 3.13892i	− 0.00894280i
$352$	0	0
$353$	454.183	1.28664	0.643319	−	0.765598i	$-0.277557\pi$
$353$	454.183	1.28664	0.643319	+	0.765598i	$0.277557\pi$
$354$	0	0
$355$	− 194.702i	− 0.548455i
$356$	0	0
$357$	360.567	1.00999
$358$	0	0
$359$	572.967i	1.59601i	0.602653	+	0.798004i	$0.294110\pi$
$359$	572.967i	1.59601i	−0.602653	+	0.798004i	$0.705890\pi$
$360$	0	0
$361$	71.7926	0.198872
$362$	0	0
$363$	115.303i	0.317638i
$364$	0	0
$365$	−197.760	−0.541809
$366$	0	0
$367$	− 313.066i	− 0.853042i	−0.904478	−	0.426521i	$-0.859739\pi$
$367$	− 313.066i	− 0.853042i	0.904478	−	0.426521i	$-0.140261\pi$
$368$	0	0
$369$	241.884	0.655513
$370$	0	0
$371$	− 199.306i	− 0.537212i
$372$	0	0
$373$	64.5178	0.172970	0.0864850	−	0.996253i	$-0.472437\pi$
$373$	64.5178	0.172970	0.0864850	+	0.996253i	$0.472437\pi$
$374$	0	0
$375$	− 49.4031i	− 0.131742i
$376$	0	0
$377$	−8.49937	−0.0225448
$378$	0	0
$379$	− 257.057i	− 0.678251i	−0.940741	−	0.339125i	$-0.889869\pi$
$379$	− 257.057i	− 0.678251i	0.940741	−	0.339125i	$-0.110131\pi$
$380$	0	0
$381$	497.773	1.30649
$382$	0	0
$383$	− 352.259i	− 0.919735i	−0.887987	−	0.459868i	$-0.847897\pi$
$383$	− 352.259i	− 0.919735i	0.887987	−	0.459868i	$-0.152103\pi$
$384$	0	0
$385$	115.348	0.299606
$386$	0	0
$387$	64.5144i	0.166704i
$388$	0	0
$389$	85.2336	0.219110	0.109555	−	0.993981i	$-0.465057\pi$
$389$	85.2336	0.219110	0.109555	+	0.993981i	$0.465057\pi$
$390$	0	0
$391$	92.0070i	0.235312i
$392$	0	0
$393$	−774.610	−1.97102
$394$	0	0
$395$	− 17.8829i	− 0.0452730i
$396$	0	0
$397$	209.006	0.526463	0.263231	−	0.964733i	$-0.415212\pi$
$397$	209.006	0.526463	0.263231	+	0.964733i	$0.415212\pi$
$398$	0	0
$399$	− 319.620i	− 0.801051i
$400$	0	0
$401$	650.604	1.62245	0.811227	−	0.584732i	$-0.198800\pi$
$401$	650.604	1.62245	0.811227	+	0.584732i	$0.198800\pi$
$402$	0	0
$403$	− 1.04328i	− 0.00258880i
$404$	0	0
$405$	−145.222	−0.358572
$406$	0	0
$407$	− 363.701i	− 0.893614i
$408$	0	0
$409$	235.653	0.576169	0.288085	−	0.957605i	$-0.406982\pi$
$409$	235.653	0.576169	0.288085	+	0.957605i	$0.406982\pi$
$410$	0	0
$411$	452.806i	1.10172i
$412$	0	0
$413$	22.7112	0.0549909
$414$	0	0
$415$	− 190.262i	− 0.458463i
$416$	0	0
$417$	334.850	0.802997
$418$	0	0
$419$	1.40161i	0.00334513i	0.999999	+	0.00167256i	$0.000532393\pi$
$419$	1.40161i	0.00334513i	−0.999999	+	0.00167256i	$0.999468\pi$
$420$	0	0
$421$	−349.595	−0.830393	−0.415197	−	0.909732i	$-0.636287\pi$
$421$	−349.595	−0.830393	−0.415197	+	0.909732i	$0.636287\pi$
$422$	0	0
$423$	129.945i	0.307198i
$424$	0	0
$425$	95.9239	0.225703
$426$	0	0
$427$	− 8.57683i	− 0.0200862i
$428$	0	0
$429$	24.9578	0.0581767
$430$	0	0
$431$	89.0519i	0.206617i	0.994649	+	0.103308i	$0.0329429\pi$
$431$	89.0519i	0.206617i	−0.994649	+	0.103308i	$0.967057\pi$
$432$	0	0
$433$	239.567	0.553273	0.276637	−	0.960975i	$-0.410780\pi$
$433$	239.567	0.553273	0.276637	+	0.960975i	$0.410780\pi$
$434$	0	0
$435$	− 180.328i	− 0.414546i
$436$	0	0
$437$	81.5584	0.186632
$438$	0	0
$439$	502.174i	1.14391i	0.820287	+	0.571953i	$0.193814\pi$
$439$	502.174i	1.14391i	−0.820287	+	0.571953i	$0.806186\pi$
$440$	0	0
$441$	−325.330	−0.737710
$442$	0	0
$443$	262.828i	0.593291i	0.954988	+	0.296645i	$0.0958679\pi$
$443$	262.828i	0.593291i	−0.954988	+	0.296645i	$0.904132\pi$
$444$	0	0
$445$	−39.6135	−0.0890191
$446$	0	0
$447$	1089.44i	2.43722i
$448$	0	0
$449$	−457.607	−1.01917	−0.509584	−	0.860421i	$-0.670201\pi$
$449$	−457.607	−1.01917	−0.509584	+	0.860421i	$0.670201\pi$
$450$	0	0
$451$	278.720i	0.618005i
$452$	0	0
$453$	466.228	1.02920
$454$	0	0
$455$	− 4.42918i	− 0.00973446i
$456$	0	0
$457$	772.045	1.68938	0.844688	−	0.535259i	$-0.179786\pi$
$457$	772.045	1.68938	0.844688	+	0.535259i	$0.179786\pi$
$458$	0	0
$459$	129.309i	0.281719i
$460$	0	0
$461$	277.725	0.602440	0.301220	−	0.953555i	$-0.402606\pi$
$461$	277.725	0.602440	0.301220	+	0.953555i	$0.402606\pi$
$462$	0	0
$463$	− 741.984i	− 1.60256i	−0.598291	−	0.801279i	$-0.704153\pi$
$463$	− 741.984i	− 1.60256i	0.598291	−	0.801279i	$-0.295847\pi$
$464$	0	0
$465$	22.1349	0.0476020
$466$	0	0
$467$	458.019i	0.980770i	0.871506	+	0.490385i	$0.163144\pi$
$467$	458.019i	0.980770i	−0.871506	+	0.490385i	$0.836856\pi$
$468$	0	0
$469$	350.241	0.746783
$470$	0	0
$471$	− 413.827i	− 0.878614i
$472$	0	0
$473$	−74.3391	−0.157165
$474$	0	0
$475$	− 85.0305i	− 0.179012i
$476$	0	0
$477$	493.205	1.03397
$478$	0	0
$479$	561.033i	1.17126i	0.810579	+	0.585629i	$0.199152\pi$
$479$	561.033i	1.17126i	−0.810579	+	0.585629i	$0.800848\pi$
$480$	0	0
$481$	−13.9655	−0.0290343
$482$	0	0
$483$	90.1348i	0.186615i
$484$	0	0
$485$	−50.8480	−0.104841
$486$	0	0
$487$	− 913.281i	− 1.87532i	−0.347554	−	0.937660i	$-0.612988\pi$
$487$	− 913.281i	− 1.87532i	0.347554	−	0.937660i	$-0.387012\pi$
$488$	0	0
$489$	334.453	0.683953
$490$	0	0
$491$	− 277.218i	− 0.564598i	−0.959326	−	0.282299i	$-0.908903\pi$
$491$	− 277.218i	− 0.564598i	0.959326	−	0.282299i	$-0.0910970\pi$
$492$	0	0
$493$	350.134	0.710211
$494$	0	0
$495$	285.443i	0.576652i
$496$	0	0
$497$	370.351	0.745173
$498$	0	0
$499$	− 9.07435i	− 0.0181851i	−0.999959	−	0.00909253i	$-0.997106\pi$
$499$	− 9.07435i	− 0.0181851i	0.999959	−	0.00909253i	$-0.00289428\pi$
$500$	0	0
$501$	−57.0495	−0.113871
$502$	0	0
$503$	− 432.849i	− 0.860534i	−0.902702	−	0.430267i	$-0.858419\pi$
$503$	− 432.849i	− 0.860534i	0.902702	−	0.430267i	$-0.141581\pi$
$504$	0	0
$505$	108.092	0.214044
$506$	0	0
$507$	745.811i	1.47103i
$508$	0	0
$509$	−548.202	−1.07702	−0.538509	−	0.842620i	$-0.681012\pi$
$509$	−548.202	−1.07702	−0.538509	+	0.842620i	$0.681012\pi$
$510$	0	0
$511$	− 376.169i	− 0.736144i
$512$	0	0
$513$	114.624	0.223439
$514$	0	0
$515$	185.446i	0.360090i
$516$	0	0
$517$	−149.733	−0.289620
$518$	0	0
$519$	− 131.634i	− 0.253630i
$520$	0	0
$521$	−785.216	−1.50713	−0.753567	−	0.657372i	$-0.771668\pi$
$521$	−785.216	−1.50713	−0.753567	+	0.657372i	$0.771668\pi$
$522$	0	0
$523$	− 435.663i	− 0.833009i	−0.909134	−	0.416504i	$-0.863255\pi$
$523$	− 435.663i	− 0.833009i	0.909134	−	0.416504i	$-0.136745\pi$
$524$	0	0
$525$	93.9720	0.178994
$526$	0	0
$527$	42.9784i	0.0815530i
$528$	0	0
$529$	−23.0000	−0.0434783
$530$	0	0
$531$	56.2016i	0.105841i
$532$	0	0
$533$	10.7024	0.0200795
$534$	0	0
$535$	− 232.525i	− 0.434626i
$536$	0	0
$537$	48.7354	0.0907550
$538$	0	0
$539$	− 374.874i	− 0.695498i
$540$	0	0
$541$	121.677	0.224912	0.112456	−	0.993657i	$-0.464128\pi$
$541$	121.677	0.224912	0.112456	+	0.993657i	$0.464128\pi$
$542$	0	0
$543$	733.815i	1.35141i
$544$	0	0
$545$	134.723	0.247198
$546$	0	0
$547$	− 388.520i	− 0.710275i	−0.934814	−	0.355138i	$-0.884434\pi$
$547$	− 388.520i	− 0.710275i	0.934814	−	0.355138i	$-0.115566\pi$
$548$	0	0
$549$	21.2244	0.0386600
$550$	0	0
$551$	− 310.372i	− 0.563288i
$552$	0	0
$553$	34.0158	0.0615114
$554$	0	0
$555$	− 296.300i	− 0.533874i
$556$	0	0
$557$	−86.1407	−0.154651	−0.0773255	−	0.997006i	$-0.524638\pi$
$557$	−86.1407	−0.154651	−0.0773255	+	0.997006i	$0.524638\pi$
$558$	0	0
$559$	2.85449i	0.00510643i
$560$	0	0
$561$	−1028.14	−1.83270
$562$	0	0
$563$	− 20.3532i	− 0.0361514i	−0.999837	−	0.0180757i	$-0.994246\pi$
$563$	− 20.3532i	− 0.0361514i	0.999837	−	0.0180757i	$-0.00575399\pi$
$564$	0	0
$565$	−106.018	−0.187643
$566$	0	0
$567$	− 276.233i	− 0.487183i
$568$	0	0
$569$	624.248	1.09710	0.548549	−	0.836119i	$-0.315181\pi$
$569$	624.248	1.09710	0.548549	+	0.836119i	$0.315181\pi$
$570$	0	0
$571$	− 520.987i	− 0.912412i	−0.889874	−	0.456206i	$-0.849208\pi$
$571$	− 520.987i	− 0.912412i	0.889874	−	0.456206i	$-0.150792\pi$
$572$	0	0
$573$	411.729	0.718550
$574$	0	0
$575$	23.9792i	0.0417029i
$576$	0	0
$577$	−549.403	−0.952171	−0.476086	−	0.879399i	$-0.657945\pi$
$577$	−549.403	−0.952171	−0.476086	+	0.879399i	$0.657945\pi$
$578$	0	0
$579$	849.796i	1.46770i
$580$	0	0
$581$	361.906	0.622902
$582$	0	0
$583$	568.314i	0.974809i
$584$	0	0
$585$	10.9605	0.0187359
$586$	0	0
$587$	683.706i	1.16475i	0.812922	+	0.582373i	$0.197876\pi$
$587$	683.706i	1.16475i	−0.812922	+	0.582373i	$0.802124\pi$
$588$	0	0
$589$	38.0976	0.0646819
$590$	0	0
$591$	− 1061.54i	− 1.79618i
$592$	0	0
$593$	952.830	1.60680	0.803398	−	0.595442i	$-0.203023\pi$
$593$	952.830	1.60680	0.803398	+	0.595442i	$0.203023\pi$
$594$	0	0
$595$	182.461i	0.306658i
$596$	0	0
$597$	45.9394	0.0769505
$598$	0	0
$599$	161.177i	0.269078i	0.990908	+	0.134539i	$0.0429553\pi$
$599$	161.177i	0.269078i	−0.990908	+	0.134539i	$0.957045\pi$
$600$	0	0
$601$	28.8340	0.0479767	0.0239883	−	0.999712i	$-0.492364\pi$
$601$	28.8340	0.0479767	0.0239883	+	0.999712i	$0.492364\pi$
$602$	0	0
$603$	866.712i	1.43733i
$604$	0	0
$605$	−58.3478	−0.0964426
$606$	0	0
$607$	278.647i	0.459057i	0.973302	+	0.229528i	$0.0737184\pi$
$607$	278.647i	0.459057i	−0.973302	+	0.229528i	$0.926282\pi$
$608$	0	0
$609$	343.010	0.563234
$610$	0	0
$611$	5.74951i	0.00941000i
$612$	0	0
$613$	−369.130	−0.602170	−0.301085	−	0.953597i	$-0.597349\pi$
$613$	−369.130	−0.602170	−0.301085	+	0.953597i	$0.597349\pi$
$614$	0	0
$615$	227.068i	0.369216i
$616$	0	0
$617$	367.774	0.596068	0.298034	−	0.954555i	$-0.403669\pi$
$617$	367.774	0.596068	0.298034	+	0.954555i	$0.403669\pi$
$618$	0	0
$619$	120.990i	0.195461i	0.995213	+	0.0977306i	$0.0311583\pi$
$619$	120.990i	0.195461i	−0.995213	+	0.0977306i	$0.968842\pi$
$620$	0	0
$621$	−32.3248	−0.0520528
$622$	0	0
$623$	− 75.3507i	− 0.120948i
$624$	0	0
$625$	25.0000	0.0400000
$626$	0	0
$627$	911.384i	1.45356i
$628$	0	0
$629$	575.313	0.914647
$630$	0	0
$631$	− 374.666i	− 0.593765i	−0.954914	−	0.296882i	$-0.904053\pi$
$631$	− 374.666i	− 0.593765i	0.954914	−	0.296882i	$-0.0959469\pi$
$632$	0	0
$633$	481.738	0.761040
$634$	0	0
$635$	251.894i	0.396683i
$636$	0	0
$637$	−14.3945	−0.0225973
$638$	0	0
$639$	916.476i	1.43424i
$640$	0	0
$641$	402.186	0.627435	0.313718	−	0.949516i	$-0.398426\pi$
$641$	402.186	0.627435	0.313718	+	0.949516i	$0.398426\pi$
$642$	0	0
$643$	− 502.899i	− 0.782113i	−0.920367	−	0.391057i	$-0.872110\pi$
$643$	− 502.899i	− 0.782113i	0.920367	−	0.391057i	$-0.127890\pi$
$644$	0	0
$645$	−60.5626	−0.0938954
$646$	0	0
$647$	1027.07i	1.58744i	0.608284	+	0.793720i	$0.291858\pi$
$647$	1027.07i	1.58744i	−0.608284	+	0.793720i	$0.708142\pi$
$648$	0	0
$649$	−64.7603	−0.0997848
$650$	0	0
$651$	42.1039i	0.0646757i
$652$	0	0
$653$	−769.028	−1.17768	−0.588842	−	0.808248i	$-0.700416\pi$
$653$	−769.028	−1.17768	−0.588842	+	0.808248i	$0.700416\pi$
$654$	0	0
$655$	− 391.984i	− 0.598450i
$656$	0	0
$657$	930.875	1.41686
$658$	0	0
$659$	− 180.406i	− 0.273757i	−0.990588	−	0.136879i	$-0.956293\pi$
$659$	− 180.406i	− 0.273757i	0.990588	−	0.136879i	$-0.0437070\pi$
$660$	0	0
$661$	394.658	0.597062	0.298531	−	0.954400i	$-0.403503\pi$
$661$	394.658	0.597062	0.298531	+	0.954400i	$0.403503\pi$
$662$	0	0
$663$	39.4790i	0.0595459i
$664$	0	0
$665$	161.740	0.243219
$666$	0	0
$667$	87.5269i	0.131225i
$668$	0	0
$669$	−99.9571	−0.149413
$670$	0	0
$671$	24.4565i	0.0364479i
$672$	0	0
$673$	582.488	0.865510	0.432755	−	0.901512i	$-0.357542\pi$
$673$	582.488	0.865510	0.432755	+	0.901512i	$0.357542\pi$
$674$	0	0
$675$	33.7009i	0.0499273i
$676$	0	0
$677$	892.223	1.31791	0.658954	−	0.752184i	$-0.270999\pi$
$677$	892.223	1.31791	0.658954	+	0.752184i	$0.270999\pi$
$678$	0	0
$679$	− 96.7204i	− 0.142445i
$680$	0	0
$681$	−1067.28	−1.56722
$682$	0	0
$683$	91.9217i	0.134585i	0.997733	+	0.0672926i	$0.0214361\pi$
$683$	91.9217i	0.134585i	−0.997733	+	0.0672926i	$0.978564\pi$
$684$	0	0
$685$	−229.139	−0.334509
$686$	0	0
$687$	773.448i	1.12583i
$688$	0	0
$689$	21.8223	0.0316724
$690$	0	0
$691$	57.4424i	0.0831294i	0.999136	+	0.0415647i	$0.0132343\pi$
$691$	57.4424i	0.0831294i	−0.999136	+	0.0415647i	$0.986766\pi$
$692$	0	0
$693$	−542.954	−0.783484
$694$	0	0
$695$	169.448i	0.243810i
$696$	0	0
$697$	−440.888	−0.632550
$698$	0	0
$699$	− 790.102i	− 1.13033i
$700$	0	0
$701$	−590.254	−0.842017	−0.421008	−	0.907057i	$-0.638324\pi$
$701$	−590.254	−0.842017	−0.421008	+	0.907057i	$0.638324\pi$
$702$	0	0
$703$	− 509.979i	− 0.725432i
$704$	0	0
$705$	−121.985	−0.173028
$706$	0	0
$707$	205.608i	0.290817i
$708$	0	0
$709$	1135.16	1.60107	0.800536	−	0.599285i	$-0.204548\pi$
$709$	1135.16	1.60107	0.800536	+	0.599285i	$0.204548\pi$
$710$	0	0
$711$	84.1761i	0.118391i
$712$	0	0
$713$	−10.7438	−0.0150684
$714$	0	0
$715$	12.6297i	0.0176639i
$716$	0	0
$717$	−1636.84	−2.28290
$718$	0	0
$719$	1213.64i	1.68795i	0.536382	+	0.843976i	$0.319791\pi$
$719$	1213.64i	1.68795i	−0.536382	+	0.843976i	$0.680209\pi$
$720$	0	0
$721$	−352.746	−0.489245
$722$	0	0
$723$	1552.25i	2.14695i
$724$	0	0
$725$	91.2531	0.125866
$726$	0	0
$727$	365.856i	0.503241i	0.967826	+	0.251620i	$0.0809634\pi$
$727$	365.856i	0.503241i	−0.967826	+	0.251620i	$0.919037\pi$
$728$	0	0
$729$	951.619	1.30538
$730$	0	0
$731$	− 117.592i	− 0.160864i
$732$	0	0
$733$	−632.802	−0.863304	−0.431652	−	0.902040i	$-0.642069\pi$
$733$	−632.802	−0.863304	−0.431652	+	0.902040i	$0.642069\pi$
$734$	0	0
$735$	− 305.402i	− 0.415513i
$736$	0	0
$737$	−998.701	−1.35509
$738$	0	0
$739$	673.511i	0.911382i	0.890138	+	0.455691i	$0.150608\pi$
$739$	673.511i	0.911382i	−0.890138	+	0.455691i	$0.849392\pi$
$740$	0	0
$741$	34.9956	0.0472275
$742$	0	0
$743$	− 1083.38i	− 1.45811i	−0.684454	−	0.729056i	$-0.739959\pi$
$743$	− 1083.38i	− 1.45811i	0.684454	−	0.729056i	$-0.260041\pi$
$744$	0	0
$745$	−551.299	−0.739998
$746$	0	0
$747$	895.579i	1.19890i
$748$	0	0
$749$	442.296	0.590516
$750$	0	0
$751$	804.016i	1.07059i	0.844664	+	0.535297i	$0.179800\pi$
$751$	804.016i	1.07059i	−0.844664	+	0.535297i	$0.820200\pi$
$752$	0	0
$753$	1642.75	2.18160
$754$	0	0
$755$	235.930i	0.312490i
$756$	0	0
$757$	737.522	0.974270	0.487135	−	0.873327i	$-0.338042\pi$
$757$	737.522	0.974270	0.487135	+	0.873327i	$0.338042\pi$
$758$	0	0
$759$	− 257.016i	− 0.338625i
$760$	0	0
$761$	−945.488	−1.24243	−0.621215	−	0.783641i	$-0.713360\pi$
$761$	−945.488	−1.24243	−0.621215	+	0.783641i	$0.713360\pi$
$762$	0	0
$763$	256.262i	0.335861i
$764$	0	0
$765$	−451.522	−0.590225
$766$	0	0
$767$	2.48669i	0.00324209i
$768$	0	0
$769$	1200.71	1.56139	0.780695	−	0.624912i	$-0.214865\pi$
$769$	1200.71	1.56139	0.780695	+	0.624912i	$0.214865\pi$
$770$	0	0
$771$	− 1171.99i	− 1.52009i
$772$	0	0
$773$	316.491	0.409433	0.204716	−	0.978821i	$-0.434373\pi$
$773$	316.491	0.409433	0.204716	+	0.978821i	$0.434373\pi$
$774$	0	0
$775$	11.2012i	0.0144531i
$776$	0	0
$777$	563.606	0.725362
$778$	0	0
$779$	390.819i	0.501693i
$780$	0	0
$781$	−1056.04	−1.35217
$782$	0	0
$783$	123.012i	0.157104i
$784$	0	0
$785$	209.413	0.266769
$786$	0	0
$787$	735.700i	0.934816i	0.884042	+	0.467408i	$0.154812\pi$
$787$	735.700i	0.934816i	−0.884042	+	0.467408i	$0.845188\pi$
$788$	0	0
$789$	244.115	0.309398
$790$	0	0
$791$	− 201.662i	− 0.254946i
$792$	0	0
$793$	0.939089	0.00118422
$794$	0	0
$795$	462.994i	0.582382i
$796$	0	0
$797$	1022.78	1.28329	0.641643	−	0.767004i	$-0.278253\pi$
$797$	1022.78	1.28329	0.641643	+	0.767004i	$0.278253\pi$
$798$	0	0
$799$	− 236.853i	− 0.296436i
$800$	0	0
$801$	186.464	0.232789
$802$	0	0
$803$	1072.63i	1.33578i
$804$	0	0
$805$	−45.6119	−0.0566607
$806$	0	0
$807$	− 183.231i	− 0.227052i
$808$	0	0
$809$	−187.931	−0.232300	−0.116150	−	0.993232i	$-0.537055\pi$
$809$	−187.931	−0.232300	−0.116150	+	0.993232i	$0.537055\pi$
$810$	0	0
$811$	974.123i	1.20114i	0.799573	+	0.600569i	$0.205059\pi$
$811$	974.123i	1.20114i	−0.799573	+	0.600569i	$0.794941\pi$
$812$	0	0
$813$	−2133.02	−2.62364
$814$	0	0
$815$	169.247i	0.207665i
$816$	0	0
$817$	−104.238	−0.127586
$818$	0	0
$819$	20.8485i	0.0254561i
$820$	0	0
$821$	−99.1291	−0.120742	−0.0603710	−	0.998176i	$-0.519228\pi$
$821$	−99.1291	−0.120742	−0.0603710	+	0.998176i	$0.519228\pi$
$822$	0	0
$823$	− 339.859i	− 0.412952i	−0.978452	−	0.206476i	$-0.933800\pi$
$823$	− 339.859i	− 0.412952i	0.978452	−	0.206476i	$-0.0661995\pi$
$824$	0	0
$825$	−267.958	−0.324798
$826$	0	0
$827$	56.9666i	0.0688834i	0.999407	+	0.0344417i	$0.0109653\pi$
$827$	56.9666i	0.0688834i	−0.999407	+	0.0344417i	$0.989035\pi$
$828$	0	0
$829$	1294.84	1.56194	0.780968	−	0.624572i	$-0.214726\pi$
$829$	1294.84	1.56194	0.780968	+	0.624572i	$0.214726\pi$
$830$	0	0
$831$	− 467.723i	− 0.562843i
$832$	0	0
$833$	592.986	0.711868
$834$	0	0
$835$	− 28.8694i	− 0.0345741i
$836$	0	0
$837$	−15.0996	−0.0180401
$838$	0	0
$839$	748.920i	0.892634i	0.894875	+	0.446317i	$0.147265\pi$
$839$	748.920i	0.892634i	−0.894875	+	0.446317i	$0.852735\pi$
$840$	0	0
$841$	−507.915	−0.603942
$842$	0	0
$843$	757.848i	0.898989i
$844$	0	0
$845$	−377.411	−0.446640
$846$	0	0
$847$	− 110.986i	− 0.131034i
$848$	0	0
$849$	995.906	1.17303
$850$	0	0
$851$	143.817i	0.168998i
$852$	0	0
$853$	−1022.97	−1.19926	−0.599630	−	0.800277i	$-0.704686\pi$
$853$	−1022.97	−1.19926	−0.599630	+	0.800277i	$0.704686\pi$
$854$	0	0
$855$	400.246i	0.468124i
$856$	0	0
$857$	1186.98	1.38504	0.692518	−	0.721401i	$-0.256501\pi$
$857$	1186.98	1.38504	0.692518	+	0.721401i	$0.256501\pi$
$858$	0	0
$859$	− 794.439i	− 0.924842i	−0.886660	−	0.462421i	$-0.846981\pi$
$859$	− 794.439i	− 0.924842i	0.886660	−	0.462421i	$-0.153019\pi$
$860$	0	0
$861$	−431.916	−0.501645
$862$	0	0
$863$	− 708.413i	− 0.820872i	−0.911889	−	0.410436i	$-0.865376\pi$
$863$	− 708.413i	− 0.820872i	0.911889	−	0.410436i	$-0.134624\pi$
$864$	0	0
$865$	66.6120	0.0770081
$866$	0	0
$867$	− 349.328i	− 0.402916i
$868$	0	0
$869$	−96.9950	−0.111617
$870$	0	0
$871$	38.3484i	0.0440280i
$872$	0	0
$873$	239.346	0.274165
$874$	0	0
$875$	47.5537i	0.0543471i
$876$	0	0
$877$	−891.331	−1.01634	−0.508171	−	0.861256i	$-0.669678\pi$
$877$	−891.331	−1.01634	−0.508171	+	0.861256i	$0.669678\pi$
$878$	0	0
$879$	− 1803.38i	− 2.05163i
$880$	0	0
$881$	536.631	0.609116	0.304558	−	0.952494i	$-0.401491\pi$
$881$	536.631	0.609116	0.304558	+	0.952494i	$0.401491\pi$
$882$	0	0
$883$	− 1275.84i	− 1.44490i	−0.691426	−	0.722448i	$-0.743017\pi$
$883$	− 1275.84i	− 1.44490i	0.691426	−	0.722448i	$-0.256983\pi$
$884$	0	0
$885$	−52.7589	−0.0596146
$886$	0	0
$887$	240.987i	0.271687i	0.990730	+	0.135844i	$0.0433745\pi$
$887$	240.987i	0.271687i	−0.990730	+	0.135844i	$0.956626\pi$
$888$	0	0
$889$	−479.139	−0.538964
$890$	0	0
$891$	787.668i	0.884027i
$892$	0	0
$893$	−209.955	−0.235112
$894$	0	0
$895$	24.6621i	0.0275554i
$896$	0	0
$897$	−9.86899	−0.0110022
$898$	0	0
$899$	40.8857i	0.0454790i
$900$	0	0
$901$	−898.975	−0.997752
$902$	0	0
$903$	− 115.199i	− 0.127574i
$904$	0	0
$905$	−371.340	−0.410321
$906$	0	0
$907$	1784.76i	1.96776i	0.178837	+	0.983879i	$0.442767\pi$
$907$	1784.76i	1.96776i	−0.178837	+	0.983879i	$0.557233\pi$
$908$	0	0
$909$	−508.800	−0.559736
$910$	0	0
$911$	− 119.363i	− 0.131024i	−0.997852	−	0.0655120i	$-0.979132\pi$
$911$	− 119.363i	− 0.131024i	0.997852	−	0.0655120i	$-0.0208681\pi$
$912$	0	0
$913$	−1031.96	−1.13030
$914$	0	0
$915$	19.9243i	0.0217751i
$916$	0	0
$917$	745.612	0.813099
$918$	0	0
$919$	− 1409.78i	− 1.53404i	−0.641623	−	0.767020i	$-0.721739\pi$
$919$	− 1409.78i	− 1.53404i	0.641623	−	0.767020i	$-0.278261\pi$
$920$	0	0
$921$	−2010.86	−2.18335
$922$	0	0
$923$	40.5503i	0.0439331i
$924$	0	0
$925$	149.940	0.162097
$926$	0	0
$927$	− 872.910i	− 0.941651i
$928$	0	0
$929$	431.248	0.464207	0.232103	−	0.972691i	$-0.425439\pi$
$929$	431.248	0.464207	0.232103	+	0.972691i	$0.425439\pi$
$930$	0	0
$931$	− 525.645i	− 0.564602i
$932$	0	0
$933$	−1247.14	−1.33669
$934$	0	0
$935$	− 520.283i	− 0.556452i
$936$	0	0
$937$	865.371	0.923555	0.461778	−	0.886996i	$-0.347212\pi$
$937$	865.371	0.923555	0.461778	+	0.886996i	$0.347212\pi$
$938$	0	0
$939$	− 1282.71i	− 1.36604i
$940$	0	0
$941$	593.857	0.631091	0.315546	−	0.948910i	$-0.397812\pi$
$941$	593.857	0.631091	0.315546	+	0.948910i	$0.397812\pi$
$942$	0	0
$943$	− 110.214i	− 0.116875i
$944$	0	0
$945$	−64.1041	−0.0678350
$946$	0	0
$947$	1149.92i	1.21428i	0.794596	+	0.607139i	$0.207683\pi$
$947$	1149.92i	1.21428i	−0.794596	+	0.607139i	$0.792317\pi$
$948$	0	0
$949$	41.1873	0.0434008
$950$	0	0
$951$	− 923.001i	− 0.970558i
$952$	0	0
$953$	517.910	0.543452	0.271726	−	0.962375i	$-0.412406\pi$
$953$	517.910	0.543452	0.271726	+	0.962375i	$0.412406\pi$
$954$	0	0
$955$	208.352i	0.218169i
$956$	0	0
$957$	−978.080	−1.02203
$958$	0	0
$959$	− 435.855i	− 0.454489i
$960$	0	0
$961$	955.981	0.994778
$962$	0	0
$963$	1094.51i	1.13657i
$964$	0	0
$965$	−430.031	−0.445628
$966$	0	0
$967$	588.004i	0.608070i	0.952661	+	0.304035i	$0.0983340\pi$
$967$	588.004i	0.608070i	−0.952661	+	0.304035i	$0.901666\pi$
$968$	0	0
$969$	−1441.65	−1.48778
$970$	0	0
$971$	− 265.211i	− 0.273132i	−0.990631	−	0.136566i	$-0.956393\pi$
$971$	− 265.211i	− 0.273132i	0.990631	−	0.136566i	$-0.0436066\pi$
$972$	0	0
$973$	−322.314	−0.331258
$974$	0	0
$975$	10.2891i	0.0105530i
$976$	0	0
$977$	1260.45	1.29012	0.645061	−	0.764131i	$-0.276832\pi$
$977$	1260.45	1.29012	0.645061	+	0.764131i	$0.276832\pi$
$978$	0	0
$979$	214.860i	0.219469i
$980$	0	0
$981$	−634.151	−0.646433
$982$	0	0
$983$	− 1438.07i	− 1.46294i	−0.681875	−	0.731468i	$-0.738835\pi$
$983$	− 1438.07i	− 1.46294i	0.681875	−	0.731468i	$-0.261165\pi$
$984$	0	0
$985$	537.184	0.545364
$986$	0	0
$987$	− 232.033i	− 0.235089i
$988$	0	0
$989$	29.3957	0.0297226
$990$	0	0
$991$	− 1426.40i	− 1.43936i	−0.694308	−	0.719678i	$-0.744289\pi$
$991$	− 1426.40i	− 1.43936i	0.694308	−	0.719678i	$-0.255711\pi$
$992$	0	0
$993$	1625.51	1.63697
$994$	0	0
$995$	23.2472i	0.0233640i
$996$	0	0
$997$	14.6189	0.0146629	0.00733144	−	0.999973i	$-0.497666\pi$
$997$	14.6189	0.0146629	0.00733144	+	0.999973i	$0.497666\pi$
$998$	0	0
$999$	202.124i	0.202327i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1840.3.c.b.1151.7	✓	56
4.3	odd	2	inner	1840.3.c.b.1151.50	yes	56

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1840.3.c.b.1151.7	✓	56	1.1	even	1	trivial
1840.3.c.b.1151.50	yes	56	4.3	odd	2	inner

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:	\( N \)	\(=\)	\( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1840.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.41875i q^{3} +2.23607 q^{5} +4.25333i q^{7} -10.5254 q^{9} +O(q^{10})\)	\(q-4.41875i q^{3} +2.23607 q^{5} +4.25333i q^{7} -10.5254 q^{9} -12.1282i q^{11} -0.465703 q^{13} -9.88063i q^{15} +19.1848 q^{17} -17.0061i q^{19} +18.7944 q^{21} +4.79583i q^{23} +5.00000 q^{25} +6.74018i q^{27} +18.2506 q^{29} +2.24023i q^{31} -53.5916 q^{33} +9.51074i q^{35} +29.9880 q^{37} +2.05783i q^{39} -22.9811 q^{41} -6.12942i q^{43} -23.5354 q^{45} -12.3459i q^{47} +30.9092 q^{49} -84.7728i q^{51} -46.8587 q^{53} -27.1195i q^{55} -75.1457 q^{57} -5.33963i q^{59} -2.01650 q^{61} -44.7678i q^{63} -1.04134 q^{65} -82.3451i q^{67} +21.1916 q^{69} -87.0732i q^{71} -88.4411 q^{73} -22.0938i q^{75} +51.5854 q^{77} -7.99745i q^{79} -64.9451 q^{81} -85.0877i q^{83} +42.8985 q^{85} -80.6449i q^{87} -17.7157 q^{89} -1.98079i q^{91} +9.89904 q^{93} -38.0268i q^{95} -22.7399 q^{97} +127.654i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 56 q - 120 q^{9}+O(q^{10}) \) 56 * q - 120 * q^9	\( 56 q - 120 q^{9} - 56 q^{13} - 96 q^{17} + 104 q^{21} + 280 q^{25} - 76 q^{29} + 240 q^{33} - 88 q^{37} - 76 q^{41} - 356 q^{49} - 88 q^{53} - 256 q^{57} + 376 q^{61} + 120 q^{65} + 192 q^{73} - 168 q^{77} - 392 q^{81} - 60 q^{85} + 368 q^{89} + 216 q^{93} + 264 q^{97}+O(q^{100}) \) 56 * q - 120 * q^9 - 56 * q^13 - 96 * q^17 + 104 * q^21 + 280 * q^25 - 76 * q^29 + 240 * q^33 - 88 * q^37 - 76 * q^41 - 356 * q^49 - 88 * q^53 - 256 * q^57 + 376 * q^61 + 120 * q^65 + 192 * q^73 - 168 * q^77 - 392 * q^81 - 60 * q^85 + 368 * q^89 + 216 * q^93 + 264 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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