LMFDB - Embedded newform 1323.2.a.w.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,2,Mod(1,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1323.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1323 = 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1323.a (trivial)

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:	yes
Analytic conductor:	$10.5642081874$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{7}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 7 $ x^2 - 7
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 189)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.64575$ of defining polynomial
Character	$\chi$	$=$	1323.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.64575 q^{2} +5.00000 q^{4} -2.64575 q^{5} -7.93725 q^{8} +7.00000 q^{10} +2.64575 q^{11} +2.00000 q^{13} +11.0000 q^{16} -7.00000 q^{19} -13.2288 q^{20} -7.00000 q^{22} +7.93725 q^{23} +2.00000 q^{25} -5.29150 q^{26} -5.29150 q^{29} -3.00000 q^{31} -13.2288 q^{32} -3.00000 q^{37} +18.5203 q^{38} +21.0000 q^{40} +2.64575 q^{41} +8.00000 q^{43} +13.2288 q^{44} -21.0000 q^{46} -5.29150 q^{50} +10.0000 q^{52} -7.00000 q^{55} +14.0000 q^{58} +8.00000 q^{61} +7.93725 q^{62} +13.0000 q^{64} -5.29150 q^{65} -2.00000 q^{67} -7.93725 q^{71} +7.93725 q^{74} -35.0000 q^{76} -4.00000 q^{79} -29.1033 q^{80} -7.00000 q^{82} +15.8745 q^{83} -21.1660 q^{86} -21.0000 q^{88} -18.5203 q^{89} +39.6863 q^{92} +18.5203 q^{95} +12.0000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 10 q^{4} + 14 q^{10} + 4 q^{13} + 22 q^{16} - 14 q^{19} - 14 q^{22} + 4 q^{25} - 6 q^{31} - 6 q^{37} + 42 q^{40} + 16 q^{43} - 42 q^{46} + 20 q^{52} - 14 q^{55} + 28 q^{58} + 16 q^{61} + 26 q^{64}+ \cdots + 24 q^{97}+O(q^{100}) $ 2 * q + 10 * q^4 + 14 * q^10 + 4 * q^13 + 22 * q^16 - 14 * q^19 - 14 * q^22 + 4 * q^25 - 6 * q^31 - 6 * q^37 + 42 * q^40 + 16 * q^43 - 42 * q^46 + 20 * q^52 - 14 * q^55 + 28 * q^58 + 16 * q^61 + 26 * q^64 - 4 * q^67 - 70 * q^76 - 8 * q^79 - 14 * q^82 - 42 * q^88 + 24 * q^97

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 $

$2$	−2.64575	−1.87083	−0.935414	−	0.353553i	$-0.884973\pi$
$2$	−2.64575	−1.87083	−0.935414	+	0.353553i	$0.884973\pi$
$3$	0	0
$3$	0	0
$4$	5.00000	2.50000
$5$	−2.64575	−1.18322	−0.591608	−	0.806226i	$-0.701507\pi$
$5$	−2.64575	−1.18322	−0.591608	+	0.806226i	$0.701507\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−7.93725	−2.80624
$9$	0	0
$10$	7.00000	2.21359
$11$	2.64575	0.797724	0.398862	−	0.917011i	$-0.369405\pi$
$11$	2.64575	0.797724	0.398862	+	0.917011i	$0.369405\pi$
$12$	0	0
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	0	0
$16$	11.0000	2.75000
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	0	0
$19$	−7.00000	−1.60591	−0.802955	−	0.596040i	$-0.796740\pi$
$19$	−7.00000	−1.60591	−0.802955	+	0.596040i	$0.796740\pi$
$20$	−13.2288	−2.95804
$21$	0	0
$22$	−7.00000	−1.49241
$23$	7.93725	1.65503	0.827516	−	0.561442i	$-0.189753\pi$
$23$	7.93725	1.65503	0.827516	+	0.561442i	$0.189753\pi$
$24$	0	0
$25$	2.00000	0.400000
$26$	−5.29150	−1.03775
$27$	0	0
$28$	0	0
$29$	−5.29150	−0.982607	−0.491304	−	0.870988i	$-0.663479\pi$
$29$	−5.29150	−0.982607	−0.491304	+	0.870988i	$0.663479\pi$
$30$	0	0
$31$	−3.00000	−0.538816	−0.269408	−	0.963026i	$-0.586828\pi$
$31$	−3.00000	−0.538816	−0.269408	+	0.963026i	$0.586828\pi$
$32$	−13.2288	−2.33854
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−3.00000	−0.493197	−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000	−0.493197	−0.246598	+	0.969118i	$0.579313\pi$
$38$	18.5203	3.00438
$39$	0	0
$40$	21.0000	3.32039
$41$	2.64575	0.413197	0.206598	−	0.978426i	$-0.433761\pi$
$41$	2.64575	0.413197	0.206598	+	0.978426i	$0.433761\pi$
$42$	0	0
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	13.2288	1.99431
$45$	0	0
$46$	−21.0000	−3.09628
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	0	0
$50$	−5.29150	−0.748331
$51$	0	0
$52$	10.0000	1.38675
$53$	0	0		−	1.00000i	$-0.5\pi$
$53$	0	0			1.00000i	$0.5\pi$
$54$	0	0
$55$	−7.00000	−0.943880
$56$	0	0
$57$	0	0
$58$	14.0000	1.83829
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	8.00000	1.02430	0.512148	−	0.858898i	$-0.328850\pi$
$61$	8.00000	1.02430	0.512148	+	0.858898i	$0.328850\pi$
$62$	7.93725	1.00803
$63$	0	0
$64$	13.0000	1.62500
$65$	−5.29150	−0.656330
$66$	0	0
$67$	−2.00000	−0.244339	−0.122169	−	0.992509i	$-0.538985\pi$
$67$	−2.00000	−0.244339	−0.122169	+	0.992509i	$0.538985\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−7.93725	−0.941979	−0.470989	−	0.882139i	$-0.656103\pi$
$71$	−7.93725	−0.941979	−0.470989	+	0.882139i	$0.656103\pi$
$72$	0	0
$73$	0	0		−	1.00000i	$-0.5\pi$
$73$	0	0			1.00000i	$0.5\pi$
$74$	7.93725	0.922687
$75$	0	0
$76$	−35.0000	−4.01478
$77$	0	0
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	−29.1033	−3.25384
$81$	0	0
$82$	−7.00000	−0.773021
$83$	15.8745	1.74245	0.871227	−	0.490881i	$-0.163325\pi$
$83$	15.8745	1.74245	0.871227	+	0.490881i	$0.163325\pi$
$84$	0	0
$85$	0	0
$86$	−21.1660	−2.28239
$87$	0	0
$88$	−21.0000	−2.23861
$89$	−18.5203	−1.96314	−0.981572	−	0.191094i	$-0.938797\pi$
$89$	−18.5203	−1.96314	−0.981572	+	0.191094i	$0.938797\pi$
$90$	0	0
$91$	0	0
$92$	39.6863	4.13758
$93$	0	0
$94$	0	0
$95$	18.5203	1.90014
$96$	0	0
$97$	12.0000	1.21842	0.609208	−	0.793011i	$-0.291488\pi$
$97$	12.0000	1.21842	0.609208	+	0.793011i	$0.291488\pi$
$98$	0	0
$99$	0	0
$100$	10.0000	1.00000
$101$	10.5830	1.05305	0.526524	−	0.850160i	$-0.323495\pi$
$101$	10.5830	1.05305	0.526524	+	0.850160i	$0.323495\pi$
$102$	0	0
$103$	13.0000	1.28093	0.640464	−	0.767988i	$-0.278742\pi$
$103$	13.0000	1.28093	0.640464	+	0.767988i	$0.278742\pi$
$104$	−15.8745	−1.55662
$105$	0	0
$106$	0	0
$107$	−5.29150	−0.511549	−0.255774	−	0.966736i	$-0.582330\pi$
$107$	−5.29150	−0.511549	−0.255774	+	0.966736i	$0.582330\pi$
$108$	0	0
$109$	9.00000	0.862044	0.431022	−	0.902342i	$-0.358153\pi$
$109$	9.00000	0.862044	0.431022	+	0.902342i	$0.358153\pi$
$110$	18.5203	1.76584
$111$	0	0
$112$	0	0
$113$	−5.29150	−0.497783	−0.248891	−	0.968531i	$-0.580066\pi$
$113$	−5.29150	−0.497783	−0.248891	+	0.968531i	$0.580066\pi$
$114$	0	0
$115$	−21.0000	−1.95826
$116$	−26.4575	−2.45652
$117$	0	0
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−4.00000	−0.363636
$122$	−21.1660	−1.91628
$123$	0	0
$124$	−15.0000	−1.34704
$125$	7.93725	0.709930
$126$	0	0
$127$	6.00000	0.532414	0.266207	−	0.963916i	$-0.414230\pi$
$127$	6.00000	0.532414	0.266207	+	0.963916i	$0.414230\pi$
$128$	−7.93725	−0.701561
$129$	0	0
$130$	14.0000	1.22788
$131$	21.1660	1.84928	0.924641	−	0.380839i	$-0.124365\pi$
$131$	21.1660	1.84928	0.924641	+	0.380839i	$0.124365\pi$
$132$	0	0
$133$	0	0
$134$	5.29150	0.457116
$135$	0	0
$136$	0	0
$137$	15.8745	1.35625	0.678125	−	0.734946i	$-0.262793\pi$
$137$	15.8745	1.35625	0.678125	+	0.734946i	$0.262793\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	0	0
$141$	0	0
$142$	21.0000	1.76228
$143$	5.29150	0.442498
$144$	0	0
$145$	14.0000	1.16264
$146$	0	0
$147$	0	0
$148$	−15.0000	−1.23299
$149$	15.8745	1.30049	0.650245	−	0.759724i	$-0.274666\pi$
$149$	15.8745	1.30049	0.650245	+	0.759724i	$0.274666\pi$
$150$	0	0
$151$	2.00000	0.162758	0.0813788	−	0.996683i	$-0.474068\pi$
$151$	2.00000	0.162758	0.0813788	+	0.996683i	$0.474068\pi$
$152$	55.5608	4.50657
$153$	0	0
$154$	0	0
$155$	7.93725	0.637536
$156$	0	0
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	10.5830	0.841939
$159$	0	0
$160$	35.0000	2.76699
$161$	0	0
$162$	0	0
$163$	10.0000	0.783260	0.391630	−	0.920123i	$-0.371911\pi$
$163$	10.0000	0.783260	0.391630	+	0.920123i	$0.371911\pi$
$164$	13.2288	1.03299
$165$	0	0
$166$	−42.0000	−3.25983
$167$	5.29150	0.409469	0.204734	−	0.978818i	$-0.434367\pi$
$167$	5.29150	0.409469	0.204734	+	0.978818i	$0.434367\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	40.0000	3.04997
$173$	−13.2288	−1.00576	−0.502882	−	0.864355i	$-0.667727\pi$
$173$	−13.2288	−1.00576	−0.502882	+	0.864355i	$0.667727\pi$
$174$	0	0
$175$	0	0
$176$	29.1033	2.19374
$177$	0	0
$178$	49.0000	3.67271
$179$	5.29150	0.395505	0.197753	−	0.980252i	$-0.436636\pi$
$179$	5.29150	0.395505	0.197753	+	0.980252i	$0.436636\pi$
$180$	0	0
$181$	14.0000	1.04061	0.520306	−	0.853980i	$-0.325818\pi$
$181$	14.0000	1.04061	0.520306	+	0.853980i	$0.325818\pi$
$182$	0	0
$183$	0	0
$184$	−63.0000	−4.64442
$185$	7.93725	0.583559
$186$	0	0
$187$	0	0
$188$	0	0
$189$	0	0
$190$	−49.0000	−3.55483
$191$	2.64575	0.191440	0.0957199	−	0.995408i	$-0.469485\pi$
$191$	2.64575	0.191440	0.0957199	+	0.995408i	$0.469485\pi$
$192$	0	0
$193$	2.00000	0.143963	0.0719816	−	0.997406i	$-0.477068\pi$
$193$	2.00000	0.143963	0.0719816	+	0.997406i	$0.477068\pi$
$194$	−31.7490	−2.27945
$195$	0	0
$196$	0	0
$197$	26.4575	1.88502	0.942510	−	0.334178i	$-0.108459\pi$
$197$	26.4575	1.88502	0.942510	+	0.334178i	$0.108459\pi$
$198$	0	0
$199$	3.00000	0.212664	0.106332	−	0.994331i	$-0.466089\pi$
$199$	3.00000	0.212664	0.106332	+	0.994331i	$0.466089\pi$
$200$	−15.8745	−1.12250
$201$	0	0
$202$	−28.0000	−1.97007
$203$	0	0
$204$	0	0
$205$	−7.00000	−0.488901
$206$	−34.3948	−2.39640
$207$	0	0
$208$	22.0000	1.52543
$209$	−18.5203	−1.28107
$210$	0	0
$211$	22.0000	1.51454	0.757271	−	0.653101i	$-0.226532\pi$
$211$	22.0000	1.51454	0.757271	+	0.653101i	$0.226532\pi$
$212$	0	0
$213$	0	0
$214$	14.0000	0.957020
$215$	−21.1660	−1.44351
$216$	0	0
$217$	0	0
$218$	−23.8118	−1.61274
$219$	0	0
$220$	−35.0000	−2.35970
$221$	0	0
$222$	0	0
$223$	−7.00000	−0.468755	−0.234377	−	0.972146i	$-0.575305\pi$
$223$	−7.00000	−0.468755	−0.234377	+	0.972146i	$0.575305\pi$
$224$	0	0
$225$	0	0
$226$	14.0000	0.931266
$227$	−15.8745	−1.05363	−0.526814	−	0.849981i	$-0.676614\pi$
$227$	−15.8745	−1.05363	−0.526814	+	0.849981i	$0.676614\pi$
$228$	0	0
$229$	−20.0000	−1.32164	−0.660819	−	0.750546i	$-0.729791\pi$
$229$	−20.0000	−1.32164	−0.660819	+	0.750546i	$0.729791\pi$
$230$	55.5608	3.66357
$231$	0	0
$232$	42.0000	2.75744
$233$	−15.8745	−1.03997	−0.519987	−	0.854174i	$-0.674063\pi$
$233$	−15.8745	−1.03997	−0.519987	+	0.854174i	$0.674063\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−15.8745	−1.02684	−0.513418	−	0.858138i	$-0.671621\pi$
$239$	−15.8745	−1.02684	−0.513418	+	0.858138i	$0.671621\pi$
$240$	0	0
$241$	4.00000	0.257663	0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000	0.257663	0.128831	+	0.991667i	$0.458877\pi$
$242$	10.5830	0.680301
$243$	0	0
$244$	40.0000	2.56074
$245$	0	0
$246$	0	0
$247$	−14.0000	−0.890799
$248$	23.8118	1.51205
$249$	0	0
$250$	−21.0000	−1.32816
$251$	−10.5830	−0.667993	−0.333997	−	0.942574i	$-0.608397\pi$
$251$	−10.5830	−0.667993	−0.333997	+	0.942574i	$0.608397\pi$
$252$	0	0
$253$	21.0000	1.32026
$254$	−15.8745	−0.996055
$255$	0	0
$256$	−5.00000	−0.312500
$257$	18.5203	1.15526	0.577631	−	0.816298i	$-0.303977\pi$
$257$	18.5203	1.15526	0.577631	+	0.816298i	$0.303977\pi$
$258$	0	0
$259$	0	0
$260$	−26.4575	−1.64083
$261$	0	0
$262$	−56.0000	−3.45969
$263$	18.5203	1.14201	0.571004	−	0.820947i	$-0.306554\pi$
$263$	18.5203	1.14201	0.571004	+	0.820947i	$0.306554\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	−10.0000	−0.610847
$269$	23.8118	1.45183	0.725914	−	0.687785i	$-0.241417\pi$
$269$	23.8118	1.45183	0.725914	+	0.687785i	$0.241417\pi$
$270$	0	0
$271$	20.0000	1.21491	0.607457	−	0.794353i	$-0.292190\pi$
$271$	20.0000	1.21491	0.607457	+	0.794353i	$0.292190\pi$
$272$	0	0
$273$	0	0
$274$	−42.0000	−2.53731
$275$	5.29150	0.319090
$276$	0	0
$277$	−23.0000	−1.38194	−0.690968	−	0.722885i	$-0.742815\pi$
$277$	−23.0000	−1.38194	−0.690968	+	0.722885i	$0.742815\pi$
$278$	−31.7490	−1.90418
$279$	0	0
$280$	0	0
$281$	−10.5830	−0.631329	−0.315665	−	0.948871i	$-0.602227\pi$
$281$	−10.5830	−0.631329	−0.315665	+	0.948871i	$0.602227\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	−39.6863	−2.35495
$285$	0	0
$286$	−14.0000	−0.827837
$287$	0	0
$288$	0	0
$289$	−17.0000	−1.00000
$290$	−37.0405	−2.17509
$291$	0	0
$292$	0	0
$293$	10.5830	0.618266	0.309133	−	0.951019i	$-0.399961\pi$
$293$	10.5830	0.618266	0.309133	+	0.951019i	$0.399961\pi$
$294$	0	0
$295$	0	0
$296$	23.8118	1.38403
$297$	0	0
$298$	−42.0000	−2.43299
$299$	15.8745	0.918046
$300$	0	0
$301$	0	0
$302$	−5.29150	−0.304492
$303$	0	0
$304$	−77.0000	−4.41625
$305$	−21.1660	−1.21196
$306$	0	0
$307$	−19.0000	−1.08439	−0.542194	−	0.840254i	$-0.682406\pi$
$307$	−19.0000	−1.08439	−0.542194	+	0.840254i	$0.682406\pi$
$308$	0	0
$309$	0	0
$310$	−21.0000	−1.19272
$311$	15.8745	0.900161	0.450080	−	0.892988i	$-0.351395\pi$
$311$	15.8745	0.900161	0.450080	+	0.892988i	$0.351395\pi$
$312$	0	0
$313$	−14.0000	−0.791327	−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000	−0.791327	−0.395663	+	0.918396i	$0.629485\pi$
$314$	−37.0405	−2.09032
$315$	0	0
$316$	−20.0000	−1.12509
$317$	0	0		−	1.00000i	$-0.5\pi$
$317$	0	0			1.00000i	$0.5\pi$
$318$	0	0
$319$	−14.0000	−0.783850
$320$	−34.3948	−1.92273
$321$	0	0
$322$	0	0
$323$	0	0
$324$	0	0
$325$	4.00000	0.221880
$326$	−26.4575	−1.46535
$327$	0	0
$328$	−21.0000	−1.15953
$329$	0	0
$330$	0	0
$331$	20.0000	1.09930	0.549650	−	0.835395i	$-0.314761\pi$
$331$	20.0000	1.09930	0.549650	+	0.835395i	$0.314761\pi$
$332$	79.3725	4.35613
$333$	0	0
$334$	−14.0000	−0.766046
$335$	5.29150	0.289106
$336$	0	0
$337$	19.0000	1.03500	0.517498	−	0.855684i	$-0.326864\pi$
$337$	19.0000	1.03500	0.517498	+	0.855684i	$0.326864\pi$
$338$	23.8118	1.29519
$339$	0	0
$340$	0	0
$341$	−7.93725	−0.429826
$342$	0	0
$343$	0	0
$344$	−63.4980	−3.42358
$345$	0	0
$346$	35.0000	1.88161
$347$	−2.64575	−0.142031	−0.0710157	−	0.997475i	$-0.522624\pi$
$347$	−2.64575	−0.142031	−0.0710157	+	0.997475i	$0.522624\pi$
$348$	0	0
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	0	0
$351$	0	0
$352$	−35.0000	−1.86551
$353$	−2.64575	−0.140819	−0.0704096	−	0.997518i	$-0.522431\pi$
$353$	−2.64575	−0.140819	−0.0704096	+	0.997518i	$0.522431\pi$
$354$	0	0
$355$	21.0000	1.11456
$356$	−92.6013	−4.90786
$357$	0	0
$358$	−14.0000	−0.739923
$359$	5.29150	0.279275	0.139637	−	0.990203i	$-0.455406\pi$
$359$	5.29150	0.279275	0.139637	+	0.990203i	$0.455406\pi$
$360$	0	0
$361$	30.0000	1.57895
$362$	−37.0405	−1.94681
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−21.0000	−1.09619	−0.548096	−	0.836416i	$-0.684647\pi$
$367$	−21.0000	−1.09619	−0.548096	+	0.836416i	$0.684647\pi$
$368$	87.3098	4.55134
$369$	0	0
$370$	−21.0000	−1.09174
$371$	0	0
$372$	0	0
$373$	29.0000	1.50156	0.750782	−	0.660551i	$-0.229677\pi$
$373$	29.0000	1.50156	0.750782	+	0.660551i	$0.229677\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−10.5830	−0.545053
$378$	0	0
$379$	36.0000	1.84920	0.924598	−	0.380945i	$-0.124401\pi$
$379$	36.0000	1.84920	0.924598	+	0.380945i	$0.124401\pi$
$380$	92.6013	4.75035
$381$	0	0
$382$	−7.00000	−0.358151
$383$	15.8745	0.811149	0.405575	−	0.914062i	$-0.367071\pi$
$383$	15.8745	0.811149	0.405575	+	0.914062i	$0.367071\pi$
$384$	0	0
$385$	0	0
$386$	−5.29150	−0.269330
$387$	0	0
$388$	60.0000	3.04604
$389$	−31.7490	−1.60974	−0.804869	−	0.593452i	$-0.797765\pi$
$389$	−31.7490	−1.60974	−0.804869	+	0.593452i	$0.797765\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	−70.0000	−3.52655
$395$	10.5830	0.532489
$396$	0	0
$397$	6.00000	0.301131	0.150566	−	0.988600i	$-0.451890\pi$
$397$	6.00000	0.301131	0.150566	+	0.988600i	$0.451890\pi$
$398$	−7.93725	−0.397859
$399$	0	0
$400$	22.0000	1.10000
$401$	−15.8745	−0.792735	−0.396368	−	0.918092i	$-0.629729\pi$
$401$	−15.8745	−0.792735	−0.396368	+	0.918092i	$0.629729\pi$
$402$	0	0
$403$	−6.00000	−0.298881
$404$	52.9150	2.63262
$405$	0	0
$406$	0	0
$407$	−7.93725	−0.393435
$408$	0	0
$409$	−32.0000	−1.58230	−0.791149	−	0.611623i	$-0.790517\pi$
$409$	−32.0000	−1.58230	−0.791149	+	0.611623i	$0.790517\pi$
$410$	18.5203	0.914650
$411$	0	0
$412$	65.0000	3.20232
$413$	0	0
$414$	0	0
$415$	−42.0000	−2.06170
$416$	−26.4575	−1.29719
$417$	0	0
$418$	49.0000	2.39667
$419$	−15.8745	−0.775520	−0.387760	−	0.921760i	$-0.626751\pi$
$419$	−15.8745	−0.775520	−0.387760	+	0.921760i	$0.626751\pi$
$420$	0	0
$421$	−19.0000	−0.926003	−0.463002	−	0.886357i	$-0.653228\pi$
$421$	−19.0000	−0.926003	−0.463002	+	0.886357i	$0.653228\pi$
$422$	−58.2065	−2.83345
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	0	0
$428$	−26.4575	−1.27887
$429$	0	0
$430$	56.0000	2.70056
$431$	−39.6863	−1.91162	−0.955810	−	0.293985i	$-0.905019\pi$
$431$	−39.6863	−1.91162	−0.955810	+	0.293985i	$0.905019\pi$
$432$	0	0
$433$	26.0000	1.24948	0.624740	−	0.780833i	$-0.285205\pi$
$433$	26.0000	1.24948	0.624740	+	0.780833i	$0.285205\pi$
$434$	0	0
$435$	0	0
$436$	45.0000	2.15511
$437$	−55.5608	−2.65783
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	55.5608	2.64876
$441$	0	0
$442$	0	0
$443$	7.93725	0.377110	0.188555	−	0.982063i	$-0.439620\pi$
$443$	7.93725	0.377110	0.188555	+	0.982063i	$0.439620\pi$
$444$	0	0
$445$	49.0000	2.32282
$446$	18.5203	0.876960
$447$	0	0
$448$	0	0
$449$	15.8745	0.749164	0.374582	−	0.927194i	$-0.377786\pi$
$449$	15.8745	0.749164	0.374582	+	0.927194i	$0.377786\pi$
$450$	0	0
$451$	7.00000	0.329617
$452$	−26.4575	−1.24446
$453$	0	0
$454$	42.0000	1.97116
$455$	0	0
$456$	0	0
$457$	25.0000	1.16945	0.584725	−	0.811231i	$-0.301202\pi$
$457$	25.0000	1.16945	0.584725	+	0.811231i	$0.301202\pi$
$458$	52.9150	2.47256
$459$	0	0
$460$	−105.000	−4.89565
$461$	18.5203	0.862574	0.431287	−	0.902215i	$-0.358060\pi$
$461$	18.5203	0.862574	0.431287	+	0.902215i	$0.358060\pi$
$462$	0	0
$463$	−26.0000	−1.20832	−0.604161	−	0.796862i	$-0.706492\pi$
$463$	−26.0000	−1.20832	−0.604161	+	0.796862i	$0.706492\pi$
$464$	−58.2065	−2.70217
$465$	0	0
$466$	42.0000	1.94561
$467$	−31.7490	−1.46917	−0.734585	−	0.678517i	$-0.762623\pi$
$467$	−31.7490	−1.46917	−0.734585	+	0.678517i	$0.762623\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	21.1660	0.973214
$474$	0	0
$475$	−14.0000	−0.642364
$476$	0	0
$477$	0	0
$478$	42.0000	1.92104
$479$	−21.1660	−0.967100	−0.483550	−	0.875317i	$-0.660653\pi$
$479$	−21.1660	−0.967100	−0.483550	+	0.875317i	$0.660653\pi$
$480$	0	0
$481$	−6.00000	−0.273576
$482$	−10.5830	−0.482043
$483$	0	0
$484$	−20.0000	−0.909091
$485$	−31.7490	−1.44165
$486$	0	0
$487$	−32.0000	−1.45006	−0.725029	−	0.688718i	$-0.758174\pi$
$487$	−32.0000	−1.45006	−0.725029	+	0.688718i	$0.758174\pi$
$488$	−63.4980	−2.87442
$489$	0	0
$490$	0	0
$491$	−2.64575	−0.119401	−0.0597005	−	0.998216i	$-0.519015\pi$
$491$	−2.64575	−0.119401	−0.0597005	+	0.998216i	$0.519015\pi$
$492$	0	0
$493$	0	0
$494$	37.0405	1.66653
$495$	0	0
$496$	−33.0000	−1.48174
$497$	0	0
$498$	0	0
$499$	4.00000	0.179065	0.0895323	−	0.995984i	$-0.471463\pi$
$499$	4.00000	0.179065	0.0895323	+	0.995984i	$0.471463\pi$
$500$	39.6863	1.77482
$501$	0	0
$502$	28.0000	1.24970
$503$	−15.8745	−0.707809	−0.353905	−	0.935282i	$-0.615146\pi$
$503$	−15.8745	−0.707809	−0.353905	+	0.935282i	$0.615146\pi$
$504$	0	0
$505$	−28.0000	−1.24598
$506$	−55.5608	−2.46998
$507$	0	0
$508$	30.0000	1.33103
$509$	10.5830	0.469083	0.234542	−	0.972106i	$-0.424641\pi$
$509$	10.5830	0.469083	0.234542	+	0.972106i	$0.424641\pi$
$510$	0	0
$511$	0	0
$512$	29.1033	1.28619
$513$	0	0
$514$	−49.0000	−2.16130
$515$	−34.3948	−1.51561
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	42.0000	1.84182
$521$	−39.6863	−1.73869	−0.869344	−	0.494208i	$-0.835458\pi$
$521$	−39.6863	−1.73869	−0.869344	+	0.494208i	$0.835458\pi$
$522$	0	0
$523$	−13.0000	−0.568450	−0.284225	−	0.958758i	$-0.591736\pi$
$523$	−13.0000	−0.568450	−0.284225	+	0.958758i	$0.591736\pi$
$524$	105.830	4.62321
$525$	0	0
$526$	−49.0000	−2.13650
$527$	0	0
$528$	0	0
$529$	40.0000	1.73913
$530$	0	0
$531$	0	0
$532$	0	0
$533$	5.29150	0.229200
$534$	0	0
$535$	14.0000	0.605273
$536$	15.8745	0.685674
$537$	0	0
$538$	−63.0000	−2.71612
$539$	0	0
$540$	0	0
$541$	29.0000	1.24681	0.623404	−	0.781900i	$-0.285749\pi$
$541$	29.0000	1.24681	0.623404	+	0.781900i	$0.285749\pi$
$542$	−52.9150	−2.27289
$543$	0	0
$544$	0	0
$545$	−23.8118	−1.01998
$546$	0	0
$547$	−8.00000	−0.342055	−0.171028	−	0.985266i	$-0.554709\pi$
$547$	−8.00000	−0.342055	−0.171028	+	0.985266i	$0.554709\pi$
$548$	79.3725	3.39063
$549$	0	0
$550$	−14.0000	−0.596962
$551$	37.0405	1.57798
$552$	0	0
$553$	0	0
$554$	60.8523	2.58537
$555$	0	0
$556$	60.0000	2.54457
$557$	−5.29150	−0.224208	−0.112104	−	0.993696i	$-0.535759\pi$
$557$	−5.29150	−0.224208	−0.112104	+	0.993696i	$0.535759\pi$
$558$	0	0
$559$	16.0000	0.676728
$560$	0	0
$561$	0	0
$562$	28.0000	1.18111
$563$	−26.4575	−1.11505	−0.557526	−	0.830160i	$-0.688249\pi$
$563$	−26.4575	−1.11505	−0.557526	+	0.830160i	$0.688249\pi$
$564$	0	0
$565$	14.0000	0.588984
$566$	10.5830	0.444837
$567$	0	0
$568$	63.0000	2.64342
$569$	26.4575	1.10916	0.554578	−	0.832132i	$-0.312880\pi$
$569$	26.4575	1.10916	0.554578	+	0.832132i	$0.312880\pi$
$570$	0	0
$571$	−10.0000	−0.418487	−0.209243	−	0.977864i	$-0.567100\pi$
$571$	−10.0000	−0.418487	−0.209243	+	0.977864i	$0.567100\pi$
$572$	26.4575	1.10624
$573$	0	0
$574$	0	0
$575$	15.8745	0.662013
$576$	0	0
$577$	38.0000	1.58196	0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000	1.58196	0.790980	+	0.611842i	$0.209571\pi$
$578$	44.9778	1.87083
$579$	0	0
$580$	70.0000	2.90659
$581$	0	0
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	−28.0000	−1.15667
$587$	−26.4575	−1.09202	−0.546009	−	0.837779i	$-0.683854\pi$
$587$	−26.4575	−1.09202	−0.546009	+	0.837779i	$0.683854\pi$
$588$	0	0
$589$	21.0000	0.865290
$590$	0	0
$591$	0	0
$592$	−33.0000	−1.35629
$593$	23.8118	0.977832	0.488916	−	0.872331i	$-0.337392\pi$
$593$	23.8118	0.977832	0.488916	+	0.872331i	$0.337392\pi$
$594$	0	0
$595$	0	0
$596$	79.3725	3.25123
$597$	0	0
$598$	−42.0000	−1.71751
$599$	−7.93725	−0.324307	−0.162154	−	0.986766i	$-0.551844\pi$
$599$	−7.93725	−0.324307	−0.162154	+	0.986766i	$0.551844\pi$
$600$	0	0
$601$	30.0000	1.22373	0.611863	−	0.790964i	$-0.290420\pi$
$601$	30.0000	1.22373	0.611863	+	0.790964i	$0.290420\pi$
$602$	0	0
$603$	0	0
$604$	10.0000	0.406894
$605$	10.5830	0.430260
$606$	0	0
$607$	28.0000	1.13648	0.568242	−	0.822861i	$-0.307624\pi$
$607$	28.0000	1.13648	0.568242	+	0.822861i	$0.307624\pi$
$608$	92.6013	3.75548
$609$	0	0
$610$	56.0000	2.26737
$611$	0	0
$612$	0	0
$613$	19.0000	0.767403	0.383701	−	0.923457i	$-0.374649\pi$
$613$	19.0000	0.767403	0.383701	+	0.923457i	$0.374649\pi$
$614$	50.2693	2.02870
$615$	0	0
$616$	0	0
$617$	0	0		−	1.00000i	$-0.5\pi$
$617$	0	0			1.00000i	$0.5\pi$
$618$	0	0
$619$	−31.0000	−1.24600	−0.622998	−	0.782224i	$-0.714085\pi$
$619$	−31.0000	−1.24600	−0.622998	+	0.782224i	$0.714085\pi$
$620$	39.6863	1.59384
$621$	0	0
$622$	−42.0000	−1.68405
$623$	0	0
$624$	0	0
$625$	−31.0000	−1.24000
$626$	37.0405	1.48044
$627$	0	0
$628$	70.0000	2.79330
$629$	0	0
$630$	0	0
$631$	20.0000	0.796187	0.398094	−	0.917345i	$-0.369672\pi$
$631$	20.0000	0.796187	0.398094	+	0.917345i	$0.369672\pi$
$632$	31.7490	1.26291
$633$	0	0
$634$	0	0
$635$	−15.8745	−0.629961
$636$	0	0
$637$	0	0
$638$	37.0405	1.46645
$639$	0	0
$640$	21.0000	0.830098
$641$	−15.8745	−0.627005	−0.313503	−	0.949587i	$-0.601502\pi$
$641$	−15.8745	−0.627005	−0.313503	+	0.949587i	$0.601502\pi$
$642$	0	0
$643$	−23.0000	−0.907031	−0.453516	−	0.891248i	$-0.649830\pi$
$643$	−23.0000	−0.907031	−0.453516	+	0.891248i	$0.649830\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	10.5830	0.416061	0.208030	−	0.978122i	$-0.433295\pi$
$647$	10.5830	0.416061	0.208030	+	0.978122i	$0.433295\pi$
$648$	0	0
$649$	0	0
$650$	−10.5830	−0.415100
$651$	0	0
$652$	50.0000	1.95815
$653$	31.7490	1.24243	0.621217	−	0.783638i	$-0.286638\pi$
$653$	31.7490	1.24243	0.621217	+	0.783638i	$0.286638\pi$
$654$	0	0
$655$	−56.0000	−2.18810
$656$	29.1033	1.13629
$657$	0	0
$658$	0	0
$659$	23.8118	0.927575	0.463787	−	0.885947i	$-0.346490\pi$
$659$	23.8118	0.927575	0.463787	+	0.885947i	$0.346490\pi$
$660$	0	0
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	−52.9150	−2.05660
$663$	0	0
$664$	−126.000	−4.88975
$665$	0	0
$666$	0	0
$667$	−42.0000	−1.62625
$668$	26.4575	1.02367
$669$	0	0
$670$	−14.0000	−0.540867
$671$	21.1660	0.817105
$672$	0	0
$673$	−2.00000	−0.0770943	−0.0385472	−	0.999257i	$-0.512273\pi$
$673$	−2.00000	−0.0770943	−0.0385472	+	0.999257i	$0.512273\pi$
$674$	−50.2693	−1.93630
$675$	0	0
$676$	−45.0000	−1.73077
$677$	−7.93725	−0.305053	−0.152527	−	0.988299i	$-0.548741\pi$
$677$	−7.93725	−0.305053	−0.152527	+	0.988299i	$0.548741\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	21.0000	0.804132
$683$	−23.8118	−0.911132	−0.455566	−	0.890202i	$-0.650563\pi$
$683$	−23.8118	−0.911132	−0.455566	+	0.890202i	$0.650563\pi$
$684$	0	0
$685$	−42.0000	−1.60474
$686$	0	0
$687$	0	0
$688$	88.0000	3.35497
$689$	0	0
$690$	0	0
$691$	−8.00000	−0.304334	−0.152167	−	0.988355i	$-0.548625\pi$
$691$	−8.00000	−0.304334	−0.152167	+	0.988355i	$0.548625\pi$
$692$	−66.1438	−2.51441
$693$	0	0
$694$	7.00000	0.265716
$695$	−31.7490	−1.20431
$696$	0	0
$697$	0	0
$698$	−5.29150	−0.200286
$699$	0	0
$700$	0	0
$701$	15.8745	0.599572	0.299786	−	0.954006i	$-0.403085\pi$
$701$	15.8745	0.599572	0.299786	+	0.954006i	$0.403085\pi$
$702$	0	0
$703$	21.0000	0.792030
$704$	34.3948	1.29630
$705$	0	0
$706$	7.00000	0.263448
$707$	0	0
$708$	0	0
$709$	−3.00000	−0.112667	−0.0563337	−	0.998412i	$-0.517941\pi$
$709$	−3.00000	−0.112667	−0.0563337	+	0.998412i	$0.517941\pi$
$710$	−55.5608	−2.08516
$711$	0	0
$712$	147.000	5.50906
$713$	−23.8118	−0.891757
$714$	0	0
$715$	−14.0000	−0.523570
$716$	26.4575	0.988764
$717$	0	0
$718$	−14.0000	−0.522475
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	0	0
$722$	−79.3725	−2.95394
$723$	0	0
$724$	70.0000	2.60153
$725$	−10.5830	−0.393043
$726$	0	0
$727$	40.0000	1.48352	0.741759	−	0.670667i	$-0.233992\pi$
$727$	40.0000	1.48352	0.741759	+	0.670667i	$0.233992\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−36.0000	−1.32969	−0.664845	−	0.746981i	$-0.731502\pi$
$733$	−36.0000	−1.32969	−0.664845	+	0.746981i	$0.731502\pi$
$734$	55.5608	2.05079
$735$	0	0
$736$	−105.000	−3.87035
$737$	−5.29150	−0.194915
$738$	0	0
$739$	−30.0000	−1.10357	−0.551784	−	0.833987i	$-0.686053\pi$
$739$	−30.0000	−1.10357	−0.551784	+	0.833987i	$0.686053\pi$
$740$	39.6863	1.45890
$741$	0	0
$742$	0	0
$743$	23.8118	0.873569	0.436784	−	0.899566i	$-0.356117\pi$
$743$	23.8118	0.873569	0.436784	+	0.899566i	$0.356117\pi$
$744$	0	0
$745$	−42.0000	−1.53876
$746$	−76.7268	−2.80917
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−8.00000	−0.291924	−0.145962	−	0.989290i	$-0.546628\pi$
$751$	−8.00000	−0.291924	−0.145962	+	0.989290i	$0.546628\pi$
$752$	0	0
$753$	0	0
$754$	28.0000	1.01970
$755$	−5.29150	−0.192577
$756$	0	0
$757$	−34.0000	−1.23575	−0.617876	−	0.786276i	$-0.712006\pi$
$757$	−34.0000	−1.23575	−0.617876	+	0.786276i	$0.712006\pi$
$758$	−95.2470	−3.45953
$759$	0	0
$760$	−147.000	−5.33225
$761$	31.7490	1.15090	0.575450	−	0.817837i	$-0.304827\pi$
$761$	31.7490	1.15090	0.575450	+	0.817837i	$0.304827\pi$
$762$	0	0
$763$	0	0
$764$	13.2288	0.478600
$765$	0	0
$766$	−42.0000	−1.51752
$767$	0	0
$768$	0	0
$769$	−14.0000	−0.504853	−0.252426	−	0.967616i	$-0.581229\pi$
$769$	−14.0000	−0.504853	−0.252426	+	0.967616i	$0.581229\pi$
$770$	0	0
$771$	0	0
$772$	10.0000	0.359908
$773$	2.64575	0.0951611	0.0475805	−	0.998867i	$-0.484849\pi$
$773$	2.64575	0.0951611	0.0475805	+	0.998867i	$0.484849\pi$
$774$	0	0
$775$	−6.00000	−0.215526
$776$	−95.2470	−3.41917
$777$	0	0
$778$	84.0000	3.01155
$779$	−18.5203	−0.663557
$780$	0	0
$781$	−21.0000	−0.751439
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−37.0405	−1.32203
$786$	0	0
$787$	−4.00000	−0.142585	−0.0712923	−	0.997455i	$-0.522712\pi$
$787$	−4.00000	−0.142585	−0.0712923	+	0.997455i	$0.522712\pi$
$788$	132.288	4.71255
$789$	0	0
$790$	−28.0000	−0.996195
$791$	0	0
$792$	0	0
$793$	16.0000	0.568177
$794$	−15.8745	−0.563365
$795$	0	0
$796$	15.0000	0.531661
$797$	44.9778	1.59319	0.796597	−	0.604510i	$-0.206631\pi$
$797$	44.9778	1.59319	0.796597	+	0.604510i	$0.206631\pi$
$798$	0	0
$799$	0	0
$800$	−26.4575	−0.935414
$801$	0	0
$802$	42.0000	1.48307
$803$	0	0
$804$	0	0
$805$	0	0
$806$	15.8745	0.559156
$807$	0	0
$808$	−84.0000	−2.95511
$809$	15.8745	0.558118	0.279059	−	0.960274i	$-0.409977\pi$
$809$	15.8745	0.558118	0.279059	+	0.960274i	$0.409977\pi$
$810$	0	0
$811$	−35.0000	−1.22902	−0.614508	−	0.788911i	$-0.710645\pi$
$811$	−35.0000	−1.22902	−0.614508	+	0.788911i	$0.710645\pi$
$812$	0	0
$813$	0	0
$814$	21.0000	0.736050
$815$	−26.4575	−0.926766
$816$	0	0
$817$	−56.0000	−1.95919
$818$	84.6640	2.96021
$819$	0	0
$820$	−35.0000	−1.22225
$821$	21.1660	0.738699	0.369349	−	0.929291i	$-0.379581\pi$
$821$	21.1660	0.738699	0.369349	+	0.929291i	$0.379581\pi$
$822$	0	0
$823$	−30.0000	−1.04573	−0.522867	−	0.852414i	$-0.675138\pi$
$823$	−30.0000	−1.04573	−0.522867	+	0.852414i	$0.675138\pi$
$824$	−103.184	−3.59460
$825$	0	0
$826$	0	0
$827$	−39.6863	−1.38003	−0.690013	−	0.723797i	$-0.742395\pi$
$827$	−39.6863	−1.38003	−0.690013	+	0.723797i	$0.742395\pi$
$828$	0	0
$829$	28.0000	0.972480	0.486240	−	0.873825i	$-0.338368\pi$
$829$	28.0000	0.972480	0.486240	+	0.873825i	$0.338368\pi$
$830$	111.122	3.85709
$831$	0	0
$832$	26.0000	0.901388
$833$	0	0
$834$	0	0
$835$	−14.0000	−0.484490
$836$	−92.6013	−3.20268
$837$	0	0
$838$	42.0000	1.45087
$839$	5.29150	0.182683	0.0913415	−	0.995820i	$-0.470885\pi$
$839$	5.29150	0.182683	0.0913415	+	0.995820i	$0.470885\pi$
$840$	0	0
$841$	−1.00000	−0.0344828
$842$	50.2693	1.73239
$843$	0	0
$844$	110.000	3.78636
$845$	23.8118	0.819150
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−23.8118	−0.816257
$852$	0	0
$853$	28.0000	0.958702	0.479351	−	0.877623i	$-0.340872\pi$
$853$	28.0000	0.958702	0.479351	+	0.877623i	$0.340872\pi$
$854$	0	0
$855$	0	0
$856$	42.0000	1.43553
$857$	13.2288	0.451886	0.225943	−	0.974141i	$-0.427454\pi$
$857$	13.2288	0.451886	0.225943	+	0.974141i	$0.427454\pi$
$858$	0	0
$859$	13.0000	0.443554	0.221777	−	0.975097i	$-0.428814\pi$
$859$	13.0000	0.443554	0.221777	+	0.975097i	$0.428814\pi$
$860$	−105.830	−3.60877
$861$	0	0
$862$	105.000	3.57631
$863$	−15.8745	−0.540375	−0.270187	−	0.962808i	$-0.587086\pi$
$863$	−15.8745	−0.540375	−0.270187	+	0.962808i	$0.587086\pi$
$864$	0	0
$865$	35.0000	1.19004
$866$	−68.7895	−2.33756
$867$	0	0
$868$	0	0
$869$	−10.5830	−0.359004
$870$	0	0
$871$	−4.00000	−0.135535
$872$	−71.4353	−2.41910
$873$	0	0
$874$	147.000	4.97235
$875$	0	0
$876$	0	0
$877$	10.0000	0.337676	0.168838	−	0.985644i	$-0.445999\pi$
$877$	10.0000	0.337676	0.168838	+	0.985644i	$0.445999\pi$
$878$	0	0
$879$	0	0
$880$	−77.0000	−2.59567
$881$	−23.8118	−0.802239	−0.401119	−	0.916026i	$-0.631379\pi$
$881$	−23.8118	−0.802239	−0.401119	+	0.916026i	$0.631379\pi$
$882$	0	0
$883$	−16.0000	−0.538443	−0.269221	−	0.963078i	$-0.586766\pi$
$883$	−16.0000	−0.538443	−0.269221	+	0.963078i	$0.586766\pi$
$884$	0	0
$885$	0	0
$886$	−21.0000	−0.705509
$887$	−37.0405	−1.24370	−0.621849	−	0.783137i	$-0.713618\pi$
$887$	−37.0405	−1.24370	−0.621849	+	0.783137i	$0.713618\pi$
$888$	0	0
$889$	0	0
$890$	−129.642	−4.34560
$891$	0	0
$892$	−35.0000	−1.17189
$893$	0	0
$894$	0	0
$895$	−14.0000	−0.467968
$896$	0	0
$897$	0	0
$898$	−42.0000	−1.40156
$899$	15.8745	0.529444
$900$	0	0
$901$	0	0
$902$	−18.5203	−0.616657
$903$	0	0
$904$	42.0000	1.39690
$905$	−37.0405	−1.23127
$906$	0	0
$907$	2.00000	0.0664089	0.0332045	−	0.999449i	$-0.489429\pi$
$907$	2.00000	0.0664089	0.0332045	+	0.999449i	$0.489429\pi$
$908$	−79.3725	−2.63407
$909$	0	0
$910$	0	0
$911$	47.6235	1.57784	0.788919	−	0.614497i	$-0.210641\pi$
$911$	47.6235	1.57784	0.788919	+	0.614497i	$0.210641\pi$
$912$	0	0
$913$	42.0000	1.39000
$914$	−66.1438	−2.18784
$915$	0	0
$916$	−100.000	−3.30409
$917$	0	0
$918$	0	0
$919$	32.0000	1.05558	0.527791	−	0.849374i	$-0.323020\pi$
$919$	32.0000	1.05558	0.527791	+	0.849374i	$0.323020\pi$
$920$	166.682	5.49535
$921$	0	0
$922$	−49.0000	−1.61373
$923$	−15.8745	−0.522516
$924$	0	0
$925$	−6.00000	−0.197279
$926$	68.7895	2.26056
$927$	0	0
$928$	70.0000	2.29786
$929$	42.3320	1.38887	0.694434	−	0.719556i	$-0.255655\pi$
$929$	42.3320	1.38887	0.694434	+	0.719556i	$0.255655\pi$
$930$	0	0
$931$	0	0
$932$	−79.3725	−2.59993
$933$	0	0
$934$	84.0000	2.74856
$935$	0	0
$936$	0	0
$937$	12.0000	0.392023	0.196011	−	0.980602i	$-0.437201\pi$
$937$	12.0000	0.392023	0.196011	+	0.980602i	$0.437201\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	50.2693	1.63873	0.819366	−	0.573271i	$-0.194326\pi$
$941$	50.2693	1.63873	0.819366	+	0.573271i	$0.194326\pi$
$942$	0	0
$943$	21.0000	0.683854
$944$	0	0
$945$	0	0
$946$	−56.0000	−1.82072
$947$	−18.5203	−0.601828	−0.300914	−	0.953651i	$-0.597292\pi$
$947$	−18.5203	−0.601828	−0.300914	+	0.953651i	$0.597292\pi$
$948$	0	0
$949$	0	0
$950$	37.0405	1.20175
$951$	0	0
$952$	0	0
$953$	−26.4575	−0.857043	−0.428521	−	0.903532i	$-0.640965\pi$
$953$	−26.4575	−0.857043	−0.428521	+	0.903532i	$0.640965\pi$
$954$	0	0
$955$	−7.00000	−0.226515
$956$	−79.3725	−2.56709
$957$	0	0
$958$	56.0000	1.80928
$959$	0	0
$960$	0	0
$961$	−22.0000	−0.709677
$962$	15.8745	0.511815
$963$	0	0
$964$	20.0000	0.644157
$965$	−5.29150	−0.170339
$966$	0	0
$967$	−50.0000	−1.60789	−0.803946	−	0.594703i	$-0.797270\pi$
$967$	−50.0000	−1.60789	−0.803946	+	0.594703i	$0.797270\pi$
$968$	31.7490	1.02045
$969$	0	0
$970$	84.0000	2.69708
$971$	−15.8745	−0.509437	−0.254719	−	0.967015i	$-0.581983\pi$
$971$	−15.8745	−0.509437	−0.254719	+	0.967015i	$0.581983\pi$
$972$	0	0
$973$	0	0
$974$	84.6640	2.71281
$975$	0	0
$976$	88.0000	2.81681
$977$	0	0		−	1.00000i	$-0.5\pi$
$977$	0	0			1.00000i	$0.5\pi$
$978$	0	0
$979$	−49.0000	−1.56605
$980$	0	0
$981$	0	0
$982$	7.00000	0.223379
$983$	31.7490	1.01264	0.506318	−	0.862347i	$-0.331006\pi$
$983$	31.7490	1.01264	0.506318	+	0.862347i	$0.331006\pi$
$984$	0	0
$985$	−70.0000	−2.23039
$986$	0	0
$987$	0	0
$988$	−70.0000	−2.22700
$989$	63.4980	2.01912
$990$	0	0
$991$	2.00000	0.0635321	0.0317660	−	0.999495i	$-0.489887\pi$
$991$	2.00000	0.0635321	0.0317660	+	0.999495i	$0.489887\pi$
$992$	39.6863	1.26004
$993$	0	0
$994$	0	0
$995$	−7.93725	−0.251628
$996$	0	0
$997$	38.0000	1.20347	0.601736	−	0.798695i	$-0.294476\pi$
$997$	38.0000	1.20347	0.601736	+	0.798695i	$0.294476\pi$
$998$	−10.5830	−0.334999
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1323.2.a.w.1.1		2
3.2	odd	2	inner	1323.2.a.w.1.2		2
7.6	odd	2		189.2.a.f.1.1	✓	2
21.20	even	2		189.2.a.f.1.2	yes	2
28.27	even	2		3024.2.a.bi.1.2		2
35.34	odd	2		4725.2.a.bb.1.2		2
63.13	odd	6		567.2.f.i.379.2		4
63.20	even	6		567.2.f.i.190.1		4
63.34	odd	6		567.2.f.i.190.2		4
63.41	even	6		567.2.f.i.379.1		4
84.83	odd	2		3024.2.a.bi.1.1		2
105.104	even	2		4725.2.a.bb.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.2.a.f.1.1	✓	2	7.6	odd	2
189.2.a.f.1.2	yes	2	21.20	even	2
567.2.f.i.190.1		4	63.20	even	6
567.2.f.i.190.2		4	63.34	odd	6
567.2.f.i.379.1		4	63.41	even	6
567.2.f.i.379.2		4	63.13	odd	6
1323.2.a.w.1.1		2	1.1	even	1	trivial
1323.2.a.w.1.2		2	3.2	odd	2	inner
3024.2.a.bi.1.1		2	84.83	odd	2
3024.2.a.bi.1.2		2	28.27	even	2
4725.2.a.bb.1.1		2	105.104	even	2
4725.2.a.bb.1.2		2	35.34	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1323 = 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1323.a (trivial)

\(f(q)\)	\(=\)	\(q-2.64575 q^{2} +5.00000 q^{4} -2.64575 q^{5} -7.93725 q^{8} +7.00000 q^{10} +2.64575 q^{11} +2.00000 q^{13} +11.0000 q^{16} -7.00000 q^{19} -13.2288 q^{20} -7.00000 q^{22} +7.93725 q^{23} +2.00000 q^{25} -5.29150 q^{26} -5.29150 q^{29} -3.00000 q^{31} -13.2288 q^{32} -3.00000 q^{37} +18.5203 q^{38} +21.0000 q^{40} +2.64575 q^{41} +8.00000 q^{43} +13.2288 q^{44} -21.0000 q^{46} -5.29150 q^{50} +10.0000 q^{52} -7.00000 q^{55} +14.0000 q^{58} +8.00000 q^{61} +7.93725 q^{62} +13.0000 q^{64} -5.29150 q^{65} -2.00000 q^{67} -7.93725 q^{71} +7.93725 q^{74} -35.0000 q^{76} -4.00000 q^{79} -29.1033 q^{80} -7.00000 q^{82} +15.8745 q^{83} -21.1660 q^{86} -21.0000 q^{88} -18.5203 q^{89} +39.6863 q^{92} +18.5203 q^{95} +12.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 10 q^{4} + 14 q^{10} + 4 q^{13} + 22 q^{16} - 14 q^{19} - 14 q^{22} + 4 q^{25} - 6 q^{31} - 6 q^{37} + 42 q^{40} + 16 q^{43} - 42 q^{46} + 20 q^{52} - 14 q^{55} + 28 q^{58} + 16 q^{61} + 26 q^{64}+ \cdots + 24 q^{97}+O(q^{100}) \) 2 * q + 10 * q^4 + 14 * q^10 + 4 * q^13 + 22 * q^16 - 14 * q^19 - 14 * q^22 + 4 * q^25 - 6 * q^31 - 6 * q^37 + 42 * q^40 + 16 * q^43 - 42 * q^46 + 20 * q^52 - 14 * q^55 + 28 * q^58 + 16 * q^61 + 26 * q^64 - 4 * q^67 - 70 * q^76 - 8 * q^79 - 14 * q^82 - 42 * q^88 + 24 * q^97

Introduction

L-functions

Modular forms

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Database

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