LMFDB - Embedded newform 3332.2.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3332,2,Mod(1,3332)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3332, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("3332.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3332 = 2^{2} \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3332.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:	$26.6061539535$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} - 8x^{6} + 36x^{5} + 17x^{4} - 76x^{3} - 20x^{2} + 44x + 17 $ x^8 - 4x^7 - 8x^6 + 36x^5 + 17x^4 - 76x^3 - 20x^2 + 44*x + 17
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$3.29150$ of defining polynomial
Character	$\chi$	$=$	3332.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.29150 q^{3} +0.255523 q^{5} +7.83400 q^{9} +O(q^{10})$	$q-3.29150 q^{3} +0.255523 q^{5} +7.83400 q^{9} +5.29125 q^{11} -5.00619 q^{13} -0.841055 q^{15} +1.00000 q^{17} -4.78151 q^{19} -2.20213 q^{23} -4.93471 q^{25} -15.9112 q^{27} -9.21615 q^{29} +10.4263 q^{31} -17.4162 q^{33} +2.45276 q^{37} +16.4779 q^{39} +3.70185 q^{41} +9.09475 q^{43} +2.00177 q^{45} +3.08561 q^{47} -3.29150 q^{51} -3.01303 q^{53} +1.35204 q^{55} +15.7384 q^{57} +8.02339 q^{59} +1.32586 q^{61} -1.27920 q^{65} -2.39116 q^{67} +7.24832 q^{69} -4.46967 q^{71} +10.1506 q^{73} +16.2426 q^{75} +3.14157 q^{79} +28.8696 q^{81} -7.79613 q^{83} +0.255523 q^{85} +30.3350 q^{87} -12.3043 q^{89} -34.3184 q^{93} -1.22179 q^{95} -11.4156 q^{97} +41.4517 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 4 q^{3} - 4 q^{5} + 8 q^{9}+O(q^{10}) $ 8 * q - 4 * q^3 - 4 * q^5 + 8 * q^9	$ 8 q - 4 q^{3} - 4 q^{5} + 8 q^{9} - 4 q^{11} - 20 q^{13} + 12 q^{15} + 8 q^{17} - 8 q^{19} + 4 q^{23} + 8 q^{25} - 28 q^{27} - 16 q^{29} + 8 q^{31} - 16 q^{33} + 8 q^{37} + 20 q^{39} - 12 q^{41} - 4 q^{43} - 20 q^{45} - 4 q^{47} - 4 q^{51} - 12 q^{55} + 16 q^{59} - 32 q^{61} - 44 q^{69} - 24 q^{73} - 24 q^{75} + 4 q^{79} + 36 q^{81} - 28 q^{83} - 4 q^{85} + 40 q^{87} - 20 q^{89} - 16 q^{93} - 20 q^{95} - 56 q^{97} - 8 q^{99}+O(q^{100}) $ 8 * q - 4 * q^3 - 4 * q^5 + 8 * q^9 - 4 * q^11 - 20 * q^13 + 12 * q^15 + 8 * q^17 - 8 * q^19 + 4 * q^23 + 8 * q^25 - 28 * q^27 - 16 * q^29 + 8 * q^31 - 16 * q^33 + 8 * q^37 + 20 * q^39 - 12 * q^41 - 4 * q^43 - 20 * q^45 - 4 * q^47 - 4 * q^51 - 12 * q^55 + 16 * q^59 - 32 * q^61 - 44 * q^69 - 24 * q^73 - 24 * q^75 + 4 * q^79 + 36 * q^81 - 28 * q^83 - 4 * q^85 + 40 * q^87 - 20 * q^89 - 16 * q^93 - 20 * q^95 - 56 * q^97 - 8 * q^99

Display 100 coefficients 10 coefficients

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
$3$	−3.29150	−1.90035	−0.950176	−	0.311715i	$-0.899097\pi$
$3$	−3.29150	−1.90035	−0.950176	+	0.311715i	$0.899097\pi$
$4$	0	0
$5$	0.255523	0.114273	0.0571367	−	0.998366i	$-0.481803\pi$
$5$	0.255523	0.114273	0.0571367	+	0.998366i	$0.481803\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	7.83400	2.61133
$10$	0	0
$11$	5.29125	1.59537	0.797686	−	0.603073i	$-0.206057\pi$
$11$	5.29125	1.59537	0.797686	+	0.603073i	$0.206057\pi$
$12$	0	0
$13$	−5.00619	−1.38847	−0.694234	−	0.719749i	$-0.744257\pi$
$13$	−5.00619	−1.38847	−0.694234	+	0.719749i	$0.744257\pi$
$14$	0	0
$15$	−0.841055	−0.217159
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−4.78151	−1.09695	−0.548477	−	0.836166i	$-0.684792\pi$
$19$	−4.78151	−1.09695	−0.548477	+	0.836166i	$0.684792\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−2.20213	−0.459176	−0.229588	−	0.973288i	$-0.573738\pi$
$23$	−2.20213	−0.459176	−0.229588	+	0.973288i	$0.573738\pi$
$24$	0	0
$25$	−4.93471	−0.986942
$26$	0	0
$27$	−15.9112	−3.06210
$28$	0	0
$29$	−9.21615	−1.71140	−0.855698	−	0.517475i	$-0.826872\pi$
$29$	−9.21615	−1.71140	−0.855698	+	0.517475i	$0.826872\pi$
$30$	0	0
$31$	10.4263	1.87263	0.936313	−	0.351166i	$-0.114215\pi$
$31$	10.4263	1.87263	0.936313	+	0.351166i	$0.114215\pi$
$32$	0	0
$33$	−17.4162	−3.03177
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.45276	0.403231	0.201616	−	0.979465i	$-0.435381\pi$
$37$	2.45276	0.403231	0.201616	+	0.979465i	$0.435381\pi$
$38$	0	0
$39$	16.4779	2.63858
$40$	0	0
$41$	3.70185	0.578131	0.289066	−	0.957309i	$-0.406655\pi$
$41$	3.70185	0.578131	0.289066	+	0.957309i	$0.406655\pi$
$42$	0	0
$43$	9.09475	1.38694	0.693468	−	0.720487i	$-0.256082\pi$
$43$	9.09475	1.38694	0.693468	+	0.720487i	$0.256082\pi$
$44$	0	0
$45$	2.00177	0.298406
$46$	0	0
$47$	3.08561	0.450083	0.225041	−	0.974349i	$-0.427748\pi$
$47$	3.08561	0.450083	0.225041	+	0.974349i	$0.427748\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−3.29150	−0.460903
$52$	0	0
$53$	−3.01303	−0.413872	−0.206936	−	0.978355i	$-0.566349\pi$
$53$	−3.01303	−0.413872	−0.206936	+	0.978355i	$0.566349\pi$
$54$	0	0
$55$	1.35204	0.182308
$56$	0	0
$57$	15.7384	2.08460
$58$	0	0
$59$	8.02339	1.04456	0.522278	−	0.852775i	$-0.325082\pi$
$59$	8.02339	1.04456	0.522278	+	0.852775i	$0.325082\pi$
$60$	0	0
$61$	1.32586	0.169759	0.0848796	−	0.996391i	$-0.472949\pi$
$61$	1.32586	0.169759	0.0848796	+	0.996391i	$0.472949\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−1.27920	−0.158665
$66$	0	0
$67$	−2.39116	−0.292126	−0.146063	−	0.989275i	$-0.546660\pi$
$67$	−2.39116	−0.292126	−0.146063	+	0.989275i	$0.546660\pi$
$68$	0	0
$69$	7.24832	0.872596
$70$	0	0
$71$	−4.46967	−0.530452	−0.265226	−	0.964186i	$-0.585447\pi$
$71$	−4.46967	−0.530452	−0.265226	+	0.964186i	$0.585447\pi$
$72$	0	0
$73$	10.1506	1.18804	0.594018	−	0.804451i	$-0.297541\pi$
$73$	10.1506	1.18804	0.594018	+	0.804451i	$0.297541\pi$
$74$	0	0
$75$	16.2426	1.87554
$76$	0	0
$77$	0	0
$78$	0	0
$79$	3.14157	0.353454	0.176727	−	0.984260i	$-0.443449\pi$
$79$	3.14157	0.353454	0.176727	+	0.984260i	$0.443449\pi$
$80$	0	0
$81$	28.8696	3.20773
$82$	0	0
$83$	−7.79613	−0.855736	−0.427868	−	0.903841i	$-0.640735\pi$
$83$	−7.79613	−0.855736	−0.427868	+	0.903841i	$0.640735\pi$
$84$	0	0
$85$	0.255523	0.0277153
$86$	0	0
$87$	30.3350	3.25225
$88$	0	0
$89$	−12.3043	−1.30425	−0.652126	−	0.758111i	$-0.726123\pi$
$89$	−12.3043	−1.30425	−0.652126	+	0.758111i	$0.726123\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−34.3184	−3.55865
$94$	0	0
$95$	−1.22179	−0.125353
$96$	0	0
$97$	−11.4156	−1.15907	−0.579537	−	0.814946i	$-0.696767\pi$
$97$	−11.4156	−1.15907	−0.579537	+	0.814946i	$0.696767\pi$
$98$	0	0
$99$	41.4517	4.16605
$100$	0	0
$101$	−8.00013	−0.796042	−0.398021	−	0.917376i	$-0.630303\pi$
$101$	−8.00013	−0.796042	−0.398021	+	0.917376i	$0.630303\pi$
$102$	0	0
$103$	−0.452364	−0.0445728	−0.0222864	−	0.999752i	$-0.507095\pi$
$103$	−0.452364	−0.0445728	−0.0222864	+	0.999752i	$0.507095\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.74202	−0.555102	−0.277551	−	0.960711i	$-0.589523\pi$
$107$	−5.74202	−0.555102	−0.277551	+	0.960711i	$0.589523\pi$
$108$	0	0
$109$	−17.6002	−1.68579	−0.842895	−	0.538077i	$-0.819151\pi$
$109$	−17.6002	−1.68579	−0.842895	+	0.538077i	$0.819151\pi$
$110$	0	0
$111$	−8.07327	−0.766281
$112$	0	0
$113$	−16.3542	−1.53848	−0.769238	−	0.638963i	$-0.779364\pi$
$113$	−16.3542	−1.53848	−0.769238	+	0.638963i	$0.779364\pi$
$114$	0	0
$115$	−0.562695	−0.0524716
$116$	0	0
$117$	−39.2185	−3.62575
$118$	0	0
$119$	0	0
$120$	0	0
$121$	16.9973	1.54521
$122$	0	0
$123$	−12.1846	−1.09865
$124$	0	0
$125$	−2.53855	−0.227054
$126$	0	0
$127$	4.13314	0.366757	0.183378	−	0.983042i	$-0.441297\pi$
$127$	4.13314	0.366757	0.183378	+	0.983042i	$0.441297\pi$
$128$	0	0
$129$	−29.9354	−2.63567
$130$	0	0
$131$	4.49275	0.392534	0.196267	−	0.980551i	$-0.437118\pi$
$131$	4.49275	0.392534	0.196267	+	0.980551i	$0.437118\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−4.06566	−0.349917
$136$	0	0
$137$	−1.05831	−0.0904179	−0.0452090	−	0.998978i	$-0.514395\pi$
$137$	−1.05831	−0.0904179	−0.0452090	+	0.998978i	$0.514395\pi$
$138$	0	0
$139$	−1.49349	−0.126676	−0.0633380	−	0.997992i	$-0.520175\pi$
$139$	−1.49349	−0.126676	−0.0633380	+	0.997992i	$0.520175\pi$
$140$	0	0
$141$	−10.1563	−0.855315
$142$	0	0
$143$	−26.4890	−2.21512
$144$	0	0
$145$	−2.35494	−0.195567
$146$	0	0
$147$	0	0
$148$	0	0
$149$	12.7843	1.04733	0.523667	−	0.851923i	$-0.324564\pi$
$149$	12.7843	1.04733	0.523667	+	0.851923i	$0.324564\pi$
$150$	0	0
$151$	18.2850	1.48801	0.744005	−	0.668175i	$-0.232924\pi$
$151$	18.2850	1.48801	0.744005	+	0.668175i	$0.232924\pi$
$152$	0	0
$153$	7.83400	0.633342
$154$	0	0
$155$	2.66417	0.213991
$156$	0	0
$157$	4.11005	0.328018	0.164009	−	0.986459i	$-0.447557\pi$
$157$	4.11005	0.328018	0.164009	+	0.986459i	$0.447557\pi$
$158$	0	0
$159$	9.91740	0.786501
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−21.8718	−1.71313	−0.856564	−	0.516040i	$-0.827406\pi$
$163$	−21.8718	−1.71313	−0.856564	+	0.516040i	$0.827406\pi$
$164$	0	0
$165$	−4.45023	−0.346450
$166$	0	0
$167$	−17.8923	−1.38455	−0.692273	−	0.721636i	$-0.743391\pi$
$167$	−17.8923	−1.38455	−0.692273	+	0.721636i	$0.743391\pi$
$168$	0	0
$169$	12.0620	0.927843
$170$	0	0
$171$	−37.4584	−2.86452
$172$	0	0
$173$	20.3589	1.54786	0.773930	−	0.633271i	$-0.218288\pi$
$173$	20.3589	1.54786	0.773930	+	0.633271i	$0.218288\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−26.4090	−1.98502
$178$	0	0
$179$	2.89511	0.216391	0.108195	−	0.994130i	$-0.465493\pi$
$179$	2.89511	0.216391	0.108195	+	0.994130i	$0.465493\pi$
$180$	0	0
$181$	−10.9778	−0.815971	−0.407985	−	0.912988i	$-0.633769\pi$
$181$	−10.9778	−0.815971	−0.407985	+	0.912988i	$0.633769\pi$
$182$	0	0
$183$	−4.36408	−0.322602
$184$	0	0
$185$	0.626736	0.0460786
$186$	0	0
$187$	5.29125	0.386935
$188$	0	0
$189$	0	0
$190$	0	0
$191$	17.8151	1.28905	0.644527	−	0.764581i	$-0.277054\pi$
$191$	17.8151	1.28905	0.644527	+	0.764581i	$0.277054\pi$
$192$	0	0
$193$	8.99469	0.647452	0.323726	−	0.946151i	$-0.395064\pi$
$193$	8.99469	0.647452	0.323726	+	0.946151i	$0.395064\pi$
$194$	0	0
$195$	4.21048	0.301519
$196$	0	0
$197$	−4.01138	−0.285799	−0.142899	−	0.989737i	$-0.545643\pi$
$197$	−4.01138	−0.285799	−0.142899	+	0.989737i	$0.545643\pi$
$198$	0	0
$199$	−26.8656	−1.90445	−0.952225	−	0.305396i	$-0.901211\pi$
$199$	−26.8656	−1.90445	−0.952225	+	0.305396i	$0.901211\pi$
$200$	0	0
$201$	7.87050	0.555142
$202$	0	0
$203$	0	0
$204$	0	0
$205$	0.945907	0.0660650
$206$	0	0
$207$	−17.2515	−1.19906
$208$	0	0
$209$	−25.3002	−1.75005
$210$	0	0
$211$	−18.5159	−1.27468	−0.637342	−	0.770581i	$-0.719966\pi$
$211$	−18.5159	−1.27468	−0.637342	+	0.770581i	$0.719966\pi$
$212$	0	0
$213$	14.7119	1.00804
$214$	0	0
$215$	2.32392	0.158490
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−33.4107	−2.25769
$220$	0	0
$221$	−5.00619	−0.336753
$222$	0	0
$223$	9.55003	0.639517	0.319759	−	0.947499i	$-0.396398\pi$
$223$	9.55003	0.639517	0.319759	+	0.947499i	$0.396398\pi$
$224$	0	0
$225$	−38.6585	−2.57724
$226$	0	0
$227$	−12.8950	−0.855874	−0.427937	−	0.903809i	$-0.640759\pi$
$227$	−12.8950	−0.855874	−0.427937	+	0.903809i	$0.640759\pi$
$228$	0	0
$229$	−1.20514	−0.0796382	−0.0398191	−	0.999207i	$-0.512678\pi$
$229$	−1.20514	−0.0796382	−0.0398191	+	0.999207i	$0.512678\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−21.0554	−1.37939	−0.689693	−	0.724102i	$-0.742254\pi$
$233$	−21.0554	−1.37939	−0.689693	+	0.724102i	$0.742254\pi$
$234$	0	0
$235$	0.788445	0.0514325
$236$	0	0
$237$	−10.3405	−0.671687
$238$	0	0
$239$	10.5525	0.682587	0.341293	−	0.939957i	$-0.389135\pi$
$239$	10.5525	0.682587	0.341293	+	0.939957i	$0.389135\pi$
$240$	0	0
$241$	3.70049	0.238369	0.119185	−	0.992872i	$-0.461972\pi$
$241$	3.70049	0.238369	0.119185	+	0.992872i	$0.461972\pi$
$242$	0	0
$243$	−47.2910	−3.03372
$244$	0	0
$245$	0	0
$246$	0	0
$247$	23.9372	1.52309
$248$	0	0
$249$	25.6610	1.62620
$250$	0	0
$251$	−4.78860	−0.302254	−0.151127	−	0.988514i	$-0.548290\pi$
$251$	−4.78860	−0.302254	−0.151127	+	0.988514i	$0.548290\pi$
$252$	0	0
$253$	−11.6520	−0.732556
$254$	0	0
$255$	−0.841055	−0.0526689
$256$	0	0
$257$	−20.3277	−1.26801	−0.634003	−	0.773331i	$-0.718589\pi$
$257$	−20.3277	−1.26801	−0.634003	+	0.773331i	$0.718589\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−72.1994	−4.46903
$262$	0	0
$263$	−17.1048	−1.05473	−0.527364	−	0.849640i	$-0.676819\pi$
$263$	−17.1048	−1.05473	−0.527364	+	0.849640i	$0.676819\pi$
$264$	0	0
$265$	−0.769898	−0.0472945
$266$	0	0
$267$	40.4996	2.47854
$268$	0	0
$269$	23.7142	1.44588	0.722940	−	0.690911i	$-0.242790\pi$
$269$	23.7142	1.44588	0.722940	+	0.690911i	$0.242790\pi$
$270$	0	0
$271$	16.9424	1.02917	0.514587	−	0.857438i	$-0.327945\pi$
$271$	16.9424	1.02917	0.514587	+	0.857438i	$0.327945\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−26.1108	−1.57454
$276$	0	0
$277$	−17.5139	−1.05231	−0.526153	−	0.850390i	$-0.676366\pi$
$277$	−17.5139	−1.05231	−0.526153	+	0.850390i	$0.676366\pi$
$278$	0	0
$279$	81.6800	4.89005
$280$	0	0
$281$	−10.5755	−0.630884	−0.315442	−	0.948945i	$-0.602153\pi$
$281$	−10.5755	−0.630884	−0.315442	+	0.948945i	$0.602153\pi$
$282$	0	0
$283$	−20.9052	−1.24268	−0.621342	−	0.783539i	$-0.713412\pi$
$283$	−20.9052	−1.24268	−0.621342	+	0.783539i	$0.713412\pi$
$284$	0	0
$285$	4.02152	0.238214
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	37.5744	2.20265
$292$	0	0
$293$	2.20380	0.128747	0.0643737	−	0.997926i	$-0.479495\pi$
$293$	2.20380	0.128747	0.0643737	+	0.997926i	$0.479495\pi$
$294$	0	0
$295$	2.05016	0.119365
$296$	0	0
$297$	−84.1899	−4.88519
$298$	0	0
$299$	11.0243	0.637551
$300$	0	0
$301$	0	0
$302$	0	0
$303$	26.3325	1.51276
$304$	0	0
$305$	0.338788	0.0193990
$306$	0	0
$307$	5.88810	0.336051	0.168026	−	0.985783i	$-0.446261\pi$
$307$	5.88810	0.336051	0.168026	+	0.985783i	$0.446261\pi$
$308$	0	0
$309$	1.48896	0.0847039
$310$	0	0
$311$	22.7831	1.29191	0.645956	−	0.763375i	$-0.276459\pi$
$311$	22.7831	1.29191	0.645956	+	0.763375i	$0.276459\pi$
$312$	0	0
$313$	−13.5499	−0.765886	−0.382943	−	0.923772i	$-0.625089\pi$
$313$	−13.5499	−0.765886	−0.382943	+	0.923772i	$0.625089\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	3.36188	0.188822	0.0944111	−	0.995533i	$-0.469903\pi$
$317$	3.36188	0.188822	0.0944111	+	0.995533i	$0.469903\pi$
$318$	0	0
$319$	−48.7650	−2.73031
$320$	0	0
$321$	18.8999	1.05489
$322$	0	0
$323$	−4.78151	−0.266051
$324$	0	0
$325$	24.7041	1.37034
$326$	0	0
$327$	57.9311	3.20359
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−22.8510	−1.25600	−0.628002	−	0.778212i	$-0.716127\pi$
$331$	−22.8510	−1.25600	−0.628002	+	0.778212i	$0.716127\pi$
$332$	0	0
$333$	19.2149	1.05297
$334$	0	0
$335$	−0.610995	−0.0333822
$336$	0	0
$337$	−16.5430	−0.901156	−0.450578	−	0.892737i	$-0.648782\pi$
$337$	−16.5430	−0.901156	−0.450578	+	0.892737i	$0.648782\pi$
$338$	0	0
$339$	53.8300	2.92364
$340$	0	0
$341$	55.1684	2.98754
$342$	0	0
$343$	0	0
$344$	0	0
$345$	1.85211	0.0997144
$346$	0	0
$347$	−19.6541	−1.05509	−0.527545	−	0.849527i	$-0.676887\pi$
$347$	−19.6541	−1.05509	−0.527545	+	0.849527i	$0.676887\pi$
$348$	0	0
$349$	−21.8638	−1.17034	−0.585170	−	0.810911i	$-0.698972\pi$
$349$	−21.8638	−1.17034	−0.585170	+	0.810911i	$0.698972\pi$
$350$	0	0
$351$	79.6543	4.25163
$352$	0	0
$353$	23.3142	1.24089	0.620445	−	0.784250i	$-0.286952\pi$
$353$	23.3142	1.24089	0.620445	+	0.784250i	$0.286952\pi$
$354$	0	0
$355$	−1.14210	−0.0606165
$356$	0	0
$357$	0	0
$358$	0	0
$359$	5.94855	0.313953	0.156976	−	0.987602i	$-0.449825\pi$
$359$	5.94855	0.313953	0.156976	+	0.987602i	$0.449825\pi$
$360$	0	0
$361$	3.86287	0.203309
$362$	0	0
$363$	−55.9468	−2.93645
$364$	0	0
$365$	2.59371	0.135761
$366$	0	0
$367$	2.56655	0.133973	0.0669864	−	0.997754i	$-0.478662\pi$
$367$	2.56655	0.133973	0.0669864	+	0.997754i	$0.478662\pi$
$368$	0	0
$369$	29.0003	1.50969
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−20.3016	−1.05118	−0.525589	−	0.850738i	$-0.676155\pi$
$373$	−20.3016	−1.05118	−0.525589	+	0.850738i	$0.676155\pi$
$374$	0	0
$375$	8.35563	0.431483
$376$	0	0
$377$	46.1378	2.37622
$378$	0	0
$379$	0.288606	0.0148247	0.00741234	−	0.999973i	$-0.497641\pi$
$379$	0.288606	0.0148247	0.00741234	+	0.999973i	$0.497641\pi$
$380$	0	0
$381$	−13.6043	−0.696967
$382$	0	0
$383$	−13.5337	−0.691540	−0.345770	−	0.938319i	$-0.612382\pi$
$383$	−13.5337	−0.691540	−0.345770	+	0.938319i	$0.612382\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	71.2483	3.62175
$388$	0	0
$389$	−18.1386	−0.919663	−0.459832	−	0.888006i	$-0.652090\pi$
$389$	−18.1386	−0.919663	−0.459832	+	0.888006i	$0.652090\pi$
$390$	0	0
$391$	−2.20213	−0.111367
$392$	0	0
$393$	−14.7879	−0.745952
$394$	0	0
$395$	0.802743	0.0403904
$396$	0	0
$397$	−13.2468	−0.664839	−0.332420	−	0.943132i	$-0.607865\pi$
$397$	−13.2468	−0.664839	−0.332420	+	0.943132i	$0.607865\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−35.3660	−1.76609	−0.883046	−	0.469286i	$-0.844511\pi$
$401$	−35.3660	−1.76609	−0.883046	+	0.469286i	$0.844511\pi$
$402$	0	0
$403$	−52.1963	−2.60008
$404$	0	0
$405$	7.37685	0.366559
$406$	0	0
$407$	12.9782	0.643304
$408$	0	0
$409$	−17.9248	−0.886323	−0.443162	−	0.896442i	$-0.646143\pi$
$409$	−17.9248	−0.886323	−0.443162	+	0.896442i	$0.646143\pi$
$410$	0	0
$411$	3.48345	0.171826
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−1.99209	−0.0977878
$416$	0	0
$417$	4.91582	0.240729
$418$	0	0
$419$	−24.6696	−1.20519	−0.602596	−	0.798047i	$-0.705867\pi$
$419$	−24.6696	−1.20519	−0.602596	+	0.798047i	$0.705867\pi$
$420$	0	0
$421$	32.7692	1.59707	0.798537	−	0.601945i	$-0.205608\pi$
$421$	32.7692	1.59707	0.798537	+	0.601945i	$0.205608\pi$
$422$	0	0
$423$	24.1727	1.17532
$424$	0	0
$425$	−4.93471	−0.239369
$426$	0	0
$427$	0	0
$428$	0	0
$429$	87.1887	4.20951
$430$	0	0
$431$	15.5168	0.747416	0.373708	−	0.927546i	$-0.378086\pi$
$431$	15.5168	0.747416	0.373708	+	0.927546i	$0.378086\pi$
$432$	0	0
$433$	−29.5889	−1.42195	−0.710976	−	0.703216i	$-0.751747\pi$
$433$	−29.5889	−1.42195	−0.710976	+	0.703216i	$0.751747\pi$
$434$	0	0
$435$	7.75129	0.371646
$436$	0	0
$437$	10.5295	0.503695
$438$	0	0
$439$	−37.5900	−1.79407	−0.897037	−	0.441955i	$-0.854285\pi$
$439$	−37.5900	−1.79407	−0.897037	+	0.441955i	$0.854285\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0.483922	0.0229918	0.0114959	−	0.999934i	$-0.496341\pi$
$443$	0.483922	0.0229918	0.0114959	+	0.999934i	$0.496341\pi$
$444$	0	0
$445$	−3.14403	−0.149041
$446$	0	0
$447$	−42.0797	−1.99030
$448$	0	0
$449$	−17.6601	−0.833430	−0.416715	−	0.909037i	$-0.636819\pi$
$449$	−17.6601	−0.833430	−0.416715	+	0.909037i	$0.636819\pi$
$450$	0	0
$451$	19.5874	0.922335
$452$	0	0
$453$	−60.1850	−2.82774
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−4.07859	−0.190789	−0.0953943	−	0.995440i	$-0.530411\pi$
$457$	−4.07859	−0.190789	−0.0953943	+	0.995440i	$0.530411\pi$
$458$	0	0
$459$	−15.9112	−0.742669
$460$	0	0
$461$	−16.9506	−0.789467	−0.394733	−	0.918796i	$-0.629163\pi$
$461$	−16.9506	−0.789467	−0.394733	+	0.918796i	$0.629163\pi$
$462$	0	0
$463$	−30.0505	−1.39657	−0.698283	−	0.715821i	$-0.746052\pi$
$463$	−30.0505	−1.39657	−0.698283	+	0.715821i	$0.746052\pi$
$464$	0	0
$465$	−8.76913	−0.406659
$466$	0	0
$467$	−23.9501	−1.10828	−0.554140	−	0.832424i	$-0.686953\pi$
$467$	−23.9501	−1.10828	−0.554140	+	0.832424i	$0.686953\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−13.5283	−0.623349
$472$	0	0
$473$	48.1226	2.21268
$474$	0	0
$475$	23.5954	1.08263
$476$	0	0
$477$	−23.6041	−1.08076
$478$	0	0
$479$	31.5188	1.44013	0.720066	−	0.693905i	$-0.244111\pi$
$479$	31.5188	1.44013	0.720066	+	0.693905i	$0.244111\pi$
$480$	0	0
$481$	−12.2790	−0.559874
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.91694	−0.132451
$486$	0	0
$487$	−14.2754	−0.646879	−0.323440	−	0.946249i	$-0.604839\pi$
$487$	−14.2754	−0.646879	−0.323440	+	0.946249i	$0.604839\pi$
$488$	0	0
$489$	71.9910	3.25555
$490$	0	0
$491$	19.3575	0.873591	0.436795	−	0.899561i	$-0.356113\pi$
$491$	19.3575	0.873591	0.436795	+	0.899561i	$0.356113\pi$
$492$	0	0
$493$	−9.21615	−0.415075
$494$	0	0
$495$	10.5919	0.476068
$496$	0	0
$497$	0	0
$498$	0	0
$499$	18.3827	0.822925	0.411462	−	0.911427i	$-0.365018\pi$
$499$	18.3827	0.822925	0.411462	+	0.911427i	$0.365018\pi$
$500$	0	0
$501$	58.8925	2.63112
$502$	0	0
$503$	35.4690	1.58149	0.790743	−	0.612148i	$-0.209694\pi$
$503$	35.4690	1.58149	0.790743	+	0.612148i	$0.209694\pi$
$504$	0	0
$505$	−2.04422	−0.0909664
$506$	0	0
$507$	−39.7020	−1.76323
$508$	0	0
$509$	−7.15587	−0.317178	−0.158589	−	0.987345i	$-0.550695\pi$
$509$	−7.15587	−0.317178	−0.158589	+	0.987345i	$0.550695\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	76.0794	3.35899
$514$	0	0
$515$	−0.115589	−0.00509348
$516$	0	0
$517$	16.3267	0.718050
$518$	0	0
$519$	−67.0115	−2.94148
$520$	0	0
$521$	4.73937	0.207636	0.103818	−	0.994596i	$-0.466894\pi$
$521$	4.73937	0.207636	0.103818	+	0.994596i	$0.466894\pi$
$522$	0	0
$523$	−1.38107	−0.0603899	−0.0301949	−	0.999544i	$-0.509613\pi$
$523$	−1.38107	−0.0603899	−0.0301949	+	0.999544i	$0.509613\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10.4263	0.454179
$528$	0	0
$529$	−18.1506	−0.789157
$530$	0	0
$531$	62.8553	2.72769
$532$	0	0
$533$	−18.5322	−0.802717
$534$	0	0
$535$	−1.46722	−0.0634333
$536$	0	0
$537$	−9.52927	−0.411218
$538$	0	0
$539$	0	0
$540$	0	0
$541$	22.1805	0.953613	0.476806	−	0.879008i	$-0.341794\pi$
$541$	22.1805	0.953613	0.476806	+	0.879008i	$0.341794\pi$
$542$	0	0
$543$	36.1334	1.55063
$544$	0	0
$545$	−4.49725	−0.192641
$546$	0	0
$547$	−2.69279	−0.115135	−0.0575676	−	0.998342i	$-0.518334\pi$
$547$	−2.69279	−0.115135	−0.0575676	+	0.998342i	$0.518334\pi$
$548$	0	0
$549$	10.3868	0.443298
$550$	0	0
$551$	44.0671	1.87732
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−2.06291	−0.0875655
$556$	0	0
$557$	19.0733	0.808163	0.404082	−	0.914723i	$-0.367591\pi$
$557$	19.0733	0.808163	0.404082	+	0.914723i	$0.367591\pi$
$558$	0	0
$559$	−45.5301	−1.92572
$560$	0	0
$561$	−17.4162	−0.735312
$562$	0	0
$563$	4.39308	0.185146	0.0925732	−	0.995706i	$-0.470491\pi$
$563$	4.39308	0.185146	0.0925732	+	0.995706i	$0.470491\pi$
$564$	0	0
$565$	−4.17888	−0.175807
$566$	0	0
$567$	0	0
$568$	0	0
$569$	22.6424	0.949219	0.474610	−	0.880196i	$-0.342589\pi$
$569$	22.6424	0.949219	0.474610	+	0.880196i	$0.342589\pi$
$570$	0	0
$571$	−19.7741	−0.827522	−0.413761	−	0.910386i	$-0.635785\pi$
$571$	−19.7741	−0.827522	−0.413761	+	0.910386i	$0.635785\pi$
$572$	0	0
$573$	−58.6385	−2.44966
$574$	0	0
$575$	10.8669	0.453180
$576$	0	0
$577$	−23.8616	−0.993372	−0.496686	−	0.867930i	$-0.665450\pi$
$577$	−23.8616	−0.993372	−0.496686	+	0.867930i	$0.665450\pi$
$578$	0	0
$579$	−29.6061	−1.23039
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−15.9427	−0.660279
$584$	0	0
$585$	−10.0212	−0.414327
$586$	0	0
$587$	−19.4335	−0.802105	−0.401052	−	0.916055i	$-0.631355\pi$
$587$	−19.4335	−0.802105	−0.401052	+	0.916055i	$0.631355\pi$
$588$	0	0
$589$	−49.8537	−2.05419
$590$	0	0
$591$	13.2035	0.543118
$592$	0	0
$593$	−15.1125	−0.620597	−0.310299	−	0.950639i	$-0.600429\pi$
$593$	−15.1125	−0.620597	−0.310299	+	0.950639i	$0.600429\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	88.4282	3.61913
$598$	0	0
$599$	−8.84472	−0.361385	−0.180693	−	0.983540i	$-0.557834\pi$
$599$	−8.84472	−0.361385	−0.180693	+	0.983540i	$0.557834\pi$
$600$	0	0
$601$	−48.2247	−1.96713	−0.983563	−	0.180568i	$-0.942206\pi$
$601$	−48.2247	−1.96713	−0.983563	+	0.180568i	$0.942206\pi$
$602$	0	0
$603$	−18.7323	−0.762839
$604$	0	0
$605$	4.34321	0.176576
$606$	0	0
$607$	1.32394	0.0537371	0.0268685	−	0.999639i	$-0.491446\pi$
$607$	1.32394	0.0537371	0.0268685	+	0.999639i	$0.491446\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−15.4472	−0.624925
$612$	0	0
$613$	−19.0413	−0.769073	−0.384536	−	0.923110i	$-0.625639\pi$
$613$	−19.0413	−0.769073	−0.384536	+	0.923110i	$0.625639\pi$
$614$	0	0
$615$	−3.11346	−0.125547
$616$	0	0
$617$	42.7409	1.72068	0.860342	−	0.509717i	$-0.170250\pi$
$617$	42.7409	1.72068	0.860342	+	0.509717i	$0.170250\pi$
$618$	0	0
$619$	2.62071	0.105335	0.0526676	−	0.998612i	$-0.483228\pi$
$619$	2.62071	0.105335	0.0526676	+	0.998612i	$0.483228\pi$
$620$	0	0
$621$	35.0384	1.40604
$622$	0	0
$623$	0	0
$624$	0	0
$625$	24.0249	0.960995
$626$	0	0
$627$	83.2757	3.32571
$628$	0	0
$629$	2.45276	0.0977980
$630$	0	0
$631$	−7.47445	−0.297553	−0.148777	−	0.988871i	$-0.547534\pi$
$631$	−7.47445	−0.297553	−0.148777	+	0.988871i	$0.547534\pi$
$632$	0	0
$633$	60.9450	2.42235
$634$	0	0
$635$	1.05611	0.0419105
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−35.0154	−1.38519
$640$	0	0
$641$	8.26131	0.326302	0.163151	−	0.986601i	$-0.447834\pi$
$641$	8.26131	0.326302	0.163151	+	0.986601i	$0.447834\pi$
$642$	0	0
$643$	−16.2308	−0.640082	−0.320041	−	0.947404i	$-0.603697\pi$
$643$	−16.2308	−0.640082	−0.320041	+	0.947404i	$0.603697\pi$
$644$	0	0
$645$	−7.64918	−0.301186
$646$	0	0
$647$	35.3766	1.39080	0.695398	−	0.718624i	$-0.255228\pi$
$647$	35.3766	1.39080	0.695398	+	0.718624i	$0.255228\pi$
$648$	0	0
$649$	42.4538	1.66646
$650$	0	0
$651$	0	0
$652$	0	0
$653$	50.3818	1.97159	0.985797	−	0.167943i	$-0.0537125\pi$
$653$	50.3818	1.97159	0.985797	+	0.167943i	$0.0537125\pi$
$654$	0	0
$655$	1.14800	0.0448561
$656$	0	0
$657$	79.5198	3.10236
$658$	0	0
$659$	−28.7225	−1.11887	−0.559435	−	0.828874i	$-0.688982\pi$
$659$	−28.7225	−1.11887	−0.559435	+	0.828874i	$0.688982\pi$
$660$	0	0
$661$	−8.34260	−0.324490	−0.162245	−	0.986751i	$-0.551873\pi$
$661$	−8.34260	−0.324490	−0.162245	+	0.986751i	$0.551873\pi$
$662$	0	0
$663$	16.4779	0.639949
$664$	0	0
$665$	0	0
$666$	0	0
$667$	20.2952	0.785832
$668$	0	0
$669$	−31.4340	−1.21531
$670$	0	0
$671$	7.01547	0.270829
$672$	0	0
$673$	28.5636	1.10105	0.550523	−	0.834820i	$-0.314428\pi$
$673$	28.5636	1.10105	0.550523	+	0.834820i	$0.314428\pi$
$674$	0	0
$675$	78.5169	3.02212
$676$	0	0
$677$	19.3306	0.742936	0.371468	−	0.928446i	$-0.378854\pi$
$677$	19.3306	0.742936	0.371468	+	0.928446i	$0.378854\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	42.4441	1.62646
$682$	0	0
$683$	−29.9730	−1.14688	−0.573442	−	0.819246i	$-0.694392\pi$
$683$	−29.9730	−1.14688	−0.573442	+	0.819246i	$0.694392\pi$
$684$	0	0
$685$	−0.270424	−0.0103324
$686$	0	0
$687$	3.96674	0.151340
$688$	0	0
$689$	15.0838	0.574647
$690$	0	0
$691$	22.6758	0.862628	0.431314	−	0.902202i	$-0.358050\pi$
$691$	22.6758	0.862628	0.431314	+	0.902202i	$0.358050\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−0.381620	−0.0144757
$696$	0	0
$697$	3.70185	0.140217
$698$	0	0
$699$	69.3040	2.62132
$700$	0	0
$701$	−44.0361	−1.66322	−0.831610	−	0.555361i	$-0.812580\pi$
$701$	−44.0361	−1.66322	−0.831610	+	0.555361i	$0.812580\pi$
$702$	0	0
$703$	−11.7279	−0.442326
$704$	0	0
$705$	−2.59517	−0.0977397
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−29.1743	−1.09567	−0.547833	−	0.836588i	$-0.684547\pi$
$709$	−29.1743	−1.09567	−0.547833	+	0.836588i	$0.684547\pi$
$710$	0	0
$711$	24.6111	0.922988
$712$	0	0
$713$	−22.9602	−0.859865
$714$	0	0
$715$	−6.76855	−0.253129
$716$	0	0
$717$	−34.7337	−1.29715
$718$	0	0
$719$	−15.2578	−0.569021	−0.284510	−	0.958673i	$-0.591831\pi$
$719$	−15.2578	−0.569021	−0.284510	+	0.958673i	$0.591831\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−12.1802	−0.452985
$724$	0	0
$725$	45.4790	1.68905
$726$	0	0
$727$	26.5752	0.985619	0.492810	−	0.870137i	$-0.335970\pi$
$727$	26.5752	0.985619	0.492810	+	0.870137i	$0.335970\pi$
$728$	0	0
$729$	69.0498	2.55740
$730$	0	0
$731$	9.09475	0.336381
$732$	0	0
$733$	11.1377	0.411382	0.205691	−	0.978617i	$-0.434056\pi$
$733$	11.1377	0.411382	0.205691	+	0.978617i	$0.434056\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−12.6522	−0.466050
$738$	0	0
$739$	−15.4519	−0.568406	−0.284203	−	0.958764i	$-0.591729\pi$
$739$	−15.4519	−0.568406	−0.284203	+	0.958764i	$0.591729\pi$
$740$	0	0
$741$	−78.7893	−2.89440
$742$	0	0
$743$	−41.8225	−1.53432	−0.767159	−	0.641457i	$-0.778330\pi$
$743$	−41.8225	−1.53432	−0.767159	+	0.641457i	$0.778330\pi$
$744$	0	0
$745$	3.26669	0.119682
$746$	0	0
$747$	−61.0749	−2.23461
$748$	0	0
$749$	0	0
$750$	0	0
$751$	38.7262	1.41314	0.706569	−	0.707644i	$-0.250242\pi$
$751$	38.7262	1.41314	0.706569	+	0.707644i	$0.250242\pi$
$752$	0	0
$753$	15.7617	0.574388
$754$	0	0
$755$	4.67223	0.170040
$756$	0	0
$757$	49.8361	1.81132	0.905662	−	0.424001i	$-0.139375\pi$
$757$	49.8361	1.81132	0.905662	+	0.424001i	$0.139375\pi$
$758$	0	0
$759$	38.3527	1.39211
$760$	0	0
$761$	40.2881	1.46044	0.730222	−	0.683210i	$-0.239417\pi$
$761$	40.2881	1.46044	0.730222	+	0.683210i	$0.239417\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	2.00177	0.0723741
$766$	0	0
$767$	−40.1666	−1.45033
$768$	0	0
$769$	6.37384	0.229847	0.114923	−	0.993374i	$-0.463338\pi$
$769$	6.37384	0.229847	0.114923	+	0.993374i	$0.463338\pi$
$770$	0	0
$771$	66.9087	2.40966
$772$	0	0
$773$	48.3293	1.73828	0.869142	−	0.494562i	$-0.164671\pi$
$773$	48.3293	1.73828	0.869142	+	0.494562i	$0.164671\pi$
$774$	0	0
$775$	−51.4510	−1.84817
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−17.7004	−0.634184
$780$	0	0
$781$	−23.6501	−0.846268
$782$	0	0
$783$	146.640	5.24047
$784$	0	0
$785$	1.05021	0.0374837
$786$	0	0
$787$	3.70691	0.132137	0.0660685	−	0.997815i	$-0.478954\pi$
$787$	3.70691	0.132137	0.0660685	+	0.997815i	$0.478954\pi$
$788$	0	0
$789$	56.3006	2.00435
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−6.63752	−0.235705
$794$	0	0
$795$	2.53412	0.0898761
$796$	0	0
$797$	−14.8274	−0.525212	−0.262606	−	0.964903i	$-0.584582\pi$
$797$	−14.8274	−0.525212	−0.262606	+	0.964903i	$0.584582\pi$
$798$	0	0
$799$	3.08561	0.109161
$800$	0	0
$801$	−96.3918	−3.40584
$802$	0	0
$803$	53.7093	1.89536
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−78.0554	−2.74768
$808$	0	0
$809$	−44.6936	−1.57134	−0.785671	−	0.618644i	$-0.787682\pi$
$809$	−44.6936	−1.57134	−0.785671	+	0.618644i	$0.787682\pi$
$810$	0	0
$811$	25.2292	0.885916	0.442958	−	0.896542i	$-0.353929\pi$
$811$	25.2292	0.885916	0.442958	+	0.896542i	$0.353929\pi$
$812$	0	0
$813$	−55.7658	−1.95579
$814$	0	0
$815$	−5.58874	−0.195765
$816$	0	0
$817$	−43.4867	−1.52141
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−15.7191	−0.548601	−0.274300	−	0.961644i	$-0.588446\pi$
$821$	−15.7191	−0.548601	−0.274300	+	0.961644i	$0.588446\pi$
$822$	0	0
$823$	43.9415	1.53170	0.765852	−	0.643017i	$-0.222318\pi$
$823$	43.9415	1.53170	0.765852	+	0.643017i	$0.222318\pi$
$824$	0	0
$825$	85.9437	2.99218
$826$	0	0
$827$	9.09896	0.316402	0.158201	−	0.987407i	$-0.449431\pi$
$827$	9.09896	0.316402	0.158201	+	0.987407i	$0.449431\pi$
$828$	0	0
$829$	−27.8080	−0.965813	−0.482907	−	0.875672i	$-0.660419\pi$
$829$	−27.8080	−0.965813	−0.482907	+	0.875672i	$0.660419\pi$
$830$	0	0
$831$	57.6469	1.99975
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−4.57189	−0.158217
$836$	0	0
$837$	−165.895	−5.73417
$838$	0	0
$839$	−27.5344	−0.950595	−0.475297	−	0.879825i	$-0.657660\pi$
$839$	−27.5344	−0.950595	−0.475297	+	0.879825i	$0.657660\pi$
$840$	0	0
$841$	55.9374	1.92888
$842$	0	0
$843$	34.8094	1.19890
$844$	0	0
$845$	3.08211	0.106028
$846$	0	0
$847$	0	0
$848$	0	0
$849$	68.8095	2.36154
$850$	0	0
$851$	−5.40130	−0.185154
$852$	0	0
$853$	−25.7996	−0.883362	−0.441681	−	0.897172i	$-0.645618\pi$
$853$	−25.7996	−0.883362	−0.441681	+	0.897172i	$0.645618\pi$
$854$	0	0
$855$	−9.57148	−0.327338
$856$	0	0
$857$	19.0031	0.649134	0.324567	−	0.945863i	$-0.394781\pi$
$857$	19.0031	0.649134	0.324567	+	0.945863i	$0.394781\pi$
$858$	0	0
$859$	52.1273	1.77856	0.889280	−	0.457363i	$-0.151206\pi$
$859$	52.1273	1.77856	0.889280	+	0.457363i	$0.151206\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−52.8951	−1.80057	−0.900286	−	0.435300i	$-0.856642\pi$
$863$	−52.8951	−1.80057	−0.900286	+	0.435300i	$0.856642\pi$
$864$	0	0
$865$	5.20217	0.176879
$866$	0	0
$867$	−3.29150	−0.111785
$868$	0	0
$869$	16.6228	0.563891
$870$	0	0
$871$	11.9706	0.405608
$872$	0	0
$873$	−89.4296	−3.02673
$874$	0	0
$875$	0	0
$876$	0	0
$877$	29.4672	0.995038	0.497519	−	0.867453i	$-0.334245\pi$
$877$	29.4672	0.995038	0.497519	+	0.867453i	$0.334245\pi$
$878$	0	0
$879$	−7.25382	−0.244665
$880$	0	0
$881$	6.31211	0.212661	0.106330	−	0.994331i	$-0.466090\pi$
$881$	6.31211	0.212661	0.106330	+	0.994331i	$0.466090\pi$
$882$	0	0
$883$	−15.1891	−0.511153	−0.255576	−	0.966789i	$-0.582265\pi$
$883$	−15.1891	−0.511153	−0.255576	+	0.966789i	$0.582265\pi$
$884$	0	0
$885$	−6.74811	−0.226835
$886$	0	0
$887$	−5.32298	−0.178728	−0.0893640	−	0.995999i	$-0.528483\pi$
$887$	−5.32298	−0.178728	−0.0893640	+	0.995999i	$0.528483\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	152.756	5.11753
$892$	0	0
$893$	−14.7539	−0.493720
$894$	0	0
$895$	0.739767	0.0247277
$896$	0	0
$897$	−36.2865	−1.21157
$898$	0	0
$899$	−96.0907	−3.20481
$900$	0	0
$901$	−3.01303	−0.100379
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−2.80507	−0.0932437
$906$	0	0
$907$	3.37784	0.112159	0.0560797	−	0.998426i	$-0.482140\pi$
$907$	3.37784	0.112159	0.0560797	+	0.998426i	$0.482140\pi$
$908$	0	0
$909$	−62.6730	−2.07873
$910$	0	0
$911$	6.94277	0.230024	0.115012	−	0.993364i	$-0.463309\pi$
$911$	6.94277	0.230024	0.115012	+	0.993364i	$0.463309\pi$
$912$	0	0
$913$	−41.2513	−1.36522
$914$	0	0
$915$	−1.11512	−0.0368648
$916$	0	0
$917$	0	0
$918$	0	0
$919$	−9.78363	−0.322732	−0.161366	−	0.986895i	$-0.551590\pi$
$919$	−9.78363	−0.322732	−0.161366	+	0.986895i	$0.551590\pi$
$920$	0	0
$921$	−19.3807	−0.638616
$922$	0	0
$923$	22.3760	0.736515
$924$	0	0
$925$	−12.1037	−0.397966
$926$	0	0
$927$	−3.54382	−0.116394
$928$	0	0
$929$	−22.9106	−0.751671	−0.375835	−	0.926686i	$-0.622644\pi$
$929$	−22.9106	−0.751671	−0.375835	+	0.926686i	$0.622644\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−74.9907	−2.45509
$934$	0	0
$935$	1.35204	0.0442163
$936$	0	0
$937$	35.4355	1.15763	0.578814	−	0.815460i	$-0.303516\pi$
$937$	35.4355	1.15763	0.578814	+	0.815460i	$0.303516\pi$
$938$	0	0
$939$	44.5996	1.45545
$940$	0	0
$941$	8.49272	0.276855	0.138427	−	0.990373i	$-0.455795\pi$
$941$	8.49272	0.276855	0.138427	+	0.990373i	$0.455795\pi$
$942$	0	0
$943$	−8.15195	−0.265464
$944$	0	0
$945$	0	0
$946$	0	0
$947$	37.2286	1.20977	0.604883	−	0.796314i	$-0.293220\pi$
$947$	37.2286	1.20977	0.604883	+	0.796314i	$0.293220\pi$
$948$	0	0
$949$	−50.8158	−1.64955
$950$	0	0
$951$	−11.0657	−0.358829
$952$	0	0
$953$	44.9065	1.45466	0.727332	−	0.686286i	$-0.240760\pi$
$953$	44.9065	1.45466	0.727332	+	0.686286i	$0.240760\pi$
$954$	0	0
$955$	4.55216	0.147305
$956$	0	0
$957$	160.510	5.18855
$958$	0	0
$959$	0	0
$960$	0	0
$961$	77.7086	2.50673
$962$	0	0
$963$	−44.9830	−1.44956
$964$	0	0
$965$	2.29835	0.0739865
$966$	0	0
$967$	36.4187	1.17115	0.585574	−	0.810619i	$-0.300869\pi$
$967$	36.4187	1.17115	0.585574	+	0.810619i	$0.300869\pi$
$968$	0	0
$969$	15.7384	0.505589
$970$	0	0
$971$	−18.3370	−0.588463	−0.294231	−	0.955734i	$-0.595064\pi$
$971$	−18.3370	−0.588463	−0.294231	+	0.955734i	$0.595064\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−81.3137	−2.60412
$976$	0	0
$977$	42.8743	1.37167	0.685835	−	0.727757i	$-0.259437\pi$
$977$	42.8743	1.37167	0.685835	+	0.727757i	$0.259437\pi$
$978$	0	0
$979$	−65.1051	−2.08077
$980$	0	0
$981$	−137.880	−4.40216
$982$	0	0
$983$	7.18466	0.229155	0.114577	−	0.993414i	$-0.463449\pi$
$983$	7.18466	0.229155	0.114577	+	0.993414i	$0.463449\pi$
$984$	0	0
$985$	−1.02500	−0.0326592
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−20.0278	−0.636848
$990$	0	0
$991$	16.2920	0.517531	0.258766	−	0.965940i	$-0.416684\pi$
$991$	16.2920	0.517531	0.258766	+	0.965940i	$0.416684\pi$
$992$	0	0
$993$	75.2142	2.38685
$994$	0	0
$995$	−6.86477	−0.217628
$996$	0	0
$997$	−55.1284	−1.74593	−0.872967	−	0.487779i	$-0.837807\pi$
$997$	−55.1284	−1.74593	−0.872967	+	0.487779i	$0.837807\pi$
$998$	0	0
$999$	−39.0262	−1.23474

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3332.2.a.t.1.1	✓	8
7.6	odd	2		3332.2.a.u.1.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3332.2.a.t.1.1	✓	8	1.1	even	1	trivial
3332.2.a.u.1.8	yes	8	7.6	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:	\( N \)	\(=\)	\( 3332 = 2^{2} \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3332.a (trivial)

\(f(q)\)	\(=\)	\(q-3.29150 q^{3} +0.255523 q^{5} +7.83400 q^{9} +O(q^{10})\)	\(q-3.29150 q^{3} +0.255523 q^{5} +7.83400 q^{9} +5.29125 q^{11} -5.00619 q^{13} -0.841055 q^{15} +1.00000 q^{17} -4.78151 q^{19} -2.20213 q^{23} -4.93471 q^{25} -15.9112 q^{27} -9.21615 q^{29} +10.4263 q^{31} -17.4162 q^{33} +2.45276 q^{37} +16.4779 q^{39} +3.70185 q^{41} +9.09475 q^{43} +2.00177 q^{45} +3.08561 q^{47} -3.29150 q^{51} -3.01303 q^{53} +1.35204 q^{55} +15.7384 q^{57} +8.02339 q^{59} +1.32586 q^{61} -1.27920 q^{65} -2.39116 q^{67} +7.24832 q^{69} -4.46967 q^{71} +10.1506 q^{73} +16.2426 q^{75} +3.14157 q^{79} +28.8696 q^{81} -7.79613 q^{83} +0.255523 q^{85} +30.3350 q^{87} -12.3043 q^{89} -34.3184 q^{93} -1.22179 q^{95} -11.4156 q^{97} +41.4517 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3} - 4 q^{5} + 8 q^{9}+O(q^{10}) \) 8 * q - 4 * q^3 - 4 * q^5 + 8 * q^9	\( 8 q - 4 q^{3} - 4 q^{5} + 8 q^{9} - 4 q^{11} - 20 q^{13} + 12 q^{15} + 8 q^{17} - 8 q^{19} + 4 q^{23} + 8 q^{25} - 28 q^{27} - 16 q^{29} + 8 q^{31} - 16 q^{33} + 8 q^{37} + 20 q^{39} - 12 q^{41} - 4 q^{43} - 20 q^{45} - 4 q^{47} - 4 q^{51} - 12 q^{55} + 16 q^{59} - 32 q^{61} - 44 q^{69} - 24 q^{73} - 24 q^{75} + 4 q^{79} + 36 q^{81} - 28 q^{83} - 4 q^{85} + 40 q^{87} - 20 q^{89} - 16 q^{93} - 20 q^{95} - 56 q^{97} - 8 q^{99}+O(q^{100}) \) 8 * q - 4 * q^3 - 4 * q^5 + 8 * q^9 - 4 * q^11 - 20 * q^13 + 12 * q^15 + 8 * q^17 - 8 * q^19 + 4 * q^23 + 8 * q^25 - 28 * q^27 - 16 * q^29 + 8 * q^31 - 16 * q^33 + 8 * q^37 + 20 * q^39 - 12 * q^41 - 4 * q^43 - 20 * q^45 - 4 * q^47 - 4 * q^51 - 12 * q^55 + 16 * q^59 - 32 * q^61 - 44 * q^69 - 24 * q^73 - 24 * q^75 + 4 * q^79 + 36 * q^81 - 28 * q^83 - 4 * q^85 + 40 * q^87 - 20 * q^89 - 16 * q^93 - 20 * q^95 - 56 * q^97 - 8 * q^99

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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