LMFDB - Embedded newform 3332.2.a.t.1.2

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			1.2
Root			$3.08774$ 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)

$f(q)$	$=$	$q-3.08774 q^{3} -3.25463 q^{5} +6.53416 q^{9} +O(q^{10})$	$q-3.08774 q^{3} -3.25463 q^{5} +6.53416 q^{9} -5.12130 q^{11} -5.13386 q^{13} +10.0495 q^{15} +1.00000 q^{17} +5.25326 q^{19} +6.97171 q^{23} +5.59260 q^{25} -10.9126 q^{27} -3.39300 q^{29} -2.35553 q^{31} +15.8132 q^{33} +2.60498 q^{37} +15.8520 q^{39} +5.70525 q^{41} +1.18641 q^{43} -21.2662 q^{45} -4.70250 q^{47} -3.08774 q^{51} +10.0116 q^{53} +16.6679 q^{55} -16.2207 q^{57} +13.0991 q^{59} -1.78135 q^{61} +16.7088 q^{65} -11.1779 q^{67} -21.5269 q^{69} +9.21864 q^{71} -14.6608 q^{73} -17.2685 q^{75} +2.05802 q^{79} +14.0927 q^{81} +15.7006 q^{83} -3.25463 q^{85} +10.4767 q^{87} +6.91321 q^{89} +7.27326 q^{93} -17.0974 q^{95} -10.5244 q^{97} -33.4634 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−3.08774	−1.78271	−0.891355	−	0.453307i	$-0.850244\pi$
$3$	−3.08774	−1.78271	−0.891355	+	0.453307i	$0.850244\pi$
$4$	0	0
$5$	−3.25463	−1.45551	−0.727757	−	0.685835i	$-0.759437\pi$
$5$	−3.25463	−1.45551	−0.727757	+	0.685835i	$0.759437\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	6.53416	2.17805
$10$	0	0
$11$	−5.12130	−1.54413	−0.772064	−	0.635544i	$-0.780776\pi$
$11$	−5.12130	−1.54413	−0.772064	+	0.635544i	$0.780776\pi$
$12$	0	0
$13$	−5.13386	−1.42388	−0.711939	−	0.702242i	$-0.752183\pi$
$13$	−5.13386	−1.42388	−0.711939	+	0.702242i	$0.752183\pi$
$14$	0	0
$15$	10.0495	2.59476
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	5.25326	1.20518	0.602590	−	0.798051i	$-0.294135\pi$
$19$	5.25326	1.20518	0.602590	+	0.798051i	$0.294135\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	6.97171	1.45370	0.726851	−	0.686795i	$-0.240983\pi$
$23$	6.97171	1.45370	0.726851	+	0.686795i	$0.240983\pi$
$24$	0	0
$25$	5.59260	1.11852
$26$	0	0
$27$	−10.9126	−2.10013
$28$	0	0
$29$	−3.39300	−0.630064	−0.315032	−	0.949081i	$-0.602015\pi$
$29$	−3.39300	−0.630064	−0.315032	+	0.949081i	$0.602015\pi$
$30$	0	0
$31$	−2.35553	−0.423065	−0.211533	−	0.977371i	$-0.567845\pi$
$31$	−2.35553	−0.423065	−0.211533	+	0.977371i	$0.567845\pi$
$32$	0	0
$33$	15.8132	2.75273
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.60498	0.428257	0.214128	−	0.976806i	$-0.431309\pi$
$37$	2.60498	0.428257	0.214128	+	0.976806i	$0.431309\pi$
$38$	0	0
$39$	15.8520	2.53836
$40$	0	0
$41$	5.70525	0.891010	0.445505	−	0.895280i	$-0.353024\pi$
$41$	5.70525	0.891010	0.445505	+	0.895280i	$0.353024\pi$
$42$	0	0
$43$	1.18641	0.180926	0.0904631	−	0.995900i	$-0.471165\pi$
$43$	1.18641	0.180926	0.0904631	+	0.995900i	$0.471165\pi$
$44$	0	0
$45$	−21.2662	−3.17019
$46$	0	0
$47$	−4.70250	−0.685930	−0.342965	−	0.939348i	$-0.611431\pi$
$47$	−4.70250	−0.685930	−0.342965	+	0.939348i	$0.611431\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	−3.08774	−0.432371
$52$	0	0
$53$	10.0116	1.37521	0.687603	−	0.726087i	$-0.258663\pi$
$53$	10.0116	1.37521	0.687603	+	0.726087i	$0.258663\pi$
$54$	0	0
$55$	16.6679	2.24750
$56$	0	0
$57$	−16.2207	−2.14849
$58$	0	0
$59$	13.0991	1.70536	0.852678	−	0.522436i	$-0.174977\pi$
$59$	13.0991	1.70536	0.852678	+	0.522436i	$0.174977\pi$
$60$	0	0
$61$	−1.78135	−0.228079	−0.114039	−	0.993476i	$-0.536379\pi$
$61$	−1.78135	−0.228079	−0.114039	+	0.993476i	$0.536379\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	16.7088	2.07247
$66$	0	0
$67$	−11.1779	−1.36559	−0.682796	−	0.730609i	$-0.739236\pi$
$67$	−11.1779	−1.36559	−0.682796	+	0.730609i	$0.739236\pi$
$68$	0	0
$69$	−21.5269	−2.59153
$70$	0	0
$71$	9.21864	1.09405	0.547026	−	0.837116i	$-0.315760\pi$
$71$	9.21864	1.09405	0.547026	+	0.837116i	$0.315760\pi$
$72$	0	0
$73$	−14.6608	−1.71591	−0.857957	−	0.513722i	$-0.828266\pi$
$73$	−14.6608	−1.71591	−0.857957	+	0.513722i	$0.828266\pi$
$74$	0	0
$75$	−17.2685	−1.99400
$76$	0	0
$77$	0	0
$78$	0	0
$79$	2.05802	0.231545	0.115773	−	0.993276i	$-0.463066\pi$
$79$	2.05802	0.231545	0.115773	+	0.993276i	$0.463066\pi$
$80$	0	0
$81$	14.0927	1.56586
$82$	0	0
$83$	15.7006	1.72336	0.861681	−	0.507450i	$-0.169412\pi$
$83$	15.7006	1.72336	0.861681	+	0.507450i	$0.169412\pi$
$84$	0	0
$85$	−3.25463	−0.353014
$86$	0	0
$87$	10.4767	1.12322
$88$	0	0
$89$	6.91321	0.732799	0.366399	−	0.930458i	$-0.380590\pi$
$89$	6.91321	0.732799	0.366399	+	0.930458i	$0.380590\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	7.27326	0.754202
$94$	0	0
$95$	−17.0974	−1.75416
$96$	0	0
$97$	−10.5244	−1.06859	−0.534295	−	0.845298i	$-0.679423\pi$
$97$	−10.5244	−1.06859	−0.534295	+	0.845298i	$0.679423\pi$
$98$	0	0
$99$	−33.4634	−3.36319
$100$	0	0
$101$	4.06349	0.404333	0.202166	−	0.979351i	$-0.435202\pi$
$101$	4.06349	0.404333	0.202166	+	0.979351i	$0.435202\pi$
$102$	0	0
$103$	14.0639	1.38576	0.692880	−	0.721053i	$-0.256342\pi$
$103$	14.0639	1.38576	0.692880	+	0.721053i	$0.256342\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	15.0690	1.45678	0.728388	−	0.685165i	$-0.240270\pi$
$107$	15.0690	1.45678	0.728388	+	0.685165i	$0.240270\pi$
$108$	0	0
$109$	2.50389	0.239829	0.119914	−	0.992784i	$-0.461738\pi$
$109$	2.50389	0.239829	0.119914	+	0.992784i	$0.461738\pi$
$110$	0	0
$111$	−8.04352	−0.763457
$112$	0	0
$113$	−16.3088	−1.53421	−0.767103	−	0.641524i	$-0.778302\pi$
$113$	−16.3088	−1.53421	−0.767103	+	0.641524i	$0.778302\pi$
$114$	0	0
$115$	−22.6903	−2.11588
$116$	0	0
$117$	−33.5455	−3.10128
$118$	0	0
$119$	0	0
$120$	0	0
$121$	15.2277	1.38433
$122$	0	0
$123$	−17.6163	−1.58841
$124$	0	0
$125$	−1.92869	−0.172507
$126$	0	0
$127$	−3.50639	−0.311142	−0.155571	−	0.987825i	$-0.549722\pi$
$127$	−3.50639	−0.311142	−0.155571	+	0.987825i	$0.549722\pi$
$128$	0	0
$129$	−3.66334	−0.322539
$130$	0	0
$131$	−9.11641	−0.796505	−0.398252	−	0.917276i	$-0.630383\pi$
$131$	−9.11641	−0.796505	−0.398252	+	0.917276i	$0.630383\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	35.5164	3.05676
$136$	0	0
$137$	−10.0279	−0.856738	−0.428369	−	0.903604i	$-0.640912\pi$
$137$	−10.0279	−0.856738	−0.428369	+	0.903604i	$0.640912\pi$
$138$	0	0
$139$	−10.1157	−0.858002	−0.429001	−	0.903304i	$-0.641134\pi$
$139$	−10.1157	−0.858002	−0.429001	+	0.903304i	$0.641134\pi$
$140$	0	0
$141$	14.5201	1.22281
$142$	0	0
$143$	26.2920	2.19865
$144$	0	0
$145$	11.0429	0.917067
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−13.5436	−1.10954	−0.554769	−	0.832005i	$-0.687193\pi$
$149$	−13.5436	−1.10954	−0.554769	+	0.832005i	$0.687193\pi$
$150$	0	0
$151$	−16.8102	−1.36799	−0.683997	−	0.729485i	$-0.739760\pi$
$151$	−16.8102	−1.36799	−0.683997	+	0.729485i	$0.739760\pi$
$152$	0	0
$153$	6.53416	0.528255
$154$	0	0
$155$	7.66636	0.615777
$156$	0	0
$157$	−17.7407	−1.41586	−0.707932	−	0.706280i	$-0.750372\pi$
$157$	−17.7407	−1.41586	−0.707932	+	0.706280i	$0.750372\pi$
$158$	0	0
$159$	−30.9134	−2.45159
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−16.3081	−1.27735	−0.638673	−	0.769478i	$-0.720516\pi$
$163$	−16.3081	−1.27735	−0.638673	+	0.769478i	$0.720516\pi$
$164$	0	0
$165$	−51.4662	−4.00664
$166$	0	0
$167$	22.3427	1.72893	0.864463	−	0.502696i	$-0.167658\pi$
$167$	22.3427	1.72893	0.864463	+	0.502696i	$0.167658\pi$
$168$	0	0
$169$	13.3565	1.02743
$170$	0	0
$171$	34.3256	2.62495
$172$	0	0
$173$	−5.13505	−0.390410	−0.195205	−	0.980762i	$-0.562537\pi$
$173$	−5.13505	−0.390410	−0.195205	+	0.980762i	$0.562537\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−40.4466	−3.04015
$178$	0	0
$179$	5.67240	0.423975	0.211987	−	0.977272i	$-0.432006\pi$
$179$	5.67240	0.423975	0.211987	+	0.977272i	$0.432006\pi$
$180$	0	0
$181$	11.7700	0.874858	0.437429	−	0.899253i	$-0.355889\pi$
$181$	11.7700	0.874858	0.437429	+	0.899253i	$0.355889\pi$
$182$	0	0
$183$	5.50036	0.406598
$184$	0	0
$185$	−8.47825	−0.623333
$186$	0	0
$187$	−5.12130	−0.374506
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−0.741528	−0.0536551	−0.0268275	−	0.999640i	$-0.508540\pi$
$191$	−0.741528	−0.0536551	−0.0268275	+	0.999640i	$0.508540\pi$
$192$	0	0
$193$	−11.7110	−0.842979	−0.421490	−	0.906833i	$-0.638493\pi$
$193$	−11.7110	−0.842979	−0.421490	+	0.906833i	$0.638493\pi$
$194$	0	0
$195$	−51.5925	−3.69462
$196$	0	0
$197$	−3.93898	−0.280641	−0.140320	−	0.990106i	$-0.544813\pi$
$197$	−3.93898	−0.280641	−0.140320	+	0.990106i	$0.544813\pi$
$198$	0	0
$199$	8.68257	0.615491	0.307746	−	0.951469i	$-0.400425\pi$
$199$	8.68257	0.615491	0.307746	+	0.951469i	$0.400425\pi$
$200$	0	0
$201$	34.5143	2.43445
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−18.5684	−1.29688
$206$	0	0
$207$	45.5543	3.16624
$208$	0	0
$209$	−26.9035	−1.86095
$210$	0	0
$211$	−9.05236	−0.623190	−0.311595	−	0.950215i	$-0.600863\pi$
$211$	−9.05236	−0.623190	−0.311595	+	0.950215i	$0.600863\pi$
$212$	0	0
$213$	−28.4648	−1.95038
$214$	0	0
$215$	−3.86133	−0.263340
$216$	0	0
$217$	0	0
$218$	0	0
$219$	45.2687	3.05897
$220$	0	0
$221$	−5.13386	−0.345341
$222$	0	0
$223$	12.2667	0.821442	0.410721	−	0.911761i	$-0.365277\pi$
$223$	12.2667	0.821442	0.410721	+	0.911761i	$0.365277\pi$
$224$	0	0
$225$	36.5429	2.43619
$226$	0	0
$227$	−17.1364	−1.13739	−0.568693	−	0.822550i	$-0.692551\pi$
$227$	−17.1364	−1.13739	−0.568693	+	0.822550i	$0.692551\pi$
$228$	0	0
$229$	−21.5237	−1.42232	−0.711162	−	0.703028i	$-0.751831\pi$
$229$	−21.5237	−1.42232	−0.711162	+	0.703028i	$0.751831\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−19.2290	−1.25974	−0.629868	−	0.776702i	$-0.716891\pi$
$233$	−19.2290	−1.25974	−0.629868	+	0.776702i	$0.716891\pi$
$234$	0	0
$235$	15.3049	0.998381
$236$	0	0
$237$	−6.35463	−0.412778
$238$	0	0
$239$	8.19767	0.530263	0.265132	−	0.964212i	$-0.414585\pi$
$239$	8.19767	0.530263	0.265132	+	0.964212i	$0.414585\pi$
$240$	0	0
$241$	25.4006	1.63620	0.818099	−	0.575078i	$-0.195028\pi$
$241$	25.4006	1.63620	0.818099	+	0.575078i	$0.195028\pi$
$242$	0	0
$243$	−10.7771	−0.691349
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−26.9695	−1.71603
$248$	0	0
$249$	−48.4794	−3.07226
$250$	0	0
$251$	−4.15836	−0.262473	−0.131237	−	0.991351i	$-0.541895\pi$
$251$	−4.15836	−0.262473	−0.131237	+	0.991351i	$0.541895\pi$
$252$	0	0
$253$	−35.7042	−2.24470
$254$	0	0
$255$	10.0495	0.629321
$256$	0	0
$257$	−17.6639	−1.10185	−0.550923	−	0.834556i	$-0.685724\pi$
$257$	−17.6639	−1.10185	−0.550923	+	0.834556i	$0.685724\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−22.1704	−1.37231
$262$	0	0
$263$	−24.6955	−1.52279	−0.761395	−	0.648288i	$-0.775485\pi$
$263$	−24.6955	−1.52279	−0.761395	+	0.648288i	$0.775485\pi$
$264$	0	0
$265$	−32.5842	−2.00163
$266$	0	0
$267$	−21.3462	−1.30637
$268$	0	0
$269$	−14.7507	−0.899365	−0.449683	−	0.893188i	$-0.648463\pi$
$269$	−14.7507	−0.899365	−0.449683	+	0.893188i	$0.648463\pi$
$270$	0	0
$271$	−8.00715	−0.486400	−0.243200	−	0.969976i	$-0.578197\pi$
$271$	−8.00715	−0.486400	−0.243200	+	0.969976i	$0.578197\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−28.6414	−1.72714
$276$	0	0
$277$	−6.21306	−0.373307	−0.186653	−	0.982426i	$-0.559764\pi$
$277$	−6.21306	−0.373307	−0.186653	+	0.982426i	$0.559764\pi$
$278$	0	0
$279$	−15.3914	−0.921458
$280$	0	0
$281$	−11.0264	−0.657777	−0.328889	−	0.944369i	$-0.606674\pi$
$281$	−11.0264	−0.657777	−0.328889	+	0.944369i	$0.606674\pi$
$282$	0	0
$283$	16.8967	1.00440	0.502201	−	0.864751i	$-0.332524\pi$
$283$	16.8967	1.00440	0.502201	+	0.864751i	$0.332524\pi$
$284$	0	0
$285$	52.7924	3.12715
$286$	0	0
$287$	0	0
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	32.4966	1.90498
$292$	0	0
$293$	22.3788	1.30738	0.653692	−	0.756761i	$-0.273219\pi$
$293$	22.3788	1.30738	0.653692	+	0.756761i	$0.273219\pi$
$294$	0	0
$295$	−42.6327	−2.48217
$296$	0	0
$297$	55.8865	3.24286
$298$	0	0
$299$	−35.7918	−2.06989
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−12.5470	−0.720808
$304$	0	0
$305$	5.79764	0.331972
$306$	0	0
$307$	−2.01060	−0.114751	−0.0573755	−	0.998353i	$-0.518273\pi$
$307$	−2.01060	−0.114751	−0.0573755	+	0.998353i	$0.518273\pi$
$308$	0	0
$309$	−43.4258	−2.47041
$310$	0	0
$311$	0.0533120	0.00302305	0.00151152	−	0.999999i	$-0.499519\pi$
$311$	0.0533120	0.00302305	0.00151152	+	0.999999i	$0.499519\pi$
$312$	0	0
$313$	−17.0370	−0.962988	−0.481494	−	0.876449i	$-0.659906\pi$
$313$	−17.0370	−0.962988	−0.481494	+	0.876449i	$0.659906\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	29.5722	1.66094	0.830471	−	0.557062i	$-0.188071\pi$
$317$	29.5722	1.66094	0.830471	+	0.557062i	$0.188071\pi$
$318$	0	0
$319$	17.3765	0.972900
$320$	0	0
$321$	−46.5292	−2.59701
$322$	0	0
$323$	5.25326	0.292299
$324$	0	0
$325$	−28.7116	−1.59263
$326$	0	0
$327$	−7.73136	−0.427545
$328$	0	0
$329$	0	0
$330$	0	0
$331$	25.0575	1.37729	0.688643	−	0.725101i	$-0.258207\pi$
$331$	25.0575	1.37729	0.688643	+	0.725101i	$0.258207\pi$
$332$	0	0
$333$	17.0214	0.932766
$334$	0	0
$335$	36.3798	1.98764
$336$	0	0
$337$	0.187963	0.0102390	0.00511949	−	0.999987i	$-0.498370\pi$
$337$	0.187963	0.0102390	0.00511949	+	0.999987i	$0.498370\pi$
$338$	0	0
$339$	50.3575	2.73504
$340$	0	0
$341$	12.0634	0.653267
$342$	0	0
$343$	0	0
$344$	0	0
$345$	70.0619	3.77201
$346$	0	0
$347$	7.39627	0.397053	0.198526	−	0.980096i	$-0.436384\pi$
$347$	7.39627	0.397053	0.198526	+	0.980096i	$0.436384\pi$
$348$	0	0
$349$	30.1736	1.61516	0.807579	−	0.589760i	$-0.200778\pi$
$349$	30.1736	1.61516	0.807579	+	0.589760i	$0.200778\pi$
$350$	0	0
$351$	56.0236	2.99032
$352$	0	0
$353$	13.5872	0.723174	0.361587	−	0.932338i	$-0.382235\pi$
$353$	13.5872	0.723174	0.361587	+	0.932338i	$0.382235\pi$
$354$	0	0
$355$	−30.0032	−1.59241
$356$	0	0
$357$	0	0
$358$	0	0
$359$	26.6480	1.40643	0.703214	−	0.710978i	$-0.251748\pi$
$359$	26.6480	1.40643	0.703214	+	0.710978i	$0.251748\pi$
$360$	0	0
$361$	8.59676	0.452461
$362$	0	0
$363$	−47.0192	−2.46787
$364$	0	0
$365$	47.7153	2.49753
$366$	0	0
$367$	25.0287	1.30649	0.653244	−	0.757147i	$-0.273408\pi$
$367$	25.0287	1.30649	0.653244	+	0.757147i	$0.273408\pi$
$368$	0	0
$369$	37.2790	1.94067
$370$	0	0
$371$	0	0
$372$	0	0
$373$	17.6096	0.911790	0.455895	−	0.890034i	$-0.349319\pi$
$373$	17.6096	0.911790	0.455895	+	0.890034i	$0.349319\pi$
$374$	0	0
$375$	5.95529	0.307530
$376$	0	0
$377$	17.4192	0.897134
$378$	0	0
$379$	−11.6143	−0.596589	−0.298295	−	0.954474i	$-0.596418\pi$
$379$	−11.6143	−0.596589	−0.298295	+	0.954474i	$0.596418\pi$
$380$	0	0
$381$	10.8268	0.554675
$382$	0	0
$383$	2.89518	0.147937	0.0739684	−	0.997261i	$-0.476434\pi$
$383$	2.89518	0.147937	0.0739684	+	0.997261i	$0.476434\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	7.75220	0.394067
$388$	0	0
$389$	26.7804	1.35782	0.678910	−	0.734221i	$-0.262452\pi$
$389$	26.7804	1.35782	0.678910	+	0.734221i	$0.262452\pi$
$390$	0	0
$391$	6.97171	0.352575
$392$	0	0
$393$	28.1491	1.41994
$394$	0	0
$395$	−6.69809	−0.337017
$396$	0	0
$397$	−27.1376	−1.36200	−0.680998	−	0.732285i	$-0.738454\pi$
$397$	−27.1376	−1.36200	−0.680998	+	0.732285i	$0.738454\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	13.9281	0.695534	0.347767	−	0.937581i	$-0.386940\pi$
$401$	13.9281	0.695534	0.347767	+	0.937581i	$0.386940\pi$
$402$	0	0
$403$	12.0930	0.602393
$404$	0	0
$405$	−45.8666	−2.27913
$406$	0	0
$407$	−13.3409	−0.661284
$408$	0	0
$409$	−10.3532	−0.511933	−0.255967	−	0.966686i	$-0.582394\pi$
$409$	−10.3532	−0.511933	−0.255967	+	0.966686i	$0.582394\pi$
$410$	0	0
$411$	30.9635	1.52732
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−51.0995	−2.50838
$416$	0	0
$417$	31.2347	1.52957
$418$	0	0
$419$	3.50968	0.171459	0.0857294	−	0.996318i	$-0.472678\pi$
$419$	3.50968	0.171459	0.0857294	+	0.996318i	$0.472678\pi$
$420$	0	0
$421$	10.5824	0.515755	0.257877	−	0.966178i	$-0.416977\pi$
$421$	10.5824	0.515755	0.257877	+	0.966178i	$0.416977\pi$
$422$	0	0
$423$	−30.7269	−1.49399
$424$	0	0
$425$	5.59260	0.271281
$426$	0	0
$427$	0	0
$428$	0	0
$429$	−81.1830	−3.91955
$430$	0	0
$431$	−0.665241	−0.0320435	−0.0160218	−	0.999872i	$-0.505100\pi$
$431$	−0.665241	−0.0320435	−0.0160218	+	0.999872i	$0.505100\pi$
$432$	0	0
$433$	−10.1221	−0.486435	−0.243217	−	0.969972i	$-0.578203\pi$
$433$	−10.1221	−0.486435	−0.243217	+	0.969972i	$0.578203\pi$
$434$	0	0
$435$	−34.0978	−1.63486
$436$	0	0
$437$	36.6242	1.75197
$438$	0	0
$439$	−15.6542	−0.747134	−0.373567	−	0.927603i	$-0.621865\pi$
$439$	−15.6542	−0.747134	−0.373567	+	0.927603i	$0.621865\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−34.1217	−1.62117	−0.810585	−	0.585621i	$-0.800851\pi$
$443$	−34.1217	−1.62117	−0.810585	+	0.585621i	$0.800851\pi$
$444$	0	0
$445$	−22.4999	−1.06660
$446$	0	0
$447$	41.8192	1.97798
$448$	0	0
$449$	9.06845	0.427967	0.213983	−	0.976837i	$-0.431356\pi$
$449$	9.06845	0.427967	0.213983	+	0.976837i	$0.431356\pi$
$450$	0	0
$451$	−29.2183	−1.37583
$452$	0	0
$453$	51.9056	2.43874
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−23.7883	−1.11277	−0.556384	−	0.830925i	$-0.687812\pi$
$457$	−23.7883	−1.11277	−0.556384	+	0.830925i	$0.687812\pi$
$458$	0	0
$459$	−10.9126	−0.509355
$460$	0	0
$461$	21.4966	1.00119	0.500597	−	0.865680i	$-0.333114\pi$
$461$	21.4966	1.00119	0.500597	+	0.865680i	$0.333114\pi$
$462$	0	0
$463$	−2.55322	−0.118658	−0.0593292	−	0.998238i	$-0.518896\pi$
$463$	−2.55322	−0.118658	−0.0593292	+	0.998238i	$0.518896\pi$
$464$	0	0
$465$	−23.6718	−1.09775
$466$	0	0
$467$	20.1634	0.933053	0.466526	−	0.884507i	$-0.345505\pi$
$467$	20.1634	0.933053	0.466526	+	0.884507i	$0.345505\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	54.7788	2.52407
$472$	0	0
$473$	−6.07597	−0.279373
$474$	0	0
$475$	29.3794	1.34802
$476$	0	0
$477$	65.4177	2.99527
$478$	0	0
$479$	−17.4499	−0.797305	−0.398652	−	0.917102i	$-0.630522\pi$
$479$	−17.4499	−0.797305	−0.398652	+	0.917102i	$0.630522\pi$
$480$	0	0
$481$	−13.3736	−0.609785
$482$	0	0
$483$	0	0
$484$	0	0
$485$	34.2529	1.55535
$486$	0	0
$487$	39.1143	1.77244	0.886219	−	0.463266i	$-0.153323\pi$
$487$	39.1143	1.77244	0.886219	+	0.463266i	$0.153323\pi$
$488$	0	0
$489$	50.3551	2.27714
$490$	0	0
$491$	21.3590	0.963917	0.481958	−	0.876194i	$-0.339926\pi$
$491$	21.3590	0.963917	0.481958	+	0.876194i	$0.339926\pi$
$492$	0	0
$493$	−3.39300	−0.152813
$494$	0	0
$495$	108.911	4.89517
$496$	0	0
$497$	0	0
$498$	0	0
$499$	31.4317	1.40708	0.703538	−	0.710657i	$-0.251602\pi$
$499$	31.4317	1.40708	0.703538	+	0.710657i	$0.251602\pi$
$500$	0	0
$501$	−68.9884	−3.08217
$502$	0	0
$503$	−20.1087	−0.896603	−0.448302	−	0.893882i	$-0.647971\pi$
$503$	−20.1087	−0.896603	−0.448302	+	0.893882i	$0.647971\pi$
$504$	0	0
$505$	−13.2252	−0.588512
$506$	0	0
$507$	−41.2416	−1.83160
$508$	0	0
$509$	−17.0898	−0.757490	−0.378745	−	0.925501i	$-0.623644\pi$
$509$	−17.0898	−0.757490	−0.378745	+	0.925501i	$0.623644\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−57.3266	−2.53103
$514$	0	0
$515$	−45.7728	−2.01699
$516$	0	0
$517$	24.0829	1.05916
$518$	0	0
$519$	15.8557	0.695988
$520$	0	0
$521$	7.41261	0.324752	0.162376	−	0.986729i	$-0.448084\pi$
$521$	7.41261	0.324752	0.162376	+	0.986729i	$0.448084\pi$
$522$	0	0
$523$	0.240819	0.0105303	0.00526515	−	0.999986i	$-0.498324\pi$
$523$	0.240819	0.0105303	0.00526515	+	0.999986i	$0.498324\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2.35553	−0.102608
$528$	0	0
$529$	25.6048	1.11325
$530$	0	0
$531$	85.5915	3.71436
$532$	0	0
$533$	−29.2899	−1.26869
$534$	0	0
$535$	−49.0440	−2.12036
$536$	0	0
$537$	−17.5149	−0.755824
$538$	0	0
$539$	0	0
$540$	0	0
$541$	26.9932	1.16053	0.580265	−	0.814428i	$-0.302949\pi$
$541$	26.9932	1.16053	0.580265	+	0.814428i	$0.302949\pi$
$542$	0	0
$543$	−36.3428	−1.55962
$544$	0	0
$545$	−8.14922	−0.349074
$546$	0	0
$547$	20.0657	0.857947	0.428973	−	0.903317i	$-0.358875\pi$
$547$	20.0657	0.857947	0.428973	+	0.903317i	$0.358875\pi$
$548$	0	0
$549$	−11.6396	−0.496768
$550$	0	0
$551$	−17.8243	−0.759341
$552$	0	0
$553$	0	0
$554$	0	0
$555$	26.1787	1.11122
$556$	0	0
$557$	2.84986	0.120753	0.0603763	−	0.998176i	$-0.480770\pi$
$557$	2.84986	0.120753	0.0603763	+	0.998176i	$0.480770\pi$
$558$	0	0
$559$	−6.09088	−0.257617
$560$	0	0
$561$	15.8132	0.667636
$562$	0	0
$563$	−29.4232	−1.24004	−0.620021	−	0.784586i	$-0.712876\pi$
$563$	−29.4232	−1.24004	−0.620021	+	0.784586i	$0.712876\pi$
$564$	0	0
$565$	53.0792	2.23306
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−35.4141	−1.48463	−0.742317	−	0.670048i	$-0.766273\pi$
$569$	−35.4141	−1.48463	−0.742317	+	0.670048i	$0.766273\pi$
$570$	0	0
$571$	−25.5965	−1.07118	−0.535590	−	0.844478i	$-0.679911\pi$
$571$	−25.5965	−1.07118	−0.535590	+	0.844478i	$0.679911\pi$
$572$	0	0
$573$	2.28965	0.0956514
$574$	0	0
$575$	38.9900	1.62599
$576$	0	0
$577$	−39.7244	−1.65375	−0.826874	−	0.562388i	$-0.809883\pi$
$577$	−39.7244	−1.65375	−0.826874	+	0.562388i	$0.809883\pi$
$578$	0	0
$579$	36.1607	1.50279
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−51.2726	−2.12349
$584$	0	0
$585$	109.178	4.51395
$586$	0	0
$587$	0.00154296	6.36846e−5 0	3.18423e−5	−	1.00000i	$-0.499990\pi$
$587$	0.00154296	6.36846e−5 0	3.18423e−5		1.00000i	$0.499990\pi$
$588$	0	0
$589$	−12.3742	−0.509870
$590$	0	0
$591$	12.1626	0.500301
$592$	0	0
$593$	22.3965	0.919714	0.459857	−	0.887993i	$-0.347901\pi$
$593$	22.3965	0.919714	0.459857	+	0.887993i	$0.347901\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−26.8096	−1.09724
$598$	0	0
$599$	10.8107	0.441713	0.220857	−	0.975306i	$-0.429115\pi$
$599$	10.8107	0.441713	0.220857	+	0.975306i	$0.429115\pi$
$600$	0	0
$601$	−19.5231	−0.796364	−0.398182	−	0.917306i	$-0.630359\pi$
$601$	−19.5231	−0.796364	−0.398182	+	0.917306i	$0.630359\pi$
$602$	0	0
$603$	−73.0379	−2.97433
$604$	0	0
$605$	−49.5604	−2.01492
$606$	0	0
$607$	−14.1715	−0.575203	−0.287601	−	0.957750i	$-0.592858\pi$
$607$	−14.1715	−0.575203	−0.287601	+	0.957750i	$0.592858\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	24.1420	0.976681
$612$	0	0
$613$	−7.61930	−0.307741	−0.153870	−	0.988091i	$-0.549174\pi$
$613$	−7.61930	−0.307741	−0.153870	+	0.988091i	$0.549174\pi$
$614$	0	0
$615$	57.3346	2.31195
$616$	0	0
$617$	−4.52135	−0.182023	−0.0910113	−	0.995850i	$-0.529010\pi$
$617$	−4.52135	−0.182023	−0.0910113	+	0.995850i	$0.529010\pi$
$618$	0	0
$619$	27.8625	1.11989	0.559943	−	0.828531i	$-0.310823\pi$
$619$	27.8625	1.11989	0.559943	+	0.828531i	$0.310823\pi$
$620$	0	0
$621$	−76.0793	−3.05296
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−21.6858	−0.867433
$626$	0	0
$627$	83.0711	3.31754
$628$	0	0
$629$	2.60498	0.103868
$630$	0	0
$631$	33.1123	1.31818	0.659090	−	0.752064i	$-0.270942\pi$
$631$	33.1123	1.31818	0.659090	+	0.752064i	$0.270942\pi$
$632$	0	0
$633$	27.9513	1.11097
$634$	0	0
$635$	11.4120	0.452871
$636$	0	0
$637$	0	0
$638$	0	0
$639$	60.2361	2.38290
$640$	0	0
$641$	−20.2853	−0.801221	−0.400610	−	0.916249i	$-0.631202\pi$
$641$	−20.2853	−0.801221	−0.400610	+	0.916249i	$0.631202\pi$
$642$	0	0
$643$	−11.6066	−0.457718	−0.228859	−	0.973460i	$-0.573500\pi$
$643$	−11.6066	−0.457718	−0.228859	+	0.973460i	$0.573500\pi$
$644$	0	0
$645$	11.9228	0.469459
$646$	0	0
$647$	−17.4179	−0.684770	−0.342385	−	0.939560i	$-0.611235\pi$
$647$	−17.4179	−0.684770	−0.342385	+	0.939560i	$0.611235\pi$
$648$	0	0
$649$	−67.0843	−2.63329
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−31.1391	−1.21857	−0.609284	−	0.792952i	$-0.708543\pi$
$653$	−31.1391	−1.21857	−0.609284	+	0.792952i	$0.708543\pi$
$654$	0	0
$655$	29.6705	1.15932
$656$	0	0
$657$	−95.7958	−3.73735
$658$	0	0
$659$	−32.6601	−1.27226	−0.636128	−	0.771583i	$-0.719465\pi$
$659$	−32.6601	−1.27226	−0.636128	+	0.771583i	$0.719465\pi$
$660$	0	0
$661$	19.1904	0.746420	0.373210	−	0.927747i	$-0.378257\pi$
$661$	19.1904	0.746420	0.373210	+	0.927747i	$0.378257\pi$
$662$	0	0
$663$	15.8520	0.615643
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−23.6550	−0.915925
$668$	0	0
$669$	−37.8766	−1.46439
$670$	0	0
$671$	9.12283	0.352183
$672$	0	0
$673$	5.61344	0.216382	0.108191	−	0.994130i	$-0.465494\pi$
$673$	5.61344	0.216382	0.108191	+	0.994130i	$0.465494\pi$
$674$	0	0
$675$	−61.0296	−2.34903
$676$	0	0
$677$	−37.4047	−1.43758	−0.718789	−	0.695228i	$-0.755303\pi$
$677$	−37.4047	−1.43758	−0.718789	+	0.695228i	$0.755303\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	52.9129	2.02763
$682$	0	0
$683$	−22.0019	−0.841880	−0.420940	−	0.907089i	$-0.638300\pi$
$683$	−22.0019	−0.841880	−0.420940	+	0.907089i	$0.638300\pi$
$684$	0	0
$685$	32.6370	1.24699
$686$	0	0
$687$	66.4596	2.53559
$688$	0	0
$689$	−51.3984	−1.95812
$690$	0	0
$691$	25.7800	0.980717	0.490359	−	0.871521i	$-0.336866\pi$
$691$	25.7800	0.980717	0.490359	+	0.871521i	$0.336866\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	32.9228	1.24883
$696$	0	0
$697$	5.70525	0.216102
$698$	0	0
$699$	59.3744	2.24574
$700$	0	0
$701$	8.82347	0.333258	0.166629	−	0.986020i	$-0.446712\pi$
$701$	8.82347	0.333258	0.166629	+	0.986020i	$0.446712\pi$
$702$	0	0
$703$	13.6847	0.516127
$704$	0	0
$705$	−47.2576	−1.77982
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−11.4925	−0.431609	−0.215805	−	0.976437i	$-0.569237\pi$
$709$	−11.4925	−0.431609	−0.215805	+	0.976437i	$0.569237\pi$
$710$	0	0
$711$	13.4474	0.504318
$712$	0	0
$713$	−16.4221	−0.615011
$714$	0	0
$715$	−85.5708	−3.20017
$716$	0	0
$717$	−25.3123	−0.945305
$718$	0	0
$719$	−9.74362	−0.363376	−0.181688	−	0.983356i	$-0.558156\pi$
$719$	−9.74362	−0.363376	−0.181688	+	0.983356i	$0.558156\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−78.4306	−2.91686
$724$	0	0
$725$	−18.9757	−0.704739
$726$	0	0
$727$	−22.2530	−0.825316	−0.412658	−	0.910886i	$-0.635400\pi$
$727$	−22.2530	−0.825316	−0.412658	+	0.910886i	$0.635400\pi$
$728$	0	0
$729$	−9.00142	−0.333386
$730$	0	0
$731$	1.18641	0.0438810
$732$	0	0
$733$	−16.2214	−0.599150	−0.299575	−	0.954073i	$-0.596845\pi$
$733$	−16.2214	−0.599150	−0.299575	+	0.954073i	$0.596845\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	57.2451	2.10865
$738$	0	0
$739$	27.6708	1.01789	0.508943	−	0.860800i	$-0.330036\pi$
$739$	27.6708	1.01789	0.508943	+	0.860800i	$0.330036\pi$
$740$	0	0
$741$	83.2750	3.05918
$742$	0	0
$743$	−16.5594	−0.607507	−0.303754	−	0.952751i	$-0.598240\pi$
$743$	−16.5594	−0.607507	−0.303754	+	0.952751i	$0.598240\pi$
$744$	0	0
$745$	44.0795	1.61495
$746$	0	0
$747$	102.590	3.75358
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−31.2453	−1.14016	−0.570078	−	0.821591i	$-0.693087\pi$
$751$	−31.2453	−1.14016	−0.570078	+	0.821591i	$0.693087\pi$
$752$	0	0
$753$	12.8399	0.467913
$754$	0	0
$755$	54.7109	1.99113
$756$	0	0
$757$	−11.7582	−0.427359	−0.213680	−	0.976904i	$-0.568545\pi$
$757$	−11.7582	−0.427359	−0.213680	+	0.976904i	$0.568545\pi$
$758$	0	0
$759$	110.245	4.00165
$760$	0	0
$761$	4.17953	0.151508	0.0757540	−	0.997127i	$-0.475864\pi$
$761$	4.17953	0.151508	0.0757540	+	0.997127i	$0.475864\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−21.2662	−0.768883
$766$	0	0
$767$	−67.2489	−2.42822
$768$	0	0
$769$	−0.863383	−0.0311344	−0.0155672	−	0.999879i	$-0.504955\pi$
$769$	−0.863383	−0.0311344	−0.0155672	+	0.999879i	$0.504955\pi$
$770$	0	0
$771$	54.5417	1.96427
$772$	0	0
$773$	−38.0952	−1.37019	−0.685094	−	0.728454i	$-0.740239\pi$
$773$	−38.0952	−1.37019	−0.685094	+	0.728454i	$0.740239\pi$
$774$	0	0
$775$	−13.1735	−0.473207
$776$	0	0
$777$	0	0
$778$	0	0
$779$	29.9711	1.07383
$780$	0	0
$781$	−47.2114	−1.68936
$782$	0	0
$783$	37.0263	1.32321
$784$	0	0
$785$	57.7395	2.06081
$786$	0	0
$787$	−48.9204	−1.74383	−0.871913	−	0.489661i	$-0.837120\pi$
$787$	−48.9204	−1.74383	−0.871913	+	0.489661i	$0.837120\pi$
$788$	0	0
$789$	76.2534	2.71469
$790$	0	0
$791$	0	0
$792$	0	0
$793$	9.14522	0.324756
$794$	0	0
$795$	100.612	3.56833
$796$	0	0
$797$	−41.4404	−1.46789	−0.733946	−	0.679208i	$-0.762324\pi$
$797$	−41.4404	−1.46789	−0.733946	+	0.679208i	$0.762324\pi$
$798$	0	0
$799$	−4.70250	−0.166363
$800$	0	0
$801$	45.1720	1.59607
$802$	0	0
$803$	75.0821	2.64959
$804$	0	0
$805$	0	0
$806$	0	0
$807$	45.5463	1.60331
$808$	0	0
$809$	12.3338	0.433633	0.216816	−	0.976212i	$-0.430433\pi$
$809$	12.3338	0.433633	0.216816	+	0.976212i	$0.430433\pi$
$810$	0	0
$811$	6.92765	0.243263	0.121631	−	0.992575i	$-0.461187\pi$
$811$	6.92765	0.243263	0.121631	+	0.992575i	$0.461187\pi$
$812$	0	0
$813$	24.7240	0.867109
$814$	0	0
$815$	53.0767	1.85920
$816$	0	0
$817$	6.23253	0.218049
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−6.22857	−0.217379	−0.108689	−	0.994076i	$-0.534665\pi$
$821$	−6.22857	−0.217379	−0.108689	+	0.994076i	$0.534665\pi$
$822$	0	0
$823$	25.2597	0.880497	0.440249	−	0.897876i	$-0.354890\pi$
$823$	25.2597	0.880497	0.440249	+	0.897876i	$0.354890\pi$
$824$	0	0
$825$	88.4371	3.07899
$826$	0	0
$827$	−21.2451	−0.738764	−0.369382	−	0.929278i	$-0.620431\pi$
$827$	−21.2451	−0.738764	−0.369382	+	0.929278i	$0.620431\pi$
$828$	0	0
$829$	−15.9143	−0.552725	−0.276362	−	0.961053i	$-0.589129\pi$
$829$	−15.9143	−0.552725	−0.276362	+	0.961053i	$0.589129\pi$
$830$	0	0
$831$	19.1843	0.665497
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−72.7170	−2.51648
$836$	0	0
$837$	25.7049	0.888490
$838$	0	0
$839$	4.58809	0.158398	0.0791992	−	0.996859i	$-0.474764\pi$
$839$	4.58809	0.158398	0.0791992	+	0.996859i	$0.474764\pi$
$840$	0	0
$841$	−17.4876	−0.603020
$842$	0	0
$843$	34.0466	1.17263
$844$	0	0
$845$	−43.4706	−1.49543
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−52.1725	−1.79056
$850$	0	0
$851$	18.1612	0.622558
$852$	0	0
$853$	40.8442	1.39848	0.699240	−	0.714887i	$-0.253522\pi$
$853$	40.8442	1.39848	0.699240	+	0.714887i	$0.253522\pi$
$854$	0	0
$855$	−111.717	−3.82065
$856$	0	0
$857$	0.0503896	0.00172128	0.000860638	−	1.00000i	$-0.499726\pi$
$857$	0.0503896	0.00172128	0.000860638		1.00000i	$0.499726\pi$
$858$	0	0
$859$	−14.7256	−0.502431	−0.251215	−	0.967931i	$-0.580830\pi$
$859$	−14.7256	−0.502431	−0.251215	+	0.967931i	$0.580830\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−25.1275	−0.855349	−0.427675	−	0.903933i	$-0.640667\pi$
$863$	−25.1275	−0.855349	−0.427675	+	0.903933i	$0.640667\pi$
$864$	0	0
$865$	16.7127	0.568247
$866$	0	0
$867$	−3.08774	−0.104865
$868$	0	0
$869$	−10.5397	−0.357536
$870$	0	0
$871$	57.3856	1.94444
$872$	0	0
$873$	−68.7680	−2.32744
$874$	0	0
$875$	0	0
$876$	0	0
$877$	19.1059	0.645161	0.322581	−	0.946542i	$-0.395450\pi$
$877$	19.1059	0.645161	0.322581	+	0.946542i	$0.395450\pi$
$878$	0	0
$879$	−69.1000	−2.33069
$880$	0	0
$881$	−22.2649	−0.750124	−0.375062	−	0.927000i	$-0.622379\pi$
$881$	−22.2649	−0.750124	−0.375062	+	0.927000i	$0.622379\pi$
$882$	0	0
$883$	20.2839	0.682607	0.341303	−	0.939953i	$-0.389132\pi$
$883$	20.2839	0.682607	0.341303	+	0.939953i	$0.389132\pi$
$884$	0	0
$885$	131.639	4.42499
$886$	0	0
$887$	39.3402	1.32092	0.660458	−	0.750863i	$-0.270362\pi$
$887$	39.3402	1.32092	0.660458	+	0.750863i	$0.270362\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−72.1731	−2.41789
$892$	0	0
$893$	−24.7035	−0.826670
$894$	0	0
$895$	−18.4615	−0.617101
$896$	0	0
$897$	110.516	3.69002
$898$	0	0
$899$	7.99230	0.266558
$900$	0	0
$901$	10.0116	0.333536
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−38.3070	−1.27337
$906$	0	0
$907$	−17.2290	−0.572080	−0.286040	−	0.958218i	$-0.592339\pi$
$907$	−17.2290	−0.572080	−0.286040	+	0.958218i	$0.592339\pi$
$908$	0	0
$909$	26.5515	0.880658
$910$	0	0
$911$	−27.5456	−0.912628	−0.456314	−	0.889819i	$-0.650831\pi$
$911$	−27.5456	−0.912628	−0.456314	+	0.889819i	$0.650831\pi$
$912$	0	0
$913$	−80.4073	−2.66109
$914$	0	0
$915$	−17.9016	−0.591809
$916$	0	0
$917$	0	0
$918$	0	0
$919$	4.05113	0.133634	0.0668172	−	0.997765i	$-0.478716\pi$
$919$	4.05113	0.133634	0.0668172	+	0.997765i	$0.478716\pi$
$920$	0	0
$921$	6.20821	0.204568
$922$	0	0
$923$	−47.3272	−1.55780
$924$	0	0
$925$	14.5686	0.479014
$926$	0	0
$927$	91.8959	3.01826
$928$	0	0
$929$	41.3767	1.35753	0.678763	−	0.734358i	$-0.262516\pi$
$929$	41.3767	1.35753	0.678763	+	0.734358i	$0.262516\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−0.164614	−0.00538922
$934$	0	0
$935$	16.6679	0.545099
$936$	0	0
$937$	−9.85113	−0.321823	−0.160911	−	0.986969i	$-0.551443\pi$
$937$	−9.85113	−0.321823	−0.160911	+	0.986969i	$0.551443\pi$
$938$	0	0
$939$	52.6059	1.71673
$940$	0	0
$941$	31.0786	1.01313	0.506567	−	0.862200i	$-0.330914\pi$
$941$	31.0786	1.01313	0.506567	+	0.862200i	$0.330914\pi$
$942$	0	0
$943$	39.7753	1.29526
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−38.8811	−1.26347	−0.631733	−	0.775186i	$-0.717656\pi$
$947$	−38.8811	−1.26347	−0.631733	+	0.775186i	$0.717656\pi$
$948$	0	0
$949$	75.2664	2.44325
$950$	0	0
$951$	−91.3115	−2.96098
$952$	0	0
$953$	56.4465	1.82848	0.914242	−	0.405170i	$-0.132788\pi$
$953$	56.4465	1.82848	0.914242	+	0.405170i	$0.132788\pi$
$954$	0	0
$955$	2.41340	0.0780957
$956$	0	0
$957$	−53.6543	−1.73440
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−25.4515	−0.821016
$962$	0	0
$963$	98.4632	3.17293
$964$	0	0
$965$	38.1151	1.22697
$966$	0	0
$967$	−14.1217	−0.454122	−0.227061	−	0.973881i	$-0.572912\pi$
$967$	−14.1217	−0.454122	−0.227061	+	0.973881i	$0.572912\pi$
$968$	0	0
$969$	−16.2207	−0.521085
$970$	0	0
$971$	−12.5180	−0.401723	−0.200861	−	0.979620i	$-0.564374\pi$
$971$	−12.5180	−0.401723	−0.200861	+	0.979620i	$0.564374\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	88.6541	2.83920
$976$	0	0
$977$	−26.0496	−0.833402	−0.416701	−	0.909044i	$-0.636814\pi$
$977$	−26.0496	−0.833402	−0.416701	+	0.909044i	$0.636814\pi$
$978$	0	0
$979$	−35.4046	−1.13154
$980$	0	0
$981$	16.3608	0.522360
$982$	0	0
$983$	13.4621	0.429374	0.214687	−	0.976683i	$-0.431127\pi$
$983$	13.4621	0.429374	0.214687	+	0.976683i	$0.431127\pi$
$984$	0	0
$985$	12.8199	0.408476
$986$	0	0
$987$	0	0
$988$	0	0
$989$	8.27132	0.263013
$990$	0	0
$991$	−20.8991	−0.663882	−0.331941	−	0.943300i	$-0.607703\pi$
$991$	−20.8991	−0.663882	−0.331941	+	0.943300i	$0.607703\pi$
$992$	0	0
$993$	−77.3712	−2.45530
$994$	0	0
$995$	−28.2585	−0.895856
$996$	0	0
$997$	−35.9728	−1.13927	−0.569636	−	0.821897i	$-0.692916\pi$
$997$	−35.9728	−1.13927	−0.569636	+	0.821897i	$0.692916\pi$
$998$	0	0
$999$	−28.4271	−0.899393

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3332.2.a.t.1.2	✓	8
7.6	odd	2		3332.2.a.u.1.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3332.2.a.t.1.2	✓	8	1.1	even	1	trivial
3332.2.a.u.1.7	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)

\(f(q)\)	\(=\)	\(q-3.08774 q^{3} -3.25463 q^{5} +6.53416 q^{9} +O(q^{10})\)	\(q-3.08774 q^{3} -3.25463 q^{5} +6.53416 q^{9} -5.12130 q^{11} -5.13386 q^{13} +10.0495 q^{15} +1.00000 q^{17} +5.25326 q^{19} +6.97171 q^{23} +5.59260 q^{25} -10.9126 q^{27} -3.39300 q^{29} -2.35553 q^{31} +15.8132 q^{33} +2.60498 q^{37} +15.8520 q^{39} +5.70525 q^{41} +1.18641 q^{43} -21.2662 q^{45} -4.70250 q^{47} -3.08774 q^{51} +10.0116 q^{53} +16.6679 q^{55} -16.2207 q^{57} +13.0991 q^{59} -1.78135 q^{61} +16.7088 q^{65} -11.1779 q^{67} -21.5269 q^{69} +9.21864 q^{71} -14.6608 q^{73} -17.2685 q^{75} +2.05802 q^{79} +14.0927 q^{81} +15.7006 q^{83} -3.25463 q^{85} +10.4767 q^{87} +6.91321 q^{89} +7.27326 q^{93} -17.0974 q^{95} -10.5244 q^{97} -33.4634 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 4 q^{3} - 4 q^{5} + 8 q^{9}+O(q^{10}) \) 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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Groups

Database

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