LMFDB - Embedded newform 3364.2.a.p.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3364,2,Mod(1,3364)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3364, 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("3364.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3364 = 2^{2} \cdot 29^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3364.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:	$26.8616752400$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.6456289.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 12x^{4} + 3x^{3} + 40x^{2} + 6x - 29 $ x^6 - x^5 - 12x^4 + 3x^3 + 40x^2 + 6x - 29
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 116)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.46835$ of defining polynomial
Character	$\chi$	$=$	3364.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.46835 q^{3} +3.45542 q^{5} -0.468352 q^{7} +3.09276 q^{9} +6.44782 q^{11} -2.67149 q^{13} +8.52918 q^{15} +3.05467 q^{17} +5.65536 q^{19} -1.15606 q^{21} -2.65625 q^{23} +6.93990 q^{25} +0.228972 q^{27} +0.162294 q^{31} +15.9155 q^{33} -1.61835 q^{35} -8.00432 q^{37} -6.59417 q^{39} -7.71247 q^{41} -6.04030 q^{43} +10.6868 q^{45} -10.2021 q^{47} -6.78065 q^{49} +7.54001 q^{51} +2.66487 q^{53} +22.2799 q^{55} +13.9594 q^{57} +3.41835 q^{59} -6.43590 q^{61} -1.44850 q^{63} -9.23110 q^{65} +2.23003 q^{67} -6.55657 q^{69} -7.96768 q^{71} -3.69941 q^{73} +17.1301 q^{75} -3.01985 q^{77} -4.67133 q^{79} -8.71311 q^{81} +8.87003 q^{83} +10.5552 q^{85} +1.02703 q^{89} +1.25120 q^{91} +0.400599 q^{93} +19.5416 q^{95} +18.5120 q^{97} +19.9416 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 5 q^{3} + 10 q^{5} + 7 q^{7} + 11 q^{9} + 9 q^{11} + 14 q^{13} + 10 q^{15} + 5 q^{17} + 9 q^{19} - 19 q^{21} + 15 q^{23} + 16 q^{25} + 20 q^{27} - 11 q^{31} + 22 q^{33} + 10 q^{35} - 16 q^{37} + 22 q^{39}+ \cdots + 8 q^{99}+O(q^{100}) $ 6 * q + 5 * q^3 + 10 * q^5 + 7 * q^7 + 11 * q^9 + 9 * q^11 + 14 * q^13 + 10 * q^15 + 5 * q^17 + 9 * q^19 - 19 * q^21 + 15 * q^23 + 16 * q^25 + 20 * q^27 - 11 * q^31 + 22 * q^33 + 10 * q^35 - 16 * q^37 + 22 * q^39 + q^41 + 15 * q^43 + 32 * q^45 - 27 * q^47 - 9 * q^49 - 20 * q^51 + 18 * q^53 + 14 * q^55 + 15 * q^57 + 18 * q^59 - 22 * q^61 - 13 * q^63 + 26 * q^65 + 9 * q^67 - 6 * q^69 - 7 * q^71 + 32 * q^73 + 22 * q^75 - 4 * q^77 - 29 * q^79 - 2 * q^81 + 3 * q^83 + 24 * q^85 - 12 * q^89 + 6 * q^91 + 14 * q^93 - 28 * q^95 + 58 * q^97 + 8 * q^99

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
$3$	2.46835	1.42510	0.712552	−	0.701619i	$-0.247539\pi$
$3$	2.46835	1.42510	0.712552	+	0.701619i	$0.247539\pi$
$4$	0	0
$5$	3.45542	1.54531	0.772655	−	0.634827i	$-0.218929\pi$
$5$	3.45542	1.54531	0.772655	+	0.634827i	$0.218929\pi$
$6$	0	0
$7$	−0.468352	−0.177021	−0.0885103	−	0.996075i	$-0.528211\pi$
$7$	−0.468352	−0.177021	−0.0885103	+	0.996075i	$0.528211\pi$
$8$	0	0
$9$	3.09276	1.03092
$10$	0	0
$11$	6.44782	1.94409	0.972045	−	0.234795i	$-0.0754418\pi$
$11$	6.44782	1.94409	0.972045	+	0.234795i	$0.0754418\pi$
$12$	0	0
$13$	−2.67149	−0.740937	−0.370468	−	0.928845i	$-0.620803\pi$
$13$	−2.67149	−0.740937	−0.370468	+	0.928845i	$0.620803\pi$
$14$	0	0
$15$	8.52918	2.20223
$16$	0	0
$17$	3.05467	0.740867	0.370434	−	0.928859i	$-0.379209\pi$
$17$	3.05467	0.740867	0.370434	+	0.928859i	$0.379209\pi$
$18$	0	0
$19$	5.65536	1.29743	0.648714	−	0.761033i	$-0.275307\pi$
$19$	5.65536	1.29743	0.648714	+	0.761033i	$0.275307\pi$
$20$	0	0
$21$	−1.15606	−0.252273
$22$	0	0
$23$	−2.65625	−0.553867	−0.276934	−	0.960889i	$-0.589318\pi$
$23$	−2.65625	−0.553867	−0.276934	+	0.960889i	$0.589318\pi$
$24$	0	0
$25$	6.93990	1.38798
$26$	0	0
$27$	0.228972	0.0440656
$28$	0	0
$29$	0	0
$29$	0	0
$30$	0	0
$31$	0.162294	0.0291489	0.0145745	−	0.999894i	$-0.495361\pi$
$31$	0.162294	0.0291489	0.0145745	+	0.999894i	$0.495361\pi$
$32$	0	0
$33$	15.9155	2.77053
$34$	0	0
$35$	−1.61835	−0.273551
$36$	0	0
$37$	−8.00432	−1.31590	−0.657951	−	0.753061i	$-0.728576\pi$
$37$	−8.00432	−1.31590	−0.657951	+	0.753061i	$0.728576\pi$
$38$	0	0
$39$	−6.59417	−1.05591
$40$	0	0
$41$	−7.71247	−1.20449	−0.602243	−	0.798313i	$-0.705726\pi$
$41$	−7.71247	−1.20449	−0.602243	+	0.798313i	$0.705726\pi$
$42$	0	0
$43$	−6.04030	−0.921137	−0.460568	−	0.887624i	$-0.652354\pi$
$43$	−6.04030	−0.921137	−0.460568	+	0.887624i	$0.652354\pi$
$44$	0	0
$45$	10.6868	1.59309
$46$	0	0
$47$	−10.2021	−1.48813	−0.744065	−	0.668107i	$-0.767105\pi$
$47$	−10.2021	−1.48813	−0.744065	+	0.668107i	$0.767105\pi$
$48$	0	0
$49$	−6.78065	−0.968664
$50$	0	0
$51$	7.54001	1.05581
$52$	0	0
$53$	2.66487	0.366049	0.183024	−	0.983108i	$-0.441411\pi$
$53$	2.66487	0.366049	0.183024	+	0.983108i	$0.441411\pi$
$54$	0	0
$55$	22.2799	3.00422
$56$	0	0
$57$	13.9594	1.84897
$58$	0	0
$59$	3.41835	0.445031	0.222516	−	0.974929i	$-0.428573\pi$
$59$	3.41835	0.445031	0.222516	+	0.974929i	$0.428573\pi$
$60$	0	0
$61$	−6.43590	−0.824033	−0.412016	−	0.911176i	$-0.635175\pi$
$61$	−6.43590	−0.824033	−0.412016	+	0.911176i	$0.635175\pi$
$62$	0	0
$63$	−1.44850	−0.182494
$64$	0	0
$65$	−9.23110	−1.14498
$66$	0	0
$67$	2.23003	0.272441	0.136221	−	0.990679i	$-0.456504\pi$
$67$	2.23003	0.272441	0.136221	+	0.990679i	$0.456504\pi$
$68$	0	0
$69$	−6.55657	−0.789318
$70$	0	0
$71$	−7.96768	−0.945590	−0.472795	−	0.881172i	$-0.656755\pi$
$71$	−7.96768	−0.945590	−0.472795	+	0.881172i	$0.656755\pi$
$72$	0	0
$73$	−3.69941	−0.432984	−0.216492	−	0.976284i	$-0.569461\pi$
$73$	−3.69941	−0.432984	−0.216492	+	0.976284i	$0.569461\pi$
$74$	0	0
$75$	17.1301	1.97802
$76$	0	0
$77$	−3.01985	−0.344144
$78$	0	0
$79$	−4.67133	−0.525566	−0.262783	−	0.964855i	$-0.584640\pi$
$79$	−4.67133	−0.525566	−0.262783	+	0.964855i	$0.584640\pi$
$80$	0	0
$81$	−8.71311	−0.968123
$82$	0	0
$83$	8.87003	0.973612	0.486806	−	0.873510i	$-0.338162\pi$
$83$	8.87003	0.973612	0.486806	+	0.873510i	$0.338162\pi$
$84$	0	0
$85$	10.5552	1.14487
$86$	0	0
$87$	0	0
$88$	0	0
$89$	1.02703	0.108865	0.0544324	−	0.998517i	$-0.482665\pi$
$89$	1.02703	0.108865	0.0544324	+	0.998517i	$0.482665\pi$
$90$	0	0
$91$	1.25120	0.131161
$92$	0	0
$93$	0.400599	0.0415402
$94$	0	0
$95$	19.5416	2.00493
$96$	0	0
$97$	18.5120	1.87961	0.939807	−	0.341706i	$-0.111005\pi$
$97$	18.5120	1.87961	0.939807	+	0.341706i	$0.111005\pi$
$98$	0	0
$99$	19.9416	2.00420
$100$	0	0
$101$	−10.9463	−1.08919	−0.544597	−	0.838698i	$-0.683317\pi$
$101$	−10.9463	−1.08919	−0.544597	+	0.838698i	$0.683317\pi$
$102$	0	0
$103$	2.04246	0.201249	0.100625	−	0.994924i	$-0.467916\pi$
$103$	2.04246	0.201249	0.100625	+	0.994924i	$0.467916\pi$
$104$	0	0
$105$	−3.99466	−0.389839
$106$	0	0
$107$	11.5539	1.11696	0.558478	−	0.829519i	$-0.311385\pi$
$107$	11.5539	1.11696	0.558478	+	0.829519i	$0.311385\pi$
$108$	0	0
$109$	3.58211	0.343103	0.171552	−	0.985175i	$-0.445122\pi$
$109$	3.58211	0.343103	0.171552	+	0.985175i	$0.445122\pi$
$110$	0	0
$111$	−19.7575	−1.87530
$112$	0	0
$113$	−5.62812	−0.529449	−0.264724	−	0.964324i	$-0.585281\pi$
$113$	−5.62812	−0.529449	−0.264724	+	0.964324i	$0.585281\pi$
$114$	0	0
$115$	−9.17846	−0.855896
$116$	0	0
$117$	−8.26227	−0.763847
$118$	0	0
$119$	−1.43066	−0.131149
$120$	0	0
$121$	30.5743	2.77949
$122$	0	0
$123$	−19.0371	−1.71652
$124$	0	0
$125$	6.70316	0.599549
$126$	0	0
$127$	−9.38911	−0.833149	−0.416575	−	0.909102i	$-0.636770\pi$
$127$	−9.38911	−0.833149	−0.416575	+	0.909102i	$0.636770\pi$
$128$	0	0
$129$	−14.9096	−1.31272
$130$	0	0
$131$	−2.20199	−0.192389	−0.0961945	−	0.995363i	$-0.530667\pi$
$131$	−2.20199	−0.192389	−0.0961945	+	0.995363i	$0.530667\pi$
$132$	0	0
$133$	−2.64870	−0.229671
$134$	0	0
$135$	0.791192	0.0680950
$136$	0	0
$137$	2.00184	0.171029	0.0855144	−	0.996337i	$-0.472747\pi$
$137$	2.00184	0.171029	0.0855144	+	0.996337i	$0.472747\pi$
$138$	0	0
$139$	9.98524	0.846937	0.423469	−	0.905911i	$-0.360812\pi$
$139$	9.98524	0.846937	0.423469	+	0.905911i	$0.360812\pi$
$140$	0	0
$141$	−25.1824	−2.12074
$142$	0	0
$143$	−17.2253	−1.44045
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−16.7370	−1.38045
$148$	0	0
$149$	8.25714	0.676451	0.338226	−	0.941065i	$-0.390173\pi$
$149$	8.25714	0.676451	0.338226	+	0.941065i	$0.390173\pi$
$150$	0	0
$151$	8.70287	0.708230	0.354115	−	0.935202i	$-0.384782\pi$
$151$	8.70287	0.708230	0.354115	+	0.935202i	$0.384782\pi$
$152$	0	0
$153$	9.44738	0.763776
$154$	0	0
$155$	0.560794	0.0450441
$156$	0	0
$157$	21.5884	1.72294	0.861472	−	0.507805i	$-0.169543\pi$
$157$	21.5884	1.72294	0.861472	+	0.507805i	$0.169543\pi$
$158$	0	0
$159$	6.57785	0.521657
$160$	0	0
$161$	1.24406	0.0980458
$162$	0	0
$163$	−6.74986	−0.528690	−0.264345	−	0.964428i	$-0.585156\pi$
$163$	−6.74986	−0.528690	−0.264345	+	0.964428i	$0.585156\pi$
$164$	0	0
$165$	54.9946	4.28133
$166$	0	0
$167$	−18.9056	−1.46296	−0.731481	−	0.681862i	$-0.761171\pi$
$167$	−18.9056	−1.46296	−0.731481	+	0.681862i	$0.761171\pi$
$168$	0	0
$169$	−5.86316	−0.451012
$170$	0	0
$171$	17.4907	1.33755
$172$	0	0
$173$	4.33780	0.329797	0.164899	−	0.986311i	$-0.447270\pi$
$173$	4.33780	0.329797	0.164899	+	0.986311i	$0.447270\pi$
$174$	0	0
$175$	−3.25032	−0.245701
$176$	0	0
$177$	8.43769	0.634216
$178$	0	0
$179$	−2.40824	−0.180000	−0.0900000	−	0.995942i	$-0.528687\pi$
$179$	−2.40824	−0.180000	−0.0900000	+	0.995942i	$0.528687\pi$
$180$	0	0
$181$	22.8390	1.69761	0.848803	−	0.528709i	$-0.177324\pi$
$181$	22.8390	1.69761	0.848803	+	0.528709i	$0.177324\pi$
$182$	0	0
$183$	−15.8861	−1.17433
$184$	0	0
$185$	−27.6582	−2.03347
$186$	0	0
$187$	19.6960	1.44031
$188$	0	0
$189$	−0.107239	−0.00780051
$190$	0	0
$191$	4.17124	0.301820	0.150910	−	0.988547i	$-0.451780\pi$
$191$	4.17124	0.301820	0.150910	+	0.988547i	$0.451780\pi$
$192$	0	0
$193$	19.4221	1.39803	0.699017	−	0.715105i	$-0.253621\pi$
$193$	19.4221	1.39803	0.699017	+	0.715105i	$0.253621\pi$
$194$	0	0
$195$	−22.7856	−1.63171
$196$	0	0
$197$	−7.09440	−0.505455	−0.252727	−	0.967538i	$-0.581328\pi$
$197$	−7.09440	−0.505455	−0.252727	+	0.967538i	$0.581328\pi$
$198$	0	0
$199$	2.33313	0.165391	0.0826956	−	0.996575i	$-0.473647\pi$
$199$	2.33313	0.165391	0.0826956	+	0.996575i	$0.473647\pi$
$200$	0	0
$201$	5.50450	0.388257
$202$	0	0
$203$	0	0
$204$	0	0
$205$	−26.6498	−1.86130
$206$	0	0
$207$	−8.21516	−0.570993
$208$	0	0
$209$	36.4647	2.52232
$210$	0	0
$211$	7.19592	0.495387	0.247694	−	0.968838i	$-0.420327\pi$
$211$	7.19592	0.495387	0.247694	+	0.968838i	$0.420327\pi$
$212$	0	0
$213$	−19.6671	−1.34756
$214$	0	0
$215$	−20.8717	−1.42344
$216$	0	0
$217$	−0.0760109	−0.00515995
$218$	0	0
$219$	−9.13146	−0.617047
$220$	0	0
$221$	−8.16052	−0.548936
$222$	0	0
$223$	−7.01578	−0.469812	−0.234906	−	0.972018i	$-0.575478\pi$
$223$	−7.01578	−0.469812	−0.234906	+	0.972018i	$0.575478\pi$
$224$	0	0
$225$	21.4635	1.43090
$226$	0	0
$227$	−21.3154	−1.41475	−0.707377	−	0.706837i	$-0.750122\pi$
$227$	−21.3154	−1.41475	−0.707377	+	0.706837i	$0.750122\pi$
$228$	0	0
$229$	−20.5594	−1.35860	−0.679300	−	0.733860i	$-0.737717\pi$
$229$	−20.5594	−1.35860	−0.679300	+	0.733860i	$0.737717\pi$
$230$	0	0
$231$	−7.45405	−0.490441
$232$	0	0
$233$	−1.22159	−0.0800287	−0.0400144	−	0.999199i	$-0.512740\pi$
$233$	−1.22159	−0.0800287	−0.0400144	+	0.999199i	$0.512740\pi$
$234$	0	0
$235$	−35.2525	−2.29962
$236$	0	0
$237$	−11.5305	−0.748986
$238$	0	0
$239$	19.4380	1.25734	0.628670	−	0.777673i	$-0.283600\pi$
$239$	19.4380	1.25734	0.628670	+	0.777673i	$0.283600\pi$
$240$	0	0
$241$	8.35978	0.538501	0.269250	−	0.963070i	$-0.413224\pi$
$241$	8.35978	0.538501	0.269250	+	0.963070i	$0.413224\pi$
$242$	0	0
$243$	−22.1939	−1.42374
$244$	0	0
$245$	−23.4300	−1.49688
$246$	0	0
$247$	−15.1082	−0.961312
$248$	0	0
$249$	21.8944	1.38750
$250$	0	0
$251$	−22.3163	−1.40859	−0.704297	−	0.709906i	$-0.748737\pi$
$251$	−22.3163	−1.40859	−0.704297	+	0.709906i	$0.748737\pi$
$252$	0	0
$253$	−17.1270	−1.07677
$254$	0	0
$255$	26.0539	1.63156
$256$	0	0
$257$	−16.5994	−1.03544	−0.517721	−	0.855549i	$-0.673220\pi$
$257$	−16.5994	−1.03544	−0.517721	+	0.855549i	$0.673220\pi$
$258$	0	0
$259$	3.74884	0.232942
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2.46065	−0.151730	−0.0758652	−	0.997118i	$-0.524172\pi$
$263$	−2.46065	−0.151730	−0.0758652	+	0.997118i	$0.524172\pi$
$264$	0	0
$265$	9.20825	0.565658
$266$	0	0
$267$	2.53507	0.155144
$268$	0	0
$269$	−5.30716	−0.323583	−0.161792	−	0.986825i	$-0.551727\pi$
$269$	−5.30716	−0.323583	−0.161792	+	0.986825i	$0.551727\pi$
$270$	0	0
$271$	2.63276	0.159929	0.0799643	−	0.996798i	$-0.474519\pi$
$271$	2.63276	0.159929	0.0799643	+	0.996798i	$0.474519\pi$
$272$	0	0
$273$	3.08839	0.186918
$274$	0	0
$275$	44.7472	2.69836
$276$	0	0
$277$	17.3377	1.04172	0.520861	−	0.853641i	$-0.325611\pi$
$277$	17.3377	1.04172	0.520861	+	0.853641i	$0.325611\pi$
$278$	0	0
$279$	0.501938	0.0300502
$280$	0	0
$281$	−13.7434	−0.819863	−0.409932	−	0.912116i	$-0.634447\pi$
$281$	−13.7434	−0.819863	−0.409932	+	0.912116i	$0.634447\pi$
$282$	0	0
$283$	−25.3515	−1.50699	−0.753495	−	0.657454i	$-0.771634\pi$
$283$	−25.3515	−1.50699	−0.753495	+	0.657454i	$0.771634\pi$
$284$	0	0
$285$	48.2356	2.85723
$286$	0	0
$287$	3.61215	0.213219
$288$	0	0
$289$	−7.66896	−0.451116
$290$	0	0
$291$	45.6943	2.67864
$292$	0	0
$293$	7.55111	0.441140	0.220570	−	0.975371i	$-0.429208\pi$
$293$	7.55111	0.441140	0.220570	+	0.975371i	$0.429208\pi$
$294$	0	0
$295$	11.8118	0.687711
$296$	0	0
$297$	1.47637	0.0856675
$298$	0	0
$299$	7.09614	0.410381
$300$	0	0
$301$	2.82899	0.163060
$302$	0	0
$303$	−27.0192	−1.55221
$304$	0	0
$305$	−22.2387	−1.27339
$306$	0	0
$307$	4.89496	0.279370	0.139685	−	0.990196i	$-0.455391\pi$
$307$	4.89496	0.279370	0.139685	+	0.990196i	$0.455391\pi$
$308$	0	0
$309$	5.04151	0.286801
$310$	0	0
$311$	−9.19809	−0.521576	−0.260788	−	0.965396i	$-0.583982\pi$
$311$	−9.19809	−0.521576	−0.260788	+	0.965396i	$0.583982\pi$
$312$	0	0
$313$	15.2320	0.860963	0.430481	−	0.902599i	$-0.358344\pi$
$313$	15.2320	0.860963	0.430481	+	0.902599i	$0.358344\pi$
$314$	0	0
$315$	−5.00518	−0.282010
$316$	0	0
$317$	14.3572	0.806379	0.403190	−	0.915116i	$-0.367902\pi$
$317$	14.3572	0.806379	0.403190	+	0.915116i	$0.367902\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	28.5191	1.59178
$322$	0	0
$323$	17.2753	0.961222
$324$	0	0
$325$	−18.5398	−1.02841
$326$	0	0
$327$	8.84190	0.488958
$328$	0	0
$329$	4.77818	0.263430
$330$	0	0
$331$	15.3588	0.844198	0.422099	−	0.906550i	$-0.361293\pi$
$331$	15.3588	0.844198	0.422099	+	0.906550i	$0.361293\pi$
$332$	0	0
$333$	−24.7554	−1.35659
$334$	0	0
$335$	7.70568	0.421006
$336$	0	0
$337$	−0.850248	−0.0463160	−0.0231580	−	0.999732i	$-0.507372\pi$
$337$	−0.850248	−0.0463160	−0.0231580	+	0.999732i	$0.507372\pi$
$338$	0	0
$339$	−13.8922	−0.754520
$340$	0	0
$341$	1.04644	0.0566681
$342$	0	0
$343$	6.45420	0.348494
$344$	0	0
$345$	−22.6557	−1.21974
$346$	0	0
$347$	−7.35642	−0.394913	−0.197457	−	0.980312i	$-0.563268\pi$
$347$	−7.35642	−0.394913	−0.197457	+	0.980312i	$0.563268\pi$
$348$	0	0
$349$	18.9497	1.01436	0.507178	−	0.861841i	$-0.330689\pi$
$349$	18.9497	1.01436	0.507178	+	0.861841i	$0.330689\pi$
$350$	0	0
$351$	−0.611694	−0.0326498
$352$	0	0
$353$	−14.4116	−0.767051	−0.383525	−	0.923530i	$-0.625290\pi$
$353$	−14.4116	−0.767051	−0.383525	+	0.923530i	$0.625290\pi$
$354$	0	0
$355$	−27.5317	−1.46123
$356$	0	0
$357$	−3.53138	−0.186901
$358$	0	0
$359$	12.8159	0.676400	0.338200	−	0.941074i	$-0.390182\pi$
$359$	12.8159	0.676400	0.338200	+	0.941074i	$0.390182\pi$
$360$	0	0
$361$	12.9830	0.683318
$362$	0	0
$363$	75.4683	3.96106
$364$	0	0
$365$	−12.7830	−0.669093
$366$	0	0
$367$	−5.89514	−0.307724	−0.153862	−	0.988092i	$-0.549171\pi$
$367$	−5.89514	−0.307724	−0.153862	+	0.988092i	$0.549171\pi$
$368$	0	0
$369$	−23.8528	−1.24173
$370$	0	0
$371$	−1.24810	−0.0647981
$372$	0	0
$373$	12.7655	0.660975	0.330487	−	0.943810i	$-0.392787\pi$
$373$	12.7655	0.660975	0.330487	+	0.943810i	$0.392787\pi$
$374$	0	0
$375$	16.5458	0.854419
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−19.5994	−1.00675	−0.503377	−	0.864067i	$-0.667909\pi$
$379$	−19.5994	−1.00675	−0.503377	+	0.864067i	$0.667909\pi$
$380$	0	0
$381$	−23.1756	−1.18732
$382$	0	0
$383$	22.1758	1.13313	0.566565	−	0.824017i	$-0.308272\pi$
$383$	22.1758	1.13313	0.566565	+	0.824017i	$0.308272\pi$
$384$	0	0
$385$	−10.4348	−0.531809
$386$	0	0
$387$	−18.6812	−0.949619
$388$	0	0
$389$	−24.1533	−1.22462	−0.612312	−	0.790617i	$-0.709760\pi$
$389$	−24.1533	−1.22462	−0.612312	+	0.790617i	$0.709760\pi$
$390$	0	0
$391$	−8.11399	−0.410342
$392$	0	0
$393$	−5.43530	−0.274174
$394$	0	0
$395$	−16.1414	−0.812162
$396$	0	0
$397$	−15.8597	−0.795976	−0.397988	−	0.917391i	$-0.630291\pi$
$397$	−15.8597	−0.795976	−0.397988	+	0.917391i	$0.630291\pi$
$398$	0	0
$399$	−6.53792	−0.327305
$400$	0	0
$401$	23.9221	1.19461	0.597307	−	0.802013i	$-0.296237\pi$
$401$	23.9221	1.19461	0.597307	+	0.802013i	$0.296237\pi$
$402$	0	0
$403$	−0.433567	−0.0215975
$404$	0	0
$405$	−30.1074	−1.49605
$406$	0	0
$407$	−51.6104	−2.55823
$408$	0	0
$409$	25.4293	1.25740	0.628699	−	0.777649i	$-0.283588\pi$
$409$	25.4293	1.25740	0.628699	+	0.777649i	$0.283588\pi$
$410$	0	0
$411$	4.94125	0.243734
$412$	0	0
$413$	−1.60099	−0.0787797
$414$	0	0
$415$	30.6496	1.50453
$416$	0	0
$417$	24.6471	1.20697
$418$	0	0
$419$	20.2925	0.991352	0.495676	−	0.868508i	$-0.334920\pi$
$419$	20.2925	0.991352	0.495676	+	0.868508i	$0.334920\pi$
$420$	0	0
$421$	−25.7015	−1.25261	−0.626306	−	0.779577i	$-0.715434\pi$
$421$	−25.7015	−1.25261	−0.626306	+	0.779577i	$0.715434\pi$
$422$	0	0
$423$	−31.5527	−1.53415
$424$	0	0
$425$	21.1991	1.02831
$426$	0	0
$427$	3.01427	0.145871
$428$	0	0
$429$	−42.5180	−2.05279
$430$	0	0
$431$	29.5645	1.42407	0.712037	−	0.702142i	$-0.247773\pi$
$431$	29.5645	1.42407	0.712037	+	0.702142i	$0.247773\pi$
$432$	0	0
$433$	−19.6937	−0.946421	−0.473211	−	0.880949i	$-0.656905\pi$
$433$	−19.6937	−0.946421	−0.473211	+	0.880949i	$0.656905\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−15.0221	−0.718602
$438$	0	0
$439$	32.3494	1.54395	0.771977	−	0.635650i	$-0.219268\pi$
$439$	32.3494	1.54395	0.771977	+	0.635650i	$0.219268\pi$
$440$	0	0
$441$	−20.9709	−0.998616
$442$	0	0
$443$	−16.8125	−0.798787	−0.399394	−	0.916779i	$-0.630779\pi$
$443$	−16.8125	−0.798787	−0.399394	+	0.916779i	$0.630779\pi$
$444$	0	0
$445$	3.54881	0.168230
$446$	0	0
$447$	20.3815	0.964013
$448$	0	0
$449$	18.4958	0.872873	0.436436	−	0.899735i	$-0.356240\pi$
$449$	18.4958	0.872873	0.436436	+	0.899735i	$0.356240\pi$
$450$	0	0
$451$	−49.7286	−2.34163
$452$	0	0
$453$	21.4818	1.00930
$454$	0	0
$455$	4.32340	0.202684
$456$	0	0
$457$	28.0543	1.31232	0.656162	−	0.754620i	$-0.272179\pi$
$457$	28.0543	1.31232	0.656162	+	0.754620i	$0.272179\pi$
$458$	0	0
$459$	0.699434	0.0326468
$460$	0	0
$461$	28.0684	1.30727	0.653637	−	0.756808i	$-0.273242\pi$
$461$	28.0684	1.30727	0.653637	+	0.756808i	$0.273242\pi$
$462$	0	0
$463$	21.0969	0.980454	0.490227	−	0.871595i	$-0.336914\pi$
$463$	21.0969	0.980454	0.490227	+	0.871595i	$0.336914\pi$
$464$	0	0
$465$	1.38424	0.0641925
$466$	0	0
$467$	9.71569	0.449589	0.224794	−	0.974406i	$-0.427829\pi$
$467$	9.71569	0.449589	0.224794	+	0.974406i	$0.427829\pi$
$468$	0	0
$469$	−1.04444	−0.0482277
$470$	0	0
$471$	53.2878	2.45537
$472$	0	0
$473$	−38.9467	−1.79077
$474$	0	0
$475$	39.2476	1.80080
$476$	0	0
$477$	8.24182	0.377367
$478$	0	0
$479$	11.5269	0.526677	0.263339	−	0.964703i	$-0.415176\pi$
$479$	11.5269	0.526677	0.263339	+	0.964703i	$0.415176\pi$
$480$	0	0
$481$	21.3834	0.975000
$482$	0	0
$483$	3.07078	0.139726
$484$	0	0
$485$	63.9668	2.90458
$486$	0	0
$487$	−18.4196	−0.834671	−0.417336	−	0.908752i	$-0.637036\pi$
$487$	−18.4196	−0.834671	−0.417336	+	0.908752i	$0.637036\pi$
$488$	0	0
$489$	−16.6610	−0.753438
$490$	0	0
$491$	−15.3585	−0.693121	−0.346560	−	0.938028i	$-0.612650\pi$
$491$	−15.3585	−0.693121	−0.346560	+	0.938028i	$0.612650\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	68.9064	3.09711
$496$	0	0
$497$	3.73168	0.167389
$498$	0	0
$499$	−35.0821	−1.57049	−0.785246	−	0.619184i	$-0.787464\pi$
$499$	−35.0821	−1.57049	−0.785246	+	0.619184i	$0.787464\pi$
$500$	0	0
$501$	−46.6658	−2.08487
$502$	0	0
$503$	−22.4221	−0.999751	−0.499875	−	0.866097i	$-0.666621\pi$
$503$	−22.4221	−0.999751	−0.499875	+	0.866097i	$0.666621\pi$
$504$	0	0
$505$	−37.8239	−1.68314
$506$	0	0
$507$	−14.4723	−0.642739
$508$	0	0
$509$	27.3980	1.21440	0.607198	−	0.794551i	$-0.292294\pi$
$509$	27.3980	1.21440	0.607198	+	0.794551i	$0.292294\pi$
$510$	0	0
$511$	1.73263	0.0766470
$512$	0	0
$513$	1.29492	0.0571719
$514$	0	0
$515$	7.05754	0.310993
$516$	0	0
$517$	−65.7814	−2.89306
$518$	0	0
$519$	10.7072	0.469995
$520$	0	0
$521$	17.6915	0.775078	0.387539	−	0.921853i	$-0.373325\pi$
$521$	17.6915	0.775078	0.387539	+	0.921853i	$0.373325\pi$
$522$	0	0
$523$	−11.8131	−0.516552	−0.258276	−	0.966071i	$-0.583154\pi$
$523$	−11.8131	−0.516552	−0.258276	+	0.966071i	$0.583154\pi$
$524$	0	0
$525$	−8.02293	−0.350149
$526$	0	0
$527$	0.495756	0.0215955
$528$	0	0
$529$	−15.9443	−0.693231
$530$	0	0
$531$	10.5721	0.458792
$532$	0	0
$533$	20.6038	0.892448
$534$	0	0
$535$	39.9235	1.72604
$536$	0	0
$537$	−5.94437	−0.256519
$538$	0	0
$539$	−43.7204	−1.88317
$540$	0	0
$541$	−5.21901	−0.224383	−0.112191	−	0.993687i	$-0.535787\pi$
$541$	−5.21901	−0.224383	−0.112191	+	0.993687i	$0.535787\pi$
$542$	0	0
$543$	56.3746	2.41926
$544$	0	0
$545$	12.3777	0.530201
$546$	0	0
$547$	−27.1921	−1.16265	−0.581324	−	0.813672i	$-0.697465\pi$
$547$	−27.1921	−1.16265	−0.581324	+	0.813672i	$0.697465\pi$
$548$	0	0
$549$	−19.9047	−0.849513
$550$	0	0
$551$	0	0
$552$	0	0
$553$	2.18783	0.0930360
$554$	0	0
$555$	−68.2703	−2.89791
$556$	0	0
$557$	11.1087	0.470691	0.235346	−	0.971912i	$-0.424378\pi$
$557$	11.1087	0.470691	0.235346	+	0.971912i	$0.424378\pi$
$558$	0	0
$559$	16.1366	0.682504
$560$	0	0
$561$	48.6166	2.05260
$562$	0	0
$563$	−15.3337	−0.646237	−0.323118	−	0.946359i	$-0.604731\pi$
$563$	−15.3337	−0.646237	−0.323118	+	0.946359i	$0.604731\pi$
$564$	0	0
$565$	−19.4475	−0.818162
$566$	0	0
$567$	4.08080	0.171378
$568$	0	0
$569$	8.08646	0.339002	0.169501	−	0.985530i	$-0.445784\pi$
$569$	8.08646	0.339002	0.169501	+	0.985530i	$0.445784\pi$
$570$	0	0
$571$	−22.8857	−0.957737	−0.478869	−	0.877887i	$-0.658953\pi$
$571$	−22.8857	−0.957737	−0.478869	+	0.877887i	$0.658953\pi$
$572$	0	0
$573$	10.2961	0.430125
$574$	0	0
$575$	−18.4341	−0.768756
$576$	0	0
$577$	−9.49247	−0.395177	−0.197588	−	0.980285i	$-0.563311\pi$
$577$	−9.49247	−0.395177	−0.197588	+	0.980285i	$0.563311\pi$
$578$	0	0
$579$	47.9406	1.99234
$580$	0	0
$581$	−4.15430	−0.172349
$582$	0	0
$583$	17.1826	0.711632
$584$	0	0
$585$	−28.5496	−1.18038
$586$	0	0
$587$	−20.0335	−0.826870	−0.413435	−	0.910534i	$-0.635671\pi$
$587$	−20.0335	−0.826870	−0.413435	+	0.910534i	$0.635671\pi$
$588$	0	0
$589$	0.917832	0.0378186
$590$	0	0
$591$	−17.5115	−0.720326
$592$	0	0
$593$	1.35116	0.0554856	0.0277428	−	0.999615i	$-0.491168\pi$
$593$	1.35116	0.0554856	0.0277428	+	0.999615i	$0.491168\pi$
$594$	0	0
$595$	−4.94354	−0.202665
$596$	0	0
$597$	5.75899	0.235700
$598$	0	0
$599$	−6.74712	−0.275680	−0.137840	−	0.990455i	$-0.544016\pi$
$599$	−6.74712	−0.275680	−0.137840	+	0.990455i	$0.544016\pi$
$600$	0	0
$601$	1.52795	0.0623264	0.0311632	−	0.999514i	$-0.490079\pi$
$601$	1.52795	0.0623264	0.0311632	+	0.999514i	$0.490079\pi$
$602$	0	0
$603$	6.89695	0.280866
$604$	0	0
$605$	105.647	4.29516
$606$	0	0
$607$	8.34258	0.338615	0.169307	−	0.985563i	$-0.445847\pi$
$607$	8.34258	0.338615	0.169307	+	0.985563i	$0.445847\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	27.2548	1.10261
$612$	0	0
$613$	−12.5594	−0.507270	−0.253635	−	0.967300i	$-0.581626\pi$
$613$	−12.5594	−0.507270	−0.253635	+	0.967300i	$0.581626\pi$
$614$	0	0
$615$	−65.7811	−2.65255
$616$	0	0
$617$	33.9771	1.36787	0.683933	−	0.729545i	$-0.260268\pi$
$617$	33.9771	1.36787	0.683933	+	0.729545i	$0.260268\pi$
$618$	0	0
$619$	14.9084	0.599221	0.299610	−	0.954062i	$-0.403143\pi$
$619$	14.9084	0.599221	0.299610	+	0.954062i	$0.403143\pi$
$620$	0	0
$621$	−0.608206	−0.0244065
$622$	0	0
$623$	−0.481011	−0.0192713
$624$	0	0
$625$	−11.5373	−0.461492
$626$	0	0
$627$	90.0077	3.59456
$628$	0	0
$629$	−24.4506	−0.974908
$630$	0	0
$631$	−31.8662	−1.26857	−0.634286	−	0.773098i	$-0.718706\pi$
$631$	−31.8662	−1.26857	−0.634286	+	0.773098i	$0.718706\pi$
$632$	0	0
$633$	17.7621	0.705978
$634$	0	0
$635$	−32.4433	−1.28747
$636$	0	0
$637$	18.1144	0.717719
$638$	0	0
$639$	−24.6422	−0.974829
$640$	0	0
$641$	−2.12216	−0.0838202	−0.0419101	−	0.999121i	$-0.513344\pi$
$641$	−2.12216	−0.0838202	−0.0419101	+	0.999121i	$0.513344\pi$
$642$	0	0
$643$	−25.5435	−1.00734	−0.503669	−	0.863897i	$-0.668017\pi$
$643$	−25.5435	−1.00734	−0.503669	+	0.863897i	$0.668017\pi$
$644$	0	0
$645$	−51.5188	−2.02855
$646$	0	0
$647$	2.30704	0.0906993	0.0453496	−	0.998971i	$-0.485560\pi$
$647$	2.30704	0.0906993	0.0453496	+	0.998971i	$0.485560\pi$
$648$	0	0
$649$	22.0409	0.865181
$650$	0	0
$651$	−0.187622	−0.00735347
$652$	0	0
$653$	−45.3683	−1.77540	−0.887699	−	0.460425i	$-0.847697\pi$
$653$	−45.3683	−1.77540	−0.887699	+	0.460425i	$0.847697\pi$
$654$	0	0
$655$	−7.60880	−0.297301
$656$	0	0
$657$	−11.4414	−0.446372
$658$	0	0
$659$	30.2677	1.17906	0.589531	−	0.807746i	$-0.299313\pi$
$659$	30.2677	1.17906	0.589531	+	0.807746i	$0.299313\pi$
$660$	0	0
$661$	−17.9479	−0.698093	−0.349047	−	0.937105i	$-0.613494\pi$
$661$	−17.9479	−0.698093	−0.349047	+	0.937105i	$0.613494\pi$
$662$	0	0
$663$	−20.1430	−0.782291
$664$	0	0
$665$	−9.15235	−0.354913
$666$	0	0
$667$	0	0
$668$	0	0
$669$	−17.3174	−0.669530
$670$	0	0
$671$	−41.4975	−1.60199
$672$	0	0
$673$	−13.9718	−0.538572	−0.269286	−	0.963060i	$-0.586788\pi$
$673$	−13.9718	−0.538572	−0.269286	+	0.963060i	$0.586788\pi$
$674$	0	0
$675$	1.58904	0.0611622
$676$	0	0
$677$	39.6603	1.52427	0.762135	−	0.647418i	$-0.224151\pi$
$677$	39.6603	1.52427	0.762135	+	0.647418i	$0.224151\pi$
$678$	0	0
$679$	−8.67016	−0.332730
$680$	0	0
$681$	−52.6140	−2.01617
$682$	0	0
$683$	33.0054	1.26291	0.631457	−	0.775411i	$-0.282457\pi$
$683$	33.0054	1.26291	0.631457	+	0.775411i	$0.282457\pi$
$684$	0	0
$685$	6.91720	0.264292
$686$	0	0
$687$	−50.7478	−1.93615
$688$	0	0
$689$	−7.11918	−0.271219
$690$	0	0
$691$	45.9365	1.74751	0.873753	−	0.486370i	$-0.161679\pi$
$691$	45.9365	1.74751	0.873753	+	0.486370i	$0.161679\pi$
$692$	0	0
$693$	−9.33968	−0.354785
$694$	0	0
$695$	34.5032	1.30878
$696$	0	0
$697$	−23.5591	−0.892364
$698$	0	0
$699$	−3.01530	−0.114049
$700$	0	0
$701$	26.2314	0.990748	0.495374	−	0.868680i	$-0.335031\pi$
$701$	26.2314	0.990748	0.495374	+	0.868680i	$0.335031\pi$
$702$	0	0
$703$	−45.2672	−1.70729
$704$	0	0
$705$	−87.0157	−3.27720
$706$	0	0
$707$	5.12671	0.192810
$708$	0	0
$709$	1.09356	0.0410695	0.0205347	−	0.999789i	$-0.493463\pi$
$709$	1.09356	0.0410695	0.0205347	+	0.999789i	$0.493463\pi$
$710$	0	0
$711$	−14.4473	−0.541817
$712$	0	0
$713$	−0.431095	−0.0161446
$714$	0	0
$715$	−59.5204	−2.22594
$716$	0	0
$717$	47.9798	1.79184
$718$	0	0
$719$	39.4380	1.47079	0.735394	−	0.677639i	$-0.236997\pi$
$719$	39.4380	1.47079	0.735394	+	0.677639i	$0.236997\pi$
$720$	0	0
$721$	−0.956590	−0.0356253
$722$	0	0
$723$	20.6349	0.767420
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−0.447183	−0.0165851	−0.00829255	−	0.999966i	$-0.502640\pi$
$727$	−0.447183	−0.0165851	−0.00829255	+	0.999966i	$0.502640\pi$
$728$	0	0
$729$	−28.6431	−1.06086
$730$	0	0
$731$	−18.4511	−0.682440
$732$	0	0
$733$	−28.4633	−1.05132	−0.525658	−	0.850696i	$-0.676181\pi$
$733$	−28.4633	−1.05132	−0.525658	+	0.850696i	$0.676181\pi$
$734$	0	0
$735$	−57.8334	−2.13322
$736$	0	0
$737$	14.3788	0.529651
$738$	0	0
$739$	−22.0518	−0.811190	−0.405595	−	0.914053i	$-0.632936\pi$
$739$	−22.0518	−0.811190	−0.405595	+	0.914053i	$0.632936\pi$
$740$	0	0
$741$	−37.2924	−1.36997
$742$	0	0
$743$	−25.1379	−0.922220	−0.461110	−	0.887343i	$-0.652549\pi$
$743$	−25.1379	−0.922220	−0.461110	+	0.887343i	$0.652549\pi$
$744$	0	0
$745$	28.5319	1.04533
$746$	0	0
$747$	27.4329	1.00372
$748$	0	0
$749$	−5.41129	−0.197724
$750$	0	0
$751$	−2.28503	−0.0833819	−0.0416909	−	0.999131i	$-0.513274\pi$
$751$	−2.28503	−0.0833819	−0.0416909	+	0.999131i	$0.513274\pi$
$752$	0	0
$753$	−55.0845	−2.00739
$754$	0	0
$755$	30.0721	1.09443
$756$	0	0
$757$	8.95161	0.325352	0.162676	−	0.986680i	$-0.447988\pi$
$757$	8.95161	0.325352	0.162676	+	0.986680i	$0.447988\pi$
$758$	0	0
$759$	−42.2756	−1.53451
$760$	0	0
$761$	47.7847	1.73220	0.866098	−	0.499875i	$-0.166621\pi$
$761$	47.7847	1.73220	0.866098	+	0.499875i	$0.166621\pi$
$762$	0	0
$763$	−1.67769	−0.0607364
$764$	0	0
$765$	32.6446	1.18027
$766$	0	0
$767$	−9.13207	−0.329740
$768$	0	0
$769$	41.2072	1.48597	0.742985	−	0.669308i	$-0.233409\pi$
$769$	41.2072	1.48597	0.742985	+	0.669308i	$0.233409\pi$
$770$	0	0
$771$	−40.9732	−1.47561
$772$	0	0
$773$	−24.4331	−0.878796	−0.439398	−	0.898292i	$-0.644808\pi$
$773$	−24.4331	−0.878796	−0.439398	+	0.898292i	$0.644808\pi$
$774$	0	0
$775$	1.12631	0.0404581
$776$	0	0
$777$	9.25346	0.331966
$778$	0	0
$779$	−43.6167	−1.56273
$780$	0	0
$781$	−51.3742	−1.83831
$782$	0	0
$783$	0	0
$784$	0	0
$785$	74.5970	2.66248
$786$	0	0
$787$	35.0439	1.24918	0.624591	−	0.780952i	$-0.285266\pi$
$787$	35.0439	1.24918	0.624591	+	0.780952i	$0.285266\pi$
$788$	0	0
$789$	−6.07376	−0.216232
$790$	0	0
$791$	2.63594	0.0937233
$792$	0	0
$793$	17.1934	0.610556
$794$	0	0
$795$	22.7292	0.806122
$796$	0	0
$797$	43.6491	1.54613	0.773066	−	0.634326i	$-0.218722\pi$
$797$	43.6491	1.54613	0.773066	+	0.634326i	$0.218722\pi$
$798$	0	0
$799$	−31.1641	−1.10251
$800$	0	0
$801$	3.17636	0.112231
$802$	0	0
$803$	−23.8531	−0.841759
$804$	0	0
$805$	4.29875	0.151511
$806$	0	0
$807$	−13.0999	−0.461140
$808$	0	0
$809$	23.0748	0.811266	0.405633	−	0.914036i	$-0.367051\pi$
$809$	23.0748	0.811266	0.405633	+	0.914036i	$0.367051\pi$
$810$	0	0
$811$	−48.5305	−1.70414	−0.852068	−	0.523431i	$-0.824652\pi$
$811$	−48.5305	−1.70414	−0.852068	+	0.523431i	$0.824652\pi$
$812$	0	0
$813$	6.49857	0.227915
$814$	0	0
$815$	−23.3236	−0.816989
$816$	0	0
$817$	−34.1600	−1.19511
$818$	0	0
$819$	3.86965	0.135217
$820$	0	0
$821$	46.7925	1.63307	0.816534	−	0.577297i	$-0.195893\pi$
$821$	46.7925	1.63307	0.816534	+	0.577297i	$0.195893\pi$
$822$	0	0
$823$	−46.0330	−1.60461	−0.802305	−	0.596914i	$-0.796393\pi$
$823$	−46.0330	−1.60461	−0.802305	+	0.596914i	$0.796393\pi$
$824$	0	0
$825$	110.452	3.84544
$826$	0	0
$827$	−34.5022	−1.19976	−0.599879	−	0.800091i	$-0.704785\pi$
$827$	−34.5022	−1.19976	−0.599879	+	0.800091i	$0.704785\pi$
$828$	0	0
$829$	−10.1549	−0.352694	−0.176347	−	0.984328i	$-0.556428\pi$
$829$	−10.1549	−0.352694	−0.176347	+	0.984328i	$0.556428\pi$
$830$	0	0
$831$	42.7956	1.48456
$832$	0	0
$833$	−20.7127	−0.717651
$834$	0	0
$835$	−65.3268	−2.26073
$836$	0	0
$837$	0.0371608	0.00128446
$838$	0	0
$839$	40.1921	1.38758	0.693792	−	0.720176i	$-0.255939\pi$
$839$	40.1921	1.38758	0.693792	+	0.720176i	$0.255939\pi$
$840$	0	0
$841$	0	0
$842$	0	0
$843$	−33.9236	−1.16839
$844$	0	0
$845$	−20.2597	−0.696953
$846$	0	0
$847$	−14.3196	−0.492026
$848$	0	0
$849$	−62.5764	−2.14762
$850$	0	0
$851$	21.2615	0.728835
$852$	0	0
$853$	−31.5341	−1.07971	−0.539854	−	0.841759i	$-0.681521\pi$
$853$	−31.5341	−1.07971	−0.539854	+	0.841759i	$0.681521\pi$
$854$	0	0
$855$	60.4375	2.06692
$856$	0	0
$857$	4.79855	0.163915	0.0819576	−	0.996636i	$-0.473883\pi$
$857$	4.79855	0.163915	0.0819576	+	0.996636i	$0.473883\pi$
$858$	0	0
$859$	0.576452	0.0196683	0.00983415	−	0.999952i	$-0.496870\pi$
$859$	0.576452	0.0196683	0.00983415	+	0.999952i	$0.496870\pi$
$860$	0	0
$861$	8.91606	0.303859
$862$	0	0
$863$	−31.9164	−1.08645	−0.543223	−	0.839588i	$-0.682796\pi$
$863$	−31.9164	−1.08645	−0.543223	+	0.839588i	$0.682796\pi$
$864$	0	0
$865$	14.9889	0.509638
$866$	0	0
$867$	−18.9297	−0.642886
$868$	0	0
$869$	−30.1199	−1.02175
$870$	0	0
$871$	−5.95749	−0.201862
$872$	0	0
$873$	57.2534	1.93773
$874$	0	0
$875$	−3.13944	−0.106132
$876$	0	0
$877$	42.0117	1.41863	0.709317	−	0.704890i	$-0.249003\pi$
$877$	42.0117	1.41863	0.709317	+	0.704890i	$0.249003\pi$
$878$	0	0
$879$	18.6388	0.628671
$880$	0	0
$881$	21.9247	0.738661	0.369331	−	0.929298i	$-0.379587\pi$
$881$	21.9247	0.738661	0.369331	+	0.929298i	$0.379587\pi$
$882$	0	0
$883$	−33.7438	−1.13557	−0.567784	−	0.823178i	$-0.692199\pi$
$883$	−33.7438	−1.13557	−0.567784	+	0.823178i	$0.692199\pi$
$884$	0	0
$885$	29.1557	0.980059
$886$	0	0
$887$	−39.5840	−1.32910	−0.664550	−	0.747244i	$-0.731377\pi$
$887$	−39.5840	−1.32910	−0.664550	+	0.747244i	$0.731377\pi$
$888$	0	0
$889$	4.39741	0.147485
$890$	0	0
$891$	−56.1805	−1.88212
$892$	0	0
$893$	−57.6966	−1.93074
$894$	0	0
$895$	−8.32146	−0.278156
$896$	0	0
$897$	17.5158	0.584835
$898$	0	0
$899$	0	0
$900$	0	0
$901$	8.14032	0.271194
$902$	0	0
$903$	6.98294	0.232378
$904$	0	0
$905$	78.9181	2.62333
$906$	0	0
$907$	−32.1363	−1.06707	−0.533534	−	0.845778i	$-0.679136\pi$
$907$	−32.1363	−1.06707	−0.533534	+	0.845778i	$0.679136\pi$
$908$	0	0
$909$	−33.8542	−1.12287
$910$	0	0
$911$	−55.7892	−1.84838	−0.924189	−	0.381936i	$-0.875257\pi$
$911$	−55.7892	−1.84838	−0.924189	+	0.381936i	$0.875257\pi$
$912$	0	0
$913$	57.1923	1.89279
$914$	0	0
$915$	−54.8930	−1.81471
$916$	0	0
$917$	1.03131	0.0340568
$918$	0	0
$919$	−12.0026	−0.395929	−0.197965	−	0.980209i	$-0.563433\pi$
$919$	−12.0026	−0.395929	−0.197965	+	0.980209i	$0.563433\pi$
$920$	0	0
$921$	12.0825	0.398132
$922$	0	0
$923$	21.2856	0.700623
$924$	0	0
$925$	−55.5491	−1.82644
$926$	0	0
$927$	6.31684	0.207472
$928$	0	0
$929$	−44.1037	−1.44700	−0.723498	−	0.690326i	$-0.757467\pi$
$929$	−44.1037	−1.44700	−0.723498	+	0.690326i	$0.757467\pi$
$930$	0	0
$931$	−38.3470	−1.25677
$932$	0	0
$933$	−22.7041	−0.743299
$934$	0	0
$935$	68.0578	2.22573
$936$	0	0
$937$	−1.83132	−0.0598265	−0.0299132	−	0.999552i	$-0.509523\pi$
$937$	−1.83132	−0.0598265	−0.0299132	+	0.999552i	$0.509523\pi$
$938$	0	0
$939$	37.5979	1.22696
$940$	0	0
$941$	−14.3133	−0.466599	−0.233300	−	0.972405i	$-0.574952\pi$
$941$	−14.3133	−0.466599	−0.233300	+	0.972405i	$0.574952\pi$
$942$	0	0
$943$	20.4863	0.667125
$944$	0	0
$945$	−0.370557	−0.0120542
$946$	0	0
$947$	34.1252	1.10892	0.554460	−	0.832210i	$-0.312925\pi$
$947$	34.1252	1.10892	0.554460	+	0.832210i	$0.312925\pi$
$948$	0	0
$949$	9.88293	0.320814
$950$	0	0
$951$	35.4386	1.14917
$952$	0	0
$953$	−29.6491	−0.960429	−0.480214	−	0.877151i	$-0.659441\pi$
$953$	−29.6491	−0.960429	−0.480214	+	0.877151i	$0.659441\pi$
$954$	0	0
$955$	14.4134	0.466405
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−0.937567	−0.0302756
$960$	0	0
$961$	−30.9737	−0.999150
$962$	0	0
$963$	35.7334	1.15149
$964$	0	0
$965$	67.1115	2.16040
$966$	0	0
$967$	−45.7185	−1.47021	−0.735104	−	0.677954i	$-0.762867\pi$
$967$	−45.7185	−1.47021	−0.735104	+	0.677954i	$0.762867\pi$
$968$	0	0
$969$	42.6414	1.36984
$970$	0	0
$971$	−26.0642	−0.836441	−0.418221	−	0.908345i	$-0.637346\pi$
$971$	−26.0642	−0.836441	−0.418221	+	0.908345i	$0.637346\pi$
$972$	0	0
$973$	−4.67661	−0.149925
$974$	0	0
$975$	−45.7629	−1.46558
$976$	0	0
$977$	10.0813	0.322528	0.161264	−	0.986911i	$-0.448443\pi$
$977$	10.0813	0.322528	0.161264	+	0.986911i	$0.448443\pi$
$978$	0	0
$979$	6.62209	0.211643
$980$	0	0
$981$	11.0786	0.353713
$982$	0	0
$983$	−32.0567	−1.02245	−0.511225	−	0.859447i	$-0.670808\pi$
$983$	−32.0567	−1.02245	−0.511225	+	0.859447i	$0.670808\pi$
$984$	0	0
$985$	−24.5141	−0.781084
$986$	0	0
$987$	11.7942	0.375415
$988$	0	0
$989$	16.0446	0.510187
$990$	0	0
$991$	52.2812	1.66077	0.830383	−	0.557193i	$-0.188122\pi$
$991$	52.2812	1.66077	0.830383	+	0.557193i	$0.188122\pi$
$992$	0	0
$993$	37.9110	1.20307
$994$	0	0
$995$	8.06194	0.255581
$996$	0	0
$997$	−40.0799	−1.26934	−0.634672	−	0.772782i	$-0.718865\pi$
$997$	−40.0799	−1.26934	−0.634672	+	0.772782i	$0.718865\pi$
$998$	0	0
$999$	−1.83276	−0.0579860

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3364.2.a.p.1.4		6
29.5	even	14		116.2.g.b.25.2	✓	12
29.6	even	14		116.2.g.b.65.2	yes	12
29.12	odd	4		3364.2.c.j.1681.10		12
29.17	odd	4		3364.2.c.j.1681.3		12
29.28	even	2		3364.2.a.m.1.3		6
87.5	odd	14		1044.2.u.c.721.2		12
87.35	odd	14		1044.2.u.c.181.2		12
116.35	odd	14		464.2.u.g.65.1		12
116.63	odd	14		464.2.u.g.257.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
116.2.g.b.25.2	✓	12	29.5	even	14
116.2.g.b.65.2	yes	12	29.6	even	14
464.2.u.g.65.1		12	116.35	odd	14
464.2.u.g.257.1		12	116.63	odd	14
1044.2.u.c.181.2		12	87.35	odd	14
1044.2.u.c.721.2		12	87.5	odd	14
3364.2.a.m.1.3		6	29.28	even	2
3364.2.a.p.1.4		6	1.1	even	1	trivial
3364.2.c.j.1681.3		12	29.17	odd	4
3364.2.c.j.1681.10		12	29.12	odd	4

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:	\( N \)	\(=\)	\( 3364 = 2^{2} \cdot 29^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3364.a (trivial)

\(f(q)\)	\(=\)	\(q+2.46835 q^{3} +3.45542 q^{5} -0.468352 q^{7} +3.09276 q^{9} +6.44782 q^{11} -2.67149 q^{13} +8.52918 q^{15} +3.05467 q^{17} +5.65536 q^{19} -1.15606 q^{21} -2.65625 q^{23} +6.93990 q^{25} +0.228972 q^{27} +0.162294 q^{31} +15.9155 q^{33} -1.61835 q^{35} -8.00432 q^{37} -6.59417 q^{39} -7.71247 q^{41} -6.04030 q^{43} +10.6868 q^{45} -10.2021 q^{47} -6.78065 q^{49} +7.54001 q^{51} +2.66487 q^{53} +22.2799 q^{55} +13.9594 q^{57} +3.41835 q^{59} -6.43590 q^{61} -1.44850 q^{63} -9.23110 q^{65} +2.23003 q^{67} -6.55657 q^{69} -7.96768 q^{71} -3.69941 q^{73} +17.1301 q^{75} -3.01985 q^{77} -4.67133 q^{79} -8.71311 q^{81} +8.87003 q^{83} +10.5552 q^{85} +1.02703 q^{89} +1.25120 q^{91} +0.400599 q^{93} +19.5416 q^{95} +18.5120 q^{97} +19.9416 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 5 q^{3} + 10 q^{5} + 7 q^{7} + 11 q^{9} + 9 q^{11} + 14 q^{13} + 10 q^{15} + 5 q^{17} + 9 q^{19} - 19 q^{21} + 15 q^{23} + 16 q^{25} + 20 q^{27} - 11 q^{31} + 22 q^{33} + 10 q^{35} - 16 q^{37} + 22 q^{39}+ \cdots + 8 q^{99}+O(q^{100}) \) 6 * q + 5 * q^3 + 10 * q^5 + 7 * q^7 + 11 * q^9 + 9 * q^11 + 14 * q^13 + 10 * q^15 + 5 * q^17 + 9 * q^19 - 19 * q^21 + 15 * q^23 + 16 * q^25 + 20 * q^27 - 11 * q^31 + 22 * q^33 + 10 * q^35 - 16 * q^37 + 22 * q^39 + q^41 + 15 * q^43 + 32 * q^45 - 27 * q^47 - 9 * q^49 - 20 * q^51 + 18 * q^53 + 14 * q^55 + 15 * q^57 + 18 * q^59 - 22 * q^61 - 13 * q^63 + 26 * q^65 + 9 * q^67 - 6 * q^69 - 7 * q^71 + 32 * q^73 + 22 * q^75 - 4 * q^77 - 29 * q^79 - 2 * q^81 + 3 * q^83 + 24 * q^85 - 12 * q^89 + 6 * q^91 + 14 * q^93 - 28 * q^95 + 58 * q^97 + 8 * q^99

Introduction

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