LMFDB - Embedded newform 1352.2.a.m.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1352,2,Mod(1,1352)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1352, 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("1352.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1352 = 2^{3} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1352.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$10.7957743533$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.3728753.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - x^{5} - 10x^{4} + 11x^{3} + 22x^{2} - 18x - 13 $ x^6 - x^5 - 10x^4 + 11x^3 + 22x^2 - 18x - 13
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.31865$ of defining polynomial
Character	$\chi$	$=$	1352.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.76369 q^{3} -4.24972 q^{5} +2.37460 q^{7} +4.63797 q^{9} +O(q^{10})$	$q+2.76369 q^{3} -4.24972 q^{5} +2.37460 q^{7} +4.63797 q^{9} +0.178841 q^{11} -11.7449 q^{15} +1.03977 q^{17} +5.25040 q^{19} +6.56264 q^{21} -0.130358 q^{23} +13.0601 q^{25} +4.52683 q^{27} +8.33214 q^{29} +2.94810 q^{31} +0.494261 q^{33} -10.0914 q^{35} +6.03464 q^{37} +0.264592 q^{41} +0.564732 q^{43} -19.7101 q^{45} -10.9224 q^{47} -1.36129 q^{49} +2.87360 q^{51} +2.41779 q^{53} -0.760024 q^{55} +14.5105 q^{57} +7.35568 q^{59} -5.06501 q^{61} +11.0133 q^{63} +1.58480 q^{67} -0.360267 q^{69} -10.3352 q^{71} +14.3011 q^{73} +36.0941 q^{75} +0.424675 q^{77} -7.26636 q^{79} -1.40316 q^{81} +12.9706 q^{83} -4.41874 q^{85} +23.0274 q^{87} -6.92320 q^{89} +8.14762 q^{93} -22.3127 q^{95} +3.63494 q^{97} +0.829459 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{3} - 2 q^{5} + 3 q^{7} + 9 q^{9}+O(q^{10}) $ 6 * q + 3 * q^3 - 2 * q^5 + 3 * q^7 + 9 * q^9	$ 6 q + 3 q^{3} - 2 q^{5} + 3 q^{7} + 9 q^{9} + 13 q^{11} - 14 q^{15} + 11 q^{17} + 15 q^{19} - 22 q^{21} + 15 q^{23} + 16 q^{25} + 21 q^{27} + 9 q^{29} - 3 q^{31} - 2 q^{33} + 14 q^{35} + 14 q^{37} - 20 q^{41} + 10 q^{43} - 9 q^{45} - 10 q^{47} + 27 q^{49} + 5 q^{51} + q^{53} + 14 q^{55} + 9 q^{57} + 50 q^{59} + 2 q^{63} + 6 q^{67} + 32 q^{69} + 9 q^{71} - 6 q^{73} + 40 q^{75} - 2 q^{77} + 13 q^{79} + 42 q^{81} + 36 q^{83} - 31 q^{85} + 22 q^{87} - 18 q^{89} - 12 q^{93} - 21 q^{95} - 14 q^{97} + 18 q^{99}+O(q^{100}) $ 6 * q + 3 * q^3 - 2 * q^5 + 3 * q^7 + 9 * q^9 + 13 * q^11 - 14 * q^15 + 11 * q^17 + 15 * q^19 - 22 * q^21 + 15 * q^23 + 16 * q^25 + 21 * q^27 + 9 * q^29 - 3 * q^31 - 2 * q^33 + 14 * q^35 + 14 * q^37 - 20 * q^41 + 10 * q^43 - 9 * q^45 - 10 * q^47 + 27 * q^49 + 5 * q^51 + q^53 + 14 * q^55 + 9 * q^57 + 50 * q^59 + 2 * q^63 + 6 * q^67 + 32 * q^69 + 9 * q^71 - 6 * q^73 + 40 * q^75 - 2 * q^77 + 13 * q^79 + 42 * q^81 + 36 * q^83 - 31 * q^85 + 22 * q^87 - 18 * q^89 - 12 * q^93 - 21 * q^95 - 14 * q^97 + 18 * 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
$3$	2.76369	1.59562	0.797808	−	0.602912i	$-0.205993\pi$
$3$	2.76369	1.59562	0.797808	+	0.602912i	$0.205993\pi$
$4$	0	0
$5$	−4.24972	−1.90053	−0.950266	−	0.311438i	$-0.899189\pi$
$5$	−4.24972	−1.90053	−0.950266	+	0.311438i	$0.899189\pi$
$6$	0	0
$7$	2.37460	0.897513	0.448756	−	0.893654i	$-0.351867\pi$
$7$	2.37460	0.897513	0.448756	+	0.893654i	$0.351867\pi$
$8$	0	0
$9$	4.63797	1.54599
$10$	0	0
$11$	0.178841	0.0539226	0.0269613	−	0.999636i	$-0.491417\pi$
$11$	0.178841	0.0539226	0.0269613	+	0.999636i	$0.491417\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−11.7449	−3.03252
$16$	0	0
$17$	1.03977	0.252182	0.126091	−	0.992019i	$-0.459757\pi$
$17$	1.03977	0.252182	0.126091	+	0.992019i	$0.459757\pi$
$18$	0	0
$19$	5.25040	1.20452	0.602262	−	0.798299i	$-0.294266\pi$
$19$	5.25040	1.20452	0.602262	+	0.798299i	$0.294266\pi$
$20$	0	0
$21$	6.56264	1.43209
$22$	0	0
$23$	−0.130358	−0.0271814	−0.0135907	−	0.999908i	$-0.504326\pi$
$23$	−0.130358	−0.0271814	−0.0135907	+	0.999908i	$0.504326\pi$
$24$	0	0
$25$	13.0601	2.61203
$26$	0	0
$27$	4.52683	0.871189
$28$	0	0
$29$	8.33214	1.54724	0.773620	−	0.633650i	$-0.218444\pi$
$29$	8.33214	1.54724	0.773620	+	0.633650i	$0.218444\pi$
$30$	0	0
$31$	2.94810	0.529494	0.264747	−	0.964318i	$-0.414712\pi$
$31$	2.94810	0.529494	0.264747	+	0.964318i	$0.414712\pi$
$32$	0	0
$33$	0.494261	0.0860397
$34$	0	0
$35$	−10.0914	−1.70575
$36$	0	0
$37$	6.03464	0.992088	0.496044	−	0.868297i	$-0.334785\pi$
$37$	6.03464	0.992088	0.496044	+	0.868297i	$0.334785\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.264592	0.0413224	0.0206612	−	0.999787i	$-0.493423\pi$
$41$	0.264592	0.0413224	0.0206612	+	0.999787i	$0.493423\pi$
$42$	0	0
$43$	0.564732	0.0861209	0.0430604	−	0.999072i	$-0.486289\pi$
$43$	0.564732	0.0861209	0.0430604	+	0.999072i	$0.486289\pi$
$44$	0	0
$45$	−19.7101	−2.93820
$46$	0	0
$47$	−10.9224	−1.59320	−0.796599	−	0.604508i	$-0.793370\pi$
$47$	−10.9224	−1.59320	−0.796599	+	0.604508i	$0.793370\pi$
$48$	0	0
$49$	−1.36129	−0.194471
$50$	0	0
$51$	2.87360	0.402385
$52$	0	0
$53$	2.41779	0.332108	0.166054	−	0.986117i	$-0.446897\pi$
$53$	2.41779	0.332108	0.166054	+	0.986117i	$0.446897\pi$
$54$	0	0
$55$	−0.760024	−0.102482
$56$	0	0
$57$	14.5105	1.92196
$58$	0	0
$59$	7.35568	0.957627	0.478814	−	0.877917i	$-0.341067\pi$
$59$	7.35568	0.957627	0.478814	+	0.877917i	$0.341067\pi$
$60$	0	0
$61$	−5.06501	−0.648508	−0.324254	−	0.945970i	$-0.605113\pi$
$61$	−5.06501	−0.648508	−0.324254	+	0.945970i	$0.605113\pi$
$62$	0	0
$63$	11.0133	1.38755
$64$	0	0
$65$	0	0
$66$	0	0
$67$	1.58480	0.193614	0.0968069	−	0.995303i	$-0.469137\pi$
$67$	1.58480	0.193614	0.0968069	+	0.995303i	$0.469137\pi$
$68$	0	0
$69$	−0.360267	−0.0433711
$70$	0	0
$71$	−10.3352	−1.22656	−0.613280	−	0.789866i	$-0.710150\pi$
$71$	−10.3352	−1.22656	−0.613280	+	0.789866i	$0.710150\pi$
$72$	0	0
$73$	14.3011	1.67382	0.836910	−	0.547340i	$-0.184360\pi$
$73$	14.3011	1.67382	0.836910	+	0.547340i	$0.184360\pi$
$74$	0	0
$75$	36.0941	4.16779
$76$	0	0
$77$	0.424675	0.0483962
$78$	0	0
$79$	−7.26636	−0.817529	−0.408764	−	0.912640i	$-0.634040\pi$
$79$	−7.26636	−0.817529	−0.408764	+	0.912640i	$0.634040\pi$
$80$	0	0
$81$	−1.40316	−0.155906
$82$	0	0
$83$	12.9706	1.42371	0.711854	−	0.702328i	$-0.247856\pi$
$83$	12.9706	1.42371	0.711854	+	0.702328i	$0.247856\pi$
$84$	0	0
$85$	−4.41874	−0.479280
$86$	0	0
$87$	23.0274	2.46880
$88$	0	0
$89$	−6.92320	−0.733858	−0.366929	−	0.930249i	$-0.619591\pi$
$89$	−6.92320	−0.733858	−0.366929	+	0.930249i	$0.619591\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	8.14762	0.844869
$94$	0	0
$95$	−22.3127	−2.28924
$96$	0	0
$97$	3.63494	0.369072	0.184536	−	0.982826i	$-0.440922\pi$
$97$	3.63494	0.369072	0.184536	+	0.982826i	$0.440922\pi$
$98$	0	0
$99$	0.829459	0.0833637
$100$	0	0
$101$	−8.25623	−0.821526	−0.410763	−	0.911742i	$-0.634738\pi$
$101$	−8.25623	−0.821526	−0.410763	+	0.911742i	$0.634738\pi$
$102$	0	0
$103$	−10.2022	−1.00525	−0.502625	−	0.864504i	$-0.667632\pi$
$103$	−10.2022	−1.00525	−0.502625	+	0.864504i	$0.667632\pi$
$104$	0	0
$105$	−27.8894	−2.72173
$106$	0	0
$107$	15.0640	1.45629	0.728145	−	0.685423i	$-0.240383\pi$
$107$	15.0640	1.45629	0.728145	+	0.685423i	$0.240383\pi$
$108$	0	0
$109$	3.50143	0.335376	0.167688	−	0.985840i	$-0.446370\pi$
$109$	3.50143	0.335376	0.167688	+	0.985840i	$0.446370\pi$
$110$	0	0
$111$	16.6778	1.58299
$112$	0	0
$113$	−14.1340	−1.32961	−0.664807	−	0.747015i	$-0.731486\pi$
$113$	−14.1340	−1.32961	−0.664807	+	0.747015i	$0.731486\pi$
$114$	0	0
$115$	0.553983	0.0516592
$116$	0	0
$117$	0	0
$118$	0	0
$119$	2.46904	0.226336
$120$	0	0
$121$	−10.9680	−0.997092
$122$	0	0
$123$	0.731250	0.0659346
$124$	0	0
$125$	−34.2533	−3.06371
$126$	0	0
$127$	−17.0668	−1.51444	−0.757218	−	0.653162i	$-0.773442\pi$
$127$	−17.0668	−1.51444	−0.757218	+	0.653162i	$0.773442\pi$
$128$	0	0
$129$	1.56074	0.137416
$130$	0	0
$131$	10.9785	0.959199	0.479599	−	0.877488i	$-0.340782\pi$
$131$	10.9785	0.959199	0.479599	+	0.877488i	$0.340782\pi$
$132$	0	0
$133$	12.4676	1.08108
$134$	0	0
$135$	−19.2378	−1.65572
$136$	0	0
$137$	−16.5900	−1.41738	−0.708689	−	0.705521i	$-0.750713\pi$
$137$	−16.5900	−1.41738	−0.708689	+	0.705521i	$0.750713\pi$
$138$	0	0
$139$	6.19057	0.525077	0.262539	−	0.964921i	$-0.415440\pi$
$139$	6.19057	0.525077	0.262539	+	0.964921i	$0.415440\pi$
$140$	0	0
$141$	−30.1861	−2.54213
$142$	0	0
$143$	0	0
$144$	0	0
$145$	−35.4093	−2.94058
$146$	0	0
$147$	−3.76219	−0.310300
$148$	0	0
$149$	−8.88398	−0.727804	−0.363902	−	0.931437i	$-0.618556\pi$
$149$	−8.88398	−0.727804	−0.363902	+	0.931437i	$0.618556\pi$
$150$	0	0
$151$	9.63748	0.784287	0.392144	−	0.919904i	$-0.371734\pi$
$151$	9.63748	0.784287	0.392144	+	0.919904i	$0.371734\pi$
$152$	0	0
$153$	4.82243	0.389870
$154$	0	0
$155$	−12.5286	−1.00632
$156$	0	0
$157$	7.92488	0.632474	0.316237	−	0.948680i	$-0.397581\pi$
$157$	7.92488	0.632474	0.316237	+	0.948680i	$0.397581\pi$
$158$	0	0
$159$	6.68200	0.529917
$160$	0	0
$161$	−0.309546	−0.0243957
$162$	0	0
$163$	−13.9125	−1.08971	−0.544857	−	0.838529i	$-0.683416\pi$
$163$	−13.9125	−1.08971	−0.544857	+	0.838529i	$0.683416\pi$
$164$	0	0
$165$	−2.10047	−0.163521
$166$	0	0
$167$	−4.14595	−0.320823	−0.160411	−	0.987050i	$-0.551282\pi$
$167$	−4.14595	−0.320823	−0.160411	+	0.987050i	$0.551282\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	24.3512	1.86218
$172$	0	0
$173$	11.5924	0.881356	0.440678	−	0.897665i	$-0.354738\pi$
$173$	11.5924	0.881356	0.440678	+	0.897665i	$0.354738\pi$
$174$	0	0
$175$	31.0125	2.34433
$176$	0	0
$177$	20.3288	1.52801
$178$	0	0
$179$	−20.3708	−1.52259	−0.761293	−	0.648407i	$-0.775435\pi$
$179$	−20.3708	−1.52259	−0.761293	+	0.648407i	$0.775435\pi$
$180$	0	0
$181$	24.3057	1.80663	0.903314	−	0.428980i	$-0.141127\pi$
$181$	24.3057	1.80663	0.903314	+	0.428980i	$0.141127\pi$
$182$	0	0
$183$	−13.9981	−1.03477
$184$	0	0
$185$	−25.6455	−1.88550
$186$	0	0
$187$	0.185954	0.0135983
$188$	0	0
$189$	10.7494	0.781904
$190$	0	0
$191$	6.89084	0.498604	0.249302	−	0.968426i	$-0.419799\pi$
$191$	6.89084	0.498604	0.249302	+	0.968426i	$0.419799\pi$
$192$	0	0
$193$	2.28762	0.164667	0.0823333	−	0.996605i	$-0.473763\pi$
$193$	2.28762	0.164667	0.0823333	+	0.996605i	$0.473763\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−0.444499	−0.0316692	−0.0158346	−	0.999875i	$-0.505041\pi$
$197$	−0.444499	−0.0316692	−0.0158346	+	0.999875i	$0.505041\pi$
$198$	0	0
$199$	16.1532	1.14507	0.572535	−	0.819880i	$-0.305960\pi$
$199$	16.1532	1.14507	0.572535	+	0.819880i	$0.305960\pi$
$200$	0	0
$201$	4.37989	0.308933
$202$	0	0
$203$	19.7855	1.38867
$204$	0	0
$205$	−1.12444	−0.0785345
$206$	0	0
$207$	−0.604594	−0.0420222
$208$	0	0
$209$	0.938986	0.0649510
$210$	0	0
$211$	−28.6933	−1.97533	−0.987665	−	0.156584i	$-0.949952\pi$
$211$	−28.6933	−1.97533	−0.987665	+	0.156584i	$0.949952\pi$
$212$	0	0
$213$	−28.5632	−1.95712
$214$	0	0
$215$	−2.39995	−0.163676
$216$	0	0
$217$	7.00054	0.475228
$218$	0	0
$219$	39.5238	2.67077
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−8.00947	−0.536354	−0.268177	−	0.963370i	$-0.586421\pi$
$223$	−8.00947	−0.536354	−0.268177	+	0.963370i	$0.586421\pi$
$224$	0	0
$225$	60.5725	4.03816
$226$	0	0
$227$	−1.26967	−0.0842708	−0.0421354	−	0.999112i	$-0.513416\pi$
$227$	−1.26967	−0.0842708	−0.0421354	+	0.999112i	$0.513416\pi$
$228$	0	0
$229$	−10.5928	−0.699993	−0.349997	−	0.936751i	$-0.613817\pi$
$229$	−10.5928	−0.699993	−0.349997	+	0.936751i	$0.613817\pi$
$230$	0	0
$231$	1.17367	0.0772218
$232$	0	0
$233$	−16.4601	−1.07833	−0.539167	−	0.842199i	$-0.681261\pi$
$233$	−16.4601	−1.07833	−0.539167	+	0.842199i	$0.681261\pi$
$234$	0	0
$235$	46.4172	3.02792
$236$	0	0
$237$	−20.0819	−1.30446
$238$	0	0
$239$	−7.73367	−0.500249	−0.250125	−	0.968214i	$-0.580472\pi$
$239$	−7.73367	−0.500249	−0.250125	+	0.968214i	$0.580472\pi$
$240$	0	0
$241$	12.9343	0.833174	0.416587	−	0.909096i	$-0.363226\pi$
$241$	12.9343	0.833174	0.416587	+	0.909096i	$0.363226\pi$
$242$	0	0
$243$	−17.4584	−1.11996
$244$	0	0
$245$	5.78512	0.369598
$246$	0	0
$247$	0	0
$248$	0	0
$249$	35.8467	2.27169
$250$	0	0
$251$	−10.6879	−0.674616	−0.337308	−	0.941394i	$-0.609516\pi$
$251$	−10.6879	−0.674616	−0.337308	+	0.941394i	$0.609516\pi$
$252$	0	0
$253$	−0.0233133	−0.00146569
$254$	0	0
$255$	−12.2120	−0.764746
$256$	0	0
$257$	11.4233	0.712565	0.356283	−	0.934378i	$-0.384044\pi$
$257$	11.4233	0.712565	0.356283	+	0.934378i	$0.384044\pi$
$258$	0	0
$259$	14.3298	0.890412
$260$	0	0
$261$	38.6442	2.39202
$262$	0	0
$263$	16.0357	0.988805	0.494402	−	0.869233i	$-0.335387\pi$
$263$	16.0357	0.988805	0.494402	+	0.869233i	$0.335387\pi$
$264$	0	0
$265$	−10.2749	−0.631183
$266$	0	0
$267$	−19.1336	−1.17095
$268$	0	0
$269$	−15.6155	−0.952092	−0.476046	−	0.879420i	$-0.657930\pi$
$269$	−15.6155	−0.952092	−0.476046	+	0.879420i	$0.657930\pi$
$270$	0	0
$271$	−1.89370	−0.115034	−0.0575170	−	0.998345i	$-0.518318\pi$
$271$	−1.89370	−0.115034	−0.0575170	+	0.998345i	$0.518318\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	2.33569	0.140847
$276$	0	0
$277$	15.0397	0.903645	0.451823	−	0.892108i	$-0.350774\pi$
$277$	15.0397	0.903645	0.451823	+	0.892108i	$0.350774\pi$
$278$	0	0
$279$	13.6732	0.818592
$280$	0	0
$281$	−7.07674	−0.422163	−0.211081	−	0.977468i	$-0.567699\pi$
$281$	−7.07674	−0.422163	−0.211081	+	0.977468i	$0.567699\pi$
$282$	0	0
$283$	−20.3692	−1.21082	−0.605412	−	0.795912i	$-0.706992\pi$
$283$	−20.3692	−1.21082	−0.605412	+	0.795912i	$0.706992\pi$
$284$	0	0
$285$	−61.6654	−3.65274
$286$	0	0
$287$	0.628300	0.0370873
$288$	0	0
$289$	−15.9189	−0.936404
$290$	0	0
$291$	10.0458	0.588897
$292$	0	0
$293$	−13.1872	−0.770407	−0.385204	−	0.922832i	$-0.625869\pi$
$293$	−13.1872	−0.770407	−0.385204	+	0.922832i	$0.625869\pi$
$294$	0	0
$295$	−31.2596	−1.82000
$296$	0	0
$297$	0.809583	0.0469768
$298$	0	0
$299$	0	0
$300$	0	0
$301$	1.34101	0.0772946
$302$	0	0
$303$	−22.8176	−1.31084
$304$	0	0
$305$	21.5249	1.23251
$306$	0	0
$307$	−24.1002	−1.37547	−0.687736	−	0.725960i	$-0.741396\pi$
$307$	−24.1002	−1.37547	−0.687736	+	0.725960i	$0.741396\pi$
$308$	0	0
$309$	−28.1956	−1.60399
$310$	0	0
$311$	22.3374	1.26664	0.633319	−	0.773891i	$-0.281692\pi$
$311$	22.3374	1.26664	0.633319	+	0.773891i	$0.281692\pi$
$312$	0	0
$313$	−0.212361	−0.0120034	−0.00600169	−	0.999982i	$-0.501910\pi$
$313$	−0.212361	−0.0120034	−0.00600169	+	0.999982i	$0.501910\pi$
$314$	0	0
$315$	−46.8035	−2.63708
$316$	0	0
$317$	14.7520	0.828553	0.414276	−	0.910151i	$-0.364035\pi$
$317$	14.7520	0.828553	0.414276	+	0.910151i	$0.364035\pi$
$318$	0	0
$319$	1.49013	0.0834312
$320$	0	0
$321$	41.6321	2.32368
$322$	0	0
$323$	5.45921	0.303759
$324$	0	0
$325$	0	0
$326$	0	0
$327$	9.67685	0.535131
$328$	0	0
$329$	−25.9363	−1.42992
$330$	0	0
$331$	26.6038	1.46228	0.731138	−	0.682229i	$-0.238989\pi$
$331$	26.6038	1.46228	0.731138	+	0.682229i	$0.238989\pi$
$332$	0	0
$333$	27.9884	1.53376
$334$	0	0
$335$	−6.73495	−0.367970
$336$	0	0
$337$	15.7138	0.855986	0.427993	−	0.903782i	$-0.359221\pi$
$337$	15.7138	0.855986	0.427993	+	0.903782i	$0.359221\pi$
$338$	0	0
$339$	−39.0619	−2.12155
$340$	0	0
$341$	0.527241	0.0285517
$342$	0	0
$343$	−19.8547	−1.07205
$344$	0	0
$345$	1.53104	0.0824282
$346$	0	0
$347$	5.19701	0.278990	0.139495	−	0.990223i	$-0.455452\pi$
$347$	5.19701	0.278990	0.139495	+	0.990223i	$0.455452\pi$
$348$	0	0
$349$	13.6677	0.731614	0.365807	−	0.930691i	$-0.380793\pi$
$349$	13.6677	0.731614	0.365807	+	0.930691i	$0.380793\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	5.88734	0.313352	0.156676	−	0.987650i	$-0.449922\pi$
$353$	5.88734	0.313352	0.156676	+	0.987650i	$0.449922\pi$
$354$	0	0
$355$	43.9216	2.33112
$356$	0	0
$357$	6.82365	0.361146
$358$	0	0
$359$	12.9742	0.684750	0.342375	−	0.939563i	$-0.388769\pi$
$359$	12.9742	0.684750	0.342375	+	0.939563i	$0.388769\pi$
$360$	0	0
$361$	8.56668	0.450878
$362$	0	0
$363$	−30.3122	−1.59098
$364$	0	0
$365$	−60.7758	−3.18115
$366$	0	0
$367$	15.9533	0.832754	0.416377	−	0.909192i	$-0.363300\pi$
$367$	15.9533	0.832754	0.416377	+	0.909192i	$0.363300\pi$
$368$	0	0
$369$	1.22717	0.0638839
$370$	0	0
$371$	5.74126	0.298072
$372$	0	0
$373$	−20.3108	−1.05165	−0.525826	−	0.850592i	$-0.676244\pi$
$373$	−20.3108	−1.05165	−0.525826	+	0.850592i	$0.676244\pi$
$374$	0	0
$375$	−94.6654	−4.88850
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−12.2946	−0.631531	−0.315766	−	0.948837i	$-0.602261\pi$
$379$	−12.2946	−0.631531	−0.315766	+	0.948837i	$0.602261\pi$
$380$	0	0
$381$	−47.1674	−2.41646
$382$	0	0
$383$	16.3629	0.836104	0.418052	−	0.908423i	$-0.362713\pi$
$383$	16.3629	0.836104	0.418052	+	0.908423i	$0.362713\pi$
$384$	0	0
$385$	−1.80475	−0.0919786
$386$	0	0
$387$	2.61921	0.133142
$388$	0	0
$389$	4.25810	0.215894	0.107947	−	0.994157i	$-0.465572\pi$
$389$	4.25810	0.215894	0.107947	+	0.994157i	$0.465572\pi$
$390$	0	0
$391$	−0.135542	−0.00685466
$392$	0	0
$393$	30.3412	1.53051
$394$	0	0
$395$	30.8800	1.55374
$396$	0	0
$397$	−14.4527	−0.725362	−0.362681	−	0.931913i	$-0.618138\pi$
$397$	−14.4527	−0.725362	−0.362681	+	0.931913i	$0.618138\pi$
$398$	0	0
$399$	34.4565	1.72498
$400$	0	0
$401$	−30.8993	−1.54304	−0.771520	−	0.636205i	$-0.780503\pi$
$401$	−30.8993	−1.54304	−0.771520	+	0.636205i	$0.780503\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	5.96302	0.296305
$406$	0	0
$407$	1.07924	0.0534959
$408$	0	0
$409$	−20.8398	−1.03046	−0.515231	−	0.857051i	$-0.672294\pi$
$409$	−20.8398	−1.03046	−0.515231	+	0.857051i	$0.672294\pi$
$410$	0	0
$411$	−45.8495	−2.26159
$412$	0	0
$413$	17.4668	0.859483
$414$	0	0
$415$	−55.1214	−2.70580
$416$	0	0
$417$	17.1088	0.837822
$418$	0	0
$419$	−24.0374	−1.17430	−0.587152	−	0.809477i	$-0.699751\pi$
$419$	−24.0374	−1.17430	−0.587152	+	0.809477i	$0.699751\pi$
$420$	0	0
$421$	−33.3068	−1.62327	−0.811637	−	0.584163i	$-0.801423\pi$
$421$	−33.3068	−1.62327	−0.811637	+	0.584163i	$0.801423\pi$
$422$	0	0
$423$	−50.6578	−2.46307
$424$	0	0
$425$	13.5795	0.658705
$426$	0	0
$427$	−12.0273	−0.582044
$428$	0	0
$429$	0	0
$430$	0	0
$431$	29.4008	1.41618	0.708092	−	0.706120i	$-0.249556\pi$
$431$	29.4008	1.41618	0.708092	+	0.706120i	$0.249556\pi$
$432$	0	0
$433$	−23.4616	−1.12749	−0.563747	−	0.825948i	$-0.690641\pi$
$433$	−23.4616	−1.12749	−0.563747	+	0.825948i	$0.690641\pi$
$434$	0	0
$435$	−97.8602	−4.69204
$436$	0	0
$437$	−0.684429	−0.0327407
$438$	0	0
$439$	0.447505	0.0213582	0.0106791	−	0.999943i	$-0.496601\pi$
$439$	0.447505	0.0213582	0.0106791	+	0.999943i	$0.496601\pi$
$440$	0	0
$441$	−6.31364	−0.300650
$442$	0	0
$443$	−35.8264	−1.70216	−0.851082	−	0.525033i	$-0.824053\pi$
$443$	−35.8264	−1.70216	−0.851082	+	0.525033i	$0.824053\pi$
$444$	0	0
$445$	29.4217	1.39472
$446$	0	0
$447$	−24.5526	−1.16130
$448$	0	0
$449$	4.90862	0.231652	0.115826	−	0.993270i	$-0.463048\pi$
$449$	4.90862	0.231652	0.115826	+	0.993270i	$0.463048\pi$
$450$	0	0
$451$	0.0473199	0.00222821
$452$	0	0
$453$	26.6350	1.25142
$454$	0	0
$455$	0	0
$456$	0	0
$457$	24.2220	1.13306	0.566528	−	0.824043i	$-0.308286\pi$
$457$	24.2220	1.13306	0.566528	+	0.824043i	$0.308286\pi$
$458$	0	0
$459$	4.70687	0.219698
$460$	0	0
$461$	22.8404	1.06378	0.531892	−	0.846812i	$-0.321481\pi$
$461$	22.8404	1.06378	0.531892	+	0.846812i	$0.321481\pi$
$462$	0	0
$463$	23.8907	1.11030	0.555148	−	0.831751i	$-0.312661\pi$
$463$	23.8907	1.11030	0.555148	+	0.831751i	$0.312661\pi$
$464$	0	0
$465$	−34.6251	−1.60570
$466$	0	0
$467$	10.3783	0.480248	0.240124	−	0.970742i	$-0.422812\pi$
$467$	10.3783	0.480248	0.240124	+	0.970742i	$0.422812\pi$
$468$	0	0
$469$	3.76325	0.173771
$470$	0	0
$471$	21.9019	1.00919
$472$	0	0
$473$	0.100997	0.00464386
$474$	0	0
$475$	68.5709	3.14625
$476$	0	0
$477$	11.2136	0.513436
$478$	0	0
$479$	−17.1365	−0.782986	−0.391493	−	0.920181i	$-0.628041\pi$
$479$	−17.1365	−0.782986	−0.391493	+	0.920181i	$0.628041\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	−0.855490	−0.0389261
$484$	0	0
$485$	−15.4475	−0.701433
$486$	0	0
$487$	−24.9093	−1.12875	−0.564373	−	0.825520i	$-0.690882\pi$
$487$	−24.9093	−1.12875	−0.564373	+	0.825520i	$0.690882\pi$
$488$	0	0
$489$	−38.4499	−1.73876
$490$	0	0
$491$	21.7352	0.980894	0.490447	−	0.871471i	$-0.336834\pi$
$491$	21.7352	0.980894	0.490447	+	0.871471i	$0.336834\pi$
$492$	0	0
$493$	8.66352	0.390185
$494$	0	0
$495$	−3.52497	−0.158436
$496$	0	0
$497$	−24.5419	−1.10085
$498$	0	0
$499$	−10.3216	−0.462060	−0.231030	−	0.972947i	$-0.574209\pi$
$499$	−10.3216	−0.462060	−0.231030	+	0.972947i	$0.574209\pi$
$500$	0	0
$501$	−11.4581	−0.511910
$502$	0	0
$503$	8.83728	0.394035	0.197017	−	0.980400i	$-0.436874\pi$
$503$	8.83728	0.394035	0.197017	+	0.980400i	$0.436874\pi$
$504$	0	0
$505$	35.0867	1.56134
$506$	0	0
$507$	0	0
$508$	0	0
$509$	6.22098	0.275740	0.137870	−	0.990450i	$-0.455974\pi$
$509$	6.22098	0.275740	0.137870	+	0.990450i	$0.455974\pi$
$510$	0	0
$511$	33.9594	1.50228
$512$	0	0
$513$	23.7677	1.04937
$514$	0	0
$515$	43.3564	1.91051
$516$	0	0
$517$	−1.95337	−0.0859093
$518$	0	0
$519$	32.0378	1.40630
$520$	0	0
$521$	−3.17145	−0.138944	−0.0694720	−	0.997584i	$-0.522131\pi$
$521$	−3.17145	−0.138944	−0.0694720	+	0.997584i	$0.522131\pi$
$522$	0	0
$523$	7.75072	0.338915	0.169458	−	0.985537i	$-0.445798\pi$
$523$	7.75072	0.338915	0.169458	+	0.985537i	$0.445798\pi$
$524$	0	0
$525$	85.7089	3.74064
$526$	0	0
$527$	3.06535	0.133529
$528$	0	0
$529$	−22.9830	−0.999261
$530$	0	0
$531$	34.1154	1.48048
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−64.0177	−2.76773
$536$	0	0
$537$	−56.2986	−2.42946
$538$	0	0
$539$	−0.243455	−0.0104864
$540$	0	0
$541$	4.89188	0.210318	0.105159	−	0.994455i	$-0.466465\pi$
$541$	4.89188	0.210318	0.105159	+	0.994455i	$0.466465\pi$
$542$	0	0
$543$	67.1733	2.88268
$544$	0	0
$545$	−14.8801	−0.637393
$546$	0	0
$547$	9.18288	0.392631	0.196316	−	0.980541i	$-0.437102\pi$
$547$	9.18288	0.392631	0.196316	+	0.980541i	$0.437102\pi$
$548$	0	0
$549$	−23.4913	−1.00259
$550$	0	0
$551$	43.7471	1.86369
$552$	0	0
$553$	−17.2547	−0.733743
$554$	0	0
$555$	−70.8762	−3.00853
$556$	0	0
$557$	−1.62368	−0.0687975	−0.0343988	−	0.999408i	$-0.510952\pi$
$557$	−1.62368	−0.0687975	−0.0343988	+	0.999408i	$0.510952\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0.513918	0.0216976
$562$	0	0
$563$	32.9459	1.38850	0.694251	−	0.719733i	$-0.255736\pi$
$563$	32.9459	1.38850	0.694251	+	0.719733i	$0.255736\pi$
$564$	0	0
$565$	60.0655	2.52698
$566$	0	0
$567$	−3.33193	−0.139928
$568$	0	0
$569$	18.2984	0.767110	0.383555	−	0.923518i	$-0.374700\pi$
$569$	18.2984	0.767110	0.383555	+	0.923518i	$0.374700\pi$
$570$	0	0
$571$	40.2247	1.68335	0.841676	−	0.539984i	$-0.181570\pi$
$571$	40.2247	1.68335	0.841676	+	0.539984i	$0.181570\pi$
$572$	0	0
$573$	19.0441	0.795580
$574$	0	0
$575$	−1.70249	−0.0709986
$576$	0	0
$577$	−2.30223	−0.0958429	−0.0479215	−	0.998851i	$-0.515260\pi$
$577$	−2.30223	−0.0958429	−0.0479215	+	0.998851i	$0.515260\pi$
$578$	0	0
$579$	6.32227	0.262744
$580$	0	0
$581$	30.7999	1.27780
$582$	0	0
$583$	0.432399	0.0179081
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14.7259	0.607802	0.303901	−	0.952704i	$-0.401711\pi$
$587$	14.7259	0.607802	0.303901	+	0.952704i	$0.401711\pi$
$588$	0	0
$589$	15.4787	0.637788
$590$	0	0
$591$	−1.22846	−0.0505319
$592$	0	0
$593$	17.8020	0.731041	0.365521	−	0.930803i	$-0.380891\pi$
$593$	17.8020	0.731041	0.365521	+	0.930803i	$0.380891\pi$
$594$	0	0
$595$	−10.4927	−0.430160
$596$	0	0
$597$	44.6424	1.82709
$598$	0	0
$599$	4.94303	0.201967	0.100983	−	0.994888i	$-0.467801\pi$
$599$	4.94303	0.201967	0.100983	+	0.994888i	$0.467801\pi$
$600$	0	0
$601$	21.3884	0.872451	0.436226	−	0.899837i	$-0.356315\pi$
$601$	21.3884	0.872451	0.436226	+	0.899837i	$0.356315\pi$
$602$	0	0
$603$	7.35024	0.299325
$604$	0	0
$605$	46.6110	1.89501
$606$	0	0
$607$	12.8970	0.523471	0.261736	−	0.965140i	$-0.415705\pi$
$607$	12.8970	0.523471	0.261736	+	0.965140i	$0.415705\pi$
$608$	0	0
$609$	54.6808	2.21578
$610$	0	0
$611$	0	0
$612$	0	0
$613$	−21.5965	−0.872273	−0.436137	−	0.899880i	$-0.643654\pi$
$613$	−21.5965	−0.872273	−0.436137	+	0.899880i	$0.643654\pi$
$614$	0	0
$615$	−3.10761	−0.125311
$616$	0	0
$617$	−28.1309	−1.13251	−0.566253	−	0.824232i	$-0.691607\pi$
$617$	−28.1309	−1.13251	−0.566253	+	0.824232i	$0.691607\pi$
$618$	0	0
$619$	−21.7346	−0.873588	−0.436794	−	0.899562i	$-0.643886\pi$
$619$	−21.7346	−0.873588	−0.436794	+	0.899562i	$0.643886\pi$
$620$	0	0
$621$	−0.590107	−0.0236802
$622$	0	0
$623$	−16.4398	−0.658647
$624$	0	0
$625$	80.2663	3.21065
$626$	0	0
$627$	2.59506	0.103637
$628$	0	0
$629$	6.27464	0.250186
$630$	0	0
$631$	−37.5557	−1.49507	−0.747534	−	0.664224i	$-0.768762\pi$
$631$	−37.5557	−1.49507	−0.747534	+	0.664224i	$0.768762\pi$
$632$	0	0
$633$	−79.2993	−3.15187
$634$	0	0
$635$	72.5293	2.87824
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−47.9342	−1.89625
$640$	0	0
$641$	24.0033	0.948072	0.474036	−	0.880505i	$-0.342797\pi$
$641$	24.0033	0.948072	0.474036	+	0.880505i	$0.342797\pi$
$642$	0	0
$643$	−38.2324	−1.50774	−0.753869	−	0.657025i	$-0.771814\pi$
$643$	−38.2324	−1.50774	−0.753869	+	0.657025i	$0.771814\pi$
$644$	0	0
$645$	−6.63273	−0.261163
$646$	0	0
$647$	−34.4831	−1.35567	−0.677835	−	0.735214i	$-0.737082\pi$
$647$	−34.4831	−1.35567	−0.677835	+	0.735214i	$0.737082\pi$
$648$	0	0
$649$	1.31550	0.0516377
$650$	0	0
$651$	19.3473	0.758281
$652$	0	0
$653$	1.42018	0.0555761	0.0277881	−	0.999614i	$-0.491154\pi$
$653$	1.42018	0.0555761	0.0277881	+	0.999614i	$0.491154\pi$
$654$	0	0
$655$	−46.6557	−1.82299
$656$	0	0
$657$	66.3282	2.58771
$658$	0	0
$659$	18.1022	0.705160	0.352580	−	0.935782i	$-0.385304\pi$
$659$	18.1022	0.705160	0.352580	+	0.935782i	$0.385304\pi$
$660$	0	0
$661$	7.08181	0.275451	0.137725	−	0.990470i	$-0.456021\pi$
$661$	7.08181	0.275451	0.137725	+	0.990470i	$0.456021\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−52.9837	−2.05462
$666$	0	0
$667$	−1.08616	−0.0420562
$668$	0	0
$669$	−22.1357	−0.855815
$670$	0	0
$671$	−0.905831	−0.0349692
$672$	0	0
$673$	−25.6958	−0.990500	−0.495250	−	0.868750i	$-0.664924\pi$
$673$	−25.6958	−0.990500	−0.495250	+	0.868750i	$0.664924\pi$
$674$	0	0
$675$	59.1210	2.27557
$676$	0	0
$677$	14.4586	0.555689	0.277845	−	0.960626i	$-0.410380\pi$
$677$	14.4586	0.555689	0.277845	+	0.960626i	$0.410380\pi$
$678$	0	0
$679$	8.63150	0.331247
$680$	0	0
$681$	−3.50896	−0.134464
$682$	0	0
$683$	−44.5985	−1.70652	−0.853258	−	0.521489i	$-0.825377\pi$
$683$	−44.5985	−1.70652	−0.853258	+	0.521489i	$0.825377\pi$
$684$	0	0
$685$	70.5028	2.69377
$686$	0	0
$687$	−29.2752	−1.11692
$688$	0	0
$689$	0	0
$690$	0	0
$691$	37.0493	1.40942	0.704711	−	0.709495i	$-0.251077\pi$
$691$	37.0493	1.40942	0.704711	+	0.709495i	$0.251077\pi$
$692$	0	0
$693$	1.96963	0.0748200
$694$	0	0
$695$	−26.3082	−0.997927
$696$	0	0
$697$	0.275115	0.0104207
$698$	0	0
$699$	−45.4905	−1.72061
$700$	0	0
$701$	16.7439	0.632409	0.316204	−	0.948691i	$-0.397591\pi$
$701$	16.7439	0.632409	0.316204	+	0.948691i	$0.397591\pi$
$702$	0	0
$703$	31.6842	1.19499
$704$	0	0
$705$	128.283	4.83140
$706$	0	0
$707$	−19.6052	−0.737330
$708$	0	0
$709$	38.5011	1.44594	0.722970	−	0.690879i	$-0.242776\pi$
$709$	38.5011	1.44594	0.722970	+	0.690879i	$0.242776\pi$
$710$	0	0
$711$	−33.7011	−1.26389
$712$	0	0
$713$	−0.384307	−0.0143924
$714$	0	0
$715$	0	0
$716$	0	0
$717$	−21.3734	−0.798206
$718$	0	0
$719$	−29.6719	−1.10658	−0.553288	−	0.832990i	$-0.686627\pi$
$719$	−29.6719	−1.10658	−0.553288	+	0.832990i	$0.686627\pi$
$720$	0	0
$721$	−24.2261	−0.902225
$722$	0	0
$723$	35.7465	1.32943
$724$	0	0
$725$	108.819	4.04143
$726$	0	0
$727$	44.9510	1.66714	0.833570	−	0.552414i	$-0.186293\pi$
$727$	44.9510	1.66714	0.833570	+	0.552414i	$0.186293\pi$
$728$	0	0
$729$	−44.0400	−1.63111
$730$	0	0
$731$	0.587193	0.0217181
$732$	0	0
$733$	−35.8327	−1.32351	−0.661756	−	0.749719i	$-0.730189\pi$
$733$	−35.8327	−1.32351	−0.661756	+	0.749719i	$0.730189\pi$
$734$	0	0
$735$	15.9883	0.589736
$736$	0	0
$737$	0.283427	0.0104402
$738$	0	0
$739$	−32.8559	−1.20862	−0.604312	−	0.796748i	$-0.706552\pi$
$739$	−32.8559	−1.20862	−0.604312	+	0.796748i	$0.706552\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	0.829458	0.0304299	0.0152149	−	0.999884i	$-0.495157\pi$
$743$	0.829458	0.0304299	0.0152149	+	0.999884i	$0.495157\pi$
$744$	0	0
$745$	37.7544	1.38322
$746$	0	0
$747$	60.1572	2.20104
$748$	0	0
$749$	35.7709	1.30704
$750$	0	0
$751$	13.0051	0.474564	0.237282	−	0.971441i	$-0.423744\pi$
$751$	13.0051	0.474564	0.237282	+	0.971441i	$0.423744\pi$
$752$	0	0
$753$	−29.5381	−1.07643
$754$	0	0
$755$	−40.9566	−1.49056
$756$	0	0
$757$	−40.3396	−1.46617	−0.733084	−	0.680138i	$-0.761920\pi$
$757$	−40.3396	−1.46617	−0.733084	+	0.680138i	$0.761920\pi$
$758$	0	0
$759$	−0.0644306	−0.00233868
$760$	0	0
$761$	−21.9457	−0.795529	−0.397765	−	0.917487i	$-0.630214\pi$
$761$	−21.9457	−0.795529	−0.397765	+	0.917487i	$0.630214\pi$
$762$	0	0
$763$	8.31447	0.301004
$764$	0	0
$765$	−20.4940	−0.740961
$766$	0	0
$767$	0	0
$768$	0	0
$769$	−25.6808	−0.926072	−0.463036	−	0.886340i	$-0.653240\pi$
$769$	−25.6808	−0.926072	−0.463036	+	0.886340i	$0.653240\pi$
$770$	0	0
$771$	31.5704	1.13698
$772$	0	0
$773$	−12.9510	−0.465814	−0.232907	−	0.972499i	$-0.574824\pi$
$773$	−12.9510	−0.465814	−0.232907	+	0.972499i	$0.574824\pi$
$774$	0	0
$775$	38.5025	1.38305
$776$	0	0
$777$	39.6031	1.42075
$778$	0	0
$779$	1.38921	0.0497738
$780$	0	0
$781$	−1.84835	−0.0661392
$782$	0	0
$783$	37.7182	1.34794
$784$	0	0
$785$	−33.6785	−1.20204
$786$	0	0
$787$	18.7237	0.667429	0.333714	−	0.942674i	$-0.391698\pi$
$787$	18.7237	0.667429	0.333714	+	0.942674i	$0.391698\pi$
$788$	0	0
$789$	44.3177	1.57775
$790$	0	0
$791$	−33.5625	−1.19335
$792$	0	0
$793$	0	0
$794$	0	0
$795$	−28.3967	−1.00713
$796$	0	0
$797$	24.3073	0.861007	0.430504	−	0.902589i	$-0.358336\pi$
$797$	24.3073	0.861007	0.430504	+	0.902589i	$0.358336\pi$
$798$	0	0
$799$	−11.3568	−0.401775
$800$	0	0
$801$	−32.1096	−1.13454
$802$	0	0
$803$	2.55763	0.0902567
$804$	0	0
$805$	1.31549	0.0463648
$806$	0	0
$807$	−43.1563	−1.51917
$808$	0	0
$809$	−28.6517	−1.00734	−0.503669	−	0.863896i	$-0.668017\pi$
$809$	−28.6517	−1.00734	−0.503669	+	0.863896i	$0.668017\pi$
$810$	0	0
$811$	46.5874	1.63590	0.817952	−	0.575287i	$-0.195109\pi$
$811$	46.5874	1.63590	0.817952	+	0.575287i	$0.195109\pi$
$812$	0	0
$813$	−5.23359	−0.183550
$814$	0	0
$815$	59.1244	2.07104
$816$	0	0
$817$	2.96507	0.103735
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−54.4547	−1.90048	−0.950242	−	0.311513i	$-0.899164\pi$
$821$	−54.4547	−1.90048	−0.950242	+	0.311513i	$0.899164\pi$
$822$	0	0
$823$	−30.5572	−1.06516	−0.532578	−	0.846381i	$-0.678777\pi$
$823$	−30.5572	−1.06516	−0.532578	+	0.846381i	$0.678777\pi$
$824$	0	0
$825$	6.45511	0.224738
$826$	0	0
$827$	12.5680	0.437031	0.218515	−	0.975834i	$-0.429879\pi$
$827$	12.5680	0.437031	0.218515	+	0.975834i	$0.429879\pi$
$828$	0	0
$829$	−23.8964	−0.829956	−0.414978	−	0.909831i	$-0.636211\pi$
$829$	−23.8964	−0.829956	−0.414978	+	0.909831i	$0.636211\pi$
$830$	0	0
$831$	41.5649	1.44187
$832$	0	0
$833$	−1.41544	−0.0490419
$834$	0	0
$835$	17.6191	0.609735
$836$	0	0
$837$	13.3455	0.461290
$838$	0	0
$839$	16.8967	0.583341	0.291670	−	0.956519i	$-0.405789\pi$
$839$	16.8967	0.583341	0.291670	+	0.956519i	$0.405789\pi$
$840$	0	0
$841$	40.4246	1.39395
$842$	0	0
$843$	−19.5579	−0.673610
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−26.0446	−0.894903
$848$	0	0
$849$	−56.2941	−1.93201
$850$	0	0
$851$	−0.786660	−0.0269664
$852$	0	0
$853$	32.8438	1.12455	0.562275	−	0.826950i	$-0.309926\pi$
$853$	32.8438	1.12455	0.562275	+	0.826950i	$0.309926\pi$
$854$	0	0
$855$	−103.486	−3.53914
$856$	0	0
$857$	−33.2239	−1.13491	−0.567453	−	0.823406i	$-0.692071\pi$
$857$	−33.2239	−1.13491	−0.567453	+	0.823406i	$0.692071\pi$
$858$	0	0
$859$	24.6389	0.840669	0.420334	−	0.907369i	$-0.361913\pi$
$859$	24.6389	0.840669	0.420334	+	0.907369i	$0.361913\pi$
$860$	0	0
$861$	1.73642	0.0591772
$862$	0	0
$863$	22.6519	0.771081	0.385540	−	0.922691i	$-0.374015\pi$
$863$	22.6519	0.771081	0.385540	+	0.922691i	$0.374015\pi$
$864$	0	0
$865$	−49.2646	−1.67505
$866$	0	0
$867$	−43.9948	−1.49414
$868$	0	0
$869$	−1.29952	−0.0440833
$870$	0	0
$871$	0	0
$872$	0	0
$873$	16.8587	0.570581
$874$	0	0
$875$	−81.3377	−2.74972
$876$	0	0
$877$	12.7793	0.431526	0.215763	−	0.976446i	$-0.430776\pi$
$877$	12.7793	0.431526	0.215763	+	0.976446i	$0.430776\pi$
$878$	0	0
$879$	−36.4454	−1.22927
$880$	0	0
$881$	−9.24001	−0.311304	−0.155652	−	0.987812i	$-0.549748\pi$
$881$	−9.24001	−0.311304	−0.155652	+	0.987812i	$0.549748\pi$
$882$	0	0
$883$	5.52457	0.185917	0.0929583	−	0.995670i	$-0.470368\pi$
$883$	5.52457	0.185917	0.0929583	+	0.995670i	$0.470368\pi$
$884$	0	0
$885$	−86.3917	−2.90402
$886$	0	0
$887$	56.0332	1.88141	0.940705	−	0.339226i	$-0.110165\pi$
$887$	56.0332	1.88141	0.940705	+	0.339226i	$0.110165\pi$
$888$	0	0
$889$	−40.5268	−1.35923
$890$	0	0
$891$	−0.250942	−0.00840687
$892$	0	0
$893$	−57.3470	−1.91904
$894$	0	0
$895$	86.5703	2.89373
$896$	0	0
$897$	0	0
$898$	0	0
$899$	24.5640	0.819254
$900$	0	0
$901$	2.51394	0.0837516
$902$	0	0
$903$	3.70614	0.123332
$904$	0	0
$905$	−103.292	−3.43356
$906$	0	0
$907$	55.8182	1.85341	0.926706	−	0.375787i	$-0.122627\pi$
$907$	55.8182	1.85341	0.926706	+	0.375787i	$0.122627\pi$
$908$	0	0
$909$	−38.2921	−1.27007
$910$	0	0
$911$	7.00193	0.231984	0.115992	−	0.993250i	$-0.462995\pi$
$911$	7.00193	0.231984	0.115992	+	0.993250i	$0.462995\pi$
$912$	0	0
$913$	2.31967	0.0767700
$914$	0	0
$915$	59.4880	1.96661
$916$	0	0
$917$	26.0696	0.860893
$918$	0	0
$919$	8.10196	0.267259	0.133629	−	0.991031i	$-0.457337\pi$
$919$	8.10196	0.267259	0.133629	+	0.991031i	$0.457337\pi$
$920$	0	0
$921$	−66.6055	−2.19473
$922$	0	0
$923$	0	0
$924$	0	0
$925$	78.8131	2.59136
$926$	0	0
$927$	−47.3174	−1.55411
$928$	0	0
$929$	−46.4284	−1.52326	−0.761632	−	0.648009i	$-0.775602\pi$
$929$	−46.4284	−1.52326	−0.761632	+	0.648009i	$0.775602\pi$
$930$	0	0
$931$	−7.14734	−0.234245
$932$	0	0
$933$	61.7336	2.02107
$934$	0	0
$935$	−0.790252	−0.0258440
$936$	0	0
$937$	21.3117	0.696222	0.348111	−	0.937453i	$-0.386823\pi$
$937$	21.3117	0.696222	0.348111	+	0.937453i	$0.386823\pi$
$938$	0	0
$939$	−0.586901	−0.0191528
$940$	0	0
$941$	11.3688	0.370611	0.185306	−	0.982681i	$-0.440673\pi$
$941$	11.3688	0.370611	0.185306	+	0.982681i	$0.440673\pi$
$942$	0	0
$943$	−0.0344916	−0.00112320
$944$	0	0
$945$	−45.6819	−1.48603
$946$	0	0
$947$	−43.1773	−1.40307	−0.701537	−	0.712633i	$-0.747502\pi$
$947$	−43.1773	−1.40307	−0.701537	+	0.712633i	$0.747502\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	40.7698	1.32205
$952$	0	0
$953$	6.69299	0.216807	0.108404	−	0.994107i	$-0.465426\pi$
$953$	6.69299	0.216807	0.108404	+	0.994107i	$0.465426\pi$
$954$	0	0
$955$	−29.2842	−0.947613
$956$	0	0
$957$	4.11825	0.133124
$958$	0	0
$959$	−39.3945	−1.27211
$960$	0	0
$961$	−22.3087	−0.719636
$962$	0	0
$963$	69.8663	2.25141
$964$	0	0
$965$	−9.72175	−0.312954
$966$	0	0
$967$	11.8674	0.381631	0.190815	−	0.981626i	$-0.438887\pi$
$967$	11.8674	0.381631	0.190815	+	0.981626i	$0.438887\pi$
$968$	0	0
$969$	15.0876	0.484682
$970$	0	0
$971$	3.26604	0.104812	0.0524062	−	0.998626i	$-0.483311\pi$
$971$	3.26604	0.104812	0.0524062	+	0.998626i	$0.483311\pi$
$972$	0	0
$973$	14.7001	0.471264
$974$	0	0
$975$	0	0
$976$	0	0
$977$	36.0457	1.15320	0.576602	−	0.817025i	$-0.304378\pi$
$977$	36.0457	1.15320	0.576602	+	0.817025i	$0.304378\pi$
$978$	0	0
$979$	−1.23815	−0.0395715
$980$	0	0
$981$	16.2395	0.518487
$982$	0	0
$983$	−19.3513	−0.617209	−0.308605	−	0.951190i	$-0.599862\pi$
$983$	−19.3513	−0.617209	−0.308605	+	0.951190i	$0.599862\pi$
$984$	0	0
$985$	1.88900	0.0601884
$986$	0	0
$987$	−71.6799	−2.28160
$988$	0	0
$989$	−0.0736171	−0.00234089
$990$	0	0
$991$	−11.4964	−0.365196	−0.182598	−	0.983188i	$-0.558451\pi$
$991$	−11.4964	−0.365196	−0.182598	+	0.983188i	$0.558451\pi$
$992$	0	0
$993$	73.5245	2.33323
$994$	0	0
$995$	−68.6466	−2.17624
$996$	0	0
$997$	−53.5272	−1.69522	−0.847612	−	0.530617i	$-0.821960\pi$
$997$	−53.5272	−1.69522	−0.847612	+	0.530617i	$0.821960\pi$
$998$	0	0
$999$	27.3178	0.864296

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1352.2.a.m.1.5	✓	6
4.3	odd	2		2704.2.a.bf.1.2		6
13.2	odd	12		1352.2.o.h.1161.4		24
13.3	even	3		1352.2.i.m.529.2		12
13.4	even	6		1352.2.i.n.1329.2		12
13.5	odd	4		1352.2.f.g.337.10		12
13.6	odd	12		1352.2.o.h.361.3		24
13.7	odd	12		1352.2.o.h.361.4		24
13.8	odd	4		1352.2.f.g.337.9		12
13.9	even	3		1352.2.i.m.1329.2		12
13.10	even	6		1352.2.i.n.529.2		12
13.11	odd	12		1352.2.o.h.1161.3		24
13.12	even	2		1352.2.a.n.1.5	yes	6
52.31	even	4		2704.2.f.r.337.4		12
52.47	even	4		2704.2.f.r.337.3		12
52.51	odd	2		2704.2.a.bg.1.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1352.2.a.m.1.5	✓	6	1.1	even	1	trivial
1352.2.a.n.1.5	yes	6	13.12	even	2
1352.2.f.g.337.9		12	13.8	odd	4
1352.2.f.g.337.10		12	13.5	odd	4
1352.2.i.m.529.2		12	13.3	even	3
1352.2.i.m.1329.2		12	13.9	even	3
1352.2.i.n.529.2		12	13.10	even	6
1352.2.i.n.1329.2		12	13.4	even	6
1352.2.o.h.361.3		24	13.6	odd	12
1352.2.o.h.361.4		24	13.7	odd	12
1352.2.o.h.1161.3		24	13.11	odd	12
1352.2.o.h.1161.4		24	13.2	odd	12
2704.2.a.bf.1.2		6	4.3	odd	2
2704.2.a.bg.1.2		6	52.51	odd	2
2704.2.f.r.337.3		12	52.47	even	4
2704.2.f.r.337.4		12	52.31	even	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:	\( N \)	\(=\)	\( 1352 = 2^{3} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1352.a (trivial)

\(f(q)\)	\(=\)	\(q+2.76369 q^{3} -4.24972 q^{5} +2.37460 q^{7} +4.63797 q^{9} +O(q^{10})\)	\(q+2.76369 q^{3} -4.24972 q^{5} +2.37460 q^{7} +4.63797 q^{9} +0.178841 q^{11} -11.7449 q^{15} +1.03977 q^{17} +5.25040 q^{19} +6.56264 q^{21} -0.130358 q^{23} +13.0601 q^{25} +4.52683 q^{27} +8.33214 q^{29} +2.94810 q^{31} +0.494261 q^{33} -10.0914 q^{35} +6.03464 q^{37} +0.264592 q^{41} +0.564732 q^{43} -19.7101 q^{45} -10.9224 q^{47} -1.36129 q^{49} +2.87360 q^{51} +2.41779 q^{53} -0.760024 q^{55} +14.5105 q^{57} +7.35568 q^{59} -5.06501 q^{61} +11.0133 q^{63} +1.58480 q^{67} -0.360267 q^{69} -10.3352 q^{71} +14.3011 q^{73} +36.0941 q^{75} +0.424675 q^{77} -7.26636 q^{79} -1.40316 q^{81} +12.9706 q^{83} -4.41874 q^{85} +23.0274 q^{87} -6.92320 q^{89} +8.14762 q^{93} -22.3127 q^{95} +3.63494 q^{97} +0.829459 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{3} - 2 q^{5} + 3 q^{7} + 9 q^{9}+O(q^{10}) \) 6 * q + 3 * q^3 - 2 * q^5 + 3 * q^7 + 9 * q^9	\( 6 q + 3 q^{3} - 2 q^{5} + 3 q^{7} + 9 q^{9} + 13 q^{11} - 14 q^{15} + 11 q^{17} + 15 q^{19} - 22 q^{21} + 15 q^{23} + 16 q^{25} + 21 q^{27} + 9 q^{29} - 3 q^{31} - 2 q^{33} + 14 q^{35} + 14 q^{37} - 20 q^{41} + 10 q^{43} - 9 q^{45} - 10 q^{47} + 27 q^{49} + 5 q^{51} + q^{53} + 14 q^{55} + 9 q^{57} + 50 q^{59} + 2 q^{63} + 6 q^{67} + 32 q^{69} + 9 q^{71} - 6 q^{73} + 40 q^{75} - 2 q^{77} + 13 q^{79} + 42 q^{81} + 36 q^{83} - 31 q^{85} + 22 q^{87} - 18 q^{89} - 12 q^{93} - 21 q^{95} - 14 q^{97} + 18 q^{99}+O(q^{100}) \) 6 * q + 3 * q^3 - 2 * q^5 + 3 * q^7 + 9 * q^9 + 13 * q^11 - 14 * q^15 + 11 * q^17 + 15 * q^19 - 22 * q^21 + 15 * q^23 + 16 * q^25 + 21 * q^27 + 9 * q^29 - 3 * q^31 - 2 * q^33 + 14 * q^35 + 14 * q^37 - 20 * q^41 + 10 * q^43 - 9 * q^45 - 10 * q^47 + 27 * q^49 + 5 * q^51 + q^53 + 14 * q^55 + 9 * q^57 + 50 * q^59 + 2 * q^63 + 6 * q^67 + 32 * q^69 + 9 * q^71 - 6 * q^73 + 40 * q^75 - 2 * q^77 + 13 * q^79 + 42 * q^81 + 36 * q^83 - 31 * q^85 + 22 * q^87 - 18 * q^89 - 12 * q^93 - 21 * q^95 - 14 * q^97 + 18 * q^99

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