LMFDB - Embedded newform 4012.2.a.j.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4012,2,Mod(1,4012)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4012, 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("4012.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4012 = 2^{2} \cdot 17 \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4012.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:	$32.0359812909$
Analytic rank:	$0$
Dimension:	$21$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	4012.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.97354 q^{3} +3.25764 q^{5} +2.36478 q^{7} +0.894844 q^{9} +O(q^{10})$	$q+1.97354 q^{3} +3.25764 q^{5} +2.36478 q^{7} +0.894844 q^{9} +4.67317 q^{11} +4.16406 q^{13} +6.42908 q^{15} +1.00000 q^{17} -3.72472 q^{19} +4.66698 q^{21} +3.29577 q^{23} +5.61224 q^{25} -4.15460 q^{27} -3.32033 q^{29} +0.358249 q^{31} +9.22268 q^{33} +7.70361 q^{35} +4.22540 q^{37} +8.21793 q^{39} -10.3537 q^{41} -7.58547 q^{43} +2.91508 q^{45} -3.72212 q^{47} -1.40781 q^{49} +1.97354 q^{51} -2.36482 q^{53} +15.2235 q^{55} -7.35087 q^{57} +1.00000 q^{59} -5.87127 q^{61} +2.11611 q^{63} +13.5650 q^{65} -8.10843 q^{67} +6.50432 q^{69} +8.07461 q^{71} -8.82546 q^{73} +11.0760 q^{75} +11.0510 q^{77} -2.58295 q^{79} -10.8838 q^{81} -12.6700 q^{83} +3.25764 q^{85} -6.55280 q^{87} -10.7252 q^{89} +9.84709 q^{91} +0.707018 q^{93} -12.1338 q^{95} +13.0852 q^{97} +4.18176 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9}+O(q^{10}) $ 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9	$ 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9} + 12 q^{11} - 4 q^{13} + 9 q^{15} + 21 q^{17} + 4 q^{19} + 8 q^{21} + 19 q^{23} + 26 q^{25} + 33 q^{27} - q^{29} + 13 q^{31} + 11 q^{33} + 15 q^{35} - 4 q^{37} + 18 q^{39} + 9 q^{41} + 7 q^{43} + 7 q^{45} + 33 q^{47} + 36 q^{49} + 9 q^{51} + 7 q^{53} + 12 q^{55} + 26 q^{57} + 21 q^{59} + 3 q^{61} + 51 q^{63} + q^{65} + 8 q^{69} + 55 q^{71} + 22 q^{73} + 14 q^{75} - 15 q^{77} + 28 q^{79} + 25 q^{81} + 54 q^{83} + q^{85} + 34 q^{87} + 30 q^{89} + 35 q^{91} - 5 q^{93} + 42 q^{95} + 9 q^{97} + 20 q^{99}+O(q^{100}) $ 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9 + 12 * q^11 - 4 * q^13 + 9 * q^15 + 21 * q^17 + 4 * q^19 + 8 * q^21 + 19 * q^23 + 26 * q^25 + 33 * q^27 - q^29 + 13 * q^31 + 11 * q^33 + 15 * q^35 - 4 * q^37 + 18 * q^39 + 9 * q^41 + 7 * q^43 + 7 * q^45 + 33 * q^47 + 36 * q^49 + 9 * q^51 + 7 * q^53 + 12 * q^55 + 26 * q^57 + 21 * q^59 + 3 * q^61 + 51 * q^63 + q^65 + 8 * q^69 + 55 * q^71 + 22 * q^73 + 14 * q^75 - 15 * q^77 + 28 * q^79 + 25 * q^81 + 54 * q^83 + q^85 + 34 * q^87 + 30 * q^89 + 35 * q^91 - 5 * q^93 + 42 * q^95 + 9 * q^97 + 20 * 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
$3$	1.97354	1.13942	0.569711	−	0.821845i	$-0.307055\pi$
$3$	1.97354	1.13942	0.569711	+	0.821845i	$0.307055\pi$
$4$	0	0
$5$	3.25764	1.45686	0.728431	−	0.685119i	$-0.240250\pi$
$5$	3.25764	1.45686	0.728431	+	0.685119i	$0.240250\pi$
$6$	0	0
$7$	2.36478	0.893803	0.446902	−	0.894583i	$-0.352527\pi$
$7$	2.36478	0.893803	0.446902	+	0.894583i	$0.352527\pi$
$8$	0	0
$9$	0.894844	0.298281
$10$	0	0
$11$	4.67317	1.40902	0.704508	−	0.709697i	$-0.251168\pi$
$11$	4.67317	1.40902	0.704508	+	0.709697i	$0.251168\pi$
$12$	0	0
$13$	4.16406	1.15490	0.577452	−	0.816425i	$-0.304047\pi$
$13$	4.16406	1.15490	0.577452	+	0.816425i	$0.304047\pi$
$14$	0	0
$15$	6.42908	1.65998
$16$	0	0
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	0	0
$19$	−3.72472	−0.854510	−0.427255	−	0.904131i	$-0.640519\pi$
$19$	−3.72472	−0.854510	−0.427255	+	0.904131i	$0.640519\pi$
$20$	0	0
$21$	4.66698	1.01842
$22$	0	0
$23$	3.29577	0.687215	0.343608	−	0.939113i	$-0.388351\pi$
$23$	3.29577	0.687215	0.343608	+	0.939113i	$0.388351\pi$
$24$	0	0
$25$	5.61224	1.12245
$26$	0	0
$27$	−4.15460	−0.799553
$28$	0	0
$29$	−3.32033	−0.616571	−0.308285	−	0.951294i	$-0.599755\pi$
$29$	−3.32033	−0.616571	−0.308285	+	0.951294i	$0.599755\pi$
$30$	0	0
$31$	0.358249	0.0643435	0.0321717	−	0.999482i	$-0.489758\pi$
$31$	0.358249	0.0643435	0.0321717	+	0.999482i	$0.489758\pi$
$32$	0	0
$33$	9.22268	1.60546
$34$	0	0
$35$	7.70361	1.30215
$36$	0	0
$37$	4.22540	0.694652	0.347326	−	0.937744i	$-0.387090\pi$
$37$	4.22540	0.694652	0.347326	+	0.937744i	$0.387090\pi$
$38$	0	0
$39$	8.21793	1.31592
$40$	0	0
$41$	−10.3537	−1.61698	−0.808490	−	0.588510i	$-0.799715\pi$
$41$	−10.3537	−1.61698	−0.808490	+	0.588510i	$0.799715\pi$
$42$	0	0
$43$	−7.58547	−1.15677	−0.578387	−	0.815763i	$-0.696318\pi$
$43$	−7.58547	−1.15677	−0.578387	+	0.815763i	$0.696318\pi$
$44$	0	0
$45$	2.91508	0.434555
$46$	0	0
$47$	−3.72212	−0.542927	−0.271463	−	0.962449i	$-0.587508\pi$
$47$	−3.72212	−0.542927	−0.271463	+	0.962449i	$0.587508\pi$
$48$	0	0
$49$	−1.40781	−0.201116
$50$	0	0
$51$	1.97354	0.276350
$52$	0	0
$53$	−2.36482	−0.324833	−0.162417	−	0.986722i	$-0.551929\pi$
$53$	−2.36482	−0.324833	−0.162417	+	0.986722i	$0.551929\pi$
$54$	0	0
$55$	15.2235	2.05274
$56$	0	0
$57$	−7.35087	−0.973647
$58$	0	0
$59$	1.00000	0.130189
$59$	1.00000	0.130189
$60$	0	0
$61$	−5.87127	−0.751739	−0.375869	−	0.926673i	$-0.622656\pi$
$61$	−5.87127	−0.751739	−0.375869	+	0.926673i	$0.622656\pi$
$62$	0	0
$63$	2.11611	0.266605
$64$	0	0
$65$	13.5650	1.68253
$66$	0	0
$67$	−8.10843	−0.990602	−0.495301	−	0.868721i	$-0.664942\pi$
$67$	−8.10843	−0.990602	−0.495301	+	0.868721i	$0.664942\pi$
$68$	0	0
$69$	6.50432	0.783028
$70$	0	0
$71$	8.07461	0.958280	0.479140	−	0.877739i	$-0.340949\pi$
$71$	8.07461	0.958280	0.479140	+	0.877739i	$0.340949\pi$
$72$	0	0
$73$	−8.82546	−1.03294	−0.516471	−	0.856305i	$-0.672755\pi$
$73$	−8.82546	−1.03294	−0.516471	+	0.856305i	$0.672755\pi$
$74$	0	0
$75$	11.0760	1.27894
$76$	0	0
$77$	11.0510	1.25938
$78$	0	0
$79$	−2.58295	−0.290604	−0.145302	−	0.989387i	$-0.546415\pi$
$79$	−2.58295	−0.290604	−0.145302	+	0.989387i	$0.546415\pi$
$80$	0	0
$81$	−10.8838	−1.20931
$82$	0	0
$83$	−12.6700	−1.39072	−0.695358	−	0.718664i	$-0.744754\pi$
$83$	−12.6700	−1.39072	−0.695358	+	0.718664i	$0.744754\pi$
$84$	0	0
$85$	3.25764	0.353341
$86$	0	0
$87$	−6.55280	−0.702534
$88$	0	0
$89$	−10.7252	−1.13687	−0.568435	−	0.822728i	$-0.692451\pi$
$89$	−10.7252	−1.13687	−0.568435	+	0.822728i	$0.692451\pi$
$90$	0	0
$91$	9.84709	1.03226
$92$	0	0
$93$	0.707018	0.0733143
$94$	0	0
$95$	−12.1338	−1.24490
$96$	0	0
$97$	13.0852	1.32860	0.664300	−	0.747466i	$-0.268730\pi$
$97$	13.0852	1.32860	0.664300	+	0.747466i	$0.268730\pi$
$98$	0	0
$99$	4.18176	0.420283
$100$	0	0
$101$	−14.0511	−1.39814	−0.699069	−	0.715054i	$-0.746402\pi$
$101$	−14.0511	−1.39814	−0.699069	+	0.715054i	$0.746402\pi$
$102$	0	0
$103$	3.77074	0.371542	0.185771	−	0.982593i	$-0.440522\pi$
$103$	3.77074	0.371542	0.185771	+	0.982593i	$0.440522\pi$
$104$	0	0
$105$	15.2034	1.48370
$106$	0	0
$107$	−0.723038	−0.0698987	−0.0349494	−	0.999389i	$-0.511127\pi$
$107$	−0.723038	−0.0698987	−0.0349494	+	0.999389i	$0.511127\pi$
$108$	0	0
$109$	10.3529	0.991625	0.495812	−	0.868430i	$-0.334870\pi$
$109$	10.3529	0.991625	0.495812	+	0.868430i	$0.334870\pi$
$110$	0	0
$111$	8.33898	0.791501
$112$	0	0
$113$	0.568725	0.0535012	0.0267506	−	0.999642i	$-0.491484\pi$
$113$	0.568725	0.0535012	0.0267506	+	0.999642i	$0.491484\pi$
$114$	0	0
$115$	10.7364	1.00118
$116$	0	0
$117$	3.72618	0.344486
$118$	0	0
$119$	2.36478	0.216779
$120$	0	0
$121$	10.8386	0.985323
$122$	0	0
$123$	−20.4334	−1.84242
$124$	0	0
$125$	1.99447	0.178391
$126$	0	0
$127$	−12.1932	−1.08197	−0.540987	−	0.841031i	$-0.681949\pi$
$127$	−12.1932	−1.08197	−0.540987	+	0.841031i	$0.681949\pi$
$128$	0	0
$129$	−14.9702	−1.31805
$130$	0	0
$131$	0.860780	0.0752067	0.0376034	−	0.999293i	$-0.488028\pi$
$131$	0.860780	0.0752067	0.0376034	+	0.999293i	$0.488028\pi$
$132$	0	0
$133$	−8.80815	−0.763764
$134$	0	0
$135$	−13.5342	−1.16484
$136$	0	0
$137$	10.3863	0.887358	0.443679	−	0.896186i	$-0.353673\pi$
$137$	10.3863	0.887358	0.443679	+	0.896186i	$0.353673\pi$
$138$	0	0
$139$	−6.58461	−0.558499	−0.279250	−	0.960219i	$-0.590086\pi$
$139$	−6.58461	−0.558499	−0.279250	+	0.960219i	$0.590086\pi$
$140$	0	0
$141$	−7.34573	−0.618622
$142$	0	0
$143$	19.4594	1.62728
$144$	0	0
$145$	−10.8165	−0.898259
$146$	0	0
$147$	−2.77837	−0.229156
$148$	0	0
$149$	−4.63899	−0.380041	−0.190021	−	0.981780i	$-0.560855\pi$
$149$	−4.63899	−0.380041	−0.190021	+	0.981780i	$0.560855\pi$
$150$	0	0
$151$	−1.87687	−0.152737	−0.0763686	−	0.997080i	$-0.524333\pi$
$151$	−1.87687	−0.152737	−0.0763686	+	0.997080i	$0.524333\pi$
$152$	0	0
$153$	0.894844	0.0723438
$154$	0	0
$155$	1.16705	0.0937396
$156$	0	0
$157$	21.5436	1.71937	0.859685	−	0.510825i	$-0.170660\pi$
$157$	21.5436	1.71937	0.859685	+	0.510825i	$0.170660\pi$
$158$	0	0
$159$	−4.66706	−0.370122
$160$	0	0
$161$	7.79377	0.614235
$162$	0	0
$163$	−14.7352	−1.15415	−0.577075	−	0.816691i	$-0.695806\pi$
$163$	−14.7352	−1.15415	−0.577075	+	0.816691i	$0.695806\pi$
$164$	0	0
$165$	30.0442	2.33894
$166$	0	0
$167$	24.0820	1.86352	0.931760	−	0.363076i	$-0.118273\pi$
$167$	24.0820	1.86352	0.931760	+	0.363076i	$0.118273\pi$
$168$	0	0
$169$	4.33941	0.333801
$170$	0	0
$171$	−3.33304	−0.254884
$172$	0	0
$173$	−6.24606	−0.474879	−0.237439	−	0.971402i	$-0.576308\pi$
$173$	−6.24606	−0.474879	−0.237439	+	0.971402i	$0.576308\pi$
$174$	0	0
$175$	13.2717	1.00325
$176$	0	0
$177$	1.97354	0.148340
$178$	0	0
$179$	0.713424	0.0533238	0.0266619	−	0.999645i	$-0.491512\pi$
$179$	0.713424	0.0533238	0.0266619	+	0.999645i	$0.491512\pi$
$180$	0	0
$181$	7.56425	0.562247	0.281123	−	0.959672i	$-0.409293\pi$
$181$	7.56425	0.562247	0.281123	+	0.959672i	$0.409293\pi$
$182$	0	0
$183$	−11.5872	−0.856547
$184$	0	0
$185$	13.7649	1.01201
$186$	0	0
$187$	4.67317	0.341736
$188$	0	0
$189$	−9.82472	−0.714643
$190$	0	0
$191$	22.6562	1.63934	0.819672	−	0.572833i	$-0.194156\pi$
$191$	22.6562	1.63934	0.819672	+	0.572833i	$0.194156\pi$
$192$	0	0
$193$	12.2782	0.883807	0.441903	−	0.897063i	$-0.354303\pi$
$193$	12.2782	0.883807	0.441903	+	0.897063i	$0.354303\pi$
$194$	0	0
$195$	26.7711	1.91712
$196$	0	0
$197$	−1.10278	−0.0785695	−0.0392847	−	0.999228i	$-0.512508\pi$
$197$	−1.10278	−0.0785695	−0.0392847	+	0.999228i	$0.512508\pi$
$198$	0	0
$199$	4.71771	0.334430	0.167215	−	0.985920i	$-0.446523\pi$
$199$	4.71771	0.334430	0.167215	+	0.985920i	$0.446523\pi$
$200$	0	0
$201$	−16.0023	−1.12871
$202$	0	0
$203$	−7.85186	−0.551093
$204$	0	0
$205$	−33.7287	−2.35572
$206$	0	0
$207$	2.94920	0.204983
$208$	0	0
$209$	−17.4063	−1.20402
$210$	0	0
$211$	12.9772	0.893390	0.446695	−	0.894686i	$-0.352601\pi$
$211$	12.9772	0.893390	0.446695	+	0.894686i	$0.352601\pi$
$212$	0	0
$213$	15.9355	1.09188
$214$	0	0
$215$	−24.7108	−1.68526
$216$	0	0
$217$	0.847181	0.0575104
$218$	0	0
$219$	−17.4174	−1.17696
$220$	0	0
$221$	4.16406	0.280105
$222$	0	0
$223$	−23.2709	−1.55833	−0.779167	−	0.626816i	$-0.784358\pi$
$223$	−23.2709	−1.55833	−0.779167	+	0.626816i	$0.784358\pi$
$224$	0	0
$225$	5.02208	0.334805
$226$	0	0
$227$	−1.23174	−0.0817536	−0.0408768	−	0.999164i	$-0.513015\pi$
$227$	−1.23174	−0.0817536	−0.0408768	+	0.999164i	$0.513015\pi$
$228$	0	0
$229$	−3.61584	−0.238941	−0.119471	−	0.992838i	$-0.538120\pi$
$229$	−3.61584	−0.238941	−0.119471	+	0.992838i	$0.538120\pi$
$230$	0	0
$231$	21.8096	1.43497
$232$	0	0
$233$	−24.1702	−1.58344	−0.791720	−	0.610884i	$-0.790814\pi$
$233$	−24.1702	−1.58344	−0.791720	+	0.610884i	$0.790814\pi$
$234$	0	0
$235$	−12.1253	−0.790969
$236$	0	0
$237$	−5.09754	−0.331121
$238$	0	0
$239$	−2.13178	−0.137894	−0.0689468	−	0.997620i	$-0.521964\pi$
$239$	−2.13178	−0.137894	−0.0689468	+	0.997620i	$0.521964\pi$
$240$	0	0
$241$	16.7689	1.08018	0.540089	−	0.841608i	$-0.318391\pi$
$241$	16.7689	1.08018	0.540089	+	0.841608i	$0.318391\pi$
$242$	0	0
$243$	−9.01574	−0.578360
$244$	0	0
$245$	−4.58615	−0.292999
$246$	0	0
$247$	−15.5100	−0.986876
$248$	0	0
$249$	−25.0047	−1.58461
$250$	0	0
$251$	−1.03852	−0.0655509	−0.0327754	−	0.999463i	$-0.510435\pi$
$251$	−1.03852	−0.0655509	−0.0327754	+	0.999463i	$0.510435\pi$
$252$	0	0
$253$	15.4017	0.968297
$254$	0	0
$255$	6.42908	0.402604
$256$	0	0
$257$	−4.09366	−0.255355	−0.127678	−	0.991816i	$-0.540752\pi$
$257$	−4.09366	−0.255355	−0.127678	+	0.991816i	$0.540752\pi$
$258$	0	0
$259$	9.99215	0.620882
$260$	0	0
$261$	−2.97118	−0.183911
$262$	0	0
$263$	13.5704	0.836790	0.418395	−	0.908265i	$-0.362593\pi$
$263$	13.5704	0.836790	0.418395	+	0.908265i	$0.362593\pi$
$264$	0	0
$265$	−7.70375	−0.473238
$266$	0	0
$267$	−21.1666	−1.29537
$268$	0	0
$269$	−8.59957	−0.524325	−0.262163	−	0.965024i	$-0.584436\pi$
$269$	−8.59957	−0.524325	−0.262163	+	0.965024i	$0.584436\pi$
$270$	0	0
$271$	16.9192	1.02777	0.513885	−	0.857859i	$-0.328206\pi$
$271$	16.9192	1.02777	0.513885	+	0.857859i	$0.328206\pi$
$272$	0	0
$273$	19.4336	1.17617
$274$	0	0
$275$	26.2270	1.58155
$276$	0	0
$277$	−12.0408	−0.723464	−0.361732	−	0.932282i	$-0.617814\pi$
$277$	−12.0408	−0.723464	−0.361732	+	0.932282i	$0.617814\pi$
$278$	0	0
$279$	0.320577	0.0191924
$280$	0	0
$281$	15.8196	0.943720	0.471860	−	0.881673i	$-0.343583\pi$
$281$	15.8196	0.943720	0.471860	+	0.881673i	$0.343583\pi$
$282$	0	0
$283$	14.1108	0.838801	0.419400	−	0.907801i	$-0.362240\pi$
$283$	14.1108	0.838801	0.419400	+	0.907801i	$0.362240\pi$
$284$	0	0
$285$	−23.9465	−1.41847
$286$	0	0
$287$	−24.4843	−1.44526
$288$	0	0
$289$	1.00000	0.0588235
$290$	0	0
$291$	25.8241	1.51384
$292$	0	0
$293$	8.34036	0.487249	0.243625	−	0.969870i	$-0.421664\pi$
$293$	8.34036	0.487249	0.243625	+	0.969870i	$0.421664\pi$
$294$	0	0
$295$	3.25764	0.189667
$296$	0	0
$297$	−19.4152	−1.12658
$298$	0	0
$299$	13.7238	0.793667
$300$	0	0
$301$	−17.9380	−1.03393
$302$	0	0
$303$	−27.7304	−1.59307
$304$	0	0
$305$	−19.1265	−1.09518
$306$	0	0
$307$	6.04109	0.344783	0.172392	−	0.985029i	$-0.444851\pi$
$307$	6.04109	0.344783	0.172392	+	0.985029i	$0.444851\pi$
$308$	0	0
$309$	7.44169	0.423343
$310$	0	0
$311$	23.1899	1.31498	0.657489	−	0.753464i	$-0.271619\pi$
$311$	23.1899	1.31498	0.657489	+	0.753464i	$0.271619\pi$
$312$	0	0
$313$	4.85844	0.274615	0.137308	−	0.990528i	$-0.456155\pi$
$313$	4.85844	0.274615	0.137308	+	0.990528i	$0.456155\pi$
$314$	0	0
$315$	6.89353	0.388406
$316$	0	0
$317$	−19.8111	−1.11270	−0.556352	−	0.830947i	$-0.687799\pi$
$317$	−19.8111	−1.11270	−0.556352	+	0.830947i	$0.687799\pi$
$318$	0	0
$319$	−15.5165	−0.868757
$320$	0	0
$321$	−1.42694	−0.0796441
$322$	0	0
$323$	−3.72472	−0.207249
$324$	0	0
$325$	23.3697	1.29632
$326$	0	0
$327$	20.4318	1.12988
$328$	0	0
$329$	−8.80199	−0.485269
$330$	0	0
$331$	23.9536	1.31661	0.658304	−	0.752752i	$-0.271274\pi$
$331$	23.9536	1.31661	0.658304	+	0.752752i	$0.271274\pi$
$332$	0	0
$333$	3.78108	0.207202
$334$	0	0
$335$	−26.4144	−1.44317
$336$	0	0
$337$	−29.7764	−1.62202	−0.811012	−	0.585030i	$-0.801083\pi$
$337$	−29.7764	−1.62202	−0.811012	+	0.585030i	$0.801083\pi$
$338$	0	0
$339$	1.12240	0.0609604
$340$	0	0
$341$	1.67416	0.0906609
$342$	0	0
$343$	−19.8826	−1.07356
$344$	0	0
$345$	21.1887	1.14076
$346$	0	0
$347$	11.2863	0.605880	0.302940	−	0.953010i	$-0.402032\pi$
$347$	11.2863	0.605880	0.302940	+	0.953010i	$0.402032\pi$
$348$	0	0
$349$	17.7509	0.950182	0.475091	−	0.879937i	$-0.342415\pi$
$349$	17.7509	0.950182	0.475091	+	0.879937i	$0.342415\pi$
$350$	0	0
$351$	−17.3000	−0.923407
$352$	0	0
$353$	−2.76123	−0.146966	−0.0734828	−	0.997296i	$-0.523411\pi$
$353$	−2.76123	−0.146966	−0.0734828	+	0.997296i	$0.523411\pi$
$354$	0	0
$355$	26.3042	1.39608
$356$	0	0
$357$	4.66698	0.247003
$358$	0	0
$359$	4.41150	0.232830	0.116415	−	0.993201i	$-0.462860\pi$
$359$	4.41150	0.232830	0.116415	+	0.993201i	$0.462860\pi$
$360$	0	0
$361$	−5.12644	−0.269813
$362$	0	0
$363$	21.3903	1.12270
$364$	0	0
$365$	−28.7502	−1.50486
$366$	0	0
$367$	23.3299	1.21781	0.608905	−	0.793243i	$-0.291609\pi$
$367$	23.3299	1.21781	0.608905	+	0.793243i	$0.291609\pi$
$368$	0	0
$369$	−9.26496	−0.482315
$370$	0	0
$371$	−5.59229	−0.290337
$372$	0	0
$373$	16.3208	0.845057	0.422529	−	0.906350i	$-0.361143\pi$
$373$	16.3208	0.845057	0.422529	+	0.906350i	$0.361143\pi$
$374$	0	0
$375$	3.93616	0.203262
$376$	0	0
$377$	−13.8261	−0.712079
$378$	0	0
$379$	19.0057	0.976259	0.488130	−	0.872771i	$-0.337679\pi$
$379$	19.0057	0.976259	0.488130	+	0.872771i	$0.337679\pi$
$380$	0	0
$381$	−24.0638	−1.23283
$382$	0	0
$383$	13.4678	0.688174	0.344087	−	0.938938i	$-0.388188\pi$
$383$	13.4678	0.688174	0.344087	+	0.938938i	$0.388188\pi$
$384$	0	0
$385$	36.0003	1.83475
$386$	0	0
$387$	−6.78781	−0.345044
$388$	0	0
$389$	30.9719	1.57034	0.785170	−	0.619281i	$-0.212576\pi$
$389$	30.9719	1.57034	0.785170	+	0.619281i	$0.212576\pi$
$390$	0	0
$391$	3.29577	0.166674
$392$	0	0
$393$	1.69878	0.0856922
$394$	0	0
$395$	−8.41433	−0.423371
$396$	0	0
$397$	27.7913	1.39480	0.697402	−	0.716680i	$-0.254339\pi$
$397$	27.7913	1.39480	0.697402	+	0.716680i	$0.254339\pi$
$398$	0	0
$399$	−17.3832	−0.870249
$400$	0	0
$401$	−14.3081	−0.714514	−0.357257	−	0.934006i	$-0.616288\pi$
$401$	−14.3081	−0.714514	−0.357257	+	0.934006i	$0.616288\pi$
$402$	0	0
$403$	1.49177	0.0743105
$404$	0	0
$405$	−35.4555	−1.76180
$406$	0	0
$407$	19.7460	0.978775
$408$	0	0
$409$	15.6156	0.772141	0.386071	−	0.922469i	$-0.373832\pi$
$409$	15.6156	0.772141	0.386071	+	0.922469i	$0.373832\pi$
$410$	0	0
$411$	20.4977	1.01108
$412$	0	0
$413$	2.36478	0.116363
$414$	0	0
$415$	−41.2744	−2.02608
$416$	0	0
$417$	−12.9950	−0.636366
$418$	0	0
$419$	−21.9929	−1.07442	−0.537212	−	0.843447i	$-0.680523\pi$
$419$	−21.9929	−1.07442	−0.537212	+	0.843447i	$0.680523\pi$
$420$	0	0
$421$	17.4560	0.850752	0.425376	−	0.905017i	$-0.360142\pi$
$421$	17.4560	0.850752	0.425376	+	0.905017i	$0.360142\pi$
$422$	0	0
$423$	−3.33071	−0.161945
$424$	0	0
$425$	5.61224	0.272234
$426$	0	0
$427$	−13.8843	−0.671906
$428$	0	0
$429$	38.4038	1.85415
$430$	0	0
$431$	17.6604	0.850669	0.425335	−	0.905036i	$-0.360156\pi$
$431$	17.6604	0.850669	0.425335	+	0.905036i	$0.360156\pi$
$432$	0	0
$433$	−24.3848	−1.17186	−0.585930	−	0.810362i	$-0.699270\pi$
$433$	−24.3848	−1.17186	−0.585930	+	0.810362i	$0.699270\pi$
$434$	0	0
$435$	−21.3467	−1.02350
$436$	0	0
$437$	−12.2758	−0.587232
$438$	0	0
$439$	2.22779	0.106327	0.0531634	−	0.998586i	$-0.483070\pi$
$439$	2.22779	0.106327	0.0531634	+	0.998586i	$0.483070\pi$
$440$	0	0
$441$	−1.25977	−0.0599892
$442$	0	0
$443$	26.8482	1.27560	0.637798	−	0.770204i	$-0.279845\pi$
$443$	26.8482	1.27560	0.637798	+	0.770204i	$0.279845\pi$
$444$	0	0
$445$	−34.9389	−1.65626
$446$	0	0
$447$	−9.15522	−0.433027
$448$	0	0
$449$	10.7672	0.508134	0.254067	−	0.967187i	$-0.418232\pi$
$449$	10.7672	0.508134	0.254067	+	0.967187i	$0.418232\pi$
$450$	0	0
$451$	−48.3847	−2.27835
$452$	0	0
$453$	−3.70406	−0.174032
$454$	0	0
$455$	32.0783	1.50385
$456$	0	0
$457$	0.800750	0.0374575	0.0187288	−	0.999825i	$-0.494038\pi$
$457$	0.800750	0.0374575	0.0187288	+	0.999825i	$0.494038\pi$
$458$	0	0
$459$	−4.15460	−0.193920
$460$	0	0
$461$	15.4897	0.721426	0.360713	−	0.932677i	$-0.382533\pi$
$461$	15.4897	0.721426	0.360713	+	0.932677i	$0.382533\pi$
$462$	0	0
$463$	−14.6114	−0.679051	−0.339526	−	0.940597i	$-0.610267\pi$
$463$	−14.6114	−0.679051	−0.339526	+	0.940597i	$0.610267\pi$
$464$	0	0
$465$	2.30321	0.106809
$466$	0	0
$467$	31.4739	1.45644	0.728219	−	0.685344i	$-0.240348\pi$
$467$	31.4739	1.45644	0.728219	+	0.685344i	$0.240348\pi$
$468$	0	0
$469$	−19.1747	−0.885403
$470$	0	0
$471$	42.5172	1.95909
$472$	0	0
$473$	−35.4482	−1.62991
$474$	0	0
$475$	−20.9041	−0.959144
$476$	0	0
$477$	−2.11615	−0.0968917
$478$	0	0
$479$	11.4654	0.523866	0.261933	−	0.965086i	$-0.415640\pi$
$479$	11.4654	0.523866	0.261933	+	0.965086i	$0.415640\pi$
$480$	0	0
$481$	17.5948	0.802256
$482$	0	0
$483$	15.3813	0.699873
$484$	0	0
$485$	42.6269	1.93559
$486$	0	0
$487$	−11.5288	−0.522420	−0.261210	−	0.965282i	$-0.584121\pi$
$487$	−11.5288	−0.522420	−0.261210	+	0.965282i	$0.584121\pi$
$488$	0	0
$489$	−29.0804	−1.31506
$490$	0	0
$491$	6.05348	0.273190	0.136595	−	0.990627i	$-0.456384\pi$
$491$	6.05348	0.273190	0.136595	+	0.990627i	$0.456384\pi$
$492$	0	0
$493$	−3.32033	−0.149540
$494$	0	0
$495$	13.6227	0.612294
$496$	0	0
$497$	19.0947	0.856513
$498$	0	0
$499$	−17.1433	−0.767442	−0.383721	−	0.923449i	$-0.625357\pi$
$499$	−17.1433	−0.767442	−0.383721	+	0.923449i	$0.625357\pi$
$500$	0	0
$501$	47.5266	2.12333
$502$	0	0
$503$	−4.76170	−0.212314	−0.106157	−	0.994349i	$-0.533855\pi$
$503$	−4.76170	−0.212314	−0.106157	+	0.994349i	$0.533855\pi$
$504$	0	0
$505$	−45.7735	−2.03690
$506$	0	0
$507$	8.56399	0.380340
$508$	0	0
$509$	−11.8727	−0.526249	−0.263125	−	0.964762i	$-0.584753\pi$
$509$	−11.8727	−0.526249	−0.263125	+	0.964762i	$0.584753\pi$
$510$	0	0
$511$	−20.8703	−0.923247
$512$	0	0
$513$	15.4747	0.683226
$514$	0	0
$515$	12.2837	0.541285
$516$	0	0
$517$	−17.3941	−0.764992
$518$	0	0
$519$	−12.3268	−0.541087
$520$	0	0
$521$	7.54227	0.330433	0.165216	−	0.986257i	$-0.447168\pi$
$521$	7.54227	0.330433	0.165216	+	0.986257i	$0.447168\pi$
$522$	0	0
$523$	36.9901	1.61746	0.808732	−	0.588177i	$-0.200154\pi$
$523$	36.9901	1.61746	0.808732	+	0.588177i	$0.200154\pi$
$524$	0	0
$525$	26.1922	1.14312
$526$	0	0
$527$	0.358249	0.0156056
$528$	0	0
$529$	−12.1379	−0.527735
$530$	0	0
$531$	0.894844	0.0388329
$532$	0	0
$533$	−43.1135	−1.86745
$534$	0	0
$535$	−2.35540	−0.101833
$536$	0	0
$537$	1.40797	0.0607583
$538$	0	0
$539$	−6.57896	−0.283376
$540$	0	0
$541$	23.1739	0.996326	0.498163	−	0.867083i	$-0.334008\pi$
$541$	23.1739	0.996326	0.498163	+	0.867083i	$0.334008\pi$
$542$	0	0
$543$	14.9283	0.640636
$544$	0	0
$545$	33.7260	1.44466
$546$	0	0
$547$	8.98934	0.384357	0.192178	−	0.981360i	$-0.438445\pi$
$547$	8.98934	0.384357	0.192178	+	0.981360i	$0.438445\pi$
$548$	0	0
$549$	−5.25386	−0.224229
$550$	0	0
$551$	12.3673	0.526866
$552$	0	0
$553$	−6.10811	−0.259743
$554$	0	0
$555$	27.1654	1.15311
$556$	0	0
$557$	42.8656	1.81627	0.908137	−	0.418672i	$-0.137504\pi$
$557$	42.8656	1.81627	0.908137	+	0.418672i	$0.137504\pi$
$558$	0	0
$559$	−31.5864	−1.33596
$560$	0	0
$561$	9.22268	0.389382
$562$	0	0
$563$	26.5091	1.11723	0.558613	−	0.829429i	$-0.311334\pi$
$563$	26.5091	1.11723	0.558613	+	0.829429i	$0.311334\pi$
$564$	0	0
$565$	1.85270	0.0779438
$566$	0	0
$567$	−25.7378	−1.08088
$568$	0	0
$569$	−33.6369	−1.41013	−0.705067	−	0.709141i	$-0.749083\pi$
$569$	−33.6369	−1.41013	−0.705067	+	0.709141i	$0.749083\pi$
$570$	0	0
$571$	−14.1578	−0.592486	−0.296243	−	0.955113i	$-0.595734\pi$
$571$	−14.1578	−0.592486	−0.296243	+	0.955113i	$0.595734\pi$
$572$	0	0
$573$	44.7128	1.86790
$574$	0	0
$575$	18.4967	0.771364
$576$	0	0
$577$	13.0908	0.544978	0.272489	−	0.962159i	$-0.412153\pi$
$577$	13.0908	0.544978	0.272489	+	0.962159i	$0.412153\pi$
$578$	0	0
$579$	24.2315	1.00703
$580$	0	0
$581$	−29.9618	−1.24303
$582$	0	0
$583$	−11.0512	−0.457695
$584$	0	0
$585$	12.1386	0.501869
$586$	0	0
$587$	−21.4389	−0.884878	−0.442439	−	0.896798i	$-0.645887\pi$
$587$	−21.4389	−0.884878	−0.442439	+	0.896798i	$0.645887\pi$
$588$	0	0
$589$	−1.33438	−0.0549821
$590$	0	0
$591$	−2.17637	−0.0895237
$592$	0	0
$593$	4.31487	0.177191	0.0885953	−	0.996068i	$-0.471762\pi$
$593$	4.31487	0.177191	0.0885953	+	0.996068i	$0.471762\pi$
$594$	0	0
$595$	7.70361	0.315817
$596$	0	0
$597$	9.31058	0.381057
$598$	0	0
$599$	4.57602	0.186971	0.0934856	−	0.995621i	$-0.470199\pi$
$599$	4.57602	0.186971	0.0934856	+	0.995621i	$0.470199\pi$
$600$	0	0
$601$	16.1206	0.657575	0.328787	−	0.944404i	$-0.393360\pi$
$601$	16.1206	0.657575	0.328787	+	0.944404i	$0.393360\pi$
$602$	0	0
$603$	−7.25578	−0.295478
$604$	0	0
$605$	35.3082	1.43548
$606$	0	0
$607$	−27.4795	−1.11536	−0.557679	−	0.830056i	$-0.688308\pi$
$607$	−27.4795	−1.11536	−0.557679	+	0.830056i	$0.688308\pi$
$608$	0	0
$609$	−15.4959	−0.627927
$610$	0	0
$611$	−15.4991	−0.627027
$612$	0	0
$613$	−22.6709	−0.915669	−0.457835	−	0.889037i	$-0.651375\pi$
$613$	−22.6709	−0.915669	−0.457835	+	0.889037i	$0.651375\pi$
$614$	0	0
$615$	−66.5649	−2.68415
$616$	0	0
$617$	22.0684	0.888441	0.444221	−	0.895917i	$-0.353481\pi$
$617$	22.0684	0.888441	0.444221	+	0.895917i	$0.353481\pi$
$618$	0	0
$619$	3.48756	0.140177	0.0700884	−	0.997541i	$-0.477672\pi$
$619$	3.48756	0.140177	0.0700884	+	0.997541i	$0.477672\pi$
$620$	0	0
$621$	−13.6926	−0.549465
$622$	0	0
$623$	−25.3627	−1.01614
$624$	0	0
$625$	−21.5639	−0.862558
$626$	0	0
$627$	−34.3519	−1.37188
$628$	0	0
$629$	4.22540	0.168478
$630$	0	0
$631$	21.9898	0.875401	0.437701	−	0.899121i	$-0.355793\pi$
$631$	21.9898	0.875401	0.437701	+	0.899121i	$0.355793\pi$
$632$	0	0
$633$	25.6110	1.01795
$634$	0	0
$635$	−39.7212	−1.57629
$636$	0	0
$637$	−5.86222	−0.232270
$638$	0	0
$639$	7.22551	0.285837
$640$	0	0
$641$	−31.7882	−1.25556	−0.627780	−	0.778391i	$-0.716036\pi$
$641$	−31.7882	−1.25556	−0.627780	+	0.778391i	$0.716036\pi$
$642$	0	0
$643$	37.0112	1.45958	0.729790	−	0.683671i	$-0.239618\pi$
$643$	37.0112	1.45958	0.729790	+	0.683671i	$0.239618\pi$
$644$	0	0
$645$	−48.7676	−1.92022
$646$	0	0
$647$	47.3211	1.86038	0.930192	−	0.367074i	$-0.119641\pi$
$647$	47.3211	1.86038	0.930192	+	0.367074i	$0.119641\pi$
$648$	0	0
$649$	4.67317	0.183438
$650$	0	0
$651$	1.67194	0.0655286
$652$	0	0
$653$	0.709347	0.0277589	0.0138794	−	0.999904i	$-0.495582\pi$
$653$	0.709347	0.0277589	0.0138794	+	0.999904i	$0.495582\pi$
$654$	0	0
$655$	2.80412	0.109566
$656$	0	0
$657$	−7.89741	−0.308107
$658$	0	0
$659$	−27.9493	−1.08875	−0.544375	−	0.838842i	$-0.683233\pi$
$659$	−27.9493	−1.08875	−0.544375	+	0.838842i	$0.683233\pi$
$660$	0	0
$661$	−19.0794	−0.742101	−0.371051	−	0.928613i	$-0.621002\pi$
$661$	−19.0794	−0.742101	−0.371051	+	0.928613i	$0.621002\pi$
$662$	0	0
$663$	8.21793	0.319158
$664$	0	0
$665$	−28.6938	−1.11270
$666$	0	0
$667$	−10.9431	−0.423717
$668$	0	0
$669$	−45.9259	−1.77560
$670$	0	0
$671$	−27.4374	−1.05921
$672$	0	0
$673$	−48.6884	−1.87680	−0.938400	−	0.345550i	$-0.887693\pi$
$673$	−48.6884	−1.87680	−0.938400	+	0.345550i	$0.887693\pi$
$674$	0	0
$675$	−23.3166	−0.897458
$676$	0	0
$677$	2.34689	0.0901983	0.0450991	−	0.998983i	$-0.485640\pi$
$677$	2.34689	0.0901983	0.0450991	+	0.998983i	$0.485640\pi$
$678$	0	0
$679$	30.9436	1.18751
$680$	0	0
$681$	−2.43089	−0.0931518
$682$	0	0
$683$	9.51278	0.363996	0.181998	−	0.983299i	$-0.441743\pi$
$683$	9.51278	0.363996	0.181998	+	0.983299i	$0.441743\pi$
$684$	0	0
$685$	33.8347	1.29276
$686$	0	0
$687$	−7.13599	−0.272255
$688$	0	0
$689$	−9.84727	−0.375151
$690$	0	0
$691$	−24.8208	−0.944228	−0.472114	−	0.881538i	$-0.656509\pi$
$691$	−24.8208	−0.944228	−0.472114	+	0.881538i	$0.656509\pi$
$692$	0	0
$693$	9.88894	0.375650
$694$	0	0
$695$	−21.4503	−0.813657
$696$	0	0
$697$	−10.3537	−0.392175
$698$	0	0
$699$	−47.7007	−1.80421
$700$	0	0
$701$	−5.38313	−0.203318	−0.101659	−	0.994819i	$-0.532415\pi$
$701$	−5.38313	−0.203318	−0.101659	+	0.994819i	$0.532415\pi$
$702$	0	0
$703$	−15.7385	−0.593587
$704$	0	0
$705$	−23.9298	−0.901247
$706$	0	0
$707$	−33.2278	−1.24966
$708$	0	0
$709$	−43.5579	−1.63585	−0.817926	−	0.575324i	$-0.804876\pi$
$709$	−43.5579	−1.63585	−0.817926	+	0.575324i	$0.804876\pi$
$710$	0	0
$711$	−2.31133	−0.0866818
$712$	0	0
$713$	1.18071	0.0442178
$714$	0	0
$715$	63.3918	2.37072
$716$	0	0
$717$	−4.20715	−0.157119
$718$	0	0
$719$	29.2358	1.09031	0.545156	−	0.838334i	$-0.316470\pi$
$719$	29.2358	1.09031	0.545156	+	0.838334i	$0.316470\pi$
$720$	0	0
$721$	8.91697	0.332085
$722$	0	0
$723$	33.0940	1.23078
$724$	0	0
$725$	−18.6345	−0.692069
$726$	0	0
$727$	−43.4853	−1.61278	−0.806390	−	0.591384i	$-0.798582\pi$
$727$	−43.4853	−1.61278	−0.806390	+	0.591384i	$0.798582\pi$
$728$	0	0
$729$	14.8585	0.550314
$730$	0	0
$731$	−7.58547	−0.280559
$732$	0	0
$733$	−13.9931	−0.516848	−0.258424	−	0.966032i	$-0.583203\pi$
$733$	−13.9931	−0.516848	−0.258424	+	0.966032i	$0.583203\pi$
$734$	0	0
$735$	−9.05094	−0.333849
$736$	0	0
$737$	−37.8921	−1.39577
$738$	0	0
$739$	−46.4555	−1.70889	−0.854447	−	0.519539i	$-0.826104\pi$
$739$	−46.4555	−1.70889	−0.854447	+	0.519539i	$0.826104\pi$
$740$	0	0
$741$	−30.6095	−1.12447
$742$	0	0
$743$	−2.57777	−0.0945693	−0.0472846	−	0.998881i	$-0.515057\pi$
$743$	−2.57777	−0.0945693	−0.0472846	+	0.998881i	$0.515057\pi$
$744$	0	0
$745$	−15.1122	−0.553668
$746$	0	0
$747$	−11.3377	−0.414824
$748$	0	0
$749$	−1.70983	−0.0624757
$750$	0	0
$751$	39.3551	1.43609	0.718044	−	0.695998i	$-0.245038\pi$
$751$	39.3551	1.43609	0.718044	+	0.695998i	$0.245038\pi$
$752$	0	0
$753$	−2.04956	−0.0746901
$754$	0	0
$755$	−6.11416	−0.222517
$756$	0	0
$757$	20.2418	0.735702	0.367851	−	0.929885i	$-0.380094\pi$
$757$	20.2418	0.735702	0.367851	+	0.929885i	$0.380094\pi$
$758$	0	0
$759$	30.3958	1.10330
$760$	0	0
$761$	12.2306	0.443358	0.221679	−	0.975120i	$-0.428846\pi$
$761$	12.2306	0.443358	0.221679	+	0.975120i	$0.428846\pi$
$762$	0	0
$763$	24.4823	0.886317
$764$	0	0
$765$	2.91508	0.105395
$766$	0	0
$767$	4.16406	0.150356
$768$	0	0
$769$	23.8700	0.860774	0.430387	−	0.902644i	$-0.358377\pi$
$769$	23.8700	0.860774	0.430387	+	0.902644i	$0.358377\pi$
$770$	0	0
$771$	−8.07898	−0.290957
$772$	0	0
$773$	−37.0435	−1.33236	−0.666181	−	0.745790i	$-0.732072\pi$
$773$	−37.0435	−1.33236	−0.666181	+	0.745790i	$0.732072\pi$
$774$	0	0
$775$	2.01058	0.0722222
$776$	0	0
$777$	19.7199	0.707446
$778$	0	0
$779$	38.5647	1.38173
$780$	0	0
$781$	37.7340	1.35023
$782$	0	0
$783$	13.7947	0.492981
$784$	0	0
$785$	70.1815	2.50489
$786$	0	0
$787$	19.1472	0.682526	0.341263	−	0.939968i	$-0.389145\pi$
$787$	19.1472	0.682526	0.341263	+	0.939968i	$0.389145\pi$
$788$	0	0
$789$	26.7818	0.953456
$790$	0	0
$791$	1.34491	0.0478195
$792$	0	0
$793$	−24.4483	−0.868185
$794$	0	0
$795$	−15.2036	−0.539217
$796$	0	0
$797$	−8.94698	−0.316918	−0.158459	−	0.987366i	$-0.550653\pi$
$797$	−8.94698	−0.316918	−0.158459	+	0.987366i	$0.550653\pi$
$798$	0	0
$799$	−3.72212	−0.131679
$800$	0	0
$801$	−9.59738	−0.339107
$802$	0	0
$803$	−41.2429	−1.45543
$804$	0	0
$805$	25.3893	0.894856
$806$	0	0
$807$	−16.9716	−0.597428
$808$	0	0
$809$	52.9284	1.86087	0.930433	−	0.366463i	$-0.119431\pi$
$809$	52.9284	1.86087	0.930433	+	0.366463i	$0.119431\pi$
$810$	0	0
$811$	32.8508	1.15355	0.576773	−	0.816904i	$-0.304312\pi$
$811$	32.8508	1.15355	0.576773	+	0.816904i	$0.304312\pi$
$812$	0	0
$813$	33.3907	1.17106
$814$	0	0
$815$	−48.0020	−1.68144
$816$	0	0
$817$	28.2538	0.988475
$818$	0	0
$819$	8.81161	0.307902
$820$	0	0
$821$	−43.0648	−1.50297	−0.751486	−	0.659749i	$-0.770663\pi$
$821$	−43.0648	−1.50297	−0.751486	+	0.659749i	$0.770663\pi$
$822$	0	0
$823$	−22.1094	−0.770686	−0.385343	−	0.922773i	$-0.625917\pi$
$823$	−22.1094	−0.770686	−0.385343	+	0.922773i	$0.625917\pi$
$824$	0	0
$825$	51.7599	1.80205
$826$	0	0
$827$	−30.2532	−1.05201	−0.526003	−	0.850483i	$-0.676310\pi$
$827$	−30.2532	−1.05201	−0.526003	+	0.850483i	$0.676310\pi$
$828$	0	0
$829$	14.2408	0.494603	0.247301	−	0.968939i	$-0.420456\pi$
$829$	14.2408	0.494603	0.247301	+	0.968939i	$0.420456\pi$
$830$	0	0
$831$	−23.7630	−0.824330
$832$	0	0
$833$	−1.40781	−0.0487778
$834$	0	0
$835$	78.4505	2.71489
$836$	0	0
$837$	−1.48838	−0.0514460
$838$	0	0
$839$	−28.8738	−0.996836	−0.498418	−	0.866937i	$-0.666085\pi$
$839$	−28.8738	−0.996836	−0.498418	+	0.866937i	$0.666085\pi$
$840$	0	0
$841$	−17.9754	−0.619841
$842$	0	0
$843$	31.2206	1.07530
$844$	0	0
$845$	14.1363	0.486302
$846$	0	0
$847$	25.6308	0.880685
$848$	0	0
$849$	27.8482	0.955748
$850$	0	0
$851$	13.9259	0.477375
$852$	0	0
$853$	−29.1800	−0.999105	−0.499552	−	0.866284i	$-0.666502\pi$
$853$	−29.1800	−0.999105	−0.499552	+	0.866284i	$0.666502\pi$
$854$	0	0
$855$	−10.8579	−0.371331
$856$	0	0
$857$	−13.5902	−0.464233	−0.232116	−	0.972688i	$-0.574565\pi$
$857$	−13.5902	−0.464233	−0.232116	+	0.972688i	$0.574565\pi$
$858$	0	0
$859$	43.4139	1.48126	0.740631	−	0.671911i	$-0.234526\pi$
$859$	43.4139	1.48126	0.740631	+	0.671911i	$0.234526\pi$
$860$	0	0
$861$	−48.3206	−1.64676
$862$	0	0
$863$	−36.2020	−1.23233	−0.616164	−	0.787618i	$-0.711314\pi$
$863$	−36.2020	−1.23233	−0.616164	+	0.787618i	$0.711314\pi$
$864$	0	0
$865$	−20.3474	−0.691833
$866$	0	0
$867$	1.97354	0.0670248
$868$	0	0
$869$	−12.0706	−0.409466
$870$	0	0
$871$	−33.7640	−1.14405
$872$	0	0
$873$	11.7092	0.396297
$874$	0	0
$875$	4.71648	0.159446
$876$	0	0
$877$	−35.8293	−1.20987	−0.604934	−	0.796276i	$-0.706801\pi$
$877$	−35.8293	−1.20987	−0.604934	+	0.796276i	$0.706801\pi$
$878$	0	0
$879$	16.4600	0.555182
$880$	0	0
$881$	−4.26561	−0.143712	−0.0718561	−	0.997415i	$-0.522892\pi$
$881$	−4.26561	−0.143712	−0.0718561	+	0.997415i	$0.522892\pi$
$882$	0	0
$883$	51.5062	1.73332	0.866661	−	0.498898i	$-0.166262\pi$
$883$	51.5062	1.73332	0.866661	+	0.498898i	$0.166262\pi$
$884$	0	0
$885$	6.42908	0.216111
$886$	0	0
$887$	29.4788	0.989801	0.494901	−	0.868950i	$-0.335204\pi$
$887$	29.4788	0.989801	0.494901	+	0.868950i	$0.335204\pi$
$888$	0	0
$889$	−28.8343	−0.967072
$890$	0	0
$891$	−50.8618	−1.70394
$892$	0	0
$893$	13.8639	0.463936
$894$	0	0
$895$	2.32408	0.0776855
$896$	0	0
$897$	27.0844	0.904321
$898$	0	0
$899$	−1.18951	−0.0396723
$900$	0	0
$901$	−2.36482	−0.0787837
$902$	0	0
$903$	−35.4012	−1.17808
$904$	0	0
$905$	24.6416	0.819116
$906$	0	0
$907$	0.271753	0.00902342	0.00451171	−	0.999990i	$-0.498564\pi$
$907$	0.271753	0.00902342	0.00451171	+	0.999990i	$0.498564\pi$
$908$	0	0
$909$	−12.5736	−0.417039
$910$	0	0
$911$	6.97746	0.231174	0.115587	−	0.993297i	$-0.463125\pi$
$911$	6.97746	0.231174	0.115587	+	0.993297i	$0.463125\pi$
$912$	0	0
$913$	−59.2092	−1.95954
$914$	0	0
$915$	−37.7468	−1.24787
$916$	0	0
$917$	2.03556	0.0672200
$918$	0	0
$919$	25.4781	0.840446	0.420223	−	0.907421i	$-0.361952\pi$
$919$	25.4781	0.840446	0.420223	+	0.907421i	$0.361952\pi$
$920$	0	0
$921$	11.9223	0.392853
$922$	0	0
$923$	33.6232	1.10672
$924$	0	0
$925$	23.7140	0.779711
$926$	0	0
$927$	3.37422	0.110824
$928$	0	0
$929$	−13.6961	−0.449354	−0.224677	−	0.974433i	$-0.572133\pi$
$929$	−13.6961	−0.449354	−0.224677	+	0.974433i	$0.572133\pi$
$930$	0	0
$931$	5.24371	0.171856
$932$	0	0
$933$	45.7661	1.49831
$934$	0	0
$935$	15.2235	0.497863
$936$	0	0
$937$	19.1495	0.625587	0.312793	−	0.949821i	$-0.398735\pi$
$937$	19.1495	0.625587	0.312793	+	0.949821i	$0.398735\pi$
$938$	0	0
$939$	9.58830	0.312902
$940$	0	0
$941$	23.4469	0.764348	0.382174	−	0.924090i	$-0.375176\pi$
$941$	23.4469	0.764348	0.382174	+	0.924090i	$0.375176\pi$
$942$	0	0
$943$	−34.1235	−1.11121
$944$	0	0
$945$	−32.0054	−1.04114
$946$	0	0
$947$	−6.57956	−0.213807	−0.106903	−	0.994269i	$-0.534094\pi$
$947$	−6.57956	−0.213807	−0.106903	+	0.994269i	$0.534094\pi$
$948$	0	0
$949$	−36.7498	−1.19295
$950$	0	0
$951$	−39.0980	−1.26784
$952$	0	0
$953$	33.9189	1.09874	0.549370	−	0.835579i	$-0.314868\pi$
$953$	33.9189	1.09874	0.549370	+	0.835579i	$0.314868\pi$
$954$	0	0
$955$	73.8058	2.38830
$956$	0	0
$957$	−30.6224	−0.989881
$958$	0	0
$959$	24.5612	0.793124
$960$	0	0
$961$	−30.8717	−0.995860
$962$	0	0
$963$	−0.647006	−0.0208495
$964$	0	0
$965$	39.9981	1.28758
$966$	0	0
$967$	−39.9107	−1.28344	−0.641721	−	0.766938i	$-0.721779\pi$
$967$	−39.9107	−1.28344	−0.641721	+	0.766938i	$0.721779\pi$
$968$	0	0
$969$	−7.35087	−0.236144
$970$	0	0
$971$	53.5247	1.71769	0.858845	−	0.512236i	$-0.171183\pi$
$971$	53.5247	1.71769	0.858845	+	0.512236i	$0.171183\pi$
$972$	0	0
$973$	−15.5712	−0.499188
$974$	0	0
$975$	46.1210	1.47705
$976$	0	0
$977$	−61.3120	−1.96154	−0.980772	−	0.195157i	$-0.937478\pi$
$977$	−61.3120	−1.96154	−0.980772	+	0.195157i	$0.937478\pi$
$978$	0	0
$979$	−50.1207	−1.60187
$980$	0	0
$981$	9.26420	0.295783
$982$	0	0
$983$	−23.9975	−0.765402	−0.382701	−	0.923872i	$-0.625006\pi$
$983$	−23.9975	−0.765402	−0.382701	+	0.923872i	$0.625006\pi$
$984$	0	0
$985$	−3.59245	−0.114465
$986$	0	0
$987$	−17.3710	−0.552926
$988$	0	0
$989$	−25.0000	−0.794952
$990$	0	0
$991$	−1.79182	−0.0569190	−0.0284595	−	0.999595i	$-0.509060\pi$
$991$	−1.79182	−0.0569190	−0.0284595	+	0.999595i	$0.509060\pi$
$992$	0	0
$993$	47.2733	1.50017
$994$	0	0
$995$	15.3686	0.487218
$996$	0	0
$997$	−16.0745	−0.509085	−0.254543	−	0.967062i	$-0.581925\pi$
$997$	−16.0745	−0.509085	−0.254543	+	0.967062i	$0.581925\pi$
$998$	0	0
$999$	−17.5549	−0.555411

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4012.2.a.j.1.15	✓	21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4012.2.a.j.1.15	✓	21	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 4012 = 2^{2} \cdot 17 \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4012.a (trivial)

\(f(q)\)	\(=\)	\(q+1.97354 q^{3} +3.25764 q^{5} +2.36478 q^{7} +0.894844 q^{9} +O(q^{10})\)	\(q+1.97354 q^{3} +3.25764 q^{5} +2.36478 q^{7} +0.894844 q^{9} +4.67317 q^{11} +4.16406 q^{13} +6.42908 q^{15} +1.00000 q^{17} -3.72472 q^{19} +4.66698 q^{21} +3.29577 q^{23} +5.61224 q^{25} -4.15460 q^{27} -3.32033 q^{29} +0.358249 q^{31} +9.22268 q^{33} +7.70361 q^{35} +4.22540 q^{37} +8.21793 q^{39} -10.3537 q^{41} -7.58547 q^{43} +2.91508 q^{45} -3.72212 q^{47} -1.40781 q^{49} +1.97354 q^{51} -2.36482 q^{53} +15.2235 q^{55} -7.35087 q^{57} +1.00000 q^{59} -5.87127 q^{61} +2.11611 q^{63} +13.5650 q^{65} -8.10843 q^{67} +6.50432 q^{69} +8.07461 q^{71} -8.82546 q^{73} +11.0760 q^{75} +11.0510 q^{77} -2.58295 q^{79} -10.8838 q^{81} -12.6700 q^{83} +3.25764 q^{85} -6.55280 q^{87} -10.7252 q^{89} +9.84709 q^{91} +0.707018 q^{93} -12.1338 q^{95} +13.0852 q^{97} +4.18176 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9}+O(q^{10}) \) 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9	\( 21 q + 9 q^{3} + q^{5} + 11 q^{7} + 26 q^{9} + 12 q^{11} - 4 q^{13} + 9 q^{15} + 21 q^{17} + 4 q^{19} + 8 q^{21} + 19 q^{23} + 26 q^{25} + 33 q^{27} - q^{29} + 13 q^{31} + 11 q^{33} + 15 q^{35} - 4 q^{37} + 18 q^{39} + 9 q^{41} + 7 q^{43} + 7 q^{45} + 33 q^{47} + 36 q^{49} + 9 q^{51} + 7 q^{53} + 12 q^{55} + 26 q^{57} + 21 q^{59} + 3 q^{61} + 51 q^{63} + q^{65} + 8 q^{69} + 55 q^{71} + 22 q^{73} + 14 q^{75} - 15 q^{77} + 28 q^{79} + 25 q^{81} + 54 q^{83} + q^{85} + 34 q^{87} + 30 q^{89} + 35 q^{91} - 5 q^{93} + 42 q^{95} + 9 q^{97} + 20 q^{99}+O(q^{100}) \) 21 * q + 9 * q^3 + q^5 + 11 * q^7 + 26 * q^9 + 12 * q^11 - 4 * q^13 + 9 * q^15 + 21 * q^17 + 4 * q^19 + 8 * q^21 + 19 * q^23 + 26 * q^25 + 33 * q^27 - q^29 + 13 * q^31 + 11 * q^33 + 15 * q^35 - 4 * q^37 + 18 * q^39 + 9 * q^41 + 7 * q^43 + 7 * q^45 + 33 * q^47 + 36 * q^49 + 9 * q^51 + 7 * q^53 + 12 * q^55 + 26 * q^57 + 21 * q^59 + 3 * q^61 + 51 * q^63 + q^65 + 8 * q^69 + 55 * q^71 + 22 * q^73 + 14 * q^75 - 15 * q^77 + 28 * q^79 + 25 * q^81 + 54 * q^83 + q^85 + 34 * q^87 + 30 * q^89 + 35 * q^91 - 5 * q^93 + 42 * q^95 + 9 * q^97 + 20 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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