LMFDB - Embedded newform 2600.2.d.k.1249.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2600,2,Mod(1249,2600)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2600, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2600.1249"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2600 = 2^{3} \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2600.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,0,0,0,0,-6,0,-4,0,0,0,0,0,0,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$20.7611045255$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{17})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 9x^{2} + 16 $ x^4 + 9*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 104)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1249.4
Root			$2.56155i$ of defining polynomial
Character	$\chi$	$=$	2600.1249
Dual form			2600.2.d.k.1249.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56155i q^{3} +2.56155i q^{7} -3.56155 q^{9} -5.12311 q^{11} +1.00000i q^{13} -5.68466i q^{17} -5.12311 q^{19} -6.56155 q^{21} -8.00000i q^{23} -1.43845i q^{27} +2.00000 q^{29} +4.00000 q^{31} -13.1231i q^{33} -9.68466i q^{37} -2.56155 q^{39} -3.12311 q^{41} +5.43845i q^{43} +0.315342i q^{47} +0.438447 q^{49} +14.5616 q^{51} +3.12311i q^{53} -13.1231i q^{57} -5.12311 q^{59} +11.1231 q^{61} -9.12311i q^{63} +5.12311i q^{67} +20.4924 q^{69} -7.68466 q^{71} -6.00000i q^{73} -13.1231i q^{77} -8.00000 q^{79} -7.00000 q^{81} +2.24621i q^{83} +5.12311i q^{87} -10.0000 q^{89} -2.56155 q^{91} +10.2462i q^{93} +8.24621i q^{97} +18.2462 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 6 q^{9} - 4 q^{11} - 4 q^{19} - 18 q^{21} + 8 q^{29} + 16 q^{31} - 2 q^{39} + 4 q^{41} + 10 q^{49} + 50 q^{51} - 4 q^{59} + 28 q^{61} + 16 q^{69} - 6 q^{71} - 32 q^{79} - 28 q^{81} - 40 q^{89} - 2 q^{91}+ \cdots + 40 q^{99}+O(q^{100}) $ 4 * q - 6 * q^9 - 4 * q^11 - 4 * q^19 - 18 * q^21 + 8 * q^29 + 16 * q^31 - 2 * q^39 + 4 * q^41 + 10 * q^49 + 50 * q^51 - 4 * q^59 + 28 * q^61 + 16 * q^69 - 6 * q^71 - 32 * q^79 - 28 * q^81 - 40 * q^89 - 2 * q^91 + 40 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2600\mathbb{Z}\right)^\times$.

$n$	$1301$	$1601$	$1951$	$1977$
$\chi(n)$	$1$	$1$	$1$	$-1$

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
$3$	2.56155i	1.47891i	0.673204	+	0.739457i	$0.264917\pi$
$3$	2.56155i	1.47891i	−0.673204	+	0.739457i	$0.735083\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	2.56155i	0.968176i	0.875019	+	0.484088i	$0.160849\pi$
$7$	2.56155i	0.968176i	−0.875019	+	0.484088i	$0.839151\pi$
$8$	0	0
$9$	−3.56155	−1.18718
$10$	0	0
$11$	−5.12311	−1.54467	−0.772337	−	0.635213i	$-0.780912\pi$
$11$	−5.12311	−1.54467	−0.772337	+	0.635213i	$0.780912\pi$
$12$	0	0
$13$	1.00000i	0.277350i
$13$	1.00000i	0.277350i
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 5.68466i	− 1.37873i	−0.724413	−	0.689366i	$-0.757889\pi$
$17$	− 5.68466i	− 1.37873i	0.724413	−	0.689366i	$-0.242111\pi$
$18$	0	0
$19$	−5.12311	−1.17532	−0.587661	−	0.809108i	$-0.699951\pi$
$19$	−5.12311	−1.17532	−0.587661	+	0.809108i	$0.699951\pi$
$20$	0	0
$21$	−6.56155	−1.43185
$22$	0	0
$23$	− 8.00000i	− 1.66812i	−0.551677	−	0.834058i	$-0.686012\pi$
$23$	− 8.00000i	− 1.66812i	0.551677	−	0.834058i	$-0.313988\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	− 1.43845i	− 0.276829i
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	0	0
$33$	− 13.1231i	− 2.28444i
$34$	0	0
$35$	0	0
$36$	0	0
$37$	− 9.68466i	− 1.59215i	−0.605199	−	0.796074i	$-0.706907\pi$
$37$	− 9.68466i	− 1.59215i	0.605199	−	0.796074i	$-0.293093\pi$
$38$	0	0
$39$	−2.56155	−0.410177
$40$	0	0
$41$	−3.12311	−0.487747	−0.243874	−	0.969807i	$-0.578418\pi$
$41$	−3.12311	−0.487747	−0.243874	+	0.969807i	$0.578418\pi$
$42$	0	0
$43$	5.43845i	0.829355i	0.909968	+	0.414678i	$0.136106\pi$
$43$	5.43845i	0.829355i	−0.909968	+	0.414678i	$0.863894\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0.315342i	0.0459973i	0.999735	+	0.0229986i	$0.00732134\pi$
$47$	0.315342i	0.0459973i	−0.999735	+	0.0229986i	$0.992679\pi$
$48$	0	0
$49$	0.438447	0.0626353
$50$	0	0
$51$	14.5616	2.03903
$52$	0	0
$53$	3.12311i	0.428992i	0.976725	+	0.214496i	$0.0688108\pi$
$53$	3.12311i	0.428992i	−0.976725	+	0.214496i	$0.931189\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	− 13.1231i	− 1.73820i
$58$	0	0
$59$	−5.12311	−0.666972	−0.333486	−	0.942755i	$-0.608225\pi$
$59$	−5.12311	−0.666972	−0.333486	+	0.942755i	$0.608225\pi$
$60$	0	0
$61$	11.1231	1.42417	0.712084	−	0.702094i	$-0.247752\pi$
$61$	11.1231	1.42417	0.712084	+	0.702094i	$0.247752\pi$
$62$	0	0
$63$	− 9.12311i	− 1.14940i
$64$	0	0
$65$	0	0
$66$	0	0
$67$	5.12311i	0.625887i	0.949772	+	0.312943i	$0.101315\pi$
$67$	5.12311i	0.625887i	−0.949772	+	0.312943i	$0.898685\pi$
$68$	0	0
$69$	20.4924	2.46700
$70$	0	0
$71$	−7.68466	−0.912001	−0.456001	−	0.889979i	$-0.650719\pi$
$71$	−7.68466	−0.912001	−0.456001	+	0.889979i	$0.650719\pi$
$72$	0	0
$73$	− 6.00000i	− 0.702247i	−0.936329	−	0.351123i	$-0.885800\pi$
$73$	− 6.00000i	− 0.702247i	0.936329	−	0.351123i	$-0.114200\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 13.1231i	− 1.49552i
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	2.24621i	0.246554i	0.992372	+	0.123277i	$0.0393403\pi$
$83$	2.24621i	0.246554i	−0.992372	+	0.123277i	$0.960660\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	5.12311i	0.549255i
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	−2.56155	−0.268524
$92$	0	0
$93$	10.2462i	1.06248i
$94$	0	0
$95$	0	0
$96$	0	0
$97$	8.24621i	0.837276i	0.908153	+	0.418638i	$0.137492\pi$
$97$	8.24621i	0.837276i	−0.908153	+	0.418638i	$0.862508\pi$
$98$	0	0
$99$	18.2462	1.83381
$100$	0	0
$101$	−7.12311	−0.708776	−0.354388	−	0.935099i	$-0.615311\pi$
$101$	−7.12311	−0.708776	−0.354388	+	0.935099i	$0.615311\pi$
$102$	0	0
$103$	2.24621i	0.221326i	0.993858	+	0.110663i	$0.0352974\pi$
$103$	2.24621i	0.221326i	−0.993858	+	0.110663i	$0.964703\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 4.00000i	− 0.386695i	−0.981130	−	0.193347i	$-0.938066\pi$
$107$	− 4.00000i	− 0.386695i	0.981130	−	0.193347i	$-0.0619344\pi$
$108$	0	0
$109$	2.80776	0.268935	0.134468	−	0.990918i	$-0.457068\pi$
$109$	2.80776	0.268935	0.134468	+	0.990918i	$0.457068\pi$
$110$	0	0
$111$	24.8078	2.35465
$112$	0	0
$113$	− 18.4924i	− 1.73962i	−0.493386	−	0.869810i	$-0.664241\pi$
$113$	− 18.4924i	− 1.73962i	0.493386	−	0.869810i	$-0.335759\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	− 3.56155i	− 0.329266i
$118$	0	0
$119$	14.5616	1.33486
$120$	0	0
$121$	15.2462	1.38602
$122$	0	0
$123$	− 8.00000i	− 0.721336i
$124$	0	0
$125$	0	0
$126$	0	0
$127$	10.2462i	0.909204i	0.890695	+	0.454602i	$0.150219\pi$
$127$	10.2462i	0.909204i	−0.890695	+	0.454602i	$0.849781\pi$
$128$	0	0
$129$	−13.9309	−1.22654
$130$	0	0
$131$	−4.80776	−0.420056	−0.210028	−	0.977695i	$-0.567356\pi$
$131$	−4.80776	−0.420056	−0.210028	+	0.977695i	$0.567356\pi$
$132$	0	0
$133$	− 13.1231i	− 1.13792i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.12311i	0.266825i	0.991061	+	0.133412i	$0.0425935\pi$
$137$	3.12311i	0.266825i	−0.991061	+	0.133412i	$0.957406\pi$
$138$	0	0
$139$	−7.68466	−0.651804	−0.325902	−	0.945404i	$-0.605668\pi$
$139$	−7.68466	−0.651804	−0.325902	+	0.945404i	$0.605668\pi$
$140$	0	0
$141$	−0.807764	−0.0680260
$142$	0	0
$143$	− 5.12311i	− 0.428416i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	1.12311i	0.0926322i
$148$	0	0
$149$	−16.2462	−1.33094	−0.665471	−	0.746424i	$-0.731769\pi$
$149$	−16.2462	−1.33094	−0.665471	+	0.746424i	$0.731769\pi$
$150$	0	0
$151$	10.5616	0.859487	0.429743	−	0.902951i	$-0.358604\pi$
$151$	10.5616	0.859487	0.429743	+	0.902951i	$0.358604\pi$
$152$	0	0
$153$	20.2462i	1.63681i
$154$	0	0
$155$	0	0
$156$	0	0
$157$	22.4924i	1.79509i	0.440922	+	0.897545i	$0.354651\pi$
$157$	22.4924i	1.79509i	−0.440922	+	0.897545i	$0.645349\pi$
$158$	0	0
$159$	−8.00000	−0.634441
$160$	0	0
$161$	20.4924	1.61503
$162$	0	0
$163$	− 20.4924i	− 1.60509i	−0.596591	−	0.802545i	$-0.703479\pi$
$163$	− 20.4924i	− 1.60509i	0.596591	−	0.802545i	$-0.296521\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 14.2462i	− 1.10240i	−0.834372	−	0.551202i	$-0.814169\pi$
$167$	− 14.2462i	− 1.10240i	0.834372	−	0.551202i	$-0.185831\pi$
$168$	0	0
$169$	−1.00000	−0.0769231
$170$	0	0
$171$	18.2462	1.39532
$172$	0	0
$173$	− 4.87689i	− 0.370783i	−0.982665	−	0.185392i	$-0.940645\pi$
$173$	− 4.87689i	− 0.370783i	0.982665	−	0.185392i	$-0.0593554\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	− 13.1231i	− 0.986393i
$178$	0	0
$179$	−7.68466	−0.574378	−0.287189	−	0.957874i	$-0.592721\pi$
$179$	−7.68466	−0.574378	−0.287189	+	0.957874i	$0.592721\pi$
$180$	0	0
$181$	−23.1231	−1.71873	−0.859363	−	0.511365i	$-0.829140\pi$
$181$	−23.1231	−1.71873	−0.859363	+	0.511365i	$0.829140\pi$
$182$	0	0
$183$	28.4924i	2.10622i
$184$	0	0
$185$	0	0
$186$	0	0
$187$	29.1231i	2.12969i
$188$	0	0
$189$	3.68466	0.268019
$190$	0	0
$191$	15.3693	1.11208	0.556042	−	0.831154i	$-0.312319\pi$
$191$	15.3693	1.11208	0.556042	+	0.831154i	$0.312319\pi$
$192$	0	0
$193$	− 21.3693i	− 1.53820i	−0.639130	−	0.769099i	$-0.720706\pi$
$193$	− 21.3693i	− 1.53820i	0.639130	−	0.769099i	$-0.279294\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 2.31534i	− 0.164961i	−0.996593	−	0.0824806i	$-0.973716\pi$
$197$	− 2.31534i	− 0.164961i	0.996593	−	0.0824806i	$-0.0262843\pi$
$198$	0	0
$199$	−5.12311	−0.363167	−0.181584	−	0.983375i	$-0.558122\pi$
$199$	−5.12311	−0.363167	−0.181584	+	0.983375i	$0.558122\pi$
$200$	0	0
$201$	−13.1231	−0.925633
$202$	0	0
$203$	5.12311i	0.359572i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	28.4924i	1.98036i
$208$	0	0
$209$	26.2462	1.81549
$210$	0	0
$211$	15.6847	1.07978	0.539888	−	0.841737i	$-0.318467\pi$
$211$	15.6847	1.07978	0.539888	+	0.841737i	$0.318467\pi$
$212$	0	0
$213$	− 19.6847i	− 1.34877i
$214$	0	0
$215$	0	0
$216$	0	0
$217$	10.2462i	0.695558i
$218$	0	0
$219$	15.3693	1.03856
$220$	0	0
$221$	5.68466	0.382392
$222$	0	0
$223$	− 15.6847i	− 1.05032i	−0.851003	−	0.525161i	$-0.824005\pi$
$223$	− 15.6847i	− 1.05032i	0.851003	−	0.525161i	$-0.175995\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	8.00000i	0.530979i	0.964114	+	0.265489i	$0.0855335\pi$
$227$	8.00000i	0.530979i	−0.964114	+	0.265489i	$0.914466\pi$
$228$	0	0
$229$	−6.80776	−0.449870	−0.224935	−	0.974374i	$-0.572217\pi$
$229$	−6.80776	−0.449870	−0.224935	+	0.974374i	$0.572217\pi$
$230$	0	0
$231$	33.6155	2.21174
$232$	0	0
$233$	− 6.80776i	− 0.445991i	−0.974819	−	0.222996i	$-0.928416\pi$
$233$	− 6.80776i	− 0.445991i	0.974819	−	0.222996i	$-0.0715836\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	− 20.4924i	− 1.33113i
$238$	0	0
$239$	2.56155	0.165693	0.0828465	−	0.996562i	$-0.473599\pi$
$239$	2.56155	0.165693	0.0828465	+	0.996562i	$0.473599\pi$
$240$	0	0
$241$	−3.75379	−0.241803	−0.120901	−	0.992665i	$-0.538578\pi$
$241$	−3.75379	−0.241803	−0.120901	+	0.992665i	$0.538578\pi$
$242$	0	0
$243$	− 22.2462i	− 1.42710i
$244$	0	0
$245$	0	0
$246$	0	0
$247$	− 5.12311i	− 0.325975i
$248$	0	0
$249$	−5.75379	−0.364632
$250$	0	0
$251$	22.2462	1.40417	0.702084	−	0.712094i	$-0.252253\pi$
$251$	22.2462	1.40417	0.702084	+	0.712094i	$0.252253\pi$
$252$	0	0
$253$	40.9848i	2.57670i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 11.4384i	− 0.713511i	−0.934198	−	0.356755i	$-0.883883\pi$
$257$	− 11.4384i	− 0.713511i	0.934198	−	0.356755i	$-0.116117\pi$
$258$	0	0
$259$	24.8078	1.54148
$260$	0	0
$261$	−7.12311	−0.440909
$262$	0	0
$263$	− 26.2462i	− 1.61841i	−0.587526	−	0.809205i	$-0.699898\pi$
$263$	− 26.2462i	− 1.61841i	0.587526	−	0.809205i	$-0.300102\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	− 25.6155i	− 1.56764i
$268$	0	0
$269$	4.87689	0.297349	0.148675	−	0.988886i	$-0.452499\pi$
$269$	4.87689	0.297349	0.148675	+	0.988886i	$0.452499\pi$
$270$	0	0
$271$	−20.8078	−1.26398	−0.631991	−	0.774976i	$-0.717762\pi$
$271$	−20.8078	−1.26398	−0.631991	+	0.774976i	$0.717762\pi$
$272$	0	0
$273$	− 6.56155i	− 0.397123i
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 29.3693i	− 1.76463i	−0.470658	−	0.882316i	$-0.655984\pi$
$277$	− 29.3693i	− 1.76463i	0.470658	−	0.882316i	$-0.344016\pi$
$278$	0	0
$279$	−14.2462	−0.852898
$280$	0	0
$281$	10.0000	0.596550	0.298275	−	0.954480i	$-0.403589\pi$
$281$	10.0000	0.596550	0.298275	+	0.954480i	$0.403589\pi$
$282$	0	0
$283$	− 8.49242i	− 0.504822i	−0.967620	−	0.252411i	$-0.918776\pi$
$283$	− 8.49242i	− 0.504822i	0.967620	−	0.252411i	$-0.0812235\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	− 8.00000i	− 0.472225i
$288$	0	0
$289$	−15.3153	−0.900902
$290$	0	0
$291$	−21.1231	−1.23826
$292$	0	0
$293$	15.4384i	0.901924i	0.892543	+	0.450962i	$0.148919\pi$
$293$	15.4384i	0.901924i	−0.892543	+	0.450962i	$0.851081\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	7.36932i	0.427611i
$298$	0	0
$299$	8.00000	0.462652
$300$	0	0
$301$	−13.9309	−0.802962
$302$	0	0
$303$	− 18.2462i	− 1.04822i
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 13.1231i	− 0.748975i	−0.927232	−	0.374488i	$-0.877819\pi$
$307$	− 13.1231i	− 0.748975i	0.927232	−	0.374488i	$-0.122181\pi$
$308$	0	0
$309$	−5.75379	−0.327322
$310$	0	0
$311$	−10.8769	−0.616772	−0.308386	−	0.951261i	$-0.599789\pi$
$311$	−10.8769	−0.616772	−0.308386	+	0.951261i	$0.599789\pi$
$312$	0	0
$313$	26.8078i	1.51526i	0.652682	+	0.757632i	$0.273644\pi$
$313$	26.8078i	1.51526i	−0.652682	+	0.757632i	$0.726356\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	12.2462i	0.687816i	0.939003	+	0.343908i	$0.111751\pi$
$317$	12.2462i	0.687816i	−0.939003	+	0.343908i	$0.888249\pi$
$318$	0	0
$319$	−10.2462	−0.573678
$320$	0	0
$321$	10.2462	0.571888
$322$	0	0
$323$	29.1231i	1.62045i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	7.19224i	0.397732i
$328$	0	0
$329$	−0.807764	−0.0445335
$330$	0	0
$331$	−4.49242	−0.246926	−0.123463	−	0.992349i	$-0.539400\pi$
$331$	−4.49242	−0.246926	−0.123463	+	0.992349i	$0.539400\pi$
$332$	0	0
$333$	34.4924i	1.89017i
$334$	0	0
$335$	0	0
$336$	0	0
$337$	13.1922i	0.718627i	0.933217	+	0.359313i	$0.116989\pi$
$337$	13.1922i	0.718627i	−0.933217	+	0.359313i	$0.883011\pi$
$338$	0	0
$339$	47.3693	2.57275
$340$	0	0
$341$	−20.4924	−1.10973
$342$	0	0
$343$	19.0540i	1.02882i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 7.68466i	− 0.412534i	−0.978496	−	0.206267i	$-0.933868\pi$
$347$	− 7.68466i	− 0.412534i	0.978496	−	0.206267i	$-0.0661316\pi$
$348$	0	0
$349$	−11.9309	−0.638645	−0.319322	−	0.947646i	$-0.603455\pi$
$349$	−11.9309	−0.638645	−0.319322	+	0.947646i	$0.603455\pi$
$350$	0	0
$351$	1.43845	0.0767786
$352$	0	0
$353$	9.36932i	0.498678i	0.968416	+	0.249339i	$0.0802134\pi$
$353$	9.36932i	0.498678i	−0.968416	+	0.249339i	$0.919787\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	37.3002i	1.97414i
$358$	0	0
$359$	−14.2462	−0.751886	−0.375943	−	0.926643i	$-0.622681\pi$
$359$	−14.2462	−0.751886	−0.375943	+	0.926643i	$0.622681\pi$
$360$	0	0
$361$	7.24621	0.381380
$362$	0	0
$363$	39.0540i	2.04980i
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 9.61553i	− 0.501926i	−0.967997	−	0.250963i	$-0.919253\pi$
$367$	− 9.61553i	− 0.501926i	0.967997	−	0.250963i	$-0.0807473\pi$
$368$	0	0
$369$	11.1231	0.579046
$370$	0	0
$371$	−8.00000	−0.415339
$372$	0	0
$373$	− 11.6155i	− 0.601429i	−0.953714	−	0.300715i	$-0.902775\pi$
$373$	− 11.6155i	− 0.601429i	0.953714	−	0.300715i	$-0.0972252\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.00000i	0.103005i
$378$	0	0
$379$	−5.75379	−0.295552	−0.147776	−	0.989021i	$-0.547212\pi$
$379$	−5.75379	−0.295552	−0.147776	+	0.989021i	$0.547212\pi$
$380$	0	0
$381$	−26.2462	−1.34463
$382$	0	0
$383$	− 11.1922i	− 0.571897i	−0.958245	−	0.285948i	$-0.907691\pi$
$383$	− 11.1922i	− 0.571897i	0.958245	−	0.285948i	$-0.0923086\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	− 19.3693i	− 0.984598i
$388$	0	0
$389$	−20.7386	−1.05149	−0.525745	−	0.850642i	$-0.676213\pi$
$389$	−20.7386	−1.05149	−0.525745	+	0.850642i	$0.676213\pi$
$390$	0	0
$391$	−45.4773	−2.29988
$392$	0	0
$393$	− 12.3153i	− 0.621227i
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.00000i	0.100377i	0.998740	+	0.0501886i	$0.0159822\pi$
$397$	2.00000i	0.100377i	−0.998740	+	0.0501886i	$0.984018\pi$
$398$	0	0
$399$	33.6155	1.68288
$400$	0	0
$401$	3.61553	0.180551	0.0902754	−	0.995917i	$-0.471225\pi$
$401$	3.61553	0.180551	0.0902754	+	0.995917i	$0.471225\pi$
$402$	0	0
$403$	4.00000i	0.199254i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	49.6155i	2.45935i
$408$	0	0
$409$	−12.8769	−0.636721	−0.318361	−	0.947970i	$-0.603132\pi$
$409$	−12.8769	−0.636721	−0.318361	+	0.947970i	$0.603132\pi$
$410$	0	0
$411$	−8.00000	−0.394611
$412$	0	0
$413$	− 13.1231i	− 0.645746i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	− 19.6847i	− 0.963962i
$418$	0	0
$419$	1.93087	0.0943292	0.0471646	−	0.998887i	$-0.484981\pi$
$419$	1.93087	0.0943292	0.0471646	+	0.998887i	$0.484981\pi$
$420$	0	0
$421$	−6.31534	−0.307791	−0.153895	−	0.988087i	$-0.549182\pi$
$421$	−6.31534	−0.307791	−0.153895	+	0.988087i	$0.549182\pi$
$422$	0	0
$423$	− 1.12311i	− 0.0546073i
$424$	0	0
$425$	0	0
$426$	0	0
$427$	28.4924i	1.37884i
$428$	0	0
$429$	13.1231	0.633590
$430$	0	0
$431$	−18.5616	−0.894079	−0.447039	−	0.894514i	$-0.647522\pi$
$431$	−18.5616	−0.894079	−0.447039	+	0.894514i	$0.647522\pi$
$432$	0	0
$433$	− 6.17708i	− 0.296852i	−0.988924	−	0.148426i	$-0.952579\pi$
$433$	− 6.17708i	− 0.296852i	0.988924	−	0.148426i	$-0.0474206\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	40.9848i	1.96057i
$438$	0	0
$439$	−21.1231	−1.00815	−0.504075	−	0.863660i	$-0.668167\pi$
$439$	−21.1231	−1.00815	−0.504075	+	0.863660i	$0.668167\pi$
$440$	0	0
$441$	−1.56155	−0.0743597
$442$	0	0
$443$	20.8078i	0.988607i	0.869289	+	0.494303i	$0.164577\pi$
$443$	20.8078i	0.988607i	−0.869289	+	0.494303i	$0.835423\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	− 41.6155i	− 1.96835i
$448$	0	0
$449$	−32.7386	−1.54503	−0.772516	−	0.634996i	$-0.781002\pi$
$449$	−32.7386	−1.54503	−0.772516	+	0.634996i	$0.781002\pi$
$450$	0	0
$451$	16.0000	0.753411
$452$	0	0
$453$	27.0540i	1.27111i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 24.7386i	− 1.15722i	−0.815603	−	0.578612i	$-0.803594\pi$
$457$	− 24.7386i	− 1.15722i	0.815603	−	0.578612i	$-0.196406\pi$
$458$	0	0
$459$	−8.17708	−0.381673
$460$	0	0
$461$	−34.8078	−1.62116	−0.810580	−	0.585628i	$-0.800848\pi$
$461$	−34.8078	−1.62116	−0.810580	+	0.585628i	$0.800848\pi$
$462$	0	0
$463$	38.2462i	1.77745i	0.458439	+	0.888726i	$0.348409\pi$
$463$	38.2462i	1.77745i	−0.458439	+	0.888726i	$0.651591\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 40.4924i	− 1.87377i	−0.349643	−	0.936883i	$-0.613697\pi$
$467$	− 40.4924i	− 1.87377i	0.349643	−	0.936883i	$-0.386303\pi$
$468$	0	0
$469$	−13.1231	−0.605969
$470$	0	0
$471$	−57.6155	−2.65478
$472$	0	0
$473$	− 27.8617i	− 1.28108i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	− 11.1231i	− 0.509292i
$478$	0	0
$479$	−11.1922	−0.511386	−0.255693	−	0.966758i	$-0.582304\pi$
$479$	−11.1922	−0.511386	−0.255693	+	0.966758i	$0.582304\pi$
$480$	0	0
$481$	9.68466	0.441582
$482$	0	0
$483$	52.4924i	2.38849i
$484$	0	0
$485$	0	0
$486$	0	0
$487$	− 1.75379i	− 0.0794718i	−0.999210	−	0.0397359i	$-0.987348\pi$
$487$	− 1.75379i	− 0.0794718i	0.999210	−	0.0397359i	$-0.0126517\pi$
$488$	0	0
$489$	52.4924	2.37379
$490$	0	0
$491$	0.946025	0.0426935	0.0213467	−	0.999772i	$-0.493205\pi$
$491$	0.946025	0.0426935	0.0213467	+	0.999772i	$0.493205\pi$
$492$	0	0
$493$	− 11.3693i	− 0.512048i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 19.6847i	− 0.882978i
$498$	0	0
$499$	−28.4924	−1.27550	−0.637748	−	0.770245i	$-0.720134\pi$
$499$	−28.4924	−1.27550	−0.637748	+	0.770245i	$0.720134\pi$
$500$	0	0
$501$	36.4924	1.63036
$502$	0	0
$503$	− 7.36932i	− 0.328582i	−0.986412	−	0.164291i	$-0.947466\pi$
$503$	− 7.36932i	− 0.328582i	0.986412	−	0.164291i	$-0.0525335\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	− 2.56155i	− 0.113763i
$508$	0	0
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	0	0
$511$	15.3693	0.679899
$512$	0	0
$513$	7.36932i	0.325363i
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 1.61553i	− 0.0710508i
$518$	0	0
$519$	12.4924	0.548356
$520$	0	0
$521$	−9.68466	−0.424293	−0.212146	−	0.977238i	$-0.568045\pi$
$521$	−9.68466	−0.424293	−0.212146	+	0.977238i	$0.568045\pi$
$522$	0	0
$523$	− 0.492423i	− 0.0215321i	−0.999942	−	0.0107661i	$-0.996573\pi$
$523$	− 0.492423i	− 0.0215321i	0.999942	−	0.0107661i	$-0.00342701\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 22.7386i	− 0.990510i
$528$	0	0
$529$	−41.0000	−1.78261
$530$	0	0
$531$	18.2462	0.791818
$532$	0	0
$533$	− 3.12311i	− 0.135277i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	− 19.6847i	− 0.849456i
$538$	0	0
$539$	−2.24621	−0.0967512
$540$	0	0
$541$	−5.68466	−0.244403	−0.122201	−	0.992505i	$-0.538995\pi$
$541$	−5.68466	−0.244403	−0.122201	+	0.992505i	$0.538995\pi$
$542$	0	0
$543$	− 59.2311i	− 2.54185i
$544$	0	0
$545$	0	0
$546$	0	0
$547$	29.4384i	1.25870i	0.777123	+	0.629349i	$0.216678\pi$
$547$	29.4384i	1.25870i	−0.777123	+	0.629349i	$0.783322\pi$
$548$	0	0
$549$	−39.6155	−1.69075
$550$	0	0
$551$	−10.2462	−0.436503
$552$	0	0
$553$	− 20.4924i	− 0.871426i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	42.1771i	1.78710i	0.448963	+	0.893550i	$0.351793\pi$
$557$	42.1771i	1.78710i	−0.448963	+	0.893550i	$0.648207\pi$
$558$	0	0
$559$	−5.43845	−0.230022
$560$	0	0
$561$	−74.6004	−3.14963
$562$	0	0
$563$	− 7.05398i	− 0.297290i	−0.988891	−	0.148645i	$-0.952509\pi$
$563$	− 7.05398i	− 0.297290i	0.988891	−	0.148645i	$-0.0474911\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	− 17.9309i	− 0.753026i
$568$	0	0
$569$	41.6847	1.74751	0.873756	−	0.486365i	$-0.161677\pi$
$569$	41.6847	1.74751	0.873756	+	0.486365i	$0.161677\pi$
$570$	0	0
$571$	28.8078	1.20557	0.602784	−	0.797905i	$-0.294058\pi$
$571$	28.8078	1.20557	0.602784	+	0.797905i	$0.294058\pi$
$572$	0	0
$573$	39.3693i	1.64468i
$574$	0	0
$575$	0	0
$576$	0	0
$577$	24.2462i	1.00938i	0.863300	+	0.504691i	$0.168394\pi$
$577$	24.2462i	1.00938i	−0.863300	+	0.504691i	$0.831606\pi$
$578$	0	0
$579$	54.7386	2.27486
$580$	0	0
$581$	−5.75379	−0.238707
$582$	0	0
$583$	− 16.0000i	− 0.662652i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	24.9848i	1.03123i	0.856819	+	0.515617i	$0.172437\pi$
$587$	24.9848i	1.03123i	−0.856819	+	0.515617i	$0.827563\pi$
$588$	0	0
$589$	−20.4924	−0.844376
$590$	0	0
$591$	5.93087	0.243963
$592$	0	0
$593$	38.4924i	1.58069i	0.612659	+	0.790347i	$0.290100\pi$
$593$	38.4924i	1.58069i	−0.612659	+	0.790347i	$0.709900\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	− 13.1231i	− 0.537093i
$598$	0	0
$599$	−23.3693	−0.954844	−0.477422	−	0.878674i	$-0.658429\pi$
$599$	−23.3693	−0.954844	−0.477422	+	0.878674i	$0.658429\pi$
$600$	0	0
$601$	34.1771	1.39411	0.697056	−	0.717017i	$-0.254493\pi$
$601$	34.1771	1.39411	0.697056	+	0.717017i	$0.254493\pi$
$602$	0	0
$603$	− 18.2462i	− 0.743043i
$604$	0	0
$605$	0	0
$606$	0	0
$607$	− 15.3693i	− 0.623821i	−0.950111	−	0.311911i	$-0.899031\pi$
$607$	− 15.3693i	− 0.623821i	0.950111	−	0.311911i	$-0.100969\pi$
$608$	0	0
$609$	−13.1231	−0.531775
$610$	0	0
$611$	−0.315342	−0.0127574
$612$	0	0
$613$	6.00000i	0.242338i	0.992632	+	0.121169i	$0.0386643\pi$
$613$	6.00000i	0.242338i	−0.992632	+	0.121169i	$0.961336\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 21.8617i	− 0.880120i	−0.897968	−	0.440060i	$-0.854957\pi$
$617$	− 21.8617i	− 0.880120i	0.897968	−	0.440060i	$-0.145043\pi$
$618$	0	0
$619$	29.7538	1.19591	0.597953	−	0.801531i	$-0.295981\pi$
$619$	29.7538	1.19591	0.597953	+	0.801531i	$0.295981\pi$
$620$	0	0
$621$	−11.5076	−0.461783
$622$	0	0
$623$	− 25.6155i	− 1.02626i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	67.2311i	2.68495i
$628$	0	0
$629$	−55.0540	−2.19515
$630$	0	0
$631$	−47.6847	−1.89830	−0.949148	−	0.314830i	$-0.898053\pi$
$631$	−47.6847	−1.89830	−0.949148	+	0.314830i	$0.898053\pi$
$632$	0	0
$633$	40.1771i	1.59690i
$634$	0	0
$635$	0	0
$636$	0	0
$637$	0.438447i	0.0173719i
$638$	0	0
$639$	27.3693	1.08271
$640$	0	0
$641$	−8.24621	−0.325706	−0.162853	−	0.986650i	$-0.552070\pi$
$641$	−8.24621	−0.325706	−0.162853	+	0.986650i	$0.552070\pi$
$642$	0	0
$643$	35.8617i	1.41425i	0.707089	+	0.707124i	$0.250008\pi$
$643$	35.8617i	1.41425i	−0.707089	+	0.707124i	$0.749992\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	− 5.12311i	− 0.201410i	−0.994916	−	0.100705i	$-0.967890\pi$
$647$	− 5.12311i	− 0.201410i	0.994916	−	0.100705i	$-0.0321098\pi$
$648$	0	0
$649$	26.2462	1.03025
$650$	0	0
$651$	−26.2462	−1.02867
$652$	0	0
$653$	− 4.87689i	− 0.190848i	−0.995437	−	0.0954238i	$-0.969579\pi$
$653$	− 4.87689i	− 0.190848i	0.995437	−	0.0954238i	$-0.0304206\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	21.3693i	0.833696i
$658$	0	0
$659$	−16.4924	−0.642454	−0.321227	−	0.947002i	$-0.604095\pi$
$659$	−16.4924	−0.642454	−0.321227	+	0.947002i	$0.604095\pi$
$660$	0	0
$661$	−40.7386	−1.58455	−0.792275	−	0.610165i	$-0.791103\pi$
$661$	−40.7386	−1.58455	−0.792275	+	0.610165i	$0.791103\pi$
$662$	0	0
$663$	14.5616i	0.565524i
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 16.0000i	− 0.619522i
$668$	0	0
$669$	40.1771	1.55334
$670$	0	0
$671$	−56.9848	−2.19988
$672$	0	0
$673$	− 19.3002i	− 0.743968i	−0.928239	−	0.371984i	$-0.878678\pi$
$673$	− 19.3002i	− 0.743968i	0.928239	−	0.371984i	$-0.121322\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	1.36932i	0.0526271i	0.999654	+	0.0263136i	$0.00837683\pi$
$677$	1.36932i	0.0526271i	−0.999654	+	0.0263136i	$0.991623\pi$
$678$	0	0
$679$	−21.1231	−0.810630
$680$	0	0
$681$	−20.4924	−0.785271
$682$	0	0
$683$	− 24.0000i	− 0.918334i	−0.888350	−	0.459167i	$-0.848148\pi$
$683$	− 24.0000i	− 0.918334i	0.888350	−	0.459167i	$-0.151852\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	− 17.4384i	− 0.665318i
$688$	0	0
$689$	−3.12311	−0.118981
$690$	0	0
$691$	16.9848	0.646134	0.323067	−	0.946376i	$-0.395286\pi$
$691$	16.9848	0.646134	0.323067	+	0.946376i	$0.395286\pi$
$692$	0	0
$693$	46.7386i	1.77545i
$694$	0	0
$695$	0	0
$696$	0	0
$697$	17.7538i	0.672473i
$698$	0	0
$699$	17.4384	0.659583
$700$	0	0
$701$	22.6307	0.854749	0.427375	−	0.904075i	$-0.359439\pi$
$701$	22.6307	0.854749	0.427375	+	0.904075i	$0.359439\pi$
$702$	0	0
$703$	49.6155i	1.87129i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 18.2462i	− 0.686219i
$708$	0	0
$709$	24.7386	0.929079	0.464539	−	0.885552i	$-0.346220\pi$
$709$	24.7386	0.929079	0.464539	+	0.885552i	$0.346220\pi$
$710$	0	0
$711$	28.4924	1.06855
$712$	0	0
$713$	− 32.0000i	− 1.19841i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	6.56155i	0.245046i
$718$	0	0
$719$	13.1231	0.489409	0.244705	−	0.969598i	$-0.421309\pi$
$719$	13.1231	0.489409	0.244705	+	0.969598i	$0.421309\pi$
$720$	0	0
$721$	−5.75379	−0.214282
$722$	0	0
$723$	− 9.61553i	− 0.357605i
$724$	0	0
$725$	0	0
$726$	0	0
$727$	46.1080i	1.71005i	0.518587	+	0.855025i	$0.326458\pi$
$727$	46.1080i	1.71005i	−0.518587	+	0.855025i	$0.673542\pi$
$728$	0	0
$729$	35.9848	1.33277
$730$	0	0
$731$	30.9157	1.14346
$732$	0	0
$733$	− 27.4384i	− 1.01346i	−0.862104	−	0.506731i	$-0.830854\pi$
$733$	− 27.4384i	− 1.01346i	0.862104	−	0.506731i	$-0.169146\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	− 26.2462i	− 0.966792i
$738$	0	0
$739$	10.2462	0.376913	0.188456	−	0.982082i	$-0.439652\pi$
$739$	10.2462	0.376913	0.188456	+	0.982082i	$0.439652\pi$
$740$	0	0
$741$	13.1231	0.482089
$742$	0	0
$743$	50.9157i	1.86792i	0.357382	+	0.933958i	$0.383669\pi$
$743$	50.9157i	1.86792i	−0.357382	+	0.933958i	$0.616331\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	− 8.00000i	− 0.292705i
$748$	0	0
$749$	10.2462	0.374388
$750$	0	0
$751$	24.9848	0.911710	0.455855	−	0.890054i	$-0.349334\pi$
$751$	24.9848	0.911710	0.455855	+	0.890054i	$0.349334\pi$
$752$	0	0
$753$	56.9848i	2.07664i
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 3.12311i	− 0.113511i	−0.998388	−	0.0567556i	$-0.981924\pi$
$757$	− 3.12311i	− 0.113511i	0.998388	−	0.0567556i	$-0.0180756\pi$
$758$	0	0
$759$	−104.985	−3.81071
$760$	0	0
$761$	46.4924	1.68535	0.842674	−	0.538423i	$-0.180980\pi$
$761$	46.4924	1.68535	0.842674	+	0.538423i	$0.180980\pi$
$762$	0	0
$763$	7.19224i	0.260376i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 5.12311i	− 0.184985i
$768$	0	0
$769$	−13.8617	−0.499867	−0.249934	−	0.968263i	$-0.580409\pi$
$769$	−13.8617	−0.499867	−0.249934	+	0.968263i	$0.580409\pi$
$770$	0	0
$771$	29.3002	1.05522
$772$	0	0
$773$	46.1771i	1.66087i	0.557112	+	0.830437i	$0.311909\pi$
$773$	46.1771i	1.66087i	−0.557112	+	0.830437i	$0.688091\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	63.5464i	2.27971i
$778$	0	0
$779$	16.0000	0.573259
$780$	0	0
$781$	39.3693	1.40874
$782$	0	0
$783$	− 2.87689i	− 0.102812i
$784$	0	0
$785$	0	0
$786$	0	0
$787$	16.0000i	0.570338i	0.958477	+	0.285169i	$0.0920498\pi$
$787$	16.0000i	0.570338i	−0.958477	+	0.285169i	$0.907950\pi$
$788$	0	0
$789$	67.2311	2.39349
$790$	0	0
$791$	47.3693	1.68426
$792$	0	0
$793$	11.1231i	0.394993i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	13.5076i	0.478463i	0.970963	+	0.239231i	$0.0768955\pi$
$797$	13.5076i	0.478463i	−0.970963	+	0.239231i	$0.923105\pi$
$798$	0	0
$799$	1.79261	0.0634180
$800$	0	0
$801$	35.6155	1.25841
$802$	0	0
$803$	30.7386i	1.08474i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	12.4924i	0.439754i
$808$	0	0
$809$	30.1771	1.06097	0.530485	−	0.847694i	$-0.322010\pi$
$809$	30.1771	1.06097	0.530485	+	0.847694i	$0.322010\pi$
$810$	0	0
$811$	−46.7386	−1.64122	−0.820608	−	0.571492i	$-0.806365\pi$
$811$	−46.7386	−1.64122	−0.820608	+	0.571492i	$0.806365\pi$
$812$	0	0
$813$	− 53.3002i	− 1.86932i
$814$	0	0
$815$	0	0
$816$	0	0
$817$	− 27.8617i	− 0.974759i
$818$	0	0
$819$	9.12311	0.318787
$820$	0	0
$821$	15.4384	0.538806	0.269403	−	0.963028i	$-0.413174\pi$
$821$	15.4384	0.538806	0.269403	+	0.963028i	$0.413174\pi$
$822$	0	0
$823$	− 9.61553i	− 0.335176i	−0.985857	−	0.167588i	$-0.946402\pi$
$823$	− 9.61553i	− 0.335176i	0.985857	−	0.167588i	$-0.0535979\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	35.8617i	1.24703i	0.781809	+	0.623517i	$0.214297\pi$
$827$	35.8617i	1.24703i	−0.781809	+	0.623517i	$0.785703\pi$
$828$	0	0
$829$	−18.4924	−0.642268	−0.321134	−	0.947034i	$-0.604064\pi$
$829$	−18.4924	−0.642268	−0.321134	+	0.947034i	$0.604064\pi$
$830$	0	0
$831$	75.2311	2.60974
$832$	0	0
$833$	− 2.49242i	− 0.0863573i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	− 5.75379i	− 0.198880i
$838$	0	0
$839$	10.7386	0.370739	0.185369	−	0.982669i	$-0.440652\pi$
$839$	10.7386	0.370739	0.185369	+	0.982669i	$0.440652\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	25.6155i	0.882246i
$844$	0	0
$845$	0	0
$846$	0	0
$847$	39.0540i	1.34191i
$848$	0	0
$849$	21.7538	0.746588
$850$	0	0
$851$	−77.4773	−2.65589
$852$	0	0
$853$	− 29.6847i	− 1.01638i	−0.861244	−	0.508192i	$-0.830314\pi$
$853$	− 29.6847i	− 1.01638i	0.861244	−	0.508192i	$-0.169686\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 40.7386i	− 1.39161i	−0.718233	−	0.695803i	$-0.755049\pi$
$857$	− 40.7386i	− 1.39161i	0.718233	−	0.695803i	$-0.244951\pi$
$858$	0	0
$859$	−12.9848	−0.443037	−0.221519	−	0.975156i	$-0.571101\pi$
$859$	−12.9848	−0.443037	−0.221519	+	0.975156i	$0.571101\pi$
$860$	0	0
$861$	20.4924	0.698380
$862$	0	0
$863$	− 49.3002i	− 1.67820i	−0.543979	−	0.839099i	$-0.683083\pi$
$863$	− 49.3002i	− 1.67820i	0.543979	−	0.839099i	$-0.316917\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	− 39.2311i	− 1.33236i
$868$	0	0
$869$	40.9848	1.39032
$870$	0	0
$871$	−5.12311	−0.173590
$872$	0	0
$873$	− 29.3693i	− 0.994001i
$874$	0	0
$875$	0	0
$876$	0	0
$877$	37.6847i	1.27252i	0.771474	+	0.636260i	$0.219520\pi$
$877$	37.6847i	1.27252i	−0.771474	+	0.636260i	$0.780480\pi$
$878$	0	0
$879$	−39.5464	−1.33387
$880$	0	0
$881$	−30.8078	−1.03794	−0.518970	−	0.854792i	$-0.673684\pi$
$881$	−30.8078	−1.03794	−0.518970	+	0.854792i	$0.673684\pi$
$882$	0	0
$883$	0.315342i	0.0106121i	0.999986	+	0.00530604i	$0.00168897\pi$
$883$	0.315342i	0.0106121i	−0.999986	+	0.00530604i	$0.998311\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	45.4773i	1.52698i	0.645821	+	0.763489i	$0.276515\pi$
$887$	45.4773i	1.52698i	−0.645821	+	0.763489i	$0.723485\pi$
$888$	0	0
$889$	−26.2462	−0.880270
$890$	0	0
$891$	35.8617	1.20141
$892$	0	0
$893$	− 1.61553i	− 0.0540616i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	20.4924i	0.684222i
$898$	0	0
$899$	8.00000	0.266815
$900$	0	0
$901$	17.7538	0.591464
$902$	0	0
$903$	− 35.6847i	− 1.18751i
$904$	0	0
$905$	0	0
$906$	0	0
$907$	20.8078i	0.690910i	0.938435	+	0.345455i	$0.112275\pi$
$907$	20.8078i	0.690910i	−0.938435	+	0.345455i	$0.887725\pi$
$908$	0	0
$909$	25.3693	0.841447
$910$	0	0
$911$	15.3693	0.509208	0.254604	−	0.967045i	$-0.418055\pi$
$911$	15.3693	0.509208	0.254604	+	0.967045i	$0.418055\pi$
$912$	0	0
$913$	− 11.5076i	− 0.380845i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 12.3153i	− 0.406688i
$918$	0	0
$919$	−24.9848	−0.824174	−0.412087	−	0.911145i	$-0.635200\pi$
$919$	−24.9848	−0.824174	−0.412087	+	0.911145i	$0.635200\pi$
$920$	0	0
$921$	33.6155	1.10767
$922$	0	0
$923$	− 7.68466i	− 0.252944i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	− 8.00000i	− 0.262754i
$928$	0	0
$929$	9.86174	0.323553	0.161777	−	0.986827i	$-0.448278\pi$
$929$	9.86174	0.323553	0.161777	+	0.986827i	$0.448278\pi$
$930$	0	0
$931$	−2.24621	−0.0736166
$932$	0	0
$933$	− 27.8617i	− 0.912152i
$934$	0	0
$935$	0	0
$936$	0	0
$937$	36.7386i	1.20020i	0.799925	+	0.600099i	$0.204872\pi$
$937$	36.7386i	1.20020i	−0.799925	+	0.600099i	$0.795128\pi$
$938$	0	0
$939$	−68.6695	−2.24094
$940$	0	0
$941$	55.7926	1.81879	0.909394	−	0.415937i	$-0.136546\pi$
$941$	55.7926	1.81879	0.909394	+	0.415937i	$0.136546\pi$
$942$	0	0
$943$	24.9848i	0.813618i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 7.36932i	− 0.239471i	−0.992806	−	0.119735i	$-0.961795\pi$
$947$	− 7.36932i	− 0.239471i	0.992806	−	0.119735i	$-0.0382046\pi$
$948$	0	0
$949$	6.00000	0.194768
$950$	0	0
$951$	−31.3693	−1.01722
$952$	0	0
$953$	− 18.6695i	− 0.604765i	−0.953187	−	0.302382i	$-0.902218\pi$
$953$	− 18.6695i	− 0.604765i	0.953187	−	0.302382i	$-0.0977819\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	− 26.2462i	− 0.848420i
$958$	0	0
$959$	−8.00000	−0.258333
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	14.2462i	0.459078i
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 2.56155i	− 0.0823740i	−0.999151	−	0.0411870i	$-0.986886\pi$
$967$	− 2.56155i	− 0.0823740i	0.999151	−	0.0411870i	$-0.0131139\pi$
$968$	0	0
$969$	−74.6004	−2.39651
$970$	0	0
$971$	−31.6847	−1.01681	−0.508405	−	0.861118i	$-0.669765\pi$
$971$	−31.6847	−1.01681	−0.508405	+	0.861118i	$0.669765\pi$
$972$	0	0
$973$	− 19.6847i	− 0.631061i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 54.4924i	− 1.74337i	−0.490069	−	0.871684i	$-0.663029\pi$
$977$	− 54.4924i	− 1.74337i	0.490069	−	0.871684i	$-0.336971\pi$
$978$	0	0
$979$	51.2311	1.63735
$980$	0	0
$981$	−10.0000	−0.319275
$982$	0	0
$983$	17.9309i	0.571906i	0.958244	+	0.285953i	$0.0923101\pi$
$983$	17.9309i	0.571906i	−0.958244	+	0.285953i	$0.907690\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	− 2.06913i	− 0.0658611i
$988$	0	0
$989$	43.5076	1.38346
$990$	0	0
$991$	−35.8617	−1.13919	−0.569593	−	0.821927i	$-0.692899\pi$
$991$	−35.8617	−1.13919	−0.569593	+	0.821927i	$0.692899\pi$
$992$	0	0
$993$	− 11.5076i	− 0.365182i
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 0.246211i	− 0.00779759i	−0.999992	−	0.00389879i	$-0.998759\pi$
$997$	− 0.246211i	− 0.00779759i	0.999992	−	0.00389879i	$-0.00124103\pi$
$998$	0	0
$999$	−13.9309	−0.440753

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2600.2.d.k.1249.4		4
5.2	odd	4		104.2.a.b.1.2	✓	2
5.3	odd	4		2600.2.a.p.1.1		2
5.4	even	2	inner	2600.2.d.k.1249.1		4
15.2	even	4		936.2.a.j.1.2		2
20.3	even	4		5200.2.a.bw.1.2		2
20.7	even	4		208.2.a.e.1.1		2
35.27	even	4		5096.2.a.m.1.1		2
40.27	even	4		832.2.a.n.1.2		2
40.37	odd	4		832.2.a.k.1.1		2
60.47	odd	4		1872.2.a.u.1.2		2
65.2	even	12		1352.2.o.d.1161.2		8
65.7	even	12		1352.2.o.d.361.1		8
65.12	odd	4		1352.2.a.g.1.2		2
65.17	odd	12		1352.2.i.d.1329.1		4
65.22	odd	12		1352.2.i.f.1329.1		4
65.32	even	12		1352.2.o.d.361.2		8
65.37	even	12		1352.2.o.d.1161.1		8
65.42	odd	12		1352.2.i.f.529.1		4
65.47	even	4		1352.2.f.c.337.3		4
65.57	even	4		1352.2.f.c.337.4		4
65.62	odd	12		1352.2.i.d.529.1		4
80.27	even	4		3328.2.b.w.1665.4		4
80.37	odd	4		3328.2.b.y.1665.1		4
80.67	even	4		3328.2.b.w.1665.1		4
80.77	odd	4		3328.2.b.y.1665.4		4
120.77	even	4		7488.2.a.cu.1.1		2
120.107	odd	4		7488.2.a.cv.1.1		2
260.47	odd	4		2704.2.f.k.337.1		4
260.187	odd	4		2704.2.f.k.337.2		4
260.207	even	4		2704.2.a.p.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.a.b.1.2	✓	2	5.2	odd	4
208.2.a.e.1.1		2	20.7	even	4
832.2.a.k.1.1		2	40.37	odd	4
832.2.a.n.1.2		2	40.27	even	4
936.2.a.j.1.2		2	15.2	even	4
1352.2.a.g.1.2		2	65.12	odd	4
1352.2.f.c.337.3		4	65.47	even	4
1352.2.f.c.337.4		4	65.57	even	4
1352.2.i.d.529.1		4	65.62	odd	12
1352.2.i.d.1329.1		4	65.17	odd	12
1352.2.i.f.529.1		4	65.42	odd	12
1352.2.i.f.1329.1		4	65.22	odd	12
1352.2.o.d.361.1		8	65.7	even	12
1352.2.o.d.361.2		8	65.32	even	12
1352.2.o.d.1161.1		8	65.37	even	12
1352.2.o.d.1161.2		8	65.2	even	12
1872.2.a.u.1.2		2	60.47	odd	4
2600.2.a.p.1.1		2	5.3	odd	4
2600.2.d.k.1249.1		4	5.4	even	2	inner
2600.2.d.k.1249.4		4	1.1	even	1	trivial
2704.2.a.p.1.1		2	260.207	even	4
2704.2.f.k.337.1		4	260.47	odd	4
2704.2.f.k.337.2		4	260.187	odd	4
3328.2.b.w.1665.1		4	80.67	even	4
3328.2.b.w.1665.4		4	80.27	even	4
3328.2.b.y.1665.1		4	80.37	odd	4
3328.2.b.y.1665.4		4	80.77	odd	4
5096.2.a.m.1.1		2	35.27	even	4
5200.2.a.bw.1.2		2	20.3	even	4
7488.2.a.cu.1.1		2	120.77	even	4
7488.2.a.cv.1.1		2	120.107	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2600 = 2^{3} \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2600.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+2.56155i q^{3} +2.56155i q^{7} -3.56155 q^{9} -5.12311 q^{11} +1.00000i q^{13} -5.68466i q^{17} -5.12311 q^{19} -6.56155 q^{21} -8.00000i q^{23} -1.43845i q^{27} +2.00000 q^{29} +4.00000 q^{31} -13.1231i q^{33} -9.68466i q^{37} -2.56155 q^{39} -3.12311 q^{41} +5.43845i q^{43} +0.315342i q^{47} +0.438447 q^{49} +14.5616 q^{51} +3.12311i q^{53} -13.1231i q^{57} -5.12311 q^{59} +11.1231 q^{61} -9.12311i q^{63} +5.12311i q^{67} +20.4924 q^{69} -7.68466 q^{71} -6.00000i q^{73} -13.1231i q^{77} -8.00000 q^{79} -7.00000 q^{81} +2.24621i q^{83} +5.12311i q^{87} -10.0000 q^{89} -2.56155 q^{91} +10.2462i q^{93} +8.24621i q^{97} +18.2462 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 6 q^{9} - 4 q^{11} - 4 q^{19} - 18 q^{21} + 8 q^{29} + 16 q^{31} - 2 q^{39} + 4 q^{41} + 10 q^{49} + 50 q^{51} - 4 q^{59} + 28 q^{61} + 16 q^{69} - 6 q^{71} - 32 q^{79} - 28 q^{81} - 40 q^{89} - 2 q^{91}+ \cdots + 40 q^{99}+O(q^{100}) \) 4 * q - 6 * q^9 - 4 * q^11 - 4 * q^19 - 18 * q^21 + 8 * q^29 + 16 * q^31 - 2 * q^39 + 4 * q^41 + 10 * q^49 + 50 * q^51 - 4 * q^59 + 28 * q^61 + 16 * q^69 - 6 * q^71 - 32 * q^79 - 28 * q^81 - 40 * q^89 - 2 * q^91 + 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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