LMFDB - Embedded newform 5096.2.a.m.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [5096,2,Mod(1,5096)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(5096, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 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("5096.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 5096 = 2^{3} \cdot 7^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5096.a (trivial)

Newform invariants

comment:select newform

sage:traces = [2,0,-1,0,-3,0,0,0,3,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$40.6917648700$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 104)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	5096.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-2.56155 q^{3} +0.561553 q^{5} +3.56155 q^{9} -5.12311 q^{11} -1.00000 q^{13} -1.43845 q^{15} -5.68466 q^{17} -5.12311 q^{19} -8.00000 q^{23} -4.68466 q^{25} -1.43845 q^{27} -2.00000 q^{29} -4.00000 q^{31} +13.1231 q^{33} +9.68466 q^{37} +2.56155 q^{39} +3.12311 q^{41} +5.43845 q^{43} +2.00000 q^{45} +0.315342 q^{47} +14.5616 q^{51} +3.12311 q^{53} -2.87689 q^{55} +13.1231 q^{57} -5.12311 q^{59} -11.1231 q^{61} -0.561553 q^{65} -5.12311 q^{67} +20.4924 q^{69} -7.68466 q^{71} +6.00000 q^{73} +12.0000 q^{75} +8.00000 q^{79} -7.00000 q^{81} -2.24621 q^{83} -3.19224 q^{85} +5.12311 q^{87} -10.0000 q^{89} +10.2462 q^{93} -2.87689 q^{95} +8.24621 q^{97} -18.2462 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - q^{3} - 3 q^{5} + 3 q^{9} - 2 q^{11} - 2 q^{13} - 7 q^{15} + q^{17} - 2 q^{19} - 16 q^{23} + 3 q^{25} - 7 q^{27} - 4 q^{29} - 8 q^{31} + 18 q^{33} + 7 q^{37} + q^{39} - 2 q^{41} + 15 q^{43} + 4 q^{45}+ \cdots - 20 q^{99}+O(q^{100}) $ 2 * q - q^3 - 3 * q^5 + 3 * q^9 - 2 * q^11 - 2 * q^13 - 7 * q^15 + q^17 - 2 * q^19 - 16 * q^23 + 3 * q^25 - 7 * q^27 - 4 * q^29 - 8 * q^31 + 18 * q^33 + 7 * q^37 + q^39 - 2 * q^41 + 15 * q^43 + 4 * q^45 + 13 * q^47 + 25 * q^51 - 2 * q^53 - 14 * q^55 + 18 * q^57 - 2 * q^59 - 14 * q^61 + 3 * q^65 - 2 * q^67 + 8 * q^69 - 3 * q^71 + 12 * q^73 + 24 * q^75 + 16 * q^79 - 14 * q^81 + 12 * q^83 - 27 * q^85 + 2 * q^87 - 20 * q^89 + 4 * q^93 - 14 * q^95 - 20 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−2.56155	−1.47891	−0.739457	−	0.673204i	$-0.764917\pi$
$3$	−2.56155	−1.47891	−0.739457	+	0.673204i	$0.764917\pi$
$4$	0	0
$5$	0.561553	0.251134	0.125567	−	0.992085i	$-0.459925\pi$
$5$	0.561553	0.251134	0.125567	+	0.992085i	$0.459925\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	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
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	0	0
$15$	−1.43845	−0.371405
$16$	0	0
$17$	−5.68466	−1.37873	−0.689366	−	0.724413i	$-0.742111\pi$
$17$	−5.68466	−1.37873	−0.689366	+	0.724413i	$0.742111\pi$
$18$	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
$21$	0	0
$22$	0	0
$23$	−8.00000	−1.66812	−0.834058	−	0.551677i	$-0.813988\pi$
$23$	−8.00000	−1.66812	−0.834058	+	0.551677i	$0.813988\pi$
$24$	0	0
$25$	−4.68466	−0.936932
$26$	0	0
$27$	−1.43845	−0.276829
$28$	0	0
$29$	−2.00000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−2.00000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−4.00000	−0.718421	−0.359211	−	0.933257i	$-0.616954\pi$
$31$	−4.00000	−0.718421	−0.359211	+	0.933257i	$0.616954\pi$
$32$	0	0
$33$	13.1231	2.28444
$34$	0	0
$35$	0	0
$36$	0	0
$37$	9.68466	1.59215	0.796074	−	0.605199i	$-0.206907\pi$
$37$	9.68466	1.59215	0.796074	+	0.605199i	$0.206907\pi$
$38$	0	0
$39$	2.56155	0.410177
$40$	0	0
$41$	3.12311	0.487747	0.243874	−	0.969807i	$-0.421582\pi$
$41$	3.12311	0.487747	0.243874	+	0.969807i	$0.421582\pi$
$42$	0	0
$43$	5.43845	0.829355	0.414678	−	0.909968i	$-0.363894\pi$
$43$	5.43845	0.829355	0.414678	+	0.909968i	$0.363894\pi$
$44$	0	0
$45$	2.00000	0.298142
$46$	0	0
$47$	0.315342	0.0459973	0.0229986	−	0.999735i	$-0.492679\pi$
$47$	0.315342	0.0459973	0.0229986	+	0.999735i	$0.492679\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	14.5616	2.03903
$52$	0	0
$53$	3.12311	0.428992	0.214496	−	0.976725i	$-0.431189\pi$
$53$	3.12311	0.428992	0.214496	+	0.976725i	$0.431189\pi$
$54$	0	0
$55$	−2.87689	−0.387920
$56$	0	0
$57$	13.1231	1.73820
$58$	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	$-0.752248\pi$
$61$	−11.1231	−1.42417	−0.712084	+	0.702094i	$0.752248\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−0.561553	−0.0696521
$66$	0	0
$67$	−5.12311	−0.625887	−0.312943	−	0.949772i	$-0.601315\pi$
$67$	−5.12311	−0.625887	−0.312943	+	0.949772i	$0.601315\pi$
$68$	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
$73$	6.00000	0.702247	0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000	0.702247	0.351123	+	0.936329i	$0.385800\pi$
$74$	0	0
$75$	12.0000	1.38564
$76$	0	0
$77$	0	0
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	−2.24621	−0.246554	−0.123277	−	0.992372i	$-0.539340\pi$
$83$	−2.24621	−0.246554	−0.123277	+	0.992372i	$0.539340\pi$
$84$	0	0
$85$	−3.19224	−0.346247
$86$	0	0
$87$	5.12311	0.549255
$88$	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
$91$	0	0
$92$	0	0
$93$	10.2462	1.06248
$94$	0	0
$95$	−2.87689	−0.295163
$96$	0	0
$97$	8.24621	0.837276	0.418638	−	0.908153i	$-0.362508\pi$
$97$	8.24621	0.837276	0.418638	+	0.908153i	$0.362508\pi$
$98$	0	0
$99$	−18.2462	−1.83381
$100$	0	0
$101$	7.12311	0.708776	0.354388	−	0.935099i	$-0.384689\pi$
$101$	7.12311	0.708776	0.354388	+	0.935099i	$0.384689\pi$
$102$	0	0
$103$	−2.24621	−0.221326	−0.110663	−	0.993858i	$-0.535297\pi$
$103$	−2.24621	−0.221326	−0.110663	+	0.993858i	$0.535297\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	4.00000	0.386695	0.193347	−	0.981130i	$-0.438066\pi$
$107$	4.00000	0.386695	0.193347	+	0.981130i	$0.438066\pi$
$108$	0	0
$109$	−2.80776	−0.268935	−0.134468	−	0.990918i	$-0.542932\pi$
$109$	−2.80776	−0.268935	−0.134468	+	0.990918i	$0.542932\pi$
$110$	0	0
$111$	−24.8078	−2.35465
$112$	0	0
$113$	−18.4924	−1.73962	−0.869810	−	0.493386i	$-0.835759\pi$
$113$	−18.4924	−1.73962	−0.869810	+	0.493386i	$0.835759\pi$
$114$	0	0
$115$	−4.49242	−0.418921
$116$	0	0
$117$	−3.56155	−0.329266
$118$	0	0
$119$	0	0
$120$	0	0
$121$	15.2462	1.38602
$122$	0	0
$123$	−8.00000	−0.721336
$124$	0	0
$125$	−5.43845	−0.486430
$126$	0	0
$127$	−10.2462	−0.909204	−0.454602	−	0.890695i	$-0.650219\pi$
$127$	−10.2462	−0.909204	−0.454602	+	0.890695i	$0.650219\pi$
$128$	0	0
$129$	−13.9309	−1.22654
$130$	0	0
$131$	4.80776	0.420056	0.210028	−	0.977695i	$-0.432644\pi$
$131$	4.80776	0.420056	0.210028	+	0.977695i	$0.432644\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−0.807764	−0.0695213
$136$	0	0
$137$	−3.12311	−0.266825	−0.133412	−	0.991061i	$-0.542594\pi$
$137$	−3.12311	−0.266825	−0.133412	+	0.991061i	$0.542594\pi$
$138$	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
$143$	5.12311	0.428416
$144$	0	0
$145$	−1.12311	−0.0932688
$146$	0	0
$147$	0	0
$148$	0	0
$149$	16.2462	1.33094	0.665471	−	0.746424i	$-0.268231\pi$
$149$	16.2462	1.33094	0.665471	+	0.746424i	$0.268231\pi$
$150$	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
$153$	−20.2462	−1.63681
$154$	0	0
$155$	−2.24621	−0.180420
$156$	0	0
$157$	22.4924	1.79509	0.897545	−	0.440922i	$-0.145349\pi$
$157$	22.4924	1.79509	0.897545	+	0.440922i	$0.145349\pi$
$158$	0	0
$159$	−8.00000	−0.634441
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−20.4924	−1.60509	−0.802545	−	0.596591i	$-0.796521\pi$
$163$	−20.4924	−1.60509	−0.802545	+	0.596591i	$0.796521\pi$
$164$	0	0
$165$	7.36932	0.573701
$166$	0	0
$167$	−14.2462	−1.10240	−0.551202	−	0.834372i	$-0.685831\pi$
$167$	−14.2462	−1.10240	−0.551202	+	0.834372i	$0.685831\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−18.2462	−1.39532
$172$	0	0
$173$	4.87689	0.370783	0.185392	−	0.982665i	$-0.440645\pi$
$173$	4.87689	0.370783	0.185392	+	0.982665i	$0.440645\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	13.1231	0.986393
$178$	0	0
$179$	7.68466	0.574378	0.287189	−	0.957874i	$-0.407279\pi$
$179$	7.68466	0.574378	0.287189	+	0.957874i	$0.407279\pi$
$180$	0	0
$181$	23.1231	1.71873	0.859363	−	0.511365i	$-0.170860\pi$
$181$	23.1231	1.71873	0.859363	+	0.511365i	$0.170860\pi$
$182$	0	0
$183$	28.4924	2.10622
$184$	0	0
$185$	5.43845	0.399843
$186$	0	0
$187$	29.1231	2.12969
$188$	0	0
$189$	0	0
$190$	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
$193$	−21.3693	−1.53820	−0.769099	−	0.639130i	$-0.779294\pi$
$193$	−21.3693	−1.53820	−0.769099	+	0.639130i	$0.779294\pi$
$194$	0	0
$195$	1.43845	0.103009
$196$	0	0
$197$	2.31534	0.164961	0.0824806	−	0.996593i	$-0.473716\pi$
$197$	2.31534	0.164961	0.0824806	+	0.996593i	$0.473716\pi$
$198$	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
$203$	0	0
$204$	0	0
$205$	1.75379	0.122490
$206$	0	0
$207$	−28.4924	−1.98036
$208$	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
$213$	19.6847	1.34877
$214$	0	0
$215$	3.05398	0.208279
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−15.3693	−1.03856
$220$	0	0
$221$	5.68466	0.382392
$222$	0	0
$223$	15.6847	1.05032	0.525161	−	0.851003i	$-0.324005\pi$
$223$	15.6847	1.05032	0.525161	+	0.851003i	$0.324005\pi$
$224$	0	0
$225$	−16.6847	−1.11231
$226$	0	0
$227$	8.00000	0.530979	0.265489	−	0.964114i	$-0.414466\pi$
$227$	8.00000	0.530979	0.265489	+	0.964114i	$0.414466\pi$
$228$	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
$231$	0	0
$232$	0	0
$233$	−6.80776	−0.445991	−0.222996	−	0.974819i	$-0.571584\pi$
$233$	−6.80776	−0.445991	−0.222996	+	0.974819i	$0.571584\pi$
$234$	0	0
$235$	0.177081	0.0115515
$236$	0	0
$237$	−20.4924	−1.33113
$238$	0	0
$239$	−2.56155	−0.165693	−0.0828465	−	0.996562i	$-0.526401\pi$
$239$	−2.56155	−0.165693	−0.0828465	+	0.996562i	$0.526401\pi$
$240$	0	0
$241$	3.75379	0.241803	0.120901	−	0.992665i	$-0.461422\pi$
$241$	3.75379	0.241803	0.120901	+	0.992665i	$0.461422\pi$
$242$	0	0
$243$	22.2462	1.42710
$244$	0	0
$245$	0	0
$246$	0	0
$247$	5.12311	0.325975
$248$	0	0
$249$	5.75379	0.364632
$250$	0	0
$251$	−22.2462	−1.40417	−0.702084	−	0.712094i	$-0.747747\pi$
$251$	−22.2462	−1.40417	−0.702084	+	0.712094i	$0.747747\pi$
$252$	0	0
$253$	40.9848	2.57670
$254$	0	0
$255$	8.17708	0.512069
$256$	0	0
$257$	−11.4384	−0.713511	−0.356755	−	0.934198i	$-0.616117\pi$
$257$	−11.4384	−0.713511	−0.356755	+	0.934198i	$0.616117\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−7.12311	−0.440909
$262$	0	0
$263$	−26.2462	−1.61841	−0.809205	−	0.587526i	$-0.800102\pi$
$263$	−26.2462	−1.61841	−0.809205	+	0.587526i	$0.800102\pi$
$264$	0	0
$265$	1.75379	0.107734
$266$	0	0
$267$	25.6155	1.56764
$268$	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	$-0.282238\pi$
$271$	20.8078	1.26398	0.631991	+	0.774976i	$0.282238\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	24.0000	1.44725
$276$	0	0
$277$	29.3693	1.76463	0.882316	−	0.470658i	$-0.155984\pi$
$277$	29.3693	1.76463	0.882316	+	0.470658i	$0.155984\pi$
$278$	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
$283$	8.49242	0.504822	0.252411	−	0.967620i	$-0.418776\pi$
$283$	8.49242	0.504822	0.252411	+	0.967620i	$0.418776\pi$
$284$	0	0
$285$	7.36932	0.436521
$286$	0	0
$287$	0	0
$288$	0	0
$289$	15.3153	0.900902
$290$	0	0
$291$	−21.1231	−1.23826
$292$	0	0
$293$	−15.4384	−0.901924	−0.450962	−	0.892543i	$-0.648919\pi$
$293$	−15.4384	−0.901924	−0.450962	+	0.892543i	$0.648919\pi$
$294$	0	0
$295$	−2.87689	−0.167499
$296$	0	0
$297$	7.36932	0.427611
$298$	0	0
$299$	8.00000	0.462652
$300$	0	0
$301$	0	0
$302$	0	0
$303$	−18.2462	−1.04822
$304$	0	0
$305$	−6.24621	−0.357657
$306$	0	0
$307$	−13.1231	−0.748975	−0.374488	−	0.927232i	$-0.622181\pi$
$307$	−13.1231	−0.748975	−0.374488	+	0.927232i	$0.622181\pi$
$308$	0	0
$309$	5.75379	0.327322
$310$	0	0
$311$	10.8769	0.616772	0.308386	−	0.951261i	$-0.400211\pi$
$311$	10.8769	0.616772	0.308386	+	0.951261i	$0.400211\pi$
$312$	0	0
$313$	−26.8078	−1.51526	−0.757632	−	0.652682i	$-0.773644\pi$
$313$	−26.8078	−1.51526	−0.757632	+	0.652682i	$0.773644\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−12.2462	−0.687816	−0.343908	−	0.939003i	$-0.611751\pi$
$317$	−12.2462	−0.687816	−0.343908	+	0.939003i	$0.611751\pi$
$318$	0	0
$319$	10.2462	0.573678
$320$	0	0
$321$	−10.2462	−0.571888
$322$	0	0
$323$	29.1231	1.62045
$324$	0	0
$325$	4.68466	0.259858
$326$	0	0
$327$	7.19224	0.397732
$328$	0	0
$329$	0	0
$330$	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
$333$	34.4924	1.89017
$334$	0	0
$335$	−2.87689	−0.157182
$336$	0	0
$337$	−13.1922	−0.718627	−0.359313	−	0.933217i	$-0.616989\pi$
$337$	−13.1922	−0.718627	−0.359313	+	0.933217i	$0.616989\pi$
$338$	0	0
$339$	47.3693	2.57275
$340$	0	0
$341$	20.4924	1.10973
$342$	0	0
$343$	0	0
$344$	0	0
$345$	11.5076	0.619547
$346$	0	0
$347$	7.68466	0.412534	0.206267	−	0.978496i	$-0.433868\pi$
$347$	7.68466	0.412534	0.206267	+	0.978496i	$0.433868\pi$
$348$	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
$353$	−9.36932	−0.498678	−0.249339	−	0.968416i	$-0.580213\pi$
$353$	−9.36932	−0.498678	−0.249339	+	0.968416i	$0.580213\pi$
$354$	0	0
$355$	−4.31534	−0.229035
$356$	0	0
$357$	0	0
$358$	0	0
$359$	14.2462	0.751886	0.375943	−	0.926643i	$-0.377319\pi$
$359$	14.2462	0.751886	0.375943	+	0.926643i	$0.377319\pi$
$360$	0	0
$361$	7.24621	0.381380
$362$	0	0
$363$	−39.0540	−2.04980
$364$	0	0
$365$	3.36932	0.176358
$366$	0	0
$367$	−9.61553	−0.501926	−0.250963	−	0.967997i	$-0.580747\pi$
$367$	−9.61553	−0.501926	−0.250963	+	0.967997i	$0.580747\pi$
$368$	0	0
$369$	11.1231	0.579046
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−11.6155	−0.601429	−0.300715	−	0.953714i	$-0.597225\pi$
$373$	−11.6155	−0.601429	−0.300715	+	0.953714i	$0.597225\pi$
$374$	0	0
$375$	13.9309	0.719387
$376$	0	0
$377$	2.00000	0.103005
$378$	0	0
$379$	5.75379	0.295552	0.147776	−	0.989021i	$-0.452788\pi$
$379$	5.75379	0.295552	0.147776	+	0.989021i	$0.452788\pi$
$380$	0	0
$381$	26.2462	1.34463
$382$	0	0
$383$	11.1922	0.571897	0.285948	−	0.958245i	$-0.407691\pi$
$383$	11.1922	0.571897	0.285948	+	0.958245i	$0.407691\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	19.3693	0.984598
$388$	0	0
$389$	20.7386	1.05149	0.525745	−	0.850642i	$-0.323787\pi$
$389$	20.7386	1.05149	0.525745	+	0.850642i	$0.323787\pi$
$390$	0	0
$391$	45.4773	2.29988
$392$	0	0
$393$	−12.3153	−0.621227
$394$	0	0
$395$	4.49242	0.226038
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	0	0
$399$	0	0
$400$	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
$403$	4.00000	0.199254
$404$	0	0
$405$	−3.93087	−0.195326
$406$	0	0
$407$	−49.6155	−2.45935
$408$	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
$413$	0	0
$414$	0	0
$415$	−1.26137	−0.0619180
$416$	0	0
$417$	19.6847	0.963962
$418$	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
$423$	1.12311	0.0546073
$424$	0	0
$425$	26.6307	1.29178
$426$	0	0
$427$	0	0
$428$	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
$433$	6.17708	0.296852	0.148426	−	0.988924i	$-0.452579\pi$
$433$	6.17708	0.296852	0.148426	+	0.988924i	$0.452579\pi$
$434$	0	0
$435$	2.87689	0.137937
$436$	0	0
$437$	40.9848	1.96057
$438$	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
$441$	0	0
$442$	0	0
$443$	20.8078	0.988607	0.494303	−	0.869289i	$-0.335423\pi$
$443$	20.8078	0.988607	0.494303	+	0.869289i	$0.335423\pi$
$444$	0	0
$445$	−5.61553	−0.266202
$446$	0	0
$447$	−41.6155	−1.96835
$448$	0	0
$449$	32.7386	1.54503	0.772516	−	0.634996i	$-0.218998\pi$
$449$	32.7386	1.54503	0.772516	+	0.634996i	$0.218998\pi$
$450$	0	0
$451$	−16.0000	−0.753411
$452$	0	0
$453$	−27.0540	−1.27111
$454$	0	0
$455$	0	0
$456$	0	0
$457$	24.7386	1.15722	0.578612	−	0.815603i	$-0.303594\pi$
$457$	24.7386	1.15722	0.578612	+	0.815603i	$0.303594\pi$
$458$	0	0
$459$	8.17708	0.381673
$460$	0	0
$461$	34.8078	1.62116	0.810580	−	0.585628i	$-0.199152\pi$
$461$	34.8078	1.62116	0.810580	+	0.585628i	$0.199152\pi$
$462$	0	0
$463$	38.2462	1.77745	0.888726	−	0.458439i	$-0.151591\pi$
$463$	38.2462	1.77745	0.888726	+	0.458439i	$0.151591\pi$
$464$	0	0
$465$	5.75379	0.266826
$466$	0	0
$467$	−40.4924	−1.87377	−0.936883	−	0.349643i	$-0.886303\pi$
$467$	−40.4924	−1.87377	−0.936883	+	0.349643i	$0.886303\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−57.6155	−2.65478
$472$	0	0
$473$	−27.8617	−1.28108
$474$	0	0
$475$	24.0000	1.10120
$476$	0	0
$477$	11.1231	0.509292
$478$	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
$483$	0	0
$484$	0	0
$485$	4.63068	0.210268
$486$	0	0
$487$	1.75379	0.0794718	0.0397359	−	0.999210i	$-0.487348\pi$
$487$	1.75379	0.0794718	0.0397359	+	0.999210i	$0.487348\pi$
$488$	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
$493$	11.3693	0.512048
$494$	0	0
$495$	−10.2462	−0.460533
$496$	0	0
$497$	0	0
$498$	0	0
$499$	28.4924	1.27550	0.637748	−	0.770245i	$-0.279866\pi$
$499$	28.4924	1.27550	0.637748	+	0.770245i	$0.279866\pi$
$500$	0	0
$501$	36.4924	1.63036
$502$	0	0
$503$	7.36932	0.328582	0.164291	−	0.986412i	$-0.447466\pi$
$503$	7.36932	0.328582	0.164291	+	0.986412i	$0.447466\pi$
$504$	0	0
$505$	4.00000	0.177998
$506$	0	0
$507$	−2.56155	−0.113763
$508$	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
$511$	0	0
$512$	0	0
$513$	7.36932	0.325363
$514$	0	0
$515$	−1.26137	−0.0555824
$516$	0	0
$517$	−1.61553	−0.0710508
$518$	0	0
$519$	−12.4924	−0.548356
$520$	0	0
$521$	9.68466	0.424293	0.212146	−	0.977238i	$-0.431955\pi$
$521$	9.68466	0.424293	0.212146	+	0.977238i	$0.431955\pi$
$522$	0	0
$523$	0.492423	0.0215321	0.0107661	−	0.999942i	$-0.496573\pi$
$523$	0.492423	0.0215321	0.0107661	+	0.999942i	$0.496573\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	22.7386	0.990510
$528$	0	0
$529$	41.0000	1.78261
$530$	0	0
$531$	−18.2462	−0.791818
$532$	0	0
$533$	−3.12311	−0.135277
$534$	0	0
$535$	2.24621	0.0971122
$536$	0	0
$537$	−19.6847	−0.849456
$538$	0	0
$539$	0	0
$540$	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
$543$	−59.2311	−2.54185
$544$	0	0
$545$	−1.57671	−0.0675387
$546$	0	0
$547$	−29.4384	−1.25870	−0.629349	−	0.777123i	$-0.716678\pi$
$547$	−29.4384	−1.25870	−0.629349	+	0.777123i	$0.716678\pi$
$548$	0	0
$549$	−39.6155	−1.69075
$550$	0	0
$551$	10.2462	0.436503
$552$	0	0
$553$	0	0
$554$	0	0
$555$	−13.9309	−0.591332
$556$	0	0
$557$	−42.1771	−1.78710	−0.893550	−	0.448963i	$-0.851793\pi$
$557$	−42.1771	−1.78710	−0.893550	+	0.448963i	$0.851793\pi$
$558$	0	0
$559$	−5.43845	−0.230022
$560$	0	0
$561$	−74.6004	−3.14963
$562$	0	0
$563$	7.05398	0.297290	0.148645	−	0.988891i	$-0.452509\pi$
$563$	7.05398	0.297290	0.148645	+	0.988891i	$0.452509\pi$
$564$	0	0
$565$	−10.3845	−0.436878
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−41.6847	−1.74751	−0.873756	−	0.486365i	$-0.838323\pi$
$569$	−41.6847	−1.74751	−0.873756	+	0.486365i	$0.838323\pi$
$570$	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
$573$	−39.3693	−1.64468
$574$	0	0
$575$	37.4773	1.56291
$576$	0	0
$577$	24.2462	1.00938	0.504691	−	0.863300i	$-0.331606\pi$
$577$	24.2462	1.00938	0.504691	+	0.863300i	$0.331606\pi$
$578$	0	0
$579$	54.7386	2.27486
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−16.0000	−0.662652
$584$	0	0
$585$	−2.00000	−0.0826898
$586$	0	0
$587$	24.9848	1.03123	0.515617	−	0.856819i	$-0.327563\pi$
$587$	24.9848	1.03123	0.515617	+	0.856819i	$0.327563\pi$
$588$	0	0
$589$	20.4924	0.844376
$590$	0	0
$591$	−5.93087	−0.243963
$592$	0	0
$593$	−38.4924	−1.58069	−0.790347	−	0.612659i	$-0.790100\pi$
$593$	−38.4924	−1.58069	−0.790347	+	0.612659i	$0.790100\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	13.1231	0.537093
$598$	0	0
$599$	23.3693	0.954844	0.477422	−	0.878674i	$-0.341571\pi$
$599$	23.3693	0.954844	0.477422	+	0.878674i	$0.341571\pi$
$600$	0	0
$601$	−34.1771	−1.39411	−0.697056	−	0.717017i	$-0.745507\pi$
$601$	−34.1771	−1.39411	−0.697056	+	0.717017i	$0.745507\pi$
$602$	0	0
$603$	−18.2462	−0.743043
$604$	0	0
$605$	8.56155	0.348077
$606$	0	0
$607$	−15.3693	−0.623821	−0.311911	−	0.950111i	$-0.600969\pi$
$607$	−15.3693	−0.623821	−0.311911	+	0.950111i	$0.600969\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−0.315342	−0.0127574
$612$	0	0
$613$	6.00000	0.242338	0.121169	−	0.992632i	$-0.461336\pi$
$613$	6.00000	0.242338	0.121169	+	0.992632i	$0.461336\pi$
$614$	0	0
$615$	−4.49242	−0.181152
$616$	0	0
$617$	21.8617	0.880120	0.440060	−	0.897968i	$-0.354957\pi$
$617$	21.8617	0.880120	0.440060	+	0.897968i	$0.354957\pi$
$618$	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
$623$	0	0
$624$	0	0
$625$	20.3693	0.814773
$626$	0	0
$627$	−67.2311	−2.68495
$628$	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
$633$	−40.1771	−1.59690
$634$	0	0
$635$	−5.75379	−0.228332
$636$	0	0
$637$	0	0
$638$	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
$643$	−35.8617	−1.41425	−0.707124	−	0.707089i	$-0.750008\pi$
$643$	−35.8617	−1.41425	−0.707124	+	0.707089i	$0.750008\pi$
$644$	0	0
$645$	−7.82292	−0.308027
$646$	0	0
$647$	−5.12311	−0.201410	−0.100705	−	0.994916i	$-0.532110\pi$
$647$	−5.12311	−0.201410	−0.100705	+	0.994916i	$0.532110\pi$
$648$	0	0
$649$	26.2462	1.03025
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−4.87689	−0.190848	−0.0954238	−	0.995437i	$-0.530421\pi$
$653$	−4.87689	−0.190848	−0.0954238	+	0.995437i	$0.530421\pi$
$654$	0	0
$655$	2.69981	0.105490
$656$	0	0
$657$	21.3693	0.833696
$658$	0	0
$659$	16.4924	0.642454	0.321227	−	0.947002i	$-0.395905\pi$
$659$	16.4924	0.642454	0.321227	+	0.947002i	$0.395905\pi$
$660$	0	0
$661$	40.7386	1.58455	0.792275	−	0.610165i	$-0.208897\pi$
$661$	40.7386	1.58455	0.792275	+	0.610165i	$0.208897\pi$
$662$	0	0
$663$	−14.5616	−0.565524
$664$	0	0
$665$	0	0
$666$	0	0
$667$	16.0000	0.619522
$668$	0	0
$669$	−40.1771	−1.55334
$670$	0	0
$671$	56.9848	2.19988
$672$	0	0
$673$	−19.3002	−0.743968	−0.371984	−	0.928239i	$-0.621322\pi$
$673$	−19.3002	−0.743968	−0.371984	+	0.928239i	$0.621322\pi$
$674$	0	0
$675$	6.73863	0.259370
$676$	0	0
$677$	1.36932	0.0526271	0.0263136	−	0.999654i	$-0.491623\pi$
$677$	1.36932	0.0526271	0.0263136	+	0.999654i	$0.491623\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−20.4924	−0.785271
$682$	0	0
$683$	−24.0000	−0.918334	−0.459167	−	0.888350i	$-0.651852\pi$
$683$	−24.0000	−0.918334	−0.459167	+	0.888350i	$0.651852\pi$
$684$	0	0
$685$	−1.75379	−0.0670088
$686$	0	0
$687$	17.4384	0.665318
$688$	0	0
$689$	−3.12311	−0.118981
$690$	0	0
$691$	−16.9848	−0.646134	−0.323067	−	0.946376i	$-0.604714\pi$
$691$	−16.9848	−0.646134	−0.323067	+	0.946376i	$0.604714\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−4.31534	−0.163690
$696$	0	0
$697$	−17.7538	−0.672473
$698$	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
$703$	−49.6155	−1.87129
$704$	0	0
$705$	−0.453602	−0.0170836
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−24.7386	−0.929079	−0.464539	−	0.885552i	$-0.653780\pi$
$709$	−24.7386	−0.929079	−0.464539	+	0.885552i	$0.653780\pi$
$710$	0	0
$711$	28.4924	1.06855
$712$	0	0
$713$	32.0000	1.19841
$714$	0	0
$715$	2.87689	0.107590
$716$	0	0
$717$	6.56155	0.245046
$718$	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
$721$	0	0
$722$	0	0
$723$	−9.61553	−0.357605
$724$	0	0
$725$	9.36932	0.347968
$726$	0	0
$727$	46.1080	1.71005	0.855025	−	0.518587i	$-0.173542\pi$
$727$	46.1080	1.71005	0.855025	+	0.518587i	$0.173542\pi$
$728$	0	0
$729$	−35.9848	−1.33277
$730$	0	0
$731$	−30.9157	−1.14346
$732$	0	0
$733$	27.4384	1.01346	0.506731	−	0.862104i	$-0.330854\pi$
$733$	27.4384	1.01346	0.506731	+	0.862104i	$0.330854\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	26.2462	0.966792
$738$	0	0
$739$	−10.2462	−0.376913	−0.188456	−	0.982082i	$-0.560348\pi$
$739$	−10.2462	−0.376913	−0.188456	+	0.982082i	$0.560348\pi$
$740$	0	0
$741$	−13.1231	−0.482089
$742$	0	0
$743$	50.9157	1.86792	0.933958	−	0.357382i	$-0.116331\pi$
$743$	50.9157	1.86792	0.933958	+	0.357382i	$0.116331\pi$
$744$	0	0
$745$	9.12311	0.334245
$746$	0	0
$747$	−8.00000	−0.292705
$748$	0	0
$749$	0	0
$750$	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
$753$	56.9848	2.07664
$754$	0	0
$755$	5.93087	0.215846
$756$	0	0
$757$	3.12311	0.113511	0.0567556	−	0.998388i	$-0.481924\pi$
$757$	3.12311	0.113511	0.0567556	+	0.998388i	$0.481924\pi$
$758$	0	0
$759$	−104.985	−3.81071
$760$	0	0
$761$	−46.4924	−1.68535	−0.842674	−	0.538423i	$-0.819020\pi$
$761$	−46.4924	−1.68535	−0.842674	+	0.538423i	$0.819020\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−11.3693	−0.411059
$766$	0	0
$767$	5.12311	0.184985
$768$	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
$773$	−46.1771	−1.66087	−0.830437	−	0.557112i	$-0.811909\pi$
$773$	−46.1771	−1.66087	−0.830437	+	0.557112i	$0.811909\pi$
$774$	0	0
$775$	18.7386	0.673112
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−16.0000	−0.573259
$780$	0	0
$781$	39.3693	1.40874
$782$	0	0
$783$	2.87689	0.102812
$784$	0	0
$785$	12.6307	0.450808
$786$	0	0
$787$	16.0000	0.570338	0.285169	−	0.958477i	$-0.407950\pi$
$787$	16.0000	0.570338	0.285169	+	0.958477i	$0.407950\pi$
$788$	0	0
$789$	67.2311	2.39349
$790$	0	0
$791$	0	0
$792$	0	0
$793$	11.1231	0.394993
$794$	0	0
$795$	−4.49242	−0.159330
$796$	0	0
$797$	13.5076	0.478463	0.239231	−	0.970963i	$-0.423105\pi$
$797$	13.5076	0.478463	0.239231	+	0.970963i	$0.423105\pi$
$798$	0	0
$799$	−1.79261	−0.0634180
$800$	0	0
$801$	−35.6155	−1.25841
$802$	0	0
$803$	−30.7386	−1.08474
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−12.4924	−0.439754
$808$	0	0
$809$	−30.1771	−1.06097	−0.530485	−	0.847694i	$-0.677990\pi$
$809$	−30.1771	−1.06097	−0.530485	+	0.847694i	$0.677990\pi$
$810$	0	0
$811$	46.7386	1.64122	0.820608	−	0.571492i	$-0.193635\pi$
$811$	46.7386	1.64122	0.820608	+	0.571492i	$0.193635\pi$
$812$	0	0
$813$	−53.3002	−1.86932
$814$	0	0
$815$	−11.5076	−0.403093
$816$	0	0
$817$	−27.8617	−0.974759
$818$	0	0
$819$	0	0
$820$	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
$823$	−9.61553	−0.335176	−0.167588	−	0.985857i	$-0.553598\pi$
$823$	−9.61553	−0.335176	−0.167588	+	0.985857i	$0.553598\pi$
$824$	0	0
$825$	−61.4773	−2.14036
$826$	0	0
$827$	−35.8617	−1.24703	−0.623517	−	0.781809i	$-0.714297\pi$
$827$	−35.8617	−1.24703	−0.623517	+	0.781809i	$0.714297\pi$
$828$	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
$833$	0	0
$834$	0	0
$835$	−8.00000	−0.276851
$836$	0	0
$837$	5.75379	0.198880
$838$	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
$843$	−25.6155	−0.882246
$844$	0	0
$845$	0.561553	0.0193180
$846$	0	0
$847$	0	0
$848$	0	0
$849$	−21.7538	−0.746588
$850$	0	0
$851$	−77.4773	−2.65589
$852$	0	0
$853$	29.6847	1.01638	0.508192	−	0.861244i	$-0.330314\pi$
$853$	29.6847	1.01638	0.508192	+	0.861244i	$0.330314\pi$
$854$	0	0
$855$	−10.2462	−0.350413
$856$	0	0
$857$	−40.7386	−1.39161	−0.695803	−	0.718233i	$-0.744951\pi$
$857$	−40.7386	−1.39161	−0.695803	+	0.718233i	$0.744951\pi$
$858$	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
$861$	0	0
$862$	0	0
$863$	−49.3002	−1.67820	−0.839099	−	0.543979i	$-0.816917\pi$
$863$	−49.3002	−1.67820	−0.839099	+	0.543979i	$0.816917\pi$
$864$	0	0
$865$	2.73863	0.0931163
$866$	0	0
$867$	−39.2311	−1.33236
$868$	0	0
$869$	−40.9848	−1.39032
$870$	0	0
$871$	5.12311	0.173590
$872$	0	0
$873$	29.3693	0.994001
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−37.6847	−1.27252	−0.636260	−	0.771474i	$-0.719520\pi$
$877$	−37.6847	−1.27252	−0.636260	+	0.771474i	$0.719520\pi$
$878$	0	0
$879$	39.5464	1.33387
$880$	0	0
$881$	30.8078	1.03794	0.518970	−	0.854792i	$-0.326316\pi$
$881$	30.8078	1.03794	0.518970	+	0.854792i	$0.326316\pi$
$882$	0	0
$883$	0.315342	0.0106121	0.00530604	−	0.999986i	$-0.498311\pi$
$883$	0.315342	0.0106121	0.00530604	+	0.999986i	$0.498311\pi$
$884$	0	0
$885$	7.36932	0.247717
$886$	0	0
$887$	45.4773	1.52698	0.763489	−	0.645821i	$-0.223485\pi$
$887$	45.4773	1.52698	0.763489	+	0.645821i	$0.223485\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	35.8617	1.20141
$892$	0	0
$893$	−1.61553	−0.0540616
$894$	0	0
$895$	4.31534	0.144246
$896$	0	0
$897$	−20.4924	−0.684222
$898$	0	0
$899$	8.00000	0.266815
$900$	0	0
$901$	−17.7538	−0.591464
$902$	0	0
$903$	0	0
$904$	0	0
$905$	12.9848	0.431631
$906$	0	0
$907$	−20.8078	−0.690910	−0.345455	−	0.938435i	$-0.612275\pi$
$907$	−20.8078	−0.690910	−0.345455	+	0.938435i	$0.612275\pi$
$908$	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
$913$	11.5076	0.380845
$914$	0	0
$915$	16.0000	0.528944
$916$	0	0
$917$	0	0
$918$	0	0
$919$	24.9848	0.824174	0.412087	−	0.911145i	$-0.364800\pi$
$919$	24.9848	0.824174	0.412087	+	0.911145i	$0.364800\pi$
$920$	0	0
$921$	33.6155	1.10767
$922$	0	0
$923$	7.68466	0.252944
$924$	0	0
$925$	−45.3693	−1.49173
$926$	0	0
$927$	−8.00000	−0.262754
$928$	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
$931$	0	0
$932$	0	0
$933$	−27.8617	−0.912152
$934$	0	0
$935$	16.3542	0.534838
$936$	0	0
$937$	36.7386	1.20020	0.600099	−	0.799925i	$-0.295128\pi$
$937$	36.7386	1.20020	0.600099	+	0.799925i	$0.295128\pi$
$938$	0	0
$939$	68.6695	2.24094
$940$	0	0
$941$	−55.7926	−1.81879	−0.909394	−	0.415937i	$-0.863454\pi$
$941$	−55.7926	−1.81879	−0.909394	+	0.415937i	$0.863454\pi$
$942$	0	0
$943$	−24.9848	−0.813618
$944$	0	0
$945$	0	0
$946$	0	0
$947$	7.36932	0.239471	0.119735	−	0.992806i	$-0.461795\pi$
$947$	7.36932	0.239471	0.119735	+	0.992806i	$0.461795\pi$
$948$	0	0
$949$	−6.00000	−0.194768
$950$	0	0
$951$	31.3693	1.01722
$952$	0	0
$953$	−18.6695	−0.604765	−0.302382	−	0.953187i	$-0.597782\pi$
$953$	−18.6695	−0.604765	−0.302382	+	0.953187i	$0.597782\pi$
$954$	0	0
$955$	8.63068	0.279282
$956$	0	0
$957$	−26.2462	−0.848420
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−15.0000	−0.483871
$962$	0	0
$963$	14.2462	0.459078
$964$	0	0
$965$	−12.0000	−0.386294
$966$	0	0
$967$	2.56155	0.0823740	0.0411870	−	0.999151i	$-0.486886\pi$
$967$	2.56155	0.0823740	0.0411870	+	0.999151i	$0.486886\pi$
$968$	0	0
$969$	−74.6004	−2.39651
$970$	0	0
$971$	31.6847	1.01681	0.508405	−	0.861118i	$-0.330235\pi$
$971$	31.6847	1.01681	0.508405	+	0.861118i	$0.330235\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−12.0000	−0.384308
$976$	0	0
$977$	54.4924	1.74337	0.871684	−	0.490069i	$-0.163029\pi$
$977$	54.4924	1.74337	0.871684	+	0.490069i	$0.163029\pi$
$978$	0	0
$979$	51.2311	1.63735
$980$	0	0
$981$	−10.0000	−0.319275
$982$	0	0
$983$	−17.9309	−0.571906	−0.285953	−	0.958244i	$-0.592310\pi$
$983$	−17.9309	−0.571906	−0.285953	+	0.958244i	$0.592310\pi$
$984$	0	0
$985$	1.30019	0.0414274
$986$	0	0
$987$	0	0
$988$	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
$993$	11.5076	0.365182
$994$	0	0
$995$	−2.87689	−0.0912037
$996$	0	0
$997$	−0.246211	−0.00779759	−0.00389879	−	0.999992i	$-0.501241\pi$
$997$	−0.246211	−0.00779759	−0.00389879	+	0.999992i	$0.501241\pi$
$998$	0	0
$999$	−13.9309	−0.440753

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5096.2.a.m.1.1		2
7.6	odd	2		104.2.a.b.1.2	✓	2
21.20	even	2		936.2.a.j.1.2		2
28.27	even	2		208.2.a.e.1.1		2
35.13	even	4		2600.2.d.k.1249.4		4
35.27	even	4		2600.2.d.k.1249.1		4
35.34	odd	2		2600.2.a.p.1.1		2
56.13	odd	2		832.2.a.k.1.1		2
56.27	even	2		832.2.a.n.1.2		2
84.83	odd	2		1872.2.a.u.1.2		2
91.6	even	12		1352.2.o.d.361.2		8
91.20	even	12		1352.2.o.d.361.1		8
91.34	even	4		1352.2.f.c.337.3		4
91.41	even	12		1352.2.o.d.1161.2		8
91.48	odd	6		1352.2.i.f.1329.1		4
91.55	odd	6		1352.2.i.f.529.1		4
91.62	odd	6		1352.2.i.d.529.1		4
91.69	odd	6		1352.2.i.d.1329.1		4
91.76	even	12		1352.2.o.d.1161.1		8
91.83	even	4		1352.2.f.c.337.4		4
91.90	odd	2		1352.2.a.g.1.2		2
112.13	odd	4		3328.2.b.y.1665.4		4
112.27	even	4		3328.2.b.w.1665.4		4
112.69	odd	4		3328.2.b.y.1665.1		4
112.83	even	4		3328.2.b.w.1665.1		4
140.139	even	2		5200.2.a.bw.1.2		2
168.83	odd	2		7488.2.a.cv.1.1		2
168.125	even	2		7488.2.a.cu.1.1		2
364.83	odd	4		2704.2.f.k.337.2		4
364.307	odd	4		2704.2.f.k.337.1		4
364.363	even	2		2704.2.a.p.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
104.2.a.b.1.2	✓	2	7.6	odd	2
208.2.a.e.1.1		2	28.27	even	2
832.2.a.k.1.1		2	56.13	odd	2
832.2.a.n.1.2		2	56.27	even	2
936.2.a.j.1.2		2	21.20	even	2
1352.2.a.g.1.2		2	91.90	odd	2
1352.2.f.c.337.3		4	91.34	even	4
1352.2.f.c.337.4		4	91.83	even	4
1352.2.i.d.529.1		4	91.62	odd	6
1352.2.i.d.1329.1		4	91.69	odd	6
1352.2.i.f.529.1		4	91.55	odd	6
1352.2.i.f.1329.1		4	91.48	odd	6
1352.2.o.d.361.1		8	91.20	even	12
1352.2.o.d.361.2		8	91.6	even	12
1352.2.o.d.1161.1		8	91.76	even	12
1352.2.o.d.1161.2		8	91.41	even	12
1872.2.a.u.1.2		2	84.83	odd	2
2600.2.a.p.1.1		2	35.34	odd	2
2600.2.d.k.1249.1		4	35.27	even	4
2600.2.d.k.1249.4		4	35.13	even	4
2704.2.a.p.1.1		2	364.363	even	2
2704.2.f.k.337.1		4	364.307	odd	4
2704.2.f.k.337.2		4	364.83	odd	4
3328.2.b.w.1665.1		4	112.83	even	4
3328.2.b.w.1665.4		4	112.27	even	4
3328.2.b.y.1665.1		4	112.69	odd	4
3328.2.b.y.1665.4		4	112.13	odd	4
5096.2.a.m.1.1		2	1.1	even	1	trivial
5200.2.a.bw.1.2		2	140.139	even	2
7488.2.a.cu.1.1		2	168.125	even	2
7488.2.a.cv.1.1		2	168.83	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 5096 = 2^{3} \cdot 7^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5096.a (trivial)

\(f(q)\)	\(=\)	\(q-2.56155 q^{3} +0.561553 q^{5} +3.56155 q^{9} -5.12311 q^{11} -1.00000 q^{13} -1.43845 q^{15} -5.68466 q^{17} -5.12311 q^{19} -8.00000 q^{23} -4.68466 q^{25} -1.43845 q^{27} -2.00000 q^{29} -4.00000 q^{31} +13.1231 q^{33} +9.68466 q^{37} +2.56155 q^{39} +3.12311 q^{41} +5.43845 q^{43} +2.00000 q^{45} +0.315342 q^{47} +14.5616 q^{51} +3.12311 q^{53} -2.87689 q^{55} +13.1231 q^{57} -5.12311 q^{59} -11.1231 q^{61} -0.561553 q^{65} -5.12311 q^{67} +20.4924 q^{69} -7.68466 q^{71} +6.00000 q^{73} +12.0000 q^{75} +8.00000 q^{79} -7.00000 q^{81} -2.24621 q^{83} -3.19224 q^{85} +5.12311 q^{87} -10.0000 q^{89} +10.2462 q^{93} -2.87689 q^{95} +8.24621 q^{97} -18.2462 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - q^{3} - 3 q^{5} + 3 q^{9} - 2 q^{11} - 2 q^{13} - 7 q^{15} + q^{17} - 2 q^{19} - 16 q^{23} + 3 q^{25} - 7 q^{27} - 4 q^{29} - 8 q^{31} + 18 q^{33} + 7 q^{37} + q^{39} - 2 q^{41} + 15 q^{43} + 4 q^{45}+ \cdots - 20 q^{99}+O(q^{100}) \) 2 * q - q^3 - 3 * q^5 + 3 * q^9 - 2 * q^11 - 2 * q^13 - 7 * q^15 + q^17 - 2 * q^19 - 16 * q^23 + 3 * q^25 - 7 * q^27 - 4 * q^29 - 8 * q^31 + 18 * q^33 + 7 * q^37 + q^39 - 2 * q^41 + 15 * q^43 + 4 * q^45 + 13 * q^47 + 25 * q^51 - 2 * q^53 - 14 * q^55 + 18 * q^57 - 2 * q^59 - 14 * q^61 + 3 * q^65 - 2 * q^67 + 8 * q^69 - 3 * q^71 + 12 * q^73 + 24 * q^75 + 16 * q^79 - 14 * q^81 + 12 * q^83 - 27 * q^85 + 2 * q^87 - 20 * q^89 + 4 * q^93 - 14 * q^95 - 20 * q^99

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