LMFDB - Embedded newform 2380.2.m.d.169.13

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2380,2,Mod(169,2380)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2380.169"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2380, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2380 = 2^{2} \cdot 5 \cdot 7 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2380.m (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [26,0,6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$19.0043956811$
Analytic rank:	$0$
Dimension:	$26$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			169.13
Character	$\chi$	$=$	2380.169
Dual form			2380.2.m.d.169.14

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+0.355118 q^{3} +(0.778855 - 2.09604i) q^{5} +1.00000 q^{7} -2.87389 q^{9} +3.29848i q^{11} +2.14706i q^{13} +(0.276586 - 0.744342i) q^{15} +(2.86942 + 2.96081i) q^{17} +4.80942 q^{19} +0.355118 q^{21} +1.16614 q^{23} +(-3.78677 - 3.26502i) q^{25} -2.08593 q^{27} +4.14905i q^{29} -0.597440i q^{31} +1.17135i q^{33} +(0.778855 - 2.09604i) q^{35} +3.57132 q^{37} +0.762461i q^{39} +5.47443i q^{41} +8.74450i q^{43} +(-2.23835 + 6.02379i) q^{45} +1.76692i q^{47} +1.00000 q^{49} +(1.01898 + 1.05144i) q^{51} -7.89948i q^{53} +(6.91374 + 2.56904i) q^{55} +1.70791 q^{57} +11.5544 q^{59} +8.59309i q^{61} -2.87389 q^{63} +(4.50033 + 1.67225i) q^{65} -13.7553i q^{67} +0.414116 q^{69} +11.3621i q^{71} +9.47179 q^{73} +(-1.34475 - 1.15947i) q^{75} +3.29848i q^{77} -6.47139i q^{79} +7.88092 q^{81} -11.8773i q^{83} +(8.44085 - 3.70838i) q^{85} +1.47340i q^{87} +5.60255 q^{89} +2.14706i q^{91} -0.212162i q^{93} +(3.74584 - 10.0807i) q^{95} -13.5262 q^{97} -9.47947i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 26 q + 6 q^{3} - 4 q^{5} + 26 q^{7} + 32 q^{9} + 10 q^{15} - 11 q^{17} + 6 q^{21} - 16 q^{23} + 4 q^{25} + 6 q^{27} - 4 q^{35} - 20 q^{37} - 18 q^{45} + 26 q^{49} + 9 q^{51} - 14 q^{55} + 32 q^{59} + 32 q^{63}+ \cdots - 46 q^{97}+O(q^{100}) $ 26 * q + 6 * q^3 - 4 * q^5 + 26 * q^7 + 32 * q^9 + 10 * q^15 - 11 * q^17 + 6 * q^21 - 16 * q^23 + 4 * q^25 + 6 * q^27 - 4 * q^35 - 20 * q^37 - 18 * q^45 + 26 * q^49 + 9 * q^51 - 14 * q^55 + 32 * q^59 + 32 * q^63 - 42 * q^65 - 8 * q^69 - 12 * q^73 + 8 * q^75 + 66 * q^81 + 11 * q^85 + 4 * q^89 + 20 * q^95 - 46 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2380\mathbb{Z}\right)^\times$.

$n$	$477$	$1191$	$1261$	$1361$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$3$	0.355118			0.205028			0.102514	−	0.994732i	$-0.467311\pi$
$3$	0.355118			0.205028			0.102514	+	0.994732i	$0.467311\pi$
$4$		0			0
$5$	0.778855	−	2.09604i	0.348315	−	0.937378i
$5$	0.778855	−	2.09604i	0.348315	−	0.937378i
$6$		0			0
$7$	1.00000			0.377964
$7$	1.00000			0.377964
$8$		0			0
$9$	−2.87389			−0.957964
$10$		0			0
$11$			3.29848i			0.994528i	0.867599	+	0.497264i	$0.165662\pi$
$11$			3.29848i			0.994528i	−0.867599	+	0.497264i	$0.834338\pi$
$12$		0			0
$13$			2.14706i			0.595488i	0.954646	+	0.297744i	$0.0962342\pi$
$13$			2.14706i			0.595488i	−0.954646	+	0.297744i	$0.903766\pi$
$14$		0			0
$15$	0.276586	−	0.744342i	0.0714141	−	0.192188i
$16$		0			0
$17$	2.86942	+	2.96081i	0.695937	+	0.718103i
$17$	2.86942	+	2.96081i	0.695937	+	0.718103i
$18$		0			0
$19$	4.80942			1.10336			0.551678	−	0.834057i	$-0.313988\pi$
$19$	4.80942			1.10336			0.551678	+	0.834057i	$0.313988\pi$
$20$		0			0
$21$	0.355118			0.0774932
$22$		0			0
$23$	1.16614			0.243156			0.121578	−	0.992582i	$-0.461205\pi$
$23$	1.16614			0.243156			0.121578	+	0.992582i	$0.461205\pi$
$24$		0			0
$25$	−3.78677	−	3.26502i	−0.757354	−	0.653005i
$26$		0			0
$27$	−2.08593			−0.401437
$28$		0			0
$29$			4.14905i			0.770460i	0.922821	+	0.385230i	$0.125878\pi$
$29$			4.14905i			0.770460i	−0.922821	+	0.385230i	$0.874122\pi$
$30$		0			0
$31$		−	0.597440i		−	0.107303i	−0.998560	−	0.0536517i	$-0.982914\pi$
$31$		−	0.597440i		−	0.107303i	0.998560	−	0.0536517i	$-0.0170861\pi$
$32$		0			0
$33$			1.17135i			0.203906i
$34$		0			0
$35$	0.778855	−	2.09604i	0.131651	−	0.354295i
$36$		0			0
$37$	3.57132			0.587122			0.293561	−	0.955940i	$-0.405160\pi$
$37$	3.57132			0.587122			0.293561	+	0.955940i	$0.405160\pi$
$38$		0			0
$39$			0.762461i			0.122091i
$40$		0			0
$41$			5.47443i			0.854962i	0.904024	+	0.427481i	$0.140599\pi$
$41$			5.47443i			0.854962i	−0.904024	+	0.427481i	$0.859401\pi$
$42$		0			0
$43$			8.74450i			1.33352i	0.745271	+	0.666762i	$0.232320\pi$
$43$			8.74450i			1.33352i	−0.745271	+	0.666762i	$0.767680\pi$
$44$		0			0
$45$	−2.23835	+	6.02379i	−0.333673	+	0.897974i
$46$		0			0
$47$			1.76692i			0.257731i	0.991662	+	0.128866i	$0.0411336\pi$
$47$			1.76692i			0.257731i	−0.991662	+	0.128866i	$0.958866\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$	1.01898	+	1.05144i	0.142686	+	0.147231i
$52$		0			0
$53$		−	7.89948i		−	1.08508i	−0.840031	−	0.542538i	$-0.817463\pi$
$53$		−	7.89948i		−	1.08508i	0.840031	−	0.542538i	$-0.182537\pi$
$54$		0			0
$55$	6.91374	+	2.56904i	0.932249	+	0.346409i
$56$		0			0
$57$	1.70791			0.226218
$58$		0			0
$59$	11.5544			1.50425			0.752127	−	0.659018i	$-0.229028\pi$
$59$	11.5544			1.50425			0.752127	+	0.659018i	$0.229028\pi$
$60$		0			0
$61$			8.59309i			1.10023i	0.835088	+	0.550116i	$0.185417\pi$
$61$			8.59309i			1.10023i	−0.835088	+	0.550116i	$0.814583\pi$
$62$		0			0
$63$	−2.87389			−0.362076
$64$		0			0
$65$	4.50033	+	1.67225i	0.558197	+	0.207417i
$66$		0			0
$67$		−	13.7553i		−	1.68048i	−0.542213	−	0.840241i	$-0.682413\pi$
$67$		−	13.7553i		−	1.68048i	0.542213	−	0.840241i	$-0.317587\pi$
$68$		0			0
$69$	0.414116			0.0498537
$70$		0			0
$71$			11.3621i			1.34843i	0.738533	+	0.674217i	$0.235519\pi$
$71$			11.3621i			1.34843i	−0.738533	+	0.674217i	$0.764481\pi$
$72$		0			0
$73$	9.47179			1.10859			0.554294	−	0.832321i	$-0.312988\pi$
$73$	9.47179			1.10859			0.554294	+	0.832321i	$0.312988\pi$
$74$		0			0
$75$	−1.34475	−	1.15947i	−0.155278	−	0.133884i
$76$		0			0
$77$			3.29848i			0.375896i
$78$		0			0
$79$		−	6.47139i		−	0.728089i	−0.931382	−	0.364044i	$-0.881396\pi$
$79$		−	6.47139i		−	0.728089i	0.931382	−	0.364044i	$-0.118604\pi$
$80$		0			0
$81$	7.88092			0.875658
$82$		0			0
$83$		−	11.8773i		−	1.30370i	−0.758349	−	0.651849i	$-0.773994\pi$
$83$		−	11.8773i		−	1.30370i	0.758349	−	0.651849i	$-0.226006\pi$
$84$		0			0
$85$	8.44085	−	3.70838i	0.915539	−	0.402230i
$86$		0			0
$87$			1.47340i			0.157966i
$88$		0			0
$89$	5.60255			0.593870			0.296935	−	0.954898i	$-0.404036\pi$
$89$	5.60255			0.593870			0.296935	+	0.954898i	$0.404036\pi$
$90$		0			0
$91$			2.14706i			0.225073i
$92$		0			0
$93$		−	0.212162i		−	0.0220002i
$94$		0			0
$95$	3.74584	−	10.0807i	0.384315	−	1.03426i
$96$		0			0
$97$	−13.5262			−1.37337			−0.686687	−	0.726954i	$-0.740936\pi$
$97$	−13.5262			−1.37337			−0.686687	+	0.726954i	$0.740936\pi$
$98$		0			0
$99$		−	9.47947i		−	0.952722i
$100$		0			0
$101$	8.51116			0.846892			0.423446	−	0.905921i	$-0.360820\pi$
$101$	8.51116			0.846892			0.423446	+	0.905921i	$0.360820\pi$
$102$		0			0
$103$			0.0327687i			0.00322880i	0.999999	+	0.00161440i	$0.000513880\pi$
$103$			0.0327687i			0.00322880i	−0.999999	+	0.00161440i	$0.999486\pi$
$104$		0			0
$105$	0.276586	−	0.744342i	0.0269920	−	0.0726404i
$106$		0			0
$107$	7.10993			0.687343			0.343672	−	0.939090i	$-0.388329\pi$
$107$	7.10993			0.687343			0.343672	+	0.939090i	$0.388329\pi$
$108$		0			0
$109$			2.71597i			0.260143i	0.991505	+	0.130071i	$0.0415207\pi$
$109$			2.71597i			0.260143i	−0.991505	+	0.130071i	$0.958479\pi$
$110$		0			0
$111$	1.26824			0.120376
$112$		0			0
$113$	−17.8811			−1.68211			−0.841055	−	0.540949i	$-0.818065\pi$
$113$	−17.8811			−1.68211			−0.841055	+	0.540949i	$0.818065\pi$
$114$		0			0
$115$	0.908251	−	2.44427i	0.0846949	−	0.227929i
$116$		0			0
$117$		−	6.17042i		−	0.570456i
$118$		0			0
$119$	2.86942	+	2.96081i	0.263039	+	0.271417i
$120$		0			0
$121$	0.120046			0.0109132
$122$		0			0
$123$			1.94407i			0.175291i
$124$		0			0
$125$	−9.79297	+	5.39424i	−0.875910	+	0.482475i
$126$		0			0
$127$		−	0.00262702i		−	0.000233111i	−1.00000		0.000116555i	$-0.999963\pi$
$127$		−	0.00262702i		−	0.000233111i	1.00000		0.000116555i	$-3.71007e-5\pi$
$128$		0			0
$129$			3.10533i			0.273409i
$130$		0			0
$131$		−	0.261389i		−	0.0228376i	−0.999935	−	0.0114188i	$-0.996365\pi$
$131$		−	0.261389i		−	0.0228376i	0.999935	−	0.0114188i	$-0.00363480\pi$
$132$		0			0
$133$	4.80942			0.417029
$134$		0			0
$135$	−1.62463	+	4.37218i	−0.139826	+	0.376298i
$136$		0			0
$137$		−	2.51963i		−	0.215266i	−0.994191	−	0.107633i	$-0.965673\pi$
$137$		−	2.51963i		−	0.215266i	0.994191	−	0.107633i	$-0.0343272\pi$
$138$		0			0
$139$		−	7.24994i		−	0.614932i	−0.951559	−	0.307466i	$-0.900519\pi$
$139$		−	7.24994i		−	0.614932i	0.951559	−	0.307466i	$-0.0994810\pi$
$140$		0			0
$141$			0.627465i			0.0528421i
$142$		0			0
$143$	−7.08204			−0.592230
$144$		0			0
$145$	8.69658	+	3.23151i	0.722212	+	0.268362i
$146$		0			0
$147$	0.355118			0.0292897
$148$		0			0
$149$	14.8200			1.21410			0.607051	−	0.794663i	$-0.292352\pi$
$149$	14.8200			1.21410			0.607051	+	0.794663i	$0.292352\pi$
$150$		0			0
$151$	−22.2442			−1.81021			−0.905103	−	0.425193i	$-0.860206\pi$
$151$	−22.2442			−1.81021			−0.905103	+	0.425193i	$0.860206\pi$
$152$		0			0
$153$	−8.24640	−	8.50906i	−0.666682	−	0.687917i
$154$		0			0
$155$	−1.25226	−	0.465319i	−0.100584	−	0.0373754i
$156$		0			0
$157$			6.48150i			0.517280i	0.965974	+	0.258640i	$0.0832743\pi$
$157$			6.48150i			0.517280i	−0.965974	+	0.258640i	$0.916726\pi$
$158$		0			0
$159$		−	2.80525i		−	0.222471i
$160$		0			0
$161$	1.16614			0.0919044
$162$		0			0
$163$	−0.677681			−0.0530801			−0.0265400	−	0.999648i	$-0.508449\pi$
$163$	−0.677681			−0.0530801			−0.0265400	+	0.999648i	$0.508449\pi$
$164$		0			0
$165$	2.45520	+	0.912312i	0.191137	+	0.0710234i
$166$		0			0
$167$	7.60691			0.588641			0.294320	−	0.955707i	$-0.404907\pi$
$167$	7.60691			0.588641			0.294320	+	0.955707i	$0.404907\pi$
$168$		0			0
$169$	8.39012			0.645394
$170$		0			0
$171$	−13.8217			−1.05698
$172$		0			0
$173$	18.8210			1.43093			0.715466	−	0.698647i	$-0.246214\pi$
$173$	18.8210			1.43093			0.715466	+	0.698647i	$0.246214\pi$
$174$		0			0
$175$	−3.78677	−	3.26502i	−0.286253	−	0.246813i
$176$		0			0
$177$	4.10318			0.308414
$178$		0			0
$179$	10.3450			0.773222			0.386611	−	0.922243i	$-0.373646\pi$
$179$	10.3450			0.773222			0.386611	+	0.922243i	$0.373646\pi$
$180$		0			0
$181$			24.1822i			1.79745i	0.438516	+	0.898723i	$0.355504\pi$
$181$			24.1822i			1.79745i	−0.438516	+	0.898723i	$0.644496\pi$
$182$		0			0
$183$			3.05156i			0.225578i
$184$		0			0
$185$	2.78154	−	7.48564i	0.204503	−	0.550355i
$186$		0			0
$187$	−9.76618	+	9.46472i	−0.714174	+	0.692129i
$188$		0			0
$189$	−2.08593			−0.151729
$190$		0			0
$191$	−0.158352			−0.0114579			−0.00572896	−	0.999984i	$-0.501824\pi$
$191$	−0.158352			−0.0114579			−0.00572896	+	0.999984i	$0.501824\pi$
$192$		0			0
$193$	21.0203			1.51307			0.756536	−	0.653952i	$-0.226890\pi$
$193$	21.0203			1.51307			0.756536	+	0.653952i	$0.226890\pi$
$194$		0			0
$195$	1.59815	+	0.593847i	0.114446	+	0.0425263i
$196$		0			0
$197$	6.07546			0.432859			0.216429	−	0.976298i	$-0.430559\pi$
$197$	6.07546			0.432859			0.216429	+	0.976298i	$0.430559\pi$
$198$		0			0
$199$			19.2707i			1.36607i	0.730388	+	0.683033i	$0.239339\pi$
$199$			19.2707i			1.36607i	−0.730388	+	0.683033i	$0.760661\pi$
$200$		0			0
$201$		−	4.88477i		−	0.344545i
$202$		0			0
$203$			4.14905i			0.291206i
$204$		0			0
$205$	11.4746	+	4.26379i	0.801422	+	0.297796i
$206$		0			0
$207$	−3.35135			−0.232935
$208$		0			0
$209$			15.8638i			1.09732i
$210$		0			0
$211$			8.69402i			0.598521i	0.954171	+	0.299261i	$0.0967400\pi$
$211$			8.69402i			0.598521i	−0.954171	+	0.299261i	$0.903260\pi$
$212$		0			0
$213$			4.03489i			0.276466i
$214$		0			0
$215$	18.3288	+	6.81070i	1.25002	+	0.464486i
$216$		0			0
$217$		−	0.597440i		−	0.0405569i
$218$		0			0
$219$	3.36360			0.227291
$220$		0			0
$221$	−6.35705	+	6.16083i	−0.427622	+	0.414422i
$222$		0			0
$223$		−	14.8220i		−	0.992553i	−0.868164	−	0.496277i	$-0.834700\pi$
$223$		−	14.8220i		−	0.992553i	0.868164	−	0.496277i	$-0.165300\pi$
$224$		0			0
$225$	10.8828	+	9.38332i	0.725517	+	0.625555i
$226$		0			0
$227$	−7.72487			−0.512717			−0.256359	−	0.966582i	$-0.582523\pi$
$227$	−7.72487			−0.512717			−0.256359	+	0.966582i	$0.582523\pi$
$228$		0			0
$229$	−26.2190			−1.73260			−0.866301	−	0.499523i	$-0.833509\pi$
$229$	−26.2190			−1.73260			−0.866301	+	0.499523i	$0.833509\pi$
$230$		0			0
$231$			1.17135i			0.0770691i
$232$		0			0
$233$	0.678913			0.0444771			0.0222385	−	0.999753i	$-0.492921\pi$
$233$	0.678913			0.0444771			0.0222385	+	0.999753i	$0.492921\pi$
$234$		0			0
$235$	3.70353	+	1.37617i	0.241592	+	0.0897717i
$236$		0			0
$237$		−	2.29811i		−	0.149278i
$238$		0			0
$239$	−0.224261			−0.0145062			−0.00725312	−	0.999974i	$-0.502309\pi$
$239$	−0.224261			−0.0145062			−0.00725312	+	0.999974i	$0.502309\pi$
$240$		0			0
$241$		−	1.44464i		−	0.0930573i	−0.998917	−	0.0465287i	$-0.985184\pi$
$241$		−	1.44464i		−	0.0930573i	0.998917	−	0.0465287i	$-0.0148159\pi$
$242$		0			0
$243$	9.05644			0.580971
$244$		0			0
$245$	0.778855	−	2.09604i	0.0497592	−	0.133911i
$246$		0			0
$247$			10.3261i			0.657035i
$248$		0			0
$249$		−	4.21783i		−	0.267294i
$250$		0			0
$251$	−17.7068			−1.11765			−0.558823	−	0.829287i	$-0.688747\pi$
$251$	−17.7068			−1.11765			−0.558823	+	0.829287i	$0.688747\pi$
$252$		0			0
$253$			3.84647i			0.241826i
$254$		0			0
$255$	2.99750	−	1.31691i	0.187711	−	0.0824682i
$256$		0			0
$257$		−	15.1319i		−	0.943903i	−0.881624	−	0.471952i	$-0.843550\pi$
$257$		−	15.1319i		−	0.943903i	0.881624	−	0.471952i	$-0.156450\pi$
$258$		0			0
$259$	3.57132			0.221911
$260$		0			0
$261$		−	11.9239i		−	0.738072i
$262$		0			0
$263$		−	8.26196i		−	0.509454i	−0.967013	−	0.254727i	$-0.918014\pi$
$263$		−	8.26196i		−	0.509454i	0.967013	−	0.254727i	$-0.0819856\pi$
$264$		0			0
$265$	−16.5576	−	6.15255i	−1.01713	−	0.377948i
$266$		0			0
$267$	1.98957			0.121760
$268$		0			0
$269$		−	3.53275i		−	0.215396i	−0.994184	−	0.107698i	$-0.965652\pi$
$269$		−	3.53275i		−	0.215396i	0.994184	−	0.107698i	$-0.0343479\pi$
$270$		0			0
$271$	−15.7283			−0.955426			−0.477713	−	0.878516i	$-0.658534\pi$
$271$	−15.7283			−0.955426			−0.477713	+	0.878516i	$0.658534\pi$
$272$		0			0
$273$			0.762461i			0.0461462i
$274$		0			0
$275$	10.7696	−	12.4906i	0.649432	−	0.753210i
$276$		0			0
$277$	−16.4543			−0.988644			−0.494322	−	0.869279i	$-0.664584\pi$
$277$	−16.4543			−0.988644			−0.494322	+	0.869279i	$0.664584\pi$
$278$		0			0
$279$			1.71698i			0.102793i
$280$		0			0
$281$	9.97526			0.595074			0.297537	−	0.954710i	$-0.403835\pi$
$281$	9.97526			0.595074			0.297537	+	0.954710i	$0.403835\pi$
$282$		0			0
$283$	−27.7913			−1.65202			−0.826009	−	0.563656i	$-0.809394\pi$
$283$	−27.7913			−1.65202			−0.826009	+	0.563656i	$0.809394\pi$
$284$		0			0
$285$	1.33022	−	3.57985i	0.0787952	−	0.212052i
$286$		0			0
$287$			5.47443i			0.323145i
$288$		0			0
$289$	−0.532847	+	16.9916i	−0.0313439	+	0.999509i
$290$		0			0
$291$	−4.80339			−0.281579
$292$		0			0
$293$			0.174404i			0.0101888i	0.999987	+	0.00509440i	$0.00162160\pi$
$293$			0.174404i			0.0101888i	−0.999987	+	0.00509440i	$0.998378\pi$
$294$		0			0
$295$	8.99921	−	24.2185i	0.523954	−	1.41005i
$296$		0			0
$297$		−	6.88038i		−	0.399240i
$298$		0			0
$299$			2.50377i			0.144797i
$300$		0			0
$301$			8.74450i			0.504025i
$302$		0			0
$303$	3.02247			0.173636
$304$		0			0
$305$	18.0115	+	6.69278i	1.03133	+	0.383227i
$306$		0			0
$307$			7.52700i			0.429588i	0.976659	+	0.214794i	$0.0689081\pi$
$307$			7.52700i			0.429588i	−0.976659	+	0.214794i	$0.931092\pi$
$308$		0			0
$309$			0.0116368i			0.000661993i
$310$		0			0
$311$			2.57441i			0.145982i	0.997333	+	0.0729908i	$0.0232544\pi$
$311$			2.57441i			0.145982i	−0.997333	+	0.0729908i	$0.976746\pi$
$312$		0			0
$313$	6.72127			0.379909			0.189954	−	0.981793i	$-0.439166\pi$
$313$	6.72127			0.379909			0.189954	+	0.981793i	$0.439166\pi$
$314$		0			0
$315$	−2.23835	+	6.02379i	−0.126116	+	0.339402i
$316$		0			0
$317$	3.48442			0.195705			0.0978524	−	0.995201i	$-0.468803\pi$
$317$	3.48442			0.195705			0.0978524	+	0.995201i	$0.468803\pi$
$318$		0			0
$319$	−13.6856			−0.766244
$320$		0			0
$321$	2.52487			0.140924
$322$		0			0
$323$	13.8002	+	14.2398i	0.767866	+	0.792323i
$324$		0			0
$325$	7.01021	−	8.13043i	0.388857	−	0.450995i
$326$		0			0
$327$			0.964491i			0.0533365i
$328$		0			0
$329$			1.76692i			0.0974133i
$330$		0			0
$331$	−28.8742			−1.58707			−0.793535	−	0.608525i	$-0.791762\pi$
$331$	−28.8742			−1.58707			−0.793535	+	0.608525i	$0.791762\pi$
$332$		0			0
$333$	−10.2636			−0.562441
$334$		0			0
$335$	−28.8317	−	10.7134i	−1.57525	−	0.585337i
$336$		0			0
$337$	28.0825			1.52975			0.764877	−	0.644176i	$-0.222800\pi$
$337$	28.0825			1.52975			0.764877	+	0.644176i	$0.222800\pi$
$338$		0			0
$339$	−6.34990			−0.344879
$340$		0			0
$341$	1.97064			0.106716
$342$		0			0
$343$	1.00000			0.0539949
$344$		0			0
$345$	0.322537	−	0.868004i	0.0173648	−	0.0467318i
$346$		0			0
$347$	−26.8782			−1.44290			−0.721448	−	0.692469i	$-0.756523\pi$
$347$	−26.8782			−1.44290			−0.721448	+	0.692469i	$0.756523\pi$
$348$		0			0
$349$	−9.64141			−0.516093			−0.258047	−	0.966132i	$-0.583079\pi$
$349$	−9.64141			−0.516093			−0.258047	+	0.966132i	$0.583079\pi$
$350$		0			0
$351$		−	4.47861i		−	0.239051i
$352$		0			0
$353$		−	1.14787i		−	0.0610948i	−0.999533	−	0.0305474i	$-0.990275\pi$
$353$		−	1.14787i		−	0.0610948i	0.999533	−	0.0305474i	$-0.00972505\pi$
$354$		0			0
$355$	23.8154	+	8.84944i	1.26399	+	0.469680i
$356$		0			0
$357$	1.01898	+	1.05144i	0.0539303	+	0.0556481i
$358$		0			0
$359$	−12.2435			−0.646189			−0.323095	−	0.946367i	$-0.604723\pi$
$359$	−12.2435			−0.646189			−0.323095	+	0.946367i	$0.604723\pi$
$360$		0			0
$361$	4.13050			0.217395
$362$		0			0
$363$	0.0426304			0.00223752
$364$		0			0
$365$	7.37715	−	19.8532i	0.386138	−	1.03917i
$366$		0			0
$367$	−12.9156			−0.674187			−0.337094	−	0.941471i	$-0.609444\pi$
$367$	−12.9156			−0.674187			−0.337094	+	0.941471i	$0.609444\pi$
$368$		0			0
$369$		−	15.7329i		−	0.819023i
$370$		0			0
$371$		−	7.89948i		−	0.410120i
$372$		0			0
$373$			23.2100i			1.20177i	0.799335	+	0.600885i	$0.205185\pi$
$373$			23.2100i			1.20177i	−0.799335	+	0.600885i	$0.794815\pi$
$374$		0			0
$375$	−3.47766	+	1.91559i	−0.179586	+	0.0989208i
$376$		0			0
$377$	−8.90827			−0.458799
$378$		0			0
$379$		−	30.8530i		−	1.58481i	−0.609996	−	0.792405i	$-0.708829\pi$
$379$		−	30.8530i		−	1.58481i	0.609996	−	0.792405i	$-0.291171\pi$
$380$		0			0
$381$		0	0.000932904i		0	4.77941e-5i
$382$		0			0
$383$			3.48763i			0.178210i	0.996022	+	0.0891048i	$0.0284006\pi$
$383$			3.48763i			0.178210i	−0.996022	+	0.0891048i	$0.971599\pi$
$384$		0			0
$385$	6.91374	+	2.56904i	0.352357	+	0.130930i
$386$		0			0
$387$		−	25.1308i		−	1.27747i
$388$		0			0
$389$	2.03369			0.103112			0.0515560	−	0.998670i	$-0.483582\pi$
$389$	2.03369			0.103112			0.0515560	+	0.998670i	$0.483582\pi$
$390$		0			0
$391$	3.34614	+	3.45271i	0.169221	+	0.174611i
$392$		0			0
$393$		−	0.0928239i		−	0.00468235i
$394$		0			0
$395$	−13.5643	−	5.04028i	−0.682494	−	0.253604i
$396$		0			0
$397$	11.5647			0.580415			0.290207	−	0.956964i	$-0.406276\pi$
$397$	11.5647			0.580415			0.290207	+	0.956964i	$0.406276\pi$
$398$		0			0
$399$	1.70791			0.0855026
$400$		0			0
$401$		−	8.53917i		−	0.426426i	−0.977006	−	0.213213i	$-0.931607\pi$
$401$		−	8.53917i		−	0.426426i	0.977006	−	0.213213i	$-0.0683928\pi$
$402$		0			0
$403$	1.28274			0.0638979
$404$		0			0
$405$	6.13810	−	16.5187i	0.305005	−	0.820822i
$406$		0			0
$407$			11.7799i			0.583909i
$408$		0			0
$409$	5.37632			0.265842			0.132921	−	0.991127i	$-0.457564\pi$
$409$	5.37632			0.265842			0.132921	+	0.991127i	$0.457564\pi$
$410$		0			0
$411$		−	0.894766i		−	0.0441356i
$412$		0			0
$413$	11.5544			0.568555
$414$		0			0
$415$	−24.8952	−	9.25066i	−1.22206	−	0.454097i
$416$		0			0
$417$		−	2.57459i		−	0.126078i
$418$		0			0
$419$		−	9.88962i		−	0.483140i	−0.970383	−	0.241570i	$-0.922338\pi$
$419$		−	9.88962i		−	0.483140i	0.970383	−	0.241570i	$-0.0776623\pi$
$420$		0			0
$421$	−37.9792			−1.85099			−0.925496	−	0.378756i	$-0.876352\pi$
$421$	−37.9792			−1.85099			−0.925496	+	0.378756i	$0.876352\pi$
$422$		0			0
$423$		−	5.07793i		−	0.246897i
$424$		0			0
$425$	−1.19870	−	20.5806i	−0.0581456	−	0.998308i
$426$		0			0
$427$			8.59309i			0.415849i
$428$		0			0
$429$	−2.51496			−0.121423
$430$		0			0
$431$			23.8422i			1.14844i	0.818702	+	0.574218i	$0.194694\pi$
$431$			23.8422i			1.14844i	−0.818702	+	0.574218i	$0.805306\pi$
$432$		0			0
$433$		−	36.3440i		−	1.74658i	−0.487200	−	0.873291i	$-0.661982\pi$
$433$		−	36.3440i		−	1.74658i	0.487200	−	0.873291i	$-0.338018\pi$
$434$		0			0
$435$	3.08831	+	1.14757i	0.148073	+	0.0550217i
$436$		0			0
$437$	5.60844			0.268288
$438$		0			0
$439$		−	32.2663i		−	1.53999i	−0.638052	−	0.769993i	$-0.720260\pi$
$439$		−	32.2663i		−	1.53999i	0.638052	−	0.769993i	$-0.279740\pi$
$440$		0			0
$441$	−2.87389			−0.136852
$442$		0			0
$443$			28.0680i			1.33355i	0.745258	+	0.666776i	$0.232326\pi$
$443$			28.0680i			1.33355i	−0.745258	+	0.666776i	$0.767674\pi$
$444$		0			0
$445$	4.36358	−	11.7432i	0.206854	−	0.556680i
$446$		0			0
$447$	5.26285			0.248924
$448$		0			0
$449$			9.53121i			0.449806i	0.974381	+	0.224903i	$0.0722065\pi$
$449$			9.53121i			0.449806i	−0.974381	+	0.224903i	$0.927794\pi$
$450$		0			0
$451$	−18.0573			−0.850284
$452$		0			0
$453$	−7.89931			−0.371142
$454$		0			0
$455$	4.50033	+	1.67225i	0.210979	+	0.0783963i
$456$		0			0
$457$		−	0.0138664i		−	0.000648642i	−1.00000		0.000324321i	$-0.999897\pi$
$457$		−	0.0138664i		−	0.000648642i	1.00000		0.000324321i	$-0.000103235\pi$
$458$		0			0
$459$	−5.98540	−	6.17604i	−0.279375	−	0.288273i
$460$		0			0
$461$	−5.43461			−0.253115			−0.126558	−	0.991959i	$-0.540393\pi$
$461$	−5.43461			−0.253115			−0.126558	+	0.991959i	$0.540393\pi$
$462$		0			0
$463$			18.6564i			0.867034i	0.901145	+	0.433517i	$0.142728\pi$
$463$			18.6564i			0.867034i	−0.901145	+	0.433517i	$0.857272\pi$
$464$		0			0
$465$	−0.444700	−	0.165243i	−0.0206225	−	0.00766298i
$466$		0			0
$467$			36.3031i			1.67991i	0.542657	+	0.839955i	$0.317419\pi$
$467$			36.3031i			1.67991i	−0.542657	+	0.839955i	$0.682581\pi$
$468$		0			0
$469$		−	13.7553i		−	0.635163i
$470$		0			0
$471$			2.30170i			0.106057i
$472$		0			0
$473$	−28.8436			−1.32623
$474$		0			0
$475$	−18.2122	−	15.7029i	−0.835631	−	0.720497i
$476$		0			0
$477$			22.7022i			1.03946i
$478$		0			0
$479$		−	17.3889i		−	0.794518i	−0.917707	−	0.397259i	$-0.869961\pi$
$479$		−	17.3889i		−	0.794518i	0.917707	−	0.397259i	$-0.130039\pi$
$480$		0			0
$481$			7.66785i			0.349624i
$482$		0			0
$483$	0.414116			0.0188429
$484$		0			0
$485$	−10.5349	+	28.3514i	−0.478366	+	1.28737i
$486$		0			0
$487$	−7.45009			−0.337596			−0.168798	−	0.985651i	$-0.553989\pi$
$487$	−7.45009			−0.337596			−0.168798	+	0.985651i	$0.553989\pi$
$488$		0			0
$489$	−0.240657			−0.0108829
$490$		0			0
$491$	2.24667			0.101391			0.0506954	−	0.998714i	$-0.483856\pi$
$491$	2.24667			0.101391			0.0506954	+	0.998714i	$0.483856\pi$
$492$		0			0
$493$	−12.2846	+	11.9054i	−0.553269	+	0.536191i
$494$		0			0
$495$	−19.8693	−	7.38313i	−0.893060	−	0.331847i
$496$		0			0
$497$			11.3621i			0.509660i
$498$		0			0
$499$		−	12.5132i		−	0.560170i	−0.959975	−	0.280085i	$-0.909637\pi$
$499$		−	12.5132i		−	0.560170i	0.959975	−	0.280085i	$-0.0903626\pi$
$500$		0			0
$501$	2.70135			0.120688
$502$		0			0
$503$	1.17869			0.0525553			0.0262777	−	0.999655i	$-0.491635\pi$
$503$	1.17869			0.0525553			0.0262777	+	0.999655i	$0.491635\pi$
$504$		0			0
$505$	6.62897	−	17.8397i	0.294985	−	0.793858i
$506$		0			0
$507$	2.97949			0.132324
$508$		0			0
$509$	17.1333			0.759419			0.379710	−	0.925106i	$-0.376024\pi$
$509$	17.1333			0.759419			0.379710	+	0.925106i	$0.376024\pi$
$510$		0			0
$511$	9.47179			0.419007
$512$		0			0
$513$	−10.0321			−0.442928
$514$		0			0
$515$	0.0686846	+	0.0255221i	0.00302661	+	0.00112464i
$516$		0			0
$517$	−5.82814			−0.256321
$518$		0			0
$519$	6.68367			0.293381
$520$		0			0
$521$			17.5140i			0.767303i	0.923478	+	0.383651i	$0.125334\pi$
$521$			17.5140i			0.767303i	−0.923478	+	0.383651i	$0.874666\pi$
$522$		0			0
$523$		−	11.8176i		−	0.516746i	−0.966045	−	0.258373i	$-0.916814\pi$
$523$		−	11.8176i		−	0.516746i	0.966045	−	0.258373i	$-0.0831864\pi$
$524$		0			0
$525$	−1.34475	−	1.15947i	−0.0586897	−	0.0506034i
$526$		0			0
$527$	1.76891	−	1.71431i	0.0770549	−	0.0746764i
$528$		0			0
$529$	−21.6401			−0.940875
$530$		0			0
$531$	−33.2061			−1.44102
$532$		0			0
$533$	−11.7539			−0.509120
$534$		0			0
$535$	5.53761	−	14.9027i	0.239412	−	0.644300i
$536$		0			0
$537$	3.67370			0.158532
$538$		0			0
$539$			3.29848i			0.142075i
$540$		0			0
$541$			21.5163i			0.925056i	0.886605	+	0.462528i	$0.153058\pi$
$541$			21.5163i			0.925056i	−0.886605	+	0.462528i	$0.846942\pi$
$542$		0			0
$543$			8.58753i			0.368526i
$544$		0			0
$545$	5.69279	+	2.11535i	0.243852	+	0.0906116i
$546$		0			0
$547$	39.2183			1.67685			0.838426	−	0.545015i	$-0.183476\pi$
$547$	39.2183			1.67685			0.838426	+	0.545015i	$0.183476\pi$
$548$		0			0
$549$		−	24.6956i		−	1.05398i
$550$		0			0
$551$			19.9545i			0.850092i
$552$		0			0
$553$		−	6.47139i		−	0.275192i
$554$		0			0
$555$	0.987777	−	2.65829i	0.0419288	−	0.112838i
$556$		0			0
$557$		−	42.2393i		−	1.78974i	−0.446328	−	0.894869i	$-0.647269\pi$
$557$		−	42.2393i		−	1.78974i	0.446328	−	0.894869i	$-0.352731\pi$
$558$		0			0
$559$	−18.7750			−0.794098
$560$		0			0
$561$	−3.46815	+	3.36110i	−0.146425	+	0.141906i
$562$		0			0
$563$		−	46.3025i		−	1.95142i	−0.219074	−	0.975708i	$-0.570303\pi$
$563$		−	46.3025i		−	1.95142i	0.219074	−	0.975708i	$-0.429697\pi$
$564$		0			0
$565$	−13.9268	+	37.4795i	−0.585904	+	1.57677i
$566$		0			0
$567$	7.88092			0.330968
$568$		0			0
$569$	38.0346			1.59449			0.797247	−	0.603654i	$-0.206289\pi$
$569$	38.0346			1.59449			0.797247	+	0.603654i	$0.206289\pi$
$570$		0			0
$571$		−	14.3752i		−	0.601582i	−0.953690	−	0.300791i	$-0.902749\pi$
$571$		−	14.3752i		−	0.601582i	0.953690	−	0.300791i	$-0.0972507\pi$
$572$		0			0
$573$	−0.0562336			−0.00234919
$574$		0			0
$575$	−4.41589	−	3.80746i	−0.184155	−	0.158782i
$576$		0			0
$577$		−	1.49609i		−	0.0622831i	−0.999515	−	0.0311415i	$-0.990086\pi$
$577$		−	1.49609i		−	0.0622831i	0.999515	−	0.0311415i	$-0.00991426\pi$
$578$		0			0
$579$	7.46468			0.310222
$580$		0			0
$581$		−	11.8773i		−	0.492752i
$582$		0			0
$583$	26.0562			1.07914
$584$		0			0
$585$	−12.9335	−	4.80587i	−0.534733	−	0.198698i
$586$		0			0
$587$		−	28.7659i		−	1.18730i	−0.804725	−	0.593648i	$-0.797687\pi$
$587$		−	28.7659i		−	1.18730i	0.804725	−	0.593648i	$-0.202313\pi$
$588$		0			0
$589$		−	2.87334i		−	0.118394i
$590$		0			0
$591$	2.15751			0.0887480
$592$		0			0
$593$		−	42.5871i		−	1.74884i	−0.485169	−	0.874420i	$-0.661242\pi$
$593$		−	42.5871i		−	1.74884i	0.485169	−	0.874420i	$-0.338758\pi$
$594$		0			0
$595$	8.44085	−	3.70838i	0.346041	−	0.152029i
$596$		0			0
$597$			6.84339i			0.280081i
$598$		0			0
$599$	25.6616			1.04851			0.524253	−	0.851562i	$-0.324345\pi$
$599$	25.6616			1.04851			0.524253	+	0.851562i	$0.324345\pi$
$600$		0			0
$601$		−	3.97407i		−	0.162106i	−0.996710	−	0.0810528i	$-0.974172\pi$
$601$		−	3.97407i		−	0.162106i	0.996710	−	0.0810528i	$-0.0258282\pi$
$602$		0			0
$603$			39.5314i			1.60984i
$604$		0			0
$605$	0.0934983	−	0.251621i	0.00380124	−	0.0102298i
$606$		0			0
$607$	−6.79462			−0.275785			−0.137892	−	0.990447i	$-0.544033\pi$
$607$	−6.79462			−0.275785			−0.137892	+	0.990447i	$0.544033\pi$
$608$		0			0
$609$			1.47340i			0.0597054i
$610$		0			0
$611$	−3.79368			−0.153476
$612$		0			0
$613$			9.29212i			0.375305i	0.982235	+	0.187653i	$0.0600879\pi$
$613$			9.29212i			0.375305i	−0.982235	+	0.187653i	$0.939912\pi$
$614$		0			0
$615$	4.07485	+	1.51415i	0.164314	+	0.0610564i
$616$		0			0
$617$	−19.1313			−0.770197			−0.385099	−	0.922875i	$-0.625833\pi$
$617$	−19.1313			−0.770197			−0.385099	+	0.922875i	$0.625833\pi$
$618$		0			0
$619$		−	15.7073i		−	0.631331i	−0.948871	−	0.315665i	$-0.897772\pi$
$619$		−	15.7073i		−	0.631331i	0.948871	−	0.315665i	$-0.102228\pi$
$620$		0			0
$621$	−2.43247			−0.0976118
$622$		0			0
$623$	5.60255			0.224462
$624$		0			0
$625$	3.67923	+	24.7278i	0.147169	+	0.989111i
$626$		0			0
$627$			5.63351i			0.224981i
$628$		0			0
$629$	10.2476	+	10.5740i	0.408600	+	0.421614i
$630$		0			0
$631$	8.92291			0.355215			0.177608	−	0.984101i	$-0.443164\pi$
$631$	8.92291			0.355215			0.177608	+	0.984101i	$0.443164\pi$
$632$		0			0
$633$			3.08741i			0.122713i
$634$		0			0
$635$	−0.00550635	−	0.00204607i	−0.000218513	−	8.11959e-5i
$636$		0			0
$637$			2.14706i			0.0850697i
$638$		0			0
$639$		−	32.6535i		−	1.29175i
$640$		0			0
$641$			32.2020i			1.27190i	0.771729	+	0.635952i	$0.219392\pi$
$641$			32.2020i			1.27190i	−0.771729	+	0.635952i	$0.780608\pi$
$642$		0			0
$643$	−41.6150			−1.64114			−0.820568	−	0.571548i	$-0.806343\pi$
$643$	−41.6150			−1.64114			−0.820568	+	0.571548i	$0.806343\pi$
$644$		0			0
$645$	6.50890	+	2.41861i	0.256288	+	0.0952325i
$646$		0			0
$647$			21.9180i			0.861686i	0.902427	+	0.430843i	$0.141784\pi$
$647$			21.9180i			0.861686i	−0.902427	+	0.430843i	$0.858216\pi$
$648$		0			0
$649$			38.1119i			1.49602i
$650$		0			0
$651$		−	0.212162i		−	0.00831528i
$652$		0			0
$653$	45.5809			1.78372			0.891859	−	0.452313i	$-0.149401\pi$
$653$	45.5809			1.78372			0.891859	+	0.452313i	$0.149401\pi$
$654$		0			0
$655$	−0.547881	−	0.203584i	−0.0214075	−	0.00795468i
$656$		0			0
$657$	−27.2209			−1.06199
$658$		0			0
$659$	−16.9857			−0.661667			−0.330834	−	0.943689i	$-0.607330\pi$
$659$	−16.9857			−0.661667			−0.330834	+	0.943689i	$0.607330\pi$
$660$		0			0
$661$	−37.8502			−1.47220			−0.736101	−	0.676872i	$-0.763335\pi$
$661$	−37.8502			−1.47220			−0.736101	+	0.676872i	$0.763335\pi$
$662$		0			0
$663$	−2.25751	+	2.18782i	−0.0876743	+	0.0849680i
$664$		0			0
$665$	3.74584	−	10.0807i	0.145257	−	0.390914i
$666$		0			0
$667$			4.83836i			0.187342i
$668$		0			0
$669$		−	5.26356i		−	0.203501i
$670$		0			0
$671$	−28.3441			−1.09421
$672$		0			0
$673$	14.9687			0.576999			0.288500	−	0.957480i	$-0.406844\pi$
$673$	14.9687			0.576999			0.288500	+	0.957480i	$0.406844\pi$
$674$		0			0
$675$	7.89892	+	6.81060i	0.304030	+	0.262140i
$676$		0			0
$677$	−21.7580			−0.836230			−0.418115	−	0.908394i	$-0.637309\pi$
$677$	−21.7580			−0.836230			−0.418115	+	0.908394i	$0.637309\pi$
$678$		0			0
$679$	−13.5262			−0.519086
$680$		0			0
$681$	−2.74324			−0.105121
$682$		0			0
$683$	−2.39193			−0.0915248			−0.0457624	−	0.998952i	$-0.514572\pi$
$683$	−2.39193			−0.0915248			−0.0457624	+	0.998952i	$0.514572\pi$
$684$		0			0
$685$	−5.28124	−	1.96243i	−0.201786	−	0.0749805i
$686$		0			0
$687$	−9.31085			−0.355231
$688$		0			0
$689$	16.9607			0.646150
$690$		0			0
$691$		−	0.227345i		−	0.00864860i	−0.999991	−	0.00432430i	$-0.998624\pi$
$691$		−	0.227345i		−	0.00864860i	0.999991	−	0.00432430i	$-0.00137647\pi$
$692$		0			0
$693$		−	9.47947i		−	0.360095i
$694$		0			0
$695$	−15.1962	−	5.64665i	−0.576423	−	0.214190i
$696$		0			0
$697$	−16.2088	+	15.7084i	−0.613951	+	0.595000i
$698$		0			0
$699$	0.241094			0.00911903
$700$		0			0
$701$	−22.4834			−0.849186			−0.424593	−	0.905384i	$-0.639583\pi$
$701$	−22.4834			−0.849186			−0.424593	+	0.905384i	$0.639583\pi$
$702$		0			0
$703$	17.1760			0.647804
$704$		0			0
$705$	1.31519	+	0.488704i	0.0495330	+	0.0184057i
$706$		0			0
$707$	8.51116			0.320095
$708$		0			0
$709$		−	46.9658i		−	1.76384i	−0.471402	−	0.881919i	$-0.656252\pi$
$709$		−	46.9658i		−	1.76384i	0.471402	−	0.881919i	$-0.343748\pi$
$710$		0			0
$711$			18.5981i			0.697483i
$712$		0			0
$713$		−	0.696696i		−	0.0260915i
$714$		0			0
$715$	−5.51588	+	14.8442i	−0.206282	+	0.555143i
$716$		0			0
$717$	−0.0796392			−0.00297418
$718$		0			0
$719$		−	30.3086i		−	1.13032i	−0.824982	−	0.565159i	$-0.808815\pi$
$719$		−	30.3086i		−	1.13032i	0.824982	−	0.565159i	$-0.191185\pi$
$720$		0			0
$721$			0.0327687i			0.00122037i
$722$		0			0
$723$		−	0.513017i		−	0.0190793i
$724$		0			0
$725$	13.5468	−	15.7115i	0.503114	−	0.583511i
$726$		0			0
$727$		−	51.7572i		−	1.91957i	−0.280740	−	0.959784i	$-0.590580\pi$
$727$		−	51.7572i		−	1.91957i	0.280740	−	0.959784i	$-0.409420\pi$
$728$		0			0
$729$	−20.4267			−0.756543
$730$		0			0
$731$	−25.8909	+	25.0917i	−0.957608	+	0.928049i
$732$		0			0
$733$			39.1675i			1.44668i	0.690490	+	0.723342i	$0.257395\pi$
$733$			39.1675i			1.44668i	−0.690490	+	0.723342i	$0.742605\pi$
$734$		0			0
$735$	0.276586	−	0.744342i	0.0102020	−	0.0274555i
$736$		0			0
$737$	45.3717			1.67129
$738$		0			0
$739$	20.1659			0.741814			0.370907	−	0.928670i	$-0.379047\pi$
$739$	20.1659			0.741814			0.370907	+	0.928670i	$0.379047\pi$
$740$		0			0
$741$			3.66699i			0.134710i
$742$		0			0
$743$	−3.29079			−0.120727			−0.0603636	−	0.998176i	$-0.519226\pi$
$743$	−3.29079			−0.120727			−0.0603636	+	0.998176i	$0.519226\pi$
$744$		0			0
$745$	11.5426	−	31.0633i	0.422890	−	1.13807i
$746$		0			0
$747$			34.1339i			1.24890i
$748$		0			0
$749$	7.10993			0.259791
$750$		0			0
$751$		−	16.7276i		−	0.610400i	−0.952288	−	0.305200i	$-0.901277\pi$
$751$		−	16.7276i		−	0.610400i	0.952288	−	0.305200i	$-0.0987233\pi$
$752$		0			0
$753$	−6.28802			−0.229148
$754$		0			0
$755$	−17.3250	+	46.6247i	−0.630521	+	1.69685i
$756$		0			0
$757$		−	22.2049i		−	0.807050i	−0.914969	−	0.403525i	$-0.867785\pi$
$757$		−	22.2049i		−	0.807050i	0.914969	−	0.403525i	$-0.132215\pi$
$758$		0			0
$759$			1.36595i			0.0495809i
$760$		0			0
$761$	20.7812			0.753317			0.376658	−	0.926352i	$-0.377073\pi$
$761$	20.7812			0.753317			0.376658	+	0.926352i	$0.377073\pi$
$762$		0			0
$763$			2.71597i			0.0983248i
$764$		0			0
$765$	−24.2581	+	10.6575i	−0.877053	+	0.385321i
$766$		0			0
$767$			24.8080i			0.895766i
$768$		0			0
$769$	2.61151			0.0941734			0.0470867	−	0.998891i	$-0.485006\pi$
$769$	2.61151			0.0941734			0.0470867	+	0.998891i	$0.485006\pi$
$770$		0			0
$771$		−	5.37362i		−	0.193526i
$772$		0			0
$773$		−	8.03760i		−	0.289092i	−0.989498	−	0.144546i	$-0.953828\pi$
$773$		−	8.03760i		−	0.289092i	0.989498	−	0.144546i	$-0.0461722\pi$
$774$		0			0
$775$	−1.95066	+	2.26237i	−0.0700697	+	0.0812666i
$776$		0			0
$777$	1.26824			0.0454979
$778$		0			0
$779$			26.3288i			0.943328i
$780$		0			0
$781$	−37.4777			−1.34106
$782$		0			0
$783$		−	8.65462i		−	0.309291i
$784$		0			0
$785$	13.5855	+	5.04815i	0.484886	+	0.180176i
$786$		0			0
$787$	−51.3631			−1.83090			−0.915449	−	0.402435i	$-0.868164\pi$
$787$	−51.3631			−1.83090			−0.915449	+	0.402435i	$0.868164\pi$
$788$		0			0
$789$		−	2.93397i		−	0.104452i
$790$		0			0
$791$	−17.8811			−0.635778
$792$		0			0
$793$	−18.4499			−0.655175
$794$		0			0
$795$	−5.87991	−	2.18488i	−0.208539	−	0.0774898i
$796$		0			0
$797$		−	42.8333i		−	1.51723i	−0.651538	−	0.758616i	$-0.725876\pi$
$797$		−	42.8333i		−	1.51723i	0.651538	−	0.758616i	$-0.274124\pi$
$798$		0			0
$799$	−5.23152	+	5.07003i	−0.185078	+	0.179365i
$800$		0			0
$801$	−16.1011			−0.568906
$802$		0			0
$803$			31.2425i			1.10252i
$804$		0			0
$805$	0.908251	−	2.44427i	0.0320117	−	0.0861491i
$806$		0			0
$807$		−	1.25454i		−	0.0441620i
$808$		0			0
$809$			33.2788i			1.17002i	0.811026	+	0.585010i	$0.198910\pi$
$809$			33.2788i			1.17002i	−0.811026	+	0.585010i	$0.801090\pi$
$810$		0			0
$811$		−	19.1048i		−	0.670861i	−0.942065	−	0.335431i	$-0.891118\pi$
$811$		−	19.1048i		−	0.670861i	0.942065	−	0.335431i	$-0.108882\pi$
$812$		0			0
$813$	−5.58540			−0.195889
$814$		0			0
$815$	−0.527815	+	1.42045i	−0.0184886	+	0.0497561i
$816$		0			0
$817$			42.0560i			1.47135i
$818$		0			0
$819$		−	6.17042i		−	0.215612i
$820$		0			0
$821$			46.5550i			1.62478i	0.583113	+	0.812391i	$0.301835\pi$
$821$			46.5550i			1.62478i	−0.583113	+	0.812391i	$0.698165\pi$
$822$		0			0
$823$	−30.2959			−1.05605			−0.528024	−	0.849229i	$-0.677067\pi$
$823$	−30.2959			−1.05605			−0.528024	+	0.849229i	$0.677067\pi$
$824$		0			0
$825$	3.82448	−	4.43563i	0.133151	−	0.154429i
$826$		0			0
$827$	52.1956			1.81502			0.907510	−	0.420030i	$-0.137980\pi$
$827$	52.1956			1.81502			0.907510	+	0.420030i	$0.137980\pi$
$828$		0			0
$829$	−35.4620			−1.23165			−0.615824	−	0.787884i	$-0.711177\pi$
$829$	−35.4620			−1.23165			−0.615824	+	0.787884i	$0.711177\pi$
$830$		0			0
$831$	−5.84323			−0.202699
$832$		0			0
$833$	2.86942	+	2.96081i	0.0994195	+	0.102586i
$834$		0			0
$835$	5.92469	−	15.9444i	0.205032	−	0.551779i
$836$		0			0
$837$			1.24622i			0.0430755i
$838$		0			0
$839$		−	44.6333i		−	1.54091i	−0.637493	−	0.770456i	$-0.720029\pi$
$839$		−	44.6333i		−	1.54091i	0.637493	−	0.770456i	$-0.279971\pi$
$840$		0			0
$841$	11.7854			0.406392
$842$		0			0
$843$	3.54240			0.122007
$844$		0			0
$845$	6.53469	−	17.5860i	0.224800	−	0.604978i
$846$		0			0
$847$	0.120046			0.00412482
$848$		0			0
$849$	−9.86918			−0.338709
$850$		0			0
$851$	4.16465			0.142762
$852$		0			0
$853$	−44.6089			−1.52738			−0.763689	−	0.645584i	$-0.776614\pi$
$853$	−44.6089			−1.52738			−0.763689	+	0.645584i	$0.776614\pi$
$854$		0			0
$855$	−10.7651	+	28.9709i	−0.368160	+	0.990785i
$856$		0			0
$857$	39.0389			1.33354			0.666772	−	0.745262i	$-0.267676\pi$
$857$	39.0389			1.33354			0.666772	+	0.745262i	$0.267676\pi$
$858$		0			0
$859$	9.24166			0.315321			0.157661	−	0.987493i	$-0.449605\pi$
$859$	9.24166			0.315321			0.157661	+	0.987493i	$0.449605\pi$
$860$		0			0
$861$			1.94407i			0.0662537i
$862$		0			0
$863$		−	21.2056i		−	0.721848i	−0.932595	−	0.360924i	$-0.882461\pi$
$863$		−	21.2056i		−	0.721848i	0.932595	−	0.360924i	$-0.117539\pi$
$864$		0			0
$865$	14.6588	−	39.4495i	0.498415	−	1.34132i
$866$		0			0
$867$	−0.189224	+	6.03404i	−0.00642637	+	0.204927i
$868$		0			0
$869$	21.3458			0.724105
$870$		0			0
$871$	29.5336			1.00071
$872$		0			0
$873$	38.8727			1.31564
$874$		0			0
$875$	−9.79297	+	5.39424i	−0.331063	+	0.182359i
$876$		0			0
$877$	11.6267			0.392605			0.196302	−	0.980543i	$-0.437107\pi$
$877$	11.6267			0.392605			0.196302	+	0.980543i	$0.437107\pi$
$878$		0			0
$879$			0.0619341i			0.00208898i
$880$		0			0
$881$			26.2424i			0.884128i	0.896983	+	0.442064i	$0.145754\pi$
$881$			26.2424i			0.884128i	−0.896983	+	0.442064i	$0.854246\pi$
$882$		0			0
$883$		−	53.1453i		−	1.78848i	−0.447587	−	0.894240i	$-0.647717\pi$
$883$		−	53.1453i		−	1.78848i	0.447587	−	0.894240i	$-0.352283\pi$
$884$		0			0
$885$	3.19578	−	8.60043i	0.107425	−	0.289100i
$886$		0			0
$887$	−27.7176			−0.930666			−0.465333	−	0.885136i	$-0.654065\pi$
$887$	−27.7176			−0.930666			−0.465333	+	0.885136i	$0.654065\pi$
$888$		0			0
$889$		−	0.00262702i		−	8.81076e-5i
$890$		0			0
$891$			25.9950i			0.870867i
$892$		0			0
$893$			8.49785i			0.284370i
$894$		0			0
$895$	8.05726	−	21.6835i	0.269325	−	0.724801i
$896$		0			0
$897$			0.889133i			0.0296873i
$898$		0			0
$899$	2.47881			0.0826730
$900$		0			0
$901$	23.3889	−	22.6669i	0.779197	−	0.755145i
$902$		0			0
$903$			3.10533i			0.103339i
$904$		0			0
$905$	50.6868	+	18.8344i	1.68489	+	0.626077i
$906$		0			0
$907$	−19.8105			−0.657797			−0.328899	−	0.944365i	$-0.606677\pi$
$907$	−19.8105			−0.657797			−0.328899	+	0.944365i	$0.606677\pi$
$908$		0			0
$909$	−24.4602			−0.811292
$910$		0			0
$911$			10.7641i			0.356631i	0.983973	+	0.178316i	$0.0570648\pi$
$911$			10.7641i			0.356631i	−0.983973	+	0.178316i	$0.942935\pi$
$912$		0			0
$913$	39.1769			1.29656
$914$		0			0
$915$	6.39620	+	2.37673i	0.211452	+	0.0785722i
$916$		0			0
$917$		−	0.261389i		−	0.00863181i
$918$		0			0
$919$	28.2712			0.932582			0.466291	−	0.884631i	$-0.345590\pi$
$919$	28.2712			0.932582			0.466291	+	0.884631i	$0.345590\pi$
$920$		0			0
$921$			2.67297i			0.0880775i
$922$		0			0
$923$	−24.3952			−0.802977
$924$		0			0
$925$	−13.5238	−	11.6605i	−0.444659	−	0.383393i
$926$		0			0
$927$		−	0.0941738i		−	0.00309307i
$928$		0			0
$929$			36.7411i			1.20544i	0.797955	+	0.602718i	$0.205915\pi$
$929$			36.7411i			1.20544i	−0.797955	+	0.602718i	$0.794085\pi$
$930$		0			0
$931$	4.80942			0.157622
$932$		0			0
$933$			0.914221i			0.0299302i
$934$		0			0
$935$	12.2320	+	27.8420i	0.400029	+	0.910529i
$936$		0			0
$937$		−	11.1941i		−	0.365695i	−0.983141	−	0.182848i	$-0.941468\pi$
$937$		−	11.1941i		−	0.365695i	0.983141	−	0.182848i	$-0.0585315\pi$
$938$		0			0
$939$	2.38684			0.0778917
$940$		0			0
$941$			16.0677i			0.523793i	0.965096	+	0.261897i	$0.0843480\pi$
$941$			16.0677i			0.523793i	−0.965096	+	0.261897i	$0.915652\pi$
$942$		0			0
$943$			6.38393i			0.207889i
$944$		0			0
$945$	−1.62463	+	4.37218i	−0.0528494	+	0.142227i
$946$		0			0
$947$	19.2299			0.624887			0.312443	−	0.949936i	$-0.398853\pi$
$947$	19.2299			0.624887			0.312443	+	0.949936i	$0.398853\pi$
$948$		0			0
$949$			20.3365i			0.660151i
$950$		0			0
$951$	1.23738			0.0401249
$952$		0			0
$953$			29.9519i			0.970239i	0.874448	+	0.485119i	$0.161224\pi$
$953$			29.9519i			0.970239i	−0.874448	+	0.485119i	$0.838776\pi$
$954$		0			0
$955$	−0.123333	+	0.331912i	−0.00399096	+	0.0107404i
$956$		0			0
$957$	−4.85999			−0.157101
$958$		0			0
$959$		−	2.51963i		−	0.0813631i
$960$		0			0
$961$	30.6431			0.988486
$962$		0			0
$963$	−20.4332			−0.658450
$964$		0			0
$965$	16.3718	−	44.0593i	0.527025	−	1.41832i
$966$		0			0
$967$		−	38.6834i		−	1.24397i	−0.783027	−	0.621987i	$-0.786325\pi$
$967$		−	38.6834i		−	1.24397i	0.783027	−	0.621987i	$-0.213675\pi$
$968$		0			0
$969$	4.90072	+	5.05681i	0.157434	+	0.162448i
$970$		0			0
$971$	−19.4526			−0.624264			−0.312132	−	0.950039i	$-0.601043\pi$
$971$	−19.4526			−0.624264			−0.312132	+	0.950039i	$0.601043\pi$
$972$		0			0
$973$		−	7.24994i		−	0.232422i
$974$		0			0
$975$	2.48945	−	2.88726i	0.0797263	−	0.0924664i
$976$		0			0
$977$		−	18.7338i		−	0.599348i	−0.954042	−	0.299674i	$-0.903122\pi$
$977$		−	18.7338i		−	0.599348i	0.954042	−	0.299674i	$-0.0968779\pi$
$978$		0			0
$979$			18.4799i			0.590620i
$980$		0			0
$981$		−	7.80541i		−	0.249208i
$982$		0			0
$983$	−39.3544			−1.25521			−0.627606	−	0.778531i	$-0.715965\pi$
$983$	−39.3544			−1.25521			−0.627606	+	0.778531i	$0.715965\pi$
$984$		0			0
$985$	4.73191	−	12.7344i	0.150771	−	0.405752i
$986$		0			0
$987$			0.627465i			0.0199724i
$988$		0			0
$989$			10.1973i			0.324255i
$990$		0			0
$991$		−	17.6529i		−	0.560763i	−0.959889	−	0.280382i	$-0.909539\pi$
$991$		−	17.6529i		−	0.560763i	0.959889	−	0.280382i	$-0.0904610\pi$
$992$		0			0
$993$	−10.2538			−0.325393
$994$		0			0
$995$	40.3922	+	15.0091i	1.28052	+	0.475821i
$996$		0			0
$997$	0.372742			0.0118049			0.00590243	−	0.999983i	$-0.498121\pi$
$997$	0.372742			0.0118049			0.00590243	+	0.999983i	$0.498121\pi$
$998$		0			0
$999$	−7.44951			−0.235692

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2380.2.m.d.169.13	yes	26
5.4	even	2		2380.2.m.c.169.13	✓	26
17.16	even	2		2380.2.m.c.169.14	yes	26
85.84	even	2	inner	2380.2.m.d.169.14	yes	26

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2380.2.m.c.169.13	✓	26	5.4	even	2
2380.2.m.c.169.14	yes	26	17.16	even	2
2380.2.m.d.169.13	yes	26	1.1	even	1	trivial
2380.2.m.d.169.14	yes	26	85.84	even	2	inner

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:	\( N \)	\(=\)	\( 2380 = 2^{2} \cdot 5 \cdot 7 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2380.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.355118 q^{3} +(0.778855 - 2.09604i) q^{5} +1.00000 q^{7} -2.87389 q^{9} +3.29848i q^{11} +2.14706i q^{13} +(0.276586 - 0.744342i) q^{15} +(2.86942 + 2.96081i) q^{17} +4.80942 q^{19} +0.355118 q^{21} +1.16614 q^{23} +(-3.78677 - 3.26502i) q^{25} -2.08593 q^{27} +4.14905i q^{29} -0.597440i q^{31} +1.17135i q^{33} +(0.778855 - 2.09604i) q^{35} +3.57132 q^{37} +0.762461i q^{39} +5.47443i q^{41} +8.74450i q^{43} +(-2.23835 + 6.02379i) q^{45} +1.76692i q^{47} +1.00000 q^{49} +(1.01898 + 1.05144i) q^{51} -7.89948i q^{53} +(6.91374 + 2.56904i) q^{55} +1.70791 q^{57} +11.5544 q^{59} +8.59309i q^{61} -2.87389 q^{63} +(4.50033 + 1.67225i) q^{65} -13.7553i q^{67} +0.414116 q^{69} +11.3621i q^{71} +9.47179 q^{73} +(-1.34475 - 1.15947i) q^{75} +3.29848i q^{77} -6.47139i q^{79} +7.88092 q^{81} -11.8773i q^{83} +(8.44085 - 3.70838i) q^{85} +1.47340i q^{87} +5.60255 q^{89} +2.14706i q^{91} -0.212162i q^{93} +(3.74584 - 10.0807i) q^{95} -13.5262 q^{97} -9.47947i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 26 q + 6 q^{3} - 4 q^{5} + 26 q^{7} + 32 q^{9} + 10 q^{15} - 11 q^{17} + 6 q^{21} - 16 q^{23} + 4 q^{25} + 6 q^{27} - 4 q^{35} - 20 q^{37} - 18 q^{45} + 26 q^{49} + 9 q^{51} - 14 q^{55} + 32 q^{59} + 32 q^{63}+ \cdots - 46 q^{97}+O(q^{100}) \) 26 * q + 6 * q^3 - 4 * q^5 + 26 * q^7 + 32 * q^9 + 10 * q^15 - 11 * q^17 + 6 * q^21 - 16 * q^23 + 4 * q^25 + 6 * q^27 - 4 * q^35 - 20 * q^37 - 18 * q^45 + 26 * q^49 + 9 * q^51 - 14 * q^55 + 32 * q^59 + 32 * q^63 - 42 * q^65 - 8 * q^69 - 12 * q^73 + 8 * q^75 + 66 * q^81 + 11 * q^85 + 4 * q^89 + 20 * q^95 - 46 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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