LMFDB - Embedded newform 980.2.e.f.589.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [980,2,Mod(589,980)] 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("980.589"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(980, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 980 = 2^{2} \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	980.e (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,1,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	no
Analytic conductor:	$7.82533939809$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-19})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} - 5x + 25 $ x^4 - x^3 - 4x^2 - 5x + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 140)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			589.4
Root			$-1.63746 - 1.52274i$ of defining polynomial
Character	$\chi$	$=$	980.589
Dual form			980.2.e.f.589.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.73205i q^{3} +(2.13746 - 0.656712i) q^{5} +2.27492 q^{11} -6.09095i q^{13} +(1.13746 + 3.70219i) q^{15} -4.77753i q^{17} +4.27492 q^{19} +0.894797i q^{23} +(4.13746 - 2.80739i) q^{25} +5.19615i q^{27} +3.27492 q^{29} -4.27492 q^{31} +3.94027i q^{33} +5.61478i q^{37} +10.5498 q^{39} -11.2749 q^{41} -6.50958i q^{43} -2.15068i q^{47} +8.27492 q^{51} +7.40437i q^{53} +(4.86254 - 1.49397i) q^{55} +7.40437i q^{57} -4.27492 q^{59} +1.54983 q^{61} +(-4.00000 - 13.0192i) q^{65} +13.9140i q^{67} -1.54983 q^{69} +10.5498 q^{71} +2.15068i q^{73} +(4.86254 + 7.16629i) q^{75} +0.274917 q^{79} -9.00000 q^{81} +5.67232i q^{83} +(-3.13746 - 10.2118i) q^{85} +5.67232i q^{87} +7.00000 q^{89} -7.40437i q^{93} +(9.13746 - 2.80739i) q^{95} +6.92820i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{5} - 6 q^{11} - 3 q^{15} + 2 q^{19} + 9 q^{25} - 2 q^{29} - 2 q^{31} + 12 q^{39} - 30 q^{41} + 18 q^{51} + 27 q^{55} - 2 q^{59} - 24 q^{61} - 16 q^{65} + 24 q^{69} + 12 q^{71} + 27 q^{75} - 14 q^{79}+ \cdots + 29 q^{95}+O(q^{100}) $ 4 * q + q^5 - 6 * q^11 - 3 * q^15 + 2 * q^19 + 9 * q^25 - 2 * q^29 - 2 * q^31 + 12 * q^39 - 30 * q^41 + 18 * q^51 + 27 * q^55 - 2 * q^59 - 24 * q^61 - 16 * q^65 + 24 * q^69 + 12 * q^71 + 27 * q^75 - 14 * q^79 - 36 * q^81 - 5 * q^85 + 28 * q^89 + 29 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/980\mathbb{Z}\right)^\times$.

$n$	$101$	$197$	$491$
$\chi(n)$	$1$	$-1$	$1$

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
$3$			1.73205i			1.00000i	0.866025	+	0.500000i	$0.166667\pi$
$3$			1.73205i			1.00000i	−0.866025	+	0.500000i	$0.833333\pi$
$4$		0			0
$5$	2.13746	−	0.656712i	0.955901	−	0.293691i
$5$	2.13746	−	0.656712i	0.955901	−	0.293691i
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.27492			0.685913			0.342957	−	0.939351i	$-0.388572\pi$
$11$	2.27492			0.685913			0.342957	+	0.939351i	$0.388572\pi$
$12$		0			0
$13$		−	6.09095i		−	1.68933i	−0.535299	−	0.844663i	$-0.679801\pi$
$13$		−	6.09095i		−	1.68933i	0.535299	−	0.844663i	$-0.320199\pi$
$14$		0			0
$15$	1.13746	+	3.70219i	0.293691	+	0.955901i
$16$		0			0
$17$		−	4.77753i		−	1.15872i	−0.815072	−	0.579360i	$-0.803303\pi$
$17$		−	4.77753i		−	1.15872i	0.815072	−	0.579360i	$-0.196697\pi$
$18$		0			0
$19$	4.27492			0.980733			0.490367	−	0.871516i	$-0.336863\pi$
$19$	4.27492			0.980733			0.490367	+	0.871516i	$0.336863\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			0.894797i			0.186578i	0.995639	+	0.0932891i	$0.0297381\pi$
$23$			0.894797i			0.186578i	−0.995639	+	0.0932891i	$0.970262\pi$
$24$		0			0
$25$	4.13746	−	2.80739i	0.827492	−	0.561478i
$26$		0			0
$27$			5.19615i			1.00000i
$28$		0			0
$29$	3.27492			0.608137			0.304068	−	0.952650i	$-0.401655\pi$
$29$	3.27492			0.608137			0.304068	+	0.952650i	$0.401655\pi$
$30$		0			0
$31$	−4.27492			−0.767798			−0.383899	−	0.923375i	$-0.625419\pi$
$31$	−4.27492			−0.767798			−0.383899	+	0.923375i	$0.625419\pi$
$32$		0			0
$33$			3.94027i			0.685913i
$34$		0			0
$35$		0			0
$36$		0			0
$37$			5.61478i			0.923064i	0.887124	+	0.461532i	$0.152700\pi$
$37$			5.61478i			0.923064i	−0.887124	+	0.461532i	$0.847300\pi$
$38$		0			0
$39$	10.5498			1.68933
$40$		0			0
$41$	−11.2749			−1.76085			−0.880423	−	0.474189i	$-0.842741\pi$
$41$	−11.2749			−1.76085			−0.880423	+	0.474189i	$0.842741\pi$
$42$		0			0
$43$		−	6.50958i		−	0.992701i	−0.868122	−	0.496351i	$-0.834673\pi$
$43$		−	6.50958i		−	0.992701i	0.868122	−	0.496351i	$-0.165327\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$		−	2.15068i		−	0.313709i	−0.987622	−	0.156854i	$-0.949865\pi$
$47$		−	2.15068i		−	0.313709i	0.987622	−	0.156854i	$-0.0501353\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	8.27492			1.15872
$52$		0			0
$53$			7.40437i			1.01707i	0.861042	+	0.508534i	$0.169813\pi$
$53$			7.40437i			1.01707i	−0.861042	+	0.508534i	$0.830187\pi$
$54$		0			0
$55$	4.86254	−	1.49397i	0.655665	−	0.201446i
$56$		0			0
$57$			7.40437i			0.980733i
$58$		0			0
$59$	−4.27492			−0.556547			−0.278273	−	0.960502i	$-0.589762\pi$
$59$	−4.27492			−0.556547			−0.278273	+	0.960502i	$0.589762\pi$
$60$		0			0
$61$	1.54983			0.198436			0.0992180	−	0.995066i	$-0.468366\pi$
$61$	1.54983			0.198436			0.0992180	+	0.995066i	$0.468366\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−4.00000	−	13.0192i	−0.496139	−	1.61483i
$66$		0			0
$67$			13.9140i			1.69986i	0.526896	+	0.849930i	$0.323356\pi$
$67$			13.9140i			1.69986i	−0.526896	+	0.849930i	$0.676644\pi$
$68$		0			0
$69$	−1.54983			−0.186578
$70$		0			0
$71$	10.5498			1.25204			0.626018	−	0.779809i	$-0.284684\pi$
$71$	10.5498			1.25204			0.626018	+	0.779809i	$0.284684\pi$
$72$		0			0
$73$			2.15068i			0.251718i	0.992048	+	0.125859i	$0.0401687\pi$
$73$			2.15068i			0.251718i	−0.992048	+	0.125859i	$0.959831\pi$
$74$		0			0
$75$	4.86254	+	7.16629i	0.561478	+	0.827492i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	0.274917			0.0309306			0.0154653	−	0.999880i	$-0.495077\pi$
$79$	0.274917			0.0309306			0.0154653	+	0.999880i	$0.495077\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$		0			0
$83$			5.67232i			0.622618i	0.950309	+	0.311309i	$0.100767\pi$
$83$			5.67232i			0.622618i	−0.950309	+	0.311309i	$0.899233\pi$
$84$		0			0
$85$	−3.13746	−	10.2118i	−0.340305	−	1.10762i
$86$		0			0
$87$			5.67232i			0.608137i
$88$		0			0
$89$	7.00000			0.741999			0.370999	−	0.928633i	$-0.379015\pi$
$89$	7.00000			0.741999			0.370999	+	0.928633i	$0.379015\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		−	7.40437i		−	0.767798i
$94$		0			0
$95$	9.13746	−	2.80739i	0.937483	−	0.288032i
$96$		0			0
$97$			6.92820i			0.703452i	0.936103	+	0.351726i	$0.114405\pi$
$97$			6.92820i			0.703452i	−0.936103	+	0.351726i	$0.885595\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.54983			0.154214			0.0771071	−	0.997023i	$-0.475432\pi$
$101$	1.54983			0.154214			0.0771071	+	0.997023i	$0.475432\pi$
$102$		0			0
$103$			2.56930i			0.253161i	0.991956	+	0.126581i	$0.0404002\pi$
$103$			2.56930i			0.253161i	−0.991956	+	0.126581i	$0.959600\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			13.9140i			1.34511i	0.740046	+	0.672556i	$0.234804\pi$
$107$			13.9140i			1.34511i	−0.740046	+	0.672556i	$0.765196\pi$
$108$		0			0
$109$	−3.54983			−0.340012			−0.170006	−	0.985443i	$-0.554379\pi$
$109$	−3.54983			−0.340012			−0.170006	+	0.985443i	$0.554379\pi$
$110$		0			0
$111$	−9.72508			−0.923064
$112$		0			0
$113$			13.0192i			1.22474i	0.790572	+	0.612369i	$0.209783\pi$
$113$			13.0192i			1.22474i	−0.790572	+	0.612369i	$0.790217\pi$
$114$		0			0
$115$	0.587624	+	1.91259i	0.0547962	+	0.178350i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−5.82475			−0.529523
$122$		0			0
$123$		−	19.5287i		−	1.76085i
$124$		0			0
$125$	7.00000	−	8.71780i	0.626099	−	0.779744i
$126$		0			0
$127$		−	1.78959i		−	0.158801i	−0.996843	−	0.0794004i	$-0.974699\pi$
$127$		−	1.78959i		−	0.158801i	0.996843	−	0.0794004i	$-0.0253006\pi$
$128$		0			0
$129$	11.2749			0.992701
$130$		0			0
$131$	−18.2749			−1.59669			−0.798343	−	0.602202i	$-0.794290\pi$
$131$	−18.2749			−1.59669			−0.798343	+	0.602202i	$0.794290\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	3.41238	+	11.1066i	0.293691	+	0.955901i
$136$		0			0
$137$		−	9.19397i		−	0.785494i	−0.919647	−	0.392747i	$-0.871525\pi$
$137$		−	9.19397i		−	0.785494i	0.919647	−	0.392747i	$-0.128475\pi$
$138$		0			0
$139$	17.0997			1.45037			0.725187	−	0.688551i	$-0.241753\pi$
$139$	17.0997			1.45037			0.725187	+	0.688551i	$0.241753\pi$
$140$		0			0
$141$	3.72508			0.313709
$142$		0			0
$143$		−	13.8564i		−	1.15873i
$144$		0			0
$145$	7.00000	−	2.15068i	0.581318	−	0.178604i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−7.54983			−0.618507			−0.309253	−	0.950980i	$-0.600079\pi$
$149$	−7.54983			−0.618507			−0.309253	+	0.950980i	$0.600079\pi$
$150$		0			0
$151$	−20.2749			−1.64995			−0.824975	−	0.565170i	$-0.808811\pi$
$151$	−20.2749			−1.64995			−0.824975	+	0.565170i	$0.808811\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−9.13746	+	2.80739i	−0.733938	+	0.225495i
$156$		0			0
$157$		−	10.8685i		−	0.867399i	−0.901058	−	0.433699i	$-0.857208\pi$
$157$		−	10.8685i		−	0.867399i	0.901058	−	0.433699i	$-0.142792\pi$
$158$		0			0
$159$	−12.8248			−1.01707
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	7.40437i		−	0.579955i	−0.957033	−	0.289978i	$-0.906352\pi$
$163$		−	7.40437i		−	0.579955i	0.957033	−	0.289978i	$-0.0936479\pi$
$164$		0			0
$165$	2.58762	+	8.42217i	0.201446	+	0.655665i
$166$		0			0
$167$		−	12.6005i		−	0.975058i	−0.873107	−	0.487529i	$-0.837898\pi$
$167$		−	12.6005i		−	0.975058i	0.873107	−	0.487529i	$-0.162102\pi$
$168$		0			0
$169$	−24.0997			−1.85382
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	18.7490i		−	1.42546i	−0.701438	−	0.712731i	$-0.747458\pi$
$173$		−	18.7490i		−	1.42546i	0.701438	−	0.712731i	$-0.252542\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	7.40437i		−	0.556547i
$178$		0			0
$179$	−0.274917			−0.0205483			−0.0102741	−	0.999947i	$-0.503270\pi$
$179$	−0.274917			−0.0205483			−0.0102741	+	0.999947i	$0.503270\pi$
$180$		0			0
$181$	−16.7251			−1.24317			−0.621583	−	0.783348i	$-0.713510\pi$
$181$	−16.7251			−1.24317			−0.621583	+	0.783348i	$0.713510\pi$
$182$		0			0
$183$			2.68439i			0.198436i
$184$		0			0
$185$	3.68729	+	12.0014i	0.271095	+	0.882357i
$186$		0			0
$187$		−	10.8685i		−	0.794782i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	22.8248			1.65154			0.825771	−	0.564006i	$-0.190741\pi$
$191$	22.8248			1.65154			0.825771	+	0.564006i	$0.190741\pi$
$192$		0			0
$193$		−	9.19397i		−	0.661796i	−0.943666	−	0.330898i	$-0.892648\pi$
$193$		−	9.19397i		−	0.661796i	0.943666	−	0.330898i	$-0.107352\pi$
$194$		0			0
$195$	22.5498	−	6.92820i	1.61483	−	0.496139i
$196$		0			0
$197$		−	26.0383i		−	1.85515i	−0.373634	−	0.927576i	$-0.621888\pi$
$197$		−	26.0383i		−	1.85515i	0.373634	−	0.927576i	$-0.378112\pi$
$198$		0			0
$199$	−9.72508			−0.689393			−0.344696	−	0.938714i	$-0.612018\pi$
$199$	−9.72508			−0.689393			−0.344696	+	0.938714i	$0.612018\pi$
$200$		0			0
$201$	−24.0997			−1.69986
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−24.0997	+	7.40437i	−1.68319	+	0.517144i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	9.72508			0.672698
$210$		0			0
$211$	−19.6495			−1.35273			−0.676364	−	0.736568i	$-0.736445\pi$
$211$	−19.6495			−1.35273			−0.676364	+	0.736568i	$0.736445\pi$
$212$		0			0
$213$			18.2728i			1.25204i
$214$		0			0
$215$	−4.27492	−	13.9140i	−0.291547	−	0.948924i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−3.72508			−0.251718
$220$		0			0
$221$	−29.0997			−1.95746
$222$		0			0
$223$			8.71780i			0.583787i	0.956451	+	0.291893i	$0.0942853\pi$
$223$			8.71780i			0.583787i	−0.956451	+	0.291893i	$0.905715\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			6.45203i			0.428236i	0.976808	+	0.214118i	$0.0686878\pi$
$227$			6.45203i			0.428236i	−0.976808	+	0.214118i	$0.931312\pi$
$228$		0			0
$229$	4.27492			0.282494			0.141247	−	0.989974i	$-0.454889\pi$
$229$	4.27492			0.282494			0.141247	+	0.989974i	$0.454889\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			18.6339i			1.22075i	0.792113	+	0.610375i	$0.208981\pi$
$233$			18.6339i			1.22075i	−0.792113	+	0.610375i	$0.791019\pi$
$234$		0			0
$235$	−1.41238	−	4.59698i	−0.0921332	−	0.299874i
$236$		0			0
$237$			0.476171i			0.0309306i
$238$		0			0
$239$	14.5498			0.941151			0.470575	−	0.882360i	$-0.344046\pi$
$239$	14.5498			0.941151			0.470575	+	0.882360i	$0.344046\pi$
$240$		0			0
$241$	−12.8248			−0.826115			−0.413057	−	0.910705i	$-0.635539\pi$
$241$	−12.8248			−0.826115			−0.413057	+	0.910705i	$0.635539\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$		−	26.0383i		−	1.65678i
$248$		0			0
$249$	−9.82475			−0.622618
$250$		0			0
$251$	−5.45017			−0.344011			−0.172006	−	0.985096i	$-0.555025\pi$
$251$	−5.45017			−0.344011			−0.172006	+	0.985096i	$0.555025\pi$
$252$		0			0
$253$			2.03559i			0.127976i
$254$		0			0
$255$	17.6873	−	5.43424i	1.10762	−	0.340305i
$256$		0			0
$257$			24.7249i			1.54230i	0.636656	+	0.771148i	$0.280317\pi$
$257$			24.7249i			1.54230i	−0.636656	+	0.771148i	$0.719683\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$			26.9331i			1.66077i	0.557193	+	0.830383i	$0.311878\pi$
$263$			26.9331i			1.66077i	−0.557193	+	0.830383i	$0.688122\pi$
$264$		0			0
$265$	4.86254	+	15.8265i	0.298704	+	0.972217i
$266$		0			0
$267$			12.1244i			0.741999i
$268$		0			0
$269$	29.5498			1.80169			0.900843	−	0.434146i	$-0.142950\pi$
$269$	29.5498			1.80169			0.900843	+	0.434146i	$0.142950\pi$
$270$		0			0
$271$	−12.8248			−0.779048			−0.389524	−	0.921016i	$-0.627361\pi$
$271$	−12.8248			−0.779048			−0.389524	+	0.921016i	$0.627361\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	9.41238	−	6.38658i	0.567588	−	0.385125i
$276$		0			0
$277$		−	18.6339i		−	1.11960i	−0.828626	−	0.559802i	$-0.810877\pi$
$277$		−	18.6339i		−	1.11960i	0.828626	−	0.559802i	$-0.189123\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	6.00000			0.357930			0.178965	−	0.983855i	$-0.442725\pi$
$281$	6.00000			0.357930			0.178965	+	0.983855i	$0.442725\pi$
$282$		0			0
$283$		−	19.5863i		−	1.16428i	−0.813087	−	0.582142i	$-0.802215\pi$
$283$		−	19.5863i		−	1.16428i	0.813087	−	0.582142i	$-0.197785\pi$
$284$		0			0
$285$	4.86254	+	15.8265i	0.288032	+	0.937483i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−5.82475			−0.342632
$290$		0			0
$291$	−12.0000			−0.703452
$292$		0			0
$293$			6.92820i			0.404750i	0.979308	+	0.202375i	$0.0648660\pi$
$293$			6.92820i			0.404750i	−0.979308	+	0.202375i	$0.935134\pi$
$294$		0			0
$295$	−9.13746	+	2.80739i	−0.532003	+	0.163453i
$296$		0			0
$297$			11.8208i			0.685913i
$298$		0			0
$299$	5.45017			0.315191
$300$		0			0
$301$		0			0
$302$		0			0
$303$			2.68439i			0.154214i
$304$		0			0
$305$	3.31271	−	1.01779i	0.189685	−	0.0582788i
$306$		0			0
$307$			3.99782i			0.228167i	0.993471	+	0.114084i	$0.0363932\pi$
$307$			3.99782i			0.228167i	−0.993471	+	0.114084i	$0.963607\pi$
$308$		0			0
$309$	−4.45017			−0.253161
$310$		0			0
$311$	12.8248			0.727225			0.363612	−	0.931550i	$-0.381543\pi$
$311$	12.8248			0.727225			0.363612	+	0.931550i	$0.381543\pi$
$312$		0			0
$313$			14.4477i			0.816630i	0.912841	+	0.408315i	$0.133884\pi$
$313$			14.4477i			0.816630i	−0.912841	+	0.408315i	$0.866116\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$		−	3.82518i		−	0.214844i	−0.994214	−	0.107422i	$-0.965740\pi$
$317$		−	3.82518i		−	0.214844i	0.994214	−	0.107422i	$-0.0342596\pi$
$318$		0			0
$319$	7.45017			0.417129
$320$		0			0
$321$	−24.0997			−1.34511
$322$		0			0
$323$		−	20.4235i		−	1.13640i
$324$		0			0
$325$	−17.0997	−	25.2011i	−0.948519	−	1.39790i
$326$		0			0
$327$		−	6.14849i		−	0.340012i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−4.82475			−0.265192			−0.132596	−	0.991170i	$-0.542331\pi$
$331$	−4.82475			−0.265192			−0.132596	+	0.991170i	$0.542331\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	9.13746	+	29.7405i	0.499233	+	1.62490i
$336$		0			0
$337$			13.0192i			0.709198i	0.935018	+	0.354599i	$0.115383\pi$
$337$			13.0192i			0.709198i	−0.935018	+	0.354599i	$0.884617\pi$
$338$		0			0
$339$	−22.5498			−1.22474
$340$		0			0
$341$	−9.72508			−0.526643
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−3.31271	+	1.01779i	−0.178350	+	0.0547962i
$346$		0			0
$347$		−	12.1244i		−	0.650870i	−0.945564	−	0.325435i	$-0.894489\pi$
$347$		−	12.1244i		−	0.650870i	0.945564	−	0.325435i	$-0.105511\pi$
$348$		0			0
$349$	11.2749			0.603532			0.301766	−	0.953382i	$-0.402424\pi$
$349$	11.2749			0.603532			0.301766	+	0.953382i	$0.402424\pi$
$350$		0			0
$351$	31.6495			1.68933
$352$		0			0
$353$			21.2608i			1.13160i	0.824543	+	0.565799i	$0.191432\pi$
$353$			21.2608i			1.13160i	−0.824543	+	0.565799i	$0.808568\pi$
$354$		0			0
$355$	22.5498	−	6.92820i	1.19682	−	0.367711i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	1.37459			0.0725479			0.0362739	−	0.999342i	$-0.488451\pi$
$359$	1.37459			0.0725479			0.0362739	+	0.999342i	$0.488451\pi$
$360$		0			0
$361$	−0.725083			−0.0381623
$362$		0			0
$363$		−	10.0888i		−	0.529523i
$364$		0			0
$365$	1.41238	+	4.59698i	0.0739271	+	0.240617i
$366$		0			0
$367$			14.7512i			0.770007i	0.922915	+	0.385003i	$0.125800\pi$
$367$			14.7512i			0.770007i	−0.922915	+	0.385003i	$0.874200\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$			5.61478i			0.290722i	0.989379	+	0.145361i	$0.0464344\pi$
$373$			5.61478i			0.290722i	−0.989379	+	0.145361i	$0.953566\pi$
$374$		0			0
$375$	15.0997	+	12.1244i	0.779744	+	0.626099i
$376$		0			0
$377$		−	19.9474i		−	1.02734i
$378$		0			0
$379$	−23.6495			−1.21479			−0.607397	−	0.794399i	$-0.707786\pi$
$379$	−23.6495			−1.21479			−0.607397	+	0.794399i	$0.707786\pi$
$380$		0			0
$381$	3.09967			0.158801
$382$		0			0
$383$			20.0049i			1.02220i	0.859520	+	0.511101i	$0.170762\pi$
$383$			20.0049i			1.02220i	−0.859520	+	0.511101i	$0.829238\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	5.37459			0.272502			0.136251	−	0.990674i	$-0.456495\pi$
$389$	5.37459			0.272502			0.136251	+	0.990674i	$0.456495\pi$
$390$		0			0
$391$	4.27492			0.216192
$392$		0			0
$393$		−	31.6531i		−	1.59669i
$394$		0			0
$395$	0.587624	−	0.180541i	0.0295666	−	0.00908403i
$396$		0			0
$397$		−	15.1698i		−	0.761352i	−0.924708	−	0.380676i	$-0.875691\pi$
$397$		−	15.1698i		−	0.761352i	0.924708	−	0.380676i	$-0.124309\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	3.00000			0.149813			0.0749064	−	0.997191i	$-0.476134\pi$
$401$	3.00000			0.149813			0.0749064	+	0.997191i	$0.476134\pi$
$402$		0			0
$403$			26.0383i			1.29706i
$404$		0			0
$405$	−19.2371	+	5.91041i	−0.955901	+	0.293691i
$406$		0			0
$407$			12.7732i			0.633142i
$408$		0			0
$409$	−10.0997			−0.499396			−0.249698	−	0.968324i	$-0.580331\pi$
$409$	−10.0997			−0.499396			−0.249698	+	0.968324i	$0.580331\pi$
$410$		0			0
$411$	15.9244			0.785494
$412$		0			0
$413$		0			0
$414$		0			0
$415$	3.72508	+	12.1244i	0.182857	+	0.595161i
$416$		0			0
$417$			29.6175i			1.45037i
$418$		0			0
$419$	−17.0997			−0.835373			−0.417687	−	0.908591i	$-0.637159\pi$
$419$	−17.0997			−0.835373			−0.417687	+	0.908591i	$0.637159\pi$
$420$		0			0
$421$	3.27492			0.159610			0.0798048	−	0.996811i	$-0.474570\pi$
$421$	3.27492			0.159610			0.0798048	+	0.996811i	$0.474570\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−13.4124	−	19.7668i	−0.650596	−	0.958831i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	24.0000			1.15873
$430$		0			0
$431$	19.3746			0.933241			0.466620	−	0.884458i	$-0.345471\pi$
$431$	19.3746			0.933241			0.466620	+	0.884458i	$0.345471\pi$
$432$		0			0
$433$		−	26.8756i		−	1.29156i	−0.763525	−	0.645778i	$-0.776533\pi$
$433$		−	26.8756i		−	1.29156i	0.763525	−	0.645778i	$-0.223467\pi$
$434$		0			0
$435$	3.72508	+	12.1244i	0.178604	+	0.581318i
$436$		0			0
$437$			3.82518i			0.182983i
$438$		0			0
$439$	−1.17525			−0.0560915			−0.0280458	−	0.999607i	$-0.508928\pi$
$439$	−1.17525			−0.0560915			−0.0280458	+	0.999607i	$0.508928\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	12.1244i		−	0.576046i	−0.957624	−	0.288023i	$-0.907002\pi$
$443$		−	12.1244i		−	0.576046i	0.957624	−	0.288023i	$-0.0929979\pi$
$444$		0			0
$445$	14.9622	−	4.59698i	0.709277	−	0.217918i
$446$		0			0
$447$		−	13.0767i		−	0.618507i
$448$		0			0
$449$	25.8248			1.21875			0.609373	−	0.792884i	$-0.291421\pi$
$449$	25.8248			1.21875			0.609373	+	0.792884i	$0.291421\pi$
$450$		0			0
$451$	−25.6495			−1.20779
$452$		0			0
$453$		−	35.1172i		−	1.64995i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	20.4235i		−	0.955372i	−0.878531	−	0.477686i	$-0.841476\pi$
$457$		−	20.4235i		−	0.955372i	0.878531	−	0.477686i	$-0.158524\pi$
$458$		0			0
$459$	24.8248			1.15872
$460$		0			0
$461$	−14.0000			−0.652045			−0.326023	−	0.945362i	$-0.605709\pi$
$461$	−14.0000			−0.652045			−0.326023	+	0.945362i	$0.605709\pi$
$462$		0			0
$463$			6.50958i			0.302526i	0.988494	+	0.151263i	$0.0483340\pi$
$463$			6.50958i			0.302526i	−0.988494	+	0.151263i	$0.951666\pi$
$464$		0			0
$465$	−4.86254	−	15.8265i	−0.225495	−	0.733938i
$466$		0			0
$467$			19.1676i			0.886973i	0.896281	+	0.443486i	$0.146259\pi$
$467$			19.1676i			0.886973i	−0.896281	+	0.443486i	$0.853741\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	18.8248			0.867399
$472$		0			0
$473$		−	14.8087i		−	0.680907i
$474$		0			0
$475$	17.6873	−	12.0014i	0.811549	−	0.550660i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	12.8248			0.585978			0.292989	−	0.956116i	$-0.405350\pi$
$479$	12.8248			0.585978			0.292989	+	0.956116i	$0.405350\pi$
$480$		0			0
$481$	34.1993			1.55936
$482$		0			0
$483$		0			0
$484$		0			0
$485$	4.54983	+	14.8087i	0.206597	+	0.672431i
$486$		0			0
$487$		−	33.4427i		−	1.51543i	−0.652584	−	0.757716i	$-0.726315\pi$
$487$		−	33.4427i		−	1.51543i	0.652584	−	0.757716i	$-0.273685\pi$
$488$		0			0
$489$	12.8248			0.579955
$490$		0			0
$491$	−13.4502			−0.606997			−0.303499	−	0.952832i	$-0.598155\pi$
$491$	−13.4502			−0.606997			−0.303499	+	0.952832i	$0.598155\pi$
$492$		0			0
$493$		−	15.6460i		−	0.704660i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−39.3746			−1.76265			−0.881324	−	0.472512i	$-0.843347\pi$
$499$	−39.3746			−1.76265			−0.881324	+	0.472512i	$0.843347\pi$
$500$		0			0
$501$	21.8248			0.975058
$502$		0			0
$503$		−	16.1797i		−	0.721418i	−0.932678	−	0.360709i	$-0.882535\pi$
$503$		−	16.1797i		−	0.721418i	0.932678	−	0.360709i	$-0.117465\pi$
$504$		0			0
$505$	3.31271	−	1.01779i	0.147414	−	0.0452913i
$506$		0			0
$507$		−	41.7419i		−	1.85382i
$508$		0			0
$509$	29.5498			1.30977			0.654887	−	0.755727i	$-0.272716\pi$
$509$	29.5498			1.30977			0.654887	+	0.755727i	$0.272716\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$			22.2131i			0.980733i
$514$		0			0
$515$	1.68729	+	5.49178i	0.0743510	+	0.241997i
$516$		0			0
$517$		−	4.89261i		−	0.215177i
$518$		0			0
$519$	32.4743			1.42546
$520$		0			0
$521$	−12.8248			−0.561863			−0.280931	−	0.959728i	$-0.590643\pi$
$521$	−12.8248			−0.561863			−0.280931	+	0.959728i	$0.590643\pi$
$522$		0			0
$523$			11.7057i			0.511856i	0.966696	+	0.255928i	$0.0823810\pi$
$523$			11.7057i			0.511856i	−0.966696	+	0.255928i	$0.917619\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			20.4235i			0.889663i
$528$		0			0
$529$	22.1993			0.965189
$530$		0			0
$531$		0			0
$532$		0			0
$533$			68.6750i			2.97464i
$534$		0			0
$535$	9.13746	+	29.7405i	0.395047	+	1.28579i
$536$		0			0
$537$		−	0.476171i		−	0.0205483i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−2.45017			−0.105341			−0.0526704	−	0.998612i	$-0.516773\pi$
$541$	−2.45017			−0.105341			−0.0526704	+	0.998612i	$0.516773\pi$
$542$		0			0
$543$		−	28.9687i		−	1.24317i
$544$		0			0
$545$	−7.58762	+	2.33122i	−0.325018	+	0.0998585i
$546$		0			0
$547$		−	36.1271i		−	1.54468i	−0.635208	−	0.772341i	$-0.719086\pi$
$547$		−	36.1271i		−	1.54468i	0.635208	−	0.772341i	$-0.280914\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	14.0000			0.596420
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−20.7870	+	6.38658i	−0.882357	+	0.271095i
$556$		0			0
$557$		−	5.61478i		−	0.237906i	−0.992900	−	0.118953i	$-0.962046\pi$
$557$		−	5.61478i		−	0.237906i	0.992900	−	0.118953i	$-0.0379538\pi$
$558$		0			0
$559$	−39.6495			−1.67700
$560$		0			0
$561$	18.8248			0.794782
$562$		0			0
$563$			12.2394i			0.515831i	0.966167	+	0.257916i	$0.0830356\pi$
$563$			12.2394i			0.515831i	−0.966167	+	0.257916i	$0.916964\pi$
$564$		0			0
$565$	8.54983	+	27.8279i	0.359694	+	1.17073i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	29.3746			1.23145			0.615723	−	0.787962i	$-0.288864\pi$
$569$	29.3746			1.23145			0.615723	+	0.787962i	$0.288864\pi$
$570$		0			0
$571$	−0.274917			−0.0115049			−0.00575246	−	0.999983i	$-0.501831\pi$
$571$	−0.274917			−0.0115049			−0.00575246	+	0.999983i	$0.501831\pi$
$572$		0			0
$573$			39.5336i			1.65154i
$574$		0			0
$575$	2.51204	+	3.70219i	0.104760	+	0.154392i
$576$		0			0
$577$			8.24163i			0.343103i	0.985175	+	0.171552i	$0.0548781\pi$
$577$			8.24163i			0.343103i	−0.985175	+	0.171552i	$0.945122\pi$
$578$		0			0
$579$	15.9244			0.661796
$580$		0			0
$581$		0			0
$582$		0			0
$583$			16.8443i			0.697621i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			13.9715i			0.576665i	0.957530	+	0.288333i	$0.0931009\pi$
$587$			13.9715i			0.576665i	−0.957530	+	0.288333i	$0.906899\pi$
$588$		0			0
$589$	−18.2749			−0.753005
$590$		0			0
$591$	45.0997			1.85515
$592$		0			0
$593$			20.3084i			0.833968i	0.908914	+	0.416984i	$0.136913\pi$
$593$			20.3084i			0.833968i	−0.908914	+	0.416984i	$0.863087\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		−	16.8443i		−	0.689393i
$598$		0			0
$599$	2.27492			0.0929506			0.0464753	−	0.998919i	$-0.485201\pi$
$599$	2.27492			0.0929506			0.0464753	+	0.998919i	$0.485201\pi$
$600$		0			0
$601$	14.0000			0.571072			0.285536	−	0.958368i	$-0.407828\pi$
$601$	14.0000			0.571072			0.285536	+	0.958368i	$0.407828\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−12.4502	+	3.82518i	−0.506171	+	0.155516i
$606$		0			0
$607$			32.1868i			1.30642i	0.757176	+	0.653211i	$0.226579\pi$
$607$			32.1868i			1.30642i	−0.757176	+	0.653211i	$0.773421\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−13.0997			−0.529956
$612$		0			0
$613$			37.0219i			1.49530i	0.664093	+	0.747650i	$0.268818\pi$
$613$			37.0219i			1.49530i	−0.664093	+	0.747650i	$0.731182\pi$
$614$		0			0
$615$	−12.8248	−	41.7419i	−0.517144	−	1.68319i
$616$		0			0
$617$		−	3.57919i		−	0.144093i	−0.997401	−	0.0720464i	$-0.977047\pi$
$617$		−	3.57919i		−	0.144093i	0.997401	−	0.0720464i	$-0.0229530\pi$
$618$		0			0
$619$	−43.9244			−1.76547			−0.882736	−	0.469870i	$-0.844301\pi$
$619$	−43.9244			−1.76547			−0.882736	+	0.469870i	$0.844301\pi$
$620$		0			0
$621$	−4.64950			−0.186578
$622$		0			0
$623$		0			0
$624$		0			0
$625$	9.23713	−	23.2309i	0.369485	−	0.929237i
$626$		0			0
$627$			16.8443i			0.672698i
$628$		0			0
$629$	26.8248			1.06957
$630$		0			0
$631$	2.90033			0.115460			0.0577302	−	0.998332i	$-0.481614\pi$
$631$	2.90033			0.115460			0.0577302	+	0.998332i	$0.481614\pi$
$632$		0			0
$633$		−	34.0339i		−	1.35273i
$634$		0			0
$635$	−1.17525	−	3.82518i	−0.0466383	−	0.151798i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	28.0997			1.10987			0.554935	−	0.831894i	$-0.312743\pi$
$641$	28.0997			1.10987			0.554935	+	0.831894i	$0.312743\pi$
$642$		0			0
$643$			38.3353i			1.51180i	0.654689	+	0.755898i	$0.272800\pi$
$643$			38.3353i			1.51180i	−0.654689	+	0.755898i	$0.727200\pi$
$644$		0			0
$645$	24.0997	−	7.40437i	0.948924	−	0.291547i
$646$		0			0
$647$		−	0.779710i		−	0.0306535i	−0.999883	−	0.0153268i	$-0.995121\pi$
$647$		−	0.779710i		−	0.0306535i	0.999883	−	0.0153268i	$-0.00487885\pi$
$648$		0			0
$649$	−9.72508			−0.381743
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	37.0219i		−	1.44878i	−0.689392	−	0.724389i	$-0.742122\pi$
$653$		−	37.0219i		−	1.44878i	0.689392	−	0.724389i	$-0.257878\pi$
$654$		0			0
$655$	−39.0619	+	12.0014i	−1.52627	+	0.468932i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	25.4502			0.991398			0.495699	−	0.868494i	$-0.334912\pi$
$659$	25.4502			0.991398			0.495699	+	0.868494i	$0.334912\pi$
$660$		0			0
$661$	−15.5498			−0.604818			−0.302409	−	0.953178i	$-0.597791\pi$
$661$	−15.5498			−0.604818			−0.302409	+	0.953178i	$0.597791\pi$
$662$		0			0
$663$		−	50.4021i		−	1.95746i
$664$		0			0
$665$		0			0
$666$		0			0
$667$			2.93039i			0.113465i
$668$		0			0
$669$	−15.0997			−0.583787
$670$		0			0
$671$	3.52575			0.136110
$672$		0			0
$673$			3.57919i			0.137968i	0.997618	+	0.0689838i	$0.0219757\pi$
$673$			3.57919i			0.137968i	−0.997618	+	0.0689838i	$0.978024\pi$
$674$		0			0
$675$	14.5876	+	21.4989i	0.561478	+	0.827492i
$676$		0			0
$677$		−	24.6098i		−	0.945831i	−0.881108	−	0.472916i	$-0.843202\pi$
$677$		−	24.6098i		−	0.945831i	0.881108	−	0.472916i	$-0.156798\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−11.1752			−0.428236
$682$		0			0
$683$		−	15.7035i		−	0.600879i	−0.953801	−	0.300440i	$-0.902867\pi$
$683$		−	15.7035i		−	0.600879i	0.953801	−	0.300440i	$-0.0971334\pi$
$684$		0			0
$685$	−6.03779	−	19.6517i	−0.230692	−	0.750854i
$686$		0			0
$687$			7.40437i			0.282494i
$688$		0			0
$689$	45.0997			1.71816
$690$		0			0
$691$	−7.37459			−0.280542			−0.140271	−	0.990113i	$-0.544797\pi$
$691$	−7.37459			−0.280542			−0.140271	+	0.990113i	$0.544797\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	36.5498	−	11.2296i	1.38641	−	0.425961i
$696$		0			0
$697$			53.8662i			2.04033i
$698$		0			0
$699$	−32.2749			−1.22075
$700$		0			0
$701$	−13.8248			−0.522154			−0.261077	−	0.965318i	$-0.584078\pi$
$701$	−13.8248			−0.522154			−0.261077	+	0.965318i	$0.584078\pi$
$702$		0			0
$703$			24.0027i			0.905280i
$704$		0			0
$705$	7.96221	−	2.44631i	0.299874	−	0.0921332i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	25.5498			0.959544			0.479772	−	0.877393i	$-0.340719\pi$
$709$	25.5498			0.959544			0.479772	+	0.877393i	$0.340719\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	3.82518i		−	0.143254i
$714$		0			0
$715$	−9.09967	−	29.6175i	−0.340308	−	1.10763i
$716$		0			0
$717$			25.2011i			0.941151i
$718$		0			0
$719$	−7.37459			−0.275026			−0.137513	−	0.990500i	$-0.543911\pi$
$719$	−7.37459			−0.275026			−0.137513	+	0.990500i	$0.543911\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		−	22.2131i		−	0.826115i
$724$		0			0
$725$	13.5498	−	9.19397i	0.503228	−	0.341455i
$726$		0			0
$727$			18.6915i			0.693228i	0.938008	+	0.346614i	$0.112669\pi$
$727$			18.6915i			0.693228i	−0.938008	+	0.346614i	$0.887331\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$	−31.0997			−1.15026
$732$		0			0
$733$			33.3276i			1.23098i	0.788144	+	0.615491i	$0.211042\pi$
$733$			33.3276i			1.23098i	−0.788144	+	0.615491i	$0.788958\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			31.6531i			1.16596i
$738$		0			0
$739$	−31.9244			−1.17436			−0.587179	−	0.809457i	$-0.699762\pi$
$739$	−31.9244			−1.17436			−0.587179	+	0.809457i	$0.699762\pi$
$740$		0			0
$741$	45.0997			1.65678
$742$		0			0
$743$		−	19.5287i		−	0.716440i	−0.933637	−	0.358220i	$-0.883384\pi$
$743$		−	19.5287i		−	0.716440i	0.933637	−	0.358220i	$-0.116616\pi$
$744$		0			0
$745$	−16.1375	+	4.95807i	−0.591231	+	0.181650i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−22.2749			−0.812823			−0.406412	−	0.913690i	$-0.633220\pi$
$751$	−22.2749			−0.812823			−0.406412	+	0.913690i	$0.633220\pi$
$752$		0			0
$753$		−	9.43996i		−	0.344011i
$754$		0			0
$755$	−43.3368	+	13.3148i	−1.57719	+	0.484575i
$756$		0			0
$757$		−	9.43996i		−	0.343101i	−0.985175	−	0.171551i	$-0.945122\pi$
$757$		−	9.43996i		−	0.343101i	0.985175	−	0.171551i	$-0.0548777\pi$
$758$		0			0
$759$	−3.52575			−0.127976
$760$		0			0
$761$	−29.9244			−1.08476			−0.542380	−	0.840133i	$-0.682477\pi$
$761$	−29.9244			−1.08476			−0.542380	+	0.840133i	$0.682477\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$			26.0383i			0.940189i
$768$		0			0
$769$	14.0000			0.504853			0.252426	−	0.967616i	$-0.418771\pi$
$769$	14.0000			0.504853			0.252426	+	0.967616i	$0.418771\pi$
$770$		0			0
$771$	−42.8248			−1.54230
$772$		0			0
$773$		−	27.2366i		−	0.979634i	−0.871825	−	0.489817i	$-0.837064\pi$
$773$		−	27.2366i		−	0.979634i	0.871825	−	0.489817i	$-0.162936\pi$
$774$		0			0
$775$	−17.6873	+	12.0014i	−0.635346	+	0.431102i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−48.1993			−1.72692
$780$		0			0
$781$	24.0000			0.858788
$782$		0			0
$783$			17.0170i			0.608137i
$784$		0			0
$785$	−7.13746	−	23.2309i	−0.254747	−	0.829147i
$786$		0			0
$787$			1.73205i			0.0617409i	0.999523	+	0.0308705i	$0.00982794\pi$
$787$			1.73205i			0.0617409i	−0.999523	+	0.0308705i	$0.990172\pi$
$788$		0			0
$789$	−46.6495			−1.66077
$790$		0			0
$791$		0			0
$792$		0			0
$793$		−	9.43996i		−	0.335223i
$794$		0			0
$795$	−27.4124	+	8.42217i	−0.972217	+	0.298704i
$796$		0			0
$797$			29.3873i			1.04095i	0.853876	+	0.520476i	$0.174246\pi$
$797$			29.3873i			1.04095i	−0.853876	+	0.520476i	$0.825754\pi$
$798$		0			0
$799$	−10.2749			−0.363500
$800$		0			0
$801$		0			0
$802$		0			0
$803$			4.89261i			0.172657i
$804$		0			0
$805$		0			0
$806$		0			0
$807$			51.1818i			1.80169i
$808$		0			0
$809$	−43.1993			−1.51881			−0.759404	−	0.650619i	$-0.774509\pi$
$809$	−43.1993			−1.51881			−0.759404	+	0.650619i	$0.774509\pi$
$810$		0			0
$811$	22.5498			0.791832			0.395916	−	0.918287i	$-0.370427\pi$
$811$	22.5498			0.791832			0.395916	+	0.918287i	$0.370427\pi$
$812$		0			0
$813$		−	22.2131i		−	0.779048i
$814$		0			0
$815$	−4.86254	−	15.8265i	−0.170327	−	0.554380i
$816$		0			0
$817$		−	27.8279i		−	0.973575i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	17.3746			0.606377			0.303189	−	0.952931i	$-0.401949\pi$
$821$	17.3746			0.606377			0.303189	+	0.952931i	$0.401949\pi$
$822$		0			0
$823$		−	32.3019i		−	1.12597i	−0.826466	−	0.562987i	$-0.809652\pi$
$823$		−	32.3019i		−	1.12597i	0.826466	−	0.562987i	$-0.190348\pi$
$824$		0			0
$825$	11.0619	+	16.3027i	0.385125	+	0.567588i
$826$		0			0
$827$			45.5670i			1.58452i	0.610183	+	0.792261i	$0.291096\pi$
$827$			45.5670i			1.58452i	−0.610183	+	0.792261i	$0.708904\pi$
$828$		0			0
$829$	−1.92442			−0.0668379			−0.0334189	−	0.999441i	$-0.510640\pi$
$829$	−1.92442			−0.0668379			−0.0334189	+	0.999441i	$0.510640\pi$
$830$		0			0
$831$	32.2749			1.11960
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−8.27492	−	26.9331i	−0.286365	−	0.932058i
$836$		0			0
$837$		−	22.2131i		−	0.767798i
$838$		0			0
$839$	−10.9003			−0.376321			−0.188161	−	0.982138i	$-0.560253\pi$
$839$	−10.9003			−0.376321			−0.188161	+	0.982138i	$0.560253\pi$
$840$		0			0
$841$	−18.2749			−0.630170
$842$		0			0
$843$			10.3923i			0.357930i
$844$		0			0
$845$	−51.5120	+	15.8265i	−1.77207	+	0.544450i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	33.9244			1.16428
$850$		0			0
$851$	−5.02409			−0.172224
$852$		0			0
$853$		−	30.4547i		−	1.04275i	−0.853327	−	0.521375i	$-0.825419\pi$
$853$		−	30.4547i		−	1.04275i	0.853327	−	0.521375i	$-0.174581\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	33.3276i		−	1.13845i	−0.822182	−	0.569224i	$-0.807244\pi$
$857$		−	33.3276i		−	1.13845i	0.822182	−	0.569224i	$-0.192756\pi$
$858$		0			0
$859$	−35.3746			−1.20697			−0.603483	−	0.797376i	$-0.706221\pi$
$859$	−35.3746			−1.20697			−0.603483	+	0.797376i	$0.706221\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	25.1435i		−	0.855895i	−0.903804	−	0.427947i	$-0.859237\pi$
$863$		−	25.1435i		−	0.855895i	0.903804	−	0.427947i	$-0.140763\pi$
$864$		0			0
$865$	−12.3127	−	40.0753i	−0.418645	−	1.36260i
$866$		0			0
$867$		−	10.0888i		−	0.342632i
$868$		0			0
$869$	0.625414			0.0212157
$870$		0			0
$871$	84.7492			2.87162
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$			44.6722i			1.50847i	0.656602	+	0.754237i	$0.271993\pi$
$877$			44.6722i			1.50847i	−0.656602	+	0.754237i	$0.728007\pi$
$878$		0			0
$879$	−12.0000			−0.404750
$880$		0			0
$881$	40.0241			1.34845			0.674223	−	0.738528i	$-0.264479\pi$
$881$	40.0241			1.34845			0.674223	+	0.738528i	$0.264479\pi$
$882$		0			0
$883$			20.6695i			0.695585i	0.937572	+	0.347792i	$0.113069\pi$
$883$			20.6695i			0.695585i	−0.937572	+	0.347792i	$0.886931\pi$
$884$		0			0
$885$	−4.86254	−	15.8265i	−0.163453	−	0.532003i
$886$		0			0
$887$		−	39.2301i		−	1.31722i	−0.752486	−	0.658609i	$-0.771145\pi$
$887$		−	39.2301i		−	1.31722i	0.752486	−	0.658609i	$-0.228855\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−20.4743			−0.685913
$892$		0			0
$893$		−	9.19397i		−	0.307664i
$894$		0			0
$895$	−0.587624	+	0.180541i	−0.0196421	+	0.00603483i
$896$		0			0
$897$			9.43996i			0.315191i
$898$		0			0
$899$	−14.0000			−0.466926
$900$		0			0
$901$	35.3746			1.17850
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−35.7492	+	10.9836i	−1.18834	+	0.365106i
$906$		0			0
$907$		−	45.3210i		−	1.50486i	−0.658672	−	0.752430i	$-0.728881\pi$
$907$		−	45.3210i		−	1.50486i	0.658672	−	0.752430i	$-0.271119\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	5.09967			0.168960			0.0844798	−	0.996425i	$-0.473077\pi$
$911$	5.09967			0.168960			0.0844798	+	0.996425i	$0.473077\pi$
$912$		0			0
$913$			12.9041i			0.427062i
$914$		0			0
$915$	1.76287	+	5.73778i	0.0582788	+	0.189685i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−5.92442			−0.195429			−0.0977143	−	0.995215i	$-0.531153\pi$
$919$	−5.92442			−0.195429			−0.0977143	+	0.995215i	$0.531153\pi$
$920$		0			0
$921$	−6.92442			−0.228167
$922$		0			0
$923$		−	64.2585i		−	2.11509i
$924$		0			0
$925$	15.7629	+	23.2309i	0.518280	+	0.763828i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	17.9003			0.587291			0.293645	−	0.955914i	$-0.405132\pi$
$929$	17.9003			0.587291			0.293645	+	0.955914i	$0.405132\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$			22.2131i			0.727225i
$934$		0			0
$935$	−7.13746	−	23.2309i	−0.233420	−	0.759732i
$936$		0			0
$937$			10.5074i			0.343262i	0.985161	+	0.171631i	$0.0549036\pi$
$937$			10.5074i			0.343262i	−0.985161	+	0.171631i	$0.945096\pi$
$938$		0			0
$939$	−25.0241			−0.816630
$940$		0			0
$941$	4.27492			0.139358			0.0696792	−	0.997569i	$-0.477802\pi$
$941$	4.27492			0.139358			0.0696792	+	0.997569i	$0.477802\pi$
$942$		0			0
$943$		−	10.0888i		−	0.328535i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	41.7419i		−	1.35643i	−0.734865	−	0.678214i	$-0.762754\pi$
$947$		−	41.7419i		−	1.35643i	0.734865	−	0.678214i	$-0.237246\pi$
$948$		0			0
$949$	13.0997			0.425233
$950$		0			0
$951$	6.62541			0.214844
$952$		0			0
$953$		−	29.6175i		−	0.959405i	−0.877431	−	0.479702i	$-0.840745\pi$
$953$		−	29.6175i		−	0.959405i	0.877431	−	0.479702i	$-0.159255\pi$
$954$		0			0
$955$	48.7870	−	14.9893i	1.57871	−	0.485042i
$956$		0			0
$957$			12.9041i			0.417129i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−12.7251			−0.410487
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−6.03779	−	19.6517i	−0.194363	−	0.632611i
$966$		0			0
$967$			6.50958i			0.209334i	0.994507	+	0.104667i	$0.0333777\pi$
$967$			6.50958i			0.209334i	−0.994507	+	0.104667i	$0.966622\pi$
$968$		0			0
$969$	35.3746			1.13640
$970$		0			0
$971$	15.9244			0.511039			0.255519	−	0.966804i	$-0.417754\pi$
$971$	15.9244			0.511039			0.255519	+	0.966804i	$0.417754\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	43.6495	−	29.6175i	1.39790	−	0.948519i
$976$		0			0
$977$			16.8443i			0.538898i	0.963015	+	0.269449i	$0.0868416\pi$
$977$			16.8443i			0.538898i	−0.963015	+	0.269449i	$0.913158\pi$
$978$		0			0
$979$	15.9244			0.508947
$980$		0			0
$981$		0			0
$982$		0			0
$983$			37.2103i			1.18682i	0.804899	+	0.593412i	$0.202220\pi$
$983$			37.2103i			1.18682i	−0.804899	+	0.593412i	$0.797780\pi$
$984$		0			0
$985$	−17.0997	−	55.6558i	−0.544841	−	1.77334i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	5.82475			0.185216
$990$		0			0
$991$	−34.4743			−1.09511			−0.547555	−	0.836769i	$-0.684441\pi$
$991$	−34.4743			−1.09511			−0.547555	+	0.836769i	$0.684441\pi$
$992$		0			0
$993$		−	8.35671i		−	0.265192i
$994$		0			0
$995$	−20.7870	+	6.38658i	−0.658991	+	0.202468i
$996$		0			0
$997$		−	4.77753i		−	0.151306i	−0.997134	−	0.0756529i	$-0.975896\pi$
$997$		−	4.77753i		−	0.151306i	0.997134	−	0.0756529i	$-0.0241041\pi$
$998$		0			0
$999$	−29.1752			−0.923064

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	980.2.e.f.589.4		4
5.2	odd	4		4900.2.a.be.1.4		4
5.3	odd	4		4900.2.a.be.1.2		4
5.4	even	2	inner	980.2.e.f.589.2		4
7.2	even	3		140.2.q.a.109.1	yes	4
7.3	odd	6		980.2.q.b.569.1		4
7.4	even	3		140.2.q.b.9.2	yes	4
7.5	odd	6		980.2.q.g.949.2		4
7.6	odd	2		980.2.e.c.589.1		4
21.2	odd	6		1260.2.bm.a.109.2		4
21.11	odd	6		1260.2.bm.b.289.1		4
28.11	odd	6		560.2.bw.a.289.2		4
28.23	odd	6		560.2.bw.e.529.1		4
35.2	odd	12		700.2.i.f.501.1		8
35.4	even	6		140.2.q.a.9.1	✓	4
35.9	even	6		140.2.q.b.109.1	yes	4
35.13	even	4		4900.2.a.bf.1.4		4
35.18	odd	12		700.2.i.f.401.4		8
35.19	odd	6		980.2.q.b.949.2		4
35.23	odd	12		700.2.i.f.501.4		8
35.24	odd	6		980.2.q.g.569.2		4
35.27	even	4		4900.2.a.bf.1.2		4
35.32	odd	12		700.2.i.f.401.1		8
35.34	odd	2		980.2.e.c.589.3		4
105.44	odd	6		1260.2.bm.b.109.2		4
105.74	odd	6		1260.2.bm.a.289.2		4
140.39	odd	6		560.2.bw.e.289.1		4
140.79	odd	6		560.2.bw.a.529.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.q.a.9.1	✓	4	35.4	even	6
140.2.q.a.109.1	yes	4	7.2	even	3
140.2.q.b.9.2	yes	4	7.4	even	3
140.2.q.b.109.1	yes	4	35.9	even	6
560.2.bw.a.289.2		4	28.11	odd	6
560.2.bw.a.529.1		4	140.79	odd	6
560.2.bw.e.289.1		4	140.39	odd	6
560.2.bw.e.529.1		4	28.23	odd	6
700.2.i.f.401.1		8	35.32	odd	12
700.2.i.f.401.4		8	35.18	odd	12
700.2.i.f.501.1		8	35.2	odd	12
700.2.i.f.501.4		8	35.23	odd	12
980.2.e.c.589.1		4	7.6	odd	2
980.2.e.c.589.3		4	35.34	odd	2
980.2.e.f.589.2		4	5.4	even	2	inner
980.2.e.f.589.4		4	1.1	even	1	trivial
980.2.q.b.569.1		4	7.3	odd	6
980.2.q.b.949.2		4	35.19	odd	6
980.2.q.g.569.2		4	35.24	odd	6
980.2.q.g.949.2		4	7.5	odd	6
1260.2.bm.a.109.2		4	21.2	odd	6
1260.2.bm.a.289.2		4	105.74	odd	6
1260.2.bm.b.109.2		4	105.44	odd	6
1260.2.bm.b.289.1		4	21.11	odd	6
4900.2.a.be.1.2		4	5.3	odd	4
4900.2.a.be.1.4		4	5.2	odd	4
4900.2.a.bf.1.2		4	35.27	even	4
4900.2.a.bf.1.4		4	35.13	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 980 = 2^{2} \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	980.e (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.73205i q^{3} +(2.13746 - 0.656712i) q^{5} +2.27492 q^{11} -6.09095i q^{13} +(1.13746 + 3.70219i) q^{15} -4.77753i q^{17} +4.27492 q^{19} +0.894797i q^{23} +(4.13746 - 2.80739i) q^{25} +5.19615i q^{27} +3.27492 q^{29} -4.27492 q^{31} +3.94027i q^{33} +5.61478i q^{37} +10.5498 q^{39} -11.2749 q^{41} -6.50958i q^{43} -2.15068i q^{47} +8.27492 q^{51} +7.40437i q^{53} +(4.86254 - 1.49397i) q^{55} +7.40437i q^{57} -4.27492 q^{59} +1.54983 q^{61} +(-4.00000 - 13.0192i) q^{65} +13.9140i q^{67} -1.54983 q^{69} +10.5498 q^{71} +2.15068i q^{73} +(4.86254 + 7.16629i) q^{75} +0.274917 q^{79} -9.00000 q^{81} +5.67232i q^{83} +(-3.13746 - 10.2118i) q^{85} +5.67232i q^{87} +7.00000 q^{89} -7.40437i q^{93} +(9.13746 - 2.80739i) q^{95} +6.92820i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{5} - 6 q^{11} - 3 q^{15} + 2 q^{19} + 9 q^{25} - 2 q^{29} - 2 q^{31} + 12 q^{39} - 30 q^{41} + 18 q^{51} + 27 q^{55} - 2 q^{59} - 24 q^{61} - 16 q^{65} + 24 q^{69} + 12 q^{71} + 27 q^{75} - 14 q^{79}+ \cdots + 29 q^{95}+O(q^{100}) \) 4 * q + q^5 - 6 * q^11 - 3 * q^15 + 2 * q^19 + 9 * q^25 - 2 * q^29 - 2 * q^31 + 12 * q^39 - 30 * q^41 + 18 * q^51 + 27 * q^55 - 2 * q^59 - 24 * q^61 - 16 * q^65 + 24 * q^69 + 12 * q^71 + 27 * q^75 - 14 * q^79 - 36 * q^81 - 5 * q^85 + 28 * q^89 + 29 * q^95

Introduction

L-functions

Modular forms

Varieties

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Database

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