LMFDB - Embedded newform 980.2.e.f.589.1

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			589.1
Root			$2.13746 - 0.656712i$ of defining polynomial
Character	$\chi$	$=$	980.589
Dual form			980.2.e.f.589.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{3} +(-1.63746 - 1.52274i) q^{5} -5.27492 q^{11} -2.62685i q^{13} +(-2.63746 + 2.83616i) q^{15} +0.418627i q^{17} -3.27492 q^{19} +7.82300i q^{23} +(0.362541 + 4.98684i) q^{25} -5.19615i q^{27} -4.27492 q^{29} +3.27492 q^{31} +9.13642i q^{33} -9.97368i q^{37} -4.54983 q^{39} -3.72508 q^{41} +2.15068i q^{43} +6.50958i q^{47} +0.725083 q^{51} +5.67232i q^{53} +(8.63746 + 8.03231i) q^{55} +5.67232i q^{57} +3.27492 q^{59} -13.5498 q^{61} +(-4.00000 + 4.30136i) q^{65} +3.52165i q^{67} +13.5498 q^{69} -4.54983 q^{71} -6.50958i q^{73} +(8.63746 - 0.627940i) q^{75} -7.27492 q^{79} -9.00000 q^{81} +7.40437i q^{83} +(0.637459 - 0.685484i) q^{85} +7.40437i q^{87} +7.00000 q^{89} -5.67232i q^{93} +(5.36254 + 4.98684i) q^{95} -6.92820i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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	$-0.833333\pi$
$3$		−	1.73205i		−	1.00000i	0.866025	−	0.500000i	$-0.166667\pi$
$4$		0			0
$5$	−1.63746	−	1.52274i	−0.732294	−	0.680989i
$5$	−1.63746	−	1.52274i	−0.732294	−	0.680989i
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−5.27492			−1.59045			−0.795224	−	0.606316i	$-0.792647\pi$
$11$	−5.27492			−1.59045			−0.795224	+	0.606316i	$0.792647\pi$
$12$		0			0
$13$		−	2.62685i		−	0.728557i	−0.931290	−	0.364278i	$-0.881316\pi$
$13$		−	2.62685i		−	0.728557i	0.931290	−	0.364278i	$-0.118684\pi$
$14$		0			0
$15$	−2.63746	+	2.83616i	−0.680989	+	0.732294i
$16$		0			0
$17$			0.418627i			0.101532i	0.998711	+	0.0507659i	$0.0161663\pi$
$17$			0.418627i			0.101532i	−0.998711	+	0.0507659i	$0.983834\pi$
$18$		0			0
$19$	−3.27492			−0.751318			−0.375659	−	0.926758i	$-0.622584\pi$
$19$	−3.27492			−0.751318			−0.375659	+	0.926758i	$0.622584\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			7.82300i			1.63121i	0.578610	+	0.815604i	$0.303595\pi$
$23$			7.82300i			1.63121i	−0.578610	+	0.815604i	$0.696405\pi$
$24$		0			0
$25$	0.362541	+	4.98684i	0.0725083	+	0.997368i
$26$		0			0
$27$		−	5.19615i		−	1.00000i
$28$		0			0
$29$	−4.27492			−0.793832			−0.396916	−	0.917855i	$-0.629920\pi$
$29$	−4.27492			−0.793832			−0.396916	+	0.917855i	$0.629920\pi$
$30$		0			0
$31$	3.27492			0.588192			0.294096	−	0.955776i	$-0.404981\pi$
$31$	3.27492			0.588192			0.294096	+	0.955776i	$0.404981\pi$
$32$		0			0
$33$			9.13642i			1.59045i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	9.97368i		−	1.63966i	−0.572605	−	0.819831i	$-0.694067\pi$
$37$		−	9.97368i		−	1.63966i	0.572605	−	0.819831i	$-0.305933\pi$
$38$		0			0
$39$	−4.54983			−0.728557
$40$		0			0
$41$	−3.72508			−0.581760			−0.290880	−	0.956760i	$-0.593948\pi$
$41$	−3.72508			−0.581760			−0.290880	+	0.956760i	$0.593948\pi$
$42$		0			0
$43$			2.15068i			0.327975i	0.986462	+	0.163988i	$0.0524357\pi$
$43$			2.15068i			0.327975i	−0.986462	+	0.163988i	$0.947564\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			6.50958i			0.949519i	0.880115	+	0.474760i	$0.157465\pi$
$47$			6.50958i			0.949519i	−0.880115	+	0.474760i	$0.842535\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	0.725083			0.101532
$52$		0			0
$53$			5.67232i			0.779153i	0.920994	+	0.389577i	$0.127379\pi$
$53$			5.67232i			0.779153i	−0.920994	+	0.389577i	$0.872621\pi$
$54$		0			0
$55$	8.63746	+	8.03231i	1.16467	+	1.08308i
$56$		0			0
$57$			5.67232i			0.751318i
$58$		0			0
$59$	3.27492			0.426358			0.213179	−	0.977013i	$-0.431618\pi$
$59$	3.27492			0.426358			0.213179	+	0.977013i	$0.431618\pi$
$60$		0			0
$61$	−13.5498			−1.73488			−0.867439	−	0.497543i	$-0.834236\pi$
$61$	−13.5498			−1.73488			−0.867439	+	0.497543i	$0.834236\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−4.00000	+	4.30136i	−0.496139	+	0.533517i
$66$		0			0
$67$			3.52165i			0.430237i	0.976588	+	0.215119i	$0.0690139\pi$
$67$			3.52165i			0.430237i	−0.976588	+	0.215119i	$0.930986\pi$
$68$		0			0
$69$	13.5498			1.63121
$70$		0			0
$71$	−4.54983			−0.539966			−0.269983	−	0.962865i	$-0.587018\pi$
$71$	−4.54983			−0.539966			−0.269983	+	0.962865i	$0.587018\pi$
$72$		0			0
$73$		−	6.50958i		−	0.761888i	−0.924598	−	0.380944i	$-0.875599\pi$
$73$		−	6.50958i		−	0.761888i	0.924598	−	0.380944i	$-0.124401\pi$
$74$		0			0
$75$	8.63746	−	0.627940i	0.997368	−	0.0725083i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−7.27492			−0.818492			−0.409246	−	0.912424i	$-0.634208\pi$
$79$	−7.27492			−0.818492			−0.409246	+	0.912424i	$0.634208\pi$
$80$		0			0
$81$	−9.00000			−1.00000
$82$		0			0
$83$			7.40437i			0.812736i	0.913710	+	0.406368i	$0.133205\pi$
$83$			7.40437i			0.812736i	−0.913710	+	0.406368i	$0.866795\pi$
$84$		0			0
$85$	0.637459	−	0.685484i	0.0691421	−	0.0743512i
$86$		0			0
$87$			7.40437i			0.793832i
$88$		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
$93$		−	5.67232i		−	0.588192i
$94$		0			0
$95$	5.36254	+	4.98684i	0.550185	+	0.511639i
$96$		0			0
$97$		−	6.92820i		−	0.703452i	−0.936103	−	0.351726i	$-0.885595\pi$
$97$		−	6.92820i		−	0.703452i	0.936103	−	0.351726i	$-0.114405\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−13.5498			−1.34826			−0.674129	−	0.738613i	$-0.735481\pi$
$101$	−13.5498			−1.34826			−0.674129	+	0.738613i	$0.735481\pi$
$102$		0			0
$103$		−	11.2871i		−	1.11215i	−0.831132	−	0.556076i	$-0.812306\pi$
$103$		−	11.2871i		−	1.11215i	0.831132	−	0.556076i	$-0.187694\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			3.52165i			0.340450i	0.985405	+	0.170225i	$0.0544495\pi$
$107$			3.52165i			0.340450i	−0.985405	+	0.170225i	$0.945550\pi$
$108$		0			0
$109$	11.5498			1.10627			0.553137	−	0.833090i	$-0.313431\pi$
$109$	11.5498			1.10627			0.553137	+	0.833090i	$0.313431\pi$
$110$		0			0
$111$	−17.2749			−1.63966
$112$		0			0
$113$		−	4.30136i		−	0.404637i	−0.979320	−	0.202319i	$-0.935152\pi$
$113$		−	4.30136i		−	0.404637i	0.979320	−	0.202319i	$-0.0648477\pi$
$114$		0			0
$115$	11.9124	−	12.8098i	1.11083	−	1.19452i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	16.8248			1.52952
$122$		0			0
$123$			6.45203i			0.581760i
$124$		0			0
$125$	7.00000	−	8.71780i	0.626099	−	0.779744i
$126$		0			0
$127$		−	15.6460i		−	1.38836i	−0.719802	−	0.694179i	$-0.755768\pi$
$127$		−	15.6460i		−	1.38836i	0.719802	−	0.694179i	$-0.244232\pi$
$128$		0			0
$129$	3.72508			0.327975
$130$		0			0
$131$	−10.7251			−0.937055			−0.468527	−	0.883449i	$-0.655215\pi$
$131$	−10.7251			−0.937055			−0.468527	+	0.883449i	$0.655215\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−7.91238	+	8.50848i	−0.680989	+	0.732294i
$136$		0			0
$137$		−	21.3183i		−	1.82135i	−0.413126	−	0.910674i	$-0.635563\pi$
$137$		−	21.3183i		−	1.82135i	0.413126	−	0.910674i	$-0.364437\pi$
$138$		0			0
$139$	−13.0997			−1.11110			−0.555550	−	0.831483i	$-0.687492\pi$
$139$	−13.0997			−1.11110			−0.555550	+	0.831483i	$0.687492\pi$
$140$		0			0
$141$	11.2749			0.949519
$142$		0			0
$143$			13.8564i			1.15873i
$144$		0			0
$145$	7.00000	+	6.50958i	0.581318	+	0.540591i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	7.54983			0.618507			0.309253	−	0.950980i	$-0.399921\pi$
$149$	7.54983			0.618507			0.309253	+	0.950980i	$0.399921\pi$
$150$		0			0
$151$	−12.7251			−1.03555			−0.517776	−	0.855516i	$-0.673240\pi$
$151$	−12.7251			−1.03555			−0.517776	+	0.855516i	$0.673240\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−5.36254	−	4.98684i	−0.430730	−	0.400553i
$156$		0			0
$157$		−	2.20822i		−	0.176235i	−0.996110	−	0.0881176i	$-0.971915\pi$
$157$		−	2.20822i		−	0.176235i	0.996110	−	0.0881176i	$-0.0280851\pi$
$158$		0			0
$159$	9.82475			0.779153
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	5.67232i		−	0.444291i	−0.975014	−	0.222145i	$-0.928694\pi$
$163$		−	5.67232i		−	0.444291i	0.975014	−	0.222145i	$-0.0713059\pi$
$164$		0			0
$165$	13.9124	−	14.9605i	1.08308	−	1.16467i
$166$		0			0
$167$		−	0.476171i		−	0.0368472i	−0.999830	−	0.0184236i	$-0.994135\pi$
$167$		−	0.476171i		−	0.0368472i	0.999830	−	0.0184236i	$-0.00586474\pi$
$168$		0			0
$169$	6.09967			0.469205
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	20.4811i		−	1.55715i	−0.627553	−	0.778573i	$-0.715944\pi$
$173$		−	20.4811i		−	1.55715i	0.627553	−	0.778573i	$-0.284056\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		−	5.67232i		−	0.426358i
$178$		0			0
$179$	7.27492			0.543753			0.271876	−	0.962332i	$-0.412356\pi$
$179$	7.27492			0.543753			0.271876	+	0.962332i	$0.412356\pi$
$180$		0			0
$181$	−24.2749			−1.80434			−0.902170	−	0.431380i	$-0.858027\pi$
$181$	−24.2749			−1.80434			−0.902170	+	0.431380i	$0.858027\pi$
$182$		0			0
$183$			23.4690i			1.73488i
$184$		0			0
$185$	−15.1873	+	16.3315i	−1.11659	+	1.20071i
$186$		0			0
$187$		−	2.20822i		−	0.161481i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	0.175248			0.0126805			0.00634026	−	0.999980i	$-0.497982\pi$
$191$	0.175248			0.0126805			0.00634026	+	0.999980i	$0.497982\pi$
$192$		0			0
$193$		−	21.3183i		−	1.53453i	−0.641332	−	0.767263i	$-0.721618\pi$
$193$		−	21.3183i		−	1.53453i	0.641332	−	0.767263i	$-0.278382\pi$
$194$		0			0
$195$	7.45017	+	6.92820i	0.533517	+	0.496139i
$196$		0			0
$197$			8.60271i			0.612918i	0.951884	+	0.306459i	$0.0991442\pi$
$197$			8.60271i			0.612918i	−0.951884	+	0.306459i	$0.900856\pi$
$198$		0			0
$199$	−17.2749			−1.22459			−0.612293	−	0.790631i	$-0.709753\pi$
$199$	−17.2749			−1.22459			−0.612293	+	0.790631i	$0.709753\pi$
$200$		0			0
$201$	6.09967			0.430237
$202$		0			0
$203$		0			0
$204$		0			0
$205$	6.09967	+	5.67232i	0.426019	+	0.396172i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	17.2749			1.19493
$210$		0			0
$211$	25.6495			1.76578			0.882892	−	0.469576i	$-0.155593\pi$
$211$	25.6495			1.76578			0.882892	+	0.469576i	$0.155593\pi$
$212$		0			0
$213$			7.88054i			0.539966i
$214$		0			0
$215$	3.27492	−	3.52165i	0.223348	−	0.240174i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−11.2749			−0.761888
$220$		0			0
$221$	1.09967			0.0739717
$222$		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
$227$		−	19.5287i		−	1.29617i	−0.761569	−	0.648084i	$-0.775571\pi$
$227$		−	19.5287i		−	1.29617i	0.761569	−	0.648084i	$-0.224429\pi$
$228$		0			0
$229$	−3.27492			−0.216413			−0.108206	−	0.994128i	$-0.534511\pi$
$229$	−3.27492			−0.216413			−0.108206	+	0.994128i	$0.534511\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	14.2750i		−	0.935189i	−0.883943	−	0.467594i	$-0.845121\pi$
$233$		−	14.2750i		−	0.935189i	0.883943	−	0.467594i	$-0.154879\pi$
$234$		0			0
$235$	9.91238	−	10.6592i	0.646612	−	0.695327i
$236$		0			0
$237$			12.6005i			0.818492i
$238$		0			0
$239$	−0.549834			−0.0355658			−0.0177829	−	0.999842i	$-0.505661\pi$
$239$	−0.549834			−0.0355658			−0.0177829	+	0.999842i	$0.505661\pi$
$240$		0			0
$241$	9.82475			0.632868			0.316434	−	0.948615i	$-0.397514\pi$
$241$	9.82475			0.632868			0.316434	+	0.948615i	$0.397514\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$			8.60271i			0.547377i
$248$		0			0
$249$	12.8248			0.812736
$250$		0			0
$251$	−20.5498			−1.29709			−0.648547	−	0.761175i	$-0.724623\pi$
$251$	−20.5498			−1.29709			−0.648547	+	0.761175i	$0.724623\pi$
$252$		0			0
$253$		−	41.2657i		−	2.59435i
$254$		0			0
$255$	−1.18729	−	1.10411i	−0.0743512	−	0.0691421i
$256$		0			0
$257$		−	11.6482i		−	0.726594i	−0.931673	−	0.363297i	$-0.881651\pi$
$257$		−	11.6482i		−	0.726594i	0.931673	−	0.363297i	$-0.118349\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	0.779710i		−	0.0480790i	−0.999711	−	0.0240395i	$-0.992347\pi$
$263$		−	0.779710i		−	0.0480790i	0.999711	−	0.0240395i	$-0.00765274\pi$
$264$		0			0
$265$	8.63746	−	9.28819i	0.530595	−	0.570569i
$266$		0			0
$267$		−	12.1244i		−	0.741999i
$268$		0			0
$269$	14.4502			0.881042			0.440521	−	0.897742i	$-0.354794\pi$
$269$	14.4502			0.881042			0.440521	+	0.897742i	$0.354794\pi$
$270$		0			0
$271$	9.82475			0.596811			0.298406	−	0.954439i	$-0.403545\pi$
$271$	9.82475			0.596811			0.298406	+	0.954439i	$0.403545\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.91238	−	26.3052i	−0.115321	−	1.58626i
$276$		0			0
$277$			14.2750i			0.857704i	0.903375	+	0.428852i	$0.141082\pi$
$277$			14.2750i			0.857704i	−0.903375	+	0.428852i	$0.858918\pi$
$278$		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
$283$		−	10.9260i		−	0.649484i	−0.945803	−	0.324742i	$-0.894722\pi$
$283$		−	10.9260i		−	0.649484i	0.945803	−	0.324742i	$-0.105278\pi$
$284$		0			0
$285$	8.63746	−	9.28819i	0.511639	−	0.550185i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	16.8248			0.989691
$290$		0			0
$291$	−12.0000			−0.703452
$292$		0			0
$293$		−	6.92820i		−	0.404750i	−0.979308	−	0.202375i	$-0.935134\pi$
$293$		−	6.92820i		−	0.404750i	0.979308	−	0.202375i	$-0.0648660\pi$
$294$		0			0
$295$	−5.36254	−	4.98684i	−0.312219	−	0.290345i
$296$		0			0
$297$			27.4093i			1.59045i
$298$		0			0
$299$	20.5498			1.18843
$300$		0			0
$301$		0			0
$302$		0			0
$303$			23.4690i			1.34826i
$304$		0			0
$305$	22.1873	+	20.6328i	1.27044	+	1.18143i
$306$		0			0
$307$			26.5145i			1.51326i	0.653843	+	0.756631i	$0.273156\pi$
$307$			26.5145i			1.51326i	−0.653843	+	0.756631i	$0.726844\pi$
$308$		0			0
$309$	−19.5498			−1.11215
$310$		0			0
$311$	−9.82475			−0.557111			−0.278555	−	0.960420i	$-0.589856\pi$
$311$	−9.82475			−0.557111			−0.278555	+	0.960420i	$0.589856\pi$
$312$		0			0
$313$			33.5002i			1.89354i	0.321904	+	0.946772i	$0.395677\pi$
$313$			33.5002i			1.89354i	−0.321904	+	0.946772i	$0.604323\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$			25.6197i			1.43894i	0.694521	+	0.719472i	$0.255616\pi$
$317$			25.6197i			1.43894i	−0.694521	+	0.719472i	$0.744384\pi$
$318$		0			0
$319$	22.5498			1.26255
$320$		0			0
$321$	6.09967			0.340450
$322$		0			0
$323$		−	1.37097i		−	0.0762827i
$324$		0			0
$325$	13.0997	−	0.952341i	0.726639	−	0.0528264i
$326$		0			0
$327$		−	20.0049i		−	1.10627i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	17.8248			0.979737			0.489868	−	0.871796i	$-0.337045\pi$
$331$	17.8248			0.979737			0.489868	+	0.871796i	$0.337045\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	5.36254	−	5.76655i	0.292987	−	0.315060i
$336$		0			0
$337$		−	4.30136i		−	0.234310i	−0.993114	−	0.117155i	$-0.962623\pi$
$337$		−	4.30136i		−	0.234310i	0.993114	−	0.117155i	$-0.0373774\pi$
$338$		0			0
$339$	−7.45017			−0.404637
$340$		0			0
$341$	−17.2749			−0.935489
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−22.1873	−	20.6328i	−1.19452	−	1.11083i
$346$		0			0
$347$			12.1244i			0.650870i	0.945564	+	0.325435i	$0.105511\pi$
$347$			12.1244i			0.650870i	−0.945564	+	0.325435i	$0.894489\pi$
$348$		0			0
$349$	3.72508			0.199399			0.0996996	−	0.995018i	$-0.468212\pi$
$349$	3.72508			0.199399			0.0996996	+	0.995018i	$0.468212\pi$
$350$		0			0
$351$	−13.6495			−0.728557
$352$		0			0
$353$		−	8.18408i		−	0.435595i	−0.975994	−	0.217797i	$-0.930113\pi$
$353$		−	8.18408i		−	0.435595i	0.975994	−	0.217797i	$-0.0698872\pi$
$354$		0			0
$355$	7.45017	+	6.92820i	0.395414	+	0.367711i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−36.3746			−1.91978			−0.959889	−	0.280382i	$-0.909539\pi$
$359$	−36.3746			−1.91978			−0.959889	+	0.280382i	$0.909539\pi$
$360$		0			0
$361$	−8.27492			−0.435522
$362$		0			0
$363$		−	29.1413i		−	1.52952i
$364$		0			0
$365$	−9.91238	+	10.6592i	−0.518837	+	0.557926i
$366$		0			0
$367$		−	6.03341i		−	0.314941i	−0.987524	−	0.157471i	$-0.949666\pi$
$367$		−	6.03341i		−	0.314941i	0.987524	−	0.157471i	$-0.0503340\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	9.97368i		−	0.516417i	−0.966089	−	0.258209i	$-0.916868\pi$
$373$		−	9.97368i		−	0.516417i	0.966089	−	0.258209i	$-0.0831322\pi$
$374$		0			0
$375$	−15.0997	−	12.1244i	−0.779744	−	0.626099i
$376$		0			0
$377$			11.2296i			0.578352i
$378$		0			0
$379$	21.6495			1.11206			0.556030	−	0.831162i	$-0.312324\pi$
$379$	21.6495			1.11206			0.556030	+	0.831162i	$0.312324\pi$
$380$		0			0
$381$	−27.0997			−1.38836
$382$		0			0
$383$			6.14849i			0.314173i	0.987585	+	0.157087i	$0.0502102\pi$
$383$			6.14849i			0.314173i	−0.987585	+	0.157087i	$0.949790\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−32.3746			−1.64146			−0.820728	−	0.571319i	$-0.806432\pi$
$389$	−32.3746			−1.64146			−0.820728	+	0.571319i	$0.806432\pi$
$390$		0			0
$391$	−3.27492			−0.165620
$392$		0			0
$393$			18.5764i			0.937055i
$394$		0			0
$395$	11.9124	+	11.0778i	0.599377	+	0.557384i
$396$		0			0
$397$			10.8109i			0.542585i	0.962497	+	0.271293i	$0.0874511\pi$
$397$			10.8109i			0.542585i	−0.962497	+	0.271293i	$0.912549\pi$
$398$		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
$403$		−	8.60271i		−	0.428532i
$404$		0			0
$405$	14.7371	+	13.7046i	0.732294	+	0.680989i
$406$		0			0
$407$			52.6103i			2.60780i
$408$		0			0
$409$	20.0997			0.993865			0.496932	−	0.867789i	$-0.334460\pi$
$409$	20.0997			0.993865			0.496932	+	0.867789i	$0.334460\pi$
$410$		0			0
$411$	−36.9244			−1.82135
$412$		0			0
$413$		0			0
$414$		0			0
$415$	11.2749	−	12.1244i	0.553464	−	0.595161i
$416$		0			0
$417$			22.6893i			1.11110i
$418$		0			0
$419$	13.0997			0.639961			0.319980	−	0.947424i	$-0.396324\pi$
$419$	13.0997			0.639961			0.319980	+	0.947424i	$0.396324\pi$
$420$		0			0
$421$	−4.27492			−0.208347			−0.104173	−	0.994559i	$-0.533220\pi$
$421$	−4.27492			−0.208347			−0.104173	+	0.994559i	$0.533220\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2.08762	+	0.151770i	−0.101265	+	0.00736190i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	24.0000			1.15873
$430$		0			0
$431$	−18.3746			−0.885073			−0.442536	−	0.896751i	$-0.645921\pi$
$431$	−18.3746			−0.885073			−0.442536	+	0.896751i	$0.645921\pi$
$432$		0			0
$433$			18.1578i			0.872606i	0.899800	+	0.436303i	$0.143712\pi$
$433$			18.1578i			0.872606i	−0.899800	+	0.436303i	$0.856288\pi$
$434$		0			0
$435$	11.2749	−	12.1244i	0.540591	−	0.581318i
$436$		0			0
$437$		−	25.6197i		−	1.22556i
$438$		0			0
$439$	−23.8248			−1.13709			−0.568547	−	0.822651i	$-0.692494\pi$
$439$	−23.8248			−1.13709			−0.568547	+	0.822651i	$0.692494\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$			12.1244i			0.576046i	0.957624	+	0.288023i	$0.0929979\pi$
$443$			12.1244i			0.576046i	−0.957624	+	0.288023i	$0.907002\pi$
$444$		0			0
$445$	−11.4622	−	10.6592i	−0.543361	−	0.505293i
$446$		0			0
$447$		−	13.0767i		−	0.618507i
$448$		0			0
$449$	3.17525			0.149849			0.0749246	−	0.997189i	$-0.476128\pi$
$449$	3.17525			0.149849			0.0749246	+	0.997189i	$0.476128\pi$
$450$		0			0
$451$	19.6495			0.925259
$452$		0			0
$453$			22.0405i			1.03555i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	1.37097i		−	0.0641312i	−0.999486	−	0.0320656i	$-0.989791\pi$
$457$		−	1.37097i		−	0.0641312i	0.999486	−	0.0320656i	$-0.0102085\pi$
$458$		0			0
$459$	2.17525			0.101532
$460$		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
$463$		−	2.15068i		−	0.0999505i	−0.998750	−	0.0499752i	$-0.984086\pi$
$463$		−	2.15068i		−	0.0999505i	0.998750	−	0.0499752i	$-0.0159142\pi$
$464$		0			0
$465$	−8.63746	+	9.28819i	−0.400553	+	0.430730i
$466$		0			0
$467$			15.7035i			0.726673i	0.931658	+	0.363337i	$0.118363\pi$
$467$			15.7035i			0.726673i	−0.931658	+	0.363337i	$0.881637\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−3.82475			−0.176235
$472$		0			0
$473$		−	11.3446i		−	0.521627i
$474$		0			0
$475$	−1.18729	−	16.3315i	−0.0544767	−	0.749340i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−9.82475			−0.448904			−0.224452	−	0.974485i	$-0.572059\pi$
$479$	−9.82475			−0.448904			−0.224452	+	0.974485i	$0.572059\pi$
$480$		0			0
$481$	−26.1993			−1.19459
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−10.5498	+	11.3446i	−0.479043	+	0.515134i
$486$		0			0
$487$			2.93039i			0.132789i	0.997793	+	0.0663943i	$0.0211495\pi$
$487$			2.93039i			0.132789i	−0.997793	+	0.0663943i	$0.978850\pi$
$488$		0			0
$489$	−9.82475			−0.444291
$490$		0			0
$491$	−28.5498			−1.28844			−0.644218	−	0.764842i	$-0.722817\pi$
$491$	−28.5498			−1.28844			−0.644218	+	0.764842i	$0.722817\pi$
$492$		0			0
$493$		−	1.78959i		−	0.0805993i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−1.62541			−0.0727635			−0.0363818	−	0.999338i	$-0.511583\pi$
$499$	−1.62541			−0.0727635			−0.0363818	+	0.999338i	$0.511583\pi$
$500$		0			0
$501$	−0.824752			−0.0368472
$502$		0			0
$503$		−	31.7682i		−	1.41647i	−0.705975	−	0.708236i	$-0.749491\pi$
$503$		−	31.7682i		−	1.41647i	0.705975	−	0.708236i	$-0.250509\pi$
$504$		0			0
$505$	22.1873	+	20.6328i	0.987322	+	0.918149i
$506$		0			0
$507$		−	10.5649i		−	0.469205i
$508$		0			0
$509$	14.4502			0.640492			0.320246	−	0.947334i	$-0.396234\pi$
$509$	14.4502			0.640492			0.320246	+	0.947334i	$0.396234\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$			17.0170i			0.751318i
$514$		0			0
$515$	−17.1873	+	18.4822i	−0.757363	+	0.814421i
$516$		0			0
$517$		−	34.3375i		−	1.51016i
$518$		0			0
$519$	−35.4743			−1.55715
$520$		0			0
$521$	9.82475			0.430430			0.215215	−	0.976567i	$-0.430955\pi$
$521$	9.82475			0.430430			0.215215	+	0.976567i	$0.430955\pi$
$522$		0			0
$523$		−	7.34683i		−	0.321254i	−0.987015	−	0.160627i	$-0.948648\pi$
$523$		−	7.34683i		−	0.321254i	0.987015	−	0.160627i	$-0.0513517\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			1.37097i			0.0597203i
$528$		0			0
$529$	−38.1993			−1.66084
$530$		0			0
$531$		0			0
$532$		0			0
$533$			9.78523i			0.423845i
$534$		0			0
$535$	5.36254	−	5.76655i	0.231843	−	0.249310i
$536$		0			0
$537$		−	12.6005i		−	0.543753i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	−17.5498			−0.754526			−0.377263	−	0.926106i	$-0.623135\pi$
$541$	−17.5498			−0.754526			−0.377263	+	0.926106i	$0.623135\pi$
$542$		0			0
$543$			42.0454i			1.80434i
$544$		0			0
$545$	−18.9124	−	17.5874i	−0.810117	−	0.753360i
$546$		0			0
$547$		−	20.5386i		−	0.878168i	−0.898446	−	0.439084i	$-0.855303\pi$
$547$		−	20.5386i		−	0.878168i	0.898446	−	0.439084i	$-0.144697\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	14.0000			0.596420
$552$		0			0
$553$		0			0
$554$		0			0
$555$	28.2870	+	26.3052i	1.20071	+	1.11659i
$556$		0			0
$557$			9.97368i			0.422598i	0.977421	+	0.211299i	$0.0677694\pi$
$557$			9.97368i			0.422598i	−0.977421	+	0.211299i	$0.932231\pi$
$558$		0			0
$559$	5.64950			0.238949
$560$		0			0
$561$	−3.82475			−0.161481
$562$		0			0
$563$			22.6317i			0.953814i	0.878954	+	0.476907i	$0.158242\pi$
$563$			22.6317i			0.953814i	−0.878954	+	0.476907i	$0.841758\pi$
$564$		0			0
$565$	−6.54983	+	7.04329i	−0.275554	+	0.296313i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−8.37459			−0.351081			−0.175540	−	0.984472i	$-0.556167\pi$
$569$	−8.37459			−0.351081			−0.175540	+	0.984472i	$0.556167\pi$
$570$		0			0
$571$	7.27492			0.304446			0.152223	−	0.988346i	$-0.451357\pi$
$571$	7.27492			0.304446			0.152223	+	0.988346i	$0.451357\pi$
$572$		0			0
$573$		−	0.303539i		−	0.0126805i
$574$		0			0
$575$	−39.0120	+	2.83616i	−1.62691	+	0.118276i
$576$		0			0
$577$		−	3.88273i		−	0.161640i	−0.996729	−	0.0808200i	$-0.974246\pi$
$577$		−	3.88273i		−	0.161640i	0.996729	−	0.0808200i	$-0.0257539\pi$
$578$		0			0
$579$	−36.9244			−1.53453
$580$		0			0
$581$		0			0
$582$		0			0
$583$		−	29.9210i		−	1.23920i
$584$		0			0
$585$		0			0
$586$		0			0
$587$			20.8997i			0.862623i	0.902203	+	0.431311i	$0.141949\pi$
$587$			20.8997i			0.862623i	−0.902203	+	0.431311i	$0.858051\pi$
$588$		0			0
$589$	−10.7251			−0.441919
$590$		0			0
$591$	14.9003			0.612918
$592$		0			0
$593$		−	33.3851i		−	1.37096i	−0.728090	−	0.685482i	$-0.759592\pi$
$593$		−	33.3851i		−	1.37096i	0.728090	−	0.685482i	$-0.240408\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$			29.9210i			1.22459i
$598$		0			0
$599$	−5.27492			−0.215527			−0.107764	−	0.994177i	$-0.534369\pi$
$599$	−5.27492			−0.215527			−0.107764	+	0.994177i	$0.534369\pi$
$600$		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
$605$	−27.5498	−	25.6197i	−1.12006	−	1.04159i
$606$		0			0
$607$			11.4022i			0.462801i	0.972859	+	0.231400i	$0.0743307\pi$
$607$			11.4022i			0.462801i	−0.972859	+	0.231400i	$0.925669\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	17.0997			0.691779
$612$		0			0
$613$			28.3616i			1.14551i	0.819725	+	0.572757i	$0.194126\pi$
$613$			28.3616i			1.14551i	−0.819725	+	0.572757i	$0.805874\pi$
$614$		0			0
$615$	9.82475	−	10.5649i	0.396172	−	0.426019i
$616$		0			0
$617$		−	31.2920i		−	1.25977i	−0.776689	−	0.629884i	$-0.783102\pi$
$617$		−	31.2920i		−	1.25977i	0.776689	−	0.629884i	$-0.216898\pi$
$618$		0			0
$619$	8.92442			0.358703			0.179351	−	0.983785i	$-0.442600\pi$
$619$	8.92442			0.358703			0.179351	+	0.983785i	$0.442600\pi$
$620$		0			0
$621$	40.6495			1.63121
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−24.7371	+	3.61587i	−0.989485	+	0.144635i
$626$		0			0
$627$		−	29.9210i		−	1.19493i
$628$		0			0
$629$	4.17525			0.166478
$630$		0			0
$631$	33.0997			1.31768			0.658839	−	0.752284i	$-0.271048\pi$
$631$	33.0997			1.31768			0.658839	+	0.752284i	$0.271048\pi$
$632$		0			0
$633$		−	44.4262i		−	1.76578i
$634$		0			0
$635$	−23.8248	+	25.6197i	−0.945456	+	1.01669i
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−2.09967			−0.0829319			−0.0414660	−	0.999140i	$-0.513203\pi$
$641$	−2.09967			−0.0829319			−0.0414660	+	0.999140i	$0.513203\pi$
$642$		0			0
$643$			31.4071i			1.23857i	0.785164	+	0.619287i	$0.212578\pi$
$643$			31.4071i			1.23857i	−0.785164	+	0.619287i	$0.787422\pi$
$644$		0			0
$645$	−6.09967	−	5.67232i	−0.240174	−	0.223348i
$646$		0			0
$647$			26.9331i			1.05885i	0.848357	+	0.529425i	$0.177592\pi$
$647$			26.9331i			1.05885i	−0.848357	+	0.529425i	$0.822408\pi$
$648$		0			0
$649$	−17.2749			−0.678100
$650$		0			0
$651$		0			0
$652$		0			0
$653$		−	28.3616i		−	1.10988i	−0.831892	−	0.554938i	$-0.812742\pi$
$653$		−	28.3616i		−	1.10988i	0.831892	−	0.554938i	$-0.187258\pi$
$654$		0			0
$655$	17.5619	+	16.3315i	0.686199	+	0.638124i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	40.5498			1.57960			0.789799	−	0.613366i	$-0.210185\pi$
$659$	40.5498			1.57960			0.789799	+	0.613366i	$0.210185\pi$
$660$		0			0
$661$	−0.450166			−0.0175094			−0.00875471	−	0.999962i	$-0.502787\pi$
$661$	−0.450166			−0.0175094			−0.00875471	+	0.999962i	$0.502787\pi$
$662$		0			0
$663$		−	1.90468i		−	0.0739717i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	33.4427i		−	1.29491i
$668$		0			0
$669$	15.0997			0.583787
$670$		0			0
$671$	71.4743			2.75923
$672$		0			0
$673$			31.2920i			1.20622i	0.797659	+	0.603109i	$0.206072\pi$
$673$			31.2920i			1.20622i	−0.797659	+	0.603109i	$0.793928\pi$
$674$		0			0
$675$	25.9124	−	1.88382i	0.997368	−	0.0725083i
$676$		0			0
$677$			46.4043i			1.78346i	0.452566	+	0.891731i	$0.350509\pi$
$677$			46.4043i			1.78346i	−0.452566	+	0.891731i	$0.649491\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−33.8248			−1.29617
$682$		0			0
$683$		−	19.1676i		−	0.733430i	−0.930333	−	0.366715i	$-0.880482\pi$
$683$		−	19.1676i		−	0.733430i	0.930333	−	0.366715i	$-0.119518\pi$
$684$		0			0
$685$	−32.4622	+	34.9079i	−1.24032	+	1.33376i
$686$		0			0
$687$			5.67232i			0.216413i
$688$		0			0
$689$	14.9003			0.567657
$690$		0			0
$691$	30.3746			1.15550			0.577752	−	0.816212i	$-0.303930\pi$
$691$	30.3746			1.15550			0.577752	+	0.816212i	$0.303930\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	21.4502	+	19.9474i	0.813651	+	0.756646i
$696$		0			0
$697$		−	1.55942i		−	0.0590672i
$698$		0			0
$699$	−24.7251			−0.935189
$700$		0			0
$701$	8.82475			0.333306			0.166653	−	0.986016i	$-0.446704\pi$
$701$	8.82475			0.333306			0.166653	+	0.986016i	$0.446704\pi$
$702$		0			0
$703$			32.6630i			1.23191i
$704$		0			0
$705$	−18.4622	−	17.1687i	−0.695327	−	0.646612i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	10.4502			0.392464			0.196232	−	0.980557i	$-0.437129\pi$
$709$	10.4502			0.392464			0.196232	+	0.980557i	$0.437129\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$			25.6197i			0.959465i
$714$		0			0
$715$	21.0997	−	22.6893i	0.789083	−	0.848531i
$716$		0			0
$717$			0.952341i			0.0355658i
$718$		0			0
$719$	30.3746			1.13278			0.566390	−	0.824137i	$-0.308339\pi$
$719$	30.3746			1.13278			0.566390	+	0.824137i	$0.308339\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		−	17.0170i		−	0.632868i
$724$		0			0
$725$	−1.54983	−	21.3183i	−0.0575594	−	0.791743i
$726$		0			0
$727$			3.10302i			0.115085i	0.998343	+	0.0575423i	$0.0183264\pi$
$727$			3.10302i			0.115085i	−0.998343	+	0.0575423i	$0.981674\pi$
$728$		0			0
$729$	−27.0000			−1.00000
$730$		0			0
$731$	−0.900331			−0.0332999
$732$		0			0
$733$		−	37.6865i		−	1.39198i	−0.718050	−	0.695991i	$-0.754965\pi$
$733$		−	37.6865i		−	1.39198i	0.718050	−	0.695991i	$-0.245035\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	18.5764i		−	0.684270i
$738$		0			0
$739$	20.9244			0.769717			0.384859	−	0.922976i	$-0.374250\pi$
$739$	20.9244			0.769717			0.384859	+	0.922976i	$0.374250\pi$
$740$		0			0
$741$	14.9003			0.547377
$742$		0			0
$743$			6.45203i			0.236702i	0.992972	+	0.118351i	$0.0377608\pi$
$743$			6.45203i			0.236702i	−0.992972	+	0.118351i	$0.962239\pi$
$744$		0			0
$745$	−12.3625	−	11.4964i	−0.452928	−	0.421196i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−14.7251			−0.537326			−0.268663	−	0.963234i	$-0.586582\pi$
$751$	−14.7251			−0.537326			−0.268663	+	0.963234i	$0.586582\pi$
$752$		0			0
$753$			35.5934i			1.29709i
$754$		0			0
$755$	20.8368	+	19.3770i	0.758329	+	0.705200i
$756$		0			0
$757$			35.5934i			1.29366i	0.762633	+	0.646831i	$0.223906\pi$
$757$			35.5934i			1.29366i	−0.762633	+	0.646831i	$0.776094\pi$
$758$		0			0
$759$	−71.4743			−2.59435
$760$		0			0
$761$	22.9244			0.831010			0.415505	−	0.909591i	$-0.363605\pi$
$761$	22.9244			0.831010			0.415505	+	0.909591i	$0.363605\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	8.60271i		−	0.310626i
$768$		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
$771$	−20.1752			−0.726594
$772$		0			0
$773$			40.3133i			1.44997i	0.688765	+	0.724985i	$0.258153\pi$
$773$			40.3133i			1.44997i	−0.688765	+	0.724985i	$0.741847\pi$
$774$		0			0
$775$	1.18729	+	16.3315i	0.0426488	+	0.586644i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	12.1993			0.437087
$780$		0			0
$781$	24.0000			0.858788
$782$		0			0
$783$			22.2131i			0.793832i
$784$		0			0
$785$	−3.36254	+	3.61587i	−0.120014	+	0.129056i
$786$		0			0
$787$		−	1.73205i		−	0.0617409i	−0.999523	−	0.0308705i	$-0.990172\pi$
$787$		−	1.73205i		−	0.0617409i	0.999523	−	0.0308705i	$-0.00982794\pi$
$788$		0			0
$789$	−1.35050			−0.0480790
$790$		0			0
$791$		0			0
$792$		0			0
$793$			35.5934i			1.26396i
$794$		0			0
$795$	−16.0876	−	14.9605i	−0.570569	−	0.530595i
$796$		0			0
$797$		−	46.8229i		−	1.65855i	−0.558839	−	0.829276i	$-0.688753\pi$
$797$		−	46.8229i		−	1.65855i	0.558839	−	0.829276i	$-0.311247\pi$
$798$		0			0
$799$	−2.72508			−0.0964065
$800$		0			0
$801$		0			0
$802$		0			0
$803$			34.3375i			1.21174i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		−	25.0284i		−	0.881042i
$808$		0			0
$809$	17.1993			0.604697			0.302348	−	0.953198i	$-0.402229\pi$
$809$	17.1993			0.604697			0.302348	+	0.953198i	$0.402229\pi$
$810$		0			0
$811$	7.45017			0.261611			0.130805	−	0.991408i	$-0.458244\pi$
$811$	7.45017			0.261611			0.130805	+	0.991408i	$0.458244\pi$
$812$		0			0
$813$		−	17.0170i		−	0.596811i
$814$		0			0
$815$	−8.63746	+	9.28819i	−0.302557	+	0.325351i
$816$		0			0
$817$		−	7.04329i		−	0.246414i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−20.3746			−0.711078			−0.355539	−	0.934661i	$-0.615703\pi$
$821$	−20.3746			−0.711078			−0.355539	+	0.934661i	$0.615703\pi$
$822$		0			0
$823$		−	46.1583i		−	1.60898i	−0.593968	−	0.804488i	$-0.702440\pi$
$823$		−	46.1583i		−	1.60898i	0.593968	−	0.804488i	$-0.297560\pi$
$824$		0			0
$825$	−45.5619	+	3.31233i	−1.58626	+	0.115321i
$826$		0			0
$827$		−	15.0547i		−	0.523505i	−0.965135	−	0.261752i	$-0.915700\pi$
$827$		−	15.0547i		−	0.523505i	0.965135	−	0.261752i	$-0.0843004\pi$
$828$		0			0
$829$	50.9244			1.76868			0.884339	−	0.466845i	$-0.154609\pi$
$829$	50.9244			1.76868			0.884339	+	0.466845i	$0.154609\pi$
$830$		0			0
$831$	24.7251			0.857704
$832$		0			0
$833$		0			0
$834$		0			0
$835$	−0.725083	+	0.779710i	−0.0250925	+	0.0269830i
$836$		0			0
$837$		−	17.0170i		−	0.588192i
$838$		0			0
$839$	−41.0997			−1.41892			−0.709459	−	0.704747i	$-0.751061\pi$
$839$	−41.0997			−1.41892			−0.709459	+	0.704747i	$0.751061\pi$
$840$		0			0
$841$	−10.7251			−0.369830
$842$		0			0
$843$		−	10.3923i		−	0.357930i
$844$		0			0
$845$	−9.98796	−	9.28819i	−0.343596	−	0.319524i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−18.9244			−0.649484
$850$		0			0
$851$	78.0241			2.67463
$852$		0			0
$853$		−	13.1342i		−	0.449708i	−0.974392	−	0.224854i	$-0.927810\pi$
$853$		−	13.1342i		−	0.449708i	0.974392	−	0.224854i	$-0.0721905\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			37.6865i			1.28735i	0.765301	+	0.643673i	$0.222590\pi$
$857$			37.6865i			1.28735i	−0.765301	+	0.643673i	$0.777410\pi$
$858$		0			0
$859$	2.37459			0.0810198			0.0405099	−	0.999179i	$-0.487102\pi$
$859$	2.37459			0.0810198			0.0405099	+	0.999179i	$0.487102\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			16.4257i			0.559138i	0.960126	+	0.279569i	$0.0901915\pi$
$863$			16.4257i			0.559138i	−0.960126	+	0.279569i	$0.909808\pi$
$864$		0			0
$865$	−31.1873	+	33.5369i	−1.06040	+	1.14029i
$866$		0			0
$867$		−	29.1413i		−	0.989691i
$868$		0			0
$869$	38.3746			1.30177
$870$		0			0
$871$	9.25083			0.313452
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	22.8777i		−	0.772527i	−0.922389	−	0.386263i	$-0.873766\pi$
$877$		−	22.8777i		−	0.772527i	0.922389	−	0.386263i	$-0.126234\pi$
$878$		0			0
$879$	−12.0000			−0.404750
$880$		0			0
$881$	−43.0241			−1.44952			−0.724759	−	0.689002i	$-0.758049\pi$
$881$	−43.0241			−1.44952			−0.724759	+	0.689002i	$0.758049\pi$
$882$		0			0
$883$		−	55.5407i		−	1.86909i	−0.355840	−	0.934547i	$-0.615805\pi$
$883$		−	55.5407i		−	1.86909i	0.355840	−	0.934547i	$-0.384195\pi$
$884$		0			0
$885$	−8.63746	+	9.28819i	−0.290345	+	0.312219i
$886$		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
$891$	47.4743			1.59045
$892$		0			0
$893$		−	21.3183i		−	0.713391i
$894$		0			0
$895$	−11.9124	−	11.0778i	−0.398187	−	0.370290i
$896$		0			0
$897$		−	35.5934i		−	1.18843i
$898$		0			0
$899$	−14.0000			−0.466926
$900$		0			0
$901$	−2.37459			−0.0791089
$902$		0			0
$903$		0			0
$904$		0			0
$905$	39.7492	+	36.9643i	1.32131	+	1.22874i
$906$		0			0
$907$		−	41.8569i		−	1.38984i	−0.719089	−	0.694918i	$-0.755440\pi$
$907$		−	41.8569i		−	1.38984i	0.719089	−	0.694918i	$-0.244560\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−25.0997			−0.831589			−0.415795	−	0.909459i	$-0.636496\pi$
$911$	−25.0997			−0.831589			−0.415795	+	0.909459i	$0.636496\pi$
$912$		0			0
$913$		−	39.0575i		−	1.29261i
$914$		0			0
$915$	35.7371	−	38.4295i	1.18143	−	1.27044i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	46.9244			1.54789			0.773947	−	0.633250i	$-0.218280\pi$
$919$	46.9244			1.54789			0.773947	+	0.633250i	$0.218280\pi$
$920$		0			0
$921$	45.9244			1.51326
$922$		0			0
$923$			11.9517i			0.393396i
$924$		0			0
$925$	49.7371	−	3.61587i	1.63535	−	0.118889i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	48.0997			1.57810			0.789049	−	0.614330i	$-0.210573\pi$
$929$	48.0997			1.57810			0.789049	+	0.614330i	$0.210573\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$			17.0170i			0.557111i
$934$		0			0
$935$	−3.36254	+	3.61587i	−0.109967	+	0.118252i
$936$		0			0
$937$			24.3638i			0.795931i	0.917400	+	0.397965i	$0.130284\pi$
$937$			24.3638i			0.795931i	−0.917400	+	0.397965i	$0.869716\pi$
$938$		0			0
$939$	58.0241			1.89354
$940$		0			0
$941$	−3.27492			−0.106759			−0.0533796	−	0.998574i	$-0.516999\pi$
$941$	−3.27492			−0.106759			−0.0533796	+	0.998574i	$0.516999\pi$
$942$		0			0
$943$		−	29.1413i		−	0.948972i
$944$		0			0
$945$		0			0
$946$		0			0
$947$		−	10.5649i		−	0.343314i	−0.985157	−	0.171657i	$-0.945088\pi$
$947$		−	10.5649i		−	0.343314i	0.985157	−	0.171657i	$-0.0549121\pi$
$948$		0			0
$949$	−17.0997			−0.555079
$950$		0			0
$951$	44.3746			1.43894
$952$		0			0
$953$		−	22.6893i		−	0.734978i	−0.930028	−	0.367489i	$-0.880218\pi$
$953$		−	22.6893i		−	0.734978i	0.930028	−	0.367489i	$-0.119782\pi$
$954$		0			0
$955$	−0.286962	−	0.266857i	−0.00928587	−	0.00863530i
$956$		0			0
$957$		−	39.0575i		−	1.26255i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−20.2749			−0.654030
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−32.4622	+	34.9079i	−1.04500	+	1.12372i
$966$		0			0
$967$		−	2.15068i		−	0.0691611i	−0.999402	−	0.0345806i	$-0.988990\pi$
$967$		−	2.15068i		−	0.0691611i	0.999402	−	0.0345806i	$-0.0110095\pi$
$968$		0			0
$969$	−2.37459			−0.0762827
$970$		0			0
$971$	−36.9244			−1.18496			−0.592481	−	0.805585i	$-0.701851\pi$
$971$	−36.9244			−1.18496			−0.592481	+	0.805585i	$0.701851\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−1.64950	−	22.6893i	−0.0528264	−	0.726639i
$976$		0			0
$977$		−	29.9210i		−	0.957259i	−0.878017	−	0.478629i	$-0.841134\pi$
$977$		−	29.9210i		−	0.957259i	0.878017	−	0.478629i	$-0.158866\pi$
$978$		0			0
$979$	−36.9244			−1.18011
$980$		0			0
$981$		0			0
$982$		0			0
$983$		−	45.9281i		−	1.46488i	−0.680832	−	0.732440i	$-0.738382\pi$
$983$		−	45.9281i		−	1.46488i	0.680832	−	0.732440i	$-0.261618\pi$
$984$		0			0
$985$	13.0997	−	14.0866i	0.417390	−	0.448836i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−16.8248			−0.534996
$990$		0			0
$991$	33.4743			1.06334			0.531672	−	0.846950i	$-0.321564\pi$
$991$	33.4743			1.06334			0.531672	+	0.846950i	$0.321564\pi$
$992$		0			0
$993$		−	30.8734i		−	0.979737i
$994$		0			0
$995$	28.2870	+	26.3052i	0.896757	+	0.833930i
$996$		0			0
$997$			0.418627i			0.0132580i	0.999978	+	0.00662902i	$0.00211010\pi$
$997$			0.418627i			0.0132580i	−0.999978	+	0.00662902i	$0.997890\pi$
$998$		0			0
$999$	−51.8248			−1.63966

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	980.2.e.f.589.1		4
5.2	odd	4		4900.2.a.be.1.1		4
5.3	odd	4		4900.2.a.be.1.3		4
5.4	even	2	inner	980.2.e.f.589.3		4
7.2	even	3		140.2.q.b.109.2	yes	4
7.3	odd	6		980.2.q.g.569.1		4
7.4	even	3		140.2.q.a.9.2	✓	4
7.5	odd	6		980.2.q.b.949.1		4
7.6	odd	2		980.2.e.c.589.4		4
21.2	odd	6		1260.2.bm.b.109.1		4
21.11	odd	6		1260.2.bm.a.289.1		4
28.11	odd	6		560.2.bw.e.289.2		4
28.23	odd	6		560.2.bw.a.529.2		4
35.2	odd	12		700.2.i.f.501.3		8
35.4	even	6		140.2.q.b.9.1	yes	4
35.9	even	6		140.2.q.a.109.2	yes	4
35.13	even	4		4900.2.a.bf.1.1		4
35.18	odd	12		700.2.i.f.401.2		8
35.19	odd	6		980.2.q.g.949.1		4
35.23	odd	12		700.2.i.f.501.2		8
35.24	odd	6		980.2.q.b.569.2		4
35.27	even	4		4900.2.a.bf.1.3		4
35.32	odd	12		700.2.i.f.401.3		8
35.34	odd	2		980.2.e.c.589.2		4
105.44	odd	6		1260.2.bm.a.109.1		4
105.74	odd	6		1260.2.bm.b.289.2		4
140.39	odd	6		560.2.bw.a.289.1		4
140.79	odd	6		560.2.bw.e.529.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
140.2.q.a.9.2	✓	4	7.4	even	3
140.2.q.a.109.2	yes	4	35.9	even	6
140.2.q.b.9.1	yes	4	35.4	even	6
140.2.q.b.109.2	yes	4	7.2	even	3
560.2.bw.a.289.1		4	140.39	odd	6
560.2.bw.a.529.2		4	28.23	odd	6
560.2.bw.e.289.2		4	28.11	odd	6
560.2.bw.e.529.2		4	140.79	odd	6
700.2.i.f.401.2		8	35.18	odd	12
700.2.i.f.401.3		8	35.32	odd	12
700.2.i.f.501.2		8	35.23	odd	12
700.2.i.f.501.3		8	35.2	odd	12
980.2.e.c.589.2		4	35.34	odd	2
980.2.e.c.589.4		4	7.6	odd	2
980.2.e.f.589.1		4	1.1	even	1	trivial
980.2.e.f.589.3		4	5.4	even	2	inner
980.2.q.b.569.2		4	35.24	odd	6
980.2.q.b.949.1		4	7.5	odd	6
980.2.q.g.569.1		4	7.3	odd	6
980.2.q.g.949.1		4	35.19	odd	6
1260.2.bm.a.109.1		4	105.44	odd	6
1260.2.bm.a.289.1		4	21.11	odd	6
1260.2.bm.b.109.1		4	21.2	odd	6
1260.2.bm.b.289.2		4	105.74	odd	6
4900.2.a.be.1.1		4	5.2	odd	4
4900.2.a.be.1.3		4	5.3	odd	4
4900.2.a.bf.1.1		4	35.13	even	4
4900.2.a.bf.1.3		4	35.27	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)

\(f(q)\)	\(=\)	\(q-1.73205i q^{3} +(-1.63746 - 1.52274i) q^{5} -5.27492 q^{11} -2.62685i q^{13} +(-2.63746 + 2.83616i) q^{15} +0.418627i q^{17} -3.27492 q^{19} +7.82300i q^{23} +(0.362541 + 4.98684i) q^{25} -5.19615i q^{27} -4.27492 q^{29} +3.27492 q^{31} +9.13642i q^{33} -9.97368i q^{37} -4.54983 q^{39} -3.72508 q^{41} +2.15068i q^{43} +6.50958i q^{47} +0.725083 q^{51} +5.67232i q^{53} +(8.63746 + 8.03231i) q^{55} +5.67232i q^{57} +3.27492 q^{59} -13.5498 q^{61} +(-4.00000 + 4.30136i) q^{65} +3.52165i q^{67} +13.5498 q^{69} -4.54983 q^{71} -6.50958i q^{73} +(8.63746 - 0.627940i) q^{75} -7.27492 q^{79} -9.00000 q^{81} +7.40437i q^{83} +(0.637459 - 0.685484i) q^{85} +7.40437i q^{87} +7.00000 q^{89} -5.67232i q^{93} +(5.36254 + 4.98684i) q^{95} -6.92820i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{5} - 6 q^{11} - 3 q^{15} + 2 q^{19} + 9 q^{25} - 2 q^{29} - 2 q^{31} + 12 q^{39} - 30 q^{41} + 18 q^{51} + 27 q^{55} - 2 q^{59} - 24 q^{61} - 16 q^{65} + 24 q^{69} + 12 q^{71} + 27 q^{75} - 14 q^{79}+ \cdots + 29 q^{95}+O(q^{100}) \) 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

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