LMFDB - Embedded newform 2940.2.f.b.1469.13

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2940,2,Mod(1469,2940)] 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("2940.1469"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2940, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2940.f (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [48] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$23.4760181943$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1469.13
Character	$\chi$	$=$	2940.1469
Dual form			2940.2.f.b.1469.15

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.03814 - 1.38645i) q^{3} +(-1.36379 - 1.77203i) q^{5} +(-0.844511 + 2.87868i) q^{9} -4.03724i q^{11} -4.67109 q^{13} +(-1.04103 + 3.73045i) q^{15} +6.20571i q^{17} -7.23153i q^{19} -2.35095 q^{23} +(-1.28017 + 4.83334i) q^{25} +(4.86788 - 1.81761i) q^{27} +5.15540i q^{29} -9.60664i q^{31} +(-5.59745 + 4.19124i) q^{33} -5.26896i q^{37} +(4.84927 + 6.47625i) q^{39} -9.85171 q^{41} +12.7596i q^{43} +(6.25284 - 2.42941i) q^{45} +7.80263i q^{47} +(8.60393 - 6.44243i) q^{51} +2.63982 q^{53} +(-7.15411 + 5.50593i) q^{55} +(-10.0262 + 7.50737i) q^{57} -4.28148 q^{59} -1.43225i q^{61} +(6.37037 + 8.27731i) q^{65} -3.05372i q^{67} +(2.44063 + 3.25949i) q^{69} +6.25374i q^{71} -2.25510 q^{73} +(8.03021 - 3.24280i) q^{75} +7.22379 q^{79} +(-7.57360 - 4.86216i) q^{81} -4.50946i q^{83} +(10.9967 - 8.46326i) q^{85} +(7.14772 - 5.35205i) q^{87} +11.8046 q^{89} +(-13.3192 + 9.97308i) q^{93} +(-12.8145 + 9.86226i) q^{95} -5.21085 q^{97} +(11.6219 + 3.40949i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 8 q^{9} - 16 q^{15} + 16 q^{25} + 56 q^{39} + 8 q^{51} + 48 q^{79} - 24 q^{81} + 32 q^{85} + 88 q^{99}+O(q^{100}) $ 48 * q - 8 * q^9 - 16 * q^15 + 16 * q^25 + 56 * q^39 + 8 * q^51 + 48 * q^79 - 24 * q^81 + 32 * q^85 + 88 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2940\mathbb{Z}\right)^\times$.

$n$	$1081$	$1177$	$1471$	$1961$
$\chi(n)$	$-1$	$-1$	$1$	$-1$

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
$3$	−1.03814	−	1.38645i	−0.599373	−	0.800470i
$3$	−1.03814	−	1.38645i	−0.599373	−	0.800470i
$4$		0			0
$5$	−1.36379	−	1.77203i	−0.609904	−	0.792475i
$5$	−1.36379	−	1.77203i	−0.609904	−	0.792475i
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−0.844511	+	2.87868i	−0.281504	+	0.959560i
$10$		0			0
$11$		−	4.03724i		−	1.21727i	−0.793449	−	0.608637i	$-0.791717\pi$
$11$		−	4.03724i		−	1.21727i	0.793449	−	0.608637i	$-0.208283\pi$
$12$		0			0
$13$	−4.67109			−1.29553			−0.647764	−	0.761841i	$-0.724296\pi$
$13$	−4.67109			−1.29553			−0.647764	+	0.761841i	$0.724296\pi$
$14$		0			0
$15$	−1.04103	+	3.73045i	−0.268793	+	0.963198i
$16$		0			0
$17$			6.20571i			1.50511i	0.658532	+	0.752553i	$0.271178\pi$
$17$			6.20571i			1.50511i	−0.658532	+	0.752553i	$0.728822\pi$
$18$		0			0
$19$		−	7.23153i		−	1.65903i	−0.558487	−	0.829513i	$-0.688618\pi$
$19$		−	7.23153i		−	1.65903i	0.558487	−	0.829513i	$-0.311382\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−2.35095			−0.490207			−0.245104	−	0.969497i	$-0.578822\pi$
$23$	−2.35095			−0.490207			−0.245104	+	0.969497i	$0.578822\pi$
$24$		0			0
$25$	−1.28017	+	4.83334i	−0.256035	+	0.966668i
$26$		0			0
$27$	4.86788	−	1.81761i	0.936825	−	0.349799i
$28$		0			0
$29$			5.15540i			0.957333i	0.877997	+	0.478667i	$0.158880\pi$
$29$			5.15540i			0.957333i	−0.877997	+	0.478667i	$0.841120\pi$
$30$		0			0
$31$		−	9.60664i		−	1.72540i	−0.505713	−	0.862702i	$-0.668770\pi$
$31$		−	9.60664i		−	1.72540i	0.505713	−	0.862702i	$-0.331230\pi$
$32$		0			0
$33$	−5.59745	+	4.19124i	−0.974391	+	0.729601i
$34$		0			0
$35$		0			0
$36$		0			0
$37$		−	5.26896i		−	0.866212i	−0.901343	−	0.433106i	$-0.857418\pi$
$37$		−	5.26896i		−	0.866212i	0.901343	−	0.433106i	$-0.142582\pi$
$38$		0			0
$39$	4.84927	+	6.47625i	0.776504	+	1.03703i
$40$		0			0
$41$	−9.85171			−1.53858			−0.769289	−	0.638900i	$-0.779390\pi$
$41$	−9.85171			−1.53858			−0.769289	+	0.638900i	$0.779390\pi$
$42$		0			0
$43$			12.7596i			1.94583i	0.231167	+	0.972914i	$0.425746\pi$
$43$			12.7596i			1.94583i	−0.231167	+	0.972914i	$0.574254\pi$
$44$		0			0
$45$	6.25284	−	2.42941i	0.932118	−	0.362155i
$46$		0			0
$47$			7.80263i			1.13813i	0.822292	+	0.569065i	$0.192695\pi$
$47$			7.80263i			1.13813i	−0.822292	+	0.569065i	$0.807305\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	8.60393	−	6.44243i	1.20479	−	0.902120i
$52$		0			0
$53$	2.63982			0.362607			0.181304	−	0.983427i	$-0.441968\pi$
$53$	2.63982			0.362607			0.181304	+	0.983427i	$0.441968\pi$
$54$		0			0
$55$	−7.15411	+	5.50593i	−0.964659	+	0.742420i
$56$		0			0
$57$	−10.0262	+	7.50737i	−1.32800	+	0.994376i
$58$		0			0
$59$	−4.28148			−0.557401			−0.278700	−	0.960378i	$-0.589904\pi$
$59$	−4.28148			−0.557401			−0.278700	+	0.960378i	$0.589904\pi$
$60$		0			0
$61$		−	1.43225i		−	0.183381i	−0.995788	−	0.0916904i	$-0.970773\pi$
$61$		−	1.43225i		−	0.183381i	0.995788	−	0.0916904i	$-0.0292270\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6.37037	+	8.27731i	0.790147	+	1.02667i
$66$		0			0
$67$		−	3.05372i		−	0.373071i	−0.982448	−	0.186535i	$-0.940274\pi$
$67$		−	3.05372i		−	0.373071i	0.982448	−	0.186535i	$-0.0597259\pi$
$68$		0			0
$69$	2.44063	+	3.25949i	0.293817	+	0.392396i
$70$		0			0
$71$			6.25374i			0.742182i	0.928596	+	0.371091i	$0.121016\pi$
$71$			6.25374i			0.742182i	−0.928596	+	0.371091i	$0.878984\pi$
$72$		0			0
$73$	−2.25510			−0.263939			−0.131970	−	0.991254i	$-0.542130\pi$
$73$	−2.25510			−0.263939			−0.131970	+	0.991254i	$0.542130\pi$
$74$		0			0
$75$	8.03021	−	3.24280i	0.927248	−	0.374447i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	7.22379			0.812740			0.406370	−	0.913709i	$-0.366794\pi$
$79$	7.22379			0.812740			0.406370	+	0.913709i	$0.366794\pi$
$80$		0			0
$81$	−7.57360	−	4.86216i	−0.841511	−	0.540239i
$82$		0			0
$83$		−	4.50946i		−	0.494977i	−0.968891	−	0.247489i	$-0.920395\pi$
$83$		−	4.50946i		−	0.494977i	0.968891	−	0.247489i	$-0.0796053\pi$
$84$		0			0
$85$	10.9967	−	8.46326i	1.19276	−	0.917970i
$86$		0			0
$87$	7.14772	−	5.35205i	0.766316	−	0.573800i
$88$		0			0
$89$	11.8046			1.25129			0.625644	−	0.780109i	$-0.284836\pi$
$89$	11.8046			1.25129			0.625644	+	0.780109i	$0.284836\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	−13.3192	+	9.97308i	−1.38113	+	1.03416i
$94$		0			0
$95$	−12.8145	+	9.86226i	−1.31474	+	1.01185i
$96$		0			0
$97$	−5.21085			−0.529082			−0.264541	−	0.964374i	$-0.585220\pi$
$97$	−5.21085			−0.529082			−0.264541	+	0.964374i	$0.585220\pi$
$98$		0			0
$99$	11.6219	+	3.40949i	1.16805	+	0.342667i
$100$		0			0
$101$	14.8957			1.48218			0.741090	−	0.671406i	$-0.234309\pi$
$101$	14.8957			1.48218			0.741090	+	0.671406i	$0.234309\pi$
$102$		0			0
$103$	2.75023			0.270988			0.135494	−	0.990778i	$-0.456738\pi$
$103$	2.75023			0.270988			0.135494	+	0.990778i	$0.456738\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2.85787			0.276280			0.138140	−	0.990413i	$-0.455888\pi$
$107$	2.85787			0.276280			0.138140	+	0.990413i	$0.455888\pi$
$108$		0			0
$109$	−4.43154			−0.424464			−0.212232	−	0.977219i	$-0.568073\pi$
$109$	−4.43154			−0.424464			−0.212232	+	0.977219i	$0.568073\pi$
$110$		0			0
$111$	−7.30517	+	5.46994i	−0.693376	+	0.519184i
$112$		0			0
$113$	2.38808			0.224651			0.112326	−	0.993671i	$-0.464170\pi$
$113$	2.38808			0.224651			0.112326	+	0.993671i	$0.464170\pi$
$114$		0			0
$115$	3.20620	+	4.16595i	0.298979	+	0.388477i
$116$		0			0
$117$	3.94479	−	13.4466i	0.364696	−	1.24314i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−5.29931			−0.481755
$122$		0			0
$123$	10.2275	+	13.6589i	0.922183	+	1.23159i
$124$		0			0
$125$	10.3107	−	4.32314i	0.922217	−	0.386673i
$126$		0			0
$127$			8.55351i			0.759002i	0.925191	+	0.379501i	$0.123904\pi$
$127$			8.55351i			0.759002i	−0.925191	+	0.379501i	$0.876096\pi$
$128$		0			0
$129$	17.6907	−	13.2464i	1.55758	−	1.16628i
$130$		0			0
$131$	−10.1699			−0.888549			−0.444274	−	0.895891i	$-0.646538\pi$
$131$	−10.1699			−0.888549			−0.444274	+	0.895891i	$0.646538\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−9.85961	−	6.14720i	−0.848580	−	0.529067i
$136$		0			0
$137$	−4.80546			−0.410558			−0.205279	−	0.978703i	$-0.565810\pi$
$137$	−4.80546			−0.410558			−0.205279	+	0.978703i	$0.565810\pi$
$138$		0			0
$139$			7.48829i			0.635148i	0.948233	+	0.317574i	$0.102868\pi$
$139$			7.48829i			0.635148i	−0.948233	+	0.317574i	$0.897132\pi$
$140$		0			0
$141$	10.8180	−	8.10026i	0.911039	−	0.682165i
$142$		0			0
$143$			18.8583i			1.57701i
$144$		0			0
$145$	9.13551	−	7.03086i	0.758663	−	0.583881i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	3.60852i		−	0.295622i	−0.989016	−	0.147811i	$-0.952777\pi$
$149$		−	3.60852i		−	0.295622i	0.989016	−	0.147811i	$-0.0472227\pi$
$150$		0			0
$151$	8.89582			0.723932			0.361966	−	0.932191i	$-0.382106\pi$
$151$	8.89582			0.723932			0.361966	+	0.932191i	$0.382106\pi$
$152$		0			0
$153$	−17.8643	−	5.24079i	−1.44424	−	0.423693i
$154$		0			0
$155$	−17.0232	+	13.1014i	−1.36734	+	1.05233i
$156$		0			0
$157$	13.8931			1.10879			0.554393	−	0.832255i	$-0.312950\pi$
$157$	13.8931			1.10879			0.554393	+	0.832255i	$0.312950\pi$
$158$		0			0
$159$	−2.74052	−	3.65999i	−0.217337	−	0.290256i
$160$		0			0
$161$		0			0
$162$		0			0
$163$		−	16.9428i		−	1.32706i	−0.748149	−	0.663531i	$-0.769057\pi$
$163$		−	16.9428i		−	1.32706i	0.748149	−	0.663531i	$-0.230943\pi$
$164$		0			0
$165$	15.0607	+	4.20289i	1.17248	+	0.327194i
$166$		0			0
$167$			5.46140i			0.422616i	0.977420	+	0.211308i	$0.0677723\pi$
$167$			5.46140i			0.422616i	−0.977420	+	0.211308i	$0.932228\pi$
$168$		0			0
$169$	8.81910			0.678392
$170$		0			0
$171$	20.8173	+	6.10711i	1.59194	+	0.467022i
$172$		0			0
$173$			21.2576i			1.61619i	0.589056	+	0.808093i	$0.299500\pi$
$173$			21.2576i			1.61619i	−0.589056	+	0.808093i	$0.700500\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	4.44479	+	5.93607i	0.334091	+	0.446183i
$178$		0			0
$179$		−	0.542133i		−	0.0405210i	−0.999795	−	0.0202605i	$-0.993550\pi$
$179$		−	0.542133i		−	0.0405210i	0.999795	−	0.0202605i	$-0.00644955\pi$
$180$		0			0
$181$			19.8749i			1.47729i	0.674094	+	0.738645i	$0.264534\pi$
$181$			19.8749i			1.47729i	−0.674094	+	0.738645i	$0.735466\pi$
$182$		0			0
$183$	−1.98575	+	1.48688i	−0.146791	+	0.109914i
$184$		0			0
$185$	−9.33675	+	7.18573i	−0.686451	+	0.528306i
$186$		0			0
$187$	25.0539			1.83213
$188$		0			0
$189$		0			0
$190$		0			0
$191$			18.0353i			1.30499i	0.757794	+	0.652494i	$0.226277\pi$
$191$			18.0353i			1.30499i	−0.757794	+	0.652494i	$0.773723\pi$
$192$		0			0
$193$		−	8.04340i		−	0.578977i	−0.957182	−	0.289488i	$-0.906515\pi$
$193$		−	8.04340i		−	0.578977i	0.957182	−	0.289488i	$-0.0934851\pi$
$194$		0			0
$195$	4.86274	−	17.4253i	0.348228	−	1.24785i
$196$		0			0
$197$	4.70893			0.335497			0.167749	−	0.985830i	$-0.446350\pi$
$197$	4.70893			0.335497			0.167749	+	0.985830i	$0.446350\pi$
$198$		0			0
$199$		−	0.712554i		−	0.0505116i	−0.999681	−	0.0252558i	$-0.991960\pi$
$199$		−	0.712554i		−	0.0505116i	0.999681	−	0.0252558i	$-0.00804003\pi$
$200$		0			0
$201$	−4.23384	+	3.17020i	−0.298632	+	0.223609i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	13.4356	+	17.4575i	0.938385	+	1.21929i
$206$		0			0
$207$	1.98540	−	6.76764i	0.137995	−	0.470383i
$208$		0			0
$209$	−29.1954			−2.01949
$210$		0			0
$211$	13.6381			0.938888			0.469444	−	0.882962i	$-0.344454\pi$
$211$	13.6381			0.938888			0.469444	+	0.882962i	$0.344454\pi$
$212$		0			0
$213$	8.67052	−	6.49228i	0.594094	−	0.444844i
$214$		0			0
$215$	22.6105	−	17.4014i	1.54202	−	1.18677i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	2.34112	+	3.12659i	0.158198	+	0.211275i
$220$		0			0
$221$		−	28.9874i		−	1.94991i
$222$		0			0
$223$	17.9291			1.20062			0.600310	−	0.799767i	$-0.295044\pi$
$223$	17.9291			1.20062			0.600310	+	0.799767i	$0.295044\pi$
$224$		0			0
$225$	−12.8325	−	7.76702i	−0.855501	−	0.517801i
$226$		0			0
$227$			8.40694i			0.557988i	0.960293	+	0.278994i	$0.0900010\pi$
$227$			8.40694i			0.557988i	−0.960293	+	0.278994i	$0.909999\pi$
$228$		0			0
$229$			21.1352i			1.39665i	0.715781	+	0.698325i	$0.246071\pi$
$229$			21.1352i			1.39665i	−0.715781	+	0.698325i	$0.753929\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	3.66241			0.239933			0.119966	−	0.992778i	$-0.461721\pi$
$233$	3.66241			0.239933			0.119966	+	0.992778i	$0.461721\pi$
$234$		0			0
$235$	13.8265	−	10.6411i	0.901941	−	0.694150i
$236$		0			0
$237$	−7.49934	−	10.0155i	−0.487135	−	0.650574i
$238$		0			0
$239$			7.74023i			0.500674i	0.968159	+	0.250337i	$0.0805414\pi$
$239$			7.74023i			0.500674i	−0.968159	+	0.250337i	$0.919459\pi$
$240$		0			0
$241$		−	18.6726i		−	1.20281i	−0.798946	−	0.601403i	$-0.794609\pi$
$241$		−	18.6726i		−	1.20281i	0.798946	−	0.601403i	$-0.205391\pi$
$242$		0			0
$243$	1.12134	+	15.5481i	0.0719339	+	0.997409i
$244$		0			0
$245$		0			0
$246$		0			0
$247$			33.7791i			2.14931i
$248$		0			0
$249$	−6.25216	+	4.68147i	−0.396214	+	0.296676i
$250$		0			0
$251$	−0.739961			−0.0467059			−0.0233530	−	0.999727i	$-0.507434\pi$
$251$	−0.739961			−0.0467059			−0.0233530	+	0.999727i	$0.507434\pi$
$252$		0			0
$253$			9.49135i			0.596716i
$254$		0			0
$255$	−23.1501	−	6.46033i	−1.44972	−	0.404561i
$256$		0			0
$257$		−	10.4422i		−	0.651369i	−0.945478	−	0.325685i	$-0.894405\pi$
$257$		−	10.4422i		−	0.651369i	0.945478	−	0.325685i	$-0.105595\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−14.8407	−	4.35379i	−0.918619	−	0.269493i
$262$		0			0
$263$	−17.0253			−1.04982			−0.524912	−	0.851157i	$-0.675902\pi$
$263$	−17.0253			−1.04982			−0.524912	+	0.851157i	$0.675902\pi$
$264$		0			0
$265$	−3.60015	−	4.67784i	−0.221156	−	0.287357i
$266$		0			0
$267$	−12.2549	−	16.3666i	−0.749989	−	1.00162i
$268$		0			0
$269$	−22.8718			−1.39452			−0.697260	−	0.716819i	$-0.745598\pi$
$269$	−22.8718			−1.39452			−0.697260	+	0.716819i	$0.745598\pi$
$270$		0			0
$271$		−	8.24933i		−	0.501111i	−0.968102	−	0.250556i	$-0.919387\pi$
$271$		−	8.24933i		−	0.501111i	0.968102	−	0.250556i	$-0.0806133\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	19.5133	+	5.16837i	1.17670	+	0.311664i
$276$		0			0
$277$			6.62029i			0.397775i	0.980022	+	0.198887i	$0.0637328\pi$
$277$			6.62029i			0.397775i	−0.980022	+	0.198887i	$0.936267\pi$
$278$		0			0
$279$	27.6545	+	8.11292i	1.65563	+	0.485708i
$280$		0			0
$281$			4.02611i			0.240178i	0.992763	+	0.120089i	$0.0383179\pi$
$281$			4.02611i			0.240178i	−0.992763	+	0.120089i	$0.961682\pi$
$282$		0			0
$283$	18.4115			1.09445			0.547224	−	0.836986i	$-0.315685\pi$
$283$	18.4115			1.09445			0.547224	+	0.836986i	$0.315685\pi$
$284$		0			0
$285$	26.9769	+	7.52823i	1.59797	+	0.445934i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−21.5108			−1.26534
$290$		0			0
$291$	5.40962	+	7.22460i	0.317117	+	0.423514i
$292$		0			0
$293$			20.7576i			1.21267i	0.795210	+	0.606335i	$0.207361\pi$
$293$			20.7576i			1.21267i	−0.795210	+	0.606335i	$0.792639\pi$
$294$		0			0
$295$	5.83902	+	7.58690i	0.339961	+	0.441727i
$296$		0			0
$297$	−7.33813	−	19.6528i	−0.425802	−	1.14037i
$298$		0			0
$299$	10.9815			0.635077
$300$		0			0
$301$		0			0
$302$		0			0
$303$	−15.4639	−	20.6522i	−0.888379	−	1.18644i
$304$		0			0
$305$	−2.53799	+	1.95328i	−0.145325	+	0.111845i
$306$		0			0
$307$	−10.9589			−0.625456			−0.312728	−	0.949843i	$-0.601243\pi$
$307$	−10.9589			−0.625456			−0.312728	+	0.949843i	$0.601243\pi$
$308$		0			0
$309$	−2.85514	−	3.81307i	−0.162423	−	0.216918i
$310$		0			0
$311$	−6.93549			−0.393276			−0.196638	−	0.980476i	$-0.563002\pi$
$311$	−6.93549			−0.393276			−0.196638	+	0.980476i	$0.563002\pi$
$312$		0			0
$313$	−28.8147			−1.62871			−0.814353	−	0.580370i	$-0.802908\pi$
$313$	−28.8147			−1.62871			−0.814353	+	0.580370i	$0.802908\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	13.3468			0.749629			0.374815	−	0.927100i	$-0.377706\pi$
$317$	13.3468			0.749629			0.374815	+	0.927100i	$0.377706\pi$
$318$		0			0
$319$	20.8136			1.16534
$320$		0			0
$321$	−2.96688	−	3.96230i	−0.165595	−	0.221154i
$322$		0			0
$323$	44.8768			2.49701
$324$		0			0
$325$	5.97981	−	22.5770i	0.331700	−	1.25234i
$326$		0			0
$327$	4.60058	+	6.14412i	0.254412	+	0.339771i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	2.06067			0.113264			0.0566322	−	0.998395i	$-0.481964\pi$
$331$	2.06067			0.113264			0.0566322	+	0.998395i	$0.481964\pi$
$332$		0			0
$333$	15.1676	+	4.44969i	0.831182	+	0.243842i
$334$		0			0
$335$	−5.41128	+	4.16462i	−0.295650	+	0.227537i
$336$		0			0
$337$		−	27.3577i		−	1.49027i	−0.666913	−	0.745136i	$-0.732385\pi$
$337$		−	27.3577i		−	1.49027i	0.666913	−	0.745136i	$-0.267615\pi$
$338$		0			0
$339$	−2.47917	−	3.31096i	−0.134650	−	0.179827i
$340$		0			0
$341$	−38.7843			−2.10029
$342$		0			0
$343$		0			0
$344$		0			0
$345$	2.44741	−	8.77011i	0.131764	−	0.472167i
$346$		0			0
$347$	−19.9356			−1.07020			−0.535100	−	0.844789i	$-0.679726\pi$
$347$	−19.9356			−1.07020			−0.535100	+	0.844789i	$0.679726\pi$
$348$		0			0
$349$		−	21.0952i		−	1.12920i	−0.825365	−	0.564600i	$-0.809031\pi$
$349$		−	21.0952i		−	1.12920i	0.825365	−	0.564600i	$-0.190969\pi$
$350$		0			0
$351$	−22.7383	+	8.49023i	−1.21368	+	0.453175i
$352$		0			0
$353$			35.8277i			1.90692i	0.301526	+	0.953458i	$0.402504\pi$
$353$			35.8277i			1.90692i	−0.301526	+	0.953458i	$0.597496\pi$
$354$		0			0
$355$	11.0818	−	8.52876i	0.588161	−	0.452660i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			21.1649i			1.11704i	0.829490	+	0.558521i	$0.188631\pi$
$359$			21.1649i			1.11704i	−0.829490	+	0.558521i	$0.811369\pi$
$360$		0			0
$361$	−33.2950			−1.75237
$362$		0			0
$363$	5.50145	+	7.34725i	0.288751	+	0.385630i
$364$		0			0
$365$	3.07547	+	3.99610i	0.160978	+	0.209165i
$366$		0			0
$367$	−24.2427			−1.26546			−0.632728	−	0.774374i	$-0.718065\pi$
$367$	−24.2427			−1.26546			−0.632728	+	0.774374i	$0.718065\pi$
$368$		0			0
$369$	8.31988	−	28.3599i	0.433116	−	1.47636i
$370$		0			0
$371$		0			0
$372$		0			0
$373$		−	9.75627i		−	0.505161i	−0.967576	−	0.252580i	$-0.918721\pi$
$373$		−	9.75627i		−	0.505161i	0.967576	−	0.252580i	$-0.0812792\pi$
$374$		0			0
$375$	−16.6978	−	9.80727i	−0.862272	−	0.506445i
$376$		0			0
$377$		−	24.0813i		−	1.24025i
$378$		0			0
$379$	−17.7885			−0.913731			−0.456866	−	0.889536i	$-0.651028\pi$
$379$	−17.7885			−0.913731			−0.456866	+	0.889536i	$0.651028\pi$
$380$		0			0
$381$	11.8591	−	8.87979i	0.607558	−	0.454925i
$382$		0			0
$383$			31.8208i			1.62597i	0.582287	+	0.812983i	$0.302158\pi$
$383$			31.8208i			1.62597i	−0.582287	+	0.812983i	$0.697842\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−36.7310	−	10.7757i	−1.86714	−	0.547758i
$388$		0			0
$389$		−	8.81090i		−	0.446730i	−0.974735	−	0.223365i	$-0.928296\pi$
$389$		−	8.81090i		−	0.446730i	0.974735	−	0.223365i	$-0.0717042\pi$
$390$		0			0
$391$		−	14.5893i		−	0.737814i
$392$		0			0
$393$	10.5578	+	14.1001i	0.532572	+	0.711256i
$394$		0			0
$395$	−9.85171	−	12.8008i	−0.495693	−	0.644077i
$396$		0			0
$397$	−36.8176			−1.84782			−0.923912	−	0.382606i	$-0.875027\pi$
$397$	−36.8176			−1.84782			−0.923912	+	0.382606i	$0.875027\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			3.41074i			0.170324i	0.996367	+	0.0851622i	$0.0271408\pi$
$401$			3.41074i			0.170324i	−0.996367	+	0.0851622i	$0.972859\pi$
$402$		0			0
$403$			44.8735i			2.23531i
$404$		0			0
$405$	1.71289	+	20.0516i	0.0851144	+	0.996371i
$406$		0			0
$407$	−21.2721			−1.05442
$408$		0			0
$409$			26.2634i			1.29864i	0.760515	+	0.649321i	$0.224947\pi$
$409$			26.2634i			1.29864i	−0.760515	+	0.649321i	$0.775053\pi$
$410$		0			0
$411$	4.98876	+	6.66255i	0.246077	+	0.328639i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−7.99089	+	6.14994i	−0.392257	+	0.301888i
$416$		0			0
$417$	10.3822	−	7.77393i	0.508417	−	0.380691i
$418$		0			0
$419$	26.3452			1.28705			0.643524	−	0.765426i	$-0.277472\pi$
$419$	26.3452			1.28705			0.643524	+	0.765426i	$0.277472\pi$
$420$		0			0
$421$	−15.3039			−0.745868			−0.372934	−	0.927858i	$-0.621648\pi$
$421$	−15.3039			−0.745868			−0.372934	+	0.927858i	$0.621648\pi$
$422$		0			0
$423$	−22.4613	−	6.58941i	−1.09211	−	0.320388i
$424$		0			0
$425$	−29.9943	−	7.94439i	−1.45494	−	0.385359i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	26.1462	−	19.5777i	1.26235	−	0.945218i
$430$		0			0
$431$		−	21.4128i		−	1.03142i	−0.856764	−	0.515709i	$-0.827528\pi$
$431$		−	21.4128i		−	1.03142i	0.856764	−	0.515709i	$-0.172472\pi$
$432$		0			0
$433$	−7.29090			−0.350378			−0.175189	−	0.984535i	$-0.556054\pi$
$433$	−7.29090			−0.350378			−0.175189	+	0.984535i	$0.556054\pi$
$434$		0			0
$435$	−19.2319	−	5.36692i	−0.922101	−	0.257324i
$436$		0			0
$437$			17.0010i			0.813267i
$438$		0			0
$439$		−	5.38824i		−	0.257167i	−0.991699	−	0.128583i	$-0.958957\pi$
$439$		−	5.38824i		−	0.257167i	0.991699	−	0.128583i	$-0.0410430\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−12.3836			−0.588362			−0.294181	−	0.955750i	$-0.595047\pi$
$443$	−12.3836			−0.588362			−0.294181	+	0.955750i	$0.595047\pi$
$444$		0			0
$445$	−16.0990	−	20.9181i	−0.763165	−	0.991615i
$446$		0			0
$447$	−5.00305	+	3.74617i	−0.236636	+	0.177188i
$448$		0			0
$449$			6.42837i			0.303374i	0.988429	+	0.151687i	$0.0484705\pi$
$449$			6.42837i			0.303374i	−0.988429	+	0.151687i	$0.951529\pi$
$450$		0			0
$451$			39.7737i			1.87287i
$452$		0			0
$453$	−9.23515	−	12.3337i	−0.433905	−	0.579485i
$454$		0			0
$455$		0			0
$456$		0			0
$457$		−	11.2063i		−	0.524206i	−0.965040	−	0.262103i	$-0.915584\pi$
$457$		−	11.2063i		−	0.524206i	0.965040	−	0.262103i	$-0.0844161\pi$
$458$		0			0
$459$	11.2796	+	30.2087i	0.526485	+	1.41002i
$460$		0			0
$461$	−6.62043			−0.308344			−0.154172	−	0.988044i	$-0.549271\pi$
$461$	−6.62043			−0.308344			−0.154172	+	0.988044i	$0.549271\pi$
$462$		0			0
$463$			24.0672i			1.11850i	0.829000	+	0.559248i	$0.188910\pi$
$463$			24.0672i			1.11850i	−0.829000	+	0.559248i	$0.811090\pi$
$464$		0			0
$465$	35.8371	+	10.0008i	1.66191	+	0.463776i
$466$		0			0
$467$		−	23.3414i		−	1.08011i	−0.841629	−	0.540055i	$-0.818403\pi$
$467$		−	23.3414i		−	1.08011i	0.841629	−	0.540055i	$-0.181597\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−14.4230	−	19.2621i	−0.664577	−	0.887550i
$472$		0			0
$473$	51.5138			2.36861
$474$		0			0
$475$	34.9524	+	9.25761i	1.60373	+	0.424768i
$476$		0			0
$477$	−2.22936	+	7.59920i	−0.102075	+	0.347944i
$478$		0			0
$479$	−10.2157			−0.466765			−0.233383	−	0.972385i	$-0.574979\pi$
$479$	−10.2157			−0.466765			−0.233383	+	0.972385i	$0.574979\pi$
$480$		0			0
$481$			24.6118i			1.12220i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	7.10649	+	9.23378i	0.322689	+	0.419284i
$486$		0			0
$487$			12.8158i			0.580738i	0.956915	+	0.290369i	$0.0937780\pi$
$487$			12.8158i			0.580738i	−0.956915	+	0.290369i	$0.906222\pi$
$488$		0			0
$489$	−23.4904	+	17.5891i	−1.06227	+	0.795405i
$490$		0			0
$491$		−	32.8628i		−	1.48308i	−0.670910	−	0.741539i	$-0.734096\pi$
$491$		−	32.8628i		−	1.48308i	0.670910	−	0.741539i	$-0.265904\pi$
$492$		0			0
$493$	−31.9929			−1.44089
$494$		0			0
$495$	−9.80810	−	25.2442i	−0.440841	−	1.13464i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	10.0650			0.450570			0.225285	−	0.974293i	$-0.427669\pi$
$499$	10.0650			0.450570			0.225285	+	0.974293i	$0.427669\pi$
$500$		0			0
$501$	7.57199	−	5.66973i	0.338291	−	0.253305i
$502$		0			0
$503$			9.82025i			0.437863i	0.975740	+	0.218932i	$0.0702572\pi$
$503$			9.82025i			0.437863i	−0.975740	+	0.218932i	$0.929743\pi$
$504$		0			0
$505$	−20.3146	−	26.3957i	−0.903987	−	1.17459i
$506$		0			0
$507$	−9.15550	−	12.2273i	−0.406610	−	0.543032i
$508$		0			0
$509$	−17.7162			−0.785258			−0.392629	−	0.919697i	$-0.628434\pi$
$509$	−17.7162			−0.785258			−0.392629	+	0.919697i	$0.628434\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−13.1441	−	35.2022i	−0.580326	−	1.55422i
$514$		0			0
$515$	−3.75073	−	4.87349i	−0.165277	−	0.214752i
$516$		0			0
$517$	31.5011			1.38542
$518$		0			0
$519$	29.4727	−	22.0685i	1.29371	−	0.968698i
$520$		0			0
$521$	−13.0312			−0.570906			−0.285453	−	0.958393i	$-0.592144\pi$
$521$	−13.0312			−0.570906			−0.285453	+	0.958393i	$0.592144\pi$
$522$		0			0
$523$	−35.2957			−1.54338			−0.771688	−	0.636002i	$-0.780587\pi$
$523$	−35.2957			−1.54338			−0.771688	+	0.636002i	$0.780587\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	59.6160			2.59692
$528$		0			0
$529$	−17.4730			−0.759697
$530$		0			0
$531$	3.61576	−	12.3250i	0.156910	−	0.534860i
$532$		0			0
$533$	46.0182			1.99327
$534$		0			0
$535$	−3.89752	−	5.06422i	−0.168504	−	0.218945i
$536$		0			0
$537$	−0.751643	+	0.562813i	−0.0324358	+	0.0242872i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	42.9924			1.84839			0.924195	−	0.381921i	$-0.124737\pi$
$541$	42.9924			1.84839			0.924195	+	0.381921i	$0.124737\pi$
$542$		0			0
$543$	27.5557	−	20.6330i	1.18253	−	0.885448i
$544$		0			0
$545$	6.04367	+	7.85281i	0.258882	+	0.336377i
$546$		0			0
$547$		−	20.4524i		−	0.874482i	−0.899344	−	0.437241i	$-0.855956\pi$
$547$		−	20.4524i		−	0.874482i	0.899344	−	0.437241i	$-0.144044\pi$
$548$		0			0
$549$	4.12299	+	1.20955i	0.175965	+	0.0516224i
$550$		0			0
$551$	37.2814			1.58824
$552$		0			0
$553$		0			0
$554$		0			0
$555$	19.6556	+	5.48514i	0.834333	+	0.232831i
$556$		0			0
$557$	−40.5528			−1.71828			−0.859140	−	0.511741i	$-0.829001\pi$
$557$	−40.5528			−1.71828			−0.859140	+	0.511741i	$0.829001\pi$
$558$		0			0
$559$		−	59.6015i		−	2.52087i
$560$		0			0
$561$	−26.0096	−	34.7361i	−1.09813	−	1.46656i
$562$		0			0
$563$		−	24.7264i		−	1.04210i	−0.853528	−	0.521048i	$-0.825541\pi$
$563$		−	24.7264i		−	1.04210i	0.853528	−	0.521048i	$-0.174459\pi$
$564$		0			0
$565$	−3.25683	−	4.23174i	−0.137016	−	0.178031i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	25.1381i		−	1.05384i	−0.849914	−	0.526922i	$-0.823346\pi$
$569$		−	25.1381i		−	1.05384i	0.849914	−	0.526922i	$-0.176654\pi$
$570$		0			0
$571$	3.42569			0.143361			0.0716804	−	0.997428i	$-0.477164\pi$
$571$	3.42569			0.143361			0.0716804	+	0.997428i	$0.477164\pi$
$572$		0			0
$573$	25.0051	−	18.7232i	1.04460	−	0.782174i
$574$		0			0
$575$	3.00963	−	11.3629i	0.125510	−	0.473867i
$576$		0			0
$577$	14.8293			0.617351			0.308676	−	0.951167i	$-0.400114\pi$
$577$	14.8293			0.617351			0.308676	+	0.951167i	$0.400114\pi$
$578$		0			0
$579$	−11.1518	+	8.35021i	−0.463453	+	0.347023i
$580$		0			0
$581$		0			0
$582$		0			0
$583$		−	10.6576i		−	0.441392i
$584$		0			0
$585$	−29.2076	+	11.3480i	−1.20758	+	0.469181i
$586$		0			0
$587$			41.4516i			1.71089i	0.517895	+	0.855444i	$0.326716\pi$
$587$			41.4516i			1.71089i	−0.517895	+	0.855444i	$0.673284\pi$
$588$		0			0
$589$	−69.4707			−2.86249
$590$		0			0
$591$	−4.88855	−	6.52871i	−0.201088	−	0.268555i
$592$		0			0
$593$			45.6599i			1.87503i	0.347946	+	0.937514i	$0.386879\pi$
$593$			45.6599i			1.87503i	−0.347946	+	0.937514i	$0.613121\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	−0.987923	+	0.739734i	−0.0404330	+	0.0302753i
$598$		0			0
$599$			9.91010i			0.404916i	0.979291	+	0.202458i	$0.0648929\pi$
$599$			9.91010i			0.404916i	−0.979291	+	0.202458i	$0.935107\pi$
$600$		0			0
$601$		−	3.39000i		−	0.138281i	−0.997607	−	0.0691404i	$-0.977974\pi$
$601$		−	3.39000i		−	0.138281i	0.997607	−	0.0691404i	$-0.0220257\pi$
$602$		0			0
$603$	8.79068	+	2.57890i	0.357984	+	0.105021i
$604$		0			0
$605$	7.22712	+	9.39052i	0.293824	+	0.381779i
$606$		0			0
$607$	−2.63410			−0.106915			−0.0534575	−	0.998570i	$-0.517024\pi$
$607$	−2.63410			−0.106915			−0.0534575	+	0.998570i	$0.517024\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	36.4468i		−	1.47448i
$612$		0			0
$613$			0.134037i			0.00541370i	0.999996	+	0.00270685i	$0.000861618\pi$
$613$			0.134037i			0.00541370i	−0.999996	+	0.00270685i	$0.999138\pi$
$614$		0			0
$615$	10.2559	−	36.7513i	0.413559	−	1.48196i
$616$		0			0
$617$	30.8209			1.24080			0.620401	−	0.784285i	$-0.286970\pi$
$617$	30.8209			1.24080			0.620401	+	0.784285i	$0.286970\pi$
$618$		0			0
$619$			33.3026i			1.33855i	0.743017	+	0.669273i	$0.233394\pi$
$619$			33.3026i			1.33855i	−0.743017	+	0.669273i	$0.766606\pi$
$620$		0			0
$621$	−11.4442	+	4.27311i	−0.459238	+	0.171474i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−21.7223	−	12.3750i	−0.868892	−	0.495001i
$626$		0			0
$627$	30.3091	+	40.4781i	1.21043	+	1.61654i
$628$		0			0
$629$	32.6976			1.30374
$630$		0			0
$631$	14.4369			0.574725			0.287362	−	0.957822i	$-0.407222\pi$
$631$	14.4369			0.574725			0.287362	+	0.957822i	$0.407222\pi$
$632$		0			0
$633$	−14.1584	−	18.9087i	−0.562745	−	0.751552i
$634$		0			0
$635$	15.1571	−	11.6652i	0.601490	−	0.462918i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−18.0025	−	5.28135i	−0.712168	−	0.208927i
$640$		0			0
$641$			37.1952i			1.46912i	0.678544	+	0.734560i	$0.262611\pi$
$641$			37.1952i			1.46912i	−0.678544	+	0.734560i	$0.737389\pi$
$642$		0			0
$643$	−29.4169			−1.16009			−0.580045	−	0.814585i	$-0.696965\pi$
$643$	−29.4169			−1.16009			−0.580045	+	0.814585i	$0.696965\pi$
$644$		0			0
$645$	−47.5992	−	13.2832i	−1.87422	−	0.523024i
$646$		0			0
$647$		−	23.2454i		−	0.913871i	−0.889500	−	0.456936i	$-0.848947\pi$
$647$		−	23.2454i		−	0.913871i	0.889500	−	0.456936i	$-0.151053\pi$
$648$		0			0
$649$			17.2854i			0.678509i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−45.1744			−1.76781			−0.883906	−	0.467664i	$-0.845096\pi$
$653$	−45.1744			−1.76781			−0.883906	+	0.467664i	$0.845096\pi$
$654$		0			0
$655$	13.8696	+	18.0214i	0.541929	+	0.704153i
$656$		0			0
$657$	1.90446	−	6.49171i	0.0742999	−	0.253266i
$658$		0			0
$659$		−	12.0020i		−	0.467533i	−0.972293	−	0.233766i	$-0.924895\pi$
$659$		−	12.0020i		−	0.467533i	0.972293	−	0.233766i	$-0.0751051\pi$
$660$		0			0
$661$		−	7.06979i		−	0.274983i	−0.990503	−	0.137491i	$-0.956096\pi$
$661$		−	7.06979i		−	0.274983i	0.990503	−	0.137491i	$-0.0439040\pi$
$662$		0			0
$663$	−40.1898	+	30.0932i	−1.56084	+	1.16872i
$664$		0			0
$665$		0			0
$666$		0			0
$667$		−	12.1201i		−	0.469292i
$668$		0			0
$669$	−18.6130	−	24.8579i	−0.719619	−	0.961060i
$670$		0			0
$671$	−5.78234			−0.223225
$672$		0			0
$673$			6.51616i			0.251179i	0.992082	+	0.125590i	$0.0400823\pi$
$673$			6.51616i			0.251179i	−0.992082	+	0.125590i	$0.959918\pi$
$674$		0			0
$675$	2.55339	+	25.8550i	0.0982801	+	0.995159i
$676$		0			0
$677$		−	37.1401i		−	1.42741i	−0.700446	−	0.713705i	$-0.747016\pi$
$677$		−	37.1401i		−	1.42741i	0.700446	−	0.713705i	$-0.252984\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	11.6558	−	8.72762i	0.446653	−	0.334443i
$682$		0			0
$683$	−33.4835			−1.28121			−0.640606	−	0.767870i	$-0.721317\pi$
$683$	−33.4835			−1.28121			−0.640606	+	0.767870i	$0.721317\pi$
$684$		0			0
$685$	6.55362	+	8.51541i	0.250401	+	0.325357i
$686$		0			0
$687$	29.3029	−	21.9414i	1.11798	−	0.837115i
$688$		0			0
$689$	−12.3308			−0.469768
$690$		0			0
$691$		−	14.3909i		−	0.547454i	−0.961807	−	0.273727i	$-0.911743\pi$
$691$		−	14.3909i		−	0.547454i	0.961807	−	0.273727i	$-0.0882566\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	13.2695	−	10.2124i	0.503339	−	0.387379i
$696$		0			0
$697$		−	61.1369i		−	2.31572i
$698$		0			0
$699$	−3.80212	−	5.07777i	−0.143809	−	0.192059i
$700$		0			0
$701$			38.0575i			1.43741i	0.695314	+	0.718707i	$0.255266\pi$
$701$			38.0575i			1.43741i	−0.695314	+	0.718707i	$0.744734\pi$
$702$		0			0
$703$	−38.1026			−1.43707
$704$		0			0
$705$	−29.1073	−	8.12277i	−1.09625	−	0.305921i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−26.2027			−0.984061			−0.492031	−	0.870578i	$-0.663745\pi$
$709$	−26.2027			−0.984061			−0.492031	+	0.870578i	$0.663745\pi$
$710$		0			0
$711$	−6.10057	+	20.7950i	−0.228789	+	0.779873i
$712$		0			0
$713$			22.5847i			0.845805i
$714$		0			0
$715$	33.4175	−	25.7187i	1.24974	−	0.961825i
$716$		0			0
$717$	10.7315	−	8.03548i	0.400774	−	0.300090i
$718$		0			0
$719$	−28.5912			−1.06627			−0.533137	−	0.846029i	$-0.678987\pi$
$719$	−28.5912			−1.06627			−0.533137	+	0.846029i	$0.678987\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	−25.8887	+	19.3848i	−0.962810	+	0.720929i
$724$		0			0
$725$	−24.9178	−	6.59980i	−0.925423	−	0.245110i
$726$		0			0
$727$	11.8333			0.438874			0.219437	−	0.975627i	$-0.429578\pi$
$727$	11.8333			0.438874			0.219437	+	0.975627i	$0.429578\pi$
$728$		0			0
$729$	20.3926	−	17.6958i	0.755281	−	0.655401i
$730$		0			0
$731$	−79.1827			−2.92868
$732$		0			0
$733$	−45.1892			−1.66910			−0.834551	−	0.550930i	$-0.814273\pi$
$733$	−45.1892			−1.66910			−0.834551	+	0.550930i	$0.814273\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−12.3286			−0.454129
$738$		0			0
$739$	−32.4888			−1.19512			−0.597560	−	0.801824i	$-0.703863\pi$
$739$	−32.4888			−1.19512			−0.597560	+	0.801824i	$0.703863\pi$
$740$		0			0
$741$	46.8332	−	35.0676i	1.72046	−	1.28824i
$742$		0			0
$743$	3.76749			0.138216			0.0691079	−	0.997609i	$-0.477985\pi$
$743$	3.76749			0.138216			0.0691079	+	0.997609i	$0.477985\pi$
$744$		0			0
$745$	−6.39440	+	4.92125i	−0.234273	+	0.180301i
$746$		0			0
$747$	12.9813	+	3.80829i	0.474960	+	0.139338i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−14.6206			−0.533512			−0.266756	−	0.963764i	$-0.585952\pi$
$751$	−14.6206			−0.533512			−0.266756	+	0.963764i	$0.585952\pi$
$752$		0			0
$753$	0.768186	+	1.02592i	0.0279943	+	0.0373867i
$754$		0			0
$755$	−12.1320	−	15.7637i	−0.441529	−	0.573698i
$756$		0			0
$757$			27.8418i			1.01193i	0.862554	+	0.505964i	$0.168863\pi$
$757$			27.8418i			1.01193i	−0.862554	+	0.505964i	$0.831137\pi$
$758$		0			0
$759$	13.1593	−	9.85340i	0.477653	−	0.357656i
$760$		0			0
$761$	10.5334			0.381837			0.190918	−	0.981606i	$-0.438853\pi$
$761$	10.5334			0.381837			0.190918	+	0.981606i	$0.438853\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	15.0762	+	38.8033i	0.545081	+	1.40294i
$766$		0			0
$767$	19.9992			0.722128
$768$		0			0
$769$			44.4440i			1.60269i	0.598201	+	0.801346i	$0.295882\pi$
$769$			44.4440i			1.60269i	−0.598201	+	0.801346i	$0.704118\pi$
$770$		0			0
$771$	−14.4777	+	10.8406i	−0.521401	+	0.390413i
$772$		0			0
$773$			22.1487i			0.796633i	0.917248	+	0.398317i	$0.130405\pi$
$773$			22.1487i			0.796633i	−0.917248	+	0.398317i	$0.869595\pi$
$774$		0			0
$775$	46.4321	+	12.2982i	1.66789	+	0.441763i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			71.2429i			2.55254i
$780$		0			0
$781$	25.2478			0.903439
$782$		0			0
$783$	9.37050	+	25.0959i	0.334874	+	0.896853i
$784$		0			0
$785$	−18.9472	−	24.6189i	−0.676253	−	0.878686i
$786$		0			0
$787$	8.31246			0.296307			0.148154	−	0.988964i	$-0.452667\pi$
$787$	8.31246			0.296307			0.148154	+	0.988964i	$0.452667\pi$
$788$		0			0
$789$	17.6747	+	23.6048i	0.629236	+	0.840352i
$790$		0			0
$791$		0			0
$792$		0			0
$793$			6.69017i			0.237575i
$794$		0			0
$795$	−2.74813	+	9.84772i	−0.0974662	+	0.349263i
$796$		0			0
$797$			7.75614i			0.274736i	0.990520	+	0.137368i	$0.0438644\pi$
$797$			7.75614i			0.274736i	−0.990520	+	0.137368i	$0.956136\pi$
$798$		0			0
$799$	−48.4209			−1.71301
$800$		0			0
$801$	−9.96914	+	33.9818i	−0.352242	+	1.20069i
$802$		0			0
$803$			9.10437i			0.321286i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	23.7443	+	31.7107i	0.835837	+	1.11627i
$808$		0			0
$809$		−	47.6690i		−	1.67595i	−0.545705	−	0.837977i	$-0.683738\pi$
$809$		−	47.6690i		−	1.67595i	0.545705	−	0.837977i	$-0.316262\pi$
$810$		0			0
$811$			10.0943i			0.354458i	0.984170	+	0.177229i	$0.0567133\pi$
$811$			10.0943i			0.354458i	−0.984170	+	0.177229i	$0.943287\pi$
$812$		0			0
$813$	−11.4373	+	8.56400i	−0.401124	+	0.300353i
$814$		0			0
$815$	−30.0231	+	23.1063i	−1.05166	+	0.809380i
$816$		0			0
$817$	92.2718			3.22818
$818$		0			0
$819$		0			0
$820$		0			0
$821$			38.3807i			1.33950i	0.742588	+	0.669749i	$0.233598\pi$
$821$			38.3807i			1.33950i	−0.742588	+	0.669749i	$0.766402\pi$
$822$		0			0
$823$			2.78821i			0.0971907i	0.998819	+	0.0485954i	$0.0154745\pi$
$823$			2.78821i			0.0971907i	−0.998819	+	0.0485954i	$0.984526\pi$
$824$		0			0
$825$	−13.0920	−	32.4199i	−0.455804	−	1.12872i
$826$		0			0
$827$	−36.9569			−1.28512			−0.642559	−	0.766236i	$-0.722127\pi$
$827$	−36.9569			−1.28512			−0.642559	+	0.766236i	$0.722127\pi$
$828$		0			0
$829$		−	2.22738i		−	0.0773601i	−0.999252	−	0.0386801i	$-0.987685\pi$
$829$		−	2.22738i		−	0.0773601i	0.999252	−	0.0386801i	$-0.0123153\pi$
$830$		0			0
$831$	9.17873	−	6.87282i	0.318407	−	0.238415i
$832$		0			0
$833$		0			0
$834$		0			0
$835$	9.67777	−	7.44819i	0.334913	−	0.257755i
$836$		0			0
$837$	−17.4611	−	46.7640i	−0.603545	−	1.61640i
$838$		0			0
$839$	−2.80173			−0.0967265			−0.0483633	−	0.998830i	$-0.515401\pi$
$839$	−2.80173			−0.0967265			−0.0483633	+	0.998830i	$0.515401\pi$
$840$		0			0
$841$	2.42188			0.0835133
$842$		0			0
$843$	5.58202	−	4.17969i	0.192255	−	0.143956i
$844$		0			0
$845$	−12.0274	−	15.6277i	−0.413754	−	0.537609i
$846$		0			0
$847$		0			0
$848$		0			0
$849$	−19.1138	−	25.5266i	−0.655982	−	0.876072i
$850$		0			0
$851$			12.3871i			0.424623i
$852$		0			0
$853$	−12.2061			−0.417928			−0.208964	−	0.977923i	$-0.567009\pi$
$853$	−12.2061			−0.417928			−0.208964	+	0.977923i	$0.567009\pi$
$854$		0			0
$855$	−17.5683	−	45.2176i	−0.600824	−	1.54641i
$856$		0			0
$857$		−	21.7856i		−	0.744180i	−0.928197	−	0.372090i	$-0.878641\pi$
$857$		−	21.7856i		−	0.744180i	0.928197	−	0.372090i	$-0.121359\pi$
$858$		0			0
$859$			31.5019i			1.07483i	0.843318	+	0.537415i	$0.180599\pi$
$859$			31.5019i			1.07483i	−0.843318	+	0.537415i	$0.819401\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−3.80236			−0.129434			−0.0647169	−	0.997904i	$-0.520614\pi$
$863$	−3.80236			−0.129434			−0.0647169	+	0.997904i	$0.520614\pi$
$864$		0			0
$865$	37.6691	−	28.9908i	1.28079	−	0.985717i
$866$		0			0
$867$	22.3314	+	29.8238i	0.758413	+	1.01287i
$868$		0			0
$869$		−	29.1642i		−	0.989327i
$870$		0			0
$871$			14.2642i			0.483324i
$872$		0			0
$873$	4.40062	−	15.0004i	0.148938	−	0.507686i
$874$		0			0
$875$		0			0
$876$		0			0
$877$		−	25.9116i		−	0.874974i	−0.899225	−	0.437487i	$-0.855869\pi$
$877$		−	25.9116i		−	0.874974i	0.899225	−	0.437487i	$-0.144131\pi$
$878$		0			0
$879$	28.7794	−	21.5493i	0.970705	−	0.726842i
$880$		0			0
$881$	22.2682			0.750236			0.375118	−	0.926977i	$-0.377602\pi$
$881$	22.2682			0.750236			0.375118	+	0.926977i	$0.377602\pi$
$882$		0			0
$883$		−	46.9783i		−	1.58095i	−0.612497	−	0.790473i	$-0.709835\pi$
$883$		−	46.9783i		−	1.58095i	0.612497	−	0.790473i	$-0.290165\pi$
$884$		0			0
$885$	4.45714	−	15.9718i	0.149825	−	0.536887i
$886$		0			0
$887$		−	9.29229i		−	0.312005i	−0.987757	−	0.156002i	$-0.950139\pi$
$887$		−	9.29229i		−	0.312005i	0.987757	−	0.156002i	$-0.0498607\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$	−19.6297	+	30.5764i	−0.657619	+	1.02435i
$892$		0			0
$893$	56.4250			1.88819
$894$		0			0
$895$	−0.960676	+	0.739354i	−0.0321119	+	0.0247139i
$896$		0			0
$897$	−11.4004	−	15.2254i	−0.380648	−	0.508360i
$898$		0			0
$899$	49.5260			1.65179
$900$		0			0
$901$			16.3820i			0.545763i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	35.2189	−	27.1051i	1.17072	−	0.901005i
$906$		0			0
$907$		−	12.4717i		−	0.414117i	−0.978329	−	0.207059i	$-0.933611\pi$
$907$		−	12.4717i		−	0.414117i	0.978329	−	0.207059i	$-0.0663891\pi$
$908$		0			0
$909$	−12.5796	+	42.8800i	−0.417239	+	1.42224i
$910$		0			0
$911$		−	10.6293i		−	0.352164i	−0.984376	−	0.176082i	$-0.943658\pi$
$911$		−	10.6293i		−	0.352164i	0.984376	−	0.176082i	$-0.0563423\pi$
$912$		0			0
$913$	−18.2058			−0.602523
$914$		0			0
$915$	5.34294	+	1.49101i	0.176632	+	0.0492914i
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−14.9765			−0.494030			−0.247015	−	0.969012i	$-0.579450\pi$
$919$	−14.9765			−0.494030			−0.247015	+	0.969012i	$0.579450\pi$
$920$		0			0
$921$	11.3769	+	15.1940i	0.374882	+	0.500659i
$922$		0			0
$923$		−	29.2118i		−	0.961517i
$924$		0			0
$925$	25.4667	+	6.74518i	0.837339	+	0.221780i
$926$		0			0
$927$	−2.32260	+	7.91704i	−0.0762843	+	0.260030i
$928$		0			0
$929$	−46.5855			−1.52842			−0.764211	−	0.644967i	$-0.776871\pi$
$929$	−46.5855			−1.52842			−0.764211	+	0.644967i	$0.776871\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$	7.20004	+	9.61574i	0.235719	+	0.314805i
$934$		0			0
$935$	−34.1682	−	44.3963i	−1.11742	−	1.45191i
$936$		0			0
$937$	28.8205			0.941526			0.470763	−	0.882260i	$-0.343979\pi$
$937$	28.8205			0.941526			0.470763	+	0.882260i	$0.343979\pi$
$938$		0			0
$939$	29.9139	+	39.9503i	0.976203	+	1.30373i
$940$		0			0
$941$	29.8353			0.972602			0.486301	−	0.873791i	$-0.338346\pi$
$941$	29.8353			0.972602			0.486301	+	0.873791i	$0.338346\pi$
$942$		0			0
$943$	23.1609			0.754223
$944$		0			0
$945$		0			0
$946$		0			0
$947$	35.2420			1.14521			0.572606	−	0.819831i	$-0.305933\pi$
$947$	35.2420			1.14521			0.572606	+	0.819831i	$0.305933\pi$
$948$		0			0
$949$	10.5338			0.341941
$950$		0			0
$951$	−13.8559	−	18.5047i	−0.449308	−	0.600056i
$952$		0			0
$953$	54.3162			1.75947			0.879737	−	0.475461i	$-0.157719\pi$
$953$	54.3162			1.75947			0.879737	+	0.475461i	$0.157719\pi$
$954$		0			0
$955$	31.9590	−	24.5963i	1.03417	−	0.795917i
$956$		0			0
$957$	−21.6075	−	28.8571i	−0.698471	−	0.932817i
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−61.2876			−1.97702
$962$		0			0
$963$	−2.41350	+	8.22689i	−0.0777740	+	0.265108i
$964$		0			0
$965$	−14.2531	+	10.9695i	−0.458825	+	0.353120i
$966$		0			0
$967$		−	20.2942i		−	0.652617i	−0.945263	−	0.326309i	$-0.894195\pi$
$967$		−	20.2942i		−	0.652617i	0.945263	−	0.326309i	$-0.105805\pi$
$968$		0			0
$969$	−46.5886	−	62.2196i	−1.49664	−	1.99878i
$970$		0			0
$971$	32.0644			1.02900			0.514498	−	0.857492i	$-0.327978\pi$
$971$	32.0644			1.02900			0.514498	+	0.857492i	$0.327978\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$	−37.5098	+	15.1474i	−1.20128	+	0.485106i
$976$		0			0
$977$	7.91539			0.253236			0.126618	−	0.991952i	$-0.459588\pi$
$977$	7.91539			0.253236			0.126618	+	0.991952i	$0.459588\pi$
$978$		0			0
$979$		−	47.6581i		−	1.52316i
$980$		0			0
$981$	3.74248	−	12.7570i	0.119488	−	0.407299i
$982$		0			0
$983$			16.0920i			0.513254i	0.966511	+	0.256627i	$0.0826111\pi$
$983$			16.0920i			0.513254i	−0.966511	+	0.256627i	$0.917389\pi$
$984$		0			0
$985$	−6.42197	−	8.34436i	−0.204621	−	0.265873i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	29.9973i		−	0.953859i
$990$		0			0
$991$	37.7840			1.20025			0.600124	−	0.799907i	$-0.295118\pi$
$991$	37.7840			1.20025			0.600124	+	0.799907i	$0.295118\pi$
$992$		0			0
$993$	−2.13927	−	2.85702i	−0.0678877	−	0.0906648i
$994$		0			0
$995$	−1.26267	+	0.971771i	−0.0400292	+	0.0308072i
$996$		0			0
$997$	−4.34448			−0.137591			−0.0687955	−	0.997631i	$-0.521916\pi$
$997$	−4.34448			−0.137591			−0.0687955	+	0.997631i	$0.521916\pi$
$998$		0			0
$999$	−9.57692	−	25.6487i	−0.303000	−	0.811488i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2940.2.f.b.1469.13	✓	48
3.2	odd	2	inner	2940.2.f.b.1469.16	yes	48
5.4	even	2	inner	2940.2.f.b.1469.35	yes	48
7.6	odd	2	inner	2940.2.f.b.1469.36	yes	48
15.14	odd	2	inner	2940.2.f.b.1469.34	yes	48
21.20	even	2	inner	2940.2.f.b.1469.33	yes	48
35.34	odd	2	inner	2940.2.f.b.1469.14	yes	48
105.104	even	2	inner	2940.2.f.b.1469.15	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2940.2.f.b.1469.13	✓	48	1.1	even	1	trivial
2940.2.f.b.1469.14	yes	48	35.34	odd	2	inner
2940.2.f.b.1469.15	yes	48	105.104	even	2	inner
2940.2.f.b.1469.16	yes	48	3.2	odd	2	inner
2940.2.f.b.1469.33	yes	48	21.20	even	2	inner
2940.2.f.b.1469.34	yes	48	15.14	odd	2	inner
2940.2.f.b.1469.35	yes	48	5.4	even	2	inner
2940.2.f.b.1469.36	yes	48	7.6	odd	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:	\( N \)	\(=\)	\( 2940 = 2^{2} \cdot 3 \cdot 5 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2940.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.03814 - 1.38645i) q^{3} +(-1.36379 - 1.77203i) q^{5} +(-0.844511 + 2.87868i) q^{9} -4.03724i q^{11} -4.67109 q^{13} +(-1.04103 + 3.73045i) q^{15} +6.20571i q^{17} -7.23153i q^{19} -2.35095 q^{23} +(-1.28017 + 4.83334i) q^{25} +(4.86788 - 1.81761i) q^{27} +5.15540i q^{29} -9.60664i q^{31} +(-5.59745 + 4.19124i) q^{33} -5.26896i q^{37} +(4.84927 + 6.47625i) q^{39} -9.85171 q^{41} +12.7596i q^{43} +(6.25284 - 2.42941i) q^{45} +7.80263i q^{47} +(8.60393 - 6.44243i) q^{51} +2.63982 q^{53} +(-7.15411 + 5.50593i) q^{55} +(-10.0262 + 7.50737i) q^{57} -4.28148 q^{59} -1.43225i q^{61} +(6.37037 + 8.27731i) q^{65} -3.05372i q^{67} +(2.44063 + 3.25949i) q^{69} +6.25374i q^{71} -2.25510 q^{73} +(8.03021 - 3.24280i) q^{75} +7.22379 q^{79} +(-7.57360 - 4.86216i) q^{81} -4.50946i q^{83} +(10.9967 - 8.46326i) q^{85} +(7.14772 - 5.35205i) q^{87} +11.8046 q^{89} +(-13.3192 + 9.97308i) q^{93} +(-12.8145 + 9.86226i) q^{95} -5.21085 q^{97} +(11.6219 + 3.40949i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 8 q^{9} - 16 q^{15} + 16 q^{25} + 56 q^{39} + 8 q^{51} + 48 q^{79} - 24 q^{81} + 32 q^{85} + 88 q^{99}+O(q^{100}) \) 48 * q - 8 * q^9 - 16 * q^15 + 16 * q^25 + 56 * q^39 + 8 * q^51 + 48 * q^79 - 24 * q^81 + 32 * q^85 + 88 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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