LMFDB - Embedded newform 756.2.k.e.109.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [756,2,Mod(109,756)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(756, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("756.109");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 756 = 2^{2} \cdot 3^{3} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	756.k (of order $3$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$6.03669039281$
Analytic rank:	$0$
Dimension:	$6$
Relative dimension:	$3$ over $\Q(\zeta_{3})$
Coefficient field:	6.0.309123.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 3x^{5} + 10x^{4} - 15x^{3} + 19x^{2} - 12x + 3 $ x^6 - 3x^5 + 10x^4 - 15x^3 + 19x^2 - 12*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			109.1
Root			$0.500000 + 0.224437i$ of defining polynomial
Character	$\chi$	$=$	756.109
Dual form			756.2.k.e.541.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.14400 - 3.71351i) q^{5} +(1.40545 - 2.24159i) q^{7} +O(q^{10})$	$q+(-2.14400 - 3.71351i) q^{5} +(1.40545 - 2.24159i) q^{7} +(-1.90545 + 3.30033i) q^{11} +3.28799 q^{13} +(-0.405446 + 0.702253i) q^{17} +(-3.54944 - 6.14781i) q^{19} +(-3.23855 - 5.60933i) q^{23} +(-6.69344 + 11.5934i) q^{25} -3.81089 q^{29} +(-1.64400 + 2.84748i) q^{31} +(-11.3374 - 0.413181i) q^{35} +(2.88255 + 4.99272i) q^{37} -2.09888 q^{41} -8.76509 q^{43} +(1.66690 + 2.88715i) q^{47} +(-3.04944 - 6.30087i) q^{49} +(4.93199 - 8.54245i) q^{53} +16.3411 q^{55} +(1.73855 - 3.01126i) q^{59} +(-2.97710 - 5.15649i) q^{61} +(-7.04944 - 12.2100i) q^{65} +(1.76145 - 3.05092i) q^{67} -6.05308 q^{71} +(5.19344 - 8.99530i) q^{73} +(4.71998 + 8.90966i) q^{77} +(2.57234 + 4.45543i) q^{79} +1.71201 q^{83} +3.47710 q^{85} +(6.26509 + 10.8515i) q^{89} +(4.62110 - 7.37033i) q^{91} +(-15.2200 + 26.3618i) q^{95} +1.04580 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - q^{5} + 2 q^{7}+O(q^{10}) $ 6 * q - q^5 + 2 * q^7	$ 6 q - q^{5} + 2 q^{7} - 5 q^{11} - 4 q^{13} + 4 q^{17} - 3 q^{19} - 14 q^{23} - 10 q^{25} - 10 q^{29} + 2 q^{31} - 26 q^{35} + 24 q^{41} - 18 q^{43} + 9 q^{47} - 6 q^{53} + 16 q^{55} + 5 q^{59} - 7 q^{61} - 24 q^{65} + 16 q^{67} + 22 q^{71} + q^{73} - 31 q^{77} + 8 q^{79} + 34 q^{83} + 10 q^{85} + 3 q^{89} + 5 q^{91} - 32 q^{95} + 28 q^{97}+O(q^{100}) $ 6 * q - q^5 + 2 * q^7 - 5 * q^11 - 4 * q^13 + 4 * q^17 - 3 * q^19 - 14 * q^23 - 10 * q^25 - 10 * q^29 + 2 * q^31 - 26 * q^35 + 24 * q^41 - 18 * q^43 + 9 * q^47 - 6 * q^53 + 16 * q^55 + 5 * q^59 - 7 * q^61 - 24 * q^65 + 16 * q^67 + 22 * q^71 + q^73 - 31 * q^77 + 8 * q^79 + 34 * q^83 + 10 * q^85 + 3 * q^89 + 5 * q^91 - 32 * q^95 + 28 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/756\mathbb{Z}\right)^\times$.

$n$	$29$	$325$	$379$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−2.14400	−	3.71351i	−0.958824	−	1.66073i	−0.725364	−	0.688366i	$-0.758328\pi$
$5$	−2.14400	−	3.71351i	−0.958824	−	1.66073i	−0.233461	−	0.972366i	$-0.575005\pi$
$6$		0			0
$7$	1.40545	−	2.24159i	0.531209	−	0.847241i
$7$	1.40545	−	2.24159i	0.531209	−	0.847241i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.90545	+	3.30033i	−0.574514	+	0.995087i	0.421581	+	0.906791i	$0.361475\pi$
$11$	−1.90545	+	3.30033i	−0.574514	+	0.995087i	−0.996094	+	0.0882959i	$0.971858\pi$
$12$		0			0
$13$	3.28799			0.911925			0.455962	−	0.889999i	$-0.349295\pi$
$13$	3.28799			0.911925			0.455962	+	0.889999i	$0.349295\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−0.405446	+	0.702253i	−0.0983351	+	0.170321i	−0.910996	−	0.412416i	$-0.864685\pi$
$17$	−0.405446	+	0.702253i	−0.0983351	+	0.170321i	0.812661	+	0.582737i	$0.198018\pi$
$18$		0			0
$19$	−3.54944	−	6.14781i	−0.814298	−	1.41041i	−0.909831	−	0.414979i	$-0.863789\pi$
$19$	−3.54944	−	6.14781i	−0.814298	−	1.41041i	0.0955331	−	0.995426i	$-0.469544\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−3.23855	−	5.60933i	−0.675284	−	1.16963i	−0.976386	−	0.216035i	$-0.930688\pi$
$23$	−3.23855	−	5.60933i	−0.675284	−	1.16963i	0.301101	−	0.953592i	$-0.402646\pi$
$24$		0			0
$25$	−6.69344	+	11.5934i	−1.33869	+	2.31868i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−3.81089			−0.707665			−0.353832	−	0.935309i	$-0.615122\pi$
$29$	−3.81089			−0.707665			−0.353832	+	0.935309i	$0.615122\pi$
$30$		0			0
$31$	−1.64400	+	2.84748i	−0.295270	+	0.511423i	−0.975048	−	0.221995i	$-0.928743\pi$
$31$	−1.64400	+	2.84748i	−0.295270	+	0.511423i	0.679777	+	0.733419i	$0.262076\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−11.3374	−	0.413181i	−1.91638	−	0.0698403i
$36$		0			0
$37$	2.88255	+	4.99272i	0.473888	+	0.820797i	0.999553	−	0.0298939i	$-0.00951695\pi$
$37$	2.88255	+	4.99272i	0.473888	+	0.820797i	−0.525665	+	0.850691i	$0.676184\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.09888			−0.327791			−0.163895	−	0.986478i	$-0.552406\pi$
$41$	−2.09888			−0.327791			−0.163895	+	0.986478i	$0.552406\pi$
$42$		0			0
$43$	−8.76509			−1.33666			−0.668332	−	0.743863i	$-0.732991\pi$
$43$	−8.76509			−1.33666			−0.668332	+	0.743863i	$0.732991\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.66690	+	2.88715i	0.243142	+	0.421134i	0.961608	−	0.274428i	$-0.0884886\pi$
$47$	1.66690	+	2.88715i	0.243142	+	0.421134i	−0.718466	+	0.695562i	$0.755155\pi$
$48$		0			0
$49$	−3.04944	−	6.30087i	−0.435635	−	0.900124i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	4.93199	−	8.54245i	0.677461	−	1.17340i	−0.298282	−	0.954478i	$-0.596414\pi$
$53$	4.93199	−	8.54245i	0.677461	−	1.17340i	0.975743	−	0.218919i	$-0.0702529\pi$
$54$		0			0
$55$	16.3411			2.20343
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.73855	−	3.01126i	0.226340	−	0.392032i	−0.730381	−	0.683040i	$-0.760657\pi$
$59$	1.73855	−	3.01126i	0.226340	−	0.392032i	0.956721	+	0.291008i	$0.0939907\pi$
$60$		0			0
$61$	−2.97710	−	5.15649i	−0.381179	−	0.660221i	0.610052	−	0.792361i	$-0.291148\pi$
$61$	−2.97710	−	5.15649i	−0.381179	−	0.660221i	−0.991231	+	0.132140i	$0.957815\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−7.04944	−	12.2100i	−0.874376	−	1.51446i
$66$		0			0
$67$	1.76145	−	3.05092i	0.215195	−	0.372729i	−0.738138	−	0.674650i	$-0.764295\pi$
$67$	1.76145	−	3.05092i	0.215195	−	0.372729i	0.953333	+	0.301921i	$0.0976278\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−6.05308			−0.718369			−0.359184	−	0.933267i	$-0.616945\pi$
$71$	−6.05308			−0.718369			−0.359184	+	0.933267i	$0.616945\pi$
$72$		0			0
$73$	5.19344	−	8.99530i	0.607846	−	1.05282i	−0.383749	−	0.923438i	$-0.625367\pi$
$73$	5.19344	−	8.99530i	0.607846	−	1.05282i	0.991595	−	0.129383i	$-0.0412995\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	4.71998	+	8.90966i	0.537892	+	1.01535i
$78$		0			0
$79$	2.57234	+	4.45543i	0.289411	+	0.501275i	0.973669	−	0.227965i	$-0.0732072\pi$
$79$	2.57234	+	4.45543i	0.289411	+	0.501275i	−0.684258	+	0.729240i	$0.739874\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1.71201			0.187917			0.0939586	−	0.995576i	$-0.470048\pi$
$83$	1.71201			0.187917			0.0939586	+	0.995576i	$0.470048\pi$
$84$		0			0
$85$	3.47710			0.377144
$86$		0			0
$87$		0			0
$88$		0			0
$89$	6.26509	+	10.8515i	0.664098	+	1.15025i	0.979529	+	0.201304i	$0.0645178\pi$
$89$	6.26509	+	10.8515i	0.664098	+	1.15025i	−0.315430	+	0.948949i	$0.602149\pi$
$90$		0			0
$91$	4.62110	−	7.37033i	0.484422	−	0.772620i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−15.2200	+	26.3618i	−1.56154	+	2.70466i
$96$		0			0
$97$	1.04580			0.106185			0.0530925	−	0.998590i	$-0.483092\pi$
$97$	1.04580			0.106185			0.0530925	+	0.998590i	$0.483092\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	8.95489	−	15.5103i	0.891045	−	1.54333i	0.0524199	−	0.998625i	$-0.483307\pi$
$101$	8.95489	−	15.5103i	0.891045	−	1.54333i	0.838625	−	0.544710i	$-0.183360\pi$
$102$		0			0
$103$	3.50000	+	6.06218i	0.344865	+	0.597324i	0.985329	−	0.170664i	$-0.0545913\pi$
$103$	3.50000	+	6.06218i	0.344865	+	0.597324i	−0.640464	+	0.767988i	$0.721258\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−3.45056	−	5.97654i	−0.333578	−	0.577774i	0.649633	−	0.760248i	$-0.274923\pi$
$107$	−3.45056	−	5.97654i	−0.333578	−	0.577774i	−0.983211	+	0.182474i	$0.941589\pi$
$108$		0			0
$109$	2.81089	−	4.86861i	0.269235	−	0.466328i	−0.699430	−	0.714701i	$-0.746563\pi$
$109$	2.81089	−	4.86861i	0.269235	−	0.466328i	0.968664	+	0.248373i	$0.0798959\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	20.1520			1.89574			0.947869	−	0.318661i	$-0.103233\pi$
$113$	20.1520			1.89574			0.947869	+	0.318661i	$0.103233\pi$
$114$		0			0
$115$	−13.8869	+	24.0528i	−1.29496	+	2.24293i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	1.00433	+	1.89582i	0.0920668	+	0.173790i
$120$		0			0
$121$	−1.76145	−	3.05092i	−0.160132	−	0.277356i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	35.9629			3.21662
$126$		0			0
$127$	−2.14468			−0.190310			−0.0951550	−	0.995462i	$-0.530335\pi$
$127$	−2.14468			−0.190310			−0.0951550	+	0.995462i	$0.530335\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1.69344	−	2.93312i	−0.147956	−	0.256268i	0.782516	−	0.622631i	$-0.213936\pi$
$131$	−1.69344	−	2.93312i	−0.147956	−	0.256268i	−0.930472	+	0.366363i	$0.880603\pi$
$132$		0			0
$133$	−18.7694	−	0.684031i	−1.62752	−	0.0593130i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.47710	+	11.2187i	−0.553376	+	0.958475i	0.444652	+	0.895703i	$0.353327\pi$
$137$	−6.47710	+	11.2187i	−0.553376	+	0.958475i	−0.998028	+	0.0627719i	$0.980006\pi$
$138$		0			0
$139$	−17.2880			−1.46635			−0.733174	−	0.680041i	$-0.761962\pi$
$139$	−17.2880			−1.46635			−0.733174	+	0.680041i	$0.761962\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−6.26509	+	10.8515i	−0.523913	+	0.907444i
$144$		0			0
$145$	8.17054	+	14.1518i	0.678526	+	1.17524i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−0.784350	−	1.35853i	−0.0642565	−	0.111295i	0.832107	−	0.554614i	$-0.187134\pi$
$149$	−0.784350	−	1.35853i	−0.0642565	−	0.111295i	−0.896364	+	0.443319i	$0.853801\pi$
$150$		0			0
$151$	0.693438	−	1.20107i	0.0564312	−	0.0977417i	−0.836430	−	0.548074i	$-0.815361\pi$
$151$	0.693438	−	1.20107i	0.0564312	−	0.0977417i	0.892861	+	0.450332i	$0.148695\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	14.0989			1.13245
$156$		0			0
$157$	8.43199	−	14.6046i	0.672946	−	1.16558i	−0.304119	−	0.952634i	$-0.598362\pi$
$157$	8.43199	−	14.6046i	0.672946	−	1.16558i	0.977065	−	0.212942i	$-0.0683047\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−17.1254	−	0.624118i	−1.34967	−	0.0491874i
$162$		0			0
$163$	−8.64833	−	14.9793i	−0.677389	−	1.17327i	−0.975764	−	0.218824i	$-0.929778\pi$
$163$	−8.64833	−	14.9793i	−0.677389	−	1.17327i	0.298375	−	0.954449i	$-0.403555\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	13.9556			1.07991			0.539957	−	0.841692i	$-0.318440\pi$
$167$	13.9556			1.07991			0.539957	+	0.841692i	$0.318440\pi$
$168$		0			0
$169$	−2.18911			−0.168393
$170$		0			0
$171$		0			0
$172$		0			0
$173$	4.57165	+	7.91834i	0.347576	+	0.602020i	0.985818	−	0.167816i	$-0.0536714\pi$
$173$	4.57165	+	7.91834i	0.347576	+	0.602020i	−0.638242	+	0.769836i	$0.720338\pi$
$174$		0			0
$175$	16.5803	+	31.2978i	1.25335	+	2.36589i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	8.53087	−	14.7759i	0.637627	−	1.10440i	−0.348325	−	0.937374i	$-0.613249\pi$
$179$	8.53087	−	14.7759i	0.637627	−	1.10440i	0.985952	−	0.167029i	$-0.0534174\pi$
$180$		0			0
$181$	−16.4400			−1.22197			−0.610986	−	0.791641i	$-0.709227\pi$
$181$	−16.4400			−1.22197			−0.610986	+	0.791641i	$0.709227\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	12.3603	−	21.4087i	0.908750	−	1.57400i
$186$		0			0
$187$	−1.54511	−	2.67621i	−0.112990	−	0.195704i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	1.48143	+	2.56591i	0.107193	+	0.185663i	0.914632	−	0.404288i	$-0.132481\pi$
$191$	1.48143	+	2.56591i	0.107193	+	0.185663i	−0.807439	+	0.589951i	$0.799147\pi$
$192$		0			0
$193$	8.71565	−	15.0959i	0.627366	−	1.08663i	−0.360712	−	0.932677i	$-0.617466\pi$
$193$	8.71565	−	15.0959i	0.627366	−	1.08663i	0.988078	−	0.153953i	$-0.0492004\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−10.2880			−0.732989			−0.366495	−	0.930420i	$-0.619442\pi$
$197$	−10.2880			−0.732989			−0.366495	+	0.930420i	$0.619442\pi$
$198$		0			0
$199$	3.50000	−	6.06218i	0.248108	−	0.429736i	−0.714893	−	0.699234i	$-0.753524\pi$
$199$	3.50000	−	6.06218i	0.248108	−	0.429736i	0.963001	+	0.269498i	$0.0868577\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−5.35600	+	8.54245i	−0.375918	+	0.599563i
$204$		0			0
$205$	4.50000	+	7.79423i	0.314294	+	0.544373i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	27.0531			1.87130
$210$		0			0
$211$	16.4327			1.13127			0.565636	−	0.824655i	$-0.308631\pi$
$211$	16.4327			1.13127			0.565636	+	0.824655i	$0.308631\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	18.7923	+	32.5493i	1.28163	+	2.21984i
$216$		0			0
$217$	4.07234	+	7.68715i	0.276449	+	0.521838i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1.33310	+	2.30900i	−0.0896743	+	0.155320i
$222$		0			0
$223$	4.13602			0.276969			0.138484	−	0.990365i	$-0.455777\pi$
$223$	4.13602			0.276969			0.138484	+	0.990365i	$0.455777\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−10.6483	+	18.4434i	−0.706754	+	1.22413i	0.259300	+	0.965797i	$0.416508\pi$
$227$	−10.6483	+	18.4434i	−0.706754	+	1.22413i	−0.966055	+	0.258338i	$0.916825\pi$
$228$		0			0
$229$	2.92766	+	5.07085i	0.193465	+	0.335091i	0.946396	−	0.323008i	$-0.104694\pi$
$229$	2.92766	+	5.07085i	0.193465	+	0.335091i	−0.752931	+	0.658099i	$0.771361\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−4.02290	−	6.96787i	−0.263549	−	0.456480i	0.703633	−	0.710563i	$-0.251560\pi$
$233$	−4.02290	−	6.96787i	−0.263549	−	0.456480i	−0.967182	+	0.254083i	$0.918226\pi$
$234$		0			0
$235$	7.14764	−	12.3801i	0.466260	−	0.807587i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	14.0989			0.911981			0.455991	−	0.889985i	$-0.349285\pi$
$239$	14.0989			0.911981			0.455991	+	0.889985i	$0.349285\pi$
$240$		0			0
$241$	13.3640	−	23.1471i	0.860849	−	1.49103i	−0.0102608	−	0.999947i	$-0.503266\pi$
$241$	13.3640	−	23.1471i	0.860849	−	1.49103i	0.871110	−	0.491088i	$-0.163400\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−16.8603	+	24.8332i	−1.07717	+	1.58653i
$246$		0			0
$247$	−11.6705	−	20.2140i	−0.742579	−	1.28618i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−2.43268			−0.153549			−0.0767746	−	0.997048i	$-0.524462\pi$
$251$	−2.43268			−0.153549			−0.0767746	+	0.997048i	$0.524462\pi$
$252$		0			0
$253$	24.6835			1.55184
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−6.26509	−	10.8515i	−0.390806	−	0.676895i	0.601750	−	0.798684i	$-0.294470\pi$
$257$	−6.26509	−	10.8515i	−0.390806	−	0.676895i	−0.992556	+	0.121789i	$0.961137\pi$
$258$		0			0
$259$	15.2429	+	0.555510i	0.947147	+	0.0345177i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−3.45056	+	5.97654i	−0.212771	+	0.368529i	−0.952581	−	0.304286i	$-0.901582\pi$
$263$	−3.45056	+	5.97654i	−0.212771	+	0.368529i	0.739810	+	0.672816i	$0.234915\pi$
$264$		0			0
$265$	−42.2967			−2.59826
$266$		0			0
$267$		0			0
$268$		0			0
$269$	10.9054	−	18.8888i	0.664917	−	1.15167i	−0.314391	−	0.949294i	$-0.601800\pi$
$269$	10.9054	−	18.8888i	0.664917	−	1.15167i	0.979308	−	0.202376i	$-0.0648663\pi$
$270$		0			0
$271$	−3.76145	−	6.51502i	−0.228492	−	0.395759i	0.728870	−	0.684653i	$-0.240046\pi$
$271$	−3.76145	−	6.51502i	−0.228492	−	0.395759i	−0.957361	+	0.288893i	$0.906713\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−25.5080	−	44.1811i	−1.53819	−	2.66422i
$276$		0			0
$277$	−4.61745	+	7.99766i	−0.277436	+	0.480533i	−0.970747	−	0.240106i	$-0.922818\pi$
$277$	−4.61745	+	7.99766i	−0.277436	+	0.480533i	0.693311	+	0.720639i	$0.256151\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	13.6749			0.815774			0.407887	−	0.913032i	$-0.366266\pi$
$281$	13.6749			0.815774			0.407887	+	0.913032i	$0.366266\pi$
$282$		0			0
$283$	−1.07165	+	1.85616i	−0.0637032	+	0.110337i	−0.896118	−	0.443816i	$-0.853624\pi$
$283$	−1.07165	+	1.85616i	−0.0637032	+	0.110337i	0.832415	+	0.554153i	$0.186958\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−2.94987	+	4.70484i	−0.174125	+	0.277718i
$288$		0			0
$289$	8.17123	+	14.1530i	0.480660	+	0.832528i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−9.29665			−0.543116			−0.271558	−	0.962422i	$-0.587539\pi$
$293$	−9.29665			−0.543116			−0.271558	+	0.962422i	$0.587539\pi$
$294$		0			0
$295$	−14.9098			−0.868081
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−10.6483	−	18.4434i	−0.615809	−	1.06661i
$300$		0			0
$301$	−12.3189	+	19.6477i	−0.710048	+	1.13248i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−12.7658	+	22.1110i	−0.730966	+	1.26607i
$306$		0			0
$307$	−10.3869			−0.592810			−0.296405	−	0.955062i	$-0.595788\pi$
$307$	−10.3869			−0.592810			−0.296405	+	0.955062i	$0.595788\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1.28799	−	2.23087i	0.0730353	−	0.126501i	−0.827195	−	0.561915i	$-0.810065\pi$
$311$	1.28799	−	2.23087i	0.0730353	−	0.126501i	0.900230	+	0.435414i	$0.143398\pi$
$312$		0			0
$313$	5.74219	+	9.94577i	0.324568	+	0.562168i	0.981425	−	0.191847i	$-0.0614478\pi$
$313$	5.74219	+	9.94577i	0.324568	+	0.562168i	−0.656857	+	0.754015i	$0.728114\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−7.59820	−	13.1605i	−0.426757	−	0.739165i	0.569826	−	0.821766i	$-0.307011\pi$
$317$	−7.59820	−	13.1605i	−0.426757	−	0.739165i	−0.996583	+	0.0826006i	$0.973677\pi$
$318$		0			0
$319$	7.26145	−	12.5772i	0.406563	−	0.704188i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	5.75643			0.320296
$324$		0			0
$325$	−22.0080	+	38.1189i	−1.22078	+	2.11446i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	8.81453	+	0.321236i	0.485961	+	0.0177103i
$330$		0			0
$331$	−7.76578	−	13.4507i	−0.426846	−	0.739319i	0.569745	−	0.821822i	$-0.307042\pi$
$331$	−7.76578	−	13.4507i	−0.426846	−	0.739319i	−0.996591	+	0.0825028i	$0.973709\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−15.1062			−0.825338
$336$		0			0
$337$	−3.00866			−0.163892			−0.0819461	−	0.996637i	$-0.526114\pi$
$337$	−3.00866			−0.163892			−0.0819461	+	0.996637i	$0.526114\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−6.26509	−	10.8515i	−0.339274	−	0.587639i
$342$		0			0
$343$	−18.4098	−	2.01993i	−0.994035	−	0.109066i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−11.4327	+	19.8020i	−0.613738	+	1.06303i	0.376866	+	0.926268i	$0.377002\pi$
$347$	−11.4327	+	19.8020i	−0.613738	+	1.06303i	−0.990604	+	0.136758i	$0.956332\pi$
$348$		0			0
$349$	9.04442			0.484137			0.242068	−	0.970259i	$-0.422174\pi$
$349$	9.04442			0.484137			0.242068	+	0.970259i	$0.422174\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	9.09820	−	15.7585i	0.484248	−	0.838742i	−0.515588	−	0.856837i	$-0.672427\pi$
$353$	9.09820	−	15.7585i	0.484248	−	0.838742i	0.999836	+	0.0180942i	$0.00575988\pi$
$354$		0			0
$355$	12.9778	+	22.4782i	0.688789	+	1.19302i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−12.1934	−	21.1197i	−0.643545	−	1.11465i	−0.984636	−	0.174622i	$-0.944130\pi$
$359$	−12.1934	−	21.1197i	−0.643545	−	1.11465i	0.341090	−	0.940030i	$-0.389204\pi$
$360$		0			0
$361$	−15.6971	+	27.1881i	−0.826162	+	1.43095i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−44.5388			−2.33127
$366$		0			0
$367$	4.43268	−	7.67762i	0.231384	−	0.400769i	−0.726832	−	0.686816i	$-0.759008\pi$
$367$	4.43268	−	7.67762i	0.231384	−	0.400769i	0.958216	+	0.286047i	$0.0923413\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−12.2170	−	23.0614i	−0.634276	−	1.19729i
$372$		0			0
$373$	1.57598	+	2.72968i	0.0816014	+	0.141338i	0.903938	−	0.427664i	$-0.140663\pi$
$373$	1.57598	+	2.72968i	0.0816014	+	0.141338i	−0.822337	+	0.569001i	$0.807330\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−12.5302			−0.645337
$378$		0			0
$379$	−2.28799			−0.117526			−0.0587631	−	0.998272i	$-0.518716\pi$
$379$	−2.28799			−0.117526			−0.0587631	+	0.998272i	$0.518716\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	14.1254	+	24.4660i	0.721776	+	1.25015i	0.960287	+	0.279012i	$0.0900071\pi$
$383$	14.1254	+	24.4660i	0.721776	+	1.25015i	−0.238512	+	0.971140i	$0.576660\pi$
$384$		0			0
$385$	22.9665	−	36.6300i	1.17048	−	1.86684i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−0.360335	+	0.624118i	−0.0182697	+	0.0316440i	−0.875016	−	0.484095i	$-0.839149\pi$
$389$	−0.360335	+	0.624118i	−0.0182697	+	0.0316440i	0.856746	+	0.515739i	$0.172482\pi$
$390$		0			0
$391$	5.25223			0.265617
$392$		0			0
$393$		0			0
$394$		0			0
$395$	11.0302	−	19.1048i	0.554989	−	0.961269i
$396$		0			0
$397$	−10.5753	−	18.3169i	−0.530759	−	0.919301i	−0.999356	−	0.0358892i	$-0.988574\pi$
$397$	−10.5753	−	18.3169i	−0.530759	−	0.919301i	0.468597	−	0.883412i	$-0.344760\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	14.2472	+	24.6769i	0.711472	+	1.23231i	0.964305	+	0.264795i	$0.0853044\pi$
$401$	14.2472	+	24.6769i	0.711472	+	1.23231i	−0.252833	+	0.967510i	$0.581362\pi$
$402$		0			0
$403$	−5.40545	+	9.36251i	−0.269264	+	0.466380i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−21.9701			−1.08902
$408$		0			0
$409$	5.38186	−	9.32165i	0.266116	−	0.460926i	−0.701740	−	0.712434i	$-0.747593\pi$
$409$	5.38186	−	9.32165i	0.266116	−	0.460926i	0.967855	+	0.251508i	$0.0809263\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−4.30656	−	8.12927i	−0.211912	−	0.400015i
$414$		0			0
$415$	−3.67054	−	6.35756i	−0.180180	−	0.312080i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	34.0604			1.66396			0.831979	−	0.554807i	$-0.187208\pi$
$419$	34.0604			1.66396			0.831979	+	0.554807i	$0.187208\pi$
$420$		0			0
$421$	20.5833			1.00317			0.501584	−	0.865109i	$-0.332751\pi$
$421$	20.5833			1.00317			0.501584	+	0.865109i	$0.332751\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−5.42766	−	9.40098i	−0.263280	−	0.456014i
$426$		0			0
$427$	−15.7429	−	0.573732i	−0.761851	−	0.0277649i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−11.9814	+	20.7524i	−0.577125	+	0.999610i	0.418682	+	0.908133i	$0.362492\pi$
$431$	−11.9814	+	20.7524i	−0.577125	+	0.999610i	−0.995807	+	0.0914772i	$0.970841\pi$
$432$		0			0
$433$	29.9642			1.43999			0.719995	−	0.693980i	$-0.244144\pi$
$433$	29.9642			1.43999			0.719995	+	0.693980i	$0.244144\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−22.9901	+	39.8200i	−1.09977	+	1.90485i
$438$		0			0
$439$	−14.4098	−	24.9585i	−0.687741	−	1.19120i	−0.972567	−	0.232623i	$-0.925269\pi$
$439$	−14.4098	−	24.9585i	−0.687741	−	1.19120i	0.284826	−	0.958579i	$-0.408064\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−16.0043	−	27.7203i	−0.760389	−	1.31703i	−0.942650	−	0.333782i	$-0.891675\pi$
$443$	−16.0043	−	27.7203i	−0.760389	−	1.31703i	0.182262	−	0.983250i	$-0.441658\pi$
$444$		0			0
$445$	26.8647	−	46.5310i	1.27351	−	2.20578i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−32.0087			−1.51058			−0.755291	−	0.655390i	$-0.772504\pi$
$449$	−32.0087			−1.51058			−0.755291	+	0.655390i	$0.772504\pi$
$450$		0			0
$451$	3.99931	−	6.92701i	0.188320	−	0.326180i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−37.2774	−	1.35853i	−1.74759	−	0.0636891i
$456$		0			0
$457$	10.4277	+	18.0612i	0.487785	+	0.844869i	0.999901	−	0.0140474i	$-0.00447158\pi$
$457$	10.4277	+	18.0612i	0.487785	+	0.844869i	−0.512116	+	0.858916i	$0.671138\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	25.2509			1.17605			0.588025	−	0.808843i	$-0.299906\pi$
$461$	25.2509			1.17605			0.588025	+	0.808843i	$0.299906\pi$
$462$		0			0
$463$	−21.1520			−0.983015			−0.491508	−	0.870873i	$-0.663554\pi$
$463$	−21.1520			−0.983015			−0.491508	+	0.870873i	$0.663554\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−7.95853	−	13.7846i	−0.368277	−	0.637874i	0.621019	−	0.783795i	$-0.286719\pi$
$467$	−7.95853	−	13.7846i	−0.368277	−	0.637874i	−0.989296	+	0.145921i	$0.953385\pi$
$468$		0			0
$469$	−4.36329	−	8.23635i	−0.201478	−	0.380319i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	16.7014	−	28.9277i	0.767932	−	1.33010i
$474$		0			0
$475$	95.0319			4.36036
$476$		0			0
$477$		0			0
$478$		0			0
$479$	0.977789	−	1.69358i	0.0446763	−	0.0773816i	−0.842823	−	0.538192i	$-0.819108\pi$
$479$	0.977789	−	1.69358i	0.0446763	−	0.0773816i	0.887499	+	0.460810i	$0.152441\pi$
$480$		0			0
$481$	9.47779	+	16.4160i	0.432150	+	0.748506i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−2.24219	−	3.88359i	−0.101813	−	0.176345i
$486$		0			0
$487$	−9.98143	+	17.2883i	−0.452302	+	0.783410i	−0.998529	−	0.0542276i	$-0.982730\pi$
$487$	−9.98143	+	17.2883i	−0.452302	+	0.783410i	0.546227	+	0.837637i	$0.316064\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	22.0975			0.997247			0.498623	−	0.866819i	$-0.333839\pi$
$491$	22.0975			0.997247			0.498623	+	0.866819i	$0.333839\pi$
$492$		0			0
$493$	1.54511	−	2.67621i	0.0695883	−	0.120531i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−8.50728	+	13.5685i	−0.381604	+	0.608632i
$498$		0			0
$499$	16.5574	+	28.6783i	0.741212	+	1.28382i	0.951944	+	0.306272i	$0.0990819\pi$
$499$	16.5574	+	28.6783i	0.741212	+	1.28382i	−0.210732	+	0.977544i	$0.567585\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−4.76509			−0.212465			−0.106232	−	0.994341i	$-0.533879\pi$
$503$	−4.76509			−0.212465			−0.106232	+	0.994341i	$0.533879\pi$
$504$		0			0
$505$	−76.7970			−3.41742
$506$		0			0
$507$		0			0
$508$		0			0
$509$	5.38255	+	9.32284i	0.238577	+	0.413228i	0.960306	−	0.278948i	$-0.0899857\pi$
$509$	5.38255	+	9.32284i	0.238577	+	0.413228i	−0.721729	+	0.692176i	$0.756652\pi$
$510$		0			0
$511$	−12.8647	−	24.2840i	−0.569099	−	1.07426i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	15.0080	−	25.9946i	0.661330	−	1.14546i
$516$		0			0
$517$	−12.7047			−0.558753
$518$		0			0
$519$		0			0
$520$		0			0
$521$	0.617454	−	1.06946i	0.0270512	−	0.0468540i	−0.852183	−	0.523244i	$-0.824722\pi$
$521$	0.617454	−	1.06946i	0.0270512	−	0.0468540i	0.879234	+	0.476390i	$0.158055\pi$
$522$		0			0
$523$	4.28435	+	7.42071i	0.187342	+	0.324485i	0.944363	−	0.328905i	$-0.106680\pi$
$523$	4.28435	+	7.42071i	0.187342	+	0.324485i	−0.757022	+	0.653390i	$0.773346\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.33310	−	2.30900i	−0.0580709	−	0.100582i
$528$		0			0
$529$	−9.47641	+	16.4136i	−0.412018	+	0.713636i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−6.90112			−0.298920
$534$		0			0
$535$	−14.7960	+	25.6274i	−0.639685	+	1.10797i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	26.6055	+	1.94180i	1.14598	+	0.0836390i
$540$		0			0
$541$	7.50433	+	12.9979i	0.322636	+	0.558823i	0.981031	−	0.193850i	$-0.0620976\pi$
$541$	7.50433	+	12.9979i	0.322636	+	0.558823i	−0.658395	+	0.752673i	$0.728764\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−24.1062			−1.03259
$546$		0			0
$547$	12.2509			0.523809			0.261904	−	0.965094i	$-0.415650\pi$
$547$	12.2509			0.523809			0.261904	+	0.965094i	$0.415650\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	13.5265	+	23.4287i	0.576250	+	0.998094i
$552$		0			0
$553$	13.6025	+	0.495729i	0.578438	+	0.0210806i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−12.8905	+	22.3270i	−0.546189	+	0.946027i	0.452342	+	0.891844i	$0.350589\pi$
$557$	−12.8905	+	22.3270i	−0.546189	+	0.946027i	−0.998531	+	0.0541823i	$0.982745\pi$
$558$		0			0
$559$	−28.8196			−1.21894
$560$		0			0
$561$		0			0
$562$		0			0
$563$	12.3418	−	21.3766i	0.520143	−	0.900915i	−0.479582	−	0.877497i	$-0.659212\pi$
$563$	12.3418	−	21.3766i	0.520143	−	0.900915i	0.999726	−	0.0234179i	$-0.00745483\pi$
$564$		0			0
$565$	−43.2057	−	74.8345i	−1.81768	−	3.14831i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	6.11677	+	10.5945i	0.256428	+	0.444147i	0.965282	−	0.261208i	$-0.0841210\pi$
$569$	6.11677	+	10.5945i	0.256428	+	0.444147i	−0.708854	+	0.705355i	$0.750788\pi$
$570$		0			0
$571$	−17.6563	+	30.5816i	−0.738893	+	1.27980i	0.214101	+	0.976812i	$0.431318\pi$
$571$	−17.6563	+	30.5816i	−0.738893	+	1.27980i	−0.952994	+	0.302989i	$0.902015\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	86.7081			3.61598
$576$		0			0
$577$	−4.52221	+	7.83270i	−0.188262	+	0.326080i	−0.944671	−	0.328020i	$-0.893619\pi$
$577$	−4.52221	+	7.83270i	−0.188262	+	0.326080i	0.756409	+	0.654099i	$0.226952\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	2.40613	−	3.83762i	0.0998233	−	0.159211i
$582$		0			0
$583$	18.7953	+	32.5544i	0.778421	+	1.34826i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−35.8196			−1.47843			−0.739216	−	0.673469i	$-0.764804\pi$
$587$	−35.8196			−1.47843			−0.739216	+	0.673469i	$0.764804\pi$
$588$		0			0
$589$	23.3411			0.961752
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−22.0309	−	38.1586i	−0.904700	−	1.56699i	−0.821320	−	0.570468i	$-0.806762\pi$
$593$	−22.0309	−	38.1586i	−0.904700	−	1.56699i	−0.0833794	−	0.996518i	$-0.526571\pi$
$594$		0			0
$595$	4.88688	−	7.79423i	0.200342	−	0.319532i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	15.0316	−	26.0354i	0.614173	−	1.06378i	−0.376356	−	0.926475i	$-0.622823\pi$
$599$	15.0316	−	26.0354i	0.614173	−	1.06378i	0.990529	−	0.137304i	$-0.0438436\pi$
$600$		0			0
$601$	29.1964			1.19095			0.595473	−	0.803375i	$-0.296965\pi$
$601$	29.1964			1.19095			0.595473	+	0.803375i	$0.296965\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−7.55308	+	13.0823i	−0.307077	+	0.531872i
$606$		0			0
$607$	8.07165	+	13.9805i	0.327618	+	0.567452i	0.982039	−	0.188679i	$-0.0604207\pi$
$607$	8.07165	+	13.9805i	0.327618	+	0.567452i	−0.654420	+	0.756131i	$0.727087\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	5.48074	+	9.49292i	0.221727	+	0.384043i
$612$		0			0
$613$	−23.5581	+	40.8038i	−0.951503	+	1.64805i	−0.209327	+	0.977846i	$0.567127\pi$
$613$	−23.5581	+	40.8038i	−0.951503	+	1.64805i	−0.742175	+	0.670206i	$0.766206\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−38.0146			−1.53041			−0.765204	−	0.643787i	$-0.777362\pi$
$617$	−38.0146			−1.53041			−0.765204	+	0.643787i	$0.777362\pi$
$618$		0			0
$619$	−6.09091	+	10.5498i	−0.244814	+	0.424031i	−0.962079	−	0.272769i	$-0.912060\pi$
$619$	−6.09091	+	10.5498i	−0.244814	+	0.424031i	0.717265	+	0.696800i	$0.245394\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	33.1298	+	1.20738i	1.32732	+	0.0483726i
$624$		0			0
$625$	−43.6370	−	75.5816i	−1.74548	−	3.02326i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−4.67487			−0.186399
$630$		0			0
$631$	−10.8640			−0.432488			−0.216244	−	0.976339i	$-0.569381\pi$
$631$	−10.8640			−0.432488			−0.216244	+	0.976339i	$0.569381\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	4.59820	+	7.96431i	0.182474	+	0.316054i
$636$		0			0
$637$	−10.0265	−	20.7172i	−0.397266	−	0.820845i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−15.2200	+	26.3618i	−0.601153	+	1.04123i	0.391494	+	0.920181i	$0.371958\pi$
$641$	−15.2200	+	26.3618i	−0.601153	+	1.04123i	−0.992647	+	0.121047i	$0.961375\pi$
$642$		0			0
$643$	−37.8640			−1.49321			−0.746605	−	0.665268i	$-0.768317\pi$
$643$	−37.8640			−1.49321			−0.746605	+	0.665268i	$0.768317\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−18.0982	+	31.3470i	−0.711513	+	1.23238i	0.252775	+	0.967525i	$0.418657\pi$
$647$	−18.0982	+	31.3470i	−0.711513	+	1.23238i	−0.964289	+	0.264853i	$0.914677\pi$
$648$		0			0
$649$	6.62543	+	11.4756i	0.260071	+	0.450456i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	11.0488	+	19.1370i	0.432371	+	0.748889i	0.997077	−	0.0764035i	$-0.0243437\pi$
$653$	11.0488	+	19.1370i	0.432371	+	0.748889i	−0.564706	+	0.825292i	$0.691010\pi$
$654$		0			0
$655$	−7.26145	+	12.5772i	−0.283728	+	0.491432i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	44.0677			1.71663			0.858316	−	0.513121i	$-0.171511\pi$
$659$	44.0677			1.71663			0.858316	+	0.513121i	$0.171511\pi$
$660$		0			0
$661$	−2.95351	+	5.11563i	−0.114878	+	0.198975i	−0.917731	−	0.397202i	$-0.869981\pi$
$661$	−2.95351	+	5.11563i	−0.114878	+	0.198975i	0.802853	+	0.596177i	$0.203314\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	37.7014	+	71.1670i	1.46200	+	2.75974i
$666$		0			0
$667$	12.3418	+	21.3766i	0.477875	+	0.827704i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	22.6908			0.875969
$672$		0			0
$673$	−2.19777			−0.0847178			−0.0423589	−	0.999102i	$-0.513487\pi$
$673$	−2.19777			−0.0847178			−0.0423589	+	0.999102i	$0.513487\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	15.0316	+	26.0354i	0.577710	+	1.00062i	0.995741	+	0.0921903i	$0.0293868\pi$
$677$	15.0316	+	26.0354i	0.577710	+	1.00062i	−0.418032	+	0.908432i	$0.637280\pi$
$678$		0			0
$679$	1.46982	−	2.34425i	0.0564064	−	0.0899642i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−4.14764	+	7.18392i	−0.158705	+	0.274885i	−0.934402	−	0.356221i	$-0.884065\pi$
$683$	−4.14764	+	7.18392i	−0.158705	+	0.274885i	0.775697	+	0.631106i	$0.217399\pi$
$684$		0			0
$685$	55.5475			2.12236
$686$		0			0
$687$		0			0
$688$		0			0
$689$	16.2163	−	28.0875i	0.617793	−	1.07005i
$690$		0			0
$691$	0.598196	+	1.03611i	0.0227564	+	0.0394153i	0.877179	−	0.480163i	$-0.159422\pi$
$691$	0.598196	+	1.03611i	0.0227564	+	0.0394153i	−0.854423	+	0.519578i	$0.826089\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	37.0654	+	64.1991i	1.40597	+	2.43521i
$696$		0			0
$697$	0.850985	−	1.47395i	0.0322333	−	0.0558298i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	31.8813			1.20414			0.602070	−	0.798443i	$-0.294343\pi$
$701$	31.8813			1.20414			0.602070	+	0.798443i	$0.294343\pi$
$702$		0			0
$703$	20.4629	−	35.4427i	0.771771	−	1.33675i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−22.1822	−	41.8721i	−0.834246	−	1.57476i
$708$		0			0
$709$	10.7163	+	18.5612i	0.402461	+	0.697082i	0.994022	−	0.109178i	$-0.0348217\pi$
$709$	10.7163	+	18.5612i	0.402461	+	0.697082i	−0.591562	+	0.806260i	$0.701488\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	21.2967			0.797566
$714$		0			0
$715$	53.7293			2.00936
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−21.6304	−	37.4650i	−0.806680	−	1.39721i	−0.915151	−	0.403110i	$-0.867929\pi$
$719$	−21.6304	−	37.4650i	−0.806680	−	1.39721i	0.108472	−	0.994100i	$-0.465404\pi$
$720$		0			0
$721$	18.5080	+	0.674503i	0.689273	+	0.0251198i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	25.5080	−	44.1811i	0.947342	−	1.64085i
$726$		0			0
$727$	12.1818			0.451799			0.225899	−	0.974151i	$-0.427468\pi$
$727$	12.1818			0.451799			0.225899	+	0.974151i	$0.427468\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	3.55377	−	6.15532i	0.131441	−	0.227663i
$732$		0			0
$733$	−6.72431	−	11.6468i	−0.248368	−	0.430186i	0.714705	−	0.699426i	$-0.246561\pi$
$733$	−6.72431	−	11.6468i	−0.248368	−	0.430186i	−0.963073	+	0.269240i	$0.913228\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	6.71270	+	11.6267i	0.247265	+	0.428276i
$738$		0			0
$739$	4.93632	−	8.54995i	0.181585	−	0.314515i	−0.760835	−	0.648945i	$-0.775210\pi$
$739$	4.93632	−	8.54995i	0.181585	−	0.314515i	0.942421	+	0.334430i	$0.108544\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	22.5687			0.827965			0.413983	−	0.910285i	$-0.364137\pi$
$743$	22.5687			0.827965			0.413983	+	0.910285i	$0.364137\pi$
$744$		0			0
$745$	−3.36329	+	5.82539i	−0.123221	+	0.213426i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−18.2465	−	0.664975i	−0.666713	−	0.0242977i
$750$		0			0
$751$	0.287992	+	0.498817i	0.0105090	+	0.0182021i	0.871232	−	0.490871i	$-0.163321\pi$
$751$	0.287992	+	0.498817i	0.0105090	+	0.0182021i	−0.860723	+	0.509073i	$0.829988\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−5.94692			−0.216430
$756$		0			0
$757$	22.8196			0.829391			0.414695	−	0.909960i	$-0.363888\pi$
$757$	22.8196			0.829391			0.414695	+	0.909960i	$0.363888\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	9.89052	+	17.1309i	0.358531	+	0.620994i	0.987716	−	0.156262i	$-0.0499444\pi$
$761$	9.89052	+	17.1309i	0.358531	+	0.620994i	−0.629185	+	0.777256i	$0.716611\pi$
$762$		0			0
$763$	−6.96286	−	13.1434i	−0.252072	−	0.475824i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	5.71634	−	9.90099i	0.206405	−	0.357504i
$768$		0			0
$769$	−28.7207			−1.03569			−0.517847	−	0.855473i	$-0.673266\pi$
$769$	−28.7207			−1.03569			−0.517847	+	0.855473i	$0.673266\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	4.10253	−	7.10578i	0.147558	−	0.255577i	−0.782767	−	0.622315i	$-0.786192\pi$
$773$	4.10253	−	7.10578i	0.147558	−	0.255577i	0.930324	+	0.366738i	$0.119525\pi$
$774$		0			0
$775$	−22.0080	−	38.1189i	−0.790550	−	1.36927i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	7.44987	+	12.9036i	0.266919	+	0.462318i
$780$		0			0
$781$	11.5338	−	19.9772i	0.412713	−	0.714839i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−72.3126			−2.58095
$786$		0			0
$787$	26.3182	−	45.5844i	0.938142	−	1.62491i	0.169208	−	0.985580i	$-0.445879\pi$
$787$	26.3182	−	45.5844i	0.938142	−	1.62491i	0.768934	−	0.639329i	$-0.220788\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	28.3225	−	45.1724i	1.00703	−	1.60615i
$792$		0			0
$793$	−9.78868	−	16.9545i	−0.347606	−	0.602072i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	17.5329			0.621049			0.310524	−	0.950565i	$-0.399495\pi$
$797$	17.5329			0.621049			0.310524	+	0.950565i	$0.399495\pi$
$798$		0			0
$799$	−2.70335			−0.0956375
$800$		0			0
$801$		0			0
$802$		0			0
$803$	19.7916	+	34.2801i	0.698432	+	1.20972i
$804$		0			0
$805$	34.3992	+	64.9336i	1.21241	+	2.28861i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−3.57530	+	6.19259i	−0.125701	+	0.217720i	−0.922007	−	0.387174i	$-0.873451\pi$
$809$	−3.57530	+	6.19259i	−0.125701	+	0.217720i	0.796306	+	0.604894i	$0.206785\pi$
$810$		0			0
$811$	−31.6835			−1.11256			−0.556280	−	0.830995i	$-0.687772\pi$
$811$	−31.6835			−1.11256			−0.556280	+	0.830995i	$0.687772\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−37.0840	+	64.2313i	−1.29899	+	2.24992i
$816$		0			0
$817$	31.1112	+	53.8862i	1.08844	+	1.88524i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−7.53451	−	13.0502i	−0.262956	−	0.455454i	0.704070	−	0.710131i	$-0.251364\pi$
$821$	−7.53451	−	13.0502i	−0.262956	−	0.455454i	−0.967026	+	0.254677i	$0.918031\pi$
$822$		0			0
$823$	10.4778	−	18.1481i	0.365233	−	0.632602i	−0.623581	−	0.781759i	$-0.714323\pi$
$823$	10.4778	−	18.1481i	0.365233	−	0.632602i	0.988813	+	0.149157i	$0.0476561\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−36.7897			−1.27930			−0.639652	−	0.768665i	$-0.720921\pi$
$827$	−36.7897			−1.27930			−0.639652	+	0.768665i	$0.720921\pi$
$828$		0			0
$829$	−23.1527	+	40.1016i	−0.804125	+	1.39279i	0.112755	+	0.993623i	$0.464032\pi$
$829$	−23.1527	+	40.1016i	−0.804125	+	1.39279i	−0.916880	+	0.399163i	$0.869301\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	5.66119	+	0.413181i	0.196149	+	0.0143159i
$834$		0			0
$835$	−29.9207	−	51.8242i	−1.03545	−	1.79345i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	25.6377			0.885113			0.442556	−	0.896741i	$-0.354072\pi$
$839$	25.6377			0.885113			0.442556	+	0.896741i	$0.354072\pi$
$840$		0			0
$841$	−14.4771			−0.499210
$842$		0			0
$843$		0			0
$844$		0			0
$845$	4.69344	+	8.12927i	0.161459	+	0.279656i
$846$		0			0
$847$	−9.31453	−	0.339458i	−0.320051	−	0.0116639i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	18.6705	−	32.3383i	0.640018	−	1.10854i
$852$		0			0
$853$	−5.43130			−0.185964			−0.0929821	−	0.995668i	$-0.529640\pi$
$853$	−5.43130			−0.185964			−0.0929821	+	0.995668i	$0.529640\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	3.61745	−	6.26561i	0.123570	−	0.214029i	−0.797603	−	0.603183i	$-0.793899\pi$
$857$	3.61745	−	6.26561i	0.123570	−	0.214029i	0.921173	+	0.389153i	$0.127232\pi$
$858$		0			0
$859$	2.71565	+	4.70364i	0.0926568	+	0.160486i	0.908628	−	0.417606i	$-0.137131\pi$
$859$	2.71565	+	4.70364i	0.0926568	+	0.160486i	−0.815971	+	0.578092i	$0.803797\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	5.35600	+	9.27687i	0.182320	+	0.315788i	0.942670	−	0.333725i	$-0.108306\pi$
$863$	5.35600	+	9.27687i	0.182320	+	0.315788i	−0.760350	+	0.649514i	$0.774972\pi$
$864$		0			0
$865$	19.6032	−	33.9538i	0.666529	−	1.15446i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−19.6058			−0.665083
$870$		0			0
$871$	5.79163	−	10.0314i	0.196242	−	0.339901i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	50.5439	−	80.6140i	1.70869	−	2.72525i
$876$		0			0
$877$	6.90840	+	11.9657i	0.233280	+	0.404053i	0.958771	−	0.284178i	$-0.0917208\pi$
$877$	6.90840	+	11.9657i	0.233280	+	0.404053i	−0.725491	+	0.688231i	$0.758387\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−23.6662			−0.797335			−0.398667	−	0.917096i	$-0.630527\pi$
$881$	−23.6662			−0.797335			−0.398667	+	0.917096i	$0.630527\pi$
$882$		0			0
$883$	18.9615			0.638105			0.319052	−	0.947737i	$-0.396635\pi$
$883$	18.9615			0.638105			0.319052	+	0.947737i	$0.396635\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	10.1978	+	17.6631i	0.342408	+	0.593067i	0.984879	−	0.173242i	$-0.0554243\pi$
$887$	10.1978	+	17.6631i	0.342408	+	0.593067i	−0.642472	+	0.766309i	$0.722091\pi$
$888$		0			0
$889$	−3.01424	+	4.80750i	−0.101094	+	0.161238i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	11.8331	−	20.4955i	0.395980	−	0.685857i
$894$		0			0
$895$	−73.1606			−2.44549
$896$		0			0
$897$		0			0
$898$		0			0
$899$	6.26509	−	10.8515i	0.208953	−	0.361916i
$900$		0			0
$901$	3.99931	+	6.92701i	0.133236	+	0.230772i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	35.2472	+	61.0500i	1.17166	+	2.02937i
$906$		0			0
$907$	−20.0073	+	34.6536i	−0.664331	+	1.15065i	0.315135	+	0.949047i	$0.397950\pi$
$907$	−20.0073	+	34.6536i	−0.664331	+	1.15065i	−0.979466	+	0.201608i	$0.935383\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−12.8268			−0.424972			−0.212486	−	0.977164i	$-0.568156\pi$
$911$	−12.8268			−0.424972			−0.212486	+	0.977164i	$0.568156\pi$
$912$		0			0
$913$	−3.26214	+	5.65019i	−0.107961	+	0.186994i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−8.95489	−	0.326351i	−0.295716	−	0.0107771i
$918$		0			0
$919$	21.9771	+	38.0655i	0.724958	+	1.25566i	0.958991	+	0.283435i	$0.0914740\pi$
$919$	21.9771	+	38.0655i	0.724958	+	1.25566i	−0.234034	+	0.972228i	$0.575193\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−19.9025			−0.655099
$924$		0			0
$925$	−77.1766			−2.53755
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−20.1175	−	34.8445i	−0.660032	−	1.14321i	−0.980607	−	0.195986i	$-0.937209\pi$
$929$	−20.1175	−	34.8445i	−0.660032	−	1.14321i	0.320574	−	0.947223i	$-0.396124\pi$
$930$		0			0
$931$	−27.9127	+	41.1120i	−0.914803	+	1.34739i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−6.62543	+	11.4756i	−0.216675	+	0.375291i
$936$		0			0
$937$	−3.06175			−0.100023			−0.0500114	−	0.998749i	$-0.515926\pi$
$937$	−3.06175			−0.100023			−0.0500114	+	0.998749i	$0.515926\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−10.0946	+	17.4843i	−0.329073	+	0.569971i	−0.982328	−	0.187167i	$-0.940070\pi$
$941$	−10.0946	+	17.4843i	−0.329073	+	0.569971i	0.653255	+	0.757138i	$0.273403\pi$
$942$		0			0
$943$	6.79734	+	11.7733i	0.221352	+	0.383393i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	0.572342	+	0.991326i	0.0185986	+	0.0322138i	0.875175	−	0.483807i	$-0.160746\pi$
$947$	0.572342	+	0.991326i	0.0185986	+	0.0322138i	−0.856576	+	0.516020i	$0.827413\pi$
$948$		0			0
$949$	17.0760	−	29.5765i	0.554310	−	0.960093i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	42.9701			1.39194			0.695970	−	0.718071i	$-0.254975\pi$
$953$	42.9701			1.39194			0.695970	+	0.718071i	$0.254975\pi$
$954$		0			0
$955$	6.35236	−	11.0026i	0.205558	−	0.356036i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	16.0444	+	30.2862i	0.518101	+	0.977993i
$960$		0			0
$961$	10.0946	+	17.4843i	0.325631	+	0.564009i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−74.7453			−2.40614
$966$		0			0
$967$	9.99862			0.321534			0.160767	−	0.986992i	$-0.448603\pi$
$967$	9.99862			0.321534			0.160767	+	0.986992i	$0.448603\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−16.3145	−	28.2576i	−0.523558	−	0.906830i	−0.999624	−	0.0274199i	$-0.991271\pi$
$971$	−16.3145	−	28.2576i	−0.523558	−	0.906830i	0.476066	−	0.879410i	$-0.342062\pi$
$972$		0			0
$973$	−24.2973	+	38.7526i	−0.778937	+	1.24235i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−20.7479	+	35.9364i	−0.663784	+	1.14971i	0.315829	+	0.948816i	$0.397717\pi$
$977$	−20.7479	+	35.9364i	−0.663784	+	1.14971i	−0.979613	+	0.200892i	$0.935616\pi$
$978$		0			0
$979$	−47.7512			−1.52613
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−17.7163	+	30.6856i	−0.565063	+	0.978719i	0.431980	+	0.901883i	$0.357815\pi$
$983$	−17.7163	+	30.6856i	−0.565063	+	0.978719i	−0.997044	+	0.0768356i	$0.975518\pi$
$984$		0			0
$985$	22.0574	+	38.2046i	0.702808	+	1.21730i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	28.3862	+	49.1663i	0.902628	+	1.56340i
$990$		0			0
$991$	24.2960	−	42.0818i	0.771787	−	1.33677i	−0.164796	−	0.986328i	$-0.552697\pi$
$991$	24.2960	−	42.0818i	0.771787	−	1.33677i	0.936583	−	0.350446i	$-0.113970\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−30.0159			−0.951569
$996$		0			0
$997$	−23.5130	+	40.7257i	−0.744664	+	1.28980i	0.205688	+	0.978618i	$0.434057\pi$
$997$	−23.5130	+	40.7257i	−0.744664	+	1.28980i	−0.950352	+	0.311178i	$0.899276\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	756.2.k.e.109.1	✓	6
3.2	odd	2		756.2.k.f.109.3	yes	6
7.2	even	3	inner	756.2.k.e.541.1	yes	6
7.3	odd	6		5292.2.a.v.1.1		3
7.4	even	3		5292.2.a.x.1.3		3
9.2	odd	6		2268.2.l.j.109.1		6
9.4	even	3		2268.2.i.j.865.1		6
9.5	odd	6		2268.2.i.k.865.3		6
9.7	even	3		2268.2.l.k.109.3		6
21.2	odd	6		756.2.k.f.541.3	yes	6
21.11	odd	6		5292.2.a.u.1.1		3
21.17	even	6		5292.2.a.w.1.3		3
63.2	odd	6		2268.2.i.k.2053.3		6
63.16	even	3		2268.2.i.j.2053.1		6
63.23	odd	6		2268.2.l.j.541.1		6
63.58	even	3		2268.2.l.k.541.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
756.2.k.e.109.1	✓	6	1.1	even	1	trivial
756.2.k.e.541.1	yes	6	7.2	even	3	inner
756.2.k.f.109.3	yes	6	3.2	odd	2
756.2.k.f.541.3	yes	6	21.2	odd	6
2268.2.i.j.865.1		6	9.4	even	3
2268.2.i.j.2053.1		6	63.16	even	3
2268.2.i.k.865.3		6	9.5	odd	6
2268.2.i.k.2053.3		6	63.2	odd	6
2268.2.l.j.109.1		6	9.2	odd	6
2268.2.l.j.541.1		6	63.23	odd	6
2268.2.l.k.109.3		6	9.7	even	3
2268.2.l.k.541.3		6	63.58	even	3
5292.2.a.u.1.1		3	21.11	odd	6
5292.2.a.v.1.1		3	7.3	odd	6
5292.2.a.w.1.3		3	21.17	even	6
5292.2.a.x.1.3		3	7.4	even	3

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	756.k (of order \(3\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-2.14400 - 3.71351i) q^{5} +(1.40545 - 2.24159i) q^{7} +O(q^{10})\)	\(q+(-2.14400 - 3.71351i) q^{5} +(1.40545 - 2.24159i) q^{7} +(-1.90545 + 3.30033i) q^{11} +3.28799 q^{13} +(-0.405446 + 0.702253i) q^{17} +(-3.54944 - 6.14781i) q^{19} +(-3.23855 - 5.60933i) q^{23} +(-6.69344 + 11.5934i) q^{25} -3.81089 q^{29} +(-1.64400 + 2.84748i) q^{31} +(-11.3374 - 0.413181i) q^{35} +(2.88255 + 4.99272i) q^{37} -2.09888 q^{41} -8.76509 q^{43} +(1.66690 + 2.88715i) q^{47} +(-3.04944 - 6.30087i) q^{49} +(4.93199 - 8.54245i) q^{53} +16.3411 q^{55} +(1.73855 - 3.01126i) q^{59} +(-2.97710 - 5.15649i) q^{61} +(-7.04944 - 12.2100i) q^{65} +(1.76145 - 3.05092i) q^{67} -6.05308 q^{71} +(5.19344 - 8.99530i) q^{73} +(4.71998 + 8.90966i) q^{77} +(2.57234 + 4.45543i) q^{79} +1.71201 q^{83} +3.47710 q^{85} +(6.26509 + 10.8515i) q^{89} +(4.62110 - 7.37033i) q^{91} +(-15.2200 + 26.3618i) q^{95} +1.04580 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - q^{5} + 2 q^{7}+O(q^{10}) \) 6 * q - q^5 + 2 * q^7	\( 6 q - q^{5} + 2 q^{7} - 5 q^{11} - 4 q^{13} + 4 q^{17} - 3 q^{19} - 14 q^{23} - 10 q^{25} - 10 q^{29} + 2 q^{31} - 26 q^{35} + 24 q^{41} - 18 q^{43} + 9 q^{47} - 6 q^{53} + 16 q^{55} + 5 q^{59} - 7 q^{61} - 24 q^{65} + 16 q^{67} + 22 q^{71} + q^{73} - 31 q^{77} + 8 q^{79} + 34 q^{83} + 10 q^{85} + 3 q^{89} + 5 q^{91} - 32 q^{95} + 28 q^{97}+O(q^{100}) \) 6 * q - q^5 + 2 * q^7 - 5 * q^11 - 4 * q^13 + 4 * q^17 - 3 * q^19 - 14 * q^23 - 10 * q^25 - 10 * q^29 + 2 * q^31 - 26 * q^35 + 24 * q^41 - 18 * q^43 + 9 * q^47 - 6 * q^53 + 16 * q^55 + 5 * q^59 - 7 * q^61 - 24 * q^65 + 16 * q^67 + 22 * q^71 + q^73 - 31 * q^77 + 8 * q^79 + 34 * q^83 + 10 * q^85 + 3 * q^89 + 5 * q^91 - 32 * q^95 + 28 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more