LMFDB - Embedded newform 3300.2.x.d.1957.10

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3300,2,Mod(1693,3300)] 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("3300.1693"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3300, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 3, 2])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3300.x (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [32,0,0,0,0,0,0,0,0,0,-24,0,0,0,0,0,0,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(19)] == traces)

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

Self dual:	no
Analytic conductor:	$26.3506326670$
Analytic rank:	$0$
Dimension:	$32$
Relative dimension:	$16$ over $\Q(i)$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1957.10
Character	$\chi$	$=$	3300.1957
Dual form			3300.2.x.d.1693.10

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.707107 - 0.707107i) q^{3} +(2.08903 - 2.08903i) q^{7} -1.00000i q^{9} +(2.54632 + 2.12514i) q^{11} +(0.270937 + 0.270937i) q^{13} +(3.69090 - 3.69090i) q^{17} -1.35260 q^{19} -2.95433i q^{21} +(1.12861 - 1.12861i) q^{23} +(-0.707107 - 0.707107i) q^{27} -1.17255 q^{29} +2.59609 q^{31} +(3.30323 - 0.297821i) q^{33} +(4.21202 + 4.21202i) q^{37} +0.383162 q^{39} -2.29068i q^{41} +(-4.84567 - 4.84567i) q^{43} +(2.47244 + 2.47244i) q^{47} -1.72808i q^{49} -5.21972i q^{51} +(2.56577 - 2.56577i) q^{53} +(-0.956432 + 0.956432i) q^{57} -2.86404i q^{59} +4.07246i q^{61} +(-2.08903 - 2.08903i) q^{63} +(4.02255 + 4.02255i) q^{67} -1.59609i q^{69} +2.49259 q^{71} +(-1.83275 - 1.83275i) q^{73} +(9.75883 - 0.879863i) q^{77} -1.86149 q^{79} -1.00000 q^{81} +(-0.432439 - 0.432439i) q^{83} +(-0.829118 + 0.829118i) q^{87} -3.35663i q^{89} +1.13199 q^{91} +(1.83571 - 1.83571i) q^{93} +(7.77050 + 7.77050i) q^{97} +(2.12514 - 2.54632i) q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 24 q^{11} + 16 q^{31} - 32 q^{71} - 32 q^{81} + 80 q^{91}+O(q^{100}) $ 32 * q - 24 * q^11 + 16 * q^31 - 32 * q^71 - 32 * q^81 + 80 * q^91

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3300\mathbb{Z}\right)^\times$.

$n$	$1201$	$1651$	$2201$	$2377$
$\chi(n)$	$-1$	$1$	$1$	$e\left(\frac{1}{4}\right)$

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
$3$	0.707107	−	0.707107i	0.408248	−	0.408248i
$3$	0.707107	−	0.707107i	0.408248	−	0.408248i
$4$		0			0
$5$		0			0
$5$		0			0
$6$		0			0
$7$	2.08903	−	2.08903i	0.789578	−	0.789578i	−0.191846	−	0.981425i	$-0.561448\pi$
$7$	2.08903	−	2.08903i	0.789578	−	0.789578i	0.981425	+	0.191846i	$0.0614476\pi$
$8$		0			0
$9$		−	1.00000i		−	0.333333i
$10$		0			0
$11$	2.54632	+	2.12514i	0.767746	+	0.640754i
$11$	2.54632	+	2.12514i	0.767746	+	0.640754i
$12$		0			0
$13$	0.270937	+	0.270937i	0.0751443	+	0.0751443i	0.743680	−	0.668536i	$-0.233079\pi$
$13$	0.270937	+	0.270937i	0.0751443	+	0.0751443i	−0.668536	+	0.743680i	$0.733079\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	3.69090	−	3.69090i	0.895175	−	0.895175i	−0.0998297	−	0.995005i	$-0.531830\pi$
$17$	3.69090	−	3.69090i	0.895175	−	0.895175i	0.995005	+	0.0998297i	$0.0318298\pi$
$18$		0			0
$19$	−1.35260			−0.310308			−0.155154	−	0.987890i	$-0.549587\pi$
$19$	−1.35260			−0.310308			−0.155154	+	0.987890i	$0.549587\pi$
$20$		0			0
$21$		−	2.95433i		−	0.644688i
$22$		0			0
$23$	1.12861	−	1.12861i	0.235331	−	0.235331i	−0.579583	−	0.814913i	$-0.696784\pi$
$23$	1.12861	−	1.12861i	0.235331	−	0.235331i	0.814913	+	0.579583i	$0.196784\pi$
$24$		0			0
$25$		0			0
$26$		0			0
$27$	−0.707107	−	0.707107i	−0.136083	−	0.136083i
$28$		0			0
$29$	−1.17255			−0.217737			−0.108869	−	0.994056i	$-0.534723\pi$
$29$	−1.17255			−0.217737			−0.108869	+	0.994056i	$0.534723\pi$
$30$		0			0
$31$	2.59609			0.466272			0.233136	−	0.972444i	$-0.425101\pi$
$31$	2.59609			0.466272			0.233136	+	0.972444i	$0.425101\pi$
$32$		0			0
$33$	3.30323	−	0.297821i	0.575018	−	0.0518440i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	4.21202	+	4.21202i	0.692451	+	0.692451i	0.962771	−	0.270320i	$-0.0871294\pi$
$37$	4.21202	+	4.21202i	0.692451	+	0.692451i	−0.270320	+	0.962771i	$0.587129\pi$
$38$		0			0
$39$	0.383162			0.0613551
$40$		0			0
$41$		−	2.29068i		−	0.357743i	−0.983872	−	0.178872i	$-0.942755\pi$
$41$		−	2.29068i		−	0.357743i	0.983872	−	0.178872i	$-0.0572447\pi$
$42$		0			0
$43$	−4.84567	−	4.84567i	−0.738958	−	0.738958i	0.233418	−	0.972376i	$-0.425009\pi$
$43$	−4.84567	−	4.84567i	−0.738958	−	0.738958i	−0.972376	+	0.233418i	$0.925009\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	2.47244	+	2.47244i	0.360643	+	0.360643i	0.864049	−	0.503407i	$-0.167920\pi$
$47$	2.47244	+	2.47244i	0.360643	+	0.360643i	−0.503407	+	0.864049i	$0.667920\pi$
$48$		0			0
$49$		−	1.72808i		−	0.246868i
$50$		0			0
$51$		−	5.21972i		−	0.730907i
$52$		0			0
$53$	2.56577	−	2.56577i	0.352436	−	0.352436i	−0.508579	−	0.861015i	$-0.669829\pi$
$53$	2.56577	−	2.56577i	0.352436	−	0.352436i	0.861015	+	0.508579i	$0.169829\pi$
$54$		0			0
$55$		0			0
$56$		0			0
$57$	−0.956432	+	0.956432i	−0.126683	+	0.126683i
$58$		0			0
$59$		−	2.86404i		−	0.372866i	−0.982468	−	0.186433i	$-0.940307\pi$
$59$		−	2.86404i		−	0.372866i	0.982468	−	0.186433i	$-0.0596928\pi$
$60$		0			0
$61$			4.07246i			0.521425i	0.965417	+	0.260712i	$0.0839574\pi$
$61$			4.07246i			0.521425i	−0.965417	+	0.260712i	$0.916043\pi$
$62$		0			0
$63$	−2.08903	−	2.08903i	−0.263193	−	0.263193i
$64$		0			0
$65$		0			0
$66$		0			0
$67$	4.02255	+	4.02255i	0.491432	+	0.491432i	0.908757	−	0.417325i	$-0.137032\pi$
$67$	4.02255	+	4.02255i	0.491432	+	0.491432i	−0.417325	+	0.908757i	$0.637032\pi$
$68$		0			0
$69$		−	1.59609i		−	0.192147i
$70$		0			0
$71$	2.49259			0.295816			0.147908	−	0.989001i	$-0.452746\pi$
$71$	2.49259			0.295816			0.147908	+	0.989001i	$0.452746\pi$
$72$		0			0
$73$	−1.83275	−	1.83275i	−0.214508	−	0.214508i	0.591672	−	0.806179i	$-0.298468\pi$
$73$	−1.83275	−	1.83275i	−0.214508	−	0.214508i	−0.806179	+	0.591672i	$0.798468\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	9.75883	−	0.879863i	1.11212	−	0.100270i
$78$		0			0
$79$	−1.86149			−0.209434			−0.104717	−	0.994502i	$-0.533394\pi$
$79$	−1.86149			−0.209434			−0.104717	+	0.994502i	$0.533394\pi$
$80$		0			0
$81$	−1.00000			−0.111111
$82$		0			0
$83$	−0.432439	−	0.432439i	−0.0474664	−	0.0474664i	0.682975	−	0.730442i	$-0.260686\pi$
$83$	−0.432439	−	0.432439i	−0.0474664	−	0.0474664i	−0.730442	+	0.682975i	$0.760686\pi$
$84$		0			0
$85$		0			0
$86$		0			0
$87$	−0.829118	+	0.829118i	−0.0888908	+	0.0888908i
$88$		0			0
$89$		−	3.35663i		−	0.355802i	−0.984048	−	0.177901i	$-0.943069\pi$
$89$		−	3.35663i		−	0.355802i	0.984048	−	0.177901i	$-0.0569306\pi$
$90$		0			0
$91$	1.13199			0.118665
$92$		0			0
$93$	1.83571	−	1.83571i	0.190355	−	0.190355i
$94$		0			0
$95$		0			0
$96$		0			0
$97$	7.77050	+	7.77050i	0.788975	+	0.788975i	0.981326	−	0.192351i	$-0.0616113\pi$
$97$	7.77050	+	7.77050i	0.788975	+	0.788975i	−0.192351	+	0.981326i	$0.561611\pi$
$98$		0			0
$99$	2.12514	−	2.54632i	0.213585	−	0.255915i
$100$		0			0
$101$		−	8.42926i		−	0.838743i	−0.907815	−	0.419371i	$-0.862251\pi$
$101$		−	8.42926i		−	0.838743i	0.907815	−	0.419371i	$-0.137749\pi$
$102$		0			0
$103$	−3.86370	+	3.86370i	−0.380702	+	0.380702i	−0.871355	−	0.490653i	$-0.836758\pi$
$103$	−3.86370	+	3.86370i	−0.380702	+	0.380702i	0.490653	+	0.871355i	$0.336758\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.14407	+	6.14407i	−0.593969	+	0.593969i	−0.938701	−	0.344732i	$-0.887970\pi$
$107$	−6.14407	+	6.14407i	−0.593969	+	0.593969i	0.344732	+	0.938701i	$0.387970\pi$
$108$		0			0
$109$	−12.6819			−1.21470			−0.607352	−	0.794433i	$-0.707768\pi$
$109$	−12.6819			−1.21470			−0.607352	+	0.794433i	$0.707768\pi$
$110$		0			0
$111$	5.95669			0.565384
$112$		0			0
$113$	7.46475	−	7.46475i	0.702225	−	0.702225i	−0.262663	−	0.964888i	$-0.584601\pi$
$113$	7.46475	−	7.46475i	0.702225	−	0.702225i	0.964888	+	0.262663i	$0.0846007\pi$
$114$		0			0
$115$		0			0
$116$		0			0
$117$	0.270937	−	0.270937i	0.0250481	−	0.0250481i
$118$		0			0
$119$		−	15.4208i		−	1.41362i
$120$		0			0
$121$	1.96754	+	10.8226i	0.178867	+	0.983873i
$122$		0			0
$123$	−1.61975	−	1.61975i	−0.146048	−	0.146048i
$124$		0			0
$125$		0			0
$126$		0			0
$127$	15.8946	−	15.8946i	1.41042	−	1.41042i	0.653452	−	0.756968i	$-0.273320\pi$
$127$	15.8946	−	15.8946i	1.41042	−	1.41042i	0.756968	−	0.653452i	$-0.226680\pi$
$128$		0			0
$129$	−6.85281			−0.603357
$130$		0			0
$131$		−	21.5656i		−	1.88420i	−0.335338	−	0.942098i	$-0.608851\pi$
$131$		−	21.5656i		−	1.88420i	0.335338	−	0.942098i	$-0.391149\pi$
$132$		0			0
$133$	−2.82562	+	2.82562i	−0.245012	+	0.245012i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	0.393683	+	0.393683i	0.0336346	+	0.0336346i	0.723724	−	0.690089i	$-0.242429\pi$
$137$	0.393683	+	0.393683i	0.0336346	+	0.0336346i	−0.690089	+	0.723724i	$0.742429\pi$
$138$		0			0
$139$	−5.90866			−0.501166			−0.250583	−	0.968095i	$-0.580622\pi$
$139$	−5.90866			−0.501166			−0.250583	+	0.968095i	$0.580622\pi$
$140$		0			0
$141$	3.49656			0.294463
$142$		0			0
$143$	0.114114	+	1.26567i	0.00954268	+	0.105841i
$144$		0			0
$145$		0			0
$146$		0			0
$147$	−1.22194	−	1.22194i	−0.100784	−	0.100784i
$148$		0			0
$149$	−18.2157			−1.49229			−0.746143	−	0.665786i	$-0.768097\pi$
$149$	−18.2157			−1.49229			−0.746143	+	0.665786i	$0.768097\pi$
$150$		0			0
$151$		−	14.8261i		−	1.20653i	−0.797541	−	0.603265i	$-0.793866\pi$
$151$		−	14.8261i		−	1.20653i	0.797541	−	0.603265i	$-0.206134\pi$
$152$		0			0
$153$	−3.69090	−	3.69090i	−0.298392	−	0.298392i
$154$		0			0
$155$		0			0
$156$		0			0
$157$	12.9719	+	12.9719i	1.03527	+	1.03527i	0.999355	+	0.0359146i	$0.0114344\pi$
$157$	12.9719	+	12.9719i	1.03527	+	1.03527i	0.0359146	+	0.999355i	$0.488566\pi$
$158$		0			0
$159$		−	3.62855i		−	0.287763i
$160$		0			0
$161$		−	4.71538i		−	0.371624i
$162$		0			0
$163$	−7.88625	+	7.88625i	−0.617699	+	0.617699i	−0.944941	−	0.327242i	$-0.893881\pi$
$163$	−7.88625	+	7.88625i	−0.617699	+	0.617699i	0.327242	+	0.944941i	$0.393881\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.12024	−	6.12024i	0.473599	−	0.473599i	−0.429478	−	0.903077i	$-0.641303\pi$
$167$	6.12024	−	6.12024i	0.473599	−	0.473599i	0.903077	+	0.429478i	$0.141303\pi$
$168$		0			0
$169$		−	12.8532i		−	0.988707i
$170$		0			0
$171$			1.35260i			0.103436i
$172$		0			0
$173$	−10.7292	−	10.7292i	−0.815723	−	0.815723i	0.169762	−	0.985485i	$-0.445700\pi$
$173$	−10.7292	−	10.7292i	−0.815723	−	0.815723i	−0.985485	+	0.169762i	$0.945700\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−2.02518	−	2.02518i	−0.152222	−	0.152222i
$178$		0			0
$179$		−	8.88489i		−	0.664088i	−0.943264	−	0.332044i	$-0.892262\pi$
$179$		−	8.88489i		−	0.664088i	0.943264	−	0.332044i	$-0.107738\pi$
$180$		0			0
$181$	−16.1135			−1.19771			−0.598854	−	0.800858i	$-0.704377\pi$
$181$	−16.1135			−1.19771			−0.598854	+	0.800858i	$0.704377\pi$
$182$		0			0
$183$	2.87966	+	2.87966i	0.212871	+	0.212871i
$184$		0			0
$185$		0			0
$186$		0			0
$187$	17.2419	−	1.55454i	1.26085	−	0.113680i
$188$		0			0
$189$	−2.95433			−0.214896
$190$		0			0
$191$	20.2415			1.46462			0.732312	−	0.680969i	$-0.238441\pi$
$191$	20.2415			1.46462			0.732312	+	0.680969i	$0.238441\pi$
$192$		0			0
$193$	17.9186	+	17.9186i	1.28981	+	1.28981i	0.934903	+	0.354904i	$0.115487\pi$
$193$	17.9186	+	17.9186i	1.28981	+	1.28981i	0.354904	+	0.934903i	$0.384513\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−2.45317	+	2.45317i	−0.174781	+	0.174781i	−0.789076	−	0.614295i	$-0.789440\pi$
$197$	−2.45317	+	2.45317i	−0.174781	+	0.174781i	0.614295	+	0.789076i	$0.289440\pi$
$198$		0			0
$199$			9.79621i			0.694435i	0.937785	+	0.347218i	$0.112873\pi$
$199$			9.79621i			0.694435i	−0.937785	+	0.347218i	$0.887127\pi$
$200$		0			0
$201$	5.68874			0.401253
$202$		0			0
$203$	−2.44949	+	2.44949i	−0.171920	+	0.171920i
$204$		0			0
$205$		0			0
$206$		0			0
$207$	−1.12861	−	1.12861i	−0.0784435	−	0.0784435i
$208$		0			0
$209$	−3.44416	−	2.87447i	−0.238237	−	0.198831i
$210$		0			0
$211$			9.16422i			0.630891i	0.948944	+	0.315446i	$0.102154\pi$
$211$			9.16422i			0.630891i	−0.948944	+	0.315446i	$0.897846\pi$
$212$		0			0
$213$	1.76253	−	1.76253i	0.120766	−	0.120766i
$214$		0			0
$215$		0			0
$216$		0			0
$217$	5.42331	−	5.42331i	0.368158	−	0.368158i
$218$		0			0
$219$	−2.59190			−0.175145
$220$		0			0
$221$	2.00000			0.134535
$222$		0			0
$223$	4.74014	−	4.74014i	0.317423	−	0.317423i	−0.530354	−	0.847777i	$-0.677941\pi$
$223$	4.74014	−	4.74014i	0.317423	−	0.317423i	0.847777	+	0.530354i	$0.177941\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−12.0986	+	12.0986i	−0.803011	+	0.803011i	−0.983565	−	0.180554i	$-0.942211\pi$
$227$	−12.0986	+	12.0986i	−0.803011	+	0.803011i	0.180554	+	0.983565i	$0.442211\pi$
$228$		0			0
$229$		−	24.5845i		−	1.62459i	−0.583248	−	0.812294i	$-0.698218\pi$
$229$		−	24.5845i		−	1.62459i	0.583248	−	0.812294i	$-0.301782\pi$
$230$		0			0
$231$	6.27838	−	7.52269i	0.413087	−	0.494957i
$232$		0			0
$233$	1.74871	+	1.74871i	0.114562	+	0.114562i	0.762064	−	0.647502i	$-0.224186\pi$
$233$	1.74871	+	1.74871i	0.114562	+	0.114562i	−0.647502	+	0.762064i	$0.724186\pi$
$234$		0			0
$235$		0			0
$236$		0			0
$237$	−1.31627	+	1.31627i	−0.0855012	+	0.0855012i
$238$		0			0
$239$	−10.0042			−0.647117			−0.323558	−	0.946208i	$-0.604879\pi$
$239$	−10.0042			−0.647117			−0.323558	+	0.946208i	$0.604879\pi$
$240$		0			0
$241$			23.7743i			1.53144i	0.643174	+	0.765720i	$0.277617\pi$
$241$			23.7743i			1.53144i	−0.643174	+	0.765720i	$0.722383\pi$
$242$		0			0
$243$	−0.707107	+	0.707107i	−0.0453609	+	0.0453609i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−0.366469	−	0.366469i	−0.0233178	−	0.0233178i
$248$		0			0
$249$	−0.611561			−0.0387561
$250$		0			0
$251$	13.2130			0.833999			0.417000	−	0.908907i	$-0.363082\pi$
$251$	13.2130			0.833999			0.417000	+	0.908907i	$0.363082\pi$
$252$		0			0
$253$	5.27225	−	0.475350i	0.331463	−	0.0298850i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−13.8483	−	13.8483i	−0.863835	−	0.863835i	0.127946	−	0.991781i	$-0.459161\pi$
$257$	−13.8483	−	13.8483i	−0.863835	−	0.863835i	−0.991781	+	0.127946i	$0.959161\pi$
$258$		0			0
$259$	17.5980			1.09349
$260$		0			0
$261$			1.17255i			0.0725790i
$262$		0			0
$263$	12.4453	+	12.4453i	0.767408	+	0.767408i	0.977649	−	0.210242i	$-0.0674251\pi$
$263$	12.4453	+	12.4453i	0.767408	+	0.767408i	−0.210242	+	0.977649i	$0.567425\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$	−2.37349	−	2.37349i	−0.145255	−	0.145255i
$268$		0			0
$269$			3.13521i			0.191157i	0.995422	+	0.0955785i	$0.0304701\pi$
$269$			3.13521i			0.191157i	−0.995422	+	0.0955785i	$0.969530\pi$
$270$		0			0
$271$		−	11.6479i		−	0.707560i	−0.935329	−	0.353780i	$-0.884896\pi$
$271$		−	11.6479i		−	0.707560i	0.935329	−	0.353780i	$-0.115104\pi$
$272$		0			0
$273$	0.800437	−	0.800437i	0.0484446	−	0.0484446i
$274$		0			0
$275$		0			0
$276$		0			0
$277$	12.8020	−	12.8020i	0.769195	−	0.769195i	−0.208770	−	0.977965i	$-0.566946\pi$
$277$	12.8020	−	12.8020i	0.769195	−	0.769195i	0.977965	+	0.208770i	$0.0669458\pi$
$278$		0			0
$279$		−	2.59609i		−	0.155424i
$280$		0			0
$281$			23.0775i			1.37669i	0.725384	+	0.688345i	$0.241662\pi$
$281$			23.0775i			1.37669i	−0.725384	+	0.688345i	$0.758338\pi$
$282$		0			0
$283$	14.5874	+	14.5874i	0.867133	+	0.867133i	0.992154	−	0.125021i	$-0.0398998\pi$
$283$	14.5874	+	14.5874i	0.867133	+	0.867133i	−0.125021	+	0.992154i	$0.539900\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−4.78529	−	4.78529i	−0.282467	−	0.282467i
$288$		0			0
$289$		−	10.2455i		−	0.602676i
$290$		0			0
$291$	10.9891			0.644195
$292$		0			0
$293$	−21.4837	−	21.4837i	−1.25509	−	1.25509i	−0.953408	−	0.301685i	$-0.902451\pi$
$293$	−21.4837	−	21.4837i	−1.25509	−	1.25509i	−0.301685	−	0.953408i	$-0.597549\pi$
$294$		0			0
$295$		0			0
$296$		0			0
$297$	−0.297821	−	3.30323i	−0.0172813	−	0.191673i
$298$		0			0
$299$	0.611561			0.0353675
$300$		0			0
$301$	−20.2455			−1.16693
$302$		0			0
$303$	−5.96039	−	5.96039i	−0.342415	−	0.342415i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	7.44246	−	7.44246i	0.424764	−	0.424764i	−0.462076	−	0.886840i	$-0.652895\pi$
$307$	7.44246	−	7.44246i	0.424764	−	0.424764i	0.886840	+	0.462076i	$0.152895\pi$
$308$		0			0
$309$			5.46410i			0.310842i
$310$		0			0
$311$	18.7212			1.06158			0.530791	−	0.847503i	$-0.321895\pi$
$311$	18.7212			1.06158			0.530791	+	0.847503i	$0.321895\pi$
$312$		0			0
$313$	−5.08283	+	5.08283i	−0.287299	+	0.287299i	−0.836011	−	0.548713i	$-0.815118\pi$
$313$	−5.08283	+	5.08283i	−0.287299	+	0.287299i	0.548713	+	0.836011i	$0.315118\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	4.88423	+	4.88423i	0.274326	+	0.274326i	0.830839	−	0.556513i	$-0.187861\pi$
$317$	4.88423	+	4.88423i	0.274326	+	0.274326i	−0.556513	+	0.830839i	$0.687861\pi$
$318$		0			0
$319$	−2.98569	−	2.49184i	−0.167167	−	0.139516i
$320$		0			0
$321$			8.68902i			0.484974i
$322$		0			0
$323$	−4.99231	+	4.99231i	−0.277780	+	0.277780i
$324$		0			0
$325$		0			0
$326$		0			0
$327$	−8.96744	+	8.96744i	−0.495901	+	0.495901i
$328$		0			0
$329$	10.3300			0.569511
$330$		0			0
$331$	−17.3854			−0.955589			−0.477795	−	0.878472i	$-0.658564\pi$
$331$	−17.3854			−0.955589			−0.477795	+	0.878472i	$0.658564\pi$
$332$		0			0
$333$	4.21202	−	4.21202i	0.230817	−	0.230817i
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−3.87136	+	3.87136i	−0.210886	+	0.210886i	−0.804644	−	0.593758i	$-0.797644\pi$
$337$	−3.87136	+	3.87136i	−0.210886	+	0.210886i	0.593758	+	0.804644i	$0.297644\pi$
$338$		0			0
$339$		−	10.5568i		−	0.573364i
$340$		0			0
$341$	6.61049	+	5.51706i	0.357978	+	0.298766i
$342$		0			0
$343$	11.0132	+	11.0132i	0.594657	+	0.594657i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1.85972	+	1.85972i	−0.0998350	+	0.0998350i	−0.755260	−	0.655425i	$-0.772489\pi$
$347$	−1.85972	+	1.85972i	−0.0998350	+	0.0998350i	0.655425	+	0.755260i	$0.272489\pi$
$348$		0			0
$349$	5.90866			0.316284			0.158142	−	0.987416i	$-0.449450\pi$
$349$	5.90866			0.316284			0.158142	+	0.987416i	$0.449450\pi$
$350$		0			0
$351$		−	0.383162i		−	0.0204517i
$352$		0			0
$353$	6.82316	−	6.82316i	0.363160	−	0.363160i	−0.501815	−	0.864975i	$-0.667334\pi$
$353$	6.82316	−	6.82316i	0.363160	−	0.363160i	0.864975	+	0.501815i	$0.167334\pi$
$354$		0			0
$355$		0			0
$356$		0			0
$357$	−10.9041	−	10.9041i	−0.577109	−	0.577109i
$358$		0			0
$359$	−25.3591			−1.33840			−0.669201	−	0.743082i	$-0.733363\pi$
$359$	−25.3591			−1.33840			−0.669201	+	0.743082i	$0.733363\pi$
$360$		0			0
$361$	−17.1705			−0.903709
$362$		0			0
$363$	9.04400	+	6.26148i	0.474687	+	0.328642i
$364$		0			0
$365$		0			0
$366$		0			0
$367$	10.3301	+	10.3301i	0.539228	+	0.539228i	0.923302	−	0.384074i	$-0.125479\pi$
$367$	10.3301	+	10.3301i	0.539228	+	0.539228i	−0.384074	+	0.923302i	$0.625479\pi$
$368$		0			0
$369$	−2.29068			−0.119248
$370$		0			0
$371$		−	10.7199i		−	0.556551i
$372$		0			0
$373$	−5.59782	−	5.59782i	−0.289844	−	0.289844i	0.547174	−	0.837019i	$-0.315703\pi$
$373$	−5.59782	−	5.59782i	−0.289844	−	0.289844i	−0.837019	+	0.547174i	$0.815703\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−0.317687	−	0.317687i	−0.0163617	−	0.0163617i
$378$		0			0
$379$			27.8300i			1.42953i	0.699365	+	0.714765i	$0.253466\pi$
$379$			27.8300i			1.42953i	−0.699365	+	0.714765i	$0.746534\pi$
$380$		0			0
$381$		−	22.4784i		−	1.15160i
$382$		0			0
$383$	−12.1771	+	12.1771i	−0.622219	+	0.622219i	−0.946098	−	0.323879i	$-0.895013\pi$
$383$	−12.1771	+	12.1771i	−0.622219	+	0.622219i	0.323879	+	0.946098i	$0.395013\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	−4.84567	+	4.84567i	−0.246319	+	0.246319i
$388$		0			0
$389$		−	4.90735i		−	0.248813i	−0.992231	−	0.124406i	$-0.960297\pi$
$389$		−	4.90735i		−	0.248813i	0.992231	−	0.124406i	$-0.0397026\pi$
$390$		0			0
$391$		−	8.33115i		−	0.421324i
$392$		0			0
$393$	−15.2492	−	15.2492i	−0.769220	−	0.769220i
$394$		0			0
$395$		0			0
$396$		0			0
$397$	2.41606	+	2.41606i	0.121258	+	0.121258i	0.765132	−	0.643874i	$-0.222674\pi$
$397$	2.41606	+	2.41606i	0.121258	+	0.121258i	−0.643874	+	0.765132i	$0.722674\pi$
$398$		0			0
$399$			3.99603i			0.200052i
$400$		0			0
$401$	−24.4918			−1.22306			−0.611532	−	0.791220i	$-0.709446\pi$
$401$	−24.4918			−1.22306			−0.611532	+	0.791220i	$0.709446\pi$
$402$		0			0
$403$	0.703376	+	0.703376i	0.0350376	+	0.0350376i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1.77403	+	19.6763i	0.0879353	+	0.975318i
$408$		0			0
$409$	−4.98910			−0.246695			−0.123348	−	0.992364i	$-0.539363\pi$
$409$	−4.98910			−0.246695			−0.123348	+	0.992364i	$0.539363\pi$
$410$		0			0
$411$	0.556752			0.0274625
$412$		0			0
$413$	−5.98306	−	5.98306i	−0.294407	−	0.294407i
$414$		0			0
$415$		0			0
$416$		0			0
$417$	−4.17806	+	4.17806i	−0.204600	+	0.204600i
$418$		0			0
$419$			34.7334i			1.69684i	0.529327	+	0.848418i	$0.322444\pi$
$419$			34.7334i			1.69684i	−0.529327	+	0.848418i	$0.677556\pi$
$420$		0			0
$421$	−21.3025			−1.03822			−0.519109	−	0.854708i	$-0.673736\pi$
$421$	−21.3025			−1.03822			−0.519109	+	0.854708i	$0.673736\pi$
$422$		0			0
$423$	2.47244	−	2.47244i	0.120214	−	0.120214i
$424$		0			0
$425$		0			0
$426$		0			0
$427$	8.50748	+	8.50748i	0.411706	+	0.411706i
$428$		0			0
$429$	0.975656	+	0.814274i	0.0471051	+	0.0393135i
$430$		0			0
$431$		−	3.96534i		−	0.191004i	−0.995429	−	0.0955019i	$-0.969554\pi$
$431$		−	3.96534i		−	0.191004i	0.995429	−	0.0955019i	$-0.0304456\pi$
$432$		0			0
$433$	−12.9719	+	12.9719i	−0.623389	+	0.623389i	−0.946396	−	0.323007i	$-0.895306\pi$
$433$	−12.9719	+	12.9719i	−0.623389	+	0.623389i	0.323007	+	0.946396i	$0.395306\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.52655	+	1.52655i	−0.0730249	+	0.0730249i
$438$		0			0
$439$	11.6373			0.555417			0.277709	−	0.960665i	$-0.410425\pi$
$439$	11.6373			0.555417			0.277709	+	0.960665i	$0.410425\pi$
$440$		0			0
$441$	−1.72808			−0.0822895
$442$		0			0
$443$	10.4870	−	10.4870i	0.498251	−	0.498251i	−0.412642	−	0.910893i	$-0.635394\pi$
$443$	10.4870	−	10.4870i	0.498251	−	0.498251i	0.910893	+	0.412642i	$0.135394\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$	−12.8804	+	12.8804i	−0.609223	+	0.609223i
$448$		0			0
$449$		−	30.2487i		−	1.42752i	−0.700388	−	0.713762i	$-0.746990\pi$
$449$		−	30.2487i		−	1.42752i	0.700388	−	0.713762i	$-0.253010\pi$
$450$		0			0
$451$	4.86801	−	5.83280i	0.229226	−	0.274656i
$452$		0			0
$453$	−10.4836	−	10.4836i	−0.492564	−	0.492564i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−3.15939	+	3.15939i	−0.147790	+	0.147790i	−0.777130	−	0.629340i	$-0.783325\pi$
$457$	−3.15939	+	3.15939i	−0.147790	+	0.147790i	0.629340	+	0.777130i	$0.283325\pi$
$458$		0			0
$459$	−5.21972			−0.243636
$460$		0			0
$461$		−	11.1676i		−	0.520128i	−0.965591	−	0.260064i	$-0.916256\pi$
$461$		−	11.1676i		−	0.520128i	0.965591	−	0.260064i	$-0.0837437\pi$
$462$		0			0
$463$	−4.94488	+	4.94488i	−0.229808	+	0.229808i	−0.812613	−	0.582804i	$-0.801955\pi$
$463$	−4.94488	+	4.94488i	−0.229808	+	0.229808i	0.582804	+	0.812613i	$0.301955\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−22.4198	−	22.4198i	−1.03746	−	1.03746i	−0.999270	−	0.0381939i	$-0.987840\pi$
$467$	−22.4198	−	22.4198i	−1.03746	−	1.03746i	−0.0381939	−	0.999270i	$-0.512160\pi$
$468$		0			0
$469$	16.8064			0.776049
$470$		0			0
$471$	18.3450			0.845294
$472$		0			0
$473$	−2.04091	−	22.6364i	−0.0938413	−	1.04082i
$474$		0			0
$475$		0			0
$476$		0			0
$477$	−2.56577	−	2.56577i	−0.117479	−	0.117479i
$478$		0			0
$479$	30.9458			1.41395			0.706975	−	0.707239i	$-0.250059\pi$
$479$	30.9458			1.41395			0.706975	+	0.707239i	$0.250059\pi$
$480$		0			0
$481$			2.28238i			0.104067i
$482$		0			0
$483$	−3.33428	−	3.33428i	−0.151715	−	0.151715i
$484$		0			0
$485$		0			0
$486$		0			0
$487$	23.7603	+	23.7603i	1.07668	+	1.07668i	0.996805	+	0.0798766i	$0.0254526\pi$
$487$	23.7603	+	23.7603i	1.07668	+	1.07668i	0.0798766	+	0.996805i	$0.474547\pi$
$488$		0			0
$489$			11.1528i			0.504349i
$490$		0			0
$491$		−	3.48841i		−	0.157430i	−0.996897	−	0.0787148i	$-0.974918\pi$
$491$		−	3.48841i		−	0.157430i	0.996897	−	0.0787148i	$-0.0250816\pi$
$492$		0			0
$493$	−4.32776	+	4.32776i	−0.194913	+	0.194913i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	5.20709	−	5.20709i	0.233570	−	0.233570i
$498$		0			0
$499$			25.8869i			1.15886i	0.815022	+	0.579429i	$0.196725\pi$
$499$			25.8869i			1.15886i	−0.815022	+	0.579429i	$0.803275\pi$
$500$		0			0
$501$		−	8.65533i		−	0.386692i
$502$		0			0
$503$	7.61590	+	7.61590i	0.339576	+	0.339576i	0.856208	−	0.516632i	$-0.172814\pi$
$503$	7.61590	+	7.61590i	0.339576	+	0.339576i	−0.516632	+	0.856208i	$0.672814\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$	−9.08858	−	9.08858i	−0.403638	−	0.403638i
$508$		0			0
$509$			20.3497i			0.901987i	0.892527	+	0.450993i	$0.148930\pi$
$509$			20.3497i			0.901987i	−0.892527	+	0.450993i	$0.851070\pi$
$510$		0			0
$511$	−7.65735			−0.338741
$512$		0			0
$513$	0.956432	+	0.956432i	0.0422275	+	0.0422275i
$514$		0			0
$515$		0			0
$516$		0			0
$517$	1.04135	+	11.5499i	0.0457985	+	0.507965i
$518$		0			0
$519$	−15.1733			−0.666035
$520$		0			0
$521$	−13.7618			−0.602917			−0.301459	−	0.953479i	$-0.597474\pi$
$521$	−13.7618			−0.602917			−0.301459	+	0.953479i	$0.597474\pi$
$522$		0			0
$523$	−5.87585	−	5.87585i	−0.256933	−	0.256933i	0.566872	−	0.823806i	$-0.308153\pi$
$523$	−5.87585	−	5.87585i	−0.256933	−	0.256933i	−0.823806	+	0.566872i	$0.808153\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	9.58191	−	9.58191i	0.417395	−	0.417395i
$528$		0			0
$529$			20.4525i			0.889239i
$530$		0			0
$531$	−2.86404			−0.124289
$532$		0			0
$533$	0.620628	−	0.620628i	0.0268824	−	0.0268824i
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−6.28257	−	6.28257i	−0.271113	−	0.271113i
$538$		0			0
$539$	3.67241	−	4.40025i	0.158182	−	0.189532i
$540$		0			0
$541$			35.7464i			1.53686i	0.639934	+	0.768430i	$0.278962\pi$
$541$			35.7464i			1.53686i	−0.639934	+	0.768430i	$0.721038\pi$
$542$		0			0
$543$	−11.3940	+	11.3940i	−0.488962	+	0.488962i
$544$		0			0
$545$		0			0
$546$		0			0
$547$	1.60781	−	1.60781i	0.0687452	−	0.0687452i	−0.671898	−	0.740643i	$-0.734521\pi$
$547$	1.60781	−	1.60781i	0.0687452	−	0.0687452i	0.740643	+	0.671898i	$0.234521\pi$
$548$		0			0
$549$	4.07246			0.173808
$550$		0			0
$551$	1.58599			0.0675654
$552$		0			0
$553$	−3.88871	+	3.88871i	−0.165365	+	0.165365i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−13.0848	+	13.0848i	−0.554422	+	0.554422i	−0.927714	−	0.373292i	$-0.878229\pi$
$557$	−13.0848	+	13.0848i	−0.554422	+	0.554422i	0.373292	+	0.927714i	$0.378229\pi$
$558$		0			0
$559$		−	2.62574i		−	0.111057i
$560$		0			0
$561$	11.0926	−	13.2911i	0.468332	−	0.561151i
$562$		0			0
$563$	14.8287	+	14.8287i	0.624955	+	0.624955i	0.946794	−	0.321840i	$-0.104301\pi$
$563$	14.8287	+	14.8287i	0.624955	+	0.624955i	−0.321840	+	0.946794i	$0.604301\pi$
$564$		0			0
$565$		0			0
$566$		0			0
$567$	−2.08903	+	2.08903i	−0.0877309	+	0.0877309i
$568$		0			0
$569$	15.7158			0.658841			0.329421	−	0.944183i	$-0.393147\pi$
$569$	15.7158			0.658841			0.329421	+	0.944183i	$0.393147\pi$
$570$		0			0
$571$			42.9824i			1.79876i	0.437170	+	0.899379i	$0.355981\pi$
$571$			42.9824i			1.79876i	−0.437170	+	0.899379i	$0.644019\pi$
$572$		0			0
$573$	14.3129	−	14.3129i	0.597931	−	0.597931i
$574$		0			0
$575$		0			0
$576$		0			0
$577$	−24.1755	−	24.1755i	−1.00644	−	1.00644i	−0.999979	−	0.00645868i	$-0.997944\pi$
$577$	−24.1755	−	24.1755i	−1.00644	−	1.00644i	−0.00645868	−	0.999979i	$-0.502056\pi$
$578$		0			0
$579$	25.3407			1.05312
$580$		0			0
$581$	−1.80676			−0.0749568
$582$		0			0
$583$	11.9859	−	1.08066i	0.496406	−	0.0447563i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−16.9771	−	16.9771i	−0.700720	−	0.700720i	0.263845	−	0.964565i	$-0.415009\pi$
$587$	−16.9771	−	16.9771i	−0.700720	−	0.700720i	−0.964565	+	0.263845i	$0.915009\pi$
$588$		0			0
$589$	−3.51147			−0.144688
$590$		0			0
$591$			3.46930i			0.142708i
$592$		0			0
$593$	23.8599	+	23.8599i	0.979810	+	0.979810i	0.999800	−	0.0199904i	$-0.00636356\pi$
$593$	23.8599	+	23.8599i	0.979810	+	0.979810i	−0.0199904	+	0.999800i	$0.506364\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	6.92697	+	6.92697i	0.283502	+	0.283502i
$598$		0			0
$599$			45.1985i			1.84676i	0.383885	+	0.923381i	$0.374586\pi$
$599$			45.1985i			1.84676i	−0.383885	+	0.923381i	$0.625414\pi$
$600$		0			0
$601$			21.0028i			0.856722i	0.903608	+	0.428361i	$0.140909\pi$
$601$			21.0028i			0.856722i	−0.903608	+	0.428361i	$0.859091\pi$
$602$		0			0
$603$	4.02255	−	4.02255i	0.163811	−	0.163811i
$604$		0			0
$605$		0			0
$606$		0			0
$607$	6.66541	−	6.66541i	0.270541	−	0.270541i	−0.558777	−	0.829318i	$-0.688729\pi$
$607$	6.66541	−	6.66541i	0.270541	−	0.270541i	0.829318	+	0.558777i	$0.188729\pi$
$608$		0			0
$609$			3.46410i			0.140372i
$610$		0			0
$611$			1.33975i			0.0542005i
$612$		0			0
$613$	−3.02593	−	3.02593i	−0.122216	−	0.122216i	0.643353	−	0.765570i	$-0.277543\pi$
$613$	−3.02593	−	3.02593i	−0.122216	−	0.122216i	−0.765570	+	0.643353i	$0.777543\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−28.3597	−	28.3597i	−1.14172	−	1.14172i	−0.988136	−	0.153581i	$-0.950919\pi$
$617$	−28.3597	−	28.3597i	−1.14172	−	1.14172i	−0.153581	−	0.988136i	$-0.549081\pi$
$618$		0			0
$619$			34.0955i			1.37041i	0.728349	+	0.685206i	$0.240288\pi$
$619$			34.0955i			1.37041i	−0.728349	+	0.685206i	$0.759712\pi$
$620$		0			0
$621$	−1.59609			−0.0640489
$622$		0			0
$623$	−7.01209	−	7.01209i	−0.280933	−	0.280933i
$624$		0			0
$625$		0			0
$626$		0			0
$627$	−4.46794	+	0.402833i	−0.178432	+	0.0160876i
$628$		0			0
$629$	31.0923			1.23973
$630$		0			0
$631$	−16.1251			−0.641930			−0.320965	−	0.947091i	$-0.604007\pi$
$631$	−16.1251			−0.641930			−0.320965	+	0.947091i	$0.604007\pi$
$632$		0			0
$633$	6.48009	+	6.48009i	0.257560	+	0.257560i
$634$		0			0
$635$		0			0
$636$		0			0
$637$	0.468200	−	0.468200i	0.0185507	−	0.0185507i
$638$		0			0
$639$		−	2.49259i		−	0.0986052i
$640$		0			0
$641$	24.6202			0.972438			0.486219	−	0.873837i	$-0.338376\pi$
$641$	24.6202			0.972438			0.486219	+	0.873837i	$0.338376\pi$
$642$		0			0
$643$	−14.5113	+	14.5113i	−0.572271	+	0.572271i	−0.932763	−	0.360491i	$-0.882609\pi$
$643$	−14.5113	+	14.5113i	−0.572271	+	0.572271i	0.360491	+	0.932763i	$0.382609\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	7.08489	+	7.08489i	0.278536	+	0.278536i	0.832524	−	0.553988i	$-0.186895\pi$
$647$	7.08489	+	7.08489i	0.278536	+	0.278536i	−0.553988	+	0.832524i	$0.686895\pi$
$648$		0			0
$649$	6.08649	−	7.29277i	0.238916	−	0.286266i
$650$		0			0
$651$		−	7.66971i		−	0.300600i
$652$		0			0
$653$	23.6106	−	23.6106i	0.923953	−	0.923953i	−0.0733530	−	0.997306i	$-0.523370\pi$
$653$	23.6106	−	23.6106i	0.923953	−	0.923953i	0.997306	+	0.0733530i	$0.0233700\pi$
$654$		0			0
$655$		0			0
$656$		0			0
$657$	−1.83275	+	1.83275i	−0.0715025	+	0.0715025i
$658$		0			0
$659$	35.4475			1.38084			0.690420	−	0.723409i	$-0.257426\pi$
$659$	35.4475			1.38084			0.690420	+	0.723409i	$0.257426\pi$
$660$		0			0
$661$	−30.3511			−1.18052			−0.590260	−	0.807214i	$-0.700975\pi$
$661$	−30.3511			−1.18052			−0.590260	+	0.807214i	$0.700975\pi$
$662$		0			0
$663$	1.41421	−	1.41421i	0.0549235	−	0.0549235i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−1.32335	+	1.32335i	−0.0512402	+	0.0512402i
$668$		0			0
$669$		−	6.70356i		−	0.259175i
$670$		0			0
$671$	−8.65455	+	10.3698i	−0.334105	+	0.400322i
$672$		0			0
$673$	−12.8801	−	12.8801i	−0.496490	−	0.496490i	0.413854	−	0.910343i	$-0.364183\pi$
$673$	−12.8801	−	12.8801i	−0.496490	−	0.496490i	−0.910343	+	0.413854i	$0.864183\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−26.4297	+	26.4297i	−1.01578	+	1.01578i	−0.0159042	+	0.999874i	$0.505063\pi$
$677$	−26.4297	+	26.4297i	−1.01578	+	1.01578i	−0.999874	+	0.0159042i	$0.994937\pi$
$678$		0			0
$679$	32.4656			1.24592
$680$		0			0
$681$			17.1100i			0.655656i
$682$		0			0
$683$	−8.06243	+	8.06243i	−0.308500	+	0.308500i	−0.844328	−	0.535827i	$-0.820000\pi$
$683$	−8.06243	+	8.06243i	−0.308500	+	0.308500i	0.535827	+	0.844328i	$0.320000\pi$
$684$		0			0
$685$		0			0
$686$		0			0
$687$	−17.3839	−	17.3839i	−0.663235	−	0.663235i
$688$		0			0
$689$	1.39032			0.0529671
$690$		0			0
$691$	25.0565			0.953195			0.476598	−	0.879122i	$-0.341870\pi$
$691$	25.0565			0.953195			0.476598	+	0.879122i	$0.341870\pi$
$692$		0			0
$693$	−0.879863	−	9.75883i	−0.0334232	−	0.370707i
$694$		0			0
$695$		0			0
$696$		0			0
$697$	−8.45466	−	8.45466i	−0.320243	−	0.320243i
$698$		0			0
$699$	2.47305			0.0935395
$700$		0			0
$701$			12.3407i			0.466102i	0.972465	+	0.233051i	$0.0748709\pi$
$701$			12.3407i			0.466102i	−0.972465	+	0.233051i	$0.925129\pi$
$702$		0			0
$703$	−5.69717	−	5.69717i	−0.214873	−	0.214873i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−17.6090	−	17.6090i	−0.662253	−	0.662253i
$708$		0			0
$709$			41.3691i			1.55365i	0.629717	+	0.776824i	$0.283171\pi$
$709$			41.3691i			1.55365i	−0.629717	+	0.776824i	$0.716829\pi$
$710$		0			0
$711$			1.86149i			0.0698114i
$712$		0			0
$713$	2.92996	−	2.92996i	0.109728	−	0.109728i
$714$		0			0
$715$		0			0
$716$		0			0
$717$	−7.07403	+	7.07403i	−0.264184	+	0.264184i
$718$		0			0
$719$		−	14.5221i		−	0.541584i	−0.962638	−	0.270792i	$-0.912714\pi$
$719$		−	14.5221i		−	0.541584i	0.962638	−	0.270792i	$-0.0872856\pi$
$720$		0			0
$721$			16.1428i			0.601188i
$722$		0			0
$723$	16.8110	+	16.8110i	0.625208	+	0.625208i
$724$		0			0
$725$		0			0
$726$		0			0
$727$	−11.2008	−	11.2008i	−0.415413	−	0.415413i	0.468206	−	0.883619i	$-0.344901\pi$
$727$	−11.2008	−	11.2008i	−0.415413	−	0.415413i	−0.883619	+	0.468206i	$0.844901\pi$
$728$		0			0
$729$			1.00000i			0.0370370i
$730$		0			0
$731$	−35.7698			−1.32299
$732$		0			0
$733$	18.9417	+	18.9417i	0.699626	+	0.699626i	0.964330	−	0.264704i	$-0.0852741\pi$
$733$	18.9417	+	18.9417i	0.699626	+	0.699626i	−0.264704	+	0.964330i	$0.585274\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1.69423	+	18.7912i	0.0624077	+	0.692183i
$738$		0			0
$739$	22.0323			0.810472			0.405236	−	0.914212i	$-0.367189\pi$
$739$	22.0323			0.810472			0.405236	+	0.914212i	$0.367189\pi$
$740$		0			0
$741$	−0.518265			−0.0190389
$742$		0			0
$743$	−15.5007	−	15.5007i	−0.568664	−	0.568664i	0.363090	−	0.931754i	$-0.381722\pi$
$743$	−15.5007	−	15.5007i	−0.568664	−	0.568664i	−0.931754	+	0.363090i	$0.881722\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	−0.432439	+	0.432439i	−0.0158221	+	0.0158221i
$748$		0			0
$749$			25.6703i			0.937971i
$750$		0			0
$751$	9.19218			0.335427			0.167714	−	0.985836i	$-0.446362\pi$
$751$	9.19218			0.335427			0.167714	+	0.985836i	$0.446362\pi$
$752$		0			0
$753$	9.34303	−	9.34303i	0.340479	−	0.340479i
$754$		0			0
$755$		0			0
$756$		0			0
$757$	−1.60255	−	1.60255i	−0.0582456	−	0.0582456i	0.677384	−	0.735630i	$-0.263114\pi$
$757$	−1.60255	−	1.60255i	−0.0582456	−	0.0582456i	−0.735630	+	0.677384i	$0.763114\pi$
$758$		0			0
$759$	3.39192	−	4.06416i	0.123119	−	0.147520i
$760$		0			0
$761$		−	23.7052i		−	0.859311i	−0.902993	−	0.429656i	$-0.858635\pi$
$761$		−	23.7052i		−	0.859311i	0.902993	−	0.429656i	$-0.141365\pi$
$762$		0			0
$763$	−26.4928	+	26.4928i	−0.959104	+	0.959104i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	0.775973	−	0.775973i	0.0280188	−	0.0280188i
$768$		0			0
$769$	−29.8422			−1.07614			−0.538069	−	0.842901i	$-0.680846\pi$
$769$	−29.8422			−1.07614			−0.538069	+	0.842901i	$0.680846\pi$
$770$		0			0
$771$	−19.5845			−0.705318
$772$		0			0
$773$	12.6870	−	12.6870i	0.456321	−	0.456321i	−0.441125	−	0.897446i	$-0.645420\pi$
$773$	12.6870	−	12.6870i	0.456321	−	0.456321i	0.897446	+	0.441125i	$0.145420\pi$
$774$		0			0
$775$		0			0
$776$		0			0
$777$	12.4437	−	12.4437i	0.446415	−	0.446415i
$778$		0			0
$779$			3.09837i			0.111010i
$780$		0			0
$781$	6.34694	+	5.29710i	0.227111	+	0.189545i
$782$		0			0
$783$	0.829118	+	0.829118i	0.0296303	+	0.0296303i
$784$		0			0
$785$		0			0
$786$		0			0
$787$	6.77964	−	6.77964i	0.241668	−	0.241668i	−0.575872	−	0.817540i	$-0.695337\pi$
$787$	6.77964	−	6.77964i	0.241668	−	0.241668i	0.817540	+	0.575872i	$0.195337\pi$
$788$		0			0
$789$	17.6003			0.626586
$790$		0			0
$791$		−	31.1882i		−	1.10892i
$792$		0			0
$793$	−1.10338	+	1.10338i	−0.0391821	+	0.0391821i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−8.96575	−	8.96575i	−0.317583	−	0.317583i	0.530255	−	0.847838i	$-0.322096\pi$
$797$	−8.96575	−	8.96575i	−0.317583	−	0.317583i	−0.847838	+	0.530255i	$0.822096\pi$
$798$		0			0
$799$	18.2511			0.645676
$800$		0			0
$801$	−3.35663			−0.118601
$802$		0			0
$803$	−0.771924	−	8.56165i	−0.0272406	−	0.302134i
$804$		0			0
$805$		0			0
$806$		0			0
$807$	2.21693	+	2.21693i	0.0780395	+	0.0780395i
$808$		0			0
$809$	−45.1829			−1.58854			−0.794272	−	0.607562i	$-0.792148\pi$
$809$	−45.1829			−1.58854			−0.794272	+	0.607562i	$0.792148\pi$
$810$		0			0
$811$			47.3482i			1.66262i	0.555808	+	0.831310i	$0.312409\pi$
$811$			47.3482i			1.66262i	−0.555808	+	0.831310i	$0.687591\pi$
$812$		0			0
$813$	−8.23631	−	8.23631i	−0.288860	−	0.288860i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	6.55425	+	6.55425i	0.229304	+	0.229304i
$818$		0			0
$819$		−	1.13199i		−	0.0395549i
$820$		0			0
$821$			48.7594i			1.70171i	0.525397	+	0.850857i	$0.323917\pi$
$821$			48.7594i			1.70171i	−0.525397	+	0.850857i	$0.676083\pi$
$822$		0			0
$823$	−4.54786	+	4.54786i	−0.158528	+	0.158528i	−0.781914	−	0.623386i	$-0.785757\pi$
$823$	−4.54786	+	4.54786i	−0.158528	+	0.158528i	0.623386	+	0.781914i	$0.285757\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−36.3759	+	36.3759i	−1.26491	+	1.26491i	−0.316233	+	0.948681i	$0.602418\pi$
$827$	−36.3759	+	36.3759i	−1.26491	+	1.26491i	−0.948681	+	0.316233i	$0.897582\pi$
$828$		0			0
$829$			4.06492i			0.141180i	0.997505	+	0.0705902i	$0.0224883\pi$
$829$			4.06492i			0.141180i	−0.997505	+	0.0705902i	$0.977512\pi$
$830$		0			0
$831$		−	18.1047i		−	0.628045i
$832$		0			0
$833$	−6.37817	−	6.37817i	−0.220990	−	0.220990i
$834$		0			0
$835$		0			0
$836$		0			0
$837$	−1.83571	−	1.83571i	−0.0634515	−	0.0634515i
$838$		0			0
$839$		−	21.1214i		−	0.729193i	−0.931166	−	0.364597i	$-0.881207\pi$
$839$		−	21.1214i		−	0.729193i	0.931166	−	0.364597i	$-0.118793\pi$
$840$		0			0
$841$	−27.6251			−0.952591
$842$		0			0
$843$	16.3183	+	16.3183i	0.562031	+	0.562031i
$844$		0			0
$845$		0			0
$846$		0			0
$847$	26.7190	+	18.4985i	0.918075	+	0.635615i
$848$		0			0
$849$	20.6298			0.708011
$850$		0			0
$851$	9.50741			0.325910
$852$		0			0
$853$	12.7043	+	12.7043i	0.434986	+	0.434986i	0.890320	−	0.455335i	$-0.150480\pi$
$853$	12.7043	+	12.7043i	0.434986	+	0.434986i	−0.455335	+	0.890320i	$0.650480\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−19.2598	+	19.2598i	−0.657902	+	0.657902i	−0.954883	−	0.296981i	$-0.904020\pi$
$857$	−19.2598	+	19.2598i	−0.657902	+	0.657902i	0.296981	+	0.954883i	$0.404020\pi$
$858$		0			0
$859$			53.1473i			1.81336i	0.421818	+	0.906681i	$0.361392\pi$
$859$			53.1473i			1.81336i	−0.421818	+	0.906681i	$0.638608\pi$
$860$		0			0
$861$	−6.76742			−0.230633
$862$		0			0
$863$	−35.3639	+	35.3639i	−1.20380	+	1.20380i	−0.230798	+	0.973002i	$0.574134\pi$
$863$	−35.3639	+	35.3639i	−1.20380	+	1.20380i	−0.973002	+	0.230798i	$0.925866\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$	−7.24466	−	7.24466i	−0.246041	−	0.246041i
$868$		0			0
$869$	−4.73997	−	3.95594i	−0.160792	−	0.134196i
$870$		0			0
$871$			2.17971i			0.0738567i
$872$		0			0
$873$	7.77050	−	7.77050i	0.262992	−	0.262992i
$874$		0			0
$875$		0			0
$876$		0			0
$877$	3.18429	−	3.18429i	0.107526	−	0.107526i	−0.651297	−	0.758823i	$-0.725775\pi$
$877$	3.18429	−	3.18429i	0.107526	−	0.107526i	0.758823	+	0.651297i	$0.225775\pi$
$878$		0			0
$879$	−30.3826			−1.02478
$880$		0			0
$881$	−7.48495			−0.252175			−0.126087	−	0.992019i	$-0.540242\pi$
$881$	−7.48495			−0.252175			−0.126087	+	0.992019i	$0.540242\pi$
$882$		0			0
$883$	3.89152	−	3.89152i	0.130960	−	0.130960i	−0.638588	−	0.769548i	$-0.720481\pi$
$883$	3.89152	−	3.89152i	0.130960	−	0.130960i	0.769548	+	0.638588i	$0.220481\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	29.9858	−	29.9858i	1.00683	−	1.00683i	0.00684878	−	0.999977i	$-0.497820\pi$
$887$	29.9858	−	29.9858i	1.00683	−	1.00683i	0.999977	−	0.00684878i	$-0.00218005\pi$
$888$		0			0
$889$		−	66.4086i		−	2.22727i
$890$		0			0
$891$	−2.54632	−	2.12514i	−0.0853051	−	0.0711949i
$892$		0			0
$893$	−3.34422	−	3.34422i	−0.111910	−	0.111910i
$894$		0			0
$895$		0			0
$896$		0			0
$897$	0.432439	−	0.432439i	0.0144387	−	0.0144387i
$898$		0			0
$899$	−3.04404			−0.101525
$900$		0			0
$901$		−	18.9400i		−	0.630983i
$902$		0			0
$903$	−14.3157	+	14.3157i	−0.476397	+	0.476397i
$904$		0			0
$905$		0			0
$906$		0			0
$907$	−27.8553	−	27.8553i	−0.924921	−	0.924921i	0.0724513	−	0.997372i	$-0.476918\pi$
$907$	−27.8553	−	27.8553i	−0.924921	−	0.924921i	−0.997372	+	0.0724513i	$0.976918\pi$
$908$		0			0
$909$	−8.42926			−0.279581
$910$		0			0
$911$	13.7561			0.455761			0.227880	−	0.973689i	$-0.426820\pi$
$911$	13.7561			0.455761			0.227880	+	0.973689i	$0.426820\pi$
$912$		0			0
$913$	−0.182136	−	2.02013i	−0.00602782	−	0.0668564i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−45.0512	−	45.0512i	−1.48772	−	1.48772i
$918$		0			0
$919$	5.46500			0.180274			0.0901369	−	0.995929i	$-0.471270\pi$
$919$	5.46500			0.180274			0.0901369	+	0.995929i	$0.471270\pi$
$920$		0			0
$921$		−	10.5252i		−	0.346818i
$922$		0			0
$923$	0.675333	+	0.675333i	0.0222289	+	0.0222289i
$924$		0			0
$925$		0			0
$926$		0			0
$927$	3.86370	+	3.86370i	0.126901	+	0.126901i
$928$		0			0
$929$			39.9976i			1.31228i	0.754639	+	0.656140i	$0.227812\pi$
$929$			39.9976i			1.31228i	−0.754639	+	0.656140i	$0.772188\pi$
$930$		0			0
$931$			2.33740i			0.0766051i
$932$		0			0
$933$	13.2379	−	13.2379i	0.433389	−	0.433389i
$934$		0			0
$935$		0			0
$936$		0			0
$937$	−23.9358	+	23.9358i	−0.781949	+	0.781949i	−0.980159	−	0.198210i	$-0.936487\pi$
$937$	−23.9358	+	23.9358i	−0.781949	+	0.781949i	0.198210	+	0.980159i	$0.436487\pi$
$938$		0			0
$939$			7.18821i			0.234578i
$940$		0			0
$941$		−	20.8750i		−	0.680507i	−0.940334	−	0.340253i	$-0.889487\pi$
$941$		−	20.8750i		−	0.680507i	0.940334	−	0.340253i	$-0.110513\pi$
$942$		0			0
$943$	−2.58527	−	2.58527i	−0.0841880	−	0.0841880i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−17.1098	−	17.1098i	−0.555994	−	0.555994i	0.372171	−	0.928164i	$-0.378614\pi$
$947$	−17.1098	−	17.1098i	−0.555994	−	0.555994i	−0.928164	+	0.372171i	$0.878614\pi$
$948$		0			0
$949$		−	0.993120i		−	0.0322380i
$950$		0			0
$951$	6.90735			0.223986
$952$		0			0
$953$	3.46868	+	3.46868i	0.112362	+	0.112362i	0.761052	−	0.648691i	$-0.224683\pi$
$953$	3.46868	+	3.46868i	0.112362	+	0.112362i	−0.648691	+	0.761052i	$0.724683\pi$
$954$		0			0
$955$		0			0
$956$		0			0
$957$	−3.87320	+	0.349210i	−0.125203	+	0.0112884i
$958$		0			0
$959$	1.64483			0.0531143
$960$		0			0
$961$	−24.2603			−0.782591
$962$		0			0
$963$	6.14407	+	6.14407i	0.197990	+	0.197990i
$964$		0			0
$965$		0			0
$966$		0			0
$967$	−13.7600	+	13.7600i	−0.442491	+	0.442491i	−0.892848	−	0.450358i	$-0.851297\pi$
$967$	−13.7600	+	13.7600i	−0.442491	+	0.442491i	0.450358	+	0.892848i	$0.351297\pi$
$968$		0			0
$969$			7.06019i			0.226806i
$970$		0			0
$971$	25.2355			0.809845			0.404923	−	0.914351i	$-0.367298\pi$
$971$	25.2355			0.809845			0.404923	+	0.914351i	$0.367298\pi$
$972$		0			0
$973$	−12.3434	+	12.3434i	−0.395710	+	0.395710i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	32.2331	+	32.2331i	1.03123	+	1.03123i	0.999496	+	0.0317318i	$0.0101022\pi$
$977$	32.2331	+	32.2331i	1.03123	+	1.03123i	0.0317318	+	0.999496i	$0.489898\pi$
$978$		0			0
$979$	7.13331	−	8.54706i	0.227982	−	0.273165i
$980$		0			0
$981$			12.6819i			0.404901i
$982$		0			0
$983$	−3.53326	+	3.53326i	−0.112693	+	0.112693i	−0.761205	−	0.648511i	$-0.775392\pi$
$983$	−3.53326	+	3.53326i	−0.112693	+	0.112693i	0.648511	+	0.761205i	$0.275392\pi$
$984$		0			0
$985$		0			0
$986$		0			0
$987$	7.30441	−	7.30441i	0.232502	−	0.232502i
$988$		0			0
$989$	−10.9377			−0.347799
$990$		0			0
$991$	−33.1167			−1.05199			−0.525994	−	0.850488i	$-0.676306\pi$
$991$	−33.1167			−1.05199			−0.525994	+	0.850488i	$0.676306\pi$
$992$		0			0
$993$	−12.2934	+	12.2934i	−0.390118	+	0.390118i
$994$		0			0
$995$		0			0
$996$		0			0
$997$	22.9030	−	22.9030i	0.725345	−	0.725345i	−0.244343	−	0.969689i	$-0.578572\pi$
$997$	22.9030	−	22.9030i	0.725345	−	0.725345i	0.969689	+	0.244343i	$0.0785724\pi$
$998$		0			0
$999$		−	5.95669i		−	0.188461i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3300.2.x.d.1957.10	yes	32
5.2	odd	4	inner	3300.2.x.d.1693.6	yes	32
5.3	odd	4	inner	3300.2.x.d.1693.9	yes	32
5.4	even	2	inner	3300.2.x.d.1957.5	yes	32
11.10	odd	2	inner	3300.2.x.d.1957.9	yes	32
55.32	even	4	inner	3300.2.x.d.1693.5	✓	32
55.43	even	4	inner	3300.2.x.d.1693.10	yes	32
55.54	odd	2	inner	3300.2.x.d.1957.6	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3300.2.x.d.1693.5	✓	32	55.32	even	4	inner
3300.2.x.d.1693.6	yes	32	5.2	odd	4	inner
3300.2.x.d.1693.9	yes	32	5.3	odd	4	inner
3300.2.x.d.1693.10	yes	32	55.43	even	4	inner
3300.2.x.d.1957.5	yes	32	5.4	even	2	inner
3300.2.x.d.1957.6	yes	32	55.54	odd	2	inner
3300.2.x.d.1957.9	yes	32	11.10	odd	2	inner
3300.2.x.d.1957.10	yes	32	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 3300 = 2^{2} \cdot 3 \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3300.x (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(0.707107 - 0.707107i) q^{3} +(2.08903 - 2.08903i) q^{7} -1.00000i q^{9} +(2.54632 + 2.12514i) q^{11} +(0.270937 + 0.270937i) q^{13} +(3.69090 - 3.69090i) q^{17} -1.35260 q^{19} -2.95433i q^{21} +(1.12861 - 1.12861i) q^{23} +(-0.707107 - 0.707107i) q^{27} -1.17255 q^{29} +2.59609 q^{31} +(3.30323 - 0.297821i) q^{33} +(4.21202 + 4.21202i) q^{37} +0.383162 q^{39} -2.29068i q^{41} +(-4.84567 - 4.84567i) q^{43} +(2.47244 + 2.47244i) q^{47} -1.72808i q^{49} -5.21972i q^{51} +(2.56577 - 2.56577i) q^{53} +(-0.956432 + 0.956432i) q^{57} -2.86404i q^{59} +4.07246i q^{61} +(-2.08903 - 2.08903i) q^{63} +(4.02255 + 4.02255i) q^{67} -1.59609i q^{69} +2.49259 q^{71} +(-1.83275 - 1.83275i) q^{73} +(9.75883 - 0.879863i) q^{77} -1.86149 q^{79} -1.00000 q^{81} +(-0.432439 - 0.432439i) q^{83} +(-0.829118 + 0.829118i) q^{87} -3.35663i q^{89} +1.13199 q^{91} +(1.83571 - 1.83571i) q^{93} +(7.77050 + 7.77050i) q^{97} +(2.12514 - 2.54632i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 24 q^{11} + 16 q^{31} - 32 q^{71} - 32 q^{81} + 80 q^{91}+O(q^{100}) \) 32 * q - 24 * q^11 + 16 * q^31 - 32 * q^71 - 32 * q^81 + 80 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more