LMFDB - Embedded newform 4620.2.m.b.1121.13

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4620,2,Mod(1121,4620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4620, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 0, 1]))

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

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

chi := DirichletCharacter("4620.1121");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4620.m (of order $2$, degree $1$, 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:	$36.8908857338$
Analytic rank:	$0$
Dimension:	$48$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1121.13
Character	$\chi$	$=$	4620.1121
Dual form			4620.2.m.b.1121.14

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.34978 - 1.08540i) q^{3} +1.00000i q^{5} -1.00000i q^{7} +(0.643830 + 2.93010i) q^{9} +O(q^{10})$	\(q+(-1.34978 - 1.08540i) q^{3} +1.00000i q^{5} -1.00000i q^{7} +(0.643830 + 2.93010i) q^{9} +(3.16726 + 0.984090i) q^{11} +4.54736i q^{13} +(1.08540 - 1.34978i) q^{15} -0.0181501 q^{17} +2.25642i q^{19} +(-1.08540 + 1.34978i) q^{21} +4.86726i q^{23} -1.00000 q^{25} +(2.31129 - 4.65381i) q^{27} -2.53551 q^{29} +4.30912 q^{31} +(-3.20699 - 4.76605i) q^{33} +1.00000 q^{35} -2.29303 q^{37} +(4.93568 - 6.13794i) q^{39} -6.83796 q^{41} +2.58312i q^{43} +(-2.93010 + 0.643830i) q^{45} +11.4487i q^{47} -1.00000 q^{49} +(0.0244988 + 0.0197001i) q^{51} -4.81302i q^{53} +(-0.984090 + 3.16726i) q^{55} +(2.44911 - 3.04568i) q^{57} -10.2435i q^{59} +5.23480i q^{61} +(2.93010 - 0.643830i) q^{63} -4.54736 q^{65} -10.5120 q^{67} +(5.28290 - 6.56974i) q^{69} +10.8778i q^{71} -15.3062i q^{73} +(1.34978 + 1.08540i) q^{75} +(0.984090 - 3.16726i) q^{77} -14.7974i q^{79} +(-8.17097 + 3.77297i) q^{81} -10.3408 q^{83} -0.0181501i q^{85} +(3.42238 + 2.75203i) q^{87} -7.52948i q^{89} +4.54736 q^{91} +(-5.81638 - 4.67711i) q^{93} -2.25642 q^{95} -15.9594 q^{97} +(-0.844304 + 9.91399i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q - 4 q^{3} + 6 q^{9}+O(q^{10}) $ 48 * q - 4 * q^3 + 6 * q^9	$ 48 q - 4 q^{3} + 6 q^{9} + 6 q^{11} + 2 q^{15} - 4 q^{17} - 2 q^{21} - 48 q^{25} - 16 q^{27} - 36 q^{29} - 16 q^{31} - 4 q^{33} + 48 q^{35} - 8 q^{37} + 18 q^{39} - 48 q^{49} + 30 q^{51} + 4 q^{55} - 16 q^{57} + 12 q^{65} + 24 q^{67} + 4 q^{69} + 4 q^{75} - 4 q^{77} - 22 q^{81} - 20 q^{83} - 44 q^{87} - 12 q^{91} - 28 q^{93} - 8 q^{95} - 56 q^{97} - 12 q^{99}+O(q^{100}) $ 48 * q - 4 * q^3 + 6 * q^9 + 6 * q^11 + 2 * q^15 - 4 * q^17 - 2 * q^21 - 48 * q^25 - 16 * q^27 - 36 * q^29 - 16 * q^31 - 4 * q^33 + 48 * q^35 - 8 * q^37 + 18 * q^39 - 48 * q^49 + 30 * q^51 + 4 * q^55 - 16 * q^57 + 12 * q^65 + 24 * q^67 + 4 * q^69 + 4 * q^75 - 4 * q^77 - 22 * q^81 - 20 * q^83 - 44 * q^87 - 12 * q^91 - 28 * q^93 - 8 * q^95 - 56 * q^97 - 12 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4620\mathbb{Z}\right)^\times$.

$n$	$661$	$1541$	$2311$	$2521$	$3697$
$\chi(n)$	$1$	$-1$	$1$	$-1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−1.34978	−	1.08540i	−0.779298	−	0.626654i
$3$	−1.34978	−	1.08540i	−0.779298	−	0.626654i
$4$		0			0
$5$			1.00000i			0.447214i
$5$			1.00000i			0.447214i
$6$		0			0
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$		0			0
$9$	0.643830	+	2.93010i	0.214610	+	0.976700i
$10$		0			0
$11$	3.16726	+	0.984090i	0.954966	+	0.296714i
$11$	3.16726	+	0.984090i	0.954966	+	0.296714i
$12$		0			0
$13$			4.54736i			1.26121i	0.776104	+	0.630605i	$0.217193\pi$
$13$			4.54736i			1.26121i	−0.776104	+	0.630605i	$0.782807\pi$
$14$		0			0
$15$	1.08540	−	1.34978i	0.280248	−	0.348513i
$16$		0			0
$17$	−0.0181501			−0.00440206			−0.00220103	−	0.999998i	$-0.500701\pi$
$17$	−0.0181501			−0.00440206			−0.00220103	+	0.999998i	$0.500701\pi$
$18$		0			0
$19$			2.25642i			0.517659i	0.965923	+	0.258830i	$0.0833368\pi$
$19$			2.25642i			0.517659i	−0.965923	+	0.258830i	$0.916663\pi$
$20$		0			0
$21$	−1.08540	+	1.34978i	−0.236853	+	0.294547i
$22$		0			0
$23$			4.86726i			1.01489i	0.861683	+	0.507447i	$0.169411\pi$
$23$			4.86726i			1.01489i	−0.861683	+	0.507447i	$0.830589\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	2.31129	−	4.65381i	0.444808	−	0.895626i
$28$		0			0
$29$	−2.53551			−0.470832			−0.235416	−	0.971895i	$-0.575645\pi$
$29$	−2.53551			−0.470832			−0.235416	+	0.971895i	$0.575645\pi$
$30$		0			0
$31$	4.30912			0.773942			0.386971	−	0.922092i	$-0.373521\pi$
$31$	4.30912			0.773942			0.386971	+	0.922092i	$0.373521\pi$
$32$		0			0
$33$	−3.20699	−	4.76605i	−0.558266	−	0.829662i
$34$		0			0
$35$	1.00000			0.169031
$36$		0			0
$37$	−2.29303			−0.376971			−0.188486	−	0.982076i	$-0.560358\pi$
$37$	−2.29303			−0.376971			−0.188486	+	0.982076i	$0.560358\pi$
$38$		0			0
$39$	4.93568	−	6.13794i	0.790342	−	0.982858i
$40$		0			0
$41$	−6.83796			−1.06791			−0.533955	−	0.845513i	$-0.679295\pi$
$41$	−6.83796			−1.06791			−0.533955	+	0.845513i	$0.679295\pi$
$42$		0			0
$43$			2.58312i			0.393922i	0.980411	+	0.196961i	$0.0631072\pi$
$43$			2.58312i			0.393922i	−0.980411	+	0.196961i	$0.936893\pi$
$44$		0			0
$45$	−2.93010	+	0.643830i	−0.436793	+	0.0959765i
$46$		0			0
$47$			11.4487i			1.66997i	0.550273	+	0.834985i	$0.314524\pi$
$47$			11.4487i			1.66997i	−0.550273	+	0.834985i	$0.685476\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	0.0244988	+	0.0197001i	0.00343051	+	0.00275857i
$52$		0			0
$53$		−	4.81302i		−	0.661120i	−0.943785	−	0.330560i	$-0.892762\pi$
$53$		−	4.81302i		−	0.661120i	0.943785	−	0.330560i	$-0.107238\pi$
$54$		0			0
$55$	−0.984090	+	3.16726i	−0.132695	+	0.427074i
$56$		0			0
$57$	2.44911	−	3.04568i	0.324393	−	0.403411i
$58$		0			0
$59$		−	10.2435i		−	1.33360i	−0.745238	−	0.666798i	$-0.767664\pi$
$59$		−	10.2435i		−	1.33360i	0.745238	−	0.666798i	$-0.232336\pi$
$60$		0			0
$61$			5.23480i			0.670248i	0.942174	+	0.335124i	$0.108778\pi$
$61$			5.23480i			0.670248i	−0.942174	+	0.335124i	$0.891222\pi$
$62$		0			0
$63$	2.93010	−	0.643830i	0.369158	−	0.0811149i
$64$		0			0
$65$	−4.54736			−0.564030
$66$		0			0
$67$	−10.5120			−1.28424			−0.642120	−	0.766604i	$-0.721945\pi$
$67$	−10.5120			−1.28424			−0.642120	+	0.766604i	$0.721945\pi$
$68$		0			0
$69$	5.28290	−	6.56974i	0.635987	−	0.790904i
$70$		0			0
$71$			10.8778i			1.29096i	0.763776	+	0.645482i	$0.223343\pi$
$71$			10.8778i			1.29096i	−0.763776	+	0.645482i	$0.776657\pi$
$72$		0			0
$73$		−	15.3062i		−	1.79145i	−0.444604	−	0.895727i	$-0.646655\pi$
$73$		−	15.3062i		−	1.79145i	0.444604	−	0.895727i	$-0.353345\pi$
$74$		0			0
$75$	1.34978	+	1.08540i	0.155860	+	0.125331i
$76$		0			0
$77$	0.984090	−	3.16726i	0.112148	−	0.360943i
$78$		0			0
$79$		−	14.7974i		−	1.66484i	−0.554147	−	0.832419i	$-0.686955\pi$
$79$		−	14.7974i		−	1.66484i	0.554147	−	0.832419i	$-0.313045\pi$
$80$		0			0
$81$	−8.17097	+	3.77297i	−0.907885	+	0.419219i
$82$		0			0
$83$	−10.3408			−1.13505			−0.567525	−	0.823356i	$-0.692099\pi$
$83$	−10.3408			−1.13505			−0.567525	+	0.823356i	$0.692099\pi$
$84$		0			0
$85$		−	0.0181501i		−	0.00196866i
$86$		0			0
$87$	3.42238	+	2.75203i	0.366918	+	0.295049i
$88$		0			0
$89$		−	7.52948i		−	0.798123i	−0.916924	−	0.399062i	$-0.869336\pi$
$89$		−	7.52948i		−	0.798123i	0.916924	−	0.399062i	$-0.130664\pi$
$90$		0			0
$91$	4.54736			0.476692
$92$		0			0
$93$	−5.81638	−	4.67711i	−0.603131	−	0.484993i
$94$		0			0
$95$	−2.25642			−0.231504
$96$		0			0
$97$	−15.9594			−1.62043			−0.810217	−	0.586130i	$-0.800651\pi$
$97$	−15.9594			−1.62043			−0.810217	+	0.586130i	$0.800651\pi$
$98$		0			0
$99$	−0.844304	+	9.91399i	−0.0848557	+	0.996393i
$100$		0			0
$101$	6.80227			0.676851			0.338426	−	0.940993i	$-0.390106\pi$
$101$	6.80227			0.676851			0.338426	+	0.940993i	$0.390106\pi$
$102$		0			0
$103$	−15.1424			−1.49203			−0.746013	−	0.665931i	$-0.768034\pi$
$103$	−15.1424			−1.49203			−0.746013	+	0.665931i	$0.768034\pi$
$104$		0			0
$105$	−1.34978	−	1.08540i	−0.131725	−	0.105924i
$106$		0			0
$107$	15.8775			1.53493			0.767467	−	0.641088i	$-0.221517\pi$
$107$	15.8775			1.53493			0.767467	+	0.641088i	$0.221517\pi$
$108$		0			0
$109$			3.41499i			0.327096i	0.986535	+	0.163548i	$0.0522939\pi$
$109$			3.41499i			0.327096i	−0.986535	+	0.163548i	$0.947706\pi$
$110$		0			0
$111$	3.09509	+	2.48884i	0.293773	+	0.236230i
$112$		0			0
$113$		−	3.73836i		−	0.351675i	−0.984419	−	0.175838i	$-0.943737\pi$
$113$		−	3.73836i		−	0.351675i	0.984419	−	0.175838i	$-0.0562634\pi$
$114$		0			0
$115$	−4.86726			−0.453874
$116$		0			0
$117$	−13.3242	+	2.92772i	−1.23182	+	0.270668i
$118$		0			0
$119$			0.0181501i			0.00166382i
$120$		0			0
$121$	9.06313	+	6.23375i	0.823921	+	0.566705i
$122$		0			0
$123$	9.22976	+	7.42189i	0.832219	+	0.669209i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$			19.5901i			1.73834i	0.494514	+	0.869170i	$0.335346\pi$
$127$			19.5901i			1.73834i	−0.494514	+	0.869170i	$0.664654\pi$
$128$		0			0
$129$	2.80371	−	3.48665i	0.246852	−	0.306982i
$130$		0			0
$131$	0.235537			0.0205790			0.0102895	−	0.999947i	$-0.496725\pi$
$131$	0.235537			0.0205790			0.0102895	+	0.999947i	$0.496725\pi$
$132$		0			0
$133$	2.25642			0.195657
$134$		0			0
$135$	4.65381	+	2.31129i	0.400536	+	0.198924i
$136$		0			0
$137$		−	14.3650i		−	1.22728i	−0.789585	−	0.613641i	$-0.789704\pi$
$137$		−	14.3650i		−	1.22728i	0.789585	−	0.613641i	$-0.210296\pi$
$138$		0			0
$139$			1.90749i			0.161791i	0.996723	+	0.0808955i	$0.0257780\pi$
$139$			1.90749i			0.161791i	−0.996723	+	0.0808955i	$0.974222\pi$
$140$		0			0
$141$	12.4264	−	15.4533i	1.04649	−	1.30140i
$142$		0			0
$143$	−4.47501	+	14.4027i	−0.374219	+	1.20441i
$144$		0			0
$145$		−	2.53551i		−	0.210562i
$146$		0			0
$147$	1.34978	+	1.08540i	0.111328	+	0.0895220i
$148$		0			0
$149$	11.0666			0.906609			0.453304	−	0.891356i	$-0.350245\pi$
$149$	11.0666			0.906609			0.453304	+	0.891356i	$0.350245\pi$
$150$		0			0
$151$			6.92114i			0.563235i	0.959527	+	0.281617i	$0.0908709\pi$
$151$			6.92114i			0.563235i	−0.959527	+	0.281617i	$0.909129\pi$
$152$		0			0
$153$	−0.0116856	−	0.0531817i	−0.000944725	−	0.00429949i
$154$		0			0
$155$			4.30912i			0.346117i
$156$		0			0
$157$	−7.01310			−0.559706			−0.279853	−	0.960043i	$-0.590286\pi$
$157$	−7.01310			−0.559706			−0.279853	+	0.960043i	$0.590286\pi$
$158$		0			0
$159$	−5.22404	+	6.49654i	−0.414293	+	0.515209i
$160$		0			0
$161$	4.86726			0.383594
$162$		0			0
$163$	−16.7352			−1.31080			−0.655400	−	0.755282i	$-0.727500\pi$
$163$	−16.7352			−1.31080			−0.655400	+	0.755282i	$0.727500\pi$
$164$		0			0
$165$	4.76605	−	3.20699i	0.371036	−	0.249664i
$166$		0			0
$167$	−0.539109			−0.0417175			−0.0208587	−	0.999782i	$-0.506640\pi$
$167$	−0.539109			−0.0417175			−0.0208587	+	0.999782i	$0.506640\pi$
$168$		0			0
$169$	−7.67845			−0.590650
$170$		0			0
$171$	−6.61155	+	1.45275i	−0.505598	+	0.111095i
$172$		0			0
$173$	−12.3261			−0.937137			−0.468568	−	0.883427i	$-0.655230\pi$
$173$	−12.3261			−0.937137			−0.468568	+	0.883427i	$0.655230\pi$
$174$		0			0
$175$			1.00000i			0.0755929i
$176$		0			0
$177$	−11.1183	+	13.8266i	−0.835703	+	1.03927i
$178$		0			0
$179$			20.8120i			1.55556i	0.628534	+	0.777782i	$0.283655\pi$
$179$			20.8120i			1.55556i	−0.628534	+	0.777782i	$0.716345\pi$
$180$		0			0
$181$	−2.17866			−0.161939			−0.0809694	−	0.996717i	$-0.525802\pi$
$181$	−2.17866			−0.161939			−0.0809694	+	0.996717i	$0.525802\pi$
$182$		0			0
$183$	5.68184	−	7.06585i	0.420013	−	0.522323i
$184$		0			0
$185$		−	2.29303i		−	0.168587i
$186$		0			0
$187$	−0.0574863	−	0.0178614i	−0.00420382	−	0.00130615i
$188$		0			0
$189$	−4.65381	−	2.31129i	−0.338515	−	0.168122i
$190$		0			0
$191$		−	2.98665i		−	0.216106i	−0.994145	−	0.108053i	$-0.965538\pi$
$191$		−	2.98665i		−	0.216106i	0.994145	−	0.108053i	$-0.0344616\pi$
$192$		0			0
$193$			23.8089i			1.71381i	0.515478	+	0.856903i	$0.327614\pi$
$193$			23.8089i			1.71381i	−0.515478	+	0.856903i	$0.672386\pi$
$194$		0			0
$195$	6.13794	+	4.93568i	0.439547	+	0.353452i
$196$		0			0
$197$	13.1597			0.937592			0.468796	−	0.883306i	$-0.344688\pi$
$197$	13.1597			0.937592			0.468796	+	0.883306i	$0.344688\pi$
$198$		0			0
$199$	−10.6783			−0.756964			−0.378482	−	0.925609i	$-0.623554\pi$
$199$	−10.6783			−0.756964			−0.378482	+	0.925609i	$0.623554\pi$
$200$		0			0
$201$	14.1889	+	11.4096i	1.00081	+	0.804774i
$202$		0			0
$203$			2.53551i			0.177958i
$204$		0			0
$205$		−	6.83796i		−	0.477584i
$206$		0			0
$207$	−14.2616	+	3.13369i	−0.991246	+	0.217806i
$208$		0			0
$209$	−2.22052	+	7.14669i	−0.153597	+	0.494347i
$210$		0			0
$211$		−	16.4284i		−	1.13097i	−0.824757	−	0.565487i	$-0.808688\pi$
$211$		−	16.4284i		−	1.13097i	0.824757	−	0.565487i	$-0.191312\pi$
$212$		0			0
$213$	11.8068	−	14.6827i	0.808987	−	1.00604i
$214$		0			0
$215$	−2.58312			−0.176167
$216$		0			0
$217$		−	4.30912i		−	0.292522i
$218$		0			0
$219$	−16.6133	+	20.6600i	−1.12262	+	1.39608i
$220$		0			0
$221$		−	0.0825352i		−	0.00555192i
$222$		0			0
$223$	−16.1189			−1.07940			−0.539701	−	0.841856i	$-0.681463\pi$
$223$	−16.1189			−1.07940			−0.539701	+	0.841856i	$0.681463\pi$
$224$		0			0
$225$	−0.643830	−	2.93010i	−0.0429220	−	0.195340i
$226$		0			0
$227$	7.75489			0.514710			0.257355	−	0.966317i	$-0.417149\pi$
$227$	7.75489			0.514710			0.257355	+	0.966317i	$0.417149\pi$
$228$		0			0
$229$	−16.7240			−1.10515			−0.552577	−	0.833462i	$-0.686355\pi$
$229$	−16.7240			−1.10515			−0.552577	+	0.833462i	$0.686355\pi$
$230$		0			0
$231$	−4.76605	+	3.20699i	−0.313583	+	0.211005i
$232$		0			0
$233$	−25.0925			−1.64387			−0.821933	−	0.569584i	$-0.807104\pi$
$233$	−25.0925			−1.64387			−0.821933	+	0.569584i	$0.807104\pi$
$234$		0			0
$235$	−11.4487			−0.746833
$236$		0			0
$237$	−16.0610	+	19.9733i	−1.04328	+	1.29740i
$238$		0			0
$239$	27.2061			1.75982			0.879908	−	0.475145i	$-0.157604\pi$
$239$	27.2061			1.75982			0.879908	+	0.475145i	$0.157604\pi$
$240$		0			0
$241$			7.12731i			0.459110i	0.973296	+	0.229555i	$0.0737271\pi$
$241$			7.12731i			0.459110i	−0.973296	+	0.229555i	$0.926273\pi$
$242$		0			0
$243$	15.1242	+	3.77604i	0.970218	+	0.242233i
$244$		0			0
$245$		−	1.00000i		−	0.0638877i
$246$		0			0
$247$	−10.2608			−0.652877
$248$		0			0
$249$	13.9578	+	11.2239i	0.884542	+	0.711283i
$250$		0			0
$251$			24.4087i			1.54066i	0.637644	+	0.770331i	$0.279909\pi$
$251$			24.4087i			1.54066i	−0.637644	+	0.770331i	$0.720091\pi$
$252$		0			0
$253$	−4.78982	+	15.4159i	−0.301134	+	0.969189i
$254$		0			0
$255$	−0.0197001	+	0.0244988i	−0.00123367	+	0.00153417i
$256$		0			0
$257$			16.1442i			1.00705i	0.863982	+	0.503523i	$0.167963\pi$
$257$			16.1442i			1.00705i	−0.863982	+	0.503523i	$0.832037\pi$
$258$		0			0
$259$			2.29303i			0.142482i
$260$		0			0
$261$	−1.63243	−	7.42929i	−0.101045	−	0.459861i
$262$		0			0
$263$	24.3020			1.49852			0.749262	−	0.662273i	$-0.230408\pi$
$263$	24.3020			1.49852			0.749262	+	0.662273i	$0.230408\pi$
$264$		0			0
$265$	4.81302			0.295662
$266$		0			0
$267$	−8.17247	+	10.1632i	−0.500147	+	0.621976i
$268$		0			0
$269$			4.67541i			0.285065i	0.989790	+	0.142532i	$0.0455245\pi$
$269$			4.67541i			0.285065i	−0.989790	+	0.142532i	$0.954476\pi$
$270$		0			0
$271$		−	12.7074i		−	0.771921i	−0.922515	−	0.385960i	$-0.873870\pi$
$271$		−	12.7074i		−	0.771921i	0.922515	−	0.385960i	$-0.126130\pi$
$272$		0			0
$273$	−6.13794	−	4.93568i	−0.371485	−	0.298721i
$274$		0			0
$275$	−3.16726	−	0.984090i	−0.190993	−	0.0593429i
$276$		0			0
$277$		−	23.2409i		−	1.39641i	−0.715897	−	0.698206i	$-0.753982\pi$
$277$		−	23.2409i		−	1.39641i	0.715897	−	0.698206i	$-0.246018\pi$
$278$		0			0
$279$	2.77434	+	12.6262i	0.166096	+	0.755909i
$280$		0			0
$281$	10.6697			0.636500			0.318250	−	0.948007i	$-0.396905\pi$
$281$	10.6697			0.636500			0.318250	+	0.948007i	$0.396905\pi$
$282$		0			0
$283$		−	9.65606i		−	0.573993i	−0.957932	−	0.286997i	$-0.907343\pi$
$283$		−	9.65606i		−	0.573993i	0.957932	−	0.286997i	$-0.0926569\pi$
$284$		0			0
$285$	3.04568	+	2.44911i	0.180411	+	0.145073i
$286$		0			0
$287$			6.83796i			0.403632i
$288$		0			0
$289$	−16.9997			−0.999981
$290$		0			0
$291$	21.5418	+	17.3223i	1.26280	+	1.01545i
$292$		0			0
$293$	13.5861			0.793711			0.396855	−	0.917881i	$-0.370101\pi$
$293$	13.5861			0.793711			0.396855	+	0.917881i	$0.370101\pi$
$294$		0			0
$295$	10.2435			0.596402
$296$		0			0
$297$	11.9002	−	12.4653i	0.690522	−	0.723312i
$298$		0			0
$299$	−22.1332			−1.27999
$300$		0			0
$301$	2.58312			0.148888
$302$		0			0
$303$	−9.18159	−	7.38316i	−0.527468	−	0.424151i
$304$		0			0
$305$	−5.23480			−0.299744
$306$		0			0
$307$			3.00625i			0.171576i	0.996313	+	0.0857878i	$0.0273407\pi$
$307$			3.00625i			0.171576i	−0.996313	+	0.0857878i	$0.972659\pi$
$308$		0			0
$309$	20.4390	+	16.4355i	1.16273	+	0.934984i
$310$		0			0
$311$			23.7509i			1.34679i	0.739284	+	0.673394i	$0.235164\pi$
$311$			23.7509i			1.34679i	−0.739284	+	0.673394i	$0.764836\pi$
$312$		0			0
$313$	−13.7445			−0.776883			−0.388442	−	0.921473i	$-0.626986\pi$
$313$	−13.7445			−0.776883			−0.388442	+	0.921473i	$0.626986\pi$
$314$		0			0
$315$	0.643830	+	2.93010i	0.0362757	+	0.165092i
$316$		0			0
$317$			35.0312i			1.96755i	0.179417	+	0.983773i	$0.442579\pi$
$317$			35.0312i			1.96755i	−0.179417	+	0.983773i	$0.557421\pi$
$318$		0			0
$319$	−8.03062	−	2.49517i	−0.449628	−	0.139703i
$320$		0			0
$321$	−21.4312	−	17.2334i	−1.19617	−	0.961873i
$322$		0			0
$323$		−	0.0409544i		−	0.00227877i
$324$		0			0
$325$		−	4.54736i		−	0.252242i
$326$		0			0
$327$	3.70661	−	4.60949i	0.204976	−	0.254905i
$328$		0			0
$329$	11.4487			0.631189
$330$		0			0
$331$	24.8085			1.36360			0.681799	−	0.731539i	$-0.261198\pi$
$331$	24.8085			1.36360			0.681799	+	0.731539i	$0.261198\pi$
$332$		0			0
$333$	−1.47632	−	6.71880i	−0.0809018	−	0.368188i
$334$		0			0
$335$		−	10.5120i		−	0.574329i
$336$		0			0
$337$			23.3020i			1.26934i	0.772783	+	0.634670i	$0.218864\pi$
$337$			23.3020i			1.26934i	−0.772783	+	0.634670i	$0.781136\pi$
$338$		0			0
$339$	−4.05760	+	5.04597i	−0.220379	+	0.274060i
$340$		0			0
$341$	13.6481	+	4.24057i	0.739088	+	0.229640i
$342$		0			0
$343$			1.00000i			0.0539949i
$344$		0			0
$345$	6.56974	+	5.28290i	0.353703	+	0.284422i
$346$		0			0
$347$	−10.0249			−0.538163			−0.269082	−	0.963117i	$-0.586720\pi$
$347$	−10.0249			−0.538163			−0.269082	+	0.963117i	$0.586720\pi$
$348$		0			0
$349$		−	7.92691i		−	0.424318i	−0.977235	−	0.212159i	$-0.931951\pi$
$349$		−	7.92691i		−	0.424318i	0.977235	−	0.212159i	$-0.0680494\pi$
$350$		0			0
$351$	21.1625	+	10.5103i	1.12957	+	0.560996i
$352$		0			0
$353$		−	12.7676i		−	0.679551i	−0.940507	−	0.339776i	$-0.889649\pi$
$353$		−	12.7676i		−	0.679551i	0.940507	−	0.339776i	$-0.110351\pi$
$354$		0			0
$355$	−10.8778			−0.577336
$356$		0			0
$357$	0.0197001	−	0.0244988i	0.00104264	−	0.00129661i
$358$		0			0
$359$	−28.9722			−1.52910			−0.764548	−	0.644566i	$-0.777038\pi$
$359$	−28.9722			−1.52910			−0.764548	+	0.644566i	$0.777038\pi$
$360$		0			0
$361$	13.9086			0.732029
$362$		0			0
$363$	−5.46717	−	18.2513i	−0.286952	−	0.957945i
$364$		0			0
$365$	15.3062			0.801163
$366$		0			0
$367$	−27.0806			−1.41360			−0.706798	−	0.707415i	$-0.749861\pi$
$367$	−27.0806			−1.41360			−0.706798	+	0.707415i	$0.749861\pi$
$368$		0			0
$369$	−4.40248	−	20.0359i	−0.229184	−	1.04303i
$370$		0			0
$371$	−4.81302			−0.249880
$372$		0			0
$373$			29.4875i			1.52680i	0.645924	+	0.763402i	$0.276472\pi$
$373$			29.4875i			1.52680i	−0.645924	+	0.763402i	$0.723528\pi$
$374$		0			0
$375$	−1.08540	+	1.34978i	−0.0560496	+	0.0697025i
$376$		0			0
$377$		−	11.5299i		−	0.593818i
$378$		0			0
$379$	−19.7840			−1.01624			−0.508119	−	0.861287i	$-0.669659\pi$
$379$	−19.7840			−1.01624			−0.508119	+	0.861287i	$0.669659\pi$
$380$		0			0
$381$	21.2630	−	26.4424i	1.08934	−	1.35468i
$382$		0			0
$383$			3.19255i			0.163132i	0.996668	+	0.0815658i	$0.0259921\pi$
$383$			3.19255i			0.163132i	−0.996668	+	0.0815658i	$0.974008\pi$
$384$		0			0
$385$	3.16726	+	0.984090i	0.161419	+	0.0501539i
$386$		0			0
$387$	−7.56879	+	1.66309i	−0.384743	+	0.0845395i
$388$		0			0
$389$		−	22.2811i		−	1.12970i	−0.825195	−	0.564848i	$-0.808935\pi$
$389$		−	22.2811i		−	1.12970i	0.825195	−	0.564848i	$-0.191065\pi$
$390$		0			0
$391$		−	0.0883415i		−	0.00446762i
$392$		0			0
$393$	−0.317924	−	0.255651i	−0.0160371	−	0.0128959i
$394$		0			0
$395$	14.7974			0.744538
$396$		0			0
$397$	−5.32173			−0.267090			−0.133545	−	0.991043i	$-0.542636\pi$
$397$	−5.32173			−0.267090			−0.133545	+	0.991043i	$0.542636\pi$
$398$		0			0
$399$	−3.04568	−	2.44911i	−0.152475	−	0.122609i
$400$		0			0
$401$			7.64856i			0.381951i	0.981595	+	0.190975i	$0.0611651\pi$
$401$			7.64856i			0.381951i	−0.981595	+	0.190975i	$0.938835\pi$
$402$		0			0
$403$			19.5951i			0.976103i
$404$		0			0
$405$	−3.77297	−	8.17097i	−0.187480	−	0.406019i
$406$		0			0
$407$	−7.26262	−	2.25655i	−0.359995	−	0.111853i
$408$		0			0
$409$			7.09382i			0.350767i	0.984500	+	0.175383i	$0.0561165\pi$
$409$			7.09382i			0.350767i	−0.984500	+	0.175383i	$0.943884\pi$
$410$		0			0
$411$	−15.5917	+	19.3896i	−0.769081	+	0.956418i
$412$		0			0
$413$	−10.2435			−0.504052
$414$		0			0
$415$		−	10.3408i		−	0.507610i
$416$		0			0
$417$	2.07038	−	2.57470i	0.101387	−	0.126083i
$418$		0			0
$419$			28.0765i			1.37163i	0.727777	+	0.685814i	$0.240554\pi$
$419$			28.0765i			1.37163i	−0.727777	+	0.685814i	$0.759446\pi$
$420$		0			0
$421$	−11.8962			−0.579783			−0.289892	−	0.957059i	$-0.593619\pi$
$421$	−11.8962			−0.579783			−0.289892	+	0.957059i	$0.593619\pi$
$422$		0			0
$423$	−33.5459	+	7.37104i	−1.63106	+	0.358392i
$424$		0			0
$425$	0.0181501			0.000880412
$426$		0			0
$427$	5.23480			0.253330
$428$		0			0
$429$	21.6729	−	14.5833i	1.04638	−	0.704090i
$430$		0			0
$431$	21.0071			1.01188			0.505939	−	0.862569i	$-0.331146\pi$
$431$	21.0071			1.01188			0.505939	+	0.862569i	$0.331146\pi$
$432$		0			0
$433$	−3.47570			−0.167031			−0.0835156	−	0.996506i	$-0.526615\pi$
$433$	−3.47570			−0.167031			−0.0835156	+	0.996506i	$0.526615\pi$
$434$		0			0
$435$	−2.75203	+	3.42238i	−0.131950	+	0.164091i
$436$		0			0
$437$	−10.9826			−0.525369
$438$		0			0
$439$			7.93048i			0.378501i	0.981929	+	0.189251i	$0.0606059\pi$
$439$			7.93048i			0.378501i	−0.981929	+	0.189251i	$0.939394\pi$
$440$		0			0
$441$	−0.643830	−	2.93010i	−0.0306586	−	0.139529i
$442$		0			0
$443$			2.38773i			0.113444i	0.998390	+	0.0567222i	$0.0180649\pi$
$443$			2.38773i			0.113444i	−0.998390	+	0.0567222i	$0.981935\pi$
$444$		0			0
$445$	7.52948			0.356932
$446$		0			0
$447$	−14.9375	−	12.0116i	−0.706518	−	0.568130i
$448$		0			0
$449$		−	4.89370i		−	0.230948i	−0.993310	−	0.115474i	$-0.963161\pi$
$449$		−	4.89370i		−	0.230948i	0.993310	−	0.115474i	$-0.0368387\pi$
$450$		0			0
$451$	−21.6576	−	6.72917i	−1.01982	−	0.316864i
$452$		0			0
$453$	7.51218	−	9.34205i	0.352953	−	0.438928i
$454$		0			0
$455$			4.54736i			0.213183i
$456$		0			0
$457$		−	11.8578i		−	0.554686i	−0.960771	−	0.277343i	$-0.910546\pi$
$457$		−	11.8578i		−	0.554686i	0.960771	−	0.277343i	$-0.0894539\pi$
$458$		0			0
$459$	−0.0419502	+	0.0844673i	−0.00195807	+	0.00394260i
$460$		0			0
$461$	2.23039			0.103879			0.0519397	−	0.998650i	$-0.483460\pi$
$461$	2.23039			0.103879			0.0519397	+	0.998650i	$0.483460\pi$
$462$		0			0
$463$	−30.5082			−1.41784			−0.708918	−	0.705291i	$-0.750816\pi$
$463$	−30.5082			−1.41784			−0.708918	+	0.705291i	$0.750816\pi$
$464$		0			0
$465$	4.67711	−	5.81638i	0.216896	−	0.269728i
$466$		0			0
$467$		−	7.84472i		−	0.363010i	−0.983390	−	0.181505i	$-0.941903\pi$
$467$		−	7.84472i		−	0.363010i	0.983390	−	0.181505i	$-0.0580969\pi$
$468$		0			0
$469$			10.5120i			0.485397i
$470$		0			0
$471$	9.46616	+	7.61199i	0.436178	+	0.350742i
$472$		0			0
$473$	−2.54202	+	8.18141i	−0.116882	+	0.376182i
$474$		0			0
$475$		−	2.25642i		−	0.103532i
$476$		0			0
$477$	14.1026	−	3.09877i	0.645715	−	0.141883i
$478$		0			0
$479$	−27.8535			−1.27266			−0.636330	−	0.771417i	$-0.719548\pi$
$479$	−27.8535			−1.27266			−0.636330	+	0.771417i	$0.719548\pi$
$480$		0			0
$481$		−	10.4272i		−	0.475440i
$482$		0			0
$483$	−6.56974	−	5.28290i	−0.298934	−	0.240380i
$484$		0			0
$485$		−	15.9594i		−	0.724680i
$486$		0			0
$487$	31.3338			1.41987			0.709935	−	0.704267i	$-0.248724\pi$
$487$	31.3338			1.41987			0.709935	+	0.704267i	$0.248724\pi$
$488$		0			0
$489$	22.5889	+	18.1643i	1.02150	+	0.821418i
$490$		0			0
$491$	38.6980			1.74642			0.873208	−	0.487347i	$-0.162035\pi$
$491$	38.6980			1.74642			0.873208	+	0.487347i	$0.162035\pi$
$492$		0			0
$493$	0.0460198			0.00207263
$494$		0			0
$495$	−9.91399	−	0.844304i	−0.445601	−	0.0379486i
$496$		0			0
$497$	10.8778			0.487938
$498$		0			0
$499$	−3.37885			−0.151258			−0.0756291	−	0.997136i	$-0.524096\pi$
$499$	−3.37885			−0.151258			−0.0756291	+	0.997136i	$0.524096\pi$
$500$		0			0
$501$	0.727680	+	0.585147i	0.0325103	+	0.0261424i
$502$		0			0
$503$	10.9016			0.486077			0.243039	−	0.970017i	$-0.421856\pi$
$503$	10.9016			0.486077			0.243039	+	0.970017i	$0.421856\pi$
$504$		0			0
$505$			6.80227i			0.302697i
$506$		0			0
$507$	10.3642	+	8.33416i	0.460292	+	0.370133i
$508$		0			0
$509$		−	23.2152i		−	1.02899i	−0.857492	−	0.514497i	$-0.827979\pi$
$509$		−	23.2152i		−	1.02899i	0.857492	−	0.514497i	$-0.172021\pi$
$510$		0			0
$511$	−15.3062			−0.677106
$512$		0			0
$513$	10.5010	+	5.21525i	0.463629	+	0.230259i
$514$		0			0
$515$		−	15.1424i		−	0.667254i
$516$		0			0
$517$	−11.2666	+	36.2612i	−0.495504	+	1.59477i
$518$		0			0
$519$	16.6376	+	13.3787i	0.730308	+	0.587260i
$520$		0			0
$521$			32.9225i			1.44236i	0.692747	+	0.721181i	$0.256400\pi$
$521$			32.9225i			1.44236i	−0.692747	+	0.721181i	$0.743600\pi$
$522$		0			0
$523$			37.5590i			1.64234i	0.570685	+	0.821169i	$0.306678\pi$
$523$			37.5590i			1.64234i	−0.570685	+	0.821169i	$0.693322\pi$
$524$		0			0
$525$	1.08540	−	1.34978i	0.0473706	−	0.0589094i
$526$		0			0
$527$	−0.0782112			−0.00340694
$528$		0			0
$529$	−0.690210			−0.0300091
$530$		0			0
$531$	30.0146	−	6.59510i	1.30252	−	0.286203i
$532$		0			0
$533$		−	31.0946i		−	1.34686i
$534$		0			0
$535$			15.8775i			0.686444i
$536$		0			0
$537$	22.5893	−	28.0917i	0.974800	−	1.21225i
$538$		0			0
$539$	−3.16726	−	0.984090i	−0.136424	−	0.0423878i
$540$		0			0
$541$		−	32.0402i		−	1.37752i	−0.724991	−	0.688758i	$-0.758156\pi$
$541$		−	32.0402i		−	1.37752i	0.724991	−	0.688758i	$-0.241844\pi$
$542$		0			0
$543$	2.94072	+	2.36471i	0.126199	+	0.101480i
$544$		0			0
$545$	−3.41499			−0.146282
$546$		0			0
$547$			3.44720i			0.147392i	0.997281	+	0.0736959i	$0.0234794\pi$
$547$			3.44720i			0.147392i	−0.997281	+	0.0736959i	$0.976521\pi$
$548$		0			0
$549$	−15.3385	+	3.37032i	−0.654631	+	0.143842i
$550$		0			0
$551$		−	5.72118i		−	0.243730i
$552$		0			0
$553$	−14.7974			−0.629250
$554$		0			0
$555$	−2.48884	+	3.09509i	−0.105645	+	0.131379i
$556$		0			0
$557$	−9.02294			−0.382314			−0.191157	−	0.981559i	$-0.561224\pi$
$557$	−9.02294			−0.382314			−0.191157	+	0.981559i	$0.561224\pi$
$558$		0			0
$559$	−11.7464			−0.496818
$560$		0			0
$561$	0.0582074	+	0.0865044i	0.00245752	+	0.00365222i
$562$		0			0
$563$	−0.150797			−0.00635534			−0.00317767	−	0.999995i	$-0.501011\pi$
$563$	−0.150797			−0.00635534			−0.00317767	+	0.999995i	$0.501011\pi$
$564$		0			0
$565$	3.73836			0.157274
$566$		0			0
$567$	3.77297	+	8.17097i	0.158450	+	0.343148i
$568$		0			0
$569$	10.5499			0.442277			0.221138	−	0.975242i	$-0.429023\pi$
$569$	10.5499			0.442277			0.221138	+	0.975242i	$0.429023\pi$
$570$		0			0
$571$			9.11634i			0.381507i	0.981638	+	0.190754i	$0.0610931\pi$
$571$			9.11634i			0.381507i	−0.981638	+	0.190754i	$0.938907\pi$
$572$		0			0
$573$	−3.24169	+	4.03132i	−0.135424	+	0.168411i
$574$		0			0
$575$		−	4.86726i		−	0.202979i
$576$		0			0
$577$	23.3082			0.970335			0.485167	−	0.874421i	$-0.338759\pi$
$577$	23.3082			0.970335			0.485167	+	0.874421i	$0.338759\pi$
$578$		0			0
$579$	25.8421	−	32.1369i	1.07396	−	1.33556i
$580$		0			0
$581$			10.3408i			0.429008i
$582$		0			0
$583$	4.73645	−	15.2441i	0.196164	−	0.631347i
$584$		0			0
$585$	−2.92772	−	13.3242i	−0.121046	−	0.550888i
$586$		0			0
$587$		−	16.8741i		−	0.696470i	−0.937407	−	0.348235i	$-0.886781\pi$
$587$		−	16.8741i		−	0.696470i	0.937407	−	0.348235i	$-0.113219\pi$
$588$		0			0
$589$			9.72321i			0.400638i
$590$		0			0
$591$	−17.7628	−	14.2835i	−0.730664	−	0.587546i
$592$		0			0
$593$	−8.68031			−0.356458			−0.178229	−	0.983989i	$-0.557037\pi$
$593$	−8.68031			−0.356458			−0.178229	+	0.983989i	$0.557037\pi$
$594$		0			0
$595$	−0.0181501			−0.000744084
$596$		0			0
$597$	14.4134	+	11.5902i	0.589901	+	0.474355i
$598$		0			0
$599$		−	0.361096i		−	0.0147540i	−0.999973	−	0.00737699i	$-0.997652\pi$
$599$		−	0.361096i		−	0.0147540i	0.999973	−	0.00737699i	$-0.00234819\pi$
$600$		0			0
$601$			5.67728i			0.231581i	0.993274	+	0.115791i	$0.0369401\pi$
$601$			5.67728i			0.231581i	−0.993274	+	0.115791i	$0.963060\pi$
$602$		0			0
$603$	−6.76791	−	30.8011i	−0.275611	−	1.25432i
$604$		0			0
$605$	−6.23375	+	9.06313i	−0.253438	+	0.368469i
$606$		0			0
$607$			36.6042i			1.48572i	0.669449	+	0.742858i	$0.266530\pi$
$607$			36.6042i			1.48572i	−0.669449	+	0.742858i	$0.733470\pi$
$608$		0			0
$609$	2.75203	−	3.42238i	0.111518	−	0.138682i
$610$		0			0
$611$	−52.0615			−2.10618
$612$		0			0
$613$		−	7.50155i		−	0.302985i	−0.988458	−	0.151492i	$-0.951592\pi$
$613$		−	7.50155i		−	0.302985i	0.988458	−	0.151492i	$-0.0484079\pi$
$614$		0			0
$615$	−7.42189	+	9.22976i	−0.299280	+	0.372180i
$616$		0			0
$617$			7.20188i			0.289937i	0.989436	+	0.144968i	$0.0463081\pi$
$617$			7.20188i			0.289937i	−0.989436	+	0.144968i	$0.953692\pi$
$618$		0			0
$619$	−11.8733			−0.477230			−0.238615	−	0.971114i	$-0.576693\pi$
$619$	−11.8733			−0.477230			−0.238615	+	0.971114i	$0.576693\pi$
$620$		0			0
$621$	22.6513	+	11.2496i	0.908965	+	0.451433i
$622$		0			0
$623$	−7.52948			−0.301662
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	10.7542	−	7.23633i	0.429482	−	0.288991i
$628$		0			0
$629$	0.0416188			0.00165945
$630$		0			0
$631$	45.8580			1.82558			0.912789	−	0.408431i	$-0.133924\pi$
$631$	45.8580			1.82558			0.912789	+	0.408431i	$0.133924\pi$
$632$		0			0
$633$	−17.8313	+	22.1747i	−0.708730	+	0.881366i
$634$		0			0
$635$	−19.5901			−0.777409
$636$		0			0
$637$		−	4.54736i		−	0.180173i
$638$		0			0
$639$	−31.8732	+	7.00348i	−1.26088	+	0.277054i
$640$		0			0
$641$		−	3.82122i		−	0.150929i	−0.997148	−	0.0754646i	$-0.975956\pi$
$641$		−	3.82122i		−	0.150929i	0.997148	−	0.0754646i	$-0.0240440\pi$
$642$		0			0
$643$	37.8290			1.49183			0.745915	−	0.666041i	$-0.232012\pi$
$643$	37.8290			1.49183			0.745915	+	0.666041i	$0.232012\pi$
$644$		0			0
$645$	3.48665	+	2.80371i	0.137287	+	0.110396i
$646$		0			0
$647$		−	8.95936i		−	0.352229i	−0.984370	−	0.176114i	$-0.943647\pi$
$647$		−	8.95936i		−	0.352229i	0.984370	−	0.176114i	$-0.0563529\pi$
$648$		0			0
$649$	10.0806	−	32.4440i	0.395697	−	1.27354i
$650$		0			0
$651$	−4.67711	+	5.81638i	−0.183310	+	0.227962i
$652$		0			0
$653$		−	43.5248i		−	1.70326i	−0.524145	−	0.851629i	$-0.675615\pi$
$653$		−	43.5248i		−	1.70326i	0.524145	−	0.851629i	$-0.324385\pi$
$654$		0			0
$655$			0.235537i			0.00920319i
$656$		0			0
$657$	44.8487	−	9.85458i	1.74971	−	0.384464i
$658$		0			0
$659$	−12.7151			−0.495311			−0.247656	−	0.968848i	$-0.579660\pi$
$659$	−12.7151			−0.495311			−0.247656	+	0.968848i	$0.579660\pi$
$660$		0			0
$661$	32.6993			1.27186			0.635928	−	0.771748i	$-0.280618\pi$
$661$	32.6993			1.27186			0.635928	+	0.771748i	$0.280618\pi$
$662$		0			0
$663$	−0.0895834	+	0.111405i	−0.00347913	+	0.00432660i
$664$		0			0
$665$			2.25642i			0.0875004i
$666$		0			0
$667$		−	12.3410i		−	0.477844i
$668$		0			0
$669$	21.7571	+	17.4954i	0.841176	+	0.676412i
$670$		0			0
$671$	−5.15152	+	16.5800i	−0.198872	+	0.640064i
$672$		0			0
$673$		−	5.79194i		−	0.223263i	−0.993750	−	0.111632i	$-0.964392\pi$
$673$		−	5.79194i		−	0.223263i	0.993750	−	0.111632i	$-0.0356076\pi$
$674$		0			0
$675$	−2.31129	+	4.65381i	−0.0889615	+	0.179125i
$676$		0			0
$677$	−25.9539			−0.997489			−0.498744	−	0.866749i	$-0.666205\pi$
$677$	−25.9539			−0.997489			−0.498744	+	0.866749i	$0.666205\pi$
$678$		0			0
$679$			15.9594i			0.612467i
$680$		0			0
$681$	−10.4674	−	8.41713i	−0.401113	−	0.322545i
$682$		0			0
$683$		−	36.3462i		−	1.39075i	−0.718649	−	0.695374i	$-0.755239\pi$
$683$		−	36.3462i		−	1.39075i	0.718649	−	0.695374i	$-0.244761\pi$
$684$		0			0
$685$	14.3650			0.548857
$686$		0			0
$687$	22.5738	+	18.1522i	0.861243	+	0.692548i
$688$		0			0
$689$	21.8865			0.833810
$690$		0			0
$691$	−14.8843			−0.566224			−0.283112	−	0.959087i	$-0.591367\pi$
$691$	−14.8843			−0.566224			−0.283112	+	0.959087i	$0.591367\pi$
$692$		0			0
$693$	9.91399	+	0.844304i	0.376601	+	0.0320724i
$694$		0			0
$695$	−1.90749			−0.0723551
$696$		0			0
$697$	0.124110			0.00470100
$698$		0			0
$699$	33.8695	+	27.2353i	1.28106	+	1.03013i
$700$		0			0
$701$	16.4030			0.619533			0.309766	−	0.950813i	$-0.399749\pi$
$701$	16.4030			0.619533			0.309766	+	0.950813i	$0.399749\pi$
$702$		0			0
$703$		−	5.17404i		−	0.195143i
$704$		0			0
$705$	15.4533	+	12.4264i	0.582006	+	0.468006i
$706$		0			0
$707$		−	6.80227i		−	0.255826i
$708$		0			0
$709$	−26.1534			−0.982213			−0.491106	−	0.871100i	$-0.663407\pi$
$709$	−26.1534			−0.982213			−0.491106	+	0.871100i	$0.663407\pi$
$710$		0			0
$711$	43.3579	−	9.52701i	1.62605	−	0.357291i
$712$		0			0
$713$			20.9736i			0.785468i
$714$		0			0
$715$	−14.4027	−	4.47501i	−0.538630	−	0.167356i
$716$		0			0
$717$	−36.7223	−	29.5294i	−1.37142	−	1.10280i
$718$		0			0
$719$			8.87504i			0.330983i	0.986211	+	0.165492i	$0.0529211\pi$
$719$			8.87504i			0.330983i	−0.986211	+	0.165492i	$0.947079\pi$
$720$		0			0
$721$			15.1424i			0.563933i
$722$		0			0
$723$	7.73595	−	9.62032i	0.287703	−	0.357783i
$724$		0			0
$725$	2.53551			0.0941663
$726$		0			0
$727$	−41.8685			−1.55282			−0.776408	−	0.630230i	$-0.782960\pi$
$727$	−41.8685			−1.55282			−0.776408	+	0.630230i	$0.782960\pi$
$728$		0			0
$729$	−16.3159	−	21.5126i	−0.604292	−	0.796763i
$730$		0			0
$731$		−	0.0468840i		−	0.00173407i
$732$		0			0
$733$		−	19.4103i		−	0.716935i	−0.933542	−	0.358468i	$-0.883299\pi$
$733$		−	19.4103i		−	0.716935i	0.933542	−	0.358468i	$-0.116701\pi$
$734$		0			0
$735$	−1.08540	+	1.34978i	−0.0400354	+	0.0497875i
$736$		0			0
$737$	−33.2941	−	10.3447i	−1.22641	−	0.381052i
$738$		0			0
$739$			3.64951i			0.134249i	0.997745	+	0.0671247i	$0.0213825\pi$
$739$			3.64951i			0.134249i	−0.997745	+	0.0671247i	$0.978617\pi$
$740$		0			0
$741$	13.8498	+	11.1370i	0.508785	+	0.409128i
$742$		0			0
$743$	14.2136			0.521445			0.260722	−	0.965414i	$-0.416039\pi$
$743$	14.2136			0.521445			0.260722	+	0.965414i	$0.416039\pi$
$744$		0			0
$745$			11.0666i			0.405448i
$746$		0			0
$747$	−6.65771	−	30.2996i	−0.243593	−	1.10860i
$748$		0			0
$749$		−	15.8775i		−	0.580151i
$750$		0			0
$751$	23.1220			0.843735			0.421868	−	0.906657i	$-0.361375\pi$
$751$	23.1220			0.843735			0.421868	+	0.906657i	$0.361375\pi$
$752$		0			0
$753$	26.4931	−	32.9464i	0.965462	−	1.20063i
$754$		0			0
$755$	−6.92114			−0.251886
$756$		0			0
$757$	18.8638			0.685617			0.342809	−	0.939405i	$-0.388622\pi$
$757$	18.8638			0.685617			0.342809	+	0.939405i	$0.388622\pi$
$758$		0			0
$759$	23.1976	−	15.6093i	0.842019	−	0.566580i
$760$		0			0
$761$	−32.6408			−1.18323			−0.591614	−	0.806221i	$-0.701509\pi$
$761$	−32.6408			−1.18323			−0.591614	+	0.806221i	$0.701509\pi$
$762$		0			0
$763$	3.41499			0.123631
$764$		0			0
$765$	0.0531817	−	0.0116856i	0.00192279	−	0.000422494i
$766$		0			0
$767$	46.5811			1.68194
$768$		0			0
$769$			14.2675i			0.514501i	0.966345	+	0.257250i	$0.0828165\pi$
$769$			14.2675i			0.514501i	−0.966345	+	0.257250i	$0.917183\pi$
$770$		0			0
$771$	17.5228	−	21.7911i	0.631069	−	0.784789i
$772$		0			0
$773$		−	18.9615i		−	0.681999i	−0.940063	−	0.341000i	$-0.889235\pi$
$773$		−	18.9615i		−	0.681999i	0.940063	−	0.341000i	$-0.110765\pi$
$774$		0			0
$775$	−4.30912			−0.154788
$776$		0			0
$777$	2.48884	−	3.09509i	0.0892867	−	0.111036i
$778$		0			0
$779$		−	15.4293i		−	0.552813i
$780$		0			0
$781$	−10.7048	+	34.4530i	−0.383047	+	1.23283i
$782$		0			0
$783$	−5.86029	+	11.7998i	−0.209430	+	0.421689i
$784$		0			0
$785$		−	7.01310i		−	0.250308i
$786$		0			0
$787$		−	22.6509i		−	0.807418i	−0.914887	−	0.403709i	$-0.867721\pi$
$787$		−	22.6509i		−	0.807418i	0.914887	−	0.403709i	$-0.132279\pi$
$788$		0			0
$789$	−32.8024	−	26.3773i	−1.16780	−	0.939056i
$790$		0			0
$791$	−3.73836			−0.132921
$792$		0			0
$793$	−23.8045			−0.845323
$794$		0			0
$795$	−6.49654	−	5.22404i	−0.230408	−	0.185278i
$796$		0			0
$797$			2.25314i			0.0798101i	0.999203	+	0.0399051i	$0.0127056\pi$
$797$			2.25314i			0.0798101i	−0.999203	+	0.0399051i	$0.987294\pi$
$798$		0			0
$799$		−	0.207796i		−	0.00735131i
$800$		0			0
$801$	22.0621	−	4.84770i	0.779527	−	0.171285i
$802$		0			0
$803$	15.0627	−	48.4788i	0.531551	−	1.71078i
$804$		0			0
$805$			4.86726i			0.171548i
$806$		0			0
$807$	5.07467	−	6.31079i	0.178637	−	0.222150i
$808$		0			0
$809$	−41.5261			−1.45998			−0.729991	−	0.683457i	$-0.760476\pi$
$809$	−41.5261			−1.45998			−0.729991	+	0.683457i	$0.760476\pi$
$810$		0			0
$811$		−	16.1679i		−	0.567732i	−0.958864	−	0.283866i	$-0.908383\pi$
$811$		−	16.1679i		−	0.567732i	0.958864	−	0.283866i	$-0.0916172\pi$
$812$		0			0
$813$	−13.7926	+	17.1523i	−0.483727	+	0.601556i
$814$		0			0
$815$		−	16.7352i		−	0.586207i
$816$		0			0
$817$	−5.82861			−0.203917
$818$		0			0
$819$	2.92772	+	13.3242i	0.102303	+	0.465585i
$820$		0			0
$821$	−10.2145			−0.356488			−0.178244	−	0.983986i	$-0.557042\pi$
$821$	−10.2145			−0.356488			−0.178244	+	0.983986i	$0.557042\pi$
$822$		0			0
$823$	−19.5040			−0.679866			−0.339933	−	0.940450i	$-0.610404\pi$
$823$	−19.5040			−0.679866			−0.339933	+	0.940450i	$0.610404\pi$
$824$		0			0
$825$	3.20699	+	4.76605i	0.111653	+	0.165932i
$826$		0			0
$827$	34.6749			1.20576			0.602881	−	0.797831i	$-0.294019\pi$
$827$	34.6749			1.20576			0.602881	+	0.797831i	$0.294019\pi$
$828$		0			0
$829$	31.6550			1.09942			0.549712	−	0.835354i	$-0.314737\pi$
$829$	31.6550			1.09942			0.549712	+	0.835354i	$0.314737\pi$
$830$		0			0
$831$	−25.2256	+	31.3702i	−0.875067	+	1.08822i
$832$		0			0
$833$	0.0181501			0.000628865
$834$		0			0
$835$		−	0.539109i		−	0.0186566i
$836$		0			0
$837$	9.95963	−	20.0538i	0.344255	−	0.693162i
$838$		0			0
$839$			12.7706i			0.440889i	0.975400	+	0.220444i	$0.0707507\pi$
$839$			12.7706i			0.440889i	−0.975400	+	0.220444i	$0.929249\pi$
$840$		0			0
$841$	−22.5712			−0.778317
$842$		0			0
$843$	−14.4018	−	11.5808i	−0.496023	−	0.398865i
$844$		0			0
$845$		−	7.67845i		−	0.264147i
$846$		0			0
$847$	6.23375	−	9.06313i	0.214194	−	0.311413i
$848$		0			0
$849$	−10.4807	+	13.0336i	−0.359695	+	0.447312i
$850$		0			0
$851$		−	11.1608i		−	0.382586i
$852$		0			0
$853$		−	30.4308i		−	1.04193i	−0.853578	−	0.520965i	$-0.825572\pi$
$853$		−	30.4308i		−	1.04193i	0.853578	−	0.520965i	$-0.174428\pi$
$854$		0			0
$855$	−1.45275	−	6.61155i	−0.0496831	−	0.226110i
$856$		0			0
$857$	−12.5922			−0.430141			−0.215070	−	0.976599i	$-0.568998\pi$
$857$	−12.5922			−0.430141			−0.215070	+	0.976599i	$0.568998\pi$
$858$		0			0
$859$	−14.8924			−0.508122			−0.254061	−	0.967188i	$-0.581766\pi$
$859$	−14.8924			−0.508122			−0.254061	+	0.967188i	$0.581766\pi$
$860$		0			0
$861$	7.42189	−	9.22976i	0.252937	−	0.314549i
$862$		0			0
$863$			23.8837i			0.813012i	0.913648	+	0.406506i	$0.133253\pi$
$863$			23.8837i			0.813012i	−0.913648	+	0.406506i	$0.866747\pi$
$864$		0			0
$865$		−	12.3261i		−	0.419100i
$866$		0			0
$867$	22.9459	+	18.4514i	0.779283	+	0.626642i
$868$		0			0
$869$	14.5620	−	46.8673i	0.493982	−	1.58986i
$870$		0			0
$871$		−	47.8016i		−	1.61970i
$872$		0			0
$873$	−10.2752	−	46.7627i	−0.347761	−	1.58268i
$874$		0			0
$875$	−1.00000			−0.0338062
$876$		0			0
$877$		−	53.7432i		−	1.81478i	−0.420290	−	0.907390i	$-0.638072\pi$
$877$		−	53.7432i		−	1.81478i	0.420290	−	0.907390i	$-0.361928\pi$
$878$		0			0
$879$	−18.3383	−	14.7463i	−0.618537	−	0.497382i
$880$		0			0
$881$		−	6.18286i		−	0.208306i	−0.994561	−	0.104153i	$-0.966787\pi$
$881$		−	6.18286i		−	0.208306i	0.994561	−	0.104153i	$-0.0332132\pi$
$882$		0			0
$883$	−1.82121			−0.0612887			−0.0306444	−	0.999530i	$-0.509756\pi$
$883$	−1.82121			−0.0612887			−0.0306444	+	0.999530i	$0.509756\pi$
$884$		0			0
$885$	−13.8266	−	11.1183i	−0.464775	−	0.373738i
$886$		0			0
$887$	18.1087			0.608030			0.304015	−	0.952667i	$-0.401673\pi$
$887$	18.1087			0.608030			0.304015	+	0.952667i	$0.401673\pi$
$888$		0			0
$889$	19.5901			0.657030
$890$		0			0
$891$	−29.5926	+	3.90903i	−0.991388	+	0.130957i
$892$		0			0
$893$	−25.8332			−0.864475
$894$		0			0
$895$	−20.8120			−0.695670
$896$		0			0
$897$	29.8750	+	24.0232i	0.997496	+	0.802113i
$898$		0			0
$899$	−10.9258			−0.364396
$900$		0			0
$901$			0.0873571i			0.00291029i
$902$		0			0
$903$	−3.48665	−	2.80371i	−0.116028	−	0.0933015i
$904$		0			0
$905$		−	2.17866i		−	0.0724212i
$906$		0			0
$907$	−39.9823			−1.32759			−0.663796	−	0.747914i	$-0.731056\pi$
$907$	−39.9823			−1.32759			−0.663796	+	0.747914i	$0.731056\pi$
$908$		0			0
$909$	4.37950	+	19.9313i	0.145259	+	0.661080i
$910$		0			0
$911$			10.1883i			0.337555i	0.985654	+	0.168777i	$0.0539819\pi$
$911$			10.1883i			0.337555i	−0.985654	+	0.168777i	$0.946018\pi$
$912$		0			0
$913$	−32.7520	−	10.1763i	−1.08393	−	0.336786i
$914$		0			0
$915$	7.06585	+	5.68184i	0.233590	+	0.187836i
$916$		0			0
$917$		−	0.235537i		−	0.00777812i
$918$		0			0
$919$			47.0194i			1.55103i	0.631330	+	0.775514i	$0.282509\pi$
$919$			47.0194i			1.55103i	−0.631330	+	0.775514i	$0.717491\pi$
$920$		0			0
$921$	3.26297	−	4.05778i	0.107519	−	0.133709i
$922$		0			0
$923$	−49.4655			−1.62818
$924$		0			0
$925$	2.29303			0.0753942
$926$		0			0
$927$	−9.74913	−	44.3688i	−0.320204	−	1.45726i
$928$		0			0
$929$			4.52071i			0.148320i	0.997246	+	0.0741598i	$0.0236275\pi$
$929$			4.52071i			0.148320i	−0.997246	+	0.0741598i	$0.976373\pi$
$930$		0			0
$931$		−	2.25642i		−	0.0739513i
$932$		0			0
$933$	25.7791	−	32.0585i	0.843970	−	1.04955i
$934$		0			0
$935$	0.0178614	−	0.0574863i	0.000584130	−	0.00188000i
$936$		0			0
$937$		−	33.0637i		−	1.08014i	−0.841619	−	0.540072i	$-0.818397\pi$
$937$		−	33.0637i		−	1.08014i	0.841619	−	0.540072i	$-0.181603\pi$
$938$		0			0
$939$	18.5521	+	14.9182i	0.605423	+	0.486837i
$940$		0			0
$941$	−4.37347			−0.142571			−0.0712856	−	0.997456i	$-0.522710\pi$
$941$	−4.37347			−0.142571			−0.0712856	+	0.997456i	$0.522710\pi$
$942$		0			0
$943$		−	33.2821i		−	1.08381i
$944$		0			0
$945$	2.31129	−	4.65381i	0.0751862	−	0.151388i
$946$		0			0
$947$		−	11.7070i		−	0.380426i	−0.981743	−	0.190213i	$-0.939082\pi$
$947$		−	11.7070i		−	0.380426i	0.981743	−	0.190213i	$-0.0609179\pi$
$948$		0			0
$949$	69.6027			2.25940
$950$		0			0
$951$	38.0227	−	47.2845i	1.23297	−	1.53330i
$952$		0			0
$953$	49.0771			1.58976			0.794881	−	0.606765i	$-0.207533\pi$
$953$	49.0771			1.58976			0.794881	+	0.606765i	$0.207533\pi$
$954$		0			0
$955$	2.98665			0.0966456
$956$		0			0
$957$	8.13135	+	12.0843i	0.262849	+	0.390631i
$958$		0			0
$959$	−14.3650			−0.463869
$960$		0			0
$961$	−12.4314			−0.401014
$962$		0			0
$963$	10.2224	+	46.5226i	0.329412	+	1.49917i
$964$		0			0
$965$	−23.8089			−0.766437
$966$		0			0
$967$			16.2835i			0.523643i	0.965116	+	0.261822i	$0.0843232\pi$
$967$			16.2835i			0.523643i	−0.965116	+	0.261822i	$0.915677\pi$
$968$		0			0
$969$	−0.0444518	+	0.0552796i	−0.00142800	+	0.00177584i
$970$		0			0
$971$		−	8.86742i		−	0.284569i	−0.989826	−	0.142284i	$-0.954555\pi$
$971$		−	8.86742i		−	0.284569i	0.989826	−	0.142284i	$-0.0454448\pi$
$972$		0			0
$973$	1.90749			0.0611513
$974$		0			0
$975$	−4.93568	+	6.13794i	−0.158068	+	0.196572i
$976$		0			0
$977$			34.5427i			1.10512i	0.833473	+	0.552560i	$0.186349\pi$
$977$			34.5427i			1.10512i	−0.833473	+	0.552560i	$0.813651\pi$
$978$		0			0
$979$	7.40969	−	23.8479i	0.236815	−	0.762181i
$980$		0			0
$981$	−10.0062	+	2.19867i	−0.319475	+	0.0701981i
$982$		0			0
$983$			35.1824i			1.12215i	0.827767	+	0.561073i	$0.189611\pi$
$983$			35.1824i			1.12215i	−0.827767	+	0.561073i	$0.810389\pi$
$984$		0			0
$985$			13.1597i			0.419304i
$986$		0			0
$987$	−15.4533	−	12.4264i	−0.491884	−	0.395537i
$988$		0			0
$989$	−12.5727			−0.399789
$990$		0			0
$991$	57.1477			1.81536			0.907678	−	0.419667i	$-0.137853\pi$
$991$	57.1477			1.81536			0.907678	+	0.419667i	$0.137853\pi$
$992$		0			0
$993$	−33.4861	−	26.9271i	−1.06265	−	0.854504i
$994$		0			0
$995$		−	10.6783i		−	0.338525i
$996$		0			0
$997$		−	36.7433i		−	1.16367i	−0.813306	−	0.581836i	$-0.802335\pi$
$997$		−	36.7433i		−	1.16367i	0.813306	−	0.581836i	$-0.197665\pi$
$998$		0			0
$999$	−5.29985	+	10.6713i	−0.167680	+	0.337625i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4620.2.m.b.1121.13	yes	48
3.2	odd	2		4620.2.m.a.1121.14	yes	48
11.10	odd	2		4620.2.m.a.1121.13	✓	48
33.32	even	2	inner	4620.2.m.b.1121.14	yes	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4620.2.m.a.1121.13	✓	48	11.10	odd	2
4620.2.m.a.1121.14	yes	48	3.2	odd	2
4620.2.m.b.1121.13	yes	48	1.1	even	1	trivial
4620.2.m.b.1121.14	yes	48	33.32	even	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4620 = 2^{2} \cdot 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4620.m (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.34978 - 1.08540i) q^{3} +1.00000i q^{5} -1.00000i q^{7} +(0.643830 + 2.93010i) q^{9} +O(q^{10})\)	\(q+(-1.34978 - 1.08540i) q^{3} +1.00000i q^{5} -1.00000i q^{7} +(0.643830 + 2.93010i) q^{9} +(3.16726 + 0.984090i) q^{11} +4.54736i q^{13} +(1.08540 - 1.34978i) q^{15} -0.0181501 q^{17} +2.25642i q^{19} +(-1.08540 + 1.34978i) q^{21} +4.86726i q^{23} -1.00000 q^{25} +(2.31129 - 4.65381i) q^{27} -2.53551 q^{29} +4.30912 q^{31} +(-3.20699 - 4.76605i) q^{33} +1.00000 q^{35} -2.29303 q^{37} +(4.93568 - 6.13794i) q^{39} -6.83796 q^{41} +2.58312i q^{43} +(-2.93010 + 0.643830i) q^{45} +11.4487i q^{47} -1.00000 q^{49} +(0.0244988 + 0.0197001i) q^{51} -4.81302i q^{53} +(-0.984090 + 3.16726i) q^{55} +(2.44911 - 3.04568i) q^{57} -10.2435i q^{59} +5.23480i q^{61} +(2.93010 - 0.643830i) q^{63} -4.54736 q^{65} -10.5120 q^{67} +(5.28290 - 6.56974i) q^{69} +10.8778i q^{71} -15.3062i q^{73} +(1.34978 + 1.08540i) q^{75} +(0.984090 - 3.16726i) q^{77} -14.7974i q^{79} +(-8.17097 + 3.77297i) q^{81} -10.3408 q^{83} -0.0181501i q^{85} +(3.42238 + 2.75203i) q^{87} -7.52948i q^{89} +4.54736 q^{91} +(-5.81638 - 4.67711i) q^{93} -2.25642 q^{95} -15.9594 q^{97} +(-0.844304 + 9.91399i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 4 q^{3} + 6 q^{9}+O(q^{10}) \) 48 * q - 4 * q^3 + 6 * q^9	\( 48 q - 4 q^{3} + 6 q^{9} + 6 q^{11} + 2 q^{15} - 4 q^{17} - 2 q^{21} - 48 q^{25} - 16 q^{27} - 36 q^{29} - 16 q^{31} - 4 q^{33} + 48 q^{35} - 8 q^{37} + 18 q^{39} - 48 q^{49} + 30 q^{51} + 4 q^{55} - 16 q^{57} + 12 q^{65} + 24 q^{67} + 4 q^{69} + 4 q^{75} - 4 q^{77} - 22 q^{81} - 20 q^{83} - 44 q^{87} - 12 q^{91} - 28 q^{93} - 8 q^{95} - 56 q^{97} - 12 q^{99}+O(q^{100}) \) 48 * q - 4 * q^3 + 6 * q^9 + 6 * q^11 + 2 * q^15 - 4 * q^17 - 2 * q^21 - 48 * q^25 - 16 * q^27 - 36 * q^29 - 16 * q^31 - 4 * q^33 + 48 * q^35 - 8 * q^37 + 18 * q^39 - 48 * q^49 + 30 * q^51 + 4 * q^55 - 16 * q^57 + 12 * q^65 + 24 * q^67 + 4 * q^69 + 4 * q^75 - 4 * q^77 - 22 * q^81 - 20 * q^83 - 44 * q^87 - 12 * q^91 - 28 * q^93 - 8 * q^95 - 56 * q^97 - 12 * q^99

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