LMFDB - Embedded newform 4620.2.m.a.1121.18

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			1121.18
Character	$\chi$	$=$	4620.1121
Dual form			4620.2.m.a.1121.17

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.839247 + 1.51514i) q^{3} +1.00000i q^{5} +1.00000i q^{7} +(-1.59133 - 2.54316i) q^{9} +O(q^{10})$	\(q+(-0.839247 + 1.51514i) q^{3} +1.00000i q^{5} +1.00000i q^{7} +(-1.59133 - 2.54316i) q^{9} +(-1.39113 - 3.01078i) q^{11} +1.12252i q^{13} +(-1.51514 - 0.839247i) q^{15} +6.62413 q^{17} -6.33775i q^{19} +(-1.51514 - 0.839247i) q^{21} +6.80175i q^{23} -1.00000 q^{25} +(5.18878 - 0.276750i) q^{27} -0.173034 q^{29} -7.28561 q^{31} +(5.72926 + 0.419028i) q^{33} -1.00000 q^{35} +5.43207 q^{37} +(-1.70079 - 0.942075i) q^{39} -6.61999 q^{41} +8.97614i q^{43} +(2.54316 - 1.59133i) q^{45} +3.13643i q^{47} -1.00000 q^{49} +(-5.55928 + 10.0365i) q^{51} +7.05474i q^{53} +(3.01078 - 1.39113i) q^{55} +(9.60260 + 5.31894i) q^{57} -0.806240i q^{59} +11.3660i q^{61} +(2.54316 - 1.59133i) q^{63} -1.12252 q^{65} +0.188894 q^{67} +(-10.3056 - 5.70835i) q^{69} -11.7395i q^{71} +7.05678i q^{73} +(0.839247 - 1.51514i) q^{75} +(3.01078 - 1.39113i) q^{77} +3.47818i q^{79} +(-3.93535 + 8.09401i) q^{81} -6.82530 q^{83} +6.62413i q^{85} +(0.145218 - 0.262172i) q^{87} -0.0346322i q^{89} -1.12252 q^{91} +(6.11443 - 11.0388i) q^{93} +6.33775 q^{95} -16.4900 q^{97} +(-5.44315 + 8.32899i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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} + 16 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} + 26 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 + 16 * 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 + 26 * 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
$3$	−0.839247	+	1.51514i	−0.484540	+	0.874769i
$3$	−0.839247	+	1.51514i	−0.484540	+	0.874769i
$4$		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
$9$	−1.59133	−	2.54316i	−0.530443	−	0.847721i
$10$		0			0
$11$	−1.39113	−	3.01078i	−0.419440	−	0.907783i
$11$	−1.39113	−	3.01078i	−0.419440	−	0.907783i
$12$		0			0
$13$			1.12252i			0.311332i	0.987810	+	0.155666i	$0.0497524\pi$
$13$			1.12252i			0.311332i	−0.987810	+	0.155666i	$0.950248\pi$
$14$		0			0
$15$	−1.51514	−	0.839247i	−0.391209	−	0.216693i
$16$		0			0
$17$	6.62413			1.60659			0.803293	−	0.595584i	$-0.203079\pi$
$17$	6.62413			1.60659			0.803293	+	0.595584i	$0.203079\pi$
$18$		0			0
$19$		−	6.33775i		−	1.45398i	−0.686649	−	0.726989i	$-0.740919\pi$
$19$		−	6.33775i		−	1.45398i	0.686649	−	0.726989i	$-0.259081\pi$
$20$		0			0
$21$	−1.51514	−	0.839247i	−0.330632	−	0.183139i
$22$		0			0
$23$			6.80175i			1.41826i	0.705076	+	0.709132i	$0.250913\pi$
$23$			6.80175i			1.41826i	−0.705076	+	0.709132i	$0.749087\pi$
$24$		0			0
$25$	−1.00000			−0.200000
$26$		0			0
$27$	5.18878	−	0.276750i	0.998581	−	0.0532605i
$28$		0			0
$29$	−0.173034			−0.0321316			−0.0160658	−	0.999871i	$-0.505114\pi$
$29$	−0.173034			−0.0321316			−0.0160658	+	0.999871i	$0.505114\pi$
$30$		0			0
$31$	−7.28561			−1.30853			−0.654267	−	0.756263i	$-0.727023\pi$
$31$	−7.28561			−1.30853			−0.654267	+	0.756263i	$0.727023\pi$
$32$		0			0
$33$	5.72926	+	0.419028i	0.997336	+	0.0729434i
$34$		0			0
$35$	−1.00000			−0.169031
$36$		0			0
$37$	5.43207			0.893027			0.446514	−	0.894777i	$-0.352665\pi$
$37$	5.43207			0.893027			0.446514	+	0.894777i	$0.352665\pi$
$38$		0			0
$39$	−1.70079	−	0.942075i	−0.272344	−	0.150853i
$40$		0			0
$41$	−6.61999			−1.03387			−0.516934	−	0.856025i	$-0.672927\pi$
$41$	−6.61999			−1.03387			−0.516934	+	0.856025i	$0.672927\pi$
$42$		0			0
$43$			8.97614i			1.36885i	0.729084	+	0.684424i	$0.239946\pi$
$43$			8.97614i			1.36885i	−0.729084	+	0.684424i	$0.760054\pi$
$44$		0			0
$45$	2.54316	−	1.59133i	0.379112	−	0.237221i
$46$		0			0
$47$			3.13643i			0.457495i	0.973486	+	0.228747i	$0.0734630\pi$
$47$			3.13643i			0.457495i	−0.973486	+	0.228747i	$0.926537\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	−5.55928	+	10.0365i	−0.778455	+	1.40539i
$52$		0			0
$53$			7.05474i			0.969043i	0.874779	+	0.484521i	$0.161006\pi$
$53$			7.05474i			0.969043i	−0.874779	+	0.484521i	$0.838994\pi$
$54$		0			0
$55$	3.01078	−	1.39113i	0.405973	−	0.187579i
$56$		0			0
$57$	9.60260	+	5.31894i	1.27190	+	0.704510i
$58$		0			0
$59$		−	0.806240i		−	0.104964i	−0.998622	−	0.0524818i	$-0.983287\pi$
$59$		−	0.806240i		−	0.104964i	0.998622	−	0.0524818i	$-0.0167131\pi$
$60$		0			0
$61$			11.3660i			1.45527i	0.685967	+	0.727633i	$0.259380\pi$
$61$			11.3660i			1.45527i	−0.685967	+	0.727633i	$0.740620\pi$
$62$		0			0
$63$	2.54316	−	1.59133i	0.320408	−	0.200488i
$64$		0			0
$65$	−1.12252			−0.139232
$66$		0			0
$67$	0.188894			0.0230770			0.0115385	−	0.999933i	$-0.496327\pi$
$67$	0.188894			0.0230770			0.0115385	+	0.999933i	$0.496327\pi$
$68$		0			0
$69$	−10.3056	−	5.70835i	−1.24065	−	0.687205i
$70$		0			0
$71$		−	11.7395i		−	1.39322i	−0.717449	−	0.696611i	$-0.754690\pi$
$71$		−	11.7395i		−	1.39322i	0.717449	−	0.696611i	$-0.245310\pi$
$72$		0			0
$73$			7.05678i			0.825933i	0.910746	+	0.412967i	$0.135507\pi$
$73$			7.05678i			0.825933i	−0.910746	+	0.412967i	$0.864493\pi$
$74$		0			0
$75$	0.839247	−	1.51514i	0.0969079	−	0.174954i
$76$		0			0
$77$	3.01078	−	1.39113i	0.343110	−	0.158534i
$78$		0			0
$79$			3.47818i			0.391326i	0.980671	+	0.195663i	$0.0626858\pi$
$79$			3.47818i			0.391326i	−0.980671	+	0.195663i	$0.937314\pi$
$80$		0			0
$81$	−3.93535	+	8.09401i	−0.437261	+	0.899335i
$82$		0			0
$83$	−6.82530			−0.749174			−0.374587	−	0.927192i	$-0.622215\pi$
$83$	−6.82530			−0.749174			−0.374587	+	0.927192i	$0.622215\pi$
$84$		0			0
$85$			6.62413i			0.718487i
$86$		0			0
$87$	0.145218	−	0.262172i	0.0155690	−	0.0281077i
$88$		0			0
$89$		−	0.0346322i		−	0.00367100i	−0.999998	−	0.00183550i	$-0.999416\pi$
$89$		−	0.0346322i		−	0.00367100i	0.999998	−	0.00183550i	$-0.000584259\pi$
$90$		0			0
$91$	−1.12252			−0.117672
$92$		0			0
$93$	6.11443	−	11.0388i	0.634037	−	1.14467i
$94$		0			0
$95$	6.33775			0.650239
$96$		0			0
$97$	−16.4900			−1.67431			−0.837154	−	0.546967i	$-0.815782\pi$
$97$	−16.4900			−1.67431			−0.837154	+	0.546967i	$0.815782\pi$
$98$		0			0
$99$	−5.44315	+	8.32899i	−0.547058	+	0.837095i
$100$		0			0
$101$	5.89346			0.586422			0.293211	−	0.956048i	$-0.405276\pi$
$101$	5.89346			0.586422			0.293211	+	0.956048i	$0.405276\pi$
$102$		0			0
$103$	−7.66948			−0.755697			−0.377848	−	0.925868i	$-0.623336\pi$
$103$	−7.66948			−0.755697			−0.377848	+	0.925868i	$0.623336\pi$
$104$		0			0
$105$	0.839247	−	1.51514i	0.0819022	−	0.147863i
$106$		0			0
$107$	−15.3431			−1.48328			−0.741639	−	0.670800i	$-0.765951\pi$
$107$	−15.3431			−1.48328			−0.741639	+	0.670800i	$0.765951\pi$
$108$		0			0
$109$			11.7701i			1.12737i	0.825988	+	0.563687i	$0.190618\pi$
$109$			11.7701i			1.12737i	−0.825988	+	0.563687i	$0.809382\pi$
$110$		0			0
$111$	−4.55885	+	8.23038i	−0.432707	+	0.781193i
$112$		0			0
$113$		−	14.2814i		−	1.34348i	−0.740788	−	0.671739i	$-0.765548\pi$
$113$		−	14.2814i		−	1.34348i	0.740788	−	0.671739i	$-0.234452\pi$
$114$		0			0
$115$	−6.80175			−0.634267
$116$		0			0
$117$	2.85476	−	1.78630i	0.263923	−	0.165144i
$118$		0			0
$119$			6.62413i			0.607233i
$120$		0			0
$121$	−7.12954	+	8.37674i	−0.648140	+	0.761521i
$122$		0			0
$123$	5.55581	−	10.0302i	0.500950	−	0.904396i
$124$		0			0
$125$		−	1.00000i		−	0.0894427i
$126$		0			0
$127$			13.6994i			1.21563i	0.794080	+	0.607813i	$0.207953\pi$
$127$			13.6994i			1.21563i	−0.794080	+	0.607813i	$0.792047\pi$
$128$		0			0
$129$	−13.6001	−	7.53320i	−1.19743	−	0.663261i
$130$		0			0
$131$	1.08443			0.0947468			0.0473734	−	0.998877i	$-0.484915\pi$
$131$	1.08443			0.0947468			0.0473734	+	0.998877i	$0.484915\pi$
$132$		0			0
$133$	6.33775			0.549552
$134$		0			0
$135$	0.276750	+	5.18878i	0.0238188	+	0.446579i
$136$		0			0
$137$			1.84087i			0.157276i	0.996903	+	0.0786382i	$0.0250572\pi$
$137$			1.84087i			0.157276i	−0.996903	+	0.0786382i	$0.974943\pi$
$138$		0			0
$139$			2.89578i			0.245616i	0.992430	+	0.122808i	$0.0391900\pi$
$139$			2.89578i			0.245616i	−0.992430	+	0.122808i	$0.960810\pi$
$140$		0			0
$141$	−4.75214	−	2.63224i	−0.400202	−	0.221674i
$142$		0			0
$143$	3.37967	−	1.56157i	0.282622	−	0.130585i
$144$		0			0
$145$		−	0.173034i		−	0.0143697i
$146$		0			0
$147$	0.839247	−	1.51514i	0.0692200	−	0.124967i
$148$		0			0
$149$	2.00288			0.164082			0.0820411	−	0.996629i	$-0.473856\pi$
$149$	2.00288			0.164082			0.0820411	+	0.996629i	$0.473856\pi$
$150$		0			0
$151$			14.1611i			1.15241i	0.817305	+	0.576206i	$0.195467\pi$
$151$			14.1611i			1.15241i	−0.817305	+	0.576206i	$0.804533\pi$
$152$		0			0
$153$	−10.5412	−	16.8462i	−0.852202	−	1.36194i
$154$		0			0
$155$		−	7.28561i		−	0.585195i
$156$		0			0
$157$	−1.05880			−0.0845012			−0.0422506	−	0.999107i	$-0.513453\pi$
$157$	−1.05880			−0.0845012			−0.0422506	+	0.999107i	$0.513453\pi$
$158$		0			0
$159$	−10.6889	−	5.92067i	−0.847689	−	0.469540i
$160$		0			0
$161$	−6.80175			−0.536053
$162$		0			0
$163$	8.88487			0.695916			0.347958	−	0.937510i	$-0.386875\pi$
$163$	8.88487			0.695916			0.347958	+	0.937510i	$0.386875\pi$
$164$		0			0
$165$	−0.419028	+	5.72926i	−0.0326213	+	0.446022i
$166$		0			0
$167$	4.49140			0.347555			0.173778	−	0.984785i	$-0.444403\pi$
$167$	4.49140			0.347555			0.173778	+	0.984785i	$0.444403\pi$
$168$		0			0
$169$	11.7399			0.903072
$170$		0			0
$171$	−16.1179	+	10.0854i	−1.23257	+	0.771252i
$172$		0			0
$173$	−21.4393			−1.63000			−0.814998	−	0.579463i	$-0.803262\pi$
$173$	−21.4393			−1.63000			−0.814998	+	0.579463i	$0.803262\pi$
$174$		0			0
$175$		−	1.00000i		−	0.0755929i
$176$		0			0
$177$	1.22157	+	0.676635i	0.0918189	+	0.0508590i
$178$		0			0
$179$		−	26.2480i		−	1.96186i	−0.194350	−	0.980932i	$-0.562260\pi$
$179$		−	26.2480i		−	1.96186i	0.194350	−	0.980932i	$-0.437740\pi$
$180$		0			0
$181$	7.03646			0.523016			0.261508	−	0.965201i	$-0.415780\pi$
$181$	7.03646			0.523016			0.261508	+	0.965201i	$0.415780\pi$
$182$		0			0
$183$	−17.2211	−	9.53888i	−1.27302	−	0.705134i
$184$		0			0
$185$			5.43207i			0.399374i
$186$		0			0
$187$	−9.21499	−	19.9438i	−0.673867	−	1.45843i
$188$		0			0
$189$	0.276750	+	5.18878i	0.0201306	+	0.377428i
$190$		0			0
$191$		−	13.1723i		−	0.953115i	−0.879143	−	0.476557i	$-0.841884\pi$
$191$		−	13.1723i		−	0.953115i	0.879143	−	0.476557i	$-0.158116\pi$
$192$		0			0
$193$		−	14.1308i		−	1.01716i	−0.861015	−	0.508579i	$-0.830171\pi$
$193$		−	14.1308i		−	1.01716i	0.861015	−	0.508579i	$-0.169829\pi$
$194$		0			0
$195$	0.942075	−	1.70079i	0.0674634	−	0.121796i
$196$		0			0
$197$	15.8100			1.12642			0.563210	−	0.826314i	$-0.309566\pi$
$197$	15.8100			1.12642			0.563210	+	0.826314i	$0.309566\pi$
$198$		0			0
$199$	21.5913			1.53056			0.765282	−	0.643695i	$-0.222600\pi$
$199$	21.5913			1.53056			0.765282	+	0.643695i	$0.222600\pi$
$200$		0			0
$201$	−0.158529	+	0.286201i	−0.0111817	+	0.0201871i
$202$		0			0
$203$		−	0.173034i		−	0.0121446i
$204$		0			0
$205$		−	6.61999i		−	0.462360i
$206$		0			0
$207$	17.2980	−	10.8238i	1.20229	−	0.752307i
$208$		0			0
$209$	−19.0815	+	8.81660i	−1.31990	+	0.609857i
$210$		0			0
$211$		−	3.64071i		−	0.250637i	−0.992117	−	0.125318i	$-0.960005\pi$
$211$		−	3.64071i		−	0.250637i	0.992117	−	0.125318i	$-0.0399953\pi$
$212$		0			0
$213$	17.7870	+	9.85234i	1.21875	+	0.675071i
$214$		0			0
$215$	−8.97614			−0.612168
$216$		0			0
$217$		−	7.28561i		−	0.494580i
$218$		0			0
$219$	−10.6920	−	5.92238i	−0.722501	−	0.400197i
$220$		0			0
$221$			7.43574i			0.500182i
$222$		0			0
$223$	−10.6950			−0.716188			−0.358094	−	0.933685i	$-0.616573\pi$
$223$	−10.6950			−0.716188			−0.358094	+	0.933685i	$0.616573\pi$
$224$		0			0
$225$	1.59133	+	2.54316i	0.106089	+	0.169544i
$226$		0			0
$227$	−17.0002			−1.12835			−0.564173	−	0.825657i	$-0.690805\pi$
$227$	−17.0002			−1.12835			−0.564173	+	0.825657i	$0.690805\pi$
$228$		0			0
$229$	−21.4972			−1.42057			−0.710287	−	0.703912i	$-0.751435\pi$
$229$	−21.4972			−1.42057			−0.710287	+	0.703912i	$0.751435\pi$
$230$		0			0
$231$	−0.419028	+	5.72926i	−0.0275700	+	0.376958i
$232$		0			0
$233$	14.2085			0.930827			0.465414	−	0.885093i	$-0.345906\pi$
$233$	14.2085			0.930827			0.465414	+	0.885093i	$0.345906\pi$
$234$		0			0
$235$	−3.13643			−0.204598
$236$		0			0
$237$	−5.26994	−	2.91905i	−0.342320	−	0.189613i
$238$		0			0
$239$	−5.63620			−0.364576			−0.182288	−	0.983245i	$-0.558350\pi$
$239$	−5.63620			−0.364576			−0.182288	+	0.983245i	$0.558350\pi$
$240$		0			0
$241$			24.3068i			1.56574i	0.622187	+	0.782868i	$0.286244\pi$
$241$			24.3068i			1.56574i	−0.622187	+	0.782868i	$0.713756\pi$
$242$		0			0
$243$	−8.96087	−	12.7555i	−0.574840	−	0.818266i
$244$		0			0
$245$		−	1.00000i		−	0.0638877i
$246$		0			0
$247$	7.11427			0.452670
$248$		0			0
$249$	5.72811	−	10.3413i	0.363004	−	0.655354i
$250$		0			0
$251$		−	29.5156i		−	1.86301i	−0.363729	−	0.931505i	$-0.618497\pi$
$251$		−	29.5156i		−	1.86301i	0.363729	−	0.931505i	$-0.381503\pi$
$252$		0			0
$253$	20.4785	−	9.46209i	1.28748	−	0.594877i
$254$		0			0
$255$	−10.0365	−	5.55928i	−0.628511	−	0.348136i
$256$		0			0
$257$			17.1851i			1.07197i	0.844226	+	0.535987i	$0.180060\pi$
$257$			17.1851i			1.07197i	−0.844226	+	0.535987i	$0.819940\pi$
$258$		0			0
$259$			5.43207i			0.337533i
$260$		0			0
$261$	0.275354	+	0.440054i	0.0170440	+	0.0272386i
$262$		0			0
$263$	−30.6097			−1.88747			−0.943736	−	0.330699i	$-0.892715\pi$
$263$	−30.6097			−1.88747			−0.943736	+	0.330699i	$0.892715\pi$
$264$		0			0
$265$	−7.05474			−0.433369
$266$		0			0
$267$	0.0524728	+	0.0290650i	0.00321128	+	0.00177875i
$268$		0			0
$269$			6.30964i			0.384705i	0.981326	+	0.192353i	$0.0616118\pi$
$269$			6.30964i			0.384705i	−0.981326	+	0.192353i	$0.938388\pi$
$270$		0			0
$271$			4.76436i			0.289414i	0.989475	+	0.144707i	$0.0462240\pi$
$271$			4.76436i			0.289414i	−0.989475	+	0.144707i	$0.953776\pi$
$272$		0			0
$273$	0.942075	−	1.70079i	0.0570170	−	0.102936i
$274$		0			0
$275$	1.39113	+	3.01078i	0.0838881	+	0.181557i
$276$		0			0
$277$			21.7028i			1.30399i	0.758222	+	0.651997i	$0.226068\pi$
$277$			21.7028i			1.30399i	−0.758222	+	0.651997i	$0.773932\pi$
$278$		0			0
$279$	11.5938	+	18.5285i	0.694103	+	1.10927i
$280$		0			0
$281$	0.740288			0.0441619			0.0220809	−	0.999756i	$-0.492971\pi$
$281$	0.740288			0.0441619			0.0220809	+	0.999756i	$0.492971\pi$
$282$		0			0
$283$			2.74893i			0.163407i	0.996657	+	0.0817035i	$0.0260361\pi$
$283$			2.74893i			0.163407i	−0.996657	+	0.0817035i	$0.973964\pi$
$284$		0			0
$285$	−5.31894	+	9.60260i	−0.315067	+	0.568809i
$286$		0			0
$287$		−	6.61999i		−	0.390765i
$288$		0			0
$289$	26.8790			1.58112
$290$		0			0
$291$	13.8392	−	24.9848i	0.811269	−	1.46463i
$292$		0			0
$293$	16.7445			0.978225			0.489113	−	0.872221i	$-0.337321\pi$
$293$	16.7445			0.978225			0.489113	+	0.872221i	$0.337321\pi$
$294$		0			0
$295$	0.806240			0.0469411
$296$		0			0
$297$	−8.05147	−	15.2372i	−0.467194	−	0.884155i
$298$		0			0
$299$	−7.63513			−0.441551
$300$		0			0
$301$	−8.97614			−0.517376
$302$		0			0
$303$	−4.94607	+	8.92945i	−0.284145	+	0.512984i
$304$		0			0
$305$	−11.3660			−0.650815
$306$		0			0
$307$			8.42361i			0.480761i	0.970679	+	0.240380i	$0.0772722\pi$
$307$			8.42361i			0.480761i	−0.970679	+	0.240380i	$0.922728\pi$
$308$		0			0
$309$	6.43659	−	11.6204i	0.366165	−	0.661060i
$310$		0			0
$311$			10.2508i			0.581272i	0.956834	+	0.290636i	$0.0938669\pi$
$311$			10.2508i			0.581272i	−0.956834	+	0.290636i	$0.906133\pi$
$312$		0			0
$313$	−15.1983			−0.859060			−0.429530	−	0.903053i	$-0.641321\pi$
$313$	−15.1983			−0.859060			−0.429530	+	0.903053i	$0.641321\pi$
$314$		0			0
$315$	1.59133	+	2.54316i	0.0896612	+	0.143291i
$316$		0			0
$317$			26.7137i			1.50039i	0.661215	+	0.750197i	$0.270041\pi$
$317$			26.7137i			1.50039i	−0.661215	+	0.750197i	$0.729959\pi$
$318$		0			0
$319$	0.240712	+	0.520967i	0.0134773	+	0.0291685i
$320$		0			0
$321$	12.8767	−	23.2471i	0.718707	−	1.29753i
$322$		0			0
$323$		−	41.9820i		−	2.33594i
$324$		0			0
$325$		−	1.12252i		−	0.0622664i
$326$		0			0
$327$	−17.8335	−	9.87805i	−0.986192	−	0.546258i
$328$		0			0
$329$	−3.13643			−0.172917
$330$		0			0
$331$	−21.8011			−1.19830			−0.599148	−	0.800638i	$-0.704494\pi$
$331$	−21.8011			−1.19830			−0.599148	+	0.800638i	$0.704494\pi$
$332$		0			0
$333$	−8.64421	−	13.8146i	−0.473700	−	0.757038i
$334$		0			0
$335$			0.188894i			0.0103204i
$336$		0			0
$337$		−	34.2955i		−	1.86819i	−0.357020	−	0.934097i	$-0.616207\pi$
$337$		−	34.2955i		−	1.86819i	0.357020	−	0.934097i	$-0.383793\pi$
$338$		0			0
$339$	21.6383	+	11.9856i	1.17523	+	0.650969i
$340$		0			0
$341$	10.1352	+	21.9353i	0.548852	+	1.18787i
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$	5.70835	−	10.3056i	0.307327	−	0.554837i
$346$		0			0
$347$	−11.2325			−0.602994			−0.301497	−	0.953467i	$-0.597486\pi$
$347$	−11.2325			−0.602994			−0.301497	+	0.953467i	$0.597486\pi$
$348$		0			0
$349$			0.777110i			0.0415978i	0.999784	+	0.0207989i	$0.00662097\pi$
$349$			0.777110i			0.0415978i	−0.999784	+	0.0207989i	$0.993379\pi$
$350$		0			0
$351$	0.310658	+	5.82452i	0.0165817	+	0.310890i
$352$		0			0
$353$		−	7.54333i		−	0.401491i	−0.979643	−	0.200746i	$-0.935664\pi$
$353$		−	7.54333i		−	0.401491i	0.979643	−	0.200746i	$-0.0643364\pi$
$354$		0			0
$355$	11.7395			0.623068
$356$		0			0
$357$	−10.0365	−	5.55928i	−0.531188	−	0.294228i
$358$		0			0
$359$	4.58129			0.241791			0.120896	−	0.992665i	$-0.461423\pi$
$359$	4.58129			0.241791			0.120896	+	0.992665i	$0.461423\pi$
$360$		0			0
$361$	−21.1670			−1.11405
$362$		0			0
$363$	−6.70852	−	17.8324i	−0.352106	−	0.935960i
$364$		0			0
$365$	−7.05678			−0.369369
$366$		0			0
$367$	−19.0172			−0.992692			−0.496346	−	0.868125i	$-0.665325\pi$
$367$	−19.0172			−0.992692			−0.496346	+	0.868125i	$0.665325\pi$
$368$		0			0
$369$	10.5346	+	16.8357i	0.548408	+	0.876432i
$370$		0			0
$371$	−7.05474			−0.366264
$372$		0			0
$373$			19.4892i			1.00911i	0.863379	+	0.504556i	$0.168344\pi$
$373$			19.4892i			1.00911i	−0.863379	+	0.504556i	$0.831656\pi$
$374$		0			0
$375$	1.51514	+	0.839247i	0.0782417	+	0.0433385i
$376$		0			0
$377$		−	0.194235i		−	0.0100036i
$378$		0			0
$379$	−28.5436			−1.46619			−0.733093	−	0.680129i	$-0.761924\pi$
$379$	−28.5436			−1.46619			−0.733093	+	0.680129i	$0.761924\pi$
$380$		0			0
$381$	−20.7566	−	11.4972i	−1.06339	−	0.589019i
$382$		0			0
$383$			9.74649i			0.498022i	0.968501	+	0.249011i	$0.0801056\pi$
$383$			9.74649i			0.498022i	−0.968501	+	0.249011i	$0.919894\pi$
$384$		0			0
$385$	1.39113	+	3.01078i	0.0708983	+	0.153443i
$386$		0			0
$387$	22.8278	−	14.2840i	1.16040	−	0.726095i
$388$		0			0
$389$		−	17.5973i		−	0.892220i	−0.894978	−	0.446110i	$-0.852809\pi$
$389$		−	17.5973i		−	0.892220i	0.894978	−	0.446110i	$-0.147191\pi$
$390$		0			0
$391$			45.0557i			2.27856i
$392$		0			0
$393$	−0.910102	+	1.64306i	−0.0459086	+	0.0828816i
$394$		0			0
$395$	−3.47818			−0.175006
$396$		0			0
$397$	−19.5816			−0.982770			−0.491385	−	0.870942i	$-0.663509\pi$
$397$	−19.5816			−0.982770			−0.491385	+	0.870942i	$0.663509\pi$
$398$		0			0
$399$	−5.31894	+	9.60260i	−0.266280	+	0.480731i
$400$		0			0
$401$			12.7639i			0.637397i	0.947856	+	0.318699i	$0.103246\pi$
$401$			12.7639i			0.637397i	−0.947856	+	0.318699i	$0.896754\pi$
$402$		0			0
$403$		−	8.17827i		−	0.407389i
$404$		0			0
$405$	−8.09401	−	3.93535i	−0.402195	−	0.195549i
$406$		0			0
$407$	−7.55670	−	16.3547i	−0.374572	−	0.810675i
$408$		0			0
$409$			0.916627i			0.0453243i	0.999743	+	0.0226621i	$0.00721420\pi$
$409$			0.916627i			0.0453243i	−0.999743	+	0.0226621i	$0.992786\pi$
$410$		0			0
$411$	−2.78919	−	1.54495i	−0.137581	−	0.0762067i
$412$		0			0
$413$	0.806240			0.0396725
$414$		0			0
$415$		−	6.82530i		−	0.335041i
$416$		0			0
$417$	−4.38752	−	2.43027i	−0.214858	−	0.119011i
$418$		0			0
$419$			26.0151i			1.27092i	0.772133	+	0.635461i	$0.219190\pi$
$419$			26.0151i			1.27092i	−0.772133	+	0.635461i	$0.780810\pi$
$420$		0			0
$421$	−7.84848			−0.382511			−0.191256	−	0.981540i	$-0.561256\pi$
$421$	−7.84848			−0.382511			−0.191256	+	0.981540i	$0.561256\pi$
$422$		0			0
$423$	7.97644	−	4.99108i	0.387828	−	0.242675i
$424$		0			0
$425$	−6.62413			−0.321317
$426$		0			0
$427$	−11.3660			−0.550039
$428$		0			0
$429$	−0.470369	+	6.43123i	−0.0227096	+	0.310503i
$430$		0			0
$431$	−37.3945			−1.80123			−0.900616	−	0.434616i	$-0.856884\pi$
$431$	−37.3945			−1.80123			−0.900616	+	0.434616i	$0.856884\pi$
$432$		0			0
$433$	−27.3799			−1.31580			−0.657898	−	0.753107i	$-0.728554\pi$
$433$	−27.3799			−1.31580			−0.657898	+	0.753107i	$0.728554\pi$
$434$		0			0
$435$	0.262172	+	0.145218i	0.0125702	+	0.00696269i
$436$		0			0
$437$	43.1078			2.06212
$438$		0			0
$439$		−	34.5335i		−	1.64819i	−0.566449	−	0.824097i	$-0.691683\pi$
$439$		−	34.5335i		−	1.64819i	0.566449	−	0.824097i	$-0.308317\pi$
$440$		0			0
$441$	1.59133	+	2.54316i	0.0757775	+	0.121103i
$442$		0			0
$443$			28.0303i			1.33176i	0.746059	+	0.665880i	$0.231944\pi$
$443$			28.0303i			1.33176i	−0.746059	+	0.665880i	$0.768056\pi$
$444$		0			0
$445$	0.0346322			0.00164172
$446$		0			0
$447$	−1.68091	+	3.03465i	−0.0795043	+	0.143534i
$448$		0			0
$449$		−	8.18479i		−	0.386264i	−0.981173	−	0.193132i	$-0.938135\pi$
$449$		−	8.18479i		−	0.386264i	0.981173	−	0.193132i	$-0.0618646\pi$
$450$		0			0
$451$	9.20923	+	19.9313i	0.433646	+	0.938528i
$452$		0			0
$453$	−21.4561	−	11.8846i	−1.00809	−	0.558389i
$454$		0			0
$455$		−	1.12252i		−	0.0526247i
$456$		0			0
$457$			27.0120i			1.26357i	0.775144	+	0.631784i	$0.217677\pi$
$457$			27.0120i			1.26357i	−0.775144	+	0.631784i	$0.782323\pi$
$458$		0			0
$459$	34.3711	−	1.83322i	1.60431	−	0.0855676i
$460$		0			0
$461$	24.5633			1.14403			0.572014	−	0.820244i	$-0.306162\pi$
$461$	24.5633			1.14403			0.572014	+	0.820244i	$0.306162\pi$
$462$		0			0
$463$	6.10594			0.283767			0.141883	−	0.989883i	$-0.454684\pi$
$463$	6.10594			0.283767			0.141883	+	0.989883i	$0.454684\pi$
$464$		0			0
$465$	11.0388	+	6.11443i	0.511910	+	0.283550i
$466$		0			0
$467$			39.3190i			1.81947i	0.415195	+	0.909733i	$0.363713\pi$
$467$			39.3190i			1.81947i	−0.415195	+	0.909733i	$0.636287\pi$
$468$		0			0
$469$			0.188894i			0.00872230i
$470$		0			0
$471$	0.888593	−	1.60423i	0.0409442	−	0.0739191i
$472$		0			0
$473$	27.0251	−	12.4869i	1.24262	−	0.574150i
$474$		0			0
$475$			6.33775i			0.290796i
$476$		0			0
$477$	17.9413	−	11.2264i	0.821478	−	0.514022i
$478$		0			0
$479$	−41.3558			−1.88960			−0.944798	−	0.327653i	$-0.893742\pi$
$479$	−41.3558			−1.88960			−0.944798	+	0.327653i	$0.893742\pi$
$480$		0			0
$481$			6.09763i			0.278028i
$482$		0			0
$483$	5.70835	−	10.3056i	0.259739	−	0.468923i
$484$		0			0
$485$		−	16.4900i		−	0.748773i
$486$		0			0
$487$	13.6890			0.620308			0.310154	−	0.950686i	$-0.399619\pi$
$487$	13.6890			0.620308			0.310154	+	0.950686i	$0.399619\pi$
$488$		0			0
$489$	−7.45660	+	13.4619i	−0.337199	+	0.608766i
$490$		0			0
$491$	−5.96283			−0.269099			−0.134549	−	0.990907i	$-0.542959\pi$
$491$	−5.96283			−0.269099			−0.134549	+	0.990907i	$0.542959\pi$
$492$		0			0
$493$	−1.14620			−0.0516222
$494$		0			0
$495$	−8.32899	−	5.44315i	−0.374360	−	0.244652i
$496$		0			0
$497$	11.7395			0.526588
$498$		0			0
$499$	23.8209			1.06637			0.533184	−	0.845999i	$-0.320995\pi$
$499$	23.8209			1.06637			0.533184	+	0.845999i	$0.320995\pi$
$500$		0			0
$501$	−3.76940	+	6.80512i	−0.168404	+	0.304030i
$502$		0			0
$503$	−18.8449			−0.840251			−0.420125	−	0.907466i	$-0.638014\pi$
$503$	−18.8449			−0.840251			−0.420125	+	0.907466i	$0.638014\pi$
$504$		0			0
$505$			5.89346i			0.262256i
$506$		0			0
$507$	−9.85271	+	17.7877i	−0.437574	+	0.789980i
$508$		0			0
$509$			2.35631i			0.104442i	0.998636	+	0.0522208i	$0.0166300\pi$
$509$			2.35631i			0.104442i	−0.998636	+	0.0522208i	$0.983370\pi$
$510$		0			0
$511$	−7.05678			−0.312173
$512$		0			0
$513$	−1.75397	−	32.8851i	−0.0774396	−	1.45191i
$514$		0			0
$515$		−	7.66948i		−	0.337958i
$516$		0			0
$517$	9.44308	−	4.36316i	0.415306	−	0.191892i
$518$		0			0
$519$	17.9928	−	32.4836i	0.789798	−	1.42587i
$520$		0			0
$521$			3.31837i			0.145380i	0.997355	+	0.0726902i	$0.0231584\pi$
$521$			3.31837i			0.145380i	−0.997355	+	0.0726902i	$0.976842\pi$
$522$		0			0
$523$			42.5306i			1.85973i	0.367895	+	0.929867i	$0.380079\pi$
$523$			42.5306i			1.85973i	−0.367895	+	0.929867i	$0.619921\pi$
$524$		0			0
$525$	1.51514	+	0.839247i	0.0661263	+	0.0366278i
$526$		0			0
$527$	−48.2608			−2.10227
$528$		0			0
$529$	−23.2638			−1.01147
$530$		0			0
$531$	−2.05040	+	1.28299i	−0.0889798	+	0.0556771i
$532$		0			0
$533$		−	7.43109i		−	0.321876i
$534$		0			0
$535$		−	15.3431i		−	0.663342i
$536$		0			0
$537$	39.7695	+	22.0285i	1.71618	+	0.950601i
$538$		0			0
$539$	1.39113	+	3.01078i	0.0599200	+	0.129683i
$540$		0			0
$541$			25.2296i			1.08471i	0.840151	+	0.542353i	$0.182466\pi$
$541$			25.2296i			1.08471i	−0.840151	+	0.542353i	$0.817534\pi$
$542$		0			0
$543$	−5.90533	+	10.6613i	−0.253422	+	0.457518i
$544$		0			0
$545$	−11.7701			−0.504177
$546$		0			0
$547$			3.67901i			0.157303i	0.996902	+	0.0786516i	$0.0250615\pi$
$547$			3.67901i			0.157303i	−0.996902	+	0.0786516i	$0.974939\pi$
$548$		0			0
$549$	28.9056	−	18.0870i	1.23366	−	0.771935i
$550$		0			0
$551$			1.09665i			0.0467187i
$552$		0			0
$553$	−3.47818			−0.147907
$554$		0			0
$555$	−8.23038	−	4.55885i	−0.349360	−	0.193512i
$556$		0			0
$557$	31.8948			1.35142			0.675712	−	0.737165i	$-0.263836\pi$
$557$	31.8948			1.35142			0.675712	+	0.737165i	$0.263836\pi$
$558$		0			0
$559$	−10.0759			−0.426166
$560$		0			0
$561$	37.9513	+	2.77569i	1.60231	+	0.117190i
$562$		0			0
$563$	12.7855			0.538843			0.269421	−	0.963022i	$-0.413168\pi$
$563$	12.7855			0.538843			0.269421	+	0.963022i	$0.413168\pi$
$564$		0			0
$565$	14.2814			0.600822
$566$		0			0
$567$	−8.09401	−	3.93535i	−0.339916	−	0.165269i
$568$		0			0
$569$	−4.42360			−0.185447			−0.0927235	−	0.995692i	$-0.529557\pi$
$569$	−4.42360			−0.185447			−0.0927235	+	0.995692i	$0.529557\pi$
$570$		0			0
$571$			25.8985i			1.08382i	0.840437	+	0.541909i	$0.182298\pi$
$571$			25.8985i			1.08382i	−0.840437	+	0.541909i	$0.817702\pi$
$572$		0			0
$573$	19.9580	+	11.0548i	0.833755	+	0.461822i
$574$		0			0
$575$		−	6.80175i		−	0.283653i
$576$		0			0
$577$	38.3648			1.59715			0.798574	−	0.601897i	$-0.205588\pi$
$577$	38.3648			1.59715			0.798574	+	0.601897i	$0.205588\pi$
$578$		0			0
$579$	21.4102	+	11.8592i	0.889778	+	0.492853i
$580$		0			0
$581$		−	6.82530i		−	0.283161i
$582$		0			0
$583$	21.2402	−	9.81403i	0.879680	−	0.406456i
$584$		0			0
$585$	1.78630	+	2.85476i	0.0738545	+	0.118030i
$586$		0			0
$587$		−	21.6739i		−	0.894576i	−0.894390	−	0.447288i	$-0.852390\pi$
$587$		−	21.6739i		−	0.894576i	0.894390	−	0.447288i	$-0.147610\pi$
$588$		0			0
$589$			46.1744i			1.90258i
$590$		0			0
$591$	−13.2685	+	23.9545i	−0.545795	+	0.985357i
$592$		0			0
$593$	−45.7902			−1.88038			−0.940190	−	0.340650i	$-0.889353\pi$
$593$	−45.7902			−1.88038			−0.940190	+	0.340650i	$0.889353\pi$
$594$		0			0
$595$	−6.62413			−0.271563
$596$		0			0
$597$	−18.1204	+	32.7139i	−0.741619	+	1.33889i
$598$		0			0
$599$		−	25.9318i		−	1.05954i	−0.848140	−	0.529772i	$-0.822278\pi$
$599$		−	25.9318i		−	1.05954i	0.848140	−	0.529772i	$-0.177722\pi$
$600$		0			0
$601$			29.2480i			1.19305i	0.802594	+	0.596526i	$0.203453\pi$
$601$			29.2480i			1.19305i	−0.802594	+	0.596526i	$0.796547\pi$
$602$		0			0
$603$	−0.300592	−	0.480387i	−0.0122410	−	0.0195629i
$604$		0			0
$605$	−8.37674	−	7.12954i	−0.340563	−	0.289857i
$606$		0			0
$607$			35.4128i			1.43736i	0.695340	+	0.718681i	$0.255254\pi$
$607$			35.4128i			1.43736i	−0.695340	+	0.718681i	$0.744746\pi$
$608$		0			0
$609$	0.262172	+	0.145218i	0.0106237	+	0.00588454i
$610$		0			0
$611$	−3.52071			−0.142433
$612$		0			0
$613$		−	47.2857i		−	1.90985i	−0.296847	−	0.954925i	$-0.595935\pi$
$613$		−	47.2857i		−	1.90985i	0.296847	−	0.954925i	$-0.404065\pi$
$614$		0			0
$615$	10.0302	+	5.55581i	0.404458	+	0.224032i
$616$		0			0
$617$		−	31.9996i		−	1.28826i	−0.764918	−	0.644128i	$-0.777220\pi$
$617$		−	31.9996i		−	1.28826i	0.764918	−	0.644128i	$-0.222780\pi$
$618$		0			0
$619$	2.96454			0.119155			0.0595775	−	0.998224i	$-0.481025\pi$
$619$	2.96454			0.119155			0.0595775	+	0.998224i	$0.481025\pi$
$620$		0			0
$621$	1.88238	+	35.2928i	0.0755374	+	1.41625i
$622$		0			0
$623$	0.0346322			0.00138751
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	2.65569	−	36.3106i	0.106058	−	1.45011i
$628$		0			0
$629$	35.9827			1.43473
$630$		0			0
$631$	−18.2221			−0.725409			−0.362705	−	0.931904i	$-0.618147\pi$
$631$	−18.2221			−0.725409			−0.362705	+	0.931904i	$0.618147\pi$
$632$		0			0
$633$	5.51621	+	3.05546i	0.219249	+	0.121444i
$634$		0			0
$635$	−13.6994			−0.543645
$636$		0			0
$637$		−	1.12252i		−	0.0444760i
$638$		0			0
$639$	−29.8555	+	18.6814i	−1.18106	+	0.739024i
$640$		0			0
$641$		−	1.87248i		−	0.0739584i	−0.999316	−	0.0369792i	$-0.988226\pi$
$641$		−	1.87248i		−	0.0739584i	0.999316	−	0.0369792i	$-0.0117735\pi$
$642$		0			0
$643$	24.9425			0.983636			0.491818	−	0.870698i	$-0.336333\pi$
$643$	24.9425			0.983636			0.491818	+	0.870698i	$0.336333\pi$
$644$		0			0
$645$	7.53320	−	13.6001i	0.296619	−	0.535505i
$646$		0			0
$647$		−	29.8502i		−	1.17353i	−0.809757	−	0.586765i	$-0.800401\pi$
$647$		−	29.8502i		−	1.17353i	0.809757	−	0.586765i	$-0.199599\pi$
$648$		0			0
$649$	−2.42741	+	1.12158i	−0.0952841	+	0.0440259i
$650$		0			0
$651$	11.0388	+	6.11443i	0.432643	+	0.239643i
$652$		0			0
$653$			38.7860i			1.51781i	0.651200	+	0.758906i	$0.274266\pi$
$653$			38.7860i			1.51781i	−0.651200	+	0.758906i	$0.725734\pi$
$654$		0			0
$655$			1.08443i			0.0423721i
$656$		0			0
$657$	17.9465	−	11.2296i	0.700161	−	0.438110i
$658$		0			0
$659$	37.8582			1.47475			0.737373	−	0.675486i	$-0.236066\pi$
$659$	37.8582			1.47475			0.737373	+	0.675486i	$0.236066\pi$
$660$		0			0
$661$	8.98317			0.349405			0.174702	−	0.984621i	$-0.444104\pi$
$661$	8.98317			0.349405			0.174702	+	0.984621i	$0.444104\pi$
$662$		0			0
$663$	−11.2662	−	6.24042i	−0.437544	−	0.242358i
$664$		0			0
$665$			6.33775i			0.245767i
$666$		0			0
$667$		−	1.17693i		−	0.0455711i
$668$		0			0
$669$	8.97573	−	16.2044i	0.347022	−	0.626500i
$670$		0			0
$671$	34.2204	−	15.8115i	1.32107	−	0.610397i
$672$		0			0
$673$		−	45.8404i		−	1.76702i	−0.468415	−	0.883508i	$-0.655175\pi$
$673$		−	45.8404i		−	1.76702i	0.468415	−	0.883508i	$-0.344825\pi$
$674$		0			0
$675$	−5.18878	+	0.276750i	−0.199716	+	0.0106521i
$676$		0			0
$677$	28.3329			1.08892			0.544461	−	0.838786i	$-0.316734\pi$
$677$	28.3329			1.08892			0.544461	+	0.838786i	$0.316734\pi$
$678$		0			0
$679$		−	16.4900i		−	0.632829i
$680$		0			0
$681$	14.2674	−	25.7578i	0.546728	−	0.987042i
$682$		0			0
$683$			23.7729i			0.909644i	0.890582	+	0.454822i	$0.150297\pi$
$683$			23.7729i			0.909644i	−0.890582	+	0.454822i	$0.849703\pi$
$684$		0			0
$685$	−1.84087			−0.0703362
$686$		0			0
$687$	18.0415	−	32.5714i	0.688325	−	1.24267i
$688$		0			0
$689$	−7.91911			−0.301694
$690$		0			0
$691$	23.1741			0.881583			0.440791	−	0.897610i	$-0.354698\pi$
$691$	23.1741			0.881583			0.440791	+	0.897610i	$0.354698\pi$
$692$		0			0
$693$	−8.32899	−	5.44315i	−0.316392	−	0.206768i
$694$		0			0
$695$	−2.89578			−0.109843
$696$		0			0
$697$	−43.8516			−1.66100
$698$		0			0
$699$	−11.9244	+	21.5279i	−0.451023	+	0.814259i
$700$		0			0
$701$	30.8577			1.16548			0.582739	−	0.812659i	$-0.301981\pi$
$701$	30.8577			1.16548			0.582739	+	0.812659i	$0.301981\pi$
$702$		0			0
$703$		−	34.4271i		−	1.29844i
$704$		0			0
$705$	2.63224	−	4.75214i	0.0991358	−	0.178976i
$706$		0			0
$707$			5.89346i			0.221647i
$708$		0			0
$709$	−11.0860			−0.416343			−0.208171	−	0.978092i	$-0.566751\pi$
$709$	−11.0860			−0.416343			−0.208171	+	0.978092i	$0.566751\pi$
$710$		0			0
$711$	8.84557	−	5.53492i	0.331735	−	0.207576i
$712$		0			0
$713$		−	49.5549i		−	1.85585i
$714$		0			0
$715$	1.56157	+	3.37967i	0.0583995	+	0.126392i
$716$		0			0
$717$	4.73017	−	8.53966i	0.176651	−	0.318920i
$718$		0			0
$719$		−	33.0241i		−	1.23159i	−0.787906	−	0.615795i	$-0.788835\pi$
$719$		−	33.0241i		−	1.23159i	0.787906	−	0.615795i	$-0.211165\pi$
$720$		0			0
$721$		−	7.66948i		−	0.285626i
$722$		0			0
$723$	−36.8283	−	20.3994i	−1.36966	−	0.758662i
$724$		0			0
$725$	0.173034			0.00642632
$726$		0			0
$727$	14.6545			0.543505			0.271752	−	0.962367i	$-0.412397\pi$
$727$	14.6545			0.543505			0.271752	+	0.962367i	$0.412397\pi$
$728$		0			0
$729$	26.8468	−	2.87199i	0.994327	−	0.106370i
$730$		0			0
$731$			59.4591i			2.19917i
$732$		0			0
$733$		−	22.0838i		−	0.815684i	−0.913053	−	0.407842i	$-0.866281\pi$
$733$		−	22.0838i		−	0.815684i	0.913053	−	0.407842i	$-0.133719\pi$
$734$		0			0
$735$	1.51514	+	0.839247i	0.0558870	+	0.0309561i
$736$		0			0
$737$	−0.262775	−	0.568717i	−0.00967944	−	0.0209489i
$738$		0			0
$739$			14.2432i			0.523945i	0.965075	+	0.261972i	$0.0843729\pi$
$739$			14.2432i			0.523945i	−0.965075	+	0.261972i	$0.915627\pi$
$740$		0			0
$741$	−5.97063	+	10.7791i	−0.219337	+	0.395982i
$742$		0			0
$743$	13.0801			0.479861			0.239930	−	0.970790i	$-0.422875\pi$
$743$	13.0801			0.479861			0.239930	+	0.970790i	$0.422875\pi$
$744$		0			0
$745$			2.00288i			0.0733797i
$746$		0			0
$747$	10.8613	+	17.3578i	0.397394	+	0.635090i
$748$		0			0
$749$		−	15.3431i		−	0.560626i
$750$		0			0
$751$	−38.4424			−1.40278			−0.701392	−	0.712776i	$-0.747438\pi$
$751$	−38.4424			−1.40278			−0.701392	+	0.712776i	$0.747438\pi$
$752$		0			0
$753$	44.7204	+	24.7709i	1.62970	+	0.902702i
$754$		0			0
$755$	−14.1611			−0.515374
$756$		0			0
$757$	33.0859			1.20253			0.601263	−	0.799051i	$-0.294664\pi$
$757$	33.0859			1.20253			0.601263	+	0.799051i	$0.294664\pi$
$758$		0			0
$759$	−2.85012	+	38.9690i	−0.103453	+	1.41449i
$760$		0			0
$761$	28.3802			1.02878			0.514390	−	0.857556i	$-0.328018\pi$
$761$	28.3802			1.02878			0.514390	+	0.857556i	$0.328018\pi$
$762$		0			0
$763$	−11.7701			−0.426107
$764$		0			0
$765$	16.8462	−	10.5412i	0.609077	−	0.381116i
$766$		0			0
$767$	0.905023			0.0326785
$768$		0			0
$769$		−	34.0510i		−	1.22791i	−0.789341	−	0.613955i	$-0.789578\pi$
$769$		−	34.0510i		−	1.22791i	0.789341	−	0.613955i	$-0.210422\pi$
$770$		0			0
$771$	−26.0378	−	14.4225i	−0.937730	−	0.519414i
$772$		0			0
$773$			23.4862i			0.844741i	0.906423	+	0.422370i	$0.138802\pi$
$773$			23.4862i			0.844741i	−0.906423	+	0.422370i	$0.861198\pi$
$774$		0			0
$775$	7.28561			0.261707
$776$		0			0
$777$	−8.23038	−	4.55885i	−0.295263	−	0.163548i
$778$		0			0
$779$			41.9558i			1.50322i
$780$		0			0
$781$	−35.3450	+	16.3311i	−1.26474	+	0.584373i
$782$		0			0
$783$	−0.897835	+	0.0478871i	−0.0320860	+	0.00171135i
$784$		0			0
$785$		−	1.05880i		−	0.0377901i
$786$		0			0
$787$			32.5955i			1.16190i	0.813938	+	0.580952i	$0.197320\pi$
$787$			32.5955i			1.16190i	−0.813938	+	0.580952i	$0.802680\pi$
$788$		0			0
$789$	25.6891	−	46.3781i	0.914555	−	1.65110i
$790$		0			0
$791$	14.2814			0.507787
$792$		0			0
$793$	−12.7586			−0.453071
$794$		0			0
$795$	5.92067	−	10.6889i	0.209985	−	0.379098i
$796$		0			0
$797$			26.4808i			0.937998i	0.883199	+	0.468999i	$0.155385\pi$
$797$			26.4808i			0.937998i	−0.883199	+	0.468999i	$0.844615\pi$
$798$		0			0
$799$			20.7761i			0.735005i
$800$		0			0
$801$	−0.0880753	+	0.0551112i	−0.00311199	+	0.00194726i
$802$		0			0
$803$	21.2464	−	9.81686i	0.749768	−	0.346430i
$804$		0			0
$805$		−	6.80175i		−	0.239730i
$806$		0			0
$807$	−9.56002	−	5.29535i	−0.336529	−	0.186405i
$808$		0			0
$809$	44.8490			1.57681			0.788403	−	0.615159i	$-0.210908\pi$
$809$	44.8490			1.57681			0.788403	+	0.615159i	$0.210908\pi$
$810$		0			0
$811$		−	40.9445i		−	1.43776i	−0.695136	−	0.718878i	$-0.744656\pi$
$811$		−	40.9445i		−	1.43776i	0.695136	−	0.718878i	$-0.255344\pi$
$812$		0			0
$813$	−7.21870	−	3.99848i	−0.253171	−	0.140233i
$814$		0			0
$815$			8.88487i			0.311223i
$816$		0			0
$817$	56.8885			1.99028
$818$		0			0
$819$	1.78630	+	2.85476i	0.0624185	+	0.0997534i
$820$		0			0
$821$	13.0145			0.454209			0.227105	−	0.973870i	$-0.427074\pi$
$821$	13.0145			0.454209			0.227105	+	0.973870i	$0.427074\pi$
$822$		0			0
$823$	31.6853			1.10448			0.552241	−	0.833685i	$-0.313773\pi$
$823$	31.6853			1.10448			0.552241	+	0.833685i	$0.313773\pi$
$824$		0			0
$825$	−5.72926	−	0.419028i	−0.199467	−	0.0145887i
$826$		0			0
$827$	0.314371			0.0109317			0.00546587	−	0.999985i	$-0.498260\pi$
$827$	0.314371			0.0109317			0.00546587	+	0.999985i	$0.498260\pi$
$828$		0			0
$829$	−11.0409			−0.383466			−0.191733	−	0.981447i	$-0.561411\pi$
$829$	−11.0409			−0.383466			−0.191733	+	0.981447i	$0.561411\pi$
$830$		0			0
$831$	−32.8829	−	18.2140i	−1.14069	−	0.631837i
$832$		0			0
$833$	−6.62413			−0.229512
$834$		0			0
$835$			4.49140i			0.155431i
$836$		0			0
$837$	−37.8034	+	2.01629i	−1.30668	+	0.0696932i
$838$		0			0
$839$		−	33.1363i		−	1.14399i	−0.820256	−	0.571997i	$-0.806169\pi$
$839$		−	33.1363i		−	1.14399i	0.820256	−	0.571997i	$-0.193831\pi$
$840$		0			0
$841$	−28.9701			−0.998968
$842$		0			0
$843$	−0.621285	+	1.12164i	−0.0213982	+	0.0386315i
$844$		0			0
$845$			11.7399i			0.403866i
$846$		0			0
$847$	−8.37674	−	7.12954i	−0.287828	−	0.244974i
$848$		0			0
$849$	−4.16503	−	2.30703i	−0.142943	−	0.0791772i
$850$		0			0
$851$			36.9476i			1.26655i
$852$		0			0
$853$		−	25.7173i		−	0.880544i	−0.897864	−	0.440272i	$-0.854882\pi$
$853$		−	25.7173i		−	0.880544i	0.897864	−	0.440272i	$-0.145118\pi$
$854$		0			0
$855$	−10.0854	−	16.1179i	−0.344914	−	0.551221i
$856$		0			0
$857$	−16.8773			−0.576518			−0.288259	−	0.957553i	$-0.593076\pi$
$857$	−16.8773			−0.576518			−0.288259	+	0.957553i	$0.593076\pi$
$858$		0			0
$859$	−16.1312			−0.550389			−0.275195	−	0.961389i	$-0.588742\pi$
$859$	−16.1312			−0.550389			−0.275195	+	0.961389i	$0.588742\pi$
$860$		0			0
$861$	10.0302	+	5.55581i	0.341830	+	0.189341i
$862$		0			0
$863$			9.92241i			0.337763i	0.985636	+	0.168881i	$0.0540155\pi$
$863$			9.92241i			0.337763i	−0.985636	+	0.168881i	$0.945985\pi$
$864$		0			0
$865$		−	21.4393i		−	0.728957i
$866$		0			0
$867$	−22.5582	+	40.7256i	−0.766115	+	1.38312i
$868$		0			0
$869$	10.4720	−	4.83858i	0.355239	−	0.164138i
$870$		0			0
$871$			0.212038i			0.00718462i
$872$		0			0
$873$	26.2410	+	41.9368i	0.888125	+	1.41935i
$874$		0			0
$875$	1.00000			0.0338062
$876$		0			0
$877$			4.15684i			0.140366i	0.997534	+	0.0701832i	$0.0223584\pi$
$877$			4.15684i			0.140366i	−0.997534	+	0.0701832i	$0.977642\pi$
$878$		0			0
$879$	−14.0528	+	25.3704i	−0.473989	+	0.855721i
$880$		0			0
$881$			4.66079i			0.157026i	0.996913	+	0.0785130i	$0.0250172\pi$
$881$			4.66079i			0.157026i	−0.996913	+	0.0785130i	$0.974983\pi$
$882$		0			0
$883$	2.30182			0.0774625			0.0387313	−	0.999250i	$-0.487668\pi$
$883$	2.30182			0.0774625			0.0387313	+	0.999250i	$0.487668\pi$
$884$		0			0
$885$	−0.676635	+	1.22157i	−0.0227448	+	0.0410626i
$886$		0			0
$887$	2.68965			0.0903097			0.0451548	−	0.998980i	$-0.485622\pi$
$887$	2.68965			0.0903097			0.0451548	+	0.998980i	$0.485622\pi$
$888$		0			0
$889$	−13.6994			−0.459464
$890$		0			0
$891$	29.8438	+	0.588672i	0.999806	+	0.0197212i
$892$		0			0
$893$	19.8779			0.665188
$894$		0			0
$895$	26.2480			0.877372
$896$		0			0
$897$	6.40776	−	11.5683i	0.213949	−	0.386255i
$898$		0			0
$899$	1.26066			0.0420453
$900$		0			0
$901$			46.7315i			1.55685i
$902$		0			0
$903$	7.53320	−	13.6001i	0.250689	−	0.452585i
$904$		0			0
$905$			7.03646i			0.233900i
$906$		0			0
$907$	−18.2461			−0.605853			−0.302926	−	0.953014i	$-0.597964\pi$
$907$	−18.2461			−0.605853			−0.302926	+	0.953014i	$0.597964\pi$
$908$		0			0
$909$	−9.37843	−	14.9880i	−0.311063	−	0.497122i
$910$		0			0
$911$			28.1450i			0.932485i	0.884657	+	0.466243i	$0.154393\pi$
$911$			28.1450i			0.932485i	−0.884657	+	0.466243i	$0.845607\pi$
$912$		0			0
$913$	9.49485	+	20.5494i	0.314234	+	0.680087i
$914$		0			0
$915$	9.53888	−	17.2211i	0.315346	−	0.569313i
$916$		0			0
$917$			1.08443i			0.0358109i
$918$		0			0
$919$			27.8363i			0.918235i	0.888375	+	0.459118i	$0.151834\pi$
$919$			27.8363i			0.918235i	−0.888375	+	0.459118i	$0.848166\pi$
$920$		0			0
$921$	−12.7630	−	7.06949i	−0.420555	−	0.232948i
$922$		0			0
$923$	13.1779			0.433755
$924$		0			0
$925$	−5.43207			−0.178605
$926$		0			0
$927$	12.2047	+	19.5047i	0.400854	+	0.640620i
$928$		0			0
$929$		−	0.849179i		−	0.0278607i	−0.999903	−	0.0139303i	$-0.995566\pi$
$929$		−	0.849179i		−	0.0278607i	0.999903	−	0.0139303i	$-0.00443431\pi$
$930$		0			0
$931$			6.33775i			0.207711i
$932$		0			0
$933$	−15.5315	−	8.60300i	−0.508479	−	0.281649i
$934$		0			0
$935$	19.9438	−	9.21499i	0.652230	−	0.301362i
$936$		0			0
$937$			26.7092i			0.872553i	0.899813	+	0.436276i	$0.143703\pi$
$937$			26.7092i			0.872553i	−0.899813	+	0.436276i	$0.856297\pi$
$938$		0			0
$939$	12.7552	−	23.0277i	0.416249	−	0.751480i
$940$		0			0
$941$	22.2437			0.725125			0.362563	−	0.931959i	$-0.381902\pi$
$941$	22.2437			0.725125			0.362563	+	0.931959i	$0.381902\pi$
$942$		0			0
$943$		−	45.0275i		−	1.46630i
$944$		0			0
$945$	−5.18878	+	0.276750i	−0.168791	+	0.00900267i
$946$		0			0
$947$		−	52.0844i		−	1.69252i	−0.532774	−	0.846258i	$-0.678850\pi$
$947$		−	52.0844i		−	1.69252i	0.532774	−	0.846258i	$-0.321150\pi$
$948$		0			0
$949$	−7.92140			−0.257139
$950$		0			0
$951$	−40.4752	−	22.4194i	−1.31250	−	0.727000i
$952$		0			0
$953$	25.2223			0.817030			0.408515	−	0.912751i	$-0.366047\pi$
$953$	25.2223			0.817030			0.408515	+	0.912751i	$0.366047\pi$
$954$		0			0
$955$	13.1723			0.426246
$956$		0			0
$957$	−0.991357	−	0.0725061i	−0.0320460	−	0.00234379i
$958$		0			0
$959$	−1.84087			−0.0594449
$960$		0			0
$961$	22.0802			0.712263
$962$		0			0
$963$	24.4160	+	39.0201i	0.786793	+	1.25740i
$964$		0			0
$965$	14.1308			0.454887
$966$		0			0
$967$			30.7545i			0.988997i	0.869179	+	0.494498i	$0.164648\pi$
$967$			30.7545i			0.988997i	−0.869179	+	0.494498i	$0.835352\pi$
$968$		0			0
$969$	63.6088	+	35.2333i	2.04341	+	1.13186i
$970$		0			0
$971$			22.7823i			0.731119i	0.930788	+	0.365559i	$0.119122\pi$
$971$			22.7823i			0.731119i	−0.930788	+	0.365559i	$0.880878\pi$
$972$		0			0
$973$	−2.89578			−0.0928343
$974$		0			0
$975$	1.70079	+	0.942075i	0.0544687	+	0.0301705i
$976$		0			0
$977$		−	23.9803i		−	0.767196i	−0.923500	−	0.383598i	$-0.874685\pi$
$977$		−	23.9803i		−	0.767196i	0.923500	−	0.383598i	$-0.125315\pi$
$978$		0			0
$979$	−0.104270	+	0.0481777i	−0.00333248	+	0.00153977i
$980$		0			0
$981$	29.9334	−	18.7301i	0.955699	−	0.598007i
$982$		0			0
$983$			47.7777i			1.52387i	0.647652	+	0.761936i	$0.275751\pi$
$983$			47.7777i			1.52387i	−0.647652	+	0.761936i	$0.724249\pi$
$984$		0			0
$985$			15.8100i			0.503750i
$986$		0			0
$987$	2.63224	−	4.75214i	0.0837851	−	0.151262i
$988$		0			0
$989$	−61.0535			−1.94139
$990$		0			0
$991$	−4.17082			−0.132490			−0.0662452	−	0.997803i	$-0.521102\pi$
$991$	−4.17082			−0.132490			−0.0662452	+	0.997803i	$0.521102\pi$
$992$		0			0
$993$	18.2965	−	33.0318i	0.580622	−	1.04823i
$994$		0			0
$995$			21.5913i			0.684489i
$996$		0			0
$997$			38.8516i			1.23044i	0.788355	+	0.615221i	$0.210933\pi$
$997$			38.8516i			1.23044i	−0.788355	+	0.615221i	$0.789067\pi$
$998$		0			0
$999$	28.1858	−	1.50332i	0.891760	−	0.0475631i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4620.2.m.a.1121.18	yes	48
3.2	odd	2		4620.2.m.b.1121.17	yes	48
11.10	odd	2		4620.2.m.b.1121.18	yes	48
33.32	even	2	inner	4620.2.m.a.1121.17	✓	48

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4620.2.m.a.1121.17	✓	48	33.32	even	2	inner
4620.2.m.a.1121.18	yes	48	1.1	even	1	trivial
4620.2.m.b.1121.17	yes	48	3.2	odd	2
4620.2.m.b.1121.18	yes	48	11.10	odd	2

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)

\(f(q)\)	\(=\)	\(q+(-0.839247 + 1.51514i) q^{3} +1.00000i q^{5} +1.00000i q^{7} +(-1.59133 - 2.54316i) q^{9} +O(q^{10})\)	\(q+(-0.839247 + 1.51514i) q^{3} +1.00000i q^{5} +1.00000i q^{7} +(-1.59133 - 2.54316i) q^{9} +(-1.39113 - 3.01078i) q^{11} +1.12252i q^{13} +(-1.51514 - 0.839247i) q^{15} +6.62413 q^{17} -6.33775i q^{19} +(-1.51514 - 0.839247i) q^{21} +6.80175i q^{23} -1.00000 q^{25} +(5.18878 - 0.276750i) q^{27} -0.173034 q^{29} -7.28561 q^{31} +(5.72926 + 0.419028i) q^{33} -1.00000 q^{35} +5.43207 q^{37} +(-1.70079 - 0.942075i) q^{39} -6.61999 q^{41} +8.97614i q^{43} +(2.54316 - 1.59133i) q^{45} +3.13643i q^{47} -1.00000 q^{49} +(-5.55928 + 10.0365i) q^{51} +7.05474i q^{53} +(3.01078 - 1.39113i) q^{55} +(9.60260 + 5.31894i) q^{57} -0.806240i q^{59} +11.3660i q^{61} +(2.54316 - 1.59133i) q^{63} -1.12252 q^{65} +0.188894 q^{67} +(-10.3056 - 5.70835i) q^{69} -11.7395i q^{71} +7.05678i q^{73} +(0.839247 - 1.51514i) q^{75} +(3.01078 - 1.39113i) q^{77} +3.47818i q^{79} +(-3.93535 + 8.09401i) q^{81} -6.82530 q^{83} +6.62413i q^{85} +(0.145218 - 0.262172i) q^{87} -0.0346322i q^{89} -1.12252 q^{91} +(6.11443 - 11.0388i) q^{93} +6.33775 q^{95} -16.4900 q^{97} +(-5.44315 + 8.32899i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q - 4 q^{3} + 6 q^{9}+O(q^{10}) \) 48 * q - 4 * q^3 + 6 * q^9	\( 48 q - 4 q^{3} + 6 q^{9} - 6 q^{11} + 2 q^{15} + 4 q^{17} + 2 q^{21} - 48 q^{25} - 16 q^{27} + 36 q^{29} - 16 q^{31} + 16 q^{33} - 48 q^{35} - 8 q^{37} - 18 q^{39} - 48 q^{49} - 30 q^{51} + 4 q^{55} + 16 q^{57} - 12 q^{65} + 24 q^{67} + 4 q^{69} + 4 q^{75} + 4 q^{77} - 22 q^{81} + 20 q^{83} + 44 q^{87} - 12 q^{91} - 28 q^{93} + 8 q^{95} - 56 q^{97} + 26 q^{99}+O(q^{100}) \) 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 + 16 * 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 + 26 * q^99

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