LMFDB - Embedded newform 624.4.c.c.337.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [624,4,Mod(337,624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(624, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("624.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 624 = 2^{4} \cdot 3 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	624.c (of order $2$, degree $1$, not 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:	$36.8171918436$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.1362828.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 23x^{2} + 48 $ x^4 + 23*x^2 + 48
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.4
Root			$-4.54739i$ of defining polynomial
Character	$\chi$	$=$	624.337
Dual form			624.4.c.c.337.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +12.9118i q^{5} +16.7289i q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +12.9118i q^{5} +16.7289i q^{7} +9.00000 q^{9} +24.9280i q^{11} +(33.7151 + 32.5621i) q^{13} -38.7355i q^{15} +134.145 q^{17} +14.9376i q^{19} -50.1866i q^{21} +72.0000 q^{23} -41.7151 q^{25} -27.0000 q^{27} -206.145 q^{29} +249.142i q^{31} -74.7841i q^{33} -216.000 q^{35} -293.955i q^{37} +(-101.145 - 97.6863i) q^{39} +250.506i q^{41} +432.145 q^{43} +116.206i q^{45} -159.889i q^{47} +63.1454 q^{49} -402.436 q^{51} -194.581 q^{53} -321.866 q^{55} -44.8129i q^{57} +232.647i q^{59} -185.006 q^{61} +150.560i q^{63} +(-420.436 + 435.324i) q^{65} +39.4393i q^{67} -216.000 q^{69} -920.460i q^{71} +549.078i q^{73} +125.145 q^{75} -417.018 q^{77} -933.140 q^{79} +81.0000 q^{81} +1095.38i q^{83} +1732.06i q^{85} +618.436 q^{87} +532.114i q^{89} +(-544.727 + 564.015i) q^{91} -747.425i q^{93} -192.872 q^{95} -362.661i q^{97} +224.352i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 12 q^{3} + 36 q^{9}+O(q^{10}) $ 4 * q - 12 * q^3 + 36 * q^9	$ 4 q - 12 q^{3} + 36 q^{9} - 12 q^{13} + 96 q^{17} + 288 q^{23} - 20 q^{25} - 108 q^{27} - 384 q^{29} - 864 q^{35} + 36 q^{39} + 1288 q^{43} - 188 q^{49} - 288 q^{51} + 984 q^{53} + 328 q^{55} + 288 q^{61} - 360 q^{65} - 864 q^{69} + 60 q^{75} + 1416 q^{77} - 4320 q^{79} + 324 q^{81} + 1152 q^{87} + 24 q^{91} + 1872 q^{95}+O(q^{100}) $ 4 * q - 12 * q^3 + 36 * q^9 - 12 * q^13 + 96 * q^17 + 288 * q^23 - 20 * q^25 - 108 * q^27 - 384 * q^29 - 864 * q^35 + 36 * q^39 + 1288 * q^43 - 188 * q^49 - 288 * q^51 + 984 * q^53 + 328 * q^55 + 288 * q^61 - 360 * q^65 - 864 * q^69 + 60 * q^75 + 1416 * q^77 - 4320 * q^79 + 324 * q^81 + 1152 * q^87 + 24 * q^91 + 1872 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/624\mathbb{Z}\right)^\times$.

$n$	$79$	$145$	$209$	$469$
$\chi(n)$	$1$	$-1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−3.00000			−0.577350
$3$	−3.00000			−0.577350
$4$		0			0
$5$			12.9118i			1.15487i	0.816437	+	0.577434i	$0.195946\pi$
$5$			12.9118i			1.15487i	−0.816437	+	0.577434i	$0.804054\pi$
$6$		0			0
$7$			16.7289i			0.903273i	0.892202	+	0.451637i	$0.149160\pi$
$7$			16.7289i			0.903273i	−0.892202	+	0.451637i	$0.850840\pi$
$8$		0			0
$9$	9.00000			0.333333
$10$		0			0
$11$			24.9280i			0.683280i	0.939831	+	0.341640i	$0.110982\pi$
$11$			24.9280i			0.683280i	−0.939831	+	0.341640i	$0.889018\pi$
$12$		0			0
$13$	33.7151	+	32.5621i	0.719299	+	0.694700i
$13$	33.7151	+	32.5621i	0.719299	+	0.694700i
$14$		0			0
$15$		−	38.7355i		−	0.666764i
$16$		0			0
$17$	134.145			1.91383			0.956913	−	0.290376i	$-0.0937804\pi$
$17$	134.145			1.91383			0.956913	+	0.290376i	$0.0937804\pi$
$18$		0			0
$19$			14.9376i			0.180365i	0.995925	+	0.0901824i	$0.0287450\pi$
$19$			14.9376i			0.180365i	−0.995925	+	0.0901824i	$0.971255\pi$
$20$		0			0
$21$		−	50.1866i		−	0.521505i
$22$		0			0
$23$	72.0000			0.652741			0.326370	−	0.945242i	$-0.394174\pi$
$23$	72.0000			0.652741			0.326370	+	0.945242i	$0.394174\pi$
$24$		0			0
$25$	−41.7151			−0.333721
$26$		0			0
$27$	−27.0000			−0.192450
$28$		0			0
$29$	−206.145			−1.32001			−0.660004	−	0.751262i	$-0.729445\pi$
$29$	−206.145			−1.32001			−0.660004	+	0.751262i	$0.729445\pi$
$30$		0			0
$31$			249.142i			1.44346i	0.692176	+	0.721728i	$0.256652\pi$
$31$			249.142i			1.44346i	−0.692176	+	0.721728i	$0.743348\pi$
$32$		0			0
$33$		−	74.7841i		−	0.394492i
$34$		0			0
$35$	−216.000			−1.04316
$36$		0			0
$37$		−	293.955i		−	1.30610i	−0.757313	−	0.653052i	$-0.773488\pi$
$37$		−	293.955i		−	1.30610i	0.757313	−	0.653052i	$-0.226512\pi$
$38$		0			0
$39$	−101.145	−	97.6863i	−0.415288	−	0.401085i
$40$		0			0
$41$			250.506i			0.954208i	0.878847	+	0.477104i	$0.158314\pi$
$41$			250.506i			0.954208i	−0.878847	+	0.477104i	$0.841686\pi$
$42$		0			0
$43$	432.145			1.53259			0.766297	−	0.642486i	$-0.222097\pi$
$43$	432.145			1.53259			0.766297	+	0.642486i	$0.222097\pi$
$44$		0			0
$45$			116.206i			0.384956i
$46$		0			0
$47$		−	159.889i		−	0.496217i	−0.968732	−	0.248109i	$-0.920191\pi$
$47$		−	159.889i		−	0.496217i	0.968732	−	0.248109i	$-0.0798090\pi$
$48$		0			0
$49$	63.1454			0.184097
$50$		0			0
$51$	−402.436			−1.10495
$52$		0			0
$53$	−194.581			−0.504298			−0.252149	−	0.967688i	$-0.581137\pi$
$53$	−194.581			−0.504298			−0.252149	+	0.967688i	$0.581137\pi$
$54$		0			0
$55$	−321.866			−0.789099
$56$		0			0
$57$		−	44.8129i		−	0.104134i
$58$		0			0
$59$			232.647i			0.513358i	0.966497	+	0.256679i	$0.0826283\pi$
$59$			232.647i			0.513358i	−0.966497	+	0.256679i	$0.917372\pi$
$60$		0			0
$61$	−185.006			−0.388321			−0.194160	−	0.980970i	$-0.562198\pi$
$61$	−185.006			−0.388321			−0.194160	+	0.980970i	$0.562198\pi$
$62$		0			0
$63$			150.560i			0.301091i
$64$		0			0
$65$	−420.436	+	435.324i	−0.802287	+	0.830696i
$66$		0			0
$67$			39.4393i			0.0719145i	0.999353	+	0.0359573i	$0.0114480\pi$
$67$			39.4393i			0.0719145i	−0.999353	+	0.0359573i	$0.988552\pi$
$68$		0			0
$69$	−216.000			−0.376860
$70$		0			0
$71$		−	920.460i		−	1.53857i	−0.638905	−	0.769286i	$-0.720612\pi$
$71$		−	920.460i		−	1.53857i	0.638905	−	0.769286i	$-0.279388\pi$
$72$		0			0
$73$			549.078i			0.880338i	0.897915	+	0.440169i	$0.145082\pi$
$73$			549.078i			0.880338i	−0.897915	+	0.440169i	$0.854918\pi$
$74$		0			0
$75$	125.145			0.192674
$76$		0			0
$77$	−417.018			−0.617189
$78$		0			0
$79$	−933.140			−1.32894			−0.664471	−	0.747314i	$-0.731343\pi$
$79$	−933.140			−1.32894			−0.664471	+	0.747314i	$0.731343\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$			1095.38i			1.44860i	0.689484	+	0.724301i	$0.257837\pi$
$83$			1095.38i			1.44860i	−0.689484	+	0.724301i	$0.742163\pi$
$84$		0			0
$85$			1732.06i			2.21022i
$86$		0			0
$87$	618.436			0.762107
$88$		0			0
$89$			532.114i			0.633753i	0.948467	+	0.316876i	$0.102634\pi$
$89$			532.114i			0.633753i	−0.948467	+	0.316876i	$0.897366\pi$
$90$		0			0
$91$	−544.727	+	564.015i	−0.627504	+	0.649724i
$92$		0			0
$93$		−	747.425i		−	0.833380i
$94$		0			0
$95$	−192.872			−0.208298
$96$		0			0
$97$		−	362.661i		−	0.379615i	−0.981821	−	0.189808i	$-0.939214\pi$
$97$		−	362.661i		−	0.379615i	0.981821	−	0.189808i	$-0.0607864\pi$
$98$		0			0
$99$			224.352i			0.227760i
$100$		0			0
$101$	−1490.58			−1.46850			−0.734249	−	0.678880i	$-0.762466\pi$
$101$	−1490.58			−1.46850			−0.734249	+	0.678880i	$0.762466\pi$
$102$		0			0
$103$	628.436			0.601181			0.300591	−	0.953753i	$-0.402816\pi$
$103$	628.436			0.601181			0.300591	+	0.953753i	$0.402816\pi$
$104$		0			0
$105$	648.000			0.602270
$106$		0			0
$107$	−477.454			−0.431376			−0.215688	−	0.976462i	$-0.569199\pi$
$107$	−477.454			−0.431376			−0.215688	+	0.976462i	$0.569199\pi$
$108$		0			0
$109$		−	378.207i		−	0.332345i	−0.986097	−	0.166173i	$-0.946859\pi$
$109$		−	378.207i		−	0.332345i	0.986097	−	0.166173i	$-0.0531409\pi$
$110$		0			0
$111$			881.864i			0.754079i
$112$		0			0
$113$	13.2732			0.0110499			0.00552495	−	0.999985i	$-0.498241\pi$
$113$	13.2732			0.0110499			0.00552495	+	0.999985i	$0.498241\pi$
$114$		0			0
$115$			929.651i			0.753830i
$116$		0			0
$117$	303.436	+	293.059i	0.239766	+	0.231567i
$118$		0			0
$119$			2244.10i			1.72871i
$120$		0			0
$121$	709.593			0.533128
$122$		0			0
$123$		−	751.519i		−	0.550912i
$124$		0			0
$125$			1075.36i			0.769465i
$126$		0			0
$127$	145.988			0.102003			0.0510015	−	0.998699i	$-0.483759\pi$
$127$	145.988			0.102003			0.0510015	+	0.998699i	$0.483759\pi$
$128$		0			0
$129$	−1296.44			−0.884844
$130$		0			0
$131$	−317.163			−0.211532			−0.105766	−	0.994391i	$-0.533729\pi$
$131$	−317.163			−0.211532			−0.105766	+	0.994391i	$0.533729\pi$
$132$		0			0
$133$	−249.890			−0.162919
$134$		0			0
$135$		−	348.619i		−	0.222255i
$136$		0			0
$137$		−	443.149i		−	0.276356i	−0.990407	−	0.138178i	$-0.955875\pi$
$137$		−	443.149i		−	0.276356i	0.990407	−	0.138178i	$-0.0441247\pi$
$138$		0			0
$139$	−785.018			−0.479024			−0.239512	−	0.970893i	$-0.576987\pi$
$139$	−785.018			−0.479024			−0.239512	+	0.970893i	$0.576987\pi$
$140$		0			0
$141$			479.667i			0.286491i
$142$		0			0
$143$	−811.709	+	840.452i	−0.474675	+	0.491483i
$144$		0			0
$145$		−	2661.71i		−	1.52444i
$146$		0			0
$147$	−189.436			−0.106289
$148$		0			0
$149$			135.420i			0.0744566i	0.999307	+	0.0372283i	$0.0118529\pi$
$149$			135.420i			0.0744566i	−0.999307	+	0.0372283i	$0.988147\pi$
$150$		0			0
$151$		−	2373.74i		−	1.27929i	−0.768672	−	0.639643i	$-0.779082\pi$
$151$		−	2373.74i		−	1.27929i	0.768672	−	0.639643i	$-0.220918\pi$
$152$		0			0
$153$	1207.31			0.637942
$154$		0			0
$155$	−3216.87			−1.66700
$156$		0			0
$157$	−1166.73			−0.593089			−0.296544	−	0.955019i	$-0.595834\pi$
$157$	−1166.73			−0.593089			−0.296544	+	0.955019i	$0.595834\pi$
$158$		0			0
$159$	583.744			0.291157
$160$		0			0
$161$			1204.48i			0.589603i
$162$		0			0
$163$		−	2309.19i		−	1.10963i	−0.831974	−	0.554815i	$-0.812789\pi$
$163$		−	2309.19i		−	1.10963i	0.831974	−	0.554815i	$-0.187211\pi$
$164$		0			0
$165$	965.599			0.455587
$166$		0			0
$167$		−	600.788i		−	0.278386i	−0.990265	−	0.139193i	$-0.955549\pi$
$167$		−	600.788i		−	0.278386i	0.990265	−	0.139193i	$-0.0444508\pi$
$168$		0			0
$169$	76.4186	+	2195.67i	0.0347831	+	0.999395i
$170$		0			0
$171$			134.439i			0.0601216i
$172$		0			0
$173$	−3430.36			−1.50755			−0.753773	−	0.657135i	$-0.771768\pi$
$173$	−3430.36			−1.50755			−0.753773	+	0.657135i	$0.771768\pi$
$174$		0			0
$175$		−	697.846i		−	0.301441i
$176$		0			0
$177$		−	697.942i		−	0.296387i
$178$		0			0
$179$	978.837			0.408725			0.204362	−	0.978895i	$-0.434488\pi$
$179$	978.837			0.408725			0.204362	+	0.978895i	$0.434488\pi$
$180$		0			0
$181$	3839.09			1.57656			0.788279	−	0.615318i	$-0.210972\pi$
$181$	3839.09			1.57656			0.788279	+	0.615318i	$0.210972\pi$
$182$		0			0
$183$	555.018			0.224197
$184$		0			0
$185$	3795.49			1.50838
$186$		0			0
$187$			3343.98i			1.30768i
$188$		0			0
$189$		−	451.679i		−	0.173835i
$190$		0			0
$191$	487.709			0.184761			0.0923806	−	0.995724i	$-0.470552\pi$
$191$	487.709			0.184761			0.0923806	+	0.995724i	$0.470552\pi$
$192$		0			0
$193$		−	4245.61i		−	1.58345i	−0.610878	−	0.791725i	$-0.709183\pi$
$193$		−	4245.61i		−	1.58345i	0.610878	−	0.791725i	$-0.290817\pi$
$194$		0			0
$195$	1261.31	−	1305.97i	0.463201	−	0.479603i
$196$		0			0
$197$		−	2712.71i		−	0.981079i	−0.871419	−	0.490539i	$-0.836800\pi$
$197$		−	2712.71i		−	0.981079i	0.871419	−	0.490539i	$-0.163200\pi$
$198$		0			0
$199$	3116.90			1.11031			0.555153	−	0.831748i	$-0.312660\pi$
$199$	3116.90			1.11031			0.555153	+	0.831748i	$0.312660\pi$
$200$		0			0
$201$		−	118.318i		−	0.0415199i
$202$		0			0
$203$		−	3448.58i		−	1.19233i
$204$		0			0
$205$	−3234.49			−1.10198
$206$		0			0
$207$	648.000			0.217580
$208$		0			0
$209$	−372.366			−0.123240
$210$		0			0
$211$	1051.22			0.342981			0.171491	−	0.985186i	$-0.445142\pi$
$211$	1051.22			0.342981			0.171491	+	0.985186i	$0.445142\pi$
$212$		0			0
$213$			2761.38i			0.888294i
$214$		0			0
$215$			5579.78i			1.76994i
$216$		0			0
$217$	−4167.85			−1.30384
$218$		0			0
$219$		−	1647.23i		−	0.508264i
$220$		0			0
$221$	4522.73	+	4368.06i	1.37661	+	1.32953i
$222$		0			0
$223$		−	5496.12i		−	1.65044i	−0.564814	−	0.825218i	$-0.691052\pi$
$223$		−	5496.12i		−	1.65044i	0.564814	−	0.825218i	$-0.308948\pi$
$224$		0			0
$225$	−375.436			−0.111240
$226$		0			0
$227$			921.570i			0.269457i	0.990883	+	0.134729i	$0.0430162\pi$
$227$			921.570i			0.269457i	−0.990883	+	0.134729i	$0.956984\pi$
$228$		0			0
$229$		−	192.941i		−	0.0556764i	−0.999612	−	0.0278382i	$-0.991138\pi$
$229$		−	192.941i		−	0.0556764i	0.999612	−	0.0278382i	$-0.00886232\pi$
$230$		0			0
$231$	1251.05			0.356334
$232$		0			0
$233$	−913.779			−0.256926			−0.128463	−	0.991714i	$-0.541004\pi$
$233$	−913.779			−0.256926			−0.128463	+	0.991714i	$0.541004\pi$
$234$		0			0
$235$	2064.46			0.573066
$236$		0			0
$237$	2799.42			0.767265
$238$		0			0
$239$			1976.86i			0.535032i	0.963553	+	0.267516i	$0.0862028\pi$
$239$			1976.86i			0.535032i	−0.963553	+	0.267516i	$0.913797\pi$
$240$		0			0
$241$		−	3904.45i		−	1.04360i	−0.853068	−	0.521800i	$-0.825261\pi$
$241$		−	3904.45i		−	1.04360i	0.853068	−	0.521800i	$-0.174739\pi$
$242$		0			0
$243$	−243.000			−0.0641500
$244$		0			0
$245$			815.322i			0.212608i
$246$		0			0
$247$	−486.401	+	503.624i	−0.125299	+	0.129736i
$248$		0			0
$249$		−	3286.15i		−	0.836350i
$250$		0			0
$251$	−942.035			−0.236895			−0.118448	−	0.992960i	$-0.537792\pi$
$251$	−942.035			−0.236895			−0.118448	+	0.992960i	$0.537792\pi$
$252$		0			0
$253$			1794.82i			0.446005i
$254$		0			0
$255$		−	5196.18i		−	1.27607i
$256$		0			0
$257$	812.616			0.197236			0.0986179	−	0.995125i	$-0.468558\pi$
$257$	812.616			0.197236			0.0986179	+	0.995125i	$0.468558\pi$
$258$		0			0
$259$	4917.52			1.17977
$260$		0			0
$261$	−1855.31			−0.440003
$262$		0			0
$263$	2608.29			0.611536			0.305768	−	0.952106i	$-0.401087\pi$
$263$	2608.29			0.611536			0.305768	+	0.952106i	$0.401087\pi$
$264$		0			0
$265$		−	2512.40i		−	0.582398i
$266$		0			0
$267$		−	1596.34i		−	0.365897i
$268$		0			0
$269$	−4791.02			−1.08592			−0.542962	−	0.839757i	$-0.682697\pi$
$269$	−4791.02			−1.08592			−0.542962	+	0.839757i	$0.682697\pi$
$270$		0			0
$271$		−	3663.62i		−	0.821214i	−0.911812	−	0.410607i	$-0.865317\pi$
$271$		−	3663.62i		−	0.821214i	0.911812	−	0.410607i	$-0.134683\pi$
$272$		0			0
$273$	1634.18	−	1692.05i	0.362290	−	0.375118i
$274$		0			0
$275$		−	1039.88i		−	0.228025i
$276$		0			0
$277$	624.326			0.135423			0.0677114	−	0.997705i	$-0.478430\pi$
$277$	624.326			0.135423			0.0677114	+	0.997705i	$0.478430\pi$
$278$		0			0
$279$			2242.27i			0.481152i
$280$		0			0
$281$		−	5535.12i		−	1.17508i	−0.809195	−	0.587540i	$-0.800096\pi$
$281$		−	5535.12i		−	1.17508i	0.809195	−	0.587540i	$-0.199904\pi$
$282$		0			0
$283$	175.151			0.0367903			0.0183952	−	0.999831i	$-0.494144\pi$
$283$	175.151			0.0367903			0.0183952	+	0.999831i	$0.494144\pi$
$284$		0			0
$285$	578.616			0.120261
$286$		0			0
$287$	−4190.69			−0.861911
$288$		0			0
$289$	13082.0			2.66273
$290$		0			0
$291$			1087.98i			0.219171i
$292$		0			0
$293$			7774.33i			1.55011i	0.631895	+	0.775054i	$0.282277\pi$
$293$			7774.33i			1.55011i	−0.631895	+	0.775054i	$0.717723\pi$
$294$		0			0
$295$	−3003.90			−0.592861
$296$		0			0
$297$		−	673.057i		−	0.131497i
$298$		0			0
$299$	2427.49	+	2344.47i	0.469516	+	0.453459i
$300$		0			0
$301$			7229.30i			1.38435i
$302$		0			0
$303$	4471.74			0.847838
$304$		0			0
$305$		−	2388.76i		−	0.448459i
$306$		0			0
$307$			8022.85i			1.49149i	0.666230	+	0.745746i	$0.267907\pi$
$307$			8022.85i			1.49149i	−0.666230	+	0.745746i	$0.732093\pi$
$308$		0			0
$309$	−1885.31			−0.347092
$310$		0			0
$311$	−9264.87			−1.68927			−0.844635	−	0.535343i	$-0.820182\pi$
$311$	−9264.87			−1.68927			−0.844635	+	0.535343i	$0.820182\pi$
$312$		0			0
$313$	7423.57			1.34059			0.670296	−	0.742094i	$-0.266167\pi$
$313$	7423.57			1.34059			0.670296	+	0.742094i	$0.266167\pi$
$314$		0			0
$315$	−1944.00			−0.347721
$316$		0			0
$317$			2641.04i			0.467935i	0.972244	+	0.233968i	$0.0751710\pi$
$317$			2641.04i			0.467935i	−0.972244	+	0.233968i	$0.924829\pi$
$318$		0			0
$319$		−	5138.80i		−	0.901936i
$320$		0			0
$321$	1432.36			0.249055
$322$		0			0
$323$			2003.82i			0.345187i
$324$		0			0
$325$	−1406.43	−	1358.33i	−0.240045	−	0.231836i
$326$		0			0
$327$			1134.62i			0.191880i
$328$		0			0
$329$	2674.76			0.448220
$330$		0			0
$331$		−	10779.5i		−	1.79001i	−0.446052	−	0.895007i	$-0.647170\pi$
$331$		−	10779.5i		−	1.79001i	0.446052	−	0.895007i	$-0.352830\pi$
$332$		0			0
$333$		−	2645.59i		−	0.435368i
$334$		0			0
$335$	−509.233			−0.0830518
$336$		0			0
$337$	313.465			0.0506693			0.0253346	−	0.999679i	$-0.491935\pi$
$337$	313.465			0.0506693			0.0253346	+	0.999679i	$0.491935\pi$
$338$		0			0
$339$	−39.8196			−0.00637966
$340$		0			0
$341$	−6210.61			−0.986286
$342$		0			0
$343$			6794.35i			1.06956i
$344$		0			0
$345$		−	2788.95i		−	0.435224i
$346$		0			0
$347$	2849.23			0.440792			0.220396	−	0.975410i	$-0.429265\pi$
$347$	2849.23			0.440792			0.220396	+	0.975410i	$0.429265\pi$
$348$		0			0
$349$		−	6466.94i		−	0.991883i	−0.868356	−	0.495941i	$-0.834823\pi$
$349$		−	6466.94i		−	0.991883i	0.868356	−	0.495941i	$-0.165177\pi$
$350$		0			0
$351$	−910.308	−	879.177i	−0.138429	−	0.133695i
$352$		0			0
$353$		−	2773.10i		−	0.418122i	−0.977903	−	0.209061i	$-0.932959\pi$
$353$		−	2773.10i		−	0.418122i	0.977903	−	0.209061i	$-0.0670408\pi$
$354$		0			0
$355$	11884.8			1.77685
$356$		0			0
$357$		−	6732.30i		−	0.998070i
$358$		0			0
$359$		−	1467.11i		−	0.215685i	−0.994168	−	0.107843i	$-0.965606\pi$
$359$		−	1467.11i		−	0.215685i	0.994168	−	0.107843i	$-0.0343942\pi$
$360$		0			0
$361$	6635.87			0.967469
$362$		0			0
$363$	−2128.78			−0.307801
$364$		0			0
$365$	−7089.59			−1.01667
$366$		0			0
$367$	−4648.22			−0.661130			−0.330565	−	0.943783i	$-0.607239\pi$
$367$	−4648.22			−0.661130			−0.330565	+	0.943783i	$0.607239\pi$
$368$		0			0
$369$			2254.56i			0.318069i
$370$		0			0
$371$		−	3255.12i		−	0.455519i
$372$		0			0
$373$	1763.72			0.244831			0.122416	−	0.992479i	$-0.460936\pi$
$373$	1763.72			0.244831			0.122416	+	0.992479i	$0.460936\pi$
$374$		0			0
$375$		−	3226.08i		−	0.444251i
$376$		0			0
$377$	−6950.22	−	6712.53i	−0.949481	−	0.917010i
$378$		0			0
$379$		−	1930.47i		−	0.261640i	−0.991406	−	0.130820i	$-0.958239\pi$
$379$		−	1930.47i		−	0.261640i	0.991406	−	0.130820i	$-0.0417611\pi$
$380$		0			0
$381$	−437.965			−0.0588914
$382$		0			0
$383$			8845.93i			1.18017i	0.807340	+	0.590086i	$0.200906\pi$
$383$			8845.93i			1.18017i	−0.807340	+	0.590086i	$0.799094\pi$
$384$		0			0
$385$		−	5384.46i		−	0.712772i
$386$		0			0
$387$	3889.31			0.510865
$388$		0			0
$389$	1598.08			0.208292			0.104146	−	0.994562i	$-0.466789\pi$
$389$	1598.08			0.208292			0.104146	+	0.994562i	$0.466789\pi$
$390$		0			0
$391$	9658.47			1.24923
$392$		0			0
$393$	951.489			0.122128
$394$		0			0
$395$		−	12048.5i		−	1.53475i
$396$		0			0
$397$		−	3578.82i		−	0.452433i	−0.974077	−	0.226217i	$-0.927364\pi$
$397$		−	3578.82i		−	0.452433i	0.974077	−	0.226217i	$-0.0726357\pi$
$398$		0			0
$399$	749.669			0.0940611
$400$		0			0
$401$			3485.99i			0.434120i	0.976158	+	0.217060i	$0.0696467\pi$
$401$			3485.99i			0.434120i	−0.976158	+	0.217060i	$0.930353\pi$
$402$		0			0
$403$	−8112.58	+	8399.84i	−1.00277	+	1.03828i
$404$		0			0
$405$			1045.86i			0.128319i
$406$		0			0
$407$	7327.71			0.892435
$408$		0			0
$409$			14709.1i			1.77828i	0.457637	+	0.889139i	$0.348696\pi$
$409$			14709.1i			1.77828i	−0.457637	+	0.889139i	$0.651304\pi$
$410$		0			0
$411$			1329.45i			0.159554i
$412$		0			0
$413$	−3891.92			−0.463702
$414$		0			0
$415$	−14143.4			−1.67294
$416$		0			0
$417$	2355.05			0.276565
$418$		0			0
$419$	−3709.01			−0.432451			−0.216226	−	0.976343i	$-0.569375\pi$
$419$	−3709.01			−0.432451			−0.216226	+	0.976343i	$0.569375\pi$
$420$		0			0
$421$			794.029i			0.0919207i	0.998943	+	0.0459603i	$0.0146348\pi$
$421$			794.029i			0.0919207i	−0.998943	+	0.0459603i	$0.985365\pi$
$422$		0			0
$423$		−	1439.00i		−	0.165406i
$424$		0			0
$425$	−5595.89			−0.638684
$426$		0			0
$427$		−	3094.94i		−	0.350760i
$428$		0			0
$429$	2435.13	−	2521.35i	0.274054	−	0.283758i
$430$		0			0
$431$		−	2891.52i		−	0.323155i	−0.986860	−	0.161577i	$-0.948342\pi$
$431$		−	2891.52i		−	0.323155i	0.986860	−	0.161577i	$-0.0516582\pi$
$432$		0			0
$433$	−5560.94			−0.617186			−0.308593	−	0.951194i	$-0.599858\pi$
$433$	−5560.94			−0.617186			−0.308593	+	0.951194i	$0.599858\pi$
$434$		0			0
$435$			7985.14i			0.880133i
$436$		0			0
$437$			1075.51i			0.117731i
$438$		0			0
$439$	15127.2			1.64460			0.822302	−	0.569051i	$-0.192689\pi$
$439$	15127.2			1.64460			0.822302	+	0.569051i	$0.192689\pi$
$440$		0			0
$441$	568.308			0.0613658
$442$		0			0
$443$	−2357.89			−0.252883			−0.126441	−	0.991974i	$-0.540356\pi$
$443$	−2357.89			−0.252883			−0.126441	+	0.991974i	$0.540356\pi$
$444$		0			0
$445$	−6870.56			−0.731901
$446$		0			0
$447$		−	406.260i		−	0.0429876i
$448$		0			0
$449$			7165.06i			0.753096i	0.926397	+	0.376548i	$0.122889\pi$
$449$			7165.06i			0.753096i	−0.926397	+	0.376548i	$0.877111\pi$
$450$		0			0
$451$	−6244.63			−0.651992
$452$		0			0
$453$			7121.22i			0.738596i
$454$		0			0
$455$	−7282.47	−	7033.41i	−0.750346	−	0.724685i
$456$		0			0
$457$			8020.96i			0.821017i	0.911857	+	0.410508i	$0.134649\pi$
$457$			8020.96i			0.821017i	−0.911857	+	0.410508i	$0.865351\pi$
$458$		0			0
$459$	−3621.92			−0.368316
$460$		0			0
$461$		−	4146.59i		−	0.418928i	−0.977816	−	0.209464i	$-0.932828\pi$
$461$		−	4146.59i		−	0.418928i	0.977816	−	0.209464i	$-0.0671720\pi$
$462$		0			0
$463$			7118.21i			0.714495i	0.934010	+	0.357248i	$0.116285\pi$
$463$			7118.21i			0.714495i	−0.934010	+	0.357248i	$0.883715\pi$
$464$		0			0
$465$	9650.62			0.962444
$466$		0			0
$467$	−2128.22			−0.210883			−0.105441	−	0.994426i	$-0.533626\pi$
$467$	−2128.22			−0.210883			−0.105441	+	0.994426i	$0.533626\pi$
$468$		0			0
$469$	−659.774			−0.0649585
$470$		0			0
$471$	3500.18			0.342420
$472$		0			0
$473$			10772.5i			1.04719i
$474$		0			0
$475$		−	623.125i		−	0.0601915i
$476$		0			0
$477$	−1751.23			−0.168099
$478$		0			0
$479$		−	3715.30i		−	0.354397i	−0.984175	−	0.177199i	$-0.943296\pi$
$479$		−	3715.30i		−	0.354397i	0.984175	−	0.177199i	$-0.0567035\pi$
$480$		0			0
$481$	9571.78	−	9910.71i	0.907350	−	0.939479i
$482$		0			0
$483$		−	3613.43i		−	0.340408i
$484$		0			0
$485$	4682.62			0.438405
$486$		0			0
$487$			8139.28i			0.757343i	0.925531	+	0.378671i	$0.123619\pi$
$487$			8139.28i			0.757343i	−0.925531	+	0.378671i	$0.876381\pi$
$488$		0			0
$489$			6927.57i			0.640645i
$490$		0			0
$491$	18081.7			1.66194			0.830972	−	0.556315i	$-0.187785\pi$
$491$	18081.7			1.66194			0.830972	+	0.556315i	$0.187785\pi$
$492$		0			0
$493$	−27653.4			−2.52626
$494$		0			0
$495$	−2896.80			−0.263033
$496$		0			0
$497$	15398.3			1.38975
$498$		0			0
$499$		−	11031.5i		−	0.989659i	−0.868990	−	0.494829i	$-0.835231\pi$
$499$		−	11031.5i		−	0.989659i	0.868990	−	0.494829i	$-0.164769\pi$
$500$		0			0
$501$			1802.37i			0.160726i
$502$		0			0
$503$	−8016.14			−0.710581			−0.355290	−	0.934756i	$-0.615618\pi$
$503$	−8016.14			−0.710581			−0.355290	+	0.934756i	$0.615618\pi$
$504$		0			0
$505$		−	19246.1i		−	1.69592i
$506$		0			0
$507$	−229.256	−	6587.01i	−0.0200821	−	0.577001i
$508$		0			0
$509$			20173.9i			1.75676i	0.477959	+	0.878382i	$0.341377\pi$
$509$			20173.9i			1.75676i	−0.477959	+	0.878382i	$0.658623\pi$
$510$		0			0
$511$	−9185.44			−0.795186
$512$		0			0
$513$		−	403.316i		−	0.0347112i
$514$		0			0
$515$			8114.25i			0.694285i
$516$		0			0
$517$	3985.72			0.339056
$518$		0			0
$519$	10291.1			0.870382
$520$		0			0
$521$	9746.95			0.819619			0.409810	−	0.912171i	$-0.365595\pi$
$521$	9746.95			0.819619			0.409810	+	0.912171i	$0.365595\pi$
$522$		0			0
$523$	18929.3			1.58264			0.791320	−	0.611402i	$-0.209394\pi$
$523$	18929.3			1.58264			0.791320	+	0.611402i	$0.209394\pi$
$524$		0			0
$525$			2093.54i			0.174037i
$526$		0			0
$527$			33421.2i			2.76252i
$528$		0			0
$529$	−6983.00			−0.573929
$530$		0			0
$531$			2093.83i			0.171119i
$532$		0			0
$533$	−8157.02	+	8445.85i	−0.662889	+	0.686361i
$534$		0			0
$535$		−	6164.80i		−	0.498182i
$536$		0			0
$537$	−2936.51			−0.235977
$538$		0			0
$539$			1574.09i			0.125790i
$540$		0			0
$541$			11366.8i			0.903321i	0.892190	+	0.451661i	$0.149168\pi$
$541$			11366.8i			0.903321i	−0.892190	+	0.451661i	$0.850832\pi$
$542$		0			0
$543$	−11517.3			−0.910227
$544$		0			0
$545$	4883.34			0.383815
$546$		0			0
$547$	−17495.4			−1.36755			−0.683775	−	0.729693i	$-0.739663\pi$
$547$	−17495.4			−1.36755			−0.683775	+	0.729693i	$0.739663\pi$
$548$		0			0
$549$	−1665.05			−0.129440
$550$		0			0
$551$		−	3079.33i		−	0.238083i
$552$		0			0
$553$		−	15610.4i		−	1.20040i
$554$		0			0
$555$	−11386.5			−0.870862
$556$		0			0
$557$			11873.1i			0.903192i	0.892223	+	0.451596i	$0.149145\pi$
$557$			11873.1i			0.903192i	−0.892223	+	0.451596i	$0.850855\pi$
$558$		0			0
$559$	14569.8	+	14071.6i	1.10239	+	1.06469i
$560$		0			0
$561$		−	10031.9i		−	0.754989i
$562$		0			0
$563$	−2829.31			−0.211796			−0.105898	−	0.994377i	$-0.533772\pi$
$563$	−2829.31			−0.211796			−0.105898	+	0.994377i	$0.533772\pi$
$564$		0			0
$565$			171.381i			0.0127612i
$566$		0			0
$567$			1355.04i			0.100364i
$568$		0			0
$569$	−16136.8			−1.18891			−0.594453	−	0.804130i	$-0.702632\pi$
$569$	−16136.8			−1.18891			−0.594453	+	0.804130i	$0.702632\pi$
$570$		0			0
$571$	−17840.5			−1.30754			−0.653769	−	0.756695i	$-0.726813\pi$
$571$	−17840.5			−1.30754			−0.653769	+	0.756695i	$0.726813\pi$
$572$		0			0
$573$	−1463.13			−0.106672
$574$		0			0
$575$	−3003.49			−0.217833
$576$		0			0
$577$			8516.17i			0.614442i	0.951638	+	0.307221i	$0.0993990\pi$
$577$			8516.17i			0.614442i	−0.951638	+	0.307221i	$0.900601\pi$
$578$		0			0
$579$			12736.8i			0.914205i
$580$		0			0
$581$	−18324.5			−1.30848
$582$		0			0
$583$		−	4850.53i		−	0.344577i
$584$		0			0
$585$	−3783.92	+	3917.91i	−0.267429	+	0.276899i
$586$		0			0
$587$			20688.3i			1.45468i	0.686277	+	0.727340i	$0.259244\pi$
$587$			20688.3i			1.45468i	−0.686277	+	0.727340i	$0.740756\pi$
$588$		0			0
$589$	−3721.59			−0.260349
$590$		0			0
$591$			8138.13i			0.566426i
$592$		0			0
$593$		−	11435.9i		−	0.791933i	−0.918265	−	0.395966i	$-0.870410\pi$
$593$		−	11435.9i		−	0.791933i	0.918265	−	0.395966i	$-0.129590\pi$
$594$		0			0
$595$	−28975.4			−1.99643
$596$		0			0
$597$	−9350.69			−0.641035
$598$		0			0
$599$	−1260.80			−0.0860016			−0.0430008	−	0.999075i	$-0.513692\pi$
$599$	−1260.80			−0.0860016			−0.0430008	+	0.999075i	$0.513692\pi$
$600$		0			0
$601$	6261.10			0.424951			0.212476	−	0.977166i	$-0.431847\pi$
$601$	6261.10			0.424951			0.212476	+	0.977166i	$0.431847\pi$
$602$		0			0
$603$			354.954i			0.0239715i
$604$		0			0
$605$			9162.14i			0.615692i
$606$		0			0
$607$	−3230.33			−0.216005			−0.108003	−	0.994151i	$-0.534445\pi$
$607$	−3230.33			−0.216005			−0.108003	+	0.994151i	$0.534445\pi$
$608$		0			0
$609$			10345.7i			0.688391i
$610$		0			0
$611$	5206.33	−	5390.68i	0.344722	−	0.356929i
$612$		0			0
$613$			14868.5i			0.979660i	0.871818	+	0.489830i	$0.162941\pi$
$613$			14868.5i			0.979660i	−0.871818	+	0.489830i	$0.837059\pi$
$614$		0			0
$615$	9703.48			0.636231
$616$		0			0
$617$		−	19952.8i		−	1.30190i	−0.759121	−	0.650949i	$-0.774371\pi$
$617$		−	19952.8i		−	1.30190i	0.759121	−	0.650949i	$-0.225629\pi$
$618$		0			0
$619$			8316.48i			0.540012i	0.962859	+	0.270006i	$0.0870257\pi$
$619$			8316.48i			0.540012i	−0.962859	+	0.270006i	$0.912974\pi$
$620$		0			0
$621$	−1944.00			−0.125620
$622$		0			0
$623$	−8901.66			−0.572452
$624$		0			0
$625$	−19099.2			−1.22235
$626$		0			0
$627$	1117.10			0.0711525
$628$		0			0
$629$		−	39432.6i		−	2.49965i
$630$		0			0
$631$		−	12605.9i		−	0.795299i	−0.917537	−	0.397649i	$-0.869826\pi$
$631$		−	12605.9i		−	0.795299i	0.917537	−	0.397649i	$-0.130174\pi$
$632$		0			0
$633$	−3153.66			−0.198020
$634$		0			0
$635$			1884.98i			0.117800i
$636$		0			0
$637$	2128.95	+	2056.15i	0.132421	+	0.127892i
$638$		0			0
$639$		−	8284.14i		−	0.512857i
$640$		0			0
$641$	9224.04			0.568374			0.284187	−	0.958769i	$-0.408276\pi$
$641$	9224.04			0.568374			0.284187	+	0.958769i	$0.408276\pi$
$642$		0			0
$643$		−	4439.16i		−	0.272260i	−0.990691	−	0.136130i	$-0.956533\pi$
$643$		−	4439.16i		−	0.272260i	0.990691	−	0.136130i	$-0.0434665\pi$
$644$		0			0
$645$		−	16739.4i		−	1.02188i
$646$		0			0
$647$	−9601.15			−0.583401			−0.291700	−	0.956510i	$-0.594221\pi$
$647$	−9601.15			−0.583401			−0.291700	+	0.956510i	$0.594221\pi$
$648$		0			0
$649$	−5799.44			−0.350767
$650$		0			0
$651$	12503.6			0.752770
$652$		0			0
$653$	27112.8			1.62482			0.812410	−	0.583087i	$-0.198155\pi$
$653$	27112.8			1.62482			0.812410	+	0.583087i	$0.198155\pi$
$654$		0			0
$655$		−	4095.15i		−	0.244291i
$656$		0			0
$657$			4941.70i			0.293446i
$658$		0			0
$659$	−5587.26			−0.330271			−0.165136	−	0.986271i	$-0.552806\pi$
$659$	−5587.26			−0.330271			−0.165136	+	0.986271i	$0.552806\pi$
$660$		0			0
$661$			3060.13i			0.180069i	0.995939	+	0.0900343i	$0.0286977\pi$
$661$			3060.13i			0.180069i	−0.995939	+	0.0900343i	$0.971302\pi$
$662$		0			0
$663$	−13568.2	−	13104.2i	−0.794788	−	0.767607i
$664$		0			0
$665$		−	3226.53i		−	0.188150i
$666$		0			0
$667$	−14842.5			−0.861623
$668$		0			0
$669$			16488.4i			0.952880i
$670$		0			0
$671$		−	4611.83i		−	0.265332i
$672$		0			0
$673$	−4121.55			−0.236069			−0.118034	−	0.993010i	$-0.537659\pi$
$673$	−4121.55			−0.236069			−0.118034	+	0.993010i	$0.537659\pi$
$674$		0			0
$675$	1126.31			0.0642246
$676$		0			0
$677$	22889.5			1.29943			0.649715	−	0.760178i	$-0.274888\pi$
$677$	22889.5			1.29943			0.649715	+	0.760178i	$0.274888\pi$
$678$		0			0
$679$	6066.91			0.342896
$680$		0			0
$681$		−	2764.71i		−	0.155571i
$682$		0			0
$683$		−	19297.9i		−	1.08113i	−0.841301	−	0.540566i	$-0.818210\pi$
$683$		−	19297.9i		−	1.08113i	0.841301	−	0.540566i	$-0.181790\pi$
$684$		0			0
$685$	5721.87			0.319155
$686$		0			0
$687$			578.823i			0.0321448i
$688$		0			0
$689$	−6560.34	−	6335.98i	−0.362742	−	0.350336i
$690$		0			0
$691$			30317.8i			1.66910i	0.550935	+	0.834548i	$0.314271\pi$
$691$			30317.8i			1.66910i	−0.550935	+	0.834548i	$0.685729\pi$
$692$		0			0
$693$	−3753.16			−0.205730
$694$		0			0
$695$		−	10136.0i		−	0.553210i
$696$		0			0
$697$			33604.3i			1.82619i
$698$		0			0
$699$	2741.34			0.148336
$700$		0			0
$701$	−9606.16			−0.517574			−0.258787	−	0.965934i	$-0.583323\pi$
$701$	−9606.16			−0.517574			−0.258787	+	0.965934i	$0.583323\pi$
$702$		0			0
$703$	4390.99			0.235575
$704$		0			0
$705$	−6193.38			−0.330860
$706$		0			0
$707$		−	24935.7i		−	1.32646i
$708$		0			0
$709$			23398.8i			1.23944i	0.784825	+	0.619718i	$0.212753\pi$
$709$			23398.8i			1.23944i	−0.784825	+	0.619718i	$0.787247\pi$
$710$		0			0
$711$	−8398.26			−0.442981
$712$		0			0
$713$			17938.2i			0.942203i
$714$		0			0
$715$	−10851.8	−	10480.6i	−0.567598	−	0.548187i
$716$		0			0
$717$		−	5930.59i		−	0.308901i
$718$		0			0
$719$	−23588.7			−1.22352			−0.611758	−	0.791045i	$-0.709537\pi$
$719$	−23588.7			−1.22352			−0.611758	+	0.791045i	$0.709537\pi$
$720$		0			0
$721$			10513.0i			0.543031i
$722$		0			0
$723$			11713.3i			0.602522i
$724$		0			0
$725$	8599.38			0.440514
$726$		0			0
$727$	15733.5			0.802644			0.401322	−	0.915937i	$-0.368551\pi$
$727$	15733.5			0.802644			0.401322	+	0.915937i	$0.368551\pi$
$728$		0			0
$729$	729.000			0.0370370
$730$		0			0
$731$	57970.3			2.93312
$732$		0			0
$733$		−	17297.1i		−	0.871598i	−0.900044	−	0.435799i	$-0.856466\pi$
$733$		−	17297.1i		−	0.871598i	0.900044	−	0.435799i	$-0.143534\pi$
$734$		0			0
$735$		−	2445.96i		−	0.122749i
$736$		0			0
$737$	−983.144			−0.0491378
$738$		0			0
$739$			38749.2i			1.92884i	0.264377	+	0.964419i	$0.414834\pi$
$739$			38749.2i			1.92884i	−0.264377	+	0.964419i	$0.585166\pi$
$740$		0			0
$741$	1459.20	−	1510.87i	0.0723417	−	0.0749033i
$742$		0			0
$743$			3139.76i			0.155029i	0.996991	+	0.0775144i	$0.0246984\pi$
$743$			3139.76i			0.155029i	−0.996991	+	0.0775144i	$0.975302\pi$
$744$		0			0
$745$	−1748.52			−0.0859876
$746$		0			0
$747$			9858.45i			0.482867i
$748$		0			0
$749$		−	7987.25i		−	0.389650i
$750$		0			0
$751$	40628.6			1.97411			0.987055	−	0.160380i	$-0.0512721\pi$
$751$	40628.6			1.97411			0.987055	+	0.160380i	$0.0512721\pi$
$752$		0			0
$753$	2826.11			0.136772
$754$		0			0
$755$	30649.3			1.47741
$756$		0			0
$757$	24004.9			1.15254			0.576271	−	0.817259i	$-0.304507\pi$
$757$	24004.9			1.15254			0.576271	+	0.817259i	$0.304507\pi$
$758$		0			0
$759$		−	5384.46i		−	0.257501i
$760$		0			0
$761$		−	29540.1i		−	1.40713i	−0.710630	−	0.703566i	$-0.751590\pi$
$761$		−	29540.1i		−	1.40713i	0.710630	−	0.703566i	$-0.248410\pi$
$762$		0			0
$763$	6326.97			0.300199
$764$		0			0
$765$			15588.5i			0.736739i
$766$		0			0
$767$	−7575.49	+	7843.73i	−0.356630	+	0.369258i
$768$		0			0
$769$			7585.63i			0.355715i	0.984056	+	0.177857i	$0.0569166\pi$
$769$			7585.63i			0.355715i	−0.984056	+	0.177857i	$0.943083\pi$
$770$		0			0
$771$	−2437.85			−0.113874
$772$		0			0
$773$			3284.29i			0.152817i	0.997077	+	0.0764086i	$0.0243453\pi$
$773$			3284.29i			0.152817i	−0.997077	+	0.0764086i	$0.975655\pi$
$774$		0			0
$775$		−	10393.0i		−	0.481712i
$776$		0			0
$777$	−14752.6			−0.681140
$778$		0			0
$779$	−3741.98			−0.172106
$780$		0			0
$781$	22945.3			1.05128
$782$		0			0
$783$	5565.92			0.254036
$784$		0			0
$785$		−	15064.6i		−	0.684939i
$786$		0			0
$787$		−	41624.1i		−	1.88531i	−0.333772	−	0.942654i	$-0.608321\pi$
$787$		−	41624.1i		−	1.88531i	0.333772	−	0.942654i	$-0.391679\pi$
$788$		0			0
$789$	−7824.87			−0.353071
$790$		0			0
$791$			222.046i			0.00998108i
$792$		0			0
$793$	−6237.49	−	6024.18i	−0.279319	−	0.269767i
$794$		0			0
$795$			7537.20i			0.336248i
$796$		0			0
$797$	30333.3			1.34813			0.674066	−	0.738671i	$-0.264546\pi$
$797$	30333.3			1.34813			0.674066	+	0.738671i	$0.264546\pi$
$798$		0			0
$799$		−	21448.4i		−	0.949674i
$800$		0			0
$801$			4789.03i			0.211251i
$802$		0			0
$803$	−13687.4			−0.601518
$804$		0			0
$805$	−15552.0			−0.680914
$806$		0			0
$807$	14373.1			0.626958
$808$		0			0
$809$	−24853.9			−1.08012			−0.540060	−	0.841627i	$-0.681598\pi$
$809$	−24853.9			−1.08012			−0.540060	+	0.841627i	$0.681598\pi$
$810$		0			0
$811$			17383.5i			0.752674i	0.926483	+	0.376337i	$0.122816\pi$
$811$			17383.5i			0.752674i	−0.926483	+	0.376337i	$0.877184\pi$
$812$		0			0
$813$			10990.9i			0.474128i
$814$		0			0
$815$	29815.8			1.28148
$816$		0			0
$817$			6455.23i			0.276426i
$818$		0			0
$819$	−4902.54	+	5076.14i	−0.209168	+	0.216575i
$820$		0			0
$821$		−	31169.4i		−	1.32499i	−0.749065	−	0.662496i	$-0.769497\pi$
$821$		−	31169.4i		−	1.32499i	0.749065	−	0.662496i	$-0.230503\pi$
$822$		0			0
$823$	5512.79			0.233492			0.116746	−	0.993162i	$-0.462754\pi$
$823$	5512.79			0.233492			0.116746	+	0.993162i	$0.462754\pi$
$824$		0			0
$825$			3119.63i			0.131650i
$826$		0			0
$827$		−	13335.8i		−	0.560738i	−0.959892	−	0.280369i	$-0.909543\pi$
$827$		−	13335.8i		−	0.560738i	0.959892	−	0.280369i	$-0.0904568\pi$
$828$		0			0
$829$	28338.5			1.18726			0.593628	−	0.804739i	$-0.297695\pi$
$829$	28338.5			1.18726			0.593628	+	0.804739i	$0.297695\pi$
$830$		0			0
$831$	−1872.98			−0.0781864
$832$		0			0
$833$	8470.66			0.352330
$834$		0			0
$835$	7757.27			0.321499
$836$		0			0
$837$		−	6726.82i		−	0.277793i
$838$		0			0
$839$		−	27149.5i		−	1.11717i	−0.829447	−	0.558585i	$-0.811344\pi$
$839$		−	27149.5i		−	1.11717i	0.829447	−	0.558585i	$-0.188656\pi$
$840$		0			0
$841$	18106.9			0.742421
$842$		0			0
$843$			16605.4i			0.678433i
$844$		0			0
$845$	−28350.1	+	986.703i	−1.15417	+	0.0401700i
$846$		0			0
$847$			11870.7i			0.481560i
$848$		0			0
$849$	−525.454			−0.0212409
$850$		0			0
$851$		−	21164.7i		−	0.852547i
$852$		0			0
$853$		−	7978.22i		−	0.320245i	−0.987097	−	0.160123i	$-0.948811\pi$
$853$		−	7978.22i		−	0.320245i	0.987097	−	0.160123i	$-0.0511889\pi$
$854$		0			0
$855$	−1735.85			−0.0694325
$856$		0			0
$857$	13614.4			0.542657			0.271329	−	0.962487i	$-0.412537\pi$
$857$	13614.4			0.542657			0.271329	+	0.962487i	$0.412537\pi$
$858$		0			0
$859$	35007.7			1.39051			0.695255	−	0.718763i	$-0.255291\pi$
$859$	35007.7			1.39051			0.695255	+	0.718763i	$0.255291\pi$
$860$		0			0
$861$	12572.1			0.497624
$862$		0			0
$863$			24461.5i			0.964867i	0.875933	+	0.482434i	$0.160247\pi$
$863$			24461.5i			0.964867i	−0.875933	+	0.482434i	$0.839753\pi$
$864$		0			0
$865$		−	44292.2i		−	1.74102i
$866$		0			0
$867$	−39245.9			−1.53733
$868$		0			0
$869$		−	23261.3i		−	0.908040i
$870$		0			0
$871$	−1284.23	+	1329.70i	−0.0499591	+	0.0517281i
$872$		0			0
$873$		−	3263.95i		−	0.126538i
$874$		0			0
$875$	−17989.5			−0.695037
$876$		0			0
$877$		−	43121.6i		−	1.66034i	−0.557514	−	0.830168i	$-0.688245\pi$
$877$		−	43121.6i		−	1.66034i	0.557514	−	0.830168i	$-0.311755\pi$
$878$		0			0
$879$		−	23323.0i		−	0.894955i
$880$		0			0
$881$	40824.5			1.56120			0.780598	−	0.625034i	$-0.214915\pi$
$881$	40824.5			1.56120			0.780598	+	0.625034i	$0.214915\pi$
$882$		0			0
$883$	8262.14			0.314885			0.157442	−	0.987528i	$-0.449675\pi$
$883$	8262.14			0.314885			0.157442	+	0.987528i	$0.449675\pi$
$884$		0			0
$885$	9011.70			0.342288
$886$		0			0
$887$	40858.6			1.54667			0.773336	−	0.633997i	$-0.218587\pi$
$887$	40858.6			1.54667			0.773336	+	0.633997i	$0.218587\pi$
$888$		0			0
$889$			2442.22i			0.0921365i
$890$		0			0
$891$			2019.17i			0.0759201i
$892$		0			0
$893$	2388.37			0.0895001
$894$		0			0
$895$			12638.6i			0.472023i
$896$		0			0
$897$	−7282.47	−	7033.41i	−0.271075	−	0.261805i
$898$		0			0
$899$		−	51359.4i		−	1.90537i
$900$		0			0
$901$	−26102.2			−0.965139
$902$		0			0
$903$		−	21687.9i		−	0.799256i
$904$		0			0
$905$			49569.6i			1.82072i
$906$		0			0
$907$	−32729.3			−1.19819			−0.599095	−	0.800678i	$-0.704473\pi$
$907$	−32729.3			−1.19819			−0.599095	+	0.800678i	$0.704473\pi$
$908$		0			0
$909$	−13415.2			−0.489500
$910$		0			0
$911$	−11065.9			−0.402447			−0.201223	−	0.979545i	$-0.564492\pi$
$911$	−11065.9			−0.402447			−0.201223	+	0.979545i	$0.564492\pi$
$912$		0			0
$913$	−27305.7			−0.989801
$914$		0			0
$915$			7166.29i			0.258918i
$916$		0			0
$917$		−	5305.77i		−	0.191071i
$918$		0			0
$919$	−50682.2			−1.81921			−0.909604	−	0.415477i	$-0.863615\pi$
$919$	−50682.2			−1.81921			−0.909604	+	0.415477i	$0.863615\pi$
$920$		0			0
$921$		−	24068.5i		−	0.861114i
$922$		0			0
$923$	29972.1	−	31033.4i	1.06885	−	1.10669i
$924$		0			0
$925$			12262.3i			0.435874i
$926$		0			0
$927$	5655.92			0.200394
$928$		0			0
$929$			41045.5i			1.44958i	0.688970	+	0.724790i	$0.258063\pi$
$929$			41045.5i			1.44958i	−0.688970	+	0.724790i	$0.741937\pi$
$930$		0			0
$931$			943.243i			0.0332047i
$932$		0			0
$933$	27794.6			0.975300
$934$		0			0
$935$	−43176.9			−1.51020
$936$		0			0
$937$	−788.985			−0.0275080			−0.0137540	−	0.999905i	$-0.504378\pi$
$937$	−788.985			−0.0275080			−0.0137540	+	0.999905i	$0.504378\pi$
$938$		0			0
$939$	−22270.7			−0.773991
$940$		0			0
$941$		−	25676.2i		−	0.889499i	−0.895655	−	0.444750i	$-0.853293\pi$
$941$		−	25676.2i		−	0.889499i	0.895655	−	0.444750i	$-0.146707\pi$
$942$		0			0
$943$			18036.5i			0.622851i
$944$		0			0
$945$	5832.00			0.200757
$946$		0			0
$947$		−	679.352i		−	0.0233115i	−0.999932	−	0.0116557i	$-0.996290\pi$
$947$		−	679.352i		−	0.0233115i	0.999932	−	0.0116557i	$-0.00371022\pi$
$948$		0			0
$949$	−17879.1	+	18512.2i	−0.611571	+	0.633227i
$950$		0			0
$951$		−	7923.12i		−	0.270163i
$952$		0			0
$953$	20238.4			0.687917			0.343958	−	0.938985i	$-0.388232\pi$
$953$	20238.4			0.687917			0.343958	+	0.938985i	$0.388232\pi$
$954$		0			0
$955$			6297.21i			0.213375i
$956$		0			0
$957$			15416.4i			0.520733i
$958$		0			0
$959$	7413.38			0.249625
$960$		0			0
$961$	−32280.6			−1.08357
$962$		0			0
$963$	−4297.08			−0.143792
$964$		0			0
$965$	54818.6			1.82868
$966$		0			0
$967$			6161.63i			0.204906i	0.994738	+	0.102453i	$0.0326692\pi$
$967$			6161.63i			0.204906i	−0.994738	+	0.102453i	$0.967331\pi$
$968$		0			0
$969$		−	6011.45i		−	0.199294i
$970$		0			0
$971$	48569.5			1.60522			0.802610	−	0.596504i	$-0.203444\pi$
$971$	48569.5			1.60522			0.802610	+	0.596504i	$0.203444\pi$
$972$		0			0
$973$		−	13132.4i		−	0.432689i
$974$		0			0
$975$	4219.29	+	4075.00i	0.138590	+	0.133851i
$976$		0			0
$977$		−	44595.8i		−	1.46033i	−0.683269	−	0.730167i	$-0.739442\pi$
$977$		−	44595.8i		−	1.46033i	0.683269	−	0.730167i	$-0.260558\pi$
$978$		0			0
$979$	−13264.6			−0.433031
$980$		0			0
$981$		−	3403.86i		−	0.110782i
$982$		0			0
$983$		−	29599.0i		−	0.960389i	−0.877162	−	0.480195i	$-0.840566\pi$
$983$		−	29599.0i		−	0.960389i	0.877162	−	0.480195i	$-0.159434\pi$
$984$		0			0
$985$	35026.0			1.13302
$986$		0			0
$987$	−8024.29			−0.258780
$988$		0			0
$989$	31114.5			1.00039
$990$		0			0
$991$	−5718.45			−0.183302			−0.0916511	−	0.995791i	$-0.529214\pi$
$991$	−5718.45			−0.183302			−0.0916511	+	0.995791i	$0.529214\pi$
$992$		0			0
$993$			32338.5i			1.03347i
$994$		0			0
$995$			40244.8i			1.28226i
$996$		0			0
$997$	−1491.44			−0.0473766			−0.0236883	−	0.999719i	$-0.507541\pi$
$997$	−1491.44			−0.0473766			−0.0236883	+	0.999719i	$0.507541\pi$
$998$		0			0
$999$			7936.77i			0.251360i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.4.c.c.337.4		4
4.3	odd	2		39.4.b.b.25.4	yes	4
12.11	even	2		117.4.b.e.64.1		4
13.12	even	2	inner	624.4.c.c.337.1		4
52.31	even	4		507.4.a.l.1.4		4
52.47	even	4		507.4.a.l.1.1		4
52.51	odd	2		39.4.b.b.25.1	✓	4
156.47	odd	4		1521.4.a.w.1.4		4
156.83	odd	4		1521.4.a.w.1.1		4
156.155	even	2		117.4.b.e.64.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.b.b.25.1	✓	4	52.51	odd	2
39.4.b.b.25.4	yes	4	4.3	odd	2
117.4.b.e.64.1		4	12.11	even	2
117.4.b.e.64.4		4	156.155	even	2
507.4.a.l.1.1		4	52.47	even	4
507.4.a.l.1.4		4	52.31	even	4
624.4.c.c.337.1		4	13.12	even	2	inner
624.4.c.c.337.4		4	1.1	even	1	trivial
1521.4.a.w.1.1		4	156.83	odd	4
1521.4.a.w.1.4		4	156.47	odd	4

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:	\( N \)	\(=\)	\( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	624.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +12.9118i q^{5} +16.7289i q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +12.9118i q^{5} +16.7289i q^{7} +9.00000 q^{9} +24.9280i q^{11} +(33.7151 + 32.5621i) q^{13} -38.7355i q^{15} +134.145 q^{17} +14.9376i q^{19} -50.1866i q^{21} +72.0000 q^{23} -41.7151 q^{25} -27.0000 q^{27} -206.145 q^{29} +249.142i q^{31} -74.7841i q^{33} -216.000 q^{35} -293.955i q^{37} +(-101.145 - 97.6863i) q^{39} +250.506i q^{41} +432.145 q^{43} +116.206i q^{45} -159.889i q^{47} +63.1454 q^{49} -402.436 q^{51} -194.581 q^{53} -321.866 q^{55} -44.8129i q^{57} +232.647i q^{59} -185.006 q^{61} +150.560i q^{63} +(-420.436 + 435.324i) q^{65} +39.4393i q^{67} -216.000 q^{69} -920.460i q^{71} +549.078i q^{73} +125.145 q^{75} -417.018 q^{77} -933.140 q^{79} +81.0000 q^{81} +1095.38i q^{83} +1732.06i q^{85} +618.436 q^{87} +532.114i q^{89} +(-544.727 + 564.015i) q^{91} -747.425i q^{93} -192.872 q^{95} -362.661i q^{97} +224.352i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{3} + 36 q^{9}+O(q^{10}) \) 4 * q - 12 * q^3 + 36 * q^9	\( 4 q - 12 q^{3} + 36 q^{9} - 12 q^{13} + 96 q^{17} + 288 q^{23} - 20 q^{25} - 108 q^{27} - 384 q^{29} - 864 q^{35} + 36 q^{39} + 1288 q^{43} - 188 q^{49} - 288 q^{51} + 984 q^{53} + 328 q^{55} + 288 q^{61} - 360 q^{65} - 864 q^{69} + 60 q^{75} + 1416 q^{77} - 4320 q^{79} + 324 q^{81} + 1152 q^{87} + 24 q^{91} + 1872 q^{95}+O(q^{100}) \) 4 * q - 12 * q^3 + 36 * q^9 - 12 * q^13 + 96 * q^17 + 288 * q^23 - 20 * q^25 - 108 * q^27 - 384 * q^29 - 864 * q^35 + 36 * q^39 + 1288 * q^43 - 188 * q^49 - 288 * q^51 + 984 * q^53 + 328 * q^55 + 288 * q^61 - 360 * q^65 - 864 * q^69 + 60 * q^75 + 1416 * q^77 - 4320 * q^79 + 324 * q^81 + 1152 * q^87 + 24 * q^91 + 1872 * q^95

Introduction

L-functions

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Database

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