LMFDB - Embedded newform 624.4.c.c.337.2

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			337.2
Root			$-1.52356i$ of defining polynomial
Character	$\chi$	$=$	624.337
Dual form			624.4.c.c.337.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} -9.65841i q^{5} -22.3639i q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} -9.65841i q^{5} -22.3639i q^{7} +9.00000 q^{9} +50.3050i q^{11} +(-39.7151 + 24.8940i) q^{13} +28.9752i q^{15} -86.1454 q^{17} +116.880i q^{19} +67.0918i q^{21} +72.0000 q^{23} +31.7151 q^{25} -27.0000 q^{27} +14.1454 q^{29} -196.215i q^{31} -150.915i q^{33} -216.000 q^{35} -154.424i q^{37} +(119.145 - 74.6819i) q^{39} +265.726i q^{41} +211.855 q^{43} -86.9257i q^{45} -67.5535i q^{47} -157.145 q^{49} +258.436 q^{51} +686.581 q^{53} +485.866 q^{55} -350.639i q^{57} +91.9304i q^{59} +329.006 q^{61} -201.275i q^{63} +(240.436 + 383.585i) q^{65} +768.370i q^{67} -216.000 q^{69} +264.969i q^{71} +771.306i q^{73} -95.1454 q^{75} +1125.02 q^{77} -1226.86 q^{79} +81.0000 q^{81} -514.019i q^{83} +832.027i q^{85} -42.4361 q^{87} +527.889i q^{89} +(556.727 + 888.186i) q^{91} +588.646i q^{93} +1128.87 q^{95} +74.2755i q^{97} +452.745i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$		−	9.65841i		−	0.863874i	−0.901904	−	0.431937i	$-0.857830\pi$
$5$		−	9.65841i		−	0.863874i	0.901904	−	0.431937i	$-0.142170\pi$
$6$		0			0
$7$		−	22.3639i		−	1.20754i	−0.797159	−	0.603769i	$-0.793665\pi$
$7$		−	22.3639i		−	1.20754i	0.797159	−	0.603769i	$-0.206335\pi$
$8$		0			0
$9$	9.00000			0.333333
$10$		0			0
$11$			50.3050i			1.37887i	0.724349	+	0.689433i	$0.242140\pi$
$11$			50.3050i			1.37887i	−0.724349	+	0.689433i	$0.757860\pi$
$12$		0			0
$13$	−39.7151	+	24.8940i	−0.847307	+	0.531103i
$13$	−39.7151	+	24.8940i	−0.847307	+	0.531103i
$14$		0			0
$15$			28.9752i			0.498758i
$16$		0			0
$17$	−86.1454			−1.22902			−0.614509	−	0.788910i	$-0.710646\pi$
$17$	−86.1454			−1.22902			−0.614509	+	0.788910i	$0.710646\pi$
$18$		0			0
$19$			116.880i			1.41127i	0.708578	+	0.705633i	$0.249337\pi$
$19$			116.880i			1.41127i	−0.708578	+	0.705633i	$0.750663\pi$
$20$		0			0
$21$			67.0918i			0.697173i
$22$		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
$25$	31.7151			0.253721
$26$		0			0
$27$	−27.0000			−0.192450
$28$		0			0
$29$	14.1454			0.0905768			0.0452884	−	0.998974i	$-0.485579\pi$
$29$	14.1454			0.0905768			0.0452884	+	0.998974i	$0.485579\pi$
$30$		0			0
$31$		−	196.215i		−	1.13682i	−0.822747	−	0.568408i	$-0.807559\pi$
$31$		−	196.215i		−	1.13682i	0.822747	−	0.568408i	$-0.192441\pi$
$32$		0			0
$33$		−	150.915i		−	0.796089i
$34$		0			0
$35$	−216.000			−1.04316
$36$		0			0
$37$		−	154.424i		−	0.686138i	−0.939310	−	0.343069i	$-0.888533\pi$
$37$		−	154.424i		−	0.686138i	0.939310	−	0.343069i	$-0.111467\pi$
$38$		0			0
$39$	119.145	−	74.6819i	0.489193	−	0.306633i
$40$		0			0
$41$			265.726i			1.01218i	0.862480	+	0.506091i	$0.168910\pi$
$41$			265.726i			1.01218i	−0.862480	+	0.506091i	$0.831090\pi$
$42$		0			0
$43$	211.855			0.751338			0.375669	−	0.926754i	$-0.377413\pi$
$43$	211.855			0.751338			0.375669	+	0.926754i	$0.377413\pi$
$44$		0			0
$45$		−	86.9257i		−	0.287958i
$46$		0			0
$47$		−	67.5535i		−	0.209653i	−0.994491	−	0.104827i	$-0.966571\pi$
$47$		−	67.5535i		−	0.209653i	0.994491	−	0.104827i	$-0.0334287\pi$
$48$		0			0
$49$	−157.145			−0.458150
$50$		0			0
$51$	258.436			0.709574
$52$		0			0
$53$	686.581			1.77942			0.889710	−	0.456527i	$-0.150907\pi$
$53$	686.581			1.77942			0.889710	+	0.456527i	$0.150907\pi$
$54$		0			0
$55$	485.866			1.19117
$56$		0			0
$57$		−	350.639i		−	0.814795i
$58$		0			0
$59$			91.9304i			0.202853i	0.994843	+	0.101426i	$0.0323406\pi$
$59$			91.9304i			0.202853i	−0.994843	+	0.101426i	$0.967659\pi$
$60$		0			0
$61$	329.006			0.690572			0.345286	−	0.938498i	$-0.387782\pi$
$61$	329.006			0.690572			0.345286	+	0.938498i	$0.387782\pi$
$62$		0			0
$63$		−	201.275i		−	0.402513i
$64$		0			0
$65$	240.436	+	383.585i	0.458807	+	0.731967i
$66$		0			0
$67$			768.370i			1.40106i	0.713621	+	0.700532i	$0.247054\pi$
$67$			768.370i			1.40106i	−0.713621	+	0.700532i	$0.752946\pi$
$68$		0			0
$69$	−216.000			−0.376860
$70$		0			0
$71$			264.969i			0.442902i	0.975171	+	0.221451i	$0.0710793\pi$
$71$			264.969i			0.442902i	−0.975171	+	0.221451i	$0.928921\pi$
$72$		0			0
$73$			771.306i			1.23664i	0.785927	+	0.618319i	$0.212186\pi$
$73$			771.306i			1.23664i	−0.785927	+	0.618319i	$0.787814\pi$
$74$		0			0
$75$	−95.1454			−0.146486
$76$		0			0
$77$	1125.02			1.66503
$78$		0			0
$79$	−1226.86			−1.74725			−0.873624	−	0.486602i	$-0.838236\pi$
$79$	−1226.86			−1.74725			−0.873624	+	0.486602i	$0.838236\pi$
$80$		0			0
$81$	81.0000			0.111111
$82$		0			0
$83$		−	514.019i		−	0.679771i	−0.940467	−	0.339885i	$-0.889612\pi$
$83$		−	514.019i		−	0.679771i	0.940467	−	0.339885i	$-0.110388\pi$
$84$		0			0
$85$			832.027i			1.06172i
$86$		0			0
$87$	−42.4361			−0.0522945
$88$		0			0
$89$			527.889i			0.628720i	0.949304	+	0.314360i	$0.101790\pi$
$89$			527.889i			0.628720i	−0.949304	+	0.314360i	$0.898210\pi$
$90$		0			0
$91$	556.727	+	888.186i	0.641328	+	1.02316i
$92$		0			0
$93$			588.646i			0.656341i
$94$		0			0
$95$	1128.87			1.21916
$96$		0			0
$97$			74.2755i			0.0777478i	0.999244	+	0.0388739i	$0.0123771\pi$
$97$			74.2755i			0.0777478i	−0.999244	+	0.0388739i	$0.987623\pi$
$98$		0			0
$99$			452.745i			0.459622i
$100$		0			0
$101$	−609.419			−0.600390			−0.300195	−	0.953878i	$-0.597052\pi$
$101$	−609.419			−0.600390			−0.300195	+	0.953878i	$0.597052\pi$
$102$		0			0
$103$	−32.4361			−0.0310293			−0.0155147	−	0.999880i	$-0.504939\pi$
$103$	−32.4361			−0.0310293			−0.0155147	+	0.999880i	$0.504939\pi$
$104$		0			0
$105$	648.000			0.602270
$106$		0			0
$107$	1725.45			1.55893			0.779467	−	0.626444i	$-0.215490\pi$
$107$	1725.45			1.55893			0.779467	+	0.626444i	$0.215490\pi$
$108$		0			0
$109$		−	1273.43i		−	1.11902i	−0.828825	−	0.559508i	$-0.810990\pi$
$109$		−	1273.43i		−	1.11902i	0.828825	−	0.559508i	$-0.189010\pi$
$110$		0			0
$111$			463.271i			0.396142i
$112$		0			0
$113$	1114.73			0.928006			0.464003	−	0.885834i	$-0.346413\pi$
$113$	1114.73			0.928006			0.464003	+	0.885834i	$0.346413\pi$
$114$		0			0
$115$		−	695.406i		−	0.563886i
$116$		0			0
$117$	−357.436	+	224.046i	−0.282436	+	0.177034i
$118$		0			0
$119$			1926.55i			1.48409i
$120$		0			0
$121$	−1199.59			−0.901272
$122$		0			0
$123$		−	797.179i		−	0.584384i
$124$		0			0
$125$		−	1513.62i		−	1.08306i
$126$		0			0
$127$	1174.01			0.820289			0.410144	−	0.912021i	$-0.365478\pi$
$127$	1174.01			0.820289			0.410144	+	0.912021i	$0.365478\pi$
$128$		0			0
$129$	−635.564			−0.433785
$130$		0			0
$131$	1445.16			0.963851			0.481925	−	0.876212i	$-0.339938\pi$
$131$	1445.16			0.963851			0.481925	+	0.876212i	$0.339938\pi$
$132$		0			0
$133$	2613.89			1.70416
$134$		0			0
$135$			260.777i			0.166253i
$136$		0			0
$137$			508.793i			0.317293i	0.987335	+	0.158646i	$0.0507130\pi$
$137$			508.793i			0.317293i	−0.987335	+	0.158646i	$0.949287\pi$
$138$		0			0
$139$	757.018			0.461938			0.230969	−	0.972961i	$-0.425810\pi$
$139$	757.018			0.461938			0.230969	+	0.972961i	$0.425810\pi$
$140$		0			0
$141$			202.661i			0.121043i
$142$		0			0
$143$	−1252.29	−	1997.87i	−0.732320	−	1.16832i
$144$		0			0
$145$		−	136.622i		−	0.0782470i
$146$		0			0
$147$	471.436			0.264513
$148$		0			0
$149$			3247.79i			1.78570i	0.450352	+	0.892851i	$0.351298\pi$
$149$			3247.79i			1.78570i	−0.450352	+	0.892851i	$0.648702\pi$
$150$		0			0
$151$		−	795.296i		−	0.428611i	−0.976767	−	0.214305i	$-0.931251\pi$
$151$		−	795.296i		−	0.428611i	0.976767	−	0.214305i	$-0.0687488\pi$
$152$		0			0
$153$	−775.308			−0.409673
$154$		0			0
$155$	−1895.13			−0.982067
$156$		0			0
$157$	−65.2732			−0.0331807			−0.0165903	−	0.999862i	$-0.505281\pi$
$157$	−65.2732			−0.0331807			−0.0165903	+	0.999862i	$0.505281\pi$
$158$		0			0
$159$	−2059.74			−1.02735
$160$		0			0
$161$		−	1610.20i		−	0.788210i
$162$		0			0
$163$			1855.39i			0.891568i	0.895140	+	0.445784i	$0.147075\pi$
$163$			1855.39i			0.891568i	−0.895140	+	0.445784i	$0.852925\pi$
$164$		0			0
$165$	−1457.60			−0.687721
$166$		0			0
$167$			3532.54i			1.63686i	0.574604	+	0.818432i	$0.305156\pi$
$167$			3532.54i			1.63686i	−0.574604	+	0.818432i	$0.694844\pi$
$168$		0			0
$169$	957.581	−	1977.33i	0.435859	−	0.900015i
$170$		0			0
$171$			1051.92i			0.470422i
$172$		0			0
$173$	3178.36			1.39680			0.698400	−	0.715708i	$-0.253896\pi$
$173$	3178.36			1.39680			0.698400	+	0.715708i	$0.253896\pi$
$174$		0			0
$175$		−	709.275i		−	0.306378i
$176$		0			0
$177$		−	275.791i		−	0.117117i
$178$		0			0
$179$	2741.16			1.14460			0.572302	−	0.820043i	$-0.306050\pi$
$179$	2741.16			1.14460			0.572302	+	0.820043i	$0.306050\pi$
$180$		0			0
$181$	−3871.09			−1.58970			−0.794850	−	0.606806i	$-0.792450\pi$
$181$	−3871.09			−1.58970			−0.794850	+	0.606806i	$0.792450\pi$
$182$		0			0
$183$	−987.018			−0.398702
$184$		0			0
$185$	−1491.49			−0.592737
$186$		0			0
$187$		−	4333.54i		−	1.69465i
$188$		0			0
$189$			603.826i			0.232391i
$190$		0			0
$191$	928.291			0.351669			0.175834	−	0.984420i	$-0.443738\pi$
$191$	928.291			0.351669			0.175834	+	0.984420i	$0.443738\pi$
$192$		0			0
$193$		−	2261.51i		−	0.843456i	−0.906722	−	0.421728i	$-0.861424\pi$
$193$		−	2261.51i		−	0.843456i	0.906722	−	0.421728i	$-0.138576\pi$
$194$		0			0
$195$	−721.308	−	1150.75i	−0.264892	−	0.422601i
$196$		0			0
$197$			2265.59i			0.819374i	0.912226	+	0.409687i	$0.134362\pi$
$197$			2265.59i			0.819374i	−0.912226	+	0.409687i	$0.865638\pi$
$198$		0			0
$199$	−260.895			−0.0929366			−0.0464683	−	0.998920i	$-0.514797\pi$
$199$	−260.895			−0.0929366			−0.0464683	+	0.998920i	$0.514797\pi$
$200$		0			0
$201$		−	2305.11i		−	0.808905i
$202$		0			0
$203$		−	316.346i		−	0.109375i
$204$		0			0
$205$	2566.49			0.874399
$206$		0			0
$207$	648.000			0.217580
$208$		0			0
$209$	−5879.63			−1.94595
$210$		0			0
$211$	−5851.22			−1.90907			−0.954537	−	0.298092i	$-0.903650\pi$
$211$	−5851.22			−1.90907			−0.954537	+	0.298092i	$0.903650\pi$
$212$		0			0
$213$		−	794.907i		−	0.255710i
$214$		0			0
$215$		−	2046.18i		−	0.649062i
$216$		0			0
$217$	−4388.15			−1.37275
$218$		0			0
$219$		−	2313.92i		−	0.713973i
$220$		0			0
$221$	3421.27	−	2144.50i	1.04136	−	0.652736i
$222$		0			0
$223$		−	3463.60i		−	1.04009i	−0.854139	−	0.520045i	$-0.825915\pi$
$223$		−	3463.60i		−	1.04009i	0.854139	−	0.520045i	$-0.174085\pi$
$224$		0			0
$225$	285.436			0.0845737
$226$		0			0
$227$			5329.15i			1.55819i	0.626908	+	0.779093i	$0.284320\pi$
$227$			5329.15i			1.55819i	−0.626908	+	0.779093i	$0.715680\pi$
$228$		0			0
$229$			4773.95i			1.37761i	0.724949	+	0.688803i	$0.241863\pi$
$229$			4773.95i			1.37761i	−0.724949	+	0.688803i	$0.758137\pi$
$230$		0			0
$231$	−3375.05			−0.961308
$232$		0			0
$233$	4813.78			1.35348			0.676741	−	0.736221i	$-0.263392\pi$
$233$	4813.78			1.35348			0.676741	+	0.736221i	$0.263392\pi$
$234$		0			0
$235$	−652.459			−0.181114
$236$		0			0
$237$	3680.58			1.00877
$238$		0			0
$239$			1683.19i			0.455549i	0.973714	+	0.227775i	$0.0731449\pi$
$239$			1683.19i			0.455549i	−0.973714	+	0.227775i	$0.926855\pi$
$240$		0			0
$241$		−	664.861i		−	0.177707i	−0.996045	−	0.0888537i	$-0.971680\pi$
$241$		−	664.861i		−	0.177707i	0.996045	−	0.0888537i	$-0.0283204\pi$
$242$		0			0
$243$	−243.000			−0.0641500
$244$		0			0
$245$			1517.77i			0.395784i
$246$		0			0
$247$	−2909.60	−	4641.89i	−0.749528	−	1.19578i
$248$		0			0
$249$			1542.06i			0.392466i
$250$		0			0
$251$	2142.04			0.538662			0.269331	−	0.963048i	$-0.413198\pi$
$251$	2142.04			0.538662			0.269331	+	0.963048i	$0.413198\pi$
$252$		0			0
$253$			3621.96i			0.900042i
$254$		0			0
$255$		−	2496.08i		−	0.612983i
$256$		0			0
$257$	−3152.62			−0.765194			−0.382597	−	0.923915i	$-0.624970\pi$
$257$	−3152.62			−0.765194			−0.382597	+	0.923915i	$0.624970\pi$
$258$		0			0
$259$	−3453.52			−0.828539
$260$		0			0
$261$	127.308			0.0301923
$262$		0			0
$263$	2167.71			0.508238			0.254119	−	0.967173i	$-0.418214\pi$
$263$	2167.71			0.508238			0.254119	+	0.967173i	$0.418214\pi$
$264$		0			0
$265$		−	6631.28i		−	1.53719i
$266$		0			0
$267$		−	1583.67i		−	0.362992i
$268$		0			0
$269$	−3248.98			−0.736409			−0.368204	−	0.929745i	$-0.620027\pi$
$269$	−3248.98			−0.736409			−0.368204	+	0.929745i	$0.620027\pi$
$270$		0			0
$271$			4897.70i			1.09784i	0.835876	+	0.548919i	$0.184960\pi$
$271$			4897.70i			1.09784i	−0.835876	+	0.548919i	$0.815040\pi$
$272$		0			0
$273$	−1670.18	−	2664.56i	−0.370271	−	0.590719i
$274$		0			0
$275$			1595.43i			0.349847i
$276$		0			0
$277$	−2900.33			−0.629111			−0.314555	−	0.949239i	$-0.601855\pi$
$277$	−2900.33			−0.629111			−0.314555	+	0.949239i	$0.601855\pi$
$278$		0			0
$279$		−	1765.94i		−	0.378939i
$280$		0			0
$281$			4396.53i			0.933364i	0.884425	+	0.466682i	$0.154551\pi$
$281$			4396.53i			0.933364i	−0.884425	+	0.466682i	$0.845449\pi$
$282$		0			0
$283$	−559.151			−0.117449			−0.0587245	−	0.998274i	$-0.518703\pi$
$283$	−559.151			−0.117449			−0.0587245	+	0.998274i	$0.518703\pi$
$284$		0			0
$285$	−3386.62			−0.703880
$286$		0			0
$287$	5942.69			1.22225
$288$		0			0
$289$	2508.02			0.510487
$290$		0			0
$291$		−	222.827i		−	0.0448877i
$292$		0			0
$293$		−	6918.65i		−	1.37950i	−0.724050	−	0.689748i	$-0.757722\pi$
$293$		−	6918.65i		−	1.37950i	0.724050	−	0.689748i	$-0.242278\pi$
$294$		0			0
$295$	887.901			0.175239
$296$		0			0
$297$		−	1358.24i		−	0.265363i
$298$		0			0
$299$	−2859.49	+	1792.37i	−0.553072	+	0.346673i
$300$		0			0
$301$		−	4737.90i		−	0.907270i
$302$		0			0
$303$	1828.26			0.346635
$304$		0			0
$305$		−	3177.67i		−	0.596567i
$306$		0			0
$307$			8980.94i			1.66961i	0.550548	+	0.834803i	$0.314419\pi$
$307$			8980.94i			1.66961i	−0.550548	+	0.834803i	$0.685581\pi$
$308$		0			0
$309$	97.3082			0.0179148
$310$		0			0
$311$	−7943.13			−1.44827			−0.724137	−	0.689656i	$-0.757762\pi$
$311$	−7943.13			−1.44827			−0.724137	+	0.689656i	$0.757762\pi$
$312$		0			0
$313$	−5059.57			−0.913686			−0.456843	−	0.889547i	$-0.651020\pi$
$313$	−5059.57			−0.913686			−0.456843	+	0.889547i	$0.651020\pi$
$314$		0			0
$315$	−1944.00			−0.347721
$316$		0			0
$317$			8702.12i			1.54183i	0.636939	+	0.770914i	$0.280200\pi$
$317$			8702.12i			1.54183i	−0.636939	+	0.770914i	$0.719800\pi$
$318$		0			0
$319$			711.582i			0.124893i
$320$		0			0
$321$	−5176.36			−0.900051
$322$		0			0
$323$		−	10068.6i		−	1.73447i
$324$		0			0
$325$	−1259.57	+	789.515i	−0.214980	+	0.134752i
$326$		0			0
$327$			3820.30i			0.646064i
$328$		0			0
$329$	−1510.76			−0.253164
$330$		0			0
$331$		−	1737.65i		−	0.288549i	−0.989538	−	0.144275i	$-0.953915\pi$
$331$		−	1737.65i		−	0.288549i	0.989538	−	0.144275i	$-0.0460848\pi$
$332$		0			0
$333$		−	1389.81i		−	0.228713i
$334$		0			0
$335$	7421.23			1.21034
$336$		0			0
$337$	−2917.47			−0.471586			−0.235793	−	0.971803i	$-0.575769\pi$
$337$	−2917.47			−0.471586			−0.235793	+	0.971803i	$0.575769\pi$
$338$		0			0
$339$	−3344.18			−0.535785
$340$		0			0
$341$	9870.61			1.56752
$342$		0			0
$343$		−	4156.44i		−	0.654305i
$344$		0			0
$345$			2086.22i			0.325560i
$346$		0			0
$347$	−5081.23			−0.786095			−0.393047	−	0.919518i	$-0.628579\pi$
$347$	−5081.23			−0.786095			−0.393047	+	0.919518i	$0.628579\pi$
$348$		0			0
$349$			4266.14i			0.654330i	0.944967	+	0.327165i	$0.106093\pi$
$349$			4266.14i			0.654330i	−0.944967	+	0.327165i	$0.893907\pi$
$350$		0			0
$351$	1072.31	−	672.137i	0.163064	−	0.102211i
$352$		0			0
$353$		−	3264.49i		−	0.492213i	−0.969243	−	0.246106i	$-0.920849\pi$
$353$		−	3264.49i		−	0.492213i	0.969243	−	0.246106i	$-0.0791513\pi$
$354$		0			0
$355$	2559.18			0.382612
$356$		0			0
$357$		−	5779.65i		−	0.856838i
$358$		0			0
$359$			4416.98i			0.649357i	0.945824	+	0.324679i	$0.105256\pi$
$359$			4416.98i			0.649357i	−0.945824	+	0.324679i	$0.894744\pi$
$360$		0			0
$361$	−6801.87			−0.991670
$362$		0			0
$363$	3598.78			0.520350
$364$		0			0
$365$	7449.59			1.06830
$366$		0			0
$367$	1740.22			0.247516			0.123758	−	0.992312i	$-0.460505\pi$
$367$	1740.22			0.247516			0.123758	+	0.992312i	$0.460505\pi$
$368$		0			0
$369$			2391.54i			0.337394i
$370$		0			0
$371$		−	15354.7i		−	2.14872i
$372$		0			0
$373$	1176.28			0.163285			0.0816427	−	0.996662i	$-0.473983\pi$
$373$	1176.28			0.163285			0.0816427	+	0.996662i	$0.473983\pi$
$374$		0			0
$375$			4540.86i			0.625304i
$376$		0			0
$377$	−561.785	+	352.134i	−0.0767464	+	0.0481056i
$378$		0			0
$379$		−	7135.54i		−	0.967092i	−0.875319	−	0.483546i	$-0.839349\pi$
$379$		−	7135.54i		−	0.967092i	0.875319	−	0.483546i	$-0.160651\pi$
$380$		0			0
$381$	−3522.04			−0.473594
$382$		0			0
$383$			1942.87i			0.259207i	0.991566	+	0.129603i	$0.0413704\pi$
$383$			1942.87i			0.259207i	−0.991566	+	0.129603i	$0.958630\pi$
$384$		0			0
$385$		−	10865.9i		−	1.43838i
$386$		0			0
$387$	1906.69			0.250446
$388$		0			0
$389$	7545.92			0.983531			0.491766	−	0.870728i	$-0.336352\pi$
$389$	7545.92			0.983531			0.491766	+	0.870728i	$0.336352\pi$
$390$		0			0
$391$	−6202.47			−0.802231
$392$		0			0
$393$	−4335.49			−0.556480
$394$		0			0
$395$			11849.5i			1.50940i
$396$		0			0
$397$		−	415.922i		−	0.0525806i	−0.999654	−	0.0262903i	$-0.991631\pi$
$397$		−	415.922i		−	0.0525806i	0.999654	−	0.0262903i	$-0.00836943\pi$
$398$		0			0
$399$	−7841.67			−0.983896
$400$		0			0
$401$			958.178i			0.119324i	0.998219	+	0.0596622i	$0.0190024\pi$
$401$			958.178i			0.119324i	−0.998219	+	0.0596622i	$0.980998\pi$
$402$		0			0
$403$	4884.58	+	7792.71i	0.603767	+	0.963233i
$404$		0			0
$405$		−	782.331i		−	0.0959860i
$406$		0			0
$407$	7768.29			0.946093
$408$		0			0
$409$			4284.83i			0.518022i	0.965874	+	0.259011i	$0.0833966\pi$
$409$			4284.83i			0.518022i	−0.965874	+	0.259011i	$0.916603\pi$
$410$		0			0
$411$		−	1526.38i		−	0.183189i
$412$		0			0
$413$	2055.92			0.244953
$414$		0			0
$415$	−4964.61			−0.587236
$416$		0			0
$417$	−2271.05			−0.266700
$418$		0			0
$419$	9949.01			1.16000			0.580001	−	0.814616i	$-0.303052\pi$
$419$	9949.01			1.16000			0.580001	+	0.814616i	$0.303052\pi$
$420$		0			0
$421$		−	377.250i		−	0.0436724i	−0.999762	−	0.0218362i	$-0.993049\pi$
$421$		−	377.250i		−	0.0436724i	0.999762	−	0.0218362i	$-0.00695122\pi$
$422$		0			0
$423$		−	607.982i		−	0.0698843i
$424$		0			0
$425$	−2732.11			−0.311828
$426$		0			0
$427$		−	7357.86i		−	0.833892i
$428$		0			0
$429$	3756.87	+	5993.61i	0.422805	+	0.674532i
$430$		0			0
$431$		−	2437.13i		−	0.272373i	−0.990683	−	0.136186i	$-0.956515\pi$
$431$		−	2437.13i		−	0.272373i	0.990683	−	0.136186i	$-0.0434846\pi$
$432$		0			0
$433$	−11215.1			−1.24471			−0.622357	−	0.782733i	$-0.713825\pi$
$433$	−11215.1			−1.24471			−0.622357	+	0.782733i	$0.713825\pi$
$434$		0			0
$435$			409.865i			0.0451759i
$436$		0			0
$437$			8415.34i			0.921191i
$438$		0			0
$439$	−1835.19			−0.199519			−0.0997596	−	0.995012i	$-0.531807\pi$
$439$	−1835.19			−0.199519			−0.0997596	+	0.995012i	$0.531807\pi$
$440$		0			0
$441$	−1414.31			−0.152717
$442$		0			0
$443$	−11610.1			−1.24518			−0.622588	−	0.782550i	$-0.713919\pi$
$443$	−11610.1			−1.24518			−0.622588	+	0.782550i	$0.713919\pi$
$444$		0			0
$445$	5098.56			0.543135
$446$		0			0
$447$		−	9743.38i		−	1.03098i
$448$		0			0
$449$		−	14087.0i		−	1.48064i	−0.672255	−	0.740319i	$-0.734674\pi$
$449$		−	14087.0i		−	1.48064i	0.672255	−	0.740319i	$-0.265326\pi$
$450$		0			0
$451$	−13367.4			−1.39566
$452$		0			0
$453$			2385.89i			0.247459i
$454$		0			0
$455$	8578.47	−	5377.10i	0.883878	−	0.554027i
$456$		0			0
$457$		−	2375.01i		−	0.243103i	−0.992585	−	0.121552i	$-0.961213\pi$
$457$		−	2375.01i		−	0.243103i	0.992585	−	0.121552i	$-0.0387870\pi$
$458$		0			0
$459$	2325.92			0.236525
$460$		0			0
$461$			6372.06i			0.643766i	0.946779	+	0.321883i	$0.104316\pi$
$461$			6372.06i			0.643766i	−0.946779	+	0.321883i	$0.895684\pi$
$462$		0			0
$463$			63.4732i			0.00637117i	0.999995	+	0.00318558i	$0.00101400\pi$
$463$			63.4732i			0.00637117i	−0.999995	+	0.00318558i	$0.998986\pi$
$464$		0			0
$465$	5685.38			0.566996
$466$		0			0
$467$	−7855.78			−0.778420			−0.389210	−	0.921149i	$-0.627252\pi$
$467$	−7855.78			−0.778420			−0.389210	+	0.921149i	$0.627252\pi$
$468$		0			0
$469$	17183.8			1.69184
$470$		0			0
$471$	195.820			0.0191569
$472$		0			0
$473$			10657.3i			1.03599i
$474$		0			0
$475$			3706.85i			0.358068i
$476$		0			0
$477$	6179.23			0.593140
$478$		0			0
$479$			13033.3i			1.24323i	0.783324	+	0.621613i	$0.213523\pi$
$479$			13033.3i			1.24323i	−0.783324	+	0.621613i	$0.786477\pi$
$480$		0			0
$481$	3844.22	+	6132.96i	0.364410	+	0.581370i
$482$		0			0
$483$			4830.61i			0.455073i
$484$		0			0
$485$	717.384			0.0671643
$486$		0			0
$487$		−	69.8976i		−	0.00650383i	−0.999995	−	0.00325191i	$-0.998965\pi$
$487$		−	69.8976i		−	0.00650383i	0.999995	−	0.00325191i	$-0.00103512\pi$
$488$		0			0
$489$		−	5566.18i		−	0.514747i
$490$		0			0
$491$	−2625.66			−0.241333			−0.120667	−	0.992693i	$-0.538503\pi$
$491$	−2625.66			−0.241333			−0.120667	+	0.992693i	$0.538503\pi$
$492$		0			0
$493$	−1218.56			−0.111321
$494$		0			0
$495$	4372.80			0.397056
$496$		0			0
$497$	5925.75			0.534821
$498$		0			0
$499$			7631.34i			0.684621i	0.939587	+	0.342310i	$0.111209\pi$
$499$			7631.34i			0.684621i	−0.939587	+	0.342310i	$0.888791\pi$
$500$		0			0
$501$		−	10597.6i		−	0.945044i
$502$		0			0
$503$	4320.14			0.382953			0.191477	−	0.981497i	$-0.438672\pi$
$503$	4320.14			0.382953			0.191477	+	0.981497i	$0.438672\pi$
$504$		0			0
$505$			5886.01i			0.518662i
$506$		0			0
$507$	−2872.74	+	5932.00i	−0.251643	+	0.519624i
$508$		0			0
$509$			12450.7i			1.08422i	0.840308	+	0.542109i	$0.182374\pi$
$509$			12450.7i			1.08422i	−0.840308	+	0.542109i	$0.817626\pi$
$510$		0			0
$511$	17249.4			1.49329
$512$		0			0
$513$		−	3155.75i		−	0.271598i
$514$		0			0
$515$			313.281i			0.0268054i
$516$		0			0
$517$	3398.28			0.289083
$518$		0			0
$519$	−9535.08			−0.806443
$520$		0			0
$521$	14373.1			1.20863			0.604314	−	0.796746i	$-0.293447\pi$
$521$	14373.1			1.20863			0.604314	+	0.796746i	$0.293447\pi$
$522$		0			0
$523$	16946.7			1.41688			0.708439	−	0.705772i	$-0.249400\pi$
$523$	16946.7			1.41688			0.708439	+	0.705772i	$0.249400\pi$
$524$		0			0
$525$			2127.82i			0.176887i
$526$		0			0
$527$			16903.0i			1.39717i
$528$		0			0
$529$	−6983.00			−0.573929
$530$		0			0
$531$			827.373i			0.0676176i
$532$		0			0
$533$	−6614.98	−	10553.4i	−0.537574	−	0.857630i
$534$		0			0
$535$		−	16665.1i		−	1.34672i
$536$		0			0
$537$	−8223.49			−0.660837
$538$		0			0
$539$		−	7905.20i		−	0.631727i
$540$		0			0
$541$			815.667i			0.0648212i	0.999475	+	0.0324106i	$0.0103184\pi$
$541$			815.667i			0.0648212i	−0.999475	+	0.0324106i	$0.989682\pi$
$542$		0			0
$543$	11613.3			0.917814
$544$		0			0
$545$	−12299.3			−0.966689
$546$		0			0
$547$	17971.4			1.40476			0.702378	−	0.711804i	$-0.252122\pi$
$547$	17971.4			1.40476			0.702378	+	0.711804i	$0.252122\pi$
$548$		0			0
$549$	2961.05			0.230191
$550$		0			0
$551$			1653.31i			0.127828i
$552$		0			0
$553$			27437.4i			2.10987i
$554$		0			0
$555$	4474.47			0.342217
$556$		0			0
$557$			6760.83i			0.514301i	0.966371	+	0.257151i	$0.0827836\pi$
$557$			6760.83i			0.514301i	−0.966371	+	0.257151i	$0.917216\pi$
$558$		0			0
$559$	−8413.83	+	5273.90i	−0.636614	+	0.399038i
$560$		0			0
$561$			13000.6i			0.978408i
$562$		0			0
$563$	−12962.7			−0.970359			−0.485179	−	0.874415i	$-0.661246\pi$
$563$	−12962.7			−0.970359			−0.485179	+	0.874415i	$0.661246\pi$
$564$		0			0
$565$		−	10766.5i		−	0.801681i
$566$		0			0
$567$		−	1811.48i		−	0.134171i
$568$		0			0
$569$	164.757			0.0121388			0.00606938	−	0.999982i	$-0.498068\pi$
$569$	164.757			0.0121388			0.00606938	+	0.999982i	$0.498068\pi$
$570$		0			0
$571$	5216.55			0.382322			0.191161	−	0.981559i	$-0.438775\pi$
$571$	5216.55			0.382322			0.191161	+	0.981559i	$0.438775\pi$
$572$		0			0
$573$	−2784.87			−0.203036
$574$		0			0
$575$	2283.49			0.165614
$576$		0			0
$577$		−	12753.3i		−	0.920151i	−0.887880	−	0.460076i	$-0.847822\pi$
$577$		−	12753.3i		−	0.920151i	0.887880	−	0.460076i	$-0.152178\pi$
$578$		0			0
$579$			6784.53i			0.486970i
$580$		0			0
$581$	−11495.5			−0.820849
$582$		0			0
$583$			34538.5i			2.45358i
$584$		0			0
$585$	2163.92	+	3452.26i	0.152936	+	0.243989i
$586$		0			0
$587$			1575.38i			0.110771i	0.998465	+	0.0553857i	$0.0176388\pi$
$587$			1575.38i			0.110771i	−0.998465	+	0.0553857i	$0.982361\pi$
$588$		0			0
$589$	22933.6			1.60435
$590$		0			0
$591$		−	6796.77i		−	0.473066i
$592$		0			0
$593$			3845.95i			0.266331i	0.991094	+	0.133165i	$0.0425142\pi$
$593$			3845.95i			0.266331i	−0.991094	+	0.133165i	$0.957486\pi$
$594$		0			0
$595$	18607.4			1.28207
$596$		0			0
$597$	782.686			0.0536570
$598$		0			0
$599$	−6107.20			−0.416583			−0.208292	−	0.978067i	$-0.566790\pi$
$599$	−6107.20			−0.416583			−0.208292	+	0.978067i	$0.566790\pi$
$600$		0			0
$601$	9638.90			0.654208			0.327104	−	0.944988i	$-0.393927\pi$
$601$	9638.90			0.654208			0.327104	+	0.944988i	$0.393927\pi$
$602$		0			0
$603$			6915.33i			0.467022i
$604$		0			0
$605$			11586.2i			0.778586i
$606$		0			0
$607$	−11821.7			−0.790489			−0.395244	−	0.918576i	$-0.629340\pi$
$607$	−11821.7			−0.790489			−0.395244	+	0.918576i	$0.629340\pi$
$608$		0			0
$609$			949.037i			0.0631477i
$610$		0			0
$611$	1681.67	+	2682.90i	0.111347	+	0.177640i
$612$		0			0
$613$		−	8107.86i		−	0.534214i	−0.963667	−	0.267107i	$-0.913932\pi$
$613$		−	8107.86i		−	0.534214i	0.963667	−	0.267107i	$-0.0860677\pi$
$614$		0			0
$615$	−7699.48			−0.504834
$616$		0			0
$617$		−	27647.6i		−	1.80397i	−0.431768	−	0.901985i	$-0.642110\pi$
$617$		−	27647.6i		−	1.80397i	0.431768	−	0.901985i	$-0.357890\pi$
$618$		0			0
$619$		−	29181.9i		−	1.89486i	−0.319956	−	0.947432i	$-0.603668\pi$
$619$		−	29181.9i		−	1.89486i	0.319956	−	0.947432i	$-0.396332\pi$
$620$		0			0
$621$	−1944.00			−0.125620
$622$		0			0
$623$	11805.7			0.759204
$624$		0			0
$625$	−10654.8			−0.681905
$626$		0			0
$627$	17638.9			1.12349
$628$		0			0
$629$			13302.9i			0.843277i
$630$		0			0
$631$			2209.34i			0.139386i	0.997569	+	0.0696928i	$0.0222019\pi$
$631$			2209.34i			0.139386i	−0.997569	+	0.0696928i	$0.977798\pi$
$632$		0			0
$633$	17553.7			1.10220
$634$		0			0
$635$		−	11339.1i		−	0.708627i
$636$		0			0
$637$	6241.05	−	3911.97i	0.388194	−	0.243325i
$638$		0			0
$639$			2384.72i			0.147634i
$640$		0			0
$641$	18256.0			1.12491			0.562455	−	0.826828i	$-0.309857\pi$
$641$	18256.0			1.12491			0.562455	+	0.826828i	$0.309857\pi$
$642$		0			0
$643$			1281.61i			0.0786033i	0.999227	+	0.0393016i	$0.0125133\pi$
$643$			1281.61i			0.0786033i	−0.999227	+	0.0393016i	$0.987487\pi$
$644$		0			0
$645$			6138.54i			0.374736i
$646$		0			0
$647$	16393.2			0.996107			0.498054	−	0.867146i	$-0.334048\pi$
$647$	16393.2			0.996107			0.498054	+	0.867146i	$0.334048\pi$
$648$		0			0
$649$	−4624.56			−0.279707
$650$		0			0
$651$	13164.4			0.792557
$652$		0			0
$653$	16759.2			1.00434			0.502172	−	0.864768i	$-0.332534\pi$
$653$	16759.2			1.00434			0.502172	+	0.864768i	$0.332534\pi$
$654$		0			0
$655$		−	13958.0i		−	0.832646i
$656$		0			0
$657$			6941.76i			0.412213i
$658$		0			0
$659$	29659.3			1.75320			0.876601	−	0.481217i	$-0.159805\pi$
$659$	29659.3			1.75320			0.876601	+	0.481217i	$0.159805\pi$
$660$		0			0
$661$		−	10386.0i		−	0.611147i	−0.952169	−	0.305573i	$-0.901152\pi$
$661$		−	10386.0i		−	0.611147i	0.952169	−	0.305573i	$-0.0988481\pi$
$662$		0			0
$663$	−10263.8	+	6433.50i	−0.601227	+	0.376857i
$664$		0			0
$665$		−	25246.0i		−	1.47218i
$666$		0			0
$667$	1018.47			0.0591232
$668$		0			0
$669$			10390.8i			0.600496i
$670$		0			0
$671$			16550.6i			0.952206i
$672$		0			0
$673$	19449.6			1.11400			0.557002	−	0.830511i	$-0.311951\pi$
$673$	19449.6			1.11400			0.557002	+	0.830511i	$0.311951\pi$
$674$		0			0
$675$	−856.308			−0.0488286
$676$		0			0
$677$	−6629.48			−0.376354			−0.188177	−	0.982135i	$-0.560258\pi$
$677$	−6629.48			−0.376354			−0.188177	+	0.982135i	$0.560258\pi$
$678$		0			0
$679$	1661.09			0.0938835
$680$		0			0
$681$		−	15987.5i		−	0.899619i
$682$		0			0
$683$			2526.40i			0.141537i	0.997493	+	0.0707687i	$0.0225452\pi$
$683$			2526.40i			0.141537i	−0.997493	+	0.0707687i	$0.977455\pi$
$684$		0			0
$685$	4914.13			0.274101
$686$		0			0
$687$		−	14321.9i		−	0.795361i
$688$		0			0
$689$	−27267.7	+	17091.7i	−1.50771	+	0.945055i
$690$		0			0
$691$			3808.76i			0.209685i	0.994489	+	0.104842i	$0.0334338\pi$
$691$			3808.76i			0.209685i	−0.994489	+	0.104842i	$0.966566\pi$
$692$		0			0
$693$	10125.2			0.555011
$694$		0			0
$695$		−	7311.59i		−	0.399056i
$696$		0			0
$697$		−	22891.1i		−	1.24399i
$698$		0			0
$699$	−14441.3			−0.781433
$700$		0			0
$701$	−33617.8			−1.81131			−0.905655	−	0.424015i	$-0.860620\pi$
$701$	−33617.8			−1.81131			−0.905655	+	0.424015i	$0.860620\pi$
$702$		0			0
$703$	18049.0			0.968323
$704$		0			0
$705$	1957.38			0.104566
$706$		0			0
$707$			13629.0i			0.724994i
$708$		0			0
$709$			26606.5i			1.40935i	0.709530	+	0.704675i	$0.248907\pi$
$709$			26606.5i			1.40935i	−0.709530	+	0.704675i	$0.751093\pi$
$710$		0			0
$711$	−11041.7			−0.582416
$712$		0			0
$713$		−	14127.5i		−	0.742046i
$714$		0			0
$715$	−19296.2	+	12095.1i	−1.00928	+	0.632633i
$716$		0			0
$717$		−	5049.56i		−	0.263011i
$718$		0			0
$719$	−16539.3			−0.857877			−0.428939	−	0.903334i	$-0.641112\pi$
$719$	−16539.3			−0.857877			−0.428939	+	0.903334i	$0.641112\pi$
$720$		0			0
$721$			725.398i			0.0374691i
$722$		0			0
$723$			1994.58i			0.102599i
$724$		0			0
$725$	448.622			0.0229812
$726$		0			0
$727$	−12757.5			−0.650823			−0.325411	−	0.945573i	$-0.605503\pi$
$727$	−12757.5			−0.650823			−0.325411	+	0.945573i	$0.605503\pi$
$728$		0			0
$729$	729.000			0.0370370
$730$		0			0
$731$	−18250.3			−0.923408
$732$		0			0
$733$			20523.4i			1.03417i	0.855933	+	0.517087i	$0.172984\pi$
$733$			20523.4i			1.03417i	−0.855933	+	0.517087i	$0.827016\pi$
$734$		0			0
$735$		−	4553.32i		−	0.228506i
$736$		0			0
$737$	−38652.9			−1.93188
$738$		0			0
$739$		−	7462.66i		−	0.371473i	−0.982600	−	0.185736i	$-0.940533\pi$
$739$		−	7462.66i		−	0.371473i	0.982600	−	0.185736i	$-0.0594670\pi$
$740$		0			0
$741$	8728.80	+	13925.7i	0.432740	+	0.690381i
$742$		0			0
$743$		−	18847.8i		−	0.930632i	−0.885145	−	0.465316i	$-0.845941\pi$
$743$		−	18847.8i		−	0.930632i	0.885145	−	0.465316i	$-0.154059\pi$
$744$		0			0
$745$	31368.5			1.54262
$746$		0			0
$747$		−	4626.17i		−	0.226590i
$748$		0			0
$749$		−	38587.9i		−	1.88247i
$750$		0			0
$751$	−20832.6			−1.01224			−0.506119	−	0.862464i	$-0.668920\pi$
$751$	−20832.6			−1.01224			−0.506119	+	0.862464i	$0.668920\pi$
$752$		0			0
$753$	−6426.11			−0.310996
$754$		0			0
$755$	−7681.29			−0.370266
$756$		0			0
$757$	−20860.9			−1.00159			−0.500795	−	0.865566i	$-0.666959\pi$
$757$	−20860.9			−1.00159			−0.500795	+	0.865566i	$0.666959\pi$
$758$		0			0
$759$		−	10865.9i		−	0.519640i
$760$		0			0
$761$			4464.86i			0.212682i	0.994330	+	0.106341i	$0.0339135\pi$
$761$			4464.86i			0.212682i	−0.994330	+	0.106341i	$0.966086\pi$
$762$		0			0
$763$	−28479.0			−1.35126
$764$		0			0
$765$			7488.24i			0.353906i
$766$		0			0
$767$	−2288.51	−	3651.03i	−0.107736	−	0.171879i
$768$		0			0
$769$			23797.7i			1.11595i	0.829857	+	0.557977i	$0.188422\pi$
$769$			23797.7i			1.11595i	−0.829857	+	0.557977i	$0.811578\pi$
$770$		0			0
$771$	9457.85			0.441785
$772$		0			0
$773$			29418.7i			1.36885i	0.729085	+	0.684423i	$0.239946\pi$
$773$			29418.7i			1.36885i	−0.729085	+	0.684423i	$0.760054\pi$
$774$		0			0
$775$		−	6222.99i		−	0.288434i
$776$		0			0
$777$	10360.6			0.478357
$778$		0			0
$779$	−31058.0			−1.42846
$780$		0			0
$781$	−13329.3			−0.610703
$782$		0			0
$783$	−381.925			−0.0174315
$784$		0			0
$785$			630.435i			0.0286640i
$786$		0			0
$787$			23896.0i			1.08234i	0.840913	+	0.541170i	$0.182018\pi$
$787$			23896.0i			1.08234i	−0.840913	+	0.541170i	$0.817982\pi$
$788$		0			0
$789$	−6503.13			−0.293432
$790$		0			0
$791$		−	24929.7i		−	1.12060i
$792$		0			0
$793$	−13066.5	+	8190.26i	−0.585126	+	0.366765i
$794$		0			0
$795$			19893.9i			0.887500i
$796$		0			0
$797$	1034.67			0.0459847			0.0229923	−	0.999736i	$-0.492681\pi$
$797$	1034.67			0.0459847			0.0229923	+	0.999736i	$0.492681\pi$
$798$		0			0
$799$			5819.42i			0.257667i
$800$		0			0
$801$			4751.00i			0.209573i
$802$		0			0
$803$	−38800.6			−1.70516
$804$		0			0
$805$	−15552.0			−0.680914
$806$		0			0
$807$	9746.95			0.425166
$808$		0			0
$809$	29557.9			1.28455			0.642275	−	0.766474i	$-0.277991\pi$
$809$	29557.9			1.28455			0.642275	+	0.766474i	$0.277991\pi$
$810$		0			0
$811$			30268.9i			1.31058i	0.755376	+	0.655292i	$0.227454\pi$
$811$			30268.9i			1.31058i	−0.755376	+	0.655292i	$0.772546\pi$
$812$		0			0
$813$		−	14693.1i		−	0.633837i
$814$		0			0
$815$	17920.2			0.770203
$816$		0			0
$817$			24761.5i			1.06034i
$818$		0			0
$819$	5010.54	+	7993.68i	0.213776	+	0.341052i
$820$		0			0
$821$		−	39292.7i		−	1.67031i	−0.550013	−	0.835156i	$-0.685377\pi$
$821$		−	39292.7i		−	1.67031i	0.550013	−	0.835156i	$-0.314623\pi$
$822$		0			0
$823$	−25988.8			−1.10074			−0.550372	−	0.834920i	$-0.685514\pi$
$823$	−25988.8			−1.10074			−0.550372	+	0.834920i	$0.685514\pi$
$824$		0			0
$825$		−	4786.29i		−	0.201984i
$826$		0			0
$827$		−	19585.1i		−	0.823508i	−0.911295	−	0.411754i	$-0.864916\pi$
$827$		−	19585.1i		−	0.823508i	0.911295	−	0.411754i	$-0.135084\pi$
$828$		0			0
$829$	−666.472			−0.0279222			−0.0139611	−	0.999903i	$-0.504444\pi$
$829$	−666.472			−0.0279222			−0.0139611	+	0.999903i	$0.504444\pi$
$830$		0			0
$831$	8700.98			0.363217
$832$		0			0
$833$	13537.3			0.563075
$834$		0			0
$835$	34118.7			1.41404
$836$		0			0
$837$			5297.81i			0.218780i
$838$		0			0
$839$		−	36379.7i		−	1.49698i	−0.663146	−	0.748490i	$-0.730779\pi$
$839$		−	36379.7i		−	1.49698i	0.663146	−	0.748490i	$-0.269221\pi$
$840$		0			0
$841$	−24188.9			−0.991796
$842$		0			0
$843$		−	13189.6i		−	0.538878i
$844$		0			0
$845$	−19097.9	−	9248.71i	−0.777500	−	0.376527i
$846$		0			0
$847$			26827.6i			1.08832i
$848$		0			0
$849$	1677.45			0.0678093
$850$		0			0
$851$		−	11118.5i		−	0.447871i
$852$		0			0
$853$			39951.4i			1.60364i	0.597563	+	0.801822i	$0.296136\pi$
$853$			39951.4i			1.60364i	−0.597563	+	0.801822i	$0.703864\pi$
$854$		0			0
$855$	10159.8			0.406385
$856$		0			0
$857$	−17226.4			−0.686629			−0.343314	−	0.939221i	$-0.611550\pi$
$857$	−17226.4			−0.686629			−0.343314	+	0.939221i	$0.611550\pi$
$858$		0			0
$859$	33392.3			1.32634			0.663172	−	0.748467i	$-0.269210\pi$
$859$	33392.3			1.32634			0.663172	+	0.748467i	$0.269210\pi$
$860$		0			0
$861$	−17828.1			−0.705666
$862$		0			0
$863$		−	17381.9i		−	0.685614i	−0.939406	−	0.342807i	$-0.888622\pi$
$863$		−	17381.9i		−	0.685614i	0.939406	−	0.342807i	$-0.111378\pi$
$864$		0			0
$865$		−	30697.9i		−	1.20666i
$866$		0			0
$867$	−7524.07			−0.294730
$868$		0			0
$869$		−	61717.2i		−	2.40922i
$870$		0			0
$871$	−19127.8	−	30515.9i	−0.744110	−	1.18713i
$872$		0			0
$873$			668.480i			0.0259159i
$874$		0			0
$875$	−33850.5			−1.30783
$876$		0			0
$877$		−	14335.6i		−	0.551970i	−0.961162	−	0.275985i	$-0.910996\pi$
$877$		−	14335.6i		−	0.551970i	0.961162	−	0.275985i	$-0.0890040\pi$
$878$		0			0
$879$			20756.0i			0.796452i
$880$		0			0
$881$	−5436.53			−0.207901			−0.103951	−	0.994582i	$-0.533148\pi$
$881$	−5436.53			−0.207901			−0.103951	+	0.994582i	$0.533148\pi$
$882$		0			0
$883$	21185.9			0.807430			0.403715	−	0.914885i	$-0.367719\pi$
$883$	21185.9			0.807430			0.403715	+	0.914885i	$0.367719\pi$
$884$		0			0
$885$	−2663.70			−0.101174
$886$		0			0
$887$	12661.4			0.479287			0.239644	−	0.970861i	$-0.422969\pi$
$887$	12661.4			0.479287			0.239644	+	0.970861i	$0.422969\pi$
$888$		0			0
$889$		−	26255.5i		−	0.990531i
$890$		0			0
$891$			4074.71i			0.153207i
$892$		0			0
$893$	7895.63			0.295876
$894$		0			0
$895$		−	26475.3i		−	0.988794i
$896$		0			0
$897$	8578.47	−	5377.10i	0.319316	−	0.200152i
$898$		0			0
$899$		−	2775.54i		−	0.102969i
$900$		0			0
$901$	−59145.8			−2.18694
$902$		0			0
$903$			14213.7i			0.523812i
$904$		0			0
$905$			37388.6i			1.37330i
$906$		0			0
$907$	3545.27			0.129789			0.0648946	−	0.997892i	$-0.479329\pi$
$907$	3545.27			0.129789			0.0648946	+	0.997892i	$0.479329\pi$
$908$		0			0
$909$	−5484.77			−0.200130
$910$		0			0
$911$	3913.88			0.142341			0.0711706	−	0.997464i	$-0.477327\pi$
$911$	3913.88			0.142341			0.0711706	+	0.997464i	$0.477327\pi$
$912$		0			0
$913$	25857.7			0.937313
$914$		0			0
$915$			9533.02i			0.344428i
$916$		0			0
$917$		−	32319.5i		−	1.16389i
$918$		0			0
$919$	−6917.79			−0.248310			−0.124155	−	0.992263i	$-0.539622\pi$
$919$	−6917.79			−0.248310			−0.124155	+	0.992263i	$0.539622\pi$
$920$		0			0
$921$		−	26942.8i		−	0.963948i
$922$		0			0
$923$	−6596.13	−	10523.3i	−0.235227	−	0.375274i
$924$		0			0
$925$		−	4897.57i		−	0.174088i
$926$		0			0
$927$	−291.925			−0.0103431
$928$		0			0
$929$			3753.03i			0.132543i	0.997802	+	0.0662717i	$0.0211104\pi$
$929$			3753.03i			0.132543i	−0.997802	+	0.0662717i	$0.978890\pi$
$930$		0			0
$931$		−	18367.1i		−	0.646571i
$932$		0			0
$933$	23829.4			0.836162
$934$		0			0
$935$	−41855.1			−1.46397
$936$		0			0
$937$	48189.0			1.68011			0.840056	−	0.542499i	$-0.182522\pi$
$937$	48189.0			1.68011			0.840056	+	0.542499i	$0.182522\pi$
$938$		0			0
$939$	15178.7			0.527517
$940$		0			0
$941$		−	26656.4i		−	0.923456i	−0.887022	−	0.461728i	$-0.847230\pi$
$941$		−	26656.4i		−	0.923456i	0.887022	−	0.461728i	$-0.152770\pi$
$942$		0			0
$943$			19132.3i			0.660693i
$944$		0			0
$945$	5832.00			0.200757
$946$		0			0
$947$		−	31258.9i		−	1.07263i	−0.844018	−	0.536314i	$-0.819816\pi$
$947$		−	31258.9i		−	1.07263i	0.844018	−	0.536314i	$-0.180184\pi$
$948$		0			0
$949$	−19200.9	−	30632.5i	−0.656783	−	1.04781i
$950$		0			0
$951$		−	26106.4i		−	0.890175i
$952$		0			0
$953$	−10602.4			−0.360382			−0.180191	−	0.983632i	$-0.557672\pi$
$953$	−10602.4			−0.360382			−0.180191	+	0.983632i	$0.557672\pi$
$954$		0			0
$955$		−	8965.81i		−	0.303798i
$956$		0			0
$957$		−	2134.75i		−	0.0721072i
$958$		0			0
$959$	11378.6			0.383144
$960$		0			0
$961$	−8709.45			−0.292352
$962$		0			0
$963$	15529.1			0.519645
$964$		0			0
$965$	−21842.6			−0.728640
$966$		0			0
$967$		−	4815.93i		−	0.160155i	−0.996789	−	0.0800774i	$-0.974483\pi$
$967$		−	4815.93i		−	0.160155i	0.996789	−	0.0800774i	$-0.0255168\pi$
$968$		0			0
$969$			30205.9i			1.00140i
$970$		0			0
$971$	−56729.5			−1.87491			−0.937454	−	0.348109i	$-0.886824\pi$
$971$	−56729.5			−1.87491			−0.937454	+	0.348109i	$0.886824\pi$
$972$		0			0
$973$		−	16929.9i		−	0.557808i
$974$		0			0
$975$	3778.71	−	2368.54i	0.124119	−	0.0777991i
$976$		0			0
$977$			55246.2i			1.80909i	0.426375	+	0.904546i	$0.359790\pi$
$977$			55246.2i			1.80909i	−0.426375	+	0.904546i	$0.640210\pi$
$978$		0			0
$979$	−26555.4			−0.866921
$980$		0			0
$981$		−	11460.9i		−	0.373005i
$982$		0			0
$983$			22820.6i			0.740451i	0.928942	+	0.370226i	$0.120720\pi$
$983$			22820.6i			0.740451i	−0.928942	+	0.370226i	$0.879280\pi$
$984$		0			0
$985$	21882.0			0.707836
$986$		0			0
$987$	4532.29			0.146164
$988$		0			0
$989$	15253.5			0.490429
$990$		0			0
$991$	46490.4			1.49023			0.745115	−	0.666936i	$-0.232395\pi$
$991$	46490.4			1.49023			0.745115	+	0.666936i	$0.232395\pi$
$992$		0			0
$993$			5212.94i			0.166594i
$994$		0			0
$995$			2519.84i			0.0802856i
$996$		0			0
$997$	24943.4			0.792344			0.396172	−	0.918176i	$-0.370338\pi$
$997$	24943.4			0.792344			0.396172	+	0.918176i	$0.370338\pi$
$998$		0			0
$999$			4169.44i			0.132047i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	624.4.c.c.337.2		4
4.3	odd	2		39.4.b.b.25.3	yes	4
12.11	even	2		117.4.b.e.64.2		4
13.12	even	2	inner	624.4.c.c.337.3		4
52.31	even	4		507.4.a.l.1.3		4
52.47	even	4		507.4.a.l.1.2		4
52.51	odd	2		39.4.b.b.25.2	✓	4
156.47	odd	4		1521.4.a.w.1.3		4
156.83	odd	4		1521.4.a.w.1.2		4
156.155	even	2		117.4.b.e.64.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.b.b.25.2	✓	4	52.51	odd	2
39.4.b.b.25.3	yes	4	4.3	odd	2
117.4.b.e.64.2		4	12.11	even	2
117.4.b.e.64.3		4	156.155	even	2
507.4.a.l.1.2		4	52.47	even	4
507.4.a.l.1.3		4	52.31	even	4
624.4.c.c.337.2		4	1.1	even	1	trivial
624.4.c.c.337.3		4	13.12	even	2	inner
1521.4.a.w.1.2		4	156.83	odd	4
1521.4.a.w.1.3		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)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} -9.65841i q^{5} -22.3639i q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} -9.65841i q^{5} -22.3639i q^{7} +9.00000 q^{9} +50.3050i q^{11} +(-39.7151 + 24.8940i) q^{13} +28.9752i q^{15} -86.1454 q^{17} +116.880i q^{19} +67.0918i q^{21} +72.0000 q^{23} +31.7151 q^{25} -27.0000 q^{27} +14.1454 q^{29} -196.215i q^{31} -150.915i q^{33} -216.000 q^{35} -154.424i q^{37} +(119.145 - 74.6819i) q^{39} +265.726i q^{41} +211.855 q^{43} -86.9257i q^{45} -67.5535i q^{47} -157.145 q^{49} +258.436 q^{51} +686.581 q^{53} +485.866 q^{55} -350.639i q^{57} +91.9304i q^{59} +329.006 q^{61} -201.275i q^{63} +(240.436 + 383.585i) q^{65} +768.370i q^{67} -216.000 q^{69} +264.969i q^{71} +771.306i q^{73} -95.1454 q^{75} +1125.02 q^{77} -1226.86 q^{79} +81.0000 q^{81} -514.019i q^{83} +832.027i q^{85} -42.4361 q^{87} +527.889i q^{89} +(556.727 + 888.186i) q^{91} +588.646i q^{93} +1128.87 q^{95} +74.2755i q^{97} +452.745i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 12 q^{3} + 36 q^{9}+O(q^{10}) \) 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

Modular forms

Varieties

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Representations

Groups

Database

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