LMFDB - Embedded newform 507.4.b.b.337.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [507,4,Mod(337,507)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(507, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("507.337");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 507 = 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	507.b (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:	$29.9139683729$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 39)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.2
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	507.337
Dual form			507.4.b.b.337.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.00000 q^{3} +8.00000 q^{4} +12.0000i q^{5} +2.00000i q^{7} +9.00000 q^{9} +O(q^{10})$	$q-3.00000 q^{3} +8.00000 q^{4} +12.0000i q^{5} +2.00000i q^{7} +9.00000 q^{9} -36.0000i q^{11} -24.0000 q^{12} -36.0000i q^{15} +64.0000 q^{16} +78.0000 q^{17} -74.0000i q^{19} +96.0000i q^{20} -6.00000i q^{21} +96.0000 q^{23} -19.0000 q^{25} -27.0000 q^{27} +16.0000i q^{28} +18.0000 q^{29} +214.000i q^{31} +108.000i q^{33} -24.0000 q^{35} +72.0000 q^{36} -286.000i q^{37} +384.000i q^{41} -524.000 q^{43} -288.000i q^{44} +108.000i q^{45} +300.000i q^{47} -192.000 q^{48} +339.000 q^{49} -234.000 q^{51} +558.000 q^{53} +432.000 q^{55} +222.000i q^{57} +576.000i q^{59} -288.000i q^{60} +74.0000 q^{61} +18.0000i q^{63} +512.000 q^{64} -38.0000i q^{67} +624.000 q^{68} -288.000 q^{69} +456.000i q^{71} -682.000i q^{73} +57.0000 q^{75} -592.000i q^{76} +72.0000 q^{77} +704.000 q^{79} +768.000i q^{80} +81.0000 q^{81} +888.000i q^{83} -48.0000i q^{84} +936.000i q^{85} -54.0000 q^{87} -1020.00i q^{89} +768.000 q^{92} -642.000i q^{93} +888.000 q^{95} -110.000i q^{97} -324.000i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{3} + 16 q^{4} + 18 q^{9}+O(q^{10}) $ 2 * q - 6 * q^3 + 16 * q^4 + 18 * q^9	$ 2 q - 6 q^{3} + 16 q^{4} + 18 q^{9} - 48 q^{12} + 128 q^{16} + 156 q^{17} + 192 q^{23} - 38 q^{25} - 54 q^{27} + 36 q^{29} - 48 q^{35} + 144 q^{36} - 1048 q^{43} - 384 q^{48} + 678 q^{49} - 468 q^{51} + 1116 q^{53} + 864 q^{55} + 148 q^{61} + 1024 q^{64} + 1248 q^{68} - 576 q^{69} + 114 q^{75} + 144 q^{77} + 1408 q^{79} + 162 q^{81} - 108 q^{87} + 1536 q^{92} + 1776 q^{95}+O(q^{100}) $ 2 * q - 6 * q^3 + 16 * q^4 + 18 * q^9 - 48 * q^12 + 128 * q^16 + 156 * q^17 + 192 * q^23 - 38 * q^25 - 54 * q^27 + 36 * q^29 - 48 * q^35 + 144 * q^36 - 1048 * q^43 - 384 * q^48 + 678 * q^49 - 468 * q^51 + 1116 * q^53 + 864 * q^55 + 148 * q^61 + 1024 * q^64 + 1248 * q^68 - 576 * q^69 + 114 * q^75 + 144 * q^77 + 1408 * q^79 + 162 * q^81 - 108 * q^87 + 1536 * q^92 + 1776 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/507\mathbb{Z}\right)^\times$.

$n$	$170$	$340$
$\chi(n)$	$1$	$-1$

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	1.00000			$0$
$2$	0	0	−1.00000			$\pi$
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	8.00000	1.00000
$5$	12.0000i	1.07331i	0.843801	+	0.536656i	$0.180313\pi$
$5$	12.0000i	1.07331i	−0.843801	+	0.536656i	$0.819687\pi$
$6$	0	0
$7$	2.00000i	0.107990i	0.998541	+	0.0539949i	$0.0171955\pi$
$7$	2.00000i	0.107990i	−0.998541	+	0.0539949i	$0.982805\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	− 36.0000i	− 0.986764i	−0.869813	−	0.493382i	$-0.835760\pi$
$11$	− 36.0000i	− 0.986764i	0.869813	−	0.493382i	$-0.164240\pi$
$12$	−24.0000	−0.577350
$13$	0	0
$13$	0	0
$14$	0	0
$15$	− 36.0000i	− 0.619677i
$16$	64.0000	1.00000
$17$	78.0000	1.11281	0.556405	−	0.830911i	$-0.312180\pi$
$17$	78.0000	1.11281	0.556405	+	0.830911i	$0.312180\pi$
$18$	0	0
$19$	− 74.0000i	− 0.893514i	−0.894655	−	0.446757i	$-0.852579\pi$
$19$	− 74.0000i	− 0.893514i	0.894655	−	0.446757i	$-0.147421\pi$
$20$	96.0000i	1.07331i
$21$	− 6.00000i	− 0.0623480i
$22$	0	0
$23$	96.0000	0.870321	0.435161	−	0.900353i	$-0.356692\pi$
$23$	96.0000	0.870321	0.435161	+	0.900353i	$0.356692\pi$
$24$	0	0
$25$	−19.0000	−0.152000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	16.0000i	0.107990i
$29$	18.0000	0.115259	0.0576296	−	0.998338i	$-0.481646\pi$
$29$	18.0000	0.115259	0.0576296	+	0.998338i	$0.481646\pi$
$30$	0	0
$31$	214.000i	1.23986i	0.784659	+	0.619928i	$0.212838\pi$
$31$	214.000i	1.23986i	−0.784659	+	0.619928i	$0.787162\pi$
$32$	0	0
$33$	108.000i	0.569709i
$34$	0	0
$35$	−24.0000	−0.115907
$36$	72.0000	0.333333
$37$	− 286.000i	− 1.27076i	−0.772200	−	0.635380i	$-0.780844\pi$
$37$	− 286.000i	− 1.27076i	0.772200	−	0.635380i	$-0.219156\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	384.000i	1.46270i	0.682002	+	0.731350i	$0.261110\pi$
$41$	384.000i	1.46270i	−0.682002	+	0.731350i	$0.738890\pi$
$42$	0	0
$43$	−524.000	−1.85835	−0.929177	−	0.369634i	$-0.879483\pi$
$43$	−524.000	−1.85835	−0.929177	+	0.369634i	$0.879483\pi$
$44$	− 288.000i	− 0.986764i
$45$	108.000i	0.357771i
$46$	0	0
$47$	300.000i	0.931053i	0.885034	+	0.465527i	$0.154135\pi$
$47$	300.000i	0.931053i	−0.885034	+	0.465527i	$0.845865\pi$
$48$	−192.000	−0.577350
$49$	339.000	0.988338
$50$	0	0
$51$	−234.000	−0.642481
$52$	0	0
$53$	558.000	1.44617	0.723087	−	0.690757i	$-0.242723\pi$
$53$	558.000	1.44617	0.723087	+	0.690757i	$0.242723\pi$
$54$	0	0
$55$	432.000	1.05911
$56$	0	0
$57$	222.000i	0.515870i
$58$	0	0
$59$	576.000i	1.27100i	0.772102	+	0.635498i	$0.219205\pi$
$59$	576.000i	1.27100i	−0.772102	+	0.635498i	$0.780795\pi$
$60$	− 288.000i	− 0.619677i
$61$	74.0000	0.155323	0.0776617	−	0.996980i	$-0.475255\pi$
$61$	74.0000	0.155323	0.0776617	+	0.996980i	$0.475255\pi$
$62$	0	0
$63$	18.0000i	0.0359966i
$64$	512.000	1.00000
$65$	0	0
$66$	0	0
$67$	− 38.0000i	− 0.0692901i	−0.999400	−	0.0346451i	$-0.988970\pi$
$67$	− 38.0000i	− 0.0692901i	0.999400	−	0.0346451i	$-0.0110301\pi$
$68$	624.000	1.11281
$69$	−288.000	−0.502480
$70$	0	0
$71$	456.000i	0.762215i	0.924531	+	0.381107i	$0.124457\pi$
$71$	456.000i	0.762215i	−0.924531	+	0.381107i	$0.875543\pi$
$72$	0	0
$73$	− 682.000i	− 1.09345i	−0.837311	−	0.546726i	$-0.815874\pi$
$73$	− 682.000i	− 1.09345i	0.837311	−	0.546726i	$-0.184126\pi$
$74$	0	0
$75$	57.0000	0.0877572
$76$	− 592.000i	− 0.893514i
$77$	72.0000	0.106561
$78$	0	0
$79$	704.000	1.00261	0.501305	−	0.865271i	$-0.332853\pi$
$79$	704.000	1.00261	0.501305	+	0.865271i	$0.332853\pi$
$80$	768.000i	1.07331i
$81$	81.0000	0.111111
$82$	0	0
$83$	888.000i	1.17435i	0.809462	+	0.587173i	$0.199759\pi$
$83$	888.000i	1.17435i	−0.809462	+	0.587173i	$0.800241\pi$
$84$	− 48.0000i	− 0.0623480i
$85$	936.000i	1.19439i
$86$	0	0
$87$	−54.0000	−0.0665449
$88$	0	0
$89$	− 1020.00i	− 1.21483i	−0.794385	−	0.607415i	$-0.792207\pi$
$89$	− 1020.00i	− 1.21483i	0.794385	−	0.607415i	$-0.207793\pi$
$90$	0	0
$91$	0	0
$92$	768.000	0.870321
$93$	− 642.000i	− 0.715831i
$94$	0	0
$95$	888.000	0.959020
$96$	0	0
$97$	− 110.000i	− 0.115142i	−0.998341	−	0.0575712i	$-0.981664\pi$
$97$	− 110.000i	− 0.115142i	0.998341	−	0.0575712i	$-0.0183356\pi$
$98$	0	0
$99$	− 324.000i	− 0.328921i
$100$	−152.000	−0.152000
$101$	990.000	0.975333	0.487667	−	0.873030i	$-0.337848\pi$
$101$	990.000	0.975333	0.487667	+	0.873030i	$0.337848\pi$
$102$	0	0
$103$	−1208.00	−1.15561	−0.577805	−	0.816175i	$-0.696090\pi$
$103$	−1208.00	−1.15561	−0.577805	+	0.816175i	$0.696090\pi$
$104$	0	0
$105$	72.0000	0.0669189
$106$	0	0
$107$	996.000	0.899878	0.449939	−	0.893059i	$-0.351446\pi$
$107$	996.000	0.899878	0.449939	+	0.893059i	$0.351446\pi$
$108$	−216.000	−0.192450
$109$	1402.00i	1.23199i	0.787749	+	0.615997i	$0.211246\pi$
$109$	1402.00i	1.23199i	−0.787749	+	0.615997i	$0.788754\pi$
$110$	0	0
$111$	858.000i	0.733673i
$112$	128.000i	0.107990i
$113$	1926.00	1.60339	0.801694	−	0.597735i	$-0.203932\pi$
$113$	1926.00	1.60339	0.801694	+	0.597735i	$0.203932\pi$
$114$	0	0
$115$	1152.00i	0.934127i
$116$	144.000	0.115259
$117$	0	0
$118$	0	0
$119$	156.000i	0.120172i
$120$	0	0
$121$	35.0000	0.0262960
$122$	0	0
$123$	− 1152.00i	− 0.844491i
$124$	1712.00i	1.23986i
$125$	1272.00i	0.910169i
$126$	0	0
$127$	988.000	0.690321	0.345161	−	0.938544i	$-0.387824\pi$
$127$	988.000	0.690321	0.345161	+	0.938544i	$0.387824\pi$
$128$	0	0
$129$	1572.00	1.07292
$130$	0	0
$131$	−2100.00	−1.40059	−0.700297	−	0.713851i	$-0.746949\pi$
$131$	−2100.00	−1.40059	−0.700297	+	0.713851i	$0.746949\pi$
$132$	864.000i	0.569709i
$133$	148.000	0.0964904
$134$	0	0
$135$	− 324.000i	− 0.206559i
$136$	0	0
$137$	− 2496.00i	− 1.55655i	−0.627922	−	0.778276i	$-0.716094\pi$
$137$	− 2496.00i	− 1.55655i	0.627922	−	0.778276i	$-0.283906\pi$
$138$	0	0
$139$	−2464.00	−1.50355	−0.751776	−	0.659418i	$-0.770803\pi$
$139$	−2464.00	−1.50355	−0.751776	+	0.659418i	$0.770803\pi$
$140$	−192.000	−0.115907
$141$	− 900.000i	− 0.537544i
$142$	0	0
$143$	0	0
$144$	576.000	0.333333
$145$	216.000i	0.123709i
$146$	0	0
$147$	−1017.00	−0.570617
$148$	− 2288.00i	− 1.27076i
$149$	− 216.000i	− 0.118761i	−0.998235	−	0.0593806i	$-0.981087\pi$
$149$	− 216.000i	− 0.118761i	0.998235	−	0.0593806i	$-0.0189125\pi$
$150$	0	0
$151$	− 898.000i	− 0.483962i	−0.970281	−	0.241981i	$-0.922203\pi$
$151$	− 898.000i	− 0.483962i	0.970281	−	0.241981i	$-0.0777971\pi$
$152$	0	0
$153$	702.000	0.370937
$154$	0	0
$155$	−2568.00	−1.33075
$156$	0	0
$157$	−1510.00	−0.767587	−0.383793	−	0.923419i	$-0.625383\pi$
$157$	−1510.00	−0.767587	−0.383793	+	0.923419i	$0.625383\pi$
$158$	0	0
$159$	−1674.00	−0.834949
$160$	0	0
$161$	192.000i	0.0939858i
$162$	0	0
$163$	− 394.000i	− 0.189328i	−0.995509	−	0.0946640i	$-0.969822\pi$
$163$	− 394.000i	− 0.189328i	0.995509	−	0.0946640i	$-0.0301777\pi$
$164$	3072.00i	1.46270i
$165$	−1296.00	−0.611476
$166$	0	0
$167$	84.0000i	0.0389228i	0.999811	+	0.0194614i	$0.00619515\pi$
$167$	84.0000i	0.0389228i	−0.999811	+	0.0194614i	$0.993805\pi$
$168$	0	0
$169$	0	0
$170$	0	0
$171$	− 666.000i	− 0.297838i
$172$	−4192.00	−1.85835
$173$	−1194.00	−0.524729	−0.262365	−	0.964969i	$-0.584502\pi$
$173$	−1194.00	−0.524729	−0.262365	+	0.964969i	$0.584502\pi$
$174$	0	0
$175$	− 38.0000i	− 0.0164145i
$176$	− 2304.00i	− 0.986764i
$177$	− 1728.00i	− 0.733810i
$178$	0	0
$179$	−3156.00	−1.31782	−0.658912	−	0.752220i	$-0.728983\pi$
$179$	−3156.00	−1.31782	−0.658912	+	0.752220i	$0.728983\pi$
$180$	864.000i	0.357771i
$181$	1078.00	0.442691	0.221346	−	0.975195i	$-0.428955\pi$
$181$	1078.00	0.442691	0.221346	+	0.975195i	$0.428955\pi$
$182$	0	0
$183$	−222.000	−0.0896760
$184$	0	0
$185$	3432.00	1.36392
$186$	0	0
$187$	− 2808.00i	− 1.09808i
$188$	2400.00i	0.931053i
$189$	− 54.0000i	− 0.0207827i
$190$	0	0
$191$	3192.00	1.20924	0.604620	−	0.796514i	$-0.293325\pi$
$191$	3192.00	1.20924	0.604620	+	0.796514i	$0.293325\pi$
$192$	−1536.00	−0.577350
$193$	722.000i	0.269278i	0.990895	+	0.134639i	$0.0429875\pi$
$193$	722.000i	0.269278i	−0.990895	+	0.134639i	$0.957012\pi$
$194$	0	0
$195$	0	0
$196$	2712.00	0.988338
$197$	− 2796.00i	− 1.01120i	−0.862767	−	0.505601i	$-0.831271\pi$
$197$	− 2796.00i	− 1.01120i	0.862767	−	0.505601i	$-0.168729\pi$
$198$	0	0
$199$	340.000	0.121115	0.0605577	−	0.998165i	$-0.480712\pi$
$199$	340.000	0.121115	0.0605577	+	0.998165i	$0.480712\pi$
$200$	0	0
$201$	114.000i	0.0400047i
$202$	0	0
$203$	36.0000i	0.0124468i
$204$	−1872.00	−0.642481
$205$	−4608.00	−1.56994
$206$	0	0
$207$	864.000	0.290107
$208$	0	0
$209$	−2664.00	−0.881688
$210$	0	0
$211$	−1924.00	−0.627742	−0.313871	−	0.949466i	$-0.601626\pi$
$211$	−1924.00	−0.627742	−0.313871	+	0.949466i	$0.601626\pi$
$212$	4464.00	1.44617
$213$	− 1368.00i	− 0.440065i
$214$	0	0
$215$	− 6288.00i	− 1.99460i
$216$	0	0
$217$	−428.000	−0.133892
$218$	0	0
$219$	2046.00i	0.631305i
$220$	3456.00	1.05911
$221$	0	0
$222$	0	0
$223$	− 5042.00i	− 1.51407i	−0.653375	−	0.757034i	$-0.726648\pi$
$223$	− 5042.00i	− 1.51407i	0.653375	−	0.757034i	$-0.273352\pi$
$224$	0	0
$225$	−171.000	−0.0506667
$226$	0	0
$227$	2676.00i	0.782433i	0.920299	+	0.391217i	$0.127946\pi$
$227$	2676.00i	0.782433i	−0.920299	+	0.391217i	$0.872054\pi$
$228$	1776.00i	0.515870i
$229$	− 2410.00i	− 0.695447i	−0.937597	−	0.347723i	$-0.886955\pi$
$229$	− 2410.00i	− 0.695447i	0.937597	−	0.347723i	$-0.113045\pi$
$230$	0	0
$231$	−216.000	−0.0615228
$232$	0	0
$233$	−3726.00	−1.04763	−0.523816	−	0.851831i	$-0.675492\pi$
$233$	−3726.00	−1.04763	−0.523816	+	0.851831i	$0.675492\pi$
$234$	0	0
$235$	−3600.00	−0.999311
$236$	4608.00i	1.27100i
$237$	−2112.00	−0.578857
$238$	0	0
$239$	− 1248.00i	− 0.337767i	−0.985636	−	0.168884i	$-0.945984\pi$
$239$	− 1248.00i	− 0.337767i	0.985636	−	0.168884i	$-0.0540162\pi$
$240$	− 2304.00i	− 0.619677i
$241$	− 4210.00i	− 1.12527i	−0.826706	−	0.562635i	$-0.809788\pi$
$241$	− 4210.00i	− 1.12527i	0.826706	−	0.562635i	$-0.190212\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	592.000	0.155323
$245$	4068.00i	1.06080i
$246$	0	0
$247$	0	0
$248$	0	0
$249$	− 2664.00i	− 0.678009i
$250$	0	0
$251$	7692.00	1.93432	0.967161	−	0.254165i	$-0.0818007\pi$
$251$	7692.00	1.93432	0.967161	+	0.254165i	$0.0818007\pi$
$252$	144.000i	0.0359966i
$253$	− 3456.00i	− 0.858802i
$254$	0	0
$255$	− 2808.00i	− 0.689583i
$256$	4096.00	1.00000
$257$	−1326.00	−0.321843	−0.160921	−	0.986967i	$-0.551447\pi$
$257$	−1326.00	−0.321843	−0.160921	+	0.986967i	$0.551447\pi$
$258$	0	0
$259$	572.000	0.137229
$260$	0	0
$261$	162.000	0.0384197
$262$	0	0
$263$	−6048.00	−1.41801	−0.709003	−	0.705205i	$-0.750855\pi$
$263$	−6048.00	−1.41801	−0.709003	+	0.705205i	$0.750855\pi$
$264$	0	0
$265$	6696.00i	1.55220i
$266$	0	0
$267$	3060.00i	0.701382i
$268$	− 304.000i	− 0.0692901i
$269$	6474.00	1.46739	0.733693	−	0.679481i	$-0.237795\pi$
$269$	6474.00	1.46739	0.733693	+	0.679481i	$0.237795\pi$
$270$	0	0
$271$	5978.00i	1.33999i	0.742365	+	0.669996i	$0.233704\pi$
$271$	5978.00i	1.33999i	−0.742365	+	0.669996i	$0.766296\pi$
$272$	4992.00	1.11281
$273$	0	0
$274$	0	0
$275$	684.000i	0.149988i
$276$	−2304.00	−0.502480
$277$	−8750.00	−1.89797	−0.948983	−	0.315327i	$-0.897886\pi$
$277$	−8750.00	−1.89797	−0.948983	+	0.315327i	$0.897886\pi$
$278$	0	0
$279$	1926.00i	0.413285i
$280$	0	0
$281$	8976.00i	1.90556i	0.303656	+	0.952782i	$0.401793\pi$
$281$	8976.00i	1.90556i	−0.303656	+	0.952782i	$0.598207\pi$
$282$	0	0
$283$	592.000	0.124349	0.0621745	−	0.998065i	$-0.480196\pi$
$283$	592.000	0.124349	0.0621745	+	0.998065i	$0.480196\pi$
$284$	3648.00i	0.762215i
$285$	−2664.00	−0.553690
$286$	0	0
$287$	−768.000	−0.157957
$288$	0	0
$289$	1171.00	0.238347
$290$	0	0
$291$	330.000i	0.0664775i
$292$	− 5456.00i	− 1.09345i
$293$	− 4608.00i	− 0.918779i	−0.888235	−	0.459389i	$-0.848068\pi$
$293$	− 4608.00i	− 0.918779i	0.888235	−	0.459389i	$-0.151932\pi$
$294$	0	0
$295$	−6912.00	−1.36418
$296$	0	0
$297$	972.000i	0.189903i
$298$	0	0
$299$	0	0
$300$	456.000	0.0877572
$301$	− 1048.00i	− 0.200683i
$302$	0	0
$303$	−2970.00	−0.563109
$304$	− 4736.00i	− 0.893514i
$305$	888.000i	0.166711i
$306$	0	0
$307$	− 3166.00i	− 0.588577i	−0.955717	−	0.294289i	$-0.904917\pi$
$307$	− 3166.00i	− 0.588577i	0.955717	−	0.294289i	$-0.0950827\pi$
$308$	576.000	0.106561
$309$	3624.00	0.667191
$310$	0	0
$311$	−2472.00	−0.450721	−0.225361	−	0.974275i	$-0.572356\pi$
$311$	−2472.00	−0.450721	−0.225361	+	0.974275i	$0.572356\pi$
$312$	0	0
$313$	−3094.00	−0.558732	−0.279366	−	0.960185i	$-0.590124\pi$
$313$	−3094.00	−0.558732	−0.279366	+	0.960185i	$0.590124\pi$
$314$	0	0
$315$	−216.000	−0.0386356
$316$	5632.00	1.00261
$317$	− 2316.00i	− 0.410345i	−0.978726	−	0.205173i	$-0.934224\pi$
$317$	− 2316.00i	− 0.410345i	0.978726	−	0.205173i	$-0.0657756\pi$
$318$	0	0
$319$	− 648.000i	− 0.113734i
$320$	6144.00i	1.07331i
$321$	−2988.00	−0.519545
$322$	0	0
$323$	− 5772.00i	− 0.994312i
$324$	648.000	0.111111
$325$	0	0
$326$	0	0
$327$	− 4206.00i	− 0.711292i
$328$	0	0
$329$	−600.000	−0.100544
$330$	0	0
$331$	4426.00i	0.734970i	0.930030	+	0.367485i	$0.119781\pi$
$331$	4426.00i	0.734970i	−0.930030	+	0.367485i	$0.880219\pi$
$332$	7104.00i	1.17435i
$333$	− 2574.00i	− 0.423587i
$334$	0	0
$335$	456.000	0.0743700
$336$	− 384.000i	− 0.0623480i
$337$	−866.000	−0.139982	−0.0699911	−	0.997548i	$-0.522297\pi$
$337$	−866.000	−0.139982	−0.0699911	+	0.997548i	$0.522297\pi$
$338$	0	0
$339$	−5778.00	−0.925716
$340$	7488.00i	1.19439i
$341$	7704.00	1.22345
$342$	0	0
$343$	1364.00i	0.214720i
$344$	0	0
$345$	− 3456.00i	− 0.539318i
$346$	0	0
$347$	−2556.00	−0.395427	−0.197714	−	0.980260i	$-0.563352\pi$
$347$	−2556.00	−0.395427	−0.197714	+	0.980260i	$0.563352\pi$
$348$	−432.000	−0.0665449
$349$	− 11014.0i	− 1.68930i	−0.535318	−	0.844650i	$-0.679808\pi$
$349$	− 11014.0i	− 1.68930i	0.535318	−	0.844650i	$-0.320192\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	9720.00i	1.46556i	0.680465	+	0.732781i	$0.261778\pi$
$353$	9720.00i	1.46556i	−0.680465	+	0.732781i	$0.738222\pi$
$354$	0	0
$355$	−5472.00	−0.818095
$356$	− 8160.00i	− 1.21483i
$357$	− 468.000i	− 0.0693815i
$358$	0	0
$359$	− 2988.00i	− 0.439277i	−0.975581	−	0.219639i	$-0.929512\pi$
$359$	− 2988.00i	− 0.439277i	0.975581	−	0.219639i	$-0.0704879\pi$
$360$	0	0
$361$	1383.00	0.201633
$362$	0	0
$363$	−105.000	−0.0151820
$364$	0	0
$365$	8184.00	1.17362
$366$	0	0
$367$	−2068.00	−0.294138	−0.147069	−	0.989126i	$-0.546984\pi$
$367$	−2068.00	−0.294138	−0.147069	+	0.989126i	$0.546984\pi$
$368$	6144.00	0.870321
$369$	3456.00i	0.487567i
$370$	0	0
$371$	1116.00i	0.156172i
$372$	− 5136.00i	− 0.715831i
$373$	902.000	0.125211	0.0626056	−	0.998038i	$-0.480059\pi$
$373$	902.000	0.125211	0.0626056	+	0.998038i	$0.480059\pi$
$374$	0	0
$375$	− 3816.00i	− 0.525486i
$376$	0	0
$377$	0	0
$378$	0	0
$379$	− 12818.0i	− 1.73725i	−0.495473	−	0.868623i	$-0.665005\pi$
$379$	− 12818.0i	− 1.73725i	0.495473	−	0.868623i	$-0.334995\pi$
$380$	7104.00	0.959020
$381$	−2964.00	−0.398557
$382$	0	0
$383$	1332.00i	0.177708i	0.996045	+	0.0888538i	$0.0283204\pi$
$383$	1332.00i	0.177708i	−0.996045	+	0.0888538i	$0.971680\pi$
$384$	0	0
$385$	864.000i	0.114373i
$386$	0	0
$387$	−4716.00	−0.619452
$388$	− 880.000i	− 0.115142i
$389$	−3054.00	−0.398056	−0.199028	−	0.979994i	$-0.563779\pi$
$389$	−3054.00	−0.398056	−0.199028	+	0.979994i	$0.563779\pi$
$390$	0	0
$391$	7488.00	0.968502
$392$	0	0
$393$	6300.00	0.808633
$394$	0	0
$395$	8448.00i	1.07611i
$396$	− 2592.00i	− 0.328921i
$397$	11162.0i	1.41110i	0.708663	+	0.705548i	$0.249299\pi$
$397$	11162.0i	1.41110i	−0.708663	+	0.705548i	$0.750701\pi$
$398$	0	0
$399$	−444.000	−0.0557088
$400$	−1216.00	−0.152000
$401$	− 14820.0i	− 1.84557i	−0.385310	−	0.922787i	$-0.625905\pi$
$401$	− 14820.0i	− 1.84557i	0.385310	−	0.922787i	$-0.374095\pi$
$402$	0	0
$403$	0	0
$404$	7920.00	0.975333
$405$	972.000i	0.119257i
$406$	0	0
$407$	−10296.0	−1.25394
$408$	0	0
$409$	9682.00i	1.17052i	0.810844	+	0.585262i	$0.199008\pi$
$409$	9682.00i	1.17052i	−0.810844	+	0.585262i	$0.800992\pi$
$410$	0	0
$411$	7488.00i	0.898676i
$412$	−9664.00	−1.15561
$413$	−1152.00	−0.137255
$414$	0	0
$415$	−10656.0	−1.26044
$416$	0	0
$417$	7392.00	0.868076
$418$	0	0
$419$	−348.000	−0.0405750	−0.0202875	−	0.999794i	$-0.506458\pi$
$419$	−348.000	−0.0405750	−0.0202875	+	0.999794i	$0.506458\pi$
$420$	576.000	0.0669189
$421$	− 2486.00i	− 0.287792i	−0.989593	−	0.143896i	$-0.954037\pi$
$421$	− 2486.00i	− 0.287792i	0.989593	−	0.143896i	$-0.0459630\pi$
$422$	0	0
$423$	2700.00i	0.310351i
$424$	0	0
$425$	−1482.00	−0.169147
$426$	0	0
$427$	148.000i	0.0167734i
$428$	7968.00	0.899878
$429$	0	0
$430$	0	0
$431$	1812.00i	0.202508i	0.994861	+	0.101254i	$0.0322855\pi$
$431$	1812.00i	0.202508i	−0.994861	+	0.101254i	$0.967715\pi$
$432$	−1728.00	−0.192450
$433$	6226.00	0.690999	0.345499	−	0.938419i	$-0.387710\pi$
$433$	6226.00	0.690999	0.345499	+	0.938419i	$0.387710\pi$
$434$	0	0
$435$	− 648.000i	− 0.0714235i
$436$	11216.0i	1.23199i
$437$	− 7104.00i	− 0.777644i
$438$	0	0
$439$	12544.0	1.36376	0.681882	−	0.731462i	$-0.261162\pi$
$439$	12544.0	1.36376	0.681882	+	0.731462i	$0.261162\pi$
$440$	0	0
$441$	3051.00	0.329446
$442$	0	0
$443$	−8556.00	−0.917625	−0.458812	−	0.888533i	$-0.651725\pi$
$443$	−8556.00	−0.917625	−0.458812	+	0.888533i	$0.651725\pi$
$444$	6864.00i	0.733673i
$445$	12240.0	1.30389
$446$	0	0
$447$	648.000i	0.0685668i
$448$	1024.00i	0.107990i
$449$	4116.00i	0.432619i	0.976325	+	0.216310i	$0.0694021\pi$
$449$	4116.00i	0.432619i	−0.976325	+	0.216310i	$0.930598\pi$
$450$	0	0
$451$	13824.0	1.44334
$452$	15408.0	1.60339
$453$	2694.00i	0.279415i
$454$	0	0
$455$	0	0
$456$	0	0
$457$	6514.00i	0.666766i	0.942791	+	0.333383i	$0.108190\pi$
$457$	6514.00i	0.666766i	−0.942791	+	0.333383i	$0.891810\pi$
$458$	0	0
$459$	−2106.00	−0.214160
$460$	9216.00i	0.934127i
$461$	− 10500.0i	− 1.06081i	−0.847744	−	0.530405i	$-0.822040\pi$
$461$	− 10500.0i	− 1.06081i	0.847744	−	0.530405i	$-0.177960\pi$
$462$	0	0
$463$	− 5542.00i	− 0.556282i	−0.960540	−	0.278141i	$-0.910282\pi$
$463$	− 5542.00i	− 0.556282i	0.960540	−	0.278141i	$-0.0897183\pi$
$464$	1152.00	0.115259
$465$	7704.00	0.768311
$466$	0	0
$467$	5220.00	0.517244	0.258622	−	0.965979i	$-0.416732\pi$
$467$	5220.00	0.517244	0.258622	+	0.965979i	$0.416732\pi$
$468$	0	0
$469$	76.0000	0.00748263
$470$	0	0
$471$	4530.00	0.443166
$472$	0	0
$473$	18864.0i	1.83376i
$474$	0	0
$475$	1406.00i	0.135814i
$476$	1248.00i	0.120172i
$477$	5022.00	0.482058
$478$	0	0
$479$	11592.0i	1.10575i	0.833266	+	0.552873i	$0.186468\pi$
$479$	11592.0i	1.10575i	−0.833266	+	0.552873i	$0.813532\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	− 576.000i	− 0.0542627i
$484$	280.000	0.0262960
$485$	1320.00	0.123584
$486$	0	0
$487$	− 12170.0i	− 1.13239i	−0.824270	−	0.566196i	$-0.808414\pi$
$487$	− 12170.0i	− 1.13239i	0.824270	−	0.566196i	$-0.191586\pi$
$488$	0	0
$489$	1182.00i	0.109309i
$490$	0	0
$491$	−1812.00	−0.166547	−0.0832733	−	0.996527i	$-0.526537\pi$
$491$	−1812.00	−0.166547	−0.0832733	+	0.996527i	$0.526537\pi$
$492$	− 9216.00i	− 0.844491i
$493$	1404.00	0.128262
$494$	0	0
$495$	3888.00	0.353036
$496$	13696.0i	1.23986i
$497$	−912.000	−0.0823115
$498$	0	0
$499$	1330.00i	0.119317i	0.998219	+	0.0596583i	$0.0190011\pi$
$499$	1330.00i	0.119317i	−0.998219	+	0.0596583i	$0.980999\pi$
$500$	10176.0i	0.910169i
$501$	− 252.000i	− 0.0224721i
$502$	0	0
$503$	−2688.00	−0.238274	−0.119137	−	0.992878i	$-0.538013\pi$
$503$	−2688.00	−0.238274	−0.119137	+	0.992878i	$0.538013\pi$
$504$	0	0
$505$	11880.0i	1.04684i
$506$	0	0
$507$	0	0
$508$	7904.00	0.690321
$509$	− 5124.00i	− 0.446203i	−0.974795	−	0.223101i	$-0.928382\pi$
$509$	− 5124.00i	− 0.446203i	0.974795	−	0.223101i	$-0.0716181\pi$
$510$	0	0
$511$	1364.00	0.118082
$512$	0	0
$513$	1998.00i	0.171957i
$514$	0	0
$515$	− 14496.0i	− 1.24033i
$516$	12576.0	1.07292
$517$	10800.0	0.918730
$518$	0	0
$519$	3582.00	0.302953
$520$	0	0
$521$	−882.000	−0.0741672	−0.0370836	−	0.999312i	$-0.511807\pi$
$521$	−882.000	−0.0741672	−0.0370836	+	0.999312i	$0.511807\pi$
$522$	0	0
$523$	−2320.00	−0.193970	−0.0969852	−	0.995286i	$-0.530920\pi$
$523$	−2320.00	−0.193970	−0.0969852	+	0.995286i	$0.530920\pi$
$524$	−16800.0	−1.40059
$525$	114.000i	0.00947689i
$526$	0	0
$527$	16692.0i	1.37972i
$528$	6912.00i	0.569709i
$529$	−2951.00	−0.242541
$530$	0	0
$531$	5184.00i	0.423666i
$532$	1184.00	0.0964904
$533$	0	0
$534$	0	0
$535$	11952.0i	0.965851i
$536$	0	0
$537$	9468.00	0.760846
$538$	0	0
$539$	− 12204.0i	− 0.975257i
$540$	− 2592.00i	− 0.206559i
$541$	21422.0i	1.70241i	0.524833	+	0.851205i	$0.324128\pi$
$541$	21422.0i	1.70241i	−0.524833	+	0.851205i	$0.675872\pi$
$542$	0	0
$543$	−3234.00	−0.255588
$544$	0	0
$545$	−16824.0	−1.32231
$546$	0	0
$547$	7040.00	0.550290	0.275145	−	0.961403i	$-0.411274\pi$
$547$	7040.00	0.550290	0.275145	+	0.961403i	$0.411274\pi$
$548$	− 19968.0i	− 1.55655i
$549$	666.000	0.0517745
$550$	0	0
$551$	− 1332.00i	− 0.102986i
$552$	0	0
$553$	1408.00i	0.108272i
$554$	0	0
$555$	−10296.0	−0.787461
$556$	−19712.0	−1.50355
$557$	− 8400.00i	− 0.638994i	−0.947587	−	0.319497i	$-0.896486\pi$
$557$	− 8400.00i	− 0.638994i	0.947587	−	0.319497i	$-0.103514\pi$
$558$	0	0
$559$	0	0
$560$	−1536.00	−0.115907
$561$	8424.00i	0.633978i
$562$	0	0
$563$	−19044.0	−1.42559	−0.712797	−	0.701371i	$-0.752572\pi$
$563$	−19044.0	−1.42559	−0.712797	+	0.701371i	$0.752572\pi$
$564$	− 7200.00i	− 0.537544i
$565$	23112.0i	1.72094i
$566$	0	0
$567$	162.000i	0.0119989i
$568$	0	0
$569$	4698.00	0.346134	0.173067	−	0.984910i	$-0.444632\pi$
$569$	4698.00	0.346134	0.173067	+	0.984910i	$0.444632\pi$
$570$	0	0
$571$	8728.00	0.639677	0.319838	−	0.947472i	$-0.396371\pi$
$571$	8728.00	0.639677	0.319838	+	0.947472i	$0.396371\pi$
$572$	0	0
$573$	−9576.00	−0.698156
$574$	0	0
$575$	−1824.00	−0.132289
$576$	4608.00	0.333333
$577$	− 2018.00i	− 0.145599i	−0.997347	−	0.0727993i	$-0.976807\pi$
$577$	− 2018.00i	− 0.145599i	0.997347	−	0.0727993i	$-0.0231933\pi$
$578$	0	0
$579$	− 2166.00i	− 0.155468i
$580$	1728.00i	0.123709i
$581$	−1776.00	−0.126817
$582$	0	0
$583$	− 20088.0i	− 1.42703i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 11376.0i	− 0.799894i	−0.916538	−	0.399947i	$-0.869029\pi$
$587$	− 11376.0i	− 0.799894i	0.916538	−	0.399947i	$-0.130971\pi$
$588$	−8136.00	−0.570617
$589$	15836.0	1.10783
$590$	0	0
$591$	8388.00i	0.583818i
$592$	− 18304.0i	− 1.27076i
$593$	− 25596.0i	− 1.77252i	−0.463192	−	0.886258i	$-0.653296\pi$
$593$	− 25596.0i	− 1.77252i	0.463192	−	0.886258i	$-0.346704\pi$
$594$	0	0
$595$	−1872.00	−0.128982
$596$	− 1728.00i	− 0.118761i
$597$	−1020.00	−0.0699260
$598$	0	0
$599$	3480.00	0.237377	0.118689	−	0.992932i	$-0.462131\pi$
$599$	3480.00	0.237377	0.118689	+	0.992932i	$0.462131\pi$
$600$	0	0
$601$	10010.0	0.679395	0.339698	−	0.940535i	$-0.389675\pi$
$601$	10010.0	0.679395	0.339698	+	0.940535i	$0.389675\pi$
$602$	0	0
$603$	− 342.000i	− 0.0230967i
$604$	− 7184.00i	− 0.483962i
$605$	420.000i	0.0282238i
$606$	0	0
$607$	3764.00	0.251690	0.125845	−	0.992050i	$-0.459836\pi$
$607$	3764.00	0.251690	0.125845	+	0.992050i	$0.459836\pi$
$608$	0	0
$609$	− 108.000i	− 0.00718618i
$610$	0	0
$611$	0	0
$612$	5616.00	0.370937
$613$	− 13610.0i	− 0.896742i	−0.893848	−	0.448371i	$-0.852004\pi$
$613$	− 13610.0i	− 0.896742i	0.893848	−	0.448371i	$-0.147996\pi$
$614$	0	0
$615$	13824.0	0.906402
$616$	0	0
$617$	− 6408.00i	− 0.418114i	−0.977903	−	0.209057i	$-0.932961\pi$
$617$	− 6408.00i	− 0.418114i	0.977903	−	0.209057i	$-0.0670394\pi$
$618$	0	0
$619$	− 6694.00i	− 0.434660i	−0.976098	−	0.217330i	$-0.930265\pi$
$619$	− 6694.00i	− 0.434660i	0.976098	−	0.217330i	$-0.0697348\pi$
$620$	−20544.0	−1.33075
$621$	−2592.00	−0.167493
$622$	0	0
$623$	2040.00	0.131189
$624$	0	0
$625$	−17639.0	−1.12890
$626$	0	0
$627$	7992.00	0.509043
$628$	−12080.0	−0.767587
$629$	− 22308.0i	− 1.41411i
$630$	0	0
$631$	− 27250.0i	− 1.71918i	−0.510981	−	0.859592i	$-0.670718\pi$
$631$	− 27250.0i	− 1.71918i	0.510981	−	0.859592i	$-0.329282\pi$
$632$	0	0
$633$	5772.00	0.362427
$634$	0	0
$635$	11856.0i	0.740931i
$636$	−13392.0	−0.834949
$637$	0	0
$638$	0	0
$639$	4104.00i	0.254072i
$640$	0	0
$641$	12630.0	0.778245	0.389122	−	0.921186i	$-0.372778\pi$
$641$	12630.0	0.778245	0.389122	+	0.921186i	$0.372778\pi$
$642$	0	0
$643$	− 14798.0i	− 0.907583i	−0.891108	−	0.453792i	$-0.850071\pi$
$643$	− 14798.0i	− 0.907583i	0.891108	−	0.453792i	$-0.149929\pi$
$644$	1536.00i	0.0939858i
$645$	18864.0i	1.15158i
$646$	0	0
$647$	−26232.0	−1.59395	−0.796976	−	0.604012i	$-0.793568\pi$
$647$	−26232.0	−1.59395	−0.796976	+	0.604012i	$0.793568\pi$
$648$	0	0
$649$	20736.0	1.25417
$650$	0	0
$651$	1284.00	0.0773025
$652$	− 3152.00i	− 0.189328i
$653$	−30390.0	−1.82121	−0.910607	−	0.413274i	$-0.864385\pi$
$653$	−30390.0	−1.82121	−0.910607	+	0.413274i	$0.864385\pi$
$654$	0	0
$655$	− 25200.0i	− 1.50328i
$656$	24576.0i	1.46270i
$657$	− 6138.00i	− 0.364484i
$658$	0	0
$659$	−28740.0	−1.69886	−0.849432	−	0.527698i	$-0.823055\pi$
$659$	−28740.0	−1.69886	−0.849432	+	0.527698i	$0.823055\pi$
$660$	−10368.0	−0.611476
$661$	− 9214.00i	− 0.542183i	−0.962554	−	0.271092i	$-0.912615\pi$
$661$	− 9214.00i	− 0.542183i	0.962554	−	0.271092i	$-0.0873846\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1776.00i	0.103564i
$666$	0	0
$667$	1728.00	0.100312
$668$	672.000i	0.0389228i
$669$	15126.0i	0.874148i
$670$	0	0
$671$	− 2664.00i	− 0.153268i
$672$	0	0
$673$	−16598.0	−0.950677	−0.475339	−	0.879803i	$-0.657674\pi$
$673$	−16598.0	−0.950677	−0.475339	+	0.879803i	$0.657674\pi$
$674$	0	0
$675$	513.000	0.0292524
$676$	0	0
$677$	−8610.00	−0.488788	−0.244394	−	0.969676i	$-0.578589\pi$
$677$	−8610.00	−0.488788	−0.244394	+	0.969676i	$0.578589\pi$
$678$	0	0
$679$	220.000	0.0124342
$680$	0	0
$681$	− 8028.00i	− 0.451738i
$682$	0	0
$683$	804.000i	0.0450428i	0.999746	+	0.0225214i	$0.00716938\pi$
$683$	804.000i	0.0450428i	−0.999746	+	0.0225214i	$0.992831\pi$
$684$	− 5328.00i	− 0.297838i
$685$	29952.0	1.67067
$686$	0	0
$687$	7230.00i	0.401516i
$688$	−33536.0	−1.85835
$689$	0	0
$690$	0	0
$691$	− 2270.00i	− 0.124971i	−0.998046	−	0.0624854i	$-0.980097\pi$
$691$	− 2270.00i	− 0.124971i	0.998046	−	0.0624854i	$-0.0199027\pi$
$692$	−9552.00	−0.524729
$693$	648.000	0.0355202
$694$	0	0
$695$	− 29568.0i	− 1.61378i
$696$	0	0
$697$	29952.0i	1.62771i
$698$	0	0
$699$	11178.0	0.604851
$700$	− 304.000i	− 0.0164145i
$701$	−1782.00	−0.0960131	−0.0480066	−	0.998847i	$-0.515287\pi$
$701$	−1782.00	−0.0960131	−0.0480066	+	0.998847i	$0.515287\pi$
$702$	0	0
$703$	−21164.0	−1.13544
$704$	− 18432.0i	− 0.986764i
$705$	10800.0	0.576953
$706$	0	0
$707$	1980.00i	0.105326i
$708$	− 13824.0i	− 0.733810i
$709$	− 10690.0i	− 0.566250i	−0.959083	−	0.283125i	$-0.908629\pi$
$709$	− 10690.0i	− 0.566250i	0.959083	−	0.283125i	$-0.0913712\pi$
$710$	0	0
$711$	6336.00	0.334203
$712$	0	0
$713$	20544.0i	1.07907i
$714$	0	0
$715$	0	0
$716$	−25248.0	−1.31782
$717$	3744.00i	0.195010i
$718$	0	0
$719$	−11568.0	−0.600019	−0.300009	−	0.953936i	$-0.596990\pi$
$719$	−11568.0	−0.600019	−0.300009	+	0.953936i	$0.596990\pi$
$720$	6912.00i	0.357771i
$721$	− 2416.00i	− 0.124794i
$722$	0	0
$723$	12630.0i	0.649675i
$724$	8624.00	0.442691
$725$	−342.000	−0.0175194
$726$	0	0
$727$	11644.0	0.594019	0.297010	−	0.954874i	$-0.404011\pi$
$727$	11644.0	0.594019	0.297010	+	0.954874i	$0.404011\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−40872.0	−2.06800
$732$	−1776.00	−0.0896760
$733$	15010.0i	0.756353i	0.925733	+	0.378177i	$0.123449\pi$
$733$	15010.0i	0.756353i	−0.925733	+	0.378177i	$0.876551\pi$
$734$	0	0
$735$	− 12204.0i	− 0.612451i
$736$	0	0
$737$	−1368.00	−0.0683730
$738$	0	0
$739$	33410.0i	1.66307i	0.555474	+	0.831534i	$0.312537\pi$
$739$	33410.0i	1.66307i	−0.555474	+	0.831534i	$0.687463\pi$
$740$	27456.0	1.36392
$741$	0	0
$742$	0	0
$743$	6504.00i	0.321142i	0.987024	+	0.160571i	$0.0513336\pi$
$743$	6504.00i	0.321142i	−0.987024	+	0.160571i	$0.948666\pi$
$744$	0	0
$745$	2592.00	0.127468
$746$	0	0
$747$	7992.00i	0.391448i
$748$	− 22464.0i	− 1.09808i
$749$	1992.00i	0.0971777i
$750$	0	0
$751$	13912.0	0.675973	0.337987	−	0.941151i	$-0.390254\pi$
$751$	13912.0	0.675973	0.337987	+	0.941151i	$0.390254\pi$
$752$	19200.0i	0.931053i
$753$	−23076.0	−1.11678
$754$	0	0
$755$	10776.0	0.519442
$756$	− 432.000i	− 0.0207827i
$757$	−23974.0	−1.15106	−0.575528	−	0.817782i	$-0.695204\pi$
$757$	−23974.0	−1.15106	−0.575528	+	0.817782i	$0.695204\pi$
$758$	0	0
$759$	10368.0i	0.495829i
$760$	0	0
$761$	− 288.000i	− 0.0137188i	−0.999976	−	0.00685939i	$-0.997817\pi$
$761$	− 288.000i	− 0.0137188i	0.999976	−	0.00685939i	$-0.00218343\pi$
$762$	0	0
$763$	−2804.00	−0.133043
$764$	25536.0	1.20924
$765$	8424.00i	0.398131i
$766$	0	0
$767$	0	0
$768$	−12288.0	−0.577350
$769$	− 1514.00i	− 0.0709964i	−0.999370	−	0.0354982i	$-0.988698\pi$
$769$	− 1514.00i	− 0.0709964i	0.999370	−	0.0354982i	$-0.0113018\pi$
$770$	0	0
$771$	3978.00	0.185816
$772$	5776.00i	0.269278i
$773$	− 15816.0i	− 0.735915i	−0.929843	−	0.367957i	$-0.880057\pi$
$773$	− 15816.0i	− 0.735915i	0.929843	−	0.367957i	$-0.119943\pi$
$774$	0	0
$775$	− 4066.00i	− 0.188458i
$776$	0	0
$777$	−1716.00	−0.0792293
$778$	0	0
$779$	28416.0	1.30694
$780$	0	0
$781$	16416.0	0.752126
$782$	0	0
$783$	−486.000	−0.0221816
$784$	21696.0	0.988338
$785$	− 18120.0i	− 0.823861i
$786$	0	0
$787$	10154.0i	0.459912i	0.973201	+	0.229956i	$0.0738583\pi$
$787$	10154.0i	0.459912i	−0.973201	+	0.229956i	$0.926142\pi$
$788$	− 22368.0i	− 1.01120i
$789$	18144.0	0.818686
$790$	0	0
$791$	3852.00i	0.173150i
$792$	0	0
$793$	0	0
$794$	0	0
$795$	− 20088.0i	− 0.896161i
$796$	2720.00	0.121115
$797$	17442.0	0.775191	0.387596	−	0.921830i	$-0.373306\pi$
$797$	17442.0	0.775191	0.387596	+	0.921830i	$0.373306\pi$
$798$	0	0
$799$	23400.0i	1.03609i
$800$	0	0
$801$	− 9180.00i	− 0.404943i
$802$	0	0
$803$	−24552.0	−1.07898
$804$	912.000i	0.0400047i
$805$	−2304.00	−0.100876
$806$	0	0
$807$	−19422.0	−0.847196
$808$	0	0
$809$	−8778.00	−0.381481	−0.190740	−	0.981641i	$-0.561089\pi$
$809$	−8778.00	−0.381481	−0.190740	+	0.981641i	$0.561089\pi$
$810$	0	0
$811$	430.000i	0.0186182i	0.999957	+	0.00930909i	$0.00296322\pi$
$811$	430.000i	0.0186182i	−0.999957	+	0.00930909i	$0.997037\pi$
$812$	288.000i	0.0124468i
$813$	− 17934.0i	− 0.773644i
$814$	0	0
$815$	4728.00	0.203208
$816$	−14976.0	−0.642481
$817$	38776.0i	1.66047i
$818$	0	0
$819$	0	0
$820$	−36864.0	−1.56994
$821$	32976.0i	1.40179i	0.713264	+	0.700895i	$0.247216\pi$
$821$	32976.0i	1.40179i	−0.713264	+	0.700895i	$0.752784\pi$
$822$	0	0
$823$	1168.00	0.0494701	0.0247351	−	0.999694i	$-0.492126\pi$
$823$	1168.00	0.0494701	0.0247351	+	0.999694i	$0.492126\pi$
$824$	0	0
$825$	− 2052.00i	− 0.0865957i
$826$	0	0
$827$	− 17172.0i	− 0.722042i	−0.932558	−	0.361021i	$-0.882428\pi$
$827$	− 17172.0i	− 0.722042i	0.932558	−	0.361021i	$-0.117572\pi$
$828$	6912.00	0.290107
$829$	−27146.0	−1.13730	−0.568649	−	0.822580i	$-0.692534\pi$
$829$	−27146.0	−1.13730	−0.568649	+	0.822580i	$0.692534\pi$
$830$	0	0
$831$	26250.0	1.09579
$832$	0	0
$833$	26442.0	1.09983
$834$	0	0
$835$	−1008.00	−0.0417764
$836$	−21312.0	−0.881688
$837$	− 5778.00i	− 0.238610i
$838$	0	0
$839$	30696.0i	1.26310i	0.775334	+	0.631552i	$0.217582\pi$
$839$	30696.0i	1.26310i	−0.775334	+	0.631552i	$0.782418\pi$
$840$	0	0
$841$	−24065.0	−0.986715
$842$	0	0
$843$	− 26928.0i	− 1.10018i
$844$	−15392.0	−0.627742
$845$	0	0
$846$	0	0
$847$	70.0000i	0.00283970i
$848$	35712.0	1.44617
$849$	−1776.00	−0.0717929
$850$	0	0
$851$	− 27456.0i	− 1.10597i
$852$	− 10944.0i	− 0.440065i
$853$	24842.0i	0.997156i	0.866845	+	0.498578i	$0.166144\pi$
$853$	24842.0i	0.997156i	−0.866845	+	0.498578i	$0.833856\pi$
$854$	0	0
$855$	7992.00	0.319673
$856$	0	0
$857$	−11406.0	−0.454634	−0.227317	−	0.973821i	$-0.572995\pi$
$857$	−11406.0	−0.454634	−0.227317	+	0.973821i	$0.572995\pi$
$858$	0	0
$859$	20540.0	0.815851	0.407925	−	0.913015i	$-0.366252\pi$
$859$	20540.0	0.815851	0.407925	+	0.913015i	$0.366252\pi$
$860$	− 50304.0i	− 1.99460i
$861$	2304.00	0.0911964
$862$	0	0
$863$	− 9108.00i	− 0.359258i	−0.983734	−	0.179629i	$-0.942510\pi$
$863$	− 9108.00i	− 0.359258i	0.983734	−	0.179629i	$-0.0574898\pi$
$864$	0	0
$865$	− 14328.0i	− 0.563198i
$866$	0	0
$867$	−3513.00	−0.137610
$868$	−3424.00	−0.133892
$869$	− 25344.0i	− 0.989340i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	− 990.000i	− 0.0383808i
$874$	0	0
$875$	−2544.00	−0.0982890
$876$	16368.0i	0.631305i
$877$	24046.0i	0.925856i	0.886396	+	0.462928i	$0.153201\pi$
$877$	24046.0i	0.925856i	−0.886396	+	0.462928i	$0.846799\pi$
$878$	0	0
$879$	13824.0i	0.530457i
$880$	27648.0	1.05911
$881$	−7998.00	−0.305856	−0.152928	−	0.988237i	$-0.548870\pi$
$881$	−7998.00	−0.305856	−0.152928	+	0.988237i	$0.548870\pi$
$882$	0	0
$883$	−24032.0	−0.915902	−0.457951	−	0.888978i	$-0.651416\pi$
$883$	−24032.0	−0.915902	−0.457951	+	0.888978i	$0.651416\pi$
$884$	0	0
$885$	20736.0	0.787608
$886$	0	0
$887$	−15648.0	−0.592343	−0.296172	−	0.955135i	$-0.595710\pi$
$887$	−15648.0	−0.592343	−0.296172	+	0.955135i	$0.595710\pi$
$888$	0	0
$889$	1976.00i	0.0745477i
$890$	0	0
$891$	− 2916.00i	− 0.109640i
$892$	− 40336.0i	− 1.51407i
$893$	22200.0	0.831909
$894$	0	0
$895$	− 37872.0i	− 1.41444i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	3852.00i	0.142905i
$900$	−1368.00	−0.0506667
$901$	43524.0	1.60932
$902$	0	0
$903$	3144.00i	0.115865i
$904$	0	0
$905$	12936.0i	0.475146i
$906$	0	0
$907$	808.000	0.0295802	0.0147901	−	0.999891i	$-0.495292\pi$
$907$	808.000	0.0295802	0.0147901	+	0.999891i	$0.495292\pi$
$908$	21408.0i	0.782433i
$909$	8910.00	0.325111
$910$	0	0
$911$	39144.0	1.42360	0.711799	−	0.702383i	$-0.247880\pi$
$911$	39144.0	1.42360	0.711799	+	0.702383i	$0.247880\pi$
$912$	14208.0i	0.515870i
$913$	31968.0	1.15880
$914$	0	0
$915$	− 2664.00i	− 0.0962504i
$916$	− 19280.0i	− 0.695447i
$917$	− 4200.00i	− 0.151250i
$918$	0	0
$919$	−38248.0	−1.37289	−0.686445	−	0.727182i	$-0.740830\pi$
$919$	−38248.0	−1.37289	−0.686445	+	0.727182i	$0.740830\pi$
$920$	0	0
$921$	9498.00i	0.339815i
$922$	0	0
$923$	0	0
$924$	−1728.00	−0.0615228
$925$	5434.00i	0.193155i
$926$	0	0
$927$	−10872.0	−0.385203
$928$	0	0
$929$	54264.0i	1.91641i	0.286084	+	0.958205i	$0.407646\pi$
$929$	54264.0i	1.91641i	−0.286084	+	0.958205i	$0.592354\pi$
$930$	0	0
$931$	− 25086.0i	− 0.883094i
$932$	−29808.0	−1.04763
$933$	7416.00	0.260224
$934$	0	0
$935$	33696.0	1.17859
$936$	0	0
$937$	12206.0	0.425563	0.212782	−	0.977100i	$-0.431748\pi$
$937$	12206.0	0.425563	0.212782	+	0.977100i	$0.431748\pi$
$938$	0	0
$939$	9282.00	0.322584
$940$	−28800.0	−0.999311
$941$	− 17664.0i	− 0.611934i	−0.952042	−	0.305967i	$-0.901020\pi$
$941$	− 17664.0i	− 0.611934i	0.952042	−	0.305967i	$-0.0989797\pi$
$942$	0	0
$943$	36864.0i	1.27302i
$944$	36864.0i	1.27100i
$945$	648.000	0.0223063
$946$	0	0
$947$	− 51984.0i	− 1.78379i	−0.452238	−	0.891897i	$-0.649374\pi$
$947$	− 51984.0i	− 1.78379i	0.452238	−	0.891897i	$-0.350626\pi$
$948$	−16896.0	−0.578857
$949$	0	0
$950$	0	0
$951$	6948.00i	0.236913i
$952$	0	0
$953$	−13782.0	−0.468460	−0.234230	−	0.972181i	$-0.575257\pi$
$953$	−13782.0	−0.468460	−0.234230	+	0.972181i	$0.575257\pi$
$954$	0	0
$955$	38304.0i	1.29789i
$956$	− 9984.00i	− 0.337767i
$957$	1944.00i	0.0656642i
$958$	0	0
$959$	4992.00	0.168092
$960$	− 18432.0i	− 0.619677i
$961$	−16005.0	−0.537243
$962$	0	0
$963$	8964.00	0.299959
$964$	− 33680.0i	− 1.12527i
$965$	−8664.00	−0.289020
$966$	0	0
$967$	− 14618.0i	− 0.486125i	−0.970011	−	0.243063i	$-0.921848\pi$
$967$	− 14618.0i	− 0.486125i	0.970011	−	0.243063i	$-0.0781521\pi$
$968$	0	0
$969$	17316.0i	0.574066i
$970$	0	0
$971$	−18708.0	−0.618299	−0.309149	−	0.951013i	$-0.600044\pi$
$971$	−18708.0	−0.618299	−0.309149	+	0.951013i	$0.600044\pi$
$972$	−1944.00	−0.0641500
$973$	− 4928.00i	− 0.162368i
$974$	0	0
$975$	0	0
$976$	4736.00	0.155323
$977$	48804.0i	1.59814i	0.601241	+	0.799068i	$0.294673\pi$
$977$	48804.0i	1.59814i	−0.601241	+	0.799068i	$0.705327\pi$
$978$	0	0
$979$	−36720.0	−1.19875
$980$	32544.0i	1.06080i
$981$	12618.0i	0.410664i
$982$	0	0
$983$	− 44736.0i	− 1.45153i	−0.687941	−	0.725766i	$-0.741485\pi$
$983$	− 44736.0i	− 1.45153i	0.687941	−	0.725766i	$-0.258515\pi$
$984$	0	0
$985$	33552.0	1.08534
$986$	0	0
$987$	1800.00	0.0580493
$988$	0	0
$989$	−50304.0	−1.61737
$990$	0	0
$991$	−21004.0	−0.673274	−0.336637	−	0.941635i	$-0.609289\pi$
$991$	−21004.0	−0.673274	−0.336637	+	0.941635i	$0.609289\pi$
$992$	0	0
$993$	− 13278.0i	− 0.424335i
$994$	0	0
$995$	4080.00i	0.129995i
$996$	− 21312.0i	− 0.678009i
$997$	9038.00	0.287098	0.143549	−	0.989643i	$-0.454149\pi$
$997$	9038.00	0.287098	0.143549	+	0.989643i	$0.454149\pi$
$998$	0	0
$999$	7722.00i	0.244558i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	507.4.b.b.337.2		2
13.5	odd	4		507.4.a.c.1.1		1
13.8	odd	4		39.4.a.a.1.1	✓	1
13.12	even	2	inner	507.4.b.b.337.1		2
39.5	even	4		1521.4.a.f.1.1		1
39.8	even	4		117.4.a.a.1.1		1
52.47	even	4		624.4.a.g.1.1		1
65.34	odd	4		975.4.a.e.1.1		1
91.34	even	4		1911.4.a.f.1.1		1
104.21	odd	4		2496.4.a.o.1.1		1
104.99	even	4		2496.4.a.f.1.1		1
156.47	odd	4		1872.4.a.m.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
39.4.a.a.1.1	✓	1	13.8	odd	4
117.4.a.a.1.1		1	39.8	even	4
507.4.a.c.1.1		1	13.5	odd	4
507.4.b.b.337.1		2	13.12	even	2	inner
507.4.b.b.337.2		2	1.1	even	1	trivial
624.4.a.g.1.1		1	52.47	even	4
975.4.a.e.1.1		1	65.34	odd	4
1521.4.a.f.1.1		1	39.5	even	4
1872.4.a.m.1.1		1	156.47	odd	4
1911.4.a.f.1.1		1	91.34	even	4
2496.4.a.f.1.1		1	104.99	even	4
2496.4.a.o.1.1		1	104.21	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:	\( N \)	\(=\)	\( 507 = 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	507.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-3.00000 q^{3} +8.00000 q^{4} +12.0000i q^{5} +2.00000i q^{7} +9.00000 q^{9} +O(q^{10})\)	\(q-3.00000 q^{3} +8.00000 q^{4} +12.0000i q^{5} +2.00000i q^{7} +9.00000 q^{9} -36.0000i q^{11} -24.0000 q^{12} -36.0000i q^{15} +64.0000 q^{16} +78.0000 q^{17} -74.0000i q^{19} +96.0000i q^{20} -6.00000i q^{21} +96.0000 q^{23} -19.0000 q^{25} -27.0000 q^{27} +16.0000i q^{28} +18.0000 q^{29} +214.000i q^{31} +108.000i q^{33} -24.0000 q^{35} +72.0000 q^{36} -286.000i q^{37} +384.000i q^{41} -524.000 q^{43} -288.000i q^{44} +108.000i q^{45} +300.000i q^{47} -192.000 q^{48} +339.000 q^{49} -234.000 q^{51} +558.000 q^{53} +432.000 q^{55} +222.000i q^{57} +576.000i q^{59} -288.000i q^{60} +74.0000 q^{61} +18.0000i q^{63} +512.000 q^{64} -38.0000i q^{67} +624.000 q^{68} -288.000 q^{69} +456.000i q^{71} -682.000i q^{73} +57.0000 q^{75} -592.000i q^{76} +72.0000 q^{77} +704.000 q^{79} +768.000i q^{80} +81.0000 q^{81} +888.000i q^{83} -48.0000i q^{84} +936.000i q^{85} -54.0000 q^{87} -1020.00i q^{89} +768.000 q^{92} -642.000i q^{93} +888.000 q^{95} -110.000i q^{97} -324.000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{3} + 16 q^{4} + 18 q^{9}+O(q^{10}) \) 2 * q - 6 * q^3 + 16 * q^4 + 18 * q^9	\( 2 q - 6 q^{3} + 16 q^{4} + 18 q^{9} - 48 q^{12} + 128 q^{16} + 156 q^{17} + 192 q^{23} - 38 q^{25} - 54 q^{27} + 36 q^{29} - 48 q^{35} + 144 q^{36} - 1048 q^{43} - 384 q^{48} + 678 q^{49} - 468 q^{51} + 1116 q^{53} + 864 q^{55} + 148 q^{61} + 1024 q^{64} + 1248 q^{68} - 576 q^{69} + 114 q^{75} + 144 q^{77} + 1408 q^{79} + 162 q^{81} - 108 q^{87} + 1536 q^{92} + 1776 q^{95}+O(q^{100}) \) 2 * q - 6 * q^3 + 16 * q^4 + 18 * q^9 - 48 * q^12 + 128 * q^16 + 156 * q^17 + 192 * q^23 - 38 * q^25 - 54 * q^27 + 36 * q^29 - 48 * q^35 + 144 * q^36 - 1048 * q^43 - 384 * q^48 + 678 * q^49 - 468 * q^51 + 1116 * q^53 + 864 * q^55 + 148 * q^61 + 1024 * q^64 + 1248 * q^68 - 576 * q^69 + 114 * q^75 + 144 * q^77 + 1408 * q^79 + 162 * q^81 - 108 * q^87 + 1536 * q^92 + 1776 * q^95

Introduction

L-functions

Modular forms

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