LMFDB - Embedded newform 1014.2.b.b.337.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1014,2,Mod(337,1014)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1014, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1014.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1014 = 2 \cdot 3 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1014.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:	$8.09683076496$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 78)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.1
Root			$1.00000i$ of defining polynomial
Character	$\chi$	$=$	1014.337
Dual form			1014.2.b.b.337.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +2.00000i q^{5} +1.00000i q^{6} -4.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +2.00000i q^{5} +1.00000i q^{6} -4.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +2.00000 q^{10} +4.00000i q^{11} +1.00000 q^{12} -4.00000 q^{14} -2.00000i q^{15} +1.00000 q^{16} -2.00000 q^{17} -1.00000i q^{18} -8.00000i q^{19} -2.00000i q^{20} +4.00000i q^{21} +4.00000 q^{22} -1.00000i q^{24} +1.00000 q^{25} -1.00000 q^{27} +4.00000i q^{28} +6.00000 q^{29} -2.00000 q^{30} -4.00000i q^{31} -1.00000i q^{32} -4.00000i q^{33} +2.00000i q^{34} +8.00000 q^{35} -1.00000 q^{36} +2.00000i q^{37} -8.00000 q^{38} -2.00000 q^{40} -10.0000i q^{41} +4.00000 q^{42} -4.00000 q^{43} -4.00000i q^{44} +2.00000i q^{45} -8.00000i q^{47} -1.00000 q^{48} -9.00000 q^{49} -1.00000i q^{50} +2.00000 q^{51} -10.0000 q^{53} +1.00000i q^{54} -8.00000 q^{55} +4.00000 q^{56} +8.00000i q^{57} -6.00000i q^{58} -4.00000i q^{59} +2.00000i q^{60} -2.00000 q^{61} -4.00000 q^{62} -4.00000i q^{63} -1.00000 q^{64} -4.00000 q^{66} -16.0000i q^{67} +2.00000 q^{68} -8.00000i q^{70} -8.00000i q^{71} +1.00000i q^{72} -2.00000i q^{73} +2.00000 q^{74} -1.00000 q^{75} +8.00000i q^{76} +16.0000 q^{77} +8.00000 q^{79} +2.00000i q^{80} +1.00000 q^{81} -10.0000 q^{82} +12.0000i q^{83} -4.00000i q^{84} -4.00000i q^{85} +4.00000i q^{86} -6.00000 q^{87} -4.00000 q^{88} -14.0000i q^{89} +2.00000 q^{90} +4.00000i q^{93} -8.00000 q^{94} +16.0000 q^{95} +1.00000i q^{96} +10.0000i q^{97} +9.00000i q^{98} +4.00000i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9	$ 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9} + 4 q^{10} + 2 q^{12} - 8 q^{14} + 2 q^{16} - 4 q^{17} + 8 q^{22} + 2 q^{25} - 2 q^{27} + 12 q^{29} - 4 q^{30} + 16 q^{35} - 2 q^{36} - 16 q^{38} - 4 q^{40} + 8 q^{42} - 8 q^{43} - 2 q^{48} - 18 q^{49} + 4 q^{51} - 20 q^{53} - 16 q^{55} + 8 q^{56} - 4 q^{61} - 8 q^{62} - 2 q^{64} - 8 q^{66} + 4 q^{68} + 4 q^{74} - 2 q^{75} + 32 q^{77} + 16 q^{79} + 2 q^{81} - 20 q^{82} - 12 q^{87} - 8 q^{88} + 4 q^{90} - 16 q^{94} + 32 q^{95}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9 + 4 * q^10 + 2 * q^12 - 8 * q^14 + 2 * q^16 - 4 * q^17 + 8 * q^22 + 2 * q^25 - 2 * q^27 + 12 * q^29 - 4 * q^30 + 16 * q^35 - 2 * q^36 - 16 * q^38 - 4 * q^40 + 8 * q^42 - 8 * q^43 - 2 * q^48 - 18 * q^49 + 4 * q^51 - 20 * q^53 - 16 * q^55 + 8 * q^56 - 4 * q^61 - 8 * q^62 - 2 * q^64 - 8 * q^66 + 4 * q^68 + 4 * q^74 - 2 * q^75 + 32 * q^77 + 16 * q^79 + 2 * q^81 - 20 * q^82 - 12 * q^87 - 8 * q^88 + 4 * q^90 - 16 * q^94 + 32 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1014\mathbb{Z}\right)^\times$.

$n$	$677$	$847$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 1.00000i	− 0.707107i
$2$	− 1.00000i	− 0.707107i
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−1.00000	−0.500000
$5$	2.00000i	0.894427i	0.894427	+	0.447214i	$0.147584\pi$
$5$	2.00000i	0.894427i	−0.894427	+	0.447214i	$0.852416\pi$
$6$	1.00000i	0.408248i
$7$	− 4.00000i	− 1.51186i	−0.654654	−	0.755929i	$-0.727186\pi$
$7$	− 4.00000i	− 1.51186i	0.654654	−	0.755929i	$-0.272814\pi$
$8$	1.00000i	0.353553i
$9$	1.00000	0.333333
$10$	2.00000	0.632456
$11$	4.00000i	1.20605i	0.797724	+	0.603023i	$0.206037\pi$
$11$	4.00000i	1.20605i	−0.797724	+	0.603023i	$0.793963\pi$
$12$	1.00000	0.288675
$13$	0	0
$13$	0	0
$14$	−4.00000	−1.06904
$15$	− 2.00000i	− 0.516398i
$16$	1.00000	0.250000
$17$	−2.00000	−0.485071	−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000	−0.485071	−0.242536	+	0.970143i	$0.577979\pi$
$18$	− 1.00000i	− 0.235702i
$19$	− 8.00000i	− 1.83533i	−0.397360	−	0.917663i	$-0.630073\pi$
$19$	− 8.00000i	− 1.83533i	0.397360	−	0.917663i	$-0.369927\pi$
$20$	− 2.00000i	− 0.447214i
$21$	4.00000i	0.872872i
$22$	4.00000	0.852803
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	− 1.00000i	− 0.204124i
$25$	1.00000	0.200000
$26$	0	0
$27$	−1.00000	−0.192450
$28$	4.00000i	0.755929i
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	−2.00000	−0.365148
$31$	− 4.00000i	− 0.718421i	−0.933257	−	0.359211i	$-0.883046\pi$
$31$	− 4.00000i	− 0.718421i	0.933257	−	0.359211i	$-0.116954\pi$
$32$	− 1.00000i	− 0.176777i
$33$	− 4.00000i	− 0.696311i
$34$	2.00000i	0.342997i
$35$	8.00000	1.35225
$36$	−1.00000	−0.166667
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	−8.00000	−1.29777
$39$	0	0
$40$	−2.00000	−0.316228
$41$	− 10.0000i	− 1.56174i	−0.624695	−	0.780869i	$-0.714777\pi$
$41$	− 10.0000i	− 1.56174i	0.624695	−	0.780869i	$-0.285223\pi$
$42$	4.00000	0.617213
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	− 4.00000i	− 0.603023i
$45$	2.00000i	0.298142i
$46$	0	0
$47$	− 8.00000i	− 1.16692i	−0.812142	−	0.583460i	$-0.801699\pi$
$47$	− 8.00000i	− 1.16692i	0.812142	−	0.583460i	$-0.198301\pi$
$48$	−1.00000	−0.144338
$49$	−9.00000	−1.28571
$50$	− 1.00000i	− 0.141421i
$51$	2.00000	0.280056
$52$	0	0
$53$	−10.0000	−1.37361	−0.686803	−	0.726844i	$-0.740986\pi$
$53$	−10.0000	−1.37361	−0.686803	+	0.726844i	$0.740986\pi$
$54$	1.00000i	0.136083i
$55$	−8.00000	−1.07872
$56$	4.00000	0.534522
$57$	8.00000i	1.05963i
$58$	− 6.00000i	− 0.787839i
$59$	− 4.00000i	− 0.520756i	−0.965507	−	0.260378i	$-0.916153\pi$
$59$	− 4.00000i	− 0.520756i	0.965507	−	0.260378i	$-0.0838471\pi$
$60$	2.00000i	0.258199i
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	−4.00000	−0.508001
$63$	− 4.00000i	− 0.503953i
$64$	−1.00000	−0.125000
$65$	0	0
$66$	−4.00000	−0.492366
$67$	− 16.0000i	− 1.95471i	−0.211604	−	0.977356i	$-0.567869\pi$
$67$	− 16.0000i	− 1.95471i	0.211604	−	0.977356i	$-0.432131\pi$
$68$	2.00000	0.242536
$69$	0	0
$70$	− 8.00000i	− 0.956183i
$71$	− 8.00000i	− 0.949425i	−0.880141	−	0.474713i	$-0.842552\pi$
$71$	− 8.00000i	− 0.949425i	0.880141	−	0.474713i	$-0.157448\pi$
$72$	1.00000i	0.117851i
$73$	− 2.00000i	− 0.234082i	−0.993127	−	0.117041i	$-0.962659\pi$
$73$	− 2.00000i	− 0.234082i	0.993127	−	0.117041i	$-0.0373409\pi$
$74$	2.00000	0.232495
$75$	−1.00000	−0.115470
$76$	8.00000i	0.917663i
$77$	16.0000	1.82337
$78$	0	0
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	2.00000i	0.223607i
$81$	1.00000	0.111111
$82$	−10.0000	−1.10432
$83$	12.0000i	1.31717i	0.752506	+	0.658586i	$0.228845\pi$
$83$	12.0000i	1.31717i	−0.752506	+	0.658586i	$0.771155\pi$
$84$	− 4.00000i	− 0.436436i
$85$	− 4.00000i	− 0.433861i
$86$	4.00000i	0.431331i
$87$	−6.00000	−0.643268
$88$	−4.00000	−0.426401
$89$	− 14.0000i	− 1.48400i	−0.670402	−	0.741999i	$-0.733878\pi$
$89$	− 14.0000i	− 1.48400i	0.670402	−	0.741999i	$-0.266122\pi$
$90$	2.00000	0.210819
$91$	0	0
$92$	0	0
$93$	4.00000i	0.414781i
$94$	−8.00000	−0.825137
$95$	16.0000	1.64157
$96$	1.00000i	0.102062i
$97$	10.0000i	1.01535i	0.861550	+	0.507673i	$0.169494\pi$
$97$	10.0000i	1.01535i	−0.861550	+	0.507673i	$0.830506\pi$
$98$	9.00000i	0.909137i
$99$	4.00000i	0.402015i
$100$	−1.00000	−0.100000
$101$	2.00000	0.199007	0.0995037	−	0.995037i	$-0.468274\pi$
$101$	2.00000	0.199007	0.0995037	+	0.995037i	$0.468274\pi$
$102$	− 2.00000i	− 0.198030i
$103$	−16.0000	−1.57653	−0.788263	−	0.615338i	$-0.789020\pi$
$103$	−16.0000	−1.57653	−0.788263	+	0.615338i	$0.789020\pi$
$104$	0	0
$105$	−8.00000	−0.780720
$106$	10.0000i	0.971286i
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	1.00000	0.0962250
$109$	− 2.00000i	− 0.191565i	−0.995402	−	0.0957826i	$-0.969465\pi$
$109$	− 2.00000i	− 0.191565i	0.995402	−	0.0957826i	$-0.0305354\pi$
$110$	8.00000i	0.762770i
$111$	− 2.00000i	− 0.189832i
$112$	− 4.00000i	− 0.377964i
$113$	−6.00000	−0.564433	−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000	−0.564433	−0.282216	+	0.959351i	$0.591070\pi$
$114$	8.00000	0.749269
$115$	0	0
$116$	−6.00000	−0.557086
$117$	0	0
$118$	−4.00000	−0.368230
$119$	8.00000i	0.733359i
$120$	2.00000	0.182574
$121$	−5.00000	−0.454545
$122$	2.00000i	0.181071i
$123$	10.0000i	0.901670i
$124$	4.00000i	0.359211i
$125$	12.0000i	1.07331i
$126$	−4.00000	−0.356348
$127$	0	0		−	1.00000i	$-0.5\pi$
$127$	0	0			1.00000i	$0.5\pi$
$128$	1.00000i	0.0883883i
$129$	4.00000	0.352180
$130$	0	0
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	4.00000i	0.348155i
$133$	−32.0000	−2.77475
$134$	−16.0000	−1.38219
$135$	− 2.00000i	− 0.172133i
$136$	− 2.00000i	− 0.171499i
$137$	10.0000i	0.854358i	0.904167	+	0.427179i	$0.140493\pi$
$137$	10.0000i	0.854358i	−0.904167	+	0.427179i	$0.859507\pi$
$138$	0	0
$139$	12.0000	1.01783	0.508913	−	0.860818i	$-0.330047\pi$
$139$	12.0000	1.01783	0.508913	+	0.860818i	$0.330047\pi$
$140$	−8.00000	−0.676123
$141$	8.00000i	0.673722i
$142$	−8.00000	−0.671345
$143$	0	0
$144$	1.00000	0.0833333
$145$	12.0000i	0.996546i
$146$	−2.00000	−0.165521
$147$	9.00000	0.742307
$148$	− 2.00000i	− 0.164399i
$149$	− 6.00000i	− 0.491539i	−0.969328	−	0.245770i	$-0.920959\pi$
$149$	− 6.00000i	− 0.491539i	0.969328	−	0.245770i	$-0.0790407\pi$
$150$	1.00000i	0.0816497i
$151$	− 12.0000i	− 0.976546i	−0.872691	−	0.488273i	$-0.837627\pi$
$151$	− 12.0000i	− 0.976546i	0.872691	−	0.488273i	$-0.162373\pi$
$152$	8.00000	0.648886
$153$	−2.00000	−0.161690
$154$	− 16.0000i	− 1.28932i
$155$	8.00000	0.642575
$156$	0	0
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	− 8.00000i	− 0.636446i
$159$	10.0000	0.793052
$160$	2.00000	0.158114
$161$	0	0
$162$	− 1.00000i	− 0.0785674i
$163$	16.0000i	1.25322i	0.779334	+	0.626608i	$0.215557\pi$
$163$	16.0000i	1.25322i	−0.779334	+	0.626608i	$0.784443\pi$
$164$	10.0000i	0.780869i
$165$	8.00000	0.622799
$166$	12.0000	0.931381
$167$	0	0	1.00000			$0$
$167$	0	0	−1.00000			$\pi$
$168$	−4.00000	−0.308607
$169$	0	0
$170$	−4.00000	−0.306786
$171$	− 8.00000i	− 0.611775i
$172$	4.00000	0.304997
$173$	10.0000	0.760286	0.380143	−	0.924928i	$-0.375875\pi$
$173$	10.0000	0.760286	0.380143	+	0.924928i	$0.375875\pi$
$174$	6.00000i	0.454859i
$175$	− 4.00000i	− 0.302372i
$176$	4.00000i	0.301511i
$177$	4.00000i	0.300658i
$178$	−14.0000	−1.04934
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	− 2.00000i	− 0.149071i
$181$	10.0000	0.743294	0.371647	−	0.928374i	$-0.378793\pi$
$181$	10.0000	0.743294	0.371647	+	0.928374i	$0.378793\pi$
$182$	0	0
$183$	2.00000	0.147844
$184$	0	0
$185$	−4.00000	−0.294086
$186$	4.00000	0.293294
$187$	− 8.00000i	− 0.585018i
$188$	8.00000i	0.583460i
$189$	4.00000i	0.290957i
$190$	− 16.0000i	− 1.16076i
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	1.00000	0.0721688
$193$	14.0000i	1.00774i	0.863779	+	0.503871i	$0.168091\pi$
$193$	14.0000i	1.00774i	−0.863779	+	0.503871i	$0.831909\pi$
$194$	10.0000	0.717958
$195$	0	0
$196$	9.00000	0.642857
$197$	18.0000i	1.28245i	0.767354	+	0.641223i	$0.221573\pi$
$197$	18.0000i	1.28245i	−0.767354	+	0.641223i	$0.778427\pi$
$198$	4.00000	0.284268
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	1.00000i	0.0707107i
$201$	16.0000i	1.12855i
$202$	− 2.00000i	− 0.140720i
$203$	− 24.0000i	− 1.68447i
$204$	−2.00000	−0.140028
$205$	20.0000	1.39686
$206$	16.0000i	1.11477i
$207$	0	0
$208$	0	0
$209$	32.0000	2.21349
$210$	8.00000i	0.552052i
$211$	12.0000	0.826114	0.413057	−	0.910705i	$-0.364461\pi$
$211$	12.0000	0.826114	0.413057	+	0.910705i	$0.364461\pi$
$212$	10.0000	0.686803
$213$	8.00000i	0.548151i
$214$	− 12.0000i	− 0.820303i
$215$	− 8.00000i	− 0.545595i
$216$	− 1.00000i	− 0.0680414i
$217$	−16.0000	−1.08615
$218$	−2.00000	−0.135457
$219$	2.00000i	0.135147i
$220$	8.00000	0.539360
$221$	0	0
$222$	−2.00000	−0.134231
$223$	− 4.00000i	− 0.267860i	−0.990991	−	0.133930i	$-0.957240\pi$
$223$	− 4.00000i	− 0.267860i	0.990991	−	0.133930i	$-0.0427597\pi$
$224$	−4.00000	−0.267261
$225$	1.00000	0.0666667
$226$	6.00000i	0.399114i
$227$	20.0000i	1.32745i	0.747978	+	0.663723i	$0.231025\pi$
$227$	20.0000i	1.32745i	−0.747978	+	0.663723i	$0.768975\pi$
$228$	− 8.00000i	− 0.529813i
$229$	− 22.0000i	− 1.45380i	−0.686743	−	0.726900i	$-0.740960\pi$
$229$	− 22.0000i	− 1.45380i	0.686743	−	0.726900i	$-0.259040\pi$
$230$	0	0
$231$	−16.0000	−1.05272
$232$	6.00000i	0.393919i
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	0	0
$235$	16.0000	1.04372
$236$	4.00000i	0.260378i
$237$	−8.00000	−0.519656
$238$	8.00000	0.518563
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	− 2.00000i	− 0.129099i
$241$	− 10.0000i	− 0.644157i	−0.946713	−	0.322078i	$-0.895619\pi$
$241$	− 10.0000i	− 0.644157i	0.946713	−	0.322078i	$-0.104381\pi$
$242$	5.00000i	0.321412i
$243$	−1.00000	−0.0641500
$244$	2.00000	0.128037
$245$	− 18.0000i	− 1.14998i
$246$	10.0000	0.637577
$247$	0	0
$248$	4.00000	0.254000
$249$	− 12.0000i	− 0.760469i
$250$	12.0000	0.758947
$251$	−4.00000	−0.252478	−0.126239	−	0.992000i	$-0.540291\pi$
$251$	−4.00000	−0.252478	−0.126239	+	0.992000i	$0.540291\pi$
$252$	4.00000i	0.251976i
$253$	0	0
$254$	0	0
$255$	4.00000i	0.250490i
$256$	1.00000	0.0625000
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	− 4.00000i	− 0.249029i
$259$	8.00000	0.497096
$260$	0	0
$261$	6.00000	0.371391
$262$	− 4.00000i	− 0.247121i
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	4.00000	0.246183
$265$	− 20.0000i	− 1.22859i
$266$	32.0000i	1.96205i
$267$	14.0000i	0.856786i
$268$	16.0000i	0.977356i
$269$	−26.0000	−1.58525	−0.792624	−	0.609711i	$-0.791286\pi$
$269$	−26.0000	−1.58525	−0.792624	+	0.609711i	$0.791286\pi$
$270$	−2.00000	−0.121716
$271$	4.00000i	0.242983i	0.992592	+	0.121491i	$0.0387677\pi$
$271$	4.00000i	0.242983i	−0.992592	+	0.121491i	$0.961232\pi$
$272$	−2.00000	−0.121268
$273$	0	0
$274$	10.0000	0.604122
$275$	4.00000i	0.241209i
$276$	0	0
$277$	−22.0000	−1.32185	−0.660926	−	0.750451i	$-0.729836\pi$
$277$	−22.0000	−1.32185	−0.660926	+	0.750451i	$0.729836\pi$
$278$	− 12.0000i	− 0.719712i
$279$	− 4.00000i	− 0.239474i
$280$	8.00000i	0.478091i
$281$	26.0000i	1.55103i	0.631329	+	0.775515i	$0.282510\pi$
$281$	26.0000i	1.55103i	−0.631329	+	0.775515i	$0.717490\pi$
$282$	8.00000	0.476393
$283$	4.00000	0.237775	0.118888	−	0.992908i	$-0.462067\pi$
$283$	4.00000	0.237775	0.118888	+	0.992908i	$0.462067\pi$
$284$	8.00000i	0.474713i
$285$	−16.0000	−0.947758
$286$	0	0
$287$	−40.0000	−2.36113
$288$	− 1.00000i	− 0.0589256i
$289$	−13.0000	−0.764706
$290$	12.0000	0.704664
$291$	− 10.0000i	− 0.586210i
$292$	2.00000i	0.117041i
$293$	− 26.0000i	− 1.51894i	−0.650545	−	0.759468i	$-0.725459\pi$
$293$	− 26.0000i	− 1.51894i	0.650545	−	0.759468i	$-0.274541\pi$
$294$	− 9.00000i	− 0.524891i
$295$	8.00000	0.465778
$296$	−2.00000	−0.116248
$297$	− 4.00000i	− 0.232104i
$298$	−6.00000	−0.347571
$299$	0	0
$300$	1.00000	0.0577350
$301$	16.0000i	0.922225i
$302$	−12.0000	−0.690522
$303$	−2.00000	−0.114897
$304$	− 8.00000i	− 0.458831i
$305$	− 4.00000i	− 0.229039i
$306$	2.00000i	0.114332i
$307$	8.00000i	0.456584i	0.973593	+	0.228292i	$0.0733141\pi$
$307$	8.00000i	0.456584i	−0.973593	+	0.228292i	$0.926686\pi$
$308$	−16.0000	−0.911685
$309$	16.0000	0.910208
$310$	− 8.00000i	− 0.454369i
$311$	0	0		−	1.00000i	$-0.5\pi$
$311$	0	0			1.00000i	$0.5\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	− 14.0000i	− 0.790066i
$315$	8.00000	0.450749
$316$	−8.00000	−0.450035
$317$	− 6.00000i	− 0.336994i	−0.985702	−	0.168497i	$-0.946109\pi$
$317$	− 6.00000i	− 0.336994i	0.985702	−	0.168497i	$-0.0538913\pi$
$318$	− 10.0000i	− 0.560772i
$319$	24.0000i	1.34374i
$320$	− 2.00000i	− 0.111803i
$321$	−12.0000	−0.669775
$322$	0	0
$323$	16.0000i	0.890264i
$324$	−1.00000	−0.0555556
$325$	0	0
$326$	16.0000	0.886158
$327$	2.00000i	0.110600i
$328$	10.0000	0.552158
$329$	−32.0000	−1.76422
$330$	− 8.00000i	− 0.440386i
$331$	8.00000i	0.439720i	0.975531	+	0.219860i	$0.0705600\pi$
$331$	8.00000i	0.439720i	−0.975531	+	0.219860i	$0.929440\pi$
$332$	− 12.0000i	− 0.658586i
$333$	2.00000i	0.109599i
$334$	0	0
$335$	32.0000	1.74835
$336$	4.00000i	0.218218i
$337$	−18.0000	−0.980522	−0.490261	−	0.871576i	$-0.663099\pi$
$337$	−18.0000	−0.980522	−0.490261	+	0.871576i	$0.663099\pi$
$338$	0	0
$339$	6.00000	0.325875
$340$	4.00000i	0.216930i
$341$	16.0000	0.866449
$342$	−8.00000	−0.432590
$343$	8.00000i	0.431959i
$344$	− 4.00000i	− 0.215666i
$345$	0	0
$346$	− 10.0000i	− 0.537603i
$347$	−12.0000	−0.644194	−0.322097	−	0.946707i	$-0.604388\pi$
$347$	−12.0000	−0.644194	−0.322097	+	0.946707i	$0.604388\pi$
$348$	6.00000	0.321634
$349$	− 6.00000i	− 0.321173i	−0.987022	−	0.160586i	$-0.948662\pi$
$349$	− 6.00000i	− 0.321173i	0.987022	−	0.160586i	$-0.0513385\pi$
$350$	−4.00000	−0.213809
$351$	0	0
$352$	4.00000	0.213201
$353$	14.0000i	0.745145i	0.928003	+	0.372572i	$0.121524\pi$
$353$	14.0000i	0.745145i	−0.928003	+	0.372572i	$0.878476\pi$
$354$	4.00000	0.212598
$355$	16.0000	0.849192
$356$	14.0000i	0.741999i
$357$	− 8.00000i	− 0.423405i
$358$	− 12.0000i	− 0.634220i
$359$	0	0	1.00000			$0$
$359$	0	0	−1.00000			$\pi$
$360$	−2.00000	−0.105409
$361$	−45.0000	−2.36842
$362$	− 10.0000i	− 0.525588i
$363$	5.00000	0.262432
$364$	0	0
$365$	4.00000	0.209370
$366$	− 2.00000i	− 0.104542i
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	0	0
$369$	− 10.0000i	− 0.520579i
$370$	4.00000i	0.207950i
$371$	40.0000i	2.07670i
$372$	− 4.00000i	− 0.207390i
$373$	6.00000	0.310668	0.155334	−	0.987862i	$-0.450355\pi$
$373$	6.00000	0.310668	0.155334	+	0.987862i	$0.450355\pi$
$374$	−8.00000	−0.413670
$375$	− 12.0000i	− 0.619677i
$376$	8.00000	0.412568
$377$	0	0
$378$	4.00000	0.205738
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	−16.0000	−0.820783
$381$	0	0
$382$	8.00000i	0.409316i
$383$	− 24.0000i	− 1.22634i	−0.789950	−	0.613171i	$-0.789894\pi$
$383$	− 24.0000i	− 1.22634i	0.789950	−	0.613171i	$-0.210106\pi$
$384$	− 1.00000i	− 0.0510310i
$385$	32.0000i	1.63087i
$386$	14.0000	0.712581
$387$	−4.00000	−0.203331
$388$	− 10.0000i	− 0.507673i
$389$	26.0000	1.31825	0.659126	−	0.752032i	$-0.270926\pi$
$389$	26.0000	1.31825	0.659126	+	0.752032i	$0.270926\pi$
$390$	0	0
$391$	0	0
$392$	− 9.00000i	− 0.454569i
$393$	−4.00000	−0.201773
$394$	18.0000	0.906827
$395$	16.0000i	0.805047i
$396$	− 4.00000i	− 0.201008i
$397$	− 6.00000i	− 0.301131i	−0.988600	−	0.150566i	$-0.951890\pi$
$397$	− 6.00000i	− 0.301131i	0.988600	−	0.150566i	$-0.0481095\pi$
$398$	− 8.00000i	− 0.401004i
$399$	32.0000	1.60200
$400$	1.00000	0.0500000
$401$	− 6.00000i	− 0.299626i	−0.988714	−	0.149813i	$-0.952133\pi$
$401$	− 6.00000i	− 0.299626i	0.988714	−	0.149813i	$-0.0478671\pi$
$402$	16.0000	0.798007
$403$	0	0
$404$	−2.00000	−0.0995037
$405$	2.00000i	0.0993808i
$406$	−24.0000	−1.19110
$407$	−8.00000	−0.396545
$408$	2.00000i	0.0990148i
$409$	2.00000i	0.0988936i	0.998777	+	0.0494468i	$0.0157458\pi$
$409$	2.00000i	0.0988936i	−0.998777	+	0.0494468i	$0.984254\pi$
$410$	− 20.0000i	− 0.987730i
$411$	− 10.0000i	− 0.493264i
$412$	16.0000	0.788263
$413$	−16.0000	−0.787309
$414$	0	0
$415$	−24.0000	−1.17811
$416$	0	0
$417$	−12.0000	−0.587643
$418$	− 32.0000i	− 1.56517i
$419$	4.00000	0.195413	0.0977064	−	0.995215i	$-0.468849\pi$
$419$	4.00000	0.195413	0.0977064	+	0.995215i	$0.468849\pi$
$420$	8.00000	0.390360
$421$	22.0000i	1.07221i	0.844150	+	0.536107i	$0.180106\pi$
$421$	22.0000i	1.07221i	−0.844150	+	0.536107i	$0.819894\pi$
$422$	− 12.0000i	− 0.584151i
$423$	− 8.00000i	− 0.388973i
$424$	− 10.0000i	− 0.485643i
$425$	−2.00000	−0.0970143
$426$	8.00000	0.387601
$427$	8.00000i	0.387147i
$428$	−12.0000	−0.580042
$429$	0	0
$430$	−8.00000	−0.385794
$431$	− 8.00000i	− 0.385346i	−0.981263	−	0.192673i	$-0.938284\pi$
$431$	− 8.00000i	− 0.385346i	0.981263	−	0.192673i	$-0.0617157\pi$
$432$	−1.00000	−0.0481125
$433$	30.0000	1.44171	0.720854	−	0.693087i	$-0.243750\pi$
$433$	30.0000	1.44171	0.720854	+	0.693087i	$0.243750\pi$
$434$	16.0000i	0.768025i
$435$	− 12.0000i	− 0.575356i
$436$	2.00000i	0.0957826i
$437$	0	0
$438$	2.00000	0.0955637
$439$	−16.0000	−0.763638	−0.381819	−	0.924237i	$-0.624702\pi$
$439$	−16.0000	−0.763638	−0.381819	+	0.924237i	$0.624702\pi$
$440$	− 8.00000i	− 0.381385i
$441$	−9.00000	−0.428571
$442$	0	0
$443$	−4.00000	−0.190046	−0.0950229	−	0.995475i	$-0.530292\pi$
$443$	−4.00000	−0.190046	−0.0950229	+	0.995475i	$0.530292\pi$
$444$	2.00000i	0.0949158i
$445$	28.0000	1.32733
$446$	−4.00000	−0.189405
$447$	6.00000i	0.283790i
$448$	4.00000i	0.188982i
$449$	− 6.00000i	− 0.283158i	−0.989927	−	0.141579i	$-0.954782\pi$
$449$	− 6.00000i	− 0.283158i	0.989927	−	0.141579i	$-0.0452178\pi$
$450$	− 1.00000i	− 0.0471405i
$451$	40.0000	1.88353
$452$	6.00000	0.282216
$453$	12.0000i	0.563809i
$454$	20.0000	0.938647
$455$	0	0
$456$	−8.00000	−0.374634
$457$	− 30.0000i	− 1.40334i	−0.712502	−	0.701670i	$-0.752438\pi$
$457$	− 30.0000i	− 1.40334i	0.712502	−	0.701670i	$-0.247562\pi$
$458$	−22.0000	−1.02799
$459$	2.00000	0.0933520
$460$	0	0
$461$	− 6.00000i	− 0.279448i	−0.990190	−	0.139724i	$-0.955378\pi$
$461$	− 6.00000i	− 0.279448i	0.990190	−	0.139724i	$-0.0446215\pi$
$462$	16.0000i	0.744387i
$463$	20.0000i	0.929479i	0.885448	+	0.464739i	$0.153852\pi$
$463$	20.0000i	0.929479i	−0.885448	+	0.464739i	$0.846148\pi$
$464$	6.00000	0.278543
$465$	−8.00000	−0.370991
$466$	18.0000i	0.833834i
$467$	4.00000	0.185098	0.0925490	−	0.995708i	$-0.470499\pi$
$467$	4.00000	0.185098	0.0925490	+	0.995708i	$0.470499\pi$
$468$	0	0
$469$	−64.0000	−2.95525
$470$	− 16.0000i	− 0.738025i
$471$	−14.0000	−0.645086
$472$	4.00000	0.184115
$473$	− 16.0000i	− 0.735681i
$474$	8.00000i	0.367452i
$475$	− 8.00000i	− 0.367065i
$476$	− 8.00000i	− 0.366679i
$477$	−10.0000	−0.457869
$478$	0	0
$479$	16.0000i	0.731059i	0.930800	+	0.365529i	$0.119112\pi$
$479$	16.0000i	0.731059i	−0.930800	+	0.365529i	$0.880888\pi$
$480$	−2.00000	−0.0912871
$481$	0	0
$482$	−10.0000	−0.455488
$483$	0	0
$484$	5.00000	0.227273
$485$	−20.0000	−0.908153
$486$	1.00000i	0.0453609i
$487$	4.00000i	0.181257i	0.995885	+	0.0906287i	$0.0288876\pi$
$487$	4.00000i	0.181257i	−0.995885	+	0.0906287i	$0.971112\pi$
$488$	− 2.00000i	− 0.0905357i
$489$	− 16.0000i	− 0.723545i
$490$	−18.0000	−0.813157
$491$	−36.0000	−1.62466	−0.812329	−	0.583200i	$-0.801800\pi$
$491$	−36.0000	−1.62466	−0.812329	+	0.583200i	$0.801800\pi$
$492$	− 10.0000i	− 0.450835i
$493$	−12.0000	−0.540453
$494$	0	0
$495$	−8.00000	−0.359573
$496$	− 4.00000i	− 0.179605i
$497$	−32.0000	−1.43540
$498$	−12.0000	−0.537733
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	− 12.0000i	− 0.536656i
$501$	0	0
$502$	4.00000i	0.178529i
$503$	40.0000	1.78351	0.891756	−	0.452517i	$-0.149474\pi$
$503$	40.0000	1.78351	0.891756	+	0.452517i	$0.149474\pi$
$504$	4.00000	0.178174
$505$	4.00000i	0.177998i
$506$	0	0
$507$	0	0
$508$	0	0
$509$	42.0000i	1.86162i	0.365507	+	0.930809i	$0.380896\pi$
$509$	42.0000i	1.86162i	−0.365507	+	0.930809i	$0.619104\pi$
$510$	4.00000	0.177123
$511$	−8.00000	−0.353899
$512$	− 1.00000i	− 0.0441942i
$513$	8.00000i	0.353209i
$514$	− 6.00000i	− 0.264649i
$515$	− 32.0000i	− 1.41009i
$516$	−4.00000	−0.176090
$517$	32.0000	1.40736
$518$	− 8.00000i	− 0.351500i
$519$	−10.0000	−0.438951
$520$	0	0
$521$	−14.0000	−0.613351	−0.306676	−	0.951814i	$-0.599217\pi$
$521$	−14.0000	−0.613351	−0.306676	+	0.951814i	$0.599217\pi$
$522$	− 6.00000i	− 0.262613i
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	−4.00000	−0.174741
$525$	4.00000i	0.174574i
$526$	− 8.00000i	− 0.348817i
$527$	8.00000i	0.348485i
$528$	− 4.00000i	− 0.174078i
$529$	−23.0000	−1.00000
$530$	−20.0000	−0.868744
$531$	− 4.00000i	− 0.173585i
$532$	32.0000	1.38738
$533$	0	0
$534$	14.0000	0.605839
$535$	24.0000i	1.03761i
$536$	16.0000	0.691095
$537$	−12.0000	−0.517838
$538$	26.0000i	1.12094i
$539$	− 36.0000i	− 1.55063i
$540$	2.00000i	0.0860663i
$541$	34.0000i	1.46177i	0.682498	+	0.730887i	$0.260893\pi$
$541$	34.0000i	1.46177i	−0.682498	+	0.730887i	$0.739107\pi$
$542$	4.00000	0.171815
$543$	−10.0000	−0.429141
$544$	2.00000i	0.0857493i
$545$	4.00000	0.171341
$546$	0	0
$547$	−28.0000	−1.19719	−0.598597	−	0.801050i	$-0.704275\pi$
$547$	−28.0000	−1.19719	−0.598597	+	0.801050i	$0.704275\pi$
$548$	− 10.0000i	− 0.427179i
$549$	−2.00000	−0.0853579
$550$	4.00000	0.170561
$551$	− 48.0000i	− 2.04487i
$552$	0	0
$553$	− 32.0000i	− 1.36078i
$554$	22.0000i	0.934690i
$555$	4.00000	0.169791
$556$	−12.0000	−0.508913
$557$	− 18.0000i	− 0.762684i	−0.924434	−	0.381342i	$-0.875462\pi$
$557$	− 18.0000i	− 0.762684i	0.924434	−	0.381342i	$-0.124538\pi$
$558$	−4.00000	−0.169334
$559$	0	0
$560$	8.00000	0.338062
$561$	8.00000i	0.337760i
$562$	26.0000	1.09674
$563$	−4.00000	−0.168580	−0.0842900	−	0.996441i	$-0.526862\pi$
$563$	−4.00000	−0.168580	−0.0842900	+	0.996441i	$0.526862\pi$
$564$	− 8.00000i	− 0.336861i
$565$	− 12.0000i	− 0.504844i
$566$	− 4.00000i	− 0.168133i
$567$	− 4.00000i	− 0.167984i
$568$	8.00000	0.335673
$569$	30.0000	1.25767	0.628833	−	0.777541i	$-0.283533\pi$
$569$	30.0000	1.25767	0.628833	+	0.777541i	$0.283533\pi$
$570$	16.0000i	0.670166i
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	8.00000	0.334205
$574$	40.0000i	1.66957i
$575$	0	0
$576$	−1.00000	−0.0416667
$577$	18.0000i	0.749350i	0.927156	+	0.374675i	$0.122246\pi$
$577$	18.0000i	0.749350i	−0.927156	+	0.374675i	$0.877754\pi$
$578$	13.0000i	0.540729i
$579$	− 14.0000i	− 0.581820i
$580$	− 12.0000i	− 0.498273i
$581$	48.0000	1.99138
$582$	−10.0000	−0.414513
$583$	− 40.0000i	− 1.65663i
$584$	2.00000	0.0827606
$585$	0	0
$586$	−26.0000	−1.07405
$587$	4.00000i	0.165098i	0.996587	+	0.0825488i	$0.0263060\pi$
$587$	4.00000i	0.165098i	−0.996587	+	0.0825488i	$0.973694\pi$
$588$	−9.00000	−0.371154
$589$	−32.0000	−1.31854
$590$	− 8.00000i	− 0.329355i
$591$	− 18.0000i	− 0.740421i
$592$	2.00000i	0.0821995i
$593$	42.0000i	1.72473i	0.506284	+	0.862367i	$0.331019\pi$
$593$	42.0000i	1.72473i	−0.506284	+	0.862367i	$0.668981\pi$
$594$	−4.00000	−0.164122
$595$	−16.0000	−0.655936
$596$	6.00000i	0.245770i
$597$	−8.00000	−0.327418
$598$	0	0
$599$	−24.0000	−0.980613	−0.490307	−	0.871550i	$-0.663115\pi$
$599$	−24.0000	−0.980613	−0.490307	+	0.871550i	$0.663115\pi$
$600$	− 1.00000i	− 0.0408248i
$601$	26.0000	1.06056	0.530281	−	0.847822i	$-0.322086\pi$
$601$	26.0000	1.06056	0.530281	+	0.847822i	$0.322086\pi$
$602$	16.0000	0.652111
$603$	− 16.0000i	− 0.651570i
$604$	12.0000i	0.488273i
$605$	− 10.0000i	− 0.406558i
$606$	2.00000i	0.0812444i
$607$	−16.0000	−0.649420	−0.324710	−	0.945814i	$-0.605267\pi$
$607$	−16.0000	−0.649420	−0.324710	+	0.945814i	$0.605267\pi$
$608$	−8.00000	−0.324443
$609$	24.0000i	0.972529i
$610$	−4.00000	−0.161955
$611$	0	0
$612$	2.00000	0.0808452
$613$	− 2.00000i	− 0.0807792i	−0.999184	−	0.0403896i	$-0.987140\pi$
$613$	− 2.00000i	− 0.0807792i	0.999184	−	0.0403896i	$-0.0128599\pi$
$614$	8.00000	0.322854
$615$	−20.0000	−0.806478
$616$	16.0000i	0.644658i
$617$	6.00000i	0.241551i	0.992680	+	0.120775i	$0.0385381\pi$
$617$	6.00000i	0.241551i	−0.992680	+	0.120775i	$0.961462\pi$
$618$	− 16.0000i	− 0.643614i
$619$	− 32.0000i	− 1.28619i	−0.765787	−	0.643094i	$-0.777650\pi$
$619$	− 32.0000i	− 1.28619i	0.765787	−	0.643094i	$-0.222350\pi$
$620$	−8.00000	−0.321288
$621$	0	0
$622$	0	0
$623$	−56.0000	−2.24359
$624$	0	0
$625$	−19.0000	−0.760000
$626$	6.00000i	0.239808i
$627$	−32.0000	−1.27796
$628$	−14.0000	−0.558661
$629$	− 4.00000i	− 0.159490i
$630$	− 8.00000i	− 0.318728i
$631$	36.0000i	1.43314i	0.697517	+	0.716569i	$0.254288\pi$
$631$	36.0000i	1.43314i	−0.697517	+	0.716569i	$0.745712\pi$
$632$	8.00000i	0.318223i
$633$	−12.0000	−0.476957
$634$	−6.00000	−0.238290
$635$	0	0
$636$	−10.0000	−0.396526
$637$	0	0
$638$	24.0000	0.950169
$639$	− 8.00000i	− 0.316475i
$640$	−2.00000	−0.0790569
$641$	−2.00000	−0.0789953	−0.0394976	−	0.999220i	$-0.512576\pi$
$641$	−2.00000	−0.0789953	−0.0394976	+	0.999220i	$0.512576\pi$
$642$	12.0000i	0.473602i
$643$	− 16.0000i	− 0.630978i	−0.948929	−	0.315489i	$-0.897831\pi$
$643$	− 16.0000i	− 0.630978i	0.948929	−	0.315489i	$-0.102169\pi$
$644$	0	0
$645$	8.00000i	0.315000i
$646$	16.0000	0.629512
$647$	−24.0000	−0.943537	−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000	−0.943537	−0.471769	+	0.881722i	$0.656384\pi$
$648$	1.00000i	0.0392837i
$649$	16.0000	0.628055
$650$	0	0
$651$	16.0000	0.627089
$652$	− 16.0000i	− 0.626608i
$653$	−10.0000	−0.391330	−0.195665	−	0.980671i	$-0.562687\pi$
$653$	−10.0000	−0.391330	−0.195665	+	0.980671i	$0.562687\pi$
$654$	2.00000	0.0782062
$655$	8.00000i	0.312586i
$656$	− 10.0000i	− 0.390434i
$657$	− 2.00000i	− 0.0780274i
$658$	32.0000i	1.24749i
$659$	−36.0000	−1.40236	−0.701180	−	0.712984i	$-0.747343\pi$
$659$	−36.0000	−1.40236	−0.701180	+	0.712984i	$0.747343\pi$
$660$	−8.00000	−0.311400
$661$	2.00000i	0.0777910i	0.999243	+	0.0388955i	$0.0123839\pi$
$661$	2.00000i	0.0777910i	−0.999243	+	0.0388955i	$0.987616\pi$
$662$	8.00000	0.310929
$663$	0	0
$664$	−12.0000	−0.465690
$665$	− 64.0000i	− 2.48181i
$666$	2.00000	0.0774984
$667$	0	0
$668$	0	0
$669$	4.00000i	0.154649i
$670$	− 32.0000i	− 1.23627i
$671$	− 8.00000i	− 0.308837i
$672$	4.00000	0.154303
$673$	14.0000	0.539660	0.269830	−	0.962908i	$-0.413032\pi$
$673$	14.0000	0.539660	0.269830	+	0.962908i	$0.413032\pi$
$674$	18.0000i	0.693334i
$675$	−1.00000	−0.0384900
$676$	0	0
$677$	38.0000	1.46046	0.730229	−	0.683202i	$-0.239413\pi$
$677$	38.0000	1.46046	0.730229	+	0.683202i	$0.239413\pi$
$678$	− 6.00000i	− 0.230429i
$679$	40.0000	1.53506
$680$	4.00000	0.153393
$681$	− 20.0000i	− 0.766402i
$682$	− 16.0000i	− 0.612672i
$683$	− 44.0000i	− 1.68361i	−0.539779	−	0.841807i	$-0.681492\pi$
$683$	− 44.0000i	− 1.68361i	0.539779	−	0.841807i	$-0.318508\pi$
$684$	8.00000i	0.305888i
$685$	−20.0000	−0.764161
$686$	8.00000	0.305441
$687$	22.0000i	0.839352i
$688$	−4.00000	−0.152499
$689$	0	0
$690$	0	0
$691$	− 32.0000i	− 1.21734i	−0.793424	−	0.608669i	$-0.791704\pi$
$691$	− 32.0000i	− 1.21734i	0.793424	−	0.608669i	$-0.208296\pi$
$692$	−10.0000	−0.380143
$693$	16.0000	0.607790
$694$	12.0000i	0.455514i
$695$	24.0000i	0.910372i
$696$	− 6.00000i	− 0.227429i
$697$	20.0000i	0.757554i
$698$	−6.00000	−0.227103
$699$	18.0000	0.680823
$700$	4.00000i	0.151186i
$701$	50.0000	1.88847	0.944237	−	0.329267i	$-0.106802\pi$
$701$	50.0000	1.88847	0.944237	+	0.329267i	$0.106802\pi$
$702$	0	0
$703$	16.0000	0.603451
$704$	− 4.00000i	− 0.150756i
$705$	−16.0000	−0.602595
$706$	14.0000	0.526897
$707$	− 8.00000i	− 0.300871i
$708$	− 4.00000i	− 0.150329i
$709$	− 6.00000i	− 0.225335i	−0.993633	−	0.112667i	$-0.964061\pi$
$709$	− 6.00000i	− 0.225335i	0.993633	−	0.112667i	$-0.0359394\pi$
$710$	− 16.0000i	− 0.600469i
$711$	8.00000	0.300023
$712$	14.0000	0.524672
$713$	0	0
$714$	−8.00000	−0.299392
$715$	0	0
$716$	−12.0000	−0.448461
$717$	0	0
$718$	0	0
$719$	24.0000	0.895049	0.447524	−	0.894272i	$-0.352306\pi$
$719$	24.0000	0.895049	0.447524	+	0.894272i	$0.352306\pi$
$720$	2.00000i	0.0745356i
$721$	64.0000i	2.38348i
$722$	45.0000i	1.67473i
$723$	10.0000i	0.371904i
$724$	−10.0000	−0.371647
$725$	6.00000	0.222834
$726$	− 5.00000i	− 0.185567i
$727$	40.0000	1.48352	0.741759	−	0.670667i	$-0.233992\pi$
$727$	40.0000	1.48352	0.741759	+	0.670667i	$0.233992\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	− 4.00000i	− 0.148047i
$731$	8.00000	0.295891
$732$	−2.00000	−0.0739221
$733$	− 2.00000i	− 0.0738717i	−0.999318	−	0.0369358i	$-0.988240\pi$
$733$	− 2.00000i	− 0.0738717i	0.999318	−	0.0369358i	$-0.0117597\pi$
$734$	− 16.0000i	− 0.590571i
$735$	18.0000i	0.663940i
$736$	0	0
$737$	64.0000	2.35747
$738$	−10.0000	−0.368105
$739$	− 40.0000i	− 1.47142i	−0.677295	−	0.735712i	$-0.736848\pi$
$739$	− 40.0000i	− 1.47142i	0.677295	−	0.735712i	$-0.263152\pi$
$740$	4.00000	0.147043
$741$	0	0
$742$	40.0000	1.46845
$743$	− 24.0000i	− 0.880475i	−0.897881	−	0.440237i	$-0.854894\pi$
$743$	− 24.0000i	− 0.880475i	0.897881	−	0.440237i	$-0.145106\pi$
$744$	−4.00000	−0.146647
$745$	12.0000	0.439646
$746$	− 6.00000i	− 0.219676i
$747$	12.0000i	0.439057i
$748$	8.00000i	0.292509i
$749$	− 48.0000i	− 1.75388i
$750$	−12.0000	−0.438178
$751$	40.0000	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$751$	40.0000	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$752$	− 8.00000i	− 0.291730i
$753$	4.00000	0.145768
$754$	0	0
$755$	24.0000	0.873449
$756$	− 4.00000i	− 0.145479i
$757$	54.0000	1.96266	0.981332	−	0.192323i	$-0.0616021\pi$
$757$	54.0000	1.96266	0.981332	+	0.192323i	$0.0616021\pi$
$758$	0	0
$759$	0	0
$760$	16.0000i	0.580381i
$761$	26.0000i	0.942499i	0.882000	+	0.471250i	$0.156197\pi$
$761$	26.0000i	0.942499i	−0.882000	+	0.471250i	$0.843803\pi$
$762$	0	0
$763$	−8.00000	−0.289619
$764$	8.00000	0.289430
$765$	− 4.00000i	− 0.144620i
$766$	−24.0000	−0.867155
$767$	0	0
$768$	−1.00000	−0.0360844
$769$	2.00000i	0.0721218i	0.999350	+	0.0360609i	$0.0114810\pi$
$769$	2.00000i	0.0721218i	−0.999350	+	0.0360609i	$0.988519\pi$
$770$	32.0000	1.15320
$771$	−6.00000	−0.216085
$772$	− 14.0000i	− 0.503871i
$773$	− 54.0000i	− 1.94225i	−0.238581	−	0.971123i	$-0.576682\pi$
$773$	− 54.0000i	− 1.94225i	0.238581	−	0.971123i	$-0.423318\pi$
$774$	4.00000i	0.143777i
$775$	− 4.00000i	− 0.143684i
$776$	−10.0000	−0.358979
$777$	−8.00000	−0.286998
$778$	− 26.0000i	− 0.932145i
$779$	−80.0000	−2.86630
$780$	0	0
$781$	32.0000	1.14505
$782$	0	0
$783$	−6.00000	−0.214423
$784$	−9.00000	−0.321429
$785$	28.0000i	0.999363i
$786$	4.00000i	0.142675i
$787$	− 40.0000i	− 1.42585i	−0.701242	−	0.712923i	$-0.747371\pi$
$787$	− 40.0000i	− 1.42585i	0.701242	−	0.712923i	$-0.252629\pi$
$788$	− 18.0000i	− 0.641223i
$789$	−8.00000	−0.284808
$790$	16.0000	0.569254
$791$	24.0000i	0.853342i
$792$	−4.00000	−0.142134
$793$	0	0
$794$	−6.00000	−0.212932
$795$	20.0000i	0.709327i
$796$	−8.00000	−0.283552
$797$	2.00000	0.0708436	0.0354218	−	0.999372i	$-0.488723\pi$
$797$	2.00000	0.0708436	0.0354218	+	0.999372i	$0.488723\pi$
$798$	− 32.0000i	− 1.13279i
$799$	16.0000i	0.566039i
$800$	− 1.00000i	− 0.0353553i
$801$	− 14.0000i	− 0.494666i
$802$	−6.00000	−0.211867
$803$	8.00000	0.282314
$804$	− 16.0000i	− 0.564276i
$805$	0	0
$806$	0	0
$807$	26.0000	0.915243
$808$	2.00000i	0.0703598i
$809$	2.00000	0.0703163	0.0351581	−	0.999382i	$-0.488807\pi$
$809$	2.00000	0.0703163	0.0351581	+	0.999382i	$0.488807\pi$
$810$	2.00000	0.0702728
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	24.0000i	0.842235i
$813$	− 4.00000i	− 0.140286i
$814$	8.00000i	0.280400i
$815$	−32.0000	−1.12091
$816$	2.00000	0.0700140
$817$	32.0000i	1.11954i
$818$	2.00000	0.0699284
$819$	0	0
$820$	−20.0000	−0.698430
$821$	42.0000i	1.46581i	0.680331	+	0.732905i	$0.261836\pi$
$821$	42.0000i	1.46581i	−0.680331	+	0.732905i	$0.738164\pi$
$822$	−10.0000	−0.348790
$823$	16.0000	0.557725	0.278862	−	0.960331i	$-0.410043\pi$
$823$	16.0000	0.557725	0.278862	+	0.960331i	$0.410043\pi$
$824$	− 16.0000i	− 0.557386i
$825$	− 4.00000i	− 0.139262i
$826$	16.0000i	0.556711i
$827$	− 28.0000i	− 0.973655i	−0.873498	−	0.486828i	$-0.838154\pi$
$827$	− 28.0000i	− 0.973655i	0.873498	−	0.486828i	$-0.161846\pi$
$828$	0	0
$829$	2.00000	0.0694629	0.0347314	−	0.999397i	$-0.488942\pi$
$829$	2.00000	0.0694629	0.0347314	+	0.999397i	$0.488942\pi$
$830$	24.0000i	0.833052i
$831$	22.0000	0.763172
$832$	0	0
$833$	18.0000	0.623663
$834$	12.0000i	0.415526i
$835$	0	0
$836$	−32.0000	−1.10674
$837$	4.00000i	0.138260i
$838$	− 4.00000i	− 0.138178i
$839$	40.0000i	1.38095i	0.723355	+	0.690477i	$0.242599\pi$
$839$	40.0000i	1.38095i	−0.723355	+	0.690477i	$0.757401\pi$
$840$	− 8.00000i	− 0.276026i
$841$	7.00000	0.241379
$842$	22.0000	0.758170
$843$	− 26.0000i	− 0.895488i
$844$	−12.0000	−0.413057
$845$	0	0
$846$	−8.00000	−0.275046
$847$	20.0000i	0.687208i
$848$	−10.0000	−0.343401
$849$	−4.00000	−0.137280
$850$	2.00000i	0.0685994i
$851$	0	0
$852$	− 8.00000i	− 0.274075i
$853$	2.00000i	0.0684787i	0.999414	+	0.0342393i	$0.0109009\pi$
$853$	2.00000i	0.0684787i	−0.999414	+	0.0342393i	$0.989099\pi$
$854$	8.00000	0.273754
$855$	16.0000	0.547188
$856$	12.0000i	0.410152i
$857$	−18.0000	−0.614868	−0.307434	−	0.951569i	$-0.599470\pi$
$857$	−18.0000	−0.614868	−0.307434	+	0.951569i	$0.599470\pi$
$858$	0	0
$859$	−44.0000	−1.50126	−0.750630	−	0.660722i	$-0.770250\pi$
$859$	−44.0000	−1.50126	−0.750630	+	0.660722i	$0.770250\pi$
$860$	8.00000i	0.272798i
$861$	40.0000	1.36320
$862$	−8.00000	−0.272481
$863$	40.0000i	1.36162i	0.732462	+	0.680808i	$0.238371\pi$
$863$	40.0000i	1.36162i	−0.732462	+	0.680808i	$0.761629\pi$
$864$	1.00000i	0.0340207i
$865$	20.0000i	0.680020i
$866$	− 30.0000i	− 1.01944i
$867$	13.0000	0.441503
$868$	16.0000	0.543075
$869$	32.0000i	1.08553i
$870$	−12.0000	−0.406838
$871$	0	0
$872$	2.00000	0.0677285
$873$	10.0000i	0.338449i
$874$	0	0
$875$	48.0000	1.62270
$876$	− 2.00000i	− 0.0675737i
$877$	22.0000i	0.742887i	0.928456	+	0.371444i	$0.121137\pi$
$877$	22.0000i	0.742887i	−0.928456	+	0.371444i	$0.878863\pi$
$878$	16.0000i	0.539974i
$879$	26.0000i	0.876958i
$880$	−8.00000	−0.269680
$881$	−26.0000	−0.875962	−0.437981	−	0.898984i	$-0.644306\pi$
$881$	−26.0000	−0.875962	−0.437981	+	0.898984i	$0.644306\pi$
$882$	9.00000i	0.303046i
$883$	−4.00000	−0.134611	−0.0673054	−	0.997732i	$-0.521440\pi$
$883$	−4.00000	−0.134611	−0.0673054	+	0.997732i	$0.521440\pi$
$884$	0	0
$885$	−8.00000	−0.268917
$886$	4.00000i	0.134383i
$887$	0	0		−	1.00000i	$-0.5\pi$
$887$	0	0			1.00000i	$0.5\pi$
$888$	2.00000	0.0671156
$889$	0	0
$890$	− 28.0000i	− 0.938562i
$891$	4.00000i	0.134005i
$892$	4.00000i	0.133930i
$893$	−64.0000	−2.14168
$894$	6.00000	0.200670
$895$	24.0000i	0.802232i
$896$	4.00000	0.133631
$897$	0	0
$898$	−6.00000	−0.200223
$899$	− 24.0000i	− 0.800445i
$900$	−1.00000	−0.0333333
$901$	20.0000	0.666297
$902$	− 40.0000i	− 1.33185i
$903$	− 16.0000i	− 0.532447i
$904$	− 6.00000i	− 0.199557i
$905$	20.0000i	0.664822i
$906$	12.0000	0.398673
$907$	−28.0000	−0.929725	−0.464862	−	0.885383i	$-0.653896\pi$
$907$	−28.0000	−0.929725	−0.464862	+	0.885383i	$0.653896\pi$
$908$	− 20.0000i	− 0.663723i
$909$	2.00000	0.0663358
$910$	0	0
$911$	40.0000	1.32526	0.662630	−	0.748947i	$-0.269440\pi$
$911$	40.0000	1.32526	0.662630	+	0.748947i	$0.269440\pi$
$912$	8.00000i	0.264906i
$913$	−48.0000	−1.58857
$914$	−30.0000	−0.992312
$915$	4.00000i	0.132236i
$916$	22.0000i	0.726900i
$917$	− 16.0000i	− 0.528367i
$918$	− 2.00000i	− 0.0660098i
$919$	−16.0000	−0.527791	−0.263896	−	0.964551i	$-0.585007\pi$
$919$	−16.0000	−0.527791	−0.263896	+	0.964551i	$0.585007\pi$
$920$	0	0
$921$	− 8.00000i	− 0.263609i
$922$	−6.00000	−0.197599
$923$	0	0
$924$	16.0000	0.526361
$925$	2.00000i	0.0657596i
$926$	20.0000	0.657241
$927$	−16.0000	−0.525509
$928$	− 6.00000i	− 0.196960i
$929$	46.0000i	1.50921i	0.656179	+	0.754606i	$0.272172\pi$
$929$	46.0000i	1.50921i	−0.656179	+	0.754606i	$0.727828\pi$
$930$	8.00000i	0.262330i
$931$	72.0000i	2.35970i
$932$	18.0000	0.589610
$933$	0	0
$934$	− 4.00000i	− 0.130884i
$935$	16.0000	0.523256
$936$	0	0
$937$	26.0000	0.849383	0.424691	−	0.905338i	$-0.360383\pi$
$937$	26.0000	0.849383	0.424691	+	0.905338i	$0.360383\pi$
$938$	64.0000i	2.08967i
$939$	6.00000	0.195803
$940$	−16.0000	−0.521862
$941$	− 46.0000i	− 1.49956i	−0.661689	−	0.749779i	$-0.730160\pi$
$941$	− 46.0000i	− 1.49956i	0.661689	−	0.749779i	$-0.269840\pi$
$942$	14.0000i	0.456145i
$943$	0	0
$944$	− 4.00000i	− 0.130189i
$945$	−8.00000	−0.260240
$946$	−16.0000	−0.520205
$947$	4.00000i	0.129983i	0.997886	+	0.0649913i	$0.0207020\pi$
$947$	4.00000i	0.129983i	−0.997886	+	0.0649913i	$0.979298\pi$
$948$	8.00000	0.259828
$949$	0	0
$950$	−8.00000	−0.259554
$951$	6.00000i	0.194563i
$952$	−8.00000	−0.259281
$953$	30.0000	0.971795	0.485898	−	0.874016i	$-0.338493\pi$
$953$	30.0000	0.971795	0.485898	+	0.874016i	$0.338493\pi$
$954$	10.0000i	0.323762i
$955$	− 16.0000i	− 0.517748i
$956$	0	0
$957$	− 24.0000i	− 0.775810i
$958$	16.0000	0.516937
$959$	40.0000	1.29167
$960$	2.00000i	0.0645497i
$961$	15.0000	0.483871
$962$	0	0
$963$	12.0000	0.386695
$964$	10.0000i	0.322078i
$965$	−28.0000	−0.901352
$966$	0	0
$967$	4.00000i	0.128631i	0.997930	+	0.0643157i	$0.0204865\pi$
$967$	4.00000i	0.128631i	−0.997930	+	0.0643157i	$0.979514\pi$
$968$	− 5.00000i	− 0.160706i
$969$	− 16.0000i	− 0.513994i
$970$	20.0000i	0.642161i
$971$	−28.0000	−0.898563	−0.449281	−	0.893390i	$-0.648320\pi$
$971$	−28.0000	−0.898563	−0.449281	+	0.893390i	$0.648320\pi$
$972$	1.00000	0.0320750
$973$	− 48.0000i	− 1.53881i
$974$	4.00000	0.128168
$975$	0	0
$976$	−2.00000	−0.0640184
$977$	6.00000i	0.191957i	0.995383	+	0.0959785i	$0.0305980\pi$
$977$	6.00000i	0.191957i	−0.995383	+	0.0959785i	$0.969402\pi$
$978$	−16.0000	−0.511624
$979$	56.0000	1.78977
$980$	18.0000i	0.574989i
$981$	− 2.00000i	− 0.0638551i
$982$	36.0000i	1.14881i
$983$	− 24.0000i	− 0.765481i	−0.923856	−	0.382741i	$-0.874980\pi$
$983$	− 24.0000i	− 0.765481i	0.923856	−	0.382741i	$-0.125020\pi$
$984$	−10.0000	−0.318788
$985$	−36.0000	−1.14706
$986$	12.0000i	0.382158i
$987$	32.0000	1.01857
$988$	0	0
$989$	0	0
$990$	8.00000i	0.254257i
$991$	−48.0000	−1.52477	−0.762385	−	0.647124i	$-0.775972\pi$
$991$	−48.0000	−1.52477	−0.762385	+	0.647124i	$0.775972\pi$
$992$	−4.00000	−0.127000
$993$	− 8.00000i	− 0.253872i
$994$	32.0000i	1.01498i
$995$	16.0000i	0.507234i
$996$	12.0000i	0.380235i
$997$	−26.0000	−0.823428	−0.411714	−	0.911313i	$-0.635070\pi$
$997$	−26.0000	−0.823428	−0.411714	+	0.911313i	$0.635070\pi$
$998$	0	0
$999$	− 2.00000i	− 0.0632772i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1014.2.b.b.337.1		2
3.2	odd	2		3042.2.b.g.1351.2		2
13.2	odd	12		1014.2.e.f.529.1		2
13.3	even	3		1014.2.i.d.823.2		4
13.4	even	6		1014.2.i.d.361.2		4
13.5	odd	4		78.2.a.a.1.1	✓	1
13.6	odd	12		1014.2.e.f.991.1		2
13.7	odd	12		1014.2.e.c.991.1		2
13.8	odd	4		1014.2.a.d.1.1		1
13.9	even	3		1014.2.i.d.361.1		4
13.10	even	6		1014.2.i.d.823.1		4
13.11	odd	12		1014.2.e.c.529.1		2
13.12	even	2	inner	1014.2.b.b.337.2		2
39.5	even	4		234.2.a.c.1.1		1
39.8	even	4		3042.2.a.f.1.1		1
39.38	odd	2		3042.2.b.g.1351.1		2
52.31	even	4		624.2.a.h.1.1		1
52.47	even	4		8112.2.a.v.1.1		1
65.18	even	4		1950.2.e.i.1249.2		2
65.44	odd	4		1950.2.a.w.1.1		1
65.57	even	4		1950.2.e.i.1249.1		2
91.83	even	4		3822.2.a.j.1.1		1
104.5	odd	4		2496.2.a.t.1.1		1
104.83	even	4		2496.2.a.b.1.1		1
117.5	even	12		2106.2.e.j.703.1		2
117.31	odd	12		2106.2.e.q.703.1		2
117.70	odd	12		2106.2.e.q.1405.1		2
117.83	even	12		2106.2.e.j.1405.1		2
143.109	even	4		9438.2.a.t.1.1		1
156.83	odd	4		1872.2.a.c.1.1		1
195.44	even	4		5850.2.a.d.1.1		1
195.83	odd	4		5850.2.e.bb.5149.1		2
195.122	odd	4		5850.2.e.bb.5149.2		2
312.5	even	4		7488.2.a.bz.1.1		1
312.83	odd	4		7488.2.a.bk.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.2.a.a.1.1	✓	1	13.5	odd	4
234.2.a.c.1.1		1	39.5	even	4
624.2.a.h.1.1		1	52.31	even	4
1014.2.a.d.1.1		1	13.8	odd	4
1014.2.b.b.337.1		2	1.1	even	1	trivial
1014.2.b.b.337.2		2	13.12	even	2	inner
1014.2.e.c.529.1		2	13.11	odd	12
1014.2.e.c.991.1		2	13.7	odd	12
1014.2.e.f.529.1		2	13.2	odd	12
1014.2.e.f.991.1		2	13.6	odd	12
1014.2.i.d.361.1		4	13.9	even	3
1014.2.i.d.361.2		4	13.4	even	6
1014.2.i.d.823.1		4	13.10	even	6
1014.2.i.d.823.2		4	13.3	even	3
1872.2.a.c.1.1		1	156.83	odd	4
1950.2.a.w.1.1		1	65.44	odd	4
1950.2.e.i.1249.1		2	65.57	even	4
1950.2.e.i.1249.2		2	65.18	even	4
2106.2.e.j.703.1		2	117.5	even	12
2106.2.e.j.1405.1		2	117.83	even	12
2106.2.e.q.703.1		2	117.31	odd	12
2106.2.e.q.1405.1		2	117.70	odd	12
2496.2.a.b.1.1		1	104.83	even	4
2496.2.a.t.1.1		1	104.5	odd	4
3042.2.a.f.1.1		1	39.8	even	4
3042.2.b.g.1351.1		2	39.38	odd	2
3042.2.b.g.1351.2		2	3.2	odd	2
3822.2.a.j.1.1		1	91.83	even	4
5850.2.a.d.1.1		1	195.44	even	4
5850.2.e.bb.5149.1		2	195.83	odd	4
5850.2.e.bb.5149.2		2	195.122	odd	4
7488.2.a.bk.1.1		1	312.83	odd	4
7488.2.a.bz.1.1		1	312.5	even	4
8112.2.a.v.1.1		1	52.47	even	4
9438.2.a.t.1.1		1	143.109	even	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:	\( N \)	\(=\)	\( 1014 = 2 \cdot 3 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1014.b (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +2.00000i q^{5} +1.00000i q^{6} -4.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000i q^{2} -1.00000 q^{3} -1.00000 q^{4} +2.00000i q^{5} +1.00000i q^{6} -4.00000i q^{7} +1.00000i q^{8} +1.00000 q^{9} +2.00000 q^{10} +4.00000i q^{11} +1.00000 q^{12} -4.00000 q^{14} -2.00000i q^{15} +1.00000 q^{16} -2.00000 q^{17} -1.00000i q^{18} -8.00000i q^{19} -2.00000i q^{20} +4.00000i q^{21} +4.00000 q^{22} -1.00000i q^{24} +1.00000 q^{25} -1.00000 q^{27} +4.00000i q^{28} +6.00000 q^{29} -2.00000 q^{30} -4.00000i q^{31} -1.00000i q^{32} -4.00000i q^{33} +2.00000i q^{34} +8.00000 q^{35} -1.00000 q^{36} +2.00000i q^{37} -8.00000 q^{38} -2.00000 q^{40} -10.0000i q^{41} +4.00000 q^{42} -4.00000 q^{43} -4.00000i q^{44} +2.00000i q^{45} -8.00000i q^{47} -1.00000 q^{48} -9.00000 q^{49} -1.00000i q^{50} +2.00000 q^{51} -10.0000 q^{53} +1.00000i q^{54} -8.00000 q^{55} +4.00000 q^{56} +8.00000i q^{57} -6.00000i q^{58} -4.00000i q^{59} +2.00000i q^{60} -2.00000 q^{61} -4.00000 q^{62} -4.00000i q^{63} -1.00000 q^{64} -4.00000 q^{66} -16.0000i q^{67} +2.00000 q^{68} -8.00000i q^{70} -8.00000i q^{71} +1.00000i q^{72} -2.00000i q^{73} +2.00000 q^{74} -1.00000 q^{75} +8.00000i q^{76} +16.0000 q^{77} +8.00000 q^{79} +2.00000i q^{80} +1.00000 q^{81} -10.0000 q^{82} +12.0000i q^{83} -4.00000i q^{84} -4.00000i q^{85} +4.00000i q^{86} -6.00000 q^{87} -4.00000 q^{88} -14.0000i q^{89} +2.00000 q^{90} +4.00000i q^{93} -8.00000 q^{94} +16.0000 q^{95} +1.00000i q^{96} +10.0000i q^{97} +9.00000i q^{98} +4.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9	\( 2 q - 2 q^{3} - 2 q^{4} + 2 q^{9} + 4 q^{10} + 2 q^{12} - 8 q^{14} + 2 q^{16} - 4 q^{17} + 8 q^{22} + 2 q^{25} - 2 q^{27} + 12 q^{29} - 4 q^{30} + 16 q^{35} - 2 q^{36} - 16 q^{38} - 4 q^{40} + 8 q^{42} - 8 q^{43} - 2 q^{48} - 18 q^{49} + 4 q^{51} - 20 q^{53} - 16 q^{55} + 8 q^{56} - 4 q^{61} - 8 q^{62} - 2 q^{64} - 8 q^{66} + 4 q^{68} + 4 q^{74} - 2 q^{75} + 32 q^{77} + 16 q^{79} + 2 q^{81} - 20 q^{82} - 12 q^{87} - 8 q^{88} + 4 q^{90} - 16 q^{94} + 32 q^{95}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^4 + 2 * q^9 + 4 * q^10 + 2 * q^12 - 8 * q^14 + 2 * q^16 - 4 * q^17 + 8 * q^22 + 2 * q^25 - 2 * q^27 + 12 * q^29 - 4 * q^30 + 16 * q^35 - 2 * q^36 - 16 * q^38 - 4 * q^40 + 8 * q^42 - 8 * q^43 - 2 * q^48 - 18 * q^49 + 4 * q^51 - 20 * q^53 - 16 * q^55 + 8 * q^56 - 4 * q^61 - 8 * q^62 - 2 * q^64 - 8 * q^66 + 4 * q^68 + 4 * q^74 - 2 * q^75 + 32 * q^77 + 16 * q^79 + 2 * q^81 - 20 * q^82 - 12 * q^87 - 8 * q^88 + 4 * q^90 - 16 * q^94 + 32 * q^95

Introduction

L-functions

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Database

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