LMFDB - Embedded newform 9610.2.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9610,2,Mod(1,9610)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9610, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("9610.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9610 = 2 \cdot 5 \cdot 31^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9610.a (trivial)

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:	yes
Analytic conductor:	$76.7362363425$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	9610.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.41421 q^{6} -4.00000 q^{7} +1.00000 q^{8} -1.00000 q^{9} +1.00000 q^{10} -4.24264 q^{11} +1.41421 q^{12} +2.82843 q^{13} -4.00000 q^{14} +1.41421 q^{15} +1.00000 q^{16} +7.07107 q^{17} -1.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} -5.65685 q^{21} -4.24264 q^{22} -2.82843 q^{23} +1.41421 q^{24} +1.00000 q^{25} +2.82843 q^{26} -5.65685 q^{27} -4.00000 q^{28} -5.65685 q^{29} +1.41421 q^{30} +1.00000 q^{32} -6.00000 q^{33} +7.07107 q^{34} -4.00000 q^{35} -1.00000 q^{36} +2.82843 q^{37} +4.00000 q^{38} +4.00000 q^{39} +1.00000 q^{40} -5.65685 q^{42} +4.24264 q^{43} -4.24264 q^{44} -1.00000 q^{45} -2.82843 q^{46} +1.41421 q^{48} +9.00000 q^{49} +1.00000 q^{50} +10.0000 q^{51} +2.82843 q^{52} +14.1421 q^{53} -5.65685 q^{54} -4.24264 q^{55} -4.00000 q^{56} +5.65685 q^{57} -5.65685 q^{58} -6.00000 q^{59} +1.41421 q^{60} +8.48528 q^{61} +4.00000 q^{63} +1.00000 q^{64} +2.82843 q^{65} -6.00000 q^{66} +12.0000 q^{67} +7.07107 q^{68} -4.00000 q^{69} -4.00000 q^{70} +4.00000 q^{71} -1.00000 q^{72} +4.24264 q^{73} +2.82843 q^{74} +1.41421 q^{75} +4.00000 q^{76} +16.9706 q^{77} +4.00000 q^{78} +8.48528 q^{79} +1.00000 q^{80} -5.00000 q^{81} +12.7279 q^{83} -5.65685 q^{84} +7.07107 q^{85} +4.24264 q^{86} -8.00000 q^{87} -4.24264 q^{88} -7.07107 q^{89} -1.00000 q^{90} -11.3137 q^{91} -2.82843 q^{92} +4.00000 q^{95} +1.41421 q^{96} -14.0000 q^{97} +9.00000 q^{98} +4.24264 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} - 8 q^{7} + 2 q^{8} - 2 q^{9} + 2 q^{10} - 8 q^{14} + 2 q^{16} - 2 q^{18} + 8 q^{19} + 2 q^{20} + 2 q^{25} - 8 q^{28} + 2 q^{32} - 12 q^{33} - 8 q^{35} - 2 q^{36} + 8 q^{38}+ \cdots + 18 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 - 8 * q^7 + 2 * q^8 - 2 * q^9 + 2 * q^10 - 8 * q^14 + 2 * q^16 - 2 * q^18 + 8 * q^19 + 2 * q^20 + 2 * q^25 - 8 * q^28 + 2 * q^32 - 12 * q^33 - 8 * q^35 - 2 * q^36 + 8 * q^38 + 8 * q^39 + 2 * q^40 - 2 * q^45 + 18 * q^49 + 2 * q^50 + 20 * q^51 - 8 * q^56 - 12 * q^59 + 8 * q^63 + 2 * q^64 - 12 * q^66 + 24 * q^67 - 8 * q^69 - 8 * q^70 + 8 * q^71 - 2 * q^72 + 8 * q^76 + 8 * q^78 + 2 * q^80 - 10 * q^81 - 16 * q^87 - 2 * q^90 + 8 * q^95 - 28 * q^97 + 18 * q^98

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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.41421	0.816497	0.408248	−	0.912871i	$-0.366140\pi$
$3$	1.41421	0.816497	0.408248	+	0.912871i	$0.366140\pi$
$4$	1.00000	0.500000
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	1.41421	0.577350
$7$	−4.00000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−4.00000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	1.00000	0.353553
$9$	−1.00000	−0.333333
$10$	1.00000	0.316228
$11$	−4.24264	−1.27920	−0.639602	−	0.768706i	$-0.720901\pi$
$11$	−4.24264	−1.27920	−0.639602	+	0.768706i	$0.720901\pi$
$12$	1.41421	0.408248
$13$	2.82843	0.784465	0.392232	−	0.919866i	$-0.371703\pi$
$13$	2.82843	0.784465	0.392232	+	0.919866i	$0.371703\pi$
$14$	−4.00000	−1.06904
$15$	1.41421	0.365148
$16$	1.00000	0.250000
$17$	7.07107	1.71499	0.857493	−	0.514496i	$-0.172021\pi$
$17$	7.07107	1.71499	0.857493	+	0.514496i	$0.172021\pi$
$18$	−1.00000	−0.235702
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	1.00000	0.223607
$21$	−5.65685	−1.23443
$22$	−4.24264	−0.904534
$23$	−2.82843	−0.589768	−0.294884	−	0.955533i	$-0.595281\pi$
$23$	−2.82843	−0.589768	−0.294884	+	0.955533i	$0.595281\pi$
$24$	1.41421	0.288675
$25$	1.00000	0.200000
$26$	2.82843	0.554700
$27$	−5.65685	−1.08866
$28$	−4.00000	−0.755929
$29$	−5.65685	−1.05045	−0.525226	−	0.850963i	$-0.676019\pi$
$29$	−5.65685	−1.05045	−0.525226	+	0.850963i	$0.676019\pi$
$30$	1.41421	0.258199
$31$	0	0
$31$	0	0
$32$	1.00000	0.176777
$33$	−6.00000	−1.04447
$34$	7.07107	1.21268
$35$	−4.00000	−0.676123
$36$	−1.00000	−0.166667
$37$	2.82843	0.464991	0.232495	−	0.972598i	$-0.425311\pi$
$37$	2.82843	0.464991	0.232495	+	0.972598i	$0.425311\pi$
$38$	4.00000	0.648886
$39$	4.00000	0.640513
$40$	1.00000	0.158114
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	−5.65685	−0.872872
$43$	4.24264	0.646997	0.323498	−	0.946229i	$-0.395141\pi$
$43$	4.24264	0.646997	0.323498	+	0.946229i	$0.395141\pi$
$44$	−4.24264	−0.639602
$45$	−1.00000	−0.149071
$46$	−2.82843	−0.417029
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	1.41421	0.204124
$49$	9.00000	1.28571
$50$	1.00000	0.141421
$51$	10.0000	1.40028
$52$	2.82843	0.392232
$53$	14.1421	1.94257	0.971286	−	0.237915i	$-0.0764641\pi$
$53$	14.1421	1.94257	0.971286	+	0.237915i	$0.0764641\pi$
$54$	−5.65685	−0.769800
$55$	−4.24264	−0.572078
$56$	−4.00000	−0.534522
$57$	5.65685	0.749269
$58$	−5.65685	−0.742781
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	1.41421	0.182574
$61$	8.48528	1.08643	0.543214	−	0.839594i	$-0.317207\pi$
$61$	8.48528	1.08643	0.543214	+	0.839594i	$0.317207\pi$
$62$	0	0
$63$	4.00000	0.503953
$64$	1.00000	0.125000
$65$	2.82843	0.350823
$66$	−6.00000	−0.738549
$67$	12.0000	1.46603	0.733017	−	0.680211i	$-0.238112\pi$
$67$	12.0000	1.46603	0.733017	+	0.680211i	$0.238112\pi$
$68$	7.07107	0.857493
$69$	−4.00000	−0.481543
$70$	−4.00000	−0.478091
$71$	4.00000	0.474713	0.237356	−	0.971423i	$-0.423719\pi$
$71$	4.00000	0.474713	0.237356	+	0.971423i	$0.423719\pi$
$72$	−1.00000	−0.117851
$73$	4.24264	0.496564	0.248282	−	0.968688i	$-0.420134\pi$
$73$	4.24264	0.496564	0.248282	+	0.968688i	$0.420134\pi$
$74$	2.82843	0.328798
$75$	1.41421	0.163299
$76$	4.00000	0.458831
$77$	16.9706	1.93398
$78$	4.00000	0.452911
$79$	8.48528	0.954669	0.477334	−	0.878722i	$-0.341603\pi$
$79$	8.48528	0.954669	0.477334	+	0.878722i	$0.341603\pi$
$80$	1.00000	0.111803
$81$	−5.00000	−0.555556
$82$	0	0
$83$	12.7279	1.39707	0.698535	−	0.715575i	$-0.253835\pi$
$83$	12.7279	1.39707	0.698535	+	0.715575i	$0.253835\pi$
$84$	−5.65685	−0.617213
$85$	7.07107	0.766965
$86$	4.24264	0.457496
$87$	−8.00000	−0.857690
$88$	−4.24264	−0.452267
$89$	−7.07107	−0.749532	−0.374766	−	0.927119i	$-0.622277\pi$
$89$	−7.07107	−0.749532	−0.374766	+	0.927119i	$0.622277\pi$
$90$	−1.00000	−0.105409
$91$	−11.3137	−1.18600
$92$	−2.82843	−0.294884
$93$	0	0
$94$	0	0
$95$	4.00000	0.410391
$96$	1.41421	0.144338
$97$	−14.0000	−1.42148	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	−14.0000	−1.42148	−0.710742	+	0.703452i	$0.751641\pi$
$98$	9.00000	0.909137
$99$	4.24264	0.426401
$100$	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	$0.468274\pi$
$102$	10.0000	0.990148
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	2.82843	0.277350
$105$	−5.65685	−0.552052
$106$	14.1421	1.37361
$107$	18.0000	1.74013	0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000	1.74013	0.870063	+	0.492941i	$0.164078\pi$
$108$	−5.65685	−0.544331
$109$	14.0000	1.34096	0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000	1.34096	0.670478	+	0.741929i	$0.266089\pi$
$110$	−4.24264	−0.404520
$111$	4.00000	0.379663
$112$	−4.00000	−0.377964
$113$	18.0000	1.69330	0.846649	−	0.532152i	$-0.178617\pi$
$113$	18.0000	1.69330	0.846649	+	0.532152i	$0.178617\pi$
$114$	5.65685	0.529813
$115$	−2.82843	−0.263752
$116$	−5.65685	−0.525226
$117$	−2.82843	−0.261488
$118$	−6.00000	−0.552345
$119$	−28.2843	−2.59281
$120$	1.41421	0.129099
$121$	7.00000	0.636364
$122$	8.48528	0.768221
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	4.00000	0.356348
$127$	−16.9706	−1.50589	−0.752947	−	0.658081i	$-0.771368\pi$
$127$	−16.9706	−1.50589	−0.752947	+	0.658081i	$0.771368\pi$
$128$	1.00000	0.0883883
$129$	6.00000	0.528271
$130$	2.82843	0.248069
$131$	12.0000	1.04844	0.524222	−	0.851581i	$-0.324356\pi$
$131$	12.0000	1.04844	0.524222	+	0.851581i	$0.324356\pi$
$132$	−6.00000	−0.522233
$133$	−16.0000	−1.38738
$134$	12.0000	1.03664
$135$	−5.65685	−0.486864
$136$	7.07107	0.606339
$137$	−15.5563	−1.32907	−0.664534	−	0.747258i	$-0.731370\pi$
$137$	−15.5563	−1.32907	−0.664534	+	0.747258i	$0.731370\pi$
$138$	−4.00000	−0.340503
$139$	−1.41421	−0.119952	−0.0599760	−	0.998200i	$-0.519102\pi$
$139$	−1.41421	−0.119952	−0.0599760	+	0.998200i	$0.519102\pi$
$140$	−4.00000	−0.338062
$141$	0	0
$142$	4.00000	0.335673
$143$	−12.0000	−1.00349
$144$	−1.00000	−0.0833333
$145$	−5.65685	−0.469776
$146$	4.24264	0.351123
$147$	12.7279	1.04978
$148$	2.82843	0.232495
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	1.41421	0.115470
$151$	−8.48528	−0.690522	−0.345261	−	0.938507i	$-0.612210\pi$
$151$	−8.48528	−0.690522	−0.345261	+	0.938507i	$0.612210\pi$
$152$	4.00000	0.324443
$153$	−7.07107	−0.571662
$154$	16.9706	1.36753
$155$	0	0
$156$	4.00000	0.320256
$157$	10.0000	0.798087	0.399043	−	0.916932i	$-0.369342\pi$
$157$	10.0000	0.798087	0.399043	+	0.916932i	$0.369342\pi$
$158$	8.48528	0.675053
$159$	20.0000	1.58610
$160$	1.00000	0.0790569
$161$	11.3137	0.891645
$162$	−5.00000	−0.392837
$163$	−18.0000	−1.40987	−0.704934	−	0.709273i	$-0.749024\pi$
$163$	−18.0000	−1.40987	−0.704934	+	0.709273i	$0.749024\pi$
$164$	0	0
$165$	−6.00000	−0.467099
$166$	12.7279	0.987878
$167$	14.1421	1.09435	0.547176	−	0.837018i	$-0.315703\pi$
$167$	14.1421	1.09435	0.547176	+	0.837018i	$0.315703\pi$
$168$	−5.65685	−0.436436
$169$	−5.00000	−0.384615
$170$	7.07107	0.542326
$171$	−4.00000	−0.305888
$172$	4.24264	0.323498
$173$	14.0000	1.06440	0.532200	−	0.846619i	$-0.321365\pi$
$173$	14.0000	1.06440	0.532200	+	0.846619i	$0.321365\pi$
$174$	−8.00000	−0.606478
$175$	−4.00000	−0.302372
$176$	−4.24264	−0.319801
$177$	−8.48528	−0.637793
$178$	−7.07107	−0.529999
$179$	−21.2132	−1.58555	−0.792775	−	0.609515i	$-0.791364\pi$
$179$	−21.2132	−1.58555	−0.792775	+	0.609515i	$0.791364\pi$
$180$	−1.00000	−0.0745356
$181$	−25.4558	−1.89212	−0.946059	−	0.323994i	$-0.894974\pi$
$181$	−25.4558	−1.89212	−0.946059	+	0.323994i	$0.894974\pi$
$182$	−11.3137	−0.838628
$183$	12.0000	0.887066
$184$	−2.82843	−0.208514
$185$	2.82843	0.207950
$186$	0	0
$187$	−30.0000	−2.19382
$188$	0	0
$189$	22.6274	1.64590
$190$	4.00000	0.290191
$191$	−12.0000	−0.868290	−0.434145	−	0.900843i	$-0.642949\pi$
$191$	−12.0000	−0.868290	−0.434145	+	0.900843i	$0.642949\pi$
$192$	1.41421	0.102062
$193$	24.0000	1.72756	0.863779	−	0.503871i	$-0.168091\pi$
$193$	24.0000	1.72756	0.863779	+	0.503871i	$0.168091\pi$
$194$	−14.0000	−1.00514
$195$	4.00000	0.286446
$196$	9.00000	0.642857
$197$	−8.48528	−0.604551	−0.302276	−	0.953221i	$-0.597746\pi$
$197$	−8.48528	−0.604551	−0.302276	+	0.953221i	$0.597746\pi$
$198$	4.24264	0.301511
$199$	16.9706	1.20301	0.601506	−	0.798869i	$-0.294568\pi$
$199$	16.9706	1.20301	0.601506	+	0.798869i	$0.294568\pi$
$200$	1.00000	0.0707107
$201$	16.9706	1.19701
$202$	2.00000	0.140720
$203$	22.6274	1.58813
$204$	10.0000	0.700140
$205$	0	0
$206$	0	0
$207$	2.82843	0.196589
$208$	2.82843	0.196116
$209$	−16.9706	−1.17388
$210$	−5.65685	−0.390360
$211$	−14.0000	−0.963800	−0.481900	−	0.876226i	$-0.660053\pi$
$211$	−14.0000	−0.963800	−0.481900	+	0.876226i	$0.660053\pi$
$212$	14.1421	0.971286
$213$	5.65685	0.387601
$214$	18.0000	1.23045
$215$	4.24264	0.289346
$216$	−5.65685	−0.384900
$217$	0	0
$218$	14.0000	0.948200
$219$	6.00000	0.405442
$220$	−4.24264	−0.286039
$221$	20.0000	1.34535
$222$	4.00000	0.268462
$223$	−11.3137	−0.757622	−0.378811	−	0.925474i	$-0.623667\pi$
$223$	−11.3137	−0.757622	−0.378811	+	0.925474i	$0.623667\pi$
$224$	−4.00000	−0.267261
$225$	−1.00000	−0.0666667
$226$	18.0000	1.19734
$227$	14.0000	0.929213	0.464606	−	0.885517i	$-0.346196\pi$
$227$	14.0000	0.929213	0.464606	+	0.885517i	$0.346196\pi$
$228$	5.65685	0.374634
$229$	−22.6274	−1.49526	−0.747631	−	0.664114i	$-0.768809\pi$
$229$	−22.6274	−1.49526	−0.747631	+	0.664114i	$0.768809\pi$
$230$	−2.82843	−0.186501
$231$	24.0000	1.57908
$232$	−5.65685	−0.371391
$233$	−18.0000	−1.17922	−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000	−1.17922	−0.589610	+	0.807688i	$0.700718\pi$
$234$	−2.82843	−0.184900
$235$	0	0
$236$	−6.00000	−0.390567
$237$	12.0000	0.779484
$238$	−28.2843	−1.83340
$239$	2.82843	0.182956	0.0914779	−	0.995807i	$-0.470841\pi$
$239$	2.82843	0.182956	0.0914779	+	0.995807i	$0.470841\pi$
$240$	1.41421	0.0912871
$241$	7.07107	0.455488	0.227744	−	0.973721i	$-0.426865\pi$
$241$	7.07107	0.455488	0.227744	+	0.973721i	$0.426865\pi$
$242$	7.00000	0.449977
$243$	9.89949	0.635053
$244$	8.48528	0.543214
$245$	9.00000	0.574989
$246$	0	0
$247$	11.3137	0.719874
$248$	0	0
$249$	18.0000	1.14070
$250$	1.00000	0.0632456
$251$	4.24264	0.267793	0.133897	−	0.990995i	$-0.457251\pi$
$251$	4.24264	0.267793	0.133897	+	0.990995i	$0.457251\pi$
$252$	4.00000	0.251976
$253$	12.0000	0.754434
$254$	−16.9706	−1.06483
$255$	10.0000	0.626224
$256$	1.00000	0.0625000
$257$	−12.0000	−0.748539	−0.374270	−	0.927320i	$-0.622107\pi$
$257$	−12.0000	−0.748539	−0.374270	+	0.927320i	$0.622107\pi$
$258$	6.00000	0.373544
$259$	−11.3137	−0.703000
$260$	2.82843	0.175412
$261$	5.65685	0.350150
$262$	12.0000	0.741362
$263$	−5.65685	−0.348817	−0.174408	−	0.984673i	$-0.555801\pi$
$263$	−5.65685	−0.348817	−0.174408	+	0.984673i	$0.555801\pi$
$264$	−6.00000	−0.369274
$265$	14.1421	0.868744
$266$	−16.0000	−0.981023
$267$	−10.0000	−0.611990
$268$	12.0000	0.733017
$269$	2.82843	0.172452	0.0862261	−	0.996276i	$-0.472519\pi$
$269$	2.82843	0.172452	0.0862261	+	0.996276i	$0.472519\pi$
$270$	−5.65685	−0.344265
$271$	−14.1421	−0.859074	−0.429537	−	0.903049i	$-0.641323\pi$
$271$	−14.1421	−0.859074	−0.429537	+	0.903049i	$0.641323\pi$
$272$	7.07107	0.428746
$273$	−16.0000	−0.968364
$274$	−15.5563	−0.939793
$275$	−4.24264	−0.255841
$276$	−4.00000	−0.240772
$277$	8.48528	0.509831	0.254916	−	0.966963i	$-0.417952\pi$
$277$	8.48528	0.509831	0.254916	+	0.966963i	$0.417952\pi$
$278$	−1.41421	−0.0848189
$279$	0	0
$280$	−4.00000	−0.239046
$281$	−8.00000	−0.477240	−0.238620	−	0.971113i	$-0.576695\pi$
$281$	−8.00000	−0.477240	−0.238620	+	0.971113i	$0.576695\pi$
$282$	0	0
$283$	18.0000	1.06999	0.534994	−	0.844856i	$-0.320314\pi$
$283$	18.0000	1.06999	0.534994	+	0.844856i	$0.320314\pi$
$284$	4.00000	0.237356
$285$	5.65685	0.335083
$286$	−12.0000	−0.709575
$287$	0	0
$288$	−1.00000	−0.0589256
$289$	33.0000	1.94118
$290$	−5.65685	−0.332182
$291$	−19.7990	−1.16064
$292$	4.24264	0.248282
$293$	26.0000	1.51894	0.759468	−	0.650545i	$-0.225459\pi$
$293$	26.0000	1.51894	0.759468	+	0.650545i	$0.225459\pi$
$294$	12.7279	0.742307
$295$	−6.00000	−0.349334
$296$	2.82843	0.164399
$297$	24.0000	1.39262
$298$	18.0000	1.04271
$299$	−8.00000	−0.462652
$300$	1.41421	0.0816497
$301$	−16.9706	−0.978167
$302$	−8.48528	−0.488273
$303$	2.82843	0.162489
$304$	4.00000	0.229416
$305$	8.48528	0.485866
$306$	−7.07107	−0.404226
$307$	2.00000	0.114146	0.0570730	−	0.998370i	$-0.481823\pi$
$307$	2.00000	0.114146	0.0570730	+	0.998370i	$0.481823\pi$
$308$	16.9706	0.966988
$309$	0	0
$310$	0	0
$311$	−16.0000	−0.907277	−0.453638	−	0.891186i	$-0.649874\pi$
$311$	−16.0000	−0.907277	−0.453638	+	0.891186i	$0.649874\pi$
$312$	4.00000	0.226455
$313$	−1.41421	−0.0799361	−0.0399680	−	0.999201i	$-0.512726\pi$
$313$	−1.41421	−0.0799361	−0.0399680	+	0.999201i	$0.512726\pi$
$314$	10.0000	0.564333
$315$	4.00000	0.225374
$316$	8.48528	0.477334
$317$	30.0000	1.68497	0.842484	−	0.538721i	$-0.181092\pi$
$317$	30.0000	1.68497	0.842484	+	0.538721i	$0.181092\pi$
$318$	20.0000	1.12154
$319$	24.0000	1.34374
$320$	1.00000	0.0559017
$321$	25.4558	1.42081
$322$	11.3137	0.630488
$323$	28.2843	1.57378
$324$	−5.00000	−0.277778
$325$	2.82843	0.156893
$326$	−18.0000	−0.996928
$327$	19.7990	1.09489
$328$	0	0
$329$	0	0
$330$	−6.00000	−0.330289
$331$	−15.5563	−0.855054	−0.427527	−	0.904003i	$-0.640615\pi$
$331$	−15.5563	−0.855054	−0.427527	+	0.904003i	$0.640615\pi$
$332$	12.7279	0.698535
$333$	−2.82843	−0.154997
$334$	14.1421	0.773823
$335$	12.0000	0.655630
$336$	−5.65685	−0.308607
$337$	−21.2132	−1.15556	−0.577778	−	0.816194i	$-0.696080\pi$
$337$	−21.2132	−1.15556	−0.577778	+	0.816194i	$0.696080\pi$
$338$	−5.00000	−0.271964
$339$	25.4558	1.38257
$340$	7.07107	0.383482
$341$	0	0
$342$	−4.00000	−0.216295
$343$	−8.00000	−0.431959
$344$	4.24264	0.228748
$345$	−4.00000	−0.215353
$346$	14.0000	0.752645
$347$	32.5269	1.74614	0.873068	−	0.487598i	$-0.162127\pi$
$347$	32.5269	1.74614	0.873068	+	0.487598i	$0.162127\pi$
$348$	−8.00000	−0.428845
$349$	26.0000	1.39175	0.695874	−	0.718164i	$-0.255017\pi$
$349$	26.0000	1.39175	0.695874	+	0.718164i	$0.255017\pi$
$350$	−4.00000	−0.213809
$351$	−16.0000	−0.854017
$352$	−4.24264	−0.226134
$353$	−9.89949	−0.526897	−0.263448	−	0.964673i	$-0.584860\pi$
$353$	−9.89949	−0.526897	−0.263448	+	0.964673i	$0.584860\pi$
$354$	−8.48528	−0.450988
$355$	4.00000	0.212298
$356$	−7.07107	−0.374766
$357$	−40.0000	−2.11702
$358$	−21.2132	−1.12115
$359$	−24.0000	−1.26667	−0.633336	−	0.773877i	$-0.718315\pi$
$359$	−24.0000	−1.26667	−0.633336	+	0.773877i	$0.718315\pi$
$360$	−1.00000	−0.0527046
$361$	−3.00000	−0.157895
$362$	−25.4558	−1.33793
$363$	9.89949	0.519589
$364$	−11.3137	−0.592999
$365$	4.24264	0.222070
$366$	12.0000	0.627250
$367$	−2.82843	−0.147643	−0.0738213	−	0.997271i	$-0.523519\pi$
$367$	−2.82843	−0.147643	−0.0738213	+	0.997271i	$0.523519\pi$
$368$	−2.82843	−0.147442
$369$	0	0
$370$	2.82843	0.147043
$371$	−56.5685	−2.93689
$372$	0	0
$373$	22.0000	1.13912	0.569558	−	0.821951i	$-0.307114\pi$
$373$	22.0000	1.13912	0.569558	+	0.821951i	$0.307114\pi$
$374$	−30.0000	−1.55126
$375$	1.41421	0.0730297
$376$	0	0
$377$	−16.0000	−0.824042
$378$	22.6274	1.16383
$379$	14.0000	0.719132	0.359566	−	0.933120i	$-0.382925\pi$
$379$	14.0000	0.719132	0.359566	+	0.933120i	$0.382925\pi$
$380$	4.00000	0.205196
$381$	−24.0000	−1.22956
$382$	−12.0000	−0.613973
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	1.41421	0.0721688
$385$	16.9706	0.864900
$386$	24.0000	1.22157
$387$	−4.24264	−0.215666
$388$	−14.0000	−0.710742
$389$	5.65685	0.286814	0.143407	−	0.989664i	$-0.454194\pi$
$389$	5.65685	0.286814	0.143407	+	0.989664i	$0.454194\pi$
$390$	4.00000	0.202548
$391$	−20.0000	−1.01144
$392$	9.00000	0.454569
$393$	16.9706	0.856052
$394$	−8.48528	−0.427482
$395$	8.48528	0.426941
$396$	4.24264	0.213201
$397$	6.00000	0.301131	0.150566	−	0.988600i	$-0.451890\pi$
$397$	6.00000	0.301131	0.150566	+	0.988600i	$0.451890\pi$
$398$	16.9706	0.850657
$399$	−22.6274	−1.13279
$400$	1.00000	0.0500000
$401$	24.0416	1.20058	0.600291	−	0.799782i	$-0.295051\pi$
$401$	24.0416	1.20058	0.600291	+	0.799782i	$0.295051\pi$
$402$	16.9706	0.846415
$403$	0	0
$404$	2.00000	0.0995037
$405$	−5.00000	−0.248452
$406$	22.6274	1.12298
$407$	−12.0000	−0.594818
$408$	10.0000	0.495074
$409$	−12.7279	−0.629355	−0.314678	−	0.949199i	$-0.601896\pi$
$409$	−12.7279	−0.629355	−0.314678	+	0.949199i	$0.601896\pi$
$410$	0	0
$411$	−22.0000	−1.08518
$412$	0	0
$413$	24.0000	1.18096
$414$	2.82843	0.139010
$415$	12.7279	0.624789
$416$	2.82843	0.138675
$417$	−2.00000	−0.0979404
$418$	−16.9706	−0.830057
$419$	30.0000	1.46560	0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000	1.46560	0.732798	+	0.680446i	$0.238214\pi$
$420$	−5.65685	−0.276026
$421$	22.0000	1.07221	0.536107	−	0.844150i	$-0.319894\pi$
$421$	22.0000	1.07221	0.536107	+	0.844150i	$0.319894\pi$
$422$	−14.0000	−0.681509
$423$	0	0
$424$	14.1421	0.686803
$425$	7.07107	0.342997
$426$	5.65685	0.274075
$427$	−33.9411	−1.64253
$428$	18.0000	0.870063
$429$	−16.9706	−0.819346
$430$	4.24264	0.204598
$431$	12.0000	0.578020	0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000	0.578020	0.289010	+	0.957326i	$0.406674\pi$
$432$	−5.65685	−0.272166
$433$	21.2132	1.01944	0.509721	−	0.860340i	$-0.329749\pi$
$433$	21.2132	1.01944	0.509721	+	0.860340i	$0.329749\pi$
$434$	0	0
$435$	−8.00000	−0.383571
$436$	14.0000	0.670478
$437$	−11.3137	−0.541208
$438$	6.00000	0.286691
$439$	32.0000	1.52728	0.763638	−	0.645644i	$-0.223411\pi$
$439$	32.0000	1.52728	0.763638	+	0.645644i	$0.223411\pi$
$440$	−4.24264	−0.202260
$441$	−9.00000	−0.428571
$442$	20.0000	0.951303
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	4.00000	0.189832
$445$	−7.07107	−0.335201
$446$	−11.3137	−0.535720
$447$	25.4558	1.20402
$448$	−4.00000	−0.188982
$449$	9.89949	0.467186	0.233593	−	0.972334i	$-0.424952\pi$
$449$	9.89949	0.467186	0.233593	+	0.972334i	$0.424952\pi$
$450$	−1.00000	−0.0471405
$451$	0	0
$452$	18.0000	0.846649
$453$	−12.0000	−0.563809
$454$	14.0000	0.657053
$455$	−11.3137	−0.530395
$456$	5.65685	0.264906
$457$	−4.24264	−0.198462	−0.0992312	−	0.995064i	$-0.531638\pi$
$457$	−4.24264	−0.198462	−0.0992312	+	0.995064i	$0.531638\pi$
$458$	−22.6274	−1.05731
$459$	−40.0000	−1.86704
$460$	−2.82843	−0.131876
$461$	−2.82843	−0.131733	−0.0658665	−	0.997828i	$-0.520981\pi$
$461$	−2.82843	−0.131733	−0.0658665	+	0.997828i	$0.520981\pi$
$462$	24.0000	1.11658
$463$	25.4558	1.18303	0.591517	−	0.806293i	$-0.298529\pi$
$463$	25.4558	1.18303	0.591517	+	0.806293i	$0.298529\pi$
$464$	−5.65685	−0.262613
$465$	0	0
$466$	−18.0000	−0.833834
$467$	−2.00000	−0.0925490	−0.0462745	−	0.998929i	$-0.514735\pi$
$467$	−2.00000	−0.0925490	−0.0462745	+	0.998929i	$0.514735\pi$
$468$	−2.82843	−0.130744
$469$	−48.0000	−2.21643
$470$	0	0
$471$	14.1421	0.651635
$472$	−6.00000	−0.276172
$473$	−18.0000	−0.827641
$474$	12.0000	0.551178
$475$	4.00000	0.183533
$476$	−28.2843	−1.29641
$477$	−14.1421	−0.647524
$478$	2.82843	0.129369
$479$	−28.0000	−1.27935	−0.639676	−	0.768644i	$-0.720932\pi$
$479$	−28.0000	−1.27935	−0.639676	+	0.768644i	$0.720932\pi$
$480$	1.41421	0.0645497
$481$	8.00000	0.364769
$482$	7.07107	0.322078
$483$	16.0000	0.728025
$484$	7.00000	0.318182
$485$	−14.0000	−0.635707
$486$	9.89949	0.449050
$487$	−16.9706	−0.769010	−0.384505	−	0.923123i	$-0.625628\pi$
$487$	−16.9706	−0.769010	−0.384505	+	0.923123i	$0.625628\pi$
$488$	8.48528	0.384111
$489$	−25.4558	−1.15115
$490$	9.00000	0.406579
$491$	−38.1838	−1.72321	−0.861605	−	0.507580i	$-0.830540\pi$
$491$	−38.1838	−1.72321	−0.861605	+	0.507580i	$0.830540\pi$
$492$	0	0
$493$	−40.0000	−1.80151
$494$	11.3137	0.509028
$495$	4.24264	0.190693
$496$	0	0
$497$	−16.0000	−0.717698
$498$	18.0000	0.806599
$499$	4.24264	0.189927	0.0949633	−	0.995481i	$-0.469727\pi$
$499$	4.24264	0.189927	0.0949633	+	0.995481i	$0.469727\pi$
$500$	1.00000	0.0447214
$501$	20.0000	0.893534
$502$	4.24264	0.189358
$503$	−28.0000	−1.24846	−0.624229	−	0.781241i	$-0.714587\pi$
$503$	−28.0000	−1.24846	−0.624229	+	0.781241i	$0.714587\pi$
$504$	4.00000	0.178174
$505$	2.00000	0.0889988
$506$	12.0000	0.533465
$507$	−7.07107	−0.314037
$508$	−16.9706	−0.752947
$509$	42.4264	1.88052	0.940259	−	0.340461i	$-0.110583\pi$
$509$	42.4264	1.88052	0.940259	+	0.340461i	$0.110583\pi$
$510$	10.0000	0.442807
$511$	−16.9706	−0.750733
$512$	1.00000	0.0441942
$513$	−22.6274	−0.999025
$514$	−12.0000	−0.529297
$515$	0	0
$516$	6.00000	0.264135
$517$	0	0
$518$	−11.3137	−0.497096
$519$	19.7990	0.869079
$520$	2.82843	0.124035
$521$	−10.0000	−0.438108	−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000	−0.438108	−0.219054	+	0.975713i	$0.570297\pi$
$522$	5.65685	0.247594
$523$	1.41421	0.0618392	0.0309196	−	0.999522i	$-0.490156\pi$
$523$	1.41421	0.0618392	0.0309196	+	0.999522i	$0.490156\pi$
$524$	12.0000	0.524222
$525$	−5.65685	−0.246885
$526$	−5.65685	−0.246651
$527$	0	0
$528$	−6.00000	−0.261116
$529$	−15.0000	−0.652174
$530$	14.1421	0.614295
$531$	6.00000	0.260378
$532$	−16.0000	−0.693688
$533$	0	0
$534$	−10.0000	−0.432742
$535$	18.0000	0.778208
$536$	12.0000	0.518321
$537$	−30.0000	−1.29460
$538$	2.82843	0.121942
$539$	−38.1838	−1.64469
$540$	−5.65685	−0.243432
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	−14.1421	−0.607457
$543$	−36.0000	−1.54491
$544$	7.07107	0.303170
$545$	14.0000	0.599694
$546$	−16.0000	−0.684737
$547$	−38.0000	−1.62476	−0.812381	−	0.583127i	$-0.801829\pi$
$547$	−38.0000	−1.62476	−0.812381	+	0.583127i	$0.801829\pi$
$548$	−15.5563	−0.664534
$549$	−8.48528	−0.362143
$550$	−4.24264	−0.180907
$551$	−22.6274	−0.963960
$552$	−4.00000	−0.170251
$553$	−33.9411	−1.44332
$554$	8.48528	0.360505
$555$	4.00000	0.169791
$556$	−1.41421	−0.0599760
$557$	−25.4558	−1.07860	−0.539299	−	0.842114i	$-0.681311\pi$
$557$	−25.4558	−1.07860	−0.539299	+	0.842114i	$0.681311\pi$
$558$	0	0
$559$	12.0000	0.507546
$560$	−4.00000	−0.169031
$561$	−42.4264	−1.79124
$562$	−8.00000	−0.337460
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	0	0
$565$	18.0000	0.757266
$566$	18.0000	0.756596
$567$	20.0000	0.839921
$568$	4.00000	0.167836
$569$	21.2132	0.889304	0.444652	−	0.895703i	$-0.353327\pi$
$569$	21.2132	0.889304	0.444652	+	0.895703i	$0.353327\pi$
$570$	5.65685	0.236940
$571$	−38.1838	−1.59794	−0.798970	−	0.601370i	$-0.794622\pi$
$571$	−38.1838	−1.59794	−0.798970	+	0.601370i	$0.794622\pi$
$572$	−12.0000	−0.501745
$573$	−16.9706	−0.708955
$574$	0	0
$575$	−2.82843	−0.117954
$576$	−1.00000	−0.0416667
$577$	32.0000	1.33218	0.666089	−	0.745873i	$-0.267967\pi$
$577$	32.0000	1.33218	0.666089	+	0.745873i	$0.267967\pi$
$578$	33.0000	1.37262
$579$	33.9411	1.41055
$580$	−5.65685	−0.234888
$581$	−50.9117	−2.11217
$582$	−19.7990	−0.820695
$583$	−60.0000	−2.48495
$584$	4.24264	0.175562
$585$	−2.82843	−0.116941
$586$	26.0000	1.07405
$587$	−46.6690	−1.92624	−0.963119	−	0.269076i	$-0.913282\pi$
$587$	−46.6690	−1.92624	−0.963119	+	0.269076i	$0.913282\pi$
$588$	12.7279	0.524891
$589$	0	0
$590$	−6.00000	−0.247016
$591$	−12.0000	−0.493614
$592$	2.82843	0.116248
$593$	−40.0000	−1.64260	−0.821302	−	0.570494i	$-0.806752\pi$
$593$	−40.0000	−1.64260	−0.821302	+	0.570494i	$0.806752\pi$
$594$	24.0000	0.984732
$595$	−28.2843	−1.15954
$596$	18.0000	0.737309
$597$	24.0000	0.982255
$598$	−8.00000	−0.327144
$599$	−28.0000	−1.14405	−0.572024	−	0.820237i	$-0.693842\pi$
$599$	−28.0000	−1.14405	−0.572024	+	0.820237i	$0.693842\pi$
$600$	1.41421	0.0577350
$601$	12.7279	0.519183	0.259591	−	0.965719i	$-0.416412\pi$
$601$	12.7279	0.519183	0.259591	+	0.965719i	$0.416412\pi$
$602$	−16.9706	−0.691669
$603$	−12.0000	−0.488678
$604$	−8.48528	−0.345261
$605$	7.00000	0.284590
$606$	2.82843	0.114897
$607$	28.0000	1.13648	0.568242	−	0.822861i	$-0.307624\pi$
$607$	28.0000	1.13648	0.568242	+	0.822861i	$0.307624\pi$
$608$	4.00000	0.162221
$609$	32.0000	1.29671
$610$	8.48528	0.343559
$611$	0	0
$612$	−7.07107	−0.285831
$613$	−16.9706	−0.685435	−0.342717	−	0.939439i	$-0.611347\pi$
$613$	−16.9706	−0.685435	−0.342717	+	0.939439i	$0.611347\pi$
$614$	2.00000	0.0807134
$615$	0	0
$616$	16.9706	0.683763
$617$	22.0000	0.885687	0.442843	−	0.896599i	$-0.353970\pi$
$617$	22.0000	0.885687	0.442843	+	0.896599i	$0.353970\pi$
$618$	0	0
$619$	1.41421	0.0568420	0.0284210	−	0.999596i	$-0.490952\pi$
$619$	1.41421	0.0568420	0.0284210	+	0.999596i	$0.490952\pi$
$620$	0	0
$621$	16.0000	0.642058
$622$	−16.0000	−0.641542
$623$	28.2843	1.13319
$624$	4.00000	0.160128
$625$	1.00000	0.0400000
$626$	−1.41421	−0.0565233
$627$	−24.0000	−0.958468
$628$	10.0000	0.399043
$629$	20.0000	0.797452
$630$	4.00000	0.159364
$631$	8.48528	0.337794	0.168897	−	0.985634i	$-0.445980\pi$
$631$	8.48528	0.337794	0.168897	+	0.985634i	$0.445980\pi$
$632$	8.48528	0.337526
$633$	−19.7990	−0.786939
$634$	30.0000	1.19145
$635$	−16.9706	−0.673456
$636$	20.0000	0.793052
$637$	25.4558	1.00860
$638$	24.0000	0.950169
$639$	−4.00000	−0.158238
$640$	1.00000	0.0395285
$641$	7.07107	0.279290	0.139645	−	0.990202i	$-0.455404\pi$
$641$	7.07107	0.279290	0.139645	+	0.990202i	$0.455404\pi$
$642$	25.4558	1.00466
$643$	−12.7279	−0.501940	−0.250970	−	0.967995i	$-0.580750\pi$
$643$	−12.7279	−0.501940	−0.250970	+	0.967995i	$0.580750\pi$
$644$	11.3137	0.445823
$645$	6.00000	0.236250
$646$	28.2843	1.11283
$647$	−11.3137	−0.444788	−0.222394	−	0.974957i	$-0.571387\pi$
$647$	−11.3137	−0.444788	−0.222394	+	0.974957i	$0.571387\pi$
$648$	−5.00000	−0.196419
$649$	25.4558	0.999229
$650$	2.82843	0.110940
$651$	0	0
$652$	−18.0000	−0.704934
$653$	−6.00000	−0.234798	−0.117399	−	0.993085i	$-0.537456\pi$
$653$	−6.00000	−0.234798	−0.117399	+	0.993085i	$0.537456\pi$
$654$	19.7990	0.774202
$655$	12.0000	0.468879
$656$	0	0
$657$	−4.24264	−0.165521
$658$	0	0
$659$	4.00000	0.155818	0.0779089	−	0.996960i	$-0.475176\pi$
$659$	4.00000	0.155818	0.0779089	+	0.996960i	$0.475176\pi$
$660$	−6.00000	−0.233550
$661$	14.0000	0.544537	0.272268	−	0.962221i	$-0.412226\pi$
$661$	14.0000	0.544537	0.272268	+	0.962221i	$0.412226\pi$
$662$	−15.5563	−0.604615
$663$	28.2843	1.09847
$664$	12.7279	0.493939
$665$	−16.0000	−0.620453
$666$	−2.82843	−0.109599
$667$	16.0000	0.619522
$668$	14.1421	0.547176
$669$	−16.0000	−0.618596
$670$	12.0000	0.463600
$671$	−36.0000	−1.38976
$672$	−5.65685	−0.218218
$673$	−7.07107	−0.272570	−0.136285	−	0.990670i	$-0.543516\pi$
$673$	−7.07107	−0.272570	−0.136285	+	0.990670i	$0.543516\pi$
$674$	−21.2132	−0.817102
$675$	−5.65685	−0.217732
$676$	−5.00000	−0.192308
$677$	−2.82843	−0.108705	−0.0543526	−	0.998522i	$-0.517310\pi$
$677$	−2.82843	−0.108705	−0.0543526	+	0.998522i	$0.517310\pi$
$678$	25.4558	0.977626
$679$	56.0000	2.14908
$680$	7.07107	0.271163
$681$	19.7990	0.758699
$682$	0	0
$683$	30.0000	1.14792	0.573959	−	0.818884i	$-0.305407\pi$
$683$	30.0000	1.14792	0.573959	+	0.818884i	$0.305407\pi$
$684$	−4.00000	−0.152944
$685$	−15.5563	−0.594378
$686$	−8.00000	−0.305441
$687$	−32.0000	−1.22088
$688$	4.24264	0.161749
$689$	40.0000	1.52388
$690$	−4.00000	−0.152277
$691$	−36.0000	−1.36950	−0.684752	−	0.728776i	$-0.740090\pi$
$691$	−36.0000	−1.36950	−0.684752	+	0.728776i	$0.740090\pi$
$692$	14.0000	0.532200
$693$	−16.9706	−0.644658
$694$	32.5269	1.23470
$695$	−1.41421	−0.0536442
$696$	−8.00000	−0.303239
$697$	0	0
$698$	26.0000	0.984115
$699$	−25.4558	−0.962828
$700$	−4.00000	−0.151186
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	−16.0000	−0.603881
$703$	11.3137	0.426705
$704$	−4.24264	−0.159901
$705$	0	0
$706$	−9.89949	−0.372572
$707$	−8.00000	−0.300871
$708$	−8.48528	−0.318896
$709$	−36.7696	−1.38091	−0.690455	−	0.723376i	$-0.742590\pi$
$709$	−36.7696	−1.38091	−0.690455	+	0.723376i	$0.742590\pi$
$710$	4.00000	0.150117
$711$	−8.48528	−0.318223
$712$	−7.07107	−0.264999
$713$	0	0
$714$	−40.0000	−1.49696
$715$	−12.0000	−0.448775
$716$	−21.2132	−0.792775
$717$	4.00000	0.149383
$718$	−24.0000	−0.895672
$719$	22.6274	0.843860	0.421930	−	0.906628i	$-0.361353\pi$
$719$	22.6274	0.843860	0.421930	+	0.906628i	$0.361353\pi$
$720$	−1.00000	−0.0372678
$721$	0	0
$722$	−3.00000	−0.111648
$723$	10.0000	0.371904
$724$	−25.4558	−0.946059
$725$	−5.65685	−0.210090
$726$	9.89949	0.367405
$727$	8.00000	0.296704	0.148352	−	0.988935i	$-0.452603\pi$
$727$	8.00000	0.296704	0.148352	+	0.988935i	$0.452603\pi$
$728$	−11.3137	−0.419314
$729$	29.0000	1.07407
$730$	4.24264	0.157027
$731$	30.0000	1.10959
$732$	12.0000	0.443533
$733$	14.0000	0.517102	0.258551	−	0.965998i	$-0.416755\pi$
$733$	14.0000	0.517102	0.258551	+	0.965998i	$0.416755\pi$
$734$	−2.82843	−0.104399
$735$	12.7279	0.469476
$736$	−2.82843	−0.104257
$737$	−50.9117	−1.87536
$738$	0	0
$739$	12.7279	0.468204	0.234102	−	0.972212i	$-0.424785\pi$
$739$	12.7279	0.468204	0.234102	+	0.972212i	$0.424785\pi$
$740$	2.82843	0.103975
$741$	16.0000	0.587775
$742$	−56.5685	−2.07670
$743$	−39.5980	−1.45271	−0.726354	−	0.687320i	$-0.758787\pi$
$743$	−39.5980	−1.45271	−0.726354	+	0.687320i	$0.758787\pi$
$744$	0	0
$745$	18.0000	0.659469
$746$	22.0000	0.805477
$747$	−12.7279	−0.465690
$748$	−30.0000	−1.09691
$749$	−72.0000	−2.63082
$750$	1.41421	0.0516398
$751$	−32.0000	−1.16770	−0.583848	−	0.811863i	$-0.698454\pi$
$751$	−32.0000	−1.16770	−0.583848	+	0.811863i	$0.698454\pi$
$752$	0	0
$753$	6.00000	0.218652
$754$	−16.0000	−0.582686
$755$	−8.48528	−0.308811
$756$	22.6274	0.822951
$757$	−28.2843	−1.02801	−0.514005	−	0.857787i	$-0.671839\pi$
$757$	−28.2843	−1.02801	−0.514005	+	0.857787i	$0.671839\pi$
$758$	14.0000	0.508503
$759$	16.9706	0.615992
$760$	4.00000	0.145095
$761$	12.7279	0.461387	0.230693	−	0.973026i	$-0.425901\pi$
$761$	12.7279	0.461387	0.230693	+	0.973026i	$0.425901\pi$
$762$	−24.0000	−0.869428
$763$	−56.0000	−2.02734
$764$	−12.0000	−0.434145
$765$	−7.07107	−0.255655
$766$	0	0
$767$	−16.9706	−0.612772
$768$	1.41421	0.0510310
$769$	−50.0000	−1.80305	−0.901523	−	0.432731i	$-0.857550\pi$
$769$	−50.0000	−1.80305	−0.901523	+	0.432731i	$0.857550\pi$
$770$	16.9706	0.611577
$771$	−16.9706	−0.611180
$772$	24.0000	0.863779
$773$	28.2843	1.01731	0.508657	−	0.860969i	$-0.330142\pi$
$773$	28.2843	1.01731	0.508657	+	0.860969i	$0.330142\pi$
$774$	−4.24264	−0.152499
$775$	0	0
$776$	−14.0000	−0.502571
$777$	−16.0000	−0.573997
$778$	5.65685	0.202808
$779$	0	0
$780$	4.00000	0.143223
$781$	−16.9706	−0.607254
$782$	−20.0000	−0.715199
$783$	32.0000	1.14359
$784$	9.00000	0.321429
$785$	10.0000	0.356915
$786$	16.9706	0.605320
$787$	38.1838	1.36110	0.680552	−	0.732700i	$-0.261740\pi$
$787$	38.1838	1.36110	0.680552	+	0.732700i	$0.261740\pi$
$788$	−8.48528	−0.302276
$789$	−8.00000	−0.284808
$790$	8.48528	0.301893
$791$	−72.0000	−2.56003
$792$	4.24264	0.150756
$793$	24.0000	0.852265
$794$	6.00000	0.212932
$795$	20.0000	0.709327
$796$	16.9706	0.601506
$797$	−2.82843	−0.100188	−0.0500940	−	0.998745i	$-0.515952\pi$
$797$	−2.82843	−0.100188	−0.0500940	+	0.998745i	$0.515952\pi$
$798$	−22.6274	−0.801002
$799$	0	0
$800$	1.00000	0.0353553
$801$	7.07107	0.249844
$802$	24.0416	0.848939
$803$	−18.0000	−0.635206
$804$	16.9706	0.598506
$805$	11.3137	0.398756
$806$	0	0
$807$	4.00000	0.140807
$808$	2.00000	0.0703598
$809$	24.0416	0.845259	0.422629	−	0.906303i	$-0.361107\pi$
$809$	24.0416	0.845259	0.422629	+	0.906303i	$0.361107\pi$
$810$	−5.00000	−0.175682
$811$	−22.0000	−0.772524	−0.386262	−	0.922389i	$-0.626234\pi$
$811$	−22.0000	−0.772524	−0.386262	+	0.922389i	$0.626234\pi$
$812$	22.6274	0.794067
$813$	−20.0000	−0.701431
$814$	−12.0000	−0.420600
$815$	−18.0000	−0.630512
$816$	10.0000	0.350070
$817$	16.9706	0.593725
$818$	−12.7279	−0.445021
$819$	11.3137	0.395333
$820$	0	0
$821$	45.2548	1.57940	0.789702	−	0.613490i	$-0.210235\pi$
$821$	45.2548	1.57940	0.789702	+	0.613490i	$0.210235\pi$
$822$	−22.0000	−0.767338
$823$	28.2843	0.985928	0.492964	−	0.870050i	$-0.335913\pi$
$823$	28.2843	0.985928	0.492964	+	0.870050i	$0.335913\pi$
$824$	0	0
$825$	−6.00000	−0.208893
$826$	24.0000	0.835067
$827$	−7.07107	−0.245885	−0.122943	−	0.992414i	$-0.539233\pi$
$827$	−7.07107	−0.245885	−0.122943	+	0.992414i	$0.539233\pi$
$828$	2.82843	0.0982946
$829$	8.48528	0.294706	0.147353	−	0.989084i	$-0.452925\pi$
$829$	8.48528	0.294706	0.147353	+	0.989084i	$0.452925\pi$
$830$	12.7279	0.441793
$831$	12.0000	0.416275
$832$	2.82843	0.0980581
$833$	63.6396	2.20498
$834$	−2.00000	−0.0692543
$835$	14.1421	0.489409
$836$	−16.9706	−0.586939
$837$	0	0
$838$	30.0000	1.03633
$839$	−20.0000	−0.690477	−0.345238	−	0.938515i	$-0.612202\pi$
$839$	−20.0000	−0.690477	−0.345238	+	0.938515i	$0.612202\pi$
$840$	−5.65685	−0.195180
$841$	3.00000	0.103448
$842$	22.0000	0.758170
$843$	−11.3137	−0.389665
$844$	−14.0000	−0.481900
$845$	−5.00000	−0.172005
$846$	0	0
$847$	−28.0000	−0.962091
$848$	14.1421	0.485643
$849$	25.4558	0.873642
$850$	7.07107	0.242536
$851$	−8.00000	−0.274236
$852$	5.65685	0.193801
$853$	−38.0000	−1.30110	−0.650548	−	0.759465i	$-0.725461\pi$
$853$	−38.0000	−1.30110	−0.650548	+	0.759465i	$0.725461\pi$
$854$	−33.9411	−1.16144
$855$	−4.00000	−0.136797
$856$	18.0000	0.615227
$857$	56.0000	1.91292	0.956462	−	0.291858i	$-0.0942733\pi$
$857$	56.0000	1.91292	0.956462	+	0.291858i	$0.0942733\pi$
$858$	−16.9706	−0.579365
$859$	−21.2132	−0.723785	−0.361893	−	0.932220i	$-0.617869\pi$
$859$	−21.2132	−0.723785	−0.361893	+	0.932220i	$0.617869\pi$
$860$	4.24264	0.144673
$861$	0	0
$862$	12.0000	0.408722
$863$	11.3137	0.385123	0.192562	−	0.981285i	$-0.438320\pi$
$863$	11.3137	0.385123	0.192562	+	0.981285i	$0.438320\pi$
$864$	−5.65685	−0.192450
$865$	14.0000	0.476014
$866$	21.2132	0.720854
$867$	46.6690	1.58496
$868$	0	0
$869$	−36.0000	−1.22122
$870$	−8.00000	−0.271225
$871$	33.9411	1.15005
$872$	14.0000	0.474100
$873$	14.0000	0.473828
$874$	−11.3137	−0.382692
$875$	−4.00000	−0.135225
$876$	6.00000	0.202721
$877$	−18.0000	−0.607817	−0.303908	−	0.952701i	$-0.598292\pi$
$877$	−18.0000	−0.607817	−0.303908	+	0.952701i	$0.598292\pi$
$878$	32.0000	1.07995
$879$	36.7696	1.24021
$880$	−4.24264	−0.143019
$881$	−7.07107	−0.238230	−0.119115	−	0.992880i	$-0.538006\pi$
$881$	−7.07107	−0.238230	−0.119115	+	0.992880i	$0.538006\pi$
$882$	−9.00000	−0.303046
$883$	4.24264	0.142776	0.0713881	−	0.997449i	$-0.477257\pi$
$883$	4.24264	0.142776	0.0713881	+	0.997449i	$0.477257\pi$
$884$	20.0000	0.672673
$885$	−8.48528	−0.285230
$886$	−36.0000	−1.20944
$887$	20.0000	0.671534	0.335767	−	0.941945i	$-0.391004\pi$
$887$	20.0000	0.671534	0.335767	+	0.941945i	$0.391004\pi$
$888$	4.00000	0.134231
$889$	67.8823	2.27670
$890$	−7.07107	−0.237023
$891$	21.2132	0.710669
$892$	−11.3137	−0.378811
$893$	0	0
$894$	25.4558	0.851371
$895$	−21.2132	−0.709079
$896$	−4.00000	−0.133631
$897$	−11.3137	−0.377754
$898$	9.89949	0.330350
$899$	0	0
$900$	−1.00000	−0.0333333
$901$	100.000	3.33148
$902$	0	0
$903$	−24.0000	−0.798670
$904$	18.0000	0.598671
$905$	−25.4558	−0.846181
$906$	−12.0000	−0.398673
$907$	38.0000	1.26177	0.630885	−	0.775877i	$-0.282692\pi$
$907$	38.0000	1.26177	0.630885	+	0.775877i	$0.282692\pi$
$908$	14.0000	0.464606
$909$	−2.00000	−0.0663358
$910$	−11.3137	−0.375046
$911$	5.65685	0.187420	0.0937100	−	0.995600i	$-0.470127\pi$
$911$	5.65685	0.187420	0.0937100	+	0.995600i	$0.470127\pi$
$912$	5.65685	0.187317
$913$	−54.0000	−1.78714
$914$	−4.24264	−0.140334
$915$	12.0000	0.396708
$916$	−22.6274	−0.747631
$917$	−48.0000	−1.58510
$918$	−40.0000	−1.32020
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	−2.82843	−0.0932505
$921$	2.82843	0.0931998
$922$	−2.82843	−0.0931493
$923$	11.3137	0.372395
$924$	24.0000	0.789542
$925$	2.82843	0.0929981
$926$	25.4558	0.836531
$927$	0	0
$928$	−5.65685	−0.185695
$929$	15.5563	0.510387	0.255194	−	0.966890i	$-0.417861\pi$
$929$	15.5563	0.510387	0.255194	+	0.966890i	$0.417861\pi$
$930$	0	0
$931$	36.0000	1.17985
$932$	−18.0000	−0.589610
$933$	−22.6274	−0.740788
$934$	−2.00000	−0.0654420
$935$	−30.0000	−0.981105
$936$	−2.82843	−0.0924500
$937$	−12.0000	−0.392023	−0.196011	−	0.980602i	$-0.562799\pi$
$937$	−12.0000	−0.392023	−0.196011	+	0.980602i	$0.562799\pi$
$938$	−48.0000	−1.56726
$939$	−2.00000	−0.0652675
$940$	0	0
$941$	39.5980	1.29086	0.645429	−	0.763821i	$-0.276679\pi$
$941$	39.5980	1.29086	0.645429	+	0.763821i	$0.276679\pi$
$942$	14.1421	0.460776
$943$	0	0
$944$	−6.00000	−0.195283
$945$	22.6274	0.736070
$946$	−18.0000	−0.585230
$947$	−7.07107	−0.229779	−0.114889	−	0.993378i	$-0.536651\pi$
$947$	−7.07107	−0.229779	−0.114889	+	0.993378i	$0.536651\pi$
$948$	12.0000	0.389742
$949$	12.0000	0.389536
$950$	4.00000	0.129777
$951$	42.4264	1.37577
$952$	−28.2843	−0.916698
$953$	−7.07107	−0.229054	−0.114527	−	0.993420i	$-0.536535\pi$
$953$	−7.07107	−0.229054	−0.114527	+	0.993420i	$0.536535\pi$
$954$	−14.1421	−0.457869
$955$	−12.0000	−0.388311
$956$	2.82843	0.0914779
$957$	33.9411	1.09716
$958$	−28.0000	−0.904639
$959$	62.2254	2.00936
$960$	1.41421	0.0456435
$961$	0	0
$962$	8.00000	0.257930
$963$	−18.0000	−0.580042
$964$	7.07107	0.227744
$965$	24.0000	0.772587
$966$	16.0000	0.514792
$967$	−11.3137	−0.363824	−0.181912	−	0.983315i	$-0.558229\pi$
$967$	−11.3137	−0.363824	−0.181912	+	0.983315i	$0.558229\pi$
$968$	7.00000	0.224989
$969$	40.0000	1.28499
$970$	−14.0000	−0.449513
$971$	−12.0000	−0.385098	−0.192549	−	0.981287i	$-0.561675\pi$
$971$	−12.0000	−0.385098	−0.192549	+	0.981287i	$0.561675\pi$
$972$	9.89949	0.317526
$973$	5.65685	0.181350
$974$	−16.9706	−0.543772
$975$	4.00000	0.128103
$976$	8.48528	0.271607
$977$	6.00000	0.191957	0.0959785	−	0.995383i	$-0.469402\pi$
$977$	6.00000	0.191957	0.0959785	+	0.995383i	$0.469402\pi$
$978$	−25.4558	−0.813988
$979$	30.0000	0.958804
$980$	9.00000	0.287494
$981$	−14.0000	−0.446986
$982$	−38.1838	−1.21849
$983$	5.65685	0.180426	0.0902128	−	0.995923i	$-0.471245\pi$
$983$	5.65685	0.180426	0.0902128	+	0.995923i	$0.471245\pi$
$984$	0	0
$985$	−8.48528	−0.270364
$986$	−40.0000	−1.27386
$987$	0	0
$988$	11.3137	0.359937
$989$	−12.0000	−0.381578
$990$	4.24264	0.134840
$991$	−45.2548	−1.43757	−0.718784	−	0.695234i	$-0.755301\pi$
$991$	−45.2548	−1.43757	−0.718784	+	0.695234i	$0.755301\pi$
$992$	0	0
$993$	−22.0000	−0.698149
$994$	−16.0000	−0.507489
$995$	16.9706	0.538003
$996$	18.0000	0.570352
$997$	38.0000	1.20347	0.601736	−	0.798695i	$-0.294476\pi$
$997$	38.0000	1.20347	0.601736	+	0.798695i	$0.294476\pi$
$998$	4.24264	0.134298
$999$	−16.0000	−0.506218

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9610.2.a.p.1.2	yes	2
31.30	odd	2	inner	9610.2.a.p.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9610.2.a.p.1.1	✓	2	31.30	odd	2	inner
9610.2.a.p.1.2	yes	2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 9610 = 2 \cdot 5 \cdot 31^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9610.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.41421 q^{6} -4.00000 q^{7} +1.00000 q^{8} -1.00000 q^{9} +1.00000 q^{10} -4.24264 q^{11} +1.41421 q^{12} +2.82843 q^{13} -4.00000 q^{14} +1.41421 q^{15} +1.00000 q^{16} +7.07107 q^{17} -1.00000 q^{18} +4.00000 q^{19} +1.00000 q^{20} -5.65685 q^{21} -4.24264 q^{22} -2.82843 q^{23} +1.41421 q^{24} +1.00000 q^{25} +2.82843 q^{26} -5.65685 q^{27} -4.00000 q^{28} -5.65685 q^{29} +1.41421 q^{30} +1.00000 q^{32} -6.00000 q^{33} +7.07107 q^{34} -4.00000 q^{35} -1.00000 q^{36} +2.82843 q^{37} +4.00000 q^{38} +4.00000 q^{39} +1.00000 q^{40} -5.65685 q^{42} +4.24264 q^{43} -4.24264 q^{44} -1.00000 q^{45} -2.82843 q^{46} +1.41421 q^{48} +9.00000 q^{49} +1.00000 q^{50} +10.0000 q^{51} +2.82843 q^{52} +14.1421 q^{53} -5.65685 q^{54} -4.24264 q^{55} -4.00000 q^{56} +5.65685 q^{57} -5.65685 q^{58} -6.00000 q^{59} +1.41421 q^{60} +8.48528 q^{61} +4.00000 q^{63} +1.00000 q^{64} +2.82843 q^{65} -6.00000 q^{66} +12.0000 q^{67} +7.07107 q^{68} -4.00000 q^{69} -4.00000 q^{70} +4.00000 q^{71} -1.00000 q^{72} +4.24264 q^{73} +2.82843 q^{74} +1.41421 q^{75} +4.00000 q^{76} +16.9706 q^{77} +4.00000 q^{78} +8.48528 q^{79} +1.00000 q^{80} -5.00000 q^{81} +12.7279 q^{83} -5.65685 q^{84} +7.07107 q^{85} +4.24264 q^{86} -8.00000 q^{87} -4.24264 q^{88} -7.07107 q^{89} -1.00000 q^{90} -11.3137 q^{91} -2.82843 q^{92} +4.00000 q^{95} +1.41421 q^{96} -14.0000 q^{97} +9.00000 q^{98} +4.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{5} - 8 q^{7} + 2 q^{8} - 2 q^{9} + 2 q^{10} - 8 q^{14} + 2 q^{16} - 2 q^{18} + 8 q^{19} + 2 q^{20} + 2 q^{25} - 8 q^{28} + 2 q^{32} - 12 q^{33} - 8 q^{35} - 2 q^{36} + 8 q^{38}+ \cdots + 18 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^5 - 8 * q^7 + 2 * q^8 - 2 * q^9 + 2 * q^10 - 8 * q^14 + 2 * q^16 - 2 * q^18 + 8 * q^19 + 2 * q^20 + 2 * q^25 - 8 * q^28 + 2 * q^32 - 12 * q^33 - 8 * q^35 - 2 * q^36 + 8 * q^38 + 8 * q^39 + 2 * q^40 - 2 * q^45 + 18 * q^49 + 2 * q^50 + 20 * q^51 - 8 * q^56 - 12 * q^59 + 8 * q^63 + 2 * q^64 - 12 * q^66 + 24 * q^67 - 8 * q^69 - 8 * q^70 + 8 * q^71 - 2 * q^72 + 8 * q^76 + 8 * q^78 + 2 * q^80 - 10 * q^81 - 16 * q^87 - 2 * q^90 + 8 * q^95 - 28 * q^97 + 18 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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