LMFDB - Embedded newform 1305.2.a.n.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1305,2,Mod(1,1305)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1305, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1305.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1305 = 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1305.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:	$10.4204774638$
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:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 145)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	1305.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.41421 q^{2} +3.82843 q^{4} -1.00000 q^{5} +0.828427 q^{7} +4.41421 q^{8} -2.41421 q^{10} +4.82843 q^{11} -2.00000 q^{13} +2.00000 q^{14} +3.00000 q^{16} +2.82843 q^{17} +0.828427 q^{19} -3.82843 q^{20} +11.6569 q^{22} +8.82843 q^{23} +1.00000 q^{25} -4.82843 q^{26} +3.17157 q^{28} -1.00000 q^{29} -10.4853 q^{31} -1.58579 q^{32} +6.82843 q^{34} -0.828427 q^{35} +8.48528 q^{37} +2.00000 q^{38} -4.41421 q^{40} +6.00000 q^{41} -6.00000 q^{43} +18.4853 q^{44} +21.3137 q^{46} +0.343146 q^{47} -6.31371 q^{49} +2.41421 q^{50} -7.65685 q^{52} -7.65685 q^{53} -4.82843 q^{55} +3.65685 q^{56} -2.41421 q^{58} +7.65685 q^{61} -25.3137 q^{62} -9.82843 q^{64} +2.00000 q^{65} -10.4853 q^{67} +10.8284 q^{68} -2.00000 q^{70} -7.31371 q^{71} -8.48528 q^{73} +20.4853 q^{74} +3.17157 q^{76} +4.00000 q^{77} +14.4853 q^{79} -3.00000 q^{80} +14.4853 q^{82} -12.8284 q^{83} -2.82843 q^{85} -14.4853 q^{86} +21.3137 q^{88} -3.65685 q^{89} -1.65685 q^{91} +33.7990 q^{92} +0.828427 q^{94} -0.828427 q^{95} +4.48528 q^{97} -15.2426 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 4 q^{7} + 6 q^{8} - 2 q^{10} + 4 q^{11} - 4 q^{13} + 4 q^{14} + 6 q^{16} - 4 q^{19} - 2 q^{20} + 12 q^{22} + 12 q^{23} + 2 q^{25} - 4 q^{26} + 12 q^{28} - 2 q^{29} - 4 q^{31}+ \cdots - 22 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 4 * q^7 + 6 * q^8 - 2 * q^10 + 4 * q^11 - 4 * q^13 + 4 * q^14 + 6 * q^16 - 4 * q^19 - 2 * q^20 + 12 * q^22 + 12 * q^23 + 2 * q^25 - 4 * q^26 + 12 * q^28 - 2 * q^29 - 4 * q^31 - 6 * q^32 + 8 * q^34 + 4 * q^35 + 4 * q^38 - 6 * q^40 + 12 * q^41 - 12 * q^43 + 20 * q^44 + 20 * q^46 + 12 * q^47 + 10 * q^49 + 2 * q^50 - 4 * q^52 - 4 * q^53 - 4 * q^55 - 4 * q^56 - 2 * q^58 + 4 * q^61 - 28 * q^62 - 14 * q^64 + 4 * q^65 - 4 * q^67 + 16 * q^68 - 4 * q^70 + 8 * q^71 + 24 * q^74 + 12 * q^76 + 8 * q^77 + 12 * q^79 - 6 * q^80 + 12 * q^82 - 20 * q^83 - 12 * q^86 + 20 * q^88 + 4 * q^89 + 8 * q^91 + 28 * q^92 - 4 * q^94 + 4 * q^95 - 8 * q^97 - 22 * 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 $

$2$	2.41421	1.70711	0.853553	−	0.521005i	$-0.174443\pi$
$2$	2.41421	1.70711	0.853553	+	0.521005i	$0.174443\pi$
$3$	0	0
$3$	0	0
$4$	3.82843	1.91421
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	0.828427	0.313116	0.156558	−	0.987669i	$-0.449960\pi$
$7$	0.828427	0.313116	0.156558	+	0.987669i	$0.449960\pi$
$8$	4.41421	1.56066
$9$	0	0
$10$	−2.41421	−0.763441
$11$	4.82843	1.45583	0.727913	−	0.685670i	$-0.240491\pi$
$11$	4.82843	1.45583	0.727913	+	0.685670i	$0.240491\pi$
$12$	0	0
$13$	−2.00000	−0.554700	−0.277350	−	0.960769i	$-0.589456\pi$
$13$	−2.00000	−0.554700	−0.277350	+	0.960769i	$0.589456\pi$
$14$	2.00000	0.534522
$15$	0	0
$16$	3.00000	0.750000
$17$	2.82843	0.685994	0.342997	−	0.939336i	$-0.388558\pi$
$17$	2.82843	0.685994	0.342997	+	0.939336i	$0.388558\pi$
$18$	0	0
$19$	0.828427	0.190054	0.0950271	−	0.995475i	$-0.469706\pi$
$19$	0.828427	0.190054	0.0950271	+	0.995475i	$0.469706\pi$
$20$	−3.82843	−0.856062
$21$	0	0
$22$	11.6569	2.48525
$23$	8.82843	1.84085	0.920427	−	0.390914i	$-0.127841\pi$
$23$	8.82843	1.84085	0.920427	+	0.390914i	$0.127841\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−4.82843	−0.946932
$27$	0	0
$28$	3.17157	0.599371
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−10.4853	−1.88321	−0.941606	−	0.336717i	$-0.890684\pi$
$31$	−10.4853	−1.88321	−0.941606	+	0.336717i	$0.890684\pi$
$32$	−1.58579	−0.280330
$33$	0	0
$34$	6.82843	1.17107
$35$	−0.828427	−0.140030
$36$	0	0
$37$	8.48528	1.39497	0.697486	−	0.716599i	$-0.254302\pi$
$37$	8.48528	1.39497	0.697486	+	0.716599i	$0.254302\pi$
$38$	2.00000	0.324443
$39$	0	0
$40$	−4.41421	−0.697948
$41$	6.00000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	6.00000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	−6.00000	−0.914991	−0.457496	−	0.889212i	$-0.651253\pi$
$43$	−6.00000	−0.914991	−0.457496	+	0.889212i	$0.651253\pi$
$44$	18.4853	2.78676
$45$	0	0
$46$	21.3137	3.14253
$47$	0.343146	0.0500530	0.0250265	−	0.999687i	$-0.492033\pi$
$47$	0.343146	0.0500530	0.0250265	+	0.999687i	$0.492033\pi$
$48$	0	0
$49$	−6.31371	−0.901958
$50$	2.41421	0.341421
$51$	0	0
$52$	−7.65685	−1.06181
$53$	−7.65685	−1.05175	−0.525875	−	0.850562i	$-0.676262\pi$
$53$	−7.65685	−1.05175	−0.525875	+	0.850562i	$0.676262\pi$
$54$	0	0
$55$	−4.82843	−0.651065
$56$	3.65685	0.488668
$57$	0	0
$58$	−2.41421	−0.317002
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	0	0
$61$	7.65685	0.980360	0.490180	−	0.871621i	$-0.336931\pi$
$61$	7.65685	0.980360	0.490180	+	0.871621i	$0.336931\pi$
$62$	−25.3137	−3.21484
$63$	0	0
$64$	−9.82843	−1.22855
$65$	2.00000	0.248069
$66$	0	0
$67$	−10.4853	−1.28098	−0.640490	−	0.767966i	$-0.721269\pi$
$67$	−10.4853	−1.28098	−0.640490	+	0.767966i	$0.721269\pi$
$68$	10.8284	1.31314
$69$	0	0
$70$	−2.00000	−0.239046
$71$	−7.31371	−0.867978	−0.433989	−	0.900918i	$-0.642894\pi$
$71$	−7.31371	−0.867978	−0.433989	+	0.900918i	$0.642894\pi$
$72$	0	0
$73$	−8.48528	−0.993127	−0.496564	−	0.868000i	$-0.665405\pi$
$73$	−8.48528	−0.993127	−0.496564	+	0.868000i	$0.665405\pi$
$74$	20.4853	2.38137
$75$	0	0
$76$	3.17157	0.363804
$77$	4.00000	0.455842
$78$	0	0
$79$	14.4853	1.62972	0.814861	−	0.579657i	$-0.196813\pi$
$79$	14.4853	1.62972	0.814861	+	0.579657i	$0.196813\pi$
$80$	−3.00000	−0.335410
$81$	0	0
$82$	14.4853	1.59963
$83$	−12.8284	−1.40810	−0.704051	−	0.710149i	$-0.748628\pi$
$83$	−12.8284	−1.40810	−0.704051	+	0.710149i	$0.748628\pi$
$84$	0	0
$85$	−2.82843	−0.306786
$86$	−14.4853	−1.56199
$87$	0	0
$88$	21.3137	2.27205
$89$	−3.65685	−0.387626	−0.193813	−	0.981039i	$-0.562085\pi$
$89$	−3.65685	−0.387626	−0.193813	+	0.981039i	$0.562085\pi$
$90$	0	0
$91$	−1.65685	−0.173686
$92$	33.7990	3.52379
$93$	0	0
$94$	0.828427	0.0854457
$95$	−0.828427	−0.0849948
$96$	0	0
$97$	4.48528	0.455411	0.227706	−	0.973730i	$-0.426878\pi$
$97$	4.48528	0.455411	0.227706	+	0.973730i	$0.426878\pi$
$98$	−15.2426	−1.53974
$99$	0	0
$100$	3.82843	0.382843
$101$	−4.34315	−0.432159	−0.216080	−	0.976376i	$-0.569327\pi$
$101$	−4.34315	−0.432159	−0.216080	+	0.976376i	$0.569327\pi$
$102$	0	0
$103$	−12.1421	−1.19640	−0.598200	−	0.801347i	$-0.704117\pi$
$103$	−12.1421	−1.19640	−0.598200	+	0.801347i	$0.704117\pi$
$104$	−8.82843	−0.865699
$105$	0	0
$106$	−18.4853	−1.79545
$107$	8.14214	0.787130	0.393565	−	0.919297i	$-0.371242\pi$
$107$	8.14214	0.787130	0.393565	+	0.919297i	$0.371242\pi$
$108$	0	0
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	−11.6569	−1.11144
$111$	0	0
$112$	2.48528	0.234837
$113$	−2.82843	−0.266076	−0.133038	−	0.991111i	$-0.542473\pi$
$113$	−2.82843	−0.266076	−0.133038	+	0.991111i	$0.542473\pi$
$114$	0	0
$115$	−8.82843	−0.823255
$116$	−3.82843	−0.355461
$117$	0	0
$118$	0	0
$119$	2.34315	0.214796
$120$	0	0
$121$	12.3137	1.11943
$122$	18.4853	1.67358
$123$	0	0
$124$	−40.1421	−3.60487
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	6.00000	0.532414	0.266207	−	0.963916i	$-0.414230\pi$
$127$	6.00000	0.532414	0.266207	+	0.963916i	$0.414230\pi$
$128$	−20.5563	−1.81694
$129$	0	0
$130$	4.82843	0.423481
$131$	−16.1421	−1.41034	−0.705172	−	0.709036i	$-0.749130\pi$
$131$	−16.1421	−1.41034	−0.705172	+	0.709036i	$0.749130\pi$
$132$	0	0
$133$	0.686292	0.0595090
$134$	−25.3137	−2.18677
$135$	0	0
$136$	12.4853	1.07060
$137$	10.8284	0.925135	0.462567	−	0.886584i	$-0.346928\pi$
$137$	10.8284	0.925135	0.462567	+	0.886584i	$0.346928\pi$
$138$	0	0
$139$	10.3431	0.877294	0.438647	−	0.898659i	$-0.355458\pi$
$139$	10.3431	0.877294	0.438647	+	0.898659i	$0.355458\pi$
$140$	−3.17157	−0.268047
$141$	0	0
$142$	−17.6569	−1.48173
$143$	−9.65685	−0.807547
$144$	0	0
$145$	1.00000	0.0830455
$146$	−20.4853	−1.69537
$147$	0	0
$148$	32.4853	2.67027
$149$	13.3137	1.09070	0.545351	−	0.838208i	$-0.316396\pi$
$149$	13.3137	1.09070	0.545351	+	0.838208i	$0.316396\pi$
$150$	0	0
$151$	−12.0000	−0.976546	−0.488273	−	0.872691i	$-0.662373\pi$
$151$	−12.0000	−0.976546	−0.488273	+	0.872691i	$0.662373\pi$
$152$	3.65685	0.296610
$153$	0	0
$154$	9.65685	0.778171
$155$	10.4853	0.842198
$156$	0	0
$157$	−16.4853	−1.31567	−0.657834	−	0.753163i	$-0.728527\pi$
$157$	−16.4853	−1.31567	−0.657834	+	0.753163i	$0.728527\pi$
$158$	34.9706	2.78211
$159$	0	0
$160$	1.58579	0.125367
$161$	7.31371	0.576401
$162$	0	0
$163$	−19.6569	−1.53964	−0.769822	−	0.638259i	$-0.779655\pi$
$163$	−19.6569	−1.53964	−0.769822	+	0.638259i	$0.779655\pi$
$164$	22.9706	1.79370
$165$	0	0
$166$	−30.9706	−2.40378
$167$	−14.4853	−1.12090	−0.560452	−	0.828187i	$-0.689373\pi$
$167$	−14.4853	−1.12090	−0.560452	+	0.828187i	$0.689373\pi$
$168$	0	0
$169$	−9.00000	−0.692308
$170$	−6.82843	−0.523716
$171$	0	0
$172$	−22.9706	−1.75149
$173$	5.31371	0.403994	0.201997	−	0.979386i	$-0.435257\pi$
$173$	5.31371	0.403994	0.201997	+	0.979386i	$0.435257\pi$
$174$	0	0
$175$	0.828427	0.0626232
$176$	14.4853	1.09187
$177$	0	0
$178$	−8.82843	−0.661719
$179$	0.686292	0.0512958	0.0256479	−	0.999671i	$-0.491835\pi$
$179$	0.686292	0.0512958	0.0256479	+	0.999671i	$0.491835\pi$
$180$	0	0
$181$	−6.00000	−0.445976	−0.222988	−	0.974821i	$-0.571581\pi$
$181$	−6.00000	−0.445976	−0.222988	+	0.974821i	$0.571581\pi$
$182$	−4.00000	−0.296500
$183$	0	0
$184$	38.9706	2.87295
$185$	−8.48528	−0.623850
$186$	0	0
$187$	13.6569	0.998688
$188$	1.31371	0.0958120
$189$	0	0
$190$	−2.00000	−0.145095
$191$	15.1716	1.09778	0.548888	−	0.835896i	$-0.315051\pi$
$191$	15.1716	1.09778	0.548888	+	0.835896i	$0.315051\pi$
$192$	0	0
$193$	−12.4853	−0.898710	−0.449355	−	0.893353i	$-0.648346\pi$
$193$	−12.4853	−0.898710	−0.449355	+	0.893353i	$0.648346\pi$
$194$	10.8284	0.777436
$195$	0	0
$196$	−24.1716	−1.72654
$197$	8.34315	0.594425	0.297212	−	0.954811i	$-0.403943\pi$
$197$	8.34315	0.594425	0.297212	+	0.954811i	$0.403943\pi$
$198$	0	0
$199$	12.0000	0.850657	0.425329	−	0.905039i	$-0.360158\pi$
$199$	12.0000	0.850657	0.425329	+	0.905039i	$0.360158\pi$
$200$	4.41421	0.312132
$201$	0	0
$202$	−10.4853	−0.737742
$203$	−0.828427	−0.0581442
$204$	0	0
$205$	−6.00000	−0.419058
$206$	−29.3137	−2.04238
$207$	0	0
$208$	−6.00000	−0.416025
$209$	4.00000	0.276686
$210$	0	0
$211$	4.82843	0.332403	0.166201	−	0.986092i	$-0.446850\pi$
$211$	4.82843	0.332403	0.166201	+	0.986092i	$0.446850\pi$
$212$	−29.3137	−2.01327
$213$	0	0
$214$	19.6569	1.34371
$215$	6.00000	0.409197
$216$	0	0
$217$	−8.68629	−0.589664
$218$	4.82843	0.327022
$219$	0	0
$220$	−18.4853	−1.24628
$221$	−5.65685	−0.380521
$222$	0	0
$223$	21.7990	1.45977	0.729884	−	0.683571i	$-0.239574\pi$
$223$	21.7990	1.45977	0.729884	+	0.683571i	$0.239574\pi$
$224$	−1.31371	−0.0877758
$225$	0	0
$226$	−6.82843	−0.454220
$227$	8.14214	0.540413	0.270206	−	0.962802i	$-0.412908\pi$
$227$	8.14214	0.540413	0.270206	+	0.962802i	$0.412908\pi$
$228$	0	0
$229$	−2.00000	−0.132164	−0.0660819	−	0.997814i	$-0.521050\pi$
$229$	−2.00000	−0.132164	−0.0660819	+	0.997814i	$0.521050\pi$
$230$	−21.3137	−1.40538
$231$	0	0
$232$	−4.41421	−0.289807
$233$	−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
$235$	−0.343146	−0.0223844
$236$	0	0
$237$	0	0
$238$	5.65685	0.366679
$239$	23.3137	1.50804	0.754019	−	0.656852i	$-0.228113\pi$
$239$	23.3137	1.50804	0.754019	+	0.656852i	$0.228113\pi$
$240$	0	0
$241$	10.0000	0.644157	0.322078	−	0.946713i	$-0.395619\pi$
$241$	10.0000	0.644157	0.322078	+	0.946713i	$0.395619\pi$
$242$	29.7279	1.91098
$243$	0	0
$244$	29.3137	1.87662
$245$	6.31371	0.403368
$246$	0	0
$247$	−1.65685	−0.105423
$248$	−46.2843	−2.93905
$249$	0	0
$250$	−2.41421	−0.152688
$251$	−3.17157	−0.200188	−0.100094	−	0.994978i	$-0.531914\pi$
$251$	−3.17157	−0.200188	−0.100094	+	0.994978i	$0.531914\pi$
$252$	0	0
$253$	42.6274	2.67996
$254$	14.4853	0.908887
$255$	0	0
$256$	−29.9706	−1.87316
$257$	−29.3137	−1.82854	−0.914269	−	0.405107i	$-0.867234\pi$
$257$	−29.3137	−1.82854	−0.914269	+	0.405107i	$0.867234\pi$
$258$	0	0
$259$	7.02944	0.436788
$260$	7.65685	0.474858
$261$	0	0
$262$	−38.9706	−2.40761
$263$	8.34315	0.514460	0.257230	−	0.966350i	$-0.417190\pi$
$263$	8.34315	0.514460	0.257230	+	0.966350i	$0.417190\pi$
$264$	0	0
$265$	7.65685	0.470357
$266$	1.65685	0.101588
$267$	0	0
$268$	−40.1421	−2.45207
$269$	−1.31371	−0.0800982	−0.0400491	−	0.999198i	$-0.512751\pi$
$269$	−1.31371	−0.0800982	−0.0400491	+	0.999198i	$0.512751\pi$
$270$	0	0
$271$	29.7990	1.81016	0.905080	−	0.425242i	$-0.139811\pi$
$271$	29.7990	1.81016	0.905080	+	0.425242i	$0.139811\pi$
$272$	8.48528	0.514496
$273$	0	0
$274$	26.1421	1.57930
$275$	4.82843	0.291165
$276$	0	0
$277$	7.65685	0.460056	0.230028	−	0.973184i	$-0.426118\pi$
$277$	7.65685	0.460056	0.230028	+	0.973184i	$0.426118\pi$
$278$	24.9706	1.49763
$279$	0	0
$280$	−3.65685	−0.218539
$281$	6.68629	0.398871	0.199435	−	0.979911i	$-0.436089\pi$
$281$	6.68629	0.398871	0.199435	+	0.979911i	$0.436089\pi$
$282$	0	0
$283$	−0.828427	−0.0492449	−0.0246224	−	0.999697i	$-0.507838\pi$
$283$	−0.828427	−0.0492449	−0.0246224	+	0.999697i	$0.507838\pi$
$284$	−28.0000	−1.66149
$285$	0	0
$286$	−23.3137	−1.37857
$287$	4.97056	0.293403
$288$	0	0
$289$	−9.00000	−0.529412
$290$	2.41421	0.141768
$291$	0	0
$292$	−32.4853	−1.90106
$293$	8.48528	0.495715	0.247858	−	0.968796i	$-0.420273\pi$
$293$	8.48528	0.495715	0.247858	+	0.968796i	$0.420273\pi$
$294$	0	0
$295$	0	0
$296$	37.4558	2.17708
$297$	0	0
$298$	32.1421	1.86194
$299$	−17.6569	−1.02112
$300$	0	0
$301$	−4.97056	−0.286498
$302$	−28.9706	−1.66707
$303$	0	0
$304$	2.48528	0.142541
$305$	−7.65685	−0.438430
$306$	0	0
$307$	10.9706	0.626123	0.313062	−	0.949733i	$-0.398645\pi$
$307$	10.9706	0.626123	0.313062	+	0.949733i	$0.398645\pi$
$308$	15.3137	0.872580
$309$	0	0
$310$	25.3137	1.43772
$311$	2.48528	0.140927	0.0704637	−	0.997514i	$-0.477552\pi$
$311$	2.48528	0.140927	0.0704637	+	0.997514i	$0.477552\pi$
$312$	0	0
$313$	−6.00000	−0.339140	−0.169570	−	0.985518i	$-0.554238\pi$
$313$	−6.00000	−0.339140	−0.169570	+	0.985518i	$0.554238\pi$
$314$	−39.7990	−2.24599
$315$	0	0
$316$	55.4558	3.11963
$317$	2.82843	0.158860	0.0794301	−	0.996840i	$-0.474690\pi$
$317$	2.82843	0.158860	0.0794301	+	0.996840i	$0.474690\pi$
$318$	0	0
$319$	−4.82843	−0.270340
$320$	9.82843	0.549426
$321$	0	0
$322$	17.6569	0.983978
$323$	2.34315	0.130376
$324$	0	0
$325$	−2.00000	−0.110940
$326$	−47.4558	−2.62834
$327$	0	0
$328$	26.4853	1.46241
$329$	0.284271	0.0156724
$330$	0	0
$331$	−17.7990	−0.978321	−0.489160	−	0.872194i	$-0.662697\pi$
$331$	−17.7990	−0.978321	−0.489160	+	0.872194i	$0.662697\pi$
$332$	−49.1127	−2.69541
$333$	0	0
$334$	−34.9706	−1.91350
$335$	10.4853	0.572872
$336$	0	0
$337$	−6.82843	−0.371968	−0.185984	−	0.982553i	$-0.559547\pi$
$337$	−6.82843	−0.371968	−0.185984	+	0.982553i	$0.559547\pi$
$338$	−21.7279	−1.18184
$339$	0	0
$340$	−10.8284	−0.587254
$341$	−50.6274	−2.74163
$342$	0	0
$343$	−11.0294	−0.595534
$344$	−26.4853	−1.42799
$345$	0	0
$346$	12.8284	0.689661
$347$	−20.1421	−1.08129	−0.540643	−	0.841252i	$-0.681819\pi$
$347$	−20.1421	−1.08129	−0.540643	+	0.841252i	$0.681819\pi$
$348$	0	0
$349$	−24.6274	−1.31828	−0.659138	−	0.752022i	$-0.729079\pi$
$349$	−24.6274	−1.31828	−0.659138	+	0.752022i	$0.729079\pi$
$350$	2.00000	0.106904
$351$	0	0
$352$	−7.65685	−0.408112
$353$	15.6569	0.833330	0.416665	−	0.909060i	$-0.363199\pi$
$353$	15.6569	0.833330	0.416665	+	0.909060i	$0.363199\pi$
$354$	0	0
$355$	7.31371	0.388171
$356$	−14.0000	−0.741999
$357$	0	0
$358$	1.65685	0.0875675
$359$	32.1421	1.69640	0.848199	−	0.529678i	$-0.177687\pi$
$359$	32.1421	1.69640	0.848199	+	0.529678i	$0.177687\pi$
$360$	0	0
$361$	−18.3137	−0.963879
$362$	−14.4853	−0.761329
$363$	0	0
$364$	−6.34315	−0.332471
$365$	8.48528	0.444140
$366$	0	0
$367$	18.0000	0.939592	0.469796	−	0.882775i	$-0.344327\pi$
$367$	18.0000	0.939592	0.469796	+	0.882775i	$0.344327\pi$
$368$	26.4853	1.38064
$369$	0	0
$370$	−20.4853	−1.06498
$371$	−6.34315	−0.329320
$372$	0	0
$373$	26.9706	1.39648	0.698241	−	0.715862i	$-0.253966\pi$
$373$	26.9706	1.39648	0.698241	+	0.715862i	$0.253966\pi$
$374$	32.9706	1.70487
$375$	0	0
$376$	1.51472	0.0781156
$377$	2.00000	0.103005
$378$	0	0
$379$	5.51472	0.283272	0.141636	−	0.989919i	$-0.454764\pi$
$379$	5.51472	0.283272	0.141636	+	0.989919i	$0.454764\pi$
$380$	−3.17157	−0.162698
$381$	0	0
$382$	36.6274	1.87402
$383$	14.4853	0.740163	0.370082	−	0.928999i	$-0.379330\pi$
$383$	14.4853	0.740163	0.370082	+	0.928999i	$0.379330\pi$
$384$	0	0
$385$	−4.00000	−0.203859
$386$	−30.1421	−1.53419
$387$	0	0
$388$	17.1716	0.871755
$389$	6.68629	0.339008	0.169504	−	0.985529i	$-0.445783\pi$
$389$	6.68629	0.339008	0.169504	+	0.985529i	$0.445783\pi$
$390$	0	0
$391$	24.9706	1.26282
$392$	−27.8701	−1.40765
$393$	0	0
$394$	20.1421	1.01475
$395$	−14.4853	−0.728834
$396$	0	0
$397$	−8.34315	−0.418730	−0.209365	−	0.977838i	$-0.567140\pi$
$397$	−8.34315	−0.418730	−0.209365	+	0.977838i	$0.567140\pi$
$398$	28.9706	1.45216
$399$	0	0
$400$	3.00000	0.150000
$401$	29.3137	1.46386	0.731928	−	0.681382i	$-0.238621\pi$
$401$	29.3137	1.46386	0.731928	+	0.681382i	$0.238621\pi$
$402$	0	0
$403$	20.9706	1.04462
$404$	−16.6274	−0.827245
$405$	0	0
$406$	−2.00000	−0.0992583
$407$	40.9706	2.03084
$408$	0	0
$409$	30.9706	1.53140	0.765698	−	0.643200i	$-0.222394\pi$
$409$	30.9706	1.53140	0.765698	+	0.643200i	$0.222394\pi$
$410$	−14.4853	−0.715377
$411$	0	0
$412$	−46.4853	−2.29017
$413$	0	0
$414$	0	0
$415$	12.8284	0.629723
$416$	3.17157	0.155499
$417$	0	0
$418$	9.65685	0.472332
$419$	4.97056	0.242828	0.121414	−	0.992602i	$-0.461257\pi$
$419$	4.97056	0.242828	0.121414	+	0.992602i	$0.461257\pi$
$420$	0	0
$421$	−14.9706	−0.729621	−0.364810	−	0.931082i	$-0.618866\pi$
$421$	−14.9706	−0.729621	−0.364810	+	0.931082i	$0.618866\pi$
$422$	11.6569	0.567447
$423$	0	0
$424$	−33.7990	−1.64142
$425$	2.82843	0.137199
$426$	0	0
$427$	6.34315	0.306966
$428$	31.1716	1.50673
$429$	0	0
$430$	14.4853	0.698542
$431$	19.3137	0.930309	0.465154	−	0.885230i	$-0.345999\pi$
$431$	19.3137	0.930309	0.465154	+	0.885230i	$0.345999\pi$
$432$	0	0
$433$	−34.8284	−1.67375	−0.836874	−	0.547396i	$-0.815619\pi$
$433$	−34.8284	−1.67375	−0.836874	+	0.547396i	$0.815619\pi$
$434$	−20.9706	−1.00662
$435$	0	0
$436$	7.65685	0.366697
$437$	7.31371	0.349862
$438$	0	0
$439$	−21.6569	−1.03363	−0.516813	−	0.856099i	$-0.672882\pi$
$439$	−21.6569	−1.03363	−0.516813	+	0.856099i	$0.672882\pi$
$440$	−21.3137	−1.01609
$441$	0	0
$442$	−13.6569	−0.649590
$443$	−3.65685	−0.173742	−0.0868712	−	0.996220i	$-0.527687\pi$
$443$	−3.65685	−0.173742	−0.0868712	+	0.996220i	$0.527687\pi$
$444$	0	0
$445$	3.65685	0.173352
$446$	52.6274	2.49198
$447$	0	0
$448$	−8.14214	−0.384680
$449$	−0.343146	−0.0161940	−0.00809702	−	0.999967i	$-0.502577\pi$
$449$	−0.343146	−0.0161940	−0.00809702	+	0.999967i	$0.502577\pi$
$450$	0	0
$451$	28.9706	1.36417
$452$	−10.8284	−0.509326
$453$	0	0
$454$	19.6569	0.922542
$455$	1.65685	0.0776745
$456$	0	0
$457$	8.34315	0.390276	0.195138	−	0.980776i	$-0.437485\pi$
$457$	8.34315	0.390276	0.195138	+	0.980776i	$0.437485\pi$
$458$	−4.82843	−0.225618
$459$	0	0
$460$	−33.7990	−1.57589
$461$	24.3431	1.13377	0.566887	−	0.823796i	$-0.308148\pi$
$461$	24.3431	1.13377	0.566887	+	0.823796i	$0.308148\pi$
$462$	0	0
$463$	−17.7990	−0.827189	−0.413595	−	0.910461i	$-0.635727\pi$
$463$	−17.7990	−0.827189	−0.413595	+	0.910461i	$0.635727\pi$
$464$	−3.00000	−0.139272
$465$	0	0
$466$	−43.4558	−2.01305
$467$	22.9706	1.06295	0.531475	−	0.847074i	$-0.321638\pi$
$467$	22.9706	1.06295	0.531475	+	0.847074i	$0.321638\pi$
$468$	0	0
$469$	−8.68629	−0.401096
$470$	−0.828427	−0.0382125
$471$	0	0
$472$	0	0
$473$	−28.9706	−1.33207
$474$	0	0
$475$	0.828427	0.0380108
$476$	8.97056	0.411165
$477$	0	0
$478$	56.2843	2.57438
$479$	−12.8284	−0.586146	−0.293073	−	0.956090i	$-0.594678\pi$
$479$	−12.8284	−0.586146	−0.293073	+	0.956090i	$0.594678\pi$
$480$	0	0
$481$	−16.9706	−0.773791
$482$	24.1421	1.09964
$483$	0	0
$484$	47.1421	2.14282
$485$	−4.48528	−0.203666
$486$	0	0
$487$	−29.7990	−1.35032	−0.675161	−	0.737671i	$-0.735926\pi$
$487$	−29.7990	−1.35032	−0.675161	+	0.737671i	$0.735926\pi$
$488$	33.7990	1.53001
$489$	0	0
$490$	15.2426	0.688592
$491$	−43.4558	−1.96113	−0.980567	−	0.196183i	$-0.937145\pi$
$491$	−43.4558	−1.96113	−0.980567	+	0.196183i	$0.937145\pi$
$492$	0	0
$493$	−2.82843	−0.127386
$494$	−4.00000	−0.179969
$495$	0	0
$496$	−31.4558	−1.41241
$497$	−6.05887	−0.271778
$498$	0	0
$499$	36.0000	1.61158	0.805791	−	0.592200i	$-0.201741\pi$
$499$	36.0000	1.61158	0.805791	+	0.592200i	$0.201741\pi$
$500$	−3.82843	−0.171212
$501$	0	0
$502$	−7.65685	−0.341742
$503$	30.0000	1.33763	0.668817	−	0.743427i	$-0.266801\pi$
$503$	30.0000	1.33763	0.668817	+	0.743427i	$0.266801\pi$
$504$	0	0
$505$	4.34315	0.193267
$506$	102.912	4.57498
$507$	0	0
$508$	22.9706	1.01915
$509$	44.6274	1.97808	0.989038	−	0.147663i	$-0.0471751\pi$
$509$	44.6274	1.97808	0.989038	+	0.147663i	$0.0471751\pi$
$510$	0	0
$511$	−7.02944	−0.310964
$512$	−31.2426	−1.38074
$513$	0	0
$514$	−70.7696	−3.12151
$515$	12.1421	0.535046
$516$	0	0
$517$	1.65685	0.0728684
$518$	16.9706	0.745644
$519$	0	0
$520$	8.82843	0.387152
$521$	−1.31371	−0.0575546	−0.0287773	−	0.999586i	$-0.509161\pi$
$521$	−1.31371	−0.0575546	−0.0287773	+	0.999586i	$0.509161\pi$
$522$	0	0
$523$	−14.4853	−0.633397	−0.316699	−	0.948526i	$-0.602574\pi$
$523$	−14.4853	−0.633397	−0.316699	+	0.948526i	$0.602574\pi$
$524$	−61.7990	−2.69970
$525$	0	0
$526$	20.1421	0.878239
$527$	−29.6569	−1.29187
$528$	0	0
$529$	54.9411	2.38874
$530$	18.4853	0.802949
$531$	0	0
$532$	2.62742	0.113913
$533$	−12.0000	−0.519778
$534$	0	0
$535$	−8.14214	−0.352015
$536$	−46.2843	−1.99918
$537$	0	0
$538$	−3.17157	−0.136736
$539$	−30.4853	−1.31309
$540$	0	0
$541$	−38.9706	−1.67548	−0.837738	−	0.546073i	$-0.816122\pi$
$541$	−38.9706	−1.67548	−0.837738	+	0.546073i	$0.816122\pi$
$542$	71.9411	3.09014
$543$	0	0
$544$	−4.48528	−0.192305
$545$	−2.00000	−0.0856706
$546$	0	0
$547$	14.4853	0.619346	0.309673	−	0.950843i	$-0.399780\pi$
$547$	14.4853	0.619346	0.309673	+	0.950843i	$0.399780\pi$
$548$	41.4558	1.77091
$549$	0	0
$550$	11.6569	0.497050
$551$	−0.828427	−0.0352922
$552$	0	0
$553$	12.0000	0.510292
$554$	18.4853	0.785364
$555$	0	0
$556$	39.5980	1.67933
$557$	−39.9411	−1.69236	−0.846180	−	0.532897i	$-0.821103\pi$
$557$	−39.9411	−1.69236	−0.846180	+	0.532897i	$0.821103\pi$
$558$	0	0
$559$	12.0000	0.507546
$560$	−2.48528	−0.105022
$561$	0	0
$562$	16.1421	0.680915
$563$	3.65685	0.154118	0.0770590	−	0.997027i	$-0.475447\pi$
$563$	3.65685	0.154118	0.0770590	+	0.997027i	$0.475447\pi$
$564$	0	0
$565$	2.82843	0.118993
$566$	−2.00000	−0.0840663
$567$	0	0
$568$	−32.2843	−1.35462
$569$	−16.3431	−0.685140	−0.342570	−	0.939492i	$-0.611297\pi$
$569$	−16.3431	−0.685140	−0.342570	+	0.939492i	$0.611297\pi$
$570$	0	0
$571$	28.0000	1.17176	0.585882	−	0.810397i	$-0.300748\pi$
$571$	28.0000	1.17176	0.585882	+	0.810397i	$0.300748\pi$
$572$	−36.9706	−1.54582
$573$	0	0
$574$	12.0000	0.500870
$575$	8.82843	0.368171
$576$	0	0
$577$	−15.7990	−0.657721	−0.328860	−	0.944379i	$-0.606665\pi$
$577$	−15.7990	−0.657721	−0.328860	+	0.944379i	$0.606665\pi$
$578$	−21.7279	−0.903762
$579$	0	0
$580$	3.82843	0.158967
$581$	−10.6274	−0.440900
$582$	0	0
$583$	−36.9706	−1.53116
$584$	−37.4558	−1.54993
$585$	0	0
$586$	20.4853	0.846239
$587$	−9.79899	−0.404448	−0.202224	−	0.979339i	$-0.564817\pi$
$587$	−9.79899	−0.404448	−0.202224	+	0.979339i	$0.564817\pi$
$588$	0	0
$589$	−8.68629	−0.357912
$590$	0	0
$591$	0	0
$592$	25.4558	1.04623
$593$	−3.65685	−0.150169	−0.0750845	−	0.997177i	$-0.523923\pi$
$593$	−3.65685	−0.150169	−0.0750845	+	0.997177i	$0.523923\pi$
$594$	0	0
$595$	−2.34315	−0.0960596
$596$	50.9706	2.08784
$597$	0	0
$598$	−42.6274	−1.74316
$599$	−1.79899	−0.0735047	−0.0367524	−	0.999324i	$-0.511701\pi$
$599$	−1.79899	−0.0735047	−0.0367524	+	0.999324i	$0.511701\pi$
$600$	0	0
$601$	2.00000	0.0815817	0.0407909	−	0.999168i	$-0.487012\pi$
$601$	2.00000	0.0815817	0.0407909	+	0.999168i	$0.487012\pi$
$602$	−12.0000	−0.489083
$603$	0	0
$604$	−45.9411	−1.86932
$605$	−12.3137	−0.500623
$606$	0	0
$607$	42.9706	1.74412	0.872061	−	0.489398i	$-0.162783\pi$
$607$	42.9706	1.74412	0.872061	+	0.489398i	$0.162783\pi$
$608$	−1.31371	−0.0532779
$609$	0	0
$610$	−18.4853	−0.748447
$611$	−0.686292	−0.0277644
$612$	0	0
$613$	−2.00000	−0.0807792	−0.0403896	−	0.999184i	$-0.512860\pi$
$613$	−2.00000	−0.0807792	−0.0403896	+	0.999184i	$0.512860\pi$
$614$	26.4853	1.06886
$615$	0	0
$616$	17.6569	0.711415
$617$	14.8284	0.596970	0.298485	−	0.954414i	$-0.403519\pi$
$617$	14.8284	0.596970	0.298485	+	0.954414i	$0.403519\pi$
$618$	0	0
$619$	29.7990	1.19772	0.598861	−	0.800853i	$-0.295620\pi$
$619$	29.7990	1.19772	0.598861	+	0.800853i	$0.295620\pi$
$620$	40.1421	1.61215
$621$	0	0
$622$	6.00000	0.240578
$623$	−3.02944	−0.121372
$624$	0	0
$625$	1.00000	0.0400000
$626$	−14.4853	−0.578948
$627$	0	0
$628$	−63.1127	−2.51847
$629$	24.0000	0.956943
$630$	0	0
$631$	3.02944	0.120600	0.0603000	−	0.998180i	$-0.480794\pi$
$631$	3.02944	0.120600	0.0603000	+	0.998180i	$0.480794\pi$
$632$	63.9411	2.54344
$633$	0	0
$634$	6.82843	0.271191
$635$	−6.00000	−0.238103
$636$	0	0
$637$	12.6274	0.500316
$638$	−11.6569	−0.461499
$639$	0	0
$640$	20.5563	0.812561
$641$	44.6274	1.76268	0.881338	−	0.472485i	$-0.156643\pi$
$641$	44.6274	1.76268	0.881338	+	0.472485i	$0.156643\pi$
$642$	0	0
$643$	−31.4558	−1.24050	−0.620249	−	0.784405i	$-0.712968\pi$
$643$	−31.4558	−1.24050	−0.620249	+	0.784405i	$0.712968\pi$
$644$	28.0000	1.10335
$645$	0	0
$646$	5.65685	0.222566
$647$	−21.1127	−0.830026	−0.415013	−	0.909816i	$-0.636223\pi$
$647$	−21.1127	−0.830026	−0.415013	+	0.909816i	$0.636223\pi$
$648$	0	0
$649$	0	0
$650$	−4.82843	−0.189386
$651$	0	0
$652$	−75.2548	−2.94721
$653$	22.8284	0.893345	0.446673	−	0.894697i	$-0.352609\pi$
$653$	22.8284	0.893345	0.446673	+	0.894697i	$0.352609\pi$
$654$	0	0
$655$	16.1421	0.630725
$656$	18.0000	0.702782
$657$	0	0
$658$	0.686292	0.0267544
$659$	37.7990	1.47244	0.736220	−	0.676743i	$-0.236609\pi$
$659$	37.7990	1.47244	0.736220	+	0.676743i	$0.236609\pi$
$660$	0	0
$661$	26.0000	1.01128	0.505641	−	0.862744i	$-0.331256\pi$
$661$	26.0000	1.01128	0.505641	+	0.862744i	$0.331256\pi$
$662$	−42.9706	−1.67010
$663$	0	0
$664$	−56.6274	−2.19757
$665$	−0.686292	−0.0266132
$666$	0	0
$667$	−8.82843	−0.341838
$668$	−55.4558	−2.14565
$669$	0	0
$670$	25.3137	0.977954
$671$	36.9706	1.42723
$672$	0	0
$673$	−10.9706	−0.422884	−0.211442	−	0.977391i	$-0.567816\pi$
$673$	−10.9706	−0.422884	−0.211442	+	0.977391i	$0.567816\pi$
$674$	−16.4853	−0.634989
$675$	0	0
$676$	−34.4558	−1.32522
$677$	−36.7696	−1.41317	−0.706584	−	0.707629i	$-0.749765\pi$
$677$	−36.7696	−1.41317	−0.706584	+	0.707629i	$0.749765\pi$
$678$	0	0
$679$	3.71573	0.142597
$680$	−12.4853	−0.478789
$681$	0	0
$682$	−122.225	−4.68025
$683$	−40.1421	−1.53600	−0.767998	−	0.640452i	$-0.778747\pi$
$683$	−40.1421	−1.53600	−0.767998	+	0.640452i	$0.778747\pi$
$684$	0	0
$685$	−10.8284	−0.413733
$686$	−26.6274	−1.01664
$687$	0	0
$688$	−18.0000	−0.686244
$689$	15.3137	0.583406
$690$	0	0
$691$	−11.0294	−0.419580	−0.209790	−	0.977747i	$-0.567278\pi$
$691$	−11.0294	−0.419580	−0.209790	+	0.977747i	$0.567278\pi$
$692$	20.3431	0.773330
$693$	0	0
$694$	−48.6274	−1.84587
$695$	−10.3431	−0.392338
$696$	0	0
$697$	16.9706	0.642806
$698$	−59.4558	−2.25044
$699$	0	0
$700$	3.17157	0.119874
$701$	−29.3137	−1.10716	−0.553582	−	0.832795i	$-0.686739\pi$
$701$	−29.3137	−1.10716	−0.553582	+	0.832795i	$0.686739\pi$
$702$	0	0
$703$	7.02944	0.265120
$704$	−47.4558	−1.78856
$705$	0	0
$706$	37.7990	1.42258
$707$	−3.59798	−0.135316
$708$	0	0
$709$	−22.0000	−0.826227	−0.413114	−	0.910679i	$-0.635559\pi$
$709$	−22.0000	−0.826227	−0.413114	+	0.910679i	$0.635559\pi$
$710$	17.6569	0.662650
$711$	0	0
$712$	−16.1421	−0.604952
$713$	−92.5685	−3.46672
$714$	0	0
$715$	9.65685	0.361146
$716$	2.62742	0.0981912
$717$	0	0
$718$	77.5980	2.89593
$719$	−10.6274	−0.396336	−0.198168	−	0.980168i	$-0.563499\pi$
$719$	−10.6274	−0.396336	−0.198168	+	0.980168i	$0.563499\pi$
$720$	0	0
$721$	−10.0589	−0.374612
$722$	−44.2132	−1.64545
$723$	0	0
$724$	−22.9706	−0.853694
$725$	−1.00000	−0.0371391
$726$	0	0
$727$	43.9411	1.62969	0.814843	−	0.579682i	$-0.196823\pi$
$727$	43.9411	1.62969	0.814843	+	0.579682i	$0.196823\pi$
$728$	−7.31371	−0.271064
$729$	0	0
$730$	20.4853	0.758194
$731$	−16.9706	−0.627679
$732$	0	0
$733$	−17.1716	−0.634247	−0.317123	−	0.948384i	$-0.602717\pi$
$733$	−17.1716	−0.634247	−0.317123	+	0.948384i	$0.602717\pi$
$734$	43.4558	1.60398
$735$	0	0
$736$	−14.0000	−0.516047
$737$	−50.6274	−1.86488
$738$	0	0
$739$	−2.48528	−0.0914226	−0.0457113	−	0.998955i	$-0.514555\pi$
$739$	−2.48528	−0.0914226	−0.0457113	+	0.998955i	$0.514555\pi$
$740$	−32.4853	−1.19418
$741$	0	0
$742$	−15.3137	−0.562184
$743$	7.37258	0.270474	0.135237	−	0.990813i	$-0.456820\pi$
$743$	7.37258	0.270474	0.135237	+	0.990813i	$0.456820\pi$
$744$	0	0
$745$	−13.3137	−0.487777
$746$	65.1127	2.38395
$747$	0	0
$748$	52.2843	1.91170
$749$	6.74517	0.246463
$750$	0	0
$751$	−12.1421	−0.443073	−0.221536	−	0.975152i	$-0.571107\pi$
$751$	−12.1421	−0.443073	−0.221536	+	0.975152i	$0.571107\pi$
$752$	1.02944	0.0375397
$753$	0	0
$754$	4.82843	0.175841
$755$	12.0000	0.436725
$756$	0	0
$757$	36.4853	1.32608	0.663040	−	0.748584i	$-0.269266\pi$
$757$	36.4853	1.32608	0.663040	+	0.748584i	$0.269266\pi$
$758$	13.3137	0.483576
$759$	0	0
$760$	−3.65685	−0.132648
$761$	36.6274	1.32774	0.663871	−	0.747847i	$-0.268912\pi$
$761$	36.6274	1.32774	0.663871	+	0.747847i	$0.268912\pi$
$762$	0	0
$763$	1.65685	0.0599822
$764$	58.0833	2.10138
$765$	0	0
$766$	34.9706	1.26354
$767$	0	0
$768$	0	0
$769$	−4.34315	−0.156618	−0.0783089	−	0.996929i	$-0.524952\pi$
$769$	−4.34315	−0.156618	−0.0783089	+	0.996929i	$0.524952\pi$
$770$	−9.65685	−0.348009
$771$	0	0
$772$	−47.7990	−1.72032
$773$	−8.48528	−0.305194	−0.152597	−	0.988288i	$-0.548764\pi$
$773$	−8.48528	−0.305194	−0.152597	+	0.988288i	$0.548764\pi$
$774$	0	0
$775$	−10.4853	−0.376642
$776$	19.7990	0.710742
$777$	0	0
$778$	16.1421	0.578724
$779$	4.97056	0.178089
$780$	0	0
$781$	−35.3137	−1.26362
$782$	60.2843	2.15576
$783$	0	0
$784$	−18.9411	−0.676469
$785$	16.4853	0.588385
$786$	0	0
$787$	−21.7990	−0.777050	−0.388525	−	0.921438i	$-0.627015\pi$
$787$	−21.7990	−0.777050	−0.388525	+	0.921438i	$0.627015\pi$
$788$	31.9411	1.13786
$789$	0	0
$790$	−34.9706	−1.24420
$791$	−2.34315	−0.0833127
$792$	0	0
$793$	−15.3137	−0.543806
$794$	−20.1421	−0.714818
$795$	0	0
$796$	45.9411	1.62834
$797$	34.1421	1.20938	0.604688	−	0.796462i	$-0.293298\pi$
$797$	34.1421	1.20938	0.604688	+	0.796462i	$0.293298\pi$
$798$	0	0
$799$	0.970563	0.0343360
$800$	−1.58579	−0.0560660
$801$	0	0
$802$	70.7696	2.49896
$803$	−40.9706	−1.44582
$804$	0	0
$805$	−7.31371	−0.257774
$806$	50.6274	1.78327
$807$	0	0
$808$	−19.1716	−0.674454
$809$	14.2843	0.502208	0.251104	−	0.967960i	$-0.419206\pi$
$809$	14.2843	0.502208	0.251104	+	0.967960i	$0.419206\pi$
$810$	0	0
$811$	26.3431	0.925033	0.462516	−	0.886611i	$-0.346947\pi$
$811$	26.3431	0.925033	0.462516	+	0.886611i	$0.346947\pi$
$812$	−3.17157	−0.111300
$813$	0	0
$814$	98.9117	3.46685
$815$	19.6569	0.688550
$816$	0	0
$817$	−4.97056	−0.173898
$818$	74.7696	2.61426
$819$	0	0
$820$	−22.9706	−0.802167
$821$	45.3137	1.58146	0.790730	−	0.612165i	$-0.209701\pi$
$821$	45.3137	1.58146	0.790730	+	0.612165i	$0.209701\pi$
$822$	0	0
$823$	2.97056	0.103547	0.0517737	−	0.998659i	$-0.483513\pi$
$823$	2.97056	0.103547	0.0517737	+	0.998659i	$0.483513\pi$
$824$	−53.5980	−1.86717
$825$	0	0
$826$	0	0
$827$	5.31371	0.184776	0.0923879	−	0.995723i	$-0.470550\pi$
$827$	5.31371	0.184776	0.0923879	+	0.995723i	$0.470550\pi$
$828$	0	0
$829$	−24.6274	−0.855346	−0.427673	−	0.903934i	$-0.640666\pi$
$829$	−24.6274	−0.855346	−0.427673	+	0.903934i	$0.640666\pi$
$830$	30.9706	1.07500
$831$	0	0
$832$	19.6569	0.681479
$833$	−17.8579	−0.618738
$834$	0	0
$835$	14.4853	0.501284
$836$	15.3137	0.529636
$837$	0	0
$838$	12.0000	0.414533
$839$	14.4853	0.500087	0.250044	−	0.968235i	$-0.419555\pi$
$839$	14.4853	0.500087	0.250044	+	0.968235i	$0.419555\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−36.1421	−1.24554
$843$	0	0
$844$	18.4853	0.636290
$845$	9.00000	0.309609
$846$	0	0
$847$	10.2010	0.350511
$848$	−22.9706	−0.788812
$849$	0	0
$850$	6.82843	0.234213
$851$	74.9117	2.56794
$852$	0	0
$853$	−11.1127	−0.380492	−0.190246	−	0.981736i	$-0.560928\pi$
$853$	−11.1127	−0.380492	−0.190246	+	0.981736i	$0.560928\pi$
$854$	15.3137	0.524024
$855$	0	0
$856$	35.9411	1.22844
$857$	−48.6274	−1.66108	−0.830540	−	0.556958i	$-0.811968\pi$
$857$	−48.6274	−1.66108	−0.830540	+	0.556958i	$0.811968\pi$
$858$	0	0
$859$	28.4264	0.969896	0.484948	−	0.874543i	$-0.338838\pi$
$859$	28.4264	0.969896	0.484948	+	0.874543i	$0.338838\pi$
$860$	22.9706	0.783290
$861$	0	0
$862$	46.6274	1.58814
$863$	7.85786	0.267485	0.133742	−	0.991016i	$-0.457301\pi$
$863$	7.85786	0.267485	0.133742	+	0.991016i	$0.457301\pi$
$864$	0	0
$865$	−5.31371	−0.180672
$866$	−84.0833	−2.85727
$867$	0	0
$868$	−33.2548	−1.12874
$869$	69.9411	2.37259
$870$	0	0
$871$	20.9706	0.710560
$872$	8.82843	0.298968
$873$	0	0
$874$	17.6569	0.597252
$875$	−0.828427	−0.0280059
$876$	0	0
$877$	−18.2843	−0.617416	−0.308708	−	0.951157i	$-0.599897\pi$
$877$	−18.2843	−0.617416	−0.308708	+	0.951157i	$0.599897\pi$
$878$	−52.2843	−1.76451
$879$	0	0
$880$	−14.4853	−0.488299
$881$	−6.68629	−0.225267	−0.112633	−	0.993637i	$-0.535929\pi$
$881$	−6.68629	−0.225267	−0.112633	+	0.993637i	$0.535929\pi$
$882$	0	0
$883$	2.48528	0.0836364	0.0418182	−	0.999125i	$-0.486685\pi$
$883$	2.48528	0.0836364	0.0418182	+	0.999125i	$0.486685\pi$
$884$	−21.6569	−0.728399
$885$	0	0
$886$	−8.82843	−0.296597
$887$	29.3137	0.984258	0.492129	−	0.870522i	$-0.336219\pi$
$887$	29.3137	0.984258	0.492129	+	0.870522i	$0.336219\pi$
$888$	0	0
$889$	4.97056	0.166707
$890$	8.82843	0.295930
$891$	0	0
$892$	83.4558	2.79431
$893$	0.284271	0.00951277
$894$	0	0
$895$	−0.686292	−0.0229402
$896$	−17.0294	−0.568914
$897$	0	0
$898$	−0.828427	−0.0276450
$899$	10.4853	0.349704
$900$	0	0
$901$	−21.6569	−0.721494
$902$	69.9411	2.32878
$903$	0	0
$904$	−12.4853	−0.415254
$905$	6.00000	0.199447
$906$	0	0
$907$	−10.0000	−0.332045	−0.166022	−	0.986122i	$-0.553092\pi$
$907$	−10.0000	−0.332045	−0.166022	+	0.986122i	$0.553092\pi$
$908$	31.1716	1.03446
$909$	0	0
$910$	4.00000	0.132599
$911$	−3.85786	−0.127817	−0.0639084	−	0.997956i	$-0.520357\pi$
$911$	−3.85786	−0.127817	−0.0639084	+	0.997956i	$0.520357\pi$
$912$	0	0
$913$	−61.9411	−2.04995
$914$	20.1421	0.666243
$915$	0	0
$916$	−7.65685	−0.252990
$917$	−13.3726	−0.441602
$918$	0	0
$919$	−36.0000	−1.18753	−0.593765	−	0.804638i	$-0.702359\pi$
$919$	−36.0000	−1.18753	−0.593765	+	0.804638i	$0.702359\pi$
$920$	−38.9706	−1.28482
$921$	0	0
$922$	58.7696	1.93547
$923$	14.6274	0.481467
$924$	0	0
$925$	8.48528	0.278994
$926$	−42.9706	−1.41210
$927$	0	0
$928$	1.58579	0.0520560
$929$	−40.6274	−1.33294	−0.666471	−	0.745531i	$-0.732196\pi$
$929$	−40.6274	−1.33294	−0.666471	+	0.745531i	$0.732196\pi$
$930$	0	0
$931$	−5.23045	−0.171421
$932$	−68.9117	−2.25728
$933$	0	0
$934$	55.4558	1.81457
$935$	−13.6569	−0.446627
$936$	0	0
$937$	8.34315	0.272559	0.136279	−	0.990670i	$-0.456486\pi$
$937$	8.34315	0.272559	0.136279	+	0.990670i	$0.456486\pi$
$938$	−20.9706	−0.684713
$939$	0	0
$940$	−1.31371	−0.0428484
$941$	−39.9411	−1.30204	−0.651022	−	0.759059i	$-0.725659\pi$
$941$	−39.9411	−1.30204	−0.651022	+	0.759059i	$0.725659\pi$
$942$	0	0
$943$	52.9706	1.72496
$944$	0	0
$945$	0	0
$946$	−69.9411	−2.27398
$947$	56.9117	1.84938	0.924691	−	0.380719i	$-0.124324\pi$
$947$	56.9117	1.84938	0.924691	+	0.380719i	$0.124324\pi$
$948$	0	0
$949$	16.9706	0.550888
$950$	2.00000	0.0648886
$951$	0	0
$952$	10.3431	0.335223
$953$	−6.68629	−0.216590	−0.108295	−	0.994119i	$-0.534539\pi$
$953$	−6.68629	−0.216590	−0.108295	+	0.994119i	$0.534539\pi$
$954$	0	0
$955$	−15.1716	−0.490941
$956$	89.2548	2.88671
$957$	0	0
$958$	−30.9706	−1.00061
$959$	8.97056	0.289675
$960$	0	0
$961$	78.9411	2.54649
$962$	−40.9706	−1.32094
$963$	0	0
$964$	38.2843	1.23305
$965$	12.4853	0.401915
$966$	0	0
$967$	−18.9706	−0.610052	−0.305026	−	0.952344i	$-0.598665\pi$
$967$	−18.9706	−0.610052	−0.305026	+	0.952344i	$0.598665\pi$
$968$	54.3553	1.74705
$969$	0	0
$970$	−10.8284	−0.347680
$971$	0.142136	0.00456135	0.00228067	−	0.999997i	$-0.499274\pi$
$971$	0.142136	0.00456135	0.00228067	+	0.999997i	$0.499274\pi$
$972$	0	0
$973$	8.56854	0.274695
$974$	−71.9411	−2.30514
$975$	0	0
$976$	22.9706	0.735270
$977$	25.3137	0.809857	0.404929	−	0.914348i	$-0.367296\pi$
$977$	25.3137	0.809857	0.404929	+	0.914348i	$0.367296\pi$
$978$	0	0
$979$	−17.6569	−0.564316
$980$	24.1716	0.772133
$981$	0	0
$982$	−104.912	−3.34787
$983$	−13.3137	−0.424641	−0.212321	−	0.977200i	$-0.568102\pi$
$983$	−13.3137	−0.424641	−0.212321	+	0.977200i	$0.568102\pi$
$984$	0	0
$985$	−8.34315	−0.265835
$986$	−6.82843	−0.217461
$987$	0	0
$988$	−6.34315	−0.201802
$989$	−52.9706	−1.68437
$990$	0	0
$991$	52.0000	1.65183	0.825917	−	0.563791i	$-0.190658\pi$
$991$	52.0000	1.65183	0.825917	+	0.563791i	$0.190658\pi$
$992$	16.6274	0.527921
$993$	0	0
$994$	−14.6274	−0.463953
$995$	−12.0000	−0.380426
$996$	0	0
$997$	1.17157	0.0371041	0.0185520	−	0.999828i	$-0.494094\pi$
$997$	1.17157	0.0371041	0.0185520	+	0.999828i	$0.494094\pi$
$998$	86.9117	2.75114
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1305.2.a.n.1.2		2
3.2	odd	2		145.2.a.b.1.1	✓	2
5.4	even	2		6525.2.a.p.1.1		2
12.11	even	2		2320.2.a.k.1.1		2
15.2	even	4		725.2.b.c.349.1		4
15.8	even	4		725.2.b.c.349.4		4
15.14	odd	2		725.2.a.c.1.2		2
21.20	even	2		7105.2.a.e.1.1		2
24.5	odd	2		9280.2.a.be.1.2		2
24.11	even	2		9280.2.a.w.1.1		2
87.86	odd	2		4205.2.a.d.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.a.b.1.1	✓	2	3.2	odd	2
725.2.a.c.1.2		2	15.14	odd	2
725.2.b.c.349.1		4	15.2	even	4
725.2.b.c.349.4		4	15.8	even	4
1305.2.a.n.1.2		2	1.1	even	1	trivial
2320.2.a.k.1.1		2	12.11	even	2
4205.2.a.d.1.2		2	87.86	odd	2
6525.2.a.p.1.1		2	5.4	even	2
7105.2.a.e.1.1		2	21.20	even	2
9280.2.a.w.1.1		2	24.11	even	2
9280.2.a.be.1.2		2	24.5	odd	2

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1305.a (trivial)

\(f(q)\)	\(=\)	\(q+2.41421 q^{2} +3.82843 q^{4} -1.00000 q^{5} +0.828427 q^{7} +4.41421 q^{8} -2.41421 q^{10} +4.82843 q^{11} -2.00000 q^{13} +2.00000 q^{14} +3.00000 q^{16} +2.82843 q^{17} +0.828427 q^{19} -3.82843 q^{20} +11.6569 q^{22} +8.82843 q^{23} +1.00000 q^{25} -4.82843 q^{26} +3.17157 q^{28} -1.00000 q^{29} -10.4853 q^{31} -1.58579 q^{32} +6.82843 q^{34} -0.828427 q^{35} +8.48528 q^{37} +2.00000 q^{38} -4.41421 q^{40} +6.00000 q^{41} -6.00000 q^{43} +18.4853 q^{44} +21.3137 q^{46} +0.343146 q^{47} -6.31371 q^{49} +2.41421 q^{50} -7.65685 q^{52} -7.65685 q^{53} -4.82843 q^{55} +3.65685 q^{56} -2.41421 q^{58} +7.65685 q^{61} -25.3137 q^{62} -9.82843 q^{64} +2.00000 q^{65} -10.4853 q^{67} +10.8284 q^{68} -2.00000 q^{70} -7.31371 q^{71} -8.48528 q^{73} +20.4853 q^{74} +3.17157 q^{76} +4.00000 q^{77} +14.4853 q^{79} -3.00000 q^{80} +14.4853 q^{82} -12.8284 q^{83} -2.82843 q^{85} -14.4853 q^{86} +21.3137 q^{88} -3.65685 q^{89} -1.65685 q^{91} +33.7990 q^{92} +0.828427 q^{94} -0.828427 q^{95} +4.48528 q^{97} -15.2426 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} - 2 q^{5} - 4 q^{7} + 6 q^{8} - 2 q^{10} + 4 q^{11} - 4 q^{13} + 4 q^{14} + 6 q^{16} - 4 q^{19} - 2 q^{20} + 12 q^{22} + 12 q^{23} + 2 q^{25} - 4 q^{26} + 12 q^{28} - 2 q^{29} - 4 q^{31}+ \cdots - 22 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 - 2 * q^5 - 4 * q^7 + 6 * q^8 - 2 * q^10 + 4 * q^11 - 4 * q^13 + 4 * q^14 + 6 * q^16 - 4 * q^19 - 2 * q^20 + 12 * q^22 + 12 * q^23 + 2 * q^25 - 4 * q^26 + 12 * q^28 - 2 * q^29 - 4 * q^31 - 6 * q^32 + 8 * q^34 + 4 * q^35 + 4 * q^38 - 6 * q^40 + 12 * q^41 - 12 * q^43 + 20 * q^44 + 20 * q^46 + 12 * q^47 + 10 * q^49 + 2 * q^50 - 4 * q^52 - 4 * q^53 - 4 * q^55 - 4 * q^56 - 2 * q^58 + 4 * q^61 - 28 * q^62 - 14 * q^64 + 4 * q^65 - 4 * q^67 + 16 * q^68 - 4 * q^70 + 8 * q^71 + 24 * q^74 + 12 * q^76 + 8 * q^77 + 12 * q^79 - 6 * q^80 + 12 * q^82 - 20 * q^83 - 12 * q^86 + 20 * q^88 + 4 * q^89 + 8 * q^91 + 28 * q^92 - 4 * q^94 + 4 * q^95 - 8 * q^97 - 22 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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