LMFDB - Embedded newform 7105.2.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7105,2,Mod(1,7105)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7105, 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("7105.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7105 = 5 \cdot 7^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7105.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:	$56.7337106361$
Analytic rank:	$1$
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			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	7105.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-2.41421 q^{2} +2.00000 q^{3} +3.82843 q^{4} -1.00000 q^{5} -4.82843 q^{6} -4.41421 q^{8} +1.00000 q^{9} +2.41421 q^{10} -4.82843 q^{11} +7.65685 q^{12} +2.00000 q^{13} -2.00000 q^{15} +3.00000 q^{16} +2.82843 q^{17} -2.41421 q^{18} -0.828427 q^{19} -3.82843 q^{20} +11.6569 q^{22} -8.82843 q^{23} -8.82843 q^{24} +1.00000 q^{25} -4.82843 q^{26} -4.00000 q^{27} +1.00000 q^{29} +4.82843 q^{30} +10.4853 q^{31} +1.58579 q^{32} -9.65685 q^{33} -6.82843 q^{34} +3.82843 q^{36} +8.48528 q^{37} +2.00000 q^{38} +4.00000 q^{39} +4.41421 q^{40} +6.00000 q^{41} -6.00000 q^{43} -18.4853 q^{44} -1.00000 q^{45} +21.3137 q^{46} +0.343146 q^{47} +6.00000 q^{48} -2.41421 q^{50} +5.65685 q^{51} +7.65685 q^{52} +7.65685 q^{53} +9.65685 q^{54} +4.82843 q^{55} -1.65685 q^{57} -2.41421 q^{58} -7.65685 q^{60} -7.65685 q^{61} -25.3137 q^{62} -9.82843 q^{64} -2.00000 q^{65} +23.3137 q^{66} -10.4853 q^{67} +10.8284 q^{68} -17.6569 q^{69} +7.31371 q^{71} -4.41421 q^{72} +8.48528 q^{73} -20.4853 q^{74} +2.00000 q^{75} -3.17157 q^{76} -9.65685 q^{78} +14.4853 q^{79} -3.00000 q^{80} -11.0000 q^{81} -14.4853 q^{82} -12.8284 q^{83} -2.82843 q^{85} +14.4853 q^{86} +2.00000 q^{87} +21.3137 q^{88} -3.65685 q^{89} +2.41421 q^{90} -33.7990 q^{92} +20.9706 q^{93} -0.828427 q^{94} +0.828427 q^{95} +3.17157 q^{96} -4.48528 q^{97} -4.82843 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 4 q^{3} + 2 q^{4} - 2 q^{5} - 4 q^{6} - 6 q^{8} + 2 q^{9} + 2 q^{10} - 4 q^{11} + 4 q^{12} + 4 q^{13} - 4 q^{15} + 6 q^{16} - 2 q^{18} + 4 q^{19} - 2 q^{20} + 12 q^{22} - 12 q^{23} - 12 q^{24}+ \cdots - 4 q^{99}+O(q^{100}) $ 2 * q - 2 * q^2 + 4 * q^3 + 2 * q^4 - 2 * q^5 - 4 * q^6 - 6 * q^8 + 2 * q^9 + 2 * q^10 - 4 * q^11 + 4 * q^12 + 4 * q^13 - 4 * q^15 + 6 * q^16 - 2 * q^18 + 4 * q^19 - 2 * q^20 + 12 * q^22 - 12 * q^23 - 12 * q^24 + 2 * q^25 - 4 * q^26 - 8 * q^27 + 2 * q^29 + 4 * q^30 + 4 * q^31 + 6 * q^32 - 8 * q^33 - 8 * q^34 + 2 * q^36 + 4 * q^38 + 8 * q^39 + 6 * q^40 + 12 * q^41 - 12 * q^43 - 20 * q^44 - 2 * q^45 + 20 * q^46 + 12 * q^47 + 12 * q^48 - 2 * q^50 + 4 * q^52 + 4 * q^53 + 8 * q^54 + 4 * q^55 + 8 * q^57 - 2 * q^58 - 4 * q^60 - 4 * q^61 - 28 * q^62 - 14 * q^64 - 4 * q^65 + 24 * q^66 - 4 * q^67 + 16 * q^68 - 24 * q^69 - 8 * q^71 - 6 * q^72 - 24 * q^74 + 4 * q^75 - 12 * q^76 - 8 * q^78 + 12 * q^79 - 6 * q^80 - 22 * q^81 - 12 * q^82 - 20 * q^83 + 12 * q^86 + 4 * q^87 + 20 * q^88 + 4 * q^89 + 2 * q^90 - 28 * q^92 + 8 * q^93 + 4 * q^94 - 4 * q^95 + 12 * q^96 + 8 * q^97 - 4 * q^99

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	$-0.825557\pi$
$2$	−2.41421	−1.70711	−0.853553	+	0.521005i	$0.825557\pi$
$3$	2.00000	1.15470	0.577350	−	0.816497i	$-0.304087\pi$
$3$	2.00000	1.15470	0.577350	+	0.816497i	$0.304087\pi$
$4$	3.82843	1.91421
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−4.82843	−1.97120
$7$	0	0
$7$	0	0
$8$	−4.41421	−1.56066
$9$	1.00000	0.333333
$10$	2.41421	0.763441
$11$	−4.82843	−1.45583	−0.727913	−	0.685670i	$-0.759509\pi$
$11$	−4.82843	−1.45583	−0.727913	+	0.685670i	$0.759509\pi$
$12$	7.65685	2.21034
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	0	0
$15$	−2.00000	−0.516398
$16$	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	$0.388558\pi$
$18$	−2.41421	−0.569036
$19$	−0.828427	−0.190054	−0.0950271	−	0.995475i	$-0.530294\pi$
$19$	−0.828427	−0.190054	−0.0950271	+	0.995475i	$0.530294\pi$
$20$	−3.82843	−0.856062
$21$	0	0
$22$	11.6569	2.48525
$23$	−8.82843	−1.84085	−0.920427	−	0.390914i	$-0.872159\pi$
$23$	−8.82843	−1.84085	−0.920427	+	0.390914i	$0.872159\pi$
$24$	−8.82843	−1.80210
$25$	1.00000	0.200000
$26$	−4.82843	−0.946932
$27$	−4.00000	−0.769800
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	4.82843	0.881546
$31$	10.4853	1.88321	0.941606	−	0.336717i	$-0.109316\pi$
$31$	10.4853	1.88321	0.941606	+	0.336717i	$0.109316\pi$
$32$	1.58579	0.280330
$33$	−9.65685	−1.68104
$34$	−6.82843	−1.17107
$35$	0	0
$36$	3.82843	0.638071
$37$	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
$39$	4.00000	0.640513
$40$	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
$45$	−1.00000	−0.149071
$46$	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	$0.492033\pi$
$48$	6.00000	0.866025
$49$	0	0
$50$	−2.41421	−0.341421
$51$	5.65685	0.792118
$52$	7.65685	1.06181
$53$	7.65685	1.05175	0.525875	−	0.850562i	$-0.323738\pi$
$53$	7.65685	1.05175	0.525875	+	0.850562i	$0.323738\pi$
$54$	9.65685	1.31413
$55$	4.82843	0.651065
$56$	0	0
$57$	−1.65685	−0.219456
$58$	−2.41421	−0.317002
$59$	0	0		−	1.00000i	$-0.5\pi$
$59$	0	0			1.00000i	$0.5\pi$
$60$	−7.65685	−0.988496
$61$	−7.65685	−0.980360	−0.490180	−	0.871621i	$-0.663069\pi$
$61$	−7.65685	−0.980360	−0.490180	+	0.871621i	$0.663069\pi$
$62$	−25.3137	−3.21484
$63$	0	0
$64$	−9.82843	−1.22855
$65$	−2.00000	−0.248069
$66$	23.3137	2.86972
$67$	−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
$69$	−17.6569	−2.12564
$70$	0	0
$71$	7.31371	0.867978	0.433989	−	0.900918i	$-0.357106\pi$
$71$	7.31371	0.867978	0.433989	+	0.900918i	$0.357106\pi$
$72$	−4.41421	−0.520220
$73$	8.48528	0.993127	0.496564	−	0.868000i	$-0.334595\pi$
$73$	8.48528	0.993127	0.496564	+	0.868000i	$0.334595\pi$
$74$	−20.4853	−2.38137
$75$	2.00000	0.230940
$76$	−3.17157	−0.363804
$77$	0	0
$78$	−9.65685	−1.09342
$79$	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
$81$	−11.0000	−1.22222
$82$	−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
$87$	2.00000	0.214423
$88$	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	$0.562085\pi$
$90$	2.41421	0.254480
$91$	0	0
$92$	−33.7990	−3.52379
$93$	20.9706	2.17455
$94$	−0.828427	−0.0854457
$95$	0.828427	0.0849948
$96$	3.17157	0.323697
$97$	−4.48528	−0.455411	−0.227706	−	0.973730i	$-0.573122\pi$
$97$	−4.48528	−0.455411	−0.227706	+	0.973730i	$0.573122\pi$
$98$	0	0
$99$	−4.82843	−0.485275
$100$	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	$0.569327\pi$
$102$	−13.6569	−1.35223
$103$	12.1421	1.19640	0.598200	−	0.801347i	$-0.295883\pi$
$103$	12.1421	1.19640	0.598200	+	0.801347i	$0.295883\pi$
$104$	−8.82843	−0.865699
$105$	0	0
$106$	−18.4853	−1.79545
$107$	−8.14214	−0.787130	−0.393565	−	0.919297i	$-0.628758\pi$
$107$	−8.14214	−0.787130	−0.393565	+	0.919297i	$0.628758\pi$
$108$	−15.3137	−1.47356
$109$	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
$111$	16.9706	1.61077
$112$	0	0
$113$	2.82843	0.266076	0.133038	−	0.991111i	$-0.457527\pi$
$113$	2.82843	0.266076	0.133038	+	0.991111i	$0.457527\pi$
$114$	4.00000	0.374634
$115$	8.82843	0.823255
$116$	3.82843	0.355461
$117$	2.00000	0.184900
$118$	0	0
$119$	0	0
$120$	8.82843	0.805921
$121$	12.3137	1.11943
$122$	18.4853	1.67358
$123$	12.0000	1.08200
$124$	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
$129$	−12.0000	−1.05654
$130$	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	$0.749130\pi$
$132$	−36.9706	−3.21787
$133$	0	0
$134$	25.3137	2.18677
$135$	4.00000	0.344265
$136$	−12.4853	−1.07060
$137$	−10.8284	−0.925135	−0.462567	−	0.886584i	$-0.653072\pi$
$137$	−10.8284	−0.925135	−0.462567	+	0.886584i	$0.653072\pi$
$138$	42.6274	3.62869
$139$	−10.3431	−0.877294	−0.438647	−	0.898659i	$-0.644542\pi$
$139$	−10.3431	−0.877294	−0.438647	+	0.898659i	$0.644542\pi$
$140$	0	0
$141$	0.686292	0.0577962
$142$	−17.6569	−1.48173
$143$	−9.65685	−0.807547
$144$	3.00000	0.250000
$145$	−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	$-0.683604\pi$
$149$	−13.3137	−1.09070	−0.545351	+	0.838208i	$0.683604\pi$
$150$	−4.82843	−0.394239
$151$	−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
$153$	2.82843	0.228665
$154$	0	0
$155$	−10.4853	−0.842198
$156$	15.3137	1.22608
$157$	16.4853	1.31567	0.657834	−	0.753163i	$-0.271473\pi$
$157$	16.4853	1.31567	0.657834	+	0.753163i	$0.271473\pi$
$158$	−34.9706	−2.78211
$159$	15.3137	1.21446
$160$	−1.58579	−0.125367
$161$	0	0
$162$	26.5563	2.08646
$163$	−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
$165$	9.65685	0.751785
$166$	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
$171$	−0.828427	−0.0633514
$172$	−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	$0.435257\pi$
$174$	−4.82843	−0.366042
$175$	0	0
$176$	−14.4853	−1.09187
$177$	0	0
$178$	8.82843	0.661719
$179$	−0.686292	−0.0512958	−0.0256479	−	0.999671i	$-0.508165\pi$
$179$	−0.686292	−0.0512958	−0.0256479	+	0.999671i	$0.508165\pi$
$180$	−3.82843	−0.285354
$181$	6.00000	0.445976	0.222988	−	0.974821i	$-0.428419\pi$
$181$	6.00000	0.445976	0.222988	+	0.974821i	$0.428419\pi$
$182$	0	0
$183$	−15.3137	−1.13202
$184$	38.9706	2.87295
$185$	−8.48528	−0.623850
$186$	−50.6274	−3.71218
$187$	−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	$-0.684949\pi$
$191$	−15.1716	−1.09778	−0.548888	+	0.835896i	$0.684949\pi$
$192$	−19.6569	−1.41861
$193$	−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
$195$	−4.00000	−0.286446
$196$	0	0
$197$	−8.34315	−0.594425	−0.297212	−	0.954811i	$-0.596057\pi$
$197$	−8.34315	−0.594425	−0.297212	+	0.954811i	$0.596057\pi$
$198$	11.6569	0.828417
$199$	−12.0000	−0.850657	−0.425329	−	0.905039i	$-0.639842\pi$
$199$	−12.0000	−0.850657	−0.425329	+	0.905039i	$0.639842\pi$
$200$	−4.41421	−0.312132
$201$	−20.9706	−1.47915
$202$	10.4853	0.737742
$203$	0	0
$204$	21.6569	1.51628
$205$	−6.00000	−0.419058
$206$	−29.3137	−2.04238
$207$	−8.82843	−0.613618
$208$	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
$213$	14.6274	1.00225
$214$	19.6569	1.34371
$215$	6.00000	0.409197
$216$	17.6569	1.20140
$217$	0	0
$218$	−4.82843	−0.327022
$219$	16.9706	1.14676
$220$	18.4853	1.24628
$221$	5.65685	0.380521
$222$	−40.9706	−2.74976
$223$	−21.7990	−1.45977	−0.729884	−	0.683571i	$-0.760426\pi$
$223$	−21.7990	−1.45977	−0.729884	+	0.683571i	$0.760426\pi$
$224$	0	0
$225$	1.00000	0.0666667
$226$	−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	$0.412908\pi$
$228$	−6.34315	−0.420085
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	−21.3137	−1.40538
$231$	0	0
$232$	−4.41421	−0.289807
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	−4.82843	−0.315644
$235$	−0.343146	−0.0223844
$236$	0	0
$237$	28.9706	1.88184
$238$	0	0
$239$	−23.3137	−1.50804	−0.754019	−	0.656852i	$-0.771887\pi$
$239$	−23.3137	−1.50804	−0.754019	+	0.656852i	$0.771887\pi$
$240$	−6.00000	−0.387298
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	−29.7279	−1.91098
$243$	−10.0000	−0.641500
$244$	−29.3137	−1.87662
$245$	0	0
$246$	−28.9706	−1.84710
$247$	−1.65685	−0.105423
$248$	−46.2843	−2.93905
$249$	−25.6569	−1.62594
$250$	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
$255$	−5.65685	−0.354246
$256$	−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	$0.867234\pi$
$258$	28.9706	1.80363
$259$	0	0
$260$	−7.65685	−0.474858
$261$	1.00000	0.0618984
$262$	38.9706	2.40761
$263$	−8.34315	−0.514460	−0.257230	−	0.966350i	$-0.582810\pi$
$263$	−8.34315	−0.514460	−0.257230	+	0.966350i	$0.582810\pi$
$264$	42.6274	2.62354
$265$	−7.65685	−0.470357
$266$	0	0
$267$	−7.31371	−0.447592
$268$	−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	$0.512751\pi$
$270$	−9.65685	−0.587697
$271$	−29.7990	−1.81016	−0.905080	−	0.425242i	$-0.860189\pi$
$271$	−29.7990	−1.81016	−0.905080	+	0.425242i	$0.860189\pi$
$272$	8.48528	0.514496
$273$	0	0
$274$	26.1421	1.57930
$275$	−4.82843	−0.291165
$276$	−67.5980	−4.06892
$277$	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
$279$	10.4853	0.627737
$280$	0	0
$281$	−6.68629	−0.398871	−0.199435	−	0.979911i	$-0.563911\pi$
$281$	−6.68629	−0.398871	−0.199435	+	0.979911i	$0.563911\pi$
$282$	−1.65685	−0.0986642
$283$	0.828427	0.0492449	0.0246224	−	0.999697i	$-0.492162\pi$
$283$	0.828427	0.0492449	0.0246224	+	0.999697i	$0.492162\pi$
$284$	28.0000	1.66149
$285$	1.65685	0.0981436
$286$	23.3137	1.37857
$287$	0	0
$288$	1.58579	0.0934434
$289$	−9.00000	−0.529412
$290$	2.41421	0.141768
$291$	−8.97056	−0.525864
$292$	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
$297$	19.3137	1.12070
$298$	32.1421	1.86194
$299$	−17.6569	−1.02112
$300$	7.65685	0.442069
$301$	0	0
$302$	28.9706	1.66707
$303$	−8.68629	−0.499014
$304$	−2.48528	−0.142541
$305$	7.65685	0.438430
$306$	−6.82843	−0.390355
$307$	−10.9706	−0.626123	−0.313062	−	0.949733i	$-0.601355\pi$
$307$	−10.9706	−0.626123	−0.313062	+	0.949733i	$0.601355\pi$
$308$	0	0
$309$	24.2843	1.38148
$310$	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	$0.477552\pi$
$312$	−17.6569	−0.999623
$313$	6.00000	0.339140	0.169570	−	0.985518i	$-0.445762\pi$
$313$	6.00000	0.339140	0.169570	+	0.985518i	$0.445762\pi$
$314$	−39.7990	−2.24599
$315$	0	0
$316$	55.4558	3.11963
$317$	−2.82843	−0.158860	−0.0794301	−	0.996840i	$-0.525310\pi$
$317$	−2.82843	−0.158860	−0.0794301	+	0.996840i	$0.525310\pi$
$318$	−36.9706	−2.07321
$319$	−4.82843	−0.270340
$320$	9.82843	0.549426
$321$	−16.2843	−0.908899
$322$	0	0
$323$	−2.34315	−0.130376
$324$	−42.1127	−2.33959
$325$	2.00000	0.110940
$326$	47.4558	2.62834
$327$	4.00000	0.221201
$328$	−26.4853	−1.46241
$329$	0	0
$330$	−23.3137	−1.28338
$331$	−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
$333$	8.48528	0.464991
$334$	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
$339$	5.65685	0.307238
$340$	−10.8284	−0.587254
$341$	−50.6274	−2.74163
$342$	2.00000	0.108148
$343$	0	0
$344$	26.4853	1.42799
$345$	17.6569	0.950613
$346$	−12.8284	−0.689661
$347$	20.1421	1.08129	0.540643	−	0.841252i	$-0.318181\pi$
$347$	20.1421	1.08129	0.540643	+	0.841252i	$0.318181\pi$
$348$	7.65685	0.410450
$349$	24.6274	1.31828	0.659138	−	0.752022i	$-0.270921\pi$
$349$	24.6274	1.31828	0.659138	+	0.752022i	$0.270921\pi$
$350$	0	0
$351$	−8.00000	−0.427008
$352$	−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	$-0.822313\pi$
$359$	−32.1421	−1.69640	−0.848199	+	0.529678i	$0.822313\pi$
$360$	4.41421	0.232649
$361$	−18.3137	−0.963879
$362$	−14.4853	−0.761329
$363$	24.6274	1.29260
$364$	0	0
$365$	−8.48528	−0.444140
$366$	36.9706	1.93248
$367$	−18.0000	−0.939592	−0.469796	−	0.882775i	$-0.655673\pi$
$367$	−18.0000	−0.939592	−0.469796	+	0.882775i	$0.655673\pi$
$368$	−26.4853	−1.38064
$369$	6.00000	0.312348
$370$	20.4853	1.06498
$371$	0	0
$372$	80.2843	4.16255
$373$	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
$375$	−2.00000	−0.103280
$376$	−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
$381$	12.0000	0.614779
$382$	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	$0.379330\pi$
$384$	41.1127	2.09802
$385$	0	0
$386$	30.1421	1.53419
$387$	−6.00000	−0.304997
$388$	−17.1716	−0.871755
$389$	−6.68629	−0.339008	−0.169504	−	0.985529i	$-0.554217\pi$
$389$	−6.68629	−0.339008	−0.169504	+	0.985529i	$0.554217\pi$
$390$	9.65685	0.488994
$391$	−24.9706	−1.26282
$392$	0	0
$393$	−32.2843	−1.62853
$394$	20.1421	1.01475
$395$	−14.4853	−0.728834
$396$	−18.4853	−0.928920
$397$	8.34315	0.418730	0.209365	−	0.977838i	$-0.432860\pi$
$397$	8.34315	0.418730	0.209365	+	0.977838i	$0.432860\pi$
$398$	28.9706	1.45216
$399$	0	0
$400$	3.00000	0.150000
$401$	−29.3137	−1.46386	−0.731928	−	0.681382i	$-0.761379\pi$
$401$	−29.3137	−1.46386	−0.731928	+	0.681382i	$0.761379\pi$
$402$	50.6274	2.52507
$403$	20.9706	1.04462
$404$	−16.6274	−0.827245
$405$	11.0000	0.546594
$406$	0	0
$407$	−40.9706	−2.03084
$408$	−24.9706	−1.23623
$409$	−30.9706	−1.53140	−0.765698	−	0.643200i	$-0.777606\pi$
$409$	−30.9706	−1.53140	−0.765698	+	0.643200i	$0.777606\pi$
$410$	14.4853	0.715377
$411$	−21.6569	−1.06825
$412$	46.4853	2.29017
$413$	0	0
$414$	21.3137	1.04751
$415$	12.8284	0.629723
$416$	3.17157	0.155499
$417$	−20.6863	−1.01301
$418$	−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
$423$	0.343146	0.0166843
$424$	−33.7990	−1.64142
$425$	2.82843	0.137199
$426$	−35.3137	−1.71095
$427$	0	0
$428$	−31.1716	−1.50673
$429$	−19.3137	−0.932475
$430$	−14.4853	−0.698542
$431$	−19.3137	−0.930309	−0.465154	−	0.885230i	$-0.654001\pi$
$431$	−19.3137	−0.930309	−0.465154	+	0.885230i	$0.654001\pi$
$432$	−12.0000	−0.577350
$433$	34.8284	1.67375	0.836874	−	0.547396i	$-0.184381\pi$
$433$	34.8284	1.67375	0.836874	+	0.547396i	$0.184381\pi$
$434$	0	0
$435$	−2.00000	−0.0958927
$436$	7.65685	0.366697
$437$	7.31371	0.349862
$438$	−40.9706	−1.95765
$439$	21.6569	1.03363	0.516813	−	0.856099i	$-0.327118\pi$
$439$	21.6569	1.03363	0.516813	+	0.856099i	$0.327118\pi$
$440$	−21.3137	−1.01609
$441$	0	0
$442$	−13.6569	−0.649590
$443$	3.65685	0.173742	0.0868712	−	0.996220i	$-0.472313\pi$
$443$	3.65685	0.173742	0.0868712	+	0.996220i	$0.472313\pi$
$444$	64.9706	3.08337
$445$	3.65685	0.173352
$446$	52.6274	2.49198
$447$	−26.6274	−1.25943
$448$	0	0
$449$	0.343146	0.0161940	0.00809702	−	0.999967i	$-0.497423\pi$
$449$	0.343146	0.0161940	0.00809702	+	0.999967i	$0.497423\pi$
$450$	−2.41421	−0.113807
$451$	−28.9706	−1.36417
$452$	10.8284	0.509326
$453$	−24.0000	−1.12762
$454$	−19.6569	−0.922542
$455$	0	0
$456$	7.31371	0.342496
$457$	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
$459$	−11.3137	−0.528079
$460$	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
$465$	−20.9706	−0.972487
$466$	−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	$0.321638\pi$
$468$	7.65685	0.353938
$469$	0	0
$470$	0.828427	0.0382125
$471$	32.9706	1.51920
$472$	0	0
$473$	28.9706	1.33207
$474$	−69.9411	−3.21250
$475$	−0.828427	−0.0380108
$476$	0	0
$477$	7.65685	0.350583
$478$	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	$0.594678\pi$
$480$	−3.17157	−0.144762
$481$	16.9706	0.773791
$482$	24.1421	1.09964
$483$	0	0
$484$	47.1421	2.14282
$485$	4.48528	0.203666
$486$	24.1421	1.09511
$487$	−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
$489$	−39.3137	−1.77783
$490$	0	0
$491$	43.4558	1.96113	0.980567	−	0.196183i	$-0.0628545\pi$
$491$	43.4558	1.96113	0.980567	+	0.196183i	$0.0628545\pi$
$492$	45.9411	2.07119
$493$	2.82843	0.127386
$494$	4.00000	0.179969
$495$	4.82843	0.217022
$496$	31.4558	1.41241
$497$	0	0
$498$	61.9411	2.77565
$499$	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
$501$	−28.9706	−1.29431
$502$	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
$507$	−18.0000	−0.799408
$508$	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	$0.0471751\pi$
$510$	13.6569	0.604736
$511$	0	0
$512$	31.2426	1.38074
$513$	3.31371	0.146304
$514$	70.7696	3.12151
$515$	−12.1421	−0.535046
$516$	−45.9411	−2.02245
$517$	−1.65685	−0.0728684
$518$	0	0
$519$	10.6274	0.466492
$520$	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	$0.509161\pi$
$522$	−2.41421	−0.105667
$523$	14.4853	0.633397	0.316699	−	0.948526i	$-0.397426\pi$
$523$	14.4853	0.633397	0.316699	+	0.948526i	$0.397426\pi$
$524$	−61.7990	−2.69970
$525$	0	0
$526$	20.1421	0.878239
$527$	29.6569	1.29187
$528$	−28.9706	−1.26078
$529$	54.9411	2.38874
$530$	18.4853	0.802949
$531$	0	0
$532$	0	0
$533$	12.0000	0.519778
$534$	17.6569	0.764087
$535$	8.14214	0.352015
$536$	46.2843	1.99918
$537$	−1.37258	−0.0592313
$538$	3.17157	0.136736
$539$	0	0
$540$	15.3137	0.658997
$541$	−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
$543$	12.0000	0.514969
$544$	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
$549$	−7.65685	−0.326787
$550$	11.6569	0.497050
$551$	−0.828427	−0.0352922
$552$	77.9411	3.31739
$553$	0	0
$554$	−18.4853	−0.785364
$555$	−16.9706	−0.720360
$556$	−39.5980	−1.67933
$557$	39.9411	1.69236	0.846180	−	0.532897i	$-0.178897\pi$
$557$	39.9411	1.69236	0.846180	+	0.532897i	$0.178897\pi$
$558$	−25.3137	−1.07161
$559$	−12.0000	−0.507546
$560$	0	0
$561$	−27.3137	−1.15319
$562$	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	$0.475447\pi$
$564$	2.62742	0.110634
$565$	−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	$-0.388703\pi$
$569$	16.3431	0.685140	0.342570	+	0.939492i	$0.388703\pi$
$570$	−4.00000	−0.167542
$571$	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
$573$	−30.3431	−1.26760
$574$	0	0
$575$	−8.82843	−0.368171
$576$	−9.82843	−0.409518
$577$	15.7990	0.657721	0.328860	−	0.944379i	$-0.393335\pi$
$577$	15.7990	0.657721	0.328860	+	0.944379i	$0.393335\pi$
$578$	21.7279	0.903762
$579$	−24.9706	−1.03774
$580$	−3.82843	−0.158967
$581$	0	0
$582$	21.6569	0.897705
$583$	−36.9706	−1.53116
$584$	−37.4558	−1.54993
$585$	−2.00000	−0.0826898
$586$	−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
$591$	−16.6863	−0.686382
$592$	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	$0.523923\pi$
$594$	−46.6274	−1.91315
$595$	0	0
$596$	−50.9706	−2.08784
$597$	−24.0000	−0.982255
$598$	42.6274	1.74316
$599$	1.79899	0.0735047	0.0367524	−	0.999324i	$-0.488299\pi$
$599$	1.79899	0.0735047	0.0367524	+	0.999324i	$0.488299\pi$
$600$	−8.82843	−0.360419
$601$	−2.00000	−0.0815817	−0.0407909	−	0.999168i	$-0.512988\pi$
$601$	−2.00000	−0.0815817	−0.0407909	+	0.999168i	$0.512988\pi$
$602$	0	0
$603$	−10.4853	−0.426994
$604$	−45.9411	−1.86932
$605$	−12.3137	−0.500623
$606$	20.9706	0.851871
$607$	−42.9706	−1.74412	−0.872061	−	0.489398i	$-0.837217\pi$
$607$	−42.9706	−1.74412	−0.872061	+	0.489398i	$0.837217\pi$
$608$	−1.31371	−0.0532779
$609$	0	0
$610$	−18.4853	−0.748447
$611$	0.686292	0.0277644
$612$	10.8284	0.437713
$613$	−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
$615$	−12.0000	−0.483887
$616$	0	0
$617$	−14.8284	−0.596970	−0.298485	−	0.954414i	$-0.596481\pi$
$617$	−14.8284	−0.596970	−0.298485	+	0.954414i	$0.596481\pi$
$618$	−58.6274	−2.35834
$619$	−29.7990	−1.19772	−0.598861	−	0.800853i	$-0.704380\pi$
$619$	−29.7990	−1.19772	−0.598861	+	0.800853i	$0.704380\pi$
$620$	−40.1421	−1.61215
$621$	35.3137	1.41709
$622$	−6.00000	−0.240578
$623$	0	0
$624$	12.0000	0.480384
$625$	1.00000	0.0400000
$626$	−14.4853	−0.578948
$627$	8.00000	0.319489
$628$	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
$633$	9.65685	0.383825
$634$	6.82843	0.271191
$635$	−6.00000	−0.238103
$636$	58.6274	2.32473
$637$	0	0
$638$	11.6569	0.461499
$639$	7.31371	0.289326
$640$	−20.5563	−0.812561
$641$	−44.6274	−1.76268	−0.881338	−	0.472485i	$-0.843357\pi$
$641$	−44.6274	−1.76268	−0.881338	+	0.472485i	$0.843357\pi$
$642$	39.3137	1.55159
$643$	31.4558	1.24050	0.620249	−	0.784405i	$-0.287032\pi$
$643$	31.4558	1.24050	0.620249	+	0.784405i	$0.287032\pi$
$644$	0	0
$645$	12.0000	0.472500
$646$	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	$0.636223\pi$
$648$	48.5563	1.90747
$649$	0	0
$650$	−4.82843	−0.189386
$651$	0	0
$652$	−75.2548	−2.94721
$653$	−22.8284	−0.893345	−0.446673	−	0.894697i	$-0.647391\pi$
$653$	−22.8284	−0.893345	−0.446673	+	0.894697i	$0.647391\pi$
$654$	−9.65685	−0.377613
$655$	16.1421	0.630725
$656$	18.0000	0.702782
$657$	8.48528	0.331042
$658$	0	0
$659$	−37.7990	−1.47244	−0.736220	−	0.676743i	$-0.763391\pi$
$659$	−37.7990	−1.47244	−0.736220	+	0.676743i	$0.763391\pi$
$660$	36.9706	1.43908
$661$	−26.0000	−1.01128	−0.505641	−	0.862744i	$-0.668744\pi$
$661$	−26.0000	−1.01128	−0.505641	+	0.862744i	$0.668744\pi$
$662$	42.9706	1.67010
$663$	11.3137	0.439388
$664$	56.6274	2.19757
$665$	0	0
$666$	−20.4853	−0.793789
$667$	−8.82843	−0.341838
$668$	−55.4558	−2.14565
$669$	−43.5980	−1.68560
$670$	−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
$675$	−4.00000	−0.153960
$676$	−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	$0.749765\pi$
$678$	−13.6569	−0.524488
$679$	0	0
$680$	12.4853	0.478789
$681$	16.2843	0.624015
$682$	122.225	4.68025
$683$	40.1421	1.53600	0.767998	−	0.640452i	$-0.221253\pi$
$683$	40.1421	1.53600	0.767998	+	0.640452i	$0.221253\pi$
$684$	−3.17157	−0.121268
$685$	10.8284	0.413733
$686$	0	0
$687$	4.00000	0.152610
$688$	−18.0000	−0.686244
$689$	15.3137	0.583406
$690$	−42.6274	−1.62280
$691$	11.0294	0.419580	0.209790	−	0.977747i	$-0.432722\pi$
$691$	11.0294	0.419580	0.209790	+	0.977747i	$0.432722\pi$
$692$	20.3431	0.773330
$693$	0	0
$694$	−48.6274	−1.84587
$695$	10.3431	0.392338
$696$	−8.82843	−0.334641
$697$	16.9706	0.642806
$698$	−59.4558	−2.25044
$699$	36.0000	1.36165
$700$	0	0
$701$	29.3137	1.10716	0.553582	−	0.832795i	$-0.313261\pi$
$701$	29.3137	1.10716	0.553582	+	0.832795i	$0.313261\pi$
$702$	19.3137	0.728949
$703$	−7.02944	−0.265120
$704$	47.4558	1.78856
$705$	−0.686292	−0.0258472
$706$	−37.7990	−1.42258
$707$	0	0
$708$	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
$711$	14.4853	0.543240
$712$	16.1421	0.604952
$713$	−92.5685	−3.46672
$714$	0	0
$715$	9.65685	0.361146
$716$	−2.62742	−0.0981912
$717$	−46.6274	−1.74133
$718$	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	$0.563499\pi$
$720$	−3.00000	−0.111803
$721$	0	0
$722$	44.2132	1.64545
$723$	−20.0000	−0.743808
$724$	22.9706	0.853694
$725$	1.00000	0.0371391
$726$	−59.4558	−2.20661
$727$	−43.9411	−1.62969	−0.814843	−	0.579682i	$-0.803177\pi$
$727$	−43.9411	−1.62969	−0.814843	+	0.579682i	$0.803177\pi$
$728$	0	0
$729$	13.0000	0.481481
$730$	20.4853	0.758194
$731$	−16.9706	−0.627679
$732$	−58.6274	−2.16693
$733$	17.1716	0.634247	0.317123	−	0.948384i	$-0.397283\pi$
$733$	17.1716	0.634247	0.317123	+	0.948384i	$0.397283\pi$
$734$	43.4558	1.60398
$735$	0	0
$736$	−14.0000	−0.516047
$737$	50.6274	1.86488
$738$	−14.4853	−0.533211
$739$	−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
$741$	−3.31371	−0.121732
$742$	0	0
$743$	−7.37258	−0.270474	−0.135237	−	0.990813i	$-0.543180\pi$
$743$	−7.37258	−0.270474	−0.135237	+	0.990813i	$0.543180\pi$
$744$	−92.5685	−3.39373
$745$	13.3137	0.487777
$746$	−65.1127	−2.38395
$747$	−12.8284	−0.469368
$748$	−52.2843	−1.91170
$749$	0	0
$750$	4.82843	0.176309
$751$	−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
$753$	−6.34315	−0.231157
$754$	−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
$759$	85.2548	3.09455
$760$	−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	$0.268912\pi$
$762$	−28.9706	−1.04949
$763$	0	0
$764$	−58.0833	−2.10138
$765$	−2.82843	−0.102262
$766$	−34.9706	−1.26354
$767$	0	0
$768$	−59.9411	−2.16294
$769$	4.34315	0.156618	0.0783089	−	0.996929i	$-0.475048\pi$
$769$	4.34315	0.156618	0.0783089	+	0.996929i	$0.475048\pi$
$770$	0	0
$771$	−58.6274	−2.11141
$772$	−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	$0.548764\pi$
$774$	14.4853	0.520663
$775$	10.4853	0.376642
$776$	19.7990	0.710742
$777$	0	0
$778$	16.1421	0.578724
$779$	−4.97056	−0.178089
$780$	−15.3137	−0.548319
$781$	−35.3137	−1.26362
$782$	60.2843	2.15576
$783$	−4.00000	−0.142948
$784$	0	0
$785$	−16.4853	−0.588385
$786$	77.9411	2.78007
$787$	21.7990	0.777050	0.388525	−	0.921438i	$-0.372985\pi$
$787$	21.7990	0.777050	0.388525	+	0.921438i	$0.372985\pi$
$788$	−31.9411	−1.13786
$789$	−16.6863	−0.594048
$790$	34.9706	1.24420
$791$	0	0
$792$	21.3137	0.757350
$793$	−15.3137	−0.543806
$794$	−20.1421	−0.714818
$795$	−15.3137	−0.543121
$796$	−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
$801$	−3.65685	−0.129209
$802$	70.7696	2.49896
$803$	−40.9706	−1.44582
$804$	−80.2843	−2.83141
$805$	0	0
$806$	−50.6274	−1.78327
$807$	−2.62742	−0.0924895
$808$	19.1716	0.674454
$809$	−14.2843	−0.502208	−0.251104	−	0.967960i	$-0.580794\pi$
$809$	−14.2843	−0.502208	−0.251104	+	0.967960i	$0.580794\pi$
$810$	−26.5563	−0.933095
$811$	−26.3431	−0.925033	−0.462516	−	0.886611i	$-0.653053\pi$
$811$	−26.3431	−0.925033	−0.462516	+	0.886611i	$0.653053\pi$
$812$	0	0
$813$	−59.5980	−2.09019
$814$	98.9117	3.46685
$815$	19.6569	0.688550
$816$	16.9706	0.594089
$817$	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	$-0.790299\pi$
$821$	−45.3137	−1.58146	−0.790730	+	0.612165i	$0.790299\pi$
$822$	52.2843	1.82362
$823$	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
$825$	−9.65685	−0.336209
$826$	0	0
$827$	−5.31371	−0.184776	−0.0923879	−	0.995723i	$-0.529450\pi$
$827$	−5.31371	−0.184776	−0.0923879	+	0.995723i	$0.529450\pi$
$828$	−33.7990	−1.17460
$829$	24.6274	0.855346	0.427673	−	0.903934i	$-0.359334\pi$
$829$	24.6274	0.855346	0.427673	+	0.903934i	$0.359334\pi$
$830$	−30.9706	−1.07500
$831$	15.3137	0.531227
$832$	−19.6569	−0.681479
$833$	0	0
$834$	49.9411	1.72932
$835$	14.4853	0.501284
$836$	15.3137	0.529636
$837$	−41.9411	−1.44970
$838$	−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
$843$	−13.3726	−0.460576
$844$	18.4853	0.636290
$845$	9.00000	0.309609
$846$	−0.828427	−0.0284819
$847$	0	0
$848$	22.9706	0.788812
$849$	1.65685	0.0568631
$850$	−6.82843	−0.234213
$851$	−74.9117	−2.56794
$852$	56.0000	1.91853
$853$	11.1127	0.380492	0.190246	−	0.981736i	$-0.439072\pi$
$853$	11.1127	0.380492	0.190246	+	0.981736i	$0.439072\pi$
$854$	0	0
$855$	0.828427	0.0283316
$856$	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	$0.811968\pi$
$858$	46.6274	1.59183
$859$	−28.4264	−0.969896	−0.484948	−	0.874543i	$-0.661162\pi$
$859$	−28.4264	−0.969896	−0.484948	+	0.874543i	$0.661162\pi$
$860$	22.9706	0.783290
$861$	0	0
$862$	46.6274	1.58814
$863$	−7.85786	−0.267485	−0.133742	−	0.991016i	$-0.542699\pi$
$863$	−7.85786	−0.267485	−0.133742	+	0.991016i	$0.542699\pi$
$864$	−6.34315	−0.215798
$865$	−5.31371	−0.180672
$866$	−84.0833	−2.85727
$867$	−18.0000	−0.611312
$868$	0	0
$869$	−69.9411	−2.37259
$870$	4.82843	0.163699
$871$	−20.9706	−0.710560
$872$	−8.82843	−0.298968
$873$	−4.48528	−0.151804
$874$	−17.6569	−0.597252
$875$	0	0
$876$	64.9706	2.19515
$877$	−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
$879$	16.9706	0.572403
$880$	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	$0.336219\pi$
$888$	−74.9117	−2.51387
$889$	0	0
$890$	−8.82843	−0.295930
$891$	53.1127	1.77934
$892$	−83.4558	−2.79431
$893$	−0.284271	−0.00951277
$894$	64.2843	2.14999
$895$	0.686292	0.0229402
$896$	0	0
$897$	−35.3137	−1.17909
$898$	−0.828427	−0.0276450
$899$	10.4853	0.349704
$900$	3.82843	0.127614
$901$	21.6569	0.721494
$902$	69.9411	2.32878
$903$	0	0
$904$	−12.4853	−0.415254
$905$	−6.00000	−0.199447
$906$	57.9411	1.92496
$907$	−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
$909$	−4.34315	−0.144053
$910$	0	0
$911$	3.85786	0.127817	0.0639084	−	0.997956i	$-0.479643\pi$
$911$	3.85786	0.127817	0.0639084	+	0.997956i	$0.479643\pi$
$912$	−4.97056	−0.164592
$913$	61.9411	2.04995
$914$	−20.1421	−0.666243
$915$	15.3137	0.506256
$916$	7.65685	0.252990
$917$	0	0
$918$	27.3137	0.901487
$919$	−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
$921$	−21.9411	−0.722985
$922$	−58.7696	−1.93547
$923$	14.6274	0.481467
$924$	0	0
$925$	8.48528	0.278994
$926$	42.9706	1.41210
$927$	12.1421	0.398800
$928$	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	$0.732196\pi$
$930$	50.6274	1.66014
$931$	0	0
$932$	68.9117	2.25728
$933$	4.97056	0.162729
$934$	−55.4558	−1.81457
$935$	13.6569	0.446627
$936$	−8.82843	−0.288566
$937$	−8.34315	−0.272559	−0.136279	−	0.990670i	$-0.543514\pi$
$937$	−8.34315	−0.272559	−0.136279	+	0.990670i	$0.543514\pi$
$938$	0	0
$939$	12.0000	0.391605
$940$	−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	$0.725659\pi$
$942$	−79.5980	−2.59344
$943$	−52.9706	−1.72496
$944$	0	0
$945$	0	0
$946$	−69.9411	−2.27398
$947$	−56.9117	−1.84938	−0.924691	−	0.380719i	$-0.875676\pi$
$947$	−56.9117	−1.84938	−0.924691	+	0.380719i	$0.875676\pi$
$948$	110.912	3.60224
$949$	16.9706	0.550888
$950$	2.00000	0.0648886
$951$	−5.65685	−0.183436
$952$	0	0
$953$	6.68629	0.216590	0.108295	−	0.994119i	$-0.465461\pi$
$953$	6.68629	0.216590	0.108295	+	0.994119i	$0.465461\pi$
$954$	−18.4853	−0.598483
$955$	15.1716	0.490941
$956$	−89.2548	−2.88671
$957$	−9.65685	−0.312162
$958$	30.9706	1.00061
$959$	0	0
$960$	19.6569	0.634422
$961$	78.9411	2.54649
$962$	−40.9706	−1.32094
$963$	−8.14214	−0.262377
$964$	−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
$969$	−4.68629	−0.150545
$970$	−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	$0.499274\pi$
$972$	−38.2843	−1.22797
$973$	0	0
$974$	71.9411	2.30514
$975$	4.00000	0.128103
$976$	−22.9706	−0.735270
$977$	−25.3137	−0.809857	−0.404929	−	0.914348i	$-0.632704\pi$
$977$	−25.3137	−0.809857	−0.404929	+	0.914348i	$0.632704\pi$
$978$	94.9117	3.03494
$979$	17.6569	0.564316
$980$	0	0
$981$	2.00000	0.0638551
$982$	−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	$0.568102\pi$
$984$	−52.9706	−1.68864
$985$	8.34315	0.265835
$986$	−6.82843	−0.217461
$987$	0	0
$988$	−6.34315	−0.201802
$989$	52.9706	1.68437
$990$	−11.6569	−0.370479
$991$	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
$993$	−35.5980	−1.12967
$994$	0	0
$995$	12.0000	0.380426
$996$	−98.2254	−3.11239
$997$	−1.17157	−0.0371041	−0.0185520	−	0.999828i	$-0.505906\pi$
$997$	−1.17157	−0.0371041	−0.0185520	+	0.999828i	$0.505906\pi$
$998$	−86.9117	−2.75114
$999$	−33.9411	−1.07385

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7105.2.a.e.1.1		2
7.6	odd	2		145.2.a.b.1.1	✓	2
21.20	even	2		1305.2.a.n.1.2		2
28.27	even	2		2320.2.a.k.1.1		2
35.13	even	4		725.2.b.c.349.4		4
35.27	even	4		725.2.b.c.349.1		4
35.34	odd	2		725.2.a.c.1.2		2
56.13	odd	2		9280.2.a.be.1.2		2
56.27	even	2		9280.2.a.w.1.1		2
105.104	even	2		6525.2.a.p.1.1		2
203.202	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	7.6	odd	2
725.2.a.c.1.2		2	35.34	odd	2
725.2.b.c.349.1		4	35.27	even	4
725.2.b.c.349.4		4	35.13	even	4
1305.2.a.n.1.2		2	21.20	even	2
2320.2.a.k.1.1		2	28.27	even	2
4205.2.a.d.1.2		2	203.202	odd	2
6525.2.a.p.1.1		2	105.104	even	2
7105.2.a.e.1.1		2	1.1	even	1	trivial
9280.2.a.w.1.1		2	56.27	even	2
9280.2.a.be.1.2		2	56.13	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 7105 = 5 \cdot 7^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7105.a (trivial)

\(f(q)\)	\(=\)	\(q-2.41421 q^{2} +2.00000 q^{3} +3.82843 q^{4} -1.00000 q^{5} -4.82843 q^{6} -4.41421 q^{8} +1.00000 q^{9} +2.41421 q^{10} -4.82843 q^{11} +7.65685 q^{12} +2.00000 q^{13} -2.00000 q^{15} +3.00000 q^{16} +2.82843 q^{17} -2.41421 q^{18} -0.828427 q^{19} -3.82843 q^{20} +11.6569 q^{22} -8.82843 q^{23} -8.82843 q^{24} +1.00000 q^{25} -4.82843 q^{26} -4.00000 q^{27} +1.00000 q^{29} +4.82843 q^{30} +10.4853 q^{31} +1.58579 q^{32} -9.65685 q^{33} -6.82843 q^{34} +3.82843 q^{36} +8.48528 q^{37} +2.00000 q^{38} +4.00000 q^{39} +4.41421 q^{40} +6.00000 q^{41} -6.00000 q^{43} -18.4853 q^{44} -1.00000 q^{45} +21.3137 q^{46} +0.343146 q^{47} +6.00000 q^{48} -2.41421 q^{50} +5.65685 q^{51} +7.65685 q^{52} +7.65685 q^{53} +9.65685 q^{54} +4.82843 q^{55} -1.65685 q^{57} -2.41421 q^{58} -7.65685 q^{60} -7.65685 q^{61} -25.3137 q^{62} -9.82843 q^{64} -2.00000 q^{65} +23.3137 q^{66} -10.4853 q^{67} +10.8284 q^{68} -17.6569 q^{69} +7.31371 q^{71} -4.41421 q^{72} +8.48528 q^{73} -20.4853 q^{74} +2.00000 q^{75} -3.17157 q^{76} -9.65685 q^{78} +14.4853 q^{79} -3.00000 q^{80} -11.0000 q^{81} -14.4853 q^{82} -12.8284 q^{83} -2.82843 q^{85} +14.4853 q^{86} +2.00000 q^{87} +21.3137 q^{88} -3.65685 q^{89} +2.41421 q^{90} -33.7990 q^{92} +20.9706 q^{93} -0.828427 q^{94} +0.828427 q^{95} +3.17157 q^{96} -4.48528 q^{97} -4.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 4 q^{3} + 2 q^{4} - 2 q^{5} - 4 q^{6} - 6 q^{8} + 2 q^{9} + 2 q^{10} - 4 q^{11} + 4 q^{12} + 4 q^{13} - 4 q^{15} + 6 q^{16} - 2 q^{18} + 4 q^{19} - 2 q^{20} + 12 q^{22} - 12 q^{23} - 12 q^{24}+ \cdots - 4 q^{99}+O(q^{100}) \) 2 * q - 2 * q^2 + 4 * q^3 + 2 * q^4 - 2 * q^5 - 4 * q^6 - 6 * q^8 + 2 * q^9 + 2 * q^10 - 4 * q^11 + 4 * q^12 + 4 * q^13 - 4 * q^15 + 6 * q^16 - 2 * q^18 + 4 * q^19 - 2 * q^20 + 12 * q^22 - 12 * q^23 - 12 * q^24 + 2 * q^25 - 4 * q^26 - 8 * q^27 + 2 * q^29 + 4 * q^30 + 4 * q^31 + 6 * q^32 - 8 * q^33 - 8 * q^34 + 2 * q^36 + 4 * q^38 + 8 * q^39 + 6 * q^40 + 12 * q^41 - 12 * q^43 - 20 * q^44 - 2 * q^45 + 20 * q^46 + 12 * q^47 + 12 * q^48 - 2 * q^50 + 4 * q^52 + 4 * q^53 + 8 * q^54 + 4 * q^55 + 8 * q^57 - 2 * q^58 - 4 * q^60 - 4 * q^61 - 28 * q^62 - 14 * q^64 - 4 * q^65 + 24 * q^66 - 4 * q^67 + 16 * q^68 - 24 * q^69 - 8 * q^71 - 6 * q^72 - 24 * q^74 + 4 * q^75 - 12 * q^76 - 8 * q^78 + 12 * q^79 - 6 * q^80 - 22 * q^81 - 12 * q^82 - 20 * q^83 + 12 * q^86 + 4 * q^87 + 20 * q^88 + 4 * q^89 + 2 * q^90 - 28 * q^92 + 8 * q^93 + 4 * q^94 - 4 * q^95 + 12 * q^96 + 8 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

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Database

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