LMFDB - Embedded newform 7105.2.a.o.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [7105,2,Mod(1,7105)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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

Newform invariants

comment:select newform

sage:traces = [3,1,-2,1,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$56.7337106361$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.148.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 3x + 1 $ x^3 - x^2 - 3*x + 1
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			$0.311108$ 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)

$f(q)$	$=$	\(q+0.311108 q^{2} -2.90321 q^{3} -1.90321 q^{4} -1.00000 q^{5} -0.903212 q^{6} -1.21432 q^{8} +5.42864 q^{9} -0.311108 q^{10} -1.52543 q^{11} +5.52543 q^{12} +0.622216 q^{13} +2.90321 q^{15} +3.42864 q^{16} +7.95407 q^{17} +1.68889 q^{18} +1.09679 q^{19} +1.90321 q^{20} -0.474572 q^{22} +7.52543 q^{23} +3.52543 q^{24} +1.00000 q^{25} +0.193576 q^{26} -7.05086 q^{27} -1.00000 q^{29} +0.903212 q^{30} +6.90321 q^{31} +3.49532 q^{32} +4.42864 q^{33} +2.47457 q^{34} -10.3319 q^{36} +3.95407 q^{37} +0.341219 q^{38} -1.80642 q^{39} +1.21432 q^{40} -3.67307 q^{41} -10.5161 q^{43} +2.90321 q^{44} -5.42864 q^{45} +2.34122 q^{46} -6.90321 q^{47} -9.95407 q^{48} +0.311108 q^{50} -23.0923 q^{51} -1.18421 q^{52} +6.42864 q^{53} -2.19358 q^{54} +1.52543 q^{55} -3.18421 q^{57} -0.311108 q^{58} +1.67307 q^{59} -5.52543 q^{60} +1.86665 q^{61} +2.14764 q^{62} -5.76986 q^{64} -0.622216 q^{65} +1.37778 q^{66} +11.5254 q^{67} -15.1383 q^{68} -21.8479 q^{69} +13.6731 q^{71} -6.59210 q^{72} -10.1891 q^{73} +1.23014 q^{74} -2.90321 q^{75} -2.08742 q^{76} -0.561993 q^{78} +9.13828 q^{79} -3.42864 q^{80} +4.18421 q^{81} -1.14272 q^{82} -10.7096 q^{83} -7.95407 q^{85} -3.27163 q^{86} +2.90321 q^{87} +1.85236 q^{88} +7.80642 q^{89} -1.68889 q^{90} -14.3225 q^{92} -20.0415 q^{93} -2.14764 q^{94} -1.09679 q^{95} -10.1476 q^{96} +4.08742 q^{97} -8.28100 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} - 2 q^{3} + q^{4} - 3 q^{5} + 4 q^{6} + 3 q^{8} + 3 q^{9} - q^{10} + 2 q^{11} + 10 q^{12} + 2 q^{13} + 2 q^{15} - 3 q^{16} + 4 q^{17} + 5 q^{18} + 10 q^{19} - q^{20} - 8 q^{22} + 16 q^{23}+ \cdots - 18 q^{99}+O(q^{100}) $ 3 * q + q^2 - 2 * q^3 + q^4 - 3 * q^5 + 4 * q^6 + 3 * q^8 + 3 * q^9 - q^10 + 2 * q^11 + 10 * q^12 + 2 * q^13 + 2 * q^15 - 3 * q^16 + 4 * q^17 + 5 * q^18 + 10 * q^19 - q^20 - 8 * q^22 + 16 * q^23 + 4 * q^24 + 3 * q^25 + 14 * q^26 - 8 * q^27 - 3 * q^29 - 4 * q^30 + 14 * q^31 - 3 * q^32 + 14 * q^34 - 11 * q^36 - 8 * q^37 + 8 * q^38 + 8 * q^39 - 3 * q^40 + 2 * q^41 + 2 * q^43 + 2 * q^44 - 3 * q^45 + 14 * q^46 - 14 * q^47 - 10 * q^48 + q^50 - 16 * q^51 + 10 * q^52 + 6 * q^53 - 20 * q^54 - 2 * q^55 + 4 * q^57 - q^58 - 8 * q^59 - 10 * q^60 + 6 * q^61 - 11 * q^64 - 2 * q^65 + 4 * q^66 + 28 * q^67 - 12 * q^68 - 12 * q^69 + 28 * q^71 - 13 * q^72 + 16 * q^73 + 10 * q^74 - 2 * q^75 + 14 * q^76 + 12 * q^78 - 6 * q^79 + 3 * q^80 - q^81 - 30 * q^82 - 12 * q^83 - 4 * q^85 + 24 * q^86 + 2 * q^87 + 12 * q^88 + 10 * q^89 - 5 * q^90 + 4 * q^92 - 20 * q^93 - 10 * q^95 - 24 * q^96 - 8 * q^97 - 18 * 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 $

$2$	0.311108	0.219986	0.109993	−	0.993932i	$-0.464917\pi$
$2$	0.311108	0.219986	0.109993	+	0.993932i	$0.464917\pi$
$3$	−2.90321	−1.67617	−0.838085	−	0.545540i	$-0.816325\pi$
$3$	−2.90321	−1.67617	−0.838085	+	0.545540i	$0.816325\pi$
$4$	−1.90321	−0.951606
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−0.903212	−0.368735
$7$	0	0
$7$	0	0
$8$	−1.21432	−0.429327
$9$	5.42864	1.80955
$10$	−0.311108	−0.0983809
$11$	−1.52543	−0.459934	−0.229967	−	0.973198i	$-0.573862\pi$
$11$	−1.52543	−0.459934	−0.229967	+	0.973198i	$0.573862\pi$
$12$	5.52543	1.59505
$13$	0.622216	0.172572	0.0862858	−	0.996270i	$-0.472500\pi$
$13$	0.622216	0.172572	0.0862858	+	0.996270i	$0.472500\pi$
$14$	0	0
$15$	2.90321	0.749606
$16$	3.42864	0.857160
$17$	7.95407	1.92914	0.964572	−	0.263819i	$-0.0849820\pi$
$17$	7.95407	1.92914	0.964572	+	0.263819i	$0.0849820\pi$
$18$	1.68889	0.398076
$19$	1.09679	0.251620	0.125810	−	0.992054i	$-0.459847\pi$
$19$	1.09679	0.251620	0.125810	+	0.992054i	$0.459847\pi$
$20$	1.90321	0.425571
$21$	0	0
$22$	−0.474572	−0.101179
$23$	7.52543	1.56916	0.784580	−	0.620028i	$-0.212879\pi$
$23$	7.52543	1.56916	0.784580	+	0.620028i	$0.212879\pi$
$24$	3.52543	0.719625
$25$	1.00000	0.200000
$26$	0.193576	0.0379634
$27$	−7.05086	−1.35694
$28$	0	0
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0.903212	0.164903
$31$	6.90321	1.23985	0.619927	−	0.784660i	$-0.287162\pi$
$31$	6.90321	1.23985	0.619927	+	0.784660i	$0.287162\pi$
$32$	3.49532	0.617890
$33$	4.42864	0.770927
$34$	2.47457	0.424386
$35$	0	0
$36$	−10.3319	−1.72198
$37$	3.95407	0.650045	0.325022	−	0.945706i	$-0.394628\pi$
$37$	3.95407	0.650045	0.325022	+	0.945706i	$0.394628\pi$
$38$	0.341219	0.0553531
$39$	−1.80642	−0.289259
$40$	1.21432	0.192001
$41$	−3.67307	−0.573637	−0.286819	−	0.957985i	$-0.592598\pi$
$41$	−3.67307	−0.573637	−0.286819	+	0.957985i	$0.592598\pi$
$42$	0	0
$43$	−10.5161	−1.60368	−0.801842	−	0.597536i	$-0.796146\pi$
$43$	−10.5161	−1.60368	−0.801842	+	0.597536i	$0.796146\pi$
$44$	2.90321	0.437676
$45$	−5.42864	−0.809254
$46$	2.34122	0.345194
$47$	−6.90321	−1.00694	−0.503468	−	0.864014i	$-0.667943\pi$
$47$	−6.90321	−1.00694	−0.503468	+	0.864014i	$0.667943\pi$
$48$	−9.95407	−1.43675
$49$	0	0
$50$	0.311108	0.0439973
$51$	−23.0923	−3.23357
$52$	−1.18421	−0.164220
$53$	6.42864	0.883042	0.441521	−	0.897251i	$-0.354439\pi$
$53$	6.42864	0.883042	0.441521	+	0.897251i	$0.354439\pi$
$54$	−2.19358	−0.298508
$55$	1.52543	0.205689
$56$	0	0
$57$	−3.18421	−0.421759
$58$	−0.311108	−0.0408505
$59$	1.67307	0.217815	0.108908	−	0.994052i	$-0.465265\pi$
$59$	1.67307	0.217815	0.108908	+	0.994052i	$0.465265\pi$
$60$	−5.52543	−0.713330
$61$	1.86665	0.239000	0.119500	−	0.992834i	$-0.461871\pi$
$61$	1.86665	0.239000	0.119500	+	0.992834i	$0.461871\pi$
$62$	2.14764	0.272751
$63$	0	0
$64$	−5.76986	−0.721232
$65$	−0.622216	−0.0771764
$66$	1.37778	0.169594
$67$	11.5254	1.40806	0.704028	−	0.710173i	$-0.251383\pi$
$67$	11.5254	1.40806	0.704028	+	0.710173i	$0.251383\pi$
$68$	−15.1383	−1.83579
$69$	−21.8479	−2.63018
$70$	0	0
$71$	13.6731	1.62269	0.811347	−	0.584564i	$-0.198734\pi$
$71$	13.6731	1.62269	0.811347	+	0.584564i	$0.198734\pi$
$72$	−6.59210	−0.776887
$73$	−10.1891	−1.19255	−0.596274	−	0.802781i	$-0.703353\pi$
$73$	−10.1891	−1.19255	−0.596274	+	0.802781i	$0.703353\pi$
$74$	1.23014	0.143001
$75$	−2.90321	−0.335234
$76$	−2.08742	−0.239444
$77$	0	0
$78$	−0.561993	−0.0636331
$79$	9.13828	1.02814	0.514068	−	0.857749i	$-0.328138\pi$
$79$	9.13828	1.02814	0.514068	+	0.857749i	$0.328138\pi$
$80$	−3.42864	−0.383334
$81$	4.18421	0.464912
$82$	−1.14272	−0.126192
$83$	−10.7096	−1.17554	−0.587768	−	0.809030i	$-0.699993\pi$
$83$	−10.7096	−1.17554	−0.587768	+	0.809030i	$0.699993\pi$
$84$	0	0
$85$	−7.95407	−0.862740
$86$	−3.27163	−0.352789
$87$	2.90321	0.311257
$88$	1.85236	0.197462
$89$	7.80642	0.827479	0.413740	−	0.910395i	$-0.364222\pi$
$89$	7.80642	0.827479	0.413740	+	0.910395i	$0.364222\pi$
$90$	−1.68889	−0.178025
$91$	0	0
$92$	−14.3225	−1.49322
$93$	−20.0415	−2.07821
$94$	−2.14764	−0.221512
$95$	−1.09679	−0.112528
$96$	−10.1476	−1.03569
$97$	4.08742	0.415015	0.207507	−	0.978233i	$-0.433465\pi$
$97$	4.08742	0.415015	0.207507	+	0.978233i	$0.433465\pi$
$98$	0	0
$99$	−8.28100	−0.832271
$100$	−1.90321	−0.190321
$101$	−13.9081	−1.38391	−0.691956	−	0.721940i	$-0.743251\pi$
$101$	−13.9081	−1.38391	−0.691956	+	0.721940i	$0.743251\pi$
$102$	−7.18421	−0.711343
$103$	12.9447	1.27548	0.637740	−	0.770252i	$-0.279870\pi$
$103$	12.9447	1.27548	0.637740	+	0.770252i	$0.279870\pi$
$104$	−0.755569	−0.0740896
$105$	0	0
$106$	2.00000	0.194257
$107$	11.0049	1.06389	0.531943	−	0.846780i	$-0.321462\pi$
$107$	11.0049	1.06389	0.531943	+	0.846780i	$0.321462\pi$
$108$	13.4193	1.29127
$109$	−18.0415	−1.72806	−0.864031	−	0.503439i	$-0.832068\pi$
$109$	−18.0415	−1.72806	−0.864031	+	0.503439i	$0.832068\pi$
$110$	0.474572	0.0452487
$111$	−11.4795	−1.08959
$112$	0	0
$113$	−10.2810	−0.967155	−0.483577	−	0.875302i	$-0.660663\pi$
$113$	−10.2810	−0.967155	−0.483577	+	0.875302i	$0.660663\pi$
$114$	−0.990632	−0.0927812
$115$	−7.52543	−0.701750
$116$	1.90321	0.176709
$117$	3.37778	0.312276
$118$	0.520505	0.0479164
$119$	0	0
$120$	−3.52543	−0.321826
$121$	−8.67307	−0.788461
$122$	0.580728	0.0525767
$123$	10.6637	0.961514
$124$	−13.1383	−1.17985
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	6.22077	0.552004	0.276002	−	0.961157i	$-0.410990\pi$
$127$	6.22077	0.552004	0.276002	+	0.961157i	$0.410990\pi$
$128$	−8.78568	−0.776552
$129$	30.5303	2.68805
$130$	−0.193576	−0.0169778
$131$	11.7605	1.02752	0.513759	−	0.857934i	$-0.328252\pi$
$131$	11.7605	1.02752	0.513759	+	0.857934i	$0.328252\pi$
$132$	−8.42864	−0.733619
$133$	0	0
$134$	3.58565	0.309753
$135$	7.05086	0.606841
$136$	−9.65878	−0.828234
$137$	−3.56691	−0.304742	−0.152371	−	0.988323i	$-0.548691\pi$
$137$	−3.56691	−0.304742	−0.152371	+	0.988323i	$0.548691\pi$
$138$	−6.79706	−0.578604
$139$	8.56199	0.726219	0.363109	−	0.931747i	$-0.381715\pi$
$139$	8.56199	0.726219	0.363109	+	0.931747i	$0.381715\pi$
$140$	0	0
$141$	20.0415	1.68780
$142$	4.25380	0.356971
$143$	−0.949145	−0.0793715
$144$	18.6128	1.55107
$145$	1.00000	0.0830455
$146$	−3.16992	−0.262344
$147$	0	0
$148$	−7.52543	−0.618586
$149$	−5.61285	−0.459822	−0.229911	−	0.973212i	$-0.573844\pi$
$149$	−5.61285	−0.459822	−0.229911	+	0.973212i	$0.573844\pi$
$150$	−0.903212	−0.0737469
$151$	10.7971	0.878652	0.439326	−	0.898328i	$-0.355217\pi$
$151$	10.7971	0.878652	0.439326	+	0.898328i	$0.355217\pi$
$152$	−1.33185	−0.108027
$153$	43.1798	3.49088
$154$	0	0
$155$	−6.90321	−0.554479
$156$	3.43801	0.275261
$157$	−2.28100	−0.182043	−0.0910217	−	0.995849i	$-0.529013\pi$
$157$	−2.28100	−0.182043	−0.0910217	+	0.995849i	$0.529013\pi$
$158$	2.84299	0.226176
$159$	−18.6637	−1.48013
$160$	−3.49532	−0.276329
$161$	0	0
$162$	1.30174	0.102274
$163$	16.3225	1.27848	0.639238	−	0.769009i	$-0.279250\pi$
$163$	16.3225	1.27848	0.639238	+	0.769009i	$0.279250\pi$
$164$	6.99063	0.545877
$165$	−4.42864	−0.344769
$166$	−3.33185	−0.258602
$167$	4.76986	0.369103	0.184551	−	0.982823i	$-0.440917\pi$
$167$	4.76986	0.369103	0.184551	+	0.982823i	$0.440917\pi$
$168$	0	0
$169$	−12.6128	−0.970219
$170$	−2.47457	−0.189791
$171$	5.95407	0.455319
$172$	20.0143	1.52608
$173$	−4.23506	−0.321986	−0.160993	−	0.986956i	$-0.551470\pi$
$173$	−4.23506	−0.321986	−0.160993	+	0.986956i	$0.551470\pi$
$174$	0.903212	0.0684723
$175$	0	0
$176$	−5.23014	−0.394237
$177$	−4.85728	−0.365095
$178$	2.42864	0.182034
$179$	9.71456	0.726100	0.363050	−	0.931770i	$-0.381735\pi$
$179$	9.71456	0.726100	0.363050	+	0.931770i	$0.381735\pi$
$180$	10.3319	0.770091
$181$	−0.326929	−0.0243005	−0.0121502	−	0.999926i	$-0.503868\pi$
$181$	−0.326929	−0.0243005	−0.0121502	+	0.999926i	$0.503868\pi$
$182$	0	0
$183$	−5.41927	−0.400604
$184$	−9.13828	−0.673683
$185$	−3.95407	−0.290709
$186$	−6.23506	−0.457177
$187$	−12.1334	−0.887279
$188$	13.1383	0.958207
$189$	0	0
$190$	−0.341219	−0.0247547
$191$	−14.9447	−1.08136	−0.540680	−	0.841228i	$-0.681833\pi$
$191$	−14.9447	−1.08136	−0.540680	+	0.841228i	$0.681833\pi$
$192$	16.7511	1.20891
$193$	−14.1476	−1.01837	−0.509185	−	0.860657i	$-0.670053\pi$
$193$	−14.1476	−1.01837	−0.509185	+	0.860657i	$0.670053\pi$
$194$	1.27163	0.0912976
$195$	1.80642	0.129361
$196$	0	0
$197$	−5.70471	−0.406444	−0.203222	−	0.979133i	$-0.565141\pi$
$197$	−5.70471	−0.406444	−0.203222	+	0.979133i	$0.565141\pi$
$198$	−2.57628	−0.183088
$199$	22.1432	1.56969	0.784845	−	0.619692i	$-0.212743\pi$
$199$	22.1432	1.56969	0.784845	+	0.619692i	$0.212743\pi$
$200$	−1.21432	−0.0858654
$201$	−33.4608	−2.36014
$202$	−4.32693	−0.304442
$203$	0	0
$204$	43.9496	3.07709
$205$	3.67307	0.256538
$206$	4.02720	0.280588
$207$	40.8528	2.83947
$208$	2.13335	0.147921
$209$	−1.67307	−0.115729
$210$	0	0
$211$	−20.8430	−1.43489	−0.717445	−	0.696615i	$-0.754689\pi$
$211$	−20.8430	−1.43489	−0.717445	+	0.696615i	$0.754689\pi$
$212$	−12.2351	−0.840308
$213$	−39.6958	−2.71991
$214$	3.42372	0.234040
$215$	10.5161	0.717189
$216$	8.56199	0.582570
$217$	0	0
$218$	−5.61285	−0.380150
$219$	29.5812	1.99891
$220$	−2.90321	−0.195735
$221$	4.94914	0.332916
$222$	−3.57136	−0.239694
$223$	−9.03657	−0.605133	−0.302567	−	0.953128i	$-0.597843\pi$
$223$	−9.03657	−0.605133	−0.302567	+	0.953128i	$0.597843\pi$
$224$	0	0
$225$	5.42864	0.361909
$226$	−3.19850	−0.212761
$227$	−19.4050	−1.28795	−0.643977	−	0.765045i	$-0.722717\pi$
$227$	−19.4050	−1.28795	−0.643977	+	0.765045i	$0.722717\pi$
$228$	6.06022	0.401348
$229$	25.6128	1.69254	0.846272	−	0.532751i	$-0.178842\pi$
$229$	25.6128	1.69254	0.846272	+	0.532751i	$0.178842\pi$
$230$	−2.34122	−0.154375
$231$	0	0
$232$	1.21432	0.0797240
$233$	−3.12399	−0.204659	−0.102330	−	0.994751i	$-0.532630\pi$
$233$	−3.12399	−0.204659	−0.102330	+	0.994751i	$0.532630\pi$
$234$	1.05086	0.0686965
$235$	6.90321	0.450316
$236$	−3.18421	−0.207274
$237$	−26.5303	−1.72333
$238$	0	0
$239$	13.9398	0.901689	0.450845	−	0.892602i	$-0.351123\pi$
$239$	13.9398	0.901689	0.450845	+	0.892602i	$0.351123\pi$
$240$	9.95407	0.642532
$241$	18.4701	1.18977	0.594883	−	0.803813i	$-0.297198\pi$
$241$	18.4701	1.18977	0.594883	+	0.803813i	$0.297198\pi$
$242$	−2.69826	−0.173451
$243$	9.00492	0.577666
$244$	−3.55262	−0.227433
$245$	0	0
$246$	3.31756	0.211520
$247$	0.682439	0.0434225
$248$	−8.38271	−0.532302
$249$	31.0923	1.97040
$250$	−0.311108	−0.0196762
$251$	13.7921	0.870552	0.435276	−	0.900297i	$-0.356651\pi$
$251$	13.7921	0.870552	0.435276	+	0.900297i	$0.356651\pi$
$252$	0	0
$253$	−11.4795	−0.721710
$254$	1.93533	0.121433
$255$	23.0923	1.44610
$256$	8.80642	0.550401
$257$	1.47949	0.0922883	0.0461442	−	0.998935i	$-0.485307\pi$
$257$	1.47949	0.0922883	0.0461442	+	0.998935i	$0.485307\pi$
$258$	9.49823	0.591334
$259$	0	0
$260$	1.18421	0.0734415
$261$	−5.42864	−0.336024
$262$	3.65878	0.226040
$263$	−0.442930	−0.0273122	−0.0136561	−	0.999907i	$-0.504347\pi$
$263$	−0.442930	−0.0273122	−0.0136561	+	0.999907i	$0.504347\pi$
$264$	−5.37778	−0.330980
$265$	−6.42864	−0.394908
$266$	0	0
$267$	−22.6637	−1.38700
$268$	−21.9353	−1.33991
$269$	−3.93978	−0.240212	−0.120106	−	0.992761i	$-0.538324\pi$
$269$	−3.93978	−0.240212	−0.120106	+	0.992761i	$0.538324\pi$
$270$	2.19358	0.133497
$271$	−6.20787	−0.377101	−0.188551	−	0.982063i	$-0.560379\pi$
$271$	−6.20787	−0.377101	−0.188551	+	0.982063i	$0.560379\pi$
$272$	27.2716	1.65359
$273$	0	0
$274$	−1.10970	−0.0670391
$275$	−1.52543	−0.0919867
$276$	41.5812	2.50289
$277$	5.57136	0.334751	0.167375	−	0.985893i	$-0.446471\pi$
$277$	5.57136	0.334751	0.167375	+	0.985893i	$0.446471\pi$
$278$	2.66370	0.159758
$279$	37.4750	2.24357
$280$	0	0
$281$	6.69535	0.399411	0.199705	−	0.979856i	$-0.436001\pi$
$281$	6.69535	0.399411	0.199705	+	0.979856i	$0.436001\pi$
$282$	6.23506	0.371293
$283$	−25.8020	−1.53377	−0.766884	−	0.641785i	$-0.778194\pi$
$283$	−25.8020	−1.53377	−0.766884	+	0.641785i	$0.778194\pi$
$284$	−26.0228	−1.54417
$285$	3.18421	0.188616
$286$	−0.295286	−0.0174607
$287$	0	0
$288$	18.9748	1.11810
$289$	46.2672	2.72160
$290$	0.311108	0.0182689
$291$	−11.8666	−0.695635
$292$	19.3921	1.13484
$293$	18.8430	1.10082	0.550410	−	0.834895i	$-0.314472\pi$
$293$	18.8430	1.10082	0.550410	+	0.834895i	$0.314472\pi$
$294$	0	0
$295$	−1.67307	−0.0974099
$296$	−4.80150	−0.279082
$297$	10.7556	0.624101
$298$	−1.74620	−0.101155
$299$	4.68244	0.270792
$300$	5.52543	0.319011
$301$	0	0
$302$	3.35905	0.193292
$303$	40.3783	2.31967
$304$	3.76049	0.215679
$305$	−1.86665	−0.106884
$306$	13.4336	0.767946
$307$	1.65878	0.0946716	0.0473358	−	0.998879i	$-0.484927\pi$
$307$	1.65878	0.0946716	0.0473358	+	0.998879i	$0.484927\pi$
$308$	0	0
$309$	−37.5812	−2.13792
$310$	−2.14764	−0.121978
$311$	−21.3002	−1.20782	−0.603912	−	0.797051i	$-0.706392\pi$
$311$	−21.3002	−1.20782	−0.603912	+	0.797051i	$0.706392\pi$
$312$	2.19358	0.124187
$313$	−8.62222	−0.487356	−0.243678	−	0.969856i	$-0.578354\pi$
$313$	−8.62222	−0.487356	−0.243678	+	0.969856i	$0.578354\pi$
$314$	−0.709636	−0.0400471
$315$	0	0
$316$	−17.3921	−0.978381
$317$	−27.5955	−1.54992	−0.774959	−	0.632012i	$-0.782229\pi$
$317$	−27.5955	−1.54992	−0.774959	+	0.632012i	$0.782229\pi$
$318$	−5.80642	−0.325608
$319$	1.52543	0.0854075
$320$	5.76986	0.322545
$321$	−31.9496	−1.78325
$322$	0	0
$323$	8.72393	0.485412
$324$	−7.96343	−0.442413
$325$	0.622216	0.0345143
$326$	5.07805	0.281247
$327$	52.3783	2.89652
$328$	4.46028	0.246278
$329$	0	0
$330$	−1.37778	−0.0758445
$331$	16.9131	0.929626	0.464813	−	0.885409i	$-0.346122\pi$
$331$	16.9131	0.929626	0.464813	+	0.885409i	$0.346122\pi$
$332$	20.3827	1.11865
$333$	21.4652	1.17629
$334$	1.48394	0.0811976
$335$	−11.5254	−0.629701
$336$	0	0
$337$	11.9956	0.653439	0.326720	−	0.945121i	$-0.394057\pi$
$337$	11.9956	0.653439	0.326720	+	0.945121i	$0.394057\pi$
$338$	−3.92396	−0.213435
$339$	29.8479	1.62112
$340$	15.1383	0.820988
$341$	−10.5303	−0.570250
$342$	1.85236	0.100164
$343$	0	0
$344$	12.7699	0.688505
$345$	21.8479	1.17625
$346$	−1.31756	−0.0708325
$347$	6.14764	0.330023	0.165011	−	0.986292i	$-0.447234\pi$
$347$	6.14764	0.330023	0.165011	+	0.986292i	$0.447234\pi$
$348$	−5.52543	−0.296194
$349$	7.12399	0.381338	0.190669	−	0.981654i	$-0.438934\pi$
$349$	7.12399	0.381338	0.190669	+	0.981654i	$0.438934\pi$
$350$	0	0
$351$	−4.38715	−0.234169
$352$	−5.33185	−0.284189
$353$	16.9175	0.900428	0.450214	−	0.892921i	$-0.351348\pi$
$353$	16.9175	0.900428	0.450214	+	0.892921i	$0.351348\pi$
$354$	−1.51114	−0.0803160
$355$	−13.6731	−0.725691
$356$	−14.8573	−0.787434
$357$	0	0
$358$	3.02227	0.159732
$359$	36.7096	1.93746	0.968730	−	0.248116i	$-0.0798115\pi$
$359$	36.7096	1.93746	0.968730	+	0.248116i	$0.0798115\pi$
$360$	6.59210	0.347434
$361$	−17.7971	−0.936687
$362$	−0.101710	−0.00534577
$363$	25.1798	1.32159
$364$	0	0
$365$	10.1891	0.533323
$366$	−1.68598	−0.0881275
$367$	−8.41435	−0.439225	−0.219613	−	0.975587i	$-0.570479\pi$
$367$	−8.41435	−0.439225	−0.219613	+	0.975587i	$0.570479\pi$
$368$	25.8020	1.34502
$369$	−19.9398	−1.03802
$370$	−1.23014	−0.0639520
$371$	0	0
$372$	38.1432	1.97763
$373$	−8.66370	−0.448590	−0.224295	−	0.974521i	$-0.572008\pi$
$373$	−8.66370	−0.448590	−0.224295	+	0.974521i	$0.572008\pi$
$374$	−3.77478	−0.195189
$375$	2.90321	0.149921
$376$	8.38271	0.432305
$377$	−0.622216	−0.0320457
$378$	0	0
$379$	−2.76986	−0.142278	−0.0711390	−	0.997466i	$-0.522663\pi$
$379$	−2.76986	−0.142278	−0.0711390	+	0.997466i	$0.522663\pi$
$380$	2.08742	0.107082
$381$	−18.0602	−0.925253
$382$	−4.64941	−0.237885
$383$	−1.67752	−0.0857171	−0.0428585	−	0.999081i	$-0.513646\pi$
$383$	−1.67752	−0.0857171	−0.0428585	+	0.999081i	$0.513646\pi$
$384$	25.5067	1.30163
$385$	0	0
$386$	−4.40144	−0.224028
$387$	−57.0879	−2.90194
$388$	−7.77923	−0.394930
$389$	−5.77478	−0.292793	−0.146397	−	0.989226i	$-0.546768\pi$
$389$	−5.77478	−0.292793	−0.146397	+	0.989226i	$0.546768\pi$
$390$	0.561993	0.0284576
$391$	59.8578	3.02714
$392$	0	0
$393$	−34.1432	−1.72230
$394$	−1.77478	−0.0894122
$395$	−9.13828	−0.459797
$396$	15.7605	0.791994
$397$	−29.9081	−1.50105	−0.750523	−	0.660844i	$-0.770198\pi$
$397$	−29.9081	−1.50105	−0.750523	+	0.660844i	$0.770198\pi$
$398$	6.88892	0.345310
$399$	0	0
$400$	3.42864	0.171432
$401$	8.53035	0.425985	0.212993	−	0.977054i	$-0.431679\pi$
$401$	8.53035	0.425985	0.212993	+	0.977054i	$0.431679\pi$
$402$	−10.4099	−0.519199
$403$	4.29529	0.213963
$404$	26.4701	1.31694
$405$	−4.18421	−0.207915
$406$	0	0
$407$	−6.03164	−0.298977
$408$	28.0415	1.38826
$409$	−5.09234	−0.251800	−0.125900	−	0.992043i	$-0.540182\pi$
$409$	−5.09234	−0.251800	−0.125900	+	0.992043i	$0.540182\pi$
$410$	1.14272	0.0564350
$411$	10.3555	0.510800
$412$	−24.6365	−1.21375
$413$	0	0
$414$	12.7096	0.624645
$415$	10.7096	0.525715
$416$	2.17484	0.106630
$417$	−24.8573	−1.21727
$418$	−0.520505	−0.0254588
$419$	24.3368	1.18893	0.594465	−	0.804122i	$-0.297364\pi$
$419$	24.3368	1.18893	0.594465	+	0.804122i	$0.297364\pi$
$420$	0	0
$421$	24.5018	1.19414	0.597072	−	0.802188i	$-0.296331\pi$
$421$	24.5018	1.19414	0.597072	+	0.802188i	$0.296331\pi$
$422$	−6.48442	−0.315656
$423$	−37.4750	−1.82210
$424$	−7.80642	−0.379113
$425$	7.95407	0.385829
$426$	−12.3497	−0.598344
$427$	0	0
$428$	−20.9447	−1.01240
$429$	2.75557	0.133040
$430$	3.27163	0.157772
$431$	4.26671	0.205520	0.102760	−	0.994706i	$-0.467233\pi$
$431$	4.26671	0.205520	0.102760	+	0.994706i	$0.467233\pi$
$432$	−24.1748	−1.16311
$433$	−27.0049	−1.29777	−0.648887	−	0.760885i	$-0.724765\pi$
$433$	−27.0049	−1.29777	−0.648887	+	0.760885i	$0.724765\pi$
$434$	0	0
$435$	−2.90321	−0.139198
$436$	34.3368	1.64443
$437$	8.25380	0.394833
$438$	9.20294	0.439734
$439$	2.03164	0.0969650	0.0484825	−	0.998824i	$-0.484561\pi$
$439$	2.03164	0.0969650	0.0484825	+	0.998824i	$0.484561\pi$
$440$	−1.85236	−0.0883076
$441$	0	0
$442$	1.53972	0.0732369
$443$	−3.46520	−0.164637	−0.0823184	−	0.996606i	$-0.526232\pi$
$443$	−3.46520	−0.164637	−0.0823184	+	0.996606i	$0.526232\pi$
$444$	21.8479	1.03686
$445$	−7.80642	−0.370060
$446$	−2.81135	−0.133121
$447$	16.2953	0.770741
$448$	0	0
$449$	37.3590	1.76308	0.881541	−	0.472107i	$-0.156506\pi$
$449$	37.3590	1.76308	0.881541	+	0.472107i	$0.156506\pi$
$450$	1.68889	0.0796151
$451$	5.60300	0.263835
$452$	19.5669	0.920350
$453$	−31.3461	−1.47277
$454$	−6.03704	−0.283332
$455$	0	0
$456$	3.86665	0.181072
$457$	13.4509	0.629207	0.314604	−	0.949223i	$-0.398128\pi$
$457$	13.4509	0.629207	0.314604	+	0.949223i	$0.398128\pi$
$458$	7.96836	0.372337
$459$	−56.0830	−2.61773
$460$	14.3225	0.667789
$461$	16.2766	0.758075	0.379037	−	0.925381i	$-0.376255\pi$
$461$	16.2766	0.758075	0.379037	+	0.925381i	$0.376255\pi$
$462$	0	0
$463$	−30.3926	−1.41246	−0.706231	−	0.707982i	$-0.749606\pi$
$463$	−30.3926	−1.41246	−0.706231	+	0.707982i	$0.749606\pi$
$464$	−3.42864	−0.159171
$465$	20.0415	0.929402
$466$	−0.971896	−0.0450222
$467$	1.18865	0.0550043	0.0275022	−	0.999622i	$-0.491245\pi$
$467$	1.18865	0.0550043	0.0275022	+	0.999622i	$0.491245\pi$
$468$	−6.42864	−0.297164
$469$	0	0
$470$	2.14764	0.0990634
$471$	6.62222	0.305136
$472$	−2.03164	−0.0935139
$473$	16.0415	0.737588
$474$	−8.25380	−0.379110
$475$	1.09679	0.0503241
$476$	0	0
$477$	34.8988	1.59790
$478$	4.33677	0.198359
$479$	−41.0464	−1.87546	−0.937729	−	0.347367i	$-0.887076\pi$
$479$	−41.0464	−1.87546	−0.937729	+	0.347367i	$0.887076\pi$
$480$	10.1476	0.463174
$481$	2.46028	0.112179
$482$	5.74620	0.261732
$483$	0	0
$484$	16.5067	0.750304
$485$	−4.08742	−0.185600
$486$	2.80150	0.127079
$487$	−10.1476	−0.459834	−0.229917	−	0.973210i	$-0.573845\pi$
$487$	−10.1476	−0.459834	−0.229917	+	0.973210i	$0.573845\pi$
$488$	−2.26671	−0.102609
$489$	−47.3876	−2.14294
$490$	0	0
$491$	−29.2083	−1.31815	−0.659077	−	0.752075i	$-0.729053\pi$
$491$	−29.2083	−1.31815	−0.659077	+	0.752075i	$0.729053\pi$
$492$	−20.2953	−0.914982
$493$	−7.95407	−0.358233
$494$	0.212312	0.00955237
$495$	8.28100	0.372203
$496$	23.6686	1.06275
$497$	0	0
$498$	9.67307	0.433461
$499$	21.9813	0.984017	0.492008	−	0.870591i	$-0.336263\pi$
$499$	21.9813	0.984017	0.492008	+	0.870591i	$0.336263\pi$
$500$	1.90321	0.0851142
$501$	−13.8479	−0.618679
$502$	4.29084	0.191510
$503$	−5.77923	−0.257683	−0.128841	−	0.991665i	$-0.541126\pi$
$503$	−5.77923	−0.257683	−0.128841	+	0.991665i	$0.541126\pi$
$504$	0	0
$505$	13.9081	0.618904
$506$	−3.57136	−0.158766
$507$	36.6178	1.62625
$508$	−11.8394	−0.525291
$509$	−13.6543	−0.605218	−0.302609	−	0.953115i	$-0.597858\pi$
$509$	−13.6543	−0.605218	−0.302609	+	0.953115i	$0.597858\pi$
$510$	7.18421	0.318122
$511$	0	0
$512$	20.3111	0.897633
$513$	−7.73329	−0.341433
$514$	0.460282	0.0203022
$515$	−12.9447	−0.570412
$516$	−58.1057	−2.55796
$517$	10.5303	0.463124
$518$	0	0
$519$	12.2953	0.539703
$520$	0.755569	0.0331339
$521$	19.6731	0.861893	0.430946	−	0.902378i	$-0.358180\pi$
$521$	19.6731	0.861893	0.430946	+	0.902378i	$0.358180\pi$
$522$	−1.68889	−0.0739208
$523$	15.1383	0.661951	0.330975	−	0.943639i	$-0.392622\pi$
$523$	15.1383	0.661951	0.330975	+	0.943639i	$0.392622\pi$
$524$	−22.3827	−0.977793
$525$	0	0
$526$	−0.137799	−0.00600832
$527$	54.9086	2.39186
$528$	15.1842	0.660808
$529$	33.6321	1.46226
$530$	−2.00000	−0.0868744
$531$	9.08250	0.394147
$532$	0	0
$533$	−2.28544	−0.0989935
$534$	−7.05086	−0.305120
$535$	−11.0049	−0.475784
$536$	−13.9956	−0.604516
$537$	−28.2034	−1.21707
$538$	−1.22570	−0.0528435
$539$	0	0
$540$	−13.4193	−0.577474
$541$	2.68244	0.115327	0.0576635	−	0.998336i	$-0.481635\pi$
$541$	2.68244	0.115327	0.0576635	+	0.998336i	$0.481635\pi$
$542$	−1.93132	−0.0829571
$543$	0.949145	0.0407317
$544$	27.8020	1.19200
$545$	18.0415	0.772812
$546$	0	0
$547$	−15.3635	−0.656896	−0.328448	−	0.944522i	$-0.606525\pi$
$547$	−15.3635	−0.656896	−0.328448	+	0.944522i	$0.606525\pi$
$548$	6.78859	0.289994
$549$	10.1334	0.432481
$550$	−0.474572	−0.0202358
$551$	−1.09679	−0.0467247
$552$	26.5303	1.12921
$553$	0	0
$554$	1.73329	0.0736406
$555$	11.4795	0.487277
$556$	−16.2953	−0.691074
$557$	−9.87955	−0.418610	−0.209305	−	0.977850i	$-0.567120\pi$
$557$	−9.87955	−0.418610	−0.209305	+	0.977850i	$0.567120\pi$
$558$	11.6588	0.493556
$559$	−6.54326	−0.276750
$560$	0	0
$561$	35.2257	1.48723
$562$	2.08297	0.0878650
$563$	27.4938	1.15872	0.579362	−	0.815070i	$-0.303302\pi$
$563$	27.4938	1.15872	0.579362	+	0.815070i	$0.303302\pi$
$564$	−38.1432	−1.60612
$565$	10.2810	0.432525
$566$	−8.02720	−0.337408
$567$	0	0
$568$	−16.6035	−0.696667
$569$	17.3590	0.727729	0.363865	−	0.931452i	$-0.381457\pi$
$569$	17.3590	0.727729	0.363865	+	0.931452i	$0.381457\pi$
$570$	0.990632	0.0414930
$571$	−25.4479	−1.06496	−0.532480	−	0.846443i	$-0.678740\pi$
$571$	−25.4479	−1.06496	−0.532480	+	0.846443i	$0.678740\pi$
$572$	1.80642	0.0755304
$573$	43.3876	1.81254
$574$	0	0
$575$	7.52543	0.313832
$576$	−31.3225	−1.30510
$577$	10.6178	0.442024	0.221012	−	0.975271i	$-0.429064\pi$
$577$	10.6178	0.442024	0.221012	+	0.975271i	$0.429064\pi$
$578$	14.3941	0.598715
$579$	41.0736	1.70696
$580$	−1.90321	−0.0790266
$581$	0	0
$582$	−3.69181	−0.153030
$583$	−9.80642	−0.406141
$584$	12.3729	0.511993
$585$	−3.37778	−0.139654
$586$	5.86220	0.242165
$587$	8.94470	0.369187	0.184594	−	0.982815i	$-0.440903\pi$
$587$	8.94470	0.369187	0.184594	+	0.982815i	$0.440903\pi$
$588$	0	0
$589$	7.57136	0.311972
$590$	−0.520505	−0.0214289
$591$	16.5620	0.681269
$592$	13.5571	0.557192
$593$	−14.1619	−0.581561	−0.290780	−	0.956790i	$-0.593915\pi$
$593$	−14.1619	−0.581561	−0.290780	+	0.956790i	$0.593915\pi$
$594$	3.34614	0.137294
$595$	0	0
$596$	10.6824	0.437570
$597$	−64.2864	−2.63107
$598$	1.45674	0.0595707
$599$	−22.5575	−0.921676	−0.460838	−	0.887484i	$-0.652451\pi$
$599$	−22.5575	−0.921676	−0.460838	+	0.887484i	$0.652451\pi$
$600$	3.52543	0.143925
$601$	40.6133	1.65665	0.828326	−	0.560246i	$-0.189294\pi$
$601$	40.6133	1.65665	0.828326	+	0.560246i	$0.189294\pi$
$602$	0	0
$603$	62.5674	2.54794
$604$	−20.5491	−0.836130
$605$	8.67307	0.352610
$606$	12.5620	0.510296
$607$	−13.5955	−0.551824	−0.275912	−	0.961183i	$-0.588980\pi$
$607$	−13.5955	−0.551824	−0.275912	+	0.961183i	$0.588980\pi$
$608$	3.83362	0.155474
$609$	0	0
$610$	−0.580728	−0.0235130
$611$	−4.29529	−0.173769
$612$	−82.1802	−3.32194
$613$	−42.0830	−1.69972	−0.849858	−	0.527012i	$-0.823312\pi$
$613$	−42.0830	−1.69972	−0.849858	+	0.527012i	$0.823312\pi$
$614$	0.516060	0.0208265
$615$	−10.6637	−0.430002
$616$	0	0
$617$	33.5067	1.34893	0.674464	−	0.738307i	$-0.264375\pi$
$617$	33.5067	1.34893	0.674464	+	0.738307i	$0.264375\pi$
$618$	−11.6918	−0.470313
$619$	14.6780	0.589958	0.294979	−	0.955504i	$-0.404687\pi$
$619$	14.6780	0.589958	0.294979	+	0.955504i	$0.404687\pi$
$620$	13.1383	0.527646
$621$	−53.0607	−2.12925
$622$	−6.62666	−0.265705
$623$	0	0
$624$	−6.19358	−0.247941
$625$	1.00000	0.0400000
$626$	−2.68244	−0.107212
$627$	4.85728	0.193981
$628$	4.34122	0.173234
$629$	31.4509	1.25403
$630$	0	0
$631$	11.3176	0.450545	0.225273	−	0.974296i	$-0.427673\pi$
$631$	11.3176	0.450545	0.225273	+	0.974296i	$0.427673\pi$
$632$	−11.0968	−0.441407
$633$	60.5116	2.40512
$634$	−8.58517	−0.340961
$635$	−6.22077	−0.246864
$636$	35.5210	1.40850
$637$	0	0
$638$	0.474572	0.0187885
$639$	74.2262	2.93634
$640$	8.78568	0.347285
$641$	34.8988	1.37842	0.689209	−	0.724562i	$-0.257958\pi$
$641$	34.8988	1.37842	0.689209	+	0.724562i	$0.257958\pi$
$642$	−9.93978	−0.392292
$643$	41.9768	1.65540	0.827702	−	0.561168i	$-0.189648\pi$
$643$	41.9768	1.65540	0.827702	+	0.561168i	$0.189648\pi$
$644$	0	0
$645$	−30.5303	−1.20213
$646$	2.71408	0.106784
$647$	−5.46520	−0.214859	−0.107430	−	0.994213i	$-0.534262\pi$
$647$	−5.46520	−0.214859	−0.107430	+	0.994213i	$0.534262\pi$
$648$	−5.08097	−0.199599
$649$	−2.55215	−0.100181
$650$	0.193576	0.00759268
$651$	0	0
$652$	−31.0651	−1.21660
$653$	−8.76986	−0.343191	−0.171596	−	0.985167i	$-0.554892\pi$
$653$	−8.76986	−0.343191	−0.171596	+	0.985167i	$0.554892\pi$
$654$	16.2953	0.637196
$655$	−11.7605	−0.459520
$656$	−12.5936	−0.491699
$657$	−55.3131	−2.15797
$658$	0	0
$659$	3.29036	0.128174	0.0640872	−	0.997944i	$-0.479586\pi$
$659$	3.29036	0.128174	0.0640872	+	0.997944i	$0.479586\pi$
$660$	8.42864	0.328084
$661$	−19.7560	−0.768421	−0.384211	−	0.923246i	$-0.625526\pi$
$661$	−19.7560	−0.768421	−0.384211	+	0.923246i	$0.625526\pi$
$662$	5.26178	0.204505
$663$	−14.3684	−0.558023
$664$	13.0049	0.504689
$665$	0	0
$666$	6.67799	0.258767
$667$	−7.52543	−0.291386
$668$	−9.07805	−0.351240
$669$	26.2351	1.01431
$670$	−3.58565	−0.138526
$671$	−2.84743	−0.109924
$672$	0	0
$673$	44.3970	1.71138	0.855689	−	0.517490i	$-0.173134\pi$
$673$	44.3970	1.71138	0.855689	+	0.517490i	$0.173134\pi$
$674$	3.73191	0.143748
$675$	−7.05086	−0.271388
$676$	24.0049	0.923266
$677$	−6.09726	−0.234337	−0.117168	−	0.993112i	$-0.537382\pi$
$677$	−6.09726	−0.234337	−0.117168	+	0.993112i	$0.537382\pi$
$678$	9.28592	0.356624
$679$	0	0
$680$	9.65878	0.370397
$681$	56.3368	2.15883
$682$	−3.27607	−0.125447
$683$	37.9224	1.45106	0.725531	−	0.688190i	$-0.241594\pi$
$683$	37.9224	1.45106	0.725531	+	0.688190i	$0.241594\pi$
$684$	−11.3319	−0.433284
$685$	3.56691	0.136285
$686$	0	0
$687$	−74.3595	−2.83699
$688$	−36.0558	−1.37461
$689$	4.00000	0.152388
$690$	6.79706	0.258759
$691$	−13.3145	−0.506507	−0.253254	−	0.967400i	$-0.581501\pi$
$691$	−13.3145	−0.506507	−0.253254	+	0.967400i	$0.581501\pi$
$692$	8.06022	0.306404
$693$	0	0
$694$	1.91258	0.0726005
$695$	−8.56199	−0.324775
$696$	−3.52543	−0.133631
$697$	−29.2159	−1.10663
$698$	2.21633	0.0838892
$699$	9.06959	0.343043
$700$	0	0
$701$	−23.4893	−0.887180	−0.443590	−	0.896230i	$-0.646295\pi$
$701$	−23.4893	−0.887180	−0.443590	+	0.896230i	$0.646295\pi$
$702$	−1.36488	−0.0515140
$703$	4.33677	0.163565
$704$	8.80150	0.331719
$705$	−20.0415	−0.754806
$706$	5.26317	0.198082
$707$	0	0
$708$	9.24443	0.347427
$709$	11.6731	0.438391	0.219196	−	0.975681i	$-0.429657\pi$
$709$	11.6731	0.438391	0.219196	+	0.975681i	$0.429657\pi$
$710$	−4.25380	−0.159642
$711$	49.6084	1.86046
$712$	−9.47949	−0.355259
$713$	51.9496	1.94553
$714$	0	0
$715$	0.949145	0.0354960
$716$	−18.4889	−0.690961
$717$	−40.4701	−1.51138
$718$	11.4207	0.426215
$719$	−29.5526	−1.10213	−0.551063	−	0.834463i	$-0.685778\pi$
$719$	−29.5526	−1.10213	−0.551063	+	0.834463i	$0.685778\pi$
$720$	−18.6128	−0.693660
$721$	0	0
$722$	−5.53680	−0.206058
$723$	−53.6227	−1.99425
$724$	0.622216	0.0231245
$725$	−1.00000	−0.0371391
$726$	7.83362	0.290733
$727$	−3.88094	−0.143936	−0.0719680	−	0.997407i	$-0.522928\pi$
$727$	−3.88094	−0.143936	−0.0719680	+	0.997407i	$0.522928\pi$
$728$	0	0
$729$	−38.6958	−1.43318
$730$	3.16992	0.117324
$731$	−83.6454	−3.09374
$732$	10.3140	0.381217
$733$	−14.8845	−0.549771	−0.274885	−	0.961477i	$-0.588640\pi$
$733$	−14.8845	−0.549771	−0.274885	+	0.961477i	$0.588640\pi$
$734$	−2.61777	−0.0966236
$735$	0	0
$736$	26.3037	0.969569
$737$	−17.5812	−0.647612
$738$	−6.20342	−0.228351
$739$	2.24935	0.0827438	0.0413719	−	0.999144i	$-0.486827\pi$
$739$	2.24935	0.0827438	0.0413719	+	0.999144i	$0.486827\pi$
$740$	7.52543	0.276640
$741$	−1.98126	−0.0727836
$742$	0	0
$743$	−3.46520	−0.127126	−0.0635630	−	0.997978i	$-0.520246\pi$
$743$	−3.46520	−0.127126	−0.0635630	+	0.997978i	$0.520246\pi$
$744$	24.3368	0.892229
$745$	5.61285	0.205639
$746$	−2.69535	−0.0986836
$747$	−58.1388	−2.12719
$748$	23.0923	0.844340
$749$	0	0
$750$	0.903212	0.0329806
$751$	3.16992	0.115672	0.0578360	−	0.998326i	$-0.481580\pi$
$751$	3.16992	0.115672	0.0578360	+	0.998326i	$0.481580\pi$
$752$	−23.6686	−0.863106
$753$	−40.0415	−1.45919
$754$	−0.193576	−0.00704963
$755$	−10.7971	−0.392945
$756$	0	0
$757$	−52.0785	−1.89283	−0.946413	−	0.322958i	$-0.895323\pi$
$757$	−52.0785	−1.89283	−0.946413	+	0.322958i	$0.895323\pi$
$758$	−0.861725	−0.0312993
$759$	33.3274	1.20971
$760$	1.33185	0.0483113
$761$	14.9777	0.542942	0.271471	−	0.962447i	$-0.412490\pi$
$761$	14.9777	0.542942	0.271471	+	0.962447i	$0.412490\pi$
$762$	−5.61868	−0.203543
$763$	0	0
$764$	28.4429	1.02903
$765$	−43.1798	−1.56117
$766$	−0.521889	−0.0188566
$767$	1.04101	0.0375887
$768$	−25.5669	−0.922567
$769$	1.90813	0.0688091	0.0344045	−	0.999408i	$-0.489047\pi$
$769$	1.90813	0.0688091	0.0344045	+	0.999408i	$0.489047\pi$
$770$	0	0
$771$	−4.29529	−0.154691
$772$	26.9260	0.969087
$773$	−21.7891	−0.783698	−0.391849	−	0.920029i	$-0.628165\pi$
$773$	−21.7891	−0.783698	−0.391849	+	0.920029i	$0.628165\pi$
$774$	−17.7605	−0.638388
$775$	6.90321	0.247971
$776$	−4.96343	−0.178177
$777$	0	0
$778$	−1.79658	−0.0644105
$779$	−4.02858	−0.144339
$780$	−3.43801	−0.123100
$781$	−20.8573	−0.746332
$782$	18.6222	0.665929
$783$	7.05086	0.251977
$784$	0	0
$785$	2.28100	0.0814122
$786$	−10.6222	−0.378882
$787$	18.1388	0.646577	0.323288	−	0.946301i	$-0.395212\pi$
$787$	18.1388	0.646577	0.323288	+	0.946301i	$0.395212\pi$
$788$	10.8573	0.386775
$789$	1.28592	0.0457799
$790$	−2.84299	−0.101149
$791$	0	0
$792$	10.0558	0.357316
$793$	1.16146	0.0412445
$794$	−9.30465	−0.330210
$795$	18.6637	0.661933
$796$	−42.1432	−1.49373
$797$	−2.96343	−0.104970	−0.0524851	−	0.998622i	$-0.516714\pi$
$797$	−2.96343	−0.104970	−0.0524851	+	0.998622i	$0.516714\pi$
$798$	0	0
$799$	−54.9086	−1.94253
$800$	3.49532	0.123578
$801$	42.3783	1.49736
$802$	2.65386	0.0937110
$803$	15.5428	0.548493
$804$	63.6829	2.24592
$805$	0	0
$806$	1.33630	0.0470691
$807$	11.4380	0.402637
$808$	16.8889	0.594150
$809$	−26.2953	−0.924493	−0.462247	−	0.886751i	$-0.652956\pi$
$809$	−26.2953	−0.924493	−0.462247	+	0.886751i	$0.652956\pi$
$810$	−1.30174	−0.0457385
$811$	−24.3783	−0.856037	−0.428018	−	0.903770i	$-0.640788\pi$
$811$	−24.3783	−0.856037	−0.428018	+	0.903770i	$0.640788\pi$
$812$	0	0
$813$	18.0228	0.632085
$814$	−1.87649	−0.0657710
$815$	−16.3225	−0.571752
$816$	−79.1753	−2.77169
$817$	−11.5339	−0.403520
$818$	−1.58427	−0.0553926
$819$	0	0
$820$	−6.99063	−0.244123
$821$	1.52987	0.0533929	0.0266965	−	0.999644i	$-0.491501\pi$
$821$	1.52987	0.0533929	0.0266965	+	0.999644i	$0.491501\pi$
$822$	3.22168	0.112369
$823$	46.7195	1.62854	0.814269	−	0.580487i	$-0.197138\pi$
$823$	46.7195	1.62854	0.814269	+	0.580487i	$0.197138\pi$
$824$	−15.7190	−0.547597
$825$	4.42864	0.154185
$826$	0	0
$827$	−29.6499	−1.03103	−0.515514	−	0.856881i	$-0.672399\pi$
$827$	−29.6499	−1.03103	−0.515514	+	0.856881i	$0.672399\pi$
$828$	−77.7516	−2.70205
$829$	8.79706	0.305534	0.152767	−	0.988262i	$-0.451182\pi$
$829$	8.79706	0.305534	0.152767	+	0.988262i	$0.451182\pi$
$830$	3.33185	0.115650
$831$	−16.1748	−0.561099
$832$	−3.59010	−0.124464
$833$	0	0
$834$	−7.73329	−0.267782
$835$	−4.76986	−0.165068
$836$	3.18421	0.110128
$837$	−48.6735	−1.68240
$838$	7.57136	0.261548
$839$	−11.3319	−0.391219	−0.195609	−	0.980682i	$-0.562668\pi$
$839$	−11.3319	−0.391219	−0.195609	+	0.980682i	$0.562668\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	7.62269	0.262695
$843$	−19.4380	−0.669481
$844$	39.6686	1.36545
$845$	12.6128	0.433895
$846$	−11.6588	−0.400837
$847$	0	0
$848$	22.0415	0.756908
$849$	74.9086	2.57086
$850$	2.47457	0.0848771
$851$	29.7560	1.02002
$852$	75.5496	2.58829
$853$	−54.8845	−1.87921	−0.939604	−	0.342263i	$-0.888807\pi$
$853$	−54.8845	−1.87921	−0.939604	+	0.342263i	$0.888807\pi$
$854$	0	0
$855$	−5.95407	−0.203625
$856$	−13.3635	−0.456755
$857$	−36.4385	−1.24471	−0.622357	−	0.782733i	$-0.713825\pi$
$857$	−36.4385	−1.24471	−0.622357	+	0.782733i	$0.713825\pi$
$858$	0.857279	0.0292670
$859$	−1.72885	−0.0589875	−0.0294938	−	0.999565i	$-0.509390\pi$
$859$	−1.72885	−0.0589875	−0.0294938	+	0.999565i	$0.509390\pi$
$860$	−20.0143	−0.682482
$861$	0	0
$862$	1.32741	0.0452116
$863$	−9.40192	−0.320045	−0.160023	−	0.987113i	$-0.551157\pi$
$863$	−9.40192	−0.320045	−0.160023	+	0.987113i	$0.551157\pi$
$864$	−24.6450	−0.838439
$865$	4.23506	0.143996
$866$	−8.40144	−0.285493
$867$	−134.323	−4.56186
$868$	0	0
$869$	−13.9398	−0.472875
$870$	−0.903212	−0.0306218
$871$	7.17130	0.242990
$872$	21.9081	0.741903
$873$	22.1891	0.750988
$874$	2.56782	0.0868579
$875$	0	0
$876$	−56.2993	−1.90218
$877$	8.91750	0.301123	0.150561	−	0.988601i	$-0.451892\pi$
$877$	8.91750	0.301123	0.150561	+	0.988601i	$0.451892\pi$
$878$	0.632060	0.0213310
$879$	−54.7052	−1.84516
$880$	5.23014	0.176308
$881$	42.1245	1.41921	0.709605	−	0.704600i	$-0.248874\pi$
$881$	42.1245	1.41921	0.709605	+	0.704600i	$0.248874\pi$
$882$	0	0
$883$	−38.4340	−1.29341	−0.646704	−	0.762741i	$-0.723853\pi$
$883$	−38.4340	−1.29341	−0.646704	+	0.762741i	$0.723853\pi$
$884$	−9.41927	−0.316804
$885$	4.85728	0.163276
$886$	−1.07805	−0.0362179
$887$	38.6365	1.29729	0.648643	−	0.761092i	$-0.275337\pi$
$887$	38.6365	1.29729	0.648643	+	0.761092i	$0.275337\pi$
$888$	13.9398	0.467788
$889$	0	0
$890$	−2.42864	−0.0814082
$891$	−6.38271	−0.213829
$892$	17.1985	0.575848
$893$	−7.57136	−0.253366
$894$	5.06959	0.169552
$895$	−9.71456	−0.324722
$896$	0	0
$897$	−13.5941	−0.453894
$898$	11.6227	0.387854
$899$	−6.90321	−0.230235
$900$	−10.3319	−0.344395
$901$	51.1338	1.70351
$902$	1.74314	0.0580402
$903$	0	0
$904$	12.4844	0.415226
$905$	0.326929	0.0108675
$906$	−9.75203	−0.323989
$907$	0.534795	0.0177576	0.00887880	−	0.999961i	$-0.497174\pi$
$907$	0.534795	0.0177576	0.00887880	+	0.999961i	$0.497174\pi$
$908$	36.9318	1.22562
$909$	−75.5022	−2.50425
$910$	0	0
$911$	−23.6686	−0.784177	−0.392088	−	0.919928i	$-0.628247\pi$
$911$	−23.6686	−0.784177	−0.392088	+	0.919928i	$0.628247\pi$
$912$	−10.9175	−0.361515
$913$	16.3368	0.540668
$914$	4.18468	0.138417
$915$	5.41927	0.179156
$916$	−48.7467	−1.61064
$917$	0	0
$918$	−17.4479	−0.575865
$919$	35.7748	1.18010	0.590051	−	0.807366i	$-0.299108\pi$
$919$	35.7748	1.18010	0.590051	+	0.807366i	$0.299108\pi$
$920$	9.13828	0.301280
$921$	−4.81579	−0.158686
$922$	5.06376	0.166766
$923$	8.50760	0.280031
$924$	0	0
$925$	3.95407	0.130009
$926$	−9.45536	−0.310722
$927$	70.2721	2.30804
$928$	−3.49532	−0.114739
$929$	52.7753	1.73150	0.865750	−	0.500477i	$-0.166842\pi$
$929$	52.7753	1.73150	0.865750	+	0.500477i	$0.166842\pi$
$930$	6.23506	0.204456
$931$	0	0
$932$	5.94561	0.194755
$933$	61.8390	2.02452
$934$	0.369800	0.0121002
$935$	12.1334	0.396803
$936$	−4.10171	−0.134069
$937$	−42.1245	−1.37615	−0.688073	−	0.725641i	$-0.741543\pi$
$937$	−42.1245	−1.37615	−0.688073	+	0.725641i	$0.741543\pi$
$938$	0	0
$939$	25.0321	0.816892
$940$	−13.1383	−0.428523
$941$	3.89829	0.127081	0.0635403	−	0.997979i	$-0.479761\pi$
$941$	3.89829	0.127081	0.0635403	+	0.997979i	$0.479761\pi$
$942$	2.06022	0.0671257
$943$	−27.6414	−0.900129
$944$	5.73636	0.186703
$945$	0	0
$946$	4.99063	0.162259
$947$	9.56691	0.310883	0.155441	−	0.987845i	$-0.450320\pi$
$947$	9.56691	0.310883	0.155441	+	0.987845i	$0.450320\pi$
$948$	50.4929	1.63993
$949$	−6.33984	−0.205800
$950$	0.341219	0.0110706
$951$	80.1156	2.59793
$952$	0	0
$953$	27.2070	0.881320	0.440660	−	0.897674i	$-0.354744\pi$
$953$	27.2070	0.881320	0.440660	+	0.897674i	$0.354744\pi$
$954$	10.8573	0.351517
$955$	14.9447	0.483599
$956$	−26.5303	−0.858053
$957$	−4.42864	−0.143158
$958$	−12.7699	−0.412575
$959$	0	0
$960$	−16.7511	−0.540640
$961$	16.6543	0.537237
$962$	0.765413	0.0246779
$963$	59.7418	1.92515
$964$	−35.1526	−1.13219
$965$	14.1476	0.455429
$966$	0	0
$967$	−16.8015	−0.540300	−0.270150	−	0.962818i	$-0.587073\pi$
$967$	−16.8015	−0.540300	−0.270150	+	0.962818i	$0.587073\pi$
$968$	10.5319	0.338507
$969$	−25.3274	−0.813633
$970$	−1.27163	−0.0408295
$971$	17.4465	0.559884	0.279942	−	0.960017i	$-0.409685\pi$
$971$	17.4465	0.559884	0.279942	+	0.960017i	$0.409685\pi$
$972$	−17.1383	−0.549710
$973$	0	0
$974$	−3.15701	−0.101157
$975$	−1.80642	−0.0578519
$976$	6.40006	0.204861
$977$	32.0513	1.02541	0.512706	−	0.858564i	$-0.328643\pi$
$977$	32.0513	1.02541	0.512706	+	0.858564i	$0.328643\pi$
$978$	−14.7427	−0.471418
$979$	−11.9081	−0.380586
$980$	0	0
$981$	−97.9407	−3.12701
$982$	−9.08694	−0.289976
$983$	16.5259	0.527094	0.263547	−	0.964646i	$-0.415108\pi$
$983$	16.5259	0.527094	0.263547	+	0.964646i	$0.415108\pi$
$984$	−12.9491	−0.412804
$985$	5.70471	0.181767
$986$	−2.47457	−0.0788064
$987$	0	0
$988$	−1.29883	−0.0413211
$989$	−79.1378	−2.51644
$990$	2.57628	0.0818796
$991$	−9.34920	−0.296987	−0.148494	−	0.988913i	$-0.547442\pi$
$991$	−9.34920	−0.296987	−0.148494	+	0.988913i	$0.547442\pi$
$992$	24.1289	0.766094
$993$	−49.1022	−1.55821
$994$	0	0
$995$	−22.1432	−0.701987
$996$	−59.1753	−1.87504
$997$	15.9956	0.506584	0.253292	−	0.967390i	$-0.418487\pi$
$997$	15.9956	0.506584	0.253292	+	0.967390i	$0.418487\pi$
$998$	6.83854	0.216470
$999$	−27.8796	−0.882070

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7105.2.a.o.1.2		3
7.6	odd	2		145.2.a.c.1.2	✓	3
21.20	even	2		1305.2.a.p.1.2		3
28.27	even	2		2320.2.a.n.1.1		3
35.13	even	4		725.2.b.e.349.3		6
35.27	even	4		725.2.b.e.349.4		6
35.34	odd	2		725.2.a.e.1.2		3
56.13	odd	2		9280.2.a.bj.1.1		3
56.27	even	2		9280.2.a.br.1.3		3
105.104	even	2		6525.2.a.be.1.2		3
203.202	odd	2		4205.2.a.f.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.a.c.1.2	✓	3	7.6	odd	2
725.2.a.e.1.2		3	35.34	odd	2
725.2.b.e.349.3		6	35.13	even	4
725.2.b.e.349.4		6	35.27	even	4
1305.2.a.p.1.2		3	21.20	even	2
2320.2.a.n.1.1		3	28.27	even	2
4205.2.a.f.1.2		3	203.202	odd	2
6525.2.a.be.1.2		3	105.104	even	2
7105.2.a.o.1.2		3	1.1	even	1	trivial
9280.2.a.bj.1.1		3	56.13	odd	2
9280.2.a.br.1.3		3	56.27	even	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:	\( N \)	\(=\)	\( 7105 = 5 \cdot 7^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7105.a (trivial)

\(f(q)\)	\(=\)	\(q+0.311108 q^{2} -2.90321 q^{3} -1.90321 q^{4} -1.00000 q^{5} -0.903212 q^{6} -1.21432 q^{8} +5.42864 q^{9} -0.311108 q^{10} -1.52543 q^{11} +5.52543 q^{12} +0.622216 q^{13} +2.90321 q^{15} +3.42864 q^{16} +7.95407 q^{17} +1.68889 q^{18} +1.09679 q^{19} +1.90321 q^{20} -0.474572 q^{22} +7.52543 q^{23} +3.52543 q^{24} +1.00000 q^{25} +0.193576 q^{26} -7.05086 q^{27} -1.00000 q^{29} +0.903212 q^{30} +6.90321 q^{31} +3.49532 q^{32} +4.42864 q^{33} +2.47457 q^{34} -10.3319 q^{36} +3.95407 q^{37} +0.341219 q^{38} -1.80642 q^{39} +1.21432 q^{40} -3.67307 q^{41} -10.5161 q^{43} +2.90321 q^{44} -5.42864 q^{45} +2.34122 q^{46} -6.90321 q^{47} -9.95407 q^{48} +0.311108 q^{50} -23.0923 q^{51} -1.18421 q^{52} +6.42864 q^{53} -2.19358 q^{54} +1.52543 q^{55} -3.18421 q^{57} -0.311108 q^{58} +1.67307 q^{59} -5.52543 q^{60} +1.86665 q^{61} +2.14764 q^{62} -5.76986 q^{64} -0.622216 q^{65} +1.37778 q^{66} +11.5254 q^{67} -15.1383 q^{68} -21.8479 q^{69} +13.6731 q^{71} -6.59210 q^{72} -10.1891 q^{73} +1.23014 q^{74} -2.90321 q^{75} -2.08742 q^{76} -0.561993 q^{78} +9.13828 q^{79} -3.42864 q^{80} +4.18421 q^{81} -1.14272 q^{82} -10.7096 q^{83} -7.95407 q^{85} -3.27163 q^{86} +2.90321 q^{87} +1.85236 q^{88} +7.80642 q^{89} -1.68889 q^{90} -14.3225 q^{92} -20.0415 q^{93} -2.14764 q^{94} -1.09679 q^{95} -10.1476 q^{96} +4.08742 q^{97} -8.28100 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} - 2 q^{3} + q^{4} - 3 q^{5} + 4 q^{6} + 3 q^{8} + 3 q^{9} - q^{10} + 2 q^{11} + 10 q^{12} + 2 q^{13} + 2 q^{15} - 3 q^{16} + 4 q^{17} + 5 q^{18} + 10 q^{19} - q^{20} - 8 q^{22} + 16 q^{23}+ \cdots - 18 q^{99}+O(q^{100}) \) 3 * q + q^2 - 2 * q^3 + q^4 - 3 * q^5 + 4 * q^6 + 3 * q^8 + 3 * q^9 - q^10 + 2 * q^11 + 10 * q^12 + 2 * q^13 + 2 * q^15 - 3 * q^16 + 4 * q^17 + 5 * q^18 + 10 * q^19 - q^20 - 8 * q^22 + 16 * q^23 + 4 * q^24 + 3 * q^25 + 14 * q^26 - 8 * q^27 - 3 * q^29 - 4 * q^30 + 14 * q^31 - 3 * q^32 + 14 * q^34 - 11 * q^36 - 8 * q^37 + 8 * q^38 + 8 * q^39 - 3 * q^40 + 2 * q^41 + 2 * q^43 + 2 * q^44 - 3 * q^45 + 14 * q^46 - 14 * q^47 - 10 * q^48 + q^50 - 16 * q^51 + 10 * q^52 + 6 * q^53 - 20 * q^54 - 2 * q^55 + 4 * q^57 - q^58 - 8 * q^59 - 10 * q^60 + 6 * q^61 - 11 * q^64 - 2 * q^65 + 4 * q^66 + 28 * q^67 - 12 * q^68 - 12 * q^69 + 28 * q^71 - 13 * q^72 + 16 * q^73 + 10 * q^74 - 2 * q^75 + 14 * q^76 + 12 * q^78 - 6 * q^79 + 3 * q^80 - q^81 - 30 * q^82 - 12 * q^83 - 4 * q^85 + 24 * q^86 + 2 * q^87 + 12 * q^88 + 10 * q^89 - 5 * q^90 + 4 * q^92 - 20 * q^93 - 10 * q^95 - 24 * q^96 - 8 * q^97 - 18 * q^99

Introduction

L-functions

Modular forms

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Database

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