LMFDB - Embedded newform 7105.2.a.s.1.3

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 = [7,3,-1,7,7] 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:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 3x^{6} - 6x^{5} + 21x^{4} + 3x^{3} - 31x^{2} + 14x - 1 $ x^7 - 3x^6 - 6x^5 + 21x^4 + 3x^3 - 31x^2 + 14x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1015)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.0885540$ 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)

$f(q)$	$=$	\(q+0.0885540 q^{2} -3.13167 q^{3} -1.99216 q^{4} +1.00000 q^{5} -0.277322 q^{6} -0.353521 q^{8} +6.80736 q^{9} +0.0885540 q^{10} +0.355768 q^{11} +6.23878 q^{12} -2.85435 q^{13} -3.13167 q^{15} +3.95301 q^{16} -2.88585 q^{17} +0.602819 q^{18} -1.92951 q^{19} -1.99216 q^{20} +0.0315046 q^{22} +5.38443 q^{23} +1.10711 q^{24} +1.00000 q^{25} -0.252764 q^{26} -11.9234 q^{27} +1.00000 q^{29} -0.277322 q^{30} +2.71940 q^{31} +1.05710 q^{32} -1.11415 q^{33} -0.255554 q^{34} -13.5613 q^{36} -1.04257 q^{37} -0.170866 q^{38} +8.93888 q^{39} -0.353521 q^{40} +1.42814 q^{41} +4.44378 q^{43} -0.708745 q^{44} +6.80736 q^{45} +0.476813 q^{46} +10.7107 q^{47} -12.3795 q^{48} +0.0885540 q^{50} +9.03754 q^{51} +5.68631 q^{52} +6.06871 q^{53} -1.05586 q^{54} +0.355768 q^{55} +6.04260 q^{57} +0.0885540 q^{58} -14.7010 q^{59} +6.23878 q^{60} -6.07233 q^{61} +0.240813 q^{62} -7.81241 q^{64} -2.85435 q^{65} -0.0986621 q^{66} -2.52730 q^{67} +5.74908 q^{68} -16.8623 q^{69} -14.7464 q^{71} -2.40655 q^{72} +7.19273 q^{73} -0.0923239 q^{74} -3.13167 q^{75} +3.84390 q^{76} +0.791573 q^{78} -5.80891 q^{79} +3.95301 q^{80} +16.9181 q^{81} +0.126468 q^{82} +13.6663 q^{83} -2.88585 q^{85} +0.393514 q^{86} -3.13167 q^{87} -0.125771 q^{88} -7.86371 q^{89} +0.602819 q^{90} -10.7266 q^{92} -8.51626 q^{93} +0.948478 q^{94} -1.92951 q^{95} -3.31048 q^{96} +2.82284 q^{97} +2.42184 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q + 3 q^{2} - q^{3} + 7 q^{4} + 7 q^{5} + 4 q^{6} + 6 q^{8} + 12 q^{9} + 3 q^{10} + 7 q^{11} + 3 q^{12} - 5 q^{13} - q^{15} + 7 q^{16} - 14 q^{17} + 29 q^{18} + 9 q^{19} + 7 q^{20} + 9 q^{22} + 12 q^{23}+ \cdots - 7 q^{99}+O(q^{100}) $ 7 * q + 3 * q^2 - q^3 + 7 * q^4 + 7 * q^5 + 4 * q^6 + 6 * q^8 + 12 * q^9 + 3 * q^10 + 7 * q^11 + 3 * q^12 - 5 * q^13 - q^15 + 7 * q^16 - 14 * q^17 + 29 * q^18 + 9 * q^19 + 7 * q^20 + 9 * q^22 + 12 * q^23 - 12 * q^24 + 7 * q^25 + 3 * q^26 - 4 * q^27 + 7 * q^29 + 4 * q^30 + 7 * q^31 + 20 * q^32 - 14 * q^33 - 17 * q^34 + 15 * q^36 + 23 * q^37 + q^38 - 5 * q^39 + 6 * q^40 - 13 * q^41 + 31 * q^43 + 6 * q^44 + 12 * q^45 + 5 * q^46 + 6 * q^47 - 12 * q^48 + 3 * q^50 - 10 * q^51 + 7 * q^52 + 23 * q^53 + 15 * q^54 + 7 * q^55 + 29 * q^57 + 3 * q^58 - 4 * q^59 + 3 * q^60 + 4 * q^61 + 5 * q^62 - 20 * q^64 - 5 * q^65 + 5 * q^66 + 13 * q^67 - 24 * q^68 + 29 * q^69 - 11 * q^71 + 53 * q^72 - 8 * q^73 + 50 * q^74 - q^75 + 43 * q^76 - 64 * q^78 + 7 * q^80 + 19 * q^81 - 32 * q^82 + 6 * q^83 - 14 * q^85 + 23 * q^86 - q^87 + 13 * q^88 + 5 * q^89 + 29 * q^90 + 11 * q^92 - 24 * q^93 + 44 * q^94 + 9 * q^95 - 13 * q^96 - 4 * q^97 - 7 * 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 $

$2$	0.0885540	0.0626171	0.0313085	−	0.999510i	$-0.490033\pi$
$2$	0.0885540	0.0626171	0.0313085	+	0.999510i	$0.490033\pi$
$3$	−3.13167	−1.80807	−0.904035	−	0.427458i	$-0.859409\pi$
$3$	−3.13167	−1.80807	−0.904035	+	0.427458i	$0.859409\pi$
$4$	−1.99216	−0.996079
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	−0.277322	−0.113216
$7$	0	0
$7$	0	0
$8$	−0.353521	−0.124989
$9$	6.80736	2.26912
$10$	0.0885540	0.0280032
$11$	0.355768	0.107268	0.0536340	−	0.998561i	$-0.482920\pi$
$11$	0.355768	0.107268	0.0536340	+	0.998561i	$0.482920\pi$
$12$	6.23878	1.80098
$13$	−2.85435	−0.791654	−0.395827	−	0.918325i	$-0.629542\pi$
$13$	−2.85435	−0.791654	−0.395827	+	0.918325i	$0.629542\pi$
$14$	0	0
$15$	−3.13167	−0.808594
$16$	3.95301	0.988253
$17$	−2.88585	−0.699922	−0.349961	−	0.936764i	$-0.613805\pi$
$17$	−2.88585	−0.699922	−0.349961	+	0.936764i	$0.613805\pi$
$18$	0.602819	0.142086
$19$	−1.92951	−0.442661	−0.221330	−	0.975199i	$-0.571040\pi$
$19$	−1.92951	−0.442661	−0.221330	+	0.975199i	$0.571040\pi$
$20$	−1.99216	−0.445460
$21$	0	0
$22$	0.0315046	0.00671681
$23$	5.38443	1.12273	0.561366	−	0.827568i	$-0.310276\pi$
$23$	5.38443	1.12273	0.561366	+	0.827568i	$0.310276\pi$
$24$	1.10711	0.225988
$25$	1.00000	0.200000
$26$	−0.252764	−0.0495711
$27$	−11.9234	−2.29466
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	−0.277322	−0.0506318
$31$	2.71940	0.488418	0.244209	−	0.969723i	$-0.421472\pi$
$31$	2.71940	0.488418	0.244209	+	0.969723i	$0.421472\pi$
$32$	1.05710	0.186870
$33$	−1.11415	−0.193948
$34$	−0.255554	−0.0438271
$35$	0	0
$36$	−13.5613	−2.26022
$37$	−1.04257	−0.171398	−0.0856990	−	0.996321i	$-0.527312\pi$
$37$	−1.04257	−0.171398	−0.0856990	+	0.996321i	$0.527312\pi$
$38$	−0.170866	−0.0277181
$39$	8.93888	1.43137
$40$	−0.353521	−0.0558966
$41$	1.42814	0.223038	0.111519	−	0.993762i	$-0.464428\pi$
$41$	1.42814	0.223038	0.111519	+	0.993762i	$0.464428\pi$
$42$	0	0
$43$	4.44378	0.677670	0.338835	−	0.940846i	$-0.389967\pi$
$43$	4.44378	0.677670	0.338835	+	0.940846i	$0.389967\pi$
$44$	−0.708745	−0.106847
$45$	6.80736	1.01478
$46$	0.476813	0.0703022
$47$	10.7107	1.56232	0.781161	−	0.624330i	$-0.214628\pi$
$47$	10.7107	1.56232	0.781161	+	0.624330i	$0.214628\pi$
$48$	−12.3795	−1.78683
$49$	0	0
$50$	0.0885540	0.0125234
$51$	9.03754	1.26551
$52$	5.68631	0.788550
$53$	6.06871	0.833601	0.416801	−	0.908998i	$-0.363151\pi$
$53$	6.06871	0.833601	0.416801	+	0.908998i	$0.363151\pi$
$54$	−1.05586	−0.143685
$55$	0.355768	0.0479717
$56$	0	0
$57$	6.04260	0.800362
$58$	0.0885540	0.0116277
$59$	−14.7010	−1.91391	−0.956954	−	0.290241i	$-0.906264\pi$
$59$	−14.7010	−1.91391	−0.956954	+	0.290241i	$0.906264\pi$
$60$	6.23878	0.805423
$61$	−6.07233	−0.777482	−0.388741	−	0.921347i	$-0.627090\pi$
$61$	−6.07233	−0.777482	−0.388741	+	0.921347i	$0.627090\pi$
$62$	0.240813	0.0305833
$63$	0	0
$64$	−7.81241	−0.976551
$65$	−2.85435	−0.354038
$66$	−0.0986621	−0.0121445
$67$	−2.52730	−0.308758	−0.154379	−	0.988012i	$-0.549338\pi$
$67$	−2.52730	−0.308758	−0.154379	+	0.988012i	$0.549338\pi$
$68$	5.74908	0.697178
$69$	−16.8623	−2.02998
$70$	0	0
$71$	−14.7464	−1.75007	−0.875037	−	0.484056i	$-0.839163\pi$
$71$	−14.7464	−1.75007	−0.875037	+	0.484056i	$0.839163\pi$
$72$	−2.40655	−0.283614
$73$	7.19273	0.841845	0.420923	−	0.907097i	$-0.361706\pi$
$73$	7.19273	0.841845	0.420923	+	0.907097i	$0.361706\pi$
$74$	−0.0923239	−0.0107324
$75$	−3.13167	−0.361614
$76$	3.84390	0.440925
$77$	0	0
$78$	0.791573	0.0896280
$79$	−5.80891	−0.653554	−0.326777	−	0.945102i	$-0.605963\pi$
$79$	−5.80891	−0.653554	−0.326777	+	0.945102i	$0.605963\pi$
$80$	3.95301	0.441960
$81$	16.9181	1.87978
$82$	0.126468	0.0139660
$83$	13.6663	1.50007	0.750035	−	0.661398i	$-0.230037\pi$
$83$	13.6663	1.50007	0.750035	+	0.661398i	$0.230037\pi$
$84$	0	0
$85$	−2.88585	−0.313015
$86$	0.393514	0.0424337
$87$	−3.13167	−0.335750
$88$	−0.125771	−0.0134073
$89$	−7.86371	−0.833552	−0.416776	−	0.909009i	$-0.636840\pi$
$89$	−7.86371	−0.833552	−0.416776	+	0.909009i	$0.636840\pi$
$90$	0.602819	0.0635427
$91$	0	0
$92$	−10.7266	−1.11833
$93$	−8.51626	−0.883095
$94$	0.948478	0.0978280
$95$	−1.92951	−0.197964
$96$	−3.31048	−0.337875
$97$	2.82284	0.286616	0.143308	−	0.989678i	$-0.454226\pi$
$97$	2.82284	0.286616	0.143308	+	0.989678i	$0.454226\pi$
$98$	0	0
$99$	2.42184	0.243404
$100$	−1.99216	−0.199216
$101$	−2.39839	−0.238649	−0.119324	−	0.992855i	$-0.538073\pi$
$101$	−2.39839	−0.238649	−0.119324	+	0.992855i	$0.538073\pi$
$102$	0.800310	0.0792425
$103$	13.3480	1.31521	0.657607	−	0.753361i	$-0.271569\pi$
$103$	13.3480	1.31521	0.657607	+	0.753361i	$0.271569\pi$
$104$	1.00907	0.0989478
$105$	0	0
$106$	0.537408	0.0521977
$107$	−4.38330	−0.423749	−0.211875	−	0.977297i	$-0.567957\pi$
$107$	−4.38330	−0.423749	−0.211875	+	0.977297i	$0.567957\pi$
$108$	23.7533	2.28566
$109$	0.453192	0.0434079	0.0217039	−	0.999764i	$-0.493091\pi$
$109$	0.453192	0.0434079	0.0217039	+	0.999764i	$0.493091\pi$
$110$	0.0315046	0.00300385
$111$	3.26499	0.309900
$112$	0	0
$113$	1.96903	0.185231	0.0926156	−	0.995702i	$-0.470477\pi$
$113$	1.96903	0.185231	0.0926156	+	0.995702i	$0.470477\pi$
$114$	0.535096	0.0501163
$115$	5.38443	0.502101
$116$	−1.99216	−0.184967
$117$	−19.4306	−1.79636
$118$	−1.30183	−0.119843
$119$	0	0
$120$	1.10711	0.101065
$121$	−10.8734	−0.988494
$122$	−0.537729	−0.0486837
$123$	−4.47247	−0.403269
$124$	−5.41747	−0.486503
$125$	1.00000	0.0894427
$126$	0	0
$127$	−3.23829	−0.287352	−0.143676	−	0.989625i	$-0.545892\pi$
$127$	−3.23829	−0.287352	−0.143676	+	0.989625i	$0.545892\pi$
$128$	−2.80601	−0.248019
$129$	−13.9164	−1.22527
$130$	−0.252764	−0.0221689
$131$	−2.39414	−0.209177	−0.104589	−	0.994516i	$-0.533353\pi$
$131$	−2.39414	−0.209177	−0.104589	+	0.994516i	$0.533353\pi$
$132$	2.21956	0.193188
$133$	0	0
$134$	−0.223802	−0.0193336
$135$	−11.9234	−1.02620
$136$	1.02021	0.0874824
$137$	−6.37997	−0.545078	−0.272539	−	0.962145i	$-0.587863\pi$
$137$	−6.37997	−0.545078	−0.272539	+	0.962145i	$0.587863\pi$
$138$	−1.49322	−0.127111
$139$	−12.4333	−1.05458	−0.527288	−	0.849687i	$-0.676791\pi$
$139$	−12.4333	−1.05458	−0.527288	+	0.849687i	$0.676791\pi$
$140$	0	0
$141$	−33.5425	−2.82479
$142$	−1.30585	−0.109585
$143$	−1.01548	−0.0849191
$144$	26.9096	2.24246
$145$	1.00000	0.0830455
$146$	0.636945	0.0527139
$147$	0	0
$148$	2.07697	0.170726
$149$	−20.6398	−1.69088	−0.845440	−	0.534070i	$-0.820662\pi$
$149$	−20.6398	−1.69088	−0.845440	+	0.534070i	$0.820662\pi$
$150$	−0.277322	−0.0226432
$151$	0.599882	0.0488177	0.0244089	−	0.999702i	$-0.492230\pi$
$151$	0.599882	0.0488177	0.0244089	+	0.999702i	$0.492230\pi$
$152$	0.682124	0.0553276
$153$	−19.6450	−1.58821
$154$	0	0
$155$	2.71940	0.218427
$156$	−17.8077	−1.42575
$157$	−13.7603	−1.09819	−0.549097	−	0.835759i	$-0.685028\pi$
$157$	−13.7603	−1.09819	−0.549097	+	0.835759i	$0.685028\pi$
$158$	−0.514402	−0.0409236
$159$	−19.0052	−1.50721
$160$	1.05710	0.0835709
$161$	0	0
$162$	1.49816	0.117707
$163$	1.08976	0.0853567	0.0426783	−	0.999089i	$-0.486411\pi$
$163$	1.08976	0.0853567	0.0426783	+	0.999089i	$0.486411\pi$
$164$	−2.84509	−0.222164
$165$	−1.11415	−0.0867362
$166$	1.21020	0.0939300
$167$	8.98266	0.695099	0.347550	−	0.937662i	$-0.387014\pi$
$167$	8.98266	0.695099	0.347550	+	0.937662i	$0.387014\pi$
$168$	0	0
$169$	−4.85269	−0.373284
$170$	−0.255554	−0.0196001
$171$	−13.1349	−1.00445
$172$	−8.85271	−0.675013
$173$	−15.9329	−1.21135	−0.605677	−	0.795710i	$-0.707098\pi$
$173$	−15.9329	−1.21135	−0.605677	+	0.795710i	$0.707098\pi$
$174$	−0.277322	−0.0210237
$175$	0	0
$176$	1.40635	0.106008
$177$	46.0387	3.46048
$178$	−0.696363	−0.0521946
$179$	16.6205	1.24227	0.621136	−	0.783703i	$-0.286671\pi$
$179$	16.6205	1.24227	0.621136	+	0.783703i	$0.286671\pi$
$180$	−13.5613	−1.01080
$181$	8.68989	0.645914	0.322957	−	0.946414i	$-0.395323\pi$
$181$	8.68989	0.645914	0.322957	+	0.946414i	$0.395323\pi$
$182$	0	0
$183$	19.0165	1.40574
$184$	−1.90351	−0.140329
$185$	−1.04257	−0.0766515
$186$	−0.754148	−0.0552968
$187$	−1.02669	−0.0750792
$188$	−21.3375	−1.55620
$189$	0	0
$190$	−0.170866	−0.0123959
$191$	21.9409	1.58759	0.793793	−	0.608188i	$-0.208103\pi$
$191$	21.9409	1.58759	0.793793	+	0.608188i	$0.208103\pi$
$192$	24.4659	1.76567
$193$	18.2580	1.31424	0.657119	−	0.753787i	$-0.271775\pi$
$193$	18.2580	1.31424	0.657119	+	0.753787i	$0.271775\pi$
$194$	0.249974	0.0179471
$195$	8.93888	0.640126
$196$	0	0
$197$	−11.2582	−0.802113	−0.401057	−	0.916053i	$-0.631357\pi$
$197$	−11.2582	−0.802113	−0.401057	+	0.916053i	$0.631357\pi$
$198$	0.214463	0.0152412
$199$	2.12307	0.150500	0.0752502	−	0.997165i	$-0.476024\pi$
$199$	2.12307	0.150500	0.0752502	+	0.997165i	$0.476024\pi$
$200$	−0.353521	−0.0249977
$201$	7.91466	0.558257
$202$	−0.212387	−0.0149435
$203$	0	0
$204$	−18.0042	−1.26055
$205$	1.42814	0.0997458
$206$	1.18202	0.0823549
$207$	36.6538	2.54761
$208$	−11.2833	−0.782354
$209$	−0.686458	−0.0474833
$210$	0	0
$211$	−8.95032	−0.616165	−0.308083	−	0.951360i	$-0.599687\pi$
$211$	−8.95032	−0.616165	−0.308083	+	0.951360i	$0.599687\pi$
$212$	−12.0898	−0.830333
$213$	46.1808	3.16426
$214$	−0.388158	−0.0265340
$215$	4.44378	0.303063
$216$	4.21517	0.286806
$217$	0	0
$218$	0.0401319	0.00271808
$219$	−22.5253	−1.52212
$220$	−0.708745	−0.0477836
$221$	8.23723	0.554096
$222$	0.289128	0.0194050
$223$	−4.62680	−0.309833	−0.154917	−	0.987928i	$-0.549511\pi$
$223$	−4.62680	−0.309833	−0.154917	+	0.987928i	$0.549511\pi$
$224$	0	0
$225$	6.80736	0.453824
$226$	0.174366	0.0115986
$227$	−22.3985	−1.48664	−0.743321	−	0.668935i	$-0.766750\pi$
$227$	−22.3985	−1.48664	−0.743321	+	0.668935i	$0.766750\pi$
$228$	−12.0378	−0.797224
$229$	29.1102	1.92365	0.961827	−	0.273660i	$-0.0882342\pi$
$229$	29.1102	1.92365	0.961827	+	0.273660i	$0.0882342\pi$
$230$	0.476813	0.0314401
$231$	0	0
$232$	−0.353521	−0.0232098
$233$	−17.4878	−1.14567	−0.572833	−	0.819672i	$-0.694156\pi$
$233$	−17.4878	−1.14567	−0.572833	+	0.819672i	$0.694156\pi$
$234$	−1.72065	−0.112483
$235$	10.7107	0.698691
$236$	29.2867	1.90640
$237$	18.1916	1.18167
$238$	0	0
$239$	−6.93435	−0.448546	−0.224273	−	0.974526i	$-0.572001\pi$
$239$	−6.93435	−0.448546	−0.224273	+	0.974526i	$0.572001\pi$
$240$	−12.3795	−0.799095
$241$	−27.8823	−1.79606	−0.898030	−	0.439935i	$-0.855002\pi$
$241$	−27.8823	−1.79606	−0.898030	+	0.439935i	$0.855002\pi$
$242$	−0.962885	−0.0618966
$243$	−17.2116	−1.10413
$244$	12.0970	0.774433
$245$	0	0
$246$	−0.396055	−0.0252515
$247$	5.50750	0.350434
$248$	−0.961365	−0.0610467
$249$	−42.7983	−2.71223
$250$	0.0885540	0.00560064
$251$	28.0329	1.76942	0.884711	−	0.466139i	$-0.154355\pi$
$251$	28.0329	1.76942	0.884711	+	0.466139i	$0.154355\pi$
$252$	0	0
$253$	1.91561	0.120433
$254$	−0.286764	−0.0179931
$255$	9.03754	0.565953
$256$	15.3763	0.961021
$257$	3.94593	0.246140	0.123070	−	0.992398i	$-0.460726\pi$
$257$	3.94593	0.246140	0.123070	+	0.992398i	$0.460726\pi$
$258$	−1.23236	−0.0767232
$259$	0	0
$260$	5.68631	0.352650
$261$	6.80736	0.421365
$262$	−0.212011	−0.0130981
$263$	14.7075	0.906903	0.453452	−	0.891281i	$-0.350192\pi$
$263$	14.7075	0.906903	0.453452	+	0.891281i	$0.350192\pi$
$264$	0.393875	0.0242413
$265$	6.06871	0.372798
$266$	0	0
$267$	24.6266	1.50712
$268$	5.03477	0.307548
$269$	−5.66631	−0.345481	−0.172741	−	0.984967i	$-0.555262\pi$
$269$	−5.66631	−0.345481	−0.172741	+	0.984967i	$0.555262\pi$
$270$	−1.05586	−0.0642578
$271$	26.1126	1.58623	0.793114	−	0.609073i	$-0.208458\pi$
$271$	26.1126	1.58623	0.793114	+	0.609073i	$0.208458\pi$
$272$	−11.4078	−0.691700
$273$	0	0
$274$	−0.564972	−0.0341312
$275$	0.355768	0.0214536
$276$	33.5923	2.02202
$277$	−23.3094	−1.40053	−0.700264	−	0.713884i	$-0.746934\pi$
$277$	−23.3094	−1.40053	−0.700264	+	0.713884i	$0.746934\pi$
$278$	−1.10101	−0.0660345
$279$	18.5119	1.10828
$280$	0	0
$281$	22.2891	1.32966	0.664828	−	0.746997i	$-0.268505\pi$
$281$	22.2891	1.32966	0.664828	+	0.746997i	$0.268505\pi$
$282$	−2.97032	−0.176880
$283$	7.59404	0.451419	0.225709	−	0.974195i	$-0.427530\pi$
$283$	7.59404	0.451419	0.225709	+	0.974195i	$0.427530\pi$
$284$	29.3771	1.74321
$285$	6.04260	0.357933
$286$	−0.0899252	−0.00531739
$287$	0	0
$288$	7.19604	0.424031
$289$	−8.67185	−0.510109
$290$	0.0885540	0.00520007
$291$	−8.84022	−0.518223
$292$	−14.3291	−0.838545
$293$	−23.3957	−1.36679	−0.683396	−	0.730048i	$-0.739498\pi$
$293$	−23.3957	−1.36679	−0.683396	+	0.730048i	$0.739498\pi$
$294$	0	0
$295$	−14.7010	−0.855925
$296$	0.368572	0.0214228
$297$	−4.24196	−0.246143
$298$	−1.82774	−0.105878
$299$	−15.3691	−0.888815
$300$	6.23878	0.360196
$301$	0	0
$302$	0.0531219	0.00305682
$303$	7.51096	0.431494
$304$	−7.62739	−0.437461
$305$	−6.07233	−0.347700
$306$	−1.73965	−0.0994489
$307$	6.44479	0.367824	0.183912	−	0.982943i	$-0.441124\pi$
$307$	6.44479	0.367824	0.183912	+	0.982943i	$0.441124\pi$
$308$	0	0
$309$	−41.8015	−2.37800
$310$	0.240813	0.0136773
$311$	24.5635	1.39287	0.696434	−	0.717621i	$-0.254769\pi$
$311$	24.5635	1.39287	0.696434	+	0.717621i	$0.254769\pi$
$312$	−3.16008	−0.178905
$313$	30.0993	1.70131	0.850657	−	0.525721i	$-0.176204\pi$
$313$	30.0993	1.70131	0.850657	+	0.525721i	$0.176204\pi$
$314$	−1.21853	−0.0687657
$315$	0	0
$316$	11.5723	0.650991
$317$	30.2492	1.69896	0.849482	−	0.527618i	$-0.176915\pi$
$317$	30.2492	1.69896	0.849482	+	0.527618i	$0.176915\pi$
$318$	−1.68299	−0.0943771
$319$	0.355768	0.0199192
$320$	−7.81241	−0.436727
$321$	13.7270	0.766169
$322$	0	0
$323$	5.56829	0.309828
$324$	−33.7035	−1.87241
$325$	−2.85435	−0.158331
$326$	0.0965026	0.00534479
$327$	−1.41925	−0.0784845
$328$	−0.504879	−0.0278773
$329$	0	0
$330$	−0.0986621	−0.00543117
$331$	32.2145	1.77067	0.885335	−	0.464955i	$-0.153929\pi$
$331$	32.2145	1.77067	0.885335	+	0.464955i	$0.153929\pi$
$332$	−27.2254	−1.49419
$333$	−7.09717	−0.388922
$334$	0.795450	0.0435251
$335$	−2.52730	−0.138081
$336$	0	0
$337$	−16.7888	−0.914546	−0.457273	−	0.889326i	$-0.651174\pi$
$337$	−16.7888	−0.914546	−0.457273	+	0.889326i	$0.651174\pi$
$338$	−0.429725	−0.0233740
$339$	−6.16637	−0.334911
$340$	5.74908	0.311787
$341$	0.967473	0.0523916
$342$	−1.16315	−0.0628958
$343$	0	0
$344$	−1.57097	−0.0847011
$345$	−16.8623	−0.907834
$346$	−1.41092	−0.0758515
$347$	7.52709	0.404075	0.202038	−	0.979378i	$-0.435244\pi$
$347$	7.52709	0.404075	0.202038	+	0.979378i	$0.435244\pi$
$348$	6.23878	0.334434
$349$	36.8662	1.97340	0.986702	−	0.162542i	$-0.0519693\pi$
$349$	36.8662	1.97340	0.986702	+	0.162542i	$0.0519693\pi$
$350$	0	0
$351$	34.0335	1.81658
$352$	0.376081	0.0200452
$353$	−3.10655	−0.165345	−0.0826725	−	0.996577i	$-0.526346\pi$
$353$	−3.10655	−0.165345	−0.0826725	+	0.996577i	$0.526346\pi$
$354$	4.07691	0.216685
$355$	−14.7464	−0.782657
$356$	15.6658	0.830284
$357$	0	0
$358$	1.47181	0.0777875
$359$	−0.949565	−0.0501161	−0.0250581	−	0.999686i	$-0.507977\pi$
$359$	−0.949565	−0.0501161	−0.0250581	+	0.999686i	$0.507977\pi$
$360$	−2.40655	−0.126836
$361$	−15.2770	−0.804052
$362$	0.769524	0.0404453
$363$	34.0520	1.78727
$364$	0	0
$365$	7.19273	0.376485
$366$	1.68399	0.0880235
$367$	23.7937	1.24202	0.621010	−	0.783802i	$-0.286723\pi$
$367$	23.7937	1.24202	0.621010	+	0.783802i	$0.286723\pi$
$368$	21.2847	1.10954
$369$	9.72188	0.506101
$370$	−0.0923239	−0.00479969
$371$	0	0
$372$	16.9657	0.879632
$373$	29.2563	1.51484	0.757418	−	0.652930i	$-0.226461\pi$
$373$	29.2563	1.51484	0.757418	+	0.652930i	$0.226461\pi$
$374$	−0.0909177	−0.00470124
$375$	−3.13167	−0.161719
$376$	−3.78647	−0.195272
$377$	−2.85435	−0.147006
$378$	0	0
$379$	−30.6820	−1.57603	−0.788014	−	0.615658i	$-0.788890\pi$
$379$	−30.6820	−1.57603	−0.788014	+	0.615658i	$0.788890\pi$
$380$	3.84390	0.197188
$381$	10.1413	0.519553
$382$	1.94295	0.0994100
$383$	1.71743	0.0877568	0.0438784	−	0.999037i	$-0.486029\pi$
$383$	1.71743	0.0877568	0.0438784	+	0.999037i	$0.486029\pi$
$384$	8.78751	0.448436
$385$	0	0
$386$	1.61682	0.0822938
$387$	30.2504	1.53771
$388$	−5.62355	−0.285493
$389$	−11.0439	−0.559947	−0.279973	−	0.960008i	$-0.590326\pi$
$389$	−11.0439	−0.559947	−0.279973	+	0.960008i	$0.590326\pi$
$390$	0.791573	0.0400829
$391$	−15.5387	−0.785825
$392$	0	0
$393$	7.49767	0.378207
$394$	−0.996958	−0.0502260
$395$	−5.80891	−0.292278
$396$	−4.82468	−0.242449
$397$	10.7207	0.538056	0.269028	−	0.963132i	$-0.413298\pi$
$397$	10.7207	0.538056	0.269028	+	0.963132i	$0.413298\pi$
$398$	0.188006	0.00942390
$399$	0	0
$400$	3.95301	0.197651
$401$	28.3632	1.41639	0.708195	−	0.706017i	$-0.249510\pi$
$401$	28.3632	1.41639	0.708195	+	0.706017i	$0.249510\pi$
$402$	0.700874	0.0349564
$403$	−7.76211	−0.386658
$404$	4.77797	0.237713
$405$	16.9181	0.840665
$406$	0	0
$407$	−0.370914	−0.0183855
$408$	−3.19496	−0.158174
$409$	38.8005	1.91856	0.959281	−	0.282452i	$-0.0911479\pi$
$409$	38.8005	1.91856	0.959281	+	0.282452i	$0.0911479\pi$
$410$	0.126468	0.00624579
$411$	19.9800	0.985539
$412$	−26.5913	−1.31006
$413$	0	0
$414$	3.24584	0.159524
$415$	13.6663	0.670852
$416$	−3.01732	−0.147937
$417$	38.9369	1.90675
$418$	−0.0607886	−0.00297327
$419$	14.7670	0.721416	0.360708	−	0.932679i	$-0.382535\pi$
$419$	14.7670	0.721416	0.360708	+	0.932679i	$0.382535\pi$
$420$	0	0
$421$	−4.87527	−0.237606	−0.118803	−	0.992918i	$-0.537906\pi$
$421$	−4.87527	−0.237606	−0.118803	+	0.992918i	$0.537906\pi$
$422$	−0.792586	−0.0385825
$423$	72.9118	3.54509
$424$	−2.14542	−0.104191
$425$	−2.88585	−0.139984
$426$	4.08949	0.198137
$427$	0	0
$428$	8.73222	0.422088
$429$	3.18016	0.153540
$430$	0.393514	0.0189769
$431$	5.25733	0.253237	0.126618	−	0.991952i	$-0.459588\pi$
$431$	5.25733	0.253237	0.126618	+	0.991952i	$0.459588\pi$
$432$	−47.1333	−2.26770
$433$	5.69676	0.273769	0.136884	−	0.990587i	$-0.456291\pi$
$433$	5.69676	0.273769	0.136884	+	0.990587i	$0.456291\pi$
$434$	0	0
$435$	−3.13167	−0.150152
$436$	−0.902829	−0.0432377
$437$	−10.3893	−0.496989
$438$	−1.99470	−0.0953105
$439$	−31.3226	−1.49495	−0.747473	−	0.664292i	$-0.768733\pi$
$439$	−31.3226	−1.49495	−0.747473	+	0.664292i	$0.768733\pi$
$440$	−0.125771	−0.00599592
$441$	0	0
$442$	0.729439	0.0346959
$443$	17.9811	0.854307	0.427154	−	0.904179i	$-0.359516\pi$
$443$	17.9811	0.854307	0.427154	+	0.904179i	$0.359516\pi$
$444$	−6.50438	−0.308684
$445$	−7.86371	−0.372776
$446$	−0.409721	−0.0194009
$447$	64.6371	3.05723
$448$	0	0
$449$	19.5462	0.922443	0.461222	−	0.887285i	$-0.347411\pi$
$449$	19.5462	0.922443	0.461222	+	0.887285i	$0.347411\pi$
$450$	0.602819	0.0284171
$451$	0.508087	0.0239249
$452$	−3.92263	−0.184505
$453$	−1.87863	−0.0882659
$454$	−1.98348	−0.0930891
$455$	0	0
$456$	−2.13619	−0.100036
$457$	30.3601	1.42019	0.710093	−	0.704108i	$-0.248653\pi$
$457$	30.3601	1.42019	0.710093	+	0.704108i	$0.248653\pi$
$458$	2.57782	0.120454
$459$	34.4092	1.60608
$460$	−10.7266	−0.500132
$461$	19.9077	0.927192	0.463596	−	0.886047i	$-0.346559\pi$
$461$	19.9077	0.927192	0.463596	+	0.886047i	$0.346559\pi$
$462$	0	0
$463$	27.2719	1.26743	0.633716	−	0.773566i	$-0.281529\pi$
$463$	27.2719	1.26743	0.633716	+	0.773566i	$0.281529\pi$
$464$	3.95301	0.183514
$465$	−8.51626	−0.394932
$466$	−1.54862	−0.0717383
$467$	−19.5504	−0.904685	−0.452342	−	0.891844i	$-0.649411\pi$
$467$	−19.5504	−0.904685	−0.452342	+	0.891844i	$0.649411\pi$
$468$	38.7088	1.78931
$469$	0	0
$470$	0.948478	0.0437500
$471$	43.0928	1.98561
$472$	5.19712	0.239217
$473$	1.58095	0.0726923
$474$	1.61094	0.0739928
$475$	−1.92951	−0.0885321
$476$	0	0
$477$	41.3119	1.89154
$478$	−0.614064	−0.0280866
$479$	−34.6951	−1.58526	−0.792629	−	0.609704i	$-0.791288\pi$
$479$	−34.6951	−1.58526	−0.792629	+	0.609704i	$0.791288\pi$
$480$	−3.31048	−0.151102
$481$	2.97587	0.135688
$482$	−2.46909	−0.112464
$483$	0	0
$484$	21.6616	0.984618
$485$	2.82284	0.128179
$486$	−1.52416	−0.0691371
$487$	−9.76149	−0.442335	−0.221168	−	0.975236i	$-0.570987\pi$
$487$	−9.76149	−0.442335	−0.221168	+	0.975236i	$0.570987\pi$
$488$	2.14670	0.0971764
$489$	−3.41277	−0.154331
$490$	0	0
$491$	−27.9310	−1.26051	−0.630254	−	0.776389i	$-0.717049\pi$
$491$	−27.9310	−1.26051	−0.630254	+	0.776389i	$0.717049\pi$
$492$	8.90987	0.401688
$493$	−2.88585	−0.129972
$494$	0.487711	0.0219432
$495$	2.42184	0.108854
$496$	10.7498	0.482681
$497$	0	0
$498$	−3.78996	−0.169832
$499$	2.25043	0.100743	0.0503715	−	0.998731i	$-0.483959\pi$
$499$	2.25043	0.100743	0.0503715	+	0.998731i	$0.483959\pi$
$500$	−1.99216	−0.0890920
$501$	−28.1307	−1.25679
$502$	2.48243	0.110796
$503$	−2.52367	−0.112525	−0.0562625	−	0.998416i	$-0.517918\pi$
$503$	−2.52367	−0.112525	−0.0562625	+	0.998416i	$0.517918\pi$
$504$	0	0
$505$	−2.39839	−0.106727
$506$	0.169635	0.00754118
$507$	15.1970	0.674924
$508$	6.45119	0.286225
$509$	−22.6098	−1.00216	−0.501081	−	0.865400i	$-0.667064\pi$
$509$	−22.6098	−1.00216	−0.501081	+	0.865400i	$0.667064\pi$
$510$	0.800310	0.0354383
$511$	0	0
$512$	6.97367	0.308195
$513$	23.0063	1.01576
$514$	0.349428	0.0154126
$515$	13.3480	0.588182
$516$	27.7238	1.22047
$517$	3.81053	0.167587
$518$	0	0
$519$	49.8965	2.19021
$520$	1.00907	0.0442508
$521$	−10.9555	−0.479971	−0.239986	−	0.970776i	$-0.577143\pi$
$521$	−10.9555	−0.479971	−0.239986	+	0.970776i	$0.577143\pi$
$522$	0.602819	0.0263847
$523$	−11.3334	−0.495576	−0.247788	−	0.968814i	$-0.579704\pi$
$523$	−11.3334	−0.495576	−0.247788	+	0.968814i	$0.579704\pi$
$524$	4.76951	0.208357
$525$	0	0
$526$	1.30241	0.0567877
$527$	−7.84778	−0.341855
$528$	−4.40423	−0.191670
$529$	5.99213	0.260527
$530$	0.537408	0.0233435
$531$	−100.075	−4.34288
$532$	0	0
$533$	−4.07642	−0.176569
$534$	2.18078	0.0943715
$535$	−4.38330	−0.189507
$536$	0.893453	0.0385913
$537$	−52.0498	−2.24612
$538$	−0.501774	−0.0216330
$539$	0	0
$540$	23.7533	1.02218
$541$	−17.8889	−0.769104	−0.384552	−	0.923103i	$-0.625644\pi$
$541$	−17.8889	−0.769104	−0.384552	+	0.923103i	$0.625644\pi$
$542$	2.31238	0.0993250
$543$	−27.2139	−1.16786
$544$	−3.05063	−0.130795
$545$	0.453192	0.0194126
$546$	0	0
$547$	14.9771	0.640374	0.320187	−	0.947354i	$-0.396254\pi$
$547$	14.9771	0.640374	0.320187	+	0.947354i	$0.396254\pi$
$548$	12.7099	0.542940
$549$	−41.3365	−1.76420
$550$	0.0315046	0.00134336
$551$	−1.92951	−0.0822000
$552$	5.96117	0.253724
$553$	0	0
$554$	−2.06414	−0.0876970
$555$	3.26499	0.138591
$556$	24.7690	1.05044
$557$	16.7249	0.708658	0.354329	−	0.935121i	$-0.384709\pi$
$557$	16.7249	0.708658	0.354329	+	0.935121i	$0.384709\pi$
$558$	1.63930	0.0693972
$559$	−12.6841	−0.536480
$560$	0	0
$561$	3.21526	0.135749
$562$	1.97379	0.0832592
$563$	−2.79378	−0.117744	−0.0588720	−	0.998266i	$-0.518750\pi$
$563$	−2.79378	−0.117744	−0.0588720	+	0.998266i	$0.518750\pi$
$564$	66.8220	2.81371
$565$	1.96903	0.0828379
$566$	0.672482	0.0282665
$567$	0	0
$568$	5.21316	0.218739
$569$	−14.3492	−0.601550	−0.300775	−	0.953695i	$-0.597245\pi$
$569$	−14.3492	−0.601550	−0.300775	+	0.953695i	$0.597245\pi$
$570$	0.535096	0.0224127
$571$	22.3583	0.935666	0.467833	−	0.883817i	$-0.345035\pi$
$571$	22.3583	0.935666	0.467833	+	0.883817i	$0.345035\pi$
$572$	2.02301	0.0845861
$573$	−68.7116	−2.87047
$574$	0	0
$575$	5.38443	0.224546
$576$	−53.1819	−2.21591
$577$	−5.91211	−0.246124	−0.123062	−	0.992399i	$-0.539271\pi$
$577$	−5.91211	−0.246124	−0.123062	+	0.992399i	$0.539271\pi$
$578$	−0.767927	−0.0319415
$579$	−57.1780	−2.37623
$580$	−1.99216	−0.0827199
$581$	0	0
$582$	−0.782836	−0.0324496
$583$	2.15905	0.0894187
$584$	−2.54278	−0.105221
$585$	−19.4306	−0.803355
$586$	−2.07178	−0.0855846
$587$	30.0285	1.23941	0.619705	−	0.784835i	$-0.287252\pi$
$587$	30.0285	1.23941	0.619705	+	0.784835i	$0.287252\pi$
$588$	0	0
$589$	−5.24711	−0.216204
$590$	−1.30183	−0.0535956
$591$	35.2570	1.45028
$592$	−4.12130	−0.169384
$593$	−6.75414	−0.277359	−0.138680	−	0.990337i	$-0.544286\pi$
$593$	−6.75414	−0.277359	−0.138680	+	0.990337i	$0.544286\pi$
$594$	−0.375642	−0.0154128
$595$	0	0
$596$	41.1178	1.68425
$597$	−6.64875	−0.272115
$598$	−1.36099	−0.0556550
$599$	−4.76656	−0.194757	−0.0973783	−	0.995247i	$-0.531046\pi$
$599$	−4.76656	−0.194757	−0.0973783	+	0.995247i	$0.531046\pi$
$600$	1.10711	0.0451977
$601$	−35.9579	−1.46675	−0.733377	−	0.679822i	$-0.762057\pi$
$601$	−35.9579	−1.46675	−0.733377	+	0.679822i	$0.762057\pi$
$602$	0	0
$603$	−17.2042	−0.700610
$604$	−1.19506	−0.0486263
$605$	−10.8734	−0.442068
$606$	0.665125	0.0270189
$607$	41.8876	1.70017	0.850083	−	0.526649i	$-0.176552\pi$
$607$	41.8876	1.70017	0.850083	+	0.526649i	$0.176552\pi$
$608$	−2.03968	−0.0827201
$609$	0	0
$610$	−0.537729	−0.0217720
$611$	−30.5722	−1.23682
$612$	39.1360	1.58198
$613$	−43.3618	−1.75137	−0.875683	−	0.482886i	$-0.839589\pi$
$613$	−43.3618	−1.75137	−0.875683	+	0.482886i	$0.839589\pi$
$614$	0.570712	0.0230320
$615$	−4.47247	−0.180347
$616$	0	0
$617$	15.2994	0.615932	0.307966	−	0.951397i	$-0.400352\pi$
$617$	15.2994	0.615932	0.307966	+	0.951397i	$0.400352\pi$
$618$	−3.70168	−0.148904
$619$	22.5748	0.907358	0.453679	−	0.891165i	$-0.350111\pi$
$619$	22.5748	0.907358	0.453679	+	0.891165i	$0.350111\pi$
$620$	−5.41747	−0.217571
$621$	−64.2007	−2.57629
$622$	2.17519	0.0872173
$623$	0	0
$624$	35.3355	1.41455
$625$	1.00000	0.0400000
$626$	2.66541	0.106531
$627$	2.14976	0.0858532
$628$	27.4127	1.09389
$629$	3.00871	0.119965
$630$	0	0
$631$	−29.5590	−1.17673	−0.588363	−	0.808597i	$-0.700228\pi$
$631$	−29.5590	−1.17673	−0.588363	+	0.808597i	$0.700228\pi$
$632$	2.05357	0.0816868
$633$	28.0294	1.11407
$634$	2.67869	0.106384
$635$	−3.23829	−0.128508
$636$	37.8614	1.50130
$637$	0	0
$638$	0.0315046	0.00124728
$639$	−100.384	−3.97113
$640$	−2.80601	−0.110917
$641$	−11.5530	−0.456315	−0.228158	−	0.973624i	$-0.573270\pi$
$641$	−11.5530	−0.456315	−0.228158	+	0.973624i	$0.573270\pi$
$642$	1.21558	0.0479753
$643$	24.0362	0.947896	0.473948	−	0.880553i	$-0.342828\pi$
$643$	24.0362	0.947896	0.473948	+	0.880553i	$0.342828\pi$
$644$	0	0
$645$	−13.9164	−0.547960
$646$	0.493094	0.0194005
$647$	−30.5386	−1.20059	−0.600297	−	0.799777i	$-0.704951\pi$
$647$	−30.5386	−1.20059	−0.600297	+	0.799777i	$0.704951\pi$
$648$	−5.98090	−0.234952
$649$	−5.23014	−0.205301
$650$	−0.252764	−0.00991421
$651$	0	0
$652$	−2.17098	−0.0850220
$653$	18.8063	0.735947	0.367974	−	0.929836i	$-0.380052\pi$
$653$	18.8063	0.735947	0.367974	+	0.929836i	$0.380052\pi$
$654$	−0.125680	−0.00491447
$655$	−2.39414	−0.0935469
$656$	5.64546	0.220418
$657$	48.9635	1.91025
$658$	0	0
$659$	27.1489	1.05757	0.528785	−	0.848756i	$-0.322648\pi$
$659$	27.1489	1.05757	0.528785	+	0.848756i	$0.322648\pi$
$660$	2.21956	0.0863961
$661$	10.8336	0.421377	0.210689	−	0.977553i	$-0.432429\pi$
$661$	10.8336	0.421377	0.210689	+	0.977553i	$0.432429\pi$
$662$	2.85272	0.110874
$663$	−25.7963	−1.00184
$664$	−4.83133	−0.187492
$665$	0	0
$666$	−0.628482	−0.0243532
$667$	5.38443	0.208486
$668$	−17.8949	−0.692374
$669$	14.4896	0.560201
$670$	−0.223802	−0.00864623
$671$	−2.16034	−0.0833989
$672$	0	0
$673$	40.9529	1.57862	0.789309	−	0.613996i	$-0.210439\pi$
$673$	40.9529	1.57862	0.789309	+	0.613996i	$0.210439\pi$
$674$	−1.48672	−0.0572662
$675$	−11.9234	−0.458932
$676$	9.66733	0.371821
$677$	9.01837	0.346604	0.173302	−	0.984869i	$-0.444556\pi$
$677$	9.01837	0.346604	0.173302	+	0.984869i	$0.444556\pi$
$678$	−0.546056	−0.0209712
$679$	0	0
$680$	1.02021	0.0391233
$681$	70.1447	2.68795
$682$	0.0856736	0.00328061
$683$	4.13735	0.158311	0.0791556	−	0.996862i	$-0.474778\pi$
$683$	4.13735	0.158311	0.0791556	+	0.996862i	$0.474778\pi$
$684$	26.1668	1.00051
$685$	−6.37997	−0.243766
$686$	0	0
$687$	−91.1634	−3.47810
$688$	17.5663	0.669709
$689$	−17.3222	−0.659924
$690$	−1.49322	−0.0568459
$691$	48.5291	1.84613	0.923067	−	0.384640i	$-0.125674\pi$
$691$	48.5291	1.84613	0.923067	+	0.384640i	$0.125674\pi$
$692$	31.7408	1.20661
$693$	0	0
$694$	0.666553	0.0253020
$695$	−12.4333	−0.471621
$696$	1.10711	0.0419650
$697$	−4.12141	−0.156110
$698$	3.26465	0.123569
$699$	54.7662	2.07145
$700$	0	0
$701$	−21.5662	−0.814546	−0.407273	−	0.913307i	$-0.633520\pi$
$701$	−21.5662	−0.814546	−0.407273	+	0.913307i	$0.633520\pi$
$702$	3.01380	0.113749
$703$	2.01166	0.0758711
$704$	−2.77940	−0.104753
$705$	−33.5425	−1.26328
$706$	−0.275097	−0.0103534
$707$	0	0
$708$	−91.7163	−3.44691
$709$	−21.4643	−0.806110	−0.403055	−	0.915176i	$-0.632052\pi$
$709$	−21.4643	−0.806110	−0.403055	+	0.915176i	$0.632052\pi$
$710$	−1.30585	−0.0490077
$711$	−39.5433	−1.48299
$712$	2.77999	0.104185
$713$	14.6424	0.548363
$714$	0	0
$715$	−1.01548	−0.0379770
$716$	−33.1106	−1.23740
$717$	21.7161	0.811002
$718$	−0.0840878	−0.00313813
$719$	44.2676	1.65090	0.825452	−	0.564473i	$-0.190921\pi$
$719$	44.2676	1.65090	0.825452	+	0.564473i	$0.190921\pi$
$720$	26.9096	1.00286
$721$	0	0
$722$	−1.35284	−0.0503474
$723$	87.3183	3.24740
$724$	−17.3116	−0.643382
$725$	1.00000	0.0371391
$726$	3.01544	0.111913
$727$	44.4337	1.64795	0.823977	−	0.566623i	$-0.191750\pi$
$727$	44.4337	1.64795	0.823977	+	0.566623i	$0.191750\pi$
$728$	0	0
$729$	3.14691	0.116552
$730$	0.636945	0.0235744
$731$	−12.8241	−0.474316
$732$	−37.8839	−1.40023
$733$	−44.3433	−1.63786	−0.818929	−	0.573894i	$-0.805432\pi$
$733$	−44.3433	−1.63786	−0.818929	+	0.573894i	$0.805432\pi$
$734$	2.10703	0.0777717
$735$	0	0
$736$	5.69187	0.209805
$737$	−0.899130	−0.0331199
$738$	0.860911	0.0316906
$739$	18.1893	0.669104	0.334552	−	0.942377i	$-0.391415\pi$
$739$	18.1893	0.669104	0.334552	+	0.942377i	$0.391415\pi$
$740$	2.07697	0.0763509
$741$	−17.2477	−0.633610
$742$	0	0
$743$	−39.1526	−1.43637	−0.718185	−	0.695852i	$-0.755027\pi$
$743$	−39.1526	−1.43637	−0.718185	+	0.695852i	$0.755027\pi$
$744$	3.01068	0.110377
$745$	−20.6398	−0.756185
$746$	2.59076	0.0948546
$747$	93.0313	3.40384
$748$	2.04533	0.0747849
$749$	0	0
$750$	−0.277322	−0.0101264
$751$	11.2040	0.408840	0.204420	−	0.978883i	$-0.434469\pi$
$751$	11.2040	0.408840	0.204420	+	0.978883i	$0.434469\pi$
$752$	42.3397	1.54397
$753$	−87.7899	−3.19924
$754$	−0.252764	−0.00920512
$755$	0.599882	0.0218319
$756$	0	0
$757$	1.30024	0.0472580	0.0236290	−	0.999721i	$-0.492478\pi$
$757$	1.30024	0.0472580	0.0236290	+	0.999721i	$0.492478\pi$
$758$	−2.71701	−0.0986862
$759$	−5.99905	−0.217752
$760$	0.682124	0.0247432
$761$	−32.6918	−1.18508	−0.592538	−	0.805542i	$-0.701874\pi$
$761$	−32.6918	−1.18508	−0.592538	+	0.805542i	$0.701874\pi$
$762$	0.898049	0.0325329
$763$	0	0
$764$	−43.7097	−1.58136
$765$	−19.6450	−0.710268
$766$	0.152086	0.00549508
$767$	41.9618	1.51515
$768$	−48.1536	−1.73759
$769$	2.18069	0.0786377	0.0393188	−	0.999227i	$-0.487481\pi$
$769$	2.18069	0.0786377	0.0393188	+	0.999227i	$0.487481\pi$
$770$	0	0
$771$	−12.3573	−0.445039
$772$	−36.3728	−1.30908
$773$	−54.5645	−1.96255	−0.981274	−	0.192618i	$-0.938302\pi$
$773$	−54.5645	−1.96255	−0.981274	+	0.192618i	$0.938302\pi$
$774$	2.67879	0.0962872
$775$	2.71940	0.0976836
$776$	−0.997936	−0.0358238
$777$	0	0
$778$	−0.977979	−0.0350622
$779$	−2.75562	−0.0987303
$780$	−17.8077	−0.637617
$781$	−5.24629	−0.187727
$782$	−1.37601	−0.0492061
$783$	−11.9234	−0.426107
$784$	0	0
$785$	−13.7603	−0.491127
$786$	0.663948	0.0236823
$787$	−12.4312	−0.443123	−0.221561	−	0.975146i	$-0.571115\pi$
$787$	−12.4312	−0.443123	−0.221561	+	0.975146i	$0.571115\pi$
$788$	22.4281	0.798968
$789$	−46.0591	−1.63975
$790$	−0.514402	−0.0183016
$791$	0	0
$792$	−0.856171	−0.0304227
$793$	17.3325	0.615496
$794$	0.949359	0.0336915
$795$	−19.0052	−0.674045
$796$	−4.22949	−0.149910
$797$	−23.6889	−0.839103	−0.419552	−	0.907732i	$-0.637813\pi$
$797$	−23.6889	−0.839103	−0.419552	+	0.907732i	$0.637813\pi$
$798$	0	0
$799$	−30.9096	−1.09350
$800$	1.05710	0.0373740
$801$	−53.5311	−1.89143
$802$	2.51167	0.0886902
$803$	2.55894	0.0903030
$804$	−15.7672	−0.556068
$805$	0	0
$806$	−0.687365	−0.0242114
$807$	17.7450	0.624654
$808$	0.847882	0.0298284
$809$	4.91542	0.172817	0.0864086	−	0.996260i	$-0.472461\pi$
$809$	4.91542	0.172817	0.0864086	+	0.996260i	$0.472461\pi$
$810$	1.49816	0.0526400
$811$	−11.8694	−0.416792	−0.208396	−	0.978045i	$-0.566824\pi$
$811$	−11.8694	−0.416792	−0.208396	+	0.978045i	$0.566824\pi$
$812$	0	0
$813$	−81.7761	−2.86801
$814$	−0.0328459	−0.00115125
$815$	1.08976	0.0381727
$816$	35.7255	1.25064
$817$	−8.57433	−0.299978
$818$	3.43594	0.120135
$819$	0	0
$820$	−2.84509	−0.0993547
$821$	28.9999	1.01210	0.506051	−	0.862503i	$-0.331105\pi$
$821$	28.9999	1.01210	0.506051	+	0.862503i	$0.331105\pi$
$822$	1.76930	0.0617116
$823$	54.0920	1.88553	0.942764	−	0.333460i	$-0.108216\pi$
$823$	54.0920	1.88553	0.942764	+	0.333460i	$0.108216\pi$
$824$	−4.71879	−0.164387
$825$	−1.11415	−0.0387896
$826$	0	0
$827$	23.9321	0.832202	0.416101	−	0.909318i	$-0.363396\pi$
$827$	23.9321	0.832202	0.416101	+	0.909318i	$0.363396\pi$
$828$	−73.0201	−2.53762
$829$	32.1573	1.11687	0.558434	−	0.829549i	$-0.311402\pi$
$829$	32.1573	1.11687	0.558434	+	0.829549i	$0.311402\pi$
$830$	1.21020	0.0420068
$831$	72.9975	2.53225
$832$	22.2993	0.773091
$833$	0	0
$834$	3.44801	0.119395
$835$	8.98266	0.310858
$836$	1.36753	0.0472971
$837$	−32.4244	−1.12075
$838$	1.30768	0.0451730
$839$	−5.17582	−0.178689	−0.0893445	−	0.996001i	$-0.528477\pi$
$839$	−5.17582	−0.178689	−0.0893445	+	0.996001i	$0.528477\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−0.431724	−0.0148782
$843$	−69.8021	−2.40411
$844$	17.8304	0.613749
$845$	−4.85269	−0.166938
$846$	6.45663	0.221984
$847$	0	0
$848$	23.9897	0.823809
$849$	−23.7820	−0.816197
$850$	−0.255554	−0.00876542
$851$	−5.61366	−0.192434
$852$	−91.9995	−3.15185
$853$	−29.7698	−1.01930	−0.509650	−	0.860382i	$-0.670225\pi$
$853$	−29.7698	−1.01930	−0.509650	+	0.860382i	$0.670225\pi$
$854$	0	0
$855$	−13.1349	−0.449204
$856$	1.54959	0.0529639
$857$	0.804388	0.0274774	0.0137387	−	0.999906i	$-0.495627\pi$
$857$	0.804388	0.0274774	0.0137387	+	0.999906i	$0.495627\pi$
$858$	0.281616	0.00961421
$859$	40.5312	1.38291	0.691454	−	0.722420i	$-0.256970\pi$
$859$	40.5312	1.38291	0.691454	+	0.722420i	$0.256970\pi$
$860$	−8.85271	−0.301875
$861$	0	0
$862$	0.465557	0.0158569
$863$	−2.69192	−0.0916341	−0.0458171	−	0.998950i	$-0.514589\pi$
$863$	−2.69192	−0.0916341	−0.0458171	+	0.998950i	$0.514589\pi$
$864$	−12.6042	−0.428803
$865$	−15.9329	−0.541734
$866$	0.504471	0.0171426
$867$	27.1574	0.922313
$868$	0	0
$869$	−2.06662	−0.0701054
$870$	−0.277322	−0.00940209
$871$	7.21378	0.244430
$872$	−0.160213	−0.00542549
$873$	19.2161	0.650367
$874$	−0.920017	−0.0311200
$875$	0	0
$876$	44.8739	1.51615
$877$	31.3922	1.06004	0.530019	−	0.847986i	$-0.322185\pi$
$877$	31.3922	1.06004	0.530019	+	0.847986i	$0.322185\pi$
$878$	−2.77374	−0.0936092
$879$	73.2677	2.47126
$880$	1.40635	0.0474082
$881$	29.2653	0.985974	0.492987	−	0.870037i	$-0.335905\pi$
$881$	29.2653	0.985974	0.492987	+	0.870037i	$0.335905\pi$
$882$	0	0
$883$	−29.5843	−0.995589	−0.497795	−	0.867295i	$-0.665857\pi$
$883$	−29.5843	−0.995589	−0.497795	+	0.867295i	$0.665857\pi$
$884$	−16.4099	−0.551924
$885$	46.0387	1.54757
$886$	1.59230	0.0534942
$887$	34.9038	1.17195	0.585977	−	0.810328i	$-0.300711\pi$
$887$	34.9038	1.17195	0.585977	+	0.810328i	$0.300711\pi$
$888$	−1.15425	−0.0387339
$889$	0	0
$890$	−0.696363	−0.0233421
$891$	6.01890	0.201641
$892$	9.21731	0.308618
$893$	−20.6665	−0.691578
$894$	5.72387	0.191435
$895$	16.6205	0.555561
$896$	0	0
$897$	48.1308	1.60704
$898$	1.73090	0.0577607
$899$	2.71940	0.0906970
$900$	−13.5613	−0.452045
$901$	−17.5134	−0.583456
$902$	0.0449931	0.00149811
$903$	0	0
$904$	−0.696096	−0.0231518
$905$	8.68989	0.288862
$906$	−0.166360	−0.00552695
$907$	15.8393	0.525936	0.262968	−	0.964805i	$-0.415299\pi$
$907$	15.8393	0.525936	0.262968	+	0.964805i	$0.415299\pi$
$908$	44.6214	1.48081
$909$	−16.3267	−0.541522
$910$	0	0
$911$	27.8073	0.921297	0.460648	−	0.887583i	$-0.347617\pi$
$911$	27.8073	0.921297	0.460648	+	0.887583i	$0.347617\pi$
$912$	23.8865	0.790960
$913$	4.86202	0.160909
$914$	2.68851	0.0889279
$915$	19.0165	0.628667
$916$	−57.9920	−1.91611
$917$	0	0
$918$	3.04707	0.100568
$919$	−17.2805	−0.570031	−0.285016	−	0.958523i	$-0.591999\pi$
$919$	−17.2805	−0.570031	−0.285016	+	0.958523i	$0.591999\pi$
$920$	−1.90351	−0.0627570
$921$	−20.1830	−0.665051
$922$	1.76290	0.0580581
$923$	42.0913	1.38545
$924$	0	0
$925$	−1.04257	−0.0342796
$926$	2.41503	0.0793629
$927$	90.8645	2.98438
$928$	1.05710	0.0347009
$929$	13.9081	0.456309	0.228155	−	0.973625i	$-0.426731\pi$
$929$	13.9081	0.456309	0.228155	+	0.973625i	$0.426731\pi$
$930$	−0.754148	−0.0247295
$931$	0	0
$932$	34.8386	1.14117
$933$	−76.9248	−2.51840
$934$	−1.73127	−0.0566487
$935$	−1.02669	−0.0335765
$936$	6.86912	0.224524
$937$	11.7921	0.385231	0.192615	−	0.981274i	$-0.438303\pi$
$937$	11.7921	0.385231	0.192615	+	0.981274i	$0.438303\pi$
$938$	0	0
$939$	−94.2612	−3.07610
$940$	−21.3375	−0.695952
$941$	16.8948	0.550755	0.275378	−	0.961336i	$-0.411197\pi$
$941$	16.8948	0.550755	0.275378	+	0.961336i	$0.411197\pi$
$942$	3.81604	0.124333
$943$	7.68974	0.250412
$944$	−58.1132	−1.89142
$945$	0	0
$946$	0.140000	0.00455178
$947$	−59.2888	−1.92663	−0.963313	−	0.268381i	$-0.913511\pi$
$947$	−59.2888	−1.92663	−0.963313	+	0.268381i	$0.913511\pi$
$948$	−36.2405	−1.17704
$949$	−20.5306	−0.666450
$950$	−0.170866	−0.00554363
$951$	−94.7305	−3.07185
$952$	0	0
$953$	55.4487	1.79616	0.898080	−	0.439831i	$-0.144962\pi$
$953$	55.4487	1.79616	0.898080	+	0.439831i	$0.144962\pi$
$954$	3.65833	0.118443
$955$	21.9409	0.709990
$956$	13.8143	0.446787
$957$	−1.11415	−0.0360153
$958$	−3.07238	−0.0992642
$959$	0	0
$960$	24.4659	0.789633
$961$	−23.6049	−0.761448
$962$	0.263525	0.00849638
$963$	−29.8387	−0.961538
$964$	55.5460	1.78902
$965$	18.2580	0.587745
$966$	0	0
$967$	−49.3255	−1.58620	−0.793100	−	0.609091i	$-0.791534\pi$
$967$	−49.3255	−1.58620	−0.793100	+	0.609091i	$0.791534\pi$
$968$	3.84399	0.123551
$969$	−17.4381	−0.560191
$970$	0.249974	0.00802618
$971$	−5.87872	−0.188657	−0.0943285	−	0.995541i	$-0.530070\pi$
$971$	−5.87872	−0.188657	−0.0943285	+	0.995541i	$0.530070\pi$
$972$	34.2883	1.09980
$973$	0	0
$974$	−0.864418	−0.0276978
$975$	8.93888	0.286273
$976$	−24.0040	−0.768348
$977$	−16.1043	−0.515222	−0.257611	−	0.966249i	$-0.582935\pi$
$977$	−16.1043	−0.515222	−0.257611	+	0.966249i	$0.582935\pi$
$978$	−0.302214	−0.00966375
$979$	−2.79765	−0.0894134
$980$	0	0
$981$	3.08504	0.0984977
$982$	−2.47340	−0.0789294
$983$	20.3035	0.647581	0.323790	−	0.946129i	$-0.395043\pi$
$983$	20.3035	0.647581	0.323790	+	0.946129i	$0.395043\pi$
$984$	1.58111	0.0504041
$985$	−11.2582	−0.358716
$986$	−0.255554	−0.00813849
$987$	0	0
$988$	−10.9718	−0.349060
$989$	23.9272	0.760842
$990$	0.214463	0.00681609
$991$	46.6807	1.48286	0.741431	−	0.671030i	$-0.234148\pi$
$991$	46.6807	1.48286	0.741431	+	0.671030i	$0.234148\pi$
$992$	2.87467	0.0912708
$993$	−100.885	−3.20149
$994$	0	0
$995$	2.12307	0.0673058
$996$	85.2610	2.70160
$997$	−28.9351	−0.916384	−0.458192	−	0.888853i	$-0.651503\pi$
$997$	−28.9351	−0.916384	−0.458192	+	0.888853i	$0.651503\pi$
$998$	0.199284	0.00630824
$999$	12.4310	0.393300

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7105.2.a.s.1.3		7
7.6	odd	2		1015.2.a.k.1.3	✓	7
21.20	even	2		9135.2.a.be.1.5		7
35.34	odd	2		5075.2.a.y.1.5		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1015.2.a.k.1.3	✓	7	7.6	odd	2
5075.2.a.y.1.5		7	35.34	odd	2
7105.2.a.s.1.3		7	1.1	even	1	trivial
9135.2.a.be.1.5		7	21.20	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)

\(f(q)\)	\(=\)	\(q+0.0885540 q^{2} -3.13167 q^{3} -1.99216 q^{4} +1.00000 q^{5} -0.277322 q^{6} -0.353521 q^{8} +6.80736 q^{9} +0.0885540 q^{10} +0.355768 q^{11} +6.23878 q^{12} -2.85435 q^{13} -3.13167 q^{15} +3.95301 q^{16} -2.88585 q^{17} +0.602819 q^{18} -1.92951 q^{19} -1.99216 q^{20} +0.0315046 q^{22} +5.38443 q^{23} +1.10711 q^{24} +1.00000 q^{25} -0.252764 q^{26} -11.9234 q^{27} +1.00000 q^{29} -0.277322 q^{30} +2.71940 q^{31} +1.05710 q^{32} -1.11415 q^{33} -0.255554 q^{34} -13.5613 q^{36} -1.04257 q^{37} -0.170866 q^{38} +8.93888 q^{39} -0.353521 q^{40} +1.42814 q^{41} +4.44378 q^{43} -0.708745 q^{44} +6.80736 q^{45} +0.476813 q^{46} +10.7107 q^{47} -12.3795 q^{48} +0.0885540 q^{50} +9.03754 q^{51} +5.68631 q^{52} +6.06871 q^{53} -1.05586 q^{54} +0.355768 q^{55} +6.04260 q^{57} +0.0885540 q^{58} -14.7010 q^{59} +6.23878 q^{60} -6.07233 q^{61} +0.240813 q^{62} -7.81241 q^{64} -2.85435 q^{65} -0.0986621 q^{66} -2.52730 q^{67} +5.74908 q^{68} -16.8623 q^{69} -14.7464 q^{71} -2.40655 q^{72} +7.19273 q^{73} -0.0923239 q^{74} -3.13167 q^{75} +3.84390 q^{76} +0.791573 q^{78} -5.80891 q^{79} +3.95301 q^{80} +16.9181 q^{81} +0.126468 q^{82} +13.6663 q^{83} -2.88585 q^{85} +0.393514 q^{86} -3.13167 q^{87} -0.125771 q^{88} -7.86371 q^{89} +0.602819 q^{90} -10.7266 q^{92} -8.51626 q^{93} +0.948478 q^{94} -1.92951 q^{95} -3.31048 q^{96} +2.82284 q^{97} +2.42184 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q + 3 q^{2} - q^{3} + 7 q^{4} + 7 q^{5} + 4 q^{6} + 6 q^{8} + 12 q^{9} + 3 q^{10} + 7 q^{11} + 3 q^{12} - 5 q^{13} - q^{15} + 7 q^{16} - 14 q^{17} + 29 q^{18} + 9 q^{19} + 7 q^{20} + 9 q^{22} + 12 q^{23}+ \cdots - 7 q^{99}+O(q^{100}) \) 7 * q + 3 * q^2 - q^3 + 7 * q^4 + 7 * q^5 + 4 * q^6 + 6 * q^8 + 12 * q^9 + 3 * q^10 + 7 * q^11 + 3 * q^12 - 5 * q^13 - q^15 + 7 * q^16 - 14 * q^17 + 29 * q^18 + 9 * q^19 + 7 * q^20 + 9 * q^22 + 12 * q^23 - 12 * q^24 + 7 * q^25 + 3 * q^26 - 4 * q^27 + 7 * q^29 + 4 * q^30 + 7 * q^31 + 20 * q^32 - 14 * q^33 - 17 * q^34 + 15 * q^36 + 23 * q^37 + q^38 - 5 * q^39 + 6 * q^40 - 13 * q^41 + 31 * q^43 + 6 * q^44 + 12 * q^45 + 5 * q^46 + 6 * q^47 - 12 * q^48 + 3 * q^50 - 10 * q^51 + 7 * q^52 + 23 * q^53 + 15 * q^54 + 7 * q^55 + 29 * q^57 + 3 * q^58 - 4 * q^59 + 3 * q^60 + 4 * q^61 + 5 * q^62 - 20 * q^64 - 5 * q^65 + 5 * q^66 + 13 * q^67 - 24 * q^68 + 29 * q^69 - 11 * q^71 + 53 * q^72 - 8 * q^73 + 50 * q^74 - q^75 + 43 * q^76 - 64 * q^78 + 7 * q^80 + 19 * q^81 - 32 * q^82 + 6 * q^83 - 14 * q^85 + 23 * q^86 - q^87 + 13 * q^88 + 5 * q^89 + 29 * q^90 + 11 * q^92 - 24 * q^93 + 44 * q^94 + 9 * q^95 - 13 * q^96 - 4 * q^97 - 7 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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