LMFDB - Embedded newform 7105.2.a.t.1.7

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 = [8,1,-4,5,-8] 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:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 10x^{6} + 11x^{5} + 25x^{4} - 25x^{3} - 16x^{2} + 9x + 4 $ x^8 - x^7 - 10x^6 + 11x^5 + 25x^4 - 25x^3 - 16x^2 + 9x + 4
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			1.7
Root			$1.90142$ 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)

$f(q)$	$=$	\(q+1.90142 q^{2} -2.54699 q^{3} +1.61538 q^{4} -1.00000 q^{5} -4.84289 q^{6} -0.731315 q^{8} +3.48717 q^{9} -1.90142 q^{10} +0.728522 q^{11} -4.11437 q^{12} -3.13413 q^{13} +2.54699 q^{15} -4.62130 q^{16} +2.84289 q^{17} +6.63056 q^{18} +5.77754 q^{19} -1.61538 q^{20} +1.38522 q^{22} -4.48221 q^{23} +1.86265 q^{24} +1.00000 q^{25} -5.95929 q^{26} -1.24082 q^{27} +1.00000 q^{29} +4.84289 q^{30} +5.44093 q^{31} -7.32439 q^{32} -1.85554 q^{33} +5.40552 q^{34} +5.63312 q^{36} +2.63378 q^{37} +10.9855 q^{38} +7.98261 q^{39} +0.731315 q^{40} -3.86096 q^{41} +6.24486 q^{43} +1.17684 q^{44} -3.48717 q^{45} -8.52254 q^{46} +1.56179 q^{47} +11.7704 q^{48} +1.90142 q^{50} -7.24083 q^{51} -5.06283 q^{52} +9.73567 q^{53} -2.35931 q^{54} -0.728522 q^{55} -14.7153 q^{57} +1.90142 q^{58} -6.37795 q^{59} +4.11437 q^{60} -13.1438 q^{61} +10.3455 q^{62} -4.68411 q^{64} +3.13413 q^{65} -3.52816 q^{66} -7.17801 q^{67} +4.59236 q^{68} +11.4162 q^{69} -0.416727 q^{71} -2.55022 q^{72} +6.24986 q^{73} +5.00792 q^{74} -2.54699 q^{75} +9.33294 q^{76} +15.1783 q^{78} -15.5074 q^{79} +4.62130 q^{80} -7.30115 q^{81} -7.34129 q^{82} +3.86824 q^{83} -2.84289 q^{85} +11.8741 q^{86} -2.54699 q^{87} -0.532780 q^{88} +3.83025 q^{89} -6.63056 q^{90} -7.24049 q^{92} -13.8580 q^{93} +2.96962 q^{94} -5.77754 q^{95} +18.6552 q^{96} +3.75306 q^{97} +2.54048 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + q^{2} - 4 q^{3} + 5 q^{4} - 8 q^{5} + 6 q^{6} - 6 q^{8} + 14 q^{9} - q^{10} + q^{11} + 7 q^{12} + q^{13} + 4 q^{15} + 3 q^{16} - 22 q^{17} + 13 q^{18} - 8 q^{19} - 5 q^{20} - 3 q^{22} + 9 q^{23}+ \cdots - 63 q^{99}+O(q^{100}) $ 8 * q + q^2 - 4 * q^3 + 5 * q^4 - 8 * q^5 + 6 * q^6 - 6 * q^8 + 14 * q^9 - q^10 + q^11 + 7 * q^12 + q^13 + 4 * q^15 + 3 * q^16 - 22 * q^17 + 13 * q^18 - 8 * q^19 - 5 * q^20 - 3 * q^22 + 9 * q^23 - 16 * q^24 + 8 * q^25 - 27 * q^26 - q^27 + 8 * q^29 - 6 * q^30 - 3 * q^31 - 26 * q^32 - 14 * q^33 - q^34 - 13 * q^36 + 9 * q^37 + 5 * q^38 + 3 * q^39 + 6 * q^40 - 24 * q^41 + 7 * q^43 + 10 * q^44 - 14 * q^45 - 27 * q^46 - 21 * q^47 + 8 * q^48 + q^50 - 8 * q^51 + 7 * q^52 - 2 * q^53 + 33 * q^54 - q^55 - 10 * q^57 + q^58 - 18 * q^59 - 7 * q^60 - 26 * q^61 - 15 * q^62 + 4 * q^64 - q^65 - 55 * q^66 - 8 * q^67 - 6 * q^68 + 2 * q^69 - 12 * q^71 + 9 * q^72 + q^73 + 14 * q^74 - 4 * q^75 - 15 * q^76 + 8 * q^78 - 14 * q^79 - 3 * q^80 + 40 * q^81 + 42 * q^82 + 10 * q^83 + 22 * q^85 + 19 * q^86 - 4 * q^87 - 29 * q^88 + 4 * q^89 - 13 * q^90 + 47 * q^92 - 24 * q^93 + 30 * q^94 + 8 * q^95 + q^96 + 11 * q^97 - 63 * 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 $

$2$	1.90142	1.34450	0.672252	−	0.740322i	$-0.265327\pi$
$2$	1.90142	1.34450	0.672252	+	0.740322i	$0.265327\pi$
$3$	−2.54699	−1.47051	−0.735253	−	0.677792i	$-0.762937\pi$
$3$	−2.54699	−1.47051	−0.735253	+	0.677792i	$0.762937\pi$
$4$	1.61538	0.807692
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−4.84289	−1.97710
$7$	0	0
$7$	0	0
$8$	−0.731315	−0.258559
$9$	3.48717	1.16239
$10$	−1.90142	−0.601281
$11$	0.728522	0.219658	0.109829	−	0.993951i	$-0.464970\pi$
$11$	0.728522	0.219658	0.109829	+	0.993951i	$0.464970\pi$
$12$	−4.11437	−1.18772
$13$	−3.13413	−0.869252	−0.434626	−	0.900611i	$-0.643119\pi$
$13$	−3.13413	−0.869252	−0.434626	+	0.900611i	$0.643119\pi$
$14$	0	0
$15$	2.54699	0.657631
$16$	−4.62130	−1.15533
$17$	2.84289	0.689503	0.344751	−	0.938694i	$-0.387963\pi$
$17$	2.84289	0.689503	0.344751	+	0.938694i	$0.387963\pi$
$18$	6.63056	1.56284
$19$	5.77754	1.32546	0.662729	−	0.748859i	$-0.269398\pi$
$19$	5.77754	1.32546	0.662729	+	0.748859i	$0.269398\pi$
$20$	−1.61538	−0.361211
$21$	0	0
$22$	1.38522	0.295331
$23$	−4.48221	−0.934605	−0.467303	−	0.884097i	$-0.654774\pi$
$23$	−4.48221	−0.934605	−0.467303	+	0.884097i	$0.654774\pi$
$24$	1.86265	0.380213
$25$	1.00000	0.200000
$26$	−5.95929	−1.16871
$27$	−1.24082	−0.238796
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	4.84289	0.884187
$31$	5.44093	0.977220	0.488610	−	0.872502i	$-0.337504\pi$
$31$	5.44093	0.977220	0.488610	+	0.872502i	$0.337504\pi$
$32$	−7.32439	−1.29478
$33$	−1.85554	−0.323008
$34$	5.40552	0.927040
$35$	0	0
$36$	5.63312	0.938853
$37$	2.63378	0.432991	0.216496	−	0.976284i	$-0.430537\pi$
$37$	2.63378	0.432991	0.216496	+	0.976284i	$0.430537\pi$
$38$	10.9855	1.78208
$39$	7.98261	1.27824
$40$	0.731315	0.115631
$41$	−3.86096	−0.602981	−0.301490	−	0.953469i	$-0.597484\pi$
$41$	−3.86096	−0.602981	−0.301490	+	0.953469i	$0.597484\pi$
$42$	0	0
$43$	6.24486	0.952332	0.476166	−	0.879355i	$-0.342026\pi$
$43$	6.24486	0.952332	0.476166	+	0.879355i	$0.342026\pi$
$44$	1.17684	0.177416
$45$	−3.48717	−0.519837
$46$	−8.52254	−1.25658
$47$	1.56179	0.227811	0.113905	−	0.993492i	$-0.463664\pi$
$47$	1.56179	0.227811	0.113905	+	0.993492i	$0.463664\pi$
$48$	11.7704	1.69891
$49$	0	0
$50$	1.90142	0.268901
$51$	−7.24083	−1.01392
$52$	−5.06283	−0.702088
$53$	9.73567	1.33730	0.668649	−	0.743578i	$-0.266873\pi$
$53$	9.73567	1.33730	0.668649	+	0.743578i	$0.266873\pi$
$54$	−2.35931	−0.321062
$55$	−0.728522	−0.0982339
$56$	0	0
$57$	−14.7153	−1.94910
$58$	1.90142	0.249668
$59$	−6.37795	−0.830338	−0.415169	−	0.909744i	$-0.636278\pi$
$59$	−6.37795	−0.830338	−0.415169	+	0.909744i	$0.636278\pi$
$60$	4.11437	0.531163
$61$	−13.1438	−1.68289	−0.841445	−	0.540343i	$-0.818295\pi$
$61$	−13.1438	−1.68289	−0.841445	+	0.540343i	$0.818295\pi$
$62$	10.3455	1.31388
$63$	0	0
$64$	−4.68411	−0.585514
$65$	3.13413	0.388741
$66$	−3.52816	−0.434286
$67$	−7.17801	−0.876933	−0.438466	−	0.898748i	$-0.644478\pi$
$67$	−7.17801	−0.876933	−0.438466	+	0.898748i	$0.644478\pi$
$68$	4.59236	0.556906
$69$	11.4162	1.37434
$70$	0	0
$71$	−0.416727	−0.0494564	−0.0247282	−	0.999694i	$-0.507872\pi$
$71$	−0.416727	−0.0494564	−0.0247282	+	0.999694i	$0.507872\pi$
$72$	−2.55022	−0.300546
$73$	6.24986	0.731490	0.365745	−	0.930715i	$-0.380814\pi$
$73$	6.24986	0.731490	0.365745	+	0.930715i	$0.380814\pi$
$74$	5.00792	0.582159
$75$	−2.54699	−0.294101
$76$	9.33294	1.07056
$77$	0	0
$78$	15.1783	1.71860
$79$	−15.5074	−1.74471	−0.872357	−	0.488869i	$-0.837409\pi$
$79$	−15.5074	−1.74471	−0.872357	+	0.488869i	$0.837409\pi$
$80$	4.62130	0.516677
$81$	−7.30115	−0.811239
$82$	−7.34129	−0.810710
$83$	3.86824	0.424595	0.212297	−	0.977205i	$-0.431905\pi$
$83$	3.86824	0.424595	0.212297	+	0.977205i	$0.431905\pi$
$84$	0	0
$85$	−2.84289	−0.308355
$86$	11.8741	1.28041
$87$	−2.54699	−0.273066
$88$	−0.532780	−0.0567945
$89$	3.83025	0.406005	0.203003	−	0.979178i	$-0.434930\pi$
$89$	3.83025	0.406005	0.203003	+	0.979178i	$0.434930\pi$
$90$	−6.63056	−0.698923
$91$	0	0
$92$	−7.24049	−0.754873
$93$	−13.8580	−1.43701
$94$	2.96962	0.306293
$95$	−5.77754	−0.592763
$96$	18.6552	1.90398
$97$	3.75306	0.381066	0.190533	−	0.981681i	$-0.438978\pi$
$97$	3.75306	0.381066	0.190533	+	0.981681i	$0.438978\pi$
$98$	0	0
$99$	2.54048	0.255328
$100$	1.61538	0.161538
$101$	−7.41208	−0.737529	−0.368765	−	0.929523i	$-0.620219\pi$
$101$	−7.41208	−0.737529	−0.368765	+	0.929523i	$0.620219\pi$
$102$	−13.7678	−1.36322
$103$	−9.80016	−0.965638	−0.482819	−	0.875720i	$-0.660387\pi$
$103$	−9.80016	−0.965638	−0.482819	+	0.875720i	$0.660387\pi$
$104$	2.29204	0.224753
$105$	0	0
$106$	18.5116	1.79800
$107$	−0.235944	−0.0228096	−0.0114048	−	0.999935i	$-0.503630\pi$
$107$	−0.235944	−0.0228096	−0.0114048	+	0.999935i	$0.503630\pi$
$108$	−2.00440	−0.192873
$109$	−2.90415	−0.278168	−0.139084	−	0.990281i	$-0.544416\pi$
$109$	−2.90415	−0.278168	−0.139084	+	0.990281i	$0.544416\pi$
$110$	−1.38522	−0.132076
$111$	−6.70823	−0.636717
$112$	0	0
$113$	0.412291	0.0387850	0.0193925	−	0.999812i	$-0.493827\pi$
$113$	0.412291	0.0387850	0.0193925	+	0.999812i	$0.493827\pi$
$114$	−27.9800	−2.62057
$115$	4.48221	0.417968
$116$	1.61538	0.149985
$117$	−10.9293	−1.01041
$118$	−12.1271	−1.11639
$119$	0	0
$120$	−1.86265	−0.170036
$121$	−10.4693	−0.951750
$122$	−24.9918	−2.26265
$123$	9.83384	0.886687
$124$	8.78919	0.789292
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	2.84410	0.252373	0.126186	−	0.992007i	$-0.459726\pi$
$127$	2.84410	0.252373	0.126186	+	0.992007i	$0.459726\pi$
$128$	5.74234	0.507556
$129$	−15.9056	−1.40041
$130$	5.95929	0.522664
$131$	−2.54530	−0.222384	−0.111192	−	0.993799i	$-0.535467\pi$
$131$	−2.54530	−0.222384	−0.111192	+	0.993799i	$0.535467\pi$
$132$	−2.99741	−0.260891
$133$	0	0
$134$	−13.6484	−1.17904
$135$	1.24082	0.106793
$136$	−2.07905	−0.178277
$137$	12.2175	1.04381	0.521907	−	0.853002i	$-0.325221\pi$
$137$	12.2175	1.04381	0.521907	+	0.853002i	$0.325221\pi$
$138$	21.7069	1.84781
$139$	2.42528	0.205710	0.102855	−	0.994696i	$-0.467202\pi$
$139$	2.42528	0.205710	0.102855	+	0.994696i	$0.467202\pi$
$140$	0	0
$141$	−3.97787	−0.334997
$142$	−0.792372	−0.0664944
$143$	−2.28329	−0.190938
$144$	−16.1153	−1.34294
$145$	−1.00000	−0.0830455
$146$	11.8836	0.983492
$147$	0	0
$148$	4.25457	0.349724
$149$	−9.60374	−0.786769	−0.393384	−	0.919374i	$-0.628696\pi$
$149$	−9.60374	−0.786769	−0.393384	+	0.919374i	$0.628696\pi$
$150$	−4.84289	−0.395421
$151$	−21.6983	−1.76578	−0.882891	−	0.469577i	$-0.844406\pi$
$151$	−21.6983	−1.76578	−0.882891	+	0.469577i	$0.844406\pi$
$152$	−4.22520	−0.342709
$153$	9.91365	0.801471
$154$	0	0
$155$	−5.44093	−0.437026
$156$	12.8950	1.03242
$157$	16.7980	1.34063	0.670314	−	0.742078i	$-0.266159\pi$
$157$	16.7980	1.34063	0.670314	+	0.742078i	$0.266159\pi$
$158$	−29.4859	−2.34578
$159$	−24.7967	−1.96651
$160$	7.32439	0.579044
$161$	0	0
$162$	−13.8825	−1.09071
$163$	23.3686	1.83037	0.915186	−	0.403031i	$-0.132043\pi$
$163$	23.3686	1.83037	0.915186	+	0.403031i	$0.132043\pi$
$164$	−6.23693	−0.487023
$165$	1.85554	0.144454
$166$	7.35514	0.570869
$167$	5.65077	0.437270	0.218635	−	0.975807i	$-0.429840\pi$
$167$	5.65077	0.437270	0.218635	+	0.975807i	$0.429840\pi$
$168$	0	0
$169$	−3.17721	−0.244401
$170$	−5.40552	−0.414585
$171$	20.1473	1.54070
$172$	10.0878	0.769191
$173$	−22.3388	−1.69839	−0.849193	−	0.528083i	$-0.822911\pi$
$173$	−22.3388	−1.69839	−0.849193	+	0.528083i	$0.822911\pi$
$174$	−4.84289	−0.367139
$175$	0	0
$176$	−3.36672	−0.253776
$177$	16.2446	1.22102
$178$	7.28290	0.545876
$179$	−4.44330	−0.332108	−0.166054	−	0.986117i	$-0.553103\pi$
$179$	−4.44330	−0.332108	−0.166054	+	0.986117i	$0.553103\pi$
$180$	−5.63312	−0.419868
$181$	−9.78079	−0.727001	−0.363500	−	0.931594i	$-0.618418\pi$
$181$	−9.78079	−0.727001	−0.363500	+	0.931594i	$0.618418\pi$
$182$	0	0
$183$	33.4771	2.47470
$184$	3.27791	0.241651
$185$	−2.63378	−0.193640
$186$	−26.3498	−1.93206
$187$	2.07111	0.151455
$188$	2.52289	0.184001
$189$	0	0
$190$	−10.9855	−0.796972
$191$	−18.9046	−1.36789	−0.683946	−	0.729533i	$-0.739738\pi$
$191$	−18.9046	−1.36789	−0.683946	+	0.729533i	$0.739738\pi$
$192$	11.9304	0.861002
$193$	15.5798	1.12146	0.560729	−	0.827999i	$-0.310521\pi$
$193$	15.5798	1.12146	0.560729	+	0.827999i	$0.310521\pi$
$194$	7.13613	0.512345
$195$	−7.98261	−0.571647
$196$	0	0
$197$	−15.5935	−1.11099	−0.555495	−	0.831520i	$-0.687471\pi$
$197$	−15.5935	−1.11099	−0.555495	+	0.831520i	$0.687471\pi$
$198$	4.83051	0.343290
$199$	0.403048	0.0285713	0.0142856	−	0.999898i	$-0.495453\pi$
$199$	0.403048	0.0285713	0.0142856	+	0.999898i	$0.495453\pi$
$200$	−0.731315	−0.0517118
$201$	18.2823	1.28954
$202$	−14.0934	−0.991611
$203$	0	0
$204$	−11.6967	−0.818934
$205$	3.86096	0.269661
$206$	−18.6342	−1.29830
$207$	−15.6302	−1.08638
$208$	14.4838	1.00427
$209$	4.20907	0.291147
$210$	0	0
$211$	23.0587	1.58742	0.793712	−	0.608294i	$-0.208146\pi$
$211$	23.0587	1.58742	0.793712	+	0.608294i	$0.208146\pi$
$212$	15.7268	1.08012
$213$	1.06140	0.0727260
$214$	−0.448629	−0.0306676
$215$	−6.24486	−0.425896
$216$	0.907430	0.0617428
$217$	0	0
$218$	−5.52201	−0.373998
$219$	−15.9183	−1.07566
$220$	−1.17684	−0.0793428
$221$	−8.91000	−0.599352
$222$	−12.7551	−0.856069
$223$	−8.59802	−0.575766	−0.287883	−	0.957666i	$-0.592951\pi$
$223$	−8.59802	−0.575766	−0.287883	+	0.957666i	$0.592951\pi$
$224$	0	0
$225$	3.48717	0.232478
$226$	0.783936	0.0521467
$227$	−11.5139	−0.764203	−0.382101	−	0.924120i	$-0.624799\pi$
$227$	−11.5139	−0.764203	−0.382101	+	0.924120i	$0.624799\pi$
$228$	−23.7709	−1.57427
$229$	−15.6307	−1.03290	−0.516452	−	0.856316i	$-0.672748\pi$
$229$	−15.6307	−1.03290	−0.516452	+	0.856316i	$0.672748\pi$
$230$	8.52254	0.561960
$231$	0	0
$232$	−0.731315	−0.0480132
$233$	4.87126	0.319127	0.159563	−	0.987188i	$-0.448991\pi$
$233$	4.87126	0.319127	0.159563	+	0.987188i	$0.448991\pi$
$234$	−20.7811	−1.35850
$235$	−1.56179	−0.101880
$236$	−10.3028	−0.670658
$237$	39.4971	2.56561
$238$	0	0
$239$	4.40727	0.285083	0.142541	−	0.989789i	$-0.454473\pi$
$239$	4.40727	0.285083	0.142541	+	0.989789i	$0.454473\pi$
$240$	−11.7704	−0.759778
$241$	−13.7374	−0.884905	−0.442452	−	0.896792i	$-0.645891\pi$
$241$	−13.7374	−0.884905	−0.442452	+	0.896792i	$0.645891\pi$
$242$	−19.9064	−1.27963
$243$	22.3184	1.43173
$244$	−21.2323	−1.35926
$245$	0	0
$246$	18.6982	1.19215
$247$	−18.1076	−1.15216
$248$	−3.97903	−0.252669
$249$	−9.85238	−0.624369
$250$	−1.90142	−0.120256
$251$	−26.3747	−1.66475	−0.832377	−	0.554209i	$-0.813021\pi$
$251$	−26.3747	−1.66475	−0.832377	+	0.554209i	$0.813021\pi$
$252$	0	0
$253$	−3.26539	−0.205293
$254$	5.40781	0.339316
$255$	7.24083	0.453438
$256$	20.2868	1.26792
$257$	−25.2286	−1.57372	−0.786860	−	0.617132i	$-0.788295\pi$
$257$	−25.2286	−1.57372	−0.786860	+	0.617132i	$0.788295\pi$
$258$	−30.2432	−1.88286
$259$	0	0
$260$	5.06283	0.313983
$261$	3.48717	0.215850
$262$	−4.83967	−0.298996
$263$	6.71900	0.414311	0.207156	−	0.978308i	$-0.433579\pi$
$263$	6.71900	0.414311	0.207156	+	0.978308i	$0.433579\pi$
$264$	1.35699	0.0835167
$265$	−9.73567	−0.598058
$266$	0	0
$267$	−9.75561	−0.597034
$268$	−11.5952	−0.708292
$269$	−16.4506	−1.00301	−0.501507	−	0.865154i	$-0.667221\pi$
$269$	−16.4506	−1.00301	−0.501507	+	0.865154i	$0.667221\pi$
$270$	2.35931	0.143583
$271$	−10.5078	−0.638302	−0.319151	−	0.947704i	$-0.603398\pi$
$271$	−10.5078	−0.638302	−0.319151	+	0.947704i	$0.603398\pi$
$272$	−13.1379	−0.796600
$273$	0	0
$274$	23.2306	1.40341
$275$	0.728522	0.0439315
$276$	18.4415	1.11005
$277$	−28.4609	−1.71005	−0.855025	−	0.518587i	$-0.826458\pi$
$277$	−28.4609	−1.71005	−0.855025	+	0.518587i	$0.826458\pi$
$278$	4.61148	0.276578
$279$	18.9734	1.13591
$280$	0	0
$281$	−7.90549	−0.471602	−0.235801	−	0.971801i	$-0.575771\pi$
$281$	−7.90549	−0.471602	−0.235801	+	0.971801i	$0.575771\pi$
$282$	−7.56359	−0.450405
$283$	−20.8806	−1.24122	−0.620612	−	0.784118i	$-0.713116\pi$
$283$	−20.8806	−1.24122	−0.620612	+	0.784118i	$0.713116\pi$
$284$	−0.673174	−0.0399455
$285$	14.7153	0.871662
$286$	−4.34148	−0.256717
$287$	0	0
$288$	−25.5414	−1.50504
$289$	−8.91796	−0.524586
$290$	−1.90142	−0.111655
$291$	−9.55902	−0.560360
$292$	10.0959	0.590819
$293$	−11.5150	−0.672714	−0.336357	−	0.941735i	$-0.609195\pi$
$293$	−11.5150	−0.672714	−0.336357	+	0.941735i	$0.609195\pi$
$294$	0	0
$295$	6.37795	0.371339
$296$	−1.92613	−0.111954
$297$	−0.903965	−0.0524533
$298$	−18.2607	−1.05781
$299$	14.0478	0.812407
$300$	−4.11437	−0.237543
$301$	0	0
$302$	−41.2575	−2.37410
$303$	18.8785	1.08454
$304$	−26.6998	−1.53134
$305$	13.1438	0.752611
$306$	18.8500	1.07758
$307$	−1.70062	−0.0970592	−0.0485296	−	0.998822i	$-0.515454\pi$
$307$	−1.70062	−0.0970592	−0.0485296	+	0.998822i	$0.515454\pi$
$308$	0	0
$309$	24.9609	1.41998
$310$	−10.3455	−0.587583
$311$	−0.169230	−0.00959613	−0.00479806	−	0.999988i	$-0.501527\pi$
$311$	−0.169230	−0.00959613	−0.00479806	+	0.999988i	$0.501527\pi$
$312$	−5.83781	−0.330501
$313$	29.9696	1.69398	0.846990	−	0.531609i	$-0.178412\pi$
$313$	29.9696	1.69398	0.846990	+	0.531609i	$0.178412\pi$
$314$	31.9400	1.80248
$315$	0	0
$316$	−25.0503	−1.40919
$317$	−17.3308	−0.973396	−0.486698	−	0.873570i	$-0.661799\pi$
$317$	−17.3308	−0.973396	−0.486698	+	0.873570i	$0.661799\pi$
$318$	−47.1488	−2.64397
$319$	0.728522	0.0407894
$320$	4.68411	0.261850
$321$	0.600949	0.0335417
$322$	0	0
$323$	16.4249	0.913907
$324$	−11.7942	−0.655231
$325$	−3.13413	−0.173850
$326$	44.4335	2.46094
$327$	7.39686	0.409047
$328$	2.82358	0.155906
$329$	0	0
$330$	3.52816	0.194219
$331$	−13.9562	−0.767104	−0.383552	−	0.923519i	$-0.625299\pi$
$331$	−13.9562	−0.767104	−0.383552	+	0.923519i	$0.625299\pi$
$332$	6.24869	0.342942
$333$	9.18445	0.503305
$334$	10.7445	0.587912
$335$	7.17801	0.392176
$336$	0	0
$337$	14.8801	0.810571	0.405286	−	0.914190i	$-0.367172\pi$
$337$	14.8801	0.810571	0.405286	+	0.914190i	$0.367172\pi$
$338$	−6.04121	−0.328598
$339$	−1.05010	−0.0570337
$340$	−4.59236	−0.249056
$341$	3.96384	0.214654
$342$	38.3083	2.07148
$343$	0	0
$344$	−4.56696	−0.246234
$345$	−11.4162	−0.614625
$346$	−42.4753	−2.28349
$347$	−29.7336	−1.59618	−0.798092	−	0.602535i	$-0.794157\pi$
$347$	−29.7336	−1.59618	−0.798092	+	0.602535i	$0.794157\pi$
$348$	−4.11437	−0.220553
$349$	12.8810	0.689503	0.344752	−	0.938694i	$-0.387963\pi$
$349$	12.8810	0.689503	0.344752	+	0.938694i	$0.387963\pi$
$350$	0	0
$351$	3.88889	0.207574
$352$	−5.33598	−0.284409
$353$	5.81067	0.309271	0.154635	−	0.987972i	$-0.450580\pi$
$353$	5.81067	0.309271	0.154635	+	0.987972i	$0.450580\pi$
$354$	30.8877	1.64166
$355$	0.416727	0.0221176
$356$	6.18732	0.327927
$357$	0	0
$358$	−8.44856	−0.446520
$359$	−36.5012	−1.92646	−0.963229	−	0.268683i	$-0.913412\pi$
$359$	−36.5012	−1.92646	−0.963229	+	0.268683i	$0.913412\pi$
$360$	2.55022	0.134408
$361$	14.3800	0.756840
$362$	−18.5974	−0.977456
$363$	26.6651	1.39956
$364$	0	0
$365$	−6.24986	−0.327132
$366$	63.6540	3.32725
$367$	−6.38217	−0.333146	−0.166573	−	0.986029i	$-0.553270\pi$
$367$	−6.38217	−0.333146	−0.166573	+	0.986029i	$0.553270\pi$
$368$	20.7136	1.07977
$369$	−13.4638	−0.700899
$370$	−5.00792	−0.260349
$371$	0	0
$372$	−22.3860	−1.16066
$373$	−10.8278	−0.560644	−0.280322	−	0.959906i	$-0.590441\pi$
$373$	−10.8278	−0.560644	−0.280322	+	0.959906i	$0.590441\pi$
$374$	3.93804	0.203631
$375$	2.54699	0.131526
$376$	−1.14216	−0.0589025
$377$	−3.13413	−0.161416
$378$	0	0
$379$	23.2489	1.19422	0.597109	−	0.802160i	$-0.296316\pi$
$379$	23.2489	1.19422	0.597109	+	0.802160i	$0.296316\pi$
$380$	−9.33294	−0.478770
$381$	−7.24389	−0.371116
$382$	−35.9456	−1.83914
$383$	17.6744	0.903120	0.451560	−	0.892241i	$-0.350868\pi$
$383$	17.6744	0.903120	0.451560	+	0.892241i	$0.350868\pi$
$384$	−14.6257	−0.746364
$385$	0	0
$386$	29.6237	1.50780
$387$	21.7769	1.10698
$388$	6.06264	0.307784
$389$	28.4890	1.44445	0.722225	−	0.691659i	$-0.243120\pi$
$389$	28.4890	1.44445	0.722225	+	0.691659i	$0.243120\pi$
$390$	−15.1783	−0.768581
$391$	−12.7424	−0.644413
$392$	0	0
$393$	6.48285	0.327017
$394$	−29.6497	−1.49373
$395$	15.5074	0.780260
$396$	4.10385	0.206226
$397$	33.7043	1.69157	0.845785	−	0.533523i	$-0.179132\pi$
$397$	33.7043	1.69157	0.845785	+	0.533523i	$0.179132\pi$
$398$	0.766361	0.0384142
$399$	0	0
$400$	−4.62130	−0.231065
$401$	−26.3551	−1.31611	−0.658055	−	0.752970i	$-0.728621\pi$
$401$	−26.3551	−1.31611	−0.658055	+	0.752970i	$0.728621\pi$
$402$	34.7623	1.73379
$403$	−17.0526	−0.849450
$404$	−11.9733	−0.595696
$405$	7.30115	0.362797
$406$	0	0
$407$	1.91877	0.0951099
$408$	5.29533	0.262158
$409$	−14.2893	−0.706558	−0.353279	−	0.935518i	$-0.614933\pi$
$409$	−14.2893	−0.706558	−0.353279	+	0.935518i	$0.614933\pi$
$410$	7.34129	0.362561
$411$	−31.1180	−1.53494
$412$	−15.8310	−0.779938
$413$	0	0
$414$	−29.7196	−1.46064
$415$	−3.86824	−0.189884
$416$	22.9556	1.12549
$417$	−6.17718	−0.302498
$418$	8.00319	0.391449
$419$	12.6926	0.620076	0.310038	−	0.950724i	$-0.399658\pi$
$419$	12.6926	0.620076	0.310038	+	0.950724i	$0.399658\pi$
$420$	0	0
$421$	24.6290	1.20035	0.600173	−	0.799870i	$-0.295098\pi$
$421$	24.6290	1.20035	0.600173	+	0.799870i	$0.295098\pi$
$422$	43.8441	2.13430
$423$	5.44624	0.264805
$424$	−7.11985	−0.345770
$425$	2.84289	0.137901
$426$	2.01816	0.0977804
$427$	0	0
$428$	−0.381141	−0.0184231
$429$	5.81551	0.280775
$430$	−11.8741	−0.572619
$431$	34.9325	1.68264	0.841320	−	0.540537i	$-0.181779\pi$
$431$	34.9325	1.68264	0.841320	+	0.540537i	$0.181779\pi$
$432$	5.73420	0.275887
$433$	−6.39096	−0.307130	−0.153565	−	0.988139i	$-0.549075\pi$
$433$	−6.39096	−0.307130	−0.153565	+	0.988139i	$0.549075\pi$
$434$	0	0
$435$	2.54699	0.122119
$436$	−4.69132	−0.224674
$437$	−25.8961	−1.23878
$438$	−30.2674	−1.44623
$439$	13.5177	0.645163	0.322582	−	0.946542i	$-0.395449\pi$
$439$	13.5177	0.645163	0.322582	+	0.946542i	$0.395449\pi$
$440$	0.532780	0.0253993
$441$	0	0
$442$	−16.9416	−0.805831
$443$	0.677780	0.0322023	0.0161011	−	0.999870i	$-0.494875\pi$
$443$	0.677780	0.0322023	0.0161011	+	0.999870i	$0.494875\pi$
$444$	−10.8364	−0.514271
$445$	−3.83025	−0.181571
$446$	−16.3484	−0.774120
$447$	24.4606	1.15695
$448$	0	0
$449$	24.4669	1.15466	0.577331	−	0.816510i	$-0.304094\pi$
$449$	24.4669	1.15466	0.577331	+	0.816510i	$0.304094\pi$
$450$	6.63056	0.312568
$451$	−2.81280	−0.132449
$452$	0.666008	0.0313264
$453$	55.2654	2.59660
$454$	−21.8927	−1.02747
$455$	0	0
$456$	10.7616	0.503956
$457$	−2.95902	−0.138417	−0.0692086	−	0.997602i	$-0.522047\pi$
$457$	−2.95902	−0.138417	−0.0692086	+	0.997602i	$0.522047\pi$
$458$	−29.7204	−1.38874
$459$	−3.52752	−0.164650
$460$	7.24049	0.337589
$461$	−19.0775	−0.888526	−0.444263	−	0.895897i	$-0.646534\pi$
$461$	−19.0775	−0.888526	−0.444263	+	0.895897i	$0.646534\pi$
$462$	0	0
$463$	−21.7261	−1.00970	−0.504848	−	0.863208i	$-0.668451\pi$
$463$	−21.7261	−1.00970	−0.504848	+	0.863208i	$0.668451\pi$
$464$	−4.62130	−0.214539
$465$	13.8580	0.642650
$466$	9.26228	0.429067
$467$	−23.5677	−1.09059	−0.545293	−	0.838246i	$-0.683581\pi$
$467$	−23.5677	−1.09059	−0.545293	+	0.838246i	$0.683581\pi$
$468$	−17.6549	−0.816100
$469$	0	0
$470$	−2.96962	−0.136978
$471$	−42.7844	−1.97140
$472$	4.66429	0.214691
$473$	4.54952	0.209187
$474$	75.1005	3.44948
$475$	5.77754	0.265092
$476$	0	0
$477$	33.9500	1.55446
$478$	8.38006	0.383295
$479$	−4.30567	−0.196731	−0.0983656	−	0.995150i	$-0.531361\pi$
$479$	−4.30567	−0.196731	−0.0983656	+	0.995150i	$0.531361\pi$
$480$	−18.6552	−0.851488
$481$	−8.25463	−0.376379
$482$	−26.1205	−1.18976
$483$	0	0
$484$	−16.9119	−0.768721
$485$	−3.75306	−0.170418
$486$	42.4366	1.92497
$487$	2.97877	0.134981	0.0674905	−	0.997720i	$-0.478501\pi$
$487$	2.97877	0.134981	0.0674905	+	0.997720i	$0.478501\pi$
$488$	9.61226	0.435126
$489$	−59.5197	−2.69158
$490$	0	0
$491$	24.0623	1.08592	0.542959	−	0.839759i	$-0.317304\pi$
$491$	24.0623	1.08592	0.542959	+	0.839759i	$0.317304\pi$
$492$	15.8854	0.716170
$493$	2.84289	0.128037
$494$	−34.4300	−1.54908
$495$	−2.54048	−0.114186
$496$	−25.1442	−1.12901
$497$	0	0
$498$	−18.7335	−0.839467
$499$	−8.35244	−0.373906	−0.186953	−	0.982369i	$-0.559861\pi$
$499$	−8.35244	−0.373906	−0.186953	+	0.982369i	$0.559861\pi$
$500$	−1.61538	−0.0722422
$501$	−14.3925	−0.643009
$502$	−50.1492	−2.23827
$503$	−23.3963	−1.04319	−0.521595	−	0.853193i	$-0.674663\pi$
$503$	−23.3963	−1.04319	−0.521595	+	0.853193i	$0.674663\pi$
$504$	0	0
$505$	7.41208	0.329833
$506$	−6.20886	−0.276018
$507$	8.09234	0.359394
$508$	4.59431	0.203839
$509$	−17.4660	−0.774167	−0.387084	−	0.922045i	$-0.626518\pi$
$509$	−17.4660	−0.774167	−0.387084	+	0.922045i	$0.626518\pi$
$510$	13.7678	0.609650
$511$	0	0
$512$	27.0890	1.19717
$513$	−7.16888	−0.316514
$514$	−47.9702	−2.11587
$515$	9.80016	0.431846
$516$	−25.6937	−1.13110
$517$	1.13780	0.0500404
$518$	0	0
$519$	56.8967	2.49749
$520$	−2.29204	−0.100513
$521$	25.5836	1.12084	0.560420	−	0.828209i	$-0.310640\pi$
$521$	25.5836	1.12084	0.560420	+	0.828209i	$0.310640\pi$
$522$	6.63056	0.290212
$523$	−23.6073	−1.03227	−0.516137	−	0.856506i	$-0.672630\pi$
$523$	−23.6073	−1.03227	−0.516137	+	0.856506i	$0.672630\pi$
$524$	−4.11163	−0.179618
$525$	0	0
$526$	12.7756	0.557043
$527$	15.4680	0.673796
$528$	8.57502	0.373180
$529$	−2.90981	−0.126513
$530$	−18.5116	−0.804091
$531$	−22.2410	−0.965177
$532$	0	0
$533$	12.1008	0.524142
$534$	−18.5495	−0.802714
$535$	0.235944	0.0102008
$536$	5.24939	0.226739
$537$	11.3170	0.488367
$538$	−31.2795	−1.34856
$539$	0	0
$540$	2.00440	0.0862556
$541$	−0.457626	−0.0196749	−0.00983743	−	0.999952i	$-0.503131\pi$
$541$	−0.457626	−0.0196749	−0.00983743	+	0.999952i	$0.503131\pi$
$542$	−19.9797	−0.858200
$543$	24.9116	1.06906
$544$	−20.8225	−0.892755
$545$	2.90415	0.124400
$546$	0	0
$547$	−26.2239	−1.12125	−0.560627	−	0.828069i	$-0.689440\pi$
$547$	−26.2239	−1.12125	−0.560627	+	0.828069i	$0.689440\pi$
$548$	19.7360	0.843081
$549$	−45.8346	−1.95617
$550$	1.38522	0.0590662
$551$	5.77754	0.246131
$552$	−8.34881	−0.355349
$553$	0	0
$554$	−54.1160	−2.29917
$555$	6.70823	0.284748
$556$	3.91777	0.166150
$557$	17.1385	0.726180	0.363090	−	0.931754i	$-0.381722\pi$
$557$	17.1385	0.726180	0.363090	+	0.931754i	$0.381722\pi$
$558$	36.0764	1.52724
$559$	−19.5722	−0.827816
$560$	0	0
$561$	−5.27510	−0.222715
$562$	−15.0316	−0.634071
$563$	42.4701	1.78990	0.894951	−	0.446164i	$-0.147210\pi$
$563$	42.4701	1.78990	0.894951	+	0.446164i	$0.147210\pi$
$564$	−6.42579	−0.270575
$565$	−0.412291	−0.0173452
$566$	−39.7028	−1.66883
$567$	0	0
$568$	0.304759	0.0127874
$569$	−15.4302	−0.646868	−0.323434	−	0.946251i	$-0.604837\pi$
$569$	−15.4302	−0.646868	−0.323434	+	0.946251i	$0.604837\pi$
$570$	27.9800	1.17195
$571$	1.34931	0.0564668	0.0282334	−	0.999601i	$-0.491012\pi$
$571$	1.34931	0.0564668	0.0282334	+	0.999601i	$0.491012\pi$
$572$	−3.68838	−0.154219
$573$	48.1500	2.01149
$574$	0	0
$575$	−4.48221	−0.186921
$576$	−16.3343	−0.680595
$577$	33.5590	1.39708	0.698540	−	0.715571i	$-0.253833\pi$
$577$	33.5590	1.39708	0.698540	+	0.715571i	$0.253833\pi$
$578$	−16.9568	−0.705308
$579$	−39.6816	−1.64911
$580$	−1.61538	−0.0670752
$581$	0	0
$582$	−18.1757	−0.753406
$583$	7.09266	0.293748
$584$	−4.57062	−0.189133
$585$	10.9293	0.451869
$586$	−21.8948	−0.904467
$587$	−25.2112	−1.04058	−0.520290	−	0.853990i	$-0.674176\pi$
$587$	−25.2112	−1.04058	−0.520290	+	0.853990i	$0.674176\pi$
$588$	0	0
$589$	31.4352	1.29526
$590$	12.1271	0.499266
$591$	39.7165	1.63372
$592$	−12.1715	−0.500246
$593$	9.68172	0.397581	0.198790	−	0.980042i	$-0.436299\pi$
$593$	9.68172	0.397581	0.198790	+	0.980042i	$0.436299\pi$
$594$	−1.71881	−0.0705238
$595$	0	0
$596$	−15.5137	−0.635467
$597$	−1.02656	−0.0420143
$598$	26.7108	1.09229
$599$	1.96995	0.0804899	0.0402449	−	0.999190i	$-0.487186\pi$
$599$	1.96995	0.0804899	0.0402449	+	0.999190i	$0.487186\pi$
$600$	1.86265	0.0760426
$601$	−5.20772	−0.212427	−0.106214	−	0.994343i	$-0.533873\pi$
$601$	−5.20772	−0.212427	−0.106214	+	0.994343i	$0.533873\pi$
$602$	0	0
$603$	−25.0309	−1.01934
$604$	−35.0511	−1.42621
$605$	10.4693	0.425636
$606$	35.8959	1.45817
$607$	33.9026	1.37606	0.688031	−	0.725681i	$-0.258475\pi$
$607$	33.9026	1.37606	0.688031	+	0.725681i	$0.258475\pi$
$608$	−42.3169	−1.71618
$609$	0	0
$610$	24.9918	1.01189
$611$	−4.89486	−0.198025
$612$	16.0144	0.647342
$613$	25.3935	1.02563	0.512816	−	0.858498i	$-0.328602\pi$
$613$	25.3935	1.02563	0.512816	+	0.858498i	$0.328602\pi$
$614$	−3.23358	−0.130497
$615$	−9.83384	−0.396539
$616$	0	0
$617$	10.7565	0.433041	0.216520	−	0.976278i	$-0.430529\pi$
$617$	10.7565	0.433041	0.216520	+	0.976278i	$0.430529\pi$
$618$	47.4611	1.90917
$619$	−0.933533	−0.0375219	−0.0187609	−	0.999824i	$-0.505972\pi$
$619$	−0.933533	−0.0375219	−0.0187609	+	0.999824i	$0.505972\pi$
$620$	−8.78919	−0.352982
$621$	5.56161	0.223180
$622$	−0.321776	−0.0129020
$623$	0	0
$624$	−36.8901	−1.47678
$625$	1.00000	0.0400000
$626$	56.9846	2.27756
$627$	−10.7205	−0.428134
$628$	27.1352	1.08281
$629$	7.48757	0.298549
$630$	0	0
$631$	−28.7515	−1.14458	−0.572290	−	0.820051i	$-0.693945\pi$
$631$	−28.7515	−1.14458	−0.572290	+	0.820051i	$0.693945\pi$
$632$	11.3408	0.451112
$633$	−58.7302	−2.33432
$634$	−32.9531	−1.30874
$635$	−2.84410	−0.112865
$636$	−40.0562	−1.58833
$637$	0	0
$638$	1.38522	0.0548416
$639$	−1.45320	−0.0574876
$640$	−5.74234	−0.226986
$641$	47.4817	1.87541	0.937706	−	0.347429i	$-0.112945\pi$
$641$	47.4817	1.87541	0.937706	+	0.347429i	$0.112945\pi$
$642$	1.14265	0.0450969
$643$	−4.20981	−0.166019	−0.0830093	−	0.996549i	$-0.526453\pi$
$643$	−4.20981	−0.166019	−0.0830093	+	0.996549i	$0.526453\pi$
$644$	0	0
$645$	15.9056	0.626282
$646$	31.2306	1.22875
$647$	−3.42384	−0.134605	−0.0673025	−	0.997733i	$-0.521439\pi$
$647$	−3.42384	−0.134605	−0.0673025	+	0.997733i	$0.521439\pi$
$648$	5.33945	0.209753
$649$	−4.64648	−0.182390
$650$	−5.95929	−0.233743
$651$	0	0
$652$	37.7493	1.47838
$653$	−20.4473	−0.800166	−0.400083	−	0.916479i	$-0.631019\pi$
$653$	−20.4473	−0.800166	−0.400083	+	0.916479i	$0.631019\pi$
$654$	14.0645	0.549966
$655$	2.54530	0.0994530
$656$	17.8427	0.696639
$657$	21.7943	0.850277
$658$	0	0
$659$	33.9663	1.32314	0.661569	−	0.749884i	$-0.269891\pi$
$659$	33.9663	1.32314	0.661569	+	0.749884i	$0.269891\pi$
$660$	2.99741	0.116674
$661$	19.2404	0.748366	0.374183	−	0.927355i	$-0.377923\pi$
$661$	19.2404	0.748366	0.374183	+	0.927355i	$0.377923\pi$
$662$	−26.5366	−1.03137
$663$	22.6937	0.881351
$664$	−2.82890	−0.109783
$665$	0	0
$666$	17.4635	0.676696
$667$	−4.48221	−0.173552
$668$	9.12817	0.353180
$669$	21.8991	0.846668
$670$	13.6484	0.527283
$671$	−9.57555	−0.369660
$672$	0	0
$673$	−47.9591	−1.84869	−0.924344	−	0.381559i	$-0.875387\pi$
$673$	−47.9591	−1.84869	−0.924344	+	0.381559i	$0.875387\pi$
$674$	28.2933	1.08982
$675$	−1.24082	−0.0477592
$676$	−5.13242	−0.197401
$677$	47.4804	1.82482	0.912411	−	0.409275i	$-0.134218\pi$
$677$	47.4804	1.82482	0.912411	+	0.409275i	$0.134218\pi$
$678$	−1.99668	−0.0766820
$679$	0	0
$680$	2.07905	0.0797280
$681$	29.3258	1.12377
$682$	7.53691	0.288603
$683$	−26.3101	−1.00673	−0.503363	−	0.864075i	$-0.667904\pi$
$683$	−26.3101	−1.00673	−0.503363	+	0.864075i	$0.667904\pi$
$684$	32.5456	1.24441
$685$	−12.2175	−0.466808
$686$	0	0
$687$	39.8112	1.51889
$688$	−28.8594	−1.10025
$689$	−30.5129	−1.16245
$690$	−21.7069	−0.826366
$691$	42.0009	1.59779	0.798894	−	0.601472i	$-0.205419\pi$
$691$	42.0009	1.59779	0.798894	+	0.601472i	$0.205419\pi$
$692$	−36.0857	−1.37177
$693$	0	0
$694$	−56.5360	−2.14608
$695$	−2.42528	−0.0919963
$696$	1.86265	0.0706037
$697$	−10.9763	−0.415757
$698$	24.4921	0.927040
$699$	−12.4070	−0.469278
$700$	0	0
$701$	6.45871	0.243942	0.121971	−	0.992534i	$-0.461079\pi$
$701$	6.45871	0.243942	0.121971	+	0.992534i	$0.461079\pi$
$702$	7.39440	0.279084
$703$	15.2168	0.573912
$704$	−3.41248	−0.128613
$705$	3.97787	0.149815
$706$	11.0485	0.415816
$707$	0	0
$708$	26.2413	0.986207
$709$	24.1580	0.907271	0.453636	−	0.891187i	$-0.350127\pi$
$709$	24.1580	0.907271	0.453636	+	0.891187i	$0.350127\pi$
$710$	0.792372	0.0297372
$711$	−54.0768	−2.02804
$712$	−2.80112	−0.104976
$713$	−24.3874	−0.913314
$714$	0	0
$715$	2.28329	0.0853900
$716$	−7.17763	−0.268241
$717$	−11.2253	−0.419216
$718$	−69.4039	−2.59013
$719$	−48.3531	−1.80327	−0.901633	−	0.432502i	$-0.857631\pi$
$719$	−48.3531	−1.80327	−0.901633	+	0.432502i	$0.857631\pi$
$720$	16.1153	0.600581
$721$	0	0
$722$	27.3423	1.01757
$723$	34.9891	1.30126
$724$	−15.7997	−0.587193
$725$	1.00000	0.0371391
$726$	50.7015	1.88171
$727$	−35.7615	−1.32632	−0.663160	−	0.748478i	$-0.730785\pi$
$727$	−35.7615	−1.32632	−0.663160	+	0.748478i	$0.730785\pi$
$728$	0	0
$729$	−34.9414	−1.29413
$730$	−11.8836	−0.439831
$731$	17.7535	0.656635
$732$	54.0784	1.99880
$733$	13.4675	0.497432	0.248716	−	0.968576i	$-0.419991\pi$
$733$	13.4675	0.497432	0.248716	+	0.968576i	$0.419991\pi$
$734$	−12.1352	−0.447917
$735$	0	0
$736$	32.8294	1.21011
$737$	−5.22934	−0.192625
$738$	−25.6003	−0.942361
$739$	14.8317	0.545595	0.272797	−	0.962072i	$-0.412051\pi$
$739$	14.8317	0.545595	0.272797	+	0.962072i	$0.412051\pi$
$740$	−4.25457	−0.156401
$741$	46.1199	1.69426
$742$	0	0
$743$	−51.6364	−1.89436	−0.947178	−	0.320709i	$-0.896079\pi$
$743$	−51.6364	−1.89436	−0.947178	+	0.320709i	$0.896079\pi$
$744$	10.1346	0.371551
$745$	9.60374	0.351854
$746$	−20.5882	−0.753789
$747$	13.4892	0.493545
$748$	3.34564	0.122329
$749$	0	0
$750$	4.84289	0.176837
$751$	−47.4696	−1.73219	−0.866095	−	0.499879i	$-0.833378\pi$
$751$	−47.4696	−1.73219	−0.866095	+	0.499879i	$0.833378\pi$
$752$	−7.21751	−0.263196
$753$	67.1761	2.44803
$754$	−5.95929	−0.217025
$755$	21.6983	0.789682
$756$	0	0
$757$	9.04485	0.328741	0.164370	−	0.986399i	$-0.447441\pi$
$757$	9.04485	0.328741	0.164370	+	0.986399i	$0.447441\pi$
$758$	44.2059	1.60563
$759$	8.31692	0.301885
$760$	4.22520	0.153264
$761$	−19.6371	−0.711846	−0.355923	−	0.934515i	$-0.615833\pi$
$761$	−19.6371	−0.711846	−0.355923	+	0.934515i	$0.615833\pi$
$762$	−13.7737	−0.498967
$763$	0	0
$764$	−30.5383	−1.10484
$765$	−9.91365	−0.358429
$766$	33.6064	1.21425
$767$	19.9893	0.721773
$768$	−51.6703	−1.86449
$769$	−8.05083	−0.290320	−0.145160	−	0.989408i	$-0.546370\pi$
$769$	−8.05083	−0.290320	−0.145160	+	0.989408i	$0.546370\pi$
$770$	0	0
$771$	64.2572	2.31417
$772$	25.1673	0.905792
$773$	41.1247	1.47915	0.739577	−	0.673072i	$-0.235026\pi$
$773$	41.1247	1.47915	0.739577	+	0.673072i	$0.235026\pi$
$774$	41.4069	1.48834
$775$	5.44093	0.195444
$776$	−2.74467	−0.0985280
$777$	0	0
$778$	54.1694	1.94207
$779$	−22.3068	−0.799226
$780$	−12.8950	−0.461714
$781$	−0.303595	−0.0108635
$782$	−24.2287	−0.866416
$783$	−1.24082	−0.0443433
$784$	0	0
$785$	−16.7980	−0.599547
$786$	12.3266	0.439675
$787$	−50.1393	−1.78727	−0.893636	−	0.448792i	$-0.851854\pi$
$787$	−50.1393	−1.78727	−0.893636	+	0.448792i	$0.851854\pi$
$788$	−25.1895	−0.897338
$789$	−17.1132	−0.609247
$790$	29.4859	1.04906
$791$	0	0
$792$	−1.85789	−0.0660174
$793$	41.1944	1.46286
$794$	64.0859	2.27432
$795$	24.7967	0.879448
$796$	0.651077	0.0230768
$797$	−3.28991	−0.116535	−0.0582673	−	0.998301i	$-0.518558\pi$
$797$	−3.28991	−0.116535	−0.0582673	+	0.998301i	$0.518558\pi$
$798$	0	0
$799$	4.44001	0.157076
$800$	−7.32439	−0.258956
$801$	13.3567	0.471937
$802$	−50.1120	−1.76952
$803$	4.55316	0.160678
$804$	29.5330	1.04155
$805$	0	0
$806$	−32.4241	−1.14209
$807$	41.8997	1.47494
$808$	5.42057	0.190695
$809$	31.3021	1.10052	0.550262	−	0.834992i	$-0.314528\pi$
$809$	31.3021	1.10052	0.550262	+	0.834992i	$0.314528\pi$
$810$	13.8825	0.487782
$811$	−22.3138	−0.783544	−0.391772	−	0.920062i	$-0.628138\pi$
$811$	−22.3138	−0.783544	−0.391772	+	0.920062i	$0.628138\pi$
$812$	0	0
$813$	26.7632	0.938628
$814$	3.64838	0.127876
$815$	−23.3686	−0.818568
$816$	33.4621	1.17141
$817$	36.0799	1.26228
$818$	−27.1698	−0.949971
$819$	0	0
$820$	6.23693	0.217803
$821$	51.5168	1.79795	0.898974	−	0.438002i	$-0.144314\pi$
$821$	51.5168	1.79795	0.898974	+	0.438002i	$0.144314\pi$
$822$	−59.1682	−2.06373
$823$	−32.1092	−1.11926	−0.559628	−	0.828744i	$-0.689056\pi$
$823$	−32.1092	−1.11926	−0.559628	+	0.828744i	$0.689056\pi$
$824$	7.16700	0.249674
$825$	−1.85554	−0.0646016
$826$	0	0
$827$	45.8979	1.59603	0.798013	−	0.602640i	$-0.205885\pi$
$827$	45.8979	1.59603	0.798013	+	0.602640i	$0.205885\pi$
$828$	−25.2488	−0.877457
$829$	36.2109	1.25766	0.628828	−	0.777544i	$-0.283535\pi$
$829$	36.2109	1.25766	0.628828	+	0.777544i	$0.283535\pi$
$830$	−7.35514	−0.255300
$831$	72.4897	2.51464
$832$	14.6806	0.508959
$833$	0	0
$834$	−11.7454	−0.406710
$835$	−5.65077	−0.195553
$836$	6.79926	0.235157
$837$	−6.75121	−0.233356
$838$	24.1340	0.833694
$839$	24.7372	0.854025	0.427012	−	0.904246i	$-0.359566\pi$
$839$	24.7372	0.854025	0.427012	+	0.904246i	$0.359566\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	46.8300	1.61387
$843$	20.1352	0.693494
$844$	37.2486	1.28215
$845$	3.17721	0.109300
$846$	10.3556	0.356032
$847$	0	0
$848$	−44.9915	−1.54501
$849$	53.1828	1.82523
$850$	5.40552	0.185408
$851$	−11.8052	−0.404676
$852$	1.71457	0.0587402
$853$	35.5893	1.21856	0.609278	−	0.792957i	$-0.291460\pi$
$853$	35.5893	1.21856	0.609278	+	0.792957i	$0.291460\pi$
$854$	0	0
$855$	−20.1473	−0.689022
$856$	0.172550	0.00589763
$857$	−36.6218	−1.25098	−0.625488	−	0.780234i	$-0.715100\pi$
$857$	−36.6218	−1.25098	−0.625488	+	0.780234i	$0.715100\pi$
$858$	11.0577	0.377504
$859$	6.91387	0.235898	0.117949	−	0.993020i	$-0.462368\pi$
$859$	6.91387	0.235898	0.117949	+	0.993020i	$0.462368\pi$
$860$	−10.0878	−0.343993
$861$	0	0
$862$	66.4213	2.26232
$863$	−49.7118	−1.69221	−0.846104	−	0.533018i	$-0.821058\pi$
$863$	−49.7118	−1.69221	−0.846104	+	0.533018i	$0.821058\pi$
$864$	9.08824	0.309188
$865$	22.3388	0.759541
$866$	−12.1519	−0.412937
$867$	22.7140	0.771407
$868$	0	0
$869$	−11.2975	−0.383240
$870$	4.84289	0.164189
$871$	22.4968	0.762276
$872$	2.12385	0.0719227
$873$	13.0876	0.442947
$874$	−49.2393	−1.66555
$875$	0	0
$876$	−25.7142	−0.868803
$877$	37.8835	1.27924	0.639618	−	0.768693i	$-0.279092\pi$
$877$	37.8835	1.27924	0.639618	+	0.768693i	$0.279092\pi$
$878$	25.7027	0.867425
$879$	29.3286	0.989230
$880$	3.36672	0.113492
$881$	−25.0332	−0.843390	−0.421695	−	0.906738i	$-0.638565\pi$
$881$	−25.0332	−0.843390	−0.421695	+	0.906738i	$0.638565\pi$
$882$	0	0
$883$	−29.0617	−0.978006	−0.489003	−	0.872282i	$-0.662639\pi$
$883$	−29.0617	−0.978006	−0.489003	+	0.872282i	$0.662639\pi$
$884$	−14.3931	−0.484091
$885$	−16.2446	−0.546056
$886$	1.28874	0.0432961
$887$	−22.8066	−0.765772	−0.382886	−	0.923796i	$-0.625070\pi$
$887$	−22.8066	−0.765772	−0.382886	+	0.923796i	$0.625070\pi$
$888$	4.90583	0.164629
$889$	0	0
$890$	−7.28290	−0.244123
$891$	−5.31905	−0.178195
$892$	−13.8891	−0.465042
$893$	9.02332	0.301954
$894$	46.5099	1.55552
$895$	4.44330	0.148523
$896$	0	0
$897$	−35.7797	−1.19465
$898$	46.5217	1.55245
$899$	5.44093	0.181465
$900$	5.63312	0.187771
$901$	27.6775	0.922070
$902$	−5.34830	−0.178079
$903$	0	0
$904$	−0.301515	−0.0100282
$905$	9.78079	0.325125
$906$	105.083	3.49113
$907$	−11.3620	−0.377268	−0.188634	−	0.982047i	$-0.560406\pi$
$907$	−11.3620	−0.377268	−0.188634	+	0.982047i	$0.560406\pi$
$908$	−18.5993	−0.617240
$909$	−25.8472	−0.857297
$910$	0	0
$911$	−36.9986	−1.22582	−0.612910	−	0.790153i	$-0.710001\pi$
$911$	−36.9986	−1.22582	−0.612910	+	0.790153i	$0.710001\pi$
$912$	68.0041	2.25184
$913$	2.81810	0.0932655
$914$	−5.62634	−0.186103
$915$	−33.4771	−1.10672
$916$	−25.2496	−0.834269
$917$	0	0
$918$	−6.70728	−0.221373
$919$	−34.2345	−1.12929	−0.564647	−	0.825333i	$-0.690987\pi$
$919$	−34.2345	−1.12929	−0.564647	+	0.825333i	$0.690987\pi$
$920$	−3.27791	−0.108069
$921$	4.33145	0.142726
$922$	−36.2742	−1.19463
$923$	1.30608	0.0429901
$924$	0	0
$925$	2.63378	0.0865983
$926$	−41.3103	−1.35754
$927$	−34.1748	−1.12245
$928$	−7.32439	−0.240435
$929$	41.7101	1.36846	0.684231	−	0.729265i	$-0.260138\pi$
$929$	41.7101	1.36846	0.684231	+	0.729265i	$0.260138\pi$
$930$	26.3498	0.864045
$931$	0	0
$932$	7.86895	0.257756
$933$	0.431026	0.0141112
$934$	−44.8121	−1.46630
$935$	−2.07111	−0.0677326
$936$	7.99273	0.261251
$937$	1.75298	0.0572674	0.0286337	−	0.999590i	$-0.490884\pi$
$937$	1.75298	0.0572674	0.0286337	+	0.999590i	$0.490884\pi$
$938$	0	0
$939$	−76.3323	−2.49101
$940$	−2.52289	−0.0822877
$941$	−5.56169	−0.181306	−0.0906530	−	0.995883i	$-0.528895\pi$
$941$	−5.56169	−0.181306	−0.0906530	+	0.995883i	$0.528895\pi$
$942$	−81.3510	−2.65056
$943$	17.3056	0.563549
$944$	29.4744	0.959311
$945$	0	0
$946$	8.65053	0.281253
$947$	−53.7143	−1.74548	−0.872741	−	0.488184i	$-0.837659\pi$
$947$	−53.7143	−1.74548	−0.872741	+	0.488184i	$0.837659\pi$
$948$	63.8030	2.07223
$949$	−19.5879	−0.635849
$950$	10.9855	0.356417
$951$	44.1415	1.43139
$952$	0	0
$953$	26.2751	0.851132	0.425566	−	0.904927i	$-0.360075\pi$
$953$	26.2751	0.851132	0.425566	+	0.904927i	$0.360075\pi$
$954$	64.5530	2.08998
$955$	18.9046	0.611740
$956$	7.11944	0.230259
$957$	−1.85554	−0.0599811
$958$	−8.18688	−0.264506
$959$	0	0
$960$	−11.9304	−0.385052
$961$	−1.39629	−0.0450416
$962$	−15.6955	−0.506043
$963$	−0.822778	−0.0265137
$964$	−22.1912	−0.714730
$965$	−15.5798	−0.501531
$966$	0	0
$967$	−10.0449	−0.323022	−0.161511	−	0.986871i	$-0.551637\pi$
$967$	−10.0449	−0.323022	−0.161511	+	0.986871i	$0.551637\pi$
$968$	7.65633	0.246084
$969$	−41.8342	−1.34391
$970$	−7.13613	−0.229127
$971$	12.9787	0.416505	0.208253	−	0.978075i	$-0.433222\pi$
$971$	12.9787	0.416505	0.208253	+	0.978075i	$0.433222\pi$
$972$	36.0529	1.15640
$973$	0	0
$974$	5.66388	0.181482
$975$	7.98261	0.255648
$976$	60.7414	1.94429
$977$	34.9178	1.11712	0.558560	−	0.829464i	$-0.311354\pi$
$977$	34.9178	1.11712	0.558560	+	0.829464i	$0.311354\pi$
$978$	−113.172	−3.61884
$979$	2.79042	0.0891822
$980$	0	0
$981$	−10.1273	−0.323339
$982$	45.7525	1.46002
$983$	32.0694	1.02286	0.511428	−	0.859326i	$-0.329117\pi$
$983$	32.0694	1.02286	0.511428	+	0.859326i	$0.329117\pi$
$984$	−7.19164	−0.229261
$985$	15.5935	0.496850
$986$	5.40552	0.172147
$987$	0	0
$988$	−29.2507	−0.930588
$989$	−27.9907	−0.890054
$990$	−4.83051	−0.153524
$991$	−39.3726	−1.25071	−0.625356	−	0.780340i	$-0.715046\pi$
$991$	−39.3726	−1.25071	−0.625356	+	0.780340i	$0.715046\pi$
$992$	−39.8515	−1.26529
$993$	35.5464	1.12803
$994$	0	0
$995$	−0.403048	−0.0127775
$996$	−15.9154	−0.504298
$997$	36.3834	1.15227	0.576137	−	0.817353i	$-0.304559\pi$
$997$	36.3834	1.15227	0.576137	+	0.817353i	$0.304559\pi$
$998$	−15.8815	−0.502719
$999$	−3.26805	−0.103397

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7105.2.a.t.1.7		8
7.6	odd	2		1015.2.a.l.1.7	✓	8
21.20	even	2		9135.2.a.bh.1.2		8
35.34	odd	2		5075.2.a.ba.1.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1015.2.a.l.1.7	✓	8	7.6	odd	2
5075.2.a.ba.1.2		8	35.34	odd	2
7105.2.a.t.1.7		8	1.1	even	1	trivial
9135.2.a.bh.1.2		8	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)

\(f(q)\)	\(=\)	\(q+1.90142 q^{2} -2.54699 q^{3} +1.61538 q^{4} -1.00000 q^{5} -4.84289 q^{6} -0.731315 q^{8} +3.48717 q^{9} -1.90142 q^{10} +0.728522 q^{11} -4.11437 q^{12} -3.13413 q^{13} +2.54699 q^{15} -4.62130 q^{16} +2.84289 q^{17} +6.63056 q^{18} +5.77754 q^{19} -1.61538 q^{20} +1.38522 q^{22} -4.48221 q^{23} +1.86265 q^{24} +1.00000 q^{25} -5.95929 q^{26} -1.24082 q^{27} +1.00000 q^{29} +4.84289 q^{30} +5.44093 q^{31} -7.32439 q^{32} -1.85554 q^{33} +5.40552 q^{34} +5.63312 q^{36} +2.63378 q^{37} +10.9855 q^{38} +7.98261 q^{39} +0.731315 q^{40} -3.86096 q^{41} +6.24486 q^{43} +1.17684 q^{44} -3.48717 q^{45} -8.52254 q^{46} +1.56179 q^{47} +11.7704 q^{48} +1.90142 q^{50} -7.24083 q^{51} -5.06283 q^{52} +9.73567 q^{53} -2.35931 q^{54} -0.728522 q^{55} -14.7153 q^{57} +1.90142 q^{58} -6.37795 q^{59} +4.11437 q^{60} -13.1438 q^{61} +10.3455 q^{62} -4.68411 q^{64} +3.13413 q^{65} -3.52816 q^{66} -7.17801 q^{67} +4.59236 q^{68} +11.4162 q^{69} -0.416727 q^{71} -2.55022 q^{72} +6.24986 q^{73} +5.00792 q^{74} -2.54699 q^{75} +9.33294 q^{76} +15.1783 q^{78} -15.5074 q^{79} +4.62130 q^{80} -7.30115 q^{81} -7.34129 q^{82} +3.86824 q^{83} -2.84289 q^{85} +11.8741 q^{86} -2.54699 q^{87} -0.532780 q^{88} +3.83025 q^{89} -6.63056 q^{90} -7.24049 q^{92} -13.8580 q^{93} +2.96962 q^{94} -5.77754 q^{95} +18.6552 q^{96} +3.75306 q^{97} +2.54048 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{2} - 4 q^{3} + 5 q^{4} - 8 q^{5} + 6 q^{6} - 6 q^{8} + 14 q^{9} - q^{10} + q^{11} + 7 q^{12} + q^{13} + 4 q^{15} + 3 q^{16} - 22 q^{17} + 13 q^{18} - 8 q^{19} - 5 q^{20} - 3 q^{22} + 9 q^{23}+ \cdots - 63 q^{99}+O(q^{100}) \) 8 * q + q^2 - 4 * q^3 + 5 * q^4 - 8 * q^5 + 6 * q^6 - 6 * q^8 + 14 * q^9 - q^10 + q^11 + 7 * q^12 + q^13 + 4 * q^15 + 3 * q^16 - 22 * q^17 + 13 * q^18 - 8 * q^19 - 5 * q^20 - 3 * q^22 + 9 * q^23 - 16 * q^24 + 8 * q^25 - 27 * q^26 - q^27 + 8 * q^29 - 6 * q^30 - 3 * q^31 - 26 * q^32 - 14 * q^33 - q^34 - 13 * q^36 + 9 * q^37 + 5 * q^38 + 3 * q^39 + 6 * q^40 - 24 * q^41 + 7 * q^43 + 10 * q^44 - 14 * q^45 - 27 * q^46 - 21 * q^47 + 8 * q^48 + q^50 - 8 * q^51 + 7 * q^52 - 2 * q^53 + 33 * q^54 - q^55 - 10 * q^57 + q^58 - 18 * q^59 - 7 * q^60 - 26 * q^61 - 15 * q^62 + 4 * q^64 - q^65 - 55 * q^66 - 8 * q^67 - 6 * q^68 + 2 * q^69 - 12 * q^71 + 9 * q^72 + q^73 + 14 * q^74 - 4 * q^75 - 15 * q^76 + 8 * q^78 - 14 * q^79 - 3 * q^80 + 40 * q^81 + 42 * q^82 + 10 * q^83 + 22 * q^85 + 19 * q^86 - 4 * q^87 - 29 * q^88 + 4 * q^89 - 13 * q^90 + 47 * q^92 - 24 * q^93 + 30 * q^94 + 8 * q^95 + q^96 + 11 * q^97 - 63 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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