LMFDB - Embedded newform 7105.2.a.t.1.4

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			1.4
Root			$-0.387528$ 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)

$f(q)$	$=$	\(q-0.387528 q^{2} -2.91250 q^{3} -1.84982 q^{4} -1.00000 q^{5} +1.12867 q^{6} +1.49191 q^{8} +5.48263 q^{9} +0.387528 q^{10} +4.25893 q^{11} +5.38760 q^{12} +6.60412 q^{13} +2.91250 q^{15} +3.12149 q^{16} -3.12867 q^{17} -2.12467 q^{18} -5.54933 q^{19} +1.84982 q^{20} -1.65045 q^{22} -7.98824 q^{23} -4.34519 q^{24} +1.00000 q^{25} -2.55928 q^{26} -7.23066 q^{27} +1.00000 q^{29} -1.12867 q^{30} -2.94888 q^{31} -4.19349 q^{32} -12.4041 q^{33} +1.21245 q^{34} -10.1419 q^{36} +5.12612 q^{37} +2.15052 q^{38} -19.2345 q^{39} -1.49191 q^{40} +3.38274 q^{41} +2.69153 q^{43} -7.87825 q^{44} -5.48263 q^{45} +3.09567 q^{46} +3.20158 q^{47} -9.09131 q^{48} -0.387528 q^{50} +9.11225 q^{51} -12.2164 q^{52} -7.50909 q^{53} +2.80208 q^{54} -4.25893 q^{55} +16.1624 q^{57} -0.387528 q^{58} -8.44024 q^{59} -5.38760 q^{60} -12.2753 q^{61} +1.14277 q^{62} -4.61788 q^{64} -6.60412 q^{65} +4.80694 q^{66} +13.3558 q^{67} +5.78749 q^{68} +23.2657 q^{69} +0.404452 q^{71} +8.17962 q^{72} +14.1367 q^{73} -1.98652 q^{74} -2.91250 q^{75} +10.2653 q^{76} +7.45390 q^{78} -10.0697 q^{79} -3.12149 q^{80} +4.61137 q^{81} -1.31091 q^{82} -13.8469 q^{83} +3.12867 q^{85} -1.04304 q^{86} -2.91250 q^{87} +6.35395 q^{88} +7.62983 q^{89} +2.12467 q^{90} +14.7768 q^{92} +8.58859 q^{93} -1.24070 q^{94} +5.54933 q^{95} +12.2135 q^{96} +13.7254 q^{97} +23.3501 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.387528	−0.274024	−0.137012	−	0.990569i	$-0.543750\pi$
$2$	−0.387528	−0.274024	−0.137012	+	0.990569i	$0.543750\pi$
$3$	−2.91250	−1.68153	−0.840765	−	0.541400i	$-0.817895\pi$
$3$	−2.91250	−1.68153	−0.840765	+	0.541400i	$0.817895\pi$
$4$	−1.84982	−0.924911
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.12867	0.460779
$7$	0	0
$7$	0	0
$8$	1.49191	0.527471
$9$	5.48263	1.82754
$10$	0.387528	0.122547
$11$	4.25893	1.28411	0.642057	−	0.766657i	$-0.278081\pi$
$11$	4.25893	1.28411	0.642057	+	0.766657i	$0.278081\pi$
$12$	5.38760	1.55527
$13$	6.60412	1.83165	0.915827	−	0.401574i	$-0.131537\pi$
$13$	6.60412	1.83165	0.915827	+	0.401574i	$0.131537\pi$
$14$	0	0
$15$	2.91250	0.752003
$16$	3.12149	0.780371
$17$	−3.12867	−0.758815	−0.379407	−	0.925230i	$-0.623872\pi$
$17$	−3.12867	−0.758815	−0.379407	+	0.925230i	$0.623872\pi$
$18$	−2.12467	−0.500791
$19$	−5.54933	−1.27310	−0.636552	−	0.771234i	$-0.719640\pi$
$19$	−5.54933	−1.27310	−0.636552	+	0.771234i	$0.719640\pi$
$20$	1.84982	0.413633
$21$	0	0
$22$	−1.65045	−0.351878
$23$	−7.98824	−1.66566	−0.832832	−	0.553526i	$-0.813282\pi$
$23$	−7.98824	−1.66566	−0.832832	+	0.553526i	$0.813282\pi$
$24$	−4.34519	−0.886959
$25$	1.00000	0.200000
$26$	−2.55928	−0.501916
$27$	−7.23066	−1.39154
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	−1.12867	−0.206067
$31$	−2.94888	−0.529634	−0.264817	−	0.964299i	$-0.585312\pi$
$31$	−2.94888	−0.529634	−0.264817	+	0.964299i	$0.585312\pi$
$32$	−4.19349	−0.741312
$33$	−12.4041	−2.15928
$34$	1.21245	0.207933
$35$	0	0
$36$	−10.1419	−1.69032
$37$	5.12612	0.842730	0.421365	−	0.906891i	$-0.361551\pi$
$37$	5.12612	0.842730	0.421365	+	0.906891i	$0.361551\pi$
$38$	2.15052	0.348860
$39$	−19.2345	−3.07998
$40$	−1.49191	−0.235892
$41$	3.38274	0.528296	0.264148	−	0.964482i	$-0.414909\pi$
$41$	3.38274	0.528296	0.264148	+	0.964482i	$0.414909\pi$
$42$	0	0
$43$	2.69153	0.410454	0.205227	−	0.978714i	$-0.434207\pi$
$43$	2.69153	0.410454	0.205227	+	0.978714i	$0.434207\pi$
$44$	−7.87825	−1.18769
$45$	−5.48263	−0.817303
$46$	3.09567	0.456431
$47$	3.20158	0.466999	0.233499	−	0.972357i	$-0.424982\pi$
$47$	3.20158	0.466999	0.233499	+	0.972357i	$0.424982\pi$
$48$	−9.09131	−1.31222
$49$	0	0
$50$	−0.387528	−0.0548047
$51$	9.11225	1.27597
$52$	−12.2164	−1.69412
$53$	−7.50909	−1.03145	−0.515726	−	0.856754i	$-0.672478\pi$
$53$	−7.50909	−1.03145	−0.515726	+	0.856754i	$0.672478\pi$
$54$	2.80208	0.381315
$55$	−4.25893	−0.574273
$56$	0	0
$57$	16.1624	2.14076
$58$	−0.387528	−0.0508849
$59$	−8.44024	−1.09883	−0.549413	−	0.835551i	$-0.685149\pi$
$59$	−8.44024	−1.09883	−0.549413	+	0.835551i	$0.685149\pi$
$60$	−5.38760	−0.695536
$61$	−12.2753	−1.57169	−0.785847	−	0.618421i	$-0.787773\pi$
$61$	−12.2753	−1.57169	−0.785847	+	0.618421i	$0.787773\pi$
$62$	1.14277	0.145132
$63$	0	0
$64$	−4.61788	−0.577234
$65$	−6.60412	−0.819140
$66$	4.80694	0.591693
$67$	13.3558	1.63167	0.815836	−	0.578283i	$-0.196277\pi$
$67$	13.3558	1.63167	0.815836	+	0.578283i	$0.196277\pi$
$68$	5.78749	0.701836
$69$	23.2657	2.80086
$70$	0	0
$71$	0.404452	0.0479996	0.0239998	−	0.999712i	$-0.492360\pi$
$71$	0.404452	0.0479996	0.0239998	+	0.999712i	$0.492360\pi$
$72$	8.17962	0.963977
$73$	14.1367	1.65457	0.827287	−	0.561779i	$-0.189883\pi$
$73$	14.1367	1.65457	0.827287	+	0.561779i	$0.189883\pi$
$74$	−1.98652	−0.230928
$75$	−2.91250	−0.336306
$76$	10.2653	1.17751
$77$	0	0
$78$	7.45390	0.843988
$79$	−10.0697	−1.13293	−0.566465	−	0.824085i	$-0.691690\pi$
$79$	−10.0697	−1.13293	−0.566465	+	0.824085i	$0.691690\pi$
$80$	−3.12149	−0.348993
$81$	4.61137	0.512375
$82$	−1.31091	−0.144766
$83$	−13.8469	−1.51989	−0.759946	−	0.649986i	$-0.774775\pi$
$83$	−13.8469	−1.51989	−0.759946	+	0.649986i	$0.774775\pi$
$84$	0	0
$85$	3.12867	0.339352
$86$	−1.04304	−0.112474
$87$	−2.91250	−0.312252
$88$	6.35395	0.677333
$89$	7.62983	0.808761	0.404380	−	0.914591i	$-0.367487\pi$
$89$	7.62983	0.808761	0.404380	+	0.914591i	$0.367487\pi$
$90$	2.12467	0.223960
$91$	0	0
$92$	14.7768	1.54059
$93$	8.58859	0.890595
$94$	−1.24070	−0.127969
$95$	5.54933	0.569349
$96$	12.2135	1.24654
$97$	13.7254	1.39360	0.696801	−	0.717265i	$-0.254606\pi$
$97$	13.7254	1.39360	0.696801	+	0.717265i	$0.254606\pi$
$98$	0	0
$99$	23.3501	2.34678
$100$	−1.84982	−0.184982
$101$	1.11539	0.110986	0.0554929	−	0.998459i	$-0.482327\pi$
$101$	1.11539	0.110986	0.0554929	+	0.998459i	$0.482327\pi$
$102$	−3.53125	−0.349646
$103$	11.6449	1.14741	0.573705	−	0.819062i	$-0.305505\pi$
$103$	11.6449	1.14741	0.573705	+	0.819062i	$0.305505\pi$
$104$	9.85278	0.966144
$105$	0	0
$106$	2.90998	0.282642
$107$	1.33922	0.129468	0.0647339	−	0.997903i	$-0.479380\pi$
$107$	1.33922	0.129468	0.0647339	+	0.997903i	$0.479380\pi$
$108$	13.3754	1.28705
$109$	−13.3541	−1.27909	−0.639546	−	0.768753i	$-0.720877\pi$
$109$	−13.3541	−1.27909	−0.639546	+	0.768753i	$0.720877\pi$
$110$	1.65045	0.157365
$111$	−14.9298	−1.41708
$112$	0	0
$113$	2.02022	0.190046	0.0950230	−	0.995475i	$-0.469708\pi$
$113$	2.02022	0.190046	0.0950230	+	0.995475i	$0.469708\pi$
$114$	−6.26338	−0.586619
$115$	7.98824	0.744907
$116$	−1.84982	−0.171752
$117$	36.2080	3.34743
$118$	3.27083	0.301104
$119$	0	0
$120$	4.34519	0.396660
$121$	7.13845	0.648950
$122$	4.75703	0.430681
$123$	−9.85223	−0.888346
$124$	5.45490	0.489864
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−13.7287	−1.21822	−0.609111	−	0.793085i	$-0.708474\pi$
$127$	−13.7287	−1.21822	−0.609111	+	0.793085i	$0.708474\pi$
$128$	10.1765	0.899487
$129$	−7.83906	−0.690191
$130$	2.55928	0.224464
$131$	−1.87495	−0.163815	−0.0819074	−	0.996640i	$-0.526101\pi$
$131$	−1.87495	−0.163815	−0.0819074	+	0.996640i	$0.526101\pi$
$132$	22.9454	1.99714
$133$	0	0
$134$	−5.17575	−0.447117
$135$	7.23066	0.622316
$136$	−4.66771	−0.400253
$137$	−10.6285	−0.908054	−0.454027	−	0.890988i	$-0.650013\pi$
$137$	−10.6285	−0.908054	−0.454027	+	0.890988i	$0.650013\pi$
$138$	−9.01612	−0.767503
$139$	−2.00407	−0.169983	−0.0849915	−	0.996382i	$-0.527086\pi$
$139$	−2.00407	−0.169983	−0.0849915	+	0.996382i	$0.527086\pi$
$140$	0	0
$141$	−9.32460	−0.785273
$142$	−0.156736	−0.0131530
$143$	28.1265	2.35205
$144$	17.1140	1.42616
$145$	−1.00000	−0.0830455
$146$	−5.47836	−0.453393
$147$	0	0
$148$	−9.48242	−0.779450
$149$	10.7388	0.879754	0.439877	−	0.898058i	$-0.355022\pi$
$149$	10.7388	0.879754	0.439877	+	0.898058i	$0.355022\pi$
$150$	1.12867	0.0921558
$151$	10.3197	0.839808	0.419904	−	0.907568i	$-0.362064\pi$
$151$	10.3197	0.839808	0.419904	+	0.907568i	$0.362064\pi$
$152$	−8.27912	−0.671525
$153$	−17.1534	−1.38677
$154$	0	0
$155$	2.94888	0.236859
$156$	35.5804	2.84871
$157$	6.66586	0.531994	0.265997	−	0.963974i	$-0.414299\pi$
$157$	6.66586	0.531994	0.265997	+	0.963974i	$0.414299\pi$
$158$	3.90230	0.310450
$159$	21.8702	1.73442
$160$	4.19349	0.331525
$161$	0	0
$162$	−1.78704	−0.140403
$163$	4.53401	0.355131	0.177566	−	0.984109i	$-0.443178\pi$
$163$	4.53401	0.355131	0.177566	+	0.984109i	$0.443178\pi$
$164$	−6.25747	−0.488627
$165$	12.4041	0.965658
$166$	5.36605	0.416486
$167$	−15.4848	−1.19825	−0.599126	−	0.800655i	$-0.704485\pi$
$167$	−15.4848	−1.19825	−0.599126	+	0.800655i	$0.704485\pi$
$168$	0	0
$169$	30.6144	2.35495
$170$	−1.21245	−0.0929906
$171$	−30.4249	−2.32665
$172$	−4.97885	−0.379634
$173$	14.6533	1.11407	0.557033	−	0.830490i	$-0.311939\pi$
$173$	14.6533	1.11407	0.557033	+	0.830490i	$0.311939\pi$
$174$	1.12867	0.0855646
$175$	0	0
$176$	13.2942	1.00209
$177$	24.5822	1.84771
$178$	−2.95677	−0.221620
$179$	−17.2343	−1.28815	−0.644077	−	0.764961i	$-0.722758\pi$
$179$	−17.2343	−1.28815	−0.644077	+	0.764961i	$0.722758\pi$
$180$	10.1419	0.755932
$181$	12.7510	0.947774	0.473887	−	0.880586i	$-0.342851\pi$
$181$	12.7510	0.947774	0.473887	+	0.880586i	$0.342851\pi$
$182$	0	0
$183$	35.7518	2.64285
$184$	−11.9178	−0.878590
$185$	−5.12612	−0.376880
$186$	−3.32832	−0.244044
$187$	−13.3248	−0.974405
$188$	−5.92236	−0.431932
$189$	0	0
$190$	−2.15052	−0.156015
$191$	1.70820	0.123601	0.0618006	−	0.998089i	$-0.480316\pi$
$191$	1.70820	0.123601	0.0618006	+	0.998089i	$0.480316\pi$
$192$	13.4495	0.970637
$193$	−26.0117	−1.87237	−0.936183	−	0.351513i	$-0.885667\pi$
$193$	−26.0117	−1.87237	−0.936183	+	0.351513i	$0.885667\pi$
$194$	−5.31897	−0.381880
$195$	19.2345	1.37741
$196$	0	0
$197$	9.48048	0.675456	0.337728	−	0.941244i	$-0.390342\pi$
$197$	9.48048	0.675456	0.337728	+	0.941244i	$0.390342\pi$
$198$	−9.04883	−0.643072
$199$	19.8037	1.40384	0.701922	−	0.712254i	$-0.252326\pi$
$199$	19.8037	1.40384	0.701922	+	0.712254i	$0.252326\pi$
$200$	1.49191	0.105494
$201$	−38.8988	−2.74371
$202$	−0.432247	−0.0304128
$203$	0	0
$204$	−16.8560	−1.18016
$205$	−3.38274	−0.236261
$206$	−4.51274	−0.314418
$207$	−43.7966	−3.04407
$208$	20.6147	1.42937
$209$	−23.6342	−1.63481
$210$	0	0
$211$	1.71986	0.118400	0.0592002	−	0.998246i	$-0.481145\pi$
$211$	1.71986	0.118400	0.0592002	+	0.998246i	$0.481145\pi$
$212$	13.8905	0.954001
$213$	−1.17796	−0.0807127
$214$	−0.518987	−0.0354772
$215$	−2.69153	−0.183561
$216$	−10.7875	−0.733998
$217$	0	0
$218$	5.17509	0.350501
$219$	−41.1731	−2.78222
$220$	7.87825	0.531152
$221$	−20.6621	−1.38989
$222$	5.78572	0.388312
$223$	−15.7503	−1.05472	−0.527360	−	0.849642i	$-0.676818\pi$
$223$	−15.7503	−1.05472	−0.527360	+	0.849642i	$0.676818\pi$
$224$	0	0
$225$	5.48263	0.365509
$226$	−0.782891	−0.0520771
$227$	2.77187	0.183976	0.0919878	−	0.995760i	$-0.470678\pi$
$227$	2.77187	0.183976	0.0919878	+	0.995760i	$0.470678\pi$
$228$	−29.8976	−1.98001
$229$	−27.5154	−1.81827	−0.909133	−	0.416506i	$-0.863255\pi$
$229$	−27.5154	−1.81827	−0.909133	+	0.416506i	$0.863255\pi$
$230$	−3.09567	−0.204122
$231$	0	0
$232$	1.49191	0.0979490
$233$	5.90211	0.386660	0.193330	−	0.981134i	$-0.438071\pi$
$233$	5.90211	0.386660	0.193330	+	0.981134i	$0.438071\pi$
$234$	−14.0316	−0.917275
$235$	−3.20158	−0.208848
$236$	15.6129	1.01632
$237$	29.3280	1.90506
$238$	0	0
$239$	−17.2425	−1.11532	−0.557661	−	0.830069i	$-0.688301\pi$
$239$	−17.2425	−1.11532	−0.557661	+	0.830069i	$0.688301\pi$
$240$	9.09131	0.586842
$241$	23.9369	1.54191	0.770955	−	0.636889i	$-0.219779\pi$
$241$	23.9369	1.54191	0.770955	+	0.636889i	$0.219779\pi$
$242$	−2.76635	−0.177828
$243$	8.26138	0.529968
$244$	22.7072	1.45368
$245$	0	0
$246$	3.81801	0.243428
$247$	−36.6484	−2.33188
$248$	−4.39947	−0.279367
$249$	40.3290	2.55574
$250$	0.387528	0.0245094
$251$	4.81069	0.303648	0.151824	−	0.988408i	$-0.451485\pi$
$251$	4.81069	0.303648	0.151824	+	0.988408i	$0.451485\pi$
$252$	0	0
$253$	−34.0213	−2.13890
$254$	5.32025	0.333822
$255$	−9.11225	−0.570631
$256$	5.29206	0.330754
$257$	29.2997	1.82767	0.913833	−	0.406091i	$-0.133108\pi$
$257$	29.2997	1.82767	0.913833	+	0.406091i	$0.133108\pi$
$258$	3.03786	0.189129
$259$	0	0
$260$	12.2164	0.757632
$261$	5.48263	0.339367
$262$	0.726594	0.0448891
$263$	3.41512	0.210586	0.105293	−	0.994441i	$-0.466422\pi$
$263$	3.41512	0.210586	0.105293	+	0.994441i	$0.466422\pi$
$264$	−18.5059	−1.13896
$265$	7.50909	0.461279
$266$	0	0
$267$	−22.2219	−1.35996
$268$	−24.7059	−1.50915
$269$	9.99894	0.609646	0.304823	−	0.952409i	$-0.401403\pi$
$269$	9.99894	0.609646	0.304823	+	0.952409i	$0.401403\pi$
$270$	−2.80208	−0.170529
$271$	24.6325	1.49632	0.748159	−	0.663520i	$-0.230938\pi$
$271$	24.6325	1.49632	0.748159	+	0.663520i	$0.230938\pi$
$272$	−9.76611	−0.592157
$273$	0	0
$274$	4.11884	0.248828
$275$	4.25893	0.256823
$276$	−43.0374	−2.59055
$277$	−7.87602	−0.473224	−0.236612	−	0.971604i	$-0.576037\pi$
$277$	−7.87602	−0.473224	−0.236612	+	0.971604i	$0.576037\pi$
$278$	0.776634	0.0465794
$279$	−16.1676	−0.967929
$280$	0	0
$281$	−15.1632	−0.904559	−0.452279	−	0.891876i	$-0.649389\pi$
$281$	−15.1632	−0.904559	−0.452279	+	0.891876i	$0.649389\pi$
$282$	3.61354	0.215183
$283$	17.0576	1.01397	0.506986	−	0.861954i	$-0.330760\pi$
$283$	17.0576	1.01397	0.506986	+	0.861954i	$0.330760\pi$
$284$	−0.748163	−0.0443953
$285$	−16.1624	−0.957378
$286$	−10.8998	−0.644518
$287$	0	0
$288$	−22.9914	−1.35478
$289$	−7.21140	−0.424200
$290$	0.387528	0.0227564
$291$	−39.9751	−2.34338
$292$	−26.1504	−1.53033
$293$	−28.1268	−1.64319	−0.821593	−	0.570075i	$-0.806914\pi$
$293$	−28.1268	−1.64319	−0.821593	+	0.570075i	$0.806914\pi$
$294$	0	0
$295$	8.44024	0.491410
$296$	7.64774	0.444516
$297$	−30.7949	−1.78690
$298$	−4.16157	−0.241073
$299$	−52.7553	−3.05092
$300$	5.38760	0.311053
$301$	0	0
$302$	−3.99919	−0.230127
$303$	−3.24858	−0.186626
$304$	−17.3221	−0.993493
$305$	12.2753	0.702883
$306$	6.64741	0.380007
$307$	−9.35594	−0.533972	−0.266986	−	0.963700i	$-0.586028\pi$
$307$	−9.35594	−0.533972	−0.266986	+	0.963700i	$0.586028\pi$
$308$	0	0
$309$	−33.9159	−1.92941
$310$	−1.14277	−0.0649051
$311$	−21.7424	−1.23290	−0.616449	−	0.787395i	$-0.711429\pi$
$311$	−21.7424	−1.23290	−0.616449	+	0.787395i	$0.711429\pi$
$312$	−28.6962	−1.62460
$313$	−15.7143	−0.888223	−0.444112	−	0.895971i	$-0.646481\pi$
$313$	−15.7143	−0.888223	−0.444112	+	0.895971i	$0.646481\pi$
$314$	−2.58321	−0.145779
$315$	0	0
$316$	18.6272	1.04786
$317$	−20.1684	−1.13277	−0.566386	−	0.824140i	$-0.691659\pi$
$317$	−20.1684	−1.13277	−0.566386	+	0.824140i	$0.691659\pi$
$318$	−8.47531	−0.475272
$319$	4.25893	0.238454
$320$	4.61788	0.258147
$321$	−3.90049	−0.217704
$322$	0	0
$323$	17.3620	0.966050
$324$	−8.53022	−0.473901
$325$	6.60412	0.366331
$326$	−1.75706	−0.0973144
$327$	38.8938	2.15083
$328$	5.04676	0.278661
$329$	0	0
$330$	−4.80694	−0.264613
$331$	−6.26857	−0.344552	−0.172276	−	0.985049i	$-0.555112\pi$
$331$	−6.26857	−0.344552	−0.172276	+	0.985049i	$0.555112\pi$
$332$	25.6143	1.40576
$333$	28.1047	1.54013
$334$	6.00081	0.328350
$335$	−13.3558	−0.729706
$336$	0	0
$337$	3.49092	0.190163	0.0950814	−	0.995470i	$-0.469689\pi$
$337$	3.49092	0.190163	0.0950814	+	0.995470i	$0.469689\pi$
$338$	−11.8639	−0.645313
$339$	−5.88387	−0.319568
$340$	−5.78749	−0.313871
$341$	−12.5590	−0.680110
$342$	11.7905	0.637558
$343$	0	0
$344$	4.01553	0.216503
$345$	−23.2657	−1.25258
$346$	−5.67855	−0.305281
$347$	7.68223	0.412404	0.206202	−	0.978509i	$-0.433890\pi$
$347$	7.68223	0.412404	0.206202	+	0.978509i	$0.433890\pi$
$348$	5.38760	0.288806
$349$	−30.4237	−1.62855	−0.814273	−	0.580482i	$-0.802864\pi$
$349$	−30.4237	−1.62855	−0.814273	+	0.580482i	$0.802864\pi$
$350$	0	0
$351$	−47.7522	−2.54882
$352$	−17.8598	−0.951929
$353$	29.0668	1.54707	0.773535	−	0.633754i	$-0.218487\pi$
$353$	29.0668	1.54707	0.773535	+	0.633754i	$0.218487\pi$
$354$	−9.52628	−0.506316
$355$	−0.404452	−0.0214661
$356$	−14.1138	−0.748032
$357$	0	0
$358$	6.67879	0.352985
$359$	9.50243	0.501519	0.250759	−	0.968049i	$-0.419320\pi$
$359$	9.50243	0.501519	0.250759	+	0.968049i	$0.419320\pi$
$360$	−8.17962	−0.431104
$361$	11.7950	0.620791
$362$	−4.94137	−0.259712
$363$	−20.7907	−1.09123
$364$	0	0
$365$	−14.1367	−0.739948
$366$	−13.8548	−0.724204
$367$	−1.25340	−0.0654267	−0.0327134	−	0.999465i	$-0.510415\pi$
$367$	−1.25340	−0.0654267	−0.0327134	+	0.999465i	$0.510415\pi$
$368$	−24.9352	−1.29984
$369$	18.5463	0.965484
$370$	1.98652	0.103274
$371$	0	0
$372$	−15.8874	−0.823721
$373$	11.1578	0.577730	0.288865	−	0.957370i	$-0.406722\pi$
$373$	11.1578	0.577730	0.288865	+	0.957370i	$0.406722\pi$
$374$	5.16373	0.267010
$375$	2.91250	0.150401
$376$	4.77649	0.246329
$377$	6.60412	0.340129
$378$	0	0
$379$	−25.5310	−1.31144	−0.655721	−	0.755004i	$-0.727635\pi$
$379$	−25.5310	−1.31144	−0.655721	+	0.755004i	$0.727635\pi$
$380$	−10.2653	−0.526597
$381$	39.9847	2.04848
$382$	−0.661976	−0.0338697
$383$	−32.1276	−1.64165	−0.820823	−	0.571183i	$-0.806485\pi$
$383$	−32.1276	−1.64165	−0.820823	+	0.571183i	$0.806485\pi$
$384$	−29.6391	−1.51252
$385$	0	0
$386$	10.0803	0.513073
$387$	14.7567	0.750123
$388$	−25.3895	−1.28896
$389$	−9.23475	−0.468220	−0.234110	−	0.972210i	$-0.575218\pi$
$389$	−9.23475	−0.468220	−0.234110	+	0.972210i	$0.575218\pi$
$390$	−7.45390	−0.377443
$391$	24.9926	1.26393
$392$	0	0
$393$	5.46077	0.275460
$394$	−3.67395	−0.185091
$395$	10.0697	0.506662
$396$	−43.1936	−2.17056
$397$	−17.0178	−0.854100	−0.427050	−	0.904228i	$-0.640447\pi$
$397$	−17.0178	−0.854100	−0.427050	+	0.904228i	$0.640447\pi$
$398$	−7.67447	−0.384687
$399$	0	0
$400$	3.12149	0.156074
$401$	−2.66135	−0.132902	−0.0664508	−	0.997790i	$-0.521168\pi$
$401$	−2.66135	−0.132902	−0.0664508	+	0.997790i	$0.521168\pi$
$402$	15.0744	0.751841
$403$	−19.4747	−0.970105
$404$	−2.06328	−0.102652
$405$	−4.61137	−0.229141
$406$	0	0
$407$	21.8318	1.08216
$408$	13.5947	0.673038
$409$	−20.8013	−1.02856	−0.514278	−	0.857624i	$-0.671940\pi$
$409$	−20.8013	−1.02856	−0.514278	+	0.857624i	$0.671940\pi$
$410$	1.31091	0.0647411
$411$	30.9554	1.52692
$412$	−21.5411	−1.06125
$413$	0	0
$414$	16.9724	0.834149
$415$	13.8469	0.679716
$416$	−27.6943	−1.35783
$417$	5.83685	0.285832
$418$	9.15891	0.447977
$419$	−16.1822	−0.790554	−0.395277	−	0.918562i	$-0.629351\pi$
$419$	−16.1822	−0.790554	−0.395277	+	0.918562i	$0.629351\pi$
$420$	0	0
$421$	−0.766529	−0.0373583	−0.0186792	−	0.999826i	$-0.505946\pi$
$421$	−0.766529	−0.0373583	−0.0186792	+	0.999826i	$0.505946\pi$
$422$	−0.666496	−0.0324445
$423$	17.5531	0.853461
$424$	−11.2029	−0.544061
$425$	−3.12867	−0.151763
$426$	0.456494	0.0221172
$427$	0	0
$428$	−2.47733	−0.119746
$429$	−81.9182	−3.95505
$430$	1.04304	0.0503000
$431$	28.4511	1.37044	0.685219	−	0.728337i	$-0.259706\pi$
$431$	28.4511	1.37044	0.685219	+	0.728337i	$0.259706\pi$
$432$	−22.5704	−1.08592
$433$	23.6539	1.13673	0.568366	−	0.822776i	$-0.307576\pi$
$433$	23.6539	1.13673	0.568366	+	0.822776i	$0.307576\pi$
$434$	0	0
$435$	2.91250	0.139644
$436$	24.7027	1.18305
$437$	44.3294	2.12056
$438$	15.9557	0.762393
$439$	−2.13176	−0.101743	−0.0508717	−	0.998705i	$-0.516200\pi$
$439$	−2.13176	−0.101743	−0.0508717	+	0.998705i	$0.516200\pi$
$440$	−6.35395	−0.302913
$441$	0	0
$442$	8.00716	0.380862
$443$	8.05174	0.382550	0.191275	−	0.981537i	$-0.438738\pi$
$443$	8.05174	0.382550	0.191275	+	0.981537i	$0.438738\pi$
$444$	27.6175	1.31067
$445$	−7.62983	−0.361689
$446$	6.10369	0.289018
$447$	−31.2766	−1.47933
$448$	0	0
$449$	10.8990	0.514354	0.257177	−	0.966364i	$-0.417208\pi$
$449$	10.8990	0.514354	0.257177	+	0.966364i	$0.417208\pi$
$450$	−2.12467	−0.100158
$451$	14.4069	0.678392
$452$	−3.73704	−0.175776
$453$	−30.0562	−1.41216
$454$	−1.07418	−0.0504137
$455$	0	0
$456$	24.1129	1.12919
$457$	−2.27850	−0.106584	−0.0532919	−	0.998579i	$-0.516971\pi$
$457$	−2.27850	−0.106584	−0.0532919	+	0.998579i	$0.516971\pi$
$458$	10.6630	0.498248
$459$	22.6224	1.05592
$460$	−14.7768	−0.688973
$461$	−20.2500	−0.943137	−0.471569	−	0.881829i	$-0.656312\pi$
$461$	−20.2500	−0.943137	−0.471569	+	0.881829i	$0.656312\pi$
$462$	0	0
$463$	20.6864	0.961380	0.480690	−	0.876891i	$-0.340386\pi$
$463$	20.6864	0.961380	0.480690	+	0.876891i	$0.340386\pi$
$464$	3.12149	0.144911
$465$	−8.58859	−0.398286
$466$	−2.28723	−0.105954
$467$	38.2072	1.76802	0.884008	−	0.467471i	$-0.154835\pi$
$467$	38.2072	1.76802	0.884008	+	0.467471i	$0.154835\pi$
$468$	−66.9783	−3.09607
$469$	0	0
$470$	1.24070	0.0572294
$471$	−19.4143	−0.894563
$472$	−12.5921	−0.579599
$473$	11.4630	0.527070
$474$	−11.3654	−0.522031
$475$	−5.54933	−0.254621
$476$	0	0
$477$	−41.1696	−1.88502
$478$	6.68194	0.305625
$479$	−12.5532	−0.573569	−0.286785	−	0.957995i	$-0.592586\pi$
$479$	−12.5532	−0.573569	−0.286785	+	0.957995i	$0.592586\pi$
$480$	−12.2135	−0.557469
$481$	33.8535	1.54359
$482$	−9.27622	−0.422520
$483$	0	0
$484$	−13.2049	−0.600221
$485$	−13.7254	−0.623238
$486$	−3.20152	−0.145224
$487$	10.4749	0.474665	0.237332	−	0.971428i	$-0.423727\pi$
$487$	10.4749	0.474665	0.237332	+	0.971428i	$0.423727\pi$
$488$	−18.3137	−0.829023
$489$	−13.2053	−0.597164
$490$	0	0
$491$	1.51276	0.0682698	0.0341349	−	0.999417i	$-0.489132\pi$
$491$	1.51276	0.0682698	0.0341349	+	0.999417i	$0.489132\pi$
$492$	18.2249	0.821641
$493$	−3.12867	−0.140908
$494$	14.2023	0.638991
$495$	−23.3501	−1.04951
$496$	−9.20487	−0.413311
$497$	0	0
$498$	−15.6286	−0.700335
$499$	−2.17338	−0.0972938	−0.0486469	−	0.998816i	$-0.515491\pi$
$499$	−2.17338	−0.0972938	−0.0486469	+	0.998816i	$0.515491\pi$
$500$	1.84982	0.0827266
$501$	45.0995	2.01490
$502$	−1.86428	−0.0832068
$503$	−16.0874	−0.717301	−0.358650	−	0.933472i	$-0.616763\pi$
$503$	−16.0874	−0.717301	−0.358650	+	0.933472i	$0.616763\pi$
$504$	0	0
$505$	−1.11539	−0.0496344
$506$	13.1842	0.586110
$507$	−89.1643	−3.95993
$508$	25.3956	1.12675
$509$	9.83915	0.436113	0.218056	−	0.975936i	$-0.430028\pi$
$509$	9.83915	0.436113	0.218056	+	0.975936i	$0.430028\pi$
$510$	3.53125	0.156367
$511$	0	0
$512$	−22.4039	−0.990122
$513$	40.1253	1.77158
$514$	−11.3545	−0.500824
$515$	−11.6449	−0.513138
$516$	14.5009	0.638365
$517$	13.6353	0.599680
$518$	0	0
$519$	−42.6776	−1.87334
$520$	−9.85278	−0.432073
$521$	−23.4942	−1.02930	−0.514650	−	0.857401i	$-0.672078\pi$
$521$	−23.4942	−1.02930	−0.514650	+	0.857401i	$0.672078\pi$
$522$	−2.12467	−0.0929945
$523$	7.59605	0.332152	0.166076	−	0.986113i	$-0.446890\pi$
$523$	7.59605	0.332152	0.166076	+	0.986113i	$0.446890\pi$
$524$	3.46832	0.151514
$525$	0	0
$526$	−1.32346	−0.0577054
$527$	9.22607	0.401894
$528$	−38.7192	−1.68504
$529$	40.8120	1.77443
$530$	−2.90998	−0.126402
$531$	−46.2748	−2.00815
$532$	0	0
$533$	22.3400	0.967655
$534$	8.61159	0.372660
$535$	−1.33922	−0.0578997
$536$	19.9257	0.860660
$537$	50.1949	2.16607
$538$	−3.87487	−0.167057
$539$	0	0
$540$	−13.3754	−0.575587
$541$	13.3251	0.572893	0.286446	−	0.958096i	$-0.407526\pi$
$541$	13.3251	0.572893	0.286446	+	0.958096i	$0.407526\pi$
$542$	−9.54578	−0.410026
$543$	−37.1372	−1.59371
$544$	13.1201	0.562518
$545$	13.3541	0.572027
$546$	0	0
$547$	27.4053	1.17177	0.585884	−	0.810395i	$-0.300747\pi$
$547$	27.4053	1.17177	0.585884	+	0.810395i	$0.300747\pi$
$548$	19.6608	0.839869
$549$	−67.3011	−2.87234
$550$	−1.65045	−0.0703756
$551$	−5.54933	−0.236409
$552$	34.7105	1.47738
$553$	0	0
$554$	3.05218	0.129675
$555$	14.9298	0.633735
$556$	3.70717	0.157219
$557$	37.9220	1.60680	0.803402	−	0.595437i	$-0.203021\pi$
$557$	37.9220	1.60680	0.803402	+	0.595437i	$0.203021\pi$
$558$	6.26540	0.265236
$559$	17.7752	0.751810
$560$	0	0
$561$	38.8084	1.63849
$562$	5.87615	0.247871
$563$	−19.3408	−0.815119	−0.407560	−	0.913179i	$-0.633620\pi$
$563$	−19.3408	−0.815119	−0.407560	+	0.913179i	$0.633620\pi$
$564$	17.2488	0.726308
$565$	−2.02022	−0.0849912
$566$	−6.61031	−0.277852
$567$	0	0
$568$	0.603407	0.0253184
$569$	13.9586	0.585175	0.292587	−	0.956239i	$-0.405484\pi$
$569$	13.9586	0.585175	0.292587	+	0.956239i	$0.405484\pi$
$570$	6.26338	0.262344
$571$	−19.2257	−0.804572	−0.402286	−	0.915514i	$-0.631784\pi$
$571$	−19.2257	−0.804572	−0.402286	+	0.915514i	$0.631784\pi$
$572$	−52.0289	−2.17544
$573$	−4.97513	−0.207839
$574$	0	0
$575$	−7.98824	−0.333133
$576$	−25.3181	−1.05492
$577$	−2.82037	−0.117414	−0.0587068	−	0.998275i	$-0.518698\pi$
$577$	−2.82037	−0.117414	−0.0587068	+	0.998275i	$0.518698\pi$
$578$	2.79462	0.116241
$579$	75.7591	3.14844
$580$	1.84982	0.0768097
$581$	0	0
$582$	15.4915	0.642143
$583$	−31.9806	−1.32450
$584$	21.0907	0.872740
$585$	−36.2080	−1.49702
$586$	10.8999	0.450272
$587$	46.3588	1.91343	0.956717	−	0.291021i	$-0.0939949\pi$
$587$	46.3588	1.91343	0.956717	+	0.291021i	$0.0939949\pi$
$588$	0	0
$589$	16.3643	0.674278
$590$	−3.27083	−0.134658
$591$	−27.6119	−1.13580
$592$	16.0011	0.657642
$593$	−16.6020	−0.681762	−0.340881	−	0.940106i	$-0.610725\pi$
$593$	−16.6020	−0.681762	−0.340881	+	0.940106i	$0.610725\pi$
$594$	11.9339	0.489653
$595$	0	0
$596$	−19.8648	−0.813694
$597$	−57.6781	−2.36061
$598$	20.4442	0.836024
$599$	−29.2006	−1.19310	−0.596552	−	0.802574i	$-0.703463\pi$
$599$	−29.2006	−1.19310	−0.596552	+	0.802574i	$0.703463\pi$
$600$	−4.34519	−0.177392
$601$	−28.9695	−1.18169	−0.590845	−	0.806785i	$-0.701205\pi$
$601$	−28.9695	−1.18169	−0.590845	+	0.806785i	$0.701205\pi$
$602$	0	0
$603$	73.2250	2.98195
$604$	−19.0897	−0.776748
$605$	−7.13845	−0.290219
$606$	1.25892	0.0511400
$607$	7.90667	0.320922	0.160461	−	0.987042i	$-0.448702\pi$
$607$	7.90667	0.320922	0.160461	+	0.987042i	$0.448702\pi$
$608$	23.2711	0.943766
$609$	0	0
$610$	−4.75703	−0.192607
$611$	21.1436	0.855380
$612$	31.7307	1.28264
$613$	9.40608	0.379908	0.189954	−	0.981793i	$-0.439166\pi$
$613$	9.40608	0.379908	0.189954	+	0.981793i	$0.439166\pi$
$614$	3.62569	0.146321
$615$	9.85223	0.397280
$616$	0	0
$617$	−0.595219	−0.0239626	−0.0119813	−	0.999928i	$-0.503814\pi$
$617$	−0.595219	−0.0239626	−0.0119813	+	0.999928i	$0.503814\pi$
$618$	13.1433	0.528703
$619$	9.33172	0.375073	0.187537	−	0.982258i	$-0.439950\pi$
$619$	9.33172	0.375073	0.187537	+	0.982258i	$0.439950\pi$
$620$	−5.45490	−0.219074
$621$	57.7603	2.31784
$622$	8.42579	0.337843
$623$	0	0
$624$	−60.0401	−2.40353
$625$	1.00000	0.0400000
$626$	6.08972	0.243394
$627$	68.8344	2.74898
$628$	−12.3307	−0.492047
$629$	−16.0380	−0.639476
$630$	0	0
$631$	−2.50182	−0.0995958	−0.0497979	−	0.998759i	$-0.515858\pi$
$631$	−2.50182	−0.0995958	−0.0497979	+	0.998759i	$0.515858\pi$
$632$	−15.0231	−0.597589
$633$	−5.00910	−0.199094
$634$	7.81584	0.310407
$635$	13.7287	0.544806
$636$	−40.4559	−1.60418
$637$	0	0
$638$	−1.65045	−0.0653421
$639$	2.21746	0.0877214
$640$	−10.1765	−0.402263
$641$	27.4893	1.08576	0.542881	−	0.839810i	$-0.317333\pi$
$641$	27.4893	1.08576	0.542881	+	0.839810i	$0.317333\pi$
$642$	1.51155	0.0596561
$643$	24.3815	0.961514	0.480757	−	0.876854i	$-0.340362\pi$
$643$	24.3815	0.961514	0.480757	+	0.876854i	$0.340362\pi$
$644$	0	0
$645$	7.83906	0.308663
$646$	−6.72828	−0.264720
$647$	−36.0128	−1.41581	−0.707904	−	0.706308i	$-0.750359\pi$
$647$	−36.0128	−1.41581	−0.707904	+	0.706308i	$0.750359\pi$
$648$	6.87977	0.270263
$649$	−35.9464	−1.41102
$650$	−2.55928	−0.100383
$651$	0	0
$652$	−8.38712	−0.328465
$653$	26.4713	1.03590	0.517952	−	0.855410i	$-0.326695\pi$
$653$	26.4713	1.03590	0.517952	+	0.855410i	$0.326695\pi$
$654$	−15.0724	−0.589379
$655$	1.87495	0.0732602
$656$	10.5592	0.412267
$657$	77.5063	3.02381
$658$	0	0
$659$	−10.1511	−0.395429	−0.197715	−	0.980260i	$-0.563352\pi$
$659$	−10.1511	−0.395429	−0.197715	+	0.980260i	$0.563352\pi$
$660$	−22.9454	−0.893148
$661$	22.0488	0.857600	0.428800	−	0.903399i	$-0.358937\pi$
$661$	22.0488	0.857600	0.428800	+	0.903399i	$0.358937\pi$
$662$	2.42925	0.0944153
$663$	60.1784	2.33714
$664$	−20.6583	−0.801699
$665$	0	0
$666$	−10.8913	−0.422031
$667$	−7.98824	−0.309306
$668$	28.6442	1.10828
$669$	45.8727	1.77354
$670$	5.17575	0.199957
$671$	−52.2797	−2.01823
$672$	0	0
$673$	17.4258	0.671714	0.335857	−	0.941913i	$-0.390974\pi$
$673$	17.4258	0.671714	0.335857	+	0.941913i	$0.390974\pi$
$674$	−1.35283	−0.0521091
$675$	−7.23066	−0.278308
$676$	−56.6312	−2.17812
$677$	−19.6693	−0.755952	−0.377976	−	0.925815i	$-0.623380\pi$
$677$	−19.6693	−0.755952	−0.377976	+	0.925815i	$0.623380\pi$
$678$	2.28017	0.0875693
$679$	0	0
$680$	4.66771	0.178999
$681$	−8.07307	−0.309361
$682$	4.86698	0.186366
$683$	39.3738	1.50660	0.753299	−	0.657678i	$-0.228461\pi$
$683$	39.3738	1.50660	0.753299	+	0.657678i	$0.228461\pi$
$684$	56.2807	2.15195
$685$	10.6285	0.406094
$686$	0	0
$687$	80.1384	3.05747
$688$	8.40157	0.320307
$689$	−49.5909	−1.88926
$690$	9.01612	0.343238
$691$	−27.5182	−1.04684	−0.523421	−	0.852074i	$-0.675344\pi$
$691$	−27.5182	−1.04684	−0.523421	+	0.852074i	$0.675344\pi$
$692$	−27.1059	−1.03041
$693$	0	0
$694$	−2.97708	−0.113008
$695$	2.00407	0.0760187
$696$	−4.34519	−0.164704
$697$	−10.5835	−0.400879
$698$	11.7901	0.446260
$699$	−17.1899	−0.650181
$700$	0	0
$701$	27.8005	1.05001	0.525005	−	0.851099i	$-0.324063\pi$
$701$	27.8005	1.05001	0.525005	+	0.851099i	$0.324063\pi$
$702$	18.5053	0.698438
$703$	−28.4465	−1.07288
$704$	−19.6672	−0.741235
$705$	9.32460	0.351185
$706$	−11.2642	−0.423934
$707$	0	0
$708$	−45.4727	−1.70897
$709$	−48.4984	−1.82139	−0.910697	−	0.413075i	$-0.864455\pi$
$709$	−48.4984	−1.82139	−0.910697	+	0.413075i	$0.864455\pi$
$710$	0.156736	0.00588221
$711$	−55.2085	−2.07048
$712$	11.3831	0.426598
$713$	23.5563	0.882192
$714$	0	0
$715$	−28.1265	−1.05187
$716$	31.8804	1.19143
$717$	50.2186	1.87545
$718$	−3.68246	−0.137428
$719$	−48.6335	−1.81372	−0.906862	−	0.421429i	$-0.861529\pi$
$719$	−48.6335	−1.81372	−0.906862	+	0.421429i	$0.861529\pi$
$720$	−17.1140	−0.637800
$721$	0	0
$722$	−4.57091	−0.170112
$723$	−69.7161	−2.59277
$724$	−23.5871	−0.876606
$725$	1.00000	0.0371391
$726$	8.05698	0.299023
$727$	−33.7689	−1.25242	−0.626209	−	0.779655i	$-0.715394\pi$
$727$	−33.7689	−1.25242	−0.626209	+	0.779655i	$0.715394\pi$
$728$	0	0
$729$	−37.8954	−1.40353
$730$	5.47836	0.202763
$731$	−8.42091	−0.311459
$732$	−66.1345	−2.44440
$733$	34.5331	1.27551	0.637755	−	0.770240i	$-0.279863\pi$
$733$	34.5331	1.27551	0.637755	+	0.770240i	$0.279863\pi$
$734$	0.485726	0.0179285
$735$	0	0
$736$	33.4986	1.23478
$737$	56.8814	2.09525
$738$	−7.18723	−0.264566
$739$	−15.0975	−0.555370	−0.277685	−	0.960672i	$-0.589567\pi$
$739$	−15.0975	−0.555370	−0.277685	+	0.960672i	$0.589567\pi$
$740$	9.48242	0.348581
$741$	106.738	3.92113
$742$	0	0
$743$	−11.5315	−0.423051	−0.211525	−	0.977372i	$-0.567843\pi$
$743$	−11.5315	−0.423051	−0.211525	+	0.977372i	$0.567843\pi$
$744$	12.8134	0.469763
$745$	−10.7388	−0.393438
$746$	−4.32397	−0.158312
$747$	−75.9173	−2.77767
$748$	24.6485	0.901238
$749$	0	0
$750$	−1.12867	−0.0412133
$751$	−22.3330	−0.814943	−0.407472	−	0.913218i	$-0.633589\pi$
$751$	−22.3330	−0.814943	−0.407472	+	0.913218i	$0.633589\pi$
$752$	9.99370	0.364433
$753$	−14.0111	−0.510593
$754$	−2.55928	−0.0932035
$755$	−10.3197	−0.375574
$756$	0	0
$757$	−11.4762	−0.417111	−0.208555	−	0.978011i	$-0.566876\pi$
$757$	−11.4762	−0.417111	−0.208555	+	0.978011i	$0.566876\pi$
$758$	9.89399	0.359366
$759$	99.0870	3.59663
$760$	8.27912	0.300315
$761$	−14.7736	−0.535544	−0.267772	−	0.963482i	$-0.586287\pi$
$761$	−14.7736	−0.535544	−0.267772	+	0.963482i	$0.586287\pi$
$762$	−15.4952	−0.561332
$763$	0	0
$764$	−3.15987	−0.114320
$765$	17.1534	0.620182
$766$	12.4504	0.449850
$767$	−55.7404	−2.01267
$768$	−15.4131	−0.556172
$769$	−15.5442	−0.560540	−0.280270	−	0.959921i	$-0.590424\pi$
$769$	−15.5442	−0.560540	−0.280270	+	0.959921i	$0.590424\pi$
$770$	0	0
$771$	−85.3353	−3.07327
$772$	48.1171	1.73177
$773$	−20.0664	−0.721738	−0.360869	−	0.932616i	$-0.617520\pi$
$773$	−20.0664	−0.721738	−0.360869	+	0.932616i	$0.617520\pi$
$774$	−5.71862	−0.205552
$775$	−2.94888	−0.105927
$776$	20.4771	0.735085
$777$	0	0
$778$	3.57873	0.128304
$779$	−18.7720	−0.672575
$780$	−35.5804	−1.27398
$781$	1.72253	0.0616369
$782$	−9.68533	−0.346347
$783$	−7.23066	−0.258403
$784$	0	0
$785$	−6.66586	−0.237915
$786$	−2.11620	−0.0754825
$787$	47.8248	1.70477	0.852384	−	0.522916i	$-0.175156\pi$
$787$	47.8248	1.70477	0.852384	+	0.522916i	$0.175156\pi$
$788$	−17.5372	−0.624737
$789$	−9.94654	−0.354106
$790$	−3.90230	−0.138837
$791$	0	0
$792$	34.8364	1.23786
$793$	−81.0677	−2.87880
$794$	6.59488	0.234044
$795$	−21.8702	−0.775655
$796$	−36.6332	−1.29843
$797$	10.6189	0.376140	0.188070	−	0.982156i	$-0.439777\pi$
$797$	10.6189	0.376140	0.188070	+	0.982156i	$0.439777\pi$
$798$	0	0
$799$	−10.0167	−0.354366
$800$	−4.19349	−0.148262
$801$	41.8316	1.47805
$802$	1.03135	0.0364182
$803$	60.2071	2.12466
$804$	71.9558	2.53768
$805$	0	0
$806$	7.54700	0.265832
$807$	−29.1219	−1.02514
$808$	1.66407	0.0585419
$809$	−10.8498	−0.381458	−0.190729	−	0.981643i	$-0.561085\pi$
$809$	−10.8498	−0.381458	−0.190729	+	0.981643i	$0.561085\pi$
$810$	1.78704	0.0627901
$811$	−54.4212	−1.91099	−0.955493	−	0.295015i	$-0.904675\pi$
$811$	−54.4212	−1.91099	−0.955493	+	0.295015i	$0.904675\pi$
$812$	0	0
$813$	−71.7420	−2.51610
$814$	−8.46043	−0.296538
$815$	−4.53401	−0.158820
$816$	28.4438	0.995731
$817$	−14.9362	−0.522550
$818$	8.06107	0.281849
$819$	0	0
$820$	6.25747	0.218520
$821$	16.0265	0.559330	0.279665	−	0.960098i	$-0.409777\pi$
$821$	16.0265	0.559330	0.279665	+	0.960098i	$0.409777\pi$
$822$	−11.9961	−0.418412
$823$	−29.3097	−1.02167	−0.510836	−	0.859678i	$-0.670664\pi$
$823$	−29.3097	−1.02167	−0.510836	+	0.859678i	$0.670664\pi$
$824$	17.3733	0.605226
$825$	−12.4041	−0.431855
$826$	0	0
$827$	19.7482	0.686713	0.343357	−	0.939205i	$-0.388436\pi$
$827$	19.7482	0.686713	0.343357	+	0.939205i	$0.388436\pi$
$828$	81.0159	2.81550
$829$	−45.5611	−1.58240	−0.791201	−	0.611556i	$-0.790544\pi$
$829$	−45.5611	−1.58240	−0.791201	+	0.611556i	$0.790544\pi$
$830$	−5.36605	−0.186258
$831$	22.9389	0.795741
$832$	−30.4970	−1.05729
$833$	0	0
$834$	−2.26194	−0.0783247
$835$	15.4848	0.535875
$836$	43.7190	1.51205
$837$	21.3223	0.737007
$838$	6.27107	0.216631
$839$	15.4477	0.533313	0.266656	−	0.963792i	$-0.414081\pi$
$839$	15.4477	0.533313	0.266656	+	0.963792i	$0.414081\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0.297052	0.0102371
$843$	44.1627	1.52104
$844$	−3.18144	−0.109510
$845$	−30.6144	−1.05317
$846$	−6.80232	−0.233869
$847$	0	0
$848$	−23.4395	−0.804916
$849$	−49.6803	−1.70502
$850$	1.21245	0.0415867
$851$	−40.9487	−1.40370
$852$	2.17902	0.0746521
$853$	−43.2823	−1.48196	−0.740978	−	0.671529i	$-0.765638\pi$
$853$	−43.2823	−1.48196	−0.740978	+	0.671529i	$0.765638\pi$
$854$	0	0
$855$	30.4249	1.04051
$856$	1.99801	0.0682905
$857$	−23.7847	−0.812470	−0.406235	−	0.913769i	$-0.633159\pi$
$857$	−23.7847	−0.812470	−0.406235	+	0.913769i	$0.633159\pi$
$858$	31.7456	1.08378
$859$	−9.22841	−0.314869	−0.157435	−	0.987529i	$-0.550322\pi$
$859$	−9.22841	−0.314869	−0.157435	+	0.987529i	$0.550322\pi$
$860$	4.97885	0.169777
$861$	0	0
$862$	−11.0256	−0.375533
$863$	−13.8765	−0.472363	−0.236182	−	0.971709i	$-0.575896\pi$
$863$	−13.8765	−0.472363	−0.236182	+	0.971709i	$0.575896\pi$
$864$	30.3217	1.03157
$865$	−14.6533	−0.498226
$866$	−9.16653	−0.311491
$867$	21.0032	0.713305
$868$	0	0
$869$	−42.8861	−1.45481
$870$	−1.12867	−0.0382656
$871$	88.2034	2.98866
$872$	−19.9232	−0.674684
$873$	75.2513	2.54687
$874$	−17.1789	−0.581084
$875$	0	0
$876$	76.1628	2.57330
$877$	8.48263	0.286438	0.143219	−	0.989691i	$-0.454255\pi$
$877$	8.48263	0.286438	0.143219	+	0.989691i	$0.454255\pi$
$878$	0.826118	0.0278801
$879$	81.9192	2.76307
$880$	−13.2942	−0.448147
$881$	20.8449	0.702283	0.351141	−	0.936322i	$-0.385794\pi$
$881$	20.8449	0.702283	0.351141	+	0.936322i	$0.385794\pi$
$882$	0	0
$883$	29.6382	0.997406	0.498703	−	0.866773i	$-0.333810\pi$
$883$	29.6382	0.997406	0.498703	+	0.866773i	$0.333810\pi$
$884$	38.2213	1.28552
$885$	−24.5822	−0.826321
$886$	−3.12028	−0.104828
$887$	−53.7769	−1.80565	−0.902826	−	0.430007i	$-0.858511\pi$
$887$	−53.7769	−1.80565	−0.902826	+	0.430007i	$0.858511\pi$
$888$	−22.2740	−0.747467
$889$	0	0
$890$	2.95677	0.0991113
$891$	19.6395	0.657948
$892$	29.1353	0.975522
$893$	−17.7666	−0.594538
$894$	12.1206	0.405372
$895$	17.2343	0.576080
$896$	0	0
$897$	153.650	5.13021
$898$	−4.22366	−0.140945
$899$	−2.94888	−0.0983505
$900$	−10.1419	−0.338063
$901$	23.4935	0.782681
$902$	−5.58306	−0.185896
$903$	0	0
$904$	3.01399	0.100244
$905$	−12.7510	−0.423857
$906$	11.6476	0.386966
$907$	−15.2739	−0.507161	−0.253580	−	0.967314i	$-0.581608\pi$
$907$	−15.2739	−0.507161	−0.253580	+	0.967314i	$0.581608\pi$
$908$	−5.12747	−0.170161
$909$	6.11530	0.202832
$910$	0	0
$911$	−34.4812	−1.14241	−0.571207	−	0.820806i	$-0.693525\pi$
$911$	−34.4812	−1.14241	−0.571207	+	0.820806i	$0.693525\pi$
$912$	50.4507	1.67059
$913$	−58.9728	−1.95171
$914$	0.882983	0.0292065
$915$	−35.7518	−1.18192
$916$	50.8985	1.68173
$917$	0	0
$918$	−8.76681	−0.289348
$919$	−26.6749	−0.879925	−0.439962	−	0.898016i	$-0.645008\pi$
$919$	−26.6749	−0.879925	−0.439962	+	0.898016i	$0.645008\pi$
$920$	11.9178	0.392917
$921$	27.2491	0.897890
$922$	7.84745	0.258442
$923$	2.67105	0.0879186
$924$	0	0
$925$	5.12612	0.168546
$926$	−8.01657	−0.263441
$927$	63.8450	2.09694
$928$	−4.19349	−0.137658
$929$	16.1623	0.530267	0.265133	−	0.964212i	$-0.414584\pi$
$929$	16.1623	0.530267	0.265133	+	0.964212i	$0.414584\pi$
$930$	3.32832	0.109140
$931$	0	0
$932$	−10.9179	−0.357626
$933$	63.3246	2.07316
$934$	−14.8063	−0.484478
$935$	13.3248	0.435767
$936$	54.0192	1.76567
$937$	4.89718	0.159984	0.0799919	−	0.996796i	$-0.474511\pi$
$937$	4.89718	0.159984	0.0799919	+	0.996796i	$0.474511\pi$
$938$	0	0
$939$	45.7678	1.49357
$940$	5.92236	0.193166
$941$	−24.3153	−0.792655	−0.396328	−	0.918109i	$-0.629716\pi$
$941$	−24.3153	−0.792655	−0.396328	+	0.918109i	$0.629716\pi$
$942$	7.52358	0.245132
$943$	−27.0222	−0.879963
$944$	−26.3461	−0.857493
$945$	0	0
$946$	−4.44224	−0.144430
$947$	34.5707	1.12340	0.561698	−	0.827342i	$-0.310148\pi$
$947$	34.5707	1.12340	0.561698	+	0.827342i	$0.310148\pi$
$948$	−54.2516	−1.76201
$949$	93.3604	3.03061
$950$	2.15052	0.0697721
$951$	58.7405	1.90479
$952$	0	0
$953$	−6.74534	−0.218503	−0.109251	−	0.994014i	$-0.534845\pi$
$953$	−6.74534	−0.218503	−0.109251	+	0.994014i	$0.534845\pi$
$954$	15.9544	0.516542
$955$	−1.70820	−0.0552761
$956$	31.8955	1.03157
$957$	−12.4041	−0.400968
$958$	4.86471	0.157172
$959$	0	0
$960$	−13.4495	−0.434082
$961$	−22.3041	−0.719488
$962$	−13.1192	−0.422980
$963$	7.34248	0.236608
$964$	−44.2790	−1.42613
$965$	26.0117	0.837348
$966$	0	0
$967$	−45.2545	−1.45529	−0.727643	−	0.685956i	$-0.759384\pi$
$967$	−45.2545	−1.45529	−0.727643	+	0.685956i	$0.759384\pi$
$968$	10.6499	0.342302
$969$	−50.5669	−1.62444
$970$	5.31897	0.170782
$971$	−21.9543	−0.704546	−0.352273	−	0.935897i	$-0.614591\pi$
$971$	−21.9543	−0.704546	−0.352273	+	0.935897i	$0.614591\pi$
$972$	−15.2821	−0.490173
$973$	0	0
$974$	−4.05933	−0.130069
$975$	−19.2345	−0.615996
$976$	−38.3172	−1.22650
$977$	−60.1513	−1.92441	−0.962206	−	0.272324i	$-0.912208\pi$
$977$	−60.1513	−1.92441	−0.962206	+	0.272324i	$0.912208\pi$
$978$	5.11742	0.163637
$979$	32.4949	1.03854
$980$	0	0
$981$	−73.2157	−2.33760
$982$	−0.586236	−0.0187075
$983$	4.51305	0.143944	0.0719720	−	0.997407i	$-0.477071\pi$
$983$	4.51305	0.143944	0.0719720	+	0.997407i	$0.477071\pi$
$984$	−14.6987	−0.468577
$985$	−9.48048	−0.302073
$986$	1.21245	0.0386122
$987$	0	0
$988$	67.7931	2.15678
$989$	−21.5006	−0.683678
$990$	9.04883	0.287591
$991$	−8.99468	−0.285725	−0.142863	−	0.989743i	$-0.545631\pi$
$991$	−8.99468	−0.285725	−0.142863	+	0.989743i	$0.545631\pi$
$992$	12.3661	0.392624
$993$	18.2572	0.579374
$994$	0	0
$995$	−19.8037	−0.627818
$996$	−74.6014	−2.36384
$997$	−1.34074	−0.0424616	−0.0212308	−	0.999775i	$-0.506758\pi$
$997$	−1.34074	−0.0424616	−0.0212308	+	0.999775i	$0.506758\pi$
$998$	0.842245	0.0266608
$999$	−37.0653	−1.17269

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7105.2.a.t.1.4		8
7.6	odd	2		1015.2.a.l.1.4	✓	8
21.20	even	2		9135.2.a.bh.1.5		8
35.34	odd	2		5075.2.a.ba.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1015.2.a.l.1.4	✓	8	7.6	odd	2
5075.2.a.ba.1.5		8	35.34	odd	2
7105.2.a.t.1.4		8	1.1	even	1	trivial
9135.2.a.bh.1.5		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)

\(f(q)\)	\(=\)	\(q-0.387528 q^{2} -2.91250 q^{3} -1.84982 q^{4} -1.00000 q^{5} +1.12867 q^{6} +1.49191 q^{8} +5.48263 q^{9} +0.387528 q^{10} +4.25893 q^{11} +5.38760 q^{12} +6.60412 q^{13} +2.91250 q^{15} +3.12149 q^{16} -3.12867 q^{17} -2.12467 q^{18} -5.54933 q^{19} +1.84982 q^{20} -1.65045 q^{22} -7.98824 q^{23} -4.34519 q^{24} +1.00000 q^{25} -2.55928 q^{26} -7.23066 q^{27} +1.00000 q^{29} -1.12867 q^{30} -2.94888 q^{31} -4.19349 q^{32} -12.4041 q^{33} +1.21245 q^{34} -10.1419 q^{36} +5.12612 q^{37} +2.15052 q^{38} -19.2345 q^{39} -1.49191 q^{40} +3.38274 q^{41} +2.69153 q^{43} -7.87825 q^{44} -5.48263 q^{45} +3.09567 q^{46} +3.20158 q^{47} -9.09131 q^{48} -0.387528 q^{50} +9.11225 q^{51} -12.2164 q^{52} -7.50909 q^{53} +2.80208 q^{54} -4.25893 q^{55} +16.1624 q^{57} -0.387528 q^{58} -8.44024 q^{59} -5.38760 q^{60} -12.2753 q^{61} +1.14277 q^{62} -4.61788 q^{64} -6.60412 q^{65} +4.80694 q^{66} +13.3558 q^{67} +5.78749 q^{68} +23.2657 q^{69} +0.404452 q^{71} +8.17962 q^{72} +14.1367 q^{73} -1.98652 q^{74} -2.91250 q^{75} +10.2653 q^{76} +7.45390 q^{78} -10.0697 q^{79} -3.12149 q^{80} +4.61137 q^{81} -1.31091 q^{82} -13.8469 q^{83} +3.12867 q^{85} -1.04304 q^{86} -2.91250 q^{87} +6.35395 q^{88} +7.62983 q^{89} +2.12467 q^{90} +14.7768 q^{92} +8.58859 q^{93} -1.24070 q^{94} +5.54933 q^{95} +12.2135 q^{96} +13.7254 q^{97} +23.3501 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + q^{2} - 4 q^{3} + 5 q^{4} - 8 q^{5} + 6 q^{6} - 6 q^{8} + 14 q^{9} - q^{10} + q^{11} + 7 q^{12} + q^{13} + 4 q^{15} + 3 q^{16} - 22 q^{17} + 13 q^{18} - 8 q^{19} - 5 q^{20} - 3 q^{22} + 9 q^{23}+ \cdots - 63 q^{99}+O(q^{100}) \) 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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