LMFDB - Embedded newform 975.2.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [975,2,Mod(1,975)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(975, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("975.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 975 = 3 \cdot 5^{2} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	975.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$7.78541419707$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.1821184.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 8x^{3} + 13x - 4 $ x^5 - 8x^3 + 13x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 195)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.47948$ of defining polynomial
Character	$\chi$	$=$	975.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.47948 q^{2} +1.00000 q^{3} +4.14785 q^{4} -2.47948 q^{6} +0.949959 q^{7} -5.32555 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.47948 q^{2} +1.00000 q^{3} +4.14785 q^{4} -2.47948 q^{6} +0.949959 q^{7} -5.32555 q^{8} +1.00000 q^{9} +3.89611 q^{11} +4.14785 q^{12} +1.00000 q^{13} -2.35541 q^{14} +4.90893 q^{16} -5.90893 q^{17} -2.47948 q^{18} +6.35541 q^{19} +0.949959 q^{21} -9.66033 q^{22} +3.30537 q^{23} -5.32555 q^{24} -2.47948 q^{26} +1.00000 q^{27} +3.94028 q^{28} -4.29569 q^{29} +7.56253 q^{31} -1.52052 q^{32} +3.89611 q^{33} +14.6511 q^{34} +4.14785 q^{36} -1.30537 q^{37} -15.7581 q^{38} +1.00000 q^{39} -6.75499 q^{41} -2.35541 q^{42} +8.29569 q^{43} +16.1604 q^{44} -8.19561 q^{46} +0.0128228 q^{47} +4.90893 q^{48} -6.09758 q^{49} -5.90893 q^{51} +4.14785 q^{52} +2.51249 q^{53} -2.47948 q^{54} -5.05905 q^{56} +6.35541 q^{57} +10.6511 q^{58} -6.30851 q^{59} +1.61324 q^{61} -18.7512 q^{62} +0.949959 q^{63} -6.04776 q^{64} -9.66033 q^{66} +4.66328 q^{67} -24.5093 q^{68} +3.30537 q^{69} +7.92175 q^{71} -5.32555 q^{72} -14.5914 q^{73} +3.23664 q^{74} +26.3612 q^{76} +3.70114 q^{77} -2.47948 q^{78} -16.6004 q^{79} +1.00000 q^{81} +16.7489 q^{82} +12.8044 q^{83} +3.94028 q^{84} -20.5690 q^{86} -4.29569 q^{87} -20.7489 q^{88} +17.6542 q^{89} +0.949959 q^{91} +13.7102 q^{92} +7.56253 q^{93} -0.0317940 q^{94} -1.52052 q^{96} +18.1226 q^{97} +15.1189 q^{98} +3.89611 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{3} + 6 q^{4} + 5 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q + 5 * q^3 + 6 * q^4 + 5 * q^7 + 5 * q^9	$ 5 q + 5 q^{3} + 6 q^{4} + 5 q^{7} + 5 q^{9} + 5 q^{11} + 6 q^{12} + 5 q^{13} + 12 q^{14} - 5 q^{17} + 8 q^{19} + 5 q^{21} - 8 q^{22} - 7 q^{23} + 5 q^{27} + 14 q^{28} + 8 q^{29} + 12 q^{31} - 20 q^{32} + 5 q^{33} + 20 q^{34} + 6 q^{36} + 17 q^{37} - 24 q^{38} + 5 q^{39} + 5 q^{41} + 12 q^{42} + 12 q^{43} + 18 q^{44} - 12 q^{46} - 10 q^{47} + 22 q^{49} - 5 q^{51} + 6 q^{52} - 13 q^{53} + 8 q^{57} + 8 q^{59} + 13 q^{61} - 40 q^{62} + 5 q^{63} - 16 q^{64} - 8 q^{66} + 28 q^{67} - 14 q^{68} - 7 q^{69} + 5 q^{71} - 14 q^{73} + 12 q^{74} - 35 q^{77} + q^{79} + 5 q^{81} + 16 q^{82} - 6 q^{83} + 14 q^{84} + 8 q^{87} - 36 q^{88} + 19 q^{89} + 5 q^{91} - 10 q^{92} + 12 q^{93} - 12 q^{94} - 20 q^{96} - 13 q^{97} + 28 q^{98} + 5 q^{99}+O(q^{100}) $ 5 * q + 5 * q^3 + 6 * q^4 + 5 * q^7 + 5 * q^9 + 5 * q^11 + 6 * q^12 + 5 * q^13 + 12 * q^14 - 5 * q^17 + 8 * q^19 + 5 * q^21 - 8 * q^22 - 7 * q^23 + 5 * q^27 + 14 * q^28 + 8 * q^29 + 12 * q^31 - 20 * q^32 + 5 * q^33 + 20 * q^34 + 6 * q^36 + 17 * q^37 - 24 * q^38 + 5 * q^39 + 5 * q^41 + 12 * q^42 + 12 * q^43 + 18 * q^44 - 12 * q^46 - 10 * q^47 + 22 * q^49 - 5 * q^51 + 6 * q^52 - 13 * q^53 + 8 * q^57 + 8 * q^59 + 13 * q^61 - 40 * q^62 + 5 * q^63 - 16 * q^64 - 8 * q^66 + 28 * q^67 - 14 * q^68 - 7 * q^69 + 5 * q^71 - 14 * q^73 + 12 * q^74 - 35 * q^77 + q^79 + 5 * q^81 + 16 * q^82 - 6 * q^83 + 14 * q^84 + 8 * q^87 - 36 * q^88 + 19 * q^89 + 5 * q^91 - 10 * q^92 + 12 * q^93 - 12 * q^94 - 20 * q^96 - 13 * q^97 + 28 * q^98 + 5 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.47948	−1.75326	−0.876630	−	0.481165i	$-0.840214\pi$
$2$	−2.47948	−1.75326	−0.876630	+	0.481165i	$0.840214\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	4.14785	2.07392
$5$	0	0
$5$	0	0
$6$	−2.47948	−1.01225
$7$	0.949959	0.359051	0.179525	−	0.983753i	$-0.442544\pi$
$7$	0.949959	0.359051	0.179525	+	0.983753i	$0.442544\pi$
$8$	−5.32555	−1.88287
$9$	1.00000	0.333333
$10$	0	0
$11$	3.89611	1.17472	0.587360	−	0.809326i	$-0.300167\pi$
$11$	3.89611	1.17472	0.587360	+	0.809326i	$0.300167\pi$
$12$	4.14785	1.19738
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−2.35541	−0.629509
$15$	0	0
$16$	4.90893	1.22723
$17$	−5.90893	−1.43313	−0.716563	−	0.697523i	$-0.754286\pi$
$17$	−5.90893	−1.43313	−0.716563	+	0.697523i	$0.754286\pi$
$18$	−2.47948	−0.584420
$19$	6.35541	1.45803	0.729015	−	0.684497i	$-0.239978\pi$
$19$	6.35541	1.45803	0.729015	+	0.684497i	$0.239978\pi$
$20$	0	0
$21$	0.949959	0.207298
$22$	−9.66033	−2.05959
$23$	3.30537	0.689217	0.344608	−	0.938747i	$-0.388012\pi$
$23$	3.30537	0.689217	0.344608	+	0.938747i	$0.388012\pi$
$24$	−5.32555	−1.08707
$25$	0	0
$26$	−2.47948	−0.486267
$27$	1.00000	0.192450
$28$	3.94028	0.744643
$29$	−4.29569	−0.797690	−0.398845	−	0.917018i	$-0.630589\pi$
$29$	−4.29569	−0.797690	−0.398845	+	0.917018i	$0.630589\pi$
$30$	0	0
$31$	7.56253	1.35827	0.679135	−	0.734013i	$-0.262355\pi$
$31$	7.56253	1.35827	0.679135	+	0.734013i	$0.262355\pi$
$32$	−1.52052	−0.268792
$33$	3.89611	0.678225
$34$	14.6511	2.51264
$35$	0	0
$36$	4.14785	0.691308
$37$	−1.30537	−0.214601	−0.107301	−	0.994227i	$-0.534221\pi$
$37$	−1.30537	−0.214601	−0.107301	+	0.994227i	$0.534221\pi$
$38$	−15.7581	−2.55631
$39$	1.00000	0.160128
$40$	0	0
$41$	−6.75499	−1.05495	−0.527476	−	0.849570i	$-0.676862\pi$
$41$	−6.75499	−1.05495	−0.527476	+	0.849570i	$0.676862\pi$
$42$	−2.35541	−0.363447
$43$	8.29569	1.26508	0.632540	−	0.774527i	$-0.282012\pi$
$43$	8.29569	1.26508	0.632540	+	0.774527i	$0.282012\pi$
$44$	16.1604	2.43628
$45$	0	0
$46$	−8.19561	−1.20838
$47$	0.0128228	0.00187040	0.000935200	−	1.00000i	$-0.499702\pi$
$47$	0.0128228	0.00187040	0.000935200		1.00000i	$0.499702\pi$
$48$	4.90893	0.708543
$49$	−6.09758	−0.871083
$50$	0	0
$51$	−5.90893	−0.827415
$52$	4.14785	0.575203
$53$	2.51249	0.345117	0.172559	−	0.984999i	$-0.444797\pi$
$53$	2.51249	0.345117	0.172559	+	0.984999i	$0.444797\pi$
$54$	−2.47948	−0.337415
$55$	0	0
$56$	−5.05905	−0.676044
$57$	6.35541	0.841794
$58$	10.6511	1.39856
$59$	−6.30851	−0.821298	−0.410649	−	0.911793i	$-0.634698\pi$
$59$	−6.30851	−0.821298	−0.410649	+	0.911793i	$0.634698\pi$
$60$	0	0
$61$	1.61324	0.206554	0.103277	−	0.994653i	$-0.467067\pi$
$61$	1.61324	0.206554	0.103277	+	0.994653i	$0.467067\pi$
$62$	−18.7512	−2.38140
$63$	0.949959	0.119684
$64$	−6.04776	−0.755970
$65$	0	0
$66$	−9.66033	−1.18911
$67$	4.66328	0.569710	0.284855	−	0.958571i	$-0.408055\pi$
$67$	4.66328	0.569710	0.284855	+	0.958571i	$0.408055\pi$
$68$	−24.5093	−2.97219
$69$	3.30537	0.397919
$70$	0	0
$71$	7.92175	0.940139	0.470069	−	0.882629i	$-0.344229\pi$
$71$	7.92175	0.940139	0.470069	+	0.882629i	$0.344229\pi$
$72$	−5.32555	−0.627622
$73$	−14.5914	−1.70779	−0.853896	−	0.520444i	$-0.825767\pi$
$73$	−14.5914	−1.70779	−0.853896	+	0.520444i	$0.825767\pi$
$74$	3.23664	0.376252
$75$	0	0
$76$	26.3612	3.02384
$77$	3.70114	0.421784
$78$	−2.47948	−0.280746
$79$	−16.6004	−1.86769	−0.933845	−	0.357678i	$-0.883569\pi$
$79$	−16.6004	−1.86769	−0.933845	+	0.357678i	$0.883569\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	16.7489	1.84961
$83$	12.8044	1.40546	0.702731	−	0.711456i	$-0.251964\pi$
$83$	12.8044	1.40546	0.702731	+	0.711456i	$0.251964\pi$
$84$	3.94028	0.429920
$85$	0	0
$86$	−20.5690	−2.21802
$87$	−4.29569	−0.460546
$88$	−20.7489	−2.21184
$89$	17.6542	1.87135	0.935673	−	0.352868i	$-0.114794\pi$
$89$	17.6542	1.87135	0.935673	+	0.352868i	$0.114794\pi$
$90$	0	0
$91$	0.949959	0.0995827
$92$	13.7102	1.42938
$93$	7.56253	0.784198
$94$	−0.0317940	−0.00327930
$95$	0	0
$96$	−1.52052	−0.155187
$97$	18.1226	1.84007	0.920033	−	0.391840i	$-0.128161\pi$
$97$	18.1226	1.84007	0.920033	+	0.391840i	$0.128161\pi$
$98$	15.1189	1.52723
$99$	3.89611	0.391573
$100$	0	0
$101$	11.6921	1.16341	0.581705	−	0.813400i	$-0.302386\pi$
$101$	11.6921	1.16341	0.581705	+	0.813400i	$0.302386\pi$
$102$	14.6511	1.45067
$103$	7.52217	0.741181	0.370591	−	0.928796i	$-0.379155\pi$
$103$	7.52217	0.741181	0.370591	+	0.928796i	$0.379155\pi$
$104$	−5.32555	−0.522213
$105$	0	0
$106$	−6.22968	−0.605080
$107$	6.00901	0.580913	0.290457	−	0.956888i	$-0.406193\pi$
$107$	6.00901	0.580913	0.290457	+	0.956888i	$0.406193\pi$
$108$	4.14785	0.399127
$109$	1.08139	0.103579	0.0517894	−	0.998658i	$-0.483508\pi$
$109$	1.08139	0.103579	0.0517894	+	0.998658i	$0.483508\pi$
$110$	0	0
$111$	−1.30537	−0.123900
$112$	4.66328	0.440639
$113$	−7.29636	−0.686383	−0.343192	−	0.939265i	$-0.611508\pi$
$113$	−7.29636	−0.686383	−0.343192	+	0.939265i	$0.611508\pi$
$114$	−15.7581	−1.47588
$115$	0	0
$116$	−17.8179	−1.65435
$117$	1.00000	0.0924500
$118$	15.6419	1.43995
$119$	−5.61324	−0.514565
$120$	0	0
$121$	4.17964	0.379967
$122$	−4.00000	−0.362143
$123$	−6.75499	−0.609077
$124$	31.3682	2.81695
$125$	0	0
$126$	−2.35541	−0.209836
$127$	10.6915	0.948714	0.474357	−	0.880333i	$-0.342681\pi$
$127$	10.6915	0.948714	0.474357	+	0.880333i	$0.342681\pi$
$128$	18.0364	1.59420
$129$	8.29569	0.730395
$130$	0	0
$131$	−0.225811	−0.0197292	−0.00986458	−	0.999951i	$-0.503140\pi$
$131$	−0.225811	−0.0197292	−0.00986458	+	0.999951i	$0.503140\pi$
$132$	16.1604	1.40659
$133$	6.03738	0.523507
$134$	−11.5625	−0.998851
$135$	0	0
$136$	31.4683	2.69838
$137$	6.63828	0.567146	0.283573	−	0.958951i	$-0.408480\pi$
$137$	6.63828	0.567146	0.283573	+	0.958951i	$0.408480\pi$
$138$	−8.19561	−0.697656
$139$	3.09758	0.262733	0.131367	−	0.991334i	$-0.458064\pi$
$139$	3.09758	0.262733	0.131367	+	0.991334i	$0.458064\pi$
$140$	0	0
$141$	0.0128228	0.00107988
$142$	−19.6419	−1.64831
$143$	3.89611	0.325809
$144$	4.90893	0.409077
$145$	0	0
$146$	36.1791	2.99420
$147$	−6.09758	−0.502920
$148$	−5.41446	−0.445066
$149$	10.9808	0.899582	0.449791	−	0.893134i	$-0.351498\pi$
$149$	10.9808	0.899582	0.449791	+	0.893134i	$0.351498\pi$
$150$	0	0
$151$	−2.78901	−0.226966	−0.113483	−	0.993540i	$-0.536201\pi$
$151$	−2.78901	−0.226966	−0.113483	+	0.993540i	$0.536201\pi$
$152$	−33.8460	−2.74528
$153$	−5.90893	−0.477709
$154$	−9.17692	−0.739497
$155$	0	0
$156$	4.14785	0.332093
$157$	−11.3843	−0.908563	−0.454281	−	0.890858i	$-0.650104\pi$
$157$	−11.3843	−0.908563	−0.454281	+	0.890858i	$0.650104\pi$
$158$	41.1604	3.27455
$159$	2.51249	0.199253
$160$	0	0
$161$	3.13996	0.247464
$162$	−2.47948	−0.194807
$163$	2.13144	0.166947	0.0834735	−	0.996510i	$-0.473399\pi$
$163$	2.13144	0.166947	0.0834735	+	0.996510i	$0.473399\pi$
$164$	−28.0187	−2.18789
$165$	0	0
$166$	−31.7482	−2.46414
$167$	−3.00587	−0.232601	−0.116300	−	0.993214i	$-0.537104\pi$
$167$	−3.00587	−0.232601	−0.116300	+	0.993214i	$0.537104\pi$
$168$	−5.05905	−0.390314
$169$	1.00000	0.0769231
$170$	0	0
$171$	6.35541	0.486010
$172$	34.4092	2.62368
$173$	0.673441	0.0512008	0.0256004	−	0.999672i	$-0.491850\pi$
$173$	0.673441	0.0512008	0.0256004	+	0.999672i	$0.491850\pi$
$174$	10.6511	0.807458
$175$	0	0
$176$	19.1257	1.44165
$177$	−6.30851	−0.474177
$178$	−43.7734	−3.28096
$179$	7.81786	0.584334	0.292167	−	0.956367i	$-0.405624\pi$
$179$	7.81786	0.584334	0.292167	+	0.956367i	$0.405624\pi$
$180$	0	0
$181$	−9.53118	−0.708447	−0.354223	−	0.935161i	$-0.615255\pi$
$181$	−9.53118	−0.708447	−0.354223	+	0.935161i	$0.615255\pi$
$182$	−2.35541	−0.174594
$183$	1.61324	0.119254
$184$	−17.6029	−1.29770
$185$	0	0
$186$	−18.7512	−1.37490
$187$	−23.0218	−1.68352
$188$	0.0531870	0.00387906
$189$	0.949959	0.0690993
$190$	0	0
$191$	−16.1073	−1.16548	−0.582740	−	0.812659i	$-0.698019\pi$
$191$	−16.1073	−1.16548	−0.582740	+	0.812659i	$0.698019\pi$
$192$	−6.04776	−0.436460
$193$	−13.8967	−1.00031	−0.500155	−	0.865936i	$-0.666724\pi$
$193$	−13.8967	−1.00031	−0.500155	+	0.865936i	$0.666724\pi$
$194$	−44.9346	−3.22612
$195$	0	0
$196$	−25.2918	−1.80656
$197$	11.7647	0.838198	0.419099	−	0.907941i	$-0.362346\pi$
$197$	11.7647	0.838198	0.419099	+	0.907941i	$0.362346\pi$
$198$	−9.66033	−0.686530
$199$	−8.41513	−0.596533	−0.298266	−	0.954483i	$-0.596408\pi$
$199$	−8.41513	−0.596533	−0.298266	+	0.954483i	$0.596408\pi$
$200$	0	0
$201$	4.66328	0.328922
$202$	−28.9905	−2.03976
$203$	−4.08073	−0.286411
$204$	−24.5093	−1.71600
$205$	0	0
$206$	−18.6511	−1.29948
$207$	3.30537	0.229739
$208$	4.90893	0.340373
$209$	24.7613	1.71278
$210$	0	0
$211$	11.5402	0.794459	0.397230	−	0.917719i	$-0.369972\pi$
$211$	11.5402	0.794459	0.397230	+	0.917719i	$0.369972\pi$
$212$	10.4214	0.715746
$213$	7.92175	0.542789
$214$	−14.8993	−1.01849
$215$	0	0
$216$	−5.32555	−0.362358
$217$	7.18409	0.487688
$218$	−2.68130	−0.181601
$219$	−14.5914	−0.985994
$220$	0	0
$221$	−5.90893	−0.397478
$222$	3.23664	0.217229
$223$	7.82173	0.523782	0.261891	−	0.965098i	$-0.415654\pi$
$223$	7.82173	0.523782	0.261891	+	0.965098i	$0.415654\pi$
$224$	−1.44443	−0.0965098
$225$	0	0
$226$	18.0912	1.20341
$227$	−21.1316	−1.40255	−0.701277	−	0.712889i	$-0.747386\pi$
$227$	−21.1316	−1.40255	−0.701277	+	0.712889i	$0.747386\pi$
$228$	26.3612	1.74582
$229$	−23.5280	−1.55477	−0.777387	−	0.629022i	$-0.783455\pi$
$229$	−23.5280	−1.55477	−0.777387	+	0.629022i	$0.783455\pi$
$230$	0	0
$231$	3.70114	0.243517
$232$	22.8769	1.50194
$233$	−6.40612	−0.419679	−0.209839	−	0.977736i	$-0.567294\pi$
$233$	−6.40612	−0.419679	−0.209839	+	0.977736i	$0.567294\pi$
$234$	−2.47948	−0.162089
$235$	0	0
$236$	−26.1667	−1.70331
$237$	−16.6004	−1.07831
$238$	13.9179	0.902166
$239$	−12.6696	−0.819530	−0.409765	−	0.912191i	$-0.634389\pi$
$239$	−12.6696	−0.819530	−0.409765	+	0.912191i	$0.634389\pi$
$240$	0	0
$241$	−0.206457	−0.0132990	−0.00664952	−	0.999978i	$-0.502117\pi$
$241$	−0.206457	−0.0132990	−0.00664952	+	0.999978i	$0.502117\pi$
$242$	−10.3634	−0.666181
$243$	1.00000	0.0641500
$244$	6.69146	0.428377
$245$	0	0
$246$	16.7489	1.06787
$247$	6.35541	0.404385
$248$	−40.2746	−2.55744
$249$	12.8044	0.811444
$250$	0	0
$251$	15.5028	0.978529	0.489264	−	0.872135i	$-0.337265\pi$
$251$	15.5028	0.978529	0.489264	+	0.872135i	$0.337265\pi$
$252$	3.94028	0.248214
$253$	12.8781	0.809637
$254$	−26.5093	−1.66334
$255$	0	0
$256$	−32.6254	−2.03909
$257$	−6.11944	−0.381720	−0.190860	−	0.981617i	$-0.561128\pi$
$257$	−6.11944	−0.381720	−0.190860	+	0.981617i	$0.561128\pi$
$258$	−20.5690	−1.28057
$259$	−1.24004	−0.0770526
$260$	0	0
$261$	−4.29569	−0.265897
$262$	0.559894	0.0345904
$263$	−17.1066	−1.05484	−0.527419	−	0.849605i	$-0.676840\pi$
$263$	−17.1066	−1.05484	−0.527419	+	0.849605i	$0.676840\pi$
$264$	−20.7489	−1.27701
$265$	0	0
$266$	−14.9696	−0.917844
$267$	17.6542	1.08042
$268$	19.3426	1.18153
$269$	27.8981	1.70098	0.850490	−	0.525992i	$-0.176306\pi$
$269$	27.8981	1.70098	0.850490	+	0.525992i	$0.176306\pi$
$270$	0	0
$271$	−28.6569	−1.74079	−0.870393	−	0.492358i	$-0.836135\pi$
$271$	−28.6569	−1.74079	−0.870393	+	0.492358i	$0.836135\pi$
$272$	−29.0065	−1.75878
$273$	0.949959	0.0574941
$274$	−16.4595	−0.994355
$275$	0	0
$276$	13.7102	0.825254
$277$	−0.433599	−0.0260525	−0.0130262	−	0.999915i	$-0.504146\pi$
$277$	−0.433599	−0.0260525	−0.0130262	+	0.999915i	$0.504146\pi$
$278$	−7.68040	−0.460640
$279$	7.56253	0.452757
$280$	0	0
$281$	−18.5747	−1.10807	−0.554036	−	0.832492i	$-0.686913\pi$
$281$	−18.5747	−1.10807	−0.554036	+	0.832492i	$0.686913\pi$
$282$	−0.0317940	−0.00189330
$283$	21.9360	1.30396	0.651979	−	0.758237i	$-0.273939\pi$
$283$	21.9360	1.30396	0.651979	+	0.758237i	$0.273939\pi$
$284$	32.8582	1.94978
$285$	0	0
$286$	−9.66033	−0.571228
$287$	−6.41696	−0.378781
$288$	−1.52052	−0.0895972
$289$	17.9154	1.05385
$290$	0	0
$291$	18.1226	1.06236
$292$	−60.5228	−3.54183
$293$	−9.26031	−0.540993	−0.270497	−	0.962721i	$-0.587188\pi$
$293$	−9.26031	−0.540993	−0.270497	+	0.962721i	$0.587188\pi$
$294$	15.1189	0.881749
$295$	0	0
$296$	6.95180	0.404065
$297$	3.89611	0.226075
$298$	−27.2267	−1.57720
$299$	3.30537	0.191154
$300$	0	0
$301$	7.88056	0.454228
$302$	6.91530	0.397931
$303$	11.6921	0.671695
$304$	31.1982	1.78934
$305$	0	0
$306$	14.6511	0.837548
$307$	9.15012	0.522225	0.261113	−	0.965308i	$-0.415911\pi$
$307$	9.15012	0.522225	0.261113	+	0.965308i	$0.415911\pi$
$308$	15.3518	0.874747
$309$	7.52217	0.427921
$310$	0	0
$311$	8.11488	0.460153	0.230076	−	0.973173i	$-0.426102\pi$
$311$	8.11488	0.460153	0.230076	+	0.973173i	$0.426102\pi$
$312$	−5.32555	−0.301500
$313$	−4.21952	−0.238501	−0.119251	−	0.992864i	$-0.538049\pi$
$313$	−4.21952	−0.238501	−0.119251	+	0.992864i	$0.538049\pi$
$314$	28.2271	1.59295
$315$	0	0
$316$	−68.8558	−3.87344
$317$	−15.4374	−0.867053	−0.433527	−	0.901141i	$-0.642731\pi$
$317$	−15.4374	−0.867053	−0.433527	+	0.901141i	$0.642731\pi$
$318$	−6.22968	−0.349343
$319$	−16.7365	−0.937062
$320$	0	0
$321$	6.00901	0.335390
$322$	−7.78549	−0.433868
$323$	−37.5537	−2.08954
$324$	4.14785	0.230436
$325$	0	0
$326$	−5.28486	−0.292701
$327$	1.08139	0.0598013
$328$	35.9740	1.98633
$329$	0.0121811	0.000671568 0
$330$	0	0
$331$	14.6619	0.805894	0.402947	−	0.915223i	$-0.367986\pi$
$331$	14.6619	0.805894	0.402947	+	0.915223i	$0.367986\pi$
$332$	53.1105	2.91482
$333$	−1.30537	−0.0715337
$334$	7.45300	0.407810
$335$	0	0
$336$	4.66328	0.254403
$337$	−26.6094	−1.44951	−0.724753	−	0.689009i	$-0.758046\pi$
$337$	−26.6094	−1.44951	−0.724753	+	0.689009i	$0.758046\pi$
$338$	−2.47948	−0.134866
$339$	−7.29636	−0.396284
$340$	0	0
$341$	29.4644	1.59559
$342$	−15.7581	−0.852103
$343$	−12.4422	−0.671813
$344$	−44.1791	−2.38198
$345$	0	0
$346$	−1.66979	−0.0897683
$347$	−3.06153	−0.164352	−0.0821759	−	0.996618i	$-0.526187\pi$
$347$	−3.06153	−0.164352	−0.0821759	+	0.996618i	$0.526187\pi$
$348$	−17.8179	−0.955137
$349$	−22.4273	−1.20050	−0.600252	−	0.799811i	$-0.704933\pi$
$349$	−22.4273	−1.20050	−0.600252	+	0.799811i	$0.704933\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	−5.92409	−0.315755
$353$	−23.7147	−1.26220	−0.631102	−	0.775700i	$-0.717397\pi$
$353$	−23.7147	−1.26220	−0.631102	+	0.775700i	$0.717397\pi$
$354$	15.6419	0.831356
$355$	0	0
$356$	73.2271	3.88103
$357$	−5.61324	−0.297084
$358$	−19.3843	−1.02449
$359$	−18.6484	−0.984227	−0.492113	−	0.870531i	$-0.663775\pi$
$359$	−18.6484	−0.984227	−0.492113	+	0.870531i	$0.663775\pi$
$360$	0	0
$361$	21.3912	1.12585
$362$	23.6324	1.24209
$363$	4.17964	0.219374
$364$	3.94028	0.206527
$365$	0	0
$366$	−4.00000	−0.209083
$367$	−10.7870	−0.563076	−0.281538	−	0.959550i	$-0.590845\pi$
$367$	−10.7870	−0.563076	−0.281538	+	0.959550i	$0.590845\pi$
$368$	16.2258	0.845829
$369$	−6.75499	−0.351651
$370$	0	0
$371$	2.38676	0.123914
$372$	31.3682	1.62637
$373$	−19.0430	−0.986009	−0.493005	−	0.870027i	$-0.664101\pi$
$373$	−19.0430	−0.986009	−0.493005	+	0.870027i	$0.664101\pi$
$374$	57.0822	2.95165
$375$	0	0
$376$	−0.0682885	−0.00352171
$377$	−4.29569	−0.221239
$378$	−2.35541	−0.121149
$379$	−15.9537	−0.819489	−0.409744	−	0.912200i	$-0.634382\pi$
$379$	−15.9537	−0.819489	−0.409744	+	0.912200i	$0.634382\pi$
$380$	0	0
$381$	10.6915	0.547740
$382$	39.9377	2.04339
$383$	−24.8044	−1.26744	−0.633722	−	0.773561i	$-0.718474\pi$
$383$	−24.8044	−1.26744	−0.633722	+	0.773561i	$0.718474\pi$
$384$	18.0364	0.920414
$385$	0	0
$386$	34.4568	1.75380
$387$	8.29569	0.421694
$388$	75.1696	3.81616
$389$	−36.2379	−1.83734	−0.918668	−	0.395030i	$-0.870734\pi$
$389$	−36.2379	−1.83734	−0.918668	+	0.395030i	$0.870734\pi$
$390$	0	0
$391$	−19.5312	−0.987734
$392$	32.4730	1.64013
$393$	−0.225811	−0.0113906
$394$	−29.1703	−1.46958
$395$	0	0
$396$	16.1604	0.812093
$397$	18.4111	0.924025	0.462013	−	0.886873i	$-0.347127\pi$
$397$	18.4111	0.924025	0.462013	+	0.886873i	$0.347127\pi$
$398$	20.8652	1.04588
$399$	6.03738	0.302247
$400$	0	0
$401$	24.0667	1.20183	0.600916	−	0.799312i	$-0.294803\pi$
$401$	24.0667	1.20183	0.600916	+	0.799312i	$0.294803\pi$
$402$	−11.5625	−0.576687
$403$	7.56253	0.376717
$404$	48.4971	2.41282
$405$	0	0
$406$	10.1181	0.502153
$407$	−5.08585	−0.252096
$408$	31.4683	1.55791
$409$	35.7102	1.76575	0.882877	−	0.469605i	$-0.155604\pi$
$409$	35.7102	1.76575	0.882877	+	0.469605i	$0.155604\pi$
$410$	0	0
$411$	6.63828	0.327442
$412$	31.2008	1.53715
$413$	−5.99283	−0.294888
$414$	−8.19561	−0.402792
$415$	0	0
$416$	−1.52052	−0.0745494
$417$	3.09758	0.151689
$418$	−61.3954	−3.00295
$419$	21.8241	1.06618	0.533090	−	0.846059i	$-0.321031\pi$
$419$	21.8241	1.06618	0.533090	+	0.846059i	$0.321031\pi$
$420$	0	0
$421$	−0.107706	−0.00524928	−0.00262464	−	0.999997i	$-0.500835\pi$
$421$	−0.107706	−0.00524928	−0.00262464	+	0.999997i	$0.500835\pi$
$422$	−28.6137	−1.39289
$423$	0.0128228	0.000623466 0
$424$	−13.3804	−0.649809
$425$	0	0
$426$	−19.6419	−0.951651
$427$	1.53251	0.0741634
$428$	24.9244	1.20477
$429$	3.89611	0.188106
$430$	0	0
$431$	32.3215	1.55687	0.778437	−	0.627723i	$-0.216013\pi$
$431$	32.3215	1.55687	0.778437	+	0.627723i	$0.216013\pi$
$432$	4.90893	0.236181
$433$	−14.5914	−0.701217	−0.350608	−	0.936522i	$-0.614025\pi$
$433$	−14.5914	−0.701217	−0.350608	+	0.936522i	$0.614025\pi$
$434$	−17.8128	−0.855044
$435$	0	0
$436$	4.48546	0.214814
$437$	21.0070	1.00490
$438$	36.1791	1.72870
$439$	−27.4117	−1.30829	−0.654146	−	0.756369i	$-0.726972\pi$
$439$	−27.4117	−1.30829	−0.654146	+	0.756369i	$0.726972\pi$
$440$	0	0
$441$	−6.09758	−0.290361
$442$	14.6511	0.696882
$443$	−7.19811	−0.341993	−0.170996	−	0.985272i	$-0.554699\pi$
$443$	−7.19811	−0.341993	−0.170996	+	0.985272i	$0.554699\pi$
$444$	−5.41446	−0.256959
$445$	0	0
$446$	−19.3939	−0.918326
$447$	10.9808	0.519374
$448$	−5.74513	−0.271432
$449$	18.9884	0.896119	0.448060	−	0.894004i	$-0.352115\pi$
$449$	18.9884	0.896119	0.448060	+	0.894004i	$0.352115\pi$
$450$	0	0
$451$	−26.3182	−1.23927
$452$	−30.2642	−1.42351
$453$	−2.78901	−0.131039
$454$	52.3955	2.45904
$455$	0	0
$456$	−33.8460	−1.58499
$457$	−26.3998	−1.23493	−0.617465	−	0.786599i	$-0.711840\pi$
$457$	−26.3998	−1.23493	−0.617465	+	0.786599i	$0.711840\pi$
$458$	58.3373	2.72592
$459$	−5.90893	−0.275805
$460$	0	0
$461$	−22.0585	−1.02737	−0.513684	−	0.857980i	$-0.671720\pi$
$461$	−22.0585	−1.02737	−0.513684	+	0.857980i	$0.671720\pi$
$462$	−9.17692	−0.426949
$463$	1.31425	0.0610781	0.0305391	−	0.999534i	$-0.490278\pi$
$463$	1.31425	0.0610781	0.0305391	+	0.999534i	$0.490278\pi$
$464$	−21.0872	−0.978950
$465$	0	0
$466$	15.8839	0.735806
$467$	14.8898	0.689017	0.344509	−	0.938783i	$-0.388046\pi$
$467$	14.8898	0.689017	0.344509	+	0.938783i	$0.388046\pi$
$468$	4.14785	0.191734
$469$	4.42992	0.204555
$470$	0	0
$471$	−11.3843	−0.524559
$472$	33.5963	1.54639
$473$	32.3209	1.48612
$474$	41.1604	1.89056
$475$	0	0
$476$	−23.2828	−1.06717
$477$	2.51249	0.115039
$478$	31.4142	1.43685
$479$	3.03653	0.138743	0.0693713	−	0.997591i	$-0.477901\pi$
$479$	3.03653	0.138743	0.0693713	+	0.997591i	$0.477901\pi$
$480$	0	0
$481$	−1.30537	−0.0595196
$482$	0.511906	0.0233167
$483$	3.13996	0.142873
$484$	17.3365	0.788022
$485$	0	0
$486$	−2.47948	−0.112472
$487$	16.2584	0.736741	0.368370	−	0.929679i	$-0.379916\pi$
$487$	16.2584	0.736741	0.368370	+	0.929679i	$0.379916\pi$
$488$	−8.59138	−0.388914
$489$	2.13144	0.0963869
$490$	0	0
$491$	−12.5098	−0.564558	−0.282279	−	0.959332i	$-0.591090\pi$
$491$	−12.5098	−0.564558	−0.282279	+	0.959332i	$0.591090\pi$
$492$	−28.0187	−1.26318
$493$	25.3829	1.14319
$494$	−15.7581	−0.708992
$495$	0	0
$496$	37.1239	1.66691
$497$	7.52534	0.337557
$498$	−31.7482	−1.42267
$499$	−0.977442	−0.0437563	−0.0218782	−	0.999761i	$-0.506965\pi$
$499$	−0.977442	−0.0437563	−0.0218782	+	0.999761i	$0.506965\pi$
$500$	0	0
$501$	−3.00587	−0.134292
$502$	−38.4390	−1.71562
$503$	22.9051	1.02129	0.510644	−	0.859792i	$-0.329407\pi$
$503$	22.9051	1.02129	0.510644	+	0.859792i	$0.329407\pi$
$504$	−5.05905	−0.225348
$505$	0	0
$506$	−31.9310	−1.41950
$507$	1.00000	0.0444116
$508$	44.3465	1.96756
$509$	−13.2463	−0.587132	−0.293566	−	0.955939i	$-0.594842\pi$
$509$	−13.2463	−0.587132	−0.293566	+	0.955939i	$0.594842\pi$
$510$	0	0
$511$	−13.8612	−0.613184
$512$	44.8214	1.98084
$513$	6.35541	0.280598
$514$	15.1731	0.669255
$515$	0	0
$516$	34.4092	1.51478
$517$	0.0499590	0.00219720
$518$	3.07467	0.135093
$519$	0.673441	0.0295608
$520$	0	0
$521$	−8.70493	−0.381370	−0.190685	−	0.981651i	$-0.561071\pi$
$521$	−8.70493	−0.381370	−0.190685	+	0.981651i	$0.561071\pi$
$522$	10.6511	0.466186
$523$	29.4014	1.28563	0.642817	−	0.766020i	$-0.277766\pi$
$523$	29.4014	1.28563	0.642817	+	0.766020i	$0.277766\pi$
$524$	−0.936627	−0.0409168
$525$	0	0
$526$	42.4155	1.84940
$527$	−44.6865	−1.94657
$528$	19.1257	0.832339
$529$	−12.0745	−0.524980
$530$	0	0
$531$	−6.30851	−0.273766
$532$	25.0421	1.08571
$533$	−6.75499	−0.292591
$534$	−43.7734	−1.89426
$535$	0	0
$536$	−24.8345	−1.07269
$537$	7.81786	0.337365
$538$	−69.1730	−2.98226
$539$	−23.7568	−1.02328
$540$	0	0
$541$	−1.40100	−0.0602335	−0.0301168	−	0.999546i	$-0.509588\pi$
$541$	−1.40100	−0.0602335	−0.0301168	+	0.999546i	$0.509588\pi$
$542$	71.0544	3.05205
$543$	−9.53118	−0.409022
$544$	8.98462	0.385212
$545$	0	0
$546$	−2.35541	−0.100802
$547$	−2.42904	−0.103858	−0.0519292	−	0.998651i	$-0.516537\pi$
$547$	−2.42904	−0.103858	−0.0519292	+	0.998651i	$0.516537\pi$
$548$	27.5345	1.17622
$549$	1.61324	0.0688513
$550$	0	0
$551$	−27.3009	−1.16306
$552$	−17.6029	−0.749229
$553$	−15.7697	−0.670595
$554$	1.07510	0.0456767
$555$	0	0
$556$	12.8483	0.544888
$557$	−14.2429	−0.603491	−0.301746	−	0.953388i	$-0.597569\pi$
$557$	−14.2429	−0.603491	−0.301746	+	0.953388i	$0.597569\pi$
$558$	−18.7512	−0.793801
$559$	8.29569	0.350870
$560$	0	0
$561$	−23.0218	−0.971982
$562$	46.0557	1.94274
$563$	−38.1924	−1.60962	−0.804810	−	0.593533i	$-0.797733\pi$
$563$	−38.1924	−1.60962	−0.804810	+	0.593533i	$0.797733\pi$
$564$	0.0531870	0.00223958
$565$	0	0
$566$	−54.3899	−2.28618
$567$	0.949959	0.0398945
$568$	−42.1877	−1.77016
$569$	−22.1387	−0.928104	−0.464052	−	0.885808i	$-0.653605\pi$
$569$	−22.1387	−0.928104	−0.464052	+	0.885808i	$0.653605\pi$
$570$	0	0
$571$	−44.8956	−1.87882	−0.939412	−	0.342791i	$-0.888628\pi$
$571$	−44.8956	−1.87882	−0.939412	+	0.342791i	$0.888628\pi$
$572$	16.1604	0.675702
$573$	−16.1073	−0.672890
$574$	15.9108	0.664103
$575$	0	0
$576$	−6.04776	−0.251990
$577$	−4.07248	−0.169540	−0.0847698	−	0.996401i	$-0.527016\pi$
$577$	−4.07248	−0.169540	−0.0847698	+	0.996401i	$0.527016\pi$
$578$	−44.4210	−1.84767
$579$	−13.8967	−0.577529
$580$	0	0
$581$	12.1636	0.504632
$582$	−44.9346	−1.86260
$583$	9.78893	0.405416
$584$	77.7071	3.21554
$585$	0	0
$586$	22.9608	0.948502
$587$	24.5786	1.01447	0.507233	−	0.861809i	$-0.330668\pi$
$587$	24.5786	1.01447	0.507233	+	0.861809i	$0.330668\pi$
$588$	−25.2918	−1.04302
$589$	48.0630	1.98040
$590$	0	0
$591$	11.7647	0.483934
$592$	−6.40795	−0.263365
$593$	0.191663	0.00787065	0.00393532	−	0.999992i	$-0.498747\pi$
$593$	0.191663	0.00787065	0.00393532	+	0.999992i	$0.498747\pi$
$594$	−9.66033	−0.396368
$595$	0	0
$596$	45.5467	1.86566
$597$	−8.41513	−0.344408
$598$	−8.19561	−0.335143
$599$	22.5977	0.923316	0.461658	−	0.887058i	$-0.347255\pi$
$599$	22.5977	0.923316	0.461658	+	0.887058i	$0.347255\pi$
$600$	0	0
$601$	17.0155	0.694077	0.347039	−	0.937851i	$-0.387187\pi$
$601$	17.0155	0.694077	0.347039	+	0.937851i	$0.387187\pi$
$602$	−19.5397	−0.796380
$603$	4.66328	0.189903
$604$	−11.5684	−0.470710
$605$	0	0
$606$	−28.9905	−1.17766
$607$	−34.4651	−1.39889	−0.699447	−	0.714684i	$-0.746570\pi$
$607$	−34.4651	−1.39889	−0.699447	+	0.714684i	$0.746570\pi$
$608$	−9.66349	−0.391906
$609$	−4.08073	−0.165359
$610$	0	0
$611$	0.0128228	0.000518755 0
$612$	−24.5093	−0.990731
$613$	−7.02682	−0.283810	−0.141905	−	0.989880i	$-0.545323\pi$
$613$	−7.02682	−0.283810	−0.141905	+	0.989880i	$0.545323\pi$
$614$	−22.6876	−0.915597
$615$	0	0
$616$	−19.7106	−0.794163
$617$	12.1122	0.487620	0.243810	−	0.969823i	$-0.421603\pi$
$617$	12.1122	0.487620	0.243810	+	0.969823i	$0.421603\pi$
$618$	−18.6511	−0.750257
$619$	30.8710	1.24081	0.620406	−	0.784281i	$-0.286968\pi$
$619$	30.8710	1.24081	0.620406	+	0.784281i	$0.286968\pi$
$620$	0	0
$621$	3.30537	0.132640
$622$	−20.1207	−0.806767
$623$	16.7708	0.671908
$624$	4.90893	0.196514
$625$	0	0
$626$	10.4622	0.418155
$627$	24.7613	0.988873
$628$	−47.2201	−1.88429
$629$	7.71332	0.307550
$630$	0	0
$631$	9.48047	0.377412	0.188706	−	0.982034i	$-0.439571\pi$
$631$	9.48047	0.377412	0.188706	+	0.982034i	$0.439571\pi$
$632$	88.4062	3.51661
$633$	11.5402	0.458681
$634$	38.2769	1.52017
$635$	0	0
$636$	10.4214	0.413236
$637$	−6.09758	−0.241595
$638$	41.4978	1.64291
$639$	7.92175	0.313380
$640$	0	0
$641$	17.3964	0.687118	0.343559	−	0.939131i	$-0.388367\pi$
$641$	17.3964	0.687118	0.343559	+	0.939131i	$0.388367\pi$
$642$	−14.8993	−0.588027
$643$	−33.9506	−1.33888	−0.669440	−	0.742866i	$-0.733466\pi$
$643$	−33.9506	−1.33888	−0.669440	+	0.742866i	$0.733466\pi$
$644$	13.0241	0.513221
$645$	0	0
$646$	93.1137	3.66351
$647$	9.18548	0.361118	0.180559	−	0.983564i	$-0.442209\pi$
$647$	9.18548	0.361118	0.180559	+	0.983564i	$0.442209\pi$
$648$	−5.32555	−0.209207
$649$	−24.5786	−0.964796
$650$	0	0
$651$	7.18409	0.281567
$652$	8.84087	0.346235
$653$	−3.54781	−0.138837	−0.0694183	−	0.997588i	$-0.522114\pi$
$653$	−3.54781	−0.138837	−0.0694183	+	0.997588i	$0.522114\pi$
$654$	−2.68130	−0.104847
$655$	0	0
$656$	−33.1598	−1.29467
$657$	−14.5914	−0.569264
$658$	−0.0302030	−0.00117743
$659$	−15.5339	−0.605115	−0.302557	−	0.953131i	$-0.597840\pi$
$659$	−15.5339	−0.605115	−0.302557	+	0.953131i	$0.597840\pi$
$660$	0	0
$661$	1.60878	0.0625745	0.0312872	−	0.999510i	$-0.490039\pi$
$661$	1.60878	0.0625745	0.0312872	+	0.999510i	$0.490039\pi$
$662$	−36.3541	−1.41294
$663$	−5.90893	−0.229484
$664$	−68.1903	−2.64630
$665$	0	0
$666$	3.23664	0.125417
$667$	−14.1988	−0.549781
$668$	−12.4679	−0.482396
$669$	7.82173	0.302405
$670$	0	0
$671$	6.28535	0.242643
$672$	−1.44443	−0.0557200
$673$	−8.18214	−0.315398	−0.157699	−	0.987487i	$-0.550408\pi$
$673$	−8.18214	−0.315398	−0.157699	+	0.987487i	$0.550408\pi$
$674$	65.9776	2.54136
$675$	0	0
$676$	4.14785	0.159533
$677$	−42.7063	−1.64134	−0.820668	−	0.571405i	$-0.806399\pi$
$677$	−42.7063	−1.64134	−0.820668	+	0.571405i	$0.806399\pi$
$678$	18.0912	0.694789
$679$	17.2157	0.660677
$680$	0	0
$681$	−21.1316	−0.809764
$682$	−73.0566	−2.79748
$683$	9.14962	0.350100	0.175050	−	0.984560i	$-0.443991\pi$
$683$	9.14962	0.350100	0.175050	+	0.984560i	$0.443991\pi$
$684$	26.3612	1.00795
$685$	0	0
$686$	30.8501	1.17786
$687$	−23.5280	−0.897649
$688$	40.7229	1.55255
$689$	2.51249	0.0957182
$690$	0	0
$691$	−16.9918	−0.646398	−0.323199	−	0.946331i	$-0.604758\pi$
$691$	−16.9918	−0.646398	−0.323199	+	0.946331i	$0.604758\pi$
$692$	2.79333	0.106186
$693$	3.70114	0.140595
$694$	7.59103	0.288152
$695$	0	0
$696$	22.8769	0.867147
$697$	39.9148	1.51188
$698$	55.6080	2.10480
$699$	−6.40612	−0.242302
$700$	0	0
$701$	−40.8320	−1.54220	−0.771101	−	0.636712i	$-0.780294\pi$
$701$	−40.8320	−1.54220	−0.771101	+	0.636712i	$0.780294\pi$
$702$	−2.47948	−0.0935821
$703$	−8.29614	−0.312895
$704$	−23.5627	−0.888053
$705$	0	0
$706$	58.8001	2.21297
$707$	11.1070	0.417723
$708$	−26.1667	−0.983406
$709$	−38.0193	−1.42784	−0.713922	−	0.700225i	$-0.753083\pi$
$709$	−38.0193	−1.42784	−0.713922	+	0.700225i	$0.753083\pi$
$710$	0	0
$711$	−16.6004	−0.622563
$712$	−94.0185	−3.52349
$713$	24.9969	0.936143
$714$	13.9179	0.520866
$715$	0	0
$716$	32.4273	1.21186
$717$	−12.6696	−0.473156
$718$	46.2385	1.72561
$719$	27.9175	1.04115	0.520573	−	0.853817i	$-0.325718\pi$
$719$	27.9175	1.04115	0.520573	+	0.853817i	$0.325718\pi$
$720$	0	0
$721$	7.14575	0.266122
$722$	−53.0392	−1.97391
$723$	−0.206457	−0.00767820
$724$	−39.5338	−1.46926
$725$	0	0
$726$	−10.3634	−0.384620
$727$	−4.31960	−0.160205	−0.0801026	−	0.996787i	$-0.525525\pi$
$727$	−4.31960	−0.160205	−0.0801026	+	0.996787i	$0.525525\pi$
$728$	−5.05905	−0.187501
$729$	1.00000	0.0370370
$730$	0	0
$731$	−49.0186	−1.81302
$732$	6.69146	0.247324
$733$	−13.1669	−0.486330	−0.243165	−	0.969985i	$-0.578186\pi$
$733$	−13.1669	−0.486330	−0.243165	+	0.969985i	$0.578186\pi$
$734$	26.7462	0.987219
$735$	0	0
$736$	−5.02586	−0.185256
$737$	18.1686	0.669250
$738$	16.7489	0.616536
$739$	2.14218	0.0788014	0.0394007	−	0.999223i	$-0.487455\pi$
$739$	2.14218	0.0788014	0.0394007	+	0.999223i	$0.487455\pi$
$740$	0	0
$741$	6.35541	0.233472
$742$	−5.91794	−0.217254
$743$	−24.4771	−0.897979	−0.448990	−	0.893537i	$-0.648216\pi$
$743$	−24.4771	−0.897979	−0.448990	+	0.893537i	$0.648216\pi$
$744$	−40.2746	−1.47654
$745$	0	0
$746$	47.2168	1.72873
$747$	12.8044	0.468487
$748$	−95.4909	−3.49149
$749$	5.70831	0.208577
$750$	0	0
$751$	−12.0058	−0.438097	−0.219049	−	0.975714i	$-0.570295\pi$
$751$	−12.0058	−0.438097	−0.219049	+	0.975714i	$0.570295\pi$
$752$	0.0629463	0.00229541
$753$	15.5028	0.564954
$754$	10.6511	0.387890
$755$	0	0
$756$	3.94028	0.143307
$757$	16.8442	0.612212	0.306106	−	0.951997i	$-0.400974\pi$
$757$	16.8442	0.612212	0.306106	+	0.951997i	$0.400974\pi$
$758$	39.5571	1.43678
$759$	12.8781	0.467444
$760$	0	0
$761$	3.97932	0.144250	0.0721251	−	0.997396i	$-0.477022\pi$
$761$	3.97932	0.144250	0.0721251	+	0.997396i	$0.477022\pi$
$762$	−26.5093	−0.960331
$763$	1.02728	0.0371900
$764$	−66.8104	−2.41712
$765$	0	0
$766$	61.5021	2.22216
$767$	−6.30851	−0.227787
$768$	−32.6254	−1.17727
$769$	15.5176	0.559579	0.279790	−	0.960061i	$-0.409735\pi$
$769$	15.5176	0.559579	0.279790	+	0.960061i	$0.409735\pi$
$770$	0	0
$771$	−6.11944	−0.220386
$772$	−57.6416	−2.07456
$773$	44.4624	1.59920	0.799601	−	0.600531i	$-0.205044\pi$
$773$	44.4624	1.59920	0.799601	+	0.600531i	$0.205044\pi$
$774$	−20.5690	−0.739339
$775$	0	0
$776$	−96.5126	−3.46460
$777$	−1.24004	−0.0444864
$778$	89.8514	3.22133
$779$	−42.9307	−1.53815
$780$	0	0
$781$	30.8640	1.10440
$782$	48.4273	1.73176
$783$	−4.29569	−0.153515
$784$	−29.9326	−1.06902
$785$	0	0
$786$	0.559894	0.0199708
$787$	32.0619	1.14288	0.571441	−	0.820643i	$-0.306384\pi$
$787$	32.0619	1.14288	0.571441	+	0.820643i	$0.306384\pi$
$788$	48.7980	1.73836
$789$	−17.1066	−0.609011
$790$	0	0
$791$	−6.93124	−0.246446
$792$	−20.7489	−0.737280
$793$	1.61324	0.0572878
$794$	−45.6500	−1.62006
$795$	0	0
$796$	−34.9046	−1.23716
$797$	24.7378	0.876260	0.438130	−	0.898912i	$-0.355641\pi$
$797$	24.7378	0.876260	0.438130	+	0.898912i	$0.355641\pi$
$798$	−14.9696	−0.529917
$799$	−0.0757691	−0.00268052
$800$	0	0
$801$	17.6542	0.623782
$802$	−59.6729	−2.10712
$803$	−56.8496	−2.00618
$804$	19.3426	0.682160
$805$	0	0
$806$	−18.7512	−0.660482
$807$	27.8981	0.982061
$808$	−62.2670	−2.19055
$809$	11.3400	0.398694	0.199347	−	0.979929i	$-0.436118\pi$
$809$	11.3400	0.398694	0.199347	+	0.979929i	$0.436118\pi$
$810$	0	0
$811$	8.96070	0.314653	0.157326	−	0.987547i	$-0.449713\pi$
$811$	8.96070	0.314653	0.157326	+	0.987547i	$0.449713\pi$
$812$	−16.9262	−0.593994
$813$	−28.6569	−1.00504
$814$	12.6103	0.441990
$815$	0	0
$816$	−29.0065	−1.01543
$817$	52.7225	1.84453
$818$	−88.5428	−3.09583
$819$	0.949959	0.0331942
$820$	0	0
$821$	5.28500	0.184448	0.0922239	−	0.995738i	$-0.470602\pi$
$821$	5.28500	0.184448	0.0922239	+	0.995738i	$0.470602\pi$
$822$	−16.4595	−0.574091
$823$	−11.2463	−0.392023	−0.196012	−	0.980602i	$-0.562799\pi$
$823$	−11.2463	−0.392023	−0.196012	+	0.980602i	$0.562799\pi$
$824$	−40.0597	−1.39554
$825$	0	0
$826$	14.8591	0.517015
$827$	−42.9625	−1.49395	−0.746976	−	0.664851i	$-0.768495\pi$
$827$	−42.9625	−1.49395	−0.746976	+	0.664851i	$0.768495\pi$
$828$	13.7102	0.476461
$829$	21.1444	0.734376	0.367188	−	0.930147i	$-0.380321\pi$
$829$	21.1444	0.734376	0.367188	+	0.930147i	$0.380321\pi$
$830$	0	0
$831$	−0.433599	−0.0150414
$832$	−6.04776	−0.209668
$833$	36.0302	1.24837
$834$	−7.68040	−0.265950
$835$	0	0
$836$	102.706	3.55217
$837$	7.56253	0.261399
$838$	−54.1126	−1.86929
$839$	−26.3671	−0.910293	−0.455146	−	0.890417i	$-0.650413\pi$
$839$	−26.3671	−0.910293	−0.455146	+	0.890417i	$0.650413\pi$
$840$	0	0
$841$	−10.5470	−0.363691
$842$	0.267056	0.00920336
$843$	−18.5747	−0.639746
$844$	47.8669	1.64765
$845$	0	0
$846$	−0.0317940	−0.00109310
$847$	3.97048	0.136427
$848$	12.3336	0.423539
$849$	21.9360	0.752840
$850$	0	0
$851$	−4.31472	−0.147907
$852$	32.8582	1.12570
$853$	52.4061	1.79435	0.897175	−	0.441676i	$-0.145616\pi$
$853$	52.4061	1.79435	0.897175	+	0.441676i	$0.145616\pi$
$854$	−3.79984	−0.130028
$855$	0	0
$856$	−32.0013	−1.09378
$857$	24.6382	0.841625	0.420813	−	0.907148i	$-0.361745\pi$
$857$	24.6382	0.841625	0.420813	+	0.907148i	$0.361745\pi$
$858$	−9.66033	−0.329798
$859$	−47.0110	−1.60399	−0.801997	−	0.597329i	$-0.796229\pi$
$859$	−47.0110	−1.60399	−0.801997	+	0.597329i	$0.796229\pi$
$860$	0	0
$861$	−6.41696	−0.218690
$862$	−80.1407	−2.72960
$863$	−14.9378	−0.508490	−0.254245	−	0.967140i	$-0.581827\pi$
$863$	−14.9378	−0.508490	−0.254245	+	0.967140i	$0.581827\pi$
$864$	−1.52052	−0.0517290
$865$	0	0
$866$	36.1791	1.22942
$867$	17.9154	0.608440
$868$	29.7985	1.01143
$869$	−64.6769	−2.19401
$870$	0	0
$871$	4.66328	0.158009
$872$	−5.75902	−0.195025
$873$	18.1226	0.613356
$874$	−52.0864	−1.76185
$875$	0	0
$876$	−60.5228	−2.04488
$877$	−29.3418	−0.990801	−0.495400	−	0.868665i	$-0.664979\pi$
$877$	−29.3418	−0.990801	−0.495400	+	0.868665i	$0.664979\pi$
$878$	67.9670	2.29378
$879$	−9.26031	−0.312342
$880$	0	0
$881$	12.1590	0.409647	0.204824	−	0.978799i	$-0.434338\pi$
$881$	12.1590	0.409647	0.204824	+	0.978799i	$0.434338\pi$
$882$	15.1189	0.509078
$883$	31.1431	1.04805	0.524024	−	0.851703i	$-0.324430\pi$
$883$	31.1431	1.04805	0.524024	+	0.851703i	$0.324430\pi$
$884$	−24.5093	−0.824338
$885$	0	0
$886$	17.8476	0.599602
$887$	−2.56117	−0.0859955	−0.0429978	−	0.999075i	$-0.513691\pi$
$887$	−2.56117	−0.0859955	−0.0429978	+	0.999075i	$0.513691\pi$
$888$	6.95180	0.233287
$889$	10.1564	0.340636
$890$	0	0
$891$	3.89611	0.130524
$892$	32.4433	1.08628
$893$	0.0814942	0.00272710
$894$	−27.2267	−0.910598
$895$	0	0
$896$	17.1338	0.572400
$897$	3.30537	0.110363
$898$	−47.0815	−1.57113
$899$	−32.4863	−1.08348
$900$	0	0
$901$	−14.8461	−0.494596
$902$	65.2555	2.17277
$903$	7.88056	0.262249
$904$	38.8571	1.29237
$905$	0	0
$906$	6.91530	0.229746
$907$	34.5273	1.14646	0.573231	−	0.819394i	$-0.305690\pi$
$907$	34.5273	1.14646	0.573231	+	0.819394i	$0.305690\pi$
$908$	−87.6506	−2.90879
$909$	11.6921	0.387803
$910$	0	0
$911$	−16.4985	−0.546619	−0.273309	−	0.961926i	$-0.588118\pi$
$911$	−16.4985	−0.546619	−0.273309	+	0.961926i	$0.588118\pi$
$912$	31.1982	1.03308
$913$	49.8872	1.65102
$914$	65.4578	2.16515
$915$	0	0
$916$	−97.5905	−3.22448
$917$	−0.214511	−0.00708377
$918$	14.6511	0.483558
$919$	29.8399	0.984327	0.492163	−	0.870503i	$-0.336206\pi$
$919$	29.8399	0.984327	0.492163	+	0.870503i	$0.336206\pi$
$920$	0	0
$921$	9.15012	0.301507
$922$	54.6938	1.80124
$923$	7.92175	0.260748
$924$	15.3518	0.505036
$925$	0	0
$926$	−3.25865	−0.107086
$927$	7.52217	0.247060
$928$	6.53166	0.214412
$929$	29.9391	0.982270	0.491135	−	0.871084i	$-0.336582\pi$
$929$	29.9391	0.982270	0.491135	+	0.871084i	$0.336582\pi$
$930$	0	0
$931$	−38.7526	−1.27007
$932$	−26.5716	−0.870381
$933$	8.11488	0.265669
$934$	−36.9190	−1.20803
$935$	0	0
$936$	−5.32555	−0.174071
$937$	−32.7542	−1.07003	−0.535016	−	0.844842i	$-0.679694\pi$
$937$	−32.7542	−1.07003	−0.535016	+	0.844842i	$0.679694\pi$
$938$	−10.9839	−0.358638
$939$	−4.21952	−0.137699
$940$	0	0
$941$	−26.6679	−0.869349	−0.434675	−	0.900588i	$-0.643137\pi$
$941$	−26.6679	−0.869349	−0.434675	+	0.900588i	$0.643137\pi$
$942$	28.2271	0.919688
$943$	−22.3277	−0.727091
$944$	−30.9680	−1.00792
$945$	0	0
$946$	−80.1391	−2.60555
$947$	−16.9685	−0.551402	−0.275701	−	0.961243i	$-0.588910\pi$
$947$	−16.9685	−0.551402	−0.275701	+	0.961243i	$0.588910\pi$
$948$	−68.8558	−2.23633
$949$	−14.5914	−0.473656
$950$	0	0
$951$	−15.4374	−0.500593
$952$	29.8936	0.968856
$953$	−45.6056	−1.47731	−0.738655	−	0.674084i	$-0.764539\pi$
$953$	−45.6056	−1.47731	−0.738655	+	0.674084i	$0.764539\pi$
$954$	−6.22968	−0.201693
$955$	0	0
$956$	−52.5517	−1.69964
$957$	−16.7365	−0.541013
$958$	−7.52903	−0.243252
$959$	6.30609	0.203634
$960$	0	0
$961$	26.1919	0.844899
$962$	3.23664	0.104353
$963$	6.00901	0.193638
$964$	−0.856350	−0.0275812
$965$	0	0
$966$	−7.78549	−0.250494
$967$	7.77794	0.250122	0.125061	−	0.992149i	$-0.460087\pi$
$967$	7.77794	0.250122	0.125061	+	0.992149i	$0.460087\pi$
$968$	−22.2589	−0.715427
$969$	−37.5537	−1.20640
$970$	0	0
$971$	14.1572	0.454327	0.227163	−	0.973857i	$-0.427055\pi$
$971$	14.1572	0.454327	0.227163	+	0.973857i	$0.427055\pi$
$972$	4.14785	0.133042
$973$	2.94257	0.0943345
$974$	−40.3126	−1.29170
$975$	0	0
$976$	7.91927	0.253490
$977$	−13.0985	−0.419058	−0.209529	−	0.977802i	$-0.567193\pi$
$977$	−13.0985	−0.419058	−0.209529	+	0.977802i	$0.567193\pi$
$978$	−5.28486	−0.168991
$979$	68.7828	2.19831
$980$	0	0
$981$	1.08139	0.0345263
$982$	31.0178	0.989817
$983$	49.1176	1.56661	0.783305	−	0.621638i	$-0.213533\pi$
$983$	49.1176	1.56661	0.783305	+	0.621638i	$0.213533\pi$
$984$	35.9740	1.14681
$985$	0	0
$986$	−62.9366	−2.00431
$987$	0.0121811	0.000387730 0
$988$	26.3612	0.838663
$989$	27.4203	0.871915
$990$	0	0
$991$	17.0358	0.541161	0.270581	−	0.962697i	$-0.412784\pi$
$991$	17.0358	0.541161	0.270581	+	0.962697i	$0.412784\pi$
$992$	−11.4989	−0.365092
$993$	14.6619	0.465283
$994$	−18.6590	−0.591826
$995$	0	0
$996$	53.1105	1.68287
$997$	−8.01802	−0.253933	−0.126967	−	0.991907i	$-0.540524\pi$
$997$	−8.01802	−0.253933	−0.126967	+	0.991907i	$0.540524\pi$
$998$	2.42355	0.0767162
$999$	−1.30537	−0.0413000

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	975.2.a.s.1.1		5
3.2	odd	2		2925.2.a.bm.1.5		5
5.2	odd	4		195.2.c.b.79.1	✓	10
5.3	odd	4		195.2.c.b.79.10	yes	10
5.4	even	2		975.2.a.r.1.5		5
15.2	even	4		585.2.c.c.469.10		10
15.8	even	4		585.2.c.c.469.1		10
15.14	odd	2		2925.2.a.bl.1.1		5
20.3	even	4		3120.2.l.p.1249.2		10
20.7	even	4		3120.2.l.p.1249.7		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.2.c.b.79.1	✓	10	5.2	odd	4
195.2.c.b.79.10	yes	10	5.3	odd	4
585.2.c.c.469.1		10	15.8	even	4
585.2.c.c.469.10		10	15.2	even	4
975.2.a.r.1.5		5	5.4	even	2
975.2.a.s.1.1		5	1.1	even	1	trivial
2925.2.a.bl.1.1		5	15.14	odd	2
2925.2.a.bm.1.5		5	3.2	odd	2
3120.2.l.p.1249.2		10	20.3	even	4
3120.2.l.p.1249.7		10	20.7	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	975.a (trivial)

\(f(q)\)	\(=\)	\(q-2.47948 q^{2} +1.00000 q^{3} +4.14785 q^{4} -2.47948 q^{6} +0.949959 q^{7} -5.32555 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.47948 q^{2} +1.00000 q^{3} +4.14785 q^{4} -2.47948 q^{6} +0.949959 q^{7} -5.32555 q^{8} +1.00000 q^{9} +3.89611 q^{11} +4.14785 q^{12} +1.00000 q^{13} -2.35541 q^{14} +4.90893 q^{16} -5.90893 q^{17} -2.47948 q^{18} +6.35541 q^{19} +0.949959 q^{21} -9.66033 q^{22} +3.30537 q^{23} -5.32555 q^{24} -2.47948 q^{26} +1.00000 q^{27} +3.94028 q^{28} -4.29569 q^{29} +7.56253 q^{31} -1.52052 q^{32} +3.89611 q^{33} +14.6511 q^{34} +4.14785 q^{36} -1.30537 q^{37} -15.7581 q^{38} +1.00000 q^{39} -6.75499 q^{41} -2.35541 q^{42} +8.29569 q^{43} +16.1604 q^{44} -8.19561 q^{46} +0.0128228 q^{47} +4.90893 q^{48} -6.09758 q^{49} -5.90893 q^{51} +4.14785 q^{52} +2.51249 q^{53} -2.47948 q^{54} -5.05905 q^{56} +6.35541 q^{57} +10.6511 q^{58} -6.30851 q^{59} +1.61324 q^{61} -18.7512 q^{62} +0.949959 q^{63} -6.04776 q^{64} -9.66033 q^{66} +4.66328 q^{67} -24.5093 q^{68} +3.30537 q^{69} +7.92175 q^{71} -5.32555 q^{72} -14.5914 q^{73} +3.23664 q^{74} +26.3612 q^{76} +3.70114 q^{77} -2.47948 q^{78} -16.6004 q^{79} +1.00000 q^{81} +16.7489 q^{82} +12.8044 q^{83} +3.94028 q^{84} -20.5690 q^{86} -4.29569 q^{87} -20.7489 q^{88} +17.6542 q^{89} +0.949959 q^{91} +13.7102 q^{92} +7.56253 q^{93} -0.0317940 q^{94} -1.52052 q^{96} +18.1226 q^{97} +15.1189 q^{98} +3.89611 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{3} + 6 q^{4} + 5 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q + 5 * q^3 + 6 * q^4 + 5 * q^7 + 5 * q^9	\( 5 q + 5 q^{3} + 6 q^{4} + 5 q^{7} + 5 q^{9} + 5 q^{11} + 6 q^{12} + 5 q^{13} + 12 q^{14} - 5 q^{17} + 8 q^{19} + 5 q^{21} - 8 q^{22} - 7 q^{23} + 5 q^{27} + 14 q^{28} + 8 q^{29} + 12 q^{31} - 20 q^{32} + 5 q^{33} + 20 q^{34} + 6 q^{36} + 17 q^{37} - 24 q^{38} + 5 q^{39} + 5 q^{41} + 12 q^{42} + 12 q^{43} + 18 q^{44} - 12 q^{46} - 10 q^{47} + 22 q^{49} - 5 q^{51} + 6 q^{52} - 13 q^{53} + 8 q^{57} + 8 q^{59} + 13 q^{61} - 40 q^{62} + 5 q^{63} - 16 q^{64} - 8 q^{66} + 28 q^{67} - 14 q^{68} - 7 q^{69} + 5 q^{71} - 14 q^{73} + 12 q^{74} - 35 q^{77} + q^{79} + 5 q^{81} + 16 q^{82} - 6 q^{83} + 14 q^{84} + 8 q^{87} - 36 q^{88} + 19 q^{89} + 5 q^{91} - 10 q^{92} + 12 q^{93} - 12 q^{94} - 20 q^{96} - 13 q^{97} + 28 q^{98} + 5 q^{99}+O(q^{100}) \) 5 * q + 5 * q^3 + 6 * q^4 + 5 * q^7 + 5 * q^9 + 5 * q^11 + 6 * q^12 + 5 * q^13 + 12 * q^14 - 5 * q^17 + 8 * q^19 + 5 * q^21 - 8 * q^22 - 7 * q^23 + 5 * q^27 + 14 * q^28 + 8 * q^29 + 12 * q^31 - 20 * q^32 + 5 * q^33 + 20 * q^34 + 6 * q^36 + 17 * q^37 - 24 * q^38 + 5 * q^39 + 5 * q^41 + 12 * q^42 + 12 * q^43 + 18 * q^44 - 12 * q^46 - 10 * q^47 + 22 * q^49 - 5 * q^51 + 6 * q^52 - 13 * q^53 + 8 * q^57 + 8 * q^59 + 13 * q^61 - 40 * q^62 + 5 * q^63 - 16 * q^64 - 8 * q^66 + 28 * q^67 - 14 * q^68 - 7 * q^69 + 5 * q^71 - 14 * q^73 + 12 * q^74 - 35 * q^77 + q^79 + 5 * q^81 + 16 * q^82 - 6 * q^83 + 14 * q^84 + 8 * q^87 - 36 * q^88 + 19 * q^89 + 5 * q^91 - 10 * q^92 + 12 * q^93 - 12 * q^94 - 20 * q^96 - 13 * q^97 + 28 * q^98 + 5 * q^99

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