LMFDB - Embedded newform 3675.2.a.bq.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3675,2,Mod(1,3675)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3675, 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("3675.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3675 = 3 \cdot 5^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3675.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:	$29.3450227428$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.88404.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 7x^{2} + 5x + 4 $ x^4 - x^3 - 7x^2 + 5x + 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 525)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.22868$ of defining polynomial
Character	$\chi$	$=$	3675.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.22868 q^{2} -1.00000 q^{3} -0.490347 q^{4} +1.22868 q^{6} +3.05984 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-1.22868 q^{2} -1.00000 q^{3} -0.490347 q^{4} +1.22868 q^{6} +3.05984 q^{8} +1.00000 q^{9} -4.74588 q^{11} +0.490347 q^{12} -6.25553 q^{13} -2.77886 q^{16} -6.28852 q^{17} -1.22868 q^{18} -4.94771 q^{19} +5.83116 q^{22} -4.98069 q^{23} -3.05984 q^{24} +7.68604 q^{26} -1.00000 q^{27} +4.00000 q^{29} -1.65919 q^{31} -2.70534 q^{32} +4.74588 q^{33} +7.72657 q^{34} -0.490347 q^{36} +4.43805 q^{37} +6.07914 q^{38} +6.25553 q^{39} -1.30782 q^{41} -10.2362 q^{43} +2.32713 q^{44} +6.11968 q^{46} +3.76518 q^{47} +2.77886 q^{48} +6.28852 q^{51} +3.06738 q^{52} -7.20323 q^{53} +1.22868 q^{54} +4.94771 q^{57} -4.91472 q^{58} -0.234819 q^{59} +5.00000 q^{61} +2.03861 q^{62} +8.88173 q^{64} -5.83116 q^{66} -2.49035 q^{67} +3.08356 q^{68} +4.98069 q^{69} +6.06598 q^{71} +3.05984 q^{72} -12.4918 q^{73} -5.45294 q^{74} +2.42609 q^{76} -7.68604 q^{78} -0.831159 q^{79} +1.00000 q^{81} +1.60689 q^{82} -1.14954 q^{83} +12.5770 q^{86} -4.00000 q^{87} -14.5216 q^{88} -2.28852 q^{89} +2.44227 q^{92} +1.65919 q^{93} -4.62620 q^{94} +2.70534 q^{96} -0.476664 q^{97} -4.74588 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} - 4 q^{3} + 7 q^{4} + q^{6} - 3 q^{8} + 4 q^{9}+O(q^{10}) $ 4 * q - q^2 - 4 * q^3 + 7 * q^4 + q^6 - 3 * q^8 + 4 * q^9	$ 4 q - q^{2} - 4 q^{3} + 7 q^{4} + q^{6} - 3 q^{8} + 4 q^{9} + 8 q^{11} - 7 q^{12} - 7 q^{13} + 17 q^{16} - 6 q^{17} - q^{18} - 3 q^{19} + 12 q^{22} - 2 q^{23} + 3 q^{24} + 19 q^{26} - 4 q^{27} + 16 q^{29} - 9 q^{31} - 17 q^{32} - 8 q^{33} - 14 q^{34} + 7 q^{36} - 8 q^{37} + 27 q^{38} + 7 q^{39} - 4 q^{41} - 5 q^{43} + 26 q^{44} - 6 q^{46} + 6 q^{47} - 17 q^{48} + 6 q^{51} - 35 q^{52} + 6 q^{53} + q^{54} + 3 q^{57} - 4 q^{58} - 10 q^{59} + 20 q^{61} + 44 q^{62} + 21 q^{64} - 12 q^{66} - q^{67} + 8 q^{68} + 2 q^{69} + 22 q^{71} - 3 q^{72} + 4 q^{73} - 21 q^{74} + 23 q^{76} - 19 q^{78} + 8 q^{79} + 4 q^{81} - 8 q^{82} + 2 q^{83} + 12 q^{86} - 16 q^{87} - 28 q^{88} + 10 q^{89} + 66 q^{92} + 9 q^{93} - 22 q^{94} + 17 q^{96} - 12 q^{97} + 8 q^{99}+O(q^{100}) $ 4 * q - q^2 - 4 * q^3 + 7 * q^4 + q^6 - 3 * q^8 + 4 * q^9 + 8 * q^11 - 7 * q^12 - 7 * q^13 + 17 * q^16 - 6 * q^17 - q^18 - 3 * q^19 + 12 * q^22 - 2 * q^23 + 3 * q^24 + 19 * q^26 - 4 * q^27 + 16 * q^29 - 9 * q^31 - 17 * q^32 - 8 * q^33 - 14 * q^34 + 7 * q^36 - 8 * q^37 + 27 * q^38 + 7 * q^39 - 4 * q^41 - 5 * q^43 + 26 * q^44 - 6 * q^46 + 6 * q^47 - 17 * q^48 + 6 * q^51 - 35 * q^52 + 6 * q^53 + q^54 + 3 * q^57 - 4 * q^58 - 10 * q^59 + 20 * q^61 + 44 * q^62 + 21 * q^64 - 12 * q^66 - q^67 + 8 * q^68 + 2 * q^69 + 22 * q^71 - 3 * q^72 + 4 * q^73 - 21 * q^74 + 23 * q^76 - 19 * q^78 + 8 * q^79 + 4 * q^81 - 8 * q^82 + 2 * q^83 + 12 * q^86 - 16 * q^87 - 28 * q^88 + 10 * q^89 + 66 * q^92 + 9 * q^93 - 22 * q^94 + 17 * q^96 - 12 * q^97 + 8 * 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 $

$2$	−1.22868	−0.868807	−0.434404	−	0.900718i	$-0.643041\pi$
$2$	−1.22868	−0.868807	−0.434404	+	0.900718i	$0.643041\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−0.490347	−0.245174
$5$	0	0
$5$	0	0
$6$	1.22868	0.501606
$7$	0	0
$7$	0	0
$8$	3.05984	1.08182
$9$	1.00000	0.333333
$10$	0	0
$11$	−4.74588	−1.43094	−0.715468	−	0.698646i	$-0.753786\pi$
$11$	−4.74588	−1.43094	−0.715468	+	0.698646i	$0.753786\pi$
$12$	0.490347	0.141551
$13$	−6.25553	−1.73497	−0.867486	−	0.497462i	$-0.834265\pi$
$13$	−6.25553	−1.73497	−0.867486	+	0.497462i	$0.834265\pi$
$14$	0	0
$15$	0	0
$16$	−2.77886	−0.694716
$17$	−6.28852	−1.52519	−0.762595	−	0.646877i	$-0.776075\pi$
$17$	−6.28852	−1.52519	−0.762595	+	0.646877i	$0.776075\pi$
$18$	−1.22868	−0.289602
$19$	−4.94771	−1.13508	−0.567541	−	0.823345i	$-0.692105\pi$
$19$	−4.94771	−1.13508	−0.567541	+	0.823345i	$0.692105\pi$
$20$	0	0
$21$	0	0
$22$	5.83116	1.24321
$23$	−4.98069	−1.03855	−0.519273	−	0.854608i	$-0.673797\pi$
$23$	−4.98069	−1.03855	−0.519273	+	0.854608i	$0.673797\pi$
$24$	−3.05984	−0.624587
$25$	0	0
$26$	7.68604	1.50736
$27$	−1.00000	−0.192450
$28$	0	0
$29$	4.00000	0.742781	0.371391	−	0.928477i	$-0.378881\pi$
$29$	4.00000	0.742781	0.371391	+	0.928477i	$0.378881\pi$
$30$	0	0
$31$	−1.65919	−0.297999	−0.149000	−	0.988837i	$-0.547605\pi$
$31$	−1.65919	−0.297999	−0.149000	+	0.988837i	$0.547605\pi$
$32$	−2.70534	−0.478242
$33$	4.74588	0.826151
$34$	7.72657	1.32510
$35$	0	0
$36$	−0.490347	−0.0817246
$37$	4.43805	0.729611	0.364806	−	0.931084i	$-0.381135\pi$
$37$	4.43805	0.729611	0.364806	+	0.931084i	$0.381135\pi$
$38$	6.07914	0.986167
$39$	6.25553	1.00169
$40$	0	0
$41$	−1.30782	−0.204248	−0.102124	−	0.994772i	$-0.532564\pi$
$41$	−1.30782	−0.204248	−0.102124	+	0.994772i	$0.532564\pi$
$42$	0	0
$43$	−10.2362	−1.56101	−0.780505	−	0.625150i	$-0.785038\pi$
$43$	−10.2362	−1.56101	−0.780505	+	0.625150i	$0.785038\pi$
$44$	2.32713	0.350828
$45$	0	0
$46$	6.11968	0.902297
$47$	3.76518	0.549208	0.274604	−	0.961557i	$-0.411453\pi$
$47$	3.76518	0.549208	0.274604	+	0.961557i	$0.411453\pi$
$48$	2.77886	0.401095
$49$	0	0
$50$	0	0
$51$	6.28852	0.880569
$52$	3.06738	0.425369
$53$	−7.20323	−0.989440	−0.494720	−	0.869052i	$-0.664729\pi$
$53$	−7.20323	−0.989440	−0.494720	+	0.869052i	$0.664729\pi$
$54$	1.22868	0.167202
$55$	0	0
$56$	0	0
$57$	4.94771	0.655340
$58$	−4.91472	−0.645334
$59$	−0.234819	−0.0305708	−0.0152854	−	0.999883i	$-0.504866\pi$
$59$	−0.234819	−0.0305708	−0.0152854	+	0.999883i	$0.504866\pi$
$60$	0	0
$61$	5.00000	0.640184	0.320092	−	0.947386i	$-0.396286\pi$
$61$	5.00000	0.640184	0.320092	+	0.947386i	$0.396286\pi$
$62$	2.03861	0.258904
$63$	0	0
$64$	8.88173	1.11022
$65$	0	0
$66$	−5.83116	−0.717766
$67$	−2.49035	−0.304244	−0.152122	−	0.988362i	$-0.548611\pi$
$67$	−2.49035	−0.304244	−0.152122	+	0.988362i	$0.548611\pi$
$68$	3.08356	0.373936
$69$	4.98069	0.599605
$70$	0	0
$71$	6.06598	0.719899	0.359950	−	0.932972i	$-0.382794\pi$
$71$	6.06598	0.719899	0.359950	+	0.932972i	$0.382794\pi$
$72$	3.05984	0.360605
$73$	−12.4918	−1.46205	−0.731024	−	0.682351i	$-0.760957\pi$
$73$	−12.4918	−1.46205	−0.731024	+	0.682351i	$0.760957\pi$
$74$	−5.45294	−0.633892
$75$	0	0
$76$	2.42609	0.278292
$77$	0	0
$78$	−7.68604	−0.870272
$79$	−0.831159	−0.0935127	−0.0467563	−	0.998906i	$-0.514888\pi$
$79$	−0.831159	−0.0935127	−0.0467563	+	0.998906i	$0.514888\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	1.60689	0.177452
$83$	−1.14954	−0.126178	−0.0630890	−	0.998008i	$-0.520095\pi$
$83$	−1.14954	−0.126178	−0.0630890	+	0.998008i	$0.520095\pi$
$84$	0	0
$85$	0	0
$86$	12.5770	1.35622
$87$	−4.00000	−0.428845
$88$	−14.5216	−1.54801
$89$	−2.28852	−0.242582	−0.121291	−	0.992617i	$-0.538703\pi$
$89$	−2.28852	−0.242582	−0.121291	+	0.992617i	$0.538703\pi$
$90$	0	0
$91$	0	0
$92$	2.44227	0.254624
$93$	1.65919	0.172050
$94$	−4.62620	−0.477156
$95$	0	0
$96$	2.70534	0.276113
$97$	−0.476664	−0.0483979	−0.0241989	−	0.999707i	$-0.507704\pi$
$97$	−0.476664	−0.0483979	−0.0241989	+	0.999707i	$0.507704\pi$
$98$	0	0
$99$	−4.74588	−0.476978
$100$	0	0
$101$	−8.28852	−0.824738	−0.412369	−	0.911017i	$-0.635299\pi$
$101$	−8.28852	−0.824738	−0.412369	+	0.911017i	$0.635299\pi$
$102$	−7.72657	−0.765044
$103$	−14.2692	−1.40599	−0.702994	−	0.711196i	$-0.748154\pi$
$103$	−14.2692	−1.40599	−0.702994	+	0.711196i	$0.748154\pi$
$104$	−19.1409	−1.87692
$105$	0	0
$106$	8.85046	0.859633
$107$	2.16884	0.209670	0.104835	−	0.994490i	$-0.466569\pi$
$107$	2.16884	0.209670	0.104835	+	0.994490i	$0.466569\pi$
$108$	0.490347	0.0471837
$109$	7.79817	0.746929	0.373465	−	0.927644i	$-0.378170\pi$
$109$	7.79817	0.746929	0.373465	+	0.927644i	$0.378170\pi$
$110$	0	0
$111$	−4.43805	−0.421241
$112$	0	0
$113$	−15.4918	−1.45734	−0.728671	−	0.684864i	$-0.759861\pi$
$113$	−15.4918	−1.45734	−0.728671	+	0.684864i	$0.759861\pi$
$114$	−6.07914	−0.569364
$115$	0	0
$116$	−1.96139	−0.182110
$117$	−6.25553	−0.578324
$118$	0.288517	0.0265602
$119$	0	0
$120$	0	0
$121$	11.5233	1.04758
$122$	−6.14340	−0.556197
$123$	1.30782	0.117922
$124$	0.813579	0.0730615
$125$	0	0
$126$	0	0
$127$	−11.7996	−1.04704	−0.523521	−	0.852013i	$-0.675382\pi$
$127$	−11.7996	−1.04704	−0.523521	+	0.852013i	$0.675382\pi$
$128$	−5.50211	−0.486322
$129$	10.2362	0.901249
$130$	0	0
$131$	3.96139	0.346108	0.173054	−	0.984912i	$-0.444636\pi$
$131$	3.96139	0.346108	0.173054	+	0.984912i	$0.444636\pi$
$132$	−2.32713	−0.202550
$133$	0	0
$134$	3.05984	0.264330
$135$	0	0
$136$	−19.2418	−1.64997
$137$	18.4724	1.57821	0.789104	−	0.614260i	$-0.210545\pi$
$137$	18.4724	1.57821	0.789104	+	0.614260i	$0.210545\pi$
$138$	−6.11968	−0.520941
$139$	−2.66059	−0.225669	−0.112834	−	0.993614i	$-0.535993\pi$
$139$	−2.66059	−0.225669	−0.112834	+	0.993614i	$0.535993\pi$
$140$	0	0
$141$	−3.76518	−0.317085
$142$	−7.45314	−0.625454
$143$	29.6880	2.48263
$144$	−2.77886	−0.231572
$145$	0	0
$146$	15.3484	1.27024
$147$	0	0
$148$	−2.17619	−0.178882
$149$	14.5876	1.19506	0.597531	−	0.801846i	$-0.296149\pi$
$149$	14.5876	1.19506	0.597531	+	0.801846i	$0.296149\pi$
$150$	0	0
$151$	−14.2569	−1.16021	−0.580106	−	0.814541i	$-0.696989\pi$
$151$	−14.2569	−1.16021	−0.580106	+	0.814541i	$0.696989\pi$
$152$	−15.1392	−1.22795
$153$	−6.28852	−0.508396
$154$	0	0
$155$	0	0
$156$	−3.06738	−0.245587
$157$	−12.1004	−0.965715	−0.482857	−	0.875699i	$-0.660401\pi$
$157$	−12.1004	−0.965715	−0.482857	+	0.875699i	$0.660401\pi$
$158$	1.02123	0.0812445
$159$	7.20323	0.571254
$160$	0	0
$161$	0	0
$162$	−1.22868	−0.0965342
$163$	6.72657	0.526866	0.263433	−	0.964678i	$-0.415145\pi$
$163$	6.72657	0.526866	0.263433	+	0.964678i	$0.415145\pi$
$164$	0.641287	0.0500761
$165$	0	0
$166$	1.41241	0.109624
$167$	0.916442	0.0709164	0.0354582	−	0.999371i	$-0.488711\pi$
$167$	0.916442	0.0709164	0.0354582	+	0.999371i	$0.488711\pi$
$168$	0	0
$169$	26.1316	2.01013
$170$	0	0
$171$	−4.94771	−0.378361
$172$	5.01931	0.382718
$173$	7.49175	0.569587	0.284794	−	0.958589i	$-0.408075\pi$
$173$	7.49175	0.569587	0.284794	+	0.958589i	$0.408075\pi$
$174$	4.91472	0.372584
$175$	0	0
$176$	13.1881	0.994094
$177$	0.234819	0.0176501
$178$	2.81185	0.210757
$179$	14.8761	1.11189	0.555946	−	0.831218i	$-0.312356\pi$
$179$	14.8761	1.11189	0.555946	+	0.831218i	$0.312356\pi$
$180$	0	0
$181$	6.62760	0.492626	0.246313	−	0.969190i	$-0.420781\pi$
$181$	6.62760	0.492626	0.246313	+	0.969190i	$0.420781\pi$
$182$	0	0
$183$	−5.00000	−0.369611
$184$	−15.2401	−1.12352
$185$	0	0
$186$	−2.03861	−0.149478
$187$	29.8445	2.18245
$188$	−1.84625	−0.134651
$189$	0	0
$190$	0	0
$191$	−23.2569	−1.68281	−0.841406	−	0.540403i	$-0.818272\pi$
$191$	−23.2569	−1.68281	−0.841406	+	0.540403i	$0.818272\pi$
$192$	−8.88173	−0.640983
$193$	7.20183	0.518399	0.259200	−	0.965824i	$-0.416541\pi$
$193$	7.20183	0.518399	0.259200	+	0.965824i	$0.416541\pi$
$194$	0.585667	0.0420484
$195$	0	0
$196$	0	0
$197$	19.0881	1.35997	0.679985	−	0.733226i	$-0.261986\pi$
$197$	19.0881	1.35997	0.679985	+	0.733226i	$0.261986\pi$
$198$	5.83116	0.414402
$199$	−20.8340	−1.47688	−0.738440	−	0.674319i	$-0.764437\pi$
$199$	−20.8340	−1.47688	−0.738440	+	0.674319i	$0.764437\pi$
$200$	0	0
$201$	2.49035	0.175656
$202$	10.1839	0.716539
$203$	0	0
$204$	−3.08356	−0.215892
$205$	0	0
$206$	17.5323	1.22153
$207$	−4.98069	−0.346182
$208$	17.3833	1.20531
$209$	23.4812	1.62423
$210$	0	0
$211$	15.5876	1.07309	0.536547	−	0.843870i	$-0.319728\pi$
$211$	15.5876	1.07309	0.536547	+	0.843870i	$0.319728\pi$
$212$	3.53209	0.242585
$213$	−6.06598	−0.415634
$214$	−2.66481	−0.182163
$215$	0	0
$216$	−3.05984	−0.208196
$217$	0	0
$218$	−9.58145	−0.648938
$219$	12.4918	0.844114
$220$	0	0
$221$	39.3380	2.64616
$222$	5.45294	0.365978
$223$	−3.57250	−0.239232	−0.119616	−	0.992820i	$-0.538166\pi$
$223$	−3.57250	−0.239232	−0.119616	+	0.992820i	$0.538166\pi$
$224$	0	0
$225$	0	0
$226$	19.0344	1.26615
$227$	15.4258	1.02384	0.511922	−	0.859032i	$-0.328934\pi$
$227$	15.4258	1.02384	0.511922	+	0.859032i	$0.328934\pi$
$228$	−2.42609	−0.160672
$229$	−15.3285	−1.01294	−0.506469	−	0.862258i	$-0.669050\pi$
$229$	−15.3285	−1.01294	−0.506469	+	0.862258i	$0.669050\pi$
$230$	0	0
$231$	0	0
$232$	12.2394	0.803553
$233$	−20.2885	−1.32914	−0.664572	−	0.747224i	$-0.731386\pi$
$233$	−20.2885	−1.32914	−0.664572	+	0.747224i	$0.731386\pi$
$234$	7.68604	0.502452
$235$	0	0
$236$	0.115143	0.00749516
$237$	0.831159	0.0539896
$238$	0	0
$239$	−11.8954	−0.769450	−0.384725	−	0.923031i	$-0.625704\pi$
$239$	−11.8954	−0.769450	−0.384725	+	0.923031i	$0.625704\pi$
$240$	0	0
$241$	−17.4847	−1.12629	−0.563145	−	0.826358i	$-0.690409\pi$
$241$	−17.4847	−1.12629	−0.563145	+	0.826358i	$0.690409\pi$
$242$	−14.1585	−0.910142
$243$	−1.00000	−0.0641500
$244$	−2.45174	−0.156956
$245$	0	0
$246$	−1.60689	−0.102452
$247$	30.9505	1.96933
$248$	−5.07685	−0.322380
$249$	1.14954	0.0728489
$250$	0	0
$251$	10.2348	0.646016	0.323008	−	0.946396i	$-0.395306\pi$
$251$	10.2348	0.646016	0.323008	+	0.946396i	$0.395306\pi$
$252$	0	0
$253$	23.6378	1.48609
$254$	14.4979	0.909679
$255$	0	0
$256$	−11.0031	−0.687696
$257$	22.3545	1.39444	0.697218	−	0.716860i	$-0.254421\pi$
$257$	22.3545	1.39444	0.697218	+	0.716860i	$0.254421\pi$
$258$	−12.5770	−0.783012
$259$	0	0
$260$	0	0
$261$	4.00000	0.247594
$262$	−4.86728	−0.300701
$263$	24.8101	1.52986	0.764929	−	0.644115i	$-0.222774\pi$
$263$	24.8101	1.52986	0.764929	+	0.644115i	$0.222774\pi$
$264$	14.5216	0.893743
$265$	0	0
$266$	0	0
$267$	2.28852	0.140055
$268$	1.22114	0.0745927
$269$	−17.7803	−1.08408	−0.542041	−	0.840352i	$-0.682348\pi$
$269$	−17.7803	−1.08408	−0.542041	+	0.840352i	$0.682348\pi$
$270$	0	0
$271$	−17.1390	−1.04112	−0.520559	−	0.853825i	$-0.674277\pi$
$271$	−17.1390	−1.04112	−0.520559	+	0.853825i	$0.674277\pi$
$272$	17.4749	1.05957
$273$	0	0
$274$	−22.6967	−1.37116
$275$	0	0
$276$	−2.44227	−0.147007
$277$	−2.88345	−0.173250	−0.0866250	−	0.996241i	$-0.527608\pi$
$277$	−2.88345	−0.173250	−0.0866250	+	0.996241i	$0.527608\pi$
$278$	3.26901	0.196062
$279$	−1.65919	−0.0993330
$280$	0	0
$281$	27.7803	1.65723	0.828616	−	0.559817i	$-0.189129\pi$
$281$	27.7803	1.65723	0.828616	+	0.559817i	$0.189129\pi$
$282$	4.62620	0.275486
$283$	8.70233	0.517300	0.258650	−	0.965971i	$-0.416722\pi$
$283$	8.70233	0.517300	0.258650	+	0.965971i	$0.416722\pi$
$284$	−2.97444	−0.176500
$285$	0	0
$286$	−36.4770	−2.15693
$287$	0	0
$288$	−2.70534	−0.159414
$289$	22.5454	1.32620
$290$	0	0
$291$	0.476664	0.0279425
$292$	6.12530	0.358456
$293$	19.8954	1.16230	0.581151	−	0.813796i	$-0.302602\pi$
$293$	19.8954	1.16230	0.581151	+	0.813796i	$0.302602\pi$
$294$	0	0
$295$	0	0
$296$	13.5797	0.789305
$297$	4.74588	0.275384
$298$	−17.9235	−1.03828
$299$	31.1569	1.80185
$300$	0	0
$301$	0	0
$302$	17.5172	1.00800
$303$	8.28852	0.476163
$304$	13.7490	0.788560
$305$	0	0
$306$	7.72657	0.441699
$307$	−5.32151	−0.303714	−0.151857	−	0.988402i	$-0.548525\pi$
$307$	−5.32151	−0.303714	−0.151857	+	0.988402i	$0.548525\pi$
$308$	0	0
$309$	14.2692	0.811747
$310$	0	0
$311$	−1.89541	−0.107479	−0.0537395	−	0.998555i	$-0.517114\pi$
$311$	−1.89541	−0.107479	−0.0537395	+	0.998555i	$0.517114\pi$
$312$	19.1409	1.08364
$313$	−26.5230	−1.49917	−0.749585	−	0.661908i	$-0.769747\pi$
$313$	−26.5230	−1.49917	−0.749585	+	0.661908i	$0.769747\pi$
$314$	14.8675	0.839020
$315$	0	0
$316$	0.407557	0.0229268
$317$	−15.6237	−0.877515	−0.438757	−	0.898606i	$-0.644581\pi$
$317$	−15.6237	−0.877515	−0.438757	+	0.898606i	$0.644581\pi$
$318$	−8.85046	−0.496309
$319$	−18.9835	−1.06287
$320$	0	0
$321$	−2.16884	−0.121053
$322$	0	0
$323$	31.1137	1.73121
$324$	−0.490347	−0.0272415
$325$	0	0
$326$	−8.26480	−0.457745
$327$	−7.79817	−0.431240
$328$	−4.00173	−0.220958
$329$	0	0
$330$	0	0
$331$	−4.02986	−0.221501	−0.110751	−	0.993848i	$-0.535325\pi$
$331$	−4.02986	−0.221501	−0.110751	+	0.993848i	$0.535325\pi$
$332$	0.563672	0.0309355
$333$	4.43805	0.243204
$334$	−1.12601	−0.0616127
$335$	0	0
$336$	0	0
$337$	−2.86415	−0.156020	−0.0780100	−	0.996953i	$-0.524857\pi$
$337$	−2.86415	−0.156020	−0.0780100	+	0.996953i	$0.524857\pi$
$338$	−32.1074	−1.74641
$339$	15.4918	0.841396
$340$	0	0
$341$	7.87430	0.426417
$342$	6.07914	0.328722
$343$	0	0
$344$	−31.3212	−1.68873
$345$	0	0
$346$	−9.20496	−0.494862
$347$	−2.51278	−0.134893	−0.0674466	−	0.997723i	$-0.521485\pi$
$347$	−2.51278	−0.134893	−0.0674466	+	0.997723i	$0.521485\pi$
$348$	1.96139	0.105142
$349$	−12.6307	−0.676108	−0.338054	−	0.941127i	$-0.609769\pi$
$349$	−12.6307	−0.676108	−0.338054	+	0.941127i	$0.609769\pi$
$350$	0	0
$351$	6.25553	0.333895
$352$	12.8392	0.684333
$353$	22.5876	1.20222	0.601108	−	0.799168i	$-0.294726\pi$
$353$	22.5876	1.20222	0.601108	+	0.799168i	$0.294726\pi$
$354$	−0.288517	−0.0153345
$355$	0	0
$356$	1.12217	0.0594748
$357$	0	0
$358$	−18.2780	−0.966020
$359$	−0.342216	−0.0180614	−0.00903072	−	0.999959i	$-0.502875\pi$
$359$	−0.342216	−0.0180614	−0.00903072	+	0.999959i	$0.502875\pi$
$360$	0	0
$361$	5.47979	0.288410
$362$	−8.14320	−0.427997
$363$	−11.5233	−0.604818
$364$	0	0
$365$	0	0
$366$	6.14340	0.321120
$367$	3.27062	0.170725	0.0853624	−	0.996350i	$-0.472795\pi$
$367$	3.27062	0.170725	0.0853624	+	0.996350i	$0.472795\pi$
$368$	13.8407	0.721495
$369$	−1.30782	−0.0680825
$370$	0	0
$371$	0	0
$372$	−0.813579	−0.0421821
$373$	−13.3559	−0.691542	−0.345771	−	0.938319i	$-0.612383\pi$
$373$	−13.3559	−0.691542	−0.345771	+	0.938319i	$0.612383\pi$
$374$	−36.6693	−1.89613
$375$	0	0
$376$	11.5208	0.594142
$377$	−25.0221	−1.28870
$378$	0	0
$379$	28.8968	1.48433	0.742165	−	0.670217i	$-0.233799\pi$
$379$	28.8968	1.48433	0.742165	+	0.670217i	$0.233799\pi$
$380$	0	0
$381$	11.7996	0.604510
$382$	28.5753	1.46204
$383$	−20.7504	−1.06030	−0.530148	−	0.847905i	$-0.677864\pi$
$383$	−20.7504	−1.06030	−0.530148	+	0.847905i	$0.677864\pi$
$384$	5.50211	0.280778
$385$	0	0
$386$	−8.84874	−0.450389
$387$	−10.2362	−0.520336
$388$	0.233731	0.0118659
$389$	8.26115	0.418857	0.209428	−	0.977824i	$-0.432840\pi$
$389$	8.26115	0.418857	0.209428	+	0.977824i	$0.432840\pi$
$390$	0	0
$391$	31.3212	1.58398
$392$	0	0
$393$	−3.96139	−0.199826
$394$	−23.4531	−1.18155
$395$	0	0
$396$	2.32713	0.116943
$397$	−24.9329	−1.25135	−0.625674	−	0.780085i	$-0.715176\pi$
$397$	−24.9329	−1.25135	−0.625674	+	0.780085i	$0.715176\pi$
$398$	25.5983	1.28312
$399$	0	0
$400$	0	0
$401$	−7.26921	−0.363007	−0.181504	−	0.983390i	$-0.558096\pi$
$401$	−7.26921	−0.363007	−0.181504	+	0.983390i	$0.558096\pi$
$402$	−3.05984	−0.152611
$403$	10.3791	0.517020
$404$	4.06425	0.202204
$405$	0	0
$406$	0	0
$407$	−21.0624	−1.04403
$408$	19.2418	0.952613
$409$	35.4096	1.75089	0.875446	−	0.483316i	$-0.160568\pi$
$409$	35.4096	1.75089	0.875446	+	0.483316i	$0.160568\pi$
$410$	0	0
$411$	−18.4724	−0.911179
$412$	6.99687	0.344711
$413$	0	0
$414$	6.11968	0.300766
$415$	0	0
$416$	16.9233	0.829735
$417$	2.66059	0.130290
$418$	−28.8509	−1.41114
$419$	−38.1119	−1.86189	−0.930945	−	0.365160i	$-0.881014\pi$
$419$	−38.1119	−1.86189	−0.930945	+	0.365160i	$0.881014\pi$
$420$	0	0
$421$	12.5454	0.611428	0.305714	−	0.952123i	$-0.401105\pi$
$421$	12.5454	0.611428	0.305714	+	0.952123i	$0.401105\pi$
$422$	−19.1521	−0.932312
$423$	3.76518	0.183069
$424$	−22.0407	−1.07039
$425$	0	0
$426$	7.45314	0.361106
$427$	0	0
$428$	−1.06349	−0.0514055
$429$	−29.6880	−1.43335
$430$	0	0
$431$	−8.51106	−0.409963	−0.204982	−	0.978766i	$-0.565713\pi$
$431$	−8.51106	−0.409963	−0.204982	+	0.978766i	$0.565713\pi$
$432$	2.77886	0.133698
$433$	−20.8010	−0.999631	−0.499816	−	0.866132i	$-0.666599\pi$
$433$	−20.8010	−0.999631	−0.499816	+	0.866132i	$0.666599\pi$
$434$	0	0
$435$	0	0
$436$	−3.82381	−0.183127
$437$	24.6430	1.17884
$438$	−15.3484	−0.733373
$439$	−10.9508	−0.522655	−0.261327	−	0.965250i	$-0.584160\pi$
$439$	−10.9508	−0.522655	−0.261327	+	0.965250i	$0.584160\pi$
$440$	0	0
$441$	0	0
$442$	−48.3338	−2.29900
$443$	38.5525	1.83168	0.915842	−	0.401540i	$-0.131525\pi$
$443$	38.5525	1.83168	0.915842	+	0.401540i	$0.131525\pi$
$444$	2.17619	0.103277
$445$	0	0
$446$	4.38946	0.207847
$447$	−14.5876	−0.689969
$448$	0	0
$449$	−21.8462	−1.03099	−0.515494	−	0.856893i	$-0.672392\pi$
$449$	−21.8462	−1.03099	−0.515494	+	0.856893i	$0.672392\pi$
$450$	0	0
$451$	6.20676	0.292265
$452$	7.59634	0.357302
$453$	14.2569	0.669849
$454$	−18.9533	−0.889524
$455$	0	0
$456$	15.1392	0.708957
$457$	3.69045	0.172632	0.0863160	−	0.996268i	$-0.472491\pi$
$457$	3.69045	0.172632	0.0863160	+	0.996268i	$0.472491\pi$
$458$	18.8338	0.880048
$459$	6.28852	0.293523
$460$	0	0
$461$	30.9449	1.44125	0.720624	−	0.693326i	$-0.243856\pi$
$461$	30.9449	1.44125	0.720624	+	0.693326i	$0.243856\pi$
$462$	0	0
$463$	−4.97717	−0.231309	−0.115654	−	0.993290i	$-0.536896\pi$
$463$	−4.97717	−0.231309	−0.115654	+	0.993290i	$0.536896\pi$
$464$	−11.1155	−0.516022
$465$	0	0
$466$	24.9281	1.15477
$467$	−7.02211	−0.324945	−0.162472	−	0.986713i	$-0.551947\pi$
$467$	−7.02211	−0.324945	−0.162472	+	0.986713i	$0.551947\pi$
$468$	3.06738	0.141790
$469$	0	0
$470$	0	0
$471$	12.1004	0.557556
$472$	−0.718508	−0.0330720
$473$	48.5798	2.23370
$474$	−1.02123	−0.0469065
$475$	0	0
$476$	0	0
$477$	−7.20323	−0.329813
$478$	14.6156	0.668504
$479$	9.22956	0.421710	0.210855	−	0.977517i	$-0.432375\pi$
$479$	9.22956	0.421710	0.210855	+	0.977517i	$0.432375\pi$
$480$	0	0
$481$	−27.7624	−1.26586
$482$	21.4831	0.978529
$483$	0	0
$484$	−5.65044	−0.256838
$485$	0	0
$486$	1.22868	0.0557340
$487$	−13.7638	−0.623696	−0.311848	−	0.950132i	$-0.600948\pi$
$487$	−13.7638	−0.623696	−0.311848	+	0.950132i	$0.600948\pi$
$488$	15.2992	0.692562
$489$	−6.72657	−0.304186
$490$	0	0
$491$	−31.0495	−1.40124	−0.700622	−	0.713533i	$-0.747094\pi$
$491$	−31.0495	−1.40124	−0.700622	+	0.713533i	$0.747094\pi$
$492$	−0.641287	−0.0289115
$493$	−25.1541	−1.13288
$494$	−38.0283	−1.71097
$495$	0	0
$496$	4.61066	0.207025
$497$	0	0
$498$	−1.41241	−0.0632916
$499$	−21.5907	−0.966533	−0.483267	−	0.875473i	$-0.660550\pi$
$499$	−21.5907	−0.966533	−0.483267	+	0.875473i	$0.660550\pi$
$500$	0	0
$501$	−0.916442	−0.0409436
$502$	−12.5753	−0.561264
$503$	33.2826	1.48400	0.741998	−	0.670402i	$-0.233878\pi$
$503$	33.2826	1.48400	0.741998	+	0.670402i	$0.233878\pi$
$504$	0	0
$505$	0	0
$506$	−29.0432	−1.29113
$507$	−26.1316	−1.16055
$508$	5.78589	0.256707
$509$	−19.2200	−0.851914	−0.425957	−	0.904744i	$-0.640062\pi$
$509$	−19.2200	−0.851914	−0.425957	+	0.904744i	$0.640062\pi$
$510$	0	0
$511$	0	0
$512$	24.5235	1.08380
$513$	4.94771	0.218447
$514$	−27.4665	−1.21150
$515$	0	0
$516$	−5.01931	−0.220963
$517$	−17.8691	−0.785881
$518$	0	0
$519$	−7.49175	−0.328851
$520$	0	0
$521$	−0.483689	−0.0211908	−0.0105954	−	0.999944i	$-0.503373\pi$
$521$	−0.483689	−0.0211908	−0.0105954	+	0.999944i	$0.503373\pi$
$522$	−4.91472	−0.215111
$523$	−6.31344	−0.276068	−0.138034	−	0.990428i	$-0.544078\pi$
$523$	−6.31344	−0.276068	−0.138034	+	0.990428i	$0.544078\pi$
$524$	−1.94246	−0.0848566
$525$	0	0
$526$	−30.4837	−1.32915
$527$	10.4338	0.454505
$528$	−13.1881	−0.573940
$529$	1.80732	0.0785791
$530$	0	0
$531$	−0.234819	−0.0101903
$532$	0	0
$533$	8.18112	0.354364
$534$	−2.81185	−0.121681
$535$	0	0
$536$	−7.62006	−0.329136
$537$	−14.8761	−0.641951
$538$	21.8462	0.941859
$539$	0	0
$540$	0	0
$541$	16.0435	0.689766	0.344883	−	0.938646i	$-0.387919\pi$
$541$	16.0435	0.689766	0.344883	+	0.938646i	$0.387919\pi$
$542$	21.0583	0.904532
$543$	−6.62760	−0.284418
$544$	17.0126	0.729409
$545$	0	0
$546$	0	0
$547$	27.2078	1.16332	0.581660	−	0.813432i	$-0.302403\pi$
$547$	27.2078	1.16332	0.581660	+	0.813432i	$0.302403\pi$
$548$	−9.05792	−0.386935
$549$	5.00000	0.213395
$550$	0	0
$551$	−19.7908	−0.843117
$552$	15.2401	0.648663
$553$	0	0
$554$	3.54284	0.150521
$555$	0	0
$556$	1.30461	0.0553280
$557$	−12.4724	−0.528475	−0.264237	−	0.964458i	$-0.585120\pi$
$557$	−12.4724	−0.528475	−0.264237	+	0.964458i	$0.585120\pi$
$558$	2.03861	0.0863013
$559$	64.0330	2.70831
$560$	0	0
$561$	−29.8445	−1.26004
$562$	−34.1330	−1.43982
$563$	15.8312	0.667204	0.333602	−	0.942714i	$-0.391736\pi$
$563$	15.8312	0.667204	0.333602	+	0.942714i	$0.391736\pi$
$564$	1.84625	0.0777410
$565$	0	0
$566$	−10.6924	−0.449434
$567$	0	0
$568$	18.5609	0.778798
$569$	22.5665	0.946036	0.473018	−	0.881053i	$-0.343165\pi$
$569$	22.5665	0.946036	0.473018	+	0.881053i	$0.343165\pi$
$570$	0	0
$571$	11.3180	0.473643	0.236821	−	0.971553i	$-0.423894\pi$
$571$	11.3180	0.473643	0.236821	+	0.971553i	$0.423894\pi$
$572$	−14.5574	−0.608676
$573$	23.2569	0.971572
$574$	0	0
$575$	0	0
$576$	8.88173	0.370072
$577$	24.8133	1.03299	0.516495	−	0.856290i	$-0.327237\pi$
$577$	24.8133	1.03299	0.516495	+	0.856290i	$0.327237\pi$
$578$	−27.7011	−1.15221
$579$	−7.20183	−0.299298
$580$	0	0
$581$	0	0
$582$	−0.585667	−0.0242767
$583$	34.1857	1.41583
$584$	−38.2227	−1.58167
$585$	0	0
$586$	−24.4451	−1.00982
$587$	3.96139	0.163504	0.0817520	−	0.996653i	$-0.473948\pi$
$587$	3.96139	0.163504	0.0817520	+	0.996653i	$0.473948\pi$
$588$	0	0
$589$	8.20918	0.338253
$590$	0	0
$591$	−19.0881	−0.785179
$592$	−12.3327	−0.506873
$593$	−6.10459	−0.250685	−0.125343	−	0.992114i	$-0.540003\pi$
$593$	−6.10459	−0.250685	−0.125343	+	0.992114i	$0.540003\pi$
$594$	−5.83116	−0.239255
$595$	0	0
$596$	−7.15299	−0.292998
$597$	20.8340	0.852678
$598$	−38.2818	−1.56546
$599$	24.7845	1.01267	0.506333	−	0.862338i	$-0.331001\pi$
$599$	24.7845	1.01267	0.506333	+	0.862338i	$0.331001\pi$
$600$	0	0
$601$	−43.9866	−1.79425	−0.897126	−	0.441775i	$-0.854349\pi$
$601$	−43.9866	−1.79425	−0.897126	+	0.441775i	$0.854349\pi$
$602$	0	0
$603$	−2.49035	−0.101415
$604$	6.99085	0.284454
$605$	0	0
$606$	−10.1839	−0.413694
$607$	−17.3008	−0.702218	−0.351109	−	0.936335i	$-0.614195\pi$
$607$	−17.3008	−0.702218	−0.351109	+	0.936335i	$0.614195\pi$
$608$	13.3852	0.542843
$609$	0	0
$610$	0	0
$611$	−23.5532	−0.952860
$612$	3.08356	0.124645
$613$	15.3994	0.621978	0.310989	−	0.950414i	$-0.399340\pi$
$613$	15.3994	0.621978	0.310989	+	0.950414i	$0.399340\pi$
$614$	6.53842	0.263869
$615$	0	0
$616$	0	0
$617$	−8.70273	−0.350359	−0.175179	−	0.984537i	$-0.556051\pi$
$617$	−8.70273	−0.350359	−0.175179	+	0.984537i	$0.556051\pi$
$618$	−17.5323	−0.705252
$619$	7.15094	0.287421	0.143710	−	0.989620i	$-0.454097\pi$
$619$	7.15094	0.287421	0.143710	+	0.989620i	$0.454097\pi$
$620$	0	0
$621$	4.98069	0.199868
$622$	2.32885	0.0933785
$623$	0	0
$624$	−17.3833	−0.695888
$625$	0	0
$626$	32.5883	1.30249
$627$	−23.4812	−0.937749
$628$	5.93339	0.236768
$629$	−27.9088	−1.11280
$630$	0	0
$631$	−40.1797	−1.59953	−0.799765	−	0.600314i	$-0.795042\pi$
$631$	−40.1797	−1.59953	−0.799765	+	0.600314i	$0.795042\pi$
$632$	−2.54321	−0.101164
$633$	−15.5876	−0.619551
$634$	19.1965	0.762391
$635$	0	0
$636$	−3.53209	−0.140056
$637$	0	0
$638$	23.3246	0.923431
$639$	6.06598	0.239966
$640$	0	0
$641$	21.1478	0.835288	0.417644	−	0.908611i	$-0.362856\pi$
$641$	21.1478	0.835288	0.417644	+	0.908611i	$0.362856\pi$
$642$	2.66481	0.105172
$643$	−29.4981	−1.16329	−0.581646	−	0.813442i	$-0.697591\pi$
$643$	−29.4981	−1.16329	−0.581646	+	0.813442i	$0.697591\pi$
$644$	0	0
$645$	0	0
$646$	−38.2288	−1.50409
$647$	−28.3836	−1.11588	−0.557938	−	0.829883i	$-0.688407\pi$
$647$	−28.3836	−1.11588	−0.557938	+	0.829883i	$0.688407\pi$
$648$	3.05984	0.120202
$649$	1.11442	0.0437449
$650$	0	0
$651$	0	0
$652$	−3.29836	−0.129174
$653$	−23.2692	−0.910595	−0.455297	−	0.890339i	$-0.650467\pi$
$653$	−23.2692	−0.910595	−0.455297	+	0.890339i	$0.650467\pi$
$654$	9.58145	0.374664
$655$	0	0
$656$	3.63426	0.141894
$657$	−12.4918	−0.487350
$658$	0	0
$659$	20.9852	0.817468	0.408734	−	0.912653i	$-0.365970\pi$
$659$	20.9852	0.817468	0.408734	+	0.912653i	$0.365970\pi$
$660$	0	0
$661$	34.7160	1.35030	0.675148	−	0.737682i	$-0.264080\pi$
$661$	34.7160	1.35030	0.675148	+	0.737682i	$0.264080\pi$
$662$	4.95140	0.192442
$663$	−39.3380	−1.52776
$664$	−3.51739	−0.136501
$665$	0	0
$666$	−5.45294	−0.211297
$667$	−19.9228	−0.771413
$668$	−0.449375	−0.0173868
$669$	3.57250	0.138121
$670$	0	0
$671$	−23.7294	−0.916063
$672$	0	0
$673$	21.4998	0.828757	0.414378	−	0.910105i	$-0.363999\pi$
$673$	21.4998	0.828757	0.414378	+	0.910105i	$0.363999\pi$
$674$	3.51912	0.135551
$675$	0	0
$676$	−12.8136	−0.492830
$677$	49.7936	1.91372	0.956862	−	0.290542i	$-0.0938355\pi$
$677$	49.7936	1.91372	0.956862	+	0.290542i	$0.0938355\pi$
$678$	−19.0344	−0.731011
$679$	0	0
$680$	0	0
$681$	−15.4258	−0.591117
$682$	−9.67499	−0.370475
$683$	−31.2597	−1.19612	−0.598060	−	0.801451i	$-0.704062\pi$
$683$	−31.2597	−1.19612	−0.598060	+	0.801451i	$0.704062\pi$
$684$	2.42609	0.0927640
$685$	0	0
$686$	0	0
$687$	15.3285	0.584820
$688$	28.4451	1.08446
$689$	45.0600	1.71665
$690$	0	0
$691$	25.1594	0.957108	0.478554	−	0.878058i	$-0.341161\pi$
$691$	25.1594	0.957108	0.478554	+	0.878058i	$0.341161\pi$
$692$	−3.67356	−0.139648
$693$	0	0
$694$	3.08740	0.117196
$695$	0	0
$696$	−12.2394	−0.463931
$697$	8.22426	0.311516
$698$	15.5191	0.587407
$699$	20.2885	0.767382
$700$	0	0
$701$	−6.21130	−0.234597	−0.117299	−	0.993097i	$-0.537423\pi$
$701$	−6.21130	−0.234597	−0.117299	+	0.993097i	$0.537423\pi$
$702$	−7.68604	−0.290091
$703$	−21.9582	−0.828169
$704$	−42.1516	−1.58865
$705$	0	0
$706$	−27.7529	−1.04449
$707$	0	0
$708$	−0.115143	−0.00432734
$709$	−14.8831	−0.558948	−0.279474	−	0.960153i	$-0.590160\pi$
$709$	−14.8831	−0.558948	−0.279474	+	0.960153i	$0.590160\pi$
$710$	0	0
$711$	−0.831159	−0.0311709
$712$	−7.00249	−0.262429
$713$	8.26391	0.309486
$714$	0	0
$715$	0	0
$716$	−7.29446	−0.272607
$717$	11.8954	0.444242
$718$	0.420473	0.0156919
$719$	−36.6167	−1.36557	−0.682787	−	0.730618i	$-0.739232\pi$
$719$	−36.6167	−1.36557	−0.682787	+	0.730618i	$0.739232\pi$
$720$	0	0
$721$	0	0
$722$	−6.73291	−0.250573
$723$	17.4847	0.650264
$724$	−3.24983	−0.120779
$725$	0	0
$726$	14.1585	0.525471
$727$	−13.8624	−0.514129	−0.257064	−	0.966394i	$-0.582755\pi$
$727$	−13.8624	−0.514129	−0.257064	+	0.966394i	$0.582755\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	64.3707	2.38084
$732$	2.45174	0.0906188
$733$	32.7697	1.21038	0.605189	−	0.796082i	$-0.293098\pi$
$733$	32.7697	1.21038	0.605189	+	0.796082i	$0.293098\pi$
$734$	−4.01854	−0.148327
$735$	0	0
$736$	13.4745	0.496676
$737$	11.8189	0.435354
$738$	1.60689	0.0591506
$739$	29.8684	1.09873	0.549363	−	0.835584i	$-0.314871\pi$
$739$	29.8684	1.09873	0.549363	+	0.835584i	$0.314871\pi$
$740$	0	0
$741$	−30.9505	−1.13700
$742$	0	0
$743$	49.5577	1.81810	0.909048	−	0.416691i	$-0.136810\pi$
$743$	49.5577	1.81810	0.909048	+	0.416691i	$0.136810\pi$
$744$	5.07685	0.186126
$745$	0	0
$746$	16.4101	0.600817
$747$	−1.14954	−0.0420593
$748$	−14.6342	−0.535079
$749$	0	0
$750$	0	0
$751$	−6.61034	−0.241215	−0.120607	−	0.992700i	$-0.538484\pi$
$751$	−6.61034	−0.241215	−0.120607	+	0.992700i	$0.538484\pi$
$752$	−10.4629	−0.381544
$753$	−10.2348	−0.372978
$754$	30.7442	1.11964
$755$	0	0
$756$	0	0
$757$	7.01509	0.254968	0.127484	−	0.991841i	$-0.459310\pi$
$757$	7.01509	0.254968	0.127484	+	0.991841i	$0.459310\pi$
$758$	−35.5049	−1.28960
$759$	−23.6378	−0.857996
$760$	0	0
$761$	−13.0088	−0.471567	−0.235783	−	0.971806i	$-0.575766\pi$
$761$	−13.0088	−0.471567	−0.235783	+	0.971806i	$0.575766\pi$
$762$	−14.4979	−0.525203
$763$	0	0
$764$	11.4040	0.412581
$765$	0	0
$766$	25.4956	0.921193
$767$	1.46892	0.0530395
$768$	11.0031	0.397041
$769$	−0.822729	−0.0296684	−0.0148342	−	0.999890i	$-0.504722\pi$
$769$	−0.822729	−0.0296684	−0.0148342	+	0.999890i	$0.504722\pi$
$770$	0	0
$771$	−22.3545	−0.805077
$772$	−3.53140	−0.127098
$773$	24.6844	0.887837	0.443919	−	0.896067i	$-0.353588\pi$
$773$	24.6844	0.887837	0.443919	+	0.896067i	$0.353588\pi$
$774$	12.5770	0.452072
$775$	0	0
$776$	−1.45851	−0.0523576
$777$	0	0
$778$	−10.1503	−0.363906
$779$	6.47072	0.231838
$780$	0	0
$781$	−28.7884	−1.03013
$782$	−38.4837	−1.37617
$783$	−4.00000	−0.142948
$784$	0	0
$785$	0	0
$786$	4.86728	0.173610
$787$	−15.7582	−0.561718	−0.280859	−	0.959749i	$-0.590619\pi$
$787$	−15.7582	−0.561718	−0.280859	+	0.959749i	$0.590619\pi$
$788$	−9.35980	−0.333429
$789$	−24.8101	−0.883264
$790$	0	0
$791$	0	0
$792$	−14.5216	−0.516003
$793$	−31.2776	−1.11070
$794$	30.6346	1.08718
$795$	0	0
$796$	10.2159	0.362092
$797$	15.8568	0.561677	0.280838	−	0.959755i	$-0.409388\pi$
$797$	15.8568	0.561677	0.280838	+	0.959755i	$0.409388\pi$
$798$	0	0
$799$	−23.6774	−0.837646
$800$	0	0
$801$	−2.28852	−0.0808608
$802$	8.93153	0.315383
$803$	59.2843	2.09210
$804$	−1.22114	−0.0430661
$805$	0	0
$806$	−12.7526	−0.449191
$807$	17.7803	0.625895
$808$	−25.3615	−0.892215
$809$	−7.85680	−0.276230	−0.138115	−	0.990416i	$-0.544104\pi$
$809$	−7.85680	−0.276230	−0.138115	+	0.990416i	$0.544104\pi$
$810$	0	0
$811$	55.8410	1.96084	0.980421	−	0.196912i	$-0.0630912\pi$
$811$	55.8410	1.96084	0.980421	+	0.196912i	$0.0630912\pi$
$812$	0	0
$813$	17.1390	0.601090
$814$	25.8790	0.907058
$815$	0	0
$816$	−17.4749	−0.611745
$817$	50.6458	1.77187
$818$	−43.5070	−1.52119
$819$	0	0
$820$	0	0
$821$	8.02806	0.280181	0.140091	−	0.990139i	$-0.455261\pi$
$821$	8.02806	0.280181	0.140091	+	0.990139i	$0.455261\pi$
$822$	22.6967	0.791639
$823$	−22.2720	−0.776354	−0.388177	−	0.921585i	$-0.626895\pi$
$823$	−22.2720	−0.776354	−0.388177	+	0.921585i	$0.626895\pi$
$824$	−43.6615	−1.52102
$825$	0	0
$826$	0	0
$827$	−53.3872	−1.85645	−0.928227	−	0.372015i	$-0.878667\pi$
$827$	−53.3872	−1.85645	−0.928227	+	0.372015i	$0.878667\pi$
$828$	2.44227	0.0848748
$829$	28.8445	1.00181	0.500906	−	0.865502i	$-0.333000\pi$
$829$	28.8445	1.00181	0.500906	+	0.865502i	$0.333000\pi$
$830$	0	0
$831$	2.88345	0.100026
$832$	−55.5599	−1.92619
$833$	0	0
$834$	−3.26901	−0.113197
$835$	0	0
$836$	−11.5139	−0.398218
$837$	1.65919	0.0573499
$838$	46.8273	1.61762
$839$	−3.38435	−0.116841	−0.0584205	−	0.998292i	$-0.518606\pi$
$839$	−3.38435	−0.116841	−0.0584205	+	0.998292i	$0.518606\pi$
$840$	0	0
$841$	−13.0000	−0.448276
$842$	−15.4143	−0.531213
$843$	−27.7803	−0.956803
$844$	−7.64333	−0.263094
$845$	0	0
$846$	−4.62620	−0.159052
$847$	0	0
$848$	20.0168	0.687380
$849$	−8.70233	−0.298663
$850$	0	0
$851$	−22.1046	−0.757736
$852$	2.97444	0.101903
$853$	32.8905	1.12615	0.563074	−	0.826406i	$-0.309618\pi$
$853$	32.8905	1.12615	0.563074	+	0.826406i	$0.309618\pi$
$854$	0	0
$855$	0	0
$856$	6.63630	0.226824
$857$	27.0193	0.922962	0.461481	−	0.887150i	$-0.347318\pi$
$857$	27.0193	0.922962	0.461481	+	0.887150i	$0.347318\pi$
$858$	36.4770	1.24530
$859$	−48.1074	−1.64140	−0.820702	−	0.571357i	$-0.806417\pi$
$859$	−48.1074	−1.64140	−0.820702	+	0.571357i	$0.806417\pi$
$860$	0	0
$861$	0	0
$862$	10.4574	0.356179
$863$	−10.1716	−0.346247	−0.173123	−	0.984900i	$-0.555386\pi$
$863$	−10.1716	−0.346247	−0.173123	+	0.984900i	$0.555386\pi$
$864$	2.70534	0.0920376
$865$	0	0
$866$	25.5577	0.868487
$867$	−22.5454	−0.765684
$868$	0	0
$869$	3.94458	0.133811
$870$	0	0
$871$	15.5784	0.527855
$872$	23.8611	0.808040
$873$	−0.476664	−0.0161326
$874$	−30.2784	−1.02418
$875$	0	0
$876$	−6.12530	−0.206955
$877$	−36.8596	−1.24466	−0.622330	−	0.782755i	$-0.713814\pi$
$877$	−36.8596	−1.24466	−0.622330	+	0.782755i	$0.713814\pi$
$878$	13.4551	0.454086
$879$	−19.8954	−0.671056
$880$	0	0
$881$	−42.1945	−1.42157	−0.710784	−	0.703410i	$-0.751660\pi$
$881$	−42.1945	−1.42157	−0.710784	+	0.703410i	$0.751660\pi$
$882$	0	0
$883$	−35.7880	−1.20436	−0.602181	−	0.798359i	$-0.705702\pi$
$883$	−35.7880	−1.20436	−0.602181	+	0.798359i	$0.705702\pi$
$884$	−19.2893	−0.648769
$885$	0	0
$886$	−47.3686	−1.59138
$887$	−42.3980	−1.42359	−0.711793	−	0.702389i	$-0.752117\pi$
$887$	−42.3980	−1.42359	−0.711793	+	0.702389i	$0.752117\pi$
$888$	−13.5797	−0.455706
$889$	0	0
$890$	0	0
$891$	−4.74588	−0.158993
$892$	1.75177	0.0586535
$893$	−18.6290	−0.623396
$894$	17.9235	0.599450
$895$	0	0
$896$	0	0
$897$	−31.1569	−1.04030
$898$	26.8420	0.895730
$899$	−6.63675	−0.221348
$900$	0	0
$901$	45.2977	1.50908
$902$	−7.62612	−0.253922
$903$	0	0
$904$	−47.4023	−1.57658
$905$	0	0
$906$	−17.5172	−0.581970
$907$	−52.5437	−1.74469	−0.872343	−	0.488895i	$-0.837400\pi$
$907$	−52.5437	−1.74469	−0.872343	+	0.488895i	$0.837400\pi$
$908$	−7.56399	−0.251020
$909$	−8.28852	−0.274913
$910$	0	0
$911$	38.7118	1.28258	0.641290	−	0.767299i	$-0.278400\pi$
$911$	38.7118	1.28258	0.641290	+	0.767299i	$0.278400\pi$
$912$	−13.7490	−0.455275
$913$	5.45555	0.180553
$914$	−4.53438	−0.149984
$915$	0	0
$916$	7.51631	0.248346
$917$	0	0
$918$	−7.72657	−0.255015
$919$	−25.8740	−0.853504	−0.426752	−	0.904369i	$-0.640342\pi$
$919$	−25.8740	−0.853504	−0.426752	+	0.904369i	$0.640342\pi$
$920$	0	0
$921$	5.32151	0.175350
$922$	−38.0213	−1.25217
$923$	−37.9459	−1.24900
$924$	0	0
$925$	0	0
$926$	6.11534	0.200963
$927$	−14.2692	−0.468662
$928$	−10.8214	−0.355229
$929$	1.77746	0.0583166	0.0291583	−	0.999575i	$-0.490717\pi$
$929$	1.77746	0.0583166	0.0291583	+	0.999575i	$0.490717\pi$
$930$	0	0
$931$	0	0
$932$	9.94842	0.325871
$933$	1.89541	0.0620530
$934$	8.62792	0.282314
$935$	0	0
$936$	−19.1409	−0.625640
$937$	58.4486	1.90943	0.954716	−	0.297518i	$-0.0961587\pi$
$937$	58.4486	1.90943	0.954716	+	0.297518i	$0.0961587\pi$
$938$	0	0
$939$	26.5230	0.865546
$940$	0	0
$941$	−3.80832	−0.124148	−0.0620739	−	0.998072i	$-0.519771\pi$
$941$	−3.80832	−0.124148	−0.0620739	+	0.998072i	$0.519771\pi$
$942$	−14.8675	−0.484408
$943$	6.51386	0.212121
$944$	0.652530	0.0212381
$945$	0	0
$946$	−59.6890	−1.94066
$947$	−54.8431	−1.78216	−0.891081	−	0.453845i	$-0.850052\pi$
$947$	−54.8431	−1.78216	−0.891081	+	0.453845i	$0.850052\pi$
$948$	−0.407557	−0.0132368
$949$	78.1425	2.53661
$950$	0	0
$951$	15.6237	0.506633
$952$	0	0
$953$	−37.4426	−1.21288	−0.606442	−	0.795128i	$-0.707404\pi$
$953$	−37.4426	−1.21288	−0.606442	+	0.795128i	$0.707404\pi$
$954$	8.85046	0.286544
$955$	0	0
$956$	5.83288	0.188649
$957$	18.9835	0.613649
$958$	−11.3402	−0.366384
$959$	0	0
$960$	0	0
$961$	−28.2471	−0.911197
$962$	34.1110	1.09978
$963$	2.16884	0.0698899
$964$	8.57359	0.276137
$965$	0	0
$966$	0	0
$967$	−32.0098	−1.02937	−0.514683	−	0.857381i	$-0.672090\pi$
$967$	−32.0098	−1.02937	−0.514683	+	0.857381i	$0.672090\pi$
$968$	35.2595	1.13328
$969$	−31.1137	−0.999517
$970$	0	0
$971$	−51.4963	−1.65259	−0.826297	−	0.563234i	$-0.809557\pi$
$971$	−51.4963	−1.65259	−0.826297	+	0.563234i	$0.809557\pi$
$972$	0.490347	0.0157279
$973$	0	0
$974$	16.9113	0.541872
$975$	0	0
$976$	−13.8943	−0.444746
$977$	3.13445	0.100280	0.0501399	−	0.998742i	$-0.484033\pi$
$977$	3.13445	0.100280	0.0501399	+	0.998742i	$0.484033\pi$
$978$	8.26480	0.264279
$979$	10.8610	0.347120
$980$	0	0
$981$	7.79817	0.248976
$982$	38.1499	1.21741
$983$	−43.1165	−1.37520	−0.687602	−	0.726088i	$-0.741337\pi$
$983$	−43.1165	−1.37520	−0.687602	+	0.726088i	$0.741337\pi$
$984$	4.00173	0.127570
$985$	0	0
$986$	30.9063	0.984257
$987$	0	0
$988$	−15.1765	−0.482829
$989$	50.9835	1.62118
$990$	0	0
$991$	−12.1552	−0.386121	−0.193061	−	0.981187i	$-0.561841\pi$
$991$	−12.1552	−0.386121	−0.193061	+	0.981187i	$0.561841\pi$
$992$	4.48867	0.142516
$993$	4.02986	0.127884
$994$	0	0
$995$	0	0
$996$	−0.563672	−0.0178606
$997$	22.7757	0.721315	0.360657	−	0.932698i	$-0.382552\pi$
$997$	22.7757	0.721315	0.360657	+	0.932698i	$0.382552\pi$
$998$	26.5281	0.839731
$999$	−4.43805	−0.140414

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3675.2.a.bq.1.2		4
5.4	even	2		3675.2.a.bx.1.3		4
7.3	odd	6		525.2.i.j.226.3	yes	8
7.5	odd	6		525.2.i.j.151.3	yes	8
7.6	odd	2		3675.2.a.br.1.2		4
35.3	even	12		525.2.r.h.499.3		16
35.12	even	12		525.2.r.h.424.3		16
35.17	even	12		525.2.r.h.499.6		16
35.19	odd	6		525.2.i.i.151.2	✓	8
35.24	odd	6		525.2.i.i.226.2	yes	8
35.33	even	12		525.2.r.h.424.6		16
35.34	odd	2		3675.2.a.bw.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
525.2.i.i.151.2	✓	8	35.19	odd	6
525.2.i.i.226.2	yes	8	35.24	odd	6
525.2.i.j.151.3	yes	8	7.5	odd	6
525.2.i.j.226.3	yes	8	7.3	odd	6
525.2.r.h.424.3		16	35.12	even	12
525.2.r.h.424.6		16	35.33	even	12
525.2.r.h.499.3		16	35.3	even	12
525.2.r.h.499.6		16	35.17	even	12
3675.2.a.bq.1.2		4	1.1	even	1	trivial
3675.2.a.br.1.2		4	7.6	odd	2
3675.2.a.bw.1.3		4	35.34	odd	2
3675.2.a.bx.1.3		4	5.4	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3675.a (trivial)

\(f(q)\)	\(=\)	\(q-1.22868 q^{2} -1.00000 q^{3} -0.490347 q^{4} +1.22868 q^{6} +3.05984 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-1.22868 q^{2} -1.00000 q^{3} -0.490347 q^{4} +1.22868 q^{6} +3.05984 q^{8} +1.00000 q^{9} -4.74588 q^{11} +0.490347 q^{12} -6.25553 q^{13} -2.77886 q^{16} -6.28852 q^{17} -1.22868 q^{18} -4.94771 q^{19} +5.83116 q^{22} -4.98069 q^{23} -3.05984 q^{24} +7.68604 q^{26} -1.00000 q^{27} +4.00000 q^{29} -1.65919 q^{31} -2.70534 q^{32} +4.74588 q^{33} +7.72657 q^{34} -0.490347 q^{36} +4.43805 q^{37} +6.07914 q^{38} +6.25553 q^{39} -1.30782 q^{41} -10.2362 q^{43} +2.32713 q^{44} +6.11968 q^{46} +3.76518 q^{47} +2.77886 q^{48} +6.28852 q^{51} +3.06738 q^{52} -7.20323 q^{53} +1.22868 q^{54} +4.94771 q^{57} -4.91472 q^{58} -0.234819 q^{59} +5.00000 q^{61} +2.03861 q^{62} +8.88173 q^{64} -5.83116 q^{66} -2.49035 q^{67} +3.08356 q^{68} +4.98069 q^{69} +6.06598 q^{71} +3.05984 q^{72} -12.4918 q^{73} -5.45294 q^{74} +2.42609 q^{76} -7.68604 q^{78} -0.831159 q^{79} +1.00000 q^{81} +1.60689 q^{82} -1.14954 q^{83} +12.5770 q^{86} -4.00000 q^{87} -14.5216 q^{88} -2.28852 q^{89} +2.44227 q^{92} +1.65919 q^{93} -4.62620 q^{94} +2.70534 q^{96} -0.476664 q^{97} -4.74588 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} - 4 q^{3} + 7 q^{4} + q^{6} - 3 q^{8} + 4 q^{9}+O(q^{10}) \) 4 * q - q^2 - 4 * q^3 + 7 * q^4 + q^6 - 3 * q^8 + 4 * q^9	\( 4 q - q^{2} - 4 q^{3} + 7 q^{4} + q^{6} - 3 q^{8} + 4 q^{9} + 8 q^{11} - 7 q^{12} - 7 q^{13} + 17 q^{16} - 6 q^{17} - q^{18} - 3 q^{19} + 12 q^{22} - 2 q^{23} + 3 q^{24} + 19 q^{26} - 4 q^{27} + 16 q^{29} - 9 q^{31} - 17 q^{32} - 8 q^{33} - 14 q^{34} + 7 q^{36} - 8 q^{37} + 27 q^{38} + 7 q^{39} - 4 q^{41} - 5 q^{43} + 26 q^{44} - 6 q^{46} + 6 q^{47} - 17 q^{48} + 6 q^{51} - 35 q^{52} + 6 q^{53} + q^{54} + 3 q^{57} - 4 q^{58} - 10 q^{59} + 20 q^{61} + 44 q^{62} + 21 q^{64} - 12 q^{66} - q^{67} + 8 q^{68} + 2 q^{69} + 22 q^{71} - 3 q^{72} + 4 q^{73} - 21 q^{74} + 23 q^{76} - 19 q^{78} + 8 q^{79} + 4 q^{81} - 8 q^{82} + 2 q^{83} + 12 q^{86} - 16 q^{87} - 28 q^{88} + 10 q^{89} + 66 q^{92} + 9 q^{93} - 22 q^{94} + 17 q^{96} - 12 q^{97} + 8 q^{99}+O(q^{100}) \) 4 * q - q^2 - 4 * q^3 + 7 * q^4 + q^6 - 3 * q^8 + 4 * q^9 + 8 * q^11 - 7 * q^12 - 7 * q^13 + 17 * q^16 - 6 * q^17 - q^18 - 3 * q^19 + 12 * q^22 - 2 * q^23 + 3 * q^24 + 19 * q^26 - 4 * q^27 + 16 * q^29 - 9 * q^31 - 17 * q^32 - 8 * q^33 - 14 * q^34 + 7 * q^36 - 8 * q^37 + 27 * q^38 + 7 * q^39 - 4 * q^41 - 5 * q^43 + 26 * q^44 - 6 * q^46 + 6 * q^47 - 17 * q^48 + 6 * q^51 - 35 * q^52 + 6 * q^53 + q^54 + 3 * q^57 - 4 * q^58 - 10 * q^59 + 20 * q^61 + 44 * q^62 + 21 * q^64 - 12 * q^66 - q^67 + 8 * q^68 + 2 * q^69 + 22 * q^71 - 3 * q^72 + 4 * q^73 - 21 * q^74 + 23 * q^76 - 19 * q^78 + 8 * q^79 + 4 * q^81 - 8 * q^82 + 2 * q^83 + 12 * q^86 - 16 * q^87 - 28 * q^88 + 10 * q^89 + 66 * q^92 + 9 * q^93 - 22 * q^94 + 17 * q^96 - 12 * q^97 + 8 * q^99

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