LMFDB - Embedded newform 3525.2.a.r.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3525,2,Mod(1,3525)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3525, 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("3525.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3525 = 3 \cdot 5^{2} \cdot 47 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3525.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:	$28.1472667125$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 705)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.56155$ of defining polynomial
Character	$\chi$	$=$	3525.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.56155 q^{2} +1.00000 q^{3} +4.56155 q^{4} +2.56155 q^{6} -0.561553 q^{7} +6.56155 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.56155 q^{2} +1.00000 q^{3} +4.56155 q^{4} +2.56155 q^{6} -0.561553 q^{7} +6.56155 q^{8} +1.00000 q^{9} -2.00000 q^{11} +4.56155 q^{12} +1.00000 q^{13} -1.43845 q^{14} +7.68466 q^{16} +4.56155 q^{17} +2.56155 q^{18} +0.561553 q^{19} -0.561553 q^{21} -5.12311 q^{22} +5.00000 q^{23} +6.56155 q^{24} +2.56155 q^{26} +1.00000 q^{27} -2.56155 q^{28} +3.43845 q^{29} +3.12311 q^{31} +6.56155 q^{32} -2.00000 q^{33} +11.6847 q^{34} +4.56155 q^{36} +1.43845 q^{38} +1.00000 q^{39} -2.56155 q^{41} -1.43845 q^{42} +0.438447 q^{43} -9.12311 q^{44} +12.8078 q^{46} -1.00000 q^{47} +7.68466 q^{48} -6.68466 q^{49} +4.56155 q^{51} +4.56155 q^{52} -1.68466 q^{53} +2.56155 q^{54} -3.68466 q^{56} +0.561553 q^{57} +8.80776 q^{58} -14.3693 q^{59} +4.12311 q^{61} +8.00000 q^{62} -0.561553 q^{63} +1.43845 q^{64} -5.12311 q^{66} +14.2462 q^{67} +20.8078 q^{68} +5.00000 q^{69} +1.00000 q^{71} +6.56155 q^{72} +2.68466 q^{73} +2.56155 q^{76} +1.12311 q^{77} +2.56155 q^{78} -13.8078 q^{79} +1.00000 q^{81} -6.56155 q^{82} +8.00000 q^{83} -2.56155 q^{84} +1.12311 q^{86} +3.43845 q^{87} -13.1231 q^{88} +3.56155 q^{89} -0.561553 q^{91} +22.8078 q^{92} +3.12311 q^{93} -2.56155 q^{94} +6.56155 q^{96} +0.876894 q^{97} -17.1231 q^{98} -2.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 2 q^{3} + 5 q^{4} + q^{6} + 3 q^{7} + 9 q^{8} + 2 q^{9}+O(q^{10}) $ 2 * q + q^2 + 2 * q^3 + 5 * q^4 + q^6 + 3 * q^7 + 9 * q^8 + 2 * q^9	$ 2 q + q^{2} + 2 q^{3} + 5 q^{4} + q^{6} + 3 q^{7} + 9 q^{8} + 2 q^{9} - 4 q^{11} + 5 q^{12} + 2 q^{13} - 7 q^{14} + 3 q^{16} + 5 q^{17} + q^{18} - 3 q^{19} + 3 q^{21} - 2 q^{22} + 10 q^{23} + 9 q^{24} + q^{26} + 2 q^{27} - q^{28} + 11 q^{29} - 2 q^{31} + 9 q^{32} - 4 q^{33} + 11 q^{34} + 5 q^{36} + 7 q^{38} + 2 q^{39} - q^{41} - 7 q^{42} + 5 q^{43} - 10 q^{44} + 5 q^{46} - 2 q^{47} + 3 q^{48} - q^{49} + 5 q^{51} + 5 q^{52} + 9 q^{53} + q^{54} + 5 q^{56} - 3 q^{57} - 3 q^{58} - 4 q^{59} + 16 q^{62} + 3 q^{63} + 7 q^{64} - 2 q^{66} + 12 q^{67} + 21 q^{68} + 10 q^{69} + 2 q^{71} + 9 q^{72} - 7 q^{73} + q^{76} - 6 q^{77} + q^{78} - 7 q^{79} + 2 q^{81} - 9 q^{82} + 16 q^{83} - q^{84} - 6 q^{86} + 11 q^{87} - 18 q^{88} + 3 q^{89} + 3 q^{91} + 25 q^{92} - 2 q^{93} - q^{94} + 9 q^{96} + 10 q^{97} - 26 q^{98} - 4 q^{99}+O(q^{100}) $ 2 * q + q^2 + 2 * q^3 + 5 * q^4 + q^6 + 3 * q^7 + 9 * q^8 + 2 * q^9 - 4 * q^11 + 5 * q^12 + 2 * q^13 - 7 * q^14 + 3 * q^16 + 5 * q^17 + q^18 - 3 * q^19 + 3 * q^21 - 2 * q^22 + 10 * q^23 + 9 * q^24 + q^26 + 2 * q^27 - q^28 + 11 * q^29 - 2 * q^31 + 9 * q^32 - 4 * q^33 + 11 * q^34 + 5 * q^36 + 7 * q^38 + 2 * q^39 - q^41 - 7 * q^42 + 5 * q^43 - 10 * q^44 + 5 * q^46 - 2 * q^47 + 3 * q^48 - q^49 + 5 * q^51 + 5 * q^52 + 9 * q^53 + q^54 + 5 * q^56 - 3 * q^57 - 3 * q^58 - 4 * q^59 + 16 * q^62 + 3 * q^63 + 7 * q^64 - 2 * q^66 + 12 * q^67 + 21 * q^68 + 10 * q^69 + 2 * q^71 + 9 * q^72 - 7 * q^73 + q^76 - 6 * q^77 + q^78 - 7 * q^79 + 2 * q^81 - 9 * q^82 + 16 * q^83 - q^84 - 6 * q^86 + 11 * q^87 - 18 * q^88 + 3 * q^89 + 3 * q^91 + 25 * q^92 - 2 * q^93 - q^94 + 9 * q^96 + 10 * q^97 - 26 * q^98 - 4 * 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 $

$2$	2.56155	1.81129	0.905646	−	0.424035i	$-0.139387\pi$
$2$	2.56155	1.81129	0.905646	+	0.424035i	$0.139387\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	4.56155	2.28078
$5$	0	0
$5$	0	0
$6$	2.56155	1.04575
$7$	−0.561553	−0.212247	−0.106124	−	0.994353i	$-0.533844\pi$
$7$	−0.561553	−0.212247	−0.106124	+	0.994353i	$0.533844\pi$
$8$	6.56155	2.31986
$9$	1.00000	0.333333
$10$	0	0
$11$	−2.00000	−0.603023	−0.301511	−	0.953463i	$-0.597491\pi$
$11$	−2.00000	−0.603023	−0.301511	+	0.953463i	$0.597491\pi$
$12$	4.56155	1.31681
$13$	1.00000	0.277350	0.138675	−	0.990338i	$-0.455716\pi$
$13$	1.00000	0.277350	0.138675	+	0.990338i	$0.455716\pi$
$14$	−1.43845	−0.384441
$15$	0	0
$16$	7.68466	1.92116
$17$	4.56155	1.10634	0.553170	−	0.833069i	$-0.313418\pi$
$17$	4.56155	1.10634	0.553170	+	0.833069i	$0.313418\pi$
$18$	2.56155	0.603764
$19$	0.561553	0.128829	0.0644145	−	0.997923i	$-0.479482\pi$
$19$	0.561553	0.128829	0.0644145	+	0.997923i	$0.479482\pi$
$20$	0	0
$21$	−0.561553	−0.122541
$22$	−5.12311	−1.09225
$23$	5.00000	1.04257	0.521286	−	0.853382i	$-0.325452\pi$
$23$	5.00000	1.04257	0.521286	+	0.853382i	$0.325452\pi$
$24$	6.56155	1.33937
$25$	0	0
$26$	2.56155	0.502362
$27$	1.00000	0.192450
$28$	−2.56155	−0.484088
$29$	3.43845	0.638504	0.319252	−	0.947670i	$-0.396568\pi$
$29$	3.43845	0.638504	0.319252	+	0.947670i	$0.396568\pi$
$30$	0	0
$31$	3.12311	0.560926	0.280463	−	0.959865i	$-0.409512\pi$
$31$	3.12311	0.560926	0.280463	+	0.959865i	$0.409512\pi$
$32$	6.56155	1.15993
$33$	−2.00000	−0.348155
$34$	11.6847	2.00390
$35$	0	0
$36$	4.56155	0.760259
$37$	0	0		−	1.00000i	$-0.5\pi$
$37$	0	0			1.00000i	$0.5\pi$
$38$	1.43845	0.233347
$39$	1.00000	0.160128
$40$	0	0
$41$	−2.56155	−0.400047	−0.200024	−	0.979791i	$-0.564102\pi$
$41$	−2.56155	−0.400047	−0.200024	+	0.979791i	$0.564102\pi$
$42$	−1.43845	−0.221957
$43$	0.438447	0.0668626	0.0334313	−	0.999441i	$-0.489357\pi$
$43$	0.438447	0.0668626	0.0334313	+	0.999441i	$0.489357\pi$
$44$	−9.12311	−1.37536
$45$	0	0
$46$	12.8078	1.88840
$47$	−1.00000	−0.145865
$47$	−1.00000	−0.145865
$48$	7.68466	1.10918
$49$	−6.68466	−0.954951
$50$	0	0
$51$	4.56155	0.638745
$52$	4.56155	0.632574
$53$	−1.68466	−0.231406	−0.115703	−	0.993284i	$-0.536912\pi$
$53$	−1.68466	−0.231406	−0.115703	+	0.993284i	$0.536912\pi$
$54$	2.56155	0.348583
$55$	0	0
$56$	−3.68466	−0.492383
$57$	0.561553	0.0743795
$58$	8.80776	1.15652
$59$	−14.3693	−1.87073	−0.935363	−	0.353690i	$-0.884927\pi$
$59$	−14.3693	−1.87073	−0.935363	+	0.353690i	$0.884927\pi$
$60$	0	0
$61$	4.12311	0.527910	0.263955	−	0.964535i	$-0.414973\pi$
$61$	4.12311	0.527910	0.263955	+	0.964535i	$0.414973\pi$
$62$	8.00000	1.01600
$63$	−0.561553	−0.0707490
$64$	1.43845	0.179806
$65$	0	0
$66$	−5.12311	−0.630611
$67$	14.2462	1.74045	0.870226	−	0.492653i	$-0.163973\pi$
$67$	14.2462	1.74045	0.870226	+	0.492653i	$0.163973\pi$
$68$	20.8078	2.52331
$69$	5.00000	0.601929
$70$	0	0
$71$	1.00000	0.118678	0.0593391	−	0.998238i	$-0.481101\pi$
$71$	1.00000	0.118678	0.0593391	+	0.998238i	$0.481101\pi$
$72$	6.56155	0.773286
$73$	2.68466	0.314216	0.157108	−	0.987581i	$-0.449783\pi$
$73$	2.68466	0.314216	0.157108	+	0.987581i	$0.449783\pi$
$74$	0	0
$75$	0	0
$76$	2.56155	0.293830
$77$	1.12311	0.127990
$78$	2.56155	0.290039
$79$	−13.8078	−1.55349	−0.776747	−	0.629812i	$-0.783132\pi$
$79$	−13.8078	−1.55349	−0.776747	+	0.629812i	$0.783132\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−6.56155	−0.724602
$83$	8.00000	0.878114	0.439057	−	0.898459i	$-0.355313\pi$
$83$	8.00000	0.878114	0.439057	+	0.898459i	$0.355313\pi$
$84$	−2.56155	−0.279488
$85$	0	0
$86$	1.12311	0.121108
$87$	3.43845	0.368640
$88$	−13.1231	−1.39893
$89$	3.56155	0.377524	0.188762	−	0.982023i	$-0.439553\pi$
$89$	3.56155	0.377524	0.188762	+	0.982023i	$0.439553\pi$
$90$	0	0
$91$	−0.561553	−0.0588667
$92$	22.8078	2.37787
$93$	3.12311	0.323851
$94$	−2.56155	−0.264204
$95$	0	0
$96$	6.56155	0.669686
$97$	0.876894	0.0890351	0.0445176	−	0.999009i	$-0.485825\pi$
$97$	0.876894	0.0890351	0.0445176	+	0.999009i	$0.485825\pi$
$98$	−17.1231	−1.72969
$99$	−2.00000	−0.201008
$100$	0	0
$101$	−3.31534	−0.329889	−0.164944	−	0.986303i	$-0.552744\pi$
$101$	−3.31534	−0.329889	−0.164944	+	0.986303i	$0.552744\pi$
$102$	11.6847	1.15695
$103$	−7.43845	−0.732932	−0.366466	−	0.930431i	$-0.619432\pi$
$103$	−7.43845	−0.732932	−0.366466	+	0.930431i	$0.619432\pi$
$104$	6.56155	0.643413
$105$	0	0
$106$	−4.31534	−0.419143
$107$	−1.56155	−0.150961	−0.0754805	−	0.997147i	$-0.524049\pi$
$107$	−1.56155	−0.150961	−0.0754805	+	0.997147i	$0.524049\pi$
$108$	4.56155	0.438936
$109$	−10.2462	−0.981409	−0.490705	−	0.871326i	$-0.663261\pi$
$109$	−10.2462	−0.981409	−0.490705	+	0.871326i	$0.663261\pi$
$110$	0	0
$111$	0	0
$112$	−4.31534	−0.407761
$113$	−13.8078	−1.29893	−0.649463	−	0.760394i	$-0.725006\pi$
$113$	−13.8078	−1.29893	−0.649463	+	0.760394i	$0.725006\pi$
$114$	1.43845	0.134723
$115$	0	0
$116$	15.6847	1.45628
$117$	1.00000	0.0924500
$118$	−36.8078	−3.38843
$119$	−2.56155	−0.234817
$120$	0	0
$121$	−7.00000	−0.636364
$122$	10.5616	0.956198
$123$	−2.56155	−0.230967
$124$	14.2462	1.27935
$125$	0	0
$126$	−1.43845	−0.128147
$127$	18.2462	1.61909	0.809545	−	0.587058i	$-0.199714\pi$
$127$	18.2462	1.61909	0.809545	+	0.587058i	$0.199714\pi$
$128$	−9.43845	−0.834249
$129$	0.438447	0.0386031
$130$	0	0
$131$	−2.43845	−0.213048	−0.106524	−	0.994310i	$-0.533972\pi$
$131$	−2.43845	−0.213048	−0.106524	+	0.994310i	$0.533972\pi$
$132$	−9.12311	−0.794064
$133$	−0.315342	−0.0273436
$134$	36.4924	3.15247
$135$	0	0
$136$	29.9309	2.56655
$137$	−10.4384	−0.891817	−0.445908	−	0.895079i	$-0.647119\pi$
$137$	−10.4384	−0.891817	−0.445908	+	0.895079i	$0.647119\pi$
$138$	12.8078	1.09027
$139$	−21.0540	−1.78577	−0.892887	−	0.450280i	$-0.851324\pi$
$139$	−21.0540	−1.78577	−0.892887	+	0.450280i	$0.851324\pi$
$140$	0	0
$141$	−1.00000	−0.0842152
$142$	2.56155	0.214961
$143$	−2.00000	−0.167248
$144$	7.68466	0.640388
$145$	0	0
$146$	6.87689	0.569136
$147$	−6.68466	−0.551341
$148$	0	0
$149$	−14.6847	−1.20301	−0.601507	−	0.798867i	$-0.705433\pi$
$149$	−14.6847	−1.20301	−0.601507	+	0.798867i	$0.705433\pi$
$150$	0	0
$151$	13.6847	1.11364	0.556821	−	0.830633i	$-0.312021\pi$
$151$	13.6847	1.11364	0.556821	+	0.830633i	$0.312021\pi$
$152$	3.68466	0.298865
$153$	4.56155	0.368780
$154$	2.87689	0.231827
$155$	0	0
$156$	4.56155	0.365217
$157$	−8.87689	−0.708453	−0.354227	−	0.935160i	$-0.615256\pi$
$157$	−8.87689	−0.708453	−0.354227	+	0.935160i	$0.615256\pi$
$158$	−35.3693	−2.81383
$159$	−1.68466	−0.133602
$160$	0	0
$161$	−2.80776	−0.221283
$162$	2.56155	0.201255
$163$	−15.8078	−1.23816	−0.619080	−	0.785328i	$-0.712494\pi$
$163$	−15.8078	−1.23816	−0.619080	+	0.785328i	$0.712494\pi$
$164$	−11.6847	−0.912419
$165$	0	0
$166$	20.4924	1.59052
$167$	13.8769	1.07383	0.536913	−	0.843638i	$-0.319590\pi$
$167$	13.8769	1.07383	0.536913	+	0.843638i	$0.319590\pi$
$168$	−3.68466	−0.284278
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0.561553	0.0429430
$172$	2.00000	0.152499
$173$	−10.8078	−0.821699	−0.410850	−	0.911703i	$-0.634768\pi$
$173$	−10.8078	−0.821699	−0.410850	+	0.911703i	$0.634768\pi$
$174$	8.80776	0.667715
$175$	0	0
$176$	−15.3693	−1.15851
$177$	−14.3693	−1.08006
$178$	9.12311	0.683806
$179$	18.2462	1.36379	0.681893	−	0.731452i	$-0.261157\pi$
$179$	18.2462	1.36379	0.681893	+	0.731452i	$0.261157\pi$
$180$	0	0
$181$	15.1231	1.12409	0.562046	−	0.827106i	$-0.310014\pi$
$181$	15.1231	1.12409	0.562046	+	0.827106i	$0.310014\pi$
$182$	−1.43845	−0.106625
$183$	4.12311	0.304789
$184$	32.8078	2.41862
$185$	0	0
$186$	8.00000	0.586588
$187$	−9.12311	−0.667148
$188$	−4.56155	−0.332685
$189$	−0.561553	−0.0408470
$190$	0	0
$191$	−8.68466	−0.628400	−0.314200	−	0.949357i	$-0.601736\pi$
$191$	−8.68466	−0.628400	−0.314200	+	0.949357i	$0.601736\pi$
$192$	1.43845	0.103811
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	2.24621	0.161269
$195$	0	0
$196$	−30.4924	−2.17803
$197$	3.19224	0.227437	0.113719	−	0.993513i	$-0.463724\pi$
$197$	3.19224	0.227437	0.113719	+	0.993513i	$0.463724\pi$
$198$	−5.12311	−0.364083
$199$	−22.8078	−1.61680	−0.808400	−	0.588634i	$-0.799666\pi$
$199$	−22.8078	−1.61680	−0.808400	+	0.588634i	$0.799666\pi$
$200$	0	0
$201$	14.2462	1.00485
$202$	−8.49242	−0.597525
$203$	−1.93087	−0.135520
$204$	20.8078	1.45683
$205$	0	0
$206$	−19.0540	−1.32755
$207$	5.00000	0.347524
$208$	7.68466	0.532835
$209$	−1.12311	−0.0776868
$210$	0	0
$211$	−20.0000	−1.37686	−0.688428	−	0.725304i	$-0.741699\pi$
$211$	−20.0000	−1.37686	−0.688428	+	0.725304i	$0.741699\pi$
$212$	−7.68466	−0.527785
$213$	1.00000	0.0685189
$214$	−4.00000	−0.273434
$215$	0	0
$216$	6.56155	0.446457
$217$	−1.75379	−0.119055
$218$	−26.2462	−1.77762
$219$	2.68466	0.181412
$220$	0	0
$221$	4.56155	0.306843
$222$	0	0
$223$	−6.93087	−0.464125	−0.232063	−	0.972701i	$-0.574547\pi$
$223$	−6.93087	−0.464125	−0.232063	+	0.972701i	$0.574547\pi$
$224$	−3.68466	−0.246192
$225$	0	0
$226$	−35.3693	−2.35273
$227$	12.8078	0.850081	0.425041	−	0.905174i	$-0.360260\pi$
$227$	12.8078	0.850081	0.425041	+	0.905174i	$0.360260\pi$
$228$	2.56155	0.169643
$229$	14.8769	0.983093	0.491546	−	0.870851i	$-0.336432\pi$
$229$	14.8769	0.983093	0.491546	+	0.870851i	$0.336432\pi$
$230$	0	0
$231$	1.12311	0.0738949
$232$	22.5616	1.48124
$233$	−4.24621	−0.278179	−0.139089	−	0.990280i	$-0.544417\pi$
$233$	−4.24621	−0.278179	−0.139089	+	0.990280i	$0.544417\pi$
$234$	2.56155	0.167454
$235$	0	0
$236$	−65.5464	−4.26671
$237$	−13.8078	−0.896911
$238$	−6.56155	−0.425322
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	−0.369317	−0.0237898	−0.0118949	−	0.999929i	$-0.503786\pi$
$241$	−0.369317	−0.0237898	−0.0118949	+	0.999929i	$0.503786\pi$
$242$	−17.9309	−1.15264
$243$	1.00000	0.0641500
$244$	18.8078	1.20404
$245$	0	0
$246$	−6.56155	−0.418349
$247$	0.561553	0.0357307
$248$	20.4924	1.30127
$249$	8.00000	0.506979
$250$	0	0
$251$	13.9309	0.879309	0.439654	−	0.898167i	$-0.355101\pi$
$251$	13.9309	0.879309	0.439654	+	0.898167i	$0.355101\pi$
$252$	−2.56155	−0.161363
$253$	−10.0000	−0.628695
$254$	46.7386	2.93264
$255$	0	0
$256$	−27.0540	−1.69087
$257$	0.192236	0.0119913	0.00599567	−	0.999982i	$-0.498092\pi$
$257$	0.192236	0.0119913	0.00599567	+	0.999982i	$0.498092\pi$
$258$	1.12311	0.0699215
$259$	0	0
$260$	0	0
$261$	3.43845	0.212835
$262$	−6.24621	−0.385892
$263$	12.8769	0.794023	0.397012	−	0.917814i	$-0.370047\pi$
$263$	12.8769	0.794023	0.397012	+	0.917814i	$0.370047\pi$
$264$	−13.1231	−0.807671
$265$	0	0
$266$	−0.807764	−0.0495272
$267$	3.56155	0.217963
$268$	64.9848	3.96958
$269$	15.1771	0.925363	0.462681	−	0.886525i	$-0.346887\pi$
$269$	15.1771	0.925363	0.462681	+	0.886525i	$0.346887\pi$
$270$	0	0
$271$	−6.24621	−0.379430	−0.189715	−	0.981839i	$-0.560756\pi$
$271$	−6.24621	−0.379430	−0.189715	+	0.981839i	$0.560756\pi$
$272$	35.0540	2.12546
$273$	−0.561553	−0.0339867
$274$	−26.7386	−1.61534
$275$	0	0
$276$	22.8078	1.37287
$277$	−17.3693	−1.04362	−0.521811	−	0.853061i	$-0.674743\pi$
$277$	−17.3693	−1.04362	−0.521811	+	0.853061i	$0.674743\pi$
$278$	−53.9309	−3.23456
$279$	3.12311	0.186975
$280$	0	0
$281$	−7.43845	−0.443741	−0.221870	−	0.975076i	$-0.571216\pi$
$281$	−7.43845	−0.443741	−0.221870	+	0.975076i	$0.571216\pi$
$282$	−2.56155	−0.152538
$283$	22.7386	1.35167	0.675836	−	0.737052i	$-0.263783\pi$
$283$	22.7386	1.35167	0.675836	+	0.737052i	$0.263783\pi$
$284$	4.56155	0.270678
$285$	0	0
$286$	−5.12311	−0.302936
$287$	1.43845	0.0849089
$288$	6.56155	0.386643
$289$	3.80776	0.223986
$290$	0	0
$291$	0.876894	0.0514045
$292$	12.2462	0.716655
$293$	−13.8078	−0.806658	−0.403329	−	0.915055i	$-0.632147\pi$
$293$	−13.8078	−0.806658	−0.403329	+	0.915055i	$0.632147\pi$
$294$	−17.1231	−0.998640
$295$	0	0
$296$	0	0
$297$	−2.00000	−0.116052
$298$	−37.6155	−2.17901
$299$	5.00000	0.289157
$300$	0	0
$301$	−0.246211	−0.0141914
$302$	35.0540	2.01713
$303$	−3.31534	−0.190461
$304$	4.31534	0.247502
$305$	0	0
$306$	11.6847	0.667967
$307$	−23.0540	−1.31576	−0.657880	−	0.753123i	$-0.728547\pi$
$307$	−23.0540	−1.31576	−0.657880	+	0.753123i	$0.728547\pi$
$308$	5.12311	0.291916
$309$	−7.43845	−0.423158
$310$	0	0
$311$	28.0000	1.58773	0.793867	−	0.608091i	$-0.208065\pi$
$311$	28.0000	1.58773	0.793867	+	0.608091i	$0.208065\pi$
$312$	6.56155	0.371475
$313$	22.8078	1.28917	0.644586	−	0.764532i	$-0.277030\pi$
$313$	22.8078	1.28917	0.644586	+	0.764532i	$0.277030\pi$
$314$	−22.7386	−1.28322
$315$	0	0
$316$	−62.9848	−3.54317
$317$	11.3153	0.635533	0.317766	−	0.948169i	$-0.397067\pi$
$317$	11.3153	0.635533	0.317766	+	0.948169i	$0.397067\pi$
$318$	−4.31534	−0.241992
$319$	−6.87689	−0.385032
$320$	0	0
$321$	−1.56155	−0.0871574
$322$	−7.19224	−0.400808
$323$	2.56155	0.142529
$324$	4.56155	0.253420
$325$	0	0
$326$	−40.4924	−2.24267
$327$	−10.2462	−0.566617
$328$	−16.8078	−0.928054
$329$	0.561553	0.0309594
$330$	0	0
$331$	34.0540	1.87178	0.935888	−	0.352298i	$-0.114600\pi$
$331$	34.0540	1.87178	0.935888	+	0.352298i	$0.114600\pi$
$332$	36.4924	2.00278
$333$	0	0
$334$	35.5464	1.94501
$335$	0	0
$336$	−4.31534	−0.235421
$337$	−2.63068	−0.143302	−0.0716512	−	0.997430i	$-0.522827\pi$
$337$	−2.63068	−0.143302	−0.0716512	+	0.997430i	$0.522827\pi$
$338$	−30.7386	−1.67196
$339$	−13.8078	−0.749935
$340$	0	0
$341$	−6.24621	−0.338251
$342$	1.43845	0.0777823
$343$	7.68466	0.414933
$344$	2.87689	0.155112
$345$	0	0
$346$	−27.6847	−1.48834
$347$	14.7386	0.791211	0.395606	−	0.918420i	$-0.370535\pi$
$347$	14.7386	0.791211	0.395606	+	0.918420i	$0.370535\pi$
$348$	15.6847	0.840786
$349$	−8.87689	−0.475169	−0.237585	−	0.971367i	$-0.576356\pi$
$349$	−8.87689	−0.475169	−0.237585	+	0.971367i	$0.576356\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	−13.1231	−0.699464
$353$	15.3693	0.818026	0.409013	−	0.912529i	$-0.365873\pi$
$353$	15.3693	0.818026	0.409013	+	0.912529i	$0.365873\pi$
$354$	−36.8078	−1.95631
$355$	0	0
$356$	16.2462	0.861047
$357$	−2.56155	−0.135572
$358$	46.7386	2.47021
$359$	33.8617	1.78715	0.893577	−	0.448910i	$-0.148188\pi$
$359$	33.8617	1.78715	0.893577	+	0.448910i	$0.148188\pi$
$360$	0	0
$361$	−18.6847	−0.983403
$362$	38.7386	2.03606
$363$	−7.00000	−0.367405
$364$	−2.56155	−0.134262
$365$	0	0
$366$	10.5616	0.552061
$367$	10.8769	0.567769	0.283885	−	0.958858i	$-0.408377\pi$
$367$	10.8769	0.567769	0.283885	+	0.958858i	$0.408377\pi$
$368$	38.4233	2.00295
$369$	−2.56155	−0.133349
$370$	0	0
$371$	0.946025	0.0491152
$372$	14.2462	0.738632
$373$	−27.1771	−1.40718	−0.703588	−	0.710608i	$-0.748420\pi$
$373$	−27.1771	−1.40718	−0.703588	+	0.710608i	$0.748420\pi$
$374$	−23.3693	−1.20840
$375$	0	0
$376$	−6.56155	−0.338386
$377$	3.43845	0.177089
$378$	−1.43845	−0.0739857
$379$	31.8617	1.63663	0.818314	−	0.574772i	$-0.194909\pi$
$379$	31.8617	1.63663	0.818314	+	0.574772i	$0.194909\pi$
$380$	0	0
$381$	18.2462	0.934782
$382$	−22.2462	−1.13822
$383$	−30.9848	−1.58325	−0.791626	−	0.611006i	$-0.790765\pi$
$383$	−30.9848	−1.58325	−0.791626	+	0.611006i	$0.790765\pi$
$384$	−9.43845	−0.481654
$385$	0	0
$386$	25.6155	1.30380
$387$	0.438447	0.0222875
$388$	4.00000	0.203069
$389$	−27.3002	−1.38417	−0.692087	−	0.721814i	$-0.743309\pi$
$389$	−27.3002	−1.38417	−0.692087	+	0.721814i	$0.743309\pi$
$390$	0	0
$391$	22.8078	1.15344
$392$	−43.8617	−2.21535
$393$	−2.43845	−0.123003
$394$	8.17708	0.411955
$395$	0	0
$396$	−9.12311	−0.458453
$397$	−36.4924	−1.83150	−0.915751	−	0.401746i	$-0.868403\pi$
$397$	−36.4924	−1.83150	−0.915751	+	0.401746i	$0.868403\pi$
$398$	−58.4233	−2.92850
$399$	−0.315342	−0.0157868
$400$	0	0
$401$	26.6847	1.33257	0.666284	−	0.745698i	$-0.267884\pi$
$401$	26.6847	1.33257	0.666284	+	0.745698i	$0.267884\pi$
$402$	36.4924	1.82008
$403$	3.12311	0.155573
$404$	−15.1231	−0.752403
$405$	0	0
$406$	−4.94602	−0.245467
$407$	0	0
$408$	29.9309	1.48180
$409$	−11.3693	−0.562177	−0.281088	−	0.959682i	$-0.590695\pi$
$409$	−11.3693	−0.562177	−0.281088	+	0.959682i	$0.590695\pi$
$410$	0	0
$411$	−10.4384	−0.514891
$412$	−33.9309	−1.67165
$413$	8.06913	0.397056
$414$	12.8078	0.629467
$415$	0	0
$416$	6.56155	0.321707
$417$	−21.0540	−1.03102
$418$	−2.87689	−0.140714
$419$	−11.6155	−0.567456	−0.283728	−	0.958905i	$-0.591571\pi$
$419$	−11.6155	−0.567456	−0.283728	+	0.958905i	$0.591571\pi$
$420$	0	0
$421$	16.8769	0.822530	0.411265	−	0.911516i	$-0.365087\pi$
$421$	16.8769	0.822530	0.411265	+	0.911516i	$0.365087\pi$
$422$	−51.2311	−2.49389
$423$	−1.00000	−0.0486217
$424$	−11.0540	−0.536828
$425$	0	0
$426$	2.56155	0.124108
$427$	−2.31534	−0.112047
$428$	−7.12311	−0.344308
$429$	−2.00000	−0.0965609
$430$	0	0
$431$	−21.9309	−1.05637	−0.528186	−	0.849128i	$-0.677128\pi$
$431$	−21.9309	−1.05637	−0.528186	+	0.849128i	$0.677128\pi$
$432$	7.68466	0.369728
$433$	−15.9309	−0.765589	−0.382794	−	0.923834i	$-0.625038\pi$
$433$	−15.9309	−0.765589	−0.382794	+	0.923834i	$0.625038\pi$
$434$	−4.49242	−0.215643
$435$	0	0
$436$	−46.7386	−2.23837
$437$	2.80776	0.134314
$438$	6.87689	0.328591
$439$	20.4384	0.975474	0.487737	−	0.872991i	$-0.337823\pi$
$439$	20.4384	0.975474	0.487737	+	0.872991i	$0.337823\pi$
$440$	0	0
$441$	−6.68466	−0.318317
$442$	11.6847	0.555783
$443$	28.1771	1.33873	0.669367	−	0.742932i	$-0.266566\pi$
$443$	28.1771	1.33873	0.669367	+	0.742932i	$0.266566\pi$
$444$	0	0
$445$	0	0
$446$	−17.7538	−0.840666
$447$	−14.6847	−0.694561
$448$	−0.807764	−0.0381633
$449$	−3.19224	−0.150651	−0.0753255	−	0.997159i	$-0.524000\pi$
$449$	−3.19224	−0.150651	−0.0753255	+	0.997159i	$0.524000\pi$
$450$	0	0
$451$	5.12311	0.241238
$452$	−62.9848	−2.96256
$453$	13.6847	0.642961
$454$	32.8078	1.53974
$455$	0	0
$456$	3.68466	0.172550
$457$	−26.2462	−1.22775	−0.613873	−	0.789405i	$-0.710389\pi$
$457$	−26.2462	−1.22775	−0.613873	+	0.789405i	$0.710389\pi$
$458$	38.1080	1.78067
$459$	4.56155	0.212915
$460$	0	0
$461$	31.3693	1.46101	0.730507	−	0.682905i	$-0.239284\pi$
$461$	31.3693	1.46101	0.730507	+	0.682905i	$0.239284\pi$
$462$	2.87689	0.133845
$463$	−2.05398	−0.0954563	−0.0477282	−	0.998860i	$-0.515198\pi$
$463$	−2.05398	−0.0954563	−0.0477282	+	0.998860i	$0.515198\pi$
$464$	26.4233	1.22667
$465$	0	0
$466$	−10.8769	−0.503862
$467$	1.63068	0.0754590	0.0377295	−	0.999288i	$-0.487987\pi$
$467$	1.63068	0.0754590	0.0377295	+	0.999288i	$0.487987\pi$
$468$	4.56155	0.210858
$469$	−8.00000	−0.369406
$470$	0	0
$471$	−8.87689	−0.409026
$472$	−94.2850	−4.33982
$473$	−0.876894	−0.0403196
$474$	−35.3693	−1.62457
$475$	0	0
$476$	−11.6847	−0.535565
$477$	−1.68466	−0.0771352
$478$	−20.4924	−0.937302
$479$	−27.7386	−1.26741	−0.633705	−	0.773575i	$-0.718467\pi$
$479$	−27.7386	−1.26741	−0.633705	+	0.773575i	$0.718467\pi$
$480$	0	0
$481$	0	0
$482$	−0.946025	−0.0430902
$483$	−2.80776	−0.127758
$484$	−31.9309	−1.45140
$485$	0	0
$486$	2.56155	0.116194
$487$	14.8078	0.671004	0.335502	−	0.942040i	$-0.391094\pi$
$487$	14.8078	0.671004	0.335502	+	0.942040i	$0.391094\pi$
$488$	27.0540	1.22468
$489$	−15.8078	−0.714852
$490$	0	0
$491$	36.8078	1.66111	0.830556	−	0.556936i	$-0.188023\pi$
$491$	36.8078	1.66111	0.830556	+	0.556936i	$0.188023\pi$
$492$	−11.6847	−0.526785
$493$	15.6847	0.706401
$494$	1.43845	0.0647188
$495$	0	0
$496$	24.0000	1.07763
$497$	−0.561553	−0.0251891
$498$	20.4924	0.918287
$499$	−14.5616	−0.651865	−0.325932	−	0.945393i	$-0.605678\pi$
$499$	−14.5616	−0.651865	−0.325932	+	0.945393i	$0.605678\pi$
$500$	0	0
$501$	13.8769	0.619974
$502$	35.6847	1.59268
$503$	−4.31534	−0.192412	−0.0962058	−	0.995361i	$-0.530671\pi$
$503$	−4.31534	−0.192412	−0.0962058	+	0.995361i	$0.530671\pi$
$504$	−3.68466	−0.164128
$505$	0	0
$506$	−25.6155	−1.13875
$507$	−12.0000	−0.532939
$508$	83.2311	3.69278
$509$	21.4384	0.950242	0.475121	−	0.879920i	$-0.342404\pi$
$509$	21.4384	0.950242	0.475121	+	0.879920i	$0.342404\pi$
$510$	0	0
$511$	−1.50758	−0.0666913
$512$	−50.4233	−2.22842
$513$	0.561553	0.0247932
$514$	0.492423	0.0217198
$515$	0	0
$516$	2.00000	0.0880451
$517$	2.00000	0.0879599
$518$	0	0
$519$	−10.8078	−0.474408
$520$	0	0
$521$	−27.4233	−1.20144	−0.600718	−	0.799461i	$-0.705119\pi$
$521$	−27.4233	−1.20144	−0.600718	+	0.799461i	$0.705119\pi$
$522$	8.80776	0.385505
$523$	20.4233	0.893048	0.446524	−	0.894772i	$-0.352662\pi$
$523$	20.4233	0.893048	0.446524	+	0.894772i	$0.352662\pi$
$524$	−11.1231	−0.485915
$525$	0	0
$526$	32.9848	1.43821
$527$	14.2462	0.620575
$528$	−15.3693	−0.668864
$529$	2.00000	0.0869565
$530$	0	0
$531$	−14.3693	−0.623575
$532$	−1.43845	−0.0623646
$533$	−2.56155	−0.110953
$534$	9.12311	0.394795
$535$	0	0
$536$	93.4773	4.03760
$537$	18.2462	0.787382
$538$	38.8769	1.67610
$539$	13.3693	0.575857
$540$	0	0
$541$	15.4924	0.666071	0.333036	−	0.942914i	$-0.391927\pi$
$541$	15.4924	0.666071	0.333036	+	0.942914i	$0.391927\pi$
$542$	−16.0000	−0.687259
$543$	15.1231	0.648995
$544$	29.9309	1.28328
$545$	0	0
$546$	−1.43845	−0.0615599
$547$	−19.3693	−0.828172	−0.414086	−	0.910238i	$-0.635899\pi$
$547$	−19.3693	−0.828172	−0.414086	+	0.910238i	$0.635899\pi$
$548$	−47.6155	−2.03403
$549$	4.12311	0.175970
$550$	0	0
$551$	1.93087	0.0822578
$552$	32.8078	1.39639
$553$	7.75379	0.329725
$554$	−44.4924	−1.89030
$555$	0	0
$556$	−96.0388	−4.07295
$557$	−23.8078	−1.00877	−0.504384	−	0.863480i	$-0.668280\pi$
$557$	−23.8078	−1.00877	−0.504384	+	0.863480i	$0.668280\pi$
$558$	8.00000	0.338667
$559$	0.438447	0.0185443
$560$	0	0
$561$	−9.12311	−0.385178
$562$	−19.0540	−0.803743
$563$	3.31534	0.139725	0.0698625	−	0.997557i	$-0.477744\pi$
$563$	3.31534	0.139725	0.0698625	+	0.997557i	$0.477744\pi$
$564$	−4.56155	−0.192076
$565$	0	0
$566$	58.2462	2.44827
$567$	−0.561553	−0.0235830
$568$	6.56155	0.275317
$569$	34.9848	1.46664	0.733320	−	0.679883i	$-0.237969\pi$
$569$	34.9848	1.46664	0.733320	+	0.679883i	$0.237969\pi$
$570$	0	0
$571$	−22.4384	−0.939020	−0.469510	−	0.882927i	$-0.655569\pi$
$571$	−22.4384	−0.939020	−0.469510	+	0.882927i	$0.655569\pi$
$572$	−9.12311	−0.381456
$573$	−8.68466	−0.362807
$574$	3.68466	0.153795
$575$	0	0
$576$	1.43845	0.0599353
$577$	31.9848	1.33155	0.665773	−	0.746154i	$-0.268102\pi$
$577$	31.9848	1.33155	0.665773	+	0.746154i	$0.268102\pi$
$578$	9.75379	0.405704
$579$	10.0000	0.415586
$580$	0	0
$581$	−4.49242	−0.186377
$582$	2.24621	0.0931085
$583$	3.36932	0.139543
$584$	17.6155	0.728936
$585$	0	0
$586$	−35.3693	−1.46109
$587$	−8.19224	−0.338130	−0.169065	−	0.985605i	$-0.554075\pi$
$587$	−8.19224	−0.338130	−0.169065	+	0.985605i	$0.554075\pi$
$588$	−30.4924	−1.25749
$589$	1.75379	0.0722636
$590$	0	0
$591$	3.19224	0.131311
$592$	0	0
$593$	−5.36932	−0.220491	−0.110246	−	0.993904i	$-0.535164\pi$
$593$	−5.36932	−0.220491	−0.110246	+	0.993904i	$0.535164\pi$
$594$	−5.12311	−0.210204
$595$	0	0
$596$	−66.9848	−2.74381
$597$	−22.8078	−0.933460
$598$	12.8078	0.523748
$599$	−31.3693	−1.28172	−0.640858	−	0.767660i	$-0.721421\pi$
$599$	−31.3693	−1.28172	−0.640858	+	0.767660i	$0.721421\pi$
$600$	0	0
$601$	45.9848	1.87576	0.937880	−	0.346959i	$-0.112786\pi$
$601$	45.9848	1.87576	0.937880	+	0.346959i	$0.112786\pi$
$602$	−0.630683	−0.0257047
$603$	14.2462	0.580151
$604$	62.4233	2.53997
$605$	0	0
$606$	−8.49242	−0.344981
$607$	−3.94602	−0.160164	−0.0800821	−	0.996788i	$-0.525518\pi$
$607$	−3.94602	−0.160164	−0.0800821	+	0.996788i	$0.525518\pi$
$608$	3.68466	0.149433
$609$	−1.93087	−0.0782428
$610$	0	0
$611$	−1.00000	−0.0404557
$612$	20.8078	0.841104
$613$	2.63068	0.106252	0.0531261	−	0.998588i	$-0.483081\pi$
$613$	2.63068	0.106252	0.0531261	+	0.998588i	$0.483081\pi$
$614$	−59.0540	−2.38322
$615$	0	0
$616$	7.36932	0.296918
$617$	−11.7538	−0.473190	−0.236595	−	0.971608i	$-0.576031\pi$
$617$	−11.7538	−0.473190	−0.236595	+	0.971608i	$0.576031\pi$
$618$	−19.0540	−0.766463
$619$	18.1922	0.731208	0.365604	−	0.930771i	$-0.380863\pi$
$619$	18.1922	0.731208	0.365604	+	0.930771i	$0.380863\pi$
$620$	0	0
$621$	5.00000	0.200643
$622$	71.7235	2.87585
$623$	−2.00000	−0.0801283
$624$	7.68466	0.307633
$625$	0	0
$626$	58.4233	2.33506
$627$	−1.12311	−0.0448525
$628$	−40.4924	−1.61582
$629$	0	0
$630$	0	0
$631$	−3.50758	−0.139634	−0.0698172	−	0.997560i	$-0.522242\pi$
$631$	−3.50758	−0.139634	−0.0698172	+	0.997560i	$0.522242\pi$
$632$	−90.6004	−3.60389
$633$	−20.0000	−0.794929
$634$	28.9848	1.15114
$635$	0	0
$636$	−7.68466	−0.304717
$637$	−6.68466	−0.264856
$638$	−17.6155	−0.697405
$639$	1.00000	0.0395594
$640$	0	0
$641$	−14.4924	−0.572416	−0.286208	−	0.958167i	$-0.592395\pi$
$641$	−14.4924	−0.572416	−0.286208	+	0.958167i	$0.592395\pi$
$642$	−4.00000	−0.157867
$643$	−24.5616	−0.968613	−0.484307	−	0.874898i	$-0.660928\pi$
$643$	−24.5616	−0.968613	−0.484307	+	0.874898i	$0.660928\pi$
$644$	−12.8078	−0.504697
$645$	0	0
$646$	6.56155	0.258161
$647$	−16.4924	−0.648384	−0.324192	−	0.945991i	$-0.605092\pi$
$647$	−16.4924	−0.648384	−0.324192	+	0.945991i	$0.605092\pi$
$648$	6.56155	0.257762
$649$	28.7386	1.12809
$650$	0	0
$651$	−1.75379	−0.0687364
$652$	−72.1080	−2.82397
$653$	47.7926	1.87027	0.935135	−	0.354292i	$-0.115278\pi$
$653$	47.7926	1.87027	0.935135	+	0.354292i	$0.115278\pi$
$654$	−26.2462	−1.02631
$655$	0	0
$656$	−19.6847	−0.768557
$657$	2.68466	0.104739
$658$	1.43845	0.0560765
$659$	18.9309	0.737442	0.368721	−	0.929540i	$-0.379796\pi$
$659$	18.9309	0.737442	0.368721	+	0.929540i	$0.379796\pi$
$660$	0	0
$661$	42.1231	1.63840	0.819199	−	0.573509i	$-0.194418\pi$
$661$	42.1231	1.63840	0.819199	+	0.573509i	$0.194418\pi$
$662$	87.2311	3.39033
$663$	4.56155	0.177156
$664$	52.4924	2.03710
$665$	0	0
$666$	0	0
$667$	17.1922	0.665686
$668$	63.3002	2.44916
$669$	−6.93087	−0.267963
$670$	0	0
$671$	−8.24621	−0.318341
$672$	−3.68466	−0.142139
$673$	−29.2462	−1.12736	−0.563679	−	0.825994i	$-0.690615\pi$
$673$	−29.2462	−1.12736	−0.563679	+	0.825994i	$0.690615\pi$
$674$	−6.73863	−0.259562
$675$	0	0
$676$	−54.7386	−2.10533
$677$	24.5464	0.943395	0.471697	−	0.881761i	$-0.343641\pi$
$677$	24.5464	0.943395	0.471697	+	0.881761i	$0.343641\pi$
$678$	−35.3693	−1.35835
$679$	−0.492423	−0.0188974
$680$	0	0
$681$	12.8078	0.490795
$682$	−16.0000	−0.612672
$683$	−37.6155	−1.43932	−0.719659	−	0.694328i	$-0.755702\pi$
$683$	−37.6155	−1.43932	−0.719659	+	0.694328i	$0.755702\pi$
$684$	2.56155	0.0979434
$685$	0	0
$686$	19.6847	0.751564
$687$	14.8769	0.567589
$688$	3.36932	0.128454
$689$	−1.68466	−0.0641804
$690$	0	0
$691$	15.4384	0.587306	0.293653	−	0.955912i	$-0.405129\pi$
$691$	15.4384	0.587306	0.293653	+	0.955912i	$0.405129\pi$
$692$	−49.3002	−1.87411
$693$	1.12311	0.0426633
$694$	37.7538	1.43311
$695$	0	0
$696$	22.5616	0.855193
$697$	−11.6847	−0.442588
$698$	−22.7386	−0.860670
$699$	−4.24621	−0.160606
$700$	0	0
$701$	−33.4384	−1.26295	−0.631476	−	0.775395i	$-0.717551\pi$
$701$	−33.4384	−1.26295	−0.631476	+	0.775395i	$0.717551\pi$
$702$	2.56155	0.0966796
$703$	0	0
$704$	−2.87689	−0.108427
$705$	0	0
$706$	39.3693	1.48168
$707$	1.86174	0.0700179
$708$	−65.5464	−2.46338
$709$	17.3153	0.650291	0.325146	−	0.945664i	$-0.394587\pi$
$709$	17.3153	0.650291	0.325146	+	0.945664i	$0.394587\pi$
$710$	0	0
$711$	−13.8078	−0.517832
$712$	23.3693	0.875802
$713$	15.6155	0.584806
$714$	−6.56155	−0.245560
$715$	0	0
$716$	83.2311	3.11049
$717$	−8.00000	−0.298765
$718$	86.7386	3.23706
$719$	−7.68466	−0.286589	−0.143295	−	0.989680i	$-0.545770\pi$
$719$	−7.68466	−0.286589	−0.143295	+	0.989680i	$0.545770\pi$
$720$	0	0
$721$	4.17708	0.155563
$722$	−47.8617	−1.78123
$723$	−0.369317	−0.0137350
$724$	68.9848	2.56380
$725$	0	0
$726$	−17.9309	−0.665477
$727$	24.9309	0.924635	0.462317	−	0.886715i	$-0.347018\pi$
$727$	24.9309	0.924635	0.462317	+	0.886715i	$0.347018\pi$
$728$	−3.68466	−0.136563
$729$	1.00000	0.0370370
$730$	0	0
$731$	2.00000	0.0739727
$732$	18.8078	0.695155
$733$	32.1080	1.18593	0.592967	−	0.805227i	$-0.297956\pi$
$733$	32.1080	1.18593	0.592967	+	0.805227i	$0.297956\pi$
$734$	27.8617	1.02840
$735$	0	0
$736$	32.8078	1.20931
$737$	−28.4924	−1.04953
$738$	−6.56155	−0.241534
$739$	−22.1922	−0.816355	−0.408177	−	0.912903i	$-0.633835\pi$
$739$	−22.1922	−0.816355	−0.408177	+	0.912903i	$0.633835\pi$
$740$	0	0
$741$	0.561553	0.0206292
$742$	2.42329	0.0889619
$743$	24.1771	0.886971	0.443486	−	0.896282i	$-0.353742\pi$
$743$	24.1771	0.886971	0.443486	+	0.896282i	$0.353742\pi$
$744$	20.4924	0.751289
$745$	0	0
$746$	−69.6155	−2.54881
$747$	8.00000	0.292705
$748$	−41.6155	−1.52161
$749$	0.876894	0.0320410
$750$	0	0
$751$	0.315342	0.0115070	0.00575349	−	0.999983i	$-0.498169\pi$
$751$	0.315342	0.0115070	0.00575349	+	0.999983i	$0.498169\pi$
$752$	−7.68466	−0.280231
$753$	13.9309	0.507669
$754$	8.80776	0.320760
$755$	0	0
$756$	−2.56155	−0.0931628
$757$	2.49242	0.0905886	0.0452943	−	0.998974i	$-0.485577\pi$
$757$	2.49242	0.0905886	0.0452943	+	0.998974i	$0.485577\pi$
$758$	81.6155	2.96441
$759$	−10.0000	−0.362977
$760$	0	0
$761$	−9.80776	−0.355531	−0.177766	−	0.984073i	$-0.556887\pi$
$761$	−9.80776	−0.355531	−0.177766	+	0.984073i	$0.556887\pi$
$762$	46.7386	1.69316
$763$	5.75379	0.208301
$764$	−39.6155	−1.43324
$765$	0	0
$766$	−79.3693	−2.86773
$767$	−14.3693	−0.518846
$768$	−27.0540	−0.976226
$769$	−48.5464	−1.75063	−0.875314	−	0.483555i	$-0.839345\pi$
$769$	−48.5464	−1.75063	−0.875314	+	0.483555i	$0.839345\pi$
$770$	0	0
$771$	0.192236	0.00692321
$772$	45.6155	1.64174
$773$	28.7386	1.03366	0.516828	−	0.856089i	$-0.327113\pi$
$773$	28.7386	1.03366	0.516828	+	0.856089i	$0.327113\pi$
$774$	1.12311	0.0403692
$775$	0	0
$776$	5.75379	0.206549
$777$	0	0
$778$	−69.9309	−2.50714
$779$	−1.43845	−0.0515377
$780$	0	0
$781$	−2.00000	−0.0715656
$782$	58.4233	2.08921
$783$	3.43845	0.122880
$784$	−51.3693	−1.83462
$785$	0	0
$786$	−6.24621	−0.222795
$787$	−36.7926	−1.31151	−0.655757	−	0.754972i	$-0.727651\pi$
$787$	−36.7926	−1.31151	−0.655757	+	0.754972i	$0.727651\pi$
$788$	14.5616	0.518734
$789$	12.8769	0.458430
$790$	0	0
$791$	7.75379	0.275693
$792$	−13.1231	−0.466309
$793$	4.12311	0.146416
$794$	−93.4773	−3.31738
$795$	0	0
$796$	−104.039	−3.68756
$797$	47.1231	1.66919	0.834593	−	0.550867i	$-0.185703\pi$
$797$	47.1231	1.66919	0.834593	+	0.550867i	$0.185703\pi$
$798$	−0.807764	−0.0285945
$799$	−4.56155	−0.161376
$800$	0	0
$801$	3.56155	0.125841
$802$	68.3542	2.41367
$803$	−5.36932	−0.189479
$804$	64.9848	2.29184
$805$	0	0
$806$	8.00000	0.281788
$807$	15.1771	0.534259
$808$	−21.7538	−0.765296
$809$	−41.1231	−1.44581	−0.722906	−	0.690947i	$-0.757194\pi$
$809$	−41.1231	−1.44581	−0.722906	+	0.690947i	$0.757194\pi$
$810$	0	0
$811$	−1.12311	−0.0394376	−0.0197188	−	0.999806i	$-0.506277\pi$
$811$	−1.12311	−0.0394376	−0.0197188	+	0.999806i	$0.506277\pi$
$812$	−8.80776	−0.309092
$813$	−6.24621	−0.219064
$814$	0	0
$815$	0	0
$816$	35.0540	1.22713
$817$	0.246211	0.00861384
$818$	−29.1231	−1.01827
$819$	−0.561553	−0.0196222
$820$	0	0
$821$	−21.1922	−0.739614	−0.369807	−	0.929109i	$-0.620576\pi$
$821$	−21.1922	−0.739614	−0.369807	+	0.929109i	$0.620576\pi$
$822$	−26.7386	−0.932617
$823$	−4.00000	−0.139431	−0.0697156	−	0.997567i	$-0.522209\pi$
$823$	−4.00000	−0.139431	−0.0697156	+	0.997567i	$0.522209\pi$
$824$	−48.8078	−1.70030
$825$	0	0
$826$	20.6695	0.719184
$827$	46.1080	1.60333	0.801665	−	0.597773i	$-0.203948\pi$
$827$	46.1080	1.60333	0.801665	+	0.597773i	$0.203948\pi$
$828$	22.8078	0.792625
$829$	−7.36932	−0.255947	−0.127973	−	0.991778i	$-0.540847\pi$
$829$	−7.36932	−0.255947	−0.127973	+	0.991778i	$0.540847\pi$
$830$	0	0
$831$	−17.3693	−0.602535
$832$	1.43845	0.0498692
$833$	−30.4924	−1.05650
$834$	−53.9309	−1.86747
$835$	0	0
$836$	−5.12311	−0.177186
$837$	3.12311	0.107950
$838$	−29.7538	−1.02783
$839$	7.36932	0.254417	0.127209	−	0.991876i	$-0.459398\pi$
$839$	7.36932	0.254417	0.127209	+	0.991876i	$0.459398\pi$
$840$	0	0
$841$	−17.1771	−0.592313
$842$	43.2311	1.48984
$843$	−7.43845	−0.256194
$844$	−91.2311	−3.14030
$845$	0	0
$846$	−2.56155	−0.0880680
$847$	3.93087	0.135066
$848$	−12.9460	−0.444568
$849$	22.7386	0.780388
$850$	0	0
$851$	0	0
$852$	4.56155	0.156276
$853$	−16.9848	−0.581550	−0.290775	−	0.956791i	$-0.593913\pi$
$853$	−16.9848	−0.581550	−0.290775	+	0.956791i	$0.593913\pi$
$854$	−5.93087	−0.202950
$855$	0	0
$856$	−10.2462	−0.350208
$857$	47.4233	1.61995	0.809974	−	0.586465i	$-0.199481\pi$
$857$	47.4233	1.61995	0.809974	+	0.586465i	$0.199481\pi$
$858$	−5.12311	−0.174900
$859$	3.19224	0.108918	0.0544588	−	0.998516i	$-0.482657\pi$
$859$	3.19224	0.108918	0.0544588	+	0.998516i	$0.482657\pi$
$860$	0	0
$861$	1.43845	0.0490221
$862$	−56.1771	−1.91340
$863$	−46.4924	−1.58262	−0.791310	−	0.611415i	$-0.790601\pi$
$863$	−46.4924	−1.58262	−0.791310	+	0.611415i	$0.790601\pi$
$864$	6.56155	0.223229
$865$	0	0
$866$	−40.8078	−1.38670
$867$	3.80776	0.129318
$868$	−8.00000	−0.271538
$869$	27.6155	0.936793
$870$	0	0
$871$	14.2462	0.482714
$872$	−67.2311	−2.27673
$873$	0.876894	0.0296784
$874$	7.19224	0.243281
$875$	0	0
$876$	12.2462	0.413761
$877$	−34.8078	−1.17537	−0.587687	−	0.809088i	$-0.699961\pi$
$877$	−34.8078	−1.17537	−0.587687	+	0.809088i	$0.699961\pi$
$878$	52.3542	1.76687
$879$	−13.8078	−0.465724
$880$	0	0
$881$	25.0540	0.844090	0.422045	−	0.906575i	$-0.361312\pi$
$881$	25.0540	0.844090	0.422045	+	0.906575i	$0.361312\pi$
$882$	−17.1231	−0.576565
$883$	9.43845	0.317629	0.158815	−	0.987308i	$-0.449233\pi$
$883$	9.43845	0.317629	0.158815	+	0.987308i	$0.449233\pi$
$884$	20.8078	0.699841
$885$	0	0
$886$	72.1771	2.42484
$887$	30.9309	1.03856	0.519278	−	0.854605i	$-0.326201\pi$
$887$	30.9309	1.03856	0.519278	+	0.854605i	$0.326201\pi$
$888$	0	0
$889$	−10.2462	−0.343647
$890$	0	0
$891$	−2.00000	−0.0670025
$892$	−31.6155	−1.05857
$893$	−0.561553	−0.0187916
$894$	−37.6155	−1.25805
$895$	0	0
$896$	5.30019	0.177067
$897$	5.00000	0.166945
$898$	−8.17708	−0.272873
$899$	10.7386	0.358153
$900$	0	0
$901$	−7.68466	−0.256013
$902$	13.1231	0.436952
$903$	−0.246211	−0.00819340
$904$	−90.6004	−3.01332
$905$	0	0
$906$	35.0540	1.16459
$907$	44.8078	1.48782	0.743909	−	0.668281i	$-0.232970\pi$
$907$	44.8078	1.48782	0.743909	+	0.668281i	$0.232970\pi$
$908$	58.4233	1.93885
$909$	−3.31534	−0.109963
$910$	0	0
$911$	−17.7538	−0.588209	−0.294105	−	0.955773i	$-0.595021\pi$
$911$	−17.7538	−0.588209	−0.294105	+	0.955773i	$0.595021\pi$
$912$	4.31534	0.142895
$913$	−16.0000	−0.529523
$914$	−67.2311	−2.22381
$915$	0	0
$916$	67.8617	2.24221
$917$	1.36932	0.0452188
$918$	11.6847	0.385651
$919$	−54.1771	−1.78714	−0.893568	−	0.448927i	$-0.851806\pi$
$919$	−54.1771	−1.78714	−0.893568	+	0.448927i	$0.851806\pi$
$920$	0	0
$921$	−23.0540	−0.759654
$922$	80.3542	2.64632
$923$	1.00000	0.0329154
$924$	5.12311	0.168538
$925$	0	0
$926$	−5.26137	−0.172899
$927$	−7.43845	−0.244311
$928$	22.5616	0.740619
$929$	4.43845	0.145621	0.0728104	−	0.997346i	$-0.476803\pi$
$929$	4.43845	0.145621	0.0728104	+	0.997346i	$0.476803\pi$
$930$	0	0
$931$	−3.75379	−0.123025
$932$	−19.3693	−0.634463
$933$	28.0000	0.916679
$934$	4.17708	0.136678
$935$	0	0
$936$	6.56155	0.214471
$937$	−31.5616	−1.03107	−0.515535	−	0.856868i	$-0.672407\pi$
$937$	−31.5616	−1.03107	−0.515535	+	0.856868i	$0.672407\pi$
$938$	−20.4924	−0.669101
$939$	22.8078	0.744303
$940$	0	0
$941$	29.8617	0.973465	0.486732	−	0.873551i	$-0.338189\pi$
$941$	29.8617	0.973465	0.486732	+	0.873551i	$0.338189\pi$
$942$	−22.7386	−0.740865
$943$	−12.8078	−0.417078
$944$	−110.423	−3.59397
$945$	0	0
$946$	−2.24621	−0.0730306
$947$	−2.00000	−0.0649913	−0.0324956	−	0.999472i	$-0.510346\pi$
$947$	−2.00000	−0.0649913	−0.0324956	+	0.999472i	$0.510346\pi$
$948$	−62.9848	−2.04565
$949$	2.68466	0.0871477
$950$	0	0
$951$	11.3153	0.366925
$952$	−16.8078	−0.544743
$953$	18.1922	0.589304	0.294652	−	0.955605i	$-0.404796\pi$
$953$	18.1922	0.589304	0.294652	+	0.955605i	$0.404796\pi$
$954$	−4.31534	−0.139714
$955$	0	0
$956$	−36.4924	−1.18025
$957$	−6.87689	−0.222298
$958$	−71.0540	−2.29565
$959$	5.86174	0.189285
$960$	0	0
$961$	−21.2462	−0.685362
$962$	0	0
$963$	−1.56155	−0.0503203
$964$	−1.68466	−0.0542592
$965$	0	0
$966$	−7.19224	−0.231406
$967$	10.6307	0.341860	0.170930	−	0.985283i	$-0.445323\pi$
$967$	10.6307	0.341860	0.170930	+	0.985283i	$0.445323\pi$
$968$	−45.9309	−1.47627
$969$	2.56155	0.0822889
$970$	0	0
$971$	52.9848	1.70036	0.850182	−	0.526488i	$-0.176492\pi$
$971$	52.9848	1.70036	0.850182	+	0.526488i	$0.176492\pi$
$972$	4.56155	0.146312
$973$	11.8229	0.379025
$974$	37.9309	1.21538
$975$	0	0
$976$	31.6847	1.01420
$977$	36.6695	1.17316	0.586581	−	0.809891i	$-0.300474\pi$
$977$	36.6695	1.17316	0.586581	+	0.809891i	$0.300474\pi$
$978$	−40.4924	−1.29480
$979$	−7.12311	−0.227655
$980$	0	0
$981$	−10.2462	−0.327136
$982$	94.2850	3.00876
$983$	21.4924	0.685502	0.342751	−	0.939426i	$-0.388641\pi$
$983$	21.4924	0.685502	0.342751	+	0.939426i	$0.388641\pi$
$984$	−16.8078	−0.535812
$985$	0	0
$986$	40.1771	1.27950
$987$	0.561553	0.0178744
$988$	2.56155	0.0814939
$989$	2.19224	0.0697090
$990$	0	0
$991$	10.3002	0.327196	0.163598	−	0.986527i	$-0.447690\pi$
$991$	10.3002	0.327196	0.163598	+	0.986527i	$0.447690\pi$
$992$	20.4924	0.650635
$993$	34.0540	1.08067
$994$	−1.43845	−0.0456248
$995$	0	0
$996$	36.4924	1.15631
$997$	−0.0691303	−0.00218938	−0.00109469	−	0.999999i	$-0.500348\pi$
$997$	−0.0691303	−0.00218938	−0.00109469	+	0.999999i	$0.500348\pi$
$998$	−37.3002	−1.18072
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3525.2.a.r.1.2		2
5.4	even	2		705.2.a.h.1.1	✓	2
15.14	odd	2		2115.2.a.m.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
705.2.a.h.1.1	✓	2	5.4	even	2
2115.2.a.m.1.2		2	15.14	odd	2
3525.2.a.r.1.2		2	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 3525 = 3 \cdot 5^{2} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3525.a (trivial)

\(f(q)\)	\(=\)	\(q+2.56155 q^{2} +1.00000 q^{3} +4.56155 q^{4} +2.56155 q^{6} -0.561553 q^{7} +6.56155 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.56155 q^{2} +1.00000 q^{3} +4.56155 q^{4} +2.56155 q^{6} -0.561553 q^{7} +6.56155 q^{8} +1.00000 q^{9} -2.00000 q^{11} +4.56155 q^{12} +1.00000 q^{13} -1.43845 q^{14} +7.68466 q^{16} +4.56155 q^{17} +2.56155 q^{18} +0.561553 q^{19} -0.561553 q^{21} -5.12311 q^{22} +5.00000 q^{23} +6.56155 q^{24} +2.56155 q^{26} +1.00000 q^{27} -2.56155 q^{28} +3.43845 q^{29} +3.12311 q^{31} +6.56155 q^{32} -2.00000 q^{33} +11.6847 q^{34} +4.56155 q^{36} +1.43845 q^{38} +1.00000 q^{39} -2.56155 q^{41} -1.43845 q^{42} +0.438447 q^{43} -9.12311 q^{44} +12.8078 q^{46} -1.00000 q^{47} +7.68466 q^{48} -6.68466 q^{49} +4.56155 q^{51} +4.56155 q^{52} -1.68466 q^{53} +2.56155 q^{54} -3.68466 q^{56} +0.561553 q^{57} +8.80776 q^{58} -14.3693 q^{59} +4.12311 q^{61} +8.00000 q^{62} -0.561553 q^{63} +1.43845 q^{64} -5.12311 q^{66} +14.2462 q^{67} +20.8078 q^{68} +5.00000 q^{69} +1.00000 q^{71} +6.56155 q^{72} +2.68466 q^{73} +2.56155 q^{76} +1.12311 q^{77} +2.56155 q^{78} -13.8078 q^{79} +1.00000 q^{81} -6.56155 q^{82} +8.00000 q^{83} -2.56155 q^{84} +1.12311 q^{86} +3.43845 q^{87} -13.1231 q^{88} +3.56155 q^{89} -0.561553 q^{91} +22.8078 q^{92} +3.12311 q^{93} -2.56155 q^{94} +6.56155 q^{96} +0.876894 q^{97} -17.1231 q^{98} -2.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 2 q^{3} + 5 q^{4} + q^{6} + 3 q^{7} + 9 q^{8} + 2 q^{9}+O(q^{10}) \) 2 * q + q^2 + 2 * q^3 + 5 * q^4 + q^6 + 3 * q^7 + 9 * q^8 + 2 * q^9	\( 2 q + q^{2} + 2 q^{3} + 5 q^{4} + q^{6} + 3 q^{7} + 9 q^{8} + 2 q^{9} - 4 q^{11} + 5 q^{12} + 2 q^{13} - 7 q^{14} + 3 q^{16} + 5 q^{17} + q^{18} - 3 q^{19} + 3 q^{21} - 2 q^{22} + 10 q^{23} + 9 q^{24} + q^{26} + 2 q^{27} - q^{28} + 11 q^{29} - 2 q^{31} + 9 q^{32} - 4 q^{33} + 11 q^{34} + 5 q^{36} + 7 q^{38} + 2 q^{39} - q^{41} - 7 q^{42} + 5 q^{43} - 10 q^{44} + 5 q^{46} - 2 q^{47} + 3 q^{48} - q^{49} + 5 q^{51} + 5 q^{52} + 9 q^{53} + q^{54} + 5 q^{56} - 3 q^{57} - 3 q^{58} - 4 q^{59} + 16 q^{62} + 3 q^{63} + 7 q^{64} - 2 q^{66} + 12 q^{67} + 21 q^{68} + 10 q^{69} + 2 q^{71} + 9 q^{72} - 7 q^{73} + q^{76} - 6 q^{77} + q^{78} - 7 q^{79} + 2 q^{81} - 9 q^{82} + 16 q^{83} - q^{84} - 6 q^{86} + 11 q^{87} - 18 q^{88} + 3 q^{89} + 3 q^{91} + 25 q^{92} - 2 q^{93} - q^{94} + 9 q^{96} + 10 q^{97} - 26 q^{98} - 4 q^{99}+O(q^{100}) \) 2 * q + q^2 + 2 * q^3 + 5 * q^4 + q^6 + 3 * q^7 + 9 * q^8 + 2 * q^9 - 4 * q^11 + 5 * q^12 + 2 * q^13 - 7 * q^14 + 3 * q^16 + 5 * q^17 + q^18 - 3 * q^19 + 3 * q^21 - 2 * q^22 + 10 * q^23 + 9 * q^24 + q^26 + 2 * q^27 - q^28 + 11 * q^29 - 2 * q^31 + 9 * q^32 - 4 * q^33 + 11 * q^34 + 5 * q^36 + 7 * q^38 + 2 * q^39 - q^41 - 7 * q^42 + 5 * q^43 - 10 * q^44 + 5 * q^46 - 2 * q^47 + 3 * q^48 - q^49 + 5 * q^51 + 5 * q^52 + 9 * q^53 + q^54 + 5 * q^56 - 3 * q^57 - 3 * q^58 - 4 * q^59 + 16 * q^62 + 3 * q^63 + 7 * q^64 - 2 * q^66 + 12 * q^67 + 21 * q^68 + 10 * q^69 + 2 * q^71 + 9 * q^72 - 7 * q^73 + q^76 - 6 * q^77 + q^78 - 7 * q^79 + 2 * q^81 - 9 * q^82 + 16 * q^83 - q^84 - 6 * q^86 + 11 * q^87 - 18 * q^88 + 3 * q^89 + 3 * q^91 + 25 * q^92 - 2 * q^93 - q^94 + 9 * q^96 + 10 * q^97 - 26 * q^98 - 4 * q^99

Introduction

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Database

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