LMFDB - Embedded newform 4425.2.a.t.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4425,2,Mod(1,4425)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4425, 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("4425.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4425 = 3 \cdot 5^{2} \cdot 59 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4425.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:	$35.3338028944$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 177)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	4425.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} -0.618034 q^{6} +2.38197 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} -0.618034 q^{6} +2.38197 q^{7} +2.23607 q^{8} +1.00000 q^{9} +2.23607 q^{11} -1.61803 q^{12} +6.23607 q^{13} -1.47214 q^{14} +1.85410 q^{16} -1.85410 q^{17} -0.618034 q^{18} +3.09017 q^{19} +2.38197 q^{21} -1.38197 q^{22} +4.61803 q^{23} +2.23607 q^{24} -3.85410 q^{26} +1.00000 q^{27} -3.85410 q^{28} +6.38197 q^{29} -10.5623 q^{31} -5.61803 q^{32} +2.23607 q^{33} +1.14590 q^{34} -1.61803 q^{36} +0.145898 q^{37} -1.90983 q^{38} +6.23607 q^{39} +8.09017 q^{41} -1.47214 q^{42} +8.70820 q^{43} -3.61803 q^{44} -2.85410 q^{46} +10.8541 q^{47} +1.85410 q^{48} -1.32624 q^{49} -1.85410 q^{51} -10.0902 q^{52} -6.23607 q^{53} -0.618034 q^{54} +5.32624 q^{56} +3.09017 q^{57} -3.94427 q^{58} -1.00000 q^{59} -3.14590 q^{61} +6.52786 q^{62} +2.38197 q^{63} -0.236068 q^{64} -1.38197 q^{66} -10.7082 q^{67} +3.00000 q^{68} +4.61803 q^{69} -7.94427 q^{71} +2.23607 q^{72} -0.854102 q^{73} -0.0901699 q^{74} -5.00000 q^{76} +5.32624 q^{77} -3.85410 q^{78} -3.00000 q^{79} +1.00000 q^{81} -5.00000 q^{82} +1.61803 q^{83} -3.85410 q^{84} -5.38197 q^{86} +6.38197 q^{87} +5.00000 q^{88} -13.7984 q^{89} +14.8541 q^{91} -7.47214 q^{92} -10.5623 q^{93} -6.70820 q^{94} -5.61803 q^{96} -3.00000 q^{97} +0.819660 q^{98} +2.23607 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{2} + 2 q^{3} - q^{4} + q^{6} + 7 q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + q^2 + 2 * q^3 - q^4 + q^6 + 7 * q^7 + 2 * q^9	$ 2 q + q^{2} + 2 q^{3} - q^{4} + q^{6} + 7 q^{7} + 2 q^{9} - q^{12} + 8 q^{13} + 6 q^{14} - 3 q^{16} + 3 q^{17} + q^{18} - 5 q^{19} + 7 q^{21} - 5 q^{22} + 7 q^{23} - q^{26} + 2 q^{27} - q^{28} + 15 q^{29} - q^{31} - 9 q^{32} + 9 q^{34} - q^{36} + 7 q^{37} - 15 q^{38} + 8 q^{39} + 5 q^{41} + 6 q^{42} + 4 q^{43} - 5 q^{44} + q^{46} + 15 q^{47} - 3 q^{48} + 13 q^{49} + 3 q^{51} - 9 q^{52} - 8 q^{53} + q^{54} - 5 q^{56} - 5 q^{57} + 10 q^{58} - 2 q^{59} - 13 q^{61} + 22 q^{62} + 7 q^{63} + 4 q^{64} - 5 q^{66} - 8 q^{67} + 6 q^{68} + 7 q^{69} + 2 q^{71} + 5 q^{73} + 11 q^{74} - 10 q^{76} - 5 q^{77} - q^{78} - 6 q^{79} + 2 q^{81} - 10 q^{82} + q^{83} - q^{84} - 13 q^{86} + 15 q^{87} + 10 q^{88} - 3 q^{89} + 23 q^{91} - 6 q^{92} - q^{93} - 9 q^{96} - 6 q^{97} + 24 q^{98}+O(q^{100}) $ 2 * q + q^2 + 2 * q^3 - q^4 + q^6 + 7 * q^7 + 2 * q^9 - q^12 + 8 * q^13 + 6 * q^14 - 3 * q^16 + 3 * q^17 + q^18 - 5 * q^19 + 7 * q^21 - 5 * q^22 + 7 * q^23 - q^26 + 2 * q^27 - q^28 + 15 * q^29 - q^31 - 9 * q^32 + 9 * q^34 - q^36 + 7 * q^37 - 15 * q^38 + 8 * q^39 + 5 * q^41 + 6 * q^42 + 4 * q^43 - 5 * q^44 + q^46 + 15 * q^47 - 3 * q^48 + 13 * q^49 + 3 * q^51 - 9 * q^52 - 8 * q^53 + q^54 - 5 * q^56 - 5 * q^57 + 10 * q^58 - 2 * q^59 - 13 * q^61 + 22 * q^62 + 7 * q^63 + 4 * q^64 - 5 * q^66 - 8 * q^67 + 6 * q^68 + 7 * q^69 + 2 * q^71 + 5 * q^73 + 11 * q^74 - 10 * q^76 - 5 * q^77 - q^78 - 6 * q^79 + 2 * q^81 - 10 * q^82 + q^83 - q^84 - 13 * q^86 + 15 * q^87 + 10 * q^88 - 3 * q^89 + 23 * q^91 - 6 * q^92 - q^93 - 9 * q^96 - 6 * q^97 + 24 * q^98

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 $

$2$	−0.618034	−0.437016	−0.218508	−	0.975835i	$-0.570119\pi$
$2$	−0.618034	−0.437016	−0.218508	+	0.975835i	$0.570119\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.61803	−0.809017
$5$	0	0
$5$	0	0
$6$	−0.618034	−0.252311
$7$	2.38197	0.900299	0.450149	−	0.892953i	$-0.351371\pi$
$7$	2.38197	0.900299	0.450149	+	0.892953i	$0.351371\pi$
$8$	2.23607	0.790569
$9$	1.00000	0.333333
$10$	0	0
$11$	2.23607	0.674200	0.337100	−	0.941469i	$-0.390554\pi$
$11$	2.23607	0.674200	0.337100	+	0.941469i	$0.390554\pi$
$12$	−1.61803	−0.467086
$13$	6.23607	1.72957	0.864787	−	0.502139i	$-0.167453\pi$
$13$	6.23607	1.72957	0.864787	+	0.502139i	$0.167453\pi$
$14$	−1.47214	−0.393445
$15$	0	0
$16$	1.85410	0.463525
$17$	−1.85410	−0.449686	−0.224843	−	0.974395i	$-0.572187\pi$
$17$	−1.85410	−0.449686	−0.224843	+	0.974395i	$0.572187\pi$
$18$	−0.618034	−0.145672
$19$	3.09017	0.708934	0.354467	−	0.935069i	$-0.384662\pi$
$19$	3.09017	0.708934	0.354467	+	0.935069i	$0.384662\pi$
$20$	0	0
$21$	2.38197	0.519788
$22$	−1.38197	−0.294636
$23$	4.61803	0.962927	0.481463	−	0.876466i	$-0.340105\pi$
$23$	4.61803	0.962927	0.481463	+	0.876466i	$0.340105\pi$
$24$	2.23607	0.456435
$25$	0	0
$26$	−3.85410	−0.755852
$27$	1.00000	0.192450
$28$	−3.85410	−0.728357
$29$	6.38197	1.18510	0.592551	−	0.805533i	$-0.298121\pi$
$29$	6.38197	1.18510	0.592551	+	0.805533i	$0.298121\pi$
$30$	0	0
$31$	−10.5623	−1.89705	−0.948523	−	0.316708i	$-0.897422\pi$
$31$	−10.5623	−1.89705	−0.948523	+	0.316708i	$0.897422\pi$
$32$	−5.61803	−0.993137
$33$	2.23607	0.389249
$34$	1.14590	0.196520
$35$	0	0
$36$	−1.61803	−0.269672
$37$	0.145898	0.0239855	0.0119927	−	0.999928i	$-0.496182\pi$
$37$	0.145898	0.0239855	0.0119927	+	0.999928i	$0.496182\pi$
$38$	−1.90983	−0.309815
$39$	6.23607	0.998570
$40$	0	0
$41$	8.09017	1.26347	0.631736	−	0.775183i	$-0.282343\pi$
$41$	8.09017	1.26347	0.631736	+	0.775183i	$0.282343\pi$
$42$	−1.47214	−0.227156
$43$	8.70820	1.32799	0.663994	−	0.747738i	$-0.268860\pi$
$43$	8.70820	1.32799	0.663994	+	0.747738i	$0.268860\pi$
$44$	−3.61803	−0.545439
$45$	0	0
$46$	−2.85410	−0.420814
$47$	10.8541	1.58323	0.791617	−	0.611018i	$-0.209240\pi$
$47$	10.8541	1.58323	0.791617	+	0.611018i	$0.209240\pi$
$48$	1.85410	0.267617
$49$	−1.32624	−0.189463
$50$	0	0
$51$	−1.85410	−0.259626
$52$	−10.0902	−1.39925
$53$	−6.23607	−0.856590	−0.428295	−	0.903639i	$-0.640886\pi$
$53$	−6.23607	−0.856590	−0.428295	+	0.903639i	$0.640886\pi$
$54$	−0.618034	−0.0841038
$55$	0	0
$56$	5.32624	0.711748
$57$	3.09017	0.409303
$58$	−3.94427	−0.517908
$59$	−1.00000	−0.130189
$59$	−1.00000	−0.130189
$60$	0	0
$61$	−3.14590	−0.402791	−0.201395	−	0.979510i	$-0.564548\pi$
$61$	−3.14590	−0.402791	−0.201395	+	0.979510i	$0.564548\pi$
$62$	6.52786	0.829040
$63$	2.38197	0.300100
$64$	−0.236068	−0.0295085
$65$	0	0
$66$	−1.38197	−0.170108
$67$	−10.7082	−1.30822	−0.654108	−	0.756401i	$-0.726956\pi$
$67$	−10.7082	−1.30822	−0.654108	+	0.756401i	$0.726956\pi$
$68$	3.00000	0.363803
$69$	4.61803	0.555946
$70$	0	0
$71$	−7.94427	−0.942812	−0.471406	−	0.881916i	$-0.656253\pi$
$71$	−7.94427	−0.942812	−0.471406	+	0.881916i	$0.656253\pi$
$72$	2.23607	0.263523
$73$	−0.854102	−0.0999651	−0.0499825	−	0.998750i	$-0.515917\pi$
$73$	−0.854102	−0.0999651	−0.0499825	+	0.998750i	$0.515917\pi$
$74$	−0.0901699	−0.0104820
$75$	0	0
$76$	−5.00000	−0.573539
$77$	5.32624	0.606981
$78$	−3.85410	−0.436391
$79$	−3.00000	−0.337526	−0.168763	−	0.985657i	$-0.553977\pi$
$79$	−3.00000	−0.337526	−0.168763	+	0.985657i	$0.553977\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−5.00000	−0.552158
$83$	1.61803	0.177602	0.0888012	−	0.996049i	$-0.471696\pi$
$83$	1.61803	0.177602	0.0888012	+	0.996049i	$0.471696\pi$
$84$	−3.85410	−0.420517
$85$	0	0
$86$	−5.38197	−0.580352
$87$	6.38197	0.684219
$88$	5.00000	0.533002
$89$	−13.7984	−1.46262	−0.731312	−	0.682043i	$-0.761092\pi$
$89$	−13.7984	−1.46262	−0.731312	+	0.682043i	$0.761092\pi$
$90$	0	0
$91$	14.8541	1.55713
$92$	−7.47214	−0.779024
$93$	−10.5623	−1.09526
$94$	−6.70820	−0.691898
$95$	0	0
$96$	−5.61803	−0.573388
$97$	−3.00000	−0.304604	−0.152302	−	0.988334i	$-0.548669\pi$
$97$	−3.00000	−0.304604	−0.152302	+	0.988334i	$0.548669\pi$
$98$	0.819660	0.0827982
$99$	2.23607	0.224733
$100$	0	0
$101$	3.70820	0.368980	0.184490	−	0.982834i	$-0.440937\pi$
$101$	3.70820	0.368980	0.184490	+	0.982834i	$0.440937\pi$
$102$	1.14590	0.113461
$103$	3.23607	0.318859	0.159430	−	0.987209i	$-0.449034\pi$
$103$	3.23607	0.318859	0.159430	+	0.987209i	$0.449034\pi$
$104$	13.9443	1.36735
$105$	0	0
$106$	3.85410	0.374343
$107$	−0.909830	−0.0879566	−0.0439783	−	0.999032i	$-0.514003\pi$
$107$	−0.909830	−0.0879566	−0.0439783	+	0.999032i	$0.514003\pi$
$108$	−1.61803	−0.155695
$109$	−4.14590	−0.397105	−0.198553	−	0.980090i	$-0.563624\pi$
$109$	−4.14590	−0.397105	−0.198553	+	0.980090i	$0.563624\pi$
$110$	0	0
$111$	0.145898	0.0138480
$112$	4.41641	0.417311
$113$	9.00000	0.846649	0.423324	−	0.905978i	$-0.360863\pi$
$113$	9.00000	0.846649	0.423324	+	0.905978i	$0.360863\pi$
$114$	−1.90983	−0.178872
$115$	0	0
$116$	−10.3262	−0.958767
$117$	6.23607	0.576525
$118$	0.618034	0.0568946
$119$	−4.41641	−0.404851
$120$	0	0
$121$	−6.00000	−0.545455
$122$	1.94427	0.176026
$123$	8.09017	0.729466
$124$	17.0902	1.53474
$125$	0	0
$126$	−1.47214	−0.131148
$127$	−15.9443	−1.41483	−0.707413	−	0.706801i	$-0.750138\pi$
$127$	−15.9443	−1.41483	−0.707413	+	0.706801i	$0.750138\pi$
$128$	11.3820	1.00603
$129$	8.70820	0.766715
$130$	0	0
$131$	−20.6525	−1.80442	−0.902208	−	0.431302i	$-0.858054\pi$
$131$	−20.6525	−1.80442	−0.902208	+	0.431302i	$0.858054\pi$
$132$	−3.61803	−0.314909
$133$	7.36068	0.638252
$134$	6.61803	0.571711
$135$	0	0
$136$	−4.14590	−0.355508
$137$	−12.2361	−1.04540	−0.522699	−	0.852517i	$-0.675075\pi$
$137$	−12.2361	−1.04540	−0.522699	+	0.852517i	$0.675075\pi$
$138$	−2.85410	−0.242957
$139$	11.7639	0.997804	0.498902	−	0.866658i	$-0.333737\pi$
$139$	11.7639	0.997804	0.498902	+	0.866658i	$0.333737\pi$
$140$	0	0
$141$	10.8541	0.914080
$142$	4.90983	0.412024
$143$	13.9443	1.16608
$144$	1.85410	0.154508
$145$	0	0
$146$	0.527864	0.0436863
$147$	−1.32624	−0.109386
$148$	−0.236068	−0.0194047
$149$	19.0902	1.56393	0.781964	−	0.623324i	$-0.214218\pi$
$149$	19.0902	1.56393	0.781964	+	0.623324i	$0.214218\pi$
$150$	0	0
$151$	2.56231	0.208517	0.104259	−	0.994550i	$-0.466753\pi$
$151$	2.56231	0.208517	0.104259	+	0.994550i	$0.466753\pi$
$152$	6.90983	0.560461
$153$	−1.85410	−0.149895
$154$	−3.29180	−0.265260
$155$	0	0
$156$	−10.0902	−0.807860
$157$	−9.00000	−0.718278	−0.359139	−	0.933284i	$-0.616930\pi$
$157$	−9.00000	−0.718278	−0.359139	+	0.933284i	$0.616930\pi$
$158$	1.85410	0.147504
$159$	−6.23607	−0.494552
$160$	0	0
$161$	11.0000	0.866921
$162$	−0.618034	−0.0485573
$163$	18.5623	1.45391	0.726956	−	0.686684i	$-0.240934\pi$
$163$	18.5623	1.45391	0.726956	+	0.686684i	$0.240934\pi$
$164$	−13.0902	−1.02217
$165$	0	0
$166$	−1.00000	−0.0776151
$167$	−7.03444	−0.544341	−0.272171	−	0.962249i	$-0.587742\pi$
$167$	−7.03444	−0.544341	−0.272171	+	0.962249i	$0.587742\pi$
$168$	5.32624	0.410928
$169$	25.8885	1.99143
$170$	0	0
$171$	3.09017	0.236311
$172$	−14.0902	−1.07437
$173$	−12.3820	−0.941383	−0.470692	−	0.882298i	$-0.655996\pi$
$173$	−12.3820	−0.941383	−0.470692	+	0.882298i	$0.655996\pi$
$174$	−3.94427	−0.299014
$175$	0	0
$176$	4.14590	0.312509
$177$	−1.00000	−0.0751646
$178$	8.52786	0.639190
$179$	−0.527864	−0.0394544	−0.0197272	−	0.999805i	$-0.506280\pi$
$179$	−0.527864	−0.0394544	−0.0197272	+	0.999805i	$0.506280\pi$
$180$	0	0
$181$	22.2705	1.65535	0.827677	−	0.561205i	$-0.189662\pi$
$181$	22.2705	1.65535	0.827677	+	0.561205i	$0.189662\pi$
$182$	−9.18034	−0.680492
$183$	−3.14590	−0.232551
$184$	10.3262	0.761260
$185$	0	0
$186$	6.52786	0.478646
$187$	−4.14590	−0.303178
$188$	−17.5623	−1.28086
$189$	2.38197	0.173263
$190$	0	0
$191$	16.4164	1.18785	0.593925	−	0.804521i	$-0.297578\pi$
$191$	16.4164	1.18785	0.593925	+	0.804521i	$0.297578\pi$
$192$	−0.236068	−0.0170367
$193$	−8.00000	−0.575853	−0.287926	−	0.957653i	$-0.592966\pi$
$193$	−8.00000	−0.575853	−0.287926	+	0.957653i	$0.592966\pi$
$194$	1.85410	0.133117
$195$	0	0
$196$	2.14590	0.153278
$197$	20.6525	1.47143	0.735714	−	0.677292i	$-0.236847\pi$
$197$	20.6525	1.47143	0.735714	+	0.677292i	$0.236847\pi$
$198$	−1.38197	−0.0982120
$199$	−16.5623	−1.17407	−0.587035	−	0.809561i	$-0.699705\pi$
$199$	−16.5623	−1.17407	−0.587035	+	0.809561i	$0.699705\pi$
$200$	0	0
$201$	−10.7082	−0.755298
$202$	−2.29180	−0.161250
$203$	15.2016	1.06694
$204$	3.00000	0.210042
$205$	0	0
$206$	−2.00000	−0.139347
$207$	4.61803	0.320976
$208$	11.5623	0.801702
$209$	6.90983	0.477963
$210$	0	0
$211$	−2.14590	−0.147730	−0.0738649	−	0.997268i	$-0.523533\pi$
$211$	−2.14590	−0.147730	−0.0738649	+	0.997268i	$0.523533\pi$
$212$	10.0902	0.692996
$213$	−7.94427	−0.544333
$214$	0.562306	0.0384384
$215$	0	0
$216$	2.23607	0.152145
$217$	−25.1591	−1.70791
$218$	2.56231	0.173541
$219$	−0.854102	−0.0577149
$220$	0	0
$221$	−11.5623	−0.777765
$222$	−0.0901699	−0.00605181
$223$	9.52786	0.638033	0.319016	−	0.947749i	$-0.396647\pi$
$223$	9.52786	0.638033	0.319016	+	0.947749i	$0.396647\pi$
$224$	−13.3820	−0.894120
$225$	0	0
$226$	−5.56231	−0.369999
$227$	22.1459	1.46987	0.734937	−	0.678135i	$-0.237211\pi$
$227$	22.1459	1.46987	0.734937	+	0.678135i	$0.237211\pi$
$228$	−5.00000	−0.331133
$229$	−9.14590	−0.604378	−0.302189	−	0.953248i	$-0.597717\pi$
$229$	−9.14590	−0.604378	−0.302189	+	0.953248i	$0.597717\pi$
$230$	0	0
$231$	5.32624	0.350441
$232$	14.2705	0.936905
$233$	8.29180	0.543214	0.271607	−	0.962408i	$-0.412445\pi$
$233$	8.29180	0.543214	0.271607	+	0.962408i	$0.412445\pi$
$234$	−3.85410	−0.251951
$235$	0	0
$236$	1.61803	0.105325
$237$	−3.00000	−0.194871
$238$	2.72949	0.176927
$239$	−1.47214	−0.0952246	−0.0476123	−	0.998866i	$-0.515161\pi$
$239$	−1.47214	−0.0952246	−0.0476123	+	0.998866i	$0.515161\pi$
$240$	0	0
$241$	−23.4164	−1.50838	−0.754192	−	0.656654i	$-0.771971\pi$
$241$	−23.4164	−1.50838	−0.754192	+	0.656654i	$0.771971\pi$
$242$	3.70820	0.238372
$243$	1.00000	0.0641500
$244$	5.09017	0.325865
$245$	0	0
$246$	−5.00000	−0.318788
$247$	19.2705	1.22615
$248$	−23.6180	−1.49975
$249$	1.61803	0.102539
$250$	0	0
$251$	15.1803	0.958175	0.479087	−	0.877767i	$-0.340968\pi$
$251$	15.1803	0.958175	0.479087	+	0.877767i	$0.340968\pi$
$252$	−3.85410	−0.242786
$253$	10.3262	0.649205
$254$	9.85410	0.618301
$255$	0	0
$256$	−6.56231	−0.410144
$257$	19.4164	1.21116	0.605581	−	0.795784i	$-0.292941\pi$
$257$	19.4164	1.21116	0.605581	+	0.795784i	$0.292941\pi$
$258$	−5.38197	−0.335067
$259$	0.347524	0.0215941
$260$	0	0
$261$	6.38197	0.395034
$262$	12.7639	0.788558
$263$	−1.61803	−0.0997722	−0.0498861	−	0.998755i	$-0.515886\pi$
$263$	−1.61803	−0.0997722	−0.0498861	+	0.998755i	$0.515886\pi$
$264$	5.00000	0.307729
$265$	0	0
$266$	−4.54915	−0.278926
$267$	−13.7984	−0.844447
$268$	17.3262	1.05837
$269$	−11.4721	−0.699468	−0.349734	−	0.936849i	$-0.613728\pi$
$269$	−11.4721	−0.699468	−0.349734	+	0.936849i	$0.613728\pi$
$270$	0	0
$271$	−7.76393	−0.471625	−0.235813	−	0.971799i	$-0.575775\pi$
$271$	−7.76393	−0.471625	−0.235813	+	0.971799i	$0.575775\pi$
$272$	−3.43769	−0.208441
$273$	14.8541	0.899011
$274$	7.56231	0.456856
$275$	0	0
$276$	−7.47214	−0.449770
$277$	5.47214	0.328789	0.164394	−	0.986395i	$-0.447433\pi$
$277$	5.47214	0.328789	0.164394	+	0.986395i	$0.447433\pi$
$278$	−7.27051	−0.436056
$279$	−10.5623	−0.632349
$280$	0	0
$281$	9.70820	0.579143	0.289571	−	0.957156i	$-0.406487\pi$
$281$	9.70820	0.579143	0.289571	+	0.957156i	$0.406487\pi$
$282$	−6.70820	−0.399468
$283$	23.2705	1.38329	0.691644	−	0.722238i	$-0.256887\pi$
$283$	23.2705	1.38329	0.691644	+	0.722238i	$0.256887\pi$
$284$	12.8541	0.762751
$285$	0	0
$286$	−8.61803	−0.509595
$287$	19.2705	1.13750
$288$	−5.61803	−0.331046
$289$	−13.5623	−0.797783
$290$	0	0
$291$	−3.00000	−0.175863
$292$	1.38197	0.0808734
$293$	−19.3820	−1.13231	−0.566153	−	0.824300i	$-0.691569\pi$
$293$	−19.3820	−1.13231	−0.566153	+	0.824300i	$0.691569\pi$
$294$	0.819660	0.0478035
$295$	0	0
$296$	0.326238	0.0189622
$297$	2.23607	0.129750
$298$	−11.7984	−0.683461
$299$	28.7984	1.66545
$300$	0	0
$301$	20.7426	1.19559
$302$	−1.58359	−0.0911255
$303$	3.70820	0.213031
$304$	5.72949	0.328609
$305$	0	0
$306$	1.14590	0.0655066
$307$	25.8885	1.47754	0.738769	−	0.673959i	$-0.235408\pi$
$307$	25.8885	1.47754	0.738769	+	0.673959i	$0.235408\pi$
$308$	−8.61803	−0.491058
$309$	3.23607	0.184093
$310$	0	0
$311$	27.4508	1.55659	0.778297	−	0.627896i	$-0.216084\pi$
$311$	27.4508	1.55659	0.778297	+	0.627896i	$0.216084\pi$
$312$	13.9443	0.789439
$313$	20.7984	1.17559	0.587797	−	0.809009i	$-0.299995\pi$
$313$	20.7984	1.17559	0.587797	+	0.809009i	$0.299995\pi$
$314$	5.56231	0.313899
$315$	0	0
$316$	4.85410	0.273065
$317$	29.1803	1.63893	0.819466	−	0.573128i	$-0.194270\pi$
$317$	29.1803	1.63893	0.819466	+	0.573128i	$0.194270\pi$
$318$	3.85410	0.216127
$319$	14.2705	0.798995
$320$	0	0
$321$	−0.909830	−0.0507818
$322$	−6.79837	−0.378859
$323$	−5.72949	−0.318797
$324$	−1.61803	−0.0898908
$325$	0	0
$326$	−11.4721	−0.635383
$327$	−4.14590	−0.229269
$328$	18.0902	0.998863
$329$	25.8541	1.42538
$330$	0	0
$331$	−11.1246	−0.611464	−0.305732	−	0.952118i	$-0.598901\pi$
$331$	−11.1246	−0.611464	−0.305732	+	0.952118i	$0.598901\pi$
$332$	−2.61803	−0.143683
$333$	0.145898	0.00799516
$334$	4.34752	0.237886
$335$	0	0
$336$	4.41641	0.240935
$337$	14.9098	0.812190	0.406095	−	0.913831i	$-0.366890\pi$
$337$	14.9098	0.812190	0.406095	+	0.913831i	$0.366890\pi$
$338$	−16.0000	−0.870285
$339$	9.00000	0.488813
$340$	0	0
$341$	−23.6180	−1.27899
$342$	−1.90983	−0.103272
$343$	−19.8328	−1.07087
$344$	19.4721	1.04987
$345$	0	0
$346$	7.65248	0.411400
$347$	−31.0344	−1.66602	−0.833008	−	0.553261i	$-0.813383\pi$
$347$	−31.0344	−1.66602	−0.833008	+	0.553261i	$0.813383\pi$
$348$	−10.3262	−0.553544
$349$	−27.0000	−1.44528	−0.722638	−	0.691226i	$-0.757071\pi$
$349$	−27.0000	−1.44528	−0.722638	+	0.691226i	$0.757071\pi$
$350$	0	0
$351$	6.23607	0.332857
$352$	−12.5623	−0.669573
$353$	16.0344	0.853427	0.426714	−	0.904387i	$-0.359671\pi$
$353$	16.0344	0.853427	0.426714	+	0.904387i	$0.359671\pi$
$354$	0.618034	0.0328481
$355$	0	0
$356$	22.3262	1.18329
$357$	−4.41641	−0.233741
$358$	0.326238	0.0172422
$359$	−21.8885	−1.15523	−0.577617	−	0.816308i	$-0.696017\pi$
$359$	−21.8885	−1.15523	−0.577617	+	0.816308i	$0.696017\pi$
$360$	0	0
$361$	−9.45085	−0.497413
$362$	−13.7639	−0.723416
$363$	−6.00000	−0.314918
$364$	−24.0344	−1.25975
$365$	0	0
$366$	1.94427	0.101629
$367$	−29.8328	−1.55726	−0.778630	−	0.627483i	$-0.784085\pi$
$367$	−29.8328	−1.55726	−0.778630	+	0.627483i	$0.784085\pi$
$368$	8.56231	0.446341
$369$	8.09017	0.421157
$370$	0	0
$371$	−14.8541	−0.771187
$372$	17.0902	0.886084
$373$	−34.3262	−1.77735	−0.888673	−	0.458542i	$-0.848372\pi$
$373$	−34.3262	−1.77735	−0.888673	+	0.458542i	$0.848372\pi$
$374$	2.56231	0.132494
$375$	0	0
$376$	24.2705	1.25166
$377$	39.7984	2.04972
$378$	−1.47214	−0.0757185
$379$	22.4164	1.15145	0.575727	−	0.817642i	$-0.304719\pi$
$379$	22.4164	1.15145	0.575727	+	0.817642i	$0.304719\pi$
$380$	0	0
$381$	−15.9443	−0.816850
$382$	−10.1459	−0.519109
$383$	18.2361	0.931820	0.465910	−	0.884832i	$-0.345727\pi$
$383$	18.2361	0.931820	0.465910	+	0.884832i	$0.345727\pi$
$384$	11.3820	0.580834
$385$	0	0
$386$	4.94427	0.251657
$387$	8.70820	0.442663
$388$	4.85410	0.246430
$389$	32.4721	1.64640	0.823201	−	0.567750i	$-0.192186\pi$
$389$	32.4721	1.64640	0.823201	+	0.567750i	$0.192186\pi$
$390$	0	0
$391$	−8.56231	−0.433014
$392$	−2.96556	−0.149783
$393$	−20.6525	−1.04178
$394$	−12.7639	−0.643038
$395$	0	0
$396$	−3.61803	−0.181813
$397$	−3.00000	−0.150566	−0.0752828	−	0.997162i	$-0.523986\pi$
$397$	−3.00000	−0.150566	−0.0752828	+	0.997162i	$0.523986\pi$
$398$	10.2361	0.513088
$399$	7.36068	0.368495
$400$	0	0
$401$	−17.0902	−0.853442	−0.426721	−	0.904383i	$-0.640331\pi$
$401$	−17.0902	−0.853442	−0.426721	+	0.904383i	$0.640331\pi$
$402$	6.61803	0.330078
$403$	−65.8673	−3.28108
$404$	−6.00000	−0.298511
$405$	0	0
$406$	−9.39512	−0.466272
$407$	0.326238	0.0161710
$408$	−4.14590	−0.205253
$409$	−10.5836	−0.523325	−0.261662	−	0.965159i	$-0.584271\pi$
$409$	−10.5836	−0.523325	−0.261662	+	0.965159i	$0.584271\pi$
$410$	0	0
$411$	−12.2361	−0.603561
$412$	−5.23607	−0.257963
$413$	−2.38197	−0.117209
$414$	−2.85410	−0.140271
$415$	0	0
$416$	−35.0344	−1.71770
$417$	11.7639	0.576082
$418$	−4.27051	−0.208877
$419$	−31.3050	−1.52935	−0.764673	−	0.644418i	$-0.777100\pi$
$419$	−31.3050	−1.52935	−0.764673	+	0.644418i	$0.777100\pi$
$420$	0	0
$421$	1.00000	0.0487370	0.0243685	−	0.999703i	$-0.492242\pi$
$421$	1.00000	0.0487370	0.0243685	+	0.999703i	$0.492242\pi$
$422$	1.32624	0.0645603
$423$	10.8541	0.527744
$424$	−13.9443	−0.677194
$425$	0	0
$426$	4.90983	0.237882
$427$	−7.49342	−0.362632
$428$	1.47214	0.0711584
$429$	13.9443	0.673236
$430$	0	0
$431$	12.3820	0.596418	0.298209	−	0.954501i	$-0.403611\pi$
$431$	12.3820	0.596418	0.298209	+	0.954501i	$0.403611\pi$
$432$	1.85410	0.0892055
$433$	−3.67376	−0.176550	−0.0882749	−	0.996096i	$-0.528135\pi$
$433$	−3.67376	−0.176550	−0.0882749	+	0.996096i	$0.528135\pi$
$434$	15.5492	0.746383
$435$	0	0
$436$	6.70820	0.321265
$437$	14.2705	0.682651
$438$	0.527864	0.0252223
$439$	−34.6180	−1.65223	−0.826114	−	0.563503i	$-0.809453\pi$
$439$	−34.6180	−1.65223	−0.826114	+	0.563503i	$0.809453\pi$
$440$	0	0
$441$	−1.32624	−0.0631542
$442$	7.14590	0.339896
$443$	−3.38197	−0.160682	−0.0803410	−	0.996767i	$-0.525601\pi$
$443$	−3.38197	−0.160682	−0.0803410	+	0.996767i	$0.525601\pi$
$444$	−0.236068	−0.0112033
$445$	0	0
$446$	−5.88854	−0.278831
$447$	19.0902	0.902934
$448$	−0.562306	−0.0265665
$449$	6.88854	0.325090	0.162545	−	0.986701i	$-0.448030\pi$
$449$	6.88854	0.325090	0.162545	+	0.986701i	$0.448030\pi$
$450$	0	0
$451$	18.0902	0.851833
$452$	−14.5623	−0.684953
$453$	2.56231	0.120388
$454$	−13.6869	−0.642359
$455$	0	0
$456$	6.90983	0.323582
$457$	−33.6869	−1.57581	−0.787904	−	0.615798i	$-0.788834\pi$
$457$	−33.6869	−1.57581	−0.787904	+	0.615798i	$0.788834\pi$
$458$	5.65248	0.264123
$459$	−1.85410	−0.0865421
$460$	0	0
$461$	23.7426	1.10581	0.552903	−	0.833246i	$-0.313520\pi$
$461$	23.7426	1.10581	0.552903	+	0.833246i	$0.313520\pi$
$462$	−3.29180	−0.153148
$463$	−4.14590	−0.192676	−0.0963381	−	0.995349i	$-0.530713\pi$
$463$	−4.14590	−0.192676	−0.0963381	+	0.995349i	$0.530713\pi$
$464$	11.8328	0.549325
$465$	0	0
$466$	−5.12461	−0.237393
$467$	−14.8328	−0.686381	−0.343190	−	0.939266i	$-0.611508\pi$
$467$	−14.8328	−0.686381	−0.343190	+	0.939266i	$0.611508\pi$
$468$	−10.0902	−0.466418
$469$	−25.5066	−1.17778
$470$	0	0
$471$	−9.00000	−0.414698
$472$	−2.23607	−0.102923
$473$	19.4721	0.895330
$474$	1.85410	0.0851617
$475$	0	0
$476$	7.14590	0.327532
$477$	−6.23607	−0.285530
$478$	0.909830	0.0416147
$479$	−14.9098	−0.681248	−0.340624	−	0.940200i	$-0.610638\pi$
$479$	−14.9098	−0.681248	−0.340624	+	0.940200i	$0.610638\pi$
$480$	0	0
$481$	0.909830	0.0414847
$482$	14.4721	0.659188
$483$	11.0000	0.500517
$484$	9.70820	0.441282
$485$	0	0
$486$	−0.618034	−0.0280346
$487$	5.74265	0.260224	0.130112	−	0.991499i	$-0.458466\pi$
$487$	5.74265	0.260224	0.130112	+	0.991499i	$0.458466\pi$
$488$	−7.03444	−0.318434
$489$	18.5623	0.839416
$490$	0	0
$491$	−11.5066	−0.519285	−0.259642	−	0.965705i	$-0.583605\pi$
$491$	−11.5066	−0.519285	−0.259642	+	0.965705i	$0.583605\pi$
$492$	−13.0902	−0.590150
$493$	−11.8328	−0.532923
$494$	−11.9098	−0.535849
$495$	0	0
$496$	−19.5836	−0.879329
$497$	−18.9230	−0.848812
$498$	−1.00000	−0.0448111
$499$	−14.4164	−0.645367	−0.322684	−	0.946507i	$-0.604585\pi$
$499$	−14.4164	−0.645367	−0.322684	+	0.946507i	$0.604585\pi$
$500$	0	0
$501$	−7.03444	−0.314276
$502$	−9.38197	−0.418738
$503$	−3.79837	−0.169361	−0.0846806	−	0.996408i	$-0.526987\pi$
$503$	−3.79837	−0.169361	−0.0846806	+	0.996408i	$0.526987\pi$
$504$	5.32624	0.237249
$505$	0	0
$506$	−6.38197	−0.283713
$507$	25.8885	1.14975
$508$	25.7984	1.14462
$509$	34.9230	1.54793	0.773967	−	0.633226i	$-0.218270\pi$
$509$	34.9230	1.54793	0.773967	+	0.633226i	$0.218270\pi$
$510$	0	0
$511$	−2.03444	−0.0899984
$512$	−18.7082	−0.826794
$513$	3.09017	0.136434
$514$	−12.0000	−0.529297
$515$	0	0
$516$	−14.0902	−0.620285
$517$	24.2705	1.06742
$518$	−0.214782	−0.00943697
$519$	−12.3820	−0.543508
$520$	0	0
$521$	23.5066	1.02984	0.514921	−	0.857238i	$-0.327821\pi$
$521$	23.5066	1.02984	0.514921	+	0.857238i	$0.327821\pi$
$522$	−3.94427	−0.172636
$523$	−14.0557	−0.614614	−0.307307	−	0.951610i	$-0.599428\pi$
$523$	−14.0557	−0.614614	−0.307307	+	0.951610i	$0.599428\pi$
$524$	33.4164	1.45980
$525$	0	0
$526$	1.00000	0.0436021
$527$	19.5836	0.853075
$528$	4.14590	0.180427
$529$	−1.67376	−0.0727723
$530$	0	0
$531$	−1.00000	−0.0433963
$532$	−11.9098	−0.516357
$533$	50.4508	2.18527
$534$	8.52786	0.369037
$535$	0	0
$536$	−23.9443	−1.03424
$537$	−0.527864	−0.0227790
$538$	7.09017	0.305679
$539$	−2.96556	−0.127736
$540$	0	0
$541$	−26.1246	−1.12318	−0.561592	−	0.827414i	$-0.689811\pi$
$541$	−26.1246	−1.12318	−0.561592	+	0.827414i	$0.689811\pi$
$542$	4.79837	0.206108
$543$	22.2705	0.955719
$544$	10.4164	0.446600
$545$	0	0
$546$	−9.18034	−0.392882
$547$	−12.4377	−0.531797	−0.265899	−	0.964001i	$-0.585669\pi$
$547$	−12.4377	−0.531797	−0.265899	+	0.964001i	$0.585669\pi$
$548$	19.7984	0.845745
$549$	−3.14590	−0.134264
$550$	0	0
$551$	19.7214	0.840158
$552$	10.3262	0.439514
$553$	−7.14590	−0.303874
$554$	−3.38197	−0.143686
$555$	0	0
$556$	−19.0344	−0.807240
$557$	25.4164	1.07693	0.538464	−	0.842649i	$-0.319005\pi$
$557$	25.4164	1.07693	0.538464	+	0.842649i	$0.319005\pi$
$558$	6.52786	0.276347
$559$	54.3050	2.29685
$560$	0	0
$561$	−4.14590	−0.175040
$562$	−6.00000	−0.253095
$563$	20.5967	0.868049	0.434025	−	0.900901i	$-0.357093\pi$
$563$	20.5967	0.868049	0.434025	+	0.900901i	$0.357093\pi$
$564$	−17.5623	−0.739506
$565$	0	0
$566$	−14.3820	−0.604519
$567$	2.38197	0.100033
$568$	−17.7639	−0.745358
$569$	−11.5066	−0.482381	−0.241190	−	0.970478i	$-0.577538\pi$
$569$	−11.5066	−0.482381	−0.241190	+	0.970478i	$0.577538\pi$
$570$	0	0
$571$	1.58359	0.0662713	0.0331356	−	0.999451i	$-0.489451\pi$
$571$	1.58359	0.0662713	0.0331356	+	0.999451i	$0.489451\pi$
$572$	−22.5623	−0.943377
$573$	16.4164	0.685805
$574$	−11.9098	−0.497107
$575$	0	0
$576$	−0.236068	−0.00983617
$577$	−12.5279	−0.521542	−0.260771	−	0.965401i	$-0.583977\pi$
$577$	−12.5279	−0.521542	−0.260771	+	0.965401i	$0.583977\pi$
$578$	8.38197	0.348644
$579$	−8.00000	−0.332469
$580$	0	0
$581$	3.85410	0.159895
$582$	1.85410	0.0768550
$583$	−13.9443	−0.577513
$584$	−1.90983	−0.0790293
$585$	0	0
$586$	11.9787	0.494836
$587$	15.3607	0.634003	0.317002	−	0.948425i	$-0.397324\pi$
$587$	15.3607	0.634003	0.317002	+	0.948425i	$0.397324\pi$
$588$	2.14590	0.0884953
$589$	−32.6393	−1.34488
$590$	0	0
$591$	20.6525	0.849529
$592$	0.270510	0.0111179
$593$	−16.0902	−0.660744	−0.330372	−	0.943851i	$-0.607174\pi$
$593$	−16.0902	−0.660744	−0.330372	+	0.943851i	$0.607174\pi$
$594$	−1.38197	−0.0567028
$595$	0	0
$596$	−30.8885	−1.26524
$597$	−16.5623	−0.677850
$598$	−17.7984	−0.727830
$599$	2.65248	0.108377	0.0541886	−	0.998531i	$-0.482743\pi$
$599$	2.65248	0.108377	0.0541886	+	0.998531i	$0.482743\pi$
$600$	0	0
$601$	23.3820	0.953770	0.476885	−	0.878966i	$-0.341766\pi$
$601$	23.3820	0.953770	0.476885	+	0.878966i	$0.341766\pi$
$602$	−12.8197	−0.522490
$603$	−10.7082	−0.436072
$604$	−4.14590	−0.168694
$605$	0	0
$606$	−2.29180	−0.0930979
$607$	13.6525	0.554137	0.277068	−	0.960850i	$-0.410637\pi$
$607$	13.6525	0.554137	0.277068	+	0.960850i	$0.410637\pi$
$608$	−17.3607	−0.704069
$609$	15.2016	0.616001
$610$	0	0
$611$	67.6869	2.73832
$612$	3.00000	0.121268
$613$	38.6525	1.56116	0.780579	−	0.625057i	$-0.214924\pi$
$613$	38.6525	1.56116	0.780579	+	0.625057i	$0.214924\pi$
$614$	−16.0000	−0.645707
$615$	0	0
$616$	11.9098	0.479861
$617$	−22.1459	−0.891560	−0.445780	−	0.895142i	$-0.647074\pi$
$617$	−22.1459	−0.891560	−0.445780	+	0.895142i	$0.647074\pi$
$618$	−2.00000	−0.0804518
$619$	12.1246	0.487329	0.243665	−	0.969860i	$-0.421650\pi$
$619$	12.1246	0.487329	0.243665	+	0.969860i	$0.421650\pi$
$620$	0	0
$621$	4.61803	0.185315
$622$	−16.9656	−0.680257
$623$	−32.8673	−1.31680
$624$	11.5623	0.462863
$625$	0	0
$626$	−12.8541	−0.513753
$627$	6.90983	0.275952
$628$	14.5623	0.581099
$629$	−0.270510	−0.0107859
$630$	0	0
$631$	−40.4164	−1.60895	−0.804476	−	0.593985i	$-0.797554\pi$
$631$	−40.4164	−1.60895	−0.804476	+	0.593985i	$0.797554\pi$
$632$	−6.70820	−0.266838
$633$	−2.14590	−0.0852918
$634$	−18.0344	−0.716239
$635$	0	0
$636$	10.0902	0.400101
$637$	−8.27051	−0.327690
$638$	−8.81966	−0.349174
$639$	−7.94427	−0.314271
$640$	0	0
$641$	−27.9443	−1.10373	−0.551866	−	0.833933i	$-0.686084\pi$
$641$	−27.9443	−1.10373	−0.551866	+	0.833933i	$0.686084\pi$
$642$	0.562306	0.0221924
$643$	19.8541	0.782969	0.391485	−	0.920185i	$-0.371962\pi$
$643$	19.8541	0.782969	0.391485	+	0.920185i	$0.371962\pi$
$644$	−17.7984	−0.701354
$645$	0	0
$646$	3.54102	0.139320
$647$	−44.9443	−1.76694	−0.883471	−	0.468486i	$-0.844800\pi$
$647$	−44.9443	−1.76694	−0.883471	+	0.468486i	$0.844800\pi$
$648$	2.23607	0.0878410
$649$	−2.23607	−0.0877733
$650$	0	0
$651$	−25.1591	−0.986061
$652$	−30.0344	−1.17624
$653$	−40.0902	−1.56885	−0.784425	−	0.620224i	$-0.787042\pi$
$653$	−40.0902	−1.56885	−0.784425	+	0.620224i	$0.787042\pi$
$654$	2.56231	0.100194
$655$	0	0
$656$	15.0000	0.585652
$657$	−0.854102	−0.0333217
$658$	−15.9787	−0.622915
$659$	1.85410	0.0722256	0.0361128	−	0.999348i	$-0.488502\pi$
$659$	1.85410	0.0722256	0.0361128	+	0.999348i	$0.488502\pi$
$660$	0	0
$661$	30.7426	1.19575	0.597875	−	0.801589i	$-0.296012\pi$
$661$	30.7426	1.19575	0.597875	+	0.801589i	$0.296012\pi$
$662$	6.87539	0.267220
$663$	−11.5623	−0.449043
$664$	3.61803	0.140407
$665$	0	0
$666$	−0.0901699	−0.00349401
$667$	29.4721	1.14117
$668$	11.3820	0.440381
$669$	9.52786	0.368369
$670$	0	0
$671$	−7.03444	−0.271562
$672$	−13.3820	−0.516221
$673$	8.47214	0.326577	0.163288	−	0.986578i	$-0.447790\pi$
$673$	8.47214	0.326577	0.163288	+	0.986578i	$0.447790\pi$
$674$	−9.21478	−0.354940
$675$	0	0
$676$	−41.8885	−1.61110
$677$	27.8197	1.06920	0.534598	−	0.845106i	$-0.320463\pi$
$677$	27.8197	1.06920	0.534598	+	0.845106i	$0.320463\pi$
$678$	−5.56231	−0.213619
$679$	−7.14590	−0.274234
$680$	0	0
$681$	22.1459	0.848633
$682$	14.5967	0.558938
$683$	−39.0000	−1.49229	−0.746147	−	0.665782i	$-0.768098\pi$
$683$	−39.0000	−1.49229	−0.746147	+	0.665782i	$0.768098\pi$
$684$	−5.00000	−0.191180
$685$	0	0
$686$	12.2574	0.467988
$687$	−9.14590	−0.348938
$688$	16.1459	0.615557
$689$	−38.8885	−1.48154
$690$	0	0
$691$	35.1246	1.33620	0.668102	−	0.744070i	$-0.267107\pi$
$691$	35.1246	1.33620	0.668102	+	0.744070i	$0.267107\pi$
$692$	20.0344	0.761595
$693$	5.32624	0.202327
$694$	19.1803	0.728076
$695$	0	0
$696$	14.2705	0.540922
$697$	−15.0000	−0.568166
$698$	16.6869	0.631609
$699$	8.29180	0.313625
$700$	0	0
$701$	18.2016	0.687466	0.343733	−	0.939067i	$-0.388308\pi$
$701$	18.2016	0.687466	0.343733	+	0.939067i	$0.388308\pi$
$702$	−3.85410	−0.145464
$703$	0.450850	0.0170041
$704$	−0.527864	−0.0198946
$705$	0	0
$706$	−9.90983	−0.372961
$707$	8.83282	0.332192
$708$	1.61803	0.0608094
$709$	−27.7082	−1.04060	−0.520302	−	0.853983i	$-0.674181\pi$
$709$	−27.7082	−1.04060	−0.520302	+	0.853983i	$0.674181\pi$
$710$	0	0
$711$	−3.00000	−0.112509
$712$	−30.8541	−1.15631
$713$	−48.7771	−1.82672
$714$	2.72949	0.102149
$715$	0	0
$716$	0.854102	0.0319193
$717$	−1.47214	−0.0549779
$718$	13.5279	0.504855
$719$	14.6180	0.545161	0.272580	−	0.962133i	$-0.412123\pi$
$719$	14.6180	0.545161	0.272580	+	0.962133i	$0.412123\pi$
$720$	0	0
$721$	7.70820	0.287069
$722$	5.84095	0.217378
$723$	−23.4164	−0.870866
$724$	−36.0344	−1.33921
$725$	0	0
$726$	3.70820	0.137624
$727$	−26.0000	−0.964287	−0.482143	−	0.876092i	$-0.660142\pi$
$727$	−26.0000	−0.964287	−0.482143	+	0.876092i	$0.660142\pi$
$728$	33.2148	1.23102
$729$	1.00000	0.0370370
$730$	0	0
$731$	−16.1459	−0.597178
$732$	5.09017	0.188138
$733$	−20.5279	−0.758214	−0.379107	−	0.925353i	$-0.623769\pi$
$733$	−20.5279	−0.758214	−0.379107	+	0.925353i	$0.623769\pi$
$734$	18.4377	0.680548
$735$	0	0
$736$	−25.9443	−0.956319
$737$	−23.9443	−0.881999
$738$	−5.00000	−0.184053
$739$	48.1033	1.76951	0.884755	−	0.466057i	$-0.154326\pi$
$739$	48.1033	1.76951	0.884755	+	0.466057i	$0.154326\pi$
$740$	0	0
$741$	19.2705	0.707920
$742$	9.18034	0.337021
$743$	24.6180	0.903148	0.451574	−	0.892234i	$-0.350863\pi$
$743$	24.6180	0.903148	0.451574	+	0.892234i	$0.350863\pi$
$744$	−23.6180	−0.865879
$745$	0	0
$746$	21.2148	0.776728
$747$	1.61803	0.0592008
$748$	6.70820	0.245276
$749$	−2.16718	−0.0791872
$750$	0	0
$751$	−4.87539	−0.177905	−0.0889527	−	0.996036i	$-0.528352\pi$
$751$	−4.87539	−0.177905	−0.0889527	+	0.996036i	$0.528352\pi$
$752$	20.1246	0.733869
$753$	15.1803	0.553202
$754$	−24.5967	−0.895761
$755$	0	0
$756$	−3.85410	−0.140172
$757$	41.3820	1.50405	0.752027	−	0.659133i	$-0.229076\pi$
$757$	41.3820	1.50405	0.752027	+	0.659133i	$0.229076\pi$
$758$	−13.8541	−0.503204
$759$	10.3262	0.374819
$760$	0	0
$761$	30.7082	1.11317	0.556586	−	0.830790i	$-0.312111\pi$
$761$	30.7082	1.11317	0.556586	+	0.830790i	$0.312111\pi$
$762$	9.85410	0.356976
$763$	−9.87539	−0.357513
$764$	−26.5623	−0.960991
$765$	0	0
$766$	−11.2705	−0.407220
$767$	−6.23607	−0.225171
$768$	−6.56231	−0.236797
$769$	10.9787	0.395903	0.197951	−	0.980212i	$-0.436571\pi$
$769$	10.9787	0.395903	0.197951	+	0.980212i	$0.436571\pi$
$770$	0	0
$771$	19.4164	0.699265
$772$	12.9443	0.465875
$773$	−28.6525	−1.03056	−0.515279	−	0.857023i	$-0.672312\pi$
$773$	−28.6525	−1.03056	−0.515279	+	0.857023i	$0.672312\pi$
$774$	−5.38197	−0.193451
$775$	0	0
$776$	−6.70820	−0.240810
$777$	0.347524	0.0124674
$778$	−20.0689	−0.719504
$779$	25.0000	0.895718
$780$	0	0
$781$	−17.7639	−0.635643
$782$	5.29180	0.189234
$783$	6.38197	0.228073
$784$	−2.45898	−0.0878207
$785$	0	0
$786$	12.7639	0.455274
$787$	4.29180	0.152986	0.0764930	−	0.997070i	$-0.475628\pi$
$787$	4.29180	0.152986	0.0764930	+	0.997070i	$0.475628\pi$
$788$	−33.4164	−1.19041
$789$	−1.61803	−0.0576035
$790$	0	0
$791$	21.4377	0.762237
$792$	5.00000	0.177667
$793$	−19.6180	−0.696657
$794$	1.85410	0.0657996
$795$	0	0
$796$	26.7984	0.949843
$797$	1.23607	0.0437838	0.0218919	−	0.999760i	$-0.493031\pi$
$797$	1.23607	0.0437838	0.0218919	+	0.999760i	$0.493031\pi$
$798$	−4.54915	−0.161038
$799$	−20.1246	−0.711958
$800$	0	0
$801$	−13.7984	−0.487542
$802$	10.5623	0.372968
$803$	−1.90983	−0.0673964
$804$	17.3262	0.611049
$805$	0	0
$806$	40.7082	1.43389
$807$	−11.4721	−0.403838
$808$	8.29180	0.291704
$809$	7.67376	0.269795	0.134898	−	0.990860i	$-0.456929\pi$
$809$	7.67376	0.269795	0.134898	+	0.990860i	$0.456929\pi$
$810$	0	0
$811$	14.6525	0.514518	0.257259	−	0.966342i	$-0.417181\pi$
$811$	14.6525	0.514518	0.257259	+	0.966342i	$0.417181\pi$
$812$	−24.5967	−0.863177
$813$	−7.76393	−0.272293
$814$	−0.201626	−0.00706699
$815$	0	0
$816$	−3.43769	−0.120343
$817$	26.9098	0.941456
$818$	6.54102	0.228701
$819$	14.8541	0.519044
$820$	0	0
$821$	−40.9230	−1.42822	−0.714111	−	0.700032i	$-0.753169\pi$
$821$	−40.9230	−1.42822	−0.714111	+	0.700032i	$0.753169\pi$
$822$	7.56231	0.263766
$823$	−11.2918	−0.393607	−0.196804	−	0.980443i	$-0.563056\pi$
$823$	−11.2918	−0.393607	−0.196804	+	0.980443i	$0.563056\pi$
$824$	7.23607	0.252080
$825$	0	0
$826$	1.47214	0.0512222
$827$	2.81966	0.0980492	0.0490246	−	0.998798i	$-0.484389\pi$
$827$	2.81966	0.0980492	0.0490246	+	0.998798i	$0.484389\pi$
$828$	−7.47214	−0.259675
$829$	−45.6869	−1.58677	−0.793386	−	0.608719i	$-0.791684\pi$
$829$	−45.6869	−1.58677	−0.793386	+	0.608719i	$0.791684\pi$
$830$	0	0
$831$	5.47214	0.189826
$832$	−1.47214	−0.0510371
$833$	2.45898	0.0851986
$834$	−7.27051	−0.251757
$835$	0	0
$836$	−11.1803	−0.386680
$837$	−10.5623	−0.365087
$838$	19.3475	0.668349
$839$	−24.8197	−0.856870	−0.428435	−	0.903573i	$-0.640935\pi$
$839$	−24.8197	−0.856870	−0.428435	+	0.903573i	$0.640935\pi$
$840$	0	0
$841$	11.7295	0.404465
$842$	−0.618034	−0.0212989
$843$	9.70820	0.334368
$844$	3.47214	0.119516
$845$	0	0
$846$	−6.70820	−0.230633
$847$	−14.2918	−0.491072
$848$	−11.5623	−0.397051
$849$	23.2705	0.798642
$850$	0	0
$851$	0.673762	0.0230963
$852$	12.8541	0.440374
$853$	3.96556	0.135778	0.0678891	−	0.997693i	$-0.478374\pi$
$853$	3.96556	0.135778	0.0678891	+	0.997693i	$0.478374\pi$
$854$	4.63119	0.158476
$855$	0	0
$856$	−2.03444	−0.0695358
$857$	28.7771	0.983007	0.491503	−	0.870876i	$-0.336448\pi$
$857$	28.7771	0.983007	0.491503	+	0.870876i	$0.336448\pi$
$858$	−8.61803	−0.294215
$859$	15.5279	0.529804	0.264902	−	0.964275i	$-0.414660\pi$
$859$	15.5279	0.529804	0.264902	+	0.964275i	$0.414660\pi$
$860$	0	0
$861$	19.2705	0.656737
$862$	−7.65248	−0.260644
$863$	−19.2016	−0.653631	−0.326815	−	0.945088i	$-0.605976\pi$
$863$	−19.2016	−0.653631	−0.326815	+	0.945088i	$0.605976\pi$
$864$	−5.61803	−0.191129
$865$	0	0
$866$	2.27051	0.0771551
$867$	−13.5623	−0.460600
$868$	40.7082	1.38173
$869$	−6.70820	−0.227560
$870$	0	0
$871$	−66.7771	−2.26266
$872$	−9.27051	−0.313939
$873$	−3.00000	−0.101535
$874$	−8.81966	−0.298329
$875$	0	0
$876$	1.38197	0.0466923
$877$	16.1115	0.544045	0.272023	−	0.962291i	$-0.412307\pi$
$877$	16.1115	0.544045	0.272023	+	0.962291i	$0.412307\pi$
$878$	21.3951	0.722050
$879$	−19.3820	−0.653737
$880$	0	0
$881$	−15.7082	−0.529223	−0.264611	−	0.964355i	$-0.585244\pi$
$881$	−15.7082	−0.529223	−0.264611	+	0.964355i	$0.585244\pi$
$882$	0.819660	0.0275994
$883$	23.4164	0.788025	0.394012	−	0.919105i	$-0.371087\pi$
$883$	23.4164	0.788025	0.394012	+	0.919105i	$0.371087\pi$
$884$	18.7082	0.629225
$885$	0	0
$886$	2.09017	0.0702206
$887$	22.5279	0.756412	0.378206	−	0.925722i	$-0.376541\pi$
$887$	22.5279	0.756412	0.378206	+	0.925722i	$0.376541\pi$
$888$	0.326238	0.0109478
$889$	−37.9787	−1.27377
$890$	0	0
$891$	2.23607	0.0749111
$892$	−15.4164	−0.516180
$893$	33.5410	1.12241
$894$	−11.7984	−0.394597
$895$	0	0
$896$	27.1115	0.905730
$897$	28.7984	0.961550
$898$	−4.25735	−0.142070
$899$	−67.4083	−2.24819
$900$	0	0
$901$	11.5623	0.385196
$902$	−11.1803	−0.372265
$903$	20.7426	0.690272
$904$	20.1246	0.669335
$905$	0	0
$906$	−1.58359	−0.0526113
$907$	19.9443	0.662239	0.331119	−	0.943589i	$-0.392574\pi$
$907$	19.9443	0.662239	0.331119	+	0.943589i	$0.392574\pi$
$908$	−35.8328	−1.18915
$909$	3.70820	0.122993
$910$	0	0
$911$	−45.1033	−1.49434	−0.747170	−	0.664633i	$-0.768588\pi$
$911$	−45.1033	−1.49434	−0.747170	+	0.664633i	$0.768588\pi$
$912$	5.72949	0.189722
$913$	3.61803	0.119739
$914$	20.8197	0.688653
$915$	0	0
$916$	14.7984	0.488952
$917$	−49.1935	−1.62451
$918$	1.14590	0.0378203
$919$	−13.5967	−0.448515	−0.224258	−	0.974530i	$-0.571996\pi$
$919$	−13.5967	−0.448515	−0.224258	+	0.974530i	$0.571996\pi$
$920$	0	0
$921$	25.8885	0.853057
$922$	−14.6738	−0.483255
$923$	−49.5410	−1.63066
$924$	−8.61803	−0.283513
$925$	0	0
$926$	2.56231	0.0842026
$927$	3.23607	0.106286
$928$	−35.8541	−1.17697
$929$	−48.7082	−1.59806	−0.799032	−	0.601288i	$-0.794654\pi$
$929$	−48.7082	−1.59806	−0.799032	+	0.601288i	$0.794654\pi$
$930$	0	0
$931$	−4.09830	−0.134316
$932$	−13.4164	−0.439469
$933$	27.4508	0.898700
$934$	9.16718	0.299959
$935$	0	0
$936$	13.9443	0.455783
$937$	−51.7214	−1.68966	−0.844832	−	0.535032i	$-0.820299\pi$
$937$	−51.7214	−1.68966	−0.844832	+	0.535032i	$0.820299\pi$
$938$	15.7639	0.514711
$939$	20.7984	0.678729
$940$	0	0
$941$	40.3050	1.31390	0.656952	−	0.753932i	$-0.271845\pi$
$941$	40.3050	1.31390	0.656952	+	0.753932i	$0.271845\pi$
$942$	5.56231	0.181230
$943$	37.3607	1.21663
$944$	−1.85410	−0.0603459
$945$	0	0
$946$	−12.0344	−0.391273
$947$	−25.0344	−0.813510	−0.406755	−	0.913537i	$-0.633340\pi$
$947$	−25.0344	−0.813510	−0.406755	+	0.913537i	$0.633340\pi$
$948$	4.85410	0.157654
$949$	−5.32624	−0.172897
$950$	0	0
$951$	29.1803	0.946237
$952$	−9.87539	−0.320063
$953$	−4.94427	−0.160161	−0.0800803	−	0.996788i	$-0.525518\pi$
$953$	−4.94427	−0.160161	−0.0800803	+	0.996788i	$0.525518\pi$
$954$	3.85410	0.124781
$955$	0	0
$956$	2.38197	0.0770383
$957$	14.2705	0.461300
$958$	9.21478	0.297716
$959$	−29.1459	−0.941170
$960$	0	0
$961$	80.5623	2.59878
$962$	−0.562306	−0.0181295
$963$	−0.909830	−0.0293189
$964$	37.8885	1.22031
$965$	0	0
$966$	−6.79837	−0.218734
$967$	−15.2705	−0.491066	−0.245533	−	0.969388i	$-0.578963\pi$
$967$	−15.2705	−0.491066	−0.245533	+	0.969388i	$0.578963\pi$
$968$	−13.4164	−0.431220
$969$	−5.72949	−0.184058
$970$	0	0
$971$	46.7984	1.50183	0.750916	−	0.660398i	$-0.229612\pi$
$971$	46.7984	1.50183	0.750916	+	0.660398i	$0.229612\pi$
$972$	−1.61803	−0.0518985
$973$	28.0213	0.898321
$974$	−3.54915	−0.113722
$975$	0	0
$976$	−5.83282	−0.186704
$977$	20.8885	0.668284	0.334142	−	0.942523i	$-0.391554\pi$
$977$	20.8885	0.668284	0.334142	+	0.942523i	$0.391554\pi$
$978$	−11.4721	−0.366838
$979$	−30.8541	−0.986101
$980$	0	0
$981$	−4.14590	−0.132368
$982$	7.11146	0.226936
$983$	9.40325	0.299917	0.149959	−	0.988692i	$-0.452086\pi$
$983$	9.40325	0.299917	0.149959	+	0.988692i	$0.452086\pi$
$984$	18.0902	0.576694
$985$	0	0
$986$	7.31308	0.232896
$987$	25.8541	0.822945
$988$	−31.1803	−0.991979
$989$	40.2148	1.27876
$990$	0	0
$991$	23.7426	0.754210	0.377105	−	0.926171i	$-0.376920\pi$
$991$	23.7426	0.754210	0.377105	+	0.926171i	$0.376920\pi$
$992$	59.3394	1.88403
$993$	−11.1246	−0.353029
$994$	11.6950	0.370944
$995$	0	0
$996$	−2.61803	−0.0829556
$997$	−53.2148	−1.68533	−0.842665	−	0.538439i	$-0.819014\pi$
$997$	−53.2148	−1.68533	−0.842665	+	0.538439i	$0.819014\pi$
$998$	8.90983	0.282036
$999$	0.145898	0.00461601

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4425.2.a.t.1.1		2
5.4	even	2		177.2.a.b.1.2	✓	2
15.14	odd	2		531.2.a.b.1.1		2
20.19	odd	2		2832.2.a.o.1.1		2
35.34	odd	2		8673.2.a.k.1.2		2
60.59	even	2		8496.2.a.bb.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
177.2.a.b.1.2	✓	2	5.4	even	2
531.2.a.b.1.1		2	15.14	odd	2
2832.2.a.o.1.1		2	20.19	odd	2
4425.2.a.t.1.1		2	1.1	even	1	trivial
8496.2.a.bb.1.2		2	60.59	even	2
8673.2.a.k.1.2		2	35.34	odd	2

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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:	\( N \)	\(=\)	\( 4425 = 3 \cdot 5^{2} \cdot 59 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4425.a (trivial)

\(f(q)\)	\(=\)	\(q-0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} -0.618034 q^{6} +2.38197 q^{7} +2.23607 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-0.618034 q^{2} +1.00000 q^{3} -1.61803 q^{4} -0.618034 q^{6} +2.38197 q^{7} +2.23607 q^{8} +1.00000 q^{9} +2.23607 q^{11} -1.61803 q^{12} +6.23607 q^{13} -1.47214 q^{14} +1.85410 q^{16} -1.85410 q^{17} -0.618034 q^{18} +3.09017 q^{19} +2.38197 q^{21} -1.38197 q^{22} +4.61803 q^{23} +2.23607 q^{24} -3.85410 q^{26} +1.00000 q^{27} -3.85410 q^{28} +6.38197 q^{29} -10.5623 q^{31} -5.61803 q^{32} +2.23607 q^{33} +1.14590 q^{34} -1.61803 q^{36} +0.145898 q^{37} -1.90983 q^{38} +6.23607 q^{39} +8.09017 q^{41} -1.47214 q^{42} +8.70820 q^{43} -3.61803 q^{44} -2.85410 q^{46} +10.8541 q^{47} +1.85410 q^{48} -1.32624 q^{49} -1.85410 q^{51} -10.0902 q^{52} -6.23607 q^{53} -0.618034 q^{54} +5.32624 q^{56} +3.09017 q^{57} -3.94427 q^{58} -1.00000 q^{59} -3.14590 q^{61} +6.52786 q^{62} +2.38197 q^{63} -0.236068 q^{64} -1.38197 q^{66} -10.7082 q^{67} +3.00000 q^{68} +4.61803 q^{69} -7.94427 q^{71} +2.23607 q^{72} -0.854102 q^{73} -0.0901699 q^{74} -5.00000 q^{76} +5.32624 q^{77} -3.85410 q^{78} -3.00000 q^{79} +1.00000 q^{81} -5.00000 q^{82} +1.61803 q^{83} -3.85410 q^{84} -5.38197 q^{86} +6.38197 q^{87} +5.00000 q^{88} -13.7984 q^{89} +14.8541 q^{91} -7.47214 q^{92} -10.5623 q^{93} -6.70820 q^{94} -5.61803 q^{96} -3.00000 q^{97} +0.819660 q^{98} +2.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{2} + 2 q^{3} - q^{4} + q^{6} + 7 q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + q^2 + 2 * q^3 - q^4 + q^6 + 7 * q^7 + 2 * q^9	\( 2 q + q^{2} + 2 q^{3} - q^{4} + q^{6} + 7 q^{7} + 2 q^{9} - q^{12} + 8 q^{13} + 6 q^{14} - 3 q^{16} + 3 q^{17} + q^{18} - 5 q^{19} + 7 q^{21} - 5 q^{22} + 7 q^{23} - q^{26} + 2 q^{27} - q^{28} + 15 q^{29} - q^{31} - 9 q^{32} + 9 q^{34} - q^{36} + 7 q^{37} - 15 q^{38} + 8 q^{39} + 5 q^{41} + 6 q^{42} + 4 q^{43} - 5 q^{44} + q^{46} + 15 q^{47} - 3 q^{48} + 13 q^{49} + 3 q^{51} - 9 q^{52} - 8 q^{53} + q^{54} - 5 q^{56} - 5 q^{57} + 10 q^{58} - 2 q^{59} - 13 q^{61} + 22 q^{62} + 7 q^{63} + 4 q^{64} - 5 q^{66} - 8 q^{67} + 6 q^{68} + 7 q^{69} + 2 q^{71} + 5 q^{73} + 11 q^{74} - 10 q^{76} - 5 q^{77} - q^{78} - 6 q^{79} + 2 q^{81} - 10 q^{82} + q^{83} - q^{84} - 13 q^{86} + 15 q^{87} + 10 q^{88} - 3 q^{89} + 23 q^{91} - 6 q^{92} - q^{93} - 9 q^{96} - 6 q^{97} + 24 q^{98}+O(q^{100}) \) 2 * q + q^2 + 2 * q^3 - q^4 + q^6 + 7 * q^7 + 2 * q^9 - q^12 + 8 * q^13 + 6 * q^14 - 3 * q^16 + 3 * q^17 + q^18 - 5 * q^19 + 7 * q^21 - 5 * q^22 + 7 * q^23 - q^26 + 2 * q^27 - q^28 + 15 * q^29 - q^31 - 9 * q^32 + 9 * q^34 - q^36 + 7 * q^37 - 15 * q^38 + 8 * q^39 + 5 * q^41 + 6 * q^42 + 4 * q^43 - 5 * q^44 + q^46 + 15 * q^47 - 3 * q^48 + 13 * q^49 + 3 * q^51 - 9 * q^52 - 8 * q^53 + q^54 - 5 * q^56 - 5 * q^57 + 10 * q^58 - 2 * q^59 - 13 * q^61 + 22 * q^62 + 7 * q^63 + 4 * q^64 - 5 * q^66 - 8 * q^67 + 6 * q^68 + 7 * q^69 + 2 * q^71 + 5 * q^73 + 11 * q^74 - 10 * q^76 - 5 * q^77 - q^78 - 6 * q^79 + 2 * q^81 - 10 * q^82 + q^83 - q^84 - 13 * q^86 + 15 * q^87 + 10 * q^88 - 3 * q^89 + 23 * q^91 - 6 * q^92 - q^93 - 9 * q^96 - 6 * q^97 + 24 * q^98

Introduction

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