LMFDB - Embedded newform 9025.2.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9025,2,Mod(1,9025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("9025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9025 = 5^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9025.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:	$72.0649878242$
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:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	9025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.61803 q^{2} +2.23607 q^{3} +4.85410 q^{4} -5.85410 q^{6} +4.23607 q^{7} -7.47214 q^{8} +2.00000 q^{9} +O(q^{10})$	\(q-2.61803 q^{2} +2.23607 q^{3} +4.85410 q^{4} -5.85410 q^{6} +4.23607 q^{7} -7.47214 q^{8} +2.00000 q^{9} +5.47214 q^{11} +10.8541 q^{12} +2.00000 q^{13} -11.0902 q^{14} +9.85410 q^{16} +4.85410 q^{17} -5.23607 q^{18} +9.47214 q^{21} -14.3262 q^{22} +1.23607 q^{23} -16.7082 q^{24} -5.23607 q^{26} -2.23607 q^{27} +20.5623 q^{28} +6.00000 q^{29} -1.85410 q^{31} -10.8541 q^{32} +12.2361 q^{33} -12.7082 q^{34} +9.70820 q^{36} -4.14590 q^{37} +4.47214 q^{39} -6.38197 q^{41} -24.7984 q^{42} -10.5623 q^{43} +26.5623 q^{44} -3.23607 q^{46} +7.61803 q^{47} +22.0344 q^{48} +10.9443 q^{49} +10.8541 q^{51} +9.70820 q^{52} +6.38197 q^{53} +5.85410 q^{54} -31.6525 q^{56} -15.7082 q^{58} -2.76393 q^{59} -5.56231 q^{61} +4.85410 q^{62} +8.47214 q^{63} +8.70820 q^{64} -32.0344 q^{66} +8.70820 q^{67} +23.5623 q^{68} +2.76393 q^{69} -4.52786 q^{71} -14.9443 q^{72} +10.7082 q^{73} +10.8541 q^{74} +23.1803 q^{77} -11.7082 q^{78} +1.00000 q^{79} -11.0000 q^{81} +16.7082 q^{82} -1.09017 q^{83} +45.9787 q^{84} +27.6525 q^{86} +13.4164 q^{87} -40.8885 q^{88} +3.70820 q^{89} +8.47214 q^{91} +6.00000 q^{92} -4.14590 q^{93} -19.9443 q^{94} -24.2705 q^{96} -7.38197 q^{97} -28.6525 q^{98} +10.9443 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 3 q^{2} + 3 q^{4} - 5 q^{6} + 4 q^{7} - 6 q^{8} + 4 q^{9}+O(q^{10}) $ 2 * q - 3 * q^2 + 3 * q^4 - 5 * q^6 + 4 * q^7 - 6 * q^8 + 4 * q^9	$ 2 q - 3 q^{2} + 3 q^{4} - 5 q^{6} + 4 q^{7} - 6 q^{8} + 4 q^{9} + 2 q^{11} + 15 q^{12} + 4 q^{13} - 11 q^{14} + 13 q^{16} + 3 q^{17} - 6 q^{18} + 10 q^{21} - 13 q^{22} - 2 q^{23} - 20 q^{24} - 6 q^{26} + 21 q^{28} + 12 q^{29} + 3 q^{31} - 15 q^{32} + 20 q^{33} - 12 q^{34} + 6 q^{36} - 15 q^{37} - 15 q^{41} - 25 q^{42} - q^{43} + 33 q^{44} - 2 q^{46} + 13 q^{47} + 15 q^{48} + 4 q^{49} + 15 q^{51} + 6 q^{52} + 15 q^{53} + 5 q^{54} - 32 q^{56} - 18 q^{58} - 10 q^{59} + 9 q^{61} + 3 q^{62} + 8 q^{63} + 4 q^{64} - 35 q^{66} + 4 q^{67} + 27 q^{68} + 10 q^{69} - 18 q^{71} - 12 q^{72} + 8 q^{73} + 15 q^{74} + 24 q^{77} - 10 q^{78} + 2 q^{79} - 22 q^{81} + 20 q^{82} + 9 q^{83} + 45 q^{84} + 24 q^{86} - 46 q^{88} - 6 q^{89} + 8 q^{91} + 12 q^{92} - 15 q^{93} - 22 q^{94} - 15 q^{96} - 17 q^{97} - 26 q^{98} + 4 q^{99}+O(q^{100}) $ 2 * q - 3 * q^2 + 3 * q^4 - 5 * q^6 + 4 * q^7 - 6 * q^8 + 4 * q^9 + 2 * q^11 + 15 * q^12 + 4 * q^13 - 11 * q^14 + 13 * q^16 + 3 * q^17 - 6 * q^18 + 10 * q^21 - 13 * q^22 - 2 * q^23 - 20 * q^24 - 6 * q^26 + 21 * q^28 + 12 * q^29 + 3 * q^31 - 15 * q^32 + 20 * q^33 - 12 * q^34 + 6 * q^36 - 15 * q^37 - 15 * q^41 - 25 * q^42 - q^43 + 33 * q^44 - 2 * q^46 + 13 * q^47 + 15 * q^48 + 4 * q^49 + 15 * q^51 + 6 * q^52 + 15 * q^53 + 5 * q^54 - 32 * q^56 - 18 * q^58 - 10 * q^59 + 9 * q^61 + 3 * q^62 + 8 * q^63 + 4 * q^64 - 35 * q^66 + 4 * q^67 + 27 * q^68 + 10 * q^69 - 18 * q^71 - 12 * q^72 + 8 * q^73 + 15 * q^74 + 24 * q^77 - 10 * q^78 + 2 * q^79 - 22 * q^81 + 20 * q^82 + 9 * q^83 + 45 * q^84 + 24 * q^86 - 46 * q^88 - 6 * q^89 + 8 * q^91 + 12 * q^92 - 15 * q^93 - 22 * q^94 - 15 * q^96 - 17 * 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 $

$2$	−2.61803	−1.85123	−0.925615	−	0.378467i	$-0.876451\pi$
$2$	−2.61803	−1.85123	−0.925615	+	0.378467i	$0.876451\pi$
$3$	2.23607	1.29099	0.645497	−	0.763763i	$-0.276650\pi$
$3$	2.23607	1.29099	0.645497	+	0.763763i	$0.276650\pi$
$4$	4.85410	2.42705
$5$	0	0
$5$	0	0
$6$	−5.85410	−2.38993
$7$	4.23607	1.60108	0.800542	−	0.599277i	$-0.204545\pi$
$7$	4.23607	1.60108	0.800542	+	0.599277i	$0.204545\pi$
$8$	−7.47214	−2.64180
$9$	2.00000	0.666667
$10$	0	0
$11$	5.47214	1.64991	0.824956	−	0.565198i	$-0.191200\pi$
$11$	5.47214	1.64991	0.824956	+	0.565198i	$0.191200\pi$
$12$	10.8541	3.13331
$13$	2.00000	0.554700	0.277350	−	0.960769i	$-0.410544\pi$
$13$	2.00000	0.554700	0.277350	+	0.960769i	$0.410544\pi$
$14$	−11.0902	−2.96397
$15$	0	0
$16$	9.85410	2.46353
$17$	4.85410	1.17729	0.588646	−	0.808391i	$-0.299661\pi$
$17$	4.85410	1.17729	0.588646	+	0.808391i	$0.299661\pi$
$18$	−5.23607	−1.23415
$19$	0	0
$19$	0	0
$20$	0	0
$21$	9.47214	2.06699
$22$	−14.3262	−3.05436
$23$	1.23607	0.257738	0.128869	−	0.991662i	$-0.458865\pi$
$23$	1.23607	0.257738	0.128869	+	0.991662i	$0.458865\pi$
$24$	−16.7082	−3.41055
$25$	0	0
$26$	−5.23607	−1.02688
$27$	−2.23607	−0.430331
$28$	20.5623	3.88591
$29$	6.00000	1.11417	0.557086	−	0.830455i	$-0.311919\pi$
$29$	6.00000	1.11417	0.557086	+	0.830455i	$0.311919\pi$
$30$	0	0
$31$	−1.85410	−0.333007	−0.166503	−	0.986041i	$-0.553248\pi$
$31$	−1.85410	−0.333007	−0.166503	+	0.986041i	$0.553248\pi$
$32$	−10.8541	−1.91875
$33$	12.2361	2.13003
$34$	−12.7082	−2.17944
$35$	0	0
$36$	9.70820	1.61803
$37$	−4.14590	−0.681581	−0.340791	−	0.940139i	$-0.610695\pi$
$37$	−4.14590	−0.681581	−0.340791	+	0.940139i	$0.610695\pi$
$38$	0	0
$39$	4.47214	0.716115
$40$	0	0
$41$	−6.38197	−0.996696	−0.498348	−	0.866977i	$-0.666060\pi$
$41$	−6.38197	−0.996696	−0.498348	+	0.866977i	$0.666060\pi$
$42$	−24.7984	−3.82647
$43$	−10.5623	−1.61074	−0.805368	−	0.592775i	$-0.798032\pi$
$43$	−10.5623	−1.61074	−0.805368	+	0.592775i	$0.798032\pi$
$44$	26.5623	4.00442
$45$	0	0
$46$	−3.23607	−0.477132
$47$	7.61803	1.11120	0.555602	−	0.831448i	$-0.312488\pi$
$47$	7.61803	1.11120	0.555602	+	0.831448i	$0.312488\pi$
$48$	22.0344	3.18040
$49$	10.9443	1.56347
$50$	0	0
$51$	10.8541	1.51988
$52$	9.70820	1.34629
$53$	6.38197	0.876630	0.438315	−	0.898821i	$-0.355575\pi$
$53$	6.38197	0.876630	0.438315	+	0.898821i	$0.355575\pi$
$54$	5.85410	0.796642
$55$	0	0
$56$	−31.6525	−4.22974
$57$	0	0
$58$	−15.7082	−2.06259
$59$	−2.76393	−0.359833	−0.179917	−	0.983682i	$-0.557583\pi$
$59$	−2.76393	−0.359833	−0.179917	+	0.983682i	$0.557583\pi$
$60$	0	0
$61$	−5.56231	−0.712180	−0.356090	−	0.934452i	$-0.615890\pi$
$61$	−5.56231	−0.712180	−0.356090	+	0.934452i	$0.615890\pi$
$62$	4.85410	0.616472
$63$	8.47214	1.06739
$64$	8.70820	1.08853
$65$	0	0
$66$	−32.0344	−3.94317
$67$	8.70820	1.06388	0.531938	−	0.846783i	$-0.321464\pi$
$67$	8.70820	1.06388	0.531938	+	0.846783i	$0.321464\pi$
$68$	23.5623	2.85735
$69$	2.76393	0.332738
$70$	0	0
$71$	−4.52786	−0.537359	−0.268679	−	0.963230i	$-0.586587\pi$
$71$	−4.52786	−0.537359	−0.268679	+	0.963230i	$0.586587\pi$
$72$	−14.9443	−1.76120
$73$	10.7082	1.25330	0.626650	−	0.779301i	$-0.284425\pi$
$73$	10.7082	1.25330	0.626650	+	0.779301i	$0.284425\pi$
$74$	10.8541	1.26176
$75$	0	0
$76$	0	0
$77$	23.1803	2.64164
$78$	−11.7082	−1.32569
$79$	1.00000	0.112509	0.0562544	−	0.998416i	$-0.482084\pi$
$79$	1.00000	0.112509	0.0562544	+	0.998416i	$0.482084\pi$
$80$	0	0
$81$	−11.0000	−1.22222
$82$	16.7082	1.84511
$83$	−1.09017	−0.119662	−0.0598308	−	0.998209i	$-0.519056\pi$
$83$	−1.09017	−0.119662	−0.0598308	+	0.998209i	$0.519056\pi$
$84$	45.9787	5.01669
$85$	0	0
$86$	27.6525	2.98184
$87$	13.4164	1.43839
$88$	−40.8885	−4.35873
$89$	3.70820	0.393069	0.196534	−	0.980497i	$-0.437031\pi$
$89$	3.70820	0.393069	0.196534	+	0.980497i	$0.437031\pi$
$90$	0	0
$91$	8.47214	0.888121
$92$	6.00000	0.625543
$93$	−4.14590	−0.429910
$94$	−19.9443	−2.05709
$95$	0	0
$96$	−24.2705	−2.47710
$97$	−7.38197	−0.749525	−0.374763	−	0.927121i	$-0.622276\pi$
$97$	−7.38197	−0.749525	−0.374763	+	0.927121i	$0.622276\pi$
$98$	−28.6525	−2.89434
$99$	10.9443	1.09994
$100$	0	0
$101$	−13.0344	−1.29698	−0.648488	−	0.761225i	$-0.724598\pi$
$101$	−13.0344	−1.29698	−0.648488	+	0.761225i	$0.724598\pi$
$102$	−28.4164	−2.81364
$103$	1.23607	0.121793	0.0608967	−	0.998144i	$-0.480604\pi$
$103$	1.23607	0.121793	0.0608967	+	0.998144i	$0.480604\pi$
$104$	−14.9443	−1.46541
$105$	0	0
$106$	−16.7082	−1.62284
$107$	19.8541	1.91937	0.959684	−	0.281080i	$-0.0906927\pi$
$107$	19.8541	1.91937	0.959684	+	0.281080i	$0.0906927\pi$
$108$	−10.8541	−1.04444
$109$	7.94427	0.760923	0.380462	−	0.924797i	$-0.375765\pi$
$109$	7.94427	0.760923	0.380462	+	0.924797i	$0.375765\pi$
$110$	0	0
$111$	−9.27051	−0.879918
$112$	41.7426	3.94431
$113$	9.32624	0.877339	0.438669	−	0.898649i	$-0.355450\pi$
$113$	9.32624	0.877339	0.438669	+	0.898649i	$0.355450\pi$
$114$	0	0
$115$	0	0
$116$	29.1246	2.70415
$117$	4.00000	0.369800
$118$	7.23607	0.666134
$119$	20.5623	1.88494
$120$	0	0
$121$	18.9443	1.72221
$122$	14.5623	1.31841
$123$	−14.2705	−1.28673
$124$	−9.00000	−0.808224
$125$	0	0
$126$	−22.1803	−1.97598
$127$	−0.854102	−0.0757893	−0.0378946	−	0.999282i	$-0.512065\pi$
$127$	−0.854102	−0.0757893	−0.0378946	+	0.999282i	$0.512065\pi$
$128$	−1.09017	−0.0963583
$129$	−23.6180	−2.07945
$130$	0	0
$131$	−12.0000	−1.04844	−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000	−1.04844	−0.524222	+	0.851581i	$0.675644\pi$
$132$	59.3951	5.16968
$133$	0	0
$134$	−22.7984	−1.96948
$135$	0	0
$136$	−36.2705	−3.11017
$137$	9.23607	0.789091	0.394545	−	0.918877i	$-0.370902\pi$
$137$	9.23607	0.789091	0.394545	+	0.918877i	$0.370902\pi$
$138$	−7.23607	−0.615975
$139$	−14.0000	−1.18746	−0.593732	−	0.804663i	$-0.702346\pi$
$139$	−14.0000	−1.18746	−0.593732	+	0.804663i	$0.702346\pi$
$140$	0	0
$141$	17.0344	1.43456
$142$	11.8541	0.994774
$143$	10.9443	0.915206
$144$	19.7082	1.64235
$145$	0	0
$146$	−28.0344	−2.32015
$147$	24.4721	2.01843
$148$	−20.1246	−1.65423
$149$	−9.47214	−0.775988	−0.387994	−	0.921662i	$-0.626832\pi$
$149$	−9.47214	−0.775988	−0.387994	+	0.921662i	$0.626832\pi$
$150$	0	0
$151$	5.79837	0.471865	0.235932	−	0.971769i	$-0.424186\pi$
$151$	5.79837	0.471865	0.235932	+	0.971769i	$0.424186\pi$
$152$	0	0
$153$	9.70820	0.784862
$154$	−60.6869	−4.89029
$155$	0	0
$156$	21.7082	1.73805
$157$	−1.81966	−0.145225	−0.0726123	−	0.997360i	$-0.523134\pi$
$157$	−1.81966	−0.145225	−0.0726123	+	0.997360i	$0.523134\pi$
$158$	−2.61803	−0.208280
$159$	14.2705	1.13173
$160$	0	0
$161$	5.23607	0.412660
$162$	28.7984	2.26261
$163$	2.76393	0.216488	0.108244	−	0.994124i	$-0.465477\pi$
$163$	2.76393	0.216488	0.108244	+	0.994124i	$0.465477\pi$
$164$	−30.9787	−2.41903
$165$	0	0
$166$	2.85410	0.221521
$167$	5.23607	0.405179	0.202590	−	0.979264i	$-0.435064\pi$
$167$	5.23607	0.405179	0.202590	+	0.979264i	$0.435064\pi$
$168$	−70.7771	−5.46057
$169$	−9.00000	−0.692308
$170$	0	0
$171$	0	0
$172$	−51.2705	−3.90934
$173$	−19.6525	−1.49415	−0.747075	−	0.664740i	$-0.768542\pi$
$173$	−19.6525	−1.49415	−0.747075	+	0.664740i	$0.768542\pi$
$174$	−35.1246	−2.66279
$175$	0	0
$176$	53.9230	4.06460
$177$	−6.18034	−0.464543
$178$	−9.70820	−0.727661
$179$	11.1803	0.835658	0.417829	−	0.908526i	$-0.362791\pi$
$179$	11.1803	0.835658	0.417829	+	0.908526i	$0.362791\pi$
$180$	0	0
$181$	−18.7082	−1.39057	−0.695285	−	0.718734i	$-0.744722\pi$
$181$	−18.7082	−1.39057	−0.695285	+	0.718734i	$0.744722\pi$
$182$	−22.1803	−1.64412
$183$	−12.4377	−0.919421
$184$	−9.23607	−0.680892
$185$	0	0
$186$	10.8541	0.795861
$187$	26.5623	1.94243
$188$	36.9787	2.69695
$189$	−9.47214	−0.688997
$190$	0	0
$191$	−17.9443	−1.29840	−0.649201	−	0.760617i	$-0.724897\pi$
$191$	−17.9443	−1.29840	−0.649201	+	0.760617i	$0.724897\pi$
$192$	19.4721	1.40528
$193$	−23.2705	−1.67505	−0.837524	−	0.546401i	$-0.815998\pi$
$193$	−23.2705	−1.67505	−0.837524	+	0.546401i	$0.815998\pi$
$194$	19.3262	1.38754
$195$	0	0
$196$	53.1246	3.79462
$197$	7.47214	0.532368	0.266184	−	0.963922i	$-0.414237\pi$
$197$	7.47214	0.532368	0.266184	+	0.963922i	$0.414237\pi$
$198$	−28.6525	−2.03624
$199$	19.4164	1.37639	0.688196	−	0.725525i	$-0.258403\pi$
$199$	19.4164	1.37639	0.688196	+	0.725525i	$0.258403\pi$
$200$	0	0
$201$	19.4721	1.37346
$202$	34.1246	2.40100
$203$	25.4164	1.78388
$204$	52.6869	3.68882
$205$	0	0
$206$	−3.23607	−0.225468
$207$	2.47214	0.171825
$208$	19.7082	1.36652
$209$	0	0
$210$	0	0
$211$	−12.2361	−0.842366	−0.421183	−	0.906976i	$-0.638385\pi$
$211$	−12.2361	−0.842366	−0.421183	+	0.906976i	$0.638385\pi$
$212$	30.9787	2.12763
$213$	−10.1246	−0.693727
$214$	−51.9787	−3.55319
$215$	0	0
$216$	16.7082	1.13685
$217$	−7.85410	−0.533171
$218$	−20.7984	−1.40864
$219$	23.9443	1.61800
$220$	0	0
$221$	9.70820	0.653044
$222$	24.2705	1.62893
$223$	−18.8885	−1.26487	−0.632435	−	0.774613i	$-0.717945\pi$
$223$	−18.8885	−1.26487	−0.632435	+	0.774613i	$0.717945\pi$
$224$	−45.9787	−3.07208
$225$	0	0
$226$	−24.4164	−1.62416
$227$	−14.4721	−0.960549	−0.480275	−	0.877118i	$-0.659463\pi$
$227$	−14.4721	−0.960549	−0.480275	+	0.877118i	$0.659463\pi$
$228$	0	0
$229$	20.6525	1.36475	0.682377	−	0.731000i	$-0.260946\pi$
$229$	20.6525	1.36475	0.682377	+	0.731000i	$0.260946\pi$
$230$	0	0
$231$	51.8328	3.41035
$232$	−44.8328	−2.94342
$233$	16.5279	1.08278	0.541388	−	0.840773i	$-0.317899\pi$
$233$	16.5279	1.08278	0.541388	+	0.840773i	$0.317899\pi$
$234$	−10.4721	−0.684585
$235$	0	0
$236$	−13.4164	−0.873334
$237$	2.23607	0.145248
$238$	−53.8328	−3.48946
$239$	0.708204	0.0458099	0.0229050	−	0.999738i	$-0.492708\pi$
$239$	0.708204	0.0458099	0.0229050	+	0.999738i	$0.492708\pi$
$240$	0	0
$241$	−13.0000	−0.837404	−0.418702	−	0.908124i	$-0.637515\pi$
$241$	−13.0000	−0.837404	−0.418702	+	0.908124i	$0.637515\pi$
$242$	−49.5967	−3.18820
$243$	−17.8885	−1.14755
$244$	−27.0000	−1.72850
$245$	0	0
$246$	37.3607	2.38203
$247$	0	0
$248$	13.8541	0.879736
$249$	−2.43769	−0.154483
$250$	0	0
$251$	5.90983	0.373025	0.186513	−	0.982453i	$-0.440281\pi$
$251$	5.90983	0.373025	0.186513	+	0.982453i	$0.440281\pi$
$252$	41.1246	2.59061
$253$	6.76393	0.425245
$254$	2.23607	0.140303
$255$	0	0
$256$	−14.5623	−0.910144
$257$	−4.20163	−0.262090	−0.131045	−	0.991376i	$-0.541833\pi$
$257$	−4.20163	−0.262090	−0.131045	+	0.991376i	$0.541833\pi$
$258$	61.8328	3.84954
$259$	−17.5623	−1.09127
$260$	0	0
$261$	12.0000	0.742781
$262$	31.4164	1.94091
$263$	22.6180	1.39469	0.697344	−	0.716737i	$-0.254365\pi$
$263$	22.6180	1.39469	0.697344	+	0.716737i	$0.254365\pi$
$264$	−91.4296	−5.62710
$265$	0	0
$266$	0	0
$267$	8.29180	0.507450
$268$	42.2705	2.58208
$269$	9.18034	0.559735	0.279868	−	0.960039i	$-0.409709\pi$
$269$	9.18034	0.559735	0.279868	+	0.960039i	$0.409709\pi$
$270$	0	0
$271$	−11.3262	−0.688020	−0.344010	−	0.938966i	$-0.611785\pi$
$271$	−11.3262	−0.688020	−0.344010	+	0.938966i	$0.611785\pi$
$272$	47.8328	2.90029
$273$	18.9443	1.14656
$274$	−24.1803	−1.46079
$275$	0	0
$276$	13.4164	0.807573
$277$	−12.0000	−0.721010	−0.360505	−	0.932757i	$-0.617396\pi$
$277$	−12.0000	−0.721010	−0.360505	+	0.932757i	$0.617396\pi$
$278$	36.6525	2.19827
$279$	−3.70820	−0.222004
$280$	0	0
$281$	4.61803	0.275489	0.137744	−	0.990468i	$-0.456015\pi$
$281$	4.61803	0.275489	0.137744	+	0.990468i	$0.456015\pi$
$282$	−44.5967	−2.65570
$283$	−23.7639	−1.41262	−0.706310	−	0.707903i	$-0.749641\pi$
$283$	−23.7639	−1.41262	−0.706310	+	0.707903i	$0.749641\pi$
$284$	−21.9787	−1.30420
$285$	0	0
$286$	−28.6525	−1.69426
$287$	−27.0344	−1.59579
$288$	−21.7082	−1.27917
$289$	6.56231	0.386018
$290$	0	0
$291$	−16.5066	−0.967633
$292$	51.9787	3.04182
$293$	13.4721	0.787051	0.393525	−	0.919314i	$-0.371255\pi$
$293$	13.4721	0.787051	0.393525	+	0.919314i	$0.371255\pi$
$294$	−64.0689	−3.73657
$295$	0	0
$296$	30.9787	1.80060
$297$	−12.2361	−0.710009
$298$	24.7984	1.43653
$299$	2.47214	0.142967
$300$	0	0
$301$	−44.7426	−2.57892
$302$	−15.1803	−0.873530
$303$	−29.1459	−1.67439
$304$	0	0
$305$	0	0
$306$	−25.4164	−1.45296
$307$	32.5623	1.85843	0.929214	−	0.369541i	$-0.120485\pi$
$307$	32.5623	1.85843	0.929214	+	0.369541i	$0.120485\pi$
$308$	112.520	6.41141
$309$	2.76393	0.157235
$310$	0	0
$311$	16.2361	0.920663	0.460331	−	0.887747i	$-0.347731\pi$
$311$	16.2361	0.920663	0.460331	+	0.887747i	$0.347731\pi$
$312$	−33.4164	−1.89183
$313$	19.9443	1.12732	0.563658	−	0.826008i	$-0.309393\pi$
$313$	19.9443	1.12732	0.563658	+	0.826008i	$0.309393\pi$
$314$	4.76393	0.268844
$315$	0	0
$316$	4.85410	0.273065
$317$	−10.9443	−0.614692	−0.307346	−	0.951598i	$-0.599441\pi$
$317$	−10.9443	−0.614692	−0.307346	+	0.951598i	$0.599441\pi$
$318$	−37.3607	−2.09508
$319$	32.8328	1.83828
$320$	0	0
$321$	44.3951	2.47789
$322$	−13.7082	−0.763928
$323$	0	0
$324$	−53.3951	−2.96640
$325$	0	0
$326$	−7.23607	−0.400769
$327$	17.7639	0.982348
$328$	47.6869	2.63307
$329$	32.2705	1.77913
$330$	0	0
$331$	−13.5623	−0.745452	−0.372726	−	0.927941i	$-0.621577\pi$
$331$	−13.5623	−0.745452	−0.372726	+	0.927941i	$0.621577\pi$
$332$	−5.29180	−0.290425
$333$	−8.29180	−0.454388
$334$	−13.7082	−0.750080
$335$	0	0
$336$	93.3394	5.09208
$337$	4.70820	0.256472	0.128236	−	0.991744i	$-0.459068\pi$
$337$	4.70820	0.256472	0.128236	+	0.991744i	$0.459068\pi$
$338$	23.5623	1.28162
$339$	20.8541	1.13264
$340$	0	0
$341$	−10.1459	−0.549431
$342$	0	0
$343$	16.7082	0.902158
$344$	78.9230	4.25524
$345$	0	0
$346$	51.4508	2.76601
$347$	33.4721	1.79688	0.898439	−	0.439098i	$-0.144702\pi$
$347$	33.4721	1.79688	0.898439	+	0.439098i	$0.144702\pi$
$348$	65.1246	3.49105
$349$	−27.0689	−1.44896	−0.724482	−	0.689294i	$-0.757921\pi$
$349$	−27.0689	−1.44896	−0.724482	+	0.689294i	$0.757921\pi$
$350$	0	0
$351$	−4.47214	−0.238705
$352$	−59.3951	−3.16577
$353$	−16.5279	−0.879689	−0.439845	−	0.898074i	$-0.644967\pi$
$353$	−16.5279	−0.879689	−0.439845	+	0.898074i	$0.644967\pi$
$354$	16.1803	0.859975
$355$	0	0
$356$	18.0000	0.953998
$357$	45.9787	2.43345
$358$	−29.2705	−1.54699
$359$	−22.5967	−1.19261	−0.596305	−	0.802758i	$-0.703365\pi$
$359$	−22.5967	−1.19261	−0.596305	+	0.802758i	$0.703365\pi$
$360$	0	0
$361$	0	0
$362$	48.9787	2.57426
$363$	42.3607	2.22336
$364$	41.1246	2.15552
$365$	0	0
$366$	32.5623	1.70206
$367$	12.4721	0.651040	0.325520	−	0.945535i	$-0.394461\pi$
$367$	12.4721	0.651040	0.325520	+	0.945535i	$0.394461\pi$
$368$	12.1803	0.634944
$369$	−12.7639	−0.664464
$370$	0	0
$371$	27.0344	1.40356
$372$	−20.1246	−1.04341
$373$	−13.1459	−0.680669	−0.340334	−	0.940304i	$-0.610540\pi$
$373$	−13.1459	−0.680669	−0.340334	+	0.940304i	$0.610540\pi$
$374$	−69.5410	−3.59588
$375$	0	0
$376$	−56.9230	−2.93558
$377$	12.0000	0.618031
$378$	24.7984	1.27549
$379$	−36.5410	−1.87699	−0.938493	−	0.345298i	$-0.887778\pi$
$379$	−36.5410	−1.87699	−0.938493	+	0.345298i	$0.887778\pi$
$380$	0	0
$381$	−1.90983	−0.0978436
$382$	46.9787	2.40364
$383$	22.0689	1.12767	0.563834	−	0.825888i	$-0.309326\pi$
$383$	22.0689	1.12767	0.563834	+	0.825888i	$0.309326\pi$
$384$	−2.43769	−0.124398
$385$	0	0
$386$	60.9230	3.10090
$387$	−21.1246	−1.07382
$388$	−35.8328	−1.81914
$389$	−4.47214	−0.226746	−0.113373	−	0.993552i	$-0.536166\pi$
$389$	−4.47214	−0.226746	−0.113373	+	0.993552i	$0.536166\pi$
$390$	0	0
$391$	6.00000	0.303433
$392$	−81.7771	−4.13037
$393$	−26.8328	−1.35354
$394$	−19.5623	−0.985535
$395$	0	0
$396$	53.1246	2.66961
$397$	9.03444	0.453426	0.226713	−	0.973962i	$-0.427202\pi$
$397$	9.03444	0.453426	0.226713	+	0.973962i	$0.427202\pi$
$398$	−50.8328	−2.54802
$399$	0	0
$400$	0	0
$401$	−5.88854	−0.294060	−0.147030	−	0.989132i	$-0.546971\pi$
$401$	−5.88854	−0.294060	−0.147030	+	0.989132i	$0.546971\pi$
$402$	−50.9787	−2.54259
$403$	−3.70820	−0.184719
$404$	−63.2705	−3.14783
$405$	0	0
$406$	−66.5410	−3.30238
$407$	−22.6869	−1.12455
$408$	−81.1033	−4.01521
$409$	16.4377	0.812792	0.406396	−	0.913697i	$-0.366785\pi$
$409$	16.4377	0.812792	0.406396	+	0.913697i	$0.366785\pi$
$410$	0	0
$411$	20.6525	1.01871
$412$	6.00000	0.295599
$413$	−11.7082	−0.576123
$414$	−6.47214	−0.318088
$415$	0	0
$416$	−21.7082	−1.06433
$417$	−31.3050	−1.53301
$418$	0	0
$419$	−9.52786	−0.465467	−0.232733	−	0.972541i	$-0.574767\pi$
$419$	−9.52786	−0.465467	−0.232733	+	0.972541i	$0.574767\pi$
$420$	0	0
$421$	−15.0000	−0.731055	−0.365528	−	0.930800i	$-0.619111\pi$
$421$	−15.0000	−0.731055	−0.365528	+	0.930800i	$0.619111\pi$
$422$	32.0344	1.55941
$423$	15.2361	0.740803
$424$	−47.6869	−2.31588
$425$	0	0
$426$	26.5066	1.28425
$427$	−23.5623	−1.14026
$428$	96.3738	4.65841
$429$	24.4721	1.18153
$430$	0	0
$431$	−10.6525	−0.513112	−0.256556	−	0.966529i	$-0.582588\pi$
$431$	−10.6525	−0.513112	−0.256556	+	0.966529i	$0.582588\pi$
$432$	−22.0344	−1.06013
$433$	−33.3607	−1.60321	−0.801606	−	0.597853i	$-0.796021\pi$
$433$	−33.3607	−1.60321	−0.801606	+	0.597853i	$0.796021\pi$
$434$	20.5623	0.987022
$435$	0	0
$436$	38.5623	1.84680
$437$	0	0
$438$	−62.6869	−2.99530
$439$	36.5066	1.74236	0.871182	−	0.490960i	$-0.163354\pi$
$439$	36.5066	1.74236	0.871182	+	0.490960i	$0.163354\pi$
$440$	0	0
$441$	21.8885	1.04231
$442$	−25.4164	−1.20894
$443$	12.6738	0.602149	0.301074	−	0.953601i	$-0.402655\pi$
$443$	12.6738	0.602149	0.301074	+	0.953601i	$0.402655\pi$
$444$	−45.0000	−2.13561
$445$	0	0
$446$	49.4508	2.34157
$447$	−21.1803	−1.00180
$448$	36.8885	1.74282
$449$	24.7984	1.17031	0.585154	−	0.810922i	$-0.301034\pi$
$449$	24.7984	1.17031	0.585154	+	0.810922i	$0.301034\pi$
$450$	0	0
$451$	−34.9230	−1.64446
$452$	45.2705	2.12935
$453$	12.9656	0.609175
$454$	37.8885	1.77820
$455$	0	0
$456$	0	0
$457$	−5.56231	−0.260194	−0.130097	−	0.991501i	$-0.541529\pi$
$457$	−5.56231	−0.260194	−0.130097	+	0.991501i	$0.541529\pi$
$458$	−54.0689	−2.52647
$459$	−10.8541	−0.506626
$460$	0	0
$461$	−8.03444	−0.374201	−0.187101	−	0.982341i	$-0.559909\pi$
$461$	−8.03444	−0.374201	−0.187101	+	0.982341i	$0.559909\pi$
$462$	−135.700	−6.31334
$463$	−41.8673	−1.94574	−0.972868	−	0.231360i	$-0.925683\pi$
$463$	−41.8673	−1.94574	−0.972868	+	0.231360i	$0.925683\pi$
$464$	59.1246	2.74479
$465$	0	0
$466$	−43.2705	−2.00447
$467$	2.72949	0.126306	0.0631529	−	0.998004i	$-0.479884\pi$
$467$	2.72949	0.126306	0.0631529	+	0.998004i	$0.479884\pi$
$468$	19.4164	0.897524
$469$	36.8885	1.70335
$470$	0	0
$471$	−4.06888	−0.187484
$472$	20.6525	0.950607
$473$	−57.7984	−2.65757
$474$	−5.85410	−0.268888
$475$	0	0
$476$	99.8115	4.57485
$477$	12.7639	0.584420
$478$	−1.85410	−0.0848047
$479$	−27.7426	−1.26759	−0.633797	−	0.773499i	$-0.718505\pi$
$479$	−27.7426	−1.26759	−0.633797	+	0.773499i	$0.718505\pi$
$480$	0	0
$481$	−8.29180	−0.378073
$482$	34.0344	1.55023
$483$	11.7082	0.532742
$484$	91.9574	4.17988
$485$	0	0
$486$	46.8328	2.12438
$487$	10.1246	0.458790	0.229395	−	0.973333i	$-0.426325\pi$
$487$	10.1246	0.458790	0.229395	+	0.973333i	$0.426325\pi$
$488$	41.5623	1.88144
$489$	6.18034	0.279485
$490$	0	0
$491$	−27.7639	−1.25297	−0.626484	−	0.779434i	$-0.715507\pi$
$491$	−27.7639	−1.25297	−0.626484	+	0.779434i	$0.715507\pi$
$492$	−69.2705	−3.12296
$493$	29.1246	1.31171
$494$	0	0
$495$	0	0
$496$	−18.2705	−0.820370
$497$	−19.1803	−0.860356
$498$	6.38197	0.285983
$499$	−10.5967	−0.474376	−0.237188	−	0.971464i	$-0.576226\pi$
$499$	−10.5967	−0.474376	−0.237188	+	0.971464i	$0.576226\pi$
$500$	0	0
$501$	11.7082	0.523084
$502$	−15.4721	−0.690555
$503$	18.7426	0.835693	0.417847	−	0.908518i	$-0.362785\pi$
$503$	18.7426	0.835693	0.417847	+	0.908518i	$0.362785\pi$
$504$	−63.3050	−2.81983
$505$	0	0
$506$	−17.7082	−0.787226
$507$	−20.1246	−0.893765
$508$	−4.14590	−0.183944
$509$	13.0344	0.577741	0.288871	−	0.957368i	$-0.406720\pi$
$509$	13.0344	0.577741	0.288871	+	0.957368i	$0.406720\pi$
$510$	0	0
$511$	45.3607	2.00664
$512$	40.3050	1.78124
$513$	0	0
$514$	11.0000	0.485189
$515$	0	0
$516$	−114.644	−5.04694
$517$	41.6869	1.83339
$518$	45.9787	2.02019
$519$	−43.9443	−1.92894
$520$	0	0
$521$	20.2361	0.886558	0.443279	−	0.896384i	$-0.353815\pi$
$521$	20.2361	0.886558	0.443279	+	0.896384i	$0.353815\pi$
$522$	−31.4164	−1.37506
$523$	−2.38197	−0.104156	−0.0520781	−	0.998643i	$-0.516584\pi$
$523$	−2.38197	−0.104156	−0.0520781	+	0.998643i	$0.516584\pi$
$524$	−58.2492	−2.54463
$525$	0	0
$526$	−59.2148	−2.58189
$527$	−9.00000	−0.392046
$528$	120.575	5.24737
$529$	−21.4721	−0.933571
$530$	0	0
$531$	−5.52786	−0.239889
$532$	0	0
$533$	−12.7639	−0.552867
$534$	−21.7082	−0.939406
$535$	0	0
$536$	−65.0689	−2.81055
$537$	25.0000	1.07883
$538$	−24.0344	−1.03620
$539$	59.8885	2.57958
$540$	0	0
$541$	7.34752	0.315895	0.157947	−	0.987448i	$-0.449512\pi$
$541$	7.34752	0.315895	0.157947	+	0.987448i	$0.449512\pi$
$542$	29.6525	1.27368
$543$	−41.8328	−1.79522
$544$	−52.6869	−2.25893
$545$	0	0
$546$	−49.5967	−2.12254
$547$	−6.41641	−0.274346	−0.137173	−	0.990547i	$-0.543802\pi$
$547$	−6.41641	−0.274346	−0.137173	+	0.990547i	$0.543802\pi$
$548$	44.8328	1.91516
$549$	−11.1246	−0.474787
$550$	0	0
$551$	0	0
$552$	−20.6525	−0.879028
$553$	4.23607	0.180136
$554$	31.4164	1.33476
$555$	0	0
$556$	−67.9574	−2.88204
$557$	−42.5066	−1.80106	−0.900531	−	0.434792i	$-0.856822\pi$
$557$	−42.5066	−1.80106	−0.900531	+	0.434792i	$0.856822\pi$
$558$	9.70820	0.410981
$559$	−21.1246	−0.893476
$560$	0	0
$561$	59.3951	2.50766
$562$	−12.0902	−0.509993
$563$	−16.5967	−0.699470	−0.349735	−	0.936849i	$-0.613728\pi$
$563$	−16.5967	−0.699470	−0.349735	+	0.936849i	$0.613728\pi$
$564$	82.6869	3.48175
$565$	0	0
$566$	62.2148	2.61508
$567$	−46.5967	−1.95688
$568$	33.8328	1.41959
$569$	24.7082	1.03582	0.517911	−	0.855435i	$-0.326710\pi$
$569$	24.7082	1.03582	0.517911	+	0.855435i	$0.326710\pi$
$570$	0	0
$571$	44.2492	1.85177	0.925886	−	0.377803i	$-0.123320\pi$
$571$	44.2492	1.85177	0.925886	+	0.377803i	$0.123320\pi$
$572$	53.1246	2.22125
$573$	−40.1246	−1.67623
$574$	70.7771	2.95418
$575$	0	0
$576$	17.4164	0.725684
$577$	−47.3050	−1.96933	−0.984665	−	0.174453i	$-0.944184\pi$
$577$	−47.3050	−1.96933	−0.984665	+	0.174453i	$0.944184\pi$
$578$	−17.1803	−0.714608
$579$	−52.0344	−2.16248
$580$	0	0
$581$	−4.61803	−0.191588
$582$	43.2148	1.79131
$583$	34.9230	1.44636
$584$	−80.0132	−3.31097
$585$	0	0
$586$	−35.2705	−1.45701
$587$	27.6738	1.14222	0.571109	−	0.820874i	$-0.306513\pi$
$587$	27.6738	1.14222	0.571109	+	0.820874i	$0.306513\pi$
$588$	118.790	4.89883
$589$	0	0
$590$	0	0
$591$	16.7082	0.687284
$592$	−40.8541	−1.67909
$593$	−12.5967	−0.517286	−0.258643	−	0.965973i	$-0.583275\pi$
$593$	−12.5967	−0.517286	−0.258643	+	0.965973i	$0.583275\pi$
$594$	32.0344	1.31439
$595$	0	0
$596$	−45.9787	−1.88336
$597$	43.4164	1.77692
$598$	−6.47214	−0.264665
$599$	−41.0689	−1.67803	−0.839015	−	0.544109i	$-0.816868\pi$
$599$	−41.0689	−1.67803	−0.839015	+	0.544109i	$0.816868\pi$
$600$	0	0
$601$	18.6869	0.762255	0.381128	−	0.924522i	$-0.375536\pi$
$601$	18.6869	0.762255	0.381128	+	0.924522i	$0.375536\pi$
$602$	117.138	4.77418
$603$	17.4164	0.709251
$604$	28.1459	1.14524
$605$	0	0
$606$	76.3050	3.09968
$607$	−26.9787	−1.09503	−0.547516	−	0.836795i	$-0.684427\pi$
$607$	−26.9787	−1.09503	−0.547516	+	0.836795i	$0.684427\pi$
$608$	0	0
$609$	56.8328	2.30298
$610$	0	0
$611$	15.2361	0.616385
$612$	47.1246	1.90490
$613$	7.50658	0.303188	0.151594	−	0.988443i	$-0.451559\pi$
$613$	7.50658	0.303188	0.151594	+	0.988443i	$0.451559\pi$
$614$	−85.2492	−3.44038
$615$	0	0
$616$	−173.207	−6.97869
$617$	−9.50658	−0.382720	−0.191360	−	0.981520i	$-0.561290\pi$
$617$	−9.50658	−0.382720	−0.191360	+	0.981520i	$0.561290\pi$
$618$	−7.23607	−0.291077
$619$	36.5066	1.46732	0.733662	−	0.679515i	$-0.237810\pi$
$619$	36.5066	1.46732	0.733662	+	0.679515i	$0.237810\pi$
$620$	0	0
$621$	−2.76393	−0.110913
$622$	−42.5066	−1.70436
$623$	15.7082	0.629336
$624$	44.0689	1.76417
$625$	0	0
$626$	−52.2148	−2.08692
$627$	0	0
$628$	−8.83282	−0.352468
$629$	−20.1246	−0.802421
$630$	0	0
$631$	−33.9787	−1.35267	−0.676336	−	0.736594i	$-0.736433\pi$
$631$	−33.9787	−1.35267	−0.676336	+	0.736594i	$0.736433\pi$
$632$	−7.47214	−0.297226
$633$	−27.3607	−1.08749
$634$	28.6525	1.13794
$635$	0	0
$636$	69.2705	2.74675
$637$	21.8885	0.867256
$638$	−85.9574	−3.40309
$639$	−9.05573	−0.358239
$640$	0	0
$641$	17.2918	0.682985	0.341492	−	0.939885i	$-0.389068\pi$
$641$	17.2918	0.682985	0.341492	+	0.939885i	$0.389068\pi$
$642$	−116.228	−4.58715
$643$	−38.3050	−1.51060	−0.755300	−	0.655379i	$-0.772509\pi$
$643$	−38.3050	−1.51060	−0.755300	+	0.655379i	$0.772509\pi$
$644$	25.4164	1.00155
$645$	0	0
$646$	0	0
$647$	21.5066	0.845511	0.422755	−	0.906244i	$-0.361063\pi$
$647$	21.5066	0.845511	0.422755	+	0.906244i	$0.361063\pi$
$648$	82.1935	3.22887
$649$	−15.1246	−0.593693
$650$	0	0
$651$	−17.5623	−0.688321
$652$	13.4164	0.525427
$653$	−14.1246	−0.552739	−0.276369	−	0.961051i	$-0.589131\pi$
$653$	−14.1246	−0.552739	−0.276369	+	0.961051i	$0.589131\pi$
$654$	−46.5066	−1.81855
$655$	0	0
$656$	−62.8885	−2.45539
$657$	21.4164	0.835534
$658$	−84.4853	−3.29358
$659$	15.0689	0.587000	0.293500	−	0.955959i	$-0.405180\pi$
$659$	15.0689	0.587000	0.293500	+	0.955959i	$0.405180\pi$
$660$	0	0
$661$	−28.4164	−1.10527	−0.552635	−	0.833423i	$-0.686378\pi$
$661$	−28.4164	−1.10527	−0.552635	+	0.833423i	$0.686378\pi$
$662$	35.5066	1.38000
$663$	21.7082	0.843077
$664$	8.14590	0.316122
$665$	0	0
$666$	21.7082	0.841176
$667$	7.41641	0.287164
$668$	25.4164	0.983390
$669$	−42.2361	−1.63294
$670$	0	0
$671$	−30.4377	−1.17503
$672$	−102.812	−3.96604
$673$	−2.81966	−0.108690	−0.0543450	−	0.998522i	$-0.517307\pi$
$673$	−2.81966	−0.108690	−0.0543450	+	0.998522i	$0.517307\pi$
$674$	−12.3262	−0.474789
$675$	0	0
$676$	−43.6869	−1.68027
$677$	−36.0902	−1.38706	−0.693529	−	0.720429i	$-0.743945\pi$
$677$	−36.0902	−1.38706	−0.693529	+	0.720429i	$0.743945\pi$
$678$	−54.5967	−2.09678
$679$	−31.2705	−1.20005
$680$	0	0
$681$	−32.3607	−1.24006
$682$	26.5623	1.01712
$683$	−36.7984	−1.40805	−0.704025	−	0.710175i	$-0.748616\pi$
$683$	−36.7984	−1.40805	−0.704025	+	0.710175i	$0.748616\pi$
$684$	0	0
$685$	0	0
$686$	−43.7426	−1.67010
$687$	46.1803	1.76189
$688$	−104.082	−3.96809
$689$	12.7639	0.486267
$690$	0	0
$691$	−7.65248	−0.291114	−0.145557	−	0.989350i	$-0.546497\pi$
$691$	−7.65248	−0.291114	−0.145557	+	0.989350i	$0.546497\pi$
$692$	−95.3951	−3.62638
$693$	46.3607	1.76110
$694$	−87.6312	−3.32643
$695$	0	0
$696$	−100.249	−3.79994
$697$	−30.9787	−1.17340
$698$	70.8673	2.68237
$699$	36.9574	1.39786
$700$	0	0
$701$	−48.9443	−1.84860	−0.924300	−	0.381667i	$-0.875350\pi$
$701$	−48.9443	−1.84860	−0.924300	+	0.381667i	$0.875350\pi$
$702$	11.7082	0.441898
$703$	0	0
$704$	47.6525	1.79597
$705$	0	0
$706$	43.2705	1.62851
$707$	−55.2148	−2.07657
$708$	−30.0000	−1.12747
$709$	8.83282	0.331723	0.165862	−	0.986149i	$-0.446959\pi$
$709$	8.83282	0.331723	0.165862	+	0.986149i	$0.446959\pi$
$710$	0	0
$711$	2.00000	0.0750059
$712$	−27.7082	−1.03841
$713$	−2.29180	−0.0858284
$714$	−120.374	−4.50488
$715$	0	0
$716$	54.2705	2.02818
$717$	1.58359	0.0591403
$718$	59.1591	2.20780
$719$	−12.2361	−0.456328	−0.228164	−	0.973623i	$-0.573272\pi$
$719$	−12.2361	−0.456328	−0.228164	+	0.973623i	$0.573272\pi$
$720$	0	0
$721$	5.23607	0.195001
$722$	0	0
$723$	−29.0689	−1.08108
$724$	−90.8115	−3.37498
$725$	0	0
$726$	−110.902	−4.11595
$727$	−40.6869	−1.50899	−0.754497	−	0.656303i	$-0.772119\pi$
$727$	−40.6869	−1.50899	−0.754497	+	0.656303i	$0.772119\pi$
$728$	−63.3050	−2.34624
$729$	−7.00000	−0.259259
$730$	0	0
$731$	−51.2705	−1.89631
$732$	−60.3738	−2.23148
$733$	50.1591	1.85267	0.926333	−	0.376705i	$-0.122943\pi$
$733$	50.1591	1.85267	0.926333	+	0.376705i	$0.122943\pi$
$734$	−32.6525	−1.20522
$735$	0	0
$736$	−13.4164	−0.494535
$737$	47.6525	1.75530
$738$	33.4164	1.23007
$739$	1.56231	0.0574704	0.0287352	−	0.999587i	$-0.490852\pi$
$739$	1.56231	0.0574704	0.0287352	+	0.999587i	$0.490852\pi$
$740$	0	0
$741$	0	0
$742$	−70.7771	−2.59831
$743$	−8.76393	−0.321517	−0.160759	−	0.986994i	$-0.551394\pi$
$743$	−8.76393	−0.321517	−0.160759	+	0.986994i	$0.551394\pi$
$744$	30.9787	1.13573
$745$	0	0
$746$	34.4164	1.26007
$747$	−2.18034	−0.0797745
$748$	128.936	4.71437
$749$	84.1033	3.07307
$750$	0	0
$751$	27.4377	1.00122	0.500608	−	0.865674i	$-0.333110\pi$
$751$	27.4377	1.00122	0.500608	+	0.865674i	$0.333110\pi$
$752$	75.0689	2.73748
$753$	13.2148	0.481573
$754$	−31.4164	−1.14412
$755$	0	0
$756$	−45.9787	−1.67223
$757$	−6.88854	−0.250368	−0.125184	−	0.992134i	$-0.539952\pi$
$757$	−6.88854	−0.250368	−0.125184	+	0.992134i	$0.539952\pi$
$758$	95.6656	3.47473
$759$	15.1246	0.548989
$760$	0	0
$761$	27.3820	0.992595	0.496298	−	0.868152i	$-0.334692\pi$
$761$	27.3820	0.992595	0.496298	+	0.868152i	$0.334692\pi$
$762$	5.00000	0.181131
$763$	33.6525	1.21830
$764$	−87.1033	−3.15129
$765$	0	0
$766$	−57.7771	−2.08757
$767$	−5.52786	−0.199600
$768$	−32.5623	−1.17499
$769$	15.2705	0.550669	0.275334	−	0.961349i	$-0.411211\pi$
$769$	15.2705	0.550669	0.275334	+	0.961349i	$0.411211\pi$
$770$	0	0
$771$	−9.39512	−0.338357
$772$	−112.957	−4.06543
$773$	−48.0344	−1.72768	−0.863839	−	0.503767i	$-0.831947\pi$
$773$	−48.0344	−1.72768	−0.863839	+	0.503767i	$0.831947\pi$
$774$	55.3050	1.98790
$775$	0	0
$776$	55.1591	1.98009
$777$	−39.2705	−1.40882
$778$	11.7082	0.419759
$779$	0	0
$780$	0	0
$781$	−24.7771	−0.886594
$782$	−15.7082	−0.561724
$783$	−13.4164	−0.479463
$784$	107.846	3.85164
$785$	0	0
$786$	70.2492	2.50571
$787$	43.5967	1.55406	0.777028	−	0.629466i	$-0.216726\pi$
$787$	43.5967	1.55406	0.777028	+	0.629466i	$0.216726\pi$
$788$	36.2705	1.29208
$789$	50.5755	1.80053
$790$	0	0
$791$	39.5066	1.40469
$792$	−81.7771	−2.90582
$793$	−11.1246	−0.395047
$794$	−23.6525	−0.839395
$795$	0	0
$796$	94.2492	3.34058
$797$	−27.8673	−0.987109	−0.493554	−	0.869715i	$-0.664303\pi$
$797$	−27.8673	−0.987109	−0.493554	+	0.869715i	$0.664303\pi$
$798$	0	0
$799$	36.9787	1.30821
$800$	0	0
$801$	7.41641	0.262046
$802$	15.4164	0.544372
$803$	58.5967	2.06783
$804$	94.5197	3.33345
$805$	0	0
$806$	9.70820	0.341957
$807$	20.5279	0.722615
$808$	97.3951	3.42635
$809$	−2.23607	−0.0786160	−0.0393080	−	0.999227i	$-0.512515\pi$
$809$	−2.23607	−0.0786160	−0.0393080	+	0.999227i	$0.512515\pi$
$810$	0	0
$811$	−9.61803	−0.337735	−0.168867	−	0.985639i	$-0.554011\pi$
$811$	−9.61803	−0.337735	−0.168867	+	0.985639i	$0.554011\pi$
$812$	123.374	4.32957
$813$	−25.3262	−0.888230
$814$	59.3951	2.08180
$815$	0	0
$816$	106.957	3.74426
$817$	0	0
$818$	−43.0344	−1.50466
$819$	16.9443	0.592081
$820$	0	0
$821$	27.9230	0.974519	0.487259	−	0.873257i	$-0.337997\pi$
$821$	27.9230	0.974519	0.487259	+	0.873257i	$0.337997\pi$
$822$	−54.0689	−1.88587
$823$	36.7426	1.28077	0.640384	−	0.768055i	$-0.278775\pi$
$823$	36.7426	1.28077	0.640384	+	0.768055i	$0.278775\pi$
$824$	−9.23607	−0.321754
$825$	0	0
$826$	30.6525	1.06654
$827$	31.2492	1.08664	0.543321	−	0.839525i	$-0.317167\pi$
$827$	31.2492	1.08664	0.543321	+	0.839525i	$0.317167\pi$
$828$	12.0000	0.417029
$829$	1.00000	0.0347314	0.0173657	−	0.999849i	$-0.494472\pi$
$829$	1.00000	0.0347314	0.0173657	+	0.999849i	$0.494472\pi$
$830$	0	0
$831$	−26.8328	−0.930820
$832$	17.4164	0.603805
$833$	53.1246	1.84066
$834$	81.9574	2.83795
$835$	0	0
$836$	0	0
$837$	4.14590	0.143303
$838$	24.9443	0.861686
$839$	54.2361	1.87244	0.936219	−	0.351418i	$-0.114301\pi$
$839$	54.2361	1.87244	0.936219	+	0.351418i	$0.114301\pi$
$840$	0	0
$841$	7.00000	0.241379
$842$	39.2705	1.35335
$843$	10.3262	0.355655
$844$	−59.3951	−2.04446
$845$	0	0
$846$	−39.8885	−1.37140
$847$	80.2492	2.75740
$848$	62.8885	2.15960
$849$	−53.1378	−1.82368
$850$	0	0
$851$	−5.12461	−0.175669
$852$	−49.1459	−1.68371
$853$	46.8541	1.60425	0.802127	−	0.597154i	$-0.203702\pi$
$853$	46.8541	1.60425	0.802127	+	0.597154i	$0.203702\pi$
$854$	61.6869	2.11088
$855$	0	0
$856$	−148.353	−5.07059
$857$	24.8197	0.847823	0.423912	−	0.905704i	$-0.360657\pi$
$857$	24.8197	0.847823	0.423912	+	0.905704i	$0.360657\pi$
$858$	−64.0689	−2.18728
$859$	11.9098	0.406358	0.203179	−	0.979142i	$-0.434873\pi$
$859$	11.9098	0.406358	0.203179	+	0.979142i	$0.434873\pi$
$860$	0	0
$861$	−60.4508	−2.06016
$862$	27.8885	0.949888
$863$	−31.9230	−1.08667	−0.543336	−	0.839516i	$-0.682839\pi$
$863$	−31.9230	−1.08667	−0.543336	+	0.839516i	$0.682839\pi$
$864$	24.2705	0.825700
$865$	0	0
$866$	87.3394	2.96791
$867$	14.6738	0.498347
$868$	−38.1246	−1.29403
$869$	5.47214	0.185629
$870$	0	0
$871$	17.4164	0.590132
$872$	−59.3607	−2.01021
$873$	−14.7639	−0.499683
$874$	0	0
$875$	0	0
$876$	116.228	3.92698
$877$	45.0344	1.52071	0.760353	−	0.649511i	$-0.225026\pi$
$877$	45.0344	1.52071	0.760353	+	0.649511i	$0.225026\pi$
$878$	−95.5755	−3.22552
$879$	30.1246	1.01608
$880$	0	0
$881$	7.47214	0.251743	0.125871	−	0.992047i	$-0.459827\pi$
$881$	7.47214	0.251743	0.125871	+	0.992047i	$0.459827\pi$
$882$	−57.3050	−1.92956
$883$	−21.7771	−0.732857	−0.366429	−	0.930446i	$-0.619420\pi$
$883$	−21.7771	−0.732857	−0.366429	+	0.930446i	$0.619420\pi$
$884$	47.1246	1.58497
$885$	0	0
$886$	−33.1803	−1.11472
$887$	−36.9443	−1.24047	−0.620234	−	0.784417i	$-0.712962\pi$
$887$	−36.9443	−1.24047	−0.620234	+	0.784417i	$0.712962\pi$
$888$	69.2705	2.32457
$889$	−3.61803	−0.121345
$890$	0	0
$891$	−60.1935	−2.01656
$892$	−91.6869	−3.06991
$893$	0	0
$894$	55.4508	1.85455
$895$	0	0
$896$	−4.61803	−0.154278
$897$	5.52786	0.184570
$898$	−64.9230	−2.16651
$899$	−11.1246	−0.371027
$900$	0	0
$901$	30.9787	1.03205
$902$	91.4296	3.04427
$903$	−100.048	−3.32938
$904$	−69.6869	−2.31775
$905$	0	0
$906$	−33.9443	−1.12772
$907$	−38.9574	−1.29356	−0.646780	−	0.762677i	$-0.723885\pi$
$907$	−38.9574	−1.29356	−0.646780	+	0.762677i	$0.723885\pi$
$908$	−70.2492	−2.33130
$909$	−26.0689	−0.864650
$910$	0	0
$911$	22.8541	0.757190	0.378595	−	0.925562i	$-0.376407\pi$
$911$	22.8541	0.757190	0.378595	+	0.925562i	$0.376407\pi$
$912$	0	0
$913$	−5.96556	−0.197431
$914$	14.5623	0.481678
$915$	0	0
$916$	100.249	3.31233
$917$	−50.8328	−1.67865
$918$	28.4164	0.937881
$919$	−30.5066	−1.00632	−0.503160	−	0.864194i	$-0.667829\pi$
$919$	−30.5066	−1.00632	−0.503160	+	0.864194i	$0.667829\pi$
$920$	0	0
$921$	72.8115	2.39922
$922$	21.0344	0.692732
$923$	−9.05573	−0.298073
$924$	251.602	8.27709
$925$	0	0
$926$	109.610	3.60200
$927$	2.47214	0.0811956
$928$	−65.1246	−2.13782
$929$	25.4721	0.835714	0.417857	−	0.908513i	$-0.362781\pi$
$929$	25.4721	0.835714	0.417857	+	0.908513i	$0.362781\pi$
$930$	0	0
$931$	0	0
$932$	80.2279	2.62795
$933$	36.3050	1.18857
$934$	−7.14590	−0.233821
$935$	0	0
$936$	−29.8885	−0.976938
$937$	26.8328	0.876590	0.438295	−	0.898831i	$-0.355583\pi$
$937$	26.8328	0.876590	0.438295	+	0.898831i	$0.355583\pi$
$938$	−96.5755	−3.15330
$939$	44.5967	1.45536
$940$	0	0
$941$	59.7426	1.94755	0.973777	−	0.227503i	$-0.0730563\pi$
$941$	59.7426	1.94755	0.973777	+	0.227503i	$0.0730563\pi$
$942$	10.6525	0.347076
$943$	−7.88854	−0.256886
$944$	−27.2361	−0.886459
$945$	0	0
$946$	151.318	4.91978
$947$	−37.6869	−1.22466	−0.612330	−	0.790602i	$-0.709768\pi$
$947$	−37.6869	−1.22466	−0.612330	+	0.790602i	$0.709768\pi$
$948$	10.8541	0.352525
$949$	21.4164	0.695206
$950$	0	0
$951$	−24.4721	−0.793563
$952$	−153.644	−4.97964
$953$	−13.4164	−0.434600	−0.217300	−	0.976105i	$-0.569725\pi$
$953$	−13.4164	−0.434600	−0.217300	+	0.976105i	$0.569725\pi$
$954$	−33.4164	−1.08190
$955$	0	0
$956$	3.43769	0.111183
$957$	73.4164	2.37322
$958$	72.6312	2.34661
$959$	39.1246	1.26340
$960$	0	0
$961$	−27.5623	−0.889107
$962$	21.7082	0.699901
$963$	39.7082	1.27958
$964$	−63.1033	−2.03242
$965$	0	0
$966$	−30.6525	−0.986227
$967$	6.09017	0.195847	0.0979233	−	0.995194i	$-0.468780\pi$
$967$	6.09017	0.195847	0.0979233	+	0.995194i	$0.468780\pi$
$968$	−141.554	−4.54972
$969$	0	0
$970$	0	0
$971$	−19.2016	−0.616210	−0.308105	−	0.951352i	$-0.599695\pi$
$971$	−19.2016	−0.616210	−0.308105	+	0.951352i	$0.599695\pi$
$972$	−86.8328	−2.78516
$973$	−59.3050	−1.90123
$974$	−26.5066	−0.849326
$975$	0	0
$976$	−54.8115	−1.75447
$977$	32.2148	1.03064	0.515321	−	0.856997i	$-0.327673\pi$
$977$	32.2148	1.03064	0.515321	+	0.856997i	$0.327673\pi$
$978$	−16.1803	−0.517390
$979$	20.2918	0.648529
$980$	0	0
$981$	15.8885	0.507282
$982$	72.6869	2.31953
$983$	12.9787	0.413957	0.206978	−	0.978346i	$-0.433637\pi$
$983$	12.9787	0.413957	0.206978	+	0.978346i	$0.433637\pi$
$984$	106.631	3.39928
$985$	0	0
$986$	−76.2492	−2.42827
$987$	72.1591	2.29685
$988$	0	0
$989$	−13.0557	−0.415148
$990$	0	0
$991$	−15.4164	−0.489718	−0.244859	−	0.969559i	$-0.578742\pi$
$991$	−15.4164	−0.489718	−0.244859	+	0.969559i	$0.578742\pi$
$992$	20.1246	0.638957
$993$	−30.3262	−0.962374
$994$	50.2148	1.59272
$995$	0	0
$996$	−11.8328	−0.374937
$997$	−6.74265	−0.213542	−0.106771	−	0.994284i	$-0.534051\pi$
$997$	−6.74265	−0.213542	−0.106771	+	0.994284i	$0.534051\pi$
$998$	27.7426	0.878178
$999$	9.27051	0.293306

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.m.1.1	yes	2
5.4	even	2		9025.2.a.t.1.2	yes	2
19.18	odd	2		9025.2.a.u.1.2	yes	2
95.94	odd	2		9025.2.a.l.1.1	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9025.2.a.l.1.1	✓	2	95.94	odd	2
9025.2.a.m.1.1	yes	2	1.1	even	1	trivial
9025.2.a.t.1.2	yes	2	5.4	even	2
9025.2.a.u.1.2	yes	2	19.18	odd	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:	\( N \)	\(=\)	\( 9025 = 5^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9025.a (trivial)

\(f(q)\)	\(=\)	\(q-2.61803 q^{2} +2.23607 q^{3} +4.85410 q^{4} -5.85410 q^{6} +4.23607 q^{7} -7.47214 q^{8} +2.00000 q^{9} +O(q^{10})\)	\(q-2.61803 q^{2} +2.23607 q^{3} +4.85410 q^{4} -5.85410 q^{6} +4.23607 q^{7} -7.47214 q^{8} +2.00000 q^{9} +5.47214 q^{11} +10.8541 q^{12} +2.00000 q^{13} -11.0902 q^{14} +9.85410 q^{16} +4.85410 q^{17} -5.23607 q^{18} +9.47214 q^{21} -14.3262 q^{22} +1.23607 q^{23} -16.7082 q^{24} -5.23607 q^{26} -2.23607 q^{27} +20.5623 q^{28} +6.00000 q^{29} -1.85410 q^{31} -10.8541 q^{32} +12.2361 q^{33} -12.7082 q^{34} +9.70820 q^{36} -4.14590 q^{37} +4.47214 q^{39} -6.38197 q^{41} -24.7984 q^{42} -10.5623 q^{43} +26.5623 q^{44} -3.23607 q^{46} +7.61803 q^{47} +22.0344 q^{48} +10.9443 q^{49} +10.8541 q^{51} +9.70820 q^{52} +6.38197 q^{53} +5.85410 q^{54} -31.6525 q^{56} -15.7082 q^{58} -2.76393 q^{59} -5.56231 q^{61} +4.85410 q^{62} +8.47214 q^{63} +8.70820 q^{64} -32.0344 q^{66} +8.70820 q^{67} +23.5623 q^{68} +2.76393 q^{69} -4.52786 q^{71} -14.9443 q^{72} +10.7082 q^{73} +10.8541 q^{74} +23.1803 q^{77} -11.7082 q^{78} +1.00000 q^{79} -11.0000 q^{81} +16.7082 q^{82} -1.09017 q^{83} +45.9787 q^{84} +27.6525 q^{86} +13.4164 q^{87} -40.8885 q^{88} +3.70820 q^{89} +8.47214 q^{91} +6.00000 q^{92} -4.14590 q^{93} -19.9443 q^{94} -24.2705 q^{96} -7.38197 q^{97} -28.6525 q^{98} +10.9443 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} + 3 q^{4} - 5 q^{6} + 4 q^{7} - 6 q^{8} + 4 q^{9}+O(q^{10}) \) 2 * q - 3 * q^2 + 3 * q^4 - 5 * q^6 + 4 * q^7 - 6 * q^8 + 4 * q^9	\( 2 q - 3 q^{2} + 3 q^{4} - 5 q^{6} + 4 q^{7} - 6 q^{8} + 4 q^{9} + 2 q^{11} + 15 q^{12} + 4 q^{13} - 11 q^{14} + 13 q^{16} + 3 q^{17} - 6 q^{18} + 10 q^{21} - 13 q^{22} - 2 q^{23} - 20 q^{24} - 6 q^{26} + 21 q^{28} + 12 q^{29} + 3 q^{31} - 15 q^{32} + 20 q^{33} - 12 q^{34} + 6 q^{36} - 15 q^{37} - 15 q^{41} - 25 q^{42} - q^{43} + 33 q^{44} - 2 q^{46} + 13 q^{47} + 15 q^{48} + 4 q^{49} + 15 q^{51} + 6 q^{52} + 15 q^{53} + 5 q^{54} - 32 q^{56} - 18 q^{58} - 10 q^{59} + 9 q^{61} + 3 q^{62} + 8 q^{63} + 4 q^{64} - 35 q^{66} + 4 q^{67} + 27 q^{68} + 10 q^{69} - 18 q^{71} - 12 q^{72} + 8 q^{73} + 15 q^{74} + 24 q^{77} - 10 q^{78} + 2 q^{79} - 22 q^{81} + 20 q^{82} + 9 q^{83} + 45 q^{84} + 24 q^{86} - 46 q^{88} - 6 q^{89} + 8 q^{91} + 12 q^{92} - 15 q^{93} - 22 q^{94} - 15 q^{96} - 17 q^{97} - 26 q^{98} + 4 q^{99}+O(q^{100}) \) 2 * q - 3 * q^2 + 3 * q^4 - 5 * q^6 + 4 * q^7 - 6 * q^8 + 4 * q^9 + 2 * q^11 + 15 * q^12 + 4 * q^13 - 11 * q^14 + 13 * q^16 + 3 * q^17 - 6 * q^18 + 10 * q^21 - 13 * q^22 - 2 * q^23 - 20 * q^24 - 6 * q^26 + 21 * q^28 + 12 * q^29 + 3 * q^31 - 15 * q^32 + 20 * q^33 - 12 * q^34 + 6 * q^36 - 15 * q^37 - 15 * q^41 - 25 * q^42 - q^43 + 33 * q^44 - 2 * q^46 + 13 * q^47 + 15 * q^48 + 4 * q^49 + 15 * q^51 + 6 * q^52 + 15 * q^53 + 5 * q^54 - 32 * q^56 - 18 * q^58 - 10 * q^59 + 9 * q^61 + 3 * q^62 + 8 * q^63 + 4 * q^64 - 35 * q^66 + 4 * q^67 + 27 * q^68 + 10 * q^69 - 18 * q^71 - 12 * q^72 + 8 * q^73 + 15 * q^74 + 24 * q^77 - 10 * q^78 + 2 * q^79 - 22 * q^81 + 20 * q^82 + 9 * q^83 + 45 * q^84 + 24 * q^86 - 46 * q^88 - 6 * q^89 + 8 * q^91 + 12 * q^92 - 15 * q^93 - 22 * q^94 - 15 * q^96 - 17 * q^97 - 26 * q^98 + 4 * q^99

Introduction

L-functions

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