LMFDB - Embedded newform 9025.2.a.v.1.2

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:	no (minimal twist has level 1805)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q+2.61803 q^{2} +2.61803 q^{3} +4.85410 q^{4} +6.85410 q^{6} -0.236068 q^{7} +7.47214 q^{8} +3.85410 q^{9} +O(q^{10})$	\(q+2.61803 q^{2} +2.61803 q^{3} +4.85410 q^{4} +6.85410 q^{6} -0.236068 q^{7} +7.47214 q^{8} +3.85410 q^{9} -4.61803 q^{11} +12.7082 q^{12} +5.00000 q^{13} -0.618034 q^{14} +9.85410 q^{16} +6.00000 q^{17} +10.0902 q^{18} -0.618034 q^{21} -12.0902 q^{22} +1.61803 q^{23} +19.5623 q^{24} +13.0902 q^{26} +2.23607 q^{27} -1.14590 q^{28} +1.85410 q^{29} -4.14590 q^{31} +10.8541 q^{32} -12.0902 q^{33} +15.7082 q^{34} +18.7082 q^{36} -1.85410 q^{37} +13.0902 q^{39} -11.1803 q^{41} -1.61803 q^{42} +2.85410 q^{43} -22.4164 q^{44} +4.23607 q^{46} +10.2361 q^{47} +25.7984 q^{48} -6.94427 q^{49} +15.7082 q^{51} +24.2705 q^{52} +5.61803 q^{53} +5.85410 q^{54} -1.76393 q^{56} +4.85410 q^{58} -0.0901699 q^{59} -3.00000 q^{61} -10.8541 q^{62} -0.909830 q^{63} +8.70820 q^{64} -31.6525 q^{66} -4.70820 q^{67} +29.1246 q^{68} +4.23607 q^{69} -3.76393 q^{71} +28.7984 q^{72} +1.00000 q^{73} -4.85410 q^{74} +1.09017 q^{77} +34.2705 q^{78} -8.00000 q^{79} -5.70820 q^{81} -29.2705 q^{82} -4.47214 q^{83} -3.00000 q^{84} +7.47214 q^{86} +4.85410 q^{87} -34.5066 q^{88} +6.70820 q^{89} -1.18034 q^{91} +7.85410 q^{92} -10.8541 q^{93} +26.7984 q^{94} +28.4164 q^{96} +6.09017 q^{97} -18.1803 q^{98} -17.7984 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 7 q^{6} + 4 q^{7} + 6 q^{8} + q^{9}+O(q^{10}) $ 2 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 + 7 * q^6 + 4 * q^7 + 6 * q^8 + q^9	$ 2 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 7 q^{6} + 4 q^{7} + 6 q^{8} + q^{9} - 7 q^{11} + 12 q^{12} + 10 q^{13} + q^{14} + 13 q^{16} + 12 q^{17} + 9 q^{18} + q^{21} - 13 q^{22} + q^{23} + 19 q^{24} + 15 q^{26} - 9 q^{28} - 3 q^{29} - 15 q^{31} + 15 q^{32} - 13 q^{33} + 18 q^{34} + 24 q^{36} + 3 q^{37} + 15 q^{39} - q^{42} - q^{43} - 18 q^{44} + 4 q^{46} + 16 q^{47} + 27 q^{48} + 4 q^{49} + 18 q^{51} + 15 q^{52} + 9 q^{53} + 5 q^{54} - 8 q^{56} + 3 q^{58} + 11 q^{59} - 6 q^{61} - 15 q^{62} - 13 q^{63} + 4 q^{64} - 32 q^{66} + 4 q^{67} + 18 q^{68} + 4 q^{69} - 12 q^{71} + 33 q^{72} + 2 q^{73} - 3 q^{74} - 9 q^{77} + 35 q^{78} - 16 q^{79} + 2 q^{81} - 25 q^{82} - 6 q^{84} + 6 q^{86} + 3 q^{87} - 31 q^{88} + 20 q^{91} + 9 q^{92} - 15 q^{93} + 29 q^{94} + 30 q^{96} + q^{97} - 14 q^{98} - 11 q^{99}+O(q^{100}) $ 2 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 + 7 * q^6 + 4 * q^7 + 6 * q^8 + q^9 - 7 * q^11 + 12 * q^12 + 10 * q^13 + q^14 + 13 * q^16 + 12 * q^17 + 9 * q^18 + q^21 - 13 * q^22 + q^23 + 19 * q^24 + 15 * q^26 - 9 * q^28 - 3 * q^29 - 15 * q^31 + 15 * q^32 - 13 * q^33 + 18 * q^34 + 24 * q^36 + 3 * q^37 + 15 * q^39 - q^42 - q^43 - 18 * q^44 + 4 * q^46 + 16 * q^47 + 27 * q^48 + 4 * q^49 + 18 * q^51 + 15 * q^52 + 9 * q^53 + 5 * q^54 - 8 * q^56 + 3 * q^58 + 11 * q^59 - 6 * q^61 - 15 * q^62 - 13 * q^63 + 4 * q^64 - 32 * q^66 + 4 * q^67 + 18 * q^68 + 4 * q^69 - 12 * q^71 + 33 * q^72 + 2 * q^73 - 3 * q^74 - 9 * q^77 + 35 * q^78 - 16 * q^79 + 2 * q^81 - 25 * q^82 - 6 * q^84 + 6 * q^86 + 3 * q^87 - 31 * q^88 + 20 * q^91 + 9 * q^92 - 15 * q^93 + 29 * q^94 + 30 * q^96 + q^97 - 14 * q^98 - 11 * 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	$-0.123549\pi$
$2$	2.61803	1.85123	0.925615	+	0.378467i	$0.123549\pi$
$3$	2.61803	1.51152	0.755761	−	0.654847i	$-0.227267\pi$
$3$	2.61803	1.51152	0.755761	+	0.654847i	$0.227267\pi$
$4$	4.85410	2.42705
$5$	0	0
$5$	0	0
$6$	6.85410	2.79818
$7$	−0.236068	−0.0892253	−0.0446127	−	0.999004i	$-0.514205\pi$
$7$	−0.236068	−0.0892253	−0.0446127	+	0.999004i	$0.514205\pi$
$8$	7.47214	2.64180
$9$	3.85410	1.28470
$10$	0	0
$11$	−4.61803	−1.39239	−0.696195	−	0.717853i	$-0.745125\pi$
$11$	−4.61803	−1.39239	−0.696195	+	0.717853i	$0.745125\pi$
$12$	12.7082	3.66854
$13$	5.00000	1.38675	0.693375	−	0.720577i	$-0.256123\pi$
$13$	5.00000	1.38675	0.693375	+	0.720577i	$0.256123\pi$
$14$	−0.618034	−0.165177
$15$	0	0
$16$	9.85410	2.46353
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	10.0902	2.37828
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−0.618034	−0.134866
$22$	−12.0902	−2.57763
$23$	1.61803	0.337383	0.168692	−	0.985669i	$-0.446046\pi$
$23$	1.61803	0.337383	0.168692	+	0.985669i	$0.446046\pi$
$24$	19.5623	3.99314
$25$	0	0
$26$	13.0902	2.56719
$27$	2.23607	0.430331
$28$	−1.14590	−0.216554
$29$	1.85410	0.344298	0.172149	−	0.985071i	$-0.444929\pi$
$29$	1.85410	0.344298	0.172149	+	0.985071i	$0.444929\pi$
$30$	0	0
$31$	−4.14590	−0.744625	−0.372313	−	0.928107i	$-0.621435\pi$
$31$	−4.14590	−0.744625	−0.372313	+	0.928107i	$0.621435\pi$
$32$	10.8541	1.91875
$33$	−12.0902	−2.10463
$34$	15.7082	2.69393
$35$	0	0
$36$	18.7082	3.11803
$37$	−1.85410	−0.304812	−0.152406	−	0.988318i	$-0.548702\pi$
$37$	−1.85410	−0.304812	−0.152406	+	0.988318i	$0.548702\pi$
$38$	0	0
$39$	13.0902	2.09610
$40$	0	0
$41$	−11.1803	−1.74608	−0.873038	−	0.487652i	$-0.837853\pi$
$41$	−11.1803	−1.74608	−0.873038	+	0.487652i	$0.837853\pi$
$42$	−1.61803	−0.249668
$43$	2.85410	0.435246	0.217623	−	0.976033i	$-0.430170\pi$
$43$	2.85410	0.435246	0.217623	+	0.976033i	$0.430170\pi$
$44$	−22.4164	−3.37940
$45$	0	0
$46$	4.23607	0.624574
$47$	10.2361	1.49308	0.746542	−	0.665338i	$-0.231713\pi$
$47$	10.2361	1.49308	0.746542	+	0.665338i	$0.231713\pi$
$48$	25.7984	3.72367
$49$	−6.94427	−0.992039
$50$	0	0
$51$	15.7082	2.19959
$52$	24.2705	3.36571
$53$	5.61803	0.771696	0.385848	−	0.922562i	$-0.373909\pi$
$53$	5.61803	0.771696	0.385848	+	0.922562i	$0.373909\pi$
$54$	5.85410	0.796642
$55$	0	0
$56$	−1.76393	−0.235715
$57$	0	0
$58$	4.85410	0.637375
$59$	−0.0901699	−0.0117391	−0.00586956	−	0.999983i	$-0.501868\pi$
$59$	−0.0901699	−0.0117391	−0.00586956	+	0.999983i	$0.501868\pi$
$60$	0	0
$61$	−3.00000	−0.384111	−0.192055	−	0.981384i	$-0.561515\pi$
$61$	−3.00000	−0.384111	−0.192055	+	0.981384i	$0.561515\pi$
$62$	−10.8541	−1.37847
$63$	−0.909830	−0.114628
$64$	8.70820	1.08853
$65$	0	0
$66$	−31.6525	−3.89615
$67$	−4.70820	−0.575199	−0.287599	−	0.957751i	$-0.592857\pi$
$67$	−4.70820	−0.575199	−0.287599	+	0.957751i	$0.592857\pi$
$68$	29.1246	3.53188
$69$	4.23607	0.509963
$70$	0	0
$71$	−3.76393	−0.446697	−0.223348	−	0.974739i	$-0.571699\pi$
$71$	−3.76393	−0.446697	−0.223348	+	0.974739i	$0.571699\pi$
$72$	28.7984	3.39392
$73$	1.00000	0.117041	0.0585206	−	0.998286i	$-0.481362\pi$
$73$	1.00000	0.117041	0.0585206	+	0.998286i	$0.481362\pi$
$74$	−4.85410	−0.564278
$75$	0	0
$76$	0	0
$77$	1.09017	0.124236
$78$	34.2705	3.88037
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	−5.70820	−0.634245
$82$	−29.2705	−3.23239
$83$	−4.47214	−0.490881	−0.245440	−	0.969412i	$-0.578933\pi$
$83$	−4.47214	−0.490881	−0.245440	+	0.969412i	$0.578933\pi$
$84$	−3.00000	−0.327327
$85$	0	0
$86$	7.47214	0.805741
$87$	4.85410	0.520414
$88$	−34.5066	−3.67841
$89$	6.70820	0.711068	0.355534	−	0.934663i	$-0.384299\pi$
$89$	6.70820	0.711068	0.355534	+	0.934663i	$0.384299\pi$
$90$	0	0
$91$	−1.18034	−0.123733
$92$	7.85410	0.818847
$93$	−10.8541	−1.12552
$94$	26.7984	2.76404
$95$	0	0
$96$	28.4164	2.90024
$97$	6.09017	0.618363	0.309182	−	0.951003i	$-0.399945\pi$
$97$	6.09017	0.618363	0.309182	+	0.951003i	$0.399945\pi$
$98$	−18.1803	−1.83649
$99$	−17.7984	−1.78880
$100$	0	0
$101$	−11.9443	−1.18850	−0.594250	−	0.804281i	$-0.702551\pi$
$101$	−11.9443	−1.18850	−0.594250	+	0.804281i	$0.702551\pi$
$102$	41.1246	4.07194
$103$	17.3262	1.70720	0.853602	−	0.520925i	$-0.174413\pi$
$103$	17.3262	1.70720	0.853602	+	0.520925i	$0.174413\pi$
$104$	37.3607	3.66352
$105$	0	0
$106$	14.7082	1.42859
$107$	3.00000	0.290021	0.145010	−	0.989430i	$-0.453678\pi$
$107$	3.00000	0.290021	0.145010	+	0.989430i	$0.453678\pi$
$108$	10.8541	1.04444
$109$	−6.23607	−0.597307	−0.298653	−	0.954362i	$-0.596537\pi$
$109$	−6.23607	−0.597307	−0.298653	+	0.954362i	$0.596537\pi$
$110$	0	0
$111$	−4.85410	−0.460731
$112$	−2.32624	−0.219809
$113$	2.94427	0.276974	0.138487	−	0.990364i	$-0.455776\pi$
$113$	2.94427	0.276974	0.138487	+	0.990364i	$0.455776\pi$
$114$	0	0
$115$	0	0
$116$	9.00000	0.835629
$117$	19.2705	1.78156
$118$	−0.236068	−0.0217318
$119$	−1.41641	−0.129842
$120$	0	0
$121$	10.3262	0.938749
$122$	−7.85410	−0.711077
$123$	−29.2705	−2.63923
$124$	−20.1246	−1.80724
$125$	0	0
$126$	−2.38197	−0.212202
$127$	−2.70820	−0.240314	−0.120157	−	0.992755i	$-0.538340\pi$
$127$	−2.70820	−0.240314	−0.120157	+	0.992755i	$0.538340\pi$
$128$	1.09017	0.0963583
$129$	7.47214	0.657885
$130$	0	0
$131$	−1.14590	−0.100118	−0.0500588	−	0.998746i	$-0.515941\pi$
$131$	−1.14590	−0.100118	−0.0500588	+	0.998746i	$0.515941\pi$
$132$	−58.6869	−5.10804
$133$	0	0
$134$	−12.3262	−1.06482
$135$	0	0
$136$	44.8328	3.84438
$137$	21.1803	1.80956	0.904779	−	0.425881i	$-0.140036\pi$
$137$	21.1803	1.80956	0.904779	+	0.425881i	$0.140036\pi$
$138$	11.0902	0.944058
$139$	14.8541	1.25991	0.629954	−	0.776632i	$-0.283074\pi$
$139$	14.8541	1.25991	0.629954	+	0.776632i	$0.283074\pi$
$140$	0	0
$141$	26.7984	2.25683
$142$	−9.85410	−0.826938
$143$	−23.0902	−1.93090
$144$	37.9787	3.16489
$145$	0	0
$146$	2.61803	0.216670
$147$	−18.1803	−1.49949
$148$	−9.00000	−0.739795
$149$	−5.38197	−0.440908	−0.220454	−	0.975397i	$-0.570754\pi$
$149$	−5.38197	−0.440908	−0.220454	+	0.975397i	$0.570754\pi$
$150$	0	0
$151$	9.61803	0.782705	0.391352	−	0.920241i	$-0.372007\pi$
$151$	9.61803	0.782705	0.391352	+	0.920241i	$0.372007\pi$
$152$	0	0
$153$	23.1246	1.86951
$154$	2.85410	0.229990
$155$	0	0
$156$	63.5410	5.08735
$157$	−5.03444	−0.401792	−0.200896	−	0.979613i	$-0.564385\pi$
$157$	−5.03444	−0.401792	−0.200896	+	0.979613i	$0.564385\pi$
$158$	−20.9443	−1.66624
$159$	14.7082	1.16644
$160$	0	0
$161$	−0.381966	−0.0301031
$162$	−14.9443	−1.17413
$163$	−17.4721	−1.36852	−0.684262	−	0.729237i	$-0.739875\pi$
$163$	−17.4721	−1.36852	−0.684262	+	0.729237i	$0.739875\pi$
$164$	−54.2705	−4.23781
$165$	0	0
$166$	−11.7082	−0.908733
$167$	3.76393	0.291262	0.145631	−	0.989339i	$-0.453479\pi$
$167$	3.76393	0.291262	0.145631	+	0.989339i	$0.453479\pi$
$168$	−4.61803	−0.356289
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	13.8541	1.05637
$173$	−21.8885	−1.66416	−0.832078	−	0.554659i	$-0.812849\pi$
$173$	−21.8885	−1.66416	−0.832078	+	0.554659i	$0.812849\pi$
$174$	12.7082	0.963406
$175$	0	0
$176$	−45.5066	−3.43019
$177$	−0.236068	−0.0177440
$178$	17.5623	1.31635
$179$	−19.4721	−1.45542	−0.727708	−	0.685887i	$-0.759414\pi$
$179$	−19.4721	−1.45542	−0.727708	+	0.685887i	$0.759414\pi$
$180$	0	0
$181$	−21.7082	−1.61356	−0.806779	−	0.590853i	$-0.798791\pi$
$181$	−21.7082	−1.61356	−0.806779	+	0.590853i	$0.798791\pi$
$182$	−3.09017	−0.229059
$183$	−7.85410	−0.580592
$184$	12.0902	0.891299
$185$	0	0
$186$	−28.4164	−2.08359
$187$	−27.7082	−2.02622
$188$	49.6869	3.62379
$189$	−0.527864	−0.0383965
$190$	0	0
$191$	23.9443	1.73255	0.866273	−	0.499570i	$-0.166509\pi$
$191$	23.9443	1.73255	0.866273	+	0.499570i	$0.166509\pi$
$192$	22.7984	1.64533
$193$	4.00000	0.287926	0.143963	−	0.989583i	$-0.454015\pi$
$193$	4.00000	0.287926	0.143963	+	0.989583i	$0.454015\pi$
$194$	15.9443	1.14473
$195$	0	0
$196$	−33.7082	−2.40773
$197$	−25.4721	−1.81481	−0.907407	−	0.420252i	$-0.861942\pi$
$197$	−25.4721	−1.81481	−0.907407	+	0.420252i	$0.861942\pi$
$198$	−46.5967	−3.31149
$199$	−13.4164	−0.951064	−0.475532	−	0.879698i	$-0.657744\pi$
$199$	−13.4164	−0.951064	−0.475532	+	0.879698i	$0.657744\pi$
$200$	0	0
$201$	−12.3262	−0.869426
$202$	−31.2705	−2.20019
$203$	−0.437694	−0.0307201
$204$	76.2492	5.33851
$205$	0	0
$206$	45.3607	3.16043
$207$	6.23607	0.433437
$208$	49.2705	3.41630
$209$	0	0
$210$	0	0
$211$	18.0902	1.24538	0.622689	−	0.782469i	$-0.286040\pi$
$211$	18.0902	1.24538	0.622689	+	0.782469i	$0.286040\pi$
$212$	27.2705	1.87295
$213$	−9.85410	−0.675192
$214$	7.85410	0.536895
$215$	0	0
$216$	16.7082	1.13685
$217$	0.978714	0.0664394
$218$	−16.3262	−1.10575
$219$	2.61803	0.176910
$220$	0	0
$221$	30.0000	2.01802
$222$	−12.7082	−0.852919
$223$	5.76393	0.385981	0.192991	−	0.981201i	$-0.438181\pi$
$223$	5.76393	0.385981	0.192991	+	0.981201i	$0.438181\pi$
$224$	−2.56231	−0.171201
$225$	0	0
$226$	7.70820	0.512742
$227$	−13.6525	−0.906147	−0.453073	−	0.891473i	$-0.649672\pi$
$227$	−13.6525	−0.906147	−0.453073	+	0.891473i	$0.649672\pi$
$228$	0	0
$229$	−19.3820	−1.28080	−0.640398	−	0.768043i	$-0.721231\pi$
$229$	−19.3820	−1.28080	−0.640398	+	0.768043i	$0.721231\pi$
$230$	0	0
$231$	2.85410	0.187786
$232$	13.8541	0.909566
$233$	29.1803	1.91167	0.955834	−	0.293908i	$-0.0949558\pi$
$233$	29.1803	1.91167	0.955834	+	0.293908i	$0.0949558\pi$
$234$	50.4508	3.29808
$235$	0	0
$236$	−0.437694	−0.0284915
$237$	−20.9443	−1.36048
$238$	−3.70820	−0.240367
$239$	3.43769	0.222366	0.111183	−	0.993800i	$-0.464536\pi$
$239$	3.43769	0.222366	0.111183	+	0.993800i	$0.464536\pi$
$240$	0	0
$241$	22.1246	1.42517	0.712586	−	0.701585i	$-0.247524\pi$
$241$	22.1246	1.42517	0.712586	+	0.701585i	$0.247524\pi$
$242$	27.0344	1.73784
$243$	−21.6525	−1.38901
$244$	−14.5623	−0.932256
$245$	0	0
$246$	−76.6312	−4.88583
$247$	0	0
$248$	−30.9787	−1.96715
$249$	−11.7082	−0.741977
$250$	0	0
$251$	−7.18034	−0.453219	−0.226610	−	0.973986i	$-0.572764\pi$
$251$	−7.18034	−0.453219	−0.226610	+	0.973986i	$0.572764\pi$
$252$	−4.41641	−0.278208
$253$	−7.47214	−0.469769
$254$	−7.09017	−0.444877
$255$	0	0
$256$	−14.5623	−0.910144
$257$	22.4721	1.40177	0.700887	−	0.713273i	$-0.252788\pi$
$257$	22.4721	1.40177	0.700887	+	0.713273i	$0.252788\pi$
$258$	19.5623	1.21790
$259$	0.437694	0.0271970
$260$	0	0
$261$	7.14590	0.442320
$262$	−3.00000	−0.185341
$263$	−15.8885	−0.979730	−0.489865	−	0.871798i	$-0.662954\pi$
$263$	−15.8885	−0.979730	−0.489865	+	0.871798i	$0.662954\pi$
$264$	−90.3394	−5.56001
$265$	0	0
$266$	0	0
$267$	17.5623	1.07480
$268$	−22.8541	−1.39604
$269$	−17.3262	−1.05640	−0.528200	−	0.849120i	$-0.677133\pi$
$269$	−17.3262	−1.05640	−0.528200	+	0.849120i	$0.677133\pi$
$270$	0	0
$271$	18.6180	1.13097	0.565483	−	0.824760i	$-0.308690\pi$
$271$	18.6180	1.13097	0.565483	+	0.824760i	$0.308690\pi$
$272$	59.1246	3.58496
$273$	−3.09017	−0.187026
$274$	55.4508	3.34991
$275$	0	0
$276$	20.5623	1.23771
$277$	−1.41641	−0.0851037	−0.0425519	−	0.999094i	$-0.513549\pi$
$277$	−1.41641	−0.0851037	−0.0425519	+	0.999094i	$0.513549\pi$
$278$	38.8885	2.33238
$279$	−15.9787	−0.956621
$280$	0	0
$281$	10.5066	0.626770	0.313385	−	0.949626i	$-0.398537\pi$
$281$	10.5066	0.626770	0.313385	+	0.949626i	$0.398537\pi$
$282$	70.1591	4.17791
$283$	−2.38197	−0.141593	−0.0707966	−	0.997491i	$-0.522554\pi$
$283$	−2.38197	−0.141593	−0.0707966	+	0.997491i	$0.522554\pi$
$284$	−18.2705	−1.08416
$285$	0	0
$286$	−60.4508	−3.57453
$287$	2.63932	0.155794
$288$	41.8328	2.46502
$289$	19.0000	1.11765
$290$	0	0
$291$	15.9443	0.934670
$292$	4.85410	0.284065
$293$	−7.03444	−0.410956	−0.205478	−	0.978662i	$-0.565875\pi$
$293$	−7.03444	−0.410956	−0.205478	+	0.978662i	$0.565875\pi$
$294$	−47.5967	−2.77590
$295$	0	0
$296$	−13.8541	−0.805253
$297$	−10.3262	−0.599189
$298$	−14.0902	−0.816222
$299$	8.09017	0.467867
$300$	0	0
$301$	−0.673762	−0.0388350
$302$	25.1803	1.44897
$303$	−31.2705	−1.79644
$304$	0	0
$305$	0	0
$306$	60.5410	3.46090
$307$	21.2705	1.21397	0.606986	−	0.794712i	$-0.292378\pi$
$307$	21.2705	1.21397	0.606986	+	0.794712i	$0.292378\pi$
$308$	5.29180	0.301528
$309$	45.3607	2.58048
$310$	0	0
$311$	−24.9443	−1.41446	−0.707230	−	0.706984i	$-0.750055\pi$
$311$	−24.9443	−1.41446	−0.707230	+	0.706984i	$0.750055\pi$
$312$	97.8115	5.53749
$313$	−14.7984	−0.836454	−0.418227	−	0.908343i	$-0.637348\pi$
$313$	−14.7984	−0.836454	−0.418227	+	0.908343i	$0.637348\pi$
$314$	−13.1803	−0.743810
$315$	0	0
$316$	−38.8328	−2.18452
$317$	−8.76393	−0.492231	−0.246116	−	0.969240i	$-0.579154\pi$
$317$	−8.76393	−0.492231	−0.246116	+	0.969240i	$0.579154\pi$
$318$	38.5066	2.15934
$319$	−8.56231	−0.479397
$320$	0	0
$321$	7.85410	0.438373
$322$	−1.00000	−0.0557278
$323$	0	0
$324$	−27.7082	−1.53934
$325$	0	0
$326$	−45.7426	−2.53345
$327$	−16.3262	−0.902843
$328$	−83.5410	−4.61278
$329$	−2.41641	−0.133221
$330$	0	0
$331$	13.7082	0.753471	0.376736	−	0.926321i	$-0.377047\pi$
$331$	13.7082	0.753471	0.376736	+	0.926321i	$0.377047\pi$
$332$	−21.7082	−1.19139
$333$	−7.14590	−0.391593
$334$	9.85410	0.539192
$335$	0	0
$336$	−6.09017	−0.332246
$337$	28.0000	1.52526	0.762629	−	0.646837i	$-0.223908\pi$
$337$	28.0000	1.52526	0.762629	+	0.646837i	$0.223908\pi$
$338$	31.4164	1.70883
$339$	7.70820	0.418652
$340$	0	0
$341$	19.1459	1.03681
$342$	0	0
$343$	3.29180	0.177740
$344$	21.3262	1.14983
$345$	0	0
$346$	−57.3050	−3.08073
$347$	−15.8885	−0.852942	−0.426471	−	0.904501i	$-0.640243\pi$
$347$	−15.8885	−0.852942	−0.426471	+	0.904501i	$0.640243\pi$
$348$	23.5623	1.26307
$349$	−20.7426	−1.11033	−0.555164	−	0.831741i	$-0.687345\pi$
$349$	−20.7426	−1.11033	−0.555164	+	0.831741i	$0.687345\pi$
$350$	0	0
$351$	11.1803	0.596762
$352$	−50.1246	−2.67165
$353$	−33.1591	−1.76488	−0.882439	−	0.470427i	$-0.844100\pi$
$353$	−33.1591	−1.76488	−0.882439	+	0.470427i	$0.844100\pi$
$354$	−0.618034	−0.0328481
$355$	0	0
$356$	32.5623	1.72580
$357$	−3.70820	−0.196259
$358$	−50.9787	−2.69431
$359$	28.6180	1.51040	0.755201	−	0.655493i	$-0.227539\pi$
$359$	28.6180	1.51040	0.755201	+	0.655493i	$0.227539\pi$
$360$	0	0
$361$	0	0
$362$	−56.8328	−2.98707
$363$	27.0344	1.41894
$364$	−5.72949	−0.300307
$365$	0	0
$366$	−20.5623	−1.07481
$367$	29.6525	1.54785	0.773923	−	0.633279i	$-0.218292\pi$
$367$	29.6525	1.54785	0.773923	+	0.633279i	$0.218292\pi$
$368$	15.9443	0.831153
$369$	−43.0902	−2.24318
$370$	0	0
$371$	−1.32624	−0.0688548
$372$	−52.6869	−2.73169
$373$	6.70820	0.347338	0.173669	−	0.984804i	$-0.444438\pi$
$373$	6.70820	0.347338	0.173669	+	0.984804i	$0.444438\pi$
$374$	−72.5410	−3.75101
$375$	0	0
$376$	76.4853	3.94443
$377$	9.27051	0.477456
$378$	−1.38197	−0.0710807
$379$	−26.8328	−1.37831	−0.689155	−	0.724614i	$-0.742018\pi$
$379$	−26.8328	−1.37831	−0.689155	+	0.724614i	$0.742018\pi$
$380$	0	0
$381$	−7.09017	−0.363240
$382$	62.6869	3.20734
$383$	−31.9230	−1.63119	−0.815594	−	0.578624i	$-0.803590\pi$
$383$	−31.9230	−1.63119	−0.815594	+	0.578624i	$0.803590\pi$
$384$	2.85410	0.145648
$385$	0	0
$386$	10.4721	0.533018
$387$	11.0000	0.559161
$388$	29.5623	1.50080
$389$	−8.67376	−0.439777	−0.219889	−	0.975525i	$-0.570569\pi$
$389$	−8.67376	−0.439777	−0.219889	+	0.975525i	$0.570569\pi$
$390$	0	0
$391$	9.70820	0.490965
$392$	−51.8885	−2.62077
$393$	−3.00000	−0.151330
$394$	−66.6869	−3.35964
$395$	0	0
$396$	−86.3951	−4.34152
$397$	−18.8885	−0.947989	−0.473994	−	0.880528i	$-0.657188\pi$
$397$	−18.8885	−0.947989	−0.473994	+	0.880528i	$0.657188\pi$
$398$	−35.1246	−1.76064
$399$	0	0
$400$	0	0
$401$	−11.2361	−0.561102	−0.280551	−	0.959839i	$-0.590517\pi$
$401$	−11.2361	−0.561102	−0.280551	+	0.959839i	$0.590517\pi$
$402$	−32.2705	−1.60951
$403$	−20.7295	−1.03261
$404$	−57.9787	−2.88455
$405$	0	0
$406$	−1.14590	−0.0568700
$407$	8.56231	0.424418
$408$	117.374	5.81087
$409$	−11.7082	−0.578933	−0.289467	−	0.957188i	$-0.593478\pi$
$409$	−11.7082	−0.578933	−0.289467	+	0.957188i	$0.593478\pi$
$410$	0	0
$411$	55.4508	2.73519
$412$	84.1033	4.14347
$413$	0.0212862	0.00104743
$414$	16.3262	0.802391
$415$	0	0
$416$	54.2705	2.66083
$417$	38.8885	1.90438
$418$	0	0
$419$	18.9443	0.925488	0.462744	−	0.886492i	$-0.346865\pi$
$419$	18.9443	0.925488	0.462744	+	0.886492i	$0.346865\pi$
$420$	0	0
$421$	14.1246	0.688391	0.344196	−	0.938898i	$-0.388152\pi$
$421$	14.1246	0.688391	0.344196	+	0.938898i	$0.388152\pi$
$422$	47.3607	2.30548
$423$	39.4508	1.91817
$424$	41.9787	2.03867
$425$	0	0
$426$	−25.7984	−1.24994
$427$	0.708204	0.0342724
$428$	14.5623	0.703896
$429$	−60.4508	−2.91859
$430$	0	0
$431$	2.81966	0.135818	0.0679091	−	0.997692i	$-0.478367\pi$
$431$	2.81966	0.135818	0.0679091	+	0.997692i	$0.478367\pi$
$432$	22.0344	1.06013
$433$	22.2148	1.06757	0.533787	−	0.845619i	$-0.320768\pi$
$433$	22.2148	1.06757	0.533787	+	0.845619i	$0.320768\pi$
$434$	2.56231	0.122995
$435$	0	0
$436$	−30.2705	−1.44969
$437$	0	0
$438$	6.85410	0.327502
$439$	5.47214	0.261171	0.130585	−	0.991437i	$-0.458314\pi$
$439$	5.47214	0.261171	0.130585	+	0.991437i	$0.458314\pi$
$440$	0	0
$441$	−26.7639	−1.27447
$442$	78.5410	3.73582
$443$	−15.3607	−0.729808	−0.364904	−	0.931045i	$-0.618898\pi$
$443$	−15.3607	−0.729808	−0.364904	+	0.931045i	$0.618898\pi$
$444$	−23.5623	−1.11822
$445$	0	0
$446$	15.0902	0.714540
$447$	−14.0902	−0.666442
$448$	−2.05573	−0.0971240
$449$	−32.5279	−1.53508	−0.767542	−	0.640998i	$-0.778521\pi$
$449$	−32.5279	−1.53508	−0.767542	+	0.640998i	$0.778521\pi$
$450$	0	0
$451$	51.6312	2.43122
$452$	14.2918	0.672230
$453$	25.1803	1.18308
$454$	−35.7426	−1.67749
$455$	0	0
$456$	0	0
$457$	6.00000	0.280668	0.140334	−	0.990104i	$-0.455182\pi$
$457$	6.00000	0.280668	0.140334	+	0.990104i	$0.455182\pi$
$458$	−50.7426	−2.37105
$459$	13.4164	0.626224
$460$	0	0
$461$	−18.9443	−0.882323	−0.441161	−	0.897428i	$-0.645433\pi$
$461$	−18.9443	−0.882323	−0.441161	+	0.897428i	$0.645433\pi$
$462$	7.47214	0.347635
$463$	−3.96556	−0.184295	−0.0921476	−	0.995745i	$-0.529373\pi$
$463$	−3.96556	−0.184295	−0.0921476	+	0.995745i	$0.529373\pi$
$464$	18.2705	0.848187
$465$	0	0
$466$	76.3951	3.53894
$467$	11.2918	0.522522	0.261261	−	0.965268i	$-0.415862\pi$
$467$	11.2918	0.522522	0.261261	+	0.965268i	$0.415862\pi$
$468$	93.5410	4.32394
$469$	1.11146	0.0513223
$470$	0	0
$471$	−13.1803	−0.607318
$472$	−0.673762	−0.0310124
$473$	−13.1803	−0.606033
$474$	−54.8328	−2.51855
$475$	0	0
$476$	−6.87539	−0.315133
$477$	21.6525	0.991399
$478$	9.00000	0.411650
$479$	−8.38197	−0.382982	−0.191491	−	0.981494i	$-0.561332\pi$
$479$	−8.38197	−0.382982	−0.191491	+	0.981494i	$0.561332\pi$
$480$	0	0
$481$	−9.27051	−0.422699
$482$	57.9230	2.63832
$483$	−1.00000	−0.0455016
$484$	50.1246	2.27839
$485$	0	0
$486$	−56.6869	−2.57137
$487$	−8.58359	−0.388960	−0.194480	−	0.980907i	$-0.562302\pi$
$487$	−8.58359	−0.388960	−0.194480	+	0.980907i	$0.562302\pi$
$488$	−22.4164	−1.01474
$489$	−45.7426	−2.06855
$490$	0	0
$491$	24.3262	1.09783	0.548914	−	0.835879i	$-0.315041\pi$
$491$	24.3262	1.09783	0.548914	+	0.835879i	$0.315041\pi$
$492$	−142.082	−6.40555
$493$	11.1246	0.501027
$494$	0	0
$495$	0	0
$496$	−40.8541	−1.83440
$497$	0.888544	0.0398566
$498$	−30.6525	−1.37357
$499$	−4.65248	−0.208273	−0.104137	−	0.994563i	$-0.533208\pi$
$499$	−4.65248	−0.208273	−0.104137	+	0.994563i	$0.533208\pi$
$500$	0	0
$501$	9.85410	0.440249
$502$	−18.7984	−0.839012
$503$	19.9443	0.889271	0.444636	−	0.895712i	$-0.353333\pi$
$503$	19.9443	0.889271	0.444636	+	0.895712i	$0.353333\pi$
$504$	−6.79837	−0.302824
$505$	0	0
$506$	−19.5623	−0.869651
$507$	31.4164	1.39525
$508$	−13.1459	−0.583255
$509$	13.7984	0.611602	0.305801	−	0.952095i	$-0.401076\pi$
$509$	13.7984	0.611602	0.305801	+	0.952095i	$0.401076\pi$
$510$	0	0
$511$	−0.236068	−0.0104430
$512$	−40.3050	−1.78124
$513$	0	0
$514$	58.8328	2.59500
$515$	0	0
$516$	36.2705	1.59672
$517$	−47.2705	−2.07895
$518$	1.14590	0.0503479
$519$	−57.3050	−2.51541
$520$	0	0
$521$	2.61803	0.114698	0.0573491	−	0.998354i	$-0.481735\pi$
$521$	2.61803	0.114698	0.0573491	+	0.998354i	$0.481735\pi$
$522$	18.7082	0.818836
$523$	−38.5967	−1.68772	−0.843859	−	0.536565i	$-0.819722\pi$
$523$	−38.5967	−1.68772	−0.843859	+	0.536565i	$0.819722\pi$
$524$	−5.56231	−0.242990
$525$	0	0
$526$	−41.5967	−1.81370
$527$	−24.8754	−1.08359
$528$	−119.138	−5.18481
$529$	−20.3820	−0.886172
$530$	0	0
$531$	−0.347524	−0.0150813
$532$	0	0
$533$	−55.9017	−2.42137
$534$	45.9787	1.98969
$535$	0	0
$536$	−35.1803	−1.51956
$537$	−50.9787	−2.19989
$538$	−45.3607	−1.95564
$539$	32.0689	1.38130
$540$	0	0
$541$	−16.5967	−0.713550	−0.356775	−	0.934190i	$-0.616124\pi$
$541$	−16.5967	−0.713550	−0.356775	+	0.934190i	$0.616124\pi$
$542$	48.7426	2.09368
$543$	−56.8328	−2.43893
$544$	65.1246	2.79219
$545$	0	0
$546$	−8.09017	−0.346227
$547$	−8.97871	−0.383902	−0.191951	−	0.981405i	$-0.561481\pi$
$547$	−8.97871	−0.383902	−0.191951	+	0.981405i	$0.561481\pi$
$548$	102.812	4.39189
$549$	−11.5623	−0.493467
$550$	0	0
$551$	0	0
$552$	31.6525	1.34722
$553$	1.88854	0.0803091
$554$	−3.70820	−0.157546
$555$	0	0
$556$	72.1033	3.05786
$557$	12.5279	0.530823	0.265411	−	0.964135i	$-0.414492\pi$
$557$	12.5279	0.530823	0.265411	+	0.964135i	$0.414492\pi$
$558$	−41.8328	−1.77092
$559$	14.2705	0.603578
$560$	0	0
$561$	−72.5410	−3.06268
$562$	27.5066	1.16029
$563$	−7.81966	−0.329559	−0.164780	−	0.986330i	$-0.552691\pi$
$563$	−7.81966	−0.329559	−0.164780	+	0.986330i	$0.552691\pi$
$564$	130.082	5.47744
$565$	0	0
$566$	−6.23607	−0.262121
$567$	1.34752	0.0565907
$568$	−28.1246	−1.18008
$569$	20.5623	0.862017	0.431008	−	0.902348i	$-0.358158\pi$
$569$	20.5623	0.862017	0.431008	+	0.902348i	$0.358158\pi$
$570$	0	0
$571$	20.1459	0.843080	0.421540	−	0.906810i	$-0.361490\pi$
$571$	20.1459	0.843080	0.421540	+	0.906810i	$0.361490\pi$
$572$	−112.082	−4.68639
$573$	62.6869	2.61878
$574$	6.90983	0.288411
$575$	0	0
$576$	33.5623	1.39843
$577$	−24.2361	−1.00896	−0.504480	−	0.863423i	$-0.668316\pi$
$577$	−24.2361	−1.00896	−0.504480	+	0.863423i	$0.668316\pi$
$578$	49.7426	2.06902
$579$	10.4721	0.435207
$580$	0	0
$581$	1.05573	0.0437990
$582$	41.7426	1.73029
$583$	−25.9443	−1.07450
$584$	7.47214	0.309199
$585$	0	0
$586$	−18.4164	−0.760775
$587$	37.7639	1.55868	0.779342	−	0.626599i	$-0.215553\pi$
$587$	37.7639	1.55868	0.779342	+	0.626599i	$0.215553\pi$
$588$	−88.2492	−3.63934
$589$	0	0
$590$	0	0
$591$	−66.6869	−2.74313
$592$	−18.2705	−0.750913
$593$	−15.6525	−0.642770	−0.321385	−	0.946949i	$-0.604148\pi$
$593$	−15.6525	−0.642770	−0.321385	+	0.946949i	$0.604148\pi$
$594$	−27.0344	−1.10924
$595$	0	0
$596$	−26.1246	−1.07011
$597$	−35.1246	−1.43755
$598$	21.1803	0.866129
$599$	37.3607	1.52652	0.763258	−	0.646094i	$-0.223599\pi$
$599$	37.3607	1.52652	0.763258	+	0.646094i	$0.223599\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	−1.76393	−0.0718925
$603$	−18.1459	−0.738958
$604$	46.6869	1.89966
$605$	0	0
$606$	−81.8673	−3.32563
$607$	3.56231	0.144590	0.0722948	−	0.997383i	$-0.476968\pi$
$607$	3.56231	0.144590	0.0722948	+	0.997383i	$0.476968\pi$
$608$	0	0
$609$	−1.14590	−0.0464341
$610$	0	0
$611$	51.1803	2.07053
$612$	112.249	4.53741
$613$	−21.9443	−0.886321	−0.443160	−	0.896442i	$-0.646143\pi$
$613$	−21.9443	−0.886321	−0.443160	+	0.896442i	$0.646143\pi$
$614$	55.6869	2.24734
$615$	0	0
$616$	8.14590	0.328208
$617$	24.0902	0.969834	0.484917	−	0.874560i	$-0.338850\pi$
$617$	24.0902	0.969834	0.484917	+	0.874560i	$0.338850\pi$
$618$	118.756	4.77706
$619$	−25.9443	−1.04279	−0.521394	−	0.853316i	$-0.674588\pi$
$619$	−25.9443	−1.04279	−0.521394	+	0.853316i	$0.674588\pi$
$620$	0	0
$621$	3.61803	0.145187
$622$	−65.3050	−2.61849
$623$	−1.58359	−0.0634453
$624$	128.992	5.16381
$625$	0	0
$626$	−38.7426	−1.54847
$627$	0	0
$628$	−24.4377	−0.975170
$629$	−11.1246	−0.443567
$630$	0	0
$631$	15.0000	0.597141	0.298570	−	0.954388i	$-0.403490\pi$
$631$	15.0000	0.597141	0.298570	+	0.954388i	$0.403490\pi$
$632$	−59.7771	−2.37780
$633$	47.3607	1.88242
$634$	−22.9443	−0.911233
$635$	0	0
$636$	71.3951	2.83100
$637$	−34.7214	−1.37571
$638$	−22.4164	−0.887474
$639$	−14.5066	−0.573871
$640$	0	0
$641$	−39.2705	−1.55109	−0.775546	−	0.631291i	$-0.782525\pi$
$641$	−39.2705	−1.55109	−0.775546	+	0.631291i	$0.782525\pi$
$642$	20.5623	0.811529
$643$	17.5967	0.693948	0.346974	−	0.937875i	$-0.387209\pi$
$643$	17.5967	0.693948	0.346974	+	0.937875i	$0.387209\pi$
$644$	−1.85410	−0.0730619
$645$	0	0
$646$	0	0
$647$	26.3050	1.03415	0.517077	−	0.855939i	$-0.327020\pi$
$647$	26.3050	1.03415	0.517077	+	0.855939i	$0.327020\pi$
$648$	−42.6525	−1.67555
$649$	0.416408	0.0163454
$650$	0	0
$651$	2.56231	0.100425
$652$	−84.8115	−3.32148
$653$	−1.14590	−0.0448425	−0.0224212	−	0.999749i	$-0.507137\pi$
$653$	−1.14590	−0.0448425	−0.0224212	+	0.999749i	$0.507137\pi$
$654$	−42.7426	−1.67137
$655$	0	0
$656$	−110.172	−4.30150
$657$	3.85410	0.150363
$658$	−6.32624	−0.246622
$659$	−33.5279	−1.30606	−0.653030	−	0.757332i	$-0.726502\pi$
$659$	−33.5279	−1.30606	−0.653030	+	0.757332i	$0.726502\pi$
$660$	0	0
$661$	−9.70820	−0.377605	−0.188803	−	0.982015i	$-0.560461\pi$
$661$	−9.70820	−0.377605	−0.188803	+	0.982015i	$0.560461\pi$
$662$	35.8885	1.39485
$663$	78.5410	3.05028
$664$	−33.4164	−1.29681
$665$	0	0
$666$	−18.7082	−0.724928
$667$	3.00000	0.116160
$668$	18.2705	0.706907
$669$	15.0902	0.583420
$670$	0	0
$671$	13.8541	0.534832
$672$	−6.70820	−0.258775
$673$	−46.8885	−1.80742	−0.903710	−	0.428145i	$-0.859167\pi$
$673$	−46.8885	−1.80742	−0.903710	+	0.428145i	$0.859167\pi$
$674$	73.3050	2.82360
$675$	0	0
$676$	58.2492	2.24035
$677$	−10.6180	−0.408084	−0.204042	−	0.978962i	$-0.565408\pi$
$677$	−10.6180	−0.408084	−0.204042	+	0.978962i	$0.565408\pi$
$678$	20.1803	0.775021
$679$	−1.43769	−0.0551736
$680$	0	0
$681$	−35.7426	−1.36966
$682$	50.1246	1.91937
$683$	14.5279	0.555893	0.277947	−	0.960597i	$-0.410346\pi$
$683$	14.5279	0.555893	0.277947	+	0.960597i	$0.410346\pi$
$684$	0	0
$685$	0	0
$686$	8.61803	0.329038
$687$	−50.7426	−1.93595
$688$	28.1246	1.07224
$689$	28.0902	1.07015
$690$	0	0
$691$	12.5279	0.476582	0.238291	−	0.971194i	$-0.423413\pi$
$691$	12.5279	0.476582	0.238291	+	0.971194i	$0.423413\pi$
$692$	−106.249	−4.03899
$693$	4.20163	0.159607
$694$	−41.5967	−1.57899
$695$	0	0
$696$	36.2705	1.37483
$697$	−67.0820	−2.54091
$698$	−54.3050	−2.05547
$699$	76.3951	2.88953
$700$	0	0
$701$	−47.5755	−1.79690	−0.898450	−	0.439075i	$-0.855306\pi$
$701$	−47.5755	−1.79690	−0.898450	+	0.439075i	$0.855306\pi$
$702$	29.2705	1.10474
$703$	0	0
$704$	−40.2148	−1.51565
$705$	0	0
$706$	−86.8115	−3.26720
$707$	2.81966	0.106044
$708$	−1.14590	−0.0430655
$709$	−2.29180	−0.0860702	−0.0430351	−	0.999074i	$-0.513703\pi$
$709$	−2.29180	−0.0860702	−0.0430351	+	0.999074i	$0.513703\pi$
$710$	0	0
$711$	−30.8328	−1.15632
$712$	50.1246	1.87850
$713$	−6.70820	−0.251224
$714$	−9.70820	−0.363320
$715$	0	0
$716$	−94.5197	−3.53237
$717$	9.00000	0.336111
$718$	74.9230	2.79610
$719$	36.7984	1.37235	0.686174	−	0.727438i	$-0.259289\pi$
$719$	36.7984	1.37235	0.686174	+	0.727438i	$0.259289\pi$
$720$	0	0
$721$	−4.09017	−0.152326
$722$	0	0
$723$	57.9230	2.15418
$724$	−105.374	−3.91619
$725$	0	0
$726$	70.7771	2.62678
$727$	37.4164	1.38770	0.693849	−	0.720121i	$-0.255914\pi$
$727$	37.4164	1.38770	0.693849	+	0.720121i	$0.255914\pi$
$728$	−8.81966	−0.326878
$729$	−39.5623	−1.46527
$730$	0	0
$731$	17.1246	0.633377
$732$	−38.1246	−1.40913
$733$	4.23607	0.156463	0.0782314	−	0.996935i	$-0.475073\pi$
$733$	4.23607	0.156463	0.0782314	+	0.996935i	$0.475073\pi$
$734$	77.6312	2.86542
$735$	0	0
$736$	17.5623	0.647355
$737$	21.7426	0.800901
$738$	−112.812	−4.15265
$739$	31.8328	1.17099	0.585495	−	0.810676i	$-0.300900\pi$
$739$	31.8328	1.17099	0.585495	+	0.810676i	$0.300900\pi$
$740$	0	0
$741$	0	0
$742$	−3.47214	−0.127466
$743$	31.7639	1.16531	0.582653	−	0.812721i	$-0.302015\pi$
$743$	31.7639	1.16531	0.582653	+	0.812721i	$0.302015\pi$
$744$	−81.1033	−2.97339
$745$	0	0
$746$	17.5623	0.643002
$747$	−17.2361	−0.630635
$748$	−134.498	−4.91775
$749$	−0.708204	−0.0258772
$750$	0	0
$751$	−17.5623	−0.640858	−0.320429	−	0.947273i	$-0.603827\pi$
$751$	−17.5623	−0.640858	−0.320429	+	0.947273i	$0.603827\pi$
$752$	100.867	3.67825
$753$	−18.7984	−0.685051
$754$	24.2705	0.883880
$755$	0	0
$756$	−2.56231	−0.0931902
$757$	16.6180	0.603993	0.301996	−	0.953309i	$-0.402347\pi$
$757$	16.6180	0.603993	0.301996	+	0.953309i	$0.402347\pi$
$758$	−70.2492	−2.55157
$759$	−19.5623	−0.710067
$760$	0	0
$761$	−7.36068	−0.266824	−0.133412	−	0.991061i	$-0.542593\pi$
$761$	−7.36068	−0.266824	−0.133412	+	0.991061i	$0.542593\pi$
$762$	−18.5623	−0.672441
$763$	1.47214	0.0532949
$764$	116.228	4.20498
$765$	0	0
$766$	−83.5755	−3.01970
$767$	−0.450850	−0.0162792
$768$	−38.1246	−1.37570
$769$	−46.6869	−1.68357	−0.841787	−	0.539810i	$-0.818496\pi$
$769$	−46.6869	−1.68357	−0.841787	+	0.539810i	$0.818496\pi$
$770$	0	0
$771$	58.8328	2.11881
$772$	19.4164	0.698812
$773$	2.74265	0.0986461	0.0493231	−	0.998783i	$-0.484294\pi$
$773$	2.74265	0.0986461	0.0493231	+	0.998783i	$0.484294\pi$
$774$	28.7984	1.03514
$775$	0	0
$776$	45.5066	1.63359
$777$	1.14590	0.0411089
$778$	−22.7082	−0.814129
$779$	0	0
$780$	0	0
$781$	17.3820	0.621976
$782$	25.4164	0.908889
$783$	4.14590	0.148162
$784$	−68.4296	−2.44391
$785$	0	0
$786$	−7.85410	−0.280147
$787$	49.4853	1.76396	0.881980	−	0.471287i	$-0.156210\pi$
$787$	49.4853	1.76396	0.881980	+	0.471287i	$0.156210\pi$
$788$	−123.644	−4.40465
$789$	−41.5967	−1.48088
$790$	0	0
$791$	−0.695048	−0.0247131
$792$	−132.992	−4.72566
$793$	−15.0000	−0.532666
$794$	−49.4508	−1.75494
$795$	0	0
$796$	−65.1246	−2.30828
$797$	20.1803	0.714824	0.357412	−	0.933947i	$-0.383659\pi$
$797$	20.1803	0.714824	0.357412	+	0.933947i	$0.383659\pi$
$798$	0	0
$799$	61.4164	2.17276
$800$	0	0
$801$	25.8541	0.913510
$802$	−29.4164	−1.03873
$803$	−4.61803	−0.162967
$804$	−59.8328	−2.11014
$805$	0	0
$806$	−54.2705	−1.91160
$807$	−45.3607	−1.59677
$808$	−89.2492	−3.13978
$809$	40.0902	1.40950	0.704748	−	0.709458i	$-0.251060\pi$
$809$	40.0902	1.40950	0.704748	+	0.709458i	$0.251060\pi$
$810$	0	0
$811$	−24.6738	−0.866413	−0.433206	−	0.901295i	$-0.642618\pi$
$811$	−24.6738	−0.866413	−0.433206	+	0.901295i	$0.642618\pi$
$812$	−2.12461	−0.0745593
$813$	48.7426	1.70948
$814$	22.4164	0.785695
$815$	0	0
$816$	154.790	5.41874
$817$	0	0
$818$	−30.6525	−1.07174
$819$	−4.54915	−0.158960
$820$	0	0
$821$	14.6180	0.510173	0.255086	−	0.966918i	$-0.417896\pi$
$821$	14.6180	0.510173	0.255086	+	0.966918i	$0.417896\pi$
$822$	145.172	5.06346
$823$	36.3607	1.26745	0.633727	−	0.773557i	$-0.281524\pi$
$823$	36.3607	1.26745	0.633727	+	0.773557i	$0.281524\pi$
$824$	129.464	4.51009
$825$	0	0
$826$	0.0557281	0.00193903
$827$	17.8328	0.620108	0.310054	−	0.950719i	$-0.399653\pi$
$827$	17.8328	0.620108	0.310054	+	0.950719i	$0.399653\pi$
$828$	30.2705	1.05197
$829$	30.3951	1.05567	0.527833	−	0.849348i	$-0.323005\pi$
$829$	30.3951	1.05567	0.527833	+	0.849348i	$0.323005\pi$
$830$	0	0
$831$	−3.70820	−0.128636
$832$	43.5410	1.50951
$833$	−41.6656	−1.44363
$834$	101.812	3.52544
$835$	0	0
$836$	0	0
$837$	−9.27051	−0.320436
$838$	49.5967	1.71329
$839$	15.6180	0.539194	0.269597	−	0.962973i	$-0.413109\pi$
$839$	15.6180	0.539194	0.269597	+	0.962973i	$0.413109\pi$
$840$	0	0
$841$	−25.5623	−0.881459
$842$	36.9787	1.27437
$843$	27.5066	0.947377
$844$	87.8115	3.02260
$845$	0	0
$846$	103.284	3.55097
$847$	−2.43769	−0.0837602
$848$	55.3607	1.90109
$849$	−6.23607	−0.214021
$850$	0	0
$851$	−3.00000	−0.102839
$852$	−47.8328	−1.63873
$853$	40.9574	1.40236	0.701178	−	0.712986i	$-0.252658\pi$
$853$	40.9574	1.40236	0.701178	+	0.712986i	$0.252658\pi$
$854$	1.85410	0.0634461
$855$	0	0
$856$	22.4164	0.766177
$857$	−30.3820	−1.03783	−0.518914	−	0.854826i	$-0.673664\pi$
$857$	−30.3820	−1.03783	−0.518914	+	0.854826i	$0.673664\pi$
$858$	−158.262	−5.40299
$859$	−35.5967	−1.21455	−0.607273	−	0.794493i	$-0.707736\pi$
$859$	−35.5967	−1.21455	−0.607273	+	0.794493i	$0.707736\pi$
$860$	0	0
$861$	6.90983	0.235486
$862$	7.38197	0.251431
$863$	26.6525	0.907261	0.453630	−	0.891190i	$-0.350129\pi$
$863$	26.6525	0.907261	0.453630	+	0.891190i	$0.350129\pi$
$864$	24.2705	0.825700
$865$	0	0
$866$	58.1591	1.97633
$867$	49.7426	1.68935
$868$	4.75078	0.161252
$869$	36.9443	1.25325
$870$	0	0
$871$	−23.5410	−0.797657
$872$	−46.5967	−1.57796
$873$	23.4721	0.794411
$874$	0	0
$875$	0	0
$876$	12.7082	0.429370
$877$	−30.1803	−1.01912	−0.509559	−	0.860436i	$-0.670191\pi$
$877$	−30.1803	−1.01912	−0.509559	+	0.860436i	$0.670191\pi$
$878$	14.3262	0.483487
$879$	−18.4164	−0.621170
$880$	0	0
$881$	−4.03444	−0.135924	−0.0679619	−	0.997688i	$-0.521650\pi$
$881$	−4.03444	−0.135924	−0.0679619	+	0.997688i	$0.521650\pi$
$882$	−70.0689	−2.35934
$883$	17.9230	0.603156	0.301578	−	0.953441i	$-0.402487\pi$
$883$	17.9230	0.603156	0.301578	+	0.953441i	$0.402487\pi$
$884$	145.623	4.89783
$885$	0	0
$886$	−40.2148	−1.35104
$887$	3.52786	0.118454	0.0592270	−	0.998245i	$-0.481136\pi$
$887$	3.52786	0.118454	0.0592270	+	0.998245i	$0.481136\pi$
$888$	−36.2705	−1.21716
$889$	0.639320	0.0214421
$890$	0	0
$891$	26.3607	0.883116
$892$	27.9787	0.936797
$893$	0	0
$894$	−36.8885	−1.23374
$895$	0	0
$896$	−0.257354	−0.00859760
$897$	21.1803	0.707191
$898$	−85.1591	−2.84179
$899$	−7.68692	−0.256373
$900$	0	0
$901$	33.7082	1.12298
$902$	135.172	4.50074
$903$	−1.76393	−0.0587000
$904$	22.0000	0.731709
$905$	0	0
$906$	65.9230	2.19014
$907$	48.2492	1.60209	0.801045	−	0.598605i	$-0.204278\pi$
$907$	48.2492	1.60209	0.801045	+	0.598605i	$0.204278\pi$
$908$	−66.2705	−2.19926
$909$	−46.0344	−1.52687
$910$	0	0
$911$	51.1033	1.69313	0.846564	−	0.532286i	$-0.178667\pi$
$911$	51.1033	1.69313	0.846564	+	0.532286i	$0.178667\pi$
$912$	0	0
$913$	20.6525	0.683497
$914$	15.7082	0.519581
$915$	0	0
$916$	−94.0820	−3.10856
$917$	0.270510	0.00893302
$918$	35.1246	1.15928
$919$	−12.8885	−0.425154	−0.212577	−	0.977144i	$-0.568186\pi$
$919$	−12.8885	−0.425154	−0.212577	+	0.977144i	$0.568186\pi$
$920$	0	0
$921$	55.6869	1.83495
$922$	−49.5967	−1.63338
$923$	−18.8197	−0.619457
$924$	13.8541	0.455766
$925$	0	0
$926$	−10.3820	−0.341173
$927$	66.7771	2.19325
$928$	20.1246	0.660623
$929$	9.38197	0.307812	0.153906	−	0.988085i	$-0.450815\pi$
$929$	9.38197	0.307812	0.153906	+	0.988085i	$0.450815\pi$
$930$	0	0
$931$	0	0
$932$	141.644	4.63971
$933$	−65.3050	−2.13799
$934$	29.5623	0.967308
$935$	0	0
$936$	143.992	4.70652
$937$	17.5623	0.573736	0.286868	−	0.957970i	$-0.407386\pi$
$937$	17.5623	0.573736	0.286868	+	0.957970i	$0.407386\pi$
$938$	2.90983	0.0950093
$939$	−38.7426	−1.26432
$940$	0	0
$941$	27.5066	0.896689	0.448344	−	0.893861i	$-0.352014\pi$
$941$	27.5066	0.896689	0.448344	+	0.893861i	$0.352014\pi$
$942$	−34.5066	−1.12429
$943$	−18.0902	−0.589097
$944$	−0.888544	−0.0289196
$945$	0	0
$946$	−34.5066	−1.12191
$947$	−48.0000	−1.55979	−0.779895	−	0.625910i	$-0.784728\pi$
$947$	−48.0000	−1.55979	−0.779895	+	0.625910i	$0.784728\pi$
$948$	−101.666	−3.30195
$949$	5.00000	0.162307
$950$	0	0
$951$	−22.9443	−0.744019
$952$	−10.5836	−0.343016
$953$	41.8328	1.35510	0.677549	−	0.735478i	$-0.263042\pi$
$953$	41.8328	1.35510	0.677549	+	0.735478i	$0.263042\pi$
$954$	56.6869	1.83531
$955$	0	0
$956$	16.6869	0.539693
$957$	−22.4164	−0.724620
$958$	−21.9443	−0.708987
$959$	−5.00000	−0.161458
$960$	0	0
$961$	−13.8115	−0.445533
$962$	−24.2705	−0.782513
$963$	11.5623	0.372590
$964$	107.395	3.45896
$965$	0	0
$966$	−2.61803	−0.0842339
$967$	−33.2361	−1.06880	−0.534400	−	0.845232i	$-0.679462\pi$
$967$	−33.2361	−1.06880	−0.534400	+	0.845232i	$0.679462\pi$
$968$	77.1591	2.47999
$969$	0	0
$970$	0	0
$971$	13.2016	0.423660	0.211830	−	0.977306i	$-0.432058\pi$
$971$	13.2016	0.423660	0.211830	+	0.977306i	$0.432058\pi$
$972$	−105.103	−3.37119
$973$	−3.50658	−0.112416
$974$	−22.4721	−0.720054
$975$	0	0
$976$	−29.5623	−0.946266
$977$	22.3475	0.714961	0.357480	−	0.933921i	$-0.383636\pi$
$977$	22.3475	0.714961	0.357480	+	0.933921i	$0.383636\pi$
$978$	−119.756	−3.82937
$979$	−30.9787	−0.990084
$980$	0	0
$981$	−24.0344	−0.767361
$982$	63.6869	2.03233
$983$	7.14590	0.227919	0.113959	−	0.993485i	$-0.463647\pi$
$983$	7.14590	0.227919	0.113959	+	0.993485i	$0.463647\pi$
$984$	−218.713	−6.97232
$985$	0	0
$986$	29.1246	0.927517
$987$	−6.32624	−0.201366
$988$	0	0
$989$	4.61803	0.146845
$990$	0	0
$991$	−24.8541	−0.789517	−0.394758	−	0.918785i	$-0.629172\pi$
$991$	−24.8541	−0.789517	−0.394758	+	0.918785i	$0.629172\pi$
$992$	−45.0000	−1.42875
$993$	35.8885	1.13889
$994$	2.32624	0.0737838
$995$	0	0
$996$	−56.8328	−1.80082
$997$	−1.23607	−0.0391467	−0.0195733	−	0.999808i	$-0.506231\pi$
$997$	−1.23607	−0.0391467	−0.0195733	+	0.999808i	$0.506231\pi$
$998$	−12.1803	−0.385562
$999$	−4.14590	−0.131170

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.v.1.2		2
5.4	even	2		1805.2.a.c.1.1	✓	2
19.18	odd	2		9025.2.a.k.1.1		2
95.94	odd	2		1805.2.a.e.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1805.2.a.c.1.1	✓	2	5.4	even	2
1805.2.a.e.1.2	yes	2	95.94	odd	2
9025.2.a.k.1.1		2	19.18	odd	2
9025.2.a.v.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:	\( N \)	\(=\)	\( 9025 = 5^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9025.a (trivial)

\(f(q)\)	\(=\)	\(q+2.61803 q^{2} +2.61803 q^{3} +4.85410 q^{4} +6.85410 q^{6} -0.236068 q^{7} +7.47214 q^{8} +3.85410 q^{9} +O(q^{10})\)	\(q+2.61803 q^{2} +2.61803 q^{3} +4.85410 q^{4} +6.85410 q^{6} -0.236068 q^{7} +7.47214 q^{8} +3.85410 q^{9} -4.61803 q^{11} +12.7082 q^{12} +5.00000 q^{13} -0.618034 q^{14} +9.85410 q^{16} +6.00000 q^{17} +10.0902 q^{18} -0.618034 q^{21} -12.0902 q^{22} +1.61803 q^{23} +19.5623 q^{24} +13.0902 q^{26} +2.23607 q^{27} -1.14590 q^{28} +1.85410 q^{29} -4.14590 q^{31} +10.8541 q^{32} -12.0902 q^{33} +15.7082 q^{34} +18.7082 q^{36} -1.85410 q^{37} +13.0902 q^{39} -11.1803 q^{41} -1.61803 q^{42} +2.85410 q^{43} -22.4164 q^{44} +4.23607 q^{46} +10.2361 q^{47} +25.7984 q^{48} -6.94427 q^{49} +15.7082 q^{51} +24.2705 q^{52} +5.61803 q^{53} +5.85410 q^{54} -1.76393 q^{56} +4.85410 q^{58} -0.0901699 q^{59} -3.00000 q^{61} -10.8541 q^{62} -0.909830 q^{63} +8.70820 q^{64} -31.6525 q^{66} -4.70820 q^{67} +29.1246 q^{68} +4.23607 q^{69} -3.76393 q^{71} +28.7984 q^{72} +1.00000 q^{73} -4.85410 q^{74} +1.09017 q^{77} +34.2705 q^{78} -8.00000 q^{79} -5.70820 q^{81} -29.2705 q^{82} -4.47214 q^{83} -3.00000 q^{84} +7.47214 q^{86} +4.85410 q^{87} -34.5066 q^{88} +6.70820 q^{89} -1.18034 q^{91} +7.85410 q^{92} -10.8541 q^{93} +26.7984 q^{94} +28.4164 q^{96} +6.09017 q^{97} -18.1803 q^{98} -17.7984 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 7 q^{6} + 4 q^{7} + 6 q^{8} + q^{9}+O(q^{10}) \) 2 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 + 7 * q^6 + 4 * q^7 + 6 * q^8 + q^9	\( 2 q + 3 q^{2} + 3 q^{3} + 3 q^{4} + 7 q^{6} + 4 q^{7} + 6 q^{8} + q^{9} - 7 q^{11} + 12 q^{12} + 10 q^{13} + q^{14} + 13 q^{16} + 12 q^{17} + 9 q^{18} + q^{21} - 13 q^{22} + q^{23} + 19 q^{24} + 15 q^{26} - 9 q^{28} - 3 q^{29} - 15 q^{31} + 15 q^{32} - 13 q^{33} + 18 q^{34} + 24 q^{36} + 3 q^{37} + 15 q^{39} - q^{42} - q^{43} - 18 q^{44} + 4 q^{46} + 16 q^{47} + 27 q^{48} + 4 q^{49} + 18 q^{51} + 15 q^{52} + 9 q^{53} + 5 q^{54} - 8 q^{56} + 3 q^{58} + 11 q^{59} - 6 q^{61} - 15 q^{62} - 13 q^{63} + 4 q^{64} - 32 q^{66} + 4 q^{67} + 18 q^{68} + 4 q^{69} - 12 q^{71} + 33 q^{72} + 2 q^{73} - 3 q^{74} - 9 q^{77} + 35 q^{78} - 16 q^{79} + 2 q^{81} - 25 q^{82} - 6 q^{84} + 6 q^{86} + 3 q^{87} - 31 q^{88} + 20 q^{91} + 9 q^{92} - 15 q^{93} + 29 q^{94} + 30 q^{96} + q^{97} - 14 q^{98} - 11 q^{99}+O(q^{100}) \) 2 * q + 3 * q^2 + 3 * q^3 + 3 * q^4 + 7 * q^6 + 4 * q^7 + 6 * q^8 + q^9 - 7 * q^11 + 12 * q^12 + 10 * q^13 + q^14 + 13 * q^16 + 12 * q^17 + 9 * q^18 + q^21 - 13 * q^22 + q^23 + 19 * q^24 + 15 * q^26 - 9 * q^28 - 3 * q^29 - 15 * q^31 + 15 * q^32 - 13 * q^33 + 18 * q^34 + 24 * q^36 + 3 * q^37 + 15 * q^39 - q^42 - q^43 - 18 * q^44 + 4 * q^46 + 16 * q^47 + 27 * q^48 + 4 * q^49 + 18 * q^51 + 15 * q^52 + 9 * q^53 + 5 * q^54 - 8 * q^56 + 3 * q^58 + 11 * q^59 - 6 * q^61 - 15 * q^62 - 13 * q^63 + 4 * q^64 - 32 * q^66 + 4 * q^67 + 18 * q^68 + 4 * q^69 - 12 * q^71 + 33 * q^72 + 2 * q^73 - 3 * q^74 - 9 * q^77 + 35 * q^78 - 16 * q^79 + 2 * q^81 - 25 * q^82 - 6 * q^84 + 6 * q^86 + 3 * q^87 - 31 * q^88 + 20 * q^91 + 9 * q^92 - 15 * q^93 + 29 * q^94 + 30 * q^96 + q^97 - 14 * q^98 - 11 * q^99

Introduction

L-functions

Modular forms

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Database

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