LMFDB - Embedded newform 9025.2.a.k.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			$-0.618034$ 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)

$f(q)$	$=$	$q-0.381966 q^{2} -0.381966 q^{3} -1.85410 q^{4} +0.145898 q^{6} +4.23607 q^{7} +1.47214 q^{8} -2.85410 q^{9} +O(q^{10})$	\(q-0.381966 q^{2} -0.381966 q^{3} -1.85410 q^{4} +0.145898 q^{6} +4.23607 q^{7} +1.47214 q^{8} -2.85410 q^{9} -2.38197 q^{11} +0.708204 q^{12} -5.00000 q^{13} -1.61803 q^{14} +3.14590 q^{16} +6.00000 q^{17} +1.09017 q^{18} -1.61803 q^{21} +0.909830 q^{22} -0.618034 q^{23} -0.562306 q^{24} +1.90983 q^{26} +2.23607 q^{27} -7.85410 q^{28} +4.85410 q^{29} +10.8541 q^{31} -4.14590 q^{32} +0.909830 q^{33} -2.29180 q^{34} +5.29180 q^{36} -4.85410 q^{37} +1.90983 q^{39} -11.1803 q^{41} +0.618034 q^{42} -3.85410 q^{43} +4.41641 q^{44} +0.236068 q^{46} +5.76393 q^{47} -1.20163 q^{48} +10.9443 q^{49} -2.29180 q^{51} +9.27051 q^{52} -3.38197 q^{53} -0.854102 q^{54} +6.23607 q^{56} -1.85410 q^{58} -11.0902 q^{59} -3.00000 q^{61} -4.14590 q^{62} -12.0902 q^{63} -4.70820 q^{64} -0.347524 q^{66} -8.70820 q^{67} -11.1246 q^{68} +0.236068 q^{69} +8.23607 q^{71} -4.20163 q^{72} +1.00000 q^{73} +1.85410 q^{74} -10.0902 q^{77} -0.729490 q^{78} +8.00000 q^{79} +7.70820 q^{81} +4.27051 q^{82} +4.47214 q^{83} +3.00000 q^{84} +1.47214 q^{86} -1.85410 q^{87} -3.50658 q^{88} +6.70820 q^{89} -21.1803 q^{91} +1.14590 q^{92} -4.14590 q^{93} -2.20163 q^{94} +1.58359 q^{96} +5.09017 q^{97} -4.18034 q^{98} +6.79837 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.381966	−0.270091	−0.135045	−	0.990839i	$-0.543118\pi$
$2$	−0.381966	−0.270091	−0.135045	+	0.990839i	$0.543118\pi$
$3$	−0.381966	−0.220528	−0.110264	−	0.993902i	$-0.535170\pi$
$3$	−0.381966	−0.220528	−0.110264	+	0.993902i	$0.535170\pi$
$4$	−1.85410	−0.927051
$5$	0	0
$5$	0	0
$6$	0.145898	0.0595626
$7$	4.23607	1.60108	0.800542	−	0.599277i	$-0.204545\pi$
$7$	4.23607	1.60108	0.800542	+	0.599277i	$0.204545\pi$
$8$	1.47214	0.520479
$9$	−2.85410	−0.951367
$10$	0	0
$11$	−2.38197	−0.718190	−0.359095	−	0.933301i	$-0.616915\pi$
$11$	−2.38197	−0.718190	−0.359095	+	0.933301i	$0.616915\pi$
$12$	0.708204	0.204441
$13$	−5.00000	−1.38675	−0.693375	−	0.720577i	$-0.743877\pi$
$13$	−5.00000	−1.38675	−0.693375	+	0.720577i	$0.743877\pi$
$14$	−1.61803	−0.432438
$15$	0	0
$16$	3.14590	0.786475
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	1.09017	0.256956
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−1.61803	−0.353084
$22$	0.909830	0.193976
$23$	−0.618034	−0.128869	−0.0644345	−	0.997922i	$-0.520524\pi$
$23$	−0.618034	−0.128869	−0.0644345	+	0.997922i	$0.520524\pi$
$24$	−0.562306	−0.114780
$25$	0	0
$26$	1.90983	0.374548
$27$	2.23607	0.430331
$28$	−7.85410	−1.48429
$29$	4.85410	0.901384	0.450692	−	0.892679i	$-0.351177\pi$
$29$	4.85410	0.901384	0.450692	+	0.892679i	$0.351177\pi$
$30$	0	0
$31$	10.8541	1.94945	0.974727	−	0.223399i	$-0.0717152\pi$
$31$	10.8541	1.94945	0.974727	+	0.223399i	$0.0717152\pi$
$32$	−4.14590	−0.732898
$33$	0.909830	0.158381
$34$	−2.29180	−0.393040
$35$	0	0
$36$	5.29180	0.881966
$37$	−4.85410	−0.798009	−0.399005	−	0.916949i	$-0.630644\pi$
$37$	−4.85410	−0.798009	−0.399005	+	0.916949i	$0.630644\pi$
$38$	0	0
$39$	1.90983	0.305818
$40$	0	0
$41$	−11.1803	−1.74608	−0.873038	−	0.487652i	$-0.837853\pi$
$41$	−11.1803	−1.74608	−0.873038	+	0.487652i	$0.837853\pi$
$42$	0.618034	0.0953647
$43$	−3.85410	−0.587745	−0.293873	−	0.955845i	$-0.594944\pi$
$43$	−3.85410	−0.587745	−0.293873	+	0.955845i	$0.594944\pi$
$44$	4.41641	0.665799
$45$	0	0
$46$	0.236068	0.0348063
$47$	5.76393	0.840756	0.420378	−	0.907349i	$-0.361897\pi$
$47$	5.76393	0.840756	0.420378	+	0.907349i	$0.361897\pi$
$48$	−1.20163	−0.173440
$49$	10.9443	1.56347
$50$	0	0
$51$	−2.29180	−0.320916
$52$	9.27051	1.28559
$53$	−3.38197	−0.464549	−0.232274	−	0.972650i	$-0.574617\pi$
$53$	−3.38197	−0.464549	−0.232274	+	0.972650i	$0.574617\pi$
$54$	−0.854102	−0.116229
$55$	0	0
$56$	6.23607	0.833330
$57$	0	0
$58$	−1.85410	−0.243456
$59$	−11.0902	−1.44382	−0.721909	−	0.691988i	$-0.756735\pi$
$59$	−11.0902	−1.44382	−0.721909	+	0.691988i	$0.756735\pi$
$60$	0	0
$61$	−3.00000	−0.384111	−0.192055	−	0.981384i	$-0.561515\pi$
$61$	−3.00000	−0.384111	−0.192055	+	0.981384i	$0.561515\pi$
$62$	−4.14590	−0.526530
$63$	−12.0902	−1.52322
$64$	−4.70820	−0.588525
$65$	0	0
$66$	−0.347524	−0.0427773
$67$	−8.70820	−1.06388	−0.531938	−	0.846783i	$-0.678536\pi$
$67$	−8.70820	−1.06388	−0.531938	+	0.846783i	$0.678536\pi$
$68$	−11.1246	−1.34906
$69$	0.236068	0.0284192
$70$	0	0
$71$	8.23607	0.977441	0.488721	−	0.872440i	$-0.337464\pi$
$71$	8.23607	0.977441	0.488721	+	0.872440i	$0.337464\pi$
$72$	−4.20163	−0.495166
$73$	1.00000	0.117041	0.0585206	−	0.998286i	$-0.481362\pi$
$73$	1.00000	0.117041	0.0585206	+	0.998286i	$0.481362\pi$
$74$	1.85410	0.215535
$75$	0	0
$76$	0	0
$77$	−10.0902	−1.14988
$78$	−0.729490	−0.0825985
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	0	0
$81$	7.70820	0.856467
$82$	4.27051	0.471599
$83$	4.47214	0.490881	0.245440	−	0.969412i	$-0.421067\pi$
$83$	4.47214	0.490881	0.245440	+	0.969412i	$0.421067\pi$
$84$	3.00000	0.327327
$85$	0	0
$86$	1.47214	0.158745
$87$	−1.85410	−0.198781
$88$	−3.50658	−0.373802
$89$	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
$91$	−21.1803	−2.22030
$92$	1.14590	0.119468
$93$	−4.14590	−0.429910
$94$	−2.20163	−0.227080
$95$	0	0
$96$	1.58359	0.161625
$97$	5.09017	0.516828	0.258414	−	0.966034i	$-0.416800\pi$
$97$	5.09017	0.516828	0.258414	+	0.966034i	$0.416800\pi$
$98$	−4.18034	−0.422278
$99$	6.79837	0.683262
$100$	0	0
$101$	5.94427	0.591477	0.295739	−	0.955269i	$-0.404434\pi$
$101$	5.94427	0.591477	0.295739	+	0.955269i	$0.404434\pi$
$102$	0.875388	0.0866763
$103$	−1.67376	−0.164921	−0.0824603	−	0.996594i	$-0.526278\pi$
$103$	−1.67376	−0.164921	−0.0824603	+	0.996594i	$0.526278\pi$
$104$	−7.36068	−0.721774
$105$	0	0
$106$	1.29180	0.125470
$107$	−3.00000	−0.290021	−0.145010	−	0.989430i	$-0.546322\pi$
$107$	−3.00000	−0.290021	−0.145010	+	0.989430i	$0.546322\pi$
$108$	−4.14590	−0.398939
$109$	1.76393	0.168954	0.0844770	−	0.996425i	$-0.473078\pi$
$109$	1.76393	0.168954	0.0844770	+	0.996425i	$0.473078\pi$
$110$	0	0
$111$	1.85410	0.175984
$112$	13.3262	1.25921
$113$	14.9443	1.40584	0.702919	−	0.711269i	$-0.251879\pi$
$113$	14.9443	1.40584	0.702919	+	0.711269i	$0.251879\pi$
$114$	0	0
$115$	0	0
$116$	−9.00000	−0.835629
$117$	14.2705	1.31931
$118$	4.23607	0.389962
$119$	25.4164	2.32992
$120$	0	0
$121$	−5.32624	−0.484203
$122$	1.14590	0.103745
$123$	4.27051	0.385059
$124$	−20.1246	−1.80724
$125$	0	0
$126$	4.61803	0.411407
$127$	−10.7082	−0.950199	−0.475100	−	0.879932i	$-0.657588\pi$
$127$	−10.7082	−0.950199	−0.475100	+	0.879932i	$0.657588\pi$
$128$	10.0902	0.891853
$129$	1.47214	0.129614
$130$	0	0
$131$	−7.85410	−0.686216	−0.343108	−	0.939296i	$-0.611480\pi$
$131$	−7.85410	−0.686216	−0.343108	+	0.939296i	$0.611480\pi$
$132$	−1.68692	−0.146827
$133$	0	0
$134$	3.32624	0.287343
$135$	0	0
$136$	8.83282	0.757408
$137$	−1.18034	−0.100843	−0.0504216	−	0.998728i	$-0.516057\pi$
$137$	−1.18034	−0.100843	−0.0504216	+	0.998728i	$0.516057\pi$
$138$	−0.0901699	−0.00767578
$139$	8.14590	0.690926	0.345463	−	0.938432i	$-0.387722\pi$
$139$	8.14590	0.690926	0.345463	+	0.938432i	$0.387722\pi$
$140$	0	0
$141$	−2.20163	−0.185410
$142$	−3.14590	−0.263998
$143$	11.9098	0.995950
$144$	−8.97871	−0.748226
$145$	0	0
$146$	−0.381966	−0.0316117
$147$	−4.18034	−0.344789
$148$	9.00000	0.739795
$149$	−7.61803	−0.624094	−0.312047	−	0.950067i	$-0.601015\pi$
$149$	−7.61803	−0.624094	−0.312047	+	0.950067i	$0.601015\pi$
$150$	0	0
$151$	−7.38197	−0.600736	−0.300368	−	0.953823i	$-0.597109\pi$
$151$	−7.38197	−0.600736	−0.300368	+	0.953823i	$0.597109\pi$
$152$	0	0
$153$	−17.1246	−1.38444
$154$	3.85410	0.310572
$155$	0	0
$156$	−3.54102	−0.283508
$157$	24.0344	1.91816	0.959079	−	0.283140i	$-0.0913761\pi$
$157$	24.0344	1.91816	0.959079	+	0.283140i	$0.0913761\pi$
$158$	−3.05573	−0.243101
$159$	1.29180	0.102446
$160$	0	0
$161$	−2.61803	−0.206330
$162$	−2.94427	−0.231324
$163$	−8.52786	−0.667954	−0.333977	−	0.942581i	$-0.608391\pi$
$163$	−8.52786	−0.667954	−0.333977	+	0.942581i	$0.608391\pi$
$164$	20.7295	1.61870
$165$	0	0
$166$	−1.70820	−0.132582
$167$	−8.23607	−0.637326	−0.318663	−	0.947868i	$-0.603234\pi$
$167$	−8.23607	−0.637326	−0.318663	+	0.947868i	$0.603234\pi$
$168$	−2.38197	−0.183773
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	7.14590	0.544870
$173$	−13.8885	−1.05593	−0.527963	−	0.849267i	$-0.677044\pi$
$173$	−13.8885	−1.05593	−0.527963	+	0.849267i	$0.677044\pi$
$174$	0.708204	0.0536888
$175$	0	0
$176$	−7.49342	−0.564838
$177$	4.23607	0.318402
$178$	−2.56231	−0.192053
$179$	10.5279	0.786890	0.393445	−	0.919348i	$-0.371283\pi$
$179$	10.5279	0.786890	0.393445	+	0.919348i	$0.371283\pi$
$180$	0	0
$181$	8.29180	0.616324	0.308162	−	0.951334i	$-0.400286\pi$
$181$	8.29180	0.616324	0.308162	+	0.951334i	$0.400286\pi$
$182$	8.09017	0.599683
$183$	1.14590	0.0847072
$184$	−0.909830	−0.0670736
$185$	0	0
$186$	1.58359	0.116115
$187$	−14.2918	−1.04512
$188$	−10.6869	−0.779424
$189$	9.47214	0.688997
$190$	0	0
$191$	6.05573	0.438177	0.219089	−	0.975705i	$-0.429692\pi$
$191$	6.05573	0.438177	0.219089	+	0.975705i	$0.429692\pi$
$192$	1.79837	0.129786
$193$	−4.00000	−0.287926	−0.143963	−	0.989583i	$-0.545985\pi$
$193$	−4.00000	−0.287926	−0.143963	+	0.989583i	$0.545985\pi$
$194$	−1.94427	−0.139591
$195$	0	0
$196$	−20.2918	−1.44941
$197$	−16.5279	−1.17756	−0.588781	−	0.808293i	$-0.700392\pi$
$197$	−16.5279	−1.17756	−0.588781	+	0.808293i	$0.700392\pi$
$198$	−2.59675	−0.184543
$199$	13.4164	0.951064	0.475532	−	0.879698i	$-0.342256\pi$
$199$	13.4164	0.951064	0.475532	+	0.879698i	$0.342256\pi$
$200$	0	0
$201$	3.32624	0.234615
$202$	−2.27051	−0.159753
$203$	20.5623	1.44319
$204$	4.24922	0.297505
$205$	0	0
$206$	0.639320	0.0445436
$207$	1.76393	0.122602
$208$	−15.7295	−1.09064
$209$	0	0
$210$	0	0
$211$	−6.90983	−0.475692	−0.237846	−	0.971303i	$-0.576441\pi$
$211$	−6.90983	−0.475692	−0.237846	+	0.971303i	$0.576441\pi$
$212$	6.27051	0.430660
$213$	−3.14590	−0.215553
$214$	1.14590	0.0783320
$215$	0	0
$216$	3.29180	0.223978
$217$	45.9787	3.12124
$218$	−0.673762	−0.0456329
$219$	−0.381966	−0.0258109
$220$	0	0
$221$	−30.0000	−2.01802
$222$	−0.708204	−0.0475315
$223$	−10.2361	−0.685458	−0.342729	−	0.939434i	$-0.611351\pi$
$223$	−10.2361	−0.685458	−0.342729	+	0.939434i	$0.611351\pi$
$224$	−17.5623	−1.17343
$225$	0	0
$226$	−5.70820	−0.379704
$227$	−17.6525	−1.17164	−0.585818	−	0.810443i	$-0.699227\pi$
$227$	−17.6525	−1.17164	−0.585818	+	0.810443i	$0.699227\pi$
$228$	0	0
$229$	−21.6180	−1.42856	−0.714280	−	0.699860i	$-0.753246\pi$
$229$	−21.6180	−1.42856	−0.714280	+	0.699860i	$0.753246\pi$
$230$	0	0
$231$	3.85410	0.253581
$232$	7.14590	0.469151
$233$	6.81966	0.446771	0.223385	−	0.974730i	$-0.428289\pi$
$233$	6.81966	0.446771	0.223385	+	0.974730i	$0.428289\pi$
$234$	−5.45085	−0.356333
$235$	0	0
$236$	20.5623	1.33849
$237$	−3.05573	−0.198491
$238$	−9.70820	−0.629289
$239$	23.5623	1.52412	0.762059	−	0.647507i	$-0.224188\pi$
$239$	23.5623	1.52412	0.762059	+	0.647507i	$0.224188\pi$
$240$	0	0
$241$	18.1246	1.16751	0.583754	−	0.811930i	$-0.301583\pi$
$241$	18.1246	1.16751	0.583754	+	0.811930i	$0.301583\pi$
$242$	2.03444	0.130779
$243$	−9.65248	−0.619207
$244$	5.56231	0.356090
$245$	0	0
$246$	−1.63119	−0.104001
$247$	0	0
$248$	15.9787	1.01465
$249$	−1.70820	−0.108253
$250$	0	0
$251$	15.1803	0.958175	0.479087	−	0.877767i	$-0.340968\pi$
$251$	15.1803	0.958175	0.479087	+	0.877767i	$0.340968\pi$
$252$	22.4164	1.41210
$253$	1.47214	0.0925524
$254$	4.09017	0.256640
$255$	0	0
$256$	5.56231	0.347644
$257$	−13.5279	−0.843845	−0.421922	−	0.906632i	$-0.638645\pi$
$257$	−13.5279	−0.843845	−0.421922	+	0.906632i	$0.638645\pi$
$258$	−0.562306	−0.0350076
$259$	−20.5623	−1.27768
$260$	0	0
$261$	−13.8541	−0.857547
$262$	3.00000	0.185341
$263$	19.8885	1.22638	0.613190	−	0.789935i	$-0.289886\pi$
$263$	19.8885	1.22638	0.613190	+	0.789935i	$0.289886\pi$
$264$	1.33939	0.0824340
$265$	0	0
$266$	0	0
$267$	−2.56231	−0.156811
$268$	16.1459	0.986268
$269$	1.67376	0.102051	0.0510255	−	0.998697i	$-0.483751\pi$
$269$	1.67376	0.102051	0.0510255	+	0.998697i	$0.483751\pi$
$270$	0	0
$271$	16.3820	0.995134	0.497567	−	0.867426i	$-0.334227\pi$
$271$	16.3820	0.995134	0.497567	+	0.867426i	$0.334227\pi$
$272$	18.8754	1.14449
$273$	8.09017	0.489639
$274$	0.450850	0.0272368
$275$	0	0
$276$	−0.437694	−0.0263461
$277$	25.4164	1.52712	0.763562	−	0.645735i	$-0.223449\pi$
$277$	25.4164	1.52712	0.763562	+	0.645735i	$0.223449\pi$
$278$	−3.11146	−0.186613
$279$	−30.9787	−1.85465
$280$	0	0
$281$	27.5066	1.64090	0.820452	−	0.571715i	$-0.193722\pi$
$281$	27.5066	1.64090	0.820452	+	0.571715i	$0.193722\pi$
$282$	0.840946	0.0500776
$283$	−4.61803	−0.274514	−0.137257	−	0.990535i	$-0.543829\pi$
$283$	−4.61803	−0.274514	−0.137257	+	0.990535i	$0.543829\pi$
$284$	−15.2705	−0.906138
$285$	0	0
$286$	−4.54915	−0.268997
$287$	−47.3607	−2.79561
$288$	11.8328	0.697255
$289$	19.0000	1.11765
$290$	0	0
$291$	−1.94427	−0.113975
$292$	−1.85410	−0.108503
$293$	−22.0344	−1.28727	−0.643633	−	0.765334i	$-0.722574\pi$
$293$	−22.0344	−1.28727	−0.643633	+	0.765334i	$0.722574\pi$
$294$	1.59675	0.0931242
$295$	0	0
$296$	−7.14590	−0.415347
$297$	−5.32624	−0.309060
$298$	2.90983	0.168562
$299$	3.09017	0.178709
$300$	0	0
$301$	−16.3262	−0.941029
$302$	2.81966	0.162253
$303$	−2.27051	−0.130437
$304$	0	0
$305$	0	0
$306$	6.54102	0.373925
$307$	12.2705	0.700315	0.350157	−	0.936691i	$-0.386128\pi$
$307$	12.2705	0.700315	0.350157	+	0.936691i	$0.386128\pi$
$308$	18.7082	1.06600
$309$	0.639320	0.0363697
$310$	0	0
$311$	−7.05573	−0.400094	−0.200047	−	0.979786i	$-0.564109\pi$
$311$	−7.05573	−0.400094	−0.200047	+	0.979786i	$0.564109\pi$
$312$	2.81153	0.159172
$313$	9.79837	0.553837	0.276918	−	0.960893i	$-0.410687\pi$
$313$	9.79837	0.553837	0.276918	+	0.960893i	$0.410687\pi$
$314$	−9.18034	−0.518077
$315$	0	0
$316$	−14.8328	−0.834411
$317$	13.2361	0.743412	0.371706	−	0.928351i	$-0.378773\pi$
$317$	13.2361	0.743412	0.371706	+	0.928351i	$0.378773\pi$
$318$	−0.493422	−0.0276697
$319$	−11.5623	−0.647365
$320$	0	0
$321$	1.14590	0.0639578
$322$	1.00000	0.0557278
$323$	0	0
$324$	−14.2918	−0.793989
$325$	0	0
$326$	3.25735	0.180408
$327$	−0.673762	−0.0372591
$328$	−16.4590	−0.908795
$329$	24.4164	1.34612
$330$	0	0
$331$	−0.291796	−0.0160386	−0.00801928	−	0.999968i	$-0.502553\pi$
$331$	−0.291796	−0.0160386	−0.00801928	+	0.999968i	$0.502553\pi$
$332$	−8.29180	−0.455071
$333$	13.8541	0.759200
$334$	3.14590	0.172136
$335$	0	0
$336$	−5.09017	−0.277692
$337$	−28.0000	−1.52526	−0.762629	−	0.646837i	$-0.776092\pi$
$337$	−28.0000	−1.52526	−0.762629	+	0.646837i	$0.776092\pi$
$338$	−4.58359	−0.249315
$339$	−5.70820	−0.310027
$340$	0	0
$341$	−25.8541	−1.40008
$342$	0	0
$343$	16.7082	0.902158
$344$	−5.67376	−0.305909
$345$	0	0
$346$	5.30495	0.285196
$347$	19.8885	1.06767	0.533836	−	0.845588i	$-0.320750\pi$
$347$	19.8885	1.06767	0.533836	+	0.845588i	$0.320750\pi$
$348$	3.43769	0.184280
$349$	21.7426	1.16386	0.581929	−	0.813240i	$-0.302298\pi$
$349$	21.7426	1.16386	0.581929	+	0.813240i	$0.302298\pi$
$350$	0	0
$351$	−11.1803	−0.596762
$352$	9.87539	0.526360
$353$	36.1591	1.92455	0.962276	−	0.272075i	$-0.0877098\pi$
$353$	36.1591	1.92455	0.962276	+	0.272075i	$0.0877098\pi$
$354$	−1.61803	−0.0859975
$355$	0	0
$356$	−12.4377	−0.659196
$357$	−9.70820	−0.513813
$358$	−4.02129	−0.212532
$359$	26.3820	1.39239	0.696193	−	0.717854i	$-0.254876\pi$
$359$	26.3820	1.39239	0.696193	+	0.717854i	$0.254876\pi$
$360$	0	0
$361$	0	0
$362$	−3.16718	−0.166464
$363$	2.03444	0.106781
$364$	39.2705	2.05833
$365$	0	0
$366$	−0.437694	−0.0228786
$367$	−1.65248	−0.0862585	−0.0431293	−	0.999070i	$-0.513733\pi$
$367$	−1.65248	−0.0862585	−0.0431293	+	0.999070i	$0.513733\pi$
$368$	−1.94427	−0.101352
$369$	31.9098	1.66116
$370$	0	0
$371$	−14.3262	−0.743781
$372$	7.68692	0.398548
$373$	6.70820	0.347338	0.173669	−	0.984804i	$-0.444438\pi$
$373$	6.70820	0.347338	0.173669	+	0.984804i	$0.444438\pi$
$374$	5.45898	0.282277
$375$	0	0
$376$	8.48529	0.437596
$377$	−24.2705	−1.24999
$378$	−3.61803	−0.186092
$379$	−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
$381$	4.09017	0.209546
$382$	−2.31308	−0.118348
$383$	−32.9230	−1.68229	−0.841143	−	0.540813i	$-0.818117\pi$
$383$	−32.9230	−1.68229	−0.841143	+	0.540813i	$0.818117\pi$
$384$	−3.85410	−0.196679
$385$	0	0
$386$	1.52786	0.0777662
$387$	11.0000	0.559161
$388$	−9.43769	−0.479126
$389$	−24.3262	−1.23339	−0.616695	−	0.787202i	$-0.711529\pi$
$389$	−24.3262	−1.23339	−0.616695	+	0.787202i	$0.711529\pi$
$390$	0	0
$391$	−3.70820	−0.187532
$392$	16.1115	0.813751
$393$	3.00000	0.151330
$394$	6.31308	0.318048
$395$	0	0
$396$	−12.6049	−0.633419
$397$	16.8885	0.847612	0.423806	−	0.905753i	$-0.360694\pi$
$397$	16.8885	0.847612	0.423806	+	0.905753i	$0.360694\pi$
$398$	−5.12461	−0.256874
$399$	0	0
$400$	0	0
$401$	6.76393	0.337775	0.168887	−	0.985635i	$-0.445983\pi$
$401$	6.76393	0.337775	0.168887	+	0.985635i	$0.445983\pi$
$402$	−1.27051	−0.0633673
$403$	−54.2705	−2.70341
$404$	−11.0213	−0.548329
$405$	0	0
$406$	−7.85410	−0.389793
$407$	11.5623	0.573122
$408$	−3.37384	−0.167030
$409$	−1.70820	−0.0844652	−0.0422326	−	0.999108i	$-0.513447\pi$
$409$	−1.70820	−0.0844652	−0.0422326	+	0.999108i	$0.513447\pi$
$410$	0	0
$411$	0.450850	0.0222388
$412$	3.10333	0.152890
$413$	−46.9787	−2.31167
$414$	−0.673762	−0.0331136
$415$	0	0
$416$	20.7295	1.01635
$417$	−3.11146	−0.152369
$418$	0	0
$419$	1.05573	0.0515757	0.0257878	−	0.999667i	$-0.491791\pi$
$419$	1.05573	0.0515757	0.0257878	+	0.999667i	$0.491791\pi$
$420$	0	0
$421$	26.1246	1.27324	0.636618	−	0.771179i	$-0.280333\pi$
$421$	26.1246	1.27324	0.636618	+	0.771179i	$0.280333\pi$
$422$	2.63932	0.128480
$423$	−16.4508	−0.799868
$424$	−4.97871	−0.241788
$425$	0	0
$426$	1.20163	0.0582190
$427$	−12.7082	−0.614993
$428$	5.56231	0.268864
$429$	−4.54915	−0.219635
$430$	0	0
$431$	−25.1803	−1.21289	−0.606447	−	0.795124i	$-0.707406\pi$
$431$	−25.1803	−1.21289	−0.606447	+	0.795124i	$0.707406\pi$
$432$	7.03444	0.338445
$433$	29.2148	1.40397	0.701986	−	0.712190i	$-0.252297\pi$
$433$	29.2148	1.40397	0.701986	+	0.712190i	$0.252297\pi$
$434$	−17.5623	−0.843018
$435$	0	0
$436$	−3.27051	−0.156629
$437$	0	0
$438$	0.145898	0.00697128
$439$	3.47214	0.165716	0.0828580	−	0.996561i	$-0.473595\pi$
$439$	3.47214	0.165716	0.0828580	+	0.996561i	$0.473595\pi$
$440$	0	0
$441$	−31.2361	−1.48743
$442$	11.4590	0.545048
$443$	29.3607	1.39497	0.697484	−	0.716600i	$-0.254303\pi$
$443$	29.3607	1.39497	0.697484	+	0.716600i	$0.254303\pi$
$444$	−3.43769	−0.163146
$445$	0	0
$446$	3.90983	0.185136
$447$	2.90983	0.137630
$448$	−19.9443	−0.942278
$449$	41.4721	1.95719	0.978596	−	0.205793i	$-0.0659773\pi$
$449$	41.4721	1.95719	0.978596	+	0.205793i	$0.0659773\pi$
$450$	0	0
$451$	26.6312	1.25401
$452$	−27.7082	−1.30328
$453$	2.81966	0.132479
$454$	6.74265	0.316448
$455$	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	$0.455182\pi$
$458$	8.25735	0.385841
$459$	13.4164	0.626224
$460$	0	0
$461$	−1.05573	−0.0491702	−0.0245851	−	0.999698i	$-0.507826\pi$
$461$	−1.05573	−0.0491702	−0.0245851	+	0.999698i	$0.507826\pi$
$462$	−1.47214	−0.0684900
$463$	−33.0344	−1.53524	−0.767620	−	0.640905i	$-0.778559\pi$
$463$	−33.0344	−1.53524	−0.767620	+	0.640905i	$0.778559\pi$
$464$	15.2705	0.708916
$465$	0	0
$466$	−2.60488	−0.120669
$467$	24.7082	1.14336	0.571680	−	0.820477i	$-0.306292\pi$
$467$	24.7082	1.14336	0.571680	+	0.820477i	$0.306292\pi$
$468$	−26.4590	−1.22307
$469$	−36.8885	−1.70335
$470$	0	0
$471$	−9.18034	−0.423008
$472$	−16.3262	−0.751476
$473$	9.18034	0.422112
$474$	1.16718	0.0536105
$475$	0	0
$476$	−47.1246	−2.15995
$477$	9.65248	0.441957
$478$	−9.00000	−0.411650
$479$	−10.6180	−0.485150	−0.242575	−	0.970133i	$-0.577992\pi$
$479$	−10.6180	−0.485150	−0.242575	+	0.970133i	$0.577992\pi$
$480$	0	0
$481$	24.2705	1.10664
$482$	−6.92299	−0.315333
$483$	1.00000	0.0455016
$484$	9.87539	0.448881
$485$	0	0
$486$	3.68692	0.167242
$487$	35.4164	1.60487	0.802435	−	0.596739i	$-0.203537\pi$
$487$	35.4164	1.60487	0.802435	+	0.596739i	$0.203537\pi$
$488$	−4.41641	−0.199921
$489$	3.25735	0.147303
$490$	0	0
$491$	8.67376	0.391441	0.195721	−	0.980660i	$-0.437295\pi$
$491$	8.67376	0.391441	0.195721	+	0.980660i	$0.437295\pi$
$492$	−7.91796	−0.356969
$493$	29.1246	1.31171
$494$	0	0
$495$	0	0
$496$	34.1459	1.53320
$497$	34.8885	1.56497
$498$	0.652476	0.0292381
$499$	26.6525	1.19313	0.596564	−	0.802565i	$-0.296532\pi$
$499$	26.6525	1.19313	0.596564	+	0.802565i	$0.296532\pi$
$500$	0	0
$501$	3.14590	0.140548
$502$	−5.79837	−0.258794
$503$	2.05573	0.0916604	0.0458302	−	0.998949i	$-0.485407\pi$
$503$	2.05573	0.0916604	0.0458302	+	0.998949i	$0.485407\pi$
$504$	−17.7984	−0.792803
$505$	0	0
$506$	−0.562306	−0.0249975
$507$	−4.58359	−0.203564
$508$	19.8541	0.880883
$509$	10.7984	0.478630	0.239315	−	0.970942i	$-0.423077\pi$
$509$	10.7984	0.478630	0.239315	+	0.970942i	$0.423077\pi$
$510$	0	0
$511$	4.23607	0.187393
$512$	−22.3050	−0.985749
$513$	0	0
$514$	5.16718	0.227915
$515$	0	0
$516$	−2.72949	−0.120159
$517$	−13.7295	−0.603822
$518$	7.85410	0.345089
$519$	5.30495	0.232862
$520$	0	0
$521$	−0.381966	−0.0167342	−0.00836712	−	0.999965i	$-0.502663\pi$
$521$	−0.381966	−0.0167342	−0.00836712	+	0.999965i	$0.502663\pi$
$522$	5.29180	0.231616
$523$	−10.5967	−0.463363	−0.231682	−	0.972792i	$-0.574423\pi$
$523$	−10.5967	−0.463363	−0.231682	+	0.972792i	$0.574423\pi$
$524$	14.5623	0.636157
$525$	0	0
$526$	−7.59675	−0.331234
$527$	65.1246	2.83687
$528$	2.86223	0.124563
$529$	−22.6180	−0.983393
$530$	0	0
$531$	31.6525	1.37360
$532$	0	0
$533$	55.9017	2.42137
$534$	0.978714	0.0423531
$535$	0	0
$536$	−12.8197	−0.553725
$537$	−4.02129	−0.173531
$538$	−0.639320	−0.0275631
$539$	−26.0689	−1.12287
$540$	0	0
$541$	32.5967	1.40144	0.700722	−	0.713435i	$-0.252861\pi$
$541$	32.5967	1.40144	0.700722	+	0.713435i	$0.252861\pi$
$542$	−6.25735	−0.268776
$543$	−3.16718	−0.135917
$544$	−24.8754	−1.06652
$545$	0	0
$546$	−3.09017	−0.132247
$547$	−37.9787	−1.62385	−0.811926	−	0.583760i	$-0.801581\pi$
$547$	−37.9787	−1.62385	−0.811926	+	0.583760i	$0.801581\pi$
$548$	2.18847	0.0934868
$549$	8.56231	0.365430
$550$	0	0
$551$	0	0
$552$	0.347524	0.0147916
$553$	33.8885	1.44109
$554$	−9.70820	−0.412462
$555$	0	0
$556$	−15.1033	−0.640524
$557$	21.4721	0.909804	0.454902	−	0.890542i	$-0.349674\pi$
$557$	21.4721	0.909804	0.454902	+	0.890542i	$0.349674\pi$
$558$	11.8328	0.500923
$559$	19.2705	0.815056
$560$	0	0
$561$	5.45898	0.230478
$562$	−10.5066	−0.443193
$563$	30.1803	1.27195	0.635975	−	0.771710i	$-0.280598\pi$
$563$	30.1803	1.27195	0.635975	+	0.771710i	$0.280598\pi$
$564$	4.08204	0.171885
$565$	0	0
$566$	1.76393	0.0741436
$567$	32.6525	1.37128
$568$	12.1246	0.508737
$569$	−0.437694	−0.0183491	−0.00917455	−	0.999958i	$-0.502920\pi$
$569$	−0.437694	−0.0183491	−0.00917455	+	0.999958i	$0.502920\pi$
$570$	0	0
$571$	26.8541	1.12381	0.561905	−	0.827202i	$-0.310069\pi$
$571$	26.8541	1.12381	0.561905	+	0.827202i	$0.310069\pi$
$572$	−22.0820	−0.923296
$573$	−2.31308	−0.0966304
$574$	18.0902	0.755069
$575$	0	0
$576$	13.4377	0.559904
$577$	−19.7639	−0.822783	−0.411392	−	0.911459i	$-0.634957\pi$
$577$	−19.7639	−0.822783	−0.411392	+	0.911459i	$0.634957\pi$
$578$	−7.25735	−0.301866
$579$	1.52786	0.0634959
$580$	0	0
$581$	18.9443	0.785941
$582$	0.742646	0.0307837
$583$	8.05573	0.333634
$584$	1.47214	0.0609174
$585$	0	0
$586$	8.41641	0.347679
$587$	42.2361	1.74327	0.871635	−	0.490156i	$-0.163060\pi$
$587$	42.2361	1.74327	0.871635	+	0.490156i	$0.163060\pi$
$588$	7.75078	0.319637
$589$	0	0
$590$	0	0
$591$	6.31308	0.259686
$592$	−15.2705	−0.627614
$593$	15.6525	0.642770	0.321385	−	0.946949i	$-0.395852\pi$
$593$	15.6525	0.642770	0.321385	+	0.946949i	$0.395852\pi$
$594$	2.03444	0.0834742
$595$	0	0
$596$	14.1246	0.578567
$597$	−5.12461	−0.209736
$598$	−1.18034	−0.0482677
$599$	7.36068	0.300749	0.150375	−	0.988629i	$-0.451952\pi$
$599$	7.36068	0.300749	0.150375	+	0.988629i	$0.451952\pi$
$600$	0	0
$601$	22.0000	0.897399	0.448699	−	0.893683i	$-0.351887\pi$
$601$	22.0000	0.897399	0.448699	+	0.893683i	$0.351887\pi$
$602$	6.23607	0.254163
$603$	24.8541	1.01214
$604$	13.6869	0.556913
$605$	0	0
$606$	0.867258	0.0352299
$607$	16.5623	0.672243	0.336122	−	0.941819i	$-0.390885\pi$
$607$	16.5623	0.672243	0.336122	+	0.941819i	$0.390885\pi$
$608$	0	0
$609$	−7.85410	−0.318264
$610$	0	0
$611$	−28.8197	−1.16592
$612$	31.7508	1.28345
$613$	−4.05573	−0.163809	−0.0819047	−	0.996640i	$-0.526100\pi$
$613$	−4.05573	−0.163809	−0.0819047	+	0.996640i	$0.526100\pi$
$614$	−4.68692	−0.189149
$615$	0	0
$616$	−14.8541	−0.598489
$617$	12.9098	0.519730	0.259865	−	0.965645i	$-0.416322\pi$
$617$	12.9098	0.519730	0.259865	+	0.965645i	$0.416322\pi$
$618$	−0.244199	−0.00982311
$619$	−8.05573	−0.323787	−0.161894	−	0.986808i	$-0.551760\pi$
$619$	−8.05573	−0.323787	−0.161894	+	0.986808i	$0.551760\pi$
$620$	0	0
$621$	−1.38197	−0.0554564
$622$	2.69505	0.108062
$623$	28.4164	1.13848
$624$	6.00813	0.240518
$625$	0	0
$626$	−3.74265	−0.149586
$627$	0	0
$628$	−44.5623	−1.77823
$629$	−29.1246	−1.16127
$630$	0	0
$631$	15.0000	0.597141	0.298570	−	0.954388i	$-0.403490\pi$
$631$	15.0000	0.597141	0.298570	+	0.954388i	$0.403490\pi$
$632$	11.7771	0.468467
$633$	2.63932	0.104904
$634$	−5.05573	−0.200789
$635$	0	0
$636$	−2.39512	−0.0949728
$637$	−54.7214	−2.16814
$638$	4.41641	0.174847
$639$	−23.5066	−0.929906
$640$	0	0
$641$	5.72949	0.226301	0.113151	−	0.993578i	$-0.463906\pi$
$641$	5.72949	0.226301	0.113151	+	0.993578i	$0.463906\pi$
$642$	−0.437694	−0.0172744
$643$	−31.5967	−1.24605	−0.623027	−	0.782200i	$-0.714097\pi$
$643$	−31.5967	−1.24605	−0.623027	+	0.782200i	$0.714097\pi$
$644$	4.85410	0.191278
$645$	0	0
$646$	0	0
$647$	−36.3050	−1.42729	−0.713647	−	0.700505i	$-0.752958\pi$
$647$	−36.3050	−1.42729	−0.713647	+	0.700505i	$0.752958\pi$
$648$	11.3475	0.445773
$649$	26.4164	1.03693
$650$	0	0
$651$	−17.5623	−0.688321
$652$	15.8115	0.619227
$653$	−7.85410	−0.307355	−0.153677	−	0.988121i	$-0.549112\pi$
$653$	−7.85410	−0.307355	−0.153677	+	0.988121i	$0.549112\pi$
$654$	0.257354	0.0100633
$655$	0	0
$656$	−35.1722	−1.37324
$657$	−2.85410	−0.111349
$658$	−9.32624	−0.363575
$659$	42.4721	1.65448	0.827240	−	0.561849i	$-0.189910\pi$
$659$	42.4721	1.65448	0.827240	+	0.561849i	$0.189910\pi$
$660$	0	0
$661$	−3.70820	−0.144232	−0.0721162	−	0.997396i	$-0.522975\pi$
$661$	−3.70820	−0.144232	−0.0721162	+	0.997396i	$0.522975\pi$
$662$	0.111456	0.00433187
$663$	11.4590	0.445030
$664$	6.58359	0.255493
$665$	0	0
$666$	−5.29180	−0.205053
$667$	−3.00000	−0.116160
$668$	15.2705	0.590834
$669$	3.90983	0.151163
$670$	0	0
$671$	7.14590	0.275864
$672$	6.70820	0.258775
$673$	11.1115	0.428315	0.214158	−	0.976799i	$-0.431299\pi$
$673$	11.1115	0.428315	0.214158	+	0.976799i	$0.431299\pi$
$674$	10.6950	0.411958
$675$	0	0
$676$	−22.2492	−0.855739
$677$	8.38197	0.322145	0.161073	−	0.986943i	$-0.448505\pi$
$677$	8.38197	0.322145	0.161073	+	0.986943i	$0.448505\pi$
$678$	2.18034	0.0837354
$679$	21.5623	0.827485
$680$	0	0
$681$	6.74265	0.258379
$682$	9.87539	0.378148
$683$	−23.4721	−0.898136	−0.449068	−	0.893498i	$-0.648244\pi$
$683$	−23.4721	−0.898136	−0.449068	+	0.893498i	$0.648244\pi$
$684$	0	0
$685$	0	0
$686$	−6.38197	−0.243665
$687$	8.25735	0.315038
$688$	−12.1246	−0.462246
$689$	16.9098	0.644213
$690$	0	0
$691$	21.4721	0.816839	0.408419	−	0.912794i	$-0.366080\pi$
$691$	21.4721	0.816839	0.408419	+	0.912794i	$0.366080\pi$
$692$	25.7508	0.978898
$693$	28.7984	1.09396
$694$	−7.59675	−0.288369
$695$	0	0
$696$	−2.72949	−0.103461
$697$	−67.0820	−2.54091
$698$	−8.30495	−0.314347
$699$	−2.60488	−0.0985255
$700$	0	0
$701$	48.5755	1.83467	0.917335	−	0.398116i	$-0.130336\pi$
$701$	48.5755	1.83467	0.917335	+	0.398116i	$0.130336\pi$
$702$	4.27051	0.161180
$703$	0	0
$704$	11.2148	0.422673
$705$	0	0
$706$	−13.8115	−0.519804
$707$	25.1803	0.947004
$708$	−7.85410	−0.295175
$709$	−15.7082	−0.589934	−0.294967	−	0.955507i	$-0.595309\pi$
$709$	−15.7082	−0.589934	−0.294967	+	0.955507i	$0.595309\pi$
$710$	0	0
$711$	−22.8328	−0.856297
$712$	9.87539	0.370096
$713$	−6.70820	−0.251224
$714$	3.70820	0.138776
$715$	0	0
$716$	−19.5197	−0.729487
$717$	−9.00000	−0.336111
$718$	−10.0770	−0.376071
$719$	12.2016	0.455044	0.227522	−	0.973773i	$-0.426938\pi$
$719$	12.2016	0.455044	0.227522	+	0.973773i	$0.426938\pi$
$720$	0	0
$721$	−7.09017	−0.264052
$722$	0	0
$723$	−6.92299	−0.257469
$724$	−15.3738	−0.571364
$725$	0	0
$726$	−0.777088	−0.0288404
$727$	10.5836	0.392524	0.196262	−	0.980552i	$-0.437120\pi$
$727$	10.5836	0.392524	0.196262	+	0.980552i	$0.437120\pi$
$728$	−31.1803	−1.15562
$729$	−19.4377	−0.719915
$730$	0	0
$731$	−23.1246	−0.855295
$732$	−2.12461	−0.0785279
$733$	−0.236068	−0.00871937	−0.00435968	−	0.999990i	$-0.501388\pi$
$733$	−0.236068	−0.00871937	−0.00435968	+	0.999990i	$0.501388\pi$
$734$	0.631190	0.0232976
$735$	0	0
$736$	2.56231	0.0944478
$737$	20.7426	0.764065
$738$	−12.1885	−0.448664
$739$	−21.8328	−0.803133	−0.401567	−	0.915830i	$-0.631534\pi$
$739$	−21.8328	−0.803133	−0.401567	+	0.915830i	$0.631534\pi$
$740$	0	0
$741$	0	0
$742$	5.47214	0.200888
$743$	−36.2361	−1.32937	−0.664686	−	0.747123i	$-0.731435\pi$
$743$	−36.2361	−1.32937	−0.664686	+	0.747123i	$0.731435\pi$
$744$	−6.10333	−0.223759
$745$	0	0
$746$	−2.56231	−0.0938127
$747$	−12.7639	−0.467008
$748$	26.4984	0.968879
$749$	−12.7082	−0.464348
$750$	0	0
$751$	−2.56231	−0.0934999	−0.0467499	−	0.998907i	$-0.514886\pi$
$751$	−2.56231	−0.0934999	−0.0467499	+	0.998907i	$0.514886\pi$
$752$	18.1327	0.661233
$753$	−5.79837	−0.211304
$754$	9.27051	0.337612
$755$	0	0
$756$	−17.5623	−0.638735
$757$	14.3820	0.522721	0.261361	−	0.965241i	$-0.415829\pi$
$757$	14.3820	0.522721	0.261361	+	0.965241i	$0.415829\pi$
$758$	10.2492	0.372269
$759$	−0.562306	−0.0204104
$760$	0	0
$761$	37.3607	1.35432	0.677162	−	0.735834i	$-0.263210\pi$
$761$	37.3607	1.35432	0.677162	+	0.735834i	$0.263210\pi$
$762$	−1.56231	−0.0565964
$763$	7.47214	0.270509
$764$	−11.2279	−0.406213
$765$	0	0
$766$	12.5755	0.454370
$767$	55.4508	2.00221
$768$	−2.12461	−0.0766653
$769$	13.6869	0.493563	0.246781	−	0.969071i	$-0.420627\pi$
$769$	13.6869	0.493563	0.246781	+	0.969071i	$0.420627\pi$
$770$	0	0
$771$	5.16718	0.186092
$772$	7.41641	0.266922
$773$	39.7426	1.42944	0.714722	−	0.699409i	$-0.246553\pi$
$773$	39.7426	1.42944	0.714722	+	0.699409i	$0.246553\pi$
$774$	−4.20163	−0.151024
$775$	0	0
$776$	7.49342	0.268998
$777$	7.85410	0.281764
$778$	9.29180	0.333127
$779$	0	0
$780$	0	0
$781$	−19.6180	−0.701988
$782$	1.41641	0.0506506
$783$	10.8541	0.387894
$784$	34.4296	1.22963
$785$	0	0
$786$	−1.14590	−0.0408728
$787$	35.4853	1.26491	0.632457	−	0.774595i	$-0.282046\pi$
$787$	35.4853	1.26491	0.632457	+	0.774595i	$0.282046\pi$
$788$	30.6443	1.09166
$789$	−7.59675	−0.270451
$790$	0	0
$791$	63.3050	2.25086
$792$	10.0081	0.355623
$793$	15.0000	0.532666
$794$	−6.45085	−0.228932
$795$	0	0
$796$	−24.8754	−0.881685
$797$	2.18034	0.0772316	0.0386158	−	0.999254i	$-0.487705\pi$
$797$	2.18034	0.0772316	0.0386158	+	0.999254i	$0.487705\pi$
$798$	0	0
$799$	34.5836	1.22348
$800$	0	0
$801$	−19.1459	−0.676487
$802$	−2.58359	−0.0912298
$803$	−2.38197	−0.0840578
$804$	−6.16718	−0.217500
$805$	0	0
$806$	20.7295	0.730165
$807$	−0.639320	−0.0225051
$808$	8.75078	0.307851
$809$	28.9098	1.01642	0.508208	−	0.861235i	$-0.330308\pi$
$809$	28.9098	1.01642	0.508208	+	0.861235i	$0.330308\pi$
$810$	0	0
$811$	40.3262	1.41605	0.708023	−	0.706190i	$-0.249587\pi$
$811$	40.3262	1.41605	0.708023	+	0.706190i	$0.249587\pi$
$812$	−38.1246	−1.33791
$813$	−6.25735	−0.219455
$814$	−4.41641	−0.154795
$815$	0	0
$816$	−7.20976	−0.252392
$817$	0	0
$818$	0.652476	0.0228133
$819$	60.4508	2.11232
$820$	0	0
$821$	12.3820	0.432134	0.216067	−	0.976379i	$-0.430677\pi$
$821$	12.3820	0.432134	0.216067	+	0.976379i	$0.430677\pi$
$822$	−0.172209	−0.00600649
$823$	−8.36068	−0.291435	−0.145717	−	0.989326i	$-0.546549\pi$
$823$	−8.36068	−0.291435	−0.145717	+	0.989326i	$0.546549\pi$
$824$	−2.46401	−0.0858377
$825$	0	0
$826$	17.9443	0.624361
$827$	35.8328	1.24603	0.623015	−	0.782210i	$-0.285908\pi$
$827$	35.8328	1.24603	0.623015	+	0.782210i	$0.285908\pi$
$828$	−3.27051	−0.113658
$829$	43.3951	1.50717	0.753587	−	0.657348i	$-0.228322\pi$
$829$	43.3951	1.50717	0.753587	+	0.657348i	$0.228322\pi$
$830$	0	0
$831$	−9.70820	−0.336774
$832$	23.5410	0.816138
$833$	65.6656	2.27518
$834$	1.18847	0.0411534
$835$	0	0
$836$	0	0
$837$	24.2705	0.838912
$838$	−0.403252	−0.0139301
$839$	−13.3820	−0.461997	−0.230998	−	0.972954i	$-0.574199\pi$
$839$	−13.3820	−0.461997	−0.230998	+	0.972954i	$0.574199\pi$
$840$	0	0
$841$	−5.43769	−0.187507
$842$	−9.97871	−0.343889
$843$	−10.5066	−0.361866
$844$	12.8115	0.440991
$845$	0	0
$846$	6.28367	0.216037
$847$	−22.5623	−0.775250
$848$	−10.6393	−0.365356
$849$	1.76393	0.0605380
$850$	0	0
$851$	3.00000	0.102839
$852$	5.83282	0.199829
$853$	−52.9574	−1.81323	−0.906614	−	0.421961i	$-0.861342\pi$
$853$	−52.9574	−1.81323	−0.906614	+	0.421961i	$0.861342\pi$
$854$	4.85410	0.166104
$855$	0	0
$856$	−4.41641	−0.150950
$857$	32.6180	1.11421	0.557105	−	0.830442i	$-0.311912\pi$
$857$	32.6180	1.11421	0.557105	+	0.830442i	$0.311912\pi$
$858$	1.73762	0.0593214
$859$	13.5967	0.463915	0.231958	−	0.972726i	$-0.425487\pi$
$859$	13.5967	0.463915	0.231958	+	0.972726i	$0.425487\pi$
$860$	0	0
$861$	18.0902	0.616511
$862$	9.61803	0.327592
$863$	4.65248	0.158372	0.0791861	−	0.996860i	$-0.474768\pi$
$863$	4.65248	0.158372	0.0791861	+	0.996860i	$0.474768\pi$
$864$	−9.27051	−0.315389
$865$	0	0
$866$	−11.1591	−0.379200
$867$	−7.25735	−0.246473
$868$	−85.2492	−2.89355
$869$	−19.0557	−0.646421
$870$	0	0
$871$	43.5410	1.47533
$872$	2.59675	0.0879370
$873$	−14.5279	−0.491694
$874$	0	0
$875$	0	0
$876$	0.708204	0.0239280
$877$	7.81966	0.264051	0.132026	−	0.991246i	$-0.457852\pi$
$877$	7.81966	0.264051	0.132026	+	0.991246i	$0.457852\pi$
$878$	−1.32624	−0.0447584
$879$	8.41641	0.283878
$880$	0	0
$881$	25.0344	0.843432	0.421716	−	0.906728i	$-0.361428\pi$
$881$	25.0344	0.843432	0.421716	+	0.906728i	$0.361428\pi$
$882$	11.9311	0.401742
$883$	−46.9230	−1.57908	−0.789542	−	0.613696i	$-0.789682\pi$
$883$	−46.9230	−1.57908	−0.789542	+	0.613696i	$0.789682\pi$
$884$	55.6231	1.87081
$885$	0	0
$886$	−11.2148	−0.376768
$887$	−12.4721	−0.418773	−0.209387	−	0.977833i	$-0.567147\pi$
$887$	−12.4721	−0.418773	−0.209387	+	0.977833i	$0.567147\pi$
$888$	2.72949	0.0915957
$889$	−45.3607	−1.52135
$890$	0	0
$891$	−18.3607	−0.615106
$892$	18.9787	0.635454
$893$	0	0
$894$	−1.11146	−0.0371727
$895$	0	0
$896$	42.7426	1.42793
$897$	−1.18034	−0.0394104
$898$	−15.8409	−0.528619
$899$	52.6869	1.75721
$900$	0	0
$901$	−20.2918	−0.676018
$902$	−10.1722	−0.338698
$903$	6.23607	0.207523
$904$	22.0000	0.731709
$905$	0	0
$906$	−1.07701	−0.0357814
$907$	32.2492	1.07082	0.535409	−	0.844593i	$-0.320158\pi$
$907$	32.2492	1.07082	0.535409	+	0.844593i	$0.320158\pi$
$908$	32.7295	1.08617
$909$	−16.9656	−0.562712
$910$	0	0
$911$	36.1033	1.19616	0.598078	−	0.801438i	$-0.295931\pi$
$911$	36.1033	1.19616	0.598078	+	0.801438i	$0.295931\pi$
$912$	0	0
$913$	−10.6525	−0.352545
$914$	−2.29180	−0.0758059
$915$	0	0
$916$	40.0820	1.32435
$917$	−33.2705	−1.09869
$918$	−5.12461	−0.169137
$919$	22.8885	0.755023	0.377512	−	0.926005i	$-0.376780\pi$
$919$	22.8885	0.755023	0.377512	+	0.926005i	$0.376780\pi$
$920$	0	0
$921$	−4.68692	−0.154439
$922$	0.403252	0.0132804
$923$	−41.1803	−1.35547
$924$	−7.14590	−0.235083
$925$	0	0
$926$	12.6180	0.414654
$927$	4.77709	0.156900
$928$	−20.1246	−0.660623
$929$	11.6180	0.381175	0.190588	−	0.981670i	$-0.438961\pi$
$929$	11.6180	0.381175	0.190588	+	0.981670i	$0.438961\pi$
$930$	0	0
$931$	0	0
$932$	−12.6443	−0.414179
$933$	2.69505	0.0882319
$934$	−9.43769	−0.308811
$935$	0	0
$936$	21.0081	0.686672
$937$	−2.56231	−0.0837069	−0.0418534	−	0.999124i	$-0.513326\pi$
$937$	−2.56231	−0.0837069	−0.0418534	+	0.999124i	$0.513326\pi$
$938$	14.0902	0.460060
$939$	−3.74265	−0.122137
$940$	0	0
$941$	10.5066	0.342505	0.171252	−	0.985227i	$-0.445219\pi$
$941$	10.5066	0.342505	0.171252	+	0.985227i	$0.445219\pi$
$942$	3.50658	0.114250
$943$	6.90983	0.225015
$944$	−34.8885	−1.13553
$945$	0	0
$946$	−3.50658	−0.114009
$947$	−48.0000	−1.55979	−0.779895	−	0.625910i	$-0.784728\pi$
$947$	−48.0000	−1.55979	−0.779895	+	0.625910i	$0.784728\pi$
$948$	5.66563	0.184011
$949$	−5.00000	−0.162307
$950$	0	0
$951$	−5.05573	−0.163943
$952$	37.4164	1.21267
$953$	11.8328	0.383302	0.191651	−	0.981463i	$-0.438616\pi$
$953$	11.8328	0.383302	0.191651	+	0.981463i	$0.438616\pi$
$954$	−3.68692	−0.119368
$955$	0	0
$956$	−43.6869	−1.41294
$957$	4.41641	0.142762
$958$	4.05573	0.131035
$959$	−5.00000	−0.161458
$960$	0	0
$961$	86.8115	2.80037
$962$	−9.27051	−0.298893
$963$	8.56231	0.275916
$964$	−33.6049	−1.08234
$965$	0	0
$966$	−0.381966	−0.0122896
$967$	−28.7639	−0.924986	−0.462493	−	0.886623i	$-0.653045\pi$
$967$	−28.7639	−0.924986	−0.462493	+	0.886623i	$0.653045\pi$
$968$	−7.84095	−0.252018
$969$	0	0
$970$	0	0
$971$	−37.7984	−1.21301	−0.606504	−	0.795081i	$-0.707428\pi$
$971$	−37.7984	−1.21301	−0.606504	+	0.795081i	$0.707428\pi$
$972$	17.8967	0.574036
$973$	34.5066	1.10623
$974$	−13.5279	−0.433461
$975$	0	0
$976$	−9.43769	−0.302093
$977$	−53.6525	−1.71649	−0.858247	−	0.513236i	$-0.828446\pi$
$977$	−53.6525	−1.71649	−0.858247	+	0.513236i	$0.828446\pi$
$978$	−1.24420	−0.0397851
$979$	−15.9787	−0.510682
$980$	0	0
$981$	−5.03444	−0.160737
$982$	−3.31308	−0.105725
$983$	−13.8541	−0.441877	−0.220939	−	0.975288i	$-0.570912\pi$
$983$	−13.8541	−0.441877	−0.220939	+	0.975288i	$0.570912\pi$
$984$	6.28677	0.200415
$985$	0	0
$986$	−11.1246	−0.354280
$987$	−9.32624	−0.296857
$988$	0	0
$989$	2.38197	0.0757421
$990$	0	0
$991$	18.1459	0.576423	0.288212	−	0.957567i	$-0.406939\pi$
$991$	18.1459	0.576423	0.288212	+	0.957567i	$0.406939\pi$
$992$	−45.0000	−1.42875
$993$	0.111456	0.00353695
$994$	−13.3262	−0.422683
$995$	0	0
$996$	3.16718	0.100356
$997$	3.23607	0.102487	0.0512437	−	0.998686i	$-0.483681\pi$
$997$	3.23607	0.102487	0.0512437	+	0.998686i	$0.483681\pi$
$998$	−10.1803	−0.322253
$999$	−10.8541	−0.343409

Twists

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

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

\(f(q)\)	\(=\)	\(q-0.381966 q^{2} -0.381966 q^{3} -1.85410 q^{4} +0.145898 q^{6} +4.23607 q^{7} +1.47214 q^{8} -2.85410 q^{9} +O(q^{10})\)	\(q-0.381966 q^{2} -0.381966 q^{3} -1.85410 q^{4} +0.145898 q^{6} +4.23607 q^{7} +1.47214 q^{8} -2.85410 q^{9} -2.38197 q^{11} +0.708204 q^{12} -5.00000 q^{13} -1.61803 q^{14} +3.14590 q^{16} +6.00000 q^{17} +1.09017 q^{18} -1.61803 q^{21} +0.909830 q^{22} -0.618034 q^{23} -0.562306 q^{24} +1.90983 q^{26} +2.23607 q^{27} -7.85410 q^{28} +4.85410 q^{29} +10.8541 q^{31} -4.14590 q^{32} +0.909830 q^{33} -2.29180 q^{34} +5.29180 q^{36} -4.85410 q^{37} +1.90983 q^{39} -11.1803 q^{41} +0.618034 q^{42} -3.85410 q^{43} +4.41641 q^{44} +0.236068 q^{46} +5.76393 q^{47} -1.20163 q^{48} +10.9443 q^{49} -2.29180 q^{51} +9.27051 q^{52} -3.38197 q^{53} -0.854102 q^{54} +6.23607 q^{56} -1.85410 q^{58} -11.0902 q^{59} -3.00000 q^{61} -4.14590 q^{62} -12.0902 q^{63} -4.70820 q^{64} -0.347524 q^{66} -8.70820 q^{67} -11.1246 q^{68} +0.236068 q^{69} +8.23607 q^{71} -4.20163 q^{72} +1.00000 q^{73} +1.85410 q^{74} -10.0902 q^{77} -0.729490 q^{78} +8.00000 q^{79} +7.70820 q^{81} +4.27051 q^{82} +4.47214 q^{83} +3.00000 q^{84} +1.47214 q^{86} -1.85410 q^{87} -3.50658 q^{88} +6.70820 q^{89} -21.1803 q^{91} +1.14590 q^{92} -4.14590 q^{93} -2.20163 q^{94} +1.58359 q^{96} +5.09017 q^{97} -4.18034 q^{98} +6.79837 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 3 q^{2} - 3 q^{3} + 3 q^{4} + 7 q^{6} + 4 q^{7} - 6 q^{8} + q^{9}+O(q^{10}) \) 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

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