LMFDB - Embedded newform 8025.2.a.bo.1.20

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8025,2,Mod(1,8025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8025 = 3 \cdot 5^{2} \cdot 107 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8025.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:	$64.0799476221$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	no (minimal twist has level 1605)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.20
Character	$\chi$	$=$	8025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.17402 q^{2} +1.00000 q^{3} +2.72637 q^{4} +2.17402 q^{6} -4.66914 q^{7} +1.57914 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+2.17402 q^{2} +1.00000 q^{3} +2.72637 q^{4} +2.17402 q^{6} -4.66914 q^{7} +1.57914 q^{8} +1.00000 q^{9} +1.06979 q^{11} +2.72637 q^{12} +1.40276 q^{13} -10.1508 q^{14} -2.01965 q^{16} +4.49347 q^{17} +2.17402 q^{18} -3.95111 q^{19} -4.66914 q^{21} +2.32574 q^{22} -7.91994 q^{23} +1.57914 q^{24} +3.04963 q^{26} +1.00000 q^{27} -12.7298 q^{28} +4.24151 q^{29} -4.93856 q^{31} -7.54905 q^{32} +1.06979 q^{33} +9.76890 q^{34} +2.72637 q^{36} +4.43858 q^{37} -8.58980 q^{38} +1.40276 q^{39} -6.98642 q^{41} -10.1508 q^{42} +1.86731 q^{43} +2.91663 q^{44} -17.2181 q^{46} -4.27804 q^{47} -2.01965 q^{48} +14.8009 q^{49} +4.49347 q^{51} +3.82444 q^{52} -9.08154 q^{53} +2.17402 q^{54} -7.37323 q^{56} -3.95111 q^{57} +9.22113 q^{58} +5.80276 q^{59} -7.50643 q^{61} -10.7365 q^{62} -4.66914 q^{63} -12.3725 q^{64} +2.32574 q^{66} +13.0711 q^{67} +12.2509 q^{68} -7.91994 q^{69} -10.3092 q^{71} +1.57914 q^{72} -7.43920 q^{73} +9.64957 q^{74} -10.7722 q^{76} -4.99498 q^{77} +3.04963 q^{78} -17.7333 q^{79} +1.00000 q^{81} -15.1886 q^{82} -9.19377 q^{83} -12.7298 q^{84} +4.05958 q^{86} +4.24151 q^{87} +1.68934 q^{88} -4.28765 q^{89} -6.54967 q^{91} -21.5927 q^{92} -4.93856 q^{93} -9.30055 q^{94} -7.54905 q^{96} +0.655434 q^{97} +32.1774 q^{98} +1.06979 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 3 q^{2} + 22 q^{3} + 13 q^{4} - 3 q^{6} - 2 q^{7} - 9 q^{8} + 22 q^{9}+O(q^{10}) $ 22 * q - 3 * q^2 + 22 * q^3 + 13 * q^4 - 3 * q^6 - 2 * q^7 - 9 * q^8 + 22 * q^9	$ 22 q - 3 q^{2} + 22 q^{3} + 13 q^{4} - 3 q^{6} - 2 q^{7} - 9 q^{8} + 22 q^{9} - 16 q^{11} + 13 q^{12} - 20 q^{14} - q^{16} - 20 q^{17} - 3 q^{18} - 16 q^{19} - 2 q^{21} - 4 q^{22} - 14 q^{23} - 9 q^{24} - 16 q^{26} + 22 q^{27} + q^{28} - 24 q^{29} - 42 q^{31} - 11 q^{32} - 16 q^{33} - 14 q^{34} + 13 q^{36} - 2 q^{37} - 21 q^{38} - 26 q^{41} - 20 q^{42} - 2 q^{43} - 24 q^{44} - 16 q^{46} - 26 q^{47} - q^{48} - 10 q^{49} - 20 q^{51} - 6 q^{52} - 22 q^{53} - 3 q^{54} - 42 q^{56} - 16 q^{57} + 69 q^{58} - 34 q^{59} - 16 q^{61} - 34 q^{62} - 2 q^{63} - 39 q^{64} - 4 q^{66} - 6 q^{68} - 14 q^{69} - 76 q^{71} - 9 q^{72} + 14 q^{73} - 12 q^{74} - 48 q^{76} - 54 q^{77} - 16 q^{78} - 72 q^{79} + 22 q^{81} - 2 q^{82} - 28 q^{83} + q^{84} - 22 q^{86} - 24 q^{87} + 19 q^{88} - 22 q^{89} - 58 q^{91} - 34 q^{92} - 42 q^{93} - 8 q^{94} - 11 q^{96} - 10 q^{97} + 3 q^{98} - 16 q^{99}+O(q^{100}) $ 22 * q - 3 * q^2 + 22 * q^3 + 13 * q^4 - 3 * q^6 - 2 * q^7 - 9 * q^8 + 22 * q^9 - 16 * q^11 + 13 * q^12 - 20 * q^14 - q^16 - 20 * q^17 - 3 * q^18 - 16 * q^19 - 2 * q^21 - 4 * q^22 - 14 * q^23 - 9 * q^24 - 16 * q^26 + 22 * q^27 + q^28 - 24 * q^29 - 42 * q^31 - 11 * q^32 - 16 * q^33 - 14 * q^34 + 13 * q^36 - 2 * q^37 - 21 * q^38 - 26 * q^41 - 20 * q^42 - 2 * q^43 - 24 * q^44 - 16 * q^46 - 26 * q^47 - q^48 - 10 * q^49 - 20 * q^51 - 6 * q^52 - 22 * q^53 - 3 * q^54 - 42 * q^56 - 16 * q^57 + 69 * q^58 - 34 * q^59 - 16 * q^61 - 34 * q^62 - 2 * q^63 - 39 * q^64 - 4 * q^66 - 6 * q^68 - 14 * q^69 - 76 * q^71 - 9 * q^72 + 14 * q^73 - 12 * q^74 - 48 * q^76 - 54 * q^77 - 16 * q^78 - 72 * q^79 + 22 * q^81 - 2 * q^82 - 28 * q^83 + q^84 - 22 * q^86 - 24 * q^87 + 19 * q^88 - 22 * q^89 - 58 * q^91 - 34 * q^92 - 42 * q^93 - 8 * q^94 - 11 * q^96 - 10 * q^97 + 3 * q^98 - 16 * 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 $

$2$	2.17402	1.53727	0.768633	−	0.639690i	$-0.220937\pi$
$2$	2.17402	1.53727	0.768633	+	0.639690i	$0.220937\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.72637	1.36318
$5$	0	0
$5$	0	0
$6$	2.17402	0.887540
$7$	−4.66914	−1.76477	−0.882384	−	0.470529i	$-0.844063\pi$
$7$	−4.66914	−1.76477	−0.882384	+	0.470529i	$0.844063\pi$
$8$	1.57914	0.558311
$9$	1.00000	0.333333
$10$	0	0
$11$	1.06979	0.322553	0.161276	−	0.986909i	$-0.448439\pi$
$11$	1.06979	0.322553	0.161276	+	0.986909i	$0.448439\pi$
$12$	2.72637	0.787035
$13$	1.40276	0.389055	0.194528	−	0.980897i	$-0.437683\pi$
$13$	1.40276	0.389055	0.194528	+	0.980897i	$0.437683\pi$
$14$	−10.1508	−2.71292
$15$	0	0
$16$	−2.01965	−0.504913
$17$	4.49347	1.08983	0.544914	−	0.838492i	$-0.316562\pi$
$17$	4.49347	1.08983	0.544914	+	0.838492i	$0.316562\pi$
$18$	2.17402	0.512422
$19$	−3.95111	−0.906447	−0.453224	−	0.891397i	$-0.649726\pi$
$19$	−3.95111	−0.906447	−0.453224	+	0.891397i	$0.649726\pi$
$20$	0	0
$21$	−4.66914	−1.01889
$22$	2.32574	0.495849
$23$	−7.91994	−1.65142	−0.825711	−	0.564094i	$-0.809226\pi$
$23$	−7.91994	−1.65142	−0.825711	+	0.564094i	$0.809226\pi$
$24$	1.57914	0.322341
$25$	0	0
$26$	3.04963	0.598081
$27$	1.00000	0.192450
$28$	−12.7298	−2.40571
$29$	4.24151	0.787629	0.393814	−	0.919190i	$-0.371155\pi$
$29$	4.24151	0.787629	0.393814	+	0.919190i	$0.371155\pi$
$30$	0	0
$31$	−4.93856	−0.886991	−0.443496	−	0.896277i	$-0.646262\pi$
$31$	−4.93856	−0.886991	−0.443496	+	0.896277i	$0.646262\pi$
$32$	−7.54905	−1.33450
$33$	1.06979	0.186226
$34$	9.76890	1.67535
$35$	0	0
$36$	2.72637	0.454395
$37$	4.43858	0.729698	0.364849	−	0.931067i	$-0.381121\pi$
$37$	4.43858	0.729698	0.364849	+	0.931067i	$0.381121\pi$
$38$	−8.58980	−1.39345
$39$	1.40276	0.224621
$40$	0	0
$41$	−6.98642	−1.09110	−0.545548	−	0.838080i	$-0.683678\pi$
$41$	−6.98642	−1.09110	−0.545548	+	0.838080i	$0.683678\pi$
$42$	−10.1508	−1.56630
$43$	1.86731	0.284763	0.142381	−	0.989812i	$-0.454524\pi$
$43$	1.86731	0.284763	0.142381	+	0.989812i	$0.454524\pi$
$44$	2.91663	0.439699
$45$	0	0
$46$	−17.2181	−2.53867
$47$	−4.27804	−0.624016	−0.312008	−	0.950079i	$-0.601002\pi$
$47$	−4.27804	−0.624016	−0.312008	+	0.950079i	$0.601002\pi$
$48$	−2.01965	−0.291512
$49$	14.8009	2.11441
$50$	0	0
$51$	4.49347	0.629212
$52$	3.82444	0.530354
$53$	−9.08154	−1.24745	−0.623723	−	0.781646i	$-0.714381\pi$
$53$	−9.08154	−1.24745	−0.623723	+	0.781646i	$0.714381\pi$
$54$	2.17402	0.295847
$55$	0	0
$56$	−7.37323	−0.985289
$57$	−3.95111	−0.523338
$58$	9.22113	1.21079
$59$	5.80276	0.755455	0.377728	−	0.925917i	$-0.376706\pi$
$59$	5.80276	0.755455	0.377728	+	0.925917i	$0.376706\pi$
$60$	0	0
$61$	−7.50643	−0.961100	−0.480550	−	0.876967i	$-0.659563\pi$
$61$	−7.50643	−0.961100	−0.480550	+	0.876967i	$0.659563\pi$
$62$	−10.7365	−1.36354
$63$	−4.66914	−0.588256
$64$	−12.3725	−1.54656
$65$	0	0
$66$	2.32574	0.286279
$67$	13.0711	1.59689	0.798443	−	0.602070i	$-0.205657\pi$
$67$	13.0711	1.59689	0.798443	+	0.602070i	$0.205657\pi$
$68$	12.2509	1.48564
$69$	−7.91994	−0.953448
$70$	0	0
$71$	−10.3092	−1.22347	−0.611736	−	0.791062i	$-0.709528\pi$
$71$	−10.3092	−1.22347	−0.611736	+	0.791062i	$0.709528\pi$
$72$	1.57914	0.186104
$73$	−7.43920	−0.870692	−0.435346	−	0.900263i	$-0.643374\pi$
$73$	−7.43920	−0.870692	−0.435346	+	0.900263i	$0.643374\pi$
$74$	9.64957	1.12174
$75$	0	0
$76$	−10.7722	−1.23565
$77$	−4.99498	−0.569231
$78$	3.04963	0.345302
$79$	−17.7333	−1.99515	−0.997575	−	0.0696020i	$-0.977827\pi$
$79$	−17.7333	−1.99515	−0.997575	+	0.0696020i	$0.977827\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	−15.1886	−1.67730
$83$	−9.19377	−1.00915	−0.504574	−	0.863369i	$-0.668350\pi$
$83$	−9.19377	−1.00915	−0.504574	+	0.863369i	$0.668350\pi$
$84$	−12.7298	−1.38893
$85$	0	0
$86$	4.05958	0.437756
$87$	4.24151	0.454738
$88$	1.68934	0.180085
$89$	−4.28765	−0.454490	−0.227245	−	0.973838i	$-0.572972\pi$
$89$	−4.28765	−0.454490	−0.227245	+	0.973838i	$0.572972\pi$
$90$	0	0
$91$	−6.54967	−0.686592
$92$	−21.5927	−2.25119
$93$	−4.93856	−0.512105
$94$	−9.30055	−0.959279
$95$	0	0
$96$	−7.54905	−0.770471
$97$	0.655434	0.0665492	0.0332746	−	0.999446i	$-0.489406\pi$
$97$	0.655434	0.0665492	0.0332746	+	0.999446i	$0.489406\pi$
$98$	32.1774	3.25041
$99$	1.06979	0.107518
$100$	0	0
$101$	−10.3839	−1.03323	−0.516617	−	0.856217i	$-0.672809\pi$
$101$	−10.3839	−1.03323	−0.516617	+	0.856217i	$0.672809\pi$
$102$	9.76890	0.967266
$103$	−9.17727	−0.904264	−0.452132	−	0.891951i	$-0.649336\pi$
$103$	−9.17727	−0.904264	−0.452132	+	0.891951i	$0.649336\pi$
$104$	2.21515	0.217214
$105$	0	0
$106$	−19.7435	−1.91766
$107$	−1.00000	−0.0966736
$107$	−1.00000	−0.0966736
$108$	2.72637	0.262345
$109$	−15.7486	−1.50844	−0.754222	−	0.656620i	$-0.771986\pi$
$109$	−15.7486	−1.50844	−0.754222	+	0.656620i	$0.771986\pi$
$110$	0	0
$111$	4.43858	0.421292
$112$	9.43004	0.891055
$113$	12.2457	1.15198	0.575992	−	0.817456i	$-0.304616\pi$
$113$	12.2457	1.15198	0.575992	+	0.817456i	$0.304616\pi$
$114$	−8.58980	−0.804509
$115$	0	0
$116$	11.5639	1.07368
$117$	1.40276	0.129685
$118$	12.6153	1.16134
$119$	−20.9806	−1.92329
$120$	0	0
$121$	−9.85556	−0.895960
$122$	−16.3191	−1.47747
$123$	−6.98642	−0.629945
$124$	−13.4643	−1.20913
$125$	0	0
$126$	−10.1508	−0.904306
$127$	0.323746	0.0287278	0.0143639	−	0.999897i	$-0.495428\pi$
$127$	0.323746	0.0287278	0.0143639	+	0.999897i	$0.495428\pi$
$128$	−11.8000	−1.04298
$129$	1.86731	0.164408
$130$	0	0
$131$	9.72051	0.849285	0.424642	−	0.905361i	$-0.360400\pi$
$131$	9.72051	0.849285	0.424642	+	0.905361i	$0.360400\pi$
$132$	2.91663	0.253860
$133$	18.4483	1.59967
$134$	28.4168	2.45484
$135$	0	0
$136$	7.09583	0.608462
$137$	8.40560	0.718139	0.359069	−	0.933311i	$-0.383094\pi$
$137$	8.40560	0.718139	0.359069	+	0.933311i	$0.383094\pi$
$138$	−17.2181	−1.46570
$139$	12.2902	1.04244	0.521222	−	0.853421i	$-0.325476\pi$
$139$	12.2902	1.04244	0.521222	+	0.853421i	$0.325476\pi$
$140$	0	0
$141$	−4.27804	−0.360276
$142$	−22.4123	−1.88080
$143$	1.50065	0.125491
$144$	−2.01965	−0.168304
$145$	0	0
$146$	−16.1730	−1.33848
$147$	14.8009	1.22075
$148$	12.1012	0.994713
$149$	7.61585	0.623914	0.311957	−	0.950096i	$-0.399015\pi$
$149$	7.61585	0.623914	0.311957	+	0.950096i	$0.399015\pi$
$150$	0	0
$151$	−13.9211	−1.13288	−0.566440	−	0.824103i	$-0.691680\pi$
$151$	−13.9211	−1.13288	−0.566440	+	0.824103i	$0.691680\pi$
$152$	−6.23936	−0.506079
$153$	4.49347	0.363276
$154$	−10.8592	−0.875059
$155$	0	0
$156$	3.82444	0.306200
$157$	6.16850	0.492300	0.246150	−	0.969232i	$-0.420834\pi$
$157$	6.16850	0.492300	0.246150	+	0.969232i	$0.420834\pi$
$158$	−38.5525	−3.06707
$159$	−9.08154	−0.720213
$160$	0	0
$161$	36.9793	2.91438
$162$	2.17402	0.170807
$163$	−5.00585	−0.392088	−0.196044	−	0.980595i	$-0.562810\pi$
$163$	−5.00585	−0.392088	−0.196044	+	0.980595i	$0.562810\pi$
$164$	−19.0476	−1.48736
$165$	0	0
$166$	−19.9875	−1.55133
$167$	23.9903	1.85642	0.928212	−	0.372051i	$-0.121345\pi$
$167$	23.9903	1.85642	0.928212	+	0.372051i	$0.121345\pi$
$168$	−7.37323	−0.568857
$169$	−11.0323	−0.848636
$170$	0	0
$171$	−3.95111	−0.302149
$172$	5.09099	0.388184
$173$	16.1145	1.22516	0.612581	−	0.790408i	$-0.290131\pi$
$173$	16.1145	1.22516	0.612581	+	0.790408i	$0.290131\pi$
$174$	9.22113	0.699052
$175$	0	0
$176$	−2.16060	−0.162861
$177$	5.80276	0.436162
$178$	−9.32144	−0.698671
$179$	6.42003	0.479855	0.239928	−	0.970791i	$-0.422876\pi$
$179$	6.42003	0.479855	0.239928	+	0.970791i	$0.422876\pi$
$180$	0	0
$181$	−24.2293	−1.80095	−0.900474	−	0.434910i	$-0.856780\pi$
$181$	−24.2293	−1.80095	−0.900474	+	0.434910i	$0.856780\pi$
$182$	−14.2391	−1.05547
$183$	−7.50643	−0.554891
$184$	−12.5067	−0.922006
$185$	0	0
$186$	−10.7365	−0.787240
$187$	4.80706	0.351527
$188$	−11.6635	−0.850649
$189$	−4.66914	−0.339630
$190$	0	0
$191$	0.331627	0.0239957	0.0119978	−	0.999928i	$-0.496181\pi$
$191$	0.331627	0.0239957	0.0119978	+	0.999928i	$0.496181\pi$
$192$	−12.3725	−0.892907
$193$	−4.10612	−0.295565	−0.147782	−	0.989020i	$-0.547214\pi$
$193$	−4.10612	−0.295565	−0.147782	+	0.989020i	$0.547214\pi$
$194$	1.42493	0.102304
$195$	0	0
$196$	40.3526	2.88233
$197$	−10.5051	−0.748459	−0.374229	−	0.927336i	$-0.622093\pi$
$197$	−10.5051	−0.748459	−0.374229	+	0.927336i	$0.622093\pi$
$198$	2.32574	0.165283
$199$	14.6229	1.03659	0.518296	−	0.855201i	$-0.326566\pi$
$199$	14.6229	1.03659	0.518296	+	0.855201i	$0.326566\pi$
$200$	0	0
$201$	13.0711	0.921963
$202$	−22.5748	−1.58835
$203$	−19.8042	−1.38998
$204$	12.2509	0.857732
$205$	0	0
$206$	−19.9516	−1.39009
$207$	−7.91994	−0.550474
$208$	−2.83308	−0.196439
$209$	−4.22685	−0.292377
$210$	0	0
$211$	15.6450	1.07705	0.538524	−	0.842610i	$-0.318982\pi$
$211$	15.6450	1.07705	0.538524	+	0.842610i	$0.318982\pi$
$212$	−24.7596	−1.70050
$213$	−10.3092	−0.706372
$214$	−2.17402	−0.148613
$215$	0	0
$216$	1.57914	0.107447
$217$	23.0588	1.56533
$218$	−34.2378	−2.31888
$219$	−7.43920	−0.502694
$220$	0	0
$221$	6.30325	0.424003
$222$	9.64957	0.647637
$223$	2.53372	0.169671	0.0848353	−	0.996395i	$-0.472964\pi$
$223$	2.53372	0.169671	0.0848353	+	0.996395i	$0.472964\pi$
$224$	35.2476	2.35508
$225$	0	0
$226$	26.6225	1.77090
$227$	8.95886	0.594621	0.297310	−	0.954781i	$-0.403910\pi$
$227$	8.95886	0.594621	0.297310	+	0.954781i	$0.403910\pi$
$228$	−10.7722	−0.713406
$229$	5.14924	0.340271	0.170136	−	0.985421i	$-0.445579\pi$
$229$	5.14924	0.340271	0.170136	+	0.985421i	$0.445579\pi$
$230$	0	0
$231$	−4.99498	−0.328646
$232$	6.69794	0.439741
$233$	24.7439	1.62102	0.810512	−	0.585722i	$-0.199189\pi$
$233$	24.7439	1.62102	0.810512	+	0.585722i	$0.199189\pi$
$234$	3.04963	0.199360
$235$	0	0
$236$	15.8205	1.02982
$237$	−17.7333	−1.15190
$238$	−45.6124	−2.95661
$239$	−16.8319	−1.08876	−0.544382	−	0.838837i	$-0.683236\pi$
$239$	−16.8319	−1.08876	−0.544382	+	0.838837i	$0.683236\pi$
$240$	0	0
$241$	28.9791	1.86671	0.933355	−	0.358956i	$-0.116867\pi$
$241$	28.9791	1.86671	0.933355	+	0.358956i	$0.116867\pi$
$242$	−21.4262	−1.37733
$243$	1.00000	0.0641500
$244$	−20.4653	−1.31016
$245$	0	0
$246$	−15.1886	−0.968392
$247$	−5.54246	−0.352658
$248$	−7.79868	−0.495217
$249$	−9.19377	−0.582631
$250$	0	0
$251$	10.7596	0.679143	0.339571	−	0.940580i	$-0.389718\pi$
$251$	10.7596	0.679143	0.339571	+	0.940580i	$0.389718\pi$
$252$	−12.7298	−0.801902
$253$	−8.47264	−0.532671
$254$	0.703830	0.0441623
$255$	0	0
$256$	−0.908379	−0.0567737
$257$	−22.9235	−1.42993	−0.714966	−	0.699160i	$-0.753558\pi$
$257$	−22.9235	−1.42993	−0.714966	+	0.699160i	$0.753558\pi$
$258$	4.05958	0.252739
$259$	−20.7244	−1.28775
$260$	0	0
$261$	4.24151	0.262543
$262$	21.1326	1.30558
$263$	−10.1790	−0.627662	−0.313831	−	0.949479i	$-0.601613\pi$
$263$	−10.1790	−0.627662	−0.313831	+	0.949479i	$0.601613\pi$
$264$	1.68934	0.103972
$265$	0	0
$266$	40.1070	2.45912
$267$	−4.28765	−0.262400
$268$	35.6366	2.17685
$269$	14.7697	0.900527	0.450264	−	0.892896i	$-0.351330\pi$
$269$	14.7697	0.900527	0.450264	+	0.892896i	$0.351330\pi$
$270$	0	0
$271$	5.46061	0.331708	0.165854	−	0.986150i	$-0.446962\pi$
$271$	5.46061	0.331708	0.165854	+	0.986150i	$0.446962\pi$
$272$	−9.07525	−0.550268
$273$	−6.54967	−0.396404
$274$	18.2740	1.10397
$275$	0	0
$276$	−21.5927	−1.29973
$277$	10.0960	0.606610	0.303305	−	0.952894i	$-0.401910\pi$
$277$	10.0960	0.606610	0.303305	+	0.952894i	$0.401910\pi$
$278$	26.7192	1.60251
$279$	−4.93856	−0.295664
$280$	0	0
$281$	11.6967	0.697769	0.348884	−	0.937166i	$-0.386561\pi$
$281$	11.6967	0.697769	0.348884	+	0.937166i	$0.386561\pi$
$282$	−9.30055	−0.553840
$283$	−28.7888	−1.71131	−0.855657	−	0.517543i	$-0.826847\pi$
$283$	−28.7888	−1.71131	−0.855657	+	0.517543i	$0.826847\pi$
$284$	−28.1066	−1.66782
$285$	0	0
$286$	3.26245	0.192913
$287$	32.6206	1.92553
$288$	−7.54905	−0.444832
$289$	3.19129	0.187723
$290$	0	0
$291$	0.655434	0.0384222
$292$	−20.2820	−1.18691
$293$	14.5644	0.850864	0.425432	−	0.904990i	$-0.360122\pi$
$293$	14.5644	0.850864	0.425432	+	0.904990i	$0.360122\pi$
$294$	32.1774	1.87662
$295$	0	0
$296$	7.00915	0.407398
$297$	1.06979	0.0620753
$298$	16.5570	0.959122
$299$	−11.1098	−0.642494
$300$	0	0
$301$	−8.71875	−0.502541
$302$	−30.2647	−1.74154
$303$	−10.3839	−0.596538
$304$	7.97987	0.457677
$305$	0	0
$306$	9.76890	0.558451
$307$	−33.5191	−1.91304	−0.956518	−	0.291673i	$-0.905788\pi$
$307$	−33.5191	−1.91304	−0.956518	+	0.291673i	$0.905788\pi$
$308$	−13.6182	−0.775967
$309$	−9.17727	−0.522077
$310$	0	0
$311$	−29.5514	−1.67571	−0.837854	−	0.545894i	$-0.816190\pi$
$311$	−29.5514	−1.67571	−0.837854	+	0.545894i	$0.816190\pi$
$312$	2.21515	0.125408
$313$	23.8120	1.34593	0.672966	−	0.739673i	$-0.265020\pi$
$313$	23.8120	1.34593	0.672966	+	0.739673i	$0.265020\pi$
$314$	13.4105	0.756796
$315$	0	0
$316$	−48.3475	−2.71976
$317$	7.58388	0.425953	0.212976	−	0.977057i	$-0.431684\pi$
$317$	7.58388	0.425953	0.212976	+	0.977057i	$0.431684\pi$
$318$	−19.7435	−1.10716
$319$	4.53751	0.254052
$320$	0	0
$321$	−1.00000	−0.0558146
$322$	80.3938	4.48017
$323$	−17.7542	−0.987871
$324$	2.72637	0.151465
$325$	0	0
$326$	−10.8828	−0.602744
$327$	−15.7486	−0.870900
$328$	−11.0325	−0.609170
$329$	19.9748	1.10124
$330$	0	0
$331$	−14.9882	−0.823826	−0.411913	−	0.911223i	$-0.635139\pi$
$331$	−14.9882	−0.823826	−0.411913	+	0.911223i	$0.635139\pi$
$332$	−25.0656	−1.37565
$333$	4.43858	0.243233
$334$	52.1554	2.85382
$335$	0	0
$336$	9.43004	0.514451
$337$	32.1753	1.75270	0.876351	−	0.481673i	$-0.159971\pi$
$337$	32.1753	1.75270	0.876351	+	0.481673i	$0.159971\pi$
$338$	−23.9844	−1.30458
$339$	12.2457	0.665098
$340$	0	0
$341$	−5.28320	−0.286101
$342$	−8.58980	−0.464483
$343$	−36.4233	−1.96668
$344$	2.94875	0.158986
$345$	0	0
$346$	35.0332	1.88340
$347$	6.76898	0.363378	0.181689	−	0.983356i	$-0.441844\pi$
$347$	6.76898	0.363378	0.181689	+	0.983356i	$0.441844\pi$
$348$	11.5639	0.619891
$349$	7.47216	0.399976	0.199988	−	0.979798i	$-0.435910\pi$
$349$	7.47216	0.399976	0.199988	+	0.979798i	$0.435910\pi$
$350$	0	0
$351$	1.40276	0.0748737
$352$	−8.07587	−0.430445
$353$	1.43249	0.0762436	0.0381218	−	0.999273i	$-0.487863\pi$
$353$	1.43249	0.0762436	0.0381218	+	0.999273i	$0.487863\pi$
$354$	12.6153	0.670497
$355$	0	0
$356$	−11.6897	−0.619553
$357$	−20.9806	−1.11041
$358$	13.9573	0.737665
$359$	36.0307	1.90163	0.950814	−	0.309763i	$-0.100250\pi$
$359$	36.0307	1.90163	0.950814	+	0.309763i	$0.100250\pi$
$360$	0	0
$361$	−3.38871	−0.178353
$362$	−52.6750	−2.76853
$363$	−9.85556	−0.517283
$364$	−17.8568	−0.935952
$365$	0	0
$366$	−16.3191	−0.853015
$367$	−2.94743	−0.153855	−0.0769273	−	0.997037i	$-0.524511\pi$
$367$	−2.94743	−0.153855	−0.0769273	+	0.997037i	$0.524511\pi$
$368$	15.9955	0.833824
$369$	−6.98642	−0.363699
$370$	0	0
$371$	42.4030	2.20145
$372$	−13.4643	−0.698093
$373$	−10.8954	−0.564144	−0.282072	−	0.959393i	$-0.591022\pi$
$373$	−10.8954	−0.564144	−0.282072	+	0.959393i	$0.591022\pi$
$374$	10.4506	0.540390
$375$	0	0
$376$	−6.75563	−0.348395
$377$	5.94981	0.306431
$378$	−10.1508	−0.522101
$379$	−15.9160	−0.817552	−0.408776	−	0.912635i	$-0.634044\pi$
$379$	−15.9160	−0.817552	−0.408776	+	0.912635i	$0.634044\pi$
$380$	0	0
$381$	0.323746	0.0165860
$382$	0.720964	0.0368877
$383$	0.461954	0.0236048	0.0118024	−	0.999930i	$-0.496243\pi$
$383$	0.461954	0.0236048	0.0118024	+	0.999930i	$0.496243\pi$
$384$	−11.8000	−0.602164
$385$	0	0
$386$	−8.92679	−0.454361
$387$	1.86731	0.0949209
$388$	1.78695	0.0907188
$389$	−25.6492	−1.30046	−0.650232	−	0.759736i	$-0.725328\pi$
$389$	−25.6492	−1.30046	−0.650232	+	0.759736i	$0.725328\pi$
$390$	0	0
$391$	−35.5880	−1.79976
$392$	23.3727	1.18050
$393$	9.72051	0.490335
$394$	−22.8384	−1.15058
$395$	0	0
$396$	2.91663	0.146566
$397$	10.3146	0.517677	0.258838	−	0.965921i	$-0.416660\pi$
$397$	10.3146	0.517677	0.258838	+	0.965921i	$0.416660\pi$
$398$	31.7906	1.59352
$399$	18.4483	0.923570
$400$	0	0
$401$	3.89198	0.194356	0.0971781	−	0.995267i	$-0.469018\pi$
$401$	3.89198	0.194356	0.0971781	+	0.995267i	$0.469018\pi$
$402$	28.4168	1.41730
$403$	−6.92760	−0.345088
$404$	−28.3103	−1.40849
$405$	0	0
$406$	−43.0548	−2.13677
$407$	4.74834	0.235366
$408$	7.09583	0.351296
$409$	−21.7781	−1.07686	−0.538429	−	0.842671i	$-0.680982\pi$
$409$	−21.7781	−1.07686	−0.538429	+	0.842671i	$0.680982\pi$
$410$	0	0
$411$	8.40560	0.414618
$412$	−25.0206	−1.23268
$413$	−27.0939	−1.33320
$414$	−17.2181	−0.846224
$415$	0	0
$416$	−10.5895	−0.519192
$417$	12.2902	0.601855
$418$	−9.18926	−0.449461
$419$	−28.7891	−1.40644	−0.703219	−	0.710973i	$-0.748255\pi$
$419$	−28.7891	−1.40644	−0.703219	+	0.710973i	$0.748255\pi$
$420$	0	0
$421$	−5.50443	−0.268270	−0.134135	−	0.990963i	$-0.542826\pi$
$421$	−5.50443	−0.268270	−0.134135	+	0.990963i	$0.542826\pi$
$422$	34.0126	1.65571
$423$	−4.27804	−0.208005
$424$	−14.3410	−0.696462
$425$	0	0
$426$	−22.4123	−1.08588
$427$	35.0486	1.69612
$428$	−2.72637	−0.131784
$429$	1.50065	0.0724522
$430$	0	0
$431$	−20.9856	−1.01084	−0.505421	−	0.862873i	$-0.668663\pi$
$431$	−20.9856	−1.01084	−0.505421	+	0.862873i	$0.668663\pi$
$432$	−2.01965	−0.0971705
$433$	−10.6734	−0.512932	−0.256466	−	0.966553i	$-0.582558\pi$
$433$	−10.6734	−0.512932	−0.256466	+	0.966553i	$0.582558\pi$
$434$	50.1304	2.40633
$435$	0	0
$436$	−42.9365	−2.05629
$437$	31.2926	1.49693
$438$	−16.1730	−0.772775
$439$	12.0803	0.576563	0.288281	−	0.957546i	$-0.406916\pi$
$439$	12.0803	0.576563	0.288281	+	0.957546i	$0.406916\pi$
$440$	0	0
$441$	14.8009	0.704803
$442$	13.7034	0.651805
$443$	24.2340	1.15139	0.575696	−	0.817664i	$-0.304731\pi$
$443$	24.2340	1.15139	0.575696	+	0.817664i	$0.304731\pi$
$444$	12.1012	0.574298
$445$	0	0
$446$	5.50837	0.260829
$447$	7.61585	0.360217
$448$	57.7689	2.72932
$449$	27.0442	1.27629	0.638147	−	0.769915i	$-0.279701\pi$
$449$	27.0442	1.27629	0.638147	+	0.769915i	$0.279701\pi$
$450$	0	0
$451$	−7.47398	−0.351936
$452$	33.3864	1.57037
$453$	−13.9211	−0.654068
$454$	19.4768	0.914090
$455$	0	0
$456$	−6.23936	−0.292185
$457$	23.8970	1.11786	0.558928	−	0.829216i	$-0.311213\pi$
$457$	23.8970	1.11786	0.558928	+	0.829216i	$0.311213\pi$
$458$	11.1946	0.523087
$459$	4.49347	0.209737
$460$	0	0
$461$	13.6001	0.633419	0.316710	−	0.948523i	$-0.397422\pi$
$461$	13.6001	0.633419	0.316710	+	0.948523i	$0.397422\pi$
$462$	−10.8592	−0.505216
$463$	6.97450	0.324132	0.162066	−	0.986780i	$-0.448184\pi$
$463$	6.97450	0.324132	0.162066	+	0.986780i	$0.448184\pi$
$464$	−8.56637	−0.397684
$465$	0	0
$466$	53.7937	2.49194
$467$	4.66690	0.215959	0.107979	−	0.994153i	$-0.465562\pi$
$467$	4.66690	0.215959	0.107979	+	0.994153i	$0.465562\pi$
$468$	3.82444	0.176785
$469$	−61.0307	−2.81813
$470$	0	0
$471$	6.16850	0.284230
$472$	9.16338	0.421779
$473$	1.99763	0.0918511
$474$	−38.5525	−1.77078
$475$	0	0
$476$	−57.2010	−2.62180
$477$	−9.08154	−0.415815
$478$	−36.5929	−1.67372
$479$	12.8530	0.587271	0.293635	−	0.955917i	$-0.405135\pi$
$479$	12.8530	0.587271	0.293635	+	0.955917i	$0.405135\pi$
$480$	0	0
$481$	6.22626	0.283893
$482$	63.0012	2.86963
$483$	36.9793	1.68262
$484$	−26.8699	−1.22136
$485$	0	0
$486$	2.17402	0.0986156
$487$	−29.4969	−1.33663	−0.668315	−	0.743878i	$-0.732984\pi$
$487$	−29.4969	−1.33663	−0.668315	+	0.743878i	$0.732984\pi$
$488$	−11.8537	−0.536592
$489$	−5.00585	−0.226372
$490$	0	0
$491$	10.5562	0.476396	0.238198	−	0.971217i	$-0.423443\pi$
$491$	10.5562	0.476396	0.238198	+	0.971217i	$0.423443\pi$
$492$	−19.0476	−0.858730
$493$	19.0591	0.858379
$494$	−12.0494	−0.542129
$495$	0	0
$496$	9.97416	0.447853
$497$	48.1349	2.15915
$498$	−19.9875	−0.895659
$499$	−1.52896	−0.0684455	−0.0342227	−	0.999414i	$-0.510896\pi$
$499$	−1.52896	−0.0684455	−0.0342227	+	0.999414i	$0.510896\pi$
$500$	0	0
$501$	23.9903	1.07181
$502$	23.3917	1.04402
$503$	30.4842	1.35922	0.679611	−	0.733573i	$-0.262149\pi$
$503$	30.4842	1.35922	0.679611	+	0.733573i	$0.262149\pi$
$504$	−7.37323	−0.328430
$505$	0	0
$506$	−18.4197	−0.818856
$507$	−11.0323	−0.489960
$508$	0.882651	0.0391613
$509$	−6.59117	−0.292148	−0.146074	−	0.989274i	$-0.546664\pi$
$509$	−6.59117	−0.292148	−0.146074	+	0.989274i	$0.546664\pi$
$510$	0	0
$511$	34.7347	1.53657
$512$	21.6251	0.955702
$513$	−3.95111	−0.174446
$514$	−49.8363	−2.19818
$515$	0	0
$516$	5.09099	0.224118
$517$	−4.57659	−0.201278
$518$	−45.0552	−1.97961
$519$	16.1145	0.707348
$520$	0	0
$521$	−27.4192	−1.20126	−0.600628	−	0.799528i	$-0.705083\pi$
$521$	−27.4192	−1.20126	−0.600628	+	0.799528i	$0.705083\pi$
$522$	9.22113	0.403598
$523$	−2.83442	−0.123941	−0.0619703	−	0.998078i	$-0.519738\pi$
$523$	−2.83442	−0.123941	−0.0619703	+	0.998078i	$0.519738\pi$
$524$	26.5017	1.15773
$525$	0	0
$526$	−22.1293	−0.964883
$527$	−22.1913	−0.966667
$528$	−2.16060	−0.0940279
$529$	39.7254	1.72719
$530$	0	0
$531$	5.80276	0.251818
$532$	50.2969	2.18065
$533$	−9.80026	−0.424496
$534$	−9.32144	−0.403378
$535$	0	0
$536$	20.6411	0.891558
$537$	6.42003	0.277045
$538$	32.1097	1.38435
$539$	15.8338	0.682009
$540$	0	0
$541$	30.4034	1.30714	0.653572	−	0.756865i	$-0.273270\pi$
$541$	30.4034	1.30714	0.653572	+	0.756865i	$0.273270\pi$
$542$	11.8715	0.509924
$543$	−24.2293	−1.03978
$544$	−33.9214	−1.45437
$545$	0	0
$546$	−14.2391	−0.609379
$547$	−42.1064	−1.80034	−0.900170	−	0.435538i	$-0.856558\pi$
$547$	−42.1064	−1.80034	−0.900170	+	0.435538i	$0.856558\pi$
$548$	22.9168	0.978956
$549$	−7.50643	−0.320367
$550$	0	0
$551$	−16.7587	−0.713944
$552$	−12.5067	−0.532320
$553$	82.7992	3.52098
$554$	21.9489	0.932520
$555$	0	0
$556$	33.5077	1.42104
$557$	−32.5940	−1.38105	−0.690527	−	0.723307i	$-0.742621\pi$
$557$	−32.5940	−1.38105	−0.690527	+	0.723307i	$0.742621\pi$
$558$	−10.7365	−0.454514
$559$	2.61939	0.110788
$560$	0	0
$561$	4.80706	0.202954
$562$	25.4290	1.07266
$563$	−38.9190	−1.64024	−0.820119	−	0.572193i	$-0.806093\pi$
$563$	−38.9190	−1.64024	−0.820119	+	0.572193i	$0.806093\pi$
$564$	−11.6635	−0.491123
$565$	0	0
$566$	−62.5874	−2.63074
$567$	−4.66914	−0.196085
$568$	−16.2796	−0.683077
$569$	−2.39160	−0.100261	−0.0501305	−	0.998743i	$-0.515964\pi$
$569$	−2.39160	−0.100261	−0.0501305	+	0.998743i	$0.515964\pi$
$570$	0	0
$571$	0.997647	0.0417502	0.0208751	−	0.999782i	$-0.493355\pi$
$571$	0.997647	0.0417502	0.0208751	+	0.999782i	$0.493355\pi$
$572$	4.09133	0.171067
$573$	0.331627	0.0138539
$574$	70.9178	2.96005
$575$	0	0
$576$	−12.3725	−0.515520
$577$	27.7402	1.15484	0.577420	−	0.816447i	$-0.304060\pi$
$577$	27.7402	1.15484	0.577420	+	0.816447i	$0.304060\pi$
$578$	6.93794	0.288580
$579$	−4.10612	−0.170644
$580$	0	0
$581$	42.9270	1.78091
$582$	1.42493	0.0590651
$583$	−9.71531	−0.402367
$584$	−11.7475	−0.486117
$585$	0	0
$586$	31.6634	1.30800
$587$	−9.48639	−0.391545	−0.195773	−	0.980649i	$-0.562721\pi$
$587$	−9.48639	−0.391545	−0.195773	+	0.980649i	$0.562721\pi$
$588$	40.3526	1.66411
$589$	19.5128	0.804011
$590$	0	0
$591$	−10.5051	−0.432123
$592$	−8.96439	−0.368434
$593$	−0.0132108	−0.000542503 0	−0.000271251	−	1.00000i	$-0.500086\pi$
$593$	−0.0132108	−0.000542503 0	−0.000271251		1.00000i	$0.500086\pi$
$594$	2.32574	0.0954262
$595$	0	0
$596$	20.7636	0.850510
$597$	14.6229	0.598477
$598$	−24.1528	−0.987683
$599$	−14.8054	−0.604932	−0.302466	−	0.953160i	$-0.597810\pi$
$599$	−14.8054	−0.604932	−0.302466	+	0.953160i	$0.597810\pi$
$600$	0	0
$601$	−17.1577	−0.699879	−0.349939	−	0.936772i	$-0.613798\pi$
$601$	−17.1577	−0.699879	−0.349939	+	0.936772i	$0.613798\pi$
$602$	−18.9548	−0.772538
$603$	13.0711	0.532295
$604$	−37.9539	−1.54432
$605$	0	0
$606$	−22.5748	−0.917037
$607$	−4.25499	−0.172705	−0.0863524	−	0.996265i	$-0.527521\pi$
$607$	−4.25499	−0.172705	−0.0863524	+	0.996265i	$0.527521\pi$
$608$	29.8271	1.20965
$609$	−19.8042	−0.802507
$610$	0	0
$611$	−6.00106	−0.242777
$612$	12.2509	0.495212
$613$	29.0134	1.17184	0.585919	−	0.810369i	$-0.300734\pi$
$613$	29.0134	1.17184	0.585919	+	0.810369i	$0.300734\pi$
$614$	−72.8712	−2.94084
$615$	0	0
$616$	−7.88778	−0.317808
$617$	16.9082	0.680697	0.340349	−	0.940299i	$-0.389455\pi$
$617$	16.9082	0.680697	0.340349	+	0.940299i	$0.389455\pi$
$618$	−19.9516	−0.802571
$619$	14.0463	0.564570	0.282285	−	0.959331i	$-0.408908\pi$
$619$	14.0463	0.564570	0.282285	+	0.959331i	$0.408908\pi$
$620$	0	0
$621$	−7.91994	−0.317816
$622$	−64.2455	−2.57601
$623$	20.0196	0.802069
$624$	−2.83308	−0.113414
$625$	0	0
$626$	51.7677	2.06905
$627$	−4.22685	−0.168804
$628$	16.8176	0.671096
$629$	19.9446	0.795245
$630$	0	0
$631$	−2.29572	−0.0913910	−0.0456955	−	0.998955i	$-0.514550\pi$
$631$	−2.29572	−0.0913910	−0.0456955	+	0.998955i	$0.514550\pi$
$632$	−28.0033	−1.11391
$633$	15.6450	0.621834
$634$	16.4875	0.654803
$635$	0	0
$636$	−24.7596	−0.981783
$637$	20.7620	0.822622
$638$	9.86464	0.390545
$639$	−10.3092	−0.407824
$640$	0	0
$641$	−30.1115	−1.18933	−0.594666	−	0.803973i	$-0.702716\pi$
$641$	−30.1115	−1.18933	−0.594666	+	0.803973i	$0.702716\pi$
$642$	−2.17402	−0.0858018
$643$	7.51571	0.296391	0.148195	−	0.988958i	$-0.452654\pi$
$643$	7.51571	0.296391	0.148195	+	0.988958i	$0.452654\pi$
$644$	100.819	3.97283
$645$	0	0
$646$	−38.5980	−1.51862
$647$	−28.6328	−1.12567	−0.562836	−	0.826568i	$-0.690290\pi$
$647$	−28.6328	−1.12567	−0.562836	+	0.826568i	$0.690290\pi$
$648$	1.57914	0.0620345
$649$	6.20772	0.243674
$650$	0	0
$651$	23.0588	0.903746
$652$	−13.6478	−0.534489
$653$	−47.2548	−1.84922	−0.924612	−	0.380910i	$-0.875611\pi$
$653$	−47.2548	−1.84922	−0.924612	+	0.380910i	$0.875611\pi$
$654$	−34.2378	−1.33880
$655$	0	0
$656$	14.1101	0.550908
$657$	−7.43920	−0.290231
$658$	43.4256	1.69291
$659$	−32.5178	−1.26672	−0.633358	−	0.773859i	$-0.718324\pi$
$659$	−32.5178	−1.26672	−0.633358	+	0.773859i	$0.718324\pi$
$660$	0	0
$661$	−2.48645	−0.0967118	−0.0483559	−	0.998830i	$-0.515398\pi$
$661$	−2.48645	−0.0967118	−0.0483559	+	0.998830i	$0.515398\pi$
$662$	−32.5847	−1.26644
$663$	6.30325	0.244798
$664$	−14.5183	−0.563418
$665$	0	0
$666$	9.64957	0.373913
$667$	−33.5925	−1.30071
$668$	65.4064	2.53065
$669$	2.53372	0.0979594
$670$	0	0
$671$	−8.03028	−0.310006
$672$	35.2476	1.35970
$673$	13.9616	0.538180	0.269090	−	0.963115i	$-0.413277\pi$
$673$	13.9616	0.538180	0.269090	+	0.963115i	$0.413277\pi$
$674$	69.9499	2.69437
$675$	0	0
$676$	−30.0780	−1.15685
$677$	−22.4345	−0.862228	−0.431114	−	0.902298i	$-0.641879\pi$
$677$	−22.4345	−0.862228	−0.431114	+	0.902298i	$0.641879\pi$
$678$	26.6225	1.02243
$679$	−3.06031	−0.117444
$680$	0	0
$681$	8.95886	0.343304
$682$	−11.4858	−0.439814
$683$	3.22700	0.123478	0.0617390	−	0.998092i	$-0.480335\pi$
$683$	3.22700	0.123478	0.0617390	+	0.998092i	$0.480335\pi$
$684$	−10.7722	−0.411885
$685$	0	0
$686$	−79.1851	−3.02330
$687$	5.14924	0.196456
$688$	−3.77133	−0.143780
$689$	−12.7392	−0.485325
$690$	0	0
$691$	14.6815	0.558511	0.279255	−	0.960217i	$-0.409912\pi$
$691$	14.6815	0.558511	0.279255	+	0.960217i	$0.409912\pi$
$692$	43.9340	1.67012
$693$	−4.99498	−0.189744
$694$	14.7159	0.558608
$695$	0	0
$696$	6.69794	0.253885
$697$	−31.3933	−1.18911
$698$	16.2446	0.614869
$699$	24.7439	0.935899
$700$	0	0
$701$	16.9415	0.639870	0.319935	−	0.947440i	$-0.396339\pi$
$701$	16.9415	0.639870	0.319935	+	0.947440i	$0.396339\pi$
$702$	3.04963	0.115101
$703$	−17.5373	−0.661433
$704$	−13.2359	−0.498848
$705$	0	0
$706$	3.11426	0.117207
$707$	48.4837	1.82342
$708$	15.8205	0.594570
$709$	42.6493	1.60173	0.800863	−	0.598847i	$-0.204374\pi$
$709$	42.6493	1.60173	0.800863	+	0.598847i	$0.204374\pi$
$710$	0	0
$711$	−17.7333	−0.665050
$712$	−6.77080	−0.253746
$713$	39.1131	1.46480
$714$	−45.6124	−1.70700
$715$	0	0
$716$	17.5034	0.654131
$717$	−16.8319	−0.628599
$718$	78.3315	2.92331
$719$	37.5010	1.39855	0.699276	−	0.714852i	$-0.253506\pi$
$719$	37.5010	1.39855	0.699276	+	0.714852i	$0.253506\pi$
$720$	0	0
$721$	42.8500	1.59582
$722$	−7.36713	−0.274176
$723$	28.9791	1.07775
$724$	−66.0579	−2.45502
$725$	0	0
$726$	−21.4262	−0.795200
$727$	−2.99876	−0.111218	−0.0556090	−	0.998453i	$-0.517710\pi$
$727$	−2.99876	−0.111218	−0.0556090	+	0.998453i	$0.517710\pi$
$728$	−10.3429	−0.383332
$729$	1.00000	0.0370370
$730$	0	0
$731$	8.39073	0.310342
$732$	−20.4653	−0.756419
$733$	28.1441	1.03953	0.519763	−	0.854310i	$-0.326020\pi$
$733$	28.1441	1.03953	0.519763	+	0.854310i	$0.326020\pi$
$734$	−6.40778	−0.236515
$735$	0	0
$736$	59.7880	2.20381
$737$	13.9833	0.515080
$738$	−15.1886	−0.559101
$739$	−11.3139	−0.416189	−0.208095	−	0.978109i	$-0.566726\pi$
$739$	−11.3139	−0.416189	−0.208095	+	0.978109i	$0.566726\pi$
$740$	0	0
$741$	−5.54246	−0.203607
$742$	92.1850	3.38422
$743$	4.78783	0.175649	0.0878243	−	0.996136i	$-0.472009\pi$
$743$	4.78783	0.175649	0.0878243	+	0.996136i	$0.472009\pi$
$744$	−7.79868	−0.285913
$745$	0	0
$746$	−23.6869	−0.867239
$747$	−9.19377	−0.336382
$748$	13.1058	0.479196
$749$	4.66914	0.170607
$750$	0	0
$751$	−13.2184	−0.482346	−0.241173	−	0.970482i	$-0.577532\pi$
$751$	−13.2184	−0.482346	−0.241173	+	0.970482i	$0.577532\pi$
$752$	8.64015	0.315074
$753$	10.7596	0.392103
$754$	12.9350	0.471066
$755$	0	0
$756$	−12.7298	−0.462978
$757$	5.23924	0.190424	0.0952118	−	0.995457i	$-0.469647\pi$
$757$	5.23924	0.190424	0.0952118	+	0.995457i	$0.469647\pi$
$758$	−34.6018	−1.25679
$759$	−8.47264	−0.307537
$760$	0	0
$761$	−27.3080	−0.989914	−0.494957	−	0.868917i	$-0.664816\pi$
$761$	−27.3080	−0.989914	−0.494957	+	0.868917i	$0.664816\pi$
$762$	0.703830	0.0254971
$763$	73.5325	2.66205
$764$	0.904137	0.0327105
$765$	0	0
$766$	1.00430	0.0362868
$767$	8.13987	0.293914
$768$	−0.908379	−0.0327783
$769$	24.8465	0.895987	0.447993	−	0.894037i	$-0.352139\pi$
$769$	24.8465	0.895987	0.447993	+	0.894037i	$0.352139\pi$
$770$	0	0
$771$	−22.9235	−0.825571
$772$	−11.1948	−0.402909
$773$	−28.9230	−1.04029	−0.520145	−	0.854078i	$-0.674122\pi$
$773$	−28.9230	−1.04029	−0.520145	+	0.854078i	$0.674122\pi$
$774$	4.05958	0.145919
$775$	0	0
$776$	1.03502	0.0371551
$777$	−20.7244	−0.743482
$778$	−55.7618	−1.99916
$779$	27.6041	0.989021
$780$	0	0
$781$	−11.0286	−0.394634
$782$	−77.3691	−2.76671
$783$	4.24151	0.151579
$784$	−29.8926	−1.06759
$785$	0	0
$786$	21.1326	0.753775
$787$	−30.8669	−1.10029	−0.550144	−	0.835070i	$-0.685427\pi$
$787$	−30.8669	−1.10029	−0.550144	+	0.835070i	$0.685427\pi$
$788$	−28.6408	−1.02029
$789$	−10.1790	−0.362381
$790$	0	0
$791$	−57.1771	−2.03298
$792$	1.68934	0.0600282
$793$	−10.5297	−0.373921
$794$	22.4242	0.795806
$795$	0	0
$796$	39.8675	1.41307
$797$	15.1196	0.535565	0.267783	−	0.963479i	$-0.413709\pi$
$797$	15.1196	0.535565	0.267783	+	0.963479i	$0.413709\pi$
$798$	40.1070	1.41977
$799$	−19.2233	−0.680070
$800$	0	0
$801$	−4.28765	−0.151497
$802$	8.46125	0.298777
$803$	−7.95835	−0.280844
$804$	35.6366	1.25680
$805$	0	0
$806$	−15.0608	−0.530492
$807$	14.7697	0.519920
$808$	−16.3976	−0.576865
$809$	−41.6969	−1.46598	−0.732992	−	0.680237i	$-0.761877\pi$
$809$	−41.6969	−1.46598	−0.732992	+	0.680237i	$0.761877\pi$
$810$	0	0
$811$	−0.467700	−0.0164232	−0.00821158	−	0.999966i	$-0.502614\pi$
$811$	−0.467700	−0.0164232	−0.00821158	+	0.999966i	$0.502614\pi$
$812$	−53.9936	−1.89480
$813$	5.46061	0.191512
$814$	10.3230	0.361820
$815$	0	0
$816$	−9.07525	−0.317697
$817$	−7.37797	−0.258123
$818$	−47.3461	−1.65542
$819$	−6.54967	−0.228864
$820$	0	0
$821$	24.3299	0.849118	0.424559	−	0.905400i	$-0.360429\pi$
$821$	24.3299	0.849118	0.424559	+	0.905400i	$0.360429\pi$
$822$	18.2740	0.637377
$823$	44.3480	1.54588	0.772938	−	0.634482i	$-0.218787\pi$
$823$	44.3480	1.54588	0.772938	+	0.634482i	$0.218787\pi$
$824$	−14.4922	−0.504860
$825$	0	0
$826$	−58.9027	−2.04949
$827$	14.9028	0.518221	0.259110	−	0.965848i	$-0.416571\pi$
$827$	14.9028	0.518221	0.259110	+	0.965848i	$0.416571\pi$
$828$	−21.5927	−0.750397
$829$	−27.7741	−0.964634	−0.482317	−	0.875997i	$-0.660205\pi$
$829$	−27.7741	−0.964634	−0.482317	+	0.875997i	$0.660205\pi$
$830$	0	0
$831$	10.0960	0.350226
$832$	−17.3556	−0.601697
$833$	66.5073	2.30434
$834$	26.7192	0.925211
$835$	0	0
$836$	−11.5239	−0.398564
$837$	−4.93856	−0.170702
$838$	−62.5881	−2.16207
$839$	30.5184	1.05361	0.526807	−	0.849985i	$-0.323389\pi$
$839$	30.5184	1.05361	0.526807	+	0.849985i	$0.323389\pi$
$840$	0	0
$841$	−11.0096	−0.379641
$842$	−11.9668	−0.412402
$843$	11.6967	0.402857
$844$	42.6541	1.46821
$845$	0	0
$846$	−9.30055	−0.319760
$847$	46.0170	1.58116
$848$	18.3416	0.629851
$849$	−28.7888	−0.988028
$850$	0	0
$851$	−35.1533	−1.20504
$852$	−28.1066	−0.962915
$853$	45.3475	1.55267	0.776334	−	0.630321i	$-0.217077\pi$
$853$	45.3475	1.55267	0.776334	+	0.630321i	$0.217077\pi$
$854$	76.1964	2.60739
$855$	0	0
$856$	−1.57914	−0.0539739
$857$	−43.9763	−1.50220	−0.751100	−	0.660188i	$-0.770477\pi$
$857$	−43.9763	−1.50220	−0.751100	+	0.660188i	$0.770477\pi$
$858$	3.26245	0.111378
$859$	−23.2635	−0.793739	−0.396870	−	0.917875i	$-0.629903\pi$
$859$	−23.2635	−0.793739	−0.396870	+	0.917875i	$0.629903\pi$
$860$	0	0
$861$	32.6206	1.11171
$862$	−45.6232	−1.55393
$863$	35.3225	1.20239	0.601197	−	0.799101i	$-0.294691\pi$
$863$	35.3225	1.20239	0.601197	+	0.799101i	$0.294691\pi$
$864$	−7.54905	−0.256824
$865$	0	0
$866$	−23.2042	−0.788512
$867$	3.19129	0.108382
$868$	62.8668	2.13384
$869$	−18.9708	−0.643541
$870$	0	0
$871$	18.3356	0.621277
$872$	−24.8693	−0.842180
$873$	0.655434	0.0221831
$874$	68.0307	2.30117
$875$	0	0
$876$	−20.2820	−0.685265
$877$	47.1264	1.59135	0.795673	−	0.605726i	$-0.207117\pi$
$877$	47.1264	1.59135	0.795673	+	0.605726i	$0.207117\pi$
$878$	26.2629	0.886329
$879$	14.5644	0.491246
$880$	0	0
$881$	−9.42768	−0.317627	−0.158813	−	0.987309i	$-0.550767\pi$
$881$	−9.42768	−0.317627	−0.158813	+	0.987309i	$0.550767\pi$
$882$	32.1774	1.08347
$883$	41.6432	1.40140	0.700702	−	0.713454i	$-0.252870\pi$
$883$	41.6432	1.40140	0.700702	+	0.713454i	$0.252870\pi$
$884$	17.1850	0.577994
$885$	0	0
$886$	52.6852	1.76999
$887$	7.55968	0.253829	0.126915	−	0.991914i	$-0.459493\pi$
$887$	7.55968	0.253829	0.126915	+	0.991914i	$0.459493\pi$
$888$	7.00915	0.235212
$889$	−1.51161	−0.0506979
$890$	0	0
$891$	1.06979	0.0358392
$892$	6.90786	0.231292
$893$	16.9030	0.565638
$894$	16.5570	0.553749
$895$	0	0
$896$	55.0956	1.84062
$897$	−11.1098	−0.370944
$898$	58.7946	1.96200
$899$	−20.9469	−0.698620
$900$	0	0
$901$	−40.8077	−1.35950
$902$	−16.2486	−0.541019
$903$	−8.71875	−0.290142
$904$	19.3378	0.643164
$905$	0	0
$906$	−30.2647	−1.00548
$907$	33.1097	1.09939	0.549694	−	0.835366i	$-0.314744\pi$
$907$	33.1097	1.09939	0.549694	+	0.835366i	$0.314744\pi$
$908$	24.4252	0.810577
$909$	−10.3839	−0.344411
$910$	0	0
$911$	−9.52490	−0.315574	−0.157787	−	0.987473i	$-0.550436\pi$
$911$	−9.52490	−0.315574	−0.157787	+	0.987473i	$0.550436\pi$
$912$	7.97987	0.264240
$913$	−9.83537	−0.325503
$914$	51.9527	1.71844
$915$	0	0
$916$	14.0387	0.463852
$917$	−45.3864	−1.49879
$918$	9.76890	0.322422
$919$	−41.6086	−1.37254	−0.686270	−	0.727347i	$-0.740753\pi$
$919$	−41.6086	−1.37254	−0.686270	+	0.727347i	$0.740753\pi$
$920$	0	0
$921$	−33.5191	−1.10449
$922$	29.5669	0.973733
$923$	−14.4613	−0.475998
$924$	−13.6182	−0.448005
$925$	0	0
$926$	15.1627	0.498277
$927$	−9.17727	−0.301421
$928$	−32.0194	−1.05109
$929$	32.7382	1.07411	0.537053	−	0.843549i	$-0.319538\pi$
$929$	32.7382	1.07411	0.537053	+	0.843549i	$0.319538\pi$
$930$	0	0
$931$	−58.4799	−1.91660
$932$	67.4609	2.20975
$933$	−29.5514	−0.967471
$934$	10.1459	0.331986
$935$	0	0
$936$	2.21515	0.0724045
$937$	−32.8854	−1.07432	−0.537159	−	0.843481i	$-0.680503\pi$
$937$	−32.8854	−1.07432	−0.537159	+	0.843481i	$0.680503\pi$
$938$	−132.682	−4.33222
$939$	23.8120	0.777074
$940$	0	0
$941$	27.8895	0.909171	0.454586	−	0.890703i	$-0.349787\pi$
$941$	27.8895	0.909171	0.454586	+	0.890703i	$0.349787\pi$
$942$	13.4105	0.436936
$943$	55.3320	1.80186
$944$	−11.7196	−0.381439
$945$	0	0
$946$	4.34289	0.141199
$947$	44.7638	1.45463	0.727313	−	0.686306i	$-0.240769\pi$
$947$	44.7638	1.45463	0.727313	+	0.686306i	$0.240769\pi$
$948$	−48.3475	−1.57025
$949$	−10.4354	−0.338747
$950$	0	0
$951$	7.58388	0.245924
$952$	−33.1314	−1.07379
$953$	6.42945	0.208270	0.104135	−	0.994563i	$-0.466793\pi$
$953$	6.42945	0.208270	0.104135	+	0.994563i	$0.466793\pi$
$954$	−19.7435	−0.639218
$955$	0	0
$956$	−45.8899	−1.48419
$957$	4.53751	0.146677
$958$	27.9428	0.902791
$959$	−39.2469	−1.26735
$960$	0	0
$961$	−6.61065	−0.213247
$962$	13.5360	0.436419
$963$	−1.00000	−0.0322245
$964$	79.0078	2.54467
$965$	0	0
$966$	80.3938	2.58663
$967$	25.3891	0.816459	0.408230	−	0.912879i	$-0.366146\pi$
$967$	25.3891	0.816459	0.408230	+	0.912879i	$0.366146\pi$
$968$	−15.5633	−0.500224
$969$	−17.7542	−0.570348
$970$	0	0
$971$	18.5128	0.594103	0.297052	−	0.954861i	$-0.403997\pi$
$971$	18.5128	0.594103	0.297052	+	0.954861i	$0.403997\pi$
$972$	2.72637	0.0874483
$973$	−57.3848	−1.83967
$974$	−64.1268	−2.05476
$975$	0	0
$976$	15.1604	0.485272
$977$	43.2939	1.38509	0.692547	−	0.721373i	$-0.256489\pi$
$977$	43.2939	1.38509	0.692547	+	0.721373i	$0.256489\pi$
$978$	−10.8828	−0.347994
$979$	−4.58687	−0.146597
$980$	0	0
$981$	−15.7486	−0.502815
$982$	22.9495	0.732347
$983$	6.81791	0.217458	0.108729	−	0.994071i	$-0.465322\pi$
$983$	6.81791	0.217458	0.108729	+	0.994071i	$0.465322\pi$
$984$	−11.0325	−0.351705
$985$	0	0
$986$	41.4349	1.31956
$987$	19.9748	0.635804
$988$	−15.1108	−0.480738
$989$	−14.7890	−0.470263
$990$	0	0
$991$	−11.9100	−0.378333	−0.189166	−	0.981945i	$-0.560579\pi$
$991$	−11.9100	−0.378333	−0.189166	+	0.981945i	$0.560579\pi$
$992$	37.2814	1.18369
$993$	−14.9882	−0.475636
$994$	104.646	3.31918
$995$	0	0
$996$	−25.0656	−0.794234
$997$	−35.9891	−1.13978	−0.569892	−	0.821719i	$-0.693015\pi$
$997$	−35.9891	−1.13978	−0.569892	+	0.821719i	$0.693015\pi$
$998$	−3.32398	−0.105219
$999$	4.43858	0.140431

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8025.2.a.bo.1.20		22
5.2	odd	4		1605.2.b.d.964.39	yes	44
5.3	odd	4		1605.2.b.d.964.6	✓	44
5.4	even	2		8025.2.a.bp.1.3		22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1605.2.b.d.964.6	✓	44	5.3	odd	4
1605.2.b.d.964.39	yes	44	5.2	odd	4
8025.2.a.bo.1.20		22	1.1	even	1	trivial
8025.2.a.bp.1.3		22	5.4	even	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:	\( N \)	\(=\)	\( 8025 = 3 \cdot 5^{2} \cdot 107 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8025.a (trivial)

\(f(q)\)	\(=\)	\(q+2.17402 q^{2} +1.00000 q^{3} +2.72637 q^{4} +2.17402 q^{6} -4.66914 q^{7} +1.57914 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+2.17402 q^{2} +1.00000 q^{3} +2.72637 q^{4} +2.17402 q^{6} -4.66914 q^{7} +1.57914 q^{8} +1.00000 q^{9} +1.06979 q^{11} +2.72637 q^{12} +1.40276 q^{13} -10.1508 q^{14} -2.01965 q^{16} +4.49347 q^{17} +2.17402 q^{18} -3.95111 q^{19} -4.66914 q^{21} +2.32574 q^{22} -7.91994 q^{23} +1.57914 q^{24} +3.04963 q^{26} +1.00000 q^{27} -12.7298 q^{28} +4.24151 q^{29} -4.93856 q^{31} -7.54905 q^{32} +1.06979 q^{33} +9.76890 q^{34} +2.72637 q^{36} +4.43858 q^{37} -8.58980 q^{38} +1.40276 q^{39} -6.98642 q^{41} -10.1508 q^{42} +1.86731 q^{43} +2.91663 q^{44} -17.2181 q^{46} -4.27804 q^{47} -2.01965 q^{48} +14.8009 q^{49} +4.49347 q^{51} +3.82444 q^{52} -9.08154 q^{53} +2.17402 q^{54} -7.37323 q^{56} -3.95111 q^{57} +9.22113 q^{58} +5.80276 q^{59} -7.50643 q^{61} -10.7365 q^{62} -4.66914 q^{63} -12.3725 q^{64} +2.32574 q^{66} +13.0711 q^{67} +12.2509 q^{68} -7.91994 q^{69} -10.3092 q^{71} +1.57914 q^{72} -7.43920 q^{73} +9.64957 q^{74} -10.7722 q^{76} -4.99498 q^{77} +3.04963 q^{78} -17.7333 q^{79} +1.00000 q^{81} -15.1886 q^{82} -9.19377 q^{83} -12.7298 q^{84} +4.05958 q^{86} +4.24151 q^{87} +1.68934 q^{88} -4.28765 q^{89} -6.54967 q^{91} -21.5927 q^{92} -4.93856 q^{93} -9.30055 q^{94} -7.54905 q^{96} +0.655434 q^{97} +32.1774 q^{98} +1.06979 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 3 q^{2} + 22 q^{3} + 13 q^{4} - 3 q^{6} - 2 q^{7} - 9 q^{8} + 22 q^{9}+O(q^{10}) \) 22 * q - 3 * q^2 + 22 * q^3 + 13 * q^4 - 3 * q^6 - 2 * q^7 - 9 * q^8 + 22 * q^9	\( 22 q - 3 q^{2} + 22 q^{3} + 13 q^{4} - 3 q^{6} - 2 q^{7} - 9 q^{8} + 22 q^{9} - 16 q^{11} + 13 q^{12} - 20 q^{14} - q^{16} - 20 q^{17} - 3 q^{18} - 16 q^{19} - 2 q^{21} - 4 q^{22} - 14 q^{23} - 9 q^{24} - 16 q^{26} + 22 q^{27} + q^{28} - 24 q^{29} - 42 q^{31} - 11 q^{32} - 16 q^{33} - 14 q^{34} + 13 q^{36} - 2 q^{37} - 21 q^{38} - 26 q^{41} - 20 q^{42} - 2 q^{43} - 24 q^{44} - 16 q^{46} - 26 q^{47} - q^{48} - 10 q^{49} - 20 q^{51} - 6 q^{52} - 22 q^{53} - 3 q^{54} - 42 q^{56} - 16 q^{57} + 69 q^{58} - 34 q^{59} - 16 q^{61} - 34 q^{62} - 2 q^{63} - 39 q^{64} - 4 q^{66} - 6 q^{68} - 14 q^{69} - 76 q^{71} - 9 q^{72} + 14 q^{73} - 12 q^{74} - 48 q^{76} - 54 q^{77} - 16 q^{78} - 72 q^{79} + 22 q^{81} - 2 q^{82} - 28 q^{83} + q^{84} - 22 q^{86} - 24 q^{87} + 19 q^{88} - 22 q^{89} - 58 q^{91} - 34 q^{92} - 42 q^{93} - 8 q^{94} - 11 q^{96} - 10 q^{97} + 3 q^{98} - 16 q^{99}+O(q^{100}) \) 22 * q - 3 * q^2 + 22 * q^3 + 13 * q^4 - 3 * q^6 - 2 * q^7 - 9 * q^8 + 22 * q^9 - 16 * q^11 + 13 * q^12 - 20 * q^14 - q^16 - 20 * q^17 - 3 * q^18 - 16 * q^19 - 2 * q^21 - 4 * q^22 - 14 * q^23 - 9 * q^24 - 16 * q^26 + 22 * q^27 + q^28 - 24 * q^29 - 42 * q^31 - 11 * q^32 - 16 * q^33 - 14 * q^34 + 13 * q^36 - 2 * q^37 - 21 * q^38 - 26 * q^41 - 20 * q^42 - 2 * q^43 - 24 * q^44 - 16 * q^46 - 26 * q^47 - q^48 - 10 * q^49 - 20 * q^51 - 6 * q^52 - 22 * q^53 - 3 * q^54 - 42 * q^56 - 16 * q^57 + 69 * q^58 - 34 * q^59 - 16 * q^61 - 34 * q^62 - 2 * q^63 - 39 * q^64 - 4 * q^66 - 6 * q^68 - 14 * q^69 - 76 * q^71 - 9 * q^72 + 14 * q^73 - 12 * q^74 - 48 * q^76 - 54 * q^77 - 16 * q^78 - 72 * q^79 + 22 * q^81 - 2 * q^82 - 28 * q^83 + q^84 - 22 * q^86 - 24 * q^87 + 19 * q^88 - 22 * q^89 - 58 * q^91 - 34 * q^92 - 42 * q^93 - 8 * q^94 - 11 * q^96 - 10 * q^97 + 3 * q^98 - 16 * q^99

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