LMFDB - Embedded newform 9025.2.a.bo.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:	$4$
Coefficient field:	4.4.2225.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 2x + 4 $ x^4 - x^3 - 5x^2 + 2x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
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			$-0.820249$ 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)

$f(q)$	$=$	$q-0.820249 q^{2} +2.32719 q^{3} -1.32719 q^{4} -1.90888 q^{6} +0.561717 q^{7} +2.72913 q^{8} +2.41582 q^{9} +O(q^{10})$	\(q-0.820249 q^{2} +2.32719 q^{3} -1.32719 q^{4} -1.90888 q^{6} +0.561717 q^{7} +2.72913 q^{8} +2.41582 q^{9} -0.111092 q^{11} -3.08863 q^{12} -6.89045 q^{13} -0.460748 q^{14} +0.415819 q^{16} +3.81167 q^{17} -1.98157 q^{18} +1.30722 q^{21} +0.0911232 q^{22} +4.04243 q^{23} +6.35120 q^{24} +5.65189 q^{26} -1.35950 q^{27} -0.745506 q^{28} +9.29239 q^{29} -4.18987 q^{31} -5.79933 q^{32} -0.258532 q^{33} -3.12652 q^{34} -3.20625 q^{36} -2.68669 q^{37} -16.0354 q^{39} -10.0988 q^{41} -1.07225 q^{42} +9.63192 q^{43} +0.147440 q^{44} -3.31580 q^{46} -2.12652 q^{47} +0.967690 q^{48} -6.68447 q^{49} +8.87048 q^{51} +9.14494 q^{52} +5.74455 q^{53} +1.11513 q^{54} +1.53300 q^{56} -7.62207 q^{58} +7.89903 q^{59} +5.56575 q^{61} +3.43674 q^{62} +1.35701 q^{63} +3.92526 q^{64} +0.212061 q^{66} +10.6534 q^{67} -5.05881 q^{68} +9.40752 q^{69} -6.64050 q^{71} +6.59307 q^{72} +8.45825 q^{73} +2.20376 q^{74} -0.0624023 q^{77} +13.1530 q^{78} -6.27087 q^{79} -10.4113 q^{81} +8.28349 q^{82} +8.16132 q^{83} -1.73493 q^{84} -7.90057 q^{86} +21.6252 q^{87} -0.303184 q^{88} +1.16741 q^{89} -3.87048 q^{91} -5.36508 q^{92} -9.75064 q^{93} +1.74428 q^{94} -13.4961 q^{96} +8.24746 q^{97} +5.48294 q^{98} -0.268378 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{2} + q^{3} + 3 q^{4} - 7 q^{6} + 11 q^{7} + 6 q^{8} + 5 q^{9}+O(q^{10}) $ 4 * q + q^2 + q^3 + 3 * q^4 - 7 * q^6 + 11 * q^7 + 6 * q^8 + 5 * q^9	$ 4 q + q^{2} + q^{3} + 3 q^{4} - 7 q^{6} + 11 q^{7} + 6 q^{8} + 5 q^{9} - 16 q^{12} - 2 q^{13} + 11 q^{14} - 3 q^{16} + 7 q^{17} + 17 q^{18} - 2 q^{21} + q^{22} + 11 q^{23} - 13 q^{24} + 9 q^{26} - 14 q^{27} + 13 q^{28} + 15 q^{29} + q^{31} + 3 q^{32} + 12 q^{33} + 22 q^{34} + 16 q^{36} - 11 q^{37} - 29 q^{39} - 22 q^{41} - 19 q^{42} + 26 q^{43} - 12 q^{44} - 10 q^{46} + 26 q^{47} - 13 q^{48} + 13 q^{49} + 11 q^{51} + 27 q^{52} - 16 q^{53} - 25 q^{54} + 8 q^{56} + 3 q^{58} + 10 q^{59} + 2 q^{61} + 31 q^{62} + 17 q^{63} + 4 q^{64} + 22 q^{66} + 3 q^{67} - 4 q^{68} - 14 q^{69} - 18 q^{71} + 29 q^{72} + 24 q^{73} - 17 q^{74} + 6 q^{77} - 15 q^{78} - 30 q^{79} - 4 q^{81} + 13 q^{82} + 12 q^{83} - 52 q^{84} + 16 q^{86} + q^{87} - 23 q^{88} - 9 q^{89} + 9 q^{91} + 25 q^{92} - 7 q^{93} + 11 q^{94} - 6 q^{96} + 19 q^{97} + 48 q^{98} - 9 q^{99}+O(q^{100}) $ 4 * q + q^2 + q^3 + 3 * q^4 - 7 * q^6 + 11 * q^7 + 6 * q^8 + 5 * q^9 - 16 * q^12 - 2 * q^13 + 11 * q^14 - 3 * q^16 + 7 * q^17 + 17 * q^18 - 2 * q^21 + q^22 + 11 * q^23 - 13 * q^24 + 9 * q^26 - 14 * q^27 + 13 * q^28 + 15 * q^29 + q^31 + 3 * q^32 + 12 * q^33 + 22 * q^34 + 16 * q^36 - 11 * q^37 - 29 * q^39 - 22 * q^41 - 19 * q^42 + 26 * q^43 - 12 * q^44 - 10 * q^46 + 26 * q^47 - 13 * q^48 + 13 * q^49 + 11 * q^51 + 27 * q^52 - 16 * q^53 - 25 * q^54 + 8 * q^56 + 3 * q^58 + 10 * q^59 + 2 * q^61 + 31 * q^62 + 17 * q^63 + 4 * q^64 + 22 * q^66 + 3 * q^67 - 4 * q^68 - 14 * q^69 - 18 * q^71 + 29 * q^72 + 24 * q^73 - 17 * q^74 + 6 * q^77 - 15 * q^78 - 30 * q^79 - 4 * q^81 + 13 * q^82 + 12 * q^83 - 52 * q^84 + 16 * q^86 + q^87 - 23 * q^88 - 9 * q^89 + 9 * q^91 + 25 * q^92 - 7 * q^93 + 11 * q^94 - 6 * q^96 + 19 * q^97 + 48 * q^98 - 9 * 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 $

$2$	−0.820249	−0.580004	−0.290002	−	0.957026i	$-0.593656\pi$
$2$	−0.820249	−0.580004	−0.290002	+	0.957026i	$0.593656\pi$
$3$	2.32719	1.34360	0.671802	−	0.740731i	$-0.265520\pi$
$3$	2.32719	1.34360	0.671802	+	0.740731i	$0.265520\pi$
$4$	−1.32719	−0.663596
$5$	0	0
$5$	0	0
$6$	−1.90888	−0.779296
$7$	0.561717	0.212309	0.106154	−	0.994350i	$-0.466146\pi$
$7$	0.561717	0.212309	0.106154	+	0.994350i	$0.466146\pi$
$8$	2.72913	0.964892
$9$	2.41582	0.805273
$10$	0	0
$11$	−0.111092	−0.0334955	−0.0167478	−	0.999860i	$-0.505331\pi$
$11$	−0.111092	−0.0334955	−0.0167478	+	0.999860i	$0.505331\pi$
$12$	−3.08863	−0.891610
$13$	−6.89045	−1.91107	−0.955534	−	0.294882i	$-0.904720\pi$
$13$	−6.89045	−1.91107	−0.955534	+	0.294882i	$0.904720\pi$
$14$	−0.460748	−0.123140
$15$	0	0
$16$	0.415819	0.103955
$17$	3.81167	0.924465	0.462233	−	0.886759i	$-0.347048\pi$
$17$	3.81167	0.924465	0.462233	+	0.886759i	$0.347048\pi$
$18$	−1.98157	−0.467061
$19$	0	0
$19$	0	0
$20$	0	0
$21$	1.30722	0.285259
$22$	0.0911232	0.0194275
$23$	4.04243	0.842906	0.421453	−	0.906850i	$-0.361520\pi$
$23$	4.04243	0.842906	0.421453	+	0.906850i	$0.361520\pi$
$24$	6.35120	1.29643
$25$	0	0
$26$	5.65189	1.10843
$27$	−1.35950	−0.261636
$28$	−0.745506	−0.140887
$29$	9.29239	1.72555	0.862776	−	0.505586i	$-0.168724\pi$
$29$	9.29239	1.72555	0.862776	+	0.505586i	$0.168724\pi$
$30$	0	0
$31$	−4.18987	−0.752524	−0.376262	−	0.926513i	$-0.622791\pi$
$31$	−4.18987	−0.752524	−0.376262	+	0.926513i	$0.622791\pi$
$32$	−5.79933	−1.02519
$33$	−0.258532	−0.0450047
$34$	−3.12652	−0.536193
$35$	0	0
$36$	−3.20625	−0.534376
$37$	−2.68669	−0.441690	−0.220845	−	0.975309i	$-0.570881\pi$
$37$	−2.68669	−0.441690	−0.220845	+	0.975309i	$0.570881\pi$
$38$	0	0
$39$	−16.0354	−2.56772
$40$	0	0
$41$	−10.0988	−1.57716	−0.788580	−	0.614932i	$-0.789183\pi$
$41$	−10.0988	−1.57716	−0.788580	+	0.614932i	$0.789183\pi$
$42$	−1.07225	−0.165451
$43$	9.63192	1.46885	0.734427	−	0.678688i	$-0.237451\pi$
$43$	9.63192	1.46885	0.734427	+	0.678688i	$0.237451\pi$
$44$	0.147440	0.0222275
$45$	0	0
$46$	−3.31580	−0.488888
$47$	−2.12652	−0.310185	−0.155092	−	0.987900i	$-0.549567\pi$
$47$	−2.12652	−0.310185	−0.155092	+	0.987900i	$0.549567\pi$
$48$	0.967690	0.139674
$49$	−6.68447	−0.954925
$50$	0	0
$51$	8.87048	1.24212
$52$	9.14494	1.26818
$53$	5.74455	0.789075	0.394537	−	0.918880i	$-0.370905\pi$
$53$	5.74455	0.789075	0.394537	+	0.918880i	$0.370905\pi$
$54$	1.11513	0.151750
$55$	0	0
$56$	1.53300	0.204855
$57$	0	0
$58$	−7.62207	−1.00083
$59$	7.89903	1.02837	0.514183	−	0.857680i	$-0.328095\pi$
$59$	7.89903	1.02837	0.514183	+	0.857680i	$0.328095\pi$
$60$	0	0
$61$	5.56575	0.712622	0.356311	−	0.934367i	$-0.384034\pi$
$61$	5.56575	0.712622	0.356311	+	0.934367i	$0.384034\pi$
$62$	3.43674	0.436467
$63$	1.35701	0.170967
$64$	3.92526	0.490657
$65$	0	0
$66$	0.212061	0.0261029
$67$	10.6534	1.30152	0.650762	−	0.759282i	$-0.274450\pi$
$67$	10.6534	1.30152	0.650762	+	0.759282i	$0.274450\pi$
$68$	−5.05881	−0.613471
$69$	9.40752	1.13253
$70$	0	0
$71$	−6.64050	−0.788082	−0.394041	−	0.919093i	$-0.628923\pi$
$71$	−6.64050	−0.788082	−0.394041	+	0.919093i	$0.628923\pi$
$72$	6.59307	0.777001
$73$	8.45825	0.989964	0.494982	−	0.868903i	$-0.335175\pi$
$73$	8.45825	0.989964	0.494982	+	0.868903i	$0.335175\pi$
$74$	2.20376	0.256182
$75$	0	0
$76$	0	0
$77$	−0.0624023	−0.00711140
$78$	13.1530	1.48929
$79$	−6.27087	−0.705528	−0.352764	−	0.935712i	$-0.614758\pi$
$79$	−6.27087	−0.705528	−0.352764	+	0.935712i	$0.614758\pi$
$80$	0	0
$81$	−10.4113	−1.15681
$82$	8.28349	0.914759
$83$	8.16132	0.895822	0.447911	−	0.894078i	$-0.352168\pi$
$83$	8.16132	0.895822	0.447911	+	0.894078i	$0.352168\pi$
$84$	−1.73493	−0.189297
$85$	0	0
$86$	−7.90057	−0.851941
$87$	21.6252	2.31846
$88$	−0.303184	−0.0323196
$89$	1.16741	0.123745	0.0618726	−	0.998084i	$-0.480293\pi$
$89$	1.16741	0.123745	0.0618726	+	0.998084i	$0.480293\pi$
$90$	0	0
$91$	−3.87048	−0.405737
$92$	−5.36508	−0.559348
$93$	−9.75064	−1.01109
$94$	1.74428	0.179908
$95$	0	0
$96$	−13.4961	−1.37744
$97$	8.24746	0.837402	0.418701	−	0.908124i	$-0.362485\pi$
$97$	8.24746	0.837402	0.418701	+	0.908124i	$0.362485\pi$
$98$	5.48294	0.553860
$99$	−0.268378	−0.0269730
$100$	0	0
$101$	−11.7400	−1.16817	−0.584087	−	0.811691i	$-0.698547\pi$
$101$	−11.7400	−1.16817	−0.584087	+	0.811691i	$0.698547\pi$
$102$	−7.27601	−0.720432
$103$	19.4676	1.91820	0.959099	−	0.283069i	$-0.0913526\pi$
$103$	19.4676	1.91820	0.959099	+	0.283069i	$0.0913526\pi$
$104$	−18.8049	−1.84397
$105$	0	0
$106$	−4.71196	−0.457666
$107$	−16.5648	−1.60138	−0.800690	−	0.599079i	$-0.795534\pi$
$107$	−16.5648	−1.60138	−0.800690	+	0.599079i	$0.795534\pi$
$108$	1.80432	0.173621
$109$	14.0588	1.34659	0.673295	−	0.739374i	$-0.264878\pi$
$109$	14.0588	1.34659	0.673295	+	0.739374i	$0.264878\pi$
$110$	0	0
$111$	−6.25245	−0.593456
$112$	0.233572	0.0220705
$113$	−8.88950	−0.836254	−0.418127	−	0.908389i	$-0.637313\pi$
$113$	−8.88950	−0.836254	−0.418127	+	0.908389i	$0.637313\pi$
$114$	0	0
$115$	0	0
$116$	−12.3328	−1.14507
$117$	−16.6461	−1.53893
$118$	−6.47917	−0.596456
$119$	2.14108	0.196272
$120$	0	0
$121$	−10.9877	−0.998878
$122$	−4.56531	−0.413323
$123$	−23.5017	−2.11908
$124$	5.56076	0.499371
$125$	0	0
$126$	−1.11308	−0.0991613
$127$	7.76469	0.689005	0.344503	−	0.938785i	$-0.388048\pi$
$127$	7.76469	0.689005	0.344503	+	0.938785i	$0.388048\pi$
$128$	8.37897	0.740603
$129$	22.4153	1.97356
$130$	0	0
$131$	−4.58042	−0.400193	−0.200097	−	0.979776i	$-0.564126\pi$
$131$	−4.58042	−0.400193	−0.200097	+	0.979776i	$0.564126\pi$
$132$	0.343122	0.0298649
$133$	0	0
$134$	−8.73847	−0.754889
$135$	0	0
$136$	10.4025	0.892009
$137$	−1.19518	−0.102111	−0.0510554	−	0.998696i	$-0.516259\pi$
$137$	−1.19518	−0.102111	−0.0510554	+	0.998696i	$0.516259\pi$
$138$	−7.71651	−0.656873
$139$	8.06036	0.683670	0.341835	−	0.939760i	$-0.388952\pi$
$139$	8.06036	0.683670	0.341835	+	0.939760i	$0.388952\pi$
$140$	0	0
$141$	−4.94881	−0.416765
$142$	5.44686	0.457091
$143$	0.765474	0.0640122
$144$	1.00454	0.0837119
$145$	0	0
$146$	−6.93787	−0.574183
$147$	−15.5560	−1.28304
$148$	3.56575	0.293103
$149$	9.89499	0.810629	0.405315	−	0.914177i	$-0.367162\pi$
$149$	9.89499	0.810629	0.405315	+	0.914177i	$0.367162\pi$
$150$	0	0
$151$	−20.7413	−1.68790	−0.843951	−	0.536421i	$-0.819776\pi$
$151$	−20.7413	−1.68790	−0.843951	+	0.536421i	$0.819776\pi$
$152$	0	0
$153$	9.20830	0.744447
$154$	0.0511854	0.00412464
$155$	0	0
$156$	21.2820	1.70393
$157$	12.9092	1.03026	0.515131	−	0.857111i	$-0.327743\pi$
$157$	12.9092	1.03026	0.515131	+	0.857111i	$0.327743\pi$
$158$	5.14368	0.409209
$159$	13.3687	1.06020
$160$	0	0
$161$	2.27070	0.178956
$162$	8.53984	0.670953
$163$	7.93134	0.621231	0.310615	−	0.950536i	$-0.399465\pi$
$163$	7.93134	0.621231	0.310615	+	0.950536i	$0.399465\pi$
$164$	13.4030	1.04660
$165$	0	0
$166$	−6.69432	−0.519580
$167$	−5.69132	−0.440408	−0.220204	−	0.975454i	$-0.570672\pi$
$167$	−5.69132	−0.440408	−0.220204	+	0.975454i	$0.570672\pi$
$168$	3.56757	0.275244
$169$	34.4783	2.65218
$170$	0	0
$171$	0	0
$172$	−12.7834	−0.974725
$173$	7.12043	0.541357	0.270678	−	0.962670i	$-0.412752\pi$
$173$	7.12043	0.541357	0.270678	+	0.962670i	$0.412752\pi$
$174$	−17.7380	−1.34472
$175$	0	0
$176$	−0.0461942	−0.00348202
$177$	18.3826	1.38172
$178$	−0.957567	−0.0717727
$179$	−2.37647	−0.177626	−0.0888129	−	0.996048i	$-0.528307\pi$
$179$	−2.37647	−0.177626	−0.0888129	+	0.996048i	$0.528307\pi$
$180$	0	0
$181$	20.1766	1.49971	0.749857	−	0.661600i	$-0.230122\pi$
$181$	20.1766	1.49971	0.749857	+	0.661600i	$0.230122\pi$
$182$	3.17476	0.235329
$183$	12.9526	0.957482
$184$	11.0323	0.813313
$185$	0	0
$186$	7.99795	0.586438
$187$	−0.423446	−0.0309655
$188$	2.82230	0.205837
$189$	−0.763655	−0.0555477
$190$	0	0
$191$	9.88950	0.715579	0.357789	−	0.933802i	$-0.383531\pi$
$191$	9.88950	0.715579	0.357789	+	0.933802i	$0.383531\pi$
$192$	9.13482	0.659249
$193$	4.55092	0.327582	0.163791	−	0.986495i	$-0.447628\pi$
$193$	4.55092	0.327582	0.163791	+	0.986495i	$0.447628\pi$
$194$	−6.76497	−0.485697
$195$	0	0
$196$	8.87158	0.633684
$197$	12.4701	0.888457	0.444229	−	0.895914i	$-0.353478\pi$
$197$	12.4701	0.888457	0.444229	+	0.895914i	$0.353478\pi$
$198$	0.220137	0.0156445
$199$	−19.9932	−1.41728	−0.708642	−	0.705569i	$-0.750692\pi$
$199$	−19.9932	−1.41728	−0.708642	+	0.705569i	$0.750692\pi$
$200$	0	0
$201$	24.7926	1.74873
$202$	9.62973	0.677546
$203$	5.21969	0.366350
$204$	−11.7728	−0.824263
$205$	0	0
$206$	−15.9683	−1.11256
$207$	9.76579	0.678769
$208$	−2.86518	−0.198664
$209$	0	0
$210$	0	0
$211$	16.8102	1.15726	0.578631	−	0.815589i	$-0.303587\pi$
$211$	16.8102	1.15726	0.578631	+	0.815589i	$0.303587\pi$
$212$	−7.62412	−0.523627
$213$	−15.4537	−1.05887
$214$	13.5873	0.928806
$215$	0	0
$216$	−3.71025	−0.252451
$217$	−2.35352	−0.159768
$218$	−11.5317	−0.781027
$219$	19.6840	1.33012
$220$	0	0
$221$	−26.2641	−1.76672
$222$	5.12857	0.344207
$223$	1.88641	0.126324	0.0631618	−	0.998003i	$-0.479882\pi$
$223$	1.88641	0.126324	0.0631618	+	0.998003i	$0.479882\pi$
$224$	−3.25758	−0.217656
$225$	0	0
$226$	7.29160	0.485030
$227$	10.7044	0.710479	0.355239	−	0.934775i	$-0.384399\pi$
$227$	10.7044	0.710479	0.355239	+	0.934775i	$0.384399\pi$
$228$	0	0
$229$	8.34388	0.551379	0.275690	−	0.961247i	$-0.411094\pi$
$229$	8.34388	0.551379	0.275690	+	0.961247i	$0.411094\pi$
$230$	0	0
$231$	−0.145222	−0.00955491
$232$	25.3601	1.66497
$233$	2.14436	0.140481	0.0702407	−	0.997530i	$-0.477623\pi$
$233$	2.14436	0.140481	0.0702407	+	0.997530i	$0.477623\pi$
$234$	13.6539	0.892586
$235$	0	0
$236$	−10.4835	−0.682419
$237$	−14.5935	−0.947951
$238$	−1.75622	−0.113839
$239$	25.0796	1.62227	0.811134	−	0.584861i	$-0.198851\pi$
$239$	25.0796	1.62227	0.811134	+	0.584861i	$0.198851\pi$
$240$	0	0
$241$	11.3527	0.731294	0.365647	−	0.930754i	$-0.380848\pi$
$241$	11.3527	0.731294	0.365647	+	0.930754i	$0.380848\pi$
$242$	9.01262	0.579353
$243$	−20.1505	−1.29266
$244$	−7.38682	−0.472893
$245$	0	0
$246$	19.2773	1.22907
$247$	0	0
$248$	−11.4347	−0.726104
$249$	18.9930	1.20363
$250$	0	0
$251$	−6.92245	−0.436941	−0.218471	−	0.975844i	$-0.570107\pi$
$251$	−6.92245	−0.436941	−0.218471	+	0.975844i	$0.570107\pi$
$252$	−1.80101	−0.113453
$253$	−0.449082	−0.0282336
$254$	−6.36898	−0.399626
$255$	0	0
$256$	−14.7234	−0.920210
$257$	−1.89721	−0.118345	−0.0591724	−	0.998248i	$-0.518846\pi$
$257$	−1.89721	−0.118345	−0.0591724	+	0.998248i	$0.518846\pi$
$258$	−18.3861	−1.14467
$259$	−1.50916	−0.0937747
$260$	0	0
$261$	22.4487	1.38954
$262$	3.75709	0.232114
$263$	11.5446	0.711868	0.355934	−	0.934511i	$-0.384163\pi$
$263$	11.5446	0.711868	0.355934	+	0.934511i	$0.384163\pi$
$264$	−0.705568	−0.0434247
$265$	0	0
$266$	0	0
$267$	2.71678	0.166265
$268$	−14.1391	−0.863685
$269$	−3.02056	−0.184167	−0.0920833	−	0.995751i	$-0.529353\pi$
$269$	−3.02056	−0.184167	−0.0920833	+	0.995751i	$0.529353\pi$
$270$	0	0
$271$	8.81150	0.535260	0.267630	−	0.963522i	$-0.413759\pi$
$271$	8.81150	0.535260	0.267630	+	0.963522i	$0.413759\pi$
$272$	1.58496	0.0961025
$273$	−9.00735	−0.545150
$274$	0.980343	0.0592247
$275$	0	0
$276$	−12.4856	−0.751543
$277$	9.68919	0.582167	0.291083	−	0.956698i	$-0.405984\pi$
$277$	9.68919	0.582167	0.291083	+	0.956698i	$0.405984\pi$
$278$	−6.61150	−0.396531
$279$	−10.1220	−0.605987
$280$	0	0
$281$	−18.6762	−1.11413	−0.557063	−	0.830470i	$-0.688072\pi$
$281$	−18.6762	−1.11413	−0.557063	+	0.830470i	$0.688072\pi$
$282$	4.05926	0.241726
$283$	7.46451	0.443719	0.221859	−	0.975079i	$-0.428787\pi$
$283$	7.46451	0.443719	0.221859	+	0.975079i	$0.428787\pi$
$284$	8.81321	0.522968
$285$	0	0
$286$	−0.627880	−0.0371273
$287$	−5.67264	−0.334845
$288$	−14.0101	−0.825554
$289$	−2.47118	−0.145364
$290$	0	0
$291$	19.1934	1.12514
$292$	−11.2257	−0.656935
$293$	3.73771	0.218359	0.109180	−	0.994022i	$-0.465178\pi$
$293$	3.73771	0.218359	0.109180	+	0.994022i	$0.465178\pi$
$294$	12.7598	0.744169
$295$	0	0
$296$	−7.33232	−0.426183
$297$	0.151030	0.00876364
$298$	−8.11636	−0.470168
$299$	−27.8542	−1.61085
$300$	0	0
$301$	5.41041	0.311851
$302$	17.0130	0.978989
$303$	−27.3212	−1.56956
$304$	0	0
$305$	0	0
$306$	−7.55310	−0.431782
$307$	−23.2825	−1.32880	−0.664402	−	0.747375i	$-0.731314\pi$
$307$	−23.2825	−1.32880	−0.664402	+	0.747375i	$0.731314\pi$
$308$	0.0828198	0.00471909
$309$	45.3048	2.57730
$310$	0	0
$311$	23.4553	1.33003	0.665013	−	0.746832i	$-0.268426\pi$
$311$	23.4553	1.33003	0.665013	+	0.746832i	$0.268426\pi$
$312$	−43.7626	−2.47757
$313$	15.0079	0.848297	0.424148	−	0.905593i	$-0.360573\pi$
$313$	15.0079	0.848297	0.424148	+	0.905593i	$0.360573\pi$
$314$	−10.5887	−0.597556
$315$	0	0
$316$	8.32265	0.468186
$317$	19.3000	1.08400	0.541998	−	0.840380i	$-0.317668\pi$
$317$	19.3000	1.08400	0.541998	+	0.840380i	$0.317668\pi$
$318$	−10.9656	−0.614923
$319$	−1.03231	−0.0577983
$320$	0	0
$321$	−38.5495	−2.15162
$322$	−1.86254	−0.103795
$323$	0	0
$324$	13.8178	0.767653
$325$	0	0
$326$	−6.50568	−0.360316
$327$	32.7175	1.80928
$328$	−27.5608	−1.52179
$329$	−1.19450	−0.0658550
$330$	0	0
$331$	19.1303	1.05150	0.525748	−	0.850641i	$-0.323786\pi$
$331$	19.1303	1.05150	0.525748	+	0.850641i	$0.323786\pi$
$332$	−10.8316	−0.594463
$333$	−6.49056	−0.355681
$334$	4.66830	0.255438
$335$	0	0
$336$	0.543568	0.0296540
$337$	−25.5100	−1.38962	−0.694810	−	0.719193i	$-0.744511\pi$
$337$	−25.5100	−1.38962	−0.694810	+	0.719193i	$0.744511\pi$
$338$	−28.2808	−1.53827
$339$	−20.6876	−1.12359
$340$	0	0
$341$	0.465462	0.0252062
$342$	0	0
$343$	−7.68680	−0.415048
$344$	26.2867	1.41728
$345$	0	0
$346$	−5.84053	−0.313989
$347$	34.3847	1.84587	0.922933	−	0.384960i	$-0.125785\pi$
$347$	34.3847	1.84587	0.922933	+	0.384960i	$0.125785\pi$
$348$	−28.7007	−1.53852
$349$	−28.6428	−1.53321	−0.766607	−	0.642117i	$-0.778057\pi$
$349$	−28.6428	−1.53321	−0.766607	+	0.642117i	$0.778057\pi$
$350$	0	0
$351$	9.36758	0.500004
$352$	0.644259	0.0343391
$353$	−23.5203	−1.25186	−0.625930	−	0.779879i	$-0.715281\pi$
$353$	−23.5203	−1.25186	−0.625930	+	0.779879i	$0.715281\pi$
$354$	−15.0783	−0.801401
$355$	0	0
$356$	−1.54938	−0.0821167
$357$	4.98270	0.263712
$358$	1.94930	0.103024
$359$	9.77437	0.515871	0.257936	−	0.966162i	$-0.416958\pi$
$359$	9.77437	0.515871	0.257936	+	0.966162i	$0.416958\pi$
$360$	0	0
$361$	0	0
$362$	−16.5498	−0.869839
$363$	−25.5704	−1.34210
$364$	5.13687	0.269245
$365$	0	0
$366$	−10.6243	−0.555343
$367$	2.03857	0.106412	0.0532062	−	0.998584i	$-0.483056\pi$
$367$	2.03857	0.106412	0.0532062	+	0.998584i	$0.483056\pi$
$368$	1.68092	0.0876240
$369$	−24.3968	−1.27004
$370$	0	0
$371$	3.22681	0.167528
$372$	12.9410	0.670958
$373$	12.9613	0.671110	0.335555	−	0.942021i	$-0.391076\pi$
$373$	12.9613	0.671110	0.335555	+	0.942021i	$0.391076\pi$
$374$	0.347331	0.0179601
$375$	0	0
$376$	−5.80354	−0.299295
$377$	−64.0287	−3.29765
$378$	0.626387	0.0322179
$379$	−6.82911	−0.350788	−0.175394	−	0.984498i	$-0.556120\pi$
$379$	−6.82911	−0.350788	−0.175394	+	0.984498i	$0.556120\pi$
$380$	0	0
$381$	18.0699	0.925750
$382$	−8.11185	−0.415038
$383$	17.1311	0.875356	0.437678	−	0.899132i	$-0.355801\pi$
$383$	17.1311	0.875356	0.437678	+	0.899132i	$0.355801\pi$
$384$	19.4995	0.995077
$385$	0	0
$386$	−3.73289	−0.189999
$387$	23.2690	1.18283
$388$	−10.9460	−0.555696
$389$	18.4370	0.934794	0.467397	−	0.884048i	$-0.345192\pi$
$389$	18.4370	0.934794	0.467397	+	0.884048i	$0.345192\pi$
$390$	0	0
$391$	15.4084	0.779237
$392$	−18.2428	−0.921399
$393$	−10.6595	−0.537701
$394$	−10.2286	−0.515308
$395$	0	0
$396$	0.356189	0.0178992
$397$	3.44622	0.172961	0.0864805	−	0.996254i	$-0.472438\pi$
$397$	3.44622	0.172961	0.0864805	+	0.996254i	$0.472438\pi$
$398$	16.3994	0.822030
$399$	0	0
$400$	0	0
$401$	−12.4272	−0.620585	−0.310293	−	0.950641i	$-0.600427\pi$
$401$	−12.4272	−0.620585	−0.310293	+	0.950641i	$0.600427\pi$
$402$	−20.3361	−1.01427
$403$	28.8701	1.43812
$404$	15.5812	0.775196
$405$	0	0
$406$	−4.28145	−0.212485
$407$	0.298470	0.0147946
$408$	24.2087	1.19851
$409$	−2.53190	−0.125194	−0.0625972	−	0.998039i	$-0.519938\pi$
$409$	−2.53190	−0.125194	−0.0625972	+	0.998039i	$0.519938\pi$
$410$	0	0
$411$	−2.78141	−0.137197
$412$	−25.8372	−1.27291
$413$	4.43702	0.218331
$414$	−8.01038	−0.393689
$415$	0	0
$416$	39.9600	1.95920
$417$	18.7580	0.918583
$418$	0	0
$419$	0.647537	0.0316342	0.0158171	−	0.999875i	$-0.494965\pi$
$419$	0.647537	0.0316342	0.0158171	+	0.999875i	$0.494965\pi$
$420$	0	0
$421$	−4.37148	−0.213053	−0.106526	−	0.994310i	$-0.533973\pi$
$421$	−4.37148	−0.213053	−0.106526	+	0.994310i	$0.533973\pi$
$422$	−13.7886	−0.671217
$423$	−5.13728	−0.249783
$424$	15.6776	0.761372
$425$	0	0
$426$	12.6759	0.614149
$427$	3.12638	0.151296
$428$	21.9847	1.06267
$429$	1.78141	0.0860071
$430$	0	0
$431$	−30.0094	−1.44550	−0.722752	−	0.691107i	$-0.757123\pi$
$431$	−30.0094	−1.44550	−0.722752	+	0.691107i	$0.757123\pi$
$432$	−0.565306	−0.0271983
$433$	17.6077	0.846173	0.423086	−	0.906089i	$-0.360947\pi$
$433$	17.6077	0.846173	0.423086	+	0.906089i	$0.360947\pi$
$434$	1.93047	0.0926658
$435$	0	0
$436$	−18.6587	−0.893591
$437$	0	0
$438$	−16.1458	−0.771474
$439$	−25.8827	−1.23532	−0.617658	−	0.786447i	$-0.711918\pi$
$439$	−25.8827	−1.23532	−0.617658	+	0.786447i	$0.711918\pi$
$440$	0	0
$441$	−16.1485	−0.768975
$442$	21.5431	1.02470
$443$	−11.8766	−0.564273	−0.282136	−	0.959374i	$-0.591043\pi$
$443$	−11.8766	−0.564273	−0.282136	+	0.959374i	$0.591043\pi$
$444$	8.29819	0.393815
$445$	0	0
$446$	−1.54733	−0.0732681
$447$	23.0275	1.08917
$448$	2.20488	0.104171
$449$	−29.4524	−1.38995	−0.694973	−	0.719035i	$-0.744584\pi$
$449$	−29.4524	−1.38995	−0.694973	+	0.719035i	$0.744584\pi$
$450$	0	0
$451$	1.12189	0.0528278
$452$	11.7981	0.554934
$453$	−48.2689	−2.26787
$454$	−8.78031	−0.412080
$455$	0	0
$456$	0	0
$457$	−24.3329	−1.13825	−0.569123	−	0.822253i	$-0.692717\pi$
$457$	−24.3329	−1.13825	−0.569123	+	0.822253i	$0.692717\pi$
$458$	−6.84406	−0.319802
$459$	−5.18197	−0.241874
$460$	0	0
$461$	34.5042	1.60702	0.803511	−	0.595290i	$-0.202963\pi$
$461$	34.5042	1.60702	0.803511	+	0.595290i	$0.202963\pi$
$462$	0.119118	0.00554188
$463$	4.89328	0.227410	0.113705	−	0.993515i	$-0.463728\pi$
$463$	4.89328	0.227410	0.113705	+	0.993515i	$0.463728\pi$
$464$	3.86395	0.179379
$465$	0	0
$466$	−1.75891	−0.0814798
$467$	−15.6840	−0.725769	−0.362885	−	0.931834i	$-0.618208\pi$
$467$	−15.6840	−0.725769	−0.362885	+	0.931834i	$0.618208\pi$
$468$	22.0925	1.02123
$469$	5.98421	0.276325
$470$	0	0
$471$	30.0421	1.38427
$472$	21.5575	0.992262
$473$	−1.07003	−0.0492000
$474$	11.9703	0.549815
$475$	0	0
$476$	−2.84162	−0.130245
$477$	13.8778	0.635421
$478$	−20.5716	−0.940921
$479$	15.1666	0.692981	0.346490	−	0.938054i	$-0.387373\pi$
$479$	15.1666	0.692981	0.346490	+	0.938054i	$0.387373\pi$
$480$	0	0
$481$	18.5125	0.844098
$482$	−9.31208	−0.424153
$483$	5.28436	0.240447
$484$	14.5827	0.662851
$485$	0	0
$486$	16.5285	0.749746
$487$	14.8366	0.672312	0.336156	−	0.941806i	$-0.390873\pi$
$487$	14.8366	0.672312	0.336156	+	0.941806i	$0.390873\pi$
$488$	15.1896	0.687603
$489$	18.4577	0.834688
$490$	0	0
$491$	21.7022	0.979408	0.489704	−	0.871889i	$-0.337105\pi$
$491$	21.7022	0.979408	0.489704	+	0.871889i	$0.337105\pi$
$492$	31.1913	1.40621
$493$	35.4195	1.59521
$494$	0	0
$495$	0	0
$496$	−1.74223	−0.0782283
$497$	−3.73008	−0.167317
$498$	−15.5790	−0.698110
$499$	−20.5764	−0.921124	−0.460562	−	0.887627i	$-0.652352\pi$
$499$	−20.5764	−0.921124	−0.460562	+	0.887627i	$0.652352\pi$
$500$	0	0
$501$	−13.2448	−0.591734
$502$	5.67813	0.253427
$503$	34.8559	1.55415	0.777074	−	0.629409i	$-0.216703\pi$
$503$	34.8559	1.55415	0.777074	+	0.629409i	$0.216703\pi$
$504$	3.70344	0.164964
$505$	0	0
$506$	0.368359	0.0163756
$507$	80.2376	3.56348
$508$	−10.3052	−0.457221
$509$	30.7078	1.36110	0.680551	−	0.732701i	$-0.261741\pi$
$509$	30.7078	1.36110	0.680551	+	0.732701i	$0.261741\pi$
$510$	0	0
$511$	4.75114	0.210178
$512$	−4.68111	−0.206878
$513$	0	0
$514$	1.55619	0.0686404
$515$	0	0
$516$	−29.7494	−1.30964
$517$	0.236239	0.0103898
$518$	1.23789	0.0543897
$519$	16.5706	0.727369
$520$	0	0
$521$	10.9533	0.479874	0.239937	−	0.970788i	$-0.422873\pi$
$521$	10.9533	0.479874	0.239937	+	0.970788i	$0.422873\pi$
$522$	−18.4135	−0.805939
$523$	−4.44485	−0.194360	−0.0971799	−	0.995267i	$-0.530982\pi$
$523$	−4.44485	−0.194360	−0.0971799	+	0.995267i	$0.530982\pi$
$524$	6.07909	0.265566
$525$	0	0
$526$	−9.46941	−0.412886
$527$	−15.9704	−0.695682
$528$	−0.107503	−0.00467845
$529$	−6.65873	−0.289510
$530$	0	0
$531$	19.0826	0.828115
$532$	0	0
$533$	69.5849	3.01406
$534$	−2.22844	−0.0964341
$535$	0	0
$536$	29.0746	1.25583
$537$	−5.53050	−0.238659
$538$	2.47761	0.106817
$539$	0.742592	0.0319857
$540$	0	0
$541$	6.38522	0.274522	0.137261	−	0.990535i	$-0.456170\pi$
$541$	6.38522	0.274522	0.137261	+	0.990535i	$0.456170\pi$
$542$	−7.22762	−0.310453
$543$	46.9548	2.01502
$544$	−22.1051	−0.947749
$545$	0	0
$546$	7.38827	0.316189
$547$	10.6627	0.455904	0.227952	−	0.973672i	$-0.426797\pi$
$547$	10.6627	0.455904	0.227952	+	0.973672i	$0.426797\pi$
$548$	1.58623	0.0677603
$549$	13.4459	0.573855
$550$	0	0
$551$	0	0
$552$	25.6743	1.09277
$553$	−3.52245	−0.149790
$554$	−7.94755	−0.337659
$555$	0	0
$556$	−10.6976	−0.453681
$557$	3.06585	0.129904	0.0649521	−	0.997888i	$-0.479311\pi$
$557$	3.06585	0.129904	0.0649521	+	0.997888i	$0.479311\pi$
$558$	8.30254	0.351475
$559$	−66.3683	−2.80708
$560$	0	0
$561$	−0.985440	−0.0416053
$562$	15.3191	0.646198
$563$	−35.1561	−1.48165	−0.740827	−	0.671696i	$-0.765566\pi$
$563$	−35.1561	−1.48165	−0.740827	+	0.671696i	$0.765566\pi$
$564$	6.56802	0.276564
$565$	0	0
$566$	−6.12276	−0.257359
$567$	−5.84819	−0.245601
$568$	−18.1228	−0.760414
$569$	42.2302	1.77038	0.885190	−	0.465229i	$-0.154028\pi$
$569$	42.2302	1.77038	0.885190	+	0.465229i	$0.154028\pi$
$570$	0	0
$571$	−24.3462	−1.01886	−0.509429	−	0.860513i	$-0.670143\pi$
$571$	−24.3462	−1.01886	−0.509429	+	0.860513i	$0.670143\pi$
$572$	−1.01593	−0.0424782
$573$	23.0148	0.961455
$574$	4.65298	0.194211
$575$	0	0
$576$	9.48271	0.395113
$577$	9.74424	0.405658	0.202829	−	0.979214i	$-0.434986\pi$
$577$	9.74424	0.405658	0.202829	+	0.979214i	$0.434986\pi$
$578$	2.02699	0.0843115
$579$	10.5909	0.440141
$580$	0	0
$581$	4.58435	0.190191
$582$	−15.7434	−0.652584
$583$	−0.638174	−0.0264305
$584$	23.0836	0.955208
$585$	0	0
$586$	−3.06585	−0.126649
$587$	−15.9708	−0.659186	−0.329593	−	0.944123i	$-0.606911\pi$
$587$	−15.9708	−0.659186	−0.329593	+	0.944123i	$0.606911\pi$
$588$	20.6459	0.851421
$589$	0	0
$590$	0	0
$591$	29.0203	1.19373
$592$	−1.11718	−0.0459157
$593$	0.404153	0.0165966	0.00829829	−	0.999966i	$-0.497359\pi$
$593$	0.404153	0.0165966	0.00829829	+	0.999966i	$0.497359\pi$
$594$	−0.123882	−0.00508294
$595$	0	0
$596$	−13.1325	−0.537930
$597$	−46.5281	−1.90427
$598$	22.8474	0.934299
$599$	−17.0426	−0.696342	−0.348171	−	0.937431i	$-0.613197\pi$
$599$	−17.0426	−0.696342	−0.348171	+	0.937431i	$0.613197\pi$
$600$	0	0
$601$	0.819125	0.0334128	0.0167064	−	0.999860i	$-0.494682\pi$
$601$	0.819125	0.0334128	0.0167064	+	0.999860i	$0.494682\pi$
$602$	−4.43788	−0.180875
$603$	25.7368	1.04808
$604$	27.5276	1.12008
$605$	0	0
$606$	22.4102	0.910353
$607$	18.2675	0.741455	0.370728	−	0.928742i	$-0.379108\pi$
$607$	18.2675	0.741455	0.370728	+	0.928742i	$0.379108\pi$
$608$	0	0
$609$	12.1472	0.492230
$610$	0	0
$611$	14.6527	0.592784
$612$	−12.2212	−0.494012
$613$	−27.6822	−1.11807	−0.559037	−	0.829142i	$-0.688829\pi$
$613$	−27.6822	−1.11807	−0.559037	+	0.829142i	$0.688829\pi$
$614$	19.0975	0.770712
$615$	0	0
$616$	−0.170304	−0.00686173
$617$	40.1940	1.61815	0.809075	−	0.587706i	$-0.199969\pi$
$617$	40.1940	1.61815	0.809075	+	0.587706i	$0.199969\pi$
$618$	−37.1612	−1.49484
$619$	46.3862	1.86442	0.932209	−	0.361920i	$-0.117879\pi$
$619$	46.3862	1.86442	0.932209	+	0.361920i	$0.117879\pi$
$620$	0	0
$621$	−5.49569	−0.220535
$622$	−19.2392	−0.771420
$623$	0.655753	0.0262722
$624$	−6.66782	−0.266926
$625$	0	0
$626$	−12.3102	−0.492015
$627$	0	0
$628$	−17.1329	−0.683678
$629$	−10.2408	−0.408327
$630$	0	0
$631$	−47.5023	−1.89104	−0.945519	−	0.325568i	$-0.894444\pi$
$631$	−47.5023	−1.89104	−0.945519	+	0.325568i	$0.894444\pi$
$632$	−17.1140	−0.680759
$633$	39.1206	1.55490
$634$	−15.8308	−0.628722
$635$	0	0
$636$	−17.7428	−0.703547
$637$	46.0590	1.82493
$638$	0.846752	0.0335232
$639$	−16.0422	−0.634621
$640$	0	0
$641$	−13.1626	−0.519891	−0.259946	−	0.965623i	$-0.583705\pi$
$641$	−13.1626	−0.519891	−0.259946	+	0.965623i	$0.583705\pi$
$642$	31.6202	1.24795
$643$	35.4538	1.39816	0.699081	−	0.715042i	$-0.253593\pi$
$643$	35.4538	1.39816	0.699081	+	0.715042i	$0.253593\pi$
$644$	−3.01366	−0.118755
$645$	0	0
$646$	0	0
$647$	−3.38283	−0.132993	−0.0664964	−	0.997787i	$-0.521182\pi$
$647$	−3.38283	−0.132993	−0.0664964	+	0.997787i	$0.521182\pi$
$648$	−28.4137	−1.11619
$649$	−0.877520	−0.0344457
$650$	0	0
$651$	−5.47710	−0.214664
$652$	−10.5264	−0.412246
$653$	0.621795	0.0243327	0.0121664	−	0.999926i	$-0.496127\pi$
$653$	0.621795	0.0243327	0.0121664	+	0.999926i	$0.496127\pi$
$654$	−26.8365	−1.04939
$655$	0	0
$656$	−4.19925	−0.163953
$657$	20.4336	0.797191
$658$	0.979789	0.0381961
$659$	−11.6203	−0.452664	−0.226332	−	0.974050i	$-0.572673\pi$
$659$	−11.6203	−0.452664	−0.226332	+	0.974050i	$0.572673\pi$
$660$	0	0
$661$	−13.2897	−0.516911	−0.258456	−	0.966023i	$-0.583214\pi$
$661$	−13.2897	−0.516911	−0.258456	+	0.966023i	$0.583214\pi$
$662$	−15.6916	−0.609871
$663$	−61.1216	−2.37377
$664$	22.2733	0.864371
$665$	0	0
$666$	5.32388	0.206296
$667$	37.5638	1.45448
$668$	7.55347	0.292253
$669$	4.39004	0.169729
$670$	0	0
$671$	−0.618311	−0.0238696
$672$	−7.58101	−0.292444
$673$	23.9351	0.922630	0.461315	−	0.887236i	$-0.347378\pi$
$673$	23.9351	0.922630	0.461315	+	0.887236i	$0.347378\pi$
$674$	20.9246	0.805985
$675$	0	0
$676$	−45.7593	−1.75997
$677$	3.71727	0.142866	0.0714331	−	0.997445i	$-0.477243\pi$
$677$	3.71727	0.142866	0.0714331	+	0.997445i	$0.477243\pi$
$678$	16.9690	0.651689
$679$	4.63273	0.177788
$680$	0	0
$681$	24.9113	0.954603
$682$	−0.381795	−0.0146197
$683$	1.40634	0.0538120	0.0269060	−	0.999638i	$-0.491435\pi$
$683$	1.40634	0.0538120	0.0269060	+	0.999638i	$0.491435\pi$
$684$	0	0
$685$	0	0
$686$	6.30509	0.240729
$687$	19.4178	0.740836
$688$	4.00513	0.152694
$689$	−39.5826	−1.50798
$690$	0	0
$691$	2.20762	0.0839820	0.0419910	−	0.999118i	$-0.486630\pi$
$691$	2.20762	0.0839820	0.0419910	+	0.999118i	$0.486630\pi$
$692$	−9.45018	−0.359242
$693$	−0.150753	−0.00572662
$694$	−28.2040	−1.07061
$695$	0	0
$696$	59.0178	2.23706
$697$	−38.4931	−1.45803
$698$	23.4942	0.889270
$699$	4.99033	0.188751
$700$	0	0
$701$	−3.95566	−0.149403	−0.0747016	−	0.997206i	$-0.523800\pi$
$701$	−3.95566	−0.149403	−0.0747016	+	0.997206i	$0.523800\pi$
$702$	−7.68375	−0.290004
$703$	0	0
$704$	−0.436065	−0.0164348
$705$	0	0
$706$	19.2925	0.726084
$707$	−6.59456	−0.248014
$708$	−24.3972	−0.916902
$709$	−31.8658	−1.19674	−0.598372	−	0.801218i	$-0.704186\pi$
$709$	−31.8658	−1.19674	−0.598372	+	0.801218i	$0.704186\pi$
$710$	0	0
$711$	−15.1493	−0.568143
$712$	3.18601	0.119401
$713$	−16.9373	−0.634306
$714$	−4.08705	−0.152954
$715$	0	0
$716$	3.15403	0.117872
$717$	58.3651	2.17969
$718$	−8.01742	−0.299207
$719$	−34.2360	−1.27679	−0.638393	−	0.769710i	$-0.720401\pi$
$719$	−34.2360	−1.27679	−0.638393	+	0.769710i	$0.720401\pi$
$720$	0	0
$721$	10.9353	0.407251
$722$	0	0
$723$	26.4200	0.982570
$724$	−26.7782	−0.995203
$725$	0	0
$726$	20.9741	0.778421
$727$	29.5181	1.09477	0.547383	−	0.836882i	$-0.315624\pi$
$727$	29.5181	1.09477	0.547383	+	0.836882i	$0.315624\pi$
$728$	−10.5630	−0.391492
$729$	−15.6603	−0.580011
$730$	0	0
$731$	36.7137	1.35790
$732$	−17.1905	−0.635381
$733$	−28.2212	−1.04237	−0.521187	−	0.853442i	$-0.674511\pi$
$733$	−28.2212	−1.04237	−0.521187	+	0.853442i	$0.674511\pi$
$734$	−1.67213	−0.0617195
$735$	0	0
$736$	−23.4434	−0.864135
$737$	−1.18351	−0.0435952
$738$	20.0114	0.736630
$739$	−24.0957	−0.886373	−0.443187	−	0.896429i	$-0.646152\pi$
$739$	−24.0957	−0.886373	−0.443187	+	0.896429i	$0.646152\pi$
$740$	0	0
$741$	0	0
$742$	−2.64679	−0.0971667
$743$	−12.9103	−0.473632	−0.236816	−	0.971554i	$-0.576104\pi$
$743$	−12.9103	−0.473632	−0.236816	+	0.971554i	$0.576104\pi$
$744$	−26.6107	−0.975596
$745$	0	0
$746$	−10.6315	−0.389247
$747$	19.7163	0.721381
$748$	0.561994	0.0205485
$749$	−9.30473	−0.339987
$750$	0	0
$751$	37.7129	1.37616	0.688082	−	0.725633i	$-0.258453\pi$
$751$	37.7129	1.37616	0.688082	+	0.725633i	$0.258453\pi$
$752$	−0.884246	−0.0322451
$753$	−16.1099	−0.587076
$754$	52.5195	1.91265
$755$	0	0
$756$	1.01352	0.0368612
$757$	34.7933	1.26458	0.632292	−	0.774730i	$-0.282114\pi$
$757$	34.7933	1.26458	0.632292	+	0.774730i	$0.282114\pi$
$758$	5.60157	0.203458
$759$	−1.04510	−0.0379347
$760$	0	0
$761$	4.00019	0.145007	0.0725034	−	0.997368i	$-0.476901\pi$
$761$	4.00019	0.145007	0.0725034	+	0.997368i	$0.476901\pi$
$762$	−14.8218	−0.536939
$763$	7.89707	0.285893
$764$	−13.1253	−0.474855
$765$	0	0
$766$	−14.0517	−0.507710
$767$	−54.4279	−1.96528
$768$	−34.2641	−1.23640
$769$	−9.59479	−0.345997	−0.172998	−	0.984922i	$-0.555346\pi$
$769$	−9.59479	−0.345997	−0.172998	+	0.984922i	$0.555346\pi$
$770$	0	0
$771$	−4.41517	−0.159009
$772$	−6.03994	−0.217382
$773$	−2.23562	−0.0804096	−0.0402048	−	0.999191i	$-0.512801\pi$
$773$	−2.23562	−0.0804096	−0.0402048	+	0.999191i	$0.512801\pi$
$774$	−19.0864	−0.686045
$775$	0	0
$776$	22.5083	0.808003
$777$	−3.51210	−0.125996
$778$	−15.1230	−0.542184
$779$	0	0
$780$	0	0
$781$	0.737707	0.0263972
$782$	−12.6387	−0.451960
$783$	−12.6330	−0.451467
$784$	−2.77953	−0.0992689
$785$	0	0
$786$	8.74346	0.311869
$787$	−19.6480	−0.700376	−0.350188	−	0.936679i	$-0.613882\pi$
$787$	−19.6480	−0.700376	−0.350188	+	0.936679i	$0.613882\pi$
$788$	−16.5502	−0.589576
$789$	26.8664	0.956469
$790$	0	0
$791$	−4.99338	−0.177544
$792$	−0.732438	−0.0260261
$793$	−38.3506	−1.36187
$794$	−2.82676	−0.100318
$795$	0	0
$796$	26.5349	0.940503
$797$	15.8851	0.562680	0.281340	−	0.959608i	$-0.409221\pi$
$797$	15.8851	0.562680	0.281340	+	0.959608i	$0.409221\pi$
$798$	0	0
$799$	−8.10558	−0.286755
$800$	0	0
$801$	2.82025	0.0996486
$802$	10.1934	0.359942
$803$	−0.939645	−0.0331593
$804$	−32.9045	−1.16045
$805$	0	0
$806$	−23.6807	−0.834117
$807$	−7.02942	−0.247447
$808$	−32.0400	−1.12716
$809$	9.82989	0.345600	0.172800	−	0.984957i	$-0.444718\pi$
$809$	9.82989	0.345600	0.172800	+	0.984957i	$0.444718\pi$
$810$	0	0
$811$	29.0667	1.02067	0.510335	−	0.859976i	$-0.329521\pi$
$811$	29.0667	1.02067	0.510335	+	0.859976i	$0.329521\pi$
$812$	−6.92752	−0.243108
$813$	20.5060	0.719178
$814$	−0.244820	−0.00858094
$815$	0	0
$816$	3.68851	0.129124
$817$	0	0
$818$	2.07679	0.0726133
$819$	−9.35038	−0.326729
$820$	0	0
$821$	9.14446	0.319144	0.159572	−	0.987186i	$-0.448989\pi$
$821$	9.14446	0.319144	0.159572	+	0.987186i	$0.448989\pi$
$822$	2.28145	0.0795746
$823$	−25.8030	−0.899436	−0.449718	−	0.893171i	$-0.648476\pi$
$823$	−25.8030	−0.899436	−0.449718	+	0.893171i	$0.648476\pi$
$824$	53.1295	1.85085
$825$	0	0
$826$	−3.63946	−0.126633
$827$	−4.28552	−0.149022	−0.0745110	−	0.997220i	$-0.523740\pi$
$827$	−4.28552	−0.149022	−0.0745110	+	0.997220i	$0.523740\pi$
$828$	−12.9611	−0.450428
$829$	5.70557	0.198163	0.0990813	−	0.995079i	$-0.468410\pi$
$829$	5.70557	0.198163	0.0990813	+	0.995079i	$0.468410\pi$
$830$	0	0
$831$	22.5486	0.782202
$832$	−27.0468	−0.937679
$833$	−25.4790	−0.882795
$834$	−15.3862	−0.532781
$835$	0	0
$836$	0	0
$837$	5.69614	0.196887
$838$	−0.531142	−0.0183480
$839$	−15.5188	−0.535769	−0.267884	−	0.963451i	$-0.586325\pi$
$839$	−15.5188	−0.535769	−0.267884	+	0.963451i	$0.586325\pi$
$840$	0	0
$841$	57.3484	1.97753
$842$	3.58570	0.123571
$843$	−43.4630	−1.49695
$844$	−22.3104	−0.767954
$845$	0	0
$846$	4.21385	0.144875
$847$	−6.17195	−0.212071
$848$	2.38869	0.0820280
$849$	17.3713	0.596183
$850$	0	0
$851$	−10.8608	−0.372303
$852$	20.5100	0.702662
$853$	49.8235	1.70592	0.852962	−	0.521972i	$-0.174804\pi$
$853$	49.8235	1.70592	0.852962	+	0.521972i	$0.174804\pi$
$854$	−2.56441	−0.0877523
$855$	0	0
$856$	−45.2074	−1.54516
$857$	43.1178	1.47288	0.736438	−	0.676505i	$-0.236506\pi$
$857$	43.1178	1.47288	0.736438	+	0.676505i	$0.236506\pi$
$858$	−1.46120	−0.0498844
$859$	17.9279	0.611691	0.305845	−	0.952081i	$-0.401061\pi$
$859$	17.9279	0.611691	0.305845	+	0.952081i	$0.401061\pi$
$860$	0	0
$861$	−13.2013	−0.449900
$862$	24.6152	0.838398
$863$	−55.6045	−1.89280	−0.946399	−	0.322999i	$-0.895309\pi$
$863$	−55.6045	−1.89280	−0.946399	+	0.322999i	$0.895309\pi$
$864$	7.88419	0.268226
$865$	0	0
$866$	−14.4427	−0.490784
$867$	−5.75091	−0.195311
$868$	3.12357	0.106021
$869$	0.696644	0.0236320
$870$	0	0
$871$	−73.4069	−2.48730
$872$	38.3683	1.29931
$873$	19.9244	0.674337
$874$	0	0
$875$	0	0
$876$	−26.1244	−0.882661
$877$	−12.1639	−0.410744	−0.205372	−	0.978684i	$-0.565840\pi$
$877$	−12.1639	−0.410744	−0.205372	+	0.978684i	$0.565840\pi$
$878$	21.2303	0.716488
$879$	8.69836	0.293388
$880$	0	0
$881$	28.8330	0.971408	0.485704	−	0.874123i	$-0.338563\pi$
$881$	28.8330	0.971408	0.485704	+	0.874123i	$0.338563\pi$
$882$	13.2458	0.446009
$883$	0.648525	0.0218246	0.0109123	−	0.999940i	$-0.496526\pi$
$883$	0.648525	0.0218246	0.0109123	+	0.999940i	$0.496526\pi$
$884$	34.8575	1.17238
$885$	0	0
$886$	9.74174	0.327280
$887$	−45.2362	−1.51888	−0.759441	−	0.650577i	$-0.774527\pi$
$887$	−45.2362	−1.51888	−0.759441	+	0.650577i	$0.774527\pi$
$888$	−17.0637	−0.572621
$889$	4.36156	0.146282
$890$	0	0
$891$	1.15661	0.0387479
$892$	−2.50363	−0.0838277
$893$	0	0
$894$	−18.8883	−0.631720
$895$	0	0
$896$	4.70661	0.157237
$897$	−64.8220	−2.16434
$898$	24.1583	0.806174
$899$	−38.9339	−1.29852
$900$	0	0
$901$	21.8963	0.729472
$902$	−0.920230	−0.0306403
$903$	12.5911	0.419004
$904$	−24.2606	−0.806894
$905$	0	0
$906$	39.5925	1.31537
$907$	15.1528	0.503142	0.251571	−	0.967839i	$-0.419053\pi$
$907$	15.1528	0.503142	0.251571	+	0.967839i	$0.419053\pi$
$908$	−14.2068	−0.471471
$909$	−28.3617	−0.940699
$910$	0	0
$911$	31.7373	1.05150	0.525752	−	0.850638i	$-0.323784\pi$
$911$	31.7373	1.05150	0.525752	+	0.850638i	$0.323784\pi$
$912$	0	0
$913$	−0.906658	−0.0300060
$914$	19.9591	0.660187
$915$	0	0
$916$	−11.0739	−0.365893
$917$	−2.57290	−0.0849646
$918$	4.25051	0.140288
$919$	5.31327	0.175269	0.0876343	−	0.996153i	$-0.472069\pi$
$919$	5.31327	0.175269	0.0876343	+	0.996153i	$0.472069\pi$
$920$	0	0
$921$	−54.1829	−1.78539
$922$	−28.3021	−0.932079
$923$	45.7560	1.50608
$924$	0.192737	0.00634059
$925$	0	0
$926$	−4.01371	−0.131899
$927$	47.0302	1.54467
$928$	−53.8896	−1.76901
$929$	−3.54458	−0.116294	−0.0581469	−	0.998308i	$-0.518519\pi$
$929$	−3.54458	−0.116294	−0.0581469	+	0.998308i	$0.518519\pi$
$930$	0	0
$931$	0	0
$932$	−2.84597	−0.0932228
$933$	54.5849	1.78703
$934$	12.8648	0.420949
$935$	0	0
$936$	−45.4292	−1.48490
$937$	−31.2800	−1.02187	−0.510936	−	0.859619i	$-0.670701\pi$
$937$	−31.2800	−1.02187	−0.510936	+	0.859619i	$0.670701\pi$
$938$	−4.90854	−0.160270
$939$	34.9263	1.13978
$940$	0	0
$941$	45.5106	1.48360	0.741802	−	0.670619i	$-0.233971\pi$
$941$	45.5106	1.48360	0.741802	+	0.670619i	$0.233971\pi$
$942$	−24.6420	−0.802879
$943$	−40.8235	−1.32940
$944$	3.28457	0.106903
$945$	0	0
$946$	0.877691	0.0285362
$947$	21.0071	0.682640	0.341320	−	0.939947i	$-0.389126\pi$
$947$	21.0071	0.682640	0.341320	+	0.939947i	$0.389126\pi$
$948$	19.3684	0.629056
$949$	−58.2812	−1.89189
$950$	0	0
$951$	44.9148	1.45646
$952$	5.84327	0.189382
$953$	32.6629	1.05806	0.529028	−	0.848604i	$-0.322557\pi$
$953$	32.6629	1.05806	0.529028	+	0.848604i	$0.322557\pi$
$954$	−11.3833	−0.368546
$955$	0	0
$956$	−33.2855	−1.07653
$957$	−2.40238	−0.0776580
$958$	−12.4404	−0.401932
$959$	−0.671351	−0.0216791
$960$	0	0
$961$	−13.4450	−0.433708
$962$	−15.1849	−0.489580
$963$	−40.0176	−1.28955
$964$	−15.0673	−0.485284
$965$	0	0
$966$	−4.33449	−0.139460
$967$	21.8080	0.701299	0.350650	−	0.936507i	$-0.385961\pi$
$967$	21.8080	0.701299	0.350650	+	0.936507i	$0.385961\pi$
$968$	−29.9867	−0.963809
$969$	0	0
$970$	0	0
$971$	55.8370	1.79189	0.895947	−	0.444162i	$-0.146498\pi$
$971$	55.8370	1.79189	0.895947	+	0.444162i	$0.146498\pi$
$972$	26.7436	0.857801
$973$	4.52764	0.145149
$974$	−12.1697	−0.389943
$975$	0	0
$976$	2.31435	0.0740804
$977$	−0.543879	−0.0174002	−0.00870012	−	0.999962i	$-0.502769\pi$
$977$	−0.543879	−0.0174002	−0.00870012	+	0.999962i	$0.502769\pi$
$978$	−15.1400	−0.484122
$979$	−0.129690	−0.00414491
$980$	0	0
$981$	33.9635	1.08437
$982$	−17.8012	−0.568060
$983$	−11.0176	−0.351407	−0.175704	−	0.984443i	$-0.556220\pi$
$983$	−11.0176	−0.351407	−0.175704	+	0.984443i	$0.556220\pi$
$984$	−64.1392	−2.04468
$985$	0	0
$986$	−29.0528	−0.925230
$987$	−2.77983	−0.0884830
$988$	0	0
$989$	38.9364	1.23811
$990$	0	0
$991$	28.1970	0.895707	0.447854	−	0.894107i	$-0.352189\pi$
$991$	28.1970	0.895707	0.447854	+	0.894107i	$0.352189\pi$
$992$	24.2984	0.771477
$993$	44.5198	1.41279
$994$	3.05959	0.0970445
$995$	0	0
$996$	−25.2073	−0.798724
$997$	30.5688	0.968123	0.484062	−	0.875034i	$-0.339161\pi$
$997$	30.5688	0.968123	0.484062	+	0.875034i	$0.339161\pi$
$998$	16.8777	0.534256
$999$	3.65256	0.115562

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.bo.1.2		4
5.4	even	2		1805.2.a.j.1.3	✓	4
19.18	odd	2		9025.2.a.bh.1.3		4
95.94	odd	2		1805.2.a.n.1.2	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1805.2.a.j.1.3	✓	4	5.4	even	2
1805.2.a.n.1.2	yes	4	95.94	odd	2
9025.2.a.bh.1.3		4	19.18	odd	2
9025.2.a.bo.1.2		4	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)

\(f(q)\)	\(=\)	\(q-0.820249 q^{2} +2.32719 q^{3} -1.32719 q^{4} -1.90888 q^{6} +0.561717 q^{7} +2.72913 q^{8} +2.41582 q^{9} +O(q^{10})\)	\(q-0.820249 q^{2} +2.32719 q^{3} -1.32719 q^{4} -1.90888 q^{6} +0.561717 q^{7} +2.72913 q^{8} +2.41582 q^{9} -0.111092 q^{11} -3.08863 q^{12} -6.89045 q^{13} -0.460748 q^{14} +0.415819 q^{16} +3.81167 q^{17} -1.98157 q^{18} +1.30722 q^{21} +0.0911232 q^{22} +4.04243 q^{23} +6.35120 q^{24} +5.65189 q^{26} -1.35950 q^{27} -0.745506 q^{28} +9.29239 q^{29} -4.18987 q^{31} -5.79933 q^{32} -0.258532 q^{33} -3.12652 q^{34} -3.20625 q^{36} -2.68669 q^{37} -16.0354 q^{39} -10.0988 q^{41} -1.07225 q^{42} +9.63192 q^{43} +0.147440 q^{44} -3.31580 q^{46} -2.12652 q^{47} +0.967690 q^{48} -6.68447 q^{49} +8.87048 q^{51} +9.14494 q^{52} +5.74455 q^{53} +1.11513 q^{54} +1.53300 q^{56} -7.62207 q^{58} +7.89903 q^{59} +5.56575 q^{61} +3.43674 q^{62} +1.35701 q^{63} +3.92526 q^{64} +0.212061 q^{66} +10.6534 q^{67} -5.05881 q^{68} +9.40752 q^{69} -6.64050 q^{71} +6.59307 q^{72} +8.45825 q^{73} +2.20376 q^{74} -0.0624023 q^{77} +13.1530 q^{78} -6.27087 q^{79} -10.4113 q^{81} +8.28349 q^{82} +8.16132 q^{83} -1.73493 q^{84} -7.90057 q^{86} +21.6252 q^{87} -0.303184 q^{88} +1.16741 q^{89} -3.87048 q^{91} -5.36508 q^{92} -9.75064 q^{93} +1.74428 q^{94} -13.4961 q^{96} +8.24746 q^{97} +5.48294 q^{98} -0.268378 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + q^{3} + 3 q^{4} - 7 q^{6} + 11 q^{7} + 6 q^{8} + 5 q^{9}+O(q^{10}) \) 4 * q + q^2 + q^3 + 3 * q^4 - 7 * q^6 + 11 * q^7 + 6 * q^8 + 5 * q^9	\( 4 q + q^{2} + q^{3} + 3 q^{4} - 7 q^{6} + 11 q^{7} + 6 q^{8} + 5 q^{9} - 16 q^{12} - 2 q^{13} + 11 q^{14} - 3 q^{16} + 7 q^{17} + 17 q^{18} - 2 q^{21} + q^{22} + 11 q^{23} - 13 q^{24} + 9 q^{26} - 14 q^{27} + 13 q^{28} + 15 q^{29} + q^{31} + 3 q^{32} + 12 q^{33} + 22 q^{34} + 16 q^{36} - 11 q^{37} - 29 q^{39} - 22 q^{41} - 19 q^{42} + 26 q^{43} - 12 q^{44} - 10 q^{46} + 26 q^{47} - 13 q^{48} + 13 q^{49} + 11 q^{51} + 27 q^{52} - 16 q^{53} - 25 q^{54} + 8 q^{56} + 3 q^{58} + 10 q^{59} + 2 q^{61} + 31 q^{62} + 17 q^{63} + 4 q^{64} + 22 q^{66} + 3 q^{67} - 4 q^{68} - 14 q^{69} - 18 q^{71} + 29 q^{72} + 24 q^{73} - 17 q^{74} + 6 q^{77} - 15 q^{78} - 30 q^{79} - 4 q^{81} + 13 q^{82} + 12 q^{83} - 52 q^{84} + 16 q^{86} + q^{87} - 23 q^{88} - 9 q^{89} + 9 q^{91} + 25 q^{92} - 7 q^{93} + 11 q^{94} - 6 q^{96} + 19 q^{97} + 48 q^{98} - 9 q^{99}+O(q^{100}) \) 4 * q + q^2 + q^3 + 3 * q^4 - 7 * q^6 + 11 * q^7 + 6 * q^8 + 5 * q^9 - 16 * q^12 - 2 * q^13 + 11 * q^14 - 3 * q^16 + 7 * q^17 + 17 * q^18 - 2 * q^21 + q^22 + 11 * q^23 - 13 * q^24 + 9 * q^26 - 14 * q^27 + 13 * q^28 + 15 * q^29 + q^31 + 3 * q^32 + 12 * q^33 + 22 * q^34 + 16 * q^36 - 11 * q^37 - 29 * q^39 - 22 * q^41 - 19 * q^42 + 26 * q^43 - 12 * q^44 - 10 * q^46 + 26 * q^47 - 13 * q^48 + 13 * q^49 + 11 * q^51 + 27 * q^52 - 16 * q^53 - 25 * q^54 + 8 * q^56 + 3 * q^58 + 10 * q^59 + 2 * q^61 + 31 * q^62 + 17 * q^63 + 4 * q^64 + 22 * q^66 + 3 * q^67 - 4 * q^68 - 14 * q^69 - 18 * q^71 + 29 * q^72 + 24 * q^73 - 17 * q^74 + 6 * q^77 - 15 * q^78 - 30 * q^79 - 4 * q^81 + 13 * q^82 + 12 * q^83 - 52 * q^84 + 16 * q^86 + q^87 - 23 * q^88 - 9 * q^89 + 9 * q^91 + 25 * q^92 - 7 * q^93 + 11 * q^94 - 6 * q^96 + 19 * q^97 + 48 * q^98 - 9 * q^99

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