LMFDB - Embedded newform 9025.2.a.bo.1.3

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			1.3
Root			$1.13856$ 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)

$f(q)$	$=$	$q+1.13856 q^{2} +1.70367 q^{3} -0.703671 q^{4} +1.93974 q^{6} +4.75660 q^{7} -3.07830 q^{8} -0.0975037 q^{9} +O(q^{10})$	\(q+1.13856 q^{2} +1.70367 q^{3} -0.703671 q^{4} +1.93974 q^{6} +4.75660 q^{7} -3.07830 q^{8} -0.0975037 q^{9} +3.46027 q^{11} -1.19882 q^{12} -1.17127 q^{13} +5.41569 q^{14} -2.09750 q^{16} +6.20500 q^{17} -0.111014 q^{18} +8.10368 q^{21} +3.93974 q^{22} -5.05910 q^{23} -5.24442 q^{24} -1.33357 q^{26} -5.27713 q^{27} -3.34708 q^{28} -1.61070 q^{29} +7.49400 q^{31} +3.76846 q^{32} +5.89516 q^{33} +7.06479 q^{34} +0.0686106 q^{36} -5.98080 q^{37} -1.99547 q^{39} +5.43374 q^{41} +9.22656 q^{42} +10.0664 q^{43} -2.43489 q^{44} -5.76011 q^{46} +8.06479 q^{47} -3.57346 q^{48} +15.6252 q^{49} +10.5713 q^{51} +0.824193 q^{52} -6.68283 q^{53} -6.00835 q^{54} -14.6423 q^{56} -1.83389 q^{58} -2.17229 q^{59} +6.20852 q^{61} +8.53240 q^{62} -0.463786 q^{63} +8.48565 q^{64} +6.71202 q^{66} -5.62257 q^{67} -4.36628 q^{68} -8.61905 q^{69} -2.72287 q^{71} +0.300146 q^{72} -3.15661 q^{73} -6.80953 q^{74} +16.4591 q^{77} -2.27197 q^{78} -12.0783 q^{79} -8.69798 q^{81} +6.18666 q^{82} +8.24958 q^{83} -5.70233 q^{84} +11.4613 q^{86} -2.74410 q^{87} -10.6518 q^{88} -8.83490 q^{89} -5.57128 q^{91} +3.55995 q^{92} +12.7673 q^{93} +9.18229 q^{94} +6.42023 q^{96} +0.707489 q^{97} +17.7903 q^{98} -0.337389 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.13856	0.805087	0.402543	−	0.915401i	$-0.368126\pi$
$2$	1.13856	0.805087	0.402543	+	0.915401i	$0.368126\pi$
$3$	1.70367	0.983615	0.491808	−	0.870704i	$-0.336336\pi$
$3$	1.70367	0.983615	0.491808	+	0.870704i	$0.336336\pi$
$4$	−0.703671	−0.351836
$5$	0	0
$5$	0	0
$6$	1.93974	0.791895
$7$	4.75660	1.79783	0.898913	−	0.438128i	$-0.144358\pi$
$7$	4.75660	1.79783	0.898913	+	0.438128i	$0.144358\pi$
$8$	−3.07830	−1.08834
$9$	−0.0975037	−0.0325012
$10$	0	0
$11$	3.46027	1.04331	0.521655	−	0.853156i	$-0.325315\pi$
$11$	3.46027	1.04331	0.521655	+	0.853156i	$0.325315\pi$
$12$	−1.19882	−0.346071
$13$	−1.17127	−0.324853	−0.162427	−	0.986721i	$-0.551932\pi$
$13$	−1.17127	−0.324853	−0.162427	+	0.986721i	$0.551932\pi$
$14$	5.41569	1.44740
$15$	0	0
$16$	−2.09750	−0.524376
$17$	6.20500	1.50493	0.752467	−	0.658630i	$-0.228864\pi$
$17$	6.20500	1.50493	0.752467	+	0.658630i	$0.228864\pi$
$18$	−0.111014	−0.0261663
$19$	0	0
$19$	0	0
$20$	0	0
$21$	8.10368	1.76837
$22$	3.93974	0.839955
$23$	−5.05910	−1.05490	−0.527448	−	0.849587i	$-0.676851\pi$
$23$	−5.05910	−1.05490	−0.527448	+	0.849587i	$0.676851\pi$
$24$	−5.24442	−1.07051
$25$	0	0
$26$	−1.33357	−0.261535
$27$	−5.27713	−1.01558
$28$	−3.34708	−0.632539
$29$	−1.61070	−0.299100	−0.149550	−	0.988754i	$-0.547782\pi$
$29$	−1.61070	−0.299100	−0.149550	+	0.988754i	$0.547782\pi$
$30$	0	0
$31$	7.49400	1.34596	0.672981	−	0.739660i	$-0.265014\pi$
$31$	7.49400	1.34596	0.672981	+	0.739660i	$0.265014\pi$
$32$	3.76846	0.666177
$33$	5.89516	1.02622
$34$	7.06479	1.21160
$35$	0	0
$36$	0.0686106	0.0114351
$37$	−5.98080	−0.983237	−0.491619	−	0.870811i	$-0.663595\pi$
$37$	−5.98080	−0.983237	−0.491619	+	0.870811i	$0.663595\pi$
$38$	0	0
$39$	−1.99547	−0.319531
$40$	0	0
$41$	5.43374	0.848607	0.424303	−	0.905520i	$-0.360519\pi$
$41$	5.43374	0.848607	0.424303	+	0.905520i	$0.360519\pi$
$42$	9.22656	1.42369
$43$	10.0664	1.53512	0.767559	−	0.640979i	$-0.221471\pi$
$43$	10.0664	1.53512	0.767559	+	0.640979i	$0.221471\pi$
$44$	−2.43489	−0.367074
$45$	0	0
$46$	−5.76011	−0.849283
$47$	8.06479	1.17637	0.588185	−	0.808726i	$-0.299842\pi$
$47$	8.06479	1.17637	0.588185	+	0.808726i	$0.299842\pi$
$48$	−3.57346	−0.515784
$49$	15.6252	2.23218
$50$	0	0
$51$	10.5713	1.48028
$52$	0.824193	0.114295
$53$	−6.68283	−0.917957	−0.458978	−	0.888447i	$-0.651784\pi$
$53$	−6.68283	−0.917957	−0.458978	+	0.888447i	$0.651784\pi$
$54$	−6.00835	−0.817633
$55$	0	0
$56$	−14.6423	−1.95665
$57$	0	0
$58$	−1.83389	−0.240801
$59$	−2.17229	−0.282808	−0.141404	−	0.989952i	$-0.545162\pi$
$59$	−2.17229	−0.282808	−0.141404	+	0.989952i	$0.545162\pi$
$60$	0	0
$61$	6.20852	0.794919	0.397460	−	0.917620i	$-0.369892\pi$
$61$	6.20852	0.794919	0.397460	+	0.917620i	$0.369892\pi$
$62$	8.53240	1.08362
$63$	−0.463786	−0.0584315
$64$	8.48565	1.06071
$65$	0	0
$66$	6.71202	0.826193
$67$	−5.62257	−0.686906	−0.343453	−	0.939170i	$-0.611597\pi$
$67$	−5.62257	−0.686906	−0.343453	+	0.939170i	$0.611597\pi$
$68$	−4.36628	−0.529490
$69$	−8.61905	−1.03761
$70$	0	0
$71$	−2.72287	−0.323145	−0.161573	−	0.986861i	$-0.551657\pi$
$71$	−2.72287	−0.323145	−0.161573	+	0.986861i	$0.551657\pi$
$72$	0.300146	0.0353725
$73$	−3.15661	−0.369453	−0.184726	−	0.982790i	$-0.559140\pi$
$73$	−3.15661	−0.369453	−0.184726	+	0.982790i	$0.559140\pi$
$74$	−6.80953	−0.791591
$75$	0	0
$76$	0	0
$77$	16.4591	1.87569
$78$	−2.27197	−0.257250
$79$	−12.0783	−1.35892	−0.679458	−	0.733715i	$-0.737785\pi$
$79$	−12.0783	−1.35892	−0.679458	+	0.733715i	$0.737785\pi$
$80$	0	0
$81$	−8.69798	−0.966442
$82$	6.18666	0.683202
$83$	8.24958	0.905509	0.452754	−	0.891635i	$-0.350441\pi$
$83$	8.24958	0.905509	0.452754	+	0.891635i	$0.350441\pi$
$84$	−5.70233	−0.622175
$85$	0	0
$86$	11.4613	1.23590
$87$	−2.74410	−0.294199
$88$	−10.6518	−1.13548
$89$	−8.83490	−0.936498	−0.468249	−	0.883597i	$-0.655115\pi$
$89$	−8.83490	−0.936498	−0.468249	+	0.883597i	$0.655115\pi$
$90$	0	0
$91$	−5.57128	−0.584029
$92$	3.55995	0.371150
$93$	12.7673	1.32391
$94$	9.18229	0.947080
$95$	0	0
$96$	6.42023	0.655262
$97$	0.707489	0.0718346	0.0359173	−	0.999355i	$-0.488565\pi$
$97$	0.707489	0.0718346	0.0359173	+	0.999355i	$0.488565\pi$
$98$	17.7903	1.79709
$99$	−0.337389	−0.0339089
$100$	0	0
$101$	−0.112658	−0.0112099	−0.00560497	−	0.999984i	$-0.501784\pi$
$101$	−0.112658	−0.0112099	−0.00560497	+	0.999984i	$0.501784\pi$
$102$	12.0361	1.19175
$103$	11.3233	1.11572	0.557861	−	0.829934i	$-0.311622\pi$
$103$	11.3233	1.11572	0.557861	+	0.829934i	$0.311622\pi$
$104$	3.60554	0.353552
$105$	0	0
$106$	−7.60883	−0.739035
$107$	−2.17861	−0.210614	−0.105307	−	0.994440i	$-0.533583\pi$
$107$	−2.17861	−0.210614	−0.105307	+	0.994440i	$0.533583\pi$
$108$	3.71336	0.357319
$109$	13.3663	1.28026	0.640129	−	0.768268i	$-0.278881\pi$
$109$	13.3663	1.28026	0.640129	+	0.768268i	$0.278881\pi$
$110$	0	0
$111$	−10.1893	−0.967127
$112$	−9.97698	−0.942736
$113$	11.8586	1.11557	0.557783	−	0.829987i	$-0.311652\pi$
$113$	11.8586	1.11557	0.557783	+	0.829987i	$0.311652\pi$
$114$	0	0
$115$	0	0
$116$	1.13340	0.105234
$117$	0.114204	0.0105581
$118$	−2.47329	−0.227685
$119$	29.5147	2.70561
$120$	0	0
$121$	0.973466	0.0884969
$122$	7.06880	0.639979
$123$	9.25730	0.834703
$124$	−5.27331	−0.473557
$125$	0	0
$126$	−0.528050	−0.0470424
$127$	20.0596	1.78000	0.890000	−	0.455960i	$-0.150704\pi$
$127$	20.0596	1.78000	0.890000	+	0.455960i	$0.150704\pi$
$128$	2.12452	0.187783
$129$	17.1499	1.50996
$130$	0	0
$131$	9.21953	0.805514	0.402757	−	0.915307i	$-0.368052\pi$
$131$	9.21953	0.805514	0.402757	+	0.915307i	$0.368052\pi$
$132$	−4.14826	−0.361059
$133$	0	0
$134$	−6.40165	−0.553019
$135$	0	0
$136$	−19.1009	−1.63789
$137$	3.46596	0.296117	0.148058	−	0.988979i	$-0.452698\pi$
$137$	3.46596	0.296117	0.148058	+	0.988979i	$0.452698\pi$
$138$	−9.81334	−0.835367
$139$	−1.92271	−0.163082	−0.0815412	−	0.996670i	$-0.525984\pi$
$139$	−1.92271	−0.163082	−0.0815412	+	0.996670i	$0.525984\pi$
$140$	0	0
$141$	13.7398	1.15710
$142$	−3.10016	−0.260160
$143$	−4.05293	−0.338923
$144$	0.204514	0.0170429
$145$	0	0
$146$	−3.59400	−0.297442
$147$	26.6203	2.19560
$148$	4.20852	0.345938
$149$	3.37579	0.276555	0.138278	−	0.990393i	$-0.455843\pi$
$149$	3.37579	0.276555	0.138278	+	0.990393i	$0.455843\pi$
$150$	0	0
$151$	−19.6888	−1.60225	−0.801125	−	0.598497i	$-0.795765\pi$
$151$	−19.6888	−1.60225	−0.801125	+	0.598497i	$0.795765\pi$
$152$	0	0
$153$	−0.605011	−0.0489122
$154$	18.7398	1.51009
$155$	0	0
$156$	1.40415	0.112422
$157$	−10.8049	−0.862321	−0.431161	−	0.902275i	$-0.641896\pi$
$157$	−10.8049	−0.862321	−0.431161	+	0.902275i	$0.641896\pi$
$158$	−13.7519	−1.09404
$159$	−11.3853	−0.902916
$160$	0	0
$161$	−24.0641	−1.89652
$162$	−9.90321	−0.778070
$163$	2.40117	0.188074	0.0940369	−	0.995569i	$-0.470023\pi$
$163$	2.40117	0.188074	0.0940369	+	0.995569i	$0.470023\pi$
$164$	−3.82356	−0.298570
$165$	0	0
$166$	9.39268	0.729013
$167$	18.8862	1.46146	0.730728	−	0.682668i	$-0.239181\pi$
$167$	18.8862	1.46146	0.730728	+	0.682668i	$0.239181\pi$
$168$	−24.9456	−1.92459
$169$	−11.6281	−0.894470
$170$	0	0
$171$	0	0
$172$	−7.08346	−0.540109
$173$	7.01969	0.533697	0.266848	−	0.963738i	$-0.414018\pi$
$173$	7.01969	0.533697	0.266848	+	0.963738i	$0.414018\pi$
$174$	−3.12434	−0.236856
$175$	0	0
$176$	−7.25793	−0.547087
$177$	−3.70087	−0.278174
$178$	−10.0591	−0.753962
$179$	9.61639	0.718763	0.359381	−	0.933191i	$-0.382988\pi$
$179$	9.61639	0.718763	0.359381	+	0.933191i	$0.382988\pi$
$180$	0	0
$181$	−18.4974	−1.37490	−0.687449	−	0.726232i	$-0.741270\pi$
$181$	−18.4974	−1.37490	−0.687449	+	0.726232i	$0.741270\pi$
$182$	−6.34326	−0.470194
$183$	10.5773	0.781895
$184$	15.5735	1.14809
$185$	0	0
$186$	14.5364	1.06586
$187$	21.4710	1.57011
$188$	−5.67496	−0.413889
$189$	−25.1012	−1.82584
$190$	0	0
$191$	−10.8586	−0.785703	−0.392852	−	0.919602i	$-0.628511\pi$
$191$	−10.8586	−0.785703	−0.392852	+	0.919602i	$0.628511\pi$
$192$	14.4568	1.04333
$193$	−12.5059	−0.900192	−0.450096	−	0.892980i	$-0.648610\pi$
$193$	−12.5059	−0.900192	−0.450096	+	0.892980i	$0.648610\pi$
$194$	0.805522	0.0578331
$195$	0	0
$196$	−10.9950	−0.785359
$197$	10.0643	0.717049	0.358525	−	0.933520i	$-0.383280\pi$
$197$	10.0643	0.717049	0.358525	+	0.933520i	$0.383280\pi$
$198$	−0.384139	−0.0272996
$199$	4.81054	0.341010	0.170505	−	0.985357i	$-0.445460\pi$
$199$	4.81054	0.341010	0.170505	+	0.985357i	$0.445460\pi$
$200$	0	0
$201$	−9.57901	−0.675651
$202$	−0.128269	−0.00902497
$203$	−7.66145	−0.537729
$204$	−7.43871	−0.520814
$205$	0	0
$206$	12.8924	0.898253
$207$	0.493281	0.0342854
$208$	2.45675	0.170345
$209$	0	0
$210$	0	0
$211$	1.42250	0.0979288	0.0489644	−	0.998801i	$-0.484408\pi$
$211$	1.42250	0.0979288	0.0489644	+	0.998801i	$0.484408\pi$
$212$	4.70251	0.322970
$213$	−4.63888	−0.317851
$214$	−2.48049	−0.169563
$215$	0	0
$216$	16.2446	1.10531
$217$	35.6459	2.41980
$218$	15.2184	1.03072
$219$	−5.37782	−0.363400
$220$	0	0
$221$	−7.26776	−0.488883
$222$	−11.6012	−0.778621
$223$	−0.280645	−0.0187934	−0.00939668	−	0.999956i	$-0.502991\pi$
$223$	−0.280645	−0.0187934	−0.00939668	+	0.999956i	$0.502991\pi$
$224$	17.9251	1.19767
$225$	0	0
$226$	13.5018	0.898128
$227$	−19.0252	−1.26275	−0.631375	−	0.775478i	$-0.717509\pi$
$227$	−19.0252	−1.26275	−0.631375	+	0.775478i	$0.717509\pi$
$228$	0	0
$229$	11.6753	0.771523	0.385762	−	0.922598i	$-0.373939\pi$
$229$	11.6753	0.771523	0.385762	+	0.922598i	$0.373939\pi$
$230$	0	0
$231$	28.0409	1.84496
$232$	4.95822	0.325523
$233$	18.1431	1.18859	0.594297	−	0.804246i	$-0.297430\pi$
$233$	18.1431	1.18859	0.594297	+	0.804246i	$0.297430\pi$
$234$	0.130028	0.00850021
$235$	0	0
$236$	1.52858	0.0995020
$237$	−20.5775	−1.33665
$238$	33.6044	2.17825
$239$	−27.2177	−1.76057	−0.880285	−	0.474446i	$-0.842648\pi$
$239$	−27.2177	−1.76057	−0.880285	+	0.474446i	$0.842648\pi$
$240$	0	0
$241$	−9.53341	−0.614101	−0.307051	−	0.951693i	$-0.599342\pi$
$241$	−9.53341	−0.614101	−0.307051	+	0.951693i	$0.599342\pi$
$242$	1.10835	0.0712477
$243$	1.01288	0.0649764
$244$	−4.36876	−0.279681
$245$	0	0
$246$	10.5400	0.672008
$247$	0	0
$248$	−23.0688	−1.46487
$249$	14.0546	0.890672
$250$	0	0
$251$	−10.1985	−0.643725	−0.321863	−	0.946786i	$-0.604309\pi$
$251$	−10.1985	−0.643725	−0.321863	+	0.946786i	$0.604309\pi$
$252$	0.326353	0.0205583
$253$	−17.5059	−1.10058
$254$	22.8391	1.43305
$255$	0	0
$256$	−14.5524	−0.909524
$257$	−20.9818	−1.30881	−0.654405	−	0.756144i	$-0.727081\pi$
$257$	−20.9818	−1.30881	−0.654405	+	0.756144i	$0.727081\pi$
$258$	19.5263	1.21565
$259$	−28.4483	−1.76769
$260$	0	0
$261$	0.157049	0.00972110
$262$	10.4970	0.648508
$263$	24.4437	1.50726	0.753632	−	0.657296i	$-0.228300\pi$
$263$	24.4437	1.50726	0.753632	+	0.657296i	$0.228300\pi$
$264$	−18.1471	−1.11688
$265$	0	0
$266$	0	0
$267$	−15.0518	−0.921153
$268$	3.95644	0.241678
$269$	28.7189	1.75102	0.875512	−	0.483197i	$-0.160525\pi$
$269$	28.7189	1.75102	0.875512	+	0.483197i	$0.160525\pi$
$270$	0	0
$271$	−20.9374	−1.27186	−0.635929	−	0.771748i	$-0.719383\pi$
$271$	−20.9374	−1.27186	−0.635929	+	0.771748i	$0.719383\pi$
$272$	−13.0150	−0.789151
$273$	−9.49164	−0.574460
$274$	3.94622	0.238400
$275$	0	0
$276$	6.06498	0.365069
$277$	18.7217	1.12488	0.562439	−	0.826839i	$-0.309863\pi$
$277$	18.7217	1.12488	0.562439	+	0.826839i	$0.309863\pi$
$278$	−2.18913	−0.131295
$279$	−0.730692	−0.0437454
$280$	0	0
$281$	19.1468	1.14220	0.571100	−	0.820880i	$-0.306517\pi$
$281$	19.1468	1.14220	0.571100	+	0.820880i	$0.306517\pi$
$282$	15.6436	0.931563
$283$	17.9013	1.06412	0.532062	−	0.846705i	$-0.321417\pi$
$283$	17.9013	1.06412	0.532062	+	0.846705i	$0.321417\pi$
$284$	1.91601	0.113694
$285$	0	0
$286$	−4.61452	−0.272862
$287$	25.8461	1.52565
$288$	−0.367439	−0.0216516
$289$	21.5020	1.26483
$290$	0	0
$291$	1.20533	0.0706576
$292$	2.22121	0.129987
$293$	−6.42187	−0.375170	−0.187585	−	0.982248i	$-0.560066\pi$
$293$	−6.42187	−0.375170	−0.187585	+	0.982248i	$0.560066\pi$
$294$	30.3089	1.76765
$295$	0	0
$296$	18.4107	1.07010
$297$	−18.2603	−1.05957
$298$	3.84355	0.222651
$299$	5.92560	0.342686
$300$	0	0
$301$	47.8820	2.75987
$302$	−22.4169	−1.28995
$303$	−0.191933	−0.0110263
$304$	0	0
$305$	0	0
$306$	−0.688844	−0.0393786
$307$	−6.15675	−0.351384	−0.175692	−	0.984445i	$-0.556216\pi$
$307$	−6.15675	−0.351384	−0.175692	+	0.984445i	$0.556216\pi$
$308$	−11.5818	−0.659935
$309$	19.2913	1.09744
$310$	0	0
$311$	3.34988	0.189954	0.0949772	−	0.995479i	$-0.469722\pi$
$311$	3.34988	0.189954	0.0949772	+	0.995479i	$0.469722\pi$
$312$	6.14265	0.347759
$313$	−24.2386	−1.37005	−0.685023	−	0.728521i	$-0.740208\pi$
$313$	−24.2386	−1.37005	−0.685023	+	0.728521i	$0.740208\pi$
$314$	−12.3020	−0.694243
$315$	0	0
$316$	8.49916	0.478115
$317$	−10.9842	−0.616933	−0.308466	−	0.951235i	$-0.599816\pi$
$317$	−10.9842	−0.616933	−0.308466	+	0.951235i	$0.599816\pi$
$318$	−12.9629	−0.726926
$319$	−5.57346	−0.312054
$320$	0	0
$321$	−3.71163	−0.207163
$322$	−27.3986	−1.52686
$323$	0	0
$324$	6.12052	0.340029
$325$	0	0
$326$	2.73388	0.151416
$327$	22.7718	1.25928
$328$	−16.7267	−0.923577
$329$	38.3610	2.11491
$330$	0	0
$331$	25.2522	1.38799	0.693994	−	0.719981i	$-0.255849\pi$
$331$	25.2522	1.38799	0.693994	+	0.719981i	$0.255849\pi$
$332$	−5.80499	−0.318590
$333$	0.583150	0.0319564
$334$	21.5031	1.17660
$335$	0	0
$336$	−16.9975	−0.927290
$337$	−8.26425	−0.450182	−0.225091	−	0.974338i	$-0.572268\pi$
$337$	−8.26425	−0.450182	−0.225091	+	0.974338i	$0.572268\pi$
$338$	−13.2394	−0.720126
$339$	20.2032	1.09729
$340$	0	0
$341$	25.9312	1.40426
$342$	0	0
$343$	41.0267	2.21524
$344$	−30.9876	−1.67074
$345$	0	0
$346$	7.99237	0.429672
$347$	−21.3088	−1.14391	−0.571957	−	0.820283i	$-0.693816\pi$
$347$	−21.3088	−1.14391	−0.571957	+	0.820283i	$0.693816\pi$
$348$	1.93095	0.103510
$349$	−23.2574	−1.24494	−0.622470	−	0.782644i	$-0.713871\pi$
$349$	−23.2574	−1.24494	−0.622470	+	0.782644i	$0.713871\pi$
$350$	0	0
$351$	6.18097	0.329916
$352$	13.0399	0.695029
$353$	−24.7741	−1.31859	−0.659296	−	0.751883i	$-0.729146\pi$
$353$	−24.7741	−1.31859	−0.659296	+	0.751883i	$0.729146\pi$
$354$	−4.21368	−0.223954
$355$	0	0
$356$	6.21687	0.329493
$357$	50.2834	2.66128
$358$	10.9489	0.578666
$359$	−3.85028	−0.203210	−0.101605	−	0.994825i	$-0.532398\pi$
$359$	−3.85028	−0.203210	−0.101605	+	0.994825i	$0.532398\pi$
$360$	0	0
$361$	0	0
$362$	−21.0604	−1.10691
$363$	1.65847	0.0870469
$364$	3.92035	0.205482
$365$	0	0
$366$	12.0429	0.629493
$367$	28.6314	1.49455	0.747274	−	0.664517i	$-0.231362\pi$
$367$	28.6314	1.49455	0.747274	+	0.664517i	$0.231362\pi$
$368$	10.6115	0.553162
$369$	−0.529809	−0.0275808
$370$	0	0
$371$	−31.7875	−1.65033
$372$	−8.98399	−0.465798
$373$	22.9648	1.18907	0.594537	−	0.804069i	$-0.297336\pi$
$373$	22.9648	1.18907	0.594537	+	0.804069i	$0.297336\pi$
$374$	24.4461	1.26408
$375$	0	0
$376$	−24.8259	−1.28030
$377$	1.88657	0.0971634
$378$	−28.5793	−1.46996
$379$	19.3472	0.993801	0.496901	−	0.867807i	$-0.334471\pi$
$379$	19.3472	0.993801	0.496901	+	0.867807i	$0.334471\pi$
$380$	0	0
$381$	34.1750	1.75084
$382$	−12.3633	−0.632559
$383$	6.13972	0.313725	0.156863	−	0.987620i	$-0.449862\pi$
$383$	6.13972	0.313725	0.156863	+	0.987620i	$0.449862\pi$
$384$	3.61949	0.184706
$385$	0	0
$386$	−14.2387	−0.724732
$387$	−0.981515	−0.0498932
$388$	−0.497840	−0.0252740
$389$	3.66728	0.185939	0.0929693	−	0.995669i	$-0.470364\pi$
$389$	3.66728	0.185939	0.0929693	+	0.995669i	$0.470364\pi$
$390$	0	0
$391$	−31.3917	−1.58755
$392$	−48.0992	−2.42938
$393$	15.7070	0.792316
$394$	11.4588	0.577287
$395$	0	0
$396$	0.237411	0.0119304
$397$	−24.5839	−1.23383	−0.616914	−	0.787030i	$-0.711618\pi$
$397$	−24.5839	−1.23383	−0.616914	+	0.787030i	$0.711618\pi$
$398$	5.47711	0.274543
$399$	0	0
$400$	0	0
$401$	−6.84605	−0.341876	−0.170938	−	0.985282i	$-0.554680\pi$
$401$	−6.84605	−0.341876	−0.170938	+	0.985282i	$0.554680\pi$
$402$	−10.9063	−0.543957
$403$	−8.77753	−0.437240
$404$	0.0792746	0.00394406
$405$	0	0
$406$	−8.72306	−0.432918
$407$	−20.6952	−1.02582
$408$	−32.5416	−1.61105
$409$	−7.92406	−0.391819	−0.195910	−	0.980622i	$-0.562766\pi$
$409$	−7.92406	−0.391819	−0.195910	+	0.980622i	$0.562766\pi$
$410$	0	0
$411$	5.90486	0.291265
$412$	−7.96792	−0.392551
$413$	−10.3327	−0.508440
$414$	0.561633	0.0276027
$415$	0	0
$416$	−4.41391	−0.216410
$417$	−3.27567	−0.160410
$418$	0	0
$419$	1.66830	0.0815018	0.0407509	−	0.999169i	$-0.487025\pi$
$419$	1.66830	0.0815018	0.0407509	+	0.999169i	$0.487025\pi$
$420$	0	0
$421$	19.0982	0.930790	0.465395	−	0.885103i	$-0.345912\pi$
$421$	19.0982	0.930790	0.465395	+	0.885103i	$0.345912\pi$
$422$	1.61960	0.0788411
$423$	−0.786347	−0.0382335
$424$	20.5718	0.999054
$425$	0	0
$426$	−5.28166	−0.255897
$427$	29.5314	1.42913
$428$	1.53302	0.0741015
$429$	−6.90486	−0.333370
$430$	0	0
$431$	18.5276	0.892442	0.446221	−	0.894923i	$-0.352769\pi$
$431$	18.5276	0.892442	0.446221	+	0.894923i	$0.352769\pi$
$432$	11.0688	0.532548
$433$	21.8081	1.04803	0.524016	−	0.851708i	$-0.324433\pi$
$433$	21.8081	1.04803	0.524016	+	0.851708i	$0.324433\pi$
$434$	40.5852	1.94815
$435$	0	0
$436$	−9.40547	−0.450440
$437$	0	0
$438$	−6.12300	−0.292568
$439$	19.6692	0.938759	0.469379	−	0.882997i	$-0.344478\pi$
$439$	19.6692	0.938759	0.469379	+	0.882997i	$0.344478\pi$
$440$	0	0
$441$	−1.52352	−0.0725485
$442$	−8.27481	−0.393593
$443$	−3.48680	−0.165663	−0.0828315	−	0.996564i	$-0.526396\pi$
$443$	−3.48680	−0.165663	−0.0828315	+	0.996564i	$0.526396\pi$
$444$	7.16993	0.340270
$445$	0	0
$446$	−0.319532	−0.0151303
$447$	5.75124	0.272024
$448$	40.3628	1.90696
$449$	−8.06276	−0.380505	−0.190253	−	0.981735i	$-0.560931\pi$
$449$	−8.06276	−0.380505	−0.190253	+	0.981735i	$0.560931\pi$
$450$	0	0
$451$	18.8022	0.885361
$452$	−8.34458	−0.392496
$453$	−33.5432	−1.57600
$454$	−21.6615	−1.01662
$455$	0	0
$456$	0	0
$457$	25.7296	1.20358	0.601790	−	0.798654i	$-0.294454\pi$
$457$	25.7296	1.20358	0.601790	+	0.798654i	$0.294454\pi$
$458$	13.2930	0.621143
$459$	−32.7446	−1.52839
$460$	0	0
$461$	7.48361	0.348547	0.174273	−	0.984697i	$-0.444242\pi$
$461$	7.48361	0.348547	0.174273	+	0.984697i	$0.444242\pi$
$462$	31.9264	1.48535
$463$	−24.4776	−1.13757	−0.568786	−	0.822485i	$-0.692587\pi$
$463$	−24.4776	−1.13757	−0.568786	+	0.822485i	$0.692587\pi$
$464$	3.37845	0.156841
$465$	0	0
$466$	20.6571	0.956921
$467$	−6.03350	−0.279197	−0.139599	−	0.990208i	$-0.544581\pi$
$467$	−6.03350	−0.279197	−0.139599	+	0.990208i	$0.544581\pi$
$468$	−0.0803618	−0.00371473
$469$	−26.7443	−1.23494
$470$	0	0
$471$	−18.4079	−0.848192
$472$	6.68697	0.307793
$473$	34.8326	1.60160
$474$	−23.4288	−1.07612
$475$	0	0
$476$	−20.7687	−0.951930
$477$	0.651600	0.0298347
$478$	−30.9891	−1.41741
$479$	22.2776	1.01789	0.508945	−	0.860799i	$-0.330036\pi$
$479$	22.2776	1.01789	0.508945	+	0.860799i	$0.330036\pi$
$480$	0	0
$481$	7.00516	0.319408
$482$	−10.8544	−0.494405
$483$	−40.9974	−1.86544
$484$	−0.685000	−0.0311364
$485$	0	0
$486$	1.15323	0.0523117
$487$	21.2868	0.964598	0.482299	−	0.876007i	$-0.339802\pi$
$487$	21.2868	0.964598	0.482299	+	0.876007i	$0.339802\pi$
$488$	−19.1117	−0.865146
$489$	4.09080	0.184992
$490$	0	0
$491$	−33.6313	−1.51776	−0.758879	−	0.651232i	$-0.774253\pi$
$491$	−33.6313	−1.51776	−0.758879	+	0.651232i	$0.774253\pi$
$492$	−6.51410	−0.293678
$493$	−9.99440	−0.450125
$494$	0	0
$495$	0	0
$496$	−15.7187	−0.705790
$497$	−12.9516	−0.580959
$498$	16.0020	0.717068
$499$	−35.2646	−1.57866	−0.789330	−	0.613969i	$-0.789572\pi$
$499$	−35.2646	−1.57866	−0.789330	+	0.613969i	$0.789572\pi$
$500$	0	0
$501$	32.1759	1.43751
$502$	−11.6117	−0.518254
$503$	−1.00821	−0.0449538	−0.0224769	−	0.999747i	$-0.507155\pi$
$503$	−1.00821	−0.0449538	−0.0224769	+	0.999747i	$0.507155\pi$
$504$	1.42767	0.0635937
$505$	0	0
$506$	−19.9315	−0.886065
$507$	−19.8105	−0.879815
$508$	−14.1154	−0.626268
$509$	−22.0570	−0.977660	−0.488830	−	0.872379i	$-0.662576\pi$
$509$	−22.0570	−0.977660	−0.488830	+	0.872379i	$0.662576\pi$
$510$	0	0
$511$	−15.0147	−0.664212
$512$	−20.8179	−0.920029
$513$	0	0
$514$	−23.8891	−1.05371
$515$	0	0
$516$	−12.0679	−0.531259
$517$	27.9064	1.22732
$518$	−32.3902	−1.42314
$519$	11.9592	0.524952
$520$	0	0
$521$	16.4008	0.718532	0.359266	−	0.933235i	$-0.383027\pi$
$521$	16.4008	0.718532	0.359266	+	0.933235i	$0.383027\pi$
$522$	0.178811	0.00782633
$523$	−17.8476	−0.780419	−0.390210	−	0.920726i	$-0.627597\pi$
$523$	−17.8476	−0.780419	−0.390210	+	0.920726i	$0.627597\pi$
$524$	−6.48752	−0.283409
$525$	0	0
$526$	27.8308	1.21348
$527$	46.5003	2.02558
$528$	−12.3651	−0.538123
$529$	2.59453	0.112806
$530$	0	0
$531$	0.211806	0.00919162
$532$	0	0
$533$	−6.36440	−0.275673
$534$	−17.1374	−0.741608
$535$	0	0
$536$	17.3080	0.747590
$537$	16.3832	0.706986
$538$	32.6983	1.40973
$539$	54.0675	2.32885
$540$	0	0
$541$	22.1825	0.953699	0.476849	−	0.878985i	$-0.341779\pi$
$541$	22.1825	0.953699	0.476849	+	0.878985i	$0.341779\pi$
$542$	−23.8386	−1.02396
$543$	−31.5134	−1.35237
$544$	23.3833	1.00255
$545$	0	0
$546$	−10.8068	−0.462490
$547$	24.9289	1.06588	0.532941	−	0.846152i	$-0.321087\pi$
$547$	24.9289	1.06588	0.532941	+	0.846152i	$0.321087\pi$
$548$	−2.43890	−0.104184
$549$	−0.605354	−0.0258359
$550$	0	0
$551$	0	0
$552$	26.5321	1.12928
$553$	−57.4516	−2.44309
$554$	21.3159	0.905625
$555$	0	0
$556$	1.35296	0.0573782
$557$	7.31171	0.309807	0.154904	−	0.987930i	$-0.450493\pi$
$557$	7.31171	0.309807	0.154904	+	0.987930i	$0.450493\pi$
$558$	−0.831940	−0.0352188
$559$	−11.7906	−0.498688
$560$	0	0
$561$	36.5795	1.54439
$562$	21.7998	0.919570
$563$	−16.5614	−0.697978	−0.348989	−	0.937127i	$-0.613475\pi$
$563$	−16.5614	−0.697978	−0.348989	+	0.937127i	$0.613475\pi$
$564$	−9.66827	−0.407108
$565$	0	0
$566$	20.3818	0.856712
$567$	−41.3728	−1.73749
$568$	8.38183	0.351694
$569$	−31.2306	−1.30926	−0.654628	−	0.755951i	$-0.727175\pi$
$569$	−31.2306	−1.30926	−0.654628	+	0.755951i	$0.727175\pi$
$570$	0	0
$571$	−1.27375	−0.0533049	−0.0266525	−	0.999645i	$-0.508485\pi$
$571$	−1.27375	−0.0533049	−0.0266525	+	0.999645i	$0.508485\pi$
$572$	2.85193	0.119245
$573$	−18.4995	−0.772830
$574$	29.4274	1.22828
$575$	0	0
$576$	−0.827382	−0.0344743
$577$	1.77096	0.0737262	0.0368631	−	0.999320i	$-0.488263\pi$
$577$	1.77096	0.0737262	0.0368631	+	0.999320i	$0.488263\pi$
$578$	24.4815	1.01829
$579$	−21.3059	−0.885442
$580$	0	0
$581$	39.2399	1.62795
$582$	1.37234	0.0568855
$583$	−23.1244	−0.957714
$584$	9.71700	0.402092
$585$	0	0
$586$	−7.31171	−0.302044
$587$	−8.25987	−0.340921	−0.170461	−	0.985364i	$-0.554526\pi$
$587$	−8.25987	−0.340921	−0.170461	+	0.985364i	$0.554526\pi$
$588$	−18.7319	−0.772491
$589$	0	0
$590$	0	0
$591$	17.1462	0.705300
$592$	12.5448	0.515586
$593$	20.8241	0.855141	0.427571	−	0.903982i	$-0.359370\pi$
$593$	20.8241	0.855141	0.427571	+	0.903982i	$0.359370\pi$
$594$	−20.7905	−0.853045
$595$	0	0
$596$	−2.37545	−0.0973021
$597$	8.19558	0.335423
$598$	6.74668	0.275892
$599$	−40.0833	−1.63776	−0.818880	−	0.573965i	$-0.805405\pi$
$599$	−40.0833	−1.63776	−0.818880	+	0.573965i	$0.805405\pi$
$600$	0	0
$601$	−48.3109	−1.97064	−0.985321	−	0.170710i	$-0.945394\pi$
$601$	−48.3109	−1.97064	−0.985321	+	0.170710i	$0.945394\pi$
$602$	54.5167	2.22194
$603$	0.548221	0.0223253
$604$	13.8544	0.563729
$605$	0	0
$606$	−0.218528	−0.00887710
$607$	−23.7640	−0.964552	−0.482276	−	0.876019i	$-0.660190\pi$
$607$	−23.7640	−0.964552	−0.482276	+	0.876019i	$0.660190\pi$
$608$	0	0
$609$	−13.0526	−0.528918
$610$	0	0
$611$	−9.44609	−0.382148
$612$	0.425729	0.0172091
$613$	−4.70479	−0.190025	−0.0950123	−	0.995476i	$-0.530289\pi$
$613$	−4.70479	−0.190025	−0.0950123	+	0.995476i	$0.530289\pi$
$614$	−7.00985	−0.282895
$615$	0	0
$616$	−50.6661	−2.04140
$617$	−2.11358	−0.0850893	−0.0425447	−	0.999095i	$-0.513546\pi$
$617$	−2.11358	−0.0850893	−0.0425447	+	0.999095i	$0.513546\pi$
$618$	21.9643	0.883536
$619$	−34.0091	−1.36694	−0.683470	−	0.729978i	$-0.739530\pi$
$619$	−34.0091	−1.36694	−0.683470	+	0.729978i	$0.739530\pi$
$620$	0	0
$621$	26.6975	1.07134
$622$	3.81406	0.152930
$623$	−42.0241	−1.68366
$624$	4.18550	0.167554
$625$	0	0
$626$	−27.5972	−1.10301
$627$	0	0
$628$	7.60307	0.303395
$629$	−37.1109	−1.47971
$630$	0	0
$631$	9.57620	0.381223	0.190611	−	0.981666i	$-0.438953\pi$
$631$	9.57620	0.381223	0.190611	+	0.981666i	$0.438953\pi$
$632$	37.1807	1.47897
$633$	2.42347	0.0963242
$634$	−12.5062	−0.496684
$635$	0	0
$636$	8.01154	0.317678
$637$	−18.3014	−0.725129
$638$	−6.34574	−0.251230
$639$	0.265490	0.0105026
$640$	0	0
$641$	−23.8257	−0.941058	−0.470529	−	0.882384i	$-0.655937\pi$
$641$	−23.8257	−0.941058	−0.470529	+	0.882384i	$0.655937\pi$
$642$	−4.22593	−0.166784
$643$	−27.3687	−1.07931	−0.539657	−	0.841885i	$-0.681446\pi$
$643$	−27.3687	−1.07931	−0.539657	+	0.841885i	$0.681446\pi$
$644$	16.9332	0.667263
$645$	0	0
$646$	0	0
$647$	38.5660	1.51619	0.758093	−	0.652147i	$-0.226131\pi$
$647$	38.5660	1.51619	0.758093	+	0.652147i	$0.226131\pi$
$648$	26.7750	1.05182
$649$	−7.51671	−0.295057
$650$	0	0
$651$	60.7290	2.38016
$652$	−1.68963	−0.0661711
$653$	14.6990	0.575216	0.287608	−	0.957748i	$-0.407140\pi$
$653$	14.6990	0.575216	0.287608	+	0.957748i	$0.407140\pi$
$654$	25.9271	1.01383
$655$	0	0
$656$	−11.3973	−0.444989
$657$	0.307781	0.0120077
$658$	43.6764	1.70268
$659$	−7.91649	−0.308383	−0.154191	−	0.988041i	$-0.549277\pi$
$659$	−7.91649	−0.308383	−0.154191	+	0.988041i	$0.549277\pi$
$660$	0	0
$661$	−33.2446	−1.29307	−0.646533	−	0.762886i	$-0.723782\pi$
$661$	−33.2446	−1.29307	−0.646533	+	0.762886i	$0.723782\pi$
$662$	28.7513	1.11745
$663$	−12.3819	−0.480872
$664$	−25.3947	−0.985506
$665$	0	0
$666$	0.663954	0.0257277
$667$	8.14870	0.315519
$668$	−13.2897	−0.514193
$669$	−0.478127	−0.0184854
$670$	0	0
$671$	21.4831	0.829348
$672$	30.5384	1.17805
$673$	34.7182	1.33829	0.669144	−	0.743133i	$-0.266661\pi$
$673$	34.7182	1.33829	0.669144	+	0.743133i	$0.266661\pi$
$674$	−9.40938	−0.362436
$675$	0	0
$676$	8.18237	0.314707
$677$	13.6369	0.524107	0.262054	−	0.965053i	$-0.415600\pi$
$677$	13.6369	0.524107	0.262054	+	0.965053i	$0.415600\pi$
$678$	23.0027	0.883412
$679$	3.36524	0.129146
$680$	0	0
$681$	−32.4128	−1.24206
$682$	29.5244	1.13055
$683$	32.0188	1.22516	0.612582	−	0.790407i	$-0.290131\pi$
$683$	32.0188	1.22516	0.612582	+	0.790407i	$0.290131\pi$
$684$	0	0
$685$	0	0
$686$	46.7116	1.78346
$687$	19.8908	0.758882
$688$	−21.1144	−0.804979
$689$	7.82743	0.298201
$690$	0	0
$691$	−42.5000	−1.61678	−0.808389	−	0.588649i	$-0.799660\pi$
$691$	−42.5000	−1.61678	−0.808389	+	0.588649i	$0.799660\pi$
$692$	−4.93955	−0.187774
$693$	−1.60482	−0.0609622
$694$	−24.2614	−0.920950
$695$	0	0
$696$	8.44719	0.320190
$697$	33.7163	1.27710
$698$	−26.4800	−1.00228
$699$	30.9099	1.16912
$700$	0	0
$701$	17.0007	0.642108	0.321054	−	0.947061i	$-0.395963\pi$
$701$	17.0007	0.642108	0.321054	+	0.947061i	$0.395963\pi$
$702$	7.03743	0.265611
$703$	0	0
$704$	29.3626	1.10665
$705$	0	0
$706$	−28.2069	−1.06158
$707$	−0.535871	−0.0201535
$708$	2.60420	0.0978717
$709$	−2.22434	−0.0835369	−0.0417685	−	0.999127i	$-0.513299\pi$
$709$	−2.22434	−0.0835369	−0.0417685	+	0.999127i	$0.513299\pi$
$710$	0	0
$711$	1.17768	0.0441664
$712$	27.1965	1.01923
$713$	−37.9129	−1.41985
$714$	57.2508	2.14256
$715$	0	0
$716$	−6.76678	−0.252886
$717$	−46.3701	−1.73172
$718$	−4.38380	−0.163602
$719$	29.4500	1.09830	0.549150	−	0.835724i	$-0.314952\pi$
$719$	29.4500	1.09830	0.549150	+	0.835724i	$0.314952\pi$
$720$	0	0
$721$	53.8606	2.00587
$722$	0	0
$723$	−16.2418	−0.604039
$724$	13.0161	0.483739
$725$	0	0
$726$	1.88827	0.0700803
$727$	5.16809	0.191674	0.0958368	−	0.995397i	$-0.469447\pi$
$727$	5.16809	0.191674	0.0958368	+	0.995397i	$0.469447\pi$
$728$	17.1501	0.635625
$729$	27.8196	1.03035
$730$	0	0
$731$	62.4623	2.31025
$732$	−7.44293	−0.275098
$733$	−6.04955	−0.223445	−0.111723	−	0.993739i	$-0.535637\pi$
$733$	−6.04955	−0.223445	−0.111723	+	0.993739i	$0.535637\pi$
$734$	32.5987	1.20324
$735$	0	0
$736$	−19.0651	−0.702747
$737$	−19.4556	−0.716656
$738$	−0.603222	−0.0222049
$739$	−27.1443	−0.998518	−0.499259	−	0.866453i	$-0.666394\pi$
$739$	−27.1443	−0.998518	−0.499259	+	0.866453i	$0.666394\pi$
$740$	0	0
$741$	0	0
$742$	−36.1921	−1.32866
$743$	−36.3675	−1.33419	−0.667097	−	0.744971i	$-0.732463\pi$
$743$	−36.3675	−1.33419	−0.667097	+	0.744971i	$0.732463\pi$
$744$	−39.3016	−1.44087
$745$	0	0
$746$	26.1469	0.957307
$747$	−0.804365	−0.0294302
$748$	−15.1085	−0.552422
$749$	−10.3628	−0.378647
$750$	0	0
$751$	−30.6467	−1.11831	−0.559157	−	0.829062i	$-0.688875\pi$
$751$	−30.6467	−1.11831	−0.559157	+	0.829062i	$0.688875\pi$
$752$	−16.9159	−0.616861
$753$	−17.3749	−0.633178
$754$	2.14798	0.0782250
$755$	0	0
$756$	17.6630	0.642396
$757$	8.24452	0.299652	0.149826	−	0.988712i	$-0.452129\pi$
$757$	8.24452	0.299652	0.149826	+	0.988712i	$0.452129\pi$
$758$	22.0281	0.800096
$759$	−29.8242	−1.08255
$760$	0	0
$761$	11.2064	0.406231	0.203116	−	0.979155i	$-0.434893\pi$
$761$	11.2064	0.406231	0.203116	+	0.979155i	$0.434893\pi$
$762$	38.9104	1.40957
$763$	63.5780	2.30168
$764$	7.64091	0.276438
$765$	0	0
$766$	6.99047	0.252576
$767$	2.54435	0.0918711
$768$	−24.7925	−0.894622
$769$	−26.1536	−0.943122	−0.471561	−	0.881833i	$-0.656309\pi$
$769$	−26.1536	−0.943122	−0.471561	+	0.881833i	$0.656309\pi$
$770$	0	0
$771$	−35.7461	−1.28737
$772$	8.80002	0.316720
$773$	14.5134	0.522010	0.261005	−	0.965337i	$-0.415946\pi$
$773$	14.5134	0.522010	0.261005	+	0.965337i	$0.415946\pi$
$774$	−1.11752	−0.0401684
$775$	0	0
$776$	−2.17787	−0.0781808
$777$	−48.4665	−1.73873
$778$	4.17544	0.149697
$779$	0	0
$780$	0	0
$781$	−9.42187	−0.337141
$782$	−35.7415	−1.27811
$783$	8.49987	0.303761
$784$	−32.7740	−1.17050
$785$	0	0
$786$	17.8835	0.637883
$787$	−48.3569	−1.72374	−0.861869	−	0.507130i	$-0.830706\pi$
$787$	−48.3569	−1.72374	−0.861869	+	0.507130i	$0.830706\pi$
$788$	−7.08193	−0.252283
$789$	41.6441	1.48257
$790$	0	0
$791$	56.4068	2.00559
$792$	1.03859	0.0369046
$793$	−7.27188	−0.258232
$794$	−27.9903	−0.993339
$795$	0	0
$796$	−3.38504	−0.119980
$797$	28.0793	0.994619	0.497310	−	0.867573i	$-0.334321\pi$
$797$	28.0793	0.994619	0.497310	+	0.867573i	$0.334321\pi$
$798$	0	0
$799$	50.0421	1.77036
$800$	0	0
$801$	0.861436	0.0304373
$802$	−7.79467	−0.275239
$803$	−10.9227	−0.385454
$804$	6.74047	0.237718
$805$	0	0
$806$	−9.99378	−0.352016
$807$	48.9276	1.72233
$808$	0.346797	0.0122003
$809$	−33.4598	−1.17638	−0.588191	−	0.808722i	$-0.700160\pi$
$809$	−33.4598	−1.17638	−0.588191	+	0.808722i	$0.700160\pi$
$810$	0	0
$811$	−10.8723	−0.381778	−0.190889	−	0.981612i	$-0.561137\pi$
$811$	−10.8723	−0.381778	−0.190889	+	0.981612i	$0.561137\pi$
$812$	5.39115	0.189192
$813$	−35.6705	−1.25102
$814$	−23.5628	−0.825875
$815$	0	0
$816$	−22.1733	−0.776221
$817$	0	0
$818$	−9.02205	−0.315448
$819$	0.543221	0.0189817
$820$	0	0
$821$	−26.8644	−0.937576	−0.468788	−	0.883311i	$-0.655309\pi$
$821$	−26.8644	−0.937576	−0.468788	+	0.883311i	$0.655309\pi$
$822$	6.72306	0.234494
$823$	26.6654	0.929496	0.464748	−	0.885443i	$-0.346145\pi$
$823$	26.6654	0.929496	0.464748	+	0.885443i	$0.346145\pi$
$824$	−34.8567	−1.21429
$825$	0	0
$826$	−11.7645	−0.409338
$827$	−20.5863	−0.715856	−0.357928	−	0.933749i	$-0.616517\pi$
$827$	−20.5863	−0.715856	−0.357928	+	0.933749i	$0.616517\pi$
$828$	−0.347108	−0.0120628
$829$	23.1471	0.803932	0.401966	−	0.915655i	$-0.368327\pi$
$829$	23.1471	0.803932	0.401966	+	0.915655i	$0.368327\pi$
$830$	0	0
$831$	31.8956	1.10645
$832$	−9.93902	−0.344574
$833$	96.9546	3.35928
$834$	−3.72956	−0.129144
$835$	0	0
$836$	0	0
$837$	−39.5468	−1.36694
$838$	1.89947	0.0656160
$839$	−41.4744	−1.43186	−0.715928	−	0.698174i	$-0.753996\pi$
$839$	−41.4744	−1.43186	−0.715928	+	0.698174i	$0.753996\pi$
$840$	0	0
$841$	−26.4056	−0.910539
$842$	21.7445	0.749367
$843$	32.6198	1.12349
$844$	−1.00097	−0.0344548
$845$	0	0
$846$	−0.895307	−0.0307813
$847$	4.63039	0.159102
$848$	14.0173	0.481355
$849$	30.4980	1.04669
$850$	0	0
$851$	30.2575	1.03721
$852$	3.26425	0.111831
$853$	−49.7956	−1.70497	−0.852484	−	0.522753i	$-0.824905\pi$
$853$	−49.7956	−1.70497	−0.852484	+	0.522753i	$0.824905\pi$
$854$	33.6234	1.15057
$855$	0	0
$856$	6.70642	0.229221
$857$	−37.3465	−1.27573	−0.637866	−	0.770148i	$-0.720182\pi$
$857$	−37.3465	−1.27573	−0.637866	+	0.770148i	$0.720182\pi$
$858$	−7.86162	−0.268391
$859$	−23.7810	−0.811397	−0.405698	−	0.914007i	$-0.632972\pi$
$859$	−23.7810	−0.811397	−0.405698	+	0.914007i	$0.632972\pi$
$860$	0	0
$861$	44.0333	1.50065
$862$	21.0948	0.718493
$863$	−28.6969	−0.976855	−0.488427	−	0.872605i	$-0.662429\pi$
$863$	−28.6969	−0.976855	−0.488427	+	0.872605i	$0.662429\pi$
$864$	−19.8867	−0.676558
$865$	0	0
$866$	24.8300	0.843757
$867$	36.6324	1.24410
$868$	−25.0830	−0.851373
$869$	−41.7942	−1.41777
$870$	0	0
$871$	6.58557	0.223144
$872$	−41.1455	−1.39336
$873$	−0.0689828	−0.00233471
$874$	0	0
$875$	0	0
$876$	3.78422	0.127857
$877$	−33.4018	−1.12790	−0.563950	−	0.825809i	$-0.690719\pi$
$877$	−33.4018	−1.12790	−0.563950	+	0.825809i	$0.690719\pi$
$878$	22.3946	0.755782
$879$	−10.9408	−0.369023
$880$	0	0
$881$	−33.0377	−1.11307	−0.556535	−	0.830824i	$-0.687869\pi$
$881$	−33.0377	−1.11307	−0.556535	+	0.830824i	$0.687869\pi$
$882$	−1.73462	−0.0584078
$883$	32.1095	1.08057	0.540286	−	0.841482i	$-0.318316\pi$
$883$	32.1095	1.08057	0.540286	+	0.841482i	$0.318316\pi$
$884$	5.11412	0.172006
$885$	0	0
$886$	−3.96995	−0.133373
$887$	−13.6924	−0.459747	−0.229873	−	0.973221i	$-0.573831\pi$
$887$	−13.6924	−0.459747	−0.229873	+	0.973221i	$0.573831\pi$
$888$	31.3658	1.05257
$889$	95.4154	3.20013
$890$	0	0
$891$	−30.0974	−1.00830
$892$	0.197482	0.00661218
$893$	0	0
$894$	6.54815	0.219003
$895$	0	0
$896$	10.1055	0.337601
$897$	10.0953	0.337071
$898$	−9.17997	−0.306340
$899$	−12.0706	−0.402576
$900$	0	0
$901$	−41.4670	−1.38146
$902$	21.4075	0.712792
$903$	81.5752	2.71465
$904$	−36.5045	−1.21412
$905$	0	0
$906$	−38.1911	−1.26881
$907$	−32.4144	−1.07630	−0.538151	−	0.842848i	$-0.680877\pi$
$907$	−32.4144	−1.07630	−0.538151	+	0.842848i	$0.680877\pi$
$908$	13.3875	0.444280
$909$	0.0109846	0.000364337 0
$910$	0	0
$911$	7.16536	0.237399	0.118699	−	0.992930i	$-0.462128\pi$
$911$	7.16536	0.237399	0.118699	+	0.992930i	$0.462128\pi$
$912$	0	0
$913$	28.5458	0.944727
$914$	29.2948	0.968987
$915$	0	0
$916$	−8.21555	−0.271449
$917$	43.8536	1.44817
$918$	−37.2818	−1.23048
$919$	−13.3921	−0.441765	−0.220883	−	0.975300i	$-0.570894\pi$
$919$	−13.3921	−0.441765	−0.220883	+	0.975300i	$0.570894\pi$
$920$	0	0
$921$	−10.4891	−0.345627
$922$	8.52058	0.280610
$923$	3.18923	0.104975
$924$	−19.7316	−0.649122
$925$	0	0
$926$	−27.8694	−0.915844
$927$	−1.10407	−0.0362624
$928$	−6.06987	−0.199253
$929$	8.49231	0.278624	0.139312	−	0.990249i	$-0.455511\pi$
$929$	8.49231	0.278624	0.139312	+	0.990249i	$0.455511\pi$
$930$	0	0
$931$	0	0
$932$	−12.7668	−0.418190
$933$	5.70710	0.186842
$934$	−6.86953	−0.224778
$935$	0	0
$936$	−0.351554	−0.0114909
$937$	50.7981	1.65950	0.829751	−	0.558134i	$-0.188483\pi$
$937$	50.7981	1.65950	0.829751	+	0.558134i	$0.188483\pi$
$938$	−30.4501	−0.994231
$939$	−41.2946	−1.34760
$940$	0	0
$941$	3.94534	0.128614	0.0643072	−	0.997930i	$-0.479516\pi$
$941$	3.94534	0.128614	0.0643072	+	0.997930i	$0.479516\pi$
$942$	−20.9586	−0.682868
$943$	−27.4898	−0.895192
$944$	4.55639	0.148298
$945$	0	0
$946$	39.6591	1.28943
$947$	−1.12607	−0.0365924	−0.0182962	−	0.999833i	$-0.505824\pi$
$947$	−1.12607	−0.0365924	−0.0182962	+	0.999833i	$0.505824\pi$
$948$	14.4798	0.470281
$949$	3.69725	0.120018
$950$	0	0
$951$	−18.7134	−0.606824
$952$	−90.8552	−2.94463
$953$	51.9998	1.68444	0.842220	−	0.539135i	$-0.181249\pi$
$953$	51.9998	1.68444	0.842220	+	0.539135i	$0.181249\pi$
$954$	0.741889	0.0240195
$955$	0	0
$956$	19.1523	0.619431
$957$	−9.49534	−0.306941
$958$	25.3645	0.819490
$959$	16.4862	0.532366
$960$	0	0
$961$	25.1600	0.811612
$962$	7.97583	0.257151
$963$	0.212422	0.00684522
$964$	6.70839	0.216063
$965$	0	0
$966$	−46.6781	−1.50184
$967$	24.6191	0.791696	0.395848	−	0.918316i	$-0.370451\pi$
$967$	24.6191	0.791696	0.395848	+	0.918316i	$0.370451\pi$
$968$	−2.99662	−0.0963152
$969$	0	0
$970$	0	0
$971$	−53.3884	−1.71332	−0.856658	−	0.515884i	$-0.827463\pi$
$971$	−53.3884	−1.71332	−0.856658	+	0.515884i	$0.827463\pi$
$972$	−0.712736	−0.0228610
$973$	−9.14557	−0.293194
$974$	24.2364	0.776585
$975$	0	0
$976$	−13.0224	−0.416837
$977$	21.4513	0.686288	0.343144	−	0.939283i	$-0.388508\pi$
$977$	21.4513	0.686288	0.343144	+	0.939283i	$0.388508\pi$
$978$	4.65764	0.148935
$979$	−30.5711	−0.977058
$980$	0	0
$981$	−1.30326	−0.0416100
$982$	−38.2914	−1.22193
$983$	−14.5902	−0.465355	−0.232677	−	0.972554i	$-0.574749\pi$
$983$	−14.5902	−0.465355	−0.232677	+	0.972554i	$0.574749\pi$
$984$	−28.4968	−0.908444
$985$	0	0
$986$	−11.3793	−0.362390
$987$	65.3545	2.08026
$988$	0	0
$989$	−50.9271	−1.61939
$990$	0	0
$991$	−5.62007	−0.178527	−0.0892636	−	0.996008i	$-0.528451\pi$
$991$	−5.62007	−0.178527	−0.0892636	+	0.996008i	$0.528451\pi$
$992$	28.2409	0.896648
$993$	43.0215	1.36525
$994$	−14.7462	−0.467722
$995$	0	0
$996$	−9.88980	−0.313370
$997$	−31.1721	−0.987230	−0.493615	−	0.869681i	$-0.664325\pi$
$997$	−31.1721	−0.987230	−0.493615	+	0.869681i	$0.664325\pi$
$998$	−40.1510	−1.27096
$999$	31.5615	0.998560

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.13856 q^{2} +1.70367 q^{3} -0.703671 q^{4} +1.93974 q^{6} +4.75660 q^{7} -3.07830 q^{8} -0.0975037 q^{9} +O(q^{10})\)	\(q+1.13856 q^{2} +1.70367 q^{3} -0.703671 q^{4} +1.93974 q^{6} +4.75660 q^{7} -3.07830 q^{8} -0.0975037 q^{9} +3.46027 q^{11} -1.19882 q^{12} -1.17127 q^{13} +5.41569 q^{14} -2.09750 q^{16} +6.20500 q^{17} -0.111014 q^{18} +8.10368 q^{21} +3.93974 q^{22} -5.05910 q^{23} -5.24442 q^{24} -1.33357 q^{26} -5.27713 q^{27} -3.34708 q^{28} -1.61070 q^{29} +7.49400 q^{31} +3.76846 q^{32} +5.89516 q^{33} +7.06479 q^{34} +0.0686106 q^{36} -5.98080 q^{37} -1.99547 q^{39} +5.43374 q^{41} +9.22656 q^{42} +10.0664 q^{43} -2.43489 q^{44} -5.76011 q^{46} +8.06479 q^{47} -3.57346 q^{48} +15.6252 q^{49} +10.5713 q^{51} +0.824193 q^{52} -6.68283 q^{53} -6.00835 q^{54} -14.6423 q^{56} -1.83389 q^{58} -2.17229 q^{59} +6.20852 q^{61} +8.53240 q^{62} -0.463786 q^{63} +8.48565 q^{64} +6.71202 q^{66} -5.62257 q^{67} -4.36628 q^{68} -8.61905 q^{69} -2.72287 q^{71} +0.300146 q^{72} -3.15661 q^{73} -6.80953 q^{74} +16.4591 q^{77} -2.27197 q^{78} -12.0783 q^{79} -8.69798 q^{81} +6.18666 q^{82} +8.24958 q^{83} -5.70233 q^{84} +11.4613 q^{86} -2.74410 q^{87} -10.6518 q^{88} -8.83490 q^{89} -5.57128 q^{91} +3.55995 q^{92} +12.7673 q^{93} +9.18229 q^{94} +6.42023 q^{96} +0.707489 q^{97} +17.7903 q^{98} -0.337389 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{2} + q^{3} + 3 q^{4} - 7 q^{6} + 11 q^{7} + 6 q^{8} + 5 q^{9}+O(q^{10}) \) 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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