LMFDB - Embedded newform 9025.2.a.cg.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:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 12x^{8} + 23x^{7} + 47x^{6} - 86x^{5} - 69x^{4} + 115x^{3} + 34x^{2} - 45x + 5 $ x^10 - 2x^9 - 12x^8 + 23x^7 + 47x^6 - 86x^5 - 69x^4 + 115x^3 + 34x^2 - 45*x + 5
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.83019$ 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)

$f(q)$	$=$	$q-1.83019 q^{2} -3.26153 q^{3} +1.34958 q^{4} +5.96920 q^{6} -2.06494 q^{7} +1.19039 q^{8} +7.63755 q^{9} +O(q^{10})$	\(q-1.83019 q^{2} -3.26153 q^{3} +1.34958 q^{4} +5.96920 q^{6} -2.06494 q^{7} +1.19039 q^{8} +7.63755 q^{9} -6.01120 q^{11} -4.40168 q^{12} +0.189304 q^{13} +3.77923 q^{14} -4.87780 q^{16} -2.84709 q^{17} -13.9781 q^{18} +6.73487 q^{21} +11.0016 q^{22} -5.37870 q^{23} -3.88250 q^{24} -0.346461 q^{26} -15.1255 q^{27} -2.78680 q^{28} +6.08122 q^{29} +5.33578 q^{31} +6.54648 q^{32} +19.6057 q^{33} +5.21070 q^{34} +10.3075 q^{36} -8.39320 q^{37} -0.617420 q^{39} +7.13091 q^{41} -12.3261 q^{42} -3.05554 q^{43} -8.11258 q^{44} +9.84402 q^{46} +0.574453 q^{47} +15.9091 q^{48} -2.73600 q^{49} +9.28585 q^{51} +0.255480 q^{52} -3.40473 q^{53} +27.6824 q^{54} -2.45810 q^{56} -11.1298 q^{58} +10.7005 q^{59} -4.01148 q^{61} -9.76546 q^{62} -15.7711 q^{63} -2.22568 q^{64} -35.8820 q^{66} +3.21439 q^{67} -3.84237 q^{68} +17.5428 q^{69} -15.6045 q^{71} +9.09169 q^{72} +12.1598 q^{73} +15.3611 q^{74} +12.4128 q^{77} +1.12999 q^{78} -3.19854 q^{79} +26.4195 q^{81} -13.0509 q^{82} -6.11296 q^{83} +9.08923 q^{84} +5.59221 q^{86} -19.8341 q^{87} -7.15570 q^{88} -15.6372 q^{89} -0.390902 q^{91} -7.25898 q^{92} -17.4028 q^{93} -1.05136 q^{94} -21.3515 q^{96} +1.68405 q^{97} +5.00739 q^{98} -45.9108 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 2 q^{2} + 8 q^{4} + 4 q^{6} - 4 q^{7} - 3 q^{8} + 12 q^{9}+O(q^{10}) $ 10 * q - 2 * q^2 + 8 * q^4 + 4 * q^6 - 4 * q^7 - 3 * q^8 + 12 * q^9	$ 10 q - 2 q^{2} + 8 q^{4} + 4 q^{6} - 4 q^{7} - 3 q^{8} + 12 q^{9} - 3 q^{11} + 14 q^{12} + 5 q^{13} - 14 q^{14} - 4 q^{16} - 11 q^{17} - 4 q^{18} + 17 q^{21} + 34 q^{22} - 32 q^{23} - 7 q^{24} - 13 q^{26} - 30 q^{27} + 12 q^{28} - 8 q^{29} - 11 q^{31} - 8 q^{32} + 19 q^{33} - 20 q^{34} - 4 q^{36} - 35 q^{37} + 20 q^{39} + 16 q^{41} - 32 q^{42} - 26 q^{43} - 44 q^{44} - 21 q^{46} - 19 q^{47} + 16 q^{48} + 18 q^{49} + 15 q^{51} + 25 q^{52} - 16 q^{53} - q^{54} - 20 q^{56} - 7 q^{58} + 2 q^{59} + 8 q^{61} - 13 q^{62} - 77 q^{63} + q^{64} + 20 q^{66} - 40 q^{68} + 12 q^{69} + 2 q^{71} - 31 q^{72} - q^{73} + 18 q^{74} - 18 q^{77} - 41 q^{78} + 18 q^{79} + 58 q^{81} - 25 q^{82} - 46 q^{83} + 24 q^{84} - 33 q^{86} - 57 q^{87} + 37 q^{88} - 13 q^{89} + 67 q^{91} - 33 q^{92} - 4 q^{93} + 25 q^{94} - 40 q^{96} - 8 q^{97} + 62 q^{98} - 62 q^{99}+O(q^{100}) $ 10 * q - 2 * q^2 + 8 * q^4 + 4 * q^6 - 4 * q^7 - 3 * q^8 + 12 * q^9 - 3 * q^11 + 14 * q^12 + 5 * q^13 - 14 * q^14 - 4 * q^16 - 11 * q^17 - 4 * q^18 + 17 * q^21 + 34 * q^22 - 32 * q^23 - 7 * q^24 - 13 * q^26 - 30 * q^27 + 12 * q^28 - 8 * q^29 - 11 * q^31 - 8 * q^32 + 19 * q^33 - 20 * q^34 - 4 * q^36 - 35 * q^37 + 20 * q^39 + 16 * q^41 - 32 * q^42 - 26 * q^43 - 44 * q^44 - 21 * q^46 - 19 * q^47 + 16 * q^48 + 18 * q^49 + 15 * q^51 + 25 * q^52 - 16 * q^53 - q^54 - 20 * q^56 - 7 * q^58 + 2 * q^59 + 8 * q^61 - 13 * q^62 - 77 * q^63 + q^64 + 20 * q^66 - 40 * q^68 + 12 * q^69 + 2 * q^71 - 31 * q^72 - q^73 + 18 * q^74 - 18 * q^77 - 41 * q^78 + 18 * q^79 + 58 * q^81 - 25 * q^82 - 46 * q^83 + 24 * q^84 - 33 * q^86 - 57 * q^87 + 37 * q^88 - 13 * q^89 + 67 * q^91 - 33 * q^92 - 4 * q^93 + 25 * q^94 - 40 * q^96 - 8 * q^97 + 62 * q^98 - 62 * 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 $

$2$	−1.83019	−1.29414	−0.647068	−	0.762432i	$-0.724005\pi$
$2$	−1.83019	−1.29414	−0.647068	+	0.762432i	$0.724005\pi$
$3$	−3.26153	−1.88304	−0.941521	−	0.336953i	$-0.890603\pi$
$3$	−3.26153	−1.88304	−0.941521	+	0.336953i	$0.890603\pi$
$4$	1.34958	0.674789
$5$	0	0
$5$	0	0
$6$	5.96920	2.43691
$7$	−2.06494	−0.780476	−0.390238	−	0.920714i	$-0.627607\pi$
$7$	−2.06494	−0.780476	−0.390238	+	0.920714i	$0.627607\pi$
$8$	1.19039	0.420868
$9$	7.63755	2.54585
$10$	0	0
$11$	−6.01120	−1.81244	−0.906222	−	0.422801i	$-0.861047\pi$
$11$	−6.01120	−1.81244	−0.906222	+	0.422801i	$0.861047\pi$
$12$	−4.40168	−1.27066
$13$	0.189304	0.0525035	0.0262517	−	0.999655i	$-0.491643\pi$
$13$	0.189304	0.0525035	0.0262517	+	0.999655i	$0.491643\pi$
$14$	3.77923	1.01004
$15$	0	0
$16$	−4.87780	−1.21945
$17$	−2.84709	−0.690521	−0.345260	−	0.938507i	$-0.612209\pi$
$17$	−2.84709	−0.690521	−0.345260	+	0.938507i	$0.612209\pi$
$18$	−13.9781	−3.29468
$19$	0	0
$19$	0	0
$20$	0	0
$21$	6.73487	1.46967
$22$	11.0016	2.34555
$23$	−5.37870	−1.12154	−0.560769	−	0.827973i	$-0.689494\pi$
$23$	−5.37870	−1.12154	−0.560769	+	0.827973i	$0.689494\pi$
$24$	−3.88250	−0.792512
$25$	0	0
$26$	−0.346461	−0.0679467
$27$	−15.1255	−2.91090
$28$	−2.78680	−0.526656
$29$	6.08122	1.12925	0.564627	−	0.825346i	$-0.309020\pi$
$29$	6.08122	1.12925	0.564627	+	0.825346i	$0.309020\pi$
$30$	0	0
$31$	5.33578	0.958334	0.479167	−	0.877724i	$-0.340939\pi$
$31$	5.33578	0.958334	0.479167	+	0.877724i	$0.340939\pi$
$32$	6.54648	1.15727
$33$	19.6057	3.41291
$34$	5.21070	0.893628
$35$	0	0
$36$	10.3075	1.71791
$37$	−8.39320	−1.37983	−0.689917	−	0.723889i	$-0.742353\pi$
$37$	−8.39320	−1.37983	−0.689917	+	0.723889i	$0.742353\pi$
$38$	0	0
$39$	−0.617420	−0.0988663
$40$	0	0
$41$	7.13091	1.11366	0.556830	−	0.830626i	$-0.312017\pi$
$41$	7.13091	1.11366	0.556830	+	0.830626i	$0.312017\pi$
$42$	−12.3261	−1.90195
$43$	−3.05554	−0.465966	−0.232983	−	0.972481i	$-0.574849\pi$
$43$	−3.05554	−0.465966	−0.232983	+	0.972481i	$0.574849\pi$
$44$	−8.11258	−1.22302
$45$	0	0
$46$	9.84402	1.45142
$47$	0.574453	0.0837926	0.0418963	−	0.999122i	$-0.486660\pi$
$47$	0.574453	0.0837926	0.0418963	+	0.999122i	$0.486660\pi$
$48$	15.9091	2.29627
$49$	−2.73600	−0.390858
$50$	0	0
$51$	9.28585	1.30028
$52$	0.255480	0.0354288
$53$	−3.40473	−0.467676	−0.233838	−	0.972276i	$-0.575128\pi$
$53$	−3.40473	−0.467676	−0.233838	+	0.972276i	$0.575128\pi$
$54$	27.6824	3.76710
$55$	0	0
$56$	−2.45810	−0.328477
$57$	0	0
$58$	−11.1298	−1.46141
$59$	10.7005	1.39309	0.696545	−	0.717513i	$-0.254719\pi$
$59$	10.7005	1.39309	0.696545	+	0.717513i	$0.254719\pi$
$60$	0	0
$61$	−4.01148	−0.513617	−0.256809	−	0.966462i	$-0.582671\pi$
$61$	−4.01148	−0.513617	−0.256809	+	0.966462i	$0.582671\pi$
$62$	−9.76546	−1.24021
$63$	−15.7711	−1.98697
$64$	−2.22568	−0.278210
$65$	0	0
$66$	−35.8820	−4.41677
$67$	3.21439	0.392700	0.196350	−	0.980534i	$-0.437091\pi$
$67$	3.21439	0.392700	0.196350	+	0.980534i	$0.437091\pi$
$68$	−3.84237	−0.465955
$69$	17.5428	2.11190
$70$	0	0
$71$	−15.6045	−1.85192	−0.925958	−	0.377628i	$-0.876740\pi$
$71$	−15.6045	−1.85192	−0.925958	+	0.377628i	$0.876740\pi$
$72$	9.09169	1.07147
$73$	12.1598	1.42320	0.711598	−	0.702587i	$-0.247972\pi$
$73$	12.1598	1.42320	0.711598	+	0.702587i	$0.247972\pi$
$74$	15.3611	1.78569
$75$	0	0
$76$	0	0
$77$	12.4128	1.41457
$78$	1.12999	0.127946
$79$	−3.19854	−0.359863	−0.179932	−	0.983679i	$-0.557588\pi$
$79$	−3.19854	−0.359863	−0.179932	+	0.983679i	$0.557588\pi$
$80$	0	0
$81$	26.4195	2.93550
$82$	−13.0509	−1.44123
$83$	−6.11296	−0.670984	−0.335492	−	0.942043i	$-0.608903\pi$
$83$	−6.11296	−0.670984	−0.335492	+	0.942043i	$0.608903\pi$
$84$	9.08923	0.991716
$85$	0	0
$86$	5.59221	0.603024
$87$	−19.8341	−2.12643
$88$	−7.15570	−0.762800
$89$	−15.6372	−1.65754	−0.828771	−	0.559587i	$-0.810960\pi$
$89$	−15.6372	−1.65754	−0.828771	+	0.559587i	$0.810960\pi$
$90$	0	0
$91$	−0.390902	−0.0409777
$92$	−7.25898	−0.756801
$93$	−17.4028	−1.80458
$94$	−1.05136	−0.108439
$95$	0	0
$96$	−21.3515	−2.17918
$97$	1.68405	0.170989	0.0854947	−	0.996339i	$-0.472753\pi$
$97$	1.68405	0.170989	0.0854947	+	0.996339i	$0.472753\pi$
$98$	5.00739	0.505823
$99$	−45.9108	−4.61421
$100$	0	0
$101$	9.43158	0.938477	0.469239	−	0.883071i	$-0.344528\pi$
$101$	9.43158	0.938477	0.469239	+	0.883071i	$0.344528\pi$
$102$	−16.9948	−1.68274
$103$	0.593377	0.0584671	0.0292336	−	0.999573i	$-0.490693\pi$
$103$	0.593377	0.0584671	0.0292336	+	0.999573i	$0.490693\pi$
$104$	0.225346	0.0220970
$105$	0	0
$106$	6.23128	0.605236
$107$	−3.96686	−0.383491	−0.191746	−	0.981445i	$-0.561415\pi$
$107$	−3.96686	−0.383491	−0.191746	+	0.981445i	$0.561415\pi$
$108$	−20.4130	−1.96424
$109$	1.14700	0.109862	0.0549312	−	0.998490i	$-0.482506\pi$
$109$	1.14700	0.109862	0.0549312	+	0.998490i	$0.482506\pi$
$110$	0	0
$111$	27.3746	2.59828
$112$	10.0724	0.951750
$113$	−8.80103	−0.827932	−0.413966	−	0.910292i	$-0.635857\pi$
$113$	−8.80103	−0.827932	−0.413966	+	0.910292i	$0.635857\pi$
$114$	0	0
$115$	0	0
$116$	8.20708	0.762008
$117$	1.44582	0.133666
$118$	−19.5840	−1.80285
$119$	5.87908	0.538935
$120$	0	0
$121$	25.1345	2.28496
$122$	7.34175	0.664691
$123$	−23.2576	−2.09707
$124$	7.20104	0.646673
$125$	0	0
$126$	28.8641	2.57141
$127$	7.19879	0.638789	0.319395	−	0.947622i	$-0.396520\pi$
$127$	7.19879	0.638789	0.319395	+	0.947622i	$0.396520\pi$
$128$	−9.01956	−0.797224
$129$	9.96574	0.877434
$130$	0	0
$131$	4.16864	0.364216	0.182108	−	0.983279i	$-0.441708\pi$
$131$	4.16864	0.364216	0.182108	+	0.983279i	$0.441708\pi$
$132$	26.4594	2.30299
$133$	0	0
$134$	−5.88293	−0.508207
$135$	0	0
$136$	−3.38916	−0.290618
$137$	3.81728	0.326132	0.163066	−	0.986615i	$-0.447862\pi$
$137$	3.81728	0.326132	0.163066	+	0.986615i	$0.447862\pi$
$138$	−32.1065	−2.73309
$139$	2.41465	0.204808	0.102404	−	0.994743i	$-0.467347\pi$
$139$	2.41465	0.204808	0.102404	+	0.994743i	$0.467347\pi$
$140$	0	0
$141$	−1.87359	−0.157785
$142$	28.5591	2.39663
$143$	−1.13794	−0.0951597
$144$	−37.2544	−3.10453
$145$	0	0
$146$	−22.2547	−1.84181
$147$	8.92354	0.736001
$148$	−11.3273	−0.931096
$149$	−21.1681	−1.73415	−0.867077	−	0.498174i	$-0.834004\pi$
$149$	−21.1681	−1.73415	−0.867077	+	0.498174i	$0.834004\pi$
$150$	0	0
$151$	−8.96248	−0.729356	−0.364678	−	0.931134i	$-0.618821\pi$
$151$	−8.96248	−0.729356	−0.364678	+	0.931134i	$0.618821\pi$
$152$	0	0
$153$	−21.7448	−1.75796
$154$	−22.7177	−1.83065
$155$	0	0
$156$	−0.833256	−0.0667139
$157$	10.3715	0.827733	0.413867	−	0.910338i	$-0.364178\pi$
$157$	10.3715	0.827733	0.413867	+	0.910338i	$0.364178\pi$
$158$	5.85391	0.465712
$159$	11.1046	0.880653
$160$	0	0
$161$	11.1067	0.875333
$162$	−48.3526	−3.79894
$163$	5.60966	0.439382	0.219691	−	0.975569i	$-0.429495\pi$
$163$	5.60966	0.439382	0.219691	+	0.975569i	$0.429495\pi$
$164$	9.62371	0.751485
$165$	0	0
$166$	11.1878	0.868345
$167$	14.5548	1.12629	0.563144	−	0.826359i	$-0.309592\pi$
$167$	14.5548	1.12629	0.563144	+	0.826359i	$0.309592\pi$
$168$	8.01715	0.618536
$169$	−12.9642	−0.997243
$170$	0	0
$171$	0	0
$172$	−4.12369	−0.314429
$173$	11.9400	0.907780	0.453890	−	0.891058i	$-0.350036\pi$
$173$	11.9400	0.907780	0.453890	+	0.891058i	$0.350036\pi$
$174$	36.3000	2.75190
$175$	0	0
$176$	29.3214	2.21018
$177$	−34.9001	−2.62325
$178$	28.6190	2.14509
$179$	0.355915	0.0266024	0.0133012	−	0.999912i	$-0.495766\pi$
$179$	0.355915	0.0266024	0.0133012	+	0.999912i	$0.495766\pi$
$180$	0	0
$181$	15.8019	1.17455	0.587274	−	0.809388i	$-0.300201\pi$
$181$	15.8019	1.17455	0.587274	+	0.809388i	$0.300201\pi$
$182$	0.715424	0.0530307
$183$	13.0835	0.967163
$184$	−6.40278	−0.472019
$185$	0	0
$186$	31.8503	2.33538
$187$	17.1144	1.25153
$188$	0.775269	0.0565423
$189$	31.2333	2.27189
$190$	0	0
$191$	−3.91338	−0.283163	−0.141581	−	0.989927i	$-0.545219\pi$
$191$	−3.91338	−0.283163	−0.141581	+	0.989927i	$0.545219\pi$
$192$	7.25911	0.523881
$193$	−7.79793	−0.561307	−0.280654	−	0.959809i	$-0.590551\pi$
$193$	−7.79793	−0.561307	−0.280654	+	0.959809i	$0.590551\pi$
$194$	−3.08212	−0.221284
$195$	0	0
$196$	−3.69245	−0.263746
$197$	16.7250	1.19161	0.595804	−	0.803130i	$-0.296834\pi$
$197$	16.7250	1.19161	0.595804	+	0.803130i	$0.296834\pi$
$198$	84.0253	5.97142
$199$	−7.12354	−0.504974	−0.252487	−	0.967600i	$-0.581249\pi$
$199$	−7.12354	−0.504974	−0.252487	+	0.967600i	$0.581249\pi$
$200$	0	0
$201$	−10.4838	−0.739471
$202$	−17.2615	−1.21452
$203$	−12.5574	−0.881356
$204$	12.5320	0.877414
$205$	0	0
$206$	−1.08599	−0.0756645
$207$	−41.0801	−2.85527
$208$	−0.923386	−0.0640253
$209$	0	0
$210$	0	0
$211$	13.8055	0.950412	0.475206	−	0.879875i	$-0.342374\pi$
$211$	13.8055	0.950412	0.475206	+	0.879875i	$0.342374\pi$
$212$	−4.59495	−0.315582
$213$	50.8945	3.48724
$214$	7.26009	0.496290
$215$	0	0
$216$	−18.0053	−1.22510
$217$	−11.0181	−0.747956
$218$	−2.09922	−0.142177
$219$	−39.6595	−2.67994
$220$	0	0
$221$	−0.538965	−0.0362547
$222$	−50.1006	−3.36253
$223$	5.52729	0.370135	0.185067	−	0.982726i	$-0.440750\pi$
$223$	5.52729	0.370135	0.185067	+	0.982726i	$0.440750\pi$
$224$	−13.5181	−0.903217
$225$	0	0
$226$	16.1075	1.07146
$227$	24.9621	1.65679	0.828396	−	0.560143i	$-0.189254\pi$
$227$	24.9621	1.65679	0.828396	+	0.560143i	$0.189254\pi$
$228$	0	0
$229$	19.9042	1.31531	0.657654	−	0.753320i	$-0.271549\pi$
$229$	19.9042	1.31531	0.657654	+	0.753320i	$0.271549\pi$
$230$	0	0
$231$	−40.4846	−2.66369
$232$	7.23905	0.475267
$233$	−3.70242	−0.242554	−0.121277	−	0.992619i	$-0.538699\pi$
$233$	−3.70242	−0.242554	−0.121277	+	0.992619i	$0.538699\pi$
$234$	−2.64612	−0.172982
$235$	0	0
$236$	14.4412	0.940042
$237$	10.4321	0.677638
$238$	−10.7598	−0.697455
$239$	−4.05743	−0.262454	−0.131227	−	0.991352i	$-0.541892\pi$
$239$	−4.05743	−0.262454	−0.131227	+	0.991352i	$0.541892\pi$
$240$	0	0
$241$	20.7387	1.33590	0.667948	−	0.744208i	$-0.267173\pi$
$241$	20.7387	1.33590	0.667948	+	0.744208i	$0.267173\pi$
$242$	−46.0008	−2.95704
$243$	−40.7914	−2.61677
$244$	−5.41380	−0.346583
$245$	0	0
$246$	42.5658	2.71389
$247$	0	0
$248$	6.35168	0.403332
$249$	19.9376	1.26349
$250$	0	0
$251$	3.50729	0.221378	0.110689	−	0.993855i	$-0.464694\pi$
$251$	3.50729	0.221378	0.110689	+	0.993855i	$0.464694\pi$
$252$	−21.2843	−1.34079
$253$	32.3325	2.03272
$254$	−13.1751	−0.826680
$255$	0	0
$256$	20.9588	1.30993
$257$	13.1273	0.818857	0.409428	−	0.912342i	$-0.365728\pi$
$257$	13.1273	0.818857	0.409428	+	0.912342i	$0.365728\pi$
$258$	−18.2391	−1.13552
$259$	17.3315	1.07693
$260$	0	0
$261$	46.4456	2.87491
$262$	−7.62938	−0.471344
$263$	−12.1683	−0.750329	−0.375164	−	0.926958i	$-0.622414\pi$
$263$	−12.1683	−0.750329	−0.375164	+	0.926958i	$0.622414\pi$
$264$	23.3385	1.43638
$265$	0	0
$266$	0	0
$267$	51.0012	3.12122
$268$	4.33807	0.264990
$269$	24.1318	1.47134	0.735672	−	0.677338i	$-0.236866\pi$
$269$	24.1318	1.47134	0.735672	+	0.677338i	$0.236866\pi$
$270$	0	0
$271$	−5.20632	−0.316261	−0.158131	−	0.987418i	$-0.550547\pi$
$271$	−5.20632	−0.316261	−0.158131	+	0.987418i	$0.550547\pi$
$272$	13.8875	0.842055
$273$	1.27494	0.0771628
$274$	−6.98632	−0.422059
$275$	0	0
$276$	23.6753	1.42509
$277$	16.6591	1.00095	0.500475	−	0.865751i	$-0.333159\pi$
$277$	16.6591	1.00095	0.500475	+	0.865751i	$0.333159\pi$
$278$	−4.41925	−0.265049
$279$	40.7522	2.43977
$280$	0	0
$281$	9.85960	0.588175	0.294087	−	0.955779i	$-0.404984\pi$
$281$	9.85960	0.588175	0.294087	+	0.955779i	$0.404984\pi$
$282$	3.42902	0.204195
$283$	−11.0347	−0.655944	−0.327972	−	0.944687i	$-0.606365\pi$
$283$	−11.0347	−0.655944	−0.327972	+	0.944687i	$0.606365\pi$
$284$	−21.0595	−1.24965
$285$	0	0
$286$	2.08265	0.123150
$287$	−14.7249	−0.869185
$288$	49.9991	2.94622
$289$	−8.89408	−0.523181
$290$	0	0
$291$	−5.49257	−0.321980
$292$	16.4106	0.960356
$293$	14.6135	0.853731	0.426866	−	0.904315i	$-0.359618\pi$
$293$	14.6135	0.853731	0.426866	+	0.904315i	$0.359618\pi$
$294$	−16.3317	−0.952486
$295$	0	0
$296$	−9.99121	−0.580727
$297$	90.9223	5.27585
$298$	38.7415	2.24423
$299$	−1.01821	−0.0588846
$300$	0	0
$301$	6.30953	0.363675
$302$	16.4030	0.943886
$303$	−30.7613	−1.76719
$304$	0	0
$305$	0	0
$306$	39.7970	2.27504
$307$	4.46546	0.254857	0.127429	−	0.991848i	$-0.459328\pi$
$307$	4.46546	0.254857	0.127429	+	0.991848i	$0.459328\pi$
$308$	16.7520	0.954535
$309$	−1.93531	−0.110096
$310$	0	0
$311$	−10.3510	−0.586951	−0.293475	−	0.955967i	$-0.594812\pi$
$311$	−10.3510	−0.586951	−0.293475	+	0.955967i	$0.594812\pi$
$312$	−0.734973	−0.0416096
$313$	11.4832	0.649067	0.324534	−	0.945874i	$-0.394793\pi$
$313$	11.4832	0.649067	0.324534	+	0.945874i	$0.394793\pi$
$314$	−18.9817	−1.07120
$315$	0	0
$316$	−4.31667	−0.242832
$317$	24.9039	1.39874	0.699370	−	0.714760i	$-0.253464\pi$
$317$	24.9039	1.39874	0.699370	+	0.714760i	$0.253464\pi$
$318$	−20.3235	−1.13968
$319$	−36.5554	−2.04671
$320$	0	0
$321$	12.9380	0.722130
$322$	−20.3274	−1.13280
$323$	0	0
$324$	35.6552	1.98084
$325$	0	0
$326$	−10.2667	−0.568620
$327$	−3.74096	−0.206875
$328$	8.48859	0.468704
$329$	−1.18621	−0.0653981
$330$	0	0
$331$	−24.9087	−1.36910	−0.684552	−	0.728964i	$-0.740002\pi$
$331$	−24.9087	−1.36910	−0.684552	+	0.728964i	$0.740002\pi$
$332$	−8.24991	−0.452772
$333$	−64.1035	−3.51285
$334$	−26.6380	−1.45757
$335$	0	0
$336$	−32.8513	−1.79219
$337$	−2.99014	−0.162883	−0.0814416	−	0.996678i	$-0.525952\pi$
$337$	−2.99014	−0.162883	−0.0814416	+	0.996678i	$0.525952\pi$
$338$	23.7268	1.29057
$339$	28.7048	1.55903
$340$	0	0
$341$	−32.0744	−1.73693
$342$	0	0
$343$	20.1043	1.08553
$344$	−3.63730	−0.196110
$345$	0	0
$346$	−21.8524	−1.17479
$347$	20.1528	1.08186	0.540930	−	0.841068i	$-0.318072\pi$
$347$	20.1528	1.08186	0.540930	+	0.841068i	$0.318072\pi$
$348$	−26.7676	−1.43489
$349$	−33.0958	−1.77158	−0.885789	−	0.464087i	$-0.846382\pi$
$349$	−33.0958	−1.77158	−0.885789	+	0.464087i	$0.846382\pi$
$350$	0	0
$351$	−2.86331	−0.152832
$352$	−39.3522	−2.09748
$353$	−12.1820	−0.648380	−0.324190	−	0.945992i	$-0.605092\pi$
$353$	−12.1820	−0.648380	−0.324190	+	0.945992i	$0.605092\pi$
$354$	63.8736	3.39484
$355$	0	0
$356$	−21.1036	−1.11849
$357$	−19.1748	−1.01484
$358$	−0.651391	−0.0344271
$359$	−23.9965	−1.26649	−0.633243	−	0.773953i	$-0.718277\pi$
$359$	−23.9965	−1.26649	−0.633243	+	0.773953i	$0.718277\pi$
$360$	0	0
$361$	0	0
$362$	−28.9205	−1.52003
$363$	−81.9769	−4.30267
$364$	−0.527553	−0.0276513
$365$	0	0
$366$	−23.9453	−1.25164
$367$	3.60201	0.188024	0.0940118	−	0.995571i	$-0.470031\pi$
$367$	3.60201	0.188024	0.0940118	+	0.995571i	$0.470031\pi$
$368$	26.2362	1.36766
$369$	54.4626	2.83521
$370$	0	0
$371$	7.03058	0.365009
$372$	−23.4864	−1.21771
$373$	27.0762	1.40195	0.700977	−	0.713184i	$-0.252748\pi$
$373$	27.0762	1.40195	0.700977	+	0.713184i	$0.252748\pi$
$374$	−31.3226	−1.61965
$375$	0	0
$376$	0.683825	0.0352656
$377$	1.15120	0.0592898
$378$	−57.1627	−2.94013
$379$	12.6705	0.650840	0.325420	−	0.945570i	$-0.394494\pi$
$379$	12.6705	0.650840	0.325420	+	0.945570i	$0.394494\pi$
$380$	0	0
$381$	−23.4790	−1.20287
$382$	7.16222	0.366451
$383$	6.91088	0.353130	0.176565	−	0.984289i	$-0.443501\pi$
$383$	6.91088	0.353130	0.176565	+	0.984289i	$0.443501\pi$
$384$	29.4175	1.50121
$385$	0	0
$386$	14.2717	0.726408
$387$	−23.3369	−1.18628
$388$	2.27276	0.115382
$389$	6.24903	0.316838	0.158419	−	0.987372i	$-0.449360\pi$
$389$	6.24903	0.316838	0.158419	+	0.987372i	$0.449360\pi$
$390$	0	0
$391$	15.3137	0.774445
$392$	−3.25692	−0.164499
$393$	−13.5961	−0.685833
$394$	−30.6099	−1.54210
$395$	0	0
$396$	−61.9602	−3.11362
$397$	31.5104	1.58146	0.790732	−	0.612162i	$-0.209700\pi$
$397$	31.5104	1.58146	0.790732	+	0.612162i	$0.209700\pi$
$398$	13.0374	0.653505
$399$	0	0
$400$	0	0
$401$	−24.0456	−1.20078	−0.600390	−	0.799708i	$-0.704988\pi$
$401$	−24.0456	−1.20078	−0.600390	+	0.799708i	$0.704988\pi$
$402$	19.1873	0.956976
$403$	1.01008	0.0503159
$404$	12.7286	0.633274
$405$	0	0
$406$	22.9823	1.14059
$407$	50.4532	2.50087
$408$	11.0538	0.547246
$409$	8.33738	0.412257	0.206129	−	0.978525i	$-0.433914\pi$
$409$	8.33738	0.412257	0.206129	+	0.978525i	$0.433914\pi$
$410$	0	0
$411$	−12.4501	−0.614120
$412$	0.800808	0.0394530
$413$	−22.0960	−1.08727
$414$	75.1842	3.69510
$415$	0	0
$416$	1.23928	0.0607605
$417$	−7.87543	−0.385662
$418$	0	0
$419$	−13.2943	−0.649468	−0.324734	−	0.945805i	$-0.605275\pi$
$419$	−13.2943	−0.649468	−0.324734	+	0.945805i	$0.605275\pi$
$420$	0	0
$421$	19.7822	0.964125	0.482063	−	0.876137i	$-0.339888\pi$
$421$	19.7822	0.964125	0.482063	+	0.876137i	$0.339888\pi$
$422$	−25.2667	−1.22996
$423$	4.38741	0.213323
$424$	−4.05297	−0.196830
$425$	0	0
$426$	−93.1464	−4.51296
$427$	8.28348	0.400866
$428$	−5.35359	−0.258775
$429$	3.71143	0.179190
$430$	0	0
$431$	6.37697	0.307168	0.153584	−	0.988136i	$-0.450918\pi$
$431$	6.37697	0.307168	0.153584	+	0.988136i	$0.450918\pi$
$432$	73.7790	3.54969
$433$	2.38654	0.114690	0.0573448	−	0.998354i	$-0.481737\pi$
$433$	2.38654	0.114690	0.0573448	+	0.998354i	$0.481737\pi$
$434$	20.1651	0.967957
$435$	0	0
$436$	1.54796	0.0741338
$437$	0	0
$438$	72.5841	3.46820
$439$	29.3204	1.39939	0.699693	−	0.714443i	$-0.253320\pi$
$439$	29.3204	1.39939	0.699693	+	0.714443i	$0.253320\pi$
$440$	0	0
$441$	−20.8964	−0.995064
$442$	0.986406	0.0469186
$443$	21.2092	1.00768	0.503840	−	0.863797i	$-0.331920\pi$
$443$	21.2092	1.00768	0.503840	+	0.863797i	$0.331920\pi$
$444$	36.9442	1.75329
$445$	0	0
$446$	−10.1160	−0.479005
$447$	69.0402	3.26549
$448$	4.59591	0.217136
$449$	−14.9025	−0.703294	−0.351647	−	0.936133i	$-0.614378\pi$
$449$	−14.9025	−0.703294	−0.351647	+	0.936133i	$0.614378\pi$
$450$	0	0
$451$	−42.8653	−2.01845
$452$	−11.8777	−0.558679
$453$	29.2314	1.37341
$454$	−45.6852	−2.14411
$455$	0	0
$456$	0	0
$457$	−10.6782	−0.499504	−0.249752	−	0.968310i	$-0.580349\pi$
$457$	−10.6782	−0.499504	−0.249752	+	0.968310i	$0.580349\pi$
$458$	−36.4284	−1.70219
$459$	43.0636	2.01004
$460$	0	0
$461$	−6.97289	−0.324760	−0.162380	−	0.986728i	$-0.551917\pi$
$461$	−6.97289	−0.324760	−0.162380	+	0.986728i	$0.551917\pi$
$462$	74.0944	3.44718
$463$	−8.06083	−0.374619	−0.187309	−	0.982301i	$-0.559977\pi$
$463$	−8.06083	−0.374619	−0.187309	+	0.982301i	$0.559977\pi$
$464$	−29.6630	−1.37707
$465$	0	0
$466$	6.77612	0.313898
$467$	−42.4416	−1.96396	−0.981982	−	0.188974i	$-0.939484\pi$
$467$	−42.4416	−1.96396	−0.981982	+	0.188974i	$0.939484\pi$
$468$	1.95124	0.0901963
$469$	−6.63754	−0.306493
$470$	0	0
$471$	−33.8268	−1.55866
$472$	12.7379	0.586307
$473$	18.3675	0.844538
$474$	−19.0927	−0.876956
$475$	0	0
$476$	7.93428	0.363667
$477$	−26.0038	−1.19063
$478$	7.42585	0.339651
$479$	9.02296	0.412269	0.206135	−	0.978524i	$-0.433912\pi$
$479$	9.02296	0.412269	0.206135	+	0.978524i	$0.433912\pi$
$480$	0	0
$481$	−1.58887	−0.0724461
$482$	−37.9556	−1.72883
$483$	−36.2249	−1.64829
$484$	33.9210	1.54186
$485$	0	0
$486$	74.6558	3.38646
$487$	−24.1910	−1.09620	−0.548099	−	0.836414i	$-0.684648\pi$
$487$	−24.1910	−1.09620	−0.548099	+	0.836414i	$0.684648\pi$
$488$	−4.77524	−0.216165
$489$	−18.2960	−0.827375
$490$	0	0
$491$	34.2414	1.54529	0.772646	−	0.634837i	$-0.218933\pi$
$491$	34.2414	1.54529	0.772646	+	0.634837i	$0.218933\pi$
$492$	−31.3880	−1.41508
$493$	−17.3138	−0.779774
$494$	0	0
$495$	0	0
$496$	−26.0268	−1.16864
$497$	32.2225	1.44537
$498$	−36.4894	−1.63513
$499$	38.6585	1.73059	0.865296	−	0.501262i	$-0.167131\pi$
$499$	38.6585	1.73059	0.865296	+	0.501262i	$0.167131\pi$
$500$	0	0
$501$	−47.4710	−2.12085
$502$	−6.41900	−0.286494
$503$	−25.5438	−1.13894	−0.569470	−	0.822012i	$-0.692852\pi$
$503$	−25.5438	−1.13894	−0.569470	+	0.822012i	$0.692852\pi$
$504$	−18.7738	−0.836253
$505$	0	0
$506$	−59.1744	−2.63062
$507$	42.2830	1.87785
$508$	9.71532	0.431048
$509$	−27.1957	−1.20543	−0.602715	−	0.797957i	$-0.705914\pi$
$509$	−27.1957	−1.20543	−0.602715	+	0.797957i	$0.705914\pi$
$510$	0	0
$511$	−25.1093	−1.11077
$512$	−20.3194	−0.897999
$513$	0	0
$514$	−24.0253	−1.05971
$515$	0	0
$516$	13.4495	0.592083
$517$	−3.45315	−0.151869
$518$	−31.7198	−1.39369
$519$	−38.9426	−1.70939
$520$	0	0
$521$	−7.96380	−0.348901	−0.174450	−	0.984666i	$-0.555815\pi$
$521$	−7.96380	−0.348901	−0.174450	+	0.984666i	$0.555815\pi$
$522$	−85.0041	−3.72053
$523$	22.9428	1.00322	0.501609	−	0.865094i	$-0.332742\pi$
$523$	22.9428	1.00322	0.501609	+	0.865094i	$0.332742\pi$
$524$	5.62590	0.245768
$525$	0	0
$526$	22.2702	0.971028
$527$	−15.1914	−0.661749
$528$	−95.6325	−4.16187
$529$	5.93046	0.257846
$530$	0	0
$531$	81.7258	3.54660
$532$	0	0
$533$	1.34991	0.0584711
$534$	−93.3417	−4.03929
$535$	0	0
$536$	3.82639	0.165275
$537$	−1.16083	−0.0500934
$538$	−44.1657	−1.90412
$539$	16.4467	0.708408
$540$	0	0
$541$	45.3710	1.95065	0.975327	−	0.220767i	$-0.0708560\pi$
$541$	45.3710	1.95065	0.975327	+	0.220767i	$0.0708560\pi$
$542$	9.52852	0.409285
$543$	−51.5384	−2.21172
$544$	−18.6384	−0.799115
$545$	0	0
$546$	−2.33337	−0.0998591
$547$	0.549415	0.0234913	0.0117457	−	0.999931i	$-0.496261\pi$
$547$	0.549415	0.0234913	0.0117457	+	0.999931i	$0.496261\pi$
$548$	5.15171	0.220070
$549$	−30.6379	−1.30759
$550$	0	0
$551$	0	0
$552$	20.8828	0.888832
$553$	6.60480	0.280865
$554$	−30.4893	−1.29537
$555$	0	0
$556$	3.25875	0.138202
$557$	−17.9440	−0.760310	−0.380155	−	0.924923i	$-0.624129\pi$
$557$	−17.9440	−0.760310	−0.380155	+	0.924923i	$0.624129\pi$
$558$	−74.5841	−3.15740
$559$	−0.578427	−0.0244649
$560$	0	0
$561$	−55.8191	−2.35669
$562$	−18.0449	−0.761178
$563$	−38.2076	−1.61026	−0.805130	−	0.593098i	$-0.797904\pi$
$563$	−38.2076	−1.61026	−0.805130	+	0.593098i	$0.797904\pi$
$564$	−2.52856	−0.106472
$565$	0	0
$566$	20.1955	0.848881
$567$	−54.5548	−2.29109
$568$	−18.5755	−0.779412
$569$	28.3751	1.18955	0.594773	−	0.803893i	$-0.297242\pi$
$569$	28.3751	1.18955	0.594773	+	0.803893i	$0.297242\pi$
$570$	0	0
$571$	10.5024	0.439513	0.219757	−	0.975555i	$-0.429474\pi$
$571$	10.5024	0.439513	0.219757	+	0.975555i	$0.429474\pi$
$572$	−1.53574	−0.0642127
$573$	12.7636	0.533207
$574$	26.9493	1.12484
$575$	0	0
$576$	−16.9987	−0.708281
$577$	24.8599	1.03493	0.517466	−	0.855704i	$-0.326876\pi$
$577$	24.8599	1.03493	0.517466	+	0.855704i	$0.326876\pi$
$578$	16.2778	0.677068
$579$	25.4331	1.05697
$580$	0	0
$581$	12.6229	0.523687
$582$	10.0524	0.416686
$583$	20.4665	0.847636
$584$	14.4749	0.598977
$585$	0	0
$586$	−26.7454	−1.10484
$587$	11.0731	0.457038	0.228519	−	0.973539i	$-0.426612\pi$
$587$	11.0731	0.457038	0.228519	+	0.973539i	$0.426612\pi$
$588$	12.0430	0.496645
$589$	0	0
$590$	0	0
$591$	−54.5490	−2.24385
$592$	40.9403	1.68264
$593$	−16.1997	−0.665244	−0.332622	−	0.943060i	$-0.607933\pi$
$593$	−16.1997	−0.665244	−0.332622	+	0.943060i	$0.607933\pi$
$594$	−166.405	−6.82766
$595$	0	0
$596$	−28.5679	−1.17019
$597$	23.2336	0.950888
$598$	1.86351	0.0762047
$599$	7.11417	0.290677	0.145339	−	0.989382i	$-0.453573\pi$
$599$	7.11417	0.290677	0.145339	+	0.989382i	$0.453573\pi$
$600$	0	0
$601$	−7.74720	−0.316015	−0.158008	−	0.987438i	$-0.550507\pi$
$601$	−7.74720	−0.316015	−0.158008	+	0.987438i	$0.550507\pi$
$602$	−11.5476	−0.470645
$603$	24.5500	0.999755
$604$	−12.0956	−0.492161
$605$	0	0
$606$	56.2989	2.28699
$607$	−26.7936	−1.08752	−0.543758	−	0.839242i	$-0.682999\pi$
$607$	−26.7936	−1.08752	−0.543758	+	0.839242i	$0.682999\pi$
$608$	0	0
$609$	40.9562	1.65963
$610$	0	0
$611$	0.108746	0.00439940
$612$	−29.3463	−1.18625
$613$	−42.1665	−1.70309	−0.851544	−	0.524283i	$-0.824333\pi$
$613$	−42.1665	−1.70309	−0.851544	+	0.524283i	$0.824333\pi$
$614$	−8.17261	−0.329820
$615$	0	0
$616$	14.7761	0.595347
$617$	36.7094	1.47787	0.738933	−	0.673779i	$-0.235330\pi$
$617$	36.7094	1.47787	0.738933	+	0.673779i	$0.235330\pi$
$618$	3.54198	0.142479
$619$	6.29155	0.252879	0.126439	−	0.991974i	$-0.459645\pi$
$619$	6.29155	0.252879	0.126439	+	0.991974i	$0.459645\pi$
$620$	0	0
$621$	81.3555	3.26468
$622$	18.9442	0.759594
$623$	32.2900	1.29367
$624$	3.01165	0.120562
$625$	0	0
$626$	−21.0163	−0.839981
$627$	0	0
$628$	13.9971	0.558545
$629$	23.8962	0.952803
$630$	0	0
$631$	−13.1570	−0.523772	−0.261886	−	0.965099i	$-0.584344\pi$
$631$	−13.1570	−0.523772	−0.261886	+	0.965099i	$0.584344\pi$
$632$	−3.80752	−0.151455
$633$	−45.0271	−1.78967
$634$	−45.5787	−1.81016
$635$	0	0
$636$	14.9865	0.594255
$637$	−0.517936	−0.0205214
$638$	66.9032	2.64872
$639$	−119.180	−4.71470
$640$	0	0
$641$	30.2129	1.19334	0.596668	−	0.802488i	$-0.296491\pi$
$641$	30.2129	1.19334	0.596668	+	0.802488i	$0.296491\pi$
$642$	−23.6790	−0.934535
$643$	12.2293	0.482275	0.241138	−	0.970491i	$-0.422479\pi$
$643$	12.2293	0.482275	0.241138	+	0.970491i	$0.422479\pi$
$644$	14.9894	0.590665
$645$	0	0
$646$	0	0
$647$	−44.8494	−1.76321	−0.881606	−	0.471987i	$-0.843537\pi$
$647$	−44.8494	−1.76321	−0.881606	+	0.471987i	$0.843537\pi$
$648$	31.4496	1.23546
$649$	−64.3230	−2.52490
$650$	0	0
$651$	35.9358	1.40843
$652$	7.57066	0.296490
$653$	−19.4871	−0.762588	−0.381294	−	0.924454i	$-0.624521\pi$
$653$	−19.4871	−0.762588	−0.381294	+	0.924454i	$0.624521\pi$
$654$	6.84664	0.267725
$655$	0	0
$656$	−34.7831	−1.35805
$657$	92.8710	3.62324
$658$	2.17099	0.0846340
$659$	−7.37699	−0.287367	−0.143683	−	0.989624i	$-0.545895\pi$
$659$	−7.37699	−0.287367	−0.143683	+	0.989624i	$0.545895\pi$
$660$	0	0
$661$	−43.4775	−1.69108	−0.845539	−	0.533914i	$-0.820720\pi$
$661$	−43.4775	−1.69108	−0.845539	+	0.533914i	$0.820720\pi$
$662$	45.5875	1.77181
$663$	1.75785	0.0682692
$664$	−7.27683	−0.282396
$665$	0	0
$666$	117.321	4.54610
$667$	−32.7091	−1.26650
$668$	19.6429	0.760006
$669$	−18.0274	−0.696979
$670$	0	0
$671$	24.1138	0.930903
$672$	44.0897	1.70080
$673$	11.8719	0.457628	0.228814	−	0.973470i	$-0.426515\pi$
$673$	11.8719	0.457628	0.228814	+	0.973470i	$0.426515\pi$
$674$	5.47251	0.210793
$675$	0	0
$676$	−17.4961	−0.672928
$677$	−40.2029	−1.54512	−0.772561	−	0.634940i	$-0.781025\pi$
$677$	−40.2029	−1.54512	−0.772561	+	0.634940i	$0.781025\pi$
$678$	−52.5351	−2.01760
$679$	−3.47747	−0.133453
$680$	0	0
$681$	−81.4145	−3.11981
$682$	58.7021	2.24782
$683$	31.0423	1.18780	0.593900	−	0.804539i	$-0.297587\pi$
$683$	31.0423	1.18780	0.593900	+	0.804539i	$0.297587\pi$
$684$	0	0
$685$	0	0
$686$	−36.7946	−1.40482
$687$	−64.9182	−2.47678
$688$	14.9043	0.568222
$689$	−0.644529	−0.0245546
$690$	0	0
$691$	5.65569	0.215153	0.107576	−	0.994197i	$-0.465691\pi$
$691$	5.65569	0.215153	0.107576	+	0.994197i	$0.465691\pi$
$692$	16.1139	0.612560
$693$	94.8033	3.60128
$694$	−36.8834	−1.40007
$695$	0	0
$696$	−23.6103	−0.894948
$697$	−20.3023	−0.769006
$698$	60.5715	2.29266
$699$	12.0755	0.456739
$700$	0	0
$701$	16.0478	0.606118	0.303059	−	0.952972i	$-0.401992\pi$
$701$	16.0478	0.606118	0.303059	+	0.952972i	$0.401992\pi$
$702$	5.24040	0.197786
$703$	0	0
$704$	13.3790	0.504240
$705$	0	0
$706$	22.2952	0.839093
$707$	−19.4757	−0.732459
$708$	−47.1003	−1.77014
$709$	−11.7298	−0.440521	−0.220260	−	0.975441i	$-0.570691\pi$
$709$	−11.7298	−0.440521	−0.220260	+	0.975441i	$0.570691\pi$
$710$	0	0
$711$	−24.4290	−0.916158
$712$	−18.6145	−0.697606
$713$	−28.6996	−1.07481
$714$	35.0934	1.31334
$715$	0	0
$716$	0.480335	0.0179510
$717$	13.2334	0.494211
$718$	43.9180	1.63901
$719$	−33.6740	−1.25583	−0.627915	−	0.778282i	$-0.716091\pi$
$719$	−33.6740	−1.25583	−0.627915	+	0.778282i	$0.716091\pi$
$720$	0	0
$721$	−1.22529	−0.0456322
$722$	0	0
$723$	−67.6397	−2.51555
$724$	21.3259	0.792572
$725$	0	0
$726$	150.033	5.56824
$727$	−50.3582	−1.86768	−0.933840	−	0.357690i	$-0.883564\pi$
$727$	−50.3582	−1.86768	−0.933840	+	0.357690i	$0.883564\pi$
$728$	−0.465328	−0.0172462
$729$	53.7838	1.99199
$730$	0	0
$731$	8.69941	0.321759
$732$	17.6572	0.652630
$733$	−28.7690	−1.06261	−0.531305	−	0.847181i	$-0.678298\pi$
$733$	−28.7690	−1.06261	−0.531305	+	0.847181i	$0.678298\pi$
$734$	−6.59235	−0.243328
$735$	0	0
$736$	−35.2116	−1.29792
$737$	−19.3223	−0.711747
$738$	−99.6767	−3.66915
$739$	18.4429	0.678435	0.339217	−	0.940708i	$-0.389838\pi$
$739$	18.4429	0.678435	0.339217	+	0.940708i	$0.389838\pi$
$740$	0	0
$741$	0	0
$742$	−12.8673	−0.472372
$743$	20.3126	0.745197	0.372599	−	0.927993i	$-0.378467\pi$
$743$	20.3126	0.745197	0.372599	+	0.927993i	$0.378467\pi$
$744$	−20.7162	−0.759491
$745$	0	0
$746$	−49.5545	−1.81432
$747$	−46.6880	−1.70822
$748$	23.0972	0.844518
$749$	8.19135	0.299305
$750$	0	0
$751$	17.6489	0.644018	0.322009	−	0.946737i	$-0.395642\pi$
$751$	17.6489	0.644018	0.322009	+	0.946737i	$0.395642\pi$
$752$	−2.80206	−0.102181
$753$	−11.4391	−0.416865
$754$	−2.10691	−0.0767291
$755$	0	0
$756$	42.1517	1.53304
$757$	22.1814	0.806195	0.403098	−	0.915157i	$-0.367933\pi$
$757$	22.1814	0.806195	0.403098	+	0.915157i	$0.367933\pi$
$758$	−23.1894	−0.842276
$759$	−105.453	−3.82771
$760$	0	0
$761$	38.3106	1.38876	0.694378	−	0.719610i	$-0.255679\pi$
$761$	38.3106	1.38876	0.694378	+	0.719610i	$0.255679\pi$
$762$	42.9710	1.55667
$763$	−2.36848	−0.0857449
$764$	−5.28141	−0.191075
$765$	0	0
$766$	−12.6482	−0.456998
$767$	2.02565	0.0731421
$768$	−68.3577	−2.46665
$769$	26.8656	0.968799	0.484400	−	0.874847i	$-0.339038\pi$
$769$	26.8656	0.968799	0.484400	+	0.874847i	$0.339038\pi$
$770$	0	0
$771$	−42.8149	−1.54194
$772$	−10.5239	−0.378764
$773$	5.13462	0.184680	0.0923398	−	0.995728i	$-0.470565\pi$
$773$	5.13462	0.184680	0.0923398	+	0.995728i	$0.470565\pi$
$774$	42.7108	1.53521
$775$	0	0
$776$	2.00468	0.0719639
$777$	−56.5271	−2.02790
$778$	−11.4369	−0.410032
$779$	0	0
$780$	0	0
$781$	93.8018	3.35649
$782$	−28.0268	−1.00224
$783$	−91.9814	−3.28715
$784$	13.3457	0.476631
$785$	0	0
$786$	24.8834	0.887562
$787$	22.2381	0.792703	0.396352	−	0.918099i	$-0.370276\pi$
$787$	22.2381	0.792703	0.396352	+	0.918099i	$0.370276\pi$
$788$	22.5717	0.804083
$789$	39.6872	1.41290
$790$	0	0
$791$	18.1736	0.646181
$792$	−54.6520	−1.94197
$793$	−0.759389	−0.0269667
$794$	−57.6700	−2.04663
$795$	0	0
$796$	−9.61376	−0.340751
$797$	−18.0766	−0.640307	−0.320153	−	0.947366i	$-0.603734\pi$
$797$	−18.0766	−0.640307	−0.320153	+	0.947366i	$0.603734\pi$
$798$	0	0
$799$	−1.63552	−0.0578605
$800$	0	0
$801$	−119.430	−4.21985
$802$	44.0079	1.55397
$803$	−73.0949	−2.57946
$804$	−14.1487	−0.498987
$805$	0	0
$806$	−1.84864	−0.0651156
$807$	−78.7066	−2.77060
$808$	11.2273	0.394975
$809$	−31.7356	−1.11576	−0.557882	−	0.829920i	$-0.688386\pi$
$809$	−31.7356	−1.11576	−0.557882	+	0.829920i	$0.688386\pi$
$810$	0	0
$811$	10.1695	0.357099	0.178549	−	0.983931i	$-0.442860\pi$
$811$	10.1695	0.357099	0.178549	+	0.983931i	$0.442860\pi$
$812$	−16.9472	−0.594729
$813$	16.9805	0.595533
$814$	−92.3387	−3.23647
$815$	0	0
$816$	−45.2945	−1.58562
$817$	0	0
$818$	−15.2590	−0.533517
$819$	−2.98554	−0.104323
$820$	0	0
$821$	13.6762	0.477304	0.238652	−	0.971105i	$-0.423295\pi$
$821$	13.6762	0.477304	0.238652	+	0.971105i	$0.423295\pi$
$822$	22.7861	0.794755
$823$	−10.9385	−0.381294	−0.190647	−	0.981659i	$-0.561059\pi$
$823$	−10.9385	−0.381294	−0.190647	+	0.981659i	$0.561059\pi$
$824$	0.706352	0.0246069
$825$	0	0
$826$	40.4398	1.40708
$827$	31.7124	1.10275	0.551374	−	0.834258i	$-0.314104\pi$
$827$	31.7124	1.10275	0.551374	+	0.834258i	$0.314104\pi$
$828$	−55.4408	−1.92670
$829$	−31.5078	−1.09431	−0.547155	−	0.837031i	$-0.684289\pi$
$829$	−31.5078	−1.09431	−0.547155	+	0.837031i	$0.684289\pi$
$830$	0	0
$831$	−54.3342	−1.88483
$832$	−0.421330	−0.0146070
$833$	7.78964	0.269895
$834$	14.4135	0.499099
$835$	0	0
$836$	0	0
$837$	−80.7062	−2.78961
$838$	24.3310	0.840501
$839$	32.1223	1.10898	0.554492	−	0.832189i	$-0.312913\pi$
$839$	32.1223	1.10898	0.554492	+	0.832189i	$0.312913\pi$
$840$	0	0
$841$	7.98127	0.275216
$842$	−36.2051	−1.24771
$843$	−32.1573	−1.10756
$844$	18.6316	0.641327
$845$	0	0
$846$	−8.02978	−0.276069
$847$	−51.9014	−1.78335
$848$	16.6076	0.570306
$849$	35.9899	1.23517
$850$	0	0
$851$	45.1445	1.54753
$852$	68.6861	2.35315
$853$	29.4353	1.00785	0.503923	−	0.863748i	$-0.331889\pi$
$853$	29.4353	1.00785	0.503923	+	0.863748i	$0.331889\pi$
$854$	−15.1603	−0.518775
$855$	0	0
$856$	−4.72213	−0.161399
$857$	6.92611	0.236591	0.118296	−	0.992978i	$-0.462257\pi$
$857$	6.92611	0.236591	0.118296	+	0.992978i	$0.462257\pi$
$858$	−6.79261	−0.231896
$859$	−49.0796	−1.67457	−0.837287	−	0.546764i	$-0.815860\pi$
$859$	−49.0796	−1.67457	−0.837287	+	0.546764i	$0.815860\pi$
$860$	0	0
$861$	48.0257	1.63671
$862$	−11.6710	−0.397517
$863$	−32.5905	−1.10939	−0.554696	−	0.832053i	$-0.687166\pi$
$863$	−32.5905	−1.10939	−0.554696	+	0.832053i	$0.687166\pi$
$864$	−99.0187	−3.36868
$865$	0	0
$866$	−4.36781	−0.148424
$867$	29.0083	0.985173
$868$	−14.8698	−0.504712
$869$	19.2270	0.652232
$870$	0	0
$871$	0.608497	0.0206181
$872$	1.36538	0.0462375
$873$	12.8620	0.435313
$874$	0	0
$875$	0	0
$876$	−53.5235	−1.80839
$877$	−14.6179	−0.493610	−0.246805	−	0.969065i	$-0.579381\pi$
$877$	−14.6179	−0.493610	−0.246805	+	0.969065i	$0.579381\pi$
$878$	−53.6618	−1.81100
$879$	−47.6624	−1.60761
$880$	0	0
$881$	1.32815	0.0447464	0.0223732	−	0.999750i	$-0.492878\pi$
$881$	1.32815	0.0447464	0.0223732	+	0.999750i	$0.492878\pi$
$882$	38.2442	1.28775
$883$	48.9774	1.64822	0.824111	−	0.566429i	$-0.191675\pi$
$883$	48.9774	1.64822	0.824111	+	0.566429i	$0.191675\pi$
$884$	−0.727375	−0.0244643
$885$	0	0
$886$	−38.8168	−1.30408
$887$	−12.3963	−0.416228	−0.208114	−	0.978105i	$-0.566733\pi$
$887$	−12.3963	−0.416228	−0.208114	+	0.978105i	$0.566733\pi$
$888$	32.5866	1.09353
$889$	−14.8651	−0.498560
$890$	0	0
$891$	−158.813	−5.32043
$892$	7.45950	0.249763
$893$	0	0
$894$	−126.356	−4.22599
$895$	0	0
$896$	18.6249	0.622214
$897$	3.32092	0.110882
$898$	27.2744	0.910158
$899$	32.4480	1.08220
$900$	0	0
$901$	9.69357	0.322940
$902$	78.4514	2.61215
$903$	−20.5787	−0.684816
$904$	−10.4767	−0.348450
$905$	0	0
$906$	−53.4988	−1.77738
$907$	−36.1123	−1.19909	−0.599545	−	0.800341i	$-0.704652\pi$
$907$	−36.1123	−1.19909	−0.599545	+	0.800341i	$0.704652\pi$
$908$	33.6883	1.11798
$909$	72.0341	2.38922
$910$	0	0
$911$	19.5978	0.649303	0.324651	−	0.945834i	$-0.394753\pi$
$911$	19.5978	0.649303	0.324651	+	0.945834i	$0.394753\pi$
$912$	0	0
$913$	36.7462	1.21612
$914$	19.5431	0.646427
$915$	0	0
$916$	26.8623	0.887555
$917$	−8.60801	−0.284261
$918$	−78.8144	−2.60126
$919$	34.8729	1.15035	0.575175	−	0.818030i	$-0.304934\pi$
$919$	34.8729	1.15035	0.575175	+	0.818030i	$0.304934\pi$
$920$	0	0
$921$	−14.5642	−0.479907
$922$	12.7617	0.420284
$923$	−2.95400	−0.0972320
$924$	−54.6372	−1.79743
$925$	0	0
$926$	14.7528	0.484808
$927$	4.53194	0.148849
$928$	39.8106	1.30685
$929$	−23.0455	−0.756098	−0.378049	−	0.925786i	$-0.623405\pi$
$929$	−23.0455	−0.756098	−0.378049	+	0.925786i	$0.623405\pi$
$930$	0	0
$931$	0	0
$932$	−4.99670	−0.163672
$933$	33.7600	1.10525
$934$	77.6760	2.54164
$935$	0	0
$936$	1.72109	0.0562557
$937$	45.7683	1.49519	0.747593	−	0.664157i	$-0.231209\pi$
$937$	45.7683	1.49519	0.747593	+	0.664157i	$0.231209\pi$
$938$	12.1479	0.396644
$939$	−37.4526	−1.22222
$940$	0	0
$941$	−48.3027	−1.57462	−0.787311	−	0.616556i	$-0.788528\pi$
$941$	−48.3027	−1.57462	−0.787311	+	0.616556i	$0.788528\pi$
$942$	61.9093	2.01711
$943$	−38.3550	−1.24901
$944$	−52.1950	−1.69880
$945$	0	0
$946$	−33.6159	−1.09295
$947$	−52.1437	−1.69444	−0.847222	−	0.531240i	$-0.821726\pi$
$947$	−52.1437	−1.69444	−0.847222	+	0.531240i	$0.821726\pi$
$948$	14.0789	0.457262
$949$	2.30190	0.0747227
$950$	0	0
$951$	−81.2246	−2.63389
$952$	6.99842	0.226820
$953$	−40.0379	−1.29695	−0.648477	−	0.761235i	$-0.724594\pi$
$953$	−40.0379	−1.29695	−0.648477	+	0.761235i	$0.724594\pi$
$954$	47.5917	1.54084
$955$	0	0
$956$	−5.47582	−0.177101
$957$	119.227	3.85405
$958$	−16.5137	−0.533533
$959$	−7.88247	−0.254538
$960$	0	0
$961$	−2.52950	−0.0815967
$962$	2.90792	0.0937551
$963$	−30.2971	−0.976310
$964$	27.9884	0.901447
$965$	0	0
$966$	66.2982	2.13311
$967$	33.0826	1.06386	0.531932	−	0.846787i	$-0.321466\pi$
$967$	33.0826	1.06386	0.531932	+	0.846787i	$0.321466\pi$
$968$	29.9200	0.961665
$969$	0	0
$970$	0	0
$971$	−22.6238	−0.726032	−0.363016	−	0.931783i	$-0.618253\pi$
$971$	−22.6238	−0.726032	−0.363016	+	0.931783i	$0.618253\pi$
$972$	−55.0512	−1.76577
$973$	−4.98611	−0.159847
$974$	44.2740	1.41863
$975$	0	0
$976$	19.5672	0.626330
$977$	−12.5594	−0.401809	−0.200905	−	0.979611i	$-0.564388\pi$
$977$	−12.5594	−0.401809	−0.200905	+	0.979611i	$0.564388\pi$
$978$	33.4851	1.07074
$979$	93.9985	3.00420
$980$	0	0
$981$	8.76024	0.279693
$982$	−62.6681	−1.99982
$983$	−50.0043	−1.59489	−0.797445	−	0.603392i	$-0.793816\pi$
$983$	−50.0043	−1.59489	−0.797445	+	0.603392i	$0.793816\pi$
$984$	−27.6857	−0.882589
$985$	0	0
$986$	31.6874	1.00913
$987$	3.86887	0.123147
$988$	0	0
$989$	16.4349	0.522599
$990$	0	0
$991$	6.54040	0.207763	0.103881	−	0.994590i	$-0.466874\pi$
$991$	6.54040	0.207763	0.103881	+	0.994590i	$0.466874\pi$
$992$	34.9306	1.10905
$993$	81.2403	2.57808
$994$	−58.9731	−1.87051
$995$	0	0
$996$	26.9073	0.852590
$997$	45.7761	1.44974	0.724872	−	0.688884i	$-0.241899\pi$
$997$	45.7761	1.44974	0.724872	+	0.688884i	$0.241899\pi$
$998$	−70.7522	−2.23962
$999$	126.951	4.01656

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.cg.1.2	✓	10
5.4	even	2		9025.2.a.cj.1.9	yes	10
19.18	odd	2		9025.2.a.ci.1.9	yes	10
95.94	odd	2		9025.2.a.ch.1.2	yes	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
9025.2.a.cg.1.2	✓	10	1.1	even	1	trivial
9025.2.a.ch.1.2	yes	10	95.94	odd	2
9025.2.a.ci.1.9	yes	10	19.18	odd	2
9025.2.a.cj.1.9	yes	10	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:	\( N \)	\(=\)	\( 9025 = 5^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9025.a (trivial)

\(f(q)\)	\(=\)	\(q-1.83019 q^{2} -3.26153 q^{3} +1.34958 q^{4} +5.96920 q^{6} -2.06494 q^{7} +1.19039 q^{8} +7.63755 q^{9} +O(q^{10})\)	\(q-1.83019 q^{2} -3.26153 q^{3} +1.34958 q^{4} +5.96920 q^{6} -2.06494 q^{7} +1.19039 q^{8} +7.63755 q^{9} -6.01120 q^{11} -4.40168 q^{12} +0.189304 q^{13} +3.77923 q^{14} -4.87780 q^{16} -2.84709 q^{17} -13.9781 q^{18} +6.73487 q^{21} +11.0016 q^{22} -5.37870 q^{23} -3.88250 q^{24} -0.346461 q^{26} -15.1255 q^{27} -2.78680 q^{28} +6.08122 q^{29} +5.33578 q^{31} +6.54648 q^{32} +19.6057 q^{33} +5.21070 q^{34} +10.3075 q^{36} -8.39320 q^{37} -0.617420 q^{39} +7.13091 q^{41} -12.3261 q^{42} -3.05554 q^{43} -8.11258 q^{44} +9.84402 q^{46} +0.574453 q^{47} +15.9091 q^{48} -2.73600 q^{49} +9.28585 q^{51} +0.255480 q^{52} -3.40473 q^{53} +27.6824 q^{54} -2.45810 q^{56} -11.1298 q^{58} +10.7005 q^{59} -4.01148 q^{61} -9.76546 q^{62} -15.7711 q^{63} -2.22568 q^{64} -35.8820 q^{66} +3.21439 q^{67} -3.84237 q^{68} +17.5428 q^{69} -15.6045 q^{71} +9.09169 q^{72} +12.1598 q^{73} +15.3611 q^{74} +12.4128 q^{77} +1.12999 q^{78} -3.19854 q^{79} +26.4195 q^{81} -13.0509 q^{82} -6.11296 q^{83} +9.08923 q^{84} +5.59221 q^{86} -19.8341 q^{87} -7.15570 q^{88} -15.6372 q^{89} -0.390902 q^{91} -7.25898 q^{92} -17.4028 q^{93} -1.05136 q^{94} -21.3515 q^{96} +1.68405 q^{97} +5.00739 q^{98} -45.9108 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 2 q^{2} + 8 q^{4} + 4 q^{6} - 4 q^{7} - 3 q^{8} + 12 q^{9}+O(q^{10}) \) 10 * q - 2 * q^2 + 8 * q^4 + 4 * q^6 - 4 * q^7 - 3 * q^8 + 12 * q^9	\( 10 q - 2 q^{2} + 8 q^{4} + 4 q^{6} - 4 q^{7} - 3 q^{8} + 12 q^{9} - 3 q^{11} + 14 q^{12} + 5 q^{13} - 14 q^{14} - 4 q^{16} - 11 q^{17} - 4 q^{18} + 17 q^{21} + 34 q^{22} - 32 q^{23} - 7 q^{24} - 13 q^{26} - 30 q^{27} + 12 q^{28} - 8 q^{29} - 11 q^{31} - 8 q^{32} + 19 q^{33} - 20 q^{34} - 4 q^{36} - 35 q^{37} + 20 q^{39} + 16 q^{41} - 32 q^{42} - 26 q^{43} - 44 q^{44} - 21 q^{46} - 19 q^{47} + 16 q^{48} + 18 q^{49} + 15 q^{51} + 25 q^{52} - 16 q^{53} - q^{54} - 20 q^{56} - 7 q^{58} + 2 q^{59} + 8 q^{61} - 13 q^{62} - 77 q^{63} + q^{64} + 20 q^{66} - 40 q^{68} + 12 q^{69} + 2 q^{71} - 31 q^{72} - q^{73} + 18 q^{74} - 18 q^{77} - 41 q^{78} + 18 q^{79} + 58 q^{81} - 25 q^{82} - 46 q^{83} + 24 q^{84} - 33 q^{86} - 57 q^{87} + 37 q^{88} - 13 q^{89} + 67 q^{91} - 33 q^{92} - 4 q^{93} + 25 q^{94} - 40 q^{96} - 8 q^{97} + 62 q^{98} - 62 q^{99}+O(q^{100}) \) 10 * q - 2 * q^2 + 8 * q^4 + 4 * q^6 - 4 * q^7 - 3 * q^8 + 12 * q^9 - 3 * q^11 + 14 * q^12 + 5 * q^13 - 14 * q^14 - 4 * q^16 - 11 * q^17 - 4 * q^18 + 17 * q^21 + 34 * q^22 - 32 * q^23 - 7 * q^24 - 13 * q^26 - 30 * q^27 + 12 * q^28 - 8 * q^29 - 11 * q^31 - 8 * q^32 + 19 * q^33 - 20 * q^34 - 4 * q^36 - 35 * q^37 + 20 * q^39 + 16 * q^41 - 32 * q^42 - 26 * q^43 - 44 * q^44 - 21 * q^46 - 19 * q^47 + 16 * q^48 + 18 * q^49 + 15 * q^51 + 25 * q^52 - 16 * q^53 - q^54 - 20 * q^56 - 7 * q^58 + 2 * q^59 + 8 * q^61 - 13 * q^62 - 77 * q^63 + q^64 + 20 * q^66 - 40 * q^68 + 12 * q^69 + 2 * q^71 - 31 * q^72 - q^73 + 18 * q^74 - 18 * q^77 - 41 * q^78 + 18 * q^79 + 58 * q^81 - 25 * q^82 - 46 * q^83 + 24 * q^84 - 33 * q^86 - 57 * q^87 + 37 * q^88 - 13 * q^89 + 67 * q^91 - 33 * q^92 - 4 * q^93 + 25 * q^94 - 40 * q^96 - 8 * q^97 + 62 * q^98 - 62 * q^99

Introduction

L-functions

Modular forms

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Database

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