LMFDB - Embedded newform 7225.2.a.bw.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7225,2,Mod(1,7225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7225, 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("7225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7225 = 5^{2} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7225.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:	$57.6919154604$
Analytic rank:	$0$
Dimension:	$15$
Coefficient field:	$\mathbb{Q}[x]/(x^{15} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{15} - 21 x^{13} - 2 x^{12} + 171 x^{11} + 30 x^{10} - 678 x^{9} - 153 x^{8} + 1350 x^{7} + 301 x^{6} + \cdots + 3 $ x^15 - 21x^13 - 2x^12 + 171x^11 + 30x^10 - 678x^9 - 153x^8 + 1350x^7 + 301x^6 - 1284x^5 - 183x^4 + 555x^3 + 18x^2 - 81*x + 3
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Root			$2.35122$ of defining polynomial
Character	$\chi$	$=$	7225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.35122 q^{2} -0.229247 q^{3} +3.52823 q^{4} -0.539011 q^{6} +2.06647 q^{7} +3.59321 q^{8} -2.94745 q^{9} +O(q^{10})$	\(q+2.35122 q^{2} -0.229247 q^{3} +3.52823 q^{4} -0.539011 q^{6} +2.06647 q^{7} +3.59321 q^{8} -2.94745 q^{9} -0.635950 q^{11} -0.808838 q^{12} +0.375698 q^{13} +4.85872 q^{14} +1.39195 q^{16} -6.93009 q^{18} +7.47683 q^{19} -0.473733 q^{21} -1.49526 q^{22} +8.34851 q^{23} -0.823733 q^{24} +0.883348 q^{26} +1.36344 q^{27} +7.29098 q^{28} +3.44036 q^{29} +6.73955 q^{31} -3.91362 q^{32} +0.145790 q^{33} -10.3993 q^{36} +4.01146 q^{37} +17.5797 q^{38} -0.0861277 q^{39} -10.9798 q^{41} -1.11385 q^{42} -9.36886 q^{43} -2.24378 q^{44} +19.6292 q^{46} -7.98824 q^{47} -0.319102 q^{48} -2.72971 q^{49} +1.32555 q^{52} +2.38051 q^{53} +3.20574 q^{54} +7.42525 q^{56} -1.71404 q^{57} +8.08904 q^{58} +0.380361 q^{59} +7.77402 q^{61} +15.8462 q^{62} -6.09080 q^{63} -11.9857 q^{64} +0.342784 q^{66} +6.63299 q^{67} -1.91387 q^{69} -0.277783 q^{71} -10.5908 q^{72} +9.98262 q^{73} +9.43183 q^{74} +26.3800 q^{76} -1.31417 q^{77} -0.202505 q^{78} +12.5539 q^{79} +8.52977 q^{81} -25.8159 q^{82} +9.85110 q^{83} -1.67144 q^{84} -22.0282 q^{86} -0.788693 q^{87} -2.28510 q^{88} +14.3107 q^{89} +0.776368 q^{91} +29.4555 q^{92} -1.54503 q^{93} -18.7821 q^{94} +0.897188 q^{96} -13.7188 q^{97} -6.41814 q^{98} +1.87443 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 15 q + 9 q^{3} + 12 q^{4} + 9 q^{6} + 12 q^{7} + 6 q^{8} + 12 q^{9}+O(q^{10}) $ 15 * q + 9 * q^3 + 12 * q^4 + 9 * q^6 + 12 * q^7 + 6 * q^8 + 12 * q^9	$ 15 q + 9 q^{3} + 12 q^{4} + 9 q^{6} + 12 q^{7} + 6 q^{8} + 12 q^{9} + 6 q^{11} + 24 q^{12} + 6 q^{16} + 12 q^{18} + 6 q^{19} + 30 q^{21} + 12 q^{22} + 36 q^{23} + 18 q^{24} + 36 q^{26} + 36 q^{27} + 24 q^{28} - 18 q^{29} + 12 q^{32} + 12 q^{33} - 9 q^{36} + 12 q^{37} - 6 q^{38} + 9 q^{39} - 18 q^{41} + 36 q^{42} - 3 q^{43} - 12 q^{44} + 21 q^{46} - 3 q^{47} - 12 q^{48} + 15 q^{49} - 27 q^{52} + 21 q^{54} - 6 q^{56} + 39 q^{57} + 18 q^{58} - 12 q^{59} - 15 q^{61} + 54 q^{62} + 60 q^{63} - 36 q^{64} + 18 q^{66} - 24 q^{67} + 42 q^{69} + 6 q^{71} + 66 q^{72} - 9 q^{73} - 36 q^{74} - 18 q^{76} + 30 q^{77} + 30 q^{78} - 9 q^{79} + 51 q^{81} - 36 q^{82} + 15 q^{83} + 9 q^{84} - 36 q^{86} - 51 q^{87} + 30 q^{88} - 24 q^{89} + 27 q^{91} + 15 q^{92} - 42 q^{93} - 57 q^{94} + 42 q^{96} + 48 q^{97} + 12 q^{99}+O(q^{100}) $ 15 * q + 9 * q^3 + 12 * q^4 + 9 * q^6 + 12 * q^7 + 6 * q^8 + 12 * q^9 + 6 * q^11 + 24 * q^12 + 6 * q^16 + 12 * q^18 + 6 * q^19 + 30 * q^21 + 12 * q^22 + 36 * q^23 + 18 * q^24 + 36 * q^26 + 36 * q^27 + 24 * q^28 - 18 * q^29 + 12 * q^32 + 12 * q^33 - 9 * q^36 + 12 * q^37 - 6 * q^38 + 9 * q^39 - 18 * q^41 + 36 * q^42 - 3 * q^43 - 12 * q^44 + 21 * q^46 - 3 * q^47 - 12 * q^48 + 15 * q^49 - 27 * q^52 + 21 * q^54 - 6 * q^56 + 39 * q^57 + 18 * q^58 - 12 * q^59 - 15 * q^61 + 54 * q^62 + 60 * q^63 - 36 * q^64 + 18 * q^66 - 24 * q^67 + 42 * q^69 + 6 * q^71 + 66 * q^72 - 9 * q^73 - 36 * q^74 - 18 * q^76 + 30 * q^77 + 30 * q^78 - 9 * q^79 + 51 * q^81 - 36 * q^82 + 15 * q^83 + 9 * q^84 - 36 * q^86 - 51 * q^87 + 30 * q^88 - 24 * q^89 + 27 * q^91 + 15 * q^92 - 42 * q^93 - 57 * q^94 + 42 * q^96 + 48 * q^97 + 12 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	2.35122	1.66256	0.831282	−	0.555852i	$-0.187608\pi$
$2$	2.35122	1.66256	0.831282	+	0.555852i	$0.187608\pi$
$3$	−0.229247	−0.132356	−0.0661780	−	0.997808i	$-0.521081\pi$
$3$	−0.229247	−0.132356	−0.0661780	+	0.997808i	$0.521081\pi$
$4$	3.52823	1.76412
$5$	0	0
$5$	0	0
$6$	−0.539011	−0.220050
$7$	2.06647	0.781052	0.390526	−	0.920592i	$-0.372293\pi$
$7$	2.06647	0.781052	0.390526	+	0.920592i	$0.372293\pi$
$8$	3.59321	1.27039
$9$	−2.94745	−0.982482
$10$	0	0
$11$	−0.635950	−0.191746	−0.0958731	−	0.995394i	$-0.530564\pi$
$11$	−0.635950	−0.191746	−0.0958731	+	0.995394i	$0.530564\pi$
$12$	−0.808838	−0.233491
$13$	0.375698	0.104200	0.0520999	−	0.998642i	$-0.483409\pi$
$13$	0.375698	0.104200	0.0520999	+	0.998642i	$0.483409\pi$
$14$	4.85872	1.29855
$15$	0	0
$16$	1.39195	0.347989
$17$	0	0
$17$	0	0
$18$	−6.93009	−1.63344
$19$	7.47683	1.71530	0.857651	−	0.514233i	$-0.171923\pi$
$19$	7.47683	1.71530	0.857651	+	0.514233i	$0.171923\pi$
$20$	0	0
$21$	−0.473733	−0.103377
$22$	−1.49526	−0.318790
$23$	8.34851	1.74078	0.870392	−	0.492360i	$-0.163866\pi$
$23$	8.34851	1.74078	0.870392	+	0.492360i	$0.163866\pi$
$24$	−0.823733	−0.168144
$25$	0	0
$26$	0.883348	0.173239
$27$	1.36344	0.262393
$28$	7.29098	1.37787
$29$	3.44036	0.638858	0.319429	−	0.947610i	$-0.396509\pi$
$29$	3.44036	0.638858	0.319429	+	0.947610i	$0.396509\pi$
$30$	0	0
$31$	6.73955	1.21046	0.605230	−	0.796051i	$-0.293081\pi$
$31$	6.73955	1.21046	0.605230	+	0.796051i	$0.293081\pi$
$32$	−3.91362	−0.691837
$33$	0.145790	0.0253788
$34$	0	0
$35$	0	0
$36$	−10.3993	−1.73321
$37$	4.01146	0.659481	0.329740	−	0.944072i	$-0.393039\pi$
$37$	4.01146	0.659481	0.329740	+	0.944072i	$0.393039\pi$
$38$	17.5797	2.85180
$39$	−0.0861277	−0.0137915
$40$	0	0
$41$	−10.9798	−1.71475	−0.857377	−	0.514689i	$-0.827908\pi$
$41$	−10.9798	−1.71475	−0.857377	+	0.514689i	$0.827908\pi$
$42$	−1.11385	−0.171871
$43$	−9.36886	−1.42874	−0.714369	−	0.699770i	$-0.753286\pi$
$43$	−9.36886	−1.42874	−0.714369	+	0.699770i	$0.753286\pi$
$44$	−2.24378	−0.338263
$45$	0	0
$46$	19.6292	2.89416
$47$	−7.98824	−1.16520	−0.582602	−	0.812757i	$-0.697966\pi$
$47$	−7.98824	−1.16520	−0.582602	+	0.812757i	$0.697966\pi$
$48$	−0.319102	−0.0460584
$49$	−2.72971	−0.389958
$50$	0	0
$51$	0	0
$52$	1.32555	0.183821
$53$	2.38051	0.326988	0.163494	−	0.986544i	$-0.447724\pi$
$53$	2.38051	0.326988	0.163494	+	0.986544i	$0.447724\pi$
$54$	3.20574	0.436246
$55$	0	0
$56$	7.42525	0.992241
$57$	−1.71404	−0.227031
$58$	8.08904	1.06214
$59$	0.380361	0.0495188	0.0247594	−	0.999693i	$-0.492118\pi$
$59$	0.380361	0.0495188	0.0247594	+	0.999693i	$0.492118\pi$
$60$	0	0
$61$	7.77402	0.995361	0.497680	−	0.867360i	$-0.334185\pi$
$61$	7.77402	0.995361	0.497680	+	0.867360i	$0.334185\pi$
$62$	15.8462	2.01247
$63$	−6.09080	−0.767369
$64$	−11.9857	−1.49821
$65$	0	0
$66$	0.342784	0.0421938
$67$	6.63299	0.810349	0.405175	−	0.914239i	$-0.367211\pi$
$67$	6.63299	0.810349	0.405175	+	0.914239i	$0.367211\pi$
$68$	0	0
$69$	−1.91387	−0.230403
$70$	0	0
$71$	−0.277783	−0.0329668	−0.0164834	−	0.999864i	$-0.505247\pi$
$71$	−0.277783	−0.0329668	−0.0164834	+	0.999864i	$0.505247\pi$
$72$	−10.5908	−1.24814
$73$	9.98262	1.16838	0.584189	−	0.811618i	$-0.301413\pi$
$73$	9.98262	1.16838	0.584189	+	0.811618i	$0.301413\pi$
$74$	9.43183	1.09643
$75$	0	0
$76$	26.3800	3.02599
$77$	−1.31417	−0.149764
$78$	−0.202505	−0.0229292
$79$	12.5539	1.41243	0.706214	−	0.707999i	$-0.250402\pi$
$79$	12.5539	1.41243	0.706214	+	0.707999i	$0.250402\pi$
$80$	0	0
$81$	8.52977	0.947753
$82$	−25.8159	−2.85089
$83$	9.85110	1.08130	0.540649	−	0.841248i	$-0.318179\pi$
$83$	9.85110	1.08130	0.540649	+	0.841248i	$0.318179\pi$
$84$	−1.67144	−0.182369
$85$	0	0
$86$	−22.0282	−2.37537
$87$	−0.788693	−0.0845568
$88$	−2.28510	−0.243593
$89$	14.3107	1.51694	0.758468	−	0.651710i	$-0.225948\pi$
$89$	14.3107	1.51694	0.758468	+	0.651710i	$0.225948\pi$
$90$	0	0
$91$	0.776368	0.0813854
$92$	29.4555	3.07094
$93$	−1.54503	−0.160212
$94$	−18.7821	−1.93723
$95$	0	0
$96$	0.897188	0.0915689
$97$	−13.7188	−1.39294	−0.696468	−	0.717587i	$-0.745246\pi$
$97$	−13.7188	−1.39294	−0.696468	+	0.717587i	$0.745246\pi$
$98$	−6.41814	−0.648330
$99$	1.87443	0.188387
$100$	0	0
$101$	4.11139	0.409099	0.204549	−	0.978856i	$-0.434427\pi$
$101$	4.11139	0.409099	0.204549	+	0.978856i	$0.434427\pi$
$102$	0	0
$103$	−3.82988	−0.377369	−0.188685	−	0.982038i	$-0.560422\pi$
$103$	−3.82988	−0.377369	−0.188685	+	0.982038i	$0.560422\pi$
$104$	1.34996	0.132374
$105$	0	0
$106$	5.59710	0.543638
$107$	−8.41769	−0.813769	−0.406884	−	0.913480i	$-0.633385\pi$
$107$	−8.41769	−0.813769	−0.406884	+	0.913480i	$0.633385\pi$
$108$	4.81052	0.462892
$109$	7.12187	0.682151	0.341076	−	0.940036i	$-0.389209\pi$
$109$	7.12187	0.682151	0.341076	+	0.940036i	$0.389209\pi$
$110$	0	0
$111$	−0.919618	−0.0872863
$112$	2.87643	0.271797
$113$	14.7882	1.39115	0.695577	−	0.718451i	$-0.255149\pi$
$113$	14.7882	1.39115	0.695577	+	0.718451i	$0.255149\pi$
$114$	−4.03009	−0.377453
$115$	0	0
$116$	12.1384	1.12702
$117$	−1.10735	−0.102374
$118$	0.894312	0.0823281
$119$	0	0
$120$	0	0
$121$	−10.5956	−0.963233
$122$	18.2784	1.65485
$123$	2.51709	0.226958
$124$	23.7787	2.13539
$125$	0	0
$126$	−14.3208	−1.27580
$127$	−10.0838	−0.894788	−0.447394	−	0.894337i	$-0.647648\pi$
$127$	−10.0838	−0.894788	−0.447394	+	0.894337i	$0.647648\pi$
$128$	−20.3538	−1.79903
$129$	2.14779	0.189102
$130$	0	0
$131$	14.5028	1.26711	0.633557	−	0.773696i	$-0.281594\pi$
$131$	14.5028	1.26711	0.633557	+	0.773696i	$0.281594\pi$
$132$	0.514381	0.0447711
$133$	15.4506	1.33974
$134$	15.5956	1.34726
$135$	0	0
$136$	0	0
$137$	4.90880	0.419387	0.209693	−	0.977767i	$-0.432753\pi$
$137$	4.90880	0.419387	0.209693	+	0.977767i	$0.432753\pi$
$138$	−4.49993	−0.383060
$139$	−14.4426	−1.22501	−0.612505	−	0.790467i	$-0.709838\pi$
$139$	−14.4426	−1.22501	−0.612505	+	0.790467i	$0.709838\pi$
$140$	0	0
$141$	1.83128	0.154222
$142$	−0.653129	−0.0548094
$143$	−0.238925	−0.0199799
$144$	−4.10271	−0.341893
$145$	0	0
$146$	23.4713	1.94250
$147$	0.625778	0.0516133
$148$	14.1534	1.16340
$149$	20.8650	1.70932	0.854662	−	0.519184i	$-0.173764\pi$
$149$	20.8650	1.70932	0.854662	+	0.519184i	$0.173764\pi$
$150$	0	0
$151$	−7.91635	−0.644223	−0.322112	−	0.946702i	$-0.604393\pi$
$151$	−7.91635	−0.644223	−0.322112	+	0.946702i	$0.604393\pi$
$152$	26.8658	2.17910
$153$	0	0
$154$	−3.08991	−0.248992
$155$	0	0
$156$	−0.303879	−0.0243298
$157$	7.38265	0.589200	0.294600	−	0.955621i	$-0.404814\pi$
$157$	7.38265	0.589200	0.294600	+	0.955621i	$0.404814\pi$
$158$	29.5170	2.34825
$159$	−0.545725	−0.0432788
$160$	0	0
$161$	17.2519	1.35964
$162$	20.0554	1.57570
$163$	19.0360	1.49102	0.745509	−	0.666496i	$-0.232206\pi$
$163$	19.0360	1.49102	0.745509	+	0.666496i	$0.232206\pi$
$164$	−38.7392	−3.02503
$165$	0	0
$166$	23.1621	1.79773
$167$	−7.30310	−0.565131	−0.282565	−	0.959248i	$-0.591185\pi$
$167$	−7.30310	−0.565131	−0.282565	+	0.959248i	$0.591185\pi$
$168$	−1.70222	−0.131329
$169$	−12.8589	−0.989142
$170$	0	0
$171$	−22.0375	−1.68525
$172$	−33.0555	−2.52046
$173$	19.1389	1.45510	0.727551	−	0.686054i	$-0.240659\pi$
$173$	19.1389	1.45510	0.727551	+	0.686054i	$0.240659\pi$
$174$	−1.85439	−0.140581
$175$	0	0
$176$	−0.885214	−0.0667255
$177$	−0.0871967	−0.00655411
$178$	33.6477	2.52200
$179$	2.05656	0.153714	0.0768572	−	0.997042i	$-0.475511\pi$
$179$	2.05656	0.153714	0.0768572	+	0.997042i	$0.475511\pi$
$180$	0	0
$181$	−1.23661	−0.0919163	−0.0459582	−	0.998943i	$-0.514634\pi$
$181$	−1.23661	−0.0919163	−0.0459582	+	0.998943i	$0.514634\pi$
$182$	1.82541	0.135308
$183$	−1.78217	−0.131742
$184$	29.9979	2.21148
$185$	0	0
$186$	−3.63269	−0.266362
$187$	0	0
$188$	−28.1844	−2.05556
$189$	2.81750	0.204943
$190$	0	0
$191$	12.3402	0.892907	0.446454	−	0.894807i	$-0.352687\pi$
$191$	12.3402	0.892907	0.446454	+	0.894807i	$0.352687\pi$
$192$	2.74769	0.198297
$193$	7.85031	0.565078	0.282539	−	0.959256i	$-0.408823\pi$
$193$	7.85031	0.565078	0.282539	+	0.959256i	$0.408823\pi$
$194$	−32.2560	−2.31585
$195$	0	0
$196$	−9.63104	−0.687931
$197$	4.26593	0.303935	0.151967	−	0.988385i	$-0.451439\pi$
$197$	4.26593	0.303935	0.151967	+	0.988385i	$0.451439\pi$
$198$	4.40719	0.313206
$199$	−6.62470	−0.469613	−0.234806	−	0.972042i	$-0.575446\pi$
$199$	−6.62470	−0.469613	−0.234806	+	0.972042i	$0.575446\pi$
$200$	0	0
$201$	−1.52060	−0.107255
$202$	9.66678	0.680152
$203$	7.10939	0.498981
$204$	0	0
$205$	0	0
$206$	−9.00488	−0.627400
$207$	−24.6068	−1.71029
$208$	0.522954	0.0362604
$209$	−4.75489	−0.328903
$210$	0	0
$211$	9.02298	0.621168	0.310584	−	0.950546i	$-0.399475\pi$
$211$	9.02298	0.621168	0.310584	+	0.950546i	$0.399475\pi$
$212$	8.39898	0.576845
$213$	0.0636810	0.00436335
$214$	−19.7918	−1.35294
$215$	0	0
$216$	4.89911	0.333342
$217$	13.9271	0.945432
$218$	16.7451	1.13412
$219$	−2.28849	−0.154642
$220$	0	0
$221$	0	0
$222$	−2.16222	−0.145119
$223$	−12.0482	−0.806805	−0.403402	−	0.915023i	$-0.632172\pi$
$223$	−12.0482	−0.806805	−0.403402	+	0.915023i	$0.632172\pi$
$224$	−8.08738	−0.540361
$225$	0	0
$226$	34.7702	2.31288
$227$	−0.129694	−0.00860812	−0.00430406	−	0.999991i	$-0.501370\pi$
$227$	−0.129694	−0.00860812	−0.00430406	+	0.999991i	$0.501370\pi$
$228$	−6.04754	−0.400508
$229$	−4.77766	−0.315716	−0.157858	−	0.987462i	$-0.550459\pi$
$229$	−4.77766	−0.315716	−0.157858	+	0.987462i	$0.550459\pi$
$230$	0	0
$231$	0.301270	0.0198221
$232$	12.3619	0.811600
$233$	21.7981	1.42804	0.714022	−	0.700123i	$-0.246872\pi$
$233$	21.7981	1.42804	0.714022	+	0.700123i	$0.246872\pi$
$234$	−2.60362	−0.170204
$235$	0	0
$236$	1.34200	0.0873568
$237$	−2.87796	−0.186943
$238$	0	0
$239$	−23.2755	−1.50556	−0.752782	−	0.658270i	$-0.771289\pi$
$239$	−23.2755	−1.50556	−0.752782	+	0.658270i	$0.771289\pi$
$240$	0	0
$241$	−19.3802	−1.24839	−0.624194	−	0.781269i	$-0.714573\pi$
$241$	−19.3802	−1.24839	−0.624194	+	0.781269i	$0.714573\pi$
$242$	−24.9125	−1.60144
$243$	−6.04574	−0.387834
$244$	27.4285	1.75593
$245$	0	0
$246$	5.91822	0.377332
$247$	2.80903	0.178734
$248$	24.2166	1.53776
$249$	−2.25834	−0.143116
$250$	0	0
$251$	−15.9579	−1.00726	−0.503628	−	0.863920i	$-0.668002\pi$
$251$	−15.9579	−1.00726	−0.503628	+	0.863920i	$0.668002\pi$
$252$	−21.4898	−1.35373
$253$	−5.30923	−0.333789
$254$	−23.7091	−1.48764
$255$	0	0
$256$	−23.8847	−1.49280
$257$	11.4165	0.712144	0.356072	−	0.934458i	$-0.384116\pi$
$257$	11.4165	0.712144	0.356072	+	0.934458i	$0.384116\pi$
$258$	5.04992	0.314394
$259$	8.28957	0.515089
$260$	0	0
$261$	−10.1403	−0.627667
$262$	34.0992	2.10666
$263$	7.43718	0.458596	0.229298	−	0.973356i	$-0.426357\pi$
$263$	7.43718	0.458596	0.229298	+	0.973356i	$0.426357\pi$
$264$	0.523854	0.0322410
$265$	0	0
$266$	36.3278	2.22740
$267$	−3.28070	−0.200776
$268$	23.4027	1.42955
$269$	−8.95192	−0.545808	−0.272904	−	0.962041i	$-0.587984\pi$
$269$	−8.95192	−0.545808	−0.272904	+	0.962041i	$0.587984\pi$
$270$	0	0
$271$	−15.1894	−0.922693	−0.461347	−	0.887220i	$-0.652634\pi$
$271$	−15.1894	−0.922693	−0.461347	+	0.887220i	$0.652634\pi$
$272$	0	0
$273$	−0.177980	−0.0107719
$274$	11.5417	0.697257
$275$	0	0
$276$	−6.75259	−0.406458
$277$	−23.7615	−1.42769	−0.713845	−	0.700304i	$-0.753048\pi$
$277$	−23.7615	−1.42769	−0.713845	+	0.700304i	$0.753048\pi$
$278$	−33.9578	−2.03666
$279$	−19.8645	−1.18925
$280$	0	0
$281$	−17.9144	−1.06868	−0.534341	−	0.845269i	$-0.679440\pi$
$281$	−17.9144	−1.06868	−0.534341	+	0.845269i	$0.679440\pi$
$282$	4.30575	0.256404
$283$	−32.1921	−1.91362	−0.956811	−	0.290709i	$-0.906109\pi$
$283$	−32.1921	−1.91362	−0.956811	+	0.290709i	$0.906109\pi$
$284$	−0.980083	−0.0581572
$285$	0	0
$286$	−0.561765	−0.0332179
$287$	−22.6894	−1.33931
$288$	11.5352	0.679718
$289$	0	0
$290$	0	0
$291$	3.14501	0.184364
$292$	35.2210	2.06115
$293$	−27.0776	−1.58189	−0.790945	−	0.611887i	$-0.790411\pi$
$293$	−27.0776	−1.58189	−0.790945	+	0.611887i	$0.790411\pi$
$294$	1.47134	0.0858104
$295$	0	0
$296$	14.4140	0.837798
$297$	−0.867078	−0.0503130
$298$	49.0581	2.84186
$299$	3.13651	0.181389
$300$	0	0
$301$	−19.3604	−1.11592
$302$	−18.6131	−1.07106
$303$	−0.942525	−0.0541467
$304$	10.4074	0.596906
$305$	0	0
$306$	0	0
$307$	−4.48828	−0.256160	−0.128080	−	0.991764i	$-0.540881\pi$
$307$	−4.48828	−0.256160	−0.128080	+	0.991764i	$0.540881\pi$
$308$	−4.63670	−0.264201
$309$	0.877990	0.0499471
$310$	0	0
$311$	−1.12576	−0.0638361	−0.0319181	−	0.999490i	$-0.510162\pi$
$311$	−1.12576	−0.0638361	−0.0319181	+	0.999490i	$0.510162\pi$
$312$	−0.309475	−0.0175206
$313$	−16.1960	−0.915451	−0.457726	−	0.889093i	$-0.651336\pi$
$313$	−16.1960	−0.915451	−0.457726	+	0.889093i	$0.651336\pi$
$314$	17.3582	0.979582
$315$	0	0
$316$	44.2932	2.49169
$317$	23.3747	1.31285	0.656427	−	0.754389i	$-0.272067\pi$
$317$	23.3747	1.31285	0.656427	+	0.754389i	$0.272067\pi$
$318$	−1.28312	−0.0719538
$319$	−2.18790	−0.122499
$320$	0	0
$321$	1.92973	0.107707
$322$	40.5631	2.26049
$323$	0	0
$324$	30.0950	1.67195
$325$	0	0
$326$	44.7579	2.47891
$327$	−1.63267	−0.0902868
$328$	−39.4526	−2.17841
$329$	−16.5075	−0.910085
$330$	0	0
$331$	10.1328	0.556949	0.278475	−	0.960444i	$-0.410171\pi$
$331$	10.1328	0.556949	0.278475	+	0.960444i	$0.410171\pi$
$332$	34.7570	1.90754
$333$	−11.8236	−0.647928
$334$	−17.1712	−0.939565
$335$	0	0
$336$	−0.659414	−0.0359740
$337$	−28.1459	−1.53320	−0.766602	−	0.642123i	$-0.778054\pi$
$337$	−28.1459	−1.53320	−0.766602	+	0.642123i	$0.778054\pi$
$338$	−30.2340	−1.64451
$339$	−3.39015	−0.184128
$340$	0	0
$341$	−4.28602	−0.232101
$342$	−51.8151	−2.80184
$343$	−20.1061	−1.08563
$344$	−33.6642	−1.81505
$345$	0	0
$346$	44.9997	2.41920
$347$	−8.56927	−0.460022	−0.230011	−	0.973188i	$-0.573876\pi$
$347$	−8.56927	−0.460022	−0.230011	+	0.973188i	$0.573876\pi$
$348$	−2.78269	−0.149168
$349$	2.37634	0.127202	0.0636012	−	0.997975i	$-0.479741\pi$
$349$	2.37634	0.127202	0.0636012	+	0.997975i	$0.479741\pi$
$350$	0	0
$351$	0.512240	0.0273413
$352$	2.48887	0.132657
$353$	10.7190	0.570515	0.285257	−	0.958451i	$-0.407921\pi$
$353$	10.7190	0.570515	0.285257	+	0.958451i	$0.407921\pi$
$354$	−0.205019	−0.0108966
$355$	0	0
$356$	50.4916	2.67605
$357$	0	0
$358$	4.83542	0.255560
$359$	−6.44127	−0.339957	−0.169979	−	0.985448i	$-0.554370\pi$
$359$	−6.44127	−0.339957	−0.169979	+	0.985448i	$0.554370\pi$
$360$	0	0
$361$	36.9029	1.94226
$362$	−2.90754	−0.152817
$363$	2.42901	0.127490
$364$	2.73920	0.143573
$365$	0	0
$366$	−4.19028	−0.219029
$367$	9.51689	0.496777	0.248389	−	0.968660i	$-0.420099\pi$
$367$	9.51689	0.496777	0.248389	+	0.968660i	$0.420099\pi$
$368$	11.6207	0.605773
$369$	32.3623	1.68472
$370$	0	0
$371$	4.91924	0.255394
$372$	−5.45121	−0.282632
$373$	−9.80017	−0.507433	−0.253717	−	0.967279i	$-0.581653\pi$
$373$	−9.80017	−0.507433	−0.253717	+	0.967279i	$0.581653\pi$
$374$	0	0
$375$	0	0
$376$	−28.7034	−1.48027
$377$	1.29253	0.0665689
$378$	6.62456	0.340730
$379$	−34.0357	−1.74829	−0.874147	−	0.485661i	$-0.838579\pi$
$379$	−34.0357	−1.74829	−0.874147	+	0.485661i	$0.838579\pi$
$380$	0	0
$381$	2.31167	0.118431
$382$	29.0146	1.48451
$383$	4.89130	0.249934	0.124967	−	0.992161i	$-0.460118\pi$
$383$	4.89130	0.249934	0.124967	+	0.992161i	$0.460118\pi$
$384$	4.66604	0.238113
$385$	0	0
$386$	18.4578	0.939477
$387$	27.6142	1.40371
$388$	−48.4032	−2.45730
$389$	−30.5964	−1.55130	−0.775649	−	0.631164i	$-0.782577\pi$
$389$	−30.5964	−1.55130	−0.775649	+	0.631164i	$0.782577\pi$
$390$	0	0
$391$	0	0
$392$	−9.80840	−0.495399
$393$	−3.32472	−0.167710
$394$	10.0301	0.505311
$395$	0	0
$396$	6.61342	0.332337
$397$	−25.4262	−1.27610	−0.638052	−	0.769993i	$-0.720260\pi$
$397$	−25.4262	−1.27610	−0.638052	+	0.769993i	$0.720260\pi$
$398$	−15.5761	−0.780761
$399$	−3.54202	−0.177323
$400$	0	0
$401$	−27.6104	−1.37880	−0.689399	−	0.724382i	$-0.742125\pi$
$401$	−27.6104	−1.37880	−0.689399	+	0.724382i	$0.742125\pi$
$402$	−3.57526	−0.178318
$403$	2.53204	0.126130
$404$	14.5059	0.721697
$405$	0	0
$406$	16.7157	0.829588
$407$	−2.55109	−0.126453
$408$	0	0
$409$	29.6761	1.46739	0.733695	−	0.679479i	$-0.237794\pi$
$409$	29.6761	1.46739	0.733695	+	0.679479i	$0.237794\pi$
$410$	0	0
$411$	−1.12533	−0.0555084
$412$	−13.5127	−0.665723
$413$	0.786004	0.0386767
$414$	−57.8559	−2.84346
$415$	0	0
$416$	−1.47034	−0.0720893
$417$	3.31094	0.162137
$418$	−11.1798	−0.546821
$419$	4.43212	0.216523	0.108262	−	0.994122i	$-0.465472\pi$
$419$	4.43212	0.216523	0.108262	+	0.994122i	$0.465472\pi$
$420$	0	0
$421$	−12.2011	−0.594643	−0.297322	−	0.954777i	$-0.596093\pi$
$421$	−12.2011	−0.594643	−0.297322	+	0.954777i	$0.596093\pi$
$422$	21.2150	1.03273
$423$	23.5449	1.14479
$424$	8.55366	0.415402
$425$	0	0
$426$	0.149728	0.00725435
$427$	16.0648	0.777428
$428$	−29.6996	−1.43558
$429$	0.0547730	0.00264446
$430$	0	0
$431$	−26.5221	−1.27752	−0.638762	−	0.769404i	$-0.720553\pi$
$431$	−26.5221	−1.27752	−0.638762	+	0.769404i	$0.720553\pi$
$432$	1.89784	0.0913100
$433$	−15.0280	−0.722201	−0.361100	−	0.932527i	$-0.617599\pi$
$433$	−15.0280	−0.722201	−0.361100	+	0.932527i	$0.617599\pi$
$434$	32.7456	1.57184
$435$	0	0
$436$	25.1276	1.20339
$437$	62.4203	2.98597
$438$	−5.38074	−0.257102
$439$	−4.32940	−0.206631	−0.103316	−	0.994649i	$-0.532945\pi$
$439$	−4.32940	−0.206631	−0.103316	+	0.994649i	$0.532945\pi$
$440$	0	0
$441$	8.04566	0.383127
$442$	0	0
$443$	12.8081	0.608532	0.304266	−	0.952587i	$-0.401589\pi$
$443$	12.8081	0.608532	0.304266	+	0.952587i	$0.401589\pi$
$444$	−3.24462	−0.153983
$445$	0	0
$446$	−28.3279	−1.34136
$447$	−4.78324	−0.226239
$448$	−24.7681	−1.17018
$449$	−20.2566	−0.955966	−0.477983	−	0.878369i	$-0.658632\pi$
$449$	−20.2566	−0.955966	−0.477983	+	0.878369i	$0.658632\pi$
$450$	0	0
$451$	6.98260	0.328798
$452$	52.1761	2.45416
$453$	1.81480	0.0852669
$454$	−0.304940	−0.0143115
$455$	0	0
$456$	−6.15891	−0.288417
$457$	10.8130	0.505811	0.252905	−	0.967491i	$-0.418614\pi$
$457$	10.8130	0.505811	0.252905	+	0.967491i	$0.418614\pi$
$458$	−11.2333	−0.524898
$459$	0	0
$460$	0	0
$461$	31.7255	1.47761	0.738803	−	0.673922i	$-0.235392\pi$
$461$	31.7255	1.47761	0.738803	+	0.673922i	$0.235392\pi$
$462$	0.708353	0.0329556
$463$	−38.9719	−1.81118	−0.905589	−	0.424156i	$-0.860571\pi$
$463$	−38.9719	−1.81118	−0.905589	+	0.424156i	$0.860571\pi$
$464$	4.78882	0.222316
$465$	0	0
$466$	51.2522	2.37421
$467$	23.4144	1.08349	0.541745	−	0.840543i	$-0.317764\pi$
$467$	23.4144	1.08349	0.541745	+	0.840543i	$0.317764\pi$
$468$	−3.90698	−0.180600
$469$	13.7069	0.632925
$470$	0	0
$471$	−1.69245	−0.0779841
$472$	1.36672	0.0629082
$473$	5.95813	0.273955
$474$	−6.76670	−0.310805
$475$	0	0
$476$	0	0
$477$	−7.01642	−0.321260
$478$	−54.7257	−2.50309
$479$	−29.6684	−1.35558	−0.677792	−	0.735254i	$-0.737063\pi$
$479$	−29.6684	−1.35558	−0.677792	+	0.735254i	$0.737063\pi$
$480$	0	0
$481$	1.50710	0.0687178
$482$	−45.5671	−2.07552
$483$	−3.95496	−0.179957
$484$	−37.3836	−1.69926
$485$	0	0
$486$	−14.2149	−0.644799
$487$	32.7621	1.48459	0.742296	−	0.670072i	$-0.233737\pi$
$487$	32.7621	1.48459	0.742296	+	0.670072i	$0.233737\pi$
$488$	27.9337	1.26450
$489$	−4.36396	−0.197345
$490$	0	0
$491$	−23.4003	−1.05604	−0.528021	−	0.849231i	$-0.677066\pi$
$491$	−23.4003	−1.05604	−0.528021	+	0.849231i	$0.677066\pi$
$492$	8.88087	0.400380
$493$	0	0
$494$	6.60464	0.297157
$495$	0	0
$496$	9.38116	0.421226
$497$	−0.574030	−0.0257488
$498$	−5.30985	−0.237940
$499$	−8.73127	−0.390865	−0.195433	−	0.980717i	$-0.562611\pi$
$499$	−8.73127	−0.390865	−0.195433	+	0.980717i	$0.562611\pi$
$500$	0	0
$501$	1.67422	0.0747985
$502$	−37.5206	−1.67463
$503$	9.08227	0.404958	0.202479	−	0.979287i	$-0.435100\pi$
$503$	9.08227	0.404958	0.202479	+	0.979287i	$0.435100\pi$
$504$	−21.8855	−0.974859
$505$	0	0
$506$	−12.4832	−0.554945
$507$	2.94786	0.130919
$508$	−35.5778	−1.57851
$509$	−9.81418	−0.435006	−0.217503	−	0.976060i	$-0.569791\pi$
$509$	−9.81418	−0.435006	−0.217503	+	0.976060i	$0.569791\pi$
$510$	0	0
$511$	20.6288	0.912563
$512$	−15.4508	−0.682833
$513$	10.1942	0.450084
$514$	26.8428	1.18398
$515$	0	0
$516$	7.57789	0.333598
$517$	5.08013	0.223424
$518$	19.4906	0.856367
$519$	−4.38754	−0.192591
$520$	0	0
$521$	8.94433	0.391858	0.195929	−	0.980618i	$-0.437228\pi$
$521$	8.94433	0.391858	0.195929	+	0.980618i	$0.437228\pi$
$522$	−23.8420	−1.04354
$523$	36.8430	1.61103	0.805516	−	0.592573i	$-0.201888\pi$
$523$	36.8430	1.61103	0.805516	+	0.592573i	$0.201888\pi$
$524$	51.1691	2.23533
$525$	0	0
$526$	17.4864	0.762445
$527$	0	0
$528$	0.202933	0.00883153
$529$	46.6975	2.03033
$530$	0	0
$531$	−1.12109	−0.0486513
$532$	54.5134	2.36346
$533$	−4.12508	−0.178677
$534$	−7.71365	−0.333802
$535$	0	0
$536$	23.8337	1.02946
$537$	−0.471461	−0.0203450
$538$	−21.0479	−0.907441
$539$	1.73596	0.0747730
$540$	0	0
$541$	−33.6263	−1.44571	−0.722854	−	0.691000i	$-0.757170\pi$
$541$	−33.6263	−1.44571	−0.722854	+	0.691000i	$0.757170\pi$
$542$	−35.7137	−1.53404
$543$	0.283489	0.0121657
$544$	0	0
$545$	0	0
$546$	−0.418471	−0.0179089
$547$	−18.1093	−0.774299	−0.387150	−	0.922017i	$-0.626540\pi$
$547$	−18.1093	−0.774299	−0.387150	+	0.922017i	$0.626540\pi$
$548$	17.3194	0.739847
$549$	−22.9135	−0.977924
$550$	0	0
$551$	25.7230	1.09583
$552$	−6.87694	−0.292702
$553$	25.9423	1.10318
$554$	−55.8685	−2.37362
$555$	0	0
$556$	−50.9570	−2.16106
$557$	−22.8689	−0.968988	−0.484494	−	0.874795i	$-0.660996\pi$
$557$	−22.8689	−0.968988	−0.484494	+	0.874795i	$0.660996\pi$
$558$	−46.7057	−1.97721
$559$	−3.51986	−0.148874
$560$	0	0
$561$	0	0
$562$	−42.1206	−1.77675
$563$	−23.9206	−1.00813	−0.504067	−	0.863665i	$-0.668164\pi$
$563$	−23.9206	−1.00813	−0.504067	+	0.863665i	$0.668164\pi$
$564$	6.46119	0.272065
$565$	0	0
$566$	−75.6907	−3.18152
$567$	17.6265	0.740244
$568$	−0.998132	−0.0418807
$569$	7.59240	0.318290	0.159145	−	0.987255i	$-0.449126\pi$
$569$	7.59240	0.318290	0.159145	+	0.987255i	$0.449126\pi$
$570$	0	0
$571$	−19.4529	−0.814078	−0.407039	−	0.913411i	$-0.633439\pi$
$571$	−19.4529	−0.814078	−0.407039	+	0.913411i	$0.633439\pi$
$572$	−0.842983	−0.0352469
$573$	−2.82896	−0.118182
$574$	−53.3477	−2.22669
$575$	0	0
$576$	35.3272	1.47197
$577$	−36.9886	−1.53986	−0.769928	−	0.638131i	$-0.779708\pi$
$577$	−36.9886	−1.53986	−0.769928	+	0.638131i	$0.779708\pi$
$578$	0	0
$579$	−1.79966	−0.0747915
$580$	0	0
$581$	20.3570	0.844550
$582$	7.39460	0.306516
$583$	−1.51388	−0.0626987
$584$	35.8696	1.48430
$585$	0	0
$586$	−63.6654	−2.62999
$587$	7.67355	0.316721	0.158361	−	0.987381i	$-0.449379\pi$
$587$	7.67355	0.316721	0.158361	+	0.987381i	$0.449379\pi$
$588$	2.20789	0.0910519
$589$	50.3905	2.07630
$590$	0	0
$591$	−0.977954	−0.0402276
$592$	5.58378	0.229492
$593$	33.1691	1.36209	0.681046	−	0.732241i	$-0.261525\pi$
$593$	33.1691	1.36209	0.681046	+	0.732241i	$0.261525\pi$
$594$	−2.03869	−0.0836485
$595$	0	0
$596$	73.6164	3.01545
$597$	1.51870	0.0621561
$598$	7.37463	0.301571
$599$	8.76149	0.357985	0.178992	−	0.983850i	$-0.442716\pi$
$599$	8.76149	0.357985	0.178992	+	0.983850i	$0.442716\pi$
$600$	0	0
$601$	20.6355	0.841741	0.420870	−	0.907121i	$-0.361725\pi$
$601$	20.6355	0.841741	0.420870	+	0.907121i	$0.361725\pi$
$602$	−45.5207	−1.85528
$603$	−19.5504	−0.796153
$604$	−27.9307	−1.13648
$605$	0	0
$606$	−2.21608	−0.0900223
$607$	−20.9037	−0.848455	−0.424228	−	0.905556i	$-0.639454\pi$
$607$	−20.9037	−0.848455	−0.424228	+	0.905556i	$0.639454\pi$
$608$	−29.2615	−1.18671
$609$	−1.62981	−0.0660432
$610$	0	0
$611$	−3.00116	−0.121414
$612$	0	0
$613$	23.1878	0.936545	0.468272	−	0.883584i	$-0.344877\pi$
$613$	23.1878	0.936545	0.468272	+	0.883584i	$0.344877\pi$
$614$	−10.5529	−0.425882
$615$	0	0
$616$	−4.72209	−0.190258
$617$	−11.8019	−0.475126	−0.237563	−	0.971372i	$-0.576349\pi$
$617$	−11.8019	−0.475126	−0.237563	+	0.971372i	$0.576349\pi$
$618$	2.06435	0.0830402
$619$	5.82234	0.234019	0.117010	−	0.993131i	$-0.462669\pi$
$619$	5.82234	0.234019	0.117010	+	0.993131i	$0.462669\pi$
$620$	0	0
$621$	11.3827	0.456770
$622$	−2.64691	−0.106132
$623$	29.5727	1.18481
$624$	−0.119886	−0.00479928
$625$	0	0
$626$	−38.0803	−1.52200
$627$	1.09005	0.0435322
$628$	26.0477	1.03942
$629$	0	0
$630$	0	0
$631$	−3.00463	−0.119613	−0.0598063	−	0.998210i	$-0.519048\pi$
$631$	−3.00463	−0.119613	−0.0598063	+	0.998210i	$0.519048\pi$
$632$	45.1089	1.79433
$633$	−2.06850	−0.0822153
$634$	54.9591	2.18270
$635$	0	0
$636$	−1.92544	−0.0763489
$637$	−1.02554	−0.0406336
$638$	−5.14423	−0.203662
$639$	0.818750	0.0323893
$640$	0	0
$641$	−2.75123	−0.108667	−0.0543335	−	0.998523i	$-0.517303\pi$
$641$	−2.75123	−0.108667	−0.0543335	+	0.998523i	$0.517303\pi$
$642$	4.53723	0.179070
$643$	17.5667	0.692763	0.346382	−	0.938094i	$-0.387410\pi$
$643$	17.5667	0.692763	0.346382	+	0.938094i	$0.387410\pi$
$644$	60.8688	2.39857
$645$	0	0
$646$	0	0
$647$	15.4224	0.606318	0.303159	−	0.952940i	$-0.401959\pi$
$647$	15.4224	0.606318	0.303159	+	0.952940i	$0.401959\pi$
$648$	30.6492	1.20402
$649$	−0.241891	−0.00949504
$650$	0	0
$651$	−3.19275	−0.125134
$652$	67.1636	2.63033
$653$	30.9554	1.21138	0.605689	−	0.795701i	$-0.292897\pi$
$653$	30.9554	1.21138	0.605689	+	0.795701i	$0.292897\pi$
$654$	−3.83876	−0.150108
$655$	0	0
$656$	−15.2834	−0.596715
$657$	−29.4232	−1.14791
$658$	−38.8126	−1.51307
$659$	37.3177	1.45369	0.726846	−	0.686801i	$-0.240986\pi$
$659$	37.3177	1.45369	0.726846	+	0.686801i	$0.240986\pi$
$660$	0	0
$661$	−27.6958	−1.07724	−0.538621	−	0.842548i	$-0.681054\pi$
$661$	−27.6958	−1.07724	−0.538621	+	0.842548i	$0.681054\pi$
$662$	23.8244	0.925963
$663$	0	0
$664$	35.3970	1.37367
$665$	0	0
$666$	−27.7998	−1.07722
$667$	28.7218	1.11211
$668$	−25.7670	−0.996956
$669$	2.76201	0.106785
$670$	0	0
$671$	−4.94389	−0.190857
$672$	1.85401	0.0715200
$673$	16.3874	0.631688	0.315844	−	0.948811i	$-0.397712\pi$
$673$	16.3874	0.631688	0.315844	+	0.948811i	$0.397712\pi$
$674$	−66.1771	−2.54905
$675$	0	0
$676$	−45.3690	−1.74496
$677$	34.9206	1.34211	0.671054	−	0.741408i	$-0.265842\pi$
$677$	34.9206	1.34211	0.671054	+	0.741408i	$0.265842\pi$
$678$	−7.97098	−0.306124
$679$	−28.3495	−1.08796
$680$	0	0
$681$	0.0297321	0.00113934
$682$	−10.0774	−0.385883
$683$	23.3673	0.894125	0.447062	−	0.894503i	$-0.352470\pi$
$683$	23.3673	0.894125	0.447062	+	0.894503i	$0.352470\pi$
$684$	−77.7535	−2.97298
$685$	0	0
$686$	−47.2739	−1.80493
$687$	1.09527	0.0417870
$688$	−13.0410	−0.497184
$689$	0.894351	0.0340721
$690$	0	0
$691$	9.90043	0.376630	0.188315	−	0.982109i	$-0.439697\pi$
$691$	9.90043	0.376630	0.188315	+	0.982109i	$0.439697\pi$
$692$	67.5264	2.56697
$693$	3.87345	0.147140
$694$	−20.1482	−0.764816
$695$	0	0
$696$	−2.83394	−0.107420
$697$	0	0
$698$	5.58729	0.211482
$699$	−4.99717	−0.189010
$700$	0	0
$701$	−50.7596	−1.91716	−0.958582	−	0.284815i	$-0.908068\pi$
$701$	−50.7596	−1.91716	−0.958582	+	0.284815i	$0.908068\pi$
$702$	1.20439	0.0454567
$703$	29.9930	1.13121
$704$	7.62231	0.287277
$705$	0	0
$706$	25.2027	0.948517
$707$	8.49606	0.319527
$708$	−0.307650	−0.0115622
$709$	−38.2672	−1.43716	−0.718578	−	0.695447i	$-0.755207\pi$
$709$	−38.2672	−1.43716	−0.718578	+	0.695447i	$0.755207\pi$
$710$	0	0
$711$	−37.0020	−1.38768
$712$	51.4215	1.92710
$713$	56.2652	2.10715
$714$	0	0
$715$	0	0
$716$	7.25602	0.271170
$717$	5.33584	0.199270
$718$	−15.1448	−0.565200
$719$	−52.7100	−1.96575	−0.982876	−	0.184270i	$-0.941008\pi$
$719$	−52.7100	−1.96575	−0.982876	+	0.184270i	$0.941008\pi$
$720$	0	0
$721$	−7.91432	−0.294745
$722$	86.7669	3.22913
$723$	4.44286	0.165232
$724$	−4.36304	−0.162151
$725$	0	0
$726$	5.71113	0.211960
$727$	23.1292	0.857816	0.428908	−	0.903348i	$-0.358898\pi$
$727$	23.1292	0.857816	0.428908	+	0.903348i	$0.358898\pi$
$728$	2.78965	0.103391
$729$	−24.2033	−0.896420
$730$	0	0
$731$	0	0
$732$	−6.28792	−0.232408
$733$	40.6341	1.50086	0.750428	−	0.660953i	$-0.229848\pi$
$733$	40.6341	1.50086	0.750428	+	0.660953i	$0.229848\pi$
$734$	22.3763	0.825924
$735$	0	0
$736$	−32.6729	−1.20434
$737$	−4.21825	−0.155381
$738$	76.0909	2.80095
$739$	−15.2983	−0.562757	−0.281378	−	0.959597i	$-0.590792\pi$
$739$	−15.2983	−0.562757	−0.281378	+	0.959597i	$0.590792\pi$
$740$	0	0
$741$	−0.643962	−0.0236565
$742$	11.5662	0.424609
$743$	−5.28603	−0.193926	−0.0969628	−	0.995288i	$-0.530913\pi$
$743$	−5.28603	−0.193926	−0.0969628	+	0.995288i	$0.530913\pi$
$744$	−5.55160	−0.203531
$745$	0	0
$746$	−23.0423	−0.843640
$747$	−29.0356	−1.06236
$748$	0	0
$749$	−17.3949	−0.635595
$750$	0	0
$751$	−0.142257	−0.00519104	−0.00259552	−	0.999997i	$-0.500826\pi$
$751$	−0.142257	−0.00519104	−0.00259552	+	0.999997i	$0.500826\pi$
$752$	−11.1193	−0.405478
$753$	3.65832	0.133316
$754$	3.03903	0.110675
$755$	0	0
$756$	9.94079	0.361543
$757$	−29.0510	−1.05588	−0.527938	−	0.849283i	$-0.677035\pi$
$757$	−29.0510	−1.05588	−0.527938	+	0.849283i	$0.677035\pi$
$758$	−80.0253	−2.90665
$759$	1.21713	0.0441790
$760$	0	0
$761$	30.8028	1.11660	0.558300	−	0.829639i	$-0.311454\pi$
$761$	30.8028	1.11660	0.558300	+	0.829639i	$0.311454\pi$
$762$	5.43525	0.196898
$763$	14.7171	0.532795
$764$	43.5392	1.57519
$765$	0	0
$766$	11.5005	0.415531
$767$	0.142901	0.00515984
$768$	5.47551	0.197581
$769$	27.0597	0.975798	0.487899	−	0.872900i	$-0.337764\pi$
$769$	27.0597	0.975798	0.487899	+	0.872900i	$0.337764\pi$
$770$	0	0
$771$	−2.61721	−0.0942566
$772$	27.6977	0.996863
$773$	−36.1757	−1.30115	−0.650574	−	0.759443i	$-0.725472\pi$
$773$	−36.1757	−1.30115	−0.650574	+	0.759443i	$0.725472\pi$
$774$	64.9270	2.33375
$775$	0	0
$776$	−49.2946	−1.76957
$777$	−1.90036	−0.0681751
$778$	−71.9388	−2.57913
$779$	−82.0939	−2.94132
$780$	0	0
$781$	0.176656	0.00632126
$782$	0	0
$783$	4.69071	0.167632
$784$	−3.79963	−0.135701
$785$	0	0
$786$	−7.81715	−0.278829
$787$	30.9429	1.10299	0.551497	−	0.834177i	$-0.314057\pi$
$787$	30.9429	1.10299	0.551497	+	0.834177i	$0.314057\pi$
$788$	15.0512	0.536177
$789$	−1.70495	−0.0606980
$790$	0	0
$791$	30.5593	1.08656
$792$	6.73521	0.239325
$793$	2.92068	0.103716
$794$	−59.7826	−2.12160
$795$	0	0
$796$	−23.3735	−0.828451
$797$	−6.80146	−0.240920	−0.120460	−	0.992718i	$-0.538437\pi$
$797$	−6.80146	−0.240920	−0.120460	+	0.992718i	$0.538437\pi$
$798$	−8.32806	−0.294810
$799$	0	0
$800$	0	0
$801$	−42.1801	−1.49036
$802$	−64.9181	−2.29234
$803$	−6.34845	−0.224032
$804$	−5.36502	−0.189210
$805$	0	0
$806$	5.95337	0.209699
$807$	2.05220	0.0722410
$808$	14.7731	0.519715
$809$	16.4527	0.578448	0.289224	−	0.957261i	$-0.406603\pi$
$809$	16.4527	0.578448	0.289224	+	0.957261i	$0.406603\pi$
$810$	0	0
$811$	−12.4637	−0.437659	−0.218830	−	0.975763i	$-0.570224\pi$
$811$	−12.4637	−0.437659	−0.218830	+	0.975763i	$0.570224\pi$
$812$	25.0836	0.880261
$813$	3.48214	0.122124
$814$	−5.99818	−0.210236
$815$	0	0
$816$	0	0
$817$	−70.0493	−2.45072
$818$	69.7750	2.43963
$819$	−2.28830	−0.0799597
$820$	0	0
$821$	−0.951303	−0.0332007	−0.0166004	−	0.999862i	$-0.505284\pi$
$821$	−0.951303	−0.0332007	−0.0166004	+	0.999862i	$0.505284\pi$
$822$	−2.64589	−0.0922862
$823$	30.8449	1.07518	0.537592	−	0.843205i	$-0.319334\pi$
$823$	30.8449	1.07518	0.537592	+	0.843205i	$0.319334\pi$
$824$	−13.7615	−0.479406
$825$	0	0
$826$	1.84807	0.0643025
$827$	−4.11063	−0.142941	−0.0714703	−	0.997443i	$-0.522769\pi$
$827$	−4.11063	−0.142941	−0.0714703	+	0.997443i	$0.522769\pi$
$828$	−86.8184	−3.01715
$829$	−8.33354	−0.289436	−0.144718	−	0.989473i	$-0.546227\pi$
$829$	−8.33354	−0.289436	−0.144718	+	0.989473i	$0.546227\pi$
$830$	0	0
$831$	5.44726	0.188963
$832$	−4.50300	−0.156113
$833$	0	0
$834$	7.78474	0.269564
$835$	0	0
$836$	−16.7764	−0.580222
$837$	9.18895	0.317617
$838$	10.4209	0.359984
$839$	34.8941	1.20468	0.602338	−	0.798241i	$-0.294236\pi$
$839$	34.8941	1.20468	0.602338	+	0.798241i	$0.294236\pi$
$840$	0	0
$841$	−17.1639	−0.591860
$842$	−28.6874	−0.988632
$843$	4.10682	0.141446
$844$	31.8352	1.09581
$845$	0	0
$846$	55.3592	1.90329
$847$	−21.8954	−0.752335
$848$	3.31356	0.113788
$849$	7.37996	0.253280
$850$	0	0
$851$	33.4897	1.14801
$852$	0.224681	0.00769746
$853$	41.9841	1.43751	0.718754	−	0.695265i	$-0.244713\pi$
$853$	41.9841	1.43751	0.718754	+	0.695265i	$0.244713\pi$
$854$	37.7718	1.29252
$855$	0	0
$856$	−30.2465	−1.03380
$857$	−22.1374	−0.756199	−0.378100	−	0.925765i	$-0.623422\pi$
$857$	−22.1374	−0.756199	−0.378100	+	0.925765i	$0.623422\pi$
$858$	0.128783	0.00439659
$859$	−45.5842	−1.55531	−0.777656	−	0.628690i	$-0.783591\pi$
$859$	−45.5842	−1.55531	−0.777656	+	0.628690i	$0.783591\pi$
$860$	0	0
$861$	5.20148	0.177266
$862$	−62.3593	−2.12397
$863$	−20.2772	−0.690245	−0.345122	−	0.938558i	$-0.612163\pi$
$863$	−20.2772	−0.690245	−0.345122	+	0.938558i	$0.612163\pi$
$864$	−5.33598	−0.181534
$865$	0	0
$866$	−35.3342	−1.20070
$867$	0	0
$868$	49.1380	1.66785
$869$	−7.98367	−0.270828
$870$	0	0
$871$	2.49200	0.0844382
$872$	25.5903	0.866598
$873$	40.4355	1.36854
$874$	146.764	4.96436
$875$	0	0
$876$	−8.07432	−0.272806
$877$	−19.5760	−0.661034	−0.330517	−	0.943800i	$-0.607223\pi$
$877$	−19.5760	−0.661034	−0.330517	+	0.943800i	$0.607223\pi$
$878$	−10.1794	−0.343537
$879$	6.20747	0.209373
$880$	0	0
$881$	−22.4477	−0.756281	−0.378141	−	0.925748i	$-0.623436\pi$
$881$	−22.4477	−0.756281	−0.378141	+	0.925748i	$0.623436\pi$
$882$	18.9171	0.636973
$883$	−24.1463	−0.812588	−0.406294	−	0.913742i	$-0.633179\pi$
$883$	−24.1463	−0.812588	−0.406294	+	0.913742i	$0.633179\pi$
$884$	0	0
$885$	0	0
$886$	30.1147	1.01172
$887$	31.8863	1.07064	0.535318	−	0.844651i	$-0.320192\pi$
$887$	31.8863	1.07064	0.535318	+	0.844651i	$0.320192\pi$
$888$	−3.30438	−0.110888
$889$	−20.8378	−0.698876
$890$	0	0
$891$	−5.42451	−0.181728
$892$	−42.5087	−1.42330
$893$	−59.7267	−1.99868
$894$	−11.2464	−0.376137
$895$	0	0
$896$	−42.0604	−1.40514
$897$	−0.719038	−0.0240080
$898$	−47.6276	−1.58935
$899$	23.1865	0.773312
$900$	0	0
$901$	0	0
$902$	16.4176	0.546647
$903$	4.43833	0.147698
$904$	53.1370	1.76731
$905$	0	0
$906$	4.26700	0.141762
$907$	51.2799	1.70272	0.851361	−	0.524581i	$-0.175778\pi$
$907$	51.2799	1.70272	0.851361	+	0.524581i	$0.175778\pi$
$908$	−0.457592	−0.0151857
$909$	−12.1181	−0.401932
$910$	0	0
$911$	13.4232	0.444729	0.222364	−	0.974964i	$-0.428623\pi$
$911$	13.4232	0.444729	0.222364	+	0.974964i	$0.428623\pi$
$912$	−2.38587	−0.0790041
$913$	−6.26481	−0.207335
$914$	25.4237	0.840942
$915$	0	0
$916$	−16.8567	−0.556960
$917$	29.9695	0.989681
$918$	0	0
$919$	−28.3439	−0.934978	−0.467489	−	0.883999i	$-0.654841\pi$
$919$	−28.3439	−0.934978	−0.467489	+	0.883999i	$0.654841\pi$
$920$	0	0
$921$	1.02893	0.0339043
$922$	74.5937	2.45661
$923$	−0.104362	−0.00343513
$924$	1.06295	0.0349685
$925$	0	0
$926$	−91.6315	−3.01120
$927$	11.2884	0.370758
$928$	−13.4643	−0.441986
$929$	52.1759	1.71184	0.855919	−	0.517111i	$-0.172992\pi$
$929$	52.1759	1.71184	0.855919	+	0.517111i	$0.172992\pi$
$930$	0	0
$931$	−20.4095	−0.668896
$932$	76.9089	2.51923
$933$	0.258078	0.00844910
$934$	55.0525	1.80137
$935$	0	0
$936$	−3.97893	−0.130055
$937$	−14.5375	−0.474921	−0.237460	−	0.971397i	$-0.576315\pi$
$937$	−14.5375	−0.474921	−0.237460	+	0.971397i	$0.576315\pi$
$938$	32.2279	1.05228
$939$	3.71289	0.121166
$940$	0	0
$941$	−51.3044	−1.67247	−0.836237	−	0.548368i	$-0.815249\pi$
$941$	−51.3044	−1.67247	−0.836237	+	0.548368i	$0.815249\pi$
$942$	−3.97933	−0.129654
$943$	−91.6648	−2.98502
$944$	0.529445	0.0172320
$945$	0	0
$946$	14.0089	0.455467
$947$	22.3683	0.726872	0.363436	−	0.931619i	$-0.381604\pi$
$947$	22.3683	0.726872	0.363436	+	0.931619i	$0.381604\pi$
$948$	−10.1541	−0.329790
$949$	3.75045	0.121745
$950$	0	0
$951$	−5.35859	−0.173764
$952$	0	0
$953$	10.2527	0.332118	0.166059	−	0.986116i	$-0.446896\pi$
$953$	10.2527	0.332118	0.166059	+	0.986116i	$0.446896\pi$
$954$	−16.4971	−0.534114
$955$	0	0
$956$	−82.1212	−2.65599
$957$	0.501570	0.0162134
$958$	−69.7569	−2.25374
$959$	10.1439	0.327563
$960$	0	0
$961$	14.4216	0.465213
$962$	3.54352	0.114248
$963$	24.8107	0.799513
$964$	−68.3778	−2.20230
$965$	0	0
$966$	−9.29897	−0.299190
$967$	−8.69544	−0.279626	−0.139813	−	0.990178i	$-0.544650\pi$
$967$	−8.69544	−0.279626	−0.139813	+	0.990178i	$0.544650\pi$
$968$	−38.0721	−1.22368
$969$	0	0
$970$	0	0
$971$	4.32983	0.138951	0.0694754	−	0.997584i	$-0.477867\pi$
$971$	4.32983	0.138951	0.0694754	+	0.997584i	$0.477867\pi$
$972$	−21.3308	−0.684184
$973$	−29.8453	−0.956796
$974$	77.0309	2.46823
$975$	0	0
$976$	10.8211	0.346374
$977$	−41.9064	−1.34070	−0.670352	−	0.742043i	$-0.733857\pi$
$977$	−41.9064	−1.34070	−0.670352	+	0.742043i	$0.733857\pi$
$978$	−10.2606	−0.328099
$979$	−9.10092	−0.290867
$980$	0	0
$981$	−20.9913	−0.670201
$982$	−55.0193	−1.75574
$983$	25.6754	0.818918	0.409459	−	0.912329i	$-0.365718\pi$
$983$	25.6754	0.818918	0.409459	+	0.912329i	$0.365718\pi$
$984$	9.04442	0.288325
$985$	0	0
$986$	0	0
$987$	3.78429	0.120455
$988$	9.91090	0.315308
$989$	−78.2160	−2.48712
$990$	0	0
$991$	−44.5299	−1.41454	−0.707269	−	0.706944i	$-0.750073\pi$
$991$	−44.5299	−1.41454	−0.707269	+	0.706944i	$0.750073\pi$
$992$	−26.3761	−0.837441
$993$	−2.32292	−0.0737156
$994$	−1.34967	−0.0428089
$995$	0	0
$996$	−7.96794	−0.252474
$997$	47.3837	1.50066	0.750328	−	0.661065i	$-0.229895\pi$
$997$	47.3837	1.50066	0.750328	+	0.661065i	$0.229895\pi$
$998$	−20.5291	−0.649838
$999$	5.46938	0.173043

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7225.2.a.bw.1.14	yes	15
5.4	even	2		7225.2.a.bu.1.2	yes	15
17.16	even	2		7225.2.a.bt.1.14	✓	15
85.84	even	2		7225.2.a.bv.1.2	yes	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7225.2.a.bt.1.14	✓	15	17.16	even	2
7225.2.a.bu.1.2	yes	15	5.4	even	2
7225.2.a.bv.1.2	yes	15	85.84	even	2
7225.2.a.bw.1.14	yes	15	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:	\( N \)	\(=\)	\( 7225 = 5^{2} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7225.a (trivial)

\(f(q)\)	\(=\)	\(q+2.35122 q^{2} -0.229247 q^{3} +3.52823 q^{4} -0.539011 q^{6} +2.06647 q^{7} +3.59321 q^{8} -2.94745 q^{9} +O(q^{10})\)	\(q+2.35122 q^{2} -0.229247 q^{3} +3.52823 q^{4} -0.539011 q^{6} +2.06647 q^{7} +3.59321 q^{8} -2.94745 q^{9} -0.635950 q^{11} -0.808838 q^{12} +0.375698 q^{13} +4.85872 q^{14} +1.39195 q^{16} -6.93009 q^{18} +7.47683 q^{19} -0.473733 q^{21} -1.49526 q^{22} +8.34851 q^{23} -0.823733 q^{24} +0.883348 q^{26} +1.36344 q^{27} +7.29098 q^{28} +3.44036 q^{29} +6.73955 q^{31} -3.91362 q^{32} +0.145790 q^{33} -10.3993 q^{36} +4.01146 q^{37} +17.5797 q^{38} -0.0861277 q^{39} -10.9798 q^{41} -1.11385 q^{42} -9.36886 q^{43} -2.24378 q^{44} +19.6292 q^{46} -7.98824 q^{47} -0.319102 q^{48} -2.72971 q^{49} +1.32555 q^{52} +2.38051 q^{53} +3.20574 q^{54} +7.42525 q^{56} -1.71404 q^{57} +8.08904 q^{58} +0.380361 q^{59} +7.77402 q^{61} +15.8462 q^{62} -6.09080 q^{63} -11.9857 q^{64} +0.342784 q^{66} +6.63299 q^{67} -1.91387 q^{69} -0.277783 q^{71} -10.5908 q^{72} +9.98262 q^{73} +9.43183 q^{74} +26.3800 q^{76} -1.31417 q^{77} -0.202505 q^{78} +12.5539 q^{79} +8.52977 q^{81} -25.8159 q^{82} +9.85110 q^{83} -1.67144 q^{84} -22.0282 q^{86} -0.788693 q^{87} -2.28510 q^{88} +14.3107 q^{89} +0.776368 q^{91} +29.4555 q^{92} -1.54503 q^{93} -18.7821 q^{94} +0.897188 q^{96} -13.7188 q^{97} -6.41814 q^{98} +1.87443 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 15 q + 9 q^{3} + 12 q^{4} + 9 q^{6} + 12 q^{7} + 6 q^{8} + 12 q^{9}+O(q^{10}) \) 15 * q + 9 * q^3 + 12 * q^4 + 9 * q^6 + 12 * q^7 + 6 * q^8 + 12 * q^9	\( 15 q + 9 q^{3} + 12 q^{4} + 9 q^{6} + 12 q^{7} + 6 q^{8} + 12 q^{9} + 6 q^{11} + 24 q^{12} + 6 q^{16} + 12 q^{18} + 6 q^{19} + 30 q^{21} + 12 q^{22} + 36 q^{23} + 18 q^{24} + 36 q^{26} + 36 q^{27} + 24 q^{28} - 18 q^{29} + 12 q^{32} + 12 q^{33} - 9 q^{36} + 12 q^{37} - 6 q^{38} + 9 q^{39} - 18 q^{41} + 36 q^{42} - 3 q^{43} - 12 q^{44} + 21 q^{46} - 3 q^{47} - 12 q^{48} + 15 q^{49} - 27 q^{52} + 21 q^{54} - 6 q^{56} + 39 q^{57} + 18 q^{58} - 12 q^{59} - 15 q^{61} + 54 q^{62} + 60 q^{63} - 36 q^{64} + 18 q^{66} - 24 q^{67} + 42 q^{69} + 6 q^{71} + 66 q^{72} - 9 q^{73} - 36 q^{74} - 18 q^{76} + 30 q^{77} + 30 q^{78} - 9 q^{79} + 51 q^{81} - 36 q^{82} + 15 q^{83} + 9 q^{84} - 36 q^{86} - 51 q^{87} + 30 q^{88} - 24 q^{89} + 27 q^{91} + 15 q^{92} - 42 q^{93} - 57 q^{94} + 42 q^{96} + 48 q^{97} + 12 q^{99}+O(q^{100}) \) 15 * q + 9 * q^3 + 12 * q^4 + 9 * q^6 + 12 * q^7 + 6 * q^8 + 12 * q^9 + 6 * q^11 + 24 * q^12 + 6 * q^16 + 12 * q^18 + 6 * q^19 + 30 * q^21 + 12 * q^22 + 36 * q^23 + 18 * q^24 + 36 * q^26 + 36 * q^27 + 24 * q^28 - 18 * q^29 + 12 * q^32 + 12 * q^33 - 9 * q^36 + 12 * q^37 - 6 * q^38 + 9 * q^39 - 18 * q^41 + 36 * q^42 - 3 * q^43 - 12 * q^44 + 21 * q^46 - 3 * q^47 - 12 * q^48 + 15 * q^49 - 27 * q^52 + 21 * q^54 - 6 * q^56 + 39 * q^57 + 18 * q^58 - 12 * q^59 - 15 * q^61 + 54 * q^62 + 60 * q^63 - 36 * q^64 + 18 * q^66 - 24 * q^67 + 42 * q^69 + 6 * q^71 + 66 * q^72 - 9 * q^73 - 36 * q^74 - 18 * q^76 + 30 * q^77 + 30 * q^78 - 9 * q^79 + 51 * q^81 - 36 * q^82 + 15 * q^83 + 9 * q^84 - 36 * q^86 - 51 * q^87 + 30 * q^88 - 24 * q^89 + 27 * q^91 + 15 * q^92 - 42 * q^93 - 57 * q^94 + 42 * q^96 + 48 * q^97 + 12 * q^99

Introduction

L-functions

Modular forms

Varieties

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Database

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