LMFDB - Embedded newform 7225.2.a.bu.1.2

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:	$1$
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			1.2
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	$-0.812392\pi$
$2$	−2.35122	−1.66256	−0.831282	+	0.555852i	$0.812392\pi$
$3$	0.229247	0.132356	0.0661780	−	0.997808i	$-0.478919\pi$
$3$	0.229247	0.132356	0.0661780	+	0.997808i	$0.478919\pi$
$4$	3.52823	1.76412
$5$	0	0
$5$	0	0
$6$	−0.539011	−0.220050
$7$	−2.06647	−0.781052	−0.390526	−	0.920592i	$-0.627707\pi$
$7$	−2.06647	−0.781052	−0.390526	+	0.920592i	$0.627707\pi$
$8$	−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	$-0.516591\pi$
$13$	−0.375698	−0.104200	−0.0520999	+	0.998642i	$0.516591\pi$
$14$	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	$-0.836134\pi$
$23$	−8.34851	−1.74078	−0.870392	+	0.492360i	$0.836134\pi$
$24$	−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	$-0.606961\pi$
$37$	−4.01146	−0.659481	−0.329740	+	0.944072i	$0.606961\pi$
$38$	−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	$-0.246714\pi$
$43$	9.36886	1.42874	0.714369	+	0.699770i	$0.246714\pi$
$44$	−2.24378	−0.338263
$45$	0	0
$46$	19.6292	2.89416
$47$	7.98824	1.16520	0.582602	−	0.812757i	$-0.302034\pi$
$47$	7.98824	1.16520	0.582602	+	0.812757i	$0.302034\pi$
$48$	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	$-0.552276\pi$
$53$	−2.38051	−0.326988	−0.163494	+	0.986544i	$0.552276\pi$
$54$	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	$-0.632789\pi$
$67$	−6.63299	−0.810349	−0.405175	+	0.914239i	$0.632789\pi$
$68$	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	$-0.698587\pi$
$73$	−9.98262	−1.16838	−0.584189	+	0.811618i	$0.698587\pi$
$74$	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	$-0.681821\pi$
$83$	−9.85110	−1.08130	−0.540649	+	0.841248i	$0.681821\pi$
$84$	−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	$-0.254754\pi$
$97$	13.7188	1.39294	0.696468	+	0.717587i	$0.254754\pi$
$98$	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	$-0.439578\pi$
$103$	3.82988	0.377369	0.188685	+	0.982038i	$0.439578\pi$
$104$	1.34996	0.132374
$105$	0	0
$106$	5.59710	0.543638
$107$	8.41769	0.813769	0.406884	−	0.913480i	$-0.366615\pi$
$107$	8.41769	0.813769	0.406884	+	0.913480i	$0.366615\pi$
$108$	−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	$-0.744851\pi$
$113$	−14.7882	−1.39115	−0.695577	+	0.718451i	$0.744851\pi$
$114$	−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	$-0.352352\pi$
$127$	10.0838	0.894788	0.447394	+	0.894337i	$0.352352\pi$
$128$	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	$-0.567247\pi$
$137$	−4.90880	−0.419387	−0.209693	+	0.977767i	$0.567247\pi$
$138$	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	$-0.595186\pi$
$157$	−7.38265	−0.589200	−0.294600	+	0.955621i	$0.595186\pi$
$158$	−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	$-0.767794\pi$
$163$	−19.0360	−1.49102	−0.745509	+	0.666496i	$0.767794\pi$
$164$	−38.7392	−3.02503
$165$	0	0
$166$	23.1621	1.79773
$167$	7.30310	0.565131	0.282565	−	0.959248i	$-0.408815\pi$
$167$	7.30310	0.565131	0.282565	+	0.959248i	$0.408815\pi$
$168$	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	$-0.759341\pi$
$173$	−19.1389	−1.45510	−0.727551	+	0.686054i	$0.759341\pi$
$174$	−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	$-0.591177\pi$
$193$	−7.85031	−0.565078	−0.282539	+	0.959256i	$0.591177\pi$
$194$	−32.2560	−2.31585
$195$	0	0
$196$	−9.63104	−0.687931
$197$	−4.26593	−0.303935	−0.151967	−	0.988385i	$-0.548561\pi$
$197$	−4.26593	−0.303935	−0.151967	+	0.988385i	$0.548561\pi$
$198$	−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	$-0.367828\pi$
$223$	12.0482	0.806805	0.403402	+	0.915023i	$0.367828\pi$
$224$	−8.08738	−0.540361
$225$	0	0
$226$	34.7702	2.31288
$227$	0.129694	0.00860812	0.00430406	−	0.999991i	$-0.498630\pi$
$227$	0.129694	0.00860812	0.00430406	+	0.999991i	$0.498630\pi$
$228$	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	$-0.753128\pi$
$233$	−21.7981	−1.42804	−0.714022	+	0.700123i	$0.753128\pi$
$234$	−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	$-0.615884\pi$
$257$	−11.4165	−0.712144	−0.356072	+	0.934458i	$0.615884\pi$
$258$	−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	$-0.573643\pi$
$263$	−7.43718	−0.458596	−0.229298	+	0.973356i	$0.573643\pi$
$264$	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	$-0.246952\pi$
$277$	23.7615	1.42769	0.713845	+	0.700304i	$0.246952\pi$
$278$	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	$-0.0938912\pi$
$283$	32.1921	1.91362	0.956811	+	0.290709i	$0.0938912\pi$
$284$	−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	$-0.209589\pi$
$293$	27.0776	1.58189	0.790945	+	0.611887i	$0.209589\pi$
$294$	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	$-0.459119\pi$
$307$	4.48828	0.256160	0.128080	+	0.991764i	$0.459119\pi$
$308$	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	$-0.348664\pi$
$313$	16.1960	0.915451	0.457726	+	0.889093i	$0.348664\pi$
$314$	17.3582	0.979582
$315$	0	0
$316$	44.2932	2.49169
$317$	−23.3747	−1.31285	−0.656427	−	0.754389i	$-0.727933\pi$
$317$	−23.3747	−1.31285	−0.656427	+	0.754389i	$0.727933\pi$
$318$	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	$-0.221946\pi$
$337$	28.1459	1.53320	0.766602	+	0.642123i	$0.221946\pi$
$338$	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	$-0.426124\pi$
$347$	8.56927	0.460022	0.230011	+	0.973188i	$0.426124\pi$
$348$	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	$-0.592079\pi$
$353$	−10.7190	−0.570515	−0.285257	+	0.958451i	$0.592079\pi$
$354$	−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	$-0.579901\pi$
$367$	−9.51689	−0.496777	−0.248389	+	0.968660i	$0.579901\pi$
$368$	−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	$-0.418347\pi$
$373$	9.80017	0.507433	0.253717	+	0.967279i	$0.418347\pi$
$374$	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	$-0.539882\pi$
$383$	−4.89130	−0.249934	−0.124967	+	0.992161i	$0.539882\pi$
$384$	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	$-0.279740\pi$
$397$	25.4262	1.27610	0.638052	+	0.769993i	$0.279740\pi$
$398$	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	$-0.382401\pi$
$433$	15.0280	0.722201	0.361100	+	0.932527i	$0.382401\pi$
$434$	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	$-0.598411\pi$
$443$	−12.8081	−0.608532	−0.304266	+	0.952587i	$0.598411\pi$
$444$	−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	$-0.581386\pi$
$457$	−10.8130	−0.505811	−0.252905	+	0.967491i	$0.581386\pi$
$458$	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	$-0.139429\pi$
$463$	38.9719	1.81118	0.905589	+	0.424156i	$0.139429\pi$
$464$	4.78882	0.222316
$465$	0	0
$466$	51.2522	2.37421
$467$	−23.4144	−1.08349	−0.541745	−	0.840543i	$-0.682236\pi$
$467$	−23.4144	−1.08349	−0.541745	+	0.840543i	$0.682236\pi$
$468$	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	$-0.766263\pi$
$487$	−32.7621	−1.48459	−0.742296	+	0.670072i	$0.766263\pi$
$488$	−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	$-0.564900\pi$
$503$	−9.08227	−0.404958	−0.202479	+	0.979287i	$0.564900\pi$
$504$	−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	$-0.798112\pi$
$523$	−36.8430	−1.61103	−0.805516	+	0.592573i	$0.798112\pi$
$524$	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	$-0.373460\pi$
$547$	18.1093	0.774299	0.387150	+	0.922017i	$0.373460\pi$
$548$	−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	$-0.339004\pi$
$557$	22.8689	0.968988	0.484494	+	0.874795i	$0.339004\pi$
$558$	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	$-0.331836\pi$
$563$	23.9206	1.00813	0.504067	+	0.863665i	$0.331836\pi$
$564$	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	$-0.220292\pi$
$577$	36.9886	1.53986	0.769928	+	0.638131i	$0.220292\pi$
$578$	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	$-0.550621\pi$
$587$	−7.67355	−0.316721	−0.158361	+	0.987381i	$0.550621\pi$
$588$	−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	$-0.738475\pi$
$593$	−33.1691	−1.36209	−0.681046	+	0.732241i	$0.738475\pi$
$594$	−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	$-0.360546\pi$
$607$	20.9037	0.848455	0.424228	+	0.905556i	$0.360546\pi$
$608$	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	$-0.655123\pi$
$613$	−23.1878	−0.936545	−0.468272	+	0.883584i	$0.655123\pi$
$614$	−10.5529	−0.425882
$615$	0	0
$616$	−4.72209	−0.190258
$617$	11.8019	0.475126	0.237563	−	0.971372i	$-0.423651\pi$
$617$	11.8019	0.475126	0.237563	+	0.971372i	$0.423651\pi$
$618$	−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	$-0.612590\pi$
$643$	−17.5667	−0.692763	−0.346382	+	0.938094i	$0.612590\pi$
$644$	60.8688	2.39857
$645$	0	0
$646$	0	0
$647$	−15.4224	−0.606318	−0.303159	−	0.952940i	$-0.598041\pi$
$647$	−15.4224	−0.606318	−0.303159	+	0.952940i	$0.598041\pi$
$648$	−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	$-0.707103\pi$
$653$	−30.9554	−1.21138	−0.605689	+	0.795701i	$0.707103\pi$
$654$	−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	$-0.602288\pi$
$673$	−16.3874	−0.631688	−0.315844	+	0.948811i	$0.602288\pi$
$674$	−66.1771	−2.54905
$675$	0	0
$676$	−45.3690	−1.74496
$677$	−34.9206	−1.34211	−0.671054	−	0.741408i	$-0.734158\pi$
$677$	−34.9206	−1.34211	−0.671054	+	0.741408i	$0.734158\pi$
$678$	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	$-0.647530\pi$
$683$	−23.3673	−0.894125	−0.447062	+	0.894503i	$0.647530\pi$
$684$	−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	$-0.641102\pi$
$727$	−23.1292	−0.857816	−0.428908	+	0.903348i	$0.641102\pi$
$728$	−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	$-0.770152\pi$
$733$	−40.6341	−1.50086	−0.750428	+	0.660953i	$0.770152\pi$
$734$	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	$-0.469087\pi$
$743$	5.28603	0.193926	0.0969628	+	0.995288i	$0.469087\pi$
$744$	−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	$-0.322965\pi$
$757$	29.0510	1.05588	0.527938	+	0.849283i	$0.322965\pi$
$758$	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	$-0.274528\pi$
$773$	36.1757	1.30115	0.650574	+	0.759443i	$0.274528\pi$
$774$	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	$-0.685943\pi$
$787$	−30.9429	−1.10299	−0.551497	+	0.834177i	$0.685943\pi$
$788$	−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	$-0.461563\pi$
$797$	6.80146	0.240920	0.120460	+	0.992718i	$0.461563\pi$
$798$	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	$-0.680666\pi$
$823$	−30.8449	−1.07518	−0.537592	+	0.843205i	$0.680666\pi$
$824$	−13.7615	−0.479406
$825$	0	0
$826$	1.84807	0.0643025
$827$	4.11063	0.142941	0.0714703	−	0.997443i	$-0.477231\pi$
$827$	4.11063	0.142941	0.0714703	+	0.997443i	$0.477231\pi$
$828$	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	$-0.755287\pi$
$853$	−41.9841	−1.43751	−0.718754	+	0.695265i	$0.755287\pi$
$854$	37.7718	1.29252
$855$	0	0
$856$	−30.2465	−1.03380
$857$	22.1374	0.756199	0.378100	−	0.925765i	$-0.376578\pi$
$857$	22.1374	0.756199	0.378100	+	0.925765i	$0.376578\pi$
$858$	−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	$-0.387837\pi$
$863$	20.2772	0.690245	0.345122	+	0.938558i	$0.387837\pi$
$864$	−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	$-0.392777\pi$
$877$	19.5760	0.661034	0.330517	+	0.943800i	$0.392777\pi$
$878$	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	$-0.366821\pi$
$883$	24.1463	0.812588	0.406294	+	0.913742i	$0.366821\pi$
$884$	0	0
$885$	0	0
$886$	30.1147	1.01172
$887$	−31.8863	−1.07064	−0.535318	−	0.844651i	$-0.679808\pi$
$887$	−31.8863	−1.07064	−0.535318	+	0.844651i	$0.679808\pi$
$888$	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	$-0.824222\pi$
$907$	−51.2799	−1.70272	−0.851361	+	0.524581i	$0.824222\pi$
$908$	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	$-0.423685\pi$
$937$	14.5375	0.474921	0.237460	+	0.971397i	$0.423685\pi$
$938$	−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	$-0.618396\pi$
$947$	−22.3683	−0.726872	−0.363436	+	0.931619i	$0.618396\pi$
$948$	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	$-0.553104\pi$
$953$	−10.2527	−0.332118	−0.166059	+	0.986116i	$0.553104\pi$
$954$	−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	$-0.455350\pi$
$967$	8.69544	0.279626	0.139813	+	0.990178i	$0.455350\pi$
$968$	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	$-0.266143\pi$
$977$	41.9064	1.34070	0.670352	+	0.742043i	$0.266143\pi$
$978$	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	$-0.634282\pi$
$983$	−25.6754	−0.818918	−0.409459	+	0.912329i	$0.634282\pi$
$984$	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	$-0.770105\pi$
$997$	−47.3837	−1.50066	−0.750328	+	0.661065i	$0.770105\pi$
$998$	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.bu.1.2	yes	15
5.4	even	2		7225.2.a.bw.1.14	yes	15
17.16	even	2		7225.2.a.bv.1.2	yes	15
85.84	even	2		7225.2.a.bt.1.14	✓	15

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7225.2.a.bt.1.14	✓	15	85.84	even	2
7225.2.a.bu.1.2	yes	15	1.1	even	1	trivial
7225.2.a.bv.1.2	yes	15	17.16	even	2
7225.2.a.bw.1.14	yes	15	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:	\( N \)	\(=\)	\( 7225 = 5^{2} \cdot 17^{2} \)
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

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