LMFDB - Embedded newform 4225.2.a.bm.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4225,2,Mod(1,4225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4225, 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("4225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4225 = 5^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4225.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:	$33.7367948540$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.1068321.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 8x^{3} - x^{2} + 12x - 3 $ x^5 - 8x^3 - x^2 + 12x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 325)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.09600$ of defining polynomial
Character	$\chi$	$=$	4225.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.09600 q^{2} -3.36467 q^{3} -0.798791 q^{4} -3.68767 q^{6} +1.59167 q^{7} -3.06747 q^{8} +8.32102 q^{9} +O(q^{10})$	\(q+1.09600 q^{2} -3.36467 q^{3} -0.798791 q^{4} -3.68767 q^{6} +1.59167 q^{7} -3.06747 q^{8} +8.32102 q^{9} -3.68767 q^{11} +2.68767 q^{12} +1.74447 q^{14} -1.76435 q^{16} -2.98488 q^{17} +9.11981 q^{18} +2.79879 q^{19} -5.35546 q^{21} -4.04167 q^{22} -0.671093 q^{23} +10.3210 q^{24} -17.9035 q^{27} -1.27141 q^{28} +2.75787 q^{29} +6.43214 q^{31} +4.20121 q^{32} +12.4078 q^{33} -3.27141 q^{34} -6.64675 q^{36} +0.958326 q^{37} +3.06747 q^{38} -2.85444 q^{41} -5.86956 q^{42} +4.88557 q^{43} +2.94568 q^{44} -0.735516 q^{46} -3.83323 q^{47} +5.93646 q^{48} -4.46658 q^{49} +10.0431 q^{51} +2.70355 q^{53} -19.6222 q^{54} -4.88240 q^{56} -9.41701 q^{57} +3.02262 q^{58} +10.3568 q^{59} +7.43214 q^{61} +7.04960 q^{62} +13.2443 q^{63} +8.13321 q^{64} +13.5989 q^{66} +11.5262 q^{67} +2.38429 q^{68} +2.25801 q^{69} -8.57770 q^{71} -25.5244 q^{72} -3.72859 q^{73} +1.05032 q^{74} -2.23565 q^{76} -5.86956 q^{77} +4.33442 q^{79} +35.2763 q^{81} -3.12846 q^{82} -15.6619 q^{83} +4.27789 q^{84} +5.35457 q^{86} -9.27934 q^{87} +11.3118 q^{88} -1.65323 q^{89} +0.536063 q^{92} -21.6420 q^{93} -4.20121 q^{94} -14.1357 q^{96} +2.25725 q^{97} -4.89536 q^{98} -30.6852 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 3 q^{3} + 6 q^{4} - 3 q^{6} - 2 q^{7} + 3 q^{8} + 4 q^{9}+O(q^{10}) $ 5 * q - 3 * q^3 + 6 * q^4 - 3 * q^6 - 2 * q^7 + 3 * q^8 + 4 * q^9	$ 5 q - 3 q^{3} + 6 q^{4} - 3 q^{6} - 2 q^{7} + 3 q^{8} + 4 q^{9} - 3 q^{11} - 2 q^{12} - 8 q^{14} + 4 q^{16} - 4 q^{17} - 2 q^{18} + 4 q^{19} + 8 q^{21} - 8 q^{22} - 15 q^{23} + 14 q^{24} - 21 q^{27} - 17 q^{28} - q^{29} + 31 q^{32} - 2 q^{33} - 27 q^{34} - 13 q^{36} + 17 q^{37} - 3 q^{38} - 6 q^{41} - 32 q^{42} - 12 q^{43} + 8 q^{44} + 7 q^{46} - 12 q^{47} + 2 q^{48} + 7 q^{49} - 8 q^{53} - 19 q^{54} - 17 q^{56} - 4 q^{57} + 38 q^{58} + 12 q^{59} + 5 q^{61} - 13 q^{62} + 26 q^{63} - 5 q^{64} + 43 q^{66} - 16 q^{67} - 25 q^{68} + 20 q^{69} - 19 q^{71} - 45 q^{72} - 8 q^{73} + 2 q^{74} - 24 q^{76} - 32 q^{77} - 14 q^{79} + 29 q^{81} + 23 q^{82} - 7 q^{83} + 34 q^{84} + 42 q^{86} - 21 q^{87} - 2 q^{88} + 10 q^{89} - 71 q^{92} - 33 q^{93} - 31 q^{94} - 17 q^{96} + 37 q^{97} - 21 q^{98} - 28 q^{99}+O(q^{100}) $ 5 * q - 3 * q^3 + 6 * q^4 - 3 * q^6 - 2 * q^7 + 3 * q^8 + 4 * q^9 - 3 * q^11 - 2 * q^12 - 8 * q^14 + 4 * q^16 - 4 * q^17 - 2 * q^18 + 4 * q^19 + 8 * q^21 - 8 * q^22 - 15 * q^23 + 14 * q^24 - 21 * q^27 - 17 * q^28 - q^29 + 31 * q^32 - 2 * q^33 - 27 * q^34 - 13 * q^36 + 17 * q^37 - 3 * q^38 - 6 * q^41 - 32 * q^42 - 12 * q^43 + 8 * q^44 + 7 * q^46 - 12 * q^47 + 2 * q^48 + 7 * q^49 - 8 * q^53 - 19 * q^54 - 17 * q^56 - 4 * q^57 + 38 * q^58 + 12 * q^59 + 5 * q^61 - 13 * q^62 + 26 * q^63 - 5 * q^64 + 43 * q^66 - 16 * q^67 - 25 * q^68 + 20 * q^69 - 19 * q^71 - 45 * q^72 - 8 * q^73 + 2 * q^74 - 24 * q^76 - 32 * q^77 - 14 * q^79 + 29 * q^81 + 23 * q^82 - 7 * q^83 + 34 * q^84 + 42 * q^86 - 21 * q^87 - 2 * q^88 + 10 * q^89 - 71 * q^92 - 33 * q^93 - 31 * q^94 - 17 * q^96 + 37 * q^97 - 21 * q^98 - 28 * 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 $

$2$	1.09600	0.774987	0.387493	−	0.921872i	$-0.373341\pi$
$2$	1.09600	0.774987	0.387493	+	0.921872i	$0.373341\pi$
$3$	−3.36467	−1.94259	−0.971297	−	0.237870i	$-0.923551\pi$
$3$	−3.36467	−1.94259	−0.971297	+	0.237870i	$0.923551\pi$
$4$	−0.798791	−0.399395
$5$	0	0
$5$	0	0
$6$	−3.68767	−1.50548
$7$	1.59167	0.601596	0.300798	−	0.953688i	$-0.402747\pi$
$7$	1.59167	0.601596	0.300798	+	0.953688i	$0.402747\pi$
$8$	−3.06747	−1.08451
$9$	8.32102	2.77367
$10$	0	0
$11$	−3.68767	−1.11187	−0.555937	−	0.831224i	$-0.687640\pi$
$11$	−3.68767	−1.11187	−0.555937	+	0.831224i	$0.687640\pi$
$12$	2.68767	0.775863
$13$	0	0
$13$	0	0
$14$	1.74447	0.466229
$15$	0	0
$16$	−1.76435	−0.441088
$17$	−2.98488	−0.723939	−0.361969	−	0.932190i	$-0.617895\pi$
$17$	−2.98488	−0.723939	−0.361969	+	0.932190i	$0.617895\pi$
$18$	9.11981	2.14956
$19$	2.79879	0.642087	0.321043	−	0.947065i	$-0.395966\pi$
$19$	2.79879	0.642087	0.321043	+	0.947065i	$0.395966\pi$
$20$	0	0
$21$	−5.35546	−1.16866
$22$	−4.04167	−0.861688
$23$	−0.671093	−0.139933	−0.0699663	−	0.997549i	$-0.522289\pi$
$23$	−0.671093	−0.139933	−0.0699663	+	0.997549i	$0.522289\pi$
$24$	10.3210	2.10677
$25$	0	0
$26$	0	0
$27$	−17.9035	−3.44553
$28$	−1.27141	−0.240275
$29$	2.75787	0.512124	0.256062	−	0.966660i	$-0.417575\pi$
$29$	2.75787	0.512124	0.256062	+	0.966660i	$0.417575\pi$
$30$	0	0
$31$	6.43214	1.15525	0.577623	−	0.816304i	$-0.303980\pi$
$31$	6.43214	1.15525	0.577623	+	0.816304i	$0.303980\pi$
$32$	4.20121	0.742676
$33$	12.4078	2.15992
$34$	−3.27141	−0.561043
$35$	0	0
$36$	−6.64675	−1.10779
$37$	0.958326	0.157548	0.0787739	−	0.996893i	$-0.474899\pi$
$37$	0.958326	0.157548	0.0787739	+	0.996893i	$0.474899\pi$
$38$	3.06747	0.497609
$39$	0	0
$40$	0	0
$41$	−2.85444	−0.445788	−0.222894	−	0.974843i	$-0.571550\pi$
$41$	−2.85444	−0.445788	−0.222894	+	0.974843i	$0.571550\pi$
$42$	−5.86956	−0.905693
$43$	4.88557	0.745043	0.372521	−	0.928024i	$-0.378493\pi$
$43$	4.88557	0.745043	0.372521	+	0.928024i	$0.378493\pi$
$44$	2.94568	0.444078
$45$	0	0
$46$	−0.735516	−0.108446
$47$	−3.83323	−0.559134	−0.279567	−	0.960126i	$-0.590191\pi$
$47$	−3.83323	−0.559134	−0.279567	+	0.960126i	$0.590191\pi$
$48$	5.93646	0.856854
$49$	−4.46658	−0.638083
$50$	0	0
$51$	10.0431	1.40632
$52$	0	0
$53$	2.70355	0.371361	0.185681	−	0.982610i	$-0.440551\pi$
$53$	2.70355	0.371361	0.185681	+	0.982610i	$0.440551\pi$
$54$	−19.6222	−2.67024
$55$	0	0
$56$	−4.88240	−0.652438
$57$	−9.41701	−1.24731
$58$	3.02262	0.396890
$59$	10.3568	1.34834	0.674169	−	0.738577i	$-0.264502\pi$
$59$	10.3568	1.34834	0.674169	+	0.738577i	$0.264502\pi$
$60$	0	0
$61$	7.43214	0.951588	0.475794	−	0.879557i	$-0.342161\pi$
$61$	7.43214	0.951588	0.475794	+	0.879557i	$0.342161\pi$
$62$	7.04960	0.895300
$63$	13.2443	1.66863
$64$	8.13321	1.01665
$65$	0	0
$66$	13.5989	1.67391
$67$	11.5262	1.40814	0.704072	−	0.710129i	$-0.251363\pi$
$67$	11.5262	1.40814	0.704072	+	0.710129i	$0.251363\pi$
$68$	2.38429	0.289138
$69$	2.25801	0.271832
$70$	0	0
$71$	−8.57770	−1.01799	−0.508993	−	0.860771i	$-0.669982\pi$
$71$	−8.57770	−1.01799	−0.508993	+	0.860771i	$0.669982\pi$
$72$	−25.5244	−3.00808
$73$	−3.72859	−0.436398	−0.218199	−	0.975904i	$-0.570018\pi$
$73$	−3.72859	−0.436398	−0.218199	+	0.975904i	$0.570018\pi$
$74$	1.05032	0.122097
$75$	0	0
$76$	−2.23565	−0.256447
$77$	−5.86956	−0.668899
$78$	0	0
$79$	4.33442	0.487661	0.243830	−	0.969818i	$-0.421596\pi$
$79$	4.33442	0.487661	0.243830	+	0.969818i	$0.421596\pi$
$80$	0	0
$81$	35.2763	3.91958
$82$	−3.12846	−0.345480
$83$	−15.6619	−1.71912	−0.859559	−	0.511036i	$-0.829262\pi$
$83$	−15.6619	−1.71912	−0.859559	+	0.511036i	$0.829262\pi$
$84$	4.27789	0.466756
$85$	0	0
$86$	5.35457	0.577398
$87$	−9.27934	−0.994850
$88$	11.3118	1.20584
$89$	−1.65323	−0.175242	−0.0876210	−	0.996154i	$-0.527926\pi$
$89$	−1.65323	−0.175242	−0.0876210	+	0.996154i	$0.527926\pi$
$90$	0	0
$91$	0	0
$92$	0.536063	0.0558884
$93$	−21.6420	−2.24417
$94$	−4.20121	−0.433322
$95$	0	0
$96$	−14.1357	−1.44272
$97$	2.25725	0.229189	0.114595	−	0.993412i	$-0.463443\pi$
$97$	2.25725	0.229189	0.114595	+	0.993412i	$0.463443\pi$
$98$	−4.89536	−0.494506
$99$	−30.6852	−3.08397
$100$	0	0
$101$	11.2866	1.12306	0.561528	−	0.827458i	$-0.310214\pi$
$101$	11.2866	1.12306	0.561528	+	0.827458i	$0.310214\pi$
$102$	11.0072	1.08988
$103$	−19.5998	−1.93122	−0.965612	−	0.259986i	$-0.916282\pi$
$103$	−19.5998	−1.93122	−0.965612	+	0.259986i	$0.916282\pi$
$104$	0	0
$105$	0	0
$106$	2.96308	0.287800
$107$	−1.76157	−0.170298	−0.0851489	−	0.996368i	$-0.527137\pi$
$107$	−1.76157	−0.170298	−0.0851489	+	0.996368i	$0.527137\pi$
$108$	14.3011	1.37613
$109$	9.33442	0.894076	0.447038	−	0.894515i	$-0.352479\pi$
$109$	9.33442	0.894076	0.447038	+	0.894515i	$0.352479\pi$
$110$	0	0
$111$	−3.22445	−0.306051
$112$	−2.80827	−0.265357
$113$	11.3389	1.06668	0.533338	−	0.845902i	$-0.320937\pi$
$113$	11.3389	1.06668	0.533338	+	0.845902i	$0.320937\pi$
$114$	−10.3210	−0.966652
$115$	0	0
$116$	−2.20297	−0.204540
$117$	0	0
$118$	11.3510	1.04494
$119$	−4.75094	−0.435518
$120$	0	0
$121$	2.59891	0.236264
$122$	8.14560	0.737468
$123$	9.60425	0.865986
$124$	−5.13793	−0.461400
$125$	0	0
$126$	14.5157	1.29317
$127$	−10.1999	−0.905097	−0.452548	−	0.891740i	$-0.649485\pi$
$127$	−10.1999	−0.905097	−0.452548	+	0.891740i	$0.649485\pi$
$128$	0.511558	0.0452158
$129$	−16.4383	−1.44732
$130$	0	0
$131$	−10.7666	−0.940679	−0.470339	−	0.882486i	$-0.655868\pi$
$131$	−10.7666	−0.940679	−0.470339	+	0.882486i	$0.655868\pi$
$132$	−9.91124	−0.862662
$133$	4.45476	0.386277
$134$	12.6326	1.09129
$135$	0	0
$136$	9.15600	0.785121
$137$	−12.7446	−1.08884	−0.544422	−	0.838811i	$-0.683251\pi$
$137$	−12.7446	−1.08884	−0.544422	+	0.838811i	$0.683251\pi$
$138$	2.47477	0.210666
$139$	−3.90344	−0.331085	−0.165543	−	0.986203i	$-0.552938\pi$
$139$	−3.90344	−0.331085	−0.165543	+	0.986203i	$0.552938\pi$
$140$	0	0
$141$	12.8976	1.08617
$142$	−9.40113	−0.788925
$143$	0	0
$144$	−14.6812	−1.22343
$145$	0	0
$146$	−4.08652	−0.338203
$147$	15.0286	1.23954
$148$	−0.765502	−0.0629239
$149$	1.45542	0.119232	0.0596162	−	0.998221i	$-0.481012\pi$
$149$	1.45542	0.119232	0.0596162	+	0.998221i	$0.481012\pi$
$150$	0	0
$151$	−18.4701	−1.50308	−0.751538	−	0.659690i	$-0.770688\pi$
$151$	−18.4701	−1.50308	−0.751538	+	0.659690i	$0.770688\pi$
$152$	−8.58520	−0.696351
$153$	−24.8372	−2.00797
$154$	−6.43302	−0.518388
$155$	0	0
$156$	0	0
$157$	−11.2891	−0.900972	−0.450486	−	0.892784i	$-0.648749\pi$
$157$	−11.2891	−0.900972	−0.450486	+	0.892784i	$0.648749\pi$
$158$	4.75051	0.377931
$159$	−9.09656	−0.721404
$160$	0	0
$161$	−1.06816	−0.0841828
$162$	38.6627	3.03763
$163$	−12.4845	−0.977860	−0.488930	−	0.872323i	$-0.662613\pi$
$163$	−12.4845	−0.977860	−0.488930	+	0.872323i	$0.662613\pi$
$164$	2.28010	0.178046
$165$	0	0
$166$	−17.1654	−1.33229
$167$	−12.5728	−0.972916	−0.486458	−	0.873704i	$-0.661711\pi$
$167$	−12.5728	−0.972916	−0.486458	+	0.873704i	$0.661711\pi$
$168$	16.4277	1.26742
$169$	0	0
$170$	0	0
$171$	23.2888	1.78094
$172$	−3.90255	−0.297567
$173$	−2.69804	−0.205128	−0.102564	−	0.994726i	$-0.532705\pi$
$173$	−2.69804	−0.205128	−0.102564	+	0.994726i	$0.532705\pi$
$174$	−10.1701	−0.770996
$175$	0	0
$176$	6.50634	0.490434
$177$	−34.8472	−2.61927
$178$	−1.81193	−0.135810
$179$	−2.31118	−0.172746	−0.0863728	−	0.996263i	$-0.527528\pi$
$179$	−2.31118	−0.172746	−0.0863728	+	0.996263i	$0.527528\pi$
$180$	0	0
$181$	4.80127	0.356875	0.178438	−	0.983951i	$-0.442896\pi$
$181$	4.80127	0.356875	0.178438	+	0.983951i	$0.442896\pi$
$182$	0	0
$183$	−25.0067	−1.84855
$184$	2.05855	0.151759
$185$	0	0
$186$	−23.7196	−1.73921
$187$	11.0072	0.804929
$188$	3.06195	0.223316
$189$	−28.4965	−2.07281
$190$	0	0
$191$	−17.8234	−1.28965	−0.644826	−	0.764329i	$-0.723070\pi$
$191$	−17.8234	−1.28965	−0.644826	+	0.764329i	$0.723070\pi$
$192$	−27.3656	−1.97494
$193$	−2.51777	−0.181233	−0.0906165	−	0.995886i	$-0.528884\pi$
$193$	−2.51777	−0.181233	−0.0906165	+	0.995886i	$0.528884\pi$
$194$	2.47394	0.177618
$195$	0	0
$196$	3.56786	0.254847
$197$	−18.5156	−1.31918	−0.659591	−	0.751624i	$-0.729271\pi$
$197$	−18.5156	−1.31918	−0.659591	+	0.751624i	$0.729271\pi$
$198$	−33.6308	−2.39004
$199$	13.2021	0.935872	0.467936	−	0.883762i	$-0.344998\pi$
$199$	13.2021	0.935872	0.467936	+	0.883762i	$0.344998\pi$
$200$	0	0
$201$	−38.7817	−2.73545
$202$	12.3701	0.870354
$203$	4.38963	0.308092
$204$	−8.02236	−0.561677
$205$	0	0
$206$	−21.4813	−1.49667
$207$	−5.58418	−0.388127
$208$	0	0
$209$	−10.3210	−0.713920
$210$	0	0
$211$	−7.93333	−0.546153	−0.273076	−	0.961992i	$-0.588041\pi$
$211$	−7.93333	−0.546153	−0.273076	+	0.961992i	$0.588041\pi$
$212$	−2.15957	−0.148320
$213$	28.8611	1.97753
$214$	−1.93068	−0.131979
$215$	0	0
$216$	54.9183	3.73672
$217$	10.2379	0.694991
$218$	10.2305	0.692897
$219$	12.5455	0.847744
$220$	0	0
$221$	0	0
$222$	−3.53399	−0.237186
$223$	3.74998	0.251117	0.125559	−	0.992086i	$-0.459928\pi$
$223$	3.74998	0.251117	0.125559	+	0.992086i	$0.459928\pi$
$224$	6.68695	0.446791
$225$	0	0
$226$	12.4274	0.826660
$227$	−12.9234	−0.857757	−0.428879	−	0.903362i	$-0.641091\pi$
$227$	−12.9234	−0.857757	−0.428879	+	0.903362i	$0.641091\pi$
$228$	7.52223	0.498172
$229$	24.5322	1.62114	0.810568	−	0.585645i	$-0.199159\pi$
$229$	24.5322	1.62114	0.810568	+	0.585645i	$0.199159\pi$
$230$	0	0
$231$	19.7492	1.29940
$232$	−8.45969	−0.555406
$233$	−8.82732	−0.578297	−0.289149	−	0.957284i	$-0.593372\pi$
$233$	−8.82732	−0.578297	−0.289149	+	0.957284i	$0.593372\pi$
$234$	0	0
$235$	0	0
$236$	−8.27290	−0.538520
$237$	−14.5839	−0.947327
$238$	−5.20702	−0.337521
$239$	−4.23582	−0.273993	−0.136996	−	0.990572i	$-0.543745\pi$
$239$	−4.23582	−0.273993	−0.136996	+	0.990572i	$0.543745\pi$
$240$	0	0
$241$	18.1357	1.16822	0.584111	−	0.811674i	$-0.301443\pi$
$241$	18.1357	1.16822	0.584111	+	0.811674i	$0.301443\pi$
$242$	2.84839	0.183102
$243$	−64.9826	−4.16864
$244$	−5.93672	−0.380060
$245$	0	0
$246$	10.5262	0.671128
$247$	0	0
$248$	−19.7304	−1.25288
$249$	52.6972	3.33955
$250$	0	0
$251$	−1.08183	−0.0682847	−0.0341424	−	0.999417i	$-0.510870\pi$
$251$	−1.08183	−0.0682847	−0.0341424	+	0.999417i	$0.510870\pi$
$252$	−10.5795	−0.666443
$253$	2.47477	0.155587
$254$	−11.1791	−0.701438
$255$	0	0
$256$	−15.7058	−0.981610
$257$	8.14560	0.508109	0.254054	−	0.967190i	$-0.418236\pi$
$257$	8.14560	0.508109	0.254054	+	0.967190i	$0.418236\pi$
$258$	−18.0164	−1.12165
$259$	1.52534	0.0947801
$260$	0	0
$261$	22.9483	1.42047
$262$	−11.8001	−0.729014
$263$	−17.2640	−1.06454	−0.532271	−	0.846574i	$-0.678661\pi$
$263$	−17.2640	−1.06454	−0.532271	+	0.846574i	$0.678661\pi$
$264$	−38.0605	−2.34246
$265$	0	0
$266$	4.88240	0.299359
$267$	5.56257	0.340424
$268$	−9.20699	−0.562406
$269$	−18.9566	−1.15580	−0.577901	−	0.816107i	$-0.696128\pi$
$269$	−18.9566	−1.15580	−0.577901	+	0.816107i	$0.696128\pi$
$270$	0	0
$271$	−23.3594	−1.41898	−0.709492	−	0.704714i	$-0.751076\pi$
$271$	−23.3594	−1.41898	−0.709492	+	0.704714i	$0.751076\pi$
$272$	5.26637	0.319320
$273$	0	0
$274$	−13.9680	−0.843840
$275$	0	0
$276$	−1.80368	−0.108569
$277$	6.00723	0.360940	0.180470	−	0.983581i	$-0.442238\pi$
$277$	6.00723	0.360940	0.180470	+	0.983581i	$0.442238\pi$
$278$	−4.27815	−0.256587
$279$	53.5219	3.20427
$280$	0	0
$281$	−18.1001	−1.07976	−0.539881	−	0.841742i	$-0.681531\pi$
$281$	−18.1001	−1.07976	−0.539881	+	0.841742i	$0.681531\pi$
$282$	14.1357	0.841768
$283$	−2.39131	−0.142149	−0.0710743	−	0.997471i	$-0.522643\pi$
$283$	−2.39131	−0.142149	−0.0710743	+	0.997471i	$0.522643\pi$
$284$	6.85179	0.406579
$285$	0	0
$286$	0	0
$287$	−4.54333	−0.268184
$288$	34.9583	2.05994
$289$	−8.09052	−0.475913
$290$	0	0
$291$	−7.59491	−0.445221
$292$	2.97836	0.174295
$293$	−14.7756	−0.863200	−0.431600	−	0.902065i	$-0.642051\pi$
$293$	−14.7756	−0.863200	−0.431600	+	0.902065i	$0.642051\pi$
$294$	16.4713	0.960624
$295$	0	0
$296$	−2.93963	−0.170863
$297$	66.0221	3.83099
$298$	1.59513	0.0924035
$299$	0	0
$300$	0	0
$301$	7.77623	0.448215
$302$	−20.2432	−1.16486
$303$	−37.9756	−2.18164
$304$	−4.93805	−0.283217
$305$	0	0
$306$	−27.2215	−1.55615
$307$	−14.6966	−0.838781	−0.419390	−	0.907806i	$-0.637756\pi$
$307$	−14.6966	−0.838781	−0.419390	+	0.907806i	$0.637756\pi$
$308$	4.68855	0.267155
$309$	65.9469	3.75159
$310$	0	0
$311$	−14.2032	−0.805392	−0.402696	−	0.915334i	$-0.631927\pi$
$311$	−14.2032	−0.805392	−0.402696	+	0.915334i	$0.631927\pi$
$312$	0	0
$313$	21.0319	1.18879	0.594397	−	0.804171i	$-0.297391\pi$
$313$	21.0319	1.18879	0.594397	+	0.804171i	$0.297391\pi$
$314$	−12.3729	−0.698241
$315$	0	0
$316$	−3.46230	−0.194769
$317$	22.4641	1.26171	0.630854	−	0.775902i	$-0.282705\pi$
$317$	22.4641	1.26171	0.630854	+	0.775902i	$0.282705\pi$
$318$	−9.96981	−0.559079
$319$	−10.1701	−0.569418
$320$	0	0
$321$	5.92712	0.330820
$322$	−1.17070	−0.0652406
$323$	−8.35404	−0.464831
$324$	−28.1784	−1.56546
$325$	0	0
$326$	−13.6829	−0.757828
$327$	−31.4073	−1.73683
$328$	8.75589	0.483463
$329$	−6.10125	−0.336373
$330$	0	0
$331$	9.32838	0.512734	0.256367	−	0.966580i	$-0.417474\pi$
$331$	9.32838	0.512734	0.256367	+	0.966580i	$0.417474\pi$
$332$	12.5106	0.686608
$333$	7.97425	0.436986
$334$	−13.7798	−0.753997
$335$	0	0
$336$	9.44890	0.515480
$337$	27.6999	1.50891	0.754455	−	0.656352i	$-0.227901\pi$
$337$	27.6999	1.50891	0.754455	+	0.656352i	$0.227901\pi$
$338$	0	0
$339$	−38.1517	−2.07212
$340$	0	0
$341$	−23.7196	−1.28449
$342$	25.5244	1.38020
$343$	−18.2510	−0.985464
$344$	−14.9863	−0.808009
$345$	0	0
$346$	−2.95704	−0.158971
$347$	0.313956	0.0168540	0.00842702	−	0.999964i	$-0.497318\pi$
$347$	0.313956	0.0168540	0.00842702	+	0.999964i	$0.497318\pi$
$348$	7.41226	0.397339
$349$	−24.7427	−1.32445	−0.662223	−	0.749307i	$-0.730387\pi$
$349$	−24.7427	−1.32445	−0.662223	+	0.749307i	$0.730387\pi$
$350$	0	0
$351$	0	0
$352$	−15.4927	−0.825762
$353$	6.24575	0.332428	0.166214	−	0.986090i	$-0.446846\pi$
$353$	6.24575	0.332428	0.166214	+	0.986090i	$0.446846\pi$
$354$	−38.1924	−2.02990
$355$	0	0
$356$	1.32058	0.0699909
$357$	15.9854	0.846035
$358$	−2.53305	−0.133876
$359$	−16.3025	−0.860411	−0.430205	−	0.902731i	$-0.641559\pi$
$359$	−16.3025	−0.860411	−0.430205	+	0.902731i	$0.641559\pi$
$360$	0	0
$361$	−11.1668	−0.587725
$362$	5.26217	0.276574
$363$	−8.74447	−0.458966
$364$	0	0
$365$	0	0
$366$	−27.4073	−1.43260
$367$	−3.97707	−0.207602	−0.103801	−	0.994598i	$-0.533100\pi$
$367$	−3.97707	−0.207602	−0.103801	+	0.994598i	$0.533100\pi$
$368$	1.18404	0.0617225
$369$	−23.7518	−1.23647
$370$	0	0
$371$	4.30317	0.223409
$372$	17.2875	0.896313
$373$	30.2382	1.56567	0.782837	−	0.622226i	$-0.213772\pi$
$373$	30.2382	1.56567	0.782837	+	0.622226i	$0.213772\pi$
$374$	12.0639	0.623809
$375$	0	0
$376$	11.7583	0.606388
$377$	0	0
$378$	−31.2320	−1.60640
$379$	8.93672	0.459049	0.229524	−	0.973303i	$-0.426283\pi$
$379$	8.93672	0.459049	0.229524	+	0.973303i	$0.426283\pi$
$380$	0	0
$381$	34.3194	1.75824
$382$	−19.5343	−0.999464
$383$	26.9347	1.37630	0.688150	−	0.725569i	$-0.258423\pi$
$383$	26.9347	1.37630	0.688150	+	0.725569i	$0.258423\pi$
$384$	−1.72122	−0.0878359
$385$	0	0
$386$	−2.75947	−0.140453
$387$	40.6529	2.06650
$388$	−1.80307	−0.0915371
$389$	29.5512	1.49831	0.749153	−	0.662397i	$-0.230461\pi$
$389$	29.5512	1.49831	0.749153	+	0.662397i	$0.230461\pi$
$390$	0	0
$391$	2.00313	0.101303
$392$	13.7011	0.692009
$393$	36.2259	1.82736
$394$	−20.2931	−1.02235
$395$	0	0
$396$	24.5110	1.23173
$397$	20.2901	1.01833	0.509165	−	0.860669i	$-0.329954\pi$
$397$	20.2901	1.01833	0.509165	+	0.860669i	$0.329954\pi$
$398$	14.4695	0.725288
$399$	−14.9888	−0.750379
$400$	0	0
$401$	−11.8484	−0.591681	−0.295840	−	0.955237i	$-0.595600\pi$
$401$	−11.8484	−0.591681	−0.295840	+	0.955237i	$0.595600\pi$
$402$	−42.5046	−2.11994
$403$	0	0
$404$	−9.01562	−0.448544
$405$	0	0
$406$	4.81102	0.238767
$407$	−3.53399	−0.175173
$408$	−30.8069	−1.52517
$409$	−28.5977	−1.41407	−0.707033	−	0.707180i	$-0.749967\pi$
$409$	−28.5977	−1.41407	−0.707033	+	0.707180i	$0.749967\pi$
$410$	0	0
$411$	42.8814	2.11518
$412$	15.6561	0.771323
$413$	16.4846	0.811154
$414$	−6.12024	−0.300793
$415$	0	0
$416$	0	0
$417$	13.1338	0.643164
$418$	−11.3118	−0.553278
$419$	−14.9206	−0.728921	−0.364461	−	0.931219i	$-0.618747\pi$
$419$	−14.9206	−0.728921	−0.364461	+	0.931219i	$0.618747\pi$
$420$	0	0
$421$	31.6984	1.54489	0.772443	−	0.635085i	$-0.219035\pi$
$421$	31.6984	1.54489	0.772443	+	0.635085i	$0.219035\pi$
$422$	−8.69490	−0.423261
$423$	−31.8964	−1.55085
$424$	−8.29305	−0.402746
$425$	0	0
$426$	31.6317	1.53256
$427$	11.8295	0.572471
$428$	1.40713	0.0680162
$429$	0	0
$430$	0	0
$431$	40.3656	1.94434	0.972172	−	0.234268i	$-0.0752693\pi$
$431$	40.3656	1.94434	0.972172	+	0.234268i	$0.0752693\pi$
$432$	31.5880	1.51978
$433$	1.11898	0.0537746	0.0268873	−	0.999638i	$-0.491440\pi$
$433$	1.11898	0.0537746	0.0268873	+	0.999638i	$0.491440\pi$
$434$	11.2207	0.538609
$435$	0	0
$436$	−7.45625	−0.357090
$437$	−1.87825	−0.0898488
$438$	13.7498	0.656991
$439$	−35.7222	−1.70493	−0.852465	−	0.522785i	$-0.824893\pi$
$439$	−35.7222	−1.70493	−0.852465	+	0.522785i	$0.824893\pi$
$440$	0	0
$441$	−37.1665	−1.76983
$442$	0	0
$443$	−26.4988	−1.25899	−0.629497	−	0.777003i	$-0.716739\pi$
$443$	−26.4988	−1.25899	−0.629497	+	0.777003i	$0.716739\pi$
$444$	2.57566	0.122236
$445$	0	0
$446$	4.10997	0.194613
$447$	−4.89700	−0.231620
$448$	12.9454	0.611613
$449$	35.0612	1.65464	0.827321	−	0.561730i	$-0.189864\pi$
$449$	35.0612	1.65464	0.827321	+	0.561730i	$0.189864\pi$
$450$	0	0
$451$	10.5262	0.495661
$452$	−9.05743	−0.426025
$453$	62.1459	2.91987
$454$	−14.1640	−0.664751
$455$	0	0
$456$	28.8864	1.35273
$457$	−11.7334	−0.548866	−0.274433	−	0.961606i	$-0.588490\pi$
$457$	−11.7334	−0.548866	−0.274433	+	0.961606i	$0.588490\pi$
$458$	26.8873	1.25636
$459$	53.4396	2.49435
$460$	0	0
$461$	−20.7028	−0.964227	−0.482114	−	0.876109i	$-0.660131\pi$
$461$	−20.7028	−0.964227	−0.482114	+	0.876109i	$0.660131\pi$
$462$	21.6450	1.00702
$463$	−10.1342	−0.470976	−0.235488	−	0.971877i	$-0.575669\pi$
$463$	−10.1342	−0.470976	−0.235488	+	0.971877i	$0.575669\pi$
$464$	−4.86586	−0.225892
$465$	0	0
$466$	−9.67472	−0.448173
$467$	23.3950	1.08259	0.541296	−	0.840832i	$-0.317934\pi$
$467$	23.3950	1.08259	0.541296	+	0.840832i	$0.317934\pi$
$468$	0	0
$469$	18.3459	0.847133
$470$	0	0
$471$	37.9843	1.75022
$472$	−31.7691	−1.46229
$473$	−18.0164	−0.828394
$474$	−15.9839	−0.734166
$475$	0	0
$476$	3.79501	0.173944
$477$	22.4963	1.03003
$478$	−4.64245	−0.212341
$479$	27.9429	1.27674	0.638372	−	0.769728i	$-0.279608\pi$
$479$	27.9429	1.27674	0.638372	+	0.769728i	$0.279608\pi$
$480$	0	0
$481$	0	0
$482$	19.8767	0.905357
$483$	3.59401	0.163533
$484$	−2.07598	−0.0943629
$485$	0	0
$486$	−71.2208	−3.23064
$487$	7.39658	0.335171	0.167586	−	0.985858i	$-0.446403\pi$
$487$	7.39658	0.335171	0.167586	+	0.985858i	$0.446403\pi$
$488$	−22.7978	−1.03201
$489$	42.0062	1.89958
$490$	0	0
$491$	−9.46851	−0.427308	−0.213654	−	0.976909i	$-0.568536\pi$
$491$	−9.46851	−0.427308	−0.213654	+	0.976909i	$0.568536\pi$
$492$	−7.67179	−0.345871
$493$	−8.23191	−0.370747
$494$	0	0
$495$	0	0
$496$	−11.3485	−0.509565
$497$	−13.6529	−0.612416
$498$	57.7560	2.58811
$499$	30.6022	1.36994	0.684971	−	0.728571i	$-0.259815\pi$
$499$	30.6022	1.36994	0.684971	+	0.728571i	$0.259815\pi$
$500$	0	0
$501$	42.3035	1.88998
$502$	−1.18569	−0.0529197
$503$	−23.9209	−1.06658	−0.533290	−	0.845933i	$-0.679044\pi$
$503$	−23.9209	−1.06658	−0.533290	+	0.845933i	$0.679044\pi$
$504$	−40.6265	−1.80965
$505$	0	0
$506$	2.71234	0.120578
$507$	0	0
$508$	8.14761	0.361492
$509$	−3.44244	−0.152584	−0.0762918	−	0.997086i	$-0.524308\pi$
$509$	−3.44244	−0.152584	−0.0762918	+	0.997086i	$0.524308\pi$
$510$	0	0
$511$	−5.93469	−0.262535
$512$	−18.2366	−0.805951
$513$	−50.1081	−2.21233
$514$	8.92755	0.393777
$515$	0	0
$516$	13.1308	0.578051
$517$	14.1357	0.621687
$518$	1.67177	0.0734533
$519$	9.07801	0.398480
$520$	0	0
$521$	−5.33204	−0.233601	−0.116800	−	0.993155i	$-0.537264\pi$
$521$	−5.33204	−0.233601	−0.116800	+	0.993155i	$0.537264\pi$
$522$	25.1513	1.10084
$523$	3.15090	0.137779	0.0688897	−	0.997624i	$-0.478054\pi$
$523$	3.15090	0.137779	0.0688897	+	0.997624i	$0.478054\pi$
$524$	8.60023	0.375703
$525$	0	0
$526$	−18.9212	−0.825005
$527$	−19.1991	−0.836327
$528$	−21.8917	−0.952714
$529$	−22.5496	−0.980419
$530$	0	0
$531$	86.1789	3.73985
$532$	−3.55842	−0.154277
$533$	0	0
$534$	6.09656	0.263824
$535$	0	0
$536$	−35.3561	−1.52715
$537$	7.77636	0.335575
$538$	−20.7763	−0.895732
$539$	16.4713	0.709468
$540$	0	0
$541$	−3.22739	−0.138757	−0.0693783	−	0.997590i	$-0.522102\pi$
$541$	−3.22739	−0.138757	−0.0693783	+	0.997590i	$0.522102\pi$
$542$	−25.6019	−1.09969
$543$	−16.1547	−0.693264
$544$	−12.5401	−0.537652
$545$	0	0
$546$	0	0
$547$	7.09772	0.303476	0.151738	−	0.988421i	$-0.451513\pi$
$547$	7.09772	0.303476	0.151738	+	0.988421i	$0.451513\pi$
$548$	10.1803	0.434880
$549$	61.8429	2.63939
$550$	0	0
$551$	7.71871	0.328828
$552$	−6.92636	−0.294806
$553$	6.89898	0.293375
$554$	6.58391	0.279724
$555$	0	0
$556$	3.11803	0.132234
$557$	−38.4073	−1.62737	−0.813685	−	0.581306i	$-0.802542\pi$
$557$	−38.4073	−1.62737	−0.813685	+	0.581306i	$0.802542\pi$
$558$	58.6599	2.48327
$559$	0	0
$560$	0	0
$561$	−37.0357	−1.56365
$562$	−19.8376	−0.836801
$563$	−28.3980	−1.19683	−0.598416	−	0.801186i	$-0.704203\pi$
$563$	−28.3980	−1.19683	−0.598416	+	0.801186i	$0.704203\pi$
$564$	−10.3025	−0.433812
$565$	0	0
$566$	−2.62087	−0.110163
$567$	56.1483	2.35801
$568$	26.3118	1.10402
$569$	−24.5349	−1.02856	−0.514279	−	0.857623i	$-0.671940\pi$
$569$	−24.5349	−1.02856	−0.514279	+	0.857623i	$0.671940\pi$
$570$	0	0
$571$	−35.1010	−1.46893	−0.734465	−	0.678646i	$-0.762567\pi$
$571$	−35.1010	−1.46893	−0.734465	+	0.678646i	$0.762567\pi$
$572$	0	0
$573$	59.9698	2.50527
$574$	−4.97948	−0.207839
$575$	0	0
$576$	67.6766	2.81986
$577$	42.0461	1.75040	0.875202	−	0.483758i	$-0.160729\pi$
$577$	42.0461	1.75040	0.875202	+	0.483758i	$0.160729\pi$
$578$	−8.86718	−0.368826
$579$	8.47146	0.352062
$580$	0	0
$581$	−24.9286	−1.03421
$582$	−8.32399	−0.345041
$583$	−9.96981	−0.412907
$584$	11.4373	0.473279
$585$	0	0
$586$	−16.1940	−0.668969
$587$	40.4856	1.67102	0.835509	−	0.549476i	$-0.185173\pi$
$587$	40.4856	1.67102	0.835509	+	0.549476i	$0.185173\pi$
$588$	−12.0047	−0.495065
$589$	18.0022	0.741768
$590$	0	0
$591$	62.2990	2.56264
$592$	−1.69082	−0.0694924
$593$	3.00502	0.123402	0.0617008	−	0.998095i	$-0.480348\pi$
$593$	3.00502	0.123402	0.0617008	+	0.998095i	$0.480348\pi$
$594$	72.3600	2.96897
$595$	0	0
$596$	−1.16257	−0.0476209
$597$	−44.4207	−1.81802
$598$	0	0
$599$	20.3595	0.831865	0.415932	−	0.909396i	$-0.363455\pi$
$599$	20.3595	0.831865	0.415932	+	0.909396i	$0.363455\pi$
$600$	0	0
$601$	−39.2246	−1.60001	−0.800003	−	0.599996i	$-0.795169\pi$
$601$	−39.2246	−1.60001	−0.800003	+	0.599996i	$0.795169\pi$
$602$	8.52273	0.347360
$603$	95.9093	3.90573
$604$	14.7538	0.600322
$605$	0	0
$606$	−41.6212	−1.69074
$607$	−15.4443	−0.626863	−0.313432	−	0.949611i	$-0.601479\pi$
$607$	−15.4443	−0.626863	−0.313432	+	0.949611i	$0.601479\pi$
$608$	11.7583	0.476862
$609$	−14.7697	−0.598497
$610$	0	0
$611$	0	0
$612$	19.8397	0.801974
$613$	−27.1583	−1.09691	−0.548457	−	0.836179i	$-0.684785\pi$
$613$	−27.1583	−1.09691	−0.548457	+	0.836179i	$0.684785\pi$
$614$	−16.1075	−0.650044
$615$	0	0
$616$	18.0047	0.725429
$617$	26.9739	1.08593	0.542964	−	0.839756i	$-0.317302\pi$
$617$	26.9739	1.08593	0.542964	+	0.839756i	$0.317302\pi$
$618$	72.2776	2.90743
$619$	−38.2116	−1.53586	−0.767928	−	0.640537i	$-0.778712\pi$
$619$	−38.2116	−1.53586	−0.767928	+	0.640537i	$0.778712\pi$
$620$	0	0
$621$	12.0149	0.482141
$622$	−15.5667	−0.624168
$623$	−2.63140	−0.105425
$624$	0	0
$625$	0	0
$626$	23.0509	0.921300
$627$	34.7268	1.38686
$628$	9.01766	0.359844
$629$	−2.86048	−0.114055
$630$	0	0
$631$	11.8977	0.473641	0.236821	−	0.971553i	$-0.423895\pi$
$631$	11.8977	0.473641	0.236821	+	0.971553i	$0.423895\pi$
$632$	−13.2957	−0.528874
$633$	26.6930	1.06095
$634$	24.6205	0.977807
$635$	0	0
$636$	7.26625	0.288126
$637$	0	0
$638$	−11.1464	−0.441291
$639$	−71.3752	−2.82356
$640$	0	0
$641$	−47.9174	−1.89262	−0.946312	−	0.323255i	$-0.895223\pi$
$641$	−47.9174	−1.89262	−0.946312	+	0.323255i	$0.895223\pi$
$642$	6.49610	0.256381
$643$	−30.7699	−1.21344	−0.606722	−	0.794914i	$-0.707516\pi$
$643$	−30.7699	−1.21344	−0.606722	+	0.794914i	$0.707516\pi$
$644$	0.853237	0.0336222
$645$	0	0
$646$	−9.15600	−0.360238
$647$	−34.3450	−1.35024	−0.675121	−	0.737707i	$-0.735908\pi$
$647$	−34.3450	−1.35024	−0.675121	+	0.737707i	$0.735908\pi$
$648$	−108.209	−4.25084
$649$	−38.1924	−1.49918
$650$	0	0
$651$	−34.4470	−1.35009
$652$	9.97249	0.390553
$653$	23.4692	0.918422	0.459211	−	0.888327i	$-0.348132\pi$
$653$	23.4692	0.918422	0.459211	+	0.888327i	$0.348132\pi$
$654$	−34.4223	−1.34602
$655$	0	0
$656$	5.03623	0.196632
$657$	−31.0256	−1.21042
$658$	−6.68695	−0.260684
$659$	48.8358	1.90237	0.951187	−	0.308615i	$-0.0998654\pi$
$659$	48.8358	1.90237	0.951187	+	0.308615i	$0.0998654\pi$
$660$	0	0
$661$	−23.8833	−0.928951	−0.464476	−	0.885586i	$-0.653757\pi$
$661$	−23.8833	−0.928951	−0.464476	+	0.885586i	$0.653757\pi$
$662$	10.2239	0.397362
$663$	0	0
$664$	48.0424	1.86441
$665$	0	0
$666$	8.73975	0.338658
$667$	−1.85079	−0.0716629
$668$	10.0431	0.388578
$669$	−12.6175	−0.487819
$670$	0	0
$671$	−27.4073	−1.05805
$672$	−22.4994	−0.867933
$673$	−28.8033	−1.11029	−0.555143	−	0.831755i	$-0.687337\pi$
$673$	−28.8033	−1.11029	−0.555143	+	0.831755i	$0.687337\pi$
$674$	30.3590	1.16938
$675$	0	0
$676$	0	0
$677$	−6.71561	−0.258102	−0.129051	−	0.991638i	$-0.541193\pi$
$677$	−6.71561	−0.258102	−0.129051	+	0.991638i	$0.541193\pi$
$678$	−41.8142	−1.60586
$679$	3.59280	0.137879
$680$	0	0
$681$	43.4831	1.66627
$682$	−25.9966	−0.995461
$683$	5.33129	0.203996	0.101998	−	0.994785i	$-0.467476\pi$
$683$	5.33129	0.203996	0.101998	+	0.994785i	$0.467476\pi$
$684$	−18.6029	−0.711299
$685$	0	0
$686$	−20.0031	−0.763721
$687$	−82.5429	−3.14921
$688$	−8.61986	−0.328629
$689$	0	0
$690$	0	0
$691$	−44.3655	−1.68774	−0.843871	−	0.536546i	$-0.819729\pi$
$691$	−44.3655	−1.68774	−0.843871	+	0.536546i	$0.819729\pi$
$692$	2.15517	0.0819272
$693$	−48.8407	−1.85531
$694$	0.344095	0.0130617
$695$	0	0
$696$	28.4641	1.07893
$697$	8.52014	0.322723
$698$	−27.1179	−1.02643
$699$	29.7010	1.12340
$700$	0	0
$701$	40.0973	1.51445	0.757227	−	0.653151i	$-0.226553\pi$
$701$	40.0973	1.51445	0.757227	+	0.653151i	$0.226553\pi$
$702$	0	0
$703$	2.68215	0.101159
$704$	−29.9926	−1.13039
$705$	0	0
$706$	6.84532	0.257627
$707$	17.9645	0.675626
$708$	27.8356	1.04613
$709$	1.60156	0.0601477	0.0300739	−	0.999548i	$-0.490426\pi$
$709$	1.60156	0.0601477	0.0300739	+	0.999548i	$0.490426\pi$
$710$	0	0
$711$	36.0668	1.35261
$712$	5.07123	0.190052
$713$	−4.31656	−0.161657
$714$	17.5199	0.655666
$715$	0	0
$716$	1.84615	0.0689938
$717$	14.2522	0.532256
$718$	−17.8674	−0.666807
$719$	−1.65323	−0.0616551	−0.0308275	−	0.999525i	$-0.509814\pi$
$719$	−1.65323	−0.0616551	−0.0308275	+	0.999525i	$0.509814\pi$
$720$	0	0
$721$	−31.1965	−1.16182
$722$	−12.2387	−0.455479
$723$	−61.0206	−2.26938
$724$	−3.83521	−0.142534
$725$	0	0
$726$	−9.58391	−0.355692
$727$	5.56875	0.206533	0.103267	−	0.994654i	$-0.467070\pi$
$727$	5.56875	0.206533	0.103267	+	0.994654i	$0.467070\pi$
$728$	0	0
$729$	112.816	4.17839
$730$	0	0
$731$	−14.5828	−0.539365
$732$	19.9751	0.738302
$733$	36.1927	1.33681	0.668403	−	0.743799i	$-0.266978\pi$
$733$	36.1927	1.33681	0.668403	+	0.743799i	$0.266978\pi$
$734$	−4.35886	−0.160888
$735$	0	0
$736$	−2.81940	−0.103925
$737$	−42.5046	−1.56568
$738$	−26.0319	−0.958249
$739$	−27.3935	−1.00768	−0.503842	−	0.863796i	$-0.668081\pi$
$739$	−27.3935	−1.00768	−0.503842	+	0.863796i	$0.668081\pi$
$740$	0	0
$741$	0	0
$742$	4.71626	0.173139
$743$	30.0245	1.10149	0.550746	−	0.834673i	$-0.314343\pi$
$743$	30.0245	1.10149	0.550746	+	0.834673i	$0.314343\pi$
$744$	66.3862	2.43384
$745$	0	0
$746$	33.1410	1.21338
$747$	−130.323	−4.76827
$748$	−8.79248	−0.321485
$749$	−2.80385	−0.102450
$750$	0	0
$751$	26.4906	0.966656	0.483328	−	0.875439i	$-0.339428\pi$
$751$	26.4906	0.966656	0.483328	+	0.875439i	$0.339428\pi$
$752$	6.76316	0.246627
$753$	3.64001	0.132649
$754$	0	0
$755$	0	0
$756$	22.7627	0.827872
$757$	−11.0584	−0.401924	−0.200962	−	0.979599i	$-0.564407\pi$
$757$	−11.0584	−0.401924	−0.200962	+	0.979599i	$0.564407\pi$
$758$	9.79462	0.355757
$759$	−8.32679	−0.302243
$760$	0	0
$761$	10.0378	0.363870	0.181935	−	0.983311i	$-0.441764\pi$
$761$	10.0378	0.363870	0.181935	+	0.983311i	$0.441764\pi$
$762$	37.6139	1.36261
$763$	14.8573	0.537872
$764$	14.2371	0.515082
$765$	0	0
$766$	29.5204	1.06661
$767$	0	0
$768$	52.8447	1.90687
$769$	5.19883	0.187474	0.0937372	−	0.995597i	$-0.470119\pi$
$769$	5.19883	0.187474	0.0937372	+	0.995597i	$0.470119\pi$
$770$	0	0
$771$	−27.4073	−0.987049
$772$	2.01117	0.0723836
$773$	2.54673	0.0915995	0.0457998	−	0.998951i	$-0.485416\pi$
$773$	2.54673	0.0915995	0.0457998	+	0.998951i	$0.485416\pi$
$774$	44.5555	1.60151
$775$	0	0
$776$	−6.92404	−0.248558
$777$	−5.13227	−0.184119
$778$	32.3880	1.16117
$779$	−7.98898	−0.286235
$780$	0	0
$781$	31.6317	1.13187
$782$	2.19542	0.0785082
$783$	−49.3755	−1.76454
$784$	7.88061	0.281450
$785$	0	0
$786$	39.7035	1.41618
$787$	40.7834	1.45377	0.726886	−	0.686758i	$-0.240967\pi$
$787$	40.7834	1.45377	0.726886	+	0.686758i	$0.240967\pi$
$788$	14.7901	0.526876
$789$	58.0875	2.06797
$790$	0	0
$791$	18.0478	0.641708
$792$	94.1257	3.34461
$793$	0	0
$794$	22.2378	0.789192
$795$	0	0
$796$	−10.5457	−0.373783
$797$	−17.7372	−0.628285	−0.314143	−	0.949376i	$-0.601717\pi$
$797$	−17.7372	−0.628285	−0.314143	+	0.949376i	$0.601717\pi$
$798$	−16.4277	−0.581534
$799$	11.4417	0.404779
$800$	0	0
$801$	−13.7565	−0.486064
$802$	−12.9858	−0.458545
$803$	13.7498	0.485220
$804$	30.9785	1.09253
$805$	0	0
$806$	0	0
$807$	63.7826	2.24526
$808$	−34.6212	−1.21797
$809$	34.0208	1.19611	0.598054	−	0.801456i	$-0.295941\pi$
$809$	34.0208	1.19611	0.598054	+	0.801456i	$0.295941\pi$
$810$	0	0
$811$	2.94630	0.103459	0.0517294	−	0.998661i	$-0.483527\pi$
$811$	2.94630	0.103459	0.0517294	+	0.998661i	$0.483527\pi$
$812$	−3.50640	−0.123051
$813$	78.5968	2.75651
$814$	−3.87324	−0.135757
$815$	0	0
$816$	−17.7196	−0.620310
$817$	13.6737	0.478382
$818$	−31.3430	−1.09588
$819$	0	0
$820$	0	0
$821$	8.51106	0.297038	0.148519	−	0.988910i	$-0.452549\pi$
$821$	8.51106	0.297038	0.148519	+	0.988910i	$0.452549\pi$
$822$	46.9979	1.63924
$823$	36.7022	1.27936	0.639678	−	0.768643i	$-0.279068\pi$
$823$	36.7022	1.27936	0.639678	+	0.768643i	$0.279068\pi$
$824$	60.1217	2.09444
$825$	0	0
$826$	18.0671	0.628634
$827$	3.32022	0.115455	0.0577277	−	0.998332i	$-0.481614\pi$
$827$	3.32022	0.115455	0.0577277	+	0.998332i	$0.481614\pi$
$828$	4.46059	0.155016
$829$	−36.1423	−1.25527	−0.627637	−	0.778506i	$-0.715978\pi$
$829$	−36.1423	−1.25527	−0.627637	+	0.778506i	$0.715978\pi$
$830$	0	0
$831$	−20.2124	−0.701159
$832$	0	0
$833$	13.3322	0.461933
$834$	14.3946	0.498444
$835$	0	0
$836$	8.24433	0.285136
$837$	−115.158	−3.98043
$838$	−16.3530	−0.564904
$839$	21.6828	0.748573	0.374286	−	0.927313i	$-0.377888\pi$
$839$	21.6828	0.748573	0.374286	+	0.927313i	$0.377888\pi$
$840$	0	0
$841$	−21.3941	−0.737729
$842$	34.7413	1.19727
$843$	60.9009	2.09754
$844$	6.33707	0.218131
$845$	0	0
$846$	−34.9583	−1.20189
$847$	4.13661	0.142136
$848$	−4.77001	−0.163803
$849$	8.04598	0.276137
$850$	0	0
$851$	−0.643126	−0.0220461
$852$	−23.0540	−0.789818
$853$	12.0199	0.411553	0.205776	−	0.978599i	$-0.434028\pi$
$853$	12.0199	0.411553	0.205776	+	0.978599i	$0.434028\pi$
$854$	12.9651	0.443658
$855$	0	0
$856$	5.40357	0.184690
$857$	−47.3024	−1.61582	−0.807910	−	0.589306i	$-0.799401\pi$
$857$	−47.3024	−1.61582	−0.807910	+	0.589306i	$0.799401\pi$
$858$	0	0
$859$	−19.5027	−0.665423	−0.332712	−	0.943029i	$-0.607964\pi$
$859$	−19.5027	−0.665423	−0.332712	+	0.943029i	$0.607964\pi$
$860$	0	0
$861$	15.2868	0.520973
$862$	44.2406	1.50684
$863$	−31.5568	−1.07420	−0.537102	−	0.843517i	$-0.680481\pi$
$863$	−31.5568	−1.07420	−0.537102	+	0.843517i	$0.680481\pi$
$864$	−75.2162	−2.55891
$865$	0	0
$866$	1.22640	0.0416746
$867$	27.2219	0.924506
$868$	−8.17791	−0.277576
$869$	−15.9839	−0.542217
$870$	0	0
$871$	0	0
$872$	−28.6330	−0.969637
$873$	18.7826	0.635695
$874$	−2.05855	−0.0696317
$875$	0	0
$876$	−10.0212	−0.338585
$877$	−10.4086	−0.351472	−0.175736	−	0.984437i	$-0.556231\pi$
$877$	−10.4086	−0.351472	−0.175736	+	0.984437i	$0.556231\pi$
$878$	−39.1515	−1.32130
$879$	49.7151	1.67685
$880$	0	0
$881$	−6.01677	−0.202710	−0.101355	−	0.994850i	$-0.532318\pi$
$881$	−6.01677	−0.202710	−0.101355	+	0.994850i	$0.532318\pi$
$882$	−40.7343	−1.37160
$883$	18.0630	0.607869	0.303934	−	0.952693i	$-0.401700\pi$
$883$	18.0630	0.607869	0.303934	+	0.952693i	$0.401700\pi$
$884$	0	0
$885$	0	0
$886$	−29.0426	−0.975704
$887$	−13.7877	−0.462947	−0.231473	−	0.972841i	$-0.574355\pi$
$887$	−13.7877	−0.462947	−0.231473	+	0.972841i	$0.574355\pi$
$888$	9.89090	0.331917
$889$	−16.2349	−0.544502
$890$	0	0
$891$	−130.087	−4.35809
$892$	−2.99545	−0.100295
$893$	−10.7284	−0.359013
$894$	−5.36709	−0.179502
$895$	0	0
$896$	0.814233	0.0272016
$897$	0	0
$898$	38.4270	1.28233
$899$	17.7390	0.591630
$900$	0	0
$901$	−8.06976	−0.268843
$902$	11.5367	0.384130
$903$	−26.1645	−0.870699
$904$	−34.7817	−1.15682
$905$	0	0
$906$	68.1117	2.26286
$907$	15.6345	0.519136	0.259568	−	0.965725i	$-0.416420\pi$
$907$	15.6345	0.519136	0.259568	+	0.965725i	$0.416420\pi$
$908$	10.3231	0.342584
$909$	93.9158	3.11499
$910$	0	0
$911$	−34.5627	−1.14511	−0.572557	−	0.819865i	$-0.694048\pi$
$911$	−34.5627	−1.14511	−0.572557	+	0.819865i	$0.694048\pi$
$912$	16.6149	0.550175
$913$	57.7560	1.91144
$914$	−12.8598	−0.425364
$915$	0	0
$916$	−19.5961	−0.647474
$917$	−17.1368	−0.565908
$918$	58.5697	1.93309
$919$	−29.4713	−0.972167	−0.486084	−	0.873912i	$-0.661575\pi$
$919$	−29.4713	−0.972167	−0.486084	+	0.873912i	$0.661575\pi$
$920$	0	0
$921$	49.4493	1.62941
$922$	−22.6902	−0.747263
$923$	0	0
$924$	−15.7754	−0.518974
$925$	0	0
$926$	−11.1070	−0.365000
$927$	−163.090	−5.35658
$928$	11.5864	0.380342
$929$	−28.4764	−0.934280	−0.467140	−	0.884183i	$-0.654716\pi$
$929$	−28.4764	−0.934280	−0.467140	+	0.884183i	$0.654716\pi$
$930$	0	0
$931$	−12.5010	−0.409704
$932$	7.05119	0.230969
$933$	47.7893	1.56455
$934$	25.6409	0.838995
$935$	0	0
$936$	0	0
$937$	−36.3202	−1.18653	−0.593264	−	0.805008i	$-0.702161\pi$
$937$	−36.3202	−1.18653	−0.593264	+	0.805008i	$0.702161\pi$
$938$	20.1070	0.656517
$939$	−70.7655	−2.30935
$940$	0	0
$941$	−46.7007	−1.52240	−0.761200	−	0.648517i	$-0.775390\pi$
$941$	−46.7007	−1.52240	−0.761200	+	0.648517i	$0.775390\pi$
$942$	41.6306	1.35640
$943$	1.91559	0.0623803
$944$	−18.2730	−0.594735
$945$	0	0
$946$	−19.7459	−0.641994
$947$	14.9627	0.486221	0.243110	−	0.969999i	$-0.421832\pi$
$947$	14.9627	0.486221	0.243110	+	0.969999i	$0.421832\pi$
$948$	11.6495	0.378358
$949$	0	0
$950$	0	0
$951$	−75.5842	−2.45099
$952$	14.5734	0.472325
$953$	−29.9584	−0.970447	−0.485224	−	0.874390i	$-0.661262\pi$
$953$	−29.9584	−0.970447	−0.485224	+	0.874390i	$0.661262\pi$
$954$	24.6559	0.798263
$955$	0	0
$956$	3.38354	0.109431
$957$	34.2191	1.10615
$958$	30.6253	0.989459
$959$	−20.2852	−0.655044
$960$	0	0
$961$	10.3724	0.334593
$962$	0	0
$963$	−14.6581	−0.472350
$964$	−14.4866	−0.466583
$965$	0	0
$966$	3.93902	0.126736
$967$	−13.5633	−0.436166	−0.218083	−	0.975930i	$-0.569980\pi$
$967$	−13.5633	−0.436166	−0.218083	+	0.975930i	$0.569980\pi$
$968$	−7.97206	−0.256232
$969$	28.1086	0.902979
$970$	0	0
$971$	−36.3429	−1.16630	−0.583150	−	0.812364i	$-0.698180\pi$
$971$	−36.3429	−1.16630	−0.583150	+	0.812364i	$0.698180\pi$
$972$	51.9075	1.66494
$973$	−6.21299	−0.199179
$974$	8.10663	0.259753
$975$	0	0
$976$	−13.1129	−0.419734
$977$	−2.76715	−0.0885291	−0.0442645	−	0.999020i	$-0.514094\pi$
$977$	−2.76715	−0.0885291	−0.0442645	+	0.999020i	$0.514094\pi$
$978$	46.0386	1.47215
$979$	6.09656	0.194847
$980$	0	0
$981$	77.6719	2.47987
$982$	−10.3775	−0.331158
$983$	52.2816	1.66752	0.833762	−	0.552124i	$-0.186183\pi$
$983$	52.2816	1.66752	0.833762	+	0.552124i	$0.186183\pi$
$984$	−29.4607	−0.939173
$985$	0	0
$986$	−9.02215	−0.287324
$987$	20.5287	0.653436
$988$	0	0
$989$	−3.27867	−0.104256
$990$	0	0
$991$	−2.59286	−0.0823650	−0.0411825	−	0.999152i	$-0.513112\pi$
$991$	−2.59286	−0.0823650	−0.0411825	+	0.999152i	$0.513112\pi$
$992$	27.0228	0.857973
$993$	−31.3869	−0.996034
$994$	−14.9635	−0.474614
$995$	0	0
$996$	−42.0941	−1.33380
$997$	17.1749	0.543935	0.271968	−	0.962306i	$-0.412326\pi$
$997$	17.1749	0.543935	0.271968	+	0.962306i	$0.412326\pi$
$998$	33.5399	1.06169
$999$	−17.1574	−0.542835

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4225.2.a.bm.1.4		5
5.4	even	2		4225.2.a.bo.1.2		5
13.4	even	6		325.2.e.d.276.4	yes	10
13.10	even	6		325.2.e.d.126.4	yes	10
13.12	even	2		4225.2.a.bn.1.2		5
65.4	even	6		325.2.e.c.276.2	yes	10
65.17	odd	12		325.2.o.c.224.4		20
65.23	odd	12		325.2.o.c.74.4		20
65.43	odd	12		325.2.o.c.224.7		20
65.49	even	6		325.2.e.c.126.2	✓	10
65.62	odd	12		325.2.o.c.74.7		20
65.64	even	2		4225.2.a.bp.1.4		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
325.2.e.c.126.2	✓	10	65.49	even	6
325.2.e.c.276.2	yes	10	65.4	even	6
325.2.e.d.126.4	yes	10	13.10	even	6
325.2.e.d.276.4	yes	10	13.4	even	6
325.2.o.c.74.4		20	65.23	odd	12
325.2.o.c.74.7		20	65.62	odd	12
325.2.o.c.224.4		20	65.17	odd	12
325.2.o.c.224.7		20	65.43	odd	12
4225.2.a.bm.1.4		5	1.1	even	1	trivial
4225.2.a.bn.1.2		5	13.12	even	2
4225.2.a.bo.1.2		5	5.4	even	2
4225.2.a.bp.1.4		5	65.64	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:	\( N \)	\(=\)	\( 4225 = 5^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4225.a (trivial)

\(f(q)\)	\(=\)	\(q+1.09600 q^{2} -3.36467 q^{3} -0.798791 q^{4} -3.68767 q^{6} +1.59167 q^{7} -3.06747 q^{8} +8.32102 q^{9} +O(q^{10})\)	\(q+1.09600 q^{2} -3.36467 q^{3} -0.798791 q^{4} -3.68767 q^{6} +1.59167 q^{7} -3.06747 q^{8} +8.32102 q^{9} -3.68767 q^{11} +2.68767 q^{12} +1.74447 q^{14} -1.76435 q^{16} -2.98488 q^{17} +9.11981 q^{18} +2.79879 q^{19} -5.35546 q^{21} -4.04167 q^{22} -0.671093 q^{23} +10.3210 q^{24} -17.9035 q^{27} -1.27141 q^{28} +2.75787 q^{29} +6.43214 q^{31} +4.20121 q^{32} +12.4078 q^{33} -3.27141 q^{34} -6.64675 q^{36} +0.958326 q^{37} +3.06747 q^{38} -2.85444 q^{41} -5.86956 q^{42} +4.88557 q^{43} +2.94568 q^{44} -0.735516 q^{46} -3.83323 q^{47} +5.93646 q^{48} -4.46658 q^{49} +10.0431 q^{51} +2.70355 q^{53} -19.6222 q^{54} -4.88240 q^{56} -9.41701 q^{57} +3.02262 q^{58} +10.3568 q^{59} +7.43214 q^{61} +7.04960 q^{62} +13.2443 q^{63} +8.13321 q^{64} +13.5989 q^{66} +11.5262 q^{67} +2.38429 q^{68} +2.25801 q^{69} -8.57770 q^{71} -25.5244 q^{72} -3.72859 q^{73} +1.05032 q^{74} -2.23565 q^{76} -5.86956 q^{77} +4.33442 q^{79} +35.2763 q^{81} -3.12846 q^{82} -15.6619 q^{83} +4.27789 q^{84} +5.35457 q^{86} -9.27934 q^{87} +11.3118 q^{88} -1.65323 q^{89} +0.536063 q^{92} -21.6420 q^{93} -4.20121 q^{94} -14.1357 q^{96} +2.25725 q^{97} -4.89536 q^{98} -30.6852 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 3 q^{3} + 6 q^{4} - 3 q^{6} - 2 q^{7} + 3 q^{8} + 4 q^{9}+O(q^{10}) \) 5 * q - 3 * q^3 + 6 * q^4 - 3 * q^6 - 2 * q^7 + 3 * q^8 + 4 * q^9	\( 5 q - 3 q^{3} + 6 q^{4} - 3 q^{6} - 2 q^{7} + 3 q^{8} + 4 q^{9} - 3 q^{11} - 2 q^{12} - 8 q^{14} + 4 q^{16} - 4 q^{17} - 2 q^{18} + 4 q^{19} + 8 q^{21} - 8 q^{22} - 15 q^{23} + 14 q^{24} - 21 q^{27} - 17 q^{28} - q^{29} + 31 q^{32} - 2 q^{33} - 27 q^{34} - 13 q^{36} + 17 q^{37} - 3 q^{38} - 6 q^{41} - 32 q^{42} - 12 q^{43} + 8 q^{44} + 7 q^{46} - 12 q^{47} + 2 q^{48} + 7 q^{49} - 8 q^{53} - 19 q^{54} - 17 q^{56} - 4 q^{57} + 38 q^{58} + 12 q^{59} + 5 q^{61} - 13 q^{62} + 26 q^{63} - 5 q^{64} + 43 q^{66} - 16 q^{67} - 25 q^{68} + 20 q^{69} - 19 q^{71} - 45 q^{72} - 8 q^{73} + 2 q^{74} - 24 q^{76} - 32 q^{77} - 14 q^{79} + 29 q^{81} + 23 q^{82} - 7 q^{83} + 34 q^{84} + 42 q^{86} - 21 q^{87} - 2 q^{88} + 10 q^{89} - 71 q^{92} - 33 q^{93} - 31 q^{94} - 17 q^{96} + 37 q^{97} - 21 q^{98} - 28 q^{99}+O(q^{100}) \) 5 * q - 3 * q^3 + 6 * q^4 - 3 * q^6 - 2 * q^7 + 3 * q^8 + 4 * q^9 - 3 * q^11 - 2 * q^12 - 8 * q^14 + 4 * q^16 - 4 * q^17 - 2 * q^18 + 4 * q^19 + 8 * q^21 - 8 * q^22 - 15 * q^23 + 14 * q^24 - 21 * q^27 - 17 * q^28 - q^29 + 31 * q^32 - 2 * q^33 - 27 * q^34 - 13 * q^36 + 17 * q^37 - 3 * q^38 - 6 * q^41 - 32 * q^42 - 12 * q^43 + 8 * q^44 + 7 * q^46 - 12 * q^47 + 2 * q^48 + 7 * q^49 - 8 * q^53 - 19 * q^54 - 17 * q^56 - 4 * q^57 + 38 * q^58 + 12 * q^59 + 5 * q^61 - 13 * q^62 + 26 * q^63 - 5 * q^64 + 43 * q^66 - 16 * q^67 - 25 * q^68 + 20 * q^69 - 19 * q^71 - 45 * q^72 - 8 * q^73 + 2 * q^74 - 24 * q^76 - 32 * q^77 - 14 * q^79 + 29 * q^81 + 23 * q^82 - 7 * q^83 + 34 * q^84 + 42 * q^86 - 21 * q^87 - 2 * q^88 + 10 * q^89 - 71 * q^92 - 33 * q^93 - 31 * q^94 - 17 * q^96 + 37 * q^97 - 21 * q^98 - 28 * q^99

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