LMFDB - Embedded newform 9025.2.a.bh.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9025,2,Mod(1,9025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("9025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9025 = 5^{2} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9025.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$72.0649878242$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.2225.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 2x + 4 $ x^4 - x^3 - 5x^2 + 2x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1805)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.75660$ of defining polynomial
Character	$\chi$	$=$	9025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.75660 q^{2} +0.0856374 q^{3} +1.08564 q^{4} +0.150431 q^{6} +1.86144 q^{7} -1.60617 q^{8} -2.99267 q^{9} +O(q^{10})$	\(q+1.75660 q^{2} +0.0856374 q^{3} +1.08564 q^{4} +0.150431 q^{6} +1.86144 q^{7} -1.60617 q^{8} -2.99267 q^{9} -1.22420 q^{11} +0.0929712 q^{12} -2.40734 q^{13} +3.26979 q^{14} -4.99267 q^{16} -6.05910 q^{17} -5.25691 q^{18} +0.159409 q^{21} -2.15043 q^{22} +7.20500 q^{23} -0.137548 q^{24} -4.22873 q^{26} -0.513197 q^{27} +2.02084 q^{28} -1.28446 q^{29} +5.87596 q^{31} -5.55777 q^{32} -0.104837 q^{33} -10.6434 q^{34} -3.24895 q^{36} -1.59883 q^{37} -0.206159 q^{39} +9.72553 q^{41} +0.280017 q^{42} +0.697495 q^{43} -1.32904 q^{44} +12.6563 q^{46} +11.6434 q^{47} -0.427559 q^{48} -3.53506 q^{49} -0.518886 q^{51} -2.61350 q^{52} +10.2614 q^{53} -0.901480 q^{54} -2.98978 q^{56} -2.25628 q^{58} -9.40836 q^{59} +3.73575 q^{61} +10.3217 q^{62} -5.57066 q^{63} +0.222557 q^{64} -0.184157 q^{66} +7.41188 q^{67} -6.57799 q^{68} +0.617018 q^{69} +8.51320 q^{71} +4.80672 q^{72} +6.21234 q^{73} -2.80851 q^{74} -2.27877 q^{77} -0.362138 q^{78} +7.39383 q^{79} +8.93405 q^{81} +17.0839 q^{82} -0.0135105 q^{83} +0.173060 q^{84} +1.22522 q^{86} -0.109998 q^{87} +1.96627 q^{88} +1.25527 q^{89} -4.48111 q^{91} +7.82202 q^{92} +0.503202 q^{93} +20.4528 q^{94} -0.475953 q^{96} +7.97800 q^{97} -6.20967 q^{98} +3.66363 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} - q^{3} + 3 q^{4} - 7 q^{6} + 11 q^{7} - 6 q^{8} + 5 q^{9}+O(q^{10}) $ 4 * q - q^2 - q^3 + 3 * q^4 - 7 * q^6 + 11 * q^7 - 6 * q^8 + 5 * q^9	$ 4 q - q^{2} - q^{3} + 3 q^{4} - 7 q^{6} + 11 q^{7} - 6 q^{8} + 5 q^{9} + 16 q^{12} + 2 q^{13} - 11 q^{14} - 3 q^{16} + 7 q^{17} - 17 q^{18} + 2 q^{21} - q^{22} + 11 q^{23} - 13 q^{24} + 9 q^{26} + 14 q^{27} + 13 q^{28} - 15 q^{29} - q^{31} - 3 q^{32} - 12 q^{33} - 22 q^{34} + 16 q^{36} + 11 q^{37} - 29 q^{39} + 22 q^{41} - 19 q^{42} + 26 q^{43} - 12 q^{44} + 10 q^{46} + 26 q^{47} + 13 q^{48} + 13 q^{49} - 11 q^{51} - 27 q^{52} + 16 q^{53} - 25 q^{54} - 8 q^{56} + 3 q^{58} - 10 q^{59} + 2 q^{61} + 31 q^{62} + 17 q^{63} + 4 q^{64} + 22 q^{66} - 3 q^{67} - 4 q^{68} + 14 q^{69} + 18 q^{71} - 29 q^{72} + 24 q^{73} - 17 q^{74} + 6 q^{77} + 15 q^{78} + 30 q^{79} - 4 q^{81} + 13 q^{82} + 12 q^{83} + 52 q^{84} - 16 q^{86} + q^{87} + 23 q^{88} + 9 q^{89} - 9 q^{91} + 25 q^{92} - 7 q^{93} - 11 q^{94} - 6 q^{96} - 19 q^{97} - 48 q^{98} - 9 q^{99}+O(q^{100}) $ 4 * q - q^2 - q^3 + 3 * q^4 - 7 * q^6 + 11 * q^7 - 6 * q^8 + 5 * q^9 + 16 * q^12 + 2 * q^13 - 11 * q^14 - 3 * q^16 + 7 * q^17 - 17 * q^18 + 2 * q^21 - q^22 + 11 * q^23 - 13 * q^24 + 9 * q^26 + 14 * q^27 + 13 * q^28 - 15 * q^29 - q^31 - 3 * q^32 - 12 * q^33 - 22 * q^34 + 16 * q^36 + 11 * q^37 - 29 * q^39 + 22 * q^41 - 19 * q^42 + 26 * q^43 - 12 * q^44 + 10 * q^46 + 26 * q^47 + 13 * q^48 + 13 * q^49 - 11 * q^51 - 27 * q^52 + 16 * q^53 - 25 * q^54 - 8 * q^56 + 3 * q^58 - 10 * q^59 + 2 * q^61 + 31 * q^62 + 17 * q^63 + 4 * q^64 + 22 * q^66 - 3 * q^67 - 4 * q^68 + 14 * q^69 + 18 * q^71 - 29 * q^72 + 24 * q^73 - 17 * q^74 + 6 * q^77 + 15 * q^78 + 30 * q^79 - 4 * q^81 + 13 * q^82 + 12 * q^83 + 52 * q^84 - 16 * q^86 + q^87 + 23 * q^88 + 9 * q^89 - 9 * q^91 + 25 * q^92 - 7 * q^93 - 11 * q^94 - 6 * q^96 - 19 * q^97 - 48 * q^98 - 9 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.75660	1.24210	0.621051	−	0.783770i	$-0.286706\pi$
$2$	1.75660	1.24210	0.621051	+	0.783770i	$0.286706\pi$
$3$	0.0856374	0.0494428	0.0247214	−	0.999694i	$-0.492130\pi$
$3$	0.0856374	0.0494428	0.0247214	+	0.999694i	$0.492130\pi$
$4$	1.08564	0.542819
$5$	0	0
$5$	0	0
$6$	0.150431	0.0614130
$7$	1.86144	0.703557	0.351778	−	0.936083i	$-0.385577\pi$
$7$	1.86144	0.703557	0.351778	+	0.936083i	$0.385577\pi$
$8$	−1.60617	−0.567866
$9$	−2.99267	−0.997555
$10$	0	0
$11$	−1.22420	−0.369111	−0.184555	−	0.982822i	$-0.559084\pi$
$11$	−1.22420	−0.369111	−0.184555	+	0.982822i	$0.559084\pi$
$12$	0.0929712	0.0268385
$13$	−2.40734	−0.667677	−0.333838	−	0.942630i	$-0.608344\pi$
$13$	−2.40734	−0.667677	−0.333838	+	0.942630i	$0.608344\pi$
$14$	3.26979	0.873889
$15$	0	0
$16$	−4.99267	−1.24817
$17$	−6.05910	−1.46955	−0.734774	−	0.678312i	$-0.762712\pi$
$17$	−6.05910	−1.46955	−0.734774	+	0.678312i	$0.762712\pi$
$18$	−5.25691	−1.23907
$19$	0	0
$19$	0	0
$20$	0	0
$21$	0.159409	0.0347858
$22$	−2.15043	−0.458473
$23$	7.20500	1.50235	0.751173	−	0.660105i	$-0.229488\pi$
$23$	7.20500	1.50235	0.751173	+	0.660105i	$0.229488\pi$
$24$	−0.137548	−0.0280769
$25$	0	0
$26$	−4.22873	−0.829323
$27$	−0.513197	−0.0987647
$28$	2.02084	0.381904
$29$	−1.28446	−0.238519	−0.119259	−	0.992863i	$-0.538052\pi$
$29$	−1.28446	−0.238519	−0.119259	+	0.992863i	$0.538052\pi$
$30$	0	0
$31$	5.87596	1.05535	0.527677	−	0.849445i	$-0.323063\pi$
$31$	5.87596	1.05535	0.527677	+	0.849445i	$0.323063\pi$
$32$	−5.55777	−0.982485
$33$	−0.104837	−0.0182499
$34$	−10.6434	−1.82533
$35$	0	0
$36$	−3.24895	−0.541492
$37$	−1.59883	−0.262847	−0.131423	−	0.991326i	$-0.541955\pi$
$37$	−1.59883	−0.262847	−0.131423	+	0.991326i	$0.541955\pi$
$38$	0	0
$39$	−0.206159	−0.0330118
$40$	0	0
$41$	9.72553	1.51887	0.759436	−	0.650581i	$-0.225475\pi$
$41$	9.72553	1.51887	0.759436	+	0.650581i	$0.225475\pi$
$42$	0.280017	0.0432075
$43$	0.697495	0.106367	0.0531835	−	0.998585i	$-0.483063\pi$
$43$	0.697495	0.106367	0.0531835	+	0.998585i	$0.483063\pi$
$44$	−1.32904	−0.200360
$45$	0	0
$46$	12.6563	1.86607
$47$	11.6434	1.69837	0.849183	−	0.528099i	$-0.177095\pi$
$47$	11.6434	1.69837	0.849183	+	0.528099i	$0.177095\pi$
$48$	−0.427559	−0.0617128
$49$	−3.53506	−0.505008
$50$	0	0
$51$	−0.518886	−0.0726586
$52$	−2.61350	−0.362427
$53$	10.2614	1.40952	0.704759	−	0.709447i	$-0.251055\pi$
$53$	10.2614	1.40952	0.704759	+	0.709447i	$0.251055\pi$
$54$	−0.901480	−0.122676
$55$	0	0
$56$	−2.98978	−0.399526
$57$	0	0
$58$	−2.25628	−0.296265
$59$	−9.40836	−1.22486	−0.612432	−	0.790523i	$-0.709809\pi$
$59$	−9.40836	−1.22486	−0.612432	+	0.790523i	$0.709809\pi$
$60$	0	0
$61$	3.73575	0.478314	0.239157	−	0.970981i	$-0.423129\pi$
$61$	3.73575	0.478314	0.239157	+	0.970981i	$0.423129\pi$
$62$	10.3217	1.31086
$63$	−5.57066	−0.701837
$64$	0.222557	0.0278197
$65$	0	0
$66$	−0.184157	−0.0226682
$67$	7.41188	0.905505	0.452752	−	0.891636i	$-0.350442\pi$
$67$	7.41188	0.905505	0.452752	+	0.891636i	$0.350442\pi$
$68$	−6.57799	−0.797698
$69$	0.617018	0.0742802
$70$	0	0
$71$	8.51320	1.01033	0.505165	−	0.863023i	$-0.331432\pi$
$71$	8.51320	1.01033	0.505165	+	0.863023i	$0.331432\pi$
$72$	4.80672	0.566478
$73$	6.21234	0.727099	0.363549	−	0.931575i	$-0.381565\pi$
$73$	6.21234	0.727099	0.363549	+	0.931575i	$0.381565\pi$
$74$	−2.80851	−0.326483
$75$	0	0
$76$	0	0
$77$	−2.27877	−0.259690
$78$	−0.362138	−0.0410040
$79$	7.39383	0.831871	0.415936	−	0.909394i	$-0.363454\pi$
$79$	7.39383	0.831871	0.415936	+	0.909394i	$0.363454\pi$
$80$	0	0
$81$	8.93405	0.992672
$82$	17.0839	1.88660
$83$	−0.0135105	−0.00148297	−0.000741487	−	1.00000i	$-0.500236\pi$
$83$	−0.0135105	−0.00148297	−0.000741487		1.00000i	$0.500236\pi$
$84$	0.173060	0.0188824
$85$	0	0
$86$	1.22522	0.132119
$87$	−0.109998	−0.0117930
$88$	1.96627	0.209605
$89$	1.25527	0.133058	0.0665291	−	0.997784i	$-0.478807\pi$
$89$	1.25527	0.133058	0.0665291	+	0.997784i	$0.478807\pi$
$90$	0	0
$91$	−4.48111	−0.469748
$92$	7.82202	0.815502
$93$	0.503202	0.0521796
$94$	20.4528	2.10954
$95$	0	0
$96$	−0.475953	−0.0485768
$97$	7.97800	0.810043	0.405022	−	0.914307i	$-0.367264\pi$
$97$	7.97800	0.810043	0.405022	+	0.914307i	$0.367264\pi$
$98$	−6.20967	−0.627272
$99$	3.66363	0.368208
$100$	0	0
$101$	18.2028	1.81125	0.905625	−	0.424080i	$-0.139403\pi$
$101$	18.2028	1.81125	0.905625	+	0.424080i	$0.139403\pi$
$102$	−0.911474	−0.0902494
$103$	3.41352	0.336344	0.168172	−	0.985758i	$-0.446214\pi$
$103$	3.41352	0.336344	0.168172	+	0.985758i	$0.446214\pi$
$104$	3.86660	0.379151
$105$	0	0
$106$	18.0252	1.75077
$107$	1.49515	0.144542	0.0722710	−	0.997385i	$-0.476975\pi$
$107$	1.49515	0.144542	0.0722710	+	0.997385i	$0.476975\pi$
$108$	−0.557145	−0.0536113
$109$	−15.5780	−1.49210	−0.746050	−	0.665889i	$-0.768052\pi$
$109$	−15.5780	−1.49210	−0.746050	+	0.665889i	$0.768052\pi$
$110$	0	0
$111$	−0.136920	−0.0129959
$112$	−9.29353	−0.878156
$113$	−13.6479	−1.28389	−0.641945	−	0.766750i	$-0.721872\pi$
$113$	−13.6479	−1.28389	−0.641945	+	0.766750i	$0.721872\pi$
$114$	0	0
$115$	0	0
$116$	−1.39446	−0.129472
$117$	7.20437	0.666045
$118$	−16.5267	−1.52141
$119$	−11.2786	−1.03391
$120$	0	0
$121$	−9.50133	−0.863757
$122$	6.56222	0.594115
$123$	0.832870	0.0750973
$124$	6.37916	0.572866
$125$	0	0
$126$	−9.78540	−0.871753
$127$	19.2055	1.70421	0.852106	−	0.523370i	$-0.175325\pi$
$127$	19.2055	1.70421	0.852106	+	0.523370i	$0.175325\pi$
$128$	11.5065	1.01704
$129$	0.0597316	0.00525908
$130$	0	0
$131$	−19.3097	−1.68710	−0.843548	−	0.537054i	$-0.819537\pi$
$131$	−19.3097	−1.68710	−0.843548	+	0.537054i	$0.819537\pi$
$132$	−0.113815	−0.00990637
$133$	0	0
$134$	13.0197	1.12473
$135$	0	0
$136$	9.73194	0.834507
$137$	14.6242	1.24943	0.624715	−	0.780853i	$-0.285215\pi$
$137$	14.6242	1.24943	0.624715	+	0.780853i	$0.285215\pi$
$138$	1.08385	0.0922636
$139$	1.39485	0.118310	0.0591548	−	0.998249i	$-0.481159\pi$
$139$	1.39485	0.118310	0.0591548	+	0.998249i	$0.481159\pi$
$140$	0	0
$141$	0.997112	0.0839720
$142$	14.9543	1.25493
$143$	2.94707	0.246447
$144$	14.9414	1.24512
$145$	0	0
$146$	10.9126	0.903131
$147$	−0.302733	−0.0249690
$148$	−1.73575	−0.142678
$149$	14.5340	1.19068	0.595338	−	0.803476i	$-0.297018\pi$
$149$	14.5340	1.19068	0.595338	+	0.803476i	$0.297018\pi$
$150$	0	0
$151$	−23.5773	−1.91870	−0.959348	−	0.282227i	$-0.908927\pi$
$151$	−23.5773	−1.91870	−0.959348	+	0.282227i	$0.908927\pi$
$152$	0	0
$153$	18.1329	1.46596
$154$	−4.00289	−0.322562
$155$	0	0
$156$	−0.223814	−0.0179194
$157$	17.0409	1.36001	0.680007	−	0.733206i	$-0.261977\pi$
$157$	17.0409	1.36001	0.680007	+	0.733206i	$0.261977\pi$
$158$	12.9880	1.03327
$159$	0.878764	0.0696905
$160$	0	0
$161$	13.4116	1.05699
$162$	15.6935	1.23300
$163$	9.98080	0.781757	0.390878	−	0.920442i	$-0.372171\pi$
$163$	9.98080	0.781757	0.390878	+	0.920442i	$0.372171\pi$
$164$	10.5584	0.824473
$165$	0	0
$166$	−0.0237326	−0.00184200
$167$	−23.1483	−1.79127	−0.895633	−	0.444794i	$-0.853277\pi$
$167$	−23.1483	−1.79127	−0.895633	+	0.444794i	$0.853277\pi$
$168$	−0.256037	−0.0197537
$169$	−7.20470	−0.554208
$170$	0	0
$171$	0	0
$172$	0.757226	0.0577380
$173$	12.4017	0.942880	0.471440	−	0.881898i	$-0.343734\pi$
$173$	12.4017	0.942880	0.471440	+	0.881898i	$0.343734\pi$
$174$	−0.193222	−0.0146482
$175$	0	0
$176$	6.11203	0.460712
$177$	−0.805708	−0.0605607
$178$	2.20500	0.165272
$179$	−22.5640	−1.68651	−0.843254	−	0.537515i	$-0.819363\pi$
$179$	−22.5640	−1.68651	−0.843254	+	0.537515i	$0.819363\pi$
$180$	0	0
$181$	−7.13669	−0.530466	−0.265233	−	0.964184i	$-0.585449\pi$
$181$	−7.13669	−0.530466	−0.265233	+	0.964184i	$0.585449\pi$
$182$	−7.87152	−0.583476
$183$	0.319920	0.0236492
$184$	−11.5724	−0.853132
$185$	0	0
$186$	0.883924	0.0648125
$187$	7.41756	0.542426
$188$	12.6405	0.921905
$189$	−0.955282	−0.0694866
$190$	0	0
$191$	−12.6479	−0.915173	−0.457587	−	0.889165i	$-0.651286\pi$
$191$	−12.6479	−0.915173	−0.457587	+	0.889165i	$0.651286\pi$
$192$	0.0190592	0.00137548
$193$	3.82038	0.274997	0.137498	−	0.990502i	$-0.456094\pi$
$193$	3.82038	0.274997	0.137498	+	0.990502i	$0.456094\pi$
$194$	14.0141	1.00616
$195$	0	0
$196$	−3.83779	−0.274128
$197$	−5.35606	−0.381603	−0.190802	−	0.981629i	$-0.561109\pi$
$197$	−5.35606	−0.381603	−0.190802	+	0.981629i	$0.561109\pi$
$198$	6.43552	0.457353
$199$	−7.19251	−0.509863	−0.254932	−	0.966959i	$-0.582053\pi$
$199$	−7.19251	−0.509863	−0.254932	+	0.966959i	$0.582053\pi$
$200$	0	0
$201$	0.634734	0.0447707
$202$	31.9751	2.24976
$203$	−2.39094	−0.167811
$204$	−0.563322	−0.0394404
$205$	0	0
$206$	5.99618	0.417774
$207$	−21.5622	−1.49867
$208$	12.0191	0.833372
$209$	0	0
$210$	0	0
$211$	23.3668	1.60863	0.804317	−	0.594200i	$-0.202531\pi$
$211$	23.3668	1.60863	0.804317	+	0.594200i	$0.202531\pi$
$212$	11.1402	0.765113
$213$	0.729048	0.0499536
$214$	2.62638	0.179536
$215$	0	0
$216$	0.824280	0.0560851
$217$	10.9377	0.742501
$218$	−27.3643	−1.85334
$219$	0.532008	0.0359498
$220$	0	0
$221$	14.5863	0.981183
$222$	−0.240513	−0.0161422
$223$	4.28166	0.286721	0.143361	−	0.989671i	$-0.454209\pi$
$223$	4.28166	0.286721	0.143361	+	0.989671i	$0.454209\pi$
$224$	−10.3454	−0.691234
$225$	0	0
$226$	−23.9740	−1.59472
$227$	−6.60883	−0.438643	−0.219322	−	0.975653i	$-0.570384\pi$
$227$	−6.60883	−0.438643	−0.219322	+	0.975653i	$0.570384\pi$
$228$	0	0
$229$	−23.3277	−1.54154	−0.770770	−	0.637113i	$-0.780128\pi$
$229$	−23.3277	−1.54154	−0.770770	+	0.637113i	$0.780128\pi$
$230$	0	0
$231$	−0.195148	−0.0128398
$232$	2.06306	0.135447
$233$	17.0372	1.11615	0.558073	−	0.829792i	$-0.311541\pi$
$233$	17.0372	1.11615	0.558073	+	0.829792i	$0.311541\pi$
$234$	12.6552	0.827296
$235$	0	0
$236$	−10.2141	−0.664879
$237$	0.633189	0.0411300
$238$	−19.8120	−1.28422
$239$	10.4194	0.673973	0.336986	−	0.941509i	$-0.390592\pi$
$239$	10.4194	0.673973	0.336986	+	0.941509i	$0.390592\pi$
$240$	0	0
$241$	3.32069	0.213904	0.106952	−	0.994264i	$-0.465891\pi$
$241$	3.32069	0.213904	0.106952	+	0.994264i	$0.465891\pi$
$242$	−16.6900	−1.07288
$243$	2.30468	0.147845
$244$	4.05567	0.259638
$245$	0	0
$246$	1.46302	0.0932786
$247$	0	0
$248$	−9.43778	−0.599300
$249$	−0.00115701	−7.33223e−5 0
$250$	0	0
$251$	−25.7802	−1.62723	−0.813616	−	0.581403i	$-0.802504\pi$
$251$	−25.7802	−1.62723	−0.813616	+	0.581403i	$0.802504\pi$
$252$	−6.04771	−0.380970
$253$	−8.82038	−0.554532
$254$	33.7363	2.11681
$255$	0	0
$256$	19.7672	1.23545
$257$	5.40015	0.336852	0.168426	−	0.985714i	$-0.446132\pi$
$257$	5.40015	0.336852	0.168426	+	0.985714i	$0.446132\pi$
$258$	0.104925	0.00653231
$259$	−2.97613	−0.184928
$260$	0	0
$261$	3.84397	0.237936
$262$	−33.9194	−2.09555
$263$	−8.76997	−0.540779	−0.270390	−	0.962751i	$-0.587153\pi$
$263$	−8.76997	−0.540779	−0.270390	+	0.962751i	$0.587153\pi$
$264$	0.168387	0.0103635
$265$	0	0
$266$	0	0
$267$	0.107498	0.00657877
$268$	8.04661	0.491525
$269$	−19.3500	−1.17979	−0.589894	−	0.807481i	$-0.700831\pi$
$269$	−19.3500	−1.17979	−0.589894	+	0.807481i	$0.700831\pi$
$270$	0	0
$271$	8.95871	0.544203	0.272101	−	0.962269i	$-0.412281\pi$
$271$	8.95871	0.544203	0.272101	+	0.962269i	$0.412281\pi$
$272$	30.2511	1.83424
$273$	−0.383751	−0.0232257
$274$	25.6889	1.55192
$275$	0	0
$276$	0.669858	0.0403207
$277$	10.4586	0.628398	0.314199	−	0.949357i	$-0.398264\pi$
$277$	10.4586	0.628398	0.314199	+	0.949357i	$0.398264\pi$
$278$	2.45019	0.146953
$279$	−17.5848	−1.05277
$280$	0	0
$281$	10.9107	0.650878	0.325439	−	0.945563i	$-0.394488\pi$
$281$	10.9107	0.650878	0.325439	+	0.945563i	$0.394488\pi$
$282$	1.75152	0.104302
$283$	0.952763	0.0566359	0.0283179	−	0.999599i	$-0.490985\pi$
$283$	0.952763	0.0566359	0.0283179	+	0.999599i	$0.490985\pi$
$284$	9.24224	0.548426
$285$	0	0
$286$	5.17682	0.306112
$287$	18.1035	1.06861
$288$	16.6326	0.980083
$289$	19.7127	1.15957
$290$	0	0
$291$	0.683215	0.0400508
$292$	6.74434	0.394683
$293$	−13.4219	−0.784114	−0.392057	−	0.919941i	$-0.628236\pi$
$293$	−13.4219	−0.784114	−0.392057	+	0.919941i	$0.628236\pi$
$294$	−0.531781	−0.0310141
$295$	0	0
$296$	2.56800	0.149262
$297$	0.628256	0.0364551
$298$	25.5305	1.47894
$299$	−17.3449	−1.00308
$300$	0	0
$301$	1.29834	0.0748352
$302$	−41.4159	−2.38322
$303$	1.55884	0.0895532
$304$	0	0
$305$	0	0
$306$	31.8522	1.82087
$307$	18.8433	1.07544	0.537721	−	0.843123i	$-0.319286\pi$
$307$	18.8433	1.07544	0.537721	+	0.843123i	$0.319286\pi$
$308$	−2.47392	−0.140965
$309$	0.292325	0.0166298
$310$	0	0
$311$	−0.912189	−0.0517255	−0.0258628	−	0.999666i	$-0.508233\pi$
$311$	−0.912189	−0.0517255	−0.0258628	+	0.999666i	$0.508233\pi$
$312$	0.331125	0.0187463
$313$	18.7665	1.06074	0.530371	−	0.847765i	$-0.322053\pi$
$313$	18.7665	1.06074	0.530371	+	0.847765i	$0.322053\pi$
$314$	29.9340	1.68928
$315$	0	0
$316$	8.02702	0.451555
$317$	−8.85956	−0.497603	−0.248801	−	0.968555i	$-0.580037\pi$
$317$	−8.85956	−0.497603	−0.248801	+	0.968555i	$0.580037\pi$
$318$	1.54363	0.0865627
$319$	1.57244	0.0880398
$320$	0	0
$321$	0.128041	0.00714656
$322$	23.5589	1.31288
$323$	0	0
$324$	9.69914	0.538841
$325$	0	0
$326$	17.5323	0.971022
$327$	−1.33406	−0.0737736
$328$	−15.6208	−0.862516
$329$	21.6735	1.19490
$330$	0	0
$331$	3.96044	0.217686	0.108843	−	0.994059i	$-0.465285\pi$
$331$	3.96044	0.217686	0.108843	+	0.994059i	$0.465285\pi$
$332$	−0.0146675	−0.000804986 0
$333$	4.78478	0.262204
$334$	−40.6622	−2.22494
$335$	0	0
$336$	−0.795874	−0.0434185
$337$	5.79148	0.315482	0.157741	−	0.987481i	$-0.449579\pi$
$337$	5.79148	0.315482	0.157741	+	0.987481i	$0.449579\pi$
$338$	−12.6558	−0.688383
$339$	−1.16877	−0.0634791
$340$	0	0
$341$	−7.19336	−0.389542
$342$	0	0
$343$	−19.6103	−1.05886
$344$	−1.12029	−0.0604022
$345$	0	0
$346$	21.7847	1.17115
$347$	−14.4126	−0.773708	−0.386854	−	0.922141i	$-0.626438\pi$
$347$	−14.4126	−0.773708	−0.386854	+	0.922141i	$0.626438\pi$
$348$	−0.119418	−0.00640148
$349$	9.11150	0.487727	0.243864	−	0.969810i	$-0.421585\pi$
$349$	9.11150	0.487727	0.243864	+	0.969810i	$0.421585\pi$
$350$	0	0
$351$	1.23544	0.0659429
$352$	6.80384	0.362646
$353$	3.59377	0.191277	0.0956386	−	0.995416i	$-0.469511\pi$
$353$	3.59377	0.191277	0.0956386	+	0.995416i	$0.469511\pi$
$354$	−1.41530	−0.0752226
$355$	0	0
$356$	1.36277	0.0722265
$357$	−0.965873	−0.0511194
$358$	−39.6358	−2.09482
$359$	−10.7465	−0.567177	−0.283588	−	0.958946i	$-0.591525\pi$
$359$	−10.7465	−0.567177	−0.283588	+	0.958946i	$0.591525\pi$
$360$	0	0
$361$	0	0
$362$	−12.5363	−0.658893
$363$	−0.813669	−0.0427066
$364$	−4.86487	−0.254988
$365$	0	0
$366$	0.561972	0.0293747
$367$	−1.68713	−0.0880676	−0.0440338	−	0.999030i	$-0.514021\pi$
$367$	−1.68713	−0.0880676	−0.0440338	+	0.999030i	$0.514021\pi$
$368$	−35.9722	−1.87518
$369$	−29.1053	−1.51516
$370$	0	0
$371$	19.1010	0.991676
$372$	0.546295	0.0283241
$373$	−34.2844	−1.77518	−0.887590	−	0.460635i	$-0.847622\pi$
$373$	−34.2844	−1.77518	−0.887590	+	0.460635i	$0.847622\pi$
$374$	13.0297	0.673749
$375$	0	0
$376$	−18.7013	−0.964444
$377$	3.09214	0.159253
$378$	−1.67805	−0.0863094
$379$	24.7637	1.27202	0.636012	−	0.771679i	$-0.280583\pi$
$379$	24.7637	1.27202	0.636012	+	0.771679i	$0.280583\pi$
$380$	0	0
$381$	1.64471	0.0842610
$382$	−22.2174	−1.13674
$383$	−17.2980	−0.883885	−0.441943	−	0.897043i	$-0.645710\pi$
$383$	−17.2980	−0.883885	−0.441943	+	0.897043i	$0.645710\pi$
$384$	0.985386	0.0502853
$385$	0	0
$386$	6.71086	0.341574
$387$	−2.08737	−0.106107
$388$	8.66121	0.439707
$389$	31.5131	1.59778	0.798888	−	0.601480i	$-0.205422\pi$
$389$	31.5131	1.59778	0.798888	+	0.601480i	$0.205422\pi$
$390$	0	0
$391$	−43.6559	−2.20777
$392$	5.67789	0.286777
$393$	−1.65363	−0.0834147
$394$	−9.40845	−0.473991
$395$	0	0
$396$	3.97737	0.199870
$397$	−27.9014	−1.40033	−0.700166	−	0.713980i	$-0.746891\pi$
$397$	−27.9014	−1.40033	−0.700166	+	0.713980i	$0.746891\pi$
$398$	−12.6343	−0.633303
$399$	0	0
$400$	0	0
$401$	−4.73460	−0.236434	−0.118217	−	0.992988i	$-0.537718\pi$
$401$	−4.73460	−0.236434	−0.118217	+	0.992988i	$0.537718\pi$
$402$	1.11497	0.0556098
$403$	−14.1455	−0.704635
$404$	19.7617	0.983180
$405$	0	0
$406$	−4.19993	−0.208439
$407$	1.95730	0.0970195
$408$	0.833418	0.0412603
$409$	8.34645	0.412706	0.206353	−	0.978478i	$-0.433841\pi$
$409$	8.34645	0.412706	0.206353	+	0.978478i	$0.433841\pi$
$410$	0	0
$411$	1.25238	0.0617753
$412$	3.70584	0.182574
$413$	−17.5131	−0.861761
$414$	−37.8761	−1.86151
$415$	0	0
$416$	13.3795	0.655982
$417$	0.119451	0.00584955
$418$	0	0
$419$	21.5120	1.05093	0.525466	−	0.850815i	$-0.323891\pi$
$419$	21.5120	1.05093	0.525466	+	0.850815i	$0.323891\pi$
$420$	0	0
$421$	−30.6789	−1.49520	−0.747598	−	0.664151i	$-0.768793\pi$
$421$	−30.6789	−1.49520	−0.747598	+	0.664151i	$0.768793\pi$
$422$	41.0460	1.99809
$423$	−34.8448	−1.69421
$424$	−16.4816	−0.800417
$425$	0	0
$426$	1.28064	0.0620474
$427$	6.95387	0.336521
$428$	1.62319	0.0784601
$429$	0.252380	0.0121850
$430$	0	0
$431$	25.5833	1.23230	0.616152	−	0.787627i	$-0.288691\pi$
$431$	25.5833	1.23230	0.616152	+	0.787627i	$0.288691\pi$
$432$	2.56222	0.123275
$433$	31.9328	1.53459	0.767295	−	0.641294i	$-0.221602\pi$
$433$	31.9328	1.53459	0.767295	+	0.641294i	$0.221602\pi$
$434$	19.2132	0.922263
$435$	0	0
$436$	−16.9120	−0.809940
$437$	0	0
$438$	0.934525	0.0446533
$439$	−9.45544	−0.451283	−0.225642	−	0.974210i	$-0.572448\pi$
$439$	−9.45544	−0.451283	−0.225642	+	0.974210i	$0.572448\pi$
$440$	0	0
$441$	10.5792	0.503774
$442$	25.6223	1.21873
$443$	−9.27713	−0.440770	−0.220385	−	0.975413i	$-0.570731\pi$
$443$	−9.27713	−0.440770	−0.220385	+	0.975413i	$0.570731\pi$
$444$	−0.148645	−0.00705440
$445$	0	0
$446$	7.52116	0.356137
$447$	1.24466	0.0588703
$448$	0.414276	0.0195727
$449$	27.6454	1.30467	0.652335	−	0.757931i	$-0.273790\pi$
$449$	27.6454	1.30467	0.652335	+	0.757931i	$0.273790\pi$
$450$	0	0
$451$	−11.9060	−0.560632
$452$	−14.8167	−0.696920
$453$	−2.01910	−0.0948656
$454$	−11.6091	−0.544840
$455$	0	0
$456$	0	0
$457$	6.99174	0.327060	0.163530	−	0.986538i	$-0.447712\pi$
$457$	6.99174	0.327060	0.163530	+	0.986538i	$0.447712\pi$
$458$	−40.9775	−1.91475
$459$	3.10951	0.145140
$460$	0	0
$461$	15.2246	0.709080	0.354540	−	0.935041i	$-0.384638\pi$
$461$	15.2246	0.709080	0.354540	+	0.935041i	$0.384638\pi$
$462$	−0.342797	−0.0159484
$463$	27.7350	1.28895	0.644477	−	0.764623i	$-0.277075\pi$
$463$	27.7350	1.28895	0.644477	+	0.764623i	$0.277075\pi$
$464$	6.41289	0.297711
$465$	0	0
$466$	29.9276	1.38637
$467$	38.3385	1.77409	0.887046	−	0.461681i	$-0.152754\pi$
$467$	38.3385	1.77409	0.887046	+	0.461681i	$0.152754\pi$
$468$	7.82134	0.361541
$469$	13.7967	0.637074
$470$	0	0
$471$	1.45934	0.0672429
$472$	15.1114	0.695559
$473$	−0.853874	−0.0392612
$474$	1.11226	0.0510877
$475$	0	0
$476$	−12.2445	−0.561226
$477$	−30.7091	−1.40607
$478$	18.3026	0.837144
$479$	−10.5137	−0.480382	−0.240191	−	0.970726i	$-0.577210\pi$
$479$	−10.5137	−0.480382	−0.240191	+	0.970726i	$0.577210\pi$
$480$	0	0
$481$	3.84894	0.175497
$482$	5.83312	0.265691
$483$	1.14854	0.0522603
$484$	−10.3150	−0.468864
$485$	0	0
$486$	4.04839	0.183639
$487$	−14.1296	−0.640273	−0.320136	−	0.947371i	$-0.603729\pi$
$487$	−14.1296	−0.640273	−0.320136	+	0.947371i	$0.603729\pi$
$488$	−6.00025	−0.271618
$489$	0.854730	0.0386522
$490$	0	0
$491$	18.7427	0.845847	0.422924	−	0.906165i	$-0.361004\pi$
$491$	18.7427	0.845847	0.422924	+	0.906165i	$0.361004\pi$
$492$	0.904194	0.0407642
$493$	7.78269	0.350515
$494$	0	0
$495$	0	0
$496$	−29.3367	−1.31726
$497$	15.8468	0.710825
$498$	−0.00203240	−9.10739e−5 0
$499$	16.2646	0.728103	0.364052	−	0.931379i	$-0.381393\pi$
$499$	16.2646	0.728103	0.364052	+	0.931379i	$0.381393\pi$
$500$	0	0
$501$	−1.98236	−0.0885652
$502$	−45.2854	−2.02119
$503$	26.1541	1.16615	0.583077	−	0.812417i	$-0.301849\pi$
$503$	26.1541	1.16615	0.583077	+	0.812417i	$0.301849\pi$
$504$	8.94741	0.398549
$505$	0	0
$506$	−15.4939	−0.688786
$507$	−0.616992	−0.0274016
$508$	20.8502	0.925078
$509$	17.3725	0.770025	0.385012	−	0.922911i	$-0.374197\pi$
$509$	17.3725	0.770025	0.385012	+	0.922911i	$0.374197\pi$
$510$	0	0
$511$	11.5639	0.511555
$512$	11.7100	0.517513
$513$	0	0
$514$	9.48589	0.418405
$515$	0	0
$516$	0.0648469	0.00285473
$517$	−14.2539	−0.626885
$518$	−5.22786	−0.229699
$519$	1.06205	0.0466186
$520$	0	0
$521$	33.7615	1.47912	0.739559	−	0.673092i	$-0.235034\pi$
$521$	33.7615	1.47912	0.739559	+	0.673092i	$0.235034\pi$
$522$	6.75231	0.295540
$523$	−28.7361	−1.25654	−0.628271	−	0.777995i	$-0.716237\pi$
$523$	−28.7361	−1.25654	−0.628271	+	0.777995i	$0.716237\pi$
$524$	−20.9633	−0.915787
$525$	0	0
$526$	−15.4053	−0.671703
$527$	−35.6031	−1.55089
$528$	0.523419	0.0227789
$529$	28.9120	1.25705
$530$	0	0
$531$	28.1561	1.22187
$532$	0	0
$533$	−23.4127	−1.01412
$534$	0.188831	0.00817150
$535$	0	0
$536$	−11.9047	−0.514205
$537$	−1.93232	−0.0833857
$538$	−33.9901	−1.46542
$539$	4.32762	0.186404
$540$	0	0
$541$	−17.7661	−0.763823	−0.381911	−	0.924199i	$-0.624734\pi$
$541$	−17.7661	−0.763823	−0.381911	+	0.924199i	$0.624734\pi$
$542$	15.7369	0.675956
$543$	−0.611168	−0.0262277
$544$	33.6751	1.44381
$545$	0	0
$546$	−0.674096	−0.0288487
$547$	−10.4531	−0.446941	−0.223471	−	0.974711i	$-0.571739\pi$
$547$	−10.4531	−0.446941	−0.223471	+	0.974711i	$0.571739\pi$
$548$	15.8766	0.678214
$549$	−11.1799	−0.477145
$550$	0	0
$551$	0	0
$552$	−0.991034	−0.0421812
$553$	13.7631	0.585268
$554$	18.3716	0.780535
$555$	0	0
$556$	1.51430	0.0642206
$557$	23.5768	0.998982	0.499491	−	0.866319i	$-0.333520\pi$
$557$	23.5768	0.998982	0.499491	+	0.866319i	$0.333520\pi$
$558$	−30.8894	−1.30765
$559$	−1.67911	−0.0710187
$560$	0	0
$561$	0.635221	0.0268191
$562$	19.1657	0.808457
$563$	−27.8106	−1.17208	−0.586038	−	0.810283i	$-0.699313\pi$
$563$	−27.8106	−1.17208	−0.586038	+	0.810283i	$0.699313\pi$
$564$	1.08250	0.0455815
$565$	0	0
$566$	1.67362	0.0703475
$567$	16.6302	0.698401
$568$	−13.6736	−0.573732
$569$	−9.72405	−0.407653	−0.203827	−	0.979007i	$-0.565338\pi$
$569$	−9.72405	−0.407653	−0.203827	+	0.979007i	$0.565338\pi$
$570$	0	0
$571$	−7.74753	−0.324224	−0.162112	−	0.986772i	$-0.551831\pi$
$571$	−7.74753	−0.324224	−0.162112	+	0.986772i	$0.551831\pi$
$572$	3.19945	0.133776
$573$	−1.08314	−0.0452487
$574$	31.8005	1.32733
$575$	0	0
$576$	−0.666040	−0.0277517
$577$	21.3536	0.888964	0.444482	−	0.895788i	$-0.353388\pi$
$577$	21.3536	0.888964	0.444482	+	0.895788i	$0.353388\pi$
$578$	34.6274	1.44031
$579$	0.327167	0.0135966
$580$	0	0
$581$	−0.0251490	−0.00104336
$582$	1.20013	0.0497472
$583$	−12.5621	−0.520268
$584$	−9.97805	−0.412895
$585$	0	0
$586$	−23.5768	−0.973950
$587$	34.7452	1.43409	0.717043	−	0.697029i	$-0.245495\pi$
$587$	34.7452	1.43409	0.717043	+	0.697029i	$0.245495\pi$
$588$	−0.328658	−0.0135536
$589$	0	0
$590$	0	0
$591$	−0.458679	−0.0188675
$592$	7.98244	0.328076
$593$	0.557914	0.0229108	0.0114554	−	0.999934i	$-0.496354\pi$
$593$	0.557914	0.0229108	0.0114554	+	0.999934i	$0.496354\pi$
$594$	1.10359	0.0452810
$595$	0	0
$596$	15.7787	0.646321
$597$	−0.615948	−0.0252091
$598$	−30.4680	−1.24593
$599$	10.1872	0.416237	0.208119	−	0.978104i	$-0.433266\pi$
$599$	10.1872	0.416237	0.208119	+	0.978104i	$0.433266\pi$
$600$	0	0
$601$	1.46619	0.0598070	0.0299035	−	0.999553i	$-0.490480\pi$
$601$	1.46619	0.0598070	0.0299035	+	0.999553i	$0.490480\pi$
$602$	2.28066	0.0929529
$603$	−22.1813	−0.903291
$604$	−25.5964	−1.04150
$605$	0	0
$606$	2.73826	0.111234
$607$	37.3950	1.51782	0.758909	−	0.651197i	$-0.225733\pi$
$607$	37.3950	1.51782	0.758909	+	0.651197i	$0.225733\pi$
$608$	0	0
$609$	−0.204754	−0.00829706
$610$	0	0
$611$	−28.0297	−1.13396
$612$	19.6857	0.795748
$613$	29.0310	1.17255	0.586276	−	0.810111i	$-0.300593\pi$
$613$	29.0310	1.17255	0.586276	+	0.810111i	$0.300593\pi$
$614$	33.1000	1.33581
$615$	0	0
$616$	3.66009	0.147469
$617$	0.259474	0.0104460	0.00522301	−	0.999986i	$-0.498337\pi$
$617$	0.259474	0.0104460	0.00522301	+	0.999986i	$0.498337\pi$
$618$	0.513498	0.0206559
$619$	23.2107	0.932917	0.466459	−	0.884543i	$-0.345530\pi$
$619$	23.2107	0.932917	0.466459	+	0.884543i	$0.345530\pi$
$620$	0	0
$621$	−3.69758	−0.148379
$622$	−1.60235	−0.0642484
$623$	2.33660	0.0936139
$624$	1.02928	0.0412042
$625$	0	0
$626$	32.9651	1.31755
$627$	0	0
$628$	18.5003	0.738241
$629$	9.68750	0.386266
$630$	0	0
$631$	−1.74339	−0.0694032	−0.0347016	−	0.999398i	$-0.511048\pi$
$631$	−1.74339	−0.0694032	−0.0347016	+	0.999398i	$0.511048\pi$
$632$	−11.8757	−0.472391
$633$	2.00107	0.0795354
$634$	−15.5627	−0.618074
$635$	0	0
$636$	0.954019	0.0378293
$637$	8.51009	0.337182
$638$	2.76215	0.109354
$639$	−25.4772	−1.00786
$640$	0	0
$641$	−9.38800	−0.370804	−0.185402	−	0.982663i	$-0.559359\pi$
$641$	−9.38800	−0.370804	−0.185402	+	0.982663i	$0.559359\pi$
$642$	0.224917	0.00887676
$643$	35.4801	1.39920	0.699600	−	0.714535i	$-0.253362\pi$
$643$	35.4801	1.39920	0.699600	+	0.714535i	$0.253362\pi$
$644$	14.5602	0.573752
$645$	0	0
$646$	0	0
$647$	9.61436	0.377979	0.188990	−	0.981979i	$-0.439479\pi$
$647$	9.61436	0.377979	0.188990	+	0.981979i	$0.439479\pi$
$648$	−14.3496	−0.563705
$649$	11.5177	0.452110
$650$	0	0
$651$	0.936679	0.0367113
$652$	10.8355	0.424352
$653$	−10.9351	−0.427922	−0.213961	−	0.976842i	$-0.568637\pi$
$653$	−10.9351	−0.427922	−0.213961	+	0.976842i	$0.568637\pi$
$654$	−2.34341	−0.0916344
$655$	0	0
$656$	−48.5563	−1.89581
$657$	−18.5914	−0.725321
$658$	38.0716	1.48418
$659$	−30.2427	−1.17809	−0.589045	−	0.808100i	$-0.700496\pi$
$659$	−30.2427	−1.17809	−0.589045	+	0.808100i	$0.700496\pi$
$660$	0	0
$661$	17.8243	0.693284	0.346642	−	0.937997i	$-0.387322\pi$
$661$	17.8243	0.693284	0.346642	+	0.937997i	$0.387322\pi$
$662$	6.95691	0.270388
$663$	1.24914	0.0485124
$664$	0.0217002	0.000842130 0
$665$	0	0
$666$	8.40493	0.325684
$667$	−9.25455	−0.358338
$668$	−25.1306	−0.972333
$669$	0.366670	0.0141763
$670$	0	0
$671$	−4.57332	−0.176551
$672$	−0.885957	−0.0341765
$673$	−16.6638	−0.642341	−0.321171	−	0.947021i	$-0.604076\pi$
$673$	−16.6638	−0.642341	−0.321171	+	0.947021i	$0.604076\pi$
$674$	10.1733	0.391861
$675$	0	0
$676$	−7.82169	−0.300834
$677$	36.5254	1.40379	0.701893	−	0.712283i	$-0.252339\pi$
$677$	36.5254	1.40379	0.701893	+	0.712283i	$0.252339\pi$
$678$	−2.05307	−0.0788476
$679$	14.8505	0.569911
$680$	0	0
$681$	−0.565963	−0.0216878
$682$	−12.6358	−0.483852
$683$	−34.2305	−1.30979	−0.654896	−	0.755719i	$-0.727288\pi$
$683$	−34.2305	−1.30979	−0.654896	+	0.755719i	$0.727288\pi$
$684$	0	0
$685$	0	0
$686$	−34.4475	−1.31521
$687$	−1.99773	−0.0762181
$688$	−3.48236	−0.132764
$689$	−24.7028	−0.941102
$690$	0	0
$691$	4.08362	0.155348	0.0776742	−	0.996979i	$-0.475251\pi$
$691$	4.08362	0.155348	0.0776742	+	0.996979i	$0.475251\pi$
$692$	13.4637	0.511813
$693$	6.81961	0.259055
$694$	−25.3171	−0.961025
$695$	0	0
$696$	0.176675	0.00669686
$697$	−58.9280	−2.23206
$698$	16.0052	0.605808
$699$	1.45903	0.0551854
$700$	0	0
$701$	25.6862	0.970154	0.485077	−	0.874471i	$-0.338791\pi$
$701$	25.6862	0.970154	0.485077	+	0.874471i	$0.338791\pi$
$702$	2.17017	0.0819078
$703$	0	0
$704$	−0.272455	−0.0102685
$705$	0	0
$706$	6.31281	0.237586
$707$	33.8834	1.27432
$708$	−0.874706	−0.0328735
$709$	−19.5953	−0.735918	−0.367959	−	0.929842i	$-0.619943\pi$
$709$	−19.5953	−0.735918	−0.367959	+	0.929842i	$0.619943\pi$
$710$	0	0
$711$	−22.1273	−0.829838
$712$	−2.01617	−0.0755592
$713$	42.3363	1.58551
$714$	−1.69665	−0.0634956
$715$	0	0
$716$	−24.4963	−0.915469
$717$	0.892288	0.0333231
$718$	−18.8772	−0.704492
$719$	−1.29094	−0.0481439	−0.0240720	−	0.999710i	$-0.507663\pi$
$719$	−1.29094	−0.0481439	−0.0240720	+	0.999710i	$0.507663\pi$
$720$	0	0
$721$	6.35405	0.236637
$722$	0	0
$723$	0.284375	0.0105760
$724$	−7.74786	−0.287947
$725$	0	0
$726$	−1.42929	−0.0530459
$727$	3.54012	0.131296	0.0656479	−	0.997843i	$-0.479089\pi$
$727$	3.54012	0.131296	0.0656479	+	0.997843i	$0.479089\pi$
$728$	7.19742	0.266754
$729$	−26.6048	−0.985362
$730$	0	0
$731$	−4.22619	−0.156311
$732$	0.347318	0.0128372
$733$	−17.2078	−0.635585	−0.317792	−	0.948160i	$-0.602941\pi$
$733$	−17.2078	−0.635585	−0.317792	+	0.948160i	$0.602941\pi$
$734$	−2.96361	−0.109389
$735$	0	0
$736$	−40.0438	−1.47603
$737$	−9.07363	−0.334231
$738$	−51.1263	−1.88198
$739$	−40.0918	−1.47480	−0.737401	−	0.675456i	$-0.763947\pi$
$739$	−40.0918	−1.47480	−0.737401	+	0.675456i	$0.763947\pi$
$740$	0	0
$741$	0	0
$742$	33.5528	1.23176
$743$	20.2637	0.743403	0.371702	−	0.928352i	$-0.378774\pi$
$743$	20.2637	0.743403	0.371702	+	0.928352i	$0.378774\pi$
$744$	−0.808227	−0.0296310
$745$	0	0
$746$	−60.2239	−2.20495
$747$	0.0404325	0.00147935
$748$	8.05279	0.294439
$749$	2.78313	0.101693
$750$	0	0
$751$	38.7484	1.41395	0.706975	−	0.707238i	$-0.250059\pi$
$751$	38.7484	1.41395	0.706975	+	0.707238i	$0.250059\pi$
$752$	−58.1317	−2.11984
$753$	−2.20775	−0.0804548
$754$	5.43165	0.197809
$755$	0	0
$756$	−1.03709	−0.0377186
$757$	−20.8068	−0.756237	−0.378118	−	0.925757i	$-0.623429\pi$
$757$	−20.8068	−0.756237	−0.378118	+	0.925757i	$0.623429\pi$
$758$	43.4998	1.57998
$759$	−0.755354	−0.0274176
$760$	0	0
$761$	48.6822	1.76473	0.882363	−	0.470569i	$-0.155951\pi$
$761$	48.6822	1.76473	0.882363	+	0.470569i	$0.155951\pi$
$762$	2.88909	0.104661
$763$	−28.9974	−1.04978
$764$	−13.7311	−0.496773
$765$	0	0
$766$	−30.3856	−1.09788
$767$	22.6491	0.817813
$768$	1.69281	0.0610840
$769$	20.0077	0.721495	0.360748	−	0.932663i	$-0.382522\pi$
$769$	20.0077	0.721495	0.360748	+	0.932663i	$0.382522\pi$
$770$	0	0
$771$	0.462455	0.0166549
$772$	4.14754	0.149273
$773$	1.59040	0.0572026	0.0286013	−	0.999591i	$-0.490895\pi$
$773$	1.59040	0.0572026	0.0286013	+	0.999591i	$0.490895\pi$
$774$	−3.66667	−0.131796
$775$	0	0
$776$	−12.8140	−0.459996
$777$	−0.254868	−0.00914333
$778$	55.3558	1.98460
$779$	0	0
$780$	0	0
$781$	−10.4219	−0.372924
$782$	−76.6858	−2.74228
$783$	0.659182	0.0235572
$784$	17.6494	0.630334
$785$	0	0
$786$	−2.90477	−0.103610
$787$	2.87913	0.102630	0.0513150	−	0.998683i	$-0.483659\pi$
$787$	2.87913	0.102630	0.0513150	+	0.998683i	$0.483659\pi$
$788$	−5.81474	−0.207142
$789$	−0.751037	−0.0267376
$790$	0	0
$791$	−25.4048	−0.903290
$792$	−5.88440	−0.209093
$793$	−8.99324	−0.319359
$794$	−49.0116	−1.73936
$795$	0	0
$796$	−7.80846	−0.276763
$797$	30.6071	1.08416	0.542080	−	0.840327i	$-0.317637\pi$
$797$	30.6071	1.08416	0.542080	+	0.840327i	$0.317637\pi$
$798$	0	0
$799$	−70.5486	−2.49583
$800$	0	0
$801$	−3.75660	−0.132733
$802$	−8.31678	−0.293676
$803$	−7.60515	−0.268380
$804$	0.689091	0.0243024
$805$	0	0
$806$	−24.8479	−0.875229
$807$	−1.65708	−0.0583320
$808$	−29.2368	−1.02855
$809$	51.0221	1.79384	0.896920	−	0.442193i	$-0.145799\pi$
$809$	51.0221	1.79384	0.896920	+	0.442193i	$0.145799\pi$
$810$	0	0
$811$	−34.3444	−1.20600	−0.602998	−	0.797743i	$-0.706027\pi$
$811$	−34.3444	−1.20600	−0.602998	+	0.797743i	$0.706027\pi$
$812$	−2.59570	−0.0910912
$813$	0.767201	0.0269069
$814$	3.43818	0.120508
$815$	0	0
$816$	2.59062	0.0906900
$817$	0	0
$818$	14.6614	0.512623
$819$	13.4105	0.468600
$820$	0	0
$821$	40.2464	1.40461	0.702305	−	0.711876i	$-0.252154\pi$
$821$	40.2464	1.40461	0.702305	+	0.711876i	$0.252154\pi$
$822$	2.19993	0.0767313
$823$	23.3478	0.813853	0.406927	−	0.913461i	$-0.366601\pi$
$823$	23.3478	0.813853	0.406927	+	0.913461i	$0.366601\pi$
$824$	−5.48268	−0.190998
$825$	0	0
$826$	−30.7634	−1.07040
$827$	−32.7322	−1.13821	−0.569105	−	0.822265i	$-0.692710\pi$
$827$	−32.7322	−1.13821	−0.569105	+	0.822265i	$0.692710\pi$
$828$	−23.4087	−0.813508
$829$	−4.83161	−0.167809	−0.0839044	−	0.996474i	$-0.526739\pi$
$829$	−4.83161	−0.167809	−0.0839044	+	0.996474i	$0.526739\pi$
$830$	0	0
$831$	0.895650	0.0310697
$832$	−0.535772	−0.0185746
$833$	21.4193	0.742134
$834$	0.209828	0.00726574
$835$	0	0
$836$	0	0
$837$	−3.01552	−0.104232
$838$	37.7880	1.30537
$839$	−37.2171	−1.28488	−0.642438	−	0.766338i	$-0.722077\pi$
$839$	−37.2171	−1.28488	−0.642438	+	0.766338i	$0.722077\pi$
$840$	0	0
$841$	−27.3502	−0.943109
$842$	−53.8904	−1.85719
$843$	0.934365	0.0321812
$844$	25.3678	0.873197
$845$	0	0
$846$	−61.2084	−2.10439
$847$	−17.6861	−0.607702
$848$	−51.2320	−1.75931
$849$	0.0815921	0.00280024
$850$	0	0
$851$	−11.5196	−0.394887
$852$	0.791482	0.0271157
$853$	12.1087	0.414593	0.207297	−	0.978278i	$-0.433533\pi$
$853$	12.1087	0.414593	0.207297	+	0.978278i	$0.433533\pi$
$854$	12.2151	0.417994
$855$	0	0
$856$	−2.40147	−0.0820805
$857$	−23.8743	−0.815532	−0.407766	−	0.913086i	$-0.633692\pi$
$857$	−23.8743	−0.815532	−0.407766	+	0.913086i	$0.633692\pi$
$858$	0.443330	0.0151350
$859$	35.4892	1.21088	0.605438	−	0.795893i	$-0.292998\pi$
$859$	35.4892	1.21088	0.605438	+	0.795893i	$0.292998\pi$
$860$	0	0
$861$	1.55033	0.0528352
$862$	44.9396	1.53065
$863$	−26.9330	−0.916810	−0.458405	−	0.888743i	$-0.651579\pi$
$863$	−26.9330	−0.916810	−0.458405	+	0.888743i	$0.651579\pi$
$864$	2.85223	0.0970348
$865$	0	0
$866$	56.0930	1.90612
$867$	1.68815	0.0573325
$868$	11.8744	0.403044
$869$	−9.05154	−0.307053
$870$	0	0
$871$	−17.8429	−0.604584
$872$	25.0209	0.847313
$873$	−23.8755	−0.808063
$874$	0	0
$875$	0	0
$876$	0.577568	0.0195142
$877$	−24.7625	−0.836170	−0.418085	−	0.908408i	$-0.637299\pi$
$877$	−24.7625	−0.836170	−0.418085	+	0.908408i	$0.637299\pi$
$878$	−16.6094	−0.560540
$879$	−1.14941	−0.0387688
$880$	0	0
$881$	53.6558	1.80771	0.903855	−	0.427840i	$-0.140725\pi$
$881$	53.6558	1.80771	0.903855	+	0.427840i	$0.140725\pi$
$882$	18.5835	0.625738
$883$	0.946198	0.0318421	0.0159210	−	0.999873i	$-0.494932\pi$
$883$	0.946198	0.0318421	0.0159210	+	0.999873i	$0.494932\pi$
$884$	15.8355	0.532605
$885$	0	0
$886$	−16.2962	−0.547481
$887$	2.27312	0.0763241	0.0381620	−	0.999272i	$-0.487850\pi$
$887$	2.27312	0.0763241	0.0381620	+	0.999272i	$0.487850\pi$
$888$	0.219917	0.00737992
$889$	35.7498	1.19901
$890$	0	0
$891$	−10.9371	−0.366406
$892$	4.64833	0.155638
$893$	0	0
$894$	2.18636	0.0731229
$895$	0	0
$896$	21.4186	0.715545
$897$	−1.48537	−0.0495952
$898$	48.5619	1.62053
$899$	−7.54745	−0.251722
$900$	0	0
$901$	−62.1752	−2.07135
$902$	−20.9141	−0.696363
$903$	0.111187	0.00370006
$904$	21.9209	0.729078
$905$	0	0
$906$	−3.54675	−0.117833
$907$	−12.3800	−0.411070	−0.205535	−	0.978650i	$-0.565893\pi$
$907$	−12.3800	−0.411070	−0.205535	+	0.978650i	$0.565893\pi$
$908$	−7.17479	−0.238104
$909$	−54.4750	−1.80682
$910$	0	0
$911$	47.9424	1.58840	0.794202	−	0.607654i	$-0.207889\pi$
$911$	47.9424	1.58840	0.794202	+	0.607654i	$0.207889\pi$
$912$	0	0
$913$	0.0165396	0.000547381 0
$914$	12.2817	0.406242
$915$	0	0
$916$	−25.3255	−0.836777
$917$	−35.9438	−1.18697
$918$	5.46216	0.180278
$919$	43.4053	1.43181	0.715904	−	0.698199i	$-0.246015\pi$
$919$	43.4053	1.43181	0.715904	+	0.698199i	$0.246015\pi$
$920$	0	0
$921$	1.61369	0.0531728
$922$	26.7435	0.880750
$923$	−20.4942	−0.674574
$924$	−0.211860	−0.00696969
$925$	0	0
$926$	48.7192	1.60101
$927$	−10.2155	−0.335522
$928$	7.13875	0.234341
$929$	−37.9300	−1.24444	−0.622222	−	0.782841i	$-0.713770\pi$
$929$	−37.9300	−1.24444	−0.622222	+	0.782841i	$0.713770\pi$
$930$	0	0
$931$	0	0
$932$	18.4963	0.605865
$933$	−0.0781175	−0.00255745
$934$	67.3453	2.20360
$935$	0	0
$936$	−11.5714	−0.378224
$937$	6.68720	0.218461	0.109231	−	0.994016i	$-0.465161\pi$
$937$	6.68720	0.218461	0.109231	+	0.994016i	$0.465161\pi$
$938$	24.2353	0.791311
$939$	1.60711	0.0524461
$940$	0	0
$941$	−4.36774	−0.142384	−0.0711921	−	0.997463i	$-0.522680\pi$
$941$	−4.36774	−0.142384	−0.0711921	+	0.997463i	$0.522680\pi$
$942$	2.56347	0.0835225
$943$	70.0725	2.28187
$944$	46.9728	1.52883
$945$	0	0
$946$	−1.49991	−0.0487664
$947$	1.50804	0.0490046	0.0245023	−	0.999700i	$-0.492200\pi$
$947$	1.50804	0.0490046	0.0245023	+	0.999700i	$0.492200\pi$
$948$	0.687413	0.0223261
$949$	−14.9552	−0.485467
$950$	0	0
$951$	−0.758710	−0.0246029
$952$	18.1154	0.587123
$953$	16.0555	0.520090	0.260045	−	0.965596i	$-0.416263\pi$
$953$	16.0555	0.520090	0.260045	+	0.965596i	$0.416263\pi$
$954$	−53.9435	−1.74649
$955$	0	0
$956$	11.3117	0.365845
$957$	0.134660	0.00435293
$958$	−18.4683	−0.596684
$959$	27.2220	0.879045
$960$	0	0
$961$	3.52693	0.113772
$962$	6.76104	0.217985
$963$	−4.47450	−0.144189
$964$	3.60506	0.116111
$965$	0	0
$966$	2.01752	0.0649127
$967$	45.0466	1.44860	0.724300	−	0.689485i	$-0.242163\pi$
$967$	45.0466	1.44860	0.724300	+	0.689485i	$0.242163\pi$
$968$	15.2607	0.490498
$969$	0	0
$970$	0	0
$971$	−15.2507	−0.489417	−0.244709	−	0.969597i	$-0.578692\pi$
$971$	−15.2507	−0.489417	−0.244709	+	0.969597i	$0.578692\pi$
$972$	2.50205	0.0802531
$973$	2.59642	0.0832374
$974$	−24.8200	−0.795284
$975$	0	0
$976$	−18.6514	−0.597016
$977$	−26.8192	−0.858023	−0.429011	−	0.903299i	$-0.641138\pi$
$977$	−26.8192	−0.858023	−0.429011	+	0.903299i	$0.641138\pi$
$978$	1.50142	0.0480100
$979$	−1.53670	−0.0491132
$980$	0	0
$981$	46.6197	1.48845
$982$	32.9234	1.05063
$983$	28.8049	0.918735	0.459367	−	0.888246i	$-0.348076\pi$
$983$	28.8049	0.918735	0.459367	+	0.888246i	$0.348076\pi$
$984$	−1.33773	−0.0426452
$985$	0	0
$986$	13.6711	0.435375
$987$	1.85606	0.0590790
$988$	0	0
$989$	5.02545	0.159800
$990$	0	0
$991$	−10.3840	−0.329859	−0.164929	−	0.986305i	$-0.552740\pi$
$991$	−10.3840	−0.329859	−0.164929	+	0.986305i	$0.552740\pi$
$992$	−32.6573	−1.03687
$993$	0.339162	0.0107630
$994$	27.8364	0.882917
$995$	0	0
$996$	−0.00125609	−3.98007e−5 0
$997$	−49.9100	−1.58066	−0.790332	−	0.612679i	$-0.790092\pi$
$997$	−49.9100	−1.58066	−0.790332	+	0.612679i	$0.790092\pi$
$998$	28.5704	0.904379
$999$	0.820516	0.0259600

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9025.2.a.bh.1.4		4
5.4	even	2		1805.2.a.n.1.1	yes	4
19.18	odd	2		9025.2.a.bo.1.1		4
95.94	odd	2		1805.2.a.j.1.4	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1805.2.a.j.1.4	✓	4	95.94	odd	2
1805.2.a.n.1.1	yes	4	5.4	even	2
9025.2.a.bh.1.4		4	1.1	even	1	trivial
9025.2.a.bo.1.1		4	19.18	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 9025 = 5^{2} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9025.a (trivial)

\(f(q)\)	\(=\)	\(q+1.75660 q^{2} +0.0856374 q^{3} +1.08564 q^{4} +0.150431 q^{6} +1.86144 q^{7} -1.60617 q^{8} -2.99267 q^{9} +O(q^{10})\)	\(q+1.75660 q^{2} +0.0856374 q^{3} +1.08564 q^{4} +0.150431 q^{6} +1.86144 q^{7} -1.60617 q^{8} -2.99267 q^{9} -1.22420 q^{11} +0.0929712 q^{12} -2.40734 q^{13} +3.26979 q^{14} -4.99267 q^{16} -6.05910 q^{17} -5.25691 q^{18} +0.159409 q^{21} -2.15043 q^{22} +7.20500 q^{23} -0.137548 q^{24} -4.22873 q^{26} -0.513197 q^{27} +2.02084 q^{28} -1.28446 q^{29} +5.87596 q^{31} -5.55777 q^{32} -0.104837 q^{33} -10.6434 q^{34} -3.24895 q^{36} -1.59883 q^{37} -0.206159 q^{39} +9.72553 q^{41} +0.280017 q^{42} +0.697495 q^{43} -1.32904 q^{44} +12.6563 q^{46} +11.6434 q^{47} -0.427559 q^{48} -3.53506 q^{49} -0.518886 q^{51} -2.61350 q^{52} +10.2614 q^{53} -0.901480 q^{54} -2.98978 q^{56} -2.25628 q^{58} -9.40836 q^{59} +3.73575 q^{61} +10.3217 q^{62} -5.57066 q^{63} +0.222557 q^{64} -0.184157 q^{66} +7.41188 q^{67} -6.57799 q^{68} +0.617018 q^{69} +8.51320 q^{71} +4.80672 q^{72} +6.21234 q^{73} -2.80851 q^{74} -2.27877 q^{77} -0.362138 q^{78} +7.39383 q^{79} +8.93405 q^{81} +17.0839 q^{82} -0.0135105 q^{83} +0.173060 q^{84} +1.22522 q^{86} -0.109998 q^{87} +1.96627 q^{88} +1.25527 q^{89} -4.48111 q^{91} +7.82202 q^{92} +0.503202 q^{93} +20.4528 q^{94} -0.475953 q^{96} +7.97800 q^{97} -6.20967 q^{98} +3.66363 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} - q^{3} + 3 q^{4} - 7 q^{6} + 11 q^{7} - 6 q^{8} + 5 q^{9}+O(q^{10}) \) 4 * q - q^2 - q^3 + 3 * q^4 - 7 * q^6 + 11 * q^7 - 6 * q^8 + 5 * q^9	\( 4 q - q^{2} - q^{3} + 3 q^{4} - 7 q^{6} + 11 q^{7} - 6 q^{8} + 5 q^{9} + 16 q^{12} + 2 q^{13} - 11 q^{14} - 3 q^{16} + 7 q^{17} - 17 q^{18} + 2 q^{21} - q^{22} + 11 q^{23} - 13 q^{24} + 9 q^{26} + 14 q^{27} + 13 q^{28} - 15 q^{29} - q^{31} - 3 q^{32} - 12 q^{33} - 22 q^{34} + 16 q^{36} + 11 q^{37} - 29 q^{39} + 22 q^{41} - 19 q^{42} + 26 q^{43} - 12 q^{44} + 10 q^{46} + 26 q^{47} + 13 q^{48} + 13 q^{49} - 11 q^{51} - 27 q^{52} + 16 q^{53} - 25 q^{54} - 8 q^{56} + 3 q^{58} - 10 q^{59} + 2 q^{61} + 31 q^{62} + 17 q^{63} + 4 q^{64} + 22 q^{66} - 3 q^{67} - 4 q^{68} + 14 q^{69} + 18 q^{71} - 29 q^{72} + 24 q^{73} - 17 q^{74} + 6 q^{77} + 15 q^{78} + 30 q^{79} - 4 q^{81} + 13 q^{82} + 12 q^{83} + 52 q^{84} - 16 q^{86} + q^{87} + 23 q^{88} + 9 q^{89} - 9 q^{91} + 25 q^{92} - 7 q^{93} - 11 q^{94} - 6 q^{96} - 19 q^{97} - 48 q^{98} - 9 q^{99}+O(q^{100}) \) 4 * q - q^2 - q^3 + 3 * q^4 - 7 * q^6 + 11 * q^7 - 6 * q^8 + 5 * q^9 + 16 * q^12 + 2 * q^13 - 11 * q^14 - 3 * q^16 + 7 * q^17 - 17 * q^18 + 2 * q^21 - q^22 + 11 * q^23 - 13 * q^24 + 9 * q^26 + 14 * q^27 + 13 * q^28 - 15 * q^29 - q^31 - 3 * q^32 - 12 * q^33 - 22 * q^34 + 16 * q^36 + 11 * q^37 - 29 * q^39 + 22 * q^41 - 19 * q^42 + 26 * q^43 - 12 * q^44 + 10 * q^46 + 26 * q^47 + 13 * q^48 + 13 * q^49 - 11 * q^51 - 27 * q^52 + 16 * q^53 - 25 * q^54 - 8 * q^56 + 3 * q^58 - 10 * q^59 + 2 * q^61 + 31 * q^62 + 17 * q^63 + 4 * q^64 + 22 * q^66 - 3 * q^67 - 4 * q^68 + 14 * q^69 + 18 * q^71 - 29 * q^72 + 24 * q^73 - 17 * q^74 + 6 * q^77 + 15 * q^78 + 30 * q^79 - 4 * q^81 + 13 * q^82 + 12 * q^83 + 52 * q^84 - 16 * q^86 + q^87 + 23 * q^88 + 9 * q^89 - 9 * q^91 + 25 * q^92 - 7 * q^93 - 11 * q^94 - 6 * q^96 - 19 * q^97 - 48 * q^98 - 9 * q^99

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