LMFDB - Embedded newform 8025.2.a.bo.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8025,2,Mod(1,8025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8025, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8025 = 3 \cdot 5^{2} \cdot 107 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8025.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:	$64.0799476221$
Analytic rank:	$1$
Dimension:	$22$
Twist minimal:	no (minimal twist has level 1605)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Character	$\chi$	$=$	8025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.13388 q^{2} +1.00000 q^{3} +2.55344 q^{4} -2.13388 q^{6} +3.51369 q^{7} -1.18098 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q-2.13388 q^{2} +1.00000 q^{3} +2.55344 q^{4} -2.13388 q^{6} +3.51369 q^{7} -1.18098 q^{8} +1.00000 q^{9} -3.29228 q^{11} +2.55344 q^{12} -1.71868 q^{13} -7.49779 q^{14} -2.58682 q^{16} -4.60454 q^{17} -2.13388 q^{18} +6.72570 q^{19} +3.51369 q^{21} +7.02533 q^{22} -4.98891 q^{23} -1.18098 q^{24} +3.66745 q^{26} +1.00000 q^{27} +8.97200 q^{28} +1.24193 q^{29} -2.61471 q^{31} +7.88192 q^{32} -3.29228 q^{33} +9.82554 q^{34} +2.55344 q^{36} +0.136021 q^{37} -14.3518 q^{38} -1.71868 q^{39} -7.67491 q^{41} -7.49779 q^{42} -0.449645 q^{43} -8.40664 q^{44} +10.6457 q^{46} +0.351126 q^{47} -2.58682 q^{48} +5.34600 q^{49} -4.60454 q^{51} -4.38855 q^{52} +2.08757 q^{53} -2.13388 q^{54} -4.14959 q^{56} +6.72570 q^{57} -2.65012 q^{58} -1.54788 q^{59} -10.4281 q^{61} +5.57947 q^{62} +3.51369 q^{63} -11.6454 q^{64} +7.02533 q^{66} +15.4553 q^{67} -11.7574 q^{68} -4.98891 q^{69} +4.13573 q^{71} -1.18098 q^{72} +2.50840 q^{73} -0.290252 q^{74} +17.1737 q^{76} -11.5680 q^{77} +3.66745 q^{78} +9.50482 q^{79} +1.00000 q^{81} +16.3773 q^{82} +14.1033 q^{83} +8.97200 q^{84} +0.959487 q^{86} +1.24193 q^{87} +3.88811 q^{88} -14.5316 q^{89} -6.03890 q^{91} -12.7389 q^{92} -2.61471 q^{93} -0.749260 q^{94} +7.88192 q^{96} -17.3609 q^{97} -11.4077 q^{98} -3.29228 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 3 q^{2} + 22 q^{3} + 13 q^{4} - 3 q^{6} - 2 q^{7} - 9 q^{8} + 22 q^{9}+O(q^{10}) $ 22 * q - 3 * q^2 + 22 * q^3 + 13 * q^4 - 3 * q^6 - 2 * q^7 - 9 * q^8 + 22 * q^9	$ 22 q - 3 q^{2} + 22 q^{3} + 13 q^{4} - 3 q^{6} - 2 q^{7} - 9 q^{8} + 22 q^{9} - 16 q^{11} + 13 q^{12} - 20 q^{14} - q^{16} - 20 q^{17} - 3 q^{18} - 16 q^{19} - 2 q^{21} - 4 q^{22} - 14 q^{23} - 9 q^{24} - 16 q^{26} + 22 q^{27} + q^{28} - 24 q^{29} - 42 q^{31} - 11 q^{32} - 16 q^{33} - 14 q^{34} + 13 q^{36} - 2 q^{37} - 21 q^{38} - 26 q^{41} - 20 q^{42} - 2 q^{43} - 24 q^{44} - 16 q^{46} - 26 q^{47} - q^{48} - 10 q^{49} - 20 q^{51} - 6 q^{52} - 22 q^{53} - 3 q^{54} - 42 q^{56} - 16 q^{57} + 69 q^{58} - 34 q^{59} - 16 q^{61} - 34 q^{62} - 2 q^{63} - 39 q^{64} - 4 q^{66} - 6 q^{68} - 14 q^{69} - 76 q^{71} - 9 q^{72} + 14 q^{73} - 12 q^{74} - 48 q^{76} - 54 q^{77} - 16 q^{78} - 72 q^{79} + 22 q^{81} - 2 q^{82} - 28 q^{83} + q^{84} - 22 q^{86} - 24 q^{87} + 19 q^{88} - 22 q^{89} - 58 q^{91} - 34 q^{92} - 42 q^{93} - 8 q^{94} - 11 q^{96} - 10 q^{97} + 3 q^{98} - 16 q^{99}+O(q^{100}) $ 22 * q - 3 * q^2 + 22 * q^3 + 13 * q^4 - 3 * q^6 - 2 * q^7 - 9 * q^8 + 22 * q^9 - 16 * q^11 + 13 * q^12 - 20 * q^14 - q^16 - 20 * q^17 - 3 * q^18 - 16 * q^19 - 2 * q^21 - 4 * q^22 - 14 * q^23 - 9 * q^24 - 16 * q^26 + 22 * q^27 + q^28 - 24 * q^29 - 42 * q^31 - 11 * q^32 - 16 * q^33 - 14 * q^34 + 13 * q^36 - 2 * q^37 - 21 * q^38 - 26 * q^41 - 20 * q^42 - 2 * q^43 - 24 * q^44 - 16 * q^46 - 26 * q^47 - q^48 - 10 * q^49 - 20 * q^51 - 6 * q^52 - 22 * q^53 - 3 * q^54 - 42 * q^56 - 16 * q^57 + 69 * q^58 - 34 * q^59 - 16 * q^61 - 34 * q^62 - 2 * q^63 - 39 * q^64 - 4 * q^66 - 6 * q^68 - 14 * q^69 - 76 * q^71 - 9 * q^72 + 14 * q^73 - 12 * q^74 - 48 * q^76 - 54 * q^77 - 16 * q^78 - 72 * q^79 + 22 * q^81 - 2 * q^82 - 28 * q^83 + q^84 - 22 * q^86 - 24 * q^87 + 19 * q^88 - 22 * q^89 - 58 * q^91 - 34 * q^92 - 42 * q^93 - 8 * q^94 - 11 * q^96 - 10 * q^97 + 3 * q^98 - 16 * 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 $

$2$	−2.13388	−1.50888	−0.754440	−	0.656369i	$-0.772092\pi$
$2$	−2.13388	−1.50888	−0.754440	+	0.656369i	$0.772092\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	2.55344	1.27672
$5$	0	0
$5$	0	0
$6$	−2.13388	−0.871153
$7$	3.51369	1.32805	0.664025	−	0.747711i	$-0.268847\pi$
$7$	3.51369	1.32805	0.664025	+	0.747711i	$0.268847\pi$
$8$	−1.18098	−0.417539
$9$	1.00000	0.333333
$10$	0	0
$11$	−3.29228	−0.992659	−0.496330	−	0.868134i	$-0.665319\pi$
$11$	−3.29228	−0.992659	−0.496330	+	0.868134i	$0.665319\pi$
$12$	2.55344	0.737115
$13$	−1.71868	−0.476676	−0.238338	−	0.971182i	$-0.576603\pi$
$13$	−1.71868	−0.476676	−0.238338	+	0.971182i	$0.576603\pi$
$14$	−7.49779	−2.00387
$15$	0	0
$16$	−2.58682	−0.646704
$17$	−4.60454	−1.11677	−0.558383	−	0.829583i	$-0.688578\pi$
$17$	−4.60454	−1.11677	−0.558383	+	0.829583i	$0.688578\pi$
$18$	−2.13388	−0.502960
$19$	6.72570	1.54298	0.771491	−	0.636241i	$-0.219511\pi$
$19$	6.72570	1.54298	0.771491	+	0.636241i	$0.219511\pi$
$20$	0	0
$21$	3.51369	0.766749
$22$	7.02533	1.49780
$23$	−4.98891	−1.04026	−0.520130	−	0.854087i	$-0.674116\pi$
$23$	−4.98891	−1.04026	−0.520130	+	0.854087i	$0.674116\pi$
$24$	−1.18098	−0.241066
$25$	0	0
$26$	3.66745	0.719247
$27$	1.00000	0.192450
$28$	8.97200	1.69555
$29$	1.24193	0.230620	0.115310	−	0.993330i	$-0.463214\pi$
$29$	1.24193	0.230620	0.115310	+	0.993330i	$0.463214\pi$
$30$	0	0
$31$	−2.61471	−0.469615	−0.234808	−	0.972042i	$-0.575446\pi$
$31$	−2.61471	−0.469615	−0.234808	+	0.972042i	$0.575446\pi$
$32$	7.88192	1.39334
$33$	−3.29228	−0.573112
$34$	9.82554	1.68507
$35$	0	0
$36$	2.55344	0.425574
$37$	0.136021	0.0223617	0.0111808	−	0.999937i	$-0.496441\pi$
$37$	0.136021	0.0223617	0.0111808	+	0.999937i	$0.496441\pi$
$38$	−14.3518	−2.32817
$39$	−1.71868	−0.275209
$40$	0	0
$41$	−7.67491	−1.19862	−0.599310	−	0.800517i	$-0.704558\pi$
$41$	−7.67491	−1.19862	−0.599310	+	0.800517i	$0.704558\pi$
$42$	−7.49779	−1.15693
$43$	−0.449645	−0.0685701	−0.0342851	−	0.999412i	$-0.510915\pi$
$43$	−0.449645	−0.0685701	−0.0342851	+	0.999412i	$0.510915\pi$
$44$	−8.40664	−1.26735
$45$	0	0
$46$	10.6457	1.56963
$47$	0.351126	0.0512169	0.0256085	−	0.999672i	$-0.491848\pi$
$47$	0.351126	0.0512169	0.0256085	+	0.999672i	$0.491848\pi$
$48$	−2.58682	−0.373375
$49$	5.34600	0.763714
$50$	0	0
$51$	−4.60454	−0.644765
$52$	−4.38855	−0.608582
$53$	2.08757	0.286749	0.143375	−	0.989668i	$-0.454205\pi$
$53$	2.08757	0.286749	0.143375	+	0.989668i	$0.454205\pi$
$54$	−2.13388	−0.290384
$55$	0	0
$56$	−4.14959	−0.554512
$57$	6.72570	0.890841
$58$	−2.65012	−0.347978
$59$	−1.54788	−0.201517	−0.100759	−	0.994911i	$-0.532127\pi$
$59$	−1.54788	−0.201517	−0.100759	+	0.994911i	$0.532127\pi$
$60$	0	0
$61$	−10.4281	−1.33518	−0.667588	−	0.744531i	$-0.732673\pi$
$61$	−10.4281	−1.33518	−0.667588	+	0.744531i	$0.732673\pi$
$62$	5.57947	0.708593
$63$	3.51369	0.442683
$64$	−11.6454	−1.45568
$65$	0	0
$66$	7.02533	0.864758
$67$	15.4553	1.88816	0.944080	−	0.329715i	$-0.106953\pi$
$67$	15.4553	1.88816	0.944080	+	0.329715i	$0.106953\pi$
$68$	−11.7574	−1.42580
$69$	−4.98891	−0.600594
$70$	0	0
$71$	4.13573	0.490821	0.245410	−	0.969419i	$-0.421077\pi$
$71$	4.13573	0.490821	0.245410	+	0.969419i	$0.421077\pi$
$72$	−1.18098	−0.139180
$73$	2.50840	0.293586	0.146793	−	0.989167i	$-0.453105\pi$
$73$	2.50840	0.293586	0.146793	+	0.989167i	$0.453105\pi$
$74$	−0.290252	−0.0337411
$75$	0	0
$76$	17.1737	1.96996
$77$	−11.5680	−1.31830
$78$	3.66745	0.415257
$79$	9.50482	1.06938	0.534688	−	0.845050i	$-0.320429\pi$
$79$	9.50482	1.06938	0.534688	+	0.845050i	$0.320429\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	16.3773	1.80857
$83$	14.1033	1.54804	0.774021	−	0.633161i	$-0.218243\pi$
$83$	14.1033	1.54804	0.774021	+	0.633161i	$0.218243\pi$
$84$	8.97200	0.978925
$85$	0	0
$86$	0.959487	0.103464
$87$	1.24193	0.133148
$88$	3.88811	0.414474
$89$	−14.5316	−1.54034	−0.770172	−	0.637836i	$-0.779830\pi$
$89$	−14.5316	−1.54034	−0.770172	+	0.637836i	$0.779830\pi$
$90$	0	0
$91$	−6.03890	−0.633049
$92$	−12.7389	−1.32812
$93$	−2.61471	−0.271133
$94$	−0.749260	−0.0772802
$95$	0	0
$96$	7.88192	0.804445
$97$	−17.3609	−1.76273	−0.881366	−	0.472434i	$-0.843376\pi$
$97$	−17.3609	−1.76273	−0.881366	+	0.472434i	$0.843376\pi$
$98$	−11.4077	−1.15235
$99$	−3.29228	−0.330886
$100$	0	0
$101$	−6.97233	−0.693773	−0.346886	−	0.937907i	$-0.612761\pi$
$101$	−6.97233	−0.693773	−0.346886	+	0.937907i	$0.612761\pi$
$102$	9.82554	0.972874
$103$	14.0868	1.38802	0.694008	−	0.719967i	$-0.255843\pi$
$103$	14.0868	1.38802	0.694008	+	0.719967i	$0.255843\pi$
$104$	2.02972	0.199031
$105$	0	0
$106$	−4.45461	−0.432670
$107$	−1.00000	−0.0966736
$107$	−1.00000	−0.0966736
$108$	2.55344	0.245705
$109$	−11.9538	−1.14497	−0.572483	−	0.819917i	$-0.694020\pi$
$109$	−11.9538	−1.14497	−0.572483	+	0.819917i	$0.694020\pi$
$110$	0	0
$111$	0.136021	0.0129105
$112$	−9.08927	−0.858855
$113$	6.10453	0.574266	0.287133	−	0.957891i	$-0.407298\pi$
$113$	6.10453	0.574266	0.287133	+	0.957891i	$0.407298\pi$
$114$	−14.3518	−1.34417
$115$	0	0
$116$	3.17118	0.294437
$117$	−1.71868	−0.158892
$118$	3.30299	0.304065
$119$	−16.1789	−1.48312
$120$	0	0
$121$	−0.160903	−0.0146276
$122$	22.2522	2.01462
$123$	−7.67491	−0.692023
$124$	−6.67650	−0.599568
$125$	0	0
$126$	−7.49779	−0.667956
$127$	0.773032	0.0685955	0.0342978	−	0.999412i	$-0.489081\pi$
$127$	0.773032	0.0685955	0.0342978	+	0.999412i	$0.489081\pi$
$128$	9.08610	0.803105
$129$	−0.449645	−0.0395890
$130$	0	0
$131$	−5.76228	−0.503453	−0.251727	−	0.967798i	$-0.580998\pi$
$131$	−5.76228	−0.503453	−0.251727	+	0.967798i	$0.580998\pi$
$132$	−8.40664	−0.731704
$133$	23.6320	2.04915
$134$	−32.9797	−2.84901
$135$	0	0
$136$	5.43787	0.466293
$137$	−4.29290	−0.366767	−0.183383	−	0.983041i	$-0.558705\pi$
$137$	−4.29290	−0.366767	−0.183383	+	0.983041i	$0.558705\pi$
$138$	10.6457	0.906225
$139$	−18.6594	−1.58267	−0.791336	−	0.611382i	$-0.790614\pi$
$139$	−18.6594	−1.58267	−0.791336	+	0.611382i	$0.790614\pi$
$140$	0	0
$141$	0.351126	0.0295701
$142$	−8.82514	−0.740590
$143$	5.65837	0.473176
$144$	−2.58682	−0.215568
$145$	0	0
$146$	−5.35263	−0.442987
$147$	5.34600	0.440931
$148$	0.347321	0.0285496
$149$	2.77489	0.227328	0.113664	−	0.993519i	$-0.463741\pi$
$149$	2.77489	0.227328	0.113664	+	0.993519i	$0.463741\pi$
$150$	0	0
$151$	−10.7481	−0.874668	−0.437334	−	0.899299i	$-0.644077\pi$
$151$	−10.7481	−0.874668	−0.437334	+	0.899299i	$0.644077\pi$
$152$	−7.94291	−0.644255
$153$	−4.60454	−0.372255
$154$	24.6848	1.98916
$155$	0	0
$156$	−4.38855	−0.351365
$157$	−11.5107	−0.918655	−0.459328	−	0.888267i	$-0.651910\pi$
$157$	−11.5107	−0.918655	−0.459328	+	0.888267i	$0.651910\pi$
$158$	−20.2821	−1.61356
$159$	2.08757	0.165555
$160$	0	0
$161$	−17.5295	−1.38152
$162$	−2.13388	−0.167653
$163$	−11.3617	−0.889916	−0.444958	−	0.895552i	$-0.646781\pi$
$163$	−11.3617	−0.889916	−0.444958	+	0.895552i	$0.646781\pi$
$164$	−19.5974	−1.53030
$165$	0	0
$166$	−30.0948	−2.33581
$167$	6.97590	0.539811	0.269905	−	0.962887i	$-0.413008\pi$
$167$	6.97590	0.539811	0.269905	+	0.962887i	$0.413008\pi$
$168$	−4.14959	−0.320148
$169$	−10.0461	−0.772780
$170$	0	0
$171$	6.72570	0.514327
$172$	−1.14814	−0.0875449
$173$	−18.0740	−1.37414	−0.687072	−	0.726590i	$-0.741104\pi$
$173$	−18.0740	−1.37414	−0.687072	+	0.726590i	$0.741104\pi$
$174$	−2.65012	−0.200905
$175$	0	0
$176$	8.51652	0.641957
$177$	−1.54788	−0.116346
$178$	31.0086	2.32420
$179$	−16.5859	−1.23969	−0.619845	−	0.784724i	$-0.712805\pi$
$179$	−16.5859	−1.23969	−0.619845	+	0.784724i	$0.712805\pi$
$180$	0	0
$181$	−12.1215	−0.900982	−0.450491	−	0.892781i	$-0.648751\pi$
$181$	−12.1215	−0.900982	−0.450491	+	0.892781i	$0.648751\pi$
$182$	12.8863	0.955195
$183$	−10.4281	−0.770864
$184$	5.89180	0.434349
$185$	0	0
$186$	5.57947	0.409107
$187$	15.1594	1.10857
$188$	0.896579	0.0653897
$189$	3.51369	0.255583
$190$	0	0
$191$	−16.3647	−1.18411	−0.592055	−	0.805897i	$-0.701683\pi$
$191$	−16.3647	−1.18411	−0.592055	+	0.805897i	$0.701683\pi$
$192$	−11.6454	−0.840436
$193$	−2.22589	−0.160223	−0.0801115	−	0.996786i	$-0.525528\pi$
$193$	−2.22589	−0.160223	−0.0801115	+	0.996786i	$0.525528\pi$
$194$	37.0461	2.65975
$195$	0	0
$196$	13.6507	0.975050
$197$	−26.4429	−1.88398	−0.941991	−	0.335638i	$-0.891048\pi$
$197$	−26.4429	−1.88398	−0.941991	+	0.335638i	$0.891048\pi$
$198$	7.02533	0.499268
$199$	2.59850	0.184203	0.0921014	−	0.995750i	$-0.470642\pi$
$199$	2.59850	0.184203	0.0921014	+	0.995750i	$0.470642\pi$
$200$	0	0
$201$	15.4553	1.09013
$202$	14.8781	1.04682
$203$	4.36374	0.306274
$204$	−11.7574	−0.823185
$205$	0	0
$206$	−30.0596	−2.09435
$207$	−4.98891	−0.346753
$208$	4.44591	0.308268
$209$	−22.1429	−1.53165
$210$	0	0
$211$	15.9985	1.10138	0.550692	−	0.834709i	$-0.314364\pi$
$211$	15.9985	1.10138	0.550692	+	0.834709i	$0.314364\pi$
$212$	5.33048	0.366099
$213$	4.13573	0.283375
$214$	2.13388	0.145869
$215$	0	0
$216$	−1.18098	−0.0803554
$217$	−9.18726	−0.623672
$218$	25.5080	1.72762
$219$	2.50840	0.169502
$220$	0	0
$221$	7.91373	0.532335
$222$	−0.290252	−0.0194804
$223$	2.86200	0.191653	0.0958267	−	0.995398i	$-0.469451\pi$
$223$	2.86200	0.191653	0.0958267	+	0.995398i	$0.469451\pi$
$224$	27.6946	1.85042
$225$	0	0
$226$	−13.0263	−0.866498
$227$	−22.9301	−1.52192	−0.760961	−	0.648798i	$-0.775272\pi$
$227$	−22.9301	−1.52192	−0.760961	+	0.648798i	$0.775272\pi$
$228$	17.1737	1.13736
$229$	7.15839	0.473040	0.236520	−	0.971627i	$-0.423993\pi$
$229$	7.15839	0.473040	0.236520	+	0.971627i	$0.423993\pi$
$230$	0	0
$231$	−11.5680	−0.761121
$232$	−1.46669	−0.0962927
$233$	26.7581	1.75298	0.876492	−	0.481416i	$-0.159877\pi$
$233$	26.7581	1.75298	0.876492	+	0.481416i	$0.159877\pi$
$234$	3.66745	0.239749
$235$	0	0
$236$	−3.95243	−0.257281
$237$	9.50482	0.617404
$238$	34.5239	2.23785
$239$	16.3324	1.05645	0.528227	−	0.849103i	$-0.322857\pi$
$239$	16.3324	1.05645	0.528227	+	0.849103i	$0.322857\pi$
$240$	0	0
$241$	3.51226	0.226244	0.113122	−	0.993581i	$-0.463915\pi$
$241$	3.51226	0.226244	0.113122	+	0.993581i	$0.463915\pi$
$242$	0.343348	0.0220713
$243$	1.00000	0.0641500
$244$	−26.6274	−1.70465
$245$	0	0
$246$	16.3773	1.04418
$247$	−11.5593	−0.735502
$248$	3.08791	0.196083
$249$	14.1033	0.893762
$250$	0	0
$251$	−1.73146	−0.109289	−0.0546444	−	0.998506i	$-0.517403\pi$
$251$	−1.73146	−0.109289	−0.0546444	+	0.998506i	$0.517403\pi$
$252$	8.97200	0.565183
$253$	16.4249	1.03262
$254$	−1.64956	−0.103502
$255$	0	0
$256$	3.90220	0.243888
$257$	30.0106	1.87201	0.936006	−	0.351984i	$-0.114493\pi$
$257$	30.0106	1.87201	0.936006	+	0.351984i	$0.114493\pi$
$258$	0.959487	0.0597351
$259$	0.477934	0.0296974
$260$	0	0
$261$	1.24193	0.0768732
$262$	12.2960	0.759651
$263$	4.73765	0.292136	0.146068	−	0.989275i	$-0.453338\pi$
$263$	4.73765	0.292136	0.146068	+	0.989275i	$0.453338\pi$
$264$	3.88811	0.239297
$265$	0	0
$266$	−50.4279	−3.09193
$267$	−14.5316	−0.889318
$268$	39.4641	2.41065
$269$	30.0891	1.83457	0.917283	−	0.398236i	$-0.130377\pi$
$269$	30.0891	1.83457	0.917283	+	0.398236i	$0.130377\pi$
$270$	0	0
$271$	−15.3409	−0.931895	−0.465948	−	0.884812i	$-0.654286\pi$
$271$	−15.3409	−0.931895	−0.465948	+	0.884812i	$0.654286\pi$
$272$	11.9111	0.722218
$273$	−6.03890	−0.365491
$274$	9.16052	0.553407
$275$	0	0
$276$	−12.7389	−0.766791
$277$	22.6003	1.35792	0.678959	−	0.734176i	$-0.262431\pi$
$277$	22.6003	1.35792	0.678959	+	0.734176i	$0.262431\pi$
$278$	39.8170	2.38806
$279$	−2.61471	−0.156538
$280$	0	0
$281$	4.59068	0.273857	0.136928	−	0.990581i	$-0.456277\pi$
$281$	4.59068	0.273857	0.136928	+	0.990581i	$0.456277\pi$
$282$	−0.749260	−0.0446178
$283$	13.0345	0.774822	0.387411	−	0.921907i	$-0.373369\pi$
$283$	13.0345	0.774822	0.387411	+	0.921907i	$0.373369\pi$
$284$	10.5603	0.626641
$285$	0	0
$286$	−12.0743	−0.713967
$287$	−26.9672	−1.59183
$288$	7.88192	0.464446
$289$	4.20183	0.247167
$290$	0	0
$291$	−17.3609	−1.01771
$292$	6.40506	0.374828
$293$	−29.0409	−1.69659	−0.848293	−	0.529527i	$-0.822369\pi$
$293$	−29.0409	−1.69659	−0.848293	+	0.529527i	$0.822369\pi$
$294$	−11.4077	−0.665312
$295$	0	0
$296$	−0.160637	−0.00933686
$297$	−3.29228	−0.191037
$298$	−5.92129	−0.343011
$299$	8.57433	0.495866
$300$	0	0
$301$	−1.57991	−0.0910645
$302$	22.9351	1.31977
$303$	−6.97233	−0.400550
$304$	−17.3982	−0.997853
$305$	0	0
$306$	9.82554	0.561689
$307$	−11.1235	−0.634853	−0.317427	−	0.948283i	$-0.602819\pi$
$307$	−11.1235	−0.634853	−0.317427	+	0.948283i	$0.602819\pi$
$308$	−29.5383	−1.68310
$309$	14.0868	0.801371
$310$	0	0
$311$	−19.7439	−1.11958	−0.559788	−	0.828636i	$-0.689117\pi$
$311$	−19.7439	−1.11958	−0.559788	+	0.828636i	$0.689117\pi$
$312$	2.02972	0.114910
$313$	2.41609	0.136565	0.0682826	−	0.997666i	$-0.478248\pi$
$313$	2.41609	0.136565	0.0682826	+	0.997666i	$0.478248\pi$
$314$	24.5625	1.38614
$315$	0	0
$316$	24.2700	1.36529
$317$	−6.42606	−0.360923	−0.180462	−	0.983582i	$-0.557759\pi$
$317$	−6.42606	−0.360923	−0.180462	+	0.983582i	$0.557759\pi$
$318$	−4.45461	−0.249802
$319$	−4.08876	−0.228927
$320$	0	0
$321$	−1.00000	−0.0558146
$322$	37.4058	2.08454
$323$	−30.9688	−1.72315
$324$	2.55344	0.141858
$325$	0	0
$326$	24.2445	1.34278
$327$	−11.9538	−0.661047
$328$	9.06390	0.500470
$329$	1.23375	0.0680186
$330$	0	0
$331$	−8.02618	−0.441159	−0.220579	−	0.975369i	$-0.570795\pi$
$331$	−8.02618	−0.441159	−0.220579	+	0.975369i	$0.570795\pi$
$332$	36.0120	1.97642
$333$	0.136021	0.00745388
$334$	−14.8857	−0.814510
$335$	0	0
$336$	−9.08927	−0.495860
$337$	−16.9208	−0.921736	−0.460868	−	0.887469i	$-0.652462\pi$
$337$	−16.9208	−0.921736	−0.460868	+	0.887469i	$0.652462\pi$
$338$	21.4373	1.16603
$339$	6.10453	0.331552
$340$	0	0
$341$	8.60834	0.466168
$342$	−14.3518	−0.776058
$343$	−5.81164	−0.313799
$344$	0.531021	0.0286307
$345$	0	0
$346$	38.5678	2.07342
$347$	3.25165	0.174558	0.0872788	−	0.996184i	$-0.472183\pi$
$347$	3.25165	0.174558	0.0872788	+	0.996184i	$0.472183\pi$
$348$	3.17118	0.169993
$349$	−6.20351	−0.332066	−0.166033	−	0.986120i	$-0.553096\pi$
$349$	−6.20351	−0.332066	−0.166033	+	0.986120i	$0.553096\pi$
$350$	0	0
$351$	−1.71868	−0.0917363
$352$	−25.9495	−1.38311
$353$	−24.6766	−1.31340	−0.656701	−	0.754151i	$-0.728049\pi$
$353$	−24.6766	−1.31340	−0.656701	+	0.754151i	$0.728049\pi$
$354$	3.30299	0.175552
$355$	0	0
$356$	−37.1055	−1.96659
$357$	−16.1789	−0.856280
$358$	35.3924	1.87054
$359$	−6.16836	−0.325553	−0.162777	−	0.986663i	$-0.552045\pi$
$359$	−6.16836	−0.325553	−0.162777	+	0.986663i	$0.552045\pi$
$360$	0	0
$361$	26.2350	1.38079
$362$	25.8658	1.35947
$363$	−0.160903	−0.00844523
$364$	−15.4200	−0.808226
$365$	0	0
$366$	22.2522	1.16314
$367$	20.8099	1.08627	0.543134	−	0.839646i	$-0.317237\pi$
$367$	20.8099	1.08627	0.543134	+	0.839646i	$0.317237\pi$
$368$	12.9054	0.672741
$369$	−7.67491	−0.399540
$370$	0	0
$371$	7.33505	0.380817
$372$	−6.67650	−0.346161
$373$	−25.2267	−1.30619	−0.653094	−	0.757277i	$-0.726529\pi$
$373$	−25.2267	−1.30619	−0.653094	+	0.757277i	$0.726529\pi$
$374$	−32.3484	−1.67270
$375$	0	0
$376$	−0.414672	−0.0213851
$377$	−2.13447	−0.109931
$378$	−7.49779	−0.385644
$379$	15.5084	0.796613	0.398306	−	0.917252i	$-0.369598\pi$
$379$	15.5084	0.796613	0.398306	+	0.917252i	$0.369598\pi$
$380$	0	0
$381$	0.773032	0.0396036
$382$	34.9204	1.78668
$383$	23.9017	1.22132	0.610661	−	0.791892i	$-0.290904\pi$
$383$	23.9017	1.22132	0.610661	+	0.791892i	$0.290904\pi$
$384$	9.08610	0.463673
$385$	0	0
$386$	4.74978	0.241757
$387$	−0.449645	−0.0228567
$388$	−44.3301	−2.25052
$389$	5.93696	0.301016	0.150508	−	0.988609i	$-0.451909\pi$
$389$	5.93696	0.301016	0.150508	+	0.988609i	$0.451909\pi$
$390$	0	0
$391$	22.9717	1.16173
$392$	−6.31351	−0.318880
$393$	−5.76228	−0.290669
$394$	56.4261	2.84270
$395$	0	0
$396$	−8.40664	−0.422450
$397$	−7.48821	−0.375822	−0.187911	−	0.982186i	$-0.560172\pi$
$397$	−7.48821	−0.375822	−0.187911	+	0.982186i	$0.560172\pi$
$398$	−5.54489	−0.277940
$399$	23.6320	1.18308
$400$	0	0
$401$	2.46398	0.123045	0.0615227	−	0.998106i	$-0.480404\pi$
$401$	2.46398	0.123045	0.0615227	+	0.998106i	$0.480404\pi$
$402$	−32.9797	−1.64488
$403$	4.49384	0.223854
$404$	−17.8034	−0.885754
$405$	0	0
$406$	−9.31169	−0.462131
$407$	−0.447818	−0.0221975
$408$	5.43787	0.269215
$409$	33.3135	1.64725	0.823624	−	0.567137i	$-0.191949\pi$
$409$	33.3135	1.64725	0.823624	+	0.567137i	$0.191949\pi$
$410$	0	0
$411$	−4.29290	−0.211753
$412$	35.9699	1.77211
$413$	−5.43877	−0.267625
$414$	10.6457	0.523209
$415$	0	0
$416$	−13.5465	−0.664171
$417$	−18.6594	−0.913756
$418$	47.2502	2.31108
$419$	−16.8152	−0.821478	−0.410739	−	0.911753i	$-0.634729\pi$
$419$	−16.8152	−0.821478	−0.410739	+	0.911753i	$0.634729\pi$
$420$	0	0
$421$	−30.9994	−1.51082	−0.755410	−	0.655252i	$-0.772562\pi$
$421$	−30.9994	−1.51082	−0.755410	+	0.655252i	$0.772562\pi$
$422$	−34.1389	−1.66186
$423$	0.351126	0.0170723
$424$	−2.46537	−0.119729
$425$	0	0
$426$	−8.82514	−0.427580
$427$	−36.6409	−1.77318
$428$	−2.55344	−0.123425
$429$	5.65837	0.273189
$430$	0	0
$431$	−25.3613	−1.22161	−0.610805	−	0.791781i	$-0.709154\pi$
$431$	−25.3613	−1.22161	−0.610805	+	0.791781i	$0.709154\pi$
$432$	−2.58682	−0.124458
$433$	−30.7176	−1.47619	−0.738096	−	0.674695i	$-0.764275\pi$
$433$	−30.7176	−1.47619	−0.738096	+	0.674695i	$0.764275\pi$
$434$	19.6045	0.941047
$435$	0	0
$436$	−30.5233	−1.46180
$437$	−33.5539	−1.60510
$438$	−5.35263	−0.255758
$439$	8.19806	0.391272	0.195636	−	0.980677i	$-0.437323\pi$
$439$	8.19806	0.391272	0.195636	+	0.980677i	$0.437323\pi$
$440$	0	0
$441$	5.34600	0.254571
$442$	−16.8870	−0.803230
$443$	−10.8363	−0.514850	−0.257425	−	0.966298i	$-0.582874\pi$
$443$	−10.8363	−0.514850	−0.257425	+	0.966298i	$0.582874\pi$
$444$	0.347321	0.0164831
$445$	0	0
$446$	−6.10715	−0.289182
$447$	2.77489	0.131248
$448$	−40.9184	−1.93321
$449$	−38.2898	−1.80701	−0.903504	−	0.428581i	$-0.859014\pi$
$449$	−38.2898	−1.80701	−0.903504	+	0.428581i	$0.859014\pi$
$450$	0	0
$451$	25.2679	1.18982
$452$	15.5876	0.733177
$453$	−10.7481	−0.504990
$454$	48.9300	2.29640
$455$	0	0
$456$	−7.94291	−0.371961
$457$	−7.68795	−0.359627	−0.179814	−	0.983701i	$-0.557549\pi$
$457$	−7.68795	−0.359627	−0.179814	+	0.983701i	$0.557549\pi$
$458$	−15.2751	−0.713761
$459$	−4.60454	−0.214922
$460$	0	0
$461$	−17.8353	−0.830674	−0.415337	−	0.909668i	$-0.636336\pi$
$461$	−17.8353	−0.830674	−0.415337	+	0.909668i	$0.636336\pi$
$462$	24.6848	1.14844
$463$	7.33726	0.340991	0.170496	−	0.985358i	$-0.445463\pi$
$463$	7.33726	0.340991	0.170496	+	0.985358i	$0.445463\pi$
$464$	−3.21263	−0.149143
$465$	0	0
$466$	−57.0987	−2.64504
$467$	27.6519	1.27958	0.639789	−	0.768551i	$-0.279022\pi$
$467$	27.6519	1.27958	0.639789	+	0.768551i	$0.279022\pi$
$468$	−4.38855	−0.202861
$469$	54.3050	2.50757
$470$	0	0
$471$	−11.5107	−0.530386
$472$	1.82802	0.0841412
$473$	1.48035	0.0680668
$474$	−20.2821	−0.931590
$475$	0	0
$476$	−41.3120	−1.89353
$477$	2.08757	0.0955831
$478$	−34.8513	−1.59406
$479$	7.28680	0.332942	0.166471	−	0.986046i	$-0.446763\pi$
$479$	7.28680	0.332942	0.166471	+	0.986046i	$0.446763\pi$
$480$	0	0
$481$	−0.233776	−0.0106593
$482$	−7.49473	−0.341376
$483$	−17.5295	−0.797619
$484$	−0.410857	−0.0186753
$485$	0	0
$486$	−2.13388	−0.0967947
$487$	4.81227	0.218065	0.109032	−	0.994038i	$-0.465225\pi$
$487$	4.81227	0.218065	0.109032	+	0.994038i	$0.465225\pi$
$488$	12.3153	0.557488
$489$	−11.3617	−0.513793
$490$	0	0
$491$	9.75084	0.440049	0.220025	−	0.975494i	$-0.429386\pi$
$491$	9.75084	0.440049	0.220025	+	0.975494i	$0.429386\pi$
$492$	−19.5974	−0.883521
$493$	−5.71850	−0.257548
$494$	24.6662	1.10978
$495$	0	0
$496$	6.76377	0.303702
$497$	14.5317	0.651834
$498$	−30.0948	−1.34858
$499$	18.1904	0.814315	0.407157	−	0.913358i	$-0.366520\pi$
$499$	18.1904	0.814315	0.407157	+	0.913358i	$0.366520\pi$
$500$	0	0
$501$	6.97590	0.311660
$502$	3.69473	0.164904
$503$	6.80923	0.303608	0.151804	−	0.988411i	$-0.451492\pi$
$503$	6.80923	0.303608	0.151804	+	0.988411i	$0.451492\pi$
$504$	−4.14959	−0.184837
$505$	0	0
$506$	−35.0487	−1.55811
$507$	−10.0461	−0.446165
$508$	1.97389	0.0875773
$509$	36.7186	1.62752	0.813761	−	0.581200i	$-0.197417\pi$
$509$	36.7186	1.62752	0.813761	+	0.581200i	$0.197417\pi$
$510$	0	0
$511$	8.81374	0.389897
$512$	−26.4990	−1.17110
$513$	6.72570	0.296947
$514$	−64.0391	−2.82464
$515$	0	0
$516$	−1.14814	−0.0505441
$517$	−1.15600	−0.0508410
$518$	−1.01985	−0.0448098
$519$	−18.0740	−0.793362
$520$	0	0
$521$	27.3546	1.19843	0.599214	−	0.800589i	$-0.295480\pi$
$521$	27.3546	1.19843	0.599214	+	0.800589i	$0.295480\pi$
$522$	−2.65012	−0.115993
$523$	43.8363	1.91683	0.958415	−	0.285380i	$-0.0921196\pi$
$523$	43.8363	1.91683	0.958415	+	0.285380i	$0.0921196\pi$
$524$	−14.7137	−0.642769
$525$	0	0
$526$	−10.1096	−0.440798
$527$	12.0395	0.524450
$528$	8.51652	0.370634
$529$	1.88923	0.0821404
$530$	0	0
$531$	−1.54788	−0.0671723
$532$	60.3430	2.61620
$533$	13.1907	0.571353
$534$	31.0086	1.34188
$535$	0	0
$536$	−18.2523	−0.788381
$537$	−16.5859	−0.715735
$538$	−64.2066	−2.76814
$539$	−17.6005	−0.758108
$540$	0	0
$541$	−6.55297	−0.281734	−0.140867	−	0.990029i	$-0.544989\pi$
$541$	−6.55297	−0.281734	−0.140867	+	0.990029i	$0.544989\pi$
$542$	32.7357	1.40612
$543$	−12.1215	−0.520182
$544$	−36.2926	−1.55603
$545$	0	0
$546$	12.8863	0.551482
$547$	6.74245	0.288286	0.144143	−	0.989557i	$-0.453957\pi$
$547$	6.74245	0.288286	0.144143	+	0.989557i	$0.453957\pi$
$548$	−10.9617	−0.468259
$549$	−10.4281	−0.445058
$550$	0	0
$551$	8.35282	0.355842
$552$	5.89180	0.250772
$553$	33.3970	1.42018
$554$	−48.2262	−2.04894
$555$	0	0
$556$	−47.6458	−2.02063
$557$	−23.9082	−1.01302	−0.506512	−	0.862233i	$-0.669065\pi$
$557$	−23.9082	−1.01302	−0.506512	+	0.862233i	$0.669065\pi$
$558$	5.57947	0.236198
$559$	0.772794	0.0326857
$560$	0	0
$561$	15.1594	0.640032
$562$	−9.79595	−0.413217
$563$	16.3349	0.688432	0.344216	−	0.938891i	$-0.388145\pi$
$563$	16.3349	0.688432	0.344216	+	0.938891i	$0.388145\pi$
$564$	0.896579	0.0377528
$565$	0	0
$566$	−27.8141	−1.16911
$567$	3.51369	0.147561
$568$	−4.88421	−0.204937
$569$	−44.5209	−1.86641	−0.933206	−	0.359341i	$-0.883002\pi$
$569$	−44.5209	−1.86641	−0.933206	+	0.359341i	$0.883002\pi$
$570$	0	0
$571$	−22.0663	−0.923444	−0.461722	−	0.887025i	$-0.652768\pi$
$571$	−22.0663	−0.923444	−0.461722	+	0.887025i	$0.652768\pi$
$572$	14.4483	0.604114
$573$	−16.3647	−0.683646
$574$	57.5448	2.40187
$575$	0	0
$576$	−11.6454	−0.485226
$577$	−19.8492	−0.826331	−0.413166	−	0.910656i	$-0.635577\pi$
$577$	−19.8492	−0.826331	−0.413166	+	0.910656i	$0.635577\pi$
$578$	−8.96621	−0.372945
$579$	−2.22589	−0.0925048
$580$	0	0
$581$	49.5547	2.05587
$582$	37.0461	1.53561
$583$	−6.87285	−0.284644
$584$	−2.96237	−0.122584
$585$	0	0
$586$	61.9697	2.55995
$587$	35.0604	1.44710	0.723549	−	0.690273i	$-0.242510\pi$
$587$	35.0604	1.44710	0.723549	+	0.690273i	$0.242510\pi$
$588$	13.6507	0.562945
$589$	−17.5857	−0.724608
$590$	0	0
$591$	−26.4429	−1.08772
$592$	−0.351861	−0.0144614
$593$	20.8471	0.856086	0.428043	−	0.903758i	$-0.359203\pi$
$593$	20.8471	0.856086	0.428043	+	0.903758i	$0.359203\pi$
$594$	7.02533	0.288253
$595$	0	0
$596$	7.08553	0.290235
$597$	2.59850	0.106350
$598$	−18.2966	−0.748203
$599$	−31.3488	−1.28088	−0.640438	−	0.768010i	$-0.721247\pi$
$599$	−31.3488	−1.28088	−0.640438	+	0.768010i	$0.721247\pi$
$600$	0	0
$601$	−18.7612	−0.765287	−0.382644	−	0.923896i	$-0.624986\pi$
$601$	−18.7612	−0.765287	−0.382644	+	0.923896i	$0.624986\pi$
$602$	3.37134	0.137405
$603$	15.4553	0.629387
$604$	−27.4446	−1.11671
$605$	0	0
$606$	14.8781	0.604382
$607$	33.9758	1.37904	0.689518	−	0.724269i	$-0.257822\pi$
$607$	33.9758	1.37904	0.689518	+	0.724269i	$0.257822\pi$
$608$	53.0114	2.14990
$609$	4.36374	0.176828
$610$	0	0
$611$	−0.603472	−0.0244139
$612$	−11.7574	−0.475266
$613$	−1.12495	−0.0454364	−0.0227182	−	0.999742i	$-0.507232\pi$
$613$	−1.12495	−0.0454364	−0.0227182	+	0.999742i	$0.507232\pi$
$614$	23.7363	0.957918
$615$	0	0
$616$	13.6616	0.550442
$617$	−29.7357	−1.19712	−0.598558	−	0.801080i	$-0.704259\pi$
$617$	−29.7357	−1.19712	−0.598558	+	0.801080i	$0.704259\pi$
$618$	−30.0596	−1.20917
$619$	−9.41350	−0.378361	−0.189180	−	0.981942i	$-0.560583\pi$
$619$	−9.41350	−0.378361	−0.189180	+	0.981942i	$0.560583\pi$
$620$	0	0
$621$	−4.98891	−0.200198
$622$	42.1312	1.68931
$623$	−51.0594	−2.04565
$624$	4.44591	0.177979
$625$	0	0
$626$	−5.15564	−0.206061
$627$	−22.1429	−0.884301
$628$	−29.3919	−1.17287
$629$	−0.626313	−0.0249727
$630$	0	0
$631$	41.4533	1.65023	0.825115	−	0.564965i	$-0.191110\pi$
$631$	41.4533	1.65023	0.825115	+	0.564965i	$0.191110\pi$
$632$	−11.2250	−0.446506
$633$	15.9985	0.635884
$634$	13.7124	0.544591
$635$	0	0
$636$	5.33048	0.211367
$637$	−9.18805	−0.364044
$638$	8.72493	0.345423
$639$	4.13573	0.163607
$640$	0	0
$641$	48.3516	1.90978	0.954888	−	0.296967i	$-0.0959752\pi$
$641$	48.3516	1.90978	0.954888	+	0.296967i	$0.0959752\pi$
$642$	2.13388	0.0842175
$643$	29.2541	1.15367	0.576835	−	0.816861i	$-0.304288\pi$
$643$	29.2541	1.15367	0.576835	+	0.816861i	$0.304288\pi$
$644$	−44.7605	−1.76381
$645$	0	0
$646$	66.0837	2.60003
$647$	−26.5624	−1.04428	−0.522138	−	0.852861i	$-0.674866\pi$
$647$	−26.5624	−1.04428	−0.522138	+	0.852861i	$0.674866\pi$
$648$	−1.18098	−0.0463932
$649$	5.09606	0.200038
$650$	0	0
$651$	−9.18726	−0.360077
$652$	−29.0114	−1.13617
$653$	−4.70553	−0.184141	−0.0920707	−	0.995752i	$-0.529349\pi$
$653$	−4.70553	−0.184141	−0.0920707	+	0.995752i	$0.529349\pi$
$654$	25.5080	0.997440
$655$	0	0
$656$	19.8536	0.775153
$657$	2.50840	0.0978621
$658$	−2.63266	−0.102632
$659$	−29.6136	−1.15358	−0.576790	−	0.816892i	$-0.695695\pi$
$659$	−29.6136	−1.15358	−0.576790	+	0.816892i	$0.695695\pi$
$660$	0	0
$661$	−10.2799	−0.399843	−0.199921	−	0.979812i	$-0.564069\pi$
$661$	−10.2799	−0.399843	−0.199921	+	0.979812i	$0.564069\pi$
$662$	17.1269	0.665656
$663$	7.91373	0.307344
$664$	−16.6557	−0.646368
$665$	0	0
$666$	−0.290252	−0.0112470
$667$	−6.19585	−0.239904
$668$	17.8125	0.689188
$669$	2.86200	0.110651
$670$	0	0
$671$	34.3321	1.32537
$672$	27.6946	1.06834
$673$	−21.8507	−0.842282	−0.421141	−	0.906995i	$-0.638370\pi$
$673$	−21.8507	−0.842282	−0.421141	+	0.906995i	$0.638370\pi$
$674$	36.1070	1.39079
$675$	0	0
$676$	−25.6522	−0.986625
$677$	16.3067	0.626718	0.313359	−	0.949635i	$-0.398546\pi$
$677$	16.3067	0.626718	0.313359	+	0.949635i	$0.398546\pi$
$678$	−13.0263	−0.500273
$679$	−61.0008	−2.34100
$680$	0	0
$681$	−22.9301	−0.878682
$682$	−18.3692	−0.703392
$683$	−30.9617	−1.18472	−0.592359	−	0.805674i	$-0.701803\pi$
$683$	−30.9617	−1.18472	−0.592359	+	0.805674i	$0.701803\pi$
$684$	17.1737	0.656652
$685$	0	0
$686$	12.4013	0.473485
$687$	7.15839	0.273110
$688$	1.16315	0.0443446
$689$	−3.58785	−0.136686
$690$	0	0
$691$	−34.4312	−1.30983	−0.654913	−	0.755705i	$-0.727295\pi$
$691$	−34.4312	−1.30983	−0.654913	+	0.755705i	$0.727295\pi$
$692$	−46.1510	−1.75440
$693$	−11.5680	−0.439433
$694$	−6.93863	−0.263387
$695$	0	0
$696$	−1.46669	−0.0555946
$697$	35.3395	1.33858
$698$	13.2375	0.501048
$699$	26.7581	1.01209
$700$	0	0
$701$	−23.3122	−0.880488	−0.440244	−	0.897878i	$-0.645108\pi$
$701$	−23.3122	−0.880488	−0.440244	+	0.897878i	$0.645108\pi$
$702$	3.66745	0.138419
$703$	0.914834	0.0345036
$704$	38.3400	1.44499
$705$	0	0
$706$	52.6569	1.98177
$707$	−24.4986	−0.921364
$708$	−3.95243	−0.148541
$709$	−41.5439	−1.56021	−0.780107	−	0.625646i	$-0.784835\pi$
$709$	−41.5439	−1.56021	−0.780107	+	0.625646i	$0.784835\pi$
$710$	0	0
$711$	9.50482	0.356459
$712$	17.1615	0.643154
$713$	13.0445	0.488522
$714$	34.5239	1.29202
$715$	0	0
$716$	−42.3512	−1.58274
$717$	16.3324	0.609944
$718$	13.1625	0.491221
$719$	−30.4073	−1.13400	−0.567001	−	0.823717i	$-0.691896\pi$
$719$	−30.4073	−1.13400	−0.567001	+	0.823717i	$0.691896\pi$
$720$	0	0
$721$	49.4967	1.84335
$722$	−55.9824	−2.08345
$723$	3.51226	0.130622
$724$	−30.9515	−1.15030
$725$	0	0
$726$	0.343348	0.0127428
$727$	21.3258	0.790930	0.395465	−	0.918481i	$-0.370583\pi$
$727$	21.3258	0.790930	0.395465	+	0.918481i	$0.370583\pi$
$728$	7.13181	0.264322
$729$	1.00000	0.0370370
$730$	0	0
$731$	2.07041	0.0765768
$732$	−26.6274	−0.984178
$733$	−12.1280	−0.447957	−0.223979	−	0.974594i	$-0.571905\pi$
$733$	−12.1280	−0.447957	−0.223979	+	0.974594i	$0.571905\pi$
$734$	−44.4059	−1.63905
$735$	0	0
$736$	−39.3222	−1.44943
$737$	−50.8830	−1.87430
$738$	16.3773	0.602858
$739$	−14.3370	−0.527394	−0.263697	−	0.964606i	$-0.584942\pi$
$739$	−14.3370	−0.527394	−0.263697	+	0.964606i	$0.584942\pi$
$740$	0	0
$741$	−11.5593	−0.424642
$742$	−15.6521	−0.574607
$743$	47.1213	1.72871	0.864357	−	0.502879i	$-0.167726\pi$
$743$	47.1213	1.72871	0.864357	+	0.502879i	$0.167726\pi$
$744$	3.08791	0.113208
$745$	0	0
$746$	53.8307	1.97088
$747$	14.1033	0.516014
$748$	38.7088	1.41533
$749$	−3.51369	−0.128387
$750$	0	0
$751$	−23.9964	−0.875643	−0.437821	−	0.899062i	$-0.644250\pi$
$751$	−23.9964	−0.875643	−0.437821	+	0.899062i	$0.644250\pi$
$752$	−0.908298	−0.0331222
$753$	−1.73146	−0.0630980
$754$	4.55470	0.165872
$755$	0	0
$756$	8.97200	0.326308
$757$	−15.8793	−0.577143	−0.288572	−	0.957458i	$-0.593180\pi$
$757$	−15.8793	−0.577143	−0.288572	+	0.957458i	$0.593180\pi$
$758$	−33.0930	−1.20199
$759$	16.4249	0.596185
$760$	0	0
$761$	−2.43618	−0.0883115	−0.0441558	−	0.999025i	$-0.514060\pi$
$761$	−2.43618	−0.0883115	−0.0441558	+	0.999025i	$0.514060\pi$
$762$	−1.64956	−0.0597572
$763$	−42.0019	−1.52057
$764$	−41.7864	−1.51178
$765$	0	0
$766$	−51.0034	−1.84283
$767$	2.66031	0.0960583
$768$	3.90220	0.140809
$769$	−31.9237	−1.15120	−0.575599	−	0.817732i	$-0.695231\pi$
$769$	−31.9237	−1.15120	−0.575599	+	0.817732i	$0.695231\pi$
$770$	0	0
$771$	30.0106	1.08081
$772$	−5.68368	−0.204560
$773$	−0.229829	−0.00826636	−0.00413318	−	0.999991i	$-0.501316\pi$
$773$	−0.229829	−0.00826636	−0.00413318	+	0.999991i	$0.501316\pi$
$774$	0.959487	0.0344881
$775$	0	0
$776$	20.5029	0.736010
$777$	0.477934	0.0171458
$778$	−12.6688	−0.454197
$779$	−51.6191	−1.84945
$780$	0	0
$781$	−13.6160	−0.487218
$782$	−49.0188	−1.75291
$783$	1.24193	0.0443828
$784$	−13.8291	−0.493897
$785$	0	0
$786$	12.2960	0.438584
$787$	9.34846	0.333237	0.166618	−	0.986021i	$-0.446715\pi$
$787$	9.34846	0.333237	0.166618	+	0.986021i	$0.446715\pi$
$788$	−67.5205	−2.40532
$789$	4.73765	0.168665
$790$	0	0
$791$	21.4494	0.762653
$792$	3.88811	0.138158
$793$	17.9225	0.636445
$794$	15.9789	0.567071
$795$	0	0
$796$	6.63512	0.235176
$797$	−31.2358	−1.10643	−0.553214	−	0.833039i	$-0.686599\pi$
$797$	−31.2358	−1.10643	−0.553214	+	0.833039i	$0.686599\pi$
$798$	−50.4279	−1.78513
$799$	−1.61677	−0.0571973
$800$	0	0
$801$	−14.5316	−0.513448
$802$	−5.25784	−0.185661
$803$	−8.25836	−0.291431
$804$	39.4641	1.39179
$805$	0	0
$806$	−9.58931	−0.337769
$807$	30.0891	1.05919
$808$	8.23417	0.289677
$809$	−1.87584	−0.0659511	−0.0329755	−	0.999456i	$-0.510498\pi$
$809$	−1.87584	−0.0659511	−0.0329755	+	0.999456i	$0.510498\pi$
$810$	0	0
$811$	20.1940	0.709107	0.354554	−	0.935036i	$-0.384633\pi$
$811$	20.1940	0.709107	0.354554	+	0.935036i	$0.384633\pi$
$812$	11.1425	0.391027
$813$	−15.3409	−0.538030
$814$	0.955589	0.0334934
$815$	0	0
$816$	11.9111	0.416973
$817$	−3.02417	−0.105802
$818$	−71.0870	−2.48550
$819$	−6.03890	−0.211016
$820$	0	0
$821$	−30.4564	−1.06293	−0.531467	−	0.847079i	$-0.678359\pi$
$821$	−30.4564	−1.06293	−0.531467	+	0.847079i	$0.678359\pi$
$822$	9.16052	0.319510
$823$	−20.9315	−0.729627	−0.364813	−	0.931081i	$-0.618867\pi$
$823$	−20.9315	−0.729627	−0.364813	+	0.931081i	$0.618867\pi$
$824$	−16.6362	−0.579551
$825$	0	0
$826$	11.6057	0.403813
$827$	−17.8008	−0.618995	−0.309498	−	0.950900i	$-0.600161\pi$
$827$	−17.8008	−0.618995	−0.309498	+	0.950900i	$0.600161\pi$
$828$	−12.7389	−0.442707
$829$	44.8805	1.55876	0.779382	−	0.626549i	$-0.215533\pi$
$829$	44.8805	1.55876	0.779382	+	0.626549i	$0.215533\pi$
$830$	0	0
$831$	22.6003	0.783994
$832$	20.0147	0.693886
$833$	−24.6159	−0.852890
$834$	39.8170	1.37875
$835$	0	0
$836$	−56.5406	−1.95550
$837$	−2.61471	−0.0903775
$838$	35.8817	1.23951
$839$	−15.8479	−0.547129	−0.273565	−	0.961854i	$-0.588203\pi$
$839$	−15.8479	−0.547129	−0.273565	+	0.961854i	$0.588203\pi$
$840$	0	0
$841$	−27.4576	−0.946815
$842$	66.1491	2.27965
$843$	4.59068	0.158111
$844$	40.8513	1.40616
$845$	0	0
$846$	−0.749260	−0.0257601
$847$	−0.565364	−0.0194261
$848$	−5.40015	−0.185442
$849$	13.0345	0.447344
$850$	0	0
$851$	−0.678595	−0.0232619
$852$	10.5603	0.361791
$853$	52.4152	1.79466	0.897331	−	0.441357i	$-0.145503\pi$
$853$	52.4152	1.79466	0.897331	+	0.441357i	$0.145503\pi$
$854$	78.1873	2.67551
$855$	0	0
$856$	1.18098	0.0403650
$857$	−38.1137	−1.30194	−0.650969	−	0.759105i	$-0.725637\pi$
$857$	−38.1137	−1.30194	−0.650969	+	0.759105i	$0.725637\pi$
$858$	−12.0743	−0.412209
$859$	34.6913	1.18365	0.591825	−	0.806066i	$-0.298407\pi$
$859$	34.6913	1.18365	0.591825	+	0.806066i	$0.298407\pi$
$860$	0	0
$861$	−26.9672	−0.919041
$862$	54.1179	1.84326
$863$	17.8945	0.609135	0.304568	−	0.952491i	$-0.401488\pi$
$863$	17.8945	0.609135	0.304568	+	0.952491i	$0.401488\pi$
$864$	7.88192	0.268148
$865$	0	0
$866$	65.5476	2.22740
$867$	4.20183	0.142702
$868$	−23.4591	−0.796255
$869$	−31.2925	−1.06153
$870$	0	0
$871$	−26.5626	−0.900040
$872$	14.1172	0.478068
$873$	−17.3609	−0.587577
$874$	71.6000	2.42191
$875$	0	0
$876$	6.40506	0.216407
$877$	27.6743	0.934494	0.467247	−	0.884127i	$-0.345246\pi$
$877$	27.6743	0.934494	0.467247	+	0.884127i	$0.345246\pi$
$878$	−17.4937	−0.590383
$879$	−29.0409	−0.979525
$880$	0	0
$881$	43.2207	1.45614	0.728071	−	0.685502i	$-0.240417\pi$
$881$	43.2207	1.45614	0.728071	+	0.685502i	$0.240417\pi$
$882$	−11.4077	−0.384118
$883$	1.10993	0.0373521	0.0186760	−	0.999826i	$-0.494055\pi$
$883$	1.10993	0.0373521	0.0186760	+	0.999826i	$0.494055\pi$
$884$	20.2073	0.679644
$885$	0	0
$886$	23.1235	0.776848
$887$	−41.5931	−1.39656	−0.698280	−	0.715824i	$-0.746051\pi$
$887$	−41.5931	−1.39656	−0.698280	+	0.715824i	$0.746051\pi$
$888$	−0.160637	−0.00539064
$889$	2.71619	0.0910982
$890$	0	0
$891$	−3.29228	−0.110295
$892$	7.30794	0.244688
$893$	2.36156	0.0790268
$894$	−5.92129	−0.198038
$895$	0	0
$896$	31.9257	1.06656
$897$	8.57433	0.286289
$898$	81.7058	2.72656
$899$	−3.24727	−0.108303
$900$	0	0
$901$	−9.61229	−0.320232
$902$	−53.9187	−1.79530
$903$	−1.57991	−0.0525761
$904$	−7.20932	−0.239778
$905$	0	0
$906$	22.9351	0.761969
$907$	42.6760	1.41703	0.708517	−	0.705693i	$-0.249364\pi$
$907$	42.6760	1.41703	0.708517	+	0.705693i	$0.249364\pi$
$908$	−58.5506	−1.94307
$909$	−6.97233	−0.231258
$910$	0	0
$911$	40.1588	1.33052	0.665261	−	0.746611i	$-0.268320\pi$
$911$	40.1588	1.33052	0.665261	+	0.746611i	$0.268320\pi$
$912$	−17.3982	−0.576111
$913$	−46.4321	−1.53668
$914$	16.4052	0.542635
$915$	0	0
$916$	18.2785	0.603940
$917$	−20.2469	−0.668610
$918$	9.82554	0.324291
$919$	−26.1659	−0.863134	−0.431567	−	0.902081i	$-0.642039\pi$
$919$	−26.1659	−0.863134	−0.431567	+	0.902081i	$0.642039\pi$
$920$	0	0
$921$	−11.1235	−0.366533
$922$	38.0585	1.25339
$923$	−7.10798	−0.233962
$924$	−29.5383	−0.971739
$925$	0	0
$926$	−15.6568	−0.514515
$927$	14.0868	0.462672
$928$	9.78875	0.321331
$929$	−2.22028	−0.0728452	−0.0364226	−	0.999336i	$-0.511596\pi$
$929$	−2.22028	−0.0728452	−0.0364226	+	0.999336i	$0.511596\pi$
$930$	0	0
$931$	35.9556	1.17840
$932$	68.3254	2.23807
$933$	−19.7439	−0.646387
$934$	−59.0058	−1.93073
$935$	0	0
$936$	2.02972	0.0663436
$937$	−53.6172	−1.75160	−0.875799	−	0.482677i	$-0.839665\pi$
$937$	−53.6172	−1.75160	−0.875799	+	0.482677i	$0.839665\pi$
$938$	−115.880	−3.78362
$939$	2.41609	0.0788460
$940$	0	0
$941$	−28.0356	−0.913933	−0.456966	−	0.889484i	$-0.651064\pi$
$941$	−28.0356	−0.913933	−0.456966	+	0.889484i	$0.651064\pi$
$942$	24.5625	0.800289
$943$	38.2894	1.24688
$944$	4.00409	0.130322
$945$	0	0
$946$	−3.15890	−0.102705
$947$	8.59619	0.279339	0.139669	−	0.990198i	$-0.455396\pi$
$947$	8.59619	0.279339	0.139669	+	0.990198i	$0.455396\pi$
$948$	24.2700	0.788253
$949$	−4.31114	−0.139945
$950$	0	0
$951$	−6.42606	−0.208379
$952$	19.1070	0.619261
$953$	9.98628	0.323487	0.161744	−	0.986833i	$-0.448288\pi$
$953$	9.98628	0.323487	0.161744	+	0.986833i	$0.448288\pi$
$954$	−4.45461	−0.144223
$955$	0	0
$956$	41.7038	1.34880
$957$	−4.08876	−0.132171
$958$	−15.5492	−0.502370
$959$	−15.0839	−0.487084
$960$	0	0
$961$	−24.1633	−0.779461
$962$	0.498849	0.0160835
$963$	−1.00000	−0.0322245
$964$	8.96834	0.288851
$965$	0	0
$966$	37.4058	1.20351
$967$	6.59999	0.212241	0.106121	−	0.994353i	$-0.466157\pi$
$967$	6.59999	0.212241	0.106121	+	0.994353i	$0.466157\pi$
$968$	0.190023	0.00610758
$969$	−30.9688	−0.994861
$970$	0	0
$971$	−1.81991	−0.0584035	−0.0292018	−	0.999574i	$-0.509297\pi$
$971$	−1.81991	−0.0584035	−0.0292018	+	0.999574i	$0.509297\pi$
$972$	2.55344	0.0819017
$973$	−65.5634	−2.10187
$974$	−10.2688	−0.329034
$975$	0	0
$976$	26.9755	0.863464
$977$	12.3409	0.394820	0.197410	−	0.980321i	$-0.436747\pi$
$977$	12.3409	0.394820	0.197410	+	0.980321i	$0.436747\pi$
$978$	24.2445	0.775252
$979$	47.8420	1.52904
$980$	0	0
$981$	−11.9538	−0.381655
$982$	−20.8071	−0.663982
$983$	−15.3195	−0.488615	−0.244307	−	0.969698i	$-0.578561\pi$
$983$	−15.3195	−0.488615	−0.244307	+	0.969698i	$0.578561\pi$
$984$	9.06390	0.288947
$985$	0	0
$986$	12.2026	0.388610
$987$	1.23375	0.0392705
$988$	−29.5160	−0.939030
$989$	2.24324	0.0713308
$990$	0	0
$991$	44.3787	1.40974	0.704868	−	0.709338i	$-0.251006\pi$
$991$	44.3787	1.40974	0.704868	+	0.709338i	$0.251006\pi$
$992$	−20.6089	−0.654333
$993$	−8.02618	−0.254703
$994$	−31.0088	−0.983539
$995$	0	0
$996$	36.0120	1.14108
$997$	−45.5308	−1.44198	−0.720988	−	0.692948i	$-0.756312\pi$
$997$	−45.5308	−1.44198	−0.720988	+	0.692948i	$0.756312\pi$
$998$	−38.8162	−1.22870
$999$	0.136021	0.00430350

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8025.2.a.bo.1.4		22
5.2	odd	4		1605.2.b.d.964.7	✓	44
5.3	odd	4		1605.2.b.d.964.38	yes	44
5.4	even	2		8025.2.a.bp.1.19		22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1605.2.b.d.964.7	✓	44	5.2	odd	4
1605.2.b.d.964.38	yes	44	5.3	odd	4
8025.2.a.bo.1.4		22	1.1	even	1	trivial
8025.2.a.bp.1.19		22	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:	\( N \)	\(=\)	\( 8025 = 3 \cdot 5^{2} \cdot 107 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8025.a (trivial)

\(f(q)\)	\(=\)	\(q-2.13388 q^{2} +1.00000 q^{3} +2.55344 q^{4} -2.13388 q^{6} +3.51369 q^{7} -1.18098 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q-2.13388 q^{2} +1.00000 q^{3} +2.55344 q^{4} -2.13388 q^{6} +3.51369 q^{7} -1.18098 q^{8} +1.00000 q^{9} -3.29228 q^{11} +2.55344 q^{12} -1.71868 q^{13} -7.49779 q^{14} -2.58682 q^{16} -4.60454 q^{17} -2.13388 q^{18} +6.72570 q^{19} +3.51369 q^{21} +7.02533 q^{22} -4.98891 q^{23} -1.18098 q^{24} +3.66745 q^{26} +1.00000 q^{27} +8.97200 q^{28} +1.24193 q^{29} -2.61471 q^{31} +7.88192 q^{32} -3.29228 q^{33} +9.82554 q^{34} +2.55344 q^{36} +0.136021 q^{37} -14.3518 q^{38} -1.71868 q^{39} -7.67491 q^{41} -7.49779 q^{42} -0.449645 q^{43} -8.40664 q^{44} +10.6457 q^{46} +0.351126 q^{47} -2.58682 q^{48} +5.34600 q^{49} -4.60454 q^{51} -4.38855 q^{52} +2.08757 q^{53} -2.13388 q^{54} -4.14959 q^{56} +6.72570 q^{57} -2.65012 q^{58} -1.54788 q^{59} -10.4281 q^{61} +5.57947 q^{62} +3.51369 q^{63} -11.6454 q^{64} +7.02533 q^{66} +15.4553 q^{67} -11.7574 q^{68} -4.98891 q^{69} +4.13573 q^{71} -1.18098 q^{72} +2.50840 q^{73} -0.290252 q^{74} +17.1737 q^{76} -11.5680 q^{77} +3.66745 q^{78} +9.50482 q^{79} +1.00000 q^{81} +16.3773 q^{82} +14.1033 q^{83} +8.97200 q^{84} +0.959487 q^{86} +1.24193 q^{87} +3.88811 q^{88} -14.5316 q^{89} -6.03890 q^{91} -12.7389 q^{92} -2.61471 q^{93} -0.749260 q^{94} +7.88192 q^{96} -17.3609 q^{97} -11.4077 q^{98} -3.29228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 3 q^{2} + 22 q^{3} + 13 q^{4} - 3 q^{6} - 2 q^{7} - 9 q^{8} + 22 q^{9}+O(q^{10}) \) 22 * q - 3 * q^2 + 22 * q^3 + 13 * q^4 - 3 * q^6 - 2 * q^7 - 9 * q^8 + 22 * q^9	\( 22 q - 3 q^{2} + 22 q^{3} + 13 q^{4} - 3 q^{6} - 2 q^{7} - 9 q^{8} + 22 q^{9} - 16 q^{11} + 13 q^{12} - 20 q^{14} - q^{16} - 20 q^{17} - 3 q^{18} - 16 q^{19} - 2 q^{21} - 4 q^{22} - 14 q^{23} - 9 q^{24} - 16 q^{26} + 22 q^{27} + q^{28} - 24 q^{29} - 42 q^{31} - 11 q^{32} - 16 q^{33} - 14 q^{34} + 13 q^{36} - 2 q^{37} - 21 q^{38} - 26 q^{41} - 20 q^{42} - 2 q^{43} - 24 q^{44} - 16 q^{46} - 26 q^{47} - q^{48} - 10 q^{49} - 20 q^{51} - 6 q^{52} - 22 q^{53} - 3 q^{54} - 42 q^{56} - 16 q^{57} + 69 q^{58} - 34 q^{59} - 16 q^{61} - 34 q^{62} - 2 q^{63} - 39 q^{64} - 4 q^{66} - 6 q^{68} - 14 q^{69} - 76 q^{71} - 9 q^{72} + 14 q^{73} - 12 q^{74} - 48 q^{76} - 54 q^{77} - 16 q^{78} - 72 q^{79} + 22 q^{81} - 2 q^{82} - 28 q^{83} + q^{84} - 22 q^{86} - 24 q^{87} + 19 q^{88} - 22 q^{89} - 58 q^{91} - 34 q^{92} - 42 q^{93} - 8 q^{94} - 11 q^{96} - 10 q^{97} + 3 q^{98} - 16 q^{99}+O(q^{100}) \) 22 * q - 3 * q^2 + 22 * q^3 + 13 * q^4 - 3 * q^6 - 2 * q^7 - 9 * q^8 + 22 * q^9 - 16 * q^11 + 13 * q^12 - 20 * q^14 - q^16 - 20 * q^17 - 3 * q^18 - 16 * q^19 - 2 * q^21 - 4 * q^22 - 14 * q^23 - 9 * q^24 - 16 * q^26 + 22 * q^27 + q^28 - 24 * q^29 - 42 * q^31 - 11 * q^32 - 16 * q^33 - 14 * q^34 + 13 * q^36 - 2 * q^37 - 21 * q^38 - 26 * q^41 - 20 * q^42 - 2 * q^43 - 24 * q^44 - 16 * q^46 - 26 * q^47 - q^48 - 10 * q^49 - 20 * q^51 - 6 * q^52 - 22 * q^53 - 3 * q^54 - 42 * q^56 - 16 * q^57 + 69 * q^58 - 34 * q^59 - 16 * q^61 - 34 * q^62 - 2 * q^63 - 39 * q^64 - 4 * q^66 - 6 * q^68 - 14 * q^69 - 76 * q^71 - 9 * q^72 + 14 * q^73 - 12 * q^74 - 48 * q^76 - 54 * q^77 - 16 * q^78 - 72 * q^79 + 22 * q^81 - 2 * q^82 - 28 * q^83 + q^84 - 22 * q^86 - 24 * q^87 + 19 * q^88 - 22 * q^89 - 58 * q^91 - 34 * q^92 - 42 * q^93 - 8 * q^94 - 11 * q^96 - 10 * q^97 + 3 * q^98 - 16 * q^99

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