LMFDB - Embedded newform 4275.2.a.z.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4275,2,Mod(1,4275)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4275, 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("4275.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4275 = 3^{2} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4275.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:	$34.1360468641$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 475)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.24698$ of defining polynomial
Character	$\chi$	$=$	4275.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.246980 q^{2} -1.93900 q^{4} +1.69202 q^{7} -0.972853 q^{8} +O(q^{10})$	$q+0.246980 q^{2} -1.93900 q^{4} +1.69202 q^{7} -0.972853 q^{8} +0.911854 q^{11} +1.55496 q^{13} +0.417895 q^{14} +3.63773 q^{16} -5.29590 q^{17} -1.00000 q^{19} +0.225209 q^{22} -4.24698 q^{23} +0.384043 q^{26} -3.28083 q^{28} -5.00969 q^{29} +1.82908 q^{31} +2.84415 q^{32} -1.30798 q^{34} +6.29590 q^{37} -0.246980 q^{38} -4.18060 q^{41} +7.31767 q^{43} -1.76809 q^{44} -1.04892 q^{46} +2.04892 q^{47} -4.13706 q^{49} -3.01507 q^{52} +2.70171 q^{53} -1.64609 q^{56} -1.23729 q^{58} -9.87800 q^{59} +0.542877 q^{61} +0.451747 q^{62} -6.57301 q^{64} -13.9976 q^{67} +10.2687 q^{68} +12.8780 q^{71} -2.80731 q^{73} +1.55496 q^{74} +1.93900 q^{76} +1.54288 q^{77} +1.59419 q^{79} -1.03252 q^{82} -12.2349 q^{83} +1.80731 q^{86} -0.887100 q^{88} -2.91723 q^{89} +2.63102 q^{91} +8.23490 q^{92} +0.506041 q^{94} +1.55496 q^{97} -1.02177 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 4 q^{2} + 4 q^{4} - 9 q^{8}+O(q^{10}) $ 3 * q - 4 * q^2 + 4 * q^4 - 9 * q^8	$ 3 q - 4 q^{2} + 4 q^{4} - 9 q^{8} - q^{11} + 5 q^{13} + 7 q^{14} + 18 q^{16} - 2 q^{17} - 3 q^{19} - q^{22} - 8 q^{23} - 9 q^{26} - 21 q^{28} + 7 q^{29} - 5 q^{31} - 27 q^{32} - 9 q^{34} + 5 q^{37} + 4 q^{38} - q^{41} + 5 q^{43} + 15 q^{44} + 6 q^{46} - 3 q^{47} - 7 q^{49} + 16 q^{52} - 19 q^{53} + 35 q^{56} - 21 q^{58} - 10 q^{59} - 17 q^{61} + 23 q^{62} + 49 q^{64} - q^{67} + 23 q^{68} + 19 q^{71} - q^{73} + 5 q^{74} - 4 q^{76} - 14 q^{77} + 18 q^{79} + 6 q^{82} - 13 q^{83} - 2 q^{86} - 46 q^{88} - 2 q^{89} - 7 q^{91} + q^{92} + 11 q^{94} + 5 q^{97}+O(q^{100}) $ 3 * q - 4 * q^2 + 4 * q^4 - 9 * q^8 - q^11 + 5 * q^13 + 7 * q^14 + 18 * q^16 - 2 * q^17 - 3 * q^19 - q^22 - 8 * q^23 - 9 * q^26 - 21 * q^28 + 7 * q^29 - 5 * q^31 - 27 * q^32 - 9 * q^34 + 5 * q^37 + 4 * q^38 - q^41 + 5 * q^43 + 15 * q^44 + 6 * q^46 - 3 * q^47 - 7 * q^49 + 16 * q^52 - 19 * q^53 + 35 * q^56 - 21 * q^58 - 10 * q^59 - 17 * q^61 + 23 * q^62 + 49 * q^64 - q^67 + 23 * q^68 + 19 * q^71 - q^73 + 5 * q^74 - 4 * q^76 - 14 * q^77 + 18 * q^79 + 6 * q^82 - 13 * q^83 - 2 * q^86 - 46 * q^88 - 2 * q^89 - 7 * q^91 + q^92 + 11 * q^94 + 5 * q^97

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 $

$2$	0.246980	0.174641	0.0873205	−	0.996180i	$-0.472170\pi$
$2$	0.246980	0.174641	0.0873205	+	0.996180i	$0.472170\pi$
$3$	0	0
$3$	0	0
$4$	−1.93900	−0.969501
$5$	0	0
$5$	0	0
$6$	0	0
$7$	1.69202	0.639524	0.319762	−	0.947498i	$-0.396397\pi$
$7$	1.69202	0.639524	0.319762	+	0.947498i	$0.396397\pi$
$8$	−0.972853	−0.343955
$9$	0	0
$10$	0	0
$11$	0.911854	0.274934	0.137467	−	0.990506i	$-0.456104\pi$
$11$	0.911854	0.274934	0.137467	+	0.990506i	$0.456104\pi$
$12$	0	0
$13$	1.55496	0.431268	0.215634	−	0.976474i	$-0.430818\pi$
$13$	1.55496	0.431268	0.215634	+	0.976474i	$0.430818\pi$
$14$	0.417895	0.111687
$15$	0	0
$16$	3.63773	0.909432
$17$	−5.29590	−1.28444	−0.642222	−	0.766519i	$-0.721987\pi$
$17$	−5.29590	−1.28444	−0.642222	+	0.766519i	$0.721987\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0.225209	0.0480148
$23$	−4.24698	−0.885556	−0.442778	−	0.896631i	$-0.646007\pi$
$23$	−4.24698	−0.885556	−0.442778	+	0.896631i	$0.646007\pi$
$24$	0	0
$25$	0	0
$26$	0.384043	0.0753170
$27$	0	0
$28$	−3.28083	−0.620019
$29$	−5.00969	−0.930276	−0.465138	−	0.885238i	$-0.653995\pi$
$29$	−5.00969	−0.930276	−0.465138	+	0.885238i	$0.653995\pi$
$30$	0	0
$31$	1.82908	0.328513	0.164257	−	0.986418i	$-0.447477\pi$
$31$	1.82908	0.328513	0.164257	+	0.986418i	$0.447477\pi$
$32$	2.84415	0.502779
$33$	0	0
$34$	−1.30798	−0.224316
$35$	0	0
$36$	0	0
$37$	6.29590	1.03504	0.517520	−	0.855671i	$-0.326855\pi$
$37$	6.29590	1.03504	0.517520	+	0.855671i	$0.326855\pi$
$38$	−0.246980	−0.0400654
$39$	0	0
$40$	0	0
$41$	−4.18060	−0.652901	−0.326450	−	0.945214i	$-0.605853\pi$
$41$	−4.18060	−0.652901	−0.326450	+	0.945214i	$0.605853\pi$
$42$	0	0
$43$	7.31767	1.11593	0.557967	−	0.829863i	$-0.311582\pi$
$43$	7.31767	1.11593	0.557967	+	0.829863i	$0.311582\pi$
$44$	−1.76809	−0.266549
$45$	0	0
$46$	−1.04892	−0.154654
$47$	2.04892	0.298865	0.149433	−	0.988772i	$-0.452255\pi$
$47$	2.04892	0.298865	0.149433	+	0.988772i	$0.452255\pi$
$48$	0	0
$49$	−4.13706	−0.591009
$50$	0	0
$51$	0	0
$52$	−3.01507	−0.418114
$53$	2.70171	0.371108	0.185554	−	0.982634i	$-0.440592\pi$
$53$	2.70171	0.371108	0.185554	+	0.982634i	$0.440592\pi$
$54$	0	0
$55$	0	0
$56$	−1.64609	−0.219968
$57$	0	0
$58$	−1.23729	−0.162464
$59$	−9.87800	−1.28601	−0.643003	−	0.765864i	$-0.722312\pi$
$59$	−9.87800	−1.28601	−0.643003	+	0.765864i	$0.722312\pi$
$60$	0	0
$61$	0.542877	0.0695082	0.0347541	−	0.999396i	$-0.488935\pi$
$61$	0.542877	0.0695082	0.0347541	+	0.999396i	$0.488935\pi$
$62$	0.451747	0.0573719
$63$	0	0
$64$	−6.57301	−0.821626
$65$	0	0
$66$	0	0
$67$	−13.9976	−1.71008	−0.855040	−	0.518562i	$-0.826467\pi$
$67$	−13.9976	−1.71008	−0.855040	+	0.518562i	$0.826467\pi$
$68$	10.2687	1.24527
$69$	0	0
$70$	0	0
$71$	12.8780	1.52834	0.764169	−	0.645016i	$-0.223149\pi$
$71$	12.8780	1.52834	0.764169	+	0.645016i	$0.223149\pi$
$72$	0	0
$73$	−2.80731	−0.328571	−0.164286	−	0.986413i	$-0.552532\pi$
$73$	−2.80731	−0.328571	−0.164286	+	0.986413i	$0.552532\pi$
$74$	1.55496	0.180760
$75$	0	0
$76$	1.93900	0.222419
$77$	1.54288	0.175827
$78$	0	0
$79$	1.59419	0.179360	0.0896800	−	0.995971i	$-0.471416\pi$
$79$	1.59419	0.179360	0.0896800	+	0.995971i	$0.471416\pi$
$80$	0	0
$81$	0	0
$82$	−1.03252	−0.114023
$83$	−12.2349	−1.34295	−0.671477	−	0.741025i	$-0.734340\pi$
$83$	−12.2349	−1.34295	−0.671477	+	0.741025i	$0.734340\pi$
$84$	0	0
$85$	0	0
$86$	1.80731	0.194888
$87$	0	0
$88$	−0.887100	−0.0945652
$89$	−2.91723	−0.309226	−0.154613	−	0.987975i	$-0.549413\pi$
$89$	−2.91723	−0.309226	−0.154613	+	0.987975i	$0.549413\pi$
$90$	0	0
$91$	2.63102	0.275806
$92$	8.23490	0.858547
$93$	0	0
$94$	0.506041	0.0521941
$95$	0	0
$96$	0	0
$97$	1.55496	0.157882	0.0789410	−	0.996879i	$-0.474846\pi$
$97$	1.55496	0.157882	0.0789410	+	0.996879i	$0.474846\pi$
$98$	−1.02177	−0.103214
$99$	0	0
$100$	0	0
$101$	16.6015	1.65191	0.825955	−	0.563737i	$-0.190637\pi$
$101$	16.6015	1.65191	0.825955	+	0.563737i	$0.190637\pi$
$102$	0	0
$103$	−4.84548	−0.477439	−0.238720	−	0.971089i	$-0.576728\pi$
$103$	−4.84548	−0.477439	−0.238720	+	0.971089i	$0.576728\pi$
$104$	−1.51275	−0.148337
$105$	0	0
$106$	0.667267	0.0648107
$107$	4.46681	0.431823	0.215912	−	0.976413i	$-0.430728\pi$
$107$	4.46681	0.431823	0.215912	+	0.976413i	$0.430728\pi$
$108$	0	0
$109$	18.8267	1.80327	0.901635	−	0.432498i	$-0.142368\pi$
$109$	18.8267	1.80327	0.901635	+	0.432498i	$0.142368\pi$
$110$	0	0
$111$	0	0
$112$	6.15511	0.581603
$113$	−20.0368	−1.88491	−0.942453	−	0.334337i	$-0.891488\pi$
$113$	−20.0368	−1.88491	−0.942453	+	0.334337i	$0.891488\pi$
$114$	0	0
$115$	0	0
$116$	9.71379	0.901903
$117$	0	0
$118$	−2.43967	−0.224589
$119$	−8.96077	−0.821433
$120$	0	0
$121$	−10.1685	−0.924411
$122$	0.134079	0.0121390
$123$	0	0
$124$	−3.54660	−0.318494
$125$	0	0
$126$	0	0
$127$	−17.8702	−1.58573	−0.792863	−	0.609399i	$-0.791411\pi$
$127$	−17.8702	−1.58573	−0.792863	+	0.609399i	$0.791411\pi$
$128$	−7.31170	−0.646269
$129$	0	0
$130$	0	0
$131$	−7.44265	−0.650267	−0.325134	−	0.945668i	$-0.605409\pi$
$131$	−7.44265	−0.650267	−0.325134	+	0.945668i	$0.605409\pi$
$132$	0	0
$133$	−1.69202	−0.146717
$134$	−3.45712	−0.298650
$135$	0	0
$136$	5.15213	0.441791
$137$	−5.68664	−0.485843	−0.242921	−	0.970046i	$-0.578106\pi$
$137$	−5.68664	−0.485843	−0.242921	+	0.970046i	$0.578106\pi$
$138$	0	0
$139$	3.61596	0.306701	0.153351	−	0.988172i	$-0.450994\pi$
$139$	3.61596	0.306701	0.153351	+	0.988172i	$0.450994\pi$
$140$	0	0
$141$	0	0
$142$	3.18060	0.266910
$143$	1.41789	0.118570
$144$	0	0
$145$	0	0
$146$	−0.693349	−0.0573820
$147$	0	0
$148$	−12.2078	−1.00347
$149$	−3.29052	−0.269570	−0.134785	−	0.990875i	$-0.543034\pi$
$149$	−3.29052	−0.269570	−0.134785	+	0.990875i	$0.543034\pi$
$150$	0	0
$151$	−10.2131	−0.831133	−0.415566	−	0.909563i	$-0.636417\pi$
$151$	−10.2131	−0.831133	−0.415566	+	0.909563i	$0.636417\pi$
$152$	0.972853	0.0789088
$153$	0	0
$154$	0.381059	0.0307066
$155$	0	0
$156$	0	0
$157$	14.3448	1.14484	0.572420	−	0.819960i	$-0.306005\pi$
$157$	14.3448	1.14484	0.572420	+	0.819960i	$0.306005\pi$
$158$	0.393732	0.0313236
$159$	0	0
$160$	0	0
$161$	−7.18598	−0.566335
$162$	0	0
$163$	−19.5308	−1.52977	−0.764885	−	0.644167i	$-0.777204\pi$
$163$	−19.5308	−1.52977	−0.764885	+	0.644167i	$0.777204\pi$
$164$	8.10620	0.632988
$165$	0	0
$166$	−3.02177	−0.234535
$167$	−11.8823	−0.919481	−0.459741	−	0.888053i	$-0.652058\pi$
$167$	−11.8823	−0.919481	−0.459741	+	0.888053i	$0.652058\pi$
$168$	0	0
$169$	−10.5821	−0.814008
$170$	0	0
$171$	0	0
$172$	−14.1890	−1.08190
$173$	−15.4722	−1.17633	−0.588164	−	0.808741i	$-0.700149\pi$
$173$	−15.4722	−1.17633	−0.588164	+	0.808741i	$0.700149\pi$
$174$	0	0
$175$	0	0
$176$	3.31708	0.250034
$177$	0	0
$178$	−0.720497	−0.0540035
$179$	−2.16421	−0.161761	−0.0808803	−	0.996724i	$-0.525773\pi$
$179$	−2.16421	−0.161761	−0.0808803	+	0.996724i	$0.525773\pi$
$180$	0	0
$181$	16.8974	1.25597	0.627986	−	0.778225i	$-0.283879\pi$
$181$	16.8974	1.25597	0.627986	+	0.778225i	$0.283879\pi$
$182$	0.649809	0.0481670
$183$	0	0
$184$	4.13169	0.304592
$185$	0	0
$186$	0	0
$187$	−4.82908	−0.353138
$188$	−3.97285	−0.289750
$189$	0	0
$190$	0	0
$191$	−5.92394	−0.428641	−0.214320	−	0.976763i	$-0.568754\pi$
$191$	−5.92394	−0.428641	−0.214320	+	0.976763i	$0.568754\pi$
$192$	0	0
$193$	−4.43535	−0.319264	−0.159632	−	0.987177i	$-0.551031\pi$
$193$	−4.43535	−0.319264	−0.159632	+	0.987177i	$0.551031\pi$
$194$	0.384043	0.0275727
$195$	0	0
$196$	8.02177	0.572984
$197$	−16.4722	−1.17359	−0.586797	−	0.809734i	$-0.699612\pi$
$197$	−16.4722	−1.17359	−0.586797	+	0.809734i	$0.699612\pi$
$198$	0	0
$199$	−24.4131	−1.73060	−0.865300	−	0.501255i	$-0.832872\pi$
$199$	−24.4131	−1.73060	−0.865300	+	0.501255i	$0.832872\pi$
$200$	0	0
$201$	0	0
$202$	4.10023	0.288491
$203$	−8.47650	−0.594934
$204$	0	0
$205$	0	0
$206$	−1.19673	−0.0833804
$207$	0	0
$208$	5.65651	0.392209
$209$	−0.911854	−0.0630743
$210$	0	0
$211$	−4.34050	−0.298813	−0.149406	−	0.988776i	$-0.547736\pi$
$211$	−4.34050	−0.298813	−0.149406	+	0.988776i	$0.547736\pi$
$212$	−5.23862	−0.359790
$213$	0	0
$214$	1.10321	0.0754140
$215$	0	0
$216$	0	0
$217$	3.09485	0.210092
$218$	4.64981	0.314925
$219$	0	0
$220$	0	0
$221$	−8.23490	−0.553939
$222$	0	0
$223$	26.2379	1.75702	0.878509	−	0.477725i	$-0.158539\pi$
$223$	26.2379	1.75702	0.878509	+	0.477725i	$0.158539\pi$
$224$	4.81236	0.321540
$225$	0	0
$226$	−4.94869	−0.329182
$227$	−14.6853	−0.974699	−0.487349	−	0.873207i	$-0.662036\pi$
$227$	−14.6853	−0.974699	−0.487349	+	0.873207i	$0.662036\pi$
$228$	0	0
$229$	−21.6407	−1.43006	−0.715029	−	0.699095i	$-0.753587\pi$
$229$	−21.6407	−1.43006	−0.715029	+	0.699095i	$0.753587\pi$
$230$	0	0
$231$	0	0
$232$	4.87369	0.319973
$233$	−27.1183	−1.77658	−0.888289	−	0.459286i	$-0.848105\pi$
$233$	−27.1183	−1.77658	−0.888289	+	0.459286i	$0.848105\pi$
$234$	0	0
$235$	0	0
$236$	19.1535	1.24678
$237$	0	0
$238$	−2.21313	−0.143456
$239$	11.5308	0.745865	0.372933	−	0.927858i	$-0.378352\pi$
$239$	11.5308	0.745865	0.372933	+	0.927858i	$0.378352\pi$
$240$	0	0
$241$	11.8194	0.761354	0.380677	−	0.924708i	$-0.375691\pi$
$241$	11.8194	0.761354	0.380677	+	0.924708i	$0.375691\pi$
$242$	−2.51142	−0.161440
$243$	0	0
$244$	−1.05264	−0.0673883
$245$	0	0
$246$	0	0
$247$	−1.55496	−0.0989396
$248$	−1.77943	−0.112994
$249$	0	0
$250$	0	0
$251$	9.66487	0.610041	0.305021	−	0.952346i	$-0.401337\pi$
$251$	9.66487	0.610041	0.305021	+	0.952346i	$0.401337\pi$
$252$	0	0
$253$	−3.87263	−0.243470
$254$	−4.41358	−0.276933
$255$	0	0
$256$	11.3402	0.708761
$257$	−14.1860	−0.884897	−0.442449	−	0.896794i	$-0.645890\pi$
$257$	−14.1860	−0.884897	−0.442449	+	0.896794i	$0.645890\pi$
$258$	0	0
$259$	10.6528	0.661932
$260$	0	0
$261$	0	0
$262$	−1.83818	−0.113563
$263$	21.7942	1.34389	0.671943	−	0.740603i	$-0.265460\pi$
$263$	21.7942	1.34389	0.671943	+	0.740603i	$0.265460\pi$
$264$	0	0
$265$	0	0
$266$	−0.417895	−0.0256228
$267$	0	0
$268$	27.1414	1.65792
$269$	24.7265	1.50760	0.753800	−	0.657104i	$-0.228219\pi$
$269$	24.7265	1.50760	0.753800	+	0.657104i	$0.228219\pi$
$270$	0	0
$271$	−13.2295	−0.803636	−0.401818	−	0.915720i	$-0.631622\pi$
$271$	−13.2295	−0.803636	−0.401818	+	0.915720i	$0.631622\pi$
$272$	−19.2650	−1.16811
$273$	0	0
$274$	−1.40449	−0.0848481
$275$	0	0
$276$	0	0
$277$	−0.560335	−0.0336673	−0.0168336	−	0.999858i	$-0.505359\pi$
$277$	−0.560335	−0.0336673	−0.0168336	+	0.999858i	$0.505359\pi$
$278$	0.893068	0.0535626
$279$	0	0
$280$	0	0
$281$	−27.6039	−1.64671	−0.823355	−	0.567527i	$-0.807900\pi$
$281$	−27.6039	−1.64671	−0.823355	+	0.567527i	$0.807900\pi$
$282$	0	0
$283$	−15.9608	−0.948769	−0.474385	−	0.880318i	$-0.657329\pi$
$283$	−15.9608	−0.948769	−0.474385	+	0.880318i	$0.657329\pi$
$284$	−24.9705	−1.48172
$285$	0	0
$286$	0.350191	0.0207072
$287$	−7.07367	−0.417546
$288$	0	0
$289$	11.0465	0.649796
$290$	0	0
$291$	0	0
$292$	5.44339	0.318550
$293$	−25.3327	−1.47995	−0.739977	−	0.672632i	$-0.765164\pi$
$293$	−25.3327	−1.47995	−0.739977	+	0.672632i	$0.765164\pi$
$294$	0	0
$295$	0	0
$296$	−6.12498	−0.356007
$297$	0	0
$298$	−0.812691	−0.0470779
$299$	−6.60388	−0.381912
$300$	0	0
$301$	12.3817	0.713666
$302$	−2.52243	−0.145150
$303$	0	0
$304$	−3.63773	−0.208638
$305$	0	0
$306$	0	0
$307$	−11.5574	−0.659613	−0.329806	−	0.944049i	$-0.606983\pi$
$307$	−11.5574	−0.659613	−0.329806	+	0.944049i	$0.606983\pi$
$308$	−2.99164	−0.170464
$309$	0	0
$310$	0	0
$311$	18.3424	1.04010	0.520052	−	0.854135i	$-0.325913\pi$
$311$	18.3424	1.04010	0.520052	+	0.854135i	$0.325913\pi$
$312$	0	0
$313$	−18.9119	−1.06896	−0.534481	−	0.845181i	$-0.679493\pi$
$313$	−18.9119	−1.06896	−0.534481	+	0.845181i	$0.679493\pi$
$314$	3.54288	0.199936
$315$	0	0
$316$	−3.09113	−0.173890
$317$	−20.2784	−1.13895	−0.569475	−	0.822008i	$-0.692854\pi$
$317$	−20.2784	−1.13895	−0.569475	+	0.822008i	$0.692854\pi$
$318$	0	0
$319$	−4.56810	−0.255765
$320$	0	0
$321$	0	0
$322$	−1.77479	−0.0989052
$323$	5.29590	0.294672
$324$	0	0
$325$	0	0
$326$	−4.82371	−0.267160
$327$	0	0
$328$	4.06711	0.224569
$329$	3.46681	0.191132
$330$	0	0
$331$	−4.77479	−0.262446	−0.131223	−	0.991353i	$-0.541890\pi$
$331$	−4.77479	−0.262446	−0.131223	+	0.991353i	$0.541890\pi$
$332$	23.7235	1.30200
$333$	0	0
$334$	−2.93469	−0.160579
$335$	0	0
$336$	0	0
$337$	24.3967	1.32897	0.664487	−	0.747300i	$-0.268650\pi$
$337$	24.3967	1.32897	0.664487	+	0.747300i	$0.268650\pi$
$338$	−2.61356	−0.142159
$339$	0	0
$340$	0	0
$341$	1.66786	0.0903196
$342$	0	0
$343$	−18.8442	−1.01749
$344$	−7.11901	−0.383832
$345$	0	0
$346$	−3.82132	−0.205435
$347$	9.99761	0.536700	0.268350	−	0.963322i	$-0.413522\pi$
$347$	9.99761	0.536700	0.268350	+	0.963322i	$0.413522\pi$
$348$	0	0
$349$	21.9584	1.17541	0.587703	−	0.809077i	$-0.300033\pi$
$349$	21.9584	1.17541	0.587703	+	0.809077i	$0.300033\pi$
$350$	0	0
$351$	0	0
$352$	2.59345	0.138231
$353$	36.8786	1.96285	0.981425	−	0.191848i	$-0.0614479\pi$
$353$	36.8786	1.96285	0.981425	+	0.191848i	$0.0614479\pi$
$354$	0	0
$355$	0	0
$356$	5.65651	0.299795
$357$	0	0
$358$	−0.534516	−0.0282500
$359$	−16.5187	−0.871824	−0.435912	−	0.899989i	$-0.643574\pi$
$359$	−16.5187	−0.871824	−0.435912	+	0.899989i	$0.643574\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	4.17331	0.219344
$363$	0	0
$364$	−5.10156	−0.267394
$365$	0	0
$366$	0	0
$367$	−6.83148	−0.356600	−0.178300	−	0.983976i	$-0.557060\pi$
$367$	−6.83148	−0.356600	−0.178300	+	0.983976i	$0.557060\pi$
$368$	−15.4494	−0.805353
$369$	0	0
$370$	0	0
$371$	4.57135	0.237333
$372$	0	0
$373$	−4.76271	−0.246604	−0.123302	−	0.992369i	$-0.539348\pi$
$373$	−4.76271	−0.246604	−0.123302	+	0.992369i	$0.539348\pi$
$374$	−1.19269	−0.0616723
$375$	0	0
$376$	−1.99330	−0.102796
$377$	−7.78986	−0.401198
$378$	0	0
$379$	14.3773	0.738514	0.369257	−	0.929327i	$-0.379612\pi$
$379$	14.3773	0.738514	0.369257	+	0.929327i	$0.379612\pi$
$380$	0	0
$381$	0	0
$382$	−1.46309	−0.0748583
$383$	30.6708	1.56721	0.783603	−	0.621261i	$-0.213379\pi$
$383$	30.6708	1.56721	0.783603	+	0.621261i	$0.213379\pi$
$384$	0	0
$385$	0	0
$386$	−1.09544	−0.0557565
$387$	0	0
$388$	−3.01507	−0.153067
$389$	12.9215	0.655148	0.327574	−	0.944825i	$-0.393769\pi$
$389$	12.9215	0.655148	0.327574	+	0.944825i	$0.393769\pi$
$390$	0	0
$391$	22.4916	1.13745
$392$	4.02475	0.203281
$393$	0	0
$394$	−4.06829	−0.204958
$395$	0	0
$396$	0	0
$397$	−29.8471	−1.49798	−0.748992	−	0.662579i	$-0.769462\pi$
$397$	−29.8471	−1.49798	−0.748992	+	0.662579i	$0.769462\pi$
$398$	−6.02954	−0.302234
$399$	0	0
$400$	0	0
$401$	28.7101	1.43371	0.716856	−	0.697221i	$-0.245580\pi$
$401$	28.7101	1.43371	0.716856	+	0.697221i	$0.245580\pi$
$402$	0	0
$403$	2.84415	0.141677
$404$	−32.1903	−1.60153
$405$	0	0
$406$	−2.09352	−0.103900
$407$	5.74094	0.284568
$408$	0	0
$409$	−13.6203	−0.673479	−0.336739	−	0.941598i	$-0.609324\pi$
$409$	−13.6203	−0.673479	−0.336739	+	0.941598i	$0.609324\pi$
$410$	0	0
$411$	0	0
$412$	9.39539	0.462878
$413$	−16.7138	−0.822432
$414$	0	0
$415$	0	0
$416$	4.42253	0.216833
$417$	0	0
$418$	−0.225209	−0.0110153
$419$	2.13946	0.104519	0.0522596	−	0.998634i	$-0.483358\pi$
$419$	2.13946	0.104519	0.0522596	+	0.998634i	$0.483358\pi$
$420$	0	0
$421$	−15.0562	−0.733795	−0.366897	−	0.930261i	$-0.619580\pi$
$421$	−15.0562	−0.733795	−0.366897	+	0.930261i	$0.619580\pi$
$422$	−1.07202	−0.0521849
$423$	0	0
$424$	−2.62837	−0.127645
$425$	0	0
$426$	0	0
$427$	0.918559	0.0444522
$428$	−8.66115	−0.418653
$429$	0	0
$430$	0	0
$431$	−4.37435	−0.210705	−0.105353	−	0.994435i	$-0.533597\pi$
$431$	−4.37435	−0.210705	−0.105353	+	0.994435i	$0.533597\pi$
$432$	0	0
$433$	−33.1564	−1.59340	−0.796698	−	0.604377i	$-0.793422\pi$
$433$	−33.1564	−1.59340	−0.796698	+	0.604377i	$0.793422\pi$
$434$	0.764365	0.0366907
$435$	0	0
$436$	−36.5050	−1.74827
$437$	4.24698	0.203161
$438$	0	0
$439$	32.2368	1.53858	0.769290	−	0.638900i	$-0.220610\pi$
$439$	32.2368	1.53858	0.769290	+	0.638900i	$0.220610\pi$
$440$	0	0
$441$	0	0
$442$	−2.03385	−0.0967405
$443$	−21.7362	−1.03272	−0.516358	−	0.856373i	$-0.672713\pi$
$443$	−21.7362	−1.03272	−0.516358	+	0.856373i	$0.672713\pi$
$444$	0	0
$445$	0	0
$446$	6.48022	0.306847
$447$	0	0
$448$	−11.1217	−0.525450
$449$	24.9584	1.17786	0.588929	−	0.808185i	$-0.299550\pi$
$449$	24.9584	1.17786	0.588929	+	0.808185i	$0.299550\pi$
$450$	0	0
$451$	−3.81210	−0.179505
$452$	38.8514	1.82742
$453$	0	0
$454$	−3.62697	−0.170222
$455$	0	0
$456$	0	0
$457$	13.1347	0.614414	0.307207	−	0.951643i	$-0.400606\pi$
$457$	13.1347	0.614414	0.307207	+	0.951643i	$0.400606\pi$
$458$	−5.34481	−0.249747
$459$	0	0
$460$	0	0
$461$	35.3726	1.64746	0.823732	−	0.566979i	$-0.191888\pi$
$461$	35.3726	1.64746	0.823732	+	0.566979i	$0.191888\pi$
$462$	0	0
$463$	14.6963	0.682997	0.341498	−	0.939882i	$-0.389066\pi$
$463$	14.6963	0.682997	0.341498	+	0.939882i	$0.389066\pi$
$464$	−18.2239	−0.846022
$465$	0	0
$466$	−6.69766	−0.310263
$467$	33.2121	1.53687	0.768435	−	0.639927i	$-0.221036\pi$
$467$	33.2121	1.53687	0.768435	+	0.639927i	$0.221036\pi$
$468$	0	0
$469$	−23.6843	−1.09364
$470$	0	0
$471$	0	0
$472$	9.60984	0.442329
$473$	6.67264	0.306809
$474$	0	0
$475$	0	0
$476$	17.3749	0.796379
$477$	0	0
$478$	2.84787	0.130259
$479$	−24.6219	−1.12500	−0.562502	−	0.826796i	$-0.690161\pi$
$479$	−24.6219	−1.12500	−0.562502	+	0.826796i	$0.690161\pi$
$480$	0	0
$481$	9.78986	0.446379
$482$	2.91915	0.132964
$483$	0	0
$484$	19.7168	0.896217
$485$	0	0
$486$	0	0
$487$	−29.5646	−1.33970	−0.669851	−	0.742496i	$-0.733642\pi$
$487$	−29.5646	−1.33970	−0.669851	+	0.742496i	$0.733642\pi$
$488$	−0.528139	−0.0239077
$489$	0	0
$490$	0	0
$491$	−36.2978	−1.63810	−0.819049	−	0.573724i	$-0.805498\pi$
$491$	−36.2978	−1.63810	−0.819049	+	0.573724i	$0.805498\pi$
$492$	0	0
$493$	26.5308	1.19489
$494$	−0.384043	−0.0172789
$495$	0	0
$496$	6.65371	0.298760
$497$	21.7899	0.977409
$498$	0	0
$499$	7.84415	0.351152	0.175576	−	0.984466i	$-0.443821\pi$
$499$	7.84415	0.351152	0.175576	+	0.984466i	$0.443821\pi$
$500$	0	0
$501$	0	0
$502$	2.38703	0.106538
$503$	20.4166	0.910330	0.455165	−	0.890407i	$-0.349580\pi$
$503$	20.4166	0.910330	0.455165	+	0.890407i	$0.349580\pi$
$504$	0	0
$505$	0	0
$506$	−0.956459	−0.0425198
$507$	0	0
$508$	34.6504	1.53736
$509$	14.7530	0.653916	0.326958	−	0.945039i	$-0.393976\pi$
$509$	14.7530	0.653916	0.326958	+	0.945039i	$0.393976\pi$
$510$	0	0
$511$	−4.75004	−0.210129
$512$	17.4242	0.770048
$513$	0	0
$514$	−3.50365	−0.154539
$515$	0	0
$516$	0	0
$517$	1.86831	0.0821683
$518$	2.63102	0.115600
$519$	0	0
$520$	0	0
$521$	−37.1487	−1.62751	−0.813756	−	0.581206i	$-0.802581\pi$
$521$	−37.1487	−1.62751	−0.813756	+	0.581206i	$0.802581\pi$
$522$	0	0
$523$	−16.3623	−0.715472	−0.357736	−	0.933823i	$-0.616451\pi$
$523$	−16.3623	−0.715472	−0.357736	+	0.933823i	$0.616451\pi$
$524$	14.4313	0.630434
$525$	0	0
$526$	5.38271	0.234698
$527$	−9.68664	−0.421957
$528$	0	0
$529$	−4.96316	−0.215790
$530$	0	0
$531$	0	0
$532$	3.28083	0.142242
$533$	−6.50066	−0.281575
$534$	0	0
$535$	0	0
$536$	13.6176	0.588191
$537$	0	0
$538$	6.10693	0.263289
$539$	−3.77240	−0.162489
$540$	0	0
$541$	−2.08947	−0.0898335	−0.0449168	−	0.998991i	$-0.514302\pi$
$541$	−2.08947	−0.0898335	−0.0449168	+	0.998991i	$0.514302\pi$
$542$	−3.26742	−0.140348
$543$	0	0
$544$	−15.0623	−0.645792
$545$	0	0
$546$	0	0
$547$	−1.86054	−0.0795511	−0.0397756	−	0.999209i	$-0.512664\pi$
$547$	−1.86054	−0.0795511	−0.0397756	+	0.999209i	$0.512664\pi$
$548$	11.0264	0.471025
$549$	0	0
$550$	0	0
$551$	5.00969	0.213420
$552$	0	0
$553$	2.69740	0.114705
$554$	−0.138391	−0.00587968
$555$	0	0
$556$	−7.01134	−0.297347
$557$	−9.80061	−0.415265	−0.207633	−	0.978207i	$-0.566576\pi$
$557$	−9.80061	−0.415265	−0.207633	+	0.978207i	$0.566576\pi$
$558$	0	0
$559$	11.3787	0.481266
$560$	0	0
$561$	0	0
$562$	−6.81759	−0.287583
$563$	38.0170	1.60222	0.801112	−	0.598514i	$-0.204242\pi$
$563$	38.0170	1.60222	0.801112	+	0.598514i	$0.204242\pi$
$564$	0	0
$565$	0	0
$566$	−3.94198	−0.165694
$567$	0	0
$568$	−12.5284	−0.525680
$569$	7.32975	0.307279	0.153640	−	0.988127i	$-0.450901\pi$
$569$	7.32975	0.307279	0.153640	+	0.988127i	$0.450901\pi$
$570$	0	0
$571$	37.8775	1.58513	0.792563	−	0.609791i	$-0.208746\pi$
$571$	37.8775	1.58513	0.792563	+	0.609791i	$0.208746\pi$
$572$	−2.74930	−0.114954
$573$	0	0
$574$	−1.74705	−0.0729206
$575$	0	0
$576$	0	0
$577$	−28.6993	−1.19477	−0.597384	−	0.801955i	$-0.703793\pi$
$577$	−28.6993	−1.19477	−0.597384	+	0.801955i	$0.703793\pi$
$578$	2.72827	0.113481
$579$	0	0
$580$	0	0
$581$	−20.7017	−0.858852
$582$	0	0
$583$	2.46357	0.102030
$584$	2.73110	0.113014
$585$	0	0
$586$	−6.25667	−0.258461
$587$	3.72348	0.153684	0.0768422	−	0.997043i	$-0.475516\pi$
$587$	3.72348	0.153684	0.0768422	+	0.997043i	$0.475516\pi$
$588$	0	0
$589$	−1.82908	−0.0753661
$590$	0	0
$591$	0	0
$592$	22.9028	0.941297
$593$	27.7399	1.13914	0.569570	−	0.821943i	$-0.307110\pi$
$593$	27.7399	1.13914	0.569570	+	0.821943i	$0.307110\pi$
$594$	0	0
$595$	0	0
$596$	6.38032	0.261348
$597$	0	0
$598$	−1.63102	−0.0666975
$599$	23.5579	0.962551	0.481276	−	0.876569i	$-0.340174\pi$
$599$	23.5579	0.962551	0.481276	+	0.876569i	$0.340174\pi$
$600$	0	0
$601$	−7.36898	−0.300587	−0.150293	−	0.988641i	$-0.548022\pi$
$601$	−7.36898	−0.300587	−0.150293	+	0.988641i	$0.548022\pi$
$602$	3.05802	0.124635
$603$	0	0
$604$	19.8033	0.805783
$605$	0	0
$606$	0	0
$607$	28.4198	1.15352	0.576762	−	0.816912i	$-0.304316\pi$
$607$	28.4198	1.15352	0.576762	+	0.816912i	$0.304316\pi$
$608$	−2.84415	−0.115346
$609$	0	0
$610$	0	0
$611$	3.18598	0.128891
$612$	0	0
$613$	6.48129	0.261777	0.130888	−	0.991397i	$-0.458217\pi$
$613$	6.48129	0.261777	0.130888	+	0.991397i	$0.458217\pi$
$614$	−2.85443	−0.115195
$615$	0	0
$616$	−1.50099	−0.0604767
$617$	24.4650	0.984924	0.492462	−	0.870334i	$-0.336097\pi$
$617$	24.4650	0.984924	0.492462	+	0.870334i	$0.336097\pi$
$618$	0	0
$619$	−22.2457	−0.894128	−0.447064	−	0.894502i	$-0.647530\pi$
$619$	−22.2457	−0.894128	−0.447064	+	0.894502i	$0.647530\pi$
$620$	0	0
$621$	0	0
$622$	4.53020	0.181645
$623$	−4.93602	−0.197757
$624$	0	0
$625$	0	0
$626$	−4.67084	−0.186684
$627$	0	0
$628$	−27.8146	−1.10992
$629$	−33.3424	−1.32945
$630$	0	0
$631$	−15.5888	−0.620581	−0.310290	−	0.950642i	$-0.600426\pi$
$631$	−15.5888	−0.620581	−0.310290	+	0.950642i	$0.600426\pi$
$632$	−1.55091	−0.0616919
$633$	0	0
$634$	−5.00836	−0.198907
$635$	0	0
$636$	0	0
$637$	−6.43296	−0.254883
$638$	−1.12823	−0.0446670
$639$	0	0
$640$	0	0
$641$	−8.17496	−0.322892	−0.161446	−	0.986882i	$-0.551616\pi$
$641$	−8.17496	−0.322892	−0.161446	+	0.986882i	$0.551616\pi$
$642$	0	0
$643$	11.1836	0.441038	0.220519	−	0.975383i	$-0.429225\pi$
$643$	11.1836	0.441038	0.220519	+	0.975383i	$0.429225\pi$
$644$	13.9336	0.549062
$645$	0	0
$646$	1.30798	0.0514617
$647$	−16.2121	−0.637362	−0.318681	−	0.947862i	$-0.603240\pi$
$647$	−16.2121	−0.637362	−0.318681	+	0.947862i	$0.603240\pi$
$648$	0	0
$649$	−9.00730	−0.353567
$650$	0	0
$651$	0	0
$652$	37.8702	1.48311
$653$	−24.7952	−0.970312	−0.485156	−	0.874427i	$-0.661237\pi$
$653$	−24.7952	−0.970312	−0.485156	+	0.874427i	$0.661237\pi$
$654$	0	0
$655$	0	0
$656$	−15.2079	−0.593769
$657$	0	0
$658$	0.856232	0.0333794
$659$	20.2868	0.790262	0.395131	−	0.918625i	$-0.370699\pi$
$659$	20.2868	0.790262	0.395131	+	0.918625i	$0.370699\pi$
$660$	0	0
$661$	20.4222	0.794332	0.397166	−	0.917747i	$-0.369994\pi$
$661$	20.4222	0.794332	0.397166	+	0.917747i	$0.369994\pi$
$662$	−1.17928	−0.0458339
$663$	0	0
$664$	11.9028	0.461917
$665$	0	0
$666$	0	0
$667$	21.2760	0.823812
$668$	23.0398	0.891437
$669$	0	0
$670$	0	0
$671$	0.495024	0.0191102
$672$	0	0
$673$	28.7254	1.10728	0.553641	−	0.832755i	$-0.313238\pi$
$673$	28.7254	1.10728	0.553641	+	0.832755i	$0.313238\pi$
$674$	6.02549	0.232093
$675$	0	0
$676$	20.5187	0.789181
$677$	−44.0062	−1.69130	−0.845648	−	0.533740i	$-0.820786\pi$
$677$	−44.0062	−1.69130	−0.845648	+	0.533740i	$0.820786\pi$
$678$	0	0
$679$	2.63102	0.100969
$680$	0	0
$681$	0	0
$682$	0.411927	0.0157735
$683$	−8.48427	−0.324642	−0.162321	−	0.986738i	$-0.551898\pi$
$683$	−8.48427	−0.324642	−0.162321	+	0.986738i	$0.551898\pi$
$684$	0	0
$685$	0	0
$686$	−4.65412	−0.177695
$687$	0	0
$688$	26.6197	1.01487
$689$	4.20105	0.160047
$690$	0	0
$691$	20.2586	0.770673	0.385336	−	0.922776i	$-0.374085\pi$
$691$	20.2586	0.770673	0.385336	+	0.922776i	$0.374085\pi$
$692$	30.0006	1.14045
$693$	0	0
$694$	2.46921	0.0937297
$695$	0	0
$696$	0	0
$697$	22.1400	0.838614
$698$	5.42327	0.205274
$699$	0	0
$700$	0	0
$701$	23.4101	0.884188	0.442094	−	0.896969i	$-0.354236\pi$
$701$	23.4101	0.884188	0.442094	+	0.896969i	$0.354236\pi$
$702$	0	0
$703$	−6.29590	−0.237454
$704$	−5.99362	−0.225893
$705$	0	0
$706$	9.10826	0.342794
$707$	28.0901	1.05644
$708$	0	0
$709$	30.3472	1.13971	0.569857	−	0.821744i	$-0.306999\pi$
$709$	30.3472	1.13971	0.569857	+	0.821744i	$0.306999\pi$
$710$	0	0
$711$	0	0
$712$	2.83804	0.106360
$713$	−7.76809	−0.290917
$714$	0	0
$715$	0	0
$716$	4.19641	0.156827
$717$	0	0
$718$	−4.07979	−0.152256
$719$	38.3230	1.42921	0.714604	−	0.699529i	$-0.246607\pi$
$719$	38.3230	1.42921	0.714604	+	0.699529i	$0.246607\pi$
$720$	0	0
$721$	−8.19865	−0.305334
$722$	0.246980	0.00919163
$723$	0	0
$724$	−32.7640	−1.21767
$725$	0	0
$726$	0	0
$727$	2.69069	0.0997923	0.0498961	−	0.998754i	$-0.484111\pi$
$727$	2.69069	0.0997923	0.0498961	+	0.998754i	$0.484111\pi$
$728$	−2.55960	−0.0948650
$729$	0	0
$730$	0	0
$731$	−38.7536	−1.43335
$732$	0	0
$733$	18.9651	0.700491	0.350246	−	0.936658i	$-0.386098\pi$
$733$	18.9651	0.700491	0.350246	+	0.936658i	$0.386098\pi$
$734$	−1.68724	−0.0622770
$735$	0	0
$736$	−12.0790	−0.445240
$737$	−12.7638	−0.470160
$738$	0	0
$739$	29.8278	1.09723	0.548616	−	0.836075i	$-0.315155\pi$
$739$	29.8278	1.09723	0.548616	+	0.836075i	$0.315155\pi$
$740$	0	0
$741$	0	0
$742$	1.12903	0.0414480
$743$	−8.78448	−0.322271	−0.161136	−	0.986932i	$-0.551516\pi$
$743$	−8.78448	−0.322271	−0.161136	+	0.986932i	$0.551516\pi$
$744$	0	0
$745$	0	0
$746$	−1.17629	−0.0430671
$747$	0	0
$748$	9.36360	0.342367
$749$	7.55794	0.276161
$750$	0	0
$751$	−18.1142	−0.660998	−0.330499	−	0.943806i	$-0.607217\pi$
$751$	−18.1142	−0.660998	−0.330499	+	0.943806i	$0.607217\pi$
$752$	7.45340	0.271798
$753$	0	0
$754$	−1.92394	−0.0700656
$755$	0	0
$756$	0	0
$757$	−0.222816	−0.00809840	−0.00404920	−	0.999992i	$-0.501289\pi$
$757$	−0.222816	−0.00809840	−0.00404920	+	0.999992i	$0.501289\pi$
$758$	3.55091	0.128975
$759$	0	0
$760$	0	0
$761$	6.07798	0.220327	0.110163	−	0.993913i	$-0.464863\pi$
$761$	6.07798	0.220327	0.110163	+	0.993913i	$0.464863\pi$
$762$	0	0
$763$	31.8552	1.15323
$764$	11.4865	0.415568
$765$	0	0
$766$	7.57507	0.273698
$767$	−15.3599	−0.554613
$768$	0	0
$769$	−14.6267	−0.527453	−0.263726	−	0.964598i	$-0.584952\pi$
$769$	−14.6267	−0.527453	−0.263726	+	0.964598i	$0.584952\pi$
$770$	0	0
$771$	0	0
$772$	8.60015	0.309526
$773$	4.05728	0.145930	0.0729651	−	0.997334i	$-0.476754\pi$
$773$	4.05728	0.145930	0.0729651	+	0.997334i	$0.476754\pi$
$774$	0	0
$775$	0	0
$776$	−1.51275	−0.0543044
$777$	0	0
$778$	3.19136	0.114416
$779$	4.18060	0.149786
$780$	0	0
$781$	11.7429	0.420192
$782$	5.55496	0.198645
$783$	0	0
$784$	−15.0495	−0.537482
$785$	0	0
$786$	0	0
$787$	19.5657	0.697442	0.348721	−	0.937227i	$-0.386616\pi$
$787$	19.5657	0.697442	0.348721	+	0.937227i	$0.386616\pi$
$788$	31.9396	1.13780
$789$	0	0
$790$	0	0
$791$	−33.9028	−1.20544
$792$	0	0
$793$	0.844150	0.0299767
$794$	−7.37163	−0.261609
$795$	0	0
$796$	47.3370	1.67782
$797$	−38.9051	−1.37809	−0.689046	−	0.724718i	$-0.741970\pi$
$797$	−38.9051	−1.37809	−0.689046	+	0.724718i	$0.741970\pi$
$798$	0	0
$799$	−10.8509	−0.383876
$800$	0	0
$801$	0	0
$802$	7.09080	0.250385
$803$	−2.55986	−0.0903355
$804$	0	0
$805$	0	0
$806$	0.702447	0.0247426
$807$	0	0
$808$	−16.1508	−0.568183
$809$	22.5730	0.793625	0.396812	−	0.917900i	$-0.370116\pi$
$809$	22.5730	0.793625	0.396812	+	0.917900i	$0.370116\pi$
$810$	0	0
$811$	−46.3605	−1.62794	−0.813968	−	0.580909i	$-0.802697\pi$
$811$	−46.3605	−1.62794	−0.813968	+	0.580909i	$0.802697\pi$
$812$	16.4359	0.576789
$813$	0	0
$814$	1.41789	0.0496972
$815$	0	0
$816$	0	0
$817$	−7.31767	−0.256013
$818$	−3.36393	−0.117617
$819$	0	0
$820$	0	0
$821$	17.8194	0.621901	0.310951	−	0.950426i	$-0.399353\pi$
$821$	17.8194	0.621901	0.310951	+	0.950426i	$0.399353\pi$
$822$	0	0
$823$	−32.5394	−1.13425	−0.567126	−	0.823631i	$-0.691945\pi$
$823$	−32.5394	−1.13425	−0.567126	+	0.823631i	$0.691945\pi$
$824$	4.71394	0.164218
$825$	0	0
$826$	−4.12797	−0.143630
$827$	−39.8256	−1.38487	−0.692436	−	0.721479i	$-0.743463\pi$
$827$	−39.8256	−1.38487	−0.692436	+	0.721479i	$0.743463\pi$
$828$	0	0
$829$	38.7525	1.34593	0.672966	−	0.739674i	$-0.265020\pi$
$829$	38.7525	1.34593	0.672966	+	0.739674i	$0.265020\pi$
$830$	0	0
$831$	0	0
$832$	−10.2208	−0.354341
$833$	21.9095	0.759118
$834$	0	0
$835$	0	0
$836$	1.76809	0.0611505
$837$	0	0
$838$	0.528402	0.0182533
$839$	−2.76749	−0.0955445	−0.0477723	−	0.998858i	$-0.515212\pi$
$839$	−2.76749	−0.0955445	−0.0477723	+	0.998858i	$0.515212\pi$
$840$	0	0
$841$	−3.90302	−0.134587
$842$	−3.71858	−0.128151
$843$	0	0
$844$	8.41624	0.289699
$845$	0	0
$846$	0	0
$847$	−17.2054	−0.591183
$848$	9.82808	0.337498
$849$	0	0
$850$	0	0
$851$	−26.7385	−0.916586
$852$	0	0
$853$	24.1390	0.826503	0.413252	−	0.910617i	$-0.364393\pi$
$853$	24.1390	0.826503	0.413252	+	0.910617i	$0.364393\pi$
$854$	0.226865	0.00776317
$855$	0	0
$856$	−4.34555	−0.148528
$857$	3.16229	0.108022	0.0540109	−	0.998540i	$-0.482799\pi$
$857$	3.16229	0.108022	0.0540109	+	0.998540i	$0.482799\pi$
$858$	0	0
$859$	4.86426	0.165967	0.0829833	−	0.996551i	$-0.473555\pi$
$859$	4.86426	0.165967	0.0829833	+	0.996551i	$0.473555\pi$
$860$	0	0
$861$	0	0
$862$	−1.08038	−0.0367978
$863$	−4.54958	−0.154870	−0.0774348	−	0.996997i	$-0.524673\pi$
$863$	−4.54958	−0.154870	−0.0774348	+	0.996997i	$0.524673\pi$
$864$	0	0
$865$	0	0
$866$	−8.18896	−0.278272
$867$	0	0
$868$	−6.00092	−0.203684
$869$	1.45367	0.0493122
$870$	0	0
$871$	−21.7657	−0.737502
$872$	−18.3156	−0.620245
$873$	0	0
$874$	1.04892	0.0354802
$875$	0	0
$876$	0	0
$877$	−18.6595	−0.630086	−0.315043	−	0.949077i	$-0.602019\pi$
$877$	−18.6595	−0.630086	−0.315043	+	0.949077i	$0.602019\pi$
$878$	7.96184	0.268699
$879$	0	0
$880$	0	0
$881$	−19.4306	−0.654632	−0.327316	−	0.944915i	$-0.606144\pi$
$881$	−19.4306	−0.654632	−0.327316	+	0.944915i	$0.606144\pi$
$882$	0	0
$883$	25.6437	0.862979	0.431490	−	0.902118i	$-0.357988\pi$
$883$	25.6437	0.862979	0.431490	+	0.902118i	$0.357988\pi$
$884$	15.9675	0.537044
$885$	0	0
$886$	−5.36839	−0.180354
$887$	31.0062	1.04109	0.520544	−	0.853835i	$-0.325729\pi$
$887$	31.0062	1.04109	0.520544	+	0.853835i	$0.325729\pi$
$888$	0	0
$889$	−30.2368	−1.01411
$890$	0	0
$891$	0	0
$892$	−50.8753	−1.70343
$893$	−2.04892	−0.0685644
$894$	0	0
$895$	0	0
$896$	−12.3716	−0.413305
$897$	0	0
$898$	6.16421	0.205702
$899$	−9.16315	−0.305608
$900$	0	0
$901$	−14.3080	−0.476668
$902$	−0.941511	−0.0313489
$903$	0	0
$904$	19.4929	0.648324
$905$	0	0
$906$	0	0
$907$	−17.7676	−0.589964	−0.294982	−	0.955503i	$-0.595314\pi$
$907$	−17.7676	−0.589964	−0.294982	+	0.955503i	$0.595314\pi$
$908$	28.4748	0.944971
$909$	0	0
$910$	0	0
$911$	−41.7313	−1.38262	−0.691309	−	0.722559i	$-0.742966\pi$
$911$	−41.7313	−1.38262	−0.691309	+	0.722559i	$0.742966\pi$
$912$	0	0
$913$	−11.1564	−0.369224
$914$	3.24400	0.107302
$915$	0	0
$916$	41.9614	1.38644
$917$	−12.5931	−0.415862
$918$	0	0
$919$	30.4088	1.00309	0.501547	−	0.865130i	$-0.332764\pi$
$919$	30.4088	1.00309	0.501547	+	0.865130i	$0.332764\pi$
$920$	0	0
$921$	0	0
$922$	8.73630	0.287715
$923$	20.0248	0.659123
$924$	0	0
$925$	0	0
$926$	3.62969	0.119279
$927$	0	0
$928$	−14.2483	−0.467724
$929$	28.8219	0.945616	0.472808	−	0.881165i	$-0.343240\pi$
$929$	28.8219	0.945616	0.472808	+	0.881165i	$0.343240\pi$
$930$	0	0
$931$	4.13706	0.135587
$932$	52.5824	1.72239
$933$	0	0
$934$	8.20270	0.268401
$935$	0	0
$936$	0	0
$937$	−50.4601	−1.64846	−0.824230	−	0.566255i	$-0.808392\pi$
$937$	−50.4601	−1.64846	−0.824230	+	0.566255i	$0.808392\pi$
$938$	−5.84953	−0.190994
$939$	0	0
$940$	0	0
$941$	−1.10082	−0.0358857	−0.0179428	−	0.999839i	$-0.505712\pi$
$941$	−1.10082	−0.0358857	−0.0179428	+	0.999839i	$0.505712\pi$
$942$	0	0
$943$	17.7549	0.578180
$944$	−35.9335	−1.16954
$945$	0	0
$946$	1.64801	0.0535813
$947$	13.9353	0.452836	0.226418	−	0.974030i	$-0.427299\pi$
$947$	13.9353	0.452836	0.226418	+	0.974030i	$0.427299\pi$
$948$	0	0
$949$	−4.36526	−0.141702
$950$	0	0
$951$	0	0
$952$	8.71751	0.282536
$953$	41.3400	1.33913	0.669567	−	0.742751i	$-0.266480\pi$
$953$	41.3400	1.33913	0.669567	+	0.742751i	$0.266480\pi$
$954$	0	0
$955$	0	0
$956$	−22.3582	−0.723117
$957$	0	0
$958$	−6.08111	−0.196472
$959$	−9.62192	−0.310708
$960$	0	0
$961$	−27.6544	−0.892079
$962$	2.41789	0.0779561
$963$	0	0
$964$	−22.9178	−0.738133
$965$	0	0
$966$	0	0
$967$	−5.26875	−0.169432	−0.0847158	−	0.996405i	$-0.526998\pi$
$967$	−5.26875	−0.169432	−0.0847158	+	0.996405i	$0.526998\pi$
$968$	9.89248	0.317956
$969$	0	0
$970$	0	0
$971$	5.15346	0.165382	0.0826911	−	0.996575i	$-0.473649\pi$
$971$	5.15346	0.165382	0.0826911	+	0.996575i	$0.473649\pi$
$972$	0	0
$973$	6.11828	0.196143
$974$	−7.30186	−0.233967
$975$	0	0
$976$	1.97484	0.0632130
$977$	−4.77612	−0.152802	−0.0764008	−	0.997077i	$-0.524343\pi$
$977$	−4.77612	−0.152802	−0.0764008	+	0.997077i	$0.524343\pi$
$978$	0	0
$979$	−2.66009	−0.0850168
$980$	0	0
$981$	0	0
$982$	−8.96482	−0.286079
$983$	28.9758	0.924186	0.462093	−	0.886832i	$-0.347099\pi$
$983$	28.9758	0.924186	0.462093	+	0.886832i	$0.347099\pi$
$984$	0	0
$985$	0	0
$986$	6.55257	0.208676
$987$	0	0
$988$	3.01507	0.0959220
$989$	−31.0780	−0.988222
$990$	0	0
$991$	−38.5042	−1.22313	−0.611564	−	0.791195i	$-0.709459\pi$
$991$	−38.5042	−1.22313	−0.611564	+	0.791195i	$0.709459\pi$
$992$	5.20219	0.165170
$993$	0	0
$994$	5.38165	0.170696
$995$	0	0
$996$	0	0
$997$	10.1491	0.321427	0.160713	−	0.987001i	$-0.448621\pi$
$997$	10.1491	0.321427	0.160713	+	0.987001i	$0.448621\pi$
$998$	1.93735	0.0613256
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4275.2.a.z.1.3		3
3.2	odd	2		475.2.a.h.1.1	yes	3
5.4	even	2		4275.2.a.bn.1.1		3
12.11	even	2		7600.2.a.bn.1.3		3
15.2	even	4		475.2.b.c.324.3		6
15.8	even	4		475.2.b.c.324.4		6
15.14	odd	2		475.2.a.d.1.3	✓	3
57.56	even	2		9025.2.a.w.1.3		3
60.59	even	2		7600.2.a.bw.1.1		3
285.284	even	2		9025.2.a.be.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
475.2.a.d.1.3	✓	3	15.14	odd	2
475.2.a.h.1.1	yes	3	3.2	odd	2
475.2.b.c.324.3		6	15.2	even	4
475.2.b.c.324.4		6	15.8	even	4
4275.2.a.z.1.3		3	1.1	even	1	trivial
4275.2.a.bn.1.1		3	5.4	even	2
7600.2.a.bn.1.3		3	12.11	even	2
7600.2.a.bw.1.1		3	60.59	even	2
9025.2.a.w.1.3		3	57.56	even	2
9025.2.a.be.1.1		3	285.284	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:	\( N \)	\(=\)	\( 4275 = 3^{2} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4275.a (trivial)

\(f(q)\)	\(=\)	\(q+0.246980 q^{2} -1.93900 q^{4} +1.69202 q^{7} -0.972853 q^{8} +O(q^{10})\)	\(q+0.246980 q^{2} -1.93900 q^{4} +1.69202 q^{7} -0.972853 q^{8} +0.911854 q^{11} +1.55496 q^{13} +0.417895 q^{14} +3.63773 q^{16} -5.29590 q^{17} -1.00000 q^{19} +0.225209 q^{22} -4.24698 q^{23} +0.384043 q^{26} -3.28083 q^{28} -5.00969 q^{29} +1.82908 q^{31} +2.84415 q^{32} -1.30798 q^{34} +6.29590 q^{37} -0.246980 q^{38} -4.18060 q^{41} +7.31767 q^{43} -1.76809 q^{44} -1.04892 q^{46} +2.04892 q^{47} -4.13706 q^{49} -3.01507 q^{52} +2.70171 q^{53} -1.64609 q^{56} -1.23729 q^{58} -9.87800 q^{59} +0.542877 q^{61} +0.451747 q^{62} -6.57301 q^{64} -13.9976 q^{67} +10.2687 q^{68} +12.8780 q^{71} -2.80731 q^{73} +1.55496 q^{74} +1.93900 q^{76} +1.54288 q^{77} +1.59419 q^{79} -1.03252 q^{82} -12.2349 q^{83} +1.80731 q^{86} -0.887100 q^{88} -2.91723 q^{89} +2.63102 q^{91} +8.23490 q^{92} +0.506041 q^{94} +1.55496 q^{97} -1.02177 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 4 q^{2} + 4 q^{4} - 9 q^{8}+O(q^{10}) \) 3 * q - 4 * q^2 + 4 * q^4 - 9 * q^8	\( 3 q - 4 q^{2} + 4 q^{4} - 9 q^{8} - q^{11} + 5 q^{13} + 7 q^{14} + 18 q^{16} - 2 q^{17} - 3 q^{19} - q^{22} - 8 q^{23} - 9 q^{26} - 21 q^{28} + 7 q^{29} - 5 q^{31} - 27 q^{32} - 9 q^{34} + 5 q^{37} + 4 q^{38} - q^{41} + 5 q^{43} + 15 q^{44} + 6 q^{46} - 3 q^{47} - 7 q^{49} + 16 q^{52} - 19 q^{53} + 35 q^{56} - 21 q^{58} - 10 q^{59} - 17 q^{61} + 23 q^{62} + 49 q^{64} - q^{67} + 23 q^{68} + 19 q^{71} - q^{73} + 5 q^{74} - 4 q^{76} - 14 q^{77} + 18 q^{79} + 6 q^{82} - 13 q^{83} - 2 q^{86} - 46 q^{88} - 2 q^{89} - 7 q^{91} + q^{92} + 11 q^{94} + 5 q^{97}+O(q^{100}) \) 3 * q - 4 * q^2 + 4 * q^4 - 9 * q^8 - q^11 + 5 * q^13 + 7 * q^14 + 18 * q^16 - 2 * q^17 - 3 * q^19 - q^22 - 8 * q^23 - 9 * q^26 - 21 * q^28 + 7 * q^29 - 5 * q^31 - 27 * q^32 - 9 * q^34 + 5 * q^37 + 4 * q^38 - q^41 + 5 * q^43 + 15 * q^44 + 6 * q^46 - 3 * q^47 - 7 * q^49 + 16 * q^52 - 19 * q^53 + 35 * q^56 - 21 * q^58 - 10 * q^59 - 17 * q^61 + 23 * q^62 + 49 * q^64 - q^67 + 23 * q^68 + 19 * q^71 - q^73 + 5 * q^74 - 4 * q^76 - 14 * q^77 + 18 * q^79 + 6 * q^82 - 13 * q^83 - 2 * q^86 - 46 * q^88 - 2 * q^89 - 7 * q^91 + q^92 + 11 * q^94 + 5 * q^97

Introduction

L-functions

Modular forms

Varieties

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Representations

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Database

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