LMFDB - Embedded newform 925.2.a.f.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [925,2,Mod(1,925)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(925, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("925.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 925 = 5^{2} \cdot 37 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	925.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,-2,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$7.38616218697$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.973904.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 8x^{3} + 6x^{2} + 19x + 6 $ x^5 - 2x^4 - 8x^3 + 6x^2 + 19x + 6
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 185)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.62871$ of defining polynomial
Character	$\chi$	$=$	925.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.728950 q^{2} -2.62871 q^{3} -1.46863 q^{4} +1.91620 q^{6} -2.55244 q^{7} +2.52846 q^{8} +3.91009 q^{9} +2.46863 q^{11} +3.86060 q^{12} -1.55854 q^{13} +1.86060 q^{14} +1.09414 q^{16} +6.83662 q^{17} -2.85026 q^{18} -7.66011 q^{19} +6.70960 q^{21} -1.79951 q^{22} +7.50003 q^{23} -6.64658 q^{24} +1.13610 q^{26} -2.39236 q^{27} +3.74859 q^{28} +3.25741 q^{29} +0.658785 q^{31} -5.85449 q^{32} -6.48930 q^{33} -4.98356 q^{34} -5.74248 q^{36} -1.00000 q^{37} +5.58384 q^{38} +4.09694 q^{39} +2.46863 q^{41} -4.89097 q^{42} -10.9579 q^{43} -3.62551 q^{44} -5.46715 q^{46} -3.11521 q^{47} -2.87617 q^{48} -0.485072 q^{49} -17.9715 q^{51} +2.28892 q^{52} -8.64184 q^{53} +1.74391 q^{54} -6.45373 q^{56} +20.1362 q^{57} -2.37449 q^{58} -6.23634 q^{59} +3.27808 q^{61} -0.480222 q^{62} -9.98026 q^{63} +2.07935 q^{64} +4.73038 q^{66} -1.47764 q^{67} -10.0405 q^{68} -19.7154 q^{69} -8.06686 q^{71} +9.88651 q^{72} +4.96199 q^{73} +0.728950 q^{74} +11.2499 q^{76} -6.30102 q^{77} -2.98647 q^{78} +12.8206 q^{79} -5.44146 q^{81} -1.79951 q^{82} -1.14934 q^{83} -9.85393 q^{84} +7.98779 q^{86} -8.56277 q^{87} +6.24184 q^{88} +11.5207 q^{89} +3.97807 q^{91} -11.0148 q^{92} -1.73175 q^{93} +2.27083 q^{94} +15.3897 q^{96} -17.2929 q^{97} +0.353594 q^{98} +9.65257 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} - 3 q^{3} + 10 q^{4} - 6 q^{6} - 11 q^{7} - 6 q^{8} + 6 q^{9} - 5 q^{11} + 2 q^{12} - 4 q^{13} - 8 q^{14} + 16 q^{16} - 2 q^{18} - 4 q^{19} + 3 q^{21} + 8 q^{22} - 4 q^{23} - 42 q^{24} - 4 q^{26}+ \cdots - 10 q^{99}+O(q^{100}) $ 5 * q - 2 * q^2 - 3 * q^3 + 10 * q^4 - 6 * q^6 - 11 * q^7 - 6 * q^8 + 6 * q^9 - 5 * q^11 + 2 * q^12 - 4 * q^13 - 8 * q^14 + 16 * q^16 - 2 * q^18 - 4 * q^19 + 3 * q^21 + 8 * q^22 - 4 * q^23 - 42 * q^24 - 4 * q^26 - 3 * q^27 - 28 * q^28 - 4 * q^29 + 8 * q^31 - 14 * q^32 - 5 * q^33 - 32 * q^34 + 16 * q^36 - 5 * q^37 + 2 * q^38 + 2 * q^39 - 5 * q^41 + 52 * q^42 - 10 * q^43 - 46 * q^44 - 7 * q^47 + 20 * q^48 + 22 * q^49 - 2 * q^51 + 32 * q^52 + q^53 - 10 * q^54 + 8 * q^56 + 8 * q^57 + 16 * q^58 - 30 * q^59 - 14 * q^61 - 24 * q^62 - 30 * q^63 + 20 * q^64 + 48 * q^66 - 24 * q^67 - 20 * q^68 + 8 * q^69 - 7 * q^71 + 10 * q^72 - 5 * q^73 + 2 * q^74 + 12 * q^76 + 17 * q^77 - 58 * q^78 + 28 * q^79 - 31 * q^81 + 8 * q^82 - 27 * q^83 - 18 * q^84 + 44 * q^86 - 36 * q^87 + 28 * q^88 + 6 * q^89 - 14 * q^91 - 56 * q^92 + 22 * q^93 + 4 * q^94 - 42 * q^96 + 26 * q^97 + 18 * q^98 - 10 * q^99

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 $

$2$	−0.728950	−0.515446	−0.257723	−	0.966219i	$-0.582972\pi$
$2$	−0.728950	−0.515446	−0.257723	+	0.966219i	$0.582972\pi$
$3$	−2.62871	−1.51768	−0.758842	−	0.651275i	$-0.774234\pi$
$3$	−2.62871	−1.51768	−0.758842	+	0.651275i	$0.774234\pi$
$4$	−1.46863	−0.734316
$5$	0	0
$5$	0	0
$6$	1.91620	0.782284
$7$	−2.55244	−0.964730	−0.482365	−	0.875970i	$-0.660222\pi$
$7$	−2.55244	−0.964730	−0.482365	+	0.875970i	$0.660222\pi$
$8$	2.52846	0.893946
$9$	3.91009	1.30336
$10$	0	0
$11$	2.46863	0.744320	0.372160	−	0.928169i	$-0.378617\pi$
$11$	2.46863	0.744320	0.372160	+	0.928169i	$0.378617\pi$
$12$	3.86060	1.11446
$13$	−1.55854	−0.432261	−0.216131	−	0.976364i	$-0.569344\pi$
$13$	−1.55854	−0.432261	−0.216131	+	0.976364i	$0.569344\pi$
$14$	1.86060	0.497266
$15$	0	0
$16$	1.09414	0.273535
$17$	6.83662	1.65812	0.829062	−	0.559156i	$-0.188875\pi$
$17$	6.83662	1.65812	0.829062	+	0.559156i	$0.188875\pi$
$18$	−2.85026	−0.671813
$19$	−7.66011	−1.75735	−0.878675	−	0.477421i	$-0.841572\pi$
$19$	−7.66011	−1.75735	−0.878675	+	0.477421i	$0.841572\pi$
$20$	0	0
$21$	6.70960	1.46415
$22$	−1.79951	−0.383657
$23$	7.50003	1.56387	0.781933	−	0.623363i	$-0.214234\pi$
$23$	7.50003	1.56387	0.781933	+	0.623363i	$0.214234\pi$
$24$	−6.64658	−1.35673
$25$	0	0
$26$	1.13610	0.222807
$27$	−2.39236	−0.460410
$28$	3.74859	0.708416
$29$	3.25741	0.604886	0.302443	−	0.953167i	$-0.402198\pi$
$29$	3.25741	0.604886	0.302443	+	0.953167i	$0.402198\pi$
$30$	0	0
$31$	0.658785	0.118321	0.0591607	−	0.998248i	$-0.481158\pi$
$31$	0.658785	0.118321	0.0591607	+	0.998248i	$0.481158\pi$
$32$	−5.85449	−1.03494
$33$	−6.48930	−1.12964
$34$	−4.98356	−0.854673
$35$	0	0
$36$	−5.74248	−0.957080
$37$	−1.00000	−0.164399
$37$	−1.00000	−0.164399
$38$	5.58384	0.905818
$39$	4.09694	0.656036
$40$	0	0
$41$	2.46863	0.385535	0.192768	−	0.981244i	$-0.438254\pi$
$41$	2.46863	0.385535	0.192768	+	0.981244i	$0.438254\pi$
$42$	−4.89097	−0.754692
$43$	−10.9579	−1.67107	−0.835535	−	0.549438i	$-0.814842\pi$
$43$	−10.9579	−1.67107	−0.835535	+	0.549438i	$0.814842\pi$
$44$	−3.62551	−0.546566
$45$	0	0
$46$	−5.46715	−0.806088
$47$	−3.11521	−0.454400	−0.227200	−	0.973848i	$-0.572957\pi$
$47$	−3.11521	−0.454400	−0.227200	+	0.973848i	$0.572957\pi$
$48$	−2.87617	−0.415140
$49$	−0.485072	−0.0692960
$50$	0	0
$51$	−17.9715	−2.51651
$52$	2.28892	0.317416
$53$	−8.64184	−1.18705	−0.593524	−	0.804816i	$-0.702264\pi$
$53$	−8.64184	−1.18705	−0.593524	+	0.804816i	$0.702264\pi$
$54$	1.74391	0.237317
$55$	0	0
$56$	−6.45373	−0.862416
$57$	20.1362	2.66710
$58$	−2.37449	−0.311786
$59$	−6.23634	−0.811903	−0.405951	−	0.913895i	$-0.633060\pi$
$59$	−6.23634	−0.811903	−0.405951	+	0.913895i	$0.633060\pi$
$60$	0	0
$61$	3.27808	0.419716	0.209858	−	0.977732i	$-0.432700\pi$
$61$	3.27808	0.419716	0.209858	+	0.977732i	$0.432700\pi$
$62$	−0.480222	−0.0609882
$63$	−9.98026	−1.25739
$64$	2.07935	0.259919
$65$	0	0
$66$	4.73038	0.582270
$67$	−1.47764	−0.180523	−0.0902615	−	0.995918i	$-0.528770\pi$
$67$	−1.47764	−0.180523	−0.0902615	+	0.995918i	$0.528770\pi$
$68$	−10.0405	−1.21759
$69$	−19.7154	−2.37345
$70$	0	0
$71$	−8.06686	−0.957360	−0.478680	−	0.877989i	$-0.658885\pi$
$71$	−8.06686	−0.957360	−0.478680	+	0.877989i	$0.658885\pi$
$72$	9.88651	1.16514
$73$	4.96199	0.580757	0.290379	−	0.956912i	$-0.406219\pi$
$73$	4.96199	0.580757	0.290379	+	0.956912i	$0.406219\pi$
$74$	0.728950	0.0847388
$75$	0	0
$76$	11.2499	1.29045
$77$	−6.30102	−0.718068
$78$	−2.98647	−0.338151
$79$	12.8206	1.44243	0.721214	−	0.692713i	$-0.243585\pi$
$79$	12.8206	1.44243	0.721214	+	0.692713i	$0.243585\pi$
$80$	0	0
$81$	−5.44146	−0.604607
$82$	−1.79951	−0.198723
$83$	−1.14934	−0.126157	−0.0630784	−	0.998009i	$-0.520092\pi$
$83$	−1.14934	−0.126157	−0.0630784	+	0.998009i	$0.520092\pi$
$84$	−9.85393	−1.07515
$85$	0	0
$86$	7.98779	0.861346
$87$	−8.56277	−0.918026
$88$	6.24184	0.665382
$89$	11.5207	1.22119	0.610596	−	0.791942i	$-0.290930\pi$
$89$	11.5207	1.22119	0.610596	+	0.791942i	$0.290930\pi$
$90$	0	0
$91$	3.97807	0.417015
$92$	−11.0148	−1.14837
$93$	−1.73175	−0.179574
$94$	2.27083	0.234218
$95$	0	0
$96$	15.3897	1.57071
$97$	−17.2929	−1.75583	−0.877916	−	0.478815i	$-0.841067\pi$
$97$	−17.2929	−1.75583	−0.877916	+	0.478815i	$0.841067\pi$
$98$	0.353594	0.0357183
$99$	9.65257	0.970120
$100$	0	0
$101$	−7.33253	−0.729614	−0.364807	−	0.931083i	$-0.618865\pi$
$101$	−7.33253	−0.729614	−0.364807	+	0.931083i	$0.618865\pi$
$102$	13.1003	1.29712
$103$	−9.26962	−0.913363	−0.456681	−	0.889630i	$-0.650962\pi$
$103$	−9.26962	−0.913363	−0.456681	+	0.889630i	$0.650962\pi$
$104$	−3.94071	−0.386418
$105$	0	0
$106$	6.29947	0.611859
$107$	12.0987	1.16962	0.584811	−	0.811170i	$-0.301169\pi$
$107$	12.0987	1.16962	0.584811	+	0.811170i	$0.301169\pi$
$108$	3.51350	0.338086
$109$	−2.24262	−0.214804	−0.107402	−	0.994216i	$-0.534253\pi$
$109$	−2.24262	−0.214804	−0.107402	+	0.994216i	$0.534253\pi$
$110$	0	0
$111$	2.62871	0.249506
$112$	−2.79272	−0.263888
$113$	11.3787	1.07042	0.535210	−	0.844719i	$-0.320232\pi$
$113$	11.3787	1.07042	0.535210	+	0.844719i	$0.320232\pi$
$114$	−14.6783	−1.37475
$115$	0	0
$116$	−4.78394	−0.444177
$117$	−6.09403	−0.563394
$118$	4.54598	0.418492
$119$	−17.4500	−1.59964
$120$	0	0
$121$	−4.90586	−0.445987
$122$	−2.38956	−0.216341
$123$	−6.48930	−0.585121
$124$	−0.967513	−0.0868852
$125$	0	0
$126$	7.27511	0.648118
$127$	−15.6642	−1.38997	−0.694985	−	0.719024i	$-0.744589\pi$
$127$	−15.6642	−1.38997	−0.694985	+	0.719024i	$0.744589\pi$
$128$	10.1932	0.900964
$129$	28.8052	2.53615
$130$	0	0
$131$	−13.6723	−1.19456	−0.597278	−	0.802034i	$-0.703751\pi$
$131$	−13.6723	−1.19456	−0.597278	+	0.802034i	$0.703751\pi$
$132$	9.53040	0.829514
$133$	19.5519	1.69537
$134$	1.07713	0.0930498
$135$	0	0
$136$	17.2861	1.48227
$137$	14.2031	1.21346	0.606728	−	0.794910i	$-0.292482\pi$
$137$	14.2031	1.21346	0.606728	+	0.794910i	$0.292482\pi$
$138$	14.3715	1.22339
$139$	−11.8416	−1.00440	−0.502198	−	0.864753i	$-0.667475\pi$
$139$	−11.8416	−1.00440	−0.502198	+	0.864753i	$0.667475\pi$
$140$	0	0
$141$	8.18896	0.689635
$142$	5.88034	0.493467
$143$	−3.84746	−0.321741
$144$	4.27819	0.356516
$145$	0	0
$146$	−3.61705	−0.299349
$147$	1.27511	0.105169
$148$	1.46863	0.120721
$149$	4.11726	0.337299	0.168649	−	0.985676i	$-0.446059\pi$
$149$	4.11726	0.337299	0.168649	+	0.985676i	$0.446059\pi$
$150$	0	0
$151$	16.9092	1.37605	0.688025	−	0.725687i	$-0.258478\pi$
$151$	16.9092	1.37605	0.688025	+	0.725687i	$0.258478\pi$
$152$	−19.3683	−1.57097
$153$	26.7318	2.16114
$154$	4.59313	0.370125
$155$	0	0
$156$	−6.01690	−0.481737
$157$	−4.02473	−0.321208	−0.160604	−	0.987019i	$-0.551344\pi$
$157$	−4.02473	−0.321208	−0.160604	+	0.987019i	$0.551344\pi$
$158$	−9.34556	−0.743493
$159$	22.7169	1.80156
$160$	0	0
$161$	−19.1434	−1.50871
$162$	3.96655	0.311642
$163$	−3.57756	−0.280216	−0.140108	−	0.990136i	$-0.544745\pi$
$163$	−3.57756	−0.280216	−0.140108	+	0.990136i	$0.544745\pi$
$164$	−3.62551	−0.283105
$165$	0	0
$166$	0.837814	0.0650270
$167$	−23.6674	−1.83144	−0.915720	−	0.401816i	$-0.868379\pi$
$167$	−23.6674	−1.83144	−0.915720	+	0.401816i	$0.868379\pi$
$168$	16.9650	1.30887
$169$	−10.5710	−0.813150
$170$	0	0
$171$	−29.9517	−2.29047
$172$	16.0932	1.22709
$173$	15.7383	1.19656	0.598279	−	0.801288i	$-0.295852\pi$
$173$	15.7383	1.19656	0.598279	+	0.801288i	$0.295852\pi$
$174$	6.24184	0.473192
$175$	0	0
$176$	2.70103	0.203598
$177$	16.3935	1.23221
$178$	−8.39802	−0.629459
$179$	10.1749	0.760510	0.380255	−	0.924882i	$-0.375836\pi$
$179$	10.1749	0.760510	0.380255	+	0.924882i	$0.375836\pi$
$180$	0	0
$181$	−2.54781	−0.189377	−0.0946886	−	0.995507i	$-0.530186\pi$
$181$	−2.54781	−0.189377	−0.0946886	+	0.995507i	$0.530186\pi$
$182$	−2.89982	−0.214949
$183$	−8.61711	−0.636995
$184$	18.9635	1.39801
$185$	0	0
$186$	1.26236	0.0925608
$187$	16.8771	1.23418
$188$	4.57509	0.333673
$189$	6.10635	0.444172
$190$	0	0
$191$	−10.4639	−0.757141	−0.378571	−	0.925572i	$-0.623584\pi$
$191$	−10.4639	−0.757141	−0.378571	+	0.925572i	$0.623584\pi$
$192$	−5.46601	−0.394475
$193$	9.81172	0.706263	0.353131	−	0.935574i	$-0.385117\pi$
$193$	9.81172	0.706263	0.353131	+	0.935574i	$0.385117\pi$
$194$	12.6057	0.905036
$195$	0	0
$196$	0.712392	0.0508852
$197$	−13.3945	−0.954317	−0.477158	−	0.878817i	$-0.658333\pi$
$197$	−13.3945	−0.954317	−0.477158	+	0.878817i	$0.658333\pi$
$198$	−7.03625	−0.500044
$199$	4.20221	0.297887	0.148943	−	0.988846i	$-0.452413\pi$
$199$	4.20221	0.297887	0.148943	+	0.988846i	$0.452413\pi$
$200$	0	0
$201$	3.88429	0.273977
$202$	5.34505	0.376077
$203$	−8.31433	−0.583552
$204$	26.3935	1.84791
$205$	0	0
$206$	6.75709	0.470789
$207$	29.3258	2.03829
$208$	−1.70526	−0.118239
$209$	−18.9100	−1.30803
$210$	0	0
$211$	20.0610	1.38106	0.690530	−	0.723304i	$-0.257377\pi$
$211$	20.0610	1.38106	0.690530	+	0.723304i	$0.257377\pi$
$212$	12.6917	0.871668
$213$	21.2054	1.45297
$214$	−8.81932	−0.602876
$215$	0	0
$216$	−6.04899	−0.411582
$217$	−1.68151	−0.114148
$218$	1.63476	0.110720
$219$	−13.0436	−0.881406
$220$	0	0
$221$	−10.6552	−0.716743
$222$	−1.91620	−0.128607
$223$	−1.72511	−0.115522	−0.0577610	−	0.998330i	$-0.518396\pi$
$223$	−1.72511	−0.115522	−0.0577610	+	0.998330i	$0.518396\pi$
$224$	14.9432	0.998436
$225$	0	0
$226$	−8.29452	−0.551744
$227$	−4.59930	−0.305266	−0.152633	−	0.988283i	$-0.548775\pi$
$227$	−4.59930	−0.305266	−0.152633	+	0.988283i	$0.548775\pi$
$228$	−29.5726	−1.95849
$229$	−23.4902	−1.55227	−0.776137	−	0.630565i	$-0.782823\pi$
$229$	−23.4902	−1.55227	−0.776137	+	0.630565i	$0.782823\pi$
$230$	0	0
$231$	16.5635	1.08980
$232$	8.23623	0.540735
$233$	−2.43465	−0.159499	−0.0797496	−	0.996815i	$-0.525412\pi$
$233$	−2.43465	−0.159499	−0.0797496	+	0.996815i	$0.525412\pi$
$234$	4.44225	0.290399
$235$	0	0
$236$	9.15889	0.596193
$237$	−33.7015	−2.18915
$238$	12.7202	0.824529
$239$	2.03195	0.131436	0.0657180	−	0.997838i	$-0.479066\pi$
$239$	2.03195	0.131436	0.0657180	+	0.997838i	$0.479066\pi$
$240$	0	0
$241$	−20.2246	−1.30278	−0.651390	−	0.758743i	$-0.725814\pi$
$241$	−20.2246	−1.30278	−0.651390	+	0.758743i	$0.725814\pi$
$242$	3.57613	0.229882
$243$	21.4811	1.37801
$244$	−4.81430	−0.308204
$245$	0	0
$246$	4.73038	0.301598
$247$	11.9386	0.759634
$248$	1.66571	0.105773
$249$	3.02128	0.191466
$250$	0	0
$251$	0.695568	0.0439038	0.0219519	−	0.999759i	$-0.493012\pi$
$251$	0.695568	0.0439038	0.0219519	+	0.999759i	$0.493012\pi$
$252$	14.6573	0.923324
$253$	18.5148	1.16402
$254$	11.4184	0.716454
$255$	0	0
$256$	−11.5891	−0.724317
$257$	−7.13610	−0.445138	−0.222569	−	0.974917i	$-0.571444\pi$
$257$	−7.13610	−0.445138	−0.222569	+	0.974917i	$0.571444\pi$
$258$	−20.9975	−1.30725
$259$	2.55244	0.158601
$260$	0	0
$261$	12.7368	0.788386
$262$	9.96644	0.615729
$263$	−21.2754	−1.31190	−0.655948	−	0.754806i	$-0.727731\pi$
$263$	−21.2754	−1.31190	−0.655948	+	0.754806i	$0.727731\pi$
$264$	−16.4079	−1.00984
$265$	0	0
$266$	−14.2524	−0.873870
$267$	−30.2845	−1.85338
$268$	2.17011	0.132561
$269$	18.3525	1.11897	0.559486	−	0.828840i	$-0.310998\pi$
$269$	18.3525	1.11897	0.559486	+	0.828840i	$0.310998\pi$
$270$	0	0
$271$	−23.2195	−1.41048	−0.705241	−	0.708967i	$-0.749161\pi$
$271$	−23.2195	−1.41048	−0.705241	+	0.708967i	$0.749161\pi$
$272$	7.48023	0.453555
$273$	−10.4572	−0.632897
$274$	−10.3534	−0.625471
$275$	0	0
$276$	28.9546	1.74286
$277$	−5.65674	−0.339880	−0.169940	−	0.985454i	$-0.554357\pi$
$277$	−5.65674	−0.339880	−0.169940	+	0.985454i	$0.554357\pi$
$278$	8.63197	0.517711
$279$	2.57591	0.154216
$280$	0	0
$281$	−24.4373	−1.45781	−0.728903	−	0.684616i	$-0.759970\pi$
$281$	−24.4373	−1.45781	−0.728903	+	0.684616i	$0.759970\pi$
$282$	−5.96935	−0.355469
$283$	−20.8498	−1.23939	−0.619697	−	0.784841i	$-0.712744\pi$
$283$	−20.8498	−1.23939	−0.619697	+	0.784841i	$0.712744\pi$
$284$	11.8472	0.703005
$285$	0	0
$286$	2.80461	0.165840
$287$	−6.30102	−0.371938
$288$	−22.8916	−1.34890
$289$	29.7394	1.74938
$290$	0	0
$291$	45.4580	2.66480
$292$	−7.28734	−0.426459
$293$	−7.72199	−0.451123	−0.225562	−	0.974229i	$-0.572422\pi$
$293$	−7.72199	−0.451123	−0.225562	+	0.974229i	$0.572422\pi$
$294$	−0.929493	−0.0542091
$295$	0	0
$296$	−2.52846	−0.146964
$297$	−5.90586	−0.342693
$298$	−3.00128	−0.173859
$299$	−11.6891	−0.675998
$300$	0	0
$301$	27.9694	1.61213
$302$	−12.3260	−0.709280
$303$	19.2751	1.10732
$304$	−8.38124	−0.480697
$305$	0	0
$306$	−19.4862	−1.11395
$307$	9.01041	0.514251	0.257126	−	0.966378i	$-0.417225\pi$
$307$	9.01041	0.514251	0.257126	+	0.966378i	$0.417225\pi$
$308$	9.25388	0.527289
$309$	24.3671	1.38620
$310$	0	0
$311$	−32.1220	−1.82147	−0.910735	−	0.412991i	$-0.864484\pi$
$311$	−32.1220	−1.82147	−0.910735	+	0.412991i	$0.864484\pi$
$312$	10.3590	0.586460
$313$	−19.8623	−1.12268	−0.561342	−	0.827584i	$-0.689715\pi$
$313$	−19.8623	−1.12268	−0.561342	+	0.827584i	$0.689715\pi$
$314$	2.93383	0.165565
$315$	0	0
$316$	−18.8287	−1.05920
$317$	8.58984	0.482453	0.241227	−	0.970469i	$-0.422450\pi$
$317$	8.58984	0.482453	0.241227	+	0.970469i	$0.422450\pi$
$318$	−16.5595	−0.928609
$319$	8.04135	0.450229
$320$	0	0
$321$	−31.8038	−1.77512
$322$	13.9546	0.777657
$323$	−52.3693	−2.91390
$324$	7.99150	0.443972
$325$	0	0
$326$	2.60786	0.144436
$327$	5.89520	0.326005
$328$	6.24184	0.344648
$329$	7.95137	0.438373
$330$	0	0
$331$	−22.9711	−1.26260	−0.631302	−	0.775537i	$-0.717479\pi$
$331$	−22.9711	−1.26260	−0.631302	+	0.775537i	$0.717479\pi$
$332$	1.68796	0.0926389
$333$	−3.91009	−0.214272
$334$	17.2524	0.944008
$335$	0	0
$336$	7.34125	0.400498
$337$	−26.2320	−1.42895	−0.714473	−	0.699663i	$-0.753334\pi$
$337$	−26.2320	−1.42895	−0.714473	+	0.699663i	$0.753334\pi$
$338$	7.70570	0.419135
$339$	−29.9113	−1.62456
$340$	0	0
$341$	1.62630	0.0880690
$342$	21.8333	1.18061
$343$	19.1052	1.03158
$344$	−27.7067	−1.49385
$345$	0	0
$346$	−11.4724	−0.616760
$347$	0.390348	0.0209550	0.0104775	−	0.999945i	$-0.496665\pi$
$347$	0.390348	0.0209550	0.0104775	+	0.999945i	$0.496665\pi$
$348$	12.5756	0.674121
$349$	−6.37205	−0.341088	−0.170544	−	0.985350i	$-0.554552\pi$
$349$	−6.37205	−0.341088	−0.170544	+	0.985350i	$0.554552\pi$
$350$	0	0
$351$	3.72859	0.199017
$352$	−14.4526	−0.770326
$353$	9.35970	0.498167	0.249083	−	0.968482i	$-0.419871\pi$
$353$	9.35970	0.498167	0.249083	+	0.968482i	$0.419871\pi$
$354$	−11.9501	−0.635138
$355$	0	0
$356$	−16.9197	−0.896741
$357$	45.8710	2.42775
$358$	−7.41702	−0.392002
$359$	−5.38443	−0.284179	−0.142090	−	0.989854i	$-0.545382\pi$
$359$	−5.38443	−0.284179	−0.142090	+	0.989854i	$0.545382\pi$
$360$	0	0
$361$	39.6773	2.08828
$362$	1.85723	0.0976137
$363$	12.8961	0.676868
$364$	−5.84232	−0.306221
$365$	0	0
$366$	6.28145	0.328337
$367$	22.1964	1.15864	0.579321	−	0.815099i	$-0.303318\pi$
$367$	22.1964	1.15864	0.579321	+	0.815099i	$0.303318\pi$
$368$	8.20609	0.427772
$369$	9.65257	0.502493
$370$	0	0
$371$	22.0577	1.14518
$372$	2.54331	0.131864
$373$	31.6626	1.63943	0.819713	−	0.572774i	$-0.194133\pi$
$373$	31.6626	1.63943	0.819713	+	0.572774i	$0.194133\pi$
$374$	−12.3026	−0.636151
$375$	0	0
$376$	−7.87668	−0.406209
$377$	−5.07680	−0.261469
$378$	−4.45123	−0.228946
$379$	−15.6622	−0.804515	−0.402257	−	0.915527i	$-0.631774\pi$
$379$	−15.6622	−0.804515	−0.402257	+	0.915527i	$0.631774\pi$
$380$	0	0
$381$	41.1765	2.10953
$382$	7.62766	0.390265
$383$	−7.84825	−0.401027	−0.200513	−	0.979691i	$-0.564261\pi$
$383$	−7.84825	−0.401027	−0.200513	+	0.979691i	$0.564261\pi$
$384$	−26.7950	−1.36738
$385$	0	0
$386$	−7.15226	−0.364040
$387$	−42.8465	−2.17801
$388$	25.3970	1.28933
$389$	17.6518	0.894981	0.447491	−	0.894289i	$-0.352318\pi$
$389$	17.6518	0.894981	0.447491	+	0.894289i	$0.352318\pi$
$390$	0	0
$391$	51.2749	2.59308
$392$	−1.22649	−0.0619469
$393$	35.9405	1.81296
$394$	9.76391	0.491899
$395$	0	0
$396$	−14.1761	−0.712374
$397$	22.7648	1.14253	0.571267	−	0.820765i	$-0.306452\pi$
$397$	22.7648	1.14253	0.571267	+	0.820765i	$0.306452\pi$
$398$	−3.06320	−0.153544
$399$	−51.3963	−2.57303
$400$	0	0
$401$	−0.168397	−0.00840934	−0.00420467	−	0.999991i	$-0.501338\pi$
$401$	−0.168397	−0.00840934	−0.00420467	+	0.999991i	$0.501338\pi$
$402$	−2.83146	−0.141220
$403$	−1.02674	−0.0511457
$404$	10.7688	0.535767
$405$	0	0
$406$	6.06073	0.300789
$407$	−2.46863	−0.122366
$408$	−45.4401	−2.24962
$409$	−21.1075	−1.04370	−0.521850	−	0.853037i	$-0.674758\pi$
$409$	−21.1075	−1.04370	−0.521850	+	0.853037i	$0.674758\pi$
$410$	0	0
$411$	−37.3359	−1.84164
$412$	13.6137	0.670697
$413$	15.9179	0.783267
$414$	−21.3771	−1.05063
$415$	0	0
$416$	9.12446	0.447364
$417$	31.1282	1.52435
$418$	13.7844	0.674219
$419$	−22.3217	−1.09049	−0.545243	−	0.838278i	$-0.683563\pi$
$419$	−22.3217	−1.09049	−0.545243	+	0.838278i	$0.683563\pi$
$420$	0	0
$421$	−27.4605	−1.33835	−0.669173	−	0.743107i	$-0.733351\pi$
$421$	−27.4605	−1.33835	−0.669173	+	0.743107i	$0.733351\pi$
$422$	−14.6235	−0.711861
$423$	−12.1807	−0.592248
$424$	−21.8506	−1.06116
$425$	0	0
$426$	−15.4577	−0.748927
$427$	−8.36710	−0.404912
$428$	−17.7685	−0.858871
$429$	10.1138	0.488301
$430$	0	0
$431$	36.7380	1.76960	0.884802	−	0.465966i	$-0.154293\pi$
$431$	36.7380	1.76960	0.884802	+	0.465966i	$0.154293\pi$
$432$	−2.61758	−0.125938
$433$	−0.370435	−0.0178020	−0.00890098	−	0.999960i	$-0.502833\pi$
$433$	−0.370435	−0.0178020	−0.00890098	+	0.999960i	$0.502833\pi$
$434$	1.22574	0.0588372
$435$	0	0
$436$	3.29359	0.157734
$437$	−57.4511	−2.74826
$438$	9.50815	0.454317
$439$	30.1768	1.44026	0.720130	−	0.693839i	$-0.244082\pi$
$439$	30.1768	1.44026	0.720130	+	0.693839i	$0.244082\pi$
$440$	0	0
$441$	−1.89668	−0.0903179
$442$	7.76708	0.369442
$443$	−38.1789	−1.81393	−0.906966	−	0.421203i	$-0.861608\pi$
$443$	−38.1789	−1.81393	−0.906966	+	0.421203i	$0.861608\pi$
$444$	−3.86060	−0.183216
$445$	0	0
$446$	1.25752	0.0595454
$447$	−10.8231	−0.511913
$448$	−5.30742	−0.250752
$449$	−3.77840	−0.178314	−0.0891569	−	0.996018i	$-0.528417\pi$
$449$	−3.77840	−0.178314	−0.0891569	+	0.996018i	$0.528417\pi$
$450$	0	0
$451$	6.09414	0.286962
$452$	−16.7111	−0.786026
$453$	−44.4493	−2.08841
$454$	3.35266	0.157348
$455$	0	0
$456$	50.9135	2.38424
$457$	−25.3343	−1.18509	−0.592544	−	0.805538i	$-0.701876\pi$
$457$	−25.3343	−1.18509	−0.592544	+	0.805538i	$0.701876\pi$
$458$	17.1232	0.800113
$459$	−16.3557	−0.763418
$460$	0	0
$461$	−14.7582	−0.687359	−0.343680	−	0.939087i	$-0.611673\pi$
$461$	−14.7582	−0.687359	−0.343680	+	0.939087i	$0.611673\pi$
$462$	−12.0740	−0.561733
$463$	−8.63700	−0.401395	−0.200698	−	0.979653i	$-0.564321\pi$
$463$	−8.63700	−0.401395	−0.200698	+	0.979653i	$0.564321\pi$
$464$	3.56407	0.165458
$465$	0	0
$466$	1.77474	0.0822132
$467$	1.64618	0.0761762	0.0380881	−	0.999274i	$-0.487873\pi$
$467$	1.64618	0.0761762	0.0380881	+	0.999274i	$0.487873\pi$
$468$	8.94989	0.413709
$469$	3.77159	0.174156
$470$	0	0
$471$	10.5798	0.487493
$472$	−15.7683	−0.725797
$473$	−27.0511	−1.24381
$474$	24.5667	1.12839
$475$	0	0
$476$	25.6277	1.17464
$477$	−33.7904	−1.54716
$478$	−1.48119	−0.0677482
$479$	16.0571	0.733669	0.366835	−	0.930286i	$-0.380442\pi$
$479$	16.0571	0.733669	0.366835	+	0.930286i	$0.380442\pi$
$480$	0	0
$481$	1.55854	0.0710633
$482$	14.7427	0.671513
$483$	50.3222	2.28974
$484$	7.20490	0.327495
$485$	0	0
$486$	−15.6586	−0.710290
$487$	6.60786	0.299431	0.149715	−	0.988729i	$-0.452164\pi$
$487$	6.60786	0.299431	0.149715	+	0.988729i	$0.452164\pi$
$488$	8.28850	0.375203
$489$	9.40435	0.425279
$490$	0	0
$491$	−18.2039	−0.821532	−0.410766	−	0.911741i	$-0.634739\pi$
$491$	−18.2039	−0.821532	−0.410766	+	0.911741i	$0.634739\pi$
$492$	9.53040	0.429663
$493$	22.2697	1.00298
$494$	−8.70264	−0.391550
$495$	0	0
$496$	0.720804	0.0323650
$497$	20.5901	0.923594
$498$	−2.20237	−0.0986904
$499$	−26.1795	−1.17196	−0.585978	−	0.810327i	$-0.699289\pi$
$499$	−26.1795	−1.17196	−0.585978	+	0.810327i	$0.699289\pi$
$500$	0	0
$501$	62.2147	2.77955
$502$	−0.507034	−0.0226301
$503$	−0.546197	−0.0243537	−0.0121769	−	0.999926i	$-0.503876\pi$
$503$	−0.546197	−0.0243537	−0.0121769	+	0.999926i	$0.503876\pi$
$504$	−25.2347	−1.12404
$505$	0	0
$506$	−13.4964	−0.599988
$507$	27.7879	1.23410
$508$	23.0049	1.02068
$509$	−27.4237	−1.21553	−0.607767	−	0.794115i	$-0.707935\pi$
$509$	−27.4237	−1.21553	−0.607767	+	0.794115i	$0.707935\pi$
$510$	0	0
$511$	−12.6652	−0.560274
$512$	−11.9386	−0.527618
$513$	18.3258	0.809102
$514$	5.20186	0.229444
$515$	0	0
$516$	−42.3042	−1.86234
$517$	−7.69030	−0.338219
$518$	−1.86060	−0.0817500
$519$	−41.3712	−1.81600
$520$	0	0
$521$	39.2211	1.71831	0.859154	−	0.511717i	$-0.170990\pi$
$521$	39.2211	1.71831	0.859154	+	0.511717i	$0.170990\pi$
$522$	−9.28448	−0.406370
$523$	2.89217	0.126466	0.0632329	−	0.997999i	$-0.479859\pi$
$523$	2.89217	0.126466	0.0632329	+	0.997999i	$0.479859\pi$
$524$	20.0796	0.877181
$525$	0	0
$526$	15.5087	0.676212
$527$	4.50387	0.196191
$528$	−7.10021	−0.308997
$529$	33.2505	1.44567
$530$	0	0
$531$	−24.3847	−1.05820
$532$	−28.7146	−1.24494
$533$	−3.84746	−0.166652
$534$	22.0759	0.955319
$535$	0	0
$536$	−3.73616	−0.161378
$537$	−26.7469	−1.15421
$538$	−13.3781	−0.576770
$539$	−1.19746	−0.0515784
$540$	0	0
$541$	−30.3727	−1.30582	−0.652912	−	0.757434i	$-0.726453\pi$
$541$	−30.3727	−1.30582	−0.652912	+	0.757434i	$0.726453\pi$
$542$	16.9258	0.727027
$543$	6.69744	0.287415
$544$	−40.0250	−1.71606
$545$	0	0
$546$	7.62277	0.326224
$547$	−18.8597	−0.806384	−0.403192	−	0.915115i	$-0.632099\pi$
$547$	−18.8597	−0.806384	−0.403192	+	0.915115i	$0.632099\pi$
$548$	−20.8592	−0.891060
$549$	12.8176	0.547042
$550$	0	0
$551$	−24.9521	−1.06300
$552$	−49.8496	−2.12174
$553$	−32.7237	−1.39155
$554$	4.12348	0.175190
$555$	0	0
$556$	17.3910	0.737543
$557$	−24.2064	−1.02566	−0.512828	−	0.858491i	$-0.671402\pi$
$557$	−24.2064	−1.02566	−0.512828	+	0.858491i	$0.671402\pi$
$558$	−1.87771	−0.0794898
$559$	17.0784	0.722339
$560$	0	0
$561$	−44.3649	−1.87309
$562$	17.8136	0.751420
$563$	−5.13444	−0.216391	−0.108196	−	0.994130i	$-0.534507\pi$
$563$	−5.13444	−0.216391	−0.108196	+	0.994130i	$0.534507\pi$
$564$	−12.0266	−0.506410
$565$	0	0
$566$	15.1985	0.638840
$567$	13.8890	0.583282
$568$	−20.3967	−0.855828
$569$	−11.9797	−0.502214	−0.251107	−	0.967959i	$-0.580795\pi$
$569$	−11.9797	−0.502214	−0.251107	+	0.967959i	$0.580795\pi$
$570$	0	0
$571$	−25.5874	−1.07080	−0.535399	−	0.844599i	$-0.679839\pi$
$571$	−25.5874	−1.07080	−0.535399	+	0.844599i	$0.679839\pi$
$572$	5.65050	0.236259
$573$	27.5065	1.14910
$574$	4.59313	0.191714
$575$	0	0
$576$	8.13047	0.338769
$577$	40.3272	1.67884	0.839421	−	0.543481i	$-0.182894\pi$
$577$	40.3272	1.67884	0.839421	+	0.543481i	$0.182894\pi$
$578$	−21.6786	−0.901709
$579$	−25.7921	−1.07188
$580$	0	0
$581$	2.93362	0.121707
$582$	−33.1367	−1.37356
$583$	−21.3335	−0.883544
$584$	12.5462	0.519165
$585$	0	0
$586$	5.62894	0.232530
$587$	31.6751	1.30737	0.653686	−	0.756766i	$-0.273222\pi$
$587$	31.6751	1.30737	0.653686	+	0.756766i	$0.273222\pi$
$588$	−1.87267	−0.0772276
$589$	−5.04637	−0.207932
$590$	0	0
$591$	35.2101	1.44835
$592$	−1.09414	−0.0449689
$593$	−6.26075	−0.257098	−0.128549	−	0.991703i	$-0.541032\pi$
$593$	−6.26075	−0.257098	−0.128549	+	0.991703i	$0.541032\pi$
$594$	4.30508	0.176640
$595$	0	0
$596$	−6.04673	−0.247684
$597$	−11.0464	−0.452098
$598$	8.52078	0.348440
$599$	34.4197	1.40635	0.703175	−	0.711017i	$-0.251765\pi$
$599$	34.4197	1.40635	0.703175	+	0.711017i	$0.251765\pi$
$600$	0	0
$601$	12.2491	0.499650	0.249825	−	0.968291i	$-0.419627\pi$
$601$	12.2491	0.499650	0.249825	+	0.968291i	$0.419627\pi$
$602$	−20.3883	−0.830966
$603$	−5.77772	−0.235287
$604$	−24.8334	−1.01046
$605$	0	0
$606$	−14.0506	−0.570765
$607$	31.0932	1.26204	0.631018	−	0.775768i	$-0.282637\pi$
$607$	31.0932	1.26204	0.631018	+	0.775768i	$0.282637\pi$
$608$	44.8461	1.81875
$609$	21.8559	0.885647
$610$	0	0
$611$	4.85518	0.196419
$612$	−39.2592	−1.58696
$613$	−15.0998	−0.609876	−0.304938	−	0.952372i	$-0.598636\pi$
$613$	−15.0998	−0.609876	−0.304938	+	0.952372i	$0.598636\pi$
$614$	−6.56814	−0.265069
$615$	0	0
$616$	−15.9319	−0.641914
$617$	27.3635	1.10161	0.550807	−	0.834632i	$-0.314320\pi$
$617$	27.3635	1.10161	0.550807	+	0.834632i	$0.314320\pi$
$618$	−17.7624	−0.714509
$619$	−11.1466	−0.448021	−0.224010	−	0.974587i	$-0.571915\pi$
$619$	−11.1466	−0.448021	−0.224010	+	0.974587i	$0.571915\pi$
$620$	0	0
$621$	−17.9428	−0.720020
$622$	23.4153	0.938869
$623$	−29.4059	−1.17812
$624$	4.48263	0.179449
$625$	0	0
$626$	14.4786	0.578683
$627$	49.7088	1.98518
$628$	5.91084	0.235868
$629$	−6.83662	−0.272594
$630$	0	0
$631$	11.9459	0.475558	0.237779	−	0.971319i	$-0.423581\pi$
$631$	11.9459	0.475558	0.237779	+	0.971319i	$0.423581\pi$
$632$	32.4163	1.28945
$633$	−52.7346	−2.09601
$634$	−6.26156	−0.248679
$635$	0	0
$636$	−33.3627	−1.32292
$637$	0.756004	0.0299540
$638$	−5.86174	−0.232069
$639$	−31.5422	−1.24779
$640$	0	0
$641$	27.9313	1.10322	0.551609	−	0.834103i	$-0.314014\pi$
$641$	27.9313	1.10322	0.551609	+	0.834103i	$0.314014\pi$
$642$	23.1834	0.914976
$643$	−39.0497	−1.53997	−0.769984	−	0.638063i	$-0.779736\pi$
$643$	−39.0497	−1.53997	−0.769984	+	0.638063i	$0.779736\pi$
$644$	28.1145	1.10787
$645$	0	0
$646$	38.1746	1.50196
$647$	3.60562	0.141752	0.0708759	−	0.997485i	$-0.477421\pi$
$647$	3.60562	0.141752	0.0708759	+	0.997485i	$0.477421\pi$
$648$	−13.7585	−0.540485
$649$	−15.3952	−0.604316
$650$	0	0
$651$	4.42019	0.173241
$652$	5.25411	0.205767
$653$	16.9343	0.662691	0.331345	−	0.943510i	$-0.392497\pi$
$653$	16.9343	0.662691	0.331345	+	0.943510i	$0.392497\pi$
$654$	−4.29731	−0.168038
$655$	0	0
$656$	2.70103	0.105458
$657$	19.4018	0.756938
$658$	−5.79615	−0.225958
$659$	31.2664	1.21797	0.608983	−	0.793183i	$-0.291578\pi$
$659$	31.2664	1.21797	0.608983	+	0.793183i	$0.291578\pi$
$660$	0	0
$661$	−39.2941	−1.52836	−0.764181	−	0.645002i	$-0.776857\pi$
$661$	−39.2941	−1.52836	−0.764181	+	0.645002i	$0.776857\pi$
$662$	16.7448	0.650804
$663$	28.0093	1.08779
$664$	−2.90607	−0.112777
$665$	0	0
$666$	2.85026	0.110445
$667$	24.4307	0.945960
$668$	34.7587	1.34486
$669$	4.53481	0.175326
$670$	0	0
$671$	8.09238	0.312403
$672$	−39.2813	−1.51531
$673$	−33.7258	−1.30004	−0.650018	−	0.759919i	$-0.725238\pi$
$673$	−33.7258	−1.30004	−0.650018	+	0.759919i	$0.725238\pi$
$674$	19.1218	0.736544
$675$	0	0
$676$	15.5248	0.597109
$677$	20.4385	0.785515	0.392758	−	0.919642i	$-0.371521\pi$
$677$	20.4385	0.785515	0.392758	+	0.919642i	$0.371521\pi$
$678$	21.8039	0.837372
$679$	44.1391	1.69390
$680$	0	0
$681$	12.0902	0.463298
$682$	−1.18549	−0.0453948
$683$	31.0965	1.18988	0.594938	−	0.803771i	$-0.297177\pi$
$683$	31.0965	1.18988	0.594938	+	0.803771i	$0.297177\pi$
$684$	43.9880	1.68192
$685$	0	0
$686$	−13.9267	−0.531725
$687$	61.7487	2.35586
$688$	−11.9895	−0.457096
$689$	13.4687	0.513115
$690$	0	0
$691$	10.8292	0.411960	0.205980	−	0.978556i	$-0.433962\pi$
$691$	10.8292	0.411960	0.205980	+	0.978556i	$0.433962\pi$
$692$	−23.1137	−0.878651
$693$	−24.6376	−0.935904
$694$	−0.284545	−0.0108012
$695$	0	0
$696$	−21.6506	−0.820665
$697$	16.8771	0.639266
$698$	4.64491	0.175812
$699$	6.39998	0.242069
$700$	0	0
$701$	3.32122	0.125441	0.0627204	−	0.998031i	$-0.480022\pi$
$701$	3.32122	0.125441	0.0627204	+	0.998031i	$0.480022\pi$
$702$	−2.71796	−0.102583
$703$	7.66011	0.288906
$704$	5.13316	0.193463
$705$	0	0
$706$	−6.82276	−0.256778
$707$	18.7158	0.703881
$708$	−24.0760	−0.904832
$709$	17.6659	0.663458	0.331729	−	0.943375i	$-0.392368\pi$
$709$	17.6659	0.663458	0.331729	+	0.943375i	$0.392368\pi$
$710$	0	0
$711$	50.1296	1.88001
$712$	29.1296	1.09168
$713$	4.94091	0.185039
$714$	−33.4377	−1.25137
$715$	0	0
$716$	−14.9432	−0.558454
$717$	−5.34140	−0.199478
$718$	3.92498	0.146479
$719$	−17.4938	−0.652410	−0.326205	−	0.945299i	$-0.605770\pi$
$719$	−17.4938	−0.652410	−0.326205	+	0.945299i	$0.605770\pi$
$720$	0	0
$721$	23.6601	0.881148
$722$	−28.9228	−1.07639
$723$	53.1645	1.97721
$724$	3.74179	0.139063
$725$	0	0
$726$	−9.40059	−0.348888
$727$	−19.2435	−0.713701	−0.356851	−	0.934161i	$-0.616149\pi$
$727$	−19.2435	−0.713701	−0.356851	+	0.934161i	$0.616149\pi$
$728$	10.0584	0.372789
$729$	−40.1430	−1.48678
$730$	0	0
$731$	−74.9153	−2.77084
$732$	12.6554	0.467756
$733$	17.2213	0.636082	0.318041	−	0.948077i	$-0.396975\pi$
$733$	17.2213	0.636082	0.318041	+	0.948077i	$0.396975\pi$
$734$	−16.1801	−0.597217
$735$	0	0
$736$	−43.9089	−1.61850
$737$	−3.64776	−0.134367
$738$	−7.03625	−0.259008
$739$	44.1590	1.62442	0.812208	−	0.583368i	$-0.198265\pi$
$739$	44.1590	1.62442	0.812208	+	0.583368i	$0.198265\pi$
$740$	0	0
$741$	−31.3830	−1.15288
$742$	−16.0790	−0.590279
$743$	15.8768	0.582461	0.291231	−	0.956653i	$-0.405935\pi$
$743$	15.8768	0.582461	0.291231	+	0.956653i	$0.405935\pi$
$744$	−4.37867	−0.160530
$745$	0	0
$746$	−23.0805	−0.845035
$747$	−4.49404	−0.164428
$748$	−24.7862	−0.906275
$749$	−30.8811	−1.12837
$750$	0	0
$751$	20.6030	0.751815	0.375907	−	0.926657i	$-0.377331\pi$
$751$	20.6030	0.751815	0.375907	+	0.926657i	$0.377331\pi$
$752$	−3.40848	−0.124294
$753$	−1.82844	−0.0666322
$754$	3.70074	0.134773
$755$	0	0
$756$	−8.96798	−0.326162
$757$	21.6914	0.788387	0.394194	−	0.919027i	$-0.371024\pi$
$757$	21.6914	0.788387	0.394194	+	0.919027i	$0.371024\pi$
$758$	11.4170	0.414684
$759$	−48.6700	−1.76661
$760$	0	0
$761$	22.5166	0.816225	0.408113	−	0.912932i	$-0.366187\pi$
$761$	22.5166	0.816225	0.408113	+	0.912932i	$0.366187\pi$
$762$	−30.0156	−1.08735
$763$	5.72415	0.207228
$764$	15.3676	0.555981
$765$	0	0
$766$	5.72098	0.206708
$767$	9.71959	0.350954
$768$	30.4643	1.09928
$769$	−0.975961	−0.0351941	−0.0175970	−	0.999845i	$-0.505602\pi$
$769$	−0.975961	−0.0351941	−0.0175970	+	0.999845i	$0.505602\pi$
$770$	0	0
$771$	18.7587	0.675578
$772$	−14.4098	−0.518620
$773$	−15.6271	−0.562066	−0.281033	−	0.959698i	$-0.590677\pi$
$773$	−15.6271	−0.562066	−0.281033	+	0.959698i	$0.590677\pi$
$774$	31.2330	1.12265
$775$	0	0
$776$	−43.7245	−1.56962
$777$	−6.70960	−0.240706
$778$	−12.8673	−0.461314
$779$	−18.9100	−0.677521
$780$	0	0
$781$	−19.9141	−0.712583
$782$	−37.3769	−1.33659
$783$	−7.79290	−0.278496
$784$	−0.530737	−0.0189549
$785$	0	0
$786$	−26.1988	−0.934482
$787$	−44.8503	−1.59874	−0.799370	−	0.600840i	$-0.794833\pi$
$787$	−44.8503	−1.59874	−0.799370	+	0.600840i	$0.794833\pi$
$788$	19.6715	0.700770
$789$	55.9267	1.99104
$790$	0	0
$791$	−29.0435	−1.03267
$792$	24.4061	0.867235
$793$	−5.10902	−0.181427
$794$	−16.5944	−0.588914
$795$	0	0
$796$	−6.17149	−0.218743
$797$	−12.4685	−0.441658	−0.220829	−	0.975313i	$-0.570876\pi$
$797$	−12.4685	−0.441658	−0.220829	+	0.975313i	$0.570876\pi$
$798$	37.4653	1.32626
$799$	−21.2975	−0.753451
$800$	0	0
$801$	45.0470	1.59166
$802$	0.122753	0.00433456
$803$	12.2493	0.432269
$804$	−5.70459	−0.201185
$805$	0	0
$806$	0.748445	0.0263628
$807$	−48.2434	−1.69825
$808$	−18.5400	−0.652236
$809$	18.8505	0.662748	0.331374	−	0.943500i	$-0.392488\pi$
$809$	18.8505	0.662748	0.331374	+	0.943500i	$0.392488\pi$
$810$	0	0
$811$	29.8244	1.04728	0.523638	−	0.851941i	$-0.324575\pi$
$811$	29.8244	1.04728	0.523638	+	0.851941i	$0.324575\pi$
$812$	12.2107	0.428511
$813$	61.0371	2.14067
$814$	1.79951	0.0630728
$815$	0	0
$816$	−19.6633	−0.688354
$817$	83.9390	2.93665
$818$	15.3863	0.537971
$819$	15.5546	0.543523
$820$	0	0
$821$	11.9059	0.415517	0.207759	−	0.978180i	$-0.433383\pi$
$821$	11.9059	0.415517	0.207759	+	0.978180i	$0.433383\pi$
$822$	27.2160	0.949267
$823$	−41.2544	−1.43804	−0.719020	−	0.694990i	$-0.755409\pi$
$823$	−41.2544	−1.43804	−0.719020	+	0.694990i	$0.755409\pi$
$824$	−23.4379	−0.816497
$825$	0	0
$826$	−11.6033	−0.403732
$827$	−27.0596	−0.940953	−0.470477	−	0.882412i	$-0.655918\pi$
$827$	−27.0596	−0.940953	−0.470477	+	0.882412i	$0.655918\pi$
$828$	−43.0688	−1.49674
$829$	−20.6787	−0.718200	−0.359100	−	0.933299i	$-0.616916\pi$
$829$	−20.6787	−0.718200	−0.359100	+	0.933299i	$0.616916\pi$
$830$	0	0
$831$	14.8699	0.515831
$832$	−3.24076	−0.112353
$833$	−3.31626	−0.114901
$834$	−22.6909	−0.785722
$835$	0	0
$836$	27.7718	0.960508
$837$	−1.57605	−0.0544763
$838$	16.2714	0.562086
$839$	−0.0974789	−0.00336534	−0.00168267	−	0.999999i	$-0.500536\pi$
$839$	−0.0974789	−0.00336534	−0.00168267	+	0.999999i	$0.500536\pi$
$840$	0	0
$841$	−18.3893	−0.634113
$842$	20.0174	0.689844
$843$	64.2385	2.21249
$844$	−29.4623	−1.01413
$845$	0	0
$846$	8.87916	0.305272
$847$	12.5219	0.430257
$848$	−9.45539	−0.324700
$849$	54.8080	1.88101
$850$	0	0
$851$	−7.50003	−0.257098
$852$	−31.1429	−1.06694
$853$	−0.919899	−0.0314967	−0.0157484	−	0.999876i	$-0.505013\pi$
$853$	−0.919899	−0.0314967	−0.0157484	+	0.999876i	$0.505013\pi$
$854$	6.09920	0.208710
$855$	0	0
$856$	30.5910	1.04558
$857$	−20.3344	−0.694608	−0.347304	−	0.937753i	$-0.612903\pi$
$857$	−20.3344	−0.694608	−0.347304	+	0.937753i	$0.612903\pi$
$858$	−7.37249	−0.251693
$859$	15.8959	0.542360	0.271180	−	0.962529i	$-0.412586\pi$
$859$	15.8959	0.542360	0.271180	+	0.962529i	$0.412586\pi$
$860$	0	0
$861$	16.5635	0.564484
$862$	−26.7801	−0.912135
$863$	37.0558	1.26139	0.630697	−	0.776029i	$-0.282769\pi$
$863$	37.0558	1.26139	0.630697	+	0.776029i	$0.282769\pi$
$864$	14.0061	0.476496
$865$	0	0
$866$	0.270029	0.00917595
$867$	−78.1762	−2.65500
$868$	2.46951	0.0838208
$869$	31.6493	1.07363
$870$	0	0
$871$	2.30297	0.0780331
$872$	−5.67039	−0.192024
$873$	−67.6170	−2.28849
$874$	41.8790	1.41658
$875$	0	0
$876$	19.1563	0.647230
$877$	25.4103	0.858045	0.429022	−	0.903294i	$-0.358858\pi$
$877$	25.4103	0.858045	0.429022	+	0.903294i	$0.358858\pi$
$878$	−21.9974	−0.742376
$879$	20.2988	0.684662
$880$	0	0
$881$	1.37793	0.0464238	0.0232119	−	0.999731i	$-0.492611\pi$
$881$	1.37793	0.0464238	0.0232119	+	0.999731i	$0.492611\pi$
$882$	1.38258	0.0465540
$883$	54.5280	1.83501	0.917506	−	0.397722i	$-0.130199\pi$
$883$	54.5280	1.83501	0.917506	+	0.397722i	$0.130199\pi$
$884$	15.6485	0.526316
$885$	0	0
$886$	27.8305	0.934984
$887$	6.43212	0.215969	0.107985	−	0.994153i	$-0.465560\pi$
$887$	6.43212	0.215969	0.107985	+	0.994153i	$0.465560\pi$
$888$	6.64658	0.223045
$889$	39.9818	1.34095
$890$	0	0
$891$	−13.4330	−0.450021
$892$	2.53355	0.0848297
$893$	23.8628	0.798539
$894$	7.88947	0.263863
$895$	0	0
$896$	−26.0176	−0.869187
$897$	30.7272	1.02595
$898$	2.75427	0.0919111
$899$	2.14593	0.0715709
$900$	0	0
$901$	−59.0810	−1.96827
$902$	−4.44233	−0.147913
$903$	−73.5234	−2.44670
$904$	28.7706	0.956897
$905$	0	0
$906$	32.4013	1.07646
$907$	33.4197	1.10968	0.554841	−	0.831957i	$-0.312779\pi$
$907$	33.4197	1.10968	0.554841	+	0.831957i	$0.312779\pi$
$908$	6.75468	0.224162
$909$	−28.6709	−0.950953
$910$	0	0
$911$	52.4122	1.73649	0.868247	−	0.496132i	$-0.165247\pi$
$911$	52.4122	1.73649	0.868247	+	0.496132i	$0.165247\pi$
$912$	22.0318	0.729546
$913$	−2.83730	−0.0939011
$914$	18.4674	0.610849
$915$	0	0
$916$	34.4984	1.13986
$917$	34.8977	1.15242
$918$	11.9225	0.393500
$919$	−11.6967	−0.385838	−0.192919	−	0.981215i	$-0.561795\pi$
$919$	−11.6967	−0.385838	−0.192919	+	0.981215i	$0.561795\pi$
$920$	0	0
$921$	−23.6857	−0.780470
$922$	10.7580	0.354297
$923$	12.5725	0.413830
$924$	−24.3257	−0.800257
$925$	0	0
$926$	6.29594	0.206898
$927$	−36.2451	−1.19044
$928$	−19.0705	−0.626020
$929$	−10.4868	−0.344059	−0.172030	−	0.985092i	$-0.555032\pi$
$929$	−10.4868	−0.344059	−0.172030	+	0.985092i	$0.555032\pi$
$930$	0	0
$931$	3.71571	0.121777
$932$	3.57560	0.117123
$933$	84.4392	2.76442
$934$	−1.19999	−0.0392647
$935$	0	0
$936$	−15.4085	−0.503643
$937$	−38.3244	−1.25200	−0.626002	−	0.779821i	$-0.715310\pi$
$937$	−38.3244	−1.25200	−0.626002	+	0.779821i	$0.715310\pi$
$938$	−2.74930	−0.0897679
$939$	52.2122	1.70388
$940$	0	0
$941$	33.4866	1.09163	0.545816	−	0.837905i	$-0.316220\pi$
$941$	33.4866	1.09163	0.545816	+	0.837905i	$0.316220\pi$
$942$	−7.71217	−0.251276
$943$	18.5148	0.602926
$944$	−6.82344	−0.222084
$945$	0	0
$946$	19.7189	0.641117
$947$	40.2997	1.30956	0.654782	−	0.755818i	$-0.272760\pi$
$947$	40.2997	1.30956	0.654782	+	0.755818i	$0.272760\pi$
$948$	49.4951	1.60753
$949$	−7.73346	−0.251039
$950$	0	0
$951$	−22.5801	−0.732211
$952$	−44.1217	−1.42999
$953$	−10.8028	−0.349939	−0.174969	−	0.984574i	$-0.555983\pi$
$953$	−10.8028	−0.349939	−0.174969	+	0.984574i	$0.555983\pi$
$954$	24.6315	0.797475
$955$	0	0
$956$	−2.98419	−0.0965156
$957$	−21.1383	−0.683305
$958$	−11.7049	−0.378167
$959$	−36.2526	−1.17066
$960$	0	0
$961$	−30.5660	−0.986000
$962$	−1.13610	−0.0366293
$963$	47.3069	1.52444
$964$	29.7025	0.956652
$965$	0	0
$966$	−36.6824	−1.18024
$967$	−46.0540	−1.48100	−0.740499	−	0.672058i	$-0.765411\pi$
$967$	−46.0540	−1.48100	−0.740499	+	0.672058i	$0.765411\pi$
$968$	−12.4043	−0.398688
$969$	137.663	4.42239
$970$	0	0
$971$	−0.231513	−0.00742961	−0.00371481	−	0.999993i	$-0.501182\pi$
$971$	−0.231513	−0.00742961	−0.00371481	+	0.999993i	$0.501182\pi$
$972$	−31.5478	−1.01190
$973$	30.2250	0.968970
$974$	−4.81680	−0.154340
$975$	0	0
$976$	3.58668	0.114807
$977$	−8.81121	−0.281896	−0.140948	−	0.990017i	$-0.545015\pi$
$977$	−8.81121	−0.281896	−0.140948	+	0.990017i	$0.545015\pi$
$978$	−6.85530	−0.219208
$979$	28.4404	0.908958
$980$	0	0
$981$	−8.76887	−0.279968
$982$	13.2698	0.423455
$983$	−18.8943	−0.602634	−0.301317	−	0.953524i	$-0.597426\pi$
$983$	−18.8943	−0.602634	−0.301317	+	0.953524i	$0.597426\pi$
$984$	−16.4079	−0.523066
$985$	0	0
$986$	−16.2335	−0.516980
$987$	−20.9018	−0.665312
$988$	−17.5334	−0.557811
$989$	−82.1849	−2.61333
$990$	0	0
$991$	30.3969	0.965589	0.482794	−	0.875734i	$-0.339622\pi$
$991$	30.3969	0.965589	0.482794	+	0.875734i	$0.339622\pi$
$992$	−3.85685	−0.122455
$993$	60.3842	1.91623
$994$	−15.0092	−0.476063
$995$	0	0
$996$	−4.43715	−0.140597
$997$	−35.2291	−1.11572	−0.557859	−	0.829936i	$-0.688377\pi$
$997$	−35.2291	−1.11572	−0.557859	+	0.829936i	$0.688377\pi$
$998$	19.0836	0.604079
$999$	2.39236	0.0756910

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	925.2.a.f.1.3		5
3.2	odd	2		8325.2.a.ch.1.3		5
5.2	odd	4		925.2.b.f.149.5		10
5.3	odd	4		925.2.b.f.149.6		10
5.4	even	2		185.2.a.e.1.3	✓	5
15.14	odd	2		1665.2.a.p.1.3		5
20.19	odd	2		2960.2.a.w.1.1		5
35.34	odd	2		9065.2.a.k.1.3		5
185.184	even	2		6845.2.a.f.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
185.2.a.e.1.3	✓	5	5.4	even	2
925.2.a.f.1.3		5	1.1	even	1	trivial
925.2.b.f.149.5		10	5.2	odd	4
925.2.b.f.149.6		10	5.3	odd	4
1665.2.a.p.1.3		5	15.14	odd	2
2960.2.a.w.1.1		5	20.19	odd	2
6845.2.a.f.1.3		5	185.184	even	2
8325.2.a.ch.1.3		5	3.2	odd	2
9065.2.a.k.1.3		5	35.34	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 925 = 5^{2} \cdot 37 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	925.a (trivial)

\(f(q)\)	\(=\)	\(q-0.728950 q^{2} -2.62871 q^{3} -1.46863 q^{4} +1.91620 q^{6} -2.55244 q^{7} +2.52846 q^{8} +3.91009 q^{9} +2.46863 q^{11} +3.86060 q^{12} -1.55854 q^{13} +1.86060 q^{14} +1.09414 q^{16} +6.83662 q^{17} -2.85026 q^{18} -7.66011 q^{19} +6.70960 q^{21} -1.79951 q^{22} +7.50003 q^{23} -6.64658 q^{24} +1.13610 q^{26} -2.39236 q^{27} +3.74859 q^{28} +3.25741 q^{29} +0.658785 q^{31} -5.85449 q^{32} -6.48930 q^{33} -4.98356 q^{34} -5.74248 q^{36} -1.00000 q^{37} +5.58384 q^{38} +4.09694 q^{39} +2.46863 q^{41} -4.89097 q^{42} -10.9579 q^{43} -3.62551 q^{44} -5.46715 q^{46} -3.11521 q^{47} -2.87617 q^{48} -0.485072 q^{49} -17.9715 q^{51} +2.28892 q^{52} -8.64184 q^{53} +1.74391 q^{54} -6.45373 q^{56} +20.1362 q^{57} -2.37449 q^{58} -6.23634 q^{59} +3.27808 q^{61} -0.480222 q^{62} -9.98026 q^{63} +2.07935 q^{64} +4.73038 q^{66} -1.47764 q^{67} -10.0405 q^{68} -19.7154 q^{69} -8.06686 q^{71} +9.88651 q^{72} +4.96199 q^{73} +0.728950 q^{74} +11.2499 q^{76} -6.30102 q^{77} -2.98647 q^{78} +12.8206 q^{79} -5.44146 q^{81} -1.79951 q^{82} -1.14934 q^{83} -9.85393 q^{84} +7.98779 q^{86} -8.56277 q^{87} +6.24184 q^{88} +11.5207 q^{89} +3.97807 q^{91} -11.0148 q^{92} -1.73175 q^{93} +2.27083 q^{94} +15.3897 q^{96} -17.2929 q^{97} +0.353594 q^{98} +9.65257 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} - 3 q^{3} + 10 q^{4} - 6 q^{6} - 11 q^{7} - 6 q^{8} + 6 q^{9} - 5 q^{11} + 2 q^{12} - 4 q^{13} - 8 q^{14} + 16 q^{16} - 2 q^{18} - 4 q^{19} + 3 q^{21} + 8 q^{22} - 4 q^{23} - 42 q^{24} - 4 q^{26}+ \cdots - 10 q^{99}+O(q^{100}) \) 5 * q - 2 * q^2 - 3 * q^3 + 10 * q^4 - 6 * q^6 - 11 * q^7 - 6 * q^8 + 6 * q^9 - 5 * q^11 + 2 * q^12 - 4 * q^13 - 8 * q^14 + 16 * q^16 - 2 * q^18 - 4 * q^19 + 3 * q^21 + 8 * q^22 - 4 * q^23 - 42 * q^24 - 4 * q^26 - 3 * q^27 - 28 * q^28 - 4 * q^29 + 8 * q^31 - 14 * q^32 - 5 * q^33 - 32 * q^34 + 16 * q^36 - 5 * q^37 + 2 * q^38 + 2 * q^39 - 5 * q^41 + 52 * q^42 - 10 * q^43 - 46 * q^44 - 7 * q^47 + 20 * q^48 + 22 * q^49 - 2 * q^51 + 32 * q^52 + q^53 - 10 * q^54 + 8 * q^56 + 8 * q^57 + 16 * q^58 - 30 * q^59 - 14 * q^61 - 24 * q^62 - 30 * q^63 + 20 * q^64 + 48 * q^66 - 24 * q^67 - 20 * q^68 + 8 * q^69 - 7 * q^71 + 10 * q^72 - 5 * q^73 + 2 * q^74 + 12 * q^76 + 17 * q^77 - 58 * q^78 + 28 * q^79 - 31 * q^81 + 8 * q^82 - 27 * q^83 - 18 * q^84 + 44 * q^86 - 36 * q^87 + 28 * q^88 + 6 * q^89 - 14 * q^91 - 56 * q^92 + 22 * q^93 + 4 * q^94 - 42 * q^96 + 26 * q^97 + 18 * q^98 - 10 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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