LMFDB - Embedded newform 7225.2.a.by.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7225,2,Mod(1,7225)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7225, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("7225.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$7225 = 5^{2} \cdot 17^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	7225.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:	$57.6919154604$
Analytic rank:	$1$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 85)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	7225.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.499161 q^{2} +0.171281 q^{3} -1.75084 q^{4} -0.0854968 q^{6} -3.87301 q^{7} +1.87227 q^{8} -2.97066 q^{9} -2.93381 q^{11} -0.299886 q^{12} -0.983677 q^{13} +1.93326 q^{14} +2.56711 q^{16} +1.48284 q^{18} +2.56860 q^{19} -0.663375 q^{21} +1.46444 q^{22} +3.00652 q^{23} +0.320685 q^{24} +0.491013 q^{26} -1.02266 q^{27} +6.78102 q^{28} +0.549546 q^{29} +7.55631 q^{31} -5.02594 q^{32} -0.502507 q^{33} +5.20115 q^{36} +9.84275 q^{37} -1.28214 q^{38} -0.168485 q^{39} -7.40387 q^{41} +0.331130 q^{42} -8.27543 q^{43} +5.13663 q^{44} -1.50073 q^{46} +10.9492 q^{47} +0.439699 q^{48} +8.00024 q^{49} +1.72226 q^{52} -3.58708 q^{53} +0.510473 q^{54} -7.25133 q^{56} +0.439953 q^{57} -0.274312 q^{58} +0.305801 q^{59} +6.80147 q^{61} -3.77181 q^{62} +11.5054 q^{63} -2.62548 q^{64} +0.250832 q^{66} -5.09621 q^{67} +0.514960 q^{69} +3.60703 q^{71} -5.56188 q^{72} +5.01077 q^{73} -4.91311 q^{74} -4.49721 q^{76} +11.3627 q^{77} +0.0841013 q^{78} +12.4437 q^{79} +8.73683 q^{81} +3.69572 q^{82} +8.12143 q^{83} +1.16146 q^{84} +4.13077 q^{86} +0.0941270 q^{87} -5.49289 q^{88} -13.2723 q^{89} +3.80980 q^{91} -5.26393 q^{92} +1.29425 q^{93} -5.46540 q^{94} -0.860850 q^{96} +4.52617 q^{97} -3.99340 q^{98} +8.71536 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$24 q + 8 q^{4} - 8 q^{9} - 8 q^{16} - 16 q^{19} - 32 q^{21} - 48 q^{26} + 8 q^{36} - 56 q^{49} - 64 q^{59} - 104 q^{64} - 16 q^{66} - 32 q^{69} - 48 q^{76} - 88 q^{81} - 96 q^{84} - 112 q^{89} - 128 q^{94}+O(q^{100})$ 24 * q + 8 * q^4 - 8 * q^9 - 8 * q^16 - 16 * q^19 - 32 * q^21 - 48 * q^26 + 8 * q^36 - 56 * q^49 - 64 * q^59 - 104 * q^64 - 16 * q^66 - 32 * q^69 - 48 * q^76 - 88 * q^81 - 96 * q^84 - 112 * q^89 - 128 * q^94

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

)

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

$2$	−0.499161	−0.352960	−0.176480	−	0.984304i	$-0.556471\pi$
$2$	−0.499161	−0.352960	−0.176480	+	0.984304i	$0.556471\pi$
$3$	0.171281	0.0988893	0.0494446	−	0.998777i	$-0.484255\pi$
$3$	0.171281	0.0988893	0.0494446	+	0.998777i	$0.484255\pi$
$4$	−1.75084	−0.875419
$5$	0	0
$5$	0	0
$6$	−0.0854968	−0.0349039
$7$	−3.87301	−1.46386	−0.731931	−	0.681379i	$-0.761381\pi$
$7$	−3.87301	−1.46386	−0.731931	+	0.681379i	$0.761381\pi$
$8$	1.87227	0.661948
$9$	−2.97066	−0.990221
$10$	0	0
$11$	−2.93381	−0.884577	−0.442289	−	0.896873i	$-0.645833\pi$
$11$	−2.93381	−0.884577	−0.442289	+	0.896873i	$0.645833\pi$
$12$	−0.299886	−0.0865696
$13$	−0.983677	−0.272823	−0.136411	−	0.990652i	$-0.543557\pi$
$13$	−0.983677	−0.272823	−0.136411	+	0.990652i	$0.543557\pi$
$14$	1.93326	0.516684
$15$	0	0
$16$	2.56711	0.641779
$17$	0	0
$17$	0	0
$18$	1.48284	0.349508
$19$	2.56860	0.589277	0.294639	−	0.955609i	$-0.404801\pi$
$19$	2.56860	0.589277	0.294639	+	0.955609i	$0.404801\pi$
$20$	0	0
$21$	−0.663375	−0.144760
$22$	1.46444	0.312220
$23$	3.00652	0.626902	0.313451	−	0.949604i	$-0.398515\pi$
$23$	3.00652	0.626902	0.313451	+	0.949604i	$0.398515\pi$
$24$	0.320685	0.0654595
$25$	0	0
$26$	0.491013	0.0962955
$27$	−1.02266	−0.196812
$28$	6.78102	1.28149
$29$	0.549546	0.102048	0.0510241	−	0.998697i	$-0.483751\pi$
$29$	0.549546	0.102048	0.0510241	+	0.998697i	$0.483751\pi$
$30$	0	0
$31$	7.55631	1.35715	0.678577	−	0.734530i	$-0.262597\pi$
$31$	7.55631	1.35715	0.678577	+	0.734530i	$0.262597\pi$
$32$	−5.02594	−0.888470
$33$	−0.502507	−0.0874752
$34$	0	0
$35$	0	0
$36$	5.20115	0.866859
$37$	9.84275	1.61814	0.809069	−	0.587714i	$-0.199972\pi$
$37$	9.84275	1.61814	0.809069	+	0.587714i	$0.199972\pi$
$38$	−1.28214	−0.207991
$39$	−0.168485	−0.0269793
$40$	0	0
$41$	−7.40387	−1.15629	−0.578145	−	0.815934i	$-0.696223\pi$
$41$	−7.40387	−1.15629	−0.578145	+	0.815934i	$0.696223\pi$
$42$	0.331130	0.0510945
$43$	−8.27543	−1.26199	−0.630996	−	0.775786i	$-0.717353\pi$
$43$	−8.27543	−1.26199	−0.630996	+	0.775786i	$0.717353\pi$
$44$	5.13663	0.774376
$45$	0	0
$46$	−1.50073	−0.221271
$47$	10.9492	1.59710	0.798551	−	0.601928i	$-0.205600\pi$
$47$	10.9492	1.59710	0.798551	+	0.601928i	$0.205600\pi$
$48$	0.439699	0.0634650
$49$	8.00024	1.14289
$50$	0	0
$51$	0	0
$52$	1.72226	0.238834
$53$	−3.58708	−0.492723	−0.246362	−	0.969178i	$-0.579235\pi$
$53$	−3.58708	−0.492723	−0.246362	+	0.969178i	$0.579235\pi$
$54$	0.510473	0.0694665
$55$	0	0
$56$	−7.25133	−0.969000
$57$	0.439953	0.0582732
$58$	−0.274312	−0.0360189
$59$	0.305801	0.0398119	0.0199059	−	0.999802i	$-0.493663\pi$
$59$	0.305801	0.0398119	0.0199059	+	0.999802i	$0.493663\pi$
$60$	0	0
$61$	6.80147	0.870839	0.435420	−	0.900228i	$-0.356600\pi$
$61$	6.80147	0.870839	0.435420	+	0.900228i	$0.356600\pi$
$62$	−3.77181	−0.479020
$63$	11.5054	1.44955
$64$	−2.62548	−0.328184
$65$	0	0
$66$	0.250832	0.0308752
$67$	−5.09621	−0.622601	−0.311301	−	0.950312i	$-0.600765\pi$
$67$	−5.09621	−0.622601	−0.311301	+	0.950312i	$0.600765\pi$
$68$	0	0
$69$	0.514960	0.0619939
$70$	0	0
$71$	3.60703	0.428076	0.214038	−	0.976825i	$-0.431338\pi$
$71$	3.60703	0.428076	0.214038	+	0.976825i	$0.431338\pi$
$72$	−5.56188	−0.655474
$73$	5.01077	0.586466	0.293233	−	0.956041i	$-0.405269\pi$
$73$	5.01077	0.586466	0.293233	+	0.956041i	$0.405269\pi$
$74$	−4.91311	−0.571138
$75$	0	0
$76$	−4.49721	−0.515865
$77$	11.3627	1.29490
$78$	0.0841013	0.00952259
$79$	12.4437	1.40002	0.700012	−	0.714131i	$-0.253178\pi$
$79$	12.4437	1.40002	0.700012	+	0.714131i	$0.253178\pi$
$80$	0	0
$81$	8.73683	0.970758
$82$	3.69572	0.408124
$83$	8.12143	0.891443	0.445721	−	0.895172i	$-0.352947\pi$
$83$	8.12143	0.891443	0.445721	+	0.895172i	$0.352947\pi$
$84$	1.16146	0.126726
$85$	0	0
$86$	4.13077	0.445432
$87$	0.0941270	0.0100915
$88$	−5.49289	−0.585544
$89$	−13.2723	−1.40687	−0.703433	−	0.710761i	$-0.748351\pi$
$89$	−13.2723	−1.40687	−0.703433	+	0.710761i	$0.748351\pi$
$90$	0	0
$91$	3.80980	0.399375
$92$	−5.26393	−0.548802
$93$	1.29425	0.134208
$94$	−5.46540	−0.563712
$95$	0	0
$96$	−0.860850	−0.0878601
$97$	4.52617	0.459563	0.229781	−	0.973242i	$-0.426199\pi$
$97$	4.52617	0.459563	0.229781	+	0.973242i	$0.426199\pi$
$98$	−3.99340	−0.403395
$99$	8.71536	0.875927
$100$	0	0
$101$	−2.62696	−0.261393	−0.130696	−	0.991422i	$-0.541721\pi$
$101$	−2.62696	−0.261393	−0.130696	+	0.991422i	$0.541721\pi$
$102$	0	0
$103$	9.21332	0.907815	0.453908	−	0.891049i	$-0.350030\pi$
$103$	9.21332	0.907815	0.453908	+	0.891049i	$0.350030\pi$
$104$	−1.84171	−0.180594
$105$	0	0
$106$	1.79053	0.173911
$107$	−5.83150	−0.563752	−0.281876	−	0.959451i	$-0.590957\pi$
$107$	−5.83150	−0.563752	−0.281876	+	0.959451i	$0.590957\pi$
$108$	1.79052	0.172293
$109$	−17.0398	−1.63212	−0.816058	−	0.577970i	$-0.803845\pi$
$109$	−17.0398	−1.63212	−0.816058	+	0.577970i	$0.803845\pi$
$110$	0	0
$111$	1.68588	0.160017
$112$	−9.94247	−0.939475
$113$	1.17074	0.110134	0.0550670	−	0.998483i	$-0.482463\pi$
$113$	1.17074	0.110134	0.0550670	+	0.998483i	$0.482463\pi$
$114$	−0.219607	−0.0205681
$115$	0	0
$116$	−0.962167	−0.0893350
$117$	2.92217	0.270155
$118$	−0.152644	−0.0140520
$119$	0	0
$120$	0	0
$121$	−2.39275	−0.217523
$122$	−3.39503	−0.307371
$123$	−1.26814	−0.114345
$124$	−13.2299	−1.18808
$125$	0	0
$126$	−5.74305	−0.511632
$127$	−6.65830	−0.590828	−0.295414	−	0.955369i	$-0.595458\pi$
$127$	−6.65830	−0.590828	−0.295414	+	0.955369i	$0.595458\pi$
$128$	11.3624	1.00431
$129$	−1.41743	−0.124797
$130$	0	0
$131$	−12.2839	−1.07325	−0.536624	−	0.843821i	$-0.680301\pi$
$131$	−12.2839	−1.07325	−0.536624	+	0.843821i	$0.680301\pi$
$132$	0.879808	0.0765775
$133$	−9.94823	−0.862621
$134$	2.54383	0.219753
$135$	0	0
$136$	0	0
$137$	−9.68972	−0.827848	−0.413924	−	0.910311i	$-0.635842\pi$
$137$	−9.68972	−0.827848	−0.413924	+	0.910311i	$0.635842\pi$
$138$	−0.257048	−0.0218814
$139$	−1.43071	−0.121351	−0.0606755	−	0.998158i	$-0.519325\pi$
$139$	−1.43071	−0.121351	−0.0606755	+	0.998158i	$0.519325\pi$
$140$	0	0
$141$	1.87539	0.157936
$142$	−1.80049	−0.151094
$143$	2.88592	0.241333
$144$	−7.62603	−0.635502
$145$	0	0
$146$	−2.50118	−0.206999
$147$	1.37029	0.113020
$148$	−17.2331	−1.41655
$149$	−9.40983	−0.770884	−0.385442	−	0.922732i	$-0.625951\pi$
$149$	−9.40983	−0.770884	−0.385442	+	0.922732i	$0.625951\pi$
$150$	0	0
$151$	8.66213	0.704914	0.352457	−	0.935828i	$-0.385346\pi$
$151$	8.66213	0.704914	0.352457	+	0.935828i	$0.385346\pi$
$152$	4.80912	0.390071
$153$	0	0
$154$	−5.67181	−0.457047
$155$	0	0
$156$	0.294991	0.0236182
$157$	17.7467	1.41634	0.708169	−	0.706043i	$-0.249521\pi$
$157$	17.7467	1.41634	0.708169	+	0.706043i	$0.249521\pi$
$158$	−6.21140	−0.494152
$159$	−0.614399	−0.0487250
$160$	0	0
$161$	−11.6443	−0.917698
$162$	−4.36108	−0.342639
$163$	−2.30071	−0.180205	−0.0901027	−	0.995932i	$-0.528720\pi$
$163$	−2.30071	−0.180205	−0.0901027	+	0.995932i	$0.528720\pi$
$164$	12.9630	1.01224
$165$	0	0
$166$	−4.05390	−0.314643
$167$	−9.36248	−0.724491	−0.362245	−	0.932083i	$-0.617990\pi$
$167$	−9.36248	−0.724491	−0.362245	+	0.932083i	$0.617990\pi$
$168$	−1.24202	−0.0958237
$169$	−12.0324	−0.925568
$170$	0	0
$171$	−7.63045	−0.583515
$172$	14.4889	1.10477
$173$	9.29988	0.707057	0.353528	−	0.935424i	$-0.384982\pi$
$173$	9.29988	0.707057	0.353528	+	0.935424i	$0.384982\pi$
$174$	−0.0469845	−0.00356188
$175$	0	0
$176$	−7.53143	−0.567703
$177$	0.0523780	0.00393697
$178$	6.62503	0.496567
$179$	−1.93778	−0.144837	−0.0724183	−	0.997374i	$-0.523072\pi$
$179$	−1.93778	−0.144837	−0.0724183	+	0.997374i	$0.523072\pi$
$180$	0	0
$181$	16.9024	1.25635	0.628173	−	0.778074i	$-0.283803\pi$
$181$	16.9024	1.25635	0.628173	+	0.778074i	$0.283803\pi$
$182$	−1.90170	−0.140963
$183$	1.16496	0.0861167
$184$	5.62901	0.414976
$185$	0	0
$186$	−0.646040	−0.0473700
$187$	0	0
$188$	−19.1702	−1.39813
$189$	3.96079	0.288105
$190$	0	0
$191$	−16.6593	−1.20543	−0.602713	−	0.797958i	$-0.705914\pi$
$191$	−16.6593	−1.20543	−0.602713	+	0.797958i	$0.705914\pi$
$192$	−0.449695	−0.0324539
$193$	−14.1648	−1.01960	−0.509801	−	0.860292i	$-0.670281\pi$
$193$	−14.1648	−1.01960	−0.509801	+	0.860292i	$0.670281\pi$
$194$	−2.25928	−0.162207
$195$	0	0
$196$	−14.0071	−1.00051
$197$	18.2978	1.30366	0.651832	−	0.758363i	$-0.274001\pi$
$197$	18.2978	1.30366	0.651832	+	0.758363i	$0.274001\pi$
$198$	−4.35036	−0.309167
$199$	−5.45781	−0.386894	−0.193447	−	0.981111i	$-0.561967\pi$
$199$	−5.45781	−0.386894	−0.193447	+	0.981111i	$0.561967\pi$
$200$	0	0
$201$	−0.872885	−0.0615686
$202$	1.31128	0.0922611
$203$	−2.12840	−0.149384
$204$	0	0
$205$	0	0
$206$	−4.59893	−0.320422
$207$	−8.93135	−0.620772
$208$	−2.52521	−0.175092
$209$	−7.53579	−0.521261
$210$	0	0
$211$	−5.50932	−0.379277	−0.189639	−	0.981854i	$-0.560732\pi$
$211$	−5.50932	−0.379277	−0.189639	+	0.981854i	$0.560732\pi$
$212$	6.28040	0.431339
$213$	0.617817	0.0423321
$214$	2.91085	0.198982
$215$	0	0
$216$	−1.91470	−0.130279
$217$	−29.2657	−1.98668
$218$	8.50559	0.576071
$219$	0.858250	0.0579952
$220$	0	0
$221$	0	0
$222$	−0.841524	−0.0564794
$223$	26.6184	1.78250	0.891249	−	0.453514i	$-0.149830\pi$
$223$	26.6184	1.78250	0.891249	+	0.453514i	$0.149830\pi$
$224$	19.4656	1.30060
$225$	0	0
$226$	−0.584387	−0.0388728
$227$	−7.66994	−0.509072	−0.254536	−	0.967063i	$-0.581923\pi$
$227$	−7.66994	−0.509072	−0.254536	+	0.967063i	$0.581923\pi$
$228$	−0.770287	−0.0510135
$229$	−9.33359	−0.616781	−0.308390	−	0.951260i	$-0.599790\pi$
$229$	−9.33359	−0.616781	−0.308390	+	0.951260i	$0.599790\pi$
$230$	0	0
$231$	1.94622	0.128052
$232$	1.02890	0.0675505
$233$	3.89898	0.255431	0.127715	−	0.991811i	$-0.459236\pi$
$233$	3.89898	0.255431	0.127715	+	0.991811i	$0.459236\pi$
$234$	−1.45863	−0.0953538
$235$	0	0
$236$	−0.535408	−0.0348521
$237$	2.13137	0.138447
$238$	0	0
$239$	−22.6299	−1.46380	−0.731902	−	0.681410i	$-0.761367\pi$
$239$	−22.6299	−1.46380	−0.731902	+	0.681410i	$0.761367\pi$
$240$	0	0
$241$	−2.89198	−0.186289	−0.0931443	−	0.995653i	$-0.529692\pi$
$241$	−2.89198	−0.186289	−0.0931443	+	0.995653i	$0.529692\pi$
$242$	1.19437	0.0767769
$243$	4.56444	0.292809
$244$	−11.9083	−0.762350
$245$	0	0
$246$	0.633007	0.0403591
$247$	−2.52667	−0.160768
$248$	14.1475	0.898364
$249$	1.39105	0.0881541
$250$	0	0
$251$	−13.3749	−0.844213	−0.422107	−	0.906546i	$-0.638709\pi$
$251$	−13.3749	−0.844213	−0.422107	+	0.906546i	$0.638709\pi$
$252$	−20.1441	−1.26896
$253$	−8.82055	−0.554543
$254$	3.32356	0.208539
$255$	0	0
$256$	−0.420720	−0.0262950
$257$	7.32959	0.457208	0.228604	−	0.973520i	$-0.426584\pi$
$257$	7.32959	0.457208	0.228604	+	0.973520i	$0.426584\pi$
$258$	0.707523	0.0440485
$259$	−38.1211	−2.36873
$260$	0	0
$261$	−1.63252	−0.101050
$262$	6.13164	0.378814
$263$	−24.1129	−1.48686	−0.743432	−	0.668811i	$-0.766803\pi$
$263$	−24.1129	−1.48686	−0.743432	+	0.668811i	$0.766803\pi$
$264$	−0.940829	−0.0579040
$265$	0	0
$266$	4.96576	0.304470
$267$	−2.27330	−0.139124
$268$	8.92264	0.545037
$269$	25.1246	1.53187	0.765935	−	0.642918i	$-0.222276\pi$
$269$	25.1246	1.53187	0.765935	+	0.642918i	$0.222276\pi$
$270$	0	0
$271$	−14.0988	−0.856440	−0.428220	−	0.903674i	$-0.640859\pi$
$271$	−14.0988	−0.856440	−0.428220	+	0.903674i	$0.640859\pi$
$272$	0	0
$273$	0.652547	0.0394939
$274$	4.83672	0.292197
$275$	0	0
$276$	−0.901612	−0.0542707
$277$	6.22200	0.373844	0.186922	−	0.982375i	$-0.440149\pi$
$277$	6.22200	0.373844	0.186922	+	0.982375i	$0.440149\pi$
$278$	0.714152	0.0428320
$279$	−22.4472	−1.34388
$280$	0	0
$281$	−5.47822	−0.326803	−0.163402	−	0.986560i	$-0.552247\pi$
$281$	−5.47822	−0.326803	−0.163402	+	0.986560i	$0.552247\pi$
$282$	−0.936120	−0.0557451
$283$	−24.7905	−1.47364	−0.736820	−	0.676089i	$-0.763673\pi$
$283$	−24.7905	−1.47364	−0.736820	+	0.676089i	$0.763673\pi$
$284$	−6.31534	−0.374746
$285$	0	0
$286$	−1.44054	−0.0851808
$287$	28.6753	1.69265
$288$	14.9304	0.879781
$289$	0	0
$290$	0	0
$291$	0.775248	0.0454458
$292$	−8.77305	−0.513404
$293$	24.3230	1.42096	0.710482	−	0.703715i	$-0.248477\pi$
$293$	24.3230	1.42096	0.710482	+	0.703715i	$0.248477\pi$
$294$	−0.683995	−0.0398914
$295$	0	0
$296$	18.4283	1.07112
$297$	3.00030	0.174095
$298$	4.69702	0.272091
$299$	−2.95744	−0.171033
$300$	0	0
$301$	32.0509	1.84738
$302$	−4.32379	−0.248806
$303$	−0.449949	−0.0258489
$304$	6.59389	0.378186
$305$	0	0
$306$	0	0
$307$	0.916042	0.0522813	0.0261406	−	0.999658i	$-0.491678\pi$
$307$	0.916042	0.0522813	0.0261406	+	0.999658i	$0.491678\pi$
$308$	−19.8942	−1.13358
$309$	1.57807	0.0897732
$310$	0	0
$311$	8.52131	0.483199	0.241600	−	0.970376i	$-0.422328\pi$
$311$	8.52131	0.483199	0.241600	+	0.970376i	$0.422328\pi$
$312$	−0.315450	−0.0178589
$313$	−16.9686	−0.959120	−0.479560	−	0.877509i	$-0.659204\pi$
$313$	−16.9686	−0.959120	−0.479560	+	0.877509i	$0.659204\pi$
$314$	−8.85844	−0.499911
$315$	0	0
$316$	−21.7869	−1.22561
$317$	24.8475	1.39558	0.697788	−	0.716304i	$-0.254168\pi$
$317$	24.8475	1.39558	0.697788	+	0.716304i	$0.254168\pi$
$318$	0.306684	0.0171980
$319$	−1.61226	−0.0902695
$320$	0	0
$321$	−0.998826	−0.0557490
$322$	5.81237	0.323911
$323$	0	0
$324$	−15.2968	−0.849821
$325$	0	0
$326$	1.14842	0.0636052
$327$	−2.91860	−0.161399
$328$	−13.8620	−0.765403
$329$	−42.4063	−2.33794
$330$	0	0
$331$	27.8980	1.53341	0.766707	−	0.641997i	$-0.221894\pi$
$331$	27.8980	1.53341	0.766707	+	0.641997i	$0.221894\pi$
$332$	−14.2193	−0.780386
$333$	−29.2395	−1.60231
$334$	4.67338	0.255716
$335$	0	0
$336$	−1.70296	−0.0929040
$337$	15.9499	0.868847	0.434424	−	0.900709i	$-0.356952\pi$
$337$	15.9499	0.868847	0.434424	+	0.900709i	$0.356952\pi$
$338$	6.00609	0.326688
$339$	0.200526	0.0108911
$340$	0	0
$341$	−22.1688	−1.20051
$342$	3.80882	0.205957
$343$	−3.87395	−0.209174
$344$	−15.4938	−0.835372
$345$	0	0
$346$	−4.64213	−0.249563
$347$	26.0814	1.40012	0.700061	−	0.714083i	$-0.253156\pi$
$347$	26.0814	1.40012	0.700061	+	0.714083i	$0.253156\pi$
$348$	−0.164801	−0.00883427
$349$	−22.9432	−1.22812	−0.614061	−	0.789259i	$-0.710465\pi$
$349$	−22.9432	−1.22812	−0.614061	+	0.789259i	$0.710465\pi$
$350$	0	0
$351$	1.00597	0.0536947
$352$	14.7452	0.785920
$353$	3.96701	0.211142	0.105571	−	0.994412i	$-0.466333\pi$
$353$	3.96701	0.211142	0.105571	+	0.994412i	$0.466333\pi$
$354$	−0.0261450	−0.00138959
$355$	0	0
$356$	23.2377	1.23160
$357$	0	0
$358$	0.967265	0.0511215
$359$	18.8044	0.992457	0.496228	−	0.868192i	$-0.334718\pi$
$359$	18.8044	0.992457	0.496228	+	0.868192i	$0.334718\pi$
$360$	0	0
$361$	−12.4023	−0.652752
$362$	−8.43701	−0.443439
$363$	−0.409834	−0.0215107
$364$	−6.67034	−0.349621
$365$	0	0
$366$	−0.581505	−0.0303957
$367$	−7.50782	−0.391905	−0.195952	−	0.980613i	$-0.562780\pi$
$367$	−7.50782	−0.391905	−0.195952	+	0.980613i	$0.562780\pi$
$368$	7.71807	0.402332
$369$	21.9944	1.14498
$370$	0	0
$371$	13.8928	0.721279
$372$	−2.26603	−0.117488
$373$	−18.5763	−0.961844	−0.480922	−	0.876763i	$-0.659698\pi$
$373$	−18.5763	−0.961844	−0.480922	+	0.876763i	$0.659698\pi$
$374$	0	0
$375$	0	0
$376$	20.4998	1.05720
$377$	−0.540576	−0.0278411
$378$	−1.97707	−0.101689
$379$	−14.6872	−0.754430	−0.377215	−	0.926126i	$-0.623118\pi$
$379$	−14.6872	−0.754430	−0.377215	+	0.926126i	$0.623118\pi$
$380$	0	0
$381$	−1.14044	−0.0584266
$382$	8.31568	0.425467
$383$	21.3063	1.08870	0.544351	−	0.838858i	$-0.316776\pi$
$383$	21.3063	1.08870	0.544351	+	0.838858i	$0.316776\pi$
$384$	1.94617	0.0993150
$385$	0	0
$386$	7.07049	0.359878
$387$	24.5835	1.24965
$388$	−7.92459	−0.402310
$389$	4.77866	0.242288	0.121144	−	0.992635i	$-0.461344\pi$
$389$	4.77866	0.242288	0.121144	+	0.992635i	$0.461344\pi$
$390$	0	0
$391$	0	0
$392$	14.9786	0.756534
$393$	−2.10400	−0.106133
$394$	−9.13354	−0.460141
$395$	0	0
$396$	−15.2592	−0.766803
$397$	32.2929	1.62073	0.810367	−	0.585922i	$-0.199268\pi$
$397$	32.2929	1.62073	0.810367	+	0.585922i	$0.199268\pi$
$398$	2.72432	0.136558
$399$	−1.70395	−0.0853040
$400$	0	0
$401$	24.8497	1.24094	0.620468	−	0.784232i	$-0.286943\pi$
$401$	24.8497	1.24094	0.620468	+	0.784232i	$0.286943\pi$
$402$	0.435710	0.0217312
$403$	−7.43297	−0.370262
$404$	4.59939	0.228828
$405$	0	0
$406$	1.06241	0.0527267
$407$	−28.8768	−1.43137
$408$	0	0
$409$	−18.4025	−0.909945	−0.454973	−	0.890505i	$-0.650351\pi$
$409$	−18.4025	−0.909945	−0.454973	+	0.890505i	$0.650351\pi$
$410$	0	0
$411$	−1.65967	−0.0818653
$412$	−16.1310	−0.794719
$413$	−1.18437	−0.0582791
$414$	4.45818	0.219107
$415$	0	0
$416$	4.94390	0.242395
$417$	−0.245053	−0.0120003
$418$	3.76157	0.183984
$419$	3.03474	0.148257	0.0741284	−	0.997249i	$-0.476383\pi$
$419$	3.03474	0.148257	0.0741284	+	0.997249i	$0.476383\pi$
$420$	0	0
$421$	−31.0514	−1.51335	−0.756677	−	0.653788i	$-0.773179\pi$
$421$	−31.0514	−1.51335	−0.756677	+	0.653788i	$0.773179\pi$
$422$	2.75004	0.133870
$423$	−32.5263	−1.58148
$424$	−6.71598	−0.326157
$425$	0	0
$426$	−0.308390	−0.0149415
$427$	−26.3422	−1.27479
$428$	10.2100	0.493519
$429$	0.494304	0.0238652
$430$	0	0
$431$	−35.5829	−1.71397	−0.856983	−	0.515344i	$-0.827664\pi$
$431$	−35.5829	−1.71397	−0.856983	+	0.515344i	$0.827664\pi$
$432$	−2.62529	−0.126309
$433$	−3.29892	−0.158536	−0.0792680	−	0.996853i	$-0.525258\pi$
$433$	−3.29892	−0.158536	−0.0792680	+	0.996853i	$0.525258\pi$
$434$	14.6083	0.701220
$435$	0	0
$436$	29.8339	1.42879
$437$	7.72254	0.369419
$438$	−0.428405	−0.0204700
$439$	3.31990	0.158450	0.0792250	−	0.996857i	$-0.474755\pi$
$439$	3.31990	0.158450	0.0792250	+	0.996857i	$0.474755\pi$
$440$	0	0
$441$	−23.7660	−1.13172
$442$	0	0
$443$	−5.62292	−0.267153	−0.133577	−	0.991039i	$-0.542646\pi$
$443$	−5.62292	−0.267153	−0.133577	+	0.991039i	$0.542646\pi$
$444$	−2.95170	−0.140082
$445$	0	0
$446$	−13.2868	−0.629150
$447$	−1.61173	−0.0762321
$448$	10.1685	0.480417
$449$	−6.13222	−0.289397	−0.144699	−	0.989476i	$-0.546221\pi$
$449$	−6.13222	−0.289397	−0.144699	+	0.989476i	$0.546221\pi$
$450$	0	0
$451$	21.7216	1.02283
$452$	−2.04978	−0.0964134
$453$	1.48366	0.0697084
$454$	3.82853	0.179682
$455$	0	0
$456$	0.823711	0.0385738
$457$	−28.3934	−1.32819	−0.664093	−	0.747650i	$-0.731182\pi$
$457$	−28.3934	−1.32819	−0.664093	+	0.747650i	$0.731182\pi$
$458$	4.65896	0.217699
$459$	0	0
$460$	0	0
$461$	−20.8024	−0.968863	−0.484432	−	0.874829i	$-0.660974\pi$
$461$	−20.8024	−0.968863	−0.484432	+	0.874829i	$0.660974\pi$
$462$	−0.971474	−0.0451971
$463$	−31.4596	−1.46205	−0.731025	−	0.682351i	$-0.760958\pi$
$463$	−31.4596	−1.46205	−0.731025	+	0.682351i	$0.760958\pi$
$464$	1.41075	0.0654923
$465$	0	0
$466$	−1.94622	−0.0901567
$467$	1.48589	0.0687586	0.0343793	−	0.999409i	$-0.489055\pi$
$467$	1.48589	0.0687586	0.0343793	+	0.999409i	$0.489055\pi$
$468$	−5.11625	−0.236499
$469$	19.7377	0.911402
$470$	0	0
$471$	3.03967	0.140061
$472$	0.572542	0.0263534
$473$	24.2785	1.11633
$474$	−1.06390	−0.0488663
$475$	0	0
$476$	0	0
$477$	10.6560	0.487905
$478$	11.2959	0.516664
$479$	14.6332	0.668608	0.334304	−	0.942465i	$-0.391499\pi$
$479$	14.6332	0.668608	0.334304	+	0.942465i	$0.391499\pi$
$480$	0	0
$481$	−9.68209	−0.441465
$482$	1.44356	0.0657524
$483$	−1.99445	−0.0907505
$484$	4.18933	0.190424
$485$	0	0
$486$	−2.27839	−0.103350
$487$	−3.37468	−0.152921	−0.0764606	−	0.997073i	$-0.524362\pi$
$487$	−3.37468	−0.152921	−0.0764606	+	0.997073i	$0.524362\pi$
$488$	12.7342	0.576450
$489$	−0.394068	−0.0178204
$490$	0	0
$491$	7.57696	0.341943	0.170972	−	0.985276i	$-0.445309\pi$
$491$	7.57696	0.341943	0.170972	+	0.985276i	$0.445309\pi$
$492$	2.22032	0.100100
$493$	0	0
$494$	1.26122	0.0567448
$495$	0	0
$496$	19.3979	0.870992
$497$	−13.9701	−0.626644
$498$	−0.694356	−0.0311149
$499$	−8.52931	−0.381825	−0.190912	−	0.981607i	$-0.561145\pi$
$499$	−8.52931	−0.381825	−0.190912	+	0.981607i	$0.561145\pi$
$500$	0	0
$501$	−1.60362	−0.0716443
$502$	6.67620	0.297973
$503$	19.2282	0.857344	0.428672	−	0.903460i	$-0.358982\pi$
$503$	19.2282	0.857344	0.428672	+	0.903460i	$0.358982\pi$
$504$	21.5413	0.959524
$505$	0	0
$506$	4.40287	0.195732
$507$	−2.06092	−0.0915287
$508$	11.6576	0.517223
$509$	−5.89076	−0.261104	−0.130552	−	0.991441i	$-0.541675\pi$
$509$	−5.89076	−0.261104	−0.130552	+	0.991441i	$0.541675\pi$
$510$	0	0
$511$	−19.4068	−0.858505
$512$	−22.5148	−0.995024
$513$	−2.62681	−0.115977
$514$	−3.65864	−0.161376
$515$	0	0
$516$	2.48168	0.109250
$517$	−32.1228	−1.41276
$518$	19.0286	0.836067
$519$	1.59290	0.0699203
$520$	0	0
$521$	−25.7911	−1.12993	−0.564965	−	0.825115i	$-0.691110\pi$
$521$	−25.7911	−1.12993	−0.564965	+	0.825115i	$0.691110\pi$
$522$	0.814888	0.0356667
$523$	23.1957	1.01428	0.507138	−	0.861865i	$-0.330703\pi$
$523$	23.1957	1.01428	0.507138	+	0.861865i	$0.330703\pi$
$524$	21.5071	0.939543
$525$	0	0
$526$	12.0362	0.524803
$527$	0	0
$528$	−1.28999	−0.0561397
$529$	−13.9609	−0.606994
$530$	0	0
$531$	−0.908431	−0.0394226
$532$	17.4177	0.755155
$533$	7.28302	0.315462
$534$	1.13474	0.0491052
$535$	0	0
$536$	−9.54148	−0.412129
$537$	−0.331906	−0.0143228
$538$	−12.5412	−0.540689
$539$	−23.4712	−1.01098
$540$	0	0
$541$	0.319010	0.0137153	0.00685766	−	0.999976i	$-0.497817\pi$
$541$	0.319010	0.0137153	0.00685766	+	0.999976i	$0.497817\pi$
$542$	7.03756	0.302289
$543$	2.89506	0.124239
$544$	0	0
$545$	0	0
$546$	−0.325725	−0.0139398
$547$	29.2638	1.25123	0.625616	−	0.780131i	$-0.284848\pi$
$547$	29.2638	1.25123	0.625616	+	0.780131i	$0.284848\pi$
$548$	16.9651	0.724714
$549$	−20.2049	−0.862323
$550$	0	0
$551$	1.41157	0.0601347
$552$	0.964144	0.0410367
$553$	−48.1946	−2.04944
$554$	−3.10578	−0.131952
$555$	0	0
$556$	2.50494	0.106233
$557$	4.51904	0.191478	0.0957390	−	0.995406i	$-0.469479\pi$
$557$	4.51904	0.191478	0.0957390	+	0.995406i	$0.469479\pi$
$558$	11.2048	0.474336
$559$	8.14035	0.344300
$560$	0	0
$561$	0	0
$562$	2.73451	0.115348
$563$	−24.5112	−1.03302	−0.516511	−	0.856280i	$-0.672770\pi$
$563$	−24.5112	−1.03302	−0.516511	+	0.856280i	$0.672770\pi$
$564$	−3.28350	−0.138260
$565$	0	0
$566$	12.3744	0.520135
$567$	−33.8379	−1.42106
$568$	6.75334	0.283364
$569$	6.12942	0.256958	0.128479	−	0.991712i	$-0.458990\pi$
$569$	6.12942	0.256958	0.128479	+	0.991712i	$0.458990\pi$
$570$	0	0
$571$	−0.524413	−0.0219460	−0.0109730	−	0.999940i	$-0.503493\pi$
$571$	−0.524413	−0.0219460	−0.0109730	+	0.999940i	$0.503493\pi$
$572$	−5.05278	−0.211268
$573$	−2.85343	−0.119204
$574$	−14.3136	−0.597437
$575$	0	0
$576$	7.79940	0.324975
$577$	39.5403	1.64608	0.823041	−	0.567982i	$-0.192276\pi$
$577$	39.5403	1.64608	0.823041	+	0.567982i	$0.192276\pi$
$578$	0	0
$579$	−2.42616	−0.100828
$580$	0	0
$581$	−31.4544	−1.30495
$582$	−0.386973	−0.0160406
$583$	10.5238	0.435852
$584$	9.38151	0.388210
$585$	0	0
$586$	−12.1411	−0.501543
$587$	−10.9375	−0.451439	−0.225719	−	0.974192i	$-0.572473\pi$
$587$	−10.9375	−0.451439	−0.225719	+	0.974192i	$0.572473\pi$
$588$	−2.39916	−0.0989397
$589$	19.4091	0.799740
$590$	0	0
$591$	3.13407	0.128918
$592$	25.2675	1.03849
$593$	−3.84749	−0.157998	−0.0789988	−	0.996875i	$-0.525172\pi$
$593$	−3.84749	−0.157998	−0.0789988	+	0.996875i	$0.525172\pi$
$594$	−1.49763	−0.0614485
$595$	0	0
$596$	16.4751	0.674847
$597$	−0.934820	−0.0382596
$598$	1.47624	0.0603679
$599$	−35.5384	−1.45206	−0.726030	−	0.687663i	$-0.758637\pi$
$599$	−35.5384	−1.45206	−0.726030	+	0.687663i	$0.758637\pi$
$600$	0	0
$601$	18.8207	0.767714	0.383857	−	0.923393i	$-0.374596\pi$
$601$	18.8207	0.767714	0.383857	+	0.923393i	$0.374596\pi$
$602$	−15.9985	−0.652051
$603$	15.1391	0.616513
$604$	−15.1660	−0.617095
$605$	0	0
$606$	0.224597	0.00912363
$607$	−28.2627	−1.14715	−0.573574	−	0.819154i	$-0.694443\pi$
$607$	−28.2627	−1.14715	−0.573574	+	0.819154i	$0.694443\pi$
$608$	−12.9096	−0.523555
$609$	−0.364555	−0.0147725
$610$	0	0
$611$	−10.7705	−0.435726
$612$	0	0
$613$	15.7155	0.634743	0.317371	−	0.948301i	$-0.397200\pi$
$613$	15.7155	0.634743	0.317371	+	0.948301i	$0.397200\pi$
$614$	−0.457252	−0.0184532
$615$	0	0
$616$	21.2740	0.857155
$617$	27.3622	1.10156	0.550781	−	0.834650i	$-0.314330\pi$
$617$	27.3622	1.10156	0.550781	+	0.834650i	$0.314330\pi$
$618$	−0.787710	−0.0316863
$619$	−7.56352	−0.304004	−0.152002	−	0.988380i	$-0.548572\pi$
$619$	−7.56352	−0.304004	−0.152002	+	0.988380i	$0.548572\pi$
$620$	0	0
$621$	−3.07465	−0.123382
$622$	−4.25350	−0.170550
$623$	51.4040	2.05946
$624$	−0.432521	−0.0173147
$625$	0	0
$626$	8.47004	0.338531
$627$	−1.29074	−0.0515472
$628$	−31.0716	−1.23989
$629$	0	0
$630$	0	0
$631$	−10.1561	−0.404306	−0.202153	−	0.979354i	$-0.564794\pi$
$631$	−10.1561	−0.404306	−0.202153	+	0.979354i	$0.564794\pi$
$632$	23.2979	0.926743
$633$	−0.943643	−0.0375065
$634$	−12.4029	−0.492582
$635$	0	0
$636$	1.07571	0.0426548
$637$	−7.86965	−0.311807
$638$	0.804779	0.0318615
$639$	−10.7153	−0.423890
$640$	0	0
$641$	−31.9499	−1.26194	−0.630972	−	0.775805i	$-0.717344\pi$
$641$	−31.9499	−1.26194	−0.630972	+	0.775805i	$0.717344\pi$
$642$	0.498574	0.0196772
$643$	−35.2573	−1.39041	−0.695207	−	0.718810i	$-0.744687\pi$
$643$	−35.2573	−1.39041	−0.695207	+	0.718810i	$0.744687\pi$
$644$	20.3873	0.803371
$645$	0	0
$646$	0	0
$647$	−7.97446	−0.313508	−0.156754	−	0.987638i	$-0.550103\pi$
$647$	−7.97446	−0.313508	−0.156754	+	0.987638i	$0.550103\pi$
$648$	16.3577	0.642591
$649$	−0.897162	−0.0352167
$650$	0	0
$651$	−5.01266	−0.196462
$652$	4.02817	0.157755
$653$	13.9977	0.547773	0.273887	−	0.961762i	$-0.411691\pi$
$653$	13.9977	0.547773	0.273887	+	0.961762i	$0.411691\pi$
$654$	1.45685	0.0569673
$655$	0	0
$656$	−19.0066	−0.742082
$657$	−14.8853	−0.580731
$658$	21.1676	0.825197
$659$	−50.5377	−1.96867	−0.984335	−	0.176309i	$-0.943584\pi$
$659$	−50.5377	−1.96867	−0.984335	+	0.176309i	$0.943584\pi$
$660$	0	0
$661$	−33.8208	−1.31547	−0.657737	−	0.753247i	$-0.728486\pi$
$661$	−33.8208	−1.31547	−0.657737	+	0.753247i	$0.728486\pi$
$662$	−13.9256	−0.541234
$663$	0	0
$664$	15.2055	0.590088
$665$	0	0
$666$	14.5952	0.565553
$667$	1.65222	0.0639742
$668$	16.3922	0.634233
$669$	4.55923	0.176270
$670$	0	0
$671$	−19.9542	−0.770325
$672$	3.33408	0.128615
$673$	0.257259	0.00991660	0.00495830	−	0.999988i	$-0.498422\pi$
$673$	0.257259	0.00991660	0.00495830	+	0.999988i	$0.498422\pi$
$674$	−7.96157	−0.306668
$675$	0	0
$676$	21.0668	0.810260
$677$	3.52328	0.135411	0.0677054	−	0.997705i	$-0.478432\pi$
$677$	3.52328	0.135411	0.0677054	+	0.997705i	$0.478432\pi$
$678$	−0.100095	−0.00384411
$679$	−17.5299	−0.672737
$680$	0	0
$681$	−1.31372	−0.0503417
$682$	11.0658	0.423731
$683$	−16.2284	−0.620961	−0.310480	−	0.950580i	$-0.600490\pi$
$683$	−16.2284	−0.620961	−0.310480	+	0.950580i	$0.600490\pi$
$684$	13.3597	0.510820
$685$	0	0
$686$	1.93372	0.0738299
$687$	−1.59867	−0.0609930
$688$	−21.2440	−0.809919
$689$	3.52853	0.134426
$690$	0	0
$691$	−13.7434	−0.522824	−0.261412	−	0.965227i	$-0.584188\pi$
$691$	−13.7434	−0.522824	−0.261412	+	0.965227i	$0.584188\pi$
$692$	−16.2826	−0.618971
$693$	−33.7547	−1.28224
$694$	−13.0188	−0.494187
$695$	0	0
$696$	0.176231	0.00668002
$697$	0	0
$698$	11.4523	0.433478
$699$	0.667822	0.0252593
$700$	0	0
$701$	−15.8410	−0.598305	−0.299153	−	0.954205i	$-0.596704\pi$
$701$	−15.8410	−0.598305	−0.299153	+	0.954205i	$0.596704\pi$
$702$	−0.502140	−0.0189521
$703$	25.2821	0.953532
$704$	7.70265	0.290305
$705$	0	0
$706$	−1.98017	−0.0745248
$707$	10.1743	0.382643
$708$	−0.0917054	−0.00344650
$709$	26.3076	0.988002	0.494001	−	0.869461i	$-0.335534\pi$
$709$	26.3076	0.988002	0.494001	+	0.869461i	$0.335534\pi$
$710$	0	0
$711$	−36.9660	−1.38633
$712$	−24.8494	−0.931272
$713$	22.7182	0.850802
$714$	0	0
$715$	0	0
$716$	3.39274	0.126793
$717$	−3.87607	−0.144755
$718$	−9.38640	−0.350297
$719$	28.4854	1.06233	0.531164	−	0.847269i	$-0.321755\pi$
$719$	28.4854	1.06233	0.531164	+	0.847269i	$0.321755\pi$
$720$	0	0
$721$	−35.6833	−1.32892
$722$	6.19073	0.230395
$723$	−0.495341	−0.0184219
$724$	−29.5934	−1.09983
$725$	0	0
$726$	0.204573	0.00759241
$727$	4.70692	0.174570	0.0872850	−	0.996183i	$-0.472181\pi$
$727$	4.70692	0.174570	0.0872850	+	0.996183i	$0.472181\pi$
$728$	7.13297	0.264365
$729$	−25.4287	−0.941803
$730$	0	0
$731$	0	0
$732$	−2.03967	−0.0753882
$733$	−26.7866	−0.989386	−0.494693	−	0.869068i	$-0.664719\pi$
$733$	−26.7866	−0.989386	−0.494693	+	0.869068i	$0.664719\pi$
$734$	3.74761	0.138327
$735$	0	0
$736$	−15.1106	−0.556984
$737$	14.9513	0.550739
$738$	−10.9787	−0.404133
$739$	−24.4868	−0.900763	−0.450381	−	0.892836i	$-0.648712\pi$
$739$	−24.4868	−0.900763	−0.450381	+	0.892836i	$0.648712\pi$
$740$	0	0
$741$	−0.432772	−0.0158983
$742$	−6.93474	−0.254582
$743$	−29.3896	−1.07820	−0.539100	−	0.842242i	$-0.681235\pi$
$743$	−29.3896	−1.07820	−0.539100	+	0.842242i	$0.681235\pi$
$744$	2.42319	0.0888386
$745$	0	0
$746$	9.27255	0.339492
$747$	−24.1260	−0.882725
$748$	0	0
$749$	22.5855	0.825255
$750$	0	0
$751$	11.5338	0.420874	0.210437	−	0.977607i	$-0.432511\pi$
$751$	11.5338	0.420874	0.210437	+	0.977607i	$0.432511\pi$
$752$	28.1078	1.02499
$753$	−2.29086	−0.0834836
$754$	0.269834	0.00982678
$755$	0	0
$756$	−6.93470	−0.252213
$757$	25.7257	0.935018	0.467509	−	0.883988i	$-0.345152\pi$
$757$	25.7257	0.935018	0.467509	+	0.883988i	$0.345152\pi$
$758$	7.33126	0.266283
$759$	−1.51080	−0.0548384
$760$	0	0
$761$	19.0450	0.690379	0.345190	−	0.938533i	$-0.387815\pi$
$761$	19.0450	0.690379	0.345190	+	0.938533i	$0.387815\pi$
$762$	0.569264	0.0206222
$763$	65.9953	2.38919
$764$	29.1678	1.05525
$765$	0	0
$766$	−10.6353	−0.384268
$767$	−0.300809	−0.0108616
$768$	−0.0720615	−0.00260029
$769$	23.2434	0.838178	0.419089	−	0.907945i	$-0.362349\pi$
$769$	23.2434	0.838178	0.419089	+	0.907945i	$0.362349\pi$
$770$	0	0
$771$	1.25542	0.0452129
$772$	24.8002	0.892579
$773$	39.0680	1.40518	0.702589	−	0.711596i	$-0.252027\pi$
$773$	39.0680	1.40518	0.702589	+	0.711596i	$0.252027\pi$
$774$	−12.2711	−0.441076
$775$	0	0
$776$	8.47421	0.304207
$777$	−6.52943	−0.234242
$778$	−2.38532	−0.0855179
$779$	−19.0176	−0.681376
$780$	0	0
$781$	−10.5824	−0.378666
$782$	0	0
$783$	−0.562000	−0.0200843
$784$	20.5375	0.733483
$785$	0	0
$786$	1.05023	0.0374606
$787$	−41.9094	−1.49391	−0.746955	−	0.664875i	$-0.768485\pi$
$787$	−41.9094	−1.49391	−0.746955	+	0.664875i	$0.768485\pi$
$788$	−32.0365	−1.14125
$789$	−4.13009	−0.147035
$790$	0	0
$791$	−4.53429	−0.161221
$792$	16.3175	0.579818
$793$	−6.69045	−0.237585
$794$	−16.1193	−0.572054
$795$	0	0
$796$	9.55574	0.338694
$797$	−40.3705	−1.43000	−0.714998	−	0.699126i	$-0.753573\pi$
$797$	−40.3705	−1.43000	−0.714998	+	0.699126i	$0.753573\pi$
$798$	0.850542	0.0301089
$799$	0	0
$800$	0	0
$801$	39.4277	1.39311
$802$	−12.4040	−0.438000
$803$	−14.7006	−0.518774
$804$	1.52828	0.0538983
$805$	0	0
$806$	3.71024	0.130688
$807$	4.30336	0.151486
$808$	−4.91839	−0.173028
$809$	26.3355	0.925906	0.462953	−	0.886383i	$-0.346790\pi$
$809$	26.3355	0.925906	0.462953	+	0.886383i	$0.346790\pi$
$810$	0	0
$811$	−17.6994	−0.621511	−0.310756	−	0.950490i	$-0.600582\pi$
$811$	−17.6994	−0.621511	−0.310756	+	0.950490i	$0.600582\pi$
$812$	3.72649	0.130774
$813$	−2.41486	−0.0846928
$814$	14.4141	0.505215
$815$	0	0
$816$	0	0
$817$	−21.2563	−0.743663
$818$	9.18580	0.321174
$819$	−11.3176	−0.395470
$820$	0	0
$821$	18.9891	0.662723	0.331362	−	0.943504i	$-0.392492\pi$
$821$	18.9891	0.662723	0.331362	+	0.943504i	$0.392492\pi$
$822$	0.828440	0.0288952
$823$	38.3769	1.33773	0.668867	−	0.743382i	$-0.266780\pi$
$823$	38.3769	1.33773	0.668867	+	0.743382i	$0.266780\pi$
$824$	17.2498	0.600926
$825$	0	0
$826$	0.591191	0.0205702
$827$	−48.6653	−1.69226	−0.846129	−	0.532978i	$-0.821073\pi$
$827$	−48.6653	−1.69226	−0.846129	+	0.532978i	$0.821073\pi$
$828$	15.6374	0.543436
$829$	51.7088	1.79592	0.897960	−	0.440076i	$-0.145049\pi$
$829$	51.7088	1.79592	0.897960	+	0.440076i	$0.145049\pi$
$830$	0	0
$831$	1.06571	0.0369691
$832$	2.58262	0.0895362
$833$	0	0
$834$	0.122321	0.00423562
$835$	0	0
$836$	13.1940	0.456322
$837$	−7.72755	−0.267103
$838$	−1.51482	−0.0523287
$839$	36.4513	1.25844	0.629219	−	0.777228i	$-0.283375\pi$
$839$	36.4513	1.25844	0.629219	+	0.777228i	$0.283375\pi$
$840$	0	0
$841$	−28.6980	−0.989586
$842$	15.4997	0.534153
$843$	−0.938316	−0.0323173
$844$	9.64593	0.332027
$845$	0	0
$846$	16.2358	0.558200
$847$	9.26717	0.318424
$848$	−9.20844	−0.316219
$849$	−4.24614	−0.145727
$850$	0	0
$851$	29.5924	1.01441
$852$	−1.08170	−0.0370584
$853$	−0.935836	−0.0320424	−0.0160212	−	0.999872i	$-0.505100\pi$
$853$	−0.935836	−0.0320424	−0.0160212	+	0.999872i	$0.505100\pi$
$854$	13.1490	0.449949
$855$	0	0
$856$	−10.9181	−0.373174
$857$	30.4722	1.04091	0.520455	−	0.853889i	$-0.325762\pi$
$857$	30.4722	1.04091	0.520455	+	0.853889i	$0.325762\pi$
$858$	−0.246737	−0.00842347
$859$	−0.987092	−0.0336792	−0.0168396	−	0.999858i	$-0.505360\pi$
$859$	−0.987092	−0.0336792	−0.0168396	+	0.999858i	$0.505360\pi$
$860$	0	0
$861$	4.91154	0.167385
$862$	17.7616	0.604961
$863$	−26.0032	−0.885159	−0.442580	−	0.896729i	$-0.645937\pi$
$863$	−26.0032	−0.885159	−0.442580	+	0.896729i	$0.645937\pi$
$864$	5.13984	0.174861
$865$	0	0
$866$	1.64669	0.0559568
$867$	0	0
$868$	51.2395	1.73918
$869$	−36.5074	−1.23843
$870$	0	0
$871$	5.01302	0.169860
$872$	−31.9031	−1.08037
$873$	−13.4457	−0.455069
$874$	−3.85479	−0.130390
$875$	0	0
$876$	−1.50266	−0.0507701
$877$	−33.8117	−1.14174	−0.570870	−	0.821040i	$-0.693394\pi$
$877$	−33.8117	−1.14174	−0.570870	+	0.821040i	$0.693394\pi$
$878$	−1.65716	−0.0559265
$879$	4.16607	0.140518
$880$	0	0
$881$	18.9539	0.638573	0.319287	−	0.947658i	$-0.396557\pi$
$881$	18.9539	0.638573	0.319287	+	0.947658i	$0.396557\pi$
$882$	11.8631	0.399450
$883$	38.1002	1.28217	0.641087	−	0.767468i	$-0.278484\pi$
$883$	38.1002	1.28217	0.641087	+	0.767468i	$0.278484\pi$
$884$	0	0
$885$	0	0
$886$	2.80674	0.0942943
$887$	−43.1234	−1.44794	−0.723972	−	0.689830i	$-0.757685\pi$
$887$	−43.1234	−1.44794	−0.723972	+	0.689830i	$0.757685\pi$
$888$	3.15642	0.105923
$889$	25.7877	0.864891
$890$	0	0
$891$	−25.6322	−0.858711
$892$	−46.6045	−1.56043
$893$	28.1241	0.941136
$894$	0.804511	0.0269069
$895$	0	0
$896$	−44.0068	−1.47016
$897$	−0.506554	−0.0169134
$898$	3.06096	0.102146
$899$	4.15254	0.138495
$900$	0	0
$901$	0	0
$902$	−10.8425	−0.361017
$903$	5.48971	0.182686
$904$	2.19194	0.0729029
$905$	0	0
$906$	−0.740585	−0.0246043
$907$	−15.1988	−0.504668	−0.252334	−	0.967640i	$-0.581198\pi$
$907$	−15.1988	−0.504668	−0.252334	+	0.967640i	$0.581198\pi$
$908$	13.4288	0.445651
$909$	7.80382	0.258836
$910$	0	0
$911$	23.3400	0.773289	0.386645	−	0.922229i	$-0.373634\pi$
$911$	23.3400	0.773289	0.386645	+	0.922229i	$0.373634\pi$
$912$	1.12941	0.0373985
$913$	−23.8267	−0.788550
$914$	14.1729	0.468796
$915$	0	0
$916$	16.3416	0.539942
$917$	47.5757	1.57109
$918$	0	0
$919$	−11.1867	−0.369017	−0.184508	−	0.982831i	$-0.559069\pi$
$919$	−11.1867	−0.369017	−0.184508	+	0.982831i	$0.559069\pi$
$920$	0	0
$921$	0.156901	0.00517006
$922$	10.3837	0.341970
$923$	−3.54816	−0.116789
$924$	−3.40751	−0.112099
$925$	0	0
$926$	15.7034	0.516045
$927$	−27.3697	−0.898938
$928$	−2.76199	−0.0906667
$929$	−0.886506	−0.0290853	−0.0145427	−	0.999894i	$-0.504629\pi$
$929$	−0.886506	−0.0290853	−0.0145427	+	0.999894i	$0.504629\pi$
$930$	0	0
$931$	20.5494	0.673480
$932$	−6.82648	−0.223609
$933$	1.45954	0.0477832
$934$	−0.741696	−0.0242690
$935$	0	0
$936$	5.47110	0.178828
$937$	7.99596	0.261217	0.130608	−	0.991434i	$-0.458307\pi$
$937$	7.99596	0.261217	0.130608	+	0.991434i	$0.458307\pi$
$938$	−9.85228	−0.321688
$939$	−2.90640	−0.0948467
$940$	0	0
$941$	46.3645	1.51144	0.755720	−	0.654894i	$-0.227287\pi$
$941$	46.3645	1.51144	0.755720	+	0.654894i	$0.227287\pi$
$942$	−1.51728	−0.0494358
$943$	−22.2599	−0.724881
$944$	0.785026	0.0255504
$945$	0	0
$946$	−12.1189	−0.394019
$947$	−31.4098	−1.02068	−0.510341	−	0.859972i	$-0.670481\pi$
$947$	−31.4098	−1.02068	−0.510341	+	0.859972i	$0.670481\pi$
$948$	−3.73169	−0.121200
$949$	−4.92898	−0.160001
$950$	0	0
$951$	4.25591	0.138007
$952$	0	0
$953$	−53.1529	−1.72179	−0.860896	−	0.508781i	$-0.830096\pi$
$953$	−53.1529	−1.72179	−0.860896	+	0.508781i	$0.830096\pi$
$954$	−5.31905	−0.172211
$955$	0	0
$956$	39.6212	1.28144
$957$	−0.276151	−0.00892669
$958$	−7.30432	−0.235992
$959$	37.5284	1.21186
$960$	0	0
$961$	26.0978	0.841864
$962$	4.83292	0.155819
$963$	17.3234	0.558239
$964$	5.06338	0.163081
$965$	0	0
$966$	0.995549	0.0320313
$967$	29.3267	0.943083	0.471541	−	0.881844i	$-0.343698\pi$
$967$	29.3267	0.943083	0.471541	+	0.881844i	$0.343698\pi$
$968$	−4.47988	−0.143989
$969$	0	0
$970$	0	0
$971$	−33.7011	−1.08152	−0.540761	−	0.841177i	$-0.681864\pi$
$971$	−33.7011	−1.08152	−0.540761	+	0.841177i	$0.681864\pi$
$972$	−7.99160	−0.256331
$973$	5.54115	0.177641
$974$	1.68451	0.0539751
$975$	0	0
$976$	17.4602	0.558886
$977$	−57.9778	−1.85488	−0.927438	−	0.373977i	$-0.877994\pi$
$977$	−57.9778	−1.85488	−0.927438	+	0.373977i	$0.877994\pi$
$978$	0.196703	0.00628988
$979$	38.9386	1.24448
$980$	0	0
$981$	50.6195	1.61615
$982$	−3.78212	−0.120692
$983$	−20.5124	−0.654243	−0.327121	−	0.944982i	$-0.606079\pi$
$983$	−20.5124	−0.654243	−0.327121	+	0.944982i	$0.606079\pi$
$984$	−2.37431	−0.0756902
$985$	0	0
$986$	0	0
$987$	−7.26341	−0.231197
$988$	4.42380	0.140740
$989$	−24.8802	−0.791145
$990$	0	0
$991$	30.6737	0.974384	0.487192	−	0.873295i	$-0.338021\pi$
$991$	30.6737	0.974384	0.487192	+	0.873295i	$0.338021\pi$
$992$	−37.9776	−1.20579
$993$	4.77841	0.151638
$994$	6.97332	0.221180
$995$	0	0
$996$	−2.43550	−0.0771718
$997$	−4.93382	−0.156256	−0.0781279	−	0.996943i	$-0.524894\pi$
$997$	−4.93382	−0.156256	−0.0781279	+	0.996943i	$0.524894\pi$
$998$	4.25750	0.134769
$999$	−10.0658	−0.318468

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7225.2.a.by.1.10		24
5.2	odd	4		1445.2.b.i.579.9		24
5.3	odd	4		1445.2.b.i.579.16		24
5.4	even	2	inner	7225.2.a.by.1.15		24
17.11	odd	16		425.2.m.e.376.3		24
17.14	odd	16		425.2.m.e.26.3		24
17.16	even	2	inner	7225.2.a.by.1.9		24
85.14	odd	16		425.2.m.e.26.4		24
85.28	even	16		85.2.m.a.19.3	yes	24
85.33	odd	4		1445.2.b.i.579.15		24
85.48	even	16		85.2.m.a.9.4	yes	24
85.62	even	16		85.2.m.a.19.4	yes	24
85.67	odd	4		1445.2.b.i.579.10		24
85.79	odd	16		425.2.m.e.376.4		24
85.82	even	16		85.2.m.a.9.3	✓	24
85.84	even	2	inner	7225.2.a.by.1.16		24
255.62	odd	16		765.2.bh.b.19.3		24
255.113	odd	16		765.2.bh.b.19.4		24
255.167	odd	16		765.2.bh.b.604.4		24
255.218	odd	16		765.2.bh.b.604.3		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
85.2.m.a.9.3	✓	24	85.82	even	16
85.2.m.a.9.4	yes	24	85.48	even	16
85.2.m.a.19.3	yes	24	85.28	even	16
85.2.m.a.19.4	yes	24	85.62	even	16
425.2.m.e.26.3		24	17.14	odd	16
425.2.m.e.26.4		24	85.14	odd	16
425.2.m.e.376.3		24	17.11	odd	16
425.2.m.e.376.4		24	85.79	odd	16
765.2.bh.b.19.3		24	255.62	odd	16
765.2.bh.b.19.4		24	255.113	odd	16
765.2.bh.b.604.3		24	255.218	odd	16
765.2.bh.b.604.4		24	255.167	odd	16
1445.2.b.i.579.9		24	5.2	odd	4
1445.2.b.i.579.10		24	85.67	odd	4
1445.2.b.i.579.15		24	85.33	odd	4
1445.2.b.i.579.16		24	5.3	odd	4
7225.2.a.by.1.9		24	17.16	even	2	inner
7225.2.a.by.1.10		24	1.1	even	1	trivial
7225.2.a.by.1.15		24	5.4	even	2	inner
7225.2.a.by.1.16		24	85.84	even	2	inner

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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	7225.2.a.by.1.10

Level	$7225$
Weight	$2$
Character	7225.1
Self dual	yes
Analytic conductor	$57.692$
Analytic rank	$1$
Dimension	$24$
Inner twists	$4$