LMFDB - Embedded newform 8281.2.a.ca.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8281,2,Mod(1,8281)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8281, 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("8281.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$8281 = 7^{2} \cdot 13^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	8281.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:	$66.1241179138$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.6995813.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{6} - x^{5} - 6x^{4} + 4x^{3} + 7x^{2} - x - 1$ x^6 - x^5 - 6x^4 + 4x^3 + 7*x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 91)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.33401$ of defining polynomial
Character	$\chi$	$=$	8281.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.90556 q^{2} -0.428448 q^{3} +1.63116 q^{4} +1.47313 q^{5} +0.816433 q^{6} +0.702849 q^{8} -2.81643 q^{9} -2.80714 q^{10} -4.39361 q^{11} -0.698866 q^{12} -0.631159 q^{15} -4.60164 q^{16} +1.20271 q^{17} +5.36688 q^{18} -3.24209 q^{19} +2.40291 q^{20} +8.37230 q^{22} -4.43710 q^{23} -0.301134 q^{24} -2.82989 q^{25} +2.49204 q^{27} +0.167561 q^{29} +1.20271 q^{30} -5.24543 q^{31} +7.36300 q^{32} +1.88243 q^{33} -2.29184 q^{34} -4.59405 q^{36} +7.05055 q^{37} +6.17800 q^{38} +1.03539 q^{40} -5.16390 q^{41} +0.0227504 q^{43} -7.16668 q^{44} -4.14897 q^{45} +8.45516 q^{46} -11.6836 q^{47} +1.97156 q^{48} +5.39252 q^{50} -0.515299 q^{51} -0.141786 q^{53} -4.74873 q^{54} -6.47236 q^{55} +1.38907 q^{57} -0.319298 q^{58} +5.34354 q^{59} -1.02952 q^{60} -11.5457 q^{61} +9.99549 q^{62} -4.82736 q^{64} -3.58709 q^{66} +4.13546 q^{67} +1.96181 q^{68} +1.90107 q^{69} -9.96971 q^{71} -1.97953 q^{72} -15.2416 q^{73} -13.4352 q^{74} +1.21246 q^{75} -5.28837 q^{76} +0.774501 q^{79} -6.77881 q^{80} +7.38159 q^{81} +9.84011 q^{82} +16.0186 q^{83} +1.77175 q^{85} -0.0433522 q^{86} -0.0717913 q^{87} -3.08805 q^{88} -6.55760 q^{89} +7.90611 q^{90} -7.23762 q^{92} +2.24739 q^{93} +22.2637 q^{94} -4.77602 q^{95} -3.15466 q^{96} -3.49166 q^{97} +12.3743 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$6 q - 2 q^{2} + q^{3} + 4 q^{4} + q^{5} - 9 q^{6} - 3 q^{8} - 3 q^{9} + 4 q^{10} - 4 q^{11} + 5 q^{12} + 2 q^{15} - 8 q^{16} + 5 q^{17} - 3 q^{18} - q^{19} - q^{20} + 5 q^{22} + q^{23} - 11 q^{24}+ \cdots - 10 q^{99}+O(q^{100})$ 6 * q - 2 * q^2 + q^3 + 4 * q^4 + q^5 - 9 * q^6 - 3 * q^8 - 3 * q^9 + 4 * q^10 - 4 * q^11 + 5 * q^12 + 2 * q^15 - 8 * q^16 + 5 * q^17 - 3 * q^18 - q^19 - q^20 + 5 * q^22 + q^23 - 11 * q^24 - 7 * q^25 + 4 * q^27 - 3 * q^29 + 5 * q^30 + 16 * q^31 - 8 * q^32 + 16 * q^33 - 16 * q^34 + 21 * q^36 + 13 * q^37 - 17 * q^38 - 5 * q^40 - 8 * q^41 + 11 * q^43 - 21 * q^44 - 7 * q^45 - 16 * q^46 - q^47 + 21 * q^48 - 6 * q^50 + 20 * q^51 + 2 * q^53 - 18 * q^54 + 9 * q^55 + 21 * q^57 + 8 * q^58 + 13 * q^59 - 20 * q^60 - 5 * q^61 + 5 * q^62 - 15 * q^64 + 18 * q^66 + 11 * q^67 + 29 * q^68 + 23 * q^69 - 6 * q^71 - 25 * q^72 - 30 * q^73 + 3 * q^74 - 3 * q^75 - 9 * q^76 - 7 * q^79 - 7 * q^80 + 6 * q^81 + q^82 + 27 * q^83 + q^85 + 7 * q^86 + 16 * q^87 + 4 * q^89 + 8 * q^90 + 27 * q^92 + 7 * q^93 + 45 * q^94 + 6 * q^95 + 19 * q^96 - 35 * q^97 - 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

)

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

$2$	−1.90556	−1.34743	−0.673717	−	0.738989i	$-0.735303\pi$
$2$	−1.90556	−1.34743	−0.673717	+	0.738989i	$0.735303\pi$
$3$	−0.428448	−0.247364	−0.123682	−	0.992322i	$-0.539470\pi$
$3$	−0.428448	−0.247364	−0.123682	+	0.992322i	$0.539470\pi$
$4$	1.63116	0.815580
$5$	1.47313	0.658804	0.329402	−	0.944190i	$-0.393153\pi$
$5$	1.47313	0.658804	0.329402	+	0.944190i	$0.393153\pi$
$6$	0.816433	0.333307
$7$	0	0
$7$	0	0
$8$	0.702849	0.248495
$9$	−2.81643	−0.938811
$10$	−2.80714	−0.887695
$11$	−4.39361	−1.32472	−0.662362	−	0.749184i	$-0.730446\pi$
$11$	−4.39361	−1.32472	−0.662362	+	0.749184i	$0.730446\pi$
$12$	−0.698866	−0.201745
$13$	0	0
$13$	0	0
$14$	0	0
$15$	−0.631159	−0.162965
$16$	−4.60164	−1.15041
$17$	1.20271	0.291700	0.145850	−	0.989307i	$-0.453408\pi$
$17$	1.20271	0.291700	0.145850	+	0.989307i	$0.453408\pi$
$18$	5.36688	1.26499
$19$	−3.24209	−0.743787	−0.371893	−	0.928275i	$-0.621291\pi$
$19$	−3.24209	−0.743787	−0.371893	+	0.928275i	$0.621291\pi$
$20$	2.40291	0.537307
$21$	0	0
$22$	8.37230	1.78498
$23$	−4.43710	−0.925200	−0.462600	−	0.886567i	$-0.653083\pi$
$23$	−4.43710	−0.925200	−0.462600	+	0.886567i	$0.653083\pi$
$24$	−0.301134	−0.0614687
$25$	−2.82989	−0.565978
$26$	0	0
$27$	2.49204	0.479593
$28$	0	0
$29$	0.167561	0.0311154	0.0155577	−	0.999879i	$-0.495048\pi$
$29$	0.167561	0.0311154	0.0155577	+	0.999879i	$0.495048\pi$
$30$	1.20271	0.219584
$31$	−5.24543	−0.942108	−0.471054	−	0.882104i	$-0.656126\pi$
$31$	−5.24543	−0.942108	−0.471054	+	0.882104i	$0.656126\pi$
$32$	7.36300	1.30161
$33$	1.88243	0.327690
$34$	−2.29184	−0.393047
$35$	0	0
$36$	−4.59405	−0.765675
$37$	7.05055	1.15910	0.579552	−	0.814936i	$-0.303228\pi$
$37$	7.05055	1.15910	0.579552	+	0.814936i	$0.303228\pi$
$38$	6.17800	1.00220
$39$	0	0
$40$	1.03539	0.163709
$41$	−5.16390	−0.806465	−0.403233	−	0.915098i	$-0.632113\pi$
$41$	−5.16390	−0.806465	−0.403233	+	0.915098i	$0.632113\pi$
$42$	0	0
$43$	0.0227504	0.00346940	0.00173470	−	0.999998i	$-0.499448\pi$
$43$	0.0227504	0.00346940	0.00173470	+	0.999998i	$0.499448\pi$
$44$	−7.16668	−1.08042
$45$	−4.14897	−0.618492
$46$	8.45516	1.24665
$47$	−11.6836	−1.70422	−0.852111	−	0.523362i	$-0.824678\pi$
$47$	−11.6836	−1.70422	−0.852111	+	0.523362i	$0.824678\pi$
$48$	1.97156	0.284570
$49$	0	0
$50$	5.39252	0.762618
$51$	−0.515299	−0.0721563
$52$	0	0
$53$	−0.141786	−0.0194758	−0.00973788	−	0.999953i	$-0.503100\pi$
$53$	−0.141786	−0.0194758	−0.00973788	+	0.999953i	$0.503100\pi$
$54$	−4.74873	−0.646220
$55$	−6.47236	−0.872734
$56$	0	0
$57$	1.38907	0.183986
$58$	−0.319298	−0.0419259
$59$	5.34354	0.695670	0.347835	−	0.937556i	$-0.386917\pi$
$59$	5.34354	0.695670	0.347835	+	0.937556i	$0.386917\pi$
$60$	−1.02952	−0.132911
$61$	−11.5457	−1.47828	−0.739141	−	0.673551i	$-0.764768\pi$
$61$	−11.5457	−1.47828	−0.739141	+	0.673551i	$0.764768\pi$
$62$	9.99549	1.26943
$63$	0	0
$64$	−4.82736	−0.603420
$65$	0	0
$66$	−3.58709	−0.441540
$67$	4.13546	0.505226	0.252613	−	0.967567i	$-0.418710\pi$
$67$	4.13546	0.505226	0.252613	+	0.967567i	$0.418710\pi$
$68$	1.96181	0.237905
$69$	1.90107	0.228861
$70$	0	0
$71$	−9.96971	−1.18319	−0.591594	−	0.806236i	$-0.701501\pi$
$71$	−9.96971	−1.18319	−0.591594	+	0.806236i	$0.701501\pi$
$72$	−1.97953	−0.233289
$73$	−15.2416	−1.78389	−0.891947	−	0.452141i	$-0.850660\pi$
$73$	−15.2416	−1.78389	−0.891947	+	0.452141i	$0.850660\pi$
$74$	−13.4352	−1.56182
$75$	1.21246	0.140003
$76$	−5.28837	−0.606617
$77$	0	0
$78$	0	0
$79$	0.774501	0.0871382	0.0435691	−	0.999050i	$-0.486127\pi$
$79$	0.774501	0.0871382	0.0435691	+	0.999050i	$0.486127\pi$
$80$	−6.77881	−0.757894
$81$	7.38159	0.820177
$82$	9.84011	1.08666
$83$	16.0186	1.75827	0.879136	−	0.476571i	$-0.158121\pi$
$83$	16.0186	1.75827	0.879136	+	0.476571i	$0.158121\pi$
$84$	0	0
$85$	1.77175	0.192173
$86$	−0.0433522	−0.00467479
$87$	−0.0717913	−0.00769683
$88$	−3.08805	−0.329187
$89$	−6.55760	−0.695104	−0.347552	−	0.937661i	$-0.612987\pi$
$89$	−6.55760	−0.695104	−0.347552	+	0.937661i	$0.612987\pi$
$90$	7.90611	0.833378
$91$	0	0
$92$	−7.23762	−0.754574
$93$	2.24739	0.233044
$94$	22.2637	2.29633
$95$	−4.77602	−0.490010
$96$	−3.15466	−0.321971
$97$	−3.49166	−0.354524	−0.177262	−	0.984164i	$-0.556724\pi$
$97$	−3.49166	−0.354524	−0.177262	+	0.984164i	$0.556724\pi$
$98$	0	0
$99$	12.3743	1.24367
$100$	−4.61600	−0.461600
$101$	−2.57780	−0.256500	−0.128250	−	0.991742i	$-0.540936\pi$
$101$	−2.57780	−0.256500	−0.128250	+	0.991742i	$0.540936\pi$
$102$	0.981933	0.0972258
$103$	16.8635	1.66161	0.830803	−	0.556567i	$-0.187882\pi$
$103$	16.8635	1.66161	0.830803	+	0.556567i	$0.187882\pi$
$104$	0	0
$105$	0	0
$106$	0.270181	0.0262423
$107$	8.68265	0.839383	0.419692	−	0.907667i	$-0.362138\pi$
$107$	8.68265	0.839383	0.419692	+	0.907667i	$0.362138\pi$
$108$	4.06491	0.391146
$109$	−12.0405	−1.15327	−0.576637	−	0.817001i	$-0.695635\pi$
$109$	−12.0405	−1.15327	−0.576637	+	0.817001i	$0.695635\pi$
$110$	12.3335	1.17595
$111$	−3.02079	−0.286721
$112$	0	0
$113$	9.37232	0.881674	0.440837	−	0.897587i	$-0.354682\pi$
$113$	9.37232	0.881674	0.440837	+	0.897587i	$0.354682\pi$
$114$	−2.64695	−0.247910
$115$	−6.53643	−0.609525
$116$	0.273319	0.0253771
$117$	0	0
$118$	−10.1824	−0.937369
$119$	0	0
$120$	−0.443609	−0.0404958
$121$	8.30385	0.754895
$122$	22.0011	1.99189
$123$	2.21246	0.199491
$124$	−8.55614	−0.768364
$125$	−11.5344	−1.03167
$126$	0	0
$127$	15.8854	1.40960	0.704800	−	0.709406i	$-0.251037\pi$
$127$	15.8854	1.40960	0.704800	+	0.709406i	$0.251037\pi$
$128$	−5.52717	−0.488537
$129$	−0.00974735	−0.000858206 0
$130$	0	0
$131$	−1.85745	−0.162286	−0.0811430	−	0.996702i	$-0.525857\pi$
$131$	−1.85745	−0.162286	−0.0811430	+	0.996702i	$0.525857\pi$
$132$	3.07055	0.267257
$133$	0	0
$134$	−7.88036	−0.680759
$135$	3.67109	0.315957
$136$	0.845324	0.0724859
$137$	−12.8002	−1.09360	−0.546798	−	0.837264i	$-0.684153\pi$
$137$	−12.8002	−1.09360	−0.546798	+	0.837264i	$0.684153\pi$
$138$	−3.62260	−0.308376
$139$	0.338729	0.0287306	0.0143653	−	0.999897i	$-0.495427\pi$
$139$	0.338729	0.0287306	0.0143653	+	0.999897i	$0.495427\pi$
$140$	0	0
$141$	5.00579	0.421564
$142$	18.9979	1.59427
$143$	0	0
$144$	12.9602	1.08002
$145$	0.246840	0.0204989
$146$	29.0438	2.40368
$147$	0	0
$148$	11.5006	0.945341
$149$	3.92316	0.321398	0.160699	−	0.987003i	$-0.448625\pi$
$149$	3.92316	0.321398	0.160699	+	0.987003i	$0.448625\pi$
$150$	−2.31041	−0.188644
$151$	−2.11879	−0.172424	−0.0862122	−	0.996277i	$-0.527476\pi$
$151$	−2.11879	−0.172424	−0.0862122	+	0.996277i	$0.527476\pi$
$152$	−2.27870	−0.184827
$153$	−3.38736	−0.273851
$154$	0	0
$155$	−7.72721	−0.620664
$156$	0	0
$157$	22.1128	1.76479	0.882397	−	0.470506i	$-0.155929\pi$
$157$	22.1128	1.76479	0.882397	+	0.470506i	$0.155929\pi$
$158$	−1.47586	−0.117413
$159$	0.0607478	0.00481761
$160$	10.8467	0.857504
$161$	0	0
$162$	−14.0661	−1.10513
$163$	3.85214	0.301723	0.150861	−	0.988555i	$-0.451795\pi$
$163$	3.85214	0.301723	0.150861	+	0.988555i	$0.451795\pi$
$164$	−8.42314	−0.657736
$165$	2.77307	0.215883
$166$	−30.5244	−2.36916
$167$	−2.13894	−0.165516	−0.0827582	−	0.996570i	$-0.526373\pi$
$167$	−2.13894	−0.165516	−0.0827582	+	0.996570i	$0.526373\pi$
$168$	0	0
$169$	0	0
$170$	−3.37618	−0.258941
$171$	9.13113	0.698275
$172$	0.0371095	0.00282957
$173$	16.6133	1.26308	0.631542	−	0.775342i	$-0.282422\pi$
$173$	16.6133	1.26308	0.631542	+	0.775342i	$0.282422\pi$
$174$	0.136803	0.0103710
$175$	0	0
$176$	20.2178	1.52398
$177$	−2.28943	−0.172084
$178$	12.4959	0.936607
$179$	−0.539496	−0.0403238	−0.0201619	−	0.999797i	$-0.506418\pi$
$179$	−0.539496	−0.0403238	−0.0201619	+	0.999797i	$0.506418\pi$
$180$	−6.76763	−0.504430
$181$	−2.77164	−0.206014	−0.103007	−	0.994681i	$-0.532846\pi$
$181$	−2.77164	−0.206014	−0.103007	+	0.994681i	$0.532846\pi$
$182$	0	0
$183$	4.94675	0.365674
$184$	−3.11861	−0.229907
$185$	10.3864	0.763621
$186$	−4.28254	−0.314011
$187$	−5.28425	−0.386423
$188$	−19.0577	−1.38993
$189$	0	0
$190$	9.10100	0.660256
$191$	−20.2407	−1.46457	−0.732284	−	0.680999i	$-0.761546\pi$
$191$	−20.2407	−1.46457	−0.732284	+	0.680999i	$0.761546\pi$
$192$	2.06827	0.149265
$193$	−16.3771	−1.17885	−0.589425	−	0.807823i	$-0.700646\pi$
$193$	−16.3771	−1.17885	−0.589425	+	0.807823i	$0.700646\pi$
$194$	6.65357	0.477698
$195$	0	0
$196$	0	0
$197$	19.7335	1.40595	0.702977	−	0.711212i	$-0.251854\pi$
$197$	19.7335	1.40595	0.702977	+	0.711212i	$0.251854\pi$
$198$	−23.5800	−1.67576
$199$	14.1175	1.00076	0.500380	−	0.865806i	$-0.333194\pi$
$199$	14.1175	1.00076	0.500380	+	0.865806i	$0.333194\pi$
$200$	−1.98898	−0.140642
$201$	−1.77183	−0.124975
$202$	4.91214	0.345617
$203$	0	0
$204$	−0.840534	−0.0588492
$205$	−7.60709	−0.531302
$206$	−32.1343	−2.23890
$207$	12.4968	0.868588
$208$	0	0
$209$	14.2445	0.985313
$210$	0	0
$211$	−4.62634	−0.318490	−0.159245	−	0.987239i	$-0.550906\pi$
$211$	−4.62634	−0.318490	−0.159245	+	0.987239i	$0.550906\pi$
$212$	−0.231275	−0.0158840
$213$	4.27150	0.292678
$214$	−16.5453	−1.13101
$215$	0.0335143	0.00228565
$216$	1.75152	0.119176
$217$	0	0
$218$	22.9439	1.55396
$219$	6.53022	0.441272
$220$	−10.5575	−0.711784
$221$	0	0
$222$	5.75630	0.386337
$223$	21.3523	1.42985	0.714926	−	0.699200i	$-0.246460\pi$
$223$	21.3523	1.42985	0.714926	+	0.699200i	$0.246460\pi$
$224$	0	0
$225$	7.97019	0.531346
$226$	−17.8595	−1.18800
$227$	10.4490	0.693526	0.346763	−	0.937953i	$-0.387281\pi$
$227$	10.4490	0.693526	0.346763	+	0.937953i	$0.387281\pi$
$228$	2.26579	0.150056
$229$	−14.4580	−0.955413	−0.477706	−	0.878520i	$-0.658532\pi$
$229$	−14.4580	−0.955413	−0.477706	+	0.878520i	$0.658532\pi$
$230$	12.4556	0.821295
$231$	0	0
$232$	0.117770	0.00773200
$233$	−9.28827	−0.608495	−0.304247	−	0.952593i	$-0.598405\pi$
$233$	−9.28827	−0.608495	−0.304247	+	0.952593i	$0.598405\pi$
$234$	0	0
$235$	−17.2114	−1.12275
$236$	8.71616	0.567374
$237$	−0.331833	−0.0215549
$238$	0	0
$239$	19.6332	1.26997	0.634983	−	0.772526i	$-0.281007\pi$
$239$	19.6332	1.26997	0.634983	+	0.772526i	$0.281007\pi$
$240$	2.90437	0.187476
$241$	−7.31105	−0.470946	−0.235473	−	0.971881i	$-0.575664\pi$
$241$	−7.31105	−0.470946	−0.235473	+	0.971881i	$0.575664\pi$
$242$	−15.8235	−1.01717
$243$	−10.6387	−0.682475
$244$	−18.8330	−1.20566
$245$	0	0
$246$	−4.21597	−0.268801
$247$	0	0
$248$	−3.68675	−0.234109
$249$	−6.86314	−0.434934
$250$	21.9796	1.39011
$251$	11.8638	0.748837	0.374419	−	0.927260i	$-0.377842\pi$
$251$	11.8638	0.748837	0.374419	+	0.927260i	$0.377842\pi$
$252$	0	0
$253$	19.4949	1.22563
$254$	−30.2706	−1.89934
$255$	−0.759102	−0.0475368
$256$	20.1871	1.26169
$257$	15.1722	0.946413	0.473206	−	0.880952i	$-0.343096\pi$
$257$	15.1722	0.946413	0.473206	+	0.880952i	$0.343096\pi$
$258$	0.0185742	0.00115638
$259$	0	0
$260$	0	0
$261$	−0.471925	−0.0292115
$262$	3.53948	0.218670
$263$	17.1964	1.06037	0.530187	−	0.847880i	$-0.322122\pi$
$263$	17.1964	1.06037	0.530187	+	0.847880i	$0.322122\pi$
$264$	1.32307	0.0814291
$265$	−0.208869	−0.0128307
$266$	0	0
$267$	2.80959	0.171944
$268$	6.74559	0.412052
$269$	−18.9220	−1.15370	−0.576849	−	0.816851i	$-0.695718\pi$
$269$	−18.9220	−1.15370	−0.576849	+	0.816851i	$0.695718\pi$
$270$	−6.99549	−0.425732
$271$	−32.1334	−1.95196	−0.975982	−	0.217853i	$-0.930095\pi$
$271$	−32.1334	−1.95196	−0.975982	+	0.217853i	$0.930095\pi$
$272$	−5.53444	−0.335575
$273$	0	0
$274$	24.3916	1.47355
$275$	12.4334	0.749764
$276$	3.10094	0.186655
$277$	18.4054	1.10587	0.552936	−	0.833224i	$-0.313507\pi$
$277$	18.4054	1.10587	0.552936	+	0.833224i	$0.313507\pi$
$278$	−0.645469	−0.0387126
$279$	14.7734	0.884461
$280$	0	0
$281$	−14.2252	−0.848603	−0.424302	−	0.905521i	$-0.639480\pi$
$281$	−14.2252	−0.848603	−0.424302	+	0.905521i	$0.639480\pi$
$282$	−9.53883	−0.568029
$283$	11.4289	0.679378	0.339689	−	0.940538i	$-0.389678\pi$
$283$	11.4289	0.679378	0.339689	+	0.940538i	$0.389678\pi$
$284$	−16.2622	−0.964983
$285$	2.04628	0.121211
$286$	0	0
$287$	0	0
$288$	−20.7374	−1.22196
$289$	−15.5535	−0.914911
$290$	−0.470368	−0.0276210
$291$	1.49599	0.0876967
$292$	−24.8615	−1.45491
$293$	13.2046	0.771422	0.385711	−	0.922620i	$-0.373956\pi$
$293$	13.2046	0.771422	0.385711	+	0.922620i	$0.373956\pi$
$294$	0	0
$295$	7.87173	0.458310
$296$	4.95547	0.288031
$297$	−10.9490	−0.635328
$298$	−7.47582	−0.433062
$299$	0	0
$300$	1.97771	0.114183
$301$	0	0
$302$	4.03748	0.232331
$303$	1.10445	0.0634490
$304$	14.9189	0.855660
$305$	−17.0084	−0.973898
$306$	6.45481	0.368997
$307$	6.65903	0.380051	0.190026	−	0.981779i	$-0.439143\pi$
$307$	6.65903	0.380051	0.190026	+	0.981779i	$0.439143\pi$
$308$	0	0
$309$	−7.22511	−0.411022
$310$	14.7247	0.836304
$311$	2.04597	0.116016	0.0580081	−	0.998316i	$-0.481525\pi$
$311$	2.04597	0.116016	0.0580081	+	0.998316i	$0.481525\pi$
$312$	0	0
$313$	−9.41767	−0.532318	−0.266159	−	0.963929i	$-0.585755\pi$
$313$	−9.41767	−0.532318	−0.266159	+	0.963929i	$0.585755\pi$
$314$	−42.1373	−2.37794
$315$	0	0
$316$	1.26333	0.0710681
$317$	−33.3713	−1.87432	−0.937159	−	0.348902i	$-0.886555\pi$
$317$	−33.3713	−1.87432	−0.937159	+	0.348902i	$0.886555\pi$
$318$	−0.115758	−0.00649141
$319$	−0.736200	−0.0412193
$320$	−7.11133	−0.397536
$321$	−3.72006	−0.207634
$322$	0	0
$323$	−3.89930	−0.216963
$324$	12.0405	0.668919
$325$	0	0
$326$	−7.34048	−0.406551
$327$	5.15873	0.285279
$328$	−3.62944	−0.200402
$329$	0	0
$330$	−5.28425	−0.290888
$331$	19.0660	1.04796	0.523980	−	0.851731i	$-0.324447\pi$
$331$	19.0660	1.04796	0.523980	+	0.851731i	$0.324447\pi$
$332$	26.1289	1.43401
$333$	−19.8574	−1.08818
$334$	4.07589	0.223023
$335$	6.09207	0.332845
$336$	0	0
$337$	−31.2849	−1.70420	−0.852098	−	0.523382i	$-0.824670\pi$
$337$	−31.2849	−1.70420	−0.852098	+	0.523382i	$0.824670\pi$
$338$	0	0
$339$	−4.01555	−0.218095
$340$	2.89001	0.156733
$341$	23.0464	1.24803
$342$	−17.3999	−0.940880
$343$	0	0
$344$	0.0159901	0.000862127 0
$345$	2.80052	0.150775
$346$	−31.6576	−1.70192
$347$	11.6752	0.626757	0.313378	−	0.949628i	$-0.398539\pi$
$347$	11.6752	0.626757	0.313378	+	0.949628i	$0.398539\pi$
$348$	−0.117103	−0.00627738
$349$	−23.9904	−1.28418	−0.642089	−	0.766631i	$-0.721932\pi$
$349$	−23.9904	−1.28418	−0.642089	+	0.766631i	$0.721932\pi$
$350$	0	0
$351$	0	0
$352$	−32.3502	−1.72427
$353$	−12.7934	−0.680922	−0.340461	−	0.940259i	$-0.610583\pi$
$353$	−12.7934	−0.680922	−0.340461	+	0.940259i	$0.610583\pi$
$354$	4.36264	0.231872
$355$	−14.6867	−0.779488
$356$	−10.6965	−0.566912
$357$	0	0
$358$	1.02804	0.0543337
$359$	12.3397	0.651265	0.325633	−	0.945496i	$-0.394423\pi$
$359$	12.3397	0.651265	0.325633	+	0.945496i	$0.394423\pi$
$360$	−2.91610	−0.153692
$361$	−8.48884	−0.446781
$362$	5.28152	0.277591
$363$	−3.55776	−0.186734
$364$	0	0
$365$	−22.4528	−1.17524
$366$	−9.42633	−0.492722
$367$	−2.03077	−0.106005	−0.0530026	−	0.998594i	$-0.516879\pi$
$367$	−2.03077	−0.106005	−0.0530026	+	0.998594i	$0.516879\pi$
$368$	20.4179	1.06436
$369$	14.5438	0.757118
$370$	−19.7919	−1.02893
$371$	0	0
$372$	3.66586	0.190066
$373$	−3.87400	−0.200588	−0.100294	−	0.994958i	$-0.531978\pi$
$373$	−3.87400	−0.200588	−0.100294	+	0.994958i	$0.531978\pi$
$374$	10.0695	0.520679
$375$	4.94190	0.255199
$376$	−8.21177	−0.423490
$377$	0	0
$378$	0	0
$379$	−14.5679	−0.748303	−0.374152	−	0.927368i	$-0.622066\pi$
$379$	−14.5679	−0.748303	−0.374152	+	0.927368i	$0.622066\pi$
$380$	−7.79045	−0.399642
$381$	−6.80606	−0.348685
$382$	38.5699	1.97341
$383$	26.7818	1.36849	0.684243	−	0.729254i	$-0.260133\pi$
$383$	26.7818	1.36849	0.684243	+	0.729254i	$0.260133\pi$
$384$	2.36810	0.120847
$385$	0	0
$386$	31.2076	1.58842
$387$	−0.0640749	−0.00325711
$388$	−5.69545	−0.289143
$389$	12.0148	0.609173	0.304586	−	0.952485i	$-0.401482\pi$
$389$	12.0148	0.609173	0.304586	+	0.952485i	$0.401482\pi$
$390$	0	0
$391$	−5.33655	−0.269881
$392$	0	0
$393$	0.795820	0.0401438
$394$	−37.6034	−1.89443
$395$	1.14094	0.0574070
$396$	20.1845	1.01431
$397$	1.65765	0.0831951	0.0415975	−	0.999134i	$-0.486755\pi$
$397$	1.65765	0.0831951	0.0415975	+	0.999134i	$0.486755\pi$
$398$	−26.9017	−1.34846
$399$	0	0
$400$	13.0221	0.651106
$401$	−20.4828	−1.02286	−0.511430	−	0.859325i	$-0.670884\pi$
$401$	−20.4828	−1.02286	−0.511430	+	0.859325i	$0.670884\pi$
$402$	3.37632	0.168396
$403$	0	0
$404$	−4.20479	−0.209196
$405$	10.8740	0.540336
$406$	0	0
$407$	−30.9774	−1.53549
$408$	−0.362177	−0.0179304
$409$	−14.8659	−0.735070	−0.367535	−	0.930010i	$-0.619798\pi$
$409$	−14.8659	−0.735070	−0.367535	+	0.930010i	$0.619798\pi$
$410$	14.4958	0.715895
$411$	5.48422	0.270517
$412$	27.5070	1.35517
$413$	0	0
$414$	−23.8134	−1.17036
$415$	23.5975	1.15836
$416$	0	0
$417$	−0.145128	−0.00710693
$418$	−27.1438	−1.32764
$419$	23.6175	1.15379	0.576895	−	0.816819i	$-0.304264\pi$
$419$	23.6175	1.15379	0.576895	+	0.816819i	$0.304264\pi$
$420$	0	0
$421$	26.0822	1.27117	0.635585	−	0.772031i	$-0.280759\pi$
$421$	26.0822	1.27117	0.635585	+	0.772031i	$0.280759\pi$
$422$	8.81576	0.429144
$423$	32.9059	1.59994
$424$	−0.0996539	−0.00483962
$425$	−3.40354	−0.165096
$426$	−8.13960	−0.394365
$427$	0	0
$428$	14.1628	0.684584
$429$	0	0
$430$	−0.0638635	−0.00307977
$431$	−13.3172	−0.641466	−0.320733	−	0.947170i	$-0.603929\pi$
$431$	−13.3172	−0.641466	−0.320733	+	0.947170i	$0.603929\pi$
$432$	−11.4675	−0.551728
$433$	−20.4221	−0.981422	−0.490711	−	0.871322i	$-0.663263\pi$
$433$	−20.4221	−0.981422	−0.490711	+	0.871322i	$0.663263\pi$
$434$	0	0
$435$	−0.105758	−0.00507070
$436$	−19.6400	−0.940586
$437$	14.3855	0.688152
$438$	−12.4437	−0.594585
$439$	9.77074	0.466332	0.233166	−	0.972437i	$-0.425091\pi$
$439$	9.77074	0.466332	0.233166	+	0.972437i	$0.425091\pi$
$440$	−4.54909	−0.216869
$441$	0	0
$442$	0	0
$443$	21.1639	1.00553	0.502763	−	0.864424i	$-0.332317\pi$
$443$	21.1639	1.00553	0.502763	+	0.864424i	$0.332317\pi$
$444$	−4.92739	−0.233844
$445$	−9.66019	−0.457937
$446$	−40.6880	−1.92663
$447$	−1.68087	−0.0795023
$448$	0	0
$449$	−18.1464	−0.856382	−0.428191	−	0.903688i	$-0.640849\pi$
$449$	−18.1464	−0.856382	−0.428191	+	0.903688i	$0.640849\pi$
$450$	−15.1877	−0.715954
$451$	22.6882	1.06834
$452$	15.2877	0.719075
$453$	0.907789	0.0426517
$454$	−19.9112	−0.934480
$455$	0	0
$456$	0.976304	0.0457196
$457$	−18.0198	−0.842932	−0.421466	−	0.906844i	$-0.638484\pi$
$457$	−18.0198	−0.842932	−0.421466	+	0.906844i	$0.638484\pi$
$458$	27.5506	1.28736
$459$	2.99720	0.139897
$460$	−10.6620	−0.497116
$461$	29.7746	1.38674	0.693370	−	0.720582i	$-0.256125\pi$
$461$	29.7746	1.38674	0.693370	+	0.720582i	$0.256125\pi$
$462$	0	0
$463$	17.7067	0.822900	0.411450	−	0.911432i	$-0.365023\pi$
$463$	17.7067	0.822900	0.411450	+	0.911432i	$0.365023\pi$
$464$	−0.771057	−0.0357954
$465$	3.31070	0.153530
$466$	17.6994	0.819907
$467$	5.82922	0.269744	0.134872	−	0.990863i	$-0.456938\pi$
$467$	5.82922	0.269744	0.134872	+	0.990863i	$0.456938\pi$
$468$	0	0
$469$	0	0
$470$	32.7973	1.51283
$471$	−9.47418	−0.436547
$472$	3.75570	0.172870
$473$	−0.0999564	−0.00459600
$474$	0.632328	0.0290438
$475$	9.17476	0.420967
$476$	0	0
$477$	0.399330	0.0182841
$478$	−37.4122	−1.71120
$479$	−14.4913	−0.662125	−0.331062	−	0.943609i	$-0.607407\pi$
$479$	−14.4913	−0.662125	−0.331062	+	0.943609i	$0.607407\pi$
$480$	−4.64722	−0.212116
$481$	0	0
$482$	13.9316	0.634569
$483$	0	0
$484$	13.5449	0.615677
$485$	−5.14367	−0.233562
$486$	20.2727	0.919591
$487$	−17.9601	−0.813851	−0.406926	−	0.913461i	$-0.633399\pi$
$487$	−17.9601	−0.813851	−0.406926	+	0.913461i	$0.633399\pi$
$488$	−8.11491	−0.367345
$489$	−1.65044	−0.0746354
$490$	0	0
$491$	−36.3009	−1.63824	−0.819119	−	0.573624i	$-0.805537\pi$
$491$	−36.3009	−1.63824	−0.819119	+	0.573624i	$0.805537\pi$
$492$	3.60887	0.162701
$493$	0.201528	0.00907637
$494$	0	0
$495$	18.2290	0.819332
$496$	24.1376	1.08381
$497$	0	0
$498$	13.0781	0.586045
$499$	23.7076	1.06130	0.530649	−	0.847591i	$-0.321948\pi$
$499$	23.7076	1.06130	0.530649	+	0.847591i	$0.321948\pi$
$500$	−18.8145	−0.841410
$501$	0.916426	0.0409429
$502$	−22.6072	−1.00901
$503$	−27.7752	−1.23843	−0.619217	−	0.785220i	$-0.712550\pi$
$503$	−27.7752	−1.23843	−0.619217	+	0.785220i	$0.712550\pi$
$504$	0	0
$505$	−3.79743	−0.168983
$506$	−37.1487	−1.65146
$507$	0	0
$508$	25.9116	1.14964
$509$	−8.70416	−0.385805	−0.192902	−	0.981218i	$-0.561790\pi$
$509$	−8.70416	−0.385805	−0.192902	+	0.981218i	$0.561790\pi$
$510$	1.44651	0.0640527
$511$	0	0
$512$	−27.4134	−1.21151
$513$	−8.07941	−0.356715
$514$	−28.9114	−1.27523
$515$	24.8421	1.09467
$516$	−0.0158995	−0.000699935 0
$517$	51.3330	2.25762
$518$	0	0
$519$	−7.11792	−0.312442
$520$	0	0
$521$	8.57146	0.375523	0.187761	−	0.982215i	$-0.439877\pi$
$521$	8.57146	0.375523	0.187761	+	0.982215i	$0.439877\pi$
$522$	0.899282	0.0393605
$523$	−29.9493	−1.30959	−0.654796	−	0.755806i	$-0.727245\pi$
$523$	−29.9493	−1.30959	−0.654796	+	0.755806i	$0.727245\pi$
$524$	−3.02980	−0.132357
$525$	0	0
$526$	−32.7688	−1.42879
$527$	−6.30874	−0.274813
$528$	−8.66228	−0.376977
$529$	−3.31212	−0.144005
$530$	0.398012	0.0172885
$531$	−15.0497	−0.653102
$532$	0	0
$533$	0	0
$534$	−5.35383	−0.231683
$535$	12.7907	0.552989
$536$	2.90660	0.125546
$537$	0.231146	0.00997467
$538$	36.0571	1.55453
$539$	0	0
$540$	5.98814	0.257688
$541$	10.4819	0.450652	0.225326	−	0.974283i	$-0.427655\pi$
$541$	10.4819	0.450652	0.225326	+	0.974283i	$0.427655\pi$
$542$	61.2321	2.63014
$543$	1.18750	0.0509606
$544$	8.85557	0.379679
$545$	−17.7373	−0.759781
$546$	0	0
$547$	15.2216	0.650829	0.325415	−	0.945571i	$-0.394496\pi$
$547$	15.2216	0.650829	0.325415	+	0.945571i	$0.394496\pi$
$548$	−20.8792	−0.891915
$549$	32.5178	1.38783
$550$	−23.6927	−1.01026
$551$	−0.543250	−0.0231432
$552$	1.33616	0.0568708
$553$	0	0
$554$	−35.0726	−1.49009
$555$	−4.45002	−0.188893
$556$	0.552521	0.0234321
$557$	11.8597	0.502513	0.251256	−	0.967921i	$-0.419156\pi$
$557$	11.8597	0.502513	0.251256	+	0.967921i	$0.419156\pi$
$558$	−28.1516	−1.19175
$559$	0	0
$560$	0	0
$561$	2.26402	0.0955872
$562$	27.1069	1.14344
$563$	−7.69349	−0.324242	−0.162121	−	0.986771i	$-0.551834\pi$
$563$	−7.69349	−0.324242	−0.162121	+	0.986771i	$0.551834\pi$
$564$	8.16524	0.343819
$565$	13.8066	0.580850
$566$	−21.7785	−0.915418
$567$	0	0
$568$	−7.00720	−0.294016
$569$	37.4196	1.56871	0.784355	−	0.620312i	$-0.212994\pi$
$569$	37.4196	1.56871	0.784355	+	0.620312i	$0.212994\pi$
$570$	−3.89930	−0.163324
$571$	14.1657	0.592816	0.296408	−	0.955061i	$-0.404211\pi$
$571$	14.1657	0.592816	0.296408	+	0.955061i	$0.404211\pi$
$572$	0	0
$573$	8.67209	0.362282
$574$	0	0
$575$	12.5565	0.523642
$576$	13.5959	0.566498
$577$	14.9755	0.623439	0.311720	−	0.950174i	$-0.399095\pi$
$577$	14.9755	0.623439	0.311720	+	0.950174i	$0.399095\pi$
$578$	29.6381	1.23278
$579$	7.01674	0.291606
$580$	0.402635	0.0167185
$581$	0	0
$582$	−2.85071	−0.118166
$583$	0.622952	0.0258000
$584$	−10.7125	−0.443288
$585$	0	0
$586$	−25.1622	−1.03944
$587$	13.1764	0.543849	0.271925	−	0.962319i	$-0.412340\pi$
$587$	13.1764	0.543849	0.271925	+	0.962319i	$0.412340\pi$
$588$	0	0
$589$	17.0062	0.700728
$590$	−15.0001	−0.617542
$591$	−8.45478	−0.347783
$592$	−32.4441	−1.33344
$593$	44.1327	1.81231	0.906156	−	0.422943i	$-0.139003\pi$
$593$	44.1327	1.81231	0.906156	+	0.422943i	$0.139003\pi$
$594$	20.8641	0.856063
$595$	0	0
$596$	6.39930	0.262125
$597$	−6.04859	−0.247552
$598$	0	0
$599$	−6.02698	−0.246256	−0.123128	−	0.992391i	$-0.539293\pi$
$599$	−6.02698	−0.246256	−0.123128	+	0.992391i	$0.539293\pi$
$600$	0.852175	0.0347899
$601$	−3.72520	−0.151954	−0.0759770	−	0.997110i	$-0.524208\pi$
$601$	−3.72520	−0.151954	−0.0759770	+	0.997110i	$0.524208\pi$
$602$	0	0
$603$	−11.6472	−0.474312
$604$	−3.45608	−0.140626
$605$	12.2326	0.497328
$606$	−2.10460	−0.0854934
$607$	6.01651	0.244203	0.122101	−	0.992518i	$-0.461037\pi$
$607$	6.01651	0.244203	0.122101	+	0.992518i	$0.461037\pi$
$608$	−23.8715	−0.968118
$609$	0	0
$610$	32.4105	1.31226
$611$	0	0
$612$	−5.52532	−0.223348
$613$	9.80825	0.396152	0.198076	−	0.980187i	$-0.436531\pi$
$613$	9.80825	0.396152	0.198076	+	0.980187i	$0.436531\pi$
$614$	−12.6892	−0.512094
$615$	3.25924	0.131425
$616$	0	0
$617$	33.7676	1.35943	0.679716	−	0.733475i	$-0.262103\pi$
$617$	33.7676	1.35943	0.679716	+	0.733475i	$0.262103\pi$
$618$	13.7679	0.553825
$619$	−4.09343	−0.164529	−0.0822644	−	0.996611i	$-0.526215\pi$
$619$	−4.09343	−0.164529	−0.0822644	+	0.996611i	$0.526215\pi$
$620$	−12.6043	−0.506201
$621$	−11.0574	−0.443719
$622$	−3.89871	−0.156324
$623$	0	0
$624$	0	0
$625$	−2.84229	−0.113692
$626$	17.9459	0.717264
$627$	−6.10302	−0.243731
$628$	36.0695	1.43933
$629$	8.47978	0.338111
$630$	0	0
$631$	−26.7736	−1.06584	−0.532921	−	0.846165i	$-0.678906\pi$
$631$	−26.7736	−1.06584	−0.532921	+	0.846165i	$0.678906\pi$
$632$	0.544357	0.0216534
$633$	1.98214	0.0787831
$634$	63.5910	2.52552
$635$	23.4012	0.928650
$636$	0.0990892	0.00392914
$637$	0	0
$638$	1.40287	0.0555403
$639$	28.0790	1.11079
$640$	−8.14224	−0.321850
$641$	−18.5722	−0.733558	−0.366779	−	0.930308i	$-0.619539\pi$
$641$	−18.5722	−0.733558	−0.366779	+	0.930308i	$0.619539\pi$
$642$	7.08880	0.279773
$643$	3.93390	0.155138	0.0775690	−	0.996987i	$-0.475284\pi$
$643$	3.93390	0.155138	0.0775690	+	0.996987i	$0.475284\pi$
$644$	0	0
$645$	−0.0143591	−0.000565389 0
$646$	7.43035	0.292343
$647$	0.197076	0.00774784	0.00387392	−	0.999992i	$-0.498767\pi$
$647$	0.197076	0.00774784	0.00387392	+	0.999992i	$0.498767\pi$
$648$	5.18814	0.203809
$649$	−23.4775	−0.921571
$650$	0	0
$651$	0	0
$652$	6.28345	0.246079
$653$	−14.4673	−0.566148	−0.283074	−	0.959098i	$-0.591354\pi$
$653$	−14.4673	−0.566148	−0.283074	+	0.959098i	$0.591354\pi$
$654$	−9.83028	−0.384394
$655$	−2.73626	−0.106915
$656$	23.7624	0.927765
$657$	42.9269	1.67474
$658$	0	0
$659$	−23.4132	−0.912048	−0.456024	−	0.889967i	$-0.650727\pi$
$659$	−23.4132	−0.912048	−0.456024	+	0.889967i	$0.650727\pi$
$660$	4.52332	0.176070
$661$	4.04817	0.157456	0.0787278	−	0.996896i	$-0.474914\pi$
$661$	4.04817	0.157456	0.0787278	+	0.996896i	$0.474914\pi$
$662$	−36.3313	−1.41206
$663$	0	0
$664$	11.2587	0.436921
$665$	0	0
$666$	37.8395	1.46625
$667$	−0.743487	−0.0287879
$668$	−3.48896	−0.134992
$669$	−9.14832	−0.353695
$670$	−11.6088	−0.448487
$671$	50.7276	1.95832
$672$	0	0
$673$	7.29407	0.281166	0.140583	−	0.990069i	$-0.455102\pi$
$673$	7.29407	0.281166	0.140583	+	0.990069i	$0.455102\pi$
$674$	59.6152	2.29629
$675$	−7.05218	−0.271439
$676$	0	0
$677$	15.7511	0.605362	0.302681	−	0.953092i	$-0.402118\pi$
$677$	15.7511	0.605362	0.302681	+	0.953092i	$0.402118\pi$
$678$	7.65187	0.293868
$679$	0	0
$680$	1.24527	0.0477540
$681$	−4.47686	−0.171553
$682$	−43.9163	−1.68164
$683$	41.4854	1.58739	0.793697	−	0.608314i	$-0.208154\pi$
$683$	41.4854	1.58739	0.793697	+	0.608314i	$0.208154\pi$
$684$	14.8943	0.569499
$685$	−18.8564	−0.720465
$686$	0	0
$687$	6.19450	0.236335
$688$	−0.104689	−0.00399123
$689$	0	0
$690$	−5.33655	−0.203159
$691$	46.8216	1.78118	0.890589	−	0.454809i	$-0.150292\pi$
$691$	46.8216	1.78118	0.890589	+	0.454809i	$0.150292\pi$
$692$	27.0989	1.03015
$693$	0	0
$694$	−22.2478	−0.844514
$695$	0.498992	0.0189278
$696$	−0.0504584	−0.00191262
$697$	−6.21068	−0.235246
$698$	45.7152	1.73034
$699$	3.97954	0.150520
$700$	0	0
$701$	29.8626	1.12790	0.563948	−	0.825810i	$-0.309282\pi$
$701$	29.8626	1.12790	0.563948	+	0.825810i	$0.309282\pi$
$702$	0	0
$703$	−22.8585	−0.862126
$704$	21.2096	0.799366
$705$	7.37418	0.277728
$706$	24.3785	0.917497
$707$	0	0
$708$	−3.73442	−0.140348
$709$	26.9332	1.01150	0.505750	−	0.862680i	$-0.331216\pi$
$709$	26.9332	1.01150	0.505750	+	0.862680i	$0.331216\pi$
$710$	27.9864	1.05031
$711$	−2.18133	−0.0818063
$712$	−4.60900	−0.172729
$713$	23.2745	0.871638
$714$	0	0
$715$	0	0
$716$	−0.880004	−0.0328873
$717$	−8.41180	−0.314144
$718$	−23.5141	−0.877537
$719$	14.4988	0.540713	0.270356	−	0.962760i	$-0.412859\pi$
$719$	14.4988	0.540713	0.270356	+	0.962760i	$0.412859\pi$
$720$	19.0921	0.711519
$721$	0	0
$722$	16.1760	0.602008
$723$	3.13240	0.116495
$724$	−4.52098	−0.168021
$725$	−0.474180	−0.0176106
$726$	6.77953	0.251612
$727$	6.26424	0.232328	0.116164	−	0.993230i	$-0.462940\pi$
$727$	6.26424	0.232328	0.116164	+	0.993230i	$0.462940\pi$
$728$	0	0
$729$	−17.5866	−0.651357
$730$	42.7852	1.58355
$731$	0.0273621	0.00101203
$732$	8.06893	0.298236
$733$	11.9838	0.442631	0.221316	−	0.975202i	$-0.428965\pi$
$733$	11.9838	0.442631	0.221316	+	0.975202i	$0.428965\pi$
$734$	3.86975	0.142835
$735$	0	0
$736$	−32.6704	−1.20425
$737$	−18.1696	−0.669286
$738$	−27.7140	−1.02017
$739$	13.5254	0.497539	0.248770	−	0.968563i	$-0.419974\pi$
$739$	13.5254	0.497539	0.248770	+	0.968563i	$0.419974\pi$
$740$	16.9418	0.622794
$741$	0	0
$742$	0	0
$743$	−38.4598	−1.41095	−0.705477	−	0.708733i	$-0.749267\pi$
$743$	−38.4598	−1.41095	−0.705477	+	0.708733i	$0.749267\pi$
$744$	1.57958	0.0579101
$745$	5.77932	0.211738
$746$	7.38214	0.270279
$747$	−45.1154	−1.65068
$748$	−8.61945	−0.315158
$749$	0	0
$750$	−9.41710	−0.343864
$751$	11.7115	0.427357	0.213679	−	0.976904i	$-0.431455\pi$
$751$	11.7115	0.427357	0.213679	+	0.976904i	$0.431455\pi$
$752$	53.7635	1.96055
$753$	−5.08302	−0.185236
$754$	0	0
$755$	−3.12125	−0.113594
$756$	0	0
$757$	9.31582	0.338589	0.169295	−	0.985566i	$-0.445851\pi$
$757$	9.31582	0.338589	0.169295	+	0.985566i	$0.445851\pi$
$758$	27.7600	1.00829
$759$	−8.35255	−0.303178
$760$	−3.35682	−0.121765
$761$	43.9381	1.59276	0.796378	−	0.604799i	$-0.206747\pi$
$761$	43.9381	1.59276	0.796378	+	0.604799i	$0.206747\pi$
$762$	12.9693	0.469830
$763$	0	0
$764$	−33.0159	−1.19447
$765$	−4.99002	−0.180414
$766$	−51.0344	−1.84395
$767$	0	0
$768$	−8.64911	−0.312098
$769$	−25.3542	−0.914294	−0.457147	−	0.889391i	$-0.651129\pi$
$769$	−25.3542	−0.914294	−0.457147	+	0.889391i	$0.651129\pi$
$770$	0	0
$771$	−6.50047	−0.234109
$772$	−26.7137	−0.961447
$773$	23.1084	0.831152	0.415576	−	0.909559i	$-0.363580\pi$
$773$	23.1084	0.831152	0.415576	+	0.909559i	$0.363580\pi$
$774$	0.122099	0.00438874
$775$	14.8440	0.533212
$776$	−2.45411	−0.0880974
$777$	0	0
$778$	−22.8949	−0.820820
$779$	16.7418	0.599838
$780$	0	0
$781$	43.8031	1.56740
$782$	10.1691	0.363647
$783$	0.417569	0.0149227
$784$	0	0
$785$	32.5750	1.16265
$786$	−1.51648	−0.0540911
$787$	24.6692	0.879364	0.439682	−	0.898154i	$-0.355091\pi$
$787$	24.6692	0.879364	0.439682	+	0.898154i	$0.355091\pi$
$788$	32.1885	1.14667
$789$	−7.36775	−0.262299
$790$	−2.17413	−0.0773521
$791$	0	0
$792$	8.69727	0.309044
$793$	0	0
$794$	−3.15875	−0.112100
$795$	0.0894893	0.00317386
$796$	23.0278	0.816199
$797$	−11.3137	−0.400752	−0.200376	−	0.979719i	$-0.564216\pi$
$797$	−11.3137	−0.400752	−0.200376	+	0.979719i	$0.564216\pi$
$798$	0	0
$799$	−14.0519	−0.497122
$800$	−20.8365	−0.736680
$801$	18.4690	0.652571
$802$	39.0311	1.37824
$803$	66.9657	2.36317
$804$	−2.89013	−0.101927
$805$	0	0
$806$	0	0
$807$	8.10710	0.285384
$808$	−1.81180	−0.0637389
$809$	16.3708	0.575566	0.287783	−	0.957696i	$-0.407082\pi$
$809$	16.3708	0.575566	0.287783	+	0.957696i	$0.407082\pi$
$810$	−20.7211	−0.728067
$811$	29.0412	1.01978	0.509888	−	0.860241i	$-0.329687\pi$
$811$	29.0412	1.01978	0.509888	+	0.860241i	$0.329687\pi$
$812$	0	0
$813$	13.7675	0.482846
$814$	59.0293	2.06898
$815$	5.67470	0.198776
$816$	2.37122	0.0830093
$817$	−0.0737588	−0.00258050
$818$	28.3278	0.990458
$819$	0	0
$820$	−12.4084	−0.433319
$821$	−13.7518	−0.479940	−0.239970	−	0.970780i	$-0.577138\pi$
$821$	−13.7518	−0.479940	−0.239970	+	0.970780i	$0.577138\pi$
$822$	−10.4505	−0.364504
$823$	−29.1153	−1.01490	−0.507448	−	0.861682i	$-0.669411\pi$
$823$	−29.1153	−1.01490	−0.507448	+	0.861682i	$0.669411\pi$
$824$	11.8525	0.412900
$825$	−5.32708	−0.185465
$826$	0	0
$827$	−22.9118	−0.796722	−0.398361	−	0.917229i	$-0.630421\pi$
$827$	−22.9118	−0.796722	−0.398361	+	0.917229i	$0.630421\pi$
$828$	20.3843	0.708402
$829$	23.3829	0.812121	0.406061	−	0.913846i	$-0.366902\pi$
$829$	23.3829	0.812121	0.406061	+	0.913846i	$0.366902\pi$
$830$	−44.9665	−1.56081
$831$	−7.88574	−0.273553
$832$	0	0
$833$	0	0
$834$	0.276550	0.00957613
$835$	−3.15094	−0.109043
$836$	23.2350	0.803601
$837$	−13.0718	−0.451828
$838$	−45.0045	−1.55466
$839$	−0.734337	−0.0253521	−0.0126761	−	0.999920i	$-0.504035\pi$
$839$	−0.734337	−0.0253521	−0.0126761	+	0.999920i	$0.504035\pi$
$840$	0	0
$841$	−28.9719	−0.999032
$842$	−49.7013	−1.71282
$843$	6.09475	0.209914
$844$	−7.54629	−0.259754
$845$	0	0
$846$	−62.7042	−2.15582
$847$	0	0
$848$	0.652447	0.0224051
$849$	−4.89669	−0.168054
$850$	6.48565	0.222456
$851$	−31.2840	−1.07240
$852$	6.96750	0.238702
$853$	−54.3567	−1.86114	−0.930569	−	0.366118i	$-0.880687\pi$
$853$	−54.3567	−1.86114	−0.930569	+	0.366118i	$0.880687\pi$
$854$	0	0
$855$	13.4513	0.460026
$856$	6.10259	0.208582
$857$	21.0211	0.718067	0.359034	−	0.933325i	$-0.383106\pi$
$857$	21.0211	0.718067	0.359034	+	0.933325i	$0.383106\pi$
$858$	0	0
$859$	51.3629	1.75248	0.876240	−	0.481875i	$-0.160044\pi$
$859$	51.3629	1.75248	0.876240	+	0.481875i	$0.160044\pi$
$860$	0.0546671	0.00186413
$861$	0	0
$862$	25.3767	0.864334
$863$	7.11319	0.242136	0.121068	−	0.992644i	$-0.461368\pi$
$863$	7.11319	0.242136	0.121068	+	0.992644i	$0.461368\pi$
$864$	18.3489	0.624241
$865$	24.4735	0.832124
$866$	38.9155	1.32240
$867$	6.66385	0.226316
$868$	0	0
$869$	−3.40286	−0.115434
$870$	0.201528	0.00683244
$871$	0	0
$872$	−8.46267	−0.286582
$873$	9.83403	0.332831
$874$	−27.4124	−0.927239
$875$	0	0
$876$	10.6518	0.359892
$877$	0.512476	0.0173051	0.00865255	−	0.999963i	$-0.497246\pi$
$877$	0.512476	0.0173051	0.00865255	+	0.999963i	$0.497246\pi$
$878$	−18.6187	−0.628352
$879$	−5.65749	−0.190822
$880$	29.7835	1.00400
$881$	37.0927	1.24969	0.624843	−	0.780751i	$-0.285163\pi$
$881$	37.0927	1.24969	0.624843	+	0.780751i	$0.285163\pi$
$882$	0	0
$883$	−15.5667	−0.523860	−0.261930	−	0.965087i	$-0.584359\pi$
$883$	−15.5667	−0.523860	−0.261930	+	0.965087i	$0.584359\pi$
$884$	0	0
$885$	−3.37262	−0.113369
$886$	−40.3290	−1.35488
$887$	−27.5799	−0.926043	−0.463022	−	0.886347i	$-0.653235\pi$
$887$	−27.5799	−0.926043	−0.463022	+	0.886347i	$0.653235\pi$
$888$	−2.12316	−0.0712485
$889$	0	0
$890$	18.4081	0.617040
$891$	−32.4319	−1.08651
$892$	34.8289	1.16616
$893$	37.8792	1.26758
$894$	3.20300	0.107124
$895$	−0.794748	−0.0265655
$896$	0	0
$897$	0	0
$898$	34.5791	1.15392
$899$	−0.878932	−0.0293140
$900$	13.0006	0.433355
$901$	−0.170527	−0.00568109
$902$	−43.2337	−1.43952
$903$	0	0
$904$	6.58732	0.219091
$905$	−4.08298	−0.135723
$906$	−1.72985	−0.0574703
$907$	−45.0471	−1.49576	−0.747882	−	0.663831i	$-0.768929\pi$
$907$	−45.0471	−1.49576	−0.747882	+	0.663831i	$0.768929\pi$
$908$	17.0440	0.565625
$909$	7.26019	0.240805
$910$	0	0
$911$	35.4678	1.17510	0.587550	−	0.809188i	$-0.300093\pi$
$911$	35.4678	1.17510	0.587550	+	0.809188i	$0.300093\pi$
$912$	−6.39198	−0.211660
$913$	−70.3796	−2.32923
$914$	34.3379	1.13580
$915$	7.28720	0.240908
$916$	−23.5833	−0.779215
$917$	0	0
$918$	−5.71135	−0.188503
$919$	17.3724	0.573064	0.286532	−	0.958071i	$-0.407497\pi$
$919$	17.3724	0.573064	0.286532	+	0.958071i	$0.407497\pi$
$920$	−4.59412	−0.151464
$921$	−2.85305	−0.0940111
$922$	−56.7372	−1.86854
$923$	0	0
$924$	0	0
$925$	−19.9523	−0.656026
$926$	−33.7412	−1.10880
$927$	−47.4948	−1.55993
$928$	1.23375	0.0405000
$929$	−10.7600	−0.353025	−0.176512	−	0.984298i	$-0.556482\pi$
$929$	−10.7600	−0.353025	−0.176512	+	0.984298i	$0.556482\pi$
$930$	−6.30874	−0.206872
$931$	0	0
$932$	−15.1506	−0.496276
$933$	−0.876590	−0.0286983
$934$	−11.1079	−0.363463
$935$	−7.78439	−0.254577
$936$	0	0
$937$	10.9816	0.358755	0.179377	−	0.983780i	$-0.442592\pi$
$937$	10.9816	0.358755	0.179377	+	0.983780i	$0.442592\pi$
$938$	0	0
$939$	4.03498	0.131677
$940$	−28.0745	−0.915690
$941$	−8.17795	−0.266594	−0.133297	−	0.991076i	$-0.542556\pi$
$941$	−8.17795	−0.266594	−0.133297	+	0.991076i	$0.542556\pi$
$942$	18.0536	0.588219
$943$	22.9127	0.746141
$944$	−24.5890	−0.800305
$945$	0	0
$946$	0.190473	0.00619281
$947$	−4.59378	−0.149278	−0.0746389	−	0.997211i	$-0.523780\pi$
$947$	−4.59378	−0.149278	−0.0746389	+	0.997211i	$0.523780\pi$
$948$	−0.541273	−0.0175797
$949$	0	0
$950$	−17.4831	−0.567225
$951$	14.2978	0.463640
$952$	0	0
$953$	21.1428	0.684883	0.342442	−	0.939539i	$-0.388746\pi$
$953$	21.1428	0.684883	0.342442	+	0.939539i	$0.388746\pi$
$954$	−0.760947	−0.0246366
$955$	−29.8172	−0.964863
$956$	32.0249	1.03576
$957$	0.315423	0.0101962
$958$	27.6141	0.892170
$959$	0	0
$960$	3.04683	0.0983361
$961$	−3.48542	−0.112433
$962$	0	0
$963$	−24.4541	−0.788022
$964$	−11.9255	−0.384094
$965$	−24.1256	−0.776631
$966$	0	0
$967$	−32.0750	−1.03146	−0.515731	−	0.856750i	$-0.672480\pi$
$967$	−32.0750	−1.03146	−0.515731	+	0.856750i	$0.672480\pi$
$968$	5.83635	0.187587
$969$	1.67065	0.0536689
$970$	9.80157	0.314710
$971$	52.0835	1.67144	0.835719	−	0.549157i	$-0.185051\pi$
$971$	52.0835	1.67144	0.835719	+	0.549157i	$0.185051\pi$
$972$	−17.3535	−0.556613
$973$	0	0
$974$	34.2241	1.09661
$975$	0	0
$976$	53.1294	1.70063
$977$	19.2580	0.616117	0.308058	−	0.951367i	$-0.400321\pi$
$977$	19.2580	0.616117	0.308058	+	0.951367i	$0.400321\pi$
$978$	3.14501	0.100566
$979$	28.8115	0.920821
$980$	0	0
$981$	33.9113	1.08271
$982$	69.1736	2.20742
$983$	16.0731	0.512653	0.256327	−	0.966590i	$-0.417488\pi$
$983$	16.0731	0.512653	0.256327	+	0.966590i	$0.417488\pi$
$984$	1.55502	0.0495723
$985$	29.0700	0.926248
$986$	−0.384024	−0.0122298
$987$	0	0
$988$	0	0
$989$	−0.100946	−0.00320989
$990$	−34.7364	−1.10400
$991$	21.4265	0.680635	0.340317	−	0.940311i	$-0.389466\pi$
$991$	21.4265	0.680635	0.340317	+	0.940311i	$0.389466\pi$
$992$	−38.6221	−1.22625
$993$	−8.16876	−0.259228
$994$	0	0
$995$	20.7968	0.659304
$996$	−11.1949	−0.354723
$997$	16.9537	0.536931	0.268465	−	0.963289i	$-0.413484\pi$
$997$	16.9537	0.536931	0.268465	+	0.963289i	$0.413484\pi$
$998$	−45.1763	−1.43003
$999$	17.5702	0.555897

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	8281.2.a.ca.1.2		6
7.3	odd	6		1183.2.e.h.170.5		12
7.5	odd	6		1183.2.e.h.508.5		12
7.6	odd	2		8281.2.a.bz.1.2		6
13.3	even	3		637.2.f.j.295.5		12
13.9	even	3		637.2.f.j.393.5		12
13.12	even	2		8281.2.a.cf.1.5		6
91.3	odd	6		91.2.g.b.9.5	✓	12
91.9	even	3		637.2.g.l.263.5		12
91.12	odd	6		1183.2.e.g.508.2		12
91.16	even	3		637.2.h.l.165.2		12
91.38	odd	6		1183.2.e.g.170.2		12
91.48	odd	6		637.2.f.k.393.5		12
91.55	odd	6		637.2.f.k.295.5		12
91.61	odd	6		91.2.g.b.81.5	yes	12
91.68	odd	6		91.2.h.b.74.2	yes	12
91.74	even	3		637.2.h.l.471.2		12
91.81	even	3		637.2.g.l.373.5		12
91.87	odd	6		91.2.h.b.16.2	yes	12
91.90	odd	2		8281.2.a.ce.1.5		6
273.68	even	6		819.2.s.d.802.5		12
273.152	even	6		819.2.n.d.172.2		12
273.185	even	6		819.2.n.d.100.2		12
273.269	even	6		819.2.s.d.289.5		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.g.b.9.5	✓	12	91.3	odd	6
91.2.g.b.81.5	yes	12	91.61	odd	6
91.2.h.b.16.2	yes	12	91.87	odd	6
91.2.h.b.74.2	yes	12	91.68	odd	6
637.2.f.j.295.5		12	13.3	even	3
637.2.f.j.393.5		12	13.9	even	3
637.2.f.k.295.5		12	91.55	odd	6
637.2.f.k.393.5		12	91.48	odd	6
637.2.g.l.263.5		12	91.9	even	3
637.2.g.l.373.5		12	91.81	even	3
637.2.h.l.165.2		12	91.16	even	3
637.2.h.l.471.2		12	91.74	even	3
819.2.n.d.100.2		12	273.185	even	6
819.2.n.d.172.2		12	273.152	even	6
819.2.s.d.289.5		12	273.269	even	6
819.2.s.d.802.5		12	273.68	even	6
1183.2.e.g.170.2		12	91.38	odd	6
1183.2.e.g.508.2		12	91.12	odd	6
1183.2.e.h.170.5		12	7.3	odd	6
1183.2.e.h.508.5		12	7.5	odd	6
8281.2.a.bz.1.2		6	7.6	odd	2
8281.2.a.ca.1.2		6	1.1	even	1	trivial
8281.2.a.ce.1.5		6	91.90	odd	2
8281.2.a.cf.1.5		6	13.12	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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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	8281.2.a.ca.1.2

Level	$8281$
Weight	$2$
Character	8281.1
Self dual	yes
Analytic conductor	$66.124$
Analytic rank	$0$
Dimension	$6$
CM	no
Inner twists	$1$