LMFDB - Embedded newform 1521.2.a.w.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1521,2,Mod(1,1521)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1521, 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("1521.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1521 = 3^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1521.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:	$12.1452461474$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.1997632.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 8x^{4} + 19x^{2} - 13 $ x^6 - 8x^4 + 19x^2 - 13
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.06082$ of defining polynomial
Character	$\chi$	$=$	1521.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.14367 q^{2} -0.692021 q^{4} +3.48695 q^{5} +3.35690 q^{7} -3.07878 q^{8} +O(q^{10})$	$q+1.14367 q^{2} -0.692021 q^{4} +3.48695 q^{5} +3.35690 q^{7} -3.07878 q^{8} +3.98792 q^{10} -2.97797 q^{11} +3.83918 q^{14} -2.13706 q^{16} +6.91796 q^{17} -0.295897 q^{19} -2.41305 q^{20} -3.40581 q^{22} +2.46899 q^{23} +7.15883 q^{25} -2.32304 q^{28} +0.609790 q^{29} +9.93900 q^{31} +3.71347 q^{32} +7.91185 q^{34} +11.7053 q^{35} -8.50365 q^{37} -0.338408 q^{38} -10.7356 q^{40} -7.30123 q^{41} -6.60388 q^{43} +2.06082 q^{44} +2.82371 q^{46} +4.63062 q^{47} +4.26875 q^{49} +8.18734 q^{50} -9.38695 q^{53} -10.3840 q^{55} -10.3351 q^{56} +0.697398 q^{58} -0.816341 q^{59} +5.08815 q^{61} +11.3669 q^{62} +8.52111 q^{64} +10.2349 q^{67} -4.78738 q^{68} +13.3870 q^{70} +10.1224 q^{71} +8.74094 q^{73} -9.72536 q^{74} +0.204767 q^{76} -9.99674 q^{77} +3.52111 q^{79} -7.45184 q^{80} -8.35019 q^{82} -11.7751 q^{83} +24.1226 q^{85} -7.55265 q^{86} +9.16852 q^{88} -7.77916 q^{89} -1.70859 q^{92} +5.29590 q^{94} -1.03178 q^{95} +2.56033 q^{97} +4.88204 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 6 q^{4} + 12 q^{7}+O(q^{10}) $ 6 * q + 6 * q^4 + 12 * q^7	$ 6 q + 6 q^{4} + 12 q^{7} - 14 q^{10} - 2 q^{16} + 26 q^{19} + 6 q^{22} + 26 q^{25} + 26 q^{28} + 40 q^{31} + 40 q^{34} + 12 q^{37} - 42 q^{40} - 22 q^{43} + 2 q^{46} + 10 q^{49} - 42 q^{55} - 12 q^{58} + 38 q^{61} + 20 q^{64} + 14 q^{67} - 28 q^{70} + 24 q^{73} + 54 q^{76} - 10 q^{79} + 2 q^{82} + 14 q^{85} - 6 q^{88} + 4 q^{94} + 10 q^{97}+O(q^{100}) $ 6 * q + 6 * q^4 + 12 * q^7 - 14 * q^10 - 2 * q^16 + 26 * q^19 + 6 * q^22 + 26 * q^25 + 26 * q^28 + 40 * q^31 + 40 * q^34 + 12 * q^37 - 42 * q^40 - 22 * q^43 + 2 * q^46 + 10 * q^49 - 42 * q^55 - 12 * q^58 + 38 * q^61 + 20 * q^64 + 14 * q^67 - 28 * q^70 + 24 * q^73 + 54 * q^76 - 10 * q^79 + 2 * q^82 + 14 * q^85 - 6 * q^88 + 4 * q^94 + 10 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

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

$2$	1.14367	0.808696	0.404348	−	0.914605i	$-0.367498\pi$
$2$	1.14367	0.808696	0.404348	+	0.914605i	$0.367498\pi$
$3$	0	0
$3$	0	0
$4$	−0.692021	−0.346011
$5$	3.48695	1.55941	0.779706	−	0.626146i	$-0.215368\pi$
$5$	3.48695	1.55941	0.779706	+	0.626146i	$0.215368\pi$
$6$	0	0
$7$	3.35690	1.26879	0.634394	−	0.773010i	$-0.281250\pi$
$7$	3.35690	1.26879	0.634394	+	0.773010i	$0.281250\pi$
$8$	−3.07878	−1.08851
$9$	0	0
$10$	3.98792	1.26109
$11$	−2.97797	−0.897892	−0.448946	−	0.893559i	$-0.648200\pi$
$11$	−2.97797	−0.897892	−0.448946	+	0.893559i	$0.648200\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	3.83918	1.02606
$15$	0	0
$16$	−2.13706	−0.534266
$17$	6.91796	1.67785	0.838926	−	0.544246i	$-0.183184\pi$
$17$	6.91796	1.67785	0.838926	+	0.544246i	$0.183184\pi$
$18$	0	0
$19$	−0.295897	−0.0678834	−0.0339417	−	0.999424i	$-0.510806\pi$
$19$	−0.295897	−0.0678834	−0.0339417	+	0.999424i	$0.510806\pi$
$20$	−2.41305	−0.539573
$21$	0	0
$22$	−3.40581	−0.726122
$23$	2.46899	0.514820	0.257410	−	0.966302i	$-0.417131\pi$
$23$	2.46899	0.514820	0.257410	+	0.966302i	$0.417131\pi$
$24$	0	0
$25$	7.15883	1.43177
$26$	0	0
$27$	0	0
$28$	−2.32304	−0.439014
$29$	0.609790	0.113235	0.0566176	−	0.998396i	$-0.481968\pi$
$29$	0.609790	0.113235	0.0566176	+	0.998396i	$0.481968\pi$
$30$	0	0
$31$	9.93900	1.78510	0.892549	−	0.450951i	$-0.148915\pi$
$31$	9.93900	1.78510	0.892549	+	0.450951i	$0.148915\pi$
$32$	3.71347	0.656455
$33$	0	0
$34$	7.91185	1.35687
$35$	11.7053	1.97856
$36$	0	0
$37$	−8.50365	−1.39799	−0.698996	−	0.715126i	$-0.746369\pi$
$37$	−8.50365	−1.39799	−0.698996	+	0.715126i	$0.746369\pi$
$38$	−0.338408	−0.0548970
$39$	0	0
$40$	−10.7356	−1.69744
$41$	−7.30123	−1.14026	−0.570130	−	0.821554i	$-0.693107\pi$
$41$	−7.30123	−1.14026	−0.570130	+	0.821554i	$0.693107\pi$
$42$	0	0
$43$	−6.60388	−1.00708	−0.503541	−	0.863971i	$-0.667970\pi$
$43$	−6.60388	−1.00708	−0.503541	+	0.863971i	$0.667970\pi$
$44$	2.06082	0.310680
$45$	0	0
$46$	2.82371	0.416333
$47$	4.63062	0.675445	0.337723	−	0.941246i	$-0.390343\pi$
$47$	4.63062	0.675445	0.337723	+	0.941246i	$0.390343\pi$
$48$	0	0
$49$	4.26875	0.609821
$50$	8.18734	1.15786
$51$	0	0
$52$	0	0
$53$	−9.38695	−1.28940	−0.644698	−	0.764437i	$-0.723017\pi$
$53$	−9.38695	−1.28940	−0.644698	+	0.764437i	$0.723017\pi$
$54$	0	0
$55$	−10.3840	−1.40018
$56$	−10.3351	−1.38109
$57$	0	0
$58$	0.697398	0.0915729
$59$	−0.816341	−0.106279	−0.0531393	−	0.998587i	$-0.516923\pi$
$59$	−0.816341	−0.106279	−0.0531393	+	0.998587i	$0.516923\pi$
$60$	0	0
$61$	5.08815	0.651470	0.325735	−	0.945461i	$-0.394388\pi$
$61$	5.08815	0.651470	0.325735	+	0.945461i	$0.394388\pi$
$62$	11.3669	1.44360
$63$	0	0
$64$	8.52111	1.06514
$65$	0	0
$66$	0	0
$67$	10.2349	1.25039	0.625196	−	0.780468i	$-0.285019\pi$
$67$	10.2349	1.25039	0.625196	+	0.780468i	$0.285019\pi$
$68$	−4.78738	−0.580555
$69$	0	0
$70$	13.3870	1.60006
$71$	10.1224	1.20131	0.600657	−	0.799507i	$-0.294906\pi$
$71$	10.1224	1.20131	0.600657	+	0.799507i	$0.294906\pi$
$72$	0	0
$73$	8.74094	1.02305	0.511525	−	0.859269i	$-0.329081\pi$
$73$	8.74094	1.02305	0.511525	+	0.859269i	$0.329081\pi$
$74$	−9.72536	−1.13055
$75$	0	0
$76$	0.204767	0.0234884
$77$	−9.99674	−1.13923
$78$	0	0
$79$	3.52111	0.396155	0.198078	−	0.980186i	$-0.436530\pi$
$79$	3.52111	0.396155	0.198078	+	0.980186i	$0.436530\pi$
$80$	−7.45184	−0.833141
$81$	0	0
$82$	−8.35019	−0.922124
$83$	−11.7751	−1.29248	−0.646242	−	0.763132i	$-0.723661\pi$
$83$	−11.7751	−1.29248	−0.646242	+	0.763132i	$0.723661\pi$
$84$	0	0
$85$	24.1226	2.61646
$86$	−7.55265	−0.814423
$87$	0	0
$88$	9.16852	0.977368
$89$	−7.77916	−0.824590	−0.412295	−	0.911050i	$-0.635273\pi$
$89$	−7.77916	−0.824590	−0.412295	+	0.911050i	$0.635273\pi$
$90$	0	0
$91$	0	0
$92$	−1.70859	−0.178133
$93$	0	0
$94$	5.29590	0.546230
$95$	−1.03178	−0.105858
$96$	0	0
$97$	2.56033	0.259963	0.129981	−	0.991516i	$-0.458508\pi$
$97$	2.56033	0.259963	0.129981	+	0.991516i	$0.458508\pi$
$98$	4.88204	0.493160
$99$	0	0
$100$	−4.95407	−0.495407
$101$	−14.7841	−1.47107	−0.735537	−	0.677484i	$-0.763070\pi$
$101$	−14.7841	−1.47107	−0.735537	+	0.677484i	$0.763070\pi$
$102$	0	0
$103$	2.09783	0.206706	0.103353	−	0.994645i	$-0.467043\pi$
$103$	2.09783	0.206706	0.103353	+	0.994645i	$0.467043\pi$
$104$	0	0
$105$	0	0
$106$	−10.7356	−1.04273
$107$	6.64658	0.642549	0.321274	−	0.946986i	$-0.395889\pi$
$107$	6.64658	0.642549	0.321274	+	0.946986i	$0.395889\pi$
$108$	0	0
$109$	3.06100	0.293191	0.146595	−	0.989197i	$-0.453169\pi$
$109$	3.06100	0.293191	0.146595	+	0.989197i	$0.453169\pi$
$110$	−11.8759	−1.13232
$111$	0	0
$112$	−7.17390	−0.677870
$113$	−5.81915	−0.547420	−0.273710	−	0.961812i	$-0.588251\pi$
$113$	−5.81915	−0.547420	−0.273710	+	0.961812i	$0.588251\pi$
$114$	0	0
$115$	8.60925	0.802817
$116$	−0.421988	−0.0391806
$117$	0	0
$118$	−0.933624	−0.0859471
$119$	23.2229	2.12884
$120$	0	0
$121$	−2.13169	−0.193790
$122$	5.81915	0.526841
$123$	0	0
$124$	−6.87800	−0.617663
$125$	7.52775	0.673302
$126$	0	0
$127$	15.9879	1.41870	0.709349	−	0.704857i	$-0.248989\pi$
$127$	15.9879	1.41870	0.709349	+	0.704857i	$0.248989\pi$
$128$	2.31839	0.204918
$129$	0	0
$130$	0	0
$131$	2.80740	0.245284	0.122642	−	0.992451i	$-0.460863\pi$
$131$	2.80740	0.245284	0.122642	+	0.992451i	$0.460863\pi$
$132$	0	0
$133$	−0.993295	−0.0861296
$134$	11.7053	1.01119
$135$	0	0
$136$	−21.2989	−1.82636
$137$	1.80326	0.154062	0.0770312	−	0.997029i	$-0.475456\pi$
$137$	1.80326	0.154062	0.0770312	+	0.997029i	$0.475456\pi$
$138$	0	0
$139$	−4.26875	−0.362071	−0.181035	−	0.983477i	$-0.557945\pi$
$139$	−4.26875	−0.362071	−0.181035	+	0.983477i	$0.557945\pi$
$140$	−8.10034	−0.684604
$141$	0	0
$142$	11.5767	0.971497
$143$	0	0
$144$	0	0
$145$	2.12631	0.176580
$146$	9.99674	0.827336
$147$	0	0
$148$	5.88471	0.483720
$149$	−21.9784	−1.80054	−0.900270	−	0.435332i	$-0.856631\pi$
$149$	−21.9784	−1.80054	−0.900270	+	0.435332i	$0.856631\pi$
$150$	0	0
$151$	7.08277	0.576388	0.288194	−	0.957572i	$-0.406945\pi$
$151$	7.08277	0.576388	0.288194	+	0.957572i	$0.406945\pi$
$152$	0.911002	0.0738920
$153$	0	0
$154$	−11.4330	−0.921294
$155$	34.6568	2.78370
$156$	0	0
$157$	−2.95646	−0.235951	−0.117976	−	0.993016i	$-0.537640\pi$
$157$	−2.95646	−0.235951	−0.117976	+	0.993016i	$0.537640\pi$
$158$	4.02698	0.320369
$159$	0	0
$160$	12.9487	1.02368
$161$	8.28815	0.653197
$162$	0	0
$163$	−6.18060	−0.484102	−0.242051	−	0.970263i	$-0.577820\pi$
$163$	−6.18060	−0.484102	−0.242051	+	0.970263i	$0.577820\pi$
$164$	5.05261	0.394542
$165$	0	0
$166$	−13.4668	−1.04523
$167$	−3.78938	−0.293231	−0.146616	−	0.989194i	$-0.546838\pi$
$167$	−3.78938	−0.293231	−0.146616	+	0.989194i	$0.546838\pi$
$168$	0	0
$169$	0	0
$170$	27.5883	2.11592
$171$	0	0
$172$	4.57002	0.348461
$173$	−10.4858	−0.797217	−0.398609	−	0.917121i	$-0.630507\pi$
$173$	−10.4858	−0.797217	−0.398609	+	0.917121i	$0.630507\pi$
$174$	0	0
$175$	24.0315	1.81661
$176$	6.36411	0.479713
$177$	0	0
$178$	−8.89679	−0.666842
$179$	−10.1473	−0.758448	−0.379224	−	0.925305i	$-0.623809\pi$
$179$	−10.1473	−0.758448	−0.379224	+	0.925305i	$0.623809\pi$
$180$	0	0
$181$	−21.4426	−1.59382	−0.796910	−	0.604098i	$-0.793533\pi$
$181$	−21.4426	−1.59382	−0.796910	+	0.604098i	$0.793533\pi$
$182$	0	0
$183$	0	0
$184$	−7.60148	−0.560389
$185$	−29.6518	−2.18004
$186$	0	0
$187$	−20.6015	−1.50653
$188$	−3.20449	−0.233711
$189$	0	0
$190$	−1.18001	−0.0856071
$191$	−21.3637	−1.54582	−0.772910	−	0.634515i	$-0.781200\pi$
$191$	−21.3637	−1.54582	−0.772910	+	0.634515i	$0.781200\pi$
$192$	0	0
$193$	−2.23191	−0.160657	−0.0803283	−	0.996768i	$-0.525597\pi$
$193$	−2.23191	−0.160657	−0.0803283	+	0.996768i	$0.525597\pi$
$194$	2.92818	0.210231
$195$	0	0
$196$	−2.95407	−0.211005
$197$	2.51386	0.179105	0.0895524	−	0.995982i	$-0.471456\pi$
$197$	2.51386	0.179105	0.0895524	+	0.995982i	$0.471456\pi$
$198$	0	0
$199$	−11.5646	−0.819796	−0.409898	−	0.912131i	$-0.634436\pi$
$199$	−11.5646	−0.819796	−0.409898	+	0.912131i	$0.634436\pi$
$200$	−22.0405	−1.55850
$201$	0	0
$202$	−16.9081	−1.18965
$203$	2.04700	0.143671
$204$	0	0
$205$	−25.4590	−1.77814
$206$	2.39923	0.167162
$207$	0	0
$208$	0	0
$209$	0.881173	0.0609520
$210$	0	0
$211$	−19.9041	−1.37025	−0.685127	−	0.728424i	$-0.740253\pi$
$211$	−19.9041	−1.37025	−0.685127	+	0.728424i	$0.740253\pi$
$212$	6.49597	0.446145
$213$	0	0
$214$	7.60148	0.519627
$215$	−23.0274	−1.57046
$216$	0	0
$217$	33.3642	2.26491
$218$	3.50077	0.237102
$219$	0	0
$220$	7.18598	0.484479
$221$	0	0
$222$	0	0
$223$	18.1468	1.21520	0.607598	−	0.794245i	$-0.292133\pi$
$223$	18.1468	1.21520	0.607598	+	0.794245i	$0.292133\pi$
$224$	12.4657	0.832902
$225$	0	0
$226$	−6.65519	−0.442696
$227$	11.5796	0.768567	0.384283	−	0.923215i	$-0.374449\pi$
$227$	11.5796	0.768567	0.384283	+	0.923215i	$0.374449\pi$
$228$	0	0
$229$	−11.8605	−0.783767	−0.391883	−	0.920015i	$-0.628176\pi$
$229$	−11.8605	−0.783767	−0.391883	+	0.920015i	$0.628176\pi$
$230$	9.84613	0.649235
$231$	0	0
$232$	−1.87741	−0.123258
$233$	−19.2331	−1.26000	−0.630001	−	0.776595i	$-0.716945\pi$
$233$	−19.2331	−1.26000	−0.630001	+	0.776595i	$0.716945\pi$
$234$	0	0
$235$	16.1468	1.05330
$236$	0.564926	0.0367735
$237$	0	0
$238$	26.5593	1.72158
$239$	1.35637	0.0877362	0.0438681	−	0.999037i	$-0.486032\pi$
$239$	1.35637	0.0877362	0.0438681	+	0.999037i	$0.486032\pi$
$240$	0	0
$241$	9.64310	0.621167	0.310583	−	0.950546i	$-0.399476\pi$
$241$	9.64310	0.621167	0.310583	+	0.950546i	$0.399476\pi$
$242$	−2.43794	−0.156717
$243$	0	0
$244$	−3.52111	−0.225416
$245$	14.8849	0.950963
$246$	0	0
$247$	0	0
$248$	−30.6000	−1.94310
$249$	0	0
$250$	8.60925	0.544497
$251$	−26.9114	−1.69863	−0.849317	−	0.527882i	$-0.822986\pi$
$251$	−26.9114	−1.69863	−0.849317	+	0.527882i	$0.822986\pi$
$252$	0	0
$253$	−7.35258	−0.462253
$254$	18.2849	1.14730
$255$	0	0
$256$	−14.3907	−0.899422
$257$	−9.72536	−0.606651	−0.303326	−	0.952887i	$-0.598097\pi$
$257$	−9.72536	−0.606651	−0.303326	+	0.952887i	$0.598097\pi$
$258$	0	0
$259$	−28.5459	−1.77375
$260$	0	0
$261$	0	0
$262$	3.21073	0.198360
$263$	15.2061	0.937649	0.468824	−	0.883291i	$-0.344678\pi$
$263$	15.2061	0.937649	0.468824	+	0.883291i	$0.344678\pi$
$264$	0	0
$265$	−32.7318	−2.01070
$266$	−1.13600	−0.0696527
$267$	0	0
$268$	−7.08277	−0.432649
$269$	−1.43721	−0.0876284	−0.0438142	−	0.999040i	$-0.513951\pi$
$269$	−1.43721	−0.0876284	−0.0438142	+	0.999040i	$0.513951\pi$
$270$	0	0
$271$	7.32006	0.444662	0.222331	−	0.974971i	$-0.428633\pi$
$271$	7.32006	0.444662	0.222331	+	0.974971i	$0.428633\pi$
$272$	−14.7841	−0.896419
$273$	0	0
$274$	2.06233	0.124590
$275$	−21.3188	−1.28557
$276$	0	0
$277$	−0.610580	−0.0366862	−0.0183431	−	0.999832i	$-0.505839\pi$
$277$	−0.610580	−0.0366862	−0.0183431	+	0.999832i	$0.505839\pi$
$278$	−4.88204	−0.292805
$279$	0	0
$280$	−36.0382	−2.15369
$281$	−11.2772	−0.672741	−0.336371	−	0.941730i	$-0.609199\pi$
$281$	−11.2772	−0.672741	−0.336371	+	0.941730i	$0.609199\pi$
$282$	0	0
$283$	−10.5894	−0.629475	−0.314737	−	0.949179i	$-0.601916\pi$
$283$	−10.5894	−0.629475	−0.314737	+	0.949179i	$0.601916\pi$
$284$	−7.00495	−0.415667
$285$	0	0
$286$	0	0
$287$	−24.5095	−1.44675
$288$	0	0
$289$	30.8582	1.81519
$290$	2.43179	0.142800
$291$	0	0
$292$	−6.04892	−0.353986
$293$	13.4028	0.783003	0.391501	−	0.920178i	$-0.371956\pi$
$293$	13.4028	0.783003	0.391501	+	0.920178i	$0.371956\pi$
$294$	0	0
$295$	−2.84654	−0.165732
$296$	26.1809	1.52173
$297$	0	0
$298$	−25.1360	−1.45609
$299$	0	0
$300$	0	0
$301$	−22.1685	−1.27777
$302$	8.10034	0.466122
$303$	0	0
$304$	0.632351	0.0362678
$305$	17.7421	1.01591
$306$	0	0
$307$	−18.0476	−1.03003	−0.515015	−	0.857181i	$-0.672214\pi$
$307$	−18.0476	−1.03003	−0.515015	+	0.857181i	$0.672214\pi$
$308$	6.91796	0.394187
$309$	0	0
$310$	39.6359	2.25117
$311$	−28.4993	−1.61604	−0.808022	−	0.589152i	$-0.799462\pi$
$311$	−28.4993	−1.61604	−0.808022	+	0.589152i	$0.799462\pi$
$312$	0	0
$313$	20.2911	1.14692	0.573461	−	0.819233i	$-0.305601\pi$
$313$	20.2911	1.14692	0.573461	+	0.819233i	$0.305601\pi$
$314$	−3.38121	−0.190813
$315$	0	0
$316$	−2.43668	−0.137074
$317$	2.54983	0.143213	0.0716065	−	0.997433i	$-0.477187\pi$
$317$	2.54983	0.143213	0.0716065	+	0.997433i	$0.477187\pi$
$318$	0	0
$319$	−1.81594	−0.101673
$320$	29.7127	1.66099
$321$	0	0
$322$	9.47889	0.528238
$323$	−2.04700	−0.113898
$324$	0	0
$325$	0	0
$326$	−7.06856	−0.391492
$327$	0	0
$328$	22.4789	1.24119
$329$	15.5445	0.856997
$330$	0	0
$331$	−0.993295	−0.0545964	−0.0272982	−	0.999627i	$-0.508690\pi$
$331$	−0.993295	−0.0545964	−0.0272982	+	0.999627i	$0.508690\pi$
$332$	8.14862	0.447214
$333$	0	0
$334$	−4.33380	−0.237135
$335$	35.6886	1.94988
$336$	0	0
$337$	20.4959	1.11648	0.558241	−	0.829679i	$-0.311477\pi$
$337$	20.4959	1.11648	0.558241	+	0.829679i	$0.311477\pi$
$338$	0	0
$339$	0	0
$340$	−16.6933	−0.905324
$341$	−29.5981	−1.60282
$342$	0	0
$343$	−9.16852	−0.495054
$344$	20.3319	1.09622
$345$	0	0
$346$	−11.9922	−0.644706
$347$	15.3939	0.826388	0.413194	−	0.910643i	$-0.364413\pi$
$347$	15.3939	0.826388	0.413194	+	0.910643i	$0.364413\pi$
$348$	0	0
$349$	3.72587	0.199441	0.0997207	−	0.995015i	$-0.468205\pi$
$349$	3.72587	0.199441	0.0997207	+	0.995015i	$0.468205\pi$
$350$	27.4840	1.46908
$351$	0	0
$352$	−11.0586	−0.589426
$353$	−13.0307	−0.693552	−0.346776	−	0.937948i	$-0.612724\pi$
$353$	−13.0307	−0.693552	−0.346776	+	0.937948i	$0.612724\pi$
$354$	0	0
$355$	35.2965	1.87334
$356$	5.38335	0.285317
$357$	0	0
$358$	−11.6052	−0.613354
$359$	−25.2339	−1.33179	−0.665897	−	0.746044i	$-0.731951\pi$
$359$	−25.2339	−1.33179	−0.665897	+	0.746044i	$0.731951\pi$
$360$	0	0
$361$	−18.9124	−0.995392
$362$	−24.5233	−1.28892
$363$	0	0
$364$	0	0
$365$	30.4792	1.59536
$366$	0	0
$367$	5.87369	0.306604	0.153302	−	0.988179i	$-0.451009\pi$
$367$	5.87369	0.306604	0.153302	+	0.988179i	$0.451009\pi$
$368$	−5.27639	−0.275051
$369$	0	0
$370$	−33.9119	−1.76299
$371$	−31.5110	−1.63597
$372$	0	0
$373$	11.4330	0.591976	0.295988	−	0.955192i	$-0.404351\pi$
$373$	11.4330	0.591976	0.295988	+	0.955192i	$0.404351\pi$
$374$	−23.5613	−1.21832
$375$	0	0
$376$	−14.2567	−0.735232
$377$	0	0
$378$	0	0
$379$	26.5773	1.36519	0.682593	−	0.730799i	$-0.260852\pi$
$379$	26.5773	1.36519	0.682593	+	0.730799i	$0.260852\pi$
$380$	0.714013	0.0366281
$381$	0	0
$382$	−24.4330	−1.25010
$383$	3.74725	0.191476	0.0957379	−	0.995407i	$-0.469479\pi$
$383$	3.74725	0.191476	0.0957379	+	0.995407i	$0.469479\pi$
$384$	0	0
$385$	−34.8582	−1.77654
$386$	−2.55257	−0.129922
$387$	0	0
$388$	−1.77181	−0.0899499
$389$	6.57955	0.333596	0.166798	−	0.985991i	$-0.446657\pi$
$389$	6.57955	0.333596	0.166798	+	0.985991i	$0.446657\pi$
$390$	0	0
$391$	17.0804	0.863792
$392$	−13.1425	−0.663799
$393$	0	0
$394$	2.87502	0.144841
$395$	12.2779	0.617770
$396$	0	0
$397$	−11.2101	−0.562621	−0.281310	−	0.959617i	$-0.590769\pi$
$397$	−11.2101	−0.562621	−0.281310	+	0.959617i	$0.590769\pi$
$398$	−13.2261	−0.662966
$399$	0	0
$400$	−15.2989	−0.764944
$401$	20.8547	1.04143	0.520717	−	0.853730i	$-0.325665\pi$
$401$	20.8547	1.04143	0.520717	+	0.853730i	$0.325665\pi$
$402$	0	0
$403$	0	0
$404$	10.2309	0.509008
$405$	0	0
$406$	2.34109	0.116187
$407$	25.3236	1.25525
$408$	0	0
$409$	36.5163	1.80562	0.902808	−	0.430044i	$-0.141502\pi$
$409$	36.5163	1.80562	0.902808	+	0.430044i	$0.141502\pi$
$410$	−29.1167	−1.43797
$411$	0	0
$412$	−1.45175	−0.0715224
$413$	−2.74037	−0.134845
$414$	0	0
$415$	−41.0592	−2.01552
$416$	0	0
$417$	0	0
$418$	1.00777	0.0492916
$419$	9.99674	0.488373	0.244186	−	0.969728i	$-0.421479\pi$
$419$	9.99674	0.488373	0.244186	+	0.969728i	$0.421479\pi$
$420$	0	0
$421$	−0.770479	−0.0375508	−0.0187754	−	0.999824i	$-0.505977\pi$
$421$	−0.770479	−0.0375508	−0.0187754	+	0.999824i	$0.505977\pi$
$422$	−22.7637	−1.10812
$423$	0	0
$424$	28.9004	1.40353
$425$	49.5245	2.40229
$426$	0	0
$427$	17.0804	0.826577
$428$	−4.59957	−0.222329
$429$	0	0
$430$	−26.3357	−1.27002
$431$	30.2838	1.45872	0.729359	−	0.684131i	$-0.239819\pi$
$431$	30.2838	1.45872	0.729359	+	0.684131i	$0.239819\pi$
$432$	0	0
$433$	−28.0713	−1.34902	−0.674510	−	0.738266i	$-0.735645\pi$
$433$	−28.0713	−1.34902	−0.674510	+	0.738266i	$0.735645\pi$
$434$	38.1576	1.83162
$435$	0	0
$436$	−2.11828	−0.101447
$437$	−0.730567	−0.0349478
$438$	0	0
$439$	−20.1347	−0.960975	−0.480488	−	0.877001i	$-0.659540\pi$
$439$	−20.1347	−0.960975	−0.480488	+	0.877001i	$0.659540\pi$
$440$	31.9702	1.52412
$441$	0	0
$442$	0	0
$443$	31.8494	1.51321	0.756606	−	0.653871i	$-0.226856\pi$
$443$	31.8494	1.51321	0.756606	+	0.653871i	$0.226856\pi$
$444$	0	0
$445$	−27.1256	−1.28588
$446$	20.7539	0.982724
$447$	0	0
$448$	28.6045	1.35143
$449$	3.47313	0.163907	0.0819537	−	0.996636i	$-0.473884\pi$
$449$	3.47313	0.163907	0.0819537	+	0.996636i	$0.473884\pi$
$450$	0	0
$451$	21.7429	1.02383
$452$	4.02698	0.189413
$453$	0	0
$454$	13.2433	0.621537
$455$	0	0
$456$	0	0
$457$	21.1021	0.987117	0.493558	−	0.869713i	$-0.335696\pi$
$457$	21.1021	0.987117	0.493558	+	0.869713i	$0.335696\pi$
$458$	−13.5645	−0.633829
$459$	0	0
$460$	−5.95779	−0.277783
$461$	7.19427	0.335071	0.167535	−	0.985866i	$-0.446419\pi$
$461$	7.19427	0.335071	0.167535	+	0.985866i	$0.446419\pi$
$462$	0	0
$463$	33.1594	1.54105	0.770525	−	0.637410i	$-0.219994\pi$
$463$	33.1594	1.54105	0.770525	+	0.637410i	$0.219994\pi$
$464$	−1.30316	−0.0604977
$465$	0	0
$466$	−21.9963	−1.01896
$467$	21.6350	1.00115	0.500575	−	0.865693i	$-0.333122\pi$
$467$	21.6350	1.00115	0.500575	+	0.865693i	$0.333122\pi$
$468$	0	0
$469$	34.3575	1.58648
$470$	18.4665	0.851798
$471$	0	0
$472$	2.51334	0.115686
$473$	19.6662	0.904251
$474$	0	0
$475$	−2.11828	−0.0971932
$476$	−16.0707	−0.736600
$477$	0	0
$478$	1.55124	0.0709519
$479$	−31.0020	−1.41652	−0.708259	−	0.705952i	$-0.750519\pi$
$479$	−31.0020	−1.41652	−0.708259	+	0.705952i	$0.750519\pi$
$480$	0	0
$481$	0	0
$482$	11.0285	0.502335
$483$	0	0
$484$	1.47517	0.0670533
$485$	8.92776	0.405389
$486$	0	0
$487$	−27.3478	−1.23925	−0.619624	−	0.784899i	$-0.712715\pi$
$487$	−27.3478	−1.23925	−0.619624	+	0.784899i	$0.712715\pi$
$488$	−15.6653	−0.709134
$489$	0	0
$490$	17.0234	0.769040
$491$	15.0017	0.677019	0.338510	−	0.940963i	$-0.390077\pi$
$491$	15.0017	0.677019	0.338510	+	0.940963i	$0.390077\pi$
$492$	0	0
$493$	4.21850	0.189992
$494$	0	0
$495$	0	0
$496$	−21.2403	−0.953716
$497$	33.9800	1.52421
$498$	0	0
$499$	5.85862	0.262268	0.131134	−	0.991365i	$-0.458138\pi$
$499$	5.85862	0.262268	0.131134	+	0.991365i	$0.458138\pi$
$500$	−5.20936	−0.232970
$501$	0	0
$502$	−30.7778	−1.37368
$503$	−25.8797	−1.15392	−0.576958	−	0.816773i	$-0.695761\pi$
$503$	−25.8797	−1.15392	−0.576958	+	0.816773i	$0.695761\pi$
$504$	0	0
$505$	−51.5515	−2.29401
$506$	−8.40892	−0.373822
$507$	0	0
$508$	−11.0640	−0.490885
$509$	−10.5306	−0.466761	−0.233381	−	0.972385i	$-0.574979\pi$
$509$	−10.5306	−0.466761	−0.233381	+	0.972385i	$0.574979\pi$
$510$	0	0
$511$	29.3424	1.29803
$512$	−21.0950	−0.932277
$513$	0	0
$514$	−11.1226	−0.490596
$515$	7.31505	0.322340
$516$	0	0
$517$	−13.7899	−0.606477
$518$	−32.6470	−1.43443
$519$	0	0
$520$	0	0
$521$	−10.5395	−0.461744	−0.230872	−	0.972984i	$-0.574158\pi$
$521$	−10.5395	−0.461744	−0.230872	+	0.972984i	$0.574158\pi$
$522$	0	0
$523$	−4.97152	−0.217390	−0.108695	−	0.994075i	$-0.534667\pi$
$523$	−4.97152	−0.217390	−0.108695	+	0.994075i	$0.534667\pi$
$524$	−1.94278	−0.0848708
$525$	0	0
$526$	17.3907	0.758273
$527$	68.7576	2.99513
$528$	0	0
$529$	−16.9041	−0.734960
$530$	−37.4344	−1.62605
$531$	0	0
$532$	0.687382	0.0298018
$533$	0	0
$534$	0	0
$535$	23.1763	1.00200
$536$	−31.5110	−1.36107
$537$	0	0
$538$	−1.64370	−0.0708647
$539$	−12.7122	−0.547554
$540$	0	0
$541$	−1.99462	−0.0857555	−0.0428778	−	0.999080i	$-0.513653\pi$
$541$	−1.99462	−0.0857555	−0.0428778	+	0.999080i	$0.513653\pi$
$542$	8.37173	0.359596
$543$	0	0
$544$	25.6896	1.10143
$545$	10.6736	0.457205
$546$	0	0
$547$	−37.6872	−1.61139	−0.805695	−	0.592331i	$-0.798208\pi$
$547$	−37.6872	−1.61139	−0.805695	+	0.592331i	$0.798208\pi$
$548$	−1.24789	−0.0533073
$549$	0	0
$550$	−24.3817	−1.03964
$551$	−0.180435	−0.00768679
$552$	0	0
$553$	11.8200	0.502637
$554$	−0.698302	−0.0296680
$555$	0	0
$556$	2.95407	0.125280
$557$	−26.4695	−1.12155	−0.560774	−	0.827969i	$-0.689496\pi$
$557$	−26.4695	−1.12155	−0.560774	+	0.827969i	$0.689496\pi$
$558$	0	0
$559$	0	0
$560$	−25.0150	−1.05708
$561$	0	0
$562$	−12.8974	−0.544043
$563$	6.79718	0.286467	0.143234	−	0.989689i	$-0.454250\pi$
$563$	6.79718	0.286467	0.143234	+	0.989689i	$0.454250\pi$
$564$	0	0
$565$	−20.2911	−0.853653
$566$	−12.1108	−0.509054
$567$	0	0
$568$	−31.1648	−1.30765
$569$	38.2784	1.60471	0.802356	−	0.596845i	$-0.203579\pi$
$569$	38.2784	1.60471	0.802356	+	0.596845i	$0.203579\pi$
$570$	0	0
$571$	29.7375	1.24447	0.622237	−	0.782829i	$-0.286224\pi$
$571$	29.7375	1.24447	0.622237	+	0.782829i	$0.286224\pi$
$572$	0	0
$573$	0	0
$574$	−28.0307	−1.16998
$575$	17.6751	0.737102
$576$	0	0
$577$	11.7549	0.489365	0.244682	−	0.969603i	$-0.421316\pi$
$577$	11.7549	0.489365	0.244682	+	0.969603i	$0.421316\pi$
$578$	35.2915	1.46793
$579$	0	0
$580$	−1.47145	−0.0610987
$581$	−39.5278	−1.63989
$582$	0	0
$583$	27.9541	1.15774
$584$	−26.9114	−1.11360
$585$	0	0
$586$	15.3284	0.633211
$587$	−18.2090	−0.751565	−0.375782	−	0.926708i	$-0.622626\pi$
$587$	−18.2090	−0.751565	−0.375782	+	0.926708i	$0.622626\pi$
$588$	0	0
$589$	−2.94092	−0.121178
$590$	−3.25550	−0.134027
$591$	0	0
$592$	18.1728	0.746899
$593$	31.5421	1.29528	0.647639	−	0.761948i	$-0.275757\pi$
$593$	31.5421	1.29528	0.647639	+	0.761948i	$0.275757\pi$
$594$	0	0
$595$	80.9770	3.31973
$596$	15.2095	0.623006
$597$	0	0
$598$	0	0
$599$	−32.4592	−1.32625	−0.663124	−	0.748510i	$-0.730770\pi$
$599$	−32.4592	−1.32625	−0.663124	+	0.748510i	$0.730770\pi$
$600$	0	0
$601$	8.47112	0.345544	0.172772	−	0.984962i	$-0.444728\pi$
$601$	8.47112	0.345544	0.172772	+	0.984962i	$0.444728\pi$
$602$	−25.3534	−1.03333
$603$	0	0
$604$	−4.90143	−0.199436
$605$	−7.43309	−0.302198
$606$	0	0
$607$	−23.9342	−0.971460	−0.485730	−	0.874109i	$-0.661446\pi$
$607$	−23.9342	−0.971460	−0.485730	+	0.874109i	$0.661446\pi$
$608$	−1.09880	−0.0445624
$609$	0	0
$610$	20.2911	0.821563
$611$	0	0
$612$	0	0
$613$	−38.8864	−1.57061	−0.785303	−	0.619112i	$-0.787493\pi$
$613$	−38.8864	−1.57061	−0.785303	+	0.619112i	$0.787493\pi$
$614$	−20.6405	−0.832981
$615$	0	0
$616$	30.7778	1.24007
$617$	−34.1927	−1.37655	−0.688273	−	0.725452i	$-0.741631\pi$
$617$	−34.1927	−1.37655	−0.688273	+	0.725452i	$0.741631\pi$
$618$	0	0
$619$	43.0398	1.72992	0.864958	−	0.501844i	$-0.167345\pi$
$619$	43.0398	1.72992	0.864958	+	0.501844i	$0.167345\pi$
$620$	−23.9833	−0.963191
$621$	0	0
$622$	−32.5937	−1.30689
$623$	−26.1138	−1.04623
$624$	0	0
$625$	−9.54527	−0.381811
$626$	23.2063	0.927511
$627$	0	0
$628$	2.04593	0.0816416
$629$	−58.8279	−2.34562
$630$	0	0
$631$	4.92394	0.196019	0.0980094	−	0.995185i	$-0.468752\pi$
$631$	4.92394	0.196019	0.0980094	+	0.995185i	$0.468752\pi$
$632$	−10.8407	−0.431221
$633$	0	0
$634$	2.91617	0.115816
$635$	55.7491	2.21234
$636$	0	0
$637$	0	0
$638$	−2.07683	−0.0822226
$639$	0	0
$640$	8.08410	0.319552
$641$	−29.8396	−1.17859	−0.589297	−	0.807917i	$-0.700595\pi$
$641$	−29.8396	−1.17859	−0.589297	+	0.807917i	$0.700595\pi$
$642$	0	0
$643$	13.7885	0.543766	0.271883	−	0.962330i	$-0.412354\pi$
$643$	13.7885	0.543766	0.271883	+	0.962330i	$0.412354\pi$
$644$	−5.73557	−0.226013
$645$	0	0
$646$	−2.34109	−0.0921091
$647$	−41.4540	−1.62972	−0.814862	−	0.579654i	$-0.803188\pi$
$647$	−41.4540	−1.62972	−0.814862	+	0.579654i	$0.803188\pi$
$648$	0	0
$649$	2.43104	0.0954267
$650$	0	0
$651$	0	0
$652$	4.27711	0.167505
$653$	39.5948	1.54946	0.774732	−	0.632290i	$-0.217885\pi$
$653$	39.5948	1.54946	0.774732	+	0.632290i	$0.217885\pi$
$654$	0	0
$655$	9.78927	0.382498
$656$	15.6032	0.609202
$657$	0	0
$658$	17.7778	0.693050
$659$	24.7437	0.963876	0.481938	−	0.876205i	$-0.339933\pi$
$659$	24.7437	0.963876	0.481938	+	0.876205i	$0.339933\pi$
$660$	0	0
$661$	21.2664	0.827165	0.413583	−	0.910467i	$-0.364277\pi$
$661$	21.2664	0.827165	0.413583	+	0.910467i	$0.364277\pi$
$662$	−1.13600	−0.0441519
$663$	0	0
$664$	36.2529	1.40689
$665$	−3.46357	−0.134312
$666$	0	0
$667$	1.50557	0.0582958
$668$	2.62233	0.101461
$669$	0	0
$670$	40.8159	1.57686
$671$	−15.1524	−0.584950
$672$	0	0
$673$	12.7821	0.492713	0.246357	−	0.969179i	$-0.420767\pi$
$673$	12.7821	0.492713	0.246357	+	0.969179i	$0.420767\pi$
$674$	23.4405	0.902894
$675$	0	0
$676$	0	0
$677$	−17.5915	−0.676097	−0.338048	−	0.941129i	$-0.609767\pi$
$677$	−17.5915	−0.676097	−0.338048	+	0.941129i	$0.609767\pi$
$678$	0	0
$679$	8.59478	0.329837
$680$	−74.2682	−2.84805
$681$	0	0
$682$	−33.8504	−1.29620
$683$	27.2609	1.04311	0.521555	−	0.853218i	$-0.325352\pi$
$683$	27.2609	1.04311	0.521555	+	0.853218i	$0.325352\pi$
$684$	0	0
$685$	6.28786	0.240247
$686$	−10.4858	−0.400348
$687$	0	0
$688$	14.1129	0.538049
$689$	0	0
$690$	0	0
$691$	27.5851	1.04939	0.524693	−	0.851291i	$-0.324180\pi$
$691$	27.5851	1.04939	0.524693	+	0.851291i	$0.324180\pi$
$692$	7.25637	0.275846
$693$	0	0
$694$	17.6055	0.668297
$695$	−14.8849	−0.564617
$696$	0	0
$697$	−50.5096	−1.91319
$698$	4.26117	0.161287
$699$	0	0
$700$	−16.6303	−0.628566
$701$	20.1813	0.762236	0.381118	−	0.924526i	$-0.375539\pi$
$701$	20.1813	0.762236	0.381118	+	0.924526i	$0.375539\pi$
$702$	0	0
$703$	2.51620	0.0949004
$704$	−25.3756	−0.956379
$705$	0	0
$706$	−14.9028	−0.560873
$707$	−49.6287	−1.86648
$708$	0	0
$709$	36.9764	1.38868	0.694339	−	0.719648i	$-0.255697\pi$
$709$	36.9764	1.38868	0.694339	+	0.719648i	$0.255697\pi$
$710$	40.3675	1.51497
$711$	0	0
$712$	23.9503	0.897577
$713$	24.5393	0.919004
$714$	0	0
$715$	0	0
$716$	7.02218	0.262431
$717$	0	0
$718$	−28.8592	−1.07702
$719$	−2.87442	−0.107198	−0.0535990	−	0.998563i	$-0.517069\pi$
$719$	−2.87442	−0.107198	−0.0535990	+	0.998563i	$0.517069\pi$
$720$	0	0
$721$	7.04221	0.262266
$722$	−21.6296	−0.804969
$723$	0	0
$724$	14.8388	0.551479
$725$	4.36539	0.162126
$726$	0	0
$727$	15.5459	0.576564	0.288282	−	0.957546i	$-0.406916\pi$
$727$	15.5459	0.576564	0.288282	+	0.957546i	$0.406916\pi$
$728$	0	0
$729$	0	0
$730$	34.8582	1.29016
$731$	−45.6853	−1.68973
$732$	0	0
$733$	37.2723	1.37668	0.688342	−	0.725386i	$-0.258339\pi$
$733$	37.2723	1.37668	0.688342	+	0.725386i	$0.258339\pi$
$734$	6.71756	0.247950
$735$	0	0
$736$	9.16852	0.337956
$737$	−30.4792	−1.12272
$738$	0	0
$739$	−29.5652	−1.08757	−0.543787	−	0.839223i	$-0.683010\pi$
$739$	−29.5652	−1.08757	−0.543787	+	0.839223i	$0.683010\pi$
$740$	20.5197	0.754319
$741$	0	0
$742$	−36.0382	−1.32300
$743$	−5.28528	−0.193898	−0.0969490	−	0.995289i	$-0.530908\pi$
$743$	−5.28528	−0.193898	−0.0969490	+	0.995289i	$0.530908\pi$
$744$	0	0
$745$	−76.6376	−2.80778
$746$	13.0755	0.478729
$747$	0	0
$748$	14.2567	0.521275
$749$	22.3119	0.815258
$750$	0	0
$751$	11.7616	0.429188	0.214594	−	0.976703i	$-0.431157\pi$
$751$	11.7616	0.429188	0.214594	+	0.976703i	$0.431157\pi$
$752$	−9.89593	−0.360867
$753$	0	0
$754$	0	0
$755$	24.6973	0.898826
$756$	0	0
$757$	−50.1226	−1.82174	−0.910868	−	0.412697i	$-0.864587\pi$
$757$	−50.1226	−1.82174	−0.910868	+	0.412697i	$0.864587\pi$
$758$	30.3957	1.10402
$759$	0	0
$760$	3.17662	0.115228
$761$	−20.5473	−0.744840	−0.372420	−	0.928064i	$-0.621472\pi$
$761$	−20.5473	−0.744840	−0.372420	+	0.928064i	$0.621472\pi$
$762$	0	0
$763$	10.2755	0.371996
$764$	14.7841	0.534871
$765$	0	0
$766$	4.28562	0.154846
$767$	0	0
$768$	0	0
$769$	−40.8745	−1.47397	−0.736987	−	0.675907i	$-0.763752\pi$
$769$	−40.8745	−1.47397	−0.736987	+	0.675907i	$0.763752\pi$
$770$	−39.8662	−1.43668
$771$	0	0
$772$	1.54453	0.0555889
$773$	−26.7857	−0.963416	−0.481708	−	0.876332i	$-0.659983\pi$
$773$	−26.7857	−0.963416	−0.481708	+	0.876332i	$0.659983\pi$
$774$	0	0
$775$	71.1517	2.55584
$776$	−7.88271	−0.282973
$777$	0	0
$778$	7.52483	0.269778
$779$	2.16041	0.0774048
$780$	0	0
$781$	−30.1444	−1.07865
$782$	19.5343	0.698545
$783$	0	0
$784$	−9.12259	−0.325807
$785$	−10.3090	−0.367945
$786$	0	0
$787$	31.3631	1.11797	0.558987	−	0.829176i	$-0.311190\pi$
$787$	31.3631	1.11797	0.558987	+	0.829176i	$0.311190\pi$
$788$	−1.73964	−0.0619722
$789$	0	0
$790$	14.0419	0.499588
$791$	−19.5343	−0.694560
$792$	0	0
$793$	0	0
$794$	−12.8207	−0.454989
$795$	0	0
$796$	8.00298	0.283658
$797$	18.4727	0.654336	0.327168	−	0.944966i	$-0.393906\pi$
$797$	18.4727	0.654336	0.327168	+	0.944966i	$0.393906\pi$
$798$	0	0
$799$	32.0344	1.13330
$800$	26.5841	0.939890
$801$	0	0
$802$	23.8509	0.842203
$803$	−26.0303	−0.918588
$804$	0	0
$805$	28.9004	1.01860
$806$	0	0
$807$	0	0
$808$	45.5171	1.60128
$809$	37.1258	1.30527	0.652637	−	0.757671i	$-0.273663\pi$
$809$	37.1258	1.30527	0.652637	+	0.757671i	$0.273663\pi$
$810$	0	0
$811$	5.15319	0.180953	0.0904766	−	0.995899i	$-0.471161\pi$
$811$	5.15319	0.180953	0.0904766	+	0.995899i	$0.471161\pi$
$812$	−1.41657	−0.0497119
$813$	0	0
$814$	28.9618	1.01511
$815$	−21.5515	−0.754915
$816$	0	0
$817$	1.95407	0.0683641
$818$	41.7626	1.46019
$819$	0	0
$820$	17.6182	0.615254
$821$	14.4368	0.503849	0.251924	−	0.967747i	$-0.418937\pi$
$821$	14.4368	0.503849	0.251924	+	0.967747i	$0.418937\pi$
$822$	0	0
$823$	29.8568	1.04074	0.520372	−	0.853940i	$-0.325793\pi$
$823$	29.8568	1.04074	0.520372	+	0.853940i	$0.325793\pi$
$824$	−6.45877	−0.225002
$825$	0	0
$826$	−3.13408	−0.109049
$827$	43.5486	1.51433	0.757167	−	0.653222i	$-0.226583\pi$
$827$	43.5486	1.51433	0.757167	+	0.653222i	$0.226583\pi$
$828$	0	0
$829$	−35.3327	−1.22716	−0.613578	−	0.789634i	$-0.710270\pi$
$829$	−35.3327	−1.22716	−0.613578	+	0.789634i	$0.710270\pi$
$830$	−46.9581	−1.62994
$831$	0	0
$832$	0	0
$833$	29.5310	1.02319
$834$	0	0
$835$	−13.2134	−0.457268
$836$	−0.609790	−0.0210900
$837$	0	0
$838$	11.4330	0.394945
$839$	36.6107	1.26394	0.631971	−	0.774992i	$-0.282246\pi$
$839$	36.6107	1.26394	0.631971	+	0.774992i	$0.282246\pi$
$840$	0	0
$841$	−28.6282	−0.987178
$842$	−0.881173	−0.0303672
$843$	0	0
$844$	13.7741	0.474122
$845$	0	0
$846$	0	0
$847$	−7.15585	−0.245878
$848$	20.0605	0.688881
$849$	0	0
$850$	56.6396	1.94272
$851$	−20.9954	−0.719714
$852$	0	0
$853$	8.47757	0.290266	0.145133	−	0.989412i	$-0.453639\pi$
$853$	8.47757	0.290266	0.145133	+	0.989412i	$0.453639\pi$
$854$	19.5343	0.668450
$855$	0	0
$856$	−20.4634	−0.699423
$857$	15.0555	0.514286	0.257143	−	0.966373i	$-0.417219\pi$
$857$	15.0555	0.514286	0.257143	+	0.966373i	$0.417219\pi$
$858$	0	0
$859$	−53.1952	−1.81500	−0.907498	−	0.420056i	$-0.862011\pi$
$859$	−53.1952	−1.81500	−0.907498	+	0.420056i	$0.862011\pi$
$860$	15.9355	0.543394
$861$	0	0
$862$	34.6346	1.17966
$863$	42.7433	1.45500	0.727500	−	0.686107i	$-0.240682\pi$
$863$	42.7433	1.45500	0.727500	+	0.686107i	$0.240682\pi$
$864$	0	0
$865$	−36.5633	−1.24319
$866$	−32.1042	−1.09095
$867$	0	0
$868$	−23.0887	−0.783683
$869$	−10.4858	−0.355705
$870$	0	0
$871$	0	0
$872$	−9.42415	−0.319142
$873$	0	0
$874$	−0.835527	−0.0282621
$875$	25.2699	0.854278
$876$	0	0
$877$	−15.7767	−0.532742	−0.266371	−	0.963871i	$-0.585825\pi$
$877$	−15.7767	−0.532742	−0.266371	+	0.963871i	$0.585825\pi$
$878$	−23.0274	−0.777137
$879$	0	0
$880$	22.1914	0.748071
$881$	−7.46072	−0.251358	−0.125679	−	0.992071i	$-0.540111\pi$
$881$	−7.46072	−0.251358	−0.125679	+	0.992071i	$0.540111\pi$
$882$	0	0
$883$	−2.54586	−0.0856750	−0.0428375	−	0.999082i	$-0.513640\pi$
$883$	−2.54586	−0.0856750	−0.0428375	+	0.999082i	$0.513640\pi$
$884$	0	0
$885$	0	0
$886$	36.4252	1.22373
$887$	−3.48422	−0.116988	−0.0584942	−	0.998288i	$-0.518630\pi$
$887$	−3.48422	−0.116988	−0.0584942	+	0.998288i	$0.518630\pi$
$888$	0	0
$889$	53.6698	1.80003
$890$	−31.0227	−1.03988
$891$	0	0
$892$	−12.5579	−0.420471
$893$	−1.37019	−0.0458515
$894$	0	0
$895$	−35.3833	−1.18273
$896$	7.78258	0.259998
$897$	0	0
$898$	3.97212	0.132551
$899$	6.06071	0.202136
$900$	0	0
$901$	−64.9385	−2.16342
$902$	24.8666	0.827968
$903$	0	0
$904$	17.9159	0.595874
$905$	−74.7695	−2.48542
$906$	0	0
$907$	40.4209	1.34215	0.671077	−	0.741388i	$-0.265832\pi$
$907$	40.4209	1.34215	0.671077	+	0.741388i	$0.265832\pi$
$908$	−8.01335	−0.265932
$909$	0	0
$910$	0	0
$911$	48.5763	1.60941	0.804703	−	0.593678i	$-0.202325\pi$
$911$	48.5763	1.60941	0.804703	+	0.593678i	$0.202325\pi$
$912$	0	0
$913$	35.0659	1.16051
$914$	24.1339	0.798277
$915$	0	0
$916$	8.20775	0.271192
$917$	9.42415	0.311213
$918$	0	0
$919$	−35.6937	−1.17743	−0.588713	−	0.808342i	$-0.700365\pi$
$919$	−35.6937	−1.17743	−0.588713	+	0.808342i	$0.700365\pi$
$920$	−26.5060	−0.873877
$921$	0	0
$922$	8.22787	0.270970
$923$	0	0
$924$	0	0
$925$	−60.8762	−2.00160
$926$	37.9234	1.24624
$927$	0	0
$928$	2.26444	0.0743338
$929$	22.5411	0.739550	0.369775	−	0.929121i	$-0.379435\pi$
$929$	22.5411	0.739550	0.369775	+	0.929121i	$0.379435\pi$
$930$	0	0
$931$	−1.26311	−0.0413968
$932$	13.3097	0.435974
$933$	0	0
$934$	24.7433	0.809627
$935$	−71.8364	−2.34930
$936$	0	0
$937$	−32.0344	−1.04652	−0.523260	−	0.852173i	$-0.675284\pi$
$937$	−32.0344	−1.04652	−0.523260	+	0.852173i	$0.675284\pi$
$938$	39.2936	1.28298
$939$	0	0
$940$	−11.1739	−0.364452
$941$	45.6156	1.48703	0.743513	−	0.668722i	$-0.233158\pi$
$941$	45.6156	1.48703	0.743513	+	0.668722i	$0.233158\pi$
$942$	0	0
$943$	−18.0267	−0.587029
$944$	1.74457	0.0567810
$945$	0	0
$946$	22.4916	0.731264
$947$	39.7494	1.29168	0.645840	−	0.763473i	$-0.276507\pi$
$947$	39.7494	1.29168	0.645840	+	0.763473i	$0.276507\pi$
$948$	0	0
$949$	0	0
$950$	−2.42261	−0.0785998
$951$	0	0
$952$	−71.4981	−2.31727
$953$	14.3621	0.465235	0.232617	−	0.972568i	$-0.425271\pi$
$953$	14.3621	0.465235	0.232617	+	0.972568i	$0.425271\pi$
$954$	0	0
$955$	−74.4941	−2.41057
$956$	−0.938637	−0.0303577
$957$	0	0
$958$	−35.4561	−1.14553
$959$	6.05334	0.195473
$960$	0	0
$961$	67.7837	2.18657
$962$	0	0
$963$	0	0
$964$	−6.67324	−0.214930
$965$	−7.78258	−0.250530
$966$	0	0
$967$	−33.2261	−1.06848	−0.534239	−	0.845333i	$-0.679402\pi$
$967$	−33.2261	−1.06848	−0.534239	+	0.845333i	$0.679402\pi$
$968$	6.56300	0.210943
$969$	0	0
$970$	10.2104	0.327836
$971$	25.0522	0.803965	0.401982	−	0.915647i	$-0.368321\pi$
$971$	25.0522	0.803965	0.401982	+	0.915647i	$0.368321\pi$
$972$	0	0
$973$	−14.3297	−0.459391
$974$	−31.2768	−1.00217
$975$	0	0
$976$	−10.8737	−0.348058
$977$	55.6599	1.78072	0.890359	−	0.455259i	$-0.150453\pi$
$977$	55.6599	1.78072	0.890359	+	0.455259i	$0.150453\pi$
$978$	0	0
$979$	23.1661	0.740393
$980$	−10.3007	−0.329043
$981$	0	0
$982$	17.1570	0.547503
$983$	−28.2878	−0.902240	−0.451120	−	0.892463i	$-0.648975\pi$
$983$	−28.2878	−0.902240	−0.451120	+	0.892463i	$0.648975\pi$
$984$	0	0
$985$	8.76569	0.279298
$986$	4.82457	0.153646
$987$	0	0
$988$	0	0
$989$	−16.3049	−0.518466
$990$	0	0
$991$	41.0863	1.30515	0.652575	−	0.757724i	$-0.273689\pi$
$991$	41.0863	1.30515	0.652575	+	0.757724i	$0.273689\pi$
$992$	36.9082	1.17184
$993$	0	0
$994$	38.8619	1.23262
$995$	−40.3254	−1.27840
$996$	0	0
$997$	45.7663	1.44943	0.724716	−	0.689047i	$-0.241971\pi$
$997$	45.7663	1.44943	0.724716	+	0.689047i	$0.241971\pi$
$998$	6.70033	0.212095
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1521.2.a.w.1.4	yes	6
3.2	odd	2	inner	1521.2.a.w.1.3	yes	6
13.5	odd	4		1521.2.b.n.1351.6		12
13.8	odd	4		1521.2.b.n.1351.7		12
13.12	even	2		1521.2.a.v.1.3	✓	6
39.5	even	4		1521.2.b.n.1351.8		12
39.8	even	4		1521.2.b.n.1351.5		12
39.38	odd	2		1521.2.a.v.1.4	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1521.2.a.v.1.3	✓	6	13.12	even	2
1521.2.a.v.1.4	yes	6	39.38	odd	2
1521.2.a.w.1.3	yes	6	3.2	odd	2	inner
1521.2.a.w.1.4	yes	6	1.1	even	1	trivial
1521.2.b.n.1351.5		12	39.8	even	4
1521.2.b.n.1351.6		12	13.5	odd	4
1521.2.b.n.1351.7		12	13.8	odd	4
1521.2.b.n.1351.8		12	39.5	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1521 = 3^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1521.a (trivial)

\(f(q)\)	\(=\)	\(q+1.14367 q^{2} -0.692021 q^{4} +3.48695 q^{5} +3.35690 q^{7} -3.07878 q^{8} +O(q^{10})\)	\(q+1.14367 q^{2} -0.692021 q^{4} +3.48695 q^{5} +3.35690 q^{7} -3.07878 q^{8} +3.98792 q^{10} -2.97797 q^{11} +3.83918 q^{14} -2.13706 q^{16} +6.91796 q^{17} -0.295897 q^{19} -2.41305 q^{20} -3.40581 q^{22} +2.46899 q^{23} +7.15883 q^{25} -2.32304 q^{28} +0.609790 q^{29} +9.93900 q^{31} +3.71347 q^{32} +7.91185 q^{34} +11.7053 q^{35} -8.50365 q^{37} -0.338408 q^{38} -10.7356 q^{40} -7.30123 q^{41} -6.60388 q^{43} +2.06082 q^{44} +2.82371 q^{46} +4.63062 q^{47} +4.26875 q^{49} +8.18734 q^{50} -9.38695 q^{53} -10.3840 q^{55} -10.3351 q^{56} +0.697398 q^{58} -0.816341 q^{59} +5.08815 q^{61} +11.3669 q^{62} +8.52111 q^{64} +10.2349 q^{67} -4.78738 q^{68} +13.3870 q^{70} +10.1224 q^{71} +8.74094 q^{73} -9.72536 q^{74} +0.204767 q^{76} -9.99674 q^{77} +3.52111 q^{79} -7.45184 q^{80} -8.35019 q^{82} -11.7751 q^{83} +24.1226 q^{85} -7.55265 q^{86} +9.16852 q^{88} -7.77916 q^{89} -1.70859 q^{92} +5.29590 q^{94} -1.03178 q^{95} +2.56033 q^{97} +4.88204 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 6 q^{4} + 12 q^{7}+O(q^{10}) \) 6 * q + 6 * q^4 + 12 * q^7	\( 6 q + 6 q^{4} + 12 q^{7} - 14 q^{10} - 2 q^{16} + 26 q^{19} + 6 q^{22} + 26 q^{25} + 26 q^{28} + 40 q^{31} + 40 q^{34} + 12 q^{37} - 42 q^{40} - 22 q^{43} + 2 q^{46} + 10 q^{49} - 42 q^{55} - 12 q^{58} + 38 q^{61} + 20 q^{64} + 14 q^{67} - 28 q^{70} + 24 q^{73} + 54 q^{76} - 10 q^{79} + 2 q^{82} + 14 q^{85} - 6 q^{88} + 4 q^{94} + 10 q^{97}+O(q^{100}) \) 6 * q + 6 * q^4 + 12 * q^7 - 14 * q^10 - 2 * q^16 + 26 * q^19 + 6 * q^22 + 26 * q^25 + 26 * q^28 + 40 * q^31 + 40 * q^34 + 12 * q^37 - 42 * q^40 - 22 * q^43 + 2 * q^46 + 10 * q^49 - 42 * q^55 - 12 * q^58 + 38 * q^61 + 20 * q^64 + 14 * q^67 - 28 * q^70 + 24 * q^73 + 54 * q^76 - 10 * q^79 + 2 * q^82 + 14 * q^85 - 6 * q^88 + 4 * q^94 + 10 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

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Database

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