LMFDB - Embedded newform 3042.2.a.y.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3042,2,Mod(1,3042)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3042, 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("3042.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$N$	$=$	$3042 = 2 \cdot 3^{2} \cdot 13^{2}$
Weight:	$k$	$=$	$2$
Character orbit:	$[\chi]$	$=$	3042.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:	$24.2904922949$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$x^{2} - 3$ x^2 - 3
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 78)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.73205$ of defining polynomial
Character	$\chi$	$=$	3042.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +3.73205 q^{5} +2.73205 q^{7} +1.00000 q^{8} +3.73205 q^{10} +1.26795 q^{11} +2.73205 q^{14} +1.00000 q^{16} +5.73205 q^{17} -4.73205 q^{19} +3.73205 q^{20} +1.26795 q^{22} -4.19615 q^{23} +8.92820 q^{25} +2.73205 q^{28} +4.46410 q^{29} -1.46410 q^{31} +1.00000 q^{32} +5.73205 q^{34} +10.1962 q^{35} -3.53590 q^{37} -4.73205 q^{38} +3.73205 q^{40} -9.39230 q^{41} -9.66025 q^{43} +1.26795 q^{44} -4.19615 q^{46} +2.19615 q^{47} +0.464102 q^{49} +8.92820 q^{50} +6.46410 q^{53} +4.73205 q^{55} +2.73205 q^{56} +4.46410 q^{58} +8.00000 q^{59} -9.19615 q^{61} -1.46410 q^{62} +1.00000 q^{64} -13.1244 q^{67} +5.73205 q^{68} +10.1962 q^{70} +4.73205 q^{71} +6.26795 q^{73} -3.53590 q^{74} -4.73205 q^{76} +3.46410 q^{77} -2.53590 q^{79} +3.73205 q^{80} -9.39230 q^{82} -0.196152 q^{83} +21.3923 q^{85} -9.66025 q^{86} +1.26795 q^{88} -9.46410 q^{89} -4.19615 q^{92} +2.19615 q^{94} -17.6603 q^{95} +6.00000 q^{97} +0.464102 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} + 2 q^{7} + 2 q^{8} + 4 q^{10} + 6 q^{11} + 2 q^{14} + 2 q^{16} + 8 q^{17} - 6 q^{19} + 4 q^{20} + 6 q^{22} + 2 q^{23} + 4 q^{25} + 2 q^{28} + 2 q^{29} + 4 q^{31} + 2 q^{32}+ \cdots - 6 q^{98}+O(q^{100})$ 2 * q + 2 * q^2 + 2 * q^4 + 4 * q^5 + 2 * q^7 + 2 * q^8 + 4 * q^10 + 6 * q^11 + 2 * q^14 + 2 * q^16 + 8 * q^17 - 6 * q^19 + 4 * q^20 + 6 * q^22 + 2 * q^23 + 4 * q^25 + 2 * q^28 + 2 * q^29 + 4 * q^31 + 2 * q^32 + 8 * q^34 + 10 * q^35 - 14 * q^37 - 6 * q^38 + 4 * q^40 + 2 * q^41 - 2 * q^43 + 6 * q^44 + 2 * q^46 - 6 * q^47 - 6 * q^49 + 4 * q^50 + 6 * q^53 + 6 * q^55 + 2 * q^56 + 2 * q^58 + 16 * q^59 - 8 * q^61 + 4 * q^62 + 2 * q^64 - 2 * q^67 + 8 * q^68 + 10 * q^70 + 6 * q^71 + 16 * q^73 - 14 * q^74 - 6 * q^76 - 12 * q^79 + 4 * q^80 + 2 * q^82 + 10 * q^83 + 22 * q^85 - 2 * q^86 + 6 * q^88 - 12 * q^89 + 2 * q^92 - 6 * q^94 - 18 * q^95 + 12 * q^97 - 6 * q^98

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

)

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

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	3.73205	1.66902	0.834512	−	0.550990i	$-0.185750\pi$
$5$	3.73205	1.66902	0.834512	+	0.550990i	$0.185750\pi$
$6$	0	0
$7$	2.73205	1.03262	0.516309	−	0.856402i	$-0.327306\pi$
$7$	2.73205	1.03262	0.516309	+	0.856402i	$0.327306\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	3.73205	1.18018
$11$	1.26795	0.382301	0.191151	−	0.981561i	$-0.438778\pi$
$11$	1.26795	0.382301	0.191151	+	0.981561i	$0.438778\pi$
$12$	0	0
$13$	0	0
$13$	0	0
$14$	2.73205	0.730171
$15$	0	0
$16$	1.00000	0.250000
$17$	5.73205	1.39023	0.695113	−	0.718900i	$-0.255354\pi$
$17$	5.73205	1.39023	0.695113	+	0.718900i	$0.255354\pi$
$18$	0	0
$19$	−4.73205	−1.08561	−0.542803	−	0.839860i	$-0.682637\pi$
$19$	−4.73205	−1.08561	−0.542803	+	0.839860i	$0.682637\pi$
$20$	3.73205	0.834512
$21$	0	0
$22$	1.26795	0.270328
$23$	−4.19615	−0.874958	−0.437479	−	0.899229i	$-0.644129\pi$
$23$	−4.19615	−0.874958	−0.437479	+	0.899229i	$0.644129\pi$
$24$	0	0
$25$	8.92820	1.78564
$26$	0	0
$27$	0	0
$28$	2.73205	0.516309
$29$	4.46410	0.828963	0.414481	−	0.910058i	$-0.363963\pi$
$29$	4.46410	0.828963	0.414481	+	0.910058i	$0.363963\pi$
$30$	0	0
$31$	−1.46410	−0.262960	−0.131480	−	0.991319i	$-0.541973\pi$
$31$	−1.46410	−0.262960	−0.131480	+	0.991319i	$0.541973\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	5.73205	0.983039
$35$	10.1962	1.72346
$36$	0	0
$37$	−3.53590	−0.581298	−0.290649	−	0.956830i	$-0.593871\pi$
$37$	−3.53590	−0.581298	−0.290649	+	0.956830i	$0.593871\pi$
$38$	−4.73205	−0.767640
$39$	0	0
$40$	3.73205	0.590089
$41$	−9.39230	−1.46683	−0.733416	−	0.679780i	$-0.762075\pi$
$41$	−9.39230	−1.46683	−0.733416	+	0.679780i	$0.762075\pi$
$42$	0	0
$43$	−9.66025	−1.47317	−0.736587	−	0.676342i	$-0.763564\pi$
$43$	−9.66025	−1.47317	−0.736587	+	0.676342i	$0.763564\pi$
$44$	1.26795	0.191151
$45$	0	0
$46$	−4.19615	−0.618689
$47$	2.19615	0.320342	0.160171	−	0.987089i	$-0.448795\pi$
$47$	2.19615	0.320342	0.160171	+	0.987089i	$0.448795\pi$
$48$	0	0
$49$	0.464102	0.0663002
$50$	8.92820	1.26264
$51$	0	0
$52$	0	0
$53$	6.46410	0.887913	0.443956	−	0.896048i	$-0.353575\pi$
$53$	6.46410	0.887913	0.443956	+	0.896048i	$0.353575\pi$
$54$	0	0
$55$	4.73205	0.638070
$56$	2.73205	0.365086
$57$	0	0
$58$	4.46410	0.586165
$59$	8.00000	1.04151	0.520756	−	0.853706i	$-0.325650\pi$
$59$	8.00000	1.04151	0.520756	+	0.853706i	$0.325650\pi$
$60$	0	0
$61$	−9.19615	−1.17745	−0.588723	−	0.808335i	$-0.700369\pi$
$61$	−9.19615	−1.17745	−0.588723	+	0.808335i	$0.700369\pi$
$62$	−1.46410	−0.185941
$63$	0	0
$64$	1.00000	0.125000
$65$	0	0
$66$	0	0
$67$	−13.1244	−1.60340	−0.801698	−	0.597730i	$-0.796070\pi$
$67$	−13.1244	−1.60340	−0.801698	+	0.597730i	$0.796070\pi$
$68$	5.73205	0.695113
$69$	0	0
$70$	10.1962	1.21867
$71$	4.73205	0.561591	0.280796	−	0.959768i	$-0.409402\pi$
$71$	4.73205	0.561591	0.280796	+	0.959768i	$0.409402\pi$
$72$	0	0
$73$	6.26795	0.733608	0.366804	−	0.930298i	$-0.380452\pi$
$73$	6.26795	0.733608	0.366804	+	0.930298i	$0.380452\pi$
$74$	−3.53590	−0.411040
$75$	0	0
$76$	−4.73205	−0.542803
$77$	3.46410	0.394771
$78$	0	0
$79$	−2.53590	−0.285311	−0.142655	−	0.989772i	$-0.545564\pi$
$79$	−2.53590	−0.285311	−0.142655	+	0.989772i	$0.545564\pi$
$80$	3.73205	0.417256
$81$	0	0
$82$	−9.39230	−1.03721
$83$	−0.196152	−0.0215305	−0.0107653	−	0.999942i	$-0.503427\pi$
$83$	−0.196152	−0.0215305	−0.0107653	+	0.999942i	$0.503427\pi$
$84$	0	0
$85$	21.3923	2.32032
$86$	−9.66025	−1.04169
$87$	0	0
$88$	1.26795	0.135164
$89$	−9.46410	−1.00319	−0.501596	−	0.865102i	$-0.667254\pi$
$89$	−9.46410	−1.00319	−0.501596	+	0.865102i	$0.667254\pi$
$90$	0	0
$91$	0	0
$92$	−4.19615	−0.437479
$93$	0	0
$94$	2.19615	0.226516
$95$	−17.6603	−1.81190
$96$	0	0
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	0.464102	0.0468813
$99$	0	0
$100$	8.92820	0.892820
$101$	−1.92820	−0.191863	−0.0959317	−	0.995388i	$-0.530583\pi$
$101$	−1.92820	−0.191863	−0.0959317	+	0.995388i	$0.530583\pi$
$102$	0	0
$103$	−15.2679	−1.50440	−0.752198	−	0.658937i	$-0.771006\pi$
$103$	−15.2679	−1.50440	−0.752198	+	0.658937i	$0.771006\pi$
$104$	0	0
$105$	0	0
$106$	6.46410	0.627849
$107$	−10.1962	−0.985699	−0.492850	−	0.870114i	$-0.664045\pi$
$107$	−10.1962	−0.985699	−0.492850	+	0.870114i	$0.664045\pi$
$108$	0	0
$109$	−1.46410	−0.140236	−0.0701178	−	0.997539i	$-0.522338\pi$
$109$	−1.46410	−0.140236	−0.0701178	+	0.997539i	$0.522338\pi$
$110$	4.73205	0.451183
$111$	0	0
$112$	2.73205	0.258155
$113$	1.33975	0.126033	0.0630163	−	0.998012i	$-0.479928\pi$
$113$	1.33975	0.126033	0.0630163	+	0.998012i	$0.479928\pi$
$114$	0	0
$115$	−15.6603	−1.46033
$116$	4.46410	0.414481
$117$	0	0
$118$	8.00000	0.736460
$119$	15.6603	1.43557
$120$	0	0
$121$	−9.39230	−0.853846
$122$	−9.19615	−0.832581
$123$	0	0
$124$	−1.46410	−0.131480
$125$	14.6603	1.31125
$126$	0	0
$127$	−9.85641	−0.874615	−0.437307	−	0.899312i	$-0.644068\pi$
$127$	−9.85641	−0.874615	−0.437307	+	0.899312i	$0.644068\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	0	0
$131$	−6.53590	−0.571044	−0.285522	−	0.958372i	$-0.592167\pi$
$131$	−6.53590	−0.571044	−0.285522	+	0.958372i	$0.592167\pi$
$132$	0	0
$133$	−12.9282	−1.12102
$134$	−13.1244	−1.13377
$135$	0	0
$136$	5.73205	0.491519
$137$	11.9282	1.01910	0.509548	−	0.860442i	$-0.329813\pi$
$137$	11.9282	1.01910	0.509548	+	0.860442i	$0.329813\pi$
$138$	0	0
$139$	17.8564	1.51456	0.757280	−	0.653090i	$-0.226528\pi$
$139$	17.8564	1.51456	0.757280	+	0.653090i	$0.226528\pi$
$140$	10.1962	0.861732
$141$	0	0
$142$	4.73205	0.397105
$143$	0	0
$144$	0	0
$145$	16.6603	1.38356
$146$	6.26795	0.518739
$147$	0	0
$148$	−3.53590	−0.290649
$149$	13.1962	1.08107	0.540535	−	0.841321i	$-0.318222\pi$
$149$	13.1962	1.08107	0.540535	+	0.841321i	$0.318222\pi$
$150$	0	0
$151$	−6.73205	−0.547847	−0.273923	−	0.961752i	$-0.588321\pi$
$151$	−6.73205	−0.547847	−0.273923	+	0.961752i	$0.588321\pi$
$152$	−4.73205	−0.383820
$153$	0	0
$154$	3.46410	0.279145
$155$	−5.46410	−0.438887
$156$	0	0
$157$	7.58846	0.605625	0.302812	−	0.953050i	$-0.402074\pi$
$157$	7.58846	0.605625	0.302812	+	0.953050i	$0.402074\pi$
$158$	−2.53590	−0.201745
$159$	0	0
$160$	3.73205	0.295045
$161$	−11.4641	−0.903498
$162$	0	0
$163$	13.4641	1.05459	0.527295	−	0.849682i	$-0.323206\pi$
$163$	13.4641	1.05459	0.527295	+	0.849682i	$0.323206\pi$
$164$	−9.39230	−0.733416
$165$	0	0
$166$	−0.196152	−0.0152244
$167$	−9.46410	−0.732354	−0.366177	−	0.930545i	$-0.619334\pi$
$167$	−9.46410	−0.732354	−0.366177	+	0.930545i	$0.619334\pi$
$168$	0	0
$169$	0	0
$170$	21.3923	1.64071
$171$	0	0
$172$	−9.66025	−0.736587
$173$	−4.39230	−0.333941	−0.166970	−	0.985962i	$-0.553398\pi$
$173$	−4.39230	−0.333941	−0.166970	+	0.985962i	$0.553398\pi$
$174$	0	0
$175$	24.3923	1.84388
$176$	1.26795	0.0955753
$177$	0	0
$178$	−9.46410	−0.709364
$179$	16.0526	1.19982	0.599912	−	0.800066i	$-0.295202\pi$
$179$	16.0526	1.19982	0.599912	+	0.800066i	$0.295202\pi$
$180$	0	0
$181$	19.1962	1.42684	0.713419	−	0.700737i	$-0.247145\pi$
$181$	19.1962	1.42684	0.713419	+	0.700737i	$0.247145\pi$
$182$	0	0
$183$	0	0
$184$	−4.19615	−0.309344
$185$	−13.1962	−0.970200
$186$	0	0
$187$	7.26795	0.531485
$188$	2.19615	0.160171
$189$	0	0
$190$	−17.6603	−1.28121
$191$	6.92820	0.501307	0.250654	−	0.968077i	$-0.419354\pi$
$191$	6.92820	0.501307	0.250654	+	0.968077i	$0.419354\pi$
$192$	0	0
$193$	11.7321	0.844491	0.422246	−	0.906481i	$-0.361242\pi$
$193$	11.7321	0.844491	0.422246	+	0.906481i	$0.361242\pi$
$194$	6.00000	0.430775
$195$	0	0
$196$	0.464102	0.0331501
$197$	−17.8564	−1.27222	−0.636108	−	0.771600i	$-0.719457\pi$
$197$	−17.8564	−1.27222	−0.636108	+	0.771600i	$0.719457\pi$
$198$	0	0
$199$	−14.1962	−1.00634	−0.503169	−	0.864188i	$-0.667833\pi$
$199$	−14.1962	−1.00634	−0.503169	+	0.864188i	$0.667833\pi$
$200$	8.92820	0.631319
$201$	0	0
$202$	−1.92820	−0.135668
$203$	12.1962	0.856002
$204$	0	0
$205$	−35.0526	−2.44818
$206$	−15.2679	−1.06377
$207$	0	0
$208$	0	0
$209$	−6.00000	−0.415029
$210$	0	0
$211$	16.3923	1.12849	0.564246	−	0.825606i	$-0.309167\pi$
$211$	16.3923	1.12849	0.564246	+	0.825606i	$0.309167\pi$
$212$	6.46410	0.443956
$213$	0	0
$214$	−10.1962	−0.696995
$215$	−36.0526	−2.45876
$216$	0	0
$217$	−4.00000	−0.271538
$218$	−1.46410	−0.0991615
$219$	0	0
$220$	4.73205	0.319035
$221$	0	0
$222$	0	0
$223$	26.9282	1.80325	0.901623	−	0.432523i	$-0.142377\pi$
$223$	26.9282	1.80325	0.901623	+	0.432523i	$0.142377\pi$
$224$	2.73205	0.182543
$225$	0	0
$226$	1.33975	0.0891186
$227$	−12.1962	−0.809487	−0.404744	−	0.914430i	$-0.632639\pi$
$227$	−12.1962	−0.809487	−0.404744	+	0.914430i	$0.632639\pi$
$228$	0	0
$229$	11.8564	0.783493	0.391747	−	0.920073i	$-0.371871\pi$
$229$	11.8564	0.783493	0.391747	+	0.920073i	$0.371871\pi$
$230$	−15.6603	−1.03261
$231$	0	0
$232$	4.46410	0.293083
$233$	7.85641	0.514690	0.257345	−	0.966320i	$-0.417152\pi$
$233$	7.85641	0.514690	0.257345	+	0.966320i	$0.417152\pi$
$234$	0	0
$235$	8.19615	0.534658
$236$	8.00000	0.520756
$237$	0	0
$238$	15.6603	1.01510
$239$	7.66025	0.495501	0.247750	−	0.968824i	$-0.420309\pi$
$239$	7.66025	0.495501	0.247750	+	0.968824i	$0.420309\pi$
$240$	0	0
$241$	−13.5885	−0.875309	−0.437655	−	0.899143i	$-0.644191\pi$
$241$	−13.5885	−0.875309	−0.437655	+	0.899143i	$0.644191\pi$
$242$	−9.39230	−0.603760
$243$	0	0
$244$	−9.19615	−0.588723
$245$	1.73205	0.110657
$246$	0	0
$247$	0	0
$248$	−1.46410	−0.0929705
$249$	0	0
$250$	14.6603	0.927196
$251$	−13.4641	−0.849847	−0.424923	−	0.905229i	$-0.639699\pi$
$251$	−13.4641	−0.849847	−0.424923	+	0.905229i	$0.639699\pi$
$252$	0	0
$253$	−5.32051	−0.334497
$254$	−9.85641	−0.618446
$255$	0	0
$256$	1.00000	0.0625000
$257$	−9.33975	−0.582597	−0.291299	−	0.956632i	$-0.594087\pi$
$257$	−9.33975	−0.582597	−0.291299	+	0.956632i	$0.594087\pi$
$258$	0	0
$259$	−9.66025	−0.600259
$260$	0	0
$261$	0	0
$262$	−6.53590	−0.403789
$263$	10.0526	0.619867	0.309934	−	0.950758i	$-0.399693\pi$
$263$	10.0526	0.619867	0.309934	+	0.950758i	$0.399693\pi$
$264$	0	0
$265$	24.1244	1.48195
$266$	−12.9282	−0.792679
$267$	0	0
$268$	−13.1244	−0.801698
$269$	−5.46410	−0.333152	−0.166576	−	0.986029i	$-0.553271\pi$
$269$	−5.46410	−0.333152	−0.166576	+	0.986029i	$0.553271\pi$
$270$	0	0
$271$	21.8564	1.32768	0.663841	−	0.747874i	$-0.268925\pi$
$271$	21.8564	1.32768	0.663841	+	0.747874i	$0.268925\pi$
$272$	5.73205	0.347557
$273$	0	0
$274$	11.9282	0.720609
$275$	11.3205	0.682652
$276$	0	0
$277$	−5.73205	−0.344406	−0.172203	−	0.985062i	$-0.555088\pi$
$277$	−5.73205	−0.344406	−0.172203	+	0.985062i	$0.555088\pi$
$278$	17.8564	1.07096
$279$	0	0
$280$	10.1962	0.609337
$281$	12.3205	0.734980	0.367490	−	0.930027i	$-0.380217\pi$
$281$	12.3205	0.734980	0.367490	+	0.930027i	$0.380217\pi$
$282$	0	0
$283$	25.6603	1.52534	0.762672	−	0.646786i	$-0.223887\pi$
$283$	25.6603	1.52534	0.762672	+	0.646786i	$0.223887\pi$
$284$	4.73205	0.280796
$285$	0	0
$286$	0	0
$287$	−25.6603	−1.51468
$288$	0	0
$289$	15.8564	0.932730
$290$	16.6603	0.978324
$291$	0	0
$292$	6.26795	0.366804
$293$	30.5167	1.78280	0.891401	−	0.453215i	$-0.149723\pi$
$293$	30.5167	1.78280	0.891401	+	0.453215i	$0.149723\pi$
$294$	0	0
$295$	29.8564	1.73831
$296$	−3.53590	−0.205520
$297$	0	0
$298$	13.1962	0.764433
$299$	0	0
$300$	0	0
$301$	−26.3923	−1.52123
$302$	−6.73205	−0.387386
$303$	0	0
$304$	−4.73205	−0.271402
$305$	−34.3205	−1.96519
$306$	0	0
$307$	22.5885	1.28919	0.644596	−	0.764524i	$-0.277026\pi$
$307$	22.5885	1.28919	0.644596	+	0.764524i	$0.277026\pi$
$308$	3.46410	0.197386
$309$	0	0
$310$	−5.46410	−0.310340
$311$	1.66025	0.0941444	0.0470722	−	0.998891i	$-0.485011\pi$
$311$	1.66025	0.0941444	0.0470722	+	0.998891i	$0.485011\pi$
$312$	0	0
$313$	6.53590	0.369431	0.184715	−	0.982792i	$-0.440864\pi$
$313$	6.53590	0.369431	0.184715	+	0.982792i	$0.440864\pi$
$314$	7.58846	0.428241
$315$	0	0
$316$	−2.53590	−0.142655
$317$	20.6603	1.16040	0.580198	−	0.814476i	$-0.302975\pi$
$317$	20.6603	1.16040	0.580198	+	0.814476i	$0.302975\pi$
$318$	0	0
$319$	5.66025	0.316913
$320$	3.73205	0.208628
$321$	0	0
$322$	−11.4641	−0.638869
$323$	−27.1244	−1.50924
$324$	0	0
$325$	0	0
$326$	13.4641	0.745708
$327$	0	0
$328$	−9.39230	−0.518603
$329$	6.00000	0.330791
$330$	0	0
$331$	−20.0000	−1.09930	−0.549650	−	0.835395i	$-0.685239\pi$
$331$	−20.0000	−1.09930	−0.549650	+	0.835395i	$0.685239\pi$
$332$	−0.196152	−0.0107653
$333$	0	0
$334$	−9.46410	−0.517853
$335$	−48.9808	−2.67610
$336$	0	0
$337$	−20.8564	−1.13612	−0.568060	−	0.822987i	$-0.692306\pi$
$337$	−20.8564	−1.13612	−0.568060	+	0.822987i	$0.692306\pi$
$338$	0	0
$339$	0	0
$340$	21.3923	1.16016
$341$	−1.85641	−0.100530
$342$	0	0
$343$	−17.8564	−0.964155
$344$	−9.66025	−0.520846
$345$	0	0
$346$	−4.39230	−0.236132
$347$	−33.1244	−1.77821	−0.889104	−	0.457705i	$-0.848672\pi$
$347$	−33.1244	−1.77821	−0.889104	+	0.457705i	$0.848672\pi$
$348$	0	0
$349$	−15.3205	−0.820088	−0.410044	−	0.912066i	$-0.634487\pi$
$349$	−15.3205	−0.820088	−0.410044	+	0.912066i	$0.634487\pi$
$350$	24.3923	1.30382
$351$	0	0
$352$	1.26795	0.0675819
$353$	−21.7846	−1.15948	−0.579739	−	0.814802i	$-0.696845\pi$
$353$	−21.7846	−1.15948	−0.579739	+	0.814802i	$0.696845\pi$
$354$	0	0
$355$	17.6603	0.937309
$356$	−9.46410	−0.501596
$357$	0	0
$358$	16.0526	0.848404
$359$	−1.12436	−0.0593412	−0.0296706	−	0.999560i	$-0.509446\pi$
$359$	−1.12436	−0.0593412	−0.0296706	+	0.999560i	$0.509446\pi$
$360$	0	0
$361$	3.39230	0.178542
$362$	19.1962	1.00893
$363$	0	0
$364$	0	0
$365$	23.3923	1.22441
$366$	0	0
$367$	−11.2679	−0.588182	−0.294091	−	0.955777i	$-0.595017\pi$
$367$	−11.2679	−0.588182	−0.294091	+	0.955777i	$0.595017\pi$
$368$	−4.19615	−0.218740
$369$	0	0
$370$	−13.1962	−0.686035
$371$	17.6603	0.916875
$372$	0	0
$373$	−13.7321	−0.711019	−0.355509	−	0.934673i	$-0.615693\pi$
$373$	−13.7321	−0.711019	−0.355509	+	0.934673i	$0.615693\pi$
$374$	7.26795	0.375817
$375$	0	0
$376$	2.19615	0.113258
$377$	0	0
$378$	0	0
$379$	5.46410	0.280672	0.140336	−	0.990104i	$-0.455182\pi$
$379$	5.46410	0.280672	0.140336	+	0.990104i	$0.455182\pi$
$380$	−17.6603	−0.905952
$381$	0	0
$382$	6.92820	0.354478
$383$	−1.46410	−0.0748121	−0.0374060	−	0.999300i	$-0.511909\pi$
$383$	−1.46410	−0.0748121	−0.0374060	+	0.999300i	$0.511909\pi$
$384$	0	0
$385$	12.9282	0.658882
$386$	11.7321	0.597146
$387$	0	0
$388$	6.00000	0.304604
$389$	−11.7846	−0.597503	−0.298752	−	0.954331i	$-0.596570\pi$
$389$	−11.7846	−0.597503	−0.298752	+	0.954331i	$0.596570\pi$
$390$	0	0
$391$	−24.0526	−1.21639
$392$	0.464102	0.0234407
$393$	0	0
$394$	−17.8564	−0.899593
$395$	−9.46410	−0.476191
$396$	0	0
$397$	−20.3923	−1.02346	−0.511730	−	0.859146i	$-0.670995\pi$
$397$	−20.3923	−1.02346	−0.511730	+	0.859146i	$0.670995\pi$
$398$	−14.1962	−0.711589
$399$	0	0
$400$	8.92820	0.446410
$401$	8.07180	0.403086	0.201543	−	0.979480i	$-0.435404\pi$
$401$	8.07180	0.403086	0.201543	+	0.979480i	$0.435404\pi$
$402$	0	0
$403$	0	0
$404$	−1.92820	−0.0959317
$405$	0	0
$406$	12.1962	0.605285
$407$	−4.48334	−0.222231
$408$	0	0
$409$	−17.7321	−0.876793	−0.438397	−	0.898782i	$-0.644454\pi$
$409$	−17.7321	−0.876793	−0.438397	+	0.898782i	$0.644454\pi$
$410$	−35.0526	−1.73112
$411$	0	0
$412$	−15.2679	−0.752198
$413$	21.8564	1.07548
$414$	0	0
$415$	−0.732051	−0.0359350
$416$	0	0
$417$	0	0
$418$	−6.00000	−0.293470
$419$	17.4641	0.853177	0.426589	−	0.904446i	$-0.359715\pi$
$419$	17.4641	0.853177	0.426589	+	0.904446i	$0.359715\pi$
$420$	0	0
$421$	22.7128	1.10695	0.553477	−	0.832864i	$-0.313301\pi$
$421$	22.7128	1.10695	0.553477	+	0.832864i	$0.313301\pi$
$422$	16.3923	0.797965
$423$	0	0
$424$	6.46410	0.313925
$425$	51.1769	2.48244
$426$	0	0
$427$	−25.1244	−1.21585
$428$	−10.1962	−0.492850
$429$	0	0
$430$	−36.0526	−1.73861
$431$	13.1244	0.632178	0.316089	−	0.948730i	$-0.397630\pi$
$431$	13.1244	0.632178	0.316089	+	0.948730i	$0.397630\pi$
$432$	0	0
$433$	12.8564	0.617839	0.308920	−	0.951088i	$-0.400033\pi$
$433$	12.8564	0.617839	0.308920	+	0.951088i	$0.400033\pi$
$434$	−4.00000	−0.192006
$435$	0	0
$436$	−1.46410	−0.0701178
$437$	19.8564	0.949861
$438$	0	0
$439$	−0.339746	−0.0162152	−0.00810760	−	0.999967i	$-0.502581\pi$
$439$	−0.339746	−0.0162152	−0.00810760	+	0.999967i	$0.502581\pi$
$440$	4.73205	0.225592
$441$	0	0
$442$	0	0
$443$	−15.6077	−0.741544	−0.370772	−	0.928724i	$-0.620907\pi$
$443$	−15.6077	−0.741544	−0.370772	+	0.928724i	$0.620907\pi$
$444$	0	0
$445$	−35.3205	−1.67435
$446$	26.9282	1.27509
$447$	0	0
$448$	2.73205	0.129077
$449$	11.3205	0.534248	0.267124	−	0.963662i	$-0.413927\pi$
$449$	11.3205	0.534248	0.267124	+	0.963662i	$0.413927\pi$
$450$	0	0
$451$	−11.9090	−0.560771
$452$	1.33975	0.0630163
$453$	0	0
$454$	−12.1962	−0.572394
$455$	0	0
$456$	0	0
$457$	−1.33975	−0.0626707	−0.0313353	−	0.999509i	$-0.509976\pi$
$457$	−1.33975	−0.0626707	−0.0313353	+	0.999509i	$0.509976\pi$
$458$	11.8564	0.554013
$459$	0	0
$460$	−15.6603	−0.730163
$461$	22.2679	1.03712	0.518561	−	0.855041i	$-0.326468\pi$
$461$	22.2679	1.03712	0.518561	+	0.855041i	$0.326468\pi$
$462$	0	0
$463$	−10.0526	−0.467182	−0.233591	−	0.972335i	$-0.575048\pi$
$463$	−10.0526	−0.467182	−0.233591	+	0.972335i	$0.575048\pi$
$464$	4.46410	0.207241
$465$	0	0
$466$	7.85641	0.363941
$467$	−18.5885	−0.860171	−0.430086	−	0.902788i	$-0.641517\pi$
$467$	−18.5885	−0.860171	−0.430086	+	0.902788i	$0.641517\pi$
$468$	0	0
$469$	−35.8564	−1.65570
$470$	8.19615	0.378060
$471$	0	0
$472$	8.00000	0.368230
$473$	−12.2487	−0.563196
$474$	0	0
$475$	−42.2487	−1.93850
$476$	15.6603	0.717787
$477$	0	0
$478$	7.66025	0.350372
$479$	33.4641	1.52901	0.764507	−	0.644616i	$-0.222983\pi$
$479$	33.4641	1.52901	0.764507	+	0.644616i	$0.222983\pi$
$480$	0	0
$481$	0	0
$482$	−13.5885	−0.618937
$483$	0	0
$484$	−9.39230	−0.426923
$485$	22.3923	1.01678
$486$	0	0
$487$	3.12436	0.141578	0.0707890	−	0.997491i	$-0.477448\pi$
$487$	3.12436	0.141578	0.0707890	+	0.997491i	$0.477448\pi$
$488$	−9.19615	−0.416290
$489$	0	0
$490$	1.73205	0.0782461
$491$	−8.73205	−0.394072	−0.197036	−	0.980396i	$-0.563132\pi$
$491$	−8.73205	−0.394072	−0.197036	+	0.980396i	$0.563132\pi$
$492$	0	0
$493$	25.5885	1.15245
$494$	0	0
$495$	0	0
$496$	−1.46410	−0.0657401
$497$	12.9282	0.579909
$498$	0	0
$499$	−32.0000	−1.43252	−0.716258	−	0.697835i	$-0.754147\pi$
$499$	−32.0000	−1.43252	−0.716258	+	0.697835i	$0.754147\pi$
$500$	14.6603	0.655626
$501$	0	0
$502$	−13.4641	−0.600932
$503$	−40.9808	−1.82724	−0.913621	−	0.406567i	$-0.866726\pi$
$503$	−40.9808	−1.82724	−0.913621	+	0.406567i	$0.866726\pi$
$504$	0	0
$505$	−7.19615	−0.320225
$506$	−5.32051	−0.236525
$507$	0	0
$508$	−9.85641	−0.437307
$509$	13.7321	0.608662	0.304331	−	0.952566i	$-0.401567\pi$
$509$	13.7321	0.608662	0.304331	+	0.952566i	$0.401567\pi$
$510$	0	0
$511$	17.1244	0.757537
$512$	1.00000	0.0441942
$513$	0	0
$514$	−9.33975	−0.411959
$515$	−56.9808	−2.51087
$516$	0	0
$517$	2.78461	0.122467
$518$	−9.66025	−0.424447
$519$	0	0
$520$	0	0
$521$	−41.4449	−1.81573	−0.907866	−	0.419260i	$-0.862290\pi$
$521$	−41.4449	−1.81573	−0.907866	+	0.419260i	$0.862290\pi$
$522$	0	0
$523$	−22.4449	−0.981445	−0.490723	−	0.871316i	$-0.663267\pi$
$523$	−22.4449	−0.981445	−0.490723	+	0.871316i	$0.663267\pi$
$524$	−6.53590	−0.285522
$525$	0	0
$526$	10.0526	0.438312
$527$	−8.39230	−0.365575
$528$	0	0
$529$	−5.39230	−0.234448
$530$	24.1244	1.04790
$531$	0	0
$532$	−12.9282	−0.560509
$533$	0	0
$534$	0	0
$535$	−38.0526	−1.64516
$536$	−13.1244	−0.566886
$537$	0	0
$538$	−5.46410	−0.235574
$539$	0.588457	0.0253466
$540$	0	0
$541$	−5.67949	−0.244180	−0.122090	−	0.992519i	$-0.538960\pi$
$541$	−5.67949	−0.244180	−0.122090	+	0.992519i	$0.538960\pi$
$542$	21.8564	0.938813
$543$	0	0
$544$	5.73205	0.245760
$545$	−5.46410	−0.234056
$546$	0	0
$547$	−4.19615	−0.179415	−0.0897073	−	0.995968i	$-0.528593\pi$
$547$	−4.19615	−0.179415	−0.0897073	+	0.995968i	$0.528593\pi$
$548$	11.9282	0.509548
$549$	0	0
$550$	11.3205	0.482708
$551$	−21.1244	−0.899928
$552$	0	0
$553$	−6.92820	−0.294617
$554$	−5.73205	−0.243532
$555$	0	0
$556$	17.8564	0.757280
$557$	42.3731	1.79540	0.897702	−	0.440603i	$-0.145235\pi$
$557$	42.3731	1.79540	0.897702	+	0.440603i	$0.145235\pi$
$558$	0	0
$559$	0	0
$560$	10.1962	0.430866
$561$	0	0
$562$	12.3205	0.519709
$563$	−34.9282	−1.47205	−0.736024	−	0.676955i	$-0.763299\pi$
$563$	−34.9282	−1.47205	−0.736024	+	0.676955i	$0.763299\pi$
$564$	0	0
$565$	5.00000	0.210352
$566$	25.6603	1.07858
$567$	0	0
$568$	4.73205	0.198552
$569$	−30.6410	−1.28454	−0.642269	−	0.766479i	$-0.722007\pi$
$569$	−30.6410	−1.28454	−0.642269	+	0.766479i	$0.722007\pi$
$570$	0	0
$571$	14.0526	0.588081	0.294041	−	0.955793i	$-0.405000\pi$
$571$	14.0526	0.588081	0.294041	+	0.955793i	$0.405000\pi$
$572$	0	0
$573$	0	0
$574$	−25.6603	−1.07104
$575$	−37.4641	−1.56236
$576$	0	0
$577$	−3.73205	−0.155367	−0.0776837	−	0.996978i	$-0.524752\pi$
$577$	−3.73205	−0.155367	−0.0776837	+	0.996978i	$0.524752\pi$
$578$	15.8564	0.659540
$579$	0	0
$580$	16.6603	0.691779
$581$	−0.535898	−0.0222328
$582$	0	0
$583$	8.19615	0.339450
$584$	6.26795	0.259370
$585$	0	0
$586$	30.5167	1.26063
$587$	16.0000	0.660391	0.330195	−	0.943913i	$-0.392885\pi$
$587$	16.0000	0.660391	0.330195	+	0.943913i	$0.392885\pi$
$588$	0	0
$589$	6.92820	0.285472
$590$	29.8564	1.22917
$591$	0	0
$592$	−3.53590	−0.145325
$593$	−9.14359	−0.375482	−0.187741	−	0.982219i	$-0.560117\pi$
$593$	−9.14359	−0.375482	−0.187741	+	0.982219i	$0.560117\pi$
$594$	0	0
$595$	58.4449	2.39601
$596$	13.1962	0.540535
$597$	0	0
$598$	0	0
$599$	2.53590	0.103614	0.0518070	−	0.998657i	$-0.483502\pi$
$599$	2.53590	0.103614	0.0518070	+	0.998657i	$0.483502\pi$
$600$	0	0
$601$	7.92820	0.323398	0.161699	−	0.986840i	$-0.448303\pi$
$601$	7.92820	0.323398	0.161699	+	0.986840i	$0.448303\pi$
$602$	−26.3923	−1.07567
$603$	0	0
$604$	−6.73205	−0.273923
$605$	−35.0526	−1.42509
$606$	0	0
$607$	−40.7846	−1.65540	−0.827698	−	0.561174i	$-0.810350\pi$
$607$	−40.7846	−1.65540	−0.827698	+	0.561174i	$0.810350\pi$
$608$	−4.73205	−0.191910
$609$	0	0
$610$	−34.3205	−1.38960
$611$	0	0
$612$	0	0
$613$	9.39230	0.379352	0.189676	−	0.981847i	$-0.439256\pi$
$613$	9.39230	0.379352	0.189676	+	0.981847i	$0.439256\pi$
$614$	22.5885	0.911596
$615$	0	0
$616$	3.46410	0.139573
$617$	−13.2487	−0.533373	−0.266687	−	0.963783i	$-0.585929\pi$
$617$	−13.2487	−0.533373	−0.266687	+	0.963783i	$0.585929\pi$
$618$	0	0
$619$	−17.4641	−0.701942	−0.350971	−	0.936386i	$-0.614148\pi$
$619$	−17.4641	−0.701942	−0.350971	+	0.936386i	$0.614148\pi$
$620$	−5.46410	−0.219444
$621$	0	0
$622$	1.66025	0.0665701
$623$	−25.8564	−1.03592
$624$	0	0
$625$	10.0718	0.402872
$626$	6.53590	0.261227
$627$	0	0
$628$	7.58846	0.302812
$629$	−20.2679	−0.808136
$630$	0	0
$631$	7.71281	0.307042	0.153521	−	0.988145i	$-0.450939\pi$
$631$	7.71281	0.307042	0.153521	+	0.988145i	$0.450939\pi$
$632$	−2.53590	−0.100873
$633$	0	0
$634$	20.6603	0.820524
$635$	−36.7846	−1.45975
$636$	0	0
$637$	0	0
$638$	5.66025	0.224092
$639$	0	0
$640$	3.73205	0.147522
$641$	25.9808	1.02618	0.513089	−	0.858335i	$-0.328501\pi$
$641$	25.9808	1.02618	0.513089	+	0.858335i	$0.328501\pi$
$642$	0	0
$643$	13.8564	0.546443	0.273222	−	0.961951i	$-0.411911\pi$
$643$	13.8564	0.546443	0.273222	+	0.961951i	$0.411911\pi$
$644$	−11.4641	−0.451749
$645$	0	0
$646$	−27.1244	−1.06719
$647$	22.2487	0.874687	0.437344	−	0.899295i	$-0.355919\pi$
$647$	22.2487	0.874687	0.437344	+	0.899295i	$0.355919\pi$
$648$	0	0
$649$	10.1436	0.398171
$650$	0	0
$651$	0	0
$652$	13.4641	0.527295
$653$	17.4641	0.683423	0.341712	−	0.939805i	$-0.388993\pi$
$653$	17.4641	0.683423	0.341712	+	0.939805i	$0.388993\pi$
$654$	0	0
$655$	−24.3923	−0.953086
$656$	−9.39230	−0.366708
$657$	0	0
$658$	6.00000	0.233904
$659$	10.2487	0.399233	0.199617	−	0.979874i	$-0.436030\pi$
$659$	10.2487	0.399233	0.199617	+	0.979874i	$0.436030\pi$
$660$	0	0
$661$	−11.3923	−0.443109	−0.221555	−	0.975148i	$-0.571113\pi$
$661$	−11.3923	−0.443109	−0.221555	+	0.975148i	$0.571113\pi$
$662$	−20.0000	−0.777322
$663$	0	0
$664$	−0.196152	−0.00761219
$665$	−48.2487	−1.87100
$666$	0	0
$667$	−18.7321	−0.725308
$668$	−9.46410	−0.366177
$669$	0	0
$670$	−48.9808	−1.89229
$671$	−11.6603	−0.450139
$672$	0	0
$673$	27.9282	1.07655	0.538277	−	0.842768i	$-0.319076\pi$
$673$	27.9282	1.07655	0.538277	+	0.842768i	$0.319076\pi$
$674$	−20.8564	−0.803359
$675$	0	0
$676$	0	0
$677$	45.4641	1.74733	0.873664	−	0.486530i	$-0.161738\pi$
$677$	45.4641	1.74733	0.873664	+	0.486530i	$0.161738\pi$
$678$	0	0
$679$	16.3923	0.629079
$680$	21.3923	0.820357
$681$	0	0
$682$	−1.85641	−0.0710855
$683$	−10.1436	−0.388134	−0.194067	−	0.980988i	$-0.562168\pi$
$683$	−10.1436	−0.388134	−0.194067	+	0.980988i	$0.562168\pi$
$684$	0	0
$685$	44.5167	1.70089
$686$	−17.8564	−0.681761
$687$	0	0
$688$	−9.66025	−0.368294
$689$	0	0
$690$	0	0
$691$	−43.6603	−1.66091	−0.830457	−	0.557082i	$-0.811921\pi$
$691$	−43.6603	−1.66091	−0.830457	+	0.557082i	$0.811921\pi$
$692$	−4.39230	−0.166970
$693$	0	0
$694$	−33.1244	−1.25738
$695$	66.6410	2.52784
$696$	0	0
$697$	−53.8372	−2.03923
$698$	−15.3205	−0.579890
$699$	0	0
$700$	24.3923	0.921942
$701$	3.32051	0.125414	0.0627069	−	0.998032i	$-0.480027\pi$
$701$	3.32051	0.125414	0.0627069	+	0.998032i	$0.480027\pi$
$702$	0	0
$703$	16.7321	0.631061
$704$	1.26795	0.0477876
$705$	0	0
$706$	−21.7846	−0.819875
$707$	−5.26795	−0.198122
$708$	0	0
$709$	−13.1436	−0.493618	−0.246809	−	0.969064i	$-0.579382\pi$
$709$	−13.1436	−0.493618	−0.246809	+	0.969064i	$0.579382\pi$
$710$	17.6603	0.662778
$711$	0	0
$712$	−9.46410	−0.354682
$713$	6.14359	0.230079
$714$	0	0
$715$	0	0
$716$	16.0526	0.599912
$717$	0	0
$718$	−1.12436	−0.0419606
$719$	29.4641	1.09883	0.549413	−	0.835551i	$-0.314851\pi$
$719$	29.4641	1.09883	0.549413	+	0.835551i	$0.314851\pi$
$720$	0	0
$721$	−41.7128	−1.55347
$722$	3.39230	0.126249
$723$	0	0
$724$	19.1962	0.713419
$725$	39.8564	1.48023
$726$	0	0
$727$	−30.9808	−1.14901	−0.574506	−	0.818500i	$-0.694806\pi$
$727$	−30.9808	−1.14901	−0.574506	+	0.818500i	$0.694806\pi$
$728$	0	0
$729$	0	0
$730$	23.3923	0.865788
$731$	−55.3731	−2.04805
$732$	0	0
$733$	−19.0000	−0.701781	−0.350891	−	0.936416i	$-0.614121\pi$
$733$	−19.0000	−0.701781	−0.350891	+	0.936416i	$0.614121\pi$
$734$	−11.2679	−0.415908
$735$	0	0
$736$	−4.19615	−0.154672
$737$	−16.6410	−0.612980
$738$	0	0
$739$	−2.92820	−0.107716	−0.0538578	−	0.998549i	$-0.517152\pi$
$739$	−2.92820	−0.107716	−0.0538578	+	0.998549i	$0.517152\pi$
$740$	−13.1962	−0.485100
$741$	0	0
$742$	17.6603	0.648328
$743$	−48.3923	−1.77534	−0.887671	−	0.460479i	$-0.847678\pi$
$743$	−48.3923	−1.77534	−0.887671	+	0.460479i	$0.847678\pi$
$744$	0	0
$745$	49.2487	1.80433
$746$	−13.7321	−0.502766
$747$	0	0
$748$	7.26795	0.265743
$749$	−27.8564	−1.01785
$750$	0	0
$751$	−49.9090	−1.82120	−0.910602	−	0.413284i	$-0.864382\pi$
$751$	−49.9090	−1.82120	−0.910602	+	0.413284i	$0.864382\pi$
$752$	2.19615	0.0800854
$753$	0	0
$754$	0	0
$755$	−25.1244	−0.914369
$756$	0	0
$757$	20.9282	0.760648	0.380324	−	0.924853i	$-0.375812\pi$
$757$	20.9282	0.760648	0.380324	+	0.924853i	$0.375812\pi$
$758$	5.46410	0.198465
$759$	0	0
$760$	−17.6603	−0.640605
$761$	11.3205	0.410368	0.205184	−	0.978723i	$-0.434221\pi$
$761$	11.3205	0.410368	0.205184	+	0.978723i	$0.434221\pi$
$762$	0	0
$763$	−4.00000	−0.144810
$764$	6.92820	0.250654
$765$	0	0
$766$	−1.46410	−0.0529001
$767$	0	0
$768$	0	0
$769$	43.8564	1.58150	0.790751	−	0.612138i	$-0.209690\pi$
$769$	43.8564	1.58150	0.790751	+	0.612138i	$0.209690\pi$
$770$	12.9282	0.465900
$771$	0	0
$772$	11.7321	0.422246
$773$	−48.9282	−1.75983	−0.879913	−	0.475136i	$-0.842399\pi$
$773$	−48.9282	−1.75983	−0.879913	+	0.475136i	$0.842399\pi$
$774$	0	0
$775$	−13.0718	−0.469553
$776$	6.00000	0.215387
$777$	0	0
$778$	−11.7846	−0.422499
$779$	44.4449	1.59240
$780$	0	0
$781$	6.00000	0.214697
$782$	−24.0526	−0.860118
$783$	0	0
$784$	0.464102	0.0165751
$785$	28.3205	1.01080
$786$	0	0
$787$	−4.67949	−0.166806	−0.0834029	−	0.996516i	$-0.526579\pi$
$787$	−4.67949	−0.166806	−0.0834029	+	0.996516i	$0.526579\pi$
$788$	−17.8564	−0.636108
$789$	0	0
$790$	−9.46410	−0.336718
$791$	3.66025	0.130144
$792$	0	0
$793$	0	0
$794$	−20.3923	−0.723696
$795$	0	0
$796$	−14.1962	−0.503169
$797$	34.0000	1.20434	0.602171	−	0.798367i	$-0.294303\pi$
$797$	34.0000	1.20434	0.602171	+	0.798367i	$0.294303\pi$
$798$	0	0
$799$	12.5885	0.445348
$800$	8.92820	0.315660
$801$	0	0
$802$	8.07180	0.285025
$803$	7.94744	0.280459
$804$	0	0
$805$	−42.7846	−1.50796
$806$	0	0
$807$	0	0
$808$	−1.92820	−0.0678340
$809$	53.5885	1.88407	0.942035	−	0.335515i	$-0.108910\pi$
$809$	53.5885	1.88407	0.942035	+	0.335515i	$0.108910\pi$
$810$	0	0
$811$	−17.1769	−0.603163	−0.301582	−	0.953440i	$-0.597515\pi$
$811$	−17.1769	−0.603163	−0.301582	+	0.953440i	$0.597515\pi$
$812$	12.1962	0.428001
$813$	0	0
$814$	−4.48334	−0.157141
$815$	50.2487	1.76014
$816$	0	0
$817$	45.7128	1.59929
$818$	−17.7321	−0.619987
$819$	0	0
$820$	−35.0526	−1.22409
$821$	−0.928203	−0.0323945	−0.0161973	−	0.999869i	$-0.505156\pi$
$821$	−0.928203	−0.0323945	−0.0161973	+	0.999869i	$0.505156\pi$
$822$	0	0
$823$	41.5692	1.44901	0.724506	−	0.689269i	$-0.242068\pi$
$823$	41.5692	1.44901	0.724506	+	0.689269i	$0.242068\pi$
$824$	−15.2679	−0.531884
$825$	0	0
$826$	21.8564	0.760482
$827$	26.5359	0.922744	0.461372	−	0.887207i	$-0.347357\pi$
$827$	26.5359	0.922744	0.461372	+	0.887207i	$0.347357\pi$
$828$	0	0
$829$	−12.1244	−0.421096	−0.210548	−	0.977583i	$-0.567525\pi$
$829$	−12.1244	−0.421096	−0.210548	+	0.977583i	$0.567525\pi$
$830$	−0.732051	−0.0254099
$831$	0	0
$832$	0	0
$833$	2.66025	0.0921723
$834$	0	0
$835$	−35.3205	−1.22232
$836$	−6.00000	−0.207514
$837$	0	0
$838$	17.4641	0.603287
$839$	41.8564	1.44504	0.722522	−	0.691348i	$-0.242983\pi$
$839$	41.8564	1.44504	0.722522	+	0.691348i	$0.242983\pi$
$840$	0	0
$841$	−9.07180	−0.312821
$842$	22.7128	0.782735
$843$	0	0
$844$	16.3923	0.564246
$845$	0	0
$846$	0	0
$847$	−25.6603	−0.881697
$848$	6.46410	0.221978
$849$	0	0
$850$	51.1769	1.75535
$851$	14.8372	0.508612
$852$	0	0
$853$	54.1769	1.85498	0.927491	−	0.373845i	$-0.121961\pi$
$853$	54.1769	1.85498	0.927491	+	0.373845i	$0.121961\pi$
$854$	−25.1244	−0.859738
$855$	0	0
$856$	−10.1962	−0.348497
$857$	−39.4449	−1.34741	−0.673705	−	0.739000i	$-0.735298\pi$
$857$	−39.4449	−1.34741	−0.673705	+	0.739000i	$0.735298\pi$
$858$	0	0
$859$	−47.1244	−1.60786	−0.803931	−	0.594722i	$-0.797262\pi$
$859$	−47.1244	−1.60786	−0.803931	+	0.594722i	$0.797262\pi$
$860$	−36.0526	−1.22938
$861$	0	0
$862$	13.1244	0.447017
$863$	−17.1244	−0.582920	−0.291460	−	0.956583i	$-0.594141\pi$
$863$	−17.1244	−0.582920	−0.291460	+	0.956583i	$0.594141\pi$
$864$	0	0
$865$	−16.3923	−0.557355
$866$	12.8564	0.436878
$867$	0	0
$868$	−4.00000	−0.135769
$869$	−3.21539	−0.109075
$870$	0	0
$871$	0	0
$872$	−1.46410	−0.0495807
$873$	0	0
$874$	19.8564	0.671653
$875$	40.0526	1.35402
$876$	0	0
$877$	−23.9282	−0.807998	−0.403999	−	0.914759i	$-0.632380\pi$
$877$	−23.9282	−0.807998	−0.403999	+	0.914759i	$0.632380\pi$
$878$	−0.339746	−0.0114659
$879$	0	0
$880$	4.73205	0.159517
$881$	27.8372	0.937858	0.468929	−	0.883236i	$-0.344640\pi$
$881$	27.8372	0.937858	0.468929	+	0.883236i	$0.344640\pi$
$882$	0	0
$883$	42.9282	1.44465	0.722325	−	0.691554i	$-0.243074\pi$
$883$	42.9282	1.44465	0.722325	+	0.691554i	$0.243074\pi$
$884$	0	0
$885$	0	0
$886$	−15.6077	−0.524351
$887$	−37.8564	−1.27109	−0.635547	−	0.772062i	$-0.719225\pi$
$887$	−37.8564	−1.27109	−0.635547	+	0.772062i	$0.719225\pi$
$888$	0	0
$889$	−26.9282	−0.903143
$890$	−35.3205	−1.18395
$891$	0	0
$892$	26.9282	0.901623
$893$	−10.3923	−0.347765
$894$	0	0
$895$	59.9090	2.00254
$896$	2.73205	0.0912714
$897$	0	0
$898$	11.3205	0.377770
$899$	−6.53590	−0.217984
$900$	0	0
$901$	37.0526	1.23440
$902$	−11.9090	−0.396525
$903$	0	0
$904$	1.33975	0.0445593
$905$	71.6410	2.38143
$906$	0	0
$907$	36.3923	1.20839	0.604193	−	0.796838i	$-0.293495\pi$
$907$	36.3923	1.20839	0.604193	+	0.796838i	$0.293495\pi$
$908$	−12.1962	−0.404744
$909$	0	0
$910$	0	0
$911$	2.53590	0.0840181	0.0420090	−	0.999117i	$-0.486624\pi$
$911$	2.53590	0.0840181	0.0420090	+	0.999117i	$0.486624\pi$
$912$	0	0
$913$	−0.248711	−0.00823114
$914$	−1.33975	−0.0443149
$915$	0	0
$916$	11.8564	0.391747
$917$	−17.8564	−0.589670
$918$	0	0
$919$	45.9615	1.51613	0.758065	−	0.652179i	$-0.226145\pi$
$919$	45.9615	1.51613	0.758065	+	0.652179i	$0.226145\pi$
$920$	−15.6603	−0.516303
$921$	0	0
$922$	22.2679	0.733356
$923$	0	0
$924$	0	0
$925$	−31.5692	−1.03799
$926$	−10.0526	−0.330348
$927$	0	0
$928$	4.46410	0.146541
$929$	−39.2487	−1.28771	−0.643854	−	0.765148i	$-0.722666\pi$
$929$	−39.2487	−1.28771	−0.643854	+	0.765148i	$0.722666\pi$
$930$	0	0
$931$	−2.19615	−0.0719760
$932$	7.85641	0.257345
$933$	0	0
$934$	−18.5885	−0.608233
$935$	27.1244	0.887061
$936$	0	0
$937$	−5.24871	−0.171468	−0.0857340	−	0.996318i	$-0.527324\pi$
$937$	−5.24871	−0.171468	−0.0857340	+	0.996318i	$0.527324\pi$
$938$	−35.8564	−1.17075
$939$	0	0
$940$	8.19615	0.267329
$941$	12.6410	0.412085	0.206043	−	0.978543i	$-0.433941\pi$
$941$	12.6410	0.412085	0.206043	+	0.978543i	$0.433941\pi$
$942$	0	0
$943$	39.4115	1.28342
$944$	8.00000	0.260378
$945$	0	0
$946$	−12.2487	−0.398240
$947$	21.0718	0.684741	0.342371	−	0.939565i	$-0.388770\pi$
$947$	21.0718	0.684741	0.342371	+	0.939565i	$0.388770\pi$
$948$	0	0
$949$	0	0
$950$	−42.2487	−1.37073
$951$	0	0
$952$	15.6603	0.507552
$953$	−41.5692	−1.34656	−0.673280	−	0.739388i	$-0.735115\pi$
$953$	−41.5692	−1.34656	−0.673280	+	0.739388i	$0.735115\pi$
$954$	0	0
$955$	25.8564	0.836694
$956$	7.66025	0.247750
$957$	0	0
$958$	33.4641	1.08118
$959$	32.5885	1.05234
$960$	0	0
$961$	−28.8564	−0.930852
$962$	0	0
$963$	0	0
$964$	−13.5885	−0.437655
$965$	43.7846	1.40948
$966$	0	0
$967$	43.1244	1.38679	0.693393	−	0.720560i	$-0.256115\pi$
$967$	43.1244	1.38679	0.693393	+	0.720560i	$0.256115\pi$
$968$	−9.39230	−0.301880
$969$	0	0
$970$	22.3923	0.718974
$971$	−30.2487	−0.970727	−0.485364	−	0.874312i	$-0.661313\pi$
$971$	−30.2487	−0.970727	−0.485364	+	0.874312i	$0.661313\pi$
$972$	0	0
$973$	48.7846	1.56396
$974$	3.12436	0.100111
$975$	0	0
$976$	−9.19615	−0.294362
$977$	45.9282	1.46937	0.734687	−	0.678407i	$-0.237329\pi$
$977$	45.9282	1.46937	0.734687	+	0.678407i	$0.237329\pi$
$978$	0	0
$979$	−12.0000	−0.383522
$980$	1.73205	0.0553283
$981$	0	0
$982$	−8.73205	−0.278651
$983$	20.7846	0.662926	0.331463	−	0.943468i	$-0.392458\pi$
$983$	20.7846	0.662926	0.331463	+	0.943468i	$0.392458\pi$
$984$	0	0
$985$	−66.6410	−2.12336
$986$	25.5885	0.814902
$987$	0	0
$988$	0	0
$989$	40.5359	1.28897
$990$	0	0
$991$	22.5885	0.717546	0.358773	−	0.933425i	$-0.383195\pi$
$991$	22.5885	0.717546	0.358773	+	0.933425i	$0.383195\pi$
$992$	−1.46410	−0.0464853
$993$	0	0
$994$	12.9282	0.410058
$995$	−52.9808	−1.67960
$996$	0	0
$997$	21.3397	0.675837	0.337918	−	0.941175i	$-0.390277\pi$
$997$	21.3397	0.675837	0.337918	+	0.941175i	$0.390277\pi$
$998$	−32.0000	−1.01294
$999$	0	0

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	3042.2.a.y.1.2		2
3.2	odd	2		1014.2.a.i.1.1		2
12.11	even	2		8112.2.a.bj.1.1		2
13.2	odd	12		234.2.l.c.199.2		4
13.5	odd	4		3042.2.b.i.1351.1		4
13.7	odd	12		234.2.l.c.127.2		4
13.8	odd	4		3042.2.b.i.1351.4		4
13.12	even	2		3042.2.a.p.1.1		2
39.2	even	12		78.2.i.a.43.1	✓	4
39.5	even	4		1014.2.b.e.337.4		4
39.8	even	4		1014.2.b.e.337.1		4
39.11	even	12		1014.2.i.a.823.2		4
39.17	odd	6		1014.2.e.g.991.2		4
39.20	even	12		78.2.i.a.49.1	yes	4
39.23	odd	6		1014.2.e.g.529.2		4
39.29	odd	6		1014.2.e.i.529.1		4
39.32	even	12		1014.2.i.a.361.2		4
39.35	odd	6		1014.2.e.i.991.1		4
39.38	odd	2		1014.2.a.k.1.2		2
52.7	even	12		1872.2.by.h.1297.2		4
52.15	even	12		1872.2.by.h.433.1		4
156.59	odd	12		624.2.bv.e.49.1		4
156.119	odd	12		624.2.bv.e.433.2		4
156.155	even	2		8112.2.a.bp.1.2		2
195.2	odd	12		1950.2.y.g.199.2		4
195.59	even	12		1950.2.bc.d.751.2		4
195.98	odd	12		1950.2.y.g.49.2		4
195.119	even	12		1950.2.bc.d.901.2		4
195.137	odd	12		1950.2.y.b.49.1		4
195.158	odd	12		1950.2.y.b.199.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
78.2.i.a.43.1	✓	4	39.2	even	12
78.2.i.a.49.1	yes	4	39.20	even	12
234.2.l.c.127.2		4	13.7	odd	12
234.2.l.c.199.2		4	13.2	odd	12
624.2.bv.e.49.1		4	156.59	odd	12
624.2.bv.e.433.2		4	156.119	odd	12
1014.2.a.i.1.1		2	3.2	odd	2
1014.2.a.k.1.2		2	39.38	odd	2
1014.2.b.e.337.1		4	39.8	even	4
1014.2.b.e.337.4		4	39.5	even	4
1014.2.e.g.529.2		4	39.23	odd	6
1014.2.e.g.991.2		4	39.17	odd	6
1014.2.e.i.529.1		4	39.29	odd	6
1014.2.e.i.991.1		4	39.35	odd	6
1014.2.i.a.361.2		4	39.32	even	12
1014.2.i.a.823.2		4	39.11	even	12
1872.2.by.h.433.1		4	52.15	even	12
1872.2.by.h.1297.2		4	52.7	even	12
1950.2.y.b.49.1		4	195.137	odd	12
1950.2.y.b.199.1		4	195.158	odd	12
1950.2.y.g.49.2		4	195.98	odd	12
1950.2.y.g.199.2		4	195.2	odd	12
1950.2.bc.d.751.2		4	195.59	even	12
1950.2.bc.d.901.2		4	195.119	even	12
3042.2.a.p.1.1		2	13.12	even	2
3042.2.a.y.1.2		2	1.1	even	1	trivial
3042.2.b.i.1351.1		4	13.5	odd	4
3042.2.b.i.1351.4		4	13.8	odd	4
8112.2.a.bj.1.1		2	12.11	even	2
8112.2.a.bp.1.2		2	156.155	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 $\F_{q}$

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Modular forms

Varieties

Fields

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Groups

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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	3042.2.a.y.1.2

Level	$3042$
Weight	$2$
Character	3042.1
Self dual	yes
Analytic conductor	$24.290$
Analytic rank	$0$
Dimension	$2$
CM	no
Inner twists	$1$