LMFDB - Embedded newform 7803.2.a.bi.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7803,2,Mod(1,7803)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7803, 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("7803.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7803 = 3^{3} \cdot 17^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7803.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:	$62.3072686972$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.29952.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 8x^{2} + 13 $ x^4 - 8*x^2 + 13
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 459)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.39417$ of defining polynomial
Character	$\chi$	$=$	7803.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.39417 q^{2} +3.73205 q^{4} +1.73205 q^{5} -1.75265 q^{7} -4.14682 q^{8} +O(q^{10})$	$q-2.39417 q^{2} +3.73205 q^{4} +1.73205 q^{5} -1.75265 q^{7} -4.14682 q^{8} -4.14682 q^{10} +2.00000 q^{11} +2.46410 q^{13} +4.19615 q^{14} +2.46410 q^{16} +3.19615 q^{19} +6.46410 q^{20} -4.78834 q^{22} -1.53590 q^{23} -2.00000 q^{25} -5.89948 q^{26} -6.54099 q^{28} -1.73205 q^{29} -9.57668 q^{31} +2.39417 q^{32} -3.03569 q^{35} +10.0463 q^{37} -7.65213 q^{38} -7.18251 q^{40} +10.6603 q^{41} -3.73205 q^{43} +7.46410 q^{44} +3.67720 q^{46} -10.0463 q^{47} -3.92820 q^{49} +4.78834 q^{50} +9.19615 q^{52} +11.3293 q^{53} +3.46410 q^{55} +7.26795 q^{56} +4.14682 q^{58} +3.03569 q^{59} -6.54099 q^{61} +22.9282 q^{62} -10.6603 q^{64} +4.26795 q^{65} +7.73205 q^{67} +7.26795 q^{70} +12.4641 q^{71} -3.03569 q^{73} -24.0526 q^{74} +11.9282 q^{76} -3.50531 q^{77} -7.82403 q^{79} +4.26795 q^{80} -25.5225 q^{82} +8.93516 q^{86} -8.29365 q^{88} +13.0820 q^{89} -4.31872 q^{91} -5.73205 q^{92} +24.0526 q^{94} +5.53590 q^{95} +12.6124 q^{97} +9.40479 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4}+O(q^{10}) $ 4 * q + 8 * q^4	$ 4 q + 8 q^{4} + 8 q^{11} - 4 q^{13} - 4 q^{14} - 4 q^{16} - 8 q^{19} + 12 q^{20} - 20 q^{23} - 8 q^{25} + 8 q^{41} - 8 q^{43} + 16 q^{44} + 12 q^{49} + 16 q^{52} + 36 q^{56} + 64 q^{62} - 8 q^{64} + 24 q^{65} + 24 q^{67} + 36 q^{70} + 36 q^{71} - 20 q^{74} + 20 q^{76} + 24 q^{80} - 16 q^{92} + 20 q^{94} + 36 q^{95}+O(q^{100}) $ 4 * q + 8 * q^4 + 8 * q^11 - 4 * q^13 - 4 * q^14 - 4 * q^16 - 8 * q^19 + 12 * q^20 - 20 * q^23 - 8 * q^25 + 8 * q^41 - 8 * q^43 + 16 * q^44 + 12 * q^49 + 16 * q^52 + 36 * q^56 + 64 * q^62 - 8 * q^64 + 24 * q^65 + 24 * q^67 + 36 * q^70 + 36 * q^71 - 20 * q^74 + 20 * q^76 + 24 * q^80 - 16 * q^92 + 20 * q^94 + 36 * q^95

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 $

$2$	−2.39417	−1.69293	−0.846467	−	0.532441i	$-0.821275\pi$
$2$	−2.39417	−1.69293	−0.846467	+	0.532441i	$0.821275\pi$
$3$	0	0
$3$	0	0
$4$	3.73205	1.86603
$5$	1.73205	0.774597	0.387298	−	0.921954i	$-0.373408\pi$
$5$	1.73205	0.774597	0.387298	+	0.921954i	$0.373408\pi$
$6$	0	0
$7$	−1.75265	−0.662441	−0.331221	−	0.943553i	$-0.607460\pi$
$7$	−1.75265	−0.662441	−0.331221	+	0.943553i	$0.607460\pi$
$8$	−4.14682	−1.46612
$9$	0	0
$10$	−4.14682	−1.31134
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	0	0
$13$	2.46410	0.683419	0.341709	−	0.939806i	$-0.388994\pi$
$13$	2.46410	0.683419	0.341709	+	0.939806i	$0.388994\pi$
$14$	4.19615	1.12147
$15$	0	0
$16$	2.46410	0.616025
$17$	0	0
$17$	0	0
$18$	0	0
$19$	3.19615	0.733248	0.366624	−	0.930369i	$-0.380514\pi$
$19$	3.19615	0.733248	0.366624	+	0.930369i	$0.380514\pi$
$20$	6.46410	1.44542
$21$	0	0
$22$	−4.78834	−1.02088
$23$	−1.53590	−0.320257	−0.160128	−	0.987096i	$-0.551191\pi$
$23$	−1.53590	−0.320257	−0.160128	+	0.987096i	$0.551191\pi$
$24$	0	0
$25$	−2.00000	−0.400000
$26$	−5.89948	−1.15698
$27$	0	0
$28$	−6.54099	−1.23613
$29$	−1.73205	−0.321634	−0.160817	−	0.986984i	$-0.551413\pi$
$29$	−1.73205	−0.321634	−0.160817	+	0.986984i	$0.551413\pi$
$30$	0	0
$31$	−9.57668	−1.72002	−0.860011	−	0.510275i	$-0.829544\pi$
$31$	−9.57668	−1.72002	−0.860011	+	0.510275i	$0.829544\pi$
$32$	2.39417	0.423233
$33$	0	0
$34$	0	0
$35$	−3.03569	−0.513125
$36$	0	0
$37$	10.0463	1.65160	0.825801	−	0.563962i	$-0.190723\pi$
$37$	10.0463	1.65160	0.825801	+	0.563962i	$0.190723\pi$
$38$	−7.65213	−1.24134
$39$	0	0
$40$	−7.18251	−1.13565
$41$	10.6603	1.66485	0.832426	−	0.554136i	$-0.186951\pi$
$41$	10.6603	1.66485	0.832426	+	0.554136i	$0.186951\pi$
$42$	0	0
$43$	−3.73205	−0.569132	−0.284566	−	0.958656i	$-0.591850\pi$
$43$	−3.73205	−0.569132	−0.284566	+	0.958656i	$0.591850\pi$
$44$	7.46410	1.12526
$45$	0	0
$46$	3.67720	0.542174
$47$	−10.0463	−1.46540	−0.732702	−	0.680550i	$-0.761741\pi$
$47$	−10.0463	−1.46540	−0.732702	+	0.680550i	$0.761741\pi$
$48$	0	0
$49$	−3.92820	−0.561172
$50$	4.78834	0.677174
$51$	0	0
$52$	9.19615	1.27528
$53$	11.3293	1.55620	0.778102	−	0.628138i	$-0.216183\pi$
$53$	11.3293	1.55620	0.778102	+	0.628138i	$0.216183\pi$
$54$	0	0
$55$	3.46410	0.467099
$56$	7.26795	0.971221
$57$	0	0
$58$	4.14682	0.544505
$59$	3.03569	0.395213	0.197606	−	0.980281i	$-0.436683\pi$
$59$	3.03569	0.395213	0.197606	+	0.980281i	$0.436683\pi$
$60$	0	0
$61$	−6.54099	−0.837489	−0.418744	−	0.908104i	$-0.637530\pi$
$61$	−6.54099	−0.837489	−0.418744	+	0.908104i	$0.637530\pi$
$62$	22.9282	2.91188
$63$	0	0
$64$	−10.6603	−1.33253
$65$	4.26795	0.529374
$66$	0	0
$67$	7.73205	0.944620	0.472310	−	0.881432i	$-0.343420\pi$
$67$	7.73205	0.944620	0.472310	+	0.881432i	$0.343420\pi$
$68$	0	0
$69$	0	0
$70$	7.26795	0.868686
$71$	12.4641	1.47922	0.739608	−	0.673037i	$-0.235011\pi$
$71$	12.4641	1.47922	0.739608	+	0.673037i	$0.235011\pi$
$72$	0	0
$73$	−3.03569	−0.355300	−0.177650	−	0.984094i	$-0.556850\pi$
$73$	−3.03569	−0.355300	−0.177650	+	0.984094i	$0.556850\pi$
$74$	−24.0526	−2.79605
$75$	0	0
$76$	11.9282	1.36826
$77$	−3.50531	−0.399467
$78$	0	0
$79$	−7.82403	−0.880272	−0.440136	−	0.897931i	$-0.645070\pi$
$79$	−7.82403	−0.880272	−0.440136	+	0.897931i	$0.645070\pi$
$80$	4.26795	0.477171
$81$	0	0
$82$	−25.5225	−2.81848
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	0	0
$86$	8.93516	0.963504
$87$	0	0
$88$	−8.29365	−0.884106
$89$	13.0820	1.38669	0.693344	−	0.720607i	$-0.256137\pi$
$89$	13.0820	1.38669	0.693344	+	0.720607i	$0.256137\pi$
$90$	0	0
$91$	−4.31872	−0.452725
$92$	−5.73205	−0.597608
$93$	0	0
$94$	24.0526	2.48083
$95$	5.53590	0.567971
$96$	0	0
$97$	12.6124	1.28059	0.640296	−	0.768128i	$-0.278812\pi$
$97$	12.6124	1.28059	0.640296	+	0.768128i	$0.278812\pi$
$98$	9.40479	0.950027
$99$	0	0
$100$	−7.46410	−0.746410
$101$	−5.25796	−0.523187	−0.261593	−	0.965178i	$-0.584248\pi$
$101$	−5.25796	−0.523187	−0.261593	+	0.965178i	$0.584248\pi$
$102$	0	0
$103$	−17.1962	−1.69439	−0.847194	−	0.531284i	$-0.821710\pi$
$103$	−17.1962	−1.69439	−0.847194	+	0.531284i	$0.821710\pi$
$104$	−10.2182	−1.00198
$105$	0	0
$106$	−27.1244	−2.63455
$107$	3.00000	0.290021	0.145010	−	0.989430i	$-0.453678\pi$
$107$	3.00000	0.290021	0.145010	+	0.989430i	$0.453678\pi$
$108$	0	0
$109$	12.6124	1.20805	0.604023	−	0.796967i	$-0.293564\pi$
$109$	12.6124	1.20805	0.604023	+	0.796967i	$0.293564\pi$
$110$	−8.29365	−0.790768
$111$	0	0
$112$	−4.31872	−0.408080
$113$	4.53590	0.426701	0.213351	−	0.976976i	$-0.431562\pi$
$113$	4.53590	0.426701	0.213351	+	0.976976i	$0.431562\pi$
$114$	0	0
$115$	−2.66025	−0.248070
$116$	−6.46410	−0.600177
$117$	0	0
$118$	−7.26795	−0.669069
$119$	0	0
$120$	0	0
$121$	−7.00000	−0.636364
$122$	15.6603	1.41781
$123$	0	0
$124$	−35.7407	−3.20961
$125$	−12.1244	−1.08444
$126$	0	0
$127$	−2.92820	−0.259836	−0.129918	−	0.991525i	$-0.541471\pi$
$127$	−2.92820	−0.259836	−0.129918	+	0.991525i	$0.541471\pi$
$128$	20.7341	1.83265
$129$	0	0
$130$	−10.2182	−0.896195
$131$	1.00000	0.0873704	0.0436852	−	0.999045i	$-0.486090\pi$
$131$	1.00000	0.0873704	0.0436852	+	0.999045i	$0.486090\pi$
$132$	0	0
$133$	−5.60175	−0.485733
$134$	−18.5118	−1.59918
$135$	0	0
$136$	0	0
$137$	13.0820	1.11767	0.558835	−	0.829279i	$-0.311249\pi$
$137$	13.0820	1.11767	0.558835	+	0.829279i	$0.311249\pi$
$138$	0	0
$139$	−7.01062	−0.594633	−0.297316	−	0.954779i	$-0.596092\pi$
$139$	−7.01062	−0.594633	−0.297316	+	0.954779i	$0.596092\pi$
$140$	−11.3293	−0.957504
$141$	0	0
$142$	−29.8412	−2.50422
$143$	4.92820	0.412117
$144$	0	0
$145$	−3.00000	−0.249136
$146$	7.26795	0.601500
$147$	0	0
$148$	37.4933	3.08193
$149$	11.3293	0.928135	0.464068	−	0.885800i	$-0.346389\pi$
$149$	11.3293	0.928135	0.464068	+	0.885800i	$0.346389\pi$
$150$	0	0
$151$	−1.73205	−0.140952	−0.0704761	−	0.997513i	$-0.522452\pi$
$151$	−1.73205	−0.140952	−0.0704761	+	0.997513i	$0.522452\pi$
$152$	−13.2539	−1.07503
$153$	0	0
$154$	8.39230	0.676271
$155$	−16.5873	−1.33232
$156$	0	0
$157$	22.9282	1.82987	0.914935	−	0.403601i	$-0.132242\pi$
$157$	22.9282	1.82987	0.914935	+	0.403601i	$0.132242\pi$
$158$	18.7321	1.49024
$159$	0	0
$160$	4.14682	0.327835
$161$	2.69190	0.212151
$162$	0	0
$163$	9.57668	0.750104	0.375052	−	0.927004i	$-0.377625\pi$
$163$	9.57668	0.750104	0.375052	+	0.927004i	$0.377625\pi$
$164$	39.7846	3.10666
$165$	0	0
$166$	0	0
$167$	6.92820	0.536120	0.268060	−	0.963402i	$-0.413617\pi$
$167$	6.92820	0.536120	0.268060	+	0.963402i	$0.413617\pi$
$168$	0	0
$169$	−6.92820	−0.532939
$170$	0	0
$171$	0	0
$172$	−13.9282	−1.06202
$173$	1.07180	0.0814872	0.0407436	−	0.999170i	$-0.487027\pi$
$173$	1.07180	0.0814872	0.0407436	+	0.999170i	$0.487027\pi$
$174$	0	0
$175$	3.50531	0.264976
$176$	4.92820	0.371477
$177$	0	0
$178$	−31.3205	−2.34757
$179$	−9.57668	−0.715795	−0.357897	−	0.933761i	$-0.616506\pi$
$179$	−9.57668	−0.715795	−0.357897	+	0.933761i	$0.616506\pi$
$180$	0	0
$181$	−9.57668	−0.711829	−0.355915	−	0.934519i	$-0.615831\pi$
$181$	−9.57668	−0.711829	−0.355915	+	0.934519i	$0.615831\pi$
$182$	10.3397	0.766433
$183$	0	0
$184$	6.36910	0.469536
$185$	17.4007	1.27933
$186$	0	0
$187$	0	0
$188$	−37.4933	−2.73448
$189$	0	0
$190$	−13.2539	−0.961538
$191$	−2.56606	−0.185674	−0.0928369	−	0.995681i	$-0.529594\pi$
$191$	−2.56606	−0.185674	−0.0928369	+	0.995681i	$0.529594\pi$
$192$	0	0
$193$	−16.5873	−1.19398	−0.596990	−	0.802249i	$-0.703637\pi$
$193$	−16.5873	−1.19398	−0.596990	+	0.802249i	$0.703637\pi$
$194$	−30.1962	−2.16796
$195$	0	0
$196$	−14.6603	−1.04716
$197$	−0.660254	−0.0470412	−0.0235206	−	0.999723i	$-0.507488\pi$
$197$	−0.660254	−0.0470412	−0.0235206	+	0.999723i	$0.507488\pi$
$198$	0	0
$199$	15.6481	1.10926	0.554631	−	0.832097i	$-0.312860\pi$
$199$	15.6481	1.10926	0.554631	+	0.832097i	$0.312860\pi$
$200$	8.29365	0.586450
$201$	0	0
$202$	12.5885	0.885721
$203$	3.03569	0.213063
$204$	0	0
$205$	18.4641	1.28959
$206$	41.1705	2.86849
$207$	0	0
$208$	6.07180	0.421003
$209$	6.39230	0.442165
$210$	0	0
$211$	24.4113	1.68054	0.840272	−	0.542164i	$-0.182395\pi$
$211$	24.4113	1.68054	0.840272	+	0.542164i	$0.182395\pi$
$212$	42.2817	2.90392
$213$	0	0
$214$	−7.18251	−0.490986
$215$	−6.46410	−0.440848
$216$	0	0
$217$	16.7846	1.13941
$218$	−30.1962	−2.04514
$219$	0	0
$220$	12.9282	0.871619
$221$	0	0
$222$	0	0
$223$	23.5885	1.57960	0.789800	−	0.613365i	$-0.210184\pi$
$223$	23.5885	1.57960	0.789800	+	0.613365i	$0.210184\pi$
$224$	−4.19615	−0.280367
$225$	0	0
$226$	−10.8597	−0.722377
$227$	23.7846	1.57864	0.789320	−	0.613982i	$-0.210433\pi$
$227$	23.7846	1.57864	0.789320	+	0.613982i	$0.210433\pi$
$228$	0	0
$229$	15.3923	1.01715	0.508576	−	0.861017i	$-0.330172\pi$
$229$	15.3923	1.01715	0.508576	+	0.861017i	$0.330172\pi$
$230$	6.36910	0.419966
$231$	0	0
$232$	7.18251	0.471555
$233$	5.33975	0.349818	0.174909	−	0.984585i	$-0.444037\pi$
$233$	5.33975	0.349818	0.174909	+	0.984585i	$0.444037\pi$
$234$	0	0
$235$	−17.4007	−1.13510
$236$	11.3293	0.737477
$237$	0	0
$238$	0	0
$239$	18.6837	1.20855	0.604275	−	0.796776i	$-0.293463\pi$
$239$	18.6837	1.20855	0.604275	+	0.796776i	$0.293463\pi$
$240$	0	0
$241$	−3.50531	−0.225797	−0.112898	−	0.993607i	$-0.536013\pi$
$241$	−3.50531	−0.225797	−0.112898	+	0.993607i	$0.536013\pi$
$242$	16.7592	1.07732
$243$	0	0
$244$	−24.4113	−1.56277
$245$	−6.80385	−0.434682
$246$	0	0
$247$	7.87564	0.501115
$248$	39.7128	2.52177
$249$	0	0
$250$	29.0278	1.83588
$251$	−16.5873	−1.04698	−0.523490	−	0.852032i	$-0.675370\pi$
$251$	−16.5873	−1.04698	−0.523490	+	0.852032i	$0.675370\pi$
$252$	0	0
$253$	−3.07180	−0.193122
$254$	7.01062	0.439885
$255$	0	0
$256$	−28.3205	−1.77003
$257$	−20.9060	−1.30408	−0.652041	−	0.758184i	$-0.726087\pi$
$257$	−20.9060	−1.30408	−0.652041	+	0.758184i	$0.726087\pi$
$258$	0	0
$259$	−17.6077	−1.09409
$260$	15.9282	0.987825
$261$	0	0
$262$	−2.39417	−0.147912
$263$	6.07137	0.374377	0.187188	−	0.982324i	$-0.440063\pi$
$263$	6.07137	0.374377	0.187188	+	0.982324i	$0.440063\pi$
$264$	0	0
$265$	19.6230	1.20543
$266$	13.4115	0.822314
$267$	0	0
$268$	28.8564	1.76269
$269$	17.3205	1.05605	0.528025	−	0.849229i	$-0.322933\pi$
$269$	17.3205	1.05605	0.528025	+	0.849229i	$0.322933\pi$
$270$	0	0
$271$	−12.0000	−0.728948	−0.364474	−	0.931214i	$-0.618751\pi$
$271$	−12.0000	−0.728948	−0.364474	+	0.931214i	$0.618751\pi$
$272$	0	0
$273$	0	0
$274$	−31.3205	−1.89214
$275$	−4.00000	−0.241209
$276$	0	0
$277$	23.1283	1.38964	0.694822	−	0.719182i	$-0.255483\pi$
$277$	23.1283	1.38964	0.694822	+	0.719182i	$0.255483\pi$
$278$	16.7846	1.00667
$279$	0	0
$280$	12.5885	0.752304
$281$	1.75265	0.104555	0.0522773	−	0.998633i	$-0.483352\pi$
$281$	1.75265	0.104555	0.0522773	+	0.998633i	$0.483352\pi$
$282$	0	0
$283$	−20.9060	−1.24273	−0.621367	−	0.783520i	$-0.713422\pi$
$283$	−20.9060	−1.24273	−0.621367	+	0.783520i	$0.713422\pi$
$284$	46.5167	2.76026
$285$	0	0
$286$	−11.7990	−0.697687
$287$	−18.6837	−1.10287
$288$	0	0
$289$	0	0
$290$	7.18251	0.421772
$291$	0	0
$292$	−11.3293	−0.662999
$293$	12.2686	0.716738	0.358369	−	0.933580i	$-0.383333\pi$
$293$	12.2686	0.716738	0.358369	+	0.933580i	$0.383333\pi$
$294$	0	0
$295$	5.25796	0.306130
$296$	−41.6603	−2.42145
$297$	0	0
$298$	−27.1244	−1.57127
$299$	−3.78461	−0.218870
$300$	0	0
$301$	6.54099	0.377017
$302$	4.14682	0.238623
$303$	0	0
$304$	7.87564	0.451699
$305$	−11.3293	−0.648716
$306$	0	0
$307$	−9.58846	−0.547242	−0.273621	−	0.961838i	$-0.588221\pi$
$307$	−9.58846	−0.547242	−0.273621	+	0.961838i	$0.588221\pi$
$308$	−13.0820	−0.745416
$309$	0	0
$310$	39.7128	2.25554
$311$	−21.7128	−1.23122	−0.615610	−	0.788051i	$-0.711090\pi$
$311$	−21.7128	−1.23122	−0.615610	+	0.788051i	$0.711090\pi$
$312$	0	0
$313$	15.6481	0.884480	0.442240	−	0.896897i	$-0.354184\pi$
$313$	15.6481	0.884480	0.442240	+	0.896897i	$0.354184\pi$
$314$	−54.8940	−3.09785
$315$	0	0
$316$	−29.1997	−1.64261
$317$	−19.5885	−1.10020	−0.550099	−	0.835100i	$-0.685410\pi$
$317$	−19.5885	−1.10020	−0.550099	+	0.835100i	$0.685410\pi$
$318$	0	0
$319$	−3.46410	−0.193952
$320$	−18.4641	−1.03217
$321$	0	0
$322$	−6.44486	−0.359158
$323$	0	0
$324$	0	0
$325$	−4.92820	−0.273368
$326$	−22.9282	−1.26988
$327$	0	0
$328$	−44.2062	−2.44088
$329$	17.6077	0.970744
$330$	0	0
$331$	2.12436	0.116765	0.0583826	−	0.998294i	$-0.481406\pi$
$331$	2.12436	0.116765	0.0583826	+	0.998294i	$0.481406\pi$
$332$	0	0
$333$	0	0
$334$	−16.5873	−0.907617
$335$	13.3923	0.731700
$336$	0	0
$337$	−29.1997	−1.59061	−0.795304	−	0.606211i	$-0.792689\pi$
$337$	−29.1997	−1.59061	−0.795304	+	0.606211i	$0.792689\pi$
$338$	16.5873	0.902230
$339$	0	0
$340$	0	0
$341$	−19.1534	−1.03721
$342$	0	0
$343$	19.1534	1.03418
$344$	15.4762	0.834419
$345$	0	0
$346$	−2.56606	−0.137952
$347$	−26.8564	−1.44173	−0.720864	−	0.693077i	$-0.756255\pi$
$347$	−26.8564	−1.44173	−0.720864	+	0.693077i	$0.756255\pi$
$348$	0	0
$349$	−31.2487	−1.67271	−0.836353	−	0.548192i	$-0.815316\pi$
$349$	−31.2487	−1.67271	−0.836353	+	0.548192i	$0.815316\pi$
$350$	−8.39230	−0.448588
$351$	0	0
$352$	4.78834	0.255219
$353$	−6.07137	−0.323147	−0.161573	−	0.986861i	$-0.551657\pi$
$353$	−6.07137	−0.323147	−0.161573	+	0.986861i	$0.551657\pi$
$354$	0	0
$355$	21.5885	1.14580
$356$	48.8226	2.58760
$357$	0	0
$358$	22.9282	1.21179
$359$	18.6837	0.986090	0.493045	−	0.870004i	$-0.335884\pi$
$359$	18.6837	0.986090	0.493045	+	0.870004i	$0.335884\pi$
$360$	0	0
$361$	−8.78461	−0.462348
$362$	22.9282	1.20508
$363$	0	0
$364$	−16.1177	−0.844796
$365$	−5.25796	−0.275214
$366$	0	0
$367$	21.8453	1.14031	0.570157	−	0.821536i	$-0.306882\pi$
$367$	21.8453	1.14031	0.570157	+	0.821536i	$0.306882\pi$
$368$	−3.78461	−0.197286
$369$	0	0
$370$	−41.6603	−2.16581
$371$	−19.8564	−1.03089
$372$	0	0
$373$	29.2487	1.51444	0.757220	−	0.653159i	$-0.226557\pi$
$373$	29.2487	1.51444	0.757220	+	0.653159i	$0.226557\pi$
$374$	0	0
$375$	0	0
$376$	41.6603	2.14846
$377$	−4.26795	−0.219811
$378$	0	0
$379$	−2.69190	−0.138274	−0.0691368	−	0.997607i	$-0.522024\pi$
$379$	−2.69190	−0.138274	−0.0691368	+	0.997607i	$0.522024\pi$
$380$	20.6603	1.05985
$381$	0	0
$382$	6.14359	0.314334
$383$	15.6481	0.799578	0.399789	−	0.916607i	$-0.369083\pi$
$383$	15.6481	0.799578	0.399789	+	0.916607i	$0.369083\pi$
$384$	0	0
$385$	−6.07137	−0.309426
$386$	39.7128	2.02133
$387$	0	0
$388$	47.0700	2.38962
$389$	−16.5873	−0.841009	−0.420505	−	0.907290i	$-0.638147\pi$
$389$	−16.5873	−0.841009	−0.420505	+	0.907290i	$0.638147\pi$
$390$	0	0
$391$	0	0
$392$	16.2896	0.822747
$393$	0	0
$394$	1.58076	0.0796376
$395$	−13.5516	−0.681856
$396$	0	0
$397$	13.0820	0.656566	0.328283	−	0.944579i	$-0.393530\pi$
$397$	13.0820	0.656566	0.328283	+	0.944579i	$0.393530\pi$
$398$	−37.4641	−1.87791
$399$	0	0
$400$	−4.92820	−0.246410
$401$	25.0526	1.25107	0.625533	−	0.780198i	$-0.284882\pi$
$401$	25.0526	1.25107	0.625533	+	0.780198i	$0.284882\pi$
$402$	0	0
$403$	−23.5979	−1.17550
$404$	−19.6230	−0.976280
$405$	0	0
$406$	−7.26795	−0.360702
$407$	20.0926	0.995953
$408$	0	0
$409$	−17.9282	−0.886493	−0.443246	−	0.896400i	$-0.646173\pi$
$409$	−17.9282	−0.886493	−0.443246	+	0.896400i	$0.646173\pi$
$410$	−44.2062	−2.18319
$411$	0	0
$412$	−64.1769	−3.16177
$413$	−5.32051	−0.261805
$414$	0	0
$415$	0	0
$416$	5.89948	0.289246
$417$	0	0
$418$	−15.3043	−0.748556
$419$	−19.9282	−0.973556	−0.486778	−	0.873526i	$-0.661828\pi$
$419$	−19.9282	−0.973556	−0.486778	+	0.873526i	$0.661828\pi$
$420$	0	0
$421$	−4.00000	−0.194948	−0.0974740	−	0.995238i	$-0.531076\pi$
$421$	−4.00000	−0.194948	−0.0974740	+	0.995238i	$0.531076\pi$
$422$	−58.4449	−2.84505
$423$	0	0
$424$	−46.9808	−2.28159
$425$	0	0
$426$	0	0
$427$	11.4641	0.554787
$428$	11.1962	0.541186
$429$	0	0
$430$	15.4762	0.746327
$431$	6.60770	0.318281	0.159141	−	0.987256i	$-0.449128\pi$
$431$	6.60770	0.318281	0.159141	+	0.987256i	$0.449128\pi$
$432$	0	0
$433$	17.0000	0.816968	0.408484	−	0.912766i	$-0.366058\pi$
$433$	17.0000	0.816968	0.408484	+	0.912766i	$0.366058\pi$
$434$	−40.1852	−1.92895
$435$	0	0
$436$	47.0700	2.25424
$437$	−4.90897	−0.234828
$438$	0	0
$439$	−27.9166	−1.33239	−0.666194	−	0.745778i	$-0.732078\pi$
$439$	−27.9166	−1.33239	−0.666194	+	0.745778i	$0.732078\pi$
$440$	−14.3650	−0.684826
$441$	0	0
$442$	0	0
$443$	−32.7050	−1.55386	−0.776930	−	0.629587i	$-0.783224\pi$
$443$	−32.7050	−1.55386	−0.776930	+	0.629587i	$0.783224\pi$
$444$	0	0
$445$	22.6587	1.07412
$446$	−56.4748	−2.67416
$447$	0	0
$448$	18.6837	0.882724
$449$	−26.9282	−1.27082	−0.635410	−	0.772175i	$-0.719169\pi$
$449$	−26.9282	−1.27082	−0.635410	+	0.772175i	$0.719169\pi$
$450$	0	0
$451$	21.3205	1.00394
$452$	16.9282	0.796236
$453$	0	0
$454$	−56.9444	−2.67253
$455$	−7.48024	−0.350679
$456$	0	0
$457$	17.8564	0.835287	0.417644	−	0.908611i	$-0.362856\pi$
$457$	17.8564	0.835287	0.417644	+	0.908611i	$0.362856\pi$
$458$	−36.8518	−1.72197
$459$	0	0
$460$	−9.92820	−0.462905
$461$	20.0926	0.935806	0.467903	−	0.883780i	$-0.345010\pi$
$461$	20.0926	0.935806	0.467903	+	0.883780i	$0.345010\pi$
$462$	0	0
$463$	−9.32051	−0.433161	−0.216580	−	0.976265i	$-0.569490\pi$
$463$	−9.32051	−0.433161	−0.216580	+	0.976265i	$0.569490\pi$
$464$	−4.26795	−0.198135
$465$	0	0
$466$	−12.7843	−0.592219
$467$	−29.1997	−1.35120	−0.675600	−	0.737269i	$-0.736115\pi$
$467$	−29.1997	−1.35120	−0.675600	+	0.737269i	$0.736115\pi$
$468$	0	0
$469$	−13.5516	−0.625755
$470$	41.6603	1.92164
$471$	0	0
$472$	−12.5885	−0.579431
$473$	−7.46410	−0.343200
$474$	0	0
$475$	−6.39230	−0.293299
$476$	0	0
$477$	0	0
$478$	−44.7321	−2.04600
$479$	−22.1769	−1.01329	−0.506645	−	0.862155i	$-0.669114\pi$
$479$	−22.1769	−1.01329	−0.506645	+	0.862155i	$0.669114\pi$
$480$	0	0
$481$	24.7551	1.12874
$482$	8.39230	0.382259
$483$	0	0
$484$	−26.1244	−1.18747
$485$	21.8453	0.991942
$486$	0	0
$487$	36.6799	1.66213	0.831063	−	0.556179i	$-0.187733\pi$
$487$	36.6799	1.66213	0.831063	+	0.556179i	$0.187733\pi$
$488$	27.1244	1.22786
$489$	0	0
$490$	16.2896	0.735888
$491$	31.2961	1.41237	0.706187	−	0.708026i	$-0.250414\pi$
$491$	31.2961	1.41237	0.706187	+	0.708026i	$0.250414\pi$
$492$	0	0
$493$	0	0
$494$	−18.8556	−0.848355
$495$	0	0
$496$	−23.5979	−1.05958
$497$	−21.8453	−0.979894
$498$	0	0
$499$	11.3293	0.507171	0.253585	−	0.967313i	$-0.418390\pi$
$499$	11.3293	0.507171	0.253585	+	0.967313i	$0.418390\pi$
$500$	−45.2487	−2.02358
$501$	0	0
$502$	39.7128	1.77247
$503$	38.3205	1.70863	0.854314	−	0.519758i	$-0.173978\pi$
$503$	38.3205	1.70863	0.854314	+	0.519758i	$0.173978\pi$
$504$	0	0
$505$	−9.10706	−0.405259
$506$	7.35440	0.326943
$507$	0	0
$508$	−10.9282	−0.484861
$509$	−10.5159	−0.466110	−0.233055	−	0.972464i	$-0.574872\pi$
$509$	−10.5159	−0.466110	−0.233055	+	0.972464i	$0.574872\pi$
$510$	0	0
$511$	5.32051	0.235365
$512$	26.3359	1.16389
$513$	0	0
$514$	50.0526	2.20772
$515$	−29.7846	−1.31247
$516$	0	0
$517$	−20.0926	−0.883672
$518$	42.1558	1.85222
$519$	0	0
$520$	−17.6984	−0.776128
$521$	−13.6077	−0.596164	−0.298082	−	0.954540i	$-0.596347\pi$
$521$	−13.6077	−0.596164	−0.298082	+	0.954540i	$0.596347\pi$
$522$	0	0
$523$	6.92820	0.302949	0.151475	−	0.988461i	$-0.451598\pi$
$523$	6.92820	0.302949	0.151475	+	0.988461i	$0.451598\pi$
$524$	3.73205	0.163035
$525$	0	0
$526$	−14.5359	−0.633795
$527$	0	0
$528$	0	0
$529$	−20.6410	−0.897435
$530$	−46.9808	−2.04071
$531$	0	0
$532$	−20.9060	−0.906391
$533$	26.2679	1.13779
$534$	0	0
$535$	5.19615	0.224649
$536$	−32.0635	−1.38493
$537$	0	0
$538$	−41.4682	−1.78782
$539$	−7.85641	−0.338399
$540$	0	0
$541$	−19.1534	−0.823467	−0.411734	−	0.911304i	$-0.635077\pi$
$541$	−19.1534	−0.823467	−0.411734	+	0.911304i	$0.635077\pi$
$542$	28.7300	1.23406
$543$	0	0
$544$	0	0
$545$	21.8453	0.935748
$546$	0	0
$547$	39.2460	1.67804	0.839018	−	0.544103i	$-0.183130\pi$
$547$	39.2460	1.67804	0.839018	+	0.544103i	$0.183130\pi$
$548$	48.8226	2.08560
$549$	0	0
$550$	9.57668	0.408351
$551$	−5.53590	−0.235837
$552$	0	0
$553$	13.7128	0.583128
$554$	−55.3731	−2.35258
$555$	0	0
$556$	−26.1640	−1.10960
$557$	−0.813410	−0.0344653	−0.0172326	−	0.999852i	$-0.505486\pi$
$557$	−0.813410	−0.0344653	−0.0172326	+	0.999852i	$0.505486\pi$
$558$	0	0
$559$	−9.19615	−0.388956
$560$	−7.48024	−0.316098
$561$	0	0
$562$	−4.19615	−0.177004
$563$	−5.60175	−0.236086	−0.118043	−	0.993009i	$-0.537662\pi$
$563$	−5.60175	−0.236086	−0.118043	+	0.993009i	$0.537662\pi$
$564$	0	0
$565$	7.85641	0.330522
$566$	50.0526	2.10387
$567$	0	0
$568$	−51.6864	−2.16871
$569$	37.4933	1.57180	0.785901	−	0.618353i	$-0.212200\pi$
$569$	37.4933	1.57180	0.785901	+	0.618353i	$0.212200\pi$
$570$	0	0
$571$	−18.3400	−0.767503	−0.383752	−	0.923436i	$-0.625368\pi$
$571$	−18.3400	−0.767503	−0.383752	+	0.923436i	$0.625368\pi$
$572$	18.3923	0.769021
$573$	0	0
$574$	44.7321	1.86708
$575$	3.07180	0.128103
$576$	0	0
$577$	−10.1436	−0.422283	−0.211142	−	0.977455i	$-0.567718\pi$
$577$	−10.1436	−0.422283	−0.211142	+	0.977455i	$0.567718\pi$
$578$	0	0
$579$	0	0
$580$	−11.1962	−0.464895
$581$	0	0
$582$	0	0
$583$	22.6587	0.938426
$584$	12.5885	0.520914
$585$	0	0
$586$	−29.3731	−1.21339
$587$	−36.2103	−1.49456	−0.747279	−	0.664510i	$-0.768640\pi$
$587$	−36.2103	−1.49456	−0.747279	+	0.664510i	$0.768640\pi$
$588$	0	0
$589$	−30.6085	−1.26120
$590$	−12.5885	−0.518259
$591$	0	0
$592$	24.7551	1.01743
$593$	−25.2247	−1.03586	−0.517928	−	0.855424i	$-0.673296\pi$
$593$	−25.2247	−1.03586	−0.517928	+	0.855424i	$0.673296\pi$
$594$	0	0
$595$	0	0
$596$	42.2817	1.73192
$597$	0	0
$598$	9.06100	0.370532
$599$	18.6837	0.763397	0.381698	−	0.924287i	$-0.375339\pi$
$599$	18.6837	0.763397	0.381698	+	0.924287i	$0.375339\pi$
$600$	0	0
$601$	26.6336	1.08641	0.543204	−	0.839601i	$-0.317211\pi$
$601$	26.6336	1.08641	0.543204	+	0.839601i	$0.317211\pi$
$602$	−15.6603	−0.638264
$603$	0	0
$604$	−6.46410	−0.263021
$605$	−12.1244	−0.492925
$606$	0	0
$607$	0.939245	0.0381228	0.0190614	−	0.999818i	$-0.493932\pi$
$607$	0.939245	0.0381228	0.0190614	+	0.999818i	$0.493932\pi$
$608$	7.65213	0.310335
$609$	0	0
$610$	27.1244	1.09823
$611$	−24.7551	−1.00148
$612$	0	0
$613$	40.1769	1.62273	0.811365	−	0.584540i	$-0.198725\pi$
$613$	40.1769	1.62273	0.811365	+	0.584540i	$0.198725\pi$
$614$	22.9564	0.926445
$615$	0	0
$616$	14.5359	0.585668
$617$	4.51666	0.181834	0.0909170	−	0.995858i	$-0.471020\pi$
$617$	4.51666	0.181834	0.0909170	+	0.995858i	$0.471020\pi$
$618$	0	0
$619$	39.2460	1.57743	0.788714	−	0.614760i	$-0.210747\pi$
$619$	39.2460	1.57743	0.788714	+	0.614760i	$0.210747\pi$
$620$	−61.9046	−2.48615
$621$	0	0
$622$	51.9842	2.08438
$623$	−22.9282	−0.918599
$624$	0	0
$625$	−11.0000	−0.440000
$626$	−37.4641	−1.49737
$627$	0	0
$628$	85.5692	3.41458
$629$	0	0
$630$	0	0
$631$	15.0526	0.599233	0.299616	−	0.954060i	$-0.403141\pi$
$631$	15.0526	0.599233	0.299616	+	0.954060i	$0.403141\pi$
$632$	32.4449	1.29059
$633$	0	0
$634$	46.8981	1.86256
$635$	−5.07180	−0.201268
$636$	0	0
$637$	−9.67949	−0.383515
$638$	8.29365	0.328349
$639$	0	0
$640$	35.9126	1.41957
$641$	19.1962	0.758202	0.379101	−	0.925355i	$-0.376233\pi$
$641$	19.1962	0.758202	0.379101	+	0.925355i	$0.376233\pi$
$642$	0	0
$643$	10.3901	0.409745	0.204873	−	0.978789i	$-0.434322\pi$
$643$	10.3901	0.409745	0.204873	+	0.978789i	$0.434322\pi$
$644$	10.0463	0.395880
$645$	0	0
$646$	0	0
$647$	−18.6837	−0.734534	−0.367267	−	0.930116i	$-0.619706\pi$
$647$	−18.6837	−0.734534	−0.367267	+	0.930116i	$0.619706\pi$
$648$	0	0
$649$	6.07137	0.238322
$650$	11.7990	0.462793
$651$	0	0
$652$	35.7407	1.39971
$653$	44.2487	1.73159	0.865793	−	0.500402i	$-0.166815\pi$
$653$	44.2487	1.73159	0.865793	+	0.500402i	$0.166815\pi$
$654$	0	0
$655$	1.73205	0.0676768
$656$	26.2679	1.02559
$657$	0	0
$658$	−42.1558	−1.64340
$659$	46.2566	1.80190	0.900950	−	0.433922i	$-0.142871\pi$
$659$	46.2566	1.80190	0.900950	+	0.433922i	$0.142871\pi$
$660$	0	0
$661$	−12.4641	−0.484797	−0.242399	−	0.970177i	$-0.577934\pi$
$661$	−12.4641	−0.484797	−0.242399	+	0.970177i	$0.577934\pi$
$662$	−5.08607	−0.197676
$663$	0	0
$664$	0	0
$665$	−9.70252	−0.376247
$666$	0	0
$667$	2.66025	0.103005
$668$	25.8564	1.00041
$669$	0	0
$670$	−32.0635	−1.23872
$671$	−13.0820	−0.505025
$672$	0	0
$673$	27.7908	1.07126	0.535628	−	0.844454i	$-0.320075\pi$
$673$	27.7908	1.07126	0.535628	+	0.844454i	$0.320075\pi$
$674$	69.9090	2.69279
$675$	0	0
$676$	−25.8564	−0.994477
$677$	34.5167	1.32658	0.663292	−	0.748361i	$-0.269159\pi$
$677$	34.5167	1.32658	0.663292	+	0.748361i	$0.269159\pi$
$678$	0	0
$679$	−22.1051	−0.848317
$680$	0	0
$681$	0	0
$682$	45.8564	1.75593
$683$	38.7846	1.48405	0.742026	−	0.670371i	$-0.233865\pi$
$683$	38.7846	1.48405	0.742026	+	0.670371i	$0.233865\pi$
$684$	0	0
$685$	22.6587	0.865743
$686$	−45.8564	−1.75081
$687$	0	0
$688$	−9.19615	−0.350600
$689$	27.9166	1.06354
$690$	0	0
$691$	15.6481	0.595280	0.297640	−	0.954678i	$-0.403800\pi$
$691$	15.6481	0.595280	0.297640	+	0.954678i	$0.403800\pi$
$692$	4.00000	0.152057
$693$	0	0
$694$	64.2988	2.44075
$695$	−12.1427	−0.460601
$696$	0	0
$697$	0	0
$698$	74.8147	2.83178
$699$	0	0
$700$	13.0820	0.494453
$701$	−17.5265	−0.661968	−0.330984	−	0.943636i	$-0.607381\pi$
$701$	−17.5265	−0.661968	−0.330984	+	0.943636i	$0.607381\pi$
$702$	0	0
$703$	32.1095	1.21103
$704$	−21.3205	−0.803547
$705$	0	0
$706$	14.5359	0.547066
$707$	9.21539	0.346580
$708$	0	0
$709$	−21.7194	−0.815690	−0.407845	−	0.913051i	$-0.633720\pi$
$709$	−21.7194	−0.815690	−0.407845	+	0.913051i	$0.633720\pi$
$710$	−51.6864	−1.93976
$711$	0	0
$712$	−54.2487	−2.03306
$713$	14.7088	0.550849
$714$	0	0
$715$	8.53590	0.319225
$716$	−35.7407	−1.33569
$717$	0	0
$718$	−44.7321	−1.66939
$719$	−22.6077	−0.843125	−0.421562	−	0.906799i	$-0.638518\pi$
$719$	−22.6077	−0.843125	−0.421562	+	0.906799i	$0.638518\pi$
$720$	0	0
$721$	30.1389	1.12243
$722$	21.0319	0.782724
$723$	0	0
$724$	−35.7407	−1.32829
$725$	3.46410	0.128654
$726$	0	0
$727$	14.8038	0.549044	0.274522	−	0.961581i	$-0.411480\pi$
$727$	14.8038	0.549044	0.274522	+	0.961581i	$0.411480\pi$
$728$	17.9090	0.663750
$729$	0	0
$730$	12.5885	0.465920
$731$	0	0
$732$	0	0
$733$	−25.5359	−0.943190	−0.471595	−	0.881815i	$-0.656322\pi$
$733$	−25.5359	−0.943190	−0.471595	+	0.881815i	$0.656322\pi$
$734$	−52.3013	−1.93048
$735$	0	0
$736$	−3.67720	−0.135543
$737$	15.4641	0.569628
$738$	0	0
$739$	3.19615	0.117572	0.0587862	−	0.998271i	$-0.481277\pi$
$739$	3.19615	0.117572	0.0587862	+	0.998271i	$0.481277\pi$
$740$	64.9403	2.38725
$741$	0	0
$742$	47.5396	1.74523
$743$	28.6410	1.05074	0.525368	−	0.850875i	$-0.323927\pi$
$743$	28.6410	1.05074	0.525368	+	0.850875i	$0.323927\pi$
$744$	0	0
$745$	19.6230	0.718930
$746$	−70.0264	−2.56385
$747$	0	0
$748$	0	0
$749$	−5.25796	−0.192122
$750$	0	0
$751$	3.50531	0.127911	0.0639553	−	0.997953i	$-0.479629\pi$
$751$	3.50531	0.127911	0.0639553	+	0.997953i	$0.479629\pi$
$752$	−24.7551	−0.902726
$753$	0	0
$754$	10.2182	0.372125
$755$	−3.00000	−0.109181
$756$	0	0
$757$	−38.1769	−1.38756	−0.693782	−	0.720185i	$-0.744057\pi$
$757$	−38.1769	−1.38756	−0.693782	+	0.720185i	$0.744057\pi$
$758$	6.44486	0.234088
$759$	0	0
$760$	−22.9564	−0.832716
$761$	44.5039	1.61327	0.806633	−	0.591052i	$-0.201287\pi$
$761$	44.5039	1.61327	0.806633	+	0.591052i	$0.201287\pi$
$762$	0	0
$763$	−22.1051	−0.800259
$764$	−9.57668	−0.346472
$765$	0	0
$766$	−37.4641	−1.35363
$767$	7.48024	0.270096
$768$	0	0
$769$	−24.0718	−0.868051	−0.434026	−	0.900901i	$-0.642907\pi$
$769$	−24.0718	−0.868051	−0.434026	+	0.900901i	$0.642907\pi$
$770$	14.5359	0.523837
$771$	0	0
$772$	−61.9046	−2.22800
$773$	−22.6587	−0.814976	−0.407488	−	0.913211i	$-0.633595\pi$
$773$	−22.6587	−0.814976	−0.407488	+	0.913211i	$0.633595\pi$
$774$	0	0
$775$	19.1534	0.688009
$776$	−52.3013	−1.87751
$777$	0	0
$778$	39.7128	1.42377
$779$	34.0718	1.22075
$780$	0	0
$781$	24.9282	0.892001
$782$	0	0
$783$	0	0
$784$	−9.67949	−0.345696
$785$	39.7128	1.41741
$786$	0	0
$787$	40.8728	1.45696	0.728479	−	0.685068i	$-0.240228\pi$
$787$	40.8728	1.45696	0.728479	+	0.685068i	$0.240228\pi$
$788$	−2.46410	−0.0877800
$789$	0	0
$790$	32.4449	1.15434
$791$	−7.94986	−0.282665
$792$	0	0
$793$	−16.1177	−0.572355
$794$	−31.3205	−1.11152
$795$	0	0
$796$	58.3993	2.06991
$797$	−9.70252	−0.343681	−0.171840	−	0.985125i	$-0.554971\pi$
$797$	−9.70252	−0.343681	−0.171840	+	0.985125i	$0.554971\pi$
$798$	0	0
$799$	0	0
$800$	−4.78834	−0.169293
$801$	0	0
$802$	−59.9801	−2.11797
$803$	−6.07137	−0.214254
$804$	0	0
$805$	4.66251	0.164332
$806$	56.4974	1.99004
$807$	0	0
$808$	21.8038	0.767057
$809$	−1.87564	−0.0659441	−0.0329721	−	0.999456i	$-0.510497\pi$
$809$	−1.87564	−0.0659441	−0.0329721	+	0.999456i	$0.510497\pi$
$810$	0	0
$811$	2.69190	0.0945254	0.0472627	−	0.998882i	$-0.484950\pi$
$811$	2.69190	0.0945254	0.0472627	+	0.998882i	$0.484950\pi$
$812$	11.3293	0.397582
$813$	0	0
$814$	−48.1051	−1.68608
$815$	16.5873	0.581028
$816$	0	0
$817$	−11.9282	−0.417315
$818$	42.9232	1.50077
$819$	0	0
$820$	68.9090	2.40641
$821$	35.4641	1.23771	0.618853	−	0.785507i	$-0.287598\pi$
$821$	35.4641	1.23771	0.618853	+	0.785507i	$0.287598\pi$
$822$	0	0
$823$	26.1640	0.912019	0.456009	−	0.889975i	$-0.349278\pi$
$823$	26.1640	0.912019	0.456009	+	0.889975i	$0.349278\pi$
$824$	71.3094	2.48418
$825$	0	0
$826$	12.7382	0.443219
$827$	0.215390	0.00748985	0.00374493	−	0.999993i	$-0.498808\pi$
$827$	0.215390	0.00748985	0.00374493	+	0.999993i	$0.498808\pi$
$828$	0	0
$829$	−16.3205	−0.566835	−0.283417	−	0.958997i	$-0.591468\pi$
$829$	−16.3205	−0.566835	−0.283417	+	0.958997i	$0.591468\pi$
$830$	0	0
$831$	0	0
$832$	−26.2679	−0.910677
$833$	0	0
$834$	0	0
$835$	12.0000	0.415277
$836$	23.8564	0.825091
$837$	0	0
$838$	47.7115	1.64817
$839$	−20.4641	−0.706499	−0.353250	−	0.935529i	$-0.614923\pi$
$839$	−20.4641	−0.706499	−0.353250	+	0.935529i	$0.614923\pi$
$840$	0	0
$841$	−26.0000	−0.896552
$842$	9.57668	0.330034
$843$	0	0
$844$	91.1043	3.13594
$845$	−12.0000	−0.412813
$846$	0	0
$847$	12.2686	0.421553
$848$	27.9166	0.958661
$849$	0	0
$850$	0	0
$851$	−15.4301	−0.528937
$852$	0	0
$853$	12.1427	0.415760	0.207880	−	0.978154i	$-0.433344\pi$
$853$	12.1427	0.415760	0.207880	+	0.978154i	$0.433344\pi$
$854$	−27.4470	−0.939217
$855$	0	0
$856$	−12.4405	−0.425207
$857$	31.4641	1.07479	0.537397	−	0.843330i	$-0.319408\pi$
$857$	31.4641	1.07479	0.537397	+	0.843330i	$0.319408\pi$
$858$	0	0
$859$	−57.8564	−1.97404	−0.987018	−	0.160612i	$-0.948653\pi$
$859$	−57.8564	−1.97404	−0.987018	+	0.160612i	$0.948653\pi$
$860$	−24.1244	−0.822634
$861$	0	0
$862$	−15.8199	−0.538830
$863$	16.5873	0.564638	0.282319	−	0.959321i	$-0.408896\pi$
$863$	16.5873	0.564638	0.282319	+	0.959321i	$0.408896\pi$
$864$	0	0
$865$	1.85641	0.0631197
$866$	−40.7009	−1.38307
$867$	0	0
$868$	62.6410	2.12617
$869$	−15.6481	−0.530824
$870$	0	0
$871$	19.0526	0.645571
$872$	−52.3013	−1.77114
$873$	0	0
$874$	11.7529	0.397548
$875$	21.2498	0.718374
$876$	0	0
$877$	44.3781	1.49854	0.749271	−	0.662264i	$-0.230404\pi$
$877$	44.3781	1.49854	0.749271	+	0.662264i	$0.230404\pi$
$878$	66.8372	2.25565
$879$	0	0
$880$	8.53590	0.287745
$881$	−7.98076	−0.268879	−0.134439	−	0.990922i	$-0.542923\pi$
$881$	−7.98076	−0.268879	−0.134439	+	0.990922i	$0.542923\pi$
$882$	0	0
$883$	−3.48334	−0.117224	−0.0586119	−	0.998281i	$-0.518667\pi$
$883$	−3.48334	−0.117224	−0.0586119	+	0.998281i	$0.518667\pi$
$884$	0	0
$885$	0	0
$886$	78.3013	2.63058
$887$	26.0000	0.872995	0.436497	−	0.899706i	$-0.356219\pi$
$887$	26.0000	0.872995	0.436497	+	0.899706i	$0.356219\pi$
$888$	0	0
$889$	5.13213	0.172126
$890$	−54.2487	−1.81842
$891$	0	0
$892$	88.0333	2.94757
$893$	−32.1095	−1.07450
$894$	0	0
$895$	−16.5873	−0.554452
$896$	−36.3397	−1.21403
$897$	0	0
$898$	64.4707	2.15142
$899$	16.5873	0.553217
$900$	0	0
$901$	0	0
$902$	−51.0449	−1.69961
$903$	0	0
$904$	−18.8096	−0.625597
$905$	−16.5873	−0.551380
$906$	0	0
$907$	−12.2686	−0.407371	−0.203686	−	0.979036i	$-0.565292\pi$
$907$	−12.2686	−0.407371	−0.203686	+	0.979036i	$0.565292\pi$
$908$	88.7654	2.94578
$909$	0	0
$910$	17.9090	0.593676
$911$	41.2487	1.36663	0.683315	−	0.730123i	$-0.260537\pi$
$911$	41.2487	1.36663	0.683315	+	0.730123i	$0.260537\pi$
$912$	0	0
$913$	0	0
$914$	−42.7513	−1.41409
$915$	0	0
$916$	57.4449	1.89803
$917$	−1.75265	−0.0578777
$918$	0	0
$919$	−13.9808	−0.461183	−0.230591	−	0.973051i	$-0.574066\pi$
$919$	−13.9808	−0.461183	−0.230591	+	0.973051i	$0.574066\pi$
$920$	11.0316	0.363701
$921$	0	0
$922$	−48.1051	−1.58426
$923$	30.7128	1.01092
$924$	0	0
$925$	−20.0926	−0.660641
$926$	22.3149	0.733313
$927$	0	0
$928$	−4.14682	−0.136126
$929$	6.12436	0.200934	0.100467	−	0.994940i	$-0.467966\pi$
$929$	6.12436	0.200934	0.100467	+	0.994940i	$0.467966\pi$
$930$	0	0
$931$	−12.5551	−0.411478
$932$	19.9282	0.652770
$933$	0	0
$934$	69.9090	2.28749
$935$	0	0
$936$	0	0
$937$	−15.1436	−0.494720	−0.247360	−	0.968924i	$-0.579563\pi$
$937$	−15.1436	−0.494720	−0.247360	+	0.968924i	$0.579563\pi$
$938$	32.4449	1.05936
$939$	0	0
$940$	−64.9403	−2.11812
$941$	−10.3923	−0.338779	−0.169390	−	0.985549i	$-0.554180\pi$
$941$	−10.3923	−0.338779	−0.169390	+	0.985549i	$0.554180\pi$
$942$	0	0
$943$	−16.3731	−0.533180
$944$	7.48024	0.243461
$945$	0	0
$946$	17.8703	0.581015
$947$	−26.7846	−0.870383	−0.435191	−	0.900338i	$-0.643319\pi$
$947$	−26.7846	−0.870383	−0.435191	+	0.900338i	$0.643319\pi$
$948$	0	0
$949$	−7.48024	−0.242819
$950$	15.3043	0.496536
$951$	0	0
$952$	0	0
$953$	4.44455	0.143973	0.0719866	−	0.997406i	$-0.477066\pi$
$953$	4.44455	0.143973	0.0719866	+	0.997406i	$0.477066\pi$
$954$	0	0
$955$	−4.44455	−0.143822
$956$	69.7287	2.25519
$957$	0	0
$958$	53.0953	1.71543
$959$	−22.9282	−0.740390
$960$	0	0
$961$	60.7128	1.95848
$962$	−59.2679	−1.91088
$963$	0	0
$964$	−13.0820	−0.421342
$965$	−28.7300	−0.924853
$966$	0	0
$967$	−3.58846	−0.115397	−0.0576985	−	0.998334i	$-0.518376\pi$
$967$	−3.58846	−0.115397	−0.0576985	+	0.998334i	$0.518376\pi$
$968$	29.0278	0.932988
$969$	0	0
$970$	−52.3013	−1.67929
$971$	−13.5516	−0.434892	−0.217446	−	0.976072i	$-0.569773\pi$
$971$	−13.5516	−0.434892	−0.217446	+	0.976072i	$0.569773\pi$
$972$	0	0
$973$	12.2872	0.393909
$974$	−87.8179	−2.81387
$975$	0	0
$976$	−16.1177	−0.515914
$977$	−7.82403	−0.250313	−0.125156	−	0.992137i	$-0.539943\pi$
$977$	−7.82403	−0.250313	−0.125156	+	0.992137i	$0.539943\pi$
$978$	0	0
$979$	26.1640	0.836204
$980$	−25.3923	−0.811127
$981$	0	0
$982$	−74.9282	−2.39106
$983$	12.3205	0.392963	0.196482	−	0.980508i	$-0.437048\pi$
$983$	12.3205	0.392963	0.196482	+	0.980508i	$0.437048\pi$
$984$	0	0
$985$	−1.14359	−0.0364379
$986$	0	0
$987$	0	0
$988$	29.3923	0.935094
$989$	5.73205	0.182269
$990$	0	0
$991$	46.1307	1.46539	0.732696	−	0.680556i	$-0.238262\pi$
$991$	46.1307	1.46539	0.732696	+	0.680556i	$0.238262\pi$
$992$	−22.9282	−0.727971
$993$	0	0
$994$	52.3013	1.65890
$995$	27.1032	0.859230
$996$	0	0
$997$	26.1640	0.828622	0.414311	−	0.910135i	$-0.364023\pi$
$997$	26.1640	0.828622	0.414311	+	0.910135i	$0.364023\pi$
$998$	−27.1244	−0.858606
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7803.2.a.bi.1.1		4
3.2	odd	2		7803.2.a.bh.1.4		4
17.4	even	4		459.2.d.b.271.7	yes	8
17.13	even	4		459.2.d.b.271.8	yes	8
17.16	even	2		7803.2.a.bh.1.1		4
51.38	odd	4		459.2.d.b.271.2	yes	8
51.47	odd	4		459.2.d.b.271.1	✓	8
51.50	odd	2	inner	7803.2.a.bi.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
459.2.d.b.271.1	✓	8	51.47	odd	4
459.2.d.b.271.2	yes	8	51.38	odd	4
459.2.d.b.271.7	yes	8	17.4	even	4
459.2.d.b.271.8	yes	8	17.13	even	4
7803.2.a.bh.1.1		4	17.16	even	2
7803.2.a.bh.1.4		4	3.2	odd	2
7803.2.a.bi.1.1		4	1.1	even	1	trivial
7803.2.a.bi.1.4		4	51.50	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 7803 = 3^{3} \cdot 17^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7803.a (trivial)

\(f(q)\)	\(=\)	\(q-2.39417 q^{2} +3.73205 q^{4} +1.73205 q^{5} -1.75265 q^{7} -4.14682 q^{8} +O(q^{10})\)	\(q-2.39417 q^{2} +3.73205 q^{4} +1.73205 q^{5} -1.75265 q^{7} -4.14682 q^{8} -4.14682 q^{10} +2.00000 q^{11} +2.46410 q^{13} +4.19615 q^{14} +2.46410 q^{16} +3.19615 q^{19} +6.46410 q^{20} -4.78834 q^{22} -1.53590 q^{23} -2.00000 q^{25} -5.89948 q^{26} -6.54099 q^{28} -1.73205 q^{29} -9.57668 q^{31} +2.39417 q^{32} -3.03569 q^{35} +10.0463 q^{37} -7.65213 q^{38} -7.18251 q^{40} +10.6603 q^{41} -3.73205 q^{43} +7.46410 q^{44} +3.67720 q^{46} -10.0463 q^{47} -3.92820 q^{49} +4.78834 q^{50} +9.19615 q^{52} +11.3293 q^{53} +3.46410 q^{55} +7.26795 q^{56} +4.14682 q^{58} +3.03569 q^{59} -6.54099 q^{61} +22.9282 q^{62} -10.6603 q^{64} +4.26795 q^{65} +7.73205 q^{67} +7.26795 q^{70} +12.4641 q^{71} -3.03569 q^{73} -24.0526 q^{74} +11.9282 q^{76} -3.50531 q^{77} -7.82403 q^{79} +4.26795 q^{80} -25.5225 q^{82} +8.93516 q^{86} -8.29365 q^{88} +13.0820 q^{89} -4.31872 q^{91} -5.73205 q^{92} +24.0526 q^{94} +5.53590 q^{95} +12.6124 q^{97} +9.40479 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4}+O(q^{10}) \) 4 * q + 8 * q^4	\( 4 q + 8 q^{4} + 8 q^{11} - 4 q^{13} - 4 q^{14} - 4 q^{16} - 8 q^{19} + 12 q^{20} - 20 q^{23} - 8 q^{25} + 8 q^{41} - 8 q^{43} + 16 q^{44} + 12 q^{49} + 16 q^{52} + 36 q^{56} + 64 q^{62} - 8 q^{64} + 24 q^{65} + 24 q^{67} + 36 q^{70} + 36 q^{71} - 20 q^{74} + 20 q^{76} + 24 q^{80} - 16 q^{92} + 20 q^{94} + 36 q^{95}+O(q^{100}) \) 4 * q + 8 * q^4 + 8 * q^11 - 4 * q^13 - 4 * q^14 - 4 * q^16 - 8 * q^19 + 12 * q^20 - 20 * q^23 - 8 * q^25 + 8 * q^41 - 8 * q^43 + 16 * q^44 + 12 * q^49 + 16 * q^52 + 36 * q^56 + 64 * q^62 - 8 * q^64 + 24 * q^65 + 24 * q^67 + 36 * q^70 + 36 * q^71 - 20 * q^74 + 20 * q^76 + 24 * q^80 - 16 * q^92 + 20 * q^94 + 36 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more