LMFDB - Embedded newform 7623.2.a.cd.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [7623,2,Mod(1,7623)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(7623, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("7623.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 7623 = 3^{2} \cdot 7 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	7623.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:	$60.8699614608$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.11491$ of defining polynomial
Character	$\chi$	$=$	7623.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.47283 q^{2} +4.11491 q^{4} +2.58774 q^{5} +1.00000 q^{7} +5.22982 q^{8} +O(q^{10})$	$q+2.47283 q^{2} +4.11491 q^{4} +2.58774 q^{5} +1.00000 q^{7} +5.22982 q^{8} +6.39905 q^{10} +5.87189 q^{13} +2.47283 q^{14} +4.70265 q^{16} +7.51396 q^{17} +2.35793 q^{19} +10.6483 q^{20} -6.94567 q^{23} +1.69641 q^{25} +14.5202 q^{26} +4.11491 q^{28} -5.87189 q^{29} -3.66152 q^{31} +1.16924 q^{32} +18.5808 q^{34} +2.58774 q^{35} +3.30359 q^{37} +5.83076 q^{38} +13.5334 q^{40} +5.28415 q^{41} -7.40530 q^{43} -17.1755 q^{46} -7.53341 q^{47} +1.00000 q^{49} +4.19493 q^{50} +24.1623 q^{52} +4.22982 q^{53} +5.22982 q^{56} -14.5202 q^{58} +0.926221 q^{59} +2.00000 q^{61} -9.05433 q^{62} -6.51396 q^{64} +15.1949 q^{65} -10.1017 q^{67} +30.9193 q^{68} +6.39905 q^{70} -4.45963 q^{71} +2.12811 q^{73} +8.16924 q^{74} +9.70265 q^{76} +4.45963 q^{79} +12.1692 q^{80} +13.0668 q^{82} -10.6894 q^{83} +19.4442 q^{85} -18.3121 q^{86} -15.8913 q^{89} +5.87189 q^{91} -28.5808 q^{92} -18.6289 q^{94} +6.10170 q^{95} +10.1212 q^{97} +2.47283 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 2 q^{2} + 6 q^{4} - 4 q^{5} + 3 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + 2 * q^2 + 6 * q^4 - 4 * q^5 + 3 * q^7 + 3 * q^8	$ 3 q + 2 q^{2} + 6 q^{4} - 4 q^{5} + 3 q^{7} + 3 q^{8} + 11 q^{10} + 4 q^{13} + 2 q^{14} - 4 q^{16} + 8 q^{17} + 8 q^{19} + 3 q^{20} - 10 q^{23} + 15 q^{25} + q^{26} + 6 q^{28} - 4 q^{29} - 2 q^{31} + 8 q^{32} - 4 q^{34} - 4 q^{35} + 13 q^{38} + 18 q^{40} + 14 q^{41} + 14 q^{43} - 28 q^{46} + 3 q^{49} - 19 q^{50} + 29 q^{52} + 3 q^{56} - q^{58} + 6 q^{61} - 38 q^{62} - 5 q^{64} + 14 q^{65} - 4 q^{67} + 42 q^{68} + 11 q^{70} + 12 q^{71} + 20 q^{73} + 29 q^{74} + 11 q^{76} - 12 q^{79} + 41 q^{80} - 6 q^{82} + 6 q^{83} + 6 q^{85} - 24 q^{86} - 26 q^{89} + 4 q^{91} - 26 q^{92} - 35 q^{94} - 8 q^{95} - 4 q^{97} + 2 q^{98}+O(q^{100}) $ 3 * q + 2 * q^2 + 6 * q^4 - 4 * q^5 + 3 * q^7 + 3 * q^8 + 11 * q^10 + 4 * q^13 + 2 * q^14 - 4 * q^16 + 8 * q^17 + 8 * q^19 + 3 * q^20 - 10 * q^23 + 15 * q^25 + q^26 + 6 * q^28 - 4 * q^29 - 2 * q^31 + 8 * q^32 - 4 * q^34 - 4 * q^35 + 13 * q^38 + 18 * q^40 + 14 * q^41 + 14 * q^43 - 28 * q^46 + 3 * q^49 - 19 * q^50 + 29 * q^52 + 3 * q^56 - q^58 + 6 * q^61 - 38 * q^62 - 5 * q^64 + 14 * q^65 - 4 * q^67 + 42 * q^68 + 11 * q^70 + 12 * q^71 + 20 * q^73 + 29 * q^74 + 11 * q^76 - 12 * q^79 + 41 * q^80 - 6 * q^82 + 6 * q^83 + 6 * q^85 - 24 * q^86 - 26 * q^89 + 4 * q^91 - 26 * q^92 - 35 * q^94 - 8 * q^95 - 4 * q^97 + 2 * q^98

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 $

$2$	2.47283	1.74856	0.874279	−	0.485424i	$-0.161335\pi$
$2$	2.47283	1.74856	0.874279	+	0.485424i	$0.161335\pi$
$3$	0	0
$3$	0	0
$4$	4.11491	2.05745
$5$	2.58774	1.15727	0.578637	−	0.815586i	$-0.303585\pi$
$5$	2.58774	1.15727	0.578637	+	0.815586i	$0.303585\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	5.22982	1.84902
$9$	0	0
$10$	6.39905	2.02356
$11$	0	0
$11$	0	0
$12$	0	0
$13$	5.87189	1.62857	0.814284	−	0.580466i	$-0.197130\pi$
$13$	5.87189	1.62857	0.814284	+	0.580466i	$0.197130\pi$
$14$	2.47283	0.660893
$15$	0	0
$16$	4.70265	1.17566
$17$	7.51396	1.82240	0.911202	−	0.411960i	$-0.135156\pi$
$17$	7.51396	1.82240	0.911202	+	0.411960i	$0.135156\pi$
$18$	0	0
$19$	2.35793	0.540945	0.270473	−	0.962728i	$-0.412820\pi$
$19$	2.35793	0.540945	0.270473	+	0.962728i	$0.412820\pi$
$20$	10.6483	2.38104
$21$	0	0
$22$	0	0
$23$	−6.94567	−1.44827	−0.724136	−	0.689657i	$-0.757761\pi$
$23$	−6.94567	−1.44827	−0.724136	+	0.689657i	$0.757761\pi$
$24$	0	0
$25$	1.69641	0.339281
$26$	14.5202	2.84765
$27$	0	0
$28$	4.11491	0.777644
$29$	−5.87189	−1.09038	−0.545191	−	0.838312i	$-0.683543\pi$
$29$	−5.87189	−1.09038	−0.545191	+	0.838312i	$0.683543\pi$
$30$	0	0
$31$	−3.66152	−0.657629	−0.328814	−	0.944395i	$-0.606649\pi$
$31$	−3.66152	−0.657629	−0.328814	+	0.944395i	$0.606649\pi$
$32$	1.16924	0.206694
$33$	0	0
$34$	18.5808	3.18658
$35$	2.58774	0.437408
$36$	0	0
$37$	3.30359	0.543108	0.271554	−	0.962423i	$-0.412463\pi$
$37$	3.30359	0.543108	0.271554	+	0.962423i	$0.412463\pi$
$38$	5.83076	0.945874
$39$	0	0
$40$	13.5334	2.13982
$41$	5.28415	0.825245	0.412623	−	0.910902i	$-0.364613\pi$
$41$	5.28415	0.825245	0.412623	+	0.910902i	$0.364613\pi$
$42$	0	0
$43$	−7.40530	−1.12930	−0.564649	−	0.825331i	$-0.690988\pi$
$43$	−7.40530	−1.12930	−0.564649	+	0.825331i	$0.690988\pi$
$44$	0	0
$45$	0	0
$46$	−17.1755	−2.53239
$47$	−7.53341	−1.09886	−0.549430	−	0.835540i	$-0.685155\pi$
$47$	−7.53341	−1.09886	−0.549430	+	0.835540i	$0.685155\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	4.19493	0.593253
$51$	0	0
$52$	24.1623	3.35071
$53$	4.22982	0.581010	0.290505	−	0.956874i	$-0.406177\pi$
$53$	4.22982	0.581010	0.290505	+	0.956874i	$0.406177\pi$
$54$	0	0
$55$	0	0
$56$	5.22982	0.698863
$57$	0	0
$58$	−14.5202	−1.90660
$59$	0.926221	0.120584	0.0602918	−	0.998181i	$-0.480797\pi$
$59$	0.926221	0.120584	0.0602918	+	0.998181i	$0.480797\pi$
$60$	0	0
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	−9.05433	−1.14990
$63$	0	0
$64$	−6.51396	−0.814245
$65$	15.1949	1.88470
$66$	0	0
$67$	−10.1017	−1.23412	−0.617060	−	0.786916i	$-0.711676\pi$
$67$	−10.1017	−1.23412	−0.617060	+	0.786916i	$0.711676\pi$
$68$	30.9193	3.74951
$69$	0	0
$70$	6.39905	0.764833
$71$	−4.45963	−0.529261	−0.264630	−	0.964350i	$-0.585250\pi$
$71$	−4.45963	−0.529261	−0.264630	+	0.964350i	$0.585250\pi$
$72$	0	0
$73$	2.12811	0.249077	0.124538	−	0.992215i	$-0.460255\pi$
$73$	2.12811	0.249077	0.124538	+	0.992215i	$0.460255\pi$
$74$	8.16924	0.949655
$75$	0	0
$76$	9.70265	1.11297
$77$	0	0
$78$	0	0
$79$	4.45963	0.501748	0.250874	−	0.968020i	$-0.419282\pi$
$79$	4.45963	0.501748	0.250874	+	0.968020i	$0.419282\pi$
$80$	12.1692	1.36056
$81$	0	0
$82$	13.0668	1.44299
$83$	−10.6894	−1.17332	−0.586660	−	0.809834i	$-0.699557\pi$
$83$	−10.6894	−1.17332	−0.586660	+	0.809834i	$0.699557\pi$
$84$	0	0
$85$	19.4442	2.10902
$86$	−18.3121	−1.97464
$87$	0	0
$88$	0	0
$89$	−15.8913	−1.68448	−0.842239	−	0.539104i	$-0.818763\pi$
$89$	−15.8913	−1.68448	−0.842239	+	0.539104i	$0.818763\pi$
$90$	0	0
$91$	5.87189	0.615541
$92$	−28.5808	−2.97975
$93$	0	0
$94$	−18.6289	−1.92142
$95$	6.10170	0.626022
$96$	0	0
$97$	10.1212	1.02765	0.513824	−	0.857896i	$-0.328229\pi$
$97$	10.1212	1.02765	0.513824	+	0.857896i	$0.328229\pi$
$98$	2.47283	0.249794
$99$	0	0
$100$	6.98055	0.698055
$101$	−13.4053	−1.33388	−0.666939	−	0.745113i	$-0.732396\pi$
$101$	−13.4053	−1.33388	−0.666939	+	0.745113i	$0.732396\pi$
$102$	0	0
$103$	−4.71585	−0.464667	−0.232333	−	0.972636i	$-0.574636\pi$
$103$	−4.71585	−0.464667	−0.232333	+	0.972636i	$0.574636\pi$
$104$	30.7089	3.01125
$105$	0	0
$106$	10.4596	1.01593
$107$	7.07378	0.683848	0.341924	−	0.939728i	$-0.388921\pi$
$107$	7.07378	0.683848	0.341924	+	0.939728i	$0.388921\pi$
$108$	0	0
$109$	−18.8370	−1.80426	−0.902129	−	0.431467i	$-0.857996\pi$
$109$	−18.8370	−1.80426	−0.902129	+	0.431467i	$0.857996\pi$
$110$	0	0
$111$	0	0
$112$	4.70265	0.444359
$113$	9.02792	0.849276	0.424638	−	0.905363i	$-0.360401\pi$
$113$	9.02792	0.849276	0.424638	+	0.905363i	$0.360401\pi$
$114$	0	0
$115$	−17.9736	−1.67605
$116$	−24.1623	−2.24341
$117$	0	0
$118$	2.29039	0.210848
$119$	7.51396	0.688804
$120$	0	0
$121$	0	0
$122$	4.94567	0.447760
$123$	0	0
$124$	−15.0668	−1.35304
$125$	−8.54885	−0.764632
$126$	0	0
$127$	10.2298	0.907749	0.453875	−	0.891066i	$-0.350041\pi$
$127$	10.2298	0.907749	0.453875	+	0.891066i	$0.350041\pi$
$128$	−18.4464	−1.63045
$129$	0	0
$130$	37.5745	3.29550
$131$	4.00000	0.349482	0.174741	−	0.984614i	$-0.444091\pi$
$131$	4.00000	0.349482	0.174741	+	0.984614i	$0.444091\pi$
$132$	0	0
$133$	2.35793	0.204458
$134$	−24.9798	−2.15793
$135$	0	0
$136$	39.2966	3.36966
$137$	10.8370	0.925868	0.462934	−	0.886393i	$-0.346797\pi$
$137$	10.8370	0.925868	0.462934	+	0.886393i	$0.346797\pi$
$138$	0	0
$139$	−0.459630	−0.0389853	−0.0194927	−	0.999810i	$-0.506205\pi$
$139$	−0.459630	−0.0389853	−0.0194927	+	0.999810i	$0.506205\pi$
$140$	10.6483	0.899947
$141$	0	0
$142$	−11.0279	−0.925443
$143$	0	0
$144$	0	0
$145$	−15.1949	−1.26187
$146$	5.26247	0.435525
$147$	0	0
$148$	13.5940	1.11742
$149$	11.0474	0.905036	0.452518	−	0.891755i	$-0.350526\pi$
$149$	11.0474	0.905036	0.452518	+	0.891755i	$0.350526\pi$
$150$	0	0
$151$	16.1212	1.31192	0.655960	−	0.754795i	$-0.272264\pi$
$151$	16.1212	1.31192	0.655960	+	0.754795i	$0.272264\pi$
$152$	12.3315	1.00022
$153$	0	0
$154$	0	0
$155$	−9.47507	−0.761056
$156$	0	0
$157$	13.0668	1.04285	0.521423	−	0.853298i	$-0.325401\pi$
$157$	13.0668	1.04285	0.521423	+	0.853298i	$0.325401\pi$
$158$	11.0279	0.877335
$159$	0	0
$160$	3.02569	0.239202
$161$	−6.94567	−0.547395
$162$	0	0
$163$	−10.1017	−0.791227	−0.395613	−	0.918417i	$-0.629468\pi$
$163$	−10.1017	−0.791227	−0.395613	+	0.918417i	$0.629468\pi$
$164$	21.7438	1.69790
$165$	0	0
$166$	−26.4332	−2.05162
$167$	−10.4860	−0.811434	−0.405717	−	0.913999i	$-0.632978\pi$
$167$	−10.4860	−0.811434	−0.405717	+	0.913999i	$0.632978\pi$
$168$	0	0
$169$	21.4791	1.65224
$170$	48.0823	3.68774
$171$	0	0
$172$	−30.4721	−2.32348
$173$	14.4985	1.10230	0.551151	−	0.834405i	$-0.314189\pi$
$173$	14.4985	1.10230	0.551151	+	0.834405i	$0.314189\pi$
$174$	0	0
$175$	1.69641	0.128236
$176$	0	0
$177$	0	0
$178$	−39.2966	−2.94541
$179$	−14.3510	−1.07264	−0.536321	−	0.844014i	$-0.680186\pi$
$179$	−14.3510	−1.07264	−0.536321	+	0.844014i	$0.680186\pi$
$180$	0	0
$181$	9.66152	0.718135	0.359068	−	0.933312i	$-0.383095\pi$
$181$	9.66152	0.718135	0.359068	+	0.933312i	$0.383095\pi$
$182$	14.5202	1.07631
$183$	0	0
$184$	−36.3246	−2.67788
$185$	8.54885	0.628524
$186$	0	0
$187$	0	0
$188$	−30.9993	−2.26086
$189$	0	0
$190$	15.0885	1.09463
$191$	9.55286	0.691220	0.345610	−	0.938378i	$-0.387672\pi$
$191$	9.55286	0.691220	0.345610	+	0.938378i	$0.387672\pi$
$192$	0	0
$193$	22.1212	1.59232	0.796158	−	0.605089i	$-0.206863\pi$
$193$	22.1212	1.59232	0.796158	+	0.605089i	$0.206863\pi$
$194$	25.0279	1.79690
$195$	0	0
$196$	4.11491	0.293922
$197$	−26.2423	−1.86969	−0.934843	−	0.355061i	$-0.884460\pi$
$197$	−26.2423	−1.86969	−0.934843	+	0.355061i	$0.884460\pi$
$198$	0	0
$199$	1.05433	0.0747396	0.0373698	−	0.999302i	$-0.488102\pi$
$199$	1.05433	0.0747396	0.0373698	+	0.999302i	$0.488102\pi$
$200$	8.87189	0.627337
$201$	0	0
$202$	−33.1491	−2.33236
$203$	−5.87189	−0.412126
$204$	0	0
$205$	13.6740	0.955034
$206$	−11.6615	−0.812497
$207$	0	0
$208$	27.6134	1.91465
$209$	0	0
$210$	0	0
$211$	8.71585	0.600024	0.300012	−	0.953935i	$-0.403009\pi$
$211$	8.71585	0.600024	0.300012	+	0.953935i	$0.403009\pi$
$212$	17.4053	1.19540
$213$	0	0
$214$	17.4923	1.19575
$215$	−19.1630	−1.30691
$216$	0	0
$217$	−3.66152	−0.248560
$218$	−46.5808	−3.15485
$219$	0	0
$220$	0	0
$221$	44.1212	2.96791
$222$	0	0
$223$	−23.7438	−1.59000	−0.795000	−	0.606609i	$-0.792529\pi$
$223$	−23.7438	−1.59000	−0.795000	+	0.606609i	$0.792529\pi$
$224$	1.16924	0.0781231
$225$	0	0
$226$	22.3246	1.48501
$227$	−13.1755	−0.874488	−0.437244	−	0.899343i	$-0.644045\pi$
$227$	−13.1755	−0.874488	−0.437244	+	0.899343i	$0.644045\pi$
$228$	0	0
$229$	13.3664	0.883277	0.441638	−	0.897193i	$-0.354397\pi$
$229$	13.3664	0.883277	0.441638	+	0.897193i	$0.354397\pi$
$230$	−44.4457	−2.93066
$231$	0	0
$232$	−30.7089	−2.01614
$233$	7.13659	0.467533	0.233767	−	0.972293i	$-0.424895\pi$
$233$	7.13659	0.467533	0.233767	+	0.972293i	$0.424895\pi$
$234$	0	0
$235$	−19.4945	−1.27168
$236$	3.81131	0.248095
$237$	0	0
$238$	18.5808	1.20441
$239$	−19.9930	−1.29324	−0.646621	−	0.762811i	$-0.723818\pi$
$239$	−19.9930	−1.29324	−0.646621	+	0.762811i	$0.723818\pi$
$240$	0	0
$241$	9.19493	0.592297	0.296149	−	0.955142i	$-0.404298\pi$
$241$	9.19493	0.592297	0.296149	+	0.955142i	$0.404298\pi$
$242$	0	0
$243$	0	0
$244$	8.22982	0.526860
$245$	2.58774	0.165325
$246$	0	0
$247$	13.8455	0.880967
$248$	−19.1491	−1.21597
$249$	0	0
$250$	−21.1399	−1.33700
$251$	9.42474	0.594885	0.297442	−	0.954740i	$-0.403866\pi$
$251$	9.42474	0.594885	0.297442	+	0.954740i	$0.403866\pi$
$252$	0	0
$253$	0	0
$254$	25.2966	1.58725
$255$	0	0
$256$	−32.5870	−2.03669
$257$	0.0194469	0.00121307	0.000606533	−	1.00000i	$-0.499807\pi$
$257$	0.0194469	0.00121307	0.000606533		1.00000i	$0.499807\pi$
$258$	0	0
$259$	3.30359	0.205275
$260$	62.5257	3.87768
$261$	0	0
$262$	9.89134	0.611089
$263$	27.9930	1.72612	0.863062	−	0.505097i	$-0.168543\pi$
$263$	27.9930	1.72612	0.863062	+	0.505097i	$0.168543\pi$
$264$	0	0
$265$	10.9457	0.672387
$266$	5.83076	0.357507
$267$	0	0
$268$	−41.5676	−2.53914
$269$	−30.9193	−1.88518	−0.942590	−	0.333951i	$-0.891618\pi$
$269$	−30.9193	−1.88518	−0.942590	+	0.333951i	$0.891618\pi$
$270$	0	0
$271$	22.3579	1.35815	0.679074	−	0.734070i	$-0.262382\pi$
$271$	22.3579	1.35815	0.679074	+	0.734070i	$0.262382\pi$
$272$	35.3355	2.14253
$273$	0	0
$274$	26.7981	1.61893
$275$	0	0
$276$	0	0
$277$	−24.7717	−1.48839	−0.744194	−	0.667964i	$-0.767166\pi$
$277$	−24.7717	−1.48839	−0.744194	+	0.667964i	$0.767166\pi$
$278$	−1.13659	−0.0681681
$279$	0	0
$280$	13.5334	0.808776
$281$	−1.19493	−0.0712835	−0.0356418	−	0.999365i	$-0.511348\pi$
$281$	−1.19493	−0.0712835	−0.0356418	+	0.999365i	$0.511348\pi$
$282$	0	0
$283$	15.5723	0.925677	0.462839	−	0.886443i	$-0.346831\pi$
$283$	15.5723	0.925677	0.462839	+	0.886443i	$0.346831\pi$
$284$	−18.3510	−1.08893
$285$	0	0
$286$	0	0
$287$	5.28415	0.311913
$288$	0	0
$289$	39.4596	2.32115
$290$	−37.5745	−2.20645
$291$	0	0
$292$	8.75698	0.512464
$293$	−9.74378	−0.569238	−0.284619	−	0.958641i	$-0.591867\pi$
$293$	−9.74378	−0.569238	−0.284619	+	0.958641i	$0.591867\pi$
$294$	0	0
$295$	2.39682	0.139548
$296$	17.2772	1.00422
$297$	0	0
$298$	27.3183	1.58251
$299$	−40.7842	−2.35861
$300$	0	0
$301$	−7.40530	−0.426834
$302$	39.8649	2.29397
$303$	0	0
$304$	11.0885	0.635969
$305$	5.17548	0.296347
$306$	0	0
$307$	−6.35097	−0.362469	−0.181234	−	0.983440i	$-0.558009\pi$
$307$	−6.35097	−0.362469	−0.181234	+	0.983440i	$0.558009\pi$
$308$	0	0
$309$	0	0
$310$	−23.4303	−1.33075
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	16.9457	0.957826	0.478913	−	0.877862i	$-0.341031\pi$
$313$	16.9457	0.957826	0.478913	+	0.877862i	$0.341031\pi$
$314$	32.3121	1.82348
$315$	0	0
$316$	18.3510	1.03232
$317$	−20.0125	−1.12401	−0.562007	−	0.827133i	$-0.689970\pi$
$317$	−20.0125	−1.12401	−0.562007	+	0.827133i	$0.689970\pi$
$318$	0	0
$319$	0	0
$320$	−16.8565	−0.942304
$321$	0	0
$322$	−17.1755	−0.957152
$323$	17.7174	0.985821
$324$	0	0
$325$	9.96111	0.552543
$326$	−24.9798	−1.38351
$327$	0	0
$328$	27.6351	1.52589
$329$	−7.53341	−0.415330
$330$	0	0
$331$	7.54037	0.414456	0.207228	−	0.978293i	$-0.433556\pi$
$331$	7.54037	0.414456	0.207228	+	0.978293i	$0.433556\pi$
$332$	−43.9861	−2.41405
$333$	0	0
$334$	−25.9302	−1.41884
$335$	−26.1406	−1.42821
$336$	0	0
$337$	12.6894	0.691238	0.345619	−	0.938375i	$-0.387669\pi$
$337$	12.6894	0.691238	0.345619	+	0.938375i	$0.387669\pi$
$338$	53.1142	2.88903
$339$	0	0
$340$	80.0111	4.33921
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	−38.7283	−2.08809
$345$	0	0
$346$	35.8524	1.92744
$347$	1.13659	0.0610153	0.0305076	−	0.999535i	$-0.490288\pi$
$347$	1.13659	0.0610153	0.0305076	+	0.999535i	$0.490288\pi$
$348$	0	0
$349$	14.7911	0.791752	0.395876	−	0.918304i	$-0.370441\pi$
$349$	14.7911	0.791752	0.395876	+	0.918304i	$0.370441\pi$
$350$	4.19493	0.224228
$351$	0	0
$352$	0	0
$353$	7.52092	0.400298	0.200149	−	0.979765i	$-0.435857\pi$
$353$	7.52092	0.400298	0.200149	+	0.979765i	$0.435857\pi$
$354$	0	0
$355$	−11.5404	−0.612499
$356$	−65.3914	−3.46574
$357$	0	0
$358$	−35.4876	−1.87558
$359$	−17.5962	−0.928693	−0.464346	−	0.885654i	$-0.653711\pi$
$359$	−17.5962	−0.928693	−0.464346	+	0.885654i	$0.653711\pi$
$360$	0	0
$361$	−13.4402	−0.707378
$362$	23.8913	1.25570
$363$	0	0
$364$	24.1623	1.26645
$365$	5.50700	0.288250
$366$	0	0
$367$	14.3121	0.747084	0.373542	−	0.927613i	$-0.378143\pi$
$367$	14.3121	0.747084	0.373542	+	0.927613i	$0.378143\pi$
$368$	−32.6630	−1.70268
$369$	0	0
$370$	21.1399	1.09901
$371$	4.22982	0.219601
$372$	0	0
$373$	16.4332	0.850880	0.425440	−	0.904987i	$-0.360119\pi$
$373$	16.4332	0.850880	0.425440	+	0.904987i	$0.360119\pi$
$374$	0	0
$375$	0	0
$376$	−39.3983	−2.03181
$377$	−34.4791	−1.77576
$378$	0	0
$379$	19.5334	1.00336	0.501682	−	0.865052i	$-0.332715\pi$
$379$	19.5334	1.00336	0.501682	+	0.865052i	$0.332715\pi$
$380$	25.1079	1.28801
$381$	0	0
$382$	23.6226	1.20864
$383$	−2.35097	−0.120129	−0.0600644	−	0.998195i	$-0.519131\pi$
$383$	−2.35097	−0.120129	−0.0600644	+	0.998195i	$0.519131\pi$
$384$	0	0
$385$	0	0
$386$	54.7019	2.78426
$387$	0	0
$388$	41.6476	2.11434
$389$	1.74378	0.0884130	0.0442065	−	0.999022i	$-0.485924\pi$
$389$	1.74378	0.0884130	0.0442065	+	0.999022i	$0.485924\pi$
$390$	0	0
$391$	−52.1895	−2.63934
$392$	5.22982	0.264146
$393$	0	0
$394$	−64.8929	−3.26925
$395$	11.5404	0.580659
$396$	0	0
$397$	6.54189	0.328328	0.164164	−	0.986433i	$-0.447507\pi$
$397$	6.54189	0.328328	0.164164	+	0.986433i	$0.447507\pi$
$398$	2.60719	0.130687
$399$	0	0
$400$	7.97760	0.398880
$401$	−5.66152	−0.282723	−0.141361	−	0.989958i	$-0.545148\pi$
$401$	−5.66152	−0.282723	−0.141361	+	0.989958i	$0.545148\pi$
$402$	0	0
$403$	−21.5000	−1.07099
$404$	−55.1616	−2.74439
$405$	0	0
$406$	−14.5202	−0.720626
$407$	0	0
$408$	0	0
$409$	5.48755	0.271342	0.135671	−	0.990754i	$-0.456681\pi$
$409$	5.48755	0.271342	0.135671	+	0.990754i	$0.456681\pi$
$410$	33.8135	1.66993
$411$	0	0
$412$	−19.4053	−0.956030
$413$	0.926221	0.0455764
$414$	0	0
$415$	−27.6615	−1.35785
$416$	6.86565	0.336616
$417$	0	0
$418$	0	0
$419$	−8.24926	−0.403003	−0.201501	−	0.979488i	$-0.564582\pi$
$419$	−8.24926	−0.403003	−0.201501	+	0.979488i	$0.564582\pi$
$420$	0	0
$421$	32.2617	1.57234	0.786171	−	0.618009i	$-0.212061\pi$
$421$	32.2617	1.57234	0.786171	+	0.618009i	$0.212061\pi$
$422$	21.5529	1.04918
$423$	0	0
$424$	22.1212	1.07430
$425$	12.7467	0.618307
$426$	0	0
$427$	2.00000	0.0967868
$428$	29.1079	1.40699
$429$	0	0
$430$	−47.3869	−2.28520
$431$	−23.2772	−1.12122	−0.560611	−	0.828079i	$-0.689434\pi$
$431$	−23.2772	−1.12122	−0.560611	+	0.828079i	$0.689434\pi$
$432$	0	0
$433$	−15.9302	−0.765558	−0.382779	−	0.923840i	$-0.625033\pi$
$433$	−15.9302	−0.765558	−0.382779	+	0.923840i	$0.625033\pi$
$434$	−9.05433	−0.434622
$435$	0	0
$436$	−77.5125	−3.71218
$437$	−16.3774	−0.783436
$438$	0	0
$439$	−22.8565	−1.09088	−0.545439	−	0.838150i	$-0.683637\pi$
$439$	−22.8565	−1.09088	−0.545439	+	0.838150i	$0.683637\pi$
$440$	0	0
$441$	0	0
$442$	109.104	5.18956
$443$	−21.5962	−1.02607	−0.513034	−	0.858368i	$-0.671478\pi$
$443$	−21.5962	−1.02607	−0.513034	+	0.858368i	$0.671478\pi$
$444$	0	0
$445$	−41.1227	−1.94940
$446$	−58.7144	−2.78021
$447$	0	0
$448$	−6.51396	−0.307756
$449$	−18.6630	−0.880763	−0.440382	−	0.897811i	$-0.645157\pi$
$449$	−18.6630	−0.880763	−0.440382	+	0.897811i	$0.645157\pi$
$450$	0	0
$451$	0	0
$452$	37.1491	1.74735
$453$	0	0
$454$	−32.5808	−1.52909
$455$	15.1949	0.712349
$456$	0	0
$457$	−36.4721	−1.70609	−0.853047	−	0.521834i	$-0.825248\pi$
$457$	−36.4721	−1.70609	−0.853047	+	0.521834i	$0.825248\pi$
$458$	33.0529	1.54446
$459$	0	0
$460$	−73.9597	−3.44839
$461$	7.17548	0.334196	0.167098	−	0.985940i	$-0.446560\pi$
$461$	7.17548	0.334196	0.167098	+	0.985940i	$0.446560\pi$
$462$	0	0
$463$	−0.452670	−0.0210373	−0.0105187	−	0.999945i	$-0.503348\pi$
$463$	−0.452670	−0.0210373	−0.0105187	+	0.999945i	$0.503348\pi$
$464$	−27.6134	−1.28192
$465$	0	0
$466$	17.6476	0.817509
$467$	9.68097	0.447982	0.223991	−	0.974591i	$-0.428091\pi$
$467$	9.68097	0.447982	0.223991	+	0.974591i	$0.428091\pi$
$468$	0	0
$469$	−10.1017	−0.466453
$470$	−48.2067	−2.22361
$471$	0	0
$472$	4.84396	0.222962
$473$	0	0
$474$	0	0
$475$	4.00000	0.183533
$476$	30.9193	1.41718
$477$	0	0
$478$	−49.4395	−2.26131
$479$	24.3635	1.11319	0.556597	−	0.830782i	$-0.312107\pi$
$479$	24.3635	1.11319	0.556597	+	0.830782i	$0.312107\pi$
$480$	0	0
$481$	19.3983	0.884488
$482$	22.7375	1.03567
$483$	0	0
$484$	0	0
$485$	26.1909	1.18927
$486$	0	0
$487$	−19.2702	−0.873217	−0.436609	−	0.899652i	$-0.643821\pi$
$487$	−19.2702	−0.873217	−0.436609	+	0.899652i	$0.643821\pi$
$488$	10.4596	0.473485
$489$	0	0
$490$	6.39905	0.289080
$491$	29.6421	1.33773	0.668864	−	0.743385i	$-0.266781\pi$
$491$	29.6421	1.33773	0.668864	+	0.743385i	$0.266781\pi$
$492$	0	0
$493$	−44.1212	−1.98712
$494$	34.2376	1.54042
$495$	0	0
$496$	−17.2188	−0.773149
$497$	−4.45963	−0.200042
$498$	0	0
$499$	29.3719	1.31487	0.657434	−	0.753512i	$-0.271642\pi$
$499$	29.3719	1.31487	0.657434	+	0.753512i	$0.271642\pi$
$500$	−35.1777	−1.57320
$501$	0	0
$502$	23.3058	1.04019
$503$	12.5947	0.561570	0.280785	−	0.959771i	$-0.409405\pi$
$503$	12.5947	0.561570	0.280785	+	0.959771i	$0.409405\pi$
$504$	0	0
$505$	−34.6894	−1.54366
$506$	0	0
$507$	0	0
$508$	42.0947	1.86765
$509$	15.2144	0.674365	0.337183	−	0.941439i	$-0.390526\pi$
$509$	15.2144	0.674365	0.337183	+	0.941439i	$0.390526\pi$
$510$	0	0
$511$	2.12811	0.0941421
$512$	−43.6894	−1.93082
$513$	0	0
$514$	0.0480890	0.00212112
$515$	−12.2034	−0.537746
$516$	0	0
$517$	0	0
$518$	8.16924	0.358936
$519$	0	0
$520$	79.4667	3.48484
$521$	−20.5180	−0.898909	−0.449454	−	0.893303i	$-0.648382\pi$
$521$	−20.5180	−0.898909	−0.449454	+	0.893303i	$0.648382\pi$
$522$	0	0
$523$	29.8844	1.30675	0.653376	−	0.757033i	$-0.273352\pi$
$523$	29.8844	1.30675	0.653376	+	0.757033i	$0.273352\pi$
$524$	16.4596	0.719042
$525$	0	0
$526$	69.2221	3.01823
$527$	−27.5125	−1.19846
$528$	0	0
$529$	25.2423	1.09749
$530$	27.0668	1.17571
$531$	0	0
$532$	9.70265	0.420663
$533$	31.0279	1.34397
$534$	0	0
$535$	18.3051	0.791399
$536$	−52.8300	−2.28191
$537$	0	0
$538$	−76.4582	−3.29635
$539$	0	0
$540$	0	0
$541$	−26.9582	−1.15902	−0.579511	−	0.814965i	$-0.696756\pi$
$541$	−26.9582	−1.15902	−0.579511	+	0.814965i	$0.696756\pi$
$542$	55.2874	2.37480
$543$	0	0
$544$	8.78562	0.376680
$545$	−48.7453	−2.08802
$546$	0	0
$547$	−26.0558	−1.11407	−0.557034	−	0.830490i	$-0.688061\pi$
$547$	−26.0558	−1.11407	−0.557034	+	0.830490i	$0.688061\pi$
$548$	44.5933	1.90493
$549$	0	0
$550$	0	0
$551$	−13.8455	−0.589837
$552$	0	0
$553$	4.45963	0.189643
$554$	−61.2563	−2.60253
$555$	0	0
$556$	−1.89134	−0.0802105
$557$	8.01945	0.339795	0.169897	−	0.985462i	$-0.445656\pi$
$557$	8.01945	0.339795	0.169897	+	0.985462i	$0.445656\pi$
$558$	0	0
$559$	−43.4831	−1.83914
$560$	12.1692	0.514244
$561$	0	0
$562$	−2.95486	−0.124643
$563$	21.9736	0.926077	0.463038	−	0.886338i	$-0.346759\pi$
$563$	21.9736	0.926077	0.463038	+	0.886338i	$0.346759\pi$
$564$	0	0
$565$	23.3619	0.982844
$566$	38.5077	1.61860
$567$	0	0
$568$	−23.3230	−0.978613
$569$	10.0000	0.419222	0.209611	−	0.977785i	$-0.432780\pi$
$569$	10.0000	0.419222	0.209611	+	0.977785i	$0.432780\pi$
$570$	0	0
$571$	−37.6740	−1.57661	−0.788304	−	0.615286i	$-0.789041\pi$
$571$	−37.6740	−1.57661	−0.788304	+	0.615286i	$0.789041\pi$
$572$	0	0
$573$	0	0
$574$	13.0668	0.545398
$575$	−11.7827	−0.491371
$576$	0	0
$577$	−7.80908	−0.325096	−0.162548	−	0.986701i	$-0.551971\pi$
$577$	−7.80908	−0.325096	−0.162548	+	0.986701i	$0.551971\pi$
$578$	97.5771	4.05867
$579$	0	0
$580$	−62.5257	−2.59624
$581$	−10.6894	−0.443473
$582$	0	0
$583$	0	0
$584$	11.1296	0.460547
$585$	0	0
$586$	−24.0947	−0.995345
$587$	43.2633	1.78567	0.892833	−	0.450388i	$-0.148714\pi$
$587$	43.2633	1.78567	0.892833	+	0.450388i	$0.148714\pi$
$588$	0	0
$589$	−8.63360	−0.355741
$590$	5.92694	0.244008
$591$	0	0
$592$	15.5356	0.638511
$593$	−29.7438	−1.22143	−0.610715	−	0.791850i	$-0.709118\pi$
$593$	−29.7438	−1.22143	−0.610715	+	0.791850i	$0.709118\pi$
$594$	0	0
$595$	19.4442	0.797134
$596$	45.4589	1.86207
$597$	0	0
$598$	−100.853	−4.12417
$599$	−11.5404	−0.471527	−0.235763	−	0.971810i	$-0.575759\pi$
$599$	−11.5404	−0.471527	−0.235763	+	0.971810i	$0.575759\pi$
$600$	0	0
$601$	−7.06129	−0.288036	−0.144018	−	0.989575i	$-0.546002\pi$
$601$	−7.06129	−0.288036	−0.144018	+	0.989575i	$0.546002\pi$
$602$	−18.3121	−0.746344
$603$	0	0
$604$	66.3370	2.69922
$605$	0	0
$606$	0	0
$607$	17.3859	0.705670	0.352835	−	0.935686i	$-0.385218\pi$
$607$	17.3859	0.705670	0.352835	+	0.935686i	$0.385218\pi$
$608$	2.75698	0.111810
$609$	0	0
$610$	12.7981	0.518180
$611$	−44.2353	−1.78957
$612$	0	0
$613$	−39.4611	−1.59382	−0.796910	−	0.604098i	$-0.793534\pi$
$613$	−39.4611	−1.59382	−0.796910	+	0.604098i	$0.793534\pi$
$614$	−15.7049	−0.633798
$615$	0	0
$616$	0	0
$617$	4.40378	0.177290	0.0886448	−	0.996063i	$-0.471746\pi$
$617$	4.40378	0.177290	0.0886448	+	0.996063i	$0.471746\pi$
$618$	0	0
$619$	16.4163	0.659825	0.329913	−	0.944011i	$-0.392981\pi$
$619$	16.4163	0.659825	0.329913	+	0.944011i	$0.392981\pi$
$620$	−38.9890	−1.56584
$621$	0	0
$622$	−19.7827	−0.793213
$623$	−15.8913	−0.636673
$624$	0	0
$625$	−30.6042	−1.22417
$626$	41.9038	1.67481
$627$	0	0
$628$	53.7688	2.14561
$629$	24.8231	0.989761
$630$	0	0
$631$	44.4596	1.76991	0.884955	−	0.465677i	$-0.154189\pi$
$631$	44.4596	1.76991	0.884955	+	0.465677i	$0.154189\pi$
$632$	23.3230	0.927741
$633$	0	0
$634$	−49.4876	−1.96540
$635$	26.4721	1.05051
$636$	0	0
$637$	5.87189	0.232653
$638$	0	0
$639$	0	0
$640$	−47.7346	−1.88688
$641$	−10.5419	−0.416380	−0.208190	−	0.978088i	$-0.566757\pi$
$641$	−10.5419	−0.416380	−0.208190	+	0.978088i	$0.566757\pi$
$642$	0	0
$643$	−23.4053	−0.923015	−0.461507	−	0.887136i	$-0.652691\pi$
$643$	−23.4053	−0.923015	−0.461507	+	0.887136i	$0.652691\pi$
$644$	−28.5808	−1.12624
$645$	0	0
$646$	43.8121	1.72376
$647$	−11.4945	−0.451896	−0.225948	−	0.974139i	$-0.572548\pi$
$647$	−11.4945	−0.451896	−0.225948	+	0.974139i	$0.572548\pi$
$648$	0	0
$649$	0	0
$650$	24.6322	0.966153
$651$	0	0
$652$	−41.5676	−1.62791
$653$	−19.8774	−0.777863	−0.388932	−	0.921267i	$-0.627156\pi$
$653$	−19.8774	−0.777863	−0.388932	+	0.921267i	$0.627156\pi$
$654$	0	0
$655$	10.3510	0.404446
$656$	24.8495	0.970210
$657$	0	0
$658$	−18.6289	−0.726229
$659$	15.0738	0.587191	0.293596	−	0.955930i	$-0.405148\pi$
$659$	15.0738	0.587191	0.293596	+	0.955930i	$0.405148\pi$
$660$	0	0
$661$	−4.74226	−0.184453	−0.0922263	−	0.995738i	$-0.529398\pi$
$661$	−4.74226	−0.184453	−0.0922263	+	0.995738i	$0.529398\pi$
$662$	18.6461	0.724701
$663$	0	0
$664$	−55.9038	−2.16949
$665$	6.10170	0.236614
$666$	0	0
$667$	40.7842	1.57917
$668$	−43.1491	−1.66949
$669$	0	0
$670$	−64.6414	−2.49731
$671$	0	0
$672$	0	0
$673$	−9.87885	−0.380802	−0.190401	−	0.981706i	$-0.560979\pi$
$673$	−9.87885	−0.380802	−0.190401	+	0.981706i	$0.560979\pi$
$674$	31.3789	1.20867
$675$	0	0
$676$	88.3844	3.39940
$677$	−29.0279	−1.11563	−0.557817	−	0.829964i	$-0.688361\pi$
$677$	−29.0279	−1.11563	−0.557817	+	0.829964i	$0.688361\pi$
$678$	0	0
$679$	10.1212	0.388414
$680$	101.690	3.89962
$681$	0	0
$682$	0	0
$683$	37.9597	1.45249	0.726243	−	0.687438i	$-0.241265\pi$
$683$	37.9597	1.45249	0.726243	+	0.687438i	$0.241265\pi$
$684$	0	0
$685$	28.0434	1.07148
$686$	2.47283	0.0944132
$687$	0	0
$688$	−34.8245	−1.32767
$689$	24.8370	0.946214
$690$	0	0
$691$	−15.5264	−0.590654	−0.295327	−	0.955396i	$-0.595429\pi$
$691$	−15.5264	−0.590654	−0.295327	+	0.955396i	$0.595429\pi$
$692$	59.6601	2.26794
$693$	0	0
$694$	2.81060	0.106689
$695$	−1.18940	−0.0451167
$696$	0	0
$697$	39.7049	1.50393
$698$	36.5761	1.38442
$699$	0	0
$700$	6.98055	0.263840
$701$	−27.4317	−1.03608	−0.518041	−	0.855356i	$-0.673338\pi$
$701$	−27.4317	−1.03608	−0.518041	+	0.855356i	$0.673338\pi$
$702$	0	0
$703$	7.78963	0.293792
$704$	0	0
$705$	0	0
$706$	18.5980	0.699945
$707$	−13.4053	−0.504158
$708$	0	0
$709$	−44.5180	−1.67191	−0.835954	−	0.548800i	$-0.815085\pi$
$709$	−44.5180	−1.67191	−0.835954	+	0.548800i	$0.815085\pi$
$710$	−28.5374	−1.07099
$711$	0	0
$712$	−83.1087	−3.11463
$713$	25.4317	0.952425
$714$	0	0
$715$	0	0
$716$	−59.0529	−2.20691
$717$	0	0
$718$	−43.5125	−1.62387
$719$	−17.3859	−0.648383	−0.324191	−	0.945992i	$-0.605092\pi$
$719$	−17.3859	−0.648383	−0.324191	+	0.945992i	$0.605092\pi$
$720$	0	0
$721$	−4.71585	−0.175628
$722$	−33.2353	−1.23689
$723$	0	0
$724$	39.7563	1.47753
$725$	−9.96111	−0.369946
$726$	0	0
$727$	−18.6506	−0.691711	−0.345855	−	0.938288i	$-0.612411\pi$
$727$	−18.6506	−0.691711	−0.345855	+	0.938288i	$0.612411\pi$
$728$	30.7089	1.13815
$729$	0	0
$730$	13.6179	0.504021
$731$	−55.6431	−2.05804
$732$	0	0
$733$	−16.3510	−0.603937	−0.301968	−	0.953318i	$-0.597644\pi$
$733$	−16.3510	−0.603937	−0.301968	+	0.953318i	$0.597644\pi$
$734$	35.3914	1.30632
$735$	0	0
$736$	−8.12115	−0.299350
$737$	0	0
$738$	0	0
$739$	32.8370	1.20793	0.603964	−	0.797011i	$-0.293587\pi$
$739$	32.8370	1.20793	0.603964	+	0.797011i	$0.293587\pi$
$740$	35.1777	1.29316
$741$	0	0
$742$	10.4596	0.383985
$743$	20.9123	0.767198	0.383599	−	0.923500i	$-0.374685\pi$
$743$	20.9123	0.767198	0.383599	+	0.923500i	$0.374685\pi$
$744$	0	0
$745$	28.5877	1.04737
$746$	40.6366	1.48781
$747$	0	0
$748$	0	0
$749$	7.07378	0.258470
$750$	0	0
$751$	−39.2633	−1.43274	−0.716368	−	0.697722i	$-0.754197\pi$
$751$	−39.2633	−1.43274	−0.716368	+	0.697722i	$0.754197\pi$
$752$	−35.4270	−1.29189
$753$	0	0
$754$	−85.2610	−3.10502
$755$	41.7174	1.51825
$756$	0	0
$757$	−10.3454	−0.376011	−0.188006	−	0.982168i	$-0.560202\pi$
$757$	−10.3454	−0.376011	−0.188006	+	0.982168i	$0.560202\pi$
$758$	48.3029	1.75444
$759$	0	0
$760$	31.9108	1.15753
$761$	30.9582	1.12223	0.561116	−	0.827737i	$-0.310372\pi$
$761$	30.9582	1.12223	0.561116	+	0.827737i	$0.310372\pi$
$762$	0	0
$763$	−18.8370	−0.681945
$764$	39.3091	1.42215
$765$	0	0
$766$	−5.81355	−0.210052
$767$	5.43867	0.196379
$768$	0	0
$769$	28.8998	1.04215	0.521077	−	0.853510i	$-0.325530\pi$
$769$	28.8998	1.04215	0.521077	+	0.853510i	$0.325530\pi$
$770$	0	0
$771$	0	0
$772$	91.0265	3.27612
$773$	26.7523	0.962212	0.481106	−	0.876663i	$-0.340235\pi$
$773$	26.7523	0.962212	0.481106	+	0.876663i	$0.340235\pi$
$774$	0	0
$775$	−6.21142	−0.223121
$776$	52.9317	1.90014
$777$	0	0
$778$	4.31207	0.154595
$779$	12.4596	0.446413
$780$	0	0
$781$	0	0
$782$	−129.056	−4.61503
$783$	0	0
$784$	4.70265	0.167952
$785$	33.8135	1.20686
$786$	0	0
$787$	−22.1406	−0.789227	−0.394614	−	0.918847i	$-0.629122\pi$
$787$	−22.1406	−0.789227	−0.394614	+	0.918847i	$0.629122\pi$
$788$	−107.985	−3.84679
$789$	0	0
$790$	28.5374	1.01532
$791$	9.02792	0.320996
$792$	0	0
$793$	11.7438	0.417034
$794$	16.1770	0.574100
$795$	0	0
$796$	4.33848	0.153773
$797$	−8.18396	−0.289891	−0.144945	−	0.989440i	$-0.546301\pi$
$797$	−8.18396	−0.289891	−0.144945	+	0.989440i	$0.546301\pi$
$798$	0	0
$799$	−56.6058	−2.00257
$800$	1.98351	0.0701275
$801$	0	0
$802$	−14.0000	−0.494357
$803$	0	0
$804$	0	0
$805$	−17.9736	−0.633486
$806$	−53.1660	−1.87269
$807$	0	0
$808$	−70.1072	−2.46636
$809$	17.6685	0.621191	0.310595	−	0.950542i	$-0.399472\pi$
$809$	17.6685	0.621191	0.310595	+	0.950542i	$0.399472\pi$
$810$	0	0
$811$	43.0210	1.51067	0.755335	−	0.655339i	$-0.227474\pi$
$811$	43.0210	1.51067	0.755335	+	0.655339i	$0.227474\pi$
$812$	−24.1623	−0.847930
$813$	0	0
$814$	0	0
$815$	−26.1406	−0.915665
$816$	0	0
$817$	−17.4611	−0.610888
$818$	13.5698	0.474457
$819$	0	0
$820$	56.2673	1.96494
$821$	18.0364	0.629475	0.314737	−	0.949179i	$-0.398084\pi$
$821$	18.0364	0.629475	0.314737	+	0.949179i	$0.398084\pi$
$822$	0	0
$823$	−8.21037	−0.286195	−0.143098	−	0.989709i	$-0.545706\pi$
$823$	−8.21037	−0.286195	−0.143098	+	0.989709i	$0.545706\pi$
$824$	−24.6630	−0.859178
$825$	0	0
$826$	2.29039	0.0796929
$827$	6.83148	0.237554	0.118777	−	0.992921i	$-0.462103\pi$
$827$	6.83148	0.237554	0.118777	+	0.992921i	$0.462103\pi$
$828$	0	0
$829$	−25.7438	−0.894118	−0.447059	−	0.894504i	$-0.647529\pi$
$829$	−25.7438	−0.894118	−0.447059	+	0.894504i	$0.647529\pi$
$830$	−68.4023	−2.37428
$831$	0	0
$832$	−38.2493	−1.32605
$833$	7.51396	0.260343
$834$	0	0
$835$	−27.1352	−0.939051
$836$	0	0
$837$	0	0
$838$	−20.3991	−0.704674
$839$	30.6002	1.05644	0.528219	−	0.849108i	$-0.322860\pi$
$839$	30.6002	1.05644	0.528219	+	0.849108i	$0.322860\pi$
$840$	0	0
$841$	5.47908	0.188934
$842$	79.7779	2.74933
$843$	0	0
$844$	35.8649	1.23452
$845$	55.5823	1.91209
$846$	0	0
$847$	0	0
$848$	19.8913	0.683071
$849$	0	0
$850$	31.5205	1.08115
$851$	−22.9457	−0.786567
$852$	0	0
$853$	30.4596	1.04292	0.521459	−	0.853276i	$-0.325388\pi$
$853$	30.4596	1.04292	0.521459	+	0.853276i	$0.325388\pi$
$854$	4.94567	0.169237
$855$	0	0
$856$	36.9946	1.26445
$857$	−41.2702	−1.40976	−0.704882	−	0.709325i	$-0.749000\pi$
$857$	−41.2702	−1.40976	−0.704882	+	0.709325i	$0.749000\pi$
$858$	0	0
$859$	−28.9193	−0.986712	−0.493356	−	0.869827i	$-0.664230\pi$
$859$	−28.9193	−0.986712	−0.493356	+	0.869827i	$0.664230\pi$
$860$	−78.8540	−2.68890
$861$	0	0
$862$	−57.5606	−1.96052
$863$	−37.0015	−1.25955	−0.629773	−	0.776779i	$-0.716852\pi$
$863$	−37.0015	−1.25955	−0.629773	+	0.776779i	$0.716852\pi$
$864$	0	0
$865$	37.5184	1.27566
$866$	−39.3928	−1.33862
$867$	0	0
$868$	−15.0668	−0.511401
$869$	0	0
$870$	0	0
$871$	−59.3161	−2.00985
$872$	−98.5140	−3.33611
$873$	0	0
$874$	−40.4985	−1.36988
$875$	−8.54885	−0.289004
$876$	0	0
$877$	6.12562	0.206847	0.103424	−	0.994637i	$-0.467020\pi$
$877$	6.12562	0.206847	0.103424	+	0.994637i	$0.467020\pi$
$878$	−56.5202	−1.90746
$879$	0	0
$880$	0	0
$881$	−25.4123	−0.856161	−0.428080	−	0.903741i	$-0.640810\pi$
$881$	−25.4123	−0.856161	−0.428080	+	0.903741i	$0.640810\pi$
$882$	0	0
$883$	−17.0040	−0.572230	−0.286115	−	0.958195i	$-0.592364\pi$
$883$	−17.0040	−0.572230	−0.286115	+	0.958195i	$0.592364\pi$
$884$	181.554	6.10634
$885$	0	0
$886$	−53.4039	−1.79414
$887$	35.2269	1.18280	0.591401	−	0.806378i	$-0.298575\pi$
$887$	35.2269	1.18280	0.591401	+	0.806378i	$0.298575\pi$
$888$	0	0
$889$	10.2298	0.343097
$890$	−101.690	−3.40864
$891$	0	0
$892$	−97.7034	−3.27135
$893$	−17.7632	−0.594424
$894$	0	0
$895$	−37.1366	−1.24134
$896$	−18.4464	−0.616252
$897$	0	0
$898$	−46.1506	−1.54007
$899$	21.5000	0.717067
$900$	0	0
$901$	31.7827	1.05883
$902$	0	0
$903$	0	0
$904$	47.2144	1.57033
$905$	25.0015	0.831079
$906$	0	0
$907$	−9.72682	−0.322974	−0.161487	−	0.986875i	$-0.551629\pi$
$907$	−9.72682	−0.322974	−0.161487	+	0.986875i	$0.551629\pi$
$908$	−54.2159	−1.79922
$909$	0	0
$910$	37.5745	1.24558
$911$	42.7717	1.41709	0.708545	−	0.705666i	$-0.249352\pi$
$911$	42.7717	1.41709	0.708545	+	0.705666i	$0.249352\pi$
$912$	0	0
$913$	0	0
$914$	−90.1895	−2.98320
$915$	0	0
$916$	55.0015	1.81730
$917$	4.00000	0.132092
$918$	0	0
$919$	−6.26871	−0.206786	−0.103393	−	0.994641i	$-0.532970\pi$
$919$	−6.26871	−0.206786	−0.103393	+	0.994641i	$0.532970\pi$
$920$	−93.9986	−3.09904
$921$	0	0
$922$	17.7438	0.584360
$923$	−26.1865	−0.861938
$924$	0	0
$925$	5.60424	0.184266
$926$	−1.11938	−0.0367850
$927$	0	0
$928$	−6.86565	−0.225376
$929$	−20.0194	−0.656817	−0.328408	−	0.944536i	$-0.606512\pi$
$929$	−20.0194	−0.656817	−0.328408	+	0.944536i	$0.606512\pi$
$930$	0	0
$931$	2.35793	0.0772779
$932$	29.3664	0.961929
$933$	0	0
$934$	23.9394	0.783322
$935$	0	0
$936$	0	0
$937$	11.2144	0.366358	0.183179	−	0.983080i	$-0.441361\pi$
$937$	11.2144	0.366358	0.183179	+	0.983080i	$0.441361\pi$
$938$	−24.9798	−0.815621
$939$	0	0
$940$	−80.2181	−2.61643
$941$	31.3400	1.02165	0.510827	−	0.859683i	$-0.329339\pi$
$941$	31.3400	1.02165	0.510827	+	0.859683i	$0.329339\pi$
$942$	0	0
$943$	−36.7019	−1.19518
$944$	4.35569	0.141766
$945$	0	0
$946$	0	0
$947$	23.2841	0.756633	0.378317	−	0.925676i	$-0.376503\pi$
$947$	23.2841	0.756633	0.378317	+	0.925676i	$0.376503\pi$
$948$	0	0
$949$	12.4960	0.405638
$950$	9.89134	0.320917
$951$	0	0
$952$	39.2966	1.27361
$953$	26.8300	0.869110	0.434555	−	0.900645i	$-0.356906\pi$
$953$	26.8300	0.869110	0.434555	+	0.900645i	$0.356906\pi$
$954$	0	0
$955$	24.7203	0.799931
$956$	−82.2695	−2.66079
$957$	0	0
$958$	60.2468	1.94648
$959$	10.8370	0.349945
$960$	0	0
$961$	−17.5933	−0.567525
$962$	47.9689	1.54658
$963$	0	0
$964$	37.8363	1.21862
$965$	57.2438	1.84274
$966$	0	0
$967$	−36.2034	−1.16422	−0.582112	−	0.813109i	$-0.697773\pi$
$967$	−36.2034	−1.16422	−0.582112	+	0.813109i	$0.697773\pi$
$968$	0	0
$969$	0	0
$970$	64.7658	2.07950
$971$	−6.06281	−0.194565	−0.0972824	−	0.995257i	$-0.531015\pi$
$971$	−6.06281	−0.194565	−0.0972824	+	0.995257i	$0.531015\pi$
$972$	0	0
$973$	−0.459630	−0.0147351
$974$	−47.6521	−1.52687
$975$	0	0
$976$	9.40530	0.301056
$977$	4.28263	0.137013	0.0685067	−	0.997651i	$-0.478177\pi$
$977$	4.28263	0.137013	0.0685067	+	0.997651i	$0.478177\pi$
$978$	0	0
$979$	0	0
$980$	10.6483	0.340148
$981$	0	0
$982$	73.2999	2.33909
$983$	17.1894	0.548257	0.274128	−	0.961693i	$-0.411611\pi$
$983$	17.1894	0.548257	0.274128	+	0.961693i	$0.411611\pi$
$984$	0	0
$985$	−67.9083	−2.16374
$986$	−109.104	−3.47459
$987$	0	0
$988$	56.9729	1.81255
$989$	51.4347	1.63553
$990$	0	0
$991$	35.4945	1.12752	0.563760	−	0.825939i	$-0.309354\pi$
$991$	35.4945	1.12752	0.563760	+	0.825939i	$0.309354\pi$
$992$	−4.28120	−0.135928
$993$	0	0
$994$	−11.0279	−0.349785
$995$	2.72834	0.0864942
$996$	0	0
$997$	−30.4068	−0.962993	−0.481497	−	0.876448i	$-0.659907\pi$
$997$	−30.4068	−0.962993	−0.481497	+	0.876448i	$0.659907\pi$
$998$	72.6319	2.29912
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.cd.1.3		3
3.2	odd	2		2541.2.a.bg.1.1		3
11.10	odd	2		693.2.a.l.1.1		3
33.32	even	2		231.2.a.e.1.3	✓	3
77.76	even	2		4851.2.a.bi.1.1		3
132.131	odd	2		3696.2.a.bo.1.1		3
165.164	even	2		5775.2.a.bp.1.1		3
231.230	odd	2		1617.2.a.t.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.a.e.1.3	✓	3	33.32	even	2
693.2.a.l.1.1		3	11.10	odd	2
1617.2.a.t.1.3		3	231.230	odd	2
2541.2.a.bg.1.1		3	3.2	odd	2
3696.2.a.bo.1.1		3	132.131	odd	2
4851.2.a.bi.1.1		3	77.76	even	2
5775.2.a.bp.1.1		3	165.164	even	2
7623.2.a.cd.1.3		3	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 7623 = 3^{2} \cdot 7 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	7623.a (trivial)

\(f(q)\)	\(=\)	\(q+2.47283 q^{2} +4.11491 q^{4} +2.58774 q^{5} +1.00000 q^{7} +5.22982 q^{8} +O(q^{10})\)	\(q+2.47283 q^{2} +4.11491 q^{4} +2.58774 q^{5} +1.00000 q^{7} +5.22982 q^{8} +6.39905 q^{10} +5.87189 q^{13} +2.47283 q^{14} +4.70265 q^{16} +7.51396 q^{17} +2.35793 q^{19} +10.6483 q^{20} -6.94567 q^{23} +1.69641 q^{25} +14.5202 q^{26} +4.11491 q^{28} -5.87189 q^{29} -3.66152 q^{31} +1.16924 q^{32} +18.5808 q^{34} +2.58774 q^{35} +3.30359 q^{37} +5.83076 q^{38} +13.5334 q^{40} +5.28415 q^{41} -7.40530 q^{43} -17.1755 q^{46} -7.53341 q^{47} +1.00000 q^{49} +4.19493 q^{50} +24.1623 q^{52} +4.22982 q^{53} +5.22982 q^{56} -14.5202 q^{58} +0.926221 q^{59} +2.00000 q^{61} -9.05433 q^{62} -6.51396 q^{64} +15.1949 q^{65} -10.1017 q^{67} +30.9193 q^{68} +6.39905 q^{70} -4.45963 q^{71} +2.12811 q^{73} +8.16924 q^{74} +9.70265 q^{76} +4.45963 q^{79} +12.1692 q^{80} +13.0668 q^{82} -10.6894 q^{83} +19.4442 q^{85} -18.3121 q^{86} -15.8913 q^{89} +5.87189 q^{91} -28.5808 q^{92} -18.6289 q^{94} +6.10170 q^{95} +10.1212 q^{97} +2.47283 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 2 q^{2} + 6 q^{4} - 4 q^{5} + 3 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + 2 * q^2 + 6 * q^4 - 4 * q^5 + 3 * q^7 + 3 * q^8	\( 3 q + 2 q^{2} + 6 q^{4} - 4 q^{5} + 3 q^{7} + 3 q^{8} + 11 q^{10} + 4 q^{13} + 2 q^{14} - 4 q^{16} + 8 q^{17} + 8 q^{19} + 3 q^{20} - 10 q^{23} + 15 q^{25} + q^{26} + 6 q^{28} - 4 q^{29} - 2 q^{31} + 8 q^{32} - 4 q^{34} - 4 q^{35} + 13 q^{38} + 18 q^{40} + 14 q^{41} + 14 q^{43} - 28 q^{46} + 3 q^{49} - 19 q^{50} + 29 q^{52} + 3 q^{56} - q^{58} + 6 q^{61} - 38 q^{62} - 5 q^{64} + 14 q^{65} - 4 q^{67} + 42 q^{68} + 11 q^{70} + 12 q^{71} + 20 q^{73} + 29 q^{74} + 11 q^{76} - 12 q^{79} + 41 q^{80} - 6 q^{82} + 6 q^{83} + 6 q^{85} - 24 q^{86} - 26 q^{89} + 4 q^{91} - 26 q^{92} - 35 q^{94} - 8 q^{95} - 4 q^{97} + 2 q^{98}+O(q^{100}) \) 3 * q + 2 * q^2 + 6 * q^4 - 4 * q^5 + 3 * q^7 + 3 * q^8 + 11 * q^10 + 4 * q^13 + 2 * q^14 - 4 * q^16 + 8 * q^17 + 8 * q^19 + 3 * q^20 - 10 * q^23 + 15 * q^25 + q^26 + 6 * q^28 - 4 * q^29 - 2 * q^31 + 8 * q^32 - 4 * q^34 - 4 * q^35 + 13 * q^38 + 18 * q^40 + 14 * q^41 + 14 * q^43 - 28 * q^46 + 3 * q^49 - 19 * q^50 + 29 * q^52 + 3 * q^56 - q^58 + 6 * q^61 - 38 * q^62 - 5 * q^64 + 14 * q^65 - 4 * q^67 + 42 * q^68 + 11 * q^70 + 12 * q^71 + 20 * q^73 + 29 * q^74 + 11 * q^76 - 12 * q^79 + 41 * q^80 - 6 * q^82 + 6 * q^83 + 6 * q^85 - 24 * q^86 - 26 * q^89 + 4 * q^91 - 26 * q^92 - 35 * q^94 - 8 * q^95 - 4 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more