LMFDB - Embedded newform 7623.2.a.cu.1.4

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:	$1$
Dimension:	$8$
Coefficient field:	8.8.6988960000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 9x^{6} + 22x^{4} - 11x^{2} + 1 $ x^8 - 9x^6 + 22x^4 - 11*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 693)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.342036$ 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)

$f(q)$	$=$	$q-0.342036 q^{2} -1.88301 q^{4} -2.14896 q^{5} -1.00000 q^{7} +1.32813 q^{8} +O(q^{10})$	$q-0.342036 q^{2} -1.88301 q^{4} -2.14896 q^{5} -1.00000 q^{7} +1.32813 q^{8} +0.735023 q^{10} +1.42874 q^{13} +0.342036 q^{14} +3.31175 q^{16} +1.38414 q^{17} -2.59251 q^{19} +4.04652 q^{20} -0.0906318 q^{23} -0.381966 q^{25} -0.488682 q^{26} +1.88301 q^{28} -7.74489 q^{29} -4.82857 q^{31} -3.78900 q^{32} -0.473428 q^{34} +2.14896 q^{35} -0.0702129 q^{37} +0.886732 q^{38} -2.85410 q^{40} +6.70278 q^{41} +6.28623 q^{43} +0.0309994 q^{46} +9.19116 q^{47} +1.00000 q^{49} +0.130646 q^{50} -2.69034 q^{52} +3.73461 q^{53} -1.32813 q^{56} +2.64903 q^{58} +9.65657 q^{59} +6.33728 q^{61} +1.65155 q^{62} -5.32753 q^{64} -3.07031 q^{65} +7.09564 q^{67} -2.60636 q^{68} -0.735023 q^{70} -8.96377 q^{71} -11.9787 q^{73} +0.0240154 q^{74} +4.88172 q^{76} +1.63043 q^{79} -7.11683 q^{80} -2.29259 q^{82} -9.90374 q^{83} -2.97447 q^{85} -2.15012 q^{86} +11.2098 q^{89} -1.42874 q^{91} +0.170661 q^{92} -3.14371 q^{94} +5.57120 q^{95} -5.60490 q^{97} -0.342036 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 2 q^{4} - 8 q^{7}+O(q^{10}) $ 8 * q + 2 * q^4 - 8 * q^7	$ 8 q + 2 q^{4} - 8 q^{7} + 14 q^{10} - 2 q^{16} - 6 q^{19} - 12 q^{25} - 2 q^{28} - 6 q^{31} - 24 q^{34} - 38 q^{37} + 4 q^{40} + 16 q^{43} - 42 q^{46} + 8 q^{49} + 2 q^{52} - 30 q^{58} + 28 q^{61} - 36 q^{64} - 36 q^{67} - 14 q^{70} + 14 q^{73} - 34 q^{76} + 22 q^{79} + 36 q^{82} - 18 q^{85} + 32 q^{94} - 48 q^{97}+O(q^{100}) $ 8 * q + 2 * q^4 - 8 * q^7 + 14 * q^10 - 2 * q^16 - 6 * q^19 - 12 * q^25 - 2 * q^28 - 6 * q^31 - 24 * q^34 - 38 * q^37 + 4 * q^40 + 16 * q^43 - 42 * q^46 + 8 * q^49 + 2 * q^52 - 30 * q^58 + 28 * q^61 - 36 * q^64 - 36 * q^67 - 14 * q^70 + 14 * q^73 - 34 * q^76 + 22 * q^79 + 36 * q^82 - 18 * q^85 + 32 * q^94 - 48 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−0.342036	−0.241856	−0.120928	−	0.992661i	$-0.538587\pi$
$2$	−0.342036	−0.241856	−0.120928	+	0.992661i	$0.538587\pi$
$3$	0	0
$3$	0	0
$4$	−1.88301	−0.941506
$5$	−2.14896	−0.961045	−0.480522	−	0.876982i	$-0.659553\pi$
$5$	−2.14896	−0.961045	−0.480522	+	0.876982i	$0.659553\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	1.32813	0.469565
$9$	0	0
$10$	0.735023	0.232435
$11$	0	0
$11$	0	0
$12$	0	0
$13$	1.42874	0.396262	0.198131	−	0.980176i	$-0.436513\pi$
$13$	1.42874	0.396262	0.198131	+	0.980176i	$0.436513\pi$
$14$	0.342036	0.0914131
$15$	0	0
$16$	3.31175	0.827938
$17$	1.38414	0.335704	0.167852	−	0.985812i	$-0.446317\pi$
$17$	1.38414	0.335704	0.167852	+	0.985812i	$0.446317\pi$
$18$	0	0
$19$	−2.59251	−0.594762	−0.297381	−	0.954759i	$-0.596113\pi$
$19$	−2.59251	−0.594762	−0.297381	+	0.954759i	$0.596113\pi$
$20$	4.04652	0.904829
$21$	0	0
$22$	0	0
$23$	−0.0906318	−0.0188980	−0.00944902	−	0.999955i	$-0.503008\pi$
$23$	−0.0906318	−0.0188980	−0.00944902	+	0.999955i	$0.503008\pi$
$24$	0	0
$25$	−0.381966	−0.0763932
$26$	−0.488682	−0.0958384
$27$	0	0
$28$	1.88301	0.355856
$29$	−7.74489	−1.43819	−0.719095	−	0.694912i	$-0.755443\pi$
$29$	−7.74489	−1.43819	−0.719095	+	0.694912i	$0.755443\pi$
$30$	0	0
$31$	−4.82857	−0.867238	−0.433619	−	0.901096i	$-0.642764\pi$
$31$	−4.82857	−0.867238	−0.433619	+	0.901096i	$0.642764\pi$
$32$	−3.78900	−0.669807
$33$	0	0
$34$	−0.473428	−0.0811922
$35$	2.14896	0.363241
$36$	0	0
$37$	−0.0702129	−0.0115429	−0.00577146	−	0.999983i	$-0.501837\pi$
$37$	−0.0702129	−0.0115429	−0.00577146	+	0.999983i	$0.501837\pi$
$38$	0.886732	0.143847
$39$	0	0
$40$	−2.85410	−0.451273
$41$	6.70278	1.04680	0.523399	−	0.852088i	$-0.324664\pi$
$41$	6.70278	1.04680	0.523399	+	0.852088i	$0.324664\pi$
$42$	0	0
$43$	6.28623	0.958640	0.479320	−	0.877640i	$-0.340883\pi$
$43$	6.28623	0.958640	0.479320	+	0.877640i	$0.340883\pi$
$44$	0	0
$45$	0	0
$46$	0.0309994	0.00457061
$47$	9.19116	1.34067	0.670334	−	0.742059i	$-0.266151\pi$
$47$	9.19116	1.34067	0.670334	+	0.742059i	$0.266151\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0.130646	0.0184762
$51$	0	0
$52$	−2.69034	−0.373083
$53$	3.73461	0.512988	0.256494	−	0.966546i	$-0.417433\pi$
$53$	3.73461	0.512988	0.256494	+	0.966546i	$0.417433\pi$
$54$	0	0
$55$	0	0
$56$	−1.32813	−0.177479
$57$	0	0
$58$	2.64903	0.347835
$59$	9.65657	1.25718	0.628589	−	0.777737i	$-0.283633\pi$
$59$	9.65657	1.25718	0.628589	+	0.777737i	$0.283633\pi$
$60$	0	0
$61$	6.33728	0.811406	0.405703	−	0.914005i	$-0.367027\pi$
$61$	6.33728	0.811406	0.405703	+	0.914005i	$0.367027\pi$
$62$	1.65155	0.209747
$63$	0	0
$64$	−5.32753	−0.665941
$65$	−3.07031	−0.380825
$66$	0	0
$67$	7.09564	0.866871	0.433435	−	0.901185i	$-0.357301\pi$
$67$	7.09564	0.866871	0.433435	+	0.901185i	$0.357301\pi$
$68$	−2.60636	−0.316068
$69$	0	0
$70$	−0.735023	−0.0878520
$71$	−8.96377	−1.06380	−0.531902	−	0.846806i	$-0.678522\pi$
$71$	−8.96377	−1.06380	−0.531902	+	0.846806i	$0.678522\pi$
$72$	0	0
$73$	−11.9787	−1.40200	−0.700998	−	0.713164i	$-0.747262\pi$
$73$	−11.9787	−1.40200	−0.700998	+	0.713164i	$0.747262\pi$
$74$	0.0240154	0.00279173
$75$	0	0
$76$	4.88172	0.559972
$77$	0	0
$78$	0	0
$79$	1.63043	0.183438	0.0917188	−	0.995785i	$-0.470764\pi$
$79$	1.63043	0.183438	0.0917188	+	0.995785i	$0.470764\pi$
$80$	−7.11683	−0.795686
$81$	0	0
$82$	−2.29259	−0.253175
$83$	−9.90374	−1.08708	−0.543538	−	0.839384i	$-0.682916\pi$
$83$	−9.90374	−1.08708	−0.543538	+	0.839384i	$0.682916\pi$
$84$	0	0
$85$	−2.97447	−0.322627
$86$	−2.15012	−0.231853
$87$	0	0
$88$	0	0
$89$	11.2098	1.18823	0.594116	−	0.804379i	$-0.297502\pi$
$89$	11.2098	1.18823	0.594116	+	0.804379i	$0.297502\pi$
$90$	0	0
$91$	−1.42874	−0.149773
$92$	0.170661	0.0177926
$93$	0	0
$94$	−3.14371	−0.324249
$95$	5.57120	0.571593
$96$	0	0
$97$	−5.60490	−0.569092	−0.284546	−	0.958662i	$-0.591843\pi$
$97$	−5.60490	−0.569092	−0.284546	+	0.958662i	$0.591843\pi$
$98$	−0.342036	−0.0345509
$99$	0	0
$100$	0.719246	0.0719246
$101$	−0.901574	−0.0897100	−0.0448550	−	0.998994i	$-0.514283\pi$
$101$	−0.901574	−0.0897100	−0.0448550	+	0.998994i	$0.514283\pi$
$102$	0	0
$103$	7.35087	0.724303	0.362151	−	0.932119i	$-0.382042\pi$
$103$	7.35087	0.724303	0.362151	+	0.932119i	$0.382042\pi$
$104$	1.89756	0.186071
$105$	0	0
$106$	−1.27737	−0.124069
$107$	−5.44055	−0.525958	−0.262979	−	0.964802i	$-0.584705\pi$
$107$	−5.44055	−0.525958	−0.262979	+	0.964802i	$0.584705\pi$
$108$	0	0
$109$	6.08042	0.582399	0.291199	−	0.956662i	$-0.405946\pi$
$109$	6.08042	0.582399	0.291199	+	0.956662i	$0.405946\pi$
$110$	0	0
$111$	0	0
$112$	−3.31175	−0.312931
$113$	2.59385	0.244009	0.122005	−	0.992530i	$-0.461068\pi$
$113$	2.59385	0.244009	0.122005	+	0.992530i	$0.461068\pi$
$114$	0	0
$115$	0.194764	0.0181619
$116$	14.5837	1.35406
$117$	0	0
$118$	−3.30290	−0.304056
$119$	−1.38414	−0.126884
$120$	0	0
$121$	0	0
$122$	−2.16758	−0.196244
$123$	0	0
$124$	9.09226	0.816509
$125$	11.5656	1.03446
$126$	0	0
$127$	15.1692	1.34605	0.673026	−	0.739619i	$-0.264994\pi$
$127$	15.1692	1.34605	0.673026	+	0.739619i	$0.264994\pi$
$128$	9.40021	0.830869
$129$	0	0
$130$	1.05016	0.0921050
$131$	20.0775	1.75418	0.877090	−	0.480327i	$-0.159482\pi$
$131$	20.0775	1.75418	0.877090	+	0.480327i	$0.159482\pi$
$132$	0	0
$133$	2.59251	0.224799
$134$	−2.42697	−0.209658
$135$	0	0
$136$	1.83833	0.157635
$137$	8.44658	0.721640	0.360820	−	0.932635i	$-0.382497\pi$
$137$	8.44658	0.721640	0.360820	+	0.932635i	$0.382497\pi$
$138$	0	0
$139$	−7.71368	−0.654265	−0.327133	−	0.944978i	$-0.606082\pi$
$139$	−7.71368	−0.654265	−0.327133	+	0.944978i	$0.606082\pi$
$140$	−4.04652	−0.341993
$141$	0	0
$142$	3.06593	0.257288
$143$	0	0
$144$	0	0
$145$	16.6435	1.38216
$146$	4.09714	0.339081
$147$	0	0
$148$	0.132212	0.0108677
$149$	15.6852	1.28498	0.642489	−	0.766295i	$-0.277902\pi$
$149$	15.6852	1.28498	0.642489	+	0.766295i	$0.277902\pi$
$150$	0	0
$151$	−13.1212	−1.06779	−0.533893	−	0.845552i	$-0.679271\pi$
$151$	−13.1212	−1.06779	−0.533893	+	0.845552i	$0.679271\pi$
$152$	−3.44319	−0.279279
$153$	0	0
$154$	0	0
$155$	10.3764	0.833454
$156$	0	0
$157$	1.43775	0.114745	0.0573727	−	0.998353i	$-0.481728\pi$
$157$	1.43775	0.114745	0.0573727	+	0.998353i	$0.481728\pi$
$158$	−0.557666	−0.0443655
$159$	0	0
$160$	8.14242	0.643715
$161$	0.0906318	0.00714279
$162$	0	0
$163$	−22.5018	−1.76248	−0.881240	−	0.472669i	$-0.843291\pi$
$163$	−22.5018	−1.76248	−0.881240	+	0.472669i	$0.843291\pi$
$164$	−12.6214	−0.985566
$165$	0	0
$166$	3.38744	0.262916
$167$	−4.91085	−0.380013	−0.190007	−	0.981783i	$-0.560851\pi$
$167$	−4.91085	−0.380013	−0.190007	+	0.981783i	$0.560851\pi$
$168$	0	0
$169$	−10.9587	−0.842977
$170$	1.01738	0.0780293
$171$	0	0
$172$	−11.8370	−0.902565
$173$	18.3873	1.39796	0.698979	−	0.715142i	$-0.253638\pi$
$173$	18.3873	1.39796	0.698979	+	0.715142i	$0.253638\pi$
$174$	0	0
$175$	0.381966	0.0288739
$176$	0	0
$177$	0	0
$178$	−3.83414	−0.287381
$179$	−26.4300	−1.97547	−0.987734	−	0.156143i	$-0.950094\pi$
$179$	−26.4300	−1.97547	−0.987734	+	0.156143i	$0.950094\pi$
$180$	0	0
$181$	−20.8451	−1.54941	−0.774704	−	0.632324i	$-0.782101\pi$
$181$	−20.8451	−1.54941	−0.774704	+	0.632324i	$0.782101\pi$
$182$	0.488682	0.0362235
$183$	0	0
$184$	−0.120371	−0.00887386
$185$	0.150885	0.0110933
$186$	0	0
$187$	0	0
$188$	−17.3071	−1.26225
$189$	0	0
$190$	−1.90555	−0.138243
$191$	−0.938968	−0.0679414	−0.0339707	−	0.999423i	$-0.510815\pi$
$191$	−0.938968	−0.0679414	−0.0339707	+	0.999423i	$0.510815\pi$
$192$	0	0
$193$	2.21482	0.159426	0.0797131	−	0.996818i	$-0.474600\pi$
$193$	2.21482	0.159426	0.0797131	+	0.996818i	$0.474600\pi$
$194$	1.91708	0.137638
$195$	0	0
$196$	−1.88301	−0.134501
$197$	1.24390	0.0886239	0.0443120	−	0.999018i	$-0.485890\pi$
$197$	1.24390	0.0886239	0.0443120	+	0.999018i	$0.485890\pi$
$198$	0	0
$199$	−13.3929	−0.949398	−0.474699	−	0.880148i	$-0.657443\pi$
$199$	−13.3929	−0.949398	−0.474699	+	0.880148i	$0.657443\pi$
$200$	−0.507301	−0.0358716
$201$	0	0
$202$	0.308371	0.0216969
$203$	7.74489	0.543585
$204$	0	0
$205$	−14.4040	−1.00602
$206$	−2.51426	−0.175177
$207$	0	0
$208$	4.73164	0.328080
$209$	0	0
$210$	0	0
$211$	19.1727	1.31990	0.659951	−	0.751309i	$-0.270577\pi$
$211$	19.1727	1.31990	0.659951	+	0.751309i	$0.270577\pi$
$212$	−7.03231	−0.482981
$213$	0	0
$214$	1.86087	0.127206
$215$	−13.5089	−0.921296
$216$	0	0
$217$	4.82857	0.327785
$218$	−2.07972	−0.140857
$219$	0	0
$220$	0	0
$221$	1.97759	0.133027
$222$	0	0
$223$	7.91656	0.530132	0.265066	−	0.964230i	$-0.414606\pi$
$223$	7.91656	0.530132	0.265066	+	0.964230i	$0.414606\pi$
$224$	3.78900	0.253163
$225$	0	0
$226$	−0.887192	−0.0590151
$227$	−16.6032	−1.10199	−0.550997	−	0.834507i	$-0.685752\pi$
$227$	−16.6032	−1.10199	−0.550997	+	0.834507i	$0.685752\pi$
$228$	0	0
$229$	−24.4568	−1.61615	−0.808076	−	0.589078i	$-0.799491\pi$
$229$	−24.4568	−1.61615	−0.808076	+	0.589078i	$0.799491\pi$
$230$	−0.0666165	−0.00439256
$231$	0	0
$232$	−10.2862	−0.675324
$233$	13.3219	0.872750	0.436375	−	0.899765i	$-0.356262\pi$
$233$	13.3219	0.872750	0.436375	+	0.899765i	$0.356262\pi$
$234$	0	0
$235$	−19.7514	−1.28844
$236$	−18.1834	−1.18364
$237$	0	0
$238$	0.473428	0.0306878
$239$	−8.18018	−0.529132	−0.264566	−	0.964368i	$-0.585229\pi$
$239$	−8.18018	−0.529132	−0.264566	+	0.964368i	$0.585229\pi$
$240$	0	0
$241$	1.51553	0.0976239	0.0488119	−	0.998808i	$-0.484457\pi$
$241$	1.51553	0.0976239	0.0488119	+	0.998808i	$0.484457\pi$
$242$	0	0
$243$	0	0
$244$	−11.9332	−0.763943
$245$	−2.14896	−0.137292
$246$	0	0
$247$	−3.70402	−0.235681
$248$	−6.41298	−0.407225
$249$	0	0
$250$	−3.95587	−0.250191
$251$	−28.2044	−1.78025	−0.890124	−	0.455718i	$-0.849382\pi$
$251$	−28.2044	−1.78025	−0.890124	+	0.455718i	$0.849382\pi$
$252$	0	0
$253$	0	0
$254$	−5.18843	−0.325551
$255$	0	0
$256$	7.43984	0.464990
$257$	−18.4980	−1.15387	−0.576936	−	0.816790i	$-0.695752\pi$
$257$	−18.4980	−1.15387	−0.576936	+	0.816790i	$0.695752\pi$
$258$	0	0
$259$	0.0702129	0.00436282
$260$	5.78143	0.358549
$261$	0	0
$262$	−6.86723	−0.424259
$263$	21.4858	1.32487	0.662437	−	0.749118i	$-0.269522\pi$
$263$	21.4858	1.32487	0.662437	+	0.749118i	$0.269522\pi$
$264$	0	0
$265$	−8.02553	−0.493004
$266$	−0.886732	−0.0543690
$267$	0	0
$268$	−13.3612	−0.816164
$269$	12.6544	0.771553	0.385776	−	0.922592i	$-0.373934\pi$
$269$	12.6544	0.771553	0.385776	+	0.922592i	$0.373934\pi$
$270$	0	0
$271$	5.91475	0.359295	0.179648	−	0.983731i	$-0.442504\pi$
$271$	5.91475	0.359295	0.179648	+	0.983731i	$0.442504\pi$
$272$	4.58395	0.277942
$273$	0	0
$274$	−2.88904	−0.174533
$275$	0	0
$276$	0	0
$277$	−18.0735	−1.08593	−0.542966	−	0.839755i	$-0.682699\pi$
$277$	−18.0735	−1.08593	−0.542966	+	0.839755i	$0.682699\pi$
$278$	2.63836	0.158238
$279$	0	0
$280$	2.85410	0.170565
$281$	−27.2162	−1.62358	−0.811791	−	0.583948i	$-0.801507\pi$
$281$	−27.2162	−1.62358	−0.811791	+	0.583948i	$0.801507\pi$
$282$	0	0
$283$	−10.1420	−0.602877	−0.301439	−	0.953486i	$-0.597467\pi$
$283$	−10.1420	−0.602877	−0.301439	+	0.953486i	$0.597467\pi$
$284$	16.8789	1.00158
$285$	0	0
$286$	0	0
$287$	−6.70278	−0.395653
$288$	0	0
$289$	−15.0841	−0.887303
$290$	−5.69267	−0.334285
$291$	0	0
$292$	22.5559	1.31999
$293$	−15.6060	−0.911715	−0.455857	−	0.890053i	$-0.650667\pi$
$293$	−15.6060	−0.911715	−0.455857	+	0.890053i	$0.650667\pi$
$294$	0	0
$295$	−20.7516	−1.20820
$296$	−0.0932519	−0.00542016
$297$	0	0
$298$	−5.36490	−0.310780
$299$	−0.129489	−0.00748857
$300$	0	0
$301$	−6.28623	−0.362332
$302$	4.48792	0.258251
$303$	0	0
$304$	−8.58574	−0.492426
$305$	−13.6186	−0.779797
$306$	0	0
$307$	5.30071	0.302528	0.151264	−	0.988493i	$-0.451666\pi$
$307$	5.30071	0.302528	0.151264	+	0.988493i	$0.451666\pi$
$308$	0	0
$309$	0	0
$310$	−3.54911	−0.201576
$311$	17.3542	0.984064	0.492032	−	0.870577i	$-0.336254\pi$
$311$	17.3542	0.984064	0.492032	+	0.870577i	$0.336254\pi$
$312$	0	0
$313$	−32.0377	−1.81088	−0.905438	−	0.424479i	$-0.860457\pi$
$313$	−32.0377	−1.81088	−0.905438	+	0.424479i	$0.860457\pi$
$314$	−0.491764	−0.0277519
$315$	0	0
$316$	−3.07012	−0.172707
$317$	30.7237	1.72561	0.862806	−	0.505535i	$-0.168705\pi$
$317$	30.7237	1.72561	0.862806	+	0.505535i	$0.168705\pi$
$318$	0	0
$319$	0	0
$320$	11.4487	0.639999
$321$	0	0
$322$	−0.0309994	−0.00172753
$323$	−3.58840	−0.199664
$324$	0	0
$325$	−0.545731	−0.0302717
$326$	7.69645	0.426267
$327$	0	0
$328$	8.90217	0.491540
$329$	−9.19116	−0.506725
$330$	0	0
$331$	14.6112	0.803103	0.401551	−	0.915837i	$-0.368471\pi$
$331$	14.6112	0.803103	0.401551	+	0.915837i	$0.368471\pi$
$332$	18.6488	1.02349
$333$	0	0
$334$	1.67969	0.0919086
$335$	−15.2483	−0.833101
$336$	0	0
$337$	4.43376	0.241522	0.120761	−	0.992682i	$-0.461467\pi$
$337$	4.43376	0.241522	0.120761	+	0.992682i	$0.461467\pi$
$338$	3.74827	0.203879
$339$	0	0
$340$	5.60097	0.303755
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	8.34893	0.450144
$345$	0	0
$346$	−6.28911	−0.338105
$347$	35.9285	1.92875	0.964373	−	0.264547i	$-0.0852225\pi$
$347$	35.9285	1.92875	0.964373	+	0.264547i	$0.0852225\pi$
$348$	0	0
$349$	−31.4747	−1.68480	−0.842401	−	0.538852i	$-0.818858\pi$
$349$	−31.4747	−1.68480	−0.842401	+	0.538852i	$0.818858\pi$
$350$	−0.130646	−0.00698334
$351$	0	0
$352$	0	0
$353$	−16.3003	−0.867575	−0.433788	−	0.901015i	$-0.642823\pi$
$353$	−16.3003	−0.867575	−0.433788	+	0.901015i	$0.642823\pi$
$354$	0	0
$355$	19.2628	1.02236
$356$	−21.1081	−1.11873
$357$	0	0
$358$	9.04001	0.477779
$359$	−1.02233	−0.0539566	−0.0269783	−	0.999636i	$-0.508589\pi$
$359$	−1.02233	−0.0539566	−0.0269783	+	0.999636i	$0.508589\pi$
$360$	0	0
$361$	−12.2789	−0.646258
$362$	7.12980	0.374734
$363$	0	0
$364$	2.69034	0.141012
$365$	25.7417	1.34738
$366$	0	0
$367$	−25.2692	−1.31904	−0.659520	−	0.751687i	$-0.729240\pi$
$367$	−25.2692	−1.31904	−0.659520	+	0.751687i	$0.729240\pi$
$368$	−0.300150	−0.0156464
$369$	0	0
$370$	−0.0516081	−0.00268298
$371$	−3.73461	−0.193891
$372$	0	0
$373$	−25.3019	−1.31008	−0.655041	−	0.755593i	$-0.727349\pi$
$373$	−25.3019	−1.31008	−0.655041	+	0.755593i	$0.727349\pi$
$374$	0	0
$375$	0	0
$376$	12.2071	0.629531
$377$	−11.0654	−0.569900
$378$	0	0
$379$	−27.6718	−1.42141	−0.710703	−	0.703492i	$-0.751623\pi$
$379$	−27.6718	−1.42141	−0.710703	+	0.703492i	$0.751623\pi$
$380$	−10.4906	−0.538158
$381$	0	0
$382$	0.321161	0.0164320
$383$	13.9689	0.713776	0.356888	−	0.934147i	$-0.383838\pi$
$383$	13.9689	0.713776	0.356888	+	0.934147i	$0.383838\pi$
$384$	0	0
$385$	0	0
$386$	−0.757549	−0.0385582
$387$	0	0
$388$	10.5541	0.535803
$389$	−31.2813	−1.58603	−0.793013	−	0.609205i	$-0.791489\pi$
$389$	−31.2813	−1.58603	−0.793013	+	0.609205i	$0.791489\pi$
$390$	0	0
$391$	−0.125448	−0.00634415
$392$	1.32813	0.0670807
$393$	0	0
$394$	−0.425458	−0.0214343
$395$	−3.50373	−0.176292
$396$	0	0
$397$	−24.7080	−1.24006	−0.620031	−	0.784578i	$-0.712880\pi$
$397$	−24.7080	−1.24006	−0.620031	+	0.784578i	$0.712880\pi$
$398$	4.58086	0.229618
$399$	0	0
$400$	−1.26498	−0.0632489
$401$	17.0929	0.853577	0.426788	−	0.904351i	$-0.359645\pi$
$401$	17.0929	0.853577	0.426788	+	0.904351i	$0.359645\pi$
$402$	0	0
$403$	−6.89879	−0.343653
$404$	1.69767	0.0844624
$405$	0	0
$406$	−2.64903	−0.131469
$407$	0	0
$408$	0	0
$409$	−14.5499	−0.719447	−0.359723	−	0.933059i	$-0.617129\pi$
$409$	−14.5499	−0.719447	−0.359723	+	0.933059i	$0.617129\pi$
$410$	4.92670	0.243312
$411$	0	0
$412$	−13.8418	−0.681935
$413$	−9.65657	−0.475169
$414$	0	0
$415$	21.2827	1.04473
$416$	−5.41351	−0.265419
$417$	0	0
$418$	0	0
$419$	−10.2910	−0.502750	−0.251375	−	0.967890i	$-0.580883\pi$
$419$	−10.2910	−0.502750	−0.251375	+	0.967890i	$0.580883\pi$
$420$	0	0
$421$	3.19049	0.155495	0.0777474	−	0.996973i	$-0.475227\pi$
$421$	3.19049	0.155495	0.0777474	+	0.996973i	$0.475227\pi$
$422$	−6.55775	−0.319226
$423$	0	0
$424$	4.96005	0.240881
$425$	−0.528696	−0.0256455
$426$	0	0
$427$	−6.33728	−0.306682
$428$	10.2446	0.495192
$429$	0	0
$430$	4.62052	0.222821
$431$	−9.16247	−0.441341	−0.220670	−	0.975348i	$-0.570825\pi$
$431$	−9.16247	−0.441341	−0.220670	+	0.975348i	$0.570825\pi$
$432$	0	0
$433$	13.3185	0.640045	0.320023	−	0.947410i	$-0.396309\pi$
$433$	13.3185	0.640045	0.320023	+	0.947410i	$0.396309\pi$
$434$	−1.65155	−0.0792769
$435$	0	0
$436$	−11.4495	−0.548331
$437$	0.234964	0.0112398
$438$	0	0
$439$	−2.75126	−0.131310	−0.0656551	−	0.997842i	$-0.520914\pi$
$439$	−2.75126	−0.131310	−0.0656551	+	0.997842i	$0.520914\pi$
$440$	0	0
$441$	0	0
$442$	−0.676406	−0.0321734
$443$	0.431953	0.0205227	0.0102614	−	0.999947i	$-0.496734\pi$
$443$	0.431953	0.0205227	0.0102614	+	0.999947i	$0.496734\pi$
$444$	0	0
$445$	−24.0893	−1.14194
$446$	−2.70775	−0.128216
$447$	0	0
$448$	5.32753	0.251702
$449$	1.71629	0.0809969	0.0404984	−	0.999180i	$-0.487105\pi$
$449$	1.71629	0.0809969	0.0404984	+	0.999180i	$0.487105\pi$
$450$	0	0
$451$	0	0
$452$	−4.88425	−0.229736
$453$	0	0
$454$	5.67890	0.266524
$455$	3.07031	0.143938
$456$	0	0
$457$	14.2291	0.665611	0.332806	−	0.942995i	$-0.392005\pi$
$457$	14.2291	0.665611	0.332806	+	0.942995i	$0.392005\pi$
$458$	8.36512	0.390876
$459$	0	0
$460$	−0.366743	−0.0170995
$461$	11.3215	0.527295	0.263647	−	0.964619i	$-0.415074\pi$
$461$	11.3215	0.527295	0.263647	+	0.964619i	$0.415074\pi$
$462$	0	0
$463$	21.7695	1.01171	0.505856	−	0.862618i	$-0.331177\pi$
$463$	21.7695	1.01171	0.505856	+	0.862618i	$0.331177\pi$
$464$	−25.6492	−1.19073
$465$	0	0
$466$	−4.55659	−0.211080
$467$	−1.42300	−0.0658487	−0.0329243	−	0.999458i	$-0.510482\pi$
$467$	−1.42300	−0.0658487	−0.0329243	+	0.999458i	$0.510482\pi$
$468$	0	0
$469$	−7.09564	−0.327646
$470$	6.75571	0.311618
$471$	0	0
$472$	12.8252	0.590327
$473$	0	0
$474$	0	0
$475$	0.990250	0.0454358
$476$	2.60636	0.119462
$477$	0	0
$478$	2.79792	0.127974
$479$	2.30945	0.105521	0.0527607	−	0.998607i	$-0.483198\pi$
$479$	2.30945	0.105521	0.0527607	+	0.998607i	$0.483198\pi$
$480$	0	0
$481$	−0.100316	−0.00457402
$482$	−0.518366	−0.0236109
$483$	0	0
$484$	0	0
$485$	12.0447	0.546922
$486$	0	0
$487$	−21.4197	−0.970621	−0.485310	−	0.874342i	$-0.661293\pi$
$487$	−21.4197	−0.970621	−0.485310	+	0.874342i	$0.661293\pi$
$488$	8.41674	0.381008
$489$	0	0
$490$	0.735023	0.0332049
$491$	4.12202	0.186024	0.0930121	−	0.995665i	$-0.470350\pi$
$491$	4.12202	0.186024	0.0930121	+	0.995665i	$0.470350\pi$
$492$	0	0
$493$	−10.7200	−0.482807
$494$	1.26691	0.0570010
$495$	0	0
$496$	−15.9910	−0.718019
$497$	8.96377	0.402080
$498$	0	0
$499$	−34.6026	−1.54903	−0.774513	−	0.632558i	$-0.782005\pi$
$499$	−34.6026	−1.54903	−0.774513	+	0.632558i	$0.782005\pi$
$500$	−21.7782	−0.973952
$501$	0	0
$502$	9.64694	0.430564
$503$	−20.2258	−0.901823	−0.450912	−	0.892569i	$-0.648901\pi$
$503$	−20.2258	−0.901823	−0.450912	+	0.892569i	$0.648901\pi$
$504$	0	0
$505$	1.93745	0.0862153
$506$	0	0
$507$	0	0
$508$	−28.5638	−1.26732
$509$	−15.2989	−0.678111	−0.339055	−	0.940766i	$-0.610107\pi$
$509$	−15.2989	−0.678111	−0.339055	+	0.940766i	$0.610107\pi$
$510$	0	0
$511$	11.9787	0.529904
$512$	−21.3451	−0.943330
$513$	0	0
$514$	6.32698	0.279071
$515$	−15.7967	−0.696087
$516$	0	0
$517$	0	0
$518$	−0.0240154	−0.00105517
$519$	0	0
$520$	−4.07778	−0.178822
$521$	33.0833	1.44941	0.724703	−	0.689061i	$-0.241977\pi$
$521$	33.0833	1.44941	0.724703	+	0.689061i	$0.241977\pi$
$522$	0	0
$523$	37.8339	1.65436	0.827182	−	0.561935i	$-0.189943\pi$
$523$	37.8339	1.65436	0.827182	+	0.561935i	$0.189943\pi$
$524$	−37.8062	−1.65157
$525$	0	0
$526$	−7.34894	−0.320429
$527$	−6.68345	−0.291135
$528$	0	0
$529$	−22.9918	−0.999643
$530$	2.74502	0.119236
$531$	0	0
$532$	−4.88172	−0.211649
$533$	9.57654	0.414806
$534$	0	0
$535$	11.6915	0.505469
$536$	9.42394	0.407052
$537$	0	0
$538$	−4.32827	−0.186605
$539$	0	0
$540$	0	0
$541$	−5.62898	−0.242009	−0.121004	−	0.992652i	$-0.538611\pi$
$541$	−5.62898	−0.242009	−0.121004	+	0.992652i	$0.538611\pi$
$542$	−2.02306	−0.0868978
$543$	0	0
$544$	−5.24453	−0.224857
$545$	−13.0666	−0.559711
$546$	0	0
$547$	15.5885	0.666515	0.333257	−	0.942836i	$-0.391852\pi$
$547$	15.5885	0.666515	0.333257	+	0.942836i	$0.391852\pi$
$548$	−15.9050	−0.679428
$549$	0	0
$550$	0	0
$551$	20.0787	0.855380
$552$	0	0
$553$	−1.63043	−0.0693329
$554$	6.18179	0.262639
$555$	0	0
$556$	14.5249	0.615995
$557$	21.7599	0.921996	0.460998	−	0.887401i	$-0.347492\pi$
$557$	21.7599	0.921996	0.460998	+	0.887401i	$0.347492\pi$
$558$	0	0
$559$	8.98140	0.379873
$560$	7.11683	0.300741
$561$	0	0
$562$	9.30892	0.392673
$563$	−34.0550	−1.43525	−0.717624	−	0.696431i	$-0.754770\pi$
$563$	−34.0550	−1.43525	−0.717624	+	0.696431i	$0.754770\pi$
$564$	0	0
$565$	−5.57409	−0.234504
$566$	3.46892	0.145810
$567$	0	0
$568$	−11.9051	−0.499525
$569$	−13.9128	−0.583257	−0.291628	−	0.956532i	$-0.594197\pi$
$569$	−13.9128	−0.583257	−0.291628	+	0.956532i	$0.594197\pi$
$570$	0	0
$571$	−3.06966	−0.128461	−0.0642306	−	0.997935i	$-0.520459\pi$
$571$	−3.06966	−0.128461	−0.0642306	+	0.997935i	$0.520459\pi$
$572$	0	0
$573$	0	0
$574$	2.29259	0.0956910
$575$	0.0346183	0.00144368
$576$	0	0
$577$	−8.00483	−0.333246	−0.166623	−	0.986021i	$-0.553286\pi$
$577$	−8.00483	−0.333246	−0.166623	+	0.986021i	$0.553286\pi$
$578$	5.15933	0.214600
$579$	0	0
$580$	−31.3398	−1.30132
$581$	9.90374	0.410876
$582$	0	0
$583$	0	0
$584$	−15.9092	−0.658328
$585$	0	0
$586$	5.33783	0.220504
$587$	37.9360	1.56579	0.782894	−	0.622155i	$-0.213743\pi$
$587$	37.9360	1.56579	0.782894	+	0.622155i	$0.213743\pi$
$588$	0	0
$589$	12.5181	0.515800
$590$	7.09780	0.292212
$591$	0	0
$592$	−0.232528	−0.00955683
$593$	−34.8764	−1.43220	−0.716101	−	0.697997i	$-0.754075\pi$
$593$	−34.8764	−1.43220	−0.716101	+	0.697997i	$0.754075\pi$
$594$	0	0
$595$	2.97447	0.121941
$596$	−29.5353	−1.20981
$597$	0	0
$598$	0.0442901	0.00181116
$599$	−4.02161	−0.164319	−0.0821593	−	0.996619i	$-0.526182\pi$
$599$	−4.02161	−0.164319	−0.0821593	+	0.996619i	$0.526182\pi$
$600$	0	0
$601$	3.89949	0.159063	0.0795317	−	0.996832i	$-0.474658\pi$
$601$	3.89949	0.159063	0.0795317	+	0.996832i	$0.474658\pi$
$602$	2.15012	0.0876323
$603$	0	0
$604$	24.7073	1.00533
$605$	0	0
$606$	0	0
$607$	43.4175	1.76226	0.881131	−	0.472872i	$-0.156783\pi$
$607$	43.4175	1.76226	0.881131	+	0.472872i	$0.156783\pi$
$608$	9.82301	0.398376
$609$	0	0
$610$	4.65805	0.188599
$611$	13.1318	0.531255
$612$	0	0
$613$	2.27638	0.0919420	0.0459710	−	0.998943i	$-0.485362\pi$
$613$	2.27638	0.0919420	0.0459710	+	0.998943i	$0.485362\pi$
$614$	−1.81304	−0.0731682
$615$	0	0
$616$	0	0
$617$	−48.3316	−1.94575	−0.972877	−	0.231321i	$-0.925695\pi$
$617$	−48.3316	−1.94575	−0.972877	+	0.231321i	$0.925695\pi$
$618$	0	0
$619$	−22.5642	−0.906933	−0.453467	−	0.891273i	$-0.649813\pi$
$619$	−22.5642	−0.906933	−0.453467	+	0.891273i	$0.649813\pi$
$620$	−19.5389	−0.784702
$621$	0	0
$622$	−5.93575	−0.238002
$623$	−11.2098	−0.449109
$624$	0	0
$625$	−22.9443	−0.917771
$626$	10.9580	0.437972
$627$	0	0
$628$	−2.70731	−0.108033
$629$	−0.0971848	−0.00387501
$630$	0	0
$631$	8.18412	0.325805	0.162902	−	0.986642i	$-0.447914\pi$
$631$	8.18412	0.325805	0.162902	+	0.986642i	$0.447914\pi$
$632$	2.16542	0.0861359
$633$	0	0
$634$	−10.5086	−0.417350
$635$	−32.5981	−1.29362
$636$	0	0
$637$	1.42874	0.0566088
$638$	0	0
$639$	0	0
$640$	−20.2007	−0.798502
$641$	2.53026	0.0999394	0.0499697	−	0.998751i	$-0.484088\pi$
$641$	2.53026	0.0999394	0.0499697	+	0.998751i	$0.484088\pi$
$642$	0	0
$643$	−46.2359	−1.82337	−0.911684	−	0.410893i	$-0.865217\pi$
$643$	−46.2359	−1.82337	−0.911684	+	0.410893i	$0.865217\pi$
$644$	−0.170661	−0.00672497
$645$	0	0
$646$	1.22736	0.0482900
$647$	−20.5421	−0.807593	−0.403796	−	0.914849i	$-0.632310\pi$
$647$	−20.5421	−0.807593	−0.403796	+	0.914849i	$0.632310\pi$
$648$	0	0
$649$	0	0
$650$	0.186660	0.00732140
$651$	0	0
$652$	42.3712	1.65939
$653$	20.0643	0.785177	0.392588	−	0.919714i	$-0.371580\pi$
$653$	20.0643	0.785177	0.392588	+	0.919714i	$0.371580\pi$
$654$	0	0
$655$	−43.1458	−1.68584
$656$	22.1980	0.866684
$657$	0	0
$658$	3.14371	0.122555
$659$	−12.5993	−0.490800	−0.245400	−	0.969422i	$-0.578919\pi$
$659$	−12.5993	−0.490800	−0.245400	+	0.969422i	$0.578919\pi$
$660$	0	0
$661$	31.5153	1.22580	0.612902	−	0.790159i	$-0.290002\pi$
$661$	31.5153	1.22580	0.612902	+	0.790159i	$0.290002\pi$
$662$	−4.99755	−0.194235
$663$	0	0
$664$	−13.1535	−0.510453
$665$	−5.57120	−0.216042
$666$	0	0
$667$	0.701933	0.0271790
$668$	9.24719	0.357785
$669$	0	0
$670$	5.21546	0.201491
$671$	0	0
$672$	0	0
$673$	41.7136	1.60794	0.803970	−	0.594670i	$-0.202717\pi$
$673$	41.7136	1.60794	0.803970	+	0.594670i	$0.202717\pi$
$674$	−1.51651	−0.0584136
$675$	0	0
$676$	20.6353	0.793667
$677$	50.6961	1.94841	0.974205	−	0.225665i	$-0.0724556\pi$
$677$	50.6961	1.94841	0.974205	+	0.225665i	$0.0724556\pi$
$678$	0	0
$679$	5.60490	0.215096
$680$	−3.95049	−0.151494
$681$	0	0
$682$	0	0
$683$	32.6596	1.24969	0.624843	−	0.780751i	$-0.285163\pi$
$683$	32.6596	1.24969	0.624843	+	0.780751i	$0.285163\pi$
$684$	0	0
$685$	−18.1514	−0.693528
$686$	0.342036	0.0130590
$687$	0	0
$688$	20.8184	0.793695
$689$	5.33579	0.203277
$690$	0	0
$691$	−20.4348	−0.777378	−0.388689	−	0.921369i	$-0.627072\pi$
$691$	−20.4348	−0.777378	−0.388689	+	0.921369i	$0.627072\pi$
$692$	−34.6234	−1.31619
$693$	0	0
$694$	−12.2889	−0.466479
$695$	16.5764	0.628778
$696$	0	0
$697$	9.27762	0.351415
$698$	10.7655	0.407480
$699$	0	0
$700$	−0.719246	−0.0271850
$701$	13.9425	0.526602	0.263301	−	0.964714i	$-0.415189\pi$
$701$	13.9425	0.526602	0.263301	+	0.964714i	$0.415189\pi$
$702$	0	0
$703$	0.182027	0.00686529
$704$	0	0
$705$	0	0
$706$	5.57528	0.209828
$707$	0.901574	0.0339072
$708$	0	0
$709$	32.6094	1.22467	0.612337	−	0.790597i	$-0.290230\pi$
$709$	32.6094	1.22467	0.612337	+	0.790597i	$0.290230\pi$
$710$	−6.58857	−0.247265
$711$	0	0
$712$	14.8880	0.557952
$713$	0.437623	0.0163891
$714$	0	0
$715$	0	0
$716$	49.7679	1.85991
$717$	0	0
$718$	0.349675	0.0130497
$719$	−3.83226	−0.142919	−0.0714595	−	0.997444i	$-0.522766\pi$
$719$	−3.83226	−0.142919	−0.0714595	+	0.997444i	$0.522766\pi$
$720$	0	0
$721$	−7.35087	−0.273761
$722$	4.19983	0.156302
$723$	0	0
$724$	39.2516	1.45878
$725$	2.95828	0.109868
$726$	0	0
$727$	−20.9072	−0.775407	−0.387703	−	0.921784i	$-0.626732\pi$
$727$	−20.9072	−0.775407	−0.387703	+	0.921784i	$0.626732\pi$
$728$	−1.89756	−0.0703281
$729$	0	0
$730$	−8.80458	−0.325872
$731$	8.70105	0.321820
$732$	0	0
$733$	10.8357	0.400225	0.200113	−	0.979773i	$-0.435869\pi$
$733$	10.8357	0.400225	0.200113	+	0.979773i	$0.435869\pi$
$734$	8.64297	0.319018
$735$	0	0
$736$	0.343404	0.0126580
$737$	0	0
$738$	0	0
$739$	−17.6110	−0.647832	−0.323916	−	0.946086i	$-0.604999\pi$
$739$	−17.6110	−0.647832	−0.323916	+	0.946086i	$0.604999\pi$
$740$	−0.284118	−0.0104444
$741$	0	0
$742$	1.27737	0.0468938
$743$	40.1071	1.47139	0.735694	−	0.677314i	$-0.236856\pi$
$743$	40.1071	1.47139	0.735694	+	0.677314i	$0.236856\pi$
$744$	0	0
$745$	−33.7068	−1.23492
$746$	8.65417	0.316852
$747$	0	0
$748$	0	0
$749$	5.44055	0.198793
$750$	0	0
$751$	10.4784	0.382364	0.191182	−	0.981555i	$-0.438768\pi$
$751$	10.4784	0.382364	0.191182	+	0.981555i	$0.438768\pi$
$752$	30.4388	1.10999
$753$	0	0
$754$	3.78479	0.137834
$755$	28.1969	1.02619
$756$	0	0
$757$	−31.8147	−1.15633	−0.578163	−	0.815921i	$-0.696230\pi$
$757$	−31.8147	−1.15633	−0.578163	+	0.815921i	$0.696230\pi$
$758$	9.46477	0.343776
$759$	0	0
$760$	7.39928	0.268400
$761$	−27.1687	−0.984866	−0.492433	−	0.870350i	$-0.663892\pi$
$761$	−27.1687	−0.984866	−0.492433	+	0.870350i	$0.663892\pi$
$762$	0	0
$763$	−6.08042	−0.220126
$764$	1.76809	0.0639672
$765$	0	0
$766$	−4.77786	−0.172631
$767$	13.7967	0.498172
$768$	0	0
$769$	22.9373	0.827141	0.413570	−	0.910472i	$-0.364282\pi$
$769$	22.9373	0.827141	0.413570	+	0.910472i	$0.364282\pi$
$770$	0	0
$771$	0	0
$772$	−4.17053	−0.150101
$773$	−5.12745	−0.184422	−0.0922108	−	0.995740i	$-0.529393\pi$
$773$	−5.12745	−0.184422	−0.0922108	+	0.995740i	$0.529393\pi$
$774$	0	0
$775$	1.84435	0.0662511
$776$	−7.44404	−0.267226
$777$	0	0
$778$	10.6993	0.383590
$779$	−17.3770	−0.622596
$780$	0	0
$781$	0	0
$782$	0.0429076	0.00153437
$783$	0	0
$784$	3.31175	0.118277
$785$	−3.08968	−0.110275
$786$	0	0
$787$	−5.17069	−0.184315	−0.0921575	−	0.995744i	$-0.529376\pi$
$787$	−5.17069	−0.184315	−0.0921575	+	0.995744i	$0.529376\pi$
$788$	−2.34227	−0.0834399
$789$	0	0
$790$	1.19840	0.0426372
$791$	−2.59385	−0.0922267
$792$	0	0
$793$	9.05434	0.321529
$794$	8.45105	0.299917
$795$	0	0
$796$	25.2190	0.893864
$797$	16.1138	0.570779	0.285390	−	0.958412i	$-0.407877\pi$
$797$	16.1138	0.570779	0.285390	+	0.958412i	$0.407877\pi$
$798$	0	0
$799$	12.7219	0.450068
$800$	1.44727	0.0511687
$801$	0	0
$802$	−5.84638	−0.206443
$803$	0	0
$804$	0	0
$805$	−0.194764	−0.00686454
$806$	2.35964	0.0831146
$807$	0	0
$808$	−1.19741	−0.0421247
$809$	−22.0178	−0.774106	−0.387053	−	0.922057i	$-0.626507\pi$
$809$	−22.0178	−0.774106	−0.387053	+	0.922057i	$0.626507\pi$
$810$	0	0
$811$	−54.3784	−1.90948	−0.954742	−	0.297436i	$-0.903868\pi$
$811$	−54.3784	−1.90948	−0.954742	+	0.297436i	$0.903868\pi$
$812$	−14.5837	−0.511788
$813$	0	0
$814$	0	0
$815$	48.3556	1.69382
$816$	0	0
$817$	−16.2971	−0.570163
$818$	4.97660	0.174003
$819$	0	0
$820$	27.1229	0.947173
$821$	−5.20805	−0.181762	−0.0908811	−	0.995862i	$-0.528968\pi$
$821$	−5.20805	−0.181762	−0.0908811	+	0.995862i	$0.528968\pi$
$822$	0	0
$823$	26.6472	0.928863	0.464431	−	0.885609i	$-0.346259\pi$
$823$	26.6472	0.928863	0.464431	+	0.885609i	$0.346259\pi$
$824$	9.76292	0.340107
$825$	0	0
$826$	3.30290	0.114923
$827$	23.7490	0.825834	0.412917	−	0.910769i	$-0.364510\pi$
$827$	23.7490	0.825834	0.412917	+	0.910769i	$0.364510\pi$
$828$	0	0
$829$	−12.4053	−0.430854	−0.215427	−	0.976520i	$-0.569114\pi$
$829$	−12.4053	−0.430854	−0.215427	+	0.976520i	$0.569114\pi$
$830$	−7.27947	−0.252674
$831$	0	0
$832$	−7.61167	−0.263887
$833$	1.38414	0.0479578
$834$	0	0
$835$	10.5532	0.365210
$836$	0	0
$837$	0	0
$838$	3.51991	0.121593
$839$	5.24545	0.181093	0.0905464	−	0.995892i	$-0.471139\pi$
$839$	5.24545	0.181093	0.0905464	+	0.995892i	$0.471139\pi$
$840$	0	0
$841$	30.9833	1.06839
$842$	−1.09126	−0.0376074
$843$	0	0
$844$	−36.1024	−1.24269
$845$	23.5498	0.810138
$846$	0	0
$847$	0	0
$848$	12.3681	0.424722
$849$	0	0
$850$	0.180833	0.00620253
$851$	0.00636352	0.000218139 0
$852$	0	0
$853$	−48.2236	−1.65114	−0.825572	−	0.564297i	$-0.809147\pi$
$853$	−48.2236	−1.65114	−0.825572	+	0.564297i	$0.809147\pi$
$854$	2.16758	0.0741731
$855$	0	0
$856$	−7.22576	−0.246972
$857$	−46.9168	−1.60265	−0.801323	−	0.598232i	$-0.795870\pi$
$857$	−46.9168	−1.60265	−0.801323	+	0.598232i	$0.795870\pi$
$858$	0	0
$859$	27.1096	0.924967	0.462483	−	0.886628i	$-0.346959\pi$
$859$	27.1096	0.924967	0.462483	+	0.886628i	$0.346959\pi$
$860$	25.4373	0.867406
$861$	0	0
$862$	3.13390	0.106741
$863$	−53.4934	−1.82094	−0.910468	−	0.413579i	$-0.864279\pi$
$863$	−53.4934	−1.82094	−0.910468	+	0.413579i	$0.864279\pi$
$864$	0	0
$865$	−39.5135	−1.34350
$866$	−4.55540	−0.154799
$867$	0	0
$868$	−9.09226	−0.308611
$869$	0	0
$870$	0	0
$871$	10.1378	0.343508
$872$	8.07559	0.273474
$873$	0	0
$874$	−0.0803661	−0.00271842
$875$	−11.5656	−0.390990
$876$	0	0
$877$	44.7852	1.51229	0.756144	−	0.654405i	$-0.227081\pi$
$877$	44.7852	1.51229	0.756144	+	0.654405i	$0.227081\pi$
$878$	0.941029	0.0317582
$879$	0	0
$880$	0	0
$881$	8.08926	0.272534	0.136267	−	0.990672i	$-0.456489\pi$
$881$	8.08926	0.272534	0.136267	+	0.990672i	$0.456489\pi$
$882$	0	0
$883$	−15.9784	−0.537715	−0.268858	−	0.963180i	$-0.586646\pi$
$883$	−15.9784	−0.537715	−0.268858	+	0.963180i	$0.586646\pi$
$884$	−3.72382	−0.125245
$885$	0	0
$886$	−0.147744	−0.00496355
$887$	−11.9660	−0.401780	−0.200890	−	0.979614i	$-0.564383\pi$
$887$	−11.9660	−0.401780	−0.200890	+	0.979614i	$0.564383\pi$
$888$	0	0
$889$	−15.1692	−0.508760
$890$	8.23943	0.276186
$891$	0	0
$892$	−14.9070	−0.499122
$893$	−23.8281	−0.797378
$894$	0	0
$895$	56.7970	1.89851
$896$	−9.40021	−0.314039
$897$	0	0
$898$	−0.587035	−0.0195896
$899$	37.3968	1.24725
$900$	0	0
$901$	5.16924	0.172212
$902$	0	0
$903$	0	0
$904$	3.44497	0.114578
$905$	44.7954	1.48905
$906$	0	0
$907$	1.50652	0.0500231	0.0250115	−	0.999687i	$-0.492038\pi$
$907$	1.50652	0.0500231	0.0250115	+	0.999687i	$0.492038\pi$
$908$	31.2640	1.03753
$909$	0	0
$910$	−1.05016	−0.0348124
$911$	−28.6169	−0.948120	−0.474060	−	0.880492i	$-0.657212\pi$
$911$	−28.6169	−0.948120	−0.474060	+	0.880492i	$0.657212\pi$
$912$	0	0
$913$	0	0
$914$	−4.86689	−0.160982
$915$	0	0
$916$	46.0525	1.52162
$917$	−20.0775	−0.663017
$918$	0	0
$919$	−19.5014	−0.643292	−0.321646	−	0.946860i	$-0.604236\pi$
$919$	−19.5014	−0.643292	−0.321646	+	0.946860i	$0.604236\pi$
$920$	0.258672	0.00852818
$921$	0	0
$922$	−3.87236	−0.127530
$923$	−12.8069	−0.421545
$924$	0	0
$925$	0.0268189	0.000881801 0
$926$	−7.44595	−0.244689
$927$	0	0
$928$	29.3454	0.963310
$929$	32.1276	1.05407	0.527036	−	0.849843i	$-0.323303\pi$
$929$	32.1276	1.05407	0.527036	+	0.849843i	$0.323303\pi$
$930$	0	0
$931$	−2.59251	−0.0849660
$932$	−25.0854	−0.821699
$933$	0	0
$934$	0.486719	0.0159259
$935$	0	0
$936$	0	0
$937$	−38.1102	−1.24501	−0.622503	−	0.782618i	$-0.713884\pi$
$937$	−38.1102	−1.24501	−0.622503	+	0.782618i	$0.713884\pi$
$938$	2.42697	0.0792433
$939$	0	0
$940$	37.1922	1.21308
$941$	−38.5091	−1.25536	−0.627681	−	0.778471i	$-0.715996\pi$
$941$	−38.5091	−1.25536	−0.627681	+	0.778471i	$0.715996\pi$
$942$	0	0
$943$	−0.607485	−0.0197824
$944$	31.9802	1.04087
$945$	0	0
$946$	0	0
$947$	−30.6307	−0.995363	−0.497681	−	0.867360i	$-0.665815\pi$
$947$	−30.6307	−0.995363	−0.497681	+	0.867360i	$0.665815\pi$
$948$	0	0
$949$	−17.1144	−0.555557
$950$	−0.338701	−0.0109889
$951$	0	0
$952$	−1.83833	−0.0595805
$953$	−22.6934	−0.735113	−0.367556	−	0.930001i	$-0.619805\pi$
$953$	−22.6934	−0.735113	−0.367556	+	0.930001i	$0.619805\pi$
$954$	0	0
$955$	2.01781	0.0652947
$956$	15.4034	0.498180
$957$	0	0
$958$	−0.789915	−0.0255210
$959$	−8.44658	−0.272754
$960$	0	0
$961$	−7.68486	−0.247899
$962$	0.0343118	0.00110626
$963$	0	0
$964$	−2.85376	−0.0919134
$965$	−4.75956	−0.153216
$966$	0	0
$967$	−47.2333	−1.51892	−0.759460	−	0.650554i	$-0.774537\pi$
$967$	−47.2333	−1.51892	−0.759460	+	0.650554i	$0.774537\pi$
$968$	0	0
$969$	0	0
$970$	−4.11973	−0.132277
$971$	−9.17271	−0.294366	−0.147183	−	0.989109i	$-0.547021\pi$
$971$	−9.17271	−0.294366	−0.147183	+	0.989109i	$0.547021\pi$
$972$	0	0
$973$	7.71368	0.247289
$974$	7.32633	0.234751
$975$	0	0
$976$	20.9875	0.671794
$977$	30.4667	0.974715	0.487358	−	0.873202i	$-0.337961\pi$
$977$	30.4667	0.974715	0.487358	+	0.873202i	$0.337961\pi$
$978$	0	0
$979$	0	0
$980$	4.04652	0.129261
$981$	0	0
$982$	−1.40988	−0.0449911
$983$	−40.1688	−1.28119	−0.640593	−	0.767881i	$-0.721311\pi$
$983$	−40.1688	−1.28119	−0.640593	+	0.767881i	$0.721311\pi$
$984$	0	0
$985$	−2.67308	−0.0851716
$986$	3.66665	0.116770
$987$	0	0
$988$	6.97472	0.221895
$989$	−0.569732	−0.0181164
$990$	0	0
$991$	2.77851	0.0882624	0.0441312	−	0.999026i	$-0.485948\pi$
$991$	2.77851	0.0882624	0.0441312	+	0.999026i	$0.485948\pi$
$992$	18.2955	0.580882
$993$	0	0
$994$	−3.06593	−0.0972455
$995$	28.7808	0.912414
$996$	0	0
$997$	−16.0755	−0.509117	−0.254559	−	0.967057i	$-0.581930\pi$
$997$	−16.0755	−0.509117	−0.254559	+	0.967057i	$0.581930\pi$
$998$	11.8354	0.374642
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.cu.1.4		8
3.2	odd	2	inner	7623.2.a.cu.1.5		8
11.5	even	5		693.2.m.h.190.3	yes	16
11.9	even	5		693.2.m.h.631.3	yes	16
11.10	odd	2		7623.2.a.cv.1.5		8
33.5	odd	10		693.2.m.h.190.2	✓	16
33.20	odd	10		693.2.m.h.631.2	yes	16
33.32	even	2		7623.2.a.cv.1.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
693.2.m.h.190.2	✓	16	33.5	odd	10
693.2.m.h.190.3	yes	16	11.5	even	5
693.2.m.h.631.2	yes	16	33.20	odd	10
693.2.m.h.631.3	yes	16	11.9	even	5
7623.2.a.cu.1.4		8	1.1	even	1	trivial
7623.2.a.cu.1.5		8	3.2	odd	2	inner
7623.2.a.cv.1.4		8	33.32	even	2
7623.2.a.cv.1.5		8	11.10	odd	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q-0.342036 q^{2} -1.88301 q^{4} -2.14896 q^{5} -1.00000 q^{7} +1.32813 q^{8} +O(q^{10})\)	\(q-0.342036 q^{2} -1.88301 q^{4} -2.14896 q^{5} -1.00000 q^{7} +1.32813 q^{8} +0.735023 q^{10} +1.42874 q^{13} +0.342036 q^{14} +3.31175 q^{16} +1.38414 q^{17} -2.59251 q^{19} +4.04652 q^{20} -0.0906318 q^{23} -0.381966 q^{25} -0.488682 q^{26} +1.88301 q^{28} -7.74489 q^{29} -4.82857 q^{31} -3.78900 q^{32} -0.473428 q^{34} +2.14896 q^{35} -0.0702129 q^{37} +0.886732 q^{38} -2.85410 q^{40} +6.70278 q^{41} +6.28623 q^{43} +0.0309994 q^{46} +9.19116 q^{47} +1.00000 q^{49} +0.130646 q^{50} -2.69034 q^{52} +3.73461 q^{53} -1.32813 q^{56} +2.64903 q^{58} +9.65657 q^{59} +6.33728 q^{61} +1.65155 q^{62} -5.32753 q^{64} -3.07031 q^{65} +7.09564 q^{67} -2.60636 q^{68} -0.735023 q^{70} -8.96377 q^{71} -11.9787 q^{73} +0.0240154 q^{74} +4.88172 q^{76} +1.63043 q^{79} -7.11683 q^{80} -2.29259 q^{82} -9.90374 q^{83} -2.97447 q^{85} -2.15012 q^{86} +11.2098 q^{89} -1.42874 q^{91} +0.170661 q^{92} -3.14371 q^{94} +5.57120 q^{95} -5.60490 q^{97} -0.342036 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{4} - 8 q^{7}+O(q^{10}) \) 8 * q + 2 * q^4 - 8 * q^7	\( 8 q + 2 q^{4} - 8 q^{7} + 14 q^{10} - 2 q^{16} - 6 q^{19} - 12 q^{25} - 2 q^{28} - 6 q^{31} - 24 q^{34} - 38 q^{37} + 4 q^{40} + 16 q^{43} - 42 q^{46} + 8 q^{49} + 2 q^{52} - 30 q^{58} + 28 q^{61} - 36 q^{64} - 36 q^{67} - 14 q^{70} + 14 q^{73} - 34 q^{76} + 22 q^{79} + 36 q^{82} - 18 q^{85} + 32 q^{94} - 48 q^{97}+O(q^{100}) \) 8 * q + 2 * q^4 - 8 * q^7 + 14 * q^10 - 2 * q^16 - 6 * q^19 - 12 * q^25 - 2 * q^28 - 6 * q^31 - 24 * q^34 - 38 * q^37 + 4 * q^40 + 16 * q^43 - 42 * q^46 + 8 * q^49 + 2 * q^52 - 30 * q^58 + 28 * q^61 - 36 * q^64 - 36 * q^67 - 14 * q^70 + 14 * q^73 - 34 * q^76 + 22 * q^79 + 36 * q^82 - 18 * q^85 + 32 * q^94 - 48 * q^97

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