LMFDB - Embedded newform 7623.2.a.dc.1.15

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 26x^{14} + 268x^{12} - 1395x^{10} + 3876x^{8} - 5635x^{6} + 4042x^{4} - 1272x^{2} + 121 $ x^16 - 26x^14 + 268x^12 - 1395x^10 + 3876x^8 - 5635x^6 + 4042x^4 - 1272*x^2 + 121
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			1.15
Root			$2.60390$ 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)

$f(q)$	$=$	$q+2.60390 q^{2} +4.78032 q^{4} -4.31860 q^{5} +1.00000 q^{7} +7.23969 q^{8} +O(q^{10})$	$q+2.60390 q^{2} +4.78032 q^{4} -4.31860 q^{5} +1.00000 q^{7} +7.23969 q^{8} -11.2452 q^{10} -2.46429 q^{13} +2.60390 q^{14} +9.29081 q^{16} -1.18654 q^{17} -3.42691 q^{19} -20.6443 q^{20} +8.32084 q^{23} +13.6503 q^{25} -6.41676 q^{26} +4.78032 q^{28} +1.45371 q^{29} +2.98816 q^{31} +9.71302 q^{32} -3.08964 q^{34} -4.31860 q^{35} +6.34373 q^{37} -8.92335 q^{38} -31.2653 q^{40} -6.16490 q^{41} +3.15289 q^{43} +21.6667 q^{46} +8.73499 q^{47} +1.00000 q^{49} +35.5440 q^{50} -11.7801 q^{52} -6.68159 q^{53} +7.23969 q^{56} +3.78533 q^{58} +11.7548 q^{59} +7.76909 q^{61} +7.78087 q^{62} +6.71015 q^{64} +10.6423 q^{65} +3.11080 q^{67} -5.67204 q^{68} -11.2452 q^{70} +5.48760 q^{71} +8.66003 q^{73} +16.5185 q^{74} -16.3817 q^{76} -8.83563 q^{79} -40.1233 q^{80} -16.0528 q^{82} -9.66363 q^{83} +5.12419 q^{85} +8.20982 q^{86} +0.985558 q^{89} -2.46429 q^{91} +39.7763 q^{92} +22.7451 q^{94} +14.7995 q^{95} +6.97238 q^{97} +2.60390 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 20 q^{4} + 16 q^{7}+O(q^{10}) $ 16 * q + 20 * q^4 + 16 * q^7	$ 16 q + 20 q^{4} + 16 q^{7} - 6 q^{10} + 32 q^{16} + 10 q^{19} + 44 q^{25} + 20 q^{28} + 30 q^{31} + 12 q^{34} + 38 q^{37} - 68 q^{40} + 16 q^{43} + 80 q^{46} + 16 q^{49} + 2 q^{52} + 18 q^{58} - 28 q^{61} + 34 q^{64} + 52 q^{67} - 6 q^{70} - 14 q^{73} - 14 q^{76} + 54 q^{79} + 64 q^{82} + 30 q^{85} - 60 q^{94} + 108 q^{97}+O(q^{100}) $ 16 * q + 20 * q^4 + 16 * q^7 - 6 * q^10 + 32 * q^16 + 10 * q^19 + 44 * q^25 + 20 * q^28 + 30 * q^31 + 12 * q^34 + 38 * q^37 - 68 * q^40 + 16 * q^43 + 80 * q^46 + 16 * q^49 + 2 * q^52 + 18 * q^58 - 28 * q^61 + 34 * q^64 + 52 * q^67 - 6 * q^70 - 14 * q^73 - 14 * q^76 + 54 * q^79 + 64 * q^82 + 30 * q^85 - 60 * q^94 + 108 * 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 $

$2$	2.60390	1.84124	0.920619	−	0.390461i	$-0.127685\pi$
$2$	2.60390	1.84124	0.920619	+	0.390461i	$0.127685\pi$
$3$	0	0
$3$	0	0
$4$	4.78032	2.39016
$5$	−4.31860	−1.93134	−0.965668	−	0.259781i	$-0.916350\pi$
$5$	−4.31860	−1.93134	−0.965668	+	0.259781i	$0.916350\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	7.23969	2.55962
$9$	0	0
$10$	−11.2452	−3.55605
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−2.46429	−0.683470	−0.341735	−	0.939796i	$-0.611014\pi$
$13$	−2.46429	−0.683470	−0.341735	+	0.939796i	$0.611014\pi$
$14$	2.60390	0.695923
$15$	0	0
$16$	9.29081	2.32270
$17$	−1.18654	−0.287778	−0.143889	−	0.989594i	$-0.545961\pi$
$17$	−1.18654	−0.287778	−0.143889	+	0.989594i	$0.545961\pi$
$18$	0	0
$19$	−3.42691	−0.786188	−0.393094	−	0.919498i	$-0.628595\pi$
$19$	−3.42691	−0.786188	−0.393094	+	0.919498i	$0.628595\pi$
$20$	−20.6443	−4.61620
$21$	0	0
$22$	0	0
$23$	8.32084	1.73501	0.867507	−	0.497425i	$-0.165721\pi$
$23$	8.32084	1.73501	0.867507	+	0.497425i	$0.165721\pi$
$24$	0	0
$25$	13.6503	2.73006
$26$	−6.41676	−1.25843
$27$	0	0
$28$	4.78032	0.903395
$29$	1.45371	0.269947	0.134974	−	0.990849i	$-0.456905\pi$
$29$	1.45371	0.269947	0.134974	+	0.990849i	$0.456905\pi$
$30$	0	0
$31$	2.98816	0.536689	0.268344	−	0.963323i	$-0.413523\pi$
$31$	2.98816	0.536689	0.268344	+	0.963323i	$0.413523\pi$
$32$	9.71302	1.71704
$33$	0	0
$34$	−3.08964	−0.529869
$35$	−4.31860	−0.729976
$36$	0	0
$37$	6.34373	1.04290	0.521452	−	0.853281i	$-0.325391\pi$
$37$	6.34373	1.04290	0.521452	+	0.853281i	$0.325391\pi$
$38$	−8.92335	−1.44756
$39$	0	0
$40$	−31.2653	−4.94348
$41$	−6.16490	−0.962795	−0.481398	−	0.876502i	$-0.659871\pi$
$41$	−6.16490	−0.962795	−0.481398	+	0.876502i	$0.659871\pi$
$42$	0	0
$43$	3.15289	0.480811	0.240406	−	0.970673i	$-0.422720\pi$
$43$	3.15289	0.480811	0.240406	+	0.970673i	$0.422720\pi$
$44$	0	0
$45$	0	0
$46$	21.6667	3.19458
$47$	8.73499	1.27413	0.637064	−	0.770811i	$-0.280149\pi$
$47$	8.73499	1.27413	0.637064	+	0.770811i	$0.280149\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	35.5440	5.02668
$51$	0	0
$52$	−11.7801	−1.63360
$53$	−6.68159	−0.917787	−0.458894	−	0.888491i	$-0.651754\pi$
$53$	−6.68159	−0.917787	−0.458894	+	0.888491i	$0.651754\pi$
$54$	0	0
$55$	0	0
$56$	7.23969	0.967444
$57$	0	0
$58$	3.78533	0.497038
$59$	11.7548	1.53035	0.765175	−	0.643822i	$-0.222652\pi$
$59$	11.7548	1.53035	0.765175	+	0.643822i	$0.222652\pi$
$60$	0	0
$61$	7.76909	0.994729	0.497365	−	0.867542i	$-0.334301\pi$
$61$	7.76909	0.994729	0.497365	+	0.867542i	$0.334301\pi$
$62$	7.78087	0.988172
$63$	0	0
$64$	6.71015	0.838769
$65$	10.6423	1.32001
$66$	0	0
$67$	3.11080	0.380045	0.190022	−	0.981780i	$-0.439144\pi$
$67$	3.11080	0.380045	0.190022	+	0.981780i	$0.439144\pi$
$68$	−5.67204	−0.687836
$69$	0	0
$70$	−11.2452	−1.34406
$71$	5.48760	0.651258	0.325629	−	0.945498i	$-0.394424\pi$
$71$	5.48760	0.651258	0.325629	+	0.945498i	$0.394424\pi$
$72$	0	0
$73$	8.66003	1.01358	0.506790	−	0.862070i	$-0.330832\pi$
$73$	8.66003	1.01358	0.506790	+	0.862070i	$0.330832\pi$
$74$	16.5185	1.92023
$75$	0	0
$76$	−16.3817	−1.87911
$77$	0	0
$78$	0	0
$79$	−8.83563	−0.994086	−0.497043	−	0.867726i	$-0.665581\pi$
$79$	−8.83563	−0.994086	−0.497043	+	0.867726i	$0.665581\pi$
$80$	−40.1233	−4.48592
$81$	0	0
$82$	−16.0528	−1.77274
$83$	−9.66363	−1.06072	−0.530360	−	0.847772i	$-0.677943\pi$
$83$	−9.66363	−1.06072	−0.530360	+	0.847772i	$0.677943\pi$
$84$	0	0
$85$	5.12419	0.555797
$86$	8.20982	0.885288
$87$	0	0
$88$	0	0
$89$	0.985558	0.104469	0.0522345	−	0.998635i	$-0.483366\pi$
$89$	0.985558	0.104469	0.0522345	+	0.998635i	$0.483366\pi$
$90$	0	0
$91$	−2.46429	−0.258327
$92$	39.7763	4.14696
$93$	0	0
$94$	22.7451	2.34597
$95$	14.7995	1.51839
$96$	0	0
$97$	6.97238	0.707938	0.353969	−	0.935257i	$-0.384832\pi$
$97$	6.97238	0.707938	0.353969	+	0.935257i	$0.384832\pi$
$98$	2.60390	0.263034
$99$	0	0
$100$	65.2527	6.52527
$101$	16.6907	1.66079	0.830394	−	0.557177i	$-0.188115\pi$
$101$	16.6907	1.66079	0.830394	+	0.557177i	$0.188115\pi$
$102$	0	0
$103$	11.5782	1.14084	0.570419	−	0.821354i	$-0.306781\pi$
$103$	11.5782	1.14084	0.570419	+	0.821354i	$0.306781\pi$
$104$	−17.8407	−1.74942
$105$	0	0
$106$	−17.3982	−1.68987
$107$	−6.01586	−0.581575	−0.290788	−	0.956788i	$-0.593917\pi$
$107$	−6.01586	−0.581575	−0.290788	+	0.956788i	$0.593917\pi$
$108$	0	0
$109$	6.68101	0.639924	0.319962	−	0.947430i	$-0.396330\pi$
$109$	6.68101	0.639924	0.319962	+	0.947430i	$0.396330\pi$
$110$	0	0
$111$	0	0
$112$	9.29081	0.877899
$113$	−18.7558	−1.76440	−0.882199	−	0.470877i	$-0.843937\pi$
$113$	−18.7558	−1.76440	−0.882199	+	0.470877i	$0.843937\pi$
$114$	0	0
$115$	−35.9343	−3.35089
$116$	6.94920	0.645217
$117$	0	0
$118$	30.6085	2.81774
$119$	−1.18654	−0.108770
$120$	0	0
$121$	0	0
$122$	20.2300	1.83153
$123$	0	0
$124$	14.2843	1.28277
$125$	−37.3571	−3.34132
$126$	0	0
$127$	3.17002	0.281294	0.140647	−	0.990060i	$-0.455082\pi$
$127$	3.17002	0.281294	0.140647	+	0.990060i	$0.455082\pi$
$128$	−1.95344	−0.172662
$129$	0	0
$130$	27.7114	2.43045
$131$	7.83376	0.684439	0.342220	−	0.939620i	$-0.388821\pi$
$131$	7.83376	0.684439	0.342220	+	0.939620i	$0.388821\pi$
$132$	0	0
$133$	−3.42691	−0.297151
$134$	8.10023	0.699753
$135$	0	0
$136$	−8.59018	−0.736602
$137$	7.18722	0.614045	0.307023	−	0.951702i	$-0.400667\pi$
$137$	7.18722	0.614045	0.307023	+	0.951702i	$0.400667\pi$
$138$	0	0
$139$	3.36801	0.285671	0.142836	−	0.989746i	$-0.454378\pi$
$139$	3.36801	0.285671	0.142836	+	0.989746i	$0.454378\pi$
$140$	−20.6443	−1.74476
$141$	0	0
$142$	14.2892	1.19912
$143$	0	0
$144$	0	0
$145$	−6.27799	−0.521359
$146$	22.5499	1.86624
$147$	0	0
$148$	30.3251	2.49271
$149$	−18.6235	−1.52569	−0.762846	−	0.646580i	$-0.776199\pi$
$149$	−18.6235	−1.52569	−0.762846	+	0.646580i	$0.776199\pi$
$150$	0	0
$151$	14.4245	1.17385	0.586924	−	0.809642i	$-0.300339\pi$
$151$	14.4245	1.17385	0.586924	+	0.809642i	$0.300339\pi$
$152$	−24.8098	−2.01234
$153$	0	0
$154$	0	0
$155$	−12.9046	−1.03653
$156$	0	0
$157$	5.31457	0.424148	0.212074	−	0.977254i	$-0.431978\pi$
$157$	5.31457	0.424148	0.212074	+	0.977254i	$0.431978\pi$
$158$	−23.0071	−1.83035
$159$	0	0
$160$	−41.9466	−3.31617
$161$	8.32084	0.655774
$162$	0	0
$163$	0.906332	0.0709894	0.0354947	−	0.999370i	$-0.488699\pi$
$163$	0.906332	0.0709894	0.0354947	+	0.999370i	$0.488699\pi$
$164$	−29.4702	−2.30123
$165$	0	0
$166$	−25.1632	−1.95304
$167$	11.6434	0.900994	0.450497	−	0.892778i	$-0.351247\pi$
$167$	11.6434	0.900994	0.450497	+	0.892778i	$0.351247\pi$
$168$	0	0
$169$	−6.92730	−0.532869
$170$	13.3429	1.02335
$171$	0	0
$172$	15.0718	1.14922
$173$	−6.85042	−0.520828	−0.260414	−	0.965497i	$-0.583859\pi$
$173$	−6.85042	−0.520828	−0.260414	+	0.965497i	$0.583859\pi$
$174$	0	0
$175$	13.6503	1.03186
$176$	0	0
$177$	0	0
$178$	2.56630	0.192352
$179$	−8.41259	−0.628787	−0.314393	−	0.949293i	$-0.601801\pi$
$179$	−8.41259	−0.628787	−0.314393	+	0.949293i	$0.601801\pi$
$180$	0	0
$181$	13.0288	0.968426	0.484213	−	0.874950i	$-0.339106\pi$
$181$	13.0288	0.968426	0.484213	+	0.874950i	$0.339106\pi$
$182$	−6.41676	−0.475642
$183$	0	0
$184$	60.2402	4.44097
$185$	−27.3960	−2.01420
$186$	0	0
$187$	0	0
$188$	41.7560	3.04537
$189$	0	0
$190$	38.5364	2.79572
$191$	3.45753	0.250178	0.125089	−	0.992145i	$-0.460078\pi$
$191$	3.45753	0.250178	0.125089	+	0.992145i	$0.460078\pi$
$192$	0	0
$193$	17.0472	1.22709	0.613544	−	0.789661i	$-0.289743\pi$
$193$	17.0472	1.22709	0.613544	+	0.789661i	$0.289743\pi$
$194$	18.1554	1.30348
$195$	0	0
$196$	4.78032	0.341451
$197$	−3.47970	−0.247918	−0.123959	−	0.992287i	$-0.539559\pi$
$197$	−3.47970	−0.247918	−0.123959	+	0.992287i	$0.539559\pi$
$198$	0	0
$199$	8.20279	0.581481	0.290740	−	0.956802i	$-0.406098\pi$
$199$	8.20279	0.581481	0.290740	+	0.956802i	$0.406098\pi$
$200$	98.8237	6.98789
$201$	0	0
$202$	43.4610	3.05791
$203$	1.45371	0.102031
$204$	0	0
$205$	26.6237	1.85948
$206$	30.1486	2.10055
$207$	0	0
$208$	−22.8952	−1.58750
$209$	0	0
$210$	0	0
$211$	22.2192	1.52963	0.764816	−	0.644248i	$-0.222830\pi$
$211$	22.2192	1.52963	0.764816	+	0.644248i	$0.222830\pi$
$212$	−31.9401	−2.19366
$213$	0	0
$214$	−15.6647	−1.07082
$215$	−13.6161	−0.928607
$216$	0	0
$217$	2.98816	0.202849
$218$	17.3967	1.17825
$219$	0	0
$220$	0	0
$221$	2.92397	0.196688
$222$	0	0
$223$	21.8353	1.46220	0.731100	−	0.682270i	$-0.239007\pi$
$223$	21.8353	1.46220	0.731100	+	0.682270i	$0.239007\pi$
$224$	9.71302	0.648979
$225$	0	0
$226$	−48.8383	−3.24868
$227$	−24.3708	−1.61755	−0.808774	−	0.588120i	$-0.799868\pi$
$227$	−24.3708	−1.61755	−0.808774	+	0.588120i	$0.799868\pi$
$228$	0	0
$229$	−2.58137	−0.170582	−0.0852910	−	0.996356i	$-0.527182\pi$
$229$	−2.58137	−0.170582	−0.0852910	+	0.996356i	$0.527182\pi$
$230$	−93.5696	−6.16980
$231$	0	0
$232$	10.5244	0.690962
$233$	−14.2545	−0.933845	−0.466923	−	0.884298i	$-0.654637\pi$
$233$	−14.2545	−0.933845	−0.466923	+	0.884298i	$0.654637\pi$
$234$	0	0
$235$	−37.7229	−2.46077
$236$	56.1919	3.65778
$237$	0	0
$238$	−3.08964	−0.200272
$239$	−11.0487	−0.714683	−0.357341	−	0.933974i	$-0.616317\pi$
$239$	−11.0487	−0.714683	−0.357341	+	0.933974i	$0.616317\pi$
$240$	0	0
$241$	−11.0131	−0.709419	−0.354709	−	0.934977i	$-0.615420\pi$
$241$	−11.0131	−0.709419	−0.354709	+	0.934977i	$0.615420\pi$
$242$	0	0
$243$	0	0
$244$	37.1387	2.37756
$245$	−4.31860	−0.275905
$246$	0	0
$247$	8.44489	0.537335
$248$	21.6333	1.37372
$249$	0	0
$250$	−97.2742	−6.15216
$251$	−11.2971	−0.713065	−0.356532	−	0.934283i	$-0.616041\pi$
$251$	−11.2971	−0.713065	−0.356532	+	0.934283i	$0.616041\pi$
$252$	0	0
$253$	0	0
$254$	8.25443	0.517929
$255$	0	0
$256$	−18.5069	−1.15668
$257$	5.10986	0.318745	0.159372	−	0.987219i	$-0.449053\pi$
$257$	5.10986	0.318745	0.159372	+	0.987219i	$0.449053\pi$
$258$	0	0
$259$	6.34373	0.394180
$260$	50.8734	3.15503
$261$	0	0
$262$	20.3984	1.26022
$263$	−6.56589	−0.404870	−0.202435	−	0.979296i	$-0.564885\pi$
$263$	−6.56589	−0.404870	−0.202435	+	0.979296i	$0.564885\pi$
$264$	0	0
$265$	28.8551	1.77256
$266$	−8.92335	−0.547126
$267$	0	0
$268$	14.8706	0.908367
$269$	−11.6093	−0.707833	−0.353916	−	0.935277i	$-0.615150\pi$
$269$	−11.6093	−0.707833	−0.353916	+	0.935277i	$0.615150\pi$
$270$	0	0
$271$	−14.6728	−0.891307	−0.445654	−	0.895205i	$-0.647029\pi$
$271$	−14.6728	−0.891307	−0.445654	+	0.895205i	$0.647029\pi$
$272$	−11.0239	−0.668424
$273$	0	0
$274$	18.7148	1.13060
$275$	0	0
$276$	0	0
$277$	−23.2754	−1.39848	−0.699241	−	0.714886i	$-0.746478\pi$
$277$	−23.2754	−1.39848	−0.699241	+	0.714886i	$0.746478\pi$
$278$	8.76999	0.525989
$279$	0	0
$280$	−31.2653	−1.86846
$281$	−10.2114	−0.609160	−0.304580	−	0.952487i	$-0.598516\pi$
$281$	−10.2114	−0.609160	−0.304580	+	0.952487i	$0.598516\pi$
$282$	0	0
$283$	−6.41587	−0.381384	−0.190692	−	0.981650i	$-0.561073\pi$
$283$	−6.41587	−0.381384	−0.190692	+	0.981650i	$0.561073\pi$
$284$	26.2325	1.55661
$285$	0	0
$286$	0	0
$287$	−6.16490	−0.363902
$288$	0	0
$289$	−15.5921	−0.917184
$290$	−16.3473	−0.959946
$291$	0	0
$292$	41.3977	2.42262
$293$	15.7851	0.922174	0.461087	−	0.887355i	$-0.347460\pi$
$293$	15.7851	0.922174	0.461087	+	0.887355i	$0.347460\pi$
$294$	0	0
$295$	−50.7644	−2.95562
$296$	45.9266	2.66943
$297$	0	0
$298$	−48.4937	−2.80916
$299$	−20.5049	−1.18583
$300$	0	0
$301$	3.15289	0.181730
$302$	37.5600	2.16134
$303$	0	0
$304$	−31.8388	−1.82608
$305$	−33.5515	−1.92116
$306$	0	0
$307$	17.4204	0.994235	0.497117	−	0.867683i	$-0.334392\pi$
$307$	17.4204	0.994235	0.497117	+	0.867683i	$0.334392\pi$
$308$	0	0
$309$	0	0
$310$	−33.6025	−1.90849
$311$	13.1950	0.748221	0.374110	−	0.927384i	$-0.377948\pi$
$311$	13.1950	0.748221	0.374110	+	0.927384i	$0.377948\pi$
$312$	0	0
$313$	18.0139	1.01821	0.509104	−	0.860705i	$-0.329977\pi$
$313$	18.0139	1.01821	0.509104	+	0.860705i	$0.329977\pi$
$314$	13.8386	0.780959
$315$	0	0
$316$	−42.2371	−2.37602
$317$	−6.70435	−0.376554	−0.188277	−	0.982116i	$-0.560290\pi$
$317$	−6.70435	−0.376554	−0.188277	+	0.982116i	$0.560290\pi$
$318$	0	0
$319$	0	0
$320$	−28.9784	−1.61994
$321$	0	0
$322$	21.6667	1.20744
$323$	4.06617	0.226248
$324$	0	0
$325$	−33.6382	−1.86591
$326$	2.36000	0.130708
$327$	0	0
$328$	−44.6319	−2.46439
$329$	8.73499	0.481575
$330$	0	0
$331$	3.96252	0.217800	0.108900	−	0.994053i	$-0.465267\pi$
$331$	3.96252	0.217800	0.108900	+	0.994053i	$0.465267\pi$
$332$	−46.1952	−2.53529
$333$	0	0
$334$	30.3183	1.65894
$335$	−13.4343	−0.733994
$336$	0	0
$337$	−12.2604	−0.667866	−0.333933	−	0.942597i	$-0.608376\pi$
$337$	−12.2604	−0.667866	−0.333933	+	0.942597i	$0.608376\pi$
$338$	−18.0380	−0.981139
$339$	0	0
$340$	24.4953	1.32844
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	22.8259	1.23069
$345$	0	0
$346$	−17.8378	−0.958968
$347$	−11.4238	−0.613261	−0.306630	−	0.951829i	$-0.599202\pi$
$347$	−11.4238	−0.613261	−0.306630	+	0.951829i	$0.599202\pi$
$348$	0	0
$349$	−29.2338	−1.56485	−0.782425	−	0.622745i	$-0.786017\pi$
$349$	−29.2338	−1.56485	−0.782425	+	0.622745i	$0.786017\pi$
$350$	35.5440	1.89991
$351$	0	0
$352$	0	0
$353$	7.76081	0.413066	0.206533	−	0.978440i	$-0.433782\pi$
$353$	7.76081	0.413066	0.206533	+	0.978440i	$0.433782\pi$
$354$	0	0
$355$	−23.6987	−1.25780
$356$	4.71128	0.249697
$357$	0	0
$358$	−21.9056	−1.15775
$359$	29.9693	1.58172	0.790860	−	0.611997i	$-0.209634\pi$
$359$	29.9693	1.58172	0.790860	+	0.611997i	$0.209634\pi$
$360$	0	0
$361$	−7.25627	−0.381909
$362$	33.9259	1.78310
$363$	0	0
$364$	−11.7801	−0.617443
$365$	−37.3992	−1.95756
$366$	0	0
$367$	−3.08710	−0.161145	−0.0805726	−	0.996749i	$-0.525675\pi$
$367$	−3.08710	−0.161145	−0.0805726	+	0.996749i	$0.525675\pi$
$368$	77.3073	4.02992
$369$	0	0
$370$	−71.3366	−3.70862
$371$	−6.68159	−0.346891
$372$	0	0
$373$	29.2948	1.51683	0.758413	−	0.651774i	$-0.225975\pi$
$373$	29.2948	1.51683	0.758413	+	0.651774i	$0.225975\pi$
$374$	0	0
$375$	0	0
$376$	63.2386	3.26128
$377$	−3.58236	−0.184501
$378$	0	0
$379$	15.6869	0.805784	0.402892	−	0.915247i	$-0.368005\pi$
$379$	15.6869	0.805784	0.402892	+	0.915247i	$0.368005\pi$
$380$	70.7461	3.62920
$381$	0	0
$382$	9.00309	0.460638
$383$	8.03590	0.410615	0.205308	−	0.978697i	$-0.434180\pi$
$383$	8.03590	0.410615	0.205308	+	0.978697i	$0.434180\pi$
$384$	0	0
$385$	0	0
$386$	44.3894	2.25936
$387$	0	0
$388$	33.3302	1.69208
$389$	−28.7874	−1.45958	−0.729789	−	0.683672i	$-0.760382\pi$
$389$	−28.7874	−1.45958	−0.729789	+	0.683672i	$0.760382\pi$
$390$	0	0
$391$	−9.87301	−0.499300
$392$	7.23969	0.365659
$393$	0	0
$394$	−9.06080	−0.456477
$395$	38.1575	1.91991
$396$	0	0
$397$	−33.7315	−1.69293	−0.846467	−	0.532441i	$-0.821275\pi$
$397$	−33.7315	−1.69293	−0.846467	+	0.532441i	$0.821275\pi$
$398$	21.3593	1.07064
$399$	0	0
$400$	126.822	6.34111
$401$	0.924552	0.0461699	0.0230850	−	0.999734i	$-0.492651\pi$
$401$	0.924552	0.0461699	0.0230850	+	0.999734i	$0.492651\pi$
$402$	0	0
$403$	−7.36367	−0.366810
$404$	79.7869	3.96955
$405$	0	0
$406$	3.78533	0.187863
$407$	0	0
$408$	0	0
$409$	25.7923	1.27535	0.637673	−	0.770307i	$-0.279897\pi$
$409$	25.7923	1.27535	0.637673	+	0.770307i	$0.279897\pi$
$410$	69.3256	3.42375
$411$	0	0
$412$	55.3477	2.72678
$413$	11.7548	0.578418
$414$	0	0
$415$	41.7333	2.04861
$416$	−23.9357	−1.17354
$417$	0	0
$418$	0	0
$419$	9.41300	0.459855	0.229928	−	0.973208i	$-0.426151\pi$
$419$	9.41300	0.459855	0.229928	+	0.973208i	$0.426151\pi$
$420$	0	0
$421$	−29.3045	−1.42821	−0.714107	−	0.700037i	$-0.753167\pi$
$421$	−29.3045	−1.42821	−0.714107	+	0.700037i	$0.753167\pi$
$422$	57.8567	2.81642
$423$	0	0
$424$	−48.3726	−2.34918
$425$	−16.1966	−0.785651
$426$	0	0
$427$	7.76909	0.375972
$428$	−28.7577	−1.39006
$429$	0	0
$430$	−35.4549	−1.70979
$431$	6.97670	0.336056	0.168028	−	0.985782i	$-0.446260\pi$
$431$	6.97670	0.336056	0.168028	+	0.985782i	$0.446260\pi$
$432$	0	0
$433$	1.32914	0.0638746	0.0319373	−	0.999490i	$-0.489832\pi$
$433$	1.32914	0.0638746	0.0319373	+	0.999490i	$0.489832\pi$
$434$	7.78087	0.373494
$435$	0	0
$436$	31.9374	1.52952
$437$	−28.5148	−1.36405
$438$	0	0
$439$	−3.63801	−0.173633	−0.0868163	−	0.996224i	$-0.527669\pi$
$439$	−3.63801	−0.173633	−0.0868163	+	0.996224i	$0.527669\pi$
$440$	0	0
$441$	0	0
$442$	7.61375	0.362149
$443$	13.4099	0.637125	0.318563	−	0.947902i	$-0.396800\pi$
$443$	13.4099	0.637125	0.318563	+	0.947902i	$0.396800\pi$
$444$	0	0
$445$	−4.25623	−0.201765
$446$	56.8570	2.69226
$447$	0	0
$448$	6.71015	0.317025
$449$	14.9993	0.707859	0.353929	−	0.935272i	$-0.384845\pi$
$449$	14.9993	0.707859	0.353929	+	0.935272i	$0.384845\pi$
$450$	0	0
$451$	0	0
$452$	−89.6587	−4.21719
$453$	0	0
$454$	−63.4593	−2.97829
$455$	10.6423	0.498917
$456$	0	0
$457$	−19.7121	−0.922094	−0.461047	−	0.887376i	$-0.652526\pi$
$457$	−19.7121	−0.922094	−0.461047	+	0.887376i	$0.652526\pi$
$458$	−6.72165	−0.314082
$459$	0	0
$460$	−171.778	−8.00917
$461$	16.3627	0.762086	0.381043	−	0.924557i	$-0.375565\pi$
$461$	16.3627	0.762086	0.381043	+	0.924557i	$0.375565\pi$
$462$	0	0
$463$	−7.48587	−0.347898	−0.173949	−	0.984755i	$-0.555653\pi$
$463$	−7.48587	−0.347898	−0.173949	+	0.984755i	$0.555653\pi$
$464$	13.5062	0.627008
$465$	0	0
$466$	−37.1174	−1.71943
$467$	−2.91022	−0.134669	−0.0673344	−	0.997730i	$-0.521449\pi$
$467$	−2.91022	−0.134669	−0.0673344	+	0.997730i	$0.521449\pi$
$468$	0	0
$469$	3.11080	0.143643
$470$	−98.2268	−4.53086
$471$	0	0
$472$	85.1014	3.91711
$473$	0	0
$474$	0	0
$475$	−46.7783	−2.14634
$476$	−5.67204	−0.259978
$477$	0	0
$478$	−28.7698	−1.31590
$479$	−26.5272	−1.21206	−0.606030	−	0.795442i	$-0.707239\pi$
$479$	−26.5272	−1.21206	−0.606030	+	0.795442i	$0.707239\pi$
$480$	0	0
$481$	−15.6328	−0.712793
$482$	−28.6772	−1.30621
$483$	0	0
$484$	0	0
$485$	−30.1109	−1.36727
$486$	0	0
$487$	−2.81722	−0.127661	−0.0638303	−	0.997961i	$-0.520332\pi$
$487$	−2.81722	−0.127661	−0.0638303	+	0.997961i	$0.520332\pi$
$488$	56.2457	2.54613
$489$	0	0
$490$	−11.2452	−0.508007
$491$	5.67888	0.256284	0.128142	−	0.991756i	$-0.459099\pi$
$491$	5.67888	0.256284	0.128142	+	0.991756i	$0.459099\pi$
$492$	0	0
$493$	−1.72489	−0.0776850
$494$	21.9897	0.989363
$495$	0	0
$496$	27.7624	1.24657
$497$	5.48760	0.246152
$498$	0	0
$499$	35.2949	1.58002	0.790008	−	0.613096i	$-0.210076\pi$
$499$	35.2949	1.58002	0.790008	+	0.613096i	$0.210076\pi$
$500$	−178.579	−7.98628
$501$	0	0
$502$	−29.4165	−1.31292
$503$	29.6610	1.32252	0.661259	−	0.750158i	$-0.270022\pi$
$503$	29.6610	1.32252	0.661259	+	0.750158i	$0.270022\pi$
$504$	0	0
$505$	−72.0805	−3.20754
$506$	0	0
$507$	0	0
$508$	15.1537	0.672337
$509$	4.71393	0.208941	0.104471	−	0.994528i	$-0.466685\pi$
$509$	4.71393	0.208941	0.104471	+	0.994528i	$0.466685\pi$
$510$	0	0
$511$	8.66003	0.383097
$512$	−44.2833	−1.95706
$513$	0	0
$514$	13.3056	0.586885
$515$	−50.0018	−2.20334
$516$	0	0
$517$	0	0
$518$	16.5185	0.725780
$519$	0	0
$520$	77.0466	3.37872
$521$	−22.6541	−0.992495	−0.496247	−	0.868181i	$-0.665289\pi$
$521$	−22.6541	−0.992495	−0.496247	+	0.868181i	$0.665289\pi$
$522$	0	0
$523$	−15.0536	−0.658247	−0.329124	−	0.944287i	$-0.606753\pi$
$523$	−15.0536	−0.658247	−0.329124	+	0.944287i	$0.606753\pi$
$524$	37.4479	1.63592
$525$	0	0
$526$	−17.0969	−0.745462
$527$	−3.54557	−0.154447
$528$	0	0
$529$	46.2363	2.01027
$530$	75.1359	3.26370
$531$	0	0
$532$	−16.3817	−0.710238
$533$	15.1921	0.658041
$534$	0	0
$535$	25.9801	1.12322
$536$	22.5212	0.972768
$537$	0	0
$538$	−30.2296	−1.30329
$539$	0	0
$540$	0	0
$541$	15.5738	0.669570	0.334785	−	0.942295i	$-0.391336\pi$
$541$	15.5738	0.669570	0.334785	+	0.942295i	$0.391336\pi$
$542$	−38.2065	−1.64111
$543$	0	0
$544$	−11.5249	−0.494126
$545$	−28.8526	−1.23591
$546$	0	0
$547$	16.9696	0.725568	0.362784	−	0.931873i	$-0.381826\pi$
$547$	16.9696	0.725568	0.362784	+	0.931873i	$0.381826\pi$
$548$	34.3572	1.46767
$549$	0	0
$550$	0	0
$551$	−4.98174	−0.212229
$552$	0	0
$553$	−8.83563	−0.375729
$554$	−60.6068	−2.57494
$555$	0	0
$556$	16.1002	0.682800
$557$	−2.65932	−0.112679	−0.0563394	−	0.998412i	$-0.517943\pi$
$557$	−2.65932	−0.112679	−0.0563394	+	0.998412i	$0.517943\pi$
$558$	0	0
$559$	−7.76962	−0.328620
$560$	−40.1233	−1.69552
$561$	0	0
$562$	−26.5895	−1.12161
$563$	−15.0516	−0.634349	−0.317174	−	0.948367i	$-0.602734\pi$
$563$	−15.0516	−0.634349	−0.317174	+	0.948367i	$0.602734\pi$
$564$	0	0
$565$	80.9987	3.40764
$566$	−16.7063	−0.702219
$567$	0	0
$568$	39.7285	1.66697
$569$	36.7508	1.54067	0.770337	−	0.637637i	$-0.220088\pi$
$569$	36.7508	1.54067	0.770337	+	0.637637i	$0.220088\pi$
$570$	0	0
$571$	−3.56292	−0.149104	−0.0745519	−	0.997217i	$-0.523753\pi$
$571$	−3.56292	−0.149104	−0.0745519	+	0.997217i	$0.523753\pi$
$572$	0	0
$573$	0	0
$574$	−16.0528	−0.670031
$575$	113.582	4.73669
$576$	0	0
$577$	19.7289	0.821325	0.410663	−	0.911787i	$-0.365297\pi$
$577$	19.7289	0.821325	0.410663	+	0.911787i	$0.365297\pi$
$578$	−40.6004	−1.68875
$579$	0	0
$580$	−30.0108	−1.24613
$581$	−9.66363	−0.400915
$582$	0	0
$583$	0	0
$584$	62.6959	2.59437
$585$	0	0
$586$	41.1028	1.69794
$587$	−8.34576	−0.344466	−0.172233	−	0.985056i	$-0.555098\pi$
$587$	−8.34576	−0.344466	−0.172233	+	0.985056i	$0.555098\pi$
$588$	0	0
$589$	−10.2402	−0.421938
$590$	−132.186	−5.44200
$591$	0	0
$592$	58.9385	2.42236
$593$	−15.7459	−0.646605	−0.323302	−	0.946296i	$-0.604793\pi$
$593$	−15.7459	−0.646605	−0.323302	+	0.946296i	$0.604793\pi$
$594$	0	0
$595$	5.12419	0.210071
$596$	−89.0261	−3.64665
$597$	0	0
$598$	−53.3928	−2.18340
$599$	0.0424211	0.00173328	0.000866640	−	1.00000i	$-0.499724\pi$
$599$	0.0424211	0.00173328	0.000866640		1.00000i	$0.499724\pi$
$600$	0	0
$601$	4.36523	0.178062	0.0890308	−	0.996029i	$-0.471623\pi$
$601$	4.36523	0.178062	0.0890308	+	0.996029i	$0.471623\pi$
$602$	8.20982	0.334607
$603$	0	0
$604$	68.9537	2.80569
$605$	0	0
$606$	0	0
$607$	−3.46475	−0.140630	−0.0703149	−	0.997525i	$-0.522400\pi$
$607$	−3.46475	−0.140630	−0.0703149	+	0.997525i	$0.522400\pi$
$608$	−33.2857	−1.34991
$609$	0	0
$610$	−87.3650	−3.53731
$611$	−21.5255	−0.870828
$612$	0	0
$613$	6.89445	0.278464	0.139232	−	0.990260i	$-0.455537\pi$
$613$	6.89445	0.278464	0.139232	+	0.990260i	$0.455537\pi$
$614$	45.3611	1.83062
$615$	0	0
$616$	0	0
$617$	13.8954	0.559409	0.279705	−	0.960086i	$-0.409763\pi$
$617$	13.8954	0.559409	0.279705	+	0.960086i	$0.409763\pi$
$618$	0	0
$619$	−30.6734	−1.23287	−0.616434	−	0.787406i	$-0.711423\pi$
$619$	−30.6734	−1.23287	−0.616434	+	0.787406i	$0.711423\pi$
$620$	−61.6883	−2.47746
$621$	0	0
$622$	34.3586	1.37765
$623$	0.985558	0.0394855
$624$	0	0
$625$	93.0787	3.72315
$626$	46.9066	1.87476
$627$	0	0
$628$	25.4053	1.01378
$629$	−7.52710	−0.300125
$630$	0	0
$631$	−45.2371	−1.80086	−0.900430	−	0.435000i	$-0.856748\pi$
$631$	−45.2371	−1.80086	−0.900430	+	0.435000i	$0.856748\pi$
$632$	−63.9672	−2.54448
$633$	0	0
$634$	−17.4575	−0.693325
$635$	−13.6900	−0.543273
$636$	0	0
$637$	−2.46429	−0.0976385
$638$	0	0
$639$	0	0
$640$	8.43614	0.333468
$641$	16.1033	0.636043	0.318022	−	0.948083i	$-0.396982\pi$
$641$	16.1033	0.636043	0.318022	+	0.948083i	$0.396982\pi$
$642$	0	0
$643$	49.3352	1.94559	0.972795	−	0.231668i	$-0.0744181\pi$
$643$	49.3352	1.94559	0.972795	+	0.231668i	$0.0744181\pi$
$644$	39.7763	1.56740
$645$	0	0
$646$	10.5879	0.416576
$647$	−35.0628	−1.37846	−0.689231	−	0.724542i	$-0.742052\pi$
$647$	−35.0628	−1.37846	−0.689231	+	0.724542i	$0.742052\pi$
$648$	0	0
$649$	0	0
$650$	−87.5906	−3.43559
$651$	0	0
$652$	4.33256	0.169676
$653$	−24.8559	−0.972688	−0.486344	−	0.873767i	$-0.661670\pi$
$653$	−24.8559	−0.972688	−0.486344	+	0.873767i	$0.661670\pi$
$654$	0	0
$655$	−33.8309	−1.32188
$656$	−57.2769	−2.23629
$657$	0	0
$658$	22.7451	0.886695
$659$	12.3854	0.482468	0.241234	−	0.970467i	$-0.422448\pi$
$659$	12.3854	0.482468	0.241234	+	0.970467i	$0.422448\pi$
$660$	0	0
$661$	−39.9739	−1.55480	−0.777402	−	0.629004i	$-0.783463\pi$
$661$	−39.9739	−1.55480	−0.777402	+	0.629004i	$0.783463\pi$
$662$	10.3180	0.401021
$663$	0	0
$664$	−69.9616	−2.71504
$665$	14.7995	0.573898
$666$	0	0
$667$	12.0961	0.468363
$668$	55.6592	2.15352
$669$	0	0
$670$	−34.9816	−1.35146
$671$	0	0
$672$	0	0
$673$	−8.31720	−0.320604	−0.160302	−	0.987068i	$-0.551247\pi$
$673$	−8.31720	−0.320604	−0.160302	+	0.987068i	$0.551247\pi$
$674$	−31.9249	−1.22970
$675$	0	0
$676$	−33.1147	−1.27364
$677$	30.7173	1.18056	0.590281	−	0.807198i	$-0.299017\pi$
$677$	30.7173	1.18056	0.590281	+	0.807198i	$0.299017\pi$
$678$	0	0
$679$	6.97238	0.267575
$680$	37.0975	1.42263
$681$	0	0
$682$	0	0
$683$	13.9802	0.534936	0.267468	−	0.963567i	$-0.413813\pi$
$683$	13.9802	0.534936	0.267468	+	0.963567i	$0.413813\pi$
$684$	0	0
$685$	−31.0387	−1.18593
$686$	2.60390	0.0994175
$687$	0	0
$688$	29.2929	1.11678
$689$	16.4653	0.627280
$690$	0	0
$691$	28.3460	1.07833	0.539167	−	0.842199i	$-0.318739\pi$
$691$	28.3460	1.07833	0.539167	+	0.842199i	$0.318739\pi$
$692$	−32.7472	−1.24486
$693$	0	0
$694$	−29.7464	−1.12916
$695$	−14.5451	−0.551727
$696$	0	0
$697$	7.31490	0.277072
$698$	−76.1220	−2.88126
$699$	0	0
$700$	65.2527	2.46632
$701$	−13.0144	−0.491546	−0.245773	−	0.969327i	$-0.579042\pi$
$701$	−13.0144	−0.491546	−0.245773	+	0.969327i	$0.579042\pi$
$702$	0	0
$703$	−21.7394	−0.819918
$704$	0	0
$705$	0	0
$706$	20.2084	0.760553
$707$	16.6907	0.627719
$708$	0	0
$709$	22.5145	0.845550	0.422775	−	0.906235i	$-0.361056\pi$
$709$	22.5145	0.845550	0.422775	+	0.906235i	$0.361056\pi$
$710$	−61.7092	−2.31590
$711$	0	0
$712$	7.13513	0.267400
$713$	24.8640	0.931163
$714$	0	0
$715$	0	0
$716$	−40.2149	−1.50290
$717$	0	0
$718$	78.0372	2.91232
$719$	13.4580	0.501899	0.250950	−	0.968000i	$-0.419257\pi$
$719$	13.4580	0.501899	0.250950	+	0.968000i	$0.419257\pi$
$720$	0	0
$721$	11.5782	0.431196
$722$	−18.8946	−0.703186
$723$	0	0
$724$	62.2820	2.31469
$725$	19.8436	0.736971
$726$	0	0
$727$	−25.2856	−0.937793	−0.468896	−	0.883253i	$-0.655348\pi$
$727$	−25.2856	−0.937793	−0.468896	+	0.883253i	$0.655348\pi$
$728$	−17.8407	−0.661219
$729$	0	0
$730$	−97.3839	−3.60434
$731$	−3.74103	−0.138367
$732$	0	0
$733$	−40.8544	−1.50899	−0.754496	−	0.656305i	$-0.772119\pi$
$733$	−40.8544	−1.50899	−0.754496	+	0.656305i	$0.772119\pi$
$734$	−8.03851	−0.296707
$735$	0	0
$736$	80.8205	2.97908
$737$	0	0
$738$	0	0
$739$	37.3128	1.37257	0.686287	−	0.727331i	$-0.259239\pi$
$739$	37.3128	1.37257	0.686287	+	0.727331i	$0.259239\pi$
$740$	−130.962	−4.81425
$741$	0	0
$742$	−17.3982	−0.638709
$743$	−39.9831	−1.46684	−0.733418	−	0.679778i	$-0.762076\pi$
$743$	−39.9831	−1.46684	−0.733418	+	0.679778i	$0.762076\pi$
$744$	0	0
$745$	80.4272	2.94662
$746$	76.2808	2.79284
$747$	0	0
$748$	0	0
$749$	−6.01586	−0.219815
$750$	0	0
$751$	29.2931	1.06892	0.534460	−	0.845194i	$-0.320515\pi$
$751$	29.2931	1.06892	0.534460	+	0.845194i	$0.320515\pi$
$752$	81.1551	2.95942
$753$	0	0
$754$	−9.32812	−0.339710
$755$	−62.2936	−2.26709
$756$	0	0
$757$	−32.8416	−1.19365	−0.596824	−	0.802372i	$-0.703571\pi$
$757$	−32.8416	−1.19365	−0.596824	+	0.802372i	$0.703571\pi$
$758$	40.8473	1.48364
$759$	0	0
$760$	107.143	3.88650
$761$	−43.3133	−1.57011	−0.785053	−	0.619429i	$-0.787364\pi$
$761$	−43.3133	−1.57011	−0.785053	+	0.619429i	$0.787364\pi$
$762$	0	0
$763$	6.68101	0.241869
$764$	16.5281	0.597966
$765$	0	0
$766$	20.9247	0.756041
$767$	−28.9673	−1.04595
$768$	0	0
$769$	31.4543	1.13427	0.567135	−	0.823625i	$-0.308052\pi$
$769$	31.4543	1.13427	0.567135	+	0.823625i	$0.308052\pi$
$770$	0	0
$771$	0	0
$772$	81.4913	2.93294
$773$	−41.8131	−1.50391	−0.751957	−	0.659212i	$-0.770890\pi$
$773$	−41.8131	−1.50391	−0.751957	+	0.659212i	$0.770890\pi$
$774$	0	0
$775$	40.7892	1.46519
$776$	50.4778	1.81205
$777$	0	0
$778$	−74.9596	−2.68743
$779$	21.1266	0.756938
$780$	0	0
$781$	0	0
$782$	−25.7084	−0.919330
$783$	0	0
$784$	9.29081	0.331815
$785$	−22.9515	−0.819173
$786$	0	0
$787$	−46.4054	−1.65418	−0.827088	−	0.562073i	$-0.810004\pi$
$787$	−46.4054	−1.65418	−0.827088	+	0.562073i	$0.810004\pi$
$788$	−16.6341	−0.592564
$789$	0	0
$790$	99.3585	3.53502
$791$	−18.7558	−0.666880
$792$	0	0
$793$	−19.1452	−0.679867
$794$	−87.8336	−3.11710
$795$	0	0
$796$	39.2120	1.38983
$797$	35.3958	1.25378	0.626891	−	0.779107i	$-0.284327\pi$
$797$	35.3958	1.25378	0.626891	+	0.779107i	$0.284327\pi$
$798$	0	0
$799$	−10.3644	−0.366667
$800$	132.585	4.68760
$801$	0	0
$802$	2.40744	0.0850098
$803$	0	0
$804$	0	0
$805$	−35.9343	−1.26652
$806$	−19.1743	−0.675386
$807$	0	0
$808$	120.836	4.25098
$809$	−23.7838	−0.836193	−0.418097	−	0.908403i	$-0.637303\pi$
$809$	−23.7838	−0.836193	−0.418097	+	0.908403i	$0.637303\pi$
$810$	0	0
$811$	−7.21740	−0.253437	−0.126719	−	0.991939i	$-0.540445\pi$
$811$	−7.21740	−0.253437	−0.126719	+	0.991939i	$0.540445\pi$
$812$	6.94920	0.243869
$813$	0	0
$814$	0	0
$815$	−3.91408	−0.137104
$816$	0	0
$817$	−10.8047	−0.378008
$818$	67.1606	2.34822
$819$	0	0
$820$	127.270	4.44446
$821$	46.4367	1.62065	0.810327	−	0.585978i	$-0.199290\pi$
$821$	46.4367	1.62065	0.810327	+	0.585978i	$0.199290\pi$
$822$	0	0
$823$	22.4195	0.781496	0.390748	−	0.920498i	$-0.372216\pi$
$823$	22.4195	0.781496	0.390748	+	0.920498i	$0.372216\pi$
$824$	83.8228	2.92011
$825$	0	0
$826$	30.6085	1.06501
$827$	30.4556	1.05904	0.529522	−	0.848296i	$-0.322371\pi$
$827$	30.4556	1.05904	0.529522	+	0.848296i	$0.322371\pi$
$828$	0	0
$829$	−43.9403	−1.52611	−0.763055	−	0.646333i	$-0.776302\pi$
$829$	−43.9403	−1.52611	−0.763055	+	0.646333i	$0.776302\pi$
$830$	108.670	3.77197
$831$	0	0
$832$	−16.5357	−0.573273
$833$	−1.18654	−0.0411112
$834$	0	0
$835$	−50.2832	−1.74012
$836$	0	0
$837$	0	0
$838$	24.5105	0.846703
$839$	2.80149	0.0967181	0.0483591	−	0.998830i	$-0.484601\pi$
$839$	2.80149	0.0967181	0.0483591	+	0.998830i	$0.484601\pi$
$840$	0	0
$841$	−26.8867	−0.927128
$842$	−76.3061	−2.62968
$843$	0	0
$844$	106.215	3.65607
$845$	29.9162	1.02915
$846$	0	0
$847$	0	0
$848$	−62.0774	−2.13175
$849$	0	0
$850$	−42.1744	−1.44657
$851$	52.7852	1.80945
$852$	0	0
$853$	−8.38258	−0.287014	−0.143507	−	0.989649i	$-0.545838\pi$
$853$	−8.38258	−0.287014	−0.143507	+	0.989649i	$0.545838\pi$
$854$	20.2300	0.692255
$855$	0	0
$856$	−43.5529	−1.48861
$857$	−53.1740	−1.81639	−0.908194	−	0.418550i	$-0.862539\pi$
$857$	−53.1740	−1.81639	−0.908194	+	0.418550i	$0.862539\pi$
$858$	0	0
$859$	39.7762	1.35715	0.678573	−	0.734533i	$-0.262599\pi$
$859$	39.7762	1.35715	0.678573	+	0.734533i	$0.262599\pi$
$860$	−65.0891	−2.21952
$861$	0	0
$862$	18.1667	0.618759
$863$	−9.11801	−0.310381	−0.155190	−	0.987885i	$-0.549599\pi$
$863$	−9.11801	−0.310381	−0.155190	+	0.987885i	$0.549599\pi$
$864$	0	0
$865$	29.5842	1.00589
$866$	3.46096	0.117608
$867$	0	0
$868$	14.2843	0.484842
$869$	0	0
$870$	0	0
$871$	−7.66590	−0.259749
$872$	48.3684	1.63796
$873$	0	0
$874$	−74.2498	−2.51154
$875$	−37.3571	−1.26290
$876$	0	0
$877$	−23.3441	−0.788274	−0.394137	−	0.919052i	$-0.628956\pi$
$877$	−23.3441	−0.788274	−0.394137	+	0.919052i	$0.628956\pi$
$878$	−9.47303	−0.319699
$879$	0	0
$880$	0	0
$881$	55.3515	1.86484	0.932419	−	0.361379i	$-0.117694\pi$
$881$	55.3515	1.86484	0.932419	+	0.361379i	$0.117694\pi$
$882$	0	0
$883$	31.8420	1.07157	0.535784	−	0.844355i	$-0.320016\pi$
$883$	31.8420	1.07157	0.535784	+	0.844355i	$0.320016\pi$
$884$	13.9775	0.470115
$885$	0	0
$886$	34.9182	1.17310
$887$	−35.2677	−1.18417	−0.592087	−	0.805874i	$-0.701696\pi$
$887$	−35.2677	−1.18417	−0.592087	+	0.805874i	$0.701696\pi$
$888$	0	0
$889$	3.17002	0.106319
$890$	−11.0828	−0.371497
$891$	0	0
$892$	104.380	3.49489
$893$	−29.9340	−1.00170
$894$	0	0
$895$	36.3306	1.21440
$896$	−1.95344	−0.0652600
$897$	0	0
$898$	39.0566	1.30334
$899$	4.34392	0.144878
$900$	0	0
$901$	7.92798	0.264119
$902$	0	0
$903$	0	0
$904$	−135.786	−4.51618
$905$	−56.2663	−1.87036
$906$	0	0
$907$	−4.60681	−0.152967	−0.0764833	−	0.997071i	$-0.524369\pi$
$907$	−4.60681	−0.152967	−0.0764833	+	0.997071i	$0.524369\pi$
$908$	−116.500	−3.86620
$909$	0	0
$910$	27.7114	0.918624
$911$	6.15440	0.203904	0.101952	−	0.994789i	$-0.467491\pi$
$911$	6.15440	0.203904	0.101952	+	0.994789i	$0.467491\pi$
$912$	0	0
$913$	0	0
$914$	−51.3285	−1.69779
$915$	0	0
$916$	−12.3398	−0.407718
$917$	7.83376	0.258694
$918$	0	0
$919$	19.9838	0.659204	0.329602	−	0.944120i	$-0.393086\pi$
$919$	19.9838	0.659204	0.329602	+	0.944120i	$0.393086\pi$
$920$	−260.153	−8.57700
$921$	0	0
$922$	42.6069	1.40318
$923$	−13.5230	−0.445115
$924$	0	0
$925$	86.5937	2.84718
$926$	−19.4925	−0.640563
$927$	0	0
$928$	14.1199	0.463509
$929$	−7.35186	−0.241207	−0.120603	−	0.992701i	$-0.538483\pi$
$929$	−7.35186	−0.241207	−0.120603	+	0.992701i	$0.538483\pi$
$930$	0	0
$931$	−3.42691	−0.112313
$932$	−68.1412	−2.23204
$933$	0	0
$934$	−7.57793	−0.247957
$935$	0	0
$936$	0	0
$937$	−22.4036	−0.731893	−0.365946	−	0.930636i	$-0.619255\pi$
$937$	−22.4036	−0.731893	−0.365946	+	0.930636i	$0.619255\pi$
$938$	8.10023	0.264482
$939$	0	0
$940$	−180.327	−5.88163
$941$	−4.86490	−0.158591	−0.0792957	−	0.996851i	$-0.525267\pi$
$941$	−4.86490	−0.158591	−0.0792957	+	0.996851i	$0.525267\pi$
$942$	0	0
$943$	−51.2971	−1.67046
$944$	109.212	3.55455
$945$	0	0
$946$	0	0
$947$	29.6187	0.962477	0.481238	−	0.876590i	$-0.340187\pi$
$947$	29.6187	0.962477	0.481238	+	0.876590i	$0.340187\pi$
$948$	0	0
$949$	−21.3408	−0.692751
$950$	−121.806	−3.95192
$951$	0	0
$952$	−8.59018	−0.278409
$953$	5.61143	0.181772	0.0908860	−	0.995861i	$-0.471030\pi$
$953$	5.61143	0.181772	0.0908860	+	0.995861i	$0.471030\pi$
$954$	0	0
$955$	−14.9317	−0.483178
$956$	−52.8165	−1.70821
$957$	0	0
$958$	−69.0744	−2.23169
$959$	7.18722	0.232087
$960$	0	0
$961$	−22.0709	−0.711965
$962$	−40.7062	−1.31242
$963$	0	0
$964$	−52.6463	−1.69562
$965$	−73.6202	−2.36992
$966$	0	0
$967$	−49.9946	−1.60772	−0.803858	−	0.594821i	$-0.797223\pi$
$967$	−49.9946	−1.60772	−0.803858	+	0.594821i	$0.797223\pi$
$968$	0	0
$969$	0	0
$970$	−78.4059	−2.51746
$971$	59.4504	1.90786	0.953928	−	0.300037i	$-0.0969990\pi$
$971$	59.4504	1.90786	0.953928	+	0.300037i	$0.0969990\pi$
$972$	0	0
$973$	3.36801	0.107974
$974$	−7.33578	−0.235054
$975$	0	0
$976$	72.1811	2.31046
$977$	20.3326	0.650497	0.325249	−	0.945629i	$-0.394552\pi$
$977$	20.3326	0.650497	0.325249	+	0.945629i	$0.394552\pi$
$978$	0	0
$979$	0	0
$980$	−20.6443	−0.659457
$981$	0	0
$982$	14.7873	0.471881
$983$	−25.0460	−0.798843	−0.399422	−	0.916767i	$-0.630789\pi$
$983$	−25.0460	−0.798843	−0.399422	+	0.916767i	$0.630789\pi$
$984$	0	0
$985$	15.0274	0.478813
$986$	−4.49144	−0.143037
$987$	0	0
$988$	40.3693	1.28432
$989$	26.2347	0.834214
$990$	0	0
$991$	10.5368	0.334713	0.167356	−	0.985896i	$-0.446477\pi$
$991$	10.5368	0.334713	0.167356	+	0.985896i	$0.446477\pi$
$992$	29.0240	0.921514
$993$	0	0
$994$	14.2892	0.453225
$995$	−35.4246	−1.12303
$996$	0	0
$997$	−34.0917	−1.07970	−0.539848	−	0.841763i	$-0.681518\pi$
$997$	−34.0917	−1.07970	−0.539848	+	0.841763i	$0.681518\pi$
$998$	91.9046	2.90919
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.dc.1.15		16
3.2	odd	2	inner	7623.2.a.dc.1.2		16
11.3	even	5		693.2.m.k.64.8	yes	32
11.4	even	5		693.2.m.k.379.8	yes	32
11.10	odd	2		7623.2.a.db.1.2		16
33.14	odd	10		693.2.m.k.64.1	✓	32
33.26	odd	10		693.2.m.k.379.1	yes	32
33.32	even	2		7623.2.a.db.1.15		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
693.2.m.k.64.1	✓	32	33.14	odd	10
693.2.m.k.64.8	yes	32	11.3	even	5
693.2.m.k.379.1	yes	32	33.26	odd	10
693.2.m.k.379.8	yes	32	11.4	even	5
7623.2.a.db.1.2		16	11.10	odd	2
7623.2.a.db.1.15		16	33.32	even	2
7623.2.a.dc.1.2		16	3.2	odd	2	inner
7623.2.a.dc.1.15		16	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)

\(f(q)\)	\(=\)	\(q+2.60390 q^{2} +4.78032 q^{4} -4.31860 q^{5} +1.00000 q^{7} +7.23969 q^{8} +O(q^{10})\)	\(q+2.60390 q^{2} +4.78032 q^{4} -4.31860 q^{5} +1.00000 q^{7} +7.23969 q^{8} -11.2452 q^{10} -2.46429 q^{13} +2.60390 q^{14} +9.29081 q^{16} -1.18654 q^{17} -3.42691 q^{19} -20.6443 q^{20} +8.32084 q^{23} +13.6503 q^{25} -6.41676 q^{26} +4.78032 q^{28} +1.45371 q^{29} +2.98816 q^{31} +9.71302 q^{32} -3.08964 q^{34} -4.31860 q^{35} +6.34373 q^{37} -8.92335 q^{38} -31.2653 q^{40} -6.16490 q^{41} +3.15289 q^{43} +21.6667 q^{46} +8.73499 q^{47} +1.00000 q^{49} +35.5440 q^{50} -11.7801 q^{52} -6.68159 q^{53} +7.23969 q^{56} +3.78533 q^{58} +11.7548 q^{59} +7.76909 q^{61} +7.78087 q^{62} +6.71015 q^{64} +10.6423 q^{65} +3.11080 q^{67} -5.67204 q^{68} -11.2452 q^{70} +5.48760 q^{71} +8.66003 q^{73} +16.5185 q^{74} -16.3817 q^{76} -8.83563 q^{79} -40.1233 q^{80} -16.0528 q^{82} -9.66363 q^{83} +5.12419 q^{85} +8.20982 q^{86} +0.985558 q^{89} -2.46429 q^{91} +39.7763 q^{92} +22.7451 q^{94} +14.7995 q^{95} +6.97238 q^{97} +2.60390 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 20 q^{4} + 16 q^{7}+O(q^{10}) \) 16 * q + 20 * q^4 + 16 * q^7	\( 16 q + 20 q^{4} + 16 q^{7} - 6 q^{10} + 32 q^{16} + 10 q^{19} + 44 q^{25} + 20 q^{28} + 30 q^{31} + 12 q^{34} + 38 q^{37} - 68 q^{40} + 16 q^{43} + 80 q^{46} + 16 q^{49} + 2 q^{52} + 18 q^{58} - 28 q^{61} + 34 q^{64} + 52 q^{67} - 6 q^{70} - 14 q^{73} - 14 q^{76} + 54 q^{79} + 64 q^{82} + 30 q^{85} - 60 q^{94} + 108 q^{97}+O(q^{100}) \) 16 * q + 20 * q^4 + 16 * q^7 - 6 * q^10 + 32 * q^16 + 10 * q^19 + 44 * q^25 + 20 * q^28 + 30 * q^31 + 12 * q^34 + 38 * q^37 - 68 * q^40 + 16 * q^43 + 80 * q^46 + 16 * q^49 + 2 * q^52 + 18 * q^58 - 28 * q^61 + 34 * q^64 + 52 * q^67 - 6 * q^70 - 14 * q^73 - 14 * q^76 + 54 * q^79 + 64 * q^82 + 30 * q^85 - 60 * q^94 + 108 * q^97

Introduction

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