LMFDB - Embedded newform 7623.2.a.cv.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:	$1$
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			1.3
Root			$-0.716111$ 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)

$f(q)$	$=$	$q-0.716111 q^{2} -1.48718 q^{4} -1.54336 q^{5} +1.00000 q^{7} +2.49721 q^{8} +O(q^{10})$	$q-0.716111 q^{2} -1.48718 q^{4} -1.54336 q^{5} +1.00000 q^{7} +2.49721 q^{8} +1.10522 q^{10} +0.301097 q^{13} -0.716111 q^{14} +1.18609 q^{16} +0.0579114 q^{17} -2.70741 q^{19} +2.29526 q^{20} +1.50757 q^{23} -2.61803 q^{25} -0.215619 q^{26} -1.48718 q^{28} -2.04096 q^{29} +4.94348 q^{31} -5.84379 q^{32} -0.0414710 q^{34} -1.54336 q^{35} -4.43195 q^{37} +1.93881 q^{38} -3.85410 q^{40} +2.69916 q^{41} -1.09671 q^{43} -1.07959 q^{46} +7.68803 q^{47} +1.00000 q^{49} +1.87480 q^{50} -0.447786 q^{52} +7.18521 q^{53} +2.49721 q^{56} +1.46155 q^{58} +0.0782427 q^{59} -7.27547 q^{61} -3.54008 q^{62} +1.81263 q^{64} -0.464701 q^{65} -6.09996 q^{67} -0.0861249 q^{68} +1.10522 q^{70} -3.72280 q^{71} -1.61278 q^{73} +3.17377 q^{74} +4.02642 q^{76} +8.86866 q^{79} -1.83056 q^{80} -1.93290 q^{82} -5.03866 q^{83} -0.0893782 q^{85} +0.785366 q^{86} -17.6900 q^{89} +0.301097 q^{91} -2.24204 q^{92} -5.50548 q^{94} +4.17852 q^{95} +7.95804 q^{97} -0.716111 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.716111	−0.506367	−0.253184	−	0.967418i	$-0.581478\pi$
$2$	−0.716111	−0.506367	−0.253184	+	0.967418i	$0.581478\pi$
$3$	0	0
$3$	0	0
$4$	−1.48718	−0.743592
$5$	−1.54336	−0.690212	−0.345106	−	0.938564i	$-0.612157\pi$
$5$	−1.54336	−0.690212	−0.345106	+	0.938564i	$0.612157\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.49721	0.882898
$9$	0	0
$10$	1.10522	0.349501
$11$	0	0
$11$	0	0
$12$	0	0
$13$	0.301097	0.0835092	0.0417546	−	0.999128i	$-0.486705\pi$
$13$	0.301097	0.0835092	0.0417546	+	0.999128i	$0.486705\pi$
$14$	−0.716111	−0.191389
$15$	0	0
$16$	1.18609	0.296522
$17$	0.0579114	0.0140456	0.00702279	−	0.999975i	$-0.497765\pi$
$17$	0.0579114	0.0140456	0.00702279	+	0.999975i	$0.497765\pi$
$18$	0	0
$19$	−2.70741	−0.621123	−0.310561	−	0.950553i	$-0.600517\pi$
$19$	−2.70741	−0.621123	−0.310561	+	0.950553i	$0.600517\pi$
$20$	2.29526	0.513237
$21$	0	0
$22$	0	0
$23$	1.50757	0.314350	0.157175	−	0.987571i	$-0.449761\pi$
$23$	1.50757	0.314350	0.157175	+	0.987571i	$0.449761\pi$
$24$	0	0
$25$	−2.61803	−0.523607
$26$	−0.215619	−0.0422863
$27$	0	0
$28$	−1.48718	−0.281052
$29$	−2.04096	−0.378997	−0.189498	−	0.981881i	$-0.560686\pi$
$29$	−2.04096	−0.378997	−0.189498	+	0.981881i	$0.560686\pi$
$30$	0	0
$31$	4.94348	0.887875	0.443938	−	0.896058i	$-0.353581\pi$
$31$	4.94348	0.887875	0.443938	+	0.896058i	$0.353581\pi$
$32$	−5.84379	−1.03305
$33$	0	0
$34$	−0.0414710	−0.00711222
$35$	−1.54336	−0.260876
$36$	0	0
$37$	−4.43195	−0.728607	−0.364304	−	0.931280i	$-0.618693\pi$
$37$	−4.43195	−0.728607	−0.364304	+	0.931280i	$0.618693\pi$
$38$	1.93881	0.314516
$39$	0	0
$40$	−3.85410	−0.609387
$41$	2.69916	0.421538	0.210769	−	0.977536i	$-0.432403\pi$
$41$	2.69916	0.421538	0.210769	+	0.977536i	$0.432403\pi$
$42$	0	0
$43$	−1.09671	−0.167247	−0.0836233	−	0.996497i	$-0.526649\pi$
$43$	−1.09671	−0.167247	−0.0836233	+	0.996497i	$0.526649\pi$
$44$	0	0
$45$	0	0
$46$	−1.07959	−0.159177
$47$	7.68803	1.12141	0.560707	−	0.828014i	$-0.310529\pi$
$47$	7.68803	1.12141	0.560707	+	0.828014i	$0.310529\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	1.87480	0.265137
$51$	0	0
$52$	−0.447786	−0.0620968
$53$	7.18521	0.986964	0.493482	−	0.869756i	$-0.335724\pi$
$53$	7.18521	0.986964	0.493482	+	0.869756i	$0.335724\pi$
$54$	0	0
$55$	0	0
$56$	2.49721	0.333704
$57$	0	0
$58$	1.46155	0.191911
$59$	0.0782427	0.0101863	0.00509317	−	0.999987i	$-0.498379\pi$
$59$	0.0782427	0.0101863	0.00509317	+	0.999987i	$0.498379\pi$
$60$	0	0
$61$	−7.27547	−0.931528	−0.465764	−	0.884909i	$-0.654220\pi$
$61$	−7.27547	−0.931528	−0.465764	+	0.884909i	$0.654220\pi$
$62$	−3.54008	−0.449591
$63$	0	0
$64$	1.81263	0.226579
$65$	−0.464701	−0.0576391
$66$	0	0
$67$	−6.09996	−0.745229	−0.372614	−	0.927986i	$-0.621539\pi$
$67$	−6.09996	−0.745229	−0.372614	+	0.927986i	$0.621539\pi$
$68$	−0.0861249	−0.0104442
$69$	0	0
$70$	1.10522	0.132099
$71$	−3.72280	−0.441815	−0.220908	−	0.975295i	$-0.570902\pi$
$71$	−3.72280	−0.441815	−0.220908	+	0.975295i	$0.570902\pi$
$72$	0	0
$73$	−1.61278	−0.188761	−0.0943805	−	0.995536i	$-0.530087\pi$
$73$	−1.61278	−0.188761	−0.0943805	+	0.995536i	$0.530087\pi$
$74$	3.17377	0.368943
$75$	0	0
$76$	4.02642	0.461862
$77$	0	0
$78$	0	0
$79$	8.86866	0.997802	0.498901	−	0.866659i	$-0.333737\pi$
$79$	8.86866	0.997802	0.498901	+	0.866659i	$0.333737\pi$
$80$	−1.83056	−0.204663
$81$	0	0
$82$	−1.93290	−0.213453
$83$	−5.03866	−0.553065	−0.276533	−	0.961005i	$-0.589185\pi$
$83$	−5.03866	−0.553065	−0.276533	+	0.961005i	$0.589185\pi$
$84$	0	0
$85$	−0.0893782	−0.00969443
$86$	0.785366	0.0846882
$87$	0	0
$88$	0	0
$89$	−17.6900	−1.87514	−0.937569	−	0.347800i	$-0.886929\pi$
$89$	−17.6900	−1.87514	−0.937569	+	0.347800i	$0.886929\pi$
$90$	0	0
$91$	0.301097	0.0315635
$92$	−2.24204	−0.233748
$93$	0	0
$94$	−5.50548	−0.567847
$95$	4.17852	0.428707
$96$	0	0
$97$	7.95804	0.808017	0.404008	−	0.914755i	$-0.367617\pi$
$97$	7.95804	0.808017	0.404008	+	0.914755i	$0.367617\pi$
$98$	−0.716111	−0.0723382
$99$	0	0
$100$	3.89350	0.389350
$101$	7.07407	0.703896	0.351948	−	0.936020i	$-0.385519\pi$
$101$	7.07407	0.703896	0.351948	+	0.936020i	$0.385519\pi$
$102$	0	0
$103$	−12.0828	−1.19056	−0.595279	−	0.803519i	$-0.702958\pi$
$103$	−12.0828	−1.19056	−0.595279	+	0.803519i	$0.702958\pi$
$104$	0.751902	0.0737301
$105$	0	0
$106$	−5.14541	−0.499766
$107$	−2.30371	−0.222708	−0.111354	−	0.993781i	$-0.535519\pi$
$107$	−2.30371	−0.222708	−0.111354	+	0.993781i	$0.535519\pi$
$108$	0	0
$109$	13.1783	1.26225	0.631125	−	0.775682i	$-0.282594\pi$
$109$	13.1783	1.26225	0.631125	+	0.775682i	$0.282594\pi$
$110$	0	0
$111$	0	0
$112$	1.18609	0.112075
$113$	14.0149	1.31841	0.659204	−	0.751964i	$-0.270893\pi$
$113$	14.0149	1.31841	0.659204	+	0.751964i	$0.270893\pi$
$114$	0	0
$115$	−2.32673	−0.216968
$116$	3.03528	0.281819
$117$	0	0
$118$	−0.0560305	−0.00515803
$119$	0.0579114	0.00530873
$120$	0	0
$121$	0	0
$122$	5.21004	0.471695
$123$	0	0
$124$	−7.35187	−0.660217
$125$	11.7574	1.05161
$126$	0	0
$127$	−9.58389	−0.850433	−0.425217	−	0.905092i	$-0.639802\pi$
$127$	−9.58389	−0.850433	−0.425217	+	0.905092i	$0.639802\pi$
$128$	10.3895	0.918315
$129$	0	0
$130$	0.332778	0.0291865
$131$	16.0393	1.40136	0.700679	−	0.713477i	$-0.252881\pi$
$131$	16.0393	1.40136	0.700679	+	0.713477i	$0.252881\pi$
$132$	0	0
$133$	−2.70741	−0.234762
$134$	4.36825	0.377359
$135$	0	0
$136$	0.144617	0.0124008
$137$	14.8017	1.26459	0.632295	−	0.774727i	$-0.282113\pi$
$137$	14.8017	1.26459	0.632295	+	0.774727i	$0.282113\pi$
$138$	0	0
$139$	−7.71799	−0.654632	−0.327316	−	0.944915i	$-0.606144\pi$
$139$	−7.71799	−0.654632	−0.327316	+	0.944915i	$0.606144\pi$
$140$	2.29526	0.193985
$141$	0	0
$142$	2.66594	0.223721
$143$	0	0
$144$	0	0
$145$	3.14994	0.261588
$146$	1.15493	0.0955824
$147$	0	0
$148$	6.59112	0.541787
$149$	5.72976	0.469400	0.234700	−	0.972068i	$-0.424589\pi$
$149$	5.72976	0.469400	0.234700	+	0.972068i	$0.424589\pi$
$150$	0	0
$151$	2.98942	0.243275	0.121638	−	0.992575i	$-0.461185\pi$
$151$	2.98942	0.243275	0.121638	+	0.992575i	$0.461185\pi$
$152$	−6.76098	−0.548388
$153$	0	0
$154$	0	0
$155$	−7.62958	−0.612823
$156$	0	0
$157$	−11.8036	−0.942032	−0.471016	−	0.882125i	$-0.656113\pi$
$157$	−11.8036	−0.942032	−0.471016	+	0.882125i	$0.656113\pi$
$158$	−6.35095	−0.505254
$159$	0	0
$160$	9.01909	0.713022
$161$	1.50757	0.118813
$162$	0	0
$163$	−6.09385	−0.477307	−0.238653	−	0.971105i	$-0.576706\pi$
$163$	−6.09385	−0.477307	−0.238653	+	0.971105i	$0.576706\pi$
$164$	−4.01415	−0.313452
$165$	0	0
$166$	3.60824	0.280054
$167$	22.8761	1.77021	0.885105	−	0.465392i	$-0.154087\pi$
$167$	22.8761	1.77021	0.885105	+	0.465392i	$0.154087\pi$
$168$	0	0
$169$	−12.9093	−0.993026
$170$	0.0640048	0.00490894
$171$	0	0
$172$	1.63101	0.124363
$173$	−25.0524	−1.90470	−0.952349	−	0.305011i	$-0.901340\pi$
$173$	−25.0524	−1.90470	−0.952349	+	0.305011i	$0.901340\pi$
$174$	0	0
$175$	−2.61803	−0.197905
$176$	0	0
$177$	0	0
$178$	12.6680	0.949508
$179$	−1.98705	−0.148519	−0.0742595	−	0.997239i	$-0.523659\pi$
$179$	−1.98705	−0.148519	−0.0742595	+	0.997239i	$0.523659\pi$
$180$	0	0
$181$	7.71750	0.573638	0.286819	−	0.957985i	$-0.407402\pi$
$181$	7.71750	0.573638	0.286819	+	0.957985i	$0.407402\pi$
$182$	−0.215619	−0.0159827
$183$	0	0
$184$	3.76472	0.277539
$185$	6.84010	0.502894
$186$	0	0
$187$	0	0
$188$	−11.4335	−0.833875
$189$	0	0
$190$	−2.99228	−0.217083
$191$	−16.2668	−1.17702	−0.588511	−	0.808489i	$-0.700286\pi$
$191$	−16.2668	−1.17702	−0.588511	+	0.808489i	$0.700286\pi$
$192$	0	0
$193$	−4.77244	−0.343528	−0.171764	−	0.985138i	$-0.554947\pi$
$193$	−4.77244	−0.343528	−0.171764	+	0.985138i	$0.554947\pi$
$194$	−5.69884	−0.409153
$195$	0	0
$196$	−1.48718	−0.106227
$197$	22.4382	1.59866	0.799329	−	0.600893i	$-0.205188\pi$
$197$	22.4382	1.59866	0.799329	+	0.600893i	$0.205188\pi$
$198$	0	0
$199$	−11.5985	−0.822193	−0.411096	−	0.911592i	$-0.634854\pi$
$199$	−11.5985	−0.822193	−0.411096	+	0.911592i	$0.634854\pi$
$200$	−6.53779	−0.462291
$201$	0	0
$202$	−5.06582	−0.356430
$203$	−2.04096	−0.143247
$204$	0	0
$205$	−4.16578	−0.290951
$206$	8.65266	0.602859
$207$	0	0
$208$	0.357127	0.0247623
$209$	0	0
$210$	0	0
$211$	16.5301	1.13798	0.568991	−	0.822344i	$-0.307334\pi$
$211$	16.5301	1.13798	0.568991	+	0.822344i	$0.307334\pi$
$212$	−10.6857	−0.733899
$213$	0	0
$214$	1.64972	0.112772
$215$	1.69262	0.115436
$216$	0	0
$217$	4.94348	0.335585
$218$	−9.43710	−0.639161
$219$	0	0
$220$	0	0
$221$	0.0174369	0.00117293
$222$	0	0
$223$	−28.3932	−1.90135	−0.950676	−	0.310186i	$-0.899609\pi$
$223$	−28.3932	−1.90135	−0.950676	+	0.310186i	$0.899609\pi$
$224$	−5.84379	−0.390455
$225$	0	0
$226$	−10.0362	−0.667599
$227$	21.9479	1.45673	0.728366	−	0.685188i	$-0.240280\pi$
$227$	21.9479	1.45673	0.728366	+	0.685188i	$0.240280\pi$
$228$	0	0
$229$	11.1717	0.738249	0.369124	−	0.929380i	$-0.379658\pi$
$229$	11.1717	0.738249	0.369124	+	0.929380i	$0.379658\pi$
$230$	1.66620	0.109866
$231$	0	0
$232$	−5.09671	−0.334615
$233$	−15.1572	−0.992981	−0.496490	−	0.868042i	$-0.665378\pi$
$233$	−15.1572	−0.992981	−0.496490	+	0.868042i	$0.665378\pi$
$234$	0	0
$235$	−11.8654	−0.774014
$236$	−0.116361	−0.00757448
$237$	0	0
$238$	−0.0414710	−0.00268817
$239$	−13.8246	−0.894238	−0.447119	−	0.894475i	$-0.647550\pi$
$239$	−13.8246	−0.894238	−0.447119	+	0.894475i	$0.647550\pi$
$240$	0	0
$241$	14.6432	0.943250	0.471625	−	0.881799i	$-0.343668\pi$
$241$	14.6432	0.943250	0.471625	+	0.881799i	$0.343668\pi$
$242$	0	0
$243$	0	0
$244$	10.8200	0.692677
$245$	−1.54336	−0.0986018
$246$	0	0
$247$	−0.815193	−0.0518695
$248$	12.3449	0.783903
$249$	0	0
$250$	−8.41959	−0.532502
$251$	19.1636	1.20959	0.604797	−	0.796380i	$-0.293254\pi$
$251$	19.1636	1.20959	0.604797	+	0.796380i	$0.293254\pi$
$252$	0	0
$253$	0	0
$254$	6.86313	0.430631
$255$	0	0
$256$	−11.0653	−0.691583
$257$	14.3335	0.894101	0.447050	−	0.894509i	$-0.352475\pi$
$257$	14.3335	0.894101	0.447050	+	0.894509i	$0.352475\pi$
$258$	0	0
$259$	−4.43195	−0.275388
$260$	0.691097	0.0428600
$261$	0	0
$262$	−11.4859	−0.709602
$263$	−27.3223	−1.68476	−0.842382	−	0.538881i	$-0.818847\pi$
$263$	−27.3223	−1.68476	−0.842382	+	0.538881i	$0.818847\pi$
$264$	0	0
$265$	−11.0894	−0.681215
$266$	1.93881	0.118876
$267$	0	0
$268$	9.07177	0.554146
$269$	−23.4293	−1.42851	−0.714256	−	0.699885i	$-0.753235\pi$
$269$	−23.4293	−1.42851	−0.714256	+	0.699885i	$0.753235\pi$
$270$	0	0
$271$	−26.1574	−1.58895	−0.794475	−	0.607297i	$-0.792254\pi$
$271$	−26.1574	−1.58895	−0.794475	+	0.607297i	$0.792254\pi$
$272$	0.0686880	0.00416482
$273$	0	0
$274$	−10.5996	−0.640347
$275$	0	0
$276$	0	0
$277$	5.06257	0.304180	0.152090	−	0.988367i	$-0.451400\pi$
$277$	5.06257	0.304180	0.152090	+	0.988367i	$0.451400\pi$
$278$	5.52694	0.331484
$279$	0	0
$280$	−3.85410	−0.230327
$281$	9.97449	0.595028	0.297514	−	0.954717i	$-0.403842\pi$
$281$	9.97449	0.595028	0.297514	+	0.954717i	$0.403842\pi$
$282$	0	0
$283$	24.6792	1.46703	0.733514	−	0.679675i	$-0.237879\pi$
$283$	24.6792	1.46703	0.733514	+	0.679675i	$0.237879\pi$
$284$	5.53650	0.328531
$285$	0	0
$286$	0	0
$287$	2.69916	0.159326
$288$	0	0
$289$	−16.9966	−0.999803
$290$	−2.25571	−0.132460
$291$	0	0
$292$	2.39849	0.140361
$293$	6.70505	0.391713	0.195857	−	0.980633i	$-0.437251\pi$
$293$	6.70505	0.391713	0.195857	+	0.980633i	$0.437251\pi$
$294$	0	0
$295$	−0.120757	−0.00703074
$296$	−11.0675	−0.643286
$297$	0	0
$298$	−4.10315	−0.237689
$299$	0.453925	0.0262511
$300$	0	0
$301$	−1.09671	−0.0632133
$302$	−2.14076	−0.123187
$303$	0	0
$304$	−3.21123	−0.184177
$305$	11.2287	0.642952
$306$	0	0
$307$	13.4156	0.765670	0.382835	−	0.923817i	$-0.374948\pi$
$307$	13.4156	0.765670	0.382835	+	0.923817i	$0.374948\pi$
$308$	0	0
$309$	0	0
$310$	5.46363	0.310313
$311$	−30.5081	−1.72995	−0.864977	−	0.501811i	$-0.832667\pi$
$311$	−30.5081	−1.72995	−0.864977	+	0.501811i	$0.832667\pi$
$312$	0	0
$313$	−18.9621	−1.07180	−0.535901	−	0.844281i	$-0.680028\pi$
$313$	−18.9621	−1.07180	−0.535901	+	0.844281i	$0.680028\pi$
$314$	8.45271	0.477014
$315$	0	0
$316$	−13.1893	−0.741958
$317$	−0.834594	−0.0468755	−0.0234377	−	0.999725i	$-0.507461\pi$
$317$	−0.834594	−0.0468755	−0.0234377	+	0.999725i	$0.507461\pi$
$318$	0	0
$319$	0	0
$320$	−2.79755	−0.156388
$321$	0	0
$322$	−1.07959	−0.0601631
$323$	−0.156790	−0.00872403
$324$	0	0
$325$	−0.788281	−0.0437260
$326$	4.36387	0.241692
$327$	0	0
$328$	6.74037	0.372175
$329$	7.68803	0.423855
$330$	0	0
$331$	−14.7431	−0.810356	−0.405178	−	0.914238i	$-0.632790\pi$
$331$	−14.7431	−0.810356	−0.405178	+	0.914238i	$0.632790\pi$
$332$	7.49342	0.411255
$333$	0	0
$334$	−16.3819	−0.896376
$335$	9.41445	0.514366
$336$	0	0
$337$	−32.3874	−1.76426	−0.882128	−	0.471010i	$-0.843889\pi$
$337$	−32.3874	−1.76426	−0.882128	+	0.471010i	$0.843889\pi$
$338$	9.24452	0.502836
$339$	0	0
$340$	0.132922	0.00720870
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	−2.73872	−0.147662
$345$	0	0
$346$	17.9403	0.964476
$347$	−12.9058	−0.692822	−0.346411	−	0.938083i	$-0.612600\pi$
$347$	−12.9058	−0.692822	−0.346411	+	0.938083i	$0.612600\pi$
$348$	0	0
$349$	−14.4321	−0.772535	−0.386267	−	0.922387i	$-0.626236\pi$
$349$	−14.4321	−0.772535	−0.386267	+	0.922387i	$0.626236\pi$
$350$	1.87480	0.100212
$351$	0	0
$352$	0	0
$353$	13.2973	0.707745	0.353872	−	0.935294i	$-0.384865\pi$
$353$	13.2973	0.707745	0.353872	+	0.935294i	$0.384865\pi$
$354$	0	0
$355$	5.74563	0.304947
$356$	26.3083	1.39434
$357$	0	0
$358$	1.42295	0.0752051
$359$	9.74033	0.514075	0.257038	−	0.966401i	$-0.417254\pi$
$359$	9.74033	0.514075	0.257038	+	0.966401i	$0.417254\pi$
$360$	0	0
$361$	−11.6699	−0.614206
$362$	−5.52659	−0.290471
$363$	0	0
$364$	−0.447786	−0.0234704
$365$	2.48910	0.130285
$366$	0	0
$367$	−16.5818	−0.865564	−0.432782	−	0.901499i	$-0.642468\pi$
$367$	−16.5818	−0.865564	−0.432782	+	0.901499i	$0.642468\pi$
$368$	1.78811	0.0932118
$369$	0	0
$370$	−4.89827	−0.254649
$371$	7.18521	0.373037
$372$	0	0
$373$	−5.52329	−0.285985	−0.142993	−	0.989724i	$-0.545673\pi$
$373$	−5.52329	−0.285985	−0.142993	+	0.989724i	$0.545673\pi$
$374$	0	0
$375$	0	0
$376$	19.1986	0.990094
$377$	−0.614526	−0.0316497
$378$	0	0
$379$	−25.2684	−1.29795	−0.648975	−	0.760810i	$-0.724802\pi$
$379$	−25.2684	−1.29795	−0.648975	+	0.760810i	$0.724802\pi$
$380$	−6.21423	−0.318783
$381$	0	0
$382$	11.6488	0.596005
$383$	−9.49680	−0.485264	−0.242632	−	0.970118i	$-0.578011\pi$
$383$	−9.49680	−0.485264	−0.242632	+	0.970118i	$0.578011\pi$
$384$	0	0
$385$	0	0
$386$	3.41760	0.173951
$387$	0	0
$388$	−11.8351	−0.600835
$389$	−5.51635	−0.279690	−0.139845	−	0.990173i	$-0.544660\pi$
$389$	−5.51635	−0.279690	−0.139845	+	0.990173i	$0.544660\pi$
$390$	0	0
$391$	0.0873055	0.00441523
$392$	2.49721	0.126128
$393$	0	0
$394$	−16.0683	−0.809508
$395$	−13.6876	−0.688696
$396$	0	0
$397$	−24.0365	−1.20635	−0.603177	−	0.797607i	$-0.706099\pi$
$397$	−24.0365	−1.20635	−0.603177	+	0.797607i	$0.706099\pi$
$398$	8.30578	0.416331
$399$	0	0
$400$	−3.10522	−0.155261
$401$	−26.7585	−1.33625	−0.668127	−	0.744047i	$-0.732904\pi$
$401$	−26.7585	−1.33625	−0.668127	+	0.744047i	$0.732904\pi$
$402$	0	0
$403$	1.48847	0.0741458
$404$	−10.5204	−0.523412
$405$	0	0
$406$	1.46155	0.0725357
$407$	0	0
$408$	0	0
$409$	2.68832	0.132929	0.0664645	−	0.997789i	$-0.478828\pi$
$409$	2.68832	0.132929	0.0664645	+	0.997789i	$0.478828\pi$
$410$	2.98316	0.147328
$411$	0	0
$412$	17.9694	0.885289
$413$	0.0782427	0.00385007
$414$	0	0
$415$	7.77648	0.381733
$416$	−1.75955	−0.0862689
$417$	0	0
$418$	0	0
$419$	25.2419	1.23315	0.616573	−	0.787297i	$-0.288520\pi$
$419$	25.2419	1.23315	0.616573	+	0.787297i	$0.288520\pi$
$420$	0	0
$421$	−9.42461	−0.459328	−0.229664	−	0.973270i	$-0.573763\pi$
$421$	−9.42461	−0.459328	−0.229664	+	0.973270i	$0.573763\pi$
$422$	−11.8374	−0.576236
$423$	0	0
$424$	17.9430	0.871389
$425$	−0.151614	−0.00735436
$426$	0	0
$427$	−7.27547	−0.352084
$428$	3.42605	0.165604
$429$	0	0
$430$	−1.21210	−0.0584529
$431$	−27.6107	−1.32996	−0.664981	−	0.746860i	$-0.731560\pi$
$431$	−27.6107	−1.32996	−0.664981	+	0.746860i	$0.731560\pi$
$432$	0	0
$433$	−36.2973	−1.74434	−0.872169	−	0.489205i	$-0.837287\pi$
$433$	−36.2973	−1.74434	−0.872169	+	0.489205i	$0.837287\pi$
$434$	−3.54008	−0.169929
$435$	0	0
$436$	−19.5985	−0.938599
$437$	−4.08162	−0.195250
$438$	0	0
$439$	−35.8022	−1.70875	−0.854374	−	0.519659i	$-0.826059\pi$
$439$	−35.8022	−1.70875	−0.854374	+	0.519659i	$0.826059\pi$
$440$	0	0
$441$	0	0
$442$	−0.0124868	−0.000593936 0
$443$	−5.99164	−0.284672	−0.142336	−	0.989818i	$-0.545461\pi$
$443$	−5.99164	−0.284672	−0.142336	+	0.989818i	$0.545461\pi$
$444$	0	0
$445$	27.3021	1.29424
$446$	20.3327	0.962782
$447$	0	0
$448$	1.81263	0.0856388
$449$	3.76704	0.177778	0.0888889	−	0.996042i	$-0.471668\pi$
$449$	3.76704	0.177778	0.0888889	+	0.996042i	$0.471668\pi$
$450$	0	0
$451$	0	0
$452$	−20.8427	−0.980359
$453$	0	0
$454$	−15.7171	−0.737641
$455$	−0.464701	−0.0217855
$456$	0	0
$457$	−16.0048	−0.748672	−0.374336	−	0.927293i	$-0.622129\pi$
$457$	−16.0048	−0.748672	−0.374336	+	0.927293i	$0.622129\pi$
$458$	−8.00020	−0.373825
$459$	0	0
$460$	3.46027	0.161336
$461$	−15.9848	−0.744486	−0.372243	−	0.928135i	$-0.621411\pi$
$461$	−15.9848	−0.744486	−0.372243	+	0.928135i	$0.621411\pi$
$462$	0	0
$463$	−9.13966	−0.424756	−0.212378	−	0.977188i	$-0.568121\pi$
$463$	−9.13966	−0.424756	−0.212378	+	0.977188i	$0.568121\pi$
$464$	−2.42076	−0.112381
$465$	0	0
$466$	10.8542	0.502813
$467$	−1.90358	−0.0880874	−0.0440437	−	0.999030i	$-0.514024\pi$
$467$	−1.90358	−0.0880874	−0.0440437	+	0.999030i	$0.514024\pi$
$468$	0	0
$469$	−6.09996	−0.281670
$470$	8.49695	0.391935
$471$	0	0
$472$	0.195389	0.00899349
$473$	0	0
$474$	0	0
$475$	7.08810	0.325224
$476$	−0.0861249	−0.00394753
$477$	0	0
$478$	9.89994	0.452813
$479$	−32.6386	−1.49130	−0.745648	−	0.666340i	$-0.767860\pi$
$479$	−32.6386	−1.49130	−0.745648	+	0.666340i	$0.767860\pi$
$480$	0	0
$481$	−1.33444	−0.0608454
$482$	−10.4861	−0.477631
$483$	0	0
$484$	0	0
$485$	−12.2821	−0.557703
$486$	0	0
$487$	−31.2014	−1.41387	−0.706936	−	0.707277i	$-0.749923\pi$
$487$	−31.2014	−1.41387	−0.706936	+	0.707277i	$0.749923\pi$
$488$	−18.1684	−0.822444
$489$	0	0
$490$	1.10522	0.0499287
$491$	−24.4486	−1.10335	−0.551676	−	0.834059i	$-0.686011\pi$
$491$	−24.4486	−1.10335	−0.551676	+	0.834059i	$0.686011\pi$
$492$	0	0
$493$	−0.118195	−0.00532323
$494$	0.583769	0.0262650
$495$	0	0
$496$	5.86340	0.263275
$497$	−3.72280	−0.166991
$498$	0	0
$499$	14.9389	0.668757	0.334379	−	0.942439i	$-0.391474\pi$
$499$	14.9389	0.668757	0.334379	+	0.942439i	$0.391474\pi$
$500$	−17.4854	−0.781971
$501$	0	0
$502$	−13.7233	−0.612499
$503$	24.2545	1.08145	0.540727	−	0.841198i	$-0.318149\pi$
$503$	24.2545	1.08145	0.540727	+	0.841198i	$0.318149\pi$
$504$	0	0
$505$	−10.9178	−0.485838
$506$	0	0
$507$	0	0
$508$	14.2530	0.632376
$509$	10.5548	0.467833	0.233916	−	0.972257i	$-0.424846\pi$
$509$	10.5548	0.467833	0.233916	+	0.972257i	$0.424846\pi$
$510$	0	0
$511$	−1.61278	−0.0713450
$512$	−12.8551	−0.568120
$513$	0	0
$514$	−10.2644	−0.452743
$515$	18.6482	0.821738
$516$	0	0
$517$	0	0
$518$	3.17377	0.139447
$519$	0	0
$520$	−1.16046	−0.0508894
$521$	21.9932	0.963540	0.481770	−	0.876298i	$-0.339994\pi$
$521$	21.9932	0.963540	0.481770	+	0.876298i	$0.339994\pi$
$522$	0	0
$523$	−5.42545	−0.237238	−0.118619	−	0.992940i	$-0.537847\pi$
$523$	−5.42545	−0.237238	−0.118619	+	0.992940i	$0.537847\pi$
$524$	−23.8534	−1.04204
$525$	0	0
$526$	19.5658	0.853109
$527$	0.286284	0.0124707
$528$	0	0
$529$	−20.7272	−0.901184
$530$	7.94123	0.344945
$531$	0	0
$532$	4.02642	0.174568
$533$	0.812708	0.0352023
$534$	0	0
$535$	3.55546	0.153716
$536$	−15.2329	−0.657961
$537$	0	0
$538$	16.7780	0.723351
$539$	0	0
$540$	0	0
$541$	−0.637614	−0.0274132	−0.0137066	−	0.999906i	$-0.504363\pi$
$541$	−0.637614	−0.0274132	−0.0137066	+	0.999906i	$0.504363\pi$
$542$	18.7316	0.804592
$543$	0	0
$544$	−0.338422	−0.0145097
$545$	−20.3388	−0.871220
$546$	0	0
$547$	2.50358	0.107045	0.0535226	−	0.998567i	$-0.482955\pi$
$547$	2.50358	0.107045	0.0535226	+	0.998567i	$0.482955\pi$
$548$	−22.0128	−0.940340
$549$	0	0
$550$	0	0
$551$	5.52572	0.235404
$552$	0	0
$553$	8.86866	0.377134
$554$	−3.62536	−0.154027
$555$	0	0
$556$	11.4781	0.486779
$557$	4.55740	0.193103	0.0965516	−	0.995328i	$-0.469219\pi$
$557$	4.55740	0.193103	0.0965516	+	0.995328i	$0.469219\pi$
$558$	0	0
$559$	−0.330216	−0.0139666
$560$	−1.83056	−0.0773554
$561$	0	0
$562$	−7.14285	−0.301303
$563$	8.98017	0.378469	0.189235	−	0.981932i	$-0.439399\pi$
$563$	8.98017	0.378469	0.189235	+	0.981932i	$0.439399\pi$
$564$	0	0
$565$	−21.6300	−0.909982
$566$	−17.6731	−0.742854
$567$	0	0
$568$	−9.29663	−0.390078
$569$	−35.8298	−1.50206	−0.751032	−	0.660266i	$-0.770444\pi$
$569$	−35.8298	−1.50206	−0.751032	+	0.660266i	$0.770444\pi$
$570$	0	0
$571$	18.5090	0.774576	0.387288	−	0.921959i	$-0.373412\pi$
$571$	18.5090	0.774576	0.387288	+	0.921959i	$0.373412\pi$
$572$	0	0
$573$	0	0
$574$	−1.93290	−0.0806776
$575$	−3.94687	−0.164596
$576$	0	0
$577$	−7.02086	−0.292282	−0.146141	−	0.989264i	$-0.546685\pi$
$577$	−7.02086	−0.292282	−0.146141	+	0.989264i	$0.546685\pi$
$578$	12.1715	0.506267
$579$	0	0
$580$	−4.68454	−0.194515
$581$	−5.03866	−0.209039
$582$	0	0
$583$	0	0
$584$	−4.02744	−0.166657
$585$	0	0
$586$	−4.80156	−0.198351
$587$	−14.7290	−0.607931	−0.303965	−	0.952683i	$-0.598311\pi$
$587$	−14.7290	−0.607931	−0.303965	+	0.952683i	$0.598311\pi$
$588$	0	0
$589$	−13.3840	−0.551480
$590$	0.0864753	0.00356013
$591$	0	0
$592$	−5.25668	−0.216048
$593$	−5.91081	−0.242728	−0.121364	−	0.992608i	$-0.538727\pi$
$593$	−5.91081	−0.242728	−0.121364	+	0.992608i	$0.538727\pi$
$594$	0	0
$595$	−0.0893782	−0.00366415
$596$	−8.52121	−0.349042
$597$	0	0
$598$	−0.325061	−0.0132927
$599$	44.8922	1.83424	0.917122	−	0.398606i	$-0.130506\pi$
$599$	44.8922	1.83424	0.917122	+	0.398606i	$0.130506\pi$
$600$	0	0
$601$	35.9081	1.46472	0.732361	−	0.680916i	$-0.238418\pi$
$601$	35.9081	1.46472	0.732361	+	0.680916i	$0.238418\pi$
$602$	0.785366	0.0320091
$603$	0	0
$604$	−4.44582	−0.180898
$605$	0	0
$606$	0	0
$607$	16.1454	0.655320	0.327660	−	0.944796i	$-0.393740\pi$
$607$	16.1454	0.655320	0.327660	+	0.944796i	$0.393740\pi$
$608$	15.8216	0.641649
$609$	0	0
$610$	−8.04098	−0.325570
$611$	2.31484	0.0936484
$612$	0	0
$613$	31.6127	1.27682	0.638412	−	0.769695i	$-0.279592\pi$
$613$	31.6127	1.27682	0.638412	+	0.769695i	$0.279592\pi$
$614$	−9.60707	−0.387710
$615$	0	0
$616$	0	0
$617$	11.3149	0.455523	0.227761	−	0.973717i	$-0.426859\pi$
$617$	11.3149	0.455523	0.227761	+	0.973717i	$0.426859\pi$
$618$	0	0
$619$	−9.92065	−0.398745	−0.199372	−	0.979924i	$-0.563890\pi$
$619$	−9.92065	−0.398745	−0.199372	+	0.979924i	$0.563890\pi$
$620$	11.3466	0.455690
$621$	0	0
$622$	21.8472	0.875992
$623$	−17.6900	−0.708735
$624$	0	0
$625$	−5.05573	−0.202229
$626$	13.5790	0.542725
$627$	0	0
$628$	17.5542	0.700488
$629$	−0.256660	−0.0102337
$630$	0	0
$631$	−9.26080	−0.368667	−0.184333	−	0.982864i	$-0.559013\pi$
$631$	−9.26080	−0.368667	−0.184333	+	0.982864i	$0.559013\pi$
$632$	22.1469	0.880958
$633$	0	0
$634$	0.597662	0.0237362
$635$	14.7914	0.586980
$636$	0	0
$637$	0.301097	0.0119299
$638$	0	0
$639$	0	0
$640$	−16.0348	−0.633832
$641$	11.4592	0.452611	0.226305	−	0.974056i	$-0.427335\pi$
$641$	11.4592	0.452611	0.226305	+	0.974056i	$0.427335\pi$
$642$	0	0
$643$	32.8446	1.29526	0.647632	−	0.761953i	$-0.275760\pi$
$643$	32.8446	1.29526	0.647632	+	0.761953i	$0.275760\pi$
$644$	−2.24204	−0.0883486
$645$	0	0
$646$	0.112279	0.00441756
$647$	−47.9724	−1.88599	−0.942995	−	0.332806i	$-0.892004\pi$
$647$	−47.9724	−1.88599	−0.942995	+	0.332806i	$0.892004\pi$
$648$	0	0
$649$	0	0
$650$	0.564497	0.0221414
$651$	0	0
$652$	9.06267	0.354922
$653$	18.1875	0.711732	0.355866	−	0.934537i	$-0.384186\pi$
$653$	18.1875	0.711732	0.355866	+	0.934537i	$0.384186\pi$
$654$	0	0
$655$	−24.7544	−0.967235
$656$	3.20144	0.124995
$657$	0	0
$658$	−5.50548	−0.214626
$659$	9.72544	0.378849	0.189425	−	0.981895i	$-0.439338\pi$
$659$	9.72544	0.378849	0.189425	+	0.981895i	$0.439338\pi$
$660$	0	0
$661$	−25.8857	−1.00684	−0.503419	−	0.864042i	$-0.667925\pi$
$661$	−25.8857	−1.00684	−0.503419	+	0.864042i	$0.667925\pi$
$662$	10.5577	0.410338
$663$	0	0
$664$	−12.5826	−0.488300
$665$	4.17852	0.162036
$666$	0	0
$667$	−3.07689	−0.119138
$668$	−34.0211	−1.31631
$669$	0	0
$670$	−6.74179	−0.260458
$671$	0	0
$672$	0	0
$673$	−5.66072	−0.218205	−0.109102	−	0.994031i	$-0.534798\pi$
$673$	−5.66072	−0.218205	−0.109102	+	0.994031i	$0.534798\pi$
$674$	23.1930	0.893361
$675$	0	0
$676$	19.1986	0.738407
$677$	−12.0626	−0.463604	−0.231802	−	0.972763i	$-0.574462\pi$
$677$	−12.0626	−0.463604	−0.231802	+	0.972763i	$0.574462\pi$
$678$	0	0
$679$	7.95804	0.305402
$680$	−0.223196	−0.00855919
$681$	0	0
$682$	0	0
$683$	−15.9182	−0.609094	−0.304547	−	0.952497i	$-0.598505\pi$
$683$	−15.9182	−0.609094	−0.304547	+	0.952497i	$0.598505\pi$
$684$	0	0
$685$	−22.8443	−0.872836
$686$	−0.716111	−0.0273413
$687$	0	0
$688$	−1.30079	−0.0495923
$689$	2.16344	0.0824206
$690$	0	0
$691$	27.7539	1.05581	0.527903	−	0.849305i	$-0.322978\pi$
$691$	27.7539	1.05581	0.527903	+	0.849305i	$0.322978\pi$
$692$	37.2575	1.41632
$693$	0	0
$694$	9.24201	0.350822
$695$	11.9117	0.451835
$696$	0	0
$697$	0.156312	0.00592074
$698$	10.3350	0.391186
$699$	0	0
$700$	3.89350	0.147160
$701$	25.5593	0.965362	0.482681	−	0.875796i	$-0.339663\pi$
$701$	25.5593	0.965362	0.482681	+	0.875796i	$0.339663\pi$
$702$	0	0
$703$	11.9991	0.452555
$704$	0	0
$705$	0	0
$706$	−9.52235	−0.358379
$707$	7.07407	0.266048
$708$	0	0
$709$	−43.8010	−1.64498	−0.822491	−	0.568778i	$-0.807416\pi$
$709$	−43.8010	−1.64498	−0.822491	+	0.568778i	$0.807416\pi$
$710$	−4.11451	−0.154415
$711$	0	0
$712$	−44.1757	−1.65555
$713$	7.45265	0.279104
$714$	0	0
$715$	0	0
$716$	2.95511	0.110438
$717$	0	0
$718$	−6.97516	−0.260311
$719$	−24.7256	−0.922109	−0.461054	−	0.887372i	$-0.652529\pi$
$719$	−24.7256	−0.922109	−0.461054	+	0.887372i	$0.652529\pi$
$720$	0	0
$721$	−12.0828	−0.449988
$722$	8.35696	0.311014
$723$	0	0
$724$	−11.4774	−0.426553
$725$	5.34330	0.198445
$726$	0	0
$727$	−15.0630	−0.558654	−0.279327	−	0.960196i	$-0.590111\pi$
$727$	−15.0630	−0.558654	−0.279327	+	0.960196i	$0.590111\pi$
$728$	0.751902	0.0278674
$729$	0	0
$730$	−1.78247	−0.0659722
$731$	−0.0635120	−0.00234908
$732$	0	0
$733$	−25.4833	−0.941249	−0.470624	−	0.882334i	$-0.655971\pi$
$733$	−25.4833	−0.941249	−0.470624	+	0.882334i	$0.655971\pi$
$734$	11.8744	0.438293
$735$	0	0
$736$	−8.80993	−0.324738
$737$	0	0
$738$	0	0
$739$	1.00152	0.0368415	0.0184207	−	0.999830i	$-0.494136\pi$
$739$	1.00152	0.0368415	0.0184207	+	0.999830i	$0.494136\pi$
$740$	−10.1725	−0.373948
$741$	0	0
$742$	−5.14541	−0.188894
$743$	−20.8622	−0.765361	−0.382680	−	0.923881i	$-0.624999\pi$
$743$	−20.8622	−0.765361	−0.382680	+	0.923881i	$0.624999\pi$
$744$	0	0
$745$	−8.84310	−0.323986
$746$	3.95529	0.144814
$747$	0	0
$748$	0	0
$749$	−2.30371	−0.0841759
$750$	0	0
$751$	39.7300	1.44977	0.724884	−	0.688871i	$-0.241893\pi$
$751$	39.7300	1.44977	0.724884	+	0.688871i	$0.241893\pi$
$752$	9.11868	0.332524
$753$	0	0
$754$	0.440069	0.0160264
$755$	−4.61375	−0.167912
$756$	0	0
$757$	−34.5382	−1.25531	−0.627656	−	0.778491i	$-0.715986\pi$
$757$	−34.5382	−1.25531	−0.627656	+	0.778491i	$0.715986\pi$
$758$	18.0950	0.657239
$759$	0	0
$760$	10.4346	0.378504
$761$	−41.2704	−1.49605	−0.748025	−	0.663670i	$-0.768998\pi$
$761$	−41.2704	−1.49605	−0.748025	+	0.663670i	$0.768998\pi$
$762$	0	0
$763$	13.1783	0.477085
$764$	24.1917	0.875225
$765$	0	0
$766$	6.80077	0.245722
$767$	0.0235586	0.000850653 0
$768$	0	0
$769$	42.6907	1.53946	0.769732	−	0.638367i	$-0.220390\pi$
$769$	42.6907	1.53946	0.769732	+	0.638367i	$0.220390\pi$
$770$	0	0
$771$	0	0
$772$	7.09750	0.255445
$773$	−19.6821	−0.707916	−0.353958	−	0.935261i	$-0.615164\pi$
$773$	−19.6821	−0.707916	−0.353958	+	0.935261i	$0.615164\pi$
$774$	0	0
$775$	−12.9422	−0.464898
$776$	19.8729	0.713396
$777$	0	0
$778$	3.95032	0.141626
$779$	−7.30774	−0.261827
$780$	0	0
$781$	0	0
$782$	−0.0625205	−0.00223573
$783$	0	0
$784$	1.18609	0.0423603
$785$	18.2173	0.650203
$786$	0	0
$787$	−4.64715	−0.165653	−0.0828266	−	0.996564i	$-0.526395\pi$
$787$	−4.64715	−0.165653	−0.0828266	+	0.996564i	$0.526395\pi$
$788$	−33.3698	−1.18875
$789$	0	0
$790$	9.80181	0.348733
$791$	14.0149	0.498312
$792$	0	0
$793$	−2.19062	−0.0777912
$794$	17.2128	0.610858
$795$	0	0
$796$	17.2490	0.611376
$797$	32.7314	1.15941	0.579703	−	0.814828i	$-0.303168\pi$
$797$	32.7314	1.15941	0.579703	+	0.814828i	$0.303168\pi$
$798$	0	0
$799$	0.445224	0.0157509
$800$	15.2993	0.540910
$801$	0	0
$802$	19.1620	0.676635
$803$	0	0
$804$	0	0
$805$	−2.32673	−0.0820064
$806$	−1.06591	−0.0375450
$807$	0	0
$808$	17.6655	0.621468
$809$	−2.51503	−0.0884237	−0.0442118	−	0.999022i	$-0.514078\pi$
$809$	−2.51503	−0.0884237	−0.0442118	+	0.999022i	$0.514078\pi$
$810$	0	0
$811$	12.1236	0.425716	0.212858	−	0.977083i	$-0.431723\pi$
$811$	12.1236	0.425716	0.212858	+	0.977083i	$0.431723\pi$
$812$	3.03528	0.106518
$813$	0	0
$814$	0	0
$815$	9.40501	0.329443
$816$	0	0
$817$	2.96925	0.103881
$818$	−1.92514	−0.0673108
$819$	0	0
$820$	6.19529	0.216349
$821$	−38.6448	−1.34871	−0.674356	−	0.738406i	$-0.735579\pi$
$821$	−38.6448	−1.34871	−0.674356	+	0.738406i	$0.735579\pi$
$822$	0	0
$823$	28.0250	0.976889	0.488445	−	0.872595i	$-0.337564\pi$
$823$	28.0250	0.976889	0.488445	+	0.872595i	$0.337564\pi$
$824$	−30.1734	−1.05114
$825$	0	0
$826$	−0.0560305	−0.00194955
$827$	−30.2810	−1.05297	−0.526487	−	0.850183i	$-0.676491\pi$
$827$	−30.2810	−1.05297	−0.526487	+	0.850183i	$0.676491\pi$
$828$	0	0
$829$	−2.34783	−0.0815434	−0.0407717	−	0.999168i	$-0.512982\pi$
$829$	−2.34783	−0.0815434	−0.0407717	+	0.999168i	$0.512982\pi$
$830$	−5.56883	−0.193297
$831$	0	0
$832$	0.545777	0.0189214
$833$	0.0579114	0.00200651
$834$	0	0
$835$	−35.3062	−1.22182
$836$	0	0
$837$	0	0
$838$	−18.0760	−0.624425
$839$	−29.4521	−1.01680	−0.508399	−	0.861121i	$-0.669763\pi$
$839$	−29.4521	−1.01680	−0.508399	+	0.861121i	$0.669763\pi$
$840$	0	0
$841$	−24.8345	−0.856361
$842$	6.74907	0.232588
$843$	0	0
$844$	−24.5834	−0.846194
$845$	19.9238	0.685399
$846$	0	0
$847$	0	0
$848$	8.52229	0.292657
$849$	0	0
$850$	0.108572	0.00372401
$851$	−6.68147	−0.229038
$852$	0	0
$853$	31.3933	1.07489	0.537443	−	0.843300i	$-0.319390\pi$
$853$	31.3933	1.07489	0.537443	+	0.843300i	$0.319390\pi$
$854$	5.21004	0.178284
$855$	0	0
$856$	−5.75286	−0.196629
$857$	12.6273	0.431339	0.215670	−	0.976466i	$-0.430807\pi$
$857$	12.6273	0.431339	0.215670	+	0.976466i	$0.430807\pi$
$858$	0	0
$859$	11.4647	0.391172	0.195586	−	0.980687i	$-0.437339\pi$
$859$	11.4647	0.391172	0.195586	+	0.980687i	$0.437339\pi$
$860$	−2.51724	−0.0858371
$861$	0	0
$862$	19.7724	0.673449
$863$	−3.68221	−0.125344	−0.0626720	−	0.998034i	$-0.519962\pi$
$863$	−3.68221	−0.125344	−0.0626720	+	0.998034i	$0.519962\pi$
$864$	0	0
$865$	38.6649	1.31465
$866$	25.9929	0.883275
$867$	0	0
$868$	−7.35187	−0.249539
$869$	0	0
$870$	0	0
$871$	−1.83668	−0.0622335
$872$	32.9089	1.11444
$873$	0	0
$874$	2.92289	0.0988682
$875$	11.7574	0.397472
$876$	0	0
$877$	45.1256	1.52378	0.761892	−	0.647704i	$-0.224271\pi$
$877$	45.1256	1.52378	0.761892	+	0.647704i	$0.224271\pi$
$878$	25.6384	0.865254
$879$	0	0
$880$	0	0
$881$	16.4792	0.555198	0.277599	−	0.960697i	$-0.410461\pi$
$881$	16.4792	0.555198	0.277599	+	0.960697i	$0.410461\pi$
$882$	0	0
$883$	48.9017	1.64567	0.822837	−	0.568278i	$-0.192390\pi$
$883$	48.9017	1.64567	0.822837	+	0.568278i	$0.192390\pi$
$884$	−0.0259319	−0.000872185 0
$885$	0	0
$886$	4.29068	0.144148
$887$	32.7168	1.09852	0.549262	−	0.835650i	$-0.314909\pi$
$887$	32.7168	1.09852	0.549262	+	0.835650i	$0.314909\pi$
$888$	0	0
$889$	−9.58389	−0.321434
$890$	−19.5513	−0.655362
$891$	0	0
$892$	42.2260	1.41383
$893$	−20.8147	−0.696536
$894$	0	0
$895$	3.06674	0.102510
$896$	10.3895	0.347090
$897$	0	0
$898$	−2.69762	−0.0900208
$899$	−10.0894	−0.336502
$900$	0	0
$901$	0.416105	0.0138625
$902$	0	0
$903$	0	0
$904$	34.9981	1.16402
$905$	−11.9109	−0.395932
$906$	0	0
$907$	−3.14064	−0.104283	−0.0521416	−	0.998640i	$-0.516605\pi$
$907$	−3.14064	−0.104283	−0.0521416	+	0.998640i	$0.516605\pi$
$908$	−32.6405	−1.08321
$909$	0	0
$910$	0.332778	0.0110315
$911$	44.8693	1.48659	0.743293	−	0.668966i	$-0.233263\pi$
$911$	44.8693	1.48659	0.743293	+	0.668966i	$0.233263\pi$
$912$	0	0
$913$	0	0
$914$	11.4612	0.379103
$915$	0	0
$916$	−16.6144	−0.548956
$917$	16.0393	0.529663
$918$	0	0
$919$	−35.4504	−1.16940	−0.584701	−	0.811249i	$-0.698788\pi$
$919$	−35.4504	−1.16940	−0.584701	+	0.811249i	$0.698788\pi$
$920$	−5.81033	−0.191561
$921$	0	0
$922$	11.4469	0.376983
$923$	−1.12092	−0.0368957
$924$	0	0
$925$	11.6030	0.381504
$926$	6.54501	0.215082
$927$	0	0
$928$	11.9270	0.391521
$929$	−9.97464	−0.327257	−0.163629	−	0.986522i	$-0.552320\pi$
$929$	−9.97464	−0.327257	−0.163629	+	0.986522i	$0.552320\pi$
$930$	0	0
$931$	−2.70741	−0.0887319
$932$	22.5415	0.738373
$933$	0	0
$934$	1.36318	0.0446045
$935$	0	0
$936$	0	0
$937$	−21.9782	−0.717997	−0.358999	−	0.933338i	$-0.616882\pi$
$937$	−21.9782	−0.717997	−0.358999	+	0.933338i	$0.616882\pi$
$938$	4.36825	0.142628
$939$	0	0
$940$	17.6461	0.575551
$941$	52.0209	1.69583	0.847916	−	0.530130i	$-0.177857\pi$
$941$	52.0209	1.69583	0.847916	+	0.530130i	$0.177857\pi$
$942$	0	0
$943$	4.06917	0.132511
$944$	0.0928028	0.00302047
$945$	0	0
$946$	0	0
$947$	3.79616	0.123359	0.0616793	−	0.998096i	$-0.480354\pi$
$947$	3.79616	0.123359	0.0616793	+	0.998096i	$0.480354\pi$
$948$	0	0
$949$	−0.485601	−0.0157633
$950$	−5.07587	−0.164683
$951$	0	0
$952$	0.144617	0.00468706
$953$	−15.7640	−0.510646	−0.255323	−	0.966856i	$-0.582182\pi$
$953$	−15.7640	−0.510646	−0.255323	+	0.966856i	$0.582182\pi$
$954$	0	0
$955$	25.1055	0.812396
$956$	20.5597	0.664948
$957$	0	0
$958$	23.3729	0.755143
$959$	14.8017	0.477970
$960$	0	0
$961$	−6.56200	−0.211678
$962$	0.955611	0.0308101
$963$	0	0
$964$	−21.7771	−0.701393
$965$	7.36560	0.237107
$966$	0	0
$967$	−15.5059	−0.498635	−0.249317	−	0.968422i	$-0.580206\pi$
$967$	−15.5059	−0.498635	−0.249317	+	0.968422i	$0.580206\pi$
$968$	0	0
$969$	0	0
$970$	8.79537	0.282402
$971$	−61.0347	−1.95870	−0.979348	−	0.202182i	$-0.935197\pi$
$971$	−61.0347	−1.95870	−0.979348	+	0.202182i	$0.935197\pi$
$972$	0	0
$973$	−7.71799	−0.247428
$974$	22.3437	0.715938
$975$	0	0
$976$	−8.62934	−0.276219
$977$	−51.7989	−1.65719	−0.828597	−	0.559845i	$-0.810861\pi$
$977$	−51.7989	−1.65719	−0.828597	+	0.559845i	$0.810861\pi$
$978$	0	0
$979$	0	0
$980$	2.29526	0.0733195
$981$	0	0
$982$	17.5079	0.558701
$983$	−12.8521	−0.409918	−0.204959	−	0.978771i	$-0.565706\pi$
$983$	−12.8521	−0.409918	−0.204959	+	0.978771i	$0.565706\pi$
$984$	0	0
$985$	−34.6303	−1.10341
$986$	0.0846406	0.00269551
$987$	0	0
$988$	1.21234	0.0385698
$989$	−1.65337	−0.0525740
$990$	0	0
$991$	14.3450	0.455685	0.227842	−	0.973698i	$-0.426833\pi$
$991$	14.3450	0.455685	0.227842	+	0.973698i	$0.426833\pi$
$992$	−28.8887	−0.917217
$993$	0	0
$994$	2.66594	0.0845585
$995$	17.9006	0.567488
$996$	0	0
$997$	31.1668	0.987063	0.493531	−	0.869728i	$-0.335706\pi$
$997$	31.1668	0.987063	0.493531	+	0.869728i	$0.335706\pi$
$998$	−10.6979	−0.338637
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.cv.1.3		8
3.2	odd	2	inner	7623.2.a.cv.1.6		8
11.7	odd	10		693.2.m.h.379.3	yes	16
11.8	odd	10		693.2.m.h.64.3	yes	16
11.10	odd	2		7623.2.a.cu.1.6		8
33.8	even	10		693.2.m.h.64.2	✓	16
33.29	even	10		693.2.m.h.379.2	yes	16
33.32	even	2		7623.2.a.cu.1.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
693.2.m.h.64.2	✓	16	33.8	even	10
693.2.m.h.64.3	yes	16	11.8	odd	10
693.2.m.h.379.2	yes	16	33.29	even	10
693.2.m.h.379.3	yes	16	11.7	odd	10
7623.2.a.cu.1.3		8	33.32	even	2
7623.2.a.cu.1.6		8	11.10	odd	2
7623.2.a.cv.1.3		8	1.1	even	1	trivial
7623.2.a.cv.1.6		8	3.2	odd	2	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.716111 q^{2} -1.48718 q^{4} -1.54336 q^{5} +1.00000 q^{7} +2.49721 q^{8} +O(q^{10})\)	\(q-0.716111 q^{2} -1.48718 q^{4} -1.54336 q^{5} +1.00000 q^{7} +2.49721 q^{8} +1.10522 q^{10} +0.301097 q^{13} -0.716111 q^{14} +1.18609 q^{16} +0.0579114 q^{17} -2.70741 q^{19} +2.29526 q^{20} +1.50757 q^{23} -2.61803 q^{25} -0.215619 q^{26} -1.48718 q^{28} -2.04096 q^{29} +4.94348 q^{31} -5.84379 q^{32} -0.0414710 q^{34} -1.54336 q^{35} -4.43195 q^{37} +1.93881 q^{38} -3.85410 q^{40} +2.69916 q^{41} -1.09671 q^{43} -1.07959 q^{46} +7.68803 q^{47} +1.00000 q^{49} +1.87480 q^{50} -0.447786 q^{52} +7.18521 q^{53} +2.49721 q^{56} +1.46155 q^{58} +0.0782427 q^{59} -7.27547 q^{61} -3.54008 q^{62} +1.81263 q^{64} -0.464701 q^{65} -6.09996 q^{67} -0.0861249 q^{68} +1.10522 q^{70} -3.72280 q^{71} -1.61278 q^{73} +3.17377 q^{74} +4.02642 q^{76} +8.86866 q^{79} -1.83056 q^{80} -1.93290 q^{82} -5.03866 q^{83} -0.0893782 q^{85} +0.785366 q^{86} -17.6900 q^{89} +0.301097 q^{91} -2.24204 q^{92} -5.50548 q^{94} +4.17852 q^{95} +7.95804 q^{97} -0.716111 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 2 q^{4} + 8 q^{7}+O(q^{10}) \) 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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