LMFDB - Embedded newform 7623.2.a.dc.1.13

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			1.13
Root			$2.08213$ 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)

$f(q)$	$=$	$q+2.08213 q^{2} +2.33525 q^{4} +1.33222 q^{5} +1.00000 q^{7} +0.698024 q^{8} +O(q^{10})$	$q+2.08213 q^{2} +2.33525 q^{4} +1.33222 q^{5} +1.00000 q^{7} +0.698024 q^{8} +2.77386 q^{10} +5.74691 q^{13} +2.08213 q^{14} -3.21712 q^{16} +4.27240 q^{17} +0.162553 q^{19} +3.11107 q^{20} +1.12680 q^{23} -3.22518 q^{25} +11.9658 q^{26} +2.33525 q^{28} +9.83547 q^{29} +9.09480 q^{31} -8.09449 q^{32} +8.89567 q^{34} +1.33222 q^{35} -8.42718 q^{37} +0.338455 q^{38} +0.929924 q^{40} +4.50712 q^{41} -8.02855 q^{43} +2.34613 q^{46} -2.79374 q^{47} +1.00000 q^{49} -6.71523 q^{50} +13.4204 q^{52} -10.4883 q^{53} +0.698024 q^{56} +20.4787 q^{58} -13.6125 q^{59} +8.06786 q^{61} +18.9365 q^{62} -10.4195 q^{64} +7.65617 q^{65} +9.10333 q^{67} +9.97710 q^{68} +2.77386 q^{70} +1.01904 q^{71} -7.64824 q^{73} -17.5464 q^{74} +0.379601 q^{76} -0.00547271 q^{79} -4.28592 q^{80} +9.38438 q^{82} +2.83233 q^{83} +5.69179 q^{85} -16.7165 q^{86} +10.5254 q^{89} +5.74691 q^{91} +2.63135 q^{92} -5.81693 q^{94} +0.216557 q^{95} +12.1613 q^{97} +2.08213 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.08213	1.47228	0.736142	−	0.676827i	$-0.236645\pi$
$2$	2.08213	1.47228	0.736142	+	0.676827i	$0.236645\pi$
$3$	0	0
$3$	0	0
$4$	2.33525	1.16762
$5$	1.33222	0.595789	0.297894	−	0.954599i	$-0.403716\pi$
$5$	1.33222	0.595789	0.297894	+	0.954599i	$0.403716\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0.698024	0.246789
$9$	0	0
$10$	2.77386	0.877171
$11$	0	0
$11$	0	0
$12$	0	0
$13$	5.74691	1.59391	0.796953	−	0.604041i	$-0.206444\pi$
$13$	5.74691	1.59391	0.796953	+	0.604041i	$0.206444\pi$
$14$	2.08213	0.556471
$15$	0	0
$16$	−3.21712	−0.804280
$17$	4.27240	1.03621	0.518105	−	0.855317i	$-0.326638\pi$
$17$	4.27240	1.03621	0.518105	+	0.855317i	$0.326638\pi$
$18$	0	0
$19$	0.162553	0.0372922	0.0186461	−	0.999826i	$-0.494064\pi$
$19$	0.162553	0.0372922	0.0186461	+	0.999826i	$0.494064\pi$
$20$	3.11107	0.695657
$21$	0	0
$22$	0	0
$23$	1.12680	0.234954	0.117477	−	0.993076i	$-0.462519\pi$
$23$	1.12680	0.234954	0.117477	+	0.993076i	$0.462519\pi$
$24$	0	0
$25$	−3.22518	−0.645036
$26$	11.9658	2.34668
$27$	0	0
$28$	2.33525	0.441320
$29$	9.83547	1.82640	0.913201	−	0.407510i	$-0.133603\pi$
$29$	9.83547	1.82640	0.913201	+	0.407510i	$0.133603\pi$
$30$	0	0
$31$	9.09480	1.63347	0.816737	−	0.577009i	$-0.195780\pi$
$31$	9.09480	1.63347	0.816737	+	0.577009i	$0.195780\pi$
$32$	−8.09449	−1.43092
$33$	0	0
$34$	8.89567	1.52559
$35$	1.33222	0.225187
$36$	0	0
$37$	−8.42718	−1.38542	−0.692710	−	0.721217i	$-0.743583\pi$
$37$	−8.42718	−1.38542	−0.692710	+	0.721217i	$0.743583\pi$
$38$	0.338455	0.0549047
$39$	0	0
$40$	0.929924	0.147034
$41$	4.50712	0.703894	0.351947	−	0.936020i	$-0.385520\pi$
$41$	4.50712	0.703894	0.351947	+	0.936020i	$0.385520\pi$
$42$	0	0
$43$	−8.02855	−1.22434	−0.612172	−	0.790725i	$-0.709704\pi$
$43$	−8.02855	−1.22434	−0.612172	+	0.790725i	$0.709704\pi$
$44$	0	0
$45$	0	0
$46$	2.34613	0.345919
$47$	−2.79374	−0.407510	−0.203755	−	0.979022i	$-0.565315\pi$
$47$	−2.79374	−0.407510	−0.203755	+	0.979022i	$0.565315\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−6.71523	−0.949677
$51$	0	0
$52$	13.4204	1.86108
$53$	−10.4883	−1.44067	−0.720336	−	0.693625i	$-0.756013\pi$
$53$	−10.4883	−1.44067	−0.720336	+	0.693625i	$0.756013\pi$
$54$	0	0
$55$	0	0
$56$	0.698024	0.0932774
$57$	0	0
$58$	20.4787	2.68898
$59$	−13.6125	−1.77220	−0.886099	−	0.463497i	$-0.846595\pi$
$59$	−13.6125	−1.77220	−0.886099	+	0.463497i	$0.846595\pi$
$60$	0	0
$61$	8.06786	1.03298	0.516492	−	0.856292i	$-0.327238\pi$
$61$	8.06786	1.03298	0.516492	+	0.856292i	$0.327238\pi$
$62$	18.9365	2.40494
$63$	0	0
$64$	−10.4195	−1.30244
$65$	7.65617	0.949631
$66$	0	0
$67$	9.10333	1.11215	0.556074	−	0.831133i	$-0.312307\pi$
$67$	9.10333	1.11215	0.556074	+	0.831133i	$0.312307\pi$
$68$	9.97710	1.20990
$69$	0	0
$70$	2.77386	0.331539
$71$	1.01904	0.120938	0.0604689	−	0.998170i	$-0.480740\pi$
$71$	1.01904	0.120938	0.0604689	+	0.998170i	$0.480740\pi$
$72$	0	0
$73$	−7.64824	−0.895159	−0.447579	−	0.894244i	$-0.647714\pi$
$73$	−7.64824	−0.895159	−0.447579	+	0.894244i	$0.647714\pi$
$74$	−17.5464	−2.03973
$75$	0	0
$76$	0.379601	0.0435432
$77$	0	0
$78$	0	0
$79$	−0.00547271	−0.000615728 0	−0.000307864	−	1.00000i	$-0.500098\pi$
$79$	−0.00547271	−0.000615728 0	−0.000307864		1.00000i	$0.500098\pi$
$80$	−4.28592	−0.479181
$81$	0	0
$82$	9.38438	1.03633
$83$	2.83233	0.310888	0.155444	−	0.987845i	$-0.450319\pi$
$83$	2.83233	0.310888	0.155444	+	0.987845i	$0.450319\pi$
$84$	0	0
$85$	5.69179	0.617362
$86$	−16.7165	−1.80258
$87$	0	0
$88$	0	0
$89$	10.5254	1.11569	0.557845	−	0.829945i	$-0.311628\pi$
$89$	10.5254	1.11569	0.557845	+	0.829945i	$0.311628\pi$
$90$	0	0
$91$	5.74691	0.602440
$92$	2.63135	0.274337
$93$	0	0
$94$	−5.81693	−0.599970
$95$	0.216557	0.0222182
$96$	0	0
$97$	12.1613	1.23479	0.617396	−	0.786652i	$-0.288188\pi$
$97$	12.1613	1.23479	0.617396	+	0.786652i	$0.288188\pi$
$98$	2.08213	0.210326
$99$	0	0
$100$	−7.53159	−0.753159
$101$	1.52188	0.151433	0.0757164	−	0.997129i	$-0.475876\pi$
$101$	1.52188	0.151433	0.0757164	+	0.997129i	$0.475876\pi$
$102$	0	0
$103$	15.4665	1.52396	0.761979	−	0.647602i	$-0.224228\pi$
$103$	15.4665	1.52396	0.761979	+	0.647602i	$0.224228\pi$
$104$	4.01148	0.393358
$105$	0	0
$106$	−21.8379	−2.12108
$107$	−8.25043	−0.797599	−0.398799	−	0.917038i	$-0.630573\pi$
$107$	−8.25043	−0.797599	−0.398799	+	0.917038i	$0.630573\pi$
$108$	0	0
$109$	0.305943	0.0293040	0.0146520	−	0.999893i	$-0.495336\pi$
$109$	0.305943	0.0293040	0.0146520	+	0.999893i	$0.495336\pi$
$110$	0	0
$111$	0	0
$112$	−3.21712	−0.303989
$113$	10.1503	0.954857	0.477428	−	0.878671i	$-0.341569\pi$
$113$	10.1503	0.954857	0.477428	+	0.878671i	$0.341569\pi$
$114$	0	0
$115$	1.50115	0.139983
$116$	22.9682	2.13255
$117$	0	0
$118$	−28.3429	−2.60918
$119$	4.27240	0.391650
$120$	0	0
$121$	0	0
$122$	16.7983	1.52085
$123$	0	0
$124$	21.2386	1.90728
$125$	−10.9578	−0.980094
$126$	0	0
$127$	3.80774	0.337882	0.168941	−	0.985626i	$-0.445965\pi$
$127$	3.80774	0.337882	0.168941	+	0.985626i	$0.445965\pi$
$128$	−5.50574	−0.486643
$129$	0	0
$130$	15.9411	1.39813
$131$	−14.6612	−1.28096	−0.640479	−	0.767976i	$-0.721264\pi$
$131$	−14.6612	−1.28096	−0.640479	+	0.767976i	$0.721264\pi$
$132$	0	0
$133$	0.162553	0.0140951
$134$	18.9543	1.63740
$135$	0	0
$136$	2.98224	0.255725
$137$	16.8306	1.43794	0.718968	−	0.695043i	$-0.244615\pi$
$137$	16.8306	1.43794	0.718968	+	0.695043i	$0.244615\pi$
$138$	0	0
$139$	−7.79467	−0.661135	−0.330568	−	0.943782i	$-0.607240\pi$
$139$	−7.79467	−0.661135	−0.330568	+	0.943782i	$0.607240\pi$
$140$	3.11107	0.262933
$141$	0	0
$142$	2.12177	0.178055
$143$	0	0
$144$	0	0
$145$	13.1031	1.08815
$146$	−15.9246	−1.31793
$147$	0	0
$148$	−19.6795	−1.61765
$149$	9.71989	0.796284	0.398142	−	0.917324i	$-0.369655\pi$
$149$	9.71989	0.796284	0.398142	+	0.917324i	$0.369655\pi$
$150$	0	0
$151$	10.1015	0.822051	0.411026	−	0.911624i	$-0.365171\pi$
$151$	10.1015	0.822051	0.411026	+	0.911624i	$0.365171\pi$
$152$	0.113466	0.00920328
$153$	0	0
$154$	0	0
$155$	12.1163	0.973206
$156$	0	0
$157$	−4.94915	−0.394985	−0.197493	−	0.980304i	$-0.563280\pi$
$157$	−4.94915	−0.394985	−0.197493	+	0.980304i	$0.563280\pi$
$158$	−0.0113949	−0.000906526 0
$159$	0	0
$160$	−10.7837	−0.852524
$161$	1.12680	0.0888041
$162$	0	0
$163$	6.73297	0.527367	0.263683	−	0.964609i	$-0.415063\pi$
$163$	6.73297	0.527367	0.263683	+	0.964609i	$0.415063\pi$
$164$	10.5252	0.821882
$165$	0	0
$166$	5.89726	0.457716
$167$	13.9874	1.08238	0.541188	−	0.840902i	$-0.317975\pi$
$167$	13.9874	1.08238	0.541188	+	0.840902i	$0.317975\pi$
$168$	0	0
$169$	20.0270	1.54054
$170$	11.8510	0.908932
$171$	0	0
$172$	−18.7486	−1.42957
$173$	−3.50877	−0.266767	−0.133383	−	0.991065i	$-0.542584\pi$
$173$	−3.50877	−0.266767	−0.133383	+	0.991065i	$0.542584\pi$
$174$	0	0
$175$	−3.22518	−0.243801
$176$	0	0
$177$	0	0
$178$	21.9152	1.64261
$179$	2.22735	0.166480	0.0832399	−	0.996530i	$-0.473473\pi$
$179$	2.22735	0.166480	0.0832399	+	0.996530i	$0.473473\pi$
$180$	0	0
$181$	1.11432	0.0828265	0.0414132	−	0.999142i	$-0.486814\pi$
$181$	1.11432	0.0828265	0.0414132	+	0.999142i	$0.486814\pi$
$182$	11.9658	0.886963
$183$	0	0
$184$	0.786532	0.0579839
$185$	−11.2269	−0.825417
$186$	0	0
$187$	0	0
$188$	−6.52408	−0.475818
$189$	0	0
$190$	0.450898	0.0327116
$191$	10.2314	0.740320	0.370160	−	0.928968i	$-0.379303\pi$
$191$	10.2314	0.740320	0.370160	+	0.928968i	$0.379303\pi$
$192$	0	0
$193$	−17.9336	−1.29089	−0.645443	−	0.763808i	$-0.723327\pi$
$193$	−17.9336	−1.29089	−0.645443	+	0.763808i	$0.723327\pi$
$194$	25.3213	1.81797
$195$	0	0
$196$	2.33525	0.166803
$197$	4.30284	0.306564	0.153282	−	0.988182i	$-0.451016\pi$
$197$	4.30284	0.306564	0.153282	+	0.988182i	$0.451016\pi$
$198$	0	0
$199$	22.0560	1.56351	0.781753	−	0.623588i	$-0.214326\pi$
$199$	22.0560	1.56351	0.781753	+	0.623588i	$0.214326\pi$
$200$	−2.25125	−0.159188
$201$	0	0
$202$	3.16875	0.222952
$203$	9.83547	0.690315
$204$	0	0
$205$	6.00449	0.419372
$206$	32.2031	2.24370
$207$	0	0
$208$	−18.4885	−1.28195
$209$	0	0
$210$	0	0
$211$	−14.8525	−1.02249	−0.511244	−	0.859435i	$-0.670815\pi$
$211$	−14.8525	−1.02249	−0.511244	+	0.859435i	$0.670815\pi$
$212$	−24.4927	−1.68216
$213$	0	0
$214$	−17.1784	−1.17429
$215$	−10.6958	−0.729450
$216$	0	0
$217$	9.09480	0.617395
$218$	0.637011	0.0431438
$219$	0	0
$220$	0	0
$221$	24.5531	1.65162
$222$	0	0
$223$	5.59782	0.374857	0.187429	−	0.982278i	$-0.439985\pi$
$223$	5.59782	0.374857	0.187429	+	0.982278i	$0.439985\pi$
$224$	−8.09449	−0.540836
$225$	0	0
$226$	21.1341	1.40582
$227$	−7.52880	−0.499704	−0.249852	−	0.968284i	$-0.580382\pi$
$227$	−7.52880	−0.499704	−0.249852	+	0.968284i	$0.580382\pi$
$228$	0	0
$229$	−27.2330	−1.79961	−0.899805	−	0.436292i	$-0.856292\pi$
$229$	−27.2330	−1.79961	−0.899805	+	0.436292i	$0.856292\pi$
$230$	3.12558	0.206094
$231$	0	0
$232$	6.86539	0.450735
$233$	−5.48986	−0.359653	−0.179826	−	0.983698i	$-0.557554\pi$
$233$	−5.48986	−0.359653	−0.179826	+	0.983698i	$0.557554\pi$
$234$	0	0
$235$	−3.72189	−0.242790
$236$	−31.7885	−2.06926
$237$	0	0
$238$	8.89567	0.576621
$239$	−24.8413	−1.60685	−0.803426	−	0.595405i	$-0.796992\pi$
$239$	−24.8413	−1.60685	−0.803426	+	0.595405i	$0.796992\pi$
$240$	0	0
$241$	17.3238	1.11592	0.557961	−	0.829867i	$-0.311584\pi$
$241$	17.3238	1.11592	0.557961	+	0.829867i	$0.311584\pi$
$242$	0	0
$243$	0	0
$244$	18.8404	1.20613
$245$	1.33222	0.0851127
$246$	0	0
$247$	0.934176	0.0594402
$248$	6.34839	0.403123
$249$	0	0
$250$	−22.8155	−1.44298
$251$	−24.6655	−1.55687	−0.778436	−	0.627724i	$-0.783987\pi$
$251$	−24.6655	−1.55687	−0.778436	+	0.627724i	$0.783987\pi$
$252$	0	0
$253$	0	0
$254$	7.92819	0.497459
$255$	0	0
$256$	9.37538	0.585961
$257$	3.87897	0.241964	0.120982	−	0.992655i	$-0.461396\pi$
$257$	3.87897	0.241964	0.120982	+	0.992655i	$0.461396\pi$
$258$	0	0
$259$	−8.42718	−0.523639
$260$	17.8790	1.10881
$261$	0	0
$262$	−30.5265	−1.88593
$263$	−5.19075	−0.320075	−0.160038	−	0.987111i	$-0.551162\pi$
$263$	−5.19075	−0.320075	−0.160038	+	0.987111i	$0.551162\pi$
$264$	0	0
$265$	−13.9727	−0.858336
$266$	0.338455	0.0207520
$267$	0	0
$268$	21.2585	1.29857
$269$	−16.8397	−1.02673	−0.513366	−	0.858170i	$-0.671602\pi$
$269$	−16.8397	−1.02673	−0.513366	+	0.858170i	$0.671602\pi$
$270$	0	0
$271$	−14.0214	−0.851739	−0.425870	−	0.904785i	$-0.640032\pi$
$271$	−14.0214	−0.851739	−0.425870	+	0.904785i	$0.640032\pi$
$272$	−13.7448	−0.833402
$273$	0	0
$274$	35.0434	2.11705
$275$	0	0
$276$	0	0
$277$	0.945328	0.0567992	0.0283996	−	0.999597i	$-0.490959\pi$
$277$	0.945328	0.0567992	0.0283996	+	0.999597i	$0.490959\pi$
$278$	−16.2295	−0.973380
$279$	0	0
$280$	0.929924	0.0555736
$281$	4.03539	0.240731	0.120366	−	0.992730i	$-0.461593\pi$
$281$	4.03539	0.240731	0.120366	+	0.992730i	$0.461593\pi$
$282$	0	0
$283$	4.80601	0.285688	0.142844	−	0.989745i	$-0.454375\pi$
$283$	4.80601	0.285688	0.142844	+	0.989745i	$0.454375\pi$
$284$	2.37971	0.141210
$285$	0	0
$286$	0	0
$287$	4.50712	0.266047
$288$	0	0
$289$	1.25340	0.0737292
$290$	27.2822	1.60207
$291$	0	0
$292$	−17.8605	−1.04521
$293$	−1.04668	−0.0611474	−0.0305737	−	0.999533i	$-0.509733\pi$
$293$	−1.04668	−0.0611474	−0.0305737	+	0.999533i	$0.509733\pi$
$294$	0	0
$295$	−18.1349	−1.05585
$296$	−5.88237	−0.341906
$297$	0	0
$298$	20.2380	1.17236
$299$	6.47561	0.374494
$300$	0	0
$301$	−8.02855	−0.462758
$302$	21.0327	1.21029
$303$	0	0
$304$	−0.522952	−0.0299933
$305$	10.7482	0.615440
$306$	0	0
$307$	−26.1929	−1.49491	−0.747455	−	0.664313i	$-0.768724\pi$
$307$	−26.1929	−1.49491	−0.747455	+	0.664313i	$0.768724\pi$
$308$	0	0
$309$	0	0
$310$	25.2277	1.43284
$311$	−22.0318	−1.24931	−0.624655	−	0.780901i	$-0.714760\pi$
$311$	−22.0318	−1.24931	−0.624655	+	0.780901i	$0.714760\pi$
$312$	0	0
$313$	30.4119	1.71898	0.859491	−	0.511151i	$-0.170781\pi$
$313$	30.4119	1.71898	0.859491	+	0.511151i	$0.170781\pi$
$314$	−10.3047	−0.581531
$315$	0	0
$316$	−0.0127801	−0.000718938 0
$317$	−19.0776	−1.07151	−0.535753	−	0.844374i	$-0.679972\pi$
$317$	−19.0776	−1.07151	−0.535753	+	0.844374i	$0.679972\pi$
$318$	0	0
$319$	0	0
$320$	−13.8811	−0.775978
$321$	0	0
$322$	2.34613	0.130745
$323$	0.694490	0.0386425
$324$	0	0
$325$	−18.5348	−1.02813
$326$	14.0189	0.776434
$327$	0	0
$328$	3.14608	0.173713
$329$	−2.79374	−0.154024
$330$	0	0
$331$	3.61239	0.198555	0.0992774	−	0.995060i	$-0.468347\pi$
$331$	3.61239	0.198555	0.0992774	+	0.995060i	$0.468347\pi$
$332$	6.61418	0.363000
$333$	0	0
$334$	29.1235	1.59357
$335$	12.1277	0.662605
$336$	0	0
$337$	−2.67839	−0.145901	−0.0729507	−	0.997336i	$-0.523242\pi$
$337$	−2.67839	−0.145901	−0.0729507	+	0.997336i	$0.523242\pi$
$338$	41.6987	2.26811
$339$	0	0
$340$	13.2917	0.720846
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	−5.60412	−0.302154
$345$	0	0
$346$	−7.30570	−0.392757
$347$	−18.4546	−0.990696	−0.495348	−	0.868695i	$-0.664959\pi$
$347$	−18.4546	−0.990696	−0.495348	+	0.868695i	$0.664959\pi$
$348$	0	0
$349$	20.3015	1.08672	0.543358	−	0.839501i	$-0.317153\pi$
$349$	20.3015	1.08672	0.543358	+	0.839501i	$0.317153\pi$
$350$	−6.71523	−0.358944
$351$	0	0
$352$	0	0
$353$	−12.7771	−0.680054	−0.340027	−	0.940416i	$-0.610436\pi$
$353$	−12.7771	−0.680054	−0.340027	+	0.940416i	$0.610436\pi$
$354$	0	0
$355$	1.35759	0.0720533
$356$	24.5794	1.30270
$357$	0	0
$358$	4.63762	0.245106
$359$	10.2976	0.543486	0.271743	−	0.962370i	$-0.412400\pi$
$359$	10.2976	0.543486	0.271743	+	0.962370i	$0.412400\pi$
$360$	0	0
$361$	−18.9736	−0.998609
$362$	2.32015	0.121944
$363$	0	0
$364$	13.4204	0.703423
$365$	−10.1892	−0.533325
$366$	0	0
$367$	2.65025	0.138342	0.0691710	−	0.997605i	$-0.477965\pi$
$367$	2.65025	0.138342	0.0691710	+	0.997605i	$0.477965\pi$
$368$	−3.62504	−0.188968
$369$	0	0
$370$	−23.3758	−1.21525
$371$	−10.4883	−0.544523
$372$	0	0
$373$	32.9030	1.70365	0.851827	−	0.523824i	$-0.175495\pi$
$373$	32.9030	1.70365	0.851827	+	0.523824i	$0.175495\pi$
$374$	0	0
$375$	0	0
$376$	−1.95010	−0.100569
$377$	56.5236	2.91111
$378$	0	0
$379$	−22.2324	−1.14200	−0.571000	−	0.820950i	$-0.693444\pi$
$379$	−22.2324	−1.14200	−0.571000	+	0.820950i	$0.693444\pi$
$380$	0.505713	0.0259425
$381$	0	0
$382$	21.3031	1.08996
$383$	25.1289	1.28402	0.642012	−	0.766694i	$-0.278100\pi$
$383$	25.1289	1.28402	0.642012	+	0.766694i	$0.278100\pi$
$384$	0	0
$385$	0	0
$386$	−37.3399	−1.90055
$387$	0	0
$388$	28.3996	1.44177
$389$	−23.0628	−1.16933	−0.584665	−	0.811275i	$-0.698774\pi$
$389$	−23.0628	−1.16933	−0.584665	+	0.811275i	$0.698774\pi$
$390$	0	0
$391$	4.81413	0.243461
$392$	0.698024	0.0352555
$393$	0	0
$394$	8.95905	0.451350
$395$	−0.00729087	−0.000366844 0
$396$	0	0
$397$	−3.78504	−0.189966	−0.0949828	−	0.995479i	$-0.530280\pi$
$397$	−3.78504	−0.189966	−0.0949828	+	0.995479i	$0.530280\pi$
$398$	45.9233	2.30193
$399$	0	0
$400$	10.3758	0.518789
$401$	−39.0596	−1.95054	−0.975271	−	0.221011i	$-0.929064\pi$
$401$	−39.0596	−1.95054	−0.975271	+	0.221011i	$0.929064\pi$
$402$	0	0
$403$	52.2670	2.60361
$404$	3.55396	0.176816
$405$	0	0
$406$	20.4787	1.01634
$407$	0	0
$408$	0	0
$409$	−29.4034	−1.45390	−0.726952	−	0.686689i	$-0.759064\pi$
$409$	−29.4034	−1.45390	−0.726952	+	0.686689i	$0.759064\pi$
$410$	12.5021	0.617435
$411$	0	0
$412$	36.1180	1.77941
$413$	−13.6125	−0.669828
$414$	0	0
$415$	3.77329	0.185224
$416$	−46.5183	−2.28075
$417$	0	0
$418$	0	0
$419$	14.5510	0.710863	0.355431	−	0.934702i	$-0.384334\pi$
$419$	14.5510	0.710863	0.355431	+	0.934702i	$0.384334\pi$
$420$	0	0
$421$	17.7574	0.865443	0.432722	−	0.901528i	$-0.357553\pi$
$421$	17.7574	0.865443	0.432722	+	0.901528i	$0.357553\pi$
$422$	−30.9248	−1.50539
$423$	0	0
$424$	−7.32105	−0.355542
$425$	−13.7793	−0.668392
$426$	0	0
$427$	8.06786	0.390431
$428$	−19.2668	−0.931295
$429$	0	0
$430$	−22.2701	−1.07396
$431$	−17.8382	−0.859234	−0.429617	−	0.903011i	$-0.641351\pi$
$431$	−17.8382	−0.859234	−0.429617	+	0.903011i	$0.641351\pi$
$432$	0	0
$433$	12.4405	0.597851	0.298925	−	0.954276i	$-0.403372\pi$
$433$	12.4405	0.597851	0.298925	+	0.954276i	$0.403372\pi$
$434$	18.9365	0.908982
$435$	0	0
$436$	0.714451	0.0342160
$437$	0.183164	0.00876193
$438$	0	0
$439$	16.3740	0.781489	0.390744	−	0.920499i	$-0.372218\pi$
$439$	16.3740	0.781489	0.390744	+	0.920499i	$0.372218\pi$
$440$	0	0
$441$	0	0
$442$	51.1226	2.43165
$443$	2.22725	0.105820	0.0529100	−	0.998599i	$-0.483150\pi$
$443$	2.22725	0.105820	0.0529100	+	0.998599i	$0.483150\pi$
$444$	0	0
$445$	14.0222	0.664715
$446$	11.6554	0.551897
$447$	0	0
$448$	−10.4195	−0.492276
$449$	31.9494	1.50779	0.753893	−	0.656997i	$-0.228174\pi$
$449$	31.9494	1.50779	0.753893	+	0.656997i	$0.228174\pi$
$450$	0	0
$451$	0	0
$452$	23.7034	1.11491
$453$	0	0
$454$	−15.6759	−0.735707
$455$	7.65617	0.358927
$456$	0	0
$457$	−22.1342	−1.03539	−0.517697	−	0.855564i	$-0.673210\pi$
$457$	−22.1342	−1.03539	−0.517697	+	0.855564i	$0.673210\pi$
$458$	−56.7026	−2.64954
$459$	0	0
$460$	3.50555	0.163447
$461$	−14.9408	−0.695865	−0.347932	−	0.937520i	$-0.613116\pi$
$461$	−14.9408	−0.695865	−0.347932	+	0.937520i	$0.613116\pi$
$462$	0	0
$463$	5.02953	0.233742	0.116871	−	0.993147i	$-0.462714\pi$
$463$	5.02953	0.233742	0.116871	+	0.993147i	$0.462714\pi$
$464$	−31.6419	−1.46894
$465$	0	0
$466$	−11.4306	−0.529512
$467$	−32.1377	−1.48716	−0.743579	−	0.668649i	$-0.766873\pi$
$467$	−32.1377	−1.48716	−0.743579	+	0.668649i	$0.766873\pi$
$468$	0	0
$469$	9.10333	0.420353
$470$	−7.74945	−0.357455
$471$	0	0
$472$	−9.50185	−0.437358
$473$	0	0
$474$	0	0
$475$	−0.524262	−0.0240548
$476$	9.97710	0.457300
$477$	0	0
$478$	−51.7228	−2.36574
$479$	−13.1491	−0.600798	−0.300399	−	0.953814i	$-0.597120\pi$
$479$	−13.1491	−0.600798	−0.300399	+	0.953814i	$0.597120\pi$
$480$	0	0
$481$	−48.4302	−2.20823
$482$	36.0703	1.64296
$483$	0	0
$484$	0	0
$485$	16.2016	0.735675
$486$	0	0
$487$	1.05225	0.0476821	0.0238411	−	0.999716i	$-0.492410\pi$
$487$	1.05225	0.0476821	0.0238411	+	0.999716i	$0.492410\pi$
$488$	5.63156	0.254929
$489$	0	0
$490$	2.77386	0.125310
$491$	27.3208	1.23297	0.616485	−	0.787367i	$-0.288556\pi$
$491$	27.3208	1.23297	0.616485	+	0.787367i	$0.288556\pi$
$492$	0	0
$493$	42.0211	1.89253
$494$	1.94507	0.0875129
$495$	0	0
$496$	−29.2591	−1.31377
$497$	1.01904	0.0457102
$498$	0	0
$499$	−23.4585	−1.05015	−0.525073	−	0.851057i	$-0.675962\pi$
$499$	−23.4585	−1.05015	−0.525073	+	0.851057i	$0.675962\pi$
$500$	−25.5891	−1.14438
$501$	0	0
$502$	−51.3567	−2.29216
$503$	−11.4795	−0.511846	−0.255923	−	0.966697i	$-0.582379\pi$
$503$	−11.4795	−0.511846	−0.255923	+	0.966697i	$0.582379\pi$
$504$	0	0
$505$	2.02749	0.0902219
$506$	0	0
$507$	0	0
$508$	8.89201	0.394519
$509$	1.40656	0.0623446	0.0311723	−	0.999514i	$-0.490076\pi$
$509$	1.40656	0.0623446	0.0311723	+	0.999514i	$0.490076\pi$
$510$	0	0
$511$	−7.64824	−0.338338
$512$	30.5322	1.34934
$513$	0	0
$514$	8.07651	0.356239
$515$	20.6048	0.907956
$516$	0	0
$517$	0	0
$518$	−17.5464	−0.770946
$519$	0	0
$520$	5.34419	0.234358
$521$	−18.5352	−0.812041	−0.406020	−	0.913864i	$-0.633084\pi$
$521$	−18.5352	−0.812041	−0.406020	+	0.913864i	$0.633084\pi$
$522$	0	0
$523$	−36.4670	−1.59459	−0.797295	−	0.603589i	$-0.793737\pi$
$523$	−36.4670	−1.59459	−0.797295	+	0.603589i	$0.793737\pi$
$524$	−34.2376	−1.49568
$525$	0	0
$526$	−10.8078	−0.471242
$527$	38.8566	1.69262
$528$	0	0
$529$	−21.7303	−0.944797
$530$	−29.0929	−1.26372
$531$	0	0
$532$	0.379601	0.0164578
$533$	25.9020	1.12194
$534$	0	0
$535$	−10.9914	−0.475200
$536$	6.35434	0.274466
$537$	0	0
$538$	−35.0623	−1.51164
$539$	0	0
$540$	0	0
$541$	22.1641	0.952910	0.476455	−	0.879199i	$-0.341921\pi$
$541$	22.1641	0.952910	0.476455	+	0.879199i	$0.341921\pi$
$542$	−29.1943	−1.25400
$543$	0	0
$544$	−34.5829	−1.48273
$545$	0.407584	0.0174590
$546$	0	0
$547$	−40.5974	−1.73582	−0.867909	−	0.496723i	$-0.834537\pi$
$547$	−40.5974	−1.73582	−0.867909	+	0.496723i	$0.834537\pi$
$548$	39.3036	1.67897
$549$	0	0
$550$	0	0
$551$	1.59878	0.0681105
$552$	0	0
$553$	−0.00547271	−0.000232723 0
$554$	1.96829	0.0836246
$555$	0	0
$556$	−18.2025	−0.771957
$557$	39.9780	1.69392	0.846961	−	0.531655i	$-0.178430\pi$
$557$	39.9780	1.69392	0.846961	+	0.531655i	$0.178430\pi$
$558$	0	0
$559$	−46.1394	−1.95149
$560$	−4.28592	−0.181113
$561$	0	0
$562$	8.40219	0.354425
$563$	−28.8313	−1.21509	−0.607546	−	0.794284i	$-0.707846\pi$
$563$	−28.8313	−1.21509	−0.607546	+	0.794284i	$0.707846\pi$
$564$	0	0
$565$	13.5224	0.568893
$566$	10.0067	0.420613
$567$	0	0
$568$	0.711314	0.0298461
$569$	−18.0183	−0.755365	−0.377683	−	0.925935i	$-0.623279\pi$
$569$	−18.0183	−0.755365	−0.377683	+	0.925935i	$0.623279\pi$
$570$	0	0
$571$	18.2208	0.762515	0.381258	−	0.924469i	$-0.375491\pi$
$571$	18.2208	0.762515	0.381258	+	0.924469i	$0.375491\pi$
$572$	0	0
$573$	0	0
$574$	9.38438	0.391697
$575$	−3.63413	−0.151554
$576$	0	0
$577$	10.5299	0.438364	0.219182	−	0.975684i	$-0.429661\pi$
$577$	10.5299	0.438364	0.219182	+	0.975684i	$0.429661\pi$
$578$	2.60973	0.108550
$579$	0	0
$580$	30.5988	1.27055
$581$	2.83233	0.117505
$582$	0	0
$583$	0	0
$584$	−5.33865	−0.220915
$585$	0	0
$586$	−2.17931	−0.0900264
$587$	−2.45140	−0.101180	−0.0505900	−	0.998720i	$-0.516110\pi$
$587$	−2.45140	−0.101180	−0.0505900	+	0.998720i	$0.516110\pi$
$588$	0	0
$589$	1.47839	0.0609158
$590$	−37.7591	−1.55452
$591$	0	0
$592$	27.1112	1.11426
$593$	27.7941	1.14137	0.570684	−	0.821170i	$-0.306678\pi$
$593$	27.7941	1.14137	0.570684	+	0.821170i	$0.306678\pi$
$594$	0	0
$595$	5.69179	0.233341
$596$	22.6983	0.929760
$597$	0	0
$598$	13.4830	0.551362
$599$	0.747037	0.0305231	0.0152616	−	0.999884i	$-0.495142\pi$
$599$	0.747037	0.0305231	0.0152616	+	0.999884i	$0.495142\pi$
$600$	0	0
$601$	−15.0794	−0.615103	−0.307551	−	0.951531i	$-0.599510\pi$
$601$	−15.0794	−0.615103	−0.307551	+	0.951531i	$0.599510\pi$
$602$	−16.7165	−0.681312
$603$	0	0
$604$	23.5896	0.959846
$605$	0	0
$606$	0	0
$607$	−4.16609	−0.169096	−0.0845481	−	0.996419i	$-0.526945\pi$
$607$	−4.16609	−0.169096	−0.0845481	+	0.996419i	$0.526945\pi$
$608$	−1.31578	−0.0533620
$609$	0	0
$610$	22.3791	0.906102
$611$	−16.0554	−0.649532
$612$	0	0
$613$	36.7033	1.48243	0.741216	−	0.671266i	$-0.234249\pi$
$613$	36.7033	1.48243	0.741216	+	0.671266i	$0.234249\pi$
$614$	−54.5370	−2.20093
$615$	0	0
$616$	0	0
$617$	−22.8780	−0.921035	−0.460518	−	0.887651i	$-0.652336\pi$
$617$	−22.8780	−0.921035	−0.460518	+	0.887651i	$0.652336\pi$
$618$	0	0
$619$	−2.59442	−0.104279	−0.0521393	−	0.998640i	$-0.516604\pi$
$619$	−2.59442	−0.104279	−0.0521393	+	0.998640i	$0.516604\pi$
$620$	28.2946	1.13634
$621$	0	0
$622$	−45.8730	−1.83934
$623$	10.5254	0.421691
$624$	0	0
$625$	1.52768	0.0611071
$626$	63.3214	2.53083
$627$	0	0
$628$	−11.5575	−0.461194
$629$	−36.0043	−1.43558
$630$	0	0
$631$	29.7938	1.18607	0.593036	−	0.805176i	$-0.297929\pi$
$631$	29.7938	1.18607	0.593036	+	0.805176i	$0.297929\pi$
$632$	−0.00382008	−0.000151955 0
$633$	0	0
$634$	−39.7220	−1.57756
$635$	5.07276	0.201306
$636$	0	0
$637$	5.74691	0.227701
$638$	0	0
$639$	0	0
$640$	−7.33488	−0.289937
$641$	32.9695	1.30222	0.651108	−	0.758985i	$-0.274305\pi$
$641$	32.9695	1.30222	0.651108	+	0.758985i	$0.274305\pi$
$642$	0	0
$643$	20.9257	0.825229	0.412614	−	0.910906i	$-0.364616\pi$
$643$	20.9257	0.825229	0.412614	+	0.910906i	$0.364616\pi$
$644$	2.63135	0.103690
$645$	0	0
$646$	1.44602	0.0568927
$647$	−12.9654	−0.509722	−0.254861	−	0.966978i	$-0.582030\pi$
$647$	−12.9654	−0.509722	−0.254861	+	0.966978i	$0.582030\pi$
$648$	0	0
$649$	0	0
$650$	−38.5918	−1.51370
$651$	0	0
$652$	15.7231	0.615765
$653$	4.59805	0.179936	0.0899678	−	0.995945i	$-0.471324\pi$
$653$	4.59805	0.179936	0.0899678	+	0.995945i	$0.471324\pi$
$654$	0	0
$655$	−19.5320	−0.763180
$656$	−14.4999	−0.566127
$657$	0	0
$658$	−5.81693	−0.226767
$659$	−27.8639	−1.08542	−0.542712	−	0.839919i	$-0.682602\pi$
$659$	−27.8639	−1.08542	−0.542712	+	0.839919i	$0.682602\pi$
$660$	0	0
$661$	36.2151	1.40861	0.704303	−	0.709900i	$-0.251260\pi$
$661$	36.2151	1.40861	0.704303	+	0.709900i	$0.251260\pi$
$662$	7.52144	0.292329
$663$	0	0
$664$	1.97703	0.0767237
$665$	0.216557	0.00839771
$666$	0	0
$667$	11.0826	0.429120
$668$	32.6640	1.26381
$669$	0	0
$670$	25.2513	0.975544
$671$	0	0
$672$	0	0
$673$	−6.97277	−0.268781	−0.134390	−	0.990928i	$-0.542908\pi$
$673$	−6.97277	−0.268781	−0.134390	+	0.990928i	$0.542908\pi$
$674$	−5.57675	−0.214808
$675$	0	0
$676$	46.7679	1.79876
$677$	−21.9071	−0.841958	−0.420979	−	0.907070i	$-0.638313\pi$
$677$	−21.9071	−0.841958	−0.420979	+	0.907070i	$0.638313\pi$
$678$	0	0
$679$	12.1613	0.466708
$680$	3.97301	0.152358
$681$	0	0
$682$	0	0
$683$	−48.9585	−1.87334	−0.936672	−	0.350208i	$-0.886111\pi$
$683$	−48.9585	−1.87334	−0.936672	+	0.350208i	$0.886111\pi$
$684$	0	0
$685$	22.4221	0.856706
$686$	2.08213	0.0794959
$687$	0	0
$688$	25.8288	0.984714
$689$	−60.2751	−2.29630
$690$	0	0
$691$	−47.8751	−1.82126	−0.910628	−	0.413227i	$-0.864402\pi$
$691$	−47.8751	−1.82126	−0.910628	+	0.413227i	$0.864402\pi$
$692$	−8.19384	−0.311483
$693$	0	0
$694$	−38.4248	−1.45859
$695$	−10.3842	−0.393897
$696$	0	0
$697$	19.2562	0.729381
$698$	42.2703	1.59995
$699$	0	0
$700$	−7.53159	−0.284667
$701$	6.95807	0.262803	0.131401	−	0.991329i	$-0.458052\pi$
$701$	6.95807	0.262803	0.131401	+	0.991329i	$0.458052\pi$
$702$	0	0
$703$	−1.36986	−0.0516653
$704$	0	0
$705$	0	0
$706$	−26.6034	−1.00123
$707$	1.52188	0.0572362
$708$	0	0
$709$	−20.3838	−0.765531	−0.382766	−	0.923845i	$-0.625028\pi$
$709$	−20.3838	−0.765531	−0.382766	+	0.923845i	$0.625028\pi$
$710$	2.82667	0.106083
$711$	0	0
$712$	7.34698	0.275340
$713$	10.2480	0.383791
$714$	0	0
$715$	0	0
$716$	5.20141	0.194386
$717$	0	0
$718$	21.4409	0.800166
$719$	−36.9536	−1.37814	−0.689068	−	0.724697i	$-0.741980\pi$
$719$	−36.9536	−1.37814	−0.689068	+	0.724697i	$0.741980\pi$
$720$	0	0
$721$	15.4665	0.576002
$722$	−39.5054	−1.47024
$723$	0	0
$724$	2.60220	0.0967101
$725$	−31.7212	−1.17809
$726$	0	0
$727$	40.8565	1.51528	0.757641	−	0.652671i	$-0.226352\pi$
$727$	40.8565	1.51528	0.757641	+	0.652671i	$0.226352\pi$
$728$	4.01148	0.148675
$729$	0	0
$730$	−21.2151	−0.785207
$731$	−34.3012	−1.26868
$732$	0	0
$733$	−17.9728	−0.663839	−0.331920	−	0.943308i	$-0.607696\pi$
$733$	−17.9728	−0.663839	−0.331920	+	0.943308i	$0.607696\pi$
$734$	5.51815	0.203679
$735$	0	0
$736$	−9.12086	−0.336199
$737$	0	0
$738$	0	0
$739$	−2.34892	−0.0864064	−0.0432032	−	0.999066i	$-0.513756\pi$
$739$	−2.34892	−0.0864064	−0.0432032	+	0.999066i	$0.513756\pi$
$740$	−26.2175	−0.963776
$741$	0	0
$742$	−21.8379	−0.801693
$743$	−38.2722	−1.40407	−0.702036	−	0.712141i	$-0.747726\pi$
$743$	−38.2722	−1.40407	−0.702036	+	0.712141i	$0.747726\pi$
$744$	0	0
$745$	12.9491	0.474417
$746$	68.5082	2.50826
$747$	0	0
$748$	0	0
$749$	−8.25043	−0.301464
$750$	0	0
$751$	−2.89080	−0.105487	−0.0527434	−	0.998608i	$-0.516797\pi$
$751$	−2.89080	−0.105487	−0.0527434	+	0.998608i	$0.516797\pi$
$752$	8.98781	0.327752
$753$	0	0
$754$	117.689	4.28599
$755$	13.4575	0.489769
$756$	0	0
$757$	24.1721	0.878551	0.439275	−	0.898353i	$-0.355235\pi$
$757$	24.1721	0.878551	0.439275	+	0.898353i	$0.355235\pi$
$758$	−46.2906	−1.68135
$759$	0	0
$760$	0.151162	0.00548321
$761$	−24.7044	−0.895533	−0.447766	−	0.894151i	$-0.647780\pi$
$761$	−24.7044	−0.895533	−0.447766	+	0.894151i	$0.647780\pi$
$762$	0	0
$763$	0.305943	0.0110759
$764$	23.8929	0.864415
$765$	0	0
$766$	52.3214	1.89045
$767$	−78.2298	−2.82472
$768$	0	0
$769$	−12.8939	−0.464967	−0.232483	−	0.972600i	$-0.574685\pi$
$769$	−12.8939	−0.464967	−0.232483	+	0.972600i	$0.574685\pi$
$770$	0	0
$771$	0	0
$772$	−41.8793	−1.50727
$773$	−32.4616	−1.16756	−0.583780	−	0.811912i	$-0.698427\pi$
$773$	−32.4616	−1.16756	−0.583780	+	0.811912i	$0.698427\pi$
$774$	0	0
$775$	−29.3324	−1.05365
$776$	8.48887	0.304733
$777$	0	0
$778$	−48.0196	−1.72159
$779$	0.732645	0.0262497
$780$	0	0
$781$	0	0
$782$	10.0236	0.358444
$783$	0	0
$784$	−3.21712	−0.114897
$785$	−6.59338	−0.235328
$786$	0	0
$787$	28.1909	1.00490	0.502448	−	0.864607i	$-0.332433\pi$
$787$	28.1909	1.00490	0.502448	+	0.864607i	$0.332433\pi$
$788$	10.0482	0.357952
$789$	0	0
$790$	−0.0151805	−0.000540098 0
$791$	10.1503	0.360902
$792$	0	0
$793$	46.3652	1.64648
$794$	−7.88093	−0.279684
$795$	0	0
$796$	51.5061	1.82559
$797$	42.5723	1.50799	0.753995	−	0.656881i	$-0.228124\pi$
$797$	42.5723	1.50799	0.753995	+	0.656881i	$0.228124\pi$
$798$	0	0
$799$	−11.9360	−0.422265
$800$	26.1062	0.922993
$801$	0	0
$802$	−81.3270	−2.87175
$803$	0	0
$804$	0	0
$805$	1.50115	0.0529085
$806$	108.826	3.83325
$807$	0	0
$808$	1.06231	0.0373719
$809$	−14.1894	−0.498873	−0.249437	−	0.968391i	$-0.580245\pi$
$809$	−14.1894	−0.498873	−0.249437	+	0.968391i	$0.580245\pi$
$810$	0	0
$811$	2.58292	0.0906987	0.0453494	−	0.998971i	$-0.485560\pi$
$811$	2.58292	0.0906987	0.0453494	+	0.998971i	$0.485560\pi$
$812$	22.9682	0.806027
$813$	0	0
$814$	0	0
$815$	8.96982	0.314199
$816$	0	0
$817$	−1.30506	−0.0456584
$818$	−61.2215	−2.14056
$819$	0	0
$820$	14.0220	0.489668
$821$	−52.5523	−1.83409	−0.917043	−	0.398787i	$-0.869431\pi$
$821$	−52.5523	−1.83409	−0.917043	+	0.398787i	$0.869431\pi$
$822$	0	0
$823$	14.6509	0.510699	0.255349	−	0.966849i	$-0.417809\pi$
$823$	14.6509	0.510699	0.255349	+	0.966849i	$0.417809\pi$
$824$	10.7960	0.376095
$825$	0	0
$826$	−28.3429	−0.986177
$827$	15.1928	0.528304	0.264152	−	0.964481i	$-0.414908\pi$
$827$	15.1928	0.528304	0.264152	+	0.964481i	$0.414908\pi$
$828$	0	0
$829$	−41.6162	−1.44539	−0.722695	−	0.691167i	$-0.757097\pi$
$829$	−41.6162	−1.44539	−0.722695	+	0.691167i	$0.757097\pi$
$830$	7.85647	0.272702
$831$	0	0
$832$	−59.8800	−2.07596
$833$	4.27240	0.148030
$834$	0	0
$835$	18.6343	0.644867
$836$	0	0
$837$	0	0
$838$	30.2970	1.04659
$839$	21.8835	0.755503	0.377752	−	0.925907i	$-0.376697\pi$
$839$	21.8835	0.755503	0.377752	+	0.925907i	$0.376697\pi$
$840$	0	0
$841$	67.7365	2.33574
$842$	36.9732	1.27418
$843$	0	0
$844$	−34.6842	−1.19388
$845$	26.6804	0.917834
$846$	0	0
$847$	0	0
$848$	33.7420	1.15870
$849$	0	0
$850$	−28.6901	−0.984063
$851$	−9.49572	−0.325509
$852$	0	0
$853$	−22.2054	−0.760299	−0.380149	−	0.924925i	$-0.624127\pi$
$853$	−22.2054	−0.760299	−0.380149	+	0.924925i	$0.624127\pi$
$854$	16.7983	0.574826
$855$	0	0
$856$	−5.75899	−0.196838
$857$	24.6876	0.843311	0.421656	−	0.906756i	$-0.361449\pi$
$857$	24.6876	0.843311	0.421656	+	0.906756i	$0.361449\pi$
$858$	0	0
$859$	10.3940	0.354639	0.177320	−	0.984153i	$-0.443257\pi$
$859$	10.3940	0.354639	0.177320	+	0.984153i	$0.443257\pi$
$860$	−24.9774	−0.851722
$861$	0	0
$862$	−37.1413	−1.26504
$863$	−45.9179	−1.56306	−0.781532	−	0.623865i	$-0.785561\pi$
$863$	−45.9179	−1.56306	−0.781532	+	0.623865i	$0.785561\pi$
$864$	0	0
$865$	−4.67447	−0.158937
$866$	25.9026	0.880207
$867$	0	0
$868$	21.2386	0.720885
$869$	0	0
$870$	0	0
$871$	52.3160	1.77266
$872$	0.213555	0.00723189
$873$	0	0
$874$	0.381371	0.0129001
$875$	−10.9578	−0.370441
$876$	0	0
$877$	34.8872	1.17806	0.589029	−	0.808112i	$-0.299510\pi$
$877$	34.8872	1.17806	0.589029	+	0.808112i	$0.299510\pi$
$878$	34.0927	1.15057
$879$	0	0
$880$	0	0
$881$	−8.85305	−0.298267	−0.149133	−	0.988817i	$-0.547648\pi$
$881$	−8.85305	−0.298267	−0.149133	+	0.988817i	$0.547648\pi$
$882$	0	0
$883$	17.3511	0.583910	0.291955	−	0.956432i	$-0.405694\pi$
$883$	17.3511	0.583910	0.291955	+	0.956432i	$0.405694\pi$
$884$	57.3375	1.92847
$885$	0	0
$886$	4.63742	0.155797
$887$	−38.0505	−1.27761	−0.638806	−	0.769368i	$-0.720571\pi$
$887$	−38.0505	−1.27761	−0.638806	+	0.769368i	$0.720571\pi$
$888$	0	0
$889$	3.80774	0.127707
$890$	29.1959	0.978650
$891$	0	0
$892$	13.0723	0.437692
$893$	−0.454131	−0.0151969
$894$	0	0
$895$	2.96733	0.0991868
$896$	−5.50574	−0.183934
$897$	0	0
$898$	66.5227	2.21989
$899$	89.4517	2.98338
$900$	0	0
$901$	−44.8100	−1.49284
$902$	0	0
$903$	0	0
$904$	7.08513	0.235648
$905$	1.48452	0.0493471
$906$	0	0
$907$	−14.1082	−0.468455	−0.234227	−	0.972182i	$-0.575256\pi$
$907$	−14.1082	−0.468455	−0.234227	+	0.972182i	$0.575256\pi$
$908$	−17.5816	−0.583466
$909$	0	0
$910$	15.9411	0.528443
$911$	29.1703	0.966457	0.483228	−	0.875494i	$-0.339464\pi$
$911$	29.1703	0.966457	0.483228	+	0.875494i	$0.339464\pi$
$912$	0	0
$913$	0	0
$914$	−46.0861	−1.52439
$915$	0	0
$916$	−63.5959	−2.10127
$917$	−14.6612	−0.484156
$918$	0	0
$919$	3.84247	0.126751	0.0633757	−	0.997990i	$-0.479813\pi$
$919$	3.84247	0.126751	0.0633757	+	0.997990i	$0.479813\pi$
$920$	1.04784	0.0345461
$921$	0	0
$922$	−31.1087	−1.02451
$923$	5.85633	0.192763
$924$	0	0
$925$	27.1792	0.893645
$926$	10.4721	0.344135
$927$	0	0
$928$	−79.6131	−2.61343
$929$	11.2254	0.368295	0.184147	−	0.982899i	$-0.441048\pi$
$929$	11.2254	0.368295	0.184147	+	0.982899i	$0.441048\pi$
$930$	0	0
$931$	0.162553	0.00532745
$932$	−12.8202	−0.419939
$933$	0	0
$934$	−66.9148	−2.18952
$935$	0	0
$936$	0	0
$937$	10.9651	0.358213	0.179106	−	0.983830i	$-0.442679\pi$
$937$	10.9651	0.358213	0.179106	+	0.983830i	$0.442679\pi$
$938$	18.9543	0.618879
$939$	0	0
$940$	−8.69154	−0.283487
$941$	−60.7290	−1.97971	−0.989855	−	0.142083i	$-0.954620\pi$
$941$	−60.7290	−1.97971	−0.989855	+	0.142083i	$0.954620\pi$
$942$	0	0
$943$	5.07861	0.165382
$944$	43.7930	1.42534
$945$	0	0
$946$	0	0
$947$	−45.2278	−1.46971	−0.734854	−	0.678226i	$-0.762749\pi$
$947$	−45.2278	−1.46971	−0.734854	+	0.678226i	$0.762749\pi$
$948$	0	0
$949$	−43.9537	−1.42680
$950$	−1.09158	−0.0354155
$951$	0	0
$952$	2.98224	0.0966548
$953$	−1.48140	−0.0479872	−0.0239936	−	0.999712i	$-0.507638\pi$
$953$	−1.48140	−0.0479872	−0.0239936	+	0.999712i	$0.507638\pi$
$954$	0	0
$955$	13.6306	0.441074
$956$	−58.0106	−1.87620
$957$	0	0
$958$	−27.3781	−0.884545
$959$	16.8306	0.543489
$960$	0	0
$961$	51.7155	1.66824
$962$	−100.838	−3.25114
$963$	0	0
$964$	40.4553	1.30298
$965$	−23.8915	−0.769095
$966$	0	0
$967$	−18.8504	−0.606187	−0.303094	−	0.952961i	$-0.598019\pi$
$967$	−18.8504	−0.606187	−0.303094	+	0.952961i	$0.598019\pi$
$968$	0	0
$969$	0	0
$970$	33.7337	1.08312
$971$	58.9217	1.89089	0.945444	−	0.325785i	$-0.105629\pi$
$971$	58.9217	1.89089	0.945444	+	0.325785i	$0.105629\pi$
$972$	0	0
$973$	−7.79467	−0.249886
$974$	2.19092	0.0702017
$975$	0	0
$976$	−25.9552	−0.830807
$977$	−43.0633	−1.37772	−0.688858	−	0.724896i	$-0.741888\pi$
$977$	−43.0633	−1.37772	−0.688858	+	0.724896i	$0.741888\pi$
$978$	0	0
$979$	0	0
$980$	3.11107	0.0993795
$981$	0	0
$982$	56.8853	1.81528
$983$	13.2499	0.422605	0.211303	−	0.977421i	$-0.432229\pi$
$983$	13.2499	0.422605	0.211303	+	0.977421i	$0.432229\pi$
$984$	0	0
$985$	5.73234	0.182648
$986$	87.4931	2.78635
$987$	0	0
$988$	2.18153	0.0694037
$989$	−9.04656	−0.287664
$990$	0	0
$991$	14.1260	0.448728	0.224364	−	0.974505i	$-0.427970\pi$
$991$	14.1260	0.448728	0.224364	+	0.974505i	$0.427970\pi$
$992$	−73.6178	−2.33737
$993$	0	0
$994$	2.12177	0.0672984
$995$	29.3835	0.931519
$996$	0	0
$997$	13.3911	0.424100	0.212050	−	0.977259i	$-0.431986\pi$
$997$	13.3911	0.424100	0.212050	+	0.977259i	$0.431986\pi$
$998$	−48.8435	−1.54611
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.dc.1.13		16
3.2	odd	2	inner	7623.2.a.dc.1.4		16
11.3	even	5		693.2.m.k.64.7	yes	32
11.4	even	5		693.2.m.k.379.7	yes	32
11.10	odd	2		7623.2.a.db.1.4		16
33.14	odd	10		693.2.m.k.64.2	✓	32
33.26	odd	10		693.2.m.k.379.2	yes	32
33.32	even	2		7623.2.a.db.1.13		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
693.2.m.k.64.2	✓	32	33.14	odd	10
693.2.m.k.64.7	yes	32	11.3	even	5
693.2.m.k.379.2	yes	32	33.26	odd	10
693.2.m.k.379.7	yes	32	11.4	even	5
7623.2.a.db.1.4		16	11.10	odd	2
7623.2.a.db.1.13		16	33.32	even	2
7623.2.a.dc.1.4		16	3.2	odd	2	inner
7623.2.a.dc.1.13		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)

\(f(q)\)	\(=\)	\(q+2.08213 q^{2} +2.33525 q^{4} +1.33222 q^{5} +1.00000 q^{7} +0.698024 q^{8} +O(q^{10})\)	\(q+2.08213 q^{2} +2.33525 q^{4} +1.33222 q^{5} +1.00000 q^{7} +0.698024 q^{8} +2.77386 q^{10} +5.74691 q^{13} +2.08213 q^{14} -3.21712 q^{16} +4.27240 q^{17} +0.162553 q^{19} +3.11107 q^{20} +1.12680 q^{23} -3.22518 q^{25} +11.9658 q^{26} +2.33525 q^{28} +9.83547 q^{29} +9.09480 q^{31} -8.09449 q^{32} +8.89567 q^{34} +1.33222 q^{35} -8.42718 q^{37} +0.338455 q^{38} +0.929924 q^{40} +4.50712 q^{41} -8.02855 q^{43} +2.34613 q^{46} -2.79374 q^{47} +1.00000 q^{49} -6.71523 q^{50} +13.4204 q^{52} -10.4883 q^{53} +0.698024 q^{56} +20.4787 q^{58} -13.6125 q^{59} +8.06786 q^{61} +18.9365 q^{62} -10.4195 q^{64} +7.65617 q^{65} +9.10333 q^{67} +9.97710 q^{68} +2.77386 q^{70} +1.01904 q^{71} -7.64824 q^{73} -17.5464 q^{74} +0.379601 q^{76} -0.00547271 q^{79} -4.28592 q^{80} +9.38438 q^{82} +2.83233 q^{83} +5.69179 q^{85} -16.7165 q^{86} +10.5254 q^{89} +5.74691 q^{91} +2.63135 q^{92} -5.81693 q^{94} +0.216557 q^{95} +12.1613 q^{97} +2.08213 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 20 q^{4} + 16 q^{7}+O(q^{10}) \) 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

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