LMFDB - Embedded newform 7623.2.a.ct.1.2

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:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{7} - 11x^{6} + 10x^{5} + 35x^{4} - 30x^{3} - 30x^{2} + 30x - 5 $ x^8 - x^7 - 11x^6 + 10x^5 + 35x^4 - 30x^3 - 30x^2 + 30x - 5
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 77)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.98451$ 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)

$f(q)$	$=$	$q-1.98451 q^{2} +1.93830 q^{4} +0.0269243 q^{5} +1.00000 q^{7} +0.122446 q^{8} +O(q^{10})$	$q-1.98451 q^{2} +1.93830 q^{4} +0.0269243 q^{5} +1.00000 q^{7} +0.122446 q^{8} -0.0534317 q^{10} -4.88112 q^{13} -1.98451 q^{14} -4.11959 q^{16} -1.67878 q^{17} +1.37394 q^{19} +0.0521874 q^{20} -8.06246 q^{23} -4.99928 q^{25} +9.68665 q^{26} +1.93830 q^{28} +6.39316 q^{29} +4.01054 q^{31} +7.93050 q^{32} +3.33156 q^{34} +0.0269243 q^{35} +0.521038 q^{37} -2.72661 q^{38} +0.00329677 q^{40} +10.5987 q^{41} +3.73968 q^{43} +16.0001 q^{46} +8.95906 q^{47} +1.00000 q^{49} +9.92114 q^{50} -9.46107 q^{52} +3.96057 q^{53} +0.122446 q^{56} -12.6873 q^{58} -9.73022 q^{59} +8.46723 q^{61} -7.95898 q^{62} -7.49902 q^{64} -0.131421 q^{65} +2.81285 q^{67} -3.25398 q^{68} -0.0534317 q^{70} -2.04551 q^{71} -10.4693 q^{73} -1.03401 q^{74} +2.66311 q^{76} +5.85521 q^{79} -0.110917 q^{80} -21.0334 q^{82} -2.60548 q^{83} -0.0452000 q^{85} -7.42146 q^{86} -1.21791 q^{89} -4.88112 q^{91} -15.6275 q^{92} -17.7794 q^{94} +0.0369924 q^{95} +3.39670 q^{97} -1.98451 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - q^{2} + 7 q^{4} - 10 q^{5} + 8 q^{7}+O(q^{10}) $ 8 * q - q^2 + 7 * q^4 - 10 * q^5 + 8 * q^7	$ 8 q - q^{2} + 7 q^{4} - 10 q^{5} + 8 q^{7} + 6 q^{10} - 6 q^{13} - q^{14} + q^{16} + 5 q^{17} - 13 q^{19} - 23 q^{20} - 16 q^{23} + 16 q^{25} + 6 q^{26} + 7 q^{28} - 9 q^{29} + 9 q^{31} - 16 q^{32} - 12 q^{34} - 10 q^{35} + 7 q^{37} + 10 q^{38} + 5 q^{40} + 10 q^{41} - 4 q^{43} + 4 q^{46} - 16 q^{47} + 8 q^{49} - 6 q^{50} - 41 q^{52} - 37 q^{53} - 15 q^{58} - q^{59} + 19 q^{61} + 18 q^{62} - 4 q^{64} + 4 q^{65} - 19 q^{67} - 9 q^{68} + 6 q^{70} - 13 q^{71} - 25 q^{73} - 33 q^{74} + 26 q^{76} - 4 q^{80} - 13 q^{82} + 25 q^{83} + 3 q^{85} - 4 q^{86} - 37 q^{89} - 6 q^{91} - 35 q^{92} - 42 q^{94} - 21 q^{95} + 15 q^{97} - q^{98}+O(q^{100}) $ 8 * q - q^2 + 7 * q^4 - 10 * q^5 + 8 * q^7 + 6 * q^10 - 6 * q^13 - q^14 + q^16 + 5 * q^17 - 13 * q^19 - 23 * q^20 - 16 * q^23 + 16 * q^25 + 6 * q^26 + 7 * q^28 - 9 * q^29 + 9 * q^31 - 16 * q^32 - 12 * q^34 - 10 * q^35 + 7 * q^37 + 10 * q^38 + 5 * q^40 + 10 * q^41 - 4 * q^43 + 4 * q^46 - 16 * q^47 + 8 * q^49 - 6 * q^50 - 41 * q^52 - 37 * q^53 - 15 * q^58 - q^59 + 19 * q^61 + 18 * q^62 - 4 * q^64 + 4 * q^65 - 19 * q^67 - 9 * q^68 + 6 * q^70 - 13 * q^71 - 25 * q^73 - 33 * q^74 + 26 * q^76 - 4 * q^80 - 13 * q^82 + 25 * q^83 + 3 * q^85 - 4 * q^86 - 37 * q^89 - 6 * q^91 - 35 * q^92 - 42 * q^94 - 21 * q^95 + 15 * q^97 - q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	−1.98451	−1.40326	−0.701632	−	0.712540i	$-0.747545\pi$
$2$	−1.98451	−1.40326	−0.701632	+	0.712540i	$0.747545\pi$
$3$	0	0
$3$	0	0
$4$	1.93830	0.969150
$5$	0.0269243	0.0120409	0.00602046	−	0.999982i	$-0.498084\pi$
$5$	0.0269243	0.0120409	0.00602046	+	0.999982i	$0.498084\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0.122446	0.0432911
$9$	0	0
$10$	−0.0534317	−0.0168966
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−4.88112	−1.35378	−0.676889	−	0.736085i	$-0.736672\pi$
$13$	−4.88112	−1.35378	−0.676889	+	0.736085i	$0.736672\pi$
$14$	−1.98451	−0.530384
$15$	0	0
$16$	−4.11959	−1.02990
$17$	−1.67878	−0.407164	−0.203582	−	0.979058i	$-0.565258\pi$
$17$	−1.67878	−0.407164	−0.203582	+	0.979058i	$0.565258\pi$
$18$	0	0
$19$	1.37394	0.315204	0.157602	−	0.987503i	$-0.449624\pi$
$19$	1.37394	0.315204	0.157602	+	0.987503i	$0.449624\pi$
$20$	0.0521874	0.0116695
$21$	0	0
$22$	0	0
$23$	−8.06246	−1.68114	−0.840570	−	0.541703i	$-0.817780\pi$
$23$	−8.06246	−1.68114	−0.840570	+	0.541703i	$0.817780\pi$
$24$	0	0
$25$	−4.99928	−0.999855
$26$	9.68665	1.89971
$27$	0	0
$28$	1.93830	0.366304
$29$	6.39316	1.18718	0.593590	−	0.804768i	$-0.297710\pi$
$29$	6.39316	1.18718	0.593590	+	0.804768i	$0.297710\pi$
$30$	0	0
$31$	4.01054	0.720315	0.360157	−	0.932892i	$-0.382723\pi$
$31$	4.01054	0.720315	0.360157	+	0.932892i	$0.382723\pi$
$32$	7.93050	1.40193
$33$	0	0
$34$	3.33156	0.571358
$35$	0.0269243	0.00455104
$36$	0	0
$37$	0.521038	0.0856581	0.0428290	−	0.999082i	$-0.486363\pi$
$37$	0.521038	0.0856581	0.0428290	+	0.999082i	$0.486363\pi$
$38$	−2.72661	−0.442314
$39$	0	0
$40$	0.00329677	0.000521265 0
$41$	10.5987	1.65525	0.827623	−	0.561285i	$-0.189693\pi$
$41$	10.5987	1.65525	0.827623	+	0.561285i	$0.189693\pi$
$42$	0	0
$43$	3.73968	0.570296	0.285148	−	0.958483i	$-0.407957\pi$
$43$	3.73968	0.570296	0.285148	+	0.958483i	$0.407957\pi$
$44$	0	0
$45$	0	0
$46$	16.0001	2.35908
$47$	8.95906	1.30681	0.653407	−	0.757007i	$-0.273339\pi$
$47$	8.95906	1.30681	0.653407	+	0.757007i	$0.273339\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	9.92114	1.40306
$51$	0	0
$52$	−9.46107	−1.31201
$53$	3.96057	0.544027	0.272013	−	0.962293i	$-0.412311\pi$
$53$	3.96057	0.544027	0.272013	+	0.962293i	$0.412311\pi$
$54$	0	0
$55$	0	0
$56$	0.122446	0.0163625
$57$	0	0
$58$	−12.6873	−1.66593
$59$	−9.73022	−1.26677	−0.633383	−	0.773838i	$-0.718334\pi$
$59$	−9.73022	−1.26677	−0.633383	+	0.773838i	$0.718334\pi$
$60$	0	0
$61$	8.46723	1.08412	0.542059	−	0.840341i	$-0.317645\pi$
$61$	8.46723	1.08412	0.542059	+	0.840341i	$0.317645\pi$
$62$	−7.95898	−1.01079
$63$	0	0
$64$	−7.49902	−0.937377
$65$	−0.131421	−0.0163008
$66$	0	0
$67$	2.81285	0.343644	0.171822	−	0.985128i	$-0.445035\pi$
$67$	2.81285	0.343644	0.171822	+	0.985128i	$0.445035\pi$
$68$	−3.25398	−0.394603
$69$	0	0
$70$	−0.0534317	−0.00638631
$71$	−2.04551	−0.242757	−0.121379	−	0.992606i	$-0.538732\pi$
$71$	−2.04551	−0.242757	−0.121379	+	0.992606i	$0.538732\pi$
$72$	0	0
$73$	−10.4693	−1.22534	−0.612671	−	0.790338i	$-0.709905\pi$
$73$	−10.4693	−1.22534	−0.612671	+	0.790338i	$0.709905\pi$
$74$	−1.03401	−0.120201
$75$	0	0
$76$	2.66311	0.305479
$77$	0	0
$78$	0	0
$79$	5.85521	0.658763	0.329381	−	0.944197i	$-0.393160\pi$
$79$	5.85521	0.658763	0.329381	+	0.944197i	$0.393160\pi$
$80$	−0.110917	−0.0124009
$81$	0	0
$82$	−21.0334	−2.32275
$83$	−2.60548	−0.285988	−0.142994	−	0.989724i	$-0.545673\pi$
$83$	−2.60548	−0.285988	−0.142994	+	0.989724i	$0.545673\pi$
$84$	0	0
$85$	−0.0452000	−0.00490263
$86$	−7.42146	−0.800276
$87$	0	0
$88$	0	0
$89$	−1.21791	−0.129099	−0.0645493	−	0.997915i	$-0.520561\pi$
$89$	−1.21791	−0.129099	−0.0645493	+	0.997915i	$0.520561\pi$
$90$	0	0
$91$	−4.88112	−0.511680
$92$	−15.6275	−1.62928
$93$	0	0
$94$	−17.7794	−1.83380
$95$	0.0369924	0.00379534
$96$	0	0
$97$	3.39670	0.344883	0.172441	−	0.985020i	$-0.444834\pi$
$97$	3.39670	0.344883	0.172441	+	0.985020i	$0.444834\pi$
$98$	−1.98451	−0.200466
$99$	0	0
$100$	−9.69009	−0.969009
$101$	4.04836	0.402826	0.201413	−	0.979506i	$-0.435447\pi$
$101$	4.04836	0.402826	0.201413	+	0.979506i	$0.435447\pi$
$102$	0	0
$103$	−3.95297	−0.389498	−0.194749	−	0.980853i	$-0.562389\pi$
$103$	−3.95297	−0.389498	−0.194749	+	0.980853i	$0.562389\pi$
$104$	−0.597672	−0.0586066
$105$	0	0
$106$	−7.85982	−0.763413
$107$	−10.3986	−1.00527	−0.502635	−	0.864498i	$-0.667636\pi$
$107$	−10.3986	−1.00527	−0.502635	+	0.864498i	$0.667636\pi$
$108$	0	0
$109$	−3.77697	−0.361768	−0.180884	−	0.983504i	$-0.557896\pi$
$109$	−3.77697	−0.361768	−0.180884	+	0.983504i	$0.557896\pi$
$110$	0	0
$111$	0	0
$112$	−4.11959	−0.389265
$113$	−6.34869	−0.597234	−0.298617	−	0.954373i	$-0.596525\pi$
$113$	−6.34869	−0.597234	−0.298617	+	0.954373i	$0.596525\pi$
$114$	0	0
$115$	−0.217076	−0.0202425
$116$	12.3919	1.15056
$117$	0	0
$118$	19.3098	1.77761
$119$	−1.67878	−0.153893
$120$	0	0
$121$	0	0
$122$	−16.8033	−1.52130
$123$	0	0
$124$	7.77363	0.698093
$125$	−0.269224	−0.0240801
$126$	0	0
$127$	−5.03128	−0.446454	−0.223227	−	0.974766i	$-0.571659\pi$
$127$	−5.03128	−0.446454	−0.223227	+	0.974766i	$0.571659\pi$
$128$	−0.979100	−0.0865410
$129$	0	0
$130$	0.260807	0.0228743
$131$	3.76357	0.328825	0.164412	−	0.986392i	$-0.447427\pi$
$131$	3.76357	0.328825	0.164412	+	0.986392i	$0.447427\pi$
$132$	0	0
$133$	1.37394	0.119136
$134$	−5.58214	−0.482223
$135$	0	0
$136$	−0.205559	−0.0176266
$137$	−19.1379	−1.63506	−0.817529	−	0.575887i	$-0.804657\pi$
$137$	−19.1379	−1.63506	−0.817529	+	0.575887i	$0.804657\pi$
$138$	0	0
$139$	3.51108	0.297806	0.148903	−	0.988852i	$-0.452426\pi$
$139$	3.51108	0.297806	0.148903	+	0.988852i	$0.452426\pi$
$140$	0.0521874	0.00441064
$141$	0	0
$142$	4.05934	0.340652
$143$	0	0
$144$	0	0
$145$	0.172132	0.0142948
$146$	20.7766	1.71948
$147$	0	0
$148$	1.00993	0.0830155
$149$	3.12643	0.256127	0.128063	−	0.991766i	$-0.459124\pi$
$149$	3.12643	0.256127	0.128063	+	0.991766i	$0.459124\pi$
$150$	0	0
$151$	19.7215	1.60491	0.802456	−	0.596712i	$-0.203526\pi$
$151$	19.7215	1.60491	0.802456	+	0.596712i	$0.203526\pi$
$152$	0.168233	0.0136455
$153$	0	0
$154$	0	0
$155$	0.107981	0.00867326
$156$	0	0
$157$	−22.9546	−1.83197	−0.915987	−	0.401207i	$-0.868591\pi$
$157$	−22.9546	−1.83197	−0.915987	+	0.401207i	$0.868591\pi$
$158$	−11.6198	−0.924418
$159$	0	0
$160$	0.213524	0.0168805
$161$	−8.06246	−0.635411
$162$	0	0
$163$	18.9891	1.48734	0.743670	−	0.668547i	$-0.233083\pi$
$163$	18.9891	1.48734	0.743670	+	0.668547i	$0.233083\pi$
$164$	20.5435	1.60418
$165$	0	0
$166$	5.17061	0.401317
$167$	3.02044	0.233728	0.116864	−	0.993148i	$-0.462716\pi$
$167$	3.02044	0.233728	0.116864	+	0.993148i	$0.462716\pi$
$168$	0	0
$169$	10.8253	0.832717
$170$	0.0897001	0.00687968
$171$	0	0
$172$	7.24863	0.552703
$173$	−8.83629	−0.671811	−0.335905	−	0.941896i	$-0.609042\pi$
$173$	−8.83629	−0.671811	−0.335905	+	0.941896i	$0.609042\pi$
$174$	0	0
$175$	−4.99928	−0.377910
$176$	0	0
$177$	0	0
$178$	2.41697	0.181159
$179$	−8.46434	−0.632654	−0.316327	−	0.948650i	$-0.602450\pi$
$179$	−8.46434	−0.632654	−0.316327	+	0.948650i	$0.602450\pi$
$180$	0	0
$181$	−24.2008	−1.79883	−0.899415	−	0.437095i	$-0.856007\pi$
$181$	−24.2008	−1.79883	−0.899415	+	0.437095i	$0.856007\pi$
$182$	9.68665	0.718022
$183$	0	0
$184$	−0.987214	−0.0727784
$185$	0.0140286	0.00103140
$186$	0	0
$187$	0	0
$188$	17.3653	1.26650
$189$	0	0
$190$	−0.0734120	−0.00532587
$191$	7.78718	0.563460	0.281730	−	0.959494i	$-0.409092\pi$
$191$	7.78718	0.563460	0.281730	+	0.959494i	$0.409092\pi$
$192$	0	0
$193$	17.0637	1.22827	0.614134	−	0.789201i	$-0.289505\pi$
$193$	17.0637	1.22827	0.614134	+	0.789201i	$0.289505\pi$
$194$	−6.74081	−0.483962
$195$	0	0
$196$	1.93830	0.138450
$197$	6.05536	0.431426	0.215713	−	0.976457i	$-0.430792\pi$
$197$	6.05536	0.431426	0.215713	+	0.976457i	$0.430792\pi$
$198$	0	0
$199$	−13.7181	−0.972451	−0.486226	−	0.873833i	$-0.661627\pi$
$199$	−13.7181	−0.972451	−0.486226	+	0.873833i	$0.661627\pi$
$200$	−0.612140	−0.0432848
$201$	0	0
$202$	−8.03402	−0.565272
$203$	6.39316	0.448712
$204$	0	0
$205$	0.285364	0.0199307
$206$	7.84472	0.546568
$207$	0	0
$208$	20.1082	1.39425
$209$	0	0
$210$	0	0
$211$	18.0352	1.24160	0.620799	−	0.783970i	$-0.286808\pi$
$211$	18.0352	1.24160	0.620799	+	0.783970i	$0.286808\pi$
$212$	7.67678	0.527243
$213$	0	0
$214$	20.6362	1.41066
$215$	0.100688	0.00686690
$216$	0	0
$217$	4.01054	0.272253
$218$	7.49546	0.507656
$219$	0	0
$220$	0	0
$221$	8.19432	0.551210
$222$	0	0
$223$	−6.53314	−0.437491	−0.218746	−	0.975782i	$-0.570197\pi$
$223$	−6.53314	−0.437491	−0.218746	+	0.975782i	$0.570197\pi$
$224$	7.93050	0.529879
$225$	0	0
$226$	12.5991	0.838077
$227$	−13.6077	−0.903176	−0.451588	−	0.892227i	$-0.649142\pi$
$227$	−13.6077	−0.903176	−0.451588	+	0.892227i	$0.649142\pi$
$228$	0	0
$229$	−19.4211	−1.28338	−0.641690	−	0.766964i	$-0.721767\pi$
$229$	−19.4211	−1.28338	−0.641690	+	0.766964i	$0.721767\pi$
$230$	0.430791	0.0284055
$231$	0	0
$232$	0.782815	0.0513943
$233$	−4.36150	−0.285731	−0.142866	−	0.989742i	$-0.545632\pi$
$233$	−4.36150	−0.285731	−0.142866	+	0.989742i	$0.545632\pi$
$234$	0	0
$235$	0.241217	0.0157353
$236$	−18.8601	−1.22769
$237$	0	0
$238$	3.33156	0.215953
$239$	−10.5901	−0.685018	−0.342509	−	0.939515i	$-0.611277\pi$
$239$	−10.5901	−0.685018	−0.342509	+	0.939515i	$0.611277\pi$
$240$	0	0
$241$	18.6887	1.20384	0.601921	−	0.798555i	$-0.294402\pi$
$241$	18.6887	1.20384	0.601921	+	0.798555i	$0.294402\pi$
$242$	0	0
$243$	0	0
$244$	16.4120	1.05067
$245$	0.0269243	0.00172013
$246$	0	0
$247$	−6.70637	−0.426716
$248$	0.491074	0.0311832
$249$	0	0
$250$	0.534279	0.0337907
$251$	−18.9832	−1.19821	−0.599103	−	0.800672i	$-0.704476\pi$
$251$	−18.9832	−1.19821	−0.599103	+	0.800672i	$0.704476\pi$
$252$	0	0
$253$	0	0
$254$	9.98465	0.626493
$255$	0	0
$256$	16.9411	1.05882
$257$	13.7090	0.855147	0.427573	−	0.903981i	$-0.359369\pi$
$257$	13.7090	0.855147	0.427573	+	0.903981i	$0.359369\pi$
$258$	0	0
$259$	0.521038	0.0323757
$260$	−0.254733	−0.0157979
$261$	0	0
$262$	−7.46887	−0.461428
$263$	−16.5767	−1.02217	−0.511083	−	0.859531i	$-0.670756\pi$
$263$	−16.5767	−1.02217	−0.511083	+	0.859531i	$0.670756\pi$
$264$	0	0
$265$	0.106636	0.00655059
$266$	−2.72661	−0.167179
$267$	0	0
$268$	5.45214	0.333043
$269$	8.54361	0.520913	0.260457	−	0.965486i	$-0.416127\pi$
$269$	8.54361	0.520913	0.260457	+	0.965486i	$0.416127\pi$
$270$	0	0
$271$	−6.34553	−0.385463	−0.192732	−	0.981251i	$-0.561735\pi$
$271$	−6.34553	−0.385463	−0.192732	+	0.981251i	$0.561735\pi$
$272$	6.91589	0.419337
$273$	0	0
$274$	37.9794	2.29442
$275$	0	0
$276$	0	0
$277$	−24.0044	−1.44228	−0.721142	−	0.692788i	$-0.756382\pi$
$277$	−24.0044	−1.44228	−0.721142	+	0.692788i	$0.756382\pi$
$278$	−6.96780	−0.417901
$279$	0	0
$280$	0.00329677	0.000197020 0
$281$	3.21144	0.191578	0.0957891	−	0.995402i	$-0.469463\pi$
$281$	3.21144	0.191578	0.0957891	+	0.995402i	$0.469463\pi$
$282$	0	0
$283$	−20.8034	−1.23663	−0.618317	−	0.785929i	$-0.712185\pi$
$283$	−20.8034	−1.23663	−0.618317	+	0.785929i	$0.712185\pi$
$284$	−3.96481	−0.235268
$285$	0	0
$286$	0	0
$287$	10.5987	0.625624
$288$	0	0
$289$	−14.1817	−0.834218
$290$	−0.341598	−0.0200593
$291$	0	0
$292$	−20.2927	−1.18754
$293$	−5.51744	−0.322332	−0.161166	−	0.986927i	$-0.551526\pi$
$293$	−5.51744	−0.322332	−0.161166	+	0.986927i	$0.551526\pi$
$294$	0	0
$295$	−0.261980	−0.0152530
$296$	0.0637988	0.00370823
$297$	0	0
$298$	−6.20444	−0.359414
$299$	39.3538	2.27589
$300$	0	0
$301$	3.73968	0.215552
$302$	−39.1376	−2.25211
$303$	0	0
$304$	−5.66008	−0.324628
$305$	0.227974	0.0130538
$306$	0	0
$307$	8.60991	0.491394	0.245697	−	0.969347i	$-0.420983\pi$
$307$	8.60991	0.491394	0.245697	+	0.969347i	$0.420983\pi$
$308$	0	0
$309$	0	0
$310$	−0.214290	−0.0121709
$311$	−19.1258	−1.08453	−0.542263	−	0.840209i	$-0.682432\pi$
$311$	−19.1258	−1.08453	−0.542263	+	0.840209i	$0.682432\pi$
$312$	0	0
$313$	−0.606755	−0.0342958	−0.0171479	−	0.999853i	$-0.505459\pi$
$313$	−0.606755	−0.0342958	−0.0171479	+	0.999853i	$0.505459\pi$
$314$	45.5537	2.57074
$315$	0	0
$316$	11.3492	0.638440
$317$	5.21283	0.292782	0.146391	−	0.989227i	$-0.453234\pi$
$317$	5.21283	0.292782	0.146391	+	0.989227i	$0.453234\pi$
$318$	0	0
$319$	0	0
$320$	−0.201906	−0.0112869
$321$	0	0
$322$	16.0001	0.891650
$323$	−2.30654	−0.128339
$324$	0	0
$325$	24.4021	1.35358
$326$	−37.6841	−2.08713
$327$	0	0
$328$	1.29777	0.0716574
$329$	8.95906	0.493929
$330$	0	0
$331$	0.669012	0.0367722	0.0183861	−	0.999831i	$-0.494147\pi$
$331$	0.669012	0.0367722	0.0183861	+	0.999831i	$0.494147\pi$
$332$	−5.05019	−0.277165
$333$	0	0
$334$	−5.99410	−0.327983
$335$	0.0757340	0.00413779
$336$	0	0
$337$	−6.60215	−0.359642	−0.179821	−	0.983699i	$-0.557552\pi$
$337$	−6.60215	−0.359642	−0.179821	+	0.983699i	$0.557552\pi$
$338$	−21.4830	−1.16852
$339$	0	0
$340$	−0.0876111	−0.00475138
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	0.457908	0.0246888
$345$	0	0
$346$	17.5358	0.942728
$347$	−12.2001	−0.654937	−0.327469	−	0.944862i	$-0.606196\pi$
$347$	−12.2001	−0.654937	−0.327469	+	0.944862i	$0.606196\pi$
$348$	0	0
$349$	−0.885800	−0.0474158	−0.0237079	−	0.999719i	$-0.507547\pi$
$349$	−0.885800	−0.0474158	−0.0237079	+	0.999719i	$0.507547\pi$
$350$	9.92114	0.530307
$351$	0	0
$352$	0	0
$353$	−5.36012	−0.285291	−0.142645	−	0.989774i	$-0.545561\pi$
$353$	−5.36012	−0.285291	−0.142645	+	0.989774i	$0.545561\pi$
$354$	0	0
$355$	−0.0550740	−0.00292302
$356$	−2.36068	−0.125116
$357$	0	0
$358$	16.7976	0.887781
$359$	−25.8450	−1.36405	−0.682023	−	0.731330i	$-0.738900\pi$
$359$	−25.8450	−1.36405	−0.682023	+	0.731330i	$0.738900\pi$
$360$	0	0
$361$	−17.1123	−0.900647
$362$	48.0268	2.52423
$363$	0	0
$364$	−9.46107	−0.495895
$365$	−0.281880	−0.0147543
$366$	0	0
$367$	−20.4504	−1.06750	−0.533752	−	0.845641i	$-0.679218\pi$
$367$	−20.4504	−1.06750	−0.533752	+	0.845641i	$0.679218\pi$
$368$	33.2141	1.73140
$369$	0	0
$370$	−0.0278400	−0.00144733
$371$	3.96057	0.205623
$372$	0	0
$373$	9.39109	0.486252	0.243126	−	0.969995i	$-0.421827\pi$
$373$	9.39109	0.486252	0.243126	+	0.969995i	$0.421827\pi$
$374$	0	0
$375$	0	0
$376$	1.09700	0.0565734
$377$	−31.2058	−1.60718
$378$	0	0
$379$	−5.63593	−0.289498	−0.144749	−	0.989468i	$-0.546238\pi$
$379$	−5.63593	−0.289498	−0.144749	+	0.989468i	$0.546238\pi$
$380$	0.0717024	0.00367826
$381$	0	0
$382$	−15.4538	−0.790684
$383$	1.25944	0.0643544	0.0321772	−	0.999482i	$-0.489756\pi$
$383$	1.25944	0.0643544	0.0321772	+	0.999482i	$0.489756\pi$
$384$	0	0
$385$	0	0
$386$	−33.8631	−1.72359
$387$	0	0
$388$	6.58383	0.334243
$389$	5.88409	0.298335	0.149168	−	0.988812i	$-0.452341\pi$
$389$	5.88409	0.298335	0.149168	+	0.988812i	$0.452341\pi$
$390$	0	0
$391$	13.5351	0.684499
$392$	0.122446	0.00618444
$393$	0	0
$394$	−12.0169	−0.605405
$395$	0.157648	0.00793212
$396$	0	0
$397$	4.86018	0.243925	0.121963	−	0.992535i	$-0.461081\pi$
$397$	4.86018	0.243925	0.121963	+	0.992535i	$0.461081\pi$
$398$	27.2238	1.36461
$399$	0	0
$400$	20.5950	1.02975
$401$	25.5322	1.27502	0.637508	−	0.770444i	$-0.279965\pi$
$401$	25.5322	1.27502	0.637508	+	0.770444i	$0.279965\pi$
$402$	0	0
$403$	−19.5759	−0.975147
$404$	7.84692	0.390399
$405$	0	0
$406$	−12.6873	−0.629661
$407$	0	0
$408$	0	0
$409$	21.0267	1.03970	0.519851	−	0.854257i	$-0.325988\pi$
$409$	21.0267	1.03970	0.519851	+	0.854257i	$0.325988\pi$
$410$	−0.566309	−0.0279680
$411$	0	0
$412$	−7.66204	−0.377481
$413$	−9.73022	−0.478793
$414$	0	0
$415$	−0.0701507	−0.00344356
$416$	−38.7097	−1.89790
$417$	0	0
$418$	0	0
$419$	31.3141	1.52980	0.764898	−	0.644151i	$-0.222789\pi$
$419$	31.3141	1.52980	0.764898	+	0.644151i	$0.222789\pi$
$420$	0	0
$421$	12.7886	0.623280	0.311640	−	0.950200i	$-0.399122\pi$
$421$	12.7886	0.623280	0.311640	+	0.950200i	$0.399122\pi$
$422$	−35.7912	−1.74229
$423$	0	0
$424$	0.484955	0.0235515
$425$	8.39268	0.407105
$426$	0	0
$427$	8.46723	0.409758
$428$	−20.1556	−0.974258
$429$	0	0
$430$	−0.199818	−0.00963607
$431$	−21.0679	−1.01480	−0.507402	−	0.861709i	$-0.669394\pi$
$431$	−21.0679	−1.01480	−0.507402	+	0.861709i	$0.669394\pi$
$432$	0	0
$433$	38.0030	1.82631	0.913155	−	0.407613i	$-0.133639\pi$
$433$	38.0030	1.82631	0.913155	+	0.407613i	$0.133639\pi$
$434$	−7.95898	−0.382043
$435$	0	0
$436$	−7.32090	−0.350608
$437$	−11.0773	−0.529901
$438$	0	0
$439$	−37.7677	−1.80256	−0.901278	−	0.433241i	$-0.857370\pi$
$439$	−37.7677	−1.80256	−0.901278	+	0.433241i	$0.857370\pi$
$440$	0	0
$441$	0	0
$442$	−16.2618	−0.773493
$443$	1.02792	0.0488381	0.0244190	−	0.999702i	$-0.492226\pi$
$443$	1.02792	0.0488381	0.0244190	+	0.999702i	$0.492226\pi$
$444$	0	0
$445$	−0.0327915	−0.00155447
$446$	12.9651	0.613916
$447$	0	0
$448$	−7.49902	−0.354295
$449$	−20.1333	−0.950149	−0.475074	−	0.879946i	$-0.657579\pi$
$449$	−20.1333	−0.950149	−0.475074	+	0.879946i	$0.657579\pi$
$450$	0	0
$451$	0	0
$452$	−12.3057	−0.578809
$453$	0	0
$454$	27.0047	1.26739
$455$	−0.131421	−0.00616111
$456$	0	0
$457$	−24.1456	−1.12948	−0.564742	−	0.825268i	$-0.691024\pi$
$457$	−24.1456	−1.12948	−0.564742	+	0.825268i	$0.691024\pi$
$458$	38.5414	1.80092
$459$	0	0
$460$	−0.420759	−0.0196180
$461$	−11.5885	−0.539731	−0.269865	−	0.962898i	$-0.586979\pi$
$461$	−11.5885	−0.539731	−0.269865	+	0.962898i	$0.586979\pi$
$462$	0	0
$463$	21.6077	1.00419	0.502097	−	0.864811i	$-0.332562\pi$
$463$	21.6077	1.00419	0.502097	+	0.864811i	$0.332562\pi$
$464$	−26.3372	−1.22268
$465$	0	0
$466$	8.65547	0.400957
$467$	−15.9149	−0.736456	−0.368228	−	0.929736i	$-0.620035\pi$
$467$	−15.9149	−0.736456	−0.368228	+	0.929736i	$0.620035\pi$
$468$	0	0
$469$	2.81285	0.129885
$470$	−0.478698	−0.0220807
$471$	0	0
$472$	−1.19142	−0.0548397
$473$	0	0
$474$	0	0
$475$	−6.86871	−0.315158
$476$	−3.25398	−0.149146
$477$	0	0
$478$	21.0162	0.961261
$479$	−5.28372	−0.241419	−0.120710	−	0.992688i	$-0.538517\pi$
$479$	−5.28372	−0.241419	−0.120710	+	0.992688i	$0.538517\pi$
$480$	0	0
$481$	−2.54325	−0.115962
$482$	−37.0879	−1.68931
$483$	0	0
$484$	0	0
$485$	0.0914540	0.00415271
$486$	0	0
$487$	−27.0656	−1.22646	−0.613229	−	0.789905i	$-0.710130\pi$
$487$	−27.0656	−1.22646	−0.613229	+	0.789905i	$0.710130\pi$
$488$	1.03678	0.0469326
$489$	0	0
$490$	−0.0534317	−0.00241380
$491$	−18.8000	−0.848431	−0.424215	−	0.905561i	$-0.639450\pi$
$491$	−18.8000	−0.848431	−0.424215	+	0.905561i	$0.639450\pi$
$492$	0	0
$493$	−10.7327	−0.483377
$494$	13.3089	0.598795
$495$	0	0
$496$	−16.5218	−0.741851
$497$	−2.04551	−0.0917536
$498$	0	0
$499$	−39.7667	−1.78020	−0.890100	−	0.455765i	$-0.849366\pi$
$499$	−39.7667	−1.78020	−0.890100	+	0.455765i	$0.849366\pi$
$500$	−0.521836	−0.0233372
$501$	0	0
$502$	37.6724	1.68140
$503$	−40.5624	−1.80859	−0.904294	−	0.426910i	$-0.859602\pi$
$503$	−40.5624	−1.80859	−0.904294	+	0.426910i	$0.859602\pi$
$504$	0	0
$505$	0.108999	0.00485040
$506$	0	0
$507$	0	0
$508$	−9.75213	−0.432681
$509$	15.4809	0.686179	0.343090	−	0.939303i	$-0.388527\pi$
$509$	15.4809	0.686179	0.343090	+	0.939303i	$0.388527\pi$
$510$	0	0
$511$	−10.4693	−0.463136
$512$	−31.6616	−1.39926
$513$	0	0
$514$	−27.2058	−1.20000
$515$	−0.106431	−0.00468991
$516$	0	0
$517$	0	0
$518$	−1.03401	−0.0454317
$519$	0	0
$520$	−0.0160919	−0.000705677 0
$521$	−37.4738	−1.64176	−0.820878	−	0.571103i	$-0.806516\pi$
$521$	−37.4738	−1.64176	−0.820878	+	0.571103i	$0.806516\pi$
$522$	0	0
$523$	−20.8115	−0.910024	−0.455012	−	0.890485i	$-0.650365\pi$
$523$	−20.8115	−0.910024	−0.455012	+	0.890485i	$0.650365\pi$
$524$	7.29493	0.318681
$525$	0	0
$526$	32.8968	1.43437
$527$	−6.73281	−0.293286
$528$	0	0
$529$	42.0033	1.82623
$530$	−0.211620	−0.00919220
$531$	0	0
$532$	2.66311	0.115460
$533$	−51.7337	−2.24084
$534$	0	0
$535$	−0.279975	−0.0121044
$536$	0.344421	0.0148767
$537$	0	0
$538$	−16.9549	−0.730979
$539$	0	0
$540$	0	0
$541$	−17.4259	−0.749196	−0.374598	−	0.927187i	$-0.622219\pi$
$541$	−17.4259	−0.749196	−0.374598	+	0.927187i	$0.622219\pi$
$542$	12.5928	0.540907
$543$	0	0
$544$	−13.3136	−0.570814
$545$	−0.101692	−0.00435602
$546$	0	0
$547$	−15.7960	−0.675386	−0.337693	−	0.941256i	$-0.609647\pi$
$547$	−15.7960	−0.675386	−0.337693	+	0.941256i	$0.609647\pi$
$548$	−37.0949	−1.58462
$549$	0	0
$550$	0	0
$551$	8.78382	0.374203
$552$	0	0
$553$	5.85521	0.248989
$554$	47.6371	2.02390
$555$	0	0
$556$	6.80553	0.288619
$557$	16.8351	0.713325	0.356663	−	0.934233i	$-0.383915\pi$
$557$	16.8351	0.713325	0.356663	+	0.934233i	$0.383915\pi$
$558$	0	0
$559$	−18.2538	−0.772055
$560$	−0.110917	−0.00468711
$561$	0	0
$562$	−6.37315	−0.268835
$563$	37.4304	1.57750	0.788751	−	0.614713i	$-0.210728\pi$
$563$	37.4304	1.57750	0.788751	+	0.614713i	$0.210728\pi$
$564$	0	0
$565$	−0.170934	−0.00719125
$566$	41.2846	1.73532
$567$	0	0
$568$	−0.250464	−0.0105092
$569$	2.76910	0.116087	0.0580433	−	0.998314i	$-0.481514\pi$
$569$	2.76910	0.116087	0.0580433	+	0.998314i	$0.481514\pi$
$570$	0	0
$571$	−8.85289	−0.370482	−0.185241	−	0.982693i	$-0.559307\pi$
$571$	−8.85289	−0.370482	−0.185241	+	0.982693i	$0.559307\pi$
$572$	0	0
$573$	0	0
$574$	−21.0334	−0.877916
$575$	40.3065	1.68090
$576$	0	0
$577$	37.1682	1.54733	0.773667	−	0.633593i	$-0.218420\pi$
$577$	37.1682	1.54733	0.773667	+	0.633593i	$0.218420\pi$
$578$	28.1438	1.17063
$579$	0	0
$580$	0.333643	0.0138538
$581$	−2.60548	−0.108093
$582$	0	0
$583$	0	0
$584$	−1.28193	−0.0530464
$585$	0	0
$586$	10.9494	0.452317
$587$	35.3189	1.45777	0.728884	−	0.684638i	$-0.240040\pi$
$587$	35.3189	1.45777	0.728884	+	0.684638i	$0.240040\pi$
$588$	0	0
$589$	5.51025	0.227046
$590$	0.519902	0.0214040
$591$	0	0
$592$	−2.14646	−0.0882191
$593$	8.09224	0.332308	0.166154	−	0.986100i	$-0.446865\pi$
$593$	8.09224	0.332308	0.166154	+	0.986100i	$0.446865\pi$
$594$	0	0
$595$	−0.0452000	−0.00185302
$596$	6.05995	0.248225
$597$	0	0
$598$	−78.0983	−3.19368
$599$	−3.83616	−0.156741	−0.0783706	−	0.996924i	$-0.524972\pi$
$599$	−3.83616	−0.156741	−0.0783706	+	0.996924i	$0.524972\pi$
$600$	0	0
$601$	−14.3366	−0.584804	−0.292402	−	0.956296i	$-0.594455\pi$
$601$	−14.3366	−0.584804	−0.292402	+	0.956296i	$0.594455\pi$
$602$	−7.42146	−0.302476
$603$	0	0
$604$	38.2261	1.55540
$605$	0	0
$606$	0	0
$607$	13.7786	0.559255	0.279627	−	0.960109i	$-0.409789\pi$
$607$	13.7786	0.559255	0.279627	+	0.960109i	$0.409789\pi$
$608$	10.8960	0.441893
$609$	0	0
$610$	−0.452419	−0.0183179
$611$	−43.7303	−1.76914
$612$	0	0
$613$	−9.54435	−0.385493	−0.192746	−	0.981249i	$-0.561739\pi$
$613$	−9.54435	−0.385493	−0.192746	+	0.981249i	$0.561739\pi$
$614$	−17.0865	−0.689555
$615$	0	0
$616$	0	0
$617$	31.5153	1.26876	0.634379	−	0.773023i	$-0.281256\pi$
$617$	31.5153	1.26876	0.634379	+	0.773023i	$0.281256\pi$
$618$	0	0
$619$	20.5297	0.825159	0.412579	−	0.910922i	$-0.364628\pi$
$619$	20.5297	0.825159	0.412579	+	0.910922i	$0.364628\pi$
$620$	0.209300	0.00840568
$621$	0	0
$622$	37.9555	1.52188
$623$	−1.21791	−0.0487947
$624$	0	0
$625$	24.9891	0.999565
$626$	1.20411	0.0481261
$627$	0	0
$628$	−44.4928	−1.77546
$629$	−0.874707	−0.0348769
$630$	0	0
$631$	35.9658	1.43178	0.715889	−	0.698214i	$-0.246022\pi$
$631$	35.9658	1.43178	0.715889	+	0.698214i	$0.246022\pi$
$632$	0.716946	0.0285186
$633$	0	0
$634$	−10.3449	−0.410850
$635$	−0.135464	−0.00537572
$636$	0	0
$637$	−4.88112	−0.193397
$638$	0	0
$639$	0	0
$640$	−0.0263616	−0.00104203
$641$	−13.6205	−0.537976	−0.268988	−	0.963144i	$-0.586689\pi$
$641$	−13.6205	−0.537976	−0.268988	+	0.963144i	$0.586689\pi$
$642$	0	0
$643$	21.9891	0.867166	0.433583	−	0.901114i	$-0.357249\pi$
$643$	21.9891	0.867166	0.433583	+	0.901114i	$0.357249\pi$
$644$	−15.6275	−0.615809
$645$	0	0
$646$	4.57737	0.180094
$647$	14.8203	0.582644	0.291322	−	0.956625i	$-0.405905\pi$
$647$	14.8203	0.582644	0.291322	+	0.956625i	$0.405905\pi$
$648$	0	0
$649$	0	0
$650$	−48.4262	−1.89943
$651$	0	0
$652$	36.8065	1.44145
$653$	−43.3776	−1.69750	−0.848748	−	0.528798i	$-0.822643\pi$
$653$	−43.3776	−1.69750	−0.848748	+	0.528798i	$0.822643\pi$
$654$	0	0
$655$	0.101332	0.00395936
$656$	−43.6625	−1.70473
$657$	0	0
$658$	−17.7794	−0.693113
$659$	−42.2093	−1.64424	−0.822121	−	0.569313i	$-0.807209\pi$
$659$	−42.2093	−1.64424	−0.822121	+	0.569313i	$0.807209\pi$
$660$	0	0
$661$	16.2794	0.633197	0.316599	−	0.948560i	$-0.397459\pi$
$661$	16.2794	0.633197	0.316599	+	0.948560i	$0.397459\pi$
$662$	−1.32766	−0.0516012
$663$	0	0
$664$	−0.319029	−0.0123807
$665$	0.0369924	0.00143451
$666$	0	0
$667$	−51.5446	−1.99582
$668$	5.85451	0.226518
$669$	0	0
$670$	−0.150295	−0.00580642
$671$	0	0
$672$	0	0
$673$	−5.31047	−0.204704	−0.102352	−	0.994748i	$-0.532637\pi$
$673$	−5.31047	−0.204704	−0.102352	+	0.994748i	$0.532637\pi$
$674$	13.1021	0.504673
$675$	0	0
$676$	20.9827	0.807027
$677$	33.7947	1.29884	0.649419	−	0.760431i	$-0.275012\pi$
$677$	33.7947	1.29884	0.649419	+	0.760431i	$0.275012\pi$
$678$	0	0
$679$	3.39670	0.130354
$680$	−0.00553455	−0.000212240 0
$681$	0	0
$682$	0	0
$683$	15.8834	0.607761	0.303880	−	0.952710i	$-0.401718\pi$
$683$	15.8834	0.607761	0.303880	+	0.952710i	$0.401718\pi$
$684$	0	0
$685$	−0.515274	−0.0196876
$686$	−1.98451	−0.0757691
$687$	0	0
$688$	−15.4060	−0.587347
$689$	−19.3320	−0.736492
$690$	0	0
$691$	25.5375	0.971490	0.485745	−	0.874100i	$-0.338548\pi$
$691$	25.5375	0.971490	0.485745	+	0.874100i	$0.338548\pi$
$692$	−17.1274	−0.651085
$693$	0	0
$694$	24.2113	0.919050
$695$	0.0945336	0.00358586
$696$	0	0
$697$	−17.7929	−0.673956
$698$	1.75788	0.0665369
$699$	0	0
$700$	−9.69009	−0.366251
$701$	3.29535	0.124464	0.0622319	−	0.998062i	$-0.480178\pi$
$701$	3.29535	0.124464	0.0622319	+	0.998062i	$0.480178\pi$
$702$	0	0
$703$	0.715875	0.0269997
$704$	0	0
$705$	0	0
$706$	10.6372	0.400338
$707$	4.04836	0.152254
$708$	0	0
$709$	−44.9026	−1.68635	−0.843176	−	0.537638i	$-0.819317\pi$
$709$	−44.9026	−1.68635	−0.843176	+	0.537638i	$0.819317\pi$
$710$	0.109295	0.00410177
$711$	0	0
$712$	−0.149128	−0.00558882
$713$	−32.3348	−1.21095
$714$	0	0
$715$	0	0
$716$	−16.4064	−0.613137
$717$	0	0
$718$	51.2898	1.91412
$719$	−3.19264	−0.119065	−0.0595327	−	0.998226i	$-0.518961\pi$
$719$	−3.19264	−0.119065	−0.0595327	+	0.998226i	$0.518961\pi$
$720$	0	0
$721$	−3.95297	−0.147216
$722$	33.9596	1.26385
$723$	0	0
$724$	−46.9084	−1.74334
$725$	−31.9612	−1.18701
$726$	0	0
$727$	20.1654	0.747891	0.373946	−	0.927451i	$-0.378005\pi$
$727$	20.1654	0.747891	0.373946	+	0.927451i	$0.378005\pi$
$728$	−0.597672	−0.0221512
$729$	0	0
$730$	0.559395	0.0207041
$731$	−6.27810	−0.232204
$732$	0	0
$733$	−19.7094	−0.727985	−0.363992	−	0.931402i	$-0.618587\pi$
$733$	−19.7094	−0.727985	−0.363992	+	0.931402i	$0.618587\pi$
$734$	40.5842	1.49799
$735$	0	0
$736$	−63.9394	−2.35684
$737$	0	0
$738$	0	0
$739$	−34.5575	−1.27122	−0.635610	−	0.772010i	$-0.719251\pi$
$739$	−34.5575	−1.27122	−0.635610	+	0.772010i	$0.719251\pi$
$740$	0.0271916	0.000999584 0
$741$	0	0
$742$	−7.85982	−0.288543
$743$	25.8381	0.947910	0.473955	−	0.880549i	$-0.342826\pi$
$743$	25.8381	0.947910	0.473955	+	0.880549i	$0.342826\pi$
$744$	0	0
$745$	0.0841770	0.00308401
$746$	−18.6368	−0.682340
$747$	0	0
$748$	0	0
$749$	−10.3986	−0.379957
$750$	0	0
$751$	−35.7199	−1.30344	−0.651719	−	0.758460i	$-0.725952\pi$
$751$	−35.7199	−1.30344	−0.651719	+	0.758460i	$0.725952\pi$
$752$	−36.9077	−1.34589
$753$	0	0
$754$	61.9283	2.25530
$755$	0.530988	0.0193246
$756$	0	0
$757$	−14.8197	−0.538629	−0.269315	−	0.963052i	$-0.586797\pi$
$757$	−14.8197	−0.538629	−0.269315	+	0.963052i	$0.586797\pi$
$758$	11.1846	0.406242
$759$	0	0
$760$	0.00452957	0.000164305 0
$761$	21.7759	0.789377	0.394689	−	0.918815i	$-0.370852\pi$
$761$	21.7759	0.789377	0.394689	+	0.918815i	$0.370852\pi$
$762$	0	0
$763$	−3.77697	−0.136736
$764$	15.0939	0.546077
$765$	0	0
$766$	−2.49938	−0.0903062
$767$	47.4943	1.71492
$768$	0	0
$769$	−35.6991	−1.28734	−0.643672	−	0.765302i	$-0.722590\pi$
$769$	−35.6991	−1.28734	−0.643672	+	0.765302i	$0.722590\pi$
$770$	0	0
$771$	0	0
$772$	33.0745	1.19038
$773$	−42.6229	−1.53304	−0.766520	−	0.642220i	$-0.778013\pi$
$773$	−42.6229	−1.53304	−0.766520	+	0.642220i	$0.778013\pi$
$774$	0	0
$775$	−20.0498	−0.720210
$776$	0.415912	0.0149304
$777$	0	0
$778$	−11.6771	−0.418643
$779$	14.5620	0.521739
$780$	0	0
$781$	0	0
$782$	−26.8606	−0.960533
$783$	0	0
$784$	−4.11959	−0.147128
$785$	−0.618037	−0.0220587
$786$	0	0
$787$	−47.0867	−1.67846	−0.839230	−	0.543777i	$-0.816994\pi$
$787$	−47.0867	−1.67846	−0.839230	+	0.543777i	$0.816994\pi$
$788$	11.7371	0.418117
$789$	0	0
$790$	−0.312854	−0.0111309
$791$	−6.34869	−0.225733
$792$	0	0
$793$	−41.3295	−1.46765
$794$	−9.64509	−0.342291
$795$	0	0
$796$	−26.5898	−0.942451
$797$	7.48263	0.265048	0.132524	−	0.991180i	$-0.457692\pi$
$797$	7.48263	0.265048	0.132524	+	0.991180i	$0.457692\pi$
$798$	0	0
$799$	−15.0403	−0.532087
$800$	−39.6468	−1.40173
$801$	0	0
$802$	−50.6690	−1.78918
$803$	0	0
$804$	0	0
$805$	−0.217076	−0.00765094
$806$	38.8487	1.36839
$807$	0	0
$808$	0.495704	0.0174388
$809$	−34.7490	−1.22171	−0.610855	−	0.791743i	$-0.709174\pi$
$809$	−34.7490	−1.22171	−0.610855	+	0.791743i	$0.709174\pi$
$810$	0	0
$811$	12.8133	0.449935	0.224968	−	0.974366i	$-0.427772\pi$
$811$	12.8133	0.449935	0.224968	+	0.974366i	$0.427772\pi$
$812$	12.3919	0.434869
$813$	0	0
$814$	0	0
$815$	0.511268	0.0179089
$816$	0	0
$817$	5.13810	0.179759
$818$	−41.7277	−1.45898
$819$	0	0
$820$	0.553121	0.0193158
$821$	54.3225	1.89587	0.947934	−	0.318466i	$-0.103168\pi$
$821$	54.3225	1.89587	0.947934	+	0.318466i	$0.103168\pi$
$822$	0	0
$823$	−19.7697	−0.689129	−0.344565	−	0.938763i	$-0.611973\pi$
$823$	−19.7697	−0.689129	−0.344565	+	0.938763i	$0.611973\pi$
$824$	−0.484024	−0.0168618
$825$	0	0
$826$	19.3098	0.671873
$827$	−12.3098	−0.428055	−0.214028	−	0.976828i	$-0.568658\pi$
$827$	−12.3098	−0.428055	−0.214028	+	0.976828i	$0.568658\pi$
$828$	0	0
$829$	−25.7259	−0.893496	−0.446748	−	0.894660i	$-0.647418\pi$
$829$	−25.7259	−0.893496	−0.446748	+	0.894660i	$0.647418\pi$
$830$	0.139215	0.00483223
$831$	0	0
$832$	36.6036	1.26900
$833$	−1.67878	−0.0581662
$834$	0	0
$835$	0.0813232	0.00281431
$836$	0	0
$837$	0	0
$838$	−62.1434	−2.14671
$839$	46.6136	1.60928	0.804640	−	0.593762i	$-0.202358\pi$
$839$	46.6136	1.60928	0.804640	+	0.593762i	$0.202358\pi$
$840$	0	0
$841$	11.8725	0.409397
$842$	−25.3793	−0.874627
$843$	0	0
$844$	34.9577	1.20329
$845$	0.291465	0.0100267
$846$	0	0
$847$	0	0
$848$	−16.3160	−0.560292
$849$	0	0
$850$	−16.6554	−0.571275
$851$	−4.20085	−0.144003
$852$	0	0
$853$	−9.80881	−0.335847	−0.167924	−	0.985800i	$-0.553706\pi$
$853$	−9.80881	−0.335847	−0.167924	+	0.985800i	$0.553706\pi$
$854$	−16.8033	−0.574998
$855$	0	0
$856$	−1.27326	−0.0435193
$857$	32.9168	1.12442	0.562208	−	0.826996i	$-0.309952\pi$
$857$	32.9168	1.12442	0.562208	+	0.826996i	$0.309952\pi$
$858$	0	0
$859$	−28.4747	−0.971543	−0.485772	−	0.874086i	$-0.661461\pi$
$859$	−28.4747	−0.971543	−0.485772	+	0.874086i	$0.661461\pi$
$860$	0.195164	0.00665505
$861$	0	0
$862$	41.8095	1.42404
$863$	−29.2770	−0.996600	−0.498300	−	0.867005i	$-0.666042\pi$
$863$	−29.2770	−0.996600	−0.498300	+	0.867005i	$0.666042\pi$
$864$	0	0
$865$	−0.237911	−0.00808922
$866$	−75.4176	−2.56279
$867$	0	0
$868$	7.77363	0.263854
$869$	0	0
$870$	0	0
$871$	−13.7298	−0.465218
$872$	−0.462474	−0.0156613
$873$	0	0
$874$	21.9832	0.743591
$875$	−0.269224	−0.00910143
$876$	0	0
$877$	−22.6025	−0.763233	−0.381616	−	0.924321i	$-0.624632\pi$
$877$	−22.6025	−0.763233	−0.381616	+	0.924321i	$0.624632\pi$
$878$	74.9506	2.52946
$879$	0	0
$880$	0	0
$881$	35.1102	1.18289	0.591447	−	0.806344i	$-0.298557\pi$
$881$	35.1102	1.18289	0.591447	+	0.806344i	$0.298557\pi$
$882$	0	0
$883$	15.2032	0.511628	0.255814	−	0.966726i	$-0.417657\pi$
$883$	15.2032	0.511628	0.255814	+	0.966726i	$0.417657\pi$
$884$	15.8830	0.534205
$885$	0	0
$886$	−2.03993	−0.0685327
$887$	38.5316	1.29377	0.646883	−	0.762589i	$-0.276072\pi$
$887$	38.5316	1.29377	0.646883	+	0.762589i	$0.276072\pi$
$888$	0	0
$889$	−5.03128	−0.168744
$890$	0.0650753	0.00218133
$891$	0	0
$892$	−12.6632	−0.423995
$893$	12.3092	0.411912
$894$	0	0
$895$	−0.227897	−0.00761774
$896$	−0.979100	−0.0327094
$897$	0	0
$898$	39.9548	1.33331
$899$	25.6400	0.855143
$900$	0	0
$901$	−6.64893	−0.221508
$902$	0	0
$903$	0	0
$904$	−0.777369	−0.0258549
$905$	−0.651590	−0.0216596
$906$	0	0
$907$	−56.4214	−1.87344	−0.936721	−	0.350076i	$-0.886156\pi$
$907$	−56.4214	−1.87344	−0.936721	+	0.350076i	$0.886156\pi$
$908$	−26.3758	−0.875312
$909$	0	0
$910$	0.260807	0.00864566
$911$	30.5904	1.01350	0.506752	−	0.862092i	$-0.330846\pi$
$911$	30.5904	1.01350	0.506752	+	0.862092i	$0.330846\pi$
$912$	0	0
$913$	0	0
$914$	47.9173	1.58496
$915$	0	0
$916$	−37.6439	−1.24379
$917$	3.76357	0.124284
$918$	0	0
$919$	32.8904	1.08495	0.542477	−	0.840071i	$-0.317487\pi$
$919$	32.8904	1.08495	0.542477	+	0.840071i	$0.317487\pi$
$920$	−0.0265801	−0.000876319 0
$921$	0	0
$922$	22.9976	0.757385
$923$	9.98437	0.328640
$924$	0	0
$925$	−2.60481	−0.0856457
$926$	−42.8808	−1.40915
$927$	0	0
$928$	50.7010	1.66434
$929$	−21.8326	−0.716305	−0.358152	−	0.933663i	$-0.616593\pi$
$929$	−21.8326	−0.716305	−0.358152	+	0.933663i	$0.616593\pi$
$930$	0	0
$931$	1.37394	0.0450291
$932$	−8.45390	−0.276917
$933$	0	0
$934$	31.5835	1.03344
$935$	0	0
$936$	0	0
$937$	−34.9523	−1.14184	−0.570921	−	0.821005i	$-0.693414\pi$
$937$	−34.9523	−1.14184	−0.570921	+	0.821005i	$0.693414\pi$
$938$	−5.58214	−0.182263
$939$	0	0
$940$	0.467550	0.0152498
$941$	20.4205	0.665689	0.332845	−	0.942982i	$-0.391992\pi$
$941$	20.4205	0.665689	0.332845	+	0.942982i	$0.391992\pi$
$942$	0	0
$943$	−85.4520	−2.78270
$944$	40.0845	1.30464
$945$	0	0
$946$	0	0
$947$	11.3122	0.367597	0.183799	−	0.982964i	$-0.441161\pi$
$947$	11.3122	0.367597	0.183799	+	0.982964i	$0.441161\pi$
$948$	0	0
$949$	51.1021	1.65884
$950$	13.6311	0.442250
$951$	0	0
$952$	−0.205559	−0.00666222
$953$	4.55919	0.147687	0.0738434	−	0.997270i	$-0.476474\pi$
$953$	4.55919	0.147687	0.0738434	+	0.997270i	$0.476474\pi$
$954$	0	0
$955$	0.209665	0.00678459
$956$	−20.5268	−0.663885
$957$	0	0
$958$	10.4856	0.338775
$959$	−19.1379	−0.617994
$960$	0	0
$961$	−14.9156	−0.481147
$962$	5.04711	0.162725
$963$	0	0
$964$	36.2242	1.16670
$965$	0.459428	0.0147895
$966$	0	0
$967$	−38.9153	−1.25143	−0.625715	−	0.780052i	$-0.715193\pi$
$967$	−38.9153	−1.25143	−0.625715	+	0.780052i	$0.715193\pi$
$968$	0	0
$969$	0	0
$970$	−0.181492	−0.00582735
$971$	−42.3409	−1.35878	−0.679392	−	0.733775i	$-0.737757\pi$
$971$	−42.3409	−1.35878	−0.679392	+	0.733775i	$0.737757\pi$
$972$	0	0
$973$	3.51108	0.112560
$974$	53.7120	1.72104
$975$	0	0
$976$	−34.8815	−1.11653
$977$	−35.6575	−1.14078	−0.570392	−	0.821372i	$-0.693209\pi$
$977$	−35.6575	−1.14078	−0.570392	+	0.821372i	$0.693209\pi$
$978$	0	0
$979$	0	0
$980$	0.0521874	0.00166707
$981$	0	0
$982$	37.3088	1.19057
$983$	29.5213	0.941583	0.470791	−	0.882245i	$-0.343968\pi$
$983$	29.5213	0.941583	0.470791	+	0.882245i	$0.343968\pi$
$984$	0	0
$985$	0.163036	0.00519477
$986$	21.2992	0.678305
$987$	0	0
$988$	−12.9989	−0.413552
$989$	−30.1511	−0.958748
$990$	0	0
$991$	−30.7292	−0.976145	−0.488072	−	0.872803i	$-0.662300\pi$
$991$	−30.7292	−0.976145	−0.488072	+	0.872803i	$0.662300\pi$
$992$	31.8056	1.00983
$993$	0	0
$994$	4.05934	0.128755
$995$	−0.369351	−0.0117092
$996$	0	0
$997$	49.9958	1.58338	0.791691	−	0.610922i	$-0.209201\pi$
$997$	49.9958	1.58338	0.791691	+	0.610922i	$0.209201\pi$
$998$	78.9175	2.49809
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.ct.1.2		8
3.2	odd	2		847.2.a.p.1.7		8
11.5	even	5		693.2.m.i.190.4		16
11.9	even	5		693.2.m.i.631.4		16
11.10	odd	2		7623.2.a.cw.1.7		8
21.20	even	2		5929.2.a.bt.1.7		8
33.2	even	10		847.2.f.x.323.4		16
33.5	odd	10		77.2.f.b.36.1	yes	16
33.8	even	10		847.2.f.v.372.1		16
33.14	odd	10		847.2.f.w.372.4		16
33.17	even	10		847.2.f.x.729.4		16
33.20	odd	10		77.2.f.b.15.1	✓	16
33.26	odd	10		847.2.f.w.148.4		16
33.29	even	10		847.2.f.v.148.1		16
33.32	even	2		847.2.a.o.1.2		8
231.5	even	30		539.2.q.f.410.4		32
231.20	even	10		539.2.f.e.246.1		16
231.38	even	30		539.2.q.f.520.1		32
231.53	odd	30		539.2.q.g.422.4		32
231.86	odd	30		539.2.q.g.312.1		32
231.104	even	10		539.2.f.e.344.1		16
231.137	odd	30		539.2.q.g.520.1		32
231.152	even	30		539.2.q.f.312.1		32
231.170	odd	30		539.2.q.g.410.4		32
231.185	even	30		539.2.q.f.422.4		32
231.230	odd	2		5929.2.a.bs.1.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.f.b.15.1	✓	16	33.20	odd	10
77.2.f.b.36.1	yes	16	33.5	odd	10
539.2.f.e.246.1		16	231.20	even	10
539.2.f.e.344.1		16	231.104	even	10
539.2.q.f.312.1		32	231.152	even	30
539.2.q.f.410.4		32	231.5	even	30
539.2.q.f.422.4		32	231.185	even	30
539.2.q.f.520.1		32	231.38	even	30
539.2.q.g.312.1		32	231.86	odd	30
539.2.q.g.410.4		32	231.170	odd	30
539.2.q.g.422.4		32	231.53	odd	30
539.2.q.g.520.1		32	231.137	odd	30
693.2.m.i.190.4		16	11.5	even	5
693.2.m.i.631.4		16	11.9	even	5
847.2.a.o.1.2		8	33.32	even	2
847.2.a.p.1.7		8	3.2	odd	2
847.2.f.v.148.1		16	33.29	even	10
847.2.f.v.372.1		16	33.8	even	10
847.2.f.w.148.4		16	33.26	odd	10
847.2.f.w.372.4		16	33.14	odd	10
847.2.f.x.323.4		16	33.2	even	10
847.2.f.x.729.4		16	33.17	even	10
5929.2.a.bs.1.2		8	231.230	odd	2
5929.2.a.bt.1.7		8	21.20	even	2
7623.2.a.ct.1.2		8	1.1	even	1	trivial
7623.2.a.cw.1.7		8	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.98451 q^{2} +1.93830 q^{4} +0.0269243 q^{5} +1.00000 q^{7} +0.122446 q^{8} +O(q^{10})\)	\(q-1.98451 q^{2} +1.93830 q^{4} +0.0269243 q^{5} +1.00000 q^{7} +0.122446 q^{8} -0.0534317 q^{10} -4.88112 q^{13} -1.98451 q^{14} -4.11959 q^{16} -1.67878 q^{17} +1.37394 q^{19} +0.0521874 q^{20} -8.06246 q^{23} -4.99928 q^{25} +9.68665 q^{26} +1.93830 q^{28} +6.39316 q^{29} +4.01054 q^{31} +7.93050 q^{32} +3.33156 q^{34} +0.0269243 q^{35} +0.521038 q^{37} -2.72661 q^{38} +0.00329677 q^{40} +10.5987 q^{41} +3.73968 q^{43} +16.0001 q^{46} +8.95906 q^{47} +1.00000 q^{49} +9.92114 q^{50} -9.46107 q^{52} +3.96057 q^{53} +0.122446 q^{56} -12.6873 q^{58} -9.73022 q^{59} +8.46723 q^{61} -7.95898 q^{62} -7.49902 q^{64} -0.131421 q^{65} +2.81285 q^{67} -3.25398 q^{68} -0.0534317 q^{70} -2.04551 q^{71} -10.4693 q^{73} -1.03401 q^{74} +2.66311 q^{76} +5.85521 q^{79} -0.110917 q^{80} -21.0334 q^{82} -2.60548 q^{83} -0.0452000 q^{85} -7.42146 q^{86} -1.21791 q^{89} -4.88112 q^{91} -15.6275 q^{92} -17.7794 q^{94} +0.0369924 q^{95} +3.39670 q^{97} -1.98451 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} + 7 q^{4} - 10 q^{5} + 8 q^{7}+O(q^{10}) \) 8 * q - q^2 + 7 * q^4 - 10 * q^5 + 8 * q^7	\( 8 q - q^{2} + 7 q^{4} - 10 q^{5} + 8 q^{7} + 6 q^{10} - 6 q^{13} - q^{14} + q^{16} + 5 q^{17} - 13 q^{19} - 23 q^{20} - 16 q^{23} + 16 q^{25} + 6 q^{26} + 7 q^{28} - 9 q^{29} + 9 q^{31} - 16 q^{32} - 12 q^{34} - 10 q^{35} + 7 q^{37} + 10 q^{38} + 5 q^{40} + 10 q^{41} - 4 q^{43} + 4 q^{46} - 16 q^{47} + 8 q^{49} - 6 q^{50} - 41 q^{52} - 37 q^{53} - 15 q^{58} - q^{59} + 19 q^{61} + 18 q^{62} - 4 q^{64} + 4 q^{65} - 19 q^{67} - 9 q^{68} + 6 q^{70} - 13 q^{71} - 25 q^{73} - 33 q^{74} + 26 q^{76} - 4 q^{80} - 13 q^{82} + 25 q^{83} + 3 q^{85} - 4 q^{86} - 37 q^{89} - 6 q^{91} - 35 q^{92} - 42 q^{94} - 21 q^{95} + 15 q^{97} - q^{98}+O(q^{100}) \) 8 * q - q^2 + 7 * q^4 - 10 * q^5 + 8 * q^7 + 6 * q^10 - 6 * q^13 - q^14 + q^16 + 5 * q^17 - 13 * q^19 - 23 * q^20 - 16 * q^23 + 16 * q^25 + 6 * q^26 + 7 * q^28 - 9 * q^29 + 9 * q^31 - 16 * q^32 - 12 * q^34 - 10 * q^35 + 7 * q^37 + 10 * q^38 + 5 * q^40 + 10 * q^41 - 4 * q^43 + 4 * q^46 - 16 * q^47 + 8 * q^49 - 6 * q^50 - 41 * q^52 - 37 * q^53 - 15 * q^58 - q^59 + 19 * q^61 + 18 * q^62 - 4 * q^64 + 4 * q^65 - 19 * q^67 - 9 * q^68 + 6 * q^70 - 13 * q^71 - 25 * q^73 - 33 * q^74 + 26 * q^76 - 4 * q^80 - 13 * q^82 + 25 * q^83 + 3 * q^85 - 4 * q^86 - 37 * q^89 - 6 * q^91 - 35 * q^92 - 42 * q^94 - 21 * q^95 + 15 * q^97 - q^98

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