LMFDB - Embedded newform 7623.2.a.ct.1.6

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			1.6
Root			$-1.40927$ 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)

$f(q)$	$=$	$q+1.40927 q^{2} -0.0139645 q^{4} +1.83139 q^{5} +1.00000 q^{7} -2.83822 q^{8} +O(q^{10})$	$q+1.40927 q^{2} -0.0139645 q^{4} +1.83139 q^{5} +1.00000 q^{7} -2.83822 q^{8} +2.58091 q^{10} -4.64706 q^{13} +1.40927 q^{14} -3.97188 q^{16} +5.47021 q^{17} -5.80118 q^{19} -0.0255744 q^{20} +0.719682 q^{23} -1.64602 q^{25} -6.54895 q^{26} -0.0139645 q^{28} +1.17247 q^{29} -1.30787 q^{31} +0.0789938 q^{32} +7.70900 q^{34} +1.83139 q^{35} +2.09474 q^{37} -8.17541 q^{38} -5.19787 q^{40} +0.916645 q^{41} +8.02379 q^{43} +1.01423 q^{46} -5.97584 q^{47} +1.00000 q^{49} -2.31969 q^{50} +0.0648939 q^{52} -10.1449 q^{53} -2.83822 q^{56} +1.65233 q^{58} +7.68081 q^{59} -6.27612 q^{61} -1.84313 q^{62} +8.05508 q^{64} -8.51056 q^{65} -15.4673 q^{67} -0.0763889 q^{68} +2.58091 q^{70} -13.9019 q^{71} +6.01462 q^{73} +2.95205 q^{74} +0.0810106 q^{76} -15.6409 q^{79} -7.27404 q^{80} +1.29180 q^{82} +4.37573 q^{83} +10.0181 q^{85} +11.3077 q^{86} -15.3437 q^{89} -4.64706 q^{91} -0.0100500 q^{92} -8.42155 q^{94} -10.6242 q^{95} +2.41124 q^{97} +1.40927 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	1.40927	0.996503	0.498251	−	0.867033i	$-0.333976\pi$
$2$	1.40927	0.996503	0.498251	+	0.867033i	$0.333976\pi$
$3$	0	0
$3$	0	0
$4$	−0.0139645	−0.00698226
$5$	1.83139	0.819021	0.409510	−	0.912305i	$-0.365700\pi$
$5$	1.83139	0.819021	0.409510	+	0.912305i	$0.365700\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.83822	−1.00346
$9$	0	0
$10$	2.58091	0.816157
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−4.64706	−1.28886	−0.644431	−	0.764663i	$-0.722906\pi$
$13$	−4.64706	−1.28886	−0.644431	+	0.764663i	$0.722906\pi$
$14$	1.40927	0.376643
$15$	0	0
$16$	−3.97188	−0.992969
$17$	5.47021	1.32672	0.663361	−	0.748300i	$-0.269129\pi$
$17$	5.47021	1.32672	0.663361	+	0.748300i	$0.269129\pi$
$18$	0	0
$19$	−5.80118	−1.33088	−0.665441	−	0.746451i	$-0.731756\pi$
$19$	−5.80118	−1.33088	−0.665441	+	0.746451i	$0.731756\pi$
$20$	−0.0255744	−0.00571861
$21$	0	0
$22$	0	0
$23$	0.719682	0.150064	0.0750321	−	0.997181i	$-0.476094\pi$
$23$	0.719682	0.150064	0.0750321	+	0.997181i	$0.476094\pi$
$24$	0	0
$25$	−1.64602	−0.329205
$26$	−6.54895	−1.28435
$27$	0	0
$28$	−0.0139645	−0.00263905
$29$	1.17247	0.217723	0.108861	−	0.994057i	$-0.465280\pi$
$29$	1.17247	0.217723	0.108861	+	0.994057i	$0.465280\pi$
$30$	0	0
$31$	−1.30787	−0.234900	−0.117450	−	0.993079i	$-0.537472\pi$
$31$	−1.30787	−0.234900	−0.117450	+	0.993079i	$0.537472\pi$
$32$	0.0789938	0.0139643
$33$	0	0
$34$	7.70900	1.32208
$35$	1.83139	0.309561
$36$	0	0
$37$	2.09474	0.344373	0.172186	−	0.985064i	$-0.444917\pi$
$37$	2.09474	0.344373	0.172186	+	0.985064i	$0.444917\pi$
$38$	−8.17541	−1.32623
$39$	0	0
$40$	−5.19787	−0.821855
$41$	0.916645	0.143156	0.0715780	−	0.997435i	$-0.477197\pi$
$41$	0.916645	0.143156	0.0715780	+	0.997435i	$0.477197\pi$
$42$	0	0
$43$	8.02379	1.22362	0.611808	−	0.791006i	$-0.290442\pi$
$43$	8.02379	1.22362	0.611808	+	0.791006i	$0.290442\pi$
$44$	0	0
$45$	0	0
$46$	1.01423	0.149539
$47$	−5.97584	−0.871665	−0.435833	−	0.900028i	$-0.643546\pi$
$47$	−5.97584	−0.871665	−0.435833	+	0.900028i	$0.643546\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−2.31969	−0.328053
$51$	0	0
$52$	0.0648939	0.00899916
$53$	−10.1449	−1.39351	−0.696757	−	0.717307i	$-0.745374\pi$
$53$	−10.1449	−1.39351	−0.696757	+	0.717307i	$0.745374\pi$
$54$	0	0
$55$	0	0
$56$	−2.83822	−0.379272
$57$	0	0
$58$	1.65233	0.216962
$59$	7.68081	0.999956	0.499978	−	0.866038i	$-0.333341\pi$
$59$	7.68081	0.999956	0.499978	+	0.866038i	$0.333341\pi$
$60$	0	0
$61$	−6.27612	−0.803575	−0.401788	−	0.915733i	$-0.631611\pi$
$61$	−6.27612	−0.803575	−0.401788	+	0.915733i	$0.631611\pi$
$62$	−1.84313	−0.234078
$63$	0	0
$64$	8.05508	1.00688
$65$	−8.51056	−1.05560
$66$	0	0
$67$	−15.4673	−1.88963	−0.944814	−	0.327608i	$-0.893758\pi$
$67$	−15.4673	−1.88963	−0.944814	+	0.327608i	$0.893758\pi$
$68$	−0.0763889	−0.00926351
$69$	0	0
$70$	2.58091	0.308478
$71$	−13.9019	−1.64985	−0.824927	−	0.565240i	$-0.808784\pi$
$71$	−13.9019	−1.64985	−0.824927	+	0.565240i	$0.808784\pi$
$72$	0	0
$73$	6.01462	0.703959	0.351979	−	0.936008i	$-0.385509\pi$
$73$	6.01462	0.703959	0.351979	+	0.936008i	$0.385509\pi$
$74$	2.95205	0.343169
$75$	0	0
$76$	0.0810106	0.00929255
$77$	0	0
$78$	0	0
$79$	−15.6409	−1.75974	−0.879872	−	0.475211i	$-0.842372\pi$
$79$	−15.6409	−1.75974	−0.879872	+	0.475211i	$0.842372\pi$
$80$	−7.27404	−0.813262
$81$	0	0
$82$	1.29180	0.142655
$83$	4.37573	0.480299	0.240149	−	0.970736i	$-0.422804\pi$
$83$	4.37573	0.480299	0.240149	+	0.970736i	$0.422804\pi$
$84$	0	0
$85$	10.0181	1.08661
$86$	11.3077	1.21934
$87$	0	0
$88$	0	0
$89$	−15.3437	−1.62643	−0.813215	−	0.581963i	$-0.802285\pi$
$89$	−15.3437	−1.62643	−0.813215	+	0.581963i	$0.802285\pi$
$90$	0	0
$91$	−4.64706	−0.487144
$92$	−0.0100500	−0.00104779
$93$	0	0
$94$	−8.42155	−0.868617
$95$	−10.6242	−1.09002
$96$	0	0
$97$	2.41124	0.244824	0.122412	−	0.992479i	$-0.460937\pi$
$97$	2.41124	0.244824	0.122412	+	0.992479i	$0.460937\pi$
$98$	1.40927	0.142358
$99$	0	0
$100$	0.0229859	0.00229859
$101$	−11.8959	−1.18368	−0.591842	−	0.806054i	$-0.701599\pi$
$101$	−11.8959	−1.18368	−0.591842	+	0.806054i	$0.701599\pi$
$102$	0	0
$103$	0.396314	0.0390500	0.0195250	−	0.999809i	$-0.493785\pi$
$103$	0.396314	0.0390500	0.0195250	+	0.999809i	$0.493785\pi$
$104$	13.1893	1.29332
$105$	0	0
$106$	−14.2969	−1.38864
$107$	3.26935	0.316060	0.158030	−	0.987434i	$-0.449486\pi$
$107$	3.26935	0.316060	0.158030	+	0.987434i	$0.449486\pi$
$108$	0	0
$109$	−2.84638	−0.272634	−0.136317	−	0.990665i	$-0.543527\pi$
$109$	−2.84638	−0.272634	−0.136317	+	0.990665i	$0.543527\pi$
$110$	0	0
$111$	0	0
$112$	−3.97188	−0.375307
$113$	−14.5445	−1.36823	−0.684117	−	0.729372i	$-0.739812\pi$
$113$	−14.5445	−1.36823	−0.684117	+	0.729372i	$0.739812\pi$
$114$	0	0
$115$	1.31802	0.122906
$116$	−0.0163730	−0.00152020
$117$	0	0
$118$	10.8243	0.996459
$119$	5.47021	0.501454
$120$	0	0
$121$	0	0
$122$	−8.84474	−0.800765
$123$	0	0
$124$	0.0182637	0.00164013
$125$	−12.1714	−1.08865
$126$	0	0
$127$	5.03287	0.446595	0.223298	−	0.974750i	$-0.428318\pi$
$127$	5.03287	0.446595	0.223298	+	0.974750i	$0.428318\pi$
$128$	11.1938	0.989399
$129$	0	0
$130$	−11.9937	−1.05191
$131$	0.180053	0.0157313	0.00786565	−	0.999969i	$-0.497496\pi$
$131$	0.180053	0.0157313	0.00786565	+	0.999969i	$0.497496\pi$
$132$	0	0
$133$	−5.80118	−0.503026
$134$	−21.7975	−1.88302
$135$	0	0
$136$	−15.5256	−1.33131
$137$	−8.32395	−0.711163	−0.355582	−	0.934645i	$-0.615717\pi$
$137$	−8.32395	−0.711163	−0.355582	+	0.934645i	$0.615717\pi$
$138$	0	0
$139$	−6.96119	−0.590441	−0.295220	−	0.955429i	$-0.595393\pi$
$139$	−6.96119	−0.590441	−0.295220	+	0.955429i	$0.595393\pi$
$140$	−0.0255744	−0.00216143
$141$	0	0
$142$	−19.5915	−1.64408
$143$	0	0
$144$	0	0
$145$	2.14725	0.178320
$146$	8.47622	0.701497
$147$	0	0
$148$	−0.0292520	−0.00240450
$149$	3.21431	0.263327	0.131663	−	0.991294i	$-0.457968\pi$
$149$	3.21431	0.263327	0.131663	+	0.991294i	$0.457968\pi$
$150$	0	0
$151$	22.2670	1.81206	0.906032	−	0.423210i	$-0.139097\pi$
$151$	22.2670	1.81206	0.906032	+	0.423210i	$0.139097\pi$
$152$	16.4650	1.33549
$153$	0	0
$154$	0	0
$155$	−2.39521	−0.192388
$156$	0	0
$157$	13.2548	1.05785	0.528923	−	0.848670i	$-0.322596\pi$
$157$	13.2548	1.05785	0.528923	+	0.848670i	$0.322596\pi$
$158$	−22.0423	−1.75359
$159$	0	0
$160$	0.144668	0.0114370
$161$	0.719682	0.0567189
$162$	0	0
$163$	13.7183	1.07450	0.537252	−	0.843422i	$-0.319462\pi$
$163$	13.7183	1.07450	0.537252	+	0.843422i	$0.319462\pi$
$164$	−0.0128005	−0.000999552 0
$165$	0	0
$166$	6.16657	0.478619
$167$	−9.30860	−0.720321	−0.360160	−	0.932890i	$-0.617278\pi$
$167$	−9.30860	−0.720321	−0.360160	+	0.932890i	$0.617278\pi$
$168$	0	0
$169$	8.59513	0.661164
$170$	14.1181	1.08281
$171$	0	0
$172$	−0.112048	−0.00854361
$173$	−10.5057	−0.798732	−0.399366	−	0.916792i	$-0.630770\pi$
$173$	−10.5057	−0.798732	−0.399366	+	0.916792i	$0.630770\pi$
$174$	0	0
$175$	−1.64602	−0.124428
$176$	0	0
$177$	0	0
$178$	−21.6234	−1.62074
$179$	8.32331	0.622113	0.311057	−	0.950391i	$-0.399317\pi$
$179$	8.32331	0.622113	0.311057	+	0.950391i	$0.399317\pi$
$180$	0	0
$181$	−14.8030	−1.10030	−0.550148	−	0.835067i	$-0.685429\pi$
$181$	−14.8030	−1.10030	−0.550148	+	0.835067i	$0.685429\pi$
$182$	−6.54895	−0.485440
$183$	0	0
$184$	−2.04261	−0.150583
$185$	3.83628	0.282049
$186$	0	0
$187$	0	0
$188$	0.0834496	0.00608619
$189$	0	0
$190$	−14.9723	−1.08621
$191$	9.60676	0.695121	0.347560	−	0.937658i	$-0.387010\pi$
$191$	9.60676	0.695121	0.347560	+	0.937658i	$0.387010\pi$
$192$	0	0
$193$	−1.48781	−0.107095	−0.0535474	−	0.998565i	$-0.517053\pi$
$193$	−1.48781	−0.107095	−0.0535474	+	0.998565i	$0.517053\pi$
$194$	3.39808	0.243968
$195$	0	0
$196$	−0.0139645	−0.000997465 0
$197$	14.0434	1.00055	0.500274	−	0.865867i	$-0.333233\pi$
$197$	14.0434	1.00055	0.500274	+	0.865867i	$0.333233\pi$
$198$	0	0
$199$	−4.28729	−0.303918	−0.151959	−	0.988387i	$-0.548558\pi$
$199$	−4.28729	−0.303918	−0.151959	+	0.988387i	$0.548558\pi$
$200$	4.67177	0.330344
$201$	0	0
$202$	−16.7645	−1.17954
$203$	1.17247	0.0822915
$204$	0	0
$205$	1.67873	0.117248
$206$	0.558512	0.0389134
$207$	0	0
$208$	18.4575	1.27980
$209$	0	0
$210$	0	0
$211$	−1.45527	−0.100185	−0.0500925	−	0.998745i	$-0.515952\pi$
$211$	−1.45527	−0.100185	−0.0500925	+	0.998745i	$0.515952\pi$
$212$	0.141669	0.00972987
$213$	0	0
$214$	4.60739	0.314955
$215$	14.6947	1.00217
$216$	0	0
$217$	−1.30787	−0.0887837
$218$	−4.01132	−0.271681
$219$	0	0
$220$	0	0
$221$	−25.4204	−1.70996
$222$	0	0
$223$	−4.85642	−0.325210	−0.162605	−	0.986691i	$-0.551990\pi$
$223$	−4.85642	−0.325210	−0.162605	+	0.986691i	$0.551990\pi$
$224$	0.0789938	0.00527799
$225$	0	0
$226$	−20.4971	−1.36345
$227$	−0.397144	−0.0263594	−0.0131797	−	0.999913i	$-0.504195\pi$
$227$	−0.397144	−0.0263594	−0.0131797	+	0.999913i	$0.504195\pi$
$228$	0	0
$229$	2.18963	0.144695	0.0723475	−	0.997379i	$-0.476951\pi$
$229$	2.18963	0.144695	0.0723475	+	0.997379i	$0.476951\pi$
$230$	1.85744	0.122476
$231$	0	0
$232$	−3.32773	−0.218476
$233$	1.26016	0.0825555	0.0412778	−	0.999148i	$-0.486857\pi$
$233$	1.26016	0.0825555	0.0412778	+	0.999148i	$0.486857\pi$
$234$	0	0
$235$	−10.9441	−0.713912
$236$	−0.107259	−0.00698195
$237$	0	0
$238$	7.70900	0.499700
$239$	11.1617	0.721987	0.360994	−	0.932568i	$-0.382438\pi$
$239$	11.1617	0.721987	0.360994	+	0.932568i	$0.382438\pi$
$240$	0	0
$241$	−21.4843	−1.38392	−0.691962	−	0.721934i	$-0.743254\pi$
$241$	−21.4843	−1.38392	−0.691962	+	0.721934i	$0.743254\pi$
$242$	0	0
$243$	0	0
$244$	0.0876430	0.00561077
$245$	1.83139	0.117003
$246$	0	0
$247$	26.9584	1.71532
$248$	3.71200	0.235712
$249$	0	0
$250$	−17.1528	−1.08484
$251$	0.423820	0.0267513	0.0133756	−	0.999911i	$-0.495742\pi$
$251$	0.423820	0.0267513	0.0133756	+	0.999911i	$0.495742\pi$
$252$	0	0
$253$	0	0
$254$	7.09267	0.445033
$255$	0	0
$256$	−0.335132	−0.0209457
$257$	−17.4401	−1.08788	−0.543941	−	0.839124i	$-0.683068\pi$
$257$	−17.4401	−1.08788	−0.543941	+	0.839124i	$0.683068\pi$
$258$	0	0
$259$	2.09474	0.130161
$260$	0.118846	0.00737050
$261$	0	0
$262$	0.253743	0.0156763
$263$	−1.51519	−0.0934307	−0.0467153	−	0.998908i	$-0.514875\pi$
$263$	−1.51519	−0.0934307	−0.0467153	+	0.998908i	$0.514875\pi$
$264$	0	0
$265$	−18.5793	−1.14132
$266$	−8.17541	−0.501267
$267$	0	0
$268$	0.215993	0.0131939
$269$	2.03103	0.123834	0.0619170	−	0.998081i	$-0.480279\pi$
$269$	2.03103	0.123834	0.0619170	+	0.998081i	$0.480279\pi$
$270$	0	0
$271$	7.60444	0.461937	0.230968	−	0.972961i	$-0.425811\pi$
$271$	7.60444	0.461937	0.230968	+	0.972961i	$0.425811\pi$
$272$	−21.7270	−1.31739
$273$	0	0
$274$	−11.7307	−0.708676
$275$	0	0
$276$	0	0
$277$	14.4268	0.866823	0.433411	−	0.901196i	$-0.357310\pi$
$277$	14.4268	0.866823	0.433411	+	0.901196i	$0.357310\pi$
$278$	−9.81019	−0.588376
$279$	0	0
$280$	−5.19787	−0.310632
$281$	−17.7496	−1.05886	−0.529428	−	0.848355i	$-0.677593\pi$
$281$	−17.7496	−1.05886	−0.529428	+	0.848355i	$0.677593\pi$
$282$	0	0
$283$	−31.1361	−1.85085	−0.925426	−	0.378929i	$-0.876292\pi$
$283$	−31.1361	−1.85085	−0.925426	+	0.378929i	$0.876292\pi$
$284$	0.194133	0.0115197
$285$	0	0
$286$	0	0
$287$	0.916645	0.0541079
$288$	0	0
$289$	12.9232	0.760190
$290$	3.02605	0.177696
$291$	0	0
$292$	−0.0839913	−0.00491522
$293$	24.0303	1.40386	0.701932	−	0.712244i	$-0.252321\pi$
$293$	24.0303	1.40386	0.701932	+	0.712244i	$0.252321\pi$
$294$	0	0
$295$	14.0665	0.818985
$296$	−5.94532	−0.345565
$297$	0	0
$298$	4.52983	0.262406
$299$	−3.34441	−0.193412
$300$	0	0
$301$	8.02379	0.462484
$302$	31.3802	1.80573
$303$	0	0
$304$	23.0416	1.32152
$305$	−11.4940	−0.658145
$306$	0	0
$307$	−5.46298	−0.311789	−0.155894	−	0.987774i	$-0.549826\pi$
$307$	−5.46298	−0.311789	−0.155894	+	0.987774i	$0.549826\pi$
$308$	0	0
$309$	0	0
$310$	−3.37549	−0.191715
$311$	−13.8884	−0.787541	−0.393770	−	0.919209i	$-0.628830\pi$
$311$	−13.8884	−0.787541	−0.393770	+	0.919209i	$0.628830\pi$
$312$	0	0
$313$	27.4486	1.55148	0.775742	−	0.631050i	$-0.217376\pi$
$313$	27.4486	1.55148	0.775742	+	0.631050i	$0.217376\pi$
$314$	18.6795	1.05415
$315$	0	0
$316$	0.218418	0.0122870
$317$	−7.82570	−0.439535	−0.219768	−	0.975552i	$-0.570530\pi$
$317$	−7.82570	−0.439535	−0.219768	+	0.975552i	$0.570530\pi$
$318$	0	0
$319$	0	0
$320$	14.7520	0.824659
$321$	0	0
$322$	1.01423	0.0565206
$323$	−31.7337	−1.76571
$324$	0	0
$325$	7.64917	0.424299
$326$	19.3328	1.07075
$327$	0	0
$328$	−2.60164	−0.143651
$329$	−5.97584	−0.329458
$330$	0	0
$331$	−28.1462	−1.54705	−0.773527	−	0.633764i	$-0.781509\pi$
$331$	−28.1462	−1.54705	−0.773527	+	0.633764i	$0.781509\pi$
$332$	−0.0611049	−0.00335357
$333$	0	0
$334$	−13.1183	−0.717802
$335$	−28.3265	−1.54764
$336$	0	0
$337$	−24.9789	−1.36069	−0.680345	−	0.732892i	$-0.738170\pi$
$337$	−24.9789	−1.36069	−0.680345	+	0.732892i	$0.738170\pi$
$338$	12.1128	0.658852
$339$	0	0
$340$	−0.139898	−0.00758701
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	−22.7732	−1.22785
$345$	0	0
$346$	−14.8053	−0.795939
$347$	−20.6492	−1.10851	−0.554254	−	0.832347i	$-0.686997\pi$
$347$	−20.6492	−1.10851	−0.554254	+	0.832347i	$0.686997\pi$
$348$	0	0
$349$	−5.99721	−0.321023	−0.160512	−	0.987034i	$-0.551314\pi$
$349$	−5.99721	−0.321023	−0.160512	+	0.987034i	$0.551314\pi$
$350$	−2.31969	−0.123993
$351$	0	0
$352$	0	0
$353$	−24.0382	−1.27942	−0.639712	−	0.768615i	$-0.720946\pi$
$353$	−24.0382	−1.27942	−0.639712	+	0.768615i	$0.720946\pi$
$354$	0	0
$355$	−25.4598	−1.35126
$356$	0.214267	0.0113562
$357$	0	0
$358$	11.7298	0.619938
$359$	10.9501	0.577925	0.288963	−	0.957340i	$-0.406690\pi$
$359$	10.9501	0.577925	0.288963	+	0.957340i	$0.406690\pi$
$360$	0	0
$361$	14.6536	0.771244
$362$	−20.8613	−1.09645
$363$	0	0
$364$	0.0648939	0.00340136
$365$	11.0151	0.576557
$366$	0	0
$367$	10.6178	0.554243	0.277121	−	0.960835i	$-0.410620\pi$
$367$	10.6178	0.554243	0.277121	+	0.960835i	$0.410620\pi$
$368$	−2.85849	−0.149009
$369$	0	0
$370$	5.40634	0.281062
$371$	−10.1449	−0.526699
$372$	0	0
$373$	36.6036	1.89526	0.947631	−	0.319367i	$-0.103470\pi$
$373$	36.6036	1.89526	0.947631	+	0.319367i	$0.103470\pi$
$374$	0	0
$375$	0	0
$376$	16.9607	0.874682
$377$	−5.44855	−0.280615
$378$	0	0
$379$	−12.6578	−0.650186	−0.325093	−	0.945682i	$-0.605396\pi$
$379$	−12.6578	−0.650186	−0.325093	+	0.945682i	$0.605396\pi$
$380$	0.148362	0.00761080
$381$	0	0
$382$	13.5385	0.692690
$383$	15.4679	0.790372	0.395186	−	0.918601i	$-0.370680\pi$
$383$	15.4679	0.790372	0.395186	+	0.918601i	$0.370680\pi$
$384$	0	0
$385$	0	0
$386$	−2.09672	−0.106720
$387$	0	0
$388$	−0.0336718	−0.00170943
$389$	12.7130	0.644572	0.322286	−	0.946642i	$-0.395549\pi$
$389$	12.7130	0.644572	0.322286	+	0.946642i	$0.395549\pi$
$390$	0	0
$391$	3.93682	0.199093
$392$	−2.83822	−0.143352
$393$	0	0
$394$	19.7909	0.997049
$395$	−28.6446	−1.44127
$396$	0	0
$397$	−18.9574	−0.951445	−0.475722	−	0.879596i	$-0.657813\pi$
$397$	−18.9574	−0.951445	−0.475722	+	0.879596i	$0.657813\pi$
$398$	−6.04193	−0.302855
$399$	0	0
$400$	6.53780	0.326890
$401$	8.68208	0.433563	0.216781	−	0.976220i	$-0.430444\pi$
$401$	8.68208	0.433563	0.216781	+	0.976220i	$0.430444\pi$
$402$	0	0
$403$	6.07772	0.302753
$404$	0.166120	0.00826478
$405$	0	0
$406$	1.65233	0.0820037
$407$	0	0
$408$	0	0
$409$	5.71406	0.282542	0.141271	−	0.989971i	$-0.454881\pi$
$409$	5.71406	0.282542	0.141271	+	0.989971i	$0.454881\pi$
$410$	2.36578	0.116838
$411$	0	0
$412$	−0.00553433	−0.000272657 0
$413$	7.68081	0.377948
$414$	0	0
$415$	8.01365	0.393375
$416$	−0.367089	−0.0179980
$417$	0	0
$418$	0	0
$419$	27.1909	1.32836	0.664181	−	0.747571i	$-0.268780\pi$
$419$	27.1909	1.32836	0.664181	+	0.747571i	$0.268780\pi$
$420$	0	0
$421$	−23.9651	−1.16799	−0.583993	−	0.811759i	$-0.698510\pi$
$421$	−23.9651	−1.16799	−0.583993	+	0.811759i	$0.698510\pi$
$422$	−2.05087	−0.0998347
$423$	0	0
$424$	28.7935	1.39834
$425$	−9.00410	−0.436763
$426$	0	0
$427$	−6.27612	−0.303723
$428$	−0.0456549	−0.00220681
$429$	0	0
$430$	20.7087	0.998663
$431$	−16.4732	−0.793484	−0.396742	−	0.917930i	$-0.629859\pi$
$431$	−16.4732	−0.793484	−0.396742	+	0.917930i	$0.629859\pi$
$432$	0	0
$433$	20.0909	0.965509	0.482754	−	0.875756i	$-0.339636\pi$
$433$	20.0909	0.965509	0.482754	+	0.875756i	$0.339636\pi$
$434$	−1.84313	−0.0884732
$435$	0	0
$436$	0.0397484	0.00190360
$437$	−4.17500	−0.199718
$438$	0	0
$439$	26.7682	1.27758	0.638788	−	0.769383i	$-0.279436\pi$
$439$	26.7682	1.27758	0.638788	+	0.769383i	$0.279436\pi$
$440$	0	0
$441$	0	0
$442$	−35.8241	−1.70398
$443$	−26.2153	−1.24553	−0.622764	−	0.782410i	$-0.713990\pi$
$443$	−26.2153	−1.24553	−0.622764	+	0.782410i	$0.713990\pi$
$444$	0	0
$445$	−28.1003	−1.33208
$446$	−6.84400	−0.324073
$447$	0	0
$448$	8.05508	0.380567
$449$	−9.74740	−0.460008	−0.230004	−	0.973190i	$-0.573874\pi$
$449$	−9.74740	−0.460008	−0.230004	+	0.973190i	$0.573874\pi$
$450$	0	0
$451$	0	0
$452$	0.203107	0.00955336
$453$	0	0
$454$	−0.559682	−0.0262672
$455$	−8.51056	−0.398981
$456$	0	0
$457$	11.8853	0.555971	0.277986	−	0.960585i	$-0.410333\pi$
$457$	11.8853	0.555971	0.277986	+	0.960585i	$0.410333\pi$
$458$	3.08578	0.144189
$459$	0	0
$460$	−0.0184055	−0.000858159 0
$461$	−9.14737	−0.426035	−0.213018	−	0.977048i	$-0.568329\pi$
$461$	−9.14737	−0.426035	−0.213018	+	0.977048i	$0.568329\pi$
$462$	0	0
$463$	38.9342	1.80943	0.904713	−	0.426021i	$-0.140085\pi$
$463$	38.9342	1.80943	0.904713	+	0.426021i	$0.140085\pi$
$464$	−4.65692	−0.216192
$465$	0	0
$466$	1.77590	0.0822668
$467$	20.7834	0.961742	0.480871	−	0.876791i	$-0.340320\pi$
$467$	20.7834	0.961742	0.480871	+	0.876791i	$0.340320\pi$
$468$	0	0
$469$	−15.4673	−0.714212
$470$	−15.4231	−0.711415
$471$	0	0
$472$	−21.7998	−1.00342
$473$	0	0
$474$	0	0
$475$	9.54887	0.438132
$476$	−0.0763889	−0.00350128
$477$	0	0
$478$	15.7298	0.719462
$479$	24.5363	1.12109	0.560546	−	0.828123i	$-0.310591\pi$
$479$	24.5363	1.12109	0.560546	+	0.828123i	$0.310591\pi$
$480$	0	0
$481$	−9.73437	−0.443849
$482$	−30.2771	−1.37908
$483$	0	0
$484$	0	0
$485$	4.41591	0.200516
$486$	0	0
$487$	−12.3713	−0.560598	−0.280299	−	0.959913i	$-0.590434\pi$
$487$	−12.3713	−0.560598	−0.280299	+	0.959913i	$0.590434\pi$
$488$	17.8130	0.806356
$489$	0	0
$490$	2.58091	0.116594
$491$	−16.5251	−0.745766	−0.372883	−	0.927878i	$-0.621631\pi$
$491$	−16.5251	−0.745766	−0.372883	+	0.927878i	$0.621631\pi$
$492$	0	0
$493$	6.41368	0.288858
$494$	37.9916	1.70932
$495$	0	0
$496$	5.19468	0.233248
$497$	−13.9019	−0.623586
$498$	0	0
$499$	−13.7410	−0.615129	−0.307565	−	0.951527i	$-0.599514\pi$
$499$	−13.7410	−0.615129	−0.307565	+	0.951527i	$0.599514\pi$
$500$	0.169968	0.00760121
$501$	0	0
$502$	0.597276	0.0266577
$503$	22.5968	1.00754	0.503770	−	0.863838i	$-0.331946\pi$
$503$	22.5968	1.00754	0.503770	+	0.863838i	$0.331946\pi$
$504$	0	0
$505$	−21.7859	−0.969461
$506$	0	0
$507$	0	0
$508$	−0.0702816	−0.00311824
$509$	21.4636	0.951358	0.475679	−	0.879619i	$-0.342202\pi$
$509$	21.4636	0.951358	0.475679	+	0.879619i	$0.342202\pi$
$510$	0	0
$511$	6.01462	0.266071
$512$	−22.8598	−1.01027
$513$	0	0
$514$	−24.5777	−1.08408
$515$	0.725804	0.0319827
$516$	0	0
$517$	0	0
$518$	2.95205	0.129706
$519$	0	0
$520$	24.1548	1.05926
$521$	−34.7116	−1.52074	−0.760371	−	0.649489i	$-0.774983\pi$
$521$	−34.7116	−1.52074	−0.760371	+	0.649489i	$0.774983\pi$
$522$	0	0
$523$	19.7281	0.862651	0.431326	−	0.902196i	$-0.358046\pi$
$523$	19.7281	0.862651	0.431326	+	0.902196i	$0.358046\pi$
$524$	−0.00251435	−0.000109840 0
$525$	0	0
$526$	−2.13531	−0.0931039
$527$	−7.15430	−0.311646
$528$	0	0
$529$	−22.4821	−0.977481
$530$	−26.1832	−1.13733
$531$	0	0
$532$	0.0810106	0.00351226
$533$	−4.25970	−0.184508
$534$	0	0
$535$	5.98745	0.258860
$536$	43.8994	1.89617
$537$	0	0
$538$	2.86226	0.123401
$539$	0	0
$540$	0	0
$541$	−5.54942	−0.238588	−0.119294	−	0.992859i	$-0.538063\pi$
$541$	−5.54942	−0.238588	−0.119294	+	0.992859i	$0.538063\pi$
$542$	10.7167	0.460321
$543$	0	0
$544$	0.432113	0.0185267
$545$	−5.21283	−0.223293
$546$	0	0
$547$	−8.44671	−0.361155	−0.180578	−	0.983561i	$-0.557797\pi$
$547$	−8.44671	−0.361155	−0.180578	+	0.983561i	$0.557797\pi$
$548$	0.116240	0.00496553
$549$	0	0
$550$	0	0
$551$	−6.80173	−0.289763
$552$	0	0
$553$	−15.6409	−0.665120
$554$	20.3312	0.863791
$555$	0	0
$556$	0.0972097	0.00412261
$557$	12.1835	0.516231	0.258115	−	0.966114i	$-0.416898\pi$
$557$	12.1835	0.516231	0.258115	+	0.966114i	$0.416898\pi$
$558$	0	0
$559$	−37.2870	−1.57707
$560$	−7.27404	−0.307384
$561$	0	0
$562$	−25.0140	−1.05515
$563$	27.3535	1.15281	0.576407	−	0.817163i	$-0.304454\pi$
$563$	27.3535	1.15281	0.576407	+	0.817163i	$0.304454\pi$
$564$	0	0
$565$	−26.6366	−1.12061
$566$	−43.8792	−1.84438
$567$	0	0
$568$	39.4566	1.65556
$569$	−7.13524	−0.299125	−0.149562	−	0.988752i	$-0.547786\pi$
$569$	−7.13524	−0.299125	−0.149562	+	0.988752i	$0.547786\pi$
$570$	0	0
$571$	32.4839	1.35941	0.679705	−	0.733486i	$-0.262108\pi$
$571$	32.4839	1.35941	0.679705	+	0.733486i	$0.262108\pi$
$572$	0	0
$573$	0	0
$574$	1.29180	0.0539186
$575$	−1.18461	−0.0494018
$576$	0	0
$577$	34.7819	1.44799	0.723995	−	0.689806i	$-0.242304\pi$
$577$	34.7819	1.44799	0.723995	+	0.689806i	$0.242304\pi$
$578$	18.2123	0.757532
$579$	0	0
$580$	−0.0299853	−0.00124507
$581$	4.37573	0.181536
$582$	0	0
$583$	0	0
$584$	−17.0708	−0.706395
$585$	0	0
$586$	33.8651	1.39896
$587$	14.4749	0.597445	0.298722	−	0.954340i	$-0.403440\pi$
$587$	14.4749	0.597445	0.298722	+	0.954340i	$0.403440\pi$
$588$	0	0
$589$	7.58716	0.312623
$590$	19.8235	0.816120
$591$	0	0
$592$	−8.32004	−0.341952
$593$	15.0291	0.617169	0.308585	−	0.951197i	$-0.400145\pi$
$593$	15.0291	0.617169	0.308585	+	0.951197i	$0.400145\pi$
$594$	0	0
$595$	10.0181	0.410701
$596$	−0.0448863	−0.00183862
$597$	0	0
$598$	−4.71316	−0.192736
$599$	−1.76045	−0.0719302	−0.0359651	−	0.999353i	$-0.511451\pi$
$599$	−1.76045	−0.0719302	−0.0359651	+	0.999353i	$0.511451\pi$
$600$	0	0
$601$	23.4365	0.955993	0.477997	−	0.878362i	$-0.341363\pi$
$601$	23.4365	0.955993	0.477997	+	0.878362i	$0.341363\pi$
$602$	11.3077	0.460866
$603$	0	0
$604$	−0.310948	−0.0126523
$605$	0	0
$606$	0	0
$607$	−25.2785	−1.02602	−0.513012	−	0.858382i	$-0.671470\pi$
$607$	−25.2785	−1.02602	−0.513012	+	0.858382i	$0.671470\pi$
$608$	−0.458257	−0.0185848
$609$	0	0
$610$	−16.1981	−0.655843
$611$	27.7700	1.12346
$612$	0	0
$613$	1.16094	0.0468900	0.0234450	−	0.999725i	$-0.492537\pi$
$613$	1.16094	0.0468900	0.0234450	+	0.999725i	$0.492537\pi$
$614$	−7.69880	−0.310698
$615$	0	0
$616$	0	0
$617$	−12.9711	−0.522197	−0.261098	−	0.965312i	$-0.584085\pi$
$617$	−12.9711	−0.522197	−0.261098	+	0.965312i	$0.584085\pi$
$618$	0	0
$619$	45.7920	1.84053	0.920267	−	0.391291i	$-0.127971\pi$
$619$	45.7920	1.84053	0.920267	+	0.391291i	$0.127971\pi$
$620$	0.0334479	0.00134330
$621$	0	0
$622$	−19.5725	−0.784787
$623$	−15.3437	−0.614733
$624$	0	0
$625$	−14.0605	−0.562419
$626$	38.6824	1.54606
$627$	0	0
$628$	−0.185097	−0.00738615
$629$	11.4587	0.456887
$630$	0	0
$631$	−12.6207	−0.502421	−0.251211	−	0.967932i	$-0.580829\pi$
$631$	−12.6207	−0.502421	−0.251211	+	0.967932i	$0.580829\pi$
$632$	44.3924	1.76583
$633$	0	0
$634$	−11.0285	−0.437998
$635$	9.21713	0.365771
$636$	0	0
$637$	−4.64706	−0.184123
$638$	0	0
$639$	0	0
$640$	20.5001	0.810338
$641$	−27.9567	−1.10422	−0.552112	−	0.833770i	$-0.686178\pi$
$641$	−27.9567	−1.10422	−0.552112	+	0.833770i	$0.686178\pi$
$642$	0	0
$643$	−49.6981	−1.95990	−0.979950	−	0.199243i	$-0.936152\pi$
$643$	−49.6981	−1.95990	−0.979950	+	0.199243i	$0.936152\pi$
$644$	−0.0100500	−0.000396026 0
$645$	0	0
$646$	−44.7212	−1.75953
$647$	11.0067	0.432718	0.216359	−	0.976314i	$-0.430582\pi$
$647$	11.0067	0.432718	0.216359	+	0.976314i	$0.430582\pi$
$648$	0	0
$649$	0	0
$650$	10.7797	0.422816
$651$	0	0
$652$	−0.191570	−0.00750246
$653$	−28.9364	−1.13237	−0.566185	−	0.824278i	$-0.691581\pi$
$653$	−28.9364	−1.13237	−0.566185	+	0.824278i	$0.691581\pi$
$654$	0	0
$655$	0.329747	0.0128843
$656$	−3.64080	−0.142149
$657$	0	0
$658$	−8.42155	−0.328306
$659$	−10.8405	−0.422288	−0.211144	−	0.977455i	$-0.567719\pi$
$659$	−10.8405	−0.422288	−0.211144	+	0.977455i	$0.567719\pi$
$660$	0	0
$661$	20.3444	0.791305	0.395652	−	0.918400i	$-0.370519\pi$
$661$	20.3444	0.791305	0.395652	+	0.918400i	$0.370519\pi$
$662$	−39.6655	−1.54164
$663$	0	0
$664$	−12.4193	−0.481961
$665$	−10.6242	−0.411989
$666$	0	0
$667$	0.843809	0.0326724
$668$	0.129990	0.00502946
$669$	0	0
$670$	−39.9197	−1.54223
$671$	0	0
$672$	0	0
$673$	−12.0788	−0.465604	−0.232802	−	0.972524i	$-0.574789\pi$
$673$	−12.0788	−0.465604	−0.232802	+	0.972524i	$0.574789\pi$
$674$	−35.2020	−1.35593
$675$	0	0
$676$	−0.120027	−0.00461642
$677$	−3.39630	−0.130530	−0.0652651	−	0.997868i	$-0.520789\pi$
$677$	−3.39630	−0.130530	−0.0652651	+	0.997868i	$0.520789\pi$
$678$	0	0
$679$	2.41124	0.0925349
$680$	−28.4335	−1.09037
$681$	0	0
$682$	0	0
$683$	4.75643	0.182000	0.0909999	−	0.995851i	$-0.470994\pi$
$683$	4.75643	0.182000	0.0909999	+	0.995851i	$0.470994\pi$
$684$	0	0
$685$	−15.2444	−0.582458
$686$	1.40927	0.0538061
$687$	0	0
$688$	−31.8695	−1.21501
$689$	47.1441	1.79605
$690$	0	0
$691$	6.63388	0.252365	0.126182	−	0.992007i	$-0.459728\pi$
$691$	6.63388	0.252365	0.126182	+	0.992007i	$0.459728\pi$
$692$	0.146707	0.00557695
$693$	0	0
$694$	−29.1003	−1.10463
$695$	−12.7486	−0.483583
$696$	0	0
$697$	5.01425	0.189928
$698$	−8.45168	−0.319901
$699$	0	0
$700$	0.0229859	0.000868786 0
$701$	−3.03003	−0.114443	−0.0572213	−	0.998362i	$-0.518224\pi$
$701$	−3.03003	−0.114443	−0.0572213	+	0.998362i	$0.518224\pi$
$702$	0	0
$703$	−12.1519	−0.458319
$704$	0	0
$705$	0	0
$706$	−33.8762	−1.27495
$707$	−11.8959	−0.447390
$708$	0	0
$709$	−13.6570	−0.512901	−0.256451	−	0.966557i	$-0.582553\pi$
$709$	−13.6570	−0.512901	−0.256451	+	0.966557i	$0.582553\pi$
$710$	−35.8796	−1.34654
$711$	0	0
$712$	43.5488	1.63206
$713$	−0.941248	−0.0352500
$714$	0	0
$715$	0	0
$716$	−0.116231	−0.00434376
$717$	0	0
$718$	15.4317	0.575904
$719$	1.90029	0.0708689	0.0354344	−	0.999372i	$-0.488719\pi$
$719$	1.90029	0.0708689	0.0354344	+	0.999372i	$0.488719\pi$
$720$	0	0
$721$	0.396314	0.0147595
$722$	20.6509	0.768547
$723$	0	0
$724$	0.206716	0.00768255
$725$	−1.92992	−0.0716754
$726$	0	0
$727$	−13.8211	−0.512595	−0.256298	−	0.966598i	$-0.582503\pi$
$727$	−13.8211	−0.512595	−0.256298	+	0.966598i	$0.582503\pi$
$728$	13.1893	0.488830
$729$	0	0
$730$	15.5232	0.574540
$731$	43.8918	1.62340
$732$	0	0
$733$	48.3744	1.78675	0.893374	−	0.449314i	$-0.148331\pi$
$733$	48.3744	1.78675	0.893374	+	0.449314i	$0.148331\pi$
$734$	14.9633	0.552304
$735$	0	0
$736$	0.0568504	0.00209553
$737$	0	0
$738$	0	0
$739$	23.2081	0.853724	0.426862	−	0.904317i	$-0.359619\pi$
$739$	23.2081	0.853724	0.426862	+	0.904317i	$0.359619\pi$
$740$	−0.0535717	−0.00196934
$741$	0	0
$742$	−14.2969	−0.524857
$743$	44.8311	1.64469	0.822347	−	0.568986i	$-0.192664\pi$
$743$	44.8311	1.64469	0.822347	+	0.568986i	$0.192664\pi$
$744$	0	0
$745$	5.88665	0.215670
$746$	51.5843	1.88863
$747$	0	0
$748$	0	0
$749$	3.26935	0.119460
$750$	0	0
$751$	−41.1856	−1.50288	−0.751442	−	0.659799i	$-0.770641\pi$
$751$	−41.1856	−1.50288	−0.751442	+	0.659799i	$0.770641\pi$
$752$	23.7353	0.865537
$753$	0	0
$754$	−7.67847	−0.279633
$755$	40.7795	1.48412
$756$	0	0
$757$	−21.8888	−0.795561	−0.397781	−	0.917481i	$-0.630219\pi$
$757$	−21.8888	−0.795561	−0.397781	+	0.917481i	$0.630219\pi$
$758$	−17.8382	−0.647912
$759$	0	0
$760$	30.1537	1.09379
$761$	35.7154	1.29468	0.647340	−	0.762201i	$-0.275881\pi$
$761$	35.7154	1.29468	0.647340	+	0.762201i	$0.275881\pi$
$762$	0	0
$763$	−2.84638	−0.103046
$764$	−0.134154	−0.00485351
$765$	0	0
$766$	21.7984	0.787608
$767$	−35.6931	−1.28880
$768$	0	0
$769$	5.30246	0.191212	0.0956058	−	0.995419i	$-0.469521\pi$
$769$	5.30246	0.191212	0.0956058	+	0.995419i	$0.469521\pi$
$770$	0	0
$771$	0	0
$772$	0.0207765	0.000747763 0
$773$	49.8912	1.79446	0.897231	−	0.441561i	$-0.145575\pi$
$773$	49.8912	1.79446	0.897231	+	0.441561i	$0.145575\pi$
$774$	0	0
$775$	2.15278	0.0773300
$776$	−6.84362	−0.245672
$777$	0	0
$778$	17.9160	0.642318
$779$	−5.31762	−0.190524
$780$	0	0
$781$	0	0
$782$	5.54803	0.198397
$783$	0	0
$784$	−3.97188	−0.141853
$785$	24.2746	0.866398
$786$	0	0
$787$	35.8570	1.27816	0.639081	−	0.769139i	$-0.279315\pi$
$787$	35.8570	1.27816	0.639081	+	0.769139i	$0.279315\pi$
$788$	−0.196109	−0.00698609
$789$	0	0
$790$	−40.3679	−1.43623
$791$	−14.5445	−0.517144
$792$	0	0
$793$	29.1655	1.03570
$794$	−26.7161	−0.948117
$795$	0	0
$796$	0.0598699	0.00212203
$797$	−14.7413	−0.522163	−0.261081	−	0.965317i	$-0.584079\pi$
$797$	−14.7413	−0.522163	−0.261081	+	0.965317i	$0.584079\pi$
$798$	0	0
$799$	−32.6891	−1.15646
$800$	−0.130026	−0.00459710
$801$	0	0
$802$	12.2354	0.432046
$803$	0	0
$804$	0	0
$805$	1.31802	0.0464540
$806$	8.56514	0.301694
$807$	0	0
$808$	33.7630	1.18778
$809$	26.9758	0.948420	0.474210	−	0.880412i	$-0.342734\pi$
$809$	26.9758	0.948420	0.474210	+	0.880412i	$0.342734\pi$
$810$	0	0
$811$	1.46440	0.0514219	0.0257109	−	0.999669i	$-0.491815\pi$
$811$	1.46440	0.0514219	0.0257109	+	0.999669i	$0.491815\pi$
$812$	−0.0163730	−0.000574581 0
$813$	0	0
$814$	0	0
$815$	25.1236	0.880041
$816$	0	0
$817$	−46.5474	−1.62849
$818$	8.05264	0.281554
$819$	0	0
$820$	−0.0234427	−0.000818654 0
$821$	−30.6287	−1.06895	−0.534474	−	0.845185i	$-0.679490\pi$
$821$	−30.6287	−1.06895	−0.534474	+	0.845185i	$0.679490\pi$
$822$	0	0
$823$	−24.5338	−0.855195	−0.427598	−	0.903969i	$-0.640640\pi$
$823$	−24.5338	−0.855195	−0.427598	+	0.903969i	$0.640640\pi$
$824$	−1.12482	−0.0391851
$825$	0	0
$826$	10.8243	0.376626
$827$	6.56686	0.228352	0.114176	−	0.993461i	$-0.463577\pi$
$827$	6.56686	0.228352	0.114176	+	0.993461i	$0.463577\pi$
$828$	0	0
$829$	−19.8442	−0.689219	−0.344610	−	0.938746i	$-0.611989\pi$
$829$	−19.8442	−0.689219	−0.344610	+	0.938746i	$0.611989\pi$
$830$	11.2934	0.391999
$831$	0	0
$832$	−37.4324	−1.29773
$833$	5.47021	0.189532
$834$	0	0
$835$	−17.0476	−0.589958
$836$	0	0
$837$	0	0
$838$	38.3193	1.32372
$839$	7.96051	0.274827	0.137414	−	0.990514i	$-0.456121\pi$
$839$	7.96051	0.274827	0.137414	+	0.990514i	$0.456121\pi$
$840$	0	0
$841$	−27.6253	−0.952597
$842$	−33.7732	−1.16390
$843$	0	0
$844$	0.0203222	0.000699518 0
$845$	15.7410	0.541507
$846$	0	0
$847$	0	0
$848$	40.2944	1.38372
$849$	0	0
$850$	−12.6892	−0.435236
$851$	1.50755	0.0516780
$852$	0	0
$853$	34.0732	1.16664	0.583322	−	0.812241i	$-0.301753\pi$
$853$	34.0732	1.16664	0.583322	+	0.812241i	$0.301753\pi$
$854$	−8.84474	−0.302661
$855$	0	0
$856$	−9.27913	−0.317154
$857$	−24.8539	−0.848992	−0.424496	−	0.905430i	$-0.639549\pi$
$857$	−24.8539	−0.848992	−0.424496	+	0.905430i	$0.639549\pi$
$858$	0	0
$859$	2.05654	0.0701683	0.0350841	−	0.999384i	$-0.488830\pi$
$859$	2.05654	0.0701683	0.0350841	+	0.999384i	$0.488830\pi$
$860$	−0.205204	−0.00699739
$861$	0	0
$862$	−23.2151	−0.790709
$863$	0.259476	0.00883265	0.00441633	−	0.999990i	$-0.498594\pi$
$863$	0.259476	0.00883265	0.00441633	+	0.999990i	$0.498594\pi$
$864$	0	0
$865$	−19.2400	−0.654178
$866$	28.3135	0.962132
$867$	0	0
$868$	0.0182637	0.000619910 0
$869$	0	0
$870$	0	0
$871$	71.8773	2.43547
$872$	8.07865	0.273578
$873$	0	0
$874$	−5.88370	−0.199019
$875$	−12.1714	−0.411470
$876$	0	0
$877$	18.5930	0.627840	0.313920	−	0.949449i	$-0.398358\pi$
$877$	18.5930	0.627840	0.313920	+	0.949449i	$0.398358\pi$
$878$	37.7235	1.27311
$879$	0	0
$880$	0	0
$881$	−6.45292	−0.217404	−0.108702	−	0.994074i	$-0.534670\pi$
$881$	−6.45292	−0.217404	−0.108702	+	0.994074i	$0.534670\pi$
$882$	0	0
$883$	−0.278487	−0.00937185	−0.00468592	−	0.999989i	$-0.501492\pi$
$883$	−0.278487	−0.00937185	−0.00468592	+	0.999989i	$0.501492\pi$
$884$	0.354983	0.0119394
$885$	0	0
$886$	−36.9444	−1.24117
$887$	−30.5570	−1.02600	−0.513001	−	0.858388i	$-0.671466\pi$
$887$	−30.5570	−1.02600	−0.513001	+	0.858388i	$0.671466\pi$
$888$	0	0
$889$	5.03287	0.168797
$890$	−39.6008	−1.32742
$891$	0	0
$892$	0.0678176	0.00227070
$893$	34.6669	1.16008
$894$	0	0
$895$	15.2432	0.509524
$896$	11.1938	0.373958
$897$	0	0
$898$	−13.7367	−0.458400
$899$	−1.53344	−0.0511430
$900$	0	0
$901$	−55.4949	−1.84880
$902$	0	0
$903$	0	0
$904$	41.2805	1.37297
$905$	−27.1099	−0.901165
$906$	0	0
$907$	31.9347	1.06037	0.530187	−	0.847881i	$-0.322122\pi$
$907$	31.9347	1.06037	0.530187	+	0.847881i	$0.322122\pi$
$908$	0.00554592	0.000184048 0
$909$	0	0
$910$	−11.9937	−0.397586
$911$	−2.80042	−0.0927821	−0.0463911	−	0.998923i	$-0.514772\pi$
$911$	−2.80042	−0.0927821	−0.0463911	+	0.998923i	$0.514772\pi$
$912$	0	0
$913$	0	0
$914$	16.7496	0.554027
$915$	0	0
$916$	−0.0305772	−0.00101030
$917$	0.180053	0.00594587
$918$	0	0
$919$	−15.8339	−0.522311	−0.261156	−	0.965297i	$-0.584104\pi$
$919$	−15.8339	−0.522311	−0.261156	+	0.965297i	$0.584104\pi$
$920$	−3.74081	−0.123331
$921$	0	0
$922$	−12.8911	−0.424545
$923$	64.6030	2.12643
$924$	0	0
$925$	−3.44799	−0.113369
$926$	54.8687	1.80310
$927$	0	0
$928$	0.0926181	0.00304034
$929$	27.4834	0.901699	0.450850	−	0.892600i	$-0.351121\pi$
$929$	27.4834	0.901699	0.450850	+	0.892600i	$0.351121\pi$
$930$	0	0
$931$	−5.80118	−0.190126
$932$	−0.0175975	−0.000576424 0
$933$	0	0
$934$	29.2894	0.958379
$935$	0	0
$936$	0	0
$937$	−12.3086	−0.402105	−0.201053	−	0.979580i	$-0.564436\pi$
$937$	−12.3086	−0.402105	−0.201053	+	0.979580i	$0.564436\pi$
$938$	−21.7975	−0.711714
$939$	0	0
$940$	0.152829	0.00498472
$941$	−1.46014	−0.0475991	−0.0237996	−	0.999717i	$-0.507576\pi$
$941$	−1.46014	−0.0475991	−0.0237996	+	0.999717i	$0.507576\pi$
$942$	0	0
$943$	0.659693	0.0214826
$944$	−30.5072	−0.992925
$945$	0	0
$946$	0	0
$947$	−11.0714	−0.359771	−0.179885	−	0.983688i	$-0.557573\pi$
$947$	−11.0714	−0.359771	−0.179885	+	0.983688i	$0.557573\pi$
$948$	0	0
$949$	−27.9503	−0.907305
$950$	13.4569	0.436600
$951$	0	0
$952$	−15.5256	−0.503189
$953$	14.8234	0.480176	0.240088	−	0.970751i	$-0.422824\pi$
$953$	14.8234	0.480176	0.240088	+	0.970751i	$0.422824\pi$
$954$	0	0
$955$	17.5937	0.569318
$956$	−0.155867	−0.00504110
$957$	0	0
$958$	34.5782	1.11717
$959$	−8.32395	−0.268794
$960$	0	0
$961$	−29.2895	−0.944822
$962$	−13.7183	−0.442297
$963$	0	0
$964$	0.300018	0.00966292
$965$	−2.72475	−0.0877129
$966$	0	0
$967$	16.5193	0.531224	0.265612	−	0.964080i	$-0.414426\pi$
$967$	16.5193	0.531224	0.265612	+	0.964080i	$0.414426\pi$
$968$	0	0
$969$	0	0
$970$	6.22320	0.199815
$971$	−40.7993	−1.30931	−0.654656	−	0.755927i	$-0.727187\pi$
$971$	−40.7993	−1.30931	−0.654656	+	0.755927i	$0.727187\pi$
$972$	0	0
$973$	−6.96119	−0.223166
$974$	−17.4345	−0.558637
$975$	0	0
$976$	24.9280	0.797925
$977$	−49.1145	−1.57131	−0.785656	−	0.618664i	$-0.787674\pi$
$977$	−49.1145	−1.57131	−0.785656	+	0.618664i	$0.787674\pi$
$978$	0	0
$979$	0	0
$980$	−0.0255744	−0.000816945 0
$981$	0	0
$982$	−23.2883	−0.743158
$983$	−1.10679	−0.0353011	−0.0176505	−	0.999844i	$-0.505619\pi$
$983$	−1.10679	−0.0353011	−0.0176505	+	0.999844i	$0.505619\pi$
$984$	0	0
$985$	25.7188	0.819470
$986$	9.03860	0.287848
$987$	0	0
$988$	−0.376461	−0.0119768
$989$	5.77458	0.183621
$990$	0	0
$991$	41.7851	1.32735	0.663674	−	0.748022i	$-0.268996\pi$
$991$	41.7851	1.32735	0.663674	+	0.748022i	$0.268996\pi$
$992$	−0.103313	−0.00328020
$993$	0	0
$994$	−19.5915	−0.621405
$995$	−7.85168	−0.248915
$996$	0	0
$997$	44.1945	1.39965	0.699827	−	0.714313i	$-0.253261\pi$
$997$	44.1945	1.39965	0.699827	+	0.714313i	$0.253261\pi$
$998$	−19.3647	−0.612978
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.ct.1.6		8
3.2	odd	2		847.2.a.p.1.3		8
11.3	even	5		693.2.m.i.64.3		16
11.4	even	5		693.2.m.i.379.3		16
11.10	odd	2		7623.2.a.cw.1.3		8
21.20	even	2		5929.2.a.bt.1.3		8
33.2	even	10		847.2.f.v.323.2		16
33.5	odd	10		847.2.f.w.729.3		16
33.8	even	10		847.2.f.x.372.3		16
33.14	odd	10		77.2.f.b.64.2	✓	16
33.17	even	10		847.2.f.v.729.2		16
33.20	odd	10		847.2.f.w.323.3		16
33.26	odd	10		77.2.f.b.71.2	yes	16
33.29	even	10		847.2.f.x.148.3		16
33.32	even	2		847.2.a.o.1.6		8
231.26	even	30		539.2.q.f.214.2		32
231.47	even	30		539.2.q.f.361.3		32
231.59	even	30		539.2.q.f.324.3		32
231.80	even	30		539.2.q.f.471.2		32
231.125	even	10		539.2.f.e.148.2		16
231.146	even	10		539.2.f.e.295.2		16
231.158	odd	30		539.2.q.g.324.3		32
231.179	odd	30		539.2.q.g.471.2		32
231.191	odd	30		539.2.q.g.214.2		32
231.212	odd	30		539.2.q.g.361.3		32
231.230	odd	2		5929.2.a.bs.1.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.f.b.64.2	✓	16	33.14	odd	10
77.2.f.b.71.2	yes	16	33.26	odd	10
539.2.f.e.148.2		16	231.125	even	10
539.2.f.e.295.2		16	231.146	even	10
539.2.q.f.214.2		32	231.26	even	30
539.2.q.f.324.3		32	231.59	even	30
539.2.q.f.361.3		32	231.47	even	30
539.2.q.f.471.2		32	231.80	even	30
539.2.q.g.214.2		32	231.191	odd	30
539.2.q.g.324.3		32	231.158	odd	30
539.2.q.g.361.3		32	231.212	odd	30
539.2.q.g.471.2		32	231.179	odd	30
693.2.m.i.64.3		16	11.3	even	5
693.2.m.i.379.3		16	11.4	even	5
847.2.a.o.1.6		8	33.32	even	2
847.2.a.p.1.3		8	3.2	odd	2
847.2.f.v.323.2		16	33.2	even	10
847.2.f.v.729.2		16	33.17	even	10
847.2.f.w.323.3		16	33.20	odd	10
847.2.f.w.729.3		16	33.5	odd	10
847.2.f.x.148.3		16	33.29	even	10
847.2.f.x.372.3		16	33.8	even	10
5929.2.a.bs.1.6		8	231.230	odd	2
5929.2.a.bt.1.3		8	21.20	even	2
7623.2.a.ct.1.6		8	1.1	even	1	trivial
7623.2.a.cw.1.3		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)

\(f(q)\)	\(=\)	\(q+1.40927 q^{2} -0.0139645 q^{4} +1.83139 q^{5} +1.00000 q^{7} -2.83822 q^{8} +O(q^{10})\)	\(q+1.40927 q^{2} -0.0139645 q^{4} +1.83139 q^{5} +1.00000 q^{7} -2.83822 q^{8} +2.58091 q^{10} -4.64706 q^{13} +1.40927 q^{14} -3.97188 q^{16} +5.47021 q^{17} -5.80118 q^{19} -0.0255744 q^{20} +0.719682 q^{23} -1.64602 q^{25} -6.54895 q^{26} -0.0139645 q^{28} +1.17247 q^{29} -1.30787 q^{31} +0.0789938 q^{32} +7.70900 q^{34} +1.83139 q^{35} +2.09474 q^{37} -8.17541 q^{38} -5.19787 q^{40} +0.916645 q^{41} +8.02379 q^{43} +1.01423 q^{46} -5.97584 q^{47} +1.00000 q^{49} -2.31969 q^{50} +0.0648939 q^{52} -10.1449 q^{53} -2.83822 q^{56} +1.65233 q^{58} +7.68081 q^{59} -6.27612 q^{61} -1.84313 q^{62} +8.05508 q^{64} -8.51056 q^{65} -15.4673 q^{67} -0.0763889 q^{68} +2.58091 q^{70} -13.9019 q^{71} +6.01462 q^{73} +2.95205 q^{74} +0.0810106 q^{76} -15.6409 q^{79} -7.27404 q^{80} +1.29180 q^{82} +4.37573 q^{83} +10.0181 q^{85} +11.3077 q^{86} -15.3437 q^{89} -4.64706 q^{91} -0.0100500 q^{92} -8.42155 q^{94} -10.6242 q^{95} +2.41124 q^{97} +1.40927 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - q^{2} + 7 q^{4} - 10 q^{5} + 8 q^{7}+O(q^{10}) \) 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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