LMFDB - Embedded newform 7623.2.a.bz.1.1

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:	$3$
Coefficient field:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 847)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.76156$ 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)

$f(q)$	$=$	$q-2.76156 q^{2} +5.62620 q^{4} +2.62620 q^{5} +1.00000 q^{7} -10.0140 q^{8} +O(q^{10})$	$q-2.76156 q^{2} +5.62620 q^{4} +2.62620 q^{5} +1.00000 q^{7} -10.0140 q^{8} -7.25240 q^{10} -2.38776 q^{13} -2.76156 q^{14} +16.4017 q^{16} +2.38776 q^{17} -1.72928 q^{19} +14.7755 q^{20} +0.626198 q^{23} +1.89692 q^{25} +6.59392 q^{26} +5.62620 q^{28} -1.72928 q^{29} -2.23844 q^{31} -25.2663 q^{32} -6.59392 q^{34} +2.62620 q^{35} -6.89692 q^{37} +4.77551 q^{38} -26.2986 q^{40} -10.3878 q^{41} +7.25240 q^{43} -1.72928 q^{46} -6.38776 q^{47} +1.00000 q^{49} -5.23844 q^{50} -13.4340 q^{52} +9.25240 q^{53} -10.0140 q^{56} +4.77551 q^{58} +1.76156 q^{59} -10.3878 q^{61} +6.18159 q^{62} +36.9711 q^{64} -6.27072 q^{65} -6.42003 q^{67} +13.4340 q^{68} -7.25240 q^{70} -8.08476 q^{71} +10.3878 q^{73} +19.0462 q^{74} -9.72928 q^{76} +15.2524 q^{79} +43.0741 q^{80} +28.6864 q^{82} -12.7755 q^{83} +6.27072 q^{85} -20.0279 q^{86} +14.1493 q^{89} -2.38776 q^{91} +3.52311 q^{92} +17.6402 q^{94} -4.54144 q^{95} -8.35548 q^{97} -2.76156 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} + 8 q^{4} - q^{5} + 3 q^{7} - 6 q^{8}+O(q^{10}) $ 3 * q - 2 * q^2 + 8 * q^4 - q^5 + 3 * q^7 - 6 * q^8	$ 3 q - 2 q^{2} + 8 q^{4} - q^{5} + 3 q^{7} - 6 q^{8} - 4 q^{10} + 8 q^{13} - 2 q^{14} + 10 q^{16} - 8 q^{17} + 14 q^{20} - 7 q^{23} + 2 q^{25} + 12 q^{26} + 8 q^{28} - 13 q^{31} - 34 q^{32} - 12 q^{34} - q^{35} - 17 q^{37} - 16 q^{38} - 36 q^{40} - 16 q^{41} + 4 q^{43} - 4 q^{47} + 3 q^{49} - 22 q^{50} + 10 q^{53} - 6 q^{56} - 16 q^{58} - q^{59} - 16 q^{61} - 4 q^{62} + 34 q^{64} - 24 q^{65} - 3 q^{67} - 4 q^{70} - 5 q^{71} + 16 q^{73} + 32 q^{74} - 24 q^{76} + 28 q^{79} + 56 q^{80} + 28 q^{82} - 8 q^{83} + 24 q^{85} - 12 q^{86} + 21 q^{89} + 8 q^{91} - 2 q^{92} + 20 q^{94} - 24 q^{95} - 11 q^{97} - 2 q^{98}+O(q^{100}) $ 3 * q - 2 * q^2 + 8 * q^4 - q^5 + 3 * q^7 - 6 * q^8 - 4 * q^10 + 8 * q^13 - 2 * q^14 + 10 * q^16 - 8 * q^17 + 14 * q^20 - 7 * q^23 + 2 * q^25 + 12 * q^26 + 8 * q^28 - 13 * q^31 - 34 * q^32 - 12 * q^34 - q^35 - 17 * q^37 - 16 * q^38 - 36 * q^40 - 16 * q^41 + 4 * q^43 - 4 * q^47 + 3 * q^49 - 22 * q^50 + 10 * q^53 - 6 * q^56 - 16 * q^58 - q^59 - 16 * q^61 - 4 * q^62 + 34 * q^64 - 24 * q^65 - 3 * q^67 - 4 * q^70 - 5 * q^71 + 16 * q^73 + 32 * q^74 - 24 * q^76 + 28 * q^79 + 56 * q^80 + 28 * q^82 - 8 * q^83 + 24 * q^85 - 12 * q^86 + 21 * q^89 + 8 * q^91 - 2 * q^92 + 20 * q^94 - 24 * q^95 - 11 * q^97 - 2 * 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 $

$2$	−2.76156	−1.95272	−0.976358	−	0.216160i	$-0.930647\pi$
$2$	−2.76156	−1.95272	−0.976358	+	0.216160i	$0.930647\pi$
$3$	0	0
$3$	0	0
$4$	5.62620	2.81310
$5$	2.62620	1.17447	0.587236	−	0.809416i	$-0.300216\pi$
$5$	2.62620	1.17447	0.587236	+	0.809416i	$0.300216\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−10.0140	−3.54047
$9$	0	0
$10$	−7.25240	−2.29341
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−2.38776	−0.662244	−0.331122	−	0.943588i	$-0.607427\pi$
$13$	−2.38776	−0.662244	−0.331122	+	0.943588i	$0.607427\pi$
$14$	−2.76156	−0.738057
$15$	0	0
$16$	16.4017	4.10043
$17$	2.38776	0.579116	0.289558	−	0.957161i	$-0.406492\pi$
$17$	2.38776	0.579116	0.289558	+	0.957161i	$0.406492\pi$
$18$	0	0
$19$	−1.72928	−0.396724	−0.198362	−	0.980129i	$-0.563562\pi$
$19$	−1.72928	−0.396724	−0.198362	+	0.980129i	$0.563562\pi$
$20$	14.7755	3.30390
$21$	0	0
$22$	0	0
$23$	0.626198	0.130571	0.0652857	−	0.997867i	$-0.479204\pi$
$23$	0.626198	0.130571	0.0652857	+	0.997867i	$0.479204\pi$
$24$	0	0
$25$	1.89692	0.379383
$26$	6.59392	1.29317
$27$	0	0
$28$	5.62620	1.06325
$29$	−1.72928	−0.321120	−0.160560	−	0.987026i	$-0.551330\pi$
$29$	−1.72928	−0.321120	−0.160560	+	0.987026i	$0.551330\pi$
$30$	0	0
$31$	−2.23844	−0.402036	−0.201018	−	0.979588i	$-0.564425\pi$
$31$	−2.23844	−0.402036	−0.201018	+	0.979588i	$0.564425\pi$
$32$	−25.2663	−4.46650
$33$	0	0
$34$	−6.59392	−1.13085
$35$	2.62620	0.443908
$36$	0	0
$37$	−6.89692	−1.13385	−0.566923	−	0.823771i	$-0.691866\pi$
$37$	−6.89692	−1.13385	−0.566923	+	0.823771i	$0.691866\pi$
$38$	4.77551	0.774690
$39$	0	0
$40$	−26.2986	−4.15818
$41$	−10.3878	−1.62229	−0.811147	−	0.584842i	$-0.801157\pi$
$41$	−10.3878	−1.62229	−0.811147	+	0.584842i	$0.801157\pi$
$42$	0	0
$43$	7.25240	1.10598	0.552990	−	0.833188i	$-0.313487\pi$
$43$	7.25240	1.10598	0.552990	+	0.833188i	$0.313487\pi$
$44$	0	0
$45$	0	0
$46$	−1.72928	−0.254969
$47$	−6.38776	−0.931750	−0.465875	−	0.884851i	$-0.654260\pi$
$47$	−6.38776	−0.931750	−0.465875	+	0.884851i	$0.654260\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−5.23844	−0.740828
$51$	0	0
$52$	−13.4340	−1.86296
$53$	9.25240	1.27091	0.635457	−	0.772136i	$-0.280812\pi$
$53$	9.25240	1.27091	0.635457	+	0.772136i	$0.280812\pi$
$54$	0	0
$55$	0	0
$56$	−10.0140	−1.33817
$57$	0	0
$58$	4.77551	0.627055
$59$	1.76156	0.229335	0.114668	−	0.993404i	$-0.463420\pi$
$59$	1.76156	0.229335	0.114668	+	0.993404i	$0.463420\pi$
$60$	0	0
$61$	−10.3878	−1.33002	−0.665008	−	0.746836i	$-0.731572\pi$
$61$	−10.3878	−1.33002	−0.665008	+	0.746836i	$0.731572\pi$
$62$	6.18159	0.785062
$63$	0	0
$64$	36.9711	4.62138
$65$	−6.27072	−0.777787
$66$	0	0
$67$	−6.42003	−0.784332	−0.392166	−	0.919895i	$-0.628274\pi$
$67$	−6.42003	−0.784332	−0.392166	+	0.919895i	$0.628274\pi$
$68$	13.4340	1.62911
$69$	0	0
$70$	−7.25240	−0.866827
$71$	−8.08476	−0.959485	−0.479742	−	0.877409i	$-0.659270\pi$
$71$	−8.08476	−0.959485	−0.479742	+	0.877409i	$0.659270\pi$
$72$	0	0
$73$	10.3878	1.21579	0.607897	−	0.794016i	$-0.292013\pi$
$73$	10.3878	1.21579	0.607897	+	0.794016i	$0.292013\pi$
$74$	19.0462	2.21408
$75$	0	0
$76$	−9.72928	−1.11603
$77$	0	0
$78$	0	0
$79$	15.2524	1.71603	0.858014	−	0.513626i	$-0.171698\pi$
$79$	15.2524	1.71603	0.858014	+	0.513626i	$0.171698\pi$
$80$	43.0741	4.81583
$81$	0	0
$82$	28.6864	3.16788
$83$	−12.7755	−1.40229	−0.701147	−	0.713017i	$-0.747328\pi$
$83$	−12.7755	−1.40229	−0.701147	+	0.713017i	$0.747328\pi$
$84$	0	0
$85$	6.27072	0.680155
$86$	−20.0279	−2.15966
$87$	0	0
$88$	0	0
$89$	14.1493	1.49982	0.749912	−	0.661538i	$-0.230096\pi$
$89$	14.1493	1.49982	0.749912	+	0.661538i	$0.230096\pi$
$90$	0	0
$91$	−2.38776	−0.250305
$92$	3.52311	0.367310
$93$	0	0
$94$	17.6402	1.81944
$95$	−4.54144	−0.465942
$96$	0	0
$97$	−8.35548	−0.848370	−0.424185	−	0.905575i	$-0.639439\pi$
$97$	−8.35548	−0.848370	−0.424185	+	0.905575i	$0.639439\pi$
$98$	−2.76156	−0.278959
$99$	0	0
$100$	10.6724	1.06724
$101$	13.8463	1.37776	0.688880	−	0.724875i	$-0.258103\pi$
$101$	13.8463	1.37776	0.688880	+	0.724875i	$0.258103\pi$
$102$	0	0
$103$	−2.92919	−0.288622	−0.144311	−	0.989532i	$-0.546097\pi$
$103$	−2.92919	−0.288622	−0.144311	+	0.989532i	$0.546097\pi$
$104$	23.9109	2.34465
$105$	0	0
$106$	−25.5510	−2.48173
$107$	−3.45856	−0.334352	−0.167176	−	0.985927i	$-0.553465\pi$
$107$	−3.45856	−0.334352	−0.167176	+	0.985927i	$0.553465\pi$
$108$	0	0
$109$	−16.2341	−1.55494	−0.777471	−	0.628919i	$-0.783498\pi$
$109$	−16.2341	−1.55494	−0.777471	+	0.628919i	$0.783498\pi$
$110$	0	0
$111$	0	0
$112$	16.4017	1.54982
$113$	−7.10308	−0.668202	−0.334101	−	0.942537i	$-0.608433\pi$
$113$	−7.10308	−0.668202	−0.334101	+	0.942537i	$0.608433\pi$
$114$	0	0
$115$	1.64452	0.153352
$116$	−9.72928	−0.903341
$117$	0	0
$118$	−4.86464	−0.447826
$119$	2.38776	0.218885
$120$	0	0
$121$	0	0
$122$	28.6864	2.59714
$123$	0	0
$124$	−12.5939	−1.13097
$125$	−8.14931	−0.728897
$126$	0	0
$127$	−4.54144	−0.402987	−0.201494	−	0.979490i	$-0.564580\pi$
$127$	−4.54144	−0.402987	−0.201494	+	0.979490i	$0.564580\pi$
$128$	−51.5650	−4.55774
$129$	0	0
$130$	17.3169	1.51880
$131$	6.27072	0.547875	0.273938	−	0.961747i	$-0.411674\pi$
$131$	6.27072	0.547875	0.273938	+	0.961747i	$0.411674\pi$
$132$	0	0
$133$	−1.72928	−0.149948
$134$	17.7293	1.53158
$135$	0	0
$136$	−23.9109	−2.05034
$137$	−9.40171	−0.803242	−0.401621	−	0.915806i	$-0.631553\pi$
$137$	−9.40171	−0.803242	−0.401621	+	0.915806i	$0.631553\pi$
$138$	0	0
$139$	15.8217	1.34198	0.670991	−	0.741465i	$-0.265869\pi$
$139$	15.8217	1.34198	0.670991	+	0.741465i	$0.265869\pi$
$140$	14.7755	1.24876
$141$	0	0
$142$	22.3265	1.87360
$143$	0	0
$144$	0	0
$145$	−4.54144	−0.377146
$146$	−28.6864	−2.37410
$147$	0	0
$148$	−38.8034	−3.18962
$149$	−3.45856	−0.283337	−0.141668	−	0.989914i	$-0.545247\pi$
$149$	−3.45856	−0.283337	−0.141668	+	0.989914i	$0.545247\pi$
$150$	0	0
$151$	−10.2986	−0.838090	−0.419045	−	0.907965i	$-0.637635\pi$
$151$	−10.2986	−0.838090	−0.419045	+	0.907965i	$0.637635\pi$
$152$	17.3169	1.40459
$153$	0	0
$154$	0	0
$155$	−5.87859	−0.472180
$156$	0	0
$157$	11.3372	0.904804	0.452402	−	0.891814i	$-0.350567\pi$
$157$	11.3372	0.904804	0.452402	+	0.891814i	$0.350567\pi$
$158$	−42.1204	−3.35092
$159$	0	0
$160$	−66.3544	−5.24578
$161$	0.626198	0.0493513
$162$	0	0
$163$	−9.72928	−0.762056	−0.381028	−	0.924563i	$-0.624430\pi$
$163$	−9.72928	−0.762056	−0.381028	+	0.924563i	$0.624430\pi$
$164$	−58.4436	−4.56368
$165$	0	0
$166$	35.2803	2.73828
$167$	−22.5048	−1.74147	−0.870737	−	0.491750i	$-0.836357\pi$
$167$	−22.5048	−1.74147	−0.870737	+	0.491750i	$0.836357\pi$
$168$	0	0
$169$	−7.29862	−0.561433
$170$	−17.3169	−1.32815
$171$	0	0
$172$	40.8034	3.11123
$173$	−6.92919	−0.526817	−0.263408	−	0.964684i	$-0.584847\pi$
$173$	−6.92919	−0.526817	−0.263408	+	0.964684i	$0.584847\pi$
$174$	0	0
$175$	1.89692	0.143393
$176$	0	0
$177$	0	0
$178$	−39.0741	−2.92873
$179$	−13.9431	−1.04216	−0.521080	−	0.853508i	$-0.674471\pi$
$179$	−13.9431	−1.04216	−0.521080	+	0.853508i	$0.674471\pi$
$180$	0	0
$181$	3.16763	0.235448	0.117724	−	0.993046i	$-0.462440\pi$
$181$	3.16763	0.235448	0.117724	+	0.993046i	$0.462440\pi$
$182$	6.59392	0.488774
$183$	0	0
$184$	−6.27072	−0.462283
$185$	−18.1127	−1.33167
$186$	0	0
$187$	0	0
$188$	−35.9388	−2.62110
$189$	0	0
$190$	12.5414	0.909851
$191$	−18.3555	−1.32816	−0.664078	−	0.747663i	$-0.731176\pi$
$191$	−18.3555	−1.32816	−0.664078	+	0.747663i	$0.731176\pi$
$192$	0	0
$193$	−3.04623	−0.219272	−0.109636	−	0.993972i	$-0.534969\pi$
$193$	−3.04623	−0.219272	−0.109636	+	0.993972i	$0.534969\pi$
$194$	23.0741	1.65663
$195$	0	0
$196$	5.62620	0.401871
$197$	−4.95377	−0.352942	−0.176471	−	0.984306i	$-0.556468\pi$
$197$	−4.95377	−0.352942	−0.176471	+	0.984306i	$0.556468\pi$
$198$	0	0
$199$	15.7047	1.11328	0.556638	−	0.830755i	$-0.312091\pi$
$199$	15.7047	1.11328	0.556638	+	0.830755i	$0.312091\pi$
$200$	−18.9956	−1.34319
$201$	0	0
$202$	−38.2374	−2.69037
$203$	−1.72928	−0.121372
$204$	0	0
$205$	−27.2803	−1.90534
$206$	8.08913	0.563596
$207$	0	0
$208$	−39.1633	−2.71548
$209$	0	0
$210$	0	0
$211$	17.5510	1.20826	0.604131	−	0.796885i	$-0.293520\pi$
$211$	17.5510	1.20826	0.604131	+	0.796885i	$0.293520\pi$
$212$	52.0558	3.57521
$213$	0	0
$214$	9.55102	0.652894
$215$	19.0462	1.29894
$216$	0	0
$217$	−2.23844	−0.151955
$218$	44.8313	3.03636
$219$	0	0
$220$	0	0
$221$	−5.70138	−0.383516
$222$	0	0
$223$	−22.2943	−1.49293	−0.746467	−	0.665423i	$-0.768251\pi$
$223$	−22.2943	−1.49293	−0.746467	+	0.665423i	$0.768251\pi$
$224$	−25.2663	−1.68818
$225$	0	0
$226$	19.6156	1.30481
$227$	−14.2707	−0.947181	−0.473590	−	0.880745i	$-0.657042\pi$
$227$	−14.2707	−0.947181	−0.473590	+	0.880745i	$0.657042\pi$
$228$	0	0
$229$	−3.87859	−0.256305	−0.128152	−	0.991754i	$-0.540905\pi$
$229$	−3.87859	−0.256305	−0.128152	+	0.991754i	$0.540905\pi$
$230$	−4.54144	−0.299453
$231$	0	0
$232$	17.3169	1.13691
$233$	−6.68305	−0.437821	−0.218911	−	0.975745i	$-0.570250\pi$
$233$	−6.68305	−0.437821	−0.218911	+	0.975745i	$0.570250\pi$
$234$	0	0
$235$	−16.7755	−1.09431
$236$	9.91087	0.645143
$237$	0	0
$238$	−6.59392	−0.427421
$239$	4.54144	0.293761	0.146881	−	0.989154i	$-0.453077\pi$
$239$	4.54144	0.293761	0.146881	+	0.989154i	$0.453077\pi$
$240$	0	0
$241$	−3.70470	−0.238641	−0.119320	−	0.992856i	$-0.538072\pi$
$241$	−3.70470	−0.238641	−0.119320	+	0.992856i	$0.538072\pi$
$242$	0	0
$243$	0	0
$244$	−58.4436	−3.74147
$245$	2.62620	0.167782
$246$	0	0
$247$	4.12910	0.262728
$248$	22.4157	1.42340
$249$	0	0
$250$	22.5048	1.42333
$251$	−8.50916	−0.537093	−0.268547	−	0.963267i	$-0.586543\pi$
$251$	−8.50916	−0.537093	−0.268547	+	0.963267i	$0.586543\pi$
$252$	0	0
$253$	0	0
$254$	12.5414	0.786920
$255$	0	0
$256$	68.4575	4.27860
$257$	2.95377	0.184251	0.0921256	−	0.995747i	$-0.470634\pi$
$257$	2.95377	0.184251	0.0921256	+	0.995747i	$0.470634\pi$
$258$	0	0
$259$	−6.89692	−0.428554
$260$	−35.2803	−2.18799
$261$	0	0
$262$	−17.3169	−1.06984
$263$	16.0000	0.986602	0.493301	−	0.869859i	$-0.335790\pi$
$263$	16.0000	0.986602	0.493301	+	0.869859i	$0.335790\pi$
$264$	0	0
$265$	24.2986	1.49265
$266$	4.77551	0.292805
$267$	0	0
$268$	−36.1204	−2.20640
$269$	6.77551	0.413110	0.206555	−	0.978435i	$-0.433775\pi$
$269$	6.77551	0.413110	0.206555	+	0.978435i	$0.433775\pi$
$270$	0	0
$271$	3.45856	0.210093	0.105046	−	0.994467i	$-0.466501\pi$
$271$	3.45856	0.210093	0.105046	+	0.994467i	$0.466501\pi$
$272$	39.1633	2.37462
$273$	0	0
$274$	25.9634	1.56850
$275$	0	0
$276$	0	0
$277$	−11.4586	−0.688478	−0.344239	−	0.938882i	$-0.611863\pi$
$277$	−11.4586	−0.688478	−0.344239	+	0.938882i	$0.611863\pi$
$278$	−43.6926	−2.62051
$279$	0	0
$280$	−26.2986	−1.57164
$281$	22.3265	1.33189	0.665945	−	0.746001i	$-0.268029\pi$
$281$	22.3265	1.33189	0.665945	+	0.746001i	$0.268029\pi$
$282$	0	0
$283$	−12.3632	−0.734915	−0.367457	−	0.930040i	$-0.619772\pi$
$283$	−12.3632	−0.734915	−0.367457	+	0.930040i	$0.619772\pi$
$284$	−45.4865	−2.69913
$285$	0	0
$286$	0	0
$287$	−10.3878	−0.613170
$288$	0	0
$289$	−11.2986	−0.664625
$290$	12.5414	0.736459
$291$	0	0
$292$	58.4436	3.42015
$293$	−2.15368	−0.125819	−0.0629097	−	0.998019i	$-0.520038\pi$
$293$	−2.15368	−0.125819	−0.0629097	+	0.998019i	$0.520038\pi$
$294$	0	0
$295$	4.62620	0.269348
$296$	69.0654	4.01434
$297$	0	0
$298$	9.55102	0.553276
$299$	−1.49521	−0.0864701
$300$	0	0
$301$	7.25240	0.418021
$302$	28.4402	1.63655
$303$	0	0
$304$	−28.3632	−1.62674
$305$	−27.2803	−1.56207
$306$	0	0
$307$	−31.8217	−1.81616	−0.908081	−	0.418794i	$-0.862453\pi$
$307$	−31.8217	−1.81616	−0.908081	+	0.418794i	$0.862453\pi$
$308$	0	0
$309$	0	0
$310$	16.2341	0.922033
$311$	10.9292	0.619738	0.309869	−	0.950779i	$-0.399715\pi$
$311$	10.9292	0.619738	0.309869	+	0.950779i	$0.399715\pi$
$312$	0	0
$313$	7.40171	0.418369	0.209185	−	0.977876i	$-0.432919\pi$
$313$	7.40171	0.418369	0.209185	+	0.977876i	$0.432919\pi$
$314$	−31.3082	−1.76682
$315$	0	0
$316$	85.8130	4.82736
$317$	8.89692	0.499701	0.249850	−	0.968284i	$-0.419619\pi$
$317$	8.89692	0.499701	0.249850	+	0.968284i	$0.419619\pi$
$318$	0	0
$319$	0	0
$320$	97.0933	5.42768
$321$	0	0
$322$	−1.72928	−0.0963691
$323$	−4.12910	−0.229749
$324$	0	0
$325$	−4.52937	−0.251244
$326$	26.8680	1.48808
$327$	0	0
$328$	104.022	5.74368
$329$	−6.38776	−0.352168
$330$	0	0
$331$	8.56165	0.470591	0.235295	−	0.971924i	$-0.424394\pi$
$331$	8.56165	0.470591	0.235295	+	0.971924i	$0.424394\pi$
$332$	−71.8776	−3.94479
$333$	0	0
$334$	62.1483	3.40060
$335$	−16.8603	−0.921175
$336$	0	0
$337$	−9.72928	−0.529988	−0.264994	−	0.964250i	$-0.585370\pi$
$337$	−9.72928	−0.529988	−0.264994	+	0.964250i	$0.585370\pi$
$338$	20.1556	1.09632
$339$	0	0
$340$	35.2803	1.91334
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	−72.6252	−3.91569
$345$	0	0
$346$	19.1354	1.02872
$347$	−14.8401	−0.796656	−0.398328	−	0.917243i	$-0.630409\pi$
$347$	−14.8401	−0.796656	−0.398328	+	0.917243i	$0.630409\pi$
$348$	0	0
$349$	16.2461	0.869636	0.434818	−	0.900518i	$-0.356813\pi$
$349$	16.2461	0.869636	0.434818	+	0.900518i	$0.356813\pi$
$350$	−5.23844	−0.280007
$351$	0	0
$352$	0	0
$353$	−30.8603	−1.64253	−0.821263	−	0.570549i	$-0.806730\pi$
$353$	−30.8603	−1.64253	−0.821263	+	0.570549i	$0.806730\pi$
$354$	0	0
$355$	−21.2322	−1.12689
$356$	79.6068	4.21915
$357$	0	0
$358$	38.5048	2.03504
$359$	11.7938	0.622455	0.311227	−	0.950335i	$-0.399260\pi$
$359$	11.7938	0.622455	0.311227	+	0.950335i	$0.399260\pi$
$360$	0	0
$361$	−16.0096	−0.842610
$362$	−8.74760	−0.459764
$363$	0	0
$364$	−13.4340	−0.704132
$365$	27.2803	1.42792
$366$	0	0
$367$	−3.55539	−0.185590	−0.0927949	−	0.995685i	$-0.529580\pi$
$367$	−3.55539	−0.185590	−0.0927949	+	0.995685i	$0.529580\pi$
$368$	10.2707	0.535398
$369$	0	0
$370$	50.0192	2.60037
$371$	9.25240	0.480360
$372$	0	0
$373$	22.5048	1.16525	0.582627	−	0.812740i	$-0.302025\pi$
$373$	22.5048	1.16525	0.582627	+	0.812740i	$0.302025\pi$
$374$	0	0
$375$	0	0
$376$	63.9667	3.29883
$377$	4.12910	0.212660
$378$	0	0
$379$	23.4296	1.20350	0.601749	−	0.798685i	$-0.294471\pi$
$379$	23.4296	1.20350	0.601749	+	0.798685i	$0.294471\pi$
$380$	−25.5510	−1.31074
$381$	0	0
$382$	50.6897	2.59351
$383$	3.42629	0.175075	0.0875376	−	0.996161i	$-0.472100\pi$
$383$	3.42629	0.175075	0.0875376	+	0.996161i	$0.472100\pi$
$384$	0	0
$385$	0	0
$386$	8.41233	0.428177
$387$	0	0
$388$	−47.0096	−2.38655
$389$	−8.05685	−0.408499	−0.204249	−	0.978919i	$-0.565475\pi$
$389$	−8.05685	−0.408499	−0.204249	+	0.978919i	$0.565475\pi$
$390$	0	0
$391$	1.49521	0.0756159
$392$	−10.0140	−0.505781
$393$	0	0
$394$	13.6801	0.689195
$395$	40.0558	2.01543
$396$	0	0
$397$	−26.3632	−1.32313	−0.661565	−	0.749888i	$-0.730107\pi$
$397$	−26.3632	−1.32313	−0.661565	+	0.749888i	$0.730107\pi$
$398$	−43.3694	−2.17391
$399$	0	0
$400$	31.1127	1.55563
$401$	22.5972	1.12845	0.564226	−	0.825620i	$-0.309175\pi$
$401$	22.5972	1.12845	0.564226	+	0.825620i	$0.309175\pi$
$402$	0	0
$403$	5.34485	0.266246
$404$	77.9021	3.87578
$405$	0	0
$406$	4.77551	0.237005
$407$	0	0
$408$	0	0
$409$	9.30488	0.460097	0.230048	−	0.973179i	$-0.426112\pi$
$409$	9.30488	0.460097	0.230048	+	0.973179i	$0.426112\pi$
$410$	75.3361	3.72059
$411$	0	0
$412$	−16.4802	−0.811922
$413$	1.76156	0.0866806
$414$	0	0
$415$	−33.5510	−1.64695
$416$	60.3299	2.95791
$417$	0	0
$418$	0	0
$419$	13.8463	0.676437	0.338218	−	0.941068i	$-0.390176\pi$
$419$	13.8463	0.676437	0.338218	+	0.941068i	$0.390176\pi$
$420$	0	0
$421$	−35.7572	−1.74270	−0.871349	−	0.490663i	$-0.836755\pi$
$421$	−35.7572	−1.74270	−0.871349	+	0.490663i	$0.836755\pi$
$422$	−48.4681	−2.35939
$423$	0	0
$424$	−92.6531	−4.49963
$425$	4.52937	0.219707
$426$	0	0
$427$	−10.3878	−0.502699
$428$	−19.4586	−0.940565
$429$	0	0
$430$	−52.5972	−2.53646
$431$	31.3082	1.50806	0.754032	−	0.656838i	$-0.228106\pi$
$431$	31.3082	1.50806	0.754032	+	0.656838i	$0.228106\pi$
$432$	0	0
$433$	21.6079	1.03841	0.519204	−	0.854650i	$-0.326228\pi$
$433$	21.6079	1.03841	0.519204	+	0.854650i	$0.326228\pi$
$434$	6.18159	0.296726
$435$	0	0
$436$	−91.3361	−4.37421
$437$	−1.08287	−0.0518008
$438$	0	0
$439$	15.5877	0.743959	0.371979	−	0.928241i	$-0.378679\pi$
$439$	15.5877	0.743959	0.371979	+	0.928241i	$0.378679\pi$
$440$	0	0
$441$	0	0
$442$	15.7447	0.748898
$443$	−23.4942	−1.11624	−0.558121	−	0.829760i	$-0.688477\pi$
$443$	−23.4942	−1.11624	−0.558121	+	0.829760i	$0.688477\pi$
$444$	0	0
$445$	37.1589	1.76150
$446$	61.5669	2.91528
$447$	0	0
$448$	36.9711	1.74672
$449$	−22.7832	−1.07521	−0.537603	−	0.843198i	$-0.680670\pi$
$449$	−22.7832	−1.07521	−0.537603	+	0.843198i	$0.680670\pi$
$450$	0	0
$451$	0	0
$452$	−39.9634	−1.87972
$453$	0	0
$454$	39.4094	1.84957
$455$	−6.27072	−0.293976
$456$	0	0
$457$	−14.5048	−0.678506	−0.339253	−	0.940695i	$-0.610174\pi$
$457$	−14.5048	−0.678506	−0.339253	+	0.940695i	$0.610174\pi$
$458$	10.7110	0.500490
$459$	0	0
$460$	9.25240	0.431395
$461$	31.1633	1.45142	0.725709	−	0.688002i	$-0.241512\pi$
$461$	31.1633	1.45142	0.725709	+	0.688002i	$0.241512\pi$
$462$	0	0
$463$	−5.87859	−0.273201	−0.136601	−	0.990626i	$-0.543618\pi$
$463$	−5.87859	−0.273201	−0.136601	+	0.990626i	$0.543618\pi$
$464$	−28.3632	−1.31673
$465$	0	0
$466$	18.4556	0.854941
$467$	23.7249	1.09786	0.548929	−	0.835869i	$-0.315036\pi$
$467$	23.7249	1.09786	0.548929	+	0.835869i	$0.315036\pi$
$468$	0	0
$469$	−6.42003	−0.296449
$470$	46.3265	2.13688
$471$	0	0
$472$	−17.6402	−0.811954
$473$	0	0
$474$	0	0
$475$	−3.28030	−0.150511
$476$	13.4340	0.615746
$477$	0	0
$478$	−12.5414	−0.573632
$479$	−8.41233	−0.384369	−0.192185	−	0.981359i	$-0.561557\pi$
$479$	−8.41233	−0.384369	−0.192185	+	0.981359i	$0.561557\pi$
$480$	0	0
$481$	16.4681	0.750883
$482$	10.2307	0.465998
$483$	0	0
$484$	0	0
$485$	−21.9431	−0.996387
$486$	0	0
$487$	24.7957	1.12360	0.561801	−	0.827273i	$-0.310109\pi$
$487$	24.7957	1.12360	0.561801	+	0.827273i	$0.310109\pi$
$488$	104.022	4.70888
$489$	0	0
$490$	−7.25240	−0.327630
$491$	−12.2062	−0.550857	−0.275428	−	0.961322i	$-0.588820\pi$
$491$	−12.2062	−0.550857	−0.275428	+	0.961322i	$0.588820\pi$
$492$	0	0
$493$	−4.12910	−0.185965
$494$	−11.4028	−0.513034
$495$	0	0
$496$	−36.7143	−1.64852
$497$	−8.08476	−0.362651
$498$	0	0
$499$	22.3265	0.999473	0.499736	−	0.866178i	$-0.333430\pi$
$499$	22.3265	0.999473	0.499736	+	0.866178i	$0.333430\pi$
$500$	−45.8496	−2.05046
$501$	0	0
$502$	23.4985	1.04879
$503$	4.54144	0.202493	0.101246	−	0.994861i	$-0.467717\pi$
$503$	4.54144	0.202493	0.101246	+	0.994861i	$0.467717\pi$
$504$	0	0
$505$	36.3632	1.61814
$506$	0	0
$507$	0	0
$508$	−25.5510	−1.13364
$509$	−4.12141	−0.182678	−0.0913391	−	0.995820i	$-0.529115\pi$
$509$	−4.12141	−0.182678	−0.0913391	+	0.995820i	$0.529115\pi$
$510$	0	0
$511$	10.3878	0.459527
$512$	−85.9194	−3.79714
$513$	0	0
$514$	−8.15701	−0.359790
$515$	−7.69264	−0.338978
$516$	0	0
$517$	0	0
$518$	19.0462	0.836843
$519$	0	0
$520$	62.7947	2.75373
$521$	−39.2880	−1.72124	−0.860619	−	0.509249i	$-0.829923\pi$
$521$	−39.2880	−1.72124	−0.860619	+	0.509249i	$0.829923\pi$
$522$	0	0
$523$	−2.14162	−0.0936464	−0.0468232	−	0.998903i	$-0.514910\pi$
$523$	−2.14162	−0.0936464	−0.0468232	+	0.998903i	$0.514910\pi$
$524$	35.2803	1.54123
$525$	0	0
$526$	−44.1849	−1.92655
$527$	−5.34485	−0.232825
$528$	0	0
$529$	−22.6079	−0.982951
$530$	−67.1020	−2.91473
$531$	0	0
$532$	−9.72928	−0.421818
$533$	24.8034	1.07436
$534$	0	0
$535$	−9.08287	−0.392687
$536$	64.2899	2.77690
$537$	0	0
$538$	−18.7110	−0.806687
$539$	0	0
$540$	0	0
$541$	−43.0462	−1.85070	−0.925351	−	0.379112i	$-0.876230\pi$
$541$	−43.0462	−1.85070	−0.925351	+	0.379112i	$0.876230\pi$
$542$	−9.55102	−0.410251
$543$	0	0
$544$	−60.3299	−2.58662
$545$	−42.6339	−1.82624
$546$	0	0
$547$	−29.0096	−1.24036	−0.620180	−	0.784459i	$-0.712940\pi$
$547$	−29.0096	−1.24036	−0.620180	+	0.784459i	$0.712940\pi$
$548$	−52.8959	−2.25960
$549$	0	0
$550$	0	0
$551$	2.99042	0.127396
$552$	0	0
$553$	15.2524	0.648598
$554$	31.6435	1.34440
$555$	0	0
$556$	89.0162	3.77513
$557$	1.49521	0.0633540	0.0316770	−	0.999498i	$-0.489915\pi$
$557$	1.49521	0.0633540	0.0316770	+	0.999498i	$0.489915\pi$
$558$	0	0
$559$	−17.3169	−0.732429
$560$	43.0741	1.82021
$561$	0	0
$562$	−61.6560	−2.60080
$563$	−6.27072	−0.264279	−0.132140	−	0.991231i	$-0.542185\pi$
$563$	−6.27072	−0.264279	−0.132140	+	0.991231i	$0.542185\pi$
$564$	0	0
$565$	−18.6541	−0.784784
$566$	34.1416	1.43508
$567$	0	0
$568$	80.9604	3.39702
$569$	−7.82174	−0.327904	−0.163952	−	0.986468i	$-0.552424\pi$
$569$	−7.82174	−0.327904	−0.163952	+	0.986468i	$0.552424\pi$
$570$	0	0
$571$	−15.1753	−0.635068	−0.317534	−	0.948247i	$-0.602855\pi$
$571$	−15.1753	−0.635068	−0.317534	+	0.948247i	$0.602855\pi$
$572$	0	0
$573$	0	0
$574$	28.6864	1.19735
$575$	1.18785	0.0495366
$576$	0	0
$577$	23.6445	0.984334	0.492167	−	0.870501i	$-0.336205\pi$
$577$	23.6445	0.984334	0.492167	+	0.870501i	$0.336205\pi$
$578$	31.2018	1.29782
$579$	0	0
$580$	−25.5510	−1.06095
$581$	−12.7755	−0.530017
$582$	0	0
$583$	0	0
$584$	−104.022	−4.30448
$585$	0	0
$586$	5.94751	0.245690
$587$	10.1537	0.419087	0.209544	−	0.977799i	$-0.432802\pi$
$587$	10.1537	0.419087	0.209544	+	0.977799i	$0.432802\pi$
$588$	0	0
$589$	3.87090	0.159498
$590$	−12.7755	−0.525959
$591$	0	0
$592$	−113.121	−4.64925
$593$	−33.7972	−1.38788	−0.693942	−	0.720031i	$-0.744127\pi$
$593$	−33.7972	−1.38788	−0.693942	+	0.720031i	$0.744127\pi$
$594$	0	0
$595$	6.27072	0.257074
$596$	−19.4586	−0.797054
$597$	0	0
$598$	4.12910	0.168852
$599$	18.3265	0.748802	0.374401	−	0.927267i	$-0.377848\pi$
$599$	18.3265	0.748802	0.374401	+	0.927267i	$0.377848\pi$
$600$	0	0
$601$	48.4802	1.97755	0.988775	−	0.149415i	$-0.0477391\pi$
$601$	48.4802	1.97755	0.988775	+	0.149415i	$0.0477391\pi$
$602$	−20.0279	−0.816277
$603$	0	0
$604$	−57.9421	−2.35763
$605$	0	0
$606$	0	0
$607$	−20.5972	−0.836017	−0.418008	−	0.908443i	$-0.637272\pi$
$607$	−20.5972	−0.836017	−0.418008	+	0.908443i	$0.637272\pi$
$608$	43.6926	1.77197
$609$	0	0
$610$	75.3361	3.05027
$611$	15.2524	0.617046
$612$	0	0
$613$	18.8122	0.759816	0.379908	−	0.925024i	$-0.375956\pi$
$613$	18.8122	0.759816	0.379908	+	0.925024i	$0.375956\pi$
$614$	87.8776	3.54645
$615$	0	0
$616$	0	0
$617$	14.2062	0.571919	0.285959	−	0.958242i	$-0.407688\pi$
$617$	14.2062	0.571919	0.285959	+	0.958242i	$0.407688\pi$
$618$	0	0
$619$	−16.0235	−0.644040	−0.322020	−	0.946733i	$-0.604362\pi$
$619$	−16.0235	−0.644040	−0.322020	+	0.946733i	$0.604362\pi$
$620$	−33.0741	−1.32829
$621$	0	0
$622$	−30.1816	−1.21017
$623$	14.1493	0.566880
$624$	0	0
$625$	−30.8863	−1.23545
$626$	−20.4402	−0.816956
$627$	0	0
$628$	63.7851	2.54530
$629$	−16.4681	−0.656628
$630$	0	0
$631$	15.1955	0.604925	0.302462	−	0.953161i	$-0.402191\pi$
$631$	15.1955	0.604925	0.302462	+	0.953161i	$0.402191\pi$
$632$	−152.737	−6.07554
$633$	0	0
$634$	−24.5693	−0.975773
$635$	−11.9267	−0.473297
$636$	0	0
$637$	−2.38776	−0.0946063
$638$	0	0
$639$	0	0
$640$	−135.420	−5.35294
$641$	10.2264	0.403918	0.201959	−	0.979394i	$-0.435269\pi$
$641$	10.2264	0.403918	0.201959	+	0.979394i	$0.435269\pi$
$642$	0	0
$643$	20.5650	0.811003	0.405502	−	0.914094i	$-0.367097\pi$
$643$	20.5650	0.811003	0.405502	+	0.914094i	$0.367097\pi$
$644$	3.52311	0.138830
$645$	0	0
$646$	11.4028	0.448635
$647$	25.4542	1.00071	0.500354	−	0.865821i	$-0.333203\pi$
$647$	25.4542	1.00071	0.500354	+	0.865821i	$0.333203\pi$
$648$	0	0
$649$	0	0
$650$	12.5081	0.490609
$651$	0	0
$652$	−54.7389	−2.14374
$653$	45.2514	1.77082	0.885411	−	0.464809i	$-0.153877\pi$
$653$	45.2514	1.77082	0.885411	+	0.464809i	$0.153877\pi$
$654$	0	0
$655$	16.4681	0.643464
$656$	−170.377	−6.65210
$657$	0	0
$658$	17.6402	0.687685
$659$	50.3544	1.96153	0.980765	−	0.195191i	$-0.0625327\pi$
$659$	50.3544	1.96153	0.980765	+	0.195191i	$0.0625327\pi$
$660$	0	0
$661$	−23.2234	−0.903287	−0.451644	−	0.892198i	$-0.649162\pi$
$661$	−23.2234	−0.903287	−0.451644	+	0.892198i	$0.649162\pi$
$662$	−23.6435	−0.918930
$663$	0	0
$664$	127.933	4.96478
$665$	−4.54144	−0.176109
$666$	0	0
$667$	−1.08287	−0.0419290
$668$	−126.616	−4.89894
$669$	0	0
$670$	46.5606	1.79879
$671$	0	0
$672$	0	0
$673$	−3.22449	−0.124295	−0.0621475	−	0.998067i	$-0.519795\pi$
$673$	−3.22449	−0.124295	−0.0621475	+	0.998067i	$0.519795\pi$
$674$	26.8680	1.03492
$675$	0	0
$676$	−41.0635	−1.57937
$677$	−42.6218	−1.63809	−0.819045	−	0.573729i	$-0.805496\pi$
$677$	−42.6218	−1.63809	−0.819045	+	0.573729i	$0.805496\pi$
$678$	0	0
$679$	−8.35548	−0.320654
$680$	−62.7947	−2.40807
$681$	0	0
$682$	0	0
$683$	−15.5877	−0.596445	−0.298223	−	0.954496i	$-0.596394\pi$
$683$	−15.5877	−0.596445	−0.298223	+	0.954496i	$0.596394\pi$
$684$	0	0
$685$	−24.6907	−0.943385
$686$	−2.76156	−0.105437
$687$	0	0
$688$	118.952	4.53499
$689$	−22.0925	−0.841656
$690$	0	0
$691$	41.9311	1.59513	0.797567	−	0.603231i	$-0.206120\pi$
$691$	41.9311	1.59513	0.797567	+	0.603231i	$0.206120\pi$
$692$	−38.9850	−1.48199
$693$	0	0
$694$	40.9817	1.55564
$695$	41.5510	1.57612
$696$	0	0
$697$	−24.8034	−0.939496
$698$	−44.8646	−1.69815
$699$	0	0
$700$	10.6724	0.403380
$701$	15.4094	0.582005	0.291003	−	0.956722i	$-0.406011\pi$
$701$	15.4094	0.582005	0.291003	+	0.956722i	$0.406011\pi$
$702$	0	0
$703$	11.9267	0.449824
$704$	0	0
$705$	0	0
$706$	85.2224	3.20739
$707$	13.8463	0.520744
$708$	0	0
$709$	2.86027	0.107420	0.0537099	−	0.998557i	$-0.482895\pi$
$709$	2.86027	0.107420	0.0537099	+	0.998557i	$0.482895\pi$
$710$	58.6339	2.20049
$711$	0	0
$712$	−141.691	−5.31008
$713$	−1.40171	−0.0524944
$714$	0	0
$715$	0	0
$716$	−78.4469	−2.93170
$717$	0	0
$718$	−32.5693	−1.21548
$719$	34.4725	1.28561	0.642804	−	0.766031i	$-0.277771\pi$
$719$	34.4725	1.28561	0.642804	+	0.766031i	$0.277771\pi$
$720$	0	0
$721$	−2.92919	−0.109089
$722$	44.2114	1.64538
$723$	0	0
$724$	17.8217	0.662340
$725$	−3.28030	−0.121827
$726$	0	0
$727$	−40.3309	−1.49579	−0.747895	−	0.663817i	$-0.768935\pi$
$727$	−40.3309	−1.49579	−0.747895	+	0.663817i	$0.768935\pi$
$728$	23.9109	0.886196
$729$	0	0
$730$	−75.3361	−2.78831
$731$	17.3169	0.640490
$732$	0	0
$733$	−28.7634	−1.06240	−0.531201	−	0.847246i	$-0.678259\pi$
$733$	−28.7634	−1.06240	−0.531201	+	0.847246i	$0.678259\pi$
$734$	9.81841	0.362404
$735$	0	0
$736$	−15.8217	−0.583197
$737$	0	0
$738$	0	0
$739$	35.1020	1.29125	0.645625	−	0.763655i	$-0.276597\pi$
$739$	35.1020	1.29125	0.645625	+	0.763655i	$0.276597\pi$
$740$	−101.905	−3.74612
$741$	0	0
$742$	−25.5510	−0.938007
$743$	32.4681	1.19114	0.595570	−	0.803303i	$-0.296926\pi$
$743$	32.4681	1.19114	0.595570	+	0.803303i	$0.296926\pi$
$744$	0	0
$745$	−9.08287	−0.332771
$746$	−62.1483	−2.27541
$747$	0	0
$748$	0	0
$749$	−3.45856	−0.126373
$750$	0	0
$751$	−28.3834	−1.03572	−0.517862	−	0.855464i	$-0.673272\pi$
$751$	−28.3834	−1.03572	−0.517862	+	0.855464i	$0.673272\pi$
$752$	−104.770	−3.82057
$753$	0	0
$754$	−11.4028	−0.415264
$755$	−27.0462	−0.984313
$756$	0	0
$757$	−17.0462	−0.619556	−0.309778	−	0.950809i	$-0.600255\pi$
$757$	−17.0462	−0.619556	−0.309778	+	0.950809i	$0.600255\pi$
$758$	−64.7022	−2.35009
$759$	0	0
$760$	45.4777	1.64965
$761$	−27.9388	−1.01278	−0.506390	−	0.862305i	$-0.669020\pi$
$761$	−27.9388	−1.01278	−0.506390	+	0.862305i	$0.669020\pi$
$762$	0	0
$763$	−16.2341	−0.587713
$764$	−103.272	−3.73623
$765$	0	0
$766$	−9.46189	−0.341872
$767$	−4.20617	−0.151876
$768$	0	0
$769$	6.69512	0.241432	0.120716	−	0.992687i	$-0.461481\pi$
$769$	6.69512	0.241432	0.120716	+	0.992687i	$0.461481\pi$
$770$	0	0
$771$	0	0
$772$	−17.1387	−0.616835
$773$	−27.9142	−1.00400	−0.502002	−	0.864866i	$-0.667403\pi$
$773$	−27.9142	−1.00400	−0.502002	+	0.864866i	$0.667403\pi$
$774$	0	0
$775$	−4.24614	−0.152526
$776$	83.6714	3.00363
$777$	0	0
$778$	22.2495	0.797682
$779$	17.9634	0.643604
$780$	0	0
$781$	0	0
$782$	−4.12910	−0.147656
$783$	0	0
$784$	16.4017	0.585775
$785$	29.7736	1.06267
$786$	0	0
$787$	21.8584	0.779167	0.389584	−	0.920991i	$-0.372619\pi$
$787$	21.8584	0.779167	0.389584	+	0.920991i	$0.372619\pi$
$788$	−27.8709	−0.992860
$789$	0	0
$790$	−110.616	−3.93556
$791$	−7.10308	−0.252557
$792$	0	0
$793$	24.8034	0.880795
$794$	72.8034	2.58370
$795$	0	0
$796$	88.3578	3.13176
$797$	34.6262	1.22652	0.613261	−	0.789880i	$-0.289857\pi$
$797$	34.6262	1.22652	0.613261	+	0.789880i	$0.289857\pi$
$798$	0	0
$799$	−15.2524	−0.539591
$800$	−47.9282	−1.69452
$801$	0	0
$802$	−62.4036	−2.20355
$803$	0	0
$804$	0	0
$805$	1.64452	0.0579617
$806$	−14.7601	−0.519903
$807$	0	0
$808$	−138.656	−4.87791
$809$	−35.0462	−1.23216	−0.616080	−	0.787684i	$-0.711280\pi$
$809$	−35.0462	−1.23216	−0.616080	+	0.787684i	$0.711280\pi$
$810$	0	0
$811$	−18.1416	−0.637038	−0.318519	−	0.947916i	$-0.603186\pi$
$811$	−18.1416	−0.637038	−0.318519	+	0.947916i	$0.603186\pi$
$812$	−9.72928	−0.341431
$813$	0	0
$814$	0	0
$815$	−25.5510	−0.895013
$816$	0	0
$817$	−12.5414	−0.438769
$818$	−25.6960	−0.898438
$819$	0	0
$820$	−153.484	−5.35991
$821$	3.45856	0.120705	0.0603524	−	0.998177i	$-0.480778\pi$
$821$	3.45856	0.120705	0.0603524	+	0.998177i	$0.480778\pi$
$822$	0	0
$823$	−33.5308	−1.16881	−0.584405	−	0.811462i	$-0.698672\pi$
$823$	−33.5308	−1.16881	−0.584405	+	0.811462i	$0.698672\pi$
$824$	29.3328	1.02186
$825$	0	0
$826$	−4.86464	−0.169263
$827$	26.2986	0.914493	0.457246	−	0.889340i	$-0.348836\pi$
$827$	26.2986	0.914493	0.457246	+	0.889340i	$0.348836\pi$
$828$	0	0
$829$	−29.0019	−1.00728	−0.503639	−	0.863914i	$-0.668006\pi$
$829$	−29.0019	−1.00728	−0.503639	+	0.863914i	$0.668006\pi$
$830$	92.6531	3.21603
$831$	0	0
$832$	−88.2778	−3.06048
$833$	2.38776	0.0827308
$834$	0	0
$835$	−59.1020	−2.04531
$836$	0	0
$837$	0	0
$838$	−38.2374	−1.32089
$839$	−29.1710	−1.00709	−0.503547	−	0.863968i	$-0.667972\pi$
$839$	−29.1710	−1.00709	−0.503547	+	0.863968i	$0.667972\pi$
$840$	0	0
$841$	−26.0096	−0.896882
$842$	98.7455	3.40300
$843$	0	0
$844$	98.7455	3.39896
$845$	−19.1676	−0.659387
$846$	0	0
$847$	0	0
$848$	151.755	5.21129
$849$	0	0
$850$	−12.5081	−0.429025
$851$	−4.31884	−0.148048
$852$	0	0
$853$	10.6218	0.363685	0.181842	−	0.983328i	$-0.441794\pi$
$853$	10.6218	0.363685	0.181842	+	0.983328i	$0.441794\pi$
$854$	28.6864	0.981628
$855$	0	0
$856$	34.6339	1.18376
$857$	−37.8463	−1.29281	−0.646403	−	0.762996i	$-0.723727\pi$
$857$	−37.8463	−1.29281	−0.646403	+	0.762996i	$0.723727\pi$
$858$	0	0
$859$	−19.0785	−0.650950	−0.325475	−	0.945551i	$-0.605524\pi$
$859$	−19.0785	−0.650950	−0.325475	+	0.945551i	$0.605524\pi$
$860$	107.158	3.65405
$861$	0	0
$862$	−86.4594	−2.94482
$863$	−11.1753	−0.380413	−0.190206	−	0.981744i	$-0.560916\pi$
$863$	−11.1753	−0.380413	−0.190206	+	0.981744i	$0.560916\pi$
$864$	0	0
$865$	−18.1974	−0.618731
$866$	−59.6714	−2.02772
$867$	0	0
$868$	−12.5939	−0.427466
$869$	0	0
$870$	0	0
$871$	15.3295	0.519419
$872$	162.567	5.50522
$873$	0	0
$874$	2.99042	0.101152
$875$	−8.14931	−0.275497
$876$	0	0
$877$	−5.85838	−0.197824	−0.0989118	−	0.995096i	$-0.531536\pi$
$877$	−5.85838	−0.197824	−0.0989118	+	0.995096i	$0.531536\pi$
$878$	−43.0462	−1.45274
$879$	0	0
$880$	0	0
$881$	25.9065	0.872812	0.436406	−	0.899750i	$-0.356251\pi$
$881$	25.9065	0.872812	0.436406	+	0.899750i	$0.356251\pi$
$882$	0	0
$883$	13.4219	0.451684	0.225842	−	0.974164i	$-0.427487\pi$
$883$	13.4219	0.451684	0.225842	+	0.974164i	$0.427487\pi$
$884$	−32.0771	−1.07887
$885$	0	0
$886$	64.8805	2.17970
$887$	49.6068	1.66563	0.832817	−	0.553548i	$-0.186726\pi$
$887$	49.6068	1.66563	0.832817	+	0.553548i	$0.186726\pi$
$888$	0	0
$889$	−4.54144	−0.152315
$890$	−102.616	−3.43971
$891$	0	0
$892$	−125.432	−4.19977
$893$	11.0462	0.369648
$894$	0	0
$895$	−36.6175	−1.22399
$896$	−51.5650	−1.72266
$897$	0	0
$898$	62.9171	2.09957
$899$	3.87090	0.129102
$900$	0	0
$901$	22.0925	0.736006
$902$	0	0
$903$	0	0
$904$	71.1299	2.36575
$905$	8.31884	0.276527
$906$	0	0
$907$	−47.8776	−1.58975	−0.794874	−	0.606775i	$-0.792463\pi$
$907$	−47.8776	−1.58975	−0.794874	+	0.606775i	$0.792463\pi$
$908$	−80.2899	−2.66451
$909$	0	0
$910$	17.3169	0.574051
$911$	−22.8122	−0.755800	−0.377900	−	0.925846i	$-0.623354\pi$
$911$	−22.8122	−0.755800	−0.377900	+	0.925846i	$0.623354\pi$
$912$	0	0
$913$	0	0
$914$	40.0558	1.32493
$915$	0	0
$916$	−21.8217	−0.721011
$917$	6.27072	0.207077
$918$	0	0
$919$	−40.0000	−1.31948	−0.659739	−	0.751495i	$-0.729333\pi$
$919$	−40.0000	−1.31948	−0.659739	+	0.751495i	$0.729333\pi$
$920$	−16.4681	−0.542939
$921$	0	0
$922$	−86.0591	−2.83421
$923$	19.3044	0.635413
$924$	0	0
$925$	−13.0829	−0.430162
$926$	16.2341	0.533485
$927$	0	0
$928$	43.6926	1.43428
$929$	16.0366	0.526145	0.263073	−	0.964776i	$-0.415264\pi$
$929$	16.0366	0.526145	0.263073	+	0.964776i	$0.415264\pi$
$930$	0	0
$931$	−1.72928	−0.0566749
$932$	−37.6002	−1.23163
$933$	0	0
$934$	−65.5177	−2.14380
$935$	0	0
$936$	0	0
$937$	−25.3049	−0.826674	−0.413337	−	0.910578i	$-0.635637\pi$
$937$	−25.3049	−0.826674	−0.413337	+	0.910578i	$0.635637\pi$
$938$	17.7293	0.578882
$939$	0	0
$940$	−94.3823	−3.07841
$941$	−20.5294	−0.669238	−0.334619	−	0.942353i	$-0.608608\pi$
$941$	−20.5294	−0.669238	−0.334619	+	0.942353i	$0.608608\pi$
$942$	0	0
$943$	−6.50479	−0.211825
$944$	28.8925	0.940372
$945$	0	0
$946$	0	0
$947$	16.6907	0.542376	0.271188	−	0.962526i	$-0.412583\pi$
$947$	16.6907	0.542376	0.271188	+	0.962526i	$0.412583\pi$
$948$	0	0
$949$	−24.8034	−0.805153
$950$	9.05874	0.293904
$951$	0	0
$952$	−23.9109	−0.774956
$953$	−56.4681	−1.82918	−0.914591	−	0.404379i	$-0.867488\pi$
$953$	−56.4681	−1.82918	−0.914591	+	0.404379i	$0.867488\pi$
$954$	0	0
$955$	−48.2051	−1.55988
$956$	25.5510	0.826379
$957$	0	0
$958$	23.2311	0.750564
$959$	−9.40171	−0.303597
$960$	0	0
$961$	−25.9894	−0.838367
$962$	−45.4777	−1.46626
$963$	0	0
$964$	−20.8434	−0.671320
$965$	−8.00000	−0.257529
$966$	0	0
$967$	47.3082	1.52133	0.760665	−	0.649145i	$-0.224873\pi$
$967$	47.3082	1.52133	0.760665	+	0.649145i	$0.224873\pi$
$968$	0	0
$969$	0	0
$970$	60.5972	1.94566
$971$	25.2355	0.809846	0.404923	−	0.914351i	$-0.367298\pi$
$971$	25.2355	0.809846	0.404923	+	0.914351i	$0.367298\pi$
$972$	0	0
$973$	15.8217	0.507222
$974$	−68.4748	−2.19407
$975$	0	0
$976$	−170.377	−5.45363
$977$	−7.10308	−0.227248	−0.113624	−	0.993524i	$-0.536246\pi$
$977$	−7.10308	−0.227248	−0.113624	+	0.993524i	$0.536246\pi$
$978$	0	0
$979$	0	0
$980$	14.7755	0.471986
$981$	0	0
$982$	33.7080	1.07567
$983$	14.1893	0.452568	0.226284	−	0.974061i	$-0.427342\pi$
$983$	14.1893	0.452568	0.226284	+	0.974061i	$0.427342\pi$
$984$	0	0
$985$	−13.0096	−0.414520
$986$	11.4028	0.363138
$987$	0	0
$988$	23.2311	0.739081
$989$	4.54144	0.144409
$990$	0	0
$991$	4.23407	0.134500	0.0672499	−	0.997736i	$-0.478578\pi$
$991$	4.23407	0.134500	0.0672499	+	0.997736i	$0.478578\pi$
$992$	56.5573	1.79570
$993$	0	0
$994$	22.3265	0.708155
$995$	41.2437	1.30751
$996$	0	0
$997$	−27.7047	−0.877417	−0.438708	−	0.898629i	$-0.644564\pi$
$997$	−27.7047	−0.877417	−0.438708	+	0.898629i	$0.644564\pi$
$998$	−61.6560	−1.95169
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.bz.1.1		3
3.2	odd	2		847.2.a.j.1.3	yes	3
11.10	odd	2		7623.2.a.ce.1.3		3
21.20	even	2		5929.2.a.y.1.3		3
33.2	even	10		847.2.f.u.323.3		12
33.5	odd	10		847.2.f.t.729.1		12
33.8	even	10		847.2.f.u.372.1		12
33.14	odd	10		847.2.f.t.372.3		12
33.17	even	10		847.2.f.u.729.3		12
33.20	odd	10		847.2.f.t.323.1		12
33.26	odd	10		847.2.f.t.148.3		12
33.29	even	10		847.2.f.u.148.1		12
33.32	even	2		847.2.a.i.1.1	✓	3
231.230	odd	2		5929.2.a.t.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
847.2.a.i.1.1	✓	3	33.32	even	2
847.2.a.j.1.3	yes	3	3.2	odd	2
847.2.f.t.148.3		12	33.26	odd	10
847.2.f.t.323.1		12	33.20	odd	10
847.2.f.t.372.3		12	33.14	odd	10
847.2.f.t.729.1		12	33.5	odd	10
847.2.f.u.148.1		12	33.29	even	10
847.2.f.u.323.3		12	33.2	even	10
847.2.f.u.372.1		12	33.8	even	10
847.2.f.u.729.3		12	33.17	even	10
5929.2.a.t.1.1		3	231.230	odd	2
5929.2.a.y.1.3		3	21.20	even	2
7623.2.a.bz.1.1		3	1.1	even	1	trivial
7623.2.a.ce.1.3		3	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)

\(f(q)\)	\(=\)	\(q-2.76156 q^{2} +5.62620 q^{4} +2.62620 q^{5} +1.00000 q^{7} -10.0140 q^{8} +O(q^{10})\)	\(q-2.76156 q^{2} +5.62620 q^{4} +2.62620 q^{5} +1.00000 q^{7} -10.0140 q^{8} -7.25240 q^{10} -2.38776 q^{13} -2.76156 q^{14} +16.4017 q^{16} +2.38776 q^{17} -1.72928 q^{19} +14.7755 q^{20} +0.626198 q^{23} +1.89692 q^{25} +6.59392 q^{26} +5.62620 q^{28} -1.72928 q^{29} -2.23844 q^{31} -25.2663 q^{32} -6.59392 q^{34} +2.62620 q^{35} -6.89692 q^{37} +4.77551 q^{38} -26.2986 q^{40} -10.3878 q^{41} +7.25240 q^{43} -1.72928 q^{46} -6.38776 q^{47} +1.00000 q^{49} -5.23844 q^{50} -13.4340 q^{52} +9.25240 q^{53} -10.0140 q^{56} +4.77551 q^{58} +1.76156 q^{59} -10.3878 q^{61} +6.18159 q^{62} +36.9711 q^{64} -6.27072 q^{65} -6.42003 q^{67} +13.4340 q^{68} -7.25240 q^{70} -8.08476 q^{71} +10.3878 q^{73} +19.0462 q^{74} -9.72928 q^{76} +15.2524 q^{79} +43.0741 q^{80} +28.6864 q^{82} -12.7755 q^{83} +6.27072 q^{85} -20.0279 q^{86} +14.1493 q^{89} -2.38776 q^{91} +3.52311 q^{92} +17.6402 q^{94} -4.54144 q^{95} -8.35548 q^{97} -2.76156 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} + 8 q^{4} - q^{5} + 3 q^{7} - 6 q^{8}+O(q^{10}) \) 3 * q - 2 * q^2 + 8 * q^4 - q^5 + 3 * q^7 - 6 * q^8	\( 3 q - 2 q^{2} + 8 q^{4} - q^{5} + 3 q^{7} - 6 q^{8} - 4 q^{10} + 8 q^{13} - 2 q^{14} + 10 q^{16} - 8 q^{17} + 14 q^{20} - 7 q^{23} + 2 q^{25} + 12 q^{26} + 8 q^{28} - 13 q^{31} - 34 q^{32} - 12 q^{34} - q^{35} - 17 q^{37} - 16 q^{38} - 36 q^{40} - 16 q^{41} + 4 q^{43} - 4 q^{47} + 3 q^{49} - 22 q^{50} + 10 q^{53} - 6 q^{56} - 16 q^{58} - q^{59} - 16 q^{61} - 4 q^{62} + 34 q^{64} - 24 q^{65} - 3 q^{67} - 4 q^{70} - 5 q^{71} + 16 q^{73} + 32 q^{74} - 24 q^{76} + 28 q^{79} + 56 q^{80} + 28 q^{82} - 8 q^{83} + 24 q^{85} - 12 q^{86} + 21 q^{89} + 8 q^{91} - 2 q^{92} + 20 q^{94} - 24 q^{95} - 11 q^{97} - 2 q^{98}+O(q^{100}) \) 3 * q - 2 * q^2 + 8 * q^4 - q^5 + 3 * q^7 - 6 * q^8 - 4 * q^10 + 8 * q^13 - 2 * q^14 + 10 * q^16 - 8 * q^17 + 14 * q^20 - 7 * q^23 + 2 * q^25 + 12 * q^26 + 8 * q^28 - 13 * q^31 - 34 * q^32 - 12 * q^34 - q^35 - 17 * q^37 - 16 * q^38 - 36 * q^40 - 16 * q^41 + 4 * q^43 - 4 * q^47 + 3 * q^49 - 22 * q^50 + 10 * q^53 - 6 * q^56 - 16 * q^58 - q^59 - 16 * q^61 - 4 * q^62 + 34 * q^64 - 24 * q^65 - 3 * q^67 - 4 * q^70 - 5 * q^71 + 16 * q^73 + 32 * q^74 - 24 * q^76 + 28 * q^79 + 56 * q^80 + 28 * q^82 - 8 * q^83 + 24 * q^85 - 12 * q^86 + 21 * q^89 + 8 * q^91 - 2 * q^92 + 20 * q^94 - 24 * q^95 - 11 * q^97 - 2 * q^98

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