LMFDB - Embedded newform 7623.2.a.co.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:	$0$
Dimension:	$4$
Coefficient field:	4.4.2525.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 5x + 5 $ x^4 - 2x^3 - 4x^2 + 5*x + 5
Coefficient ring:	$\Z[a_1, a_2]$
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			$-0.777484$ of defining polynomial
Character	$\chi$	$=$	7623.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.777484 q^{2} -1.39552 q^{4} +0.222516 q^{5} +1.00000 q^{7} +2.63996 q^{8} +O(q^{10})$	$q-0.777484 q^{2} -1.39552 q^{4} +0.222516 q^{5} +1.00000 q^{7} +2.63996 q^{8} -0.173002 q^{10} -6.52954 q^{13} -0.777484 q^{14} +0.738508 q^{16} -4.33461 q^{17} -2.91501 q^{19} -0.310525 q^{20} +3.89796 q^{23} -4.95049 q^{25} +5.07662 q^{26} -1.39552 q^{28} -3.77399 q^{29} -6.88958 q^{31} -5.85410 q^{32} +3.37009 q^{34} +0.222516 q^{35} -5.65351 q^{37} +2.26637 q^{38} +0.587433 q^{40} +1.33811 q^{41} +4.70820 q^{43} -3.03060 q^{46} -6.04554 q^{47} +1.00000 q^{49} +3.84893 q^{50} +9.11210 q^{52} +1.71792 q^{53} +2.63996 q^{56} +2.93422 q^{58} +9.53304 q^{59} +9.62943 q^{61} +5.35654 q^{62} +3.07446 q^{64} -1.45293 q^{65} +1.27155 q^{67} +6.04903 q^{68} -0.173002 q^{70} +9.30919 q^{71} -5.58911 q^{73} +4.39552 q^{74} +4.06794 q^{76} +4.52954 q^{79} +0.164330 q^{80} -1.04036 q^{82} +11.2838 q^{83} -0.964520 q^{85} -3.66055 q^{86} -7.92157 q^{89} -6.52954 q^{91} -5.43967 q^{92} +4.70031 q^{94} -0.648635 q^{95} -9.05391 q^{97} -0.777484 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 4 q^{4} + 6 q^{5} + 4 q^{7} + 9 q^{8}+O(q^{10}) $ 4 * q + 2 * q^2 + 4 * q^4 + 6 * q^5 + 4 * q^7 + 9 * q^8	$ 4 q + 2 q^{2} + 4 q^{4} + 6 q^{5} + 4 q^{7} + 9 q^{8} + 14 q^{10} + 2 q^{14} - 4 q^{16} - 3 q^{17} - 3 q^{19} + 17 q^{20} + 8 q^{23} + 12 q^{26} + 4 q^{28} + 3 q^{29} - 3 q^{31} - 10 q^{32} - 12 q^{34} + 6 q^{35} - 7 q^{37} + 20 q^{38} + 13 q^{40} + 4 q^{41} - 8 q^{43} + 3 q^{46} + 14 q^{47} + 4 q^{49} + 33 q^{50} + 17 q^{52} + 9 q^{53} + 9 q^{56} + 3 q^{58} + 25 q^{59} + 19 q^{61} + 10 q^{62} + 3 q^{64} + 12 q^{65} - 15 q^{67} - q^{68} + 14 q^{70} + 7 q^{71} + 11 q^{73} + 8 q^{74} + 26 q^{76} - 8 q^{79} + 4 q^{80} + 3 q^{82} - q^{83} - 15 q^{85} - 4 q^{86} + 17 q^{89} + 17 q^{92} + 20 q^{94} + 17 q^{95} - 15 q^{97} + 2 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + 4 * q^4 + 6 * q^5 + 4 * q^7 + 9 * q^8 + 14 * q^10 + 2 * q^14 - 4 * q^16 - 3 * q^17 - 3 * q^19 + 17 * q^20 + 8 * q^23 + 12 * q^26 + 4 * q^28 + 3 * q^29 - 3 * q^31 - 10 * q^32 - 12 * q^34 + 6 * q^35 - 7 * q^37 + 20 * q^38 + 13 * q^40 + 4 * q^41 - 8 * q^43 + 3 * q^46 + 14 * q^47 + 4 * q^49 + 33 * q^50 + 17 * q^52 + 9 * q^53 + 9 * q^56 + 3 * q^58 + 25 * q^59 + 19 * q^61 + 10 * q^62 + 3 * q^64 + 12 * q^65 - 15 * q^67 - q^68 + 14 * q^70 + 7 * q^71 + 11 * q^73 + 8 * q^74 + 26 * q^76 - 8 * q^79 + 4 * q^80 + 3 * q^82 - q^83 - 15 * q^85 - 4 * q^86 + 17 * q^89 + 17 * q^92 + 20 * q^94 + 17 * q^95 - 15 * 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 $

$2$	−0.777484	−0.549764	−0.274882	−	0.961478i	$-0.588639\pi$
$2$	−0.777484	−0.549764	−0.274882	+	0.961478i	$0.588639\pi$
$3$	0	0
$3$	0	0
$4$	−1.39552	−0.697759
$5$	0.222516	0.0995121	0.0497560	−	0.998761i	$-0.484156\pi$
$5$	0.222516	0.0995121	0.0497560	+	0.998761i	$0.484156\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	2.63996	0.933368
$9$	0	0
$10$	−0.173002	−0.0547082
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−6.52954	−1.81097	−0.905485	−	0.424379i	$-0.860493\pi$
$13$	−6.52954	−1.81097	−0.905485	+	0.424379i	$0.860493\pi$
$14$	−0.777484	−0.207791
$15$	0	0
$16$	0.738508	0.184627
$17$	−4.33461	−1.05130	−0.525649	−	0.850701i	$-0.676178\pi$
$17$	−4.33461	−1.05130	−0.525649	+	0.850701i	$0.676178\pi$
$18$	0	0
$19$	−2.91501	−0.668748	−0.334374	−	0.942440i	$-0.608525\pi$
$19$	−2.91501	−0.668748	−0.334374	+	0.942440i	$0.608525\pi$
$20$	−0.310525	−0.0694355
$21$	0	0
$22$	0	0
$23$	3.89796	0.812780	0.406390	−	0.913700i	$-0.366787\pi$
$23$	3.89796	0.812780	0.406390	+	0.913700i	$0.366787\pi$
$24$	0	0
$25$	−4.95049	−0.990097
$26$	5.07662	0.995607
$27$	0	0
$28$	−1.39552	−0.263728
$29$	−3.77399	−0.700812	−0.350406	−	0.936598i	$-0.613956\pi$
$29$	−3.77399	−0.700812	−0.350406	+	0.936598i	$0.613956\pi$
$30$	0	0
$31$	−6.88958	−1.23741	−0.618703	−	0.785625i	$-0.712341\pi$
$31$	−6.88958	−1.23741	−0.618703	+	0.785625i	$0.712341\pi$
$32$	−5.85410	−1.03487
$33$	0	0
$34$	3.37009	0.577966
$35$	0.222516	0.0376120
$36$	0	0
$37$	−5.65351	−0.929432	−0.464716	−	0.885460i	$-0.653844\pi$
$37$	−5.65351	−0.929432	−0.464716	+	0.885460i	$0.653844\pi$
$38$	2.26637	0.367654
$39$	0	0
$40$	0.587433	0.0928813
$41$	1.33811	0.208978	0.104489	−	0.994526i	$-0.466679\pi$
$41$	1.33811	0.208978	0.104489	+	0.994526i	$0.466679\pi$
$42$	0	0
$43$	4.70820	0.717994	0.358997	−	0.933339i	$-0.383119\pi$
$43$	4.70820	0.717994	0.358997	+	0.933339i	$0.383119\pi$
$44$	0	0
$45$	0	0
$46$	−3.03060	−0.446838
$47$	−6.04554	−0.881832	−0.440916	−	0.897548i	$-0.645346\pi$
$47$	−6.04554	−0.881832	−0.440916	+	0.897548i	$0.645346\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	3.84893	0.544320
$51$	0	0
$52$	9.11210	1.26362
$53$	1.71792	0.235974	0.117987	−	0.993015i	$-0.462356\pi$
$53$	1.71792	0.235974	0.117987	+	0.993015i	$0.462356\pi$
$54$	0	0
$55$	0	0
$56$	2.63996	0.352780
$57$	0	0
$58$	2.93422	0.385281
$59$	9.53304	1.24110	0.620548	−	0.784168i	$-0.286910\pi$
$59$	9.53304	1.24110	0.620548	+	0.784168i	$0.286910\pi$
$60$	0	0
$61$	9.62943	1.23292	0.616461	−	0.787386i	$-0.288566\pi$
$61$	9.62943	1.23292	0.616461	+	0.787386i	$0.288566\pi$
$62$	5.35654	0.680281
$63$	0	0
$64$	3.07446	0.384307
$65$	−1.45293	−0.180213
$66$	0	0
$67$	1.27155	0.155344	0.0776722	−	0.996979i	$-0.475251\pi$
$67$	1.27155	0.155344	0.0776722	+	0.996979i	$0.475251\pi$
$68$	6.04903	0.733553
$69$	0	0
$70$	−0.173002	−0.0206778
$71$	9.30919	1.10480	0.552399	−	0.833580i	$-0.313713\pi$
$71$	9.30919	1.10480	0.552399	+	0.833580i	$0.313713\pi$
$72$	0	0
$73$	−5.58911	−0.654156	−0.327078	−	0.944997i	$-0.606064\pi$
$73$	−5.58911	−0.654156	−0.327078	+	0.944997i	$0.606064\pi$
$74$	4.39552	0.510969
$75$	0	0
$76$	4.06794	0.466625
$77$	0	0
$78$	0	0
$79$	4.52954	0.509614	0.254807	−	0.966992i	$-0.417988\pi$
$79$	4.52954	0.509614	0.254807	+	0.966992i	$0.417988\pi$
$80$	0.164330	0.0183726
$81$	0	0
$82$	−1.04036	−0.114888
$83$	11.2838	1.23855	0.619277	−	0.785173i	$-0.287426\pi$
$83$	11.2838	1.23855	0.619277	+	0.785173i	$0.287426\pi$
$84$	0	0
$85$	−0.964520	−0.104617
$86$	−3.66055	−0.394728
$87$	0	0
$88$	0	0
$89$	−7.92157	−0.839684	−0.419842	−	0.907597i	$-0.637915\pi$
$89$	−7.92157	−0.839684	−0.419842	+	0.907597i	$0.637915\pi$
$90$	0	0
$91$	−6.52954	−0.684482
$92$	−5.43967	−0.567125
$93$	0	0
$94$	4.70031	0.484800
$95$	−0.648635	−0.0665485
$96$	0	0
$97$	−9.05391	−0.919285	−0.459643	−	0.888104i	$-0.652023\pi$
$97$	−9.05391	−0.919285	−0.459643	+	0.888104i	$0.652023\pi$
$98$	−0.777484	−0.0785378
$99$	0	0
$100$	6.90849	0.690849
$101$	−19.2102	−1.91148	−0.955741	−	0.294208i	$-0.904944\pi$
$101$	−19.2102	−1.91148	−0.955741	+	0.294208i	$0.904944\pi$
$102$	0	0
$103$	16.2383	1.60001	0.800004	−	0.599995i	$-0.204831\pi$
$103$	16.2383	1.60001	0.800004	+	0.599995i	$0.204831\pi$
$104$	−17.2377	−1.69030
$105$	0	0
$106$	−1.33565	−0.129730
$107$	−5.39336	−0.521396	−0.260698	−	0.965420i	$-0.583953\pi$
$107$	−5.39336	−0.521396	−0.260698	+	0.965420i	$0.583953\pi$
$108$	0	0
$109$	−5.39901	−0.517132	−0.258566	−	0.965994i	$-0.583250\pi$
$109$	−5.39901	−0.517132	−0.258566	+	0.965994i	$0.583250\pi$
$110$	0	0
$111$	0	0
$112$	0.738508	0.0697824
$113$	16.4962	1.55183	0.775917	−	0.630835i	$-0.217287\pi$
$113$	16.4962	1.55183	0.775917	+	0.630835i	$0.217287\pi$
$114$	0	0
$115$	0.867357	0.0808815
$116$	5.26667	0.488998
$117$	0	0
$118$	−7.41179	−0.682310
$119$	−4.33461	−0.397353
$120$	0	0
$121$	0	0
$122$	−7.48673	−0.677816
$123$	0	0
$124$	9.61454	0.863411
$125$	−2.21414	−0.198039
$126$	0	0
$127$	−1.99728	−0.177230	−0.0886150	−	0.996066i	$-0.528244\pi$
$127$	−1.99728	−0.177230	−0.0886150	+	0.996066i	$0.528244\pi$
$128$	9.31786	0.823590
$129$	0	0
$130$	1.12963	0.0990749
$131$	1.37009	0.119706	0.0598528	−	0.998207i	$-0.480937\pi$
$131$	1.37009	0.119706	0.0598528	+	0.998207i	$0.480937\pi$
$132$	0	0
$133$	−2.91501	−0.252763
$134$	−0.988609	−0.0854028
$135$	0	0
$136$	−11.4432	−0.981248
$137$	−2.49924	−0.213525	−0.106762	−	0.994285i	$-0.534048\pi$
$137$	−2.49924	−0.213525	−0.106762	+	0.994285i	$0.534048\pi$
$138$	0	0
$139$	4.76260	0.403958	0.201979	−	0.979390i	$-0.435263\pi$
$139$	4.76260	0.403958	0.201979	+	0.979390i	$0.435263\pi$
$140$	−0.310525	−0.0262441
$141$	0	0
$142$	−7.23775	−0.607378
$143$	0	0
$144$	0	0
$145$	−0.839772	−0.0697392
$146$	4.34545	0.359632
$147$	0	0
$148$	7.88958	0.648520
$149$	−7.22985	−0.592293	−0.296146	−	0.955143i	$-0.595702\pi$
$149$	−7.22985	−0.592293	−0.296146	+	0.955143i	$0.595702\pi$
$150$	0	0
$151$	−8.13968	−0.662398	−0.331199	−	0.943561i	$-0.607453\pi$
$151$	−8.13968	−0.662398	−0.331199	+	0.943561i	$0.607453\pi$
$152$	−7.69551	−0.624188
$153$	0	0
$154$	0	0
$155$	−1.53304	−0.123137
$156$	0	0
$157$	20.1514	1.60826	0.804129	−	0.594455i	$-0.202632\pi$
$157$	20.1514	1.60826	0.804129	+	0.594455i	$0.202632\pi$
$158$	−3.52165	−0.280167
$159$	0	0
$160$	−1.30263	−0.102982
$161$	3.89796	0.307202
$162$	0	0
$163$	7.43449	0.582315	0.291157	−	0.956675i	$-0.405960\pi$
$163$	7.43449	0.582315	0.291157	+	0.956675i	$0.405960\pi$
$164$	−1.86736	−0.145816
$165$	0	0
$166$	−8.77295	−0.680913
$167$	−2.21112	−0.171102	−0.0855510	−	0.996334i	$-0.527265\pi$
$167$	−2.21112	−0.171102	−0.0855510	+	0.996334i	$0.527265\pi$
$168$	0	0
$169$	29.6349	2.27961
$170$	0.749899	0.0575146
$171$	0	0
$172$	−6.57038	−0.500987
$173$	−10.3622	−0.787823	−0.393912	−	0.919148i	$-0.628878\pi$
$173$	−10.3622	−0.787823	−0.393912	+	0.919148i	$0.628878\pi$
$174$	0	0
$175$	−4.95049	−0.374222
$176$	0	0
$177$	0	0
$178$	6.15889	0.461629
$179$	23.4094	1.74970	0.874851	−	0.484392i	$-0.160959\pi$
$179$	23.4094	1.74970	0.874851	+	0.484392i	$0.160959\pi$
$180$	0	0
$181$	−13.7161	−1.01951	−0.509755	−	0.860320i	$-0.670264\pi$
$181$	−13.7161	−1.01951	−0.509755	+	0.860320i	$0.670264\pi$
$182$	5.07662	0.376304
$183$	0	0
$184$	10.2905	0.758623
$185$	−1.25800	−0.0924897
$186$	0	0
$187$	0	0
$188$	8.43666	0.615306
$189$	0	0
$190$	0.504303	0.0365860
$191$	12.0249	0.870094	0.435047	−	0.900408i	$-0.356732\pi$
$191$	12.0249	0.870094	0.435047	+	0.900408i	$0.356732\pi$
$192$	0	0
$193$	−15.8839	−1.14335	−0.571675	−	0.820480i	$-0.693706\pi$
$193$	−15.8839	−1.14335	−0.571675	+	0.820480i	$0.693706\pi$
$194$	7.03927	0.505390
$195$	0	0
$196$	−1.39552	−0.0996799
$197$	12.3035	0.876590	0.438295	−	0.898831i	$-0.355582\pi$
$197$	12.3035	0.876590	0.438295	+	0.898831i	$0.355582\pi$
$198$	0	0
$199$	15.2615	1.08186	0.540929	−	0.841068i	$-0.318073\pi$
$199$	15.2615	1.08186	0.540929	+	0.841068i	$0.318073\pi$
$200$	−13.0691	−0.924125
$201$	0	0
$202$	14.9356	1.05087
$203$	−3.77399	−0.264882
$204$	0	0
$205$	0.297751	0.0207958
$206$	−12.6250	−0.879627
$207$	0	0
$208$	−4.82212	−0.334354
$209$	0	0
$210$	0	0
$211$	−8.93344	−0.615003	−0.307502	−	0.951548i	$-0.599493\pi$
$211$	−8.93344	−0.615003	−0.307502	+	0.951548i	$0.599493\pi$
$212$	−2.39738	−0.164653
$213$	0	0
$214$	4.19325	0.286645
$215$	1.04765	0.0714491
$216$	0	0
$217$	−6.88958	−0.467695
$218$	4.19765	0.284301
$219$	0	0
$220$	0	0
$221$	28.3031	1.90387
$222$	0	0
$223$	0.715244	0.0478963	0.0239481	−	0.999713i	$-0.492376\pi$
$223$	0.715244	0.0478963	0.0239481	+	0.999713i	$0.492376\pi$
$224$	−5.85410	−0.391144
$225$	0	0
$226$	−12.8256	−0.853143
$227$	−5.32162	−0.353208	−0.176604	−	0.984282i	$-0.556511\pi$
$227$	−5.32162	−0.353208	−0.176604	+	0.984282i	$0.556511\pi$
$228$	0	0
$229$	−6.60832	−0.436690	−0.218345	−	0.975872i	$-0.570066\pi$
$229$	−6.60832	−0.436690	−0.218345	+	0.975872i	$0.570066\pi$
$230$	−0.674356	−0.0444657
$231$	0	0
$232$	−9.96318	−0.654115
$233$	−9.55713	−0.626108	−0.313054	−	0.949735i	$-0.601352\pi$
$233$	−9.55713	−0.626108	−0.313054	+	0.949735i	$0.601352\pi$
$234$	0	0
$235$	−1.34523	−0.0877529
$236$	−13.3035	−0.865986
$237$	0	0
$238$	3.37009	0.218451
$239$	26.8466	1.73656	0.868282	−	0.496071i	$-0.165224\pi$
$239$	26.8466	1.73656	0.868282	+	0.496071i	$0.165224\pi$
$240$	0	0
$241$	−18.8663	−1.21529	−0.607643	−	0.794210i	$-0.707885\pi$
$241$	−18.8663	−1.21529	−0.607643	+	0.794210i	$0.707885\pi$
$242$	0	0
$243$	0	0
$244$	−13.4380	−0.860282
$245$	0.222516	0.0142160
$246$	0	0
$247$	19.0337	1.21108
$248$	−18.1882	−1.15495
$249$	0	0
$250$	1.72146	0.108875
$251$	−29.3732	−1.85402	−0.927009	−	0.375040i	$-0.877629\pi$
$251$	−29.3732	−1.85402	−0.927009	+	0.375040i	$0.877629\pi$
$252$	0	0
$253$	0	0
$254$	1.55285	0.0974348
$255$	0	0
$256$	−13.3934	−0.837088
$257$	16.9164	1.05522	0.527608	−	0.849488i	$-0.323089\pi$
$257$	16.9164	1.05522	0.527608	+	0.849488i	$0.323089\pi$
$258$	0	0
$259$	−5.65351	−0.351292
$260$	2.02759	0.125746
$261$	0	0
$262$	−1.06523	−0.0658099
$263$	−8.18034	−0.504421	−0.252211	−	0.967672i	$-0.581158\pi$
$263$	−8.18034	−0.504421	−0.252211	+	0.967672i	$0.581158\pi$
$264$	0	0
$265$	0.382263	0.0234822
$266$	2.26637	0.138960
$267$	0	0
$268$	−1.77447	−0.108393
$269$	6.24716	0.380896	0.190448	−	0.981697i	$-0.439006\pi$
$269$	6.24716	0.380896	0.190448	+	0.981697i	$0.439006\pi$
$270$	0	0
$271$	−7.86688	−0.477879	−0.238939	−	0.971034i	$-0.576800\pi$
$271$	−7.86688	−0.477879	−0.238939	+	0.971034i	$0.576800\pi$
$272$	−3.20115	−0.194098
$273$	0	0
$274$	1.94312	0.117388
$275$	0	0
$276$	0	0
$277$	−12.0476	−0.723873	−0.361937	−	0.932203i	$-0.617884\pi$
$277$	−12.0476	−0.723873	−0.361937	+	0.932203i	$0.617884\pi$
$278$	−3.70284	−0.222082
$279$	0	0
$280$	0.587433	0.0351058
$281$	21.1549	1.26200	0.630998	−	0.775784i	$-0.282646\pi$
$281$	21.1549	1.26200	0.630998	+	0.775784i	$0.282646\pi$
$282$	0	0
$283$	25.0034	1.48630	0.743148	−	0.669127i	$-0.233332\pi$
$283$	25.0034	1.48630	0.743148	+	0.669127i	$0.233332\pi$
$284$	−12.9911	−0.770883
$285$	0	0
$286$	0	0
$287$	1.33811	0.0789861
$288$	0	0
$289$	1.78888	0.105228
$290$	0.652909	0.0383402
$291$	0	0
$292$	7.79971	0.456443
$293$	−0.455087	−0.0265865	−0.0132932	−	0.999912i	$-0.504231\pi$
$293$	−0.455087	−0.0265865	−0.0132932	+	0.999912i	$0.504231\pi$
$294$	0	0
$295$	2.12125	0.123504
$296$	−14.9251	−0.867502
$297$	0	0
$298$	5.62110	0.325621
$299$	−25.4519	−1.47192
$300$	0	0
$301$	4.70820	0.271376
$302$	6.32848	0.364163
$303$	0	0
$304$	−2.15275	−0.123469
$305$	2.14270	0.122691
$306$	0	0
$307$	8.03578	0.458626	0.229313	−	0.973353i	$-0.426352\pi$
$307$	8.03578	0.458626	0.229313	+	0.973353i	$0.426352\pi$
$308$	0	0
$309$	0	0
$310$	1.19191	0.0676962
$311$	−0.136965	−0.00776656	−0.00388328	−	0.999992i	$-0.501236\pi$
$311$	−0.136965	−0.00776656	−0.00388328	+	0.999992i	$0.501236\pi$
$312$	0	0
$313$	−15.2899	−0.864238	−0.432119	−	0.901817i	$-0.642234\pi$
$313$	−15.2899	−0.864238	−0.432119	+	0.901817i	$0.642234\pi$
$314$	−15.6674	−0.884163
$315$	0	0
$316$	−6.32106	−0.355587
$317$	32.9925	1.85304	0.926522	−	0.376239i	$-0.122783\pi$
$317$	32.9925	1.85304	0.926522	+	0.376239i	$0.122783\pi$
$318$	0	0
$319$	0	0
$320$	0.684115	0.0382432
$321$	0	0
$322$	−3.03060	−0.168889
$323$	12.6354	0.703054
$324$	0	0
$325$	32.3244	1.79304
$326$	−5.78020	−0.320136
$327$	0	0
$328$	3.53256	0.195053
$329$	−6.04554	−0.333301
$330$	0	0
$331$	29.5335	1.62331	0.811653	−	0.584140i	$-0.198568\pi$
$331$	29.5335	1.62331	0.811653	+	0.584140i	$0.198568\pi$
$332$	−15.7467	−0.864212
$333$	0	0
$334$	1.71911	0.0940658
$335$	0.282939	0.0154586
$336$	0	0
$337$	−5.63025	−0.306699	−0.153350	−	0.988172i	$-0.549006\pi$
$337$	−5.63025	−0.306699	−0.153350	+	0.988172i	$0.549006\pi$
$338$	−23.0407	−1.25325
$339$	0	0
$340$	1.34600	0.0729974
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	12.4295	0.670153
$345$	0	0
$346$	8.05645	0.433117
$347$	8.73061	0.468684	0.234342	−	0.972154i	$-0.424706\pi$
$347$	8.73061	0.468684	0.234342	+	0.972154i	$0.424706\pi$
$348$	0	0
$349$	−18.4670	−0.988514	−0.494257	−	0.869316i	$-0.664560\pi$
$349$	−18.4670	−0.988514	−0.494257	+	0.869316i	$0.664560\pi$
$350$	3.84893	0.205734
$351$	0	0
$352$	0	0
$353$	23.4857	1.25002	0.625009	−	0.780618i	$-0.285095\pi$
$353$	23.4857	1.25002	0.625009	+	0.780618i	$0.285095\pi$
$354$	0	0
$355$	2.07144	0.109941
$356$	11.0547	0.585897
$357$	0	0
$358$	−18.2005	−0.961924
$359$	14.8812	0.785400	0.392700	−	0.919667i	$-0.371541\pi$
$359$	14.8812	0.785400	0.392700	+	0.919667i	$0.371541\pi$
$360$	0	0
$361$	−10.5027	−0.552776
$362$	10.6641	0.560490
$363$	0	0
$364$	9.11210	0.477604
$365$	−1.24367	−0.0650964
$366$	0	0
$367$	−34.2259	−1.78658	−0.893289	−	0.449482i	$-0.851609\pi$
$367$	−34.2259	−1.78658	−0.893289	+	0.449482i	$0.851609\pi$
$368$	2.87867	0.150061
$369$	0	0
$370$	0.978072	0.0508475
$371$	1.71792	0.0891897
$372$	0	0
$373$	−3.01739	−0.156235	−0.0781173	−	0.996944i	$-0.524891\pi$
$373$	−3.01739	−0.156235	−0.0781173	+	0.996944i	$0.524891\pi$
$374$	0	0
$375$	0	0
$376$	−15.9600	−0.823073
$377$	24.6424	1.26915
$378$	0	0
$379$	−6.12431	−0.314585	−0.157292	−	0.987552i	$-0.550277\pi$
$379$	−6.12431	−0.314585	−0.157292	+	0.987552i	$0.550277\pi$
$380$	0.905182	0.0464348
$381$	0	0
$382$	−9.34920	−0.478347
$383$	−4.39098	−0.224369	−0.112184	−	0.993687i	$-0.535785\pi$
$383$	−4.39098	−0.224369	−0.112184	+	0.993687i	$0.535785\pi$
$384$	0	0
$385$	0	0
$386$	12.3495	0.628573
$387$	0	0
$388$	12.6349	0.641440
$389$	−10.4960	−0.532169	−0.266084	−	0.963950i	$-0.585730\pi$
$389$	−10.4960	−0.532169	−0.266084	+	0.963950i	$0.585730\pi$
$390$	0	0
$391$	−16.8961	−0.854475
$392$	2.63996	0.133338
$393$	0	0
$394$	−9.56580	−0.481918
$395$	1.00789	0.0507127
$396$	0	0
$397$	11.3888	0.571589	0.285794	−	0.958291i	$-0.407743\pi$
$397$	11.3888	0.571589	0.285794	+	0.958291i	$0.407743\pi$
$398$	−11.8656	−0.594767
$399$	0	0
$400$	−3.65597	−0.182799
$401$	4.72496	0.235953	0.117977	−	0.993016i	$-0.462359\pi$
$401$	4.72496	0.235953	0.117977	+	0.993016i	$0.462359\pi$
$402$	0	0
$403$	44.9858	2.24090
$404$	26.8081	1.33375
$405$	0	0
$406$	2.93422	0.145623
$407$	0	0
$408$	0	0
$409$	−2.46544	−0.121908	−0.0609541	−	0.998141i	$-0.519414\pi$
$409$	−2.46544	−0.121908	−0.0609541	+	0.998141i	$0.519414\pi$
$410$	−0.231496	−0.0114328
$411$	0	0
$412$	−22.6609	−1.11642
$413$	9.53304	0.469090
$414$	0	0
$415$	2.51082	0.123251
$416$	38.2246	1.87412
$417$	0	0
$418$	0	0
$419$	14.3399	0.700548	0.350274	−	0.936647i	$-0.386088\pi$
$419$	14.3399	0.700548	0.350274	+	0.936647i	$0.386088\pi$
$420$	0	0
$421$	−17.3157	−0.843918	−0.421959	−	0.906615i	$-0.638657\pi$
$421$	−17.3157	−0.843918	−0.421959	+	0.906615i	$0.638657\pi$
$422$	6.94561	0.338107
$423$	0	0
$424$	4.53523	0.220250
$425$	21.4584	1.04089
$426$	0	0
$427$	9.62943	0.466001
$428$	7.52653	0.363808
$429$	0	0
$430$	−0.814531	−0.0392802
$431$	27.8923	1.34352	0.671762	−	0.740767i	$-0.265538\pi$
$431$	27.8923	1.34352	0.671762	+	0.740767i	$0.265538\pi$
$432$	0	0
$433$	−29.5470	−1.41994	−0.709969	−	0.704232i	$-0.751291\pi$
$433$	−29.5470	−1.41994	−0.709969	+	0.704232i	$0.751291\pi$
$434$	5.35654	0.257122
$435$	0	0
$436$	7.53442	0.360833
$437$	−11.3626	−0.543546
$438$	0	0
$439$	33.6655	1.60677	0.803384	−	0.595461i	$-0.203030\pi$
$439$	33.6655	1.60677	0.803384	+	0.595461i	$0.203030\pi$
$440$	0	0
$441$	0	0
$442$	−22.0052	−1.04668
$443$	−10.3267	−0.490637	−0.245319	−	0.969443i	$-0.578893\pi$
$443$	−10.3267	−0.490637	−0.245319	+	0.969443i	$0.578893\pi$
$444$	0	0
$445$	−1.76267	−0.0835587
$446$	−0.556091	−0.0263317
$447$	0	0
$448$	3.07446	0.145254
$449$	2.75785	0.130151	0.0650756	−	0.997880i	$-0.479271\pi$
$449$	2.75785	0.130151	0.0650756	+	0.997880i	$0.479271\pi$
$450$	0	0
$451$	0	0
$452$	−23.0208	−1.08281
$453$	0	0
$454$	4.13747	0.194181
$455$	−1.45293	−0.0681142
$456$	0	0
$457$	40.3305	1.88658	0.943291	−	0.331968i	$-0.107713\pi$
$457$	40.3305	1.88658	0.943291	+	0.331968i	$0.107713\pi$
$458$	5.13787	0.240077
$459$	0	0
$460$	−1.21041	−0.0564358
$461$	−34.2251	−1.59402	−0.797011	−	0.603965i	$-0.793587\pi$
$461$	−34.2251	−1.59402	−0.797011	+	0.603965i	$0.793587\pi$
$462$	0	0
$463$	0.707349	0.0328733	0.0164367	−	0.999865i	$-0.494768\pi$
$463$	0.707349	0.0328733	0.0164367	+	0.999865i	$0.494768\pi$
$464$	−2.78712	−0.129389
$465$	0	0
$466$	7.43052	0.344212
$467$	28.5924	1.32310	0.661550	−	0.749901i	$-0.269899\pi$
$467$	28.5924	1.32310	0.661550	+	0.749901i	$0.269899\pi$
$468$	0	0
$469$	1.27155	0.0587146
$470$	1.04589	0.0482434
$471$	0	0
$472$	25.1669	1.15840
$473$	0	0
$474$	0	0
$475$	14.4307	0.662126
$476$	6.04903	0.277257
$477$	0	0
$478$	−20.8728	−0.954701
$479$	−5.09716	−0.232895	−0.116448	−	0.993197i	$-0.537151\pi$
$479$	−5.09716	−0.232895	−0.116448	+	0.993197i	$0.537151\pi$
$480$	0	0
$481$	36.9149	1.68317
$482$	14.6683	0.668121
$483$	0	0
$484$	0	0
$485$	−2.01464	−0.0914800
$486$	0	0
$487$	29.1883	1.32265	0.661324	−	0.750101i	$-0.269995\pi$
$487$	29.1883	1.32265	0.661324	+	0.750101i	$0.269995\pi$
$488$	25.4213	1.15077
$489$	0	0
$490$	−0.173002	−0.00781546
$491$	30.1439	1.36038	0.680189	−	0.733037i	$-0.261898\pi$
$491$	30.1439	1.36038	0.680189	+	0.733037i	$0.261898\pi$
$492$	0	0
$493$	16.3588	0.736762
$494$	−14.7984	−0.665810
$495$	0	0
$496$	−5.08801	−0.228458
$497$	9.30919	0.417574
$498$	0	0
$499$	5.95104	0.266405	0.133203	−	0.991089i	$-0.457474\pi$
$499$	5.95104	0.266405	0.133203	+	0.991089i	$0.457474\pi$
$500$	3.08987	0.138183
$501$	0	0
$502$	22.8372	1.01927
$503$	−4.97855	−0.221983	−0.110991	−	0.993821i	$-0.535403\pi$
$503$	−4.97855	−0.221983	−0.110991	+	0.993821i	$0.535403\pi$
$504$	0	0
$505$	−4.27456	−0.190216
$506$	0	0
$507$	0	0
$508$	2.78724	0.123664
$509$	21.3285	0.945370	0.472685	−	0.881231i	$-0.343285\pi$
$509$	21.3285	0.945370	0.472685	+	0.881231i	$0.343285\pi$
$510$	0	0
$511$	−5.58911	−0.247248
$512$	−8.22256	−0.363389
$513$	0	0
$514$	−13.1522	−0.580120
$515$	3.61328	0.159220
$516$	0	0
$517$	0	0
$518$	4.39552	0.193128
$519$	0	0
$520$	−3.83567	−0.168205
$521$	22.1383	0.969899	0.484949	−	0.874542i	$-0.338838\pi$
$521$	22.1383	0.969899	0.484949	+	0.874542i	$0.338838\pi$
$522$	0	0
$523$	−4.91146	−0.214763	−0.107382	−	0.994218i	$-0.534247\pi$
$523$	−4.91146	−0.214763	−0.107382	+	0.994218i	$0.534247\pi$
$524$	−1.91199	−0.0835257
$525$	0	0
$526$	6.36009	0.277313
$527$	29.8637	1.30088
$528$	0	0
$529$	−7.80592	−0.339388
$530$	−0.297204	−0.0129097
$531$	0	0
$532$	4.06794	0.176368
$533$	−8.73725	−0.378452
$534$	0	0
$535$	−1.20011	−0.0518851
$536$	3.35684	0.144993
$537$	0	0
$538$	−4.85707	−0.209403
$539$	0	0
$540$	0	0
$541$	−19.3845	−0.833404	−0.416702	−	0.909043i	$-0.636814\pi$
$541$	−19.3845	−0.833404	−0.416702	+	0.909043i	$0.636814\pi$
$542$	6.11637	0.262721
$543$	0	0
$544$	25.3753	1.08796
$545$	−1.20137	−0.0514609
$546$	0	0
$547$	−14.4501	−0.617840	−0.308920	−	0.951088i	$-0.599968\pi$
$547$	−14.4501	−0.617840	−0.308920	+	0.951088i	$0.599968\pi$
$548$	3.48774	0.148989
$549$	0	0
$550$	0	0
$551$	11.0012	0.468667
$552$	0	0
$553$	4.52954	0.192616
$554$	9.36686	0.397960
$555$	0	0
$556$	−6.64629	−0.281865
$557$	18.4708	0.782634	0.391317	−	0.920256i	$-0.372020\pi$
$557$	18.4708	0.782634	0.391317	+	0.920256i	$0.372020\pi$
$558$	0	0
$559$	−30.7424	−1.30027
$560$	0.164330	0.00694419
$561$	0	0
$562$	−16.4476	−0.693801
$563$	31.2838	1.31846	0.659228	−	0.751943i	$-0.270883\pi$
$563$	31.2838	1.31846	0.659228	+	0.751943i	$0.270883\pi$
$564$	0	0
$565$	3.67067	0.154426
$566$	−19.4397	−0.817112
$567$	0	0
$568$	24.5759	1.03118
$569$	1.73605	0.0727790	0.0363895	−	0.999338i	$-0.488414\pi$
$569$	1.73605	0.0727790	0.0363895	+	0.999338i	$0.488414\pi$
$570$	0	0
$571$	−12.5309	−0.524403	−0.262201	−	0.965013i	$-0.584449\pi$
$571$	−12.5309	−0.524403	−0.262201	+	0.965013i	$0.584449\pi$
$572$	0	0
$573$	0	0
$574$	−1.04036	−0.0434238
$575$	−19.2968	−0.804732
$576$	0	0
$577$	20.1896	0.840505	0.420252	−	0.907407i	$-0.361942\pi$
$577$	20.1896	0.840505	0.420252	+	0.907407i	$0.361942\pi$
$578$	−1.39082	−0.0578506
$579$	0	0
$580$	1.17192	0.0486612
$581$	11.2838	0.468129
$582$	0	0
$583$	0	0
$584$	−14.7550	−0.610568
$585$	0	0
$586$	0.353823	0.0146163
$587$	0.00837574	0.000345704 0	0.000172852	−	1.00000i	$-0.499945\pi$
$587$	0.00837574	0.000345704 0	0.000172852		1.00000i	$0.499945\pi$
$588$	0	0
$589$	20.0832	0.827513
$590$	−1.64924	−0.0678981
$591$	0	0
$592$	−4.17516	−0.171598
$593$	−0.439298	−0.0180398	−0.00901989	−	0.999959i	$-0.502871\pi$
$593$	−0.439298	−0.0180398	−0.00901989	+	0.999959i	$0.502871\pi$
$594$	0	0
$595$	−0.964520	−0.0395415
$596$	10.0894	0.413278
$597$	0	0
$598$	19.7884	0.809210
$599$	−45.1664	−1.84545	−0.922724	−	0.385462i	$-0.874042\pi$
$599$	−45.1664	−1.84545	−0.922724	+	0.385462i	$0.874042\pi$
$600$	0	0
$601$	−11.1437	−0.454559	−0.227280	−	0.973830i	$-0.572983\pi$
$601$	−11.1437	−0.454559	−0.227280	+	0.973830i	$0.572983\pi$
$602$	−3.66055	−0.149193
$603$	0	0
$604$	11.3591	0.462194
$605$	0	0
$606$	0	0
$607$	35.6998	1.44901	0.724504	−	0.689270i	$-0.242069\pi$
$607$	35.6998	1.44901	0.724504	+	0.689270i	$0.242069\pi$
$608$	17.0647	0.692067
$609$	0	0
$610$	−1.66591	−0.0674509
$611$	39.4746	1.59697
$612$	0	0
$613$	−17.1980	−0.694620	−0.347310	−	0.937750i	$-0.612905\pi$
$613$	−17.1980	−0.694620	−0.347310	+	0.937750i	$0.612905\pi$
$614$	−6.24769	−0.252136
$615$	0	0
$616$	0	0
$617$	16.8852	0.679774	0.339887	−	0.940466i	$-0.389611\pi$
$617$	16.8852	0.679774	0.339887	+	0.940466i	$0.389611\pi$
$618$	0	0
$619$	31.0692	1.24878	0.624389	−	0.781114i	$-0.285348\pi$
$619$	31.0692	1.24878	0.624389	+	0.781114i	$0.285348\pi$
$620$	2.13939	0.0859198
$621$	0	0
$622$	0.106488	0.00426978
$623$	−7.92157	−0.317371
$624$	0	0
$625$	24.2598	0.970390
$626$	11.8877	0.475127
$627$	0	0
$628$	−28.1217	−1.12218
$629$	24.5058	0.977110
$630$	0	0
$631$	7.25364	0.288763	0.144382	−	0.989522i	$-0.453881\pi$
$631$	7.25364	0.288763	0.144382	+	0.989522i	$0.453881\pi$
$632$	11.9578	0.475657
$633$	0	0
$634$	−25.6512	−1.01874
$635$	−0.444427	−0.0176365
$636$	0	0
$637$	−6.52954	−0.258710
$638$	0	0
$639$	0	0
$640$	2.07337	0.0819572
$641$	20.7499	0.819572	0.409786	−	0.912182i	$-0.365603\pi$
$641$	20.7499	0.819572	0.409786	+	0.912182i	$0.365603\pi$
$642$	0	0
$643$	7.11127	0.280441	0.140221	−	0.990120i	$-0.455219\pi$
$643$	7.11127	0.280441	0.140221	+	0.990120i	$0.455219\pi$
$644$	−5.43967	−0.214353
$645$	0	0
$646$	−9.82385	−0.386514
$647$	−27.5789	−1.08424	−0.542119	−	0.840302i	$-0.682378\pi$
$647$	−27.5789	−1.08424	−0.542119	+	0.840302i	$0.682378\pi$
$648$	0	0
$649$	0	0
$650$	−25.1317	−0.985748
$651$	0	0
$652$	−10.3750	−0.406315
$653$	−16.1541	−0.632160	−0.316080	−	0.948732i	$-0.602367\pi$
$653$	−16.1541	−0.632160	−0.316080	+	0.948732i	$0.602367\pi$
$654$	0	0
$655$	0.304867	0.0119122
$656$	0.988205	0.0385829
$657$	0	0
$658$	4.70031	0.183237
$659$	−13.2085	−0.514531	−0.257266	−	0.966341i	$-0.582822\pi$
$659$	−13.2085	−0.514531	−0.257266	+	0.966341i	$0.582822\pi$
$660$	0	0
$661$	−4.90660	−0.190845	−0.0954223	−	0.995437i	$-0.530420\pi$
$661$	−4.90660	−0.190845	−0.0954223	+	0.995437i	$0.530420\pi$
$662$	−22.9618	−0.892436
$663$	0	0
$664$	29.7887	1.15603
$665$	−0.648635	−0.0251530
$666$	0	0
$667$	−14.7108	−0.569606
$668$	3.08566	0.119388
$669$	0	0
$670$	−0.219981	−0.00849861
$671$	0	0
$672$	0	0
$673$	29.9872	1.15592	0.577960	−	0.816065i	$-0.303849\pi$
$673$	29.9872	1.15592	0.577960	+	0.816065i	$0.303849\pi$
$674$	4.37743	0.168612
$675$	0	0
$676$	−41.3561	−1.59062
$677$	12.5889	0.483832	0.241916	−	0.970297i	$-0.422224\pi$
$677$	12.5889	0.483832	0.241916	+	0.970297i	$0.422224\pi$
$678$	0	0
$679$	−9.05391	−0.347457
$680$	−2.54630	−0.0976460
$681$	0	0
$682$	0	0
$683$	28.5342	1.09183	0.545916	−	0.837840i	$-0.316182\pi$
$683$	28.5342	1.09183	0.545916	+	0.837840i	$0.316182\pi$
$684$	0	0
$685$	−0.556120	−0.0212483
$686$	−0.777484	−0.0296845
$687$	0	0
$688$	3.47704	0.132561
$689$	−11.2172	−0.427341
$690$	0	0
$691$	25.3742	0.965281	0.482641	−	0.875818i	$-0.339678\pi$
$691$	25.3742	0.965281	0.482641	+	0.875818i	$0.339678\pi$
$692$	14.4606	0.549711
$693$	0	0
$694$	−6.78791	−0.257666
$695$	1.05975	0.0401987
$696$	0	0
$697$	−5.80019	−0.219698
$698$	14.3578	0.543450
$699$	0	0
$700$	6.90849	0.261117
$701$	−45.7021	−1.72615	−0.863073	−	0.505079i	$-0.831463\pi$
$701$	−45.7021	−1.72615	−0.863073	+	0.505079i	$0.831463\pi$
$702$	0	0
$703$	16.4800	0.621556
$704$	0	0
$705$	0	0
$706$	−18.2598	−0.687215
$707$	−19.2102	−0.722473
$708$	0	0
$709$	38.5304	1.44704	0.723520	−	0.690304i	$-0.242523\pi$
$709$	38.5304	1.44704	0.723520	+	0.690304i	$0.242523\pi$
$710$	−1.61051	−0.0604415
$711$	0	0
$712$	−20.9126	−0.783734
$713$	−26.8553	−1.00574
$714$	0	0
$715$	0	0
$716$	−32.6683	−1.22087
$717$	0	0
$718$	−11.5699	−0.431785
$719$	−5.70213	−0.212653	−0.106327	−	0.994331i	$-0.533909\pi$
$719$	−5.70213	−0.212653	−0.106327	+	0.994331i	$0.533909\pi$
$720$	0	0
$721$	16.2383	0.604746
$722$	8.16571	0.303896
$723$	0	0
$724$	19.1411	0.711372
$725$	18.6831	0.693872
$726$	0	0
$727$	11.8221	0.438458	0.219229	−	0.975673i	$-0.429646\pi$
$727$	11.8221	0.438458	0.219229	+	0.975673i	$0.429646\pi$
$728$	−17.2377	−0.638873
$729$	0	0
$730$	0.966931	0.0357877
$731$	−20.4082	−0.754826
$732$	0	0
$733$	−8.81193	−0.325476	−0.162738	−	0.986669i	$-0.552033\pi$
$733$	−8.81193	−0.325476	−0.162738	+	0.986669i	$0.552033\pi$
$734$	26.6101	0.982197
$735$	0	0
$736$	−22.8190	−0.841121
$737$	0	0
$738$	0	0
$739$	−0.480809	−0.0176868	−0.00884342	−	0.999961i	$-0.502815\pi$
$739$	−0.480809	−0.0176868	−0.00884342	+	0.999961i	$0.502815\pi$
$740$	1.75556	0.0645355
$741$	0	0
$742$	−1.33565	−0.0490333
$743$	33.2443	1.21961	0.609807	−	0.792550i	$-0.291247\pi$
$743$	33.2443	1.21961	0.609807	+	0.792550i	$0.291247\pi$
$744$	0	0
$745$	−1.60876	−0.0589403
$746$	2.34598	0.0858923
$747$	0	0
$748$	0	0
$749$	−5.39336	−0.197069
$750$	0	0
$751$	28.0066	1.02198	0.510988	−	0.859588i	$-0.329280\pi$
$751$	28.0066	1.02198	0.510988	+	0.859588i	$0.329280\pi$
$752$	−4.46467	−0.162810
$753$	0	0
$754$	−19.1591	−0.697733
$755$	−1.81121	−0.0659166
$756$	0	0
$757$	−22.2468	−0.808575	−0.404288	−	0.914632i	$-0.632481\pi$
$757$	−22.2468	−0.808575	−0.404288	+	0.914632i	$0.632481\pi$
$758$	4.76156	0.172948
$759$	0	0
$760$	−1.71237	−0.0621142
$761$	−48.2254	−1.74817	−0.874085	−	0.485774i	$-0.838538\pi$
$761$	−48.2254	−1.74817	−0.874085	+	0.485774i	$0.838538\pi$
$762$	0	0
$763$	−5.39901	−0.195457
$764$	−16.7810	−0.607116
$765$	0	0
$766$	3.41392	0.123350
$767$	−62.2464	−2.24759
$768$	0	0
$769$	−43.6883	−1.57544	−0.787721	−	0.616032i	$-0.788739\pi$
$769$	−43.6883	−1.57544	−0.787721	+	0.616032i	$0.788739\pi$
$770$	0	0
$771$	0	0
$772$	22.1663	0.797783
$773$	0.585569	0.0210615	0.0105307	−	0.999945i	$-0.496648\pi$
$773$	0.585569	0.0210615	0.0105307	+	0.999945i	$0.496648\pi$
$774$	0	0
$775$	34.1068	1.22515
$776$	−23.9020	−0.858031
$777$	0	0
$778$	8.16048	0.292567
$779$	−3.90060	−0.139753
$780$	0	0
$781$	0	0
$782$	13.1365	0.469760
$783$	0	0
$784$	0.738508	0.0263753
$785$	4.48401	0.160041
$786$	0	0
$787$	−30.7166	−1.09493	−0.547464	−	0.836829i	$-0.684407\pi$
$787$	−30.7166	−1.09493	−0.547464	+	0.836829i	$0.684407\pi$
$788$	−17.1698	−0.611649
$789$	0	0
$790$	−0.783622	−0.0278800
$791$	16.4962	0.586538
$792$	0	0
$793$	−62.8758	−2.23278
$794$	−8.85463	−0.314239
$795$	0	0
$796$	−21.2977	−0.754877
$797$	10.8333	0.383735	0.191867	−	0.981421i	$-0.438546\pi$
$797$	10.8333	0.383735	0.191867	+	0.981421i	$0.438546\pi$
$798$	0	0
$799$	26.2051	0.927068
$800$	28.9807	1.02462
$801$	0	0
$802$	−3.67358	−0.129719
$803$	0	0
$804$	0	0
$805$	0.867357	0.0305703
$806$	−34.9758	−1.23197
$807$	0	0
$808$	−50.7141	−1.78412
$809$	−38.5901	−1.35675	−0.678377	−	0.734714i	$-0.737316\pi$
$809$	−38.5901	−1.35675	−0.678377	+	0.734714i	$0.737316\pi$
$810$	0	0
$811$	50.8935	1.78711	0.893556	−	0.448951i	$-0.148202\pi$
$811$	50.8935	1.78711	0.893556	+	0.448951i	$0.148202\pi$
$812$	5.26667	0.184824
$813$	0	0
$814$	0	0
$815$	1.65429	0.0579473
$816$	0	0
$817$	−13.7244	−0.480158
$818$	1.91684	0.0670208
$819$	0	0
$820$	−0.415516	−0.0145105
$821$	−40.3094	−1.40681	−0.703404	−	0.710790i	$-0.748338\pi$
$821$	−40.3094	−1.40681	−0.703404	+	0.710790i	$0.748338\pi$
$822$	0	0
$823$	25.6817	0.895209	0.447605	−	0.894232i	$-0.352277\pi$
$823$	25.6817	0.895209	0.447605	+	0.894232i	$0.352277\pi$
$824$	42.8685	1.49340
$825$	0	0
$826$	−7.41179	−0.257889
$827$	−31.9557	−1.11121	−0.555604	−	0.831447i	$-0.687513\pi$
$827$	−31.9557	−1.11121	−0.555604	+	0.831447i	$0.687513\pi$
$828$	0	0
$829$	48.4002	1.68101	0.840504	−	0.541805i	$-0.182259\pi$
$829$	48.4002	1.68101	0.840504	+	0.541805i	$0.182259\pi$
$830$	−1.95212	−0.0677591
$831$	0	0
$832$	−20.0748	−0.695969
$833$	−4.33461	−0.150185
$834$	0	0
$835$	−0.492010	−0.0170267
$836$	0	0
$837$	0	0
$838$	−11.1490	−0.385137
$839$	37.7165	1.30212	0.651059	−	0.759027i	$-0.274325\pi$
$839$	37.7165	1.30212	0.651059	+	0.759027i	$0.274325\pi$
$840$	0	0
$841$	−14.7570	−0.508863
$842$	13.4627	0.463956
$843$	0	0
$844$	12.4668	0.429124
$845$	6.59424	0.226849
$846$	0	0
$847$	0	0
$848$	1.26869	0.0435671
$849$	0	0
$850$	−16.6836	−0.572243
$851$	−22.0372	−0.755424
$852$	0	0
$853$	−9.28864	−0.318037	−0.159019	−	0.987276i	$-0.550833\pi$
$853$	−9.28864	−0.318037	−0.159019	+	0.987276i	$0.550833\pi$
$854$	−7.48673	−0.256191
$855$	0	0
$856$	−14.2383	−0.486654
$857$	−29.7644	−1.01673	−0.508365	−	0.861141i	$-0.669750\pi$
$857$	−29.7644	−1.01673	−0.508365	+	0.861141i	$0.669750\pi$
$858$	0	0
$859$	33.2611	1.13485	0.567427	−	0.823424i	$-0.307939\pi$
$859$	33.2611	1.13485	0.567427	+	0.823424i	$0.307939\pi$
$860$	−1.46201	−0.0498543
$861$	0	0
$862$	−21.6858	−0.738622
$863$	18.5677	0.632051	0.316025	−	0.948751i	$-0.397652\pi$
$863$	18.5677	0.632051	0.316025	+	0.948751i	$0.397652\pi$
$864$	0	0
$865$	−2.30575	−0.0783979
$866$	22.9723	0.780632
$867$	0	0
$868$	9.61454	0.326339
$869$	0	0
$870$	0	0
$871$	−8.30263	−0.281324
$872$	−14.2532	−0.482674
$873$	0	0
$874$	8.83422	0.298822
$875$	−2.21414	−0.0748516
$876$	0	0
$877$	22.4921	0.759505	0.379752	−	0.925088i	$-0.376009\pi$
$877$	22.4921	0.759505	0.379752	+	0.925088i	$0.376009\pi$
$878$	−26.1744	−0.883344
$879$	0	0
$880$	0	0
$881$	−7.06565	−0.238048	−0.119024	−	0.992891i	$-0.537977\pi$
$881$	−7.06565	−0.238048	−0.119024	+	0.992891i	$0.537977\pi$
$882$	0	0
$883$	−19.9382	−0.670974	−0.335487	−	0.942045i	$-0.608901\pi$
$883$	−19.9382	−0.670974	−0.335487	+	0.942045i	$0.608901\pi$
$884$	−39.4974	−1.32844
$885$	0	0
$886$	8.02886	0.269735
$887$	6.95481	0.233520	0.116760	−	0.993160i	$-0.462749\pi$
$887$	6.95481	0.233520	0.116760	+	0.993160i	$0.462749\pi$
$888$	0	0
$889$	−1.99728	−0.0669867
$890$	1.37045	0.0459376
$891$	0	0
$892$	−0.998136	−0.0334201
$893$	17.6228	0.589724
$894$	0	0
$895$	5.20896	0.174116
$896$	9.31786	0.311288
$897$	0	0
$898$	−2.14419	−0.0715525
$899$	26.0012	0.867189
$900$	0	0
$901$	−7.44650	−0.248079
$902$	0	0
$903$	0	0
$904$	43.5494	1.44843
$905$	−3.05205	−0.101454
$906$	0	0
$907$	−23.4401	−0.778317	−0.389159	−	0.921171i	$-0.627234\pi$
$907$	−23.4401	−0.778317	−0.389159	+	0.921171i	$0.627234\pi$
$908$	7.42642	0.246454
$909$	0	0
$910$	1.12963	0.0374468
$911$	−46.8096	−1.55087	−0.775436	−	0.631426i	$-0.782470\pi$
$911$	−46.8096	−1.55087	−0.775436	+	0.631426i	$0.782470\pi$
$912$	0	0
$913$	0	0
$914$	−31.3563	−1.03718
$915$	0	0
$916$	9.22203	0.304705
$917$	1.37009	0.0452445
$918$	0	0
$919$	−10.6273	−0.350563	−0.175281	−	0.984518i	$-0.556084\pi$
$919$	−10.6273	−0.350563	−0.175281	+	0.984518i	$0.556084\pi$
$920$	2.28979	0.0754921
$921$	0	0
$922$	26.6095	0.876336
$923$	−60.7848	−2.00075
$924$	0	0
$925$	27.9876	0.920228
$926$	−0.549953	−0.0180726
$927$	0	0
$928$	22.0933	0.725248
$929$	36.6710	1.20313	0.601567	−	0.798822i	$-0.294543\pi$
$929$	36.6710	1.20313	0.601567	+	0.798822i	$0.294543\pi$
$930$	0	0
$931$	−2.91501	−0.0955355
$932$	13.3371	0.436873
$933$	0	0
$934$	−22.2302	−0.727393
$935$	0	0
$936$	0	0
$937$	40.2174	1.31384	0.656922	−	0.753959i	$-0.271858\pi$
$937$	40.2174	1.31384	0.656922	+	0.753959i	$0.271858\pi$
$938$	−0.988609	−0.0322792
$939$	0	0
$940$	1.87729	0.0612304
$941$	24.9097	0.812032	0.406016	−	0.913866i	$-0.366918\pi$
$941$	24.9097	0.812032	0.406016	+	0.913866i	$0.366918\pi$
$942$	0	0
$943$	5.21590	0.169853
$944$	7.04022	0.229140
$945$	0	0
$946$	0	0
$947$	32.2061	1.04656	0.523279	−	0.852161i	$-0.324708\pi$
$947$	32.2061	1.04656	0.523279	+	0.852161i	$0.324708\pi$
$948$	0	0
$949$	36.4944	1.18466
$950$	−11.2196	−0.364013
$951$	0	0
$952$	−11.4432	−0.370877
$953$	45.0611	1.45967	0.729837	−	0.683621i	$-0.239596\pi$
$953$	45.0611	1.45967	0.729837	+	0.683621i	$0.239596\pi$
$954$	0	0
$955$	2.67574	0.0865849
$956$	−37.4650	−1.21170
$957$	0	0
$958$	3.96296	0.128038
$959$	−2.49924	−0.0807047
$960$	0	0
$961$	16.4663	0.531172
$962$	−28.7007	−0.925349
$963$	0	0
$964$	26.3283	0.847977
$965$	−3.53442	−0.113777
$966$	0	0
$967$	−1.81387	−0.0583300	−0.0291650	−	0.999575i	$-0.509285\pi$
$967$	−1.81387	−0.0583300	−0.0291650	+	0.999575i	$0.509285\pi$
$968$	0	0
$969$	0	0
$970$	1.56635	0.0502924
$971$	−23.0539	−0.739835	−0.369918	−	0.929065i	$-0.620614\pi$
$971$	−23.0539	−0.739835	−0.369918	+	0.929065i	$0.620614\pi$
$972$	0	0
$973$	4.76260	0.152682
$974$	−22.6934	−0.727144
$975$	0	0
$976$	7.11140	0.227630
$977$	7.09449	0.226973	0.113486	−	0.993540i	$-0.463798\pi$
$977$	7.09449	0.226973	0.113486	+	0.993540i	$0.463798\pi$
$978$	0	0
$979$	0	0
$980$	−0.310525	−0.00991935
$981$	0	0
$982$	−23.4364	−0.747887
$983$	−38.3502	−1.22318	−0.611591	−	0.791174i	$-0.709470\pi$
$983$	−38.3502	−1.22318	−0.611591	+	0.791174i	$0.709470\pi$
$984$	0	0
$985$	2.73773	0.0872313
$986$	−12.7187	−0.405046
$987$	0	0
$988$	−26.5618	−0.845044
$989$	18.3524	0.583572
$990$	0	0
$991$	20.2722	0.643967	0.321984	−	0.946745i	$-0.395650\pi$
$991$	20.2722	0.643967	0.321984	+	0.946745i	$0.395650\pi$
$992$	40.3323	1.28055
$993$	0	0
$994$	−7.23775	−0.229567
$995$	3.39592	0.107658
$996$	0	0
$997$	−15.4107	−0.488062	−0.244031	−	0.969767i	$-0.578470\pi$
$997$	−15.4107	−0.488062	−0.244031	+	0.969767i	$0.578470\pi$
$998$	−4.62684	−0.146460
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.co.1.2		4
3.2	odd	2		847.2.a.k.1.3		4
11.7	odd	10		693.2.m.g.379.2		8
11.8	odd	10		693.2.m.g.64.2		8
11.10	odd	2		7623.2.a.ch.1.3		4
21.20	even	2		5929.2.a.bb.1.3		4
33.2	even	10		847.2.f.p.323.2		8
33.5	odd	10		847.2.f.s.729.1		8
33.8	even	10		77.2.f.a.64.1	✓	8
33.14	odd	10		847.2.f.q.372.2		8
33.17	even	10		847.2.f.p.729.2		8
33.20	odd	10		847.2.f.s.323.1		8
33.26	odd	10		847.2.f.q.148.2		8
33.29	even	10		77.2.f.a.71.1	yes	8
33.32	even	2		847.2.a.l.1.2		4
231.41	odd	10		539.2.f.d.295.1		8
231.62	odd	10		539.2.f.d.148.1		8
231.74	even	30		539.2.q.c.471.1		16
231.95	even	30		539.2.q.c.324.2		16
231.107	even	30		539.2.q.c.361.2		16
231.128	even	30		539.2.q.c.214.1		16
231.173	odd	30		539.2.q.b.361.2		16
231.194	odd	30		539.2.q.b.214.1		16
231.206	odd	30		539.2.q.b.471.1		16
231.227	odd	30		539.2.q.b.324.2		16
231.230	odd	2		5929.2.a.bi.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
77.2.f.a.64.1	✓	8	33.8	even	10
77.2.f.a.71.1	yes	8	33.29	even	10
539.2.f.d.148.1		8	231.62	odd	10
539.2.f.d.295.1		8	231.41	odd	10
539.2.q.b.214.1		16	231.194	odd	30
539.2.q.b.324.2		16	231.227	odd	30
539.2.q.b.361.2		16	231.173	odd	30
539.2.q.b.471.1		16	231.206	odd	30
539.2.q.c.214.1		16	231.128	even	30
539.2.q.c.324.2		16	231.95	even	30
539.2.q.c.361.2		16	231.107	even	30
539.2.q.c.471.1		16	231.74	even	30
693.2.m.g.64.2		8	11.8	odd	10
693.2.m.g.379.2		8	11.7	odd	10
847.2.a.k.1.3		4	3.2	odd	2
847.2.a.l.1.2		4	33.32	even	2
847.2.f.p.323.2		8	33.2	even	10
847.2.f.p.729.2		8	33.17	even	10
847.2.f.q.148.2		8	33.26	odd	10
847.2.f.q.372.2		8	33.14	odd	10
847.2.f.s.323.1		8	33.20	odd	10
847.2.f.s.729.1		8	33.5	odd	10
5929.2.a.bb.1.3		4	21.20	even	2
5929.2.a.bi.1.2		4	231.230	odd	2
7623.2.a.ch.1.3		4	11.10	odd	2
7623.2.a.co.1.2		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-0.777484 q^{2} -1.39552 q^{4} +0.222516 q^{5} +1.00000 q^{7} +2.63996 q^{8} +O(q^{10})\)	\(q-0.777484 q^{2} -1.39552 q^{4} +0.222516 q^{5} +1.00000 q^{7} +2.63996 q^{8} -0.173002 q^{10} -6.52954 q^{13} -0.777484 q^{14} +0.738508 q^{16} -4.33461 q^{17} -2.91501 q^{19} -0.310525 q^{20} +3.89796 q^{23} -4.95049 q^{25} +5.07662 q^{26} -1.39552 q^{28} -3.77399 q^{29} -6.88958 q^{31} -5.85410 q^{32} +3.37009 q^{34} +0.222516 q^{35} -5.65351 q^{37} +2.26637 q^{38} +0.587433 q^{40} +1.33811 q^{41} +4.70820 q^{43} -3.03060 q^{46} -6.04554 q^{47} +1.00000 q^{49} +3.84893 q^{50} +9.11210 q^{52} +1.71792 q^{53} +2.63996 q^{56} +2.93422 q^{58} +9.53304 q^{59} +9.62943 q^{61} +5.35654 q^{62} +3.07446 q^{64} -1.45293 q^{65} +1.27155 q^{67} +6.04903 q^{68} -0.173002 q^{70} +9.30919 q^{71} -5.58911 q^{73} +4.39552 q^{74} +4.06794 q^{76} +4.52954 q^{79} +0.164330 q^{80} -1.04036 q^{82} +11.2838 q^{83} -0.964520 q^{85} -3.66055 q^{86} -7.92157 q^{89} -6.52954 q^{91} -5.43967 q^{92} +4.70031 q^{94} -0.648635 q^{95} -9.05391 q^{97} -0.777484 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 4 q^{4} + 6 q^{5} + 4 q^{7} + 9 q^{8}+O(q^{10}) \) 4 * q + 2 * q^2 + 4 * q^4 + 6 * q^5 + 4 * q^7 + 9 * q^8	\( 4 q + 2 q^{2} + 4 q^{4} + 6 q^{5} + 4 q^{7} + 9 q^{8} + 14 q^{10} + 2 q^{14} - 4 q^{16} - 3 q^{17} - 3 q^{19} + 17 q^{20} + 8 q^{23} + 12 q^{26} + 4 q^{28} + 3 q^{29} - 3 q^{31} - 10 q^{32} - 12 q^{34} + 6 q^{35} - 7 q^{37} + 20 q^{38} + 13 q^{40} + 4 q^{41} - 8 q^{43} + 3 q^{46} + 14 q^{47} + 4 q^{49} + 33 q^{50} + 17 q^{52} + 9 q^{53} + 9 q^{56} + 3 q^{58} + 25 q^{59} + 19 q^{61} + 10 q^{62} + 3 q^{64} + 12 q^{65} - 15 q^{67} - q^{68} + 14 q^{70} + 7 q^{71} + 11 q^{73} + 8 q^{74} + 26 q^{76} - 8 q^{79} + 4 q^{80} + 3 q^{82} - q^{83} - 15 q^{85} - 4 q^{86} + 17 q^{89} + 17 q^{92} + 20 q^{94} + 17 q^{95} - 15 q^{97} + 2 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 4 * q^4 + 6 * q^5 + 4 * q^7 + 9 * q^8 + 14 * q^10 + 2 * q^14 - 4 * q^16 - 3 * q^17 - 3 * q^19 + 17 * q^20 + 8 * q^23 + 12 * q^26 + 4 * q^28 + 3 * q^29 - 3 * q^31 - 10 * q^32 - 12 * q^34 + 6 * q^35 - 7 * q^37 + 20 * q^38 + 13 * q^40 + 4 * q^41 - 8 * q^43 + 3 * q^46 + 14 * q^47 + 4 * q^49 + 33 * q^50 + 17 * q^52 + 9 * q^53 + 9 * q^56 + 3 * q^58 + 25 * q^59 + 19 * q^61 + 10 * q^62 + 3 * q^64 + 12 * q^65 - 15 * q^67 - q^68 + 14 * q^70 + 7 * q^71 + 11 * q^73 + 8 * q^74 + 26 * q^76 - 8 * q^79 + 4 * q^80 + 3 * q^82 - q^83 - 15 * q^85 - 4 * q^86 + 17 * q^89 + 17 * q^92 + 20 * q^94 + 17 * q^95 - 15 * q^97 + 2 * q^98

Introduction

L-functions

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