LMFDB - Embedded newform 7623.2.a.cb.1.3

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:	$3$
Coefficient field:	3.3.837.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 6x - 1 $ x^3 - 6*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.52892$ 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)

$f(q)$	$=$	$q+2.52892 q^{2} +4.39543 q^{4} -0.133492 q^{5} +1.00000 q^{7} +6.05784 q^{8} +O(q^{10})$	$q+2.52892 q^{2} +4.39543 q^{4} -0.133492 q^{5} +1.00000 q^{7} +6.05784 q^{8} -0.337590 q^{10} -0.133492 q^{13} +2.52892 q^{14} +6.52892 q^{16} -5.05784 q^{17} +0.924344 q^{19} -0.586754 q^{20} +7.05784 q^{23} -4.98218 q^{25} -0.337590 q^{26} +4.39543 q^{28} +3.86651 q^{29} +2.79085 q^{31} +4.39543 q^{32} -12.7909 q^{34} -0.133492 q^{35} +9.98218 q^{37} +2.33759 q^{38} -0.808672 q^{40} +11.8487 q^{41} +3.05784 q^{43} +17.8487 q^{46} +3.07566 q^{47} +1.00000 q^{49} -12.5995 q^{50} -0.586754 q^{52} +4.79085 q^{53} +6.05784 q^{56} +9.77808 q^{58} +12.6574 q^{59} -6.00000 q^{61} +7.05784 q^{62} -1.94216 q^{64} +0.0178201 q^{65} +8.92434 q^{67} -22.2313 q^{68} -0.337590 q^{70} +6.11567 q^{71} -7.86651 q^{73} +25.2441 q^{74} +4.06289 q^{76} -14.1157 q^{79} -0.871558 q^{80} +29.9644 q^{82} -1.20915 q^{83} +0.675180 q^{85} +7.73302 q^{86} -15.5817 q^{89} -0.133492 q^{91} +31.0222 q^{92} +7.77808 q^{94} -0.123392 q^{95} +12.7909 q^{97} +2.52892 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 6 q^{4} + 3 q^{7} + 3 q^{8}+O(q^{10}) $ 3 * q + 6 * q^4 + 3 * q^7 + 3 * q^8	$ 3 q + 6 q^{4} + 3 q^{7} + 3 q^{8} - 9 q^{10} + 12 q^{16} - 12 q^{19} + 21 q^{20} + 6 q^{23} + 15 q^{25} - 9 q^{26} + 6 q^{28} + 12 q^{29} - 6 q^{31} + 6 q^{32} - 24 q^{34} + 15 q^{38} - 18 q^{40} + 6 q^{41} - 6 q^{43} + 24 q^{46} + 24 q^{47} + 3 q^{49} - 39 q^{50} + 21 q^{52} + 3 q^{56} - 9 q^{58} + 24 q^{59} - 18 q^{61} + 6 q^{62} - 21 q^{64} + 30 q^{65} + 12 q^{67} - 6 q^{68} - 9 q^{70} - 12 q^{71} - 24 q^{73} + 39 q^{74} + 3 q^{76} - 12 q^{79} - 9 q^{80} + 30 q^{82} - 18 q^{83} + 18 q^{85} + 24 q^{86} - 18 q^{89} + 18 q^{92} - 15 q^{94} + 12 q^{95} + 24 q^{97}+O(q^{100}) $ 3 * q + 6 * q^4 + 3 * q^7 + 3 * q^8 - 9 * q^10 + 12 * q^16 - 12 * q^19 + 21 * q^20 + 6 * q^23 + 15 * q^25 - 9 * q^26 + 6 * q^28 + 12 * q^29 - 6 * q^31 + 6 * q^32 - 24 * q^34 + 15 * q^38 - 18 * q^40 + 6 * q^41 - 6 * q^43 + 24 * q^46 + 24 * q^47 + 3 * q^49 - 39 * q^50 + 21 * q^52 + 3 * q^56 - 9 * q^58 + 24 * q^59 - 18 * q^61 + 6 * q^62 - 21 * q^64 + 30 * q^65 + 12 * q^67 - 6 * q^68 - 9 * q^70 - 12 * q^71 - 24 * q^73 + 39 * q^74 + 3 * q^76 - 12 * q^79 - 9 * q^80 + 30 * q^82 - 18 * q^83 + 18 * q^85 + 24 * q^86 - 18 * q^89 + 18 * q^92 - 15 * q^94 + 12 * q^95 + 24 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	2.52892	1.78822	0.894108	−	0.447852i	$-0.147811\pi$
$2$	2.52892	1.78822	0.894108	+	0.447852i	$0.147811\pi$
$3$	0	0
$3$	0	0
$4$	4.39543	2.19771
$5$	−0.133492	−0.0596994	−0.0298497	−	0.999554i	$-0.509503\pi$
$5$	−0.133492	−0.0596994	−0.0298497	+	0.999554i	$0.509503\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	6.05784	2.14177
$9$	0	0
$10$	−0.337590	−0.106755
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−0.133492	−0.0370240	−0.0185120	−	0.999829i	$-0.505893\pi$
$13$	−0.133492	−0.0370240	−0.0185120	+	0.999829i	$0.505893\pi$
$14$	2.52892	0.675882
$15$	0	0
$16$	6.52892	1.63223
$17$	−5.05784	−1.22671	−0.613353	−	0.789809i	$-0.710180\pi$
$17$	−5.05784	−1.22671	−0.613353	+	0.789809i	$0.710180\pi$
$18$	0	0
$19$	0.924344	0.212059	0.106030	−	0.994363i	$-0.466186\pi$
$19$	0.924344	0.212059	0.106030	+	0.994363i	$0.466186\pi$
$20$	−0.586754	−0.131202
$21$	0	0
$22$	0	0
$23$	7.05784	1.47166	0.735830	−	0.677166i	$-0.236792\pi$
$23$	7.05784	1.47166	0.735830	+	0.677166i	$0.236792\pi$
$24$	0	0
$25$	−4.98218	−0.996436
$26$	−0.337590	−0.0662069
$27$	0	0
$28$	4.39543	0.830657
$29$	3.86651	0.717993	0.358996	−	0.933339i	$-0.383119\pi$
$29$	3.86651	0.717993	0.358996	+	0.933339i	$0.383119\pi$
$30$	0	0
$31$	2.79085	0.501252	0.250626	−	0.968084i	$-0.419364\pi$
$31$	2.79085	0.501252	0.250626	+	0.968084i	$0.419364\pi$
$32$	4.39543	0.777009
$33$	0	0
$34$	−12.7909	−2.19361
$35$	−0.133492	−0.0225643
$36$	0	0
$37$	9.98218	1.64106	0.820530	−	0.571603i	$-0.193678\pi$
$37$	9.98218	1.64106	0.820530	+	0.571603i	$0.193678\pi$
$38$	2.33759	0.379207
$39$	0	0
$40$	−0.808672	−0.127862
$41$	11.8487	1.85045	0.925227	−	0.379414i	$-0.123874\pi$
$41$	11.8487	1.85045	0.925227	+	0.379414i	$0.123874\pi$
$42$	0	0
$43$	3.05784	0.466316	0.233158	−	0.972439i	$-0.425094\pi$
$43$	3.05784	0.466316	0.233158	+	0.972439i	$0.425094\pi$
$44$	0	0
$45$	0	0
$46$	17.8487	2.63165
$47$	3.07566	0.448631	0.224315	−	0.974517i	$-0.427985\pi$
$47$	3.07566	0.448631	0.224315	+	0.974517i	$0.427985\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−12.5995	−1.78184
$51$	0	0
$52$	−0.586754	−0.0813681
$53$	4.79085	0.658074	0.329037	−	0.944317i	$-0.393276\pi$
$53$	4.79085	0.658074	0.329037	+	0.944317i	$0.393276\pi$
$54$	0	0
$55$	0	0
$56$	6.05784	0.809512
$57$	0	0
$58$	9.77808	1.28393
$59$	12.6574	1.64785	0.823924	−	0.566700i	$-0.191780\pi$
$59$	12.6574	1.64785	0.823924	+	0.566700i	$0.191780\pi$
$60$	0	0
$61$	−6.00000	−0.768221	−0.384111	−	0.923287i	$-0.625492\pi$
$61$	−6.00000	−0.768221	−0.384111	+	0.923287i	$0.625492\pi$
$62$	7.05784	0.896346
$63$	0	0
$64$	−1.94216	−0.242771
$65$	0.0178201	0.00221031
$66$	0	0
$67$	8.92434	1.09028	0.545141	−	0.838344i	$-0.316476\pi$
$67$	8.92434	1.09028	0.545141	+	0.838344i	$0.316476\pi$
$68$	−22.2313	−2.69595
$69$	0	0
$70$	−0.337590	−0.0403497
$71$	6.11567	0.725797	0.362898	−	0.931829i	$-0.381787\pi$
$71$	6.11567	0.725797	0.362898	+	0.931829i	$0.381787\pi$
$72$	0	0
$73$	−7.86651	−0.920705	−0.460353	−	0.887736i	$-0.652277\pi$
$73$	−7.86651	−0.920705	−0.460353	+	0.887736i	$0.652277\pi$
$74$	25.2441	2.93457
$75$	0	0
$76$	4.06289	0.466045
$77$	0	0
$78$	0	0
$79$	−14.1157	−1.58814	−0.794069	−	0.607828i	$-0.792041\pi$
$79$	−14.1157	−1.58814	−0.794069	+	0.607828i	$0.792041\pi$
$80$	−0.871558	−0.0974431
$81$	0	0
$82$	29.9644	3.30901
$83$	−1.20915	−0.132721	−0.0663606	−	0.997796i	$-0.521139\pi$
$83$	−1.20915	−0.132721	−0.0663606	+	0.997796i	$0.521139\pi$
$84$	0	0
$85$	0.675180	0.0732336
$86$	7.73302	0.833873
$87$	0	0
$88$	0	0
$89$	−15.5817	−1.65166	−0.825829	−	0.563921i	$-0.809292\pi$
$89$	−15.5817	−1.65166	−0.825829	+	0.563921i	$0.809292\pi$
$90$	0	0
$91$	−0.133492	−0.0139938
$92$	31.0222	3.23429
$93$	0	0
$94$	7.77808	0.802248
$95$	−0.123392	−0.0126598
$96$	0	0
$97$	12.7909	1.29871	0.649357	−	0.760484i	$-0.275038\pi$
$97$	12.7909	1.29871	0.649357	+	0.760484i	$0.275038\pi$
$98$	2.52892	0.255459
$99$	0	0
$100$	−21.8988	−2.18988
$101$	−9.59180	−0.954420	−0.477210	−	0.878789i	$-0.658352\pi$
$101$	−9.59180	−0.954420	−0.477210	+	0.878789i	$0.658352\pi$
$102$	0	0
$103$	9.84869	0.970420	0.485210	−	0.874398i	$-0.338743\pi$
$103$	9.84869	0.970420	0.485210	+	0.874398i	$0.338743\pi$
$104$	−0.808672	−0.0792968
$105$	0	0
$106$	12.1157	1.17678
$107$	0.924344	0.0893597	0.0446799	−	0.999001i	$-0.485773\pi$
$107$	0.924344	0.0893597	0.0446799	+	0.999001i	$0.485773\pi$
$108$	0	0
$109$	8.52387	0.816439	0.408219	−	0.912884i	$-0.366150\pi$
$109$	8.52387	0.816439	0.408219	+	0.912884i	$0.366150\pi$
$110$	0	0
$111$	0	0
$112$	6.52892	0.616925
$113$	12.1157	1.13975	0.569873	−	0.821733i	$-0.306992\pi$
$113$	12.1157	1.13975	0.569873	+	0.821733i	$0.306992\pi$
$114$	0	0
$115$	−0.942164	−0.0878573
$116$	16.9950	1.57794
$117$	0	0
$118$	32.0094	2.94671
$119$	−5.05784	−0.463651
$120$	0	0
$121$	0	0
$122$	−15.1735	−1.37374
$123$	0	0
$124$	12.2670	1.10161
$125$	1.33254	0.119186
$126$	0	0
$127$	8.90652	0.790326	0.395163	−	0.918611i	$-0.370688\pi$
$127$	8.90652	0.790326	0.395163	+	0.918611i	$0.370688\pi$
$128$	−13.7024	−1.21113
$129$	0	0
$130$	0.0450656	0.00395251
$131$	−15.6974	−1.37149	−0.685743	−	0.727844i	$-0.740523\pi$
$131$	−15.6974	−1.37149	−0.685743	+	0.727844i	$0.740523\pi$
$132$	0	0
$133$	0.924344	0.0801508
$134$	22.5689	1.94966
$135$	0	0
$136$	−30.6395	−2.62732
$137$	−14.6395	−1.25074	−0.625370	−	0.780328i	$-0.715052\pi$
$137$	−14.6395	−1.25074	−0.625370	+	0.780328i	$0.715052\pi$
$138$	0	0
$139$	−18.6496	−1.58184	−0.790921	−	0.611918i	$-0.790398\pi$
$139$	−18.6496	−1.58184	−0.790921	+	0.611918i	$0.790398\pi$
$140$	−0.586754	−0.0495898
$141$	0	0
$142$	15.4660	1.29788
$143$	0	0
$144$	0	0
$145$	−0.516148	−0.0428637
$146$	−19.8938	−1.64642
$147$	0	0
$148$	43.8759	3.60658
$149$	11.8665	0.972142	0.486071	−	0.873919i	$-0.338430\pi$
$149$	11.8665	0.972142	0.486071	+	0.873919i	$0.338430\pi$
$150$	0	0
$151$	11.3248	0.921601	0.460800	−	0.887504i	$-0.347562\pi$
$151$	11.3248	0.921601	0.460800	+	0.887504i	$0.347562\pi$
$152$	5.59952	0.454181
$153$	0	0
$154$	0	0
$155$	−0.372556	−0.0299244
$156$	0	0
$157$	−20.3827	−1.62671	−0.813357	−	0.581766i	$-0.802362\pi$
$157$	−20.3827	−1.62671	−0.813357	+	0.581766i	$0.802362\pi$
$158$	−35.6974	−2.83993
$159$	0	0
$160$	−0.586754	−0.0463870
$161$	7.05784	0.556235
$162$	0	0
$163$	−19.3070	−1.51224	−0.756120	−	0.654432i	$-0.772908\pi$
$163$	−19.3070	−1.51224	−0.756120	+	0.654432i	$0.772908\pi$
$164$	52.0800	4.06677
$165$	0	0
$166$	−3.05784	−0.237334
$167$	12.6395	0.978077	0.489038	−	0.872262i	$-0.337348\pi$
$167$	12.6395	0.978077	0.489038	+	0.872262i	$0.337348\pi$
$168$	0	0
$169$	−12.9822	−0.998629
$170$	1.70748	0.130957
$171$	0	0
$172$	13.4405	1.02483
$173$	−19.8487	−1.50907	−0.754534	−	0.656261i	$-0.772137\pi$
$173$	−19.8487	−1.50907	−0.754534	+	0.656261i	$0.772137\pi$
$174$	0	0
$175$	−4.98218	−0.376617
$176$	0	0
$177$	0	0
$178$	−39.4049	−2.95352
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	2.67518	0.198845	0.0994223	−	0.995045i	$-0.468301\pi$
$181$	2.67518	0.198845	0.0994223	+	0.995045i	$0.468301\pi$
$182$	−0.337590	−0.0250238
$183$	0	0
$184$	42.7552	3.15196
$185$	−1.33254	−0.0979703
$186$	0	0
$187$	0	0
$188$	13.5188	0.985961
$189$	0	0
$190$	−0.312049	−0.0226384
$191$	−3.32482	−0.240576	−0.120288	−	0.992739i	$-0.538382\pi$
$191$	−3.32482	−0.240576	−0.120288	+	0.992739i	$0.538382\pi$
$192$	0	0
$193$	−18.3726	−1.32249	−0.661243	−	0.750172i	$-0.729971\pi$
$193$	−18.3726	−1.32249	−0.661243	+	0.750172i	$0.729971\pi$
$194$	32.3470	2.32238
$195$	0	0
$196$	4.39543	0.313959
$197$	−8.11567	−0.578218	−0.289109	−	0.957296i	$-0.593359\pi$
$197$	−8.11567	−0.578218	−0.289109	+	0.957296i	$0.593359\pi$
$198$	0	0
$199$	−6.52387	−0.462465	−0.231232	−	0.972899i	$-0.574276\pi$
$199$	−6.52387	−0.462465	−0.231232	+	0.972899i	$0.574276\pi$
$200$	−30.1812	−2.13414
$201$	0	0
$202$	−24.2569	−1.70671
$203$	3.86651	0.271376
$204$	0	0
$205$	−1.58170	−0.110471
$206$	24.9065	1.73532
$207$	0	0
$208$	−0.871558	−0.0604317
$209$	0	0
$210$	0	0
$211$	−13.8487	−0.953383	−0.476691	−	0.879071i	$-0.658164\pi$
$211$	−13.8487	−0.953383	−0.476691	+	0.879071i	$0.658164\pi$
$212$	21.0578	1.44626
$213$	0	0
$214$	2.33759	0.159794
$215$	−0.408196	−0.0278388
$216$	0	0
$217$	2.79085	0.189455
$218$	21.5562	1.45997
$219$	0	0
$220$	0	0
$221$	0.675180	0.0454175
$222$	0	0
$223$	−24.4983	−1.64053	−0.820265	−	0.571984i	$-0.806174\pi$
$223$	−24.4983	−1.64053	−0.820265	+	0.571984i	$0.806174\pi$
$224$	4.39543	0.293682
$225$	0	0
$226$	30.6395	2.03811
$227$	−7.73302	−0.513258	−0.256629	−	0.966510i	$-0.582612\pi$
$227$	−7.73302	−0.513258	−0.256629	+	0.966510i	$0.582612\pi$
$228$	0	0
$229$	−4.79085	−0.316588	−0.158294	−	0.987392i	$-0.550599\pi$
$229$	−4.79085	−0.316588	−0.158294	+	0.987392i	$0.550599\pi$
$230$	−2.38266	−0.157108
$231$	0	0
$232$	23.4227	1.53777
$233$	−9.69738	−0.635296	−0.317648	−	0.948209i	$-0.602893\pi$
$233$	−9.69738	−0.635296	−0.317648	+	0.948209i	$0.602893\pi$
$234$	0	0
$235$	−0.410575	−0.0267830
$236$	55.6345	3.62150
$237$	0	0
$238$	−12.7909	−0.829108
$239$	23.3070	1.50760	0.753802	−	0.657101i	$-0.228218\pi$
$239$	23.3070	1.50760	0.753802	+	0.657101i	$0.228218\pi$
$240$	0	0
$241$	−9.71520	−0.625811	−0.312905	−	0.949784i	$-0.601302\pi$
$241$	−9.71520	−0.625811	−0.312905	+	0.949784i	$0.601302\pi$
$242$	0	0
$243$	0	0
$244$	−26.3726	−1.68833
$245$	−0.133492	−0.00852849
$246$	0	0
$247$	−0.123392	−0.00785127
$248$	16.9065	1.07357
$249$	0	0
$250$	3.36989	0.213130
$251$	15.0400	0.949317	0.474659	−	0.880170i	$-0.342572\pi$
$251$	15.0400	0.949317	0.474659	+	0.880170i	$0.342572\pi$
$252$	0	0
$253$	0	0
$254$	22.5239	1.41327
$255$	0	0
$256$	−30.7680	−1.92300
$257$	−15.8309	−0.987502	−0.493751	−	0.869603i	$-0.664375\pi$
$257$	−15.8309	−0.987502	−0.493751	+	0.869603i	$0.664375\pi$
$258$	0	0
$259$	9.98218	0.620262
$260$	0.0783269	0.00485763
$261$	0	0
$262$	−39.6974	−2.45251
$263$	−3.07566	−0.189653	−0.0948265	−	0.995494i	$-0.530230\pi$
$263$	−3.07566	−0.189653	−0.0948265	+	0.995494i	$0.530230\pi$
$264$	0	0
$265$	−0.639540	−0.0392866
$266$	2.33759	0.143327
$267$	0	0
$268$	39.2263	2.39613
$269$	10.5340	0.642267	0.321134	−	0.947034i	$-0.395936\pi$
$269$	10.5340	0.642267	0.321134	+	0.947034i	$0.395936\pi$
$270$	0	0
$271$	4.92434	0.299133	0.149566	−	0.988752i	$-0.452212\pi$
$271$	4.92434	0.299133	0.149566	+	0.988752i	$0.452212\pi$
$272$	−33.0222	−2.00226
$273$	0	0
$274$	−37.0222	−2.23659
$275$	0	0
$276$	0	0
$277$	0.151312	0.00909146	0.00454573	−	0.999990i	$-0.498553\pi$
$277$	0.151312	0.00909146	0.00454573	+	0.999990i	$0.498553\pi$
$278$	−47.1634	−2.82867
$279$	0	0
$280$	−0.808672	−0.0483274
$281$	6.51615	0.388721	0.194360	−	0.980930i	$-0.437737\pi$
$281$	6.51615	0.388721	0.194360	+	0.980930i	$0.437737\pi$
$282$	0	0
$283$	0.390376	0.0232055	0.0116027	−	0.999933i	$-0.496307\pi$
$283$	0.390376	0.0232055	0.0116027	+	0.999933i	$0.496307\pi$
$284$	26.8810	1.59509
$285$	0	0
$286$	0	0
$287$	11.8487	0.699406
$288$	0	0
$289$	8.58170	0.504806
$290$	−1.30529	−0.0766496
$291$	0	0
$292$	−34.5767	−2.02345
$293$	29.4304	1.71934	0.859671	−	0.510848i	$-0.170669\pi$
$293$	29.4304	1.71934	0.859671	+	0.510848i	$0.170669\pi$
$294$	0	0
$295$	−1.68966	−0.0983755
$296$	60.4704	3.51477
$297$	0	0
$298$	30.0094	1.73840
$299$	−0.942164	−0.0544868
$300$	0	0
$301$	3.05784	0.176251
$302$	28.6395	1.64802
$303$	0	0
$304$	6.03497	0.346129
$305$	0.800952	0.0458624
$306$	0	0
$307$	−23.6974	−1.35248	−0.676240	−	0.736681i	$-0.736392\pi$
$307$	−23.6974	−1.35248	−0.676240	+	0.736681i	$0.736392\pi$
$308$	0	0
$309$	0	0
$310$	−0.942164	−0.0535113
$311$	12.2313	0.693576	0.346788	−	0.937944i	$-0.387272\pi$
$311$	12.2313	0.693576	0.346788	+	0.937944i	$0.387272\pi$
$312$	0	0
$313$	−19.1735	−1.08375	−0.541875	−	0.840459i	$-0.682286\pi$
$313$	−19.1735	−1.08375	−0.541875	+	0.840459i	$0.682286\pi$
$314$	−51.5461	−2.90891
$315$	0	0
$316$	−62.0444	−3.49027
$317$	−29.5562	−1.66004	−0.830020	−	0.557734i	$-0.811671\pi$
$317$	−29.5562	−1.66004	−0.830020	+	0.557734i	$0.811671\pi$
$318$	0	0
$319$	0	0
$320$	0.259263	0.0144933
$321$	0	0
$322$	17.8487	0.994668
$323$	−4.67518	−0.260134
$324$	0	0
$325$	0.665081	0.0368920
$326$	−48.8258	−2.70421
$327$	0	0
$328$	71.7774	3.96324
$329$	3.07566	0.169566
$330$	0	0
$331$	18.6496	1.02508	0.512538	−	0.858664i	$-0.328705\pi$
$331$	18.6496	1.02508	0.512538	+	0.858664i	$0.328705\pi$
$332$	−5.31472	−0.291683
$333$	0	0
$334$	31.9644	1.74901
$335$	−1.19133	−0.0650892
$336$	0	0
$337$	21.0222	1.14515	0.572576	−	0.819852i	$-0.305944\pi$
$337$	21.0222	1.14515	0.572576	+	0.819852i	$0.305944\pi$
$338$	−32.8309	−1.78576
$339$	0	0
$340$	2.96770	0.160946
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	18.5239	0.998740
$345$	0	0
$346$	−50.1957	−2.69854
$347$	23.1634	1.24348	0.621738	−	0.783225i	$-0.286427\pi$
$347$	23.1634	1.24348	0.621738	+	0.783225i	$0.286427\pi$
$348$	0	0
$349$	28.0979	1.50404	0.752022	−	0.659138i	$-0.229079\pi$
$349$	28.0979	1.50404	0.752022	+	0.659138i	$0.229079\pi$
$350$	−12.5995	−0.673473
$351$	0	0
$352$	0	0
$353$	33.4126	1.77837	0.889186	−	0.457546i	$-0.151272\pi$
$353$	33.4126	1.77837	0.889186	+	0.457546i	$0.151272\pi$
$354$	0	0
$355$	−0.816393	−0.0433296
$356$	−68.4882	−3.62987
$357$	0	0
$358$	30.3470	1.60389
$359$	13.5817	0.716815	0.358407	−	0.933565i	$-0.383320\pi$
$359$	13.5817	0.716815	0.358407	+	0.933565i	$0.383320\pi$
$360$	0	0
$361$	−18.1456	−0.955031
$362$	6.76531	0.355577
$363$	0	0
$364$	−0.586754	−0.0307543
$365$	1.05012	0.0549655
$366$	0	0
$367$	−3.73302	−0.194862	−0.0974309	−	0.995242i	$-0.531063\pi$
$367$	−3.73302	−0.194862	−0.0974309	+	0.995242i	$0.531063\pi$
$368$	46.0800	2.40209
$369$	0	0
$370$	−3.36989	−0.175192
$371$	4.79085	0.248729
$372$	0	0
$373$	−6.94216	−0.359452	−0.179726	−	0.983717i	$-0.557521\pi$
$373$	−6.94216	−0.359452	−0.179726	+	0.983717i	$0.557521\pi$
$374$	0	0
$375$	0	0
$376$	18.6318	0.960863
$377$	−0.516148	−0.0265830
$378$	0	0
$379$	−0.390376	−0.0200523	−0.0100261	−	0.999950i	$-0.503191\pi$
$379$	−0.390376	−0.0200523	−0.0100261	+	0.999950i	$0.503191\pi$
$380$	−0.542362	−0.0278226
$381$	0	0
$382$	−8.40820	−0.430201
$383$	0.533968	0.0272845	0.0136422	−	0.999907i	$-0.495657\pi$
$383$	0.533968	0.0272845	0.0136422	+	0.999907i	$0.495657\pi$
$384$	0	0
$385$	0	0
$386$	−46.4627	−2.36489
$387$	0	0
$388$	56.2212	2.85420
$389$	10.4983	0.532286	0.266143	−	0.963934i	$-0.414251\pi$
$389$	10.4983	0.532286	0.266143	+	0.963934i	$0.414251\pi$
$390$	0	0
$391$	−35.6974	−1.80529
$392$	6.05784	0.305967
$393$	0	0
$394$	−20.5239	−1.03398
$395$	1.88433	0.0948108
$396$	0	0
$397$	−20.7552	−1.04167	−0.520837	−	0.853656i	$-0.674380\pi$
$397$	−20.7552	−1.04167	−0.520837	+	0.853656i	$0.674380\pi$
$398$	−16.4983	−0.826986
$399$	0	0
$400$	−32.5282	−1.62641
$401$	−21.3248	−1.06491	−0.532455	−	0.846458i	$-0.678731\pi$
$401$	−21.3248	−1.06491	−0.532455	+	0.846458i	$0.678731\pi$
$402$	0	0
$403$	−0.372556	−0.0185583
$404$	−42.1601	−2.09754
$405$	0	0
$406$	9.77808	0.485278
$407$	0	0
$408$	0	0
$409$	−33.9287	−1.67767	−0.838834	−	0.544388i	$-0.816762\pi$
$409$	−33.9287	−1.67767	−0.838834	+	0.544388i	$0.816762\pi$
$410$	−4.00000	−0.197546
$411$	0	0
$412$	43.2892	2.13270
$413$	12.6574	0.622828
$414$	0	0
$415$	0.161411	0.00792338
$416$	−0.586754	−0.0287680
$417$	0	0
$418$	0	0
$419$	9.99228	0.488155	0.244077	−	0.969756i	$-0.421515\pi$
$419$	9.99228	0.488155	0.244077	+	0.969756i	$0.421515\pi$
$420$	0	0
$421$	−29.9822	−1.46124	−0.730621	−	0.682783i	$-0.760769\pi$
$421$	−29.9822	−1.46124	−0.730621	+	0.682783i	$0.760769\pi$
$422$	−35.0222	−1.70485
$423$	0	0
$424$	29.0222	1.40944
$425$	25.1990	1.22233
$426$	0	0
$427$	−6.00000	−0.290360
$428$	4.06289	0.196387
$429$	0	0
$430$	−1.03230	−0.0497817
$431$	−2.54169	−0.122429	−0.0612144	−	0.998125i	$-0.519497\pi$
$431$	−2.54169	−0.122429	−0.0612144	+	0.998125i	$0.519497\pi$
$432$	0	0
$433$	20.3827	0.979528	0.489764	−	0.871855i	$-0.337083\pi$
$433$	20.3827	0.979528	0.489764	+	0.871855i	$0.337083\pi$
$434$	7.05784	0.338787
$435$	0	0
$436$	37.4660	1.79430
$437$	6.52387	0.312079
$438$	0	0
$439$	5.45831	0.260511	0.130256	−	0.991480i	$-0.458420\pi$
$439$	5.45831	0.260511	0.130256	+	0.991480i	$0.458420\pi$
$440$	0	0
$441$	0	0
$442$	1.70748	0.0812163
$443$	10.6496	0.505980	0.252990	−	0.967469i	$-0.418586\pi$
$443$	10.6496	0.505980	0.252990	+	0.967469i	$0.418586\pi$
$444$	0	0
$445$	2.08003	0.0986030
$446$	−61.9543	−2.93362
$447$	0	0
$448$	−1.94216	−0.0917586
$449$	−27.0323	−1.27573	−0.637866	−	0.770147i	$-0.720183\pi$
$449$	−27.0323	−1.27573	−0.637866	+	0.770147i	$0.720183\pi$
$450$	0	0
$451$	0	0
$452$	53.2535	2.50484
$453$	0	0
$454$	−19.5562	−0.917816
$455$	0.0178201	0.000835419 0
$456$	0	0
$457$	−4.79085	−0.224107	−0.112053	−	0.993702i	$-0.535743\pi$
$457$	−4.79085	−0.224107	−0.112053	+	0.993702i	$0.535743\pi$
$458$	−12.1157	−0.566128
$459$	0	0
$460$	−4.14121	−0.193085
$461$	22.4983	1.04785	0.523926	−	0.851764i	$-0.324467\pi$
$461$	22.4983	1.04785	0.523926	+	0.851764i	$0.324467\pi$
$462$	0	0
$463$	25.6897	1.19390	0.596950	−	0.802279i	$-0.296379\pi$
$463$	25.6897	1.19390	0.596950	+	0.802279i	$0.296379\pi$
$464$	25.2441	1.17193
$465$	0	0
$466$	−24.5239	−1.13605
$467$	−4.62172	−0.213868	−0.106934	−	0.994266i	$-0.534103\pi$
$467$	−4.62172	−0.213868	−0.106934	+	0.994266i	$0.534103\pi$
$468$	0	0
$469$	8.92434	0.412088
$470$	−1.03831	−0.0478937
$471$	0	0
$472$	76.6762	3.52931
$473$	0	0
$474$	0	0
$475$	−4.60525	−0.211303
$476$	−22.2313	−1.01897
$477$	0	0
$478$	58.9415	2.69592
$479$	−0.372556	−0.0170225	−0.00851126	−	0.999964i	$-0.502709\pi$
$479$	−0.372556	−0.0170225	−0.00851126	+	0.999964i	$0.502709\pi$
$480$	0	0
$481$	−1.33254	−0.0607586
$482$	−24.5689	−1.11908
$483$	0	0
$484$	0	0
$485$	−1.70748	−0.0775325
$486$	0	0
$487$	4.30262	0.194971	0.0974853	−	0.995237i	$-0.468920\pi$
$487$	4.30262	0.194971	0.0974853	+	0.995237i	$0.468920\pi$
$488$	−36.3470	−1.64535
$489$	0	0
$490$	−0.337590	−0.0152508
$491$	25.4583	1.14892	0.574459	−	0.818534i	$-0.305213\pi$
$491$	25.4583	1.14892	0.574459	+	0.818534i	$0.305213\pi$
$492$	0	0
$493$	−19.5562	−0.880765
$494$	−0.312049	−0.0140398
$495$	0	0
$496$	18.2212	0.818158
$497$	6.11567	0.274325
$498$	0	0
$499$	−8.39038	−0.375605	−0.187802	−	0.982207i	$-0.560136\pi$
$499$	−8.39038	−0.375605	−0.187802	+	0.982207i	$0.560136\pi$
$500$	5.85708	0.261937
$501$	0	0
$502$	38.0350	1.69758
$503$	−26.7552	−1.19296	−0.596478	−	0.802629i	$-0.703434\pi$
$503$	−26.7552	−1.19296	−0.596478	+	0.802629i	$0.703434\pi$
$504$	0	0
$505$	1.28043	0.0569783
$506$	0	0
$507$	0	0
$508$	39.1480	1.73691
$509$	18.0000	0.797836	0.398918	−	0.916987i	$-0.369386\pi$
$509$	18.0000	0.797836	0.398918	+	0.916987i	$0.369386\pi$
$510$	0	0
$511$	−7.86651	−0.347994
$512$	−50.4049	−2.22760
$513$	0	0
$514$	−40.0350	−1.76587
$515$	−1.31472	−0.0579335
$516$	0	0
$517$	0	0
$518$	25.2441	1.10916
$519$	0	0
$520$	0.107951	0.00473397
$521$	1.44821	0.0634473	0.0317237	−	0.999497i	$-0.489900\pi$
$521$	1.44821	0.0634473	0.0317237	+	0.999497i	$0.489900\pi$
$522$	0	0
$523$	−23.5740	−1.03082	−0.515409	−	0.856944i	$-0.672360\pi$
$523$	−23.5740	−1.03082	−0.515409	+	0.856944i	$0.672360\pi$
$524$	−68.9967	−3.01413
$525$	0	0
$526$	−7.77808	−0.339140
$527$	−14.1157	−0.614888
$528$	0	0
$529$	26.8130	1.16578
$530$	−1.61734	−0.0702529
$531$	0	0
$532$	4.06289	0.176148
$533$	−1.58170	−0.0685112
$534$	0	0
$535$	−0.123392	−0.00533472
$536$	54.0622	2.33513
$537$	0	0
$538$	26.6395	1.14851
$539$	0	0
$540$	0	0
$541$	18.4983	0.795305	0.397653	−	0.917536i	$-0.369825\pi$
$541$	18.4983	0.795305	0.397653	+	0.917536i	$0.369825\pi$
$542$	12.4533	0.534913
$543$	0	0
$544$	−22.2313	−0.953161
$545$	−1.13787	−0.0487409
$546$	0	0
$547$	−20.0000	−0.855138	−0.427569	−	0.903983i	$-0.640630\pi$
$547$	−20.0000	−0.855138	−0.427569	+	0.903983i	$0.640630\pi$
$548$	−64.3470	−2.74877
$549$	0	0
$550$	0	0
$551$	3.57398	0.152257
$552$	0	0
$553$	−14.1157	−0.600259
$554$	0.382656	0.0162575
$555$	0	0
$556$	−81.9731	−3.47643
$557$	1.98218	0.0839877	0.0419938	−	0.999118i	$-0.486629\pi$
$557$	1.98218	0.0839877	0.0419938	+	0.999118i	$0.486629\pi$
$558$	0	0
$559$	−0.408196	−0.0172649
$560$	−0.871558	−0.0368300
$561$	0	0
$562$	16.4788	0.695116
$563$	1.98990	0.0838643	0.0419322	−	0.999120i	$-0.486649\pi$
$563$	1.98990	0.0838643	0.0419322	+	0.999120i	$0.486649\pi$
$564$	0	0
$565$	−1.61734	−0.0680422
$566$	0.987230	0.0414964
$567$	0	0
$568$	37.0477	1.55449
$569$	−14.5340	−0.609296	−0.304648	−	0.952465i	$-0.598539\pi$
$569$	−14.5340	−0.609296	−0.304648	+	0.952465i	$0.598539\pi$
$570$	0	0
$571$	29.2791	1.22529	0.612646	−	0.790358i	$-0.290105\pi$
$571$	29.2791	1.22529	0.612646	+	0.790358i	$0.290105\pi$
$572$	0	0
$573$	0	0
$574$	29.9644	1.25069
$575$	−35.1634	−1.46642
$576$	0	0
$577$	−5.59180	−0.232790	−0.116395	−	0.993203i	$-0.537134\pi$
$577$	−5.59180	−0.232790	−0.116395	+	0.993203i	$0.537134\pi$
$578$	21.7024	0.902702
$579$	0	0
$580$	−2.26869	−0.0942022
$581$	−1.20915	−0.0501639
$582$	0	0
$583$	0	0
$584$	−47.6540	−1.97194
$585$	0	0
$586$	74.4270	3.07455
$587$	2.80867	0.115926	0.0579632	−	0.998319i	$-0.481539\pi$
$587$	2.80867	0.115926	0.0579632	+	0.998319i	$0.481539\pi$
$588$	0	0
$589$	2.57971	0.106295
$590$	−4.27300	−0.175917
$591$	0	0
$592$	65.1728	2.67859
$593$	−42.1957	−1.73277	−0.866385	−	0.499377i	$-0.833562\pi$
$593$	−42.1957	−1.73277	−0.866385	+	0.499377i	$0.833562\pi$
$594$	0	0
$595$	0.675180	0.0276797
$596$	52.1584	2.13649
$597$	0	0
$598$	−2.38266	−0.0974340
$599$	−34.3470	−1.40338	−0.701691	−	0.712482i	$-0.747571\pi$
$599$	−34.3470	−1.40338	−0.701691	+	0.712482i	$0.747571\pi$
$600$	0	0
$601$	−5.98218	−0.244018	−0.122009	−	0.992529i	$-0.538934\pi$
$601$	−5.98218	−0.244018	−0.122009	+	0.992529i	$0.538934\pi$
$602$	7.73302	0.315174
$603$	0	0
$604$	49.7774	2.02541
$605$	0	0
$606$	0	0
$607$	−8.12339	−0.329718	−0.164859	−	0.986317i	$-0.552717\pi$
$607$	−8.12339	−0.329718	−0.164859	+	0.986317i	$0.552717\pi$
$608$	4.06289	0.164772
$609$	0	0
$610$	2.02554	0.0820117
$611$	−0.410575	−0.0166101
$612$	0	0
$613$	16.5239	0.667393	0.333696	−	0.942681i	$-0.391704\pi$
$613$	16.5239	0.667393	0.333696	+	0.942681i	$0.391704\pi$
$614$	−59.9287	−2.41853
$615$	0	0
$616$	0	0
$617$	−11.8843	−0.478445	−0.239223	−	0.970965i	$-0.576893\pi$
$617$	−11.8843	−0.478445	−0.239223	+	0.970965i	$0.576893\pi$
$618$	0	0
$619$	43.0222	1.72921	0.864604	−	0.502454i	$-0.167569\pi$
$619$	43.0222	1.72921	0.864604	+	0.502454i	$0.167569\pi$
$620$	−1.63754	−0.0657653
$621$	0	0
$622$	30.9321	1.24026
$623$	−15.5817	−0.624268
$624$	0	0
$625$	24.7330	0.989321
$626$	−48.4882	−1.93798
$627$	0	0
$628$	−89.5905	−3.57505
$629$	−50.4882	−2.01310
$630$	0	0
$631$	13.5817	0.540679	0.270340	−	0.962765i	$-0.412864\pi$
$631$	13.5817	0.540679	0.270340	+	0.962765i	$0.412864\pi$
$632$	−85.5104	−3.40142
$633$	0	0
$634$	−74.7451	−2.96851
$635$	−1.18895	−0.0471820
$636$	0	0
$637$	−0.133492	−0.00528914
$638$	0	0
$639$	0	0
$640$	1.82916	0.0723040
$641$	−36.7552	−1.45174	−0.725872	−	0.687830i	$-0.758563\pi$
$641$	−36.7552	−1.45174	−0.725872	+	0.687830i	$0.758563\pi$
$642$	0	0
$643$	−44.3369	−1.74848	−0.874239	−	0.485496i	$-0.838639\pi$
$643$	−44.3369	−1.74848	−0.874239	+	0.485496i	$0.838639\pi$
$644$	31.0222	1.22245
$645$	0	0
$646$	−11.8231	−0.465176
$647$	17.9721	0.706555	0.353278	−	0.935519i	$-0.385067\pi$
$647$	17.9721	0.706555	0.353278	+	0.935519i	$0.385067\pi$
$648$	0	0
$649$	0	0
$650$	1.68193	0.0659709
$651$	0	0
$652$	−84.8625	−3.32347
$653$	22.1957	0.868585	0.434292	−	0.900772i	$-0.356998\pi$
$653$	22.1957	0.868585	0.434292	+	0.900772i	$0.356998\pi$
$654$	0	0
$655$	2.09547	0.0818769
$656$	77.3591	3.02037
$657$	0	0
$658$	7.77808	0.303221
$659$	9.99228	0.389244	0.194622	−	0.980878i	$-0.437652\pi$
$659$	9.99228	0.389244	0.194622	+	0.980878i	$0.437652\pi$
$660$	0	0
$661$	27.9543	1.08729	0.543647	−	0.839314i	$-0.317043\pi$
$661$	27.9543	1.08729	0.543647	+	0.839314i	$0.317043\pi$
$662$	47.1634	1.83306
$663$	0	0
$664$	−7.32482	−0.284258
$665$	−0.123392	−0.00478495
$666$	0	0
$667$	27.2892	1.05664
$668$	55.5562	2.14953
$669$	0	0
$670$	−3.01277	−0.116393
$671$	0	0
$672$	0	0
$673$	43.4203	1.67373	0.836865	−	0.547410i	$-0.184386\pi$
$673$	43.4203	1.67373	0.836865	+	0.547410i	$0.184386\pi$
$674$	53.1634	2.04778
$675$	0	0
$676$	−57.0622	−2.19470
$677$	−12.3470	−0.474534	−0.237267	−	0.971444i	$-0.576252\pi$
$677$	−12.3470	−0.474534	−0.237267	+	0.971444i	$0.576252\pi$
$678$	0	0
$679$	12.7909	0.490868
$680$	4.09013	0.156849
$681$	0	0
$682$	0	0
$683$	27.8386	1.06521	0.532607	−	0.846363i	$-0.321212\pi$
$683$	27.8386	1.06521	0.532607	+	0.846363i	$0.321212\pi$
$684$	0	0
$685$	1.95426	0.0746684
$686$	2.52892	0.0965545
$687$	0	0
$688$	19.9644	0.761134
$689$	−0.639540	−0.0243645
$690$	0	0
$691$	−16.8010	−0.639138	−0.319569	−	0.947563i	$-0.603538\pi$
$691$	−16.8010	−0.639138	−0.319569	+	0.947563i	$0.603538\pi$
$692$	−87.2434	−3.31650
$693$	0	0
$694$	58.5784	2.22360
$695$	2.48958	0.0944350
$696$	0	0
$697$	−59.9287	−2.26996
$698$	71.0572	2.68955
$699$	0	0
$700$	−21.8988	−0.827697
$701$	9.16341	0.346097	0.173049	−	0.984913i	$-0.444638\pi$
$701$	9.16341	0.346097	0.173049	+	0.984913i	$0.444638\pi$
$702$	0	0
$703$	9.22697	0.348002
$704$	0	0
$705$	0	0
$706$	84.4977	3.18011
$707$	−9.59180	−0.360737
$708$	0	0
$709$	−34.7475	−1.30497	−0.652485	−	0.757802i	$-0.726273\pi$
$709$	−34.7475	−1.30497	−0.652485	+	0.757802i	$0.726273\pi$
$710$	−2.06459	−0.0774827
$711$	0	0
$712$	−94.3914	−3.53747
$713$	19.6974	0.737673
$714$	0	0
$715$	0	0
$716$	52.7451	1.97118
$717$	0	0
$718$	34.3470	1.28182
$719$	25.1557	0.938149	0.469074	−	0.883159i	$-0.344588\pi$
$719$	25.1557	0.938149	0.469074	+	0.883159i	$0.344588\pi$
$720$	0	0
$721$	9.84869	0.366784
$722$	−45.8887	−1.70780
$723$	0	0
$724$	11.7586	0.437003
$725$	−19.2636	−0.715434
$726$	0	0
$727$	−23.5918	−0.874972	−0.437486	−	0.899225i	$-0.644131\pi$
$727$	−23.5918	−0.874972	−0.437486	+	0.899225i	$0.644131\pi$
$728$	−0.808672	−0.0299714
$729$	0	0
$730$	2.65566	0.0982902
$731$	−15.4660	−0.572032
$732$	0	0
$733$	−21.9287	−0.809956	−0.404978	−	0.914326i	$-0.632721\pi$
$733$	−21.9287	−0.809956	−0.404978	+	0.914326i	$0.632721\pi$
$734$	−9.44049	−0.348455
$735$	0	0
$736$	31.0222	1.14349
$737$	0	0
$738$	0	0
$739$	−35.0578	−1.28962	−0.644812	−	0.764341i	$-0.723064\pi$
$739$	−35.0578	−1.28962	−0.644812	+	0.764341i	$0.723064\pi$
$740$	−5.85708	−0.215311
$741$	0	0
$742$	12.1157	0.444780
$743$	−31.3070	−1.14854	−0.574271	−	0.818665i	$-0.694715\pi$
$743$	−31.3070	−1.14854	−0.574271	+	0.818665i	$0.694715\pi$
$744$	0	0
$745$	−1.58408	−0.0580363
$746$	−17.5562	−0.642777
$747$	0	0
$748$	0	0
$749$	0.924344	0.0337748
$750$	0	0
$751$	−4.42602	−0.161508	−0.0807538	−	0.996734i	$-0.525733\pi$
$751$	−4.42602	−0.161508	−0.0807538	+	0.996734i	$0.525733\pi$
$752$	20.0807	0.732268
$753$	0	0
$754$	−1.30529	−0.0475360
$755$	−1.51177	−0.0550190
$756$	0	0
$757$	2.51615	0.0914509	0.0457255	−	0.998954i	$-0.485440\pi$
$757$	2.51615	0.0914509	0.0457255	+	0.998954i	$0.485440\pi$
$758$	−0.987230	−0.0358578
$759$	0	0
$760$	−0.747491	−0.0271144
$761$	13.7330	0.497821	0.248911	−	0.968526i	$-0.419927\pi$
$761$	13.7330	0.497821	0.248911	+	0.968526i	$0.419927\pi$
$762$	0	0
$763$	8.52387	0.308585
$764$	−14.6140	−0.528716
$765$	0	0
$766$	1.35036	0.0487905
$767$	−1.68966	−0.0610099
$768$	0	0
$769$	−46.4805	−1.67613	−0.838065	−	0.545570i	$-0.816313\pi$
$769$	−46.4805	−1.67613	−0.838065	+	0.545570i	$0.816313\pi$
$770$	0	0
$771$	0	0
$772$	−80.7552	−2.90644
$773$	45.9822	1.65386	0.826932	−	0.562302i	$-0.190084\pi$
$773$	45.9822	1.65386	0.826932	+	0.562302i	$0.190084\pi$
$774$	0	0
$775$	−13.9045	−0.499465
$776$	77.4849	2.78155
$777$	0	0
$778$	26.5494	0.951842
$779$	10.9523	0.392406
$780$	0	0
$781$	0	0
$782$	−90.2757	−3.22825
$783$	0	0
$784$	6.52892	0.233176
$785$	2.72092	0.0971138
$786$	0	0
$787$	−40.9243	−1.45880	−0.729398	−	0.684090i	$-0.760200\pi$
$787$	−40.9243	−1.45880	−0.729398	+	0.684090i	$0.760200\pi$
$788$	−35.6718	−1.27076
$789$	0	0
$790$	4.76531	0.169542
$791$	12.1157	0.430784
$792$	0	0
$793$	0.800952	0.0284426
$794$	−52.4882	−1.86274
$795$	0	0
$796$	−28.6752	−1.01636
$797$	28.8786	1.02293	0.511466	−	0.859303i	$-0.329102\pi$
$797$	28.8786	1.02293	0.511466	+	0.859303i	$0.329102\pi$
$798$	0	0
$799$	−15.5562	−0.550338
$800$	−21.8988	−0.774240
$801$	0	0
$802$	−53.9287	−1.90429
$803$	0	0
$804$	0	0
$805$	−0.942164	−0.0332069
$806$	−0.942164	−0.0331863
$807$	0	0
$808$	−58.1056	−2.04415
$809$	6.51615	0.229096	0.114548	−	0.993418i	$-0.463458\pi$
$809$	6.51615	0.229096	0.114548	+	0.993418i	$0.463458\pi$
$810$	0	0
$811$	−38.5060	−1.35213	−0.676065	−	0.736842i	$-0.736316\pi$
$811$	−38.5060	−1.35213	−0.676065	+	0.736842i	$0.736316\pi$
$812$	16.9950	0.596406
$813$	0	0
$814$	0	0
$815$	2.57733	0.0902799
$816$	0	0
$817$	2.82649	0.0988864
$818$	−85.8029	−3.00003
$819$	0	0
$820$	−6.95226	−0.242784
$821$	−37.3769	−1.30446	−0.652232	−	0.758019i	$-0.726167\pi$
$821$	−37.3769	−1.30446	−0.652232	+	0.758019i	$0.726167\pi$
$822$	0	0
$823$	43.0400	1.50028	0.750140	−	0.661279i	$-0.229986\pi$
$823$	43.0400	1.50028	0.750140	+	0.661279i	$0.229986\pi$
$824$	59.6617	2.07842
$825$	0	0
$826$	32.0094	1.11375
$827$	11.3426	0.394422	0.197211	−	0.980361i	$-0.436812\pi$
$827$	11.3426	0.394422	0.197211	+	0.980361i	$0.436812\pi$
$828$	0	0
$829$	14.4983	0.503548	0.251774	−	0.967786i	$-0.418986\pi$
$829$	14.4983	0.503548	0.251774	+	0.967786i	$0.418986\pi$
$830$	0.408196	0.0141687
$831$	0	0
$832$	0.259263	0.00898833
$833$	−5.05784	−0.175244
$834$	0	0
$835$	−1.68728	−0.0583906
$836$	0	0
$837$	0	0
$838$	25.2697	0.872926
$839$	3.60962	0.124618	0.0623090	−	0.998057i	$-0.480154\pi$
$839$	3.60962	0.124618	0.0623090	+	0.998057i	$0.480154\pi$
$840$	0	0
$841$	−14.0501	−0.484487
$842$	−75.8225	−2.61301
$843$	0	0
$844$	−60.8709	−2.09526
$845$	1.73302	0.0596176
$846$	0	0
$847$	0	0
$848$	31.2791	1.07413
$849$	0	0
$850$	63.7263	2.18579
$851$	70.4526	2.41508
$852$	0	0
$853$	24.6496	0.843988	0.421994	−	0.906599i	$-0.361330\pi$
$853$	24.6496	0.843988	0.421994	+	0.906599i	$0.361330\pi$
$854$	−15.1735	−0.519227
$855$	0	0
$856$	5.59952	0.191388
$857$	17.9287	0.612433	0.306217	−	0.951962i	$-0.400937\pi$
$857$	17.9287	0.612433	0.306217	+	0.951962i	$0.400937\pi$
$858$	0	0
$859$	−15.6974	−0.535588	−0.267794	−	0.963476i	$-0.586295\pi$
$859$	−15.6974	−0.535588	−0.267794	+	0.963476i	$0.586295\pi$
$860$	−1.79420	−0.0611816
$861$	0	0
$862$	−6.42772	−0.218929
$863$	15.0578	0.512575	0.256287	−	0.966601i	$-0.417501\pi$
$863$	15.0578	0.512575	0.256287	+	0.966601i	$0.417501\pi$
$864$	0	0
$865$	2.64964	0.0900904
$866$	51.5461	1.75161
$867$	0	0
$868$	12.2670	0.416369
$869$	0	0
$870$	0	0
$871$	−1.19133	−0.0403666
$872$	51.6362	1.74862
$873$	0	0
$874$	16.4983	0.558064
$875$	1.33254	0.0450481
$876$	0	0
$877$	43.8487	1.48066	0.740332	−	0.672241i	$-0.234668\pi$
$877$	43.8487	1.48066	0.740332	+	0.672241i	$0.234668\pi$
$878$	13.8036	0.465850
$879$	0	0
$880$	0	0
$881$	−11.0656	−0.372808	−0.186404	−	0.982473i	$-0.559683\pi$
$881$	−11.0656	−0.372808	−0.186404	+	0.982473i	$0.559683\pi$
$882$	0	0
$883$	5.72530	0.192672	0.0963358	−	0.995349i	$-0.469288\pi$
$883$	5.72530	0.192672	0.0963358	+	0.995349i	$0.469288\pi$
$884$	2.96770	0.0998147
$885$	0	0
$886$	26.9321	0.904800
$887$	15.0935	0.506789	0.253395	−	0.967363i	$-0.418453\pi$
$887$	15.0935	0.506789	0.253395	+	0.967363i	$0.418453\pi$
$888$	0	0
$889$	8.90652	0.298715
$890$	5.26023	0.176323
$891$	0	0
$892$	−107.681	−3.60541
$893$	2.84296	0.0951362
$894$	0	0
$895$	−1.60190	−0.0535457
$896$	−13.7024	−0.457766
$897$	0	0
$898$	−68.3625	−2.28128
$899$	10.7909	0.359895
$900$	0	0
$901$	−24.2313	−0.807263
$902$	0	0
$903$	0	0
$904$	73.3948	2.44107
$905$	−0.357115	−0.0118709
$906$	0	0
$907$	41.2993	1.37132	0.685660	−	0.727922i	$-0.259514\pi$
$907$	41.2993	1.37132	0.685660	+	0.727922i	$0.259514\pi$
$908$	−33.9899	−1.12799
$909$	0	0
$910$	0.0450656	0.00149391
$911$	−3.45059	−0.114323	−0.0571616	−	0.998365i	$-0.518205\pi$
$911$	−3.45059	−0.114323	−0.0571616	+	0.998365i	$0.518205\pi$
$912$	0	0
$913$	0	0
$914$	−12.1157	−0.400751
$915$	0	0
$916$	−21.0578	−0.695770
$917$	−15.6974	−0.518373
$918$	0	0
$919$	10.8265	0.357133	0.178567	−	0.983928i	$-0.442854\pi$
$919$	10.8265	0.357133	0.178567	+	0.983928i	$0.442854\pi$
$920$	−5.70748	−0.188170
$921$	0	0
$922$	56.8964	1.87378
$923$	−0.816393	−0.0268719
$924$	0	0
$925$	−49.7330	−1.63521
$926$	64.9670	2.13495
$927$	0	0
$928$	16.9950	0.557887
$929$	−17.4684	−0.573120	−0.286560	−	0.958062i	$-0.592512\pi$
$929$	−17.4684	−0.573120	−0.286560	+	0.958062i	$0.592512\pi$
$930$	0	0
$931$	0.924344	0.0302942
$932$	−42.6241	−1.39620
$933$	0	0
$934$	−11.6880	−0.382441
$935$	0	0
$936$	0	0
$937$	−10.5340	−0.344130	−0.172065	−	0.985086i	$-0.555044\pi$
$937$	−10.5340	−0.344130	−0.172065	+	0.985086i	$0.555044\pi$
$938$	22.5689	0.736902
$939$	0	0
$940$	−1.80465	−0.0588613
$941$	−16.1513	−0.526518	−0.263259	−	0.964725i	$-0.584797\pi$
$941$	−16.1513	−0.526518	−0.263259	+	0.964725i	$0.584797\pi$
$942$	0	0
$943$	83.6261	2.72324
$944$	82.6389	2.68967
$945$	0	0
$946$	0	0
$947$	6.91662	0.224760	0.112380	−	0.993665i	$-0.464153\pi$
$947$	6.91662	0.224760	0.112380	+	0.993665i	$0.464153\pi$
$948$	0	0
$949$	1.05012	0.0340882
$950$	−11.6463	−0.377856
$951$	0	0
$952$	−30.6395	−0.993033
$953$	−16.1335	−0.522615	−0.261308	−	0.965256i	$-0.584154\pi$
$953$	−16.1335	−0.522615	−0.261308	+	0.965256i	$0.584154\pi$
$954$	0	0
$955$	0.443837	0.0143622
$956$	102.444	3.31328
$957$	0	0
$958$	−0.942164	−0.0304399
$959$	−14.6395	−0.472735
$960$	0	0
$961$	−23.2111	−0.748747
$962$	−3.36989	−0.108649
$963$	0	0
$964$	−42.7024	−1.37535
$965$	2.45259	0.0789516
$966$	0	0
$967$	−17.8487	−0.573975	−0.286988	−	0.957934i	$-0.592654\pi$
$967$	−17.8487	−0.573975	−0.286988	+	0.957934i	$0.592654\pi$
$968$	0	0
$969$	0	0
$970$	−4.31807	−0.138645
$971$	−33.9566	−1.08972	−0.544860	−	0.838527i	$-0.683417\pi$
$971$	−33.9566	−1.08972	−0.544860	+	0.838527i	$0.683417\pi$
$972$	0	0
$973$	−18.6496	−0.597880
$974$	10.8810	0.348649
$975$	0	0
$976$	−39.1735	−1.25391
$977$	−41.0222	−1.31242	−0.656208	−	0.754580i	$-0.727841\pi$
$977$	−41.0222	−1.31242	−0.656208	+	0.754580i	$0.727841\pi$
$978$	0	0
$979$	0	0
$980$	−0.586754	−0.0187432
$981$	0	0
$982$	64.3820	2.05451
$983$	−42.3470	−1.35066	−0.675330	−	0.737516i	$-0.735999\pi$
$983$	−42.3470	−1.35066	−0.675330	+	0.737516i	$0.735999\pi$
$984$	0	0
$985$	1.08338	0.0345192
$986$	−49.4559	−1.57500
$987$	0	0
$988$	−0.542362	−0.0172548
$989$	21.5817	0.686258
$990$	0	0
$991$	50.9687	1.61908	0.809538	−	0.587068i	$-0.199718\pi$
$991$	50.9687	1.61908	0.809538	+	0.587068i	$0.199718\pi$
$992$	12.2670	0.389477
$993$	0	0
$994$	15.4660	0.490553
$995$	0.870884	0.0276089
$996$	0	0
$997$	32.8608	1.04071	0.520356	−	0.853950i	$-0.325799\pi$
$997$	32.8608	1.04071	0.520356	+	0.853950i	$0.325799\pi$
$998$	−21.2186	−0.671662
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.cb.1.3		3
3.2	odd	2		2541.2.a.bi.1.1		3
11.10	odd	2		693.2.a.m.1.1		3
33.32	even	2		231.2.a.d.1.3	✓	3
77.76	even	2		4851.2.a.bp.1.1		3
132.131	odd	2		3696.2.a.bp.1.2		3
165.164	even	2		5775.2.a.bw.1.1		3
231.230	odd	2		1617.2.a.s.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.a.d.1.3	✓	3	33.32	even	2
693.2.a.m.1.1		3	11.10	odd	2
1617.2.a.s.1.3		3	231.230	odd	2
2541.2.a.bi.1.1		3	3.2	odd	2
3696.2.a.bp.1.2		3	132.131	odd	2
4851.2.a.bp.1.1		3	77.76	even	2
5775.2.a.bw.1.1		3	165.164	even	2
7623.2.a.cb.1.3		3	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+2.52892 q^{2} +4.39543 q^{4} -0.133492 q^{5} +1.00000 q^{7} +6.05784 q^{8} +O(q^{10})\)	\(q+2.52892 q^{2} +4.39543 q^{4} -0.133492 q^{5} +1.00000 q^{7} +6.05784 q^{8} -0.337590 q^{10} -0.133492 q^{13} +2.52892 q^{14} +6.52892 q^{16} -5.05784 q^{17} +0.924344 q^{19} -0.586754 q^{20} +7.05784 q^{23} -4.98218 q^{25} -0.337590 q^{26} +4.39543 q^{28} +3.86651 q^{29} +2.79085 q^{31} +4.39543 q^{32} -12.7909 q^{34} -0.133492 q^{35} +9.98218 q^{37} +2.33759 q^{38} -0.808672 q^{40} +11.8487 q^{41} +3.05784 q^{43} +17.8487 q^{46} +3.07566 q^{47} +1.00000 q^{49} -12.5995 q^{50} -0.586754 q^{52} +4.79085 q^{53} +6.05784 q^{56} +9.77808 q^{58} +12.6574 q^{59} -6.00000 q^{61} +7.05784 q^{62} -1.94216 q^{64} +0.0178201 q^{65} +8.92434 q^{67} -22.2313 q^{68} -0.337590 q^{70} +6.11567 q^{71} -7.86651 q^{73} +25.2441 q^{74} +4.06289 q^{76} -14.1157 q^{79} -0.871558 q^{80} +29.9644 q^{82} -1.20915 q^{83} +0.675180 q^{85} +7.73302 q^{86} -15.5817 q^{89} -0.133492 q^{91} +31.0222 q^{92} +7.77808 q^{94} -0.123392 q^{95} +12.7909 q^{97} +2.52892 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 6 q^{4} + 3 q^{7} + 3 q^{8}+O(q^{10}) \) 3 * q + 6 * q^4 + 3 * q^7 + 3 * q^8	\( 3 q + 6 q^{4} + 3 q^{7} + 3 q^{8} - 9 q^{10} + 12 q^{16} - 12 q^{19} + 21 q^{20} + 6 q^{23} + 15 q^{25} - 9 q^{26} + 6 q^{28} + 12 q^{29} - 6 q^{31} + 6 q^{32} - 24 q^{34} + 15 q^{38} - 18 q^{40} + 6 q^{41} - 6 q^{43} + 24 q^{46} + 24 q^{47} + 3 q^{49} - 39 q^{50} + 21 q^{52} + 3 q^{56} - 9 q^{58} + 24 q^{59} - 18 q^{61} + 6 q^{62} - 21 q^{64} + 30 q^{65} + 12 q^{67} - 6 q^{68} - 9 q^{70} - 12 q^{71} - 24 q^{73} + 39 q^{74} + 3 q^{76} - 12 q^{79} - 9 q^{80} + 30 q^{82} - 18 q^{83} + 18 q^{85} + 24 q^{86} - 18 q^{89} + 18 q^{92} - 15 q^{94} + 12 q^{95} + 24 q^{97}+O(q^{100}) \) 3 * q + 6 * q^4 + 3 * q^7 + 3 * q^8 - 9 * q^10 + 12 * q^16 - 12 * q^19 + 21 * q^20 + 6 * q^23 + 15 * q^25 - 9 * q^26 + 6 * q^28 + 12 * q^29 - 6 * q^31 + 6 * q^32 - 24 * q^34 + 15 * q^38 - 18 * q^40 + 6 * q^41 - 6 * q^43 + 24 * q^46 + 24 * q^47 + 3 * q^49 - 39 * q^50 + 21 * q^52 + 3 * q^56 - 9 * q^58 + 24 * q^59 - 18 * q^61 + 6 * q^62 - 21 * q^64 + 30 * q^65 + 12 * q^67 - 6 * q^68 - 9 * q^70 - 12 * q^71 - 24 * q^73 + 39 * q^74 + 3 * q^76 - 12 * q^79 - 9 * q^80 + 30 * q^82 - 18 * q^83 + 18 * q^85 + 24 * q^86 - 18 * q^89 + 18 * q^92 - 15 * q^94 + 12 * q^95 + 24 * q^97

Introduction

L-functions

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Database

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