LMFDB - Embedded newform 7623.2.a.cn.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:	$1$
Dimension:	$4$
Coefficient field:	4.4.7488.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 2x + 1 $ x^4 - 2x^3 - 4x^2 + 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2541)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.05896$ of defining polynomial
Character	$\chi$	$=$	7623.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.32691 q^{2} -0.239314 q^{4} -0.326909 q^{5} +1.00000 q^{7} -2.97136 q^{8} +O(q^{10})$	$q+1.32691 q^{2} -0.239314 q^{4} -0.326909 q^{5} +1.00000 q^{7} -2.97136 q^{8} -0.433778 q^{10} +0.0589594 q^{13} +1.32691 q^{14} -3.46410 q^{16} +6.28375 q^{17} -2.43378 q^{19} +0.0782337 q^{20} -5.93756 q^{23} -4.89313 q^{25} +0.0782337 q^{26} -0.239314 q^{28} +9.43072 q^{29} +1.98589 q^{31} +1.34618 q^{32} +8.33796 q^{34} -0.326909 q^{35} -6.53242 q^{37} -3.22940 q^{38} +0.971364 q^{40} -0.0782337 q^{41} -4.92177 q^{43} -7.87861 q^{46} -6.01621 q^{47} +1.00000 q^{49} -6.49274 q^{50} -0.0141098 q^{52} +5.94273 q^{53} -2.97136 q^{56} +12.5137 q^{58} -1.13719 q^{59} -12.2807 q^{61} +2.63509 q^{62} +8.71446 q^{64} -0.0192743 q^{65} +8.75721 q^{67} -1.50379 q^{68} -0.433778 q^{70} +1.65857 q^{71} -9.40514 q^{73} -8.66793 q^{74} +0.582436 q^{76} +13.0717 q^{79} +1.13244 q^{80} -0.103809 q^{82} -7.49443 q^{83} -2.05421 q^{85} -6.53073 q^{86} -10.6013 q^{89} +0.0589594 q^{91} +1.42094 q^{92} -7.98297 q^{94} +0.795623 q^{95} -6.71278 q^{97} +1.32691 q^{98} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 2 q^{2} + 4 q^{4} + 2 q^{5} + 4 q^{7}+O(q^{10}) $ 4 * q + 2 * q^2 + 4 * q^4 + 2 * q^5 + 4 * q^7	$ 4 q + 2 q^{2} + 4 q^{4} + 2 q^{5} + 4 q^{7} - 10 q^{10} - 10 q^{13} + 2 q^{14} + 6 q^{17} - 18 q^{19} + 2 q^{23} - 8 q^{25} + 4 q^{28} + 6 q^{29} + 12 q^{32} - 2 q^{34} + 2 q^{35} - 4 q^{37} - 8 q^{40} - 20 q^{43} - 16 q^{46} + 6 q^{47} + 4 q^{49} - 24 q^{50} - 8 q^{52} + 24 q^{58} + 6 q^{59} + 10 q^{61} - 16 q^{64} - 10 q^{65} + 4 q^{67} + 28 q^{68} - 10 q^{70} + 6 q^{71} - 34 q^{73} - 36 q^{74} - 36 q^{76} - 24 q^{79} - 12 q^{80} + 28 q^{82} + 6 q^{83} + 8 q^{85} - 38 q^{86} - 18 q^{89} - 10 q^{91} - 24 q^{92} + 6 q^{94} - 18 q^{95} - 10 q^{97} + 2 q^{98}+O(q^{100}) $ 4 * q + 2 * q^2 + 4 * q^4 + 2 * q^5 + 4 * q^7 - 10 * q^10 - 10 * q^13 + 2 * q^14 + 6 * q^17 - 18 * q^19 + 2 * q^23 - 8 * q^25 + 4 * q^28 + 6 * q^29 + 12 * q^32 - 2 * q^34 + 2 * q^35 - 4 * q^37 - 8 * q^40 - 20 * q^43 - 16 * q^46 + 6 * q^47 + 4 * q^49 - 24 * q^50 - 8 * q^52 + 24 * q^58 + 6 * q^59 + 10 * q^61 - 16 * q^64 - 10 * q^65 + 4 * q^67 + 28 * q^68 - 10 * q^70 + 6 * q^71 - 34 * q^73 - 36 * q^74 - 36 * q^76 - 24 * q^79 - 12 * q^80 + 28 * q^82 + 6 * q^83 + 8 * q^85 - 38 * q^86 - 18 * q^89 - 10 * q^91 - 24 * q^92 + 6 * q^94 - 18 * q^95 - 10 * 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 $

$2$	1.32691	0.938266	0.469133	−	0.883128i	$-0.344566\pi$
$2$	1.32691	0.938266	0.469133	+	0.883128i	$0.344566\pi$
$3$	0	0
$3$	0	0
$4$	−0.239314	−0.119657
$5$	−0.326909	−0.146198	−0.0730990	−	0.997325i	$-0.523289\pi$
$5$	−0.326909	−0.146198	−0.0730990	+	0.997325i	$0.523289\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	−2.97136	−1.05054
$9$	0	0
$10$	−0.433778	−0.137173
$11$	0	0
$11$	0	0
$12$	0	0
$13$	0.0589594	0.0163524	0.00817619	−	0.999967i	$-0.497397\pi$
$13$	0.0589594	0.0163524	0.00817619	+	0.999967i	$0.497397\pi$
$14$	1.32691	0.354631
$15$	0	0
$16$	−3.46410	−0.866025
$17$	6.28375	1.52403	0.762016	−	0.647558i	$-0.224210\pi$
$17$	6.28375	1.52403	0.762016	+	0.647558i	$0.224210\pi$
$18$	0	0
$19$	−2.43378	−0.558347	−0.279173	−	0.960241i	$-0.590060\pi$
$19$	−2.43378	−0.558347	−0.279173	+	0.960241i	$0.590060\pi$
$20$	0.0782337	0.0174936
$21$	0	0
$22$	0	0
$23$	−5.93756	−1.23807	−0.619034	−	0.785364i	$-0.712476\pi$
$23$	−5.93756	−1.23807	−0.619034	+	0.785364i	$0.712476\pi$
$24$	0	0
$25$	−4.89313	−0.978626
$26$	0.0782337	0.0153429
$27$	0	0
$28$	−0.239314	−0.0452260
$29$	9.43072	1.75124	0.875620	−	0.483000i	$-0.160453\pi$
$29$	9.43072	1.75124	0.875620	+	0.483000i	$0.160453\pi$
$30$	0	0
$31$	1.98589	0.356676	0.178338	−	0.983969i	$-0.442928\pi$
$31$	1.98589	0.356676	0.178338	+	0.983969i	$0.442928\pi$
$32$	1.34618	0.237974
$33$	0	0
$34$	8.33796	1.42995
$35$	−0.326909	−0.0552576
$36$	0	0
$37$	−6.53242	−1.07392	−0.536962	−	0.843607i	$-0.680428\pi$
$37$	−6.53242	−1.07392	−0.536962	+	0.843607i	$0.680428\pi$
$38$	−3.22940	−0.523878
$39$	0	0
$40$	0.971364	0.153586
$41$	−0.0782337	−0.0122180	−0.00610902	−	0.999981i	$-0.501945\pi$
$41$	−0.0782337	−0.0122180	−0.00610902	+	0.999981i	$0.501945\pi$
$42$	0	0
$43$	−4.92177	−0.750562	−0.375281	−	0.926911i	$-0.622454\pi$
$43$	−4.92177	−0.750562	−0.375281	+	0.926911i	$0.622454\pi$
$44$	0	0
$45$	0	0
$46$	−7.87861	−1.16164
$47$	−6.01621	−0.877555	−0.438778	−	0.898596i	$-0.644588\pi$
$47$	−6.01621	−0.877555	−0.438778	+	0.898596i	$0.644588\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	−6.49274	−0.918212
$51$	0	0
$52$	−0.0141098	−0.00195667
$53$	5.94273	0.816297	0.408148	−	0.912916i	$-0.366175\pi$
$53$	5.94273	0.816297	0.408148	+	0.912916i	$0.366175\pi$
$54$	0	0
$55$	0	0
$56$	−2.97136	−0.397065
$57$	0	0
$58$	12.5137	1.64313
$59$	−1.13719	−0.148050	−0.0740250	−	0.997256i	$-0.523584\pi$
$59$	−1.13719	−0.148050	−0.0740250	+	0.997256i	$0.523584\pi$
$60$	0	0
$61$	−12.2807	−1.57238	−0.786190	−	0.617984i	$-0.787950\pi$
$61$	−12.2807	−1.57238	−0.786190	+	0.617984i	$0.787950\pi$
$62$	2.63509	0.334657
$63$	0	0
$64$	8.71446	1.08931
$65$	−0.0192743	−0.00239069
$66$	0	0
$67$	8.75721	1.06986	0.534932	−	0.844895i	$-0.320337\pi$
$67$	8.75721	1.06986	0.534932	+	0.844895i	$0.320337\pi$
$68$	−1.50379	−0.182361
$69$	0	0
$70$	−0.433778	−0.0518464
$71$	1.65857	0.196836	0.0984178	−	0.995145i	$-0.468622\pi$
$71$	1.65857	0.196836	0.0984178	+	0.995145i	$0.468622\pi$
$72$	0	0
$73$	−9.40514	−1.10079	−0.550394	−	0.834905i	$-0.685523\pi$
$73$	−9.40514	−1.10079	−0.550394	+	0.834905i	$0.685523\pi$
$74$	−8.66793	−1.00763
$75$	0	0
$76$	0.582436	0.0668100
$77$	0	0
$78$	0	0
$79$	13.0717	1.47068	0.735340	−	0.677698i	$-0.237022\pi$
$79$	13.0717	1.47068	0.735340	+	0.677698i	$0.237022\pi$
$80$	1.13244	0.126611
$81$	0	0
$82$	−0.103809	−0.0114638
$83$	−7.49443	−0.822620	−0.411310	−	0.911496i	$-0.634929\pi$
$83$	−7.49443	−0.822620	−0.411310	+	0.911496i	$0.634929\pi$
$84$	0	0
$85$	−2.05421	−0.222810
$86$	−6.53073	−0.704227
$87$	0	0
$88$	0	0
$89$	−10.6013	−1.12373	−0.561867	−	0.827227i	$-0.689917\pi$
$89$	−10.6013	−1.12373	−0.561867	+	0.827227i	$0.689917\pi$
$90$	0	0
$91$	0.0589594	0.00618062
$92$	1.42094	0.148143
$93$	0	0
$94$	−7.98297	−0.823380
$95$	0.795623	0.0816292
$96$	0	0
$97$	−6.71278	−0.681579	−0.340790	−	0.940140i	$-0.610694\pi$
$97$	−6.71278	−0.681579	−0.340790	+	0.940140i	$0.610694\pi$
$98$	1.32691	0.134038
$99$	0	0
$100$	1.17099	0.117099
$101$	8.44830	0.840638	0.420319	−	0.907376i	$-0.361918\pi$
$101$	8.44830	0.840638	0.420319	+	0.907376i	$0.361918\pi$
$102$	0	0
$103$	−15.9631	−1.57289	−0.786447	−	0.617657i	$-0.788082\pi$
$103$	−15.9631	−1.57289	−0.786447	+	0.617657i	$0.788082\pi$
$104$	−0.175190	−0.0171788
$105$	0	0
$106$	7.88546	0.765903
$107$	−5.17044	−0.499845	−0.249923	−	0.968266i	$-0.580405\pi$
$107$	−5.17044	−0.499845	−0.249923	+	0.968266i	$0.580405\pi$
$108$	0	0
$109$	−3.02444	−0.289689	−0.144844	−	0.989454i	$-0.546268\pi$
$109$	−3.02444	−0.289689	−0.144844	+	0.989454i	$0.546268\pi$
$110$	0	0
$111$	0	0
$112$	−3.46410	−0.327327
$113$	6.23584	0.586618	0.293309	−	0.956018i	$-0.405243\pi$
$113$	6.23584	0.586618	0.293309	+	0.956018i	$0.405243\pi$
$114$	0	0
$115$	1.94104	0.181003
$116$	−2.25690	−0.209548
$117$	0	0
$118$	−1.50895	−0.138910
$119$	6.28375	0.576030
$120$	0	0
$121$	0	0
$122$	−16.2953	−1.47531
$123$	0	0
$124$	−0.475251	−0.0426788
$125$	3.23415	0.289271
$126$	0	0
$127$	−7.40039	−0.656679	−0.328339	−	0.944560i	$-0.606489\pi$
$127$	−7.40039	−0.656679	−0.328339	+	0.944560i	$0.606489\pi$
$128$	8.87093	0.784087
$129$	0	0
$130$	−0.0255753	−0.00224310
$131$	10.9479	0.956522	0.478261	−	0.878218i	$-0.341267\pi$
$131$	10.9479	0.956522	0.478261	+	0.878218i	$0.341267\pi$
$132$	0	0
$133$	−2.43378	−0.211035
$134$	11.6200	1.00382
$135$	0	0
$136$	−18.6713	−1.60105
$137$	−4.96620	−0.424291	−0.212146	−	0.977238i	$-0.568045\pi$
$137$	−4.96620	−0.424291	−0.212146	+	0.977238i	$0.568045\pi$
$138$	0	0
$139$	6.19560	0.525504	0.262752	−	0.964863i	$-0.415370\pi$
$139$	6.19560	0.525504	0.262752	+	0.964863i	$0.415370\pi$
$140$	0.0782337	0.00661195
$141$	0	0
$142$	2.20077	0.184684
$143$	0	0
$144$	0	0
$145$	−3.08298	−0.256028
$146$	−12.4798	−1.03283
$147$	0	0
$148$	1.56330	0.128502
$149$	−20.3158	−1.66433	−0.832166	−	0.554527i	$-0.812899\pi$
$149$	−20.3158	−1.66433	−0.832166	+	0.554527i	$0.812899\pi$
$150$	0	0
$151$	−9.49959	−0.773066	−0.386533	−	0.922276i	$-0.626327\pi$
$151$	−9.49959	−0.773066	−0.386533	+	0.922276i	$0.626327\pi$
$152$	7.23164	0.586564
$153$	0	0
$154$	0	0
$155$	−0.649205	−0.0521454
$156$	0	0
$157$	−17.4500	−1.39266	−0.696330	−	0.717721i	$-0.745185\pi$
$157$	−17.4500	−1.39266	−0.696330	+	0.717721i	$0.745185\pi$
$158$	17.3449	1.37989
$159$	0	0
$160$	−0.440079	−0.0347913
$161$	−5.93756	−0.467946
$162$	0	0
$163$	−7.35304	−0.575934	−0.287967	−	0.957640i	$-0.592979\pi$
$163$	−7.35304	−0.575934	−0.287967	+	0.957640i	$0.592979\pi$
$164$	0.0187224	0.00146197
$165$	0	0
$166$	−9.94442	−0.771836
$167$	−4.74141	−0.366901	−0.183451	−	0.983029i	$-0.558727\pi$
$167$	−4.74141	−0.366901	−0.183451	+	0.983029i	$0.558727\pi$
$168$	0	0
$169$	−12.9965	−0.999733
$170$	−2.72575	−0.209055
$171$	0	0
$172$	1.17785	0.0898099
$173$	−8.04791	−0.611871	−0.305936	−	0.952052i	$-0.598969\pi$
$173$	−8.04791	−0.611871	−0.305936	+	0.952052i	$0.598969\pi$
$174$	0	0
$175$	−4.89313	−0.369886
$176$	0	0
$177$	0	0
$178$	−14.0669	−1.05436
$179$	10.5393	0.787742	0.393871	−	0.919166i	$-0.371136\pi$
$179$	10.5393	0.787742	0.393871	+	0.919166i	$0.371136\pi$
$180$	0	0
$181$	−23.5389	−1.74963	−0.874815	−	0.484457i	$-0.839017\pi$
$181$	−23.5389	−1.74963	−0.874815	+	0.484457i	$0.839017\pi$
$182$	0.0782337	0.00579907
$183$	0	0
$184$	17.6427	1.30063
$185$	2.13550	0.157005
$186$	0	0
$187$	0	0
$188$	1.43976	0.105005
$189$	0	0
$190$	1.05572	0.0765899
$191$	−17.9188	−1.29656	−0.648281	−	0.761401i	$-0.724512\pi$
$191$	−17.9188	−1.29656	−0.648281	+	0.761401i	$0.724512\pi$
$192$	0	0
$193$	−11.1962	−0.805917	−0.402958	−	0.915218i	$-0.632018\pi$
$193$	−11.1962	−0.805917	−0.402958	+	0.915218i	$0.632018\pi$
$194$	−8.90724	−0.639503
$195$	0	0
$196$	−0.239314	−0.0170938
$197$	5.32512	0.379399	0.189700	−	0.981842i	$-0.439249\pi$
$197$	5.32512	0.379399	0.189700	+	0.981842i	$0.439249\pi$
$198$	0	0
$199$	9.05938	0.642202	0.321101	−	0.947045i	$-0.395947\pi$
$199$	9.05938	0.642202	0.321101	+	0.947045i	$0.395947\pi$
$200$	14.5393	1.02808
$201$	0	0
$202$	11.2101	0.788742
$203$	9.43072	0.661907
$204$	0	0
$205$	0.0255753	0.00178625
$206$	−21.1816	−1.47579
$207$	0	0
$208$	−0.204241	−0.0141616
$209$	0	0
$210$	0	0
$211$	−27.8370	−1.91638	−0.958189	−	0.286136i	$-0.907629\pi$
$211$	−27.8370	−1.91638	−0.958189	+	0.286136i	$0.907629\pi$
$212$	−1.42218	−0.0976755
$213$	0	0
$214$	−6.86070	−0.468988
$215$	1.60897	0.109731
$216$	0	0
$217$	1.98589	0.134811
$218$	−4.01315	−0.271805
$219$	0	0
$220$	0	0
$221$	0.370486	0.0249216
$222$	0	0
$223$	19.3064	1.29285	0.646426	−	0.762977i	$-0.276263\pi$
$223$	19.3064	1.29285	0.646426	+	0.762977i	$0.276263\pi$
$224$	1.34618	0.0899456
$225$	0	0
$226$	8.27439	0.550404
$227$	1.31586	0.0873366	0.0436683	−	0.999046i	$-0.486096\pi$
$227$	1.31586	0.0873366	0.0436683	+	0.999046i	$0.486096\pi$
$228$	0	0
$229$	−0.440079	−0.0290812	−0.0145406	−	0.999894i	$-0.504629\pi$
$229$	−0.440079	−0.0290812	−0.0145406	+	0.999894i	$0.504629\pi$
$230$	2.57558	0.169829
$231$	0	0
$232$	−28.0221	−1.83974
$233$	−5.29786	−0.347074	−0.173537	−	0.984827i	$-0.555520\pi$
$233$	−5.29786	−0.347074	−0.173537	+	0.984827i	$0.555520\pi$
$234$	0	0
$235$	1.96675	0.128297
$236$	0.272146	0.0177152
$237$	0	0
$238$	8.33796	0.540470
$239$	24.3293	1.57373	0.786866	−	0.617123i	$-0.211702\pi$
$239$	24.3293	1.57373	0.786866	+	0.617123i	$0.211702\pi$
$240$	0	0
$241$	26.5165	1.70808	0.854040	−	0.520208i	$-0.174145\pi$
$241$	26.5165	1.70808	0.854040	+	0.520208i	$0.174145\pi$
$242$	0	0
$243$	0	0
$244$	2.93894	0.188146
$245$	−0.326909	−0.0208854
$246$	0	0
$247$	−0.143494	−0.00913030
$248$	−5.90080	−0.374701
$249$	0	0
$250$	4.29142	0.271413
$251$	19.1828	1.21081	0.605403	−	0.795919i	$-0.293012\pi$
$251$	19.1828	1.21081	0.605403	+	0.795919i	$0.293012\pi$
$252$	0	0
$253$	0	0
$254$	−9.81965	−0.616139
$255$	0	0
$256$	−5.65801	−0.353626
$257$	−12.2417	−0.763618	−0.381809	−	0.924241i	$-0.624699\pi$
$257$	−12.2417	−0.763618	−0.381809	+	0.924241i	$0.624699\pi$
$258$	0	0
$259$	−6.53242	−0.405905
$260$	0.00461261	0.000286062 0
$261$	0	0
$262$	14.5269	0.897472
$263$	−18.2025	−1.12241	−0.561206	−	0.827676i	$-0.689662\pi$
$263$	−18.2025	−1.12241	−0.561206	+	0.827676i	$0.689662\pi$
$264$	0	0
$265$	−1.94273	−0.119341
$266$	−3.22940	−0.198007
$267$	0	0
$268$	−2.09572	−0.128016
$269$	−9.40683	−0.573545	−0.286772	−	0.957999i	$-0.592582\pi$
$269$	−9.40683	−0.573545	−0.286772	+	0.957999i	$0.592582\pi$
$270$	0	0
$271$	14.5252	0.882341	0.441170	−	0.897423i	$-0.354563\pi$
$271$	14.5252	0.882341	0.441170	+	0.897423i	$0.354563\pi$
$272$	−21.7675	−1.31985
$273$	0	0
$274$	−6.58969	−0.398098
$275$	0	0
$276$	0	0
$277$	−8.26447	−0.496564	−0.248282	−	0.968688i	$-0.579866\pi$
$277$	−8.26447	−0.496564	−0.248282	+	0.968688i	$0.579866\pi$
$278$	8.22100	0.493063
$279$	0	0
$280$	0.971364	0.0580501
$281$	−24.0080	−1.43220	−0.716098	−	0.697999i	$-0.754074\pi$
$281$	−24.0080	−1.43220	−0.716098	+	0.697999i	$0.754074\pi$
$282$	0	0
$283$	−16.0512	−0.954142	−0.477071	−	0.878865i	$-0.658302\pi$
$283$	−16.0512	−0.954142	−0.477071	+	0.878865i	$0.658302\pi$
$284$	−0.396917	−0.0235527
$285$	0	0
$286$	0	0
$287$	−0.0782337	−0.00461799
$288$	0	0
$289$	22.4855	1.32268
$290$	−4.09084	−0.240222
$291$	0	0
$292$	2.25078	0.131717
$293$	7.60088	0.444048	0.222024	−	0.975041i	$-0.428734\pi$
$293$	7.60088	0.444048	0.222024	+	0.975041i	$0.428734\pi$
$294$	0	0
$295$	0.371758	0.0216446
$296$	19.4102	1.12820
$297$	0	0
$298$	−26.9572	−1.56159
$299$	−0.350075	−0.0202454
$300$	0	0
$301$	−4.92177	−0.283686
$302$	−12.6051	−0.725341
$303$	0	0
$304$	8.43085	0.483543
$305$	4.01466	0.229879
$306$	0	0
$307$	−19.7345	−1.12631	−0.563153	−	0.826353i	$-0.690412\pi$
$307$	−19.7345	−1.12631	−0.563153	+	0.826353i	$0.690412\pi$
$308$	0	0
$309$	0	0
$310$	−0.861435	−0.0489262
$311$	30.3127	1.71888	0.859438	−	0.511240i	$-0.170814\pi$
$311$	30.3127	1.71888	0.859438	+	0.511240i	$0.170814\pi$
$312$	0	0
$313$	−19.7850	−1.11832	−0.559158	−	0.829061i	$-0.688875\pi$
$313$	−19.7850	−1.11832	−0.559158	+	0.829061i	$0.688875\pi$
$314$	−23.1545	−1.30669
$315$	0	0
$316$	−3.12824	−0.175977
$317$	18.9639	1.06512	0.532558	−	0.846393i	$-0.321231\pi$
$317$	18.9639	1.06512	0.532558	+	0.846393i	$0.321231\pi$
$318$	0	0
$319$	0	0
$320$	−2.84883	−0.159255
$321$	0	0
$322$	−7.87861	−0.439057
$323$	−15.2932	−0.850939
$324$	0	0
$325$	−0.288496	−0.0160029
$326$	−9.75681	−0.540380
$327$	0	0
$328$	0.232461	0.0128355
$329$	−6.01621	−0.331685
$330$	0	0
$331$	−3.98895	−0.219253	−0.109626	−	0.993973i	$-0.534965\pi$
$331$	−3.98895	−0.219253	−0.109626	+	0.993973i	$0.534965\pi$
$332$	1.79352	0.0984321
$333$	0	0
$334$	−6.29142	−0.344251
$335$	−2.86281	−0.156412
$336$	0	0
$337$	−27.5690	−1.50178	−0.750891	−	0.660426i	$-0.770376\pi$
$337$	−27.5690	−1.50178	−0.750891	+	0.660426i	$0.770376\pi$
$338$	−17.2452	−0.938015
$339$	0	0
$340$	0.491601	0.0266608
$341$	0	0
$342$	0	0
$343$	1.00000	0.0539949
$344$	14.6244	0.788493
$345$	0	0
$346$	−10.6788	−0.574098
$347$	−32.5280	−1.74619	−0.873097	−	0.487547i	$-0.837892\pi$
$347$	−32.5280	−1.74619	−0.873097	+	0.487547i	$0.837892\pi$
$348$	0	0
$349$	−28.7099	−1.53680	−0.768402	−	0.639968i	$-0.778948\pi$
$349$	−28.7099	−1.53680	−0.768402	+	0.639968i	$0.778948\pi$
$350$	−6.49274	−0.347051
$351$	0	0
$352$	0	0
$353$	0.421356	0.0224265	0.0112133	−	0.999937i	$-0.496431\pi$
$353$	0.421356	0.0224265	0.0112133	+	0.999937i	$0.496431\pi$
$354$	0	0
$355$	−0.542199	−0.0287770
$356$	2.53703	0.134463
$357$	0	0
$358$	13.9847	0.739112
$359$	8.17340	0.431376	0.215688	−	0.976462i	$-0.430801\pi$
$359$	8.17340	0.431376	0.215688	+	0.976462i	$0.430801\pi$
$360$	0	0
$361$	−13.0767	−0.688249
$362$	−31.2339	−1.64162
$363$	0	0
$364$	−0.0141098	−0.000739554 0
$365$	3.07462	0.160933
$366$	0	0
$367$	−10.9415	−0.571139	−0.285570	−	0.958358i	$-0.592183\pi$
$367$	−10.9415	−0.571139	−0.285570	+	0.958358i	$0.592183\pi$
$368$	20.5683	1.07220
$369$	0	0
$370$	2.83362	0.147313
$371$	5.94273	0.308531
$372$	0	0
$373$	−15.2953	−0.791963	−0.395982	−	0.918258i	$-0.629596\pi$
$373$	−15.2953	−0.791963	−0.395982	+	0.918258i	$0.629596\pi$
$374$	0	0
$375$	0	0
$376$	17.8764	0.921903
$377$	0.556029	0.0286370
$378$	0	0
$379$	32.1698	1.65245	0.826226	−	0.563339i	$-0.190484\pi$
$379$	32.1698	1.65245	0.826226	+	0.563339i	$0.190484\pi$
$380$	−0.190403	−0.00976749
$381$	0	0
$382$	−23.7767	−1.21652
$383$	20.5485	1.04998	0.524991	−	0.851108i	$-0.324069\pi$
$383$	20.5485	1.04998	0.524991	+	0.851108i	$0.324069\pi$
$384$	0	0
$385$	0	0
$386$	−14.8563	−0.756164
$387$	0	0
$388$	1.60646	0.0815556
$389$	−16.9025	−0.856992	−0.428496	−	0.903544i	$-0.640956\pi$
$389$	−16.9025	−0.856992	−0.428496	+	0.903544i	$0.640956\pi$
$390$	0	0
$391$	−37.3102	−1.88686
$392$	−2.97136	−0.150077
$393$	0	0
$394$	7.06595	0.355977
$395$	−4.27325	−0.215011
$396$	0	0
$397$	17.7606	0.891378	0.445689	−	0.895188i	$-0.352959\pi$
$397$	17.7606	0.891378	0.445689	+	0.895188i	$0.352959\pi$
$398$	12.0210	0.602556
$399$	0	0
$400$	16.9503	0.847515
$401$	−8.19963	−0.409470	−0.204735	−	0.978817i	$-0.565633\pi$
$401$	−8.19963	−0.409470	−0.204735	+	0.978817i	$0.565633\pi$
$402$	0	0
$403$	0.117087	0.00583251
$404$	−2.02179	−0.100588
$405$	0	0
$406$	12.5137	0.621044
$407$	0	0
$408$	0	0
$409$	36.5772	1.80862	0.904312	−	0.426871i	$-0.140384\pi$
$409$	36.5772	1.80862	0.904312	+	0.426871i	$0.140384\pi$
$410$	0.0339360	0.00167598
$411$	0	0
$412$	3.82020	0.188208
$413$	−1.13719	−0.0559576
$414$	0	0
$415$	2.44999	0.120265
$416$	0.0793701	0.00389144
$417$	0	0
$418$	0	0
$419$	13.8296	0.675618	0.337809	−	0.941215i	$-0.390314\pi$
$419$	13.8296	0.675618	0.337809	+	0.941215i	$0.390314\pi$
$420$	0	0
$421$	−30.8503	−1.50355	−0.751775	−	0.659419i	$-0.770802\pi$
$421$	−30.8503	−1.50355	−0.751775	+	0.659419i	$0.770802\pi$
$422$	−36.9371	−1.79807
$423$	0	0
$424$	−17.6580	−0.857549
$425$	−30.7472	−1.49146
$426$	0	0
$427$	−12.2807	−0.594304
$428$	1.23736	0.0598099
$429$	0	0
$430$	2.13495	0.102957
$431$	−3.61107	−0.173939	−0.0869696	−	0.996211i	$-0.527718\pi$
$431$	−3.61107	−0.173939	−0.0869696	+	0.996211i	$0.527718\pi$
$432$	0	0
$433$	38.0051	1.82641	0.913203	−	0.407504i	$-0.133601\pi$
$433$	38.0051	1.82641	0.913203	+	0.407504i	$0.133601\pi$
$434$	2.63509	0.126489
$435$	0	0
$436$	0.723790	0.0346632
$437$	14.4507	0.691271
$438$	0	0
$439$	25.8855	1.23545	0.617723	−	0.786396i	$-0.288055\pi$
$439$	25.8855	1.23545	0.617723	+	0.786396i	$0.288055\pi$
$440$	0	0
$441$	0	0
$442$	0.491601	0.0233831
$443$	−14.8247	−0.704342	−0.352171	−	0.935936i	$-0.614556\pi$
$443$	−14.8247	−0.704342	−0.352171	+	0.935936i	$0.614556\pi$
$444$	0	0
$445$	3.46565	0.164288
$446$	25.6178	1.21304
$447$	0	0
$448$	8.71446	0.411720
$449$	−37.8071	−1.78423	−0.892114	−	0.451810i	$-0.850779\pi$
$449$	−37.8071	−1.78423	−0.892114	+	0.451810i	$0.850779\pi$
$450$	0	0
$451$	0	0
$452$	−1.49232	−0.0701929
$453$	0	0
$454$	1.74602	0.0819450
$455$	−0.0192743	−0.000903594 0
$456$	0	0
$457$	23.5114	1.09982	0.549908	−	0.835226i	$-0.314663\pi$
$457$	23.5114	1.09982	0.549908	+	0.835226i	$0.314663\pi$
$458$	−0.583944	−0.0272859
$459$	0	0
$460$	−0.464518	−0.0216582
$461$	−40.9211	−1.90589	−0.952943	−	0.303149i	$-0.901962\pi$
$461$	−40.9211	−1.90589	−0.952943	+	0.303149i	$0.901962\pi$
$462$	0	0
$463$	0.0607472	0.00282316	0.00141158	−	0.999999i	$-0.499551\pi$
$463$	0.0607472	0.00282316	0.00141158	+	0.999999i	$0.499551\pi$
$464$	−32.6690	−1.51662
$465$	0	0
$466$	−7.02977	−0.325648
$467$	37.4362	1.73234	0.866170	−	0.499749i	$-0.166575\pi$
$467$	37.4362	1.73234	0.866170	+	0.499749i	$0.166575\pi$
$468$	0	0
$469$	8.75721	0.404370
$470$	2.60970	0.120376
$471$	0	0
$472$	3.37902	0.155532
$473$	0	0
$474$	0	0
$475$	11.9088	0.546413
$476$	−1.50379	−0.0689259
$477$	0	0
$478$	32.2828	1.47658
$479$	31.3978	1.43460	0.717300	−	0.696764i	$-0.245378\pi$
$479$	31.3978	1.43460	0.717300	+	0.696764i	$0.245378\pi$
$480$	0	0
$481$	−0.385147	−0.0175612
$482$	35.1850	1.60263
$483$	0	0
$484$	0	0
$485$	2.19446	0.0996455
$486$	0	0
$487$	−20.3881	−0.923873	−0.461937	−	0.886913i	$-0.652845\pi$
$487$	−20.3881	−0.923873	−0.461937	+	0.886913i	$0.652845\pi$
$488$	36.4904	1.65184
$489$	0	0
$490$	−0.433778	−0.0195961
$491$	11.6120	0.524043	0.262022	−	0.965062i	$-0.415611\pi$
$491$	11.6120	0.524043	0.262022	+	0.965062i	$0.415611\pi$
$492$	0	0
$493$	59.2602	2.66895
$494$	−0.190403	−0.00856665
$495$	0	0
$496$	−6.87933	−0.308891
$497$	1.65857	0.0743968
$498$	0	0
$499$	38.1606	1.70830	0.854151	−	0.520025i	$-0.174078\pi$
$499$	38.1606	1.70830	0.854151	+	0.520025i	$0.174078\pi$
$500$	−0.773976	−0.0346133
$501$	0	0
$502$	25.4538	1.13606
$503$	−32.5721	−1.45232	−0.726160	−	0.687526i	$-0.758697\pi$
$503$	−32.5721	−1.45232	−0.726160	+	0.687526i	$0.758697\pi$
$504$	0	0
$505$	−2.76182	−0.122899
$506$	0	0
$507$	0	0
$508$	1.77102	0.0785761
$509$	23.4898	1.04117	0.520584	−	0.853811i	$-0.325714\pi$
$509$	23.4898	1.04117	0.520584	+	0.853811i	$0.325714\pi$
$510$	0	0
$511$	−9.40514	−0.416059
$512$	−25.2495	−1.11588
$513$	0	0
$514$	−16.2436	−0.716477
$515$	5.21849	0.229954
$516$	0	0
$517$	0	0
$518$	−8.66793	−0.380847
$519$	0	0
$520$	0.0572710	0.00251150
$521$	34.5918	1.51549	0.757747	−	0.652548i	$-0.226300\pi$
$521$	34.5918	1.51549	0.757747	+	0.652548i	$0.226300\pi$
$522$	0	0
$523$	−8.30429	−0.363121	−0.181561	−	0.983380i	$-0.558115\pi$
$523$	−8.30429	−0.363121	−0.181561	+	0.983380i	$0.558115\pi$
$524$	−2.61998	−0.114454
$525$	0	0
$526$	−24.1530	−1.05312
$527$	12.4788	0.543586
$528$	0	0
$529$	12.2547	0.532812
$530$	−2.57782	−0.111974
$531$	0	0
$532$	0.582436	0.0252518
$533$	−0.00461261	−0.000199794 0
$534$	0	0
$535$	1.69026	0.0730764
$536$	−26.0209	−1.12393
$537$	0	0
$538$	−12.4820	−0.538137
$539$	0	0
$540$	0	0
$541$	23.3943	1.00580	0.502899	−	0.864345i	$-0.332267\pi$
$541$	23.3943	1.00580	0.502899	+	0.864345i	$0.332267\pi$
$542$	19.2736	0.827871
$543$	0	0
$544$	8.45907	0.362680
$545$	0.988715	0.0423519
$546$	0	0
$547$	19.1606	0.819247	0.409623	−	0.912255i	$-0.365660\pi$
$547$	19.1606	0.819247	0.409623	+	0.912255i	$0.365660\pi$
$548$	1.18848	0.0507693
$549$	0	0
$550$	0	0
$551$	−22.9523	−0.977800
$552$	0	0
$553$	13.0717	0.555865
$554$	−10.9662	−0.465909
$555$	0	0
$556$	−1.48269	−0.0628801
$557$	25.9067	1.09770	0.548852	−	0.835920i	$-0.315065\pi$
$557$	25.9067	1.09770	0.548852	+	0.835920i	$0.315065\pi$
$558$	0	0
$559$	−0.290184	−0.0122735
$560$	1.13244	0.0478545
$561$	0	0
$562$	−31.8564	−1.34378
$563$	−2.58381	−0.108895	−0.0544473	−	0.998517i	$-0.517340\pi$
$563$	−2.58381	−0.108895	−0.0544473	+	0.998517i	$0.517340\pi$
$564$	0	0
$565$	−2.03855	−0.0857624
$566$	−21.2984	−0.895239
$567$	0	0
$568$	−4.92820	−0.206783
$569$	25.6769	1.07643	0.538215	−	0.842807i	$-0.319099\pi$
$569$	25.6769	1.07643	0.538215	+	0.842807i	$0.319099\pi$
$570$	0	0
$571$	−1.12907	−0.0472500	−0.0236250	−	0.999721i	$-0.507521\pi$
$571$	−1.12907	−0.0472500	−0.0236250	+	0.999721i	$0.507521\pi$
$572$	0	0
$573$	0	0
$574$	−0.103809	−0.00433290
$575$	29.0533	1.21161
$576$	0	0
$577$	−33.9141	−1.41186	−0.705932	−	0.708280i	$-0.749472\pi$
$577$	−33.9141	−1.41186	−0.705932	+	0.708280i	$0.749472\pi$
$578$	29.8362	1.24102
$579$	0	0
$580$	0.737800	0.0306355
$581$	−7.49443	−0.310921
$582$	0	0
$583$	0	0
$584$	27.9461	1.15642
$585$	0	0
$586$	10.0857	0.416635
$587$	−42.7729	−1.76543	−0.882713	−	0.469912i	$-0.844286\pi$
$587$	−42.7729	−1.76543	−0.882713	+	0.469912i	$0.844286\pi$
$588$	0	0
$589$	−4.83322	−0.199149
$590$	0.493289	0.0203084
$591$	0	0
$592$	22.6290	0.930045
$593$	−30.9085	−1.26926	−0.634630	−	0.772816i	$-0.718848\pi$
$593$	−30.9085	−1.26926	−0.634630	+	0.772816i	$0.718848\pi$
$594$	0	0
$595$	−2.05421	−0.0842144
$596$	4.86184	0.199149
$597$	0	0
$598$	−0.464518	−0.0189955
$599$	−10.2428	−0.418509	−0.209255	−	0.977861i	$-0.567104\pi$
$599$	−10.2428	−0.418509	−0.209255	+	0.977861i	$0.567104\pi$
$600$	0	0
$601$	−27.5452	−1.12359	−0.561795	−	0.827276i	$-0.689889\pi$
$601$	−27.5452	−1.12359	−0.561795	+	0.827276i	$0.689889\pi$
$602$	−6.53073	−0.266173
$603$	0	0
$604$	2.27338	0.0925026
$605$	0	0
$606$	0	0
$607$	17.3982	0.706171	0.353085	−	0.935591i	$-0.385133\pi$
$607$	17.3982	0.706171	0.353085	+	0.935591i	$0.385133\pi$
$608$	−3.27631	−0.132872
$609$	0	0
$610$	5.32709	0.215688
$611$	−0.354712	−0.0143501
$612$	0	0
$613$	−0.178250	−0.00719945	−0.00359973	−	0.999994i	$-0.501146\pi$
$613$	−0.178250	−0.00719945	−0.00359973	+	0.999994i	$0.501146\pi$
$614$	−26.1858	−1.05677
$615$	0	0
$616$	0	0
$617$	3.66139	0.147402	0.0737010	−	0.997280i	$-0.476519\pi$
$617$	3.66139	0.147402	0.0737010	+	0.997280i	$0.476519\pi$
$618$	0	0
$619$	23.7370	0.954070	0.477035	−	0.878884i	$-0.341712\pi$
$619$	23.7370	0.954070	0.477035	+	0.878884i	$0.341712\pi$
$620$	0.155364	0.00623955
$621$	0	0
$622$	40.2222	1.61276
$623$	−10.6013	−0.424732
$624$	0	0
$625$	23.4084	0.936335
$626$	−26.2529	−1.04928
$627$	0	0
$628$	4.17602	0.166641
$629$	−41.0481	−1.63669
$630$	0	0
$631$	16.0214	0.637801	0.318901	−	0.947788i	$-0.396686\pi$
$631$	16.0214	0.637801	0.318901	+	0.947788i	$0.396686\pi$
$632$	−38.8408	−1.54500
$633$	0	0
$634$	25.1633	0.999363
$635$	2.41925	0.0960051
$636$	0	0
$637$	0.0589594	0.00233606
$638$	0	0
$639$	0	0
$640$	−2.89998	−0.114632
$641$	26.3387	1.04032	0.520158	−	0.854070i	$-0.325873\pi$
$641$	26.3387	1.04032	0.520158	+	0.854070i	$0.325873\pi$
$642$	0	0
$643$	32.9889	1.30095	0.650477	−	0.759526i	$-0.274569\pi$
$643$	32.9889	1.30095	0.650477	+	0.759526i	$0.274569\pi$
$644$	1.42094	0.0559929
$645$	0	0
$646$	−20.2927	−0.798407
$647$	15.1898	0.597171	0.298585	−	0.954383i	$-0.403485\pi$
$647$	15.1898	0.597171	0.298585	+	0.954383i	$0.403485\pi$
$648$	0	0
$649$	0	0
$650$	−0.382808	−0.0150150
$651$	0	0
$652$	1.75968	0.0689145
$653$	−34.1357	−1.33583	−0.667916	−	0.744236i	$-0.732814\pi$
$653$	−34.1357	−1.33583	−0.667916	+	0.744236i	$0.732814\pi$
$654$	0	0
$655$	−3.57896	−0.139842
$656$	0.271009	0.0105811
$657$	0	0
$658$	−7.98297	−0.311208
$659$	−11.4960	−0.447820	−0.223910	−	0.974610i	$-0.571882\pi$
$659$	−11.4960	−0.447820	−0.223910	+	0.974610i	$0.571882\pi$
$660$	0	0
$661$	8.84312	0.343957	0.171979	−	0.985101i	$-0.444984\pi$
$661$	8.84312	0.343957	0.171979	+	0.985101i	$0.444984\pi$
$662$	−5.29297	−0.205717
$663$	0	0
$664$	22.2687	0.864192
$665$	0.795623	0.0308529
$666$	0	0
$667$	−55.9955	−2.16815
$668$	1.13468	0.0439023
$669$	0	0
$670$	−3.79868	−0.146756
$671$	0	0
$672$	0	0
$673$	6.96022	0.268297	0.134148	−	0.990961i	$-0.457170\pi$
$673$	6.96022	0.268297	0.134148	+	0.990961i	$0.457170\pi$
$674$	−36.5816	−1.40907
$675$	0	0
$676$	3.11025	0.119625
$677$	−24.6140	−0.945993	−0.472996	−	0.881064i	$-0.656828\pi$
$677$	−24.6140	−0.945993	−0.472996	+	0.881064i	$0.656828\pi$
$678$	0	0
$679$	−6.71278	−0.257613
$680$	6.10381	0.234070
$681$	0	0
$682$	0	0
$683$	2.65760	0.101690	0.0508451	−	0.998707i	$-0.483809\pi$
$683$	2.65760	0.101690	0.0508451	+	0.998707i	$0.483809\pi$
$684$	0	0
$685$	1.62349	0.0620305
$686$	1.32691	0.0506616
$687$	0	0
$688$	17.0495	0.650006
$689$	0.350380	0.0133484
$690$	0	0
$691$	6.06065	0.230558	0.115279	−	0.993333i	$-0.463224\pi$
$691$	6.06065	0.230558	0.115279	+	0.993333i	$0.463224\pi$
$692$	1.92597	0.0732146
$693$	0	0
$694$	−43.1617	−1.63839
$695$	−2.02539	−0.0768276
$696$	0	0
$697$	−0.491601	−0.0186207
$698$	−38.0953	−1.44193
$699$	0	0
$700$	1.17099	0.0442594
$701$	−29.2016	−1.10293	−0.551465	−	0.834198i	$-0.685931\pi$
$701$	−29.2016	−1.10293	−0.551465	+	0.834198i	$0.685931\pi$
$702$	0	0
$703$	15.8985	0.599622
$704$	0	0
$705$	0	0
$706$	0.559101	0.0210421
$707$	8.44830	0.317731
$708$	0	0
$709$	5.34783	0.200842	0.100421	−	0.994945i	$-0.467981\pi$
$709$	5.34783	0.200842	0.100421	+	0.994945i	$0.467981\pi$
$710$	−0.719449	−0.0270004
$711$	0	0
$712$	31.5003	1.18052
$713$	−11.7914	−0.441590
$714$	0	0
$715$	0	0
$716$	−2.52219	−0.0942588
$717$	0	0
$718$	10.8454	0.404745
$719$	−39.3239	−1.46653	−0.733267	−	0.679941i	$-0.762005\pi$
$719$	−39.3239	−1.46653	−0.733267	+	0.679941i	$0.762005\pi$
$720$	0	0
$721$	−15.9631	−0.594498
$722$	−17.3516	−0.645760
$723$	0	0
$724$	5.63317	0.209355
$725$	−46.1457	−1.71381
$726$	0	0
$727$	−3.52389	−0.130694	−0.0653470	−	0.997863i	$-0.520815\pi$
$727$	−3.52389	−0.130694	−0.0653470	+	0.997863i	$0.520815\pi$
$728$	−0.175190	−0.00649296
$729$	0	0
$730$	4.07974	0.150998
$731$	−30.9271	−1.14388
$732$	0	0
$733$	−3.58578	−0.132444	−0.0662218	−	0.997805i	$-0.521095\pi$
$733$	−3.58578	−0.132444	−0.0662218	+	0.997805i	$0.521095\pi$
$734$	−14.5183	−0.535881
$735$	0	0
$736$	−7.99305	−0.294628
$737$	0	0
$738$	0	0
$739$	20.4213	0.751208	0.375604	−	0.926780i	$-0.377435\pi$
$739$	20.4213	0.751208	0.375604	+	0.926780i	$0.377435\pi$
$740$	−0.511055	−0.0187868
$741$	0	0
$742$	7.88546	0.289484
$743$	26.7582	0.981662	0.490831	−	0.871255i	$-0.336693\pi$
$743$	26.7582	0.981662	0.490831	+	0.871255i	$0.336693\pi$
$744$	0	0
$745$	6.64140	0.243322
$746$	−20.2955	−0.743072
$747$	0	0
$748$	0	0
$749$	−5.17044	−0.188924
$750$	0	0
$751$	40.7545	1.48715	0.743576	−	0.668652i	$-0.233128\pi$
$751$	40.7545	1.48715	0.743576	+	0.668652i	$0.233128\pi$
$752$	20.8408	0.759985
$753$	0	0
$754$	0.737800	0.0268691
$755$	3.10550	0.113021
$756$	0	0
$757$	10.4923	0.381350	0.190675	−	0.981653i	$-0.438932\pi$
$757$	10.4923	0.381350	0.190675	+	0.981653i	$0.438932\pi$
$758$	42.6864	1.55044
$759$	0	0
$760$	−2.36409	−0.0857544
$761$	2.38294	0.0863816	0.0431908	−	0.999067i	$-0.486248\pi$
$761$	2.38294	0.0863816	0.0431908	+	0.999067i	$0.486248\pi$
$762$	0	0
$763$	−3.02444	−0.109492
$764$	4.28822	0.155142
$765$	0	0
$766$	27.2660	0.985162
$767$	−0.0670482	−0.00242097
$768$	0	0
$769$	−23.0495	−0.831186	−0.415593	−	0.909551i	$-0.636426\pi$
$769$	−23.0495	−0.831186	−0.415593	+	0.909551i	$0.636426\pi$
$770$	0	0
$771$	0	0
$772$	2.67939	0.0964334
$773$	30.2708	1.08876	0.544382	−	0.838837i	$-0.316764\pi$
$773$	30.2708	1.08876	0.544382	+	0.838837i	$0.316764\pi$
$774$	0	0
$775$	−9.71722	−0.349053
$776$	19.9461	0.716023
$777$	0	0
$778$	−22.4281	−0.804087
$779$	0.190403	0.00682191
$780$	0	0
$781$	0	0
$782$	−49.5072	−1.77037
$783$	0	0
$784$	−3.46410	−0.123718
$785$	5.70455	0.203604
$786$	0	0
$787$	7.41144	0.264189	0.132095	−	0.991237i	$-0.457830\pi$
$787$	7.41144	0.264189	0.132095	+	0.991237i	$0.457830\pi$
$788$	−1.27437	−0.0453977
$789$	0	0
$790$	−5.67021	−0.201737
$791$	6.23584	0.221721
$792$	0	0
$793$	−0.724062	−0.0257122
$794$	23.5667	0.836350
$795$	0	0
$796$	−2.16803	−0.0768439
$797$	33.5597	1.18875	0.594373	−	0.804190i	$-0.297400\pi$
$797$	33.5597	1.18875	0.594373	+	0.804190i	$0.297400\pi$
$798$	0	0
$799$	−37.8044	−1.33742
$800$	−6.58705	−0.232887
$801$	0	0
$802$	−10.8802	−0.384192
$803$	0	0
$804$	0	0
$805$	1.94104	0.0684127
$806$	0.155364	0.00547245
$807$	0	0
$808$	−25.1030	−0.883120
$809$	42.2328	1.48483	0.742413	−	0.669942i	$-0.233681\pi$
$809$	42.2328	1.48483	0.742413	+	0.669942i	$0.233681\pi$
$810$	0	0
$811$	10.1590	0.356730	0.178365	−	0.983964i	$-0.442919\pi$
$811$	10.1590	0.356730	0.178365	+	0.983964i	$0.442919\pi$
$812$	−2.25690	−0.0792017
$813$	0	0
$814$	0	0
$815$	2.40377	0.0842004
$816$	0	0
$817$	11.9785	0.419074
$818$	48.5346	1.69697
$819$	0	0
$820$	−0.00612051	−0.000213737 0
$821$	−25.6507	−0.895214	−0.447607	−	0.894230i	$-0.647724\pi$
$821$	−25.6507	−0.895214	−0.447607	+	0.894230i	$0.647724\pi$
$822$	0	0
$823$	−47.8879	−1.66927	−0.834634	−	0.550805i	$-0.814321\pi$
$823$	−47.8879	−1.66927	−0.834634	+	0.550805i	$0.814321\pi$
$824$	47.4323	1.65238
$825$	0	0
$826$	−1.50895	−0.0525031
$827$	−13.3901	−0.465619	−0.232810	−	0.972522i	$-0.574792\pi$
$827$	−13.3901	−0.465619	−0.232810	+	0.972522i	$0.574792\pi$
$828$	0	0
$829$	−19.7867	−0.687221	−0.343610	−	0.939112i	$-0.611650\pi$
$829$	−19.7867	−0.687221	−0.343610	+	0.939112i	$0.611650\pi$
$830$	3.25092	0.112841
$831$	0	0
$832$	0.513799	0.0178128
$833$	6.28375	0.217719
$834$	0	0
$835$	1.55001	0.0536402
$836$	0	0
$837$	0	0
$838$	18.3506	0.633910
$839$	−46.1948	−1.59482	−0.797410	−	0.603437i	$-0.793797\pi$
$839$	−46.1948	−1.59482	−0.797410	+	0.603437i	$0.793797\pi$
$840$	0	0
$841$	59.9384	2.06684
$842$	−40.9355	−1.41073
$843$	0	0
$844$	6.66177	0.229308
$845$	4.24867	0.146159
$846$	0	0
$847$	0	0
$848$	−20.5862	−0.706934
$849$	0	0
$850$	−40.7987	−1.39938
$851$	38.7867	1.32959
$852$	0	0
$853$	37.6343	1.28857	0.644287	−	0.764784i	$-0.277154\pi$
$853$	37.6343	1.28857	0.644287	+	0.764784i	$0.277154\pi$
$854$	−16.2953	−0.557615
$855$	0	0
$856$	15.3633	0.525106
$857$	−12.9962	−0.443943	−0.221972	−	0.975053i	$-0.571249\pi$
$857$	−12.9962	−0.443943	−0.221972	+	0.975053i	$0.571249\pi$
$858$	0	0
$859$	−26.0560	−0.889020	−0.444510	−	0.895774i	$-0.646622\pi$
$859$	−26.0560	−0.889020	−0.444510	+	0.895774i	$0.646622\pi$
$860$	−0.385048	−0.0131300
$861$	0	0
$862$	−4.79156	−0.163201
$863$	19.3134	0.657434	0.328717	−	0.944428i	$-0.393384\pi$
$863$	19.3134	0.657434	0.328717	+	0.944428i	$0.393384\pi$
$864$	0	0
$865$	2.63093	0.0894543
$866$	50.4292	1.71366
$867$	0	0
$868$	−0.475251	−0.0161311
$869$	0	0
$870$	0	0
$871$	0.516320	0.0174948
$872$	8.98671	0.304328
$873$	0	0
$874$	19.1748	0.648596
$875$	3.23415	0.109334
$876$	0	0
$877$	22.9946	0.776472	0.388236	−	0.921560i	$-0.373084\pi$
$877$	22.9946	0.776472	0.388236	+	0.921560i	$0.373084\pi$
$878$	34.3476	1.15918
$879$	0	0
$880$	0	0
$881$	50.4759	1.70058	0.850288	−	0.526318i	$-0.176428\pi$
$881$	50.4759	1.70058	0.850288	+	0.526318i	$0.176428\pi$
$882$	0	0
$883$	14.8564	0.499958	0.249979	−	0.968251i	$-0.419576\pi$
$883$	14.8564	0.499958	0.249979	+	0.968251i	$0.419576\pi$
$884$	−0.0886623	−0.00298204
$885$	0	0
$886$	−19.6710	−0.660860
$887$	−46.0622	−1.54662	−0.773309	−	0.634029i	$-0.781400\pi$
$887$	−46.0622	−1.54662	−0.773309	+	0.634029i	$0.781400\pi$
$888$	0	0
$889$	−7.40039	−0.248201
$890$	4.59861	0.154146
$891$	0	0
$892$	−4.62029	−0.154699
$893$	14.6421	0.489980
$894$	0	0
$895$	−3.44538	−0.115166
$896$	8.87093	0.296357
$897$	0	0
$898$	−50.1666	−1.67408
$899$	18.7284	0.624626
$900$	0	0
$901$	37.3426	1.24406
$902$	0	0
$903$	0	0
$904$	−18.5289	−0.616264
$905$	7.69505	0.255792
$906$	0	0
$907$	28.5358	0.947516	0.473758	−	0.880655i	$-0.342897\pi$
$907$	28.5358	0.947516	0.473758	+	0.880655i	$0.342897\pi$
$908$	−0.314903	−0.0104504
$909$	0	0
$910$	−0.0255753	−0.000847812 0
$911$	−31.5458	−1.04516	−0.522580	−	0.852590i	$-0.675030\pi$
$911$	−31.5458	−1.04516	−0.522580	+	0.852590i	$0.675030\pi$
$912$	0	0
$913$	0	0
$914$	31.1974	1.03192
$915$	0	0
$916$	0.105317	0.00347977
$917$	10.9479	0.361531
$918$	0	0
$919$	40.0921	1.32252	0.661259	−	0.750158i	$-0.270022\pi$
$919$	40.0921	1.32252	0.661259	+	0.750158i	$0.270022\pi$
$920$	−5.76754	−0.190150
$921$	0	0
$922$	−54.2986	−1.78823
$923$	0.0977880	0.00321873
$924$	0	0
$925$	31.9640	1.05097
$926$	0.0806060	0.00264888
$927$	0	0
$928$	12.6955	0.416749
$929$	−4.40597	−0.144555	−0.0722777	−	0.997385i	$-0.523027\pi$
$929$	−4.40597	−0.144555	−0.0722777	+	0.997385i	$0.523027\pi$
$930$	0	0
$931$	−2.43378	−0.0797638
$932$	1.26785	0.0415298
$933$	0	0
$934$	49.6744	1.62540
$935$	0	0
$936$	0	0
$937$	−0.902908	−0.0294967	−0.0147484	−	0.999891i	$-0.504695\pi$
$937$	−0.902908	−0.0294967	−0.0147484	+	0.999891i	$0.504695\pi$
$938$	11.6200	0.379407
$939$	0	0
$940$	−0.470671	−0.0153516
$941$	34.8442	1.13589	0.567943	−	0.823068i	$-0.307739\pi$
$941$	34.8442	1.13589	0.567943	+	0.823068i	$0.307739\pi$
$942$	0	0
$943$	0.464518	0.0151268
$944$	3.93935	0.128215
$945$	0	0
$946$	0	0
$947$	−25.9556	−0.843444	−0.421722	−	0.906725i	$-0.638574\pi$
$947$	−25.9556	−0.843444	−0.421722	+	0.906725i	$0.638574\pi$
$948$	0	0
$949$	−0.554521	−0.0180005
$950$	15.8019	0.512681
$951$	0	0
$952$	−18.6713	−0.605140
$953$	1.69616	0.0549440	0.0274720	−	0.999623i	$-0.491254\pi$
$953$	1.69616	0.0549440	0.0274720	+	0.999623i	$0.491254\pi$
$954$	0	0
$955$	5.85782	0.189555
$956$	−5.82234	−0.188308
$957$	0	0
$958$	41.6620	1.34604
$959$	−4.96620	−0.160367
$960$	0	0
$961$	−27.0562	−0.872782
$962$	−0.511055	−0.0164771
$963$	0	0
$964$	−6.34577	−0.204383
$965$	3.66012	0.117823
$966$	0	0
$967$	−25.0337	−0.805028	−0.402514	−	0.915414i	$-0.631864\pi$
$967$	−25.0337	−0.805028	−0.402514	+	0.915414i	$0.631864\pi$
$968$	0	0
$969$	0	0
$970$	2.91185	0.0934940
$971$	57.3356	1.83999	0.919994	−	0.391933i	$-0.128193\pi$
$971$	57.3356	1.83999	0.919994	+	0.391933i	$0.128193\pi$
$972$	0	0
$973$	6.19560	0.198622
$974$	−27.0532	−0.866839
$975$	0	0
$976$	42.5415	1.36172
$977$	−19.6196	−0.627687	−0.313844	−	0.949475i	$-0.601617\pi$
$977$	−19.6196	−0.627687	−0.313844	+	0.949475i	$0.601617\pi$
$978$	0	0
$979$	0	0
$980$	0.0782337	0.00249908
$981$	0	0
$982$	15.4081	0.491692
$983$	−32.4742	−1.03577	−0.517883	−	0.855451i	$-0.673280\pi$
$983$	−32.4742	−1.03577	−0.517883	+	0.855451i	$0.673280\pi$
$984$	0	0
$985$	−1.74083	−0.0554674
$986$	78.6329	2.50418
$987$	0	0
$988$	0.0343401	0.00109250
$989$	29.2233	0.929247
$990$	0	0
$991$	21.7384	0.690543	0.345271	−	0.938503i	$-0.387787\pi$
$991$	21.7384	0.690543	0.345271	+	0.938503i	$0.387787\pi$
$992$	2.67337	0.0848796
$993$	0	0
$994$	2.20077	0.0698040
$995$	−2.96159	−0.0938886
$996$	0	0
$997$	−7.14889	−0.226408	−0.113204	−	0.993572i	$-0.536111\pi$
$997$	−7.14889	−0.226408	−0.113204	+	0.993572i	$0.536111\pi$
$998$	50.6356	1.60284
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	7623.2.a.cn.1.3		4
3.2	odd	2		2541.2.a.bl.1.2	✓	4
11.10	odd	2		7623.2.a.cg.1.2		4
33.32	even	2		2541.2.a.bp.1.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2541.2.a.bl.1.2	✓	4	3.2	odd	2
2541.2.a.bp.1.3	yes	4	33.32	even	2
7623.2.a.cg.1.2		4	11.10	odd	2
7623.2.a.cn.1.3		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)

\(f(q)\)	\(=\)	\(q+1.32691 q^{2} -0.239314 q^{4} -0.326909 q^{5} +1.00000 q^{7} -2.97136 q^{8} +O(q^{10})\)	\(q+1.32691 q^{2} -0.239314 q^{4} -0.326909 q^{5} +1.00000 q^{7} -2.97136 q^{8} -0.433778 q^{10} +0.0589594 q^{13} +1.32691 q^{14} -3.46410 q^{16} +6.28375 q^{17} -2.43378 q^{19} +0.0782337 q^{20} -5.93756 q^{23} -4.89313 q^{25} +0.0782337 q^{26} -0.239314 q^{28} +9.43072 q^{29} +1.98589 q^{31} +1.34618 q^{32} +8.33796 q^{34} -0.326909 q^{35} -6.53242 q^{37} -3.22940 q^{38} +0.971364 q^{40} -0.0782337 q^{41} -4.92177 q^{43} -7.87861 q^{46} -6.01621 q^{47} +1.00000 q^{49} -6.49274 q^{50} -0.0141098 q^{52} +5.94273 q^{53} -2.97136 q^{56} +12.5137 q^{58} -1.13719 q^{59} -12.2807 q^{61} +2.63509 q^{62} +8.71446 q^{64} -0.0192743 q^{65} +8.75721 q^{67} -1.50379 q^{68} -0.433778 q^{70} +1.65857 q^{71} -9.40514 q^{73} -8.66793 q^{74} +0.582436 q^{76} +13.0717 q^{79} +1.13244 q^{80} -0.103809 q^{82} -7.49443 q^{83} -2.05421 q^{85} -6.53073 q^{86} -10.6013 q^{89} +0.0589594 q^{91} +1.42094 q^{92} -7.98297 q^{94} +0.795623 q^{95} -6.71278 q^{97} +1.32691 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 2 q^{2} + 4 q^{4} + 2 q^{5} + 4 q^{7}+O(q^{10}) \) 4 * q + 2 * q^2 + 4 * q^4 + 2 * q^5 + 4 * q^7	\( 4 q + 2 q^{2} + 4 q^{4} + 2 q^{5} + 4 q^{7} - 10 q^{10} - 10 q^{13} + 2 q^{14} + 6 q^{17} - 18 q^{19} + 2 q^{23} - 8 q^{25} + 4 q^{28} + 6 q^{29} + 12 q^{32} - 2 q^{34} + 2 q^{35} - 4 q^{37} - 8 q^{40} - 20 q^{43} - 16 q^{46} + 6 q^{47} + 4 q^{49} - 24 q^{50} - 8 q^{52} + 24 q^{58} + 6 q^{59} + 10 q^{61} - 16 q^{64} - 10 q^{65} + 4 q^{67} + 28 q^{68} - 10 q^{70} + 6 q^{71} - 34 q^{73} - 36 q^{74} - 36 q^{76} - 24 q^{79} - 12 q^{80} + 28 q^{82} + 6 q^{83} + 8 q^{85} - 38 q^{86} - 18 q^{89} - 10 q^{91} - 24 q^{92} + 6 q^{94} - 18 q^{95} - 10 q^{97} + 2 q^{98}+O(q^{100}) \) 4 * q + 2 * q^2 + 4 * q^4 + 2 * q^5 + 4 * q^7 - 10 * q^10 - 10 * q^13 + 2 * q^14 + 6 * q^17 - 18 * q^19 + 2 * q^23 - 8 * q^25 + 4 * q^28 + 6 * q^29 + 12 * q^32 - 2 * q^34 + 2 * q^35 - 4 * q^37 - 8 * q^40 - 20 * q^43 - 16 * q^46 + 6 * q^47 + 4 * q^49 - 24 * q^50 - 8 * q^52 + 24 * q^58 + 6 * q^59 + 10 * q^61 - 16 * q^64 - 10 * q^65 + 4 * q^67 + 28 * q^68 - 10 * q^70 + 6 * q^71 - 34 * q^73 - 36 * q^74 - 36 * q^76 - 24 * q^79 - 12 * q^80 + 28 * q^82 + 6 * q^83 + 8 * q^85 - 38 * q^86 - 18 * q^89 - 10 * q^91 - 24 * q^92 + 6 * q^94 - 18 * q^95 - 10 * q^97 + 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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