LMFDB - Embedded newform 9882.2.a.bk.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [9882,2,Mod(1,9882)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(9882, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("9882.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 9882 = 2 \cdot 3^{4} \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	9882.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:	$78.9081672774$
Analytic rank:	$0$
Dimension:	$17$
Coefficient field:	$\mathbb{Q}[x]/(x^{17} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{17} - x^{16} - 49 x^{15} + 46 x^{14} + 980 x^{13} - 821 x^{12} - 10276 x^{11} + 7017 x^{10} + \cdots + 3888 $ x^17 - x^16 - 49x^15 + 46x^14 + 980x^13 - 821x^12 - 10276x^11 + 7017x^10 + 60475x^9 - 27723x^8 - 199170x^7 + 32292x^6 + 345105x^5 + 52515x^4 - 250992x^3 - 101736x^2 + 19440*x + 3888
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{3} $
Twist minimal:	no (minimal twist has level 1098)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-1.18694$ of defining polynomial
Character	$\chi$	$=$	9882.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -4.04279 q^{5} -2.62852 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -4.04279 q^{5} -2.62852 q^{7} -1.00000 q^{8} +4.04279 q^{10} +4.21434 q^{11} +6.12870 q^{13} +2.62852 q^{14} +1.00000 q^{16} -5.44390 q^{17} -4.44023 q^{19} -4.04279 q^{20} -4.21434 q^{22} +1.64031 q^{23} +11.3441 q^{25} -6.12870 q^{26} -2.62852 q^{28} +7.57561 q^{29} +0.216906 q^{31} -1.00000 q^{32} +5.44390 q^{34} +10.6265 q^{35} +8.34941 q^{37} +4.44023 q^{38} +4.04279 q^{40} -8.10654 q^{41} +4.21626 q^{43} +4.21434 q^{44} -1.64031 q^{46} +6.34729 q^{47} -0.0908960 q^{49} -11.3441 q^{50} +6.12870 q^{52} -5.78041 q^{53} -17.0377 q^{55} +2.62852 q^{56} -7.57561 q^{58} -3.87515 q^{59} -1.00000 q^{61} -0.216906 q^{62} +1.00000 q^{64} -24.7770 q^{65} +6.56062 q^{67} -5.44390 q^{68} -10.6265 q^{70} -14.7541 q^{71} -7.90683 q^{73} -8.34941 q^{74} -4.44023 q^{76} -11.0775 q^{77} +11.7270 q^{79} -4.04279 q^{80} +8.10654 q^{82} -11.5452 q^{83} +22.0085 q^{85} -4.21626 q^{86} -4.21434 q^{88} +13.5586 q^{89} -16.1094 q^{91} +1.64031 q^{92} -6.34729 q^{94} +17.9509 q^{95} -3.98544 q^{97} +0.0908960 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 17 q - 17 q^{2} + 17 q^{4} - 5 q^{5} + 11 q^{7} - 17 q^{8}+O(q^{10}) $ 17 * q - 17 * q^2 + 17 * q^4 - 5 * q^5 + 11 * q^7 - 17 * q^8	$ 17 q - 17 q^{2} + 17 q^{4} - 5 q^{5} + 11 q^{7} - 17 q^{8} + 5 q^{10} - 3 q^{11} + 10 q^{13} - 11 q^{14} + 17 q^{16} - 8 q^{17} + 11 q^{19} - 5 q^{20} + 3 q^{22} - 5 q^{23} + 32 q^{25} - 10 q^{26} + 11 q^{28} - 6 q^{29} + 20 q^{31} - 17 q^{32} + 8 q^{34} + 8 q^{35} + 29 q^{37} - 11 q^{38} + 5 q^{40} - 20 q^{41} + 30 q^{43} - 3 q^{44} + 5 q^{46} - 10 q^{47} + 42 q^{49} - 32 q^{50} + 10 q^{52} - 2 q^{53} + 15 q^{55} - 11 q^{56} + 6 q^{58} + 3 q^{59} - 17 q^{61} - 20 q^{62} + 17 q^{64} - 28 q^{65} + 11 q^{67} - 8 q^{68} - 8 q^{70} - 63 q^{71} + 28 q^{73} - 29 q^{74} + 11 q^{76} - 17 q^{77} + 17 q^{79} - 5 q^{80} + 20 q^{82} + 20 q^{83} + 33 q^{85} - 30 q^{86} + 3 q^{88} - 2 q^{89} + 30 q^{91} - 5 q^{92} + 10 q^{94} - q^{95} + 32 q^{97} - 42 q^{98}+O(q^{100}) $ 17 * q - 17 * q^2 + 17 * q^4 - 5 * q^5 + 11 * q^7 - 17 * q^8 + 5 * q^10 - 3 * q^11 + 10 * q^13 - 11 * q^14 + 17 * q^16 - 8 * q^17 + 11 * q^19 - 5 * q^20 + 3 * q^22 - 5 * q^23 + 32 * q^25 - 10 * q^26 + 11 * q^28 - 6 * q^29 + 20 * q^31 - 17 * q^32 + 8 * q^34 + 8 * q^35 + 29 * q^37 - 11 * q^38 + 5 * q^40 - 20 * q^41 + 30 * q^43 - 3 * q^44 + 5 * q^46 - 10 * q^47 + 42 * q^49 - 32 * q^50 + 10 * q^52 - 2 * q^53 + 15 * q^55 - 11 * q^56 + 6 * q^58 + 3 * q^59 - 17 * q^61 - 20 * q^62 + 17 * q^64 - 28 * q^65 + 11 * q^67 - 8 * q^68 - 8 * q^70 - 63 * q^71 + 28 * q^73 - 29 * q^74 + 11 * q^76 - 17 * q^77 + 17 * q^79 - 5 * q^80 + 20 * q^82 + 20 * q^83 + 33 * q^85 - 30 * q^86 + 3 * q^88 - 2 * q^89 + 30 * q^91 - 5 * q^92 + 10 * q^94 - q^95 + 32 * q^97 - 42 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−4.04279	−1.80799	−0.903994	−	0.427544i	$-0.859379\pi$
$5$	−4.04279	−1.80799	−0.903994	+	0.427544i	$0.859379\pi$
$6$	0	0
$7$	−2.62852	−0.993486	−0.496743	−	0.867898i	$-0.665471\pi$
$7$	−2.62852	−0.993486	−0.496743	+	0.867898i	$0.665471\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	4.04279	1.27844
$11$	4.21434	1.27067	0.635336	−	0.772236i	$-0.280861\pi$
$11$	4.21434	1.27067	0.635336	+	0.772236i	$0.280861\pi$
$12$	0	0
$13$	6.12870	1.69979	0.849897	−	0.526949i	$-0.176664\pi$
$13$	6.12870	1.69979	0.849897	+	0.526949i	$0.176664\pi$
$14$	2.62852	0.702501
$15$	0	0
$16$	1.00000	0.250000
$17$	−5.44390	−1.32034	−0.660169	−	0.751117i	$-0.729516\pi$
$17$	−5.44390	−1.32034	−0.660169	+	0.751117i	$0.729516\pi$
$18$	0	0
$19$	−4.44023	−1.01866	−0.509329	−	0.860572i	$-0.670106\pi$
$19$	−4.44023	−1.01866	−0.509329	+	0.860572i	$0.670106\pi$
$20$	−4.04279	−0.903994
$21$	0	0
$22$	−4.21434	−0.898501
$23$	1.64031	0.342027	0.171014	−	0.985269i	$-0.445296\pi$
$23$	1.64031	0.342027	0.171014	+	0.985269i	$0.445296\pi$
$24$	0	0
$25$	11.3441	2.26882
$26$	−6.12870	−1.20194
$27$	0	0
$28$	−2.62852	−0.496743
$29$	7.57561	1.40676	0.703378	−	0.710816i	$-0.251674\pi$
$29$	7.57561	1.40676	0.703378	+	0.710816i	$0.251674\pi$
$30$	0	0
$31$	0.216906	0.0389574	0.0194787	−	0.999810i	$-0.493799\pi$
$31$	0.216906	0.0389574	0.0194787	+	0.999810i	$0.493799\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	5.44390	0.933621
$35$	10.6265	1.79621
$36$	0	0
$37$	8.34941	1.37263	0.686317	−	0.727303i	$-0.259226\pi$
$37$	8.34941	1.37263	0.686317	+	0.727303i	$0.259226\pi$
$38$	4.44023	0.720300
$39$	0	0
$40$	4.04279	0.639221
$41$	−8.10654	−1.26603	−0.633015	−	0.774140i	$-0.718183\pi$
$41$	−8.10654	−1.26603	−0.633015	+	0.774140i	$0.718183\pi$
$42$	0	0
$43$	4.21626	0.642973	0.321487	−	0.946914i	$-0.395817\pi$
$43$	4.21626	0.642973	0.321487	+	0.946914i	$0.395817\pi$
$44$	4.21434	0.635336
$45$	0	0
$46$	−1.64031	−0.241850
$47$	6.34729	0.925848	0.462924	−	0.886398i	$-0.346800\pi$
$47$	6.34729	0.925848	0.462924	+	0.886398i	$0.346800\pi$
$48$	0	0
$49$	−0.0908960	−0.0129851
$50$	−11.3441	−1.60430
$51$	0	0
$52$	6.12870	0.849897
$53$	−5.78041	−0.794000	−0.397000	−	0.917819i	$-0.629949\pi$
$53$	−5.78041	−0.794000	−0.397000	+	0.917819i	$0.629949\pi$
$54$	0	0
$55$	−17.0377	−2.29736
$56$	2.62852	0.351250
$57$	0	0
$58$	−7.57561	−0.994726
$59$	−3.87515	−0.504501	−0.252251	−	0.967662i	$-0.581171\pi$
$59$	−3.87515	−0.504501	−0.252251	+	0.967662i	$0.581171\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−0.216906	−0.0275471
$63$	0	0
$64$	1.00000	0.125000
$65$	−24.7770	−3.07321
$66$	0	0
$67$	6.56062	0.801508	0.400754	−	0.916186i	$-0.368748\pi$
$67$	6.56062	0.801508	0.400754	+	0.916186i	$0.368748\pi$
$68$	−5.44390	−0.660169
$69$	0	0
$70$	−10.6265	−1.27011
$71$	−14.7541	−1.75099	−0.875494	−	0.483229i	$-0.839464\pi$
$71$	−14.7541	−1.75099	−0.875494	+	0.483229i	$0.839464\pi$
$72$	0	0
$73$	−7.90683	−0.925425	−0.462713	−	0.886508i	$-0.653124\pi$
$73$	−7.90683	−0.925425	−0.462713	+	0.886508i	$0.653124\pi$
$74$	−8.34941	−0.970599
$75$	0	0
$76$	−4.44023	−0.509329
$77$	−11.0775	−1.26240
$78$	0	0
$79$	11.7270	1.31939	0.659695	−	0.751533i	$-0.270685\pi$
$79$	11.7270	1.31939	0.659695	+	0.751533i	$0.270685\pi$
$80$	−4.04279	−0.451997
$81$	0	0
$82$	8.10654	0.895218
$83$	−11.5452	−1.26724	−0.633622	−	0.773642i	$-0.718433\pi$
$83$	−11.5452	−1.26724	−0.633622	+	0.773642i	$0.718433\pi$
$84$	0	0
$85$	22.0085	2.38716
$86$	−4.21626	−0.454651
$87$	0	0
$88$	−4.21434	−0.449251
$89$	13.5586	1.43721	0.718603	−	0.695421i	$-0.244782\pi$
$89$	13.5586	1.43721	0.718603	+	0.695421i	$0.244782\pi$
$90$	0	0
$91$	−16.1094	−1.68872
$92$	1.64031	0.171014
$93$	0	0
$94$	−6.34729	−0.654673
$95$	17.9509	1.84172
$96$	0	0
$97$	−3.98544	−0.404660	−0.202330	−	0.979317i	$-0.564851\pi$
$97$	−3.98544	−0.404660	−0.202330	+	0.979317i	$0.564851\pi$
$98$	0.0908960	0.00918188
$99$	0	0
$100$	11.3441	1.13441
$101$	−2.84004	−0.282595	−0.141297	−	0.989967i	$-0.545127\pi$
$101$	−2.84004	−0.282595	−0.141297	+	0.989967i	$0.545127\pi$
$102$	0	0
$103$	−3.95419	−0.389618	−0.194809	−	0.980841i	$-0.562409\pi$
$103$	−3.95419	−0.389618	−0.194809	+	0.980841i	$0.562409\pi$
$104$	−6.12870	−0.600968
$105$	0	0
$106$	5.78041	0.561443
$107$	−17.1906	−1.66188	−0.830938	−	0.556366i	$-0.812195\pi$
$107$	−17.1906	−1.66188	−0.830938	+	0.556366i	$0.812195\pi$
$108$	0	0
$109$	−2.56099	−0.245299	−0.122649	−	0.992450i	$-0.539139\pi$
$109$	−2.56099	−0.245299	−0.122649	+	0.992450i	$0.539139\pi$
$110$	17.0377	1.62448
$111$	0	0
$112$	−2.62852	−0.248372
$113$	5.71813	0.537917	0.268958	−	0.963152i	$-0.413321\pi$
$113$	5.71813	0.537917	0.268958	+	0.963152i	$0.413321\pi$
$114$	0	0
$115$	−6.63140	−0.618382
$116$	7.57561	0.703378
$117$	0	0
$118$	3.87515	0.356736
$119$	14.3094	1.31174
$120$	0	0
$121$	6.76070	0.614609
$122$	1.00000	0.0905357
$123$	0	0
$124$	0.216906	0.0194787
$125$	−25.6479	−2.29402
$126$	0	0
$127$	−0.859611	−0.0762781	−0.0381391	−	0.999272i	$-0.512143\pi$
$127$	−0.859611	−0.0762781	−0.0381391	+	0.999272i	$0.512143\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	24.7770	2.17309
$131$	5.85031	0.511144	0.255572	−	0.966790i	$-0.417736\pi$
$131$	5.85031	0.511144	0.255572	+	0.966790i	$0.417736\pi$
$132$	0	0
$133$	11.6712	1.01202
$134$	−6.56062	−0.566751
$135$	0	0
$136$	5.44390	0.466810
$137$	9.80269	0.837500	0.418750	−	0.908101i	$-0.362468\pi$
$137$	9.80269	0.837500	0.418750	+	0.908101i	$0.362468\pi$
$138$	0	0
$139$	−2.96901	−0.251828	−0.125914	−	0.992041i	$-0.540186\pi$
$139$	−2.96901	−0.251828	−0.125914	+	0.992041i	$0.540186\pi$
$140$	10.6265	0.898106
$141$	0	0
$142$	14.7541	1.23814
$143$	25.8284	2.15988
$144$	0	0
$145$	−30.6266	−2.54340
$146$	7.90683	0.654374
$147$	0	0
$148$	8.34941	0.686317
$149$	−20.4599	−1.67614	−0.838070	−	0.545562i	$-0.816316\pi$
$149$	−20.4599	−1.67614	−0.838070	+	0.545562i	$0.816316\pi$
$150$	0	0
$151$	−0.665480	−0.0541560	−0.0270780	−	0.999633i	$-0.508620\pi$
$151$	−0.665480	−0.0541560	−0.0270780	+	0.999633i	$0.508620\pi$
$152$	4.44023	0.360150
$153$	0	0
$154$	11.0775	0.892649
$155$	−0.876904	−0.0704346
$156$	0	0
$157$	−8.20803	−0.655072	−0.327536	−	0.944839i	$-0.606218\pi$
$157$	−8.20803	−0.655072	−0.327536	+	0.944839i	$0.606218\pi$
$158$	−11.7270	−0.932950
$159$	0	0
$160$	4.04279	0.319610
$161$	−4.31157	−0.339799
$162$	0	0
$163$	12.3611	0.968195	0.484097	−	0.875014i	$-0.339148\pi$
$163$	12.3611	0.968195	0.484097	+	0.875014i	$0.339148\pi$
$164$	−8.10654	−0.633015
$165$	0	0
$166$	11.5452	0.896077
$167$	25.4108	1.96634	0.983172	−	0.182681i	$-0.0584774\pi$
$167$	25.4108	1.96634	0.983172	+	0.182681i	$0.0584774\pi$
$168$	0	0
$169$	24.5609	1.88930
$170$	−22.0085	−1.68798
$171$	0	0
$172$	4.21626	0.321487
$173$	−16.7503	−1.27350	−0.636752	−	0.771069i	$-0.719722\pi$
$173$	−16.7503	−1.27350	−0.636752	+	0.771069i	$0.719722\pi$
$174$	0	0
$175$	−29.8182	−2.25405
$176$	4.21434	0.317668
$177$	0	0
$178$	−13.5586	−1.01626
$179$	10.9587	0.819088	0.409544	−	0.912290i	$-0.365688\pi$
$179$	10.9587	0.819088	0.409544	+	0.912290i	$0.365688\pi$
$180$	0	0
$181$	−6.70682	−0.498514	−0.249257	−	0.968437i	$-0.580186\pi$
$181$	−6.70682	−0.498514	−0.249257	+	0.968437i	$0.580186\pi$
$182$	16.1094	1.19411
$183$	0	0
$184$	−1.64031	−0.120925
$185$	−33.7549	−2.48171
$186$	0	0
$187$	−22.9425	−1.67772
$188$	6.34729	0.462924
$189$	0	0
$190$	−17.9509	−1.30229
$191$	−20.4119	−1.47695	−0.738475	−	0.674280i	$-0.764454\pi$
$191$	−20.4119	−1.47695	−0.738475	+	0.674280i	$0.764454\pi$
$192$	0	0
$193$	13.8831	0.999326	0.499663	−	0.866220i	$-0.333457\pi$
$193$	13.8831	0.999326	0.499663	+	0.866220i	$0.333457\pi$
$194$	3.98544	0.286138
$195$	0	0
$196$	−0.0908960	−0.00649257
$197$	19.8807	1.41644	0.708221	−	0.705991i	$-0.249498\pi$
$197$	19.8807	1.41644	0.708221	+	0.705991i	$0.249498\pi$
$198$	0	0
$199$	15.2050	1.07785	0.538926	−	0.842353i	$-0.318830\pi$
$199$	15.2050	1.07785	0.538926	+	0.842353i	$0.318830\pi$
$200$	−11.3441	−0.802150
$201$	0	0
$202$	2.84004	0.199825
$203$	−19.9126	−1.39759
$204$	0	0
$205$	32.7730	2.28897
$206$	3.95419	0.275502
$207$	0	0
$208$	6.12870	0.424949
$209$	−18.7127	−1.29438
$210$	0	0
$211$	7.37516	0.507727	0.253864	−	0.967240i	$-0.418299\pi$
$211$	7.37516	0.507727	0.253864	+	0.967240i	$0.418299\pi$
$212$	−5.78041	−0.397000
$213$	0	0
$214$	17.1906	1.17512
$215$	−17.0454	−1.16249
$216$	0	0
$217$	−0.570141	−0.0387037
$218$	2.56099	0.173452
$219$	0	0
$220$	−17.0377	−1.14868
$221$	−33.3640	−2.24430
$222$	0	0
$223$	4.00942	0.268491	0.134245	−	0.990948i	$-0.457139\pi$
$223$	4.00942	0.268491	0.134245	+	0.990948i	$0.457139\pi$
$224$	2.62852	0.175625
$225$	0	0
$226$	−5.71813	−0.380365
$227$	−6.48523	−0.430440	−0.215220	−	0.976566i	$-0.569047\pi$
$227$	−6.48523	−0.430440	−0.215220	+	0.976566i	$0.569047\pi$
$228$	0	0
$229$	−9.62511	−0.636045	−0.318023	−	0.948083i	$-0.603019\pi$
$229$	−9.62511	−0.636045	−0.318023	+	0.948083i	$0.603019\pi$
$230$	6.63140	0.437262
$231$	0	0
$232$	−7.57561	−0.497363
$233$	−27.2350	−1.78422	−0.892112	−	0.451813i	$-0.850777\pi$
$233$	−27.2350	−1.78422	−0.892112	+	0.451813i	$0.850777\pi$
$234$	0	0
$235$	−25.6608	−1.67392
$236$	−3.87515	−0.252251
$237$	0	0
$238$	−14.3094	−0.927539
$239$	1.68728	0.109141	0.0545706	−	0.998510i	$-0.482621\pi$
$239$	1.68728	0.109141	0.0545706	+	0.998510i	$0.482621\pi$
$240$	0	0
$241$	15.4374	0.994413	0.497206	−	0.867632i	$-0.334359\pi$
$241$	15.4374	0.994413	0.497206	+	0.867632i	$0.334359\pi$
$242$	−6.76070	−0.434594
$243$	0	0
$244$	−1.00000	−0.0640184
$245$	0.367473	0.0234770
$246$	0	0
$247$	−27.2128	−1.73151
$248$	−0.216906	−0.0137735
$249$	0	0
$250$	25.6479	1.62212
$251$	−14.3588	−0.906318	−0.453159	−	0.891430i	$-0.649703\pi$
$251$	−14.3588	−0.906318	−0.453159	+	0.891430i	$0.649703\pi$
$252$	0	0
$253$	6.91281	0.434605
$254$	0.859611	0.0539368
$255$	0	0
$256$	1.00000	0.0625000
$257$	10.3987	0.648654	0.324327	−	0.945945i	$-0.394862\pi$
$257$	10.3987	0.648654	0.324327	+	0.945945i	$0.394862\pi$
$258$	0	0
$259$	−21.9466	−1.36369
$260$	−24.7770	−1.53660
$261$	0	0
$262$	−5.85031	−0.361433
$263$	7.75780	0.478367	0.239183	−	0.970974i	$-0.423120\pi$
$263$	7.75780	0.478367	0.239183	+	0.970974i	$0.423120\pi$
$264$	0	0
$265$	23.3690	1.43554
$266$	−11.6712	−0.715608
$267$	0	0
$268$	6.56062	0.400754
$269$	−9.58542	−0.584433	−0.292217	−	0.956352i	$-0.594393\pi$
$269$	−9.58542	−0.584433	−0.292217	+	0.956352i	$0.594393\pi$
$270$	0	0
$271$	1.47114	0.0893653	0.0446827	−	0.999001i	$-0.485772\pi$
$271$	1.47114	0.0893653	0.0446827	+	0.999001i	$0.485772\pi$
$272$	−5.44390	−0.330085
$273$	0	0
$274$	−9.80269	−0.592202
$275$	47.8080	2.88293
$276$	0	0
$277$	−21.0115	−1.26246	−0.631229	−	0.775596i	$-0.717449\pi$
$277$	−21.0115	−1.26246	−0.631229	+	0.775596i	$0.717449\pi$
$278$	2.96901	0.178070
$279$	0	0
$280$	−10.6265	−0.635057
$281$	30.8848	1.84243	0.921217	−	0.389050i	$-0.127197\pi$
$281$	30.8848	1.84243	0.921217	+	0.389050i	$0.127197\pi$
$282$	0	0
$283$	−12.4742	−0.741511	−0.370756	−	0.928730i	$-0.620901\pi$
$283$	−12.4742	−0.741511	−0.370756	+	0.928730i	$0.620901\pi$
$284$	−14.7541	−0.875494
$285$	0	0
$286$	−25.8284	−1.52727
$287$	21.3082	1.25778
$288$	0	0
$289$	12.6360	0.743295
$290$	30.6266	1.79845
$291$	0	0
$292$	−7.90683	−0.462713
$293$	6.61766	0.386608	0.193304	−	0.981139i	$-0.438080\pi$
$293$	6.61766	0.386608	0.193304	+	0.981139i	$0.438080\pi$
$294$	0	0
$295$	15.6664	0.912133
$296$	−8.34941	−0.485299
$297$	0	0
$298$	20.4599	1.18521
$299$	10.0529	0.581376
$300$	0	0
$301$	−11.0825	−0.638785
$302$	0.665480	0.0382941
$303$	0	0
$304$	−4.44023	−0.254665
$305$	4.04279	0.231489
$306$	0	0
$307$	7.81948	0.446282	0.223141	−	0.974786i	$-0.428369\pi$
$307$	7.81948	0.446282	0.223141	+	0.974786i	$0.428369\pi$
$308$	−11.0775	−0.631198
$309$	0	0
$310$	0.876904	0.0498048
$311$	−20.3382	−1.15328	−0.576638	−	0.817000i	$-0.695636\pi$
$311$	−20.3382	−1.15328	−0.576638	+	0.817000i	$0.695636\pi$
$312$	0	0
$313$	−27.1325	−1.53362	−0.766809	−	0.641875i	$-0.778157\pi$
$313$	−27.1325	−1.53362	−0.766809	+	0.641875i	$0.778157\pi$
$314$	8.20803	0.463206
$315$	0	0
$316$	11.7270	0.659695
$317$	−8.58918	−0.482416	−0.241208	−	0.970473i	$-0.577544\pi$
$317$	−8.58918	−0.482416	−0.241208	+	0.970473i	$0.577544\pi$
$318$	0	0
$319$	31.9262	1.78753
$320$	−4.04279	−0.225999
$321$	0	0
$322$	4.31157	0.240274
$323$	24.1721	1.34497
$324$	0	0
$325$	69.5247	3.85653
$326$	−12.3611	−0.684617
$327$	0	0
$328$	8.10654	0.447609
$329$	−16.6840	−0.919817
$330$	0	0
$331$	−3.43123	−0.188598	−0.0942988	−	0.995544i	$-0.530061\pi$
$331$	−3.43123	−0.188598	−0.0942988	+	0.995544i	$0.530061\pi$
$332$	−11.5452	−0.633622
$333$	0	0
$334$	−25.4108	−1.39042
$335$	−26.5232	−1.44912
$336$	0	0
$337$	9.14365	0.498086	0.249043	−	0.968492i	$-0.419884\pi$
$337$	9.14365	0.498086	0.249043	+	0.968492i	$0.419884\pi$
$338$	−24.5609	−1.33594
$339$	0	0
$340$	22.0085	1.19358
$341$	0.914116	0.0495021
$342$	0	0
$343$	18.6385	1.00639
$344$	−4.21626	−0.227325
$345$	0	0
$346$	16.7503	0.900503
$347$	36.7802	1.97447	0.987233	−	0.159283i	$-0.0509184\pi$
$347$	36.7802	1.97447	0.987233	+	0.159283i	$0.0509184\pi$
$348$	0	0
$349$	17.0096	0.910502	0.455251	−	0.890363i	$-0.349550\pi$
$349$	17.0096	0.910502	0.455251	+	0.890363i	$0.349550\pi$
$350$	29.8182	1.59385
$351$	0	0
$352$	−4.21434	−0.224625
$353$	26.2902	1.39929	0.699644	−	0.714492i	$-0.253342\pi$
$353$	26.2902	1.39929	0.699644	+	0.714492i	$0.253342\pi$
$354$	0	0
$355$	59.6476	3.16577
$356$	13.5586	0.718603
$357$	0	0
$358$	−10.9587	−0.579183
$359$	17.1316	0.904173	0.452087	−	0.891974i	$-0.350680\pi$
$359$	17.1316	0.904173	0.452087	+	0.891974i	$0.350680\pi$
$360$	0	0
$361$	0.715635	0.0376650
$362$	6.70682	0.352502
$363$	0	0
$364$	−16.1094	−0.844361
$365$	31.9656	1.67316
$366$	0	0
$367$	−7.12297	−0.371816	−0.185908	−	0.982567i	$-0.559523\pi$
$367$	−7.12297	−0.371816	−0.185908	+	0.982567i	$0.559523\pi$
$368$	1.64031	0.0855068
$369$	0	0
$370$	33.7549	1.75483
$371$	15.1939	0.788828
$372$	0	0
$373$	−13.0150	−0.673889	−0.336944	−	0.941525i	$-0.609393\pi$
$373$	−13.0150	−0.673889	−0.336944	+	0.941525i	$0.609393\pi$
$374$	22.9425	1.18633
$375$	0	0
$376$	−6.34729	−0.327337
$377$	46.4286	2.39119
$378$	0	0
$379$	4.71603	0.242246	0.121123	−	0.992638i	$-0.461350\pi$
$379$	4.71603	0.242246	0.121123	+	0.992638i	$0.461350\pi$
$380$	17.9509	0.920862
$381$	0	0
$382$	20.4119	1.04436
$383$	4.26041	0.217697	0.108848	−	0.994058i	$-0.465284\pi$
$383$	4.26041	0.217697	0.108848	+	0.994058i	$0.465284\pi$
$384$	0	0
$385$	44.7839	2.28240
$386$	−13.8831	−0.706630
$387$	0	0
$388$	−3.98544	−0.202330
$389$	26.5071	1.34396	0.671982	−	0.740568i	$-0.265443\pi$
$389$	26.5071	1.34396	0.671982	+	0.740568i	$0.265443\pi$
$390$	0	0
$391$	−8.92965	−0.451592
$392$	0.0908960	0.00459094
$393$	0	0
$394$	−19.8807	−1.00158
$395$	−47.4098	−2.38544
$396$	0	0
$397$	8.28900	0.416013	0.208006	−	0.978127i	$-0.433303\pi$
$397$	8.28900	0.416013	0.208006	+	0.978127i	$0.433303\pi$
$398$	−15.2050	−0.762156
$399$	0	0
$400$	11.3441	0.567206
$401$	−21.8269	−1.08998	−0.544992	−	0.838441i	$-0.683467\pi$
$401$	−21.8269	−1.08998	−0.544992	+	0.838441i	$0.683467\pi$
$402$	0	0
$403$	1.32935	0.0662196
$404$	−2.84004	−0.141297
$405$	0	0
$406$	19.9126	0.988247
$407$	35.1873	1.74417
$408$	0	0
$409$	−10.0393	−0.496414	−0.248207	−	0.968707i	$-0.579841\pi$
$409$	−10.0393	−0.496414	−0.248207	+	0.968707i	$0.579841\pi$
$410$	−32.7730	−1.61854
$411$	0	0
$412$	−3.95419	−0.194809
$413$	10.1859	0.501215
$414$	0	0
$415$	46.6746	2.29116
$416$	−6.12870	−0.300484
$417$	0	0
$418$	18.7127	0.915266
$419$	−0.889810	−0.0434701	−0.0217350	−	0.999764i	$-0.506919\pi$
$419$	−0.889810	−0.0434701	−0.0217350	+	0.999764i	$0.506919\pi$
$420$	0	0
$421$	20.1983	0.984403	0.492202	−	0.870481i	$-0.336192\pi$
$421$	20.1983	0.984403	0.492202	+	0.870481i	$0.336192\pi$
$422$	−7.37516	−0.359017
$423$	0	0
$424$	5.78041	0.280722
$425$	−61.7562	−2.99562
$426$	0	0
$427$	2.62852	0.127203
$428$	−17.1906	−0.830938
$429$	0	0
$430$	17.0454	0.822004
$431$	12.4423	0.599322	0.299661	−	0.954046i	$-0.403126\pi$
$431$	12.4423	0.599322	0.299661	+	0.954046i	$0.403126\pi$
$432$	0	0
$433$	4.03198	0.193765	0.0968823	−	0.995296i	$-0.469113\pi$
$433$	4.03198	0.193765	0.0968823	+	0.995296i	$0.469113\pi$
$434$	0.570141	0.0273676
$435$	0	0
$436$	−2.56099	−0.122649
$437$	−7.28333	−0.348409
$438$	0	0
$439$	−15.1413	−0.722656	−0.361328	−	0.932439i	$-0.617677\pi$
$439$	−15.1413	−0.722656	−0.361328	+	0.932439i	$0.617677\pi$
$440$	17.0377	0.812240
$441$	0	0
$442$	33.3640	1.58696
$443$	1.03565	0.0492050	0.0246025	−	0.999697i	$-0.492168\pi$
$443$	1.03565	0.0492050	0.0246025	+	0.999697i	$0.492168\pi$
$444$	0	0
$445$	−54.8144	−2.59845
$446$	−4.00942	−0.189852
$447$	0	0
$448$	−2.62852	−0.124186
$449$	22.6531	1.06907	0.534534	−	0.845147i	$-0.320487\pi$
$449$	22.6531	1.06907	0.534534	+	0.845147i	$0.320487\pi$
$450$	0	0
$451$	−34.1638	−1.60871
$452$	5.71813	0.268958
$453$	0	0
$454$	6.48523	0.304367
$455$	65.1268	3.05319
$456$	0	0
$457$	25.9301	1.21296	0.606479	−	0.795100i	$-0.292582\pi$
$457$	25.9301	1.21296	0.606479	+	0.795100i	$0.292582\pi$
$458$	9.62511	0.449752
$459$	0	0
$460$	−6.63140	−0.309191
$461$	12.5283	0.583501	0.291750	−	0.956494i	$-0.405762\pi$
$461$	12.5283	0.583501	0.291750	+	0.956494i	$0.405762\pi$
$462$	0	0
$463$	−9.38177	−0.436008	−0.218004	−	0.975948i	$-0.569955\pi$
$463$	−9.38177	−0.436008	−0.218004	+	0.975948i	$0.569955\pi$
$464$	7.57561	0.351689
$465$	0	0
$466$	27.2350	1.26164
$467$	−13.0532	−0.604031	−0.302015	−	0.953303i	$-0.597659\pi$
$467$	−13.0532	−0.604031	−0.302015	+	0.953303i	$0.597659\pi$
$468$	0	0
$469$	−17.2447	−0.796287
$470$	25.6608	1.18364
$471$	0	0
$472$	3.87515	0.178368
$473$	17.7688	0.817008
$474$	0	0
$475$	−50.3705	−2.31116
$476$	14.3094	0.655869
$477$	0	0
$478$	−1.68728	−0.0771744
$479$	41.1150	1.87859	0.939295	−	0.343110i	$-0.111480\pi$
$479$	41.1150	1.87859	0.939295	+	0.343110i	$0.111480\pi$
$480$	0	0
$481$	51.1710	2.33320
$482$	−15.4374	−0.703156
$483$	0	0
$484$	6.76070	0.307304
$485$	16.1123	0.731621
$486$	0	0
$487$	6.86990	0.311305	0.155652	−	0.987812i	$-0.450252\pi$
$487$	6.86990	0.311305	0.155652	+	0.987812i	$0.450252\pi$
$488$	1.00000	0.0452679
$489$	0	0
$490$	−0.367473	−0.0166007
$491$	20.2204	0.912535	0.456267	−	0.889843i	$-0.349186\pi$
$491$	20.2204	0.912535	0.456267	+	0.889843i	$0.349186\pi$
$492$	0	0
$493$	−41.2408	−1.85739
$494$	27.2128	1.22436
$495$	0	0
$496$	0.216906	0.00973936
$497$	38.7814	1.73958
$498$	0	0
$499$	11.2610	0.504111	0.252055	−	0.967713i	$-0.418894\pi$
$499$	11.2610	0.504111	0.252055	+	0.967713i	$0.418894\pi$
$500$	−25.6479	−1.14701
$501$	0	0
$502$	14.3588	0.640863
$503$	15.6779	0.699041	0.349520	−	0.936929i	$-0.386345\pi$
$503$	15.6779	0.699041	0.349520	+	0.936929i	$0.386345\pi$
$504$	0	0
$505$	11.4817	0.510928
$506$	−6.91281	−0.307312
$507$	0	0
$508$	−0.859611	−0.0381391
$509$	−22.7855	−1.00995	−0.504975	−	0.863134i	$-0.668498\pi$
$509$	−22.7855	−1.00995	−0.504975	+	0.863134i	$0.668498\pi$
$510$	0	0
$511$	20.7833	0.919397
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−10.3987	−0.458668
$515$	15.9860	0.704425
$516$	0	0
$517$	26.7497	1.17645
$518$	21.9466	0.964276
$519$	0	0
$520$	24.7770	1.08654
$521$	29.8673	1.30851	0.654254	−	0.756275i	$-0.272983\pi$
$521$	29.8673	1.30851	0.654254	+	0.756275i	$0.272983\pi$
$522$	0	0
$523$	−19.4600	−0.850927	−0.425464	−	0.904976i	$-0.639889\pi$
$523$	−19.4600	−0.850927	−0.425464	+	0.904976i	$0.639889\pi$
$524$	5.85031	0.255572
$525$	0	0
$526$	−7.75780	−0.338256
$527$	−1.18081	−0.0514370
$528$	0	0
$529$	−20.3094	−0.883017
$530$	−23.3690	−1.01508
$531$	0	0
$532$	11.6712	0.506012
$533$	−49.6825	−2.15199
$534$	0	0
$535$	69.4978	3.00465
$536$	−6.56062	−0.283376
$537$	0	0
$538$	9.58542	0.413257
$539$	−0.383067	−0.0164999
$540$	0	0
$541$	−22.9150	−0.985193	−0.492596	−	0.870258i	$-0.663952\pi$
$541$	−22.9150	−0.985193	−0.492596	+	0.870258i	$0.663952\pi$
$542$	−1.47114	−0.0631908
$543$	0	0
$544$	5.44390	0.233405
$545$	10.3536	0.443497
$546$	0	0
$547$	4.70369	0.201115	0.100558	−	0.994931i	$-0.467937\pi$
$547$	4.70369	0.201115	0.100558	+	0.994931i	$0.467937\pi$
$548$	9.80269	0.418750
$549$	0	0
$550$	−47.8080	−2.03854
$551$	−33.6374	−1.43300
$552$	0	0
$553$	−30.8246	−1.31080
$554$	21.0115	0.892693
$555$	0	0
$556$	−2.96901	−0.125914
$557$	−27.2033	−1.15264	−0.576320	−	0.817224i	$-0.695512\pi$
$557$	−27.2033	−1.15264	−0.576320	+	0.817224i	$0.695512\pi$
$558$	0	0
$559$	25.8402	1.09292
$560$	10.6265	0.449053
$561$	0	0
$562$	−30.8848	−1.30280
$563$	17.1837	0.724208	0.362104	−	0.932138i	$-0.382059\pi$
$563$	17.1837	0.724208	0.362104	+	0.932138i	$0.382059\pi$
$564$	0	0
$565$	−23.1172	−0.972548
$566$	12.4742	0.524328
$567$	0	0
$568$	14.7541	0.619068
$569$	4.82747	0.202378	0.101189	−	0.994867i	$-0.467735\pi$
$569$	4.82747	0.202378	0.101189	+	0.994867i	$0.467735\pi$
$570$	0	0
$571$	34.0832	1.42634	0.713169	−	0.700992i	$-0.247259\pi$
$571$	34.0832	1.42634	0.713169	+	0.700992i	$0.247259\pi$
$572$	25.8284	1.07994
$573$	0	0
$574$	−21.3082	−0.889387
$575$	18.6078	0.776000
$576$	0	0
$577$	46.1916	1.92298	0.961490	−	0.274839i	$-0.0886247\pi$
$577$	46.1916	1.92298	0.961490	+	0.274839i	$0.0886247\pi$
$578$	−12.6360	−0.525589
$579$	0	0
$580$	−30.6266	−1.27170
$581$	30.3466	1.25899
$582$	0	0
$583$	−24.3606	−1.00891
$584$	7.90683	0.327187
$585$	0	0
$586$	−6.61766	−0.273373
$587$	11.1741	0.461206	0.230603	−	0.973048i	$-0.425930\pi$
$587$	11.1741	0.461206	0.230603	+	0.973048i	$0.425930\pi$
$588$	0	0
$589$	−0.963111	−0.0396843
$590$	−15.6664	−0.644975
$591$	0	0
$592$	8.34941	0.343158
$593$	29.7071	1.21992	0.609962	−	0.792431i	$-0.291185\pi$
$593$	29.7071	1.21992	0.609962	+	0.792431i	$0.291185\pi$
$594$	0	0
$595$	−57.8498	−2.37161
$596$	−20.4599	−0.838070
$597$	0	0
$598$	−10.0529	−0.411095
$599$	−40.2135	−1.64308	−0.821539	−	0.570152i	$-0.806884\pi$
$599$	−40.2135	−1.64308	−0.821539	+	0.570152i	$0.806884\pi$
$600$	0	0
$601$	17.5947	0.717704	0.358852	−	0.933394i	$-0.383168\pi$
$601$	17.5947	0.717704	0.358852	+	0.933394i	$0.383168\pi$
$602$	11.0825	0.451689
$603$	0	0
$604$	−0.665480	−0.0270780
$605$	−27.3320	−1.11121
$606$	0	0
$607$	−0.841142	−0.0341409	−0.0170705	−	0.999854i	$-0.505434\pi$
$607$	−0.841142	−0.0341409	−0.0170705	+	0.999854i	$0.505434\pi$
$608$	4.44023	0.180075
$609$	0	0
$610$	−4.04279	−0.163688
$611$	38.9006	1.57375
$612$	0	0
$613$	24.2957	0.981294	0.490647	−	0.871358i	$-0.336760\pi$
$613$	24.2957	0.981294	0.490647	+	0.871358i	$0.336760\pi$
$614$	−7.81948	−0.315569
$615$	0	0
$616$	11.0775	0.446324
$617$	−4.35711	−0.175411	−0.0877054	−	0.996146i	$-0.527953\pi$
$617$	−4.35711	−0.175411	−0.0877054	+	0.996146i	$0.527953\pi$
$618$	0	0
$619$	−24.6418	−0.990439	−0.495220	−	0.868768i	$-0.664912\pi$
$619$	−24.6418	−0.990439	−0.495220	+	0.868768i	$0.664912\pi$
$620$	−0.876904	−0.0352173
$621$	0	0
$622$	20.3382	0.815489
$623$	−35.6389	−1.42784
$624$	0	0
$625$	46.9685	1.87874
$626$	27.1325	1.08443
$627$	0	0
$628$	−8.20803	−0.327536
$629$	−45.4533	−1.81234
$630$	0	0
$631$	7.91509	0.315095	0.157547	−	0.987511i	$-0.449641\pi$
$631$	7.91509	0.315095	0.157547	+	0.987511i	$0.449641\pi$
$632$	−11.7270	−0.466475
$633$	0	0
$634$	8.58918	0.341120
$635$	3.47522	0.137910
$636$	0	0
$637$	−0.557074	−0.0220721
$638$	−31.9262	−1.26397
$639$	0	0
$640$	4.04279	0.159805
$641$	3.94286	0.155733	0.0778667	−	0.996964i	$-0.475189\pi$
$641$	3.94286	0.155733	0.0778667	+	0.996964i	$0.475189\pi$
$642$	0	0
$643$	3.83005	0.151042	0.0755211	−	0.997144i	$-0.475938\pi$
$643$	3.83005	0.151042	0.0755211	+	0.997144i	$0.475938\pi$
$644$	−4.31157	−0.169900
$645$	0	0
$646$	−24.1721	−0.951040
$647$	−33.0472	−1.29922	−0.649610	−	0.760268i	$-0.725068\pi$
$647$	−33.0472	−1.29922	−0.649610	+	0.760268i	$0.725068\pi$
$648$	0	0
$649$	−16.3312	−0.641056
$650$	−69.5247	−2.72698
$651$	0	0
$652$	12.3611	0.484097
$653$	45.5129	1.78106	0.890528	−	0.454928i	$-0.150335\pi$
$653$	45.5129	1.78106	0.890528	+	0.454928i	$0.150335\pi$
$654$	0	0
$655$	−23.6515	−0.924142
$656$	−8.10654	−0.316507
$657$	0	0
$658$	16.6840	0.650409
$659$	−31.3395	−1.22081	−0.610407	−	0.792088i	$-0.708994\pi$
$659$	−31.3395	−1.22081	−0.610407	+	0.792088i	$0.708994\pi$
$660$	0	0
$661$	−11.0984	−0.431678	−0.215839	−	0.976429i	$-0.569249\pi$
$661$	−11.0984	−0.431678	−0.215839	+	0.976429i	$0.569249\pi$
$662$	3.43123	0.133359
$663$	0	0
$664$	11.5452	0.448039
$665$	−47.1842	−1.82973
$666$	0	0
$667$	12.4263	0.481149
$668$	25.4108	0.983172
$669$	0	0
$670$	26.5232	1.02468
$671$	−4.21434	−0.162693
$672$	0	0
$673$	38.6166	1.48856	0.744281	−	0.667867i	$-0.232793\pi$
$673$	38.6166	1.48856	0.744281	+	0.667867i	$0.232793\pi$
$674$	−9.14365	−0.352200
$675$	0	0
$676$	24.5609	0.944650
$677$	30.8307	1.18492	0.592461	−	0.805599i	$-0.298157\pi$
$677$	30.8307	1.18492	0.592461	+	0.805599i	$0.298157\pi$
$678$	0	0
$679$	10.4758	0.402024
$680$	−22.0085	−0.843988
$681$	0	0
$682$	−0.914116	−0.0350033
$683$	20.2866	0.776245	0.388122	−	0.921608i	$-0.373124\pi$
$683$	20.2866	0.776245	0.388122	+	0.921608i	$0.373124\pi$
$684$	0	0
$685$	−39.6302	−1.51419
$686$	−18.6385	−0.711623
$687$	0	0
$688$	4.21626	0.160743
$689$	−35.4264	−1.34964
$690$	0	0
$691$	22.1783	0.843701	0.421850	−	0.906665i	$-0.361381\pi$
$691$	22.1783	0.843701	0.421850	+	0.906665i	$0.361381\pi$
$692$	−16.7503	−0.636752
$693$	0	0
$694$	−36.7802	−1.39616
$695$	12.0031	0.455303
$696$	0	0
$697$	44.1312	1.67159
$698$	−17.0096	−0.643822
$699$	0	0
$700$	−29.8182	−1.12702
$701$	−14.5406	−0.549189	−0.274595	−	0.961560i	$-0.588544\pi$
$701$	−14.5406	−0.549189	−0.274595	+	0.961560i	$0.588544\pi$
$702$	0	0
$703$	−37.0733	−1.39825
$704$	4.21434	0.158834
$705$	0	0
$706$	−26.2902	−0.989446
$707$	7.46510	0.280754
$708$	0	0
$709$	−21.0942	−0.792209	−0.396105	−	0.918205i	$-0.629638\pi$
$709$	−21.0942	−0.792209	−0.396105	+	0.918205i	$0.629638\pi$
$710$	−59.6476	−2.23853
$711$	0	0
$712$	−13.5586	−0.508129
$713$	0.355792	0.0133245
$714$	0	0
$715$	−104.419	−3.90504
$716$	10.9587	0.409544
$717$	0	0
$718$	−17.1316	−0.639347
$719$	−24.8266	−0.925877	−0.462938	−	0.886390i	$-0.653205\pi$
$719$	−24.8266	−0.925877	−0.462938	+	0.886390i	$0.653205\pi$
$720$	0	0
$721$	10.3937	0.387080
$722$	−0.715635	−0.0266332
$723$	0	0
$724$	−6.70682	−0.249257
$725$	85.9386	3.19168
$726$	0	0
$727$	12.8137	0.475235	0.237617	−	0.971359i	$-0.423634\pi$
$727$	12.8137	0.475235	0.237617	+	0.971359i	$0.423634\pi$
$728$	16.1094	0.597053
$729$	0	0
$730$	−31.9656	−1.18310
$731$	−22.9529	−0.848943
$732$	0	0
$733$	9.28851	0.343079	0.171539	−	0.985177i	$-0.445126\pi$
$733$	9.28851	0.343079	0.171539	+	0.985177i	$0.445126\pi$
$734$	7.12297	0.262914
$735$	0	0
$736$	−1.64031	−0.0604625
$737$	27.6487	1.01845
$738$	0	0
$739$	−17.2041	−0.632861	−0.316431	−	0.948616i	$-0.602484\pi$
$739$	−17.2041	−0.632861	−0.316431	+	0.948616i	$0.602484\pi$
$740$	−33.7549	−1.24085
$741$	0	0
$742$	−15.1939	−0.557786
$743$	−0.0369162	−0.00135432	−0.000677162	−	1.00000i	$-0.500216\pi$
$743$	−0.0369162	−0.00135432	−0.000677162		1.00000i	$0.500216\pi$
$744$	0	0
$745$	82.7150	3.03044
$746$	13.0150	0.476511
$747$	0	0
$748$	−22.9425	−0.838859
$749$	45.1857	1.65105
$750$	0	0
$751$	−9.70522	−0.354148	−0.177074	−	0.984198i	$-0.556663\pi$
$751$	−9.70522	−0.354148	−0.177074	+	0.984198i	$0.556663\pi$
$752$	6.34729	0.231462
$753$	0	0
$754$	−46.4286	−1.69083
$755$	2.69039	0.0979135
$756$	0	0
$757$	−44.2726	−1.60912	−0.804558	−	0.593874i	$-0.797598\pi$
$757$	−44.2726	−1.60912	−0.804558	+	0.593874i	$0.797598\pi$
$758$	−4.71603	−0.171294
$759$	0	0
$760$	−17.9509	−0.651147
$761$	−12.1563	−0.440666	−0.220333	−	0.975425i	$-0.570714\pi$
$761$	−12.1563	−0.440666	−0.220333	+	0.975425i	$0.570714\pi$
$762$	0	0
$763$	6.73162	0.243701
$764$	−20.4119	−0.738475
$765$	0	0
$766$	−4.26041	−0.153935
$767$	−23.7496	−0.857549
$768$	0	0
$769$	−16.2052	−0.584374	−0.292187	−	0.956361i	$-0.594383\pi$
$769$	−16.2052	−0.584374	−0.292187	+	0.956361i	$0.594383\pi$
$770$	−44.7839	−1.61390
$771$	0	0
$772$	13.8831	0.499663
$773$	23.7724	0.855032	0.427516	−	0.904008i	$-0.359389\pi$
$773$	23.7724	0.855032	0.427516	+	0.904008i	$0.359389\pi$
$774$	0	0
$775$	2.46061	0.0883875
$776$	3.98544	0.143069
$777$	0	0
$778$	−26.5071	−0.950326
$779$	35.9949	1.28965
$780$	0	0
$781$	−62.1788	−2.22493
$782$	8.92965	0.319324
$783$	0	0
$784$	−0.0908960	−0.00324628
$785$	33.1833	1.18436
$786$	0	0
$787$	48.9615	1.74529	0.872645	−	0.488356i	$-0.162403\pi$
$787$	48.9615	1.74529	0.872645	+	0.488356i	$0.162403\pi$
$788$	19.8807	0.708221
$789$	0	0
$790$	47.4098	1.68676
$791$	−15.0302	−0.534413
$792$	0	0
$793$	−6.12870	−0.217636
$794$	−8.28900	−0.294165
$795$	0	0
$796$	15.2050	0.538926
$797$	8.21735	0.291073	0.145537	−	0.989353i	$-0.453509\pi$
$797$	8.21735	0.291073	0.145537	+	0.989353i	$0.453509\pi$
$798$	0	0
$799$	−34.5540	−1.22243
$800$	−11.3441	−0.401075
$801$	0	0
$802$	21.8269	0.770736
$803$	−33.3221	−1.17591
$804$	0	0
$805$	17.4308	0.614354
$806$	−1.32935	−0.0468243
$807$	0	0
$808$	2.84004	0.0999123
$809$	−26.7693	−0.941159	−0.470580	−	0.882358i	$-0.655955\pi$
$809$	−26.7693	−0.941159	−0.470580	+	0.882358i	$0.655955\pi$
$810$	0	0
$811$	18.3883	0.645702	0.322851	−	0.946450i	$-0.395359\pi$
$811$	18.3883	0.645702	0.322851	+	0.946450i	$0.395359\pi$
$812$	−19.9126	−0.698796
$813$	0	0
$814$	−35.1873	−1.23331
$815$	−49.9732	−1.75049
$816$	0	0
$817$	−18.7212	−0.654970
$818$	10.0393	0.351017
$819$	0	0
$820$	32.7730	1.14448
$821$	11.2810	0.393711	0.196855	−	0.980433i	$-0.436927\pi$
$821$	11.2810	0.393711	0.196855	+	0.980433i	$0.436927\pi$
$822$	0	0
$823$	−21.0116	−0.732418	−0.366209	−	0.930533i	$-0.619345\pi$
$823$	−21.0116	−0.732418	−0.366209	+	0.930533i	$0.619345\pi$
$824$	3.95419	0.137751
$825$	0	0
$826$	−10.1859	−0.354413
$827$	38.2016	1.32840	0.664200	−	0.747555i	$-0.268772\pi$
$827$	38.2016	1.32840	0.664200	+	0.747555i	$0.268772\pi$
$828$	0	0
$829$	36.5596	1.26977	0.634885	−	0.772607i	$-0.281048\pi$
$829$	36.5596	1.26977	0.634885	+	0.772607i	$0.281048\pi$
$830$	−46.6746	−1.62010
$831$	0	0
$832$	6.12870	0.212474
$833$	0.494828	0.0171448
$834$	0	0
$835$	−102.730	−3.55513
$836$	−18.7127	−0.647191
$837$	0	0
$838$	0.889810	0.0307380
$839$	2.14689	0.0741187	0.0370594	−	0.999313i	$-0.488201\pi$
$839$	2.14689	0.0741187	0.0370594	+	0.999313i	$0.488201\pi$
$840$	0	0
$841$	28.3899	0.978960
$842$	−20.1983	−0.696078
$843$	0	0
$844$	7.37516	0.253864
$845$	−99.2945	−3.41584
$846$	0	0
$847$	−17.7706	−0.610605
$848$	−5.78041	−0.198500
$849$	0	0
$850$	61.7562	2.11822
$851$	13.6956	0.469478
$852$	0	0
$853$	44.8284	1.53490	0.767448	−	0.641111i	$-0.221526\pi$
$853$	44.8284	1.53490	0.767448	+	0.641111i	$0.221526\pi$
$854$	−2.62852	−0.0899460
$855$	0	0
$856$	17.1906	0.587562
$857$	48.5764	1.65934	0.829669	−	0.558256i	$-0.188529\pi$
$857$	48.5764	1.65934	0.829669	+	0.558256i	$0.188529\pi$
$858$	0	0
$859$	−26.6196	−0.908249	−0.454125	−	0.890938i	$-0.650048\pi$
$859$	−26.6196	−0.908249	−0.454125	+	0.890938i	$0.650048\pi$
$860$	−17.0454	−0.581244
$861$	0	0
$862$	−12.4423	−0.423785
$863$	28.3414	0.964753	0.482377	−	0.875964i	$-0.339774\pi$
$863$	28.3414	0.964753	0.482377	+	0.875964i	$0.339774\pi$
$864$	0	0
$865$	67.7180	2.30248
$866$	−4.03198	−0.137012
$867$	0	0
$868$	−0.570141	−0.0193518
$869$	49.4216	1.67651
$870$	0	0
$871$	40.2081	1.36240
$872$	2.56099	0.0867262
$873$	0	0
$874$	7.28333	0.246362
$875$	67.4160	2.27908
$876$	0	0
$877$	17.5281	0.591881	0.295940	−	0.955206i	$-0.404367\pi$
$877$	17.5281	0.591881	0.295940	+	0.955206i	$0.404367\pi$
$878$	15.1413	0.510995
$879$	0	0
$880$	−17.0377	−0.574340
$881$	−8.16640	−0.275133	−0.137566	−	0.990493i	$-0.543928\pi$
$881$	−8.16640	−0.275133	−0.137566	+	0.990493i	$0.543928\pi$
$882$	0	0
$883$	−17.8946	−0.602200	−0.301100	−	0.953592i	$-0.597354\pi$
$883$	−17.8946	−0.602200	−0.301100	+	0.953592i	$0.597354\pi$
$884$	−33.3640	−1.12215
$885$	0	0
$886$	−1.03565	−0.0347932
$887$	−6.86131	−0.230380	−0.115190	−	0.993343i	$-0.536748\pi$
$887$	−6.86131	−0.230380	−0.115190	+	0.993343i	$0.536748\pi$
$888$	0	0
$889$	2.25950	0.0757813
$890$	54.8144	1.83738
$891$	0	0
$892$	4.00942	0.134245
$893$	−28.1834	−0.943123
$894$	0	0
$895$	−44.3035	−1.48090
$896$	2.62852	0.0878126
$897$	0	0
$898$	−22.6531	−0.755945
$899$	1.64319	0.0548036
$900$	0	0
$901$	31.4680	1.04835
$902$	34.1638	1.13753
$903$	0	0
$904$	−5.71813	−0.190182
$905$	27.1142	0.901307
$906$	0	0
$907$	−14.5217	−0.482187	−0.241093	−	0.970502i	$-0.577506\pi$
$907$	−14.5217	−0.482187	−0.241093	+	0.970502i	$0.577506\pi$
$908$	−6.48523	−0.215220
$909$	0	0
$910$	−65.1268	−2.15893
$911$	−41.9025	−1.38829	−0.694147	−	0.719834i	$-0.744218\pi$
$911$	−41.9025	−1.38829	−0.694147	+	0.719834i	$0.744218\pi$
$912$	0	0
$913$	−48.6552	−1.61025
$914$	−25.9301	−0.857690
$915$	0	0
$916$	−9.62511	−0.318023
$917$	−15.3776	−0.507814
$918$	0	0
$919$	−48.6533	−1.60493	−0.802463	−	0.596702i	$-0.796477\pi$
$919$	−48.6533	−1.60493	−0.802463	+	0.596702i	$0.796477\pi$
$920$	6.63140	0.218631
$921$	0	0
$922$	−12.5283	−0.412597
$923$	−90.4233	−2.97632
$924$	0	0
$925$	94.7167	3.11426
$926$	9.38177	0.308304
$927$	0	0
$928$	−7.57561	−0.248682
$929$	4.02234	0.131969	0.0659844	−	0.997821i	$-0.478981\pi$
$929$	4.02234	0.131969	0.0659844	+	0.997821i	$0.478981\pi$
$930$	0	0
$931$	0.403599	0.0132274
$932$	−27.2350	−0.892112
$933$	0	0
$934$	13.0532	0.427114
$935$	92.7514	3.03330
$936$	0	0
$937$	−17.1767	−0.561138	−0.280569	−	0.959834i	$-0.590523\pi$
$937$	−17.1767	−0.561138	−0.280569	+	0.959834i	$0.590523\pi$
$938$	17.2447	0.563060
$939$	0	0
$940$	−25.6608	−0.836961
$941$	−33.9535	−1.10685	−0.553426	−	0.832898i	$-0.686680\pi$
$941$	−33.9535	−1.10685	−0.553426	+	0.832898i	$0.686680\pi$
$942$	0	0
$943$	−13.2972	−0.433017
$944$	−3.87515	−0.126125
$945$	0	0
$946$	−17.7688	−0.577712
$947$	31.7235	1.03088	0.515438	−	0.856927i	$-0.327629\pi$
$947$	31.7235	1.03088	0.515438	+	0.856927i	$0.327629\pi$
$948$	0	0
$949$	−48.4586	−1.57303
$950$	50.3705	1.63423
$951$	0	0
$952$	−14.3094	−0.463770
$953$	−2.97694	−0.0964327	−0.0482163	−	0.998837i	$-0.515354\pi$
$953$	−2.97694	−0.0964327	−0.0482163	+	0.998837i	$0.515354\pi$
$954$	0	0
$955$	82.5208	2.67031
$956$	1.68728	0.0545706
$957$	0	0
$958$	−41.1150	−1.32836
$959$	−25.7665	−0.832045
$960$	0	0
$961$	−30.9530	−0.998482
$962$	−51.1710	−1.64982
$963$	0	0
$964$	15.4374	0.497206
$965$	−56.1263	−1.80677
$966$	0	0
$967$	−20.9656	−0.674208	−0.337104	−	0.941467i	$-0.609447\pi$
$967$	−20.9656	−0.674208	−0.337104	+	0.941467i	$0.609447\pi$
$968$	−6.76070	−0.217297
$969$	0	0
$970$	−16.1123	−0.517334
$971$	46.8628	1.50390	0.751950	−	0.659220i	$-0.229113\pi$
$971$	46.8628	1.50390	0.751950	+	0.659220i	$0.229113\pi$
$972$	0	0
$973$	7.80410	0.250188
$974$	−6.86990	−0.220126
$975$	0	0
$976$	−1.00000	−0.0320092
$977$	25.7293	0.823152	0.411576	−	0.911375i	$-0.364978\pi$
$977$	25.7293	0.823152	0.411576	+	0.911375i	$0.364978\pi$
$978$	0	0
$979$	57.1405	1.82622
$980$	0.367473	0.0117385
$981$	0	0
$982$	−20.2204	−0.645259
$983$	33.4789	1.06781	0.533905	−	0.845544i	$-0.320724\pi$
$983$	33.4789	1.06781	0.533905	+	0.845544i	$0.320724\pi$
$984$	0	0
$985$	−80.3735	−2.56091
$986$	41.2408	1.31338
$987$	0	0
$988$	−27.2128	−0.865755
$989$	6.91595	0.219914
$990$	0	0
$991$	1.68880	0.0536466	0.0268233	−	0.999640i	$-0.491461\pi$
$991$	1.68880	0.0536466	0.0268233	+	0.999640i	$0.491461\pi$
$992$	−0.216906	−0.00688677
$993$	0	0
$994$	−38.7814	−1.23007
$995$	−61.4704	−1.94874
$996$	0	0
$997$	27.2050	0.861591	0.430795	−	0.902450i	$-0.358233\pi$
$997$	27.2050	0.861591	0.430795	+	0.902450i	$0.358233\pi$
$998$	−11.2610	−0.356460
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	9882.2.a.bk.1.2		17
3.2	odd	2		9882.2.a.bm.1.16		17
9.2	odd	6		1098.2.e.f.733.1	yes	34
9.5	odd	6		1098.2.e.f.367.1	✓	34

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1098.2.e.f.367.1	✓	34	9.5	odd	6
1098.2.e.f.733.1	yes	34	9.2	odd	6
9882.2.a.bk.1.2		17	1.1	even	1	trivial
9882.2.a.bm.1.16		17	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 9882 = 2 \cdot 3^{4} \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	9882.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -4.04279 q^{5} -2.62852 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -4.04279 q^{5} -2.62852 q^{7} -1.00000 q^{8} +4.04279 q^{10} +4.21434 q^{11} +6.12870 q^{13} +2.62852 q^{14} +1.00000 q^{16} -5.44390 q^{17} -4.44023 q^{19} -4.04279 q^{20} -4.21434 q^{22} +1.64031 q^{23} +11.3441 q^{25} -6.12870 q^{26} -2.62852 q^{28} +7.57561 q^{29} +0.216906 q^{31} -1.00000 q^{32} +5.44390 q^{34} +10.6265 q^{35} +8.34941 q^{37} +4.44023 q^{38} +4.04279 q^{40} -8.10654 q^{41} +4.21626 q^{43} +4.21434 q^{44} -1.64031 q^{46} +6.34729 q^{47} -0.0908960 q^{49} -11.3441 q^{50} +6.12870 q^{52} -5.78041 q^{53} -17.0377 q^{55} +2.62852 q^{56} -7.57561 q^{58} -3.87515 q^{59} -1.00000 q^{61} -0.216906 q^{62} +1.00000 q^{64} -24.7770 q^{65} +6.56062 q^{67} -5.44390 q^{68} -10.6265 q^{70} -14.7541 q^{71} -7.90683 q^{73} -8.34941 q^{74} -4.44023 q^{76} -11.0775 q^{77} +11.7270 q^{79} -4.04279 q^{80} +8.10654 q^{82} -11.5452 q^{83} +22.0085 q^{85} -4.21626 q^{86} -4.21434 q^{88} +13.5586 q^{89} -16.1094 q^{91} +1.64031 q^{92} -6.34729 q^{94} +17.9509 q^{95} -3.98544 q^{97} +0.0908960 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 17 q - 17 q^{2} + 17 q^{4} - 5 q^{5} + 11 q^{7} - 17 q^{8}+O(q^{10}) \) 17 * q - 17 * q^2 + 17 * q^4 - 5 * q^5 + 11 * q^7 - 17 * q^8	\( 17 q - 17 q^{2} + 17 q^{4} - 5 q^{5} + 11 q^{7} - 17 q^{8} + 5 q^{10} - 3 q^{11} + 10 q^{13} - 11 q^{14} + 17 q^{16} - 8 q^{17} + 11 q^{19} - 5 q^{20} + 3 q^{22} - 5 q^{23} + 32 q^{25} - 10 q^{26} + 11 q^{28} - 6 q^{29} + 20 q^{31} - 17 q^{32} + 8 q^{34} + 8 q^{35} + 29 q^{37} - 11 q^{38} + 5 q^{40} - 20 q^{41} + 30 q^{43} - 3 q^{44} + 5 q^{46} - 10 q^{47} + 42 q^{49} - 32 q^{50} + 10 q^{52} - 2 q^{53} + 15 q^{55} - 11 q^{56} + 6 q^{58} + 3 q^{59} - 17 q^{61} - 20 q^{62} + 17 q^{64} - 28 q^{65} + 11 q^{67} - 8 q^{68} - 8 q^{70} - 63 q^{71} + 28 q^{73} - 29 q^{74} + 11 q^{76} - 17 q^{77} + 17 q^{79} - 5 q^{80} + 20 q^{82} + 20 q^{83} + 33 q^{85} - 30 q^{86} + 3 q^{88} - 2 q^{89} + 30 q^{91} - 5 q^{92} + 10 q^{94} - q^{95} + 32 q^{97} - 42 q^{98}+O(q^{100}) \) 17 * q - 17 * q^2 + 17 * q^4 - 5 * q^5 + 11 * q^7 - 17 * q^8 + 5 * q^10 - 3 * q^11 + 10 * q^13 - 11 * q^14 + 17 * q^16 - 8 * q^17 + 11 * q^19 - 5 * q^20 + 3 * q^22 - 5 * q^23 + 32 * q^25 - 10 * q^26 + 11 * q^28 - 6 * q^29 + 20 * q^31 - 17 * q^32 + 8 * q^34 + 8 * q^35 + 29 * q^37 - 11 * q^38 + 5 * q^40 - 20 * q^41 + 30 * q^43 - 3 * q^44 + 5 * q^46 - 10 * q^47 + 42 * q^49 - 32 * q^50 + 10 * q^52 - 2 * q^53 + 15 * q^55 - 11 * q^56 + 6 * q^58 + 3 * q^59 - 17 * q^61 - 20 * q^62 + 17 * q^64 - 28 * q^65 + 11 * q^67 - 8 * q^68 - 8 * q^70 - 63 * q^71 + 28 * q^73 - 29 * q^74 + 11 * q^76 - 17 * q^77 + 17 * q^79 - 5 * q^80 + 20 * q^82 + 20 * q^83 + 33 * q^85 - 30 * q^86 + 3 * q^88 - 2 * q^89 + 30 * q^91 - 5 * q^92 + 10 * q^94 - q^95 + 32 * q^97 - 42 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more