LMFDB - Embedded newform 6864.2.a.bz.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6864,2,Mod(1,6864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6864, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("6864.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6864.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:	$54.8093159474$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.8468.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 3x + 4 $ x^4 - x^3 - 5x^2 + 3x + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 429)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.31743$ of defining polynomial
Character	$\chi$	$=$	6864.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -2.71878 q^{5} +4.30059 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -2.71878 q^{5} +4.30059 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} -2.71878 q^{15} -6.11056 q^{17} +0.300590 q^{19} +4.30059 q^{21} -1.49789 q^{23} +2.39178 q^{25} +1.00000 q^{27} -0.312085 q^{29} -4.55094 q^{31} +1.00000 q^{33} -11.6924 q^{35} -7.96632 q^{37} +1.00000 q^{39} +1.03087 q^{41} -12.4906 q^{43} -2.71878 q^{45} -13.1299 q^{47} +11.4951 q^{49} -6.11056 q^{51} +8.49507 q^{53} -2.71878 q^{55} +0.300590 q^{57} -9.22535 q^{59} -9.86021 q^{61} +4.30059 q^{63} -2.71878 q^{65} +8.05024 q^{67} -1.49789 q^{69} +0.696596 q^{71} +4.57454 q^{73} +2.39178 q^{75} +4.30059 q^{77} +17.0918 q^{79} +1.00000 q^{81} -8.76902 q^{83} +16.6133 q^{85} -0.312085 q^{87} -3.24754 q^{89} +4.30059 q^{91} -4.55094 q^{93} -0.817238 q^{95} +7.63063 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} - 2 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 - 2 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} - 2 q^{7} + 4 q^{9} + 4 q^{11} + 4 q^{13} - 8 q^{17} - 18 q^{19} - 2 q^{21} + 4 q^{25} + 4 q^{27} - 10 q^{29} - 12 q^{31} + 4 q^{33} - 22 q^{35} - 2 q^{37} + 4 q^{39} + 2 q^{41} - 28 q^{43} - 6 q^{47} + 8 q^{49} - 8 q^{51} - 4 q^{53} - 18 q^{57} - 16 q^{59} - 10 q^{61} - 2 q^{63} - 10 q^{71} - 6 q^{73} + 4 q^{75} - 2 q^{77} + 8 q^{79} + 4 q^{81} + 8 q^{83} - 18 q^{85} - 10 q^{87} + 6 q^{89} - 2 q^{91} - 12 q^{93} - 22 q^{95} + 10 q^{97} + 4 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 - 2 * q^7 + 4 * q^9 + 4 * q^11 + 4 * q^13 - 8 * q^17 - 18 * q^19 - 2 * q^21 + 4 * q^25 + 4 * q^27 - 10 * q^29 - 12 * q^31 + 4 * q^33 - 22 * q^35 - 2 * q^37 + 4 * q^39 + 2 * q^41 - 28 * q^43 - 6 * q^47 + 8 * q^49 - 8 * q^51 - 4 * q^53 - 18 * q^57 - 16 * q^59 - 10 * q^61 - 2 * q^63 - 10 * q^71 - 6 * q^73 + 4 * q^75 - 2 * q^77 + 8 * q^79 + 4 * q^81 + 8 * q^83 - 18 * q^85 - 10 * q^87 + 6 * q^89 - 2 * q^91 - 12 * q^93 - 22 * q^95 + 10 * q^97 + 4 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−2.71878	−1.21588	−0.607938	−	0.793984i	$-0.708003\pi$
$5$	−2.71878	−1.21588	−0.607938	+	0.793984i	$0.708003\pi$
$6$	0	0
$7$	4.30059	1.62547	0.812735	−	0.582633i	$-0.197978\pi$
$7$	4.30059	1.62547	0.812735	+	0.582633i	$0.197978\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−2.71878	−0.701987
$16$	0	0
$17$	−6.11056	−1.48203	−0.741014	−	0.671489i	$-0.765655\pi$
$17$	−6.11056	−1.48203	−0.741014	+	0.671489i	$0.765655\pi$
$18$	0	0
$19$	0.300590	0.0689600	0.0344800	−	0.999405i	$-0.489023\pi$
$19$	0.300590	0.0689600	0.0344800	+	0.999405i	$0.489023\pi$
$20$	0	0
$21$	4.30059	0.938466
$22$	0	0
$23$	−1.49789	−0.312331	−0.156165	−	0.987731i	$-0.549913\pi$
$23$	−1.49789	−0.312331	−0.156165	+	0.987731i	$0.549913\pi$
$24$	0	0
$25$	2.39178	0.478356
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−0.312085	−0.0579527	−0.0289763	−	0.999580i	$-0.509225\pi$
$29$	−0.312085	−0.0579527	−0.0289763	+	0.999580i	$0.509225\pi$
$30$	0	0
$31$	−4.55094	−0.817373	−0.408686	−	0.912675i	$-0.634013\pi$
$31$	−4.55094	−0.817373	−0.408686	+	0.912675i	$0.634013\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	−11.6924	−1.97637
$36$	0	0
$37$	−7.96632	−1.30965	−0.654827	−	0.755779i	$-0.727259\pi$
$37$	−7.96632	−1.30965	−0.654827	+	0.755779i	$0.727259\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	1.03087	0.160994	0.0804972	−	0.996755i	$-0.474349\pi$
$41$	1.03087	0.160994	0.0804972	+	0.996755i	$0.474349\pi$
$42$	0	0
$43$	−12.4906	−1.90480	−0.952401	−	0.304849i	$-0.901394\pi$
$43$	−12.4906	−1.90480	−0.952401	+	0.304849i	$0.901394\pi$
$44$	0	0
$45$	−2.71878	−0.405292
$46$	0	0
$47$	−13.1299	−1.91520	−0.957599	−	0.288105i	$-0.906975\pi$
$47$	−13.1299	−1.91520	−0.957599	+	0.288105i	$0.906975\pi$
$48$	0	0
$49$	11.4951	1.64215
$50$	0	0
$51$	−6.11056	−0.855650
$52$	0	0
$53$	8.49507	1.16689	0.583444	−	0.812153i	$-0.301705\pi$
$53$	8.49507	1.16689	0.583444	+	0.812153i	$0.301705\pi$
$54$	0	0
$55$	−2.71878	−0.366601
$56$	0	0
$57$	0.300590	0.0398141
$58$	0	0
$59$	−9.22535	−1.20104	−0.600519	−	0.799610i	$-0.705039\pi$
$59$	−9.22535	−1.20104	−0.600519	+	0.799610i	$0.705039\pi$
$60$	0	0
$61$	−9.86021	−1.26247	−0.631235	−	0.775591i	$-0.717452\pi$
$61$	−9.86021	−1.26247	−0.631235	+	0.775591i	$0.717452\pi$
$62$	0	0
$63$	4.30059	0.541823
$64$	0	0
$65$	−2.71878	−0.337223
$66$	0	0
$67$	8.05024	0.983493	0.491747	−	0.870738i	$-0.336359\pi$
$67$	8.05024	0.983493	0.491747	+	0.870738i	$0.336359\pi$
$68$	0	0
$69$	−1.49789	−0.180324
$70$	0	0
$71$	0.696596	0.0826707	0.0413353	−	0.999145i	$-0.486839\pi$
$71$	0.696596	0.0826707	0.0413353	+	0.999145i	$0.486839\pi$
$72$	0	0
$73$	4.57454	0.535409	0.267705	−	0.963501i	$-0.413735\pi$
$73$	4.57454	0.535409	0.267705	+	0.963501i	$0.413735\pi$
$74$	0	0
$75$	2.39178	0.276179
$76$	0	0
$77$	4.30059	0.490098
$78$	0	0
$79$	17.0918	1.92298	0.961489	−	0.274844i	$-0.0886263\pi$
$79$	17.0918	1.92298	0.961489	+	0.274844i	$0.0886263\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−8.76902	−0.962525	−0.481263	−	0.876576i	$-0.659822\pi$
$83$	−8.76902	−0.962525	−0.481263	+	0.876576i	$0.659822\pi$
$84$	0	0
$85$	16.6133	1.80196
$86$	0	0
$87$	−0.312085	−0.0334590
$88$	0	0
$89$	−3.24754	−0.344238	−0.172119	−	0.985076i	$-0.555061\pi$
$89$	−3.24754	−0.344238	−0.172119	+	0.985076i	$0.555061\pi$
$90$	0	0
$91$	4.30059	0.450824
$92$	0	0
$93$	−4.55094	−0.471910
$94$	0	0
$95$	−0.817238	−0.0838468
$96$	0	0
$97$	7.63063	0.774773	0.387387	−	0.921917i	$-0.373378\pi$
$97$	7.63063	0.774773	0.387387	+	0.921917i	$0.373378\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	3.83661	0.381757	0.190878	−	0.981614i	$-0.438866\pi$
$101$	3.83661	0.381757	0.190878	+	0.981614i	$0.438866\pi$
$102$	0	0
$103$	−11.3315	−1.11652	−0.558261	−	0.829666i	$-0.688531\pi$
$103$	−11.3315	−1.11652	−0.558261	+	0.829666i	$0.688531\pi$
$104$	0	0
$105$	−11.6924	−1.14106
$106$	0	0
$107$	15.5526	1.50352	0.751762	−	0.659434i	$-0.229204\pi$
$107$	15.5526	1.50352	0.751762	+	0.659434i	$0.229204\pi$
$108$	0	0
$109$	−0.0266392	−0.00255157	−0.00127578	−	0.999999i	$-0.500406\pi$
$109$	−0.0266392	−0.00255157	−0.00127578	+	0.999999i	$0.500406\pi$
$110$	0	0
$111$	−7.96632	−0.756129
$112$	0	0
$113$	−18.4333	−1.73406	−0.867031	−	0.498254i	$-0.833975\pi$
$113$	−18.4333	−1.73406	−0.867031	+	0.498254i	$0.833975\pi$
$114$	0	0
$115$	4.07243	0.379756
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−26.2790	−2.40899
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	1.03087	0.0929502
$124$	0	0
$125$	7.09119	0.634255
$126$	0	0
$127$	8.01514	0.711229	0.355615	−	0.934633i	$-0.384272\pi$
$127$	8.01514	0.711229	0.355615	+	0.934633i	$0.384272\pi$
$128$	0	0
$129$	−12.4906	−1.09974
$130$	0	0
$131$	−11.5437	−1.00858	−0.504288	−	0.863536i	$-0.668245\pi$
$131$	−11.5437	−1.00858	−0.504288	+	0.863536i	$0.668245\pi$
$132$	0	0
$133$	1.29271	0.112092
$134$	0	0
$135$	−2.71878	−0.233996
$136$	0	0
$137$	−5.97781	−0.510719	−0.255360	−	0.966846i	$-0.582194\pi$
$137$	−5.97781	−0.510719	−0.255360	+	0.966846i	$0.582194\pi$
$138$	0	0
$139$	−9.85576	−0.835954	−0.417977	−	0.908458i	$-0.637261\pi$
$139$	−9.85576	−0.835954	−0.417977	+	0.908458i	$0.637261\pi$
$140$	0	0
$141$	−13.1299	−1.10574
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	0.848491	0.0704633
$146$	0	0
$147$	11.4951	0.948097
$148$	0	0
$149$	−11.5703	−0.947877	−0.473938	−	0.880558i	$-0.657168\pi$
$149$	−11.5703	−0.947877	−0.473938	+	0.880558i	$0.657168\pi$
$150$	0	0
$151$	3.24308	0.263918	0.131959	−	0.991255i	$-0.457873\pi$
$151$	3.24308	0.263918	0.131959	+	0.991255i	$0.457873\pi$
$152$	0	0
$153$	−6.11056	−0.494010
$154$	0	0
$155$	12.3730	0.993825
$156$	0	0
$157$	8.21363	0.655519	0.327759	−	0.944761i	$-0.393706\pi$
$157$	8.21363	0.655519	0.327759	+	0.944761i	$0.393706\pi$
$158$	0	0
$159$	8.49507	0.673703
$160$	0	0
$161$	−6.44179	−0.507684
$162$	0	0
$163$	2.80574	0.219763	0.109881	−	0.993945i	$-0.464953\pi$
$163$	2.80574	0.219763	0.109881	+	0.993945i	$0.464953\pi$
$164$	0	0
$165$	−2.71878	−0.211657
$166$	0	0
$167$	1.93827	0.149987	0.0749937	−	0.997184i	$-0.476106\pi$
$167$	1.93827	0.149987	0.0749937	+	0.997184i	$0.476106\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	0.300590	0.0229867
$172$	0	0
$173$	−8.55235	−0.650223	−0.325112	−	0.945676i	$-0.605402\pi$
$173$	−8.55235	−0.650223	−0.325112	+	0.945676i	$0.605402\pi$
$174$	0	0
$175$	10.2861	0.777553
$176$	0	0
$177$	−9.22535	−0.693420
$178$	0	0
$179$	−17.9312	−1.34024	−0.670121	−	0.742252i	$-0.733758\pi$
$179$	−17.9312	−1.34024	−0.670121	+	0.742252i	$0.733758\pi$
$180$	0	0
$181$	0.334272	0.0248462	0.0124231	−	0.999923i	$-0.496046\pi$
$181$	0.334272	0.0248462	0.0124231	+	0.999923i	$0.496046\pi$
$182$	0	0
$183$	−9.86021	−0.728888
$184$	0	0
$185$	21.6587	1.59238
$186$	0	0
$187$	−6.11056	−0.446848
$188$	0	0
$189$	4.30059	0.312822
$190$	0	0
$191$	20.1538	1.45827	0.729137	−	0.684367i	$-0.239922\pi$
$191$	20.1538	1.45827	0.729137	+	0.684367i	$0.239922\pi$
$192$	0	0
$193$	22.5689	1.62455	0.812273	−	0.583278i	$-0.198230\pi$
$193$	22.5689	1.62455	0.812273	+	0.583278i	$0.198230\pi$
$194$	0	0
$195$	−2.71878	−0.194696
$196$	0	0
$197$	−26.6133	−1.89612	−0.948059	−	0.318095i	$-0.896957\pi$
$197$	−26.6133	−1.89612	−0.948059	+	0.318095i	$0.896957\pi$
$198$	0	0
$199$	−4.81340	−0.341213	−0.170606	−	0.985339i	$-0.554573\pi$
$199$	−4.81340	−0.341213	−0.170606	+	0.985339i	$0.554573\pi$
$200$	0	0
$201$	8.05024	0.567820
$202$	0	0
$203$	−1.34215	−0.0942004
$204$	0	0
$205$	−2.80270	−0.195749
$206$	0	0
$207$	−1.49789	−0.104110
$208$	0	0
$209$	0.300590	0.0207922
$210$	0	0
$211$	−25.2596	−1.73895	−0.869473	−	0.493981i	$-0.835541\pi$
$211$	−25.2596	−1.73895	−0.869473	+	0.493981i	$0.835541\pi$
$212$	0	0
$213$	0.696596	0.0477299
$214$	0	0
$215$	33.9593	2.31600
$216$	0	0
$217$	−19.5717	−1.32862
$218$	0	0
$219$	4.57454	0.309119
$220$	0	0
$221$	−6.11056	−0.411041
$222$	0	0
$223$	11.8291	0.792136	0.396068	−	0.918221i	$-0.370374\pi$
$223$	11.8291	0.792136	0.396068	+	0.918221i	$0.370374\pi$
$224$	0	0
$225$	2.39178	0.159452
$226$	0	0
$227$	−21.1370	−1.40291	−0.701455	−	0.712714i	$-0.747466\pi$
$227$	−21.1370	−1.40291	−0.701455	+	0.712714i	$0.747466\pi$
$228$	0	0
$229$	−11.6778	−0.771693	−0.385846	−	0.922563i	$-0.626091\pi$
$229$	−11.6778	−0.771693	−0.385846	+	0.922563i	$0.626091\pi$
$230$	0	0
$231$	4.30059	0.282958
$232$	0	0
$233$	10.1156	0.662696	0.331348	−	0.943509i	$-0.392497\pi$
$233$	10.1156	0.662696	0.331348	+	0.943509i	$0.392497\pi$
$234$	0	0
$235$	35.6974	2.32864
$236$	0	0
$237$	17.0918	1.11023
$238$	0	0
$239$	−2.65081	−0.171467	−0.0857333	−	0.996318i	$-0.527323\pi$
$239$	−2.65081	−0.171467	−0.0857333	+	0.996318i	$0.527323\pi$
$240$	0	0
$241$	4.15939	0.267930	0.133965	−	0.990986i	$-0.457229\pi$
$241$	4.15939	0.267930	0.133965	+	0.990986i	$0.457229\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−31.2526	−1.99666
$246$	0	0
$247$	0.300590	0.0191261
$248$	0	0
$249$	−8.76902	−0.555714
$250$	0	0
$251$	−15.7115	−0.991702	−0.495851	−	0.868408i	$-0.665144\pi$
$251$	−15.7115	−0.991702	−0.495851	+	0.868408i	$0.665144\pi$
$252$	0	0
$253$	−1.49789	−0.0941713
$254$	0	0
$255$	16.6133	1.04036
$256$	0	0
$257$	4.53944	0.283163	0.141581	−	0.989927i	$-0.454781\pi$
$257$	4.53944	0.283163	0.141581	+	0.989927i	$0.454781\pi$
$258$	0	0
$259$	−34.2599	−2.12880
$260$	0	0
$261$	−0.312085	−0.0193176
$262$	0	0
$263$	−8.38006	−0.516737	−0.258368	−	0.966046i	$-0.583185\pi$
$263$	−8.38006	−0.516737	−0.258368	+	0.966046i	$0.583185\pi$
$264$	0	0
$265$	−23.0963	−1.41879
$266$	0	0
$267$	−3.24754	−0.198746
$268$	0	0
$269$	−4.95140	−0.301892	−0.150946	−	0.988542i	$-0.548232\pi$
$269$	−4.95140	−0.301892	−0.150946	+	0.988542i	$0.548232\pi$
$270$	0	0
$271$	24.5750	1.49282	0.746412	−	0.665484i	$-0.231775\pi$
$271$	24.5750	1.49282	0.746412	+	0.665484i	$0.231775\pi$
$272$	0	0
$273$	4.30059	0.260284
$274$	0	0
$275$	2.39178	0.144230
$276$	0	0
$277$	27.5913	1.65780	0.828901	−	0.559395i	$-0.188967\pi$
$277$	27.5913	1.65780	0.828901	+	0.559395i	$0.188967\pi$
$278$	0	0
$279$	−4.55094	−0.272458
$280$	0	0
$281$	10.3478	0.617297	0.308649	−	0.951176i	$-0.400123\pi$
$281$	10.3478	0.617297	0.308649	+	0.951176i	$0.400123\pi$
$282$	0	0
$283$	1.97640	0.117485	0.0587424	−	0.998273i	$-0.481291\pi$
$283$	1.97640	0.117485	0.0587424	+	0.998273i	$0.481291\pi$
$284$	0	0
$285$	−0.817238	−0.0484090
$286$	0	0
$287$	4.43334	0.261692
$288$	0	0
$289$	20.3389	1.19641
$290$	0	0
$291$	7.63063	0.447316
$292$	0	0
$293$	13.5877	0.793800	0.396900	−	0.917862i	$-0.370086\pi$
$293$	13.5877	0.793800	0.396900	+	0.917862i	$0.370086\pi$
$294$	0	0
$295$	25.0817	1.46031
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	−1.49789	−0.0866250
$300$	0	0
$301$	−53.7170	−3.09620
$302$	0	0
$303$	3.83661	0.220407
$304$	0	0
$305$	26.8078	1.53501
$306$	0	0
$307$	−31.8302	−1.81664	−0.908322	−	0.418271i	$-0.862636\pi$
$307$	−31.8302	−1.81664	−0.908322	+	0.418271i	$0.862636\pi$
$308$	0	0
$309$	−11.3315	−0.644624
$310$	0	0
$311$	−6.64697	−0.376915	−0.188457	−	0.982081i	$-0.560349\pi$
$311$	−6.64697	−0.376915	−0.188457	+	0.982081i	$0.560349\pi$
$312$	0	0
$313$	−22.6171	−1.27840	−0.639198	−	0.769042i	$-0.720733\pi$
$313$	−22.6171	−1.27840	−0.639198	+	0.769042i	$0.720733\pi$
$314$	0	0
$315$	−11.6924	−0.658790
$316$	0	0
$317$	26.3971	1.48261	0.741303	−	0.671170i	$-0.234208\pi$
$317$	26.3971	1.48261	0.741303	+	0.671170i	$0.234208\pi$
$318$	0	0
$319$	−0.312085	−0.0174734
$320$	0	0
$321$	15.5526	0.868060
$322$	0	0
$323$	−1.83677	−0.102201
$324$	0	0
$325$	2.39178	0.132672
$326$	0	0
$327$	−0.0266392	−0.00147315
$328$	0	0
$329$	−56.4665	−3.11310
$330$	0	0
$331$	−20.1227	−1.10604	−0.553021	−	0.833167i	$-0.686525\pi$
$331$	−20.1227	−1.10604	−0.553021	+	0.833167i	$0.686525\pi$
$332$	0	0
$333$	−7.96632	−0.436552
$334$	0	0
$335$	−21.8868	−1.19581
$336$	0	0
$337$	13.5526	0.738256	0.369128	−	0.929379i	$-0.379656\pi$
$337$	13.5526	0.738256	0.369128	+	0.929379i	$0.379656\pi$
$338$	0	0
$339$	−18.4333	−1.00116
$340$	0	0
$341$	−4.55094	−0.246447
$342$	0	0
$343$	19.3315	1.04380
$344$	0	0
$345$	4.07243	0.219252
$346$	0	0
$347$	2.24590	0.120566	0.0602831	−	0.998181i	$-0.480800\pi$
$347$	2.24590	0.120566	0.0602831	+	0.998181i	$0.480800\pi$
$348$	0	0
$349$	28.7634	1.53967	0.769835	−	0.638244i	$-0.220339\pi$
$349$	28.7634	1.53967	0.769835	+	0.638244i	$0.220339\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−24.4920	−1.30358	−0.651790	−	0.758400i	$-0.725982\pi$
$353$	−24.4920	−1.30358	−0.651790	+	0.758400i	$0.725982\pi$
$354$	0	0
$355$	−1.89389	−0.100517
$356$	0	0
$357$	−26.2790	−1.39083
$358$	0	0
$359$	−7.12852	−0.376229	−0.188114	−	0.982147i	$-0.560238\pi$
$359$	−7.12852	−0.376229	−0.188114	+	0.982147i	$0.560238\pi$
$360$	0	0
$361$	−18.9096	−0.995245
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	−12.4372	−0.650992
$366$	0	0
$367$	−6.56244	−0.342556	−0.171278	−	0.985223i	$-0.554790\pi$
$367$	−6.56244	−0.342556	−0.171278	+	0.985223i	$0.554790\pi$
$368$	0	0
$369$	1.03087	0.0536648
$370$	0	0
$371$	36.5338	1.89674
$372$	0	0
$373$	19.0345	0.985570	0.492785	−	0.870151i	$-0.335979\pi$
$373$	19.0345	0.985570	0.492785	+	0.870151i	$0.335979\pi$
$374$	0	0
$375$	7.09119	0.366187
$376$	0	0
$377$	−0.312085	−0.0160732
$378$	0	0
$379$	−9.46866	−0.486372	−0.243186	−	0.969980i	$-0.578193\pi$
$379$	−9.46866	−0.486372	−0.243186	+	0.969980i	$0.578193\pi$
$380$	0	0
$381$	8.01514	0.410628
$382$	0	0
$383$	−1.89389	−0.0967734	−0.0483867	−	0.998829i	$-0.515408\pi$
$383$	−1.89389	−0.0967734	−0.0483867	+	0.998829i	$0.515408\pi$
$384$	0	0
$385$	−11.6924	−0.595898
$386$	0	0
$387$	−12.4906	−0.634934
$388$	0	0
$389$	−6.31269	−0.320066	−0.160033	−	0.987112i	$-0.551160\pi$
$389$	−6.31269	−0.320066	−0.160033	+	0.987112i	$0.551160\pi$
$390$	0	0
$391$	9.15292	0.462883
$392$	0	0
$393$	−11.5437	−0.582301
$394$	0	0
$395$	−46.4689	−2.33810
$396$	0	0
$397$	−15.2949	−0.767631	−0.383816	−	0.923410i	$-0.625390\pi$
$397$	−15.2949	−0.767631	−0.383816	+	0.923410i	$0.625390\pi$
$398$	0	0
$399$	1.29271	0.0647166
$400$	0	0
$401$	2.45329	0.122511	0.0612557	−	0.998122i	$-0.480489\pi$
$401$	2.45329	0.122511	0.0612557	+	0.998122i	$0.480489\pi$
$402$	0	0
$403$	−4.55094	−0.226698
$404$	0	0
$405$	−2.71878	−0.135097
$406$	0	0
$407$	−7.96632	−0.394876
$408$	0	0
$409$	−3.76114	−0.185977	−0.0929883	−	0.995667i	$-0.529642\pi$
$409$	−3.76114	−0.185977	−0.0929883	+	0.995667i	$0.529642\pi$
$410$	0	0
$411$	−5.97781	−0.294864
$412$	0	0
$413$	−39.6744	−1.95225
$414$	0	0
$415$	23.8411	1.17031
$416$	0	0
$417$	−9.85576	−0.482639
$418$	0	0
$419$	−32.2080	−1.57346	−0.786732	−	0.617295i	$-0.788229\pi$
$419$	−32.2080	−1.57346	−0.786732	+	0.617295i	$0.788229\pi$
$420$	0	0
$421$	−26.3216	−1.28284	−0.641418	−	0.767191i	$-0.721654\pi$
$421$	−26.3216	−1.28284	−0.641418	+	0.767191i	$0.721654\pi$
$422$	0	0
$423$	−13.1299	−0.638399
$424$	0	0
$425$	−14.6151	−0.708937
$426$	0	0
$427$	−42.4047	−2.05211
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	−2.71619	−0.130834	−0.0654172	−	0.997858i	$-0.520838\pi$
$431$	−2.71619	−0.130834	−0.0654172	+	0.997858i	$0.520838\pi$
$432$	0	0
$433$	6.54790	0.314672	0.157336	−	0.987545i	$-0.449709\pi$
$433$	6.54790	0.314672	0.157336	+	0.987545i	$0.449709\pi$
$434$	0	0
$435$	0.848491	0.0406820
$436$	0	0
$437$	−0.450249	−0.0215383
$438$	0	0
$439$	−18.9413	−0.904020	−0.452010	−	0.892013i	$-0.649293\pi$
$439$	−18.9413	−0.904020	−0.452010	+	0.892013i	$0.649293\pi$
$440$	0	0
$441$	11.4951	0.547384
$442$	0	0
$443$	33.8568	1.60859	0.804293	−	0.594233i	$-0.202544\pi$
$443$	33.8568	1.60859	0.804293	+	0.594233i	$0.202544\pi$
$444$	0	0
$445$	8.82934	0.418551
$446$	0	0
$447$	−11.5703	−0.547257
$448$	0	0
$449$	−2.62721	−0.123986	−0.0619928	−	0.998077i	$-0.519746\pi$
$449$	−2.62721	−0.123986	−0.0619928	+	0.998077i	$0.519746\pi$
$450$	0	0
$451$	1.03087	0.0485416
$452$	0	0
$453$	3.24308	0.152373
$454$	0	0
$455$	−11.6924	−0.548147
$456$	0	0
$457$	19.3230	0.903892	0.451946	−	0.892045i	$-0.350730\pi$
$457$	19.3230	0.903892	0.451946	+	0.892045i	$0.350730\pi$
$458$	0	0
$459$	−6.11056	−0.285217
$460$	0	0
$461$	−6.17995	−0.287829	−0.143914	−	0.989590i	$-0.545969\pi$
$461$	−6.17995	−0.287829	−0.143914	+	0.989590i	$0.545969\pi$
$462$	0	0
$463$	3.36773	0.156512	0.0782558	−	0.996933i	$-0.475065\pi$
$463$	3.36773	0.156512	0.0782558	+	0.996933i	$0.475065\pi$
$464$	0	0
$465$	12.3730	0.573785
$466$	0	0
$467$	−37.3632	−1.72896	−0.864480	−	0.502667i	$-0.832352\pi$
$467$	−37.3632	−1.72896	−0.864480	+	0.502667i	$0.832352\pi$
$468$	0	0
$469$	34.6208	1.59864
$470$	0	0
$471$	8.21363	0.378464
$472$	0	0
$473$	−12.4906	−0.574319
$474$	0	0
$475$	0.718944	0.0329874
$476$	0	0
$477$	8.49507	0.388963
$478$	0	0
$479$	20.6133	0.941845	0.470922	−	0.882175i	$-0.343921\pi$
$479$	20.6133	0.941845	0.470922	+	0.882175i	$0.343921\pi$
$480$	0	0
$481$	−7.96632	−0.363233
$482$	0	0
$483$	−6.44179	−0.293112
$484$	0	0
$485$	−20.7460	−0.942029
$486$	0	0
$487$	−7.47711	−0.338820	−0.169410	−	0.985546i	$-0.554186\pi$
$487$	−7.47711	−0.338820	−0.169410	+	0.985546i	$0.554186\pi$
$488$	0	0
$489$	2.80574	0.126880
$490$	0	0
$491$	11.6250	0.524629	0.262315	−	0.964982i	$-0.415514\pi$
$491$	11.6250	0.524629	0.262315	+	0.964982i	$0.415514\pi$
$492$	0	0
$493$	1.90701	0.0858875
$494$	0	0
$495$	−2.71878	−0.122200
$496$	0	0
$497$	2.99577	0.134379
$498$	0	0
$499$	−24.2714	−1.08654	−0.543268	−	0.839560i	$-0.682813\pi$
$499$	−24.2714	−1.08654	−0.543268	+	0.839560i	$0.682813\pi$
$500$	0	0
$501$	1.93827	0.0865953
$502$	0	0
$503$	31.7648	1.41632	0.708161	−	0.706051i	$-0.249525\pi$
$503$	31.7648	1.41632	0.708161	+	0.706051i	$0.249525\pi$
$504$	0	0
$505$	−10.4309	−0.464169
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−8.06984	−0.357689	−0.178845	−	0.983877i	$-0.557236\pi$
$509$	−8.06984	−0.357689	−0.178845	+	0.983877i	$0.557236\pi$
$510$	0	0
$511$	19.6732	0.870292
$512$	0	0
$513$	0.300590	0.0132714
$514$	0	0
$515$	30.8078	1.35755
$516$	0	0
$517$	−13.1299	−0.577454
$518$	0	0
$519$	−8.55235	−0.375407
$520$	0	0
$521$	16.9986	0.744722	0.372361	−	0.928088i	$-0.378548\pi$
$521$	16.9986	0.744722	0.372361	+	0.928088i	$0.378548\pi$
$522$	0	0
$523$	−7.38591	−0.322963	−0.161482	−	0.986876i	$-0.551627\pi$
$523$	−7.38591	−0.322963	−0.161482	+	0.986876i	$0.551627\pi$
$524$	0	0
$525$	10.2861	0.448920
$526$	0	0
$527$	27.8088	1.21137
$528$	0	0
$529$	−20.7563	−0.902449
$530$	0	0
$531$	−9.22535	−0.400346
$532$	0	0
$533$	1.03087	0.0446518
$534$	0	0
$535$	−42.2841	−1.82810
$536$	0	0
$537$	−17.9312	−0.773789
$538$	0	0
$539$	11.4951	0.495128
$540$	0	0
$541$	34.9901	1.50434	0.752172	−	0.658967i	$-0.229006\pi$
$541$	34.9901	1.50434	0.752172	+	0.658967i	$0.229006\pi$
$542$	0	0
$543$	0.334272	0.0143450
$544$	0	0
$545$	0.0724261	0.00310239
$546$	0	0
$547$	−9.93203	−0.424663	−0.212331	−	0.977198i	$-0.568106\pi$
$547$	−9.93203	−0.424663	−0.212331	+	0.977198i	$0.568106\pi$
$548$	0	0
$549$	−9.86021	−0.420824
$550$	0	0
$551$	−0.0938094	−0.00399642
$552$	0	0
$553$	73.5048	3.12574
$554$	0	0
$555$	21.6587	0.919360
$556$	0	0
$557$	4.53624	0.192207	0.0961034	−	0.995371i	$-0.469362\pi$
$557$	4.53624	0.192207	0.0961034	+	0.995371i	$0.469362\pi$
$558$	0	0
$559$	−12.4906	−0.528297
$560$	0	0
$561$	−6.11056	−0.257988
$562$	0	0
$563$	24.0916	1.01534	0.507669	−	0.861552i	$-0.330507\pi$
$563$	24.0916	1.01534	0.507669	+	0.861552i	$0.330507\pi$
$564$	0	0
$565$	50.1162	2.10841
$566$	0	0
$567$	4.30059	0.180608
$568$	0	0
$569$	2.72583	0.114273	0.0571363	−	0.998366i	$-0.481803\pi$
$569$	2.72583	0.114273	0.0571363	+	0.998366i	$0.481803\pi$
$570$	0	0
$571$	−37.1221	−1.55351	−0.776755	−	0.629802i	$-0.783136\pi$
$571$	−37.1221	−1.55351	−0.776755	+	0.629802i	$0.783136\pi$
$572$	0	0
$573$	20.1538	0.841935
$574$	0	0
$575$	−3.58261	−0.149405
$576$	0	0
$577$	8.06173	0.335614	0.167807	−	0.985820i	$-0.446331\pi$
$577$	8.06173	0.335614	0.167807	+	0.985820i	$0.446331\pi$
$578$	0	0
$579$	22.5689	0.937932
$580$	0	0
$581$	−37.7120	−1.56456
$582$	0	0
$583$	8.49507	0.351830
$584$	0	0
$585$	−2.71878	−0.112408
$586$	0	0
$587$	−1.16924	−0.0482599	−0.0241299	−	0.999709i	$-0.507682\pi$
$587$	−1.16924	−0.0482599	−0.0241299	+	0.999709i	$0.507682\pi$
$588$	0	0
$589$	−1.36797	−0.0563660
$590$	0	0
$591$	−26.6133	−1.09472
$592$	0	0
$593$	13.6550	0.560745	0.280373	−	0.959891i	$-0.409542\pi$
$593$	13.6550	0.560745	0.280373	+	0.959891i	$0.409542\pi$
$594$	0	0
$595$	71.4469	2.92904
$596$	0	0
$597$	−4.81340	−0.196999
$598$	0	0
$599$	23.0289	0.940935	0.470467	−	0.882417i	$-0.344085\pi$
$599$	23.0289	0.940935	0.470467	+	0.882417i	$0.344085\pi$
$600$	0	0
$601$	1.70261	0.0694509	0.0347255	−	0.999397i	$-0.488944\pi$
$601$	1.70261	0.0694509	0.0347255	+	0.999397i	$0.488944\pi$
$602$	0	0
$603$	8.05024	0.327831
$604$	0	0
$605$	−2.71878	−0.110534
$606$	0	0
$607$	−8.51584	−0.345647	−0.172824	−	0.984953i	$-0.555289\pi$
$607$	−8.51584	−0.345647	−0.172824	+	0.984953i	$0.555289\pi$
$608$	0	0
$609$	−1.34215	−0.0543866
$610$	0	0
$611$	−13.1299	−0.531180
$612$	0	0
$613$	−0.185577	−0.00749538	−0.00374769	−	0.999993i	$-0.501193\pi$
$613$	−0.185577	−0.00749538	−0.00374769	+	0.999993i	$0.501193\pi$
$614$	0	0
$615$	−2.80270	−0.113016
$616$	0	0
$617$	−11.7230	−0.471951	−0.235975	−	0.971759i	$-0.575828\pi$
$617$	−11.7230	−0.471951	−0.235975	+	0.971759i	$0.575828\pi$
$618$	0	0
$619$	44.5649	1.79121	0.895607	−	0.444845i	$-0.146741\pi$
$619$	44.5649	1.79121	0.895607	+	0.444845i	$0.146741\pi$
$620$	0	0
$621$	−1.49789	−0.0601081
$622$	0	0
$623$	−13.9663	−0.559549
$624$	0	0
$625$	−31.2383	−1.24953
$626$	0	0
$627$	0.300590	0.0120044
$628$	0	0
$629$	48.6787	1.94095
$630$	0	0
$631$	5.29191	0.210668	0.105334	−	0.994437i	$-0.466409\pi$
$631$	5.29191	0.210668	0.105334	+	0.994437i	$0.466409\pi$
$632$	0	0
$633$	−25.2596	−1.00398
$634$	0	0
$635$	−21.7914	−0.864767
$636$	0	0
$637$	11.4951	0.455451
$638$	0	0
$639$	0.696596	0.0275569
$640$	0	0
$641$	13.1406	0.519023	0.259512	−	0.965740i	$-0.416438\pi$
$641$	13.1406	0.519023	0.259512	+	0.965740i	$0.416438\pi$
$642$	0	0
$643$	−23.8594	−0.940923	−0.470462	−	0.882420i	$-0.655913\pi$
$643$	−23.8594	−0.940923	−0.470462	+	0.882420i	$0.655913\pi$
$644$	0	0
$645$	33.9593	1.33715
$646$	0	0
$647$	12.0472	0.473624	0.236812	−	0.971555i	$-0.423897\pi$
$647$	12.0472	0.473624	0.236812	+	0.971555i	$0.423897\pi$
$648$	0	0
$649$	−9.22535	−0.362127
$650$	0	0
$651$	−19.5717	−0.767076
$652$	0	0
$653$	2.21222	0.0865707	0.0432854	−	0.999063i	$-0.486218\pi$
$653$	2.21222	0.0865707	0.0432854	+	0.999063i	$0.486218\pi$
$654$	0	0
$655$	31.3847	1.22630
$656$	0	0
$657$	4.57454	0.178470
$658$	0	0
$659$	−28.6764	−1.11708	−0.558538	−	0.829479i	$-0.688637\pi$
$659$	−28.6764	−1.11708	−0.558538	+	0.829479i	$0.688637\pi$
$660$	0	0
$661$	−27.9500	−1.08713	−0.543564	−	0.839367i	$-0.682926\pi$
$661$	−27.9500	−1.08713	−0.543564	+	0.839367i	$0.682926\pi$
$662$	0	0
$663$	−6.11056	−0.237315
$664$	0	0
$665$	−3.51460	−0.136291
$666$	0	0
$667$	0.467467	0.0181004
$668$	0	0
$669$	11.8291	0.457340
$670$	0	0
$671$	−9.86021	−0.380649
$672$	0	0
$673$	−28.5534	−1.10065	−0.550327	−	0.834949i	$-0.685497\pi$
$673$	−28.5534	−1.10065	−0.550327	+	0.834949i	$0.685497\pi$
$674$	0	0
$675$	2.39178	0.0920596
$676$	0	0
$677$	7.52290	0.289129	0.144564	−	0.989495i	$-0.453822\pi$
$677$	7.52290	0.289129	0.144564	+	0.989495i	$0.453822\pi$
$678$	0	0
$679$	32.8162	1.25937
$680$	0	0
$681$	−21.1370	−0.809971
$682$	0	0
$683$	1.33324	0.0510152	0.0255076	−	0.999675i	$-0.491880\pi$
$683$	1.33324	0.0510152	0.0255076	+	0.999675i	$0.491880\pi$
$684$	0	0
$685$	16.2524	0.620971
$686$	0	0
$687$	−11.6778	−0.445537
$688$	0	0
$689$	8.49507	0.323636
$690$	0	0
$691$	−8.96796	−0.341157	−0.170579	−	0.985344i	$-0.554564\pi$
$691$	−8.96796	−0.341157	−0.170579	+	0.985344i	$0.554564\pi$
$692$	0	0
$693$	4.30059	0.163366
$694$	0	0
$695$	26.7957	1.01642
$696$	0	0
$697$	−6.29918	−0.238598
$698$	0	0
$699$	10.1156	0.382608
$700$	0	0
$701$	7.66269	0.289416	0.144708	−	0.989474i	$-0.453776\pi$
$701$	7.66269	0.289416	0.144708	+	0.989474i	$0.453776\pi$
$702$	0	0
$703$	−2.39459	−0.0903138
$704$	0	0
$705$	35.6974	1.34444
$706$	0	0
$707$	16.4997	0.620535
$708$	0	0
$709$	17.4253	0.654420	0.327210	−	0.944952i	$-0.393892\pi$
$709$	17.4253	0.654420	0.327210	+	0.944952i	$0.393892\pi$
$710$	0	0
$711$	17.0918	0.640993
$712$	0	0
$713$	6.81679	0.255291
$714$	0	0
$715$	−2.71878	−0.101677
$716$	0	0
$717$	−2.65081	−0.0989963
$718$	0	0
$719$	−6.79764	−0.253509	−0.126755	−	0.991934i	$-0.540456\pi$
$719$	−6.79764	−0.253509	−0.126755	+	0.991934i	$0.540456\pi$
$720$	0	0
$721$	−48.7319	−1.81487
$722$	0	0
$723$	4.15939	0.154689
$724$	0	0
$725$	−0.746437	−0.0277220
$726$	0	0
$727$	30.6932	1.13835	0.569174	−	0.822217i	$-0.307263\pi$
$727$	30.6932	1.13835	0.569174	+	0.822217i	$0.307263\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	76.3247	2.82297
$732$	0	0
$733$	−16.3482	−0.603836	−0.301918	−	0.953334i	$-0.597627\pi$
$733$	−16.3482	−0.603836	−0.301918	+	0.953334i	$0.597627\pi$
$734$	0	0
$735$	−31.2526	−1.15277
$736$	0	0
$737$	8.05024	0.296534
$738$	0	0
$739$	19.9677	0.734525	0.367262	−	0.930117i	$-0.380295\pi$
$739$	19.9677	0.734525	0.367262	+	0.930117i	$0.380295\pi$
$740$	0	0
$741$	0.300590	0.0110424
$742$	0	0
$743$	−18.9756	−0.696148	−0.348074	−	0.937467i	$-0.613164\pi$
$743$	−18.9756	−0.696148	−0.348074	+	0.937467i	$0.613164\pi$
$744$	0	0
$745$	31.4572	1.15250
$746$	0	0
$747$	−8.76902	−0.320842
$748$	0	0
$749$	66.8853	2.44393
$750$	0	0
$751$	−5.94717	−0.217015	−0.108508	−	0.994096i	$-0.534607\pi$
$751$	−5.94717	−0.217015	−0.108508	+	0.994096i	$0.534607\pi$
$752$	0	0
$753$	−15.7115	−0.572559
$754$	0	0
$755$	−8.81724	−0.320892
$756$	0	0
$757$	−21.7275	−0.789698	−0.394849	−	0.918746i	$-0.629203\pi$
$757$	−21.7275	−0.789698	−0.394849	+	0.918746i	$0.629203\pi$
$758$	0	0
$759$	−1.49789	−0.0543698
$760$	0	0
$761$	−11.7241	−0.424997	−0.212499	−	0.977161i	$-0.568160\pi$
$761$	−11.7241	−0.424997	−0.212499	+	0.977161i	$0.568160\pi$
$762$	0	0
$763$	−0.114564	−0.00414750
$764$	0	0
$765$	16.6133	0.600655
$766$	0	0
$767$	−9.22535	−0.333108
$768$	0	0
$769$	27.0676	0.976084	0.488042	−	0.872820i	$-0.337711\pi$
$769$	27.0676	0.976084	0.488042	+	0.872820i	$0.337711\pi$
$770$	0	0
$771$	4.53944	0.163484
$772$	0	0
$773$	17.6574	0.635094	0.317547	−	0.948243i	$-0.397141\pi$
$773$	17.6574	0.635094	0.317547	+	0.948243i	$0.397141\pi$
$774$	0	0
$775$	−10.8848	−0.390995
$776$	0	0
$777$	−34.2599	−1.22907
$778$	0	0
$779$	0.309868	0.0111022
$780$	0	0
$781$	0.696596	0.0249262
$782$	0	0
$783$	−0.312085	−0.0111530
$784$	0	0
$785$	−22.3311	−0.797030
$786$	0	0
$787$	19.9677	0.711773	0.355886	−	0.934529i	$-0.384179\pi$
$787$	19.9677	0.711773	0.355886	+	0.934529i	$0.384179\pi$
$788$	0	0
$789$	−8.38006	−0.298338
$790$	0	0
$791$	−79.2742	−2.81867
$792$	0	0
$793$	−9.86021	−0.350146
$794$	0	0
$795$	−23.0963	−0.819140
$796$	0	0
$797$	15.2343	0.539625	0.269812	−	0.962913i	$-0.413038\pi$
$797$	15.2343	0.539625	0.269812	+	0.962913i	$0.413038\pi$
$798$	0	0
$799$	80.2312	2.83838
$800$	0	0
$801$	−3.24754	−0.114746
$802$	0	0
$803$	4.57454	0.161432
$804$	0	0
$805$	17.5138	0.617282
$806$	0	0
$807$	−4.95140	−0.174297
$808$	0	0
$809$	−34.4551	−1.21138	−0.605689	−	0.795701i	$-0.707103\pi$
$809$	−34.4551	−1.21138	−0.605689	+	0.795701i	$0.707103\pi$
$810$	0	0
$811$	−4.75409	−0.166939	−0.0834693	−	0.996510i	$-0.526600\pi$
$811$	−4.75409	−0.166939	−0.0834693	+	0.996510i	$0.526600\pi$
$812$	0	0
$813$	24.5750	0.861882
$814$	0	0
$815$	−7.62820	−0.267204
$816$	0	0
$817$	−3.75455	−0.131355
$818$	0	0
$819$	4.30059	0.150275
$820$	0	0
$821$	7.36700	0.257110	0.128555	−	0.991702i	$-0.458966\pi$
$821$	7.36700	0.257110	0.128555	+	0.991702i	$0.458966\pi$
$822$	0	0
$823$	−35.5385	−1.23879	−0.619397	−	0.785078i	$-0.712623\pi$
$823$	−35.5385	−1.23879	−0.619397	+	0.785078i	$0.712623\pi$
$824$	0	0
$825$	2.39178	0.0832710
$826$	0	0
$827$	16.1506	0.561610	0.280805	−	0.959765i	$-0.409399\pi$
$827$	16.1506	0.561610	0.280805	+	0.959765i	$0.409399\pi$
$828$	0	0
$829$	28.9986	1.00716	0.503581	−	0.863948i	$-0.332015\pi$
$829$	28.9986	1.00716	0.503581	+	0.863948i	$0.332015\pi$
$830$	0	0
$831$	27.5913	0.957132
$832$	0	0
$833$	−70.2413	−2.43372
$834$	0	0
$835$	−5.26972	−0.182366
$836$	0	0
$837$	−4.55094	−0.157303
$838$	0	0
$839$	12.7246	0.439304	0.219652	−	0.975578i	$-0.429508\pi$
$839$	12.7246	0.439304	0.219652	+	0.975578i	$0.429508\pi$
$840$	0	0
$841$	−28.9026	−0.996641
$842$	0	0
$843$	10.3478	0.356397
$844$	0	0
$845$	−2.71878	−0.0935290
$846$	0	0
$847$	4.30059	0.147770
$848$	0	0
$849$	1.97640	0.0678299
$850$	0	0
$851$	11.9326	0.409045
$852$	0	0
$853$	31.0434	1.06291	0.531453	−	0.847088i	$-0.321646\pi$
$853$	31.0434	1.06291	0.531453	+	0.847088i	$0.321646\pi$
$854$	0	0
$855$	−0.817238	−0.0279489
$856$	0	0
$857$	36.4658	1.24565	0.622825	−	0.782361i	$-0.285985\pi$
$857$	36.4658	1.24565	0.622825	+	0.782361i	$0.285985\pi$
$858$	0	0
$859$	44.3200	1.51218	0.756089	−	0.654468i	$-0.227108\pi$
$859$	44.3200	1.51218	0.756089	+	0.654468i	$0.227108\pi$
$860$	0	0
$861$	4.43334	0.151088
$862$	0	0
$863$	35.1467	1.19641	0.598204	−	0.801344i	$-0.295881\pi$
$863$	35.1467	1.19641	0.598204	+	0.801344i	$0.295881\pi$
$864$	0	0
$865$	23.2520	0.790591
$866$	0	0
$867$	20.3389	0.690747
$868$	0	0
$869$	17.0918	0.579800
$870$	0	0
$871$	8.05024	0.272772
$872$	0	0
$873$	7.63063	0.258258
$874$	0	0
$875$	30.4963	1.03096
$876$	0	0
$877$	−11.2927	−0.381328	−0.190664	−	0.981655i	$-0.561064\pi$
$877$	−11.2927	−0.381328	−0.190664	+	0.981655i	$0.561064\pi$
$878$	0	0
$879$	13.5877	0.458301
$880$	0	0
$881$	31.7503	1.06969	0.534847	−	0.844949i	$-0.320369\pi$
$881$	31.7503	1.06969	0.534847	+	0.844949i	$0.320369\pi$
$882$	0	0
$883$	−42.8824	−1.44311	−0.721554	−	0.692358i	$-0.756572\pi$
$883$	−42.8824	−1.44311	−0.721554	+	0.692358i	$0.756572\pi$
$884$	0	0
$885$	25.0817	0.843113
$886$	0	0
$887$	−32.8290	−1.10229	−0.551144	−	0.834410i	$-0.685809\pi$
$887$	−32.8290	−1.10229	−0.551144	+	0.834410i	$0.685809\pi$
$888$	0	0
$889$	34.4698	1.15608
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	−3.94672	−0.132072
$894$	0	0
$895$	48.7511	1.62957
$896$	0	0
$897$	−1.49789	−0.0500130
$898$	0	0
$899$	1.42028	0.0473690
$900$	0	0
$901$	−51.9096	−1.72936
$902$	0	0
$903$	−53.7170	−1.78759
$904$	0	0
$905$	−0.908812	−0.0302099
$906$	0	0
$907$	−43.1597	−1.43309	−0.716546	−	0.697539i	$-0.754278\pi$
$907$	−43.1597	−1.43309	−0.716546	+	0.697539i	$0.754278\pi$
$908$	0	0
$909$	3.83661	0.127252
$910$	0	0
$911$	28.7690	0.953160	0.476580	−	0.879131i	$-0.341876\pi$
$911$	28.7690	0.953160	0.476580	+	0.879131i	$0.341876\pi$
$912$	0	0
$913$	−8.76902	−0.290212
$914$	0	0
$915$	26.8078	0.886237
$916$	0	0
$917$	−49.6446	−1.63941
$918$	0	0
$919$	−26.5804	−0.876807	−0.438403	−	0.898778i	$-0.644456\pi$
$919$	−26.5804	−0.876807	−0.438403	+	0.898778i	$0.644456\pi$
$920$	0	0
$921$	−31.8302	−1.04884
$922$	0	0
$923$	0.696596	0.0229287
$924$	0	0
$925$	−19.0537	−0.626481
$926$	0	0
$927$	−11.3315	−0.372174
$928$	0	0
$929$	28.1614	0.923946	0.461973	−	0.886894i	$-0.347142\pi$
$929$	28.1614	0.923946	0.461973	+	0.886894i	$0.347142\pi$
$930$	0	0
$931$	3.45530	0.113243
$932$	0	0
$933$	−6.64697	−0.217612
$934$	0	0
$935$	16.6133	0.543312
$936$	0	0
$937$	−35.6727	−1.16537	−0.582687	−	0.812696i	$-0.697999\pi$
$937$	−35.6727	−1.16537	−0.582687	+	0.812696i	$0.697999\pi$
$938$	0	0
$939$	−22.6171	−0.738082
$940$	0	0
$941$	18.0210	0.587468	0.293734	−	0.955887i	$-0.405102\pi$
$941$	18.0210	0.587468	0.293734	+	0.955887i	$0.405102\pi$
$942$	0	0
$943$	−1.54412	−0.0502835
$944$	0	0
$945$	−11.6924	−0.380353
$946$	0	0
$947$	25.0093	0.812693	0.406346	−	0.913719i	$-0.366803\pi$
$947$	25.0093	0.812693	0.406346	+	0.913719i	$0.366803\pi$
$948$	0	0
$949$	4.57454	0.148496
$950$	0	0
$951$	26.3971	0.855983
$952$	0	0
$953$	18.2302	0.590534	0.295267	−	0.955415i	$-0.404591\pi$
$953$	18.2302	0.590534	0.295267	+	0.955415i	$0.404591\pi$
$954$	0	0
$955$	−54.7937	−1.77308
$956$	0	0
$957$	−0.312085	−0.0100883
$958$	0	0
$959$	−25.7081	−0.830159
$960$	0	0
$961$	−10.2889	−0.331901
$962$	0	0
$963$	15.5526	0.501175
$964$	0	0
$965$	−61.3600	−1.97525
$966$	0	0
$967$	−21.5792	−0.693941	−0.346970	−	0.937876i	$-0.612790\pi$
$967$	−21.5792	−0.693941	−0.346970	+	0.937876i	$0.612790\pi$
$968$	0	0
$969$	−1.83677	−0.0590056
$970$	0	0
$971$	−52.4263	−1.68244	−0.841220	−	0.540693i	$-0.818162\pi$
$971$	−52.4263	−1.68244	−0.841220	+	0.540693i	$0.818162\pi$
$972$	0	0
$973$	−42.3856	−1.35882
$974$	0	0
$975$	2.39178	0.0765982
$976$	0	0
$977$	50.5189	1.61624	0.808122	−	0.589015i	$-0.200484\pi$
$977$	50.5189	1.61624	0.808122	+	0.589015i	$0.200484\pi$
$978$	0	0
$979$	−3.24754	−0.103792
$980$	0	0
$981$	−0.0266392	−0.000850523 0
$982$	0	0
$983$	29.2556	0.933110	0.466555	−	0.884492i	$-0.345495\pi$
$983$	29.2556	0.933110	0.466555	+	0.884492i	$0.345495\pi$
$984$	0	0
$985$	72.3557	2.30545
$986$	0	0
$987$	−56.4665	−1.79735
$988$	0	0
$989$	18.7095	0.594928
$990$	0	0
$991$	12.1117	0.384742	0.192371	−	0.981322i	$-0.438382\pi$
$991$	12.1117	0.384742	0.192371	+	0.981322i	$0.438382\pi$
$992$	0	0
$993$	−20.1227	−0.638573
$994$	0	0
$995$	13.0866	0.414872
$996$	0	0
$997$	−47.9395	−1.51826	−0.759129	−	0.650940i	$-0.774375\pi$
$997$	−47.9395	−1.51826	−0.759129	+	0.650940i	$0.774375\pi$
$998$	0	0
$999$	−7.96632	−0.252043

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.bz.1.1		4
4.3	odd	2		429.2.a.h.1.1	✓	4
12.11	even	2		1287.2.a.m.1.4		4
44.43	even	2		4719.2.a.z.1.4		4
52.51	odd	2		5577.2.a.m.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
429.2.a.h.1.1	✓	4	4.3	odd	2
1287.2.a.m.1.4		4	12.11	even	2
4719.2.a.z.1.4		4	44.43	even	2
5577.2.a.m.1.4		4	52.51	odd	2
6864.2.a.bz.1.1		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:	\( N \)	\(=\)	\( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6864.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -2.71878 q^{5} +4.30059 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -2.71878 q^{5} +4.30059 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} -2.71878 q^{15} -6.11056 q^{17} +0.300590 q^{19} +4.30059 q^{21} -1.49789 q^{23} +2.39178 q^{25} +1.00000 q^{27} -0.312085 q^{29} -4.55094 q^{31} +1.00000 q^{33} -11.6924 q^{35} -7.96632 q^{37} +1.00000 q^{39} +1.03087 q^{41} -12.4906 q^{43} -2.71878 q^{45} -13.1299 q^{47} +11.4951 q^{49} -6.11056 q^{51} +8.49507 q^{53} -2.71878 q^{55} +0.300590 q^{57} -9.22535 q^{59} -9.86021 q^{61} +4.30059 q^{63} -2.71878 q^{65} +8.05024 q^{67} -1.49789 q^{69} +0.696596 q^{71} +4.57454 q^{73} +2.39178 q^{75} +4.30059 q^{77} +17.0918 q^{79} +1.00000 q^{81} -8.76902 q^{83} +16.6133 q^{85} -0.312085 q^{87} -3.24754 q^{89} +4.30059 q^{91} -4.55094 q^{93} -0.817238 q^{95} +7.63063 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} - 2 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 - 2 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} - 2 q^{7} + 4 q^{9} + 4 q^{11} + 4 q^{13} - 8 q^{17} - 18 q^{19} - 2 q^{21} + 4 q^{25} + 4 q^{27} - 10 q^{29} - 12 q^{31} + 4 q^{33} - 22 q^{35} - 2 q^{37} + 4 q^{39} + 2 q^{41} - 28 q^{43} - 6 q^{47} + 8 q^{49} - 8 q^{51} - 4 q^{53} - 18 q^{57} - 16 q^{59} - 10 q^{61} - 2 q^{63} - 10 q^{71} - 6 q^{73} + 4 q^{75} - 2 q^{77} + 8 q^{79} + 4 q^{81} + 8 q^{83} - 18 q^{85} - 10 q^{87} + 6 q^{89} - 2 q^{91} - 12 q^{93} - 22 q^{95} + 10 q^{97} + 4 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 - 2 * q^7 + 4 * q^9 + 4 * q^11 + 4 * q^13 - 8 * q^17 - 18 * q^19 - 2 * q^21 + 4 * q^25 + 4 * q^27 - 10 * q^29 - 12 * q^31 + 4 * q^33 - 22 * q^35 - 2 * q^37 + 4 * q^39 + 2 * q^41 - 28 * q^43 - 6 * q^47 + 8 * q^49 - 8 * q^51 - 4 * q^53 - 18 * q^57 - 16 * q^59 - 10 * q^61 - 2 * q^63 - 10 * q^71 - 6 * q^73 + 4 * q^75 - 2 * q^77 + 8 * q^79 + 4 * q^81 + 8 * q^83 - 18 * q^85 - 10 * q^87 + 6 * q^89 - 2 * q^91 - 12 * q^93 - 22 * q^95 + 10 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more