LMFDB - Embedded newform 6864.2.a.cb.1.2

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:	$0$
Dimension:	$4$
Coefficient field:	4.4.90996.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 11x^{2} - 3x + 2 $ x^4 - x^3 - 11x^2 - 3x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1716)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.309233$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +0.472388 q^{5} +5.15838 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +0.472388 q^{5} +5.15838 q^{7} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} +0.472388 q^{15} +6.24924 q^{17} +5.15838 q^{19} +5.15838 q^{21} +6.83207 q^{23} -4.77685 q^{25} +1.00000 q^{27} +8.68600 q^{29} -1.52761 q^{31} -1.00000 q^{33} +2.43676 q^{35} -0.436758 q^{37} +1.00000 q^{39} -4.39532 q^{41} -6.06753 q^{43} +0.472388 q^{45} -13.0457 q^{47} +19.6089 q^{49} +6.24924 q^{51} -1.05522 q^{53} -0.472388 q^{55} +5.15838 q^{57} +0.944776 q^{59} +5.38153 q^{61} +5.15838 q^{63} +0.472388 q^{65} -13.0813 q^{67} +6.83207 q^{69} -3.67369 q^{71} -4.21361 q^{73} -4.77685 q^{75} -5.15838 q^{77} -9.12275 q^{79} +1.00000 q^{81} +0.292156 q^{83} +2.95206 q^{85} +8.68600 q^{87} +10.9091 q^{89} +5.15838 q^{91} -1.52761 q^{93} +2.43676 q^{95} -18.3072 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 4 q^{3} + 2 q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q + 4 * q^3 + 2 * q^5 - 2 * q^7 + 4 * q^9	$ 4 q + 4 q^{3} + 2 q^{5} - 2 q^{7} + 4 q^{9} - 4 q^{11} + 4 q^{13} + 2 q^{15} + 2 q^{17} - 2 q^{19} - 2 q^{21} + 4 q^{23} + 4 q^{25} + 4 q^{27} + 12 q^{29} - 6 q^{31} - 4 q^{33} + 10 q^{35} - 2 q^{37} + 4 q^{39} + 6 q^{41} - 2 q^{43} + 2 q^{45} - 6 q^{47} + 32 q^{49} + 2 q^{51} - 4 q^{53} - 2 q^{55} - 2 q^{57} + 4 q^{59} + 22 q^{61} - 2 q^{63} + 2 q^{65} - 6 q^{67} + 4 q^{69} - 14 q^{71} + 6 q^{73} + 4 q^{75} + 2 q^{77} - 14 q^{79} + 4 q^{81} + 34 q^{85} + 12 q^{87} + 44 q^{89} - 2 q^{91} - 6 q^{93} + 10 q^{95} + 18 q^{97} - 4 q^{99}+O(q^{100}) $ 4 * q + 4 * q^3 + 2 * q^5 - 2 * q^7 + 4 * q^9 - 4 * q^11 + 4 * q^13 + 2 * q^15 + 2 * q^17 - 2 * q^19 - 2 * q^21 + 4 * q^23 + 4 * q^25 + 4 * q^27 + 12 * q^29 - 6 * q^31 - 4 * q^33 + 10 * q^35 - 2 * q^37 + 4 * q^39 + 6 * q^41 - 2 * q^43 + 2 * q^45 - 6 * q^47 + 32 * q^49 + 2 * q^51 - 4 * q^53 - 2 * q^55 - 2 * q^57 + 4 * q^59 + 22 * q^61 - 2 * q^63 + 2 * q^65 - 6 * q^67 + 4 * q^69 - 14 * q^71 + 6 * q^73 + 4 * q^75 + 2 * q^77 - 14 * q^79 + 4 * q^81 + 34 * q^85 + 12 * q^87 + 44 * q^89 - 2 * q^91 - 6 * q^93 + 10 * q^95 + 18 * 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
$5$	0.472388	0.211258	0.105629	−	0.994406i	$-0.466314\pi$
$5$	0.472388	0.211258	0.105629	+	0.994406i	$0.466314\pi$
$6$	0	0
$7$	5.15838	1.94969	0.974843	−	0.222893i	$-0.0715501\pi$
$7$	5.15838	1.94969	0.974843	+	0.222893i	$0.0715501\pi$
$8$	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
$15$	0.472388	0.121970
$16$	0	0
$17$	6.24924	1.51566	0.757831	−	0.652450i	$-0.226259\pi$
$17$	6.24924	1.51566	0.757831	+	0.652450i	$0.226259\pi$
$18$	0	0
$19$	5.15838	1.18341	0.591707	−	0.806153i	$-0.298454\pi$
$19$	5.15838	1.18341	0.591707	+	0.806153i	$0.298454\pi$
$20$	0	0
$21$	5.15838	1.12565
$22$	0	0
$23$	6.83207	1.42459	0.712293	−	0.701882i	$-0.247657\pi$
$23$	6.83207	1.42459	0.712293	+	0.701882i	$0.247657\pi$
$24$	0	0
$25$	−4.77685	−0.955370
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	8.68600	1.61295	0.806474	−	0.591269i	$-0.201373\pi$
$29$	8.68600	1.61295	0.806474	+	0.591269i	$0.201373\pi$
$30$	0	0
$31$	−1.52761	−0.274367	−0.137184	−	0.990546i	$-0.543805\pi$
$31$	−1.52761	−0.274367	−0.137184	+	0.990546i	$0.543805\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	2.43676	0.411887
$36$	0	0
$37$	−0.436758	−0.0718026	−0.0359013	−	0.999355i	$-0.511430\pi$
$37$	−0.436758	−0.0718026	−0.0359013	+	0.999355i	$0.511430\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−4.39532	−0.686433	−0.343216	−	0.939256i	$-0.611516\pi$
$41$	−4.39532	−0.686433	−0.343216	+	0.939256i	$0.611516\pi$
$42$	0	0
$43$	−6.06753	−0.925290	−0.462645	−	0.886544i	$-0.653099\pi$
$43$	−6.06753	−0.925290	−0.462645	+	0.886544i	$0.653099\pi$
$44$	0	0
$45$	0.472388	0.0704194
$46$	0	0
$47$	−13.0457	−1.90291	−0.951454	−	0.307791i	$-0.900410\pi$
$47$	−13.0457	−1.90291	−0.951454	+	0.307791i	$0.900410\pi$
$48$	0	0
$49$	19.6089	2.80127
$50$	0	0
$51$	6.24924	0.875068
$52$	0	0
$53$	−1.05522	−0.144946	−0.0724731	−	0.997370i	$-0.523089\pi$
$53$	−1.05522	−0.144946	−0.0724731	+	0.997370i	$0.523089\pi$
$54$	0	0
$55$	−0.472388	−0.0636968
$56$	0	0
$57$	5.15838	0.683245
$58$	0	0
$59$	0.944776	0.122999	0.0614997	−	0.998107i	$-0.480412\pi$
$59$	0.944776	0.122999	0.0614997	+	0.998107i	$0.480412\pi$
$60$	0	0
$61$	5.38153	0.689035	0.344517	−	0.938780i	$-0.388043\pi$
$61$	5.38153	0.689035	0.344517	+	0.938780i	$0.388043\pi$
$62$	0	0
$63$	5.15838	0.649895
$64$	0	0
$65$	0.472388	0.0585925
$66$	0	0
$67$	−13.0813	−1.59814	−0.799068	−	0.601240i	$-0.794673\pi$
$67$	−13.0813	−1.59814	−0.799068	+	0.601240i	$0.794673\pi$
$68$	0	0
$69$	6.83207	0.822485
$70$	0	0
$71$	−3.67369	−0.435987	−0.217993	−	0.975950i	$-0.569951\pi$
$71$	−3.67369	−0.435987	−0.217993	+	0.975950i	$0.569951\pi$
$72$	0	0
$73$	−4.21361	−0.493166	−0.246583	−	0.969122i	$-0.579308\pi$
$73$	−4.21361	−0.493166	−0.246583	+	0.969122i	$0.579308\pi$
$74$	0	0
$75$	−4.77685	−0.551583
$76$	0	0
$77$	−5.15838	−0.587852
$78$	0	0
$79$	−9.12275	−1.02639	−0.513195	−	0.858272i	$-0.671538\pi$
$79$	−9.12275	−1.02639	−0.513195	+	0.858272i	$0.671538\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	0.292156	0.0320683	0.0160341	−	0.999871i	$-0.494896\pi$
$83$	0.292156	0.0320683	0.0160341	+	0.999871i	$0.494896\pi$
$84$	0	0
$85$	2.95206	0.320196
$86$	0	0
$87$	8.68600	0.931237
$88$	0	0
$89$	10.9091	1.15637	0.578184	−	0.815907i	$-0.303762\pi$
$89$	10.9091	1.15637	0.578184	+	0.815907i	$0.303762\pi$
$90$	0	0
$91$	5.15838	0.540746
$92$	0	0
$93$	−1.52761	−0.158406
$94$	0	0
$95$	2.43676	0.250006
$96$	0	0
$97$	−18.3072	−1.85882	−0.929409	−	0.369053i	$-0.879682\pi$
$97$	−18.3072	−1.85882	−0.929409	+	0.369053i	$0.879682\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	12.0675	1.20076	0.600382	−	0.799713i	$-0.295015\pi$
$101$	12.0675	1.20076	0.600382	+	0.799713i	$0.295015\pi$
$102$	0	0
$103$	−12.4272	−1.22449	−0.612245	−	0.790668i	$-0.709733\pi$
$103$	−12.4272	−1.22449	−0.612245	+	0.790668i	$0.709733\pi$
$104$	0	0
$105$	2.43676	0.237803
$106$	0	0
$107$	−3.05522	−0.295360	−0.147680	−	0.989035i	$-0.547181\pi$
$107$	−3.05522	−0.295360	−0.147680	+	0.989035i	$0.547181\pi$
$108$	0	0
$109$	10.1032	0.967707	0.483854	−	0.875149i	$-0.339237\pi$
$109$	10.1032	0.967707	0.483854	+	0.875149i	$0.339237\pi$
$110$	0	0
$111$	−0.436758	−0.0414553
$112$	0	0
$113$	−9.26154	−0.871253	−0.435626	−	0.900128i	$-0.643473\pi$
$113$	−9.26154	−0.871253	−0.435626	+	0.900128i	$0.643473\pi$
$114$	0	0
$115$	3.22739	0.300956
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	32.2360	2.95507
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−4.39532	−0.396312
$124$	0	0
$125$	−4.61847	−0.413088
$126$	0	0
$127$	−12.5755	−1.11590	−0.557950	−	0.829875i	$-0.688412\pi$
$127$	−12.5755	−1.11590	−0.557950	+	0.829875i	$0.688412\pi$
$128$	0	0
$129$	−6.06753	−0.534216
$130$	0	0
$131$	−3.55370	−0.310488	−0.155244	−	0.987876i	$-0.549616\pi$
$131$	−3.55370	−0.310488	−0.155244	+	0.987876i	$0.549616\pi$
$132$	0	0
$133$	26.6089	2.30729
$134$	0	0
$135$	0.472388	0.0406567
$136$	0	0
$137$	−21.7244	−1.85604	−0.928020	−	0.372531i	$-0.878490\pi$
$137$	−21.7244	−1.85604	−0.928020	+	0.372531i	$0.878490\pi$
$138$	0	0
$139$	5.74122	0.486964	0.243482	−	0.969905i	$-0.421710\pi$
$139$	5.74122	0.486964	0.243482	+	0.969905i	$0.421710\pi$
$140$	0	0
$141$	−13.0457	−1.09864
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	4.10316	0.340749
$146$	0	0
$147$	19.6089	1.61732
$148$	0	0
$149$	12.3953	1.01546	0.507732	−	0.861515i	$-0.330484\pi$
$149$	12.3953	1.01546	0.507732	+	0.861515i	$0.330484\pi$
$150$	0	0
$151$	6.03190	0.490869	0.245435	−	0.969413i	$-0.421069\pi$
$151$	6.03190	0.490869	0.245435	+	0.969413i	$0.421069\pi$
$152$	0	0
$153$	6.24924	0.505221
$154$	0	0
$155$	−0.721626	−0.0579624
$156$	0	0
$157$	−20.0936	−1.60365	−0.801823	−	0.597562i	$-0.796136\pi$
$157$	−20.0936	−1.60365	−0.801823	+	0.597562i	$0.796136\pi$
$158$	0	0
$159$	−1.05522	−0.0836847
$160$	0	0
$161$	35.2425	2.77749
$162$	0	0
$163$	−14.5733	−1.14147	−0.570734	−	0.821135i	$-0.693341\pi$
$163$	−14.5733	−1.14147	−0.570734	+	0.821135i	$0.693341\pi$
$164$	0	0
$165$	−0.472388	−0.0367754
$166$	0	0
$167$	2.31677	0.179277	0.0896384	−	0.995974i	$-0.471429\pi$
$167$	2.31677	0.179277	0.0896384	+	0.995974i	$0.471429\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	5.15838	0.394471
$172$	0	0
$173$	17.7317	1.34811	0.674057	−	0.738679i	$-0.264550\pi$
$173$	17.7317	1.34811	0.674057	+	0.738679i	$0.264550\pi$
$174$	0	0
$175$	−24.6408	−1.86267
$176$	0	0
$177$	0.944776	0.0710137
$178$	0	0
$179$	11.1242	0.831464	0.415732	−	0.909487i	$-0.363525\pi$
$179$	11.1242	0.831464	0.415732	+	0.909487i	$0.363525\pi$
$180$	0	0
$181$	24.0936	1.79086	0.895432	−	0.445198i	$-0.146867\pi$
$181$	24.0936	1.79086	0.895432	+	0.445198i	$0.146867\pi$
$182$	0	0
$183$	5.38153	0.397814
$184$	0	0
$185$	−0.206319	−0.0151689
$186$	0	0
$187$	−6.24924	−0.456990
$188$	0	0
$189$	5.15838	0.375217
$190$	0	0
$191$	16.9448	1.22608	0.613040	−	0.790052i	$-0.289946\pi$
$191$	16.9448	1.22608	0.613040	+	0.790052i	$0.289946\pi$
$192$	0	0
$193$	9.22964	0.664364	0.332182	−	0.943215i	$-0.392215\pi$
$193$	9.22964	0.664364	0.332182	+	0.943215i	$0.392215\pi$
$194$	0	0
$195$	0.472388	0.0338284
$196$	0	0
$197$	−1.78639	−0.127275	−0.0636376	−	0.997973i	$-0.520270\pi$
$197$	−1.78639	−0.127275	−0.0636376	+	0.997973i	$0.520270\pi$
$198$	0	0
$199$	−17.0798	−1.21076	−0.605379	−	0.795938i	$-0.706978\pi$
$199$	−17.0798	−1.21076	−0.605379	+	0.795938i	$0.706978\pi$
$200$	0	0
$201$	−13.0813	−0.922685
$202$	0	0
$203$	44.8057	3.14474
$204$	0	0
$205$	−2.07629	−0.145015
$206$	0	0
$207$	6.83207	0.474862
$208$	0	0
$209$	−5.15838	−0.356813
$210$	0	0
$211$	0.541393	0.0372711	0.0186355	−	0.999826i	$-0.494068\pi$
$211$	0.541393	0.0372711	0.0186355	+	0.999826i	$0.494068\pi$
$212$	0	0
$213$	−3.67369	−0.251717
$214$	0	0
$215$	−2.86623	−0.195475
$216$	0	0
$217$	−7.88001	−0.534930
$218$	0	0
$219$	−4.21361	−0.284729
$220$	0	0
$221$	6.24924	0.420369
$222$	0	0
$223$	−15.1059	−1.01157	−0.505784	−	0.862660i	$-0.668797\pi$
$223$	−15.1059	−1.01157	−0.505784	+	0.862660i	$0.668797\pi$
$224$	0	0
$225$	−4.77685	−0.318457
$226$	0	0
$227$	30.0595	1.99512	0.997558	−	0.0698384i	$-0.0222484\pi$
$227$	30.0595	1.99512	0.997558	+	0.0698384i	$0.0222484\pi$
$228$	0	0
$229$	−14.1722	−0.936523	−0.468262	−	0.883590i	$-0.655119\pi$
$229$	−14.1722	−0.936523	−0.468262	+	0.883590i	$0.655119\pi$
$230$	0	0
$231$	−5.15838	−0.339397
$232$	0	0
$233$	12.9782	0.850227	0.425113	−	0.905140i	$-0.360234\pi$
$233$	12.9782	0.850227	0.425113	+	0.905140i	$0.360234\pi$
$234$	0	0
$235$	−6.16262	−0.402005
$236$	0	0
$237$	−9.12275	−0.592587
$238$	0	0
$239$	−19.9490	−1.29039	−0.645197	−	0.764016i	$-0.723225\pi$
$239$	−19.9490	−1.29039	−0.645197	+	0.764016i	$0.723225\pi$
$240$	0	0
$241$	−20.2455	−1.30413	−0.652064	−	0.758164i	$-0.726097\pi$
$241$	−20.2455	−1.30413	−0.652064	+	0.758164i	$0.726097\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	9.26302	0.591793
$246$	0	0
$247$	5.15838	0.328220
$248$	0	0
$249$	0.292156	0.0185146
$250$	0	0
$251$	−15.6888	−0.990266	−0.495133	−	0.868817i	$-0.664881\pi$
$251$	−15.6888	−0.990266	−0.495133	+	0.868817i	$0.664881\pi$
$252$	0	0
$253$	−6.83207	−0.429529
$254$	0	0
$255$	2.95206	0.184865
$256$	0	0
$257$	−8.76307	−0.546625	−0.273313	−	0.961925i	$-0.588119\pi$
$257$	−8.76307	−0.546625	−0.273313	+	0.961925i	$0.588119\pi$
$258$	0	0
$259$	−2.25297	−0.139993
$260$	0	0
$261$	8.68600	0.537650
$262$	0	0
$263$	4.07126	0.251045	0.125522	−	0.992091i	$-0.459939\pi$
$263$	4.07126	0.251045	0.125522	+	0.992091i	$0.459939\pi$
$264$	0	0
$265$	−0.498475	−0.0306211
$266$	0	0
$267$	10.9091	0.667629
$268$	0	0
$269$	−15.1511	−0.923779	−0.461889	−	0.886938i	$-0.652828\pi$
$269$	−15.1511	−0.923779	−0.461889	+	0.886938i	$0.652828\pi$
$270$	0	0
$271$	−29.8778	−1.81494	−0.907472	−	0.420112i	$-0.861991\pi$
$271$	−29.8778	−1.81494	−0.907472	+	0.420112i	$0.861991\pi$
$272$	0	0
$273$	5.15838	0.312200
$274$	0	0
$275$	4.77685	0.288055
$276$	0	0
$277$	10.4272	0.626511	0.313255	−	0.949669i	$-0.398580\pi$
$277$	10.4272	0.626511	0.313255	+	0.949669i	$0.398580\pi$
$278$	0	0
$279$	−1.52761	−0.0914557
$280$	0	0
$281$	2.21361	0.132053	0.0660264	−	0.997818i	$-0.478968\pi$
$281$	2.21361	0.132053	0.0660264	+	0.997818i	$0.478968\pi$
$282$	0	0
$283$	−1.31400	−0.0781094	−0.0390547	−	0.999237i	$-0.512435\pi$
$283$	−1.31400	−0.0781094	−0.0390547	+	0.999237i	$0.512435\pi$
$284$	0	0
$285$	2.43676	0.144341
$286$	0	0
$287$	−22.6727	−1.33833
$288$	0	0
$289$	22.0530	1.29723
$290$	0	0
$291$	−18.3072	−1.07319
$292$	0	0
$293$	−13.7864	−0.805410	−0.402705	−	0.915330i	$-0.631930\pi$
$293$	−13.7864	−0.805410	−0.402705	+	0.915330i	$0.631930\pi$
$294$	0	0
$295$	0.446301	0.0259846
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	6.83207	0.395109
$300$	0	0
$301$	−31.2986	−1.80402
$302$	0	0
$303$	12.0675	0.693261
$304$	0	0
$305$	2.54217	0.145564
$306$	0	0
$307$	19.4752	1.11151	0.555753	−	0.831348i	$-0.312430\pi$
$307$	19.4752	1.11151	0.555753	+	0.831348i	$0.312430\pi$
$308$	0	0
$309$	−12.4272	−0.706960
$310$	0	0
$311$	−5.59514	−0.317271	−0.158636	−	0.987337i	$-0.550710\pi$
$311$	−5.59514	−0.317271	−0.158636	+	0.987337i	$0.550710\pi$
$312$	0	0
$313$	−5.48469	−0.310013	−0.155007	−	0.987913i	$-0.549540\pi$
$313$	−5.48469	−0.310013	−0.155007	+	0.987913i	$0.549540\pi$
$314$	0	0
$315$	2.43676	0.137296
$316$	0	0
$317$	−17.9644	−1.00898	−0.504490	−	0.863418i	$-0.668319\pi$
$317$	−17.9644	−1.00898	−0.504490	+	0.863418i	$0.668319\pi$
$318$	0	0
$319$	−8.68600	−0.486322
$320$	0	0
$321$	−3.05522	−0.170526
$322$	0	0
$323$	32.2360	1.79366
$324$	0	0
$325$	−4.77685	−0.264972
$326$	0	0
$327$	10.1032	0.558706
$328$	0	0
$329$	−67.2946	−3.71007
$330$	0	0
$331$	25.8348	1.42001	0.710006	−	0.704196i	$-0.248692\pi$
$331$	25.8348	1.42001	0.710006	+	0.704196i	$0.248692\pi$
$332$	0	0
$333$	−0.436758	−0.0239342
$334$	0	0
$335$	−6.17945	−0.337620
$336$	0	0
$337$	−6.94478	−0.378306	−0.189153	−	0.981948i	$-0.560574\pi$
$337$	−6.94478	−0.378306	−0.189153	+	0.981948i	$0.560574\pi$
$338$	0	0
$339$	−9.26154	−0.503018
$340$	0	0
$341$	1.52761	0.0827248
$342$	0	0
$343$	65.0417	3.51192
$344$	0	0
$345$	3.22739	0.173757
$346$	0	0
$347$	10.0095	0.537340	0.268670	−	0.963232i	$-0.413416\pi$
$347$	10.0095	0.537340	0.268670	+	0.963232i	$0.413416\pi$
$348$	0	0
$349$	29.4824	1.57816	0.789079	−	0.614291i	$-0.210558\pi$
$349$	29.4824	1.57816	0.789079	+	0.614291i	$0.210558\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	4.76454	0.253591	0.126796	−	0.991929i	$-0.459531\pi$
$353$	4.76454	0.253591	0.126796	+	0.991929i	$0.459531\pi$
$354$	0	0
$355$	−1.73541	−0.0921058
$356$	0	0
$357$	32.2360	1.70611
$358$	0	0
$359$	−3.37928	−0.178352	−0.0891758	−	0.996016i	$-0.528423\pi$
$359$	−3.37928	−0.178352	−0.0891758	+	0.996016i	$0.528423\pi$
$360$	0	0
$361$	7.60892	0.400470
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	−1.99046	−0.104185
$366$	0	0
$367$	24.9257	1.30111	0.650555	−	0.759459i	$-0.274536\pi$
$367$	24.9257	1.30111	0.650555	+	0.759459i	$0.274536\pi$
$368$	0	0
$369$	−4.39532	−0.228811
$370$	0	0
$371$	−5.44325	−0.282600
$372$	0	0
$373$	−15.8183	−0.819040	−0.409520	−	0.912301i	$-0.634304\pi$
$373$	−15.8183	−0.819040	−0.409520	+	0.912301i	$0.634304\pi$
$374$	0	0
$375$	−4.61847	−0.238497
$376$	0	0
$377$	8.68600	0.447352
$378$	0	0
$379$	−6.06024	−0.311294	−0.155647	−	0.987813i	$-0.549746\pi$
$379$	−6.06024	−0.311294	−0.155647	+	0.987813i	$0.549746\pi$
$380$	0	0
$381$	−12.5755	−0.644265
$382$	0	0
$383$	12.4985	0.638642	0.319321	−	0.947647i	$-0.396545\pi$
$383$	12.4985	0.638642	0.319321	+	0.947647i	$0.396545\pi$
$384$	0	0
$385$	−2.43676	−0.124189
$386$	0	0
$387$	−6.06753	−0.308430
$388$	0	0
$389$	−34.0522	−1.72651	−0.863257	−	0.504765i	$-0.831579\pi$
$389$	−34.0522	−1.72651	−0.863257	+	0.504765i	$0.831579\pi$
$390$	0	0
$391$	42.6953	2.15919
$392$	0	0
$393$	−3.55370	−0.179260
$394$	0	0
$395$	−4.30948	−0.216833
$396$	0	0
$397$	−22.9598	−1.15232	−0.576161	−	0.817336i	$-0.695450\pi$
$397$	−22.9598	−1.15232	−0.576161	+	0.817336i	$0.695450\pi$
$398$	0	0
$399$	26.6089	1.33211
$400$	0	0
$401$	14.4970	0.723946	0.361973	−	0.932189i	$-0.382103\pi$
$401$	14.4970	0.723946	0.361973	+	0.932189i	$0.382103\pi$
$402$	0	0
$403$	−1.52761	−0.0760958
$404$	0	0
$405$	0.472388	0.0234731
$406$	0	0
$407$	0.436758	0.0216493
$408$	0	0
$409$	29.2251	1.44509	0.722545	−	0.691324i	$-0.242972\pi$
$409$	29.2251	1.44509	0.722545	+	0.691324i	$0.242972\pi$
$410$	0	0
$411$	−21.7244	−1.07159
$412$	0	0
$413$	4.87352	0.239810
$414$	0	0
$415$	0.138011	0.00677469
$416$	0	0
$417$	5.74122	0.281149
$418$	0	0
$419$	−0.586565	−0.0286556	−0.0143278	−	0.999897i	$-0.504561\pi$
$419$	−0.586565	−0.0286556	−0.0143278	+	0.999897i	$0.504561\pi$
$420$	0	0
$421$	−15.0798	−0.734946	−0.367473	−	0.930034i	$-0.619777\pi$
$421$	−15.0798	−0.734946	−0.367473	+	0.930034i	$0.619777\pi$
$422$	0	0
$423$	−13.0457	−0.634303
$424$	0	0
$425$	−29.8517	−1.44802
$426$	0	0
$427$	27.7600	1.34340
$428$	0	0
$429$	−1.00000	−0.0482805
$430$	0	0
$431$	6.53766	0.314908	0.157454	−	0.987526i	$-0.449671\pi$
$431$	6.53766	0.314908	0.157454	+	0.987526i	$0.449671\pi$
$432$	0	0
$433$	22.8544	1.09831	0.549157	−	0.835719i	$-0.314949\pi$
$433$	22.8544	1.09831	0.549157	+	0.835719i	$0.314949\pi$
$434$	0	0
$435$	4.10316	0.196731
$436$	0	0
$437$	35.2425	1.68588
$438$	0	0
$439$	−5.43399	−0.259350	−0.129675	−	0.991557i	$-0.541393\pi$
$439$	−5.43399	−0.259350	−0.129675	+	0.991557i	$0.541393\pi$
$440$	0	0
$441$	19.6089	0.933758
$442$	0	0
$443$	−8.63354	−0.410192	−0.205096	−	0.978742i	$-0.565751\pi$
$443$	−8.63354	−0.410192	−0.205096	+	0.978742i	$0.565751\pi$
$444$	0	0
$445$	5.15335	0.244292
$446$	0	0
$447$	12.3953	0.586278
$448$	0	0
$449$	15.4418	0.728742	0.364371	−	0.931254i	$-0.381284\pi$
$449$	15.4418	0.728742	0.364371	+	0.931254i	$0.381284\pi$
$450$	0	0
$451$	4.39532	0.206967
$452$	0	0
$453$	6.03190	0.283403
$454$	0	0
$455$	2.43676	0.114237
$456$	0	0
$457$	−4.33585	−0.202823	−0.101411	−	0.994845i	$-0.532336\pi$
$457$	−4.33585	−0.202823	−0.101411	+	0.994845i	$0.532336\pi$
$458$	0	0
$459$	6.24924	0.291689
$460$	0	0
$461$	17.1859	0.800429	0.400215	−	0.916421i	$-0.368936\pi$
$461$	17.1859	0.800429	0.400215	+	0.916421i	$0.368936\pi$
$462$	0	0
$463$	−14.1461	−0.657424	−0.328712	−	0.944430i	$-0.606615\pi$
$463$	−14.1461	−0.657424	−0.328712	+	0.944430i	$0.606615\pi$
$464$	0	0
$465$	−0.721626	−0.0334646
$466$	0	0
$467$	−14.7492	−0.682511	−0.341256	−	0.939971i	$-0.610852\pi$
$467$	−14.7492	−0.682511	−0.341256	+	0.939971i	$0.610852\pi$
$468$	0	0
$469$	−67.4784	−3.11586
$470$	0	0
$471$	−20.0936	−0.925865
$472$	0	0
$473$	6.06753	0.278985
$474$	0	0
$475$	−24.6408	−1.13060
$476$	0	0
$477$	−1.05522	−0.0483154
$478$	0	0
$479$	−21.0479	−0.961705	−0.480852	−	0.876802i	$-0.659673\pi$
$479$	−21.0479	−0.961705	−0.480852	+	0.876802i	$0.659673\pi$
$480$	0	0
$481$	−0.436758	−0.0199145
$482$	0	0
$483$	35.2425	1.60359
$484$	0	0
$485$	−8.64811	−0.392691
$486$	0	0
$487$	0.702827	0.0318481	0.0159241	−	0.999873i	$-0.494931\pi$
$487$	0.702827	0.0318481	0.0159241	+	0.999873i	$0.494931\pi$
$488$	0	0
$489$	−14.5733	−0.659027
$490$	0	0
$491$	−2.14755	−0.0969177	−0.0484589	−	0.998825i	$-0.515431\pi$
$491$	−2.14755	−0.0969177	−0.0484589	+	0.998825i	$0.515431\pi$
$492$	0	0
$493$	54.2809	2.44469
$494$	0	0
$495$	−0.472388	−0.0212323
$496$	0	0
$497$	−18.9503	−0.850037
$498$	0	0
$499$	31.7585	1.42171	0.710854	−	0.703340i	$-0.248309\pi$
$499$	31.7585	1.42171	0.710854	+	0.703340i	$0.248309\pi$
$500$	0	0
$501$	2.31677	0.103506
$502$	0	0
$503$	−2.69181	−0.120022	−0.0600109	−	0.998198i	$-0.519114\pi$
$503$	−2.69181	−0.120022	−0.0600109	+	0.998198i	$0.519114\pi$
$504$	0	0
$505$	5.70056	0.253671
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	23.8348	1.05646	0.528230	−	0.849101i	$-0.322856\pi$
$509$	23.8348	1.05646	0.528230	+	0.849101i	$0.322856\pi$
$510$	0	0
$511$	−21.7354	−0.961518
$512$	0	0
$513$	5.15838	0.227748
$514$	0	0
$515$	−5.87047	−0.258684
$516$	0	0
$517$	13.0457	0.573748
$518$	0	0
$519$	17.7317	0.778334
$520$	0	0
$521$	28.5231	1.24962	0.624810	−	0.780777i	$-0.285177\pi$
$521$	28.5231	1.24962	0.624810	+	0.780777i	$0.285177\pi$
$522$	0	0
$523$	−29.5500	−1.29213	−0.646065	−	0.763282i	$-0.723587\pi$
$523$	−29.5500	−1.29213	−0.646065	+	0.763282i	$0.723587\pi$
$524$	0	0
$525$	−24.6408	−1.07541
$526$	0	0
$527$	−9.54641	−0.415848
$528$	0	0
$529$	23.6772	1.02944
$530$	0	0
$531$	0.944776	0.0409998
$532$	0	0
$533$	−4.39532	−0.190382
$534$	0	0
$535$	−1.44325	−0.0623972
$536$	0	0
$537$	11.1242	0.480046
$538$	0	0
$539$	−19.6089	−0.844616
$540$	0	0
$541$	31.8514	1.36940	0.684699	−	0.728826i	$-0.259934\pi$
$541$	31.8514	1.36940	0.684699	+	0.728826i	$0.259934\pi$
$542$	0	0
$543$	24.0936	1.03396
$544$	0	0
$545$	4.77261	0.204436
$546$	0	0
$547$	7.47663	0.319677	0.159839	−	0.987143i	$-0.448903\pi$
$547$	7.47663	0.319677	0.159839	+	0.987143i	$0.448903\pi$
$548$	0	0
$549$	5.38153	0.229678
$550$	0	0
$551$	44.8057	1.90879
$552$	0	0
$553$	−47.0587	−2.00114
$554$	0	0
$555$	−0.206319	−0.00875777
$556$	0	0
$557$	26.7759	1.13453	0.567265	−	0.823535i	$-0.308001\pi$
$557$	26.7759	1.13453	0.567265	+	0.823535i	$0.308001\pi$
$558$	0	0
$559$	−6.06753	−0.256629
$560$	0	0
$561$	−6.24924	−0.263843
$562$	0	0
$563$	−0.809716	−0.0341255	−0.0170627	−	0.999854i	$-0.505431\pi$
$563$	−0.809716	−0.0341255	−0.0170627	+	0.999854i	$0.505431\pi$
$564$	0	0
$565$	−4.37504	−0.184059
$566$	0	0
$567$	5.15838	0.216632
$568$	0	0
$569$	32.2738	1.35299	0.676495	−	0.736447i	$-0.263498\pi$
$569$	32.2738	1.35299	0.676495	+	0.736447i	$0.263498\pi$
$570$	0	0
$571$	10.2055	0.427089	0.213544	−	0.976933i	$-0.431499\pi$
$571$	10.2055	0.427089	0.213544	+	0.976933i	$0.431499\pi$
$572$	0	0
$573$	16.9448	0.707878
$574$	0	0
$575$	−32.6358	−1.36101
$576$	0	0
$577$	7.73246	0.321906	0.160953	−	0.986962i	$-0.448543\pi$
$577$	7.73246	0.321906	0.160953	+	0.986962i	$0.448543\pi$
$578$	0	0
$579$	9.22964	0.383571
$580$	0	0
$581$	1.50705	0.0625230
$582$	0	0
$583$	1.05522	0.0437029
$584$	0	0
$585$	0.472388	0.0195308
$586$	0	0
$587$	12.0713	0.498234	0.249117	−	0.968473i	$-0.419860\pi$
$587$	12.0713	0.498234	0.249117	+	0.968473i	$0.419860\pi$
$588$	0	0
$589$	−7.88001	−0.324690
$590$	0	0
$591$	−1.78639	−0.0734824
$592$	0	0
$593$	−13.1830	−0.541361	−0.270680	−	0.962669i	$-0.587249\pi$
$593$	−13.1830	−0.541361	−0.270680	+	0.962669i	$0.587249\pi$
$594$	0	0
$595$	15.2279	0.624282
$596$	0	0
$597$	−17.0798	−0.699031
$598$	0	0
$599$	42.7962	1.74860	0.874302	−	0.485383i	$-0.161320\pi$
$599$	42.7962	1.74860	0.874302	+	0.485383i	$0.161320\pi$
$600$	0	0
$601$	46.3936	1.89243	0.946216	−	0.323535i	$-0.104871\pi$
$601$	46.3936	1.89243	0.946216	+	0.323535i	$0.104871\pi$
$602$	0	0
$603$	−13.0813	−0.532712
$604$	0	0
$605$	0.472388	0.0192053
$606$	0	0
$607$	−23.9038	−0.970227	−0.485114	−	0.874451i	$-0.661222\pi$
$607$	−23.9038	−0.970227	−0.485114	+	0.874451i	$0.661222\pi$
$608$	0	0
$609$	44.8057	1.81562
$610$	0	0
$611$	−13.0457	−0.527772
$612$	0	0
$613$	17.6378	0.712383	0.356191	−	0.934413i	$-0.384075\pi$
$613$	17.6378	0.712383	0.356191	+	0.934413i	$0.384075\pi$
$614$	0	0
$615$	−2.07629	−0.0837243
$616$	0	0
$617$	30.5683	1.23063	0.615316	−	0.788281i	$-0.289028\pi$
$617$	30.5683	1.23063	0.615316	+	0.788281i	$0.289028\pi$
$618$	0	0
$619$	3.38301	0.135975	0.0679873	−	0.997686i	$-0.478342\pi$
$619$	3.38301	0.135975	0.0679873	+	0.997686i	$0.478342\pi$
$620$	0	0
$621$	6.83207	0.274162
$622$	0	0
$623$	56.2736	2.25455
$624$	0	0
$625$	21.7025	0.868102
$626$	0	0
$627$	−5.15838	−0.206006
$628$	0	0
$629$	−2.72941	−0.108829
$630$	0	0
$631$	−7.67221	−0.305426	−0.152713	−	0.988271i	$-0.548801\pi$
$631$	−7.67221	−0.305426	−0.152713	+	0.988271i	$0.548801\pi$
$632$	0	0
$633$	0.541393	0.0215185
$634$	0	0
$635$	−5.94054	−0.235743
$636$	0	0
$637$	19.6089	0.776934
$638$	0	0
$639$	−3.67369	−0.145329
$640$	0	0
$641$	−39.6451	−1.56589	−0.782943	−	0.622094i	$-0.786282\pi$
$641$	−39.6451	−1.56589	−0.782943	+	0.622094i	$0.786282\pi$
$642$	0	0
$643$	−26.6596	−1.05135	−0.525676	−	0.850685i	$-0.676188\pi$
$643$	−26.6596	−1.05135	−0.525676	+	0.850685i	$0.676188\pi$
$644$	0	0
$645$	−2.86623	−0.112858
$646$	0	0
$647$	41.7992	1.64330	0.821648	−	0.569995i	$-0.193055\pi$
$647$	41.7992	1.64330	0.821648	+	0.569995i	$0.193055\pi$
$648$	0	0
$649$	−0.944776	−0.0370857
$650$	0	0
$651$	−7.88001	−0.308842
$652$	0	0
$653$	−19.2862	−0.754726	−0.377363	−	0.926066i	$-0.623169\pi$
$653$	−19.2862	−0.754726	−0.377363	+	0.926066i	$0.623169\pi$
$654$	0	0
$655$	−1.67872	−0.0655932
$656$	0	0
$657$	−4.21361	−0.164389
$658$	0	0
$659$	−36.2169	−1.41081	−0.705405	−	0.708805i	$-0.749235\pi$
$659$	−36.2169	−1.41081	−0.705405	+	0.708805i	$0.749235\pi$
$660$	0	0
$661$	8.84890	0.344183	0.172091	−	0.985081i	$-0.444948\pi$
$661$	8.84890	0.344183	0.172091	+	0.985081i	$0.444948\pi$
$662$	0	0
$663$	6.24924	0.242700
$664$	0	0
$665$	12.5697	0.487433
$666$	0	0
$667$	59.3434	2.29778
$668$	0	0
$669$	−15.1059	−0.584029
$670$	0	0
$671$	−5.38153	−0.207752
$672$	0	0
$673$	−16.2114	−0.624902	−0.312451	−	0.949934i	$-0.601150\pi$
$673$	−16.2114	−0.624902	−0.312451	+	0.949934i	$0.601150\pi$
$674$	0	0
$675$	−4.77685	−0.183861
$676$	0	0
$677$	−28.0826	−1.07930	−0.539651	−	0.841889i	$-0.681444\pi$
$677$	−28.0826	−1.07930	−0.539651	+	0.841889i	$0.681444\pi$
$678$	0	0
$679$	−94.4357	−3.62411
$680$	0	0
$681$	30.0595	1.15188
$682$	0	0
$683$	10.2159	0.390899	0.195450	−	0.980714i	$-0.437383\pi$
$683$	10.2159	0.390899	0.195450	+	0.980714i	$0.437383\pi$
$684$	0	0
$685$	−10.2623	−0.392104
$686$	0	0
$687$	−14.1722	−0.540702
$688$	0	0
$689$	−1.05522	−0.0402008
$690$	0	0
$691$	20.1069	0.764902	0.382451	−	0.923976i	$-0.375080\pi$
$691$	20.1069	0.764902	0.382451	+	0.923976i	$0.375080\pi$
$692$	0	0
$693$	−5.15838	−0.195951
$694$	0	0
$695$	2.71208	0.102875
$696$	0	0
$697$	−27.4674	−1.04040
$698$	0	0
$699$	12.9782	0.490879
$700$	0	0
$701$	5.32355	0.201068	0.100534	−	0.994934i	$-0.467945\pi$
$701$	5.32355	0.201068	0.100534	+	0.994934i	$0.467945\pi$
$702$	0	0
$703$	−2.25297	−0.0849723
$704$	0	0
$705$	−6.16262	−0.232098
$706$	0	0
$707$	62.2490	2.34111
$708$	0	0
$709$	−35.4538	−1.33150	−0.665748	−	0.746177i	$-0.731887\pi$
$709$	−35.4538	−1.33150	−0.665748	+	0.746177i	$0.731887\pi$
$710$	0	0
$711$	−9.12275	−0.342130
$712$	0	0
$713$	−10.4368	−0.390860
$714$	0	0
$715$	−0.472388	−0.0176663
$716$	0	0
$717$	−19.9490	−0.745010
$718$	0	0
$719$	21.0753	0.785977	0.392989	−	0.919543i	$-0.371441\pi$
$719$	21.0753	0.785977	0.392989	+	0.919543i	$0.371441\pi$
$720$	0	0
$721$	−64.1043	−2.38737
$722$	0	0
$723$	−20.2455	−0.752939
$724$	0	0
$725$	−41.4917	−1.54096
$726$	0	0
$727$	−50.2264	−1.86279	−0.931397	−	0.364004i	$-0.881410\pi$
$727$	−50.2264	−1.86279	−0.931397	+	0.364004i	$0.881410\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−37.9174	−1.40243
$732$	0	0
$733$	−2.70656	−0.0999689	−0.0499845	−	0.998750i	$-0.515917\pi$
$733$	−2.70656	−0.0999689	−0.0499845	+	0.998750i	$0.515917\pi$
$734$	0	0
$735$	9.26302	0.341672
$736$	0	0
$737$	13.0813	0.481856
$738$	0	0
$739$	−18.9796	−0.698177	−0.349088	−	0.937090i	$-0.613509\pi$
$739$	−18.9796	−0.698177	−0.349088	+	0.937090i	$0.613509\pi$
$740$	0	0
$741$	5.15838	0.189498
$742$	0	0
$743$	10.8981	0.399814	0.199907	−	0.979815i	$-0.435936\pi$
$743$	10.8981	0.399814	0.199907	+	0.979815i	$0.435936\pi$
$744$	0	0
$745$	5.85540	0.214525
$746$	0	0
$747$	0.292156	0.0106894
$748$	0	0
$749$	−15.7600	−0.575859
$750$	0	0
$751$	−3.41864	−0.124748	−0.0623740	−	0.998053i	$-0.519867\pi$
$751$	−3.41864	−0.124748	−0.0623740	+	0.998053i	$0.519867\pi$
$752$	0	0
$753$	−15.6888	−0.571730
$754$	0	0
$755$	2.84940	0.103700
$756$	0	0
$757$	−24.0745	−0.875004	−0.437502	−	0.899217i	$-0.644137\pi$
$757$	−24.0745	−0.875004	−0.437502	+	0.899217i	$0.644137\pi$
$758$	0	0
$759$	−6.83207	−0.247989
$760$	0	0
$761$	21.5464	0.781057	0.390528	−	0.920591i	$-0.372292\pi$
$761$	21.5464	0.781057	0.390528	+	0.920591i	$0.372292\pi$
$762$	0	0
$763$	52.1160	1.88672
$764$	0	0
$765$	2.95206	0.106732
$766$	0	0
$767$	0.944776	0.0341139
$768$	0	0
$769$	−24.8865	−0.897430	−0.448715	−	0.893675i	$-0.648118\pi$
$769$	−24.8865	−0.897430	−0.448715	+	0.893675i	$0.648118\pi$
$770$	0	0
$771$	−8.76307	−0.315594
$772$	0	0
$773$	50.3057	1.80937	0.904686	−	0.426079i	$-0.140105\pi$
$773$	50.3057	1.80937	0.904686	+	0.426079i	$0.140105\pi$
$774$	0	0
$775$	7.29717	0.262122
$776$	0	0
$777$	−2.25297	−0.0808247
$778$	0	0
$779$	−22.6727	−0.812335
$780$	0	0
$781$	3.67369	0.131455
$782$	0	0
$783$	8.68600	0.310412
$784$	0	0
$785$	−9.49198	−0.338783
$786$	0	0
$787$	3.71513	0.132430	0.0662151	−	0.997805i	$-0.478908\pi$
$787$	3.71513	0.132430	0.0662151	+	0.997805i	$0.478908\pi$
$788$	0	0
$789$	4.07126	0.144941
$790$	0	0
$791$	−47.7746	−1.69867
$792$	0	0
$793$	5.38153	0.191104
$794$	0	0
$795$	−0.498475	−0.0176791
$796$	0	0
$797$	0.267544	0.00947690	0.00473845	−	0.999989i	$-0.498492\pi$
$797$	0.267544	0.00947690	0.00473845	+	0.999989i	$0.498492\pi$
$798$	0	0
$799$	−81.5256	−2.88417
$800$	0	0
$801$	10.9091	0.385456
$802$	0	0
$803$	4.21361	0.148695
$804$	0	0
$805$	16.6481	0.586769
$806$	0	0
$807$	−15.1511	−0.533344
$808$	0	0
$809$	36.4264	1.28069	0.640343	−	0.768089i	$-0.278792\pi$
$809$	36.4264	1.28069	0.640343	+	0.768089i	$0.278792\pi$
$810$	0	0
$811$	4.36775	0.153373	0.0766863	−	0.997055i	$-0.475566\pi$
$811$	4.36775	0.153373	0.0766863	+	0.997055i	$0.475566\pi$
$812$	0	0
$813$	−29.8778	−1.04786
$814$	0	0
$815$	−6.88425	−0.241145
$816$	0	0
$817$	−31.2986	−1.09500
$818$	0	0
$819$	5.15838	0.180249
$820$	0	0
$821$	−46.7833	−1.63275	−0.816375	−	0.577522i	$-0.804020\pi$
$821$	−46.7833	−1.63275	−0.816375	+	0.577522i	$0.804020\pi$
$822$	0	0
$823$	−30.8790	−1.07638	−0.538188	−	0.842825i	$-0.680891\pi$
$823$	−30.8790	−1.07638	−0.538188	+	0.842825i	$0.680891\pi$
$824$	0	0
$825$	4.77685	0.166309
$826$	0	0
$827$	−1.55804	−0.0541783	−0.0270891	−	0.999633i	$-0.508624\pi$
$827$	−1.55804	−0.0541783	−0.0270891	+	0.999633i	$0.508624\pi$
$828$	0	0
$829$	31.2178	1.08424	0.542120	−	0.840301i	$-0.317622\pi$
$829$	31.2178	1.08424	0.542120	+	0.840301i	$0.317622\pi$
$830$	0	0
$831$	10.4272	0.361716
$832$	0	0
$833$	122.541	4.24579
$834$	0	0
$835$	1.09441	0.0378737
$836$	0	0
$837$	−1.52761	−0.0528020
$838$	0	0
$839$	−41.7876	−1.44267	−0.721334	−	0.692588i	$-0.756471\pi$
$839$	−41.7876	−1.44267	−0.721334	+	0.692588i	$0.756471\pi$
$840$	0	0
$841$	46.4465	1.60160
$842$	0	0
$843$	2.21361	0.0762407
$844$	0	0
$845$	0.472388	0.0162506
$846$	0	0
$847$	5.15838	0.177244
$848$	0	0
$849$	−1.31400	−0.0450965
$850$	0	0
$851$	−2.98396	−0.102289
$852$	0	0
$853$	−33.5346	−1.14820	−0.574102	−	0.818784i	$-0.694649\pi$
$853$	−33.5346	−1.14820	−0.574102	+	0.818784i	$0.694649\pi$
$854$	0	0
$855$	2.43676	0.0833354
$856$	0	0
$857$	−42.6915	−1.45831	−0.729157	−	0.684346i	$-0.760088\pi$
$857$	−42.6915	−1.45831	−0.729157	+	0.684346i	$0.760088\pi$
$858$	0	0
$859$	23.4824	0.801211	0.400605	−	0.916251i	$-0.368800\pi$
$859$	23.4824	0.801211	0.400605	+	0.916251i	$0.368800\pi$
$860$	0	0
$861$	−22.6727	−0.772684
$862$	0	0
$863$	17.5145	0.596201	0.298100	−	0.954535i	$-0.403647\pi$
$863$	17.5145	0.596201	0.298100	+	0.954535i	$0.403647\pi$
$864$	0	0
$865$	8.37623	0.284800
$866$	0	0
$867$	22.0530	0.748958
$868$	0	0
$869$	9.12275	0.309468
$870$	0	0
$871$	−13.0813	−0.443243
$872$	0	0
$873$	−18.3072	−0.619606
$874$	0	0
$875$	−23.8238	−0.805392
$876$	0	0
$877$	−20.7514	−0.700726	−0.350363	−	0.936614i	$-0.613942\pi$
$877$	−20.7514	−0.700726	−0.350363	+	0.936614i	$0.613942\pi$
$878$	0	0
$879$	−13.7864	−0.465003
$880$	0	0
$881$	45.1074	1.51971	0.759853	−	0.650094i	$-0.225271\pi$
$881$	45.1074	1.51971	0.759853	+	0.650094i	$0.225271\pi$
$882$	0	0
$883$	−21.2224	−0.714189	−0.357095	−	0.934068i	$-0.616233\pi$
$883$	−21.2224	−0.714189	−0.357095	+	0.934068i	$0.616233\pi$
$884$	0	0
$885$	0.446301	0.0150022
$886$	0	0
$887$	−53.3012	−1.78968	−0.894840	−	0.446387i	$-0.852710\pi$
$887$	−53.3012	−1.78968	−0.894840	+	0.446387i	$0.852710\pi$
$888$	0	0
$889$	−64.8695	−2.17565
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	−67.2946	−2.25193
$894$	0	0
$895$	5.25495	0.175654
$896$	0	0
$897$	6.83207	0.228116
$898$	0	0
$899$	−13.2688	−0.442540
$900$	0	0
$901$	−6.59435	−0.219690
$902$	0	0
$903$	−31.2986	−1.04155
$904$	0	0
$905$	11.3815	0.378335
$906$	0	0
$907$	37.9226	1.25920	0.629600	−	0.776919i	$-0.283219\pi$
$907$	37.9226	1.25920	0.629600	+	0.776919i	$0.283219\pi$
$908$	0	0
$909$	12.0675	0.400255
$910$	0	0
$911$	−32.5623	−1.07884	−0.539418	−	0.842038i	$-0.681356\pi$
$911$	−32.5623	−1.07884	−0.539418	+	0.842038i	$0.681356\pi$
$912$	0	0
$913$	−0.292156	−0.00966895
$914$	0	0
$915$	2.54217	0.0840416
$916$	0	0
$917$	−18.3313	−0.605354
$918$	0	0
$919$	11.7879	0.388846	0.194423	−	0.980918i	$-0.437717\pi$
$919$	11.7879	0.388846	0.194423	+	0.980918i	$0.437717\pi$
$920$	0	0
$921$	19.4752	0.641728
$922$	0	0
$923$	−3.67369	−0.120921
$924$	0	0
$925$	2.08633	0.0685981
$926$	0	0
$927$	−12.4272	−0.408163
$928$	0	0
$929$	43.0964	1.41395	0.706973	−	0.707240i	$-0.250060\pi$
$929$	43.0964	1.41395	0.706973	+	0.707240i	$0.250060\pi$
$930$	0	0
$931$	101.150	3.31507
$932$	0	0
$933$	−5.59514	−0.183177
$934$	0	0
$935$	−2.95206	−0.0965428
$936$	0	0
$937$	−16.8002	−0.548838	−0.274419	−	0.961610i	$-0.588485\pi$
$937$	−16.8002	−0.548838	−0.274419	+	0.961610i	$0.588485\pi$
$938$	0	0
$939$	−5.48469	−0.178986
$940$	0	0
$941$	−57.8833	−1.88694	−0.943471	−	0.331456i	$-0.892460\pi$
$941$	−57.8833	−1.88694	−0.943471	+	0.331456i	$0.892460\pi$
$942$	0	0
$943$	−30.0291	−0.977883
$944$	0	0
$945$	2.43676	0.0792678
$946$	0	0
$947$	13.2711	0.431252	0.215626	−	0.976476i	$-0.430821\pi$
$947$	13.2711	0.431252	0.215626	+	0.976476i	$0.430821\pi$
$948$	0	0
$949$	−4.21361	−0.136780
$950$	0	0
$951$	−17.9644	−0.582535
$952$	0	0
$953$	−1.63077	−0.0528259	−0.0264129	−	0.999651i	$-0.508408\pi$
$953$	−1.63077	−0.0528259	−0.0264129	+	0.999651i	$0.508408\pi$
$954$	0	0
$955$	8.00451	0.259020
$956$	0	0
$957$	−8.68600	−0.280778
$958$	0	0
$959$	−112.063	−3.61869
$960$	0	0
$961$	−28.6664	−0.924723
$962$	0	0
$963$	−3.05522	−0.0984532
$964$	0	0
$965$	4.35997	0.140352
$966$	0	0
$967$	−13.6759	−0.439789	−0.219894	−	0.975524i	$-0.570571\pi$
$967$	−13.6759	−0.439789	−0.219894	+	0.975524i	$0.570571\pi$
$968$	0	0
$969$	32.2360	1.03557
$970$	0	0
$971$	22.6750	0.727675	0.363837	−	0.931463i	$-0.381466\pi$
$971$	22.6750	0.727675	0.363837	+	0.931463i	$0.381466\pi$
$972$	0	0
$973$	29.6154	0.949427
$974$	0	0
$975$	−4.77685	−0.152982
$976$	0	0
$977$	7.18222	0.229779	0.114890	−	0.993378i	$-0.463349\pi$
$977$	7.18222	0.229779	0.114890	+	0.993378i	$0.463349\pi$
$978$	0	0
$979$	−10.9091	−0.348658
$980$	0	0
$981$	10.1032	0.322569
$982$	0	0
$983$	−35.6771	−1.13792	−0.568962	−	0.822364i	$-0.692655\pi$
$983$	−35.6771	−1.13792	−0.568962	+	0.822364i	$0.692655\pi$
$984$	0	0
$985$	−0.843870	−0.0268879
$986$	0	0
$987$	−67.2946	−2.14201
$988$	0	0
$989$	−41.4538	−1.31815
$990$	0	0
$991$	18.0201	0.572427	0.286214	−	0.958166i	$-0.407603\pi$
$991$	18.0201	0.572427	0.286214	+	0.958166i	$0.407603\pi$
$992$	0	0
$993$	25.8348	0.819844
$994$	0	0
$995$	−8.06831	−0.255783
$996$	0	0
$997$	56.7962	1.79875	0.899376	−	0.437176i	$-0.144021\pi$
$997$	56.7962	1.79875	0.899376	+	0.437176i	$0.144021\pi$
$998$	0	0
$999$	−0.436758	−0.0138184

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.cb.1.2		4
4.3	odd	2		1716.2.a.i.1.2	✓	4
12.11	even	2		5148.2.a.q.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1716.2.a.i.1.2	✓	4	4.3	odd	2
5148.2.a.q.1.3		4	12.11	even	2
6864.2.a.cb.1.2		4	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6864.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +0.472388 q^{5} +5.15838 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +0.472388 q^{5} +5.15838 q^{7} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} +0.472388 q^{15} +6.24924 q^{17} +5.15838 q^{19} +5.15838 q^{21} +6.83207 q^{23} -4.77685 q^{25} +1.00000 q^{27} +8.68600 q^{29} -1.52761 q^{31} -1.00000 q^{33} +2.43676 q^{35} -0.436758 q^{37} +1.00000 q^{39} -4.39532 q^{41} -6.06753 q^{43} +0.472388 q^{45} -13.0457 q^{47} +19.6089 q^{49} +6.24924 q^{51} -1.05522 q^{53} -0.472388 q^{55} +5.15838 q^{57} +0.944776 q^{59} +5.38153 q^{61} +5.15838 q^{63} +0.472388 q^{65} -13.0813 q^{67} +6.83207 q^{69} -3.67369 q^{71} -4.21361 q^{73} -4.77685 q^{75} -5.15838 q^{77} -9.12275 q^{79} +1.00000 q^{81} +0.292156 q^{83} +2.95206 q^{85} +8.68600 q^{87} +10.9091 q^{89} +5.15838 q^{91} -1.52761 q^{93} +2.43676 q^{95} -18.3072 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 2 q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q + 4 * q^3 + 2 * q^5 - 2 * q^7 + 4 * q^9	\( 4 q + 4 q^{3} + 2 q^{5} - 2 q^{7} + 4 q^{9} - 4 q^{11} + 4 q^{13} + 2 q^{15} + 2 q^{17} - 2 q^{19} - 2 q^{21} + 4 q^{23} + 4 q^{25} + 4 q^{27} + 12 q^{29} - 6 q^{31} - 4 q^{33} + 10 q^{35} - 2 q^{37} + 4 q^{39} + 6 q^{41} - 2 q^{43} + 2 q^{45} - 6 q^{47} + 32 q^{49} + 2 q^{51} - 4 q^{53} - 2 q^{55} - 2 q^{57} + 4 q^{59} + 22 q^{61} - 2 q^{63} + 2 q^{65} - 6 q^{67} + 4 q^{69} - 14 q^{71} + 6 q^{73} + 4 q^{75} + 2 q^{77} - 14 q^{79} + 4 q^{81} + 34 q^{85} + 12 q^{87} + 44 q^{89} - 2 q^{91} - 6 q^{93} + 10 q^{95} + 18 q^{97} - 4 q^{99}+O(q^{100}) \) 4 * q + 4 * q^3 + 2 * q^5 - 2 * q^7 + 4 * q^9 - 4 * q^11 + 4 * q^13 + 2 * q^15 + 2 * q^17 - 2 * q^19 - 2 * q^21 + 4 * q^23 + 4 * q^25 + 4 * q^27 + 12 * q^29 - 6 * q^31 - 4 * q^33 + 10 * q^35 - 2 * q^37 + 4 * q^39 + 6 * q^41 - 2 * q^43 + 2 * q^45 - 6 * q^47 + 32 * q^49 + 2 * q^51 - 4 * q^53 - 2 * q^55 - 2 * q^57 + 4 * q^59 + 22 * q^61 - 2 * q^63 + 2 * q^65 - 6 * q^67 + 4 * q^69 - 14 * q^71 + 6 * q^73 + 4 * q^75 + 2 * q^77 - 14 * q^79 + 4 * q^81 + 34 * q^85 + 12 * q^87 + 44 * q^89 - 2 * q^91 - 6 * q^93 + 10 * q^95 + 18 * q^97 - 4 * q^99

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