LMFDB - Embedded newform 6864.2.a.bh.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:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 858)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.56155$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} -3.56155 q^{5} -1.56155 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -3.56155 q^{5} -1.56155 q^{7} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} -3.56155 q^{15} +5.12311 q^{17} +3.12311 q^{19} -1.56155 q^{21} +2.43845 q^{23} +7.68466 q^{25} +1.00000 q^{27} -10.6847 q^{29} -10.2462 q^{31} -1.00000 q^{33} +5.56155 q^{35} -8.24621 q^{37} +1.00000 q^{39} -7.56155 q^{41} +1.56155 q^{43} -3.56155 q^{45} +8.00000 q^{47} -4.56155 q^{49} +5.12311 q^{51} +12.2462 q^{53} +3.56155 q^{55} +3.12311 q^{57} +9.56155 q^{59} +0.438447 q^{61} -1.56155 q^{63} -3.56155 q^{65} +10.4384 q^{67} +2.43845 q^{69} -9.36932 q^{71} -11.5616 q^{73} +7.68466 q^{75} +1.56155 q^{77} -4.87689 q^{79} +1.00000 q^{81} +8.87689 q^{83} -18.2462 q^{85} -10.6847 q^{87} +7.36932 q^{89} -1.56155 q^{91} -10.2462 q^{93} -11.1231 q^{95} -18.4924 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} - 3 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) $ 2 * q + 2 * q^3 - 3 * q^5 + q^7 + 2 * q^9	$ 2 q + 2 q^{3} - 3 q^{5} + q^{7} + 2 q^{9} - 2 q^{11} + 2 q^{13} - 3 q^{15} + 2 q^{17} - 2 q^{19} + q^{21} + 9 q^{23} + 3 q^{25} + 2 q^{27} - 9 q^{29} - 4 q^{31} - 2 q^{33} + 7 q^{35} + 2 q^{39} - 11 q^{41} - q^{43} - 3 q^{45} + 16 q^{47} - 5 q^{49} + 2 q^{51} + 8 q^{53} + 3 q^{55} - 2 q^{57} + 15 q^{59} + 5 q^{61} + q^{63} - 3 q^{65} + 25 q^{67} + 9 q^{69} + 6 q^{71} - 19 q^{73} + 3 q^{75} - q^{77} - 18 q^{79} + 2 q^{81} + 26 q^{83} - 20 q^{85} - 9 q^{87} - 10 q^{89} + q^{91} - 4 q^{93} - 14 q^{95} - 4 q^{97} - 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 - 3 * q^5 + q^7 + 2 * q^9 - 2 * q^11 + 2 * q^13 - 3 * q^15 + 2 * q^17 - 2 * q^19 + q^21 + 9 * q^23 + 3 * q^25 + 2 * q^27 - 9 * q^29 - 4 * q^31 - 2 * q^33 + 7 * q^35 + 2 * q^39 - 11 * q^41 - q^43 - 3 * q^45 + 16 * q^47 - 5 * q^49 + 2 * q^51 + 8 * q^53 + 3 * q^55 - 2 * q^57 + 15 * q^59 + 5 * q^61 + q^63 - 3 * q^65 + 25 * q^67 + 9 * q^69 + 6 * q^71 - 19 * q^73 + 3 * q^75 - q^77 - 18 * q^79 + 2 * q^81 + 26 * q^83 - 20 * q^85 - 9 * q^87 - 10 * q^89 + q^91 - 4 * q^93 - 14 * q^95 - 4 * q^97 - 2 * 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
$5$	−3.56155	−1.59277	−0.796387	−	0.604787i	$-0.793258\pi$
$5$	−3.56155	−1.59277	−0.796387	+	0.604787i	$0.793258\pi$
$6$	0	0
$7$	−1.56155	−0.590211	−0.295106	−	0.955465i	$-0.595355\pi$
$7$	−1.56155	−0.590211	−0.295106	+	0.955465i	$0.595355\pi$
$8$	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
$15$	−3.56155	−0.919589
$16$	0	0
$17$	5.12311	1.24254	0.621268	−	0.783598i	$-0.286618\pi$
$17$	5.12311	1.24254	0.621268	+	0.783598i	$0.286618\pi$
$18$	0	0
$19$	3.12311	0.716490	0.358245	−	0.933628i	$-0.383375\pi$
$19$	3.12311	0.716490	0.358245	+	0.933628i	$0.383375\pi$
$20$	0	0
$21$	−1.56155	−0.340759
$22$	0	0
$23$	2.43845	0.508451	0.254226	−	0.967145i	$-0.418179\pi$
$23$	2.43845	0.508451	0.254226	+	0.967145i	$0.418179\pi$
$24$	0	0
$25$	7.68466	1.53693
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−10.6847	−1.98409	−0.992046	−	0.125879i	$-0.959825\pi$
$29$	−10.6847	−1.98409	−0.992046	+	0.125879i	$0.959825\pi$
$30$	0	0
$31$	−10.2462	−1.84027	−0.920137	−	0.391597i	$-0.871923\pi$
$31$	−10.2462	−1.84027	−0.920137	+	0.391597i	$0.871923\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	5.56155	0.940074
$36$	0	0
$37$	−8.24621	−1.35567	−0.677834	−	0.735215i	$-0.737081\pi$
$37$	−8.24621	−1.35567	−0.677834	+	0.735215i	$0.737081\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−7.56155	−1.18092	−0.590458	−	0.807068i	$-0.701053\pi$
$41$	−7.56155	−1.18092	−0.590458	+	0.807068i	$0.701053\pi$
$42$	0	0
$43$	1.56155	0.238135	0.119067	−	0.992886i	$-0.462010\pi$
$43$	1.56155	0.238135	0.119067	+	0.992886i	$0.462010\pi$
$44$	0	0
$45$	−3.56155	−0.530925
$46$	0	0
$47$	8.00000	1.16692	0.583460	−	0.812142i	$-0.301699\pi$
$47$	8.00000	1.16692	0.583460	+	0.812142i	$0.301699\pi$
$48$	0	0
$49$	−4.56155	−0.651650
$50$	0	0
$51$	5.12311	0.717378
$52$	0	0
$53$	12.2462	1.68215	0.841073	−	0.540921i	$-0.181924\pi$
$53$	12.2462	1.68215	0.841073	+	0.540921i	$0.181924\pi$
$54$	0	0
$55$	3.56155	0.480240
$56$	0	0
$57$	3.12311	0.413665
$58$	0	0
$59$	9.56155	1.24481	0.622404	−	0.782696i	$-0.286156\pi$
$59$	9.56155	1.24481	0.622404	+	0.782696i	$0.286156\pi$
$60$	0	0
$61$	0.438447	0.0561374	0.0280687	−	0.999606i	$-0.491064\pi$
$61$	0.438447	0.0561374	0.0280687	+	0.999606i	$0.491064\pi$
$62$	0	0
$63$	−1.56155	−0.196737
$64$	0	0
$65$	−3.56155	−0.441756
$66$	0	0
$67$	10.4384	1.27526	0.637630	−	0.770343i	$-0.279915\pi$
$67$	10.4384	1.27526	0.637630	+	0.770343i	$0.279915\pi$
$68$	0	0
$69$	2.43845	0.293555
$70$	0	0
$71$	−9.36932	−1.11193	−0.555967	−	0.831205i	$-0.687652\pi$
$71$	−9.36932	−1.11193	−0.555967	+	0.831205i	$0.687652\pi$
$72$	0	0
$73$	−11.5616	−1.35318	−0.676589	−	0.736361i	$-0.736542\pi$
$73$	−11.5616	−1.35318	−0.676589	+	0.736361i	$0.736542\pi$
$74$	0	0
$75$	7.68466	0.887348
$76$	0	0
$77$	1.56155	0.177955
$78$	0	0
$79$	−4.87689	−0.548693	−0.274347	−	0.961631i	$-0.588462\pi$
$79$	−4.87689	−0.548693	−0.274347	+	0.961631i	$0.588462\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	8.87689	0.974366	0.487183	−	0.873300i	$-0.338025\pi$
$83$	8.87689	0.974366	0.487183	+	0.873300i	$0.338025\pi$
$84$	0	0
$85$	−18.2462	−1.97908
$86$	0	0
$87$	−10.6847	−1.14552
$88$	0	0
$89$	7.36932	0.781146	0.390573	−	0.920572i	$-0.372277\pi$
$89$	7.36932	0.781146	0.390573	+	0.920572i	$0.372277\pi$
$90$	0	0
$91$	−1.56155	−0.163695
$92$	0	0
$93$	−10.2462	−1.06248
$94$	0	0
$95$	−11.1231	−1.14121
$96$	0	0
$97$	−18.4924	−1.87762	−0.938811	−	0.344434i	$-0.888071\pi$
$97$	−18.4924	−1.87762	−0.938811	+	0.344434i	$0.888071\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	−0.246211	−0.0244989	−0.0122495	−	0.999925i	$-0.503899\pi$
$101$	−0.246211	−0.0244989	−0.0122495	+	0.999925i	$0.503899\pi$
$102$	0	0
$103$	13.5616	1.33626	0.668130	−	0.744045i	$-0.267095\pi$
$103$	13.5616	1.33626	0.668130	+	0.744045i	$0.267095\pi$
$104$	0	0
$105$	5.56155	0.542752
$106$	0	0
$107$	17.5616	1.69774	0.848870	−	0.528602i	$-0.177284\pi$
$107$	17.5616	1.69774	0.848870	+	0.528602i	$0.177284\pi$
$108$	0	0
$109$	15.3693	1.47211	0.736057	−	0.676920i	$-0.236686\pi$
$109$	15.3693	1.47211	0.736057	+	0.676920i	$0.236686\pi$
$110$	0	0
$111$	−8.24621	−0.782696
$112$	0	0
$113$	7.56155	0.711331	0.355666	−	0.934613i	$-0.384254\pi$
$113$	7.56155	0.711331	0.355666	+	0.934613i	$0.384254\pi$
$114$	0	0
$115$	−8.68466	−0.809849
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−8.00000	−0.733359
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−7.56155	−0.681802
$124$	0	0
$125$	−9.56155	−0.855211
$126$	0	0
$127$	−3.12311	−0.277131	−0.138565	−	0.990353i	$-0.544249\pi$
$127$	−3.12311	−0.277131	−0.138565	+	0.990353i	$0.544249\pi$
$128$	0	0
$129$	1.56155	0.137487
$130$	0	0
$131$	4.68466	0.409301	0.204650	−	0.978835i	$-0.434394\pi$
$131$	4.68466	0.409301	0.204650	+	0.978835i	$0.434394\pi$
$132$	0	0
$133$	−4.87689	−0.422880
$134$	0	0
$135$	−3.56155	−0.306530
$136$	0	0
$137$	−14.4924	−1.23817	−0.619086	−	0.785324i	$-0.712497\pi$
$137$	−14.4924	−1.23817	−0.619086	+	0.785324i	$0.712497\pi$
$138$	0	0
$139$	−10.2462	−0.869072	−0.434536	−	0.900654i	$-0.643088\pi$
$139$	−10.2462	−0.869072	−0.434536	+	0.900654i	$0.643088\pi$
$140$	0	0
$141$	8.00000	0.673722
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	38.0540	3.16021
$146$	0	0
$147$	−4.56155	−0.376231
$148$	0	0
$149$	−1.12311	−0.0920084	−0.0460042	−	0.998941i	$-0.514649\pi$
$149$	−1.12311	−0.0920084	−0.0460042	+	0.998941i	$0.514649\pi$
$150$	0	0
$151$	10.2462	0.833825	0.416912	−	0.908947i	$-0.363112\pi$
$151$	10.2462	0.833825	0.416912	+	0.908947i	$0.363112\pi$
$152$	0	0
$153$	5.12311	0.414179
$154$	0	0
$155$	36.4924	2.93114
$156$	0	0
$157$	−8.24621	−0.658119	−0.329060	−	0.944309i	$-0.606732\pi$
$157$	−8.24621	−0.658119	−0.329060	+	0.944309i	$0.606732\pi$
$158$	0	0
$159$	12.2462	0.971188
$160$	0	0
$161$	−3.80776	−0.300094
$162$	0	0
$163$	11.8078	0.924855	0.462428	−	0.886657i	$-0.346978\pi$
$163$	11.8078	0.924855	0.462428	+	0.886657i	$0.346978\pi$
$164$	0	0
$165$	3.56155	0.277267
$166$	0	0
$167$	14.9309	1.15539	0.577693	−	0.816254i	$-0.303953\pi$
$167$	14.9309	1.15539	0.577693	+	0.816254i	$0.303953\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	3.12311	0.238830
$172$	0	0
$173$	22.6847	1.72468	0.862341	−	0.506327i	$-0.168997\pi$
$173$	22.6847	1.72468	0.862341	+	0.506327i	$0.168997\pi$
$174$	0	0
$175$	−12.0000	−0.907115
$176$	0	0
$177$	9.56155	0.718690
$178$	0	0
$179$	−4.00000	−0.298974	−0.149487	−	0.988764i	$-0.547762\pi$
$179$	−4.00000	−0.298974	−0.149487	+	0.988764i	$0.547762\pi$
$180$	0	0
$181$	12.2462	0.910254	0.455127	−	0.890427i	$-0.349594\pi$
$181$	12.2462	0.910254	0.455127	+	0.890427i	$0.349594\pi$
$182$	0	0
$183$	0.438447	0.0324109
$184$	0	0
$185$	29.3693	2.15928
$186$	0	0
$187$	−5.12311	−0.374639
$188$	0	0
$189$	−1.56155	−0.113586
$190$	0	0
$191$	−10.4384	−0.755300	−0.377650	−	0.925949i	$-0.623268\pi$
$191$	−10.4384	−0.755300	−0.377650	+	0.925949i	$0.623268\pi$
$192$	0	0
$193$	8.24621	0.593575	0.296788	−	0.954944i	$-0.404085\pi$
$193$	8.24621	0.593575	0.296788	+	0.954944i	$0.404085\pi$
$194$	0	0
$195$	−3.56155	−0.255048
$196$	0	0
$197$	25.6155	1.82503	0.912515	−	0.409042i	$-0.134137\pi$
$197$	25.6155	1.82503	0.912515	+	0.409042i	$0.134137\pi$
$198$	0	0
$199$	22.9309	1.62553	0.812763	−	0.582594i	$-0.197962\pi$
$199$	22.9309	1.62553	0.812763	+	0.582594i	$0.197962\pi$
$200$	0	0
$201$	10.4384	0.736271
$202$	0	0
$203$	16.6847	1.17103
$204$	0	0
$205$	26.9309	1.88093
$206$	0	0
$207$	2.43845	0.169484
$208$	0	0
$209$	−3.12311	−0.216030
$210$	0	0
$211$	−4.00000	−0.275371	−0.137686	−	0.990476i	$-0.543966\pi$
$211$	−4.00000	−0.275371	−0.137686	+	0.990476i	$0.543966\pi$
$212$	0	0
$213$	−9.36932	−0.641975
$214$	0	0
$215$	−5.56155	−0.379295
$216$	0	0
$217$	16.0000	1.08615
$218$	0	0
$219$	−11.5616	−0.781257
$220$	0	0
$221$	5.12311	0.344617
$222$	0	0
$223$	8.49242	0.568695	0.284347	−	0.958721i	$-0.408223\pi$
$223$	8.49242	0.568695	0.284347	+	0.958721i	$0.408223\pi$
$224$	0	0
$225$	7.68466	0.512311
$226$	0	0
$227$	−10.2462	−0.680065	−0.340032	−	0.940414i	$-0.610438\pi$
$227$	−10.2462	−0.680065	−0.340032	+	0.940414i	$0.610438\pi$
$228$	0	0
$229$	−6.19224	−0.409194	−0.204597	−	0.978846i	$-0.565588\pi$
$229$	−6.19224	−0.409194	−0.204597	+	0.978846i	$0.565588\pi$
$230$	0	0
$231$	1.56155	0.102743
$232$	0	0
$233$	16.2462	1.06432	0.532162	−	0.846642i	$-0.321380\pi$
$233$	16.2462	1.06432	0.532162	+	0.846642i	$0.321380\pi$
$234$	0	0
$235$	−28.4924	−1.85864
$236$	0	0
$237$	−4.87689	−0.316788
$238$	0	0
$239$	1.06913	0.0691563	0.0345781	−	0.999402i	$-0.488991\pi$
$239$	1.06913	0.0691563	0.0345781	+	0.999402i	$0.488991\pi$
$240$	0	0
$241$	0.246211	0.0158599	0.00792993	−	0.999969i	$-0.497476\pi$
$241$	0.246211	0.0158599	0.00792993	+	0.999969i	$0.497476\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	16.2462	1.03793
$246$	0	0
$247$	3.12311	0.198718
$248$	0	0
$249$	8.87689	0.562550
$250$	0	0
$251$	−8.87689	−0.560305	−0.280152	−	0.959956i	$-0.590385\pi$
$251$	−8.87689	−0.560305	−0.280152	+	0.959956i	$0.590385\pi$
$252$	0	0
$253$	−2.43845	−0.153304
$254$	0	0
$255$	−18.2462	−1.14262
$256$	0	0
$257$	−3.56155	−0.222164	−0.111082	−	0.993811i	$-0.535432\pi$
$257$	−3.56155	−0.222164	−0.111082	+	0.993811i	$0.535432\pi$
$258$	0	0
$259$	12.8769	0.800131
$260$	0	0
$261$	−10.6847	−0.661364
$262$	0	0
$263$	20.4924	1.26362	0.631808	−	0.775125i	$-0.282313\pi$
$263$	20.4924	1.26362	0.631808	+	0.775125i	$0.282313\pi$
$264$	0	0
$265$	−43.6155	−2.67928
$266$	0	0
$267$	7.36932	0.450995
$268$	0	0
$269$	−17.6155	−1.07404	−0.537019	−	0.843570i	$-0.680450\pi$
$269$	−17.6155	−1.07404	−0.537019	+	0.843570i	$0.680450\pi$
$270$	0	0
$271$	−5.75379	−0.349518	−0.174759	−	0.984611i	$-0.555915\pi$
$271$	−5.75379	−0.349518	−0.174759	+	0.984611i	$0.555915\pi$
$272$	0	0
$273$	−1.56155	−0.0945095
$274$	0	0
$275$	−7.68466	−0.463402
$276$	0	0
$277$	8.43845	0.507017	0.253509	−	0.967333i	$-0.418415\pi$
$277$	8.43845	0.507017	0.253509	+	0.967333i	$0.418415\pi$
$278$	0	0
$279$	−10.2462	−0.613425
$280$	0	0
$281$	3.56155	0.212464	0.106232	−	0.994341i	$-0.466121\pi$
$281$	3.56155	0.212464	0.106232	+	0.994341i	$0.466121\pi$
$282$	0	0
$283$	3.31534	0.197077	0.0985383	−	0.995133i	$-0.468583\pi$
$283$	3.31534	0.197077	0.0985383	+	0.995133i	$0.468583\pi$
$284$	0	0
$285$	−11.1231	−0.658876
$286$	0	0
$287$	11.8078	0.696990
$288$	0	0
$289$	9.24621	0.543895
$290$	0	0
$291$	−18.4924	−1.08405
$292$	0	0
$293$	16.2462	0.949114	0.474557	−	0.880225i	$-0.342608\pi$
$293$	16.2462	0.949114	0.474557	+	0.880225i	$0.342608\pi$
$294$	0	0
$295$	−34.0540	−1.98270
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	2.43845	0.141019
$300$	0	0
$301$	−2.43845	−0.140550
$302$	0	0
$303$	−0.246211	−0.0141445
$304$	0	0
$305$	−1.56155	−0.0894143
$306$	0	0
$307$	25.3693	1.44790	0.723952	−	0.689851i	$-0.242324\pi$
$307$	25.3693	1.44790	0.723952	+	0.689851i	$0.242324\pi$
$308$	0	0
$309$	13.5616	0.771490
$310$	0	0
$311$	16.0000	0.907277	0.453638	−	0.891186i	$-0.350126\pi$
$311$	16.0000	0.907277	0.453638	+	0.891186i	$0.350126\pi$
$312$	0	0
$313$	24.9309	1.40918	0.704588	−	0.709617i	$-0.251132\pi$
$313$	24.9309	1.40918	0.704588	+	0.709617i	$0.251132\pi$
$314$	0	0
$315$	5.56155	0.313358
$316$	0	0
$317$	15.5616	0.874024	0.437012	−	0.899456i	$-0.356037\pi$
$317$	15.5616	0.874024	0.437012	+	0.899456i	$0.356037\pi$
$318$	0	0
$319$	10.6847	0.598226
$320$	0	0
$321$	17.5616	0.980190
$322$	0	0
$323$	16.0000	0.890264
$324$	0	0
$325$	7.68466	0.426268
$326$	0	0
$327$	15.3693	0.849925
$328$	0	0
$329$	−12.4924	−0.688730
$330$	0	0
$331$	−10.0540	−0.552616	−0.276308	−	0.961069i	$-0.589111\pi$
$331$	−10.0540	−0.552616	−0.276308	+	0.961069i	$0.589111\pi$
$332$	0	0
$333$	−8.24621	−0.451890
$334$	0	0
$335$	−37.1771	−2.03120
$336$	0	0
$337$	−20.2462	−1.10288	−0.551441	−	0.834214i	$-0.685922\pi$
$337$	−20.2462	−1.10288	−0.551441	+	0.834214i	$0.685922\pi$
$338$	0	0
$339$	7.56155	0.410687
$340$	0	0
$341$	10.2462	0.554863
$342$	0	0
$343$	18.0540	0.974823
$344$	0	0
$345$	−8.68466	−0.467566
$346$	0	0
$347$	24.4924	1.31482	0.657411	−	0.753532i	$-0.271652\pi$
$347$	24.4924	1.31482	0.657411	+	0.753532i	$0.271652\pi$
$348$	0	0
$349$	−23.8617	−1.27729	−0.638645	−	0.769502i	$-0.720505\pi$
$349$	−23.8617	−1.27729	−0.638645	+	0.769502i	$0.720505\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	−28.7386	−1.52960	−0.764802	−	0.644266i	$-0.777163\pi$
$353$	−28.7386	−1.52960	−0.764802	+	0.644266i	$0.777163\pi$
$354$	0	0
$355$	33.3693	1.77106
$356$	0	0
$357$	−8.00000	−0.423405
$358$	0	0
$359$	−13.5616	−0.715751	−0.357876	−	0.933769i	$-0.616499\pi$
$359$	−13.5616	−0.715751	−0.357876	+	0.933769i	$0.616499\pi$
$360$	0	0
$361$	−9.24621	−0.486643
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	41.1771	2.15531
$366$	0	0
$367$	16.0000	0.835193	0.417597	−	0.908633i	$-0.362873\pi$
$367$	16.0000	0.835193	0.417597	+	0.908633i	$0.362873\pi$
$368$	0	0
$369$	−7.56155	−0.393639
$370$	0	0
$371$	−19.1231	−0.992822
$372$	0	0
$373$	−4.05398	−0.209907	−0.104953	−	0.994477i	$-0.533469\pi$
$373$	−4.05398	−0.209907	−0.104953	+	0.994477i	$0.533469\pi$
$374$	0	0
$375$	−9.56155	−0.493756
$376$	0	0
$377$	−10.6847	−0.550288
$378$	0	0
$379$	20.4924	1.05263	0.526313	−	0.850291i	$-0.323574\pi$
$379$	20.4924	1.05263	0.526313	+	0.850291i	$0.323574\pi$
$380$	0	0
$381$	−3.12311	−0.160002
$382$	0	0
$383$	31.6155	1.61548	0.807739	−	0.589540i	$-0.200691\pi$
$383$	31.6155	1.61548	0.807739	+	0.589540i	$0.200691\pi$
$384$	0	0
$385$	−5.56155	−0.283443
$386$	0	0
$387$	1.56155	0.0793782
$388$	0	0
$389$	2.49242	0.126371	0.0631854	−	0.998002i	$-0.479874\pi$
$389$	2.49242	0.126371	0.0631854	+	0.998002i	$0.479874\pi$
$390$	0	0
$391$	12.4924	0.631769
$392$	0	0
$393$	4.68466	0.236310
$394$	0	0
$395$	17.3693	0.873945
$396$	0	0
$397$	−28.4384	−1.42728	−0.713642	−	0.700510i	$-0.752956\pi$
$397$	−28.4384	−1.42728	−0.713642	+	0.700510i	$0.752956\pi$
$398$	0	0
$399$	−4.87689	−0.244150
$400$	0	0
$401$	−6.87689	−0.343416	−0.171708	−	0.985148i	$-0.554929\pi$
$401$	−6.87689	−0.343416	−0.171708	+	0.985148i	$0.554929\pi$
$402$	0	0
$403$	−10.2462	−0.510400
$404$	0	0
$405$	−3.56155	−0.176975
$406$	0	0
$407$	8.24621	0.408750
$408$	0	0
$409$	5.80776	0.287175	0.143588	−	0.989638i	$-0.454136\pi$
$409$	5.80776	0.287175	0.143588	+	0.989638i	$0.454136\pi$
$410$	0	0
$411$	−14.4924	−0.714858
$412$	0	0
$413$	−14.9309	−0.734700
$414$	0	0
$415$	−31.6155	−1.55195
$416$	0	0
$417$	−10.2462	−0.501759
$418$	0	0
$419$	12.0000	0.586238	0.293119	−	0.956076i	$-0.405307\pi$
$419$	12.0000	0.586238	0.293119	+	0.956076i	$0.405307\pi$
$420$	0	0
$421$	3.56155	0.173579	0.0867897	−	0.996227i	$-0.472339\pi$
$421$	3.56155	0.173579	0.0867897	+	0.996227i	$0.472339\pi$
$422$	0	0
$423$	8.00000	0.388973
$424$	0	0
$425$	39.3693	1.90969
$426$	0	0
$427$	−0.684658	−0.0331329
$428$	0	0
$429$	−1.00000	−0.0482805
$430$	0	0
$431$	−8.00000	−0.385346	−0.192673	−	0.981263i	$-0.561716\pi$
$431$	−8.00000	−0.385346	−0.192673	+	0.981263i	$0.561716\pi$
$432$	0	0
$433$	18.3002	0.879451	0.439725	−	0.898132i	$-0.355076\pi$
$433$	18.3002	0.879451	0.439725	+	0.898132i	$0.355076\pi$
$434$	0	0
$435$	38.0540	1.82455
$436$	0	0
$437$	7.61553	0.364300
$438$	0	0
$439$	−26.7386	−1.27617	−0.638083	−	0.769968i	$-0.720272\pi$
$439$	−26.7386	−1.27617	−0.638083	+	0.769968i	$0.720272\pi$
$440$	0	0
$441$	−4.56155	−0.217217
$442$	0	0
$443$	−5.75379	−0.273371	−0.136685	−	0.990615i	$-0.543645\pi$
$443$	−5.75379	−0.273371	−0.136685	+	0.990615i	$0.543645\pi$
$444$	0	0
$445$	−26.2462	−1.24419
$446$	0	0
$447$	−1.12311	−0.0531211
$448$	0	0
$449$	31.3693	1.48041	0.740205	−	0.672381i	$-0.234729\pi$
$449$	31.3693	1.48041	0.740205	+	0.672381i	$0.234729\pi$
$450$	0	0
$451$	7.56155	0.356060
$452$	0	0
$453$	10.2462	0.481409
$454$	0	0
$455$	5.56155	0.260730
$456$	0	0
$457$	15.5616	0.727939	0.363969	−	0.931411i	$-0.381421\pi$
$457$	15.5616	0.727939	0.363969	+	0.931411i	$0.381421\pi$
$458$	0	0
$459$	5.12311	0.239126
$460$	0	0
$461$	−4.63068	−0.215672	−0.107836	−	0.994169i	$-0.534392\pi$
$461$	−4.63068	−0.215672	−0.107836	+	0.994169i	$0.534392\pi$
$462$	0	0
$463$	−17.8617	−0.830105	−0.415053	−	0.909797i	$-0.636237\pi$
$463$	−17.8617	−0.830105	−0.415053	+	0.909797i	$0.636237\pi$
$464$	0	0
$465$	36.4924	1.69230
$466$	0	0
$467$	42.2462	1.95492	0.977461	−	0.211117i	$-0.0677102\pi$
$467$	42.2462	1.95492	0.977461	+	0.211117i	$0.0677102\pi$
$468$	0	0
$469$	−16.3002	−0.752673
$470$	0	0
$471$	−8.24621	−0.379965
$472$	0	0
$473$	−1.56155	−0.0718003
$474$	0	0
$475$	24.0000	1.10120
$476$	0	0
$477$	12.2462	0.560715
$478$	0	0
$479$	−32.3002	−1.47583	−0.737917	−	0.674892i	$-0.764190\pi$
$479$	−32.3002	−1.47583	−0.737917	+	0.674892i	$0.764190\pi$
$480$	0	0
$481$	−8.24621	−0.375995
$482$	0	0
$483$	−3.80776	−0.173259
$484$	0	0
$485$	65.8617	2.99063
$486$	0	0
$487$	−13.3693	−0.605822	−0.302911	−	0.953019i	$-0.597958\pi$
$487$	−13.3693	−0.605822	−0.302911	+	0.953019i	$0.597958\pi$
$488$	0	0
$489$	11.8078	0.533966
$490$	0	0
$491$	17.1771	0.775191	0.387595	−	0.921830i	$-0.373306\pi$
$491$	17.1771	0.775191	0.387595	+	0.921830i	$0.373306\pi$
$492$	0	0
$493$	−54.7386	−2.46530
$494$	0	0
$495$	3.56155	0.160080
$496$	0	0
$497$	14.6307	0.656276
$498$	0	0
$499$	24.3002	1.08783	0.543913	−	0.839142i	$-0.316942\pi$
$499$	24.3002	1.08783	0.543913	+	0.839142i	$0.316942\pi$
$500$	0	0
$501$	14.9309	0.667062
$502$	0	0
$503$	−4.49242	−0.200307	−0.100154	−	0.994972i	$-0.531933\pi$
$503$	−4.49242	−0.200307	−0.100154	+	0.994972i	$0.531933\pi$
$504$	0	0
$505$	0.876894	0.0390213
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	2.00000	0.0886484	0.0443242	−	0.999017i	$-0.485887\pi$
$509$	2.00000	0.0886484	0.0443242	+	0.999017i	$0.485887\pi$
$510$	0	0
$511$	18.0540	0.798661
$512$	0	0
$513$	3.12311	0.137888
$514$	0	0
$515$	−48.3002	−2.12836
$516$	0	0
$517$	−8.00000	−0.351840
$518$	0	0
$519$	22.6847	0.995746
$520$	0	0
$521$	14.1922	0.621773	0.310887	−	0.950447i	$-0.399374\pi$
$521$	14.1922	0.621773	0.310887	+	0.950447i	$0.399374\pi$
$522$	0	0
$523$	−4.00000	−0.174908	−0.0874539	−	0.996169i	$-0.527873\pi$
$523$	−4.00000	−0.174908	−0.0874539	+	0.996169i	$0.527873\pi$
$524$	0	0
$525$	−12.0000	−0.523723
$526$	0	0
$527$	−52.4924	−2.28661
$528$	0	0
$529$	−17.0540	−0.741477
$530$	0	0
$531$	9.56155	0.414936
$532$	0	0
$533$	−7.56155	−0.327527
$534$	0	0
$535$	−62.5464	−2.70412
$536$	0	0
$537$	−4.00000	−0.172613
$538$	0	0
$539$	4.56155	0.196480
$540$	0	0
$541$	−4.73863	−0.203730	−0.101865	−	0.994798i	$-0.532481\pi$
$541$	−4.73863	−0.203730	−0.101865	+	0.994798i	$0.532481\pi$
$542$	0	0
$543$	12.2462	0.525535
$544$	0	0
$545$	−54.7386	−2.34475
$546$	0	0
$547$	−26.9309	−1.15148	−0.575740	−	0.817633i	$-0.695286\pi$
$547$	−26.9309	−1.15148	−0.575740	+	0.817633i	$0.695286\pi$
$548$	0	0
$549$	0.438447	0.0187125
$550$	0	0
$551$	−33.3693	−1.42158
$552$	0	0
$553$	7.61553	0.323845
$554$	0	0
$555$	29.3693	1.24666
$556$	0	0
$557$	8.24621	0.349403	0.174702	−	0.984621i	$-0.444104\pi$
$557$	8.24621	0.349403	0.174702	+	0.984621i	$0.444104\pi$
$558$	0	0
$559$	1.56155	0.0660466
$560$	0	0
$561$	−5.12311	−0.216298
$562$	0	0
$563$	−38.7386	−1.63264	−0.816319	−	0.577601i	$-0.803989\pi$
$563$	−38.7386	−1.63264	−0.816319	+	0.577601i	$0.803989\pi$
$564$	0	0
$565$	−26.9309	−1.13299
$566$	0	0
$567$	−1.56155	−0.0655791
$568$	0	0
$569$	6.49242	0.272177	0.136088	−	0.990697i	$-0.456547\pi$
$569$	6.49242	0.272177	0.136088	+	0.990697i	$0.456547\pi$
$570$	0	0
$571$	−22.4384	−0.939020	−0.469510	−	0.882927i	$-0.655569\pi$
$571$	−22.4384	−0.939020	−0.469510	+	0.882927i	$0.655569\pi$
$572$	0	0
$573$	−10.4384	−0.436072
$574$	0	0
$575$	18.7386	0.781455
$576$	0	0
$577$	13.1231	0.546322	0.273161	−	0.961968i	$-0.411931\pi$
$577$	13.1231	0.546322	0.273161	+	0.961968i	$0.411931\pi$
$578$	0	0
$579$	8.24621	0.342701
$580$	0	0
$581$	−13.8617	−0.575082
$582$	0	0
$583$	−12.2462	−0.507186
$584$	0	0
$585$	−3.56155	−0.147252
$586$	0	0
$587$	14.4384	0.595938	0.297969	−	0.954575i	$-0.403691\pi$
$587$	14.4384	0.595938	0.297969	+	0.954575i	$0.403691\pi$
$588$	0	0
$589$	−32.0000	−1.31854
$590$	0	0
$591$	25.6155	1.05368
$592$	0	0
$593$	−22.4924	−0.923653	−0.461827	−	0.886970i	$-0.652806\pi$
$593$	−22.4924	−0.923653	−0.461827	+	0.886970i	$0.652806\pi$
$594$	0	0
$595$	28.4924	1.16808
$596$	0	0
$597$	22.9309	0.938498
$598$	0	0
$599$	−11.8078	−0.482452	−0.241226	−	0.970469i	$-0.577550\pi$
$599$	−11.8078	−0.482452	−0.241226	+	0.970469i	$0.577550\pi$
$600$	0	0
$601$	23.8617	0.973341	0.486670	−	0.873586i	$-0.338211\pi$
$601$	23.8617	0.973341	0.486670	+	0.873586i	$0.338211\pi$
$602$	0	0
$603$	10.4384	0.425086
$604$	0	0
$605$	−3.56155	−0.144798
$606$	0	0
$607$	3.12311	0.126763	0.0633815	−	0.997989i	$-0.479812\pi$
$607$	3.12311	0.126763	0.0633815	+	0.997989i	$0.479812\pi$
$608$	0	0
$609$	16.6847	0.676096
$610$	0	0
$611$	8.00000	0.323645
$612$	0	0
$613$	27.8617	1.12533	0.562663	−	0.826687i	$-0.309777\pi$
$613$	27.8617	1.12533	0.562663	+	0.826687i	$0.309777\pi$
$614$	0	0
$615$	26.9309	1.08596
$616$	0	0
$617$	4.24621	0.170946	0.0854730	−	0.996340i	$-0.472760\pi$
$617$	4.24621	0.170946	0.0854730	+	0.996340i	$0.472760\pi$
$618$	0	0
$619$	38.9309	1.56476	0.782382	−	0.622799i	$-0.214005\pi$
$619$	38.9309	1.56476	0.782382	+	0.622799i	$0.214005\pi$
$620$	0	0
$621$	2.43845	0.0978515
$622$	0	0
$623$	−11.5076	−0.461041
$624$	0	0
$625$	−4.36932	−0.174773
$626$	0	0
$627$	−3.12311	−0.124725
$628$	0	0
$629$	−42.2462	−1.68447
$630$	0	0
$631$	−8.87689	−0.353384	−0.176692	−	0.984266i	$-0.556540\pi$
$631$	−8.87689	−0.353384	−0.176692	+	0.984266i	$0.556540\pi$
$632$	0	0
$633$	−4.00000	−0.158986
$634$	0	0
$635$	11.1231	0.441407
$636$	0	0
$637$	−4.56155	−0.180735
$638$	0	0
$639$	−9.36932	−0.370644
$640$	0	0
$641$	10.6847	0.422019	0.211009	−	0.977484i	$-0.432325\pi$
$641$	10.6847	0.422019	0.211009	+	0.977484i	$0.432325\pi$
$642$	0	0
$643$	17.7538	0.700141	0.350071	−	0.936723i	$-0.386158\pi$
$643$	17.7538	0.700141	0.350071	+	0.936723i	$0.386158\pi$
$644$	0	0
$645$	−5.56155	−0.218986
$646$	0	0
$647$	−48.9848	−1.92579	−0.962896	−	0.269871i	$-0.913019\pi$
$647$	−48.9848	−1.92579	−0.962896	+	0.269871i	$0.913019\pi$
$648$	0	0
$649$	−9.56155	−0.375324
$650$	0	0
$651$	16.0000	0.627089
$652$	0	0
$653$	−6.49242	−0.254068	−0.127034	−	0.991898i	$-0.540546\pi$
$653$	−6.49242	−0.254068	−0.127034	+	0.991898i	$0.540546\pi$
$654$	0	0
$655$	−16.6847	−0.651924
$656$	0	0
$657$	−11.5616	−0.451059
$658$	0	0
$659$	−21.7538	−0.847407	−0.423704	−	0.905801i	$-0.639270\pi$
$659$	−21.7538	−0.847407	−0.423704	+	0.905801i	$0.639270\pi$
$660$	0	0
$661$	−8.24621	−0.320740	−0.160370	−	0.987057i	$-0.551269\pi$
$661$	−8.24621	−0.320740	−0.160370	+	0.987057i	$0.551269\pi$
$662$	0	0
$663$	5.12311	0.198965
$664$	0	0
$665$	17.3693	0.673553
$666$	0	0
$667$	−26.0540	−1.00881
$668$	0	0
$669$	8.49242	0.328336
$670$	0	0
$671$	−0.438447	−0.0169261
$672$	0	0
$673$	−31.3693	−1.20920	−0.604599	−	0.796530i	$-0.706667\pi$
$673$	−31.3693	−1.20920	−0.604599	+	0.796530i	$0.706667\pi$
$674$	0	0
$675$	7.68466	0.295783
$676$	0	0
$677$	14.0000	0.538064	0.269032	−	0.963131i	$-0.413296\pi$
$677$	14.0000	0.538064	0.269032	+	0.963131i	$0.413296\pi$
$678$	0	0
$679$	28.8769	1.10819
$680$	0	0
$681$	−10.2462	−0.392636
$682$	0	0
$683$	−42.9309	−1.64270	−0.821352	−	0.570422i	$-0.806780\pi$
$683$	−42.9309	−1.64270	−0.821352	+	0.570422i	$0.806780\pi$
$684$	0	0
$685$	51.6155	1.97213
$686$	0	0
$687$	−6.19224	−0.236249
$688$	0	0
$689$	12.2462	0.466543
$690$	0	0
$691$	−40.0000	−1.52167	−0.760836	−	0.648944i	$-0.775211\pi$
$691$	−40.0000	−1.52167	−0.760836	+	0.648944i	$0.775211\pi$
$692$	0	0
$693$	1.56155	0.0593185
$694$	0	0
$695$	36.4924	1.38424
$696$	0	0
$697$	−38.7386	−1.46733
$698$	0	0
$699$	16.2462	0.614488
$700$	0	0
$701$	−24.9309	−0.941626	−0.470813	−	0.882233i	$-0.656039\pi$
$701$	−24.9309	−0.941626	−0.470813	+	0.882233i	$0.656039\pi$
$702$	0	0
$703$	−25.7538	−0.971323
$704$	0	0
$705$	−28.4924	−1.07309
$706$	0	0
$707$	0.384472	0.0144596
$708$	0	0
$709$	9.80776	0.368338	0.184169	−	0.982895i	$-0.441041\pi$
$709$	9.80776	0.368338	0.184169	+	0.982895i	$0.441041\pi$
$710$	0	0
$711$	−4.87689	−0.182898
$712$	0	0
$713$	−24.9848	−0.935690
$714$	0	0
$715$	3.56155	0.133195
$716$	0	0
$717$	1.06913	0.0399274
$718$	0	0
$719$	18.0540	0.673300	0.336650	−	0.941630i	$-0.390706\pi$
$719$	18.0540	0.673300	0.336650	+	0.941630i	$0.390706\pi$
$720$	0	0
$721$	−21.1771	−0.788676
$722$	0	0
$723$	0.246211	0.00915669
$724$	0	0
$725$	−82.1080	−3.04941
$726$	0	0
$727$	33.7538	1.25186	0.625929	−	0.779880i	$-0.284720\pi$
$727$	33.7538	1.25186	0.625929	+	0.779880i	$0.284720\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	8.00000	0.295891
$732$	0	0
$733$	−2.00000	−0.0738717	−0.0369358	−	0.999318i	$-0.511760\pi$
$733$	−2.00000	−0.0738717	−0.0369358	+	0.999318i	$0.511760\pi$
$734$	0	0
$735$	16.2462	0.599251
$736$	0	0
$737$	−10.4384	−0.384505
$738$	0	0
$739$	−26.7386	−0.983597	−0.491798	−	0.870709i	$-0.663660\pi$
$739$	−26.7386	−0.983597	−0.491798	+	0.870709i	$0.663660\pi$
$740$	0	0
$741$	3.12311	0.114730
$742$	0	0
$743$	26.4384	0.969933	0.484966	−	0.874533i	$-0.338832\pi$
$743$	26.4384	0.969933	0.484966	+	0.874533i	$0.338832\pi$
$744$	0	0
$745$	4.00000	0.146549
$746$	0	0
$747$	8.87689	0.324789
$748$	0	0
$749$	−27.4233	−1.00203
$750$	0	0
$751$	−25.0691	−0.914786	−0.457393	−	0.889265i	$-0.651217\pi$
$751$	−25.0691	−0.914786	−0.457393	+	0.889265i	$0.651217\pi$
$752$	0	0
$753$	−8.87689	−0.323492
$754$	0	0
$755$	−36.4924	−1.32810
$756$	0	0
$757$	22.0000	0.799604	0.399802	−	0.916602i	$-0.369079\pi$
$757$	22.0000	0.799604	0.399802	+	0.916602i	$0.369079\pi$
$758$	0	0
$759$	−2.43845	−0.0885100
$760$	0	0
$761$	−24.9309	−0.903743	−0.451872	−	0.892083i	$-0.649244\pi$
$761$	−24.9309	−0.903743	−0.451872	+	0.892083i	$0.649244\pi$
$762$	0	0
$763$	−24.0000	−0.868858
$764$	0	0
$765$	−18.2462	−0.659693
$766$	0	0
$767$	9.56155	0.345248
$768$	0	0
$769$	−11.5616	−0.416920	−0.208460	−	0.978031i	$-0.566845\pi$
$769$	−11.5616	−0.416920	−0.208460	+	0.978031i	$0.566845\pi$
$770$	0	0
$771$	−3.56155	−0.128266
$772$	0	0
$773$	52.7386	1.89688	0.948438	−	0.316961i	$-0.102663\pi$
$773$	52.7386	1.89688	0.948438	+	0.316961i	$0.102663\pi$
$774$	0	0
$775$	−78.7386	−2.82838
$776$	0	0
$777$	12.8769	0.461956
$778$	0	0
$779$	−23.6155	−0.846114
$780$	0	0
$781$	9.36932	0.335261
$782$	0	0
$783$	−10.6847	−0.381839
$784$	0	0
$785$	29.3693	1.04824
$786$	0	0
$787$	−24.0000	−0.855508	−0.427754	−	0.903895i	$-0.640695\pi$
$787$	−24.0000	−0.855508	−0.427754	+	0.903895i	$0.640695\pi$
$788$	0	0
$789$	20.4924	0.729550
$790$	0	0
$791$	−11.8078	−0.419836
$792$	0	0
$793$	0.438447	0.0155697
$794$	0	0
$795$	−43.6155	−1.54688
$796$	0	0
$797$	−47.8617	−1.69535	−0.847675	−	0.530516i	$-0.821998\pi$
$797$	−47.8617	−1.69535	−0.847675	+	0.530516i	$0.821998\pi$
$798$	0	0
$799$	40.9848	1.44994
$800$	0	0
$801$	7.36932	0.260382
$802$	0	0
$803$	11.5616	0.407998
$804$	0	0
$805$	13.5616	0.477982
$806$	0	0
$807$	−17.6155	−0.620096
$808$	0	0
$809$	28.7386	1.01040	0.505198	−	0.863003i	$-0.331419\pi$
$809$	28.7386	1.01040	0.505198	+	0.863003i	$0.331419\pi$
$810$	0	0
$811$	2.73863	0.0961664	0.0480832	−	0.998843i	$-0.484689\pi$
$811$	2.73863	0.0961664	0.0480832	+	0.998843i	$0.484689\pi$
$812$	0	0
$813$	−5.75379	−0.201794
$814$	0	0
$815$	−42.0540	−1.47309
$816$	0	0
$817$	4.87689	0.170621
$818$	0	0
$819$	−1.56155	−0.0545651
$820$	0	0
$821$	35.7538	1.24782	0.623908	−	0.781498i	$-0.285544\pi$
$821$	35.7538	1.24782	0.623908	+	0.781498i	$0.285544\pi$
$822$	0	0
$823$	21.1771	0.738187	0.369093	−	0.929392i	$-0.379668\pi$
$823$	21.1771	0.738187	0.369093	+	0.929392i	$0.379668\pi$
$824$	0	0
$825$	−7.68466	−0.267545
$826$	0	0
$827$	38.7386	1.34707	0.673537	−	0.739153i	$-0.264774\pi$
$827$	38.7386	1.34707	0.673537	+	0.739153i	$0.264774\pi$
$828$	0	0
$829$	22.9848	0.798297	0.399148	−	0.916886i	$-0.369306\pi$
$829$	22.9848	0.798297	0.399148	+	0.916886i	$0.369306\pi$
$830$	0	0
$831$	8.43845	0.292726
$832$	0	0
$833$	−23.3693	−0.809699
$834$	0	0
$835$	−53.1771	−1.84027
$836$	0	0
$837$	−10.2462	−0.354161
$838$	0	0
$839$	−12.8769	−0.444560	−0.222280	−	0.974983i	$-0.571350\pi$
$839$	−12.8769	−0.444560	−0.222280	+	0.974983i	$0.571350\pi$
$840$	0	0
$841$	85.1619	2.93662
$842$	0	0
$843$	3.56155	0.122666
$844$	0	0
$845$	−3.56155	−0.122521
$846$	0	0
$847$	−1.56155	−0.0536556
$848$	0	0
$849$	3.31534	0.113782
$850$	0	0
$851$	−20.1080	−0.689292
$852$	0	0
$853$	37.6155	1.28793	0.643966	−	0.765054i	$-0.277288\pi$
$853$	37.6155	1.28793	0.643966	+	0.765054i	$0.277288\pi$
$854$	0	0
$855$	−11.1231	−0.380402
$856$	0	0
$857$	49.6155	1.69483	0.847417	−	0.530928i	$-0.178156\pi$
$857$	49.6155	1.69483	0.847417	+	0.530928i	$0.178156\pi$
$858$	0	0
$859$	15.5076	0.529112	0.264556	−	0.964370i	$-0.414775\pi$
$859$	15.5076	0.529112	0.264556	+	0.964370i	$0.414775\pi$
$860$	0	0
$861$	11.8078	0.402408
$862$	0	0
$863$	14.2462	0.484947	0.242473	−	0.970158i	$-0.422041\pi$
$863$	14.2462	0.484947	0.242473	+	0.970158i	$0.422041\pi$
$864$	0	0
$865$	−80.7926	−2.74703
$866$	0	0
$867$	9.24621	0.314018
$868$	0	0
$869$	4.87689	0.165437
$870$	0	0
$871$	10.4384	0.353693
$872$	0	0
$873$	−18.4924	−0.625874
$874$	0	0
$875$	14.9309	0.504756
$876$	0	0
$877$	39.3693	1.32941	0.664704	−	0.747107i	$-0.268558\pi$
$877$	39.3693	1.32941	0.664704	+	0.747107i	$0.268558\pi$
$878$	0	0
$879$	16.2462	0.547971
$880$	0	0
$881$	−14.6847	−0.494739	−0.247369	−	0.968921i	$-0.579566\pi$
$881$	−14.6847	−0.494739	−0.247369	+	0.968921i	$0.579566\pi$
$882$	0	0
$883$	−2.63068	−0.0885295	−0.0442648	−	0.999020i	$-0.514095\pi$
$883$	−2.63068	−0.0885295	−0.0442648	+	0.999020i	$0.514095\pi$
$884$	0	0
$885$	−34.0540	−1.14471
$886$	0	0
$887$	−16.0000	−0.537227	−0.268614	−	0.963248i	$-0.586566\pi$
$887$	−16.0000	−0.537227	−0.268614	+	0.963248i	$0.586566\pi$
$888$	0	0
$889$	4.87689	0.163566
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	24.9848	0.836086
$894$	0	0
$895$	14.2462	0.476198
$896$	0	0
$897$	2.43845	0.0814174
$898$	0	0
$899$	109.477	3.65127
$900$	0	0
$901$	62.7386	2.09013
$902$	0	0
$903$	−2.43845	−0.0811464
$904$	0	0
$905$	−43.6155	−1.44983
$906$	0	0
$907$	−18.2462	−0.605856	−0.302928	−	0.953014i	$-0.597964\pi$
$907$	−18.2462	−0.605856	−0.302928	+	0.953014i	$0.597964\pi$
$908$	0	0
$909$	−0.246211	−0.00816631
$910$	0	0
$911$	−8.00000	−0.265052	−0.132526	−	0.991180i	$-0.542309\pi$
$911$	−8.00000	−0.265052	−0.132526	+	0.991180i	$0.542309\pi$
$912$	0	0
$913$	−8.87689	−0.293782
$914$	0	0
$915$	−1.56155	−0.0516233
$916$	0	0
$917$	−7.31534	−0.241574
$918$	0	0
$919$	−1.75379	−0.0578522	−0.0289261	−	0.999582i	$-0.509209\pi$
$919$	−1.75379	−0.0578522	−0.0289261	+	0.999582i	$0.509209\pi$
$920$	0	0
$921$	25.3693	0.835947
$922$	0	0
$923$	−9.36932	−0.308395
$924$	0	0
$925$	−63.3693	−2.08357
$926$	0	0
$927$	13.5616	0.445420
$928$	0	0
$929$	−14.8769	−0.488095	−0.244048	−	0.969763i	$-0.578475\pi$
$929$	−14.8769	−0.488095	−0.244048	+	0.969763i	$0.578475\pi$
$930$	0	0
$931$	−14.2462	−0.466901
$932$	0	0
$933$	16.0000	0.523816
$934$	0	0
$935$	18.2462	0.596715
$936$	0	0
$937$	23.8617	0.779529	0.389765	−	0.920915i	$-0.372556\pi$
$937$	23.8617	0.779529	0.389765	+	0.920915i	$0.372556\pi$
$938$	0	0
$939$	24.9309	0.813588
$940$	0	0
$941$	31.8617	1.03866	0.519332	−	0.854573i	$-0.326181\pi$
$941$	31.8617	1.03866	0.519332	+	0.854573i	$0.326181\pi$
$942$	0	0
$943$	−18.4384	−0.600438
$944$	0	0
$945$	5.56155	0.180917
$946$	0	0
$947$	12.9848	0.421951	0.210975	−	0.977491i	$-0.432336\pi$
$947$	12.9848	0.421951	0.210975	+	0.977491i	$0.432336\pi$
$948$	0	0
$949$	−11.5616	−0.375304
$950$	0	0
$951$	15.5616	0.504618
$952$	0	0
$953$	−45.6155	−1.47763	−0.738816	−	0.673907i	$-0.764615\pi$
$953$	−45.6155	−1.47763	−0.738816	+	0.673907i	$0.764615\pi$
$954$	0	0
$955$	37.1771	1.20302
$956$	0	0
$957$	10.6847	0.345386
$958$	0	0
$959$	22.6307	0.730783
$960$	0	0
$961$	73.9848	2.38661
$962$	0	0
$963$	17.5616	0.565913
$964$	0	0
$965$	−29.3693	−0.945432
$966$	0	0
$967$	−50.9309	−1.63783	−0.818913	−	0.573917i	$-0.805423\pi$
$967$	−50.9309	−1.63783	−0.818913	+	0.573917i	$0.805423\pi$
$968$	0	0
$969$	16.0000	0.513994
$970$	0	0
$971$	4.38447	0.140704	0.0703522	−	0.997522i	$-0.477588\pi$
$971$	4.38447	0.140704	0.0703522	+	0.997522i	$0.477588\pi$
$972$	0	0
$973$	16.0000	0.512936
$974$	0	0
$975$	7.68466	0.246106
$976$	0	0
$977$	47.3693	1.51548	0.757739	−	0.652557i	$-0.226304\pi$
$977$	47.3693	1.51548	0.757739	+	0.652557i	$0.226304\pi$
$978$	0	0
$979$	−7.36932	−0.235524
$980$	0	0
$981$	15.3693	0.490705
$982$	0	0
$983$	9.36932	0.298835	0.149417	−	0.988774i	$-0.452260\pi$
$983$	9.36932	0.298835	0.149417	+	0.988774i	$0.452260\pi$
$984$	0	0
$985$	−91.2311	−2.90686
$986$	0	0
$987$	−12.4924	−0.397638
$988$	0	0
$989$	3.80776	0.121080
$990$	0	0
$991$	16.3002	0.517792	0.258896	−	0.965905i	$-0.416641\pi$
$991$	16.3002	0.517792	0.258896	+	0.965905i	$0.416641\pi$
$992$	0	0
$993$	−10.0540	−0.319053
$994$	0	0
$995$	−81.6695	−2.58910
$996$	0	0
$997$	−31.1771	−0.987388	−0.493694	−	0.869636i	$-0.664354\pi$
$997$	−31.1771	−0.987388	−0.493694	+	0.869636i	$0.664354\pi$
$998$	0	0
$999$	−8.24621	−0.260899

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.bh.1.1		2
4.3	odd	2		858.2.a.n.1.1	✓	2
12.11	even	2		2574.2.a.bg.1.2		2
44.43	even	2		9438.2.a.bv.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
858.2.a.n.1.1	✓	2	4.3	odd	2
2574.2.a.bg.1.2		2	12.11	even	2
6864.2.a.bh.1.1		2	1.1	even	1	trivial
9438.2.a.bv.1.1		2	44.43	even	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:	\( N \)	\(=\)	\( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6864.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -3.56155 q^{5} -1.56155 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -3.56155 q^{5} -1.56155 q^{7} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} -3.56155 q^{15} +5.12311 q^{17} +3.12311 q^{19} -1.56155 q^{21} +2.43845 q^{23} +7.68466 q^{25} +1.00000 q^{27} -10.6847 q^{29} -10.2462 q^{31} -1.00000 q^{33} +5.56155 q^{35} -8.24621 q^{37} +1.00000 q^{39} -7.56155 q^{41} +1.56155 q^{43} -3.56155 q^{45} +8.00000 q^{47} -4.56155 q^{49} +5.12311 q^{51} +12.2462 q^{53} +3.56155 q^{55} +3.12311 q^{57} +9.56155 q^{59} +0.438447 q^{61} -1.56155 q^{63} -3.56155 q^{65} +10.4384 q^{67} +2.43845 q^{69} -9.36932 q^{71} -11.5616 q^{73} +7.68466 q^{75} +1.56155 q^{77} -4.87689 q^{79} +1.00000 q^{81} +8.87689 q^{83} -18.2462 q^{85} -10.6847 q^{87} +7.36932 q^{89} -1.56155 q^{91} -10.2462 q^{93} -11.1231 q^{95} -18.4924 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} - 3 q^{5} + q^{7} + 2 q^{9}+O(q^{10}) \) 2 * q + 2 * q^3 - 3 * q^5 + q^7 + 2 * q^9	\( 2 q + 2 q^{3} - 3 q^{5} + q^{7} + 2 q^{9} - 2 q^{11} + 2 q^{13} - 3 q^{15} + 2 q^{17} - 2 q^{19} + q^{21} + 9 q^{23} + 3 q^{25} + 2 q^{27} - 9 q^{29} - 4 q^{31} - 2 q^{33} + 7 q^{35} + 2 q^{39} - 11 q^{41} - q^{43} - 3 q^{45} + 16 q^{47} - 5 q^{49} + 2 q^{51} + 8 q^{53} + 3 q^{55} - 2 q^{57} + 15 q^{59} + 5 q^{61} + q^{63} - 3 q^{65} + 25 q^{67} + 9 q^{69} + 6 q^{71} - 19 q^{73} + 3 q^{75} - q^{77} - 18 q^{79} + 2 q^{81} + 26 q^{83} - 20 q^{85} - 9 q^{87} - 10 q^{89} + q^{91} - 4 q^{93} - 14 q^{95} - 4 q^{97} - 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 - 3 * q^5 + q^7 + 2 * q^9 - 2 * q^11 + 2 * q^13 - 3 * q^15 + 2 * q^17 - 2 * q^19 + q^21 + 9 * q^23 + 3 * q^25 + 2 * q^27 - 9 * q^29 - 4 * q^31 - 2 * q^33 + 7 * q^35 + 2 * q^39 - 11 * q^41 - q^43 - 3 * q^45 + 16 * q^47 - 5 * q^49 + 2 * q^51 + 8 * q^53 + 3 * q^55 - 2 * q^57 + 15 * q^59 + 5 * q^61 + q^63 - 3 * q^65 + 25 * q^67 + 9 * q^69 + 6 * q^71 - 19 * q^73 + 3 * q^75 - q^77 - 18 * q^79 + 2 * q^81 + 26 * q^83 - 20 * q^85 - 9 * q^87 - 10 * q^89 + q^91 - 4 * q^93 - 14 * q^95 - 4 * q^97 - 2 * q^99

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