LMFDB - Embedded newform 6864.2.a.bt.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:	$3$
Coefficient field:	3.3.404.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x - 1 $ x^3 - x^2 - 5*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1716)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.65544$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -3.70682 q^{5} +2.25951 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -3.70682 q^{5} +2.25951 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} +3.70682 q^{15} +2.65544 q^{17} +7.84324 q^{19} -2.25951 q^{21} -7.05137 q^{23} +8.74049 q^{25} -1.00000 q^{27} +5.44731 q^{29} -8.91495 q^{31} -1.00000 q^{33} -8.37559 q^{35} -0.791864 q^{37} -1.00000 q^{39} +4.25951 q^{41} +9.44731 q^{43} -3.70682 q^{45} -2.51902 q^{47} -1.89461 q^{49} -2.65544 q^{51} +7.82991 q^{53} -3.70682 q^{55} -7.84324 q^{57} -7.41363 q^{59} +6.62177 q^{61} +2.25951 q^{63} -3.70682 q^{65} +10.2258 q^{67} +7.05137 q^{69} -2.79186 q^{71} +6.53235 q^{73} -8.74049 q^{75} +2.25951 q^{77} -14.1718 q^{79} +1.00000 q^{81} -7.41363 q^{83} -9.84324 q^{85} -5.44731 q^{87} +4.66877 q^{89} +2.25951 q^{91} +8.91495 q^{93} -29.0734 q^{95} +3.41363 q^{97} +1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} + 4 q^{5} + 4 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q - 3 * q^3 + 4 * q^5 + 4 * q^7 + 3 * q^9	$ 3 q - 3 q^{3} + 4 q^{5} + 4 q^{7} + 3 q^{9} + 3 q^{11} + 3 q^{13} - 4 q^{15} + 2 q^{17} + 8 q^{19} - 4 q^{21} - 12 q^{23} + 29 q^{25} - 3 q^{27} + 4 q^{29} - 18 q^{31} - 3 q^{33} - 6 q^{35} + 4 q^{37} - 3 q^{39} + 10 q^{41} + 16 q^{43} + 4 q^{45} - 2 q^{47} + 19 q^{49} - 2 q^{51} + 6 q^{53} + 4 q^{55} - 8 q^{57} + 8 q^{59} - 4 q^{61} + 4 q^{63} + 4 q^{65} + 10 q^{67} + 12 q^{69} - 2 q^{71} + 16 q^{73} - 29 q^{75} + 4 q^{77} + 12 q^{79} + 3 q^{81} + 8 q^{83} - 14 q^{85} - 4 q^{87} + 10 q^{89} + 4 q^{91} + 18 q^{93} - 22 q^{95} - 20 q^{97} + 3 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 + 4 * q^5 + 4 * q^7 + 3 * q^9 + 3 * q^11 + 3 * q^13 - 4 * q^15 + 2 * q^17 + 8 * q^19 - 4 * q^21 - 12 * q^23 + 29 * q^25 - 3 * q^27 + 4 * q^29 - 18 * q^31 - 3 * q^33 - 6 * q^35 + 4 * q^37 - 3 * q^39 + 10 * q^41 + 16 * q^43 + 4 * q^45 - 2 * q^47 + 19 * q^49 - 2 * q^51 + 6 * q^53 + 4 * q^55 - 8 * q^57 + 8 * q^59 - 4 * q^61 + 4 * q^63 + 4 * q^65 + 10 * q^67 + 12 * q^69 - 2 * q^71 + 16 * q^73 - 29 * q^75 + 4 * q^77 + 12 * q^79 + 3 * q^81 + 8 * q^83 - 14 * q^85 - 4 * q^87 + 10 * q^89 + 4 * q^91 + 18 * q^93 - 22 * q^95 - 20 * q^97 + 3 * 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
$5$	−3.70682	−1.65774	−0.828869	−	0.559442i	$-0.811015\pi$
$5$	−3.70682	−1.65774	−0.828869	+	0.559442i	$0.811015\pi$
$6$	0	0
$7$	2.25951	0.854015	0.427007	−	0.904248i	$-0.359568\pi$
$7$	2.25951	0.854015	0.427007	+	0.904248i	$0.359568\pi$
$8$	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
$15$	3.70682	0.957096
$16$	0	0
$17$	2.65544	0.644039	0.322020	−	0.946733i	$-0.395638\pi$
$17$	2.65544	0.644039	0.322020	+	0.946733i	$0.395638\pi$
$18$	0	0
$19$	7.84324	1.79936	0.899681	−	0.436548i	$-0.143799\pi$
$19$	7.84324	1.79936	0.899681	+	0.436548i	$0.143799\pi$
$20$	0	0
$21$	−2.25951	−0.493066
$22$	0	0
$23$	−7.05137	−1.47031	−0.735157	−	0.677897i	$-0.762891\pi$
$23$	−7.05137	−1.47031	−0.735157	+	0.677897i	$0.762891\pi$
$24$	0	0
$25$	8.74049	1.74810
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	5.44731	1.01154	0.505770	−	0.862669i	$-0.331209\pi$
$29$	5.44731	1.01154	0.505770	+	0.862669i	$0.331209\pi$
$30$	0	0
$31$	−8.91495	−1.60117	−0.800586	−	0.599217i	$-0.795479\pi$
$31$	−8.91495	−1.60117	−0.800586	+	0.599217i	$0.795479\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	−8.37559	−1.41573
$36$	0	0
$37$	−0.791864	−0.130182	−0.0650908	−	0.997879i	$-0.520734\pi$
$37$	−0.791864	−0.130182	−0.0650908	+	0.997879i	$0.520734\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	4.25951	0.665224	0.332612	−	0.943064i	$-0.392070\pi$
$41$	4.25951	0.665224	0.332612	+	0.943064i	$0.392070\pi$
$42$	0	0
$43$	9.44731	1.44070	0.720350	−	0.693610i	$-0.243981\pi$
$43$	9.44731	1.44070	0.720350	+	0.693610i	$0.243981\pi$
$44$	0	0
$45$	−3.70682	−0.552580
$46$	0	0
$47$	−2.51902	−0.367437	−0.183718	−	0.982979i	$-0.558813\pi$
$47$	−2.51902	−0.367437	−0.183718	+	0.982979i	$0.558813\pi$
$48$	0	0
$49$	−1.89461	−0.270659
$50$	0	0
$51$	−2.65544	−0.371836
$52$	0	0
$53$	7.82991	1.07552	0.537760	−	0.843098i	$-0.319271\pi$
$53$	7.82991	1.07552	0.537760	+	0.843098i	$0.319271\pi$
$54$	0	0
$55$	−3.70682	−0.499827
$56$	0	0
$57$	−7.84324	−1.03886
$58$	0	0
$59$	−7.41363	−0.965173	−0.482586	−	0.875848i	$-0.660303\pi$
$59$	−7.41363	−0.965173	−0.482586	+	0.875848i	$0.660303\pi$
$60$	0	0
$61$	6.62177	0.847831	0.423915	−	0.905702i	$-0.360655\pi$
$61$	6.62177	0.847831	0.423915	+	0.905702i	$0.360655\pi$
$62$	0	0
$63$	2.25951	0.284672
$64$	0	0
$65$	−3.70682	−0.459774
$66$	0	0
$67$	10.2258	1.24928	0.624642	−	0.780911i	$-0.285245\pi$
$67$	10.2258	1.24928	0.624642	+	0.780911i	$0.285245\pi$
$68$	0	0
$69$	7.05137	0.848886
$70$	0	0
$71$	−2.79186	−0.331333	−0.165667	−	0.986182i	$-0.552978\pi$
$71$	−2.79186	−0.331333	−0.165667	+	0.986182i	$0.552978\pi$
$72$	0	0
$73$	6.53235	0.764554	0.382277	−	0.924048i	$-0.375140\pi$
$73$	6.53235	0.764554	0.382277	+	0.924048i	$0.375140\pi$
$74$	0	0
$75$	−8.74049	−1.00926
$76$	0	0
$77$	2.25951	0.257495
$78$	0	0
$79$	−14.1718	−1.59445	−0.797227	−	0.603679i	$-0.793701\pi$
$79$	−14.1718	−1.59445	−0.797227	+	0.603679i	$0.793701\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−7.41363	−0.813752	−0.406876	−	0.913483i	$-0.633382\pi$
$83$	−7.41363	−0.813752	−0.406876	+	0.913483i	$0.633382\pi$
$84$	0	0
$85$	−9.84324	−1.06765
$86$	0	0
$87$	−5.44731	−0.584013
$88$	0	0
$89$	4.66877	0.494889	0.247445	−	0.968902i	$-0.420409\pi$
$89$	4.66877	0.494889	0.247445	+	0.968902i	$0.420409\pi$
$90$	0	0
$91$	2.25951	0.236861
$92$	0	0
$93$	8.91495	0.924438
$94$	0	0
$95$	−29.0734	−2.98287
$96$	0	0
$97$	3.41363	0.346602	0.173301	−	0.984869i	$-0.444557\pi$
$97$	3.41363	0.346602	0.173301	+	0.984869i	$0.444557\pi$
$98$	0	0
$99$	1.00000	0.100504
$100$	0	0
$101$	−10.3419	−1.02906	−0.514530	−	0.857473i	$-0.672033\pi$
$101$	−10.3419	−1.02906	−0.514530	+	0.857473i	$0.672033\pi$
$102$	0	0
$103$	−5.06471	−0.499040	−0.249520	−	0.968370i	$-0.580273\pi$
$103$	−5.06471	−0.499040	−0.249520	+	0.968370i	$0.580273\pi$
$104$	0	0
$105$	8.37559	0.817374
$106$	0	0
$107$	−8.79186	−0.849942	−0.424971	−	0.905207i	$-0.639716\pi$
$107$	−8.79186	−0.849942	−0.424971	+	0.905207i	$0.639716\pi$
$108$	0	0
$109$	−13.1541	−1.25994	−0.629968	−	0.776621i	$-0.716932\pi$
$109$	−13.1541	−1.25994	−0.629968	+	0.776621i	$0.716932\pi$
$110$	0	0
$111$	0.791864	0.0751604
$112$	0	0
$113$	−15.5164	−1.45966	−0.729829	−	0.683630i	$-0.760400\pi$
$113$	−15.5164	−1.45966	−0.729829	+	0.683630i	$0.760400\pi$
$114$	0	0
$115$	26.1382	2.43740
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	6.00000	0.550019
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−4.25951	−0.384067
$124$	0	0
$125$	−13.8653	−1.24015
$126$	0	0
$127$	−0.758191	−0.0672786	−0.0336393	−	0.999434i	$-0.510710\pi$
$127$	−0.758191	−0.0672786	−0.0336393	+	0.999434i	$0.510710\pi$
$128$	0	0
$129$	−9.44731	−0.831789
$130$	0	0
$131$	−2.10275	−0.183718	−0.0918590	−	0.995772i	$-0.529281\pi$
$131$	−2.10275	−0.183718	−0.0918590	+	0.995772i	$0.529281\pi$
$132$	0	0
$133$	17.7219	1.53668
$134$	0	0
$135$	3.70682	0.319032
$136$	0	0
$137$	3.29054	0.281130	0.140565	−	0.990071i	$-0.455108\pi$
$137$	3.29054	0.281130	0.140565	+	0.990071i	$0.455108\pi$
$138$	0	0
$139$	17.5501	1.48858	0.744288	−	0.667859i	$-0.232789\pi$
$139$	17.5501	1.48858	0.744288	+	0.667859i	$0.232789\pi$
$140$	0	0
$141$	2.51902	0.212140
$142$	0	0
$143$	1.00000	0.0836242
$144$	0	0
$145$	−20.1922	−1.67687
$146$	0	0
$147$	1.89461	0.156265
$148$	0	0
$149$	1.32422	0.108484	0.0542420	−	0.998528i	$-0.482726\pi$
$149$	1.32422	0.108484	0.0542420	+	0.998528i	$0.482726\pi$
$150$	0	0
$151$	−2.77853	−0.226114	−0.113057	−	0.993589i	$-0.536064\pi$
$151$	−2.77853	−0.226114	−0.113057	+	0.993589i	$0.536064\pi$
$152$	0	0
$153$	2.65544	0.214680
$154$	0	0
$155$	33.0461	2.65433
$156$	0	0
$157$	−3.94599	−0.314924	−0.157462	−	0.987525i	$-0.550331\pi$
$157$	−3.94599	−0.314924	−0.157462	+	0.987525i	$0.550331\pi$
$158$	0	0
$159$	−7.82991	−0.620952
$160$	0	0
$161$	−15.9327	−1.25567
$162$	0	0
$163$	−21.2232	−1.66233	−0.831165	−	0.556026i	$-0.812325\pi$
$163$	−21.2232	−1.66233	−0.831165	+	0.556026i	$0.812325\pi$
$164$	0	0
$165$	3.70682	0.288575
$166$	0	0
$167$	−3.48098	−0.269366	−0.134683	−	0.990889i	$-0.543002\pi$
$167$	−3.48098	−0.269366	−0.134683	+	0.990889i	$0.543002\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	7.84324	0.599787
$172$	0	0
$173$	21.3446	1.62280	0.811398	−	0.584494i	$-0.198707\pi$
$173$	21.3446	1.62280	0.811398	+	0.584494i	$0.198707\pi$
$174$	0	0
$175$	19.7492	1.49290
$176$	0	0
$177$	7.41363	0.557243
$178$	0	0
$179$	16.5678	1.23833	0.619166	−	0.785260i	$-0.287471\pi$
$179$	16.5678	1.23833	0.619166	+	0.785260i	$0.287471\pi$
$180$	0	0
$181$	24.9840	1.85705	0.928524	−	0.371272i	$-0.121078\pi$
$181$	24.9840	1.85705	0.928524	+	0.371272i	$0.121078\pi$
$182$	0	0
$183$	−6.62177	−0.489495
$184$	0	0
$185$	2.93529	0.215807
$186$	0	0
$187$	2.65544	0.194185
$188$	0	0
$189$	−2.25951	−0.164355
$190$	0	0
$191$	12.9974	0.940456	0.470228	−	0.882545i	$-0.344172\pi$
$191$	12.9974	0.940456	0.470228	+	0.882545i	$0.344172\pi$
$192$	0	0
$193$	4.42960	0.318850	0.159425	−	0.987210i	$-0.449036\pi$
$193$	4.42960	0.318850	0.159425	+	0.987210i	$0.449036\pi$
$194$	0	0
$195$	3.70682	0.265451
$196$	0	0
$197$	−1.05137	−0.0749073	−0.0374537	−	0.999298i	$-0.511925\pi$
$197$	−1.05137	−0.0749073	−0.0374537	+	0.999298i	$0.511925\pi$
$198$	0	0
$199$	12.5190	0.887450	0.443725	−	0.896163i	$-0.353657\pi$
$199$	12.5190	0.887450	0.443725	+	0.896163i	$0.353657\pi$
$200$	0	0
$201$	−10.2258	−0.721275
$202$	0	0
$203$	12.3082	0.863869
$204$	0	0
$205$	−15.7892	−1.10277
$206$	0	0
$207$	−7.05137	−0.490104
$208$	0	0
$209$	7.84324	0.542528
$210$	0	0
$211$	9.72015	0.669163	0.334581	−	0.942367i	$-0.391405\pi$
$211$	9.72015	0.669163	0.334581	+	0.942367i	$0.391405\pi$
$212$	0	0
$213$	2.79186	0.191295
$214$	0	0
$215$	−35.0194	−2.38831
$216$	0	0
$217$	−20.1434	−1.36743
$218$	0	0
$219$	−6.53235	−0.441416
$220$	0	0
$221$	2.65544	0.178624
$222$	0	0
$223$	15.5367	1.04042	0.520208	−	0.854040i	$-0.325854\pi$
$223$	15.5367	1.04042	0.520208	+	0.854040i	$0.325854\pi$
$224$	0	0
$225$	8.74049	0.582699
$226$	0	0
$227$	−11.9460	−0.792883	−0.396441	−	0.918060i	$-0.629755\pi$
$227$	−11.9460	−0.792883	−0.396441	+	0.918060i	$0.629755\pi$
$228$	0	0
$229$	24.9300	1.64742	0.823711	−	0.567010i	$-0.191900\pi$
$229$	24.9300	1.64742	0.823711	+	0.567010i	$0.191900\pi$
$230$	0	0
$231$	−2.25951	−0.148665
$232$	0	0
$233$	2.23917	0.146693	0.0733464	−	0.997307i	$-0.476632\pi$
$233$	2.23917	0.146693	0.0733464	+	0.997307i	$0.476632\pi$
$234$	0	0
$235$	9.33755	0.609115
$236$	0	0
$237$	14.1718	0.920559
$238$	0	0
$239$	17.5297	1.13390	0.566951	−	0.823751i	$-0.308123\pi$
$239$	17.5297	1.13390	0.566951	+	0.823751i	$0.308123\pi$
$240$	0	0
$241$	17.1001	1.10151	0.550757	−	0.834665i	$-0.314339\pi$
$241$	17.1001	1.10151	0.550757	+	0.834665i	$0.314339\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	7.02298	0.448682
$246$	0	0
$247$	7.84324	0.499053
$248$	0	0
$249$	7.41363	0.469820
$250$	0	0
$251$	−14.3756	−0.907379	−0.453690	−	0.891160i	$-0.649893\pi$
$251$	−14.3756	−0.907379	−0.453690	+	0.891160i	$0.649893\pi$
$252$	0	0
$253$	−7.05137	−0.443316
$254$	0	0
$255$	9.84324	0.616407
$256$	0	0
$257$	6.00000	0.374270	0.187135	−	0.982334i	$-0.440080\pi$
$257$	6.00000	0.374270	0.187135	+	0.982334i	$0.440080\pi$
$258$	0	0
$259$	−1.78922	−0.111177
$260$	0	0
$261$	5.44731	0.337180
$262$	0	0
$263$	19.4136	1.19710	0.598548	−	0.801087i	$-0.295745\pi$
$263$	19.4136	1.19710	0.598548	+	0.801087i	$0.295745\pi$
$264$	0	0
$265$	−29.0240	−1.78293
$266$	0	0
$267$	−4.66877	−0.285724
$268$	0	0
$269$	−21.9327	−1.33726	−0.668629	−	0.743596i	$-0.733118\pi$
$269$	−21.9327	−1.33726	−0.668629	+	0.743596i	$0.733118\pi$
$270$	0	0
$271$	15.2569	0.926789	0.463394	−	0.886152i	$-0.346631\pi$
$271$	15.2569	0.926789	0.463394	+	0.886152i	$0.346631\pi$
$272$	0	0
$273$	−2.25951	−0.136752
$274$	0	0
$275$	8.74049	0.527071
$276$	0	0
$277$	13.3463	0.801901	0.400950	−	0.916100i	$-0.368680\pi$
$277$	13.3463	0.801901	0.400950	+	0.916100i	$0.368680\pi$
$278$	0	0
$279$	−8.91495	−0.533724
$280$	0	0
$281$	5.36490	0.320043	0.160022	−	0.987114i	$-0.448844\pi$
$281$	5.36490	0.320043	0.160022	+	0.987114i	$0.448844\pi$
$282$	0	0
$283$	24.4180	1.45150	0.725750	−	0.687959i	$-0.241493\pi$
$283$	24.4180	1.45150	0.725750	+	0.687959i	$0.241493\pi$
$284$	0	0
$285$	29.0734	1.72216
$286$	0	0
$287$	9.62441	0.568111
$288$	0	0
$289$	−9.94863	−0.585213
$290$	0	0
$291$	−3.41363	−0.200111
$292$	0	0
$293$	19.3596	1.13100	0.565501	−	0.824748i	$-0.308683\pi$
$293$	19.3596	1.13100	0.565501	+	0.824748i	$0.308683\pi$
$294$	0	0
$295$	27.4810	1.60000
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	−7.05137	−0.407792
$300$	0	0
$301$	21.3463	1.23038
$302$	0	0
$303$	10.3419	0.594128
$304$	0	0
$305$	−24.5457	−1.40548
$306$	0	0
$307$	−4.88128	−0.278589	−0.139295	−	0.990251i	$-0.544484\pi$
$307$	−4.88128	−0.278589	−0.139295	+	0.990251i	$0.544484\pi$
$308$	0	0
$309$	5.06471	0.288121
$310$	0	0
$311$	−33.1541	−1.88000	−0.939999	−	0.341177i	$-0.889174\pi$
$311$	−33.1541	−1.88000	−0.939999	+	0.341177i	$0.889174\pi$
$312$	0	0
$313$	22.6084	1.27790	0.638952	−	0.769246i	$-0.279368\pi$
$313$	22.6084	1.27790	0.638952	+	0.769246i	$0.279368\pi$
$314$	0	0
$315$	−8.37559	−0.471911
$316$	0	0
$317$	16.6688	0.936212	0.468106	−	0.883672i	$-0.344937\pi$
$317$	16.6688	0.936212	0.468106	+	0.883672i	$0.344937\pi$
$318$	0	0
$319$	5.44731	0.304991
$320$	0	0
$321$	8.79186	0.490714
$322$	0	0
$323$	20.8273	1.15886
$324$	0	0
$325$	8.74049	0.484835
$326$	0	0
$327$	13.1541	0.727425
$328$	0	0
$329$	−5.69175	−0.313797
$330$	0	0
$331$	−24.1585	−1.32787	−0.663935	−	0.747790i	$-0.731115\pi$
$331$	−24.1585	−1.32787	−0.663935	+	0.747790i	$0.731115\pi$
$332$	0	0
$333$	−0.791864	−0.0433939
$334$	0	0
$335$	−37.9053	−2.07099
$336$	0	0
$337$	−26.6572	−1.45211	−0.726054	−	0.687637i	$-0.758648\pi$
$337$	−26.6572	−1.45211	−0.726054	+	0.687637i	$0.758648\pi$
$338$	0	0
$339$	15.5164	0.842734
$340$	0	0
$341$	−8.91495	−0.482772
$342$	0	0
$343$	−20.0975	−1.08516
$344$	0	0
$345$	−26.1382	−1.40723
$346$	0	0
$347$	24.7599	1.32918	0.664591	−	0.747207i	$-0.268606\pi$
$347$	24.7599	1.32918	0.664591	+	0.747207i	$0.268606\pi$
$348$	0	0
$349$	6.47834	0.346778	0.173389	−	0.984853i	$-0.444528\pi$
$349$	6.47834	0.346778	0.173389	+	0.984853i	$0.444528\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	18.5341	0.986470	0.493235	−	0.869896i	$-0.335814\pi$
$353$	18.5341	0.986470	0.493235	+	0.869896i	$0.335814\pi$
$354$	0	0
$355$	10.3489	0.549264
$356$	0	0
$357$	−6.00000	−0.317554
$358$	0	0
$359$	−22.5678	−1.19108	−0.595540	−	0.803325i	$-0.703062\pi$
$359$	−22.5678	−1.19108	−0.595540	+	0.803325i	$0.703062\pi$
$360$	0	0
$361$	42.5164	2.23770
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	−24.2142	−1.26743
$366$	0	0
$367$	−24.6572	−1.28709	−0.643547	−	0.765407i	$-0.722538\pi$
$367$	−24.6572	−1.28709	−0.643547	+	0.765407i	$0.722538\pi$
$368$	0	0
$369$	4.25951	0.221741
$370$	0	0
$371$	17.6918	0.918510
$372$	0	0
$373$	−22.9974	−1.19076	−0.595379	−	0.803445i	$-0.702998\pi$
$373$	−22.9974	−1.19076	−0.595379	+	0.803445i	$0.702998\pi$
$374$	0	0
$375$	13.8653	0.716001
$376$	0	0
$377$	5.44731	0.280551
$378$	0	0
$379$	33.4287	1.71712	0.858558	−	0.512716i	$-0.171361\pi$
$379$	33.4287	1.71712	0.858558	+	0.512716i	$0.171361\pi$
$380$	0	0
$381$	0.758191	0.0388433
$382$	0	0
$383$	22.8946	1.16986	0.584930	−	0.811084i	$-0.301122\pi$
$383$	22.8946	1.16986	0.584930	+	0.811084i	$0.301122\pi$
$384$	0	0
$385$	−8.37559	−0.426860
$386$	0	0
$387$	9.44731	0.480234
$388$	0	0
$389$	20.3756	1.03308	0.516542	−	0.856262i	$-0.327219\pi$
$389$	20.3756	1.03308	0.516542	+	0.856262i	$0.327219\pi$
$390$	0	0
$391$	−18.7245	−0.946940
$392$	0	0
$393$	2.10275	0.106070
$394$	0	0
$395$	52.5324	2.64319
$396$	0	0
$397$	−0.246178	−0.0123553	−0.00617767	−	0.999981i	$-0.501966\pi$
$397$	−0.246178	−0.0123553	−0.00617767	+	0.999981i	$0.501966\pi$
$398$	0	0
$399$	−17.7219	−0.887204
$400$	0	0
$401$	4.43133	0.221290	0.110645	−	0.993860i	$-0.464708\pi$
$401$	4.43133	0.221290	0.110645	+	0.993860i	$0.464708\pi$
$402$	0	0
$403$	−8.91495	−0.444085
$404$	0	0
$405$	−3.70682	−0.184193
$406$	0	0
$407$	−0.791864	−0.0392512
$408$	0	0
$409$	12.9486	0.640268	0.320134	−	0.947372i	$-0.396272\pi$
$409$	12.9486	0.640268	0.320134	+	0.947372i	$0.396272\pi$
$410$	0	0
$411$	−3.29054	−0.162311
$412$	0	0
$413$	−16.7512	−0.824272
$414$	0	0
$415$	27.4810	1.34899
$416$	0	0
$417$	−17.5501	−0.859430
$418$	0	0
$419$	11.5297	0.563263	0.281632	−	0.959523i	$-0.409124\pi$
$419$	11.5297	0.563263	0.281632	+	0.959523i	$0.409124\pi$
$420$	0	0
$421$	17.4810	0.851971	0.425985	−	0.904730i	$-0.359927\pi$
$421$	17.4810	0.851971	0.425985	+	0.904730i	$0.359927\pi$
$422$	0	0
$423$	−2.51902	−0.122479
$424$	0	0
$425$	23.2099	1.12584
$426$	0	0
$427$	14.9620	0.724060
$428$	0	0
$429$	−1.00000	−0.0482805
$430$	0	0
$431$	31.6865	1.52628	0.763142	−	0.646231i	$-0.223656\pi$
$431$	31.6865	1.52628	0.763142	+	0.646231i	$0.223656\pi$
$432$	0	0
$433$	−3.03804	−0.145999	−0.0729995	−	0.997332i	$-0.523257\pi$
$433$	−3.03804	−0.145999	−0.0729995	+	0.997332i	$0.523257\pi$
$434$	0	0
$435$	20.1922	0.968140
$436$	0	0
$437$	−55.3056	−2.64563
$438$	0	0
$439$	25.6528	1.22434	0.612171	−	0.790726i	$-0.290297\pi$
$439$	25.6528	1.22434	0.612171	+	0.790726i	$0.290297\pi$
$440$	0	0
$441$	−1.89461	−0.0902196
$442$	0	0
$443$	−37.4490	−1.77926	−0.889629	−	0.456685i	$-0.849037\pi$
$443$	−37.4490	−1.77926	−0.889629	+	0.456685i	$0.849037\pi$
$444$	0	0
$445$	−17.3063	−0.820397
$446$	0	0
$447$	−1.32422	−0.0626333
$448$	0	0
$449$	−17.8096	−0.840485	−0.420243	−	0.907412i	$-0.638055\pi$
$449$	−17.8096	−0.840485	−0.420243	+	0.907412i	$0.638055\pi$
$450$	0	0
$451$	4.25951	0.200573
$452$	0	0
$453$	2.77853	0.130547
$454$	0	0
$455$	−8.37559	−0.392654
$456$	0	0
$457$	−12.9353	−0.605088	−0.302544	−	0.953135i	$-0.597836\pi$
$457$	−12.9353	−0.605088	−0.302544	+	0.953135i	$0.597836\pi$
$458$	0	0
$459$	−2.65544	−0.123945
$460$	0	0
$461$	9.29755	0.433030	0.216515	−	0.976279i	$-0.430531\pi$
$461$	9.29755	0.433030	0.216515	+	0.976279i	$0.430531\pi$
$462$	0	0
$463$	−3.46064	−0.160829	−0.0804147	−	0.996761i	$-0.525624\pi$
$463$	−3.46064	−0.160829	−0.0804147	+	0.996761i	$0.525624\pi$
$464$	0	0
$465$	−33.0461	−1.53248
$466$	0	0
$467$	4.94863	0.228995	0.114498	−	0.993424i	$-0.463474\pi$
$467$	4.94863	0.228995	0.114498	+	0.993424i	$0.463474\pi$
$468$	0	0
$469$	23.1054	1.06691
$470$	0	0
$471$	3.94599	0.181821
$472$	0	0
$473$	9.44731	0.434388
$474$	0	0
$475$	68.5537	3.14546
$476$	0	0
$477$	7.82991	0.358507
$478$	0	0
$479$	−40.9840	−1.87261	−0.936304	−	0.351190i	$-0.885777\pi$
$479$	−40.9840	−1.87261	−0.936304	+	0.351190i	$0.885777\pi$
$480$	0	0
$481$	−0.791864	−0.0361059
$482$	0	0
$483$	15.9327	0.724961
$484$	0	0
$485$	−12.6537	−0.574575
$486$	0	0
$487$	9.35789	0.424046	0.212023	−	0.977265i	$-0.431995\pi$
$487$	9.35789	0.424046	0.212023	+	0.977265i	$0.431995\pi$
$488$	0	0
$489$	21.2232	0.959746
$490$	0	0
$491$	−15.6244	−0.705120	−0.352560	−	0.935789i	$-0.614689\pi$
$491$	−15.6244	−0.705120	−0.352560	+	0.935789i	$0.614689\pi$
$492$	0	0
$493$	14.4650	0.651471
$494$	0	0
$495$	−3.70682	−0.166609
$496$	0	0
$497$	−6.30825	−0.282963
$498$	0	0
$499$	25.7068	1.15080	0.575398	−	0.817874i	$-0.304847\pi$
$499$	25.7068	1.15080	0.575398	+	0.817874i	$0.304847\pi$
$500$	0	0
$501$	3.48098	0.155519
$502$	0	0
$503$	31.4136	1.40066	0.700332	−	0.713817i	$-0.253035\pi$
$503$	31.4136	1.40066	0.700332	+	0.713817i	$0.253035\pi$
$504$	0	0
$505$	38.3356	1.70591
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	41.1204	1.82263	0.911316	−	0.411708i	$-0.135068\pi$
$509$	41.1204	1.82263	0.911316	+	0.411708i	$0.135068\pi$
$510$	0	0
$511$	14.7599	0.652940
$512$	0	0
$513$	−7.84324	−0.346287
$514$	0	0
$515$	18.7739	0.827279
$516$	0	0
$517$	−2.51902	−0.110786
$518$	0	0
$519$	−21.3446	−0.936922
$520$	0	0
$521$	6.54569	0.286772	0.143386	−	0.989667i	$-0.454201\pi$
$521$	6.54569	0.286772	0.143386	+	0.989667i	$0.454201\pi$
$522$	0	0
$523$	13.3800	0.585065	0.292532	−	0.956256i	$-0.405502\pi$
$523$	13.3800	0.585065	0.292532	+	0.956256i	$0.405502\pi$
$524$	0	0
$525$	−19.7492	−0.861927
$526$	0	0
$527$	−23.6731	−1.03122
$528$	0	0
$529$	26.7219	1.16182
$530$	0	0
$531$	−7.41363	−0.321724
$532$	0	0
$533$	4.25951	0.184500
$534$	0	0
$535$	32.5898	1.40898
$536$	0	0
$537$	−16.5678	−0.714951
$538$	0	0
$539$	−1.89461	−0.0816067
$540$	0	0
$541$	−31.7892	−1.36673	−0.683363	−	0.730079i	$-0.739483\pi$
$541$	−31.7892	−1.36673	−0.683363	+	0.730079i	$0.739483\pi$
$542$	0	0
$543$	−24.9840	−1.07217
$544$	0	0
$545$	48.7599	2.08865
$546$	0	0
$547$	−10.0957	−0.431663	−0.215831	−	0.976431i	$-0.569246\pi$
$547$	−10.0957	−0.431663	−0.215831	+	0.976431i	$0.569246\pi$
$548$	0	0
$549$	6.62177	0.282610
$550$	0	0
$551$	42.7245	1.82013
$552$	0	0
$553$	−32.0214	−1.36169
$554$	0	0
$555$	−2.93529	−0.124596
$556$	0	0
$557$	−31.0868	−1.31719	−0.658595	−	0.752498i	$-0.728849\pi$
$557$	−31.0868	−1.31719	−0.658595	+	0.752498i	$0.728849\pi$
$558$	0	0
$559$	9.44731	0.399578
$560$	0	0
$561$	−2.65544	−0.112113
$562$	0	0
$563$	0.653712	0.0275507	0.0137753	−	0.999905i	$-0.495615\pi$
$563$	0.653712	0.0275507	0.0137753	+	0.999905i	$0.495615\pi$
$564$	0	0
$565$	57.5164	2.41973
$566$	0	0
$567$	2.25951	0.0948905
$568$	0	0
$569$	5.75555	0.241285	0.120643	−	0.992696i	$-0.461504\pi$
$569$	5.75555	0.241285	0.120643	+	0.992696i	$0.461504\pi$
$570$	0	0
$571$	11.8229	0.494773	0.247386	−	0.968917i	$-0.420428\pi$
$571$	11.8229	0.494773	0.247386	+	0.968917i	$0.420428\pi$
$572$	0	0
$573$	−12.9974	−0.542973
$574$	0	0
$575$	−61.6325	−2.57025
$576$	0	0
$577$	−45.8920	−1.91051	−0.955254	−	0.295787i	$-0.904418\pi$
$577$	−45.8920	−1.91051	−0.955254	+	0.295787i	$0.904418\pi$
$578$	0	0
$579$	−4.42960	−0.184088
$580$	0	0
$581$	−16.7512	−0.694956
$582$	0	0
$583$	7.82991	0.324282
$584$	0	0
$585$	−3.70682	−0.153258
$586$	0	0
$587$	−16.6572	−0.687515	−0.343758	−	0.939058i	$-0.611700\pi$
$587$	−16.6572	−0.687515	−0.343758	+	0.939058i	$0.611700\pi$
$588$	0	0
$589$	−69.9221	−2.88109
$590$	0	0
$591$	1.05137	0.0432478
$592$	0	0
$593$	10.9486	0.449606	0.224803	−	0.974404i	$-0.427826\pi$
$593$	10.9486	0.449606	0.224803	+	0.974404i	$0.427826\pi$
$594$	0	0
$595$	−22.2409	−0.911788
$596$	0	0
$597$	−12.5190	−0.512369
$598$	0	0
$599$	−47.0734	−1.92337	−0.961684	−	0.274159i	$-0.911601\pi$
$599$	−47.0734	−1.92337	−0.961684	+	0.274159i	$0.911601\pi$
$600$	0	0
$601$	29.3730	1.19815	0.599074	−	0.800694i	$-0.295536\pi$
$601$	29.3730	1.19815	0.599074	+	0.800694i	$0.295536\pi$
$602$	0	0
$603$	10.2258	0.416428
$604$	0	0
$605$	−3.70682	−0.150704
$606$	0	0
$607$	−26.2072	−1.06372	−0.531859	−	0.846833i	$-0.678506\pi$
$607$	−26.2072	−1.06372	−0.531859	+	0.846833i	$0.678506\pi$
$608$	0	0
$609$	−12.3082	−0.498755
$610$	0	0
$611$	−2.51902	−0.101909
$612$	0	0
$613$	2.39766	0.0968406	0.0484203	−	0.998827i	$-0.484581\pi$
$613$	2.39766	0.0968406	0.0484203	+	0.998827i	$0.484581\pi$
$614$	0	0
$615$	15.7892	0.636683
$616$	0	0
$617$	−9.39857	−0.378372	−0.189186	−	0.981941i	$-0.560585\pi$
$617$	−9.39857	−0.378372	−0.189186	+	0.981941i	$0.560585\pi$
$618$	0	0
$619$	−32.1178	−1.29092	−0.645462	−	0.763792i	$-0.723335\pi$
$619$	−32.1178	−1.29092	−0.645462	+	0.763792i	$0.723335\pi$
$620$	0	0
$621$	7.05137	0.282962
$622$	0	0
$623$	10.5491	0.422643
$624$	0	0
$625$	7.69371	0.307748
$626$	0	0
$627$	−7.84324	−0.313229
$628$	0	0
$629$	−2.10275	−0.0838421
$630$	0	0
$631$	23.3666	0.930211	0.465105	−	0.885255i	$-0.346016\pi$
$631$	23.3666	0.930211	0.465105	+	0.885255i	$0.346016\pi$
$632$	0	0
$633$	−9.72015	−0.386341
$634$	0	0
$635$	2.81047	0.111530
$636$	0	0
$637$	−1.89461	−0.0750673
$638$	0	0
$639$	−2.79186	−0.110444
$640$	0	0
$641$	34.9300	1.37965	0.689826	−	0.723975i	$-0.257687\pi$
$641$	34.9300	1.37965	0.689826	+	0.723975i	$0.257687\pi$
$642$	0	0
$643$	0.601429	0.0237180	0.0118590	−	0.999930i	$-0.496225\pi$
$643$	0.601429	0.0237180	0.0118590	+	0.999930i	$0.496225\pi$
$644$	0	0
$645$	35.0194	1.37889
$646$	0	0
$647$	44.3170	1.74228	0.871140	−	0.491034i	$-0.163381\pi$
$647$	44.3170	1.74228	0.871140	+	0.491034i	$0.163381\pi$
$648$	0	0
$649$	−7.41363	−0.291011
$650$	0	0
$651$	20.1434	0.789483
$652$	0	0
$653$	28.6218	1.12006	0.560028	−	0.828474i	$-0.310790\pi$
$653$	28.6218	1.12006	0.560028	+	0.828474i	$0.310790\pi$
$654$	0	0
$655$	7.79450	0.304556
$656$	0	0
$657$	6.53235	0.254851
$658$	0	0
$659$	8.82727	0.343861	0.171931	−	0.985109i	$-0.444999\pi$
$659$	8.82727	0.343861	0.171931	+	0.985109i	$0.444999\pi$
$660$	0	0
$661$	−1.10011	−0.0427893	−0.0213946	−	0.999771i	$-0.506811\pi$
$661$	−1.10011	−0.0427893	−0.0213946	+	0.999771i	$0.506811\pi$
$662$	0	0
$663$	−2.65544	−0.103129
$664$	0	0
$665$	−65.6918	−2.54742
$666$	0	0
$667$	−38.4110	−1.48728
$668$	0	0
$669$	−15.5367	−0.600684
$670$	0	0
$671$	6.62177	0.255631
$672$	0	0
$673$	−14.3489	−0.553110	−0.276555	−	0.960998i	$-0.589193\pi$
$673$	−14.3489	−0.553110	−0.276555	+	0.960998i	$0.589193\pi$
$674$	0	0
$675$	−8.74049	−0.336422
$676$	0	0
$677$	−35.0310	−1.34635	−0.673176	−	0.739482i	$-0.735070\pi$
$677$	−35.0310	−1.34635	−0.673176	+	0.739482i	$0.735070\pi$
$678$	0	0
$679$	7.71314	0.296003
$680$	0	0
$681$	11.9460	0.457771
$682$	0	0
$683$	28.7866	1.10149	0.550744	−	0.834674i	$-0.314344\pi$
$683$	28.7866	1.10149	0.550744	+	0.834674i	$0.314344\pi$
$684$	0	0
$685$	−12.1974	−0.466040
$686$	0	0
$687$	−24.9300	−0.951139
$688$	0	0
$689$	7.82991	0.298296
$690$	0	0
$691$	8.53936	0.324853	0.162426	−	0.986721i	$-0.448068\pi$
$691$	8.53936	0.324853	0.162426	+	0.986721i	$0.448068\pi$
$692$	0	0
$693$	2.25951	0.0858317
$694$	0	0
$695$	−65.0548	−2.46767
$696$	0	0
$697$	11.3109	0.428430
$698$	0	0
$699$	−2.23917	−0.0846932
$700$	0	0
$701$	34.0691	1.28677	0.643386	−	0.765542i	$-0.277529\pi$
$701$	34.0691	1.28677	0.643386	+	0.765542i	$0.277529\pi$
$702$	0	0
$703$	−6.21078	−0.234244
$704$	0	0
$705$	−9.33755	−0.351672
$706$	0	0
$707$	−23.3677	−0.878832
$708$	0	0
$709$	20.6165	0.774269	0.387134	−	0.922023i	$-0.373465\pi$
$709$	20.6165	0.774269	0.387134	+	0.922023i	$0.373465\pi$
$710$	0	0
$711$	−14.1718	−0.531485
$712$	0	0
$713$	62.8627	2.35423
$714$	0	0
$715$	−3.70682	−0.138627
$716$	0	0
$717$	−17.5297	−0.654659
$718$	0	0
$719$	15.3730	0.573314	0.286657	−	0.958033i	$-0.407456\pi$
$719$	15.3730	0.573314	0.286657	+	0.958033i	$0.407456\pi$
$720$	0	0
$721$	−11.4438	−0.426188
$722$	0	0
$723$	−17.1001	−0.635960
$724$	0	0
$725$	47.6121	1.76827
$726$	0	0
$727$	45.3817	1.68311	0.841557	−	0.540169i	$-0.181640\pi$
$727$	45.3817	1.68311	0.841557	+	0.540169i	$0.181640\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	25.0868	0.927868
$732$	0	0
$733$	−9.12746	−0.337130	−0.168565	−	0.985691i	$-0.553913\pi$
$733$	−9.12746	−0.337130	−0.168565	+	0.985691i	$0.553913\pi$
$734$	0	0
$735$	−7.02298	−0.259047
$736$	0	0
$737$	10.2258	0.376674
$738$	0	0
$739$	−52.6652	−1.93732	−0.968661	−	0.248387i	$-0.920099\pi$
$739$	−52.6652	−1.93732	−0.968661	+	0.248387i	$0.920099\pi$
$740$	0	0
$741$	−7.84324	−0.288129
$742$	0	0
$743$	−23.7272	−0.870465	−0.435232	−	0.900318i	$-0.643334\pi$
$743$	−23.7272	−0.870465	−0.435232	+	0.900318i	$0.643334\pi$
$744$	0	0
$745$	−4.90863	−0.179838
$746$	0	0
$747$	−7.41363	−0.271251
$748$	0	0
$749$	−19.8653	−0.725863
$750$	0	0
$751$	−33.0682	−1.20667	−0.603337	−	0.797486i	$-0.706163\pi$
$751$	−33.0682	−1.20667	−0.603337	+	0.797486i	$0.706163\pi$
$752$	0	0
$753$	14.3756	0.523876
$754$	0	0
$755$	10.2995	0.374837
$756$	0	0
$757$	46.8813	1.70393	0.851965	−	0.523599i	$-0.175411\pi$
$757$	46.8813	1.70393	0.851965	+	0.523599i	$0.175411\pi$
$758$	0	0
$759$	7.05137	0.255949
$760$	0	0
$761$	8.55902	0.310264	0.155132	−	0.987894i	$-0.450420\pi$
$761$	8.55902	0.310264	0.155132	+	0.987894i	$0.450420\pi$
$762$	0	0
$763$	−29.7219	−1.07600
$764$	0	0
$765$	−9.84324	−0.355883
$766$	0	0
$767$	−7.41363	−0.267691
$768$	0	0
$769$	34.4110	1.24089	0.620446	−	0.784249i	$-0.286952\pi$
$769$	34.4110	1.24089	0.620446	+	0.784249i	$0.286952\pi$
$770$	0	0
$771$	−6.00000	−0.216085
$772$	0	0
$773$	18.9504	0.681597	0.340798	−	0.940136i	$-0.389303\pi$
$773$	18.9504	0.681597	0.340798	+	0.940136i	$0.389303\pi$
$774$	0	0
$775$	−77.9211	−2.79901
$776$	0	0
$777$	1.78922	0.0641881
$778$	0	0
$779$	33.4084	1.19698
$780$	0	0
$781$	−2.79186	−0.0999007
$782$	0	0
$783$	−5.44731	−0.194671
$784$	0	0
$785$	14.6270	0.522062
$786$	0	0
$787$	6.28617	0.224078	0.112039	−	0.993704i	$-0.464262\pi$
$787$	6.28617	0.224078	0.112039	+	0.993704i	$0.464262\pi$
$788$	0	0
$789$	−19.4136	−0.691144
$790$	0	0
$791$	−35.0594	−1.24657
$792$	0	0
$793$	6.62177	0.235146
$794$	0	0
$795$	29.0240	1.02938
$796$	0	0
$797$	−43.2702	−1.53271	−0.766355	−	0.642418i	$-0.777931\pi$
$797$	−43.2702	−1.53271	−0.766355	+	0.642418i	$0.777931\pi$
$798$	0	0
$799$	−6.68912	−0.236644
$800$	0	0
$801$	4.66877	0.164963
$802$	0	0
$803$	6.53235	0.230522
$804$	0	0
$805$	59.0594	2.08157
$806$	0	0
$807$	21.9327	0.772066
$808$	0	0
$809$	−20.3100	−0.714061	−0.357030	−	0.934093i	$-0.616211\pi$
$809$	−20.3100	−0.714061	−0.357030	+	0.934093i	$0.616211\pi$
$810$	0	0
$811$	−45.7033	−1.60486	−0.802429	−	0.596747i	$-0.796460\pi$
$811$	−45.7033	−1.60486	−0.802429	+	0.596747i	$0.796460\pi$
$812$	0	0
$813$	−15.2569	−0.535082
$814$	0	0
$815$	78.6705	2.75571
$816$	0	0
$817$	74.0975	2.59234
$818$	0	0
$819$	2.25951	0.0789537
$820$	0	0
$821$	−8.01333	−0.279667	−0.139834	−	0.990175i	$-0.544657\pi$
$821$	−8.01333	−0.279667	−0.139834	+	0.990175i	$0.544657\pi$
$822$	0	0
$823$	−21.6511	−0.754709	−0.377354	−	0.926069i	$-0.623166\pi$
$823$	−21.6511	−0.754709	−0.377354	+	0.926069i	$0.623166\pi$
$824$	0	0
$825$	−8.74049	−0.304305
$826$	0	0
$827$	−8.57303	−0.298114	−0.149057	−	0.988829i	$-0.547624\pi$
$827$	−8.57303	−0.298114	−0.149057	+	0.988829i	$0.547624\pi$
$828$	0	0
$829$	21.0328	0.730498	0.365249	−	0.930910i	$-0.380984\pi$
$829$	21.0328	0.730498	0.365249	+	0.930910i	$0.380984\pi$
$830$	0	0
$831$	−13.3463	−0.462978
$832$	0	0
$833$	−5.03103	−0.174315
$834$	0	0
$835$	12.9034	0.446539
$836$	0	0
$837$	8.91495	0.308146
$838$	0	0
$839$	−11.2755	−0.389273	−0.194636	−	0.980875i	$-0.562353\pi$
$839$	−11.2755	−0.389273	−0.194636	+	0.980875i	$0.562353\pi$
$840$	0	0
$841$	0.673144	0.0232119
$842$	0	0
$843$	−5.36490	−0.184777
$844$	0	0
$845$	−3.70682	−0.127518
$846$	0	0
$847$	2.25951	0.0776377
$848$	0	0
$849$	−24.4180	−0.838024
$850$	0	0
$851$	5.58373	0.191408
$852$	0	0
$853$	−38.3843	−1.31425	−0.657127	−	0.753780i	$-0.728229\pi$
$853$	−38.3843	−1.31425	−0.657127	+	0.753780i	$0.728229\pi$
$854$	0	0
$855$	−29.0734	−0.994291
$856$	0	0
$857$	29.5181	1.00832	0.504160	−	0.863610i	$-0.331802\pi$
$857$	29.5181	1.00832	0.504160	+	0.863610i	$0.331802\pi$
$858$	0	0
$859$	−33.7219	−1.15058	−0.575288	−	0.817951i	$-0.695110\pi$
$859$	−33.7219	−1.15058	−0.575288	+	0.817951i	$0.695110\pi$
$860$	0	0
$861$	−9.62441	−0.327999
$862$	0	0
$863$	−22.6078	−0.769577	−0.384788	−	0.923005i	$-0.625726\pi$
$863$	−22.6078	−0.769577	−0.384788	+	0.923005i	$0.625726\pi$
$864$	0	0
$865$	−79.1204	−2.69017
$866$	0	0
$867$	9.94863	0.337873
$868$	0	0
$869$	−14.1718	−0.480746
$870$	0	0
$871$	10.2258	0.346489
$872$	0	0
$873$	3.41363	0.115534
$874$	0	0
$875$	−31.3288	−1.05911
$876$	0	0
$877$	−35.7219	−1.20624	−0.603121	−	0.797650i	$-0.706076\pi$
$877$	−35.7219	−1.20624	−0.603121	+	0.797650i	$0.706076\pi$
$878$	0	0
$879$	−19.3596	−0.652984
$880$	0	0
$881$	39.8600	1.34292	0.671459	−	0.741041i	$-0.265668\pi$
$881$	39.8600	1.34292	0.671459	+	0.741041i	$0.265668\pi$
$882$	0	0
$883$	43.5518	1.46563	0.732817	−	0.680426i	$-0.238205\pi$
$883$	43.5518	1.46563	0.732817	+	0.680426i	$0.238205\pi$
$884$	0	0
$885$	−27.4810	−0.923763
$886$	0	0
$887$	20.8273	0.699311	0.349656	−	0.936878i	$-0.386299\pi$
$887$	20.8273	0.699311	0.349656	+	0.936878i	$0.386299\pi$
$888$	0	0
$889$	−1.71314	−0.0574569
$890$	0	0
$891$	1.00000	0.0335013
$892$	0	0
$893$	−19.7573	−0.661152
$894$	0	0
$895$	−61.4136	−2.05283
$896$	0	0
$897$	7.05137	0.235439
$898$	0	0
$899$	−48.5625	−1.61965
$900$	0	0
$901$	20.7919	0.692677
$902$	0	0
$903$	−21.3463	−0.710360
$904$	0	0
$905$	−92.6112	−3.07850
$906$	0	0
$907$	−0.586367	−0.0194700	−0.00973499	−	0.999953i	$-0.503099\pi$
$907$	−0.586367	−0.0194700	−0.00973499	+	0.999953i	$0.503099\pi$
$908$	0	0
$909$	−10.3419	−0.343020
$910$	0	0
$911$	54.1062	1.79262	0.896309	−	0.443429i	$-0.146238\pi$
$911$	54.1062	1.79262	0.896309	+	0.443429i	$0.146238\pi$
$912$	0	0
$913$	−7.41363	−0.245355
$914$	0	0
$915$	24.5457	0.811455
$916$	0	0
$917$	−4.75118	−0.156898
$918$	0	0
$919$	−29.7962	−0.982887	−0.491444	−	0.870909i	$-0.663531\pi$
$919$	−29.7962	−0.982887	−0.491444	+	0.870909i	$0.663531\pi$
$920$	0	0
$921$	4.88128	0.160844
$922$	0	0
$923$	−2.79186	−0.0918953
$924$	0	0
$925$	−6.92128	−0.227570
$926$	0	0
$927$	−5.06471	−0.166347
$928$	0	0
$929$	−8.05574	−0.264300	−0.132150	−	0.991230i	$-0.542188\pi$
$929$	−8.05574	−0.264300	−0.132150	+	0.991230i	$0.542188\pi$
$930$	0	0
$931$	−14.8599	−0.487013
$932$	0	0
$933$	33.1541	1.08542
$934$	0	0
$935$	−9.84324	−0.321908
$936$	0	0
$937$	−23.1268	−0.755519	−0.377759	−	0.925904i	$-0.623305\pi$
$937$	−23.1268	−0.755519	−0.377759	+	0.925904i	$0.623305\pi$
$938$	0	0
$939$	−22.6084	−0.737798
$940$	0	0
$941$	−8.57303	−0.279473	−0.139736	−	0.990189i	$-0.544626\pi$
$941$	−8.57303	−0.279473	−0.139736	+	0.990189i	$0.544626\pi$
$942$	0	0
$943$	−30.0354	−0.978087
$944$	0	0
$945$	8.37559	0.272458
$946$	0	0
$947$	−11.8706	−0.385742	−0.192871	−	0.981224i	$-0.561780\pi$
$947$	−11.8706	−0.385742	−0.192871	+	0.981224i	$0.561780\pi$
$948$	0	0
$949$	6.53235	0.212049
$950$	0	0
$951$	−16.6688	−0.540522
$952$	0	0
$953$	47.7556	1.54695	0.773477	−	0.633824i	$-0.218516\pi$
$953$	47.7556	1.54695	0.773477	+	0.633824i	$0.218516\pi$
$954$	0	0
$955$	−48.1788	−1.55903
$956$	0	0
$957$	−5.44731	−0.176086
$958$	0	0
$959$	7.43502	0.240089
$960$	0	0
$961$	48.4764	1.56375
$962$	0	0
$963$	−8.79186	−0.283314
$964$	0	0
$965$	−16.4197	−0.528570
$966$	0	0
$967$	22.2188	0.714509	0.357255	−	0.934007i	$-0.383713\pi$
$967$	22.2188	0.714509	0.357255	+	0.934007i	$0.383713\pi$
$968$	0	0
$969$	−20.8273	−0.669068
$970$	0	0
$971$	−31.7759	−1.01974	−0.509868	−	0.860252i	$-0.670306\pi$
$971$	−31.7759	−1.01974	−0.509868	+	0.860252i	$0.670306\pi$
$972$	0	0
$973$	39.6545	1.27127
$974$	0	0
$975$	−8.74049	−0.279920
$976$	0	0
$977$	−37.3666	−1.19546	−0.597732	−	0.801696i	$-0.703931\pi$
$977$	−37.3666	−1.19546	−0.597732	+	0.801696i	$0.703931\pi$
$978$	0	0
$979$	4.66877	0.149215
$980$	0	0
$981$	−13.1541	−0.419979
$982$	0	0
$983$	−47.3463	−1.51011	−0.755056	−	0.655660i	$-0.772390\pi$
$983$	−47.3463	−1.51011	−0.755056	+	0.655660i	$0.772390\pi$
$984$	0	0
$985$	3.89725	0.124177
$986$	0	0
$987$	5.69175	0.181171
$988$	0	0
$989$	−66.6165	−2.11828
$990$	0	0
$991$	−42.2409	−1.34183	−0.670913	−	0.741536i	$-0.734098\pi$
$991$	−42.2409	−1.34183	−0.670913	+	0.741536i	$0.734098\pi$
$992$	0	0
$993$	24.1585	0.766647
$994$	0	0
$995$	−46.4057	−1.47116
$996$	0	0
$997$	−9.54832	−0.302398	−0.151199	−	0.988503i	$-0.548314\pi$
$997$	−9.54832	−0.302398	−0.151199	+	0.988503i	$0.548314\pi$
$998$	0	0
$999$	0.791864	0.0250535

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.bt.1.1		3
4.3	odd	2		1716.2.a.h.1.1	✓	3
12.11	even	2		5148.2.a.l.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1716.2.a.h.1.1	✓	3	4.3	odd	2
5148.2.a.l.1.3		3	12.11	even	2
6864.2.a.bt.1.1		3	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -3.70682 q^{5} +2.25951 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -3.70682 q^{5} +2.25951 q^{7} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} +3.70682 q^{15} +2.65544 q^{17} +7.84324 q^{19} -2.25951 q^{21} -7.05137 q^{23} +8.74049 q^{25} -1.00000 q^{27} +5.44731 q^{29} -8.91495 q^{31} -1.00000 q^{33} -8.37559 q^{35} -0.791864 q^{37} -1.00000 q^{39} +4.25951 q^{41} +9.44731 q^{43} -3.70682 q^{45} -2.51902 q^{47} -1.89461 q^{49} -2.65544 q^{51} +7.82991 q^{53} -3.70682 q^{55} -7.84324 q^{57} -7.41363 q^{59} +6.62177 q^{61} +2.25951 q^{63} -3.70682 q^{65} +10.2258 q^{67} +7.05137 q^{69} -2.79186 q^{71} +6.53235 q^{73} -8.74049 q^{75} +2.25951 q^{77} -14.1718 q^{79} +1.00000 q^{81} -7.41363 q^{83} -9.84324 q^{85} -5.44731 q^{87} +4.66877 q^{89} +2.25951 q^{91} +8.91495 q^{93} -29.0734 q^{95} +3.41363 q^{97} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} + 4 q^{5} + 4 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q - 3 * q^3 + 4 * q^5 + 4 * q^7 + 3 * q^9	\( 3 q - 3 q^{3} + 4 q^{5} + 4 q^{7} + 3 q^{9} + 3 q^{11} + 3 q^{13} - 4 q^{15} + 2 q^{17} + 8 q^{19} - 4 q^{21} - 12 q^{23} + 29 q^{25} - 3 q^{27} + 4 q^{29} - 18 q^{31} - 3 q^{33} - 6 q^{35} + 4 q^{37} - 3 q^{39} + 10 q^{41} + 16 q^{43} + 4 q^{45} - 2 q^{47} + 19 q^{49} - 2 q^{51} + 6 q^{53} + 4 q^{55} - 8 q^{57} + 8 q^{59} - 4 q^{61} + 4 q^{63} + 4 q^{65} + 10 q^{67} + 12 q^{69} - 2 q^{71} + 16 q^{73} - 29 q^{75} + 4 q^{77} + 12 q^{79} + 3 q^{81} + 8 q^{83} - 14 q^{85} - 4 q^{87} + 10 q^{89} + 4 q^{91} + 18 q^{93} - 22 q^{95} - 20 q^{97} + 3 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 + 4 * q^5 + 4 * q^7 + 3 * q^9 + 3 * q^11 + 3 * q^13 - 4 * q^15 + 2 * q^17 + 8 * q^19 - 4 * q^21 - 12 * q^23 + 29 * q^25 - 3 * q^27 + 4 * q^29 - 18 * q^31 - 3 * q^33 - 6 * q^35 + 4 * q^37 - 3 * q^39 + 10 * q^41 + 16 * q^43 + 4 * q^45 - 2 * q^47 + 19 * q^49 - 2 * q^51 + 6 * q^53 + 4 * q^55 - 8 * q^57 + 8 * q^59 - 4 * q^61 + 4 * q^63 + 4 * q^65 + 10 * q^67 + 12 * q^69 - 2 * q^71 + 16 * q^73 - 29 * q^75 + 4 * q^77 + 12 * q^79 + 3 * q^81 + 8 * q^83 - 14 * q^85 - 4 * q^87 + 10 * q^89 + 4 * q^91 + 18 * q^93 - 22 * q^95 - 20 * q^97 + 3 * q^99

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