LMFDB - Embedded newform 6864.2.a.cb.1.3

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			1.3
Root			$3.94698$ 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)

$f(q)$	$=$	$q+1.00000 q^{3} +1.59571 q^{5} -4.44027 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +1.59571 q^{5} -4.44027 q^{7} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} +1.59571 q^{15} +5.04941 q^{17} -4.44027 q^{19} -4.44027 q^{21} +2.26227 q^{23} -2.45370 q^{25} +1.00000 q^{27} -2.03598 q^{29} -0.404287 q^{31} -1.00000 q^{33} -7.08539 q^{35} +9.08539 q^{37} +1.00000 q^{39} -9.34767 q^{41} +11.9299 q^{43} +1.59571 q^{45} +3.36942 q^{47} +12.7160 q^{49} +5.04941 q^{51} +1.19143 q^{53} -1.59571 q^{55} -4.44027 q^{57} +3.19143 q^{59} -1.89397 q^{61} -4.44027 q^{63} +1.59571 q^{65} -7.31169 q^{67} +2.26227 q^{69} -8.70254 q^{71} +7.63169 q^{73} -2.45370 q^{75} +4.44027 q^{77} +11.1214 q^{79} +1.00000 q^{81} +12.5965 q^{83} +8.05741 q^{85} -2.03598 q^{87} +2.51032 q^{89} -4.44027 q^{91} -0.404287 q^{93} -7.08539 q^{95} +15.0585 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	1.59571	0.713625	0.356812	−	0.934176i	$-0.383864\pi$
$5$	1.59571	0.713625	0.356812	+	0.934176i	$0.383864\pi$
$6$	0	0
$7$	−4.44027	−1.67826	−0.839132	−	0.543928i	$-0.816936\pi$
$7$	−4.44027	−1.67826	−0.839132	+	0.543928i	$0.816936\pi$
$8$	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
$15$	1.59571	0.412011
$16$	0	0
$17$	5.04941	1.22466	0.612331	−	0.790601i	$-0.290232\pi$
$17$	5.04941	1.22466	0.612331	+	0.790601i	$0.290232\pi$
$18$	0	0
$19$	−4.44027	−1.01867	−0.509334	−	0.860569i	$-0.670108\pi$
$19$	−4.44027	−1.01867	−0.509334	+	0.860569i	$0.670108\pi$
$20$	0	0
$21$	−4.44027	−0.968946
$22$	0	0
$23$	2.26227	0.471717	0.235858	−	0.971787i	$-0.424210\pi$
$23$	2.26227	0.471717	0.235858	+	0.971787i	$0.424210\pi$
$24$	0	0
$25$	−2.45370	−0.490740
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−2.03598	−0.378072	−0.189036	−	0.981970i	$-0.560536\pi$
$29$	−2.03598	−0.378072	−0.189036	+	0.981970i	$0.560536\pi$
$30$	0	0
$31$	−0.404287	−0.0726121	−0.0363060	−	0.999341i	$-0.511559\pi$
$31$	−0.404287	−0.0726121	−0.0363060	+	0.999341i	$0.511559\pi$
$32$	0	0
$33$	−1.00000	−0.174078
$34$	0	0
$35$	−7.08539	−1.19765
$36$	0	0
$37$	9.08539	1.49363	0.746815	−	0.665032i	$-0.231582\pi$
$37$	9.08539	1.49363	0.746815	+	0.665032i	$0.231582\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	−9.34767	−1.45986	−0.729930	−	0.683522i	$-0.760447\pi$
$41$	−9.34767	−1.45986	−0.729930	+	0.683522i	$0.760447\pi$
$42$	0	0
$43$	11.9299	1.81930	0.909650	−	0.415376i	$-0.136350\pi$
$43$	11.9299	1.81930	0.909650	+	0.415376i	$0.136350\pi$
$44$	0	0
$45$	1.59571	0.237875
$46$	0	0
$47$	3.36942	0.491481	0.245740	−	0.969336i	$-0.420969\pi$
$47$	3.36942	0.491481	0.245740	+	0.969336i	$0.420969\pi$
$48$	0	0
$49$	12.7160	1.81657
$50$	0	0
$51$	5.04941	0.707059
$52$	0	0
$53$	1.19143	0.163655	0.0818275	−	0.996647i	$-0.473924\pi$
$53$	1.19143	0.163655	0.0818275	+	0.996647i	$0.473924\pi$
$54$	0	0
$55$	−1.59571	−0.215166
$56$	0	0
$57$	−4.44027	−0.588128
$58$	0	0
$59$	3.19143	0.415488	0.207744	−	0.978183i	$-0.433388\pi$
$59$	3.19143	0.415488	0.207744	+	0.978183i	$0.433388\pi$
$60$	0	0
$61$	−1.89397	−0.242498	−0.121249	−	0.992622i	$-0.538690\pi$
$61$	−1.89397	−0.242498	−0.121249	+	0.992622i	$0.538690\pi$
$62$	0	0
$63$	−4.44027	−0.559421
$64$	0	0
$65$	1.59571	0.197924
$66$	0	0
$67$	−7.31169	−0.893265	−0.446632	−	0.894718i	$-0.647377\pi$
$67$	−7.31169	−0.893265	−0.446632	+	0.894718i	$0.647377\pi$
$68$	0	0
$69$	2.26227	0.272346
$70$	0	0
$71$	−8.70254	−1.03280	−0.516401	−	0.856347i	$-0.672728\pi$
$71$	−8.70254	−1.03280	−0.516401	+	0.856347i	$0.672728\pi$
$72$	0	0
$73$	7.63169	0.893222	0.446611	−	0.894728i	$-0.352631\pi$
$73$	7.63169	0.893222	0.446611	+	0.894728i	$0.352631\pi$
$74$	0	0
$75$	−2.45370	−0.283329
$76$	0	0
$77$	4.44027	0.506015
$78$	0	0
$79$	11.1214	1.25125	0.625626	−	0.780123i	$-0.284844\pi$
$79$	11.1214	1.25125	0.625626	+	0.780123i	$0.284844\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	12.5965	1.38265	0.691323	−	0.722546i	$-0.257028\pi$
$83$	12.5965	1.38265	0.691323	+	0.722546i	$0.257028\pi$
$84$	0	0
$85$	8.05741	0.873949
$86$	0	0
$87$	−2.03598	−0.218280
$88$	0	0
$89$	2.51032	0.266093	0.133047	−	0.991110i	$-0.457524\pi$
$89$	2.51032	0.266093	0.133047	+	0.991110i	$0.457524\pi$
$90$	0	0
$91$	−4.44027	−0.465466
$92$	0	0
$93$	−0.404287	−0.0419226
$94$	0	0
$95$	−7.08539	−0.726946
$96$	0	0
$97$	15.0585	1.52896	0.764481	−	0.644646i	$-0.222995\pi$
$97$	15.0585	1.52896	0.764481	+	0.644646i	$0.222995\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	−5.92995	−0.590052	−0.295026	−	0.955489i	$-0.595328\pi$
$101$	−5.92995	−0.590052	−0.295026	+	0.955489i	$0.595328\pi$
$102$	0	0
$103$	11.2634	1.10981	0.554907	−	0.831912i	$-0.312754\pi$
$103$	11.2634	1.10981	0.554907	+	0.831912i	$0.312754\pi$
$104$	0	0
$105$	−7.08539	−0.691464
$106$	0	0
$107$	−0.808574	−0.0781678	−0.0390839	−	0.999236i	$-0.512444\pi$
$107$	−0.808574	−0.0781678	−0.0390839	+	0.999236i	$0.512444\pi$
$108$	0	0
$109$	2.75116	0.263513	0.131757	−	0.991282i	$-0.457938\pi$
$109$	2.75116	0.263513	0.131757	+	0.991282i	$0.457938\pi$
$110$	0	0
$111$	9.08539	0.862347
$112$	0	0
$113$	7.68911	0.723330	0.361665	−	0.932308i	$-0.382208\pi$
$113$	7.68911	0.723330	0.361665	+	0.932308i	$0.382208\pi$
$114$	0	0
$115$	3.60994	0.336629
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−22.4207	−2.05531
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−9.34767	−0.842851
$124$	0	0
$125$	−11.8940	−1.06383
$126$	0	0
$127$	−6.34687	−0.563194	−0.281597	−	0.959533i	$-0.590864\pi$
$127$	−6.34687	−0.563194	−0.281597	+	0.959533i	$0.590864\pi$
$128$	0	0
$129$	11.9299	1.05037
$130$	0	0
$131$	1.09260	0.0954610	0.0477305	−	0.998860i	$-0.484801\pi$
$131$	1.09260	0.0954610	0.0477305	+	0.998860i	$0.484801\pi$
$132$	0	0
$133$	19.7160	1.70959
$134$	0	0
$135$	1.59571	0.137337
$136$	0	0
$137$	8.27139	0.706672	0.353336	−	0.935496i	$-0.385047\pi$
$137$	8.27139	0.706672	0.353336	+	0.935496i	$0.385047\pi$
$138$	0	0
$139$	−7.22741	−0.613021	−0.306510	−	0.951867i	$-0.599161\pi$
$139$	−7.22741	−0.613021	−0.306510	+	0.951867i	$0.599161\pi$
$140$	0	0
$141$	3.36942	0.283756
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−3.24884	−0.269802
$146$	0	0
$147$	12.7160	1.04880
$148$	0	0
$149$	17.3477	1.42118	0.710588	−	0.703608i	$-0.248429\pi$
$149$	17.3477	1.42118	0.710588	+	0.703608i	$0.248429\pi$
$150$	0	0
$151$	−22.6111	−1.84006	−0.920031	−	0.391846i	$-0.871837\pi$
$151$	−22.6111	−1.84006	−0.920031	+	0.391846i	$0.871837\pi$
$152$	0	0
$153$	5.04941	0.408221
$154$	0	0
$155$	−0.645126	−0.0518177
$156$	0	0
$157$	1.42683	0.113874	0.0569369	−	0.998378i	$-0.481867\pi$
$157$	1.42683	0.113874	0.0569369	+	0.998378i	$0.481867\pi$
$158$	0	0
$159$	1.19143	0.0944863
$160$	0	0
$161$	−10.0451	−0.791664
$162$	0	0
$163$	2.96513	0.232247	0.116124	−	0.993235i	$-0.462953\pi$
$163$	2.96513	0.232247	0.116124	+	0.993235i	$0.462953\pi$
$164$	0	0
$165$	−1.59571	−0.124226
$166$	0	0
$167$	−16.8805	−1.30625	−0.653127	−	0.757248i	$-0.726543\pi$
$167$	−16.8805	−1.30625	−0.653127	+	0.757248i	$0.726543\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−4.44027	−0.339556
$172$	0	0
$173$	−9.40540	−0.715079	−0.357540	−	0.933898i	$-0.616384\pi$
$173$	−9.40540	−0.715079	−0.357540	+	0.933898i	$0.616384\pi$
$174$	0	0
$175$	10.8951	0.823591
$176$	0	0
$177$	3.19143	0.239882
$178$	0	0
$179$	18.8588	1.40957	0.704786	−	0.709420i	$-0.251043\pi$
$179$	18.8588	1.40957	0.704786	+	0.709420i	$0.251043\pi$
$180$	0	0
$181$	2.57317	0.191262	0.0956309	−	0.995417i	$-0.469513\pi$
$181$	2.57317	0.191262	0.0956309	+	0.995417i	$0.469513\pi$
$182$	0	0
$183$	−1.89397	−0.140006
$184$	0	0
$185$	14.4977	1.06589
$186$	0	0
$187$	−5.04941	−0.369250
$188$	0	0
$189$	−4.44027	−0.322982
$190$	0	0
$191$	19.1914	1.38864	0.694321	−	0.719665i	$-0.255705\pi$
$191$	19.1914	1.38864	0.694321	+	0.719665i	$0.255705\pi$
$192$	0	0
$193$	20.9219	1.50599	0.752997	−	0.658024i	$-0.228607\pi$
$193$	20.9219	1.50599	0.752997	+	0.658024i	$0.228607\pi$
$194$	0	0
$195$	1.59571	0.114271
$196$	0	0
$197$	−13.6317	−0.971218	−0.485609	−	0.874176i	$-0.661402\pi$
$197$	−13.6317	−0.971218	−0.485609	+	0.874176i	$0.661402\pi$
$198$	0	0
$199$	16.6685	1.18160	0.590798	−	0.806819i	$-0.298813\pi$
$199$	16.6685	1.18160	0.590798	+	0.806819i	$0.298813\pi$
$200$	0	0
$201$	−7.31169	−0.515727
$202$	0	0
$203$	9.04030	0.634504
$204$	0	0
$205$	−14.9162	−1.04179
$206$	0	0
$207$	2.26227	0.157239
$208$	0	0
$209$	4.44027	0.307140
$210$	0	0
$211$	11.6459	0.801738	0.400869	−	0.916135i	$-0.368708\pi$
$211$	11.6459	0.801738	0.400869	+	0.916135i	$0.368708\pi$
$212$	0	0
$213$	−8.70254	−0.596288
$214$	0	0
$215$	19.0368	1.29830
$216$	0	0
$217$	1.79514	0.121862
$218$	0	0
$219$	7.63169	0.515702
$220$	0	0
$221$	5.04941	0.339660
$222$	0	0
$223$	22.1654	1.48430	0.742151	−	0.670233i	$-0.233806\pi$
$223$	22.1654	1.48430	0.742151	+	0.670233i	$0.233806\pi$
$224$	0	0
$225$	−2.45370	−0.163580
$226$	0	0
$227$	25.8722	1.71720	0.858600	−	0.512647i	$-0.171335\pi$
$227$	25.8722	1.71720	0.858600	+	0.512647i	$0.171335\pi$
$228$	0	0
$229$	−16.8014	−1.11027	−0.555133	−	0.831762i	$-0.687333\pi$
$229$	−16.8014	−1.11027	−0.555133	+	0.831762i	$0.687333\pi$
$230$	0	0
$231$	4.44027	0.292148
$232$	0	0
$233$	14.5605	0.953892	0.476946	−	0.878933i	$-0.341744\pi$
$233$	14.5605	0.953892	0.476946	+	0.878933i	$0.341744\pi$
$234$	0	0
$235$	5.37663	0.350733
$236$	0	0
$237$	11.1214	0.722411
$238$	0	0
$239$	−20.2551	−1.31019	−0.655096	−	0.755546i	$-0.727372\pi$
$239$	−20.2551	−1.31019	−0.655096	+	0.755546i	$0.727372\pi$
$240$	0	0
$241$	20.2427	1.30395	0.651975	−	0.758240i	$-0.273941\pi$
$241$	20.2427	1.30395	0.651975	+	0.758240i	$0.273941\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	20.2910	1.29635
$246$	0	0
$247$	−4.44027	−0.282527
$248$	0	0
$249$	12.5965	0.798271
$250$	0	0
$251$	24.9525	1.57499	0.787494	−	0.616323i	$-0.211378\pi$
$251$	24.9525	1.57499	0.787494	+	0.616323i	$0.211378\pi$
$252$	0	0
$253$	−2.26227	−0.142228
$254$	0	0
$255$	8.05741	0.504575
$256$	0	0
$257$	5.78793	0.361041	0.180521	−	0.983571i	$-0.442222\pi$
$257$	5.78793	0.361041	0.180521	+	0.983571i	$0.442222\pi$
$258$	0	0
$259$	−40.3416	−2.50670
$260$	0	0
$261$	−2.03598	−0.126024
$262$	0	0
$263$	25.3622	1.56390	0.781951	−	0.623341i	$-0.214225\pi$
$263$	25.3622	1.56390	0.781951	+	0.623341i	$0.214225\pi$
$264$	0	0
$265$	1.90117	0.116788
$266$	0	0
$267$	2.51032	0.153629
$268$	0	0
$269$	−2.69374	−0.164241	−0.0821203	−	0.996622i	$-0.526169\pi$
$269$	−2.69374	−0.164241	−0.0821203	+	0.996622i	$0.526169\pi$
$270$	0	0
$271$	−8.89285	−0.540202	−0.270101	−	0.962832i	$-0.587057\pi$
$271$	−8.89285	−0.540202	−0.270101	+	0.962832i	$0.587057\pi$
$272$	0	0
$273$	−4.44027	−0.268737
$274$	0	0
$275$	2.45370	0.147964
$276$	0	0
$277$	−13.2634	−0.796920	−0.398460	−	0.917186i	$-0.630455\pi$
$277$	−13.2634	−0.796920	−0.398460	+	0.917186i	$0.630455\pi$
$278$	0	0
$279$	−0.404287	−0.0242040
$280$	0	0
$281$	−9.63169	−0.574579	−0.287289	−	0.957844i	$-0.592754\pi$
$281$	−9.63169	−0.574579	−0.287289	+	0.957844i	$0.592754\pi$
$282$	0	0
$283$	−12.0360	−0.715465	−0.357732	−	0.933824i	$-0.616450\pi$
$283$	−12.0360	−0.715465	−0.357732	+	0.933824i	$0.616450\pi$
$284$	0	0
$285$	−7.08539	−0.419702
$286$	0	0
$287$	41.5061	2.45003
$288$	0	0
$289$	8.49657	0.499798
$290$	0	0
$291$	15.0585	0.882747
$292$	0	0
$293$	−25.6317	−1.49742	−0.748710	−	0.662898i	$-0.769326\pi$
$293$	−25.6317	−1.49742	−0.748710	+	0.662898i	$0.769326\pi$
$294$	0	0
$295$	5.09260	0.296503
$296$	0	0
$297$	−1.00000	−0.0580259
$298$	0	0
$299$	2.26227	0.130831
$300$	0	0
$301$	−52.9722	−3.05326
$302$	0	0
$303$	−5.92995	−0.340667
$304$	0	0
$305$	−3.02223	−0.173052
$306$	0	0
$307$	−9.32080	−0.531966	−0.265983	−	0.963978i	$-0.585697\pi$
$307$	−9.32080	−0.531966	−0.265983	+	0.963978i	$0.585697\pi$
$308$	0	0
$309$	11.2634	0.640752
$310$	0	0
$311$	13.5257	0.766970	0.383485	−	0.923547i	$-0.374724\pi$
$311$	13.5257	0.766970	0.383485	+	0.923547i	$0.374724\pi$
$312$	0	0
$313$	9.14281	0.516782	0.258391	−	0.966040i	$-0.416808\pi$
$313$	9.14281	0.516782	0.258391	+	0.966040i	$0.416808\pi$
$314$	0	0
$315$	−7.08539	−0.399217
$316$	0	0
$317$	−7.31889	−0.411070	−0.205535	−	0.978650i	$-0.565893\pi$
$317$	−7.31889	−0.411070	−0.205535	+	0.978650i	$0.565893\pi$
$318$	0	0
$319$	2.03598	0.113993
$320$	0	0
$321$	−0.808574	−0.0451302
$322$	0	0
$323$	−22.4207	−1.24752
$324$	0	0
$325$	−2.45370	−0.136107
$326$	0	0
$327$	2.75116	0.152139
$328$	0	0
$329$	−14.9611	−0.824834
$330$	0	0
$331$	−8.65424	−0.475680	−0.237840	−	0.971304i	$-0.576439\pi$
$331$	−8.65424	−0.475680	−0.237840	+	0.971304i	$0.576439\pi$
$332$	0	0
$333$	9.08539	0.497876
$334$	0	0
$335$	−11.6674	−0.637456
$336$	0	0
$337$	−9.19143	−0.500689	−0.250344	−	0.968157i	$-0.580544\pi$
$337$	−9.19143	−0.500689	−0.250344	+	0.968157i	$0.580544\pi$
$338$	0	0
$339$	7.68911	0.417615
$340$	0	0
$341$	0.404287	0.0218934
$342$	0	0
$343$	−25.3804	−1.37042
$344$	0	0
$345$	3.60994	0.194353
$346$	0	0
$347$	24.1780	1.29794	0.648971	−	0.760813i	$-0.275199\pi$
$347$	24.1780	1.29794	0.648971	+	0.760813i	$0.275199\pi$
$348$	0	0
$349$	3.54519	0.189769	0.0948847	−	0.995488i	$-0.469752\pi$
$349$	3.54519	0.189769	0.0948847	+	0.995488i	$0.469752\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	18.1922	0.968274	0.484137	−	0.874992i	$-0.339134\pi$
$353$	18.1922	0.968274	0.484137	+	0.874992i	$0.339134\pi$
$354$	0	0
$355$	−13.8868	−0.737033
$356$	0	0
$357$	−22.4207	−1.18663
$358$	0	0
$359$	15.2060	0.802541	0.401270	−	0.915960i	$-0.368569\pi$
$359$	15.2060	0.802541	0.401270	+	0.915960i	$0.368569\pi$
$360$	0	0
$361$	0.715972	0.0376828
$362$	0	0
$363$	1.00000	0.0524864
$364$	0	0
$365$	12.1780	0.637425
$366$	0	0
$367$	−1.16456	−0.0607896	−0.0303948	−	0.999538i	$-0.509676\pi$
$367$	−1.16456	−0.0607896	−0.0303948	+	0.999538i	$0.509676\pi$
$368$	0	0
$369$	−9.34767	−0.486620
$370$	0	0
$371$	−5.29025	−0.274656
$372$	0	0
$373$	0.979359	0.0507093	0.0253547	−	0.999679i	$-0.491928\pi$
$373$	0.979359	0.0507093	0.0253547	+	0.999679i	$0.491928\pi$
$374$	0	0
$375$	−11.8940	−0.614202
$376$	0	0
$377$	−2.03598	−0.104858
$378$	0	0
$379$	14.7959	0.760016	0.380008	−	0.924983i	$-0.375921\pi$
$379$	14.7959	0.760016	0.380008	+	0.924983i	$0.375921\pi$
$380$	0	0
$381$	−6.34687	−0.325160
$382$	0	0
$383$	10.0988	0.516026	0.258013	−	0.966141i	$-0.416932\pi$
$383$	10.0988	0.516026	0.258013	+	0.966141i	$0.416932\pi$
$384$	0	0
$385$	7.08539	0.361105
$386$	0	0
$387$	11.9299	0.606433
$388$	0	0
$389$	−27.0062	−1.36927	−0.684635	−	0.728886i	$-0.740038\pi$
$389$	−27.0062	−1.36927	−0.684635	+	0.728886i	$0.740038\pi$
$390$	0	0
$391$	11.4231	0.577694
$392$	0	0
$393$	1.09260	0.0551144
$394$	0	0
$395$	17.7465	0.892924
$396$	0	0
$397$	20.4636	1.02704	0.513520	−	0.858078i	$-0.328341\pi$
$397$	20.4636	1.02704	0.513520	+	0.858078i	$0.328341\pi$
$398$	0	0
$399$	19.7160	0.987033
$400$	0	0
$401$	−15.8813	−0.793076	−0.396538	−	0.918018i	$-0.629788\pi$
$401$	−15.8813	−0.793076	−0.396538	+	0.918018i	$0.629788\pi$
$402$	0	0
$403$	−0.404287	−0.0201390
$404$	0	0
$405$	1.59571	0.0792916
$406$	0	0
$407$	−9.08539	−0.450346
$408$	0	0
$409$	18.2979	0.904775	0.452387	−	0.891822i	$-0.350572\pi$
$409$	18.2979	0.904775	0.452387	+	0.891822i	$0.350572\pi$
$410$	0	0
$411$	8.27139	0.407998
$412$	0	0
$413$	−14.1708	−0.697299
$414$	0	0
$415$	20.1004	0.986690
$416$	0	0
$417$	−7.22741	−0.353928
$418$	0	0
$419$	−36.5050	−1.78339	−0.891693	−	0.452640i	$-0.850482\pi$
$419$	−36.5050	−1.78339	−0.891693	+	0.452640i	$0.850482\pi$
$420$	0	0
$421$	18.6685	0.909845	0.454923	−	0.890531i	$-0.349667\pi$
$421$	18.6685	0.909845	0.454923	+	0.890531i	$0.349667\pi$
$422$	0	0
$423$	3.36942	0.163827
$424$	0	0
$425$	−12.3897	−0.600991
$426$	0	0
$427$	8.40972	0.406975
$428$	0	0
$429$	−1.00000	−0.0482805
$430$	0	0
$431$	−21.6462	−1.04266	−0.521331	−	0.853354i	$-0.674564\pi$
$431$	−21.6462	−1.04266	−0.521331	+	0.853354i	$0.674564\pi$
$432$	0	0
$433$	−24.5268	−1.17868	−0.589341	−	0.807885i	$-0.700612\pi$
$433$	−24.5268	−1.17868	−0.589341	+	0.807885i	$0.700612\pi$
$434$	0	0
$435$	−3.24884	−0.155770
$436$	0	0
$437$	−10.0451	−0.480522
$438$	0	0
$439$	−25.8311	−1.23285	−0.616426	−	0.787413i	$-0.711420\pi$
$439$	−25.8311	−1.23285	−0.616426	+	0.787413i	$0.711420\pi$
$440$	0	0
$441$	12.7160	0.605522
$442$	0	0
$443$	29.7611	1.41399	0.706996	−	0.707218i	$-0.250050\pi$
$443$	29.7611	1.41399	0.706996	+	0.707218i	$0.250050\pi$
$444$	0	0
$445$	4.00575	0.189891
$446$	0	0
$447$	17.3477	0.820516
$448$	0	0
$449$	−12.6899	−0.598873	−0.299437	−	0.954116i	$-0.596799\pi$
$449$	−12.6899	−0.598873	−0.299437	+	0.954116i	$0.596799\pi$
$450$	0	0
$451$	9.34767	0.440164
$452$	0	0
$453$	−22.6111	−1.06236
$454$	0	0
$455$	−7.08539	−0.332168
$456$	0	0
$457$	−13.4755	−0.630355	−0.315178	−	0.949033i	$-0.602064\pi$
$457$	−13.4755	−0.630355	−0.315178	+	0.949033i	$0.602064\pi$
$458$	0	0
$459$	5.04941	0.235686
$460$	0	0
$461$	32.0430	1.49239	0.746196	−	0.665727i	$-0.231878\pi$
$461$	32.0430	1.49239	0.746196	+	0.665727i	$0.231878\pi$
$462$	0	0
$463$	−20.2983	−0.943340	−0.471670	−	0.881775i	$-0.656349\pi$
$463$	−20.2983	−0.943340	−0.471670	+	0.881775i	$0.656349\pi$
$464$	0	0
$465$	−0.645126	−0.0299170
$466$	0	0
$467$	−39.1284	−1.81065	−0.905323	−	0.424724i	$-0.860371\pi$
$467$	−39.1284	−1.81065	−0.905323	+	0.424724i	$0.860371\pi$
$468$	0	0
$469$	32.4658	1.49913
$470$	0	0
$471$	1.42683	0.0657451
$472$	0	0
$473$	−11.9299	−0.548540
$474$	0	0
$475$	10.8951	0.499901
$476$	0	0
$477$	1.19143	0.0545517
$478$	0	0
$479$	−15.9426	−0.728435	−0.364218	−	0.931314i	$-0.618664\pi$
$479$	−15.9426	−0.728435	−0.364218	+	0.931314i	$0.618664\pi$
$480$	0	0
$481$	9.08539	0.414258
$482$	0	0
$483$	−10.0451	−0.457068
$484$	0	0
$485$	24.0291	1.09110
$486$	0	0
$487$	7.00800	0.317563	0.158781	−	0.987314i	$-0.449243\pi$
$487$	7.00800	0.317563	0.158781	+	0.987314i	$0.449243\pi$
$488$	0	0
$489$	2.96513	0.134088
$490$	0	0
$491$	−36.2784	−1.63722	−0.818611	−	0.574349i	$-0.805255\pi$
$491$	−36.2784	−1.63722	−0.818611	+	0.574349i	$0.805255\pi$
$492$	0	0
$493$	−10.2805	−0.463011
$494$	0	0
$495$	−1.59571	−0.0717220
$496$	0	0
$497$	38.6416	1.73331
$498$	0	0
$499$	−15.5704	−0.697028	−0.348514	−	0.937303i	$-0.613314\pi$
$499$	−15.5704	−0.697028	−0.348514	+	0.937303i	$0.613314\pi$
$500$	0	0
$501$	−16.8805	−0.754167
$502$	0	0
$503$	33.1501	1.47809	0.739046	−	0.673655i	$-0.235277\pi$
$503$	33.1501	1.47809	0.739046	+	0.673655i	$0.235277\pi$
$504$	0	0
$505$	−9.46249	−0.421075
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−10.6542	−0.472241	−0.236120	−	0.971724i	$-0.575876\pi$
$509$	−10.6542	−0.472241	−0.236120	+	0.971724i	$0.575876\pi$
$510$	0	0
$511$	−33.8868	−1.49906
$512$	0	0
$513$	−4.44027	−0.196043
$514$	0	0
$515$	17.9731	0.791991
$516$	0	0
$517$	−3.36942	−0.148187
$518$	0	0
$519$	−9.40540	−0.412851
$520$	0	0
$521$	−5.37822	−0.235624	−0.117812	−	0.993036i	$-0.537588\pi$
$521$	−5.37822	−0.235624	−0.117812	+	0.993036i	$0.537588\pi$
$522$	0	0
$523$	14.3848	0.629002	0.314501	−	0.949257i	$-0.398163\pi$
$523$	14.3848	0.629002	0.314501	+	0.949257i	$0.398163\pi$
$524$	0	0
$525$	10.8951	0.475500
$526$	0	0
$527$	−2.04141	−0.0889253
$528$	0	0
$529$	−17.8821	−0.777484
$530$	0	0
$531$	3.19143	0.138496
$532$	0	0
$533$	−9.34767	−0.404892
$534$	0	0
$535$	−1.29025	−0.0557824
$536$	0	0
$537$	18.8588	0.813816
$538$	0	0
$539$	−12.7160	−0.547716
$540$	0	0
$541$	−20.3291	−0.874017	−0.437009	−	0.899457i	$-0.643962\pi$
$541$	−20.3291	−0.874017	−0.437009	+	0.899457i	$0.643962\pi$
$542$	0	0
$543$	2.57317	0.110425
$544$	0	0
$545$	4.39006	0.188050
$546$	0	0
$547$	6.65935	0.284733	0.142367	−	0.989814i	$-0.454529\pi$
$547$	6.65935	0.284733	0.142367	+	0.989814i	$0.454529\pi$
$548$	0	0
$549$	−1.89397	−0.0808325
$550$	0	0
$551$	9.04030	0.385130
$552$	0	0
$553$	−49.3819	−2.09993
$554$	0	0
$555$	14.4977	0.615392
$556$	0	0
$557$	−44.7550	−1.89633	−0.948165	−	0.317780i	$-0.897063\pi$
$557$	−44.7550	−1.89633	−0.948165	+	0.317780i	$0.897063\pi$
$558$	0	0
$559$	11.9299	0.504583
$560$	0	0
$561$	−5.04941	−0.213186
$562$	0	0
$563$	−39.0513	−1.64582	−0.822908	−	0.568174i	$-0.807650\pi$
$563$	−39.0513	−1.64582	−0.822908	+	0.568174i	$0.807650\pi$
$564$	0	0
$565$	12.2696	0.516186
$566$	0	0
$567$	−4.44027	−0.186474
$568$	0	0
$569$	−0.427628	−0.0179271	−0.00896356	−	0.999960i	$-0.502853\pi$
$569$	−0.427628	−0.0179271	−0.00896356	+	0.999960i	$0.502853\pi$
$570$	0	0
$571$	12.1705	0.509318	0.254659	−	0.967031i	$-0.418037\pi$
$571$	12.1705	0.509318	0.254659	+	0.967031i	$0.418037\pi$
$572$	0	0
$573$	19.1914	0.801733
$574$	0	0
$575$	−5.55094	−0.231490
$576$	0	0
$577$	−36.0735	−1.50176	−0.750881	−	0.660438i	$-0.770371\pi$
$577$	−36.0735	−1.50176	−0.750881	+	0.660438i	$0.770371\pi$
$578$	0	0
$579$	20.9219	0.869486
$580$	0	0
$581$	−55.9319	−2.32044
$582$	0	0
$583$	−1.19143	−0.0493438
$584$	0	0
$585$	1.59571	0.0659746
$586$	0	0
$587$	33.3622	1.37701	0.688503	−	0.725234i	$-0.258268\pi$
$587$	33.3622	1.37701	0.688503	+	0.725234i	$0.258268\pi$
$588$	0	0
$589$	1.79514	0.0739675
$590$	0	0
$591$	−13.6317	−0.560733
$592$	0	0
$593$	27.9173	1.14643	0.573213	−	0.819406i	$-0.305697\pi$
$593$	27.9173	1.14643	0.573213	+	0.819406i	$0.305697\pi$
$594$	0	0
$595$	−35.7771	−1.46672
$596$	0	0
$597$	16.6685	0.682195
$598$	0	0
$599$	−7.13770	−0.291638	−0.145819	−	0.989311i	$-0.546582\pi$
$599$	−7.13770	−0.291638	−0.145819	+	0.989311i	$0.546582\pi$
$600$	0	0
$601$	−11.3514	−0.463031	−0.231516	−	0.972831i	$-0.574368\pi$
$601$	−11.3514	−0.463031	−0.231516	+	0.972831i	$0.574368\pi$
$602$	0	0
$603$	−7.31169	−0.297755
$604$	0	0
$605$	1.59571	0.0648750
$606$	0	0
$607$	0.604036	0.0245171	0.0122585	−	0.999925i	$-0.496098\pi$
$607$	0.604036	0.0245171	0.0122585	+	0.999925i	$0.496098\pi$
$608$	0	0
$609$	9.04030	0.366331
$610$	0	0
$611$	3.36942	0.136312
$612$	0	0
$613$	−22.6974	−0.916741	−0.458370	−	0.888761i	$-0.651567\pi$
$613$	−22.6974	−0.916741	−0.458370	+	0.888761i	$0.651567\pi$
$614$	0	0
$615$	−14.9162	−0.601479
$616$	0	0
$617$	21.4809	0.864788	0.432394	−	0.901685i	$-0.357669\pi$
$617$	21.4809	0.864788	0.432394	+	0.901685i	$0.357669\pi$
$618$	0	0
$619$	24.0862	0.968106	0.484053	−	0.875039i	$-0.339164\pi$
$619$	24.0862	0.968106	0.484053	+	0.875039i	$0.339164\pi$
$620$	0	0
$621$	2.26227	0.0907819
$622$	0	0
$623$	−11.1465	−0.446575
$624$	0	0
$625$	−6.71086	−0.268435
$626$	0	0
$627$	4.44027	0.177327
$628$	0	0
$629$	45.8759	1.82919
$630$	0	0
$631$	15.2776	0.608192	0.304096	−	0.952641i	$-0.401646\pi$
$631$	15.2776	0.608192	0.304096	+	0.952641i	$0.401646\pi$
$632$	0	0
$633$	11.6459	0.462884
$634$	0	0
$635$	−10.1278	−0.401909
$636$	0	0
$637$	12.7160	0.503825
$638$	0	0
$639$	−8.70254	−0.344267
$640$	0	0
$641$	−2.16856	−0.0856529	−0.0428264	−	0.999083i	$-0.513636\pi$
$641$	−2.16856	−0.0856529	−0.0428264	+	0.999083i	$0.513636\pi$
$642$	0	0
$643$	15.2580	0.601715	0.300858	−	0.953669i	$-0.402727\pi$
$643$	15.2580	0.601715	0.300858	+	0.953669i	$0.402727\pi$
$644$	0	0
$645$	19.0368	0.749572
$646$	0	0
$647$	−3.33535	−0.131126	−0.0655630	−	0.997848i	$-0.520884\pi$
$647$	−3.33535	−0.131126	−0.0655630	+	0.997848i	$0.520884\pi$
$648$	0	0
$649$	−3.19143	−0.125274
$650$	0	0
$651$	1.79514	0.0703571
$652$	0	0
$653$	29.1661	1.14136	0.570680	−	0.821173i	$-0.306680\pi$
$653$	29.1661	1.14136	0.570680	+	0.821173i	$0.306680\pi$
$654$	0	0
$655$	1.74348	0.0681233
$656$	0	0
$657$	7.63169	0.297741
$658$	0	0
$659$	46.7767	1.82216	0.911081	−	0.412227i	$-0.135249\pi$
$659$	46.7767	1.82216	0.911081	+	0.412227i	$0.135249\pi$
$660$	0	0
$661$	21.3063	0.828717	0.414359	−	0.910114i	$-0.364006\pi$
$661$	21.3063	0.828717	0.414359	+	0.910114i	$0.364006\pi$
$662$	0	0
$663$	5.04941	0.196103
$664$	0	0
$665$	31.4610	1.22001
$666$	0	0
$667$	−4.60594	−0.178343
$668$	0	0
$669$	22.1654	0.856962
$670$	0	0
$671$	1.89397	0.0731158
$672$	0	0
$673$	6.94370	0.267660	0.133830	−	0.991004i	$-0.457272\pi$
$673$	6.94370	0.267660	0.133830	+	0.991004i	$0.457272\pi$
$674$	0	0
$675$	−2.45370	−0.0944429
$676$	0	0
$677$	35.5850	1.36764	0.683821	−	0.729650i	$-0.260317\pi$
$677$	35.5850	1.36764	0.683821	+	0.729650i	$0.260317\pi$
$678$	0	0
$679$	−66.8639	−2.56600
$680$	0	0
$681$	25.8722	0.991425
$682$	0	0
$683$	9.68031	0.370407	0.185203	−	0.982700i	$-0.440706\pi$
$683$	9.68031	0.370407	0.185203	+	0.982700i	$0.440706\pi$
$684$	0	0
$685$	13.1988	0.504299
$686$	0	0
$687$	−16.8014	−0.641012
$688$	0	0
$689$	1.19143	0.0453897
$690$	0	0
$691$	52.0433	1.97982	0.989911	−	0.141694i	$-0.0452548\pi$
$691$	52.0433	1.97982	0.989911	+	0.141694i	$0.0452548\pi$
$692$	0	0
$693$	4.44027	0.168672
$694$	0	0
$695$	−11.5329	−0.437467
$696$	0	0
$697$	−47.2002	−1.78784
$698$	0	0
$699$	14.5605	0.550730
$700$	0	0
$701$	30.2140	1.14117	0.570583	−	0.821240i	$-0.306717\pi$
$701$	30.2140	1.14117	0.570583	+	0.821240i	$0.306717\pi$
$702$	0	0
$703$	−40.3416	−1.52151
$704$	0	0
$705$	5.37663	0.202496
$706$	0	0
$707$	26.3305	0.990262
$708$	0	0
$709$	32.9888	1.23892	0.619460	−	0.785028i	$-0.287352\pi$
$709$	32.9888	1.23892	0.619460	+	0.785028i	$0.287352\pi$
$710$	0	0
$711$	11.1214	0.417084
$712$	0	0
$713$	−0.914607	−0.0342523
$714$	0	0
$715$	−1.59571	−0.0596763
$716$	0	0
$717$	−20.2551	−0.756439
$718$	0	0
$719$	−35.2925	−1.31619	−0.658094	−	0.752936i	$-0.728637\pi$
$719$	−35.2925	−1.31619	−0.658094	+	0.752936i	$0.728637\pi$
$720$	0	0
$721$	−50.0124	−1.86256
$722$	0	0
$723$	20.2427	0.752836
$724$	0	0
$725$	4.99568	0.185535
$726$	0	0
$727$	18.5987	0.689789	0.344894	−	0.938641i	$-0.387915\pi$
$727$	18.5987	0.689789	0.344894	+	0.938641i	$0.387915\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	60.2392	2.22803
$732$	0	0
$733$	−48.3002	−1.78401	−0.892004	−	0.452028i	$-0.850701\pi$
$733$	−48.3002	−1.78401	−0.892004	+	0.452028i	$0.850701\pi$
$734$	0	0
$735$	20.2910	0.748446
$736$	0	0
$737$	7.31169	0.269329
$738$	0	0
$739$	−48.5407	−1.78560	−0.892799	−	0.450456i	$-0.851262\pi$
$739$	−48.5407	−1.78560	−0.892799	+	0.450456i	$0.851262\pi$
$740$	0	0
$741$	−4.44027	−0.163117
$742$	0	0
$743$	−39.6478	−1.45454	−0.727269	−	0.686353i	$-0.759211\pi$
$743$	−39.6478	−1.45454	−0.727269	+	0.686353i	$0.759211\pi$
$744$	0	0
$745$	27.6819	1.01419
$746$	0	0
$747$	12.5965	0.460882
$748$	0	0
$749$	3.59028	0.131186
$750$	0	0
$751$	−34.7673	−1.26868	−0.634338	−	0.773056i	$-0.718727\pi$
$751$	−34.7673	−1.26868	−0.634338	+	0.773056i	$0.718727\pi$
$752$	0	0
$753$	24.9525	0.909319
$754$	0	0
$755$	−36.0808	−1.31311
$756$	0	0
$757$	25.7828	0.937093	0.468546	−	0.883439i	$-0.344778\pi$
$757$	25.7828	0.937093	0.468546	+	0.883439i	$0.344778\pi$
$758$	0	0
$759$	−2.26227	−0.0821153
$760$	0	0
$761$	14.0414	0.509001	0.254500	−	0.967073i	$-0.418089\pi$
$761$	14.0414	0.509001	0.254500	+	0.967073i	$0.418089\pi$
$762$	0	0
$763$	−12.2159	−0.442245
$764$	0	0
$765$	8.05741	0.291316
$766$	0	0
$767$	3.19143	0.115236
$768$	0	0
$769$	−24.5805	−0.886396	−0.443198	−	0.896424i	$-0.646156\pi$
$769$	−24.5805	−0.886396	−0.443198	+	0.896424i	$0.646156\pi$
$770$	0	0
$771$	5.78793	0.208447
$772$	0	0
$773$	−11.0387	−0.397034	−0.198517	−	0.980097i	$-0.563612\pi$
$773$	−11.0387	−0.397034	−0.198517	+	0.980097i	$0.563612\pi$
$774$	0	0
$775$	0.991998	0.0356336
$776$	0	0
$777$	−40.3416	−1.44725
$778$	0	0
$779$	41.5061	1.48711
$780$	0	0
$781$	8.70254	0.311401
$782$	0	0
$783$	−2.03598	−0.0727600
$784$	0	0
$785$	2.27682	0.0812632
$786$	0	0
$787$	−5.73052	−0.204271	−0.102135	−	0.994770i	$-0.532568\pi$
$787$	−5.73052	−0.204271	−0.102135	+	0.994770i	$0.532568\pi$
$788$	0	0
$789$	25.3622	0.902919
$790$	0	0
$791$	−34.1417	−1.21394
$792$	0	0
$793$	−1.89397	−0.0672567
$794$	0	0
$795$	1.90117	0.0674277
$796$	0	0
$797$	44.0735	1.56117	0.780583	−	0.625053i	$-0.214923\pi$
$797$	44.0735	1.56117	0.780583	+	0.625053i	$0.214923\pi$
$798$	0	0
$799$	17.0136	0.601898
$800$	0	0
$801$	2.51032	0.0886978
$802$	0	0
$803$	−7.63169	−0.269317
$804$	0	0
$805$	−16.0291	−0.564951
$806$	0	0
$807$	−2.69374	−0.0948243
$808$	0	0
$809$	29.4048	1.03382	0.516908	−	0.856041i	$-0.327083\pi$
$809$	29.4048	1.03382	0.516908	+	0.856041i	$0.327083\pi$
$810$	0	0
$811$	−15.1356	−0.531483	−0.265741	−	0.964044i	$-0.585617\pi$
$811$	−15.1356	−0.531483	−0.265741	+	0.964044i	$0.585617\pi$
$812$	0	0
$813$	−8.89285	−0.311886
$814$	0	0
$815$	4.73150	0.165737
$816$	0	0
$817$	−52.9722	−1.85326
$818$	0	0
$819$	−4.44027	−0.155155
$820$	0	0
$821$	−53.8293	−1.87866	−0.939329	−	0.343019i	$-0.888551\pi$
$821$	−53.8293	−1.87866	−0.939329	+	0.343019i	$0.888551\pi$
$822$	0	0
$823$	48.0038	1.67331	0.836654	−	0.547732i	$-0.184509\pi$
$823$	48.0038	1.67331	0.836654	+	0.547732i	$0.184509\pi$
$824$	0	0
$825$	2.45370	0.0854268
$826$	0	0
$827$	56.1869	1.95381	0.976905	−	0.213673i	$-0.0685426\pi$
$827$	56.1869	1.95381	0.976905	+	0.213673i	$0.0685426\pi$
$828$	0	0
$829$	17.4319	0.605436	0.302718	−	0.953080i	$-0.402106\pi$
$829$	17.4319	0.605436	0.302718	+	0.953080i	$0.402106\pi$
$830$	0	0
$831$	−13.2634	−0.460102
$832$	0	0
$833$	64.2082	2.22468
$834$	0	0
$835$	−26.9365	−0.932176
$836$	0	0
$837$	−0.404287	−0.0139742
$838$	0	0
$839$	−46.8930	−1.61893	−0.809463	−	0.587171i	$-0.800242\pi$
$839$	−46.8930	−1.61893	−0.809463	+	0.587171i	$0.800242\pi$
$840$	0	0
$841$	−24.8548	−0.857062
$842$	0	0
$843$	−9.63169	−0.331733
$844$	0	0
$845$	1.59571	0.0548942
$846$	0	0
$847$	−4.44027	−0.152569
$848$	0	0
$849$	−12.0360	−0.413074
$850$	0	0
$851$	20.5536	0.704570
$852$	0	0
$853$	−0.551410	−0.0188799	−0.00943997	−	0.999955i	$-0.503005\pi$
$853$	−0.551410	−0.0188799	−0.00943997	+	0.999955i	$0.503005\pi$
$854$	0	0
$855$	−7.08539	−0.242315
$856$	0	0
$857$	27.8690	0.951987	0.475994	−	0.879449i	$-0.342089\pi$
$857$	27.8690	0.951987	0.475994	+	0.879449i	$0.342089\pi$
$858$	0	0
$859$	−2.45481	−0.0837572	−0.0418786	−	0.999123i	$-0.513334\pi$
$859$	−2.45481	−0.0837572	−0.0418786	+	0.999123i	$0.513334\pi$
$860$	0	0
$861$	41.5061	1.41453
$862$	0	0
$863$	38.6525	1.31575	0.657873	−	0.753129i	$-0.271457\pi$
$863$	38.6525	1.31575	0.657873	+	0.753129i	$0.271457\pi$
$864$	0	0
$865$	−15.0083	−0.510298
$866$	0	0
$867$	8.49657	0.288559
$868$	0	0
$869$	−11.1214	−0.377267
$870$	0	0
$871$	−7.31169	−0.247747
$872$	0	0
$873$	15.0585	0.509654
$874$	0	0
$875$	52.8124	1.78538
$876$	0	0
$877$	−56.4404	−1.90586	−0.952928	−	0.303195i	$-0.901946\pi$
$877$	−56.4404	−1.90586	−0.952928	+	0.303195i	$0.901946\pi$
$878$	0	0
$879$	−25.6317	−0.864536
$880$	0	0
$881$	35.8148	1.20663	0.603316	−	0.797503i	$-0.293846\pi$
$881$	35.8148	1.20663	0.603316	+	0.797503i	$0.293846\pi$
$882$	0	0
$883$	−30.0560	−1.01146	−0.505732	−	0.862691i	$-0.668777\pi$
$883$	−30.0560	−1.01146	−0.505732	+	0.862691i	$0.668777\pi$
$884$	0	0
$885$	5.09260	0.171186
$886$	0	0
$887$	40.8212	1.37064	0.685321	−	0.728241i	$-0.259662\pi$
$887$	40.8212	1.37064	0.685321	+	0.728241i	$0.259662\pi$
$888$	0	0
$889$	28.1818	0.945188
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	−14.9611	−0.500655
$894$	0	0
$895$	30.0932	1.00590
$896$	0	0
$897$	2.26227	0.0755351
$898$	0	0
$899$	0.823120	0.0274526
$900$	0	0
$901$	6.01600	0.200422
$902$	0	0
$903$	−52.9722	−1.76280
$904$	0	0
$905$	4.10603	0.136489
$906$	0	0
$907$	7.03309	0.233530	0.116765	−	0.993160i	$-0.462748\pi$
$907$	7.03309	0.233530	0.116765	+	0.993160i	$0.462748\pi$
$908$	0	0
$909$	−5.92995	−0.196684
$910$	0	0
$911$	27.1233	0.898634	0.449317	−	0.893372i	$-0.351667\pi$
$911$	27.1233	0.898634	0.449317	+	0.893372i	$0.351667\pi$
$912$	0	0
$913$	−12.5965	−0.416884
$914$	0	0
$915$	−3.02223	−0.0999118
$916$	0	0
$917$	−4.85144	−0.160209
$918$	0	0
$919$	51.6118	1.70252	0.851259	−	0.524746i	$-0.175840\pi$
$919$	51.6118	1.70252	0.851259	+	0.524746i	$0.175840\pi$
$920$	0	0
$921$	−9.32080	−0.307131
$922$	0	0
$923$	−8.70254	−0.286448
$924$	0	0
$925$	−22.2928	−0.732983
$926$	0	0
$927$	11.2634	0.369938
$928$	0	0
$929$	−8.34335	−0.273736	−0.136868	−	0.990589i	$-0.543704\pi$
$929$	−8.34335	−0.273736	−0.136868	+	0.990589i	$0.543704\pi$
$930$	0	0
$931$	−56.4623	−1.85048
$932$	0	0
$933$	13.5257	0.442810
$934$	0	0
$935$	−8.05741	−0.263506
$936$	0	0
$937$	−40.8733	−1.33527	−0.667637	−	0.744487i	$-0.732694\pi$
$937$	−40.8733	−1.33527	−0.667637	+	0.744487i	$0.732694\pi$
$938$	0	0
$939$	9.14281	0.298364
$940$	0	0
$941$	22.9402	0.747828	0.373914	−	0.927463i	$-0.378015\pi$
$941$	22.9402	0.747828	0.373914	+	0.927463i	$0.378015\pi$
$942$	0	0
$943$	−21.1470	−0.688640
$944$	0	0
$945$	−7.08539	−0.230488
$946$	0	0
$947$	10.4889	0.340843	0.170422	−	0.985371i	$-0.445487\pi$
$947$	10.4889	0.340843	0.170422	+	0.985371i	$0.445487\pi$
$948$	0	0
$949$	7.63169	0.247735
$950$	0	0
$951$	−7.31889	−0.237331
$952$	0	0
$953$	6.84455	0.221717	0.110858	−	0.993836i	$-0.464640\pi$
$953$	6.84455	0.221717	0.110858	+	0.993836i	$0.464640\pi$
$954$	0	0
$955$	30.6240	0.990970
$956$	0	0
$957$	2.03598	0.0658139
$958$	0	0
$959$	−36.7272	−1.18598
$960$	0	0
$961$	−30.8366	−0.994727
$962$	0	0
$963$	−0.808574	−0.0260559
$964$	0	0
$965$	33.3854	1.07471
$966$	0	0
$967$	−30.0145	−0.965203	−0.482601	−	0.875840i	$-0.660308\pi$
$967$	−30.0145	−0.965203	−0.482601	+	0.875840i	$0.660308\pi$
$968$	0	0
$969$	−22.4207	−0.720258
$970$	0	0
$971$	−30.1941	−0.968976	−0.484488	−	0.874798i	$-0.660994\pi$
$971$	−30.1941	−0.968976	−0.484488	+	0.874798i	$0.660994\pi$
$972$	0	0
$973$	32.0916	1.02881
$974$	0	0
$975$	−2.45370	−0.0785813
$976$	0	0
$977$	−17.2492	−0.551850	−0.275925	−	0.961179i	$-0.588984\pi$
$977$	−17.2492	−0.551850	−0.275925	+	0.961179i	$0.588984\pi$
$978$	0	0
$979$	−2.51032	−0.0802302
$980$	0	0
$981$	2.75116	0.0878378
$982$	0	0
$983$	−45.2758	−1.44407	−0.722037	−	0.691854i	$-0.756794\pi$
$983$	−45.2758	−1.44407	−0.722037	+	0.691854i	$0.756794\pi$
$984$	0	0
$985$	−21.7523	−0.693085
$986$	0	0
$987$	−14.9611	−0.476218
$988$	0	0
$989$	26.9888	0.858194
$990$	0	0
$991$	−36.1011	−1.14679	−0.573394	−	0.819280i	$-0.694374\pi$
$991$	−36.1011	−1.14679	−0.573394	+	0.819280i	$0.694374\pi$
$992$	0	0
$993$	−8.65424	−0.274634
$994$	0	0
$995$	26.5981	0.843216
$996$	0	0
$997$	6.86230	0.217331	0.108666	−	0.994078i	$-0.465342\pi$
$997$	6.86230	0.217331	0.108666	+	0.994078i	$0.465342\pi$
$998$	0	0
$999$	9.08539	0.287449

Twists

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

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

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +1.59571 q^{5} -4.44027 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +1.59571 q^{5} -4.44027 q^{7} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} +1.59571 q^{15} +5.04941 q^{17} -4.44027 q^{19} -4.44027 q^{21} +2.26227 q^{23} -2.45370 q^{25} +1.00000 q^{27} -2.03598 q^{29} -0.404287 q^{31} -1.00000 q^{33} -7.08539 q^{35} +9.08539 q^{37} +1.00000 q^{39} -9.34767 q^{41} +11.9299 q^{43} +1.59571 q^{45} +3.36942 q^{47} +12.7160 q^{49} +5.04941 q^{51} +1.19143 q^{53} -1.59571 q^{55} -4.44027 q^{57} +3.19143 q^{59} -1.89397 q^{61} -4.44027 q^{63} +1.59571 q^{65} -7.31169 q^{67} +2.26227 q^{69} -8.70254 q^{71} +7.63169 q^{73} -2.45370 q^{75} +4.44027 q^{77} +11.1214 q^{79} +1.00000 q^{81} +12.5965 q^{83} +8.05741 q^{85} -2.03598 q^{87} +2.51032 q^{89} -4.44027 q^{91} -0.404287 q^{93} -7.08539 q^{95} +15.0585 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 4 q^{3} + 2 q^{5} - 2 q^{7} + 4 q^{9}+O(q^{10}) \) 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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more