LMFDB - Embedded newform 6864.2.a.ce.1.4

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:	$5$
Coefficient field:	5.5.46437524.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 20x^{3} + 8x^{2} + 70x + 28 $ x^5 - x^4 - 20x^3 + 8x^2 + 70*x + 28
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 3432)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$2.49314$ of defining polynomial
Character	$\chi$	$=$	6864.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{3} +2.49314 q^{5} +2.46927 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} +2.49314 q^{5} +2.46927 q^{7} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} -2.49314 q^{15} +2.02386 q^{17} +6.62551 q^{19} -2.46927 q^{21} -0.469274 q^{23} +1.21573 q^{25} -1.00000 q^{27} +10.4336 q^{29} -3.66310 q^{31} +1.00000 q^{33} +6.15623 q^{35} +4.93855 q^{37} -1.00000 q^{39} +4.46927 q^{41} -3.19383 q^{43} +2.49314 q^{45} -4.25355 q^{47} -0.902687 q^{49} -2.02386 q^{51} +0.685000 q^{53} -2.49314 q^{55} -6.62551 q^{57} -10.8412 q^{59} -0.830037 q^{61} +2.46927 q^{63} +2.49314 q^{65} +0.493136 q^{67} +0.469274 q^{69} -8.01120 q^{71} +5.78427 q^{73} -1.21573 q^{75} -2.46927 q^{77} +15.9599 q^{79} +1.00000 q^{81} +4.98627 q^{83} +5.04576 q^{85} -10.4336 q^{87} +6.26172 q^{89} +2.46927 q^{91} +3.66310 q^{93} +16.5183 q^{95} +1.69873 q^{97} -1.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 5 q^{3} + q^{5} + 3 q^{7} + 5 q^{9}+O(q^{10}) $ 5 * q - 5 * q^3 + q^5 + 3 * q^7 + 5 * q^9	$ 5 q - 5 q^{3} + q^{5} + 3 q^{7} + 5 q^{9} - 5 q^{11} + 5 q^{13} - q^{15} + 8 q^{17} - 10 q^{19} - 3 q^{21} + 7 q^{23} + 16 q^{25} - 5 q^{27} - 3 q^{29} + 4 q^{31} + 5 q^{33} - 3 q^{35} + 6 q^{37} - 5 q^{39} + 13 q^{41} - 3 q^{43} + q^{45} - 2 q^{47} + 10 q^{49} - 8 q^{51} + 4 q^{53} - q^{55} + 10 q^{57} - 21 q^{59} - 15 q^{61} + 3 q^{63} + q^{65} - 9 q^{67} - 7 q^{69} + 4 q^{71} + 19 q^{73} - 16 q^{75} - 3 q^{77} - 8 q^{79} + 5 q^{81} + 2 q^{83} + 46 q^{85} + 3 q^{87} + 12 q^{89} + 3 q^{91} - 4 q^{93} + 32 q^{97} - 5 q^{99}+O(q^{100}) $ 5 * q - 5 * q^3 + q^5 + 3 * q^7 + 5 * q^9 - 5 * q^11 + 5 * q^13 - q^15 + 8 * q^17 - 10 * q^19 - 3 * q^21 + 7 * q^23 + 16 * q^25 - 5 * q^27 - 3 * q^29 + 4 * q^31 + 5 * q^33 - 3 * q^35 + 6 * q^37 - 5 * q^39 + 13 * q^41 - 3 * q^43 + q^45 - 2 * q^47 + 10 * q^49 - 8 * q^51 + 4 * q^53 - q^55 + 10 * q^57 - 21 * q^59 - 15 * q^61 + 3 * q^63 + q^65 - 9 * q^67 - 7 * q^69 + 4 * q^71 + 19 * q^73 - 16 * q^75 - 3 * q^77 - 8 * q^79 + 5 * q^81 + 2 * q^83 + 46 * q^85 + 3 * q^87 + 12 * q^89 + 3 * q^91 - 4 * q^93 + 32 * q^97 - 5 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	0	0
$5$	2.49314	1.11496	0.557482	−	0.830189i	$-0.311768\pi$
$5$	2.49314	1.11496	0.557482	+	0.830189i	$0.311768\pi$
$6$	0	0
$7$	2.46927	0.933298	0.466649	−	0.884443i	$-0.345461\pi$
$7$	2.46927	0.933298	0.466649	+	0.884443i	$0.345461\pi$
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	−2.49314	−0.643725
$16$	0	0
$17$	2.02386	0.490859	0.245429	−	0.969414i	$-0.421071\pi$
$17$	2.02386	0.490859	0.245429	+	0.969414i	$0.421071\pi$
$18$	0	0
$19$	6.62551	1.52000	0.759998	−	0.649925i	$-0.225200\pi$
$19$	6.62551	1.52000	0.759998	+	0.649925i	$0.225200\pi$
$20$	0	0
$21$	−2.46927	−0.538840
$22$	0	0
$23$	−0.469274	−0.0978503	−0.0489252	−	0.998802i	$-0.515580\pi$
$23$	−0.469274	−0.0978503	−0.0489252	+	0.998802i	$0.515580\pi$
$24$	0	0
$25$	1.21573	0.243145
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	10.4336	1.93748	0.968740	−	0.248080i	$-0.0797996\pi$
$29$	10.4336	1.93748	0.968740	+	0.248080i	$0.0797996\pi$
$30$	0	0
$31$	−3.66310	−0.657912	−0.328956	−	0.944345i	$-0.606697\pi$
$31$	−3.66310	−0.657912	−0.328956	+	0.944345i	$0.606697\pi$
$32$	0	0
$33$	1.00000	0.174078
$34$	0	0
$35$	6.15623	1.04059
$36$	0	0
$37$	4.93855	0.811892	0.405946	−	0.913897i	$-0.366942\pi$
$37$	4.93855	0.811892	0.405946	+	0.913897i	$0.366942\pi$
$38$	0	0
$39$	−1.00000	−0.160128
$40$	0	0
$41$	4.46927	0.697983	0.348992	−	0.937126i	$-0.386524\pi$
$41$	4.46927	0.697983	0.348992	+	0.937126i	$0.386524\pi$
$42$	0	0
$43$	−3.19383	−0.487054	−0.243527	−	0.969894i	$-0.578304\pi$
$43$	−3.19383	−0.487054	−0.243527	+	0.969894i	$0.578304\pi$
$44$	0	0
$45$	2.49314	0.371655
$46$	0	0
$47$	−4.25355	−0.620444	−0.310222	−	0.950664i	$-0.600403\pi$
$47$	−4.25355	−0.620444	−0.310222	+	0.950664i	$0.600403\pi$
$48$	0	0
$49$	−0.902687	−0.128955
$50$	0	0
$51$	−2.02386	−0.283397
$52$	0	0
$53$	0.685000	0.0940920	0.0470460	−	0.998893i	$-0.485019\pi$
$53$	0.685000	0.0940920	0.0470460	+	0.998893i	$0.485019\pi$
$54$	0	0
$55$	−2.49314	−0.336174
$56$	0	0
$57$	−6.62551	−0.877570
$58$	0	0
$59$	−10.8412	−1.41141	−0.705704	−	0.708507i	$-0.749369\pi$
$59$	−10.8412	−1.41141	−0.705704	+	0.708507i	$0.749369\pi$
$60$	0	0
$61$	−0.830037	−0.106275	−0.0531377	−	0.998587i	$-0.516922\pi$
$61$	−0.830037	−0.106275	−0.0531377	+	0.998587i	$0.516922\pi$
$62$	0	0
$63$	2.46927	0.311099
$64$	0	0
$65$	2.49314	0.309235
$66$	0	0
$67$	0.493136	0.0602461	0.0301231	−	0.999546i	$-0.490410\pi$
$67$	0.493136	0.0602461	0.0301231	+	0.999546i	$0.490410\pi$
$68$	0	0
$69$	0.469274	0.0564939
$70$	0	0
$71$	−8.01120	−0.950754	−0.475377	−	0.879782i	$-0.657688\pi$
$71$	−8.01120	−0.950754	−0.475377	+	0.879782i	$0.657688\pi$
$72$	0	0
$73$	5.78427	0.676998	0.338499	−	0.940967i	$-0.390081\pi$
$73$	5.78427	0.676998	0.338499	+	0.940967i	$0.390081\pi$
$74$	0	0
$75$	−1.21573	−0.140380
$76$	0	0
$77$	−2.46927	−0.281400
$78$	0	0
$79$	15.9599	1.79563	0.897813	−	0.440376i	$-0.145155\pi$
$79$	15.9599	1.79563	0.897813	+	0.440376i	$0.145155\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	4.98627	0.547314	0.273657	−	0.961827i	$-0.411767\pi$
$83$	4.98627	0.547314	0.273657	+	0.961827i	$0.411767\pi$
$84$	0	0
$85$	5.04576	0.547290
$86$	0	0
$87$	−10.4336	−1.11860
$88$	0	0
$89$	6.26172	0.663741	0.331871	−	0.943325i	$-0.392320\pi$
$89$	6.26172	0.663741	0.331871	+	0.943325i	$0.392320\pi$
$90$	0	0
$91$	2.46927	0.258850
$92$	0	0
$93$	3.66310	0.379846
$94$	0	0
$95$	16.5183	1.69474
$96$	0	0
$97$	1.69873	0.172480	0.0862399	−	0.996274i	$-0.472515\pi$
$97$	1.69873	0.172480	0.0862399	+	0.996274i	$0.472515\pi$
$98$	0	0
$99$	−1.00000	−0.100504
$100$	0	0
$101$	5.64741	0.561938	0.280969	−	0.959717i	$-0.409344\pi$
$101$	5.64741	0.561938	0.280969	+	0.959717i	$0.409344\pi$
$102$	0	0
$103$	1.85496	0.182775	0.0913875	−	0.995815i	$-0.470870\pi$
$103$	1.85496	0.182775	0.0913875	+	0.995815i	$0.470870\pi$
$104$	0	0
$105$	−6.15623	−0.600787
$106$	0	0
$107$	−8.10851	−0.783879	−0.391940	−	0.919991i	$-0.628196\pi$
$107$	−8.10851	−0.783879	−0.391940	+	0.919991i	$0.628196\pi$
$108$	0	0
$109$	9.94051	0.952128	0.476064	−	0.879411i	$-0.342063\pi$
$109$	9.94051	0.952128	0.476064	+	0.879411i	$0.342063\pi$
$110$	0	0
$111$	−4.93855	−0.468746
$112$	0	0
$113$	−0.841235	−0.0791367	−0.0395684	−	0.999217i	$-0.512598\pi$
$113$	−0.841235	−0.0791367	−0.0395684	+	0.999217i	$0.512598\pi$
$114$	0	0
$115$	−1.16996	−0.109100
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	4.99747	0.458117
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−4.46927	−0.402981
$124$	0	0
$125$	−9.43471	−0.843866
$126$	0	0
$127$	1.08531	0.0963061	0.0481531	−	0.998840i	$-0.484666\pi$
$127$	1.08531	0.0963061	0.0481531	+	0.998840i	$0.484666\pi$
$128$	0	0
$129$	3.19383	0.281201
$130$	0	0
$131$	14.1288	1.23444	0.617219	−	0.786792i	$-0.288259\pi$
$131$	14.1288	1.23444	0.617219	+	0.786792i	$0.288259\pi$
$132$	0	0
$133$	16.3602	1.41861
$134$	0	0
$135$	−2.49314	−0.214575
$136$	0	0
$137$	12.6016	1.07663	0.538316	−	0.842743i	$-0.319061\pi$
$137$	12.6016	1.07663	0.538316	+	0.842743i	$0.319061\pi$
$138$	0	0
$139$	10.5310	0.893224	0.446612	−	0.894728i	$-0.352630\pi$
$139$	10.5310	0.893224	0.446612	+	0.894728i	$0.352630\pi$
$140$	0	0
$141$	4.25355	0.358213
$142$	0	0
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	26.0125	2.16022
$146$	0	0
$147$	0.902687	0.0744524
$148$	0	0
$149$	−21.2353	−1.73967	−0.869833	−	0.493346i	$-0.835773\pi$
$149$	−21.2353	−1.73967	−0.869833	+	0.493346i	$0.835773\pi$
$150$	0	0
$151$	−8.32424	−0.677417	−0.338708	−	0.940891i	$-0.609990\pi$
$151$	−8.32424	−0.677417	−0.338708	+	0.940891i	$0.609990\pi$
$152$	0	0
$153$	2.02386	0.163620
$154$	0	0
$155$	−9.13260	−0.733548
$156$	0	0
$157$	−8.55033	−0.682390	−0.341195	−	0.939992i	$-0.610832\pi$
$157$	−8.55033	−0.682390	−0.341195	+	0.939992i	$0.610832\pi$
$158$	0	0
$159$	−0.685000	−0.0543240
$160$	0	0
$161$	−1.15877	−0.0913235
$162$	0	0
$163$	−16.7467	−1.31170	−0.655851	−	0.754891i	$-0.727690\pi$
$163$	−16.7467	−1.31170	−0.655851	+	0.754891i	$0.727690\pi$
$164$	0	0
$165$	2.49314	0.194090
$166$	0	0
$167$	−15.1060	−1.16894	−0.584468	−	0.811417i	$-0.698697\pi$
$167$	−15.1060	−1.16894	−0.584468	+	0.811417i	$0.698697\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	6.62551	0.506665
$172$	0	0
$173$	−6.80617	−0.517464	−0.258732	−	0.965949i	$-0.583305\pi$
$173$	−6.80617	−0.517464	−0.258732	+	0.965949i	$0.583305\pi$
$174$	0	0
$175$	3.00196	0.226927
$176$	0	0
$177$	10.8412	0.814877
$178$	0	0
$179$	−14.5503	−1.08754	−0.543771	−	0.839233i	$-0.683004\pi$
$179$	−14.5503	−1.08754	−0.543771	+	0.839233i	$0.683004\pi$
$180$	0	0
$181$	−25.8013	−1.91780	−0.958899	−	0.283746i	$-0.908423\pi$
$181$	−25.8013	−1.91780	−0.958899	+	0.283746i	$0.908423\pi$
$182$	0	0
$183$	0.830037	0.0613581
$184$	0	0
$185$	12.3125	0.905231
$186$	0	0
$187$	−2.02386	−0.147999
$188$	0	0
$189$	−2.46927	−0.179613
$190$	0	0
$191$	−4.20396	−0.304188	−0.152094	−	0.988366i	$-0.548602\pi$
$191$	−4.20396	−0.304188	−0.152094	+	0.988366i	$0.548602\pi$
$192$	0	0
$193$	−1.61178	−0.116018	−0.0580092	−	0.998316i	$-0.518475\pi$
$193$	−1.61178	−0.116018	−0.0580092	+	0.998316i	$0.518475\pi$
$194$	0	0
$195$	−2.49314	−0.178537
$196$	0	0
$197$	−23.9131	−1.70373	−0.851867	−	0.523758i	$-0.824530\pi$
$197$	−23.9131	−1.70373	−0.851867	+	0.523758i	$0.824530\pi$
$198$	0	0
$199$	−7.82751	−0.554877	−0.277439	−	0.960743i	$-0.589486\pi$
$199$	−7.82751	−0.554877	−0.277439	+	0.960743i	$0.589486\pi$
$200$	0	0
$201$	−0.493136	−0.0347831
$202$	0	0
$203$	25.7635	1.80824
$204$	0	0
$205$	11.1425	0.778226
$206$	0	0
$207$	−0.469274	−0.0326168
$208$	0	0
$209$	−6.62551	−0.458296
$210$	0	0
$211$	−2.95121	−0.203170	−0.101585	−	0.994827i	$-0.532391\pi$
$211$	−2.95121	−0.203170	−0.101585	+	0.994827i	$0.532391\pi$
$212$	0	0
$213$	8.01120	0.548918
$214$	0	0
$215$	−7.96264	−0.543048
$216$	0	0
$217$	−9.04519	−0.614028
$218$	0	0
$219$	−5.78427	−0.390865
$220$	0	0
$221$	2.02386	0.136140
$222$	0	0
$223$	16.1702	1.08284	0.541418	−	0.840753i	$-0.317888\pi$
$223$	16.1702	1.08284	0.541418	+	0.840753i	$0.317888\pi$
$224$	0	0
$225$	1.21573	0.0810484
$226$	0	0
$227$	−3.04969	−0.202415	−0.101207	−	0.994865i	$-0.532271\pi$
$227$	−3.04969	−0.202415	−0.101207	+	0.994865i	$0.532271\pi$
$228$	0	0
$229$	6.74579	0.445774	0.222887	−	0.974844i	$-0.428452\pi$
$229$	6.74579	0.445774	0.222887	+	0.974844i	$0.428452\pi$
$230$	0	0
$231$	2.46927	0.162466
$232$	0	0
$233$	−5.98734	−0.392243	−0.196122	−	0.980580i	$-0.562835\pi$
$233$	−5.98734	−0.392243	−0.196122	+	0.980580i	$0.562835\pi$
$234$	0	0
$235$	−10.6047	−0.691772
$236$	0	0
$237$	−15.9599	−1.03671
$238$	0	0
$239$	28.6588	1.85379	0.926893	−	0.375326i	$-0.122469\pi$
$239$	28.6588	1.85379	0.926893	+	0.375326i	$0.122469\pi$
$240$	0	0
$241$	7.31500	0.471201	0.235600	−	0.971850i	$-0.424294\pi$
$241$	7.31500	0.471201	0.235600	+	0.971850i	$0.424294\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−2.25052	−0.143781
$246$	0	0
$247$	6.62551	0.421571
$248$	0	0
$249$	−4.98627	−0.315992
$250$	0	0
$251$	3.10918	0.196250	0.0981248	−	0.995174i	$-0.468716\pi$
$251$	3.10918	0.196250	0.0981248	+	0.995174i	$0.468716\pi$
$252$	0	0
$253$	0.469274	0.0295030
$254$	0	0
$255$	−5.04576	−0.315978
$256$	0	0
$257$	14.7661	0.921081	0.460541	−	0.887639i	$-0.347656\pi$
$257$	14.7661	0.921081	0.460541	+	0.887639i	$0.347656\pi$
$258$	0	0
$259$	12.1946	0.757737
$260$	0	0
$261$	10.4336	0.645826
$262$	0	0
$263$	29.8058	1.83791	0.918953	−	0.394366i	$-0.129036\pi$
$263$	29.8058	1.83791	0.918953	+	0.394366i	$0.129036\pi$
$264$	0	0
$265$	1.70780	0.104909
$266$	0	0
$267$	−6.26172	−0.383211
$268$	0	0
$269$	15.1204	0.921906	0.460953	−	0.887425i	$-0.347508\pi$
$269$	15.1204	0.921906	0.460953	+	0.887425i	$0.347508\pi$
$270$	0	0
$271$	15.6504	0.950696	0.475348	−	0.879798i	$-0.342322\pi$
$271$	15.6504	0.950696	0.475348	+	0.879798i	$0.342322\pi$
$272$	0	0
$273$	−2.46927	−0.149447
$274$	0	0
$275$	−1.21573	−0.0733111
$276$	0	0
$277$	−17.1060	−1.02780	−0.513899	−	0.857850i	$-0.671800\pi$
$277$	−17.1060	−1.02780	−0.513899	+	0.857850i	$0.671800\pi$
$278$	0	0
$279$	−3.66310	−0.219304
$280$	0	0
$281$	21.4779	1.28127	0.640633	−	0.767847i	$-0.278672\pi$
$281$	21.4779	1.28127	0.640633	+	0.767847i	$0.278672\pi$
$282$	0	0
$283$	10.4297	0.619983	0.309991	−	0.950739i	$-0.399674\pi$
$283$	10.4297	0.619983	0.309991	+	0.950739i	$0.399674\pi$
$284$	0	0
$285$	−16.5183	−0.978459
$286$	0	0
$287$	11.0359	0.651426
$288$	0	0
$289$	−12.9040	−0.759058
$290$	0	0
$291$	−1.69873	−0.0995812
$292$	0	0
$293$	−17.6504	−1.03115	−0.515575	−	0.856845i	$-0.672422\pi$
$293$	−17.6504	−1.03115	−0.515575	+	0.856845i	$0.672422\pi$
$294$	0	0
$295$	−27.0287	−1.57367
$296$	0	0
$297$	1.00000	0.0580259
$298$	0	0
$299$	−0.469274	−0.0271388
$300$	0	0
$301$	−7.88643	−0.454566
$302$	0	0
$303$	−5.64741	−0.324435
$304$	0	0
$305$	−2.06939	−0.118493
$306$	0	0
$307$	−14.5301	−0.829274	−0.414637	−	0.909987i	$-0.636091\pi$
$307$	−14.5301	−0.829274	−0.414637	+	0.909987i	$0.636091\pi$
$308$	0	0
$309$	−1.85496	−0.105525
$310$	0	0
$311$	−9.33078	−0.529100	−0.264550	−	0.964372i	$-0.585223\pi$
$311$	−9.33078	−0.529100	−0.264550	+	0.964372i	$0.585223\pi$
$312$	0	0
$313$	10.1719	0.574951	0.287476	−	0.957788i	$-0.407184\pi$
$313$	10.1719	0.574951	0.287476	+	0.957788i	$0.407184\pi$
$314$	0	0
$315$	6.15623	0.346865
$316$	0	0
$317$	−6.93579	−0.389553	−0.194776	−	0.980848i	$-0.562398\pi$
$317$	−6.93579	−0.389553	−0.194776	+	0.980848i	$0.562398\pi$
$318$	0	0
$319$	−10.4336	−0.584172
$320$	0	0
$321$	8.10851	0.452573
$322$	0	0
$323$	13.4091	0.746103
$324$	0	0
$325$	1.21573	0.0674364
$326$	0	0
$327$	−9.94051	−0.549711
$328$	0	0
$329$	−10.5032	−0.579059
$330$	0	0
$331$	0.191864	0.0105458	0.00527290	−	0.999986i	$-0.498322\pi$
$331$	0.191864	0.0105458	0.00527290	+	0.999986i	$0.498322\pi$
$332$	0	0
$333$	4.93855	0.270631
$334$	0	0
$335$	1.22945	0.0671723
$336$	0	0
$337$	18.6962	1.01845	0.509223	−	0.860634i	$-0.329933\pi$
$337$	18.6962	1.01845	0.509223	+	0.860634i	$0.329933\pi$
$338$	0	0
$339$	0.841235	0.0456896
$340$	0	0
$341$	3.66310	0.198368
$342$	0	0
$343$	−19.5139	−1.05365
$344$	0	0
$345$	1.16996	0.0629887
$346$	0	0
$347$	24.1896	1.29856	0.649282	−	0.760548i	$-0.275070\pi$
$347$	24.1896	1.29856	0.649282	+	0.760548i	$0.275070\pi$
$348$	0	0
$349$	19.6347	1.05102	0.525512	−	0.850786i	$-0.323874\pi$
$349$	19.6347	1.05102	0.525512	+	0.850786i	$0.323874\pi$
$350$	0	0
$351$	−1.00000	−0.0533761
$352$	0	0
$353$	10.8990	0.580095	0.290048	−	0.957012i	$-0.406329\pi$
$353$	10.8990	0.580095	0.290048	+	0.957012i	$0.406329\pi$
$354$	0	0
$355$	−19.9730	−1.06006
$356$	0	0
$357$	−4.99747	−0.264494
$358$	0	0
$359$	−4.75429	−0.250922	−0.125461	−	0.992099i	$-0.540041\pi$
$359$	−4.75429	−0.250922	−0.125461	+	0.992099i	$0.540041\pi$
$360$	0	0
$361$	24.8974	1.31039
$362$	0	0
$363$	−1.00000	−0.0524864
$364$	0	0
$365$	14.4210	0.754829
$366$	0	0
$367$	−21.7745	−1.13662	−0.568309	−	0.822815i	$-0.692402\pi$
$367$	−21.7745	−1.13662	−0.568309	+	0.822815i	$0.692402\pi$
$368$	0	0
$369$	4.46927	0.232661
$370$	0	0
$371$	1.69145	0.0878158
$372$	0	0
$373$	18.8138	0.974141	0.487070	−	0.873363i	$-0.338066\pi$
$373$	18.8138	0.974141	0.487070	+	0.873363i	$0.338066\pi$
$374$	0	0
$375$	9.43471	0.487206
$376$	0	0
$377$	10.4336	0.537360
$378$	0	0
$379$	12.4787	0.640990	0.320495	−	0.947250i	$-0.396151\pi$
$379$	12.4787	0.640990	0.320495	+	0.947250i	$0.396151\pi$
$380$	0	0
$381$	−1.08531	−0.0556024
$382$	0	0
$383$	−6.29220	−0.321516	−0.160758	−	0.986994i	$-0.551394\pi$
$383$	−6.29220	−0.321516	−0.160758	+	0.986994i	$0.551394\pi$
$384$	0	0
$385$	−6.15623	−0.313751
$386$	0	0
$387$	−3.19383	−0.162351
$388$	0	0
$389$	−3.28362	−0.166486	−0.0832431	−	0.996529i	$-0.526528\pi$
$389$	−3.28362	−0.166486	−0.0832431	+	0.996529i	$0.526528\pi$
$390$	0	0
$391$	−0.949745	−0.0480307
$392$	0	0
$393$	−14.1288	−0.712703
$394$	0	0
$395$	39.7901	2.00206
$396$	0	0
$397$	26.7386	1.34197	0.670986	−	0.741470i	$-0.265871\pi$
$397$	26.7386	1.34197	0.670986	+	0.741470i	$0.265871\pi$
$398$	0	0
$399$	−16.3602	−0.819034
$400$	0	0
$401$	6.71335	0.335249	0.167624	−	0.985851i	$-0.446390\pi$
$401$	6.71335	0.335249	0.167624	+	0.985851i	$0.446390\pi$
$402$	0	0
$403$	−3.66310	−0.182472
$404$	0	0
$405$	2.49314	0.123885
$406$	0	0
$407$	−4.93855	−0.244795
$408$	0	0
$409$	22.0967	1.09261	0.546307	−	0.837585i	$-0.316033\pi$
$409$	22.0967	1.09261	0.546307	+	0.837585i	$0.316033\pi$
$410$	0	0
$411$	−12.6016	−0.621593
$412$	0	0
$413$	−26.7700	−1.31726
$414$	0	0
$415$	12.4315	0.610236
$416$	0	0
$417$	−10.5310	−0.515703
$418$	0	0
$419$	−13.9478	−0.681394	−0.340697	−	0.940173i	$-0.610663\pi$
$419$	−13.9478	−0.681394	−0.340697	+	0.940173i	$0.610663\pi$
$420$	0	0
$421$	3.81631	0.185996	0.0929978	−	0.995666i	$-0.470355\pi$
$421$	3.81631	0.185996	0.0929978	+	0.995666i	$0.470355\pi$
$422$	0	0
$423$	−4.25355	−0.206815
$424$	0	0
$425$	2.46046	0.119350
$426$	0	0
$427$	−2.04959	−0.0991865
$428$	0	0
$429$	1.00000	0.0482805
$430$	0	0
$431$	−25.8536	−1.24532	−0.622661	−	0.782492i	$-0.713948\pi$
$431$	−25.8536	−1.24532	−0.622661	+	0.782492i	$0.713948\pi$
$432$	0	0
$433$	15.6092	0.750128	0.375064	−	0.926999i	$-0.377621\pi$
$433$	15.6092	0.750128	0.375064	+	0.926999i	$0.377621\pi$
$434$	0	0
$435$	−26.0125	−1.24720
$436$	0	0
$437$	−3.10918	−0.148732
$438$	0	0
$439$	11.2194	0.535474	0.267737	−	0.963492i	$-0.413724\pi$
$439$	11.2194	0.535474	0.267737	+	0.963492i	$0.413724\pi$
$440$	0	0
$441$	−0.902687	−0.0429851
$442$	0	0
$443$	24.6025	1.16890	0.584451	−	0.811429i	$-0.301310\pi$
$443$	24.6025	1.16890	0.584451	+	0.811429i	$0.301310\pi$
$444$	0	0
$445$	15.6113	0.740047
$446$	0	0
$447$	21.2353	1.00440
$448$	0	0
$449$	−29.1199	−1.37425	−0.687127	−	0.726537i	$-0.741129\pi$
$449$	−29.1199	−1.37425	−0.687127	+	0.726537i	$0.741129\pi$
$450$	0	0
$451$	−4.46927	−0.210450
$452$	0	0
$453$	8.32424	0.391107
$454$	0	0
$455$	6.15623	0.288609
$456$	0	0
$457$	−20.1288	−0.941585	−0.470792	−	0.882244i	$-0.656032\pi$
$457$	−20.1288	−0.941585	−0.470792	+	0.882244i	$0.656032\pi$
$458$	0	0
$459$	−2.02386	−0.0944658
$460$	0	0
$461$	−23.9608	−1.11596	−0.557982	−	0.829853i	$-0.688424\pi$
$461$	−23.9608	−1.11596	−0.557982	+	0.829853i	$0.688424\pi$
$462$	0	0
$463$	30.2017	1.40359	0.701795	−	0.712379i	$-0.252382\pi$
$463$	30.2017	1.40359	0.701795	+	0.712379i	$0.252382\pi$
$464$	0	0
$465$	9.13260	0.423514
$466$	0	0
$467$	11.2555	0.520843	0.260421	−	0.965495i	$-0.416139\pi$
$467$	11.2555	0.520843	0.260421	+	0.965495i	$0.416139\pi$
$468$	0	0
$469$	1.21769	0.0562276
$470$	0	0
$471$	8.55033	0.393978
$472$	0	0
$473$	3.19383	0.146852
$474$	0	0
$475$	8.05481	0.369580
$476$	0	0
$477$	0.685000	0.0313640
$478$	0	0
$479$	43.5699	1.99076	0.995380	−	0.0960094i	$-0.0306079\pi$
$479$	43.5699	1.99076	0.995380	+	0.0960094i	$0.0306079\pi$
$480$	0	0
$481$	4.93855	0.225178
$482$	0	0
$483$	1.15877	0.0527256
$484$	0	0
$485$	4.23516	0.192309
$486$	0	0
$487$	14.3997	0.652515	0.326257	−	0.945281i	$-0.394212\pi$
$487$	14.3997	0.652515	0.326257	+	0.945281i	$0.394212\pi$
$488$	0	0
$489$	16.7467	0.757311
$490$	0	0
$491$	−4.02605	−0.181693	−0.0908466	−	0.995865i	$-0.528957\pi$
$491$	−4.02605	−0.181693	−0.0908466	+	0.995865i	$0.528957\pi$
$492$	0	0
$493$	21.1163	0.951028
$494$	0	0
$495$	−2.49314	−0.112058
$496$	0	0
$497$	−19.7818	−0.887337
$498$	0	0
$499$	28.1795	1.26149	0.630744	−	0.775991i	$-0.282750\pi$
$499$	28.1795	1.26149	0.630744	+	0.775991i	$0.282750\pi$
$500$	0	0
$501$	15.1060	0.674885
$502$	0	0
$503$	1.35365	0.0603565	0.0301782	−	0.999545i	$-0.490393\pi$
$503$	1.35365	0.0603565	0.0301782	+	0.999545i	$0.490393\pi$
$504$	0	0
$505$	14.0798	0.626541
$506$	0	0
$507$	−1.00000	−0.0444116
$508$	0	0
$509$	−38.5193	−1.70734	−0.853668	−	0.520817i	$-0.825627\pi$
$509$	−38.5193	−1.70734	−0.853668	+	0.520817i	$0.825627\pi$
$510$	0	0
$511$	14.2830	0.631841
$512$	0	0
$513$	−6.62551	−0.292523
$514$	0	0
$515$	4.62468	0.203788
$516$	0	0
$517$	4.25355	0.187071
$518$	0	0
$519$	6.80617	0.298758
$520$	0	0
$521$	4.65213	0.203814	0.101907	−	0.994794i	$-0.467506\pi$
$521$	4.65213	0.203814	0.101907	+	0.994794i	$0.467506\pi$
$522$	0	0
$523$	−15.5762	−0.681098	−0.340549	−	0.940227i	$-0.610613\pi$
$523$	−15.5762	−0.681098	−0.340549	+	0.940227i	$0.610613\pi$
$524$	0	0
$525$	−3.00196	−0.131016
$526$	0	0
$527$	−7.41361	−0.322942
$528$	0	0
$529$	−22.7798	−0.990425
$530$	0	0
$531$	−10.8412	−0.470470
$532$	0	0
$533$	4.46927	0.193586
$534$	0	0
$535$	−20.2156	−0.873997
$536$	0	0
$537$	14.5503	0.627893
$538$	0	0
$539$	0.902687	0.0388815
$540$	0	0
$541$	23.8771	1.02656	0.513278	−	0.858222i	$-0.328431\pi$
$541$	23.8771	1.02656	0.513278	+	0.858222i	$0.328431\pi$
$542$	0	0
$543$	25.8013	1.10724
$544$	0	0
$545$	24.7830	1.06159
$546$	0	0
$547$	23.7762	1.01660	0.508298	−	0.861181i	$-0.330275\pi$
$547$	23.7762	1.01660	0.508298	+	0.861181i	$0.330275\pi$
$548$	0	0
$549$	−0.830037	−0.0354251
$550$	0	0
$551$	69.1282	2.94496
$552$	0	0
$553$	39.4093	1.67585
$554$	0	0
$555$	−12.3125	−0.522635
$556$	0	0
$557$	22.9268	0.971439	0.485719	−	0.874115i	$-0.338558\pi$
$557$	22.9268	0.971439	0.485719	+	0.874115i	$0.338558\pi$
$558$	0	0
$559$	−3.19383	−0.135084
$560$	0	0
$561$	2.02386	0.0854475
$562$	0	0
$563$	0.0386545	0.00162909	0.000814547	−	1.00000i	$-0.499741\pi$
$563$	0.0386545	0.00162909	0.000814547		1.00000i	$0.499741\pi$
$564$	0	0
$565$	−2.09731	−0.0882346
$566$	0	0
$567$	2.46927	0.103700
$568$	0	0
$569$	−24.8709	−1.04264	−0.521321	−	0.853361i	$-0.674561\pi$
$569$	−24.8709	−1.04264	−0.521321	+	0.853361i	$0.674561\pi$
$570$	0	0
$571$	18.6120	0.778888	0.389444	−	0.921050i	$-0.372667\pi$
$571$	18.6120	0.778888	0.389444	+	0.921050i	$0.372667\pi$
$572$	0	0
$573$	4.20396	0.175623
$574$	0	0
$575$	−0.570508	−0.0237918
$576$	0	0
$577$	18.9385	0.788422	0.394211	−	0.919020i	$-0.371018\pi$
$577$	18.9385	0.788422	0.394211	+	0.919020i	$0.371018\pi$
$578$	0	0
$579$	1.61178	0.0669833
$580$	0	0
$581$	12.3125	0.510807
$582$	0	0
$583$	−0.685000	−0.0283698
$584$	0	0
$585$	2.49314	0.103078
$586$	0	0
$587$	−47.6608	−1.96717	−0.983586	−	0.180441i	$-0.942248\pi$
$587$	−47.6608	−1.96717	−0.983586	+	0.180441i	$0.942248\pi$
$588$	0	0
$589$	−24.2699	−1.00002
$590$	0	0
$591$	23.9131	0.983652
$592$	0	0
$593$	−8.36804	−0.343634	−0.171817	−	0.985129i	$-0.554964\pi$
$593$	−8.36804	−0.343634	−0.171817	+	0.985129i	$0.554964\pi$
$594$	0	0
$595$	12.4594	0.510784
$596$	0	0
$597$	7.82751	0.320359
$598$	0	0
$599$	10.3269	0.421944	0.210972	−	0.977492i	$-0.432337\pi$
$599$	10.3269	0.421944	0.210972	+	0.977492i	$0.432337\pi$
$600$	0	0
$601$	−16.1896	−0.660386	−0.330193	−	0.943913i	$-0.607114\pi$
$601$	−16.1896	−0.660386	−0.330193	+	0.943913i	$0.607114\pi$
$602$	0	0
$603$	0.493136	0.0200820
$604$	0	0
$605$	2.49314	0.101360
$606$	0	0
$607$	−39.5396	−1.60486	−0.802432	−	0.596744i	$-0.796461\pi$
$607$	−39.5396	−1.60486	−0.802432	+	0.596744i	$0.796461\pi$
$608$	0	0
$609$	−25.7635	−1.04399
$610$	0	0
$611$	−4.25355	−0.172080
$612$	0	0
$613$	12.0320	0.485969	0.242985	−	0.970030i	$-0.421874\pi$
$613$	12.0320	0.485969	0.242985	+	0.970030i	$0.421874\pi$
$614$	0	0
$615$	−11.1425	−0.449309
$616$	0	0
$617$	−21.5055	−0.865777	−0.432889	−	0.901447i	$-0.642506\pi$
$617$	−21.5055	−0.865777	−0.432889	+	0.901447i	$0.642506\pi$
$618$	0	0
$619$	16.0252	0.644105	0.322053	−	0.946722i	$-0.395627\pi$
$619$	16.0252	0.644105	0.322053	+	0.946722i	$0.395627\pi$
$620$	0	0
$621$	0.469274	0.0188313
$622$	0	0
$623$	15.4619	0.619468
$624$	0	0
$625$	−29.6006	−1.18403
$626$	0	0
$627$	6.62551	0.264597
$628$	0	0
$629$	9.99494	0.398524
$630$	0	0
$631$	16.8317	0.670058	0.335029	−	0.942208i	$-0.391254\pi$
$631$	16.8317	0.670058	0.335029	+	0.942208i	$0.391254\pi$
$632$	0	0
$633$	2.95121	0.117300
$634$	0	0
$635$	2.70584	0.107378
$636$	0	0
$637$	−0.902687	−0.0357658
$638$	0	0
$639$	−8.01120	−0.316918
$640$	0	0
$641$	−20.9332	−0.826813	−0.413406	−	0.910547i	$-0.635661\pi$
$641$	−20.9332	−0.826813	−0.413406	+	0.910547i	$0.635661\pi$
$642$	0	0
$643$	31.4010	1.23834	0.619168	−	0.785259i	$-0.287470\pi$
$643$	31.4010	1.23834	0.619168	+	0.785259i	$0.287470\pi$
$644$	0	0
$645$	7.96264	0.313529
$646$	0	0
$647$	−21.3060	−0.837626	−0.418813	−	0.908073i	$-0.637554\pi$
$647$	−21.3060	−0.837626	−0.418813	+	0.908073i	$0.637554\pi$
$648$	0	0
$649$	10.8412	0.425556
$650$	0	0
$651$	9.04519	0.354509
$652$	0	0
$653$	−4.08425	−0.159829	−0.0799145	−	0.996802i	$-0.525465\pi$
$653$	−4.08425	−0.159829	−0.0799145	+	0.996802i	$0.525465\pi$
$654$	0	0
$655$	35.2250	1.37635
$656$	0	0
$657$	5.78427	0.225666
$658$	0	0
$659$	−10.2647	−0.399858	−0.199929	−	0.979810i	$-0.564071\pi$
$659$	−10.2647	−0.399858	−0.199929	+	0.979810i	$0.564071\pi$
$660$	0	0
$661$	25.7946	1.00330	0.501648	−	0.865072i	$-0.332727\pi$
$661$	25.7946	1.00330	0.501648	+	0.865072i	$0.332727\pi$
$662$	0	0
$663$	−2.02386	−0.0786003
$664$	0	0
$665$	40.7882	1.58170
$666$	0	0
$667$	−4.89623	−0.189583
$668$	0	0
$669$	−16.1702	−0.625176
$670$	0	0
$671$	0.830037	0.0320432
$672$	0	0
$673$	−27.8424	−1.07324	−0.536622	−	0.843823i	$-0.680300\pi$
$673$	−27.8424	−1.07324	−0.536622	+	0.843823i	$0.680300\pi$
$674$	0	0
$675$	−1.21573	−0.0467933
$676$	0	0
$677$	−24.8469	−0.954943	−0.477472	−	0.878647i	$-0.658447\pi$
$677$	−24.8469	−0.954943	−0.477472	+	0.878647i	$0.658447\pi$
$678$	0	0
$679$	4.19463	0.160975
$680$	0	0
$681$	3.04969	0.116864
$682$	0	0
$683$	−24.3458	−0.931566	−0.465783	−	0.884899i	$-0.654227\pi$
$683$	−24.3458	−0.931566	−0.465783	+	0.884899i	$0.654227\pi$
$684$	0	0
$685$	31.4176	1.20041
$686$	0	0
$687$	−6.74579	−0.257368
$688$	0	0
$689$	0.685000	0.0260964
$690$	0	0
$691$	−14.8385	−0.564482	−0.282241	−	0.959344i	$-0.591078\pi$
$691$	−14.8385	−0.564482	−0.282241	+	0.959344i	$0.591078\pi$
$692$	0	0
$693$	−2.46927	−0.0938000
$694$	0	0
$695$	26.2551	0.995913
$696$	0	0
$697$	9.04519	0.342611
$698$	0	0
$699$	5.98734	0.226462
$700$	0	0
$701$	−36.8768	−1.39282	−0.696408	−	0.717646i	$-0.745220\pi$
$701$	−36.8768	−1.39282	−0.696408	+	0.717646i	$0.745220\pi$
$702$	0	0
$703$	32.7204	1.23407
$704$	0	0
$705$	10.6047	0.399395
$706$	0	0
$707$	13.9450	0.524456
$708$	0	0
$709$	34.5495	1.29753	0.648767	−	0.760987i	$-0.275285\pi$
$709$	34.5495	1.29753	0.648767	+	0.760987i	$0.275285\pi$
$710$	0	0
$711$	15.9599	0.598542
$712$	0	0
$713$	1.71900	0.0643769
$714$	0	0
$715$	−2.49314	−0.0932380
$716$	0	0
$717$	−28.6588	−1.07028
$718$	0	0
$719$	−3.43471	−0.128093	−0.0640465	−	0.997947i	$-0.520401\pi$
$719$	−3.43471	−0.128093	−0.0640465	+	0.997947i	$0.520401\pi$
$720$	0	0
$721$	4.58041	0.170583
$722$	0	0
$723$	−7.31500	−0.272048
$724$	0	0
$725$	12.6845	0.471089
$726$	0	0
$727$	50.0427	1.85598	0.927990	−	0.372606i	$-0.121536\pi$
$727$	50.0427	1.85598	0.927990	+	0.372606i	$0.121536\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−6.46386	−0.239075
$732$	0	0
$733$	9.59666	0.354461	0.177230	−	0.984169i	$-0.443286\pi$
$733$	9.59666	0.354461	0.177230	+	0.984169i	$0.443286\pi$
$734$	0	0
$735$	2.25052	0.0830118
$736$	0	0
$737$	−0.493136	−0.0181649
$738$	0	0
$739$	0.307978	0.0113292	0.00566458	−	0.999984i	$-0.498197\pi$
$739$	0.307978	0.0113292	0.00566458	+	0.999984i	$0.498197\pi$
$740$	0	0
$741$	−6.62551	−0.243394
$742$	0	0
$743$	48.7902	1.78994	0.894970	−	0.446126i	$-0.147197\pi$
$743$	48.7902	1.78994	0.894970	+	0.446126i	$0.147197\pi$
$744$	0	0
$745$	−52.9426	−1.93967
$746$	0	0
$747$	4.98627	0.182438
$748$	0	0
$749$	−20.0221	−0.731593
$750$	0	0
$751$	−9.92508	−0.362171	−0.181086	−	0.983467i	$-0.557961\pi$
$751$	−9.92508	−0.362171	−0.181086	+	0.983467i	$0.557961\pi$
$752$	0	0
$753$	−3.10918	−0.113305
$754$	0	0
$755$	−20.7535	−0.755295
$756$	0	0
$757$	−24.4773	−0.889642	−0.444821	−	0.895620i	$-0.646733\pi$
$757$	−24.4773	−0.889642	−0.444821	+	0.895620i	$0.646733\pi$
$758$	0	0
$759$	−0.469274	−0.0170336
$760$	0	0
$761$	−30.9236	−1.12098	−0.560490	−	0.828162i	$-0.689387\pi$
$761$	−30.9236	−1.12098	−0.560490	+	0.828162i	$0.689387\pi$
$762$	0	0
$763$	24.5458	0.888619
$764$	0	0
$765$	5.04576	0.182430
$766$	0	0
$767$	−10.8412	−0.391454
$768$	0	0
$769$	−34.4373	−1.24184	−0.620921	−	0.783873i	$-0.713241\pi$
$769$	−34.4373	−1.24184	−0.620921	+	0.783873i	$0.713241\pi$
$770$	0	0
$771$	−14.7661	−0.531786
$772$	0	0
$773$	−33.1329	−1.19171	−0.595854	−	0.803093i	$-0.703186\pi$
$773$	−33.1329	−1.19171	−0.595854	+	0.803093i	$0.703186\pi$
$774$	0	0
$775$	−4.45333	−0.159968
$776$	0	0
$777$	−12.1946	−0.437480
$778$	0	0
$779$	29.6112	1.06093
$780$	0	0
$781$	8.01120	0.286663
$782$	0	0
$783$	−10.4336	−0.372868
$784$	0	0
$785$	−21.3171	−0.760841
$786$	0	0
$787$	2.31304	0.0824509	0.0412255	−	0.999150i	$-0.486874\pi$
$787$	2.31304	0.0824509	0.0412255	+	0.999150i	$0.486874\pi$
$788$	0	0
$789$	−29.8058	−1.06112
$790$	0	0
$791$	−2.07724	−0.0738581
$792$	0	0
$793$	−0.830037	−0.0294755
$794$	0	0
$795$	−1.70780	−0.0605694
$796$	0	0
$797$	−1.52989	−0.0541916	−0.0270958	−	0.999633i	$-0.508626\pi$
$797$	−1.52989	−0.0541916	−0.0270958	+	0.999633i	$0.508626\pi$
$798$	0	0
$799$	−8.60859	−0.304550
$800$	0	0
$801$	6.26172	0.221247
$802$	0	0
$803$	−5.78427	−0.204123
$804$	0	0
$805$	−2.88896	−0.101822
$806$	0	0
$807$	−15.1204	−0.532262
$808$	0	0
$809$	16.2612	0.571712	0.285856	−	0.958273i	$-0.407722\pi$
$809$	16.2612	0.571712	0.285856	+	0.958273i	$0.407722\pi$
$810$	0	0
$811$	25.6870	0.901991	0.450996	−	0.892526i	$-0.351069\pi$
$811$	25.6870	0.901991	0.450996	+	0.892526i	$0.351069\pi$
$812$	0	0
$813$	−15.6504	−0.548885
$814$	0	0
$815$	−41.7518	−1.46250
$816$	0	0
$817$	−21.1607	−0.740320
$818$	0	0
$819$	2.46927	0.0862834
$820$	0	0
$821$	28.1301	0.981746	0.490873	−	0.871231i	$-0.336678\pi$
$821$	28.1301	0.981746	0.490873	+	0.871231i	$0.336678\pi$
$822$	0	0
$823$	26.0424	0.907781	0.453890	−	0.891057i	$-0.350036\pi$
$823$	26.0424	0.907781	0.453890	+	0.891057i	$0.350036\pi$
$824$	0	0
$825$	1.21573	0.0423262
$826$	0	0
$827$	−45.0446	−1.56636	−0.783178	−	0.621798i	$-0.786402\pi$
$827$	−45.0446	−1.56636	−0.783178	+	0.621798i	$0.786402\pi$
$828$	0	0
$829$	−35.1096	−1.21941	−0.609704	−	0.792629i	$-0.708712\pi$
$829$	−35.1096	−1.21941	−0.609704	+	0.792629i	$0.708712\pi$
$830$	0	0
$831$	17.1060	0.593400
$832$	0	0
$833$	−1.82691	−0.0632988
$834$	0	0
$835$	−37.6613	−1.30332
$836$	0	0
$837$	3.66310	0.126615
$838$	0	0
$839$	37.1518	1.28262	0.641312	−	0.767280i	$-0.278390\pi$
$839$	37.1518	1.28262	0.641312	+	0.767280i	$0.278390\pi$
$840$	0	0
$841$	79.8609	2.75383
$842$	0	0
$843$	−21.4779	−0.739740
$844$	0	0
$845$	2.49314	0.0857665
$846$	0	0
$847$	2.46927	0.0848452
$848$	0	0
$849$	−10.4297	−0.357947
$850$	0	0
$851$	−2.31753	−0.0794439
$852$	0	0
$853$	−52.1806	−1.78663	−0.893315	−	0.449432i	$-0.851626\pi$
$853$	−52.1806	−1.78663	−0.893315	+	0.449432i	$0.851626\pi$
$854$	0	0
$855$	16.5183	0.564914
$856$	0	0
$857$	34.0630	1.16357	0.581785	−	0.813343i	$-0.302355\pi$
$857$	34.0630	1.16357	0.581785	+	0.813343i	$0.302355\pi$
$858$	0	0
$859$	8.04772	0.274585	0.137292	−	0.990531i	$-0.456160\pi$
$859$	8.04772	0.274585	0.137292	+	0.990531i	$0.456160\pi$
$860$	0	0
$861$	−11.0359	−0.376101
$862$	0	0
$863$	−11.2948	−0.384480	−0.192240	−	0.981348i	$-0.561575\pi$
$863$	−11.2948	−0.384480	−0.192240	+	0.981348i	$0.561575\pi$
$864$	0	0
$865$	−16.9687	−0.576954
$866$	0	0
$867$	12.9040	0.438242
$868$	0	0
$869$	−15.9599	−0.541402
$870$	0	0
$871$	0.493136	0.0167093
$872$	0	0
$873$	1.69873	0.0574932
$874$	0	0
$875$	−23.2969	−0.787578
$876$	0	0
$877$	50.0595	1.69039	0.845194	−	0.534459i	$-0.179485\pi$
$877$	50.0595	1.69039	0.845194	+	0.534459i	$0.179485\pi$
$878$	0	0
$879$	17.6504	0.595334
$880$	0	0
$881$	−38.0221	−1.28100	−0.640499	−	0.767959i	$-0.721272\pi$
$881$	−38.0221	−1.28100	−0.640499	+	0.767959i	$0.721272\pi$
$882$	0	0
$883$	−37.1763	−1.25108	−0.625541	−	0.780191i	$-0.715122\pi$
$883$	−37.1763	−1.25108	−0.625541	+	0.780191i	$0.715122\pi$
$884$	0	0
$885$	27.0287	0.908559
$886$	0	0
$887$	−3.61963	−0.121535	−0.0607676	−	0.998152i	$-0.519355\pi$
$887$	−3.61963	−0.121535	−0.0607676	+	0.998152i	$0.519355\pi$
$888$	0	0
$889$	2.67994	0.0898823
$890$	0	0
$891$	−1.00000	−0.0335013
$892$	0	0
$893$	−28.1819	−0.943072
$894$	0	0
$895$	−36.2759	−1.21257
$896$	0	0
$897$	0.469274	0.0156686
$898$	0	0
$899$	−38.2195	−1.27469
$900$	0	0
$901$	1.38635	0.0461859
$902$	0	0
$903$	7.88643	0.262444
$904$	0	0
$905$	−64.3263	−2.13828
$906$	0	0
$907$	18.3365	0.608852	0.304426	−	0.952536i	$-0.401535\pi$
$907$	18.3365	0.608852	0.304426	+	0.952536i	$0.401535\pi$
$908$	0	0
$909$	5.64741	0.187313
$910$	0	0
$911$	20.6889	0.685455	0.342727	−	0.939435i	$-0.388649\pi$
$911$	20.6889	0.685455	0.342727	+	0.939435i	$0.388649\pi$
$912$	0	0
$913$	−4.98627	−0.165022
$914$	0	0
$915$	2.06939	0.0684121
$916$	0	0
$917$	34.8878	1.15210
$918$	0	0
$919$	54.4344	1.79562	0.897812	−	0.440379i	$-0.145156\pi$
$919$	54.4344	1.79562	0.897812	+	0.440379i	$0.145156\pi$
$920$	0	0
$921$	14.5301	0.478782
$922$	0	0
$923$	−8.01120	−0.263692
$924$	0	0
$925$	6.00392	0.197408
$926$	0	0
$927$	1.85496	0.0609250
$928$	0	0
$929$	7.06056	0.231649	0.115825	−	0.993270i	$-0.463049\pi$
$929$	7.06056	0.231649	0.115825	+	0.993270i	$0.463049\pi$
$930$	0	0
$931$	−5.98076	−0.196012
$932$	0	0
$933$	9.33078	0.305476
$934$	0	0
$935$	−5.04576	−0.165014
$936$	0	0
$937$	−50.3130	−1.64365	−0.821827	−	0.569737i	$-0.807045\pi$
$937$	−50.3130	−1.64365	−0.821827	+	0.569737i	$0.807045\pi$
$938$	0	0
$939$	−10.1719	−0.331948
$940$	0	0
$941$	−12.1111	−0.394812	−0.197406	−	0.980322i	$-0.563252\pi$
$941$	−12.1111	−0.394812	−0.197406	+	0.980322i	$0.563252\pi$
$942$	0	0
$943$	−2.09731	−0.0682979
$944$	0	0
$945$	−6.15623	−0.200262
$946$	0	0
$947$	−10.3451	−0.336170	−0.168085	−	0.985773i	$-0.553758\pi$
$947$	−10.3451	−0.336170	−0.168085	+	0.985773i	$0.553758\pi$
$948$	0	0
$949$	5.78427	0.187765
$950$	0	0
$951$	6.93579	0.224908
$952$	0	0
$953$	−20.4704	−0.663102	−0.331551	−	0.943437i	$-0.607572\pi$
$953$	−20.4704	−0.663102	−0.331551	+	0.943437i	$0.607572\pi$
$954$	0	0
$955$	−10.4810	−0.339159
$956$	0	0
$957$	10.4336	0.337272
$958$	0	0
$959$	31.1169	1.00482
$960$	0	0
$961$	−17.5817	−0.567152
$962$	0	0
$963$	−8.10851	−0.261293
$964$	0	0
$965$	−4.01839	−0.129356
$966$	0	0
$967$	−46.8647	−1.50707	−0.753534	−	0.657409i	$-0.771653\pi$
$967$	−46.8647	−1.50707	−0.753534	+	0.657409i	$0.771653\pi$
$968$	0	0
$969$	−13.4091	−0.430763
$970$	0	0
$971$	1.05696	0.0339195	0.0169597	−	0.999856i	$-0.494601\pi$
$971$	1.05696	0.0339195	0.0169597	+	0.999856i	$0.494601\pi$
$972$	0	0
$973$	26.0038	0.833644
$974$	0	0
$975$	−1.21573	−0.0389344
$976$	0	0
$977$	−3.53627	−0.113135	−0.0565677	−	0.998399i	$-0.518016\pi$
$977$	−3.53627	−0.113135	−0.0565677	+	0.998399i	$0.518016\pi$
$978$	0	0
$979$	−6.26172	−0.200125
$980$	0	0
$981$	9.94051	0.317376
$982$	0	0
$983$	−25.1294	−0.801505	−0.400752	−	0.916186i	$-0.631251\pi$
$983$	−25.1294	−0.801505	−0.400752	+	0.916186i	$0.631251\pi$
$984$	0	0
$985$	−59.6185	−1.89960
$986$	0	0
$987$	10.5032	0.334320
$988$	0	0
$989$	1.49878	0.0476584
$990$	0	0
$991$	27.0583	0.859534	0.429767	−	0.902940i	$-0.358596\pi$
$991$	27.0583	0.859534	0.429767	+	0.902940i	$0.358596\pi$
$992$	0	0
$993$	−0.191864	−0.00608863
$994$	0	0
$995$	−19.5150	−0.618668
$996$	0	0
$997$	−8.69087	−0.275243	−0.137621	−	0.990485i	$-0.543946\pi$
$997$	−8.69087	−0.275243	−0.137621	+	0.990485i	$0.543946\pi$
$998$	0	0
$999$	−4.93855	−0.156249

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6864.2.a.ce.1.4		5
4.3	odd	2		3432.2.a.x.1.4	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
3432.2.a.x.1.4	✓	5	4.3	odd	2
6864.2.a.ce.1.4		5	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} +2.49314 q^{5} +2.46927 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} +2.49314 q^{5} +2.46927 q^{7} +1.00000 q^{9} -1.00000 q^{11} +1.00000 q^{13} -2.49314 q^{15} +2.02386 q^{17} +6.62551 q^{19} -2.46927 q^{21} -0.469274 q^{23} +1.21573 q^{25} -1.00000 q^{27} +10.4336 q^{29} -3.66310 q^{31} +1.00000 q^{33} +6.15623 q^{35} +4.93855 q^{37} -1.00000 q^{39} +4.46927 q^{41} -3.19383 q^{43} +2.49314 q^{45} -4.25355 q^{47} -0.902687 q^{49} -2.02386 q^{51} +0.685000 q^{53} -2.49314 q^{55} -6.62551 q^{57} -10.8412 q^{59} -0.830037 q^{61} +2.46927 q^{63} +2.49314 q^{65} +0.493136 q^{67} +0.469274 q^{69} -8.01120 q^{71} +5.78427 q^{73} -1.21573 q^{75} -2.46927 q^{77} +15.9599 q^{79} +1.00000 q^{81} +4.98627 q^{83} +5.04576 q^{85} -10.4336 q^{87} +6.26172 q^{89} +2.46927 q^{91} +3.66310 q^{93} +16.5183 q^{95} +1.69873 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 5 q^{3} + q^{5} + 3 q^{7} + 5 q^{9}+O(q^{10}) \) 5 * q - 5 * q^3 + q^5 + 3 * q^7 + 5 * q^9	\( 5 q - 5 q^{3} + q^{5} + 3 q^{7} + 5 q^{9} - 5 q^{11} + 5 q^{13} - q^{15} + 8 q^{17} - 10 q^{19} - 3 q^{21} + 7 q^{23} + 16 q^{25} - 5 q^{27} - 3 q^{29} + 4 q^{31} + 5 q^{33} - 3 q^{35} + 6 q^{37} - 5 q^{39} + 13 q^{41} - 3 q^{43} + q^{45} - 2 q^{47} + 10 q^{49} - 8 q^{51} + 4 q^{53} - q^{55} + 10 q^{57} - 21 q^{59} - 15 q^{61} + 3 q^{63} + q^{65} - 9 q^{67} - 7 q^{69} + 4 q^{71} + 19 q^{73} - 16 q^{75} - 3 q^{77} - 8 q^{79} + 5 q^{81} + 2 q^{83} + 46 q^{85} + 3 q^{87} + 12 q^{89} + 3 q^{91} - 4 q^{93} + 32 q^{97} - 5 q^{99}+O(q^{100}) \) 5 * q - 5 * q^3 + q^5 + 3 * q^7 + 5 * q^9 - 5 * q^11 + 5 * q^13 - q^15 + 8 * q^17 - 10 * q^19 - 3 * q^21 + 7 * q^23 + 16 * q^25 - 5 * q^27 - 3 * q^29 + 4 * q^31 + 5 * q^33 - 3 * q^35 + 6 * q^37 - 5 * q^39 + 13 * q^41 - 3 * q^43 + q^45 - 2 * q^47 + 10 * q^49 - 8 * q^51 + 4 * q^53 - q^55 + 10 * q^57 - 21 * q^59 - 15 * q^61 + 3 * q^63 + q^65 - 9 * q^67 - 7 * q^69 + 4 * q^71 + 19 * q^73 - 16 * q^75 - 3 * q^77 - 8 * q^79 + 5 * q^81 + 2 * q^83 + 46 * q^85 + 3 * q^87 + 12 * q^89 + 3 * q^91 - 4 * q^93 + 32 * q^97 - 5 * q^99

Introduction

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