LMFDB - Embedded newform 6840.2.a.z.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6840,2,Mod(1,6840)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6840, 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("6840.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6840.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:	$54.6176749826$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 760)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	6840.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -2.82843 q^{7} +O(q^{10})$	$q+1.00000 q^{5} -2.82843 q^{7} +0.828427 q^{11} +3.41421 q^{13} -4.82843 q^{17} -1.00000 q^{19} -4.00000 q^{23} +1.00000 q^{25} +0.828427 q^{29} -2.82843 q^{35} +10.2426 q^{37} +0.828427 q^{41} +2.82843 q^{43} +8.48528 q^{47} +1.00000 q^{49} -13.0711 q^{53} +0.828427 q^{55} -2.82843 q^{59} -1.65685 q^{61} +3.41421 q^{65} -9.41421 q^{67} -15.3137 q^{71} +12.1421 q^{73} -2.34315 q^{77} -9.17157 q^{79} -8.00000 q^{83} -4.82843 q^{85} +3.17157 q^{89} -9.65685 q^{91} -1.00000 q^{95} -2.24264 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5}+O(q^{10}) $ 2 * q + 2 * q^5	$ 2 q + 2 q^{5} - 4 q^{11} + 4 q^{13} - 4 q^{17} - 2 q^{19} - 8 q^{23} + 2 q^{25} - 4 q^{29} + 12 q^{37} - 4 q^{41} + 2 q^{49} - 12 q^{53} - 4 q^{55} + 8 q^{61} + 4 q^{65} - 16 q^{67} - 8 q^{71} - 4 q^{73} - 16 q^{77} - 24 q^{79} - 16 q^{83} - 4 q^{85} + 12 q^{89} - 8 q^{91} - 2 q^{95} + 4 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 - 4 * q^11 + 4 * q^13 - 4 * q^17 - 2 * q^19 - 8 * q^23 + 2 * q^25 - 4 * q^29 + 12 * q^37 - 4 * q^41 + 2 * q^49 - 12 * q^53 - 4 * q^55 + 8 * q^61 + 4 * q^65 - 16 * q^67 - 8 * q^71 - 4 * q^73 - 16 * q^77 - 24 * q^79 - 16 * q^83 - 4 * q^85 + 12 * q^89 - 8 * q^91 - 2 * q^95 + 4 * q^97

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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−2.82843	−1.06904	−0.534522	−	0.845154i	$-0.679509\pi$
$7$	−2.82843	−1.06904	−0.534522	+	0.845154i	$0.679509\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.828427	0.249780	0.124890	−	0.992171i	$-0.460142\pi$
$11$	0.828427	0.249780	0.124890	+	0.992171i	$0.460142\pi$
$12$	0	0
$13$	3.41421	0.946932	0.473466	−	0.880812i	$-0.343003\pi$
$13$	3.41421	0.946932	0.473466	+	0.880812i	$0.343003\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.82843	−1.17107	−0.585533	−	0.810649i	$-0.699115\pi$
$17$	−4.82843	−1.17107	−0.585533	+	0.810649i	$0.699115\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0.828427	0.153835	0.0769175	−	0.997037i	$-0.475492\pi$
$29$	0.828427	0.153835	0.0769175	+	0.997037i	$0.475492\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−2.82843	−0.478091
$36$	0	0
$37$	10.2426	1.68388	0.841940	−	0.539571i	$-0.181414\pi$
$37$	10.2426	1.68388	0.841940	+	0.539571i	$0.181414\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.828427	0.129379	0.0646893	−	0.997905i	$-0.479394\pi$
$41$	0.828427	0.129379	0.0646893	+	0.997905i	$0.479394\pi$
$42$	0	0
$43$	2.82843	0.431331	0.215666	−	0.976467i	$-0.430808\pi$
$43$	2.82843	0.431331	0.215666	+	0.976467i	$0.430808\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.48528	1.23771	0.618853	−	0.785507i	$-0.287598\pi$
$47$	8.48528	1.23771	0.618853	+	0.785507i	$0.287598\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−13.0711	−1.79545	−0.897725	−	0.440557i	$-0.854781\pi$
$53$	−13.0711	−1.79545	−0.897725	+	0.440557i	$0.854781\pi$
$54$	0	0
$55$	0.828427	0.111705
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.82843	−0.368230	−0.184115	−	0.982905i	$-0.558942\pi$
$59$	−2.82843	−0.368230	−0.184115	+	0.982905i	$0.558942\pi$
$60$	0	0
$61$	−1.65685	−0.212138	−0.106069	−	0.994359i	$-0.533827\pi$
$61$	−1.65685	−0.212138	−0.106069	+	0.994359i	$0.533827\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	3.41421	0.423481
$66$	0	0
$67$	−9.41421	−1.15013	−0.575065	−	0.818108i	$-0.695023\pi$
$67$	−9.41421	−1.15013	−0.575065	+	0.818108i	$0.695023\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−15.3137	−1.81740	−0.908701	−	0.417447i	$-0.862925\pi$
$71$	−15.3137	−1.81740	−0.908701	+	0.417447i	$0.862925\pi$
$72$	0	0
$73$	12.1421	1.42113	0.710565	−	0.703632i	$-0.248440\pi$
$73$	12.1421	1.42113	0.710565	+	0.703632i	$0.248440\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−2.34315	−0.267026
$78$	0	0
$79$	−9.17157	−1.03188	−0.515941	−	0.856624i	$-0.672558\pi$
$79$	−9.17157	−1.03188	−0.515941	+	0.856624i	$0.672558\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−8.00000	−0.878114	−0.439057	−	0.898459i	$-0.644687\pi$
$83$	−8.00000	−0.878114	−0.439057	+	0.898459i	$0.644687\pi$
$84$	0	0
$85$	−4.82843	−0.523716
$86$	0	0
$87$	0	0
$88$	0	0
$89$	3.17157	0.336186	0.168093	−	0.985771i	$-0.446239\pi$
$89$	3.17157	0.336186	0.168093	+	0.985771i	$0.446239\pi$
$90$	0	0
$91$	−9.65685	−1.01231
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	−2.24264	−0.227706	−0.113853	−	0.993498i	$-0.536319\pi$
$97$	−2.24264	−0.227706	−0.113853	+	0.993498i	$0.536319\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−17.6569	−1.75692	−0.878461	−	0.477813i	$-0.841429\pi$
$101$	−17.6569	−1.75692	−0.878461	+	0.477813i	$0.841429\pi$
$102$	0	0
$103$	8.24264	0.812172	0.406086	−	0.913835i	$-0.366893\pi$
$103$	8.24264	0.812172	0.406086	+	0.913835i	$0.366893\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1.41421	−0.136717	−0.0683586	−	0.997661i	$-0.521776\pi$
$107$	−1.41421	−0.136717	−0.0683586	+	0.997661i	$0.521776\pi$
$108$	0	0
$109$	3.65685	0.350263	0.175132	−	0.984545i	$-0.443965\pi$
$109$	3.65685	0.350263	0.175132	+	0.984545i	$0.443965\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	14.7279	1.38549	0.692743	−	0.721184i	$-0.256402\pi$
$113$	14.7279	1.38549	0.692743	+	0.721184i	$0.256402\pi$
$114$	0	0
$115$	−4.00000	−0.373002
$116$	0	0
$117$	0	0
$118$	0	0
$119$	13.6569	1.25192
$120$	0	0
$121$	−10.3137	−0.937610
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−13.4142	−1.19032	−0.595159	−	0.803608i	$-0.702911\pi$
$127$	−13.4142	−1.19032	−0.595159	+	0.803608i	$0.702911\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−7.31371	−0.639002	−0.319501	−	0.947586i	$-0.603515\pi$
$131$	−7.31371	−0.639002	−0.319501	+	0.947586i	$0.603515\pi$
$132$	0	0
$133$	2.82843	0.245256
$134$	0	0
$135$	0	0
$136$	0	0
$137$	7.17157	0.612709	0.306354	−	0.951918i	$-0.400891\pi$
$137$	7.17157	0.612709	0.306354	+	0.951918i	$0.400891\pi$
$138$	0	0
$139$	−3.17157	−0.269009	−0.134505	−	0.990913i	$-0.542944\pi$
$139$	−3.17157	−0.269009	−0.134505	+	0.990913i	$0.542944\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	2.82843	0.236525
$144$	0	0
$145$	0.828427	0.0687971
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−1.65685	−0.135735	−0.0678674	−	0.997694i	$-0.521619\pi$
$149$	−1.65685	−0.135735	−0.0678674	+	0.997694i	$0.521619\pi$
$150$	0	0
$151$	6.14214	0.499840	0.249920	−	0.968267i	$-0.419596\pi$
$151$	6.14214	0.499840	0.249920	+	0.968267i	$0.419596\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−10.4853	−0.836817	−0.418408	−	0.908259i	$-0.637412\pi$
$157$	−10.4853	−0.836817	−0.418408	+	0.908259i	$0.637412\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	11.3137	0.891645
$162$	0	0
$163$	14.8284	1.16145	0.580726	−	0.814099i	$-0.302769\pi$
$163$	14.8284	1.16145	0.580726	+	0.814099i	$0.302769\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−2.10051	−0.162542	−0.0812710	−	0.996692i	$-0.525898\pi$
$167$	−2.10051	−0.162542	−0.0812710	+	0.996692i	$0.525898\pi$
$168$	0	0
$169$	−1.34315	−0.103319
$170$	0	0
$171$	0	0
$172$	0	0
$173$	10.7279	0.815629	0.407814	−	0.913065i	$-0.366291\pi$
$173$	10.7279	0.815629	0.407814	+	0.913065i	$0.366291\pi$
$174$	0	0
$175$	−2.82843	−0.213809
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−8.48528	−0.634220	−0.317110	−	0.948389i	$-0.602712\pi$
$179$	−8.48528	−0.634220	−0.317110	+	0.948389i	$0.602712\pi$
$180$	0	0
$181$	−5.31371	−0.394965	−0.197482	−	0.980306i	$-0.563277\pi$
$181$	−5.31371	−0.394965	−0.197482	+	0.980306i	$0.563277\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	10.2426	0.753054
$186$	0	0
$187$	−4.00000	−0.292509
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3.31371	−0.239772	−0.119886	−	0.992788i	$-0.538253\pi$
$191$	−3.31371	−0.239772	−0.119886	+	0.992788i	$0.538253\pi$
$192$	0	0
$193$	−6.92893	−0.498755	−0.249378	−	0.968406i	$-0.580226\pi$
$193$	−6.92893	−0.498755	−0.249378	+	0.968406i	$0.580226\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−25.3137	−1.80353	−0.901764	−	0.432230i	$-0.857727\pi$
$197$	−25.3137	−1.80353	−0.901764	+	0.432230i	$0.857727\pi$
$198$	0	0
$199$	−5.65685	−0.401004	−0.200502	−	0.979693i	$-0.564257\pi$
$199$	−5.65685	−0.401004	−0.200502	+	0.979693i	$0.564257\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−2.34315	−0.164457
$204$	0	0
$205$	0.828427	0.0578599
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−0.828427	−0.0573035
$210$	0	0
$211$	−16.4853	−1.13489	−0.567447	−	0.823410i	$-0.692069\pi$
$211$	−16.4853	−1.13489	−0.567447	+	0.823410i	$0.692069\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2.82843	0.192897
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−16.4853	−1.10892
$222$	0	0
$223$	13.4142	0.898282	0.449141	−	0.893461i	$-0.351730\pi$
$223$	13.4142	0.898282	0.449141	+	0.893461i	$0.351730\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−24.0416	−1.59570	−0.797850	−	0.602857i	$-0.794029\pi$
$227$	−24.0416	−1.59570	−0.797850	+	0.602857i	$0.794029\pi$
$228$	0	0
$229$	11.3137	0.747631	0.373815	−	0.927503i	$-0.378049\pi$
$229$	11.3137	0.747631	0.373815	+	0.927503i	$0.378049\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	0	0
$235$	8.48528	0.553519
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−8.00000	−0.517477	−0.258738	−	0.965947i	$-0.583307\pi$
$239$	−8.00000	−0.517477	−0.258738	+	0.965947i	$0.583307\pi$
$240$	0	0
$241$	8.14214	0.524481	0.262241	−	0.965003i	$-0.415539\pi$
$241$	8.14214	0.524481	0.262241	+	0.965003i	$0.415539\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	1.00000	0.0638877
$246$	0	0
$247$	−3.41421	−0.217241
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−28.9706	−1.82861	−0.914303	−	0.405031i	$-0.867261\pi$
$251$	−28.9706	−1.82861	−0.914303	+	0.405031i	$0.867261\pi$
$252$	0	0
$253$	−3.31371	−0.208331
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4.10051	0.255782	0.127891	−	0.991788i	$-0.459179\pi$
$257$	4.10051	0.255782	0.127891	+	0.991788i	$0.459179\pi$
$258$	0	0
$259$	−28.9706	−1.80014
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−27.3137	−1.68424	−0.842118	−	0.539294i	$-0.818691\pi$
$263$	−27.3137	−1.68424	−0.842118	+	0.539294i	$0.818691\pi$
$264$	0	0
$265$	−13.0711	−0.802949
$266$	0	0
$267$	0	0
$268$	0	0
$269$	29.3137	1.78729	0.893644	−	0.448776i	$-0.148140\pi$
$269$	29.3137	1.78729	0.893644	+	0.448776i	$0.148140\pi$
$270$	0	0
$271$	−9.51472	−0.577978	−0.288989	−	0.957332i	$-0.593319\pi$
$271$	−9.51472	−0.577978	−0.288989	+	0.957332i	$0.593319\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.828427	0.0499560
$276$	0	0
$277$	19.1716	1.15191	0.575954	−	0.817482i	$-0.304631\pi$
$277$	19.1716	1.15191	0.575954	+	0.817482i	$0.304631\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−11.1716	−0.666440	−0.333220	−	0.942849i	$-0.608135\pi$
$281$	−11.1716	−0.666440	−0.333220	+	0.942849i	$0.608135\pi$
$282$	0	0
$283$	6.34315	0.377061	0.188530	−	0.982067i	$-0.439628\pi$
$283$	6.34315	0.377061	0.188530	+	0.982067i	$0.439628\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.34315	−0.138312
$288$	0	0
$289$	6.31371	0.371395
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−32.3848	−1.89194	−0.945969	−	0.324256i	$-0.894886\pi$
$293$	−32.3848	−1.89194	−0.945969	+	0.324256i	$0.894886\pi$
$294$	0	0
$295$	−2.82843	−0.164677
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−13.6569	−0.789796
$300$	0	0
$301$	−8.00000	−0.461112
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1.65685	−0.0948712
$306$	0	0
$307$	1.41421	0.0807134	0.0403567	−	0.999185i	$-0.487151\pi$
$307$	1.41421	0.0807134	0.0403567	+	0.999185i	$0.487151\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−19.1716	−1.08712	−0.543560	−	0.839370i	$-0.682924\pi$
$311$	−19.1716	−1.08712	−0.543560	+	0.839370i	$0.682924\pi$
$312$	0	0
$313$	−5.31371	−0.300349	−0.150174	−	0.988660i	$-0.547983\pi$
$313$	−5.31371	−0.300349	−0.150174	+	0.988660i	$0.547983\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−0.100505	−0.00564493	−0.00282246	−	0.999996i	$-0.500898\pi$
$317$	−0.100505	−0.00564493	−0.00282246	+	0.999996i	$0.500898\pi$
$318$	0	0
$319$	0.686292	0.0384249
$320$	0	0
$321$	0	0
$322$	0	0
$323$	4.82843	0.268661
$324$	0	0
$325$	3.41421	0.189386
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−24.0000	−1.32316
$330$	0	0
$331$	−17.4558	−0.959460	−0.479730	−	0.877416i	$-0.659265\pi$
$331$	−17.4558	−0.959460	−0.479730	+	0.877416i	$0.659265\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−9.41421	−0.514353
$336$	0	0
$337$	−20.3848	−1.11043	−0.555215	−	0.831707i	$-0.687364\pi$
$337$	−20.3848	−1.11043	−0.555215	+	0.831707i	$0.687364\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	0	0
$342$	0	0
$343$	16.9706	0.916324
$344$	0	0
$345$	0	0
$346$	0	0
$347$	20.2843	1.08892	0.544458	−	0.838788i	$-0.316735\pi$
$347$	20.2843	1.08892	0.544458	+	0.838788i	$0.316735\pi$
$348$	0	0
$349$	12.3431	0.660713	0.330357	−	0.943856i	$-0.392831\pi$
$349$	12.3431	0.660713	0.330357	+	0.943856i	$0.392831\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	12.1421	0.646261	0.323130	−	0.946354i	$-0.395265\pi$
$353$	12.1421	0.646261	0.323130	+	0.946354i	$0.395265\pi$
$354$	0	0
$355$	−15.3137	−0.812767
$356$	0	0
$357$	0	0
$358$	0	0
$359$	19.4558	1.02684	0.513420	−	0.858137i	$-0.328378\pi$
$359$	19.4558	1.02684	0.513420	+	0.858137i	$0.328378\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	12.1421	0.635548
$366$	0	0
$367$	31.1127	1.62407	0.812035	−	0.583609i	$-0.198360\pi$
$367$	31.1127	1.62407	0.812035	+	0.583609i	$0.198360\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	36.9706	1.91942
$372$	0	0
$373$	−33.5563	−1.73748	−0.868741	−	0.495267i	$-0.835070\pi$
$373$	−33.5563	−1.73748	−0.868741	+	0.495267i	$0.835070\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.82843	0.145671
$378$	0	0
$379$	9.65685	0.496039	0.248020	−	0.968755i	$-0.420220\pi$
$379$	9.65685	0.496039	0.248020	+	0.968755i	$0.420220\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−32.7279	−1.67232	−0.836159	−	0.548487i	$-0.815204\pi$
$383$	−32.7279	−1.67232	−0.836159	+	0.548487i	$0.815204\pi$
$384$	0	0
$385$	−2.34315	−0.119418
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−9.31371	−0.472224	−0.236112	−	0.971726i	$-0.575873\pi$
$389$	−9.31371	−0.472224	−0.236112	+	0.971726i	$0.575873\pi$
$390$	0	0
$391$	19.3137	0.976736
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−9.17157	−0.461472
$396$	0	0
$397$	36.8284	1.84837	0.924183	−	0.381950i	$-0.124747\pi$
$397$	36.8284	1.84837	0.924183	+	0.381950i	$0.124747\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−14.9706	−0.747594	−0.373797	−	0.927510i	$-0.621944\pi$
$401$	−14.9706	−0.747594	−0.373797	+	0.927510i	$0.621944\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8.48528	0.420600
$408$	0	0
$409$	−30.2843	−1.49746	−0.748730	−	0.662875i	$-0.769336\pi$
$409$	−30.2843	−1.49746	−0.748730	+	0.662875i	$0.769336\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	8.00000	0.393654
$414$	0	0
$415$	−8.00000	−0.392705
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−1.65685	−0.0809426	−0.0404713	−	0.999181i	$-0.512886\pi$
$419$	−1.65685	−0.0809426	−0.0404713	+	0.999181i	$0.512886\pi$
$420$	0	0
$421$	−20.8284	−1.01512	−0.507558	−	0.861618i	$-0.669452\pi$
$421$	−20.8284	−1.01512	−0.507558	+	0.861618i	$0.669452\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−4.82843	−0.234213
$426$	0	0
$427$	4.68629	0.226786
$428$	0	0
$429$	0	0
$430$	0	0
$431$	14.8284	0.714260	0.357130	−	0.934055i	$-0.383755\pi$
$431$	14.8284	0.714260	0.357130	+	0.934055i	$0.383755\pi$
$432$	0	0
$433$	−18.0416	−0.867025	−0.433513	−	0.901147i	$-0.642726\pi$
$433$	−18.0416	−0.867025	−0.433513	+	0.901147i	$0.642726\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	4.00000	0.191346
$438$	0	0
$439$	14.1421	0.674967	0.337484	−	0.941331i	$-0.390424\pi$
$439$	14.1421	0.674967	0.337484	+	0.941331i	$0.390424\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−0.686292	−0.0326067	−0.0163033	−	0.999867i	$-0.505190\pi$
$443$	−0.686292	−0.0326067	−0.0163033	+	0.999867i	$0.505190\pi$
$444$	0	0
$445$	3.17157	0.150347
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−26.2843	−1.24043	−0.620216	−	0.784431i	$-0.712955\pi$
$449$	−26.2843	−1.24043	−0.620216	+	0.784431i	$0.712955\pi$
$450$	0	0
$451$	0.686292	0.0323162
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−9.65685	−0.452720
$456$	0	0
$457$	−34.2843	−1.60375	−0.801875	−	0.597491i	$-0.796164\pi$
$457$	−34.2843	−1.60375	−0.801875	+	0.597491i	$0.796164\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	8.62742	0.401819	0.200909	−	0.979610i	$-0.435610\pi$
$461$	8.62742	0.401819	0.200909	+	0.979610i	$0.435610\pi$
$462$	0	0
$463$	26.6274	1.23748	0.618741	−	0.785595i	$-0.287643\pi$
$463$	26.6274	1.23748	0.618741	+	0.785595i	$0.287643\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−24.0000	−1.11059	−0.555294	−	0.831654i	$-0.687394\pi$
$467$	−24.0000	−1.11059	−0.555294	+	0.831654i	$0.687394\pi$
$468$	0	0
$469$	26.6274	1.22954
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2.34315	0.107738
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−23.4558	−1.07172	−0.535862	−	0.844305i	$-0.680013\pi$
$479$	−23.4558	−1.07172	−0.535862	+	0.844305i	$0.680013\pi$
$480$	0	0
$481$	34.9706	1.59452
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−2.24264	−0.101833
$486$	0	0
$487$	27.5563	1.24870	0.624349	−	0.781146i	$-0.285364\pi$
$487$	27.5563	1.24870	0.624349	+	0.781146i	$0.285364\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	12.9706	0.585353	0.292677	−	0.956211i	$-0.405454\pi$
$491$	12.9706	0.585353	0.292677	+	0.956211i	$0.405454\pi$
$492$	0	0
$493$	−4.00000	−0.180151
$494$	0	0
$495$	0	0
$496$	0	0
$497$	43.3137	1.94289
$498$	0	0
$499$	4.14214	0.185427	0.0927137	−	0.995693i	$-0.470446\pi$
$499$	4.14214	0.185427	0.0927137	+	0.995693i	$0.470446\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1.65685	0.0738755	0.0369377	−	0.999318i	$-0.488240\pi$
$503$	1.65685	0.0738755	0.0369377	+	0.999318i	$0.488240\pi$
$504$	0	0
$505$	−17.6569	−0.785720
$506$	0	0
$507$	0	0
$508$	0	0
$509$	15.4558	0.685068	0.342534	−	0.939505i	$-0.388715\pi$
$509$	15.4558	0.685068	0.342534	+	0.939505i	$0.388715\pi$
$510$	0	0
$511$	−34.3431	−1.51925
$512$	0	0
$513$	0	0
$514$	0	0
$515$	8.24264	0.363214
$516$	0	0
$517$	7.02944	0.309154
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−14.6863	−0.643418	−0.321709	−	0.946839i	$-0.604257\pi$
$521$	−14.6863	−0.643418	−0.321709	+	0.946839i	$0.604257\pi$
$522$	0	0
$523$	−36.2426	−1.58478	−0.792390	−	0.610015i	$-0.791163\pi$
$523$	−36.2426	−1.58478	−0.792390	+	0.610015i	$0.791163\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	2.82843	0.122513
$534$	0	0
$535$	−1.41421	−0.0611418
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0.828427	0.0356829
$540$	0	0
$541$	36.2843	1.55998	0.779991	−	0.625790i	$-0.215223\pi$
$541$	36.2843	1.55998	0.779991	+	0.625790i	$0.215223\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3.65685	0.156642
$546$	0	0
$547$	−39.0711	−1.67056	−0.835279	−	0.549826i	$-0.814694\pi$
$547$	−39.0711	−1.67056	−0.835279	+	0.549826i	$0.814694\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−0.828427	−0.0352922
$552$	0	0
$553$	25.9411	1.10313
$554$	0	0
$555$	0	0
$556$	0	0
$557$	5.79899	0.245711	0.122856	−	0.992425i	$-0.460795\pi$
$557$	5.79899	0.245711	0.122856	+	0.992425i	$0.460795\pi$
$558$	0	0
$559$	9.65685	0.408441
$560$	0	0
$561$	0	0
$562$	0	0
$563$	4.24264	0.178806	0.0894030	−	0.995996i	$-0.471504\pi$
$563$	4.24264	0.178806	0.0894030	+	0.995996i	$0.471504\pi$
$564$	0	0
$565$	14.7279	0.619608
$566$	0	0
$567$	0	0
$568$	0	0
$569$	31.9411	1.33904	0.669521	−	0.742793i	$-0.266499\pi$
$569$	31.9411	1.33904	0.669521	+	0.742793i	$0.266499\pi$
$570$	0	0
$571$	−30.4853	−1.27577	−0.637885	−	0.770132i	$-0.720190\pi$
$571$	−30.4853	−1.27577	−0.637885	+	0.770132i	$0.720190\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−4.00000	−0.166812
$576$	0	0
$577$	44.6274	1.85786	0.928932	−	0.370251i	$-0.120728\pi$
$577$	44.6274	1.85786	0.928932	+	0.370251i	$0.120728\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	22.6274	0.938743
$582$	0	0
$583$	−10.8284	−0.448468
$584$	0	0
$585$	0	0
$586$	0	0
$587$	5.65685	0.233483	0.116742	−	0.993162i	$-0.462755\pi$
$587$	5.65685	0.233483	0.116742	+	0.993162i	$0.462755\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−16.6274	−0.682806	−0.341403	−	0.939917i	$-0.610902\pi$
$593$	−16.6274	−0.682806	−0.341403	+	0.939917i	$0.610902\pi$
$594$	0	0
$595$	13.6569	0.559876
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−6.14214	−0.250961	−0.125480	−	0.992096i	$-0.540047\pi$
$599$	−6.14214	−0.250961	−0.125480	+	0.992096i	$0.540047\pi$
$600$	0	0
$601$	−14.4853	−0.590867	−0.295433	−	0.955363i	$-0.595464\pi$
$601$	−14.4853	−0.590867	−0.295433	+	0.955363i	$0.595464\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−10.3137	−0.419312
$606$	0	0
$607$	−7.75736	−0.314862	−0.157431	−	0.987530i	$-0.550321\pi$
$607$	−7.75736	−0.314862	−0.157431	+	0.987530i	$0.550321\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	28.9706	1.17202
$612$	0	0
$613$	17.7990	0.718894	0.359447	−	0.933165i	$-0.382965\pi$
$613$	17.7990	0.718894	0.359447	+	0.933165i	$0.382965\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	37.7990	1.52173	0.760865	−	0.648910i	$-0.224775\pi$
$617$	37.7990	1.52173	0.760865	+	0.648910i	$0.224775\pi$
$618$	0	0
$619$	40.1421	1.61345	0.806724	−	0.590928i	$-0.201238\pi$
$619$	40.1421	1.61345	0.806724	+	0.590928i	$0.201238\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−8.97056	−0.359398
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−49.4558	−1.97193
$630$	0	0
$631$	−34.4853	−1.37284	−0.686419	−	0.727207i	$-0.740818\pi$
$631$	−34.4853	−1.37284	−0.686419	+	0.727207i	$0.740818\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−13.4142	−0.532327
$636$	0	0
$637$	3.41421	0.135276
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−27.4558	−1.08444	−0.542220	−	0.840236i	$-0.682416\pi$
$641$	−27.4558	−1.08444	−0.542220	+	0.840236i	$0.682416\pi$
$642$	0	0
$643$	−35.3137	−1.39264	−0.696318	−	0.717733i	$-0.745180\pi$
$643$	−35.3137	−1.39264	−0.696318	+	0.717733i	$0.745180\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	16.6863	0.656006	0.328003	−	0.944677i	$-0.393624\pi$
$647$	16.6863	0.656006	0.328003	+	0.944677i	$0.393624\pi$
$648$	0	0
$649$	−2.34315	−0.0919765
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−12.1421	−0.475158	−0.237579	−	0.971368i	$-0.576354\pi$
$653$	−12.1421	−0.475158	−0.237579	+	0.971368i	$0.576354\pi$
$654$	0	0
$655$	−7.31371	−0.285770
$656$	0	0
$657$	0	0
$658$	0	0
$659$	28.4853	1.10963	0.554815	−	0.831974i	$-0.312789\pi$
$659$	28.4853	1.10963	0.554815	+	0.831974i	$0.312789\pi$
$660$	0	0
$661$	8.82843	0.343386	0.171693	−	0.985151i	$-0.445076\pi$
$661$	8.82843	0.343386	0.171693	+	0.985151i	$0.445076\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	2.82843	0.109682
$666$	0	0
$667$	−3.31371	−0.128307
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−1.37258	−0.0529880
$672$	0	0
$673$	8.38478	0.323209	0.161605	−	0.986856i	$-0.448333\pi$
$673$	8.38478	0.323209	0.161605	+	0.986856i	$0.448333\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	7.61522	0.292677	0.146338	−	0.989235i	$-0.453251\pi$
$677$	7.61522	0.292677	0.146338	+	0.989235i	$0.453251\pi$
$678$	0	0
$679$	6.34315	0.243428
$680$	0	0
$681$	0	0
$682$	0	0
$683$	9.89949	0.378794	0.189397	−	0.981901i	$-0.439347\pi$
$683$	9.89949	0.378794	0.189397	+	0.981901i	$0.439347\pi$
$684$	0	0
$685$	7.17157	0.274012
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−44.6274	−1.70017
$690$	0	0
$691$	−35.1716	−1.33799	−0.668995	−	0.743267i	$-0.733275\pi$
$691$	−35.1716	−1.33799	−0.668995	+	0.743267i	$0.733275\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−3.17157	−0.120305
$696$	0	0
$697$	−4.00000	−0.151511
$698$	0	0
$699$	0	0
$700$	0	0
$701$	24.0000	0.906467	0.453234	−	0.891392i	$-0.350270\pi$
$701$	24.0000	0.906467	0.453234	+	0.891392i	$0.350270\pi$
$702$	0	0
$703$	−10.2426	−0.386309
$704$	0	0
$705$	0	0
$706$	0	0
$707$	49.9411	1.87823
$708$	0	0
$709$	−18.2843	−0.686680	−0.343340	−	0.939211i	$-0.611558\pi$
$709$	−18.2843	−0.686680	−0.343340	+	0.939211i	$0.611558\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	2.82843	0.105777
$716$	0	0
$717$	0	0
$718$	0	0
$719$	31.4558	1.17311	0.586553	−	0.809911i	$-0.300485\pi$
$719$	31.4558	1.17311	0.586553	+	0.809911i	$0.300485\pi$
$720$	0	0
$721$	−23.3137	−0.868248
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0.828427	0.0307670
$726$	0	0
$727$	−41.4558	−1.53751	−0.768756	−	0.639542i	$-0.779124\pi$
$727$	−41.4558	−1.53751	−0.768756	+	0.639542i	$0.779124\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−13.6569	−0.505117
$732$	0	0
$733$	4.34315	0.160418	0.0802089	−	0.996778i	$-0.474441\pi$
$733$	4.34315	0.160418	0.0802089	+	0.996778i	$0.474441\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−7.79899	−0.287279
$738$	0	0
$739$	0.686292	0.0252456	0.0126228	−	0.999920i	$-0.495982\pi$
$739$	0.686292	0.0252456	0.0126228	+	0.999920i	$0.495982\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	15.7574	0.578081	0.289041	−	0.957317i	$-0.406664\pi$
$743$	15.7574	0.578081	0.289041	+	0.957317i	$0.406664\pi$
$744$	0	0
$745$	−1.65685	−0.0607024
$746$	0	0
$747$	0	0
$748$	0	0
$749$	4.00000	0.146157
$750$	0	0
$751$	−44.4853	−1.62329	−0.811645	−	0.584150i	$-0.801428\pi$
$751$	−44.4853	−1.62329	−0.811645	+	0.584150i	$0.801428\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	6.14214	0.223535
$756$	0	0
$757$	−22.2843	−0.809936	−0.404968	−	0.914331i	$-0.632717\pi$
$757$	−22.2843	−0.809936	−0.404968	+	0.914331i	$0.632717\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−1.37258	−0.0497561	−0.0248780	−	0.999690i	$-0.507920\pi$
$761$	−1.37258	−0.0497561	−0.0248780	+	0.999690i	$0.507920\pi$
$762$	0	0
$763$	−10.3431	−0.374447
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−9.65685	−0.348689
$768$	0	0
$769$	40.0000	1.44244	0.721218	−	0.692708i	$-0.243582\pi$
$769$	40.0000	1.44244	0.721218	+	0.692708i	$0.243582\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	48.8701	1.75773	0.878867	−	0.477067i	$-0.158300\pi$
$773$	48.8701	1.75773	0.878867	+	0.477067i	$0.158300\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−0.828427	−0.0296815
$780$	0	0
$781$	−12.6863	−0.453951
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−10.4853	−0.374236
$786$	0	0
$787$	−1.41421	−0.0504113	−0.0252056	−	0.999682i	$-0.508024\pi$
$787$	−1.41421	−0.0504113	−0.0252056	+	0.999682i	$0.508024\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−41.6569	−1.48115
$792$	0	0
$793$	−5.65685	−0.200881
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−27.6985	−0.981131	−0.490565	−	0.871404i	$-0.663210\pi$
$797$	−27.6985	−0.981131	−0.490565	+	0.871404i	$0.663210\pi$
$798$	0	0
$799$	−40.9706	−1.44943
$800$	0	0
$801$	0	0
$802$	0	0
$803$	10.0589	0.354970
$804$	0	0
$805$	11.3137	0.398756
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−38.0000	−1.33601	−0.668004	−	0.744157i	$-0.732851\pi$
$809$	−38.0000	−1.33601	−0.668004	+	0.744157i	$0.732851\pi$
$810$	0	0
$811$	−0.686292	−0.0240990	−0.0120495	−	0.999927i	$-0.503836\pi$
$811$	−0.686292	−0.0240990	−0.0120495	+	0.999927i	$0.503836\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	14.8284	0.519417
$816$	0	0
$817$	−2.82843	−0.0989541
$818$	0	0
$819$	0	0
$820$	0	0
$821$	49.5980	1.73098	0.865491	−	0.500925i	$-0.167007\pi$
$821$	49.5980	1.73098	0.865491	+	0.500925i	$0.167007\pi$
$822$	0	0
$823$	−16.4853	−0.574641	−0.287320	−	0.957835i	$-0.592764\pi$
$823$	−16.4853	−0.574641	−0.287320	+	0.957835i	$0.592764\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−13.6985	−0.476343	−0.238171	−	0.971223i	$-0.576548\pi$
$827$	−13.6985	−0.476343	−0.238171	+	0.971223i	$0.576548\pi$
$828$	0	0
$829$	−49.5980	−1.72261	−0.861305	−	0.508089i	$-0.830352\pi$
$829$	−49.5980	−1.72261	−0.861305	+	0.508089i	$0.830352\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−4.82843	−0.167295
$834$	0	0
$835$	−2.10051	−0.0726910
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−0.201010	−0.00693964	−0.00346982	−	0.999994i	$-0.501104\pi$
$839$	−0.201010	−0.00693964	−0.00346982	+	0.999994i	$0.501104\pi$
$840$	0	0
$841$	−28.3137	−0.976335
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−1.34315	−0.0462056
$846$	0	0
$847$	29.1716	1.00235
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−40.9706	−1.40445
$852$	0	0
$853$	−27.9411	−0.956686	−0.478343	−	0.878173i	$-0.658762\pi$
$853$	−27.9411	−0.956686	−0.478343	+	0.878173i	$0.658762\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−31.2132	−1.06622	−0.533111	−	0.846045i	$-0.678977\pi$
$857$	−31.2132	−1.06622	−0.533111	+	0.846045i	$0.678977\pi$
$858$	0	0
$859$	6.34315	0.216425	0.108213	−	0.994128i	$-0.465487\pi$
$859$	6.34315	0.216425	0.108213	+	0.994128i	$0.465487\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	44.0416	1.49919	0.749597	−	0.661894i	$-0.230247\pi$
$863$	44.0416	1.49919	0.749597	+	0.661894i	$0.230247\pi$
$864$	0	0
$865$	10.7279	0.364760
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−7.59798	−0.257744
$870$	0	0
$871$	−32.1421	−1.08909
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−2.82843	−0.0956183
$876$	0	0
$877$	18.7279	0.632397	0.316198	−	0.948693i	$-0.397593\pi$
$877$	18.7279	0.632397	0.316198	+	0.948693i	$0.397593\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	1.65685	0.0558208	0.0279104	−	0.999610i	$-0.491115\pi$
$881$	1.65685	0.0558208	0.0279104	+	0.999610i	$0.491115\pi$
$882$	0	0
$883$	−10.8284	−0.364406	−0.182203	−	0.983261i	$-0.558323\pi$
$883$	−10.8284	−0.364406	−0.182203	+	0.983261i	$0.558323\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	15.2721	0.512786	0.256393	−	0.966573i	$-0.417466\pi$
$887$	15.2721	0.512786	0.256393	+	0.966573i	$0.417466\pi$
$888$	0	0
$889$	37.9411	1.27250
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−8.48528	−0.283949
$894$	0	0
$895$	−8.48528	−0.283632
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	63.1127	2.10259
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−5.31371	−0.176634
$906$	0	0
$907$	48.0416	1.59520	0.797598	−	0.603189i	$-0.206104\pi$
$907$	48.0416	1.59520	0.797598	+	0.603189i	$0.206104\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	9.17157	0.303868	0.151934	−	0.988391i	$-0.451450\pi$
$911$	9.17157	0.303868	0.151934	+	0.988391i	$0.451450\pi$
$912$	0	0
$913$	−6.62742	−0.219335
$914$	0	0
$915$	0	0
$916$	0	0
$917$	20.6863	0.683122
$918$	0	0
$919$	−17.9411	−0.591823	−0.295912	−	0.955215i	$-0.595623\pi$
$919$	−17.9411	−0.591823	−0.295912	+	0.955215i	$0.595623\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−52.2843	−1.72096
$924$	0	0
$925$	10.2426	0.336776
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−41.3137	−1.35546	−0.677729	−	0.735311i	$-0.737036\pi$
$929$	−41.3137	−1.35546	−0.677729	+	0.735311i	$0.737036\pi$
$930$	0	0
$931$	−1.00000	−0.0327737
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−4.00000	−0.130814
$936$	0	0
$937$	55.4558	1.81166	0.905832	−	0.423638i	$-0.139247\pi$
$937$	55.4558	1.81166	0.905832	+	0.423638i	$0.139247\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−21.5980	−0.704074	−0.352037	−	0.935986i	$-0.614511\pi$
$941$	−21.5980	−0.704074	−0.352037	+	0.935986i	$0.614511\pi$
$942$	0	0
$943$	−3.31371	−0.107909
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−47.3137	−1.53749	−0.768744	−	0.639556i	$-0.779118\pi$
$947$	−47.3137	−1.53749	−0.768744	+	0.639556i	$0.779118\pi$
$948$	0	0
$949$	41.4558	1.34571
$950$	0	0
$951$	0	0
$952$	0	0
$953$	34.7279	1.12495	0.562474	−	0.826815i	$-0.309850\pi$
$953$	34.7279	1.12495	0.562474	+	0.826815i	$0.309850\pi$
$954$	0	0
$955$	−3.31371	−0.107229
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−20.2843	−0.655013
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−6.92893	−0.223050
$966$	0	0
$967$	37.6569	1.21096	0.605481	−	0.795859i	$-0.292981\pi$
$967$	37.6569	1.21096	0.605481	+	0.795859i	$0.292981\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−25.9411	−0.832490	−0.416245	−	0.909252i	$-0.636654\pi$
$971$	−25.9411	−0.832490	−0.416245	+	0.909252i	$0.636654\pi$
$972$	0	0
$973$	8.97056	0.287583
$974$	0	0
$975$	0	0
$976$	0	0
$977$	53.8406	1.72251	0.861257	−	0.508170i	$-0.169678\pi$
$977$	53.8406	1.72251	0.861257	+	0.508170i	$0.169678\pi$
$978$	0	0
$979$	2.62742	0.0839726
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−4.04163	−0.128908	−0.0644540	−	0.997921i	$-0.520531\pi$
$983$	−4.04163	−0.128908	−0.0644540	+	0.997921i	$0.520531\pi$
$984$	0	0
$985$	−25.3137	−0.806562
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−11.3137	−0.359755
$990$	0	0
$991$	44.7696	1.42215	0.711076	−	0.703115i	$-0.248208\pi$
$991$	44.7696	1.42215	0.711076	+	0.703115i	$0.248208\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−5.65685	−0.179334
$996$	0	0
$997$	12.8284	0.406280	0.203140	−	0.979150i	$-0.434885\pi$
$997$	12.8284	0.406280	0.203140	+	0.979150i	$0.434885\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6840.2.a.z.1.1		2
3.2	odd	2		760.2.a.f.1.1	✓	2
12.11	even	2		1520.2.a.m.1.2		2
15.2	even	4		3800.2.d.i.3649.3		4
15.8	even	4		3800.2.d.i.3649.1		4
15.14	odd	2		3800.2.a.n.1.2		2
24.5	odd	2		6080.2.a.bf.1.2		2
24.11	even	2		6080.2.a.bg.1.1		2
60.59	even	2		7600.2.a.ba.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.a.f.1.1	✓	2	3.2	odd	2
1520.2.a.m.1.2		2	12.11	even	2
3800.2.a.n.1.2		2	15.14	odd	2
3800.2.d.i.3649.1		4	15.8	even	4
3800.2.d.i.3649.3		4	15.2	even	4
6080.2.a.bf.1.2		2	24.5	odd	2
6080.2.a.bg.1.1		2	24.11	even	2
6840.2.a.z.1.1		2	1.1	even	1	trivial
7600.2.a.ba.1.1		2	60.59	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6840.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -2.82843 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -2.82843 q^{7} +0.828427 q^{11} +3.41421 q^{13} -4.82843 q^{17} -1.00000 q^{19} -4.00000 q^{23} +1.00000 q^{25} +0.828427 q^{29} -2.82843 q^{35} +10.2426 q^{37} +0.828427 q^{41} +2.82843 q^{43} +8.48528 q^{47} +1.00000 q^{49} -13.0711 q^{53} +0.828427 q^{55} -2.82843 q^{59} -1.65685 q^{61} +3.41421 q^{65} -9.41421 q^{67} -15.3137 q^{71} +12.1421 q^{73} -2.34315 q^{77} -9.17157 q^{79} -8.00000 q^{83} -4.82843 q^{85} +3.17157 q^{89} -9.65685 q^{91} -1.00000 q^{95} -2.24264 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5}+O(q^{10}) \) 2 * q + 2 * q^5	\( 2 q + 2 q^{5} - 4 q^{11} + 4 q^{13} - 4 q^{17} - 2 q^{19} - 8 q^{23} + 2 q^{25} - 4 q^{29} + 12 q^{37} - 4 q^{41} + 2 q^{49} - 12 q^{53} - 4 q^{55} + 8 q^{61} + 4 q^{65} - 16 q^{67} - 8 q^{71} - 4 q^{73} - 16 q^{77} - 24 q^{79} - 16 q^{83} - 4 q^{85} + 12 q^{89} - 8 q^{91} - 2 q^{95} + 4 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 - 4 * q^11 + 4 * q^13 - 4 * q^17 - 2 * q^19 - 8 * q^23 + 2 * q^25 - 4 * q^29 + 12 * q^37 - 4 * q^41 + 2 * q^49 - 12 * q^53 - 4 * q^55 + 8 * q^61 + 4 * q^65 - 16 * q^67 - 8 * q^71 - 4 * q^73 - 16 * q^77 - 24 * q^79 - 16 * q^83 - 4 * q^85 + 12 * q^89 - 8 * q^91 - 2 * q^95 + 4 * q^97

Introduction

L-functions

Modular forms

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