LMFDB - Embedded newform 6840.2.a.x.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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2280)
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)

$f(q)$	$=$	$q+1.00000 q^{5} -3.41421 q^{7} +O(q^{10})$	$q+1.00000 q^{5} -3.41421 q^{7} +1.41421 q^{11} +4.24264 q^{13} -2.82843 q^{17} -1.00000 q^{19} +4.82843 q^{23} +1.00000 q^{25} -2.24264 q^{29} -8.82843 q^{31} -3.41421 q^{35} -7.07107 q^{37} -2.24264 q^{41} -1.75736 q^{43} -4.82843 q^{47} +4.65685 q^{49} +12.4853 q^{53} +1.41421 q^{55} +2.82843 q^{59} -8.00000 q^{61} +4.24264 q^{65} -11.3137 q^{67} +5.17157 q^{71} -3.65685 q^{73} -4.82843 q^{77} -2.34315 q^{79} +6.00000 q^{83} -2.82843 q^{85} +13.5563 q^{89} -14.4853 q^{91} -1.00000 q^{95} +9.89949 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} - 4 q^{7}+O(q^{10}) $ 2 * q + 2 * q^5 - 4 * q^7	$ 2 q + 2 q^{5} - 4 q^{7} - 2 q^{19} + 4 q^{23} + 2 q^{25} + 4 q^{29} - 12 q^{31} - 4 q^{35} + 4 q^{41} - 12 q^{43} - 4 q^{47} - 2 q^{49} + 8 q^{53} - 16 q^{61} + 16 q^{71} + 4 q^{73} - 4 q^{77} - 16 q^{79} + 12 q^{83} - 4 q^{89} - 12 q^{91} - 2 q^{95}+O(q^{100}) $ 2 * q + 2 * q^5 - 4 * q^7 - 2 * q^19 + 4 * q^23 + 2 * q^25 + 4 * q^29 - 12 * q^31 - 4 * q^35 + 4 * q^41 - 12 * q^43 - 4 * q^47 - 2 * q^49 + 8 * q^53 - 16 * q^61 + 16 * q^71 + 4 * q^73 - 4 * q^77 - 16 * q^79 + 12 * q^83 - 4 * q^89 - 12 * q^91 - 2 * q^95

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
$7$	−3.41421	−1.29045	−0.645226	−	0.763992i	$-0.723237\pi$
$7$	−3.41421	−1.29045	−0.645226	+	0.763992i	$0.723237\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	1.41421	0.426401	0.213201	−	0.977008i	$-0.431611\pi$
$11$	1.41421	0.426401	0.213201	+	0.977008i	$0.431611\pi$
$12$	0	0
$13$	4.24264	1.17670	0.588348	−	0.808608i	$-0.299778\pi$
$13$	4.24264	1.17670	0.588348	+	0.808608i	$0.299778\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.82843	−0.685994	−0.342997	−	0.939336i	$-0.611442\pi$
$17$	−2.82843	−0.685994	−0.342997	+	0.939336i	$0.611442\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	0	0
$22$	0	0
$23$	4.82843	1.00680	0.503398	−	0.864054i	$-0.332083\pi$
$23$	4.82843	1.00680	0.503398	+	0.864054i	$0.332083\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−2.24264	−0.416448	−0.208224	−	0.978081i	$-0.566768\pi$
$29$	−2.24264	−0.416448	−0.208224	+	0.978081i	$0.566768\pi$
$30$	0	0
$31$	−8.82843	−1.58563	−0.792816	−	0.609461i	$-0.791386\pi$
$31$	−8.82843	−1.58563	−0.792816	+	0.609461i	$0.791386\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−3.41421	−0.577107
$36$	0	0
$37$	−7.07107	−1.16248	−0.581238	−	0.813733i	$-0.697432\pi$
$37$	−7.07107	−1.16248	−0.581238	+	0.813733i	$0.697432\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−2.24264	−0.350242	−0.175121	−	0.984547i	$-0.556032\pi$
$41$	−2.24264	−0.350242	−0.175121	+	0.984547i	$0.556032\pi$
$42$	0	0
$43$	−1.75736	−0.267995	−0.133997	−	0.990982i	$-0.542781\pi$
$43$	−1.75736	−0.267995	−0.133997	+	0.990982i	$0.542781\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.82843	−0.704298	−0.352149	−	0.935944i	$-0.614549\pi$
$47$	−4.82843	−0.704298	−0.352149	+	0.935944i	$0.614549\pi$
$48$	0	0
$49$	4.65685	0.665265
$50$	0	0
$51$	0	0
$52$	0	0
$53$	12.4853	1.71499	0.857493	−	0.514496i	$-0.172021\pi$
$53$	12.4853	1.71499	0.857493	+	0.514496i	$0.172021\pi$
$54$	0	0
$55$	1.41421	0.190693
$56$	0	0
$57$	0	0
$58$	0	0
$59$	2.82843	0.368230	0.184115	−	0.982905i	$-0.441058\pi$
$59$	2.82843	0.368230	0.184115	+	0.982905i	$0.441058\pi$
$60$	0	0
$61$	−8.00000	−1.02430	−0.512148	−	0.858898i	$-0.671150\pi$
$61$	−8.00000	−1.02430	−0.512148	+	0.858898i	$0.671150\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.24264	0.526235
$66$	0	0
$67$	−11.3137	−1.38219	−0.691095	−	0.722764i	$-0.742871\pi$
$67$	−11.3137	−1.38219	−0.691095	+	0.722764i	$0.742871\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	5.17157	0.613753	0.306876	−	0.951749i	$-0.400716\pi$
$71$	5.17157	0.613753	0.306876	+	0.951749i	$0.400716\pi$
$72$	0	0
$73$	−3.65685	−0.428002	−0.214001	−	0.976833i	$-0.568650\pi$
$73$	−3.65685	−0.428002	−0.214001	+	0.976833i	$0.568650\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−4.82843	−0.550250
$78$	0	0
$79$	−2.34315	−0.263624	−0.131812	−	0.991275i	$-0.542080\pi$
$79$	−2.34315	−0.263624	−0.131812	+	0.991275i	$0.542080\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	6.00000	0.658586	0.329293	−	0.944228i	$-0.393190\pi$
$83$	6.00000	0.658586	0.329293	+	0.944228i	$0.393190\pi$
$84$	0	0
$85$	−2.82843	−0.306786
$86$	0	0
$87$	0	0
$88$	0	0
$89$	13.5563	1.43697	0.718485	−	0.695542i	$-0.244836\pi$
$89$	13.5563	1.43697	0.718485	+	0.695542i	$0.244836\pi$
$90$	0	0
$91$	−14.4853	−1.51847
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.00000	−0.102598
$96$	0	0
$97$	9.89949	1.00514	0.502571	−	0.864536i	$-0.332388\pi$
$97$	9.89949	1.00514	0.502571	+	0.864536i	$0.332388\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	12.8284	1.27648	0.638238	−	0.769839i	$-0.279664\pi$
$101$	12.8284	1.27648	0.638238	+	0.769839i	$0.279664\pi$
$102$	0	0
$103$	−8.00000	−0.788263	−0.394132	−	0.919054i	$-0.628955\pi$
$103$	−8.00000	−0.788263	−0.394132	+	0.919054i	$0.628955\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−13.6569	−1.32026	−0.660129	−	0.751152i	$-0.729498\pi$
$107$	−13.6569	−1.32026	−0.660129	+	0.751152i	$0.729498\pi$
$108$	0	0
$109$	12.1421	1.16301	0.581503	−	0.813544i	$-0.302465\pi$
$109$	12.1421	1.16301	0.581503	+	0.813544i	$0.302465\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−9.65685	−0.908440	−0.454220	−	0.890889i	$-0.650082\pi$
$113$	−9.65685	−0.908440	−0.454220	+	0.890889i	$0.650082\pi$
$114$	0	0
$115$	4.82843	0.450253
$116$	0	0
$117$	0	0
$118$	0	0
$119$	9.65685	0.885242
$120$	0	0
$121$	−9.00000	−0.818182
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	16.9706	1.50589	0.752947	−	0.658081i	$-0.228632\pi$
$127$	16.9706	1.50589	0.752947	+	0.658081i	$0.228632\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−4.92893	−0.430643	−0.215321	−	0.976543i	$-0.569080\pi$
$131$	−4.92893	−0.430643	−0.215321	+	0.976543i	$0.569080\pi$
$132$	0	0
$133$	3.41421	0.296050
$134$	0	0
$135$	0	0
$136$	0	0
$137$	14.9706	1.27902	0.639511	−	0.768782i	$-0.279137\pi$
$137$	14.9706	1.27902	0.639511	+	0.768782i	$0.279137\pi$
$138$	0	0
$139$	−12.4853	−1.05899	−0.529494	−	0.848314i	$-0.677618\pi$
$139$	−12.4853	−1.05899	−0.529494	+	0.848314i	$0.677618\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	6.00000	0.501745
$144$	0	0
$145$	−2.24264	−0.186241
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−11.6569	−0.954967	−0.477483	−	0.878641i	$-0.658451\pi$
$149$	−11.6569	−0.954967	−0.477483	+	0.878641i	$0.658451\pi$
$150$	0	0
$151$	−11.1716	−0.909130	−0.454565	−	0.890714i	$-0.650205\pi$
$151$	−11.1716	−0.909130	−0.454565	+	0.890714i	$0.650205\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−8.82843	−0.709116
$156$	0	0
$157$	−8.14214	−0.649813	−0.324907	−	0.945746i	$-0.605333\pi$
$157$	−8.14214	−0.649813	−0.324907	+	0.945746i	$0.605333\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−16.4853	−1.29922
$162$	0	0
$163$	−0.100505	−0.00787216	−0.00393608	−	0.999992i	$-0.501253\pi$
$163$	−0.100505	−0.00787216	−0.00393608	+	0.999992i	$0.501253\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−15.6569	−1.21156	−0.605782	−	0.795631i	$-0.707140\pi$
$167$	−15.6569	−1.21156	−0.605782	+	0.795631i	$0.707140\pi$
$168$	0	0
$169$	5.00000	0.384615
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−13.6569	−1.03831	−0.519156	−	0.854680i	$-0.673754\pi$
$173$	−13.6569	−1.03831	−0.519156	+	0.854680i	$0.673754\pi$
$174$	0	0
$175$	−3.41421	−0.258090
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−23.7990	−1.77882	−0.889410	−	0.457110i	$-0.848884\pi$
$179$	−23.7990	−1.77882	−0.889410	+	0.457110i	$0.848884\pi$
$180$	0	0
$181$	4.82843	0.358894	0.179447	−	0.983768i	$-0.442569\pi$
$181$	4.82843	0.358894	0.179447	+	0.983768i	$0.442569\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−7.07107	−0.519875
$186$	0	0
$187$	−4.00000	−0.292509
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−13.4142	−0.970618	−0.485309	−	0.874343i	$-0.661293\pi$
$191$	−13.4142	−0.970618	−0.485309	+	0.874343i	$0.661293\pi$
$192$	0	0
$193$	23.0711	1.66069	0.830346	−	0.557248i	$-0.188143\pi$
$193$	23.0711	1.66069	0.830346	+	0.557248i	$0.188143\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−8.48528	−0.604551	−0.302276	−	0.953221i	$-0.597746\pi$
$197$	−8.48528	−0.604551	−0.302276	+	0.953221i	$0.597746\pi$
$198$	0	0
$199$	−14.1421	−1.00251	−0.501255	−	0.865300i	$-0.667128\pi$
$199$	−14.1421	−1.00251	−0.501255	+	0.865300i	$0.667128\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	7.65685	0.537406
$204$	0	0
$205$	−2.24264	−0.156633
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−1.41421	−0.0978232
$210$	0	0
$211$	−12.0000	−0.826114	−0.413057	−	0.910705i	$-0.635539\pi$
$211$	−12.0000	−0.826114	−0.413057	+	0.910705i	$0.635539\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1.75736	−0.119851
$216$	0	0
$217$	30.1421	2.04618
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−12.0000	−0.807207
$222$	0	0
$223$	−27.3137	−1.82906	−0.914531	−	0.404517i	$-0.867440\pi$
$223$	−27.3137	−1.82906	−0.914531	+	0.404517i	$0.867440\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5.31371	0.352683	0.176342	−	0.984329i	$-0.443574\pi$
$227$	5.31371	0.352683	0.176342	+	0.984329i	$0.443574\pi$
$228$	0	0
$229$	−18.6274	−1.23093	−0.615467	−	0.788163i	$-0.711033\pi$
$229$	−18.6274	−1.23093	−0.615467	+	0.788163i	$0.711033\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−7.65685	−0.501617	−0.250809	−	0.968037i	$-0.580696\pi$
$233$	−7.65685	−0.501617	−0.250809	+	0.968037i	$0.580696\pi$
$234$	0	0
$235$	−4.82843	−0.314972
$236$	0	0
$237$	0	0
$238$	0	0
$239$	12.2426	0.791911	0.395955	−	0.918270i	$-0.370414\pi$
$239$	12.2426	0.791911	0.395955	+	0.918270i	$0.370414\pi$
$240$	0	0
$241$	−14.9706	−0.964339	−0.482169	−	0.876078i	$-0.660151\pi$
$241$	−14.9706	−0.964339	−0.482169	+	0.876078i	$0.660151\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4.65685	0.297516
$246$	0	0
$247$	−4.24264	−0.269953
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−24.7279	−1.56081	−0.780406	−	0.625273i	$-0.784988\pi$
$251$	−24.7279	−1.56081	−0.780406	+	0.625273i	$0.784988\pi$
$252$	0	0
$253$	6.82843	0.429300
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−17.6569	−1.10140	−0.550702	−	0.834702i	$-0.685640\pi$
$257$	−17.6569	−1.10140	−0.550702	+	0.834702i	$0.685640\pi$
$258$	0	0
$259$	24.1421	1.50012
$260$	0	0
$261$	0	0
$262$	0	0
$263$	19.6569	1.21209	0.606047	−	0.795429i	$-0.292754\pi$
$263$	19.6569	1.21209	0.606047	+	0.795429i	$0.292754\pi$
$264$	0	0
$265$	12.4853	0.766965
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−4.10051	−0.250012	−0.125006	−	0.992156i	$-0.539895\pi$
$269$	−4.10051	−0.250012	−0.125006	+	0.992156i	$0.539895\pi$
$270$	0	0
$271$	2.14214	0.130125	0.0650627	−	0.997881i	$-0.479275\pi$
$271$	2.14214	0.130125	0.0650627	+	0.997881i	$0.479275\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	1.41421	0.0852803
$276$	0	0
$277$	−5.31371	−0.319270	−0.159635	−	0.987176i	$-0.551032\pi$
$277$	−5.31371	−0.319270	−0.159635	+	0.987176i	$0.551032\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−14.2426	−0.849645	−0.424822	−	0.905277i	$-0.639663\pi$
$281$	−14.2426	−0.849645	−0.424822	+	0.905277i	$0.639663\pi$
$282$	0	0
$283$	−4.10051	−0.243750	−0.121875	−	0.992545i	$-0.538891\pi$
$283$	−4.10051	−0.243750	−0.121875	+	0.992545i	$0.538891\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	7.65685	0.451970
$288$	0	0
$289$	−9.00000	−0.529412
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−4.00000	−0.233682	−0.116841	−	0.993151i	$-0.537277\pi$
$293$	−4.00000	−0.233682	−0.116841	+	0.993151i	$0.537277\pi$
$294$	0	0
$295$	2.82843	0.164677
$296$	0	0
$297$	0	0
$298$	0	0
$299$	20.4853	1.18469
$300$	0	0
$301$	6.00000	0.345834
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−8.00000	−0.458079
$306$	0	0
$307$	1.17157	0.0668652	0.0334326	−	0.999441i	$-0.489356\pi$
$307$	1.17157	0.0668652	0.0334326	+	0.999441i	$0.489356\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	9.41421	0.533831	0.266916	−	0.963720i	$-0.413996\pi$
$311$	9.41421	0.533831	0.266916	+	0.963720i	$0.413996\pi$
$312$	0	0
$313$	29.7990	1.68434	0.842169	−	0.539213i	$-0.181278\pi$
$313$	29.7990	1.68434	0.842169	+	0.539213i	$0.181278\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−8.48528	−0.476581	−0.238290	−	0.971194i	$-0.576587\pi$
$317$	−8.48528	−0.476581	−0.238290	+	0.971194i	$0.576587\pi$
$318$	0	0
$319$	−3.17157	−0.177574
$320$	0	0
$321$	0	0
$322$	0	0
$323$	2.82843	0.157378
$324$	0	0
$325$	4.24264	0.235339
$326$	0	0
$327$	0	0
$328$	0	0
$329$	16.4853	0.908863
$330$	0	0
$331$	−12.8284	−0.705114	−0.352557	−	0.935790i	$-0.614688\pi$
$331$	−12.8284	−0.705114	−0.352557	+	0.935790i	$0.614688\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−11.3137	−0.618134
$336$	0	0
$337$	−1.89949	−0.103472	−0.0517360	−	0.998661i	$-0.516475\pi$
$337$	−1.89949	−0.103472	−0.0517360	+	0.998661i	$0.516475\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−12.4853	−0.676116
$342$	0	0
$343$	8.00000	0.431959
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−0.343146	−0.0184210	−0.00921051	−	0.999958i	$-0.502932\pi$
$347$	−0.343146	−0.0184210	−0.00921051	+	0.999958i	$0.502932\pi$
$348$	0	0
$349$	−22.0000	−1.17763	−0.588817	−	0.808267i	$-0.700406\pi$
$349$	−22.0000	−1.17763	−0.588817	+	0.808267i	$0.700406\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−14.0000	−0.745145	−0.372572	−	0.928003i	$-0.621524\pi$
$353$	−14.0000	−0.745145	−0.372572	+	0.928003i	$0.621524\pi$
$354$	0	0
$355$	5.17157	0.274479
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−20.0416	−1.05776	−0.528878	−	0.848698i	$-0.677387\pi$
$359$	−20.0416	−1.05776	−0.528878	+	0.848698i	$0.677387\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−3.65685	−0.191408
$366$	0	0
$367$	−5.07107	−0.264708	−0.132354	−	0.991203i	$-0.542254\pi$
$367$	−5.07107	−0.264708	−0.132354	+	0.991203i	$0.542254\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−42.6274	−2.21311
$372$	0	0
$373$	−28.0416	−1.45194	−0.725970	−	0.687726i	$-0.758609\pi$
$373$	−28.0416	−1.45194	−0.725970	+	0.687726i	$0.758609\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−9.51472	−0.490033
$378$	0	0
$379$	−10.4853	−0.538593	−0.269296	−	0.963057i	$-0.586791\pi$
$379$	−10.4853	−0.538593	−0.269296	+	0.963057i	$0.586791\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	23.3137	1.19127	0.595637	−	0.803253i	$-0.296900\pi$
$383$	23.3137	1.19127	0.595637	+	0.803253i	$0.296900\pi$
$384$	0	0
$385$	−4.82843	−0.246079
$386$	0	0
$387$	0	0
$388$	0	0
$389$	23.6569	1.19945	0.599725	−	0.800206i	$-0.295277\pi$
$389$	23.6569	1.19945	0.599725	+	0.800206i	$0.295277\pi$
$390$	0	0
$391$	−13.6569	−0.690657
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−2.34315	−0.117896
$396$	0	0
$397$	26.0000	1.30490	0.652451	−	0.757831i	$-0.273741\pi$
$397$	26.0000	1.30490	0.652451	+	0.757831i	$0.273741\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	11.2132	0.559961	0.279980	−	0.960006i	$-0.409672\pi$
$401$	11.2132	0.559961	0.279980	+	0.960006i	$0.409672\pi$
$402$	0	0
$403$	−37.4558	−1.86581
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−10.0000	−0.495682
$408$	0	0
$409$	−36.1421	−1.78711	−0.893557	−	0.448950i	$-0.851798\pi$
$409$	−36.1421	−1.78711	−0.893557	+	0.448950i	$0.851798\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−9.65685	−0.475183
$414$	0	0
$415$	6.00000	0.294528
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−28.2426	−1.37974	−0.689872	−	0.723932i	$-0.742333\pi$
$419$	−28.2426	−1.37974	−0.689872	+	0.723932i	$0.742333\pi$
$420$	0	0
$421$	25.3137	1.23371	0.616857	−	0.787075i	$-0.288406\pi$
$421$	25.3137	1.23371	0.616857	+	0.787075i	$0.288406\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2.82843	−0.137199
$426$	0	0
$427$	27.3137	1.32180
$428$	0	0
$429$	0	0
$430$	0	0
$431$	34.1421	1.64457	0.822284	−	0.569077i	$-0.192699\pi$
$431$	34.1421	1.64457	0.822284	+	0.569077i	$0.192699\pi$
$432$	0	0
$433$	11.7574	0.565023	0.282511	−	0.959264i	$-0.408833\pi$
$433$	11.7574	0.565023	0.282511	+	0.959264i	$0.408833\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.82843	−0.230975
$438$	0	0
$439$	−4.68629	−0.223664	−0.111832	−	0.993727i	$-0.535672\pi$
$439$	−4.68629	−0.223664	−0.111832	+	0.993727i	$0.535672\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−4.14214	−0.196799	−0.0983994	−	0.995147i	$-0.531372\pi$
$443$	−4.14214	−0.196799	−0.0983994	+	0.995147i	$0.531372\pi$
$444$	0	0
$445$	13.5563	0.642633
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−26.5269	−1.25188	−0.625941	−	0.779870i	$-0.715285\pi$
$449$	−26.5269	−1.25188	−0.625941	+	0.779870i	$0.715285\pi$
$450$	0	0
$451$	−3.17157	−0.149344
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−14.4853	−0.679080
$456$	0	0
$457$	20.1421	0.942209	0.471105	−	0.882077i	$-0.343855\pi$
$457$	20.1421	0.942209	0.471105	+	0.882077i	$0.343855\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	20.6274	0.960715	0.480357	−	0.877073i	$-0.340507\pi$
$461$	20.6274	0.960715	0.480357	+	0.877073i	$0.340507\pi$
$462$	0	0
$463$	−25.7574	−1.19705	−0.598523	−	0.801106i	$-0.704245\pi$
$463$	−25.7574	−1.19705	−0.598523	+	0.801106i	$0.704245\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	20.1421	0.932067	0.466033	−	0.884767i	$-0.345683\pi$
$467$	20.1421	0.932067	0.466033	+	0.884767i	$0.345683\pi$
$468$	0	0
$469$	38.6274	1.78365
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−2.48528	−0.114273
$474$	0	0
$475$	−1.00000	−0.0458831
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−1.61522	−0.0738015	−0.0369007	−	0.999319i	$-0.511749\pi$
$479$	−1.61522	−0.0738015	−0.0369007	+	0.999319i	$0.511749\pi$
$480$	0	0
$481$	−30.0000	−1.36788
$482$	0	0
$483$	0	0
$484$	0	0
$485$	9.89949	0.449513
$486$	0	0
$487$	18.8284	0.853197	0.426599	−	0.904441i	$-0.359712\pi$
$487$	18.8284	0.853197	0.426599	+	0.904441i	$0.359712\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−31.7574	−1.43319	−0.716595	−	0.697490i	$-0.754300\pi$
$491$	−31.7574	−1.43319	−0.716595	+	0.697490i	$0.754300\pi$
$492$	0	0
$493$	6.34315	0.285681
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−17.6569	−0.792018
$498$	0	0
$499$	14.8284	0.663812	0.331906	−	0.943313i	$-0.392308\pi$
$499$	14.8284	0.663812	0.331906	+	0.943313i	$0.392308\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	26.2843	1.17196	0.585979	−	0.810326i	$-0.300710\pi$
$503$	26.2843	1.17196	0.585979	+	0.810326i	$0.300710\pi$
$504$	0	0
$505$	12.8284	0.570858
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−13.0711	−0.579365	−0.289682	−	0.957123i	$-0.593550\pi$
$509$	−13.0711	−0.579365	−0.289682	+	0.957123i	$0.593550\pi$
$510$	0	0
$511$	12.4853	0.552316
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−8.00000	−0.352522
$516$	0	0
$517$	−6.82843	−0.300314
$518$	0	0
$519$	0	0
$520$	0	0
$521$	20.3848	0.893073	0.446537	−	0.894765i	$-0.352657\pi$
$521$	20.3848	0.893073	0.446537	+	0.894765i	$0.352657\pi$
$522$	0	0
$523$	38.1421	1.66784	0.833920	−	0.551886i	$-0.186092\pi$
$523$	38.1421	1.66784	0.833920	+	0.551886i	$0.186092\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	24.9706	1.08773
$528$	0	0
$529$	0.313708	0.0136395
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−9.51472	−0.412128
$534$	0	0
$535$	−13.6569	−0.590437
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6.58579	0.283670
$540$	0	0
$541$	18.6863	0.803386	0.401693	−	0.915774i	$-0.368422\pi$
$541$	18.6863	0.803386	0.401693	+	0.915774i	$0.368422\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	12.1421	0.520112
$546$	0	0
$547$	28.4853	1.21794	0.608971	−	0.793192i	$-0.291582\pi$
$547$	28.4853	1.21794	0.608971	+	0.793192i	$0.291582\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	2.24264	0.0955397
$552$	0	0
$553$	8.00000	0.340195
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−10.9706	−0.464838	−0.232419	−	0.972616i	$-0.574664\pi$
$557$	−10.9706	−0.464838	−0.232419	+	0.972616i	$0.574664\pi$
$558$	0	0
$559$	−7.45584	−0.315349
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−33.3137	−1.40401	−0.702003	−	0.712174i	$-0.747711\pi$
$563$	−33.3137	−1.40401	−0.702003	+	0.712174i	$0.747711\pi$
$564$	0	0
$565$	−9.65685	−0.406267
$566$	0	0
$567$	0	0
$568$	0	0
$569$	4.10051	0.171902	0.0859511	−	0.996299i	$-0.472607\pi$
$569$	4.10051	0.171902	0.0859511	+	0.996299i	$0.472607\pi$
$570$	0	0
$571$	0.485281	0.0203084	0.0101542	−	0.999948i	$-0.496768\pi$
$571$	0.485281	0.0203084	0.0101542	+	0.999948i	$0.496768\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4.82843	0.201359
$576$	0	0
$577$	1.51472	0.0630586	0.0315293	−	0.999503i	$-0.489962\pi$
$577$	1.51472	0.0630586	0.0315293	+	0.999503i	$0.489962\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−20.4853	−0.849873
$582$	0	0
$583$	17.6569	0.731272
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−7.17157	−0.296002	−0.148001	−	0.988987i	$-0.547284\pi$
$587$	−7.17157	−0.296002	−0.148001	+	0.988987i	$0.547284\pi$
$588$	0	0
$589$	8.82843	0.363769
$590$	0	0
$591$	0	0
$592$	0	0
$593$	21.3137	0.875249	0.437625	−	0.899158i	$-0.355820\pi$
$593$	21.3137	0.875249	0.437625	+	0.899158i	$0.355820\pi$
$594$	0	0
$595$	9.65685	0.395892
$596$	0	0
$597$	0	0
$598$	0	0
$599$	16.9706	0.693398	0.346699	−	0.937976i	$-0.387302\pi$
$599$	16.9706	0.693398	0.346699	+	0.937976i	$0.387302\pi$
$600$	0	0
$601$	−40.1421	−1.63743	−0.818716	−	0.574199i	$-0.805314\pi$
$601$	−40.1421	−1.63743	−0.818716	+	0.574199i	$0.805314\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−9.00000	−0.365902
$606$	0	0
$607$	34.1421	1.38579	0.692893	−	0.721040i	$-0.256336\pi$
$607$	34.1421	1.38579	0.692893	+	0.721040i	$0.256336\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−20.4853	−0.828746
$612$	0	0
$613$	0.142136	0.00574080	0.00287040	−	0.999996i	$-0.499086\pi$
$613$	0.142136	0.00574080	0.00287040	+	0.999996i	$0.499086\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	31.1127	1.25255	0.626275	−	0.779602i	$-0.284579\pi$
$617$	31.1127	1.25255	0.626275	+	0.779602i	$0.284579\pi$
$618$	0	0
$619$	−5.17157	−0.207863	−0.103932	−	0.994584i	$-0.533142\pi$
$619$	−5.17157	−0.207863	−0.103932	+	0.994584i	$0.533142\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−46.2843	−1.85434
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	20.0000	0.797452
$630$	0	0
$631$	8.97056	0.357112	0.178556	−	0.983930i	$-0.442857\pi$
$631$	8.97056	0.357112	0.178556	+	0.983930i	$0.442857\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	16.9706	0.673456
$636$	0	0
$637$	19.7574	0.782815
$638$	0	0
$639$	0	0
$640$	0	0
$641$	34.2426	1.35250	0.676251	−	0.736671i	$-0.263603\pi$
$641$	34.2426	1.35250	0.676251	+	0.736671i	$0.263603\pi$
$642$	0	0
$643$	1.75736	0.0693035	0.0346517	−	0.999399i	$-0.488968\pi$
$643$	1.75736	0.0693035	0.0346517	+	0.999399i	$0.488968\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−20.3431	−0.799772	−0.399886	−	0.916565i	$-0.630950\pi$
$647$	−20.3431	−0.799772	−0.399886	+	0.916565i	$0.630950\pi$
$648$	0	0
$649$	4.00000	0.157014
$650$	0	0
$651$	0	0
$652$	0	0
$653$	12.4853	0.488587	0.244293	−	0.969701i	$-0.421444\pi$
$653$	12.4853	0.488587	0.244293	+	0.969701i	$0.421444\pi$
$654$	0	0
$655$	−4.92893	−0.192589
$656$	0	0
$657$	0	0
$658$	0	0
$659$	6.34315	0.247094	0.123547	−	0.992339i	$-0.460573\pi$
$659$	6.34315	0.247094	0.123547	+	0.992339i	$0.460573\pi$
$660$	0	0
$661$	25.1127	0.976771	0.488385	−	0.872628i	$-0.337586\pi$
$661$	25.1127	0.976771	0.488385	+	0.872628i	$0.337586\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	3.41421	0.132398
$666$	0	0
$667$	−10.8284	−0.419278
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−11.3137	−0.436761
$672$	0	0
$673$	−29.8995	−1.15254	−0.576270	−	0.817259i	$-0.695492\pi$
$673$	−29.8995	−1.15254	−0.576270	+	0.817259i	$0.695492\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−39.7990	−1.52960	−0.764800	−	0.644268i	$-0.777162\pi$
$677$	−39.7990	−1.52960	−0.764800	+	0.644268i	$0.777162\pi$
$678$	0	0
$679$	−33.7990	−1.29709
$680$	0	0
$681$	0	0
$682$	0	0
$683$	20.6863	0.791539	0.395769	−	0.918350i	$-0.370478\pi$
$683$	20.6863	0.791539	0.395769	+	0.918350i	$0.370478\pi$
$684$	0	0
$685$	14.9706	0.571996
$686$	0	0
$687$	0	0
$688$	0	0
$689$	52.9706	2.01802
$690$	0	0
$691$	−31.1127	−1.18358	−0.591791	−	0.806091i	$-0.701579\pi$
$691$	−31.1127	−1.18358	−0.591791	+	0.806091i	$0.701579\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−12.4853	−0.473594
$696$	0	0
$697$	6.34315	0.240264
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−0.343146	−0.0129604	−0.00648022	−	0.999979i	$-0.502063\pi$
$701$	−0.343146	−0.0129604	−0.00648022	+	0.999979i	$0.502063\pi$
$702$	0	0
$703$	7.07107	0.266690
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−43.7990	−1.64723
$708$	0	0
$709$	−18.3431	−0.688891	−0.344446	−	0.938806i	$-0.611933\pi$
$709$	−18.3431	−0.688891	−0.344446	+	0.938806i	$0.611933\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−42.6274	−1.59641
$714$	0	0
$715$	6.00000	0.224387
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−46.3848	−1.72986	−0.864930	−	0.501892i	$-0.832637\pi$
$719$	−46.3848	−1.72986	−0.864930	+	0.501892i	$0.832637\pi$
$720$	0	0
$721$	27.3137	1.01722
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−2.24264	−0.0832896
$726$	0	0
$727$	8.38478	0.310974	0.155487	−	0.987838i	$-0.450305\pi$
$727$	8.38478	0.310974	0.155487	+	0.987838i	$0.450305\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	4.97056	0.183843
$732$	0	0
$733$	26.0000	0.960332	0.480166	−	0.877178i	$-0.340576\pi$
$733$	26.0000	0.960332	0.480166	+	0.877178i	$0.340576\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−16.0000	−0.589368
$738$	0	0
$739$	8.68629	0.319530	0.159765	−	0.987155i	$-0.448926\pi$
$739$	8.68629	0.319530	0.159765	+	0.987155i	$0.448926\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−45.9411	−1.68542	−0.842708	−	0.538371i	$-0.819040\pi$
$743$	−45.9411	−1.68542	−0.842708	+	0.538371i	$0.819040\pi$
$744$	0	0
$745$	−11.6569	−0.427074
$746$	0	0
$747$	0	0
$748$	0	0
$749$	46.6274	1.70373
$750$	0	0
$751$	−0.142136	−0.00518660	−0.00259330	−	0.999997i	$-0.500825\pi$
$751$	−0.142136	−0.00518660	−0.00259330	+	0.999997i	$0.500825\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−11.1716	−0.406575
$756$	0	0
$757$	46.7696	1.69987	0.849934	−	0.526889i	$-0.176642\pi$
$757$	46.7696	1.69987	0.849934	+	0.526889i	$0.176642\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−14.9706	−0.542682	−0.271341	−	0.962483i	$-0.587467\pi$
$761$	−14.9706	−0.542682	−0.271341	+	0.962483i	$0.587467\pi$
$762$	0	0
$763$	−41.4558	−1.50080
$764$	0	0
$765$	0	0
$766$	0	0
$767$	12.0000	0.433295
$768$	0	0
$769$	−19.6569	−0.708844	−0.354422	−	0.935086i	$-0.615322\pi$
$769$	−19.6569	−0.708844	−0.354422	+	0.935086i	$0.615322\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−24.4853	−0.880674	−0.440337	−	0.897832i	$-0.645141\pi$
$773$	−24.4853	−0.880674	−0.440337	+	0.897832i	$0.645141\pi$
$774$	0	0
$775$	−8.82843	−0.317126
$776$	0	0
$777$	0	0
$778$	0	0
$779$	2.24264	0.0803509
$780$	0	0
$781$	7.31371	0.261705
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−8.14214	−0.290605
$786$	0	0
$787$	−8.48528	−0.302468	−0.151234	−	0.988498i	$-0.548325\pi$
$787$	−8.48528	−0.302468	−0.151234	+	0.988498i	$0.548325\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	32.9706	1.17230
$792$	0	0
$793$	−33.9411	−1.20528
$794$	0	0
$795$	0	0
$796$	0	0
$797$	40.9706	1.45125	0.725626	−	0.688089i	$-0.241550\pi$
$797$	40.9706	1.45125	0.725626	+	0.688089i	$0.241550\pi$
$798$	0	0
$799$	13.6569	0.483145
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−5.17157	−0.182501
$804$	0	0
$805$	−16.4853	−0.581030
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−2.68629	−0.0944450	−0.0472225	−	0.998884i	$-0.515037\pi$
$809$	−2.68629	−0.0944450	−0.0472225	+	0.998884i	$0.515037\pi$
$810$	0	0
$811$	−48.2843	−1.69549	−0.847745	−	0.530404i	$-0.822040\pi$
$811$	−48.2843	−1.69549	−0.847745	+	0.530404i	$0.822040\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−0.100505	−0.00352054
$816$	0	0
$817$	1.75736	0.0614822
$818$	0	0
$819$	0	0
$820$	0	0
$821$	22.4853	0.784742	0.392371	−	0.919807i	$-0.371655\pi$
$821$	22.4853	0.784742	0.392371	+	0.919807i	$0.371655\pi$
$822$	0	0
$823$	19.2132	0.669730	0.334865	−	0.942266i	$-0.391309\pi$
$823$	19.2132	0.669730	0.334865	+	0.942266i	$0.391309\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−26.0000	−0.904109	−0.452054	−	0.891990i	$-0.649309\pi$
$827$	−26.0000	−0.904109	−0.452054	+	0.891990i	$0.649309\pi$
$828$	0	0
$829$	−1.79899	−0.0624815	−0.0312408	−	0.999512i	$-0.509946\pi$
$829$	−1.79899	−0.0624815	−0.0312408	+	0.999512i	$0.509946\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−13.1716	−0.456368
$834$	0	0
$835$	−15.6569	−0.541828
$836$	0	0
$837$	0	0
$838$	0	0
$839$	23.5147	0.811818	0.405909	−	0.913913i	$-0.366955\pi$
$839$	23.5147	0.811818	0.405909	+	0.913913i	$0.366955\pi$
$840$	0	0
$841$	−23.9706	−0.826571
$842$	0	0
$843$	0	0
$844$	0	0
$845$	5.00000	0.172005
$846$	0	0
$847$	30.7279	1.05582
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−34.1421	−1.17038
$852$	0	0
$853$	−47.4558	−1.62486	−0.812429	−	0.583061i	$-0.801855\pi$
$853$	−47.4558	−1.62486	−0.812429	+	0.583061i	$0.801855\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−44.0833	−1.50586	−0.752928	−	0.658103i	$-0.771359\pi$
$857$	−44.0833	−1.50586	−0.752928	+	0.658103i	$0.771359\pi$
$858$	0	0
$859$	49.9411	1.70397	0.851985	−	0.523567i	$-0.175399\pi$
$859$	49.9411	1.70397	0.851985	+	0.523567i	$0.175399\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	36.0000	1.22545	0.612727	−	0.790295i	$-0.290072\pi$
$863$	36.0000	1.22545	0.612727	+	0.790295i	$0.290072\pi$
$864$	0	0
$865$	−13.6569	−0.464347
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−3.31371	−0.112410
$870$	0	0
$871$	−48.0000	−1.62642
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−3.41421	−0.115421
$876$	0	0
$877$	37.4142	1.26339	0.631694	−	0.775218i	$-0.282360\pi$
$877$	37.4142	1.26339	0.631694	+	0.775218i	$0.282360\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−33.1127	−1.11560	−0.557798	−	0.829977i	$-0.688353\pi$
$881$	−33.1127	−1.11560	−0.557798	+	0.829977i	$0.688353\pi$
$882$	0	0
$883$	9.27208	0.312030	0.156015	−	0.987755i	$-0.450135\pi$
$883$	9.27208	0.312030	0.156015	+	0.987755i	$0.450135\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−16.9706	−0.569816	−0.284908	−	0.958555i	$-0.591963\pi$
$887$	−16.9706	−0.569816	−0.284908	+	0.958555i	$0.591963\pi$
$888$	0	0
$889$	−57.9411	−1.94328
$890$	0	0
$891$	0	0
$892$	0	0
$893$	4.82843	0.161577
$894$	0	0
$895$	−23.7990	−0.795512
$896$	0	0
$897$	0	0
$898$	0	0
$899$	19.7990	0.660333
$900$	0	0
$901$	−35.3137	−1.17647
$902$	0	0
$903$	0	0
$904$	0	0
$905$	4.82843	0.160502
$906$	0	0
$907$	−13.8579	−0.460143	−0.230071	−	0.973174i	$-0.573896\pi$
$907$	−13.8579	−0.460143	−0.230071	+	0.973174i	$0.573896\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	20.0000	0.662630	0.331315	−	0.943520i	$-0.392508\pi$
$911$	20.0000	0.662630	0.331315	+	0.943520i	$0.392508\pi$
$912$	0	0
$913$	8.48528	0.280822
$914$	0	0
$915$	0	0
$916$	0	0
$917$	16.8284	0.555724
$918$	0	0
$919$	−17.6569	−0.582446	−0.291223	−	0.956655i	$-0.594062\pi$
$919$	−17.6569	−0.582446	−0.291223	+	0.956655i	$0.594062\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	21.9411	0.722201
$924$	0	0
$925$	−7.07107	−0.232495
$926$	0	0
$927$	0	0
$928$	0	0
$929$	24.8284	0.814594	0.407297	−	0.913296i	$-0.366471\pi$
$929$	24.8284	0.814594	0.407297	+	0.913296i	$0.366471\pi$
$930$	0	0
$931$	−4.65685	−0.152622
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−4.00000	−0.130814
$936$	0	0
$937$	−13.1127	−0.428373	−0.214187	−	0.976793i	$-0.568710\pi$
$937$	−13.1127	−0.428373	−0.214187	+	0.976793i	$0.568710\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−22.0416	−0.718537	−0.359268	−	0.933234i	$-0.616974\pi$
$941$	−22.0416	−0.718537	−0.359268	+	0.933234i	$0.616974\pi$
$942$	0	0
$943$	−10.8284	−0.352622
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−25.3137	−0.822585	−0.411292	−	0.911503i	$-0.634923\pi$
$947$	−25.3137	−0.822585	−0.411292	+	0.911503i	$0.634923\pi$
$948$	0	0
$949$	−15.5147	−0.503629
$950$	0	0
$951$	0	0
$952$	0	0
$953$	21.9411	0.710743	0.355371	−	0.934725i	$-0.384354\pi$
$953$	21.9411	0.710743	0.355371	+	0.934725i	$0.384354\pi$
$954$	0	0
$955$	−13.4142	−0.434074
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−51.1127	−1.65052
$960$	0	0
$961$	46.9411	1.51423
$962$	0	0
$963$	0	0
$964$	0	0
$965$	23.0711	0.742684
$966$	0	0
$967$	−7.89949	−0.254031	−0.127015	−	0.991901i	$-0.540540\pi$
$967$	−7.89949	−0.254031	−0.127015	+	0.991901i	$0.540540\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−0.686292	−0.0220241	−0.0110121	−	0.999939i	$-0.503505\pi$
$971$	−0.686292	−0.0220241	−0.0110121	+	0.999939i	$0.503505\pi$
$972$	0	0
$973$	42.6274	1.36657
$974$	0	0
$975$	0	0
$976$	0	0
$977$	20.2010	0.646288	0.323144	−	0.946350i	$-0.395260\pi$
$977$	20.2010	0.646288	0.323144	+	0.946350i	$0.395260\pi$
$978$	0	0
$979$	19.1716	0.612726
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−21.5980	−0.688869	−0.344434	−	0.938810i	$-0.611929\pi$
$983$	−21.5980	−0.688869	−0.344434	+	0.938810i	$0.611929\pi$
$984$	0	0
$985$	−8.48528	−0.270364
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−8.48528	−0.269816
$990$	0	0
$991$	−15.0294	−0.477426	−0.238713	−	0.971090i	$-0.576725\pi$
$991$	−15.0294	−0.477426	−0.238713	+	0.971090i	$0.576725\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−14.1421	−0.448336
$996$	0	0
$997$	49.7990	1.57715	0.788575	−	0.614939i	$-0.210819\pi$
$997$	49.7990	1.57715	0.788575	+	0.614939i	$0.210819\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6840.2.a.x.1.1		2
3.2	odd	2		2280.2.a.o.1.1	✓	2
12.11	even	2		4560.2.a.bg.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2280.2.a.o.1.1	✓	2	3.2	odd	2
4560.2.a.bg.1.2		2	12.11	even	2
6840.2.a.x.1.1		2	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:	\( N \)	\(=\)	\( 6840 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6840.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -3.41421 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -3.41421 q^{7} +1.41421 q^{11} +4.24264 q^{13} -2.82843 q^{17} -1.00000 q^{19} +4.82843 q^{23} +1.00000 q^{25} -2.24264 q^{29} -8.82843 q^{31} -3.41421 q^{35} -7.07107 q^{37} -2.24264 q^{41} -1.75736 q^{43} -4.82843 q^{47} +4.65685 q^{49} +12.4853 q^{53} +1.41421 q^{55} +2.82843 q^{59} -8.00000 q^{61} +4.24264 q^{65} -11.3137 q^{67} +5.17157 q^{71} -3.65685 q^{73} -4.82843 q^{77} -2.34315 q^{79} +6.00000 q^{83} -2.82843 q^{85} +13.5563 q^{89} -14.4853 q^{91} -1.00000 q^{95} +9.89949 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} - 4 q^{7}+O(q^{10}) \) 2 * q + 2 * q^5 - 4 * q^7	\( 2 q + 2 q^{5} - 4 q^{7} - 2 q^{19} + 4 q^{23} + 2 q^{25} + 4 q^{29} - 12 q^{31} - 4 q^{35} + 4 q^{41} - 12 q^{43} - 4 q^{47} - 2 q^{49} + 8 q^{53} - 16 q^{61} + 16 q^{71} + 4 q^{73} - 4 q^{77} - 16 q^{79} + 12 q^{83} - 4 q^{89} - 12 q^{91} - 2 q^{95}+O(q^{100}) \) 2 * q + 2 * q^5 - 4 * q^7 - 2 * q^19 + 4 * q^23 + 2 * q^25 + 4 * q^29 - 12 * q^31 - 4 * q^35 + 4 * q^41 - 12 * q^43 - 4 * q^47 - 2 * q^49 + 8 * q^53 - 16 * q^61 + 16 * q^71 + 4 * q^73 - 4 * q^77 - 16 * q^79 + 12 * q^83 - 4 * q^89 - 12 * q^91 - 2 * q^95

Introduction

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