LMFDB - Embedded newform 8624.2.a.bn.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8624,2,Mod(1,8624)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8624, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("8624.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8624 = 2^{4} \cdot 7^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8624.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:	$68.8629867032$
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 4312)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	8624.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843 q^{5} -3.00000 q^{9} +O(q^{10})$	$q-2.82843 q^{5} -3.00000 q^{9} -1.00000 q^{11} -1.41421 q^{13} +2.82843 q^{17} +4.24264 q^{19} +2.00000 q^{23} +3.00000 q^{25} +1.41421 q^{31} +2.00000 q^{37} -8.48528 q^{41} -2.00000 q^{43} +8.48528 q^{45} +7.07107 q^{47} +6.00000 q^{53} +2.82843 q^{55} +2.82843 q^{59} +1.41421 q^{61} +4.00000 q^{65} +4.00000 q^{67} -8.00000 q^{71} +2.82843 q^{73} -8.00000 q^{79} +9.00000 q^{81} +1.41421 q^{83} -8.00000 q^{85} +7.07107 q^{89} -12.0000 q^{95} -12.7279 q^{97} +3.00000 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 6 q^{9}+O(q^{10}) $ 2 * q - 6 * q^9	$ 2 q - 6 q^{9} - 2 q^{11} + 4 q^{23} + 6 q^{25} + 4 q^{37} - 4 q^{43} + 12 q^{53} + 8 q^{65} + 8 q^{67} - 16 q^{71} - 16 q^{79} + 18 q^{81} - 16 q^{85} - 24 q^{95} + 6 q^{99}+O(q^{100}) $ 2 * q - 6 * q^9 - 2 * q^11 + 4 * q^23 + 6 * q^25 + 4 * q^37 - 4 * q^43 + 12 * q^53 + 8 * q^65 + 8 * q^67 - 16 * q^71 - 16 * q^79 + 18 * q^81 - 16 * q^85 - 24 * q^95 + 6 * 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
$3$	0	0		−	1.00000i	$-0.5\pi$
$3$	0	0			1.00000i	$0.5\pi$
$4$	0	0
$5$	−2.82843	−1.26491	−0.632456	−	0.774597i	$-0.717953\pi$
$5$	−2.82843	−1.26491	−0.632456	+	0.774597i	$0.717953\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	−3.00000	−1.00000
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.41421	−0.392232	−0.196116	−	0.980581i	$-0.562833\pi$
$13$	−1.41421	−0.392232	−0.196116	+	0.980581i	$0.562833\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	2.82843	0.685994	0.342997	−	0.939336i	$-0.388558\pi$
$17$	2.82843	0.685994	0.342997	+	0.939336i	$0.388558\pi$
$18$	0	0
$19$	4.24264	0.973329	0.486664	−	0.873589i	$-0.338214\pi$
$19$	4.24264	0.973329	0.486664	+	0.873589i	$0.338214\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	2.00000	0.417029	0.208514	−	0.978019i	$-0.433137\pi$
$23$	2.00000	0.417029	0.208514	+	0.978019i	$0.433137\pi$
$24$	0	0
$25$	3.00000	0.600000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	0	0
$31$	1.41421	0.254000	0.127000	−	0.991903i	$-0.459465\pi$
$31$	1.41421	0.254000	0.127000	+	0.991903i	$0.459465\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.00000	0.328798	0.164399	−	0.986394i	$-0.447432\pi$
$37$	2.00000	0.328798	0.164399	+	0.986394i	$0.447432\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.48528	−1.32518	−0.662589	−	0.748983i	$-0.730542\pi$
$41$	−8.48528	−1.32518	−0.662589	+	0.748983i	$0.730542\pi$
$42$	0	0
$43$	−2.00000	−0.304997	−0.152499	−	0.988304i	$-0.548732\pi$
$43$	−2.00000	−0.304997	−0.152499	+	0.988304i	$0.548732\pi$
$44$	0	0
$45$	8.48528	1.26491
$46$	0	0
$47$	7.07107	1.03142	0.515711	−	0.856763i	$-0.327528\pi$
$47$	7.07107	1.03142	0.515711	+	0.856763i	$0.327528\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.00000	0.824163	0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000	0.824163	0.412082	+	0.911147i	$0.364802\pi$
$54$	0	0
$55$	2.82843	0.381385
$56$	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
$61$	1.41421	0.181071	0.0905357	−	0.995893i	$-0.471142\pi$
$61$	1.41421	0.181071	0.0905357	+	0.995893i	$0.471142\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.00000	0.496139
$66$	0	0
$67$	4.00000	0.488678	0.244339	−	0.969690i	$-0.421429\pi$
$67$	4.00000	0.488678	0.244339	+	0.969690i	$0.421429\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8.00000	−0.949425	−0.474713	−	0.880141i	$-0.657448\pi$
$71$	−8.00000	−0.949425	−0.474713	+	0.880141i	$0.657448\pi$
$72$	0	0
$73$	2.82843	0.331042	0.165521	−	0.986206i	$-0.447069\pi$
$73$	2.82843	0.331042	0.165521	+	0.986206i	$0.447069\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−8.00000	−0.900070	−0.450035	−	0.893011i	$-0.648589\pi$
$79$	−8.00000	−0.900070	−0.450035	+	0.893011i	$0.648589\pi$
$80$	0	0
$81$	9.00000	1.00000
$82$	0	0
$83$	1.41421	0.155230	0.0776151	−	0.996983i	$-0.475269\pi$
$83$	1.41421	0.155230	0.0776151	+	0.996983i	$0.475269\pi$
$84$	0	0
$85$	−8.00000	−0.867722
$86$	0	0
$87$	0	0
$88$	0	0
$89$	7.07107	0.749532	0.374766	−	0.927119i	$-0.377723\pi$
$89$	7.07107	0.749532	0.374766	+	0.927119i	$0.377723\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−12.0000	−1.23117
$96$	0	0
$97$	−12.7279	−1.29232	−0.646162	−	0.763200i	$-0.723627\pi$
$97$	−12.7279	−1.29232	−0.646162	+	0.763200i	$0.723627\pi$
$98$	0	0
$99$	3.00000	0.301511
$100$	0	0
$101$	15.5563	1.54791	0.773957	−	0.633238i	$-0.218274\pi$
$101$	15.5563	1.54791	0.773957	+	0.633238i	$0.218274\pi$
$102$	0	0
$103$	7.07107	0.696733	0.348367	−	0.937358i	$-0.386736\pi$
$103$	7.07107	0.696733	0.348367	+	0.937358i	$0.386736\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−10.0000	−0.966736	−0.483368	−	0.875417i	$-0.660587\pi$
$107$	−10.0000	−0.966736	−0.483368	+	0.875417i	$0.660587\pi$
$108$	0	0
$109$	−2.00000	−0.191565	−0.0957826	−	0.995402i	$-0.530535\pi$
$109$	−2.00000	−0.191565	−0.0957826	+	0.995402i	$0.530535\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	8.00000	0.752577	0.376288	−	0.926503i	$-0.377200\pi$
$113$	8.00000	0.752577	0.376288	+	0.926503i	$0.377200\pi$
$114$	0	0
$115$	−5.65685	−0.527504
$116$	0	0
$117$	4.24264	0.392232
$118$	0	0
$119$	0	0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	0	0
$124$	0	0
$125$	5.65685	0.505964
$126$	0	0
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	9.89949	0.864923	0.432461	−	0.901652i	$-0.357645\pi$
$131$	9.89949	0.864923	0.432461	+	0.901652i	$0.357645\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	−7.07107	−0.599760	−0.299880	−	0.953977i	$-0.596947\pi$
$139$	−7.07107	−0.599760	−0.299880	+	0.953977i	$0.596947\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	1.41421	0.118262
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	12.0000	0.983078	0.491539	−	0.870855i	$-0.336434\pi$
$149$	12.0000	0.983078	0.491539	+	0.870855i	$0.336434\pi$
$150$	0	0
$151$	4.00000	0.325515	0.162758	−	0.986666i	$-0.447961\pi$
$151$	4.00000	0.325515	0.162758	+	0.986666i	$0.447961\pi$
$152$	0	0
$153$	−8.48528	−0.685994
$154$	0	0
$155$	−4.00000	−0.321288
$156$	0	0
$157$	16.9706	1.35440	0.677199	−	0.735800i	$-0.263194\pi$
$157$	16.9706	1.35440	0.677199	+	0.735800i	$0.263194\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−24.0000	−1.87983	−0.939913	−	0.341415i	$-0.889094\pi$
$163$	−24.0000	−1.87983	−0.939913	+	0.341415i	$0.889094\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	2.82843	0.218870	0.109435	−	0.993994i	$-0.465096\pi$
$167$	2.82843	0.218870	0.109435	+	0.993994i	$0.465096\pi$
$168$	0	0
$169$	−11.0000	−0.846154
$170$	0	0
$171$	−12.7279	−0.973329
$172$	0	0
$173$	1.41421	0.107521	0.0537603	−	0.998554i	$-0.482879\pi$
$173$	1.41421	0.107521	0.0537603	+	0.998554i	$0.482879\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−12.0000	−0.896922	−0.448461	−	0.893802i	$-0.648028\pi$
$179$	−12.0000	−0.896922	−0.448461	+	0.893802i	$0.648028\pi$
$180$	0	0
$181$	8.48528	0.630706	0.315353	−	0.948974i	$-0.397877\pi$
$181$	8.48528	0.630706	0.315353	+	0.948974i	$0.397877\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−5.65685	−0.415900
$186$	0	0
$187$	−2.82843	−0.206835
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	0	0
$193$	−14.0000	−1.00774	−0.503871	−	0.863779i	$-0.668091\pi$
$193$	−14.0000	−1.00774	−0.503871	+	0.863779i	$0.668091\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−22.0000	−1.56744	−0.783718	−	0.621117i	$-0.786679\pi$
$197$	−22.0000	−1.56744	−0.783718	+	0.621117i	$0.786679\pi$
$198$	0	0
$199$	−15.5563	−1.10276	−0.551380	−	0.834254i	$-0.685899\pi$
$199$	−15.5563	−1.10276	−0.551380	+	0.834254i	$0.685899\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0	0
$205$	24.0000	1.67623
$206$	0	0
$207$	−6.00000	−0.417029
$208$	0	0
$209$	−4.24264	−0.293470
$210$	0	0
$211$	−10.0000	−0.688428	−0.344214	−	0.938891i	$-0.611855\pi$
$211$	−10.0000	−0.688428	−0.344214	+	0.938891i	$0.611855\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	5.65685	0.385794
$216$	0	0
$217$	0	0
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4.00000	−0.269069
$222$	0	0
$223$	−21.2132	−1.42054	−0.710271	−	0.703929i	$-0.751427\pi$
$223$	−21.2132	−1.42054	−0.710271	+	0.703929i	$0.751427\pi$
$224$	0	0
$225$	−9.00000	−0.600000
$226$	0	0
$227$	1.41421	0.0938647	0.0469323	−	0.998898i	$-0.485055\pi$
$227$	1.41421	0.0938647	0.0469323	+	0.998898i	$0.485055\pi$
$228$	0	0
$229$	−8.48528	−0.560723	−0.280362	−	0.959894i	$-0.590454\pi$
$229$	−8.48528	−0.560723	−0.280362	+	0.959894i	$0.590454\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6.00000	−0.393073	−0.196537	−	0.980497i	$-0.562969\pi$
$233$	−6.00000	−0.393073	−0.196537	+	0.980497i	$0.562969\pi$
$234$	0	0
$235$	−20.0000	−1.30466
$236$	0	0
$237$	0	0
$238$	0	0
$239$	28.0000	1.81117	0.905585	−	0.424165i	$-0.139432\pi$
$239$	28.0000	1.81117	0.905585	+	0.424165i	$0.139432\pi$
$240$	0	0
$241$	−8.48528	−0.546585	−0.273293	−	0.961931i	$-0.588113\pi$
$241$	−8.48528	−0.546585	−0.273293	+	0.961931i	$0.588113\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−6.00000	−0.381771
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−25.4558	−1.60676	−0.803379	−	0.595468i	$-0.796967\pi$
$251$	−25.4558	−1.60676	−0.803379	+	0.595468i	$0.796967\pi$
$252$	0	0
$253$	−2.00000	−0.125739
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−15.5563	−0.970378	−0.485189	−	0.874409i	$-0.661249\pi$
$257$	−15.5563	−0.970378	−0.485189	+	0.874409i	$0.661249\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−28.0000	−1.72655	−0.863277	−	0.504730i	$-0.831592\pi$
$263$	−28.0000	−1.72655	−0.863277	+	0.504730i	$0.831592\pi$
$264$	0	0
$265$	−16.9706	−1.04249
$266$	0	0
$267$	0	0
$268$	0	0
$269$	31.1127	1.89697	0.948487	−	0.316815i	$-0.102613\pi$
$269$	31.1127	1.89697	0.948487	+	0.316815i	$0.102613\pi$
$270$	0	0
$271$	8.48528	0.515444	0.257722	−	0.966219i	$-0.417028\pi$
$271$	8.48528	0.515444	0.257722	+	0.966219i	$0.417028\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.00000	−0.180907
$276$	0	0
$277$	−20.0000	−1.20168	−0.600842	−	0.799368i	$-0.705168\pi$
$277$	−20.0000	−1.20168	−0.600842	+	0.799368i	$0.705168\pi$
$278$	0	0
$279$	−4.24264	−0.254000
$280$	0	0
$281$	30.0000	1.78965	0.894825	−	0.446417i	$-0.147300\pi$
$281$	30.0000	1.78965	0.894825	+	0.446417i	$0.147300\pi$
$282$	0	0
$283$	−12.7279	−0.756596	−0.378298	−	0.925684i	$-0.623491\pi$
$283$	−12.7279	−0.756596	−0.378298	+	0.925684i	$0.623491\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0	0
$288$	0	0
$289$	−9.00000	−0.529412
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−9.89949	−0.578335	−0.289167	−	0.957279i	$-0.593378\pi$
$293$	−9.89949	−0.578335	−0.289167	+	0.957279i	$0.593378\pi$
$294$	0	0
$295$	−8.00000	−0.465778
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.82843	−0.163572
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−4.00000	−0.229039
$306$	0	0
$307$	29.6985	1.69498	0.847491	−	0.530810i	$-0.178112\pi$
$307$	29.6985	1.69498	0.847491	+	0.530810i	$0.178112\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−9.89949	−0.561349	−0.280674	−	0.959803i	$-0.590558\pi$
$311$	−9.89949	−0.561349	−0.280674	+	0.959803i	$0.590558\pi$
$312$	0	0
$313$	−12.7279	−0.719425	−0.359712	−	0.933063i	$-0.617125\pi$
$313$	−12.7279	−0.719425	−0.359712	+	0.933063i	$0.617125\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2.00000	0.112331	0.0561656	−	0.998421i	$-0.482113\pi$
$317$	2.00000	0.112331	0.0561656	+	0.998421i	$0.482113\pi$
$318$	0	0
$319$	0	0
$320$	0	0
$321$	0	0
$322$	0	0
$323$	12.0000	0.667698
$324$	0	0
$325$	−4.24264	−0.235339
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	16.0000	0.879440	0.439720	−	0.898135i	$-0.355078\pi$
$331$	16.0000	0.879440	0.439720	+	0.898135i	$0.355078\pi$
$332$	0	0
$333$	−6.00000	−0.328798
$334$	0	0
$335$	−11.3137	−0.618134
$336$	0	0
$337$	6.00000	0.326841	0.163420	−	0.986557i	$-0.447747\pi$
$337$	6.00000	0.326841	0.163420	+	0.986557i	$0.447747\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1.41421	−0.0765840
$342$	0	0
$343$	0	0
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−18.0000	−0.966291	−0.483145	−	0.875540i	$-0.660506\pi$
$347$	−18.0000	−0.966291	−0.483145	+	0.875540i	$0.660506\pi$
$348$	0	0
$349$	−26.8701	−1.43832	−0.719161	−	0.694844i	$-0.755473\pi$
$349$	−26.8701	−1.43832	−0.719161	+	0.694844i	$0.755473\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	15.5563	0.827981	0.413990	−	0.910281i	$-0.364135\pi$
$353$	15.5563	0.827981	0.413990	+	0.910281i	$0.364135\pi$
$354$	0	0
$355$	22.6274	1.20094
$356$	0	0
$357$	0	0
$358$	0	0
$359$	8.00000	0.422224	0.211112	−	0.977462i	$-0.432292\pi$
$359$	8.00000	0.422224	0.211112	+	0.977462i	$0.432292\pi$
$360$	0	0
$361$	−1.00000	−0.0526316
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−8.00000	−0.418739
$366$	0	0
$367$	21.2132	1.10732	0.553660	−	0.832743i	$-0.313231\pi$
$367$	21.2132	1.10732	0.553660	+	0.832743i	$0.313231\pi$
$368$	0	0
$369$	25.4558	1.32518
$370$	0	0
$371$	0	0
$372$	0	0
$373$	26.0000	1.34623	0.673114	−	0.739538i	$-0.264956\pi$
$373$	26.0000	1.34623	0.673114	+	0.739538i	$0.264956\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	0	0
$378$	0	0
$379$	−28.0000	−1.43826	−0.719132	−	0.694874i	$-0.755460\pi$
$379$	−28.0000	−1.43826	−0.719132	+	0.694874i	$0.755460\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−24.0416	−1.22847	−0.614235	−	0.789123i	$-0.710535\pi$
$383$	−24.0416	−1.22847	−0.614235	+	0.789123i	$0.710535\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	6.00000	0.304997
$388$	0	0
$389$	6.00000	0.304212	0.152106	−	0.988364i	$-0.451394\pi$
$389$	6.00000	0.304212	0.152106	+	0.988364i	$0.451394\pi$
$390$	0	0
$391$	5.65685	0.286079
$392$	0	0
$393$	0	0
$394$	0	0
$395$	22.6274	1.13851
$396$	0	0
$397$	−5.65685	−0.283909	−0.141955	−	0.989873i	$-0.545339\pi$
$397$	−5.65685	−0.283909	−0.141955	+	0.989873i	$0.545339\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−30.0000	−1.49813	−0.749064	−	0.662497i	$-0.769497\pi$
$401$	−30.0000	−1.49813	−0.749064	+	0.662497i	$0.769497\pi$
$402$	0	0
$403$	−2.00000	−0.0996271
$404$	0	0
$405$	−25.4558	−1.26491
$406$	0	0
$407$	−2.00000	−0.0991363
$408$	0	0
$409$	0	0		−	1.00000i	$-0.5\pi$
$409$	0	0			1.00000i	$0.5\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−4.00000	−0.196352
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−11.3137	−0.552711	−0.276355	−	0.961056i	$-0.589127\pi$
$419$	−11.3137	−0.552711	−0.276355	+	0.961056i	$0.589127\pi$
$420$	0	0
$421$	2.00000	0.0974740	0.0487370	−	0.998812i	$-0.484480\pi$
$421$	2.00000	0.0974740	0.0487370	+	0.998812i	$0.484480\pi$
$422$	0	0
$423$	−21.2132	−1.03142
$424$	0	0
$425$	8.48528	0.411597
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−36.0000	−1.73406	−0.867029	−	0.498257i	$-0.833974\pi$
$431$	−36.0000	−1.73406	−0.867029	+	0.498257i	$0.833974\pi$
$432$	0	0
$433$	−21.2132	−1.01944	−0.509721	−	0.860340i	$-0.670251\pi$
$433$	−21.2132	−1.01944	−0.509721	+	0.860340i	$0.670251\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	8.48528	0.405906
$438$	0	0
$439$	19.7990	0.944954	0.472477	−	0.881343i	$-0.343360\pi$
$439$	19.7990	0.944954	0.472477	+	0.881343i	$0.343360\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	20.0000	0.950229	0.475114	−	0.879924i	$-0.342407\pi$
$443$	20.0000	0.950229	0.475114	+	0.879924i	$0.342407\pi$
$444$	0	0
$445$	−20.0000	−0.948091
$446$	0	0
$447$	0	0
$448$	0	0
$449$	40.0000	1.88772	0.943858	−	0.330350i	$-0.107167\pi$
$449$	40.0000	1.88772	0.943858	+	0.330350i	$0.107167\pi$
$450$	0	0
$451$	8.48528	0.399556
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−10.0000	−0.467780	−0.233890	−	0.972263i	$-0.575146\pi$
$457$	−10.0000	−0.467780	−0.233890	+	0.972263i	$0.575146\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	1.41421	0.0658665	0.0329332	−	0.999458i	$-0.489515\pi$
$461$	1.41421	0.0658665	0.0329332	+	0.999458i	$0.489515\pi$
$462$	0	0
$463$	−24.0000	−1.11537	−0.557687	−	0.830051i	$-0.688311\pi$
$463$	−24.0000	−1.11537	−0.557687	+	0.830051i	$0.688311\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	2.00000	0.0919601
$474$	0	0
$475$	12.7279	0.583997
$476$	0	0
$477$	−18.0000	−0.824163
$478$	0	0
$479$	11.3137	0.516937	0.258468	−	0.966020i	$-0.416782\pi$
$479$	11.3137	0.516937	0.258468	+	0.966020i	$0.416782\pi$
$480$	0	0
$481$	−2.82843	−0.128965
$482$	0	0
$483$	0	0
$484$	0	0
$485$	36.0000	1.63468
$486$	0	0
$487$	34.0000	1.54069	0.770344	−	0.637629i	$-0.220085\pi$
$487$	34.0000	1.54069	0.770344	+	0.637629i	$0.220085\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−20.0000	−0.902587	−0.451294	−	0.892375i	$-0.649037\pi$
$491$	−20.0000	−0.902587	−0.451294	+	0.892375i	$0.649037\pi$
$492$	0	0
$493$	0	0
$494$	0	0
$495$	−8.48528	−0.381385
$496$	0	0
$497$	0	0
$498$	0	0
$499$	16.0000	0.716258	0.358129	−	0.933672i	$-0.383415\pi$
$499$	16.0000	0.716258	0.358129	+	0.933672i	$0.383415\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−19.7990	−0.882793	−0.441397	−	0.897312i	$-0.645517\pi$
$503$	−19.7990	−0.882793	−0.441397	+	0.897312i	$0.645517\pi$
$504$	0	0
$505$	−44.0000	−1.95797
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−14.1421	−0.626839	−0.313420	−	0.949615i	$-0.601475\pi$
$509$	−14.1421	−0.626839	−0.313420	+	0.949615i	$0.601475\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−20.0000	−0.881305
$516$	0	0
$517$	−7.07107	−0.310985
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−18.3848	−0.805452	−0.402726	−	0.915321i	$-0.631937\pi$
$521$	−18.3848	−0.805452	−0.402726	+	0.915321i	$0.631937\pi$
$522$	0	0
$523$	−21.2132	−0.927589	−0.463794	−	0.885943i	$-0.653512\pi$
$523$	−21.2132	−0.927589	−0.463794	+	0.885943i	$0.653512\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.00000	0.174243
$528$	0	0
$529$	−19.0000	−0.826087
$530$	0	0
$531$	−8.48528	−0.368230
$532$	0	0
$533$	12.0000	0.519778
$534$	0	0
$535$	28.2843	1.22284
$536$	0	0
$537$	0	0
$538$	0	0
$539$	0	0
$540$	0	0
$541$	22.0000	0.945854	0.472927	−	0.881102i	$-0.343197\pi$
$541$	22.0000	0.945854	0.472927	+	0.881102i	$0.343197\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	5.65685	0.242313
$546$	0	0
$547$	12.0000	0.513083	0.256541	−	0.966533i	$-0.417417\pi$
$547$	12.0000	0.513083	0.256541	+	0.966533i	$0.417417\pi$
$548$	0	0
$549$	−4.24264	−0.181071
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0	0
$556$	0	0
$557$	14.0000	0.593199	0.296600	−	0.955002i	$-0.404147\pi$
$557$	14.0000	0.593199	0.296600	+	0.955002i	$0.404147\pi$
$558$	0	0
$559$	2.82843	0.119630
$560$	0	0
$561$	0	0
$562$	0	0
$563$	26.8701	1.13244	0.566219	−	0.824255i	$-0.308406\pi$
$563$	26.8701	1.13244	0.566219	+	0.824255i	$0.308406\pi$
$564$	0	0
$565$	−22.6274	−0.951943
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−42.0000	−1.76073	−0.880366	−	0.474295i	$-0.842703\pi$
$569$	−42.0000	−1.76073	−0.880366	+	0.474295i	$0.842703\pi$
$570$	0	0
$571$	−12.0000	−0.502184	−0.251092	−	0.967963i	$-0.580790\pi$
$571$	−12.0000	−0.502184	−0.251092	+	0.967963i	$0.580790\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	6.00000	0.250217
$576$	0	0
$577$	1.41421	0.0588745	0.0294372	−	0.999567i	$-0.490628\pi$
$577$	1.41421	0.0588745	0.0294372	+	0.999567i	$0.490628\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−6.00000	−0.248495
$584$	0	0
$585$	−12.0000	−0.496139
$586$	0	0
$587$	−14.1421	−0.583708	−0.291854	−	0.956463i	$-0.594272\pi$
$587$	−14.1421	−0.583708	−0.291854	+	0.956463i	$0.594272\pi$
$588$	0	0
$589$	6.00000	0.247226
$590$	0	0
$591$	0	0
$592$	0	0
$593$	19.7990	0.813047	0.406524	−	0.913640i	$-0.366741\pi$
$593$	19.7990	0.813047	0.406524	+	0.913640i	$0.366741\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	8.00000	0.326871	0.163436	−	0.986554i	$-0.447742\pi$
$599$	8.00000	0.326871	0.163436	+	0.986554i	$0.447742\pi$
$600$	0	0
$601$	−5.65685	−0.230748	−0.115374	−	0.993322i	$-0.536807\pi$
$601$	−5.65685	−0.230748	−0.115374	+	0.993322i	$0.536807\pi$
$602$	0	0
$603$	−12.0000	−0.488678
$604$	0	0
$605$	−2.82843	−0.114992
$606$	0	0
$607$	−11.3137	−0.459209	−0.229605	−	0.973284i	$-0.573743\pi$
$607$	−11.3137	−0.459209	−0.229605	+	0.973284i	$0.573743\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−10.0000	−0.404557
$612$	0	0
$613$	4.00000	0.161558	0.0807792	−	0.996732i	$-0.474259\pi$
$613$	4.00000	0.161558	0.0807792	+	0.996732i	$0.474259\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−4.00000	−0.161034	−0.0805170	−	0.996753i	$-0.525657\pi$
$617$	−4.00000	−0.161034	−0.0805170	+	0.996753i	$0.525657\pi$
$618$	0	0
$619$	5.65685	0.227368	0.113684	−	0.993517i	$-0.463735\pi$
$619$	5.65685	0.227368	0.113684	+	0.993517i	$0.463735\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−31.0000	−1.24000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	5.65685	0.225554
$630$	0	0
$631$	14.0000	0.557331	0.278666	−	0.960388i	$-0.410108\pi$
$631$	14.0000	0.557331	0.278666	+	0.960388i	$0.410108\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	45.2548	1.79588
$636$	0	0
$637$	0	0
$638$	0	0
$639$	24.0000	0.949425
$640$	0	0
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	0	0
$643$	−25.4558	−1.00388	−0.501940	−	0.864902i	$-0.667380\pi$
$643$	−25.4558	−1.00388	−0.501940	+	0.864902i	$0.667380\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−15.5563	−0.611583	−0.305792	−	0.952098i	$-0.598921\pi$
$647$	−15.5563	−0.611583	−0.305792	+	0.952098i	$0.598921\pi$
$648$	0	0
$649$	−2.82843	−0.111025
$650$	0	0
$651$	0	0
$652$	0	0
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	0	0
$655$	−28.0000	−1.09405
$656$	0	0
$657$	−8.48528	−0.331042
$658$	0	0
$659$	42.0000	1.63609	0.818044	−	0.575156i	$-0.195059\pi$
$659$	42.0000	1.63609	0.818044	+	0.575156i	$0.195059\pi$
$660$	0	0
$661$	8.48528	0.330039	0.165020	−	0.986290i	$-0.447231\pi$
$661$	8.48528	0.330039	0.165020	+	0.986290i	$0.447231\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−1.41421	−0.0545951
$672$	0	0
$673$	−22.0000	−0.848038	−0.424019	−	0.905653i	$-0.639381\pi$
$673$	−22.0000	−0.848038	−0.424019	+	0.905653i	$0.639381\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	49.4975	1.90234	0.951171	−	0.308664i	$-0.0998818\pi$
$677$	49.4975	1.90234	0.951171	+	0.308664i	$0.0998818\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	0	0
$682$	0	0
$683$	16.0000	0.612223	0.306111	−	0.951996i	$-0.400972\pi$
$683$	16.0000	0.612223	0.306111	+	0.951996i	$0.400972\pi$
$684$	0	0
$685$	16.9706	0.648412
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−8.48528	−0.323263
$690$	0	0
$691$	−42.4264	−1.61398	−0.806988	−	0.590567i	$-0.798904\pi$
$691$	−42.4264	−1.61398	−0.806988	+	0.590567i	$0.798904\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	20.0000	0.758643
$696$	0	0
$697$	−24.0000	−0.909065
$698$	0	0
$699$	0	0
$700$	0	0
$701$	18.0000	0.679851	0.339925	−	0.940452i	$-0.389598\pi$
$701$	18.0000	0.679851	0.339925	+	0.940452i	$0.389598\pi$
$702$	0	0
$703$	8.48528	0.320028
$704$	0	0
$705$	0	0
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−50.0000	−1.87779	−0.938895	−	0.344204i	$-0.888149\pi$
$709$	−50.0000	−1.87779	−0.938895	+	0.344204i	$0.888149\pi$
$710$	0	0
$711$	24.0000	0.900070
$712$	0	0
$713$	2.82843	0.105925
$714$	0	0
$715$	−4.00000	−0.149592
$716$	0	0
$717$	0	0
$718$	0	0
$719$	18.3848	0.685636	0.342818	−	0.939402i	$-0.388619\pi$
$719$	18.3848	0.685636	0.342818	+	0.939402i	$0.388619\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−4.24264	−0.157351	−0.0786754	−	0.996900i	$-0.525069\pi$
$727$	−4.24264	−0.157351	−0.0786754	+	0.996900i	$0.525069\pi$
$728$	0	0
$729$	−27.0000	−1.00000
$730$	0	0
$731$	−5.65685	−0.209226
$732$	0	0
$733$	7.07107	0.261176	0.130588	−	0.991437i	$-0.458314\pi$
$733$	7.07107	0.261176	0.130588	+	0.991437i	$0.458314\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4.00000	−0.147342
$738$	0	0
$739$	36.0000	1.32428	0.662141	−	0.749380i	$-0.269648\pi$
$739$	36.0000	1.32428	0.662141	+	0.749380i	$0.269648\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−44.0000	−1.61420	−0.807102	−	0.590412i	$-0.798965\pi$
$743$	−44.0000	−1.61420	−0.807102	+	0.590412i	$0.798965\pi$
$744$	0	0
$745$	−33.9411	−1.24351
$746$	0	0
$747$	−4.24264	−0.155230
$748$	0	0
$749$	0	0
$750$	0	0
$751$	10.0000	0.364905	0.182453	−	0.983215i	$-0.441596\pi$
$751$	10.0000	0.364905	0.182453	+	0.983215i	$0.441596\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−11.3137	−0.411748
$756$	0	0
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	5.65685	0.205061	0.102530	−	0.994730i	$-0.467306\pi$
$761$	5.65685	0.205061	0.102530	+	0.994730i	$0.467306\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	24.0000	0.867722
$766$	0	0
$767$	−4.00000	−0.144432
$768$	0	0
$769$	33.9411	1.22395	0.611974	−	0.790878i	$-0.290376\pi$
$769$	33.9411	1.22395	0.611974	+	0.790878i	$0.290376\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	28.2843	1.01731	0.508657	−	0.860969i	$-0.330142\pi$
$773$	28.2843	1.01731	0.508657	+	0.860969i	$0.330142\pi$
$774$	0	0
$775$	4.24264	0.152400
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−36.0000	−1.28983
$780$	0	0
$781$	8.00000	0.286263
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−48.0000	−1.71319
$786$	0	0
$787$	−9.89949	−0.352879	−0.176439	−	0.984311i	$-0.556458\pi$
$787$	−9.89949	−0.352879	−0.176439	+	0.984311i	$0.556458\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−2.00000	−0.0710221
$794$	0	0
$795$	0	0
$796$	0	0
$797$	22.6274	0.801504	0.400752	−	0.916187i	$-0.368749\pi$
$797$	22.6274	0.801504	0.400752	+	0.916187i	$0.368749\pi$
$798$	0	0
$799$	20.0000	0.707549
$800$	0	0
$801$	−21.2132	−0.749532
$802$	0	0
$803$	−2.82843	−0.0998130
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−26.0000	−0.914111	−0.457056	−	0.889438i	$-0.651096\pi$
$809$	−26.0000	−0.914111	−0.457056	+	0.889438i	$0.651096\pi$
$810$	0	0
$811$	−46.6690	−1.63877	−0.819386	−	0.573242i	$-0.805685\pi$
$811$	−46.6690	−1.63877	−0.819386	+	0.573242i	$0.805685\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	67.8823	2.37781
$816$	0	0
$817$	−8.48528	−0.296862
$818$	0	0
$819$	0	0
$820$	0	0
$821$	4.00000	0.139601	0.0698005	−	0.997561i	$-0.477764\pi$
$821$	4.00000	0.139601	0.0698005	+	0.997561i	$0.477764\pi$
$822$	0	0
$823$	50.0000	1.74289	0.871445	−	0.490493i	$-0.163183\pi$
$823$	50.0000	1.74289	0.871445	+	0.490493i	$0.163183\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−44.0000	−1.53003	−0.765015	−	0.644013i	$-0.777268\pi$
$827$	−44.0000	−1.53003	−0.765015	+	0.644013i	$0.777268\pi$
$828$	0	0
$829$	31.1127	1.08059	0.540294	−	0.841476i	$-0.318313\pi$
$829$	31.1127	1.08059	0.540294	+	0.841476i	$0.318313\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	0	0
$834$	0	0
$835$	−8.00000	−0.276851
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−41.0122	−1.41590	−0.707949	−	0.706263i	$-0.750379\pi$
$839$	−41.0122	−1.41590	−0.707949	+	0.706263i	$0.750379\pi$
$840$	0	0
$841$	−29.0000	−1.00000
$842$	0	0
$843$	0	0
$844$	0	0
$845$	31.1127	1.07031
$846$	0	0
$847$	0	0
$848$	0	0
$849$	0	0
$850$	0	0
$851$	4.00000	0.137118
$852$	0	0
$853$	−21.2132	−0.726326	−0.363163	−	0.931726i	$-0.618303\pi$
$853$	−21.2132	−0.726326	−0.363163	+	0.931726i	$0.618303\pi$
$854$	0	0
$855$	36.0000	1.23117
$856$	0	0
$857$	−39.5980	−1.35264	−0.676321	−	0.736607i	$-0.736427\pi$
$857$	−39.5980	−1.35264	−0.676321	+	0.736607i	$0.736427\pi$
$858$	0	0
$859$	−2.82843	−0.0965047	−0.0482523	−	0.998835i	$-0.515365\pi$
$859$	−2.82843	−0.0965047	−0.0482523	+	0.998835i	$0.515365\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−22.0000	−0.748889	−0.374444	−	0.927249i	$-0.622167\pi$
$863$	−22.0000	−0.748889	−0.374444	+	0.927249i	$0.622167\pi$
$864$	0	0
$865$	−4.00000	−0.136004
$866$	0	0
$867$	0	0
$868$	0	0
$869$	8.00000	0.271381
$870$	0	0
$871$	−5.65685	−0.191675
$872$	0	0
$873$	38.1838	1.29232
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	55.1543	1.85820	0.929098	−	0.369833i	$-0.120585\pi$
$881$	55.1543	1.85820	0.929098	+	0.369833i	$0.120585\pi$
$882$	0	0
$883$	44.0000	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−28.2843	−0.949693	−0.474846	−	0.880069i	$-0.657496\pi$
$887$	−28.2843	−0.949693	−0.474846	+	0.880069i	$0.657496\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	−9.00000	−0.301511
$892$	0	0
$893$	30.0000	1.00391
$894$	0	0
$895$	33.9411	1.13453
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	0	0
$901$	16.9706	0.565371
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−24.0000	−0.797787
$906$	0	0
$907$	12.0000	0.398453	0.199227	−	0.979953i	$-0.436157\pi$
$907$	12.0000	0.398453	0.199227	+	0.979953i	$0.436157\pi$
$908$	0	0
$909$	−46.6690	−1.54791
$910$	0	0
$911$	2.00000	0.0662630	0.0331315	−	0.999451i	$-0.489452\pi$
$911$	2.00000	0.0662630	0.0331315	+	0.999451i	$0.489452\pi$
$912$	0	0
$913$	−1.41421	−0.0468036
$914$	0	0
$915$	0	0
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0		−	1.00000i	$-0.5\pi$
$919$	0	0			1.00000i	$0.5\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	11.3137	0.372395
$924$	0	0
$925$	6.00000	0.197279
$926$	0	0
$927$	−21.2132	−0.696733
$928$	0	0
$929$	−9.89949	−0.324792	−0.162396	−	0.986726i	$-0.551922\pi$
$929$	−9.89949	−0.324792	−0.162396	+	0.986726i	$0.551922\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	0	0
$934$	0	0
$935$	8.00000	0.261628
$936$	0	0
$937$	42.4264	1.38601	0.693005	−	0.720933i	$-0.256286\pi$
$937$	42.4264	1.38601	0.693005	+	0.720933i	$0.256286\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−46.6690	−1.52137	−0.760684	−	0.649123i	$-0.775136\pi$
$941$	−46.6690	−1.52137	−0.760684	+	0.649123i	$0.775136\pi$
$942$	0	0
$943$	−16.9706	−0.552638
$944$	0	0
$945$	0	0
$946$	0	0
$947$	4.00000	0.129983	0.0649913	−	0.997886i	$-0.479298\pi$
$947$	4.00000	0.129983	0.0649913	+	0.997886i	$0.479298\pi$
$948$	0	0
$949$	−4.00000	−0.129845
$950$	0	0
$951$	0	0
$952$	0	0
$953$	34.0000	1.10137	0.550684	−	0.834714i	$-0.314367\pi$
$953$	34.0000	1.10137	0.550684	+	0.834714i	$0.314367\pi$
$954$	0	0
$955$	50.9117	1.64746
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−29.0000	−0.935484
$962$	0	0
$963$	30.0000	0.966736
$964$	0	0
$965$	39.5980	1.27470
$966$	0	0
$967$	−40.0000	−1.28631	−0.643157	−	0.765735i	$-0.722376\pi$
$967$	−40.0000	−1.28631	−0.643157	+	0.765735i	$0.722376\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	22.6274	0.726148	0.363074	−	0.931760i	$-0.381727\pi$
$971$	22.6274	0.726148	0.363074	+	0.931760i	$0.381727\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−8.00000	−0.255943	−0.127971	−	0.991778i	$-0.540847\pi$
$977$	−8.00000	−0.255943	−0.127971	+	0.991778i	$0.540847\pi$
$978$	0	0
$979$	−7.07107	−0.225992
$980$	0	0
$981$	6.00000	0.191565
$982$	0	0
$983$	21.2132	0.676596	0.338298	−	0.941039i	$-0.390149\pi$
$983$	21.2132	0.676596	0.338298	+	0.941039i	$0.390149\pi$
$984$	0	0
$985$	62.2254	1.98267
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−4.00000	−0.127193
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	44.0000	1.39489
$996$	0	0
$997$	18.3848	0.582252	0.291126	−	0.956685i	$-0.405970\pi$
$997$	18.3848	0.582252	0.291126	+	0.956685i	$0.405970\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8624.2.a.bn.1.1		2
4.3	odd	2		4312.2.a.p.1.1	✓	2
7.6	odd	2	inner	8624.2.a.bn.1.2		2
28.27	even	2		4312.2.a.p.1.2	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4312.2.a.p.1.1	✓	2	4.3	odd	2
4312.2.a.p.1.2	yes	2	28.27	even	2
8624.2.a.bn.1.1		2	1.1	even	1	trivial
8624.2.a.bn.1.2		2	7.6	odd	2	inner

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:	\( N \)	\(=\)	\( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8624.a (trivial)

\(f(q)\)	\(=\)	\(q-2.82843 q^{5} -3.00000 q^{9} +O(q^{10})\)	\(q-2.82843 q^{5} -3.00000 q^{9} -1.00000 q^{11} -1.41421 q^{13} +2.82843 q^{17} +4.24264 q^{19} +2.00000 q^{23} +3.00000 q^{25} +1.41421 q^{31} +2.00000 q^{37} -8.48528 q^{41} -2.00000 q^{43} +8.48528 q^{45} +7.07107 q^{47} +6.00000 q^{53} +2.82843 q^{55} +2.82843 q^{59} +1.41421 q^{61} +4.00000 q^{65} +4.00000 q^{67} -8.00000 q^{71} +2.82843 q^{73} -8.00000 q^{79} +9.00000 q^{81} +1.41421 q^{83} -8.00000 q^{85} +7.07107 q^{89} -12.0000 q^{95} -12.7279 q^{97} +3.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 6 q^{9}+O(q^{10}) \) 2 * q - 6 * q^9	\( 2 q - 6 q^{9} - 2 q^{11} + 4 q^{23} + 6 q^{25} + 4 q^{37} - 4 q^{43} + 12 q^{53} + 8 q^{65} + 8 q^{67} - 16 q^{71} - 16 q^{79} + 18 q^{81} - 16 q^{85} - 24 q^{95} + 6 q^{99}+O(q^{100}) \) 2 * q - 6 * q^9 - 2 * q^11 + 4 * q^23 + 6 * q^25 + 4 * q^37 - 4 * q^43 + 12 * q^53 + 8 * q^65 + 8 * q^67 - 16 * q^71 - 16 * q^79 + 18 * q^81 - 16 * q^85 - 24 * q^95 + 6 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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