LMFDB - Embedded newform 8100.2.d.c.649.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8100,2,Mod(649,8100)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8100, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("8100.649");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8100.d (of order $2$, degree $1$, not minimal)

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:	no
Analytic conductor:	$64.6788256372$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(i)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 1 $ x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 36)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.1
Root			$-1.00000i$ of defining polynomial
Character	$\chi$	$=$	8100.649
Dual form			8100.2.d.c.649.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{7} +O(q^{10})$	$q-1.00000i q^{7} -3.00000 q^{11} +1.00000i q^{13} -6.00000i q^{17} +4.00000 q^{19} -3.00000i q^{23} +3.00000 q^{29} +5.00000 q^{31} +2.00000i q^{37} -3.00000 q^{41} +1.00000i q^{43} +9.00000i q^{47} +6.00000 q^{49} -6.00000i q^{53} -3.00000 q^{59} -13.0000 q^{61} -7.00000i q^{67} +12.0000 q^{71} +10.0000i q^{73} +3.00000i q^{77} -11.0000 q^{79} -9.00000i q^{83} +6.00000 q^{89} +1.00000 q^{91} +11.0000i q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q+O(q^{10}) $ 2 * q	$ 2 q - 6 q^{11} + 8 q^{19} + 6 q^{29} + 10 q^{31} - 6 q^{41} + 12 q^{49} - 6 q^{59} - 26 q^{61} + 24 q^{71} - 22 q^{79} + 12 q^{89} + 2 q^{91}+O(q^{100}) $ 2 * q - 6 * q^11 + 8 * q^19 + 6 * q^29 + 10 * q^31 - 6 * q^41 + 12 * q^49 - 6 * q^59 - 26 * q^61 + 24 * q^71 - 22 * q^79 + 12 * q^89 + 2 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/8100\mathbb{Z}\right)^\times$.

$n$	$4051$	$6401$	$7777$
$\chi(n)$	$1$	$1$	$-1$

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
$5$	0	0
$5$	0	0
$6$	0	0
$7$	− 1.00000i	− 0.377964i	−0.981981	−	0.188982i	$-0.939481\pi$
$7$	− 1.00000i	− 0.377964i	0.981981	−	0.188982i	$-0.0605189\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.00000	−0.904534	−0.452267	−	0.891883i	$-0.649385\pi$
$11$	−3.00000	−0.904534	−0.452267	+	0.891883i	$0.649385\pi$
$12$	0	0
$13$	1.00000i	0.277350i	0.990338	+	0.138675i	$0.0442844\pi$
$13$	1.00000i	0.277350i	−0.990338	+	0.138675i	$0.955716\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 6.00000i	− 1.45521i	−0.685994	−	0.727607i	$-0.740633\pi$
$17$	− 6.00000i	− 1.45521i	0.685994	−	0.727607i	$-0.259367\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	− 3.00000i	− 0.625543i	−0.949828	−	0.312772i	$-0.898743\pi$
$23$	− 3.00000i	− 0.625543i	0.949828	−	0.312772i	$-0.101257\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.00000	0.557086	0.278543	−	0.960424i	$-0.410149\pi$
$29$	3.00000	0.557086	0.278543	+	0.960424i	$0.410149\pi$
$30$	0	0
$31$	5.00000	0.898027	0.449013	−	0.893525i	$-0.351776\pi$
$31$	5.00000	0.898027	0.449013	+	0.893525i	$0.351776\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0	0
$36$	0	0
$37$	2.00000i	0.328798i	0.986394	+	0.164399i	$0.0525685\pi$
$37$	2.00000i	0.328798i	−0.986394	+	0.164399i	$0.947432\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−3.00000	−0.468521	−0.234261	−	0.972174i	$-0.575267\pi$
$41$	−3.00000	−0.468521	−0.234261	+	0.972174i	$0.575267\pi$
$42$	0	0
$43$	1.00000i	0.152499i	0.997089	+	0.0762493i	$0.0242945\pi$
$43$	1.00000i	0.152499i	−0.997089	+	0.0762493i	$0.975706\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.00000i	1.31278i	0.754420	+	0.656392i	$0.227918\pi$
$47$	9.00000i	1.31278i	−0.754420	+	0.656392i	$0.772082\pi$
$48$	0	0
$49$	6.00000	0.857143
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 6.00000i	− 0.824163i	−0.911147	−	0.412082i	$-0.864802\pi$
$53$	− 6.00000i	− 0.824163i	0.911147	−	0.412082i	$-0.135198\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−3.00000	−0.390567	−0.195283	−	0.980747i	$-0.562563\pi$
$59$	−3.00000	−0.390567	−0.195283	+	0.980747i	$0.562563\pi$
$60$	0	0
$61$	−13.0000	−1.66448	−0.832240	−	0.554416i	$-0.812942\pi$
$61$	−13.0000	−1.66448	−0.832240	+	0.554416i	$0.812942\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0	0
$66$	0	0
$67$	− 7.00000i	− 0.855186i	−0.903971	−	0.427593i	$-0.859362\pi$
$67$	− 7.00000i	− 0.855186i	0.903971	−	0.427593i	$-0.140638\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	12.0000	1.42414	0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000	1.42414	0.712069	+	0.702109i	$0.247758\pi$
$72$	0	0
$73$	10.0000i	1.17041i	0.810885	+	0.585206i	$0.198986\pi$
$73$	10.0000i	1.17041i	−0.810885	+	0.585206i	$0.801014\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	3.00000i	0.341882i
$78$	0	0
$79$	−11.0000	−1.23760	−0.618798	−	0.785550i	$-0.712380\pi$
$79$	−11.0000	−1.23760	−0.618798	+	0.785550i	$0.712380\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 9.00000i	− 0.987878i	−0.869496	−	0.493939i	$-0.835557\pi$
$83$	− 9.00000i	− 0.987878i	0.869496	−	0.493939i	$-0.164443\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	0	0
$88$	0	0
$89$	6.00000	0.635999	0.317999	−	0.948091i	$-0.396989\pi$
$89$	6.00000	0.635999	0.317999	+	0.948091i	$0.396989\pi$
$90$	0	0
$91$	1.00000	0.104828
$92$	0	0
$93$	0	0
$94$	0	0
$95$	0	0
$96$	0	0
$97$	11.0000i	1.11688i	0.829545	+	0.558440i	$0.188600\pi$
$97$	11.0000i	1.11688i	−0.829545	+	0.558440i	$0.811400\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−15.0000	−1.49256	−0.746278	−	0.665635i	$-0.768161\pi$
$101$	−15.0000	−1.49256	−0.746278	+	0.665635i	$0.768161\pi$
$102$	0	0
$103$	7.00000i	0.689730i	0.938652	+	0.344865i	$0.112075\pi$
$103$	7.00000i	0.689730i	−0.938652	+	0.344865i	$0.887925\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 12.0000i	− 1.16008i	−0.814587	−	0.580042i	$-0.803036\pi$
$107$	− 12.0000i	− 1.16008i	0.814587	−	0.580042i	$-0.196964\pi$
$108$	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
$113$	− 9.00000i	− 0.846649i	−0.905978	−	0.423324i	$-0.860863\pi$
$113$	− 9.00000i	− 0.846649i	0.905978	−	0.423324i	$-0.139137\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.00000	−0.550019
$120$	0	0
$121$	−2.00000	−0.181818
$122$	0	0
$123$	0	0
$124$	0	0
$125$	0	0
$126$	0	0
$127$	− 16.0000i	− 1.41977i	−0.704317	−	0.709885i	$-0.748747\pi$
$127$	− 16.0000i	− 1.41977i	0.704317	−	0.709885i	$-0.251253\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−21.0000	−1.83478	−0.917389	−	0.397991i	$-0.869707\pi$
$131$	−21.0000	−1.83478	−0.917389	+	0.397991i	$0.869707\pi$
$132$	0	0
$133$	− 4.00000i	− 0.346844i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 3.00000i	− 0.256307i	−0.991754	−	0.128154i	$-0.959095\pi$
$137$	− 3.00000i	− 0.256307i	0.991754	−	0.128154i	$-0.0409051\pi$
$138$	0	0
$139$	−5.00000	−0.424094	−0.212047	−	0.977259i	$-0.568013\pi$
$139$	−5.00000	−0.424094	−0.212047	+	0.977259i	$0.568013\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	− 3.00000i	− 0.250873i
$144$	0	0
$145$	0	0
$146$	0	0
$147$	0	0
$148$	0	0
$149$	15.0000	1.22885	0.614424	−	0.788976i	$-0.289388\pi$
$149$	15.0000	1.22885	0.614424	+	0.788976i	$0.289388\pi$
$150$	0	0
$151$	−13.0000	−1.05792	−0.528962	−	0.848645i	$-0.677419\pi$
$151$	−13.0000	−1.05792	−0.528962	+	0.848645i	$0.677419\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0	0
$156$	0	0
$157$	− 13.0000i	− 1.03751i	−0.854922	−	0.518756i	$-0.826395\pi$
$157$	− 13.0000i	− 1.03751i	0.854922	−	0.518756i	$-0.173605\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−3.00000	−0.236433
$162$	0	0
$163$	− 20.0000i	− 1.56652i	−0.621694	−	0.783260i	$-0.713555\pi$
$163$	− 20.0000i	− 1.56652i	0.621694	−	0.783260i	$-0.286445\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	− 9.00000i	− 0.696441i	−0.937413	−	0.348220i	$-0.886786\pi$
$167$	− 9.00000i	− 0.696441i	0.937413	−	0.348220i	$-0.113214\pi$
$168$	0	0
$169$	12.0000	0.923077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 9.00000i	− 0.684257i	−0.939653	−	0.342129i	$-0.888852\pi$
$173$	− 9.00000i	− 0.684257i	0.939653	−	0.342129i	$-0.111148\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	0	0
$178$	0	0
$179$	12.0000	0.896922	0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000	0.896922	0.448461	+	0.893802i	$0.351972\pi$
$180$	0	0
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	0	0
$186$	0	0
$187$	18.0000i	1.31629i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−15.0000	−1.08536	−0.542681	−	0.839939i	$-0.682591\pi$
$191$	−15.0000	−1.08536	−0.542681	+	0.839939i	$0.682591\pi$
$192$	0	0
$193$	− 11.0000i	− 0.791797i	−0.918294	−	0.395899i	$-0.870433\pi$
$193$	− 11.0000i	− 0.791797i	0.918294	−	0.395899i	$-0.129567\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	− 6.00000i	− 0.427482i	−0.976890	−	0.213741i	$-0.931435\pi$
$197$	− 6.00000i	− 0.427482i	0.976890	−	0.213741i	$-0.0685649\pi$
$198$	0	0
$199$	4.00000	0.283552	0.141776	−	0.989899i	$-0.454719\pi$
$199$	4.00000	0.283552	0.141776	+	0.989899i	$0.454719\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 3.00000i	− 0.210559i
$204$	0	0
$205$	0	0
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−12.0000	−0.830057
$210$	0	0
$211$	17.0000	1.17033	0.585164	−	0.810915i	$-0.301030\pi$
$211$	17.0000	1.17033	0.585164	+	0.810915i	$0.301030\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	− 5.00000i	− 0.339422i
$218$	0	0
$219$	0	0
$220$	0	0
$221$	6.00000	0.403604
$222$	0	0
$223$	1.00000i	0.0669650i	0.999439	+	0.0334825i	$0.0106598\pi$
$223$	1.00000i	0.0669650i	−0.999439	+	0.0334825i	$0.989340\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 27.0000i	− 1.79205i	−0.444001	−	0.896026i	$-0.646441\pi$
$227$	− 27.0000i	− 1.79205i	0.444001	−	0.896026i	$-0.353559\pi$
$228$	0	0
$229$	13.0000	0.859064	0.429532	−	0.903052i	$-0.358679\pi$
$229$	13.0000	0.859064	0.429532	+	0.903052i	$0.358679\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 6.00000i	− 0.393073i	−0.980497	−	0.196537i	$-0.937031\pi$
$233$	− 6.00000i	− 0.393073i	0.980497	−	0.196537i	$-0.0629694\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−27.0000	−1.74648	−0.873242	−	0.487286i	$-0.837987\pi$
$239$	−27.0000	−1.74648	−0.873242	+	0.487286i	$0.837987\pi$
$240$	0	0
$241$	−1.00000	−0.0644157	−0.0322078	−	0.999481i	$-0.510254\pi$
$241$	−1.00000	−0.0644157	−0.0322078	+	0.999481i	$0.510254\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4.00000i	0.254514i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−12.0000	−0.757433	−0.378717	−	0.925513i	$-0.623635\pi$
$251$	−12.0000	−0.757433	−0.378717	+	0.925513i	$0.623635\pi$
$252$	0	0
$253$	9.00000i	0.565825i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	9.00000i	0.561405i	0.959795	+	0.280702i	$0.0905674\pi$
$257$	9.00000i	0.561405i	−0.959795	+	0.280702i	$0.909433\pi$
$258$	0	0
$259$	2.00000	0.124274
$260$	0	0
$261$	0	0
$262$	0	0
$263$	− 21.0000i	− 1.29492i	−0.762101	−	0.647458i	$-0.775832\pi$
$263$	− 21.0000i	− 1.29492i	0.762101	−	0.647458i	$-0.224168\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−6.00000	−0.365826	−0.182913	−	0.983129i	$-0.558553\pi$
$269$	−6.00000	−0.365826	−0.182913	+	0.983129i	$0.558553\pi$
$270$	0	0
$271$	8.00000	0.485965	0.242983	−	0.970031i	$-0.421874\pi$
$271$	8.00000	0.485965	0.242983	+	0.970031i	$0.421874\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	− 1.00000i	− 0.0600842i	−0.999549	−	0.0300421i	$-0.990436\pi$
$277$	− 1.00000i	− 0.0600842i	0.999549	−	0.0300421i	$-0.00956413\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−3.00000	−0.178965	−0.0894825	−	0.995988i	$-0.528521\pi$
$281$	−3.00000	−0.178965	−0.0894825	+	0.995988i	$0.528521\pi$
$282$	0	0
$283$	− 5.00000i	− 0.297219i	−0.988896	−	0.148610i	$-0.952520\pi$
$283$	− 5.00000i	− 0.297219i	0.988896	−	0.148610i	$-0.0474798\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	3.00000i	0.177084i
$288$	0	0
$289$	−19.0000	−1.11765
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 21.0000i	− 1.22683i	−0.789760	−	0.613417i	$-0.789795\pi$
$293$	− 21.0000i	− 1.22683i	0.789760	−	0.613417i	$-0.210205\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.00000	0.173494
$300$	0	0
$301$	1.00000	0.0576390
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	20.0000i	1.14146i	0.821138	+	0.570730i	$0.193340\pi$
$307$	20.0000i	1.14146i	−0.821138	+	0.570730i	$0.806660\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−21.0000	−1.19080	−0.595400	−	0.803429i	$-0.703007\pi$
$311$	−21.0000	−1.19080	−0.595400	+	0.803429i	$0.703007\pi$
$312$	0	0
$313$	1.00000i	0.0565233i	0.999601	+	0.0282617i	$0.00899717\pi$
$313$	1.00000i	0.0565233i	−0.999601	+	0.0282617i	$0.991003\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	21.0000i	1.17948i	0.807594	+	0.589739i	$0.200769\pi$
$317$	21.0000i	1.17948i	−0.807594	+	0.589739i	$0.799231\pi$
$318$	0	0
$319$	−9.00000	−0.503903
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 24.0000i	− 1.33540i
$324$	0	0
$325$	0	0
$326$	0	0
$327$	0	0
$328$	0	0
$329$	9.00000	0.496186
$330$	0	0
$331$	11.0000	0.604615	0.302307	−	0.953211i	$-0.402243\pi$
$331$	11.0000	0.604615	0.302307	+	0.953211i	$0.402243\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	23.0000i	1.25289i	0.779466	+	0.626445i	$0.215491\pi$
$337$	23.0000i	1.25289i	−0.779466	+	0.626445i	$0.784509\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−15.0000	−0.812296
$342$	0	0
$343$	− 13.0000i	− 0.701934i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 9.00000i	− 0.483145i	−0.970383	−	0.241573i	$-0.922337\pi$
$347$	− 9.00000i	− 0.483145i	0.970383	−	0.241573i	$-0.0776632\pi$
$348$	0	0
$349$	1.00000	0.0535288	0.0267644	−	0.999642i	$-0.491480\pi$
$349$	1.00000	0.0535288	0.0267644	+	0.999642i	$0.491480\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.00000i	0.159674i	0.996808	+	0.0798369i	$0.0254400\pi$
$353$	3.00000i	0.159674i	−0.996808	+	0.0798369i	$0.974560\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0	0		−	1.00000i	$-0.5\pi$
$359$	0	0			1.00000i	$0.5\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	0	0
$364$	0	0
$365$	0	0
$366$	0	0
$367$	− 13.0000i	− 0.678594i	−0.940679	−	0.339297i	$-0.889811\pi$
$367$	− 13.0000i	− 0.678594i	0.940679	−	0.339297i	$-0.110189\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−6.00000	−0.311504
$372$	0	0
$373$	1.00000i	0.0517780i	0.999665	+	0.0258890i	$0.00824165\pi$
$373$	1.00000i	0.0517780i	−0.999665	+	0.0258890i	$0.991758\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.00000i	0.154508i
$378$	0	0
$379$	16.0000	0.821865	0.410932	−	0.911666i	$-0.365203\pi$
$379$	16.0000	0.821865	0.410932	+	0.911666i	$0.365203\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	− 15.0000i	− 0.766464i	−0.923652	−	0.383232i	$-0.874811\pi$
$383$	− 15.0000i	− 0.766464i	0.923652	−	0.383232i	$-0.125189\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	0	0
$389$	15.0000	0.760530	0.380265	−	0.924878i	$-0.375833\pi$
$389$	15.0000	0.760530	0.380265	+	0.924878i	$0.375833\pi$
$390$	0	0
$391$	−18.0000	−0.910299
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	2.00000i	0.100377i	0.998740	+	0.0501886i	$0.0159822\pi$
$397$	2.00000i	0.100377i	−0.998740	+	0.0501886i	$0.984018\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−3.00000	−0.149813	−0.0749064	−	0.997191i	$-0.523866\pi$
$401$	−3.00000	−0.149813	−0.0749064	+	0.997191i	$0.523866\pi$
$402$	0	0
$403$	5.00000i	0.249068i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	− 6.00000i	− 0.297409i
$408$	0	0
$409$	−23.0000	−1.13728	−0.568638	−	0.822588i	$-0.692530\pi$
$409$	−23.0000	−1.13728	−0.568638	+	0.822588i	$0.692530\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	3.00000i	0.147620i
$414$	0	0
$415$	0	0
$416$	0	0
$417$	0	0
$418$	0	0
$419$	9.00000	0.439679	0.219839	−	0.975536i	$-0.429447\pi$
$419$	9.00000	0.439679	0.219839	+	0.975536i	$0.429447\pi$
$420$	0	0
$421$	35.0000	1.70580	0.852898	−	0.522078i	$-0.174843\pi$
$421$	35.0000	1.70580	0.852898	+	0.522078i	$0.174843\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	0	0
$426$	0	0
$427$	13.0000i	0.629114i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−24.0000	−1.15604	−0.578020	−	0.816023i	$-0.696174\pi$
$431$	−24.0000	−1.15604	−0.578020	+	0.816023i	$0.696174\pi$
$432$	0	0
$433$	34.0000i	1.63394i	0.576683	+	0.816968i	$0.304347\pi$
$433$	34.0000i	1.63394i	−0.576683	+	0.816968i	$0.695653\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 12.0000i	− 0.574038i
$438$	0	0
$439$	−35.0000	−1.67046	−0.835229	−	0.549902i	$-0.814665\pi$
$439$	−35.0000	−1.67046	−0.835229	+	0.549902i	$0.814665\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 9.00000i	− 0.427603i	−0.976877	−	0.213801i	$-0.931415\pi$
$443$	− 9.00000i	− 0.427603i	0.976877	−	0.213801i	$-0.0685846\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−18.0000	−0.849473	−0.424736	−	0.905317i	$-0.639633\pi$
$449$	−18.0000	−0.849473	−0.424736	+	0.905317i	$0.639633\pi$
$450$	0	0
$451$	9.00000	0.423793
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	− 37.0000i	− 1.73079i	−0.501093	−	0.865393i	$-0.667069\pi$
$457$	− 37.0000i	− 1.73079i	0.501093	−	0.865393i	$-0.332931\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−3.00000	−0.139724	−0.0698620	−	0.997557i	$-0.522256\pi$
$461$	−3.00000	−0.139724	−0.0698620	+	0.997557i	$0.522256\pi$
$462$	0	0
$463$	19.0000i	0.883005i	0.897260	+	0.441502i	$0.145554\pi$
$463$	19.0000i	0.883005i	−0.897260	+	0.441502i	$0.854446\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 12.0000i	− 0.555294i	−0.960683	−	0.277647i	$-0.910445\pi$
$467$	− 12.0000i	− 0.555294i	0.960683	−	0.277647i	$-0.0895545\pi$
$468$	0	0
$469$	−7.00000	−0.323230
$470$	0	0
$471$	0	0
$472$	0	0
$473$	− 3.00000i	− 0.137940i
$474$	0	0
$475$	0	0
$476$	0	0
$477$	0	0
$478$	0	0
$479$	27.0000	1.23366	0.616831	−	0.787096i	$-0.288416\pi$
$479$	27.0000	1.23366	0.616831	+	0.787096i	$0.288416\pi$
$480$	0	0
$481$	−2.00000	−0.0911922
$482$	0	0
$483$	0	0
$484$	0	0
$485$	0	0
$486$	0	0
$487$	32.0000i	1.45006i	0.688718	+	0.725029i	$0.258174\pi$
$487$	32.0000i	1.45006i	−0.688718	+	0.725029i	$0.741826\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	3.00000	0.135388	0.0676941	−	0.997706i	$-0.478436\pi$
$491$	3.00000	0.135388	0.0676941	+	0.997706i	$0.478436\pi$
$492$	0	0
$493$	− 18.0000i	− 0.810679i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	− 12.0000i	− 0.538274i
$498$	0	0
$499$	−5.00000	−0.223831	−0.111915	−	0.993718i	$-0.535699\pi$
$499$	−5.00000	−0.223831	−0.111915	+	0.993718i	$0.535699\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	− 36.0000i	− 1.60516i	−0.596544	−	0.802580i	$-0.703460\pi$
$503$	− 36.0000i	− 1.60516i	0.596544	−	0.802580i	$-0.296540\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0	0
$508$	0	0
$509$	39.0000	1.72864	0.864322	−	0.502938i	$-0.167748\pi$
$509$	39.0000	1.72864	0.864322	+	0.502938i	$0.167748\pi$
$510$	0	0
$511$	10.0000	0.442374
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	− 27.0000i	− 1.18746i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−42.0000	−1.84005	−0.920027	−	0.391856i	$-0.871833\pi$
$521$	−42.0000	−1.84005	−0.920027	+	0.391856i	$0.871833\pi$
$522$	0	0
$523$	− 8.00000i	− 0.349816i	−0.984585	−	0.174908i	$-0.944037\pi$
$523$	− 8.00000i	− 0.349816i	0.984585	−	0.174908i	$-0.0559627\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	− 30.0000i	− 1.30682i
$528$	0	0
$529$	14.0000	0.608696
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 3.00000i	− 0.129944i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−18.0000	−0.775315
$540$	0	0
$541$	−34.0000	−1.46177	−0.730887	−	0.682498i	$-0.760893\pi$
$541$	−34.0000	−1.46177	−0.730887	+	0.682498i	$0.760893\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	0	0
$546$	0	0
$547$	− 13.0000i	− 0.555840i	−0.960604	−	0.277920i	$-0.910355\pi$
$547$	− 13.0000i	− 0.555840i	0.960604	−	0.277920i	$-0.0896450\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	12.0000	0.511217
$552$	0	0
$553$	11.0000i	0.467768i
$554$	0	0
$555$	0	0
$556$	0	0
$557$	30.0000i	1.27114i	0.772043	+	0.635570i	$0.219235\pi$
$557$	30.0000i	1.27114i	−0.772043	+	0.635570i	$0.780765\pi$
$558$	0	0
$559$	−1.00000	−0.0422955
$560$	0	0
$561$	0	0
$562$	0	0
$563$	9.00000i	0.379305i	0.981851	+	0.189652i	$0.0607361\pi$
$563$	9.00000i	0.379305i	−0.981851	+	0.189652i	$0.939264\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	0	0
$568$	0	0
$569$	15.0000	0.628833	0.314416	−	0.949285i	$-0.398191\pi$
$569$	15.0000	0.628833	0.314416	+	0.949285i	$0.398191\pi$
$570$	0	0
$571$	−31.0000	−1.29731	−0.648655	−	0.761083i	$-0.724668\pi$
$571$	−31.0000	−1.29731	−0.648655	+	0.761083i	$0.724668\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	0	0
$576$	0	0
$577$	− 10.0000i	− 0.416305i	−0.978096	−	0.208153i	$-0.933255\pi$
$577$	− 10.0000i	− 0.416305i	0.978096	−	0.208153i	$-0.0667451\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−9.00000	−0.373383
$582$	0	0
$583$	18.0000i	0.745484i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	− 15.0000i	− 0.619116i	−0.950881	−	0.309558i	$-0.899819\pi$
$587$	− 15.0000i	− 0.619116i	0.950881	−	0.309558i	$-0.100181\pi$
$588$	0	0
$589$	20.0000	0.824086
$590$	0	0
$591$	0	0
$592$	0	0
$593$	6.00000i	0.246390i	0.992382	+	0.123195i	$0.0393141\pi$
$593$	6.00000i	0.246390i	−0.992382	+	0.123195i	$0.960686\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−39.0000	−1.59350	−0.796748	−	0.604311i	$-0.793448\pi$
$599$	−39.0000	−1.59350	−0.796748	+	0.604311i	$0.793448\pi$
$600$	0	0
$601$	35.0000	1.42768	0.713840	−	0.700309i	$-0.246954\pi$
$601$	35.0000	1.42768	0.713840	+	0.700309i	$0.246954\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0	0
$606$	0	0
$607$	41.0000i	1.66414i	0.554672	+	0.832069i	$0.312844\pi$
$607$	41.0000i	1.66414i	−0.554672	+	0.832069i	$0.687156\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−9.00000	−0.364101
$612$	0	0
$613$	− 26.0000i	− 1.05013i	−0.851062	−	0.525065i	$-0.824041\pi$
$613$	− 26.0000i	− 1.05013i	0.851062	−	0.525065i	$-0.175959\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 3.00000i	− 0.120775i	−0.998175	−	0.0603877i	$-0.980766\pi$
$617$	− 3.00000i	− 0.120775i	0.998175	−	0.0603877i	$-0.0192337\pi$
$618$	0	0
$619$	13.0000	0.522514	0.261257	−	0.965269i	$-0.415863\pi$
$619$	13.0000	0.522514	0.261257	+	0.965269i	$0.415863\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	− 6.00000i	− 0.240385i
$624$	0	0
$625$	0	0
$626$	0	0
$627$	0	0
$628$	0	0
$629$	12.0000	0.478471
$630$	0	0
$631$	−16.0000	−0.636950	−0.318475	−	0.947931i	$-0.603171\pi$
$631$	−16.0000	−0.636950	−0.318475	+	0.947931i	$0.603171\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	0	0
$636$	0	0
$637$	6.00000i	0.237729i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	33.0000	1.30342	0.651711	−	0.758468i	$-0.274052\pi$
$641$	33.0000	1.30342	0.651711	+	0.758468i	$0.274052\pi$
$642$	0	0
$643$	− 41.0000i	− 1.61688i	−0.588577	−	0.808441i	$-0.700312\pi$
$643$	− 41.0000i	− 1.61688i	0.588577	−	0.808441i	$-0.299688\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	9.00000	0.353281
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 21.0000i	− 0.821794i	−0.911682	−	0.410897i	$-0.865216\pi$
$653$	− 21.0000i	− 0.821794i	0.911682	−	0.410897i	$-0.134784\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−21.0000	−0.818044	−0.409022	−	0.912525i	$-0.634130\pi$
$659$	−21.0000	−0.818044	−0.409022	+	0.912525i	$0.634130\pi$
$660$	0	0
$661$	11.0000	0.427850	0.213925	−	0.976850i	$-0.431375\pi$
$661$	11.0000	0.427850	0.213925	+	0.976850i	$0.431375\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	− 9.00000i	− 0.348481i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	39.0000	1.50558
$672$	0	0
$673$	− 11.0000i	− 0.424019i	−0.977268	−	0.212009i	$-0.931999\pi$
$673$	− 11.0000i	− 0.424019i	0.977268	−	0.212009i	$-0.0680008\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 15.0000i	− 0.576497i	−0.957556	−	0.288248i	$-0.906927\pi$
$677$	− 15.0000i	− 0.576497i	0.957556	−	0.288248i	$-0.0930729\pi$
$678$	0	0
$679$	11.0000	0.422141
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 36.0000i	− 1.37750i	−0.724998	−	0.688751i	$-0.758159\pi$
$683$	− 36.0000i	− 1.37750i	0.724998	−	0.688751i	$-0.241841\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	6.00000	0.228582
$690$	0	0
$691$	−1.00000	−0.0380418	−0.0190209	−	0.999819i	$-0.506055\pi$
$691$	−1.00000	−0.0380418	−0.0190209	+	0.999819i	$0.506055\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	0	0
$696$	0	0
$697$	18.0000i	0.681799i
$698$	0	0
$699$	0	0
$700$	0	0
$701$	6.00000	0.226617	0.113308	−	0.993560i	$-0.463855\pi$
$701$	6.00000	0.226617	0.113308	+	0.993560i	$0.463855\pi$
$702$	0	0
$703$	8.00000i	0.301726i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	15.0000i	0.564133i
$708$	0	0
$709$	25.0000	0.938895	0.469447	−	0.882960i	$-0.344453\pi$
$709$	25.0000	0.938895	0.469447	+	0.882960i	$0.344453\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	− 15.0000i	− 0.561754i
$714$	0	0
$715$	0	0
$716$	0	0
$717$	0	0
$718$	0	0
$719$	0	0		−	1.00000i	$-0.5\pi$
$719$	0	0			1.00000i	$0.5\pi$
$720$	0	0
$721$	7.00000	0.260694
$722$	0	0
$723$	0	0
$724$	0	0
$725$	0	0
$726$	0	0
$727$	− 37.0000i	− 1.37225i	−0.727482	−	0.686127i	$-0.759309\pi$
$727$	− 37.0000i	− 1.37225i	0.727482	−	0.686127i	$-0.240691\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	6.00000	0.221918
$732$	0	0
$733$	− 23.0000i	− 0.849524i	−0.905305	−	0.424762i	$-0.860358\pi$
$733$	− 23.0000i	− 0.849524i	0.905305	−	0.424762i	$-0.139642\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	21.0000i	0.773545i
$738$	0	0
$739$	16.0000	0.588570	0.294285	−	0.955718i	$-0.404919\pi$
$739$	16.0000	0.588570	0.294285	+	0.955718i	$0.404919\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	9.00000i	0.330178i	0.986279	+	0.165089i	$0.0527911\pi$
$743$	9.00000i	0.330178i	−0.986279	+	0.165089i	$0.947209\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−12.0000	−0.438470
$750$	0	0
$751$	−31.0000	−1.13121	−0.565603	−	0.824678i	$-0.691357\pi$
$751$	−31.0000	−1.13121	−0.565603	+	0.824678i	$0.691357\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0	0
$757$	− 10.0000i	− 0.363456i	−0.983349	−	0.181728i	$-0.941831\pi$
$757$	− 10.0000i	− 0.363456i	0.983349	−	0.181728i	$-0.0581691\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−27.0000	−0.978749	−0.489375	−	0.872074i	$-0.662775\pi$
$761$	−27.0000	−0.978749	−0.489375	+	0.872074i	$0.662775\pi$
$762$	0	0
$763$	2.00000i	0.0724049i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	− 3.00000i	− 0.108324i
$768$	0	0
$769$	1.00000	0.0360609	0.0180305	−	0.999837i	$-0.494260\pi$
$769$	1.00000	0.0360609	0.0180305	+	0.999837i	$0.494260\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	18.0000i	0.647415i	0.946157	+	0.323708i	$0.104929\pi$
$773$	18.0000i	0.647415i	−0.946157	+	0.323708i	$0.895071\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−12.0000	−0.429945
$780$	0	0
$781$	−36.0000	−1.28818
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0	0
$786$	0	0
$787$	− 43.0000i	− 1.53278i	−0.642373	−	0.766392i	$-0.722050\pi$
$787$	− 43.0000i	− 1.53278i	0.642373	−	0.766392i	$-0.277950\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−9.00000	−0.320003
$792$	0	0
$793$	− 13.0000i	− 0.461644i
$794$	0	0
$795$	0	0
$796$	0	0
$797$	9.00000i	0.318796i	0.987214	+	0.159398i	$0.0509554\pi$
$797$	9.00000i	0.318796i	−0.987214	+	0.159398i	$0.949045\pi$
$798$	0	0
$799$	54.0000	1.91038
$800$	0	0
$801$	0	0
$802$	0	0
$803$	− 30.0000i	− 1.05868i
$804$	0	0
$805$	0	0
$806$	0	0
$807$	0	0
$808$	0	0
$809$	6.00000	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	6.00000	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	20.0000	0.702295	0.351147	−	0.936320i	$-0.385792\pi$
$811$	20.0000	0.702295	0.351147	+	0.936320i	$0.385792\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	0	0
$816$	0	0
$817$	4.00000i	0.139942i
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−51.0000	−1.77991	−0.889956	−	0.456046i	$-0.849265\pi$
$821$	−51.0000	−1.77991	−0.889956	+	0.456046i	$0.849265\pi$
$822$	0	0
$823$	19.0000i	0.662298i	0.943578	+	0.331149i	$0.107436\pi$
$823$	19.0000i	0.662298i	−0.943578	+	0.331149i	$0.892564\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12.0000i	0.417281i	0.977992	+	0.208640i	$0.0669038\pi$
$827$	12.0000i	0.417281i	−0.977992	+	0.208640i	$0.933096\pi$
$828$	0	0
$829$	−50.0000	−1.73657	−0.868286	−	0.496064i	$-0.834778\pi$
$829$	−50.0000	−1.73657	−0.868286	+	0.496064i	$0.834778\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 36.0000i	− 1.24733i
$834$	0	0
$835$	0	0
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−9.00000	−0.310715	−0.155357	−	0.987858i	$-0.549653\pi$
$839$	−9.00000	−0.310715	−0.155357	+	0.987858i	$0.549653\pi$
$840$	0	0
$841$	−20.0000	−0.689655
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0	0
$846$	0	0
$847$	2.00000i	0.0687208i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	6.00000	0.205677
$852$	0	0
$853$	13.0000i	0.445112i	0.974920	+	0.222556i	$0.0714399\pi$
$853$	13.0000i	0.445112i	−0.974920	+	0.222556i	$0.928560\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	− 27.0000i	− 0.922302i	−0.887322	−	0.461151i	$-0.847437\pi$
$857$	− 27.0000i	− 0.922302i	0.887322	−	0.461151i	$-0.152563\pi$
$858$	0	0
$859$	−41.0000	−1.39890	−0.699451	−	0.714681i	$-0.746572\pi$
$859$	−41.0000	−1.39890	−0.699451	+	0.714681i	$0.746572\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	24.0000i	0.816970i	0.912765	+	0.408485i	$0.133943\pi$
$863$	24.0000i	0.816970i	−0.912765	+	0.408485i	$0.866057\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	0	0
$868$	0	0
$869$	33.0000	1.11945
$870$	0	0
$871$	7.00000	0.237186
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0	0
$876$	0	0
$877$	23.0000i	0.776655i	0.921521	+	0.388327i	$0.126947\pi$
$877$	23.0000i	0.776655i	−0.921521	+	0.388327i	$0.873053\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−30.0000	−1.01073	−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000	−1.01073	−0.505363	+	0.862907i	$0.668641\pi$
$882$	0	0
$883$	4.00000i	0.134611i	0.997732	+	0.0673054i	$0.0214402\pi$
$883$	4.00000i	0.134611i	−0.997732	+	0.0673054i	$0.978560\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	− 21.0000i	− 0.705111i	−0.935791	−	0.352555i	$-0.885313\pi$
$887$	− 21.0000i	− 0.705111i	0.935791	−	0.352555i	$-0.114687\pi$
$888$	0	0
$889$	−16.0000	−0.536623
$890$	0	0
$891$	0	0
$892$	0	0
$893$	36.0000i	1.20469i
$894$	0	0
$895$	0	0
$896$	0	0
$897$	0	0
$898$	0	0
$899$	15.0000	0.500278
$900$	0	0
$901$	−36.0000	−1.19933
$902$	0	0
$903$	0	0
$904$	0	0
$905$	0	0
$906$	0	0
$907$	47.0000i	1.56061i	0.625400	+	0.780305i	$0.284936\pi$
$907$	47.0000i	1.56061i	−0.625400	+	0.780305i	$0.715064\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	45.0000	1.49092	0.745458	−	0.666552i	$-0.232231\pi$
$911$	45.0000	1.49092	0.745458	+	0.666552i	$0.232231\pi$
$912$	0	0
$913$	27.0000i	0.893570i
$914$	0	0
$915$	0	0
$916$	0	0
$917$	21.0000i	0.693481i
$918$	0	0
$919$	16.0000	0.527791	0.263896	−	0.964551i	$-0.414993\pi$
$919$	16.0000	0.527791	0.263896	+	0.964551i	$0.414993\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	12.0000i	0.394985i
$924$	0	0
$925$	0	0
$926$	0	0
$927$	0	0
$928$	0	0
$929$	27.0000	0.885841	0.442921	−	0.896561i	$-0.353942\pi$
$929$	27.0000	0.885841	0.442921	+	0.896561i	$0.353942\pi$
$930$	0	0
$931$	24.0000	0.786568
$932$	0	0
$933$	0	0
$934$	0	0
$935$	0	0
$936$	0	0
$937$	− 34.0000i	− 1.11073i	−0.831606	−	0.555366i	$-0.812578\pi$
$937$	− 34.0000i	− 1.11073i	0.831606	−	0.555366i	$-0.187422\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	21.0000	0.684580	0.342290	−	0.939594i	$-0.388797\pi$
$941$	21.0000	0.684580	0.342290	+	0.939594i	$0.388797\pi$
$942$	0	0
$943$	9.00000i	0.293080i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	− 27.0000i	− 0.877382i	−0.898638	−	0.438691i	$-0.855442\pi$
$947$	− 27.0000i	− 0.877382i	0.898638	−	0.438691i	$-0.144558\pi$
$948$	0	0
$949$	−10.0000	−0.324614
$950$	0	0
$951$	0	0
$952$	0	0
$953$	54.0000i	1.74923i	0.484817	+	0.874616i	$0.338886\pi$
$953$	54.0000i	1.74923i	−0.484817	+	0.874616i	$0.661114\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−3.00000	−0.0968751
$960$	0	0
$961$	−6.00000	−0.193548
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	− 43.0000i	− 1.38279i	−0.722478	−	0.691393i	$-0.756997\pi$
$967$	− 43.0000i	− 1.38279i	0.722478	−	0.691393i	$-0.243003\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	0	0		−	1.00000i	$-0.5\pi$
$971$	0	0			1.00000i	$0.5\pi$
$972$	0	0
$973$	5.00000i	0.160293i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	57.0000i	1.82359i	0.410644	+	0.911796i	$0.365304\pi$
$977$	57.0000i	1.82359i	−0.410644	+	0.911796i	$0.634696\pi$
$978$	0	0
$979$	−18.0000	−0.575282
$980$	0	0
$981$	0	0
$982$	0	0
$983$	51.0000i	1.62665i	0.581811	+	0.813324i	$0.302344\pi$
$983$	51.0000i	1.62665i	−0.581811	+	0.813324i	$0.697656\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0	0
$988$	0	0
$989$	3.00000	0.0953945
$990$	0	0
$991$	8.00000	0.254128	0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000	0.254128	0.127064	+	0.991894i	$0.459445\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	0	0
$996$	0	0
$997$	− 1.00000i	− 0.0316703i	−0.999875	−	0.0158352i	$-0.994959\pi$
$997$	− 1.00000i	− 0.0316703i	0.999875	−	0.0158352i	$-0.00504070\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8100.2.d.c.649.1		2
3.2	odd	2		8100.2.d.h.649.1		2
5.2	odd	4		8100.2.a.g.1.1		1
5.3	odd	4		324.2.a.a.1.1		1
5.4	even	2	inner	8100.2.d.c.649.2		2
9.2	odd	6		900.2.s.b.49.1		4
9.4	even	3		2700.2.s.b.2449.2		4
9.5	odd	6		900.2.s.b.349.2		4
9.7	even	3		2700.2.s.b.1549.1		4
15.2	even	4		8100.2.a.j.1.1		1
15.8	even	4		324.2.a.c.1.1		1
15.14	odd	2		8100.2.d.h.649.2		2
20.3	even	4		1296.2.a.b.1.1		1
40.3	even	4		5184.2.a.bb.1.1		1
40.13	odd	4		5184.2.a.ba.1.1		1
45.2	even	12		900.2.i.b.301.1		2
45.4	even	6		2700.2.s.b.2449.1		4
45.7	odd	12		2700.2.i.b.901.1		2
45.13	odd	12		108.2.e.a.73.1		2
45.14	odd	6		900.2.s.b.349.1		4
45.22	odd	12		2700.2.i.b.1801.1		2
45.23	even	12		36.2.e.a.25.1	yes	2
45.29	odd	6		900.2.s.b.49.2		4
45.32	even	12		900.2.i.b.601.1		2
45.34	even	6		2700.2.s.b.1549.2		4
45.38	even	12		36.2.e.a.13.1	✓	2
45.43	odd	12		108.2.e.a.37.1		2
60.23	odd	4		1296.2.a.k.1.1		1
120.53	even	4		5184.2.a.e.1.1		1
120.83	odd	4		5184.2.a.f.1.1		1
180.23	odd	12		144.2.i.a.97.1		2
180.43	even	12		432.2.i.c.145.1		2
180.83	odd	12		144.2.i.a.49.1		2
180.103	even	12		432.2.i.c.289.1		2
315.13	even	12		5292.2.j.a.3529.1		2
315.23	even	12		1764.2.i.a.1537.1		2
315.38	odd	12		1764.2.i.c.373.1		2
315.58	odd	12		5292.2.i.c.2125.1		2
315.68	odd	12		1764.2.i.c.1537.1		2
315.83	odd	12		1764.2.j.b.589.1		2
315.88	odd	12		5292.2.i.c.1549.1		2
315.103	even	12		5292.2.i.a.2125.1		2
315.128	even	12		1764.2.l.c.949.1		2
315.158	even	12		1764.2.l.c.961.1		2
315.173	odd	12		1764.2.l.a.949.1		2
315.178	even	12		5292.2.i.a.1549.1		2
315.193	odd	12		5292.2.l.a.3313.1		2
315.223	even	12		5292.2.j.a.1765.1		2
315.248	odd	12		1764.2.l.a.961.1		2
315.263	even	12		1764.2.i.a.373.1		2
315.268	odd	12		5292.2.l.a.361.1		2
315.283	even	12		5292.2.l.c.3313.1		2
315.293	odd	12		1764.2.j.b.1177.1		2
315.313	even	12		5292.2.l.c.361.1		2
360.13	odd	12		1728.2.i.d.1153.1		2
360.43	even	12		1728.2.i.c.577.1		2
360.83	odd	12		576.2.i.e.193.1		2
360.133	odd	12		1728.2.i.d.577.1		2
360.173	even	12		576.2.i.f.193.1		2
360.203	odd	12		576.2.i.e.385.1		2
360.283	even	12		1728.2.i.c.1153.1		2
360.293	even	12		576.2.i.f.385.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
36.2.e.a.13.1	✓	2	45.38	even	12
36.2.e.a.25.1	yes	2	45.23	even	12
108.2.e.a.37.1		2	45.43	odd	12
108.2.e.a.73.1		2	45.13	odd	12
144.2.i.a.49.1		2	180.83	odd	12
144.2.i.a.97.1		2	180.23	odd	12
324.2.a.a.1.1		1	5.3	odd	4
324.2.a.c.1.1		1	15.8	even	4
432.2.i.c.145.1		2	180.43	even	12
432.2.i.c.289.1		2	180.103	even	12
576.2.i.e.193.1		2	360.83	odd	12
576.2.i.e.385.1		2	360.203	odd	12
576.2.i.f.193.1		2	360.173	even	12
576.2.i.f.385.1		2	360.293	even	12
900.2.i.b.301.1		2	45.2	even	12
900.2.i.b.601.1		2	45.32	even	12
900.2.s.b.49.1		4	9.2	odd	6
900.2.s.b.49.2		4	45.29	odd	6
900.2.s.b.349.1		4	45.14	odd	6
900.2.s.b.349.2		4	9.5	odd	6
1296.2.a.b.1.1		1	20.3	even	4
1296.2.a.k.1.1		1	60.23	odd	4
1728.2.i.c.577.1		2	360.43	even	12
1728.2.i.c.1153.1		2	360.283	even	12
1728.2.i.d.577.1		2	360.133	odd	12
1728.2.i.d.1153.1		2	360.13	odd	12
1764.2.i.a.373.1		2	315.263	even	12
1764.2.i.a.1537.1		2	315.23	even	12
1764.2.i.c.373.1		2	315.38	odd	12
1764.2.i.c.1537.1		2	315.68	odd	12
1764.2.j.b.589.1		2	315.83	odd	12
1764.2.j.b.1177.1		2	315.293	odd	12
1764.2.l.a.949.1		2	315.173	odd	12
1764.2.l.a.961.1		2	315.248	odd	12
1764.2.l.c.949.1		2	315.128	even	12
1764.2.l.c.961.1		2	315.158	even	12
2700.2.i.b.901.1		2	45.7	odd	12
2700.2.i.b.1801.1		2	45.22	odd	12
2700.2.s.b.1549.1		4	9.7	even	3
2700.2.s.b.1549.2		4	45.34	even	6
2700.2.s.b.2449.1		4	45.4	even	6
2700.2.s.b.2449.2		4	9.4	even	3
5184.2.a.e.1.1		1	120.53	even	4
5184.2.a.f.1.1		1	120.83	odd	4
5184.2.a.ba.1.1		1	40.13	odd	4
5184.2.a.bb.1.1		1	40.3	even	4
5292.2.i.a.1549.1		2	315.178	even	12
5292.2.i.a.2125.1		2	315.103	even	12
5292.2.i.c.1549.1		2	315.88	odd	12
5292.2.i.c.2125.1		2	315.58	odd	12
5292.2.j.a.1765.1		2	315.223	even	12
5292.2.j.a.3529.1		2	315.13	even	12
5292.2.l.a.361.1		2	315.268	odd	12
5292.2.l.a.3313.1		2	315.193	odd	12
5292.2.l.c.361.1		2	315.313	even	12
5292.2.l.c.3313.1		2	315.283	even	12
8100.2.a.g.1.1		1	5.2	odd	4
8100.2.a.j.1.1		1	15.2	even	4
8100.2.d.c.649.1		2	1.1	even	1	trivial
8100.2.d.c.649.2		2	5.4	even	2	inner
8100.2.d.h.649.1		2	3.2	odd	2
8100.2.d.h.649.2		2	15.14	odd	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:	\( N \)	\(=\)	\( 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8100.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{7} +O(q^{10})\)	\(q-1.00000i q^{7} -3.00000 q^{11} +1.00000i q^{13} -6.00000i q^{17} +4.00000 q^{19} -3.00000i q^{23} +3.00000 q^{29} +5.00000 q^{31} +2.00000i q^{37} -3.00000 q^{41} +1.00000i q^{43} +9.00000i q^{47} +6.00000 q^{49} -6.00000i q^{53} -3.00000 q^{59} -13.0000 q^{61} -7.00000i q^{67} +12.0000 q^{71} +10.0000i q^{73} +3.00000i q^{77} -11.0000 q^{79} -9.00000i q^{83} +6.00000 q^{89} +1.00000 q^{91} +11.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q+O(q^{10}) \) 2 * q	\( 2 q - 6 q^{11} + 8 q^{19} + 6 q^{29} + 10 q^{31} - 6 q^{41} + 12 q^{49} - 6 q^{59} - 26 q^{61} + 24 q^{71} - 22 q^{79} + 12 q^{89} + 2 q^{91}+O(q^{100}) \) 2 * q - 6 * q^11 + 8 * q^19 + 6 * q^29 + 10 * q^31 - 6 * q^41 + 12 * q^49 - 6 * q^59 - 26 * q^61 + 24 * q^71 - 22 * q^79 + 12 * q^89 + 2 * q^91

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more