LMFDB - Embedded newform 2099.1.b.a.2098.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2099,1,Mod(2098,2099)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2099, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 1, names="a")

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

chi := DirichletCharacter("2099.2098");

S:= CuspForms(chi, 1);

N := Newforms(S);

Level:	$ N $	$=$	$ 2099 $
Weight:	$ k $	$=$	$ 1 $
Character orbit:	$[\chi]$	$=$	2099.b (of order $2$, degree $1$, 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:	yes
Analytic conductor:	$1.04753746152$
Analytic rank:	$0$
Dimension:	$9$
Coefficient field:	$\Q(\zeta_{38})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - x^{8} - 8x^{7} + 7x^{6} + 21x^{5} - 15x^{4} - 20x^{3} + 10x^{2} + 5x - 1 $ x^9 - x^8 - 8x^7 + 7x^6 + 21x^5 - 15x^4 - 20x^3 + 10x^2 + 5*x - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Projective image:	$D_{19}$
Projective field:	Galois closure of $\mathbb{Q}[x]/(x^{19} - \cdots)$

Embedding invariants

Embedding label			2098.5
Root			$1.97272$ of defining polynomial
Character	$\chi$	$=$	2099.2098

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.165159 q^{3} +1.00000 q^{4} +1.09390 q^{5} +1.89163 q^{7} -0.972723 q^{9} -0.165159 q^{12} -0.180666 q^{15} +1.00000 q^{16} -1.35456 q^{17} +1.09390 q^{20} -0.312420 q^{21} +0.196609 q^{25} +0.325812 q^{27} +1.89163 q^{28} -1.35456 q^{31} +2.06925 q^{35} -0.972723 q^{36} -1.75895 q^{37} -1.75895 q^{41} -1.06406 q^{45} -0.803391 q^{47} -0.165159 q^{48} +2.57828 q^{49} +0.223718 q^{51} -1.97272 q^{59} -0.180666 q^{60} +0.490971 q^{61} -1.84004 q^{63} +1.00000 q^{64} -1.35456 q^{68} -1.97272 q^{73} -0.0324717 q^{75} +1.09390 q^{80} +0.918912 q^{81} +1.89163 q^{83} -0.312420 q^{84} -1.48175 q^{85} +0.223718 q^{93} +1.57828 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q - q^{3} + 9 q^{4} - q^{5} - q^{7} + 8 q^{9} - q^{12} - 2 q^{15} + 9 q^{16} - q^{17} - q^{20} - 2 q^{21} + 8 q^{25} - 2 q^{27} - q^{28} - q^{31} - 2 q^{35} + 8 q^{36} - q^{37} - q^{41} - 3 q^{45}+ \cdots - q^{97}+O(q^{100}) $ 9 * q - q^3 + 9 * q^4 - q^5 - q^7 + 8 * q^9 - q^12 - 2 * q^15 + 9 * q^16 - q^17 - q^20 - 2 * q^21 + 8 * q^25 - 2 * q^27 - q^28 - q^31 - 2 * q^35 + 8 * q^36 - q^37 - q^41 - 3 * q^45 - q^47 - q^48 + 8 * q^49 - 2 * q^51 - q^59 - 2 * q^60 - q^61 - 3 * q^63 + 9 * q^64 - q^68 - q^73 - 3 * q^75 - q^80 + 7 * q^81 - q^83 - 2 * q^84 - 2 * q^85 - 2 * q^93 - q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2099\mathbb{Z}\right)^\times$.

$n$	$2$
$\chi(n)$	$-1$

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	1.00000			$0$
$2$	0	0	−1.00000			$\pi$
$3$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$3$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$4$	1.00000	1.00000
$5$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$5$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$6$	0	0
$7$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$7$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$8$	0	0
$9$	−0.972723	−0.972723
$10$	0	0
$11$	0	0	1.00000			$0$
$11$	0	0	−1.00000			$\pi$
$12$	−0.165159	−0.165159
$13$	0	0	1.00000			$0$
$13$	0	0	−1.00000			$\pi$
$14$	0	0
$15$	−0.180666	−0.180666
$16$	1.00000	1.00000
$17$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$17$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$18$	0	0
$19$	0	0	1.00000			$0$
$19$	0	0	−1.00000			$\pi$
$20$	1.09390	1.09390
$21$	−0.312420	−0.312420
$22$	0	0
$23$	0	0	1.00000			$0$
$23$	0	0	−1.00000			$\pi$
$24$	0	0
$25$	0.196609	0.196609
$26$	0	0
$27$	0.325812	0.325812
$28$	1.89163	1.89163
$29$	0	0	1.00000			$0$
$29$	0	0	−1.00000			$\pi$
$30$	0	0
$31$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$31$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	2.06925	2.06925
$36$	−0.972723	−0.972723
$37$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$37$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$41$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$42$	0	0
$43$	0	0	1.00000			$0$
$43$	0	0	−1.00000			$\pi$
$44$	0	0
$45$	−1.06406	−1.06406
$46$	0	0
$47$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$47$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$48$	−0.165159	−0.165159
$49$	2.57828	2.57828
$50$	0	0
$51$	0.223718	0.223718
$52$	0	0
$53$	0	0	1.00000			$0$
$53$	0	0	−1.00000			$\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$59$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$60$	−0.180666	−0.180666
$61$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$61$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$62$	0	0
$63$	−1.84004	−1.84004
$64$	1.00000	1.00000
$65$	0	0
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	−1.35456	−1.35456
$69$	0	0
$70$	0	0
$71$	0	0	1.00000			$0$
$71$	0	0	−1.00000			$\pi$
$72$	0	0
$73$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$73$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$74$	0	0
$75$	−0.0324717	−0.0324717
$76$	0	0
$77$	0	0
$78$	0	0
$79$	0	0	1.00000			$0$
$79$	0	0	−1.00000			$\pi$
$80$	1.09390	1.09390
$81$	0.918912	0.918912
$82$	0	0
$83$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$83$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$84$	−0.312420	−0.312420
$85$	−1.48175	−1.48175
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0	0	1.00000			$0$
$89$	0	0	−1.00000			$\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	0.223718	0.223718
$94$	0	0
$95$	0	0
$96$	0	0
$97$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$97$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$98$	0	0
$99$	0	0
$100$	0.196609	0.196609
$101$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$101$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$102$	0	0
$103$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$103$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$104$	0	0
$105$	−0.341755	−0.341755
$106$	0	0
$107$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$107$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$108$	0.325812	0.325812
$109$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$109$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$110$	0	0
$111$	0.290505	0.290505
$112$	1.89163	1.89163
$113$	0	0	1.00000			$0$
$113$	0	0	−1.00000			$\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−2.56234	−2.56234
$120$	0	0
$121$	1.00000	1.00000
$122$	0	0
$123$	0.290505	0.290505
$124$	−1.35456	−1.35456
$125$	−0.878826	−0.878826
$126$	0	0
$127$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$127$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	0	0	1.00000			$0$
$131$	0	0	−1.00000			$\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	0.356405	0.356405
$136$	0	0
$137$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$137$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$138$	0	0
$139$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$139$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$140$	2.06925	2.06925
$141$	0.132687	0.132687
$142$	0	0
$143$	0	0
$144$	−0.972723	−0.972723
$145$	0	0
$146$	0	0
$147$	−0.425826	−0.425826
$148$	−1.75895	−1.75895
$149$	0	0	1.00000			$0$
$149$	0	0	−1.00000			$\pi$
$150$	0	0
$151$	0	0	1.00000			$0$
$151$	0	0	−1.00000			$\pi$
$152$	0	0
$153$	1.31761	1.31761
$154$	0	0
$155$	−1.48175	−1.48175
$156$	0	0
$157$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$157$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	0	0
$162$	0	0
$163$	0	0	1.00000			$0$
$163$	0	0	−1.00000			$\pi$
$164$	−1.75895	−1.75895
$165$	0	0
$166$	0	0
$167$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$167$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$168$	0	0
$169$	1.00000	1.00000
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$173$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$174$	0	0
$175$	0.371913	0.371913
$176$	0	0
$177$	0.325812	0.325812
$178$	0	0
$179$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$179$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$180$	−1.06406	−1.06406
$181$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$181$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$182$	0	0
$183$	−0.0810881	−0.0810881
$184$	0	0
$185$	−1.92411	−1.92411
$186$	0	0
$187$	0	0
$188$	−0.803391	−0.803391
$189$	0.616318	0.616318
$190$	0	0
$191$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$191$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$192$	−0.165159	−0.165159
$193$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$193$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$194$	0	0
$195$	0	0
$196$	2.57828	2.57828
$197$	0	0	1.00000			$0$
$197$	0	0	−1.00000			$\pi$
$198$	0	0
$199$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$199$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	0	0
$204$	0.223718	0.223718
$205$	−1.92411	−1.92411
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$211$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−2.56234	−2.56234
$218$	0	0
$219$	0.325812	0.325812
$220$	0	0
$221$	0	0
$222$	0	0
$223$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$223$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$224$	0	0
$225$	−0.191246	−0.191246
$226$	0	0
$227$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$227$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$228$	0	0
$229$	0	0	1.00000			$0$
$229$	0	0	−1.00000			$\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$233$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$234$	0	0
$235$	−0.878826	−0.878826
$236$	−1.97272	−1.97272
$237$	0	0
$238$	0	0
$239$	0	0	1.00000			$0$
$239$	0	0	−1.00000			$\pi$
$240$	−0.180666	−0.180666
$241$	0	0	1.00000			$0$
$241$	0	0	−1.00000			$\pi$
$242$	0	0
$243$	−0.477579	−0.477579
$244$	0.490971	0.490971
$245$	2.82037	2.82037
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−0.312420	−0.312420
$250$	0	0
$251$	0	0	1.00000			$0$
$251$	0	0	−1.00000			$\pi$
$252$	−1.84004	−1.84004
$253$	0	0
$254$	0	0
$255$	0.244724	0.244724
$256$	1.00000	1.00000
$257$	0	0	1.00000			$0$
$257$	0	0	−1.00000			$\pi$
$258$	0	0
$259$	−3.32729	−3.32729
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0	0	1.00000			$0$
$263$	0	0	−1.00000			$\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	0	0
$268$	0	0
$269$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$269$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$270$	0	0
$271$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$271$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$272$	−1.35456	−1.35456
$273$	0	0
$274$	0	0
$275$	0	0
$276$	0	0
$277$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$277$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$278$	0	0
$279$	1.31761	1.31761
$280$	0	0
$281$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$281$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$282$	0	0
$283$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$283$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−3.32729	−3.32729
$288$	0	0
$289$	0.834841	0.834841
$290$	0	0
$291$	−0.260667	−0.260667
$292$	−1.97272	−1.97272
$293$	0	0	1.00000			$0$
$293$	0	0	−1.00000			$\pi$
$294$	0	0
$295$	−2.15795	−2.15795
$296$	0	0
$297$	0	0
$298$	0	0
$299$	0	0
$300$	−0.0324717	−0.0324717
$301$	0	0
$302$	0	0
$303$	−0.312420	−0.312420
$304$	0	0
$305$	0.537071	0.537071
$306$	0	0
$307$	0	0	1.00000			$0$
$307$	0	0	−1.00000			$\pi$
$308$	0	0
$309$	−0.0810881	−0.0810881
$310$	0	0
$311$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$311$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$312$	0	0
$313$	0	0	1.00000			$0$
$313$	0	0	−1.00000			$\pi$
$314$	0	0
$315$	−2.01281	−2.01281
$316$	0	0
$317$	0	0	1.00000			$0$
$317$	0	0	−1.00000			$\pi$
$318$	0	0
$319$	0	0
$320$	1.09390	1.09390
$321$	0.132687	0.132687
$322$	0	0
$323$	0	0
$324$	0.918912	0.918912
$325$	0	0
$326$	0	0
$327$	−0.180666	−0.180666
$328$	0	0
$329$	−1.51972	−1.51972
$330$	0	0
$331$	0	0	1.00000			$0$
$331$	0	0	−1.00000			$\pi$
$332$	1.89163	1.89163
$333$	1.71097	1.71097
$334$	0	0
$335$	0	0
$336$	−0.312420	−0.312420
$337$	0	0	1.00000			$0$
$337$	0	0	−1.00000			$\pi$
$338$	0	0
$339$	0	0
$340$	−1.48175	−1.48175
$341$	0	0
$342$	0	0
$343$	2.98553	2.98553
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$347$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$348$	0	0
$349$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$349$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$353$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	0.423192	0.423192
$358$	0	0
$359$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$359$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$360$	0	0
$361$	1.00000	1.00000
$362$	0	0
$363$	−0.165159	−0.165159
$364$	0	0
$365$	−2.15795	−2.15795
$366$	0	0
$367$	0	0	1.00000			$0$
$367$	0	0	−1.00000			$\pi$
$368$	0	0
$369$	1.71097	1.71097
$370$	0	0
$371$	0	0
$372$	0.223718	0.223718
$373$	0	0	1.00000			$0$
$373$	0	0	−1.00000			$\pi$
$374$	0	0
$375$	0.145146	0.145146
$376$	0	0
$377$	0	0
$378$	0	0
$379$	0	0	1.00000			$0$
$379$	0	0	−1.00000			$\pi$
$380$	0	0
$381$	−0.180666	−0.180666
$382$	0	0
$383$	0	0	1.00000			$0$
$383$	0	0	−1.00000			$\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0	0
$388$	1.57828	1.57828
$389$	0	0	1.00000			$0$
$389$	0	0	−1.00000			$\pi$
$390$	0	0
$391$	0	0
$392$	0	0
$393$	0	0
$394$	0	0
$395$	0	0
$396$	0	0
$397$	0	0	1.00000			$0$
$397$	0	0	−1.00000			$\pi$
$398$	0	0
$399$	0	0
$400$	0.196609	0.196609
$401$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$401$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$402$	0	0
$403$	0	0
$404$	1.89163	1.89163
$405$	1.00519	1.00519
$406$	0	0
$407$	0	0
$408$	0	0
$409$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$409$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$410$	0	0
$411$	−0.0810881	−0.0810881
$412$	0.490971	0.490971
$413$	−3.73167	−3.73167
$414$	0	0
$415$	2.06925	2.06925
$416$	0	0
$417$	0.290505	0.290505
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	−0.341755	−0.341755
$421$	0	0	1.00000			$0$
$421$	0	0	−1.00000			$\pi$
$422$	0	0
$423$	0.781476	0.781476
$424$	0	0
$425$	−0.266320	−0.266320
$426$	0	0
$427$	0.928738	0.928738
$428$	−0.803391	−0.803391
$429$	0	0
$430$	0	0
$431$	0	0	1.00000			$0$
$431$	0	0	−1.00000			$\pi$
$432$	0.325812	0.325812
$433$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$433$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$434$	0	0
$435$	0	0
$436$	1.09390	1.09390
$437$	0	0
$438$	0	0
$439$	0	0	1.00000			$0$
$439$	0	0	−1.00000			$\pi$
$440$	0	0
$441$	−2.50795	−2.50795
$442$	0	0
$443$	0	0	1.00000			$0$
$443$	0	0	−1.00000			$\pi$
$444$	0.290505	0.290505
$445$	0	0
$446$	0	0
$447$	0	0
$448$	1.89163	1.89163
$449$	0	0	1.00000			$0$
$449$	0	0	−1.00000			$\pi$
$450$	0	0
$451$	0	0
$452$	0	0
$453$	0	0
$454$	0	0
$455$	0	0
$456$	0	0
$457$	0	0	1.00000			$0$
$457$	0	0	−1.00000			$\pi$
$458$	0	0
$459$	−0.441333	−0.441333
$460$	0	0
$461$	0	0	1.00000			$0$
$461$	0	0	−1.00000			$\pi$
$462$	0	0
$463$	0	0	1.00000			$0$
$463$	0	0	−1.00000			$\pi$
$464$	0	0
$465$	0.244724	0.244724
$466$	0	0
$467$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$467$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−0.260667	−0.260667
$472$	0	0
$473$	0	0
$474$	0	0
$475$	0	0
$476$	−2.56234	−2.56234
$477$	0	0
$478$	0	0
$479$	0	0	1.00000			$0$
$479$	0	0	−1.00000			$\pi$
$480$	0	0
$481$	0	0
$482$	0	0
$483$	0	0
$484$	1.00000	1.00000
$485$	1.72648	1.72648
$486$	0	0
$487$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$487$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0	0	1.00000			$0$
$491$	0	0	−1.00000			$\pi$
$492$	0.290505	0.290505
$493$	0	0
$494$	0	0
$495$	0	0
$496$	−1.35456	−1.35456
$497$	0	0
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	−0.878826	−0.878826
$501$	−0.260667	−0.260667
$502$	0	0
$503$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$503$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$504$	0	0
$505$	2.06925	2.06925
$506$	0	0
$507$	−0.165159	−0.165159
$508$	1.09390	1.09390
$509$	0	0	1.00000			$0$
$509$	0	0	−1.00000			$\pi$
$510$	0	0
$511$	−3.73167	−3.73167
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0.537071	0.537071
$516$	0	0
$517$	0	0
$518$	0	0
$519$	−0.260667	−0.260667
$520$	0	0
$521$	0	0	1.00000			$0$
$521$	0	0	−1.00000			$\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	−0.0614246	−0.0614246
$526$	0	0
$527$	1.83484	1.83484
$528$	0	0
$529$	1.00000	1.00000
$530$	0	0
$531$	1.91891	1.91891
$532$	0	0
$533$	0	0
$534$	0	0
$535$	−0.878826	−0.878826
$536$	0	0
$537$	0.0272774	0.0272774
$538$	0	0
$539$	0	0
$540$	0.356405	0.356405
$541$	0	0	1.00000			$0$
$541$	0	0	−1.00000			$\pi$
$542$	0	0
$543$	−0.180666	−0.180666
$544$	0	0
$545$	1.19661	1.19661
$546$	0	0
$547$	0	0	1.00000			$0$
$547$	0	0	−1.00000			$\pi$
$548$	0.490971	0.490971
$549$	−0.477579	−0.477579
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0	0
$554$	0	0
$555$	0.317783	0.317783
$556$	−1.75895	−1.75895
$557$	0	0	1.00000			$0$
$557$	0	0	−1.00000			$\pi$
$558$	0	0
$559$	0	0
$560$	2.06925	2.06925
$561$	0	0
$562$	0	0
$563$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$563$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$564$	0.132687	0.132687
$565$	0	0
$566$	0	0
$567$	1.73825	1.73825
$568$	0	0
$569$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$569$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$570$	0	0
$571$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$571$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$572$	0	0
$573$	−0.312420	−0.312420
$574$	0	0
$575$	0	0
$576$	−0.972723	−0.972723
$577$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$577$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$578$	0	0
$579$	0.223718	0.223718
$580$	0	0
$581$	3.57828	3.57828
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$587$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$588$	−0.425826	−0.425826
$589$	0	0
$590$	0	0
$591$	0	0
$592$	−1.75895	−1.75895
$593$	0	0	1.00000			$0$
$593$	0	0	−1.00000			$\pi$
$594$	0	0
$595$	−2.80293	−2.80293
$596$	0	0
$597$	0.0272774	0.0272774
$598$	0	0
$599$	0	0	1.00000			$0$
$599$	0	0	−1.00000			$\pi$
$600$	0	0
$601$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$601$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	1.09390	1.09390
$606$	0	0
$607$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$607$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	1.31761	1.31761
$613$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$613$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$614$	0	0
$615$	0.317783	0.317783
$616$	0	0
$617$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$617$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$618$	0	0
$619$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$619$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$620$	−1.48175	−1.48175
$621$	0	0
$622$	0	0
$623$	0	0
$624$	0	0
$625$	−1.15795	−1.15795
$626$	0	0
$627$	0	0
$628$	1.57828	1.57828
$629$	2.38261	2.38261
$630$	0	0
$631$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$631$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$632$	0	0
$633$	0.132687	0.132687
$634$	0	0
$635$	1.19661	1.19661
$636$	0	0
$637$	0	0
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$641$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$642$	0	0
$643$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$643$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0	0	1.00000			$0$
$647$	0	0	−1.00000			$\pi$
$648$	0	0
$649$	0	0
$650$	0	0
$651$	0.423192	0.423192
$652$	0	0
$653$	0	0	1.00000			$0$
$653$	0	0	−1.00000			$\pi$
$654$	0	0
$655$	0	0
$656$	−1.75895	−1.75895
$657$	1.91891	1.91891
$658$	0	0
$659$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$659$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$660$	0	0
$661$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$661$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0	0
$668$	1.57828	1.57828
$669$	0.325812	0.325812
$670$	0	0
$671$	0	0
$672$	0	0
$673$	0	0	1.00000			$0$
$673$	0	0	−1.00000			$\pi$
$674$	0	0
$675$	0.0640577	0.0640577
$676$	1.00000	1.00000
$677$	0	0	1.00000			$0$
$677$	0	0	−1.00000			$\pi$
$678$	0	0
$679$	2.98553	2.98553
$680$	0	0
$681$	0.0272774	0.0272774
$682$	0	0
$683$	1.57828	1.57828	0.789141	−	0.614213i	$-0.210526\pi$
$683$	1.57828	1.57828	0.789141	+	0.614213i	$0.210526\pi$
$684$	0	0
$685$	0.537071	0.537071
$686$	0	0
$687$	0	0
$688$	0	0
$689$	0	0
$690$	0	0
$691$	0	0	1.00000			$0$
$691$	0	0	−1.00000			$\pi$
$692$	1.57828	1.57828
$693$	0	0
$694$	0	0
$695$	−1.92411	−1.92411
$696$	0	0
$697$	2.38261	2.38261
$698$	0	0
$699$	0.290505	0.290505
$700$	0.371913	0.371913
$701$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$701$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	0.145146	0.145146
$706$	0	0
$707$	3.57828	3.57828
$708$	0.325812	0.325812
$709$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$709$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	0	0
$714$	0	0
$715$	0	0
$716$	−0.165159	−0.165159
$717$	0	0
$718$	0	0
$719$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$719$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$720$	−1.06406	−1.06406
$721$	0.928738	0.928738
$722$	0	0
$723$	0	0
$724$	1.09390	1.09390
$725$	0	0
$726$	0	0
$727$	0	0	1.00000			$0$
$727$	0	0	−1.00000			$\pi$
$728$	0	0
$729$	−0.840036	−0.840036
$730$	0	0
$731$	0	0
$732$	−0.0810881	−0.0810881
$733$	−0.165159	−0.165159	−0.0825793	−	0.996584i	$-0.526316\pi$
$733$	−0.165159	−0.165159	−0.0825793	+	0.996584i	$0.526316\pi$
$734$	0	0
$735$	−0.465809	−0.465809
$736$	0	0
$737$	0	0
$738$	0	0
$739$	0	0	1.00000			$0$
$739$	0	0	−1.00000			$\pi$
$740$	−1.92411	−1.92411
$741$	0	0
$742$	0	0
$743$	−1.97272	−1.97272	−0.986361	−	0.164595i	$-0.947368\pi$
$743$	−1.97272	−1.97272	−0.986361	+	0.164595i	$0.947368\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−1.84004	−1.84004
$748$	0	0
$749$	−1.51972	−1.51972
$750$	0	0
$751$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$751$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$752$	−0.803391	−0.803391
$753$	0	0
$754$	0	0
$755$	0	0
$756$	0.616318	0.616318
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	0.490971	0.490971	0.245485	−	0.969400i	$-0.421053\pi$
$761$	0.490971	0.490971	0.245485	+	0.969400i	$0.421053\pi$
$762$	0	0
$763$	2.06925	2.06925
$764$	1.89163	1.89163
$765$	1.44133	1.44133
$766$	0	0
$767$	0	0
$768$	−0.165159	−0.165159
$769$	0	0	1.00000			$0$
$769$	0	0	−1.00000			$\pi$
$770$	0	0
$771$	0	0
$772$	−1.35456	−1.35456
$773$	0	0	1.00000			$0$
$773$	0	0	−1.00000			$\pi$
$774$	0	0
$775$	−0.266320	−0.266320
$776$	0	0
$777$	0.549530	0.549530
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0	0
$782$	0	0
$783$	0	0
$784$	2.57828	2.57828
$785$	1.72648	1.72648
$786$	0	0
$787$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$787$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	0	0
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	−0.165159	−0.165159
$797$	2.00000	2.00000	1.00000			$0$
$797$	2.00000	2.00000	1.00000			$0$
$798$	0	0
$799$	1.08824	1.08824
$800$	0	0
$801$	0	0
$802$	0	0
$803$	0	0
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−0.260667	−0.260667
$808$	0	0
$809$	0	0	1.00000			$0$
$809$	0	0	−1.00000			$\pi$
$810$	0	0
$811$	0	0	1.00000			$0$
$811$	0	0	−1.00000			$\pi$
$812$	0	0
$813$	−0.260667	−0.260667
$814$	0	0
$815$	0	0
$816$	0.223718	0.223718
$817$	0	0
$818$	0	0
$819$	0	0
$820$	−1.92411	−1.92411
$821$	0	0	1.00000			$0$
$821$	0	0	−1.00000			$\pi$
$822$	0	0
$823$	0	0	1.00000			$0$
$823$	0	0	−1.00000			$\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0	1.00000			$0$
$827$	0	0	−1.00000			$\pi$
$828$	0	0
$829$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$829$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$830$	0	0
$831$	0.0272774	0.0272774
$832$	0	0
$833$	−3.49244	−3.49244
$834$	0	0
$835$	1.72648	1.72648
$836$	0	0
$837$	−0.441333	−0.441333
$838$	0	0
$839$	0	0	1.00000			$0$
$839$	0	0	−1.00000			$\pi$
$840$	0	0
$841$	1.00000	1.00000
$842$	0	0
$843$	−0.260667	−0.260667
$844$	−0.803391	−0.803391
$845$	1.09390	1.09390
$846$	0	0
$847$	1.89163	1.89163
$848$	0	0
$849$	0.132687	0.132687
$850$	0	0
$851$	0	0
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	0	0	1.00000			$0$
$857$	0	0	−1.00000			$\pi$
$858$	0	0
$859$	−1.75895	−1.75895	−0.879474	−	0.475947i	$-0.842105\pi$
$859$	−1.75895	−1.75895	−0.879474	+	0.475947i	$0.842105\pi$
$860$	0	0
$861$	0.549530	0.549530
$862$	0	0
$863$	0	0	1.00000			$0$
$863$	0	0	−1.00000			$\pi$
$864$	0	0
$865$	1.72648	1.72648
$866$	0	0
$867$	−0.137881	−0.137881
$868$	−2.56234	−2.56234
$869$	0	0
$870$	0	0
$871$	0	0
$872$	0	0
$873$	−1.53523	−1.53523
$874$	0	0
$875$	−1.66242	−1.66242
$876$	0.325812	0.325812
$877$	0	0	1.00000			$0$
$877$	0	0	−1.00000			$\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	0	0	1.00000			$0$
$881$	0	0	−1.00000			$\pi$
$882$	0	0
$883$	−1.35456	−1.35456	−0.677282	−	0.735724i	$-0.736842\pi$
$883$	−1.35456	−1.35456	−0.677282	+	0.735724i	$0.736842\pi$
$884$	0	0
$885$	0.356405	0.356405
$886$	0	0
$887$	0	0	1.00000			$0$
$887$	0	0	−1.00000			$\pi$
$888$	0	0
$889$	2.06925	2.06925
$890$	0	0
$891$	0	0
$892$	−1.97272	−1.97272
$893$	0	0
$894$	0	0
$895$	−0.180666	−0.180666
$896$	0	0
$897$	0	0
$898$	0	0
$899$	0	0
$900$	−0.191246	−0.191246
$901$	0	0
$902$	0	0
$903$	0	0
$904$	0	0
$905$	1.19661	1.19661
$906$	0	0
$907$	0	0	1.00000			$0$
$907$	0	0	−1.00000			$\pi$
$908$	−0.165159	−0.165159
$909$	−1.84004	−1.84004
$910$	0	0
$911$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$911$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$912$	0	0
$913$	0	0
$914$	0	0
$915$	−0.0887020	−0.0887020
$916$	0	0
$917$	0	0
$918$	0	0
$919$	0	0	1.00000			$0$
$919$	0	0	−1.00000			$\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	0	0
$924$	0	0
$925$	−0.345825	−0.345825
$926$	0	0
$927$	−0.477579	−0.477579
$928$	0	0
$929$	0	0	1.00000			$0$
$929$	0	0	−1.00000			$\pi$
$930$	0	0
$931$	0	0
$932$	−1.75895	−1.75895
$933$	−0.0810881	−0.0810881
$934$	0	0
$935$	0	0
$936$	0	0
$937$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$937$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$938$	0	0
$939$	0	0
$940$	−0.878826	−0.878826
$941$	−0.803391	−0.803391	−0.401695	−	0.915773i	$-0.631579\pi$
$941$	−0.803391	−0.803391	−0.401695	+	0.915773i	$0.631579\pi$
$942$	0	0
$943$	0	0
$944$	−1.97272	−1.97272
$945$	0.674188	0.674188
$946$	0	0
$947$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$947$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$948$	0	0
$949$	0	0
$950$	0	0
$951$	0	0
$952$	0	0
$953$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$953$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$954$	0	0
$955$	2.06925	2.06925
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0.928738	0.928738
$960$	−0.180666	−0.180666
$961$	0.834841	0.834841
$962$	0	0
$963$	0.781476	0.781476
$964$	0	0
$965$	−1.48175	−1.48175
$966$	0	0
$967$	0	0	1.00000			$0$
$967$	0	0	−1.00000			$\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	1.09390	1.09390	0.546948	−	0.837166i	$-0.315789\pi$
$971$	1.09390	1.09390	0.546948	+	0.837166i	$0.315789\pi$
$972$	−0.477579	−0.477579
$973$	−3.32729	−3.32729
$974$	0	0
$975$	0	0
$976$	0.490971	0.490971
$977$	0	0	1.00000			$0$
$977$	0	0	−1.00000			$\pi$
$978$	0	0
$979$	0	0
$980$	2.82037	2.82037
$981$	−1.06406	−1.06406
$982$	0	0
$983$	0	0	1.00000			$0$
$983$	0	0	−1.00000			$\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	0.250995	0.250995
$988$	0	0
$989$	0	0
$990$	0	0
$991$	0	0	1.00000			$0$
$991$	0	0	−1.00000			$\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−0.180666	−0.180666
$996$	−0.312420	−0.312420
$997$	1.89163	1.89163	0.945817	−	0.324699i	$-0.105263\pi$
$997$	1.89163	1.89163	0.945817	+	0.324699i	$0.105263\pi$
$998$	0	0
$999$	−0.573087	−0.573087

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2099.1.b.a.2098.5	✓	9
2099.2098	odd	2	CM	2099.1.b.a.2098.5	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2099.1.b.a.2098.5	✓	9	1.1	even	1	trivial
2099.1.b.a.2098.5	✓	9	2099.2098	odd	2	CM

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:	\( N \)	\(=\)	\( 2099 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2099.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-0.165159 q^{3} +1.00000 q^{4} +1.09390 q^{5} +1.89163 q^{7} -0.972723 q^{9} -0.165159 q^{12} -0.180666 q^{15} +1.00000 q^{16} -1.35456 q^{17} +1.09390 q^{20} -0.312420 q^{21} +0.196609 q^{25} +0.325812 q^{27} +1.89163 q^{28} -1.35456 q^{31} +2.06925 q^{35} -0.972723 q^{36} -1.75895 q^{37} -1.75895 q^{41} -1.06406 q^{45} -0.803391 q^{47} -0.165159 q^{48} +2.57828 q^{49} +0.223718 q^{51} -1.97272 q^{59} -0.180666 q^{60} +0.490971 q^{61} -1.84004 q^{63} +1.00000 q^{64} -1.35456 q^{68} -1.97272 q^{73} -0.0324717 q^{75} +1.09390 q^{80} +0.918912 q^{81} +1.89163 q^{83} -0.312420 q^{84} -1.48175 q^{85} +0.223718 q^{93} +1.57828 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q - q^{3} + 9 q^{4} - q^{5} - q^{7} + 8 q^{9} - q^{12} - 2 q^{15} + 9 q^{16} - q^{17} - q^{20} - 2 q^{21} + 8 q^{25} - 2 q^{27} - q^{28} - q^{31} - 2 q^{35} + 8 q^{36} - q^{37} - q^{41} - 3 q^{45}+ \cdots - q^{97}+O(q^{100}) \) 9 * q - q^3 + 9 * q^4 - q^5 - q^7 + 8 * q^9 - q^12 - 2 * q^15 + 9 * q^16 - q^17 - q^20 - 2 * q^21 + 8 * q^25 - 2 * q^27 - q^28 - q^31 - 2 * q^35 + 8 * q^36 - q^37 - q^41 - 3 * q^45 - q^47 - q^48 + 8 * q^49 - 2 * q^51 - q^59 - 2 * q^60 - q^61 - 3 * q^63 + 9 * q^64 - q^68 - q^73 - 3 * q^75 - q^80 + 7 * q^81 - q^83 - 2 * q^84 - 2 * q^85 - 2 * q^93 - q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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