Properties

Label 99.13.c.b
Level $99$
Weight $13$
Character orbit 99.c
Analytic conductor $90.485$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [99,13,Mod(10,99)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(99, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("99.10");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 99.c (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: no
Analytic conductor: \(90.4853879104\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 30654x^{8} + 318945120x^{6} + 1305642637440x^{4} + 2049564619929600x^{2} + 957721368231936000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{3}\cdot 11^{3} \)
Twist minimal: no (minimal twist has level 11)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{3} + \beta_{2} - 2035) q^{4} + (\beta_{7} - \beta_{3} - 2650) q^{5} + ( - \beta_{8} + \beta_{5} - 247 \beta_1) q^{7} + (\beta_{5} + \beta_{4} - 1652 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{3} + \beta_{2} - 2035) q^{4} + (\beta_{7} - \beta_{3} - 2650) q^{5} + ( - \beta_{8} + \beta_{5} - 247 \beta_1) q^{7} + (\beta_{5} + \beta_{4} - 1652 \beta_1) q^{8} + ( - \beta_{9} - \beta_{8} + 6 \beta_{5} + \beta_{4} - 3814 \beta_1) q^{10} + ( - \beta_{9} - 2 \beta_{8} + 11 \beta_{7} - 33 \beta_{6} + 7 \beta_{5} + \cdots + 171754) q^{11}+ \cdots + ( - 457844 \beta_{9} + 2494412 \beta_{8} + \cdots + 3576789453 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 20348 q^{4} - 26492 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 20348 q^{4} - 26492 q^{5} + 1716374 q^{11} + 15139368 q^{14} + 17974408 q^{16} + 125399668 q^{20} + 83533560 q^{22} - 330297476 q^{23} - 438477018 q^{25} + 372191832 q^{26} - 1921955548 q^{31} + 7677299352 q^{34} + 1788323996 q^{37} - 11254769640 q^{38} - 6124969708 q^{44} + 24975510124 q^{47} - 6325710998 q^{49} + 16325502124 q^{53} + 14298843812 q^{55} - 82892128176 q^{56} - 84518430720 q^{58} - 62339390564 q^{59} + 95192926864 q^{64} - 90035301244 q^{67} - 382808641560 q^{70} + 359910119740 q^{71} - 425991883680 q^{77} - 1147768798712 q^{80} + 625030365960 q^{82} + 1219447545552 q^{86} + 134692485840 q^{88} - 670996780412 q^{89} + 1356772643808 q^{91} + 3181666532764 q^{92} - 6250704684964 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 30654x^{8} + 318945120x^{6} + 1305642637440x^{4} + 2049564619929600x^{2} + 957721368231936000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4809 \nu^{8} + 116860406 \nu^{6} + 910996482960 \nu^{4} + \cdots + 53\!\cdots\!00 ) / 568951057152000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4809 \nu^{8} + 116860406 \nu^{6} + 910996482960 \nu^{4} + \cdots + 18\!\cdots\!00 ) / 568951057152000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 4809 \nu^{9} + 116860406 \nu^{7} + 910996482960 \nu^{5} + \cdots + 73\!\cdots\!00 \nu ) / 568951057152000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 4809 \nu^{9} - 116860406 \nu^{7} - 910996482960 \nu^{5} + \cdots - 17\!\cdots\!00 \nu ) / 568951057152000 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 53843 \nu^{8} + 3109065762 \nu^{6} + 42940625799120 \nu^{4} + \cdots + 16\!\cdots\!00 ) / 853426585728000 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 76993 \nu^{8} + 2278331262 \nu^{6} + 20369889642720 \nu^{4} + \cdots + 27\!\cdots\!00 ) / 853426585728000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 35681 \nu^{9} - 1047127134 \nu^{7} - 10033340422560 \nu^{5} + \cdots - 32\!\cdots\!00 \nu ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 668371 \nu^{9} - 18600313914 \nu^{7} - 156520246211040 \nu^{5} + \cdots - 89\!\cdots\!00 \nu ) / 68\!\cdots\!00 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{3} + \beta_{2} - 6131 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} - 9844\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -526\beta_{7} + 119\beta_{6} + 18023\beta_{3} - 13297\beta_{2} + 60360303 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 526\beta_{9} - 1378\beta_{8} - 16250\beta_{5} - 14418\beta_{4} + 110601102\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 9564084\beta_{7} - 1689786\beta_{6} - 249635562\beta_{3} + 160166598\beta_{2} - 678206513082 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -9564084\beta_{9} + 17472492\beta_{8} + 234933900\beta_{5} + 178179612\beta_{4} - 1288838972148\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 132767433816 \beta_{7} + 18519545964 \beta_{6} + 3298338354828 \beta_{3} - 1901174467092 \beta_{2} + 79\!\cdots\!08 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 132767433816 \beta_{9} - 163545301608 \beta_{8} - 3276952323240 \beta_{5} - 2126539630728 \beta_{4} + 15\!\cdots\!12 \beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/99\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(56\)
\(\chi(n)\) \(-1\) \(1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
10.1
110.069i
107.361i
64.4927i
45.2549i
28.3748i
28.3748i
45.2549i
64.4927i
107.361i
110.069i
110.069i 0 −8019.20 −14011.2 0 131085.i 431823.i 0 1.54220e6i
10.2 107.361i 0 −7430.36 −423.935 0 67177.7i 357980.i 0 45514.0i
10.3 64.4927i 0 −63.3083 22798.6 0 28764.7i 260079.i 0 1.47035e6i
10.4 45.2549i 0 2048.00 −15889.4 0 93839.3i 278046.i 0 719072.i
10.5 28.3748i 0 3290.87 −5720.10 0 202583.i 209601.i 0 162307.i
10.6 28.3748i 0 3290.87 −5720.10 0 202583.i 209601.i 0 162307.i
10.7 45.2549i 0 2048.00 −15889.4 0 93839.3i 278046.i 0 719072.i
10.8 64.4927i 0 −63.3083 22798.6 0 28764.7i 260079.i 0 1.47035e6i
10.9 107.361i 0 −7430.36 −423.935 0 67177.7i 357980.i 0 45514.0i
10.10 110.069i 0 −8019.20 −14011.2 0 131085.i 431823.i 0 1.54220e6i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 10.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.13.c.b 10
3.b odd 2 1 11.13.b.b 10
11.b odd 2 1 inner 99.13.c.b 10
12.b even 2 1 176.13.h.c 10
33.d even 2 1 11.13.b.b 10
132.d odd 2 1 176.13.h.c 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
11.13.b.b 10 3.b odd 2 1
11.13.b.b 10 33.d even 2 1
99.13.c.b 10 1.a even 1 1 trivial
99.13.c.b 10 11.b odd 2 1 inner
176.13.h.c 10 12.b even 2 1
176.13.h.c 10 132.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} + 30654 T_{2}^{8} + 318945120 T_{2}^{6} + 1305642637440 T_{2}^{4} + \cdots + 95\!\cdots\!00 \) acting on \(S_{13}^{\mathrm{new}}(99, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + 30654 T^{8} + \cdots + 95\!\cdots\!00 \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( (T^{5} + 13246 T^{4} + \cdots - 12\!\cdots\!50)^{2} \) Copy content Toggle raw display
$7$ \( T^{10} + 72369291504 T^{8} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{10} - 1716374 T^{9} + \cdots + 30\!\cdots\!01 \) Copy content Toggle raw display
$13$ \( T^{10} + 169803521725584 T^{8} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{5} + 165148738 T^{4} + \cdots + 96\!\cdots\!50)^{2} \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{5} + 960977774 T^{4} + \cdots - 13\!\cdots\!18)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} - 894161998 T^{4} + \cdots + 13\!\cdots\!50)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{10} + \cdots + 87\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T^{5} - 12487755062 T^{4} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{5} - 8162751062 T^{4} + \cdots + 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{5} + 31169695282 T^{4} + \cdots + 87\!\cdots\!62)^{2} \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{5} + 45017650622 T^{4} + \cdots + 58\!\cdots\!50)^{2} \) Copy content Toggle raw display
$71$ \( (T^{5} - 179955059870 T^{4} + \cdots - 79\!\cdots\!78)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{10} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{5} + 335498390206 T^{4} + \cdots + 11\!\cdots\!98)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} + 3125352342482 T^{4} + \cdots - 21\!\cdots\!50)^{2} \) Copy content Toggle raw display
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