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Properties

Label 99.2.g.b
Level $99$
Weight $2$
Character orbit 99.g
Analytic conductor $0.791$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [99,2,Mod(32,99)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(99, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5, 3]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("99.32");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 99 = 3^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 99.g (of order \(6\), degree \(2\), 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: \(0.790518980011\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 15x^{14} + 150x^{12} + 837x^{10} + 3372x^{8} + 8010x^{6} + 13761x^{4} + 13392x^{2} + 8649 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{14} + \beta_{13} + \cdots + \beta_{6}) q^{3}+ \cdots + ( - 2 \beta_{13} - \beta_{9} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{14} + \beta_{13} + \cdots + \beta_{6}) q^{3}+ \cdots + ( - \beta_{15} - \beta_{14} + \beta_{13} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 6 q^{3} - 14 q^{4} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 6 q^{3} - 14 q^{4} + 6 q^{9} - 12 q^{11} + 12 q^{12} - 6 q^{14} - 30 q^{15} - 2 q^{16} + 36 q^{20} + 6 q^{22} + 12 q^{23} - 12 q^{25} + 18 q^{27} - 4 q^{31} + 18 q^{33} - 18 q^{36} - 28 q^{37} + 66 q^{38} - 54 q^{42} - 42 q^{45} - 30 q^{47} + 42 q^{48} + 10 q^{49} + 20 q^{55} - 120 q^{56} - 6 q^{58} - 36 q^{59} + 30 q^{60} + 40 q^{64} + 54 q^{66} + 8 q^{67} + 96 q^{69} + 24 q^{75} + 72 q^{77} - 42 q^{78} + 30 q^{81} + 12 q^{82} - 72 q^{86} - 6 q^{88} - 12 q^{91} + 18 q^{92} - 24 q^{93} - 4 q^{97} - 36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 15x^{14} + 150x^{12} + 837x^{10} + 3372x^{8} + 8010x^{6} + 13761x^{4} + 13392x^{2} + 8649 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 2126395 \nu^{14} - 29523650 \nu^{12} - 288594750 \nu^{10} - 1489331264 \nu^{8} + \cdots - 22396351233 ) / 13131109983 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2126395 \nu^{15} + 29523650 \nu^{13} + 288594750 \nu^{11} + 1489331264 \nu^{9} + \cdots + 9265241250 \nu ) / 13131109983 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4873266 \nu^{15} - 160857665 \nu^{13} - 1890026425 \nu^{11} - 14824095809 \nu^{9} + \cdots - 290195282991 \nu ) / 13131109983 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 47500907 \nu^{15} + 812018893 \nu^{13} + 8261503794 \nu^{11} + 49665627558 \nu^{9} + \cdots + 551339033751 \nu ) / 39393329949 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 49032332 \nu^{14} - 864786042 \nu^{12} - 8872595664 \nu^{10} - 54770706318 \nu^{8} + \cdots - 671327583345 ) / 39393329949 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 54617732 \nu^{14} - 903112393 \nu^{12} - 9132887847 \nu^{10} - 53705137428 \nu^{8} + \cdots - 448939285020 ) / 39393329949 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 54617732 \nu^{15} + 903112393 \nu^{13} + 9132887847 \nu^{11} + 53705137428 \nu^{9} + \cdots + 409545955071 \nu ) / 39393329949 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 58397849 \nu^{14} - 683729698 \nu^{12} - 6054561519 \nu^{10} - 23065315299 \nu^{8} + \cdots + 208063920222 ) / 39393329949 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 64777034 \nu^{15} + 772300648 \nu^{13} + 6920345769 \nu^{11} + 27533309091 \nu^{9} + \cdots - 180268196472 \nu ) / 39393329949 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 70362434 \nu^{15} + 810626999 \nu^{13} + 7180637952 \nu^{11} + 26467740201 \nu^{9} + \cdots - 442049824746 \nu ) / 39393329949 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 93878026 \nu^{14} - 1321129241 \nu^{12} - 12590096616 \nu^{10} - 63366471351 \nu^{8} + \cdots - 157488193548 ) / 39393329949 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 105898756 \nu^{14} + 1495748591 \nu^{12} + 14316065313 \nu^{10} + 72730942857 \nu^{8} + \cdots + 303881783580 ) / 39393329949 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 131099776 \nu^{14} - 1868178342 \nu^{12} - 17994696072 \nu^{10} - 92965220511 \nu^{8} + \cdots - 446373561123 ) / 39393329949 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 145719574 \nu^{15} + 2350751337 \nu^{13} + 23664775347 \nu^{11} + 137437507938 \nu^{9} + \cdots + 1277566080147 \nu ) / 39393329949 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{12} + \beta_{9} + \beta_{7} - 3\beta_{2} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{10} + \beta_{5} + \beta_{4} + 5\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{14} + 9\beta_{13} + 8\beta_{12} - \beta_{7} + \beta_{6} + 12\beta_{2} \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{15} + \beta_{11} + 8\beta_{10} - 8\beta_{8} + \beta_{5} - 9\beta_{4} - 29\beta_{3} - 29\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 13\beta_{14} - 67\beta_{13} - 14\beta_{12} - 68\beta_{9} - 41\beta_{7} - 26\beta_{6} + 57 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -13\beta_{15} - 26\beta_{11} + 27\beta_{10} + 81\beta_{8} - 53\beta_{5} + \beta_{4} + 13\beta_{3} + 166\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -240\beta_{14} - 15\beta_{13} - 234\beta_{12} + 474\beta_{9} + 369\beta_{7} + 120\beta_{6} - 294\beta_{2} - 294 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 240 \beta_{15} + 120 \beta_{11} - 609 \beta_{10} - 255 \beta_{8} + 219 \beta_{5} + 474 \beta_{4} + \cdots + 120 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 969\beta_{14} + 3438\beta_{13} + 2469\beta_{12} + 150\beta_{9} - 1119\beta_{7} + 969\beta_{6} + 1584\beta_{2} \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 969 \beta_{15} + 969 \beta_{11} + 2319 \beta_{10} - 2319 \beta_{8} + 969 \beta_{5} + \cdots - 7341 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 7314\beta_{14} - 22581\beta_{13} - 8583\beta_{12} - 23850\beta_{9} - 7953\beta_{7} - 14628\beta_{6} + 8802 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 7314 \beta_{15} - 14628 \beta_{11} + 15897 \beta_{10} + 31164 \beta_{8} - 13998 \beta_{5} + \cdots + 40605 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 106212 \beta_{14} - 9852 \beta_{13} - 47928 \beta_{12} + 154140 \beta_{9} + 110886 \beta_{7} + \cdots - 50265 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 106212 \beta_{15} + 53106 \beta_{11} - 217098 \beta_{10} - 116064 \beta_{8} + 38076 \beta_{5} + \cdots + 53106 \beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

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

\(n\) \(46\) \(56\)
\(\chi(n)\) \(-1\) \(1 + \beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
32.1
−1.29716 − 2.24675i
−1.10617 − 1.91594i
−0.679041 − 1.17613i
−0.618600 − 1.07145i
0.618600 + 1.07145i
0.679041 + 1.17613i
1.10617 + 1.91594i
1.29716 + 2.24675i
−1.29716 + 2.24675i
−1.10617 + 1.91594i
−0.679041 + 1.17613i
−0.618600 + 1.07145i
0.618600 − 1.07145i
0.679041 − 1.17613i
1.10617 − 1.91594i
1.29716 − 2.24675i
−1.29716 − 2.24675i −1.72785 + 0.120512i −2.36526 + 4.09675i −0.137404 − 0.0793301i 2.51207 + 3.72573i −2.91814 + 1.68479i 7.08384 2.97095 − 0.416454i 0.411616i
32.2 −1.10617 − 1.91594i 1.59999 − 0.663339i −1.44722 + 2.50665i −2.54721 − 1.47063i −3.04078 − 2.33173i 1.72189 − 0.994132i 1.97879 2.11996 − 2.12268i 6.50707i
32.3 −0.679041 − 1.17613i −0.323333 − 1.70160i 0.0778064 − 0.134765i 0.901139 + 0.520273i −1.78176 + 1.53574i 0.600962 − 0.346965i −2.92750 −2.79091 + 1.10037i − 1.41315i
32.4 −0.618600 − 1.07145i −1.04881 + 1.37841i 0.234668 − 0.406456i 1.78348 + 1.02969i 2.12568 + 0.271060i 3.59283 − 2.07432i −3.05506 −0.800005 − 2.89137i − 2.54787i
32.5 0.618600 + 1.07145i −1.04881 + 1.37841i 0.234668 − 0.406456i 1.78348 + 1.02969i −2.12568 − 0.271060i −3.59283 + 2.07432i 3.05506 −0.800005 − 2.89137i 2.54787i
32.6 0.679041 + 1.17613i −0.323333 − 1.70160i 0.0778064 − 0.134765i 0.901139 + 0.520273i 1.78176 − 1.53574i −0.600962 + 0.346965i 2.92750 −2.79091 + 1.10037i 1.41315i
32.7 1.10617 + 1.91594i 1.59999 − 0.663339i −1.44722 + 2.50665i −2.54721 − 1.47063i 3.04078 + 2.33173i −1.72189 + 0.994132i −1.97879 2.11996 − 2.12268i − 6.50707i
32.8 1.29716 + 2.24675i −1.72785 + 0.120512i −2.36526 + 4.09675i −0.137404 − 0.0793301i −2.51207 − 3.72573i 2.91814 − 1.68479i −7.08384 2.97095 − 0.416454i − 0.411616i
65.1 −1.29716 + 2.24675i −1.72785 − 0.120512i −2.36526 − 4.09675i −0.137404 + 0.0793301i 2.51207 − 3.72573i −2.91814 − 1.68479i 7.08384 2.97095 + 0.416454i − 0.411616i
65.2 −1.10617 + 1.91594i 1.59999 + 0.663339i −1.44722 − 2.50665i −2.54721 + 1.47063i −3.04078 + 2.33173i 1.72189 + 0.994132i 1.97879 2.11996 + 2.12268i − 6.50707i
65.3 −0.679041 + 1.17613i −0.323333 + 1.70160i 0.0778064 + 0.134765i 0.901139 − 0.520273i −1.78176 − 1.53574i 0.600962 + 0.346965i −2.92750 −2.79091 − 1.10037i 1.41315i
65.4 −0.618600 + 1.07145i −1.04881 − 1.37841i 0.234668 + 0.406456i 1.78348 − 1.02969i 2.12568 − 0.271060i 3.59283 + 2.07432i −3.05506 −0.800005 + 2.89137i 2.54787i
65.5 0.618600 − 1.07145i −1.04881 − 1.37841i 0.234668 + 0.406456i 1.78348 − 1.02969i −2.12568 + 0.271060i −3.59283 − 2.07432i 3.05506 −0.800005 + 2.89137i − 2.54787i
65.6 0.679041 − 1.17613i −0.323333 + 1.70160i 0.0778064 + 0.134765i 0.901139 − 0.520273i 1.78176 + 1.53574i −0.600962 − 0.346965i 2.92750 −2.79091 − 1.10037i − 1.41315i
65.7 1.10617 − 1.91594i 1.59999 + 0.663339i −1.44722 − 2.50665i −2.54721 + 1.47063i 3.04078 − 2.33173i −1.72189 − 0.994132i −1.97879 2.11996 + 2.12268i 6.50707i
65.8 1.29716 − 2.24675i −1.72785 − 0.120512i −2.36526 − 4.09675i −0.137404 + 0.0793301i −2.51207 + 3.72573i 2.91814 + 1.68479i −7.08384 2.97095 + 0.416454i 0.411616i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 32.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner
11.b odd 2 1 inner
99.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 99.2.g.b ✓ 16
3.b odd 2 1 297.2.g.b 16
9.c even 3 1 297.2.g.b 16
9.c even 3 1 891.2.d.b 16
9.d odd 6 1 inner 99.2.g.b ✓ 16
9.d odd 6 1 891.2.d.b 16
11.b odd 2 1 inner 99.2.g.b ✓ 16
33.d even 2 1 297.2.g.b 16
99.g even 6 1 inner 99.2.g.b ✓ 16
99.g even 6 1 891.2.d.b 16
99.h odd 6 1 297.2.g.b 16
99.h odd 6 1 891.2.d.b 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
99.2.g.b ✓ 16 1.a even 1 1 trivial
99.2.g.b ✓ 16 9.d odd 6 1 inner
99.2.g.b ✓ 16 11.b odd 2 1 inner
99.2.g.b ✓ 16 99.g even 6 1 inner
297.2.g.b 16 3.b odd 2 1
297.2.g.b 16 9.c even 3 1
297.2.g.b 16 33.d even 2 1
297.2.g.b 16 99.h odd 6 1
891.2.d.b 16 9.c even 3 1
891.2.d.b 16 9.d odd 6 1
891.2.d.b 16 99.g even 6 1
891.2.d.b 16 99.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 15T_{2}^{14} + 150T_{2}^{12} + 837T_{2}^{10} + 3372T_{2}^{8} + 8010T_{2}^{6} + 13761T_{2}^{4} + 13392T_{2}^{2} + 8649 \) acting on \(S_{2}^{\mathrm{new}}(99, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 15 T^{14} + \cdots + 8649 \) Copy content Toggle raw display
$3$ \( (T^{8} + 3 T^{7} + 3 T^{6} + \cdots + 81)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} - 7 T^{6} + 48 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} - 33 T^{14} + \cdots + 138384 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 214358881 \) Copy content Toggle raw display
$13$ \( T^{16} - 51 T^{14} + \cdots + 138384 \) Copy content Toggle raw display
$17$ \( (T^{8} - 93 T^{6} + \cdots + 214272)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 63 T^{6} + \cdots + 3348)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 6 T^{7} + \cdots + 119716)^{2} \) Copy content Toggle raw display
$29$ \( T^{16} + 30 T^{14} + \cdots + 2214144 \) Copy content Toggle raw display
$31$ \( (T^{8} + 2 T^{7} + 19 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 7 T^{3} - 51 T^{2} + \cdots + 31)^{4} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 18034341264 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 54056250000 \) Copy content Toggle raw display
$47$ \( (T^{8} + 15 T^{7} + \cdots + 6889)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 146 T^{6} + \cdots + 105625)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 18 T^{7} + \cdots + 265225)^{2} \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 16\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( (T^{8} - 4 T^{7} + \cdots + 18769)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 53 T^{6} + \cdots + 361)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + 408 T^{6} + \cdots + 64686708)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 9069133824 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 162096184890000 \) Copy content Toggle raw display
$89$ \( (T^{8} + 308 T^{6} + \cdots + 1893376)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 2 T^{7} + \cdots + 564001)^{2} \) Copy content Toggle raw display
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