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Properties

Label 96.7.b.a
Level $96$
Weight $7$
Character orbit 96.b
Analytic conductor $22.085$
Analytic rank $0$
Dimension $12$
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] = [96,7,Mod(79,96)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(96, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("96.79");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 96 = 2^{5} \cdot 3 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 96.b (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: \(22.0851920275\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 31 x^{10} - 1286 x^{9} + 7702 x^{8} - 174032 x^{7} + 1952056 x^{6} + \cdots + 767595744 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{60}\cdot 3^{11} \)
Twist minimal: no (minimal twist has level 24)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{2} q^{5} + (\beta_{4} - \beta_{2}) q^{7} + 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{2} q^{5} + (\beta_{4} - \beta_{2}) q^{7} + 243 q^{9} + ( - \beta_{7} - 227) q^{11} + ( - 3 \beta_{4} + \beta_{3} + 3 \beta_{2}) q^{13} + ( - \beta_{10} + 5 \beta_{2}) q^{15} + (\beta_{8} + \beta_{7} - 2 \beta_{6} + \cdots - 407) q^{17}+ \cdots + ( - 243 \beta_{7} - 55161) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2916 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2916 q^{9} - 2720 q^{11} - 4888 q^{17} - 3936 q^{19} - 27204 q^{25} - 162336 q^{35} - 54280 q^{41} + 49824 q^{43} - 304644 q^{49} - 80352 q^{51} - 136080 q^{57} + 886144 q^{59} + 473376 q^{65} - 1565952 q^{67} + 555480 q^{73} + 1073088 q^{75} + 708588 q^{81} - 2497760 q^{83} + 367400 q^{89} + 4475808 q^{91} - 1165656 q^{97} - 660960 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 31 x^{10} - 1286 x^{9} + 7702 x^{8} - 174032 x^{7} + 1952056 x^{6} + \cdots + 767595744 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 78\!\cdots\!93 \nu^{11} + \cdots + 48\!\cdots\!44 ) / 31\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 69\!\cdots\!69 \nu^{11} + \cdots + 11\!\cdots\!92 ) / 18\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13\!\cdots\!80 \nu^{11} + \cdots + 16\!\cdots\!20 ) / 18\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 17\!\cdots\!33 \nu^{11} + \cdots - 29\!\cdots\!76 ) / 18\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 40\!\cdots\!02 \nu^{11} + \cdots - 73\!\cdots\!64 ) / 18\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 60\!\cdots\!69 \nu^{11} + \cdots - 41\!\cdots\!68 ) / 23\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 10\!\cdots\!87 \nu^{11} + \cdots + 60\!\cdots\!36 ) / 39\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 11\!\cdots\!15 \nu^{11} + \cdots + 42\!\cdots\!56 ) / 31\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 30\!\cdots\!09 \nu^{11} + \cdots + 52\!\cdots\!64 ) / 63\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 15\!\cdots\!93 \nu^{11} + \cdots + 24\!\cdots\!44 ) / 18\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 44\!\cdots\!05 \nu^{11} + \cdots + 27\!\cdots\!84 ) / 31\!\cdots\!52 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 6 \beta_{11} - 10 \beta_{10} - 3 \beta_{9} - 6 \beta_{8} - 18 \beta_{7} - 15 \beta_{5} - 9 \beta_{4} + \cdots + 1530 ) / 9216 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 6 \beta_{11} + 68 \beta_{10} - 153 \beta_{9} - 90 \beta_{8} + 90 \beta_{7} - 108 \beta_{6} + \cdots - 44514 ) / 9216 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 114 \beta_{11} - 405 \beta_{10} + 765 \beta_{9} - 78 \beta_{8} + 642 \beta_{7} - 90 \beta_{6} + \cdots + 942294 ) / 3072 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 12066 \beta_{11} + 458 \beta_{10} - 5157 \beta_{9} + 3198 \beta_{8} - 62550 \beta_{7} + \cdots - 14672754 ) / 9216 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 93066 \beta_{11} + 99953 \beta_{10} - 296511 \beta_{9} - 8118 \beta_{8} + 483282 \beta_{7} + \cdots + 482412870 ) / 9216 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 51882 \beta_{11} - 889746 \beta_{10} + 1191303 \beta_{9} - 292362 \beta_{8} - 269790 \beta_{7} + \cdots - 1110565194 ) / 3072 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 9862386 \beta_{11} + 25703459 \beta_{10} - 30334773 \beta_{9} - 1840434 \beta_{8} + \cdots - 38877402366 ) / 9216 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 206530998 \beta_{11} - 142980694 \beta_{10} + 177351351 \beta_{9} + 95490774 \beta_{8} + \cdots + 1300209250950 ) / 9216 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 320139798 \beta_{11} - 252652515 \beta_{10} + 196769985 \beta_{9} - 37938518 \beta_{8} + \cdots - 1390347762026 ) / 1024 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 18204604194 \beta_{11} + 61624082618 \beta_{10} - 94203899835 \beta_{9} - 7785963006 \beta_{8} + \cdots + 41367648684546 ) / 9216 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 179379311550 \beta_{11} - 835608873109 \beta_{10} + 1367568695907 \beta_{9} + \cdots + 806078260932498 ) / 9216 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/96\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(37\) \(65\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
79.1
8.16014 + 0.886446i
−9.37784 + 8.67520i
−0.0143493 − 10.5849i
−0.0143493 + 10.5849i
−9.37784 − 8.67520i
8.16014 − 0.886446i
3.47372 − 1.02840i
0.630233 − 2.92643i
−1.87190 + 1.33932i
−1.87190 − 1.33932i
0.630233 + 2.92643i
3.47372 + 1.02840i
0 −15.5885 0 − 232.265i 0 − 483.645i 0 243.000 0
79.2 0 −15.5885 0 − 100.822i 0 277.765i 0 243.000 0
79.3 0 −15.5885 0 − 82.3007i 0 351.467i 0 243.000 0
79.4 0 −15.5885 0 82.3007i 0 − 351.467i 0 243.000 0
79.5 0 −15.5885 0 100.822i 0 − 277.765i 0 243.000 0
79.6 0 −15.5885 0 232.265i 0 483.645i 0 243.000 0
79.7 0 15.5885 0 − 128.353i 0 − 534.624i 0 243.000 0
79.8 0 15.5885 0 − 111.403i 0 106.838i 0 243.000 0
79.9 0 15.5885 0 − 87.0704i 0 355.505i 0 243.000 0
79.10 0 15.5885 0 87.0704i 0 − 355.505i 0 243.000 0
79.11 0 15.5885 0 111.403i 0 − 106.838i 0 243.000 0
79.12 0 15.5885 0 128.353i 0 534.624i 0 243.000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 79.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 96.7.b.a 12
3.b odd 2 1 288.7.b.d 12
4.b odd 2 1 24.7.b.a ✓ 12
8.b even 2 1 24.7.b.a ✓ 12
8.d odd 2 1 inner 96.7.b.a 12
12.b even 2 1 72.7.b.c 12
16.e even 4 2 768.7.g.l 24
16.f odd 4 2 768.7.g.l 24
24.f even 2 1 288.7.b.d 12
24.h odd 2 1 72.7.b.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.7.b.a ✓ 12 4.b odd 2 1
24.7.b.a ✓ 12 8.b even 2 1
72.7.b.c 12 12.b even 2 1
72.7.b.c 12 24.h odd 2 1
96.7.b.a 12 1.a even 1 1 trivial
96.7.b.a 12 8.d odd 2 1 inner
288.7.b.d 12 3.b odd 2 1
288.7.b.d 12 24.f even 2 1
768.7.g.l 24 16.e even 4 2
768.7.g.l 24 16.f odd 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(96, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( (T^{2} - 243)^{6} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 91\!\cdots\!36 \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots + 23\!\cdots\!12)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 36\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots - 15\!\cdots\!24)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 24\!\cdots\!16)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 12\!\cdots\!56 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 28\!\cdots\!04 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 84\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 11\!\cdots\!36)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 19\!\cdots\!32)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 41\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 33\!\cdots\!64 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 12\!\cdots\!28)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 34\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots - 54\!\cdots\!96)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 49\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 11\!\cdots\!24)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 65\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 53\!\cdots\!04)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 62\!\cdots\!04)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots - 50\!\cdots\!12)^{2} \) Copy content Toggle raw display
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