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Properties

Label 24.7.b.a
Level $24$
Weight $7$
Character orbit 24.b
Analytic conductor $5.521$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [24,7,Mod(19,24)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(24, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 0])) N = Newforms(chi, 7, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("24.19"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Level: \( N \) \(=\) \( 24 = 2^{3} \cdot 3 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 24.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(5.52129800688\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content 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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{26}\cdot 3^{11} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q + (\beta_1 + 1) q^{2} - \beta_{2} q^{3} + ( - \beta_{4} + \beta_1 + 2) q^{4} + ( - \beta_{6} - \beta_{4} - \beta_{2} + \cdots - 1) q^{5} + (\beta_{3} - \beta_{2} + 13) q^{6} + ( - \beta_{9} - \beta_{7} + \beta_{6} + \cdots + 2) q^{7}+ \cdots + ( - 1458 \beta_{9} - 486 \beta_{8} + \cdots + 52488) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 10 q^{2} + 24 q^{4} + 162 q^{6} + 796 q^{8} + 2916 q^{9} + 2172 q^{10} + 2720 q^{11} - 972 q^{12} - 6444 q^{14} + 11640 q^{16} - 4888 q^{17} + 2430 q^{18} + 3936 q^{19} - 31608 q^{20} - 60432 q^{22}+ \cdots + 660960 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( - 52\!\cdots\!05 \nu^{11} + \cdots + 30\!\cdots\!40 ) / 24\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 78\!\cdots\!93 \nu^{11} + \cdots + 48\!\cdots\!44 ) / 31\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 52\!\cdots\!77 \nu^{11} + \cdots - 95\!\cdots\!12 ) / 31\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 59\!\cdots\!55 \nu^{11} + \cdots + 40\!\cdots\!44 ) / 24\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 15\!\cdots\!79 \nu^{11} + \cdots + 68\!\cdots\!68 ) / 18\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 64\!\cdots\!07 \nu^{11} + \cdots + 77\!\cdots\!52 ) / 18\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 80\!\cdots\!95 \nu^{11} + \cdots - 20\!\cdots\!76 ) / 18\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 82\!\cdots\!09 \nu^{11} + \cdots + 20\!\cdots\!44 ) / 18\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 49\!\cdots\!09 \nu^{11} + \cdots - 39\!\cdots\!68 ) / 94\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 32\!\cdots\!86 \nu^{11} + \cdots + 39\!\cdots\!60 ) / 47\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 55\!\cdots\!81 \nu^{11} + \cdots + 81\!\cdots\!96 ) / 18\!\cdots\!12 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} + 9\beta _1 + 4 ) / 18 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{10} + 3\beta_{9} - 18\beta_{4} - 2\beta_{3} - \beta_{2} - 39\beta _1 - 95 ) / 18 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 3 \beta_{11} - 6 \beta_{10} - 12 \beta_{9} - 18 \beta_{8} + 3 \beta_{7} + 36 \beta_{6} - 18 \beta_{5} + \cdots + 3693 ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 357 \beta_{11} - 540 \beta_{10} + 1950 \beta_{9} + 1230 \beta_{8} + 219 \beta_{7} - 300 \beta_{6} + \cdots - 56521 ) / 36 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 3807 \beta_{11} + 8472 \beta_{10} - 16326 \beta_{9} - 6060 \beta_{8} - 4419 \beta_{7} + \cdots + 1867349 ) / 36 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 4611 \beta_{11} - 12396 \beta_{10} - 29058 \beta_{9} - 30546 \beta_{8} + 3699 \beta_{7} + \cdots - 4133025 ) / 12 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 600495 \beta_{11} + 227424 \beta_{10} + 4118718 \beta_{9} + 1967148 \beta_{8} + 358995 \beta_{7} + \cdots - 158232205 ) / 36 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 7537527 \beta_{11} + 986508 \beta_{10} - 43767474 \beta_{9} - 19962258 \beta_{8} - 5710113 \beta_{7} + \cdots + 5074286099 ) / 36 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 6404373 \beta_{11} - 15184456 \beta_{10} + 27634658 \beta_{9} + 16440372 \beta_{8} + 5476273 \beta_{7} + \cdots - 5313412983 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 190121853 \beta_{11} + 2641917588 \beta_{10} + 892893546 \beta_{9} + 357314610 \beta_{8} + \cdots + 146770389593 ) / 36 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 9134185545 \beta_{11} - 17219981064 \beta_{10} - 64522083162 \beta_{9} - 40612660812 \beta_{8} + \cdots + 3274465816475 ) / 36 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/24\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(13\) \(17\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
3.47372 + 1.02840i
3.47372 − 1.02840i
−9.37784 − 8.67520i
−9.37784 + 8.67520i
0.630233 + 2.92643i
0.630233 − 2.92643i
−0.0143493 − 10.5849i
−0.0143493 + 10.5849i
−1.87190 + 1.33932i
−1.87190 − 1.33932i
8.16014 − 0.886446i
8.16014 + 0.886446i
−7.49039 − 2.80965i −15.5885 48.2118 + 42.0907i 128.353i 116.764 + 43.7981i − 534.624i −242.865 − 450.734i 243.000 360.627 − 961.414i
19.2 −7.49039 + 2.80965i −15.5885 48.2118 − 42.0907i − 128.353i 116.764 − 43.7981i 534.624i −242.865 + 450.734i 243.000 360.627 + 961.414i
19.3 −4.86506 − 6.35069i 15.5885 −16.6624 + 61.7929i 100.822i −75.8387 − 98.9974i 277.765i 473.491 − 194.808i 243.000 640.287 − 490.503i
19.4 −4.86506 + 6.35069i 15.5885 −16.6624 − 61.7929i − 100.822i −75.8387 + 98.9974i − 277.765i 473.491 + 194.808i 243.000 640.287 + 490.503i
19.5 0.278171 − 7.99516i −15.5885 −63.8452 − 4.44805i 111.403i −4.33626 + 124.632i 106.838i −53.3228 + 509.216i 243.000 890.684 + 30.9891i
19.6 0.278171 + 7.99516i −15.5885 −63.8452 + 4.44805i − 111.403i −4.33626 − 124.632i − 106.838i −53.3228 − 509.216i 243.000 890.684 − 30.9891i
19.7 1.98950 − 7.74867i 15.5885 −56.0838 − 30.8319i − 82.3007i 31.0132 − 120.790i − 351.467i −350.485 + 373.235i 243.000 −637.721 − 163.737i
19.8 1.98950 + 7.74867i 15.5885 −56.0838 + 30.8319i 82.3007i 31.0132 + 120.790i 351.467i −350.485 − 373.235i 243.000 −637.721 + 163.737i
19.9 7.11414 − 3.65910i −15.5885 37.2219 − 52.0627i − 87.0704i −110.898 + 57.0398i − 355.505i 74.2991 − 506.580i 243.000 −318.599 − 619.431i
19.10 7.11414 + 3.65910i −15.5885 37.2219 + 52.0627i 87.0704i −110.898 − 57.0398i 355.505i 74.2991 + 506.580i 243.000 −318.599 + 619.431i
19.11 7.97364 − 0.648923i 15.5885 63.1578 − 10.3486i 232.265i 124.297 − 10.1157i − 483.645i 496.882 − 123.500i 243.000 150.722 + 1852.00i
19.12 7.97364 + 0.648923i 15.5885 63.1578 + 10.3486i − 232.265i 124.297 + 10.1157i 483.645i 496.882 + 123.500i 243.000 150.722 − 1852.00i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.12
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 24.7.b.a ✓ 12
3.b odd 2 1 72.7.b.c 12
4.b odd 2 1 96.7.b.a 12
8.b even 2 1 96.7.b.a 12
8.d odd 2 1 inner 24.7.b.a ✓ 12
12.b even 2 1 288.7.b.d 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 72.7.b.c 12
24.h odd 2 1 288.7.b.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.7.b.a ✓ 12 1.a even 1 1 trivial
24.7.b.a ✓ 12 8.d odd 2 1 inner
72.7.b.c 12 3.b odd 2 1
72.7.b.c 12 24.f even 2 1
96.7.b.a 12 4.b odd 2 1
96.7.b.a 12 8.b even 2 1
288.7.b.d 12 12.b even 2 1
288.7.b.d 12 24.h odd 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 \(S_{7}^{\mathrm{new}}(24, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 68719476736 \) 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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