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Properties

Label 72.7.b.c
Level $72$
Weight $7$
Character orbit 72.b
Analytic conductor $16.564$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [72,7,Mod(19,72)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(72, 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("72.19"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 72.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,-10] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(16.5638940206\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{30}\cdot 3^{11} \)
Twist minimal: no (minimal twist has level 24)
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.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{2} + (\beta_{5} + \beta_1 + 2) q^{4} + ( - \beta_{5} + \beta_{4} + 3 \beta_1 + 1) q^{5} + (\beta_{9} + \beta_{5} + \beta_{4} + \cdots + 2) q^{7} + ( - \beta_{9} + \beta_{7} - \beta_{6} + \cdots - 67) q^{8}+ \cdots + ( - 192 \beta_{11} + 512 \beta_{10} + \cdots - 10485) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 10 q^{2} + 24 q^{4} - 796 q^{8} + 2172 q^{10} - 2720 q^{11} + 6444 q^{14} + 11640 q^{16} + 4888 q^{17} + 3936 q^{19} + 31608 q^{20} - 60432 q^{22} - 27204 q^{25} - 53952 q^{26} - 57072 q^{28} - 109480 q^{32}+ \cdots - 182674 q^{98}+O(q^{100}) \) 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

\(\beta_{1}\)\(=\) \( ( 12\!\cdots\!35 \nu^{11} + \cdots - 14\!\cdots\!36 ) / 39\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 31\!\cdots\!93 \nu^{11} + \cdots - 43\!\cdots\!40 ) / 75\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 70\!\cdots\!15 \nu^{11} + \cdots + 56\!\cdots\!96 ) / 75\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 46\!\cdots\!77 \nu^{11} + \cdots - 80\!\cdots\!88 ) / 37\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 60\!\cdots\!33 \nu^{11} + \cdots + 12\!\cdots\!52 ) / 39\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 55\!\cdots\!57 \nu^{11} + \cdots - 66\!\cdots\!08 ) / 18\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 12\!\cdots\!36 \nu^{11} + \cdots + 40\!\cdots\!68 ) / 37\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 56\!\cdots\!77 \nu^{11} + \cdots + 10\!\cdots\!88 ) / 75\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 19\!\cdots\!99 \nu^{11} + \cdots - 37\!\cdots\!56 ) / 18\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 13\!\cdots\!83 \nu^{11} + \cdots + 79\!\cdots\!16 ) / 12\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 13\!\cdots\!41 \nu^{11} + \cdots - 25\!\cdots\!80 ) / 75\!\cdots\!48 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3\beta_{11} - 2\beta_{10} - 3\beta_{9} - 6\beta_{6} + 12\beta_{5} + 3\beta_{2} - 24\beta _1 + 42 ) / 288 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 3 \beta_{11} + 46 \beta_{10} - 9 \beta_{9} - 12 \beta_{8} - 36 \beta_{7} + 54 \beta_{6} + \cdots - 1206 ) / 288 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 33 \beta_{11} - 42 \beta_{10} + 57 \beta_{9} - 168 \beta_{8} + 72 \beta_{7} + 54 \beta_{6} + \cdots + 28965 ) / 96 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3177 \beta_{11} + 3910 \beta_{10} - 2373 \beta_{9} + 5220 \beta_{8} + 2412 \beta_{7} - 8346 \beta_{6} + \cdots - 427620 ) / 288 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 16077 \beta_{11} - 20114 \beta_{10} - 1467 \beta_{9} + 4536 \beta_{8} - 55128 \beta_{7} + \cdots + 14919327 ) / 288 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 62829 \beta_{11} - 200466 \beta_{10} - 26409 \beta_{9} - 249468 \beta_{8} + 129228 \beta_{7} + \cdots - 35834472 ) / 96 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 127233 \beta_{11} + 15815530 \beta_{10} - 1056489 \beta_{9} + 6833112 \beta_{8} - 2822136 \beta_{7} + \cdots - 1128380043 ) / 288 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 42965283 \beta_{11} - 124959842 \beta_{10} + 43704567 \beta_{9} - 61508076 \beta_{8} + \cdots + 39833211204 ) / 288 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 108834915 \beta_{11} - 14175762 \beta_{10} - 80984555 \beta_{9} + 54900152 \beta_{8} + \cdots - 43230321565 ) / 32 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 7581327273 \beta_{11} + 16323434890 \beta_{10} + 1920596565 \beta_{9} + 4891236204 \beta_{8} + \cdots + 1360648800456 ) / 288 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 16616171415 \beta_{11} - 285454786022 \beta_{10} + 63370238031 \beta_{9} - 239403789192 \beta_{8} + \cdots + 23495738443101 ) / 288 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/72\mathbb{Z}\right)^\times\).

\(n\) \(37\) \(55\) \(65\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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.

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
8.16014 − 0.886446i
8.16014 + 0.886446i
−1.87190 + 1.33932i
−1.87190 − 1.33932i
−0.0143493 − 10.5849i
−0.0143493 + 10.5849i
0.630233 + 2.92643i
0.630233 − 2.92643i
−9.37784 − 8.67520i
−9.37784 + 8.67520i
3.47372 + 1.02840i
3.47372 − 1.02840i
−7.97364 − 0.648923i 0 63.1578 + 10.3486i 232.265i 0 483.645i −496.882 − 123.500i 0 150.722 − 1852.00i
19.2 −7.97364 + 0.648923i 0 63.1578 − 10.3486i − 232.265i 0 − 483.645i −496.882 + 123.500i 0 150.722 + 1852.00i
19.3 −7.11414 − 3.65910i 0 37.2219 + 52.0627i − 87.0704i 0 355.505i −74.2991 − 506.580i 0 −318.599 + 619.431i
19.4 −7.11414 + 3.65910i 0 37.2219 − 52.0627i 87.0704i 0 − 355.505i −74.2991 + 506.580i 0 −318.599 − 619.431i
19.5 −1.98950 − 7.74867i 0 −56.0838 + 30.8319i − 82.3007i 0 351.467i 350.485 + 373.235i 0 −637.721 + 163.737i
19.6 −1.98950 + 7.74867i 0 −56.0838 − 30.8319i 82.3007i 0 − 351.467i 350.485 − 373.235i 0 −637.721 − 163.737i
19.7 −0.278171 − 7.99516i 0 −63.8452 + 4.44805i 111.403i 0 − 106.838i 53.3228 + 509.216i 0 890.684 − 30.9891i
19.8 −0.278171 + 7.99516i 0 −63.8452 − 4.44805i − 111.403i 0 106.838i 53.3228 − 509.216i 0 890.684 + 30.9891i
19.9 4.86506 − 6.35069i 0 −16.6624 − 61.7929i 100.822i 0 − 277.765i −473.491 − 194.808i 0 640.287 + 490.503i
19.10 4.86506 + 6.35069i 0 −16.6624 + 61.7929i − 100.822i 0 277.765i −473.491 + 194.808i 0 640.287 − 490.503i
19.11 7.49039 − 2.80965i 0 48.2118 − 42.0907i 128.353i 0 534.624i 242.865 − 450.734i 0 360.627 + 961.414i
19.12 7.49039 + 2.80965i 0 48.2118 + 42.0907i − 128.353i 0 − 534.624i 242.865 + 450.734i 0 360.627 − 961.414i
\(n\): 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 72.7.b.c 12
3.b odd 2 1 24.7.b.a ✓ 12
4.b odd 2 1 288.7.b.d 12
8.b even 2 1 288.7.b.d 12
8.d odd 2 1 inner 72.7.b.c 12
12.b even 2 1 96.7.b.a 12
24.f even 2 1 24.7.b.a ✓ 12
24.h odd 2 1 96.7.b.a 12
48.i odd 4 2 768.7.g.l 24
48.k even 4 2 768.7.g.l 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
24.7.b.a ✓ 12 3.b odd 2 1
24.7.b.a ✓ 12 24.f even 2 1
72.7.b.c 12 1.a even 1 1 trivial
72.7.b.c 12 8.d odd 2 1 inner
96.7.b.a 12 12.b even 2 1
96.7.b.a 12 24.h odd 2 1
288.7.b.d 12 4.b odd 2 1
288.7.b.d 12 8.b even 2 1
768.7.g.l 24 48.i odd 4 2
768.7.g.l 24 48.k even 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 107352 T_{5}^{10} + 3991016688 T_{5}^{8} + 71113492512000 T_{5}^{6} + \cdots + 57\!\cdots\!00 \) acting on \(S_{7}^{\mathrm{new}}(72, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 68719476736 \) Copy content Toggle raw display
$3$ \( T^{12} \) 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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