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Properties

Label 912.3.h.c
Level $912$
Weight $3$
Character orbit 912.h
Analytic conductor $24.850$
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] = [912,3,Mod(305,912)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("912.305"); S:= CuspForms(chi, 3); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(912, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 1, 0])) N = Newforms(chi, 3, names="a")
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 912.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,4,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(24.8502001097\)
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: \( x^{12} + 52x^{10} + 1030x^{8} + 9728x^{6} + 44345x^{4} + 84236x^{2} + 34596 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{3} \)
Twist minimal: no (minimal twist has level 114)
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.

\(f(q)\) \(=\) \( q + \beta_{6} q^{3} + \beta_{3} q^{5} + ( - \beta_{6} - \beta_{5} + \beta_1 + 1) q^{7} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \cdots - 1) q^{9} + ( - \beta_{3} - \beta_{2}) q^{11} + ( - \beta_{9} - \beta_{7} - \beta_{6} + \cdots + 3) q^{13}+ \cdots + (\beta_{10} + 2 \beta_{9} + \beta_{8} + \cdots + 23) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 4 q^{3} + 8 q^{7} - 4 q^{9} + 32 q^{13} + 4 q^{15} - 60 q^{21} - 108 q^{25} - 116 q^{27} + 88 q^{31} + 4 q^{33} + 112 q^{37} - 168 q^{39} - 56 q^{43} - 224 q^{45} + 156 q^{49} + 28 q^{51} + 264 q^{55}+ \cdots + 304 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 52x^{10} + 1030x^{8} + 9728x^{6} + 44345x^{4} + 84236x^{2} + 34596 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{10} - 62\nu^{8} - 1338\nu^{6} - 11876\nu^{4} - 38617\nu^{2} - 26118 ) / 1872 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 19\nu^{11} + 554\nu^{9} + 2334\nu^{7} - 37996\nu^{5} - 181685\nu^{3} + 248202\nu ) / 87048 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -7\nu^{11} - 395\nu^{9} - 7923\nu^{7} - 67259\nu^{5} - 230032\nu^{3} - 269484\nu ) / 29016 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 19\nu^{11} + 554\nu^{9} + 2334\nu^{7} - 47668\nu^{5} - 394469\nu^{3} - 573918\nu ) / 58032 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 65 \nu^{11} - 31 \nu^{10} - 2977 \nu^{9} - 713 \nu^{8} - 49218 \nu^{7} + 3255 \nu^{6} + \cdots + 24552 ) / 87048 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 130 \nu^{11} - 155 \nu^{10} + 5954 \nu^{9} - 7192 \nu^{8} + 98436 \nu^{7} - 117924 \nu^{6} + \cdots - 813006 ) / 174096 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 83 \nu^{11} - 155 \nu^{10} - 3820 \nu^{9} - 7192 \nu^{8} - 64410 \nu^{7} - 117924 \nu^{6} + \cdots - 900054 ) / 87048 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 130 \nu^{11} - 899 \nu^{10} + 5954 \nu^{9} - 38812 \nu^{8} + 98436 \nu^{7} - 605616 \nu^{6} + \cdots - 11104758 ) / 174096 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 224 \nu^{11} - 155 \nu^{10} - 10222 \nu^{9} - 7192 \nu^{8} - 166488 \nu^{7} - 117924 \nu^{6} + \cdots - 813006 ) / 174096 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -47\nu^{11} - 2134\nu^{9} - 34026\nu^{7} - 221368\nu^{5} - 467627\nu^{3} + 134334\nu ) / 29016 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 271 \nu^{11} + 899 \nu^{10} - 12356 \nu^{9} + 38812 \nu^{8} - 200514 \nu^{7} + 591108 \nu^{6} + \cdots + 2661102 ) / 174096 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{10} - 2\beta_{9} + \beta_{7} + 1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} + \beta_{7} - 3\beta_{5} - 3\beta _1 - 26 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -8\beta_{10} + 22\beta_{9} - 11\beta_{7} + 6\beta_{4} - 12\beta_{3} - 11 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -6\beta_{11} + 3\beta_{10} - 20\beta_{9} - 3\beta_{8} - 20\beta_{7} - 12\beta_{6} + 51\beta_{5} + 57\beta _1 + 325 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 91\beta_{10} - 314\beta_{9} + 157\beta_{7} - 168\beta_{4} + 264\beta_{3} + 54\beta_{2} + 157 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 168 \beta_{11} - 84 \beta_{10} + 367 \beta_{9} + 66 \beta_{8} + 367 \beta_{7} + 408 \beta_{6} + \cdots - 4658 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1124 \beta_{10} + 4978 \beta_{9} - 2597 \beta_{7} + 216 \beta_{6} + 3546 \beta_{4} - 4788 \beta_{3} + \cdots - 2597 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 3654 \beta_{11} + 1827 \beta_{10} - 6632 \beta_{9} - 1161 \beta_{8} - 6632 \beta_{7} - 9864 \beta_{6} + \cdots + 71947 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 14365 \beta_{10} - 82886 \beta_{9} + 45439 \beta_{7} - 7992 \beta_{6} - 69024 \beta_{4} + 82536 \beta_{3} + \cdots + 45439 ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 73020 \beta_{11} - 36510 \beta_{10} + 119041 \beta_{9} + 19302 \beta_{8} + 119041 \beta_{7} + \cdots - 1162322 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 188360 \beta_{10} + 1413838 \beta_{9} - 810167 \beta_{7} + 206496 \beta_{6} + 1298406 \beta_{4} + \cdots - 810167 ) / 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/912\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(229\) \(305\) \(799\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
305.1
− 4.20796i
4.20796i
− 2.53753i
2.53753i
− 0.743998i
0.743998i
2.03439i
− 2.03439i
3.20774i
− 3.20774i
− 3.58778i
3.58778i
0 −2.97547 − 0.382826i 0 − 8.20051i 0 6.96186 0 8.70689 + 2.27818i 0
305.2 0 −2.97547 + 0.382826i 0 8.20051i 0 6.96186 0 8.70689 − 2.27818i 0
305.3 0 −1.79430 − 2.40426i 0 5.53299i 0 −1.56824 0 −2.56095 + 8.62795i 0
305.4 0 −1.79430 + 2.40426i 0 − 5.53299i 0 −1.56824 0 −2.56095 − 8.62795i 0
305.5 0 0.526086 − 2.95351i 0 4.13985i 0 −0.323502 0 −8.44647 − 3.10760i 0
305.6 0 0.526086 + 2.95351i 0 − 4.13985i 0 −0.323502 0 −8.44647 + 3.10760i 0
305.7 0 1.43853 − 2.63261i 0 − 1.06860i 0 11.7020 0 −4.86127 − 7.57417i 0
305.8 0 1.43853 + 2.63261i 0 1.06860i 0 11.7020 0 −4.86127 + 7.57417i 0
305.9 0 2.26822 − 1.96346i 0 − 1.39314i 0 −13.5439 0 1.28962 − 8.90713i 0
305.10 0 2.26822 + 1.96346i 0 1.39314i 0 −13.5439 0 1.28962 + 8.90713i 0
305.11 0 2.53694 − 1.60122i 0 − 9.26912i 0 0.771732 0 3.87218 − 8.12442i 0
305.12 0 2.53694 + 1.60122i 0 9.26912i 0 0.771732 0 3.87218 + 8.12442i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 305.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 912.3.h.c 12
3.b odd 2 1 inner 912.3.h.c 12
4.b odd 2 1 114.3.c.a ✓ 12
12.b even 2 1 114.3.c.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
114.3.c.a ✓ 12 4.b odd 2 1
114.3.c.a ✓ 12 12.b even 2 1
912.3.h.c 12 1.a even 1 1 trivial
912.3.h.c 12 3.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 204T_{5}^{10} + 14238T_{5}^{8} + 398684T_{5}^{6} + 4159881T_{5}^{4} + 10134720T_{5}^{2} + 6718464 \) acting on \(S_{3}^{\mathrm{new}}(912, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 4 T^{11} + \cdots + 531441 \) Copy content Toggle raw display
$5$ \( T^{12} + 204 T^{10} + \cdots + 6718464 \) Copy content Toggle raw display
$7$ \( (T^{6} - 4 T^{5} + \cdots - 432)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + 440 T^{10} + \cdots + 88623396 \) Copy content Toggle raw display
$13$ \( (T^{6} - 16 T^{5} + \cdots - 415920)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 5578855041600 \) Copy content Toggle raw display
$19$ \( (T^{2} - 19)^{6} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 22\!\cdots\!44 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 71\!\cdots\!04 \) Copy content Toggle raw display
$31$ \( (T^{6} - 44 T^{5} + \cdots + 33120576)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 56 T^{5} + \cdots - 2675966000)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 89\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{6} + 28 T^{5} + \cdots - 4439438448)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 42\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 17\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( (T^{6} + 36 T^{5} + \cdots + 478201744)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 184 T^{5} + \cdots + 622909440)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 87\!\cdots\!04 \) Copy content Toggle raw display
$73$ \( (T^{6} + 28 T^{5} + \cdots - 33631788672)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 8 T^{5} + \cdots - 13515530752)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 41\!\cdots\!16 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 68\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( (T^{6} + 264 T^{5} + \cdots + 658054160)^{2} \) Copy content Toggle raw display
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