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Properties

Label 1875.2.b.g
Level $1875$
Weight $2$
Character orbit 1875.b
Analytic conductor $14.972$
Analytic rank $0$
Dimension $16$
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] = [1875,2,Mod(1249,1875)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1875, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1875.1249"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1875 = 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1875.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,-18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.9719503790\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 25x^{14} + 246x^{12} + 1220x^{10} + 3281x^{8} + 4880x^{6} + 3936x^{4} + 1600x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1) q^{2} + \beta_{11} q^{3} + ( - \beta_{8} + \beta_{7} - \beta_{3} - 1) q^{4} + ( - \beta_{13} + \beta_{9} + \beta_{6} + 1) q^{6} + ( - \beta_{11} + \beta_{10} + \beta_{2}) q^{7}+ \cdots + ( - \beta_{15} - \beta_{13} - \beta_{8} + \cdots - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 18 q^{4} + 2 q^{6} - 16 q^{9} + 24 q^{11} - 32 q^{14} + 30 q^{16} - 32 q^{19} + 24 q^{21} - 6 q^{24} - 68 q^{26} - 4 q^{29} + 26 q^{31} + 74 q^{34} + 18 q^{36} - 28 q^{39} - 24 q^{41} - 94 q^{44}+ \cdots - 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 25x^{14} + 246x^{12} + 1220x^{10} + 3281x^{8} + 4880x^{6} + 3936x^{4} + 1600x^{2} + 256 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{15} + 25\nu^{13} + 246\nu^{11} + 1220\nu^{9} + 3281\nu^{7} + 4880\nu^{5} + 3936\nu^{3} + 1472\nu ) / 128 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{14} + 25\nu^{12} + 246\nu^{10} + 1220\nu^{8} + 3281\nu^{6} + 4880\nu^{4} + 3872\nu^{2} + 1216 ) / 64 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{15} - 75\nu^{13} - 738\nu^{11} - 3660\nu^{9} - 9843\nu^{7} - 14640\nu^{5} - 11680\nu^{3} - 3904\nu ) / 128 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 4\nu^{15} + 99\nu^{13} + 959\nu^{11} + 4634\nu^{9} + 11904\nu^{7} + 16239\nu^{5} + 10896\nu^{3} + 2752\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -13\nu^{14} - 321\nu^{12} - 3098\nu^{10} - 14876\nu^{8} - 37773\nu^{6} - 50380\nu^{4} - 32416\nu^{2} - 7744 ) / 64 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 9\nu^{14} + 221\nu^{12} + 2114\nu^{10} + 9996\nu^{8} + 24681\nu^{6} + 31308\nu^{4} + 18592\nu^{2} + 4032 ) / 32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 9\nu^{14} + 221\nu^{12} + 2114\nu^{10} + 9996\nu^{8} + 24681\nu^{6} + 31308\nu^{4} + 18624\nu^{2} + 4128 ) / 32 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -5\nu^{14} - 123\nu^{12} - 1180\nu^{10} - 5608\nu^{8} - 13981\nu^{6} - 18078\nu^{4} - 11120\nu^{2} - 2528 ) / 16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 21\nu^{15} + 519\nu^{13} + 5016\nu^{11} + 24144\nu^{9} + 61581\nu^{7} + 82858\nu^{5} + 54208\nu^{3} + 13248\nu ) / 64 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 79 \nu^{15} + 1935 \nu^{13} + 18450 \nu^{11} + 86940 \nu^{9} + 214335 \nu^{7} + 273672 \nu^{5} + \cdots + 37440 \nu ) / 128 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 23\nu^{15} + 561\nu^{13} + 5316\nu^{11} + 24816\nu^{9} + 60319\nu^{7} + 75490\nu^{5} + 44712\nu^{3} + 9792\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 23\nu^{14} + 561\nu^{12} + 5316\nu^{10} + 24816\nu^{8} + 60319\nu^{6} + 75490\nu^{4} + 44712\nu^{2} + 9824 ) / 32 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 151 \nu^{15} + 3683 \nu^{13} + 34910 \nu^{11} + 163124 \nu^{9} + 397463 \nu^{7} + 500052 \nu^{5} + \cdots + 66304 \nu ) / 128 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( -23\nu^{14} - 560\nu^{12} - 5295\nu^{10} - 24654\nu^{8} - 59763\nu^{6} - 74673\nu^{4} - 44268\nu^{2} - 9744 ) / 16 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - \beta_{7} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + 3\beta_{2} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{9} - 7\beta_{8} + 8\beta_{7} + 2\beta_{3} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{10} - 3\beta_{5} - 10\beta_{4} - 24\beta_{2} + 22\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -14\beta_{9} + 46\beta_{8} - 59\beta_{7} + 2\beta_{6} - 20\beta_{3} - 90 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -\beta_{12} + 2\beta_{11} - 16\beta_{10} + 42\beta_{5} + 79\beta_{4} + 171\beta_{2} - 136\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( \beta_{15} + 3\beta_{13} + 139\beta_{9} - 307\beta_{8} + 427\beta_{7} - 32\beta_{6} + 158\beta_{3} + 577 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 2 \beta_{14} + 23 \beta_{12} - 38 \beta_{11} + 172 \beta_{10} - 412 \beta_{5} - 585 \beta_{4} + \cdots + 887 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( -25\beta_{15} - 71\beta_{13} - 1207\beta_{9} + 2086\beta_{8} - 3060\beta_{7} + 344\beta_{6} - 1170\beta_{3} - 3814 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 50 \beta_{14} - 329 \beta_{12} + 466 \beta_{11} - 1576 \beta_{10} + 3524 \beta_{5} + 4230 \beta_{4} + \cdots - 5971 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 379 \beta_{15} + 1037 \beta_{13} + 9796 \beta_{9} - 14373 \beta_{8} + 21798 \beta_{7} - 3152 \beta_{6} + \cdots + 25657 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 758 \beta_{14} + 3787 \beta_{12} - 4742 \beta_{11} + 13327 \beta_{10} - 28177 \beta_{5} + \cdots + 41067 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 4545 \beta_{15} - 12119 \beta_{13} - 76504 \beta_{9} + 100071 \beta_{8} - 154719 \beta_{7} + \cdots - 174631 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 9090 \beta_{14} - 38520 \beta_{12} + 43712 \beta_{11} - 107703 \beta_{10} + 216999 \beta_{5} + \cdots - 286821 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1875\mathbb{Z}\right)^\times\).

\(n\) \(626\) \(1252\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1249.1
− 0.741379i
− 0.770071i
− 0.895394i
− 1.31354i
− 1.52260i
− 2.23365i
− 2.59716i
− 2.69767i
2.69767i
2.59716i
2.23365i
1.52260i
1.31354i
0.895394i
0.770071i
0.741379i
− 2.69767i − 1.00000i −5.27745 0 −2.69767 − 3.56649i 8.84149i −1.00000 0
1249.2 − 2.59716i 1.00000i −4.74525 0 2.59716 − 3.28414i 7.12986i −1.00000 0
1249.3 − 2.23365i 1.00000i −2.98921 0 2.23365 1.03143i 2.20956i −1.00000 0
1249.4 − 1.52260i − 1.00000i −0.318310 0 −1.52260 − 0.990985i − 2.56054i −1.00000 0
1249.5 − 1.31354i 1.00000i 0.274605 0 1.31354 − 4.19091i − 2.98779i −1.00000 0
1249.6 − 0.895394i − 1.00000i 1.19827 0 −0.895394 5.08992i − 2.86371i −1.00000 0
1249.7 − 0.770071i − 1.00000i 1.40699 0 −0.770071 3.98808i − 2.62363i −1.00000 0
1249.8 − 0.741379i 1.00000i 1.45036 0 0.741379 − 1.03586i − 2.55802i −1.00000 0
1249.9 0.741379i − 1.00000i 1.45036 0 0.741379 1.03586i 2.55802i −1.00000 0
1249.10 0.770071i 1.00000i 1.40699 0 −0.770071 − 3.98808i 2.62363i −1.00000 0
1249.11 0.895394i 1.00000i 1.19827 0 −0.895394 − 5.08992i 2.86371i −1.00000 0
1249.12 1.31354i − 1.00000i 0.274605 0 1.31354 4.19091i 2.98779i −1.00000 0
1249.13 1.52260i 1.00000i −0.318310 0 −1.52260 0.990985i 2.56054i −1.00000 0
1249.14 2.23365i − 1.00000i −2.98921 0 2.23365 − 1.03143i − 2.20956i −1.00000 0
1249.15 2.59716i − 1.00000i −4.74525 0 2.59716 3.28414i − 7.12986i −1.00000 0
1249.16 2.69767i 1.00000i −5.27745 0 −2.69767 3.56649i − 8.84149i −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1249.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1875.2.b.g 16
5.b even 2 1 inner 1875.2.b.g 16
5.c odd 4 1 1875.2.a.n ✓ 8
5.c odd 4 1 1875.2.a.o yes 8
15.e even 4 1 5625.2.a.u 8
15.e even 4 1 5625.2.a.bc 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1875.2.a.n ✓ 8 5.c odd 4 1
1875.2.a.o yes 8 5.c odd 4 1
1875.2.b.g 16 1.a even 1 1 trivial
1875.2.b.g 16 5.b even 2 1 inner
5625.2.a.u 8 15.e even 4 1
5625.2.a.bc 8 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{16} + 25T_{2}^{14} + 246T_{2}^{12} + 1220T_{2}^{10} + 3281T_{2}^{8} + 4880T_{2}^{6} + 3936T_{2}^{4} + 1600T_{2}^{2} + 256 \) acting on \(S_{2}^{\mathrm{new}}(1875, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 25 T^{14} + \cdots + 256 \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + 86 T^{14} + \cdots + 1113025 \) Copy content Toggle raw display
$11$ \( (T^{8} - 12 T^{7} + \cdots + 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + 110 T^{14} + \cdots + 76230361 \) Copy content Toggle raw display
$17$ \( T^{16} + 177 T^{14} + \cdots + 74926336 \) Copy content Toggle raw display
$19$ \( (T^{8} + 16 T^{7} + \cdots - 14975)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 824838400 \) Copy content Toggle raw display
$29$ \( (T^{8} + 2 T^{7} + \cdots + 57520)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} - 13 T^{7} + \cdots + 801025)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + 126 T^{14} + \cdots + 625 \) Copy content Toggle raw display
$41$ \( (T^{8} + 12 T^{7} + \cdots - 48080)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 5530748161 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 65445918976 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 824838400 \) Copy content Toggle raw display
$59$ \( (T^{8} + 14 T^{7} + \cdots + 11920)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 10 T^{7} + \cdots - 1093919)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 408210546216841 \) Copy content Toggle raw display
$71$ \( (T^{8} - 21 T^{7} + \cdots + 67696)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 144137919025 \) Copy content Toggle raw display
$79$ \( (T^{8} + 10 T^{7} + \cdots + 6951025)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 12937697536 \) Copy content Toggle raw display
$89$ \( (T^{8} - 9 T^{7} + \cdots - 12105680)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 161554959721 \) Copy content Toggle raw display
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