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Properties

Label 238.2.q.a
Level $238$
Weight $2$
Character orbit 238.q
Analytic conductor $1.900$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [238,2,Mod(9,238)] 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("238.9"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(238, base_ring=CyclotomicField(24)) chi = DirichletCharacter(H, H._module([8, 3])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 238 = 2 \cdot 7 \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 238.q (of order \(24\), degree \(8\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.90043956811\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(2\) over \(\Q(\zeta_{24})\)
Coefficient field: 16.0.2353561680715186176.2
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} + 2 x^{14} + 41 x^{12} - 92 x^{11} + 66 x^{10} - 104 x^{9} + 291 x^{8} - 388 x^{7} + \cdots + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{24}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{12} q^{2} + ( - \beta_{13} - \beta_{6}) q^{3} - \beta_{10} q^{4} + (\beta_{11} + \beta_{9} + 1) q^{5} + (\beta_{9} - \beta_{3} + \beta_{2}) q^{6} + (\beta_{15} + \beta_{14} - \beta_{13} + \cdots + 1) q^{7}+ \cdots + (4 \beta_{14} - \beta_{13} - 6 \beta_{12} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{3} + 8 q^{5} - 8 q^{6} + 12 q^{7} - 4 q^{9} - 8 q^{10} + 4 q^{11} - 12 q^{14} + 8 q^{15} + 8 q^{16} + 8 q^{18} + 12 q^{19} - 16 q^{22} + 8 q^{23} - 8 q^{25} - 12 q^{26} - 8 q^{27} - 40 q^{29}+ \cdots + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} + 2 x^{14} + 41 x^{12} - 92 x^{11} + 66 x^{10} - 104 x^{9} + 291 x^{8} - 388 x^{7} + \cdots + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 280271512 \nu^{15} + 10072934636 \nu^{14} - 27673760232 \nu^{13} - 10701549751 \nu^{12} + \cdots + 74617805308 ) / 20109322702 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 831625517 \nu^{15} - 2708910402 \nu^{14} - 8563703729 \nu^{13} + 20937841398 \nu^{12} + \cdots + 123058003068 ) / 20109322702 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1826463756 \nu^{15} + 18123523334 \nu^{14} - 38648985300 \nu^{13} - 9591752753 \nu^{12} + \cdots + 117948954108 ) / 20109322702 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 2216645857 \nu^{15} + 12643497460 \nu^{14} - 16166209892 \nu^{13} - 4714908696 \nu^{12} + \cdots + 57670039504 ) / 20109322702 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7009520438 \nu^{15} - 20878761079 \nu^{14} - 14259775673 \nu^{13} + 10277818617 \nu^{12} + \cdots + 83233832390 ) / 20109322702 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7052378620 \nu^{15} - 23952235516 \nu^{14} + 2982690857 \nu^{13} - 8174997110 \nu^{12} + \cdots - 107302166388 ) / 20109322702 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 7296601091 \nu^{15} - 19650361232 \nu^{14} - 28389521900 \nu^{13} + 26672489204 \nu^{12} + \cdots + 240511973136 ) / 20109322702 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 8028432850 \nu^{15} - 45535125048 \nu^{14} + 57542760768 \nu^{13} + 16715212937 \nu^{12} + \cdots - 206965565620 ) / 20109322702 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 11175608166 \nu^{15} - 37005033535 \nu^{14} - 2381595084 \nu^{13} - 3936897701 \nu^{12} + \cdots - 87372551998 ) / 20109322702 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 13063453792 \nu^{15} + 39476919321 \nu^{14} + 23925370606 \nu^{13} - 16616446465 \nu^{12} + \cdots - 126792138422 ) / 20109322702 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 13713559086 \nu^{15} + 60674191409 \nu^{14} - 43587993102 \nu^{13} - 12906229613 \nu^{12} + \cdots + 205052174448 ) / 20109322702 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 17762428845 \nu^{15} - 68761071716 \nu^{14} + 27970521720 \nu^{13} + 2908848677 \nu^{12} + \cdots - 193048533492 ) / 20109322702 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 18895033199 \nu^{15} + 56326890157 \nu^{14} + 28972065305 \nu^{13} - 3008003257 \nu^{12} + \cdots - 75904936782 ) / 20109322702 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 20623572503 \nu^{15} + 72838186054 \nu^{14} - 10805072210 \nu^{13} + 4140769327 \nu^{12} + \cdots + 166351026428 ) / 20109322702 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 20772477263 \nu^{15} - 64898826643 \nu^{14} - 20002733393 \nu^{13} - 1575594495 \nu^{12} + \cdots - 23651732226 ) / 20109322702 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{14} - \beta_{11} + \beta_{9} - \beta_{8} - \beta_{4} - \beta_{3} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{15} + 3\beta_{13} - 2\beta_{10} - 4\beta_{9} - \beta_{7} + \beta_{6} + 2\beta_{5} - \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{15} + 5 \beta_{14} - \beta_{13} + 4 \beta_{12} - \beta_{11} + 3 \beta_{9} - 5 \beta_{8} + \cdots + 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -\beta_{14} - 6\beta_{12} - 15\beta_{11} - 5\beta_{9} - 11\beta_{8} - 5\beta_{4} - 9\beta_{3} + 11\beta _1 - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 17\beta_{15} + 11\beta_{13} - 10\beta_{10} - 18\beta_{9} - 23\beta_{7} - 13\beta_{6} + 26\beta_{5} + 3\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 29 \beta_{15} + 15 \beta_{14} - 59 \beta_{13} - 6 \beta_{12} - 59 \beta_{11} + 52 \beta_{10} + \cdots - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 91 \beta_{14} - 136 \beta_{12} - 113 \beta_{11} - 83 \beta_{9} - 7 \beta_{8} + 31 \beta_{4} + \cdots - 83 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 23 \beta_{15} - 159 \beta_{13} + 160 \beta_{10} + 114 \beta_{9} - 151 \beta_{7} + \cdots + 181 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 499 \beta_{15} - 281 \beta_{14} - 689 \beta_{13} - 554 \beta_{12} - 689 \beta_{11} + 630 \beta_{10} + \cdots - 295 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 969 \beta_{14} - 1070 \beta_{12} + 45 \beta_{11} - 557 \beta_{9} + 1113 \beta_{8} + 963 \beta_{4} + \cdots - 557 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 1907 \beta_{15} - 3263 \beta_{13} + 3146 \beta_{10} + 3062 \beta_{9} + 381 \beta_{7} + \cdots + 2249 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 3715 \beta_{15} - 4909 \beta_{14} - 3067 \beta_{13} - 6654 \beta_{12} - 3067 \beta_{11} + 2464 \beta_{10} + \cdots - 3289 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 3033 \beta_{14} + 654 \beta_{12} + 12183 \beta_{11} + 625 \beta_{9} + 15435 \beta_{8} + 9275 \beta_{4} + \cdots + 625 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 22439 \beta_{15} - 26549 \beta_{13} + 24056 \beta_{10} + 27160 \beta_{9} + 20031 \beta_{7} + \cdots + 11333 \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 1765 \beta_{15} - 34093 \beta_{14} + 30449 \beta_{13} - 29144 \beta_{12} + 30449 \beta_{11} + \cdots - 12445 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/238\mathbb{Z}\right)^\times\).

\(n\) \(71\) \(171\)
\(\chi(n)\) \(-\beta_{11} - \beta_{13}\) \(-1 - \beta_{9}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
1.50047 + 0.288947i
0.224274 − 0.447866i
2.07391 − 0.620024i
−0.349168 + 0.778942i
1.50047 − 0.288947i
0.224274 + 0.447866i
0.849168 − 0.0870829i
−1.57391 − 1.48605i
−1.00047 − 1.15497i
0.275726 − 0.418160i
0.849168 + 0.0870829i
−1.57391 + 1.48605i
−1.00047 + 1.15497i
0.275726 + 0.418160i
2.07391 + 0.620024i
−0.349168 − 0.778942i
0.258819 + 0.965926i −0.0571148 − 0.433830i −0.866025 + 0.500000i −0.465926 − 0.607206i 0.404265 − 0.167452i 2.01009 + 1.72033i −0.707107 − 0.707107i 2.71283 − 0.726901i 0.465926 − 0.607206i
9.2 0.258819 + 0.965926i 0.298296 + 2.26578i −0.866025 + 0.500000i −0.465926 − 0.607206i −2.11137 + 0.874559i −1.37612 + 2.25971i −0.707107 − 0.707107i −2.14701 + 0.575288i 0.465926 − 0.607206i
25.1 0.965926 + 0.258819i −0.884777 − 1.15306i 0.866025 + 0.500000i 0.241181 + 1.83195i −0.556194 − 1.34277i 2.36988 − 1.17630i 0.707107 + 0.707107i 0.229731 − 0.857367i −0.241181 + 1.83195i
25.2 0.965926 + 0.258819i 0.418852 + 0.545858i 0.866025 + 0.500000i 0.241181 + 1.83195i 0.263301 + 0.635665i −0.00385258 + 2.64575i 0.707107 + 0.707107i 0.653933 − 2.44051i −0.241181 + 1.83195i
53.1 0.258819 − 0.965926i −0.0571148 + 0.433830i −0.866025 − 0.500000i −0.465926 + 0.607206i 0.404265 + 0.167452i 2.01009 − 1.72033i −0.707107 + 0.707107i 2.71283 + 0.726901i 0.465926 + 0.607206i
53.2 0.258819 − 0.965926i 0.298296 − 2.26578i −0.866025 − 0.500000i −0.465926 + 0.607206i −2.11137 − 0.874559i −1.37612 − 2.25971i −0.707107 + 0.707107i −2.14701 − 0.575288i 0.465926 + 0.607206i
93.1 −0.258819 − 0.965926i −0.682153 + 0.0898071i −0.866025 + 0.500000i 1.46593 − 1.12484i 0.263301 + 0.635665i −1.87355 − 1.86810i 0.707107 + 0.707107i −2.44051 + 0.653933i −1.46593 − 1.12484i
93.2 −0.258819 − 0.965926i 1.44097 − 0.189708i −0.866025 + 0.500000i 1.46593 − 1.12484i −0.556194 − 1.34277i 2.50753 − 0.843988i 0.707107 + 0.707107i −0.857367 + 0.229731i −1.46593 − 1.12484i
121.1 −0.965926 + 0.258819i −0.347150 − 0.266378i 0.866025 − 0.500000i 0.758819 + 0.0999004i 0.404265 + 0.167452i −0.204895 − 2.63781i −0.707107 + 0.707107i −0.726901 − 2.71283i −0.758819 + 0.0999004i
121.2 −0.965926 + 0.258819i 1.81308 + 1.39122i 0.866025 − 0.500000i 0.758819 + 0.0999004i −2.11137 − 0.874559i 2.57092 − 0.624795i −0.707107 + 0.707107i 0.575288 + 2.14701i −0.758819 + 0.0999004i
151.1 −0.258819 + 0.965926i −0.682153 − 0.0898071i −0.866025 − 0.500000i 1.46593 + 1.12484i 0.263301 − 0.635665i −1.87355 + 1.86810i 0.707107 − 0.707107i −2.44051 − 0.653933i −1.46593 + 1.12484i
151.2 −0.258819 + 0.965926i 1.44097 + 0.189708i −0.866025 − 0.500000i 1.46593 + 1.12484i −0.556194 + 1.34277i 2.50753 + 0.843988i 0.707107 − 0.707107i −0.857367 − 0.229731i −1.46593 + 1.12484i
179.1 −0.965926 − 0.258819i −0.347150 + 0.266378i 0.866025 + 0.500000i 0.758819 − 0.0999004i 0.404265 − 0.167452i −0.204895 + 2.63781i −0.707107 − 0.707107i −0.726901 + 2.71283i −0.758819 − 0.0999004i
179.2 −0.965926 − 0.258819i 1.81308 − 1.39122i 0.866025 + 0.500000i 0.758819 − 0.0999004i −2.11137 + 0.874559i 2.57092 + 0.624795i −0.707107 − 0.707107i 0.575288 − 2.14701i −0.758819 − 0.0999004i
219.1 0.965926 − 0.258819i −0.884777 + 1.15306i 0.866025 − 0.500000i 0.241181 − 1.83195i −0.556194 + 1.34277i 2.36988 + 1.17630i 0.707107 − 0.707107i 0.229731 + 0.857367i −0.241181 − 1.83195i
219.2 0.965926 − 0.258819i 0.418852 − 0.545858i 0.866025 − 0.500000i 0.241181 − 1.83195i 0.263301 − 0.635665i −0.00385258 − 2.64575i 0.707107 − 0.707107i 0.653933 + 2.44051i −0.241181 − 1.83195i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner
17.d even 8 1 inner
119.q even 24 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 238.2.q.a ✓ 16
7.c even 3 1 inner 238.2.q.a ✓ 16
17.d even 8 1 inner 238.2.q.a ✓ 16
119.q even 24 1 inner 238.2.q.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
238.2.q.a ✓ 16 1.a even 1 1 trivial
238.2.q.a ✓ 16 7.c even 3 1 inner
238.2.q.a ✓ 16 17.d even 8 1 inner
238.2.q.a ✓ 16 119.q even 24 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 4 T_{3}^{15} + 10 T_{3}^{14} - 16 T_{3}^{13} + 18 T_{3}^{12} - 4 T_{3}^{11} + 32 T_{3}^{10} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(238, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - T^{4} + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} - 4 T^{15} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( (T^{8} - 4 T^{7} + 10 T^{6} + \cdots + 4)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} - 12 T^{15} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( T^{16} - 4 T^{15} + \cdots + 5764801 \) Copy content Toggle raw display
$13$ \( (T^{8} + 32 T^{6} + \cdots + 529)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 6975757441 \) Copy content Toggle raw display
$19$ \( T^{16} - 12 T^{15} + \cdots + 38416 \) Copy content Toggle raw display
$23$ \( T^{16} - 8 T^{15} + \cdots + 38416 \) Copy content Toggle raw display
$29$ \( (T^{8} + 20 T^{7} + \cdots + 470596)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} - 20 T^{15} + \cdots + 92236816 \) Copy content Toggle raw display
$37$ \( (T^{8} + 12 T^{7} + \cdots + 419904)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 12 T^{7} + \cdots + 31684)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} - 8 T^{7} + \cdots + 246016)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 4876129574416 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 146493284040481 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 68415660933376 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 13476652155136 \) Copy content Toggle raw display
$67$ \( (T^{8} + 12 T^{7} + \cdots + 9604)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + 36 T^{7} + \cdots + 13300609)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 76973319621136 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 104821185121 \) Copy content Toggle raw display
$83$ \( (T^{8} - 40 T^{7} + \cdots + 418775296)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 81108054012001 \) Copy content Toggle raw display
$97$ \( (T^{8} - 12 T^{7} + \cdots + 15376)^{2} \) Copy content Toggle raw display
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