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Properties

Label 1922.4.a.m
Level $1922$
Weight $4$
Character orbit 1922.a
Self dual yes
Analytic conductor $113.402$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

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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] = [1922,4,Mod(1,1922)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1922, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1922.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 1922 = 2 \cdot 31^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1922.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [8,-16,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(113.401671031\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 115x^{6} + 275x^{5} + 4001x^{4} - 11438x^{3} - 39019x^{2} + 152582x - 111476 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 62)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 q^{2} + ( - \beta_{3} + \beta_1 + 1) q^{3} + 4 q^{4} + (\beta_{4} - 2 \beta_{3} - 1) q^{5} + (2 \beta_{3} - 2 \beta_1 - 2) q^{6} + ( - \beta_{6} + \beta_{3} + 2 \beta_{2} + \cdots + 1) q^{7} - 8 q^{8}+ \cdots + (56 \beta_{7} + 23 \beta_{6} + \cdots + 500) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{2} + 14 q^{3} + 32 q^{4} - q^{5} - 28 q^{6} - 3 q^{7} - 64 q^{8} + 42 q^{9} + 2 q^{10} + 110 q^{11} + 56 q^{12} + 61 q^{13} + 6 q^{14} - 50 q^{15} + 128 q^{16} - 25 q^{17} - 84 q^{18} - 166 q^{19}+ \cdots + 5277 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 115x^{6} + 275x^{5} + 4001x^{4} - 11438x^{3} - 39019x^{2} + 152582x - 111476 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 871 \nu^{7} - 1700902 \nu^{6} + 7813647 \nu^{5} + 146336783 \nu^{4} - 731412859 \nu^{3} + \cdots - 12362822134 ) / 376128270 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 29327 \nu^{7} - 28084 \nu^{6} + 3316619 \nu^{5} + 2777511 \nu^{4} - 113343473 \nu^{3} + \cdots - 819605838 ) / 376128270 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 42862 \nu^{7} - 285439 \nu^{6} + 2120499 \nu^{5} + 25662221 \nu^{4} + 54738737 \nu^{3} + \cdots + 6394674947 ) / 188064135 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1399 \nu^{7} - 903 \nu^{6} + 174878 \nu^{5} + 64417 \nu^{4} - 6384361 \nu^{3} + 121113 \nu^{2} + \cdots - 58796531 ) / 6066585 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 223844 \nu^{7} - 322882 \nu^{6} - 22719033 \nu^{5} + 36157808 \nu^{4} + 642297356 \nu^{3} + \cdots + 5252189291 ) / 188064135 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 924331 \nu^{7} - 409192 \nu^{6} + 102898977 \nu^{5} + 144383 \nu^{4} - 3469745719 \nu^{3} + \cdots - 46661600644 ) / 376128270 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} + 2\beta_{6} - \beta_{5} + 2\beta_{3} - \beta_{2} - \beta _1 + 30 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 3\beta_{6} + 12\beta_{5} + 2\beta_{4} - 27\beta_{3} - 2\beta_{2} + 50\beta _1 - 36 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 54\beta_{7} + 123\beta_{6} - 78\beta_{5} - 3\beta_{4} + 417\beta_{3} - 63\beta_{2} - 81\beta _1 + 1448 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 24\beta_{7} + 135\beta_{6} + 1074\beta_{5} + 120\beta_{4} - 2220\beta_{3} - 96\beta_{2} + 2699\beta _1 - 2457 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2867 \beta_{7} + 6991 \beta_{6} - 5516 \beta_{5} - 570 \beta_{4} + 34450 \beta_{3} - 3761 \beta_{2} + \cdots + 77802 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 841 \beta_{7} + 4509 \beta_{6} + 75036 \beta_{5} + 6103 \beta_{4} - 157152 \beta_{3} - 3433 \beta_{2} + \cdots - 165819 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−7.85130
−5.69079
−5.26578
1.09953
2.65107
3.64904
7.33677
6.07146
−2.00000 −7.46933 4.00000 0.873836 14.9387 −2.38883 −8.00000 28.7909 −1.74767
1.2 −2.00000 −3.07276 4.00000 −1.13633 6.14552 31.6021 −8.00000 −17.5582 2.27266
1.3 −2.00000 −2.64775 4.00000 6.50831 5.29550 −22.7916 −8.00000 −19.9894 −13.0166
1.4 −2.00000 1.48149 4.00000 14.2740 −2.96298 1.74388 −8.00000 −24.8052 −28.5480
1.5 −2.00000 3.03304 4.00000 −14.4653 −6.06607 31.4695 −8.00000 −17.8007 28.9306
1.6 −2.00000 6.26708 4.00000 −18.8033 −12.5342 −4.56899 −8.00000 12.2763 37.6065
1.7 −2.00000 7.71874 4.00000 −9.00877 −15.4375 −31.2065 −8.00000 32.5789 18.0175
1.8 −2.00000 8.68950 4.00000 20.7575 −17.3790 −6.85959 −8.00000 48.5074 −41.5151
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \( +1 \)
\(31\) \( +1 \)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1922.4.a.m 8
31.b odd 2 1 1922.4.a.k 8
31.d even 5 2 62.4.d.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
62.4.d.b ✓ 16 31.d even 5 2
1922.4.a.k 8 31.b odd 2 1
1922.4.a.m 8 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 14T_{3}^{7} - 31T_{3}^{6} + 1039T_{3}^{5} - 1919T_{3}^{4} - 15548T_{3}^{3} + 32094T_{3}^{2} + 65643T_{3} - 114781 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1922))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 2)^{8} \) Copy content Toggle raw display
$3$ \( T^{8} - 14 T^{7} + \cdots - 114781 \) Copy content Toggle raw display
$5$ \( T^{8} + T^{7} + \cdots + 4691916 \) Copy content Toggle raw display
$7$ \( T^{8} + 3 T^{7} + \cdots - 92351709 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots - 556565730804 \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 431793745939 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots - 578666936252799 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 11735966661795 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 247723106274261 \) Copy content Toggle raw display
$29$ \( T^{8} + \cdots - 171123992359605 \) Copy content Toggle raw display
$31$ \( T^{8} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots - 43\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( T^{8} + \cdots + 91\!\cdots\!24 \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots - 83\!\cdots\!41 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots - 14\!\cdots\!49 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots - 53\!\cdots\!49 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 62\!\cdots\!75 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots - 12\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 89\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 76\!\cdots\!59 \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 25\!\cdots\!71 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 10\!\cdots\!20 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots - 61\!\cdots\!89 \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 21\!\cdots\!45 \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots - 54\!\cdots\!81 \) Copy content Toggle raw display
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