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Properties

Label 975.4.a.bc
Level $975$
Weight $4$
Character orbit 975.a
Self dual yes
Analytic conductor $57.527$
Analytic rank $0$
Dimension $11$
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] = [975,4,Mod(1,975)] 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("975.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(975, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 975 = 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 975.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [11,3,-33,55,0,-9,-20] 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: yes
Analytic conductor: \(57.5268622556\)
Analytic rank: \(0\)
Dimension: \(11\)
Coefficient field: \(\mathbb{Q}[x]/(x^{11} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{11} - 3 x^{10} - 67 x^{9} + 177 x^{8} + 1527 x^{7} - 3289 x^{6} - 14195 x^{5} + 20777 x^{4} + \cdots - 5840 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 5^{3} \)
Twist minimal: no (minimal twist has level 195)
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 \(1,\beta_1,\ldots,\beta_{10}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - 3 q^{3} + (\beta_{2} + 5) q^{4} - 3 \beta_1 q^{6} + ( - \beta_{9} + \beta_1 - 2) q^{7} + (\beta_{3} + 7 \beta_1 + 3) q^{8} + 9 q^{9} + ( - \beta_{10} + \beta_{9} - \beta_{4} + \cdots + 10) q^{11}+ \cdots + ( - 9 \beta_{10} + 9 \beta_{9} + \cdots + 90) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 11 q + 3 q^{2} - 33 q^{3} + 55 q^{4} - 9 q^{6} - 20 q^{7} + 51 q^{8} + 99 q^{9} + 100 q^{11} - 165 q^{12} + 143 q^{13} + 150 q^{14} + 511 q^{16} + 56 q^{17} + 27 q^{18} + 44 q^{19} + 60 q^{21} - 262 q^{22}+ \cdots + 900 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{11} - 3 x^{10} - 67 x^{9} + 177 x^{8} + 1527 x^{7} - 3289 x^{6} - 14195 x^{5} + 20777 x^{4} + \cdots - 5840 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 23\nu - 3 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 51054 \nu^{10} + 362146 \nu^{9} + 3300097 \nu^{8} - 22542636 \nu^{7} - 75601881 \nu^{6} + \cdots + 2137216970 ) / 64082618 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 839753 \nu^{10} + 8470848 \nu^{9} + 45116291 \nu^{8} - 533152920 \nu^{7} - 683551071 \nu^{6} + \cdots + 8378916592 ) / 512660944 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 614799 \nu^{10} + 251824 \nu^{9} - 39214905 \nu^{8} - 32066840 \nu^{7} + 785226805 \nu^{6} + \cdots + 2231106368 ) / 256330472 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1376581 \nu^{10} - 2863868 \nu^{9} + 98511343 \nu^{8} + 201136876 \nu^{7} + \cdots + 22079626096 ) / 512660944 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 824933 \nu^{10} - 3056258 \nu^{9} - 51531339 \nu^{8} + 179596086 \nu^{7} + 1042342951 \nu^{6} + \cdots - 27499358736 ) / 256330472 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 1666787 \nu^{10} - 5397840 \nu^{9} - 104671697 \nu^{8} + 310518776 \nu^{7} + 2095025125 \nu^{6} + \cdots - 19882912176 ) / 512660944 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 2908221 \nu^{10} - 8613188 \nu^{9} - 181333599 \nu^{8} + 489369860 \nu^{7} + 3574895979 \nu^{6} + \cdots + 5786063168 ) / 512660944 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 23\beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} - \beta_{9} - \beta_{8} - \beta_{4} + \beta_{3} + 32\beta_{2} + 2\beta _1 + 295 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 3\beta_{10} - 4\beta_{9} + \beta_{8} - 2\beta_{6} + \beta_{5} + \beta_{4} + 35\beta_{3} + 9\beta_{2} + 601\beta _1 + 108 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 40 \beta_{10} - 51 \beta_{9} - 35 \beta_{8} + \beta_{7} + 6 \beta_{6} - 50 \beta_{4} + 44 \beta_{3} + \cdots + 7727 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 159 \beta_{10} - 261 \beta_{9} + 62 \beta_{8} - 23 \beta_{7} - 99 \beta_{6} + 17 \beta_{5} + 62 \beta_{4} + \cdots + 4115 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1389 \beta_{10} - 2160 \beta_{9} - 958 \beta_{8} + 59 \beta_{7} + 368 \beta_{6} - 42 \beta_{5} + \cdots + 213073 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 6372 \beta_{10} - 11931 \beta_{9} + 2803 \beta_{8} - 1493 \beta_{7} - 3611 \beta_{6} - 240 \beta_{5} + \cdots + 151254 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 46610 \beta_{10} - 83218 \beta_{9} - 23992 \beta_{8} + 1898 \beta_{7} + 15670 \beta_{6} - 3258 \beta_{5} + \cdots + 6016893 \) 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
−5.38219
−4.60776
−2.70833
−1.71052
−1.29959
−0.0799779
1.66800
3.05864
3.33847
5.17836
5.54491
−5.38219 −3.00000 20.9680 0 16.1466 −12.4545 −69.7962 9.00000 0
1.2 −4.60776 −3.00000 13.2315 0 13.8233 −15.5130 −24.1055 9.00000 0
1.3 −2.70833 −3.00000 −0.664940 0 8.12499 −5.41772 23.4675 9.00000 0
1.4 −1.71052 −3.00000 −5.07413 0 5.13155 20.6809 22.3635 9.00000 0
1.5 −1.29959 −3.00000 −6.31107 0 3.89877 −34.4577 18.5985 9.00000 0
1.6 −0.0799779 −3.00000 −7.99360 0 0.239934 28.1840 1.27914 9.00000 0
1.7 1.66800 −3.00000 −5.21778 0 −5.00400 1.83536 −22.0473 9.00000 0
1.8 3.05864 −3.00000 1.35528 0 −9.17592 23.3910 −20.3238 9.00000 0
1.9 3.33847 −3.00000 3.14536 0 −10.0154 −22.0097 −16.2070 9.00000 0
1.10 5.17836 −3.00000 18.8154 0 −15.5351 −32.8406 56.0060 9.00000 0
1.11 5.54491 −3.00000 22.7460 0 −16.6347 28.6020 81.7651 9.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.11
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(5\) \( -1 \)
\(13\) \( -1 \)

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 975.4.a.bc 11
5.b even 2 1 975.4.a.bb 11
5.c odd 4 2 195.4.c.c ✓ 22
15.e even 4 2 585.4.c.e 22
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
195.4.c.c ✓ 22 5.c odd 4 2
585.4.c.e 22 15.e even 4 2
975.4.a.bb 11 5.b even 2 1
975.4.a.bc 11 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(975))\):

\( T_{2}^{11} - 3 T_{2}^{10} - 67 T_{2}^{9} + 177 T_{2}^{8} + 1527 T_{2}^{7} - 3289 T_{2}^{6} - 14195 T_{2}^{5} + \cdots - 5840 \) Copy content Toggle raw display
\( T_{7}^{11} + 20 T_{7}^{10} - 2683 T_{7}^{9} - 47866 T_{7}^{8} + 2586888 T_{7}^{7} + \cdots - 18658986063872 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{11} - 3 T^{10} + \cdots - 5840 \) Copy content Toggle raw display
$3$ \( (T + 3)^{11} \) Copy content Toggle raw display
$5$ \( T^{11} \) Copy content Toggle raw display
$7$ \( T^{11} + \cdots - 18658986063872 \) Copy content Toggle raw display
$11$ \( T^{11} + \cdots + 22\!\cdots\!20 \) Copy content Toggle raw display
$13$ \( (T - 13)^{11} \) Copy content Toggle raw display
$17$ \( T^{11} + \cdots + 59\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{11} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{11} + \cdots - 11\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{11} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{11} + \cdots - 19\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( T^{11} + \cdots - 56\!\cdots\!52 \) Copy content Toggle raw display
$41$ \( T^{11} + \cdots - 11\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( T^{11} + \cdots - 45\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{11} + \cdots - 86\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{11} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{11} + \cdots - 80\!\cdots\!80 \) Copy content Toggle raw display
$61$ \( T^{11} + \cdots - 19\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{11} + \cdots - 13\!\cdots\!96 \) Copy content Toggle raw display
$71$ \( T^{11} + \cdots - 86\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{11} + \cdots + 61\!\cdots\!88 \) Copy content Toggle raw display
$79$ \( T^{11} + \cdots + 93\!\cdots\!88 \) Copy content Toggle raw display
$83$ \( T^{11} + \cdots + 56\!\cdots\!04 \) Copy content Toggle raw display
$89$ \( T^{11} + \cdots - 14\!\cdots\!60 \) Copy content Toggle raw display
$97$ \( T^{11} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
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