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Properties

Label 7935.2.a.bs
Level $7935$
Weight $2$
Character orbit 7935.a
Self dual yes
Analytic conductor $63.361$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7935,2,Mod(1,7935)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7935, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("7935.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7935 = 3 \cdot 5 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7935.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(63.3612940039\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 24 x^{14} + 228 x^{12} - 4 x^{11} - 1098 x^{10} + 56 x^{9} + 2836 x^{8} - 276 x^{7} - 3812 x^{6} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + q^{5} + \beta_1 q^{6} + (\beta_{15} + 1) q^{7} + (\beta_{14} - \beta_{12} + \cdots - \beta_1) q^{8}+ \cdots + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - q^{3} + (\beta_{2} + 1) q^{4} + q^{5} + \beta_1 q^{6} + (\beta_{15} + 1) q^{7} + (\beta_{14} - \beta_{12} + \cdots - \beta_1) q^{8}+ \cdots + ( - \beta_{10} + \beta_{8} + \cdots - \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{3} + 16 q^{4} + 16 q^{5} + 12 q^{7} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{3} + 16 q^{4} + 16 q^{5} + 12 q^{7} + 16 q^{9} - 8 q^{11} - 16 q^{12} - 8 q^{13} - 16 q^{15} + 16 q^{16} + 20 q^{17} + 16 q^{19} + 16 q^{20} - 12 q^{21} + 16 q^{22} + 16 q^{25} + 20 q^{26} - 16 q^{27} - 16 q^{28} + 40 q^{29} + 16 q^{31} - 20 q^{32} + 8 q^{33} + 16 q^{34} + 12 q^{35} + 16 q^{36} + 28 q^{37} + 56 q^{38} + 8 q^{39} + 12 q^{41} + 4 q^{43} - 48 q^{44} + 16 q^{45} - 20 q^{47} - 16 q^{48} + 20 q^{49} - 20 q^{51} - 36 q^{52} + 4 q^{53} - 8 q^{55} - 8 q^{56} - 16 q^{57} - 16 q^{58} + 20 q^{59} - 16 q^{60} + 32 q^{61} + 28 q^{62} + 12 q^{63} + 28 q^{64} - 8 q^{65} - 16 q^{66} + 4 q^{67} - 24 q^{68} + 24 q^{71} - 36 q^{73} - 100 q^{74} - 16 q^{75} + 88 q^{76} + 4 q^{77} - 20 q^{78} + 24 q^{79} + 16 q^{80} + 16 q^{81} - 20 q^{82} + 44 q^{83} + 16 q^{84} + 20 q^{85} + 52 q^{86} - 40 q^{87} + 48 q^{88} + 16 q^{89} + 24 q^{91} - 16 q^{93} - 20 q^{94} + 16 q^{95} + 20 q^{96} + 64 q^{97} + 4 q^{98} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 24 x^{14} + 228 x^{12} - 4 x^{11} - 1098 x^{10} + 56 x^{9} + 2836 x^{8} - 276 x^{7} - 3812 x^{6} + \cdots + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 10877 \nu^{15} - 55304 \nu^{14} - 271950 \nu^{13} + 1240936 \nu^{12} + 2774228 \nu^{11} + \cdots + 4556728 ) / 769976 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 27652 \nu^{15} - 5451 \nu^{14} + 620468 \nu^{13} + 147136 \nu^{12} - 5401924 \nu^{11} + \cdots - 348064 ) / 384988 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 112713 \nu^{15} - 169280 \nu^{14} + 2705936 \nu^{13} + 3773320 \nu^{12} - 25571924 \nu^{11} + \cdots + 27520784 ) / 1539952 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 113257 \nu^{15} + 64188 \nu^{14} - 2637280 \nu^{13} - 1535120 \nu^{12} + 24005924 \nu^{11} + \cdots - 8243040 ) / 769976 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 257595 \nu^{15} - 226514 \nu^{14} + 6053904 \nu^{13} + 5274560 \nu^{12} - 55661420 \nu^{11} + \cdots + 22126240 ) / 1539952 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 1267 \nu^{15} - 1236 \nu^{14} + 30072 \nu^{13} + 29024 \nu^{12} - 279340 \nu^{11} - 261940 \nu^{10} + \cdots + 171136 ) / 7064 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 282043 \nu^{15} - 259290 \nu^{14} + 6523900 \nu^{13} + 6142880 \nu^{12} - 58838892 \nu^{11} + \cdots + 52466736 ) / 1539952 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 317873 \nu^{15} - 172466 \nu^{14} + 7408760 \nu^{13} + 4169512 \nu^{12} - 67563236 \nu^{11} + \cdots + 27124112 ) / 1539952 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 263054 \nu^{15} + 41685 \nu^{14} - 6177384 \nu^{13} - 992332 \nu^{12} + 56860720 \nu^{11} + \cdots + 2679944 ) / 769976 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 2674 \nu^{15} - 1267 \nu^{14} + 62940 \nu^{13} + 30072 \nu^{12} - 580648 \nu^{11} - 268644 \nu^{10} + \cdots + 142120 ) / 7064 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 608579 \nu^{15} - 604590 \nu^{14} + 14206612 \nu^{13} + 14163704 \nu^{12} - 129440892 \nu^{11} + \cdots + 78764400 ) / 1539952 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 673891 \nu^{15} - 556094 \nu^{14} + 15882956 \nu^{13} + 12810888 \nu^{12} - 146604732 \nu^{11} + \cdots + 67616400 ) / 1539952 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 195701 \nu^{15} - 139820 \nu^{14} + 4568963 \nu^{13} + 3301770 \nu^{12} - 41662932 \nu^{11} + \cdots + 17959896 ) / 384988 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{14} + \beta_{12} + \beta_{5} + \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{13} - \beta_{10} + \beta_{9} + \beta_{8} + \beta_{7} + \beta_{4} + 6\beta_{2} + 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - \beta_{15} - 8 \beta_{14} - \beta_{13} + 11 \beta_{12} + \beta_{11} + \beta_{10} - 2 \beta_{7} + \cdots + 28 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{15} - 13 \beta_{13} + \beta_{12} + \beta_{11} - 11 \beta_{10} + 11 \beta_{9} + 11 \beta_{8} + \cdots + 87 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 10 \beta_{15} - 58 \beta_{14} - 14 \beta_{13} + 96 \beta_{12} + 14 \beta_{11} + 12 \beta_{10} + \cdots + 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 16 \beta_{15} - \beta_{14} - 126 \beta_{13} + 21 \beta_{12} + 17 \beta_{11} - 96 \beta_{10} + 96 \beta_{9} + \cdots + 551 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 79 \beta_{15} - 419 \beta_{14} - 142 \beta_{13} + 779 \beta_{12} + 140 \beta_{11} + 111 \beta_{10} + \cdots + 36 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 170 \beta_{15} - 24 \beta_{14} - 1092 \beta_{13} + 276 \beta_{12} + 194 \beta_{11} - 774 \beta_{10} + \cdots + 3714 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 580 \beta_{15} - 3061 \beta_{14} - 1287 \beta_{13} + 6137 \beta_{12} + 1235 \beta_{11} + 923 \beta_{10} + \cdots + 466 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1536 \beta_{15} - 358 \beta_{14} - 8974 \beta_{13} + 2978 \beta_{12} + 1888 \beta_{11} - 6020 \beta_{10} + \cdots + 26138 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 4136 \beta_{15} - 22628 \beta_{14} - 11096 \beta_{13} + 47704 \beta_{12} + 10280 \beta_{11} + \cdots + 5220 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 12812 \beta_{15} - 4318 \beta_{14} - 71744 \beta_{13} + 28990 \beta_{12} + 16954 \beta_{11} + \cdots + 189332 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 29142 \beta_{15} - 168876 \beta_{14} - 93240 \beta_{13} + 368468 \beta_{12} + 83068 \beta_{11} + \cdots + 53676 \) 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.

comment: embeddings in the coefficient field
 
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
2.80216
2.25106
2.05186
1.90386
1.30963
0.767196
0.703835
0.546587
−0.395158
−0.497130
−0.856558
−1.69096
−1.87840
−2.10132
−2.21254
−2.70413
−2.80216 −1.00000 5.85211 1.00000 2.80216 1.26956 −10.7942 1.00000 −2.80216
1.2 −2.25106 −1.00000 3.06727 1.00000 2.25106 −3.50886 −2.40248 1.00000 −2.25106
1.3 −2.05186 −1.00000 2.21014 1.00000 2.05186 −0.292845 −0.431172 1.00000 −2.05186
1.4 −1.90386 −1.00000 1.62470 1.00000 1.90386 2.21087 0.714520 1.00000 −1.90386
1.5 −1.30963 −1.00000 −0.284864 1.00000 1.30963 3.58187 2.99233 1.00000 −1.30963
1.6 −0.767196 −1.00000 −1.41141 1.00000 0.767196 4.52413 2.61722 1.00000 −0.767196
1.7 −0.703835 −1.00000 −1.50462 1.00000 0.703835 −3.01305 2.46667 1.00000 −0.703835
1.8 −0.546587 −1.00000 −1.70124 1.00000 0.546587 −1.58407 2.02305 1.00000 −0.546587
1.9 0.395158 −1.00000 −1.84385 1.00000 −0.395158 4.18181 −1.51893 1.00000 0.395158
1.10 0.497130 −1.00000 −1.75286 1.00000 −0.497130 2.87397 −1.86566 1.00000 0.497130
1.11 0.856558 −1.00000 −1.26631 1.00000 −0.856558 1.30416 −2.79778 1.00000 0.856558
1.12 1.69096 −1.00000 0.859342 1.00000 −1.69096 4.77523 −1.92881 1.00000 1.69096
1.13 1.87840 −1.00000 1.52838 1.00000 −1.87840 −3.70522 −0.885884 1.00000 1.87840
1.14 2.10132 −1.00000 2.41554 1.00000 −2.10132 −1.86076 0.873183 1.00000 2.10132
1.15 2.21254 −1.00000 2.89534 1.00000 −2.21254 0.655648 1.98097 1.00000 2.21254
1.16 2.70413 −1.00000 5.31234 1.00000 −2.70413 0.587543 8.95699 1.00000 2.70413
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(1\)
\(5\) \(-1\)
\(23\) \(1\)

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 7935.2.a.bs yes 16
23.b odd 2 1 7935.2.a.br ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7935.2.a.br ✓ 16 23.b odd 2 1
7935.2.a.bs yes 16 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7935))\):

\( T_{2}^{16} - 24 T_{2}^{14} + 228 T_{2}^{12} + 4 T_{2}^{11} - 1098 T_{2}^{10} - 56 T_{2}^{9} + 2836 T_{2}^{8} + \cdots + 64 \) Copy content Toggle raw display
\( T_{7}^{16} - 12 T_{7}^{15} + 6 T_{7}^{14} + 428 T_{7}^{13} - 1285 T_{7}^{12} - 4812 T_{7}^{11} + \cdots + 44344 \) Copy content Toggle raw display
\( T_{11}^{16} + 8 T_{11}^{15} - 72 T_{11}^{14} - 584 T_{11}^{13} + 2265 T_{11}^{12} + 16928 T_{11}^{11} + \cdots - 1492208 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 24 T^{14} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( (T + 1)^{16} \) Copy content Toggle raw display
$5$ \( (T - 1)^{16} \) Copy content Toggle raw display
$7$ \( T^{16} - 12 T^{15} + \cdots + 44344 \) Copy content Toggle raw display
$11$ \( T^{16} + 8 T^{15} + \cdots - 1492208 \) Copy content Toggle raw display
$13$ \( T^{16} + 8 T^{15} + \cdots + 2360398 \) Copy content Toggle raw display
$17$ \( T^{16} - 20 T^{15} + \cdots - 241184 \) Copy content Toggle raw display
$19$ \( T^{16} - 16 T^{15} + \cdots + 47883172 \) Copy content Toggle raw display
$23$ \( T^{16} \) Copy content Toggle raw display
$29$ \( T^{16} - 40 T^{15} + \cdots + 71756032 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 1041866512 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 8998446680638 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots - 28249448576 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 52476159118 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 261987509248 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 21627663552736 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 660356992 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 4162567921 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 42811047424 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots - 4300690810400 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots - 46817921024 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots - 4343225187836 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots - 261887021600 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 64566294400 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 16478913617464 \) Copy content Toggle raw display
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