Properties

Label 7935.2.a.bl
Level $7935$
Weight $2$
Character orbit 7935.a
Self dual yes
Analytic conductor $63.361$
Analytic rank $0$
Dimension $12$
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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 16x^{10} + 92x^{8} - 4x^{7} - 234x^{6} + 32x^{5} + 252x^{4} - 68x^{3} - 76x^{2} + 32x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + q^{3} + ( - \beta_{8} + \beta_{7} + 1) q^{4} - q^{5} + \beta_1 q^{6} + ( - \beta_{6} + \beta_{5} + \beta_1) q^{7} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_1) q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + q^{3} + ( - \beta_{8} + \beta_{7} + 1) q^{4} - q^{5} + \beta_1 q^{6} + ( - \beta_{6} + \beta_{5} + \beta_1) q^{7} + (\beta_{5} - \beta_{4} - \beta_{3} + \beta_1) q^{8} + q^{9} - \beta_1 q^{10} + (\beta_{9} + \beta_{6} + 2) q^{11} + ( - \beta_{8} + \beta_{7} + 1) q^{12} + ( - \beta_{11} - \beta_{7} - \beta_1 - 1) q^{13} + ( - \beta_{11} - \beta_{10} - \beta_{9} - 2 \beta_{8} - \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} + \cdots + 2) q^{14}+ \cdots + (\beta_{9} + \beta_{6} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{3} + 8 q^{4} - 12 q^{5} + 4 q^{7} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{3} + 8 q^{4} - 12 q^{5} + 4 q^{7} + 12 q^{9} + 24 q^{11} + 8 q^{12} - 8 q^{13} + 16 q^{14} - 12 q^{15} + 28 q^{17} + 16 q^{19} - 8 q^{20} + 4 q^{21} + 12 q^{25} - 36 q^{26} + 12 q^{27} + 8 q^{28} - 16 q^{29} + 20 q^{32} + 24 q^{33} + 16 q^{34} - 4 q^{35} + 8 q^{36} + 20 q^{37} + 16 q^{38} - 8 q^{39} - 4 q^{41} + 16 q^{42} - 12 q^{43} + 16 q^{44} - 12 q^{45} + 4 q^{47} + 24 q^{49} + 28 q^{51} - 36 q^{52} + 28 q^{53} - 24 q^{55} + 56 q^{56} + 16 q^{57} + 20 q^{59} - 8 q^{60} + 32 q^{61} + 12 q^{62} + 4 q^{63} - 4 q^{64} + 8 q^{65} - 4 q^{67} + 64 q^{68} - 16 q^{70} - 8 q^{71} + 4 q^{73} - 36 q^{74} + 12 q^{75} + 8 q^{76} - 28 q^{77} - 36 q^{78} + 40 q^{79} + 12 q^{81} - 28 q^{82} + 100 q^{83} + 8 q^{84} - 28 q^{85} - 20 q^{86} - 16 q^{87} + 80 q^{89} - 24 q^{91} + 44 q^{94} - 16 q^{95} + 20 q^{96} - 8 q^{97} + 28 q^{98} + 24 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 16x^{10} + 92x^{8} - 4x^{7} - 234x^{6} + 32x^{5} + 252x^{4} - 68x^{3} - 76x^{2} + 32x - 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{11} - 4 \nu^{10} + 30 \nu^{9} + 45 \nu^{8} - 257 \nu^{7} - 154 \nu^{6} + 818 \nu^{5} + 195 \nu^{4} - 912 \nu^{3} - 70 \nu^{2} + 201 \nu - 8 ) / 45 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 4 \nu^{11} - \nu^{10} + 60 \nu^{9} + 30 \nu^{8} - 323 \nu^{7} - 241 \nu^{6} + 782 \nu^{5} + 690 \nu^{4} - 813 \nu^{3} - 640 \nu^{2} + 234 \nu + 73 ) / 45 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 8 \nu^{11} + 8 \nu^{10} + 110 \nu^{9} - 100 \nu^{8} - 506 \nu^{7} + 413 \nu^{6} + 924 \nu^{5} - 660 \nu^{4} - 606 \nu^{3} + 310 \nu^{2} + 148 \nu - 39 ) / 15 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 28 \nu^{11} + 23 \nu^{10} + 390 \nu^{9} - 270 \nu^{8} - 1841 \nu^{7} + 998 \nu^{6} + 3554 \nu^{5} - 1290 \nu^{4} - 2586 \nu^{3} + 290 \nu^{2} + 453 \nu - 44 ) / 45 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 44 \nu^{11} - 4 \nu^{10} - 645 \nu^{9} + 3253 \nu^{7} + 251 \nu^{6} - 6787 \nu^{5} - 885 \nu^{4} + 5343 \nu^{3} + 920 \nu^{2} - 924 \nu - 98 ) / 45 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 22 \nu^{11} + 28 \nu^{10} - 375 \nu^{9} - 390 \nu^{8} + 2294 \nu^{7} + 1753 \nu^{6} - 6146 \nu^{5} - 2850 \nu^{4} + 6834 \nu^{3} + 1135 \nu^{2} - 1962 \nu + 71 ) / 45 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 22 \nu^{11} + 28 \nu^{10} - 375 \nu^{9} - 390 \nu^{8} + 2294 \nu^{7} + 1753 \nu^{6} - 6146 \nu^{5} - 2850 \nu^{4} + 6834 \nu^{3} + 1090 \nu^{2} - 1962 \nu + 206 ) / 45 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 46 \nu^{11} + 26 \nu^{10} + 675 \nu^{9} - 300 \nu^{8} - 3437 \nu^{7} + 1091 \nu^{6} + 7358 \nu^{5} - 1380 \nu^{4} - 6132 \nu^{3} + 245 \nu^{2} + 1311 \nu - 23 ) / 45 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 38 \nu^{11} - 17 \nu^{10} + 600 \nu^{9} + 270 \nu^{8} - 3376 \nu^{7} - 1352 \nu^{6} + 8314 \nu^{5} + 2325 \nu^{4} - 8556 \nu^{3} - 725 \nu^{2} + 2418 \nu - 349 ) / 45 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 71 \nu^{11} + 14 \nu^{10} - 1095 \nu^{9} - 270 \nu^{8} + 5962 \nu^{7} + 1574 \nu^{6} - 14023 \nu^{5} - 3270 \nu^{4} + 13587 \nu^{3} + 1910 \nu^{2} - 3696 \nu + 298 ) / 45 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{8} + \beta_{7} + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - \beta_{4} - \beta_{3} + 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -\beta_{10} + \beta_{9} - 7\beta_{8} + 6\beta_{7} - \beta_{5} - 2\beta_{2} + 14 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 2 \beta_{11} - \beta_{10} + 2 \beta_{8} - \beta_{7} + \beta_{6} + 8 \beta_{5} - 9 \beta_{4} - 11 \beta_{3} - 2 \beta_{2} + 28 \beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -8\beta_{10} + 10\beta_{9} - 44\beta_{8} + 36\beta_{7} - 10\beta_{5} - \beta_{4} - 2\beta_{3} - 20\beta_{2} + 75 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 22 \beta_{11} - 10 \beta_{10} + \beta_{9} + 22 \beta_{8} - 9 \beta_{7} + 10 \beta_{6} + 54 \beta_{5} - 68 \beta_{4} - 90 \beta_{3} - 22 \beta_{2} + 167 \beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 3 \beta_{11} - 54 \beta_{10} + 78 \beta_{9} - 272 \beta_{8} + 222 \beta_{7} + \beta_{6} - 76 \beta_{5} - 17 \beta_{4} - 28 \beta_{3} - 157 \beta_{2} + 3 \beta _1 + 437 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 180 \beta_{11} - 78 \beta_{10} + 18 \beta_{9} + 175 \beta_{8} - 60 \beta_{7} + 78 \beta_{6} + 348 \beta_{5} - 483 \beta_{4} - 662 \beta_{3} - 186 \beta_{2} + 1034 \beta _1 + 185 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 48 \beta_{11} - 354 \beta_{10} + 561 \beta_{9} - 1688 \beta_{8} + 1397 \beta_{7} + 18 \beta_{6} - 523 \beta_{5} - 186 \beta_{4} - 282 \beta_{3} - 1136 \beta_{2} + 58 \beta _1 + 2682 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 1325 \beta_{11} - 565 \beta_{10} + 204 \beta_{9} + 1225 \beta_{8} - 347 \beta_{7} + 561 \beta_{6} + 2211 \beta_{5} - 3331 \beta_{4} - 4640 \beta_{3} - 1438 \beta_{2} + 6549 \beta _1 + 1431 \) Copy content Toggle raw display

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
−2.51193
−1.85426
−1.80514
−1.64536
−0.756768
0.0774485
0.430223
0.525898
1.27924
1.56791
2.08171
2.61103
−2.51193 1.00000 4.30981 −1.00000 −2.51193 −1.59040 −5.80210 1.00000 2.51193
1.2 −1.85426 1.00000 1.43827 −1.00000 −1.85426 −4.25122 1.04159 1.00000 1.85426
1.3 −1.80514 1.00000 1.25854 −1.00000 −1.80514 3.38925 1.33844 1.00000 1.80514
1.4 −1.64536 1.00000 0.707219 −1.00000 −1.64536 −0.989641 2.12709 1.00000 1.64536
1.5 −0.756768 1.00000 −1.42730 −1.00000 −0.756768 2.44646 2.59367 1.00000 0.756768
1.6 0.0774485 1.00000 −1.99400 −1.00000 0.0774485 3.48186 −0.309330 1.00000 −0.0774485
1.7 0.430223 1.00000 −1.81491 −1.00000 0.430223 1.05251 −1.64126 1.00000 −0.430223
1.8 0.525898 1.00000 −1.72343 −1.00000 0.525898 −3.99872 −1.95814 1.00000 −0.525898
1.9 1.27924 1.00000 −0.363534 −1.00000 1.27924 2.12712 −3.02354 1.00000 −1.27924
1.10 1.56791 1.00000 0.458355 −1.00000 1.56791 −3.42407 −2.41717 1.00000 −1.56791
1.11 2.08171 1.00000 2.33352 −1.00000 2.08171 1.02026 0.694295 1.00000 −2.08171
1.12 2.61103 1.00000 4.81746 −1.00000 2.61103 4.73661 7.35645 1.00000 −2.61103
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
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.bl 12
23.b odd 2 1 7935.2.a.bm yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7935.2.a.bl 12 1.a even 1 1 trivial
7935.2.a.bm yes 12 23.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(7935))\):

\( T_{2}^{12} - 16T_{2}^{10} + 92T_{2}^{8} - 4T_{2}^{7} - 234T_{2}^{6} + 32T_{2}^{5} + 252T_{2}^{4} - 68T_{2}^{3} - 76T_{2}^{2} + 32T_{2} - 2 \) Copy content Toggle raw display
\( T_{7}^{12} - 4 T_{7}^{11} - 46 T_{7}^{10} + 196 T_{7}^{9} + 682 T_{7}^{8} - 3384 T_{7}^{7} - 2944 T_{7}^{6} + 23600 T_{7}^{5} - 7052 T_{7}^{4} - 54720 T_{7}^{3} + 38328 T_{7}^{2} + 33936 T_{7} - 28616 \) Copy content Toggle raw display
\( T_{11}^{12} - 24 T_{11}^{11} + 224 T_{11}^{10} - 912 T_{11}^{9} + 193 T_{11}^{8} + 14264 T_{11}^{7} - 66566 T_{11}^{6} + 155060 T_{11}^{5} - 211661 T_{11}^{4} + 171036 T_{11}^{3} - 76898 T_{11}^{2} + 16408 T_{11} - 1127 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 16 T^{10} + 92 T^{8} - 4 T^{7} + \cdots - 2 \) Copy content Toggle raw display
$3$ \( (T - 1)^{12} \) Copy content Toggle raw display
$5$ \( (T + 1)^{12} \) Copy content Toggle raw display
$7$ \( T^{12} - 4 T^{11} - 46 T^{10} + \cdots - 28616 \) Copy content Toggle raw display
$11$ \( T^{12} - 24 T^{11} + 224 T^{10} + \cdots - 1127 \) Copy content Toggle raw display
$13$ \( T^{12} + 8 T^{11} - 38 T^{10} - 404 T^{9} + \cdots + 526 \) Copy content Toggle raw display
$17$ \( T^{12} - 28 T^{11} + 288 T^{10} + \cdots + 673726 \) Copy content Toggle raw display
$19$ \( T^{12} - 16 T^{11} + 4 T^{10} + \cdots + 1460833 \) Copy content Toggle raw display
$23$ \( T^{12} \) Copy content Toggle raw display
$29$ \( T^{12} + 16 T^{11} - 1208 T^{9} + \cdots + 1794448 \) Copy content Toggle raw display
$31$ \( T^{12} - 202 T^{10} + \cdots + 74973313 \) Copy content Toggle raw display
$37$ \( T^{12} - 20 T^{11} + 42 T^{10} + \cdots - 30121250 \) Copy content Toggle raw display
$41$ \( T^{12} + 4 T^{11} - 248 T^{10} + \cdots - 21275759 \) Copy content Toggle raw display
$43$ \( T^{12} + 12 T^{11} - 236 T^{10} + \cdots - 254978 \) Copy content Toggle raw display
$47$ \( T^{12} - 4 T^{11} + \cdots - 1766783264 \) Copy content Toggle raw display
$53$ \( T^{12} - 28 T^{11} + \cdots + 2593939678 \) Copy content Toggle raw display
$59$ \( T^{12} - 20 T^{11} + \cdots + 110786308 \) Copy content Toggle raw display
$61$ \( T^{12} - 32 T^{11} + 212 T^{10} + \cdots + 62664529 \) Copy content Toggle raw display
$67$ \( T^{12} + 4 T^{11} + \cdots - 105209106944 \) Copy content Toggle raw display
$71$ \( T^{12} + 8 T^{11} - 302 T^{10} + \cdots + 1210537 \) Copy content Toggle raw display
$73$ \( T^{12} - 4 T^{11} + \cdots - 2306547272 \) Copy content Toggle raw display
$79$ \( T^{12} - 40 T^{11} + \cdots - 541785409775 \) Copy content Toggle raw display
$83$ \( T^{12} - 100 T^{11} + \cdots - 68861906 \) Copy content Toggle raw display
$89$ \( T^{12} - 80 T^{11} + 2726 T^{10} + \cdots + 18487492 \) Copy content Toggle raw display
$97$ \( T^{12} + 8 T^{11} - 640 T^{10} + \cdots - 146161634 \) Copy content Toggle raw display
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