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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [7935,2,Mod(1,7935)] 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("7935.1"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage: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")
 
Level: \( N \) \(=\) \( 7935 = 3 \cdot 5 \cdot 23^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7935.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [15,-8,15,12,15,-8,-13] 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: \(63.3612940039\)
Analytic rank: \(1\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 7 x^{14} + 4 x^{13} + 66 x^{12} - 105 x^{11} - 231 x^{10} + 486 x^{9} + 370 x^{8} - 947 x^{7} + \cdots + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 345)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1) q^{2} + q^{3} + (\beta_{13} + \beta_{10} + \beta_{9} + \cdots + 2) q^{4} + q^{5} + (\beta_1 - 1) q^{6} + ( - \beta_{6} - 1) q^{7} + (\beta_{14} - 2 \beta_{13} + \beta_{11} + \cdots - 3) q^{8}+ \cdots + ( - \beta_{14} - \beta_{13} - \beta_{12} + \cdots - 1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - 8 q^{2} + 15 q^{3} + 12 q^{4} + 15 q^{5} - 8 q^{6} - 13 q^{7} - 24 q^{8} + 15 q^{9} - 8 q^{10} - 3 q^{11} + 12 q^{12} - 14 q^{13} - q^{14} + 15 q^{15} + 18 q^{16} + 4 q^{17} - 8 q^{18} - 3 q^{19}+ \cdots - 3 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 7 x^{14} + 4 x^{13} + 66 x^{12} - 105 x^{11} - 231 x^{10} + 486 x^{9} + 370 x^{8} - 947 x^{7} + \cdots + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 178 \nu^{14} + 438 \nu^{13} - 17731 \nu^{12} + 50463 \nu^{11} + 147281 \nu^{10} - 618198 \nu^{9} + \cdots - 7013 ) / 28313 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 586 \nu^{14} - 8738 \nu^{13} + 33883 \nu^{12} + 24566 \nu^{11} - 356568 \nu^{10} + 276613 \nu^{9} + \cdots + 63760 ) / 28313 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1981 \nu^{14} - 30119 \nu^{13} + 109180 \nu^{12} + 122762 \nu^{11} - 1163070 \nu^{10} + 534131 \nu^{9} + \cdots + 38225 ) / 28313 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3189 \nu^{14} - 28419 \nu^{13} + 49928 \nu^{12} + 210413 \nu^{11} - 675724 \nu^{10} - 438370 \nu^{9} + \cdots + 54574 ) / 28313 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 4365 \nu^{14} - 24889 \nu^{13} - 12817 \nu^{12} + 260679 \nu^{11} - 126099 \nu^{10} - 1080229 \nu^{9} + \cdots - 66200 ) / 28313 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4458 \nu^{14} - 31977 \nu^{13} + 31841 \nu^{12} + 244575 \nu^{11} - 559260 \nu^{10} - 432625 \nu^{9} + \cdots - 77340 ) / 28313 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 6333 \nu^{14} - 52495 \nu^{13} + 74594 \nu^{12} + 422225 \nu^{11} - 1139114 \nu^{10} - 1029352 \nu^{9} + \cdots + 373 ) / 28313 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 6385 \nu^{14} + 34234 \nu^{13} + 36521 \nu^{12} - 408654 \nu^{11} + 68867 \nu^{10} + 1899323 \nu^{9} + \cdots - 28546 ) / 28313 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 7522 \nu^{14} + 51478 \nu^{13} - 33618 \nu^{12} - 433707 \nu^{11} + 739544 \nu^{10} + 1187957 \nu^{9} + \cdots + 7184 ) / 28313 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 8010 \nu^{14} - 65229 \nu^{13} + 79808 \nu^{12} + 572055 \nu^{11} - 1328308 \nu^{10} - 1799263 \nu^{9} + \cdots + 24171 ) / 28313 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 10461 \nu^{14} - 62061 \nu^{13} - 12756 \nu^{12} + 601558 \nu^{11} - 424388 \nu^{10} - 2260251 \nu^{9} + \cdots - 6385 ) / 28313 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 12158 \nu^{14} - 83017 \nu^{13} + 47728 \nu^{12} + 728745 \nu^{11} - 1151523 \nu^{10} - 2238780 \nu^{9} + \cdots - 34911 ) / 28313 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 13722 \nu^{14} - 80441 \nu^{13} - 18673 \nu^{12} + 775757 \nu^{11} - 567806 \nu^{10} - 2777210 \nu^{9} + \cdots + 108977 ) / 28313 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{13} + \beta_{10} + \beta_{9} + \beta_{8} - \beta_{5} - \beta_{4} + \beta_{3} + \beta _1 + 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{14} + \beta_{13} + \beta_{11} + 2 \beta_{10} + 3 \beta_{9} + 2 \beta_{8} - 3 \beta_{5} - 2 \beta_{4} + \cdots + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2 \beta_{14} + 7 \beta_{13} + 2 \beta_{11} + 9 \beta_{10} + 10 \beta_{9} + 8 \beta_{8} - \beta_{7} + \cdots + 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9 \beta_{14} + 14 \beta_{13} + 9 \beta_{11} + 24 \beta_{10} + 29 \beta_{9} + 20 \beta_{8} - 3 \beta_{7} + \cdots + 32 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 23 \beta_{14} + 55 \beta_{13} + \beta_{12} + 23 \beta_{11} + 84 \beta_{10} + 87 \beta_{9} + 64 \beta_{8} + \cdots + 113 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 78 \beta_{14} + 141 \beta_{13} + 3 \beta_{12} + 76 \beta_{11} + 246 \beta_{10} + 251 \beta_{9} + \cdots + 290 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 219 \beta_{14} + 463 \beta_{13} + 15 \beta_{12} + 211 \beta_{11} + 789 \beta_{10} + 735 \beta_{9} + \cdots + 905 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 682 \beta_{14} + 1303 \beta_{13} + 44 \beta_{12} + 639 \beta_{11} + 2369 \beta_{10} + 2124 \beta_{9} + \cdots + 2536 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1985 \beta_{14} + 4018 \beta_{13} + 153 \beta_{12} + 1827 \beta_{11} + 7328 \beta_{10} + 6182 \beta_{9} + \cdots + 7606 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 6003 \beta_{14} + 11709 \beta_{13} + 430 \beta_{12} + 5386 \beta_{11} + 22089 \beta_{10} + 17917 \beta_{9} + \cdots + 21988 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 17712 \beta_{14} + 35295 \beta_{13} + 1281 \beta_{12} + 15577 \beta_{11} + 67188 \beta_{10} + \cdots + 65122 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 53007 \beta_{14} + 104204 \beta_{13} + 3411 \beta_{12} + 45568 \beta_{11} + 202170 \beta_{10} + \cdots + 190482 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 157211 \beta_{14} + 311532 \beta_{13} + 9148 \beta_{12} + 132503 \beta_{11} + 609646 \beta_{10} + \cdots + 562218 \) 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
−1.79819
−1.61357
−1.51951
−1.09310
−0.575465
−0.468895
−0.0341759
0.461809
0.575387
1.43449
1.66540
1.71400
2.54877
2.71945
2.98360
−2.79819 1.00000 5.82986 1.00000 −2.79819 1.77415 −10.7167 1.00000 −2.79819
1.2 −2.61357 1.00000 4.83075 1.00000 −2.61357 −1.46136 −7.39836 1.00000 −2.61357
1.3 −2.51951 1.00000 4.34792 1.00000 −2.51951 −3.39486 −5.91560 1.00000 −2.51951
1.4 −2.09310 1.00000 2.38107 1.00000 −2.09310 −2.81599 −0.797611 1.00000 −2.09310
1.5 −1.57547 1.00000 0.482090 1.00000 −1.57547 −0.121150 2.39141 1.00000 −1.57547
1.6 −1.46889 1.00000 0.157651 1.00000 −1.46889 0.0551329 2.70622 1.00000 −1.46889
1.7 −1.03418 1.00000 −0.930480 1.00000 −1.03418 1.07500 3.03063 1.00000 −1.03418
1.8 −0.538191 1.00000 −1.71035 1.00000 −0.538191 1.89409 1.99688 1.00000 −0.538191
1.9 −0.424613 1.00000 −1.81970 1.00000 −0.424613 0.618206 1.62190 1.00000 −0.424613
1.10 0.434487 1.00000 −1.81122 1.00000 0.434487 −4.25480 −1.65593 1.00000 0.434487
1.11 0.665398 1.00000 −1.55725 1.00000 0.665398 −1.92036 −2.36698 1.00000 0.665398
1.12 0.714001 1.00000 −1.49020 1.00000 0.714001 0.712390 −2.49201 1.00000 0.714001
1.13 1.54877 1.00000 0.398697 1.00000 1.54877 −1.99732 −2.48005 1.00000 1.54877
1.14 1.71945 1.00000 0.956498 1.00000 1.71945 0.125509 −1.79425 1.00000 1.71945
1.15 1.98360 1.00000 1.93467 1.00000 1.98360 −3.28863 −0.129580 1.00000 1.98360
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.15
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.bo 15
23.b odd 2 1 7935.2.a.bn 15
23.c even 11 2 345.2.m.b ✓ 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
345.2.m.b ✓ 30 23.c even 11 2
7935.2.a.bn 15 23.b odd 2 1
7935.2.a.bo 15 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 \(S_{2}^{\mathrm{new}}(\Gamma_0(7935))\):

\( T_{2}^{15} + 8 T_{2}^{14} + 11 T_{2}^{13} - 64 T_{2}^{12} - 184 T_{2}^{11} + 110 T_{2}^{10} + 772 T_{2}^{9} + \cdots + 23 \) Copy content Toggle raw display
\( T_{7}^{15} + 13 T_{7}^{14} + 51 T_{7}^{13} - 10 T_{7}^{12} - 483 T_{7}^{11} - 638 T_{7}^{10} + 1462 T_{7}^{9} + \cdots - 1 \) Copy content Toggle raw display
\( T_{11}^{15} + 3 T_{11}^{14} - 76 T_{11}^{13} - 207 T_{11}^{12} + 2038 T_{11}^{11} + 5201 T_{11}^{10} + \cdots + 8119 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} + 8 T^{14} + \cdots + 23 \) Copy content Toggle raw display
$3$ \( (T - 1)^{15} \) Copy content Toggle raw display
$5$ \( (T - 1)^{15} \) Copy content Toggle raw display
$7$ \( T^{15} + 13 T^{14} + \cdots - 1 \) Copy content Toggle raw display
$11$ \( T^{15} + 3 T^{14} + \cdots + 8119 \) Copy content Toggle raw display
$13$ \( T^{15} + 14 T^{14} + \cdots - 2881 \) Copy content Toggle raw display
$17$ \( T^{15} - 4 T^{14} + \cdots + 79548743 \) Copy content Toggle raw display
$19$ \( T^{15} + \cdots + 157228259 \) Copy content Toggle raw display
$23$ \( T^{15} \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots - 970813691 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots - 2206752241 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots - 812233333 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots + 16618486433 \) Copy content Toggle raw display
$43$ \( T^{15} + 28 T^{14} + \cdots + 97351 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 1024937143 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots - 97577096189 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots - 4428458387 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots - 745845592691 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 38590951871 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots - 138003607169 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots + 2140984691 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots + 2611907497 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots + 11004712265293 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots - 16208958107 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 74093472223 \) Copy content Toggle raw display
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