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Properties

Label 325.4.c.f
Level $325$
Weight $4$
Character orbit 325.c
Analytic conductor $19.176$
Analytic rank $0$
Dimension $14$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [325,4,Mod(51,325)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) 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("325.51"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 325.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14,0,12] 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: no
Analytic conductor: \(19.1756207519\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 84x^{12} + 2674x^{10} + 40708x^{8} + 309729x^{6} + 1120672x^{4} + 1546596x^{2} + 83136 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{3} + 1) q^{3} + (\beta_{2} - 4) q^{4} + ( - \beta_{4} + 2 \beta_1) q^{6} + (\beta_{7} + \beta_1) q^{7} + (\beta_{5} + \beta_{4} - 4 \beta_1) q^{8} + ( - \beta_{9} + \beta_{3} - \beta_{2} + 11) q^{9}+ \cdots + (\beta_{12} + 6 \beta_{11} + \cdots + 60 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 12 q^{3} - 56 q^{4} + 158 q^{9} - 300 q^{12} - 46 q^{13} - 92 q^{14} + 280 q^{16} - 106 q^{17} - 192 q^{22} - 148 q^{23} - 284 q^{26} + 456 q^{27} + 18 q^{29} - 2084 q^{36} - 740 q^{38} - 560 q^{39}+ \cdots - 5924 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 84x^{12} + 2674x^{10} + 40708x^{8} + 309729x^{6} + 1120672x^{4} + 1546596x^{2} + 83136 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 535 \nu^{12} - 48501 \nu^{10} - 1663369 \nu^{8} - 26660115 \nu^{6} - 196298852 \nu^{4} + \cdots - 143226944 ) / 24705280 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 535 \nu^{13} + 48501 \nu^{11} + 1663369 \nu^{9} + 26660115 \nu^{7} + 196298852 \nu^{5} + \cdots + 167932224 \nu ) / 24705280 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 535 \nu^{13} - 48501 \nu^{11} - 1663369 \nu^{9} - 26660115 \nu^{7} - 196298852 \nu^{5} + \cdots + 326173376 \nu ) / 24705280 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 7841 \nu^{12} - 620787 \nu^{10} - 18188287 \nu^{8} - 243874757 \nu^{6} - 1468393340 \nu^{4} + \cdots + 998611008 ) / 24705280 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 979 \nu^{13} + 77785 \nu^{11} + 2258205 \nu^{9} + 29156943 \nu^{7} + 159936804 \nu^{5} + \cdots - 180984576 \nu ) / 4941056 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 20115 \nu^{12} + 1604201 \nu^{10} + 46827469 \nu^{8} + 609798975 \nu^{6} + 3395034932 \nu^{4} + \cdots - 2290611136 ) / 24705280 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 20495 \nu^{12} + 1609789 \nu^{10} + 46104081 \nu^{8} + 589108555 \nu^{6} + 3294336068 \nu^{4} + \cdots + 555715136 ) / 24705280 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 4125 \nu^{13} + 51661 \nu^{12} + 345095 \nu^{11} + 4125783 \nu^{10} + 10920195 \nu^{9} + \cdots + 695203776 ) / 49410560 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 4125 \nu^{13} + 51661 \nu^{12} - 345095 \nu^{11} + 4125783 \nu^{10} - 10920195 \nu^{9} + \cdots + 695203776 ) / 49410560 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 13743 \nu^{13} - 1117165 \nu^{11} - 33800417 \nu^{9} - 473642651 \nu^{7} + \cdots - 7316472192 \nu ) / 24705280 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 3815 \nu^{13} - 297871 \nu^{11} - 8424199 \nu^{9} - 104234025 \nu^{7} - 524873862 \nu^{5} + \cdots + 1439030136 \nu ) / 3088160 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} - 20\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + \beta_{10} - \beta_{9} - \beta_{8} + \beta_{6} + 6\beta_{3} - 29\beta_{2} + 244 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{13} + 7\beta_{7} - 32\beta_{5} - 39\beta_{4} + 478\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -46\beta_{11} - 46\beta_{10} + 54\beta_{9} + 39\beta_{8} - 41\beta_{6} - 306\beta_{3} + 786\beta_{2} - 5853 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 54 \beta_{13} + 10 \beta_{12} - 18 \beta_{11} + 18 \beta_{10} - 364 \beta_{7} + 895 \beta_{5} + \cdots - 12320 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1655 \beta_{11} + 1655 \beta_{10} - 2095 \beta_{9} - 1309 \beta_{8} + 1285 \beta_{6} + 11506 \beta_{3} + \cdots + 150690 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2095 \beta_{13} - 740 \beta_{12} + 1180 \beta_{11} - 1180 \beta_{10} + 13549 \beta_{7} + \cdots + 329598 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 54392 \beta_{11} - 54392 \beta_{10} + 71592 \beta_{9} + 42023 \beta_{8} - 36837 \beta_{6} + \cdots - 4024499 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 71592 \beta_{13} + 35110 \beta_{12} - 52310 \beta_{11} + 52310 \beta_{10} - 446710 \beta_{7} + \cdots - 9013972 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1710761 \beta_{11} + 1710761 \beta_{10} - 2300697 \beta_{9} - 1316361 \beta_{8} + 1020501 \beta_{6} + \cdots + 109921244 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 2300697 \beta_{13} - 1380520 \beta_{12} + 1970456 \beta_{11} - 1970456 \beta_{10} + \cdots + 250173346 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/325\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1
− 5.39779i
− 4.79494i
− 3.86564i
− 2.90404i
− 2.16518i
− 1.93695i
− 0.236624i
0.236624i
1.93695i
2.16518i
2.90404i
3.86564i
4.79494i
5.39779i
− 5.39779i 9.57032 −21.1361 0 − 51.6586i − 19.5662i 70.9061i 64.5911 0
51.2 − 4.79494i −3.41049 −14.9915 0 16.3531i − 3.51137i 33.5237i −15.3686 0
51.3 − 3.86564i 2.17714 −6.94315 0 − 8.41604i 30.3578i − 4.08541i −22.2601 0
51.4 − 2.90404i −8.97961 −0.433472 0 26.0772i − 4.22545i − 21.9735i 53.6333 0
51.5 − 2.16518i 2.56039 3.31200 0 − 5.54370i − 25.8792i − 24.4925i −20.4444 0
51.6 − 1.93695i 7.72075 4.24823 0 − 14.9547i 13.0366i − 23.7242i 32.6099 0
51.7 − 0.236624i −3.63850 7.94401 0 0.860956i 9.09368i − 3.77273i −13.7613 0
51.8 0.236624i −3.63850 7.94401 0 − 0.860956i − 9.09368i 3.77273i −13.7613 0
51.9 1.93695i 7.72075 4.24823 0 14.9547i − 13.0366i 23.7242i 32.6099 0
51.10 2.16518i 2.56039 3.31200 0 5.54370i 25.8792i 24.4925i −20.4444 0
51.11 2.90404i −8.97961 −0.433472 0 − 26.0772i 4.22545i 21.9735i 53.6333 0
51.12 3.86564i 2.17714 −6.94315 0 8.41604i − 30.3578i 4.08541i −22.2601 0
51.13 4.79494i −3.41049 −14.9915 0 − 16.3531i 3.51137i − 33.5237i −15.3686 0
51.14 5.39779i 9.57032 −21.1361 0 51.6586i 19.5662i − 70.9061i 64.5911 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.14
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.4.c.f yes 14
5.b even 2 1 325.4.c.d ✓ 14
5.c odd 4 2 325.4.d.e 28
13.b even 2 1 inner 325.4.c.f yes 14
65.d even 2 1 325.4.c.d ✓ 14
65.h odd 4 2 325.4.d.e 28
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
325.4.c.d ✓ 14 5.b even 2 1
325.4.c.d ✓ 14 65.d even 2 1
325.4.c.f yes 14 1.a even 1 1 trivial
325.4.c.f yes 14 13.b even 2 1 inner
325.4.d.e 28 5.c odd 4 2
325.4.d.e 28 65.h odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(325, [\chi])\):

\( T_{2}^{14} + 84T_{2}^{12} + 2674T_{2}^{10} + 40708T_{2}^{8} + 309729T_{2}^{6} + 1120672T_{2}^{4} + 1546596T_{2}^{2} + 83136 \) Copy content Toggle raw display
\( T_{3}^{7} - 6T_{3}^{6} - 116T_{3}^{5} + 584T_{3}^{4} + 3019T_{3}^{3} - 9214T_{3}^{2} - 18564T_{3} + 45896 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + 84 T^{12} + \cdots + 83136 \) Copy content Toggle raw display
$3$ \( (T^{7} - 6 T^{6} + \cdots + 45896)^{2} \) Copy content Toggle raw display
$5$ \( T^{14} \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 731077251960000 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 24\!\cdots\!13 \) Copy content Toggle raw display
$17$ \( (T^{7} + 53 T^{6} + \cdots - 25559275500)^{2} \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( (T^{7} + \cdots + 46060010222592)^{2} \) Copy content Toggle raw display
$29$ \( (T^{7} + \cdots + 20750230556400)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 88\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 17\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( (T^{7} + \cdots - 62\!\cdots\!56)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 32\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( (T^{7} + \cdots + 12\!\cdots\!56)^{2} \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 65\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( (T^{7} + \cdots - 11\!\cdots\!94)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 67\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 56\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( (T^{7} + \cdots - 94\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 15\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 30\!\cdots\!04 \) Copy content Toggle raw display
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