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Properties

Label 325.6.b.f
Level $325$
Weight $6$
Character orbit 325.b
Analytic conductor $52.125$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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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] = [325,6,Mod(274,325)] 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("325.274"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(325, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 325.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,-268,0,-104] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(52.1247414392\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 326x^{10} + 40401x^{8} + 2365264x^{6} + 65636064x^{4} + 738923264x^{2} + 2250553600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{11} \)
Twist minimal: no (minimal twist has level 65)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - 3 \beta_{8} - \beta_{7}) q^{3} + (\beta_{5} + \beta_{3} - \beta_{2} - 23) q^{4} + (2 \beta_{6} + 3 \beta_{5} + \beta_{4} + \cdots - 10) q^{6} + (\beta_{11} + \beta_{9} + \cdots + 6 \beta_1) q^{7}+ \cdots + ( - 3106 \beta_{6} + 1323 \beta_{5} + \cdots - 46690) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 268 q^{4} - 104 q^{6} - 2068 q^{9} + 1600 q^{11} - 4216 q^{14} + 644 q^{16} - 10608 q^{19} + 2144 q^{21} - 9512 q^{24} - 676 q^{26} + 7328 q^{29} + 33328 q^{31} - 2760 q^{34} + 3636 q^{36} + 6760 q^{39}+ \cdots - 602528 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 326x^{10} + 40401x^{8} + 2365264x^{6} + 65636064x^{4} + 738923264x^{2} + 2250553600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 433\nu^{10} + 125290\nu^{8} + 12609065\nu^{6} + 514595636\nu^{4} + 7581670960\nu^{2} + 27729818560 ) / 1015329792 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1621 \nu^{10} - 409504 \nu^{8} - 34914705 \nu^{6} - 1109046834 \nu^{4} - 7905208736 \nu^{2} + 57572090080 ) / 1269162240 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 23\nu^{10} + 5926\nu^{8} + 551647\nu^{6} + 23670316\nu^{4} + 494495312\nu^{2} + 3203264064 ) / 15154176 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 8649 \nu^{10} + 2264466 \nu^{8} + 202704145 \nu^{6} + 7009165516 \nu^{4} + 74605838704 \nu^{2} + 187576425280 ) / 5076648960 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 3983 \nu^{10} - 1091430 \nu^{8} - 106093495 \nu^{6} - 4299802012 \nu^{4} + \cdots - 215961478720 ) / 1269162240 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 45345 \nu^{11} - 20244854 \nu^{9} - 6623091303 \nu^{7} - 540210631500 \nu^{5} + \cdots - 18709825579072 \nu ) / 12041811333120 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 146131 \nu^{11} - 42503326 \nu^{9} - 4417899131 \nu^{7} - 196094882684 \nu^{5} + \cdots - 18060977905984 \nu ) / 12041811333120 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 1028933 \nu^{11} - 264616098 \nu^{9} - 22208846909 \nu^{7} - 604717356212 \nu^{5} + \cdots + 12800153679424 \nu ) / 6020905666560 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 1473585 \nu^{11} + 415749338 \nu^{9} + 43408925001 \nu^{7} + 2070951836580 \nu^{5} + \cdots + 328905061230784 \nu ) / 6020905666560 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 176943 \nu^{11} - 46523158 \nu^{9} - 4191810103 \nu^{7} - 145144038412 \nu^{5} + \cdots + 3731837243328 \nu ) / 179728527360 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{3} - \beta_{2} - 55 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{11} + 2\beta_{10} - 5\beta_{9} + 29\beta_{8} - 2\beta_{7} - 79\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 10\beta_{6} - 105\beta_{5} + 10\beta_{4} - 105\beta_{3} + 143\beta_{2} + 4325 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -105\beta_{11} - 248\beta_{10} + 645\beta_{9} - 5415\beta_{8} + 488\beta_{7} + 6993\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -1960\beta_{6} + 10001\beta_{5} - 1492\beta_{4} + 10481\beta_{3} - 17677\beta_{2} - 380327 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 10025\beta_{11} + 29550\beta_{10} - 70285\beta_{9} + 750985\beta_{8} - 69150\beta_{7} - 654039\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 267270\beta_{6} - 966669\beta_{5} + 170670\beta_{4} - 1044429\beta_{3} + 2093199\beta_{2} + 35330245 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 928989 \beta_{11} - 3446268 \beta_{10} + 7307865 \beta_{9} - 94493151 \beta_{8} + \cdots + 63089401 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 32144220 \beta_{6} + 95754205 \beta_{5} - 17820960 \beta_{4} + 104276605 \beta_{3} + \cdots - 3388735735 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 84152485 \beta_{11} + 394055642 \beta_{10} - 746836265 \beta_{9} + 11271713585 \beta_{8} + \cdots - 6198569407 \beta_1 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
274.1
− 10.1784i
− 9.62003i
− 7.62628i
− 7.05260i
− 4.15349i
− 2.16876i
2.16876i
4.15349i
7.05260i
7.62628i
9.62003i
10.1784i
− 10.1784i 10.4132i −71.6007 0 105.990 − 125.896i 403.073i 134.565 0
274.2 − 9.62003i 10.7175i −60.5450 0 103.103 − 18.4289i 274.604i 128.135 0
274.3 − 7.62628i − 29.7832i −26.1601 0 −227.135 − 171.199i − 44.5366i −644.042 0
274.4 − 7.05260i − 22.8190i −17.7392 0 −160.933 141.347i − 100.576i −277.708 0
274.5 − 4.15349i 29.2293i 14.7485 0 121.404 − 42.5171i − 194.170i −611.349 0
274.6 − 2.16876i 2.56930i 27.2965 0 5.57220 − 75.5966i − 128.600i 236.399 0
274.7 2.16876i − 2.56930i 27.2965 0 5.57220 75.5966i 128.600i 236.399 0
274.8 4.15349i − 29.2293i 14.7485 0 121.404 42.5171i 194.170i −611.349 0
274.9 7.05260i 22.8190i −17.7392 0 −160.933 − 141.347i 100.576i −277.708 0
274.10 7.62628i 29.7832i −26.1601 0 −227.135 171.199i 44.5366i −644.042 0
274.11 9.62003i − 10.7175i −60.5450 0 103.103 18.4289i − 274.604i 128.135 0
274.12 10.1784i − 10.4132i −71.6007 0 105.990 125.896i − 403.073i 134.565 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 274.12
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 325.6.b.f 12
5.b even 2 1 inner 325.6.b.f 12
5.c odd 4 1 65.6.a.e ✓ 6
5.c odd 4 1 325.6.a.f 6
15.e even 4 1 585.6.a.k 6
20.e even 4 1 1040.6.a.r 6
65.h odd 4 1 845.6.a.g 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
65.6.a.e ✓ 6 5.c odd 4 1
325.6.a.f 6 5.c odd 4 1
325.6.b.f 12 1.a even 1 1 trivial
325.6.b.f 12 5.b even 2 1 inner
585.6.a.k 6 15.e even 4 1
845.6.a.g 6 65.h odd 4 1
1040.6.a.r 6 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} + 326T_{2}^{10} + 40401T_{2}^{8} + 2365264T_{2}^{6} + 65636064T_{2}^{4} + 738923264T_{2}^{2} + 2250553600 \) acting on \(S_{6}^{\mathrm{new}}(325, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 2250553600 \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 32445965330496 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 32\!\cdots\!04 \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots - 674919089487832)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 28561)^{6} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots + 58\!\cdots\!40)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 70\!\cdots\!04 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 24\!\cdots\!20)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 47\!\cdots\!84)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 15\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 78\!\cdots\!32)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 26\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 29\!\cdots\!80)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 11\!\cdots\!68)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 97\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots - 14\!\cdots\!12)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 41\!\cdots\!64 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 22\!\cdots\!80)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 49\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
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