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Properties

Label 325.10.a.h
Level $325$
Weight $10$
Character orbit 325.a
Self dual yes
Analytic conductor $167.387$
Analytic rank $1$
Dimension $17$
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] = [325,10,Mod(1,325)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(325, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("325.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 325 = 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 325.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: \(167.386646753\)
Analytic rank: \(1\)
Dimension: \(17\)
Coefficient field: \(\mathbb{Q}[x]/(x^{17} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{17} - x^{16} - 6453 x^{15} - 11965 x^{14} + 16673200 x^{13} + 68278926 x^{12} - 22023799708 x^{11} + \cdots + 22\!\cdots\!80 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{4}\cdot 5^{8} \)
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 \(1,\beta_1,\ldots,\beta_{16}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 2) q^{2} + ( - \beta_{3} - 4) q^{3} + (\beta_{2} + 251) q^{4} + ( - \beta_{5} - 7 \beta_{3} + \cdots - 63) q^{6} + (\beta_{6} + 2 \beta_{2} + 24 \beta_1 + 626) q^{7} + ( - \beta_{4} - 4 \beta_{3} + \cdots - 659) q^{8}+ \cdots + (1934 \beta_{16} - 3910 \beta_{15} + \cdots - 178130303) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 17 q + 33 q^{2} - 73 q^{3} + 4267 q^{4} - 1103 q^{6} + 10670 q^{7} - 11481 q^{8} + 47590 q^{9} - 130917 q^{11} - 32239 q^{12} - 485537 q^{13} - 292206 q^{14} + 1064251 q^{16} + 193953 q^{17} + 2026286 q^{18}+ \cdots - 3023832936 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{17} - x^{16} - 6453 x^{15} - 11965 x^{14} + 16673200 x^{13} + 68278926 x^{12} - 22023799708 x^{11} + \cdots + 22\!\cdots\!80 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4\nu - 759 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 23\!\cdots\!71 \nu^{16} + \cdots + 98\!\cdots\!20 ) / 39\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 23\!\cdots\!71 \nu^{16} + \cdots - 62\!\cdots\!20 ) / 99\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 38\!\cdots\!31 \nu^{16} + \cdots + 35\!\cdots\!80 ) / 39\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 23\!\cdots\!99 \nu^{16} + \cdots - 63\!\cdots\!00 ) / 44\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 27\!\cdots\!35 \nu^{16} + \cdots - 18\!\cdots\!20 ) / 33\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 62\!\cdots\!51 \nu^{16} + \cdots + 14\!\cdots\!60 ) / 66\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 23\!\cdots\!19 \nu^{16} + \cdots - 17\!\cdots\!20 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 17\!\cdots\!09 \nu^{16} + \cdots + 44\!\cdots\!40 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 69\!\cdots\!51 \nu^{16} + \cdots + 14\!\cdots\!80 ) / 49\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 90\!\cdots\!67 \nu^{16} + \cdots - 42\!\cdots\!40 ) / 39\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 10\!\cdots\!27 \nu^{16} + \cdots - 57\!\cdots\!60 ) / 39\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 14\!\cdots\!55 \nu^{16} + \cdots + 93\!\cdots\!20 ) / 39\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 16\!\cdots\!83 \nu^{16} + \cdots + 33\!\cdots\!60 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 40\!\cdots\!77 \nu^{16} + \cdots + 11\!\cdots\!20 ) / 39\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4\beta _1 + 759 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{4} + 4\beta_{3} + 9\beta_{2} + 1289\beta _1 + 3173 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{16} - \beta_{15} - 2 \beta_{12} - 3 \beta_{11} - 2 \beta_{8} - 11 \beta_{5} + 15 \beta_{4} + \cdots + 979399 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 42 \beta_{16} - 16 \beta_{15} + 21 \beta_{14} - 73 \beta_{13} + 22 \beta_{12} + 11 \beta_{11} + \cdots + 9855909 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 3583 \beta_{16} - 3023 \beta_{15} + 104 \beta_{14} - 212 \beta_{13} - 5926 \beta_{12} + \cdots + 1482284811 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 156122 \beta_{16} - 67060 \beta_{15} + 48215 \beta_{14} - 199779 \beta_{13} + 39810 \beta_{12} + \cdots + 24145260029 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 9358991 \beta_{16} - 6937831 \beta_{15} + 391748 \beta_{14} - 1127576 \beta_{13} - 13607806 \beta_{12} + \cdots + 2446213626667 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 419684982 \beta_{16} - 195352416 \beta_{15} + 83794423 \beta_{14} - 418523443 \beta_{13} + \cdots + 54033229558469 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 21855852387 \beta_{16} - 14786754339 \beta_{15} + 907756048 \beta_{14} - 3761389716 \beta_{13} + \cdots + 42\!\cdots\!23 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 998996362906 \beta_{16} - 490312250156 \beta_{15} + 128805747539 \beta_{14} - 808236954127 \beta_{13} + \cdots + 11\!\cdots\!69 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 48620319227551 \beta_{16} - 30867905424551 \beta_{15} + 1606771497484 \beta_{14} - 10234724463120 \beta_{13} + \cdots + 78\!\cdots\!19 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 22\!\cdots\!46 \beta_{16} + \cdots + 24\!\cdots\!61 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 10\!\cdots\!39 \beta_{16} + \cdots + 14\!\cdots\!67 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 49\!\cdots\!18 \beta_{16} + \cdots + 50\!\cdots\!57 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 22\!\cdots\!91 \beta_{16} + \cdots + 28\!\cdots\!91 \) 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
45.5365
41.9783
32.7069
32.3076
22.0262
13.6975
12.0171
8.33068
−6.66848
−8.60741
−11.1845
−15.0241
−27.7222
−30.4531
−33.1624
−35.1331
−39.6455
−43.5365 −112.372 1383.43 0 4892.27 6926.24 −37939.1 −7055.63 0
1.2 −39.9783 152.057 1086.26 0 −6078.99 3359.20 −22958.1 3438.43 0
1.3 −30.7069 −116.164 430.914 0 3567.03 −1957.66 2489.89 −6188.99 0
1.4 −30.3076 7.32501 406.551 0 −222.003 −9118.61 3195.89 −19629.3 0
1.5 −20.0262 143.807 −110.950 0 −2879.91 11821.9 12475.3 997.339 0
1.6 −11.6975 −207.589 −375.168 0 2428.27 5519.45 10377.7 23410.1 0
1.7 −10.0171 174.740 −411.658 0 −1750.39 −417.700 9252.35 10851.2 0
1.8 −6.33068 −33.0585 −471.922 0 209.283 180.411 6228.90 −18590.1 0
1.9 8.66848 −65.6463 −436.857 0 −569.054 −11239.1 −8225.15 −15373.6 0
1.10 10.6074 39.6173 −399.483 0 420.237 7059.84 −9668.47 −18113.5 0
1.11 13.1845 −236.239 −338.168 0 −3114.70 −5988.15 −11209.1 36125.9 0
1.12 17.0241 228.743 −222.181 0 3894.14 −5864.90 −12498.8 32640.5 0
1.13 29.7222 192.679 371.411 0 5726.85 7108.54 −4178.62 17442.2 0
1.14 32.4531 −140.165 541.204 0 −4548.78 −1977.44 947.752 −36.8521 0
1.15 35.1624 −237.080 724.396 0 −8336.29 11662.6 7468.37 36523.7 0
1.16 37.1331 92.8669 866.866 0 3448.44 −3610.83 13177.3 −11058.7 0
1.17 41.6455 43.4764 1222.35 0 1810.60 −2793.78 29582.9 −17792.8 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.17
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(13\) \( +1 \)

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 325.10.a.h yes 17
5.b even 2 1 325.10.a.g ✓ 17
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
325.10.a.g ✓ 17 5.b even 2 1
325.10.a.h yes 17 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{17} - 33 T_{2}^{16} - 5941 T_{2}^{15} + 200595 T_{2}^{14} + 13661520 T_{2}^{13} + \cdots - 26\!\cdots\!00 \) acting on \(S_{10}^{\mathrm{new}}(\Gamma_0(325))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{17} + \cdots - 26\!\cdots\!00 \) Copy content Toggle raw display
$3$ \( T^{17} + \cdots - 91\!\cdots\!92 \) Copy content Toggle raw display
$5$ \( T^{17} \) Copy content Toggle raw display
$7$ \( T^{17} + \cdots + 94\!\cdots\!48 \) Copy content Toggle raw display
$11$ \( T^{17} + \cdots - 16\!\cdots\!64 \) Copy content Toggle raw display
$13$ \( (T + 28561)^{17} \) Copy content Toggle raw display
$17$ \( T^{17} + \cdots + 10\!\cdots\!72 \) Copy content Toggle raw display
$19$ \( T^{17} + \cdots - 34\!\cdots\!60 \) Copy content Toggle raw display
$23$ \( T^{17} + \cdots - 13\!\cdots\!52 \) Copy content Toggle raw display
$29$ \( T^{17} + \cdots + 11\!\cdots\!60 \) Copy content Toggle raw display
$31$ \( T^{17} + \cdots - 13\!\cdots\!20 \) Copy content Toggle raw display
$37$ \( T^{17} + \cdots - 49\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{17} + \cdots - 23\!\cdots\!12 \) Copy content Toggle raw display
$43$ \( T^{17} + \cdots - 14\!\cdots\!40 \) Copy content Toggle raw display
$47$ \( T^{17} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{17} + \cdots - 10\!\cdots\!72 \) Copy content Toggle raw display
$59$ \( T^{17} + \cdots + 74\!\cdots\!80 \) Copy content Toggle raw display
$61$ \( T^{17} + \cdots - 73\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{17} + \cdots - 24\!\cdots\!72 \) Copy content Toggle raw display
$71$ \( T^{17} + \cdots + 66\!\cdots\!32 \) Copy content Toggle raw display
$73$ \( T^{17} + \cdots + 11\!\cdots\!92 \) Copy content Toggle raw display
$79$ \( T^{17} + \cdots - 29\!\cdots\!80 \) Copy content Toggle raw display
$83$ \( T^{17} + \cdots + 33\!\cdots\!88 \) Copy content Toggle raw display
$89$ \( T^{17} + \cdots - 16\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{17} + \cdots + 64\!\cdots\!00 \) Copy content Toggle raw display
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