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Properties

Label 425.6.a.k
Level $425$
Weight $6$
Character orbit 425.a
Self dual yes
Analytic conductor $68.163$
Analytic rank $0$
Dimension $15$
CM no
Inner twists $1$

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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] = [425,6,Mod(1,425)] 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("425.1"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 6, names="a")
 
Level: \( N \) \(=\) \( 425 = 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 425.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [15,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(68.1631234205\)
Analytic rank: \(0\)
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: \( x^{15} - x^{14} - 378 x^{13} + 106 x^{12} + 55677 x^{11} + 23739 x^{10} - 4018640 x^{9} + \cdots - 45034730496 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{9}\cdot 5^{4} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{6} + 1) q^{3} + (\beta_{2} + \beta_1 + 18) q^{4} + ( - \beta_{8} + 2 \beta_{6} + \cdots + 12) q^{6} + ( - \beta_{6} + \beta_{3} + 5 \beta_1 + 11) q^{7} + (\beta_{6} + \beta_{5} + \beta_{2} + \cdots + 49) q^{8}+ \cdots + (412 \beta_{14} + 105 \beta_{13} + \cdots - 12712) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q + q^{2} + 9 q^{3} + 277 q^{4} + 169 q^{6} + 181 q^{7} + 753 q^{8} + 1826 q^{9} + 172 q^{11} - 2109 q^{12} - 389 q^{13} + 3635 q^{14} + 6837 q^{16} + 4335 q^{17} + 6742 q^{18} + 5150 q^{19} - 6891 q^{21}+ \cdots - 183214 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - x^{14} - 378 x^{13} + 106 x^{12} + 55677 x^{11} + 23739 x^{10} - 4018640 x^{9} + \cdots - 45034730496 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - \nu - 50 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 37\!\cdots\!97 \nu^{14} + \cdots - 43\!\cdots\!72 ) / 90\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 51\!\cdots\!97 \nu^{14} + \cdots - 94\!\cdots\!12 ) / 90\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10\!\cdots\!99 \nu^{14} + \cdots - 61\!\cdots\!16 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 10\!\cdots\!99 \nu^{14} + \cdots + 63\!\cdots\!16 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 34\!\cdots\!33 \nu^{14} + \cdots - 13\!\cdots\!32 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 23\!\cdots\!67 \nu^{14} + \cdots + 27\!\cdots\!68 ) / 90\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 97\!\cdots\!23 \nu^{14} + \cdots + 26\!\cdots\!04 ) / 36\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 57\!\cdots\!37 \nu^{14} + \cdots + 21\!\cdots\!08 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 57\!\cdots\!89 \nu^{14} + \cdots + 13\!\cdots\!96 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 45\!\cdots\!13 \nu^{14} + \cdots - 64\!\cdots\!08 ) / 90\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 99\!\cdots\!81 \nu^{14} + \cdots + 12\!\cdots\!36 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 15\!\cdots\!97 \nu^{14} + \cdots - 36\!\cdots\!28 ) / 22\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + \beta _1 + 50 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{5} + \beta_{2} + 84\beta _1 + 49 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - \beta_{14} - 2 \beta_{13} - 2 \beta_{12} + \beta_{10} - 5 \beta_{8} - \beta_{7} - \beta_{6} + \cdots + 4220 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 15 \beta_{14} + 4 \beta_{13} + 10 \beta_{12} + 2 \beta_{11} - 3 \beta_{10} + 10 \beta_{9} - 3 \beta_{8} + \cdots + 7407 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 126 \beta_{14} - 390 \beta_{13} - 434 \beta_{12} + 16 \beta_{11} + 198 \beta_{10} + 14 \beta_{9} + \cdots + 408720 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2786 \beta_{14} + 632 \beta_{13} + 2058 \beta_{12} + 296 \beta_{11} - 456 \beta_{10} + 1960 \beta_{9} + \cdots + 909079 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 12699 \beta_{14} - 54514 \beta_{13} - 65564 \beta_{12} + 2284 \beta_{11} + 29861 \beta_{10} + \cdots + 41901122 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 383773 \beta_{14} + 79332 \beta_{13} + 309904 \beta_{12} + 31954 \beta_{11} - 50787 \beta_{10} + \cdots + 106641721 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1176916 \beta_{14} - 6785854 \beta_{13} - 8698192 \beta_{12} + 169300 \beta_{11} + 4042114 \beta_{10} + \cdots + 4441061942 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 47603288 \beta_{14} + 9654048 \beta_{13} + 41538948 \beta_{12} + 2864728 \beta_{11} - 4862420 \beta_{10} + \cdots + 12406704993 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 103772941 \beta_{14} - 801024930 \beta_{13} - 1085329974 \beta_{12} - 612640 \beta_{11} + \cdots + 481394100456 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 5636314563 \beta_{14} + 1193003588 \beta_{13} + 5253790318 \beta_{12} + 196330266 \beta_{11} + \cdots + 1444389141643 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 8774389810 \beta_{14} - 91932739830 \beta_{13} - 130967331406 \beta_{12} - 2864205664 \beta_{11} + \cdots + 53015883730140 \) 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
−10.6499
−9.28897
−8.07389
−6.08782
−5.24710
−3.60757
−3.03936
−0.712026
2.29863
3.10924
3.75815
8.32336
8.93277
10.4296
10.8549
−10.6499 −11.0256 81.4211 0 117.422 161.064 −526.332 −121.436 0
1.2 −9.28897 −25.4370 54.2850 0 236.284 −206.180 −207.004 404.041 0
1.3 −8.07389 18.0213 33.1877 0 −145.502 −142.473 −9.58900 81.7682 0
1.4 −6.08782 13.3684 5.06160 0 −81.3842 212.745 163.996 −64.2872 0
1.5 −5.24710 −2.93232 −4.46796 0 15.3862 −50.2912 191.351 −234.402 0
1.6 −3.60757 26.6100 −18.9854 0 −95.9974 −224.656 183.934 465.091 0
1.7 −3.03936 −24.6948 −22.7623 0 75.0563 −18.9561 166.442 366.834 0
1.8 −0.712026 −8.79140 −31.4930 0 6.25971 199.419 45.2087 −165.711 0
1.9 2.29863 −1.07628 −26.7163 0 −2.47397 17.5205 −134.967 −241.842 0
1.10 3.10924 28.9596 −22.3326 0 90.0423 107.101 −168.933 595.658 0
1.11 3.75815 4.76153 −17.8763 0 17.8945 −137.455 −187.443 −220.328 0
1.12 8.32336 −15.0453 37.2783 0 −125.228 −79.5600 43.9329 −16.6383 0
1.13 8.93277 10.3024 47.7943 0 92.0293 112.726 141.087 −136.860 0
1.14 10.4296 −30.2240 76.7771 0 −315.225 254.023 467.009 670.489 0
1.15 10.8549 26.2035 85.8289 0 284.436 −24.0281 584.307 443.624 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.15
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(5\) \( +1 \)
\(17\) \( -1 \)

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 425.6.a.k yes 15
5.b even 2 1 425.6.a.j ✓ 15
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
425.6.a.j ✓ 15 5.b even 2 1
425.6.a.k yes 15 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{15} - T_{2}^{14} - 378 T_{2}^{13} + 106 T_{2}^{12} + 55677 T_{2}^{11} + 23739 T_{2}^{10} + \cdots - 45034730496 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(425))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} + \cdots - 45034730496 \) Copy content Toggle raw display
$3$ \( T^{15} + \cdots - 20\!\cdots\!75 \) Copy content Toggle raw display
$5$ \( T^{15} \) Copy content Toggle raw display
$7$ \( T^{15} + \cdots - 60\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{15} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{15} + \cdots - 80\!\cdots\!09 \) Copy content Toggle raw display
$17$ \( (T - 289)^{15} \) Copy content Toggle raw display
$19$ \( T^{15} + \cdots - 25\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots - 19\!\cdots\!40 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots + 14\!\cdots\!85 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots + 42\!\cdots\!52 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots - 24\!\cdots\!60 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots + 60\!\cdots\!52 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots - 44\!\cdots\!85 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots - 26\!\cdots\!60 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots + 32\!\cdots\!99 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots - 47\!\cdots\!52 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots + 10\!\cdots\!65 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots - 33\!\cdots\!84 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots - 11\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
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