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Properties

Label 676.6.a.j
Level $676$
Weight $6$
Character orbit 676.a
Self dual yes
Analytic conductor $108.419$
Analytic rank $0$
Dimension $15$
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] = [676,6,Mod(1,676)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(676, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("676.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 676 = 2^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 676.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: \(108.419462194\)
Analytic rank: \(0\)
Dimension: \(15\)
Coefficient field: \(\mathbb{Q}[x]/(x^{15} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{15} - 2 x^{14} - 2256 x^{13} + 4815 x^{12} + 1898132 x^{11} - 3668041 x^{10} - 739402084 x^{9} + \cdots - 69\!\cdots\!63 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{12}\cdot 13^{7} \)
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 \(1,\beta_1,\ldots,\beta_{14}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 1) q^{3} + (\beta_{3} + \beta_{2} + 9) q^{5} + (\beta_{5} - 2 \beta_{4} + \beta_{2} + \cdots - 1) q^{7}+ \cdots + (\beta_{6} + 3 \beta_{4} - 2 \beta_{2} + \cdots + 64) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 1) q^{3} + (\beta_{3} + \beta_{2} + 9) q^{5} + (\beta_{5} - 2 \beta_{4} + \beta_{2} + \cdots - 1) q^{7}+ \cdots + ( - 12 \beta_{14} - 125 \beta_{13} + \cdots + 33663) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 15 q - 12 q^{3} + 132 q^{5} - 21 q^{7} + 983 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 15 q - 12 q^{3} + 132 q^{5} - 21 q^{7} + 983 q^{9} + 418 q^{11} + 1640 q^{15} + 1212 q^{17} + 1648 q^{19} + 3482 q^{21} + 1231 q^{23} + 6203 q^{25} - 4053 q^{27} - 2947 q^{29} - 3391 q^{31} + 13754 q^{33} + 14580 q^{35} + 7744 q^{37} + 24953 q^{41} - 26615 q^{43} - 12795 q^{45} - 50336 q^{47} + 43666 q^{49} + 89785 q^{51} - 35821 q^{53} + 33824 q^{55} + 34595 q^{57} + 84301 q^{59} - 44040 q^{61} - 41610 q^{63} + 22131 q^{67} - 109390 q^{69} + 43731 q^{71} + 111093 q^{73} + 63795 q^{75} + 22944 q^{77} - 210043 q^{79} + 216995 q^{81} + 197863 q^{83} + 221031 q^{85} + 90262 q^{87} + 343797 q^{89} + 447674 q^{93} - 421577 q^{95} + 369183 q^{97} + 509249 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{15} - 2 x^{14} - 2256 x^{13} + 4815 x^{12} + 1898132 x^{11} - 3668041 x^{10} - 739402084 x^{9} + \cdots - 69\!\cdots\!63 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 63\!\cdots\!32 \nu^{14} + \cdots + 86\!\cdots\!39 ) / 16\!\cdots\!61 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 31\!\cdots\!16 \nu^{14} + \cdots - 46\!\cdots\!62 ) / 12\!\cdots\!97 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 12\!\cdots\!93 \nu^{14} + \cdots - 24\!\cdots\!91 ) / 79\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 20\!\cdots\!27 \nu^{14} + \cdots - 17\!\cdots\!22 ) / 12\!\cdots\!97 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 20\!\cdots\!03 \nu^{14} + \cdots - 87\!\cdots\!66 ) / 11\!\cdots\!93 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 67\!\cdots\!39 \nu^{14} + \cdots + 45\!\cdots\!96 ) / 23\!\cdots\!23 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 50\!\cdots\!61 \nu^{14} + \cdots + 92\!\cdots\!66 ) / 11\!\cdots\!93 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 96\!\cdots\!57 \nu^{14} + \cdots - 27\!\cdots\!37 ) / 16\!\cdots\!61 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 10\!\cdots\!01 \nu^{14} + \cdots - 90\!\cdots\!86 ) / 79\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 17\!\cdots\!78 \nu^{14} + \cdots - 73\!\cdots\!41 ) / 79\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 17\!\cdots\!51 \nu^{14} + \cdots + 12\!\cdots\!49 ) / 79\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 17\!\cdots\!78 \nu^{14} + \cdots + 44\!\cdots\!70 ) / 79\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 20\!\cdots\!84 \nu^{14} + \cdots - 16\!\cdots\!27 ) / 79\!\cdots\!51 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 24\!\cdots\!20 \nu^{14} + \cdots - 41\!\cdots\!29 ) / 79\!\cdots\!51 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{2} + 13\beta _1 + 5 ) / 13 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -4\beta_{8} + 13\beta_{6} + 43\beta_{4} - 30\beta_{2} - 7\beta _1 + 3887 ) / 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 32 \beta_{14} - 6 \beta_{13} + 54 \beta_{12} - 20 \beta_{10} + 31 \beta_{9} + 64 \beta_{8} + \cdots + 1161 ) / 13 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 342 \beta_{14} - 906 \beta_{13} + 107 \beta_{12} - 13 \beta_{11} - 251 \beta_{10} + 612 \beta_{9} + \cdots + 2204619 ) / 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 31202 \beta_{14} - 8803 \beta_{13} + 40721 \beta_{12} + 3276 \beta_{11} - 18926 \beta_{10} + \cdots - 2181728 ) / 13 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 394061 \beta_{14} - 1275893 \beta_{13} + 80100 \beta_{12} + 69537 \beta_{11} - 342061 \beta_{10} + \cdots + 1437329434 ) / 13 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 26574279 \beta_{14} - 8166552 \beta_{13} + 27820751 \beta_{12} + 2829463 \beta_{11} - 16645300 \beta_{10} + \cdots - 2941239450 ) / 13 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 374390838 \beta_{14} - 1317517354 \beta_{13} + 21388756 \beta_{12} + 154686064 \beta_{11} + \cdots + 984348189588 ) / 13 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 21744877018 \beta_{14} - 6257525151 \beta_{13} + 19396451748 \beta_{12} + 1602921762 \beta_{11} + \cdots - 2884821977728 ) / 13 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 330971976348 \beta_{14} - 1207731839106 \beta_{13} - 26312160641 \beta_{12} + 214561411012 \beta_{11} + \cdots + 693291462208800 ) / 13 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 17443254646013 \beta_{14} - 4198883027132 \beta_{13} + 14019571278163 \beta_{12} + \cdots - 25\!\cdots\!21 ) / 13 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 281889738437978 \beta_{14} + \cdots + 49\!\cdots\!15 ) / 13 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 13\!\cdots\!22 \beta_{14} + \cdots - 22\!\cdots\!80 ) / 13 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 23\!\cdots\!93 \beta_{14} + \cdots + 36\!\cdots\!32 ) / 13 \) 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
27.4961
24.4334
22.2539
13.3137
9.19169
13.4202
2.04877
−7.55343
−5.21297
−2.53104
−8.25269
−10.3642
−25.8734
−22.2239
−28.1461
0 −28.3862 0 −23.0985 0 −256.288 0 562.776 0
1.2 0 −28.0373 0 −46.8617 0 29.1552 0 543.088 0
1.3 0 −19.7600 0 −20.8172 0 −36.7614 0 147.456 0
1.4 0 −14.2038 0 3.76406 0 171.012 0 −41.2525 0
1.5 0 −12.7956 0 76.6891 0 185.570 0 −79.2735 0
1.6 0 −10.9262 0 108.780 0 188.253 0 −123.618 0
1.7 0 −5.65264 0 30.6741 0 −192.577 0 −211.048 0
1.8 0 3.94955 0 −84.5341 0 51.3164 0 −227.401 0
1.9 0 4.32288 0 97.4703 0 −104.767 0 −224.313 0
1.10 0 5.02500 0 −5.20100 0 −197.895 0 −217.749 0
1.11 0 7.36260 0 −61.8265 0 −70.8871 0 −188.792 0
1.12 0 12.8581 0 −58.0981 0 139.619 0 −77.6683 0
1.13 0 22.2696 0 26.6718 0 17.0181 0 252.933 0
1.14 0 24.7179 0 81.3204 0 −50.9986 0 367.973 0
1.15 0 27.2560 0 7.06768 0 107.231 0 499.890 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
\(2\) \(-1\)
\(13\) \(1\)

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 676.6.a.j yes 15
13.b even 2 1 676.6.a.i ✓ 15
13.d odd 4 2 676.6.d.g 30
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
676.6.a.i ✓ 15 13.b even 2 1
676.6.a.j yes 15 1.a even 1 1 trivial
676.6.d.g 30 13.d odd 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(676))\):

\( T_{3}^{15} + 12 T_{3}^{14} - 2242 T_{3}^{13} - 24185 T_{3}^{12} + 1874826 T_{3}^{11} + \cdots + 21\!\cdots\!87 \) Copy content Toggle raw display
\( T_{5}^{15} - 132 T_{5}^{14} - 17827 T_{5}^{13} + 2423377 T_{5}^{12} + 127441867 T_{5}^{11} + \cdots + 51\!\cdots\!08 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{15} \) Copy content Toggle raw display
$3$ \( T^{15} + \cdots + 21\!\cdots\!87 \) Copy content Toggle raw display
$5$ \( T^{15} + \cdots + 51\!\cdots\!08 \) Copy content Toggle raw display
$7$ \( T^{15} + \cdots + 30\!\cdots\!76 \) Copy content Toggle raw display
$11$ \( T^{15} + \cdots + 30\!\cdots\!93 \) Copy content Toggle raw display
$13$ \( T^{15} \) Copy content Toggle raw display
$17$ \( T^{15} + \cdots + 11\!\cdots\!89 \) Copy content Toggle raw display
$19$ \( T^{15} + \cdots - 66\!\cdots\!73 \) Copy content Toggle raw display
$23$ \( T^{15} + \cdots + 61\!\cdots\!12 \) Copy content Toggle raw display
$29$ \( T^{15} + \cdots - 13\!\cdots\!52 \) Copy content Toggle raw display
$31$ \( T^{15} + \cdots + 47\!\cdots\!28 \) Copy content Toggle raw display
$37$ \( T^{15} + \cdots - 13\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( T^{15} + \cdots + 25\!\cdots\!01 \) Copy content Toggle raw display
$43$ \( T^{15} + \cdots - 71\!\cdots\!67 \) Copy content Toggle raw display
$47$ \( T^{15} + \cdots + 83\!\cdots\!32 \) Copy content Toggle raw display
$53$ \( T^{15} + \cdots + 97\!\cdots\!88 \) Copy content Toggle raw display
$59$ \( T^{15} + \cdots + 18\!\cdots\!13 \) Copy content Toggle raw display
$61$ \( T^{15} + \cdots - 49\!\cdots\!92 \) Copy content Toggle raw display
$67$ \( T^{15} + \cdots - 77\!\cdots\!79 \) Copy content Toggle raw display
$71$ \( T^{15} + \cdots - 15\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{15} + \cdots + 89\!\cdots\!99 \) Copy content Toggle raw display
$79$ \( T^{15} + \cdots - 55\!\cdots\!88 \) Copy content Toggle raw display
$83$ \( T^{15} + \cdots - 23\!\cdots\!29 \) Copy content Toggle raw display
$89$ \( T^{15} + \cdots - 13\!\cdots\!41 \) Copy content Toggle raw display
$97$ \( T^{15} + \cdots - 11\!\cdots\!23 \) Copy content Toggle raw display
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