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Properties

Label 49.8.c.h
Level $49$
Weight $8$
Character orbit 49.c
Analytic conductor $15.307$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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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] = [49,8,Mod(18,49)] 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("49.18"); S:= CuspForms(chi, 8); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(49, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([4])) N = Newforms(chi, 8, names="a")
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 49.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,-30,0] 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: \(15.3068662487\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 1124 x^{14} - 4480 x^{13} + 503818 x^{12} + 3794560 x^{11} - 106136536 x^{10} + \cdots + 33\!\cdots\!76 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{12}\cdot 7^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{2} - 4 \beta_1 - 4) q^{2} + ( - \beta_{10} - \beta_{7}) q^{3} + (\beta_{11} - 5 \beta_{3} + \cdots + 59 \beta_1) q^{4} + (2 \beta_{14} + \beta_{10} - \beta_{5}) q^{5} + (\beta_{14} - 2 \beta_{9} + \cdots + \beta_{4}) q^{6}+ \cdots + (9305 \beta_{15} + 41070 \beta_{6} + \cdots + 4522440) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 30 q^{2} - 458 q^{4} + 8580 q^{8} - 7600 q^{9} - 17760 q^{11} + 58432 q^{15} - 33762 q^{16} - 244850 q^{18} + 861320 q^{22} - 96000 q^{23} - 468992 q^{25} + 472560 q^{29} + 193752 q^{30} - 1064370 q^{32}+ \cdots + 72185600 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 1124 x^{14} - 4480 x^{13} + 503818 x^{12} + 3794560 x^{11} - 106136536 x^{10} + \cdots + 33\!\cdots\!76 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 6085623659520 \nu^{15} - 51806943112998 \nu^{14} + \cdots + 23\!\cdots\!60 ) / 34\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 57\!\cdots\!83 \nu^{15} + \cdots + 25\!\cdots\!64 ) / 15\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 13\!\cdots\!19 \nu^{15} + \cdots - 37\!\cdots\!40 ) / 79\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 10\!\cdots\!35 \nu^{15} + \cdots - 61\!\cdots\!60 ) / 15\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 50\!\cdots\!33 \nu^{15} + \cdots - 13\!\cdots\!80 ) / 65\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 63\!\cdots\!59 \nu^{15} + \cdots - 17\!\cdots\!60 ) / 79\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 21\!\cdots\!87 \nu^{15} + \cdots - 49\!\cdots\!12 ) / 26\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 20\!\cdots\!33 \nu^{15} + \cdots - 85\!\cdots\!48 ) / 22\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 24\!\cdots\!52 \nu^{15} + \cdots - 62\!\cdots\!32 ) / 22\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 31\!\cdots\!63 \nu^{15} + \cdots - 13\!\cdots\!88 ) / 26\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 13\!\cdots\!42 \nu^{15} + \cdots - 54\!\cdots\!68 ) / 79\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 65\!\cdots\!89 \nu^{15} + \cdots - 19\!\cdots\!60 ) / 32\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 57\!\cdots\!83 \nu^{15} + \cdots + 24\!\cdots\!04 ) / 15\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 44\!\cdots\!71 \nu^{15} + \cdots - 15\!\cdots\!76 ) / 79\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 17\!\cdots\!29 \nu^{15} + \cdots - 49\!\cdots\!40 ) / 79\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 7\beta_{6} + 2\beta_{5} - 42\beta_{3} - 14 ) / 98 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 7 \beta_{15} - 14 \beta_{14} + 28 \beta_{10} + 14 \beta_{8} + 14 \beta_{6} + 2 \beta_{5} - 791 \beta_{3} + \cdots + 13664 ) / 98 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 49 \beta_{15} - 84 \beta_{14} - 4 \beta_{12} + 42 \beta_{11} - 126 \beta_{10} + 336 \beta_{8} + \cdots + 77728 ) / 98 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 1897 \beta_{15} - 5600 \beta_{14} + 84 \beta_{13} - 8 \beta_{12} + 168 \beta_{11} + 8568 \beta_{10} + \cdots + 3142692 ) / 98 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 20727 \beta_{15} - 78260 \beta_{14} + 980 \beta_{13} - 6392 \beta_{12} + 26180 \beta_{11} + \cdots + 37402148 ) / 98 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 505491 \beta_{15} - 2229080 \beta_{14} + 59010 \beta_{13} - 91096 \beta_{12} + 353220 \beta_{11} + \cdots + 876993012 ) / 98 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 7096033 \beta_{15} - 39794468 \beta_{14} + 904834 \beta_{13} - 5379080 \beta_{12} + 13991852 \beta_{11} + \cdots + 13532437492 ) / 98 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 138235769 \beta_{15} - 879288256 \beta_{14} + 31080896 \beta_{13} - 111024704 \beta_{12} + \cdots + 266721979636 ) / 98 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 2165520063 \beta_{15} - 16714147044 \beta_{14} + 565616016 \beta_{13} - 3672963872 \beta_{12} + \cdots + 4526844925204 ) / 98 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 37674725095 \beta_{15} - 336393660960 \beta_{14} + 14813751330 \beta_{13} - 83442469216 \beta_{12} + \cdots + 83823878881876 ) / 98 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 606843936517 \beta_{15} - 6427347808420 \beta_{14} + 291054712410 \beta_{13} - 2206489935784 \beta_{12} + \cdots + 14\!\cdots\!20 ) / 98 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 9844681874125 \beta_{15} - 124048367706816 \beta_{14} + 6654351538296 \beta_{13} + \cdots + 26\!\cdots\!36 ) / 98 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 154975675405523 \beta_{15} + \cdots + 47\!\cdots\!36 ) / 98 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 23\!\cdots\!31 \beta_{15} + \cdots + 83\!\cdots\!56 ) / 98 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 34\!\cdots\!73 \beta_{15} + \cdots + 14\!\cdots\!44 ) / 98 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/49\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1 - \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
18.1
2.78924 − 1.22474i
4.20346 + 1.22474i
−10.4785 + 1.22474i
−11.8927 − 1.22474i
−10.9886 − 1.22474i
−9.57442 + 1.22474i
17.2637 − 1.22474i
18.6779 + 1.22474i
2.78924 + 1.22474i
4.20346 − 1.22474i
−10.4785 − 1.22474i
−11.8927 + 1.22474i
−10.9886 + 1.22474i
−9.57442 − 1.22474i
17.2637 + 1.22474i
18.6779 − 1.22474i
−10.2199 − 17.7014i −30.4918 + 52.8133i −144.894 + 250.963i 133.959 + 232.023i 1246.49 0 3306.90 −765.996 − 1326.74i 2738.10 − 4742.52i
18.2 −10.2199 − 17.7014i 30.4918 − 52.8133i −144.894 + 250.963i −133.959 − 232.023i −1246.49 0 3306.90 −765.996 − 1326.74i −2738.10 + 4742.52i
18.3 −4.85070 − 8.40166i −45.3030 + 78.4672i 16.9415 − 29.3435i −115.940 − 200.814i 879.006 0 −1570.49 −3011.23 − 5215.61i −1124.78 + 1948.18i
18.4 −4.85070 − 8.40166i 45.3030 − 78.4672i 16.9415 − 29.3435i 115.940 + 200.814i −879.006 0 −1570.49 −3011.23 − 5215.61i 1124.78 − 1948.18i
18.5 0.00266545 + 0.00461669i −12.1595 + 21.0609i 64.0000 − 110.851i −249.565 − 432.259i −0.129642 0 1.36471 797.792 + 1381.82i 1.33040 − 2.30433i
18.6 0.00266545 + 0.00461669i 12.1595 − 21.0609i 64.0000 − 110.851i 249.565 + 432.259i 0.129642 0 1.36471 797.792 + 1381.82i −1.33040 + 2.30433i
18.7 7.56795 + 13.1081i −2.65175 + 4.59297i −50.5479 + 87.5515i 207.556 + 359.497i −80.2734 0 407.221 1079.44 + 1869.64i −3141.54 + 5441.31i
18.8 7.56795 + 13.1081i 2.65175 − 4.59297i −50.5479 + 87.5515i −207.556 − 359.497i 80.2734 0 407.221 1079.44 + 1869.64i 3141.54 − 5441.31i
30.1 −10.2199 + 17.7014i −30.4918 − 52.8133i −144.894 − 250.963i 133.959 − 232.023i 1246.49 0 3306.90 −765.996 + 1326.74i 2738.10 + 4742.52i
30.2 −10.2199 + 17.7014i 30.4918 + 52.8133i −144.894 − 250.963i −133.959 + 232.023i −1246.49 0 3306.90 −765.996 + 1326.74i −2738.10 − 4742.52i
30.3 −4.85070 + 8.40166i −45.3030 − 78.4672i 16.9415 + 29.3435i −115.940 + 200.814i 879.006 0 −1570.49 −3011.23 + 5215.61i −1124.78 − 1948.18i
30.4 −4.85070 + 8.40166i 45.3030 + 78.4672i 16.9415 + 29.3435i 115.940 − 200.814i −879.006 0 −1570.49 −3011.23 + 5215.61i 1124.78 + 1948.18i
30.5 0.00266545 − 0.00461669i −12.1595 − 21.0609i 64.0000 + 110.851i −249.565 + 432.259i −0.129642 0 1.36471 797.792 − 1381.82i 1.33040 + 2.30433i
30.6 0.00266545 − 0.00461669i 12.1595 + 21.0609i 64.0000 + 110.851i 249.565 − 432.259i 0.129642 0 1.36471 797.792 − 1381.82i −1.33040 − 2.30433i
30.7 7.56795 − 13.1081i −2.65175 − 4.59297i −50.5479 − 87.5515i 207.556 − 359.497i −80.2734 0 407.221 1079.44 − 1869.64i −3141.54 − 5441.31i
30.8 7.56795 − 13.1081i 2.65175 + 4.59297i −50.5479 − 87.5515i −207.556 + 359.497i 80.2734 0 407.221 1079.44 − 1869.64i 3141.54 + 5441.31i
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 18.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
7.c even 3 1 inner
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.8.c.h 16
7.b odd 2 1 inner 49.8.c.h 16
7.c even 3 1 49.8.a.g ✓ 8
7.c even 3 1 inner 49.8.c.h 16
7.d odd 6 1 49.8.a.g ✓ 8
7.d odd 6 1 inner 49.8.c.h 16
21.g even 6 1 441.8.a.ba 8
21.h odd 6 1 441.8.a.ba 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
49.8.a.g ✓ 8 7.c even 3 1
49.8.a.g ✓ 8 7.d odd 6 1
49.8.c.h 16 1.a even 1 1 trivial
49.8.c.h 16 7.b odd 2 1 inner
49.8.c.h 16 7.c even 3 1 inner
49.8.c.h 16 7.d odd 6 1 inner
441.8.a.ba 8 21.g even 6 1
441.8.a.ba 8 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{8}^{\mathrm{new}}(49, [\chi])\):

\( T_{2}^{8} + 15T_{2}^{7} + 483T_{2}^{6} + 2130T_{2}^{5} + 111548T_{2}^{4} + 773520T_{2}^{3} + 9004128T_{2}^{2} - 48000T_{2} + 256 \) Copy content Toggle raw display
\( T_{3}^{16} + 12548 T_{3}^{14} + 119514540 T_{3}^{12} + 437815743248 T_{3}^{10} + \cdots + 25\!\cdots\!96 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} + 15 T^{7} + \cdots + 256)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 25\!\cdots\!96 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 43\!\cdots\!96)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 79\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 35\!\cdots\!36 \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 52\!\cdots\!56)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 28\!\cdots\!00)^{4} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 25\!\cdots\!16 \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 13\!\cdots\!16)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 22\!\cdots\!56)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 96\!\cdots\!00)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 53\!\cdots\!96)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 20\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 30\!\cdots\!36 \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 93\!\cdots\!56)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 32\!\cdots\!44)^{4} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 40\!\cdots\!76 \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 22\!\cdots\!76)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 12\!\cdots\!96)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 19\!\cdots\!56 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 23\!\cdots\!96)^{2} \) Copy content Toggle raw display
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