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Properties

Label 49.14.c.g
Level $49$
Weight $14$
Character orbit 49.c
Analytic conductor $52.543$
Analytic rank $0$
Dimension $16$
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] = [49,14,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, 14); 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, 14, names="a")
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 49.c (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,-2,-728] 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: \(52.5431551864\)
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: \( x^{16} - 6 x^{15} - 24508 x^{14} + 79772 x^{13} + 237177405 x^{12} + 118141132 x^{11} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{34}\cdot 3^{8}\cdot 7^{12} \)
Twist minimal: no (minimal twist has level 7)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + \beta_{2}) q^{2} + ( - \beta_{6} + \beta_{4} + 91 \beta_1 - 91) q^{3} + (\beta_{8} - \beta_{6} - \beta_{5} + \cdots - 4075) q^{4} + ( - \beta_{8} + \beta_{7} + \cdots - 5864 \beta_1) q^{5}+ \cdots + ( - 3888775 \beta_{13} + \cdots + 116618821539) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 2 q^{2} - 728 q^{3} - 32588 q^{4} - 46928 q^{5} + 30212 q^{6} - 1093248 q^{8} - 2910808 q^{9} - 879018 q^{10} + 3746000 q^{11} - 13786276 q^{12} + 63556528 q^{13} + 179729504 q^{15} - 158020976 q^{16}+ \cdots + 1803523183088 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 6 x^{15} - 24508 x^{14} + 79772 x^{13} + 237177405 x^{12} + 118141132 x^{11} + \cdots + 13\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 19\!\cdots\!79 \nu^{15} + \cdots + 55\!\cdots\!30 ) / 92\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 19\!\cdots\!79 \nu^{15} + \cdots - 55\!\cdots\!30 ) / 46\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 35\!\cdots\!87 \nu^{15} + \cdots + 12\!\cdots\!70 ) / 23\!\cdots\!90 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 40\!\cdots\!48 \nu^{15} + \cdots - 97\!\cdots\!60 ) / 15\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 59\!\cdots\!44 \nu^{15} + \cdots + 78\!\cdots\!80 ) / 15\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 78\!\cdots\!41 \nu^{15} + \cdots + 15\!\cdots\!00 ) / 46\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 49\!\cdots\!87 \nu^{15} + \cdots - 92\!\cdots\!40 ) / 22\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 43\!\cdots\!11 \nu^{15} + \cdots - 11\!\cdots\!60 ) / 46\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 15\!\cdots\!76 \nu^{15} + \cdots - 33\!\cdots\!00 ) / 11\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 14\!\cdots\!97 \nu^{15} + \cdots - 35\!\cdots\!80 ) / 74\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 34\!\cdots\!92 \nu^{15} + \cdots + 50\!\cdots\!00 ) / 78\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 26\!\cdots\!01 \nu^{15} + \cdots + 73\!\cdots\!00 ) / 11\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 16\!\cdots\!52 \nu^{15} + \cdots + 48\!\cdots\!00 ) / 33\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 25\!\cdots\!13 \nu^{15} + \cdots - 12\!\cdots\!20 ) / 23\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 39\!\cdots\!87 \nu^{15} + \cdots + 24\!\cdots\!20 ) / 33\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + 2\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - \beta_{4} - 4\beta_{3} + 10\beta_{2} + 4\beta _1 + 12263 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{11} + 3 \beta_{9} + 6 \beta_{8} - 6 \beta_{6} - 22 \beta_{5} - 40 \beta_{4} - 48 \beta_{3} + \cdots + 69403 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 4 \beta_{14} + 8 \beta_{13} + 12 \beta_{12} - 9 \beta_{11} + 708 \beta_{10} - 27 \beta_{9} + \cdots + 126894403 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 40 \beta_{15} + 55 \beta_{14} + 552 \beta_{13} - 105 \beta_{12} + 8225 \beta_{11} - 5076 \beta_{10} + \cdots - 11098762 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 3432 \beta_{15} - 49545 \beta_{14} + 63752 \beta_{13} + 191079 \beta_{12} - 119907 \beta_{11} + \cdots + 778854171150 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 229348 \beta_{15} + 506667 \beta_{14} + 2738356 \beta_{13} - 76419 \beta_{12} + 29143236 \beta_{11} + \cdots - 2480325262183 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 22848264 \beta_{15} - 235519413 \beta_{14} + 223601196 \beta_{13} + 1043425851 \beta_{12} + \cdots + 25\!\cdots\!07 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 4238713944 \beta_{15} + 12355365726 \beta_{14} + 42325435200 \beta_{13} + 5809844610 \beta_{12} + \cdots - 67\!\cdots\!59 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 445476309816 \beta_{15} - 4097440325140 \beta_{14} + 3061919616320 \beta_{13} + 19643195719992 \beta_{12} + \cdots + 34\!\cdots\!65 ) / 8 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 36247845539752 \beta_{15} + 127599974237857 \beta_{14} + 301798420149360 \beta_{13} + \cdots - 67\!\cdots\!94 ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 38\!\cdots\!76 \beta_{15} + \cdots + 23\!\cdots\!88 ) / 8 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 14\!\cdots\!84 \beta_{15} + \cdots - 29\!\cdots\!78 ) / 4 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 15\!\cdots\!32 \beta_{15} + \cdots + 81\!\cdots\!39 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 23\!\cdots\!96 \beta_{15} + \cdots - 49\!\cdots\!61 ) / 8 \) Copy content Toggle raw display

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

Character values

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

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

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
−84.9382 + 0.866025i
−47.5374 + 0.866025i
−41.1086 + 0.866025i
−23.8715 + 0.866025i
14.1123 + 0.866025i
37.1577 + 0.866025i
67.8886 + 0.866025i
81.2971 + 0.866025i
−84.9382 − 0.866025i
−47.5374 − 0.866025i
−41.1086 − 0.866025i
−23.8715 − 0.866025i
14.1123 − 0.866025i
37.1577 − 0.866025i
67.8886 − 0.866025i
81.2971 − 0.866025i
−85.4382 − 147.983i −275.475 + 477.137i −10503.4 + 18192.4i −18457.8 − 31969.8i 94144.4 0 2.18974e6 645388. + 1.11785e6i −3.15400e6 + 5.46289e6i
18.2 −48.0374 − 83.2032i −45.1151 + 78.1417i −519.184 + 899.253i 28325.9 + 49061.9i 8668.85 0 −687284. 793091. + 1.37367e6i 2.72141e6 − 4.71362e6i
18.3 −41.6086 − 72.0682i 951.376 − 1647.83i 633.448 − 1097.16i −8420.78 − 14585.2i −158342. 0 −787143. −1.01307e6 − 1.75469e6i −700754. + 1.21374e6i
18.4 −24.3715 − 42.2127i −1099.13 + 1903.74i 2908.06 − 5036.91i −12992.6 − 22503.8i 107149. 0 −682798. −1.61900e6 − 2.80418e6i −633297. + 1.09690e6i
18.5 13.6123 + 23.5772i 273.794 − 474.225i 3725.41 − 6452.60i −16862.5 − 29206.7i 14907.8 0 425869. 647235. + 1.12104e6i 459074. − 795140.i
18.6 36.6577 + 63.4930i −306.948 + 531.649i 1408.43 − 2439.47i 16706.4 + 28936.3i −45008.0 0 807118. 608728. + 1.05435e6i −1.22484e6 + 2.12148e6i
18.7 67.3886 + 116.721i 948.121 − 1642.19i −4986.46 + 8636.79i 7164.19 + 12408.7i 255570. 0 −240026. −1.00071e6 − 1.73328e6i −965570. + 1.67242e6i
18.8 80.7971 + 139.945i −810.627 + 1404.05i −8960.34 + 15519.8i −18926.8 − 32782.3i −261985. 0 −1.57210e6 −517071. − 895593.i 3.05847e6 − 5.29742e6i
30.1 −85.4382 + 147.983i −275.475 − 477.137i −10503.4 − 18192.4i −18457.8 + 31969.8i 94144.4 0 2.18974e6 645388. − 1.11785e6i −3.15400e6 − 5.46289e6i
30.2 −48.0374 + 83.2032i −45.1151 − 78.1417i −519.184 − 899.253i 28325.9 − 49061.9i 8668.85 0 −687284. 793091. − 1.37367e6i 2.72141e6 + 4.71362e6i
30.3 −41.6086 + 72.0682i 951.376 + 1647.83i 633.448 + 1097.16i −8420.78 + 14585.2i −158342. 0 −787143. −1.01307e6 + 1.75469e6i −700754. − 1.21374e6i
30.4 −24.3715 + 42.2127i −1099.13 − 1903.74i 2908.06 + 5036.91i −12992.6 + 22503.8i 107149. 0 −682798. −1.61900e6 + 2.80418e6i −633297. − 1.09690e6i
30.5 13.6123 − 23.5772i 273.794 + 474.225i 3725.41 + 6452.60i −16862.5 + 29206.7i 14907.8 0 425869. 647235. − 1.12104e6i 459074. + 795140.i
30.6 36.6577 − 63.4930i −306.948 − 531.649i 1408.43 + 2439.47i 16706.4 − 28936.3i −45008.0 0 807118. 608728. − 1.05435e6i −1.22484e6 − 2.12148e6i
30.7 67.3886 − 116.721i 948.121 + 1642.19i −4986.46 − 8636.79i 7164.19 − 12408.7i 255570. 0 −240026. −1.00071e6 + 1.73328e6i −965570. − 1.67242e6i
30.8 80.7971 − 139.945i −810.627 − 1404.05i −8960.34 − 15519.8i −18926.8 + 32782.3i −261985. 0 −1.57210e6 −517071. + 895593.i 3.05847e6 + 5.29742e6i
\(n\): 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.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 49.14.c.g 16
7.b odd 2 1 7.14.c.a ✓ 16
7.c even 3 1 49.14.a.f 8
7.c even 3 1 inner 49.14.c.g 16
7.d odd 6 1 7.14.c.a ✓ 16
7.d odd 6 1 49.14.a.e 8
21.c even 2 1 63.14.e.c 16
21.g even 6 1 63.14.e.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.14.c.a ✓ 16 7.b odd 2 1
7.14.c.a ✓ 16 7.d odd 6 1
49.14.a.e 8 7.d odd 6 1
49.14.a.f 8 7.c even 3 1
49.14.c.g 16 1.a even 1 1 trivial
49.14.c.g 16 7.c even 3 1 inner
63.14.e.c 16 21.c even 2 1
63.14.e.c 16 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{14}^{\mathrm{new}}(49, [\chi])\):

\( T_{2}^{16} + 2 T_{2}^{15} + 49064 T_{2}^{14} + 440696 T_{2}^{13} + 1712484560 T_{2}^{12} + \cdots + 83\!\cdots\!36 \) Copy content Toggle raw display
\( T_{3}^{16} + 728 T_{3}^{15} + 8097688 T_{3}^{14} + 4825207296 T_{3}^{13} + 46232939954934 T_{3}^{12} + \cdots + 46\!\cdots\!21 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 83\!\cdots\!36 \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 46\!\cdots\!21 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 31\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 43\!\cdots\!25 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots - 26\!\cdots\!64)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 36\!\cdots\!61 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 14\!\cdots\!29 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 16\!\cdots\!61 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots - 85\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 15\!\cdots\!41 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 53\!\cdots\!25 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 45\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots - 51\!\cdots\!92)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 99\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 68\!\cdots\!69 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 20\!\cdots\!25 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 49\!\cdots\!09 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 49\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 83\!\cdots\!76)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 11\!\cdots\!41 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 26\!\cdots\!21 \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots - 22\!\cdots\!24)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 37\!\cdots\!09 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 40\!\cdots\!36)^{2} \) Copy content Toggle raw display
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