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Properties

Label 49.13.d.b
Level $49$
Weight $13$
Character orbit 49.d
Analytic conductor $44.786$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [49,13,Mod(19,49)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(49, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("49.19");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 49.d (of order \(6\), degree \(2\), not minimal)

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: no
Analytic conductor: \(44.7856970465\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} + 86506 x^{10} - 9621431 x^{9} + 6476445009 x^{8} - 430204837720 x^{7} + \cdots + 47\!\cdots\!56 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{24}\cdot 3^{5}\cdot 7^{6} \)
Twist minimal: no (minimal twist has level 7)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{4} q^{2} + ( - \beta_{7} + \beta_{5}) q^{3} + (\beta_{6} + 25 \beta_{2} + 2715 \beta_1) q^{4} + (\beta_{9} + 8 \beta_{7}) q^{5} + ( - \beta_{10} + \beta_{8} + 26 \beta_{5}) q^{6} + ( - 2024 \beta_{4} + 2024 \beta_{2} - 172800) q^{8} + (3150 \beta_{4} - 24 \beta_{3} + \cdots + 280119) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{4} q^{2} + ( - \beta_{7} + \beta_{5}) q^{3} + (\beta_{6} + 25 \beta_{2} + 2715 \beta_1) q^{4} + (\beta_{9} + 8 \beta_{7}) q^{5} + ( - \beta_{10} + \beta_{8} + 26 \beta_{5}) q^{6} + ( - 2024 \beta_{4} + 2024 \beta_{2} - 172800) q^{8} + (3150 \beta_{4} - 24 \beta_{3} + \cdots + 280119) q^{9}+ \cdots + (80026815 \beta_{6} + \cdots + 10902022989) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 16288 q^{4} - 2073600 q^{8} + 1680762 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 16288 q^{4} - 2073600 q^{8} + 1680762 q^{9} - 1704780 q^{11} - 81053760 q^{15} - 15993088 q^{16} - 128356800 q^{18} - 166355200 q^{22} + 266735700 q^{23} + 1115987130 q^{25} + 4180370904 q^{29} - 3469961280 q^{30} + 666316800 q^{32} - 10221951168 q^{36} + 4824866900 q^{37} + 9660671328 q^{39} - 57448378600 q^{43} - 40254705216 q^{44} + 56843757056 q^{46} + 61179373440 q^{50} - 74552249088 q^{51} + 104521857300 q^{53} + 13890686400 q^{57} - 169979689600 q^{58} + 621243786240 q^{60} - 809060642816 q^{64} - 420221360160 q^{65} + 129746603700 q^{67} - 30035209320 q^{71} - 550229836800 q^{72} + 538689695616 q^{74} - 673549699200 q^{78} + 172527631668 q^{79} + 1648796648346 q^{81} + 125273456640 q^{85} - 1808205928704 q^{86} + 462937446400 q^{88} + 4026183580800 q^{92} + 2143380662400 q^{93} - 3657881967840 q^{95} + 131144383128 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 86506 x^{10} - 9621431 x^{9} + 6476445009 x^{8} - 430204837720 x^{7} + \cdots + 47\!\cdots\!56 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 92\!\cdots\!72 \nu^{11} + \cdots - 26\!\cdots\!04 ) / 23\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 26\!\cdots\!03 \nu^{11} + \cdots - 37\!\cdots\!24 ) / 15\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 40\!\cdots\!38 \nu^{11} + \cdots - 19\!\cdots\!84 ) / 25\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 13\!\cdots\!85 \nu^{11} + \cdots - 35\!\cdots\!36 ) / 76\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 24\!\cdots\!56 \nu^{11} + \cdots - 15\!\cdots\!48 ) / 24\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 29\!\cdots\!27 \nu^{11} + \cdots - 42\!\cdots\!64 ) / 25\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 94\!\cdots\!31 \nu^{11} + \cdots - 71\!\cdots\!28 ) / 50\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 34\!\cdots\!04 \nu^{11} + \cdots - 25\!\cdots\!56 ) / 22\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 13\!\cdots\!93 \nu^{11} + \cdots - 10\!\cdots\!64 ) / 45\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 75\!\cdots\!08 \nu^{11} + \cdots - 38\!\cdots\!00 ) / 22\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 16\!\cdots\!53 \nu^{11} + \cdots + 50\!\cdots\!64 ) / 45\!\cdots\!20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - \beta_{11} - \beta_{10} - 8 \beta_{9} - 8 \beta_{8} + 105 \beta_{7} - 18 \beta_{6} + \cdots - 342 \beta_1 ) / 2016 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 43 \beta_{11} - 86 \beta_{10} - 160 \beta_{9} + 320 \beta_{8} + 5285 \beta_{7} - 10570 \beta_{5} + \cdots - 19375950 ) / 672 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 502 \beta_{11} - 251 \beta_{10} + 75296 \beta_{9} - 37648 \beta_{8} - 1058358 \beta_{7} + \cdots + 343199124 ) / 144 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2463361 \beta_{11} + 2463361 \beta_{10} - 27700864 \beta_{9} - 27700864 \beta_{8} + \cdots + 1220807506662 \beta_1 ) / 672 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 925943099 \beta_{11} - 1851886198 \beta_{10} - 39749193536 \beta_{9} + 79498387072 \beta_{8} + \cdots - 673749481980438 ) / 2016 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 11979646028 \beta_{11} + 5989823014 \beta_{10} + 217219442048 \beta_{9} - 108609721024 \beta_{8} + \cdots + 33\!\cdots\!75 ) / 24 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 115241375656763 \beta_{11} + 115241375656763 \beta_{10} + \cdots + 70\!\cdots\!86 \beta_1 ) / 2016 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 13\!\cdots\!95 \beta_{11} + \cdots - 80\!\cdots\!82 ) / 672 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 16\!\cdots\!46 \beta_{11} + \cdots + 47\!\cdots\!92 ) / 144 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 11\!\cdots\!41 \beta_{11} + \cdots + 71\!\cdots\!30 \beta_1 ) / 672 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 10\!\cdots\!43 \beta_{11} + \cdots - 62\!\cdots\!46 ) / 2016 \) 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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
19.1
−35.1641 − 60.9061i
87.9185 + 152.279i
100.367 + 173.842i
−150.825 − 261.237i
26.1484 + 45.2904i
−27.9449 − 48.4020i
−35.1641 + 60.9061i
87.9185 − 152.279i
100.367 − 173.842i
−150.825 + 261.237i
26.1484 − 45.2904i
−27.9449 + 48.4020i
−45.6118 + 79.0020i −570.009 + 329.095i −2112.88 − 3659.61i 17571.1 + 10144.7i − 60042.4i 0 11836.7 −49113.9 + 85067.8i −1.60290e6 + 925437.i
19.2 −45.6118 + 79.0020i 570.009 − 329.095i −2112.88 − 3659.61i −17571.1 − 10144.7i 60042.4i 0 11836.7 −49113.9 + 85067.8i 1.60290e6 − 925437.i
19.3 −8.71664 + 15.0977i −822.658 + 474.962i 1896.04 + 3284.04i −15924.7 − 9194.15i − 16560.3i 0 −137515. 185457. − 321221.i 277620. − 160284.i
19.4 −8.71664 + 15.0977i 822.658 − 474.962i 1896.04 + 3284.04i 15924.7 + 9194.15i 16560.3i 0 −137515. 185457. − 321221.i −277620. + 160284.i
19.5 54.3285 − 94.0997i −907.938 + 524.198i −3855.16 − 6677.34i 20136.3 + 11625.7i 113916.i 0 −392722. 283847. − 491638.i 2.18794e6 − 1.26321e6i
19.6 54.3285 − 94.0997i 907.938 − 524.198i −3855.16 − 6677.34i −20136.3 − 11625.7i − 113916.i 0 −392722. 283847. − 491638.i −2.18794e6 + 1.26321e6i
31.1 −45.6118 − 79.0020i −570.009 − 329.095i −2112.88 + 3659.61i 17571.1 − 10144.7i 60042.4i 0 11836.7 −49113.9 − 85067.8i −1.60290e6 − 925437.i
31.2 −45.6118 − 79.0020i 570.009 + 329.095i −2112.88 + 3659.61i −17571.1 + 10144.7i − 60042.4i 0 11836.7 −49113.9 − 85067.8i 1.60290e6 + 925437.i
31.3 −8.71664 − 15.0977i −822.658 − 474.962i 1896.04 − 3284.04i −15924.7 + 9194.15i 16560.3i 0 −137515. 185457. + 321221.i 277620. + 160284.i
31.4 −8.71664 − 15.0977i 822.658 + 474.962i 1896.04 − 3284.04i 15924.7 − 9194.15i − 16560.3i 0 −137515. 185457. + 321221.i −277620. − 160284.i
31.5 54.3285 + 94.0997i −907.938 − 524.198i −3855.16 + 6677.34i 20136.3 − 11625.7i − 113916.i 0 −392722. 283847. + 491638.i 2.18794e6 + 1.26321e6i
31.6 54.3285 + 94.0997i 907.938 + 524.198i −3855.16 + 6677.34i −20136.3 + 11625.7i 113916.i 0 −392722. 283847. + 491638.i −2.18794e6 − 1.26321e6i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 19.6
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 49.13.d.b 12
7.b odd 2 1 inner 49.13.d.b 12
7.c even 3 1 7.13.b.b ✓ 6
7.c even 3 1 inner 49.13.d.b 12
7.d odd 6 1 7.13.b.b ✓ 6
7.d odd 6 1 inner 49.13.d.b 12
21.g even 6 1 63.13.d.d 6
21.h odd 6 1 63.13.d.d 6
28.f even 6 1 112.13.c.b 6
28.g odd 6 1 112.13.c.b 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
7.13.b.b ✓ 6 7.c even 3 1
7.13.b.b ✓ 6 7.d odd 6 1
49.13.d.b 12 1.a even 1 1 trivial
49.13.d.b 12 7.b odd 2 1 inner
49.13.d.b 12 7.c even 3 1 inner
49.13.d.b 12 7.d odd 6 1 inner
63.13.d.d 6 21.g even 6 1
63.13.d.d 6 21.h odd 6 1
112.13.c.b 6 28.f even 6 1
112.13.c.b 6 28.g odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 10216T_{2}^{4} + 345600T_{2}^{3} + 104366656T_{2}^{2} + 1765324800T_{2} + 29859840000 \) acting on \(S_{13}^{\mathrm{new}}(49, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + 10216 T^{4} + \cdots + 29859840000)^{2} \) Copy content Toggle raw display
$3$ \( T^{12} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots + 50\!\cdots\!04)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 33\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 58\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 54\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} + \cdots + 41\!\cdots\!92)^{4} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 49\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + \cdots + 36\!\cdots\!00)^{4} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 74\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 94\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 58\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} + \cdots - 18\!\cdots\!12)^{4} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 59\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots + 44\!\cdots\!24)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 30\!\cdots\!00)^{2} \) Copy content Toggle raw display
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