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Properties

Label 912.6.a.ba
Level $912$
Weight $6$
Character orbit 912.a
Self dual yes
Analytic conductor $146.270$
Analytic rank $0$
Dimension $7$
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] = [912,6,Mod(1,912)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(912, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("912.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 912 = 2^{4} \cdot 3 \cdot 19 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 912.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: \(146.270043669\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 2x^{6} - 13283x^{5} + 151552x^{4} + 37807562x^{3} - 618683214x^{2} - 3628923248x + 52902120264 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12} \)
Twist minimal: no (minimal twist has level 456)
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 \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 9 q^{3} + ( - \beta_1 + 8) q^{5} + ( - \beta_{3} - \beta_1 - 17) q^{7} + 81 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 9 q^{3} + ( - \beta_1 + 8) q^{5} + ( - \beta_{3} - \beta_1 - 17) q^{7} + 81 q^{9} + (\beta_{5} - \beta_{3} + \beta_{2} + \cdots - 33) q^{11}+ \cdots + (81 \beta_{5} - 81 \beta_{3} + \cdots - 2673) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 63 q^{3} + 55 q^{5} - 119 q^{7} + 567 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 63 q^{3} + 55 q^{5} - 119 q^{7} + 567 q^{9} - 233 q^{11} + 686 q^{13} + 495 q^{15} - 1219 q^{17} - 2527 q^{19} - 1071 q^{21} - 992 q^{23} + 6504 q^{25} + 5103 q^{27} + 1374 q^{29} + 9578 q^{31} - 2097 q^{33} + 23841 q^{35} - 6556 q^{37} + 6174 q^{39} - 360 q^{41} + 28921 q^{43} + 4455 q^{45} + 4177 q^{47} + 20026 q^{49} - 10971 q^{51} + 24814 q^{53} + 28615 q^{55} - 22743 q^{57} - 3324 q^{59} + 49065 q^{61} - 9639 q^{63} + 31042 q^{65} - 45868 q^{67} - 8928 q^{69} - 48584 q^{71} + 90007 q^{73} + 58536 q^{75} + 202797 q^{77} + 10808 q^{79} + 45927 q^{81} + 5316 q^{83} + 288085 q^{85} + 12366 q^{87} + 279150 q^{89} - 172946 q^{91} + 86202 q^{93} - 19855 q^{95} + 382822 q^{97} - 18873 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 2x^{6} - 13283x^{5} + 151552x^{4} + 37807562x^{3} - 618683214x^{2} - 3628923248x + 52902120264 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 50244274848831 \nu^{6} - 615449390109152 \nu^{5} + \cdots - 70\!\cdots\!08 ) / 93\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 283017519438119 \nu^{6} + \cdots - 55\!\cdots\!12 ) / 93\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 383506069135781 \nu^{6} + \cdots + 19\!\cdots\!20 ) / 93\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 325171907054507 \nu^{6} + \cdots + 18\!\cdots\!72 ) / 46\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 305646707388351 \nu^{6} - 448960353434068 \nu^{5} + \cdots - 92\!\cdots\!04 ) / 31\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 35\!\cdots\!93 \nu^{6} + \cdots - 13\!\cdots\!28 ) / 93\!\cdots\!04 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{3} + \beta_{2} + 2\beta _1 + 2 ) / 8 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 12\beta_{6} - 2\beta_{5} - 24\beta_{4} + 83\beta_{3} - 31\beta_{2} + 298\beta _1 + 30330 ) / 8 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -79\beta_{6} - 2734\beta_{5} + 1527\beta_{4} + 2462\beta_{3} + 6824\beta_{2} + 16127\beta _1 - 432361 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 152879 \beta_{6} + 35114 \beta_{5} - 230469 \beta_{4} + 1136364 \beta_{3} - 349272 \beta_{2} + \cdots + 229144111 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2106649 \beta_{6} - 36224802 \beta_{5} + 19114329 \beta_{4} - 17194936 \beta_{3} + 57762278 \beta_{2} + \cdots - 6424371851 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1592083171 \beta_{6} + 893270102 \beta_{5} - 2317718157 \beta_{4} + 12125820016 \beta_{3} + \cdots + 1975788222999 ) / 8 \) 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
88.8772
61.3383
−63.5345
−101.866
−9.57914
18.1320
8.63252
0 9.00000 0 −69.3034 0 −234.208 0 81.0000 0
1.2 0 9.00000 0 −66.7486 0 −51.5844 0 81.0000 0
1.3 0 9.00000 0 −24.8152 0 159.996 0 81.0000 0
1.4 0 9.00000 0 8.02190 0 −107.486 0 81.0000 0
1.5 0 9.00000 0 18.6009 0 −9.94695 0 81.0000 0
1.6 0 9.00000 0 84.9316 0 194.767 0 81.0000 0
1.7 0 9.00000 0 104.313 0 −70.5370 0 81.0000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.7
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(3\) \(-1\)
\(19\) \(1\)

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 912.6.a.ba 7
4.b odd 2 1 456.6.a.g ✓ 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
456.6.a.g ✓ 7 4.b odd 2 1
912.6.a.ba 7 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{7} - 55 T_{5}^{6} - 12677 T_{5}^{5} + 382991 T_{5}^{4} + 46460696 T_{5}^{3} - 288214972 T_{5}^{2} + \cdots + 151750763136 \) acting on \(S_{6}^{\mathrm{new}}(\Gamma_0(912))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} \) Copy content Toggle raw display
$3$ \( (T - 9)^{7} \) Copy content Toggle raw display
$5$ \( T^{7} + \cdots + 151750763136 \) Copy content Toggle raw display
$7$ \( T^{7} + \cdots + 28392559137792 \) Copy content Toggle raw display
$11$ \( T^{7} + \cdots - 93\!\cdots\!08 \) Copy content Toggle raw display
$13$ \( T^{7} + \cdots + 93\!\cdots\!52 \) Copy content Toggle raw display
$17$ \( T^{7} + \cdots - 17\!\cdots\!80 \) Copy content Toggle raw display
$19$ \( (T + 361)^{7} \) Copy content Toggle raw display
$23$ \( T^{7} + \cdots + 20\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( T^{7} + \cdots - 28\!\cdots\!52 \) Copy content Toggle raw display
$31$ \( T^{7} + \cdots + 27\!\cdots\!40 \) Copy content Toggle raw display
$37$ \( T^{7} + \cdots + 51\!\cdots\!88 \) Copy content Toggle raw display
$41$ \( T^{7} + \cdots - 24\!\cdots\!08 \) Copy content Toggle raw display
$43$ \( T^{7} + \cdots - 96\!\cdots\!28 \) Copy content Toggle raw display
$47$ \( T^{7} + \cdots + 87\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{7} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{7} + \cdots + 49\!\cdots\!04 \) Copy content Toggle raw display
$61$ \( T^{7} + \cdots + 10\!\cdots\!20 \) Copy content Toggle raw display
$67$ \( T^{7} + \cdots + 16\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( T^{7} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{7} + \cdots + 95\!\cdots\!52 \) Copy content Toggle raw display
$79$ \( T^{7} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{7} + \cdots - 25\!\cdots\!72 \) Copy content Toggle raw display
$89$ \( T^{7} + \cdots - 86\!\cdots\!20 \) Copy content Toggle raw display
$97$ \( T^{7} + \cdots - 71\!\cdots\!84 \) Copy content Toggle raw display
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