Properties

Label 1336.2.a.e
Level $1336$
Weight $2$
Character orbit 1336.a
Self dual yes
Analytic conductor $10.668$
Analytic rank $0$
Dimension $12$
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] = [1336,2,Mod(1,1336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1336, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1336.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1336 = 2^{3} \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1336.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: \(10.6680137100\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5 x^{11} - 15 x^{10} + 100 x^{9} + 36 x^{8} - 641 x^{7} + 129 x^{6} + 1804 x^{5} - 433 x^{4} - 2399 x^{3} + 148 x^{2} + 1224 x + 224 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{9} q^{5} + (\beta_{11} + 1) q^{7} + (\beta_{11} - \beta_{9} - \beta_{7} + \beta_{5} - \beta_{3} + \beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{9} q^{5} + (\beta_{11} + 1) q^{7} + (\beta_{11} - \beta_{9} - \beta_{7} + \beta_{5} - \beta_{3} + \beta_{2} + 2) q^{9} + (\beta_{10} - \beta_{6} + \beta_{2} + 1) q^{11} + (\beta_{7} + \beta_1 - 1) q^{13} + ( - \beta_{10} + \beta_{9} - \beta_{7} + \beta_{6} + \beta_{5} - \beta_1 + 2) q^{15} + ( - \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{17} + ( - \beta_{11} - \beta_{8} - \beta_{3}) q^{19} + ( - \beta_{6} - 2 \beta_{5} + \beta_{4} + \beta_{3} + \beta_1 - 1) q^{21} + ( - \beta_{10} + \beta_{8} + \beta_{7} + \beta_{6} - \beta_{2} + 2) q^{23} + ( - \beta_{11} + \beta_{10} + \beta_{9} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} + \beta_1) q^{25} + (\beta_{9} + 2 \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 2) q^{27} + ( - \beta_{9} - \beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_1) q^{29} + ( - \beta_{11} - \beta_{8} - \beta_{7} - 2 \beta_{6} + \beta_{4} + \beta_{3} - \beta_1) q^{31} + (\beta_{10} + 2 \beta_{8} + \beta_{7} - \beta_{4} + \beta_{3} + 2 \beta_1 + 1) q^{33} + ( - 2 \beta_{11} + 2 \beta_{9} - \beta_{8} - \beta_{5} + \beta_{4} + \beta_{3} - 2 \beta_{2}) q^{35} + ( - \beta_{10} + \beta_{6} + \beta_{5} - 2 \beta_{4} - 1) q^{37} + ( - \beta_{9} - \beta_{8} - \beta_{7} + \beta_{5} + \beta_{2} - 2 \beta_1 + 3) q^{39} + ( - \beta_{10} - 3 \beta_{9} - 2 \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2}) q^{41} + (2 \beta_{8} + \beta_{7} - \beta_{4} + \beta_{3} + \beta_{2} + 3) q^{43} + ( - 2 \beta_{11} - 2 \beta_{10} + 3 \beta_{9} + 2 \beta_{6} - \beta_{4} + \beta_{3} - 3 \beta_{2} + \cdots - 2) q^{45}+ \cdots + (2 \beta_{11} + \beta_{10} - 3 \beta_{9} + 2 \beta_{8} + \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} + \cdots + 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 5 q^{3} - 2 q^{5} + 10 q^{7} + 19 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 5 q^{3} - 2 q^{5} + 10 q^{7} + 19 q^{9} + 14 q^{11} - 9 q^{13} + 8 q^{15} + 4 q^{17} + 9 q^{19} + 7 q^{21} + 15 q^{23} + 18 q^{25} + 20 q^{27} + 17 q^{29} + 11 q^{31} + 8 q^{33} + 19 q^{35} - 29 q^{37} + 26 q^{39} + 14 q^{41} + 15 q^{43} - 26 q^{45} + 17 q^{47} + 20 q^{49} + 36 q^{51} - 7 q^{53} + 13 q^{55} + 3 q^{57} + 32 q^{59} - 8 q^{61} + 50 q^{63} + 37 q^{65} + 39 q^{67} + q^{69} + 35 q^{71} - 12 q^{73} + 29 q^{75} + 13 q^{77} + 36 q^{79} + 44 q^{81} + 25 q^{83} - 38 q^{85} + 14 q^{87} + 23 q^{89} - 2 q^{91} - 25 q^{93} + 38 q^{95} - 2 q^{97} + 43 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 5 x^{11} - 15 x^{10} + 100 x^{9} + 36 x^{8} - 641 x^{7} + 129 x^{6} + 1804 x^{5} - 433 x^{4} - 2399 x^{3} + 148 x^{2} + 1224 x + 224 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 4134 \nu^{11} + 110987 \nu^{10} - 343875 \nu^{9} - 1750092 \nu^{8} + 7226582 \nu^{7} + 6214148 \nu^{6} - 38734849 \nu^{5} - 4221695 \nu^{4} + \cdots - 7879652 ) / 243358 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 26659 \nu^{11} + 333527 \nu^{10} - 500007 \nu^{9} - 5651176 \nu^{8} + 15580284 \nu^{7} + 25075571 \nu^{6} - 91240819 \nu^{5} - 37971272 \nu^{4} + \cdots - 23226840 ) / 973432 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 28693 \nu^{11} - 134357 \nu^{10} - 387059 \nu^{9} + 2394592 \nu^{8} + 500524 \nu^{7} - 11991941 \nu^{6} + 3606689 \nu^{5} + 22278164 \nu^{4} - 3899557 \nu^{3} + \cdots + 3831112 ) / 973432 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 41307 \nu^{11} - 307651 \nu^{10} - 114673 \nu^{9} + 5338476 \nu^{8} - 7428812 \nu^{7} - 25143667 \nu^{6} + 48730959 \nu^{5} + 42238968 \nu^{4} + \cdots + 13360032 ) / 973432 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 12411 \nu^{11} - 65032 \nu^{10} - 134173 \nu^{9} + 1111702 \nu^{8} - 358916 \nu^{7} - 4950043 \nu^{6} + 4244680 \nu^{5} + 6497698 \nu^{4} - 5950306 \nu^{3} + \cdots - 173194 ) / 243358 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 15928 \nu^{11} - 102912 \nu^{10} - 108909 \nu^{9} + 1828134 \nu^{8} - 1744024 \nu^{7} - 9086400 \nu^{6} + 13627972 \nu^{5} + 16669301 \nu^{4} + \cdots + 5796342 ) / 243358 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 70349 \nu^{11} + 489671 \nu^{10} + 295481 \nu^{9} - 8486560 \nu^{8} + 11005132 \nu^{7} + 39588421 \nu^{6} - 76436047 \nu^{5} - 62809378 \nu^{4} + \cdots - 18920800 ) / 486716 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 194845 \nu^{11} - 951441 \nu^{10} - 2586835 \nu^{9} + 17268420 \nu^{8} + 1949556 \nu^{7} - 89411357 \nu^{6} + 40411013 \nu^{5} + 168228596 \nu^{4} + \cdots + 13689440 ) / 973432 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 49825 \nu^{11} - 360853 \nu^{10} - 146621 \nu^{9} + 6201508 \nu^{8} - 8855778 \nu^{7} - 28321401 \nu^{6} + 58720147 \nu^{5} + 42956708 \nu^{4} + \cdots + 12190868 ) / 243358 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 207127 \nu^{11} - 1165859 \nu^{10} - 2032305 \nu^{9} + 20591672 \nu^{8} - 10923772 \nu^{7} - 100394311 \nu^{6} + 109890519 \nu^{5} + 171582340 \nu^{4} + \cdots + 26939384 ) / 973432 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} - \beta_{9} - \beta_{7} + \beta_{5} - \beta_{3} + \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + 2\beta_{8} + \beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + 7\beta _1 + 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 12\beta_{11} - 13\beta_{9} + \beta_{8} - 9\beta_{7} + 3\beta_{6} + 11\beta_{5} - 14\beta_{3} + 13\beta_{2} + 42 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 4 \beta_{11} + 10 \beta_{9} + 28 \beta_{8} + 15 \beta_{7} + 13 \beta_{6} - 12 \beta_{5} - 17 \beta_{4} - 13 \beta_{3} - 12 \beta_{2} + 64 \beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 131 \beta_{11} + 5 \beta_{10} - 149 \beta_{9} + 14 \beta_{8} - 86 \beta_{7} + 41 \beta_{6} + 113 \beta_{5} - 2 \beta_{4} - 157 \beta_{3} + 150 \beta_{2} - 3 \beta _1 + 409 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 62 \beta_{11} + 11 \beta_{10} + 98 \beta_{9} + 333 \beta_{8} + 191 \beta_{7} + 137 \beta_{6} - 142 \beta_{5} - 212 \beta_{4} - 132 \beta_{3} - 132 \beta_{2} + 649 \beta _1 + 208 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1418 \beta_{11} + 114 \beta_{10} - 1661 \beta_{9} + 168 \beta_{8} - 863 \beta_{7} + 433 \beta_{6} + 1163 \beta_{5} - 23 \beta_{4} - 1675 \beta_{3} + 1675 \beta_{2} - 70 \beta _1 + 4196 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 731 \beta_{11} + 268 \beta_{10} + 1008 \beta_{9} + 3765 \beta_{8} + 2278 \beta_{7} + 1331 \beta_{6} - 1718 \beta_{5} - 2388 \beta_{4} - 1229 \beta_{3} - 1452 \beta_{2} + 6834 \beta _1 + 1859 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 15372 \beta_{11} + 1796 \beta_{10} - 18337 \beta_{9} + 1900 \beta_{8} - 8924 \beta_{7} + 4192 \beta_{6} + 12015 \beta_{5} - 84 \beta_{4} - 17624 \beta_{3} + 18443 \beta_{2} - 1159 \beta _1 + 43958 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 7975 \beta_{11} + 4409 \beta_{10} + 10718 \beta_{9} + 41683 \beta_{8} + 26303 \beta_{7} + 12366 \beta_{6} - 20812 \beta_{5} - 25790 \beta_{4} - 10887 \beta_{3} - 16117 \beta_{2} + 73015 \beta _1 + 15931 \) Copy content Toggle raw display

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
−3.31023
−2.02356
−1.26807
−1.22013
−0.945593
−0.203398
1.18487
1.85311
1.94180
2.51502
3.16421
3.31199
0 −3.31023 0 −2.56191 0 1.56163 0 7.95764 0
1.2 0 −2.02356 0 −0.0986065 0 4.09952 0 1.09480 0
1.3 0 −1.26807 0 −0.339536 0 −4.08348 0 −1.39199 0
1.4 0 −1.22013 0 3.37141 0 3.01986 0 −1.51128 0
1.5 0 −0.945593 0 −3.79127 0 −1.93894 0 −2.10585 0
1.6 0 −0.203398 0 1.32800 0 0.559903 0 −2.95863 0
1.7 0 1.18487 0 −0.696822 0 2.51795 0 −1.59609 0
1.8 0 1.85311 0 3.00198 0 0.497697 0 0.434028 0
1.9 0 1.94180 0 −3.68233 0 −2.51484 0 0.770577 0
1.10 0 2.51502 0 3.24915 0 3.64134 0 3.32531 0
1.11 0 3.16421 0 1.18478 0 −2.23231 0 7.01222 0
1.12 0 3.31199 0 −2.96484 0 4.87167 0 7.96927 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(167\) \(-1\)

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 1336.2.a.e 12
4.b odd 2 1 2672.2.a.n 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1336.2.a.e 12 1.a even 1 1 trivial
2672.2.a.n 12 4.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{12} - 5 T_{3}^{11} - 15 T_{3}^{10} + 100 T_{3}^{9} + 36 T_{3}^{8} - 641 T_{3}^{7} + 129 T_{3}^{6} + 1804 T_{3}^{5} - 433 T_{3}^{4} - 2399 T_{3}^{3} + 148 T_{3}^{2} + 1224 T_{3} + 224 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(1336))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 5 T^{11} - 15 T^{10} + 100 T^{9} + \cdots + 224 \) Copy content Toggle raw display
$5$ \( T^{12} + 2 T^{11} - 37 T^{10} - 62 T^{9} + \cdots - 128 \) Copy content Toggle raw display
$7$ \( T^{12} - 10 T^{11} - 2 T^{10} + \cdots + 10696 \) Copy content Toggle raw display
$11$ \( T^{12} - 14 T^{11} + 28 T^{10} + \cdots + 8192 \) Copy content Toggle raw display
$13$ \( T^{12} + 9 T^{11} - 32 T^{10} + \cdots + 14336 \) Copy content Toggle raw display
$17$ \( T^{12} - 4 T^{11} - 107 T^{10} + \cdots - 152320 \) Copy content Toggle raw display
$19$ \( T^{12} - 9 T^{11} - 101 T^{10} + \cdots + 256 \) Copy content Toggle raw display
$23$ \( T^{12} - 15 T^{11} - 8 T^{10} + \cdots - 3584 \) Copy content Toggle raw display
$29$ \( T^{12} - 17 T^{11} - 40 T^{10} + \cdots + 86030192 \) Copy content Toggle raw display
$31$ \( T^{12} - 11 T^{11} - 104 T^{10} + \cdots - 10048 \) Copy content Toggle raw display
$37$ \( T^{12} + 29 T^{11} + 187 T^{10} + \cdots + 32082176 \) Copy content Toggle raw display
$41$ \( T^{12} - 14 T^{11} + \cdots - 401387264 \) Copy content Toggle raw display
$43$ \( T^{12} - 15 T^{11} - 258 T^{10} + \cdots + 226048 \) Copy content Toggle raw display
$47$ \( T^{12} - 17 T^{11} - 148 T^{10} + \cdots - 23909920 \) Copy content Toggle raw display
$53$ \( T^{12} + 7 T^{11} + \cdots + 109447774208 \) Copy content Toggle raw display
$59$ \( T^{12} - 32 T^{11} + \cdots - 25602668800 \) Copy content Toggle raw display
$61$ \( T^{12} + 8 T^{11} - 139 T^{10} + \cdots + 50608 \) Copy content Toggle raw display
$67$ \( T^{12} - 39 T^{11} + \cdots + 4208021440 \) Copy content Toggle raw display
$71$ \( T^{12} - 35 T^{11} + \cdots - 7322525696 \) Copy content Toggle raw display
$73$ \( T^{12} + 12 T^{11} + \cdots - 21136762880 \) Copy content Toggle raw display
$79$ \( T^{12} - 36 T^{11} + \cdots - 744310784 \) Copy content Toggle raw display
$83$ \( T^{12} - 25 T^{11} + \cdots - 6437720512 \) Copy content Toggle raw display
$89$ \( T^{12} - 23 T^{11} + \cdots - 947344672 \) Copy content Toggle raw display
$97$ \( T^{12} + 2 T^{11} + \cdots + 5304470056 \) Copy content Toggle raw display
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