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Properties

Label 402.2.d.a
Level $402$
Weight $2$
Character orbit 402.d
Analytic conductor $3.210$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [402,2,Mod(401,402)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(402, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("402.401");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 402 = 2 \cdot 3 \cdot 67 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 402.d (of order \(2\), degree \(1\), 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: \(3.20998616126\)
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: \( x^{12} - x^{11} - 2 x^{10} + 4 x^{9} + 3 x^{8} - 11 x^{7} + 12 x^{6} - 33 x^{5} + 27 x^{4} + 108 x^{3} + \cdots + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - q^{2} + \beta_{8} q^{3} + q^{4} - \beta_{7} q^{5} - \beta_{8} q^{6} - \beta_{5} q^{7} - q^{8} - \beta_{9} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - q^{2} + \beta_{8} q^{3} + q^{4} - \beta_{7} q^{5} - \beta_{8} q^{6} - \beta_{5} q^{7} - q^{8} - \beta_{9} q^{9} + \beta_{7} q^{10} + ( - \beta_{11} - \beta_{10} + 1) q^{11} + \beta_{8} q^{12} + \beta_{3} q^{13} + \beta_{5} q^{14} + ( - \beta_{11} - \beta_{7} + \cdots + \beta_{4}) q^{15}+ \cdots + (\beta_{10} + \beta_{7} + \beta_{6} + \cdots + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} + q^{3} + 12 q^{4} - q^{6} - 12 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{2} + q^{3} + 12 q^{4} - q^{6} - 12 q^{8} + 5 q^{9} + 6 q^{11} + q^{12} - 4 q^{15} + 12 q^{16} - 5 q^{18} - 20 q^{19} - 6 q^{22} - q^{24} + 24 q^{25} - 5 q^{27} + 4 q^{30} - 12 q^{32} + 5 q^{36} - 22 q^{37} + 20 q^{38} + 6 q^{39} + 14 q^{41} + 6 q^{44} - 2 q^{45} + q^{48} - 2 q^{49} - 24 q^{50} + 6 q^{51} + 5 q^{54} - 8 q^{55} - 4 q^{57} - 4 q^{60} + 8 q^{63} + 12 q^{64} - 8 q^{67} + 23 q^{69} - 5 q^{72} - 20 q^{73} + 22 q^{74} + 42 q^{75} - 20 q^{76} - 6 q^{78} - 11 q^{81} - 14 q^{82} + 14 q^{87} - 6 q^{88} + 2 q^{90} - 8 q^{91} - 12 q^{93} + 4 q^{95} - q^{96} + 2 q^{98} + 19 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} - 2 x^{10} + 4 x^{9} + 3 x^{8} - 11 x^{7} + 12 x^{6} - 33 x^{5} + 27 x^{4} + 108 x^{3} + \cdots + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{11} - 2 \nu^{10} - 22 \nu^{9} - 52 \nu^{8} + 39 \nu^{7} - 106 \nu^{6} - 222 \nu^{5} + \cdots - 3159 ) / 1944 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} + 20 \nu^{10} + 40 \nu^{9} - 74 \nu^{8} + 15 \nu^{7} - 74 \nu^{6} + 402 \nu^{5} + 417 \nu^{4} + \cdots - 243 ) / 2916 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 19 \nu^{11} - 2 \nu^{10} + 50 \nu^{9} - 52 \nu^{8} + 309 \nu^{7} + 2 \nu^{6} + 30 \nu^{5} + \cdots - 5589 ) / 5832 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 23 \nu^{11} - 44 \nu^{10} + 20 \nu^{9} + 62 \nu^{8} - 159 \nu^{7} + 242 \nu^{6} + 210 \nu^{5} + \cdots + 6075 ) / 5832 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 11 \nu^{11} + 23 \nu^{10} + 19 \nu^{9} + 31 \nu^{8} - 30 \nu^{7} - 104 \nu^{6} - 210 \nu^{5} + \cdots + 1944 ) / 2916 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - \nu^{11} + \nu^{10} + 2 \nu^{9} - 4 \nu^{8} - 3 \nu^{7} + 11 \nu^{6} - 12 \nu^{5} + 33 \nu^{4} + \cdots + 243 ) / 243 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{11} + 2 \nu^{10} - 5 \nu^{9} - 2 \nu^{8} + 15 \nu^{7} - 2 \nu^{6} - 21 \nu^{5} + 3 \nu^{4} + \cdots - 729 ) / 243 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 25 \nu^{11} + 4 \nu^{10} - 100 \nu^{9} + 86 \nu^{8} + 129 \nu^{7} - 94 \nu^{6} - 1014 \nu^{5} + \cdots + 243 ) / 5832 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{11} - 2 \nu^{10} + 6 \nu^{9} - 4 \nu^{8} - 3 \nu^{7} + 2 \nu^{6} + 14 \nu^{5} - 23 \nu^{4} + \cdots + 243 ) / 216 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} + \beta_{6} + \beta_{4} + \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + 2\beta_{10} - \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -3\beta_{11} - \beta_{10} - 2\beta_{9} - 2\beta_{8} - 3\beta_{7} + \beta_{5} + 3\beta_{4} - \beta_{3} + \beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - \beta_{11} - \beta_{10} + 3 \beta_{9} + 9 \beta_{8} - 5 \beta_{7} + 2 \beta_{6} + \beta_{5} + \cdots - 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{11} + \beta_{10} + 5 \beta_{9} - 4 \beta_{8} - \beta_{7} + 13 \beta_{5} - \beta_{4} - 3 \beta_{3} + \cdots + 26 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - \beta_{11} - \beta_{10} + \beta_{9} - 14 \beta_{8} + 13 \beta_{7} + 5 \beta_{5} - 11 \beta_{4} + \cdots - 11 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 25 \beta_{11} - 9 \beta_{10} + 5 \beta_{9} + 2 \beta_{8} + 29 \beta_{7} - 4 \beta_{6} + 13 \beta_{5} + \cdots - 33 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 23 \beta_{11} - 14 \beta_{10} + 37 \beta_{9} + 22 \beta_{8} + 53 \beta_{7} + 35 \beta_{6} - 9 \beta_{5} + \cdots + 20 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 86 \beta_{11} + 9 \beta_{10} + 85 \beta_{9} - 26 \beta_{8} + 10 \beta_{7} - 11 \beta_{6} - 76 \beta_{5} + \cdots + 57 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/402\mathbb{Z}\right)^\times\).

\(n\) \(269\) \(337\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
401.1
−1.62157 + 0.608708i
−1.62157 − 0.608708i
−1.45422 + 0.940871i
−1.45422 − 0.940871i
−0.322781 + 1.70171i
−0.322781 − 1.70171i
0.873148 + 1.49586i
0.873148 − 1.49586i
1.33579 + 1.10258i
1.33579 − 1.10258i
1.68963 + 0.380970i
1.68963 − 0.380970i
−1.00000 −1.62157 − 0.608708i 1.00000 2.31596 1.62157 + 0.608708i − 3.80471i −1.00000 2.25895 + 1.97412i −2.31596
401.2 −1.00000 −1.62157 + 0.608708i 1.00000 2.31596 1.62157 − 0.608708i 3.80471i −1.00000 2.25895 − 1.97412i −2.31596
401.3 −1.00000 −1.45422 − 0.940871i 1.00000 −1.82975 1.45422 + 0.940871i 1.94474i −1.00000 1.22952 + 2.73647i 1.82975
401.4 −1.00000 −1.45422 + 0.940871i 1.00000 −1.82975 1.45422 − 0.940871i − 1.94474i −1.00000 1.22952 − 2.73647i 1.82975
401.5 −1.00000 −0.322781 − 1.70171i 1.00000 −0.751820 0.322781 + 1.70171i 0.441803i −1.00000 −2.79162 + 1.09856i 0.751820
401.6 −1.00000 −0.322781 + 1.70171i 1.00000 −0.751820 0.322781 − 1.70171i − 0.441803i −1.00000 −2.79162 − 1.09856i 0.751820
401.7 −1.00000 0.873148 − 1.49586i 1.00000 3.78807 −0.873148 + 1.49586i − 2.53236i −1.00000 −1.47522 − 2.61222i −3.78807
401.8 −1.00000 0.873148 + 1.49586i 1.00000 3.78807 −0.873148 − 1.49586i 2.53236i −1.00000 −1.47522 + 2.61222i −3.78807
401.9 −1.00000 1.33579 − 1.10258i 1.00000 −4.22799 −1.33579 + 1.10258i 3.83465i −1.00000 0.568651 − 2.94561i 4.22799
401.10 −1.00000 1.33579 + 1.10258i 1.00000 −4.22799 −1.33579 − 1.10258i − 3.83465i −1.00000 0.568651 + 2.94561i 4.22799
401.11 −1.00000 1.68963 − 0.380970i 1.00000 0.705530 −1.68963 + 0.380970i − 1.85193i −1.00000 2.70972 − 1.28740i −0.705530
401.12 −1.00000 1.68963 + 0.380970i 1.00000 0.705530 −1.68963 − 0.380970i 1.85193i −1.00000 2.70972 + 1.28740i −0.705530
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 401.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
201.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 402.2.d.a ✓ 12
3.b odd 2 1 402.2.d.b yes 12
67.b odd 2 1 402.2.d.b yes 12
201.d even 2 1 inner 402.2.d.a ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
402.2.d.a ✓ 12 1.a even 1 1 trivial
402.2.d.a ✓ 12 201.d even 2 1 inner
402.2.d.b yes 12 3.b odd 2 1
402.2.d.b yes 12 67.b odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - 21T_{5}^{4} + 5T_{5}^{3} + 79T_{5}^{2} - 36 \) acting on \(S_{2}^{\mathrm{new}}(402, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T + 1)^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - T^{11} + \cdots + 729 \) Copy content Toggle raw display
$5$ \( (T^{6} - 21 T^{4} + \cdots - 36)^{2} \) Copy content Toggle raw display
$7$ \( T^{12} + 43 T^{10} + \cdots + 3456 \) Copy content Toggle raw display
$11$ \( (T^{6} - 3 T^{5} + \cdots - 144)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + 79 T^{10} + \cdots + 279936 \) Copy content Toggle raw display
$17$ \( T^{12} + 127 T^{10} + \cdots + 13824 \) Copy content Toggle raw display
$19$ \( (T^{6} + 10 T^{5} + \cdots + 4608)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 210 T^{10} + \cdots + 2181654 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 2660130816 \) Copy content Toggle raw display
$31$ \( T^{12} + 163 T^{10} + \cdots + 279936 \) Copy content Toggle raw display
$37$ \( (T^{6} + 11 T^{5} + \cdots + 30304)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} - 7 T^{5} + \cdots - 1008)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 493807104 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 1382958744 \) Copy content Toggle raw display
$53$ \( (T^{6} - 133 T^{4} + \cdots + 5724)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + 235 T^{10} + \cdots + 6144 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 177300576 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 90458382169 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 17694849816 \) Copy content Toggle raw display
$73$ \( (T^{6} + 10 T^{5} + \cdots - 2224)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 146658816 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 161243136 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 5709594624 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 3251306594304 \) Copy content Toggle raw display
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