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Properties

Label 624.2.bz.g
Level $624$
Weight $2$
Character orbit 624.bz
Analytic conductor $4.983$
Analytic rank $0$
Dimension $16$
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] = [624,2,Mod(95,624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(624, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("624.95");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 624 = 2^{4} \cdot 3 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 624.bz (of order \(6\), degree \(2\), 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: \(4.98266508613\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 3 x^{15} + 5 x^{14} - 6 x^{13} + 9 x^{12} + 3 x^{11} - 46 x^{10} + 141 x^{9} - 266 x^{8} + \cdots + 6561 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + ( - \beta_{15} + \beta_{14} - \beta_{13} + \cdots + 2) q^{5}+ \cdots + ( - \beta_{10} + \beta_{7} + \cdots - \beta_{4}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + ( - \beta_{15} + \beta_{14} - \beta_{13} + \cdots + 2) q^{5}+ \cdots + ( - 2 \beta_{14} - \beta_{13} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 3 q^{3} - 6 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 3 q^{3} - 6 q^{7} - q^{9} + 14 q^{13} + 6 q^{15} - 6 q^{19} + 28 q^{25} - 24 q^{31} + 9 q^{33} + 12 q^{37} + 27 q^{39} - 6 q^{43} + 30 q^{45} + 14 q^{49} + 24 q^{55} + 8 q^{61} + 21 q^{63} - 30 q^{67} + 9 q^{69} - 15 q^{75} - 13 q^{81} - 30 q^{85} + 39 q^{87} - 30 q^{91} - 72 q^{93} - 54 q^{97} - 42 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 3 x^{15} + 5 x^{14} - 6 x^{13} + 9 x^{12} + 3 x^{11} - 46 x^{10} + 141 x^{9} - 266 x^{8} + \cdots + 6561 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5 \nu^{15} + 3 \nu^{14} - 34 \nu^{13} + 402 \nu^{12} - 765 \nu^{11} + 417 \nu^{10} + \cdots + 199017 ) / 78732 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 8 \nu^{15} + 63 \nu^{14} - 148 \nu^{13} + 162 \nu^{12} - 18 \nu^{11} + 3 \nu^{10} + 566 \nu^{9} + \cdots - 41553 ) / 26244 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 13 \nu^{15} - 75 \nu^{14} + 200 \nu^{13} - 474 \nu^{12} + 1035 \nu^{11} - 1851 \nu^{10} + \cdots + 80919 ) / 26244 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 7 \nu^{15} - 15 \nu^{14} + 32 \nu^{13} - 84 \nu^{12} + 129 \nu^{11} - 231 \nu^{10} + 209 \nu^{9} + \cdots - 2187 ) / 8748 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 26 \nu^{15} + 57 \nu^{14} - 112 \nu^{13} + 294 \nu^{12} - 576 \nu^{11} + 1029 \nu^{10} + \cdots - 85293 ) / 26244 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 50 \nu^{15} + 219 \nu^{14} - 583 \nu^{13} + 1239 \nu^{12} - 2547 \nu^{11} + 4008 \nu^{10} + \cdots - 201204 ) / 39366 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 4 \nu^{15} + 15 \nu^{14} - 44 \nu^{13} + 102 \nu^{12} - 210 \nu^{11} + 357 \nu^{10} - 428 \nu^{9} + \cdots - 9477 ) / 2916 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 131 \nu^{15} - 300 \nu^{14} + 664 \nu^{13} - 996 \nu^{12} + 1899 \nu^{11} - 2712 \nu^{10} + \cdots + 214326 ) / 78732 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 124 \nu^{15} + 627 \nu^{14} - 1718 \nu^{13} + 3774 \nu^{12} - 7632 \nu^{11} + 12561 \nu^{10} + \cdots - 369603 ) / 78732 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 139 \nu^{15} + 780 \nu^{14} - 2360 \nu^{13} + 5052 \nu^{12} - 10359 \nu^{11} + 18186 \nu^{10} + \cdots - 887922 ) / 78732 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 83 \nu^{15} + 174 \nu^{14} - 235 \nu^{13} + 231 \nu^{12} - 360 \nu^{11} + 183 \nu^{10} + \cdots + 67797 ) / 39366 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 155 \nu^{15} + 786 \nu^{14} - 2134 \nu^{13} + 4992 \nu^{12} - 9513 \nu^{11} + 15762 \nu^{10} + \cdots - 542376 ) / 78732 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 181 \nu^{15} - 879 \nu^{14} + 2732 \nu^{13} - 5574 \nu^{12} + 11061 \nu^{11} - 19113 \nu^{10} + \cdots + 1130679 ) / 78732 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 227 \nu^{15} + 1128 \nu^{14} - 3142 \nu^{13} + 6378 \nu^{12} - 12753 \nu^{11} + 20244 \nu^{10} + \cdots - 804816 ) / 78732 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{10} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + \beta_{13} - 2 \beta_{12} - 3 \beta_{11} + 3 \beta_{10} - 2 \beta_{9} - \beta_{8} + \cdots - 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{15} - 2 \beta_{14} - \beta_{12} - 5 \beta_{11} + 7 \beta_{10} + \beta_{9} - 2 \beta_{8} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2 \beta_{15} + 3 \beta_{13} + \beta_{12} - 2 \beta_{11} + \beta_{10} + 2 \beta_{9} + \beta_{8} + \cdots - 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 5 \beta_{15} + 2 \beta_{14} - 3 \beta_{13} - 5 \beta_{12} - 7 \beta_{11} + 9 \beta_{10} + 2 \beta_{9} + \cdots + 1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 6 \beta_{15} - 13 \beta_{13} - 2 \beta_{12} - \beta_{11} + 11 \beta_{10} + 6 \beta_{9} - 11 \beta_{7} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 6 \beta_{15} + 12 \beta_{14} - 27 \beta_{13} + 15 \beta_{12} + 54 \beta_{11} - 43 \beta_{10} + \cdots + 42 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 23 \beta_{15} + 54 \beta_{14} - 20 \beta_{13} + 7 \beta_{12} + 105 \beta_{11} - 51 \beta_{10} + \cdots + 13 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 86 \beta_{15} + 28 \beta_{14} - 72 \beta_{13} + 59 \beta_{12} + 148 \beta_{11} - 80 \beta_{10} + \cdots + 74 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 58 \beta_{15} + 63 \beta_{13} + 28 \beta_{12} + 61 \beta_{11} - 116 \beta_{10} - 4 \beta_{9} + \cdots + 57 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 26 \beta_{15} - 40 \beta_{14} + 132 \beta_{13} - 170 \beta_{12} - 268 \beta_{11} + 228 \beta_{10} + \cdots + 223 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 72 \beta_{15} + 90 \beta_{14} + 218 \beta_{13} + 70 \beta_{12} - 46 \beta_{11} - 34 \beta_{10} + \cdots - 610 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 156 \beta_{15} + 312 \beta_{14} + 252 \beta_{13} + 372 \beta_{12} + 432 \beta_{11} - 697 \beta_{10} + \cdots + 120 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 115 \beta_{15} + 270 \beta_{14} + 589 \beta_{13} - 398 \beta_{12} - 129 \beta_{11} + 1227 \beta_{10} + \cdots - 152 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/624\mathbb{Z}\right)^\times\).

\(n\) \(79\) \(145\) \(209\) \(469\)
\(\chi(n)\) \(-1\) \(\beta_{4}\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
95.1
1.65688 + 0.504714i
1.46152 − 0.929486i
1.26554 + 1.18255i
0.756936 − 1.55790i
−0.0741962 + 1.73046i
−0.863924 − 1.50121i
−0.970712 + 1.43447i
−1.73205 + 0.00242576i
1.65688 − 0.504714i
1.46152 + 0.929486i
1.26554 − 1.18255i
0.756936 + 1.55790i
−0.0741962 − 1.73046i
−0.863924 + 1.50121i
−0.970712 − 1.43447i
−1.73205 − 0.00242576i
0 −1.65688 − 0.504714i 0 −3.51957 0 −0.196951 − 0.341129i 0 2.49053 + 1.67250i 0
95.2 0 −1.46152 + 0.929486i 0 1.88845 0 −0.897962 − 1.55532i 0 1.27211 − 2.71693i 0
95.3 0 −1.26554 − 1.18255i 0 3.51957 0 −0.196951 − 0.341129i 0 0.203168 + 2.99311i 0
95.4 0 −0.756936 + 1.55790i 0 −1.03002 0 1.26763 + 2.19560i 0 −1.85410 − 2.35846i 0
95.5 0 0.0741962 − 1.73046i 0 −1.88845 0 −0.897962 − 1.55532i 0 −2.98899 − 0.256787i 0
95.6 0 0.863924 + 1.50121i 0 −3.15997 0 −1.67272 − 2.89723i 0 −1.50727 + 2.59386i 0
95.7 0 0.970712 − 1.43447i 0 1.03002 0 1.26763 + 2.19560i 0 −1.11544 − 2.78492i 0
95.8 0 1.73205 − 0.00242576i 0 3.15997 0 −1.67272 − 2.89723i 0 2.99999 − 0.00840307i 0
335.1 0 −1.65688 + 0.504714i 0 −3.51957 0 −0.196951 + 0.341129i 0 2.49053 − 1.67250i 0
335.2 0 −1.46152 − 0.929486i 0 1.88845 0 −0.897962 + 1.55532i 0 1.27211 + 2.71693i 0
335.3 0 −1.26554 + 1.18255i 0 3.51957 0 −0.196951 + 0.341129i 0 0.203168 − 2.99311i 0
335.4 0 −0.756936 − 1.55790i 0 −1.03002 0 1.26763 − 2.19560i 0 −1.85410 + 2.35846i 0
335.5 0 0.0741962 + 1.73046i 0 −1.88845 0 −0.897962 + 1.55532i 0 −2.98899 + 0.256787i 0
335.6 0 0.863924 − 1.50121i 0 −3.15997 0 −1.67272 + 2.89723i 0 −1.50727 − 2.59386i 0
335.7 0 0.970712 + 1.43447i 0 1.03002 0 1.26763 − 2.19560i 0 −1.11544 + 2.78492i 0
335.8 0 1.73205 + 0.00242576i 0 3.15997 0 −1.67272 + 2.89723i 0 2.99999 + 0.00840307i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 95.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
52.i odd 6 1 inner
156.r even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 624.2.bz.g ✓ 16
3.b odd 2 1 inner 624.2.bz.g ✓ 16
4.b odd 2 1 624.2.bz.h yes 16
12.b even 2 1 624.2.bz.h yes 16
13.e even 6 1 624.2.bz.h yes 16
39.h odd 6 1 624.2.bz.h yes 16
52.i odd 6 1 inner 624.2.bz.g ✓ 16
156.r even 6 1 inner 624.2.bz.g ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
624.2.bz.g ✓ 16 1.a even 1 1 trivial
624.2.bz.g ✓ 16 3.b odd 2 1 inner
624.2.bz.g ✓ 16 52.i odd 6 1 inner
624.2.bz.g ✓ 16 156.r even 6 1 inner
624.2.bz.h yes 16 4.b odd 2 1
624.2.bz.h yes 16 12.b even 2 1
624.2.bz.h yes 16 13.e even 6 1
624.2.bz.h yes 16 39.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(624, [\chi])\):

\( T_{5}^{8} - 27T_{5}^{6} + 231T_{5}^{4} - 657T_{5}^{2} + 468 \) Copy content Toggle raw display
\( T_{7}^{8} + 3T_{7}^{7} + 15T_{7}^{6} + 18T_{7}^{5} + 96T_{7}^{4} + 144T_{7}^{3} + 288T_{7}^{2} + 108T_{7} + 36 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + 3 T^{15} + \cdots + 6561 \) Copy content Toggle raw display
$5$ \( (T^{8} - 27 T^{6} + \cdots + 468)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} + 3 T^{7} + 15 T^{6} + \cdots + 36)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} - 27 T^{14} + \cdots + 219024 \) Copy content Toggle raw display
$13$ \( (T^{8} - 7 T^{7} + \cdots + 28561)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 316733209 \) Copy content Toggle raw display
$19$ \( (T^{8} + 3 T^{7} + 15 T^{6} + \cdots + 36)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 1437016464 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 119299087609 \) Copy content Toggle raw display
$31$ \( (T^{4} + 6 T^{3} + \cdots - 864)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} - 6 T^{7} + \cdots + 4761)^{2} \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 28176700597929 \) Copy content Toggle raw display
$43$ \( (T^{8} + 3 T^{7} + \cdots + 171396)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 216 T^{6} + \cdots + 151632)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 101 T^{6} + \cdots + 140608)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 15690725167104 \) Copy content Toggle raw display
$61$ \( (T^{8} - 4 T^{7} + \cdots + 2128681)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 15 T^{7} + \cdots + 26244)^{2} \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 602328808742544 \) Copy content Toggle raw display
$73$ \( (T^{8} + 129 T^{6} + \cdots + 197136)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 252 T^{6} + \cdots + 746496)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + 684 T^{6} + \cdots + 212601168)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + 141 T^{14} + \cdots + 56070144 \) Copy content Toggle raw display
$97$ \( (T^{8} + 27 T^{7} + \cdots + 9809424)^{2} \) Copy content Toggle raw display
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