Properties

Label 7700.2.e.t
Level $7700$
Weight $2$
Character orbit 7700.e
Analytic conductor $61.485$
Analytic rank $0$
Dimension $8$
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] = [7700,2,Mod(1849,7700)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7700, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 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("7700.1849");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7700 = 2^{2} \cdot 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7700.e (of order \(2\), degree \(1\), 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: \(61.4848095564\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 20x^{6} + 108x^{4} + 84x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 1540)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} - \beta_{4} q^{7} + (\beta_{2} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} - \beta_{4} q^{7} + (\beta_{2} - 2) q^{9} + q^{11} + ( - \beta_{5} - \beta_{4}) q^{13} + (2 \beta_{4} - \beta_1) q^{17} + (\beta_{6} + \beta_{3} - 2) q^{19} + \beta_{3} q^{21} + (\beta_{7} + \beta_{5} + 2 \beta_{4} - \beta_1) q^{23} + (\beta_{5} - \beta_{4} - 3 \beta_1) q^{27} - 2 q^{29} + (\beta_{3} + \beta_{2} + 1) q^{31} + \beta_1 q^{33} + ( - \beta_{7} - \beta_{5} + 2 \beta_{4} + \beta_1) q^{37} + (\beta_{2} - 1) q^{39} + ( - \beta_{3} + \beta_{2} + 3) q^{41} + (\beta_{7} + \beta_{5} + \beta_1) q^{43} + (\beta_{5} - \beta_{4} - 2 \beta_1) q^{47} - q^{49} + ( - 2 \beta_{3} - \beta_{2} + 5) q^{51} + ( - \beta_{7} - \beta_{5} - 2 \beta_{4} - 3 \beta_1) q^{53} + (4 \beta_{4} - 2 \beta_1) q^{57} + ( - 3 \beta_{3} - \beta_{2} - 1) q^{59} + ( - \beta_{6} - 2 \beta_{3} - \beta_{2} + 7) q^{61} + (\beta_{7} + 2 \beta_{4}) q^{63} + (\beta_{7} + \beta_{5} - 2 \beta_{4} + \beta_1) q^{67} + (\beta_{6} - 5 \beta_{3} - 2 \beta_{2} + 4) q^{69} + (2 \beta_{6} - 2 \beta_{3}) q^{71} + ( - 6 \beta_{4} - \beta_1) q^{73} - \beta_{4} q^{77} + ( - 2 \beta_{6} - 4 \beta_{3} + \beta_{2} - 1) q^{79} + (2 \beta_{3} - \beta_{2} + 10) q^{81} + ( - 2 \beta_{7} + \beta_{5} + \beta_{4} - \beta_1) q^{83} - 2 \beta_1 q^{87} + ( - \beta_{6} + \beta_{3} + 2 \beta_{2} + 2) q^{89} - \beta_{6} q^{91} + (\beta_{7} + \beta_{5} + 4 \beta_{4} - 3 \beta_1) q^{93} + ( - \beta_{5} - \beta_{4} + \beta_1) q^{97} + (\beta_{2} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{9} + 8 q^{11} - 12 q^{19} - 16 q^{29} + 8 q^{31} - 8 q^{39} + 24 q^{41} - 8 q^{49} + 40 q^{51} - 8 q^{59} + 52 q^{61} + 36 q^{69} + 8 q^{71} - 16 q^{79} + 80 q^{81} + 12 q^{89} - 4 q^{91} - 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 20x^{6} + 108x^{4} + 84x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} + 10\nu^{2} + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 20\nu^{5} + 104\nu^{3} + 44\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{7} + 20\nu^{5} + 112\nu^{3} + 116\nu ) / 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{6} + 19\nu^{4} + 94\nu^{2} + 40 ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -5\nu^{7} - 96\nu^{5} - 480\nu^{3} - 204\nu ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} - \beta_{4} - 9\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{3} - 10\beta_{2} + 46 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 2\beta_{7} - 10\beta_{5} + 20\beta_{4} + 86\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2\beta_{6} - 38\beta_{3} + 96\beta_{2} - 444 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -40\beta_{7} + 96\beta_{5} - 288\beta_{4} - 828\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/7700\mathbb{Z}\right)^\times\).

\(n\) \(2201\) \(3851\) \(5601\) \(6777\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1849.1
3.19911i
2.97964i
0.766757i
0.547280i
0.547280i
0.766757i
2.97964i
3.19911i
0 3.19911i 0 0 0 1.00000i 0 −7.23433 0
1849.2 0 2.97964i 0 0 0 1.00000i 0 −5.87824 0
1849.3 0 0.766757i 0 0 0 1.00000i 0 2.41208 0
1849.4 0 0.547280i 0 0 0 1.00000i 0 2.70049 0
1849.5 0 0.547280i 0 0 0 1.00000i 0 2.70049 0
1849.6 0 0.766757i 0 0 0 1.00000i 0 2.41208 0
1849.7 0 2.97964i 0 0 0 1.00000i 0 −5.87824 0
1849.8 0 3.19911i 0 0 0 1.00000i 0 −7.23433 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1849.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 7700.2.e.t 8
5.b even 2 1 inner 7700.2.e.t 8
5.c odd 4 1 1540.2.a.i 4
5.c odd 4 1 7700.2.a.bb 4
20.e even 4 1 6160.2.a.bs 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1540.2.a.i 4 5.c odd 4 1
6160.2.a.bs 4 20.e even 4 1
7700.2.a.bb 4 5.c odd 4 1
7700.2.e.t 8 1.a even 1 1 trivial
7700.2.e.t 8 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(7700, [\chi])\):

\( T_{3}^{8} + 20T_{3}^{6} + 108T_{3}^{4} + 84T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{13}^{8} + 72T_{13}^{6} + 1340T_{13}^{4} + 6532T_{13}^{2} + 9216 \) Copy content Toggle raw display
\( T_{17}^{8} + 36T_{17}^{6} + 308T_{17}^{4} + 772T_{17}^{2} + 576 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 20 T^{6} + 108 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$11$ \( (T - 1)^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + 72 T^{6} + 1340 T^{4} + \cdots + 9216 \) Copy content Toggle raw display
$17$ \( T^{8} + 36 T^{6} + 308 T^{4} + \cdots + 576 \) Copy content Toggle raw display
$19$ \( (T^{4} + 6 T^{3} - 28 T^{2} - 152 T - 96)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 148 T^{6} + 5600 T^{4} + \cdots + 33856 \) Copy content Toggle raw display
$29$ \( (T + 2)^{8} \) Copy content Toggle raw display
$31$ \( (T^{4} - 4 T^{3} - 40 T^{2} + 134 T + 176)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 164 T^{6} + 5376 T^{4} + \cdots + 69696 \) Copy content Toggle raw display
$41$ \( (T^{4} - 12 T^{3} - 4 T^{2} + 370 T - 668)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 148 T^{6} + 4112 T^{4} + \cdots + 20736 \) Copy content Toggle raw display
$47$ \( T^{8} + 176 T^{6} + 8884 T^{4} + \cdots + 1089936 \) Copy content Toggle raw display
$53$ \( T^{8} + 324 T^{6} + 28320 T^{4} + \cdots + 2585664 \) Copy content Toggle raw display
$59$ \( (T^{4} + 4 T^{3} - 108 T^{2} - 810 T - 1448)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 26 T^{3} + 140 T^{2} + 738 T - 5636)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 172 T^{6} + 2000 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$71$ \( (T^{4} - 4 T^{3} - 192 T^{2} + 384 T + 8704)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 164 T^{6} + 8532 T^{4} + \cdots + 906304 \) Copy content Toggle raw display
$79$ \( (T^{4} + 8 T^{3} - 280 T^{2} - 1908 T + 10528)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 524 T^{6} + \cdots + 15492096 \) Copy content Toggle raw display
$89$ \( (T^{4} - 6 T^{3} - 160 T^{2} + 352 T + 2592)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 100 T^{6} + 3584 T^{4} + \cdots + 295936 \) Copy content Toggle raw display
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