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Properties

Label 240.7.c.c
Level $240$
Weight $7$
Character orbit 240.c
Analytic conductor $55.213$
Analytic rank $0$
Dimension $8$
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] = [240,7,Mod(209,240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("240.209");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 240 = 2^{4} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 240.c (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: \(55.2129800688\)
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: \( x^{8} + 60x^{6} + 7774x^{4} + 206220x^{2} + 5736025 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{7}\cdot 3^{6}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 15)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_1) q^{3} + ( - \beta_{6} - 8 \beta_{5} + 3 \beta_{3}) q^{5} + (\beta_{4} - 3 \beta_{3} - 6 \beta_1) q^{7} + (3 \beta_{7} - 3 \beta_{6} + \cdots - 249) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_1) q^{3} + ( - \beta_{6} - 8 \beta_{5} + 3 \beta_{3}) q^{5} + (\beta_{4} - 3 \beta_{3} - 6 \beta_1) q^{7} + (3 \beta_{7} - 3 \beta_{6} + \cdots - 249) q^{9}+ \cdots + ( - 1134 \beta_{7} - 2556 \beta_{6} + \cdots - 789723) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 1980 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 1980 q^{9} - 1980 q^{15} + 28520 q^{19} + 24228 q^{21} + 800 q^{25} - 5896 q^{31} - 233568 q^{39} - 159300 q^{45} - 1041472 q^{49} - 66888 q^{51} - 442800 q^{55} + 1493416 q^{61} + 1502532 q^{69} + 2574000 q^{75} + 1286360 q^{79} - 1752192 q^{81} + 1651240 q^{85} - 383616 q^{91} - 6322320 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 60x^{6} + 7774x^{4} + 206220x^{2} + 5736025 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 13 \nu^{7} + 1437 \nu^{6} + 1615 \nu^{5} + 64665 \nu^{4} + 198313 \nu^{3} + 11638263 \nu^{2} + \cdots + 164874195 ) / 10346400 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} + 30\nu^{2} + 3437 ) / 36 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 13\nu^{7} - 1615\nu^{5} - 198313\nu^{3} - 1771445\nu ) / 5173200 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} - 45\nu^{4} - 4724\nu^{2} - 64110 ) / 225 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 73\nu^{7} + 1985\nu^{5} + 411827\nu^{3} - 606845\nu ) / 5173200 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{7} + 60\nu^{5} + 10169\nu^{3} + 666060\nu ) / 43110 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 61\nu^{7} + 6055\nu^{5} + 4790\nu^{4} + 433499\nu^{3} + 143700\nu^{2} + 20279345\nu + 16635670 ) / 344880 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} - 2\beta_{5} + 2\beta_{3} ) / 15 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} + 16\beta_{3} + 32\beta _1 - 225 ) / 15 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -10\beta_{7} + 33\beta_{6} + 119\beta_{5} - 269\beta_{3} + 5\beta_{2} + 5 ) / 15 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -2\beta_{4} - 32\beta_{3} + 36\beta_{2} - 64\beta _1 - 2987 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1380\beta_{7} - 5399\beta_{6} - 7712\beta_{5} - 3988\beta_{3} - 690\beta_{2} - 690 ) / 15 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -6749\beta_{4} - 53984\beta_{3} - 24300\beta_{2} - 107968\beta _1 + 2117475 ) / 15 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 18890\beta_{7} - 31047\beta_{6} + 584729\beta_{5} + 1642621\beta_{3} - 9445\beta_{2} - 9445 ) / 15 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/240\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(97\) \(161\) \(181\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
209.1
5.55992 − 6.77589i
5.55992 + 6.77589i
−2.84382 + 4.80492i
−2.84382 − 4.80492i
2.84382 − 4.80492i
2.84382 + 4.80492i
−5.55992 + 6.77589i
−5.55992 − 6.77589i
0 −19.8701 − 18.2805i 0 72.7643 + 101.638i 0 − 435.542i 0 60.6432 + 726.473i 0
209.2 0 −19.8701 + 18.2805i 0 72.7643 − 101.638i 0 435.542i 0 60.6432 − 726.473i 0
209.3 0 −9.31012 − 25.3441i 0 −102.129 − 72.0738i 0 553.145i 0 −555.643 + 471.913i 0
209.4 0 −9.31012 + 25.3441i 0 −102.129 + 72.0738i 0 − 553.145i 0 −555.643 − 471.913i 0
209.5 0 9.31012 − 25.3441i 0 102.129 + 72.0738i 0 553.145i 0 −555.643 − 471.913i 0
209.6 0 9.31012 + 25.3441i 0 102.129 − 72.0738i 0 − 553.145i 0 −555.643 + 471.913i 0
209.7 0 19.8701 − 18.2805i 0 −72.7643 − 101.638i 0 − 435.542i 0 60.6432 − 726.473i 0
209.8 0 19.8701 + 18.2805i 0 −72.7643 + 101.638i 0 435.542i 0 60.6432 + 726.473i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 209.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 240.7.c.c 8
3.b odd 2 1 inner 240.7.c.c 8
4.b odd 2 1 15.7.d.c ✓ 8
5.b even 2 1 inner 240.7.c.c 8
12.b even 2 1 15.7.d.c ✓ 8
15.d odd 2 1 inner 240.7.c.c 8
20.d odd 2 1 15.7.d.c ✓ 8
20.e even 4 2 75.7.c.e 8
60.h even 2 1 15.7.d.c ✓ 8
60.l odd 4 2 75.7.c.e 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.7.d.c ✓ 8 4.b odd 2 1
15.7.d.c ✓ 8 12.b even 2 1
15.7.d.c ✓ 8 20.d odd 2 1
15.7.d.c ✓ 8 60.h even 2 1
75.7.c.e 8 20.e even 4 2
75.7.c.e 8 60.l odd 4 2
240.7.c.c 8 1.a even 1 1 trivial
240.7.c.c 8 3.b odd 2 1 inner
240.7.c.c 8 5.b even 2 1 inner
240.7.c.c 8 15.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{7}^{\mathrm{new}}(240, [\chi])\):

\( T_{7}^{4} + 495666T_{7}^{2} + 58041360000 \) Copy content Toggle raw display
\( T_{17}^{4} - 10866836T_{17}^{2} + 28015260981760 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 282429536481 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 59\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{4} + 495666 T^{2} + 58041360000)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 1754154144000)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots + 3379669401600)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 28015260981760)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 7130 T + 12704536)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} + \cdots + 11\!\cdots\!40)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 85\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 1474 T - 159937856)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + \cdots + 30\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + \cdots + 56\!\cdots\!60)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 13\!\cdots\!60)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 64\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 373354 T + 19740227104)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 32\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 37\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 321590 T - 331589130704)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 80\!\cdots\!60)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 39\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 25\!\cdots\!00)^{2} \) Copy content Toggle raw display
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