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Properties

Label 245.4.b.c
Level $245$
Weight $4$
Character orbit 245.b
Analytic conductor $14.455$
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] = [245,4,Mod(99,245)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(245, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.99");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 245.b (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: \(14.4554679514\)
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: \( x^{8} + 202x^{6} + 12253x^{4} + 210844x^{2} + 592900 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 5 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} + \beta_{2} q^{3} + ( - \beta_1 - 9) q^{4} + ( - \beta_{3} + \beta_{2}) q^{5} + ( - \beta_{5} + 2 \beta_{3}) q^{6} + (8 \beta_{6} + \beta_{4}) q^{8} + ( - 3 \beta_1 - 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} + \beta_{2} q^{3} + ( - \beta_1 - 9) q^{4} + ( - \beta_{3} + \beta_{2}) q^{5} + ( - \beta_{5} + 2 \beta_{3}) q^{6} + (8 \beta_{6} + \beta_{4}) q^{8} + ( - 3 \beta_1 - 7) q^{9} + (\beta_{7} - \beta_{5} + \cdots + 5 \beta_{2}) q^{10}+ \cdots + (14 \beta_1 - 280) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 76 q^{4} - 68 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 76 q^{4} - 68 q^{9} - 60 q^{11} - 256 q^{15} + 500 q^{16} + 412 q^{25} - 916 q^{29} - 1388 q^{30} + 3340 q^{36} + 276 q^{39} - 328 q^{44} - 1096 q^{46} - 1364 q^{50} + 2844 q^{51} + 4228 q^{60} - 5764 q^{64} - 1016 q^{65} + 2632 q^{71} - 8984 q^{74} + 1468 q^{79} + 992 q^{81} + 3196 q^{85} - 3920 q^{86} + 1508 q^{95} - 2184 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 202x^{6} + 12253x^{4} + 210844x^{2} + 592900 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{6} + 311\nu^{4} + 12612\nu^{2} + 97115 ) / 5135 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -211\nu^{7} - 53864\nu^{5} - 3938119\nu^{3} - 68723526\nu ) / 12652640 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 53 \nu^{7} + 308 \nu^{6} - 9782 \nu^{5} + 47894 \nu^{4} - 505727 \nu^{3} + 1151458 \nu^{2} + \cdots - 25374580 ) / 3163160 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 16\nu^{7} + 1461\nu^{5} + 118355\nu^{3} + 9602958\nu ) / 790790 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 212 \nu^{7} + 539 \nu^{6} - 39128 \nu^{5} + 281512 \nu^{4} - 2022908 \nu^{3} + 23564079 \nu^{2} + \cdots + 238301910 ) / 6326320 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 53\nu^{7} + 9782\nu^{5} + 505727\nu^{3} + 6138778\nu ) / 1581580 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1697 \nu^{7} - 1232 \nu^{6} - 298288 \nu^{5} - 191576 \nu^{4} - 14268053 \nu^{3} + \cdots + 101498320 ) / 12652640 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + 5\beta_{6} + \beta_{3} + \beta_{2} ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{7} - 18\beta_{3} + 2\beta_{2} + 5\beta _1 - 255 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -77\beta_{7} - 420\beta_{6} + 25\beta_{4} - 77\beta_{3} - 117\beta_{2} ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -236\beta_{7} + 160\beta_{5} + 1804\beta_{3} - 236\beta_{2} - 545\beta _1 + 20645 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 1393\beta_{7} + 7446\beta_{6} - 645\beta_{4} + 1393\beta_{3} + 1577\beta_{2} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 24086\beta_{7} - 24880\beta_{5} - 167014\beta_{3} + 24086\beta_{2} + 66055\beta _1 - 1845055 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -666593\beta_{7} - 3293670\beta_{6} + 356675\beta_{4} - 666593\beta_{3} - 454713\beta_{2} ) / 5 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/245\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(197\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
1.86383i
8.73708i
9.98913i
− 4.73359i
4.73359i
− 9.98913i
− 8.73708i
− 1.86383i
− 5.30045i − 8.20271i −20.0948 9.10783 − 6.48440i −43.4781 0 64.1080i −40.2844 −34.3702 − 48.2756i
99.2 − 5.30045i 8.20271i −20.0948 −9.10783 + 6.48440i 43.4781 0 64.1080i −40.2844 34.3702 + 48.2756i
99.3 − 2.62777i − 1.92758i 1.09481 −9.67199 − 5.60826i −5.06525 0 − 23.8991i 23.2844 −14.7372 + 25.4158i
99.4 − 2.62777i 1.92758i 1.09481 9.67199 + 5.60826i 5.06525 0 − 23.8991i 23.2844 14.7372 − 25.4158i
99.5 2.62777i − 1.92758i 1.09481 9.67199 − 5.60826i 5.06525 0 23.8991i 23.2844 14.7372 + 25.4158i
99.6 2.62777i 1.92758i 1.09481 −9.67199 + 5.60826i −5.06525 0 23.8991i 23.2844 −14.7372 − 25.4158i
99.7 5.30045i − 8.20271i −20.0948 −9.10783 − 6.48440i 43.4781 0 − 64.1080i −40.2844 34.3702 − 48.2756i
99.8 5.30045i 8.20271i −20.0948 9.10783 + 6.48440i −43.4781 0 − 64.1080i −40.2844 −34.3702 + 48.2756i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
7.b odd 2 1 inner
35.c odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 245.4.b.c ✓ 8
5.b even 2 1 inner 245.4.b.c ✓ 8
5.c odd 4 2 1225.4.a.bm 8
7.b odd 2 1 inner 245.4.b.c ✓ 8
7.c even 3 2 245.4.j.c 16
7.d odd 6 2 245.4.j.c 16
35.c odd 2 1 inner 245.4.b.c ✓ 8
35.f even 4 2 1225.4.a.bm 8
35.i odd 6 2 245.4.j.c 16
35.j even 6 2 245.4.j.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.4.b.c ✓ 8 1.a even 1 1 trivial
245.4.b.c ✓ 8 5.b even 2 1 inner
245.4.b.c ✓ 8 7.b odd 2 1 inner
245.4.b.c ✓ 8 35.c odd 2 1 inner
245.4.j.c 16 7.c even 3 2
245.4.j.c 16 7.d odd 6 2
245.4.j.c 16 35.i odd 6 2
245.4.j.c 16 35.j even 6 2
1225.4.a.bm 8 5.c odd 4 2
1225.4.a.bm 8 35.f even 4 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(245, [\chi])\):

\( T_{2}^{4} + 35T_{2}^{2} + 194 \) Copy content Toggle raw display
\( T_{19}^{4} - 20546T_{19}^{2} + 93896000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 35 T^{2} + 194)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} + 71 T^{2} + 250)^{2} \) Copy content Toggle raw display
$5$ \( T^{8} - 206 T^{6} + \cdots + 244140625 \) Copy content Toggle raw display
$7$ \( T^{8} \) Copy content Toggle raw display
$11$ \( (T^{2} + 15 T - 56)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} + 6391 T^{2} + 3906250)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 8701 T^{2} + 17956000)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 20546 T^{2} + 93896000)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 12100 T^{2} + 34876544)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 229 T + 4018)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} - 45736 T^{2} + 3104000)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 154084 T^{2} + 5825897600)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 156776 T^{2} + 3802400000)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 27440 T^{2} + 119243264)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} + 379749 T^{2} + 35988001000)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 261700 T^{2} + 16420160000)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 316226 T^{2} + 24586784000)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 645250 T^{2} + 23765000000)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} + 1025728 T^{2} + 40946180096)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 658 T - 53848)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 610044 T^{2} + 91508356000)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 367 T - 119998)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 1018386 T^{2} + 135536164000)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 1818144 T^{2} + 486756864000)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 1264725 T^{2} + 392832400000)^{2} \) Copy content Toggle raw display
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