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Properties

Label 245.4.b.g
Level $245$
Weight $4$
Character orbit 245.b
Analytic conductor $14.455$
Analytic rank $0$
Dimension $12$
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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 58x^{10} + 1421x^{8} + 14900x^{6} + 72364x^{4} - 114800x^{2} + 810000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 7^{2} \)
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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} - \beta_{2} q^{3} + (\beta_{5} - 4) q^{4} + ( - \beta_{9} - \beta_{8} - \beta_{3}) q^{5} + (\beta_{8} - \beta_{7}) q^{6} + ( - \beta_{11} + 3 \beta_{4} - \beta_1) q^{8} + ( - \beta_{6} - \beta_{5} + 12) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} - \beta_{2} q^{3} + (\beta_{5} - 4) q^{4} + ( - \beta_{9} - \beta_{8} - \beta_{3}) q^{5} + (\beta_{8} - \beta_{7}) q^{6} + ( - \beta_{11} + 3 \beta_{4} - \beta_1) q^{8} + ( - \beta_{6} - \beta_{5} + 12) q^{9} + (3 \beta_{10} + 2 \beta_{9} + \cdots - 4 \beta_{2}) q^{10}+ \cdots + (8 \beta_{6} + 60 \beta_{5} - 312) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 44 q^{4} + 140 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 44 q^{4} + 140 q^{9} - 24 q^{11} - 136 q^{15} + 84 q^{16} + 584 q^{25} + 296 q^{29} - 632 q^{30} - 1100 q^{36} - 184 q^{39} + 2760 q^{44} - 2440 q^{46} + 540 q^{50} - 1928 q^{51} - 1000 q^{60} + 2332 q^{64} - 588 q^{65} - 1840 q^{71} + 8656 q^{74} + 5032 q^{79} - 1284 q^{81} + 516 q^{85} - 1680 q^{86} - 6416 q^{95} - 3504 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 58x^{10} + 1421x^{8} + 14900x^{6} + 72364x^{4} - 114800x^{2} + 810000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 23191 \nu^{11} + 13342472 \nu^{9} + 1120067089 \nu^{7} + 30761688250 \nu^{5} + \cdots + 389884119200 \nu ) / 234565218000 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 108063 \nu^{10} + 11332414 \nu^{8} + 410195783 \nu^{6} + 6694410740 \nu^{4} + \cdots - 2213425400 ) / 36487922800 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 382323 \nu^{10} + 24645994 \nu^{8} + 687990643 \nu^{6} + 8545099440 \nu^{4} + \cdots - 5163889000 ) / 36487922800 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 555757 \nu^{11} - 30470806 \nu^{9} - 704143397 \nu^{7} - 6494955200 \nu^{5} + \cdots - 77286349000 \nu ) / 234565218000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -1959\nu^{10} - 95097\nu^{8} - 1984249\nu^{6} - 13219205\nu^{4} + 26449604\nu^{2} + 1324214840 ) / 130314010 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 23679\nu^{10} + 1349027\nu^{8} + 29571929\nu^{6} + 203688005\nu^{4} - 412300804\nu^{2} - 7215061200 ) / 651570050 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 3890299 \nu^{11} - 213295642 \nu^{9} - 4929003779 \nu^{7} - 45464686400 \nu^{5} + \cdots + 1100952083000 \nu ) / 234565218000 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 4239029 \nu^{11} - 256167332 \nu^{9} - 6443014309 \nu^{7} - 67425529150 \nu^{5} + \cdots + 1377957332800 \nu ) / 234565218000 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 12475771 \nu^{11} + 16578720 \nu^{10} + 703404793 \nu^{9} + 948476160 \nu^{8} + \cdots - 37170171000 ) / 820978263000 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 12475771 \nu^{11} + 16578720 \nu^{10} - 703404793 \nu^{9} + 948476160 \nu^{8} + \cdots - 37170171000 ) / 820978263000 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 3179203 \nu^{11} - 198233749 \nu^{9} - 5274803063 \nu^{7} - 65731946975 \nu^{5} + \cdots - 800334405100 \nu ) / 117282609000 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - 7\beta_{4} ) / 7 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + 2\beta_{3} - 2\beta_{2} - 10 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -7\beta_{11} + 21\beta_{10} - 21\beta_{9} - 40\beta_{7} + 91\beta_{4} - 7\beta_1 ) / 7 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 12\beta_{10} + 12\beta_{9} - 5\beta_{6} - 15\beta_{5} - 64\beta_{3} + 48\beta_{2} + 92 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 49\beta_{11} - 805\beta_{10} + 805\beta_{9} + 350\beta_{8} + 1054\beta_{7} - 287\beta_{4} + 119\beta_1 ) / 7 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -426\beta_{10} - 426\beta_{9} + 25\beta_{6} - 83\beta_{5} + 1668\beta_{3} - 820\beta_{2} + 1268 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2289 \beta_{11} + 20041 \beta_{10} - 20041 \beta_{9} - 12250 \beta_{8} - 21846 \beta_{7} + \cdots + 2779 \beta_1 ) / 7 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 9288\beta_{10} + 9288\beta_{9} + 3665\beta_{6} + 13981\beta_{5} - 31696\beta_{3} + 11472\beta_{2} - 104436 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 133371 \beta_{11} - 319053 \beta_{10} + 319053 \beta_{9} + 220290 \beta_{8} + 314470 \beta_{7} + \cdots - 208201 \beta_1 ) / 7 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 100358 \beta_{10} - 100358 \beta_{9} - 169495 \beta_{6} - 546419 \beta_{5} + 308012 \beta_{3} + \cdots + 3705500 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 4196297 \beta_{11} + 438669 \beta_{10} - 438669 \beta_{9} - 647570 \beta_{8} + 18666 \beta_{7} + \cdots + 6860147 \beta_1 ) / 7 \) 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
1.41421 − 4.85259i
−1.41421 − 4.85259i
1.41421 − 3.26688i
−1.41421 − 3.26688i
−1.41421 − 0.883124i
1.41421 − 0.883124i
1.41421 + 0.883124i
−1.41421 + 0.883124i
−1.41421 + 3.26688i
1.41421 + 3.26688i
−1.41421 + 4.85259i
1.41421 + 4.85259i
− 4.85259i − 1.14521i −15.5476 −11.0934 − 1.39125i −5.55723 0 36.6254i 25.6885 −6.75118 + 53.8319i
99.2 − 4.85259i 1.14521i −15.5476 11.0934 + 1.39125i 5.55723 0 36.6254i 25.6885 6.75118 − 53.8319i
99.3 − 3.26688i − 5.72526i −2.67248 11.1571 + 0.720232i −18.7037 0 − 17.4043i −5.77855 2.35291 − 36.4489i
99.4 − 3.26688i 5.72526i −2.67248 −11.1571 − 0.720232i 18.7037 0 − 17.4043i −5.77855 −2.35291 + 36.4489i
99.5 − 0.883124i − 3.45108i 7.22009 3.59921 − 10.5852i −3.04773 0 − 13.4412i 15.0901 −9.34802 − 3.17855i
99.6 − 0.883124i 3.45108i 7.22009 −3.59921 + 10.5852i 3.04773 0 − 13.4412i 15.0901 9.34802 + 3.17855i
99.7 0.883124i − 3.45108i 7.22009 −3.59921 − 10.5852i 3.04773 0 13.4412i 15.0901 9.34802 − 3.17855i
99.8 0.883124i 3.45108i 7.22009 3.59921 + 10.5852i −3.04773 0 13.4412i 15.0901 −9.34802 + 3.17855i
99.9 3.26688i − 5.72526i −2.67248 −11.1571 + 0.720232i 18.7037 0 17.4043i −5.77855 −2.35291 − 36.4489i
99.10 3.26688i 5.72526i −2.67248 11.1571 − 0.720232i −18.7037 0 17.4043i −5.77855 2.35291 + 36.4489i
99.11 4.85259i − 1.14521i −15.5476 11.0934 − 1.39125i 5.55723 0 − 36.6254i 25.6885 6.75118 + 53.8319i
99.12 4.85259i 1.14521i −15.5476 −11.0934 + 1.39125i −5.55723 0 − 36.6254i 25.6885 −6.75118 − 53.8319i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.12
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.g ✓ 12
5.b even 2 1 inner 245.4.b.g ✓ 12
5.c odd 4 2 1225.4.a.bs 12
7.b odd 2 1 inner 245.4.b.g ✓ 12
7.c even 3 2 245.4.j.g 24
7.d odd 6 2 245.4.j.g 24
35.c odd 2 1 inner 245.4.b.g ✓ 12
35.f even 4 2 1225.4.a.bs 12
35.i odd 6 2 245.4.j.g 24
35.j even 6 2 245.4.j.g 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.4.b.g ✓ 12 1.a even 1 1 trivial
245.4.b.g ✓ 12 5.b even 2 1 inner
245.4.b.g ✓ 12 7.b odd 2 1 inner
245.4.b.g ✓ 12 35.c odd 2 1 inner
245.4.j.g 24 7.c even 3 2
245.4.j.g 24 7.d odd 6 2
245.4.j.g 24 35.i odd 6 2
245.4.j.g 24 35.j even 6 2
1225.4.a.bs 12 5.c odd 4 2
1225.4.a.bs 12 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 \(S_{4}^{\mathrm{new}}(245, [\chi])\):

\( T_{2}^{6} + 35T_{2}^{4} + 278T_{2}^{2} + 196 \) Copy content Toggle raw display
\( T_{19}^{6} - 19440T_{19}^{4} + 1000512T_{19}^{2} - 4917248 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + 35 T^{4} + \cdots + 196)^{2} \) Copy content Toggle raw display
$3$ \( (T^{6} + 46 T^{4} + \cdots + 512)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 3814697265625 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{3} + 6 T^{2} + \cdots + 11340)^{4} \) Copy content Toggle raw display
$13$ \( (T^{6} + 4280 T^{4} + \cdots + 85621698)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + 15704 T^{4} + \cdots + 1365240258)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} - 19440 T^{4} + \cdots - 4917248)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 1936238854144)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} - 74 T^{2} + \cdots - 1629180)^{4} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 1421945497728)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 313315913335824)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 25\!\cdots\!48)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 59\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 23\!\cdots\!84)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 41\!\cdots\!92)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} - 69794 T^{4} + \cdots - 10292107392)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 323613907089984)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} + 460 T^{2} + \cdots - 23030496)^{4} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 10\!\cdots\!52)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 1258 T^{2} + \cdots + 79678656)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 63\!\cdots\!08)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 50\!\cdots\!58)^{2} \) Copy content Toggle raw display
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