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Properties

Label 245.6.b.c
Level $245$
Weight $6$
Character orbit 245.b
Analytic conductor $39.294$
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] = [245,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("245.99");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 245 = 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
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: \(39.2940358542\)
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: \( x^{12} + 2408 x^{10} + 1934778 x^{8} + 685997236 x^{6} + 109856760265 x^{4} + 6693959319900 x^{2} + 67265422402500 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 5^{3}\cdot 11^{2} \)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} - \beta_{4} q^{3} + (\beta_1 - 20) q^{4} - \beta_{7} q^{5} + ( - \beta_{7} - \beta_{6}) q^{6} + (\beta_{10} + 5 \beta_{3}) q^{8} + (\beta_{2} - 3 \beta_1 - 106) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} - \beta_{4} q^{3} + (\beta_1 - 20) q^{4} - \beta_{7} q^{5} + ( - \beta_{7} - \beta_{6}) q^{6} + (\beta_{10} + 5 \beta_{3}) q^{8} + (\beta_{2} - 3 \beta_1 - 106) q^{9} + ( - \beta_{9} - \beta_{8} + \cdots + 7 \beta_{4}) q^{10}+ \cdots + ( - 346 \beta_{2} + 262 \beta_1 + 81702) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 236 q^{4} - 1280 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 236 q^{4} - 1280 q^{9} - 12 q^{11} - 820 q^{15} - 4524 q^{16} + 9260 q^{25} + 8372 q^{29} + 27820 q^{30} - 8996 q^{36} + 11484 q^{39} + 51464 q^{44} - 36376 q^{46} + 45700 q^{50} - 198516 q^{51} - 90620 q^{60} + 64156 q^{64} + 137140 q^{65} - 466736 q^{71} + 56824 q^{74} + 107876 q^{79} + 369068 q^{81} + 356780 q^{85} - 76400 q^{86} - 436480 q^{95} + 980088 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 2408 x^{10} + 1934778 x^{8} + 685997236 x^{6} + 109856760265 x^{4} + 6693959319900 x^{2} + 67265422402500 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 263634983 \nu^{10} - 573801260176 \nu^{8} - 380082755735263 \nu^{6} + \cdots - 17\!\cdots\!00 ) / 19\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 20370187 \nu^{10} - 40734113464 \nu^{8} - 22933286415907 \nu^{6} + \cdots + 75\!\cdots\!00 ) / 56\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 64673274406 \nu^{11} - 151492415719877 \nu^{9} + \cdots - 17\!\cdots\!00 \nu ) / 45\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 64673274406 \nu^{11} - 151492415719877 \nu^{9} + \cdots - 12\!\cdots\!00 \nu ) / 45\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 22181094430091 \nu^{11} + 129155610821553 \nu^{10} + \cdots - 44\!\cdots\!00 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 110905472150455 \nu^{11} + 214218646941627 \nu^{10} + \cdots + 90\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 22181094430091 \nu^{11} + 129155610821553 \nu^{10} + \cdots - 44\!\cdots\!00 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 291457994842259 \nu^{11} + \cdots - 82\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 277100527924127 \nu^{11} + \cdots - 63\!\cdots\!00 \nu ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 919401732771 \nu^{11} + \cdots - 20\!\cdots\!00 \nu ) / 30\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 10131448417 \nu^{11} - 24061436453524 \nu^{9} + \cdots - 18\!\cdots\!50 \nu ) / 10\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_{4} - \beta_{3} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 2\beta_{7} + 2\beta_{6} + \beta_{2} - 2\beta _1 - 401 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -6\beta_{11} - 26\beta_{10} + 33\beta_{9} + 21\beta_{7} - 21\beta_{5} - 784\beta_{4} + 951\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 72 \beta_{9} - 144 \beta_{8} - 3892 \beta_{7} - 2508 \beta_{6} - 1384 \beta_{5} + 72 \beta_{4} + \cdots + 321511 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 10266 \beta_{11} + 36396 \beta_{10} - 46955 \beta_{9} - 26981 \beta_{7} + 26981 \beta_{5} + \cdots - 1172961 \beta_{3} \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 98364 \beta_{9} + 196728 \beta_{8} + 5463762 \beta_{7} + 3159574 \beta_{6} + 2304188 \beta_{5} + \cdots - 340828661 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 13902066 \beta_{11} - 46242206 \beta_{10} + 58910967 \beta_{9} + 34053771 \beta_{7} + \cdots + 1474843071 \beta_{3} \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 121877616 \beta_{9} - 243755232 \beta_{8} - 7039593032 \beta_{7} - 3958247320 \beta_{6} + \cdots + 400631424911 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 17694850666 \beta_{11} + 57717602616 \beta_{10} - 72774041139 \beta_{9} - 42481645161 \beta_{7} + \cdots - 1842600196281 \beta_{3} \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 150360651828 \beta_{9} + 300721303656 \beta_{8} + 8828952635502 \beta_{7} + 4927171066746 \beta_{6} + \cdots - 487363129183461 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 22096490279066 \beta_{11} - 71652944822626 \beta_{10} + 89844868070771 \beta_{9} + \cdots + 22\!\cdots\!91 \beta_{3} \) 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
− 3.51767i
− 15.4014i
21.1807i
− 35.1720i
10.7536i
− 18.8969i
18.8969i
− 10.7536i
35.1720i
− 21.1807i
15.4014i
3.51767i
− 9.45952i − 5.94185i −57.4825 20.1522 + 52.1429i −56.2070 0 241.053i 207.694 493.247 − 190.631i
99.2 − 9.45952i 5.94185i −57.4825 −20.1522 − 52.1429i 56.2070 0 241.053i 207.694 −493.247 + 190.631i
99.3 − 6.99564i − 28.1764i −16.9390 −55.7986 − 3.39325i −197.112 0 − 105.361i −550.907 −23.7380 + 390.347i
99.4 − 6.99564i 28.1764i −16.9390 55.7986 + 3.39325i 197.112 0 − 105.361i −550.907 23.7380 − 390.347i
99.5 − 4.07166i − 14.8252i 15.4216 48.2224 − 28.2772i −60.3633 0 − 193.085i 23.2123 −115.135 − 196.345i
99.6 − 4.07166i 14.8252i 15.4216 −48.2224 + 28.2772i 60.3633 0 − 193.085i 23.2123 115.135 + 196.345i
99.7 4.07166i − 14.8252i 15.4216 −48.2224 − 28.2772i 60.3633 0 193.085i 23.2123 115.135 − 196.345i
99.8 4.07166i 14.8252i 15.4216 48.2224 + 28.2772i −60.3633 0 193.085i 23.2123 −115.135 + 196.345i
99.9 6.99564i − 28.1764i −16.9390 55.7986 − 3.39325i 197.112 0 105.361i −550.907 23.7380 + 390.347i
99.10 6.99564i 28.1764i −16.9390 −55.7986 + 3.39325i −197.112 0 105.361i −550.907 −23.7380 − 390.347i
99.11 9.45952i − 5.94185i −57.4825 −20.1522 + 52.1429i 56.2070 0 − 241.053i 207.694 −493.247 − 190.631i
99.12 9.45952i 5.94185i −57.4825 20.1522 − 52.1429i −56.2070 0 − 241.053i 207.694 493.247 + 190.631i
\(n\): 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 245.6.b.c ✓ 12
5.b even 2 1 inner 245.6.b.c ✓ 12
7.b odd 2 1 inner 245.6.b.c ✓ 12
35.c odd 2 1 inner 245.6.b.c ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
245.6.b.c ✓ 12 1.a even 1 1 trivial
245.6.b.c ✓ 12 5.b even 2 1 inner
245.6.b.c ✓ 12 7.b odd 2 1 inner
245.6.b.c ✓ 12 35.c 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 \(S_{6}^{\mathrm{new}}(245, [\chi])\):

\( T_{2}^{6} + 155T_{2}^{4} + 6674T_{2}^{2} + 72600 \) Copy content Toggle raw display
\( T_{19}^{6} - 5927156T_{19}^{4} + 8181417424100T_{19}^{2} - 381436409323200000 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} + 155 T^{4} + \cdots + 72600)^{2} \) Copy content Toggle raw display
$3$ \( (T^{6} + 1049 T^{4} + \cdots + 6160500)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 93\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{12} \) Copy content Toggle raw display
$11$ \( (T^{3} + 3 T^{2} + \cdots + 1878912)^{4} \) Copy content Toggle raw display
$13$ \( (T^{6} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots - 38\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{3} - 2093 T^{2} + \cdots + 11335647708)^{4} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots - 42\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots + 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 52\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots + 34\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} + \cdots - 83054594482416)^{4} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 60\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} + \cdots + 100888489425988)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 92\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
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