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Properties

Label 504.2.ch.a
Level $504$
Weight $2$
Character orbit 504.ch
Analytic conductor $4.024$
Analytic rank $0$
Dimension $8$
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [504,2,Mod(269,504)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(504, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 3, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("504.269");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 504 = 2^{3} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 504.ch (of order \(6\), degree \(2\), 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: \(4.02446026187\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.4857532416.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 7x^{6} - 2x^{5} + 98x^{4} - 98x^{3} + 67x^{2} - 30x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} + ( - 2 \beta_{3} + 2) q^{4} - \beta_{5} q^{5} + ( - \beta_{3} + 3) q^{7} + 2 \beta_{4} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} + ( - 2 \beta_{3} + 2) q^{4} - \beta_{5} q^{5} + ( - \beta_{3} + 3) q^{7} + 2 \beta_{4} q^{8} + (\beta_{7} - \beta_1) q^{10} + ( - \beta_{6} + \beta_{5}) q^{11} + \beta_1 q^{13} + (\beta_{4} + 2 \beta_{2}) q^{14} - 4 \beta_{3} q^{16} + ( - \beta_{4} - \beta_{2}) q^{17} - \beta_{7} q^{19} + 2 \beta_{6} q^{20} + ( - 2 \beta_{7} + \beta_1) q^{22} - \beta_{2} q^{23} + ( - 6 \beta_{3} + 6) q^{25} + 2 \beta_{5} q^{26} + ( - 6 \beta_{3} + 4) q^{28} + (\beta_{6} + 2 \beta_{5}) q^{29} + (3 \beta_{3} + 3) q^{31} + (4 \beta_{4} - 4 \beta_{2}) q^{32} + (4 \beta_{3} - 2) q^{34} + (\beta_{6} - 2 \beta_{5}) q^{35} + (\beta_{7} - 2 \beta_1) q^{37} + ( - 2 \beta_{6} - 2 \beta_{5}) q^{38} + 2 \beta_{7} q^{40} + (4 \beta_{4} - 8 \beta_{2}) q^{41} + (2 \beta_{7} - \beta_1) q^{43} + ( - 4 \beta_{6} - 2 \beta_{5}) q^{44} + (2 \beta_{3} - 2) q^{46} + ( - 6 \beta_{4} + 3 \beta_{2}) q^{47} + ( - 5 \beta_{3} + 8) q^{49} + 6 \beta_{4} q^{50} + ( - 2 \beta_{7} + 2 \beta_1) q^{52} + (\beta_{6} - \beta_{5}) q^{53} + (22 \beta_{3} - 11) q^{55} + (6 \beta_{4} - 2 \beta_{2}) q^{56} + ( - \beta_{7} + 2 \beta_1) q^{58} + (\beta_{6} + \beta_{5}) q^{59} - \beta_{7} q^{61} + ( - 3 \beta_{4} + 6 \beta_{2}) q^{62} - 8 q^{64} - 11 \beta_{2} q^{65} + ( - 4 \beta_{4} + 2 \beta_{2}) q^{68} + (3 \beta_{7} - 2 \beta_1) q^{70} + \beta_{4} q^{71} + ( - 2 \beta_{3} - 2) q^{73} + (2 \beta_{6} - 2 \beta_{5}) q^{74} - 2 \beta_1 q^{76} + ( - 4 \beta_{6} + \beta_{5}) q^{77} + 7 \beta_{3} q^{79} + (4 \beta_{6} + 4 \beta_{5}) q^{80} + (8 \beta_{3} - 16) q^{82} - \beta_{6} q^{83} + ( - 2 \beta_{7} + \beta_1) q^{85} + (4 \beta_{6} + 2 \beta_{5}) q^{86} + ( - 2 \beta_{7} - 2 \beta_1) q^{88} + ( - 8 \beta_{4} + 4 \beta_{2}) q^{89} + ( - \beta_{7} + 3 \beta_1) q^{91} - 2 \beta_{4} q^{92} + (6 \beta_{3} + 6) q^{94} + ( - 11 \beta_{4} + 11 \beta_{2}) q^{95} + ( - 2 \beta_{3} + 1) q^{97} + (5 \beta_{4} + 3 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{4} + 20 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{4} + 20 q^{7} - 16 q^{16} + 24 q^{25} + 8 q^{28} + 36 q^{31} - 8 q^{46} + 44 q^{49} - 64 q^{64} - 24 q^{73} + 28 q^{79} - 96 q^{82} + 72 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 7x^{6} - 2x^{5} + 98x^{4} - 98x^{3} + 67x^{2} - 30x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 33\nu^{7} + 61\nu^{6} - 282\nu^{5} - 294\nu^{4} + 388\nu^{3} - 312\nu^{2} + 153\nu + 44177 ) / 9430 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 3823 \nu^{7} + 12079 \nu^{6} + 22382 \nu^{5} - 30236 \nu^{4} - 414148 \nu^{3} + 781972 \nu^{2} + \cdots + 135243 ) / 84870 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 1508 \nu^{7} + 2499 \nu^{6} + 11172 \nu^{5} + 7434 \nu^{4} - 143178 \nu^{3} + 100842 \nu^{2} + \cdots + 42843 ) / 28290 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 209\nu^{7} - 227\nu^{6} - 1786\nu^{5} - 1862\nu^{4} + 19784\nu^{3} - 1976\nu^{2} + 969\nu + 261 ) / 2070 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 9068 \nu^{7} + 27959 \nu^{6} + 51772 \nu^{5} - 65806 \nu^{4} - 976178 \nu^{3} + 1837142 \nu^{2} + \cdots + 317583 ) / 84870 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -88\nu^{7} + 97\nu^{6} + 752\nu^{5} + 784\nu^{4} - 8278\nu^{3} + 832\nu^{2} - 408\nu - 108 ) / 369 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 7023 \nu^{7} + 11879 \nu^{6} + 51442 \nu^{5} + 32564 \nu^{4} - 668948 \nu^{3} + 471032 \nu^{2} + \cdots + 200643 ) / 28290 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} + \beta_{5} + 2\beta_{4} - \beta_{3} - 2\beta_{2} + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{7} + \beta_{5} + 9\beta_{3} - 2\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 8\beta_{6} + 19\beta_{4} - 3\beta _1 + 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -16\beta_{7} + 17\beta_{6} + 17\beta_{5} + 40\beta_{4} + 75\beta_{3} - 40\beta_{2} + 16\beta _1 - 75 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -45\beta_{7} - 61\beta_{5} + 211\beta_{3} + 143\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 113\beta_{6} + 265\beta_{4} + 55\beta _1 - 258 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -546\beta_{7} - 365\beta_{6} - 365\beta_{5} - 856\beta_{4} + 2561\beta_{3} + 856\beta_{2} + 546\beta _1 - 2561 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/504\mathbb{Z}\right)^\times\).

\(n\) \(73\) \(127\) \(253\) \(281\)
\(\chi(n)\) \(\beta_{3}\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
269.1
2.91089 + 1.10325i
0.0386042 − 0.555062i
0.461396 − 0.310963i
−2.41089 − 1.96928i
2.91089 − 1.10325i
0.0386042 + 0.555062i
0.461396 + 0.310963i
−2.41089 + 1.96928i
−1.22474 + 0.707107i 0 1.00000 − 1.73205i −2.87228 + 1.65831i 0 2.50000 − 0.866025i 2.82843i 0 2.34521 − 4.06202i
269.2 −1.22474 + 0.707107i 0 1.00000 − 1.73205i 2.87228 − 1.65831i 0 2.50000 − 0.866025i 2.82843i 0 −2.34521 + 4.06202i
269.3 1.22474 − 0.707107i 0 1.00000 − 1.73205i −2.87228 + 1.65831i 0 2.50000 − 0.866025i − 2.82843i 0 −2.34521 + 4.06202i
269.4 1.22474 − 0.707107i 0 1.00000 − 1.73205i 2.87228 − 1.65831i 0 2.50000 − 0.866025i − 2.82843i 0 2.34521 − 4.06202i
341.1 −1.22474 − 0.707107i 0 1.00000 + 1.73205i −2.87228 − 1.65831i 0 2.50000 + 0.866025i − 2.82843i 0 2.34521 + 4.06202i
341.2 −1.22474 − 0.707107i 0 1.00000 + 1.73205i 2.87228 + 1.65831i 0 2.50000 + 0.866025i − 2.82843i 0 −2.34521 − 4.06202i
341.3 1.22474 + 0.707107i 0 1.00000 + 1.73205i −2.87228 − 1.65831i 0 2.50000 + 0.866025i 2.82843i 0 −2.34521 − 4.06202i
341.4 1.22474 + 0.707107i 0 1.00000 + 1.73205i 2.87228 + 1.65831i 0 2.50000 + 0.866025i 2.82843i 0 2.34521 + 4.06202i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 269.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
8.b even 2 1 inner
21.g even 6 1 inner
24.h odd 2 1 inner
56.j odd 6 1 inner
168.ba even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 504.2.ch.a ✓ 8
3.b odd 2 1 inner 504.2.ch.a ✓ 8
4.b odd 2 1 2016.2.cp.a 8
7.d odd 6 1 inner 504.2.ch.a ✓ 8
8.b even 2 1 inner 504.2.ch.a ✓ 8
8.d odd 2 1 2016.2.cp.a 8
12.b even 2 1 2016.2.cp.a 8
21.g even 6 1 inner 504.2.ch.a ✓ 8
24.f even 2 1 2016.2.cp.a 8
24.h odd 2 1 inner 504.2.ch.a ✓ 8
28.f even 6 1 2016.2.cp.a 8
56.j odd 6 1 inner 504.2.ch.a ✓ 8
56.m even 6 1 2016.2.cp.a 8
84.j odd 6 1 2016.2.cp.a 8
168.ba even 6 1 inner 504.2.ch.a ✓ 8
168.be odd 6 1 2016.2.cp.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
504.2.ch.a ✓ 8 1.a even 1 1 trivial
504.2.ch.a ✓ 8 3.b odd 2 1 inner
504.2.ch.a ✓ 8 7.d odd 6 1 inner
504.2.ch.a ✓ 8 8.b even 2 1 inner
504.2.ch.a ✓ 8 21.g even 6 1 inner
504.2.ch.a ✓ 8 24.h odd 2 1 inner
504.2.ch.a ✓ 8 56.j odd 6 1 inner
504.2.ch.a ✓ 8 168.ba even 6 1 inner
2016.2.cp.a 8 4.b odd 2 1
2016.2.cp.a 8 8.d odd 2 1
2016.2.cp.a 8 12.b even 2 1
2016.2.cp.a 8 24.f even 2 1
2016.2.cp.a 8 28.f even 6 1
2016.2.cp.a 8 56.m even 6 1
2016.2.cp.a 8 84.j odd 6 1
2016.2.cp.a 8 168.be odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} - 11T_{5}^{2} + 121 \) acting on \(S_{2}^{\mathrm{new}}(504, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( (T^{4} - 11 T^{2} + 121)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} - 5 T + 7)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 33 T^{2} + 1089)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} - 22)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 6 T^{2} + 36)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 22 T^{2} + 484)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} - 2 T^{2} + 4)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} - 33)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 9 T + 27)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 66 T^{2} + 4356)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 96)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 66)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} + 54 T^{2} + 2916)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 33 T^{2} + 1089)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 11 T^{2} + 121)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 22 T^{2} + 484)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} \) Copy content Toggle raw display
$71$ \( (T^{2} + 2)^{4} \) Copy content Toggle raw display
$73$ \( (T^{2} + 6 T + 12)^{4} \) Copy content Toggle raw display
$79$ \( (T^{2} - 7 T + 49)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 11)^{4} \) Copy content Toggle raw display
$89$ \( (T^{4} + 96 T^{2} + 9216)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} + 3)^{4} \) Copy content Toggle raw display
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