Properties

Label 475.2.b.d
Level $475$
Weight $2$
Character orbit 475.b
Analytic conductor $3.793$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [475,2,Mod(324,475)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(475, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("475.324");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 475 = 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 475.b (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: \(3.79289409601\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.350464.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} + 2x^{4} + 2x^{3} + 4x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 95)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} + ( - \beta_{4} + \beta_{2}) q^{3} + \beta_1 q^{4} + (\beta_1 - 1) q^{6} + ( - \beta_{5} + \beta_{2}) q^{7} + \beta_{5} q^{8} + (\beta_{3} - \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} + ( - \beta_{4} + \beta_{2}) q^{3} + \beta_1 q^{4} + (\beta_1 - 1) q^{6} + ( - \beta_{5} + \beta_{2}) q^{7} + \beta_{5} q^{8} + (\beta_{3} - \beta_1 - 1) q^{9} + (\beta_{3} + \beta_1 - 2) q^{11} + (\beta_{5} + \beta_{4} + 2 \beta_{2}) q^{12} + (2 \beta_{5} - \beta_{4} + \beta_{2}) q^{13} + (\beta_{3} + \beta_1 + 2) q^{14} + ( - \beta_{3} + \beta_1 - 1) q^{16} + (\beta_{5} - 2 \beta_{4} - \beta_{2}) q^{17} + (\beta_{5} - \beta_{4} + \beta_{2}) q^{18} + q^{19} + (2 \beta_{3} - 2 \beta_1 - 4) q^{21} + (3 \beta_{5} + 4 \beta_{4} + \beta_{2}) q^{22} + ( - \beta_{5} + 2 \beta_{4} - \beta_{2}) q^{23} + ( - \beta_{3} + 1) q^{24} + ( - 2 \beta_{3} - \beta_1 - 3) q^{26} + ( - 2 \beta_{4} - 2 \beta_{2}) q^{27} + (\beta_{5} + 3 \beta_{2}) q^{28} + (2 \beta_1 + 4) q^{29} + ( - 2 \beta_{3} - 2 \beta_1) q^{31} + (\beta_{5} + 3 \beta_{4} - \beta_{2}) q^{32} + (2 \beta_{5} + 4 \beta_{4} - 2 \beta_{2}) q^{33} + ( - \beta_{3} + \beta_1 - 6) q^{34} + (\beta_{3} - 2 \beta_1 - 4) q^{36} + ( - 4 \beta_{5} + \beta_{4} - 3 \beta_{2}) q^{37} - \beta_{4} q^{38} + ( - \beta_{3} - \beta_1 - 2) q^{39} + 2 \beta_{3} q^{41} + (2 \beta_{5} + 2 \beta_{2}) q^{42} + ( - 3 \beta_{5} - 4 \beta_{4} + 3 \beta_{2}) q^{43} + ( - \beta_{3} - 5 \beta_1 + 2) q^{44} + (\beta_{3} - \beta_1 + 4) q^{46} + ( - \beta_{5} + \beta_{2}) q^{47} + (\beta_{4} + 3 \beta_{2}) q^{48} + ( - 4 \beta_1 - 5) q^{49} + ( - 2 \beta_{3} + 4 \beta_1 + 2) q^{51} + ( - \beta_{5} - \beta_{4}) q^{52} + (2 \beta_{5} + \beta_{4} + 3 \beta_{2}) q^{53} + (2 \beta_1 - 6) q^{54} + (\beta_{3} + \beta_1 + 6) q^{56} + ( - \beta_{4} + \beta_{2}) q^{57} + 2 \beta_{5} q^{58} + ( - 2 \beta_1 + 6) q^{59} + ( - 3 \beta_{3} - \beta_1 - 2) q^{61} + ( - 6 \beta_{5} - 4 \beta_{4} - 2 \beta_{2}) q^{62} + ( - 3 \beta_{5} + 4 \beta_{4} - 9 \beta_{2}) q^{63} + ( - 3 \beta_{3} - 2 \beta_1 + 2) q^{64} + ( - 2 \beta_{3} - 6 \beta_1 + 4) q^{66} + ( - 5 \beta_{4} + \beta_{2}) q^{67} + (\beta_{5} + 4 \beta_{4} - 3 \beta_{2}) q^{68} + 4 q^{69} + 4 \beta_1 q^{71} + (2 \beta_{5} - 2 \beta_{4} + 3 \beta_{2}) q^{72} + (\beta_{5} + 2 \beta_{4} - \beta_{2}) q^{73} + (4 \beta_{3} + 3 \beta_1 + 3) q^{74} + \beta_1 q^{76} + (4 \beta_{4} - 4 \beta_{2}) q^{77} + ( - 3 \beta_{5} - \beta_{2}) q^{78} + (6 \beta_1 + 2) q^{79} + (\beta_{3} + 3 \beta_1 + 1) q^{81} + (4 \beta_{5} + 2 \beta_{2}) q^{82} + ( - 5 \beta_{5} - 2 \beta_{4} - 5 \beta_{2}) q^{83} + (2 \beta_{3} - 6 \beta_1 - 8) q^{84} + (3 \beta_{3} + 7 \beta_1 - 2) q^{86} + (2 \beta_{5} - 2 \beta_{4} + 8 \beta_{2}) q^{87} + ( - \beta_{5} - 4 \beta_{4} + \beta_{2}) q^{88} + ( - 2 \beta_{3} + 4 \beta_1) q^{89} + (4 \beta_{3} + 8) q^{91} + ( - \beta_{5} - 2 \beta_{4} - \beta_{2}) q^{92} + ( - 4 \beta_{5} - 4 \beta_{4}) q^{93} + (\beta_{3} + \beta_1 + 2) q^{94} + ( - 2 \beta_{3} - \beta_1 + 7) q^{96} + ( - 2 \beta_{5} + 7 \beta_{4} - 7 \beta_{2}) q^{97} + ( - 4 \beta_{5} - 3 \beta_{4}) q^{98} + ( - \beta_{3} + 3 \beta_1 + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{4} - 8 q^{6} - 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{4} - 8 q^{6} - 6 q^{9} - 16 q^{11} + 8 q^{14} - 6 q^{16} + 6 q^{19} - 24 q^{21} + 8 q^{24} - 12 q^{26} + 20 q^{29} + 8 q^{31} - 36 q^{34} - 22 q^{36} - 8 q^{39} - 4 q^{41} + 24 q^{44} + 24 q^{46} - 22 q^{49} + 8 q^{51} - 40 q^{54} + 32 q^{56} + 40 q^{59} - 4 q^{61} + 22 q^{64} + 40 q^{66} + 24 q^{69} - 8 q^{71} + 4 q^{74} - 2 q^{76} - 2 q^{81} - 40 q^{84} - 32 q^{86} - 4 q^{89} + 40 q^{91} + 8 q^{94} + 48 q^{96} + 32 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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

\(\beta_{1}\)\(=\) \( ( -4\nu^{5} + 9\nu^{4} - 16\nu^{3} - 4\nu^{2} + 8\nu - 9 ) / 23 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -7\nu^{5} + 10\nu^{4} - 5\nu^{3} - 7\nu^{2} - 32\nu + 13 ) / 23 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -6\nu^{5} + 25\nu^{4} - 24\nu^{3} - 6\nu^{2} + 12\nu + 67 ) / 23 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -11\nu^{5} + 19\nu^{4} - 21\nu^{3} - 11\nu^{2} - 70\nu + 27 ) / 23 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 21\nu^{5} - 30\nu^{4} + 15\nu^{3} + 67\nu^{2} + 96\nu - 39 ) / 23 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{4} + \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} + 3\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{4} + \beta_{3} + 4\beta_{2} - 3\beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{3} - 3\beta _1 - 7 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -\beta_{5} + 6\beta_{4} + 5\beta_{3} - 17\beta_{2} - 11\beta _1 - 18 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/475\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
324.1
−0.854638 0.854638i
0.403032 0.403032i
1.45161 + 1.45161i
1.45161 1.45161i
0.403032 + 0.403032i
−0.854638 + 0.854638i
2.17009i 1.70928i −2.70928 0 −3.70928 1.07838i 1.53919i 0.0783777 0
324.2 1.48119i 0.806063i −0.193937 0 −1.19394 3.35026i 2.67513i 2.35026 0
324.3 0.311108i 2.90321i 1.90321 0 0.903212 4.42864i 1.21432i −5.42864 0
324.4 0.311108i 2.90321i 1.90321 0 0.903212 4.42864i 1.21432i −5.42864 0
324.5 1.48119i 0.806063i −0.193937 0 −1.19394 3.35026i 2.67513i 2.35026 0
324.6 2.17009i 1.70928i −2.70928 0 −3.70928 1.07838i 1.53919i 0.0783777 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 324.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 475.2.b.d 6
5.b even 2 1 inner 475.2.b.d 6
5.c odd 4 1 95.2.a.a 3
5.c odd 4 1 475.2.a.f 3
15.e even 4 1 855.2.a.i 3
15.e even 4 1 4275.2.a.bk 3
20.e even 4 1 1520.2.a.p 3
20.e even 4 1 7600.2.a.bx 3
35.f even 4 1 4655.2.a.u 3
40.i odd 4 1 6080.2.a.bo 3
40.k even 4 1 6080.2.a.by 3
95.g even 4 1 1805.2.a.f 3
95.g even 4 1 9025.2.a.bb 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
95.2.a.a 3 5.c odd 4 1
475.2.a.f 3 5.c odd 4 1
475.2.b.d 6 1.a even 1 1 trivial
475.2.b.d 6 5.b even 2 1 inner
855.2.a.i 3 15.e even 4 1
1520.2.a.p 3 20.e even 4 1
1805.2.a.f 3 95.g even 4 1
4275.2.a.bk 3 15.e even 4 1
4655.2.a.u 3 35.f even 4 1
6080.2.a.bo 3 40.i odd 4 1
6080.2.a.by 3 40.k even 4 1
7600.2.a.bx 3 20.e even 4 1
9025.2.a.bb 3 95.g even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 7T_{2}^{4} + 11T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(475, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 7 T^{4} + 11 T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{6} + 12 T^{4} + 32 T^{2} + 16 \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + 32 T^{4} + 256 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$11$ \( (T^{3} + 8 T^{2} + 8 T - 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + 40 T^{4} + 80 T^{2} + 16 \) Copy content Toggle raw display
$17$ \( T^{6} + 76 T^{4} + 1712 T^{2} + \cdots + 10816 \) Copy content Toggle raw display
$19$ \( (T - 1)^{6} \) Copy content Toggle raw display
$23$ \( T^{6} + 32 T^{4} + 192 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$29$ \( (T^{3} - 10 T^{2} + 12 T + 40)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} - 4 T^{2} - 48 T + 64)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + 152 T^{4} + 5616 T^{2} + \cdots + 59536 \) Copy content Toggle raw display
$41$ \( (T^{3} + 2 T^{2} - 36 T - 104)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + 304 T^{4} + 25472 T^{2} + \cdots + 350464 \) Copy content Toggle raw display
$47$ \( T^{6} + 32 T^{4} + 256 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$53$ \( T^{6} + 104 T^{4} + 2832 T^{2} + \cdots + 8464 \) Copy content Toggle raw display
$59$ \( (T^{3} - 20 T^{2} + 112 T - 160)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + 2 T^{2} - 84 T + 232)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + 156 T^{4} + 5312 T^{2} + \cdots + 13456 \) Copy content Toggle raw display
$71$ \( (T^{3} + 4 T^{2} - 80 T - 64)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + 44 T^{4} + 432 T^{2} + \cdots + 64 \) Copy content Toggle raw display
$79$ \( (T^{3} - 192 T + 160)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + 368 T^{4} + 38976 T^{2} + \cdots + 1149184 \) Copy content Toggle raw display
$89$ \( (T^{3} + 2 T^{2} - 132 T - 680)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + 520 T^{4} + 73520 T^{2} + \cdots + 3055504 \) Copy content Toggle raw display
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