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Properties

Label 675.2.f.g
Level $675$
Weight $2$
Character orbit 675.f
Analytic conductor $5.390$
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] = [675,2,Mod(107,675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(675, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([2, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("675.107");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.f (of order \(4\), 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: \(5.38990213644\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(i)\)
Coefficient field: 8.0.33973862400.8
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 44x^{4} + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 135)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{5} + \beta_{4} + 2 \beta_{2}) q^{4} + (\beta_{4} + \beta_{2} - 1) q^{7} + (\beta_{7} + \beta_{6} + 2 \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{5} + \beta_{4} + 2 \beta_{2}) q^{4} + (\beta_{4} + \beta_{2} - 1) q^{7} + (\beta_{7} + \beta_{6} + 2 \beta_{3}) q^{8} + \beta_{7} q^{11} + ( - \beta_{5} + 2 \beta_{2} + 2) q^{13} + (\beta_{6} + 2 \beta_{3} - 2 \beta_1) q^{14} + ( - 2 \beta_{5} + 2 \beta_{4} - 2) q^{16} + (\beta_{7} - \beta_{6}) q^{17} - 3 \beta_{2} q^{19} + (2 \beta_{4} - \beta_{2} + 1) q^{22} + \beta_{3} q^{23} + ( - \beta_{7} + \beta_{3} + \beta_1) q^{26} + ( - 4 \beta_{5} - 5 \beta_{2} - 5) q^{28} + (\beta_{6} - 2 \beta_{3} + 2 \beta_1) q^{29} + ( - 3 \beta_{5} + 3 \beta_{4} + 2) q^{31} - 2 \beta_1 q^{32} + (2 \beta_{5} + 2 \beta_{4} - 2 \beta_{2}) q^{34} + ( - 3 \beta_{4} - 2 \beta_{2} + 2) q^{37} - 3 \beta_{3} q^{38} + (\beta_{7} + 3 \beta_{3} + 3 \beta_1) q^{41} + ( - 2 \beta_{5} - \beta_{2} - 1) q^{43} + (\beta_{3} - \beta_1) q^{44} + ( - \beta_{5} + \beta_{4} - 4) q^{46} + 4 \beta_1 q^{47} + ( - 2 \beta_{5} - 2 \beta_{4} + 2 \beta_{2}) q^{49} + (2 \beta_{4} + \beta_{2} - 1) q^{52} + ( - 2 \beta_{7} - 2 \beta_{6} - 3 \beta_{3}) q^{53} + ( - 2 \beta_{7} - 5 \beta_{3} - 5 \beta_1) q^{56} + (2 \beta_{5} + 9 \beta_{2} + 9) q^{58} + ( - 4 \beta_{6} - \beta_{3} + \beta_1) q^{59} - q^{61} + ( - 3 \beta_{7} + 3 \beta_{6} - 4 \beta_1) q^{62} + (2 \beta_{5} + 2 \beta_{4} - 4 \beta_{2}) q^{64} + ( - \beta_{4} - 5 \beta_{2} + 5) q^{67} + 2 \beta_{3} q^{68} + (\beta_{7} - 3 \beta_{3} - 3 \beta_1) q^{71} + (3 \beta_{5} - \beta_{2} - 1) q^{73} + ( - 3 \beta_{6} - 5 \beta_{3} + 5 \beta_1) q^{74} + (3 \beta_{5} - 3 \beta_{4} + 6) q^{76} - \beta_1 q^{77} + ( - \beta_{5} - \beta_{4} + 3 \beta_{2}) q^{79} + (8 \beta_{4} + 11 \beta_{2} - 11) q^{82} + ( - \beta_{7} - \beta_{6} - \beta_{3}) q^{83} + ( - 2 \beta_{7} - 3 \beta_{3} - 3 \beta_1) q^{86} + (2 \beta_{5} - 6 \beta_{2} - 6) q^{88} + (3 \beta_{3} - 3 \beta_1) q^{89} + ( - \beta_{5} + \beta_{4} - 1) q^{91} + ( - \beta_{7} + \beta_{6} - 4 \beta_1) q^{92} + (4 \beta_{5} + 4 \beta_{4} + 16 \beta_{2}) q^{94} + ( - 5 \beta_{4} + 4 \beta_{2} - 4) q^{97} + ( - 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{7} + 16 q^{13} - 16 q^{16} + 8 q^{22} - 40 q^{28} + 16 q^{31} + 16 q^{37} - 8 q^{43} - 32 q^{46} - 8 q^{52} + 72 q^{58} - 8 q^{61} + 40 q^{67} - 8 q^{73} + 48 q^{76} - 88 q^{82} - 48 q^{88} - 8 q^{91} - 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 44x^{4} + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 54\nu^{2} ) / 80 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 54\nu^{3} ) / 80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{6} + 5\nu^{4} - 68\nu^{2} + 110 ) / 80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{6} - 5\nu^{4} - 68\nu^{2} - 110 ) / 80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -3\nu^{7} + 5\nu^{5} - 122\nu^{3} + 190\nu ) / 80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -3\nu^{7} - 5\nu^{5} - 122\nu^{3} - 190\nu ) / 80 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} + \beta_{4} + 4\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{6} + 6\beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{5} + 8\beta_{4} - 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{7} + 8\beta_{6} - 38\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -54\beta_{5} - 54\beta_{4} - 136\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -54\beta_{7} - 54\beta_{6} - 244\beta_{3} \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/675\mathbb{Z}\right)^\times\).

\(n\) \(326\) \(352\)
\(\chi(n)\) \(-1\) \(-\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
107.1
−1.79576 + 1.79576i
−0.880486 + 0.880486i
0.880486 − 0.880486i
1.79576 − 1.79576i
−1.79576 − 1.79576i
−0.880486 − 0.880486i
0.880486 + 0.880486i
1.79576 + 1.79576i
−1.79576 + 1.79576i 0 − 4.44949i 0 0 −2.22474 − 2.22474i 4.39869 + 4.39869i 0 0
107.2 −0.880486 + 0.880486i 0 0.449490i 0 0 0.224745 + 0.224745i −2.15674 − 2.15674i 0 0
107.3 0.880486 − 0.880486i 0 0.449490i 0 0 0.224745 + 0.224745i 2.15674 + 2.15674i 0 0
107.4 1.79576 − 1.79576i 0 − 4.44949i 0 0 −2.22474 − 2.22474i −4.39869 − 4.39869i 0 0
593.1 −1.79576 − 1.79576i 0 4.44949i 0 0 −2.22474 + 2.22474i 4.39869 − 4.39869i 0 0
593.2 −0.880486 − 0.880486i 0 − 0.449490i 0 0 0.224745 − 0.224745i −2.15674 + 2.15674i 0 0
593.3 0.880486 + 0.880486i 0 − 0.449490i 0 0 0.224745 − 0.224745i 2.15674 − 2.15674i 0 0
593.4 1.79576 + 1.79576i 0 4.44949i 0 0 −2.22474 + 2.22474i −4.39869 + 4.39869i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.c odd 4 1 inner
15.e even 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.2.f.g 8
3.b odd 2 1 inner 675.2.f.g 8
5.b even 2 1 135.2.f.b ✓ 8
5.c odd 4 1 135.2.f.b ✓ 8
5.c odd 4 1 inner 675.2.f.g 8
15.d odd 2 1 135.2.f.b ✓ 8
15.e even 4 1 135.2.f.b ✓ 8
15.e even 4 1 inner 675.2.f.g 8
20.d odd 2 1 2160.2.w.a 8
20.e even 4 1 2160.2.w.a 8
45.h odd 6 2 405.2.m.b 16
45.j even 6 2 405.2.m.b 16
45.k odd 12 2 405.2.m.b 16
45.l even 12 2 405.2.m.b 16
60.h even 2 1 2160.2.w.a 8
60.l odd 4 1 2160.2.w.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
135.2.f.b ✓ 8 5.b even 2 1
135.2.f.b ✓ 8 5.c odd 4 1
135.2.f.b ✓ 8 15.d odd 2 1
135.2.f.b ✓ 8 15.e even 4 1
405.2.m.b 16 45.h odd 6 2
405.2.m.b 16 45.j even 6 2
405.2.m.b 16 45.k odd 12 2
405.2.m.b 16 45.l even 12 2
675.2.f.g 8 1.a even 1 1 trivial
675.2.f.g 8 3.b odd 2 1 inner
675.2.f.g 8 5.c odd 4 1 inner
675.2.f.g 8 15.e even 4 1 inner
2160.2.w.a 8 20.d odd 2 1
2160.2.w.a 8 20.e even 4 1
2160.2.w.a 8 60.h even 2 1
2160.2.w.a 8 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(675, [\chi])\):

\( T_{2}^{8} + 44T_{2}^{4} + 100 \) Copy content Toggle raw display
\( T_{7}^{4} + 4T_{7}^{3} + 8T_{7}^{2} - 4T_{7} + 1 \) Copy content Toggle raw display
\( T_{29}^{4} - 96T_{29}^{2} + 2250 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 44T^{4} + 100 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{4} + 4 T^{3} + 8 T^{2} + \cdots + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 16 T^{2} + 10)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} - 8 T^{3} + 32 T^{2} + \cdots + 25)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + 944T^{4} + 1600 \) Copy content Toggle raw display
$19$ \( (T^{2} + 9)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} + 44T^{4} + 100 \) Copy content Toggle raw display
$29$ \( (T^{4} - 96 T^{2} + 2250)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 4 T - 50)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 8 T^{3} + \cdots + 361)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 136 T^{2} + 250)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 4 T^{3} + \cdots + 100)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + 11264 T^{4} + 6553600 \) Copy content Toggle raw display
$53$ \( T^{8} + 12524 T^{4} + 27984100 \) Copy content Toggle raw display
$59$ \( (T^{4} - 240 T^{2} + 9000)^{2} \) Copy content Toggle raw display
$61$ \( (T + 1)^{8} \) Copy content Toggle raw display
$67$ \( (T^{4} - 20 T^{3} + \cdots + 2209)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 184 T^{2} + 8410)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 4 T^{3} + \cdots + 625)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 30 T^{2} + 9)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 524 T^{4} + 62500 \) Copy content Toggle raw display
$89$ \( (T^{4} - 144 T^{2} + 3240)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 16 T^{3} + \cdots + 1849)^{2} \) Copy content Toggle raw display
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