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Properties

Label 675.2.e.e
Level $675$
Weight $2$
Character orbit 675.e
Analytic conductor $5.390$
Analytic rank $0$
Dimension $8$
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] = [675,2,Mod(226,675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(675, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([2, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("675.226");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 675.e (of order \(3\), degree \(2\), 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: \(5.38990213644\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.1223810289.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} + 8x^{6} - 2x^{5} + 23x^{4} - 8x^{3} + 37x^{2} + 15x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 225)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{6} - \beta_{3} - \beta_{2} + \beta_1) q^{4} + \beta_{4} q^{7} + ( - \beta_{7} - \beta_{3} - \beta_{2} - 1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + (\beta_{6} - \beta_{3} - \beta_{2} + \beta_1) q^{4} + \beta_{4} q^{7} + ( - \beta_{7} - \beta_{3} - \beta_{2} - 1) q^{8} + (2 \beta_{6} - \beta_{5} - \beta_{4}) q^{11} + (2 \beta_{6} - 2 \beta_{3} + \beta_{2} - \beta_1) q^{13} + ( - \beta_{7} + \beta_{6} - 2 \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 2) q^{14} + (\beta_{5} - 2 \beta_{4} - 3 \beta_1) q^{16} + (\beta_{7} - 2 \beta_{3} - 2) q^{17} + ( - \beta_{3} + 2 \beta_{2}) q^{19} + ( - \beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} - 2 \beta_{2} + 2 \beta_1) q^{22} + ( - \beta_{7} + \beta_{6} - 5 \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 5) q^{23} + ( - 2 \beta_{7} + \beta_{3} - 3 \beta_{2} + 4) q^{26} + ( - 2 \beta_{3} - 2 \beta_{2} + 1) q^{28} + ( - \beta_{6} + 2 \beta_{5} - \beta_{4} - 2 \beta_1) q^{29} + (2 \beta_{7} + 2 \beta_{6} - \beta_{5} - 2 \beta_{4} - 2 \beta_{3} + \beta_{2} - \beta_1 + 1) q^{31} + ( - 3 \beta_{6} + 3 \beta_{3} + 2 \beta_{2} - 2 \beta_1) q^{32} + (\beta_{6} - \beta_{4} - 5 \beta_1) q^{34} + (\beta_{7} + \beta_{3} + 4 \beta_{2} - 3) q^{37} + (2 \beta_{6} + 5 \beta_{5} - \beta_{4}) q^{38} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 - 1) q^{41} + (2 \beta_{6} + \beta_{5} - 2 \beta_{4} + 2 \beta_1) q^{43} + ( - 2 \beta_{7} + \beta_{3} - \beta_{2} - 5) q^{44} + ( - 2 \beta_{7} - 2 \beta_{3} + \beta_{2} + 1) q^{46} + ( - \beta_{6} + 5 \beta_{5} + 2 \beta_{4}) q^{47} + (\beta_{7} - 2 \beta_{6} - \beta_{4} + 2 \beta_{3} - \beta_{2} + \beta_1) q^{49} + ( - \beta_{6} - 3 \beta_{5} - \beta_{4} - \beta_1) q^{52} + (3 \beta_{3} - \beta_{2} - 6) q^{53} + ( - 6 \beta_{5} - 3 \beta_1) q^{56} + (2 \beta_{7} - 3 \beta_{6} - \beta_{5} - 2 \beta_{4} + 3 \beta_{3} + 3 \beta_{2} - 3 \beta_1 + 1) q^{58} + (3 \beta_{7} - 3 \beta_{5} - 3 \beta_{4} + \beta_{2} - \beta_1 + 3) q^{59} + (2 \beta_{6} + \beta_{5} + \beta_{4} + 2 \beta_1) q^{61} + 3 \beta_{3} q^{62} + ( - \beta_{7} + 2 \beta_{3} + 2 \beta_{2} + 3) q^{64} + (\beta_{7} + 3 \beta_{5} - \beta_{4} - 3 \beta_{2} + 3 \beta_1 - 3) q^{67} + (2 \beta_{7} - 2 \beta_{6} - 5 \beta_{5} - 2 \beta_{4} + 2 \beta_{3} + 4 \beta_{2} + \cdots + 5) q^{68}+ \cdots + (3 \beta_{7} + 4 \beta_{2} - 6) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} - 4 q^{4} + q^{7} - 18 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{2} - 4 q^{4} + q^{7} - 18 q^{8} - q^{11} - 2 q^{13} + 3 q^{14} - 4 q^{16} - 22 q^{17} + 4 q^{19} - 3 q^{22} + 15 q^{23} + 20 q^{26} - 8 q^{28} + q^{29} + 4 q^{31} + 10 q^{32} - 9 q^{34} - 2 q^{37} + 23 q^{38} - 5 q^{41} + 10 q^{43} - 44 q^{44} + 20 q^{47} + 3 q^{49} - 17 q^{52} - 40 q^{53} - 30 q^{56} + 18 q^{58} + 17 q^{59} + 13 q^{61} + 12 q^{62} + 38 q^{64} - 17 q^{67} + 34 q^{68} + 16 q^{71} + 4 q^{73} + 40 q^{74} - 11 q^{76} + 12 q^{77} + 7 q^{79} + 24 q^{82} + 30 q^{83} - 34 q^{86} - 9 q^{88} + 18 q^{89} - 34 q^{91} - 12 q^{92} - 3 q^{94} + 19 q^{97} - 26 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} + 8x^{6} - 2x^{5} + 23x^{4} - 8x^{3} + 37x^{2} + 15x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -76\nu^{7} - 32\nu^{6} - 361\nu^{5} - 722\nu^{4} - 3496\nu^{3} - 1691\nu^{2} - 741\nu - 2934 ) / 6165 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -12\nu^{7} + 31\nu^{6} - 57\nu^{5} - 114\nu^{4} + 133\nu^{3} - 267\nu^{2} - 117\nu - 968 ) / 685 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -263\nu^{7} + 1079\nu^{6} - 3818\nu^{5} + 4694\nu^{4} - 10043\nu^{3} + 9047\nu^{2} - 29793\nu + 2988 ) / 6165 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -326\nu^{7} + 728\nu^{6} - 2576\nu^{5} + 1013\nu^{4} - 6776\nu^{3} + 6104\nu^{2} - 10371\nu + 2016 ) / 6165 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 156\nu^{7} - 403\nu^{6} + 1426\nu^{5} - 1258\nu^{4} + 3751\nu^{3} - 3379\nu^{2} + 4261\nu - 1116 ) / 2055 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 488\nu^{7} - 119\nu^{6} + 2318\nu^{5} + 4636\nu^{4} + 10118\nu^{3} + 10858\nu^{2} + 4758\nu + 17217 ) / 6165 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + 2\beta_{5} - \beta_{3} - \beta_{2} + \beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{7} - \beta_{3} - 5\beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -6\beta_{6} - 7\beta_{5} - 2\beta_{4} - 9\beta_1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 8\beta_{7} - 11\beta_{6} - 8\beta_{5} - 8\beta_{4} + 11\beta_{3} + 30\beta_{2} - 30\beta _1 + 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 19\beta_{7} + 38\beta_{3} + 68\beta_{2} + 33 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 87\beta_{6} + 60\beta_{5} + 57\beta_{4} + 196\beta_1 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(326\) \(352\)
\(\chi(n)\) \(-1 + \beta_{5}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
226.1
−0.816862 − 1.41485i
−0.236627 − 0.409850i
0.736627 + 1.27588i
1.31686 + 2.28087i
−0.816862 + 1.41485i
−0.236627 + 0.409850i
0.736627 − 1.27588i
1.31686 − 2.28087i
−0.816862 − 1.41485i 0 −0.334526 + 0.579416i 0 0 0.252674 + 0.437645i −2.17440 0 0
226.2 −0.236627 − 0.409850i 0 0.888015 − 1.53809i 0 0 1.28153 + 2.21967i −1.78702 0 0
226.3 0.736627 + 1.27588i 0 −0.0852394 + 0.147639i 0 0 −1.93291 − 3.34791i 2.69535 0 0
226.4 1.31686 + 2.28087i 0 −2.46825 + 4.27513i 0 0 0.898714 + 1.55662i −7.73393 0 0
451.1 −0.816862 + 1.41485i 0 −0.334526 − 0.579416i 0 0 0.252674 − 0.437645i −2.17440 0 0
451.2 −0.236627 + 0.409850i 0 0.888015 + 1.53809i 0 0 1.28153 − 2.21967i −1.78702 0 0
451.3 0.736627 − 1.27588i 0 −0.0852394 − 0.147639i 0 0 −1.93291 + 3.34791i 2.69535 0 0
451.4 1.31686 − 2.28087i 0 −2.46825 − 4.27513i 0 0 0.898714 − 1.55662i −7.73393 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 451.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 675.2.e.e 8
3.b odd 2 1 225.2.e.c ✓ 8
5.b even 2 1 675.2.e.c 8
5.c odd 4 2 675.2.k.c 16
9.c even 3 1 inner 675.2.e.e 8
9.c even 3 1 2025.2.a.p 4
9.d odd 6 1 225.2.e.c ✓ 8
9.d odd 6 1 2025.2.a.y 4
15.d odd 2 1 225.2.e.e yes 8
15.e even 4 2 225.2.k.c 16
45.h odd 6 1 225.2.e.e yes 8
45.h odd 6 1 2025.2.a.q 4
45.j even 6 1 675.2.e.c 8
45.j even 6 1 2025.2.a.z 4
45.k odd 12 2 675.2.k.c 16
45.k odd 12 2 2025.2.b.o 8
45.l even 12 2 225.2.k.c 16
45.l even 12 2 2025.2.b.n 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
225.2.e.c ✓ 8 3.b odd 2 1
225.2.e.c ✓ 8 9.d odd 6 1
225.2.e.e yes 8 15.d odd 2 1
225.2.e.e yes 8 45.h odd 6 1
225.2.k.c 16 15.e even 4 2
225.2.k.c 16 45.l even 12 2
675.2.e.c 8 5.b even 2 1
675.2.e.c 8 45.j even 6 1
675.2.e.e 8 1.a even 1 1 trivial
675.2.e.e 8 9.c even 3 1 inner
675.2.k.c 16 5.c odd 4 2
675.2.k.c 16 45.k odd 12 2
2025.2.a.p 4 9.c even 3 1
2025.2.a.q 4 45.h odd 6 1
2025.2.a.y 4 9.d odd 6 1
2025.2.a.z 4 45.j even 6 1
2025.2.b.n 8 45.l even 12 2
2025.2.b.o 8 45.k odd 12 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} - 2T_{2}^{7} + 8T_{2}^{6} - 2T_{2}^{5} + 23T_{2}^{4} - 8T_{2}^{3} + 37T_{2}^{2} + 15T_{2} + 9 \) acting on \(S_{2}^{\mathrm{new}}(675, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} - 2 T^{7} + 8 T^{6} - 2 T^{5} + \cdots + 9 \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - T^{7} + 13 T^{6} - 36 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$11$ \( T^{8} + T^{7} + 26 T^{6} - 107 T^{5} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( T^{8} + 2 T^{7} + 34 T^{6} + \cdots + 11449 \) Copy content Toggle raw display
$17$ \( (T^{4} + 11 T^{3} + 20 T^{2} - 110 T - 303)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} - 2 T^{3} - 27 T^{2} + 80 T - 25)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} - 15 T^{7} + 168 T^{6} + \cdots + 59049 \) Copy content Toggle raw display
$29$ \( T^{8} - T^{7} + 41 T^{6} - 244 T^{5} + \cdots + 16641 \) Copy content Toggle raw display
$31$ \( T^{8} - 4 T^{7} + 58 T^{6} + \cdots + 59049 \) Copy content Toggle raw display
$37$ \( (T^{4} + T^{3} - 99 T^{2} - 503 T - 647)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} + 5 T^{7} + 50 T^{6} + \cdots + 42849 \) Copy content Toggle raw display
$43$ \( T^{8} - 10 T^{7} + 148 T^{6} + \cdots + 452929 \) Copy content Toggle raw display
$47$ \( T^{8} - 20 T^{7} + 293 T^{6} + \cdots + 145161 \) Copy content Toggle raw display
$53$ \( (T^{4} + 20 T^{3} + 86 T^{2} - 179 T - 471)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 17 T^{7} + 287 T^{6} + \cdots + 5349969 \) Copy content Toggle raw display
$61$ \( T^{8} - 13 T^{7} + 172 T^{6} - 143 T^{5} + \cdots + 1 \) Copy content Toggle raw display
$67$ \( T^{8} + 17 T^{7} + 253 T^{6} + \cdots + 59049 \) Copy content Toggle raw display
$71$ \( (T^{4} - 8 T^{3} - 40 T^{2} + 263 T + 381)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 2 T^{3} - 96 T^{2} - 241 T + 113)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} - 7 T^{7} + 82 T^{6} + \cdots + 42849 \) Copy content Toggle raw display
$83$ \( T^{8} - 30 T^{7} + 612 T^{6} + \cdots + 531441 \) Copy content Toggle raw display
$89$ \( (T^{4} - 9 T^{3} - 99 T^{2} + 405 T + 2025)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} - 19 T^{7} + 280 T^{6} + \cdots + 908209 \) Copy content Toggle raw display
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