Properties

Label 1375.2.b.b
Level $1375$
Weight $2$
Character orbit 1375.b
Analytic conductor $10.979$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1375,2,Mod(749,1375)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1375, 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("1375.749");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1375 = 5^{3} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1375.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: \(10.9794302779\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.6988960000.10
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 9x^{6} + 22x^{4} + 11x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} - \beta_{7} q^{3} + \beta_{2} q^{4} + (\beta_{3} - 1) q^{6} + ( - \beta_{7} + \beta_{6} - 2 \beta_1) q^{7} + (\beta_{6} + \beta_{5} - \beta_1) q^{8} + (\beta_{3} - \beta_{2}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} - \beta_{7} q^{3} + \beta_{2} q^{4} + (\beta_{3} - 1) q^{6} + ( - \beta_{7} + \beta_{6} - 2 \beta_1) q^{7} + (\beta_{6} + \beta_{5} - \beta_1) q^{8} + (\beta_{3} - \beta_{2}) q^{9} - q^{11} + ( - \beta_{7} + \beta_{5} - \beta_1) q^{12} + (\beta_{6} - 2 \beta_{5} + \beta_1) q^{13} + (\beta_{3} - 2 \beta_{2} + 2) q^{14} + (\beta_{4} - \beta_{3} + \beta_{2}) q^{16} + ( - 2 \beta_{7} + \beta_{6} - 2 \beta_{5} + \beta_1) q^{17} + (\beta_{7} - \beta_{6} + 3 \beta_1) q^{18} + (2 \beta_{4} - \beta_{2} + 1) q^{19} + (\beta_{4} + \beta_{3} - \beta_{2} - 2) q^{21} - \beta_1 q^{22} + ( - \beta_{7} + 2 \beta_{6} - \beta_{5} - 2 \beta_1) q^{23} + (\beta_{4} + 2 \beta_{3} - \beta_{2} - 2) q^{24} + ( - 2 \beta_{4} + 2 \beta_{3} + \beta_{2} - 1) q^{26} + ( - 2 \beta_{7} - \beta_{6} - \beta_{5} + \beta_1) q^{27} + ( - \beta_{7} - \beta_{5} + 4 \beta_1) q^{28} + ( - \beta_{4} + 5) q^{29} + (3 \beta_{4} + \beta_{3} - 4) q^{31} + ( - 2 \beta_{7} + 3 \beta_{6} - 4 \beta_1) q^{32} + \beta_{7} q^{33} + ( - 2 \beta_{4} + 4 \beta_{3} + \beta_{2} - 3) q^{34} + (\beta_{3} + \beta_{2} - 4) q^{36} + ( - 2 \beta_{7} - 2 \beta_{6} - 3 \beta_{5} + \beta_1) q^{37} + ( - 2 \beta_{7} - \beta_{6} - 5 \beta_{5} + 6 \beta_1) q^{38} + (\beta_{4} + \beta_{3} + 2 \beta_{2} + 2) q^{39} + ( - \beta_{4} + 2 \beta_{3} + \beta_{2} - 6) q^{41} + ( - \beta_{6} - 2 \beta_{5} + 2 \beta_1) q^{42} + (3 \beta_{7} + 2 \beta_1) q^{43} - \beta_{2} q^{44} + ( - \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 2) q^{46} + ( - \beta_{7} - 2 \beta_{6} - 4 \beta_{5} + 4 \beta_1) q^{47} + ( - \beta_{7} - \beta_{6} + \beta_{5}) q^{48} + (\beta_{4} + 2 \beta_{2}) q^{49} + (\beta_{4} + 3 \beta_{3} - 4) q^{51} + (4 \beta_{7} + 3 \beta_{6} + 3 \beta_{5} - 4 \beta_1) q^{52} + ( - 6 \beta_{7} - 2 \beta_{6} - 2 \beta_{5}) q^{53} + ( - \beta_{4} + 3 \beta_{3} + \beta_{2} - 2) q^{54} + ( - \beta_{4} + 4 \beta_{3} - 4) q^{56} + ( - 4 \beta_{6} - \beta_{5} + 3 \beta_1) q^{57} + (\beta_{7} + 2 \beta_{5} + 4 \beta_1) q^{58} + ( - 3 \beta_{4} - \beta_{3} + 3 \beta_{2} + 1) q^{59} + ( - 3 \beta_{4} + 4 \beta_{3} - 4 \beta_{2} - 4) q^{61} + ( - 2 \beta_{7} - 5 \beta_{5} - \beta_1) q^{62} + ( - \beta_{5} - 4 \beta_1) q^{63} + (2 \beta_{4} - 2 \beta_{2} + 3) q^{64} + ( - \beta_{3} + 1) q^{66} + (5 \beta_{6} + 4 \beta_{5} - \beta_1) q^{67} + (2 \beta_{7} + 3 \beta_{6} + 5 \beta_{5} - 6 \beta_1) q^{68} + (2 \beta_{4} + 2 \beta_{3} - 1) q^{69} + ( - 2 \beta_{4} - 4 \beta_{3} - \beta_{2} + 5) q^{71} + (3 \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_1) q^{72} + ( - 2 \beta_{7} - 5 \beta_{6} - \beta_{5} + 4 \beta_1) q^{73} + ( - 3 \beta_{4} + 5 \beta_{3} + \beta_{2} + 1) q^{74} + ( - \beta_{4} + 7 \beta_{3} + 4 \beta_{2} - 6) q^{76} + (\beta_{7} - \beta_{6} + 2 \beta_1) q^{77} + (2 \beta_{6} + \beta_{5} - 3 \beta_1) q^{78} + ( - 2 \beta_{4} - 4 \beta_{3} - \beta_{2} + 7) q^{79} + ( - \beta_{4} + 3 \beta_{3} - 4 \beta_{2} - 4) q^{81} + (3 \beta_{7} + \beta_{6} + 5 \beta_{5} - 10 \beta_1) q^{82} + (\beta_{7} + 3 \beta_{6} + 3 \beta_{5} - 7 \beta_1) q^{83} + (4 \beta_{3} - 5) q^{84} + ( - 3 \beta_{3} + 2 \beta_{2} - 1) q^{86} + ( - 5 \beta_{7} + 2 \beta_{6} - \beta_1) q^{87} + ( - \beta_{6} - \beta_{5} + \beta_1) q^{88} + ( - 2 \beta_{4} + \beta_{3} + 2 \beta_{2} + 10) q^{89} + (2 \beta_{4} - 6 \beta_{3} - \beta_{2} + 3) q^{91} + (\beta_{7} + 2 \beta_{6} + 3 \beta_1) q^{92} + (4 \beta_{7} - 7 \beta_{6} + 3 \beta_1) q^{93} + ( - 4 \beta_{4} + 5 \beta_{3} + 4 \beta_{2} - 3) q^{94} + (3 \beta_{4} + 4 \beta_{3} - 2 \beta_{2} - 5) q^{96} + (8 \beta_{7} - 6 \beta_{6} + 6 \beta_{5}) q^{97} + ( - \beta_{7} + 2 \beta_{6} - 5 \beta_1) q^{98} + ( - \beta_{3} + \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{4} - 4 q^{6} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{4} - 4 q^{6} + 6 q^{9} - 8 q^{11} + 24 q^{14} - 2 q^{16} + 18 q^{19} - 6 q^{21} - 2 q^{24} - 10 q^{26} + 36 q^{29} - 16 q^{31} - 18 q^{34} - 30 q^{36} + 20 q^{39} - 46 q^{41} + 2 q^{44} + 24 q^{46} - 16 q^{51} - 10 q^{54} - 20 q^{56} - 14 q^{59} - 20 q^{61} + 36 q^{64} + 4 q^{66} + 8 q^{69} + 18 q^{71} + 14 q^{74} - 32 q^{76} + 34 q^{79} - 16 q^{81} - 24 q^{84} - 24 q^{86} + 72 q^{89} + 10 q^{91} - 28 q^{94} - 8 q^{96} - 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 9x^{6} + 22x^{4} + 11x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{6} + 8\nu^{4} + 16\nu^{2} + 5 ) / 2 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{6} + 10\nu^{4} + 26\nu^{2} + 9 ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\nu^{7} - 9\nu^{5} - 21\nu^{3} - 6\nu \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{7} + 9\nu^{5} + 22\nu^{3} + 11\nu \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{7} + 26\nu^{5} + 58\nu^{3} + 17\nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{6} + \beta_{5} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} - \beta_{3} - 5\beta_{2} + 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -2\beta_{7} - 5\beta_{6} - 8\beta_{5} + 24\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -8\beta_{4} + 10\beta_{3} + 24\beta_{2} - 37 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 18\beta_{7} + 24\beta_{6} + 50\beta_{5} - 117\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1375\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(1002\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
749.1
2.25947i
1.80692i
0.716111i
0.342036i
0.342036i
0.716111i
1.80692i
2.25947i
2.25947i 0.716111i −3.10522 0 −1.61803 3.36025i 2.49721i 2.48718 0
749.2 1.80692i 0.342036i −1.26498 0 0.618034 3.40246i 1.32813i 2.88301 0
749.3 0.716111i 2.25947i 1.48718 0 −1.61803 2.22368i 2.49721i −2.10522 0
749.4 0.342036i 1.80692i 1.88301 0 0.618034 0.432668i 1.32813i −0.264977 0
749.5 0.342036i 1.80692i 1.88301 0 0.618034 0.432668i 1.32813i −0.264977 0
749.6 0.716111i 2.25947i 1.48718 0 −1.61803 2.22368i 2.49721i −2.10522 0
749.7 1.80692i 0.342036i −1.26498 0 0.618034 3.40246i 1.32813i 2.88301 0
749.8 2.25947i 0.716111i −3.10522 0 −1.61803 3.36025i 2.49721i 2.48718 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 749.8
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 1375.2.b.b 8
5.b even 2 1 inner 1375.2.b.b 8
5.c odd 4 2 1375.2.a.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1375.2.a.c 8 5.c odd 4 2
1375.2.b.b 8 1.a even 1 1 trivial
1375.2.b.b 8 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 9T_{2}^{6} + 22T_{2}^{4} + 11T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1375, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 9 T^{6} + 22 T^{4} + 11 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{8} + 9 T^{6} + 22 T^{4} + 11 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 28 T^{6} + 249 T^{4} + \cdots + 121 \) Copy content Toggle raw display
$11$ \( (T + 1)^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + 60 T^{6} + 1075 T^{4} + \cdots + 10000 \) Copy content Toggle raw display
$17$ \( T^{8} + 56 T^{6} + 987 T^{4} + \cdots + 15376 \) Copy content Toggle raw display
$19$ \( (T^{4} - 9 T^{3} - 9 T^{2} + 116 T + 176)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 59 T^{6} + 1062 T^{4} + \cdots + 10201 \) Copy content Toggle raw display
$29$ \( (T^{4} - 18 T^{3} + 113 T^{2} - 288 T + 251)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 8 T^{3} - 55 T^{2} - 374 T + 484)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 121 T^{6} + 4807 T^{4} + \cdots + 364816 \) Copy content Toggle raw display
$41$ \( (T^{4} + 23 T^{3} + 168 T^{2} + 483 T + 461)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 93 T^{6} + 2974 T^{4} + \cdots + 128881 \) Copy content Toggle raw display
$47$ \( T^{8} + 181 T^{6} + 6846 T^{4} + \cdots + 109561 \) Copy content Toggle raw display
$53$ \( T^{8} + 344 T^{6} + \cdots + 16128256 \) Copy content Toggle raw display
$59$ \( (T^{4} + 7 T^{3} - 115 T^{2} - 316 T + 844)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 10 T^{3} - 191 T^{2} - 1840 T + 359)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 404 T^{6} + \cdots + 15335056 \) Copy content Toggle raw display
$71$ \( (T^{4} - 9 T^{3} - 65 T^{2} + 828 T - 1936)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 327 T^{6} + 20579 T^{4} + \cdots + 55696 \) Copy content Toggle raw display
$79$ \( (T^{4} - 17 T^{3} + 13 T^{2} + 948 T - 3764)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 331 T^{6} + 10311 T^{4} + \cdots + 274576 \) Copy content Toggle raw display
$89$ \( (T^{4} - 36 T^{3} + 417 T^{2} - 1444 T - 1159)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 732 T^{6} + \cdots + 939790336 \) Copy content Toggle raw display
show more
show less