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Properties

Label 375.2.b.c
Level $375$
Weight $2$
Character orbit 375.b
Analytic conductor $2.994$
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] = [375,2,Mod(124,375)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(375, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("375.124");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 375 = 3 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 375.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: \(2.99439007580\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.1632160000.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 12x^{6} + 46x^{4} + 65x^{2} + 25 \) 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.

\(f(q)\) \(=\) \( q + (\beta_{7} + \beta_{4}) q^{2} - \beta_{4} q^{3} + (\beta_{6} + \beta_{5} + \beta_{2} - 1) q^{4} + ( - \beta_{2} + 1) q^{6} + 2 \beta_1 q^{7} + ( - \beta_{7} - 2 \beta_{4} + \beta_1) q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} + \beta_{4}) q^{2} - \beta_{4} q^{3} + (\beta_{6} + \beta_{5} + \beta_{2} - 1) q^{4} + ( - \beta_{2} + 1) q^{6} + 2 \beta_1 q^{7} + ( - \beta_{7} - 2 \beta_{4} + \beta_1) q^{8} - q^{9} + (2 \beta_{6} + 2 \beta_{2} + 2) q^{11} + (\beta_{7} + \beta_{4} + \beta_{3} - \beta_1) q^{12} + (2 \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{13} + (4 \beta_{6} - 2 \beta_{5} - 2 \beta_{2} + 2) q^{14} + (3 \beta_{6} - \beta_{2} + 3) q^{16} + ( - \beta_{4} + 3 \beta_{3}) q^{17} + ( - \beta_{7} - \beta_{4}) q^{18} + ( - \beta_{6} - 2 \beta_{2} - 2) q^{19} + 2 \beta_{5} q^{21} + ( - 2 \beta_{4} - 4 \beta_{3}) q^{22} + (2 \beta_{7} - 3 \beta_{4} + \beta_{3} - 2 \beta_1) q^{23} + (\beta_{5} + \beta_{2} - 2) q^{24} + (6 \beta_{6} - 4 \beta_{5} + 2 \beta_{2}) q^{26} + \beta_{4} q^{27} + (8 \beta_{4} - 6 \beta_{3} - 4 \beta_1) q^{28} + (2 \beta_{6} - 2 \beta_{5} + 2 \beta_{2} - 2) q^{29} + ( - 3 \beta_{6} + 2 \beta_{5} - 2 \beta_{2} + 1) q^{31} + ( - 2 \beta_{7} + \beta_{4} - 2 \beta_{3} - 2 \beta_1) q^{32} + (2 \beta_{7} - 2 \beta_{4} + 2 \beta_{3}) q^{33} + (3 \beta_{6} - 3 \beta_{5} + 2 \beta_{2} + 1) q^{34} + ( - \beta_{6} - \beta_{5} - \beta_{2} + 1) q^{36} + ( - 2 \beta_{7} - 6 \beta_{4} + 2 \beta_{3} - 2 \beta_1) q^{37} + ( - \beta_{7} + 2 \beta_{4} + 3 \beta_{3} - \beta_1) q^{38} + ( - 2 \beta_{6} + 2 \beta_{5} + 2) q^{39} + ( - 2 \beta_{6} - 2 \beta_{5} + 4) q^{41} + ( - 2 \beta_{7} - 2 \beta_{4} + 4 \beta_{3} + 2 \beta_1) q^{42} + (2 \beta_{7} + 6 \beta_{4} - 4 \beta_{3} + 2 \beta_1) q^{43} + (4 \beta_{5} - 2 \beta_{2} + 6) q^{44} + ( - \beta_{6} + 3 \beta_{5} - 3) q^{46} + ( - 2 \beta_{7} - \beta_{3}) q^{47} + ( - \beta_{7} - 3 \beta_{4} + 3 \beta_{3}) q^{48} + ( - 4 \beta_{6} + 4 \beta_{5} - 5) q^{49} + ( - 3 \beta_{6} - 1) q^{51} + ( - 2 \beta_{7} + 4 \beta_{4} - 12 \beta_{3} - 4 \beta_1) q^{52} + (2 \beta_{4} + \beta_{3} + 4 \beta_1) q^{53} + (\beta_{2} - 1) q^{54} + ( - 6 \beta_{6} + 6 \beta_{5} + 2 \beta_{2} - 8) q^{56} + ( - 2 \beta_{7} + 2 \beta_{4} - \beta_{3}) q^{57} + ( - 2 \beta_{7} - 4 \beta_{4} - 8 \beta_{3} - 2 \beta_1) q^{58} + ( - 6 \beta_{6} + 2 \beta_{2} - 4) q^{59} + ( - \beta_{6} - 4 \beta_{5}) q^{61} + (2 \beta_{7} + 3 \beta_{4} + 9 \beta_{3} + 3 \beta_1) q^{62} - 2 \beta_1 q^{63} + ( - 2 \beta_{6} + 2 \beta_{5} - \beta_{2} + 7) q^{64} + (4 \beta_{6} - 2) q^{66} + (4 \beta_{7} - 2 \beta_{4} - 2 \beta_1) q^{67} + (\beta_{7} - 2 \beta_{4} - 5 \beta_{3} - 4 \beta_1) q^{68} + ( - \beta_{6} - 2 \beta_{5} - 2 \beta_{2} - 3) q^{69} + ( - 6 \beta_{6} - 2 \beta_{5} - 4 \beta_{2} - 6) q^{71} + (\beta_{7} + 2 \beta_{4} - \beta_1) q^{72} - 2 \beta_{7} q^{73} + ( - 4 \beta_{6} - 2 \beta_{5} - 2 \beta_{2} + 8) q^{74} + ( - 2 \beta_{6} - 3 \beta_{5} + 2 \beta_{2} - 5) q^{76} + (4 \beta_{4} - 8 \beta_{3} + 4 \beta_1) q^{77} + (2 \beta_{7} + 6 \beta_{3} + 4 \beta_1) q^{78} + ( - \beta_{6} - 4 \beta_{5} + 2 \beta_{2} + 2) q^{79} + q^{81} + (8 \beta_{7} + 6 \beta_{4} - 2 \beta_{3}) q^{82} + (2 \beta_{7} - 8 \beta_{4} + 5 \beta_{3}) q^{83} + (6 \beta_{6} - 4 \beta_{5} + 8) q^{84} + (2 \beta_{6} + 4 \beta_{5} - 8) q^{86} + (2 \beta_{7} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{87} + (2 \beta_{7} + 2 \beta_{4} + 2 \beta_{3} + 2 \beta_1) q^{88} + (2 \beta_{6} - 4 \beta_{5} - 8) q^{89} + ( - 4 \beta_{6} + 4 \beta_{2} - 12) q^{91} + ( - \beta_{7} - 12 \beta_{4} + 9 \beta_{3}) q^{92} + ( - 2 \beta_{7} - \beta_{4} - 3 \beta_{3} - 2 \beta_1) q^{93} + ( - 3 \beta_{6} - \beta_{5} - \beta_{2} + 4) q^{94} + (2 \beta_{6} - 2 \beta_{5} + 2 \beta_{2} + 1) q^{96} + (2 \beta_{7} - 6 \beta_{4} + 8 \beta_{3} + 4 \beta_1) q^{97} + ( - 5 \beta_{7} - 9 \beta_{4} + 12 \beta_{3} + 8 \beta_1) q^{98} + ( - 2 \beta_{6} - 2 \beta_{2} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 14 q^{4} + 6 q^{6} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 14 q^{4} + 6 q^{6} - 8 q^{9} + 12 q^{11} + 4 q^{14} + 10 q^{16} - 16 q^{19} - 8 q^{21} - 18 q^{24} - 4 q^{26} - 12 q^{29} + 8 q^{31} + 12 q^{34} + 14 q^{36} + 16 q^{39} + 48 q^{41} + 28 q^{44} - 32 q^{46} - 40 q^{49} + 4 q^{51} - 6 q^{54} - 60 q^{56} - 4 q^{59} + 20 q^{61} + 54 q^{64} - 32 q^{66} - 16 q^{69} - 24 q^{71} + 84 q^{74} - 16 q^{76} + 40 q^{79} + 8 q^{81} + 56 q^{84} - 88 q^{86} - 56 q^{89} - 72 q^{91} + 46 q^{94} + 12 q^{96} - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 12x^{6} + 46x^{4} + 65x^{2} + 25 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 7\nu^{4} + 6\nu^{2} - 5 ) / 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{7} + 17\nu^{5} + 81\nu^{3} + 95\nu ) / 25 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -3\nu^{7} - 26\nu^{5} - 43\nu^{3} + 15\nu ) / 25 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{6} - 19\nu^{4} - 42\nu^{2} - 15 ) / 5 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -2\nu^{6} - 19\nu^{4} - 47\nu^{2} - 30 ) / 5 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2\nu^{7} + 19\nu^{5} + 47\nu^{3} + 30\nu ) / 5 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( -\beta_{6} + \beta_{5} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + 3\beta_{4} - \beta_{3} - 4\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 6\beta_{6} - 7\beta_{5} - 2\beta_{2} + 13 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -8\beta_{7} - 23\beta_{4} + 11\beta_{3} + 20\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -36\beta_{6} + 43\beta_{5} + 19\beta_{2} - 68 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 55\beta_{7} + 148\beta_{4} - 81\beta_{3} - 111\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/375\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(251\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
124.1
2.46673i
− 0.777484i
− 1.77748i
− 1.46673i
1.46673i
1.77748i
0.777484i
− 2.46673i
− 2.52452i 1.00000i −4.37322 0 2.52452 4.93346i 5.99126i −1.00000 0
124.2 − 2.25800i 1.00000i −3.09855 0 2.25800 − 1.55497i 2.48051i −1.00000 0
124.3 − 1.87603i − 1.00000i −1.51949 0 −1.87603 − 3.55497i − 0.901454i −1.00000 0
124.4 − 0.0935099i 1.00000i 1.99126 0 0.0935099 − 2.93346i − 0.373222i −1.00000 0
124.5 0.0935099i − 1.00000i 1.99126 0 0.0935099 2.93346i 0.373222i −1.00000 0
124.6 1.87603i 1.00000i −1.51949 0 −1.87603 3.55497i 0.901454i −1.00000 0
124.7 2.25800i − 1.00000i −3.09855 0 2.25800 1.55497i − 2.48051i −1.00000 0
124.8 2.52452i − 1.00000i −4.37322 0 2.52452 − 4.93346i − 5.99126i −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 124.8
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 375.2.b.c 8
3.b odd 2 1 1125.2.b.g 8
4.b odd 2 1 6000.2.f.o 8
5.b even 2 1 inner 375.2.b.c 8
5.c odd 4 1 375.2.a.e ✓ 4
5.c odd 4 1 375.2.a.f yes 4
15.d odd 2 1 1125.2.b.g 8
15.e even 4 1 1125.2.a.h 4
15.e even 4 1 1125.2.a.l 4
20.d odd 2 1 6000.2.f.o 8
20.e even 4 1 6000.2.a.bg 4
20.e even 4 1 6000.2.a.bh 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
375.2.a.e ✓ 4 5.c odd 4 1
375.2.a.f yes 4 5.c odd 4 1
375.2.b.c 8 1.a even 1 1 trivial
375.2.b.c 8 5.b even 2 1 inner
1125.2.a.h 4 15.e even 4 1
1125.2.a.l 4 15.e even 4 1
1125.2.b.g 8 3.b odd 2 1
1125.2.b.g 8 15.d odd 2 1
6000.2.a.bg 4 20.e even 4 1
6000.2.a.bh 4 20.e even 4 1
6000.2.f.o 8 4.b odd 2 1
6000.2.f.o 8 20.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 15T_{2}^{6} + 73T_{2}^{4} + 115T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(375, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 15 T^{6} + 73 T^{4} + 115 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 48 T^{6} + 736 T^{4} + \cdots + 6400 \) Copy content Toggle raw display
$11$ \( (T^{4} - 6 T^{3} - 12 T^{2} + 88 T - 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 80 T^{6} + 1888 T^{4} + \cdots + 30976 \) Copy content Toggle raw display
$17$ \( (T^{4} + 23 T^{2} + 121)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 8 T^{3} + T^{2} - 60 T + 25)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 102 T^{6} + 2491 T^{4} + \cdots + 3025 \) Copy content Toggle raw display
$29$ \( (T^{4} + 6 T^{3} - 36 T^{2} - 40 T + 80)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 4 T^{3} - 51 T^{2} - 80 T + 25)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 188 T^{6} + 12816 T^{4} + \cdots + 4000000 \) Copy content Toggle raw display
$41$ \( (T^{4} - 24 T^{3} + 184 T^{2} - 440 T - 80)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 244 T^{6} + 19936 T^{4} + \cdots + 4946176 \) Copy content Toggle raw display
$47$ \( (T^{4} + 23 T^{2} + 101)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 182 T^{6} + 11491 T^{4} + \cdots + 2088025 \) Copy content Toggle raw display
$59$ \( (T^{4} + 2 T^{3} - 144 T^{2} - 320 T + 2320)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} - 10 T^{3} - 53 T^{2} + 470 T + 781)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 256 T^{6} + 21856 T^{4} + \cdots + 1290496 \) Copy content Toggle raw display
$71$ \( (T^{4} + 12 T^{3} - 96 T^{2} - 392 T + 656)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 52 T^{6} + 736 T^{4} + \cdots + 6400 \) Copy content Toggle raw display
$79$ \( (T^{4} - 20 T^{3} + 25 T^{2} + 500 T + 725)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 270 T^{6} + 19963 T^{4} + \cdots + 2343961 \) Copy content Toggle raw display
$89$ \( (T^{4} + 28 T^{3} + 196 T^{2} - 160 T - 2320)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 556 T^{6} + \cdots + 82882816 \) Copy content Toggle raw display
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