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Properties

Label 1875.2.b.d
Level $1875$
Weight $2$
Character orbit 1875.b
Analytic conductor $14.972$
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] = [1875,2,Mod(1249,1875)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1875, 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("1875.1249");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1875 = 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1875.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: \(14.9719503790\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.324000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 9x^{6} + 26x^{4} + 24x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 9x^{6} + 26x^{4} + 24x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} + 3\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} + 4\nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} + 5\nu^{3} + 5\nu \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{6} + 6\nu^{4} + 9\nu^{2} + 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( \nu^{7} + 7\nu^{5} + 14\nu^{3} + 7\nu \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} - 3\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} - 4\beta_{2} + 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} - 5\beta_{3} + 10\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{6} - 6\beta_{4} + 15\beta_{2} - 20 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( \beta_{7} - 7\beta_{5} + 21\beta_{3} - 35\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1875\mathbb{Z}\right)^\times\).

\(n\) \(626\) \(1252\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1249.1
0.209057i
− 1.95630i
1.82709i
− 1.33826i
1.33826i
− 1.82709i
1.95630i
− 0.209057i
− 1.95630i 1.00000i −1.82709 0 1.95630 4.57433i − 0.338261i −1.00000 0
1249.2 − 1.82709i − 1.00000i −1.33826 0 −1.82709 1.44512i − 1.20906i −1.00000 0
1249.3 − 1.33826i − 1.00000i 0.209057 0 −1.33826 − 1.27977i − 2.95630i −1.00000 0
1249.4 − 0.209057i 1.00000i 1.95630 0 0.209057 0.591023i − 0.827091i −1.00000 0
1249.5 0.209057i − 1.00000i 1.95630 0 0.209057 − 0.591023i 0.827091i −1.00000 0
1249.6 1.33826i 1.00000i 0.209057 0 −1.33826 1.27977i 2.95630i −1.00000 0
1249.7 1.82709i 1.00000i −1.33826 0 −1.82709 − 1.44512i 1.20906i −1.00000 0
1249.8 1.95630i − 1.00000i −1.82709 0 1.95630 − 4.57433i 0.338261i −1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1249.8
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 1875.2.b.d 8
5.b even 2 1 inner 1875.2.b.d 8
5.c odd 4 1 1875.2.a.f ✓ 4
5.c odd 4 1 1875.2.a.g yes 4
15.e even 4 1 5625.2.a.j 4
15.e even 4 1 5625.2.a.m 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1875.2.a.f ✓ 4 5.c odd 4 1
1875.2.a.g yes 4 5.c odd 4 1
1875.2.b.d 8 1.a even 1 1 trivial
1875.2.b.d 8 5.b even 2 1 inner
5625.2.a.j 4 15.e even 4 1
5625.2.a.m 4 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 9T_{2}^{6} + 26T_{2}^{4} + 24T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1875, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 9 T^{6} + 26 T^{4} + 24 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 25 T^{6} + 90 T^{4} + 100 T^{2} + \cdots + 25 \) Copy content Toggle raw display
$11$ \( (T^{4} + 6 T^{3} + 6 T^{2} - 9 T - 9)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 61 T^{6} + 1146 T^{4} + \cdots + 7921 \) Copy content Toggle raw display
$17$ \( T^{8} + 81 T^{6} + 2366 T^{4} + \cdots + 128881 \) Copy content Toggle raw display
$19$ \( (T^{4} - 9 T^{3} + 6 T^{2} + 36 T - 9)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 60 T^{6} + 590 T^{4} + \cdots + 25 \) Copy content Toggle raw display
$29$ \( (T^{4} - 28 T^{3} + 284 T^{2} - 1217 T + 1801)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 10 T^{3} - 125 T - 125)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 280 T^{6} + 25650 T^{4} + \cdots + 2175625 \) Copy content Toggle raw display
$41$ \( (T^{4} - 70 T^{2} - 135 T + 145)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 159 T^{6} + 4721 T^{4} + \cdots + 175561 \) Copy content Toggle raw display
$47$ \( T^{8} + 351 T^{6} + \cdots + 37075921 \) Copy content Toggle raw display
$53$ \( T^{8} + 290 T^{6} + 27015 T^{4} + \cdots + 8970025 \) Copy content Toggle raw display
$59$ \( (T^{4} + 4 T^{3} - 154 T^{2} - 421 T + 1531)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 43 T^{3} + 669 T^{2} + \cdots + 10261)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 186 T^{6} + 11351 T^{4} + \cdots + 22801 \) Copy content Toggle raw display
$71$ \( (T^{4} + 27 T^{3} + 134 T^{2} - 672 T + 271)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 345 T^{6} + 36090 T^{4} + \cdots + 7535025 \) Copy content Toggle raw display
$79$ \( (T^{4} + 10 T^{3} - 105 T^{2} - 1370 T - 3155)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 501 T^{6} + \cdots + 182007081 \) Copy content Toggle raw display
$89$ \( (T^{4} - 9 T^{3} - 4 T^{2} + 96 T + 61)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 601 T^{6} + \cdots + 216119401 \) Copy content Toggle raw display
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