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Properties

Label 1875.2.b.c
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: 8.0.6724000000.12
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 16x^{6} + 86x^{4} + 181x^{2} + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 75)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{6} + \beta_{5} - 2) q^{4} + ( - \beta_{5} + 1) q^{6} + (2 \beta_{4} + \beta_{3} + \beta_1) q^{7} + (\beta_{7} + \beta_{4} + 4 \beta_{3} - \beta_1) q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{3} q^{3} + (\beta_{6} + \beta_{5} - 2) q^{4} + ( - \beta_{5} + 1) q^{6} + (2 \beta_{4} + \beta_{3} + \beta_1) q^{7} + (\beta_{7} + \beta_{4} + 4 \beta_{3} - \beta_1) q^{8} - q^{9} + (3 \beta_{6} - \beta_{2}) q^{11} + ( - \beta_{4} - 2 \beta_{3} + \beta_1) q^{12} + ( - \beta_{7} + 2 \beta_{4} + \beta_{3}) q^{13} + (\beta_{6} - 2 \beta_{2} - 3) q^{14} + ( - 2 \beta_{6} - 3 \beta_{5} - 2 \beta_{2} + 2) q^{16} + ( - \beta_{3} - \beta_1) q^{17} - \beta_1 q^{18} + (2 \beta_{5} - \beta_{2} - 2) q^{19} + (2 \beta_{6} - \beta_{5} + 2) q^{21} + (2 \beta_{7} - 3 \beta_{4} - \beta_{3} - 3 \beta_1) q^{22} + (\beta_{7} - 2 \beta_{4} + \beta_1) q^{23} + (\beta_{6} + \beta_{5} + \beta_{2} - 4) q^{24} + (3 \beta_{6} - \beta_{5} - \beta_{2} + 3) q^{26} - \beta_{3} q^{27} + ( - \beta_{7} - 2 \beta_{4} - 2 \beta_1) q^{28} + ( - \beta_{6} - \beta_{5} + 5) q^{29} + (2 \beta_{6} - \beta_{2} + 7) q^{31} + ( - 2 \beta_{7} - 7 \beta_{4} - 6 \beta_{3} + 2 \beta_1) q^{32} + (\beta_{7} - 3 \beta_{4} - 3 \beta_{3}) q^{33} + ( - \beta_{6} + 3) q^{34} + ( - \beta_{6} - \beta_{5} + 2) q^{36} + ( - 2 \beta_{7} + \beta_{4} - \beta_{3} - \beta_1) q^{37} + ( - \beta_{7} - \beta_{4} + 7 \beta_{3} - 2 \beta_1) q^{38} + (2 \beta_{6} - \beta_{2} + 1) q^{39} + ( - 4 \beta_{6} - 5) q^{41} + (2 \beta_{7} - \beta_{4} - 4 \beta_{3}) q^{42} + (2 \beta_{7} - \beta_{4} + 4 \beta_{3}) q^{43} + ( - 3 \beta_{6} - 2 \beta_{5} - \beta_{2} + 7) q^{44} + ( - 2 \beta_{6} + \beta_{5} + \beta_{2} - 6) q^{46} + (2 \beta_{7} - \beta_{4} - \beta_{3} - \beta_1) q^{47} + (2 \beta_{7} + 2 \beta_{4} + \beta_{3} - 3 \beta_1) q^{48} + (\beta_{6} - \beta_{5} - 4 \beta_{2}) q^{49} + \beta_{5} q^{51} - 3 \beta_{3} q^{52} + ( - 2 \beta_{7} - \beta_{3} + \beta_1) q^{53} + (\beta_{5} - 1) q^{54} + (3 \beta_{6} - 2 \beta_{5} - \beta_{2} + 4) q^{56} + (\beta_{7} + 2 \beta_1) q^{57} + ( - \beta_{7} - \beta_{4} - 4 \beta_{3} + 6 \beta_1) q^{58} + ( - 3 \beta_{6} - 3 \beta_{2} + 3) q^{59} + ( - 4 \beta_{6} + 2 \beta_{2} - 3) q^{61} + (\beta_{7} - 3 \beta_{4} - \beta_{3} + 5 \beta_1) q^{62} + ( - 2 \beta_{4} - \beta_{3} - \beta_1) q^{63} + (4 \beta_{6} + 2 \beta_{5} + 5 \beta_{2} - 6) q^{64} + ( - 3 \beta_{6} + 3 \beta_{5} + 2 \beta_{2} - 5) q^{66} + ( - \beta_{4} - 2 \beta_{3} - 2 \beta_1) q^{67} + ( - \beta_{7} - 2 \beta_{3} + 2 \beta_1) q^{68} + ( - 2 \beta_{6} - \beta_{5} + \beta_{2} - 1) q^{69} + ( - 2 \beta_{6} - 3 \beta_{5} - 4 \beta_{2} + 1) q^{71} + ( - \beta_{7} - \beta_{4} - 4 \beta_{3} + \beta_1) q^{72} + ( - 2 \beta_{7} - \beta_{4} + 5 \beta_{3} + 4 \beta_1) q^{73} + (5 \beta_{6} + \beta_{2} + 7) q^{74} + (\beta_{6} - 5 \beta_{5} + 13) q^{76} + (\beta_{7} - 6 \beta_{4} - 10 \beta_{3} - 5 \beta_1) q^{77} + (\beta_{7} - 3 \beta_{4} - \beta_{3} - \beta_1) q^{78} + ( - \beta_{6} - 3 \beta_{5} - 3 \beta_{2} - 7) q^{79} + q^{81} + ( - 4 \beta_{7} - \beta_1) q^{82} + (4 \beta_{7} + 5 \beta_{3}) q^{83} + ( - 2 \beta_{6} + 2 \beta_{5} - \beta_{2} - 4) q^{84} + ( - 6 \beta_{6} - 4 \beta_{5} - \beta_{2}) q^{86} + (\beta_{4} + 5 \beta_{3} - \beta_1) q^{87} + ( - 11 \beta_{4} - 11 \beta_{3} + 4 \beta_1) q^{88} + (4 \beta_{6} - \beta_{2} + 11) q^{89} + (3 \beta_{6} - 3 \beta_{5}) q^{91} + (\beta_{7} + 5 \beta_{3} - 2 \beta_1) q^{92} + (\beta_{7} - 2 \beta_{4} + 5 \beta_{3}) q^{93} + ( - 7 \beta_{6} - \beta_{2} - 1) q^{94} + ( - 7 \beta_{6} - 2 \beta_{5} - 2 \beta_{2} + 1) q^{96} + ( - 4 \beta_{7} + 2 \beta_{4} - 3 \beta_{3}) q^{97} + ( - 3 \beta_{7} - 13 \beta_{4} - 8 \beta_{3} - \beta_1) q^{98} + ( - 3 \beta_{6} + \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{4} + 4 q^{6} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{4} + 4 q^{6} - 8 q^{9} - 14 q^{11} - 32 q^{14} + 8 q^{16} - 10 q^{19} + 4 q^{21} - 30 q^{24} + 6 q^{26} + 40 q^{29} + 46 q^{31} + 28 q^{34} + 16 q^{36} - 2 q^{39} - 24 q^{41} + 58 q^{44} - 34 q^{46} - 16 q^{49} + 4 q^{51} - 4 q^{54} + 10 q^{56} + 30 q^{59} - 4 q^{61} - 46 q^{64} - 12 q^{66} - 2 q^{69} - 4 q^{71} + 38 q^{74} + 80 q^{76} - 70 q^{79} + 8 q^{81} - 18 q^{84} + 6 q^{86} + 70 q^{89} - 24 q^{91} + 18 q^{94} + 24 q^{96} + 14 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 16x^{6} + 86x^{4} + 181x^{2} + 121 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 8\nu^{4} + 4\nu^{2} - 22 ) / 9 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -5\nu^{7} - 58\nu^{5} - 155\nu^{3} - 25\nu ) / 99 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -2\nu^{7} - 43\nu^{5} - 260\nu^{3} - 406\nu ) / 99 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -2\nu^{6} - 25\nu^{4} - 80\nu^{2} - 46 ) / 9 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2\nu^{6} + 25\nu^{4} + 89\nu^{2} + 82 ) / 9 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2\nu^{7} + 25\nu^{5} + 89\nu^{3} + 91\nu ) / 9 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{6} + \beta_{5} - 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{7} + \beta_{4} + 4\beta_{3} - 5\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{6} - 9\beta_{5} - 2\beta_{2} + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -10\beta_{7} - 15\beta_{4} - 38\beta_{3} + 30\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 60\beta_{6} + 68\beta_{5} + 25\beta_{2} - 138 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 85\beta_{7} + 143\beta_{4} + 297\beta_{3} - 198\beta_1 \) 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
− 2.70636i
− 2.12233i
− 1.70636i
− 1.12233i
1.12233i
1.70636i
2.12233i
2.70636i
− 2.70636i 1.00000i −5.32440 0 2.70636 − 0.470294i 8.99702i −1.00000 0
1249.2 − 2.12233i 1.00000i −2.50430 0 2.12233 − 4.35840i 1.07029i −1.00000 0
1249.3 − 1.70636i − 1.00000i −0.911672 0 −1.70636 − 3.94243i − 1.85708i −1.00000 0
1249.4 − 1.12233i − 1.00000i 0.740367 0 −1.12233 1.11373i − 3.07561i −1.00000 0
1249.5 1.12233i 1.00000i 0.740367 0 −1.12233 − 1.11373i 3.07561i −1.00000 0
1249.6 1.70636i 1.00000i −0.911672 0 −1.70636 3.94243i 1.85708i −1.00000 0
1249.7 2.12233i − 1.00000i −2.50430 0 2.12233 4.35840i − 1.07029i −1.00000 0
1249.8 2.70636i − 1.00000i −5.32440 0 2.70636 0.470294i − 8.99702i −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.c 8
5.b even 2 1 inner 1875.2.b.c 8
5.c odd 4 1 1875.2.a.e 4
5.c odd 4 1 1875.2.a.h 4
15.e even 4 1 5625.2.a.i 4
15.e even 4 1 5625.2.a.n 4
25.d even 5 2 375.2.i.b 16
25.e even 10 2 375.2.i.b 16
25.f odd 20 2 75.2.g.b ✓ 8
25.f odd 20 2 375.2.g.b 8
75.l even 20 2 225.2.h.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.g.b ✓ 8 25.f odd 20 2
225.2.h.c 8 75.l even 20 2
375.2.g.b 8 25.f odd 20 2
375.2.i.b 16 25.d even 5 2
375.2.i.b 16 25.e even 10 2
1875.2.a.e 4 5.c odd 4 1
1875.2.a.h 4 5.c odd 4 1
1875.2.b.c 8 1.a even 1 1 trivial
1875.2.b.c 8 5.b even 2 1 inner
5625.2.a.i 4 15.e even 4 1
5625.2.a.n 4 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 16T_{2}^{6} + 86T_{2}^{4} + 181T_{2}^{2} + 121 \) acting on \(S_{2}^{\mathrm{new}}(1875, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 16 T^{6} + 86 T^{4} + \cdots + 121 \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 36 T^{6} + 346 T^{4} + \cdots + 81 \) Copy content Toggle raw display
$11$ \( (T^{4} + 7 T^{3} - 6 T^{2} - 92 T - 109)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 29 T^{6} + 226 T^{4} + \cdots + 81 \) Copy content Toggle raw display
$17$ \( T^{8} + 16 T^{6} + 86 T^{4} + \cdots + 121 \) Copy content Toggle raw display
$19$ \( (T^{4} + 5 T^{3} - 35 T^{2} - 75 T + 275)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 49 T^{6} + 446 T^{4} + \cdots + 361 \) Copy content Toggle raw display
$29$ \( (T^{4} - 20 T^{3} + 140 T^{2} - 405 T + 405)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 23 T^{3} + 184 T^{2} - 612 T + 711)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 96 T^{6} + 2306 T^{4} + \cdots + 1 \) Copy content Toggle raw display
$41$ \( (T^{2} + 6 T - 11)^{4} \) Copy content Toggle raw display
$43$ \( T^{8} + 134 T^{6} + 4291 T^{4} + \cdots + 9801 \) Copy content Toggle raw display
$47$ \( T^{8} + 76 T^{6} + 2066 T^{4} + \cdots + 96721 \) Copy content Toggle raw display
$53$ \( T^{8} + 84 T^{6} + 2086 T^{4} + \cdots + 68121 \) Copy content Toggle raw display
$59$ \( (T^{4} - 15 T^{3} - 45 T^{2} + 1215 T - 3645)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 2 T^{3} - 56 T^{2} + 78 T - 9)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 66 T^{6} + 1351 T^{4} + \cdots + 29241 \) Copy content Toggle raw display
$71$ \( (T^{4} + 2 T^{3} - 216 T^{2} - 1237 T + 911)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 354 T^{6} + \cdots + 23707161 \) Copy content Toggle raw display
$79$ \( (T^{4} + 35 T^{3} + 320 T^{2} - 350 T - 9845)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 364 T^{6} + 42686 T^{4} + \cdots + 958441 \) Copy content Toggle raw display
$89$ \( (T^{4} - 35 T^{3} + 420 T^{2} - 2030 T + 3305)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 316 T^{6} + 29166 T^{4} + \cdots + 4414201 \) Copy content Toggle raw display
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