Properties

Label 1925.2.b.p
Level $1925$
Weight $2$
Character orbit 1925.b
Analytic conductor $15.371$
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] = [1925,2,Mod(1849,1925)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1925, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1925.1849");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1925 = 5^{2} \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1925.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: \(15.3712023891\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 11x^{6} + 31x^{4} + 21x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 385)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{6} + \beta_{4}) q^{2} - \beta_{7} q^{3} + ( - \beta_{5} + \beta_{2} - 2) q^{4} + ( - \beta_{5} + \beta_{3} + 2) q^{6} + \beta_{6} q^{7} + (\beta_{7} + 5 \beta_{6} - 2 \beta_{4} + \beta_1) q^{8} + (\beta_{5} - \beta_{2} - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{6} + \beta_{4}) q^{2} - \beta_{7} q^{3} + ( - \beta_{5} + \beta_{2} - 2) q^{4} + ( - \beta_{5} + \beta_{3} + 2) q^{6} + \beta_{6} q^{7} + (\beta_{7} + 5 \beta_{6} - 2 \beta_{4} + \beta_1) q^{8} + (\beta_{5} - \beta_{2} - 2) q^{9} - q^{11} + (\beta_{7} + 4 \beta_{6} + \beta_1) q^{12} + ( - \beta_{7} - 2 \beta_{6} + \beta_1) q^{13} + \beta_{5} q^{14} + (4 \beta_{5} - 2 \beta_{3} - 2 \beta_{2} + 1) q^{16} + (\beta_{7} + 2 \beta_{6}) q^{17} + ( - \beta_{7} - 3 \beta_{6} - \beta_1) q^{18} + (2 \beta_{5} - \beta_{3} - 3) q^{19} - \beta_{2} q^{21} + (\beta_{6} - \beta_{4}) q^{22} + (\beta_{7} + 4 \beta_{6} + \beta_{4} + \beta_1) q^{23} + (3 \beta_{5} - 2 \beta_{2} + 1) q^{24} + ( - 3 \beta_{5} - 2 \beta_{2} + 1) q^{26} + ( - 4 \beta_{6} - \beta_1) q^{27} + ( - \beta_{7} - 3 \beta_{6} + \beta_{4}) q^{28} - 2 \beta_{3} q^{29} + (3 \beta_{5} + 1) q^{31} + ( - 6 \beta_{7} - 13 \beta_{6} + 5 \beta_{4} - 2 \beta_1) q^{32} + \beta_{7} q^{33} + (3 \beta_{5} - \beta_{3} - 2) q^{34} + ( - 2 \beta_{5} + 2 \beta_{3} - 1) q^{36} + (\beta_{7} + 2 \beta_{6} - \beta_{4} + \beta_1) q^{37} + ( - 4 \beta_{7} - 8 \beta_{6} - \beta_1) q^{38} + (3 \beta_{5} + 2 \beta_{3} + \beta_{2} - 5) q^{39} + ( - \beta_{5} - 2 \beta_{2} - 1) q^{41} + (\beta_{4} - \beta_1) q^{42} + (\beta_{7} - 4 \beta_{6} + \beta_{4} - \beta_1) q^{43} + (\beta_{5} - \beta_{2} + 2) q^{44} + (5 \beta_{5} - 2 \beta_{3} - \beta_{2} - 7) q^{46} + (3 \beta_{7} - 4 \beta_{4} + \beta_1) q^{47} + ( - \beta_{7} - 8 \beta_{6} + 6 \beta_{4}) q^{48} - q^{49} + ( - \beta_{5} - \beta_{2} + 5) q^{51} + (\beta_{7} + 10 \beta_{6}) q^{52} + (\beta_{7} - 2 \beta_{6} + 3 \beta_{4} - 3 \beta_1) q^{53} + ( - 4 \beta_{5} + \beta_{3} + 2 \beta_{2} + 1) q^{54} + ( - 2 \beta_{5} + \beta_{3} + \beta_{2} - 2) q^{56} + (2 \beta_{7} + 4 \beta_{4}) q^{57} + ( - 4 \beta_{7} - 2 \beta_{6} + 2 \beta_{4} - 2 \beta_1) q^{58} + (\beta_{5} + 2 \beta_{3} + 2 \beta_{2} - 7) q^{59} + (\beta_{5} - \beta_{3} - 4) q^{61} + ( - 3 \beta_{7} - 16 \beta_{6} + 4 \beta_{4}) q^{62} + (\beta_{7} - \beta_{6} - \beta_{4}) q^{63} + ( - 11 \beta_{5} + 4 \beta_{3} + 5 \beta_{2} - 4) q^{64} + (\beta_{5} - \beta_{3} - 2) q^{66} + ( - \beta_{7} + 5 \beta_{4} + \beta_1) q^{67} + ( - 3 \beta_{7} - 10 \beta_{6} + 2 \beta_{4} - \beta_1) q^{68} + (3 \beta_{3} - 4 \beta_{2} + 7) q^{69} + 8 q^{71} + (4 \beta_{7} + 7 \beta_{6} - 5 \beta_{4}) q^{72} + (\beta_{7} - 2 \beta_{6} + 2 \beta_1) q^{73} + (3 \beta_{5} - 2 \beta_{3} - 3 \beta_{2} + 1) q^{74} + ( - 8 \beta_{5} + 3 \beta_{3} + 2 \beta_{2} + 3) q^{76} - \beta_{6} q^{77} + (\beta_{7} - 8 \beta_{6} - 5 \beta_{4} + 3 \beta_1) q^{78} + (\beta_{5} - 2 \beta_{3} + \beta_{2} - 7) q^{79} + (\beta_{5} - 2 \beta_{3} + \beta_{2} - 6) q^{81} + (\beta_{7} + 6 \beta_{6} - 2 \beta_1) q^{82} + (2 \beta_{7} - 2 \beta_{4} + \beta_1) q^{83} + (\beta_{3} + \beta_{2} - 3) q^{84} + ( - 3 \beta_{5} + 3 \beta_{2} - 5) q^{86} + ( - 2 \beta_{7} + 4 \beta_{4} + 4 \beta_1) q^{87} + ( - \beta_{7} - 5 \beta_{6} + 2 \beta_{4} - \beta_1) q^{88} + (\beta_{3} + 3) q^{89} + (\beta_{3} - \beta_{2} + 3) q^{91} + ( - 7 \beta_{7} - 12 \beta_{6} + 3 \beta_{4} - \beta_1) q^{92} + ( - \beta_{7} + 3 \beta_{4} - 3 \beta_1) q^{93} + (3 \beta_{5} - 4 \beta_{3} - 6 \beta_{2} + 9) q^{94} + ( - 3 \beta_{5} + \beta_{3} + 2 \beta_{2} - 20) q^{96} + (2 \beta_{6} + 3 \beta_1) q^{97} + (\beta_{6} - \beta_{4}) q^{98} + ( - \beta_{5} + \beta_{2} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 16 q^{4} + 8 q^{6} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 16 q^{4} + 8 q^{6} - 16 q^{9} - 8 q^{11} + 4 q^{14} + 24 q^{16} - 12 q^{19} - 4 q^{21} + 12 q^{24} - 12 q^{26} + 8 q^{29} + 20 q^{31} - 24 q^{36} - 32 q^{39} - 20 q^{41} + 16 q^{44} - 32 q^{46} - 8 q^{49} + 32 q^{51} - 4 q^{54} - 24 q^{56} - 52 q^{59} - 24 q^{61} - 72 q^{64} - 8 q^{66} + 28 q^{69} + 64 q^{71} + 16 q^{74} - 12 q^{76} - 40 q^{79} - 32 q^{81} - 24 q^{84} - 40 q^{86} + 20 q^{89} + 16 q^{91} + 76 q^{94} - 168 q^{96} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 11x^{6} + 31x^{4} + 21x^{2} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 10\nu^{4} + 21\nu^{2} + 4 ) / 2 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\nu^{6} - 10\nu^{4} - 23\nu^{2} - 9 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{7} + 10\nu^{5} + 21\nu^{3} ) / 2 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{6} + 32\nu^{4} + 81\nu^{2} + 30 ) / 2 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 2\nu^{7} + 21\nu^{5} + 52\nu^{3} + 19\nu ) / 2 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 3\nu^{7} + 32\nu^{5} + 81\nu^{3} + 28\nu ) / 2 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{3} - 2\beta_{2} - 5 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -2\beta_{7} + 4\beta_{6} - 2\beta_{4} - 5\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{5} + 9\beta_{3} + 12\beta_{2} + 27 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 20\beta_{7} - 36\beta_{6} + 12\beta_{4} + 31\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -20\beta_{5} - 69\beta_{3} - 74\beta_{2} - 173 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -158\beta_{7} + 276\beta_{6} - 74\beta_{4} - 205\beta_1 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1925\mathbb{Z}\right)^\times\).

\(n\) \(276\) \(1002\) \(1751\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1849.1
0.589216i
0.723742i
2.64119i
1.77571i
1.77571i
2.64119i
0.723742i
0.589216i
2.80513i 1.06361i −5.86874 0 −2.98356 1.00000i 10.8523i 1.86874 0
1849.2 2.03967i 2.19994i −2.16027 0 4.48716 1.00000i 0.326891i −1.83973 0
1849.3 1.88395i 2.33468i −1.54927 0 4.39842 1.00000i 0.849150i −2.45073 0
1849.4 0.649405i 2.92887i 1.57827 0 −1.90202 1.00000i 2.32375i −5.57827 0
1849.5 0.649405i 2.92887i 1.57827 0 −1.90202 1.00000i 2.32375i −5.57827 0
1849.6 1.88395i 2.33468i −1.54927 0 4.39842 1.00000i 0.849150i −2.45073 0
1849.7 2.03967i 2.19994i −2.16027 0 4.48716 1.00000i 0.326891i −1.83973 0
1849.8 2.80513i 1.06361i −5.86874 0 −2.98356 1.00000i 10.8523i 1.86874 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1849.8
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 1925.2.b.p 8
5.b even 2 1 inner 1925.2.b.p 8
5.c odd 4 1 385.2.a.h 4
5.c odd 4 1 1925.2.a.x 4
15.e even 4 1 3465.2.a.bk 4
20.e even 4 1 6160.2.a.br 4
35.f even 4 1 2695.2.a.l 4
55.e even 4 1 4235.2.a.r 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
385.2.a.h 4 5.c odd 4 1
1925.2.a.x 4 5.c odd 4 1
1925.2.b.p 8 1.a even 1 1 trivial
1925.2.b.p 8 5.b even 2 1 inner
2695.2.a.l 4 35.f even 4 1
3465.2.a.bk 4 15.e even 4 1
4235.2.a.r 4 55.e even 4 1
6160.2.a.br 4 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1925, [\chi])\):

\( T_{2}^{8} + 16T_{2}^{6} + 82T_{2}^{4} + 148T_{2}^{2} + 49 \) Copy content Toggle raw display
\( T_{3}^{8} + 20T_{3}^{6} + 136T_{3}^{4} + 356T_{3}^{2} + 256 \) Copy content Toggle raw display
\( T_{19}^{4} + 6T_{19}^{3} - 28T_{19}^{2} - 120T_{19} + 32 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 16 T^{6} + 82 T^{4} + 148 T^{2} + \cdots + 49 \) Copy content Toggle raw display
$3$ \( T^{8} + 20 T^{6} + 136 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$11$ \( (T + 1)^{8} \) Copy content Toggle raw display
$13$ \( T^{8} + 80 T^{6} + 2184 T^{4} + \cdots + 55696 \) Copy content Toggle raw display
$17$ \( T^{8} + 28 T^{6} + 192 T^{4} + \cdots + 16 \) Copy content Toggle raw display
$19$ \( (T^{4} + 6 T^{3} - 28 T^{2} - 120 T + 32)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 140 T^{6} + 6784 T^{4} + \cdots + 952576 \) Copy content Toggle raw display
$29$ \( (T^{4} - 4 T^{3} - 80 T^{2} + 272 T + 304)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} - 10 T^{3} - 30 T^{2} + 302 T + 304)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 60 T^{6} + 880 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$41$ \( (T^{4} + 10 T^{3} - 14 T^{2} - 302 T - 428)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 124 T^{6} + 4000 T^{4} + \cdots + 73984 \) Copy content Toggle raw display
$47$ \( T^{8} + 400 T^{6} + \cdots + 70694464 \) Copy content Toggle raw display
$53$ \( T^{8} + 476 T^{6} + \cdots + 162001984 \) Copy content Toggle raw display
$59$ \( (T^{4} + 26 T^{3} + 110 T^{2} + \cdots - 11848)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 12 T^{3} + 30 T^{2} - 38 T - 4)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + 484 T^{6} + \cdots + 124724224 \) Copy content Toggle raw display
$71$ \( (T - 8)^{8} \) Copy content Toggle raw display
$73$ \( T^{8} + 236 T^{6} + 3168 T^{4} + \cdots + 15376 \) Copy content Toggle raw display
$79$ \( (T^{4} + 20 T^{3} + 52 T^{2} - 572 T - 544)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 156 T^{6} + 8240 T^{4} + \cdots + 984064 \) Copy content Toggle raw display
$89$ \( (T^{4} - 10 T^{3} + 16 T^{2} + 32 T - 32)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 388 T^{6} + 43584 T^{4} + \cdots + 5161984 \) Copy content Toggle raw display
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