↑

Properties

Label 3750.2.a.v
Level $3750$
Weight $2$
Character orbit 3750.a
Self dual yes
Analytic conductor $29.944$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3750,2,Mod(1,3750)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3750, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("3750.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3750 = 2 \cdot 3 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3750.a (trivial)

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: yes
Analytic conductor: \(29.9439007580\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.8.71684000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 2x^{7} - 18x^{6} + 10x^{5} + 101x^{4} + 40x^{3} - 132x^{2} - 96x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: no (minimal twist has level 150)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

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

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 2x^{7} - 18x^{6} + 10x^{5} + 101x^{4} + 40x^{3} - 132x^{2} - 96x - 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -3\nu^{7} + 11\nu^{6} + 32\nu^{5} - 70\nu^{4} - 145\nu^{3} + 21\nu^{2} + 208\nu + 106 ) / 20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4\nu^{7} - 19\nu^{6} - 26\nu^{5} + 144\nu^{4} + 48\nu^{3} - 247\nu^{2} + 10\nu + 86 ) / 20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 7\nu^{7} - 27\nu^{6} - 78\nu^{5} + 222\nu^{4} + 319\nu^{3} - 361\nu^{2} - 340\nu + 18 ) / 20 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -14\nu^{7} + 47\nu^{6} + 186\nu^{5} - 386\nu^{4} - 862\nu^{3} + 569\nu^{2} + 998\nu + 22 ) / 20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16\nu^{7} - 57\nu^{6} - 194\nu^{5} + 450\nu^{4} + 840\nu^{3} - 587\nu^{2} - 866\nu - 162 ) / 20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 9\nu^{7} - 32\nu^{6} - 111\nu^{5} + 256\nu^{4} + 502\nu^{3} - 364\nu^{2} - 578\nu - 42 ) / 10 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 28\nu^{7} - 103\nu^{6} - 332\nu^{5} + 838\nu^{4} + 1446\nu^{3} - 1309\nu^{2} - 1650\nu - 18 ) / 20 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} + 2\beta_{4} - 4\beta_{3} + 2\beta _1 + 4 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5\beta_{7} - 2\beta_{6} + 3\beta_{5} + 6\beta_{4} - 7\beta_{3} - 4\beta_{2} + \beta _1 + 32 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 27\beta_{7} - 3\beta_{6} + 12\beta_{5} + 29\beta_{4} - 58\beta_{3} - 10\beta_{2} + 14\beta _1 + 98 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 119\beta_{7} - 50\beta_{6} + 55\beta_{5} + 110\beta_{4} - 205\beta_{3} - 58\beta_{2} + 35\beta _1 + 470 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 549\beta_{7} - 199\beta_{6} + 236\beta_{5} + 487\beta_{4} - 1054\beta_{3} - 206\beta_{2} + 182\beta _1 + 1904 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2411 \beta_{7} - 1082 \beta_{6} + 1063 \beta_{5} + 2016 \beta_{4} - 4417 \beta_{3} - 956 \beta_{2} + \cdots + 8442 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 10719 \beta_{7} - 4723 \beta_{6} + 4642 \beta_{5} + 8799 \beta_{4} - 20178 \beta_{3} - 3894 \beta_{2} + \cdots + 36238 ) / 5 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
2.75978
−0.852282
−1.74919
1.37243
4.37243
−0.0444111
−1.65651
−2.20224
1.00000 1.00000 1.00000 0 1.00000 −4.80694 1.00000 1.00000 0
1.2 1.00000 1.00000 1.00000 0 1.00000 −3.52206 1.00000 1.00000 0
1.3 1.00000 1.00000 1.00000 0 1.00000 −0.533559 1.00000 1.00000 0
1.4 1.00000 1.00000 1.00000 0 1.00000 −0.329315 1.00000 1.00000 0
1.5 1.00000 1.00000 1.00000 0 1.00000 2.61995 1.00000 1.00000 0
1.6 1.00000 1.00000 1.00000 0 1.00000 2.70913 1.00000 1.00000 0
1.7 1.00000 1.00000 1.00000 0 1.00000 3.23143 1.00000 1.00000 0
1.8 1.00000 1.00000 1.00000 0 1.00000 4.63137 1.00000 1.00000 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.8
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(3\) \(-1\)
\(5\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3750.2.a.v 8
5.b even 2 1 3750.2.a.u 8
5.c odd 4 2 3750.2.c.k 16
25.d even 5 2 750.2.g.f 16
25.e even 10 2 750.2.g.g 16
25.f odd 20 2 150.2.h.b ✓ 16
25.f odd 20 2 750.2.h.d 16
75.l even 20 2 450.2.l.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
150.2.h.b ✓ 16 25.f odd 20 2
450.2.l.c 16 75.l even 20 2
750.2.g.f 16 25.d even 5 2
750.2.g.g 16 25.e even 10 2
750.2.h.d 16 25.f odd 20 2
3750.2.a.u 8 5.b even 2 1
3750.2.a.v 8 1.a even 1 1 trivial
3750.2.c.k 16 5.c odd 4 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} - 4T_{7}^{7} - 33T_{7}^{6} + 148T_{7}^{5} + 205T_{7}^{4} - 1328T_{7}^{3} + 592T_{7}^{2} + 1304T_{7} + 316 \) acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3750))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{8} \) Copy content Toggle raw display
$3$ \( (T - 1)^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} - 4 T^{7} + \cdots + 316 \) Copy content Toggle raw display
$11$ \( T^{8} - 6 T^{7} + \cdots - 724 \) Copy content Toggle raw display
$13$ \( T^{8} - 2 T^{7} + \cdots - 64 \) Copy content Toggle raw display
$17$ \( T^{8} - 14 T^{7} + \cdots + 1936 \) Copy content Toggle raw display
$19$ \( T^{8} - 10 T^{7} + \cdots - 1600 \) Copy content Toggle raw display
$23$ \( T^{8} - 12 T^{7} + \cdots + 143296 \) Copy content Toggle raw display
$29$ \( T^{8} - 10 T^{7} + \cdots + 6400 \) Copy content Toggle raw display
$31$ \( T^{8} - 16 T^{7} + \cdots - 14324 \) Copy content Toggle raw display
$37$ \( T^{8} + 6 T^{7} + \cdots - 64 \) Copy content Toggle raw display
$41$ \( T^{8} - 6 T^{7} + \cdots + 279376 \) Copy content Toggle raw display
$43$ \( T^{8} - 2 T^{7} + \cdots + 122816 \) Copy content Toggle raw display
$47$ \( T^{8} - 14 T^{7} + \cdots + 13122496 \) Copy content Toggle raw display
$53$ \( T^{8} - 12 T^{7} + \cdots + 10561 \) Copy content Toggle raw display
$59$ \( T^{8} - 225 T^{6} + \cdots - 40000 \) Copy content Toggle raw display
$61$ \( T^{8} - 16 T^{7} + \cdots - 3184 \) Copy content Toggle raw display
$67$ \( T^{8} + 6 T^{7} + \cdots + 14212096 \) Copy content Toggle raw display
$71$ \( T^{8} - 6 T^{7} + \cdots + 1843456 \) Copy content Toggle raw display
$73$ \( T^{8} + 8 T^{7} + \cdots - 2185984 \) Copy content Toggle raw display
$79$ \( T^{8} - 10 T^{7} + \cdots - 10529600 \) Copy content Toggle raw display
$83$ \( T^{8} - 22 T^{7} + \cdots + 159916 \) Copy content Toggle raw display
$89$ \( T^{8} - 20 T^{7} + \cdots - 190000 \) Copy content Toggle raw display
$97$ \( T^{8} + 16 T^{7} + \cdots - 9576059 \) Copy content Toggle raw display
show more
show less