Properties

Label 3700.2.d.i
Level $3700$
Weight $2$
Character orbit 3700.d
Analytic conductor $29.545$
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] = [3700,2,Mod(149,3700)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3700, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3700.149");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3700 = 2^{2} \cdot 5^{2} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3700.d (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: \(29.5446487479\)
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: \( x^{8} + 23x^{6} + 139x^{4} + 273x^{2} + 144 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{37}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 740)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + \beta_1) q^{3} + ( - \beta_{4} - \beta_1) q^{7} + (\beta_{5} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + \beta_1) q^{3} + ( - \beta_{4} - \beta_1) q^{7} + (\beta_{5} - 3) q^{9} + (\beta_{6} + \beta_{5} + 2) q^{11} + (2 \beta_{4} - \beta_{3}) q^{13} + (2 \beta_{4} + \beta_{3}) q^{17} + ( - \beta_{6} - \beta_{5} + \beta_{2} - 1) q^{19} + ( - \beta_{5} + 2 \beta_{2} + 4) q^{21} - 2 \beta_1 q^{23} + (\beta_{7} + 2 \beta_{4} + \beta_{3} - 2 \beta_1) q^{27} - 2 \beta_{2} q^{29} + ( - \beta_{6} + \beta_{5} - \beta_{2} - 1) q^{31} + (3 \beta_{7} + 3 \beta_{3} + 2 \beta_1) q^{33} + \beta_{4} q^{37} + (2 \beta_{6} + 2 \beta_{5} - 4 \beta_{2} + 2) q^{39} + (\beta_{6} + \beta_{5} + 8) q^{41} + ( - 2 \beta_{7} - 2 \beta_1) q^{43} + ( - \beta_{4} - 2 \beta_{3} - \beta_1) q^{47} + (\beta_{5} - 4 \beta_{2} + 1) q^{49} + ( - 2 \beta_{6} - 2 \beta_{5} + 2) q^{51} + ( - \beta_{7} + 2 \beta_{4} - \beta_{3} - 2 \beta_1) q^{53} + ( - 2 \beta_{7} + 4 \beta_{4} - 3 \beta_{3}) q^{57} + (\beta_{6} - \beta_{5} - \beta_{2} + 3) q^{59} + 2 q^{61} + (\beta_{7} + \beta_{4} - \beta_{3} + 5 \beta_1) q^{63} + (\beta_{7} + 3 \beta_{4} + \beta_{3} - \beta_1) q^{67} + ( - 2 \beta_{5} + 2 \beta_{2} + 10) q^{69} + (\beta_{6} + \beta_{5} + 2 \beta_{2} - 4) q^{71} + ( - \beta_{7} - \beta_{3} - 4 \beta_1) q^{73} + ( - \beta_{7} - 4 \beta_{4} - \beta_{3} - 2 \beta_1) q^{77} + ( - \beta_{6} - \beta_{5} + \beta_{2} + 5) q^{79} + ( - 3 \beta_{6} - 2 \beta_{5} + 5) q^{81} + ( - 2 \beta_{7} - 7 \beta_{4} - \beta_1) q^{83} + ( - 2 \beta_{7} - 10 \beta_{4} - 2 \beta_1) q^{87} + (2 \beta_{6} + 2) q^{89} + ( - 2 \beta_{5} + 4 \beta_{2} + 2) q^{91} + ( - 2 \beta_{7} - 2 \beta_{4} - \beta_{3} - 6 \beta_1) q^{93} + ( - 2 \beta_{7} + 4 \beta_{4} - 2 \beta_1) q^{97} + ( - 6 \beta_{6} - 4 \beta_{5} - 2 \beta_{2} + 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 26 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 26 q^{9} + 10 q^{11} + 38 q^{21} - 4 q^{29} - 8 q^{31} - 4 q^{39} + 58 q^{41} - 2 q^{49} + 28 q^{51} + 20 q^{59} + 16 q^{61} + 88 q^{69} - 34 q^{71} + 48 q^{79} + 56 q^{81} + 8 q^{89} + 28 q^{91} + 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 23x^{6} + 139x^{4} + 273x^{2} + 144 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 26\nu^{4} + 163\nu^{2} + 168 ) / 54 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + \nu^{5} + 323\nu^{3} + 1209\nu ) / 162 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 7\nu^{7} + 155\nu^{5} + 817\nu^{3} + 933\nu ) / 324 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{6} + 26\nu^{4} + 190\nu^{2} + 303 ) / 27 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -5\nu^{6} - 103\nu^{4} - 464\nu^{2} - 462 ) / 27 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -29\nu^{7} - 619\nu^{5} - 3107\nu^{3} - 4305\nu ) / 324 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{5} - 2\beta_{2} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{7} - 8\beta_{4} + \beta_{3} - 11\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{6} - 13\beta_{5} + 36\beta_{2} + 51 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 38\beta_{7} + 154\beta_{4} - 12\beta_{3} + 151\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -26\beta_{6} + 175\beta_{5} - 556\beta_{2} - 679 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -608\beta_{7} - 2430\beta_{4} + 149\beta_{3} - 2193\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3700\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1777\) \(1851\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
2.24418i
3.84956i
1.51951i
0.914132i
0.914132i
1.51951i
3.84956i
2.24418i
0 3.24418i 0 0 0 1.24418i 0 −7.52471 0
149.2 0 2.84956i 0 0 0 4.84956i 0 −5.11999 0
149.3 0 2.51951i 0 0 0 0.519509i 0 −3.34793 0
149.4 0 0.0858680i 0 0 0 1.91413i 0 2.99263 0
149.5 0 0.0858680i 0 0 0 1.91413i 0 2.99263 0
149.6 0 2.51951i 0 0 0 0.519509i 0 −3.34793 0
149.7 0 2.84956i 0 0 0 4.84956i 0 −5.11999 0
149.8 0 3.24418i 0 0 0 1.24418i 0 −7.52471 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 149.8
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 3700.2.d.i 8
5.b even 2 1 inner 3700.2.d.i 8
5.c odd 4 1 740.2.a.f 4
5.c odd 4 1 3700.2.a.j 4
15.e even 4 1 6660.2.a.r 4
20.e even 4 1 2960.2.a.v 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
740.2.a.f 4 5.c odd 4 1
2960.2.a.v 4 20.e even 4 1
3700.2.a.j 4 5.c odd 4 1
3700.2.d.i 8 1.a even 1 1 trivial
3700.2.d.i 8 5.b even 2 1 inner
6660.2.a.r 4 15.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 \(S_{2}^{\mathrm{new}}(3700, [\chi])\):

\( T_{3}^{8} + 25T_{3}^{6} + 204T_{3}^{4} + 544T_{3}^{2} + 4 \) Copy content Toggle raw display
\( T_{7}^{8} + 29T_{7}^{6} + 136T_{7}^{4} + 168T_{7}^{2} + 36 \) Copy content Toggle raw display
\( T_{13}^{8} + 76T_{13}^{6} + 1728T_{13}^{4} + 13456T_{13}^{2} + 25600 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 25 T^{6} + 204 T^{4} + 544 T^{2} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + 29 T^{6} + 136 T^{4} + \cdots + 36 \) Copy content Toggle raw display
$11$ \( (T^{4} - 5 T^{3} - 32 T^{2} + 192 T - 240)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + 76 T^{6} + 1728 T^{4} + \cdots + 25600 \) Copy content Toggle raw display
$17$ \( T^{8} + 92 T^{6} + 2272 T^{4} + \cdots + 2304 \) Copy content Toggle raw display
$19$ \( (T^{4} - 38 T^{2} + 54 T + 64)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + 92 T^{6} + 2224 T^{4} + \cdots + 36864 \) Copy content Toggle raw display
$29$ \( (T^{4} + 2 T^{3} - 44 T^{2} + 24 T + 192)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 4 T^{3} - 90 T^{2} - 350 T + 148)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 1)^{4} \) Copy content Toggle raw display
$41$ \( (T^{4} - 29 T^{3} + 274 T^{2} - 828 T - 168)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 272 T^{6} + 21280 T^{4} + \cdots + 1679616 \) Copy content Toggle raw display
$47$ \( T^{8} + 285 T^{6} + 23472 T^{4} + \cdots + 2916 \) Copy content Toggle raw display
$53$ \( T^{8} + 117 T^{6} + 2844 T^{4} + \cdots + 5184 \) Copy content Toggle raw display
$59$ \( (T^{4} - 10 T^{3} - 62 T^{2} + 846 T - 1968)^{2} \) Copy content Toggle raw display
$61$ \( (T - 2)^{8} \) Copy content Toggle raw display
$67$ \( T^{8} + 184 T^{6} + 636 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$71$ \( (T^{4} + 17 T^{3} - 8 T^{2} - 240 T - 288)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + 341 T^{6} + 26908 T^{4} + \cdots + 129600 \) Copy content Toggle raw display
$79$ \( (T^{4} - 24 T^{3} + 178 T^{2} - 354 T - 332)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 473 T^{6} + \cdots + 27436644 \) Copy content Toggle raw display
$89$ \( (T^{4} - 4 T^{3} - 128 T^{2} + 240 T + 3504)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 304 T^{6} + 23328 T^{4} + \cdots + 160000 \) Copy content Toggle raw display
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