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Properties

Label 4.37.b.b
Level $4$
Weight $37$
Character orbit 4.b
Analytic conductor $32.837$
Analytic rank $0$
Dimension $16$
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] = [4,37,Mod(3,4)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 37, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4.3");
 
S:= CuspForms(chi, 37);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 37 \)
Character orbit: \([\chi]\) \(=\) 4.b (of order \(2\), degree \(1\), 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: \(32.8365034637\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{240}\cdot 3^{24}\cdot 5^{6}\cdot 7^{2} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 11077) q^{2} + (\beta_{2} + 198 \beta_1 + 50) q^{3} + (\beta_{3} + 12 \beta_{2} + \cdots - 350666438) q^{4}+ \cdots + (\beta_{9} - \beta_{8} + \cdots - 58\!\cdots\!27) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 11077) q^{2} + (\beta_{2} + 198 \beta_1 + 50) q^{3} + (\beta_{3} + 12 \beta_{2} + \cdots - 350666438) q^{4}+ \cdots + ( - 13\!\cdots\!96 \beta_{15} + \cdots - 43\!\cdots\!22) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 177228 q^{2} - 5610707696 q^{4} + 5816089539360 q^{5} - 217628996575488 q^{6} + 11\!\cdots\!28 q^{8}+ \cdots - 93\!\cdots\!68 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 177228 q^{2} - 5610707696 q^{4} + 5816089539360 q^{5} - 217628996575488 q^{6} + 11\!\cdots\!28 q^{8}+ \cdots + 10\!\cdots\!28 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + \cdots + 51\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 11\!\cdots\!75 \nu^{15} + \cdots + 71\!\cdots\!00 ) / 46\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11\!\cdots\!75 \nu^{15} + \cdots - 71\!\cdots\!00 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 41\!\cdots\!39 \nu^{15} + \cdots - 51\!\cdots\!00 ) / 77\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 53\!\cdots\!33 \nu^{15} + \cdots - 59\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 39\!\cdots\!05 \nu^{15} + \cdots + 12\!\cdots\!00 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 90\!\cdots\!09 \nu^{15} + \cdots + 90\!\cdots\!00 ) / 46\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 81\!\cdots\!23 \nu^{15} + \cdots + 34\!\cdots\!00 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 19\!\cdots\!71 \nu^{15} + \cdots - 77\!\cdots\!00 ) / 14\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 40\!\cdots\!11 \nu^{15} + \cdots + 82\!\cdots\!00 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 12\!\cdots\!51 \nu^{15} + \cdots + 65\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 56\!\cdots\!63 \nu^{15} + \cdots - 42\!\cdots\!00 ) / 46\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 12\!\cdots\!89 \nu^{15} + \cdots + 64\!\cdots\!00 ) / 97\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 15\!\cdots\!95 \nu^{15} + \cdots + 17\!\cdots\!00 ) / 46\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 34\!\cdots\!39 \nu^{15} + \cdots + 32\!\cdots\!00 ) / 51\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 32\!\cdots\!63 \nu^{15} + \cdots - 56\!\cdots\!00 ) / 46\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} + 198\beta _1 + 50 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} - \beta_{8} - 89 \beta_{7} + 2378 \beta_{5} + 6644 \beta_{4} - 31791 \beta_{3} + \cdots - 20\!\cdots\!48 ) / 256 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 407791 \beta_{15} + 21026577 \beta_{14} - 26417792 \beta_{13} - 96107592 \beta_{12} + \cdots - 52\!\cdots\!84 ) / 4096 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 27294115667058 \beta_{13} + \cdots + 17\!\cdots\!32 ) / 16384 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 18\!\cdots\!30 \beta_{15} + \cdots - 78\!\cdots\!94 ) / 262144 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 37\!\cdots\!34 \beta_{13} + \cdots - 16\!\cdots\!14 ) / 1048576 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 45\!\cdots\!64 \beta_{15} + \cdots + 20\!\cdots\!82 ) / 16777216 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 11\!\cdots\!24 \beta_{13} + \cdots + 23\!\cdots\!83 ) / 8388608 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 59\!\cdots\!00 \beta_{15} + \cdots - 26\!\cdots\!85 ) / 8388608 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 32\!\cdots\!00 \beta_{13} + \cdots - 41\!\cdots\!67 ) / 8388608 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 27\!\cdots\!96 \beta_{15} + \cdots + 12\!\cdots\!48 ) / 16777216 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 38\!\cdots\!28 \beta_{13} + \cdots + 38\!\cdots\!59 ) / 4194304 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 14\!\cdots\!12 \beta_{15} + \cdots - 67\!\cdots\!91 ) / 4194304 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 87\!\cdots\!24 \beta_{13} + \cdots - 73\!\cdots\!61 ) / 4194304 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 12\!\cdots\!48 \beta_{15} + \cdots + 57\!\cdots\!94 ) / 16777216 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
− 1.10776e7i
1.10776e7i
4.46418e7i
− 4.46418e7i
− 3.83971e7i
3.83971e7i
8.47589e6i
− 8.47589e6i
− 2.99676e7i
2.99676e7i
2.95956e7i
− 2.95956e7i
− 2.71780e7i
2.71780e7i
− 1.85096e7i
1.85096e7i
−227674. − 129939.i − 1.77242e8i 3.49511e10 + 5.91675e10i 5.55508e12 −2.30307e13 + 4.03533e13i − 2.49971e15i −2.69262e14 − 1.80124e16i 1.18680e17 −1.26475e18 − 7.21823e17i
3.2 −227674. + 129939.i 1.77242e8i 3.49511e10 − 5.91675e10i 5.55508e12 −2.30307e13 − 4.03533e13i 2.49971e15i −2.69262e14 + 1.80124e16i 1.18680e17 −1.26475e18 + 7.21823e17i
3.3 −210329. − 156464.i 7.14268e8i 1.97575e10 + 6.58180e10i 8.84656e11 1.11757e14 − 1.50232e14i 1.08208e15i 6.14256e15 − 1.69348e16i −3.60085e17 −1.86069e17 − 1.38417e17i
3.4 −210329. + 156464.i − 7.14268e8i 1.97575e10 − 6.58180e10i 8.84656e11 1.11757e14 + 1.50232e14i − 1.08208e15i 6.14256e15 + 1.69348e16i −3.60085e17 −1.86069e17 + 1.38417e17i
3.5 −167207. − 201894.i − 6.14354e8i −1.28029e10 + 6.75163e10i −3.33360e12 −1.24034e14 + 1.02724e14i 2.74304e15i 1.57719e16 − 8.70438e15i −2.27336e17 5.57402e17 + 6.73034e17i
3.6 −167207. + 201894.i 6.14354e8i −1.28029e10 − 6.75163e10i −3.33360e12 −1.24034e14 − 1.02724e14i − 2.74304e15i 1.57719e16 + 8.70438e15i −2.27336e17 5.57402e17 − 6.73034e17i
3.7 −39525.7 − 259147.i 1.35614e8i −6.55949e10 + 2.04859e10i −1.24080e12 3.51440e13 − 5.36025e12i − 4.20496e14i 7.90156e15 + 1.61890e16i 1.31703e17 4.90435e16 + 3.21549e17i
3.8 −39525.7 + 259147.i − 1.35614e8i −6.55949e10 − 2.04859e10i −1.24080e12 3.51440e13 + 5.36025e12i 4.20496e14i 7.90156e15 − 1.61890e16i 1.31703e17 4.90435e16 − 3.21549e17i
3.9 98385.8 − 242981.i − 4.79482e8i −4.93600e10 − 4.78117e10i 6.60636e12 −1.16505e14 − 4.71742e13i 2.27014e14i −1.64737e16 + 7.28954e15i −7.98083e16 6.49972e17 − 1.60522e18i
3.10 98385.8 + 242981.i 4.79482e8i −4.93600e10 + 4.78117e10i 6.60636e12 −1.16505e14 + 4.71742e13i − 2.27014e14i −1.64737e16 − 7.28954e15i −7.98083e16 6.49972e17 + 1.60522e18i
3.11 178949. − 191564.i 4.73530e8i −4.67423e9 − 6.85603e10i 4.97998e11 9.07113e13 + 8.47375e13i 9.39063e14i −1.39702e16 − 1.13734e16i −7.41357e16 8.91161e16 − 9.53986e16i
3.12 178949. + 191564.i − 4.73530e8i −4.67423e9 + 6.85603e10i 4.97998e11 9.07113e13 − 8.47375e13i − 9.39063e14i −1.39702e16 + 1.13734e16i −7.41357e16 8.91161e16 + 9.53986e16i
3.13 194816. − 175403.i − 4.34848e8i 7.18697e9 − 6.83426e10i −7.52461e12 −7.62737e13 − 8.47153e13i − 2.26538e15i −1.05874e16 − 1.45748e16i −3.89984e16 −1.46591e18 + 1.31984e18i
3.14 194816. + 175403.i 4.34848e8i 7.18697e9 + 6.83426e10i −7.52461e12 −7.62737e13 + 8.47153e13i 2.26538e15i −1.05874e16 + 1.45748e16i −3.89984e16 −1.46591e18 − 1.31984e18i
3.15 261200. − 22229.7i − 2.96154e8i 6.77312e10 − 1.16128e10i 1.46296e12 −6.58343e12 − 7.73554e13i 2.38729e15i 1.74332e16 − 4.53891e15i 6.23874e16 3.82125e17 − 3.25212e16i
3.16 261200. + 22229.7i 2.96154e8i 6.77312e10 + 1.16128e10i 1.46296e12 −6.58343e12 + 7.73554e13i − 2.38729e15i 1.74332e16 + 4.53891e15i 6.23874e16 3.82125e17 + 3.25212e16i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 4.37.b.b ✓ 16
4.b odd 2 1 inner 4.37.b.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.37.b.b ✓ 16 1.a even 1 1 trivial
4.37.b.b ✓ 16 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + \cdots + 95\!\cdots\!00 \) acting on \(S_{37}^{\mathrm{new}}(4, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + \cdots + 49\!\cdots\!56 \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 95\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( (T^{8} + \cdots - 73\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 80\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots - 88\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 52\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots - 79\!\cdots\!76)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots - 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 11\!\cdots\!56)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots - 30\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots - 43\!\cdots\!64)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 55\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 41\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 48\!\cdots\!64)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
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