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Properties

Label 4.33.b.b
Level $4$
Weight $33$
Character orbit 4.b
Analytic conductor $25.947$
Analytic rank $0$
Dimension $14$
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] = [4,33,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, 33, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4.3");
 
S:= CuspForms(chi, 33);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 33 \)
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: \(25.9466620569\)
Analytic rank: \(0\)
Dimension: \(14\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 72511313626452 x^{12} + \cdots + 62\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{182}\cdot 3^{20}\cdot 5^{6} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 - 1699) q^{2} + (\beta_{2} - 21 \beta_1 + 9) q^{3} + (\beta_{3} + 10 \beta_{2} + \cdots - 208774120) q^{4}+ \cdots + ( - \beta_{12} + \cdots - 798819458325515) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 - 1699) q^{2} + (\beta_{2} - 21 \beta_1 + 9) q^{3} + (\beta_{3} + 10 \beta_{2} + \cdots - 208774120) q^{4}+ \cdots + (17\!\cdots\!56 \beta_{13} + \cdots - 23\!\cdots\!65) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 23780 q^{2} - 2922848368 q^{4} + 138121491740 q^{5} + 1262734959552 q^{6} - 191366550113600 q^{8} - 11\!\cdots\!50 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 23780 q^{2} - 2922848368 q^{4} + 138121491740 q^{5} + 1262734959552 q^{6} - 191366550113600 q^{8} - 11\!\cdots\!50 q^{9}+ \cdots - 46\!\cdots\!00 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 72511313626452 x^{12} + \cdots + 62\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 39\!\cdots\!85 \nu^{13} + \cdots + 11\!\cdots\!60 ) / 75\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 39\!\cdots\!85 \nu^{13} + \cdots + 11\!\cdots\!80 ) / 36\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 98\!\cdots\!11 \nu^{13} + \cdots + 11\!\cdots\!20 ) / 67\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 15\!\cdots\!77 \nu^{13} + \cdots - 35\!\cdots\!40 ) / 17\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 93\!\cdots\!97 \nu^{13} + \cdots + 51\!\cdots\!80 ) / 75\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 15\!\cdots\!97 \nu^{13} + \cdots + 40\!\cdots\!00 ) / 75\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 40\!\cdots\!87 \nu^{13} + \cdots - 16\!\cdots\!20 ) / 75\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 41\!\cdots\!21 \nu^{13} + \cdots - 86\!\cdots\!60 ) / 10\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 19\!\cdots\!09 \nu^{13} + \cdots + 85\!\cdots\!60 ) / 25\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 14\!\cdots\!83 \nu^{13} + \cdots + 12\!\cdots\!20 ) / 18\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 76\!\cdots\!49 \nu^{13} + \cdots + 40\!\cdots\!40 ) / 75\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 19\!\cdots\!31 \nu^{13} + \cdots + 11\!\cdots\!80 ) / 75\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 36\!\cdots\!69 \nu^{13} + \cdots + 99\!\cdots\!00 ) / 28\!\cdots\!60 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} - 21\beta _1 + 9 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{12} - 2 \beta_{11} + 4 \beta_{10} + 2 \beta_{9} - \beta_{8} - 7 \beta_{6} + \cdots - 26\!\cdots\!56 ) / 256 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2368120 \beta_{13} + 18284465 \beta_{12} - 199408558 \beta_{11} + 65384769 \beta_{10} + \cdots + 68\!\cdots\!19 ) / 4096 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 13\!\cdots\!90 \beta_{12} + \cdots + 28\!\cdots\!14 ) / 16384 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 29\!\cdots\!40 \beta_{13} + \cdots - 70\!\cdots\!67 ) / 131072 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 19\!\cdots\!08 \beta_{12} + \cdots - 36\!\cdots\!74 ) / 1048576 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 27\!\cdots\!80 \beta_{13} + \cdots + 12\!\cdots\!45 ) / 524288 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 21\!\cdots\!12 \beta_{12} + \cdots + 40\!\cdots\!23 ) / 524288 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 55\!\cdots\!60 \beta_{13} + \cdots - 40\!\cdots\!47 ) / 524288 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 10\!\cdots\!64 \beta_{12} + \cdots - 18\!\cdots\!64 ) / 1048576 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 51\!\cdots\!00 \beta_{13} + \cdots + 59\!\cdots\!58 ) / 262144 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 60\!\cdots\!72 \beta_{12} + \cdots + 11\!\cdots\!19 ) / 262144 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 90\!\cdots\!80 \beta_{13} + \cdots - 16\!\cdots\!43 ) / 262144 \) 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
2.77300e6i
− 2.77300e6i
− 2.55660e6i
2.55660e6i
− 1.78456e6i
1.78456e6i
4.36932e6i
− 4.36932e6i
895767.i
− 895767.i
− 5.00408e6i
5.00408e6i
3.18854e6i
− 3.18854e6i
−61401.1 − 22910.1i 4.43679e7i 3.24522e9 + 2.81341e9i 1.39224e11 1.01647e12 − 2.72424e12i − 2.23454e13i −1.34805e14 − 2.47095e14i −1.15494e14 −8.54848e15 − 3.18962e15i
3.2 −61401.1 + 22910.1i − 4.43679e7i 3.24522e9 − 2.81341e9i 1.39224e11 1.01647e12 + 2.72424e12i 2.23454e13i −1.34805e14 + 2.47095e14i −1.15494e14 −8.54848e15 + 3.18962e15i
3.3 −56022.2 − 34007.1i − 4.09055e7i 1.98200e9 + 3.81031e9i −2.17196e11 −1.39108e12 + 2.29162e12i − 2.81808e12i 1.85418e13 − 2.80864e14i 1.79757e14 1.21678e16 + 7.38622e15i
3.4 −56022.2 + 34007.1i 4.09055e7i 1.98200e9 − 3.81031e9i −2.17196e11 −1.39108e12 − 2.29162e12i 2.81808e12i 1.85418e13 + 2.80864e14i 1.79757e14 1.21678e16 − 7.38622e15i
3.5 −18058.8 − 62998.8i − 2.85529e7i −3.64273e9 + 2.27536e9i 1.21258e11 −1.79880e12 + 5.15631e11i 3.17263e10i 2.09128e14 + 1.88397e14i 1.03775e15 −2.18978e15 − 7.63913e15i
3.6 −18058.8 + 62998.8i 2.85529e7i −3.64273e9 − 2.27536e9i 1.21258e11 −1.79880e12 − 5.15631e11i − 3.17263e10i 2.09128e14 − 1.88397e14i 1.03775e15 −2.18978e15 + 7.63913e15i
3.7 −15224.0 − 63743.2i 6.99091e7i −3.83143e9 + 1.94085e9i −1.46681e11 4.45623e12 − 1.06430e12i 5.71644e13i 1.82046e14 + 2.14680e14i −3.03426e15 2.23307e15 + 9.34993e15i
3.8 −15224.0 + 63743.2i − 6.99091e7i −3.83143e9 − 1.94085e9i −1.46681e11 4.45623e12 + 1.06430e12i − 5.71644e13i 1.82046e14 − 2.14680e14i −3.03426e15 2.23307e15 − 9.34993e15i
3.9 32011.4 − 57186.0i 1.43323e7i −2.24550e9 − 3.66121e9i −4.45746e10 8.19605e11 + 4.58796e11i − 4.42477e13i −2.81252e14 + 1.12108e13i 1.64761e15 −1.42690e15 + 2.54904e15i
3.10 32011.4 + 57186.0i − 1.43323e7i −2.24550e9 + 3.66121e9i −4.45746e10 8.19605e11 − 4.58796e11i 4.42477e13i −2.81252e14 − 1.12108e13i 1.64761e15 −1.42690e15 − 2.54904e15i
3.11 46092.7 − 46587.9i − 8.00653e7i −4.59006e7 − 4.29472e9i −3.25403e10 −3.73007e12 − 3.69042e12i 4.74477e13i −2.02198e14 − 1.95817e14i −4.55743e15 −1.49987e15 + 1.51599e15i
3.12 46092.7 + 46587.9i 8.00653e7i −4.59006e7 + 4.29472e9i −3.25403e10 −3.73007e12 + 3.69042e12i − 4.74477e13i −2.02198e14 + 1.95817e14i −4.55743e15 −1.49987e15 − 1.51599e15i
3.13 60712.0 − 24678.5i 5.10167e7i 3.07691e9 − 2.99655e9i 2.49571e11 1.25901e12 + 3.09732e12i 4.65014e13i 1.12855e14 − 2.57860e14i −7.49683e14 1.51520e16 − 6.15903e15i
3.14 60712.0 + 24678.5i − 5.10167e7i 3.07691e9 + 2.99655e9i 2.49571e11 1.25901e12 − 3.09732e12i − 4.65014e13i 1.12855e14 + 2.57860e14i −7.49683e14 1.51520e16 + 6.15903e15i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.14
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 4.33.b.b ✓ 14
3.b odd 2 1 36.33.d.b 14
4.b odd 2 1 inner 4.33.b.b ✓ 14
12.b even 2 1 36.33.d.b 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.33.b.b ✓ 14 1.a even 1 1 trivial
4.33.b.b ✓ 14 4.b odd 2 1 inner
36.33.d.b 14 3.b odd 2 1
36.33.d.b 14 12.b even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{14} + \cdots + 44\!\cdots\!00 \) acting on \(S_{33}^{\mathrm{new}}(4, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} + \cdots + 26\!\cdots\!16 \) Copy content Toggle raw display
$3$ \( T^{14} + \cdots + 44\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( (T^{7} + \cdots - 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( T^{14} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{14} + \cdots + 47\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{7} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{7} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( T^{14} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{7} + \cdots - 57\!\cdots\!52)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{7} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{7} + \cdots + 13\!\cdots\!52)^{2} \) Copy content Toggle raw display
$43$ \( T^{14} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{14} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{7} + \cdots - 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{14} + \cdots + 31\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{7} + \cdots - 24\!\cdots\!68)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{14} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{7} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{14} + \cdots + 97\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 91\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{7} + \cdots + 15\!\cdots\!28)^{2} \) Copy content Toggle raw display
$97$ \( (T^{7} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
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