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Properties

Label 4.39.b.a
Level $4$
Weight $39$
Character orbit 4.b
Analytic conductor $36.585$
Analytic rank $0$
Dimension $18$
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] = [4,39,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, 39, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4.3");
 
S:= CuspForms(chi, 39);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 39 \)
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: \(36.5853876134\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 9 x^{17} + \cdots + 98\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{306}\cdot 3^{34}\cdot 5^{10}\cdot 19^{2} \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 20235) q^{2} + ( - \beta_{2} + 165 \beta_1 - 18) q^{3} + ( - \beta_{3} + 6 \beta_{2} + \cdots + 11142631926) q^{4}+ \cdots + (\beta_{8} - \beta_{7} + \cdots - 41\!\cdots\!00) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 20235) q^{2} + ( - \beta_{2} + 165 \beta_1 - 18) q^{3} + ( - \beta_{3} + 6 \beta_{2} + \cdots + 11142631926) q^{4}+ \cdots + ( - 13\!\cdots\!66 \beta_{17} + \cdots - 28\!\cdots\!54) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 364228 q^{2} + 200567335824 q^{4} - 8991287507020 q^{5} + 817599417526752 q^{6} + 90\!\cdots\!32 q^{8}+ \cdots - 75\!\cdots\!14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 364228 q^{2} + 200567335824 q^{4} - 8991287507020 q^{5} + 817599417526752 q^{6} + 90\!\cdots\!32 q^{8}+ \cdots + 79\!\cdots\!72 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 9 x^{17} + \cdots + 98\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 46\!\cdots\!75 \nu^{17} + \cdots - 16\!\cdots\!00 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 46\!\cdots\!75 \nu^{17} + \cdots - 16\!\cdots\!00 ) / 22\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 14\!\cdots\!97 \nu^{17} + \cdots - 27\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 25\!\cdots\!26 \nu^{17} + \cdots + 64\!\cdots\!00 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 41\!\cdots\!87 \nu^{17} + \cdots + 18\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 22\!\cdots\!25 \nu^{17} + \cdots - 63\!\cdots\!00 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 47\!\cdots\!09 \nu^{17} + \cdots + 67\!\cdots\!00 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 47\!\cdots\!53 \nu^{17} + \cdots + 29\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 21\!\cdots\!59 \nu^{17} + \cdots + 59\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 15\!\cdots\!77 \nu^{17} + \cdots + 32\!\cdots\!00 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 47\!\cdots\!55 \nu^{17} + \cdots - 37\!\cdots\!00 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 20\!\cdots\!25 \nu^{17} + \cdots + 35\!\cdots\!00 ) / 93\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 13\!\cdots\!91 \nu^{17} + \cdots + 46\!\cdots\!00 ) / 41\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 31\!\cdots\!09 \nu^{17} + \cdots + 28\!\cdots\!00 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 20\!\cdots\!41 \nu^{17} + \cdots - 57\!\cdots\!00 ) / 18\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 84\!\cdots\!57 \nu^{17} + \cdots + 89\!\cdots\!00 ) / 37\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 38\!\cdots\!89 \nu^{17} + \cdots + 11\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} - 165\beta _1 + 26 ) / 16 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{8} - \beta_{7} + 93 \beta_{6} + 184 \beta_{5} - 787 \beta_{4} + 338301 \beta_{3} + \cdots - 17\!\cdots\!37 ) / 256 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 229606 \beta_{17} + 238184 \beta_{16} + 1952544 \beta_{15} - 87240 \beta_{14} + \cdots - 10\!\cdots\!82 ) / 4096 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 102656040740096 \beta_{17} - 70908194140864 \beta_{16} - 282262760903076 \beta_{15} + \cdots + 29\!\cdots\!42 ) / 32768 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 27\!\cdots\!95 \beta_{17} + \cdots + 14\!\cdots\!74 ) / 262144 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 11\!\cdots\!08 \beta_{17} + \cdots - 15\!\cdots\!90 ) / 1048576 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 31\!\cdots\!33 \beta_{17} + \cdots - 16\!\cdots\!96 ) / 16777216 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 90\!\cdots\!84 \beta_{17} + \cdots + 86\!\cdots\!54 ) / 33554432 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 45\!\cdots\!40 \beta_{17} + \cdots + 22\!\cdots\!97 ) / 134217728 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 82\!\cdots\!52 \beta_{17} + \cdots - 64\!\cdots\!19 ) / 134217728 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 82\!\cdots\!36 \beta_{17} + \cdots - 40\!\cdots\!05 ) / 134217728 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 17\!\cdots\!08 \beta_{17} + \cdots + 12\!\cdots\!55 ) / 134217728 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 76\!\cdots\!60 \beta_{17} + \cdots + 38\!\cdots\!57 ) / 67108864 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 18\!\cdots\!56 \beta_{17} + \cdots - 12\!\cdots\!71 ) / 67108864 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 14\!\cdots\!00 \beta_{17} + \cdots - 75\!\cdots\!71 ) / 67108864 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 39\!\cdots\!56 \beta_{17} + \cdots + 23\!\cdots\!37 ) / 67108864 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 56\!\cdots\!56 \beta_{17} + \cdots + 30\!\cdots\!31 ) / 134217728 \) 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
0.500000 + 9.26410e7i
0.500000 − 9.26410e7i
0.500000 − 4.81893e7i
0.500000 + 4.81893e7i
0.500000 + 7.90224e7i
0.500000 − 7.90224e7i
0.500000 − 5.46285e7i
0.500000 + 5.46285e7i
0.500000 − 1.22705e8i
0.500000 + 1.22705e8i
0.500000 + 7.59483e7i
0.500000 − 7.59483e7i
0.500000 − 2.95082e7i
0.500000 + 2.95082e7i
0.500000 − 4.14425e6i
0.500000 + 4.14425e6i
0.500000 + 1.42659e8i
0.500000 − 1.42659e8i
−521279. − 56087.1i − 1.48226e9i 2.68586e11 + 5.84741e10i 1.90217e13 −8.31355e13 + 7.72670e14i 1.45250e16i −1.36729e17 − 4.55456e16i −8.46232e17 −9.91563e18 − 1.06687e18i
3.2 −521279. + 56087.1i 1.48226e9i 2.68586e11 − 5.84741e10i 1.90217e13 −8.31355e13 − 7.72670e14i − 1.45250e16i −1.36729e17 + 4.55456e16i −8.46232e17 −9.91563e18 + 1.06687e18i
3.3 −453236. − 263543.i 7.71029e8i 1.35968e11 + 2.38895e11i −2.10588e13 2.03199e14 − 3.49458e14i 5.85221e15i 1.33358e15 − 1.44109e17i 7.56367e17 9.54463e18 + 5.54992e18i
3.4 −453236. + 263543.i − 7.71029e8i 1.35968e11 − 2.38895e11i −2.10588e13 2.03199e14 + 3.49458e14i − 5.85221e15i 1.33358e15 + 1.44109e17i 7.56367e17 9.54463e18 − 5.54992e18i
3.5 −306058. − 425684.i − 1.26436e9i −8.75354e10 + 2.60567e11i −4.90299e12 −5.38217e14 + 3.86966e14i − 1.75338e16i 1.37710e17 − 4.24863e16i −2.47749e17 1.50060e18 + 2.08712e18i
3.6 −306058. + 425684.i 1.26436e9i −8.75354e10 − 2.60567e11i −4.90299e12 −5.38217e14 − 3.86966e14i 1.75338e16i 1.37710e17 + 4.24863e16i −2.47749e17 1.50060e18 − 2.08712e18i
3.7 −173179. − 494861.i 8.74057e8i −2.14896e11 + 1.71399e11i 2.48589e13 4.32536e14 − 1.51368e14i 8.29070e15i 1.22034e17 + 7.66608e16i 5.86877e17 −4.30504e18 − 1.23017e19i
3.8 −173179. + 494861.i − 8.74057e8i −2.14896e11 − 1.71399e11i 2.48589e13 4.32536e14 + 1.51368e14i − 8.29070e15i 1.22034e17 − 7.66608e16i 5.86877e17 −4.30504e18 + 1.23017e19i
3.9 112582. − 512058.i 1.96328e9i −2.49528e11 − 1.15297e11i −3.09501e13 1.00531e15 + 2.21030e14i − 9.14517e15i −8.71314e16 + 1.14792e17i −2.50360e18 −3.48443e18 + 1.58482e19i
3.10 112582. + 512058.i − 1.96328e9i −2.49528e11 + 1.15297e11i −3.09501e13 1.00531e15 − 2.21030e14i 9.14517e15i −8.71314e16 − 1.14792e17i −2.50360e18 −3.48443e18 − 1.58482e19i
3.11 130787. − 507713.i − 1.21517e9i −2.40667e11 − 1.32805e11i −7.30752e12 −6.16959e14 − 1.58929e14i 7.26783e15i −9.89028e16 + 1.04821e17i −1.25791e17 −9.55728e17 + 3.71012e18i
3.12 130787. + 507713.i 1.21517e9i −2.40667e11 + 1.32805e11i −7.30752e12 −6.16959e14 + 1.58929e14i − 7.26783e15i −9.89028e16 − 1.04821e17i −1.25791e17 −9.55728e17 − 3.71012e18i
3.13 383569. − 357425.i 4.72132e8i 1.93720e10 − 2.74194e11i 2.09077e13 1.68752e14 + 1.81095e14i − 1.08901e16i −9.05736e16 − 1.12096e17i 1.12794e18 8.01953e18 − 7.47293e18i
3.14 383569. + 357425.i − 4.72132e8i 1.93720e10 + 2.74194e11i 2.09077e13 1.68752e14 − 1.81095e14i 1.08901e16i −9.05736e16 + 1.12096e17i 1.12794e18 8.01953e18 + 7.47293e18i
3.15 490276. − 185761.i 6.63080e7i 2.05863e11 − 1.82149e11i −1.96572e13 1.23175e13 + 3.25092e13i 1.51470e16i 6.70938e16 − 1.27545e17i 1.34645e18 −9.63744e18 + 3.65154e18i
3.16 490276. + 185761.i − 6.63080e7i 2.05863e11 + 1.82149e11i −1.96572e13 1.23175e13 − 3.25092e13i − 1.51470e16i 6.70938e16 + 1.27545e17i 1.34645e18 −9.63744e18 − 3.65154e18i
3.17 518652. − 76670.9i − 2.28254e9i 2.63121e11 − 7.95309e10i 1.45926e13 −1.75005e14 − 1.18385e15i − 8.90761e15i 1.30370e17 − 6.14226e16i −3.85916e18 7.56849e18 − 1.11883e18i
3.18 518652. + 76670.9i 2.28254e9i 2.63121e11 + 7.95309e10i 1.45926e13 −1.75005e14 + 1.18385e15i 8.90761e15i 1.30370e17 + 6.14226e16i −3.85916e18 7.56849e18 + 1.11883e18i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.18
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 4.39.b.a ✓ 18
4.b odd 2 1 inner 4.39.b.a ✓ 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.39.b.a ✓ 18 1.a even 1 1 trivial
4.39.b.a ✓ 18 4.b odd 2 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{39}^{\mathrm{new}}(4, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} + \cdots + 89\!\cdots\!04 \) Copy content Toggle raw display
$3$ \( T^{18} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( (T^{9} + \cdots + 66\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{18} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{9} + \cdots + 67\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{9} + \cdots - 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( T^{18} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{9} + \cdots + 12\!\cdots\!88)^{2} \) Copy content Toggle raw display
$31$ \( T^{18} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{9} + \cdots - 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{9} + \cdots - 57\!\cdots\!72)^{2} \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 68\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{9} + \cdots + 66\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{18} + \cdots + 28\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{9} + \cdots - 22\!\cdots\!92)^{2} \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{18} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{9} + \cdots - 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{18} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{9} + \cdots + 44\!\cdots\!68)^{2} \) Copy content Toggle raw display
$97$ \( (T^{9} + \cdots - 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
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