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Properties

Label 4.21.b.b
Level $4$
Weight $21$
Character orbit 4.b
Analytic conductor $10.141$
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] = [4,21,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, 21, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4.3");
 
S:= CuspForms(chi, 21);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4 = 2^{2} \)
Weight: \( k \) \(=\) \( 21 \)
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: \(10.1405506041\)
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: \( x^{8} - 3 x^{7} + 41744 x^{6} - 12461904 x^{5} + 2339673984 x^{4} - 888904297728 x^{3} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{56}\cdot 3^{5}\cdot 5^{2} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_1 + 49) q^{2} + ( - \beta_{2} - 12 \beta_1 + 6) q^{3} + (\beta_{3} - \beta_{2} + 57 \beta_1 - 164539) q^{4} + ( - \beta_{5} + 4 \beta_{4} + \cdots + 2322148) q^{5}+ \cdots + (28 \beta_{7} + 28 \beta_{6} + \cdots - 1301762023) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 49) q^{2} + ( - \beta_{2} - 12 \beta_1 + 6) q^{3} + (\beta_{3} - \beta_{2} + 57 \beta_1 - 164539) q^{4} + ( - \beta_{5} + 4 \beta_{4} + \cdots + 2322148) q^{5}+ \cdots + (526662398304 \beta_{7} + \cdots + 18\!\cdots\!38) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 396 q^{2} - 1316080 q^{4} + 18568080 q^{5} + 99460608 q^{6} + 2177274816 q^{8} - 10411362168 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 396 q^{2} - 1316080 q^{4} + 18568080 q^{5} + 99460608 q^{6} + 2177274816 q^{8} - 10411362168 q^{9} - 18158047400 q^{10} - 41779799040 q^{12} - 43179394928 q^{13} - 200556776448 q^{14} - 1029373411072 q^{16} - 2434881831408 q^{17} + 5206289577036 q^{18} + 20485893906720 q^{20} + 33281721747456 q^{21} - 31324969489920 q^{22} - 47563142934528 q^{24} - 277943780989800 q^{25} + 132190787676888 q^{26} + 385881741772800 q^{28} + 100761472221840 q^{29} - 10\!\cdots\!40 q^{30}+ \cdots - 19\!\cdots\!84 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3 x^{7} + 41744 x^{6} - 12461904 x^{5} + 2339673984 x^{4} - 888904297728 x^{3} + \cdots + 18\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu - 1 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 18041 \nu^{7} - 3891479 \nu^{6} + 787005348 \nu^{5} + 302936330112 \nu^{4} + \cdots - 25\!\cdots\!76 ) / 36\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 18041 \nu^{7} - 3891479 \nu^{6} + 787005348 \nu^{5} + 302936330112 \nu^{4} + \cdots + 61\!\cdots\!24 ) / 36\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 249607 \nu^{7} - 123241 \nu^{6} + 10433869404 \nu^{5} - 3772990508928 \nu^{4} + \cdots - 58\!\cdots\!36 ) / 18\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 492821 \nu^{7} + 351728581 \nu^{6} - 22119906540 \nu^{5} + 10760216187264 \nu^{4} + \cdots + 43\!\cdots\!72 ) / 18\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1155067 \nu^{7} + 11657291 \nu^{6} + 211723652396 \nu^{5} - 32382180710784 \nu^{4} + \cdots + 10\!\cdots\!36 ) / 12\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 616355 \nu^{7} + 258986637 \nu^{6} + 168407965236 \nu^{5} + 14964065291648 \nu^{4} + \cdots - 41\!\cdots\!68 ) / 30\!\cdots\!88 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - \beta_{2} - 39\beta _1 - 166939 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{7} - 17\beta_{6} - 16\beta_{5} + 218\beta_{4} + 19\beta_{3} - 682\beta_{2} - 179149\beta _1 + 296169917 ) / 64 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 617 \beta_{7} - 659 \beta_{6} + 14224 \beta_{5} - 125170 \beta_{4} - 68455 \beta_{3} + \cdots - 45843578153 ) / 64 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 829803 \beta_{7} + 323639 \beta_{6} - 3021648 \beta_{5} + 11725770 \beta_{4} + 27176379 \beta_{3} + \cdots + 14687099299349 ) / 64 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 103862433 \beta_{7} - 125334907 \beta_{6} + 2054718480 \beta_{5} + 3844046494 \beta_{4} + \cdots - 617296274243617 ) / 64 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 32717086077 \beta_{7} + 18427121327 \beta_{6} + 381838456368 \beta_{5} + 1797322630810 \beta_{4} + \cdots - 11\!\cdots\!59 ) / 64 \) 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
−205.349 − 167.786i
−205.349 + 167.786i
−98.5328 − 240.932i
−98.5328 + 240.932i
77.2239 − 239.948i
77.2239 + 239.948i
228.158 − 88.6584i
228.158 + 88.6584i
−773.395 − 671.146i − 44571.4i 147703. + 1.03812e6i 1.43704e7 −2.99139e7 + 3.44713e7i 2.43808e8i 5.82498e8 − 9.02008e8i 1.50018e9 −1.11140e10 − 9.64462e9i
3.2 −773.395 + 671.146i 44571.4i 147703. − 1.03812e6i 1.43704e7 −2.99139e7 − 3.44713e7i − 2.43808e8i 5.82498e8 + 9.02008e8i 1.50018e9 −1.11140e10 + 9.64462e9i
3.3 −346.131 − 963.727i 109031.i −808962. + 667152.i −3.29975e6 1.05076e8 − 3.77390e7i − 3.27338e8i 9.22959e8 + 5.48696e8i −8.40096e9 1.14215e9 + 3.18005e9i
3.4 −346.131 + 963.727i − 109031.i −808962. − 667152.i −3.29975e6 1.05076e8 + 3.77390e7i 3.27338e8i 9.22959e8 − 5.48696e8i −8.40096e9 1.14215e9 − 3.18005e9i
3.5 356.896 − 959.792i − 46975.6i −793827. − 685091.i −4.32154e6 −4.50868e7 − 1.67654e7i − 1.10972e8i −9.40859e8 + 5.17403e8i 1.28008e9 −1.54234e9 + 4.14778e9i
3.6 356.896 + 959.792i 46975.6i −793827. + 685091.i −4.32154e6 −4.50868e7 + 1.67654e7i 1.10972e8i −9.40859e8 − 5.17403e8i 1.28008e9 −1.54234e9 − 4.14778e9i
3.7 960.631 − 354.634i 55423.5i 797046. − 681344.i 2.53495e6 1.96550e7 + 5.32415e7i 4.45714e8i 5.24039e8 − 9.37179e8i 4.15025e8 2.43515e9 − 8.98977e8i
3.8 960.631 + 354.634i − 55423.5i 797046. + 681344.i 2.53495e6 1.96550e7 − 5.32415e7i − 4.45714e8i 5.24039e8 + 9.37179e8i 4.15025e8 2.43515e9 + 8.98977e8i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.8
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 4.21.b.b ✓ 8
3.b odd 2 1 36.21.d.b 8
4.b odd 2 1 inner 4.21.b.b ✓ 8
8.b even 2 1 64.21.c.d 8
8.d odd 2 1 64.21.c.d 8
12.b even 2 1 36.21.d.b 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
4.21.b.b ✓ 8 1.a even 1 1 trivial
4.21.b.b ✓ 8 4.b odd 2 1 inner
36.21.d.b 8 3.b odd 2 1
36.21.d.b 8 12.b even 2 1
64.21.c.d 8 8.b even 2 1
64.21.c.d 8 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 19152818688 T_{3}^{6} + \cdots + 16\!\cdots\!80 \) acting on \(S_{21}^{\mathrm{new}}(4, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + \cdots + 12\!\cdots\!76 \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 16\!\cdots\!80 \) Copy content Toggle raw display
$5$ \( (T^{4} + \cdots + 51\!\cdots\!00)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{4} + \cdots - 53\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + \cdots + 52\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 22\!\cdots\!80 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 61\!\cdots\!56)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots - 87\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 42\!\cdots\!64)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 27\!\cdots\!80 \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 69\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots - 95\!\cdots\!44)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 24\!\cdots\!80 \) Copy content Toggle raw display
$71$ \( T^{8} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 91\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 51\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 16\!\cdots\!80 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots - 67\!\cdots\!84)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
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