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Properties

Label 8.22.b.a
Level $8$
Weight $22$
Character orbit 8.b
Analytic conductor $22.358$
Analytic rank $0$
Dimension $20$
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] = [8,22,Mod(5,8)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8.5");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 8.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: \(22.3581875430\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 10 x^{19} + 8499037025 x^{18} - 76491332940 x^{17} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: multiple of \( 2^{190}\cdot 3^{14}\cdot 5^{4}\cdot 7^{7} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 + 14) q^{2} + ( - \beta_{2} + 3 \beta_1 + 1) q^{3} + ( - \beta_{3} - 14 \beta_1 + 20490) q^{4} + (\beta_{5} - \beta_{3} - 9 \beta_{2} + \cdots - 365) q^{5}+ \cdots + (\beta_{9} + \beta_{7} + \cdots - 3137816611) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 + 14) q^{2} + ( - \beta_{2} + 3 \beta_1 + 1) q^{3} + ( - \beta_{3} - 14 \beta_1 + 20490) q^{4} + (\beta_{5} - \beta_{3} - 9 \beta_{2} + \cdots - 365) q^{5}+ \cdots + (18680248 \beta_{19} + \cdots - 11\!\cdots\!99) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 286 q^{2} + 409876 q^{4} + 118236748 q^{6} + 564950496 q^{7} + 3649699336 q^{8} - 62762119220 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 286 q^{2} + 409876 q^{4} + 118236748 q^{6} + 564950496 q^{7} + 3649699336 q^{8} - 62762119220 q^{9} - 51269339528 q^{10} + 65316900136 q^{12} - 1924309104464 q^{14} - 2285680856096 q^{15} + 7793508571920 q^{16} + 1740714497000 q^{17} - 41325344023822 q^{18} - 78999934156016 q^{20} + 294957216952508 q^{22} + 335749622520224 q^{23} + 786737698607504 q^{24} - 16\!\cdots\!04 q^{25}+ \cdots + 13\!\cdots\!30 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 10 x^{19} + 8499037025 x^{18} - 76491332940 x^{17} + \cdots + 35\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 29\!\cdots\!75 \nu^{19} + \cdots + 36\!\cdots\!00 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 29\!\cdots\!75 \nu^{19} + \cdots + 36\!\cdots\!00 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 14\!\cdots\!18 \nu^{19} + \cdots + 80\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 62\!\cdots\!41 \nu^{19} + \cdots - 68\!\cdots\!00 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 26\!\cdots\!78 \nu^{19} + \cdots + 42\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 19\!\cdots\!79 \nu^{19} + \cdots - 28\!\cdots\!00 ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 99\!\cdots\!17 \nu^{19} + \cdots - 14\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 31\!\cdots\!63 \nu^{19} + \cdots - 64\!\cdots\!00 ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 10\!\cdots\!53 \nu^{19} + \cdots - 26\!\cdots\!00 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 40\!\cdots\!99 \nu^{19} + \cdots + 77\!\cdots\!00 ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 24\!\cdots\!83 \nu^{19} + \cdots - 17\!\cdots\!00 ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 16\!\cdots\!83 \nu^{19} + \cdots + 45\!\cdots\!00 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 17\!\cdots\!97 \nu^{19} + \cdots + 34\!\cdots\!00 ) / 40\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 54\!\cdots\!50 \nu^{19} + \cdots - 59\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 56\!\cdots\!29 \nu^{19} + \cdots - 34\!\cdots\!00 ) / 12\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 12\!\cdots\!95 \nu^{19} + \cdots + 35\!\cdots\!00 ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 87\!\cdots\!49 \nu^{19} + \cdots + 16\!\cdots\!00 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( - 83\!\cdots\!59 \nu^{19} + \cdots + 14\!\cdots\!00 ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 32\!\cdots\!15 \nu^{19} + \cdots + 53\!\cdots\!00 ) / 48\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{2} - 3\beta _1 + 1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} + \beta_{7} - 4\beta_{6} - \beta_{5} - 467\beta_{3} + 315\beta_{2} + 963976\beta _1 - 13598169814 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 216 \beta_{19} + 128 \beta_{18} - 148 \beta_{17} + 330 \beta_{16} + 1961 \beta_{15} + \cdots - 65924238814 ) / 64 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6479142 \beta_{19} - 6477766 \beta_{18} + 5751370 \beta_{17} - 6368267 \beta_{16} + \cdots + 15\!\cdots\!16 ) / 128 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 4492587005148 \beta_{19} + 1507086129732 \beta_{18} + 3284020257464 \beta_{17} + \cdots + 16\!\cdots\!30 ) / 512 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 95\!\cdots\!96 \beta_{19} + \cdots - 10\!\cdots\!90 ) / 512 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 24\!\cdots\!23 \beta_{19} + \cdots - 11\!\cdots\!98 ) / 128 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 12\!\cdots\!16 \beta_{19} + \cdots + 10\!\cdots\!62 ) / 256 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 20\!\cdots\!12 \beta_{19} + \cdots + 11\!\cdots\!94 ) / 512 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 55\!\cdots\!64 \beta_{19} + \cdots - 38\!\cdots\!78 ) / 512 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 20\!\cdots\!58 \beta_{19} + \cdots - 13\!\cdots\!78 ) / 256 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 29\!\cdots\!82 \beta_{19} + \cdots + 18\!\cdots\!64 ) / 128 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 83\!\cdots\!96 \beta_{19} + \cdots + 64\!\cdots\!30 ) / 512 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 23\!\cdots\!64 \beta_{19} + \cdots - 15\!\cdots\!62 ) / 512 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 42\!\cdots\!79 \beta_{19} + \cdots - 36\!\cdots\!92 ) / 128 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 23\!\cdots\!64 \beta_{19} + \cdots + 15\!\cdots\!18 ) / 256 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 35\!\cdots\!92 \beta_{19} + \cdots + 33\!\cdots\!50 ) / 512 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 94\!\cdots\!56 \beta_{19} + \cdots - 60\!\cdots\!58 ) / 512 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 36\!\cdots\!66 \beta_{19} + \cdots - 37\!\cdots\!90 ) / 256 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
0.500000 + 19382.6i
0.500000 − 19382.6i
0.500000 − 32835.5i
0.500000 + 32835.5i
0.500000 + 21396.5i
0.500000 − 21396.5i
0.500000 − 6359.74i
0.500000 + 6359.74i
0.500000 − 37074.8i
0.500000 + 37074.8i
0.500000 + 45387.1i
0.500000 − 45387.1i
0.500000 − 6610.70i
0.500000 + 6610.70i
0.500000 − 44665.4i
0.500000 + 44665.4i
0.500000 + 15078.6i
0.500000 − 15078.6i
0.500000 + 29091.7i
0.500000 − 29091.7i
−1394.30 − 391.258i − 77530.5i 1.79099e6 + 1.09106e6i − 2.27953e7i −3.03344e7 + 1.08101e8i 1.10984e9 −2.07029e9 − 2.22200e9i 4.44938e9 −8.91883e9 + 3.17834e10i
5.2 −1394.30 + 391.258i 77530.5i 1.79099e6 − 1.09106e6i 2.27953e7i −3.03344e7 − 1.08101e8i 1.10984e9 −2.07029e9 + 2.22200e9i 4.44938e9 −8.91883e9 − 3.17834e10i
5.3 −1358.45 − 501.755i 131342.i 1.59363e6 + 1.36322e6i − 4.61121e6i 6.59016e7 − 1.78422e8i −9.67293e8 −1.48087e9 − 2.65149e9i −6.79041e9 −2.31370e9 + 6.26411e9i
5.4 −1358.45 + 501.755i − 131342.i 1.59363e6 − 1.36322e6i 4.61121e6i 6.59016e7 + 1.78422e8i −9.67293e8 −1.48087e9 + 2.65149e9i −6.79041e9 −2.31370e9 − 6.26411e9i
5.5 −962.766 − 1081.77i − 85585.8i −243314. + 2.08299e6i 2.31314e7i −9.25844e7 + 8.23991e7i −3.19239e8 2.48758e9 − 1.74222e9i 3.13542e9 2.50229e10 − 2.22701e10i
5.6 −962.766 + 1081.77i 85585.8i −243314. − 2.08299e6i − 2.31314e7i −9.25844e7 − 8.23991e7i −3.19239e8 2.48758e9 + 1.74222e9i 3.13542e9 2.50229e10 + 2.22701e10i
5.7 −476.143 − 1367.64i 25439.0i −1.64373e6 + 1.30238e6i − 3.81460e7i 3.47914e7 − 1.21126e7i 7.30446e7 2.56384e9 + 1.62791e9i 9.81321e9 −5.21701e10 + 1.81630e10i
5.8 −476.143 + 1367.64i − 25439.0i −1.64373e6 − 1.30238e6i 3.81460e7i 3.47914e7 + 1.21126e7i 7.30446e7 2.56384e9 − 1.62791e9i 9.81321e9 −5.21701e10 − 1.81630e10i
5.9 −377.837 − 1398.00i 148299.i −1.81163e6 + 1.05643e6i 2.51716e7i 2.07321e8 − 5.60329e7i 8.96078e8 2.16139e9 + 2.13349e9i −1.15323e10 3.51898e10 − 9.51079e9i
5.10 −377.837 + 1398.00i − 148299.i −1.81163e6 − 1.05643e6i − 2.51716e7i 2.07321e8 + 5.60329e7i 8.96078e8 2.16139e9 − 2.13349e9i −1.15323e10 3.51898e10 + 9.51079e9i
5.11 228.196 − 1430.06i − 181549.i −1.99301e6 − 652669.i − 6.89272e6i −2.59626e8 − 4.14287e7i 1.45891e8 −1.38815e9 + 2.70119e9i −2.24995e10 −9.85702e9 − 1.57289e9i
5.12 228.196 + 1430.06i 181549.i −1.99301e6 + 652669.i 6.89272e6i −2.59626e8 + 4.14287e7i 1.45891e8 −1.38815e9 − 2.70119e9i −2.24995e10 −9.85702e9 + 1.57289e9i
5.13 582.905 − 1325.66i 26442.8i −1.41760e6 − 1.54547e6i 7.17621e6i 3.50542e7 + 1.54136e7i −5.46394e8 −2.87509e9 + 9.78391e8i 9.76113e9 9.51321e9 + 4.18305e9i
5.14 582.905 + 1325.66i − 26442.8i −1.41760e6 + 1.54547e6i − 7.17621e6i 3.50542e7 − 1.54136e7i −5.46394e8 −2.87509e9 − 9.78391e8i 9.76113e9 9.51321e9 − 4.18305e9i
5.15 1174.53 − 847.130i 178662.i 661894. − 1.98996e6i − 3.12530e7i 1.51350e8 + 2.09844e8i 2.98846e8 −9.08340e8 − 2.89798e9i −2.14596e10 −2.64753e10 − 3.67076e10i
5.16 1174.53 + 847.130i − 178662.i 661894. + 1.98996e6i 3.12530e7i 1.51350e8 − 2.09844e8i 2.98846e8 −9.08340e8 + 2.89798e9i −2.14596e10 −2.64753e10 + 3.67076e10i
5.17 1282.51 − 672.555i − 60314.6i 1.19249e6 − 1.72511e6i 1.19399e7i −4.05649e7 − 7.73538e7i 8.57498e8 3.69144e8 − 3.01448e9i 6.82251e9 8.03026e9 + 1.53130e10i
5.18 1282.51 + 672.555i 60314.6i 1.19249e6 + 1.72511e6i − 1.19399e7i −4.05649e7 + 7.73538e7i 8.57498e8 3.69144e8 + 3.01448e9i 6.82251e9 8.03026e9 − 1.53130e10i
5.19 1444.36 − 104.758i − 116367.i 2.07520e6 − 302616.i − 3.48989e7i −1.21903e7 − 1.68076e8i −1.26580e9 2.96564e9 − 6.54481e8i −3.08088e9 −3.65593e9 − 5.04066e10i
5.20 1444.36 + 104.758i 116367.i 2.07520e6 + 302616.i 3.48989e7i −1.21903e7 + 1.68076e8i −1.26580e9 2.96564e9 + 6.54481e8i −3.08088e9 −3.65593e9 + 5.04066e10i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.20
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8.22.b.a ✓ 20
4.b odd 2 1 32.22.b.a 20
8.b even 2 1 inner 8.22.b.a ✓ 20
8.d odd 2 1 32.22.b.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.22.b.a ✓ 20 1.a even 1 1 trivial
8.22.b.a ✓ 20 8.b even 2 1 inner
32.22.b.a 20 4.b odd 2 1
32.22.b.a 20 8.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{22}^{\mathrm{new}}(8, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{20} + \cdots + 16\!\cdots\!24 \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 39\!\cdots\!48 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots + 58\!\cdots\!52)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots + 48\!\cdots\!24)^{2} \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 21\!\cdots\!28 \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots - 42\!\cdots\!24)^{2} \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 50\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots - 19\!\cdots\!72)^{2} \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 33\!\cdots\!12 \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 21\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 63\!\cdots\!92 \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots + 61\!\cdots\!64)^{2} \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 48\!\cdots\!28 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 11\!\cdots\!92 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 54\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 43\!\cdots\!08 \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots - 18\!\cdots\!16)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots - 39\!\cdots\!52)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 55\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 55\!\cdots\!32 \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 11\!\cdots\!96)^{2} \) Copy content Toggle raw display
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