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Properties

Label 164.4.b.a
Level $164$
Weight $4$
Character orbit 164.b
Analytic conductor $9.676$
Analytic rank $0$
Dimension $10$
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] = [164,4,Mod(81,164)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(164, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("164.81");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 164 = 2^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 164.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: \(9.67631324094\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 136x^{8} + 6902x^{6} + 158078x^{4} + 1517385x^{2} + 3808800 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{18} \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{3} - \beta_{3} q^{5} + (\beta_{2} - \beta_1) q^{7} + ( - \beta_{4} - 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{3} - \beta_{3} q^{5} + (\beta_{2} - \beta_1) q^{7} + ( - \beta_{4} - 5) q^{9} + (\beta_{7} + \beta_{2}) q^{11} + (\beta_{8} + \beta_{2}) q^{13} + ( - \beta_{9} - \beta_{8} + \beta_{2}) q^{15} + ( - \beta_{9} + 2 \beta_{2} + \beta_1) q^{17} + (\beta_{8} + \beta_{7} + \beta_1) q^{19} + ( - \beta_{6} + 2 \beta_{4} + 46) q^{21} + (\beta_{5} + \beta_{4} + \beta_{3} - 23) q^{23} + ( - \beta_{6} - \beta_{5} - 2 \beta_{4} + \cdots + 52) q^{25}+ \cdots + (4 \beta_{9} + 8 \beta_{8} + \cdots + 40 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 54 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 54 q^{9} + 468 q^{21} - 224 q^{23} + 510 q^{25} + 48 q^{31} + 356 q^{33} - 208 q^{37} + 224 q^{39} - 302 q^{41} - 952 q^{43} + 384 q^{45} - 1534 q^{49} + 464 q^{51} - 212 q^{57} - 712 q^{59} + 1476 q^{61} + 232 q^{73} - 1172 q^{77} - 386 q^{81} + 584 q^{83} + 1392 q^{87} - 1344 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 136x^{8} + 6902x^{6} + 158078x^{4} + 1517385x^{2} + 3808800 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 4\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 67\nu^{9} + 6292\nu^{7} + 196814\nu^{5} + 2320886\nu^{3} + 7803735\nu ) / 1221300 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -48\nu^{8} - 4948\nu^{6} - 169506\nu^{4} - 2094164\nu^{2} - 5784960 ) / 20355 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -56\nu^{8} - 5576\nu^{6} - 182122\nu^{4} - 2137768\nu^{2} - 6274170 ) / 20355 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -144\nu^{8} - 13664\nu^{6} - 414708\nu^{4} - 4124272\nu^{2} - 5691465 ) / 20355 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 44\nu^{8} + 4044\nu^{6} + 123078\nu^{4} + 1373212\nu^{2} + 3674480 ) / 6785 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -76\nu^{9} - 8326\nu^{7} - 309242\nu^{5} - 4118258\nu^{3} - 8922930\nu ) / 305325 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -7\nu^{9} - 732\nu^{7} - 25694\nu^{5} - 336006\nu^{3} - 1169235\nu ) / 6900 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -623\nu^{9} - 61148\nu^{7} - 1926766\nu^{5} - 20091334\nu^{3} - 27892515\nu ) / 610650 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{5} - 6\beta_{4} + 4\beta_{3} - 433 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{9} - 4\beta_{8} + 10\beta_{7} - 10\beta_{2} - 69\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{6} - 4\beta_{5} + 39\beta_{4} - 28\beta_{3} + 1862 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{9} + 203\beta_{8} - 565\beta_{7} + 1209\beta_{2} + 2552\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -1272\beta_{6} + 991\beta_{5} - 13830\beta_{4} + 9664\beta_{3} - 550423 ) / 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -1322\beta_{9} - 9685\beta_{8} + 26317\beta_{7} - 84277\beta_{2} - 98487\beta_1 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 18655\beta_{6} - 8195\beta_{5} + 146406\beta_{4} - 96617\beta_{3} + 5274715 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 43286\beta_{9} + 225883\beta_{8} - 579071\beta_{7} + 2427627\beta_{2} + 1954810\beta_1 ) / 4 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/164\mathbb{Z}\right)^\times\).

\(n\) \(83\) \(129\)
\(\chi(n)\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
81.1
− 6.03715i
6.60440i
− 5.71260i
1.93812i
4.42093i
− 4.42093i
− 1.93812i
5.71260i
− 6.60440i
6.03715i
0 − 8.61911i 0 −11.3719 0 32.7677i 0 −47.2890 0
81.2 0 − 7.52588i 0 3.25192 0 − 18.8917i 0 −29.6389 0
81.3 0 − 4.76524i 0 21.5929 0 27.6156i 0 4.29250 0
81.4 0 − 2.44806i 0 2.83148 0 − 5.30444i 0 21.0070 0
81.5 0 − 1.53998i 0 −16.3044 0 − 16.1437i 0 24.6284 0
81.6 0 1.53998i 0 −16.3044 0 16.1437i 0 24.6284 0
81.7 0 2.44806i 0 2.83148 0 5.30444i 0 21.0070 0
81.8 0 4.76524i 0 21.5929 0 − 27.6156i 0 4.29250 0
81.9 0 7.52588i 0 3.25192 0 18.8917i 0 −29.6389 0
81.10 0 8.61911i 0 −11.3719 0 − 32.7677i 0 −47.2890 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 81.10
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 164.4.b.a ✓ 10
3.b odd 2 1 1476.4.f.c 10
4.b odd 2 1 656.4.d.e 10
41.b even 2 1 inner 164.4.b.a ✓ 10
123.b odd 2 1 1476.4.f.c 10
164.d odd 2 1 656.4.d.e 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
164.4.b.a ✓ 10 1.a even 1 1 trivial
164.4.b.a ✓ 10 41.b even 2 1 inner
656.4.d.e 10 4.b odd 2 1
656.4.d.e 10 164.d odd 2 1
1476.4.f.c 10 3.b odd 2 1
1476.4.f.c 10 123.b odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(164, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} + 162 T^{8} + \cdots + 1357952 \) Copy content Toggle raw display
$5$ \( (T^{5} - 440 T^{3} + \cdots - 36864)^{2} \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 2143046201472 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 217717680799872 \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 647585346355200 \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 359876372245632 \) Copy content Toggle raw display
$23$ \( (T^{5} + 112 T^{4} + \cdots + 29561905152)^{2} \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 64\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{5} - 24 T^{4} + \cdots - 4367253504)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} + 104 T^{4} + \cdots + 16368582528)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 15\!\cdots\!01 \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots + 1216215346176)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 12\!\cdots\!92 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 93\!\cdots\!32 \) Copy content Toggle raw display
$59$ \( (T^{5} + \cdots + 4702128178176)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots - 1365056985248)^{2} \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 16\!\cdots\!32 \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots + 2619352663360)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 16\!\cdots\!48 \) Copy content Toggle raw display
$83$ \( (T^{5} + \cdots - 44122441651200)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 45\!\cdots\!88 \) Copy content Toggle raw display
$97$ \( T^{10} + \cdots + 53\!\cdots\!00 \) Copy content Toggle raw display
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