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Properties

Label 162.7.d.f
Level $162$
Weight $7$
Character orbit 162.d
Analytic conductor $37.269$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [162,7,Mod(53,162)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(162, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([5]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("162.53");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 162 = 2 \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 162.d (of order \(6\), degree \(2\), not 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: \(37.2687615464\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + 4 \beta_{3} q^{2} + ( - 32 \beta_1 + 32) q^{4} + ( - 25 \beta_{6} + 95 \beta_{5}) q^{5} + (\beta_{2} + \beta_1) q^{7} + ( - 128 \beta_{5} + 128 \beta_{3}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 4 \beta_{3} q^{2} + ( - 32 \beta_1 + 32) q^{4} + ( - 25 \beta_{6} + 95 \beta_{5}) q^{5} + (\beta_{2} + \beta_1) q^{7} + ( - 128 \beta_{5} + 128 \beta_{3}) q^{8} + ( - 100 \beta_{4} + 660) q^{10} + (290 \beta_{7} - 290 \beta_{6} + 991 \beta_{3}) q^{11} + (362 \beta_{4} - 362 \beta_{2} + 959 \beta_1 - 959) q^{13} + 8 \beta_{6} q^{14} - 1024 \beta_1 q^{16} + ( - 193 \beta_{7} - 1795 \beta_{5} + 1795 \beta_{3}) q^{17} + (547 \beta_{4} + 2561) q^{19} + (800 \beta_{7} - 800 \beta_{6} + 3040 \beta_{3}) q^{20} + ( - 1160 \beta_{4} + 1160 \beta_{2} - 6768 \beta_1 + 6768) q^{22} + ( - 2600 \beta_{6} + 6385 \beta_{5}) q^{23} + ( - 4125 \beta_{2} + 6425 \beta_1) q^{25} + ( - 2896 \beta_{7} + 5284 \beta_{5} - 5284 \beta_{3}) q^{26} + (32 \beta_{4} + 32) q^{28} + ( - 9805 \beta_{7} + 9805 \beta_{6} - 5672 \beta_{3}) q^{29} + ( - 2826 \beta_{4} + 2826 \beta_{2} - 10024 \beta_1 + 10024) q^{31} - 4096 \beta_{5} q^{32} + ( - 772 \beta_{2} - 15132 \beta_1) q^{34} + (140 \beta_{7} - 325 \beta_{5} + 325 \beta_{3}) q^{35} + (2303 \beta_{4} - 5050) q^{37} + ( - 4376 \beta_{7} + 4376 \beta_{6} + 8056 \beta_{3}) q^{38} + ( - 3200 \beta_{4} + 3200 \beta_{2} - 21120 \beta_1 + 21120) q^{40} + ( - 23146 \beta_{6} + 33629 \beta_{5}) q^{41} + ( - 14799 \beta_{2} - 46235 \beta_1) q^{43} + (9280 \beta_{7} - 31712 \beta_{5} + 31712 \beta_{3}) q^{44} + ( - 10400 \beta_{4} + 40680) q^{46} + ( - 2502 \beta_{7} + 2502 \beta_{6} + 60168 \beta_{3}) q^{47} + ( - 2 \beta_{4} + 2 \beta_{2} - 117621 \beta_1 + 117621) q^{49} + ( - 33000 \beta_{6} + 42200 \beta_{5}) q^{50} + ( - 11584 \beta_{2} + 30688 \beta_1) q^{52} + (36874 \beta_{7} - 20681 \beta_{5} + 20681 \beta_{3}) q^{53} + ( - 45075 \beta_{4} + 237465) q^{55} + ( - 256 \beta_{7} + 256 \beta_{6}) q^{56} + (39220 \beta_{4} - 39220 \beta_{2} + 6156 \beta_1 - 6156) q^{58} + (16858 \beta_{6} - 72788 \beta_{5}) q^{59} + ( - 32409 \beta_{2} - 152264 \beta_1) q^{61} + (22608 \beta_{7} - 51400 \beta_{5} + 51400 \beta_{3}) q^{62} - 32768 q^{64} + ( - 83705 \beta_{7} + 83705 \beta_{6} - 243145 \beta_{3}) q^{65} + (5307 \beta_{4} - 5307 \beta_{2} - 502243 \beta_1 + 502243) q^{67} + ( - 6176 \beta_{6} - 57440 \beta_{5}) q^{68} + (560 \beta_{2} - 2040 \beta_1) q^{70} + (65072 \beta_{7} + 242831 \beta_{5} - 242831 \beta_{3}) q^{71} + ( - 38535 \beta_{4} + 131456) q^{73} + ( - 18424 \beta_{7} + 18424 \beta_{6} - 29412 \beta_{3}) q^{74} + (17504 \beta_{4} - 17504 \beta_{2} - 81952 \beta_1 + 81952) q^{76} + (1402 \beta_{6} - 3770 \beta_{5}) q^{77} + ( - 131623 \beta_{2} + 212179 \beta_1) q^{79} + (25600 \beta_{7} - 97280 \beta_{5} + 97280 \beta_{3}) q^{80} + ( - 92584 \beta_{4} + 176448) q^{82} + ( - 133176 \beta_{7} + 133176 \beta_{6} - 388866 \beta_{3}) q^{83} + ( - 31365 \beta_{4} + 31365 \beta_{2} - 246960 \beta_1 + 246960) q^{85} + ( - 118392 \beta_{6} - 125744 \beta_{5}) q^{86} + (37120 \beta_{2} - 216576 \beta_1) q^{88} + (245729 \beta_{7} - 43270 \beta_{5} + 43270 \beta_{3}) q^{89} + ( - 597 \beta_{4} + 8815) q^{91} + (83200 \beta_{7} - 83200 \beta_{6} + 204320 \beta_{3}) q^{92} + (10008 \beta_{4} - 10008 \beta_{2} - 491352 \beta_1 + 491352) q^{94} + (26230 \beta_{6} + 13555 \beta_{5}) q^{95} + (93770 \beta_{2} - 905432 \beta_1) q^{97} + (16 \beta_{7} - 470492 \beta_{5} + 470492 \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 128 q^{4} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 128 q^{4} + 4 q^{7} + 5280 q^{10} - 3836 q^{13} - 4096 q^{16} + 20488 q^{19} + 27072 q^{22} + 25700 q^{25} + 256 q^{28} + 40096 q^{31} - 60528 q^{34} - 40400 q^{37} + 84480 q^{40} - 184940 q^{43} + 325440 q^{46} + 470484 q^{49} + 122752 q^{52} + 1899720 q^{55} - 24624 q^{58} - 609056 q^{61} - 262144 q^{64} + 2008972 q^{67} - 8160 q^{70} + 1051648 q^{73} + 327808 q^{76} + 848716 q^{79} + 1411584 q^{82} + 987840 q^{85} - 866304 q^{88} + 70520 q^{91} + 1965408 q^{94} - 3621728 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{24}^{4} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 3\zeta_{24}^{6} + 3\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -3\zeta_{24}^{6} + 6\zeta_{24}^{2} \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( -\zeta_{24}^{7} + \zeta_{24}^{5} + \zeta_{24}^{3} \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \zeta_{24}^{7} - \zeta_{24}^{5} + 2\zeta_{24}^{3} + 3\zeta_{24} \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( 3\zeta_{24}^{7} + 2\zeta_{24}^{5} - \zeta_{24}^{3} + \zeta_{24} \) Copy content Toggle raw display

Display \(\zeta_{24}^j\) in terms of \(\beta_i\)

\(\zeta_{24}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} + 5\beta_{3} ) / 9 \) Copy content Toggle raw display
\(\zeta_{24}^{2}\)\(=\) \( ( \beta_{4} + \beta_{2} ) / 9 \) Copy content Toggle raw display
\(\zeta_{24}^{3}\)\(=\) \( ( -\beta_{7} + 2\beta_{6} + 4\beta_{5} - 5\beta_{3} ) / 9 \) Copy content Toggle raw display
\(\zeta_{24}^{4}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{24}^{5}\)\(=\) \( ( 2\beta_{7} - \beta_{6} + 4\beta_{5} + \beta_{3} ) / 9 \) Copy content Toggle raw display
\(\zeta_{24}^{6}\)\(=\) \( ( -\beta_{4} + 2\beta_{2} ) / 9 \) Copy content Toggle raw display
\(\zeta_{24}^{7}\)\(=\) \( ( \beta_{7} + \beta_{6} - \beta_{5} - 4\beta_{3} ) / 9 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{24}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/162\mathbb{Z}\right)^\times\).

\(n\) \(83\)
\(\chi(n)\) \(1 - \beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
53.1
−0.258819 + 0.965926i
−0.965926 − 0.258819i
0.965926 + 0.258819i
0.258819 − 0.965926i
−0.258819 − 0.965926i
−0.965926 + 0.258819i
0.965926 − 0.258819i
0.258819 + 0.965926i
−4.89898 + 2.82843i 0 16.0000 − 27.7128i −180.591 − 104.264i 0 −2.09808 − 3.63397i 181.019i 0 1179.62
53.2 −4.89898 + 2.82843i 0 16.0000 − 27.7128i −21.4919 − 12.4084i 0 3.09808 + 5.36603i 181.019i 0 140.385
53.3 4.89898 − 2.82843i 0 16.0000 − 27.7128i 21.4919 + 12.4084i 0 3.09808 + 5.36603i − 181.019i 0 140.385
53.4 4.89898 − 2.82843i 0 16.0000 − 27.7128i 180.591 + 104.264i 0 −2.09808 − 3.63397i − 181.019i 0 1179.62
107.1 −4.89898 − 2.82843i 0 16.0000 + 27.7128i −180.591 + 104.264i 0 −2.09808 + 3.63397i − 181.019i 0 1179.62
107.2 −4.89898 − 2.82843i 0 16.0000 + 27.7128i −21.4919 + 12.4084i 0 3.09808 − 5.36603i − 181.019i 0 140.385
107.3 4.89898 + 2.82843i 0 16.0000 + 27.7128i 21.4919 − 12.4084i 0 3.09808 − 5.36603i 181.019i 0 140.385
107.4 4.89898 + 2.82843i 0 16.0000 + 27.7128i 180.591 − 104.264i 0 −2.09808 + 3.63397i 181.019i 0 1179.62
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 107.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
9.c even 3 1 inner
9.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 162.7.d.f 8
3.b odd 2 1 inner 162.7.d.f 8
9.c even 3 1 162.7.b.a ✓ 4
9.c even 3 1 inner 162.7.d.f 8
9.d odd 6 1 162.7.b.a ✓ 4
9.d odd 6 1 inner 162.7.d.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
162.7.b.a ✓ 4 9.c even 3 1
162.7.b.a ✓ 4 9.d odd 6 1
162.7.d.f 8 1.a even 1 1 trivial
162.7.d.f 8 3.b odd 2 1 inner
162.7.d.f 8 9.c even 3 1 inner
162.7.d.f 8 9.d odd 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} - 44100T_{5}^{6} + 1918029375T_{5}^{4} - 1181025562500T_{5}^{2} + 717201875390625 \) acting on \(S_{7}^{\mathrm{new}}(162, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 32 T^{2} + 1024)^{2} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 717201875390625 \) Copy content Toggle raw display
$7$ \( (T^{4} - 2 T^{3} + 30 T^{2} + 52 T + 676)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} - 5133564 T^{6} + \cdots + 76\!\cdots\!76 \) Copy content Toggle raw display
$13$ \( (T^{4} + 1918 T^{3} + \cdots + 6856578909049)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} + 15316812 T^{2} + \cdots + 44258191098489)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 5122 T - 1519922)^{4} \) Copy content Toggle raw display
$23$ \( T^{8} - 285948900 T^{6} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{8} - 2598095196 T^{6} + \cdots + 28\!\cdots\!41 \) Copy content Toggle raw display
$31$ \( (T^{4} - 20048 T^{3} + \cdots + 13\!\cdots\!76)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 10100 T - 117700343)^{4} \) Copy content Toggle raw display
$41$ \( T^{8} - 16410776076 T^{6} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$43$ \( (T^{4} + 92470 T^{3} + \cdots + 14\!\cdots\!04)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} - 15258194352 T^{6} + \cdots + 30\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( (T^{4} + 36731822796 T^{2} + \cdots + 33\!\cdots\!16)^{2} \) Copy content Toggle raw display
$59$ \( T^{8} - 24241511952 T^{6} + \cdots + 39\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( (T^{4} + 304528 T^{3} + \cdots + 26\!\cdots\!81)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 1004486 T^{3} + \cdots + 63\!\cdots\!76)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 417635798724 T^{2} + \cdots + 89\!\cdots\!36)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} - 262912 T - 22812868139)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} - 424358 T^{3} + \cdots + 17\!\cdots\!64)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} - 894320305488 T^{6} + \cdots + 10\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{4} + 1655675156628 T^{2} + \cdots + 64\!\cdots\!49)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 1810864 T^{3} + \cdots + 33\!\cdots\!76)^{2} \) Copy content Toggle raw display
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