Properties

Label 32.2.g.b
Level $32$
Weight $2$
Character orbit 32.g
Analytic conductor $0.256$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 32 = 2^{5} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 32.g (of order \(8\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.255521286468\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{8})\)
Coefficient field: 8.0.18939904.2
Defining polynomial: \( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{8}]$

$q$-expansion

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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + (\beta_{7} + \beta_{5} + \beta_{3} + \beta_1) q^{3} + (\beta_{6} - \beta_{3} + \beta_{2}) q^{4} + ( - \beta_{7} - \beta_{6}) q^{5} + ( - 2 \beta_{7} - \beta_{5} + \beta_{2} - \beta_1 - 2) q^{6} + (\beta_{7} - 2 \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 - 2) q^{7} + (\beta_{7} - 2 \beta_{6} - \beta_{4} - \beta_{2}) q^{8} + ( - \beta_{7} + \beta_{6} - 2 \beta_{4} - 2 \beta_1 - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + (\beta_{7} + \beta_{5} + \beta_{3} + \beta_1) q^{3} + (\beta_{6} - \beta_{3} + \beta_{2}) q^{4} + ( - \beta_{7} - \beta_{6}) q^{5} + ( - 2 \beta_{7} - \beta_{5} + \beta_{2} - \beta_1 - 2) q^{6} + (\beta_{7} - 2 \beta_{5} + \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 - 2) q^{7} + (\beta_{7} - 2 \beta_{6} - \beta_{4} - \beta_{2}) q^{8} + ( - \beta_{7} + \beta_{6} - 2 \beta_{4} - 2 \beta_1 - 1) q^{9} + (\beta_{7} + \beta_{6} + \beta_{4} + \beta_{3}) q^{10} + ( - \beta_{7} + \beta_{6} + \beta_{4} - \beta_{2} + 1) q^{11} + (2 \beta_{7} - \beta_{6} + \beta_{5} + 2 \beta_{4} + \beta_{3} + \beta_1 + 2) q^{12} + ( - \beta_{7} + \beta_{5} - 2 \beta_{3} + 2 \beta_1) q^{13} + ( - \beta_{6} + 3 \beta_{5} - 2 \beta_{4} + \beta_{3} + \beta_1 + 2) q^{14} + ( - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - \beta_{2} + \beta_1 + 1) q^{15} + ( - \beta_{7} + 3 \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} + \beta_{2} + \beta_1) q^{16} + (2 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{4} + 2 \beta_{3}) q^{17} + (\beta_{7} - \beta_{6} - 2 \beta_{5} + \beta_{4} - \beta_{3} - \beta_{2} + 2 \beta_1 + 4) q^{18} + ( - \beta_{7} + \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_1) q^{19} + ( - 2 \beta_{7} - \beta_{6} + \beta_{5} - \beta_{3} - \beta_1) q^{20} + (3 \beta_{7} - 3 \beta_{6} - \beta_{5} + 2 \beta_{4} - 2 \beta_{2} + 1) q^{21} + (\beta_{7} + \beta_{5} + \beta_{4} - 2 \beta_{3} - \beta_1) q^{22} + ( - \beta_{7} + 2 \beta_{6} - \beta_{4} + \beta_{3} + \beta_{2} - \beta_1 - 2) q^{23} + ( - 3 \beta_{7} + \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - 3 \beta_1 - 2) q^{24} + (\beta_{7} + 3 \beta_{5} - 1) q^{25} + (3 \beta_{7} - \beta_{5} - \beta_{4} + 2 \beta_{2} - \beta_1 - 4) q^{26} + ( - 3 \beta_{7} - 3 \beta_{6} - \beta_{5} - 2 \beta_{3} + 1) q^{27} + ( - \beta_{7} + \beta_{6} - 5 \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 2) q^{28} + ( - \beta_{7} + 2 \beta_{6} - \beta_{5} + 4 \beta_{2} - 2) q^{29} + (\beta_{7} + 2 \beta_{6} - 2 \beta_{5} - \beta_{4} + \beta_{2} - 2) q^{30} + ( - 2 \beta_{6} - 2 \beta_{5} + 4) q^{31} + ( - 4 \beta_{6} + 2 \beta_{4} - 2 \beta_{3} + 2 \beta_1 - 2) q^{32} + ( - \beta_{6} - \beta_{5} + 2 \beta_{2} + 2 \beta_1 - 2) q^{33} + ( - 4 \beta_{7} - 2 \beta_{6} + 4 \beta_{5} - 2 \beta_{3}) q^{34} + (\beta_{7} + 3 \beta_{6} + \beta_{5} + 2 \beta_{2} + 1) q^{35} + (2 \beta_{6} + 3 \beta_{5} - 2 \beta_{4} - \beta_{2} + \beta_1 - 4) q^{36} + (3 \beta_{7} + 3 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} + 4 \beta_{3} - 2 \beta_{2}) q^{37} + ( - 3 \beta_{5} + 2 \beta_{4} - \beta_{2} + \beta_1 + 2) q^{38} + ( - \beta_{7} + 4 \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 2) q^{39} + (3 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 + 2) q^{40} + (\beta_{7} - 4 \beta_{6} + 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 + 3) q^{41} + ( - 3 \beta_{7} + 5 \beta_{6} + 3 \beta_{5} - 3 \beta_{4} + \beta_{3} + \beta_{2} - \beta_1) q^{42} + (3 \beta_{7} - 3 \beta_{6} - 2 \beta_{5} + \beta_{4} - \beta_{2} - 1) q^{43} + (\beta_{7} - 3 \beta_{4} - \beta_{2} - 2 \beta_1 + 2) q^{44} + (\beta_{6} + 2 \beta_{3} - 2 \beta_1 - 1) q^{45} + ( - 3 \beta_{6} - \beta_{5} + 2 \beta_{4} - \beta_{3} + \beta_1 + 2) q^{46} + ( - 2 \beta_{7} - 2 \beta_{2} + 2 \beta_1 + 2) q^{47} + (2 \beta_{7} - 2 \beta_{5} + 4 \beta_{4} - 2 \beta_{3} + 6) q^{48} + ( - 3 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} - 2 \beta_{4} - 2 \beta_{3} - 2 \beta_{2} + \cdots + 2) q^{49}+ \cdots + (\beta_{6} + \beta_{3} - 6 \beta_{2} + \beta_1 + 7) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{2} - 4 q^{3} + 4 q^{4} - 8 q^{6} - 8 q^{7} - 4 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{2} - 4 q^{3} + 4 q^{4} - 8 q^{6} - 8 q^{7} - 4 q^{8} + 4 q^{11} + 12 q^{12} - 8 q^{13} + 12 q^{14} + 20 q^{18} + 4 q^{19} + 4 q^{20} + 4 q^{22} - 8 q^{23} - 8 q^{24} - 8 q^{25} - 20 q^{26} + 8 q^{27} - 16 q^{28} - 12 q^{30} + 32 q^{31} - 24 q^{32} - 16 q^{33} + 16 q^{35} - 40 q^{36} - 8 q^{37} + 8 q^{38} + 16 q^{39} + 16 q^{40} + 8 q^{41} + 8 q^{42} - 12 q^{43} + 20 q^{44} + 12 q^{46} + 48 q^{48} + 16 q^{50} - 16 q^{51} + 12 q^{52} + 8 q^{53} - 8 q^{54} - 16 q^{55} + 8 q^{56} + 16 q^{57} - 12 q^{58} - 20 q^{59} - 8 q^{60} + 24 q^{61} - 24 q^{62} - 40 q^{63} - 8 q^{64} - 28 q^{66} - 36 q^{67} + 16 q^{68} + 32 q^{69} - 8 q^{70} - 24 q^{71} + 12 q^{72} - 32 q^{73} + 8 q^{74} - 12 q^{75} - 20 q^{76} + 16 q^{77} + 28 q^{78} + 8 q^{80} - 20 q^{82} + 20 q^{83} + 8 q^{84} + 8 q^{85} + 4 q^{86} + 56 q^{87} + 8 q^{88} - 16 q^{89} + 28 q^{90} + 40 q^{91} - 16 q^{92} - 16 q^{93} - 24 q^{94} - 8 q^{95} - 16 q^{96} + 32 q^{97} - 24 q^{98} + 28 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} + 14x^{6} - 28x^{5} + 43x^{4} - 44x^{3} + 30x^{2} - 12x + 2 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu^{7} - 7\nu^{6} + 24\nu^{5} - 42\nu^{4} + 59\nu^{3} - 48\nu^{2} + 24\nu - 5 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -2\nu^{7} + 7\nu^{6} - 24\nu^{5} + 43\nu^{4} - 61\nu^{3} + 54\nu^{2} - 29\nu + 8 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -3\nu^{7} + 10\nu^{6} - 35\nu^{5} + 60\nu^{4} - 87\nu^{3} + 73\nu^{2} - 42\nu + 11 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( -3\nu^{7} + 11\nu^{6} - 38\nu^{5} + 70\nu^{4} - 102\nu^{3} + 91\nu^{2} - 53\nu + 13 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( 5\nu^{7} - 17\nu^{6} + 60\nu^{5} - 105\nu^{4} + 155\nu^{3} - 133\nu^{2} + 77\nu - 19 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( -5\nu^{7} + 18\nu^{6} - 63\nu^{5} + 115\nu^{4} - 170\nu^{3} + 152\nu^{2} - 89\nu + 23 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( -8\nu^{7} + 28\nu^{6} - 98\nu^{5} + 175\nu^{4} - 256\nu^{3} + 223\nu^{2} - 126\nu + 31 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} - \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{7} + 3\beta_{6} + \beta_{5} - 3\beta_{4} + \beta_{3} + \beta_{2} - \beta _1 - 4 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{7} - \beta_{6} + 7\beta_{5} - \beta_{4} + 5\beta_{3} - 3\beta_{2} + 3\beta _1 - 2 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 11\beta_{7} - 15\beta_{6} + 3\beta_{5} + 11\beta_{4} - \beta_{3} - 5\beta_{2} + 9\beta _1 + 14 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -13\beta_{7} - 11\beta_{6} - 29\beta_{5} + 17\beta_{4} - 23\beta_{3} + 13\beta_{2} - 3\beta _1 + 18 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -67\beta_{7} + 59\beta_{6} - 41\beta_{5} - 29\beta_{4} - 15\beta_{3} + 37\beta_{2} - 47\beta _1 - 48 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -7\beta_{7} + 113\beta_{6} + 97\beta_{5} - 105\beta_{4} + 91\beta_{3} - 31\beta_{2} - 39\beta _1 - 122 ) / 2 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/32\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(31\)
\(\chi(n)\) \(-\beta_{5}\) \(1\)

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.

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 + 2.10607i
0.500000 0.691860i
0.500000 2.10607i
0.500000 + 0.691860i
0.500000 + 0.0297061i
0.500000 1.44392i
0.500000 0.0297061i
0.500000 + 1.44392i
−1.26330 0.635665i −1.07947 2.60607i 1.19186 + 1.60607i 0.707107 + 0.292893i −0.292893 + 3.97844i 1.68554 + 1.68554i −0.484753 2.78658i −3.50504 + 3.50504i −0.707107 0.819496i
5.2 −0.443806 + 1.34277i 0.0794708 + 0.191860i −1.60607 1.19186i 0.707107 + 0.292893i −0.292893 + 0.0215628i −2.27133 2.27133i 2.31318 1.62764i 2.09083 2.09083i −0.707107 + 0.819496i
13.1 −1.26330 + 0.635665i −1.07947 + 2.60607i 1.19186 1.60607i 0.707107 0.292893i −0.292893 3.97844i 1.68554 1.68554i −0.484753 + 2.78658i −3.50504 3.50504i −0.707107 + 0.819496i
13.2 −0.443806 1.34277i 0.0794708 0.191860i −1.60607 + 1.19186i 0.707107 0.292893i −0.292893 0.0215628i −2.27133 + 2.27133i 2.31318 + 1.62764i 2.09083 + 2.09083i −0.707107 0.819496i
21.1 −1.40426 + 0.167452i 1.27882 0.529706i 1.94392 0.470294i −0.707107 + 1.70711i −1.70711 + 0.957989i −2.74912 2.74912i −2.65103 + 0.985930i −0.766519 + 0.766519i 0.707107 2.51564i
21.2 1.11137 0.874559i −2.27882 + 0.943920i 0.470294 1.94392i −0.707107 + 1.70711i −1.70711 + 3.04201i −0.665096 0.665096i −1.17740 2.57172i 2.18073 2.18073i 0.707107 + 2.51564i
29.1 −1.40426 0.167452i 1.27882 + 0.529706i 1.94392 + 0.470294i −0.707107 1.70711i −1.70711 0.957989i −2.74912 + 2.74912i −2.65103 0.985930i −0.766519 0.766519i 0.707107 + 2.51564i
29.2 1.11137 + 0.874559i −2.27882 0.943920i 0.470294 + 1.94392i −0.707107 1.70711i −1.70711 3.04201i −0.665096 + 0.665096i −1.17740 + 2.57172i 2.18073 + 2.18073i 0.707107 2.51564i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
32.g even 8 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 32.2.g.b 8
3.b odd 2 1 288.2.v.b 8
4.b odd 2 1 128.2.g.b 8
5.b even 2 1 800.2.y.b 8
5.c odd 4 1 800.2.ba.c 8
5.c odd 4 1 800.2.ba.d 8
8.b even 2 1 256.2.g.d 8
8.d odd 2 1 256.2.g.c 8
12.b even 2 1 1152.2.v.b 8
16.e even 4 1 512.2.g.e 8
16.e even 4 1 512.2.g.h 8
16.f odd 4 1 512.2.g.f 8
16.f odd 4 1 512.2.g.g 8
32.g even 8 1 inner 32.2.g.b 8
32.g even 8 1 256.2.g.d 8
32.g even 8 1 512.2.g.e 8
32.g even 8 1 512.2.g.h 8
32.h odd 8 1 128.2.g.b 8
32.h odd 8 1 256.2.g.c 8
32.h odd 8 1 512.2.g.f 8
32.h odd 8 1 512.2.g.g 8
64.i even 16 2 4096.2.a.k 8
64.j odd 16 2 4096.2.a.q 8
96.o even 8 1 1152.2.v.b 8
96.p odd 8 1 288.2.v.b 8
160.v odd 8 1 800.2.ba.c 8
160.z even 8 1 800.2.y.b 8
160.bb odd 8 1 800.2.ba.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
32.2.g.b 8 1.a even 1 1 trivial
32.2.g.b 8 32.g even 8 1 inner
128.2.g.b 8 4.b odd 2 1
128.2.g.b 8 32.h odd 8 1
256.2.g.c 8 8.d odd 2 1
256.2.g.c 8 32.h odd 8 1
256.2.g.d 8 8.b even 2 1
256.2.g.d 8 32.g even 8 1
288.2.v.b 8 3.b odd 2 1
288.2.v.b 8 96.p odd 8 1
512.2.g.e 8 16.e even 4 1
512.2.g.e 8 32.g even 8 1
512.2.g.f 8 16.f odd 4 1
512.2.g.f 8 32.h odd 8 1
512.2.g.g 8 16.f odd 4 1
512.2.g.g 8 32.h odd 8 1
512.2.g.h 8 16.e even 4 1
512.2.g.h 8 32.g even 8 1
800.2.y.b 8 5.b even 2 1
800.2.y.b 8 160.z even 8 1
800.2.ba.c 8 5.c odd 4 1
800.2.ba.c 8 160.v odd 8 1
800.2.ba.d 8 5.c odd 4 1
800.2.ba.d 8 160.bb odd 8 1
1152.2.v.b 8 12.b even 2 1
1152.2.v.b 8 96.o even 8 1
4096.2.a.k 8 64.i even 16 2
4096.2.a.q 8 64.j odd 16 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + 4T_{3}^{7} + 8T_{3}^{6} - 32T_{3}^{4} - 24T_{3}^{3} + 96T_{3}^{2} - 16T_{3} + 4 \) acting on \(S_{2}^{\mathrm{new}}(32, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + 4 T^{7} + 6 T^{6} + 4 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$3$ \( T^{8} + 4 T^{7} + 8 T^{6} - 32 T^{4} + \cdots + 4 \) Copy content Toggle raw display
$5$ \( (T^{4} + 2 T^{2} - 4 T + 2)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 8 T^{7} + 32 T^{6} + 48 T^{5} + \cdots + 784 \) Copy content Toggle raw display
$11$ \( T^{8} - 4 T^{7} + 8 T^{6} - 64 T^{5} + \cdots + 4 \) Copy content Toggle raw display
$13$ \( T^{8} + 8 T^{7} + 36 T^{6} + \cdots + 6724 \) Copy content Toggle raw display
$17$ \( T^{8} + 64 T^{6} + 1056 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( T^{8} - 4 T^{7} - 8 T^{6} + 48 T^{5} + \cdots + 196 \) Copy content Toggle raw display
$23$ \( T^{8} + 8 T^{7} + 32 T^{6} + 16 T^{5} + \cdots + 16 \) Copy content Toggle raw display
$29$ \( T^{8} - 12 T^{6} + 168 T^{5} + \cdots + 188356 \) Copy content Toggle raw display
$31$ \( (T^{2} - 8 T + 8)^{4} \) Copy content Toggle raw display
$37$ \( T^{8} + 8 T^{7} - 44 T^{6} + \cdots + 64516 \) Copy content Toggle raw display
$41$ \( T^{8} - 8 T^{7} + 32 T^{6} + \cdots + 26896 \) Copy content Toggle raw display
$43$ \( T^{8} + 12 T^{7} + 56 T^{6} + \cdots + 31684 \) Copy content Toggle raw display
$47$ \( T^{8} + 64 T^{6} + 544 T^{4} + \cdots + 256 \) Copy content Toggle raw display
$53$ \( T^{8} - 8 T^{7} + 100 T^{6} + \cdots + 158404 \) Copy content Toggle raw display
$59$ \( T^{8} + 20 T^{7} + 136 T^{6} + \cdots + 643204 \) Copy content Toggle raw display
$61$ \( T^{8} - 24 T^{7} + 132 T^{6} + \cdots + 42436 \) Copy content Toggle raw display
$67$ \( T^{8} + 36 T^{7} + 504 T^{6} + \cdots + 1285956 \) Copy content Toggle raw display
$71$ \( T^{8} + 24 T^{7} + 288 T^{6} + \cdots + 21196816 \) Copy content Toggle raw display
$73$ \( T^{8} + 32 T^{7} + 512 T^{6} + \cdots + 38416 \) Copy content Toggle raw display
$79$ \( T^{8} + 512 T^{6} + \cdots + 99361024 \) Copy content Toggle raw display
$83$ \( T^{8} - 20 T^{7} + \cdots + 138250564 \) Copy content Toggle raw display
$89$ \( T^{8} + 16 T^{7} + 128 T^{6} + \cdots + 17007376 \) Copy content Toggle raw display
$97$ \( (T^{4} - 16 T^{3} + 40 T^{2} + 288 T - 992)^{2} \) Copy content Toggle raw display
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