Properties

Label 304.2.u.b
Level $304$
Weight $2$
Character orbit 304.u
Analytic conductor $2.427$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 304 = 2^{4} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 304.u (of order \(9\), degree \(6\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.42745222145\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \( x^{6} - x^{3} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 19)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{18}^{5} - \zeta_{18}^{3} - \zeta_{18} + 1) q^{3} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{2} - 1) q^{5} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{2}) q^{7} + (2 \zeta_{18}^{4} - 3 \zeta_{18}^{3} - 3 \zeta_{18} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{18}^{5} - \zeta_{18}^{3} - \zeta_{18} + 1) q^{3} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{2} - 1) q^{5} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + \zeta_{18}^{2}) q^{7} + (2 \zeta_{18}^{4} - 3 \zeta_{18}^{3} - 3 \zeta_{18} + 2) q^{9} + ( - 2 \zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{2} - 2 \zeta_{18}) q^{11} + ( - \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 2 \zeta_{18} - 2) q^{13} + ( - \zeta_{18}^{5} + \zeta_{18}^{4} + \zeta_{18}^{3} - 1) q^{15} + ( - \zeta_{18}^{5} - 2 \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} + 2 \zeta_{18} + 1) q^{17} + ( - \zeta_{18}^{5} - 2 \zeta_{18}^{4} - \zeta_{18}^{2} - 2 \zeta_{18} + 2) q^{19} + (\zeta_{18}^{4} - \zeta_{18}) q^{21} + ( - 2 \zeta_{18}^{5} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 2 \zeta_{18}) q^{23} + ( - 2 \zeta_{18}^{5} + \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{2} + \zeta_{18} + 1) q^{25} + (3 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 3 \zeta_{18}^{2}) q^{27} + ( - \zeta_{18}^{4} + \zeta_{18}^{3} - 5 \zeta_{18}^{2} + \zeta_{18} - 1) q^{29} + (\zeta_{18}^{5} + \zeta_{18}^{4} + 3 \zeta_{18}^{3} - 3 \zeta_{18}^{2} + 2 \zeta_{18} - 3) q^{31} + ( - 2 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} - 1) q^{33} + (2 \zeta_{18}^{5} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + \zeta_{18}) q^{35} + (3 \zeta_{18}^{5} - \zeta_{18}^{4} - 2 \zeta_{18}^{2} - 2 \zeta_{18}) q^{37} + (2 \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{2} - \zeta_{18} + 4) q^{39} + (\zeta_{18}^{5} - \zeta_{18}^{3} + 3 \zeta_{18}^{2} + 4 \zeta_{18} + 4) q^{41} + (3 \zeta_{18}^{5} + 5 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 2) q^{43} + ( - \zeta_{18}^{5} - \zeta_{18}^{4} + 5 \zeta_{18}^{3} + \zeta_{18}^{2} - 5) q^{45} + (2 \zeta_{18}^{4} - 3 \zeta_{18}^{3} + 2 \zeta_{18}^{2} - 3 \zeta_{18} + 2) q^{47} + (\zeta_{18}^{5} - \zeta_{18}^{4} + 5 \zeta_{18}^{3} - \zeta_{18}^{2} + \zeta_{18}) q^{49} + (\zeta_{18}^{5} + \zeta_{18}^{3} - \zeta_{18}^{2} - \zeta_{18} - 1) q^{51} + (2 \zeta_{18}^{5} - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{3} + 2 \zeta_{18} + 1) q^{53} + (3 \zeta_{18}^{5} - 3 \zeta_{18}^{4} + 3 \zeta_{18} - 3) q^{55} + (5 \zeta_{18}^{5} + \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 4 \zeta_{18} + 5) q^{57} + (2 \zeta_{18}^{5} - 7 \zeta_{18}^{4} + 7 \zeta_{18} - 2) q^{59} + ( - 3 \zeta_{18}^{5} + 4 \zeta_{18}^{4} + 4 \zeta_{18}^{3} - 4) q^{61} + ( - \zeta_{18}^{5} + \zeta_{18}^{4} + 2 \zeta_{18}^{3} + \zeta_{18}^{2} - 3 \zeta_{18} - 3) q^{63} + (5 \zeta_{18}^{5} - 4 \zeta_{18}^{3} + 5 \zeta_{18}) q^{65} + (4 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 6 \zeta_{18}^{2} + 2 \zeta_{18} + 4) q^{67} + (4 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 2 \zeta_{18} - 4) q^{69} + (10 \zeta_{18}^{4} - 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 2) q^{71} + (4 \zeta_{18}^{5} - 4 \zeta_{18}^{3} - 4 \zeta_{18}^{2}) q^{73} + ( - 4 \zeta_{18}^{5} + 3 \zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} - 5) q^{75} + (\zeta_{18}^{5} - 2 \zeta_{18}^{4} + \zeta_{18}^{2} + \zeta_{18} - 3) q^{77} + ( - \zeta_{18}^{5} + \zeta_{18}^{3} + 7 \zeta_{18}^{2} - 3 \zeta_{18} + 6) q^{79} + (\zeta_{18}^{5} + 5 \zeta_{18}^{4} - \zeta_{18}^{2} + 1) q^{81} + ( - 6 \zeta_{18}^{5} - 6 \zeta_{18}^{4} - 3 \zeta_{18}^{2} + 9 \zeta_{18}) q^{83} + (\zeta_{18}^{4} - 2 \zeta_{18}^{3} - 3 \zeta_{18}^{2} - 2 \zeta_{18} + 1) q^{85} + (6 \zeta_{18}^{5} - 7 \zeta_{18}^{4} + 7 \zeta_{18}^{3} - 7 \zeta_{18}^{2} + 6 \zeta_{18}) q^{87} + ( - 5 \zeta_{18}^{5} + 3 \zeta_{18}^{4} - 2 \zeta_{18}^{3} + 5 \zeta_{18}^{2} - \zeta_{18} - 1) q^{89} + (5 \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} + 3 \zeta_{18} - 2) q^{91} + (2 \zeta_{18}^{5} - 8 \zeta_{18}^{4} + 7 \zeta_{18}^{3} - 7 \zeta_{18}^{2} + 8 \zeta_{18} - 2) q^{93} + (3 \zeta_{18}^{5} - \zeta_{18}^{4} - \zeta_{18}^{3} + \zeta_{18}^{2} + \zeta_{18} - 6) q^{95} + ( - 2 \zeta_{18}^{5} - 5 \zeta_{18}^{4} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{2} + 5 \zeta_{18} + 2) q^{97} + ( - 6 \zeta_{18}^{5} + 5 \zeta_{18}^{4} + 5 \zeta_{18}^{3} - \zeta_{18} - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 3 q^{3} - 6 q^{5} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 3 q^{3} - 6 q^{5} + 3 q^{9} - 3 q^{13} - 3 q^{15} + 3 q^{17} + 12 q^{19} - 6 q^{23} - 6 q^{27} - 3 q^{29} - 9 q^{31} - 9 q^{33} - 6 q^{35} + 24 q^{39} + 21 q^{41} + 3 q^{43} - 15 q^{45} + 3 q^{47} + 15 q^{49} - 3 q^{51} - 3 q^{53} - 18 q^{55} + 24 q^{57} - 12 q^{59} - 12 q^{61} - 12 q^{63} - 12 q^{65} + 30 q^{67} - 12 q^{69} + 6 q^{71} - 12 q^{73} - 30 q^{75} - 18 q^{77} + 39 q^{79} + 6 q^{81} + 21 q^{87} - 12 q^{89} - 15 q^{91} + 9 q^{93} - 39 q^{95} + 18 q^{97} - 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/304\mathbb{Z}\right)^\times\).

\(n\) \(97\) \(191\) \(229\)
\(\chi(n)\) \(-\zeta_{18}^{5}\) \(1\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
−0.766044 0.642788i
−0.173648 + 0.984808i
−0.766044 + 0.642788i
0.939693 + 0.342020i
0.939693 0.342020i
−0.173648 0.984808i
0 2.20574 + 1.85083i 0 −0.826352 + 0.300767i 0 0.173648 + 0.300767i 0 0.918748 + 5.21048i 0
81.1 0 −0.0923963 + 0.524005i 0 −1.93969 1.62760i 0 −0.939693 1.62760i 0 2.55303 + 0.929228i 0
161.1 0 2.20574 1.85083i 0 −0.826352 0.300767i 0 0.173648 0.300767i 0 0.918748 5.21048i 0
177.1 0 −0.613341 0.223238i 0 −0.233956 1.32683i 0 0.766044 1.32683i 0 −1.97178 1.65452i 0
225.1 0 −0.613341 + 0.223238i 0 −0.233956 + 1.32683i 0 0.766044 + 1.32683i 0 −1.97178 + 1.65452i 0
289.1 0 −0.0923963 0.524005i 0 −1.93969 + 1.62760i 0 −0.939693 + 1.62760i 0 2.55303 0.929228i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 289.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 304.2.u.b 6
4.b odd 2 1 19.2.e.a 6
12.b even 2 1 171.2.u.c 6
19.e even 9 1 inner 304.2.u.b 6
19.e even 9 1 5776.2.a.br 3
19.f odd 18 1 5776.2.a.bi 3
20.d odd 2 1 475.2.l.a 6
20.e even 4 2 475.2.u.a 12
28.d even 2 1 931.2.w.a 6
28.f even 6 1 931.2.v.a 6
28.f even 6 1 931.2.x.b 6
28.g odd 6 1 931.2.v.b 6
28.g odd 6 1 931.2.x.a 6
76.d even 2 1 361.2.e.h 6
76.f even 6 1 361.2.e.a 6
76.f even 6 1 361.2.e.b 6
76.g odd 6 1 361.2.e.f 6
76.g odd 6 1 361.2.e.g 6
76.k even 18 1 361.2.a.h 3
76.k even 18 2 361.2.c.h 6
76.k even 18 1 361.2.e.a 6
76.k even 18 1 361.2.e.b 6
76.k even 18 1 361.2.e.h 6
76.l odd 18 1 19.2.e.a 6
76.l odd 18 1 361.2.a.g 3
76.l odd 18 2 361.2.c.i 6
76.l odd 18 1 361.2.e.f 6
76.l odd 18 1 361.2.e.g 6
228.u odd 18 1 3249.2.a.s 3
228.v even 18 1 171.2.u.c 6
228.v even 18 1 3249.2.a.z 3
380.ba odd 18 1 475.2.l.a 6
380.ba odd 18 1 9025.2.a.bd 3
380.bb even 18 1 9025.2.a.x 3
380.bj even 36 2 475.2.u.a 12
532.br even 18 1 931.2.v.a 6
532.bt odd 18 1 931.2.v.b 6
532.cc even 18 1 931.2.w.a 6
532.cd even 18 1 931.2.x.b 6
532.cf odd 18 1 931.2.x.a 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
19.2.e.a 6 4.b odd 2 1
19.2.e.a 6 76.l odd 18 1
171.2.u.c 6 12.b even 2 1
171.2.u.c 6 228.v even 18 1
304.2.u.b 6 1.a even 1 1 trivial
304.2.u.b 6 19.e even 9 1 inner
361.2.a.g 3 76.l odd 18 1
361.2.a.h 3 76.k even 18 1
361.2.c.h 6 76.k even 18 2
361.2.c.i 6 76.l odd 18 2
361.2.e.a 6 76.f even 6 1
361.2.e.a 6 76.k even 18 1
361.2.e.b 6 76.f even 6 1
361.2.e.b 6 76.k even 18 1
361.2.e.f 6 76.g odd 6 1
361.2.e.f 6 76.l odd 18 1
361.2.e.g 6 76.g odd 6 1
361.2.e.g 6 76.l odd 18 1
361.2.e.h 6 76.d even 2 1
361.2.e.h 6 76.k even 18 1
475.2.l.a 6 20.d odd 2 1
475.2.l.a 6 380.ba odd 18 1
475.2.u.a 12 20.e even 4 2
475.2.u.a 12 380.bj even 36 2
931.2.v.a 6 28.f even 6 1
931.2.v.a 6 532.br even 18 1
931.2.v.b 6 28.g odd 6 1
931.2.v.b 6 532.bt odd 18 1
931.2.w.a 6 28.d even 2 1
931.2.w.a 6 532.cc even 18 1
931.2.x.a 6 28.g odd 6 1
931.2.x.a 6 532.cf odd 18 1
931.2.x.b 6 28.f even 6 1
931.2.x.b 6 532.cd even 18 1
3249.2.a.s 3 228.u odd 18 1
3249.2.a.z 3 228.v even 18 1
5776.2.a.bi 3 19.f odd 18 1
5776.2.a.br 3 19.e even 9 1
9025.2.a.x 3 380.bb even 18 1
9025.2.a.bd 3 380.ba odd 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 3T_{3}^{5} + 3T_{3}^{4} + 8T_{3}^{3} + 6T_{3}^{2} + 3T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(304, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} \) Copy content Toggle raw display
$3$ \( T^{6} - 3 T^{5} + 3 T^{4} + 8 T^{3} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{6} + 6 T^{5} + 18 T^{4} + 30 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$7$ \( T^{6} + 3 T^{4} - 2 T^{3} + 9 T^{2} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{6} + 9 T^{4} + 18 T^{3} + 81 T^{2} + \cdots + 81 \) Copy content Toggle raw display
$13$ \( T^{6} + 3 T^{5} + 24 T^{4} + \cdots + 1369 \) Copy content Toggle raw display
$17$ \( T^{6} - 3 T^{5} + 30 T^{3} + 36 T^{2} + \cdots + 9 \) Copy content Toggle raw display
$19$ \( T^{6} - 12 T^{5} + 78 T^{4} + \cdots + 6859 \) Copy content Toggle raw display
$23$ \( T^{6} + 6 T^{5} + 36 T^{4} + 192 T^{3} + \cdots + 576 \) Copy content Toggle raw display
$29$ \( T^{6} + 3 T^{5} + 36 T^{4} + \cdots + 12321 \) Copy content Toggle raw display
$31$ \( T^{6} + 9 T^{5} + 75 T^{4} + \cdots + 2809 \) Copy content Toggle raw display
$37$ \( (T^{3} - 21 T - 17)^{2} \) Copy content Toggle raw display
$41$ \( T^{6} - 21 T^{5} + 162 T^{4} + \cdots + 12321 \) Copy content Toggle raw display
$43$ \( T^{6} - 3 T^{5} + 60 T^{4} + \cdots + 26569 \) Copy content Toggle raw display
$47$ \( T^{6} - 3 T^{5} + 54 T^{4} - 24 T^{3} + \cdots + 9 \) Copy content Toggle raw display
$53$ \( T^{6} + 3 T^{5} - 84 T^{3} + \cdots + 2601 \) Copy content Toggle raw display
$59$ \( T^{6} + 12 T^{5} + 18 T^{4} + \cdots + 71289 \) Copy content Toggle raw display
$61$ \( T^{6} + 12 T^{5} + 24 T^{4} + \cdots + 32761 \) Copy content Toggle raw display
$67$ \( T^{6} - 30 T^{5} + 348 T^{4} + \cdots + 179776 \) Copy content Toggle raw display
$71$ \( T^{6} - 6 T^{5} - 36 T^{4} + \cdots + 788544 \) Copy content Toggle raw display
$73$ \( T^{6} + 12 T^{5} + 96 T^{4} + \cdots + 4096 \) Copy content Toggle raw display
$79$ \( T^{6} - 39 T^{5} + 708 T^{4} + \cdots + 654481 \) Copy content Toggle raw display
$83$ \( T^{6} + 189 T^{4} - 918 T^{3} + \cdots + 210681 \) Copy content Toggle raw display
$89$ \( T^{6} + 12 T^{5} + 54 T^{4} + \cdots + 3249 \) Copy content Toggle raw display
$97$ \( T^{6} - 18 T^{5} + 234 T^{4} + \cdots + 16129 \) Copy content Toggle raw display
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