Properties

Label 2352.2.bl.o
Level $2352$
Weight $2$
Character orbit 2352.bl
Analytic conductor $18.781$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2352.bl (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(18.7808145554\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.339738624.1
Defining polynomial: \(x^{8} - 4 x^{6} + 14 x^{4} - 8 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{4} q^{3} + ( \beta_{2} - \beta_{3} - \beta_{6} ) q^{5} + ( -1 - \beta_{4} ) q^{9} +O(q^{10})\) \( q + \beta_{4} q^{3} + ( \beta_{2} - \beta_{3} - \beta_{6} ) q^{5} + ( -1 - \beta_{4} ) q^{9} + ( -2 \beta_{2} + \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} ) q^{11} + ( 2 \beta_{1} - 2 \beta_{5} - \beta_{7} ) q^{13} + ( \beta_{2} + 2 \beta_{6} - \beta_{7} ) q^{15} + ( -2 \beta_{2} + \beta_{3} - 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{17} + ( \beta_{1} - 2 \beta_{2} + \beta_{3} - 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{19} + ( -2 + \beta_{1} - \beta_{2} - \beta_{3} + 2 \beta_{4} + \beta_{6} ) q^{23} + ( -4 \beta_{1} - 2 \beta_{3} - 3 \beta_{4} + 2 \beta_{5} - \beta_{6} + 2 \beta_{7} ) q^{25} + q^{27} + ( 2 + \beta_{1} - \beta_{2} - 2 \beta_{3} + \beta_{5} - \beta_{7} ) q^{29} + ( -2 \beta_{1} - \beta_{3} - 4 \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{31} + ( -\beta_{1} + \beta_{2} - \beta_{3} - \beta_{6} ) q^{33} + ( -2 \beta_{1} - \beta_{2} - 2 \beta_{3} + 4 \beta_{5} - \beta_{6} - 4 \beta_{7} ) q^{37} + ( \beta_{3} + 2 \beta_{5} + \beta_{7} ) q^{39} + ( -2 + \beta_{1} - \beta_{2} - 4 \beta_{4} - \beta_{5} - 2 \beta_{6} ) q^{41} + ( -2 + 2 \beta_{1} - 4 \beta_{4} - 2 \beta_{5} + 6 \beta_{7} ) q^{43} + ( -2 \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} ) q^{45} + ( 2 - \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{47} + ( 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{6} ) q^{51} + ( -4 \beta_{1} - 2 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{53} + ( -8 + \beta_{1} - 2 \beta_{2} + 6 \beta_{3} + \beta_{5} + 3 \beta_{7} ) q^{55} + ( \beta_{1} + 2 \beta_{2} - 2 \beta_{3} + \beta_{5} - \beta_{7} ) q^{57} + ( 6 \beta_{1} - \beta_{3} + 6 \beta_{4} - 3 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{59} + ( -4 + 2 \beta_{2} + \beta_{3} + 4 \beta_{4} - 2 \beta_{6} ) q^{61} + ( 2 - \beta_{1} + 3 \beta_{2} - 3 \beta_{3} + 2 \beta_{4} + 2 \beta_{5} + 3 \beta_{6} - 6 \beta_{7} ) q^{65} + ( 8 + 2 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} + 2 \beta_{7} ) q^{67} + ( -2 - \beta_{1} - \beta_{2} - 4 \beta_{4} + \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{69} + ( 4 + 3 \beta_{1} - \beta_{2} + 8 \beta_{4} - 3 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} ) q^{71} + ( -8 - 4 \beta_{2} - \beta_{3} - 4 \beta_{4} + 2 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{73} + ( 3 + 2 \beta_{1} + \beta_{2} - 2 \beta_{3} + 3 \beta_{4} - 4 \beta_{5} + \beta_{6} - 4 \beta_{7} ) q^{75} + ( 2 - 4 \beta_{1} - 4 \beta_{3} - 2 \beta_{4} ) q^{79} + \beta_{4} q^{81} + ( 2 \beta_{1} - 2 \beta_{2} - 8 \beta_{3} + 2 \beta_{5} - 4 \beta_{7} ) q^{83} + ( -8 + 4 \beta_{1} + \beta_{2} + 4 \beta_{5} ) q^{85} + ( -2 \beta_{1} + \beta_{3} + 2 \beta_{4} + \beta_{5} - \beta_{6} - \beta_{7} ) q^{87} + ( -4 + 2 \beta_{1} + 3 \beta_{2} + 7 \beta_{3} + 4 \beta_{4} - 3 \beta_{6} ) q^{89} + ( 4 + \beta_{1} - 2 \beta_{2} - \beta_{3} + 4 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{93} + ( -12 + 2 \beta_{3} - 6 \beta_{4} + 8 \beta_{5} + 2 \beta_{7} ) q^{95} + ( -4 - 6 \beta_{1} - 2 \beta_{2} - 8 \beta_{4} + 6 \beta_{5} - 4 \beta_{6} - \beta_{7} ) q^{97} + ( \beta_{1} + \beta_{2} - \beta_{5} + 2 \beta_{6} - \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q - 4q^{3} - 4q^{9} + O(q^{10}) \) \( 8q - 4q^{3} - 4q^{9} - 24q^{23} + 12q^{25} + 8q^{27} + 16q^{29} + 16q^{31} + 8q^{47} - 8q^{53} - 64q^{55} - 24q^{59} - 48q^{61} + 8q^{65} + 48q^{67} - 48q^{73} + 12q^{75} + 24q^{79} - 4q^{81} - 64q^{85} - 8q^{87} - 48q^{89} + 16q^{93} - 72q^{95} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 4 x^{6} + 14 x^{4} - 8 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{6} + 20 \)\()/14\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{7} + 34 \nu \)\()/14\)
\(\beta_{4}\)\(=\)\((\)\( -2 \nu^{6} + 7 \nu^{4} - 28 \nu^{2} + 2 \)\()/14\)
\(\beta_{5}\)\(=\)\((\)\( -2 \nu^{7} + 7 \nu^{5} - 28 \nu^{3} + 16 \nu \)\()/14\)
\(\beta_{6}\)\(=\)\((\)\( -2 \nu^{6} + 7 \nu^{4} - 21 \nu^{2} + 2 \)\()/7\)
\(\beta_{7}\)\(=\)\((\)\( -6 \nu^{7} + 21 \nu^{5} - 70 \nu^{3} + 6 \nu \)\()/14\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{6} - 2 \beta_{4}\)
\(\nu^{3}\)\(=\)\(\beta_{7} - 3 \beta_{5} + 3 \beta_{1}\)
\(\nu^{4}\)\(=\)\(4 \beta_{6} - 6 \beta_{4} + 4 \beta_{2} - 6\)
\(\nu^{5}\)\(=\)\(4 \beta_{7} - 10 \beta_{5} + 4 \beta_{3}\)
\(\nu^{6}\)\(=\)\(14 \beta_{2} - 20\)
\(\nu^{7}\)\(=\)\(14 \beta_{3} - 34 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2352\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1471\) \(1765\) \(2257\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1 + \beta_{4}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
−1.60021 + 0.923880i
1.60021 0.923880i
0.662827 0.382683i
−0.662827 + 0.382683i
−1.60021 0.923880i
1.60021 + 0.923880i
0.662827 + 0.382683i
−0.662827 0.382683i
0 −0.500000 + 0.866025i 0 −2.78415 + 1.60743i 0 0 0 −0.500000 0.866025i 0
31.2 0 −0.500000 + 0.866025i 0 −1.45849 + 0.842061i 0 0 0 −0.500000 0.866025i 0
31.3 0 −0.500000 + 0.866025i 0 0.521114 0.300865i 0 0 0 −0.500000 0.866025i 0
31.4 0 −0.500000 + 0.866025i 0 3.72153 2.14862i 0 0 0 −0.500000 0.866025i 0
607.1 0 −0.500000 0.866025i 0 −2.78415 1.60743i 0 0 0 −0.500000 + 0.866025i 0
607.2 0 −0.500000 0.866025i 0 −1.45849 0.842061i 0 0 0 −0.500000 + 0.866025i 0
607.3 0 −0.500000 0.866025i 0 0.521114 + 0.300865i 0 0 0 −0.500000 + 0.866025i 0
607.4 0 −0.500000 0.866025i 0 3.72153 + 2.14862i 0 0 0 −0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 607.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
28.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2352.2.bl.o 8
4.b odd 2 1 2352.2.bl.t 8
7.b odd 2 1 2352.2.bl.r 8
7.c even 3 1 2352.2.b.l yes 8
7.c even 3 1 2352.2.bl.q 8
7.d odd 6 1 2352.2.b.k 8
7.d odd 6 1 2352.2.bl.t 8
21.g even 6 1 7056.2.b.x 8
21.h odd 6 1 7056.2.b.w 8
28.d even 2 1 2352.2.bl.q 8
28.f even 6 1 2352.2.b.l yes 8
28.f even 6 1 inner 2352.2.bl.o 8
28.g odd 6 1 2352.2.b.k 8
28.g odd 6 1 2352.2.bl.r 8
84.j odd 6 1 7056.2.b.w 8
84.n even 6 1 7056.2.b.x 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2352.2.b.k 8 7.d odd 6 1
2352.2.b.k 8 28.g odd 6 1
2352.2.b.l yes 8 7.c even 3 1
2352.2.b.l yes 8 28.f even 6 1
2352.2.bl.o 8 1.a even 1 1 trivial
2352.2.bl.o 8 28.f even 6 1 inner
2352.2.bl.q 8 7.c even 3 1
2352.2.bl.q 8 28.d even 2 1
2352.2.bl.r 8 7.b odd 2 1
2352.2.bl.r 8 28.g odd 6 1
2352.2.bl.t 8 4.b odd 2 1
2352.2.bl.t 8 7.d odd 6 1
7056.2.b.w 8 21.h odd 6 1
7056.2.b.w 8 84.j odd 6 1
7056.2.b.x 8 21.g even 6 1
7056.2.b.x 8 84.n even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2352, [\chi])\):

\( T_{5}^{8} - 16 T_{5}^{6} + 242 T_{5}^{4} + 384 T_{5}^{3} - 32 T_{5}^{2} - 336 T_{5} + 196 \)
\( T_{11}^{8} - 20 T_{11}^{6} + 404 T_{11}^{4} - 960 T_{11}^{3} + 848 T_{11}^{2} - 192 T_{11} + 16 \)
\( T_{17}^{8} - 32 T_{17}^{6} + 1106 T_{17}^{4} + 5376 T_{17}^{3} + 12032 T_{17}^{2} + 13776 T_{17} + 6724 \)
\( T_{19}^{8} + 40 T_{19}^{6} + 192 T_{19}^{5} + 1656 T_{19}^{4} + 3840 T_{19}^{3} + 6976 T_{19}^{2} + 5376 T_{19} + 3136 \)
\(T_{31}^{8} - \cdots\)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \)
$3$ \( ( 1 + T + T^{2} )^{4} \)
$5$ \( 196 - 336 T - 32 T^{2} + 384 T^{3} + 242 T^{4} - 16 T^{6} + T^{8} \)
$7$ \( T^{8} \)
$11$ \( 16 - 192 T + 848 T^{2} - 960 T^{3} + 404 T^{4} - 20 T^{6} + T^{8} \)
$13$ \( ( 98 + 20 T^{2} + T^{4} )^{2} \)
$17$ \( 6724 + 13776 T + 12032 T^{2} + 5376 T^{3} + 1106 T^{4} - 32 T^{6} + T^{8} \)
$19$ \( 3136 + 5376 T + 6976 T^{2} + 3840 T^{3} + 1656 T^{4} + 192 T^{5} + 40 T^{6} + T^{8} \)
$23$ \( 38416 - 10192 T^{2} + 2900 T^{4} + 1248 T^{5} + 244 T^{6} + 24 T^{7} + T^{8} \)
$29$ \( ( 28 + 32 T - 4 T^{2} - 8 T^{3} + T^{4} )^{2} \)
$31$ \( 678976 - 131840 T + 71744 T^{2} - 17408 T^{3} + 6520 T^{4} - 1216 T^{5} + 200 T^{6} - 16 T^{7} + T^{8} \)
$37$ \( 929296 - 185088 T + 133264 T^{2} + 19200 T^{3} + 9036 T^{4} + 384 T^{5} + 100 T^{6} + T^{8} \)
$41$ \( 6724 + 10912 T^{2} + 1820 T^{4} + 80 T^{6} + T^{8} \)
$43$ \( 20214016 + 1679104 T^{2} + 42848 T^{4} + 368 T^{6} + T^{8} \)
$47$ \( 153664 - 87808 T + 43904 T^{2} - 9856 T^{3} + 2440 T^{4} - 320 T^{5} + 80 T^{6} - 8 T^{7} + T^{8} \)
$53$ \( 614656 - 25088 T + 70016 T^{2} - 9728 T^{3} + 7216 T^{4} - 640 T^{5} + 152 T^{6} + 8 T^{7} + T^{8} \)
$59$ \( 71503936 + 12176640 T + 2750080 T^{2} + 290688 T^{3} + 49416 T^{4} + 4800 T^{5} + 496 T^{6} + 24 T^{7} + T^{8} \)
$61$ \( 9604 + 98784 T + 361816 T^{2} + 237888 T^{3} + 71726 T^{4} + 11328 T^{5} + 1004 T^{6} + 48 T^{7} + T^{8} \)
$67$ \( 802816 - 1376256 T + 1015808 T^{2} - 393216 T^{3} + 89216 T^{4} - 12288 T^{5} + 1024 T^{6} - 48 T^{7} + T^{8} \)
$71$ \( 10265616 + 1314144 T^{2} + 39816 T^{4} + 360 T^{6} + T^{8} \)
$73$ \( 4866436 - 741216 T - 447688 T^{2} + 73920 T^{3} + 55982 T^{4} + 10560 T^{5} + 988 T^{6} + 48 T^{7} + T^{8} \)
$79$ \( 430336 + 818688 T + 482432 T^{2} - 69888 T^{3} - 7504 T^{4} + 1344 T^{5} + 136 T^{6} - 24 T^{7} + T^{8} \)
$83$ \( ( 12256 - 192 T - 256 T^{2} + T^{4} )^{2} \)
$89$ \( 1766857156 + 446905488 T + 39024896 T^{2} + 340224 T^{3} - 127054 T^{4} - 1536 T^{5} + 736 T^{6} + 48 T^{7} + T^{8} \)
$97$ \( 325658116 + 10546576 T^{2} + 121364 T^{4} + 584 T^{6} + T^{8} \)
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