Properties

Label 256.12.b.e
Level $256$
Weight $12$
Character orbit 256.b
Analytic conductor $196.696$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 256 = 2^{8} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 256.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(196.695854223\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 126 \beta q^{3} - 2415 \beta q^{5} + 16744 q^{7} + 113643 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 126 \beta q^{3} - 2415 \beta q^{5} + 16744 q^{7} + 113643 q^{9} - 267306 \beta q^{11} - 288869 \beta q^{13} + 1217160 q^{15} - 6905934 q^{17} + 5330710 \beta q^{19} + 2109744 \beta q^{21} - 18643272 q^{23} + 25499225 q^{25} + 36639540 \beta q^{27} + 64203315 \beta q^{29} - 52843168 q^{31} + 134722224 q^{33} - 40436760 \beta q^{35} + 91106657 \beta q^{37} + 145589976 q^{39} - 308120442 q^{41} + 8562854 \beta q^{43} - 274447845 \beta q^{45} + 2687348496 q^{47} - 1696965207 q^{49} - 870147684 \beta q^{51} + 798027849 \beta q^{53} - 2582175960 q^{55} - 2686677840 q^{57} + 2594601870 \beta q^{59} + 3478239331 \beta q^{61} + 1902838392 q^{63} - 2790474540 q^{65} - 7740913442 \beta q^{67} - 2349052272 \beta q^{69} - 9791485272 q^{71} - 1463791322 q^{73} + 3212902350 \beta q^{75} - 4475771664 \beta q^{77} + 38116845680 q^{79} + 1665188361 q^{81} - 14667549834 \beta q^{83} + 16677830610 \beta q^{85} - 32358470760 q^{87} + 24992917110 q^{89} - 4836822536 \beta q^{91} - 6658239168 \beta q^{93} + 51494658600 q^{95} + 75013568546 q^{97} - 30377455758 \beta q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 33488 q^{7} + 227286 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 33488 q^{7} + 227286 q^{9} + 2434320 q^{15} - 13811868 q^{17} - 37286544 q^{23} + 50998450 q^{25} - 105686336 q^{31} + 269444448 q^{33} + 291179952 q^{39} - 616240884 q^{41} + 5374696992 q^{47} - 3393930414 q^{49} - 5164351920 q^{55} - 5373355680 q^{57} + 3805676784 q^{63} - 5580949080 q^{65} - 19582970544 q^{71} - 2927582644 q^{73} + 76233691360 q^{79} + 3330376722 q^{81} - 64716941520 q^{87} + 49985834220 q^{89} + 102989317200 q^{95} + 150027137092 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/256\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(255\)
\(\chi(n)\) \(-1\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
1.00000i
1.00000i
0 252.000i 0 4830.00i 0 16744.0 0 113643. 0
129.2 0 252.000i 0 4830.00i 0 16744.0 0 113643. 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 256.12.b.e 2
4.b odd 2 1 256.12.b.c 2
8.b even 2 1 inner 256.12.b.e 2
8.d odd 2 1 256.12.b.c 2
16.e even 4 1 1.12.a.a 1
16.e even 4 1 64.12.a.b 1
16.f odd 4 1 16.12.a.a 1
16.f odd 4 1 64.12.a.f 1
48.i odd 4 1 9.12.a.b 1
48.k even 4 1 144.12.a.d 1
80.i odd 4 1 25.12.b.b 2
80.q even 4 1 25.12.a.b 1
80.t odd 4 1 25.12.b.b 2
112.l odd 4 1 49.12.a.a 1
112.w even 12 2 49.12.c.b 2
112.x odd 12 2 49.12.c.c 2
144.w odd 12 2 81.12.c.b 2
144.x even 12 2 81.12.c.d 2
176.l odd 4 1 121.12.a.b 1
208.p even 4 1 169.12.a.a 1
240.bb even 4 1 225.12.b.d 2
240.bf even 4 1 225.12.b.d 2
240.bm odd 4 1 225.12.a.b 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1.12.a.a 1 16.e even 4 1
9.12.a.b 1 48.i odd 4 1
16.12.a.a 1 16.f odd 4 1
25.12.a.b 1 80.q even 4 1
25.12.b.b 2 80.i odd 4 1
25.12.b.b 2 80.t odd 4 1
49.12.a.a 1 112.l odd 4 1
49.12.c.b 2 112.w even 12 2
49.12.c.c 2 112.x odd 12 2
64.12.a.b 1 16.e even 4 1
64.12.a.f 1 16.f odd 4 1
81.12.c.b 2 144.w odd 12 2
81.12.c.d 2 144.x even 12 2
121.12.a.b 1 176.l odd 4 1
144.12.a.d 1 48.k even 4 1
169.12.a.a 1 208.p even 4 1
225.12.a.b 1 240.bm odd 4 1
225.12.b.d 2 240.bb even 4 1
225.12.b.d 2 240.bf even 4 1
256.12.b.c 2 4.b odd 2 1
256.12.b.c 2 8.d odd 2 1
256.12.b.e 2 1.a even 1 1 trivial
256.12.b.e 2 8.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{12}^{\mathrm{new}}(256, [\chi])\):

\( T_{3}^{2} + 63504 \) Copy content Toggle raw display
\( T_{7} - 16744 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 63504 \) Copy content Toggle raw display
$5$ \( T^{2} + 23328900 \) Copy content Toggle raw display
$7$ \( (T - 16744)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} + 285809990544 \) Copy content Toggle raw display
$13$ \( T^{2} + 333781196644 \) Copy content Toggle raw display
$17$ \( (T + 6905934)^{2} \) Copy content Toggle raw display
$19$ \( T^{2} + \cdots + 113665876416400 \) Copy content Toggle raw display
$23$ \( (T + 18643272)^{2} \) Copy content Toggle raw display
$29$ \( T^{2} + 16\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T + 52843168)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 33\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T + 308120442)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + \cdots + 293289874501264 \) Copy content Toggle raw display
$47$ \( (T - 2687348496)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 25\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{2} + 26\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{2} + 48\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{2} + 23\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T + 9791485272)^{2} \) Copy content Toggle raw display
$73$ \( (T + 1463791322)^{2} \) Copy content Toggle raw display
$79$ \( (T - 38116845680)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 86\!\cdots\!24 \) Copy content Toggle raw display
$89$ \( (T - 24992917110)^{2} \) Copy content Toggle raw display
$97$ \( (T - 75013568546)^{2} \) Copy content Toggle raw display
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