Properties

Label 3456.2.i.e
Level $3456$
Weight $2$
Character orbit 3456.i
Analytic conductor $27.596$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3456 = 2^{7} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3456.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(27.5962989386\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: 10.0.8528759163648.1
Defining polynomial: \( x^{10} - 2x^{9} + x^{8} + 9x^{6} - 36x^{5} + 27x^{4} + 27x^{2} - 162x + 243 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 1152)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{9} + \beta_{8} - \beta_{2}) q^{5} + (\beta_{9} + \beta_1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{9} + \beta_{8} - \beta_{2}) q^{5} + (\beta_{9} + \beta_1) q^{7} + (\beta_{7} - \beta_{6} + \beta_{4} + \beta_{3}) q^{11} + (\beta_{4} + \beta_1 + 1) q^{13} + (\beta_{7} - \beta_{6} - \beta_{5} + 1) q^{17} + ( - \beta_{7} + \beta_{5} + \beta_{3} - \beta_{2} - 2) q^{19} + ( - \beta_{9} - \beta_{8} + \beta_{5} + \beta_{4} + \beta_{2} - \beta_1 - 1) q^{23} + ( - \beta_{8} - \beta_{7}) q^{25} + (2 \beta_{9} + 2 \beta_{8} + 2 \beta_{7} - \beta_{6} + \beta_{4} + \beta_{3} - \beta_1) q^{29} + (3 \beta_{9} + \beta_{8} + \beta_{5} - \beta_{4} - 2 \beta_{3} - \beta_{2} - \beta_1 - 1) q^{31} + (\beta_{7} - 2 \beta_{6} - \beta_{5} + 2 \beta_{3} + 2 \beta_{2} - 2) q^{35} + ( - 2 \beta_{7} + 3 \beta_{6} + 2 \beta_{5} - \beta_{3} - \beta_{2} - 3) q^{37} + (\beta_{9} - 3 \beta_{8} - \beta_{4} - 4 \beta_{3} + 3 \beta_{2} + 2 \beta_1 + 2) q^{41} + ( - 2 \beta_{9} - 3 \beta_{8} - 2 \beta_{7} + 2 \beta_{6} - 2 \beta_{4} - 2 \beta_{3} + \cdots - 2 \beta_1) q^{43}+ \cdots + ( - \beta_{9} - 5 \beta_{8} + 3 \beta_{6} - 3 \beta_{4} - 3 \beta_{3} - 4 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 4 q^{7} - q^{11} + 6 q^{13} + 6 q^{17} - 18 q^{19} - 4 q^{23} + q^{25} + 4 q^{29} - 8 q^{31} - 24 q^{35} - 20 q^{37} + 5 q^{41} + 13 q^{43} + 6 q^{47} + 3 q^{49} + 12 q^{55} - 13 q^{59} + 10 q^{61} + 17 q^{67} - 8 q^{71} - 34 q^{73} - 8 q^{77} - 6 q^{79} + 12 q^{83} + 18 q^{85} - 44 q^{89} - 36 q^{91} + 6 q^{95} + 27 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 2x^{9} + x^{8} + 9x^{6} - 36x^{5} + 27x^{4} + 27x^{2} - 162x + 243 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{9} + \nu^{8} + 4\nu^{7} + 12\nu^{6} + 45\nu^{5} - 63\nu^{4} + 27\nu - 1053 ) / 486 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{9} - \nu^{8} - 4\nu^{7} + 15\nu^{6} - 18\nu^{5} + 9\nu^{4} + 81\nu^{3} + 162\nu^{2} - 270\nu + 81 ) / 243 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{9} - 8\nu^{8} - 5\nu^{7} + 3\nu^{6} + 45\nu^{5} - 36\nu^{4} + 81\nu^{3} + 162\nu^{2} + 270\nu - 324 ) / 243 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{9} + 5\nu^{8} + 2\nu^{7} + 12\nu^{6} - 27\nu^{5} + 63\nu^{4} - 54\nu^{3} - 351\nu + 243 ) / 162 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2\nu^{9} + 2\nu^{8} + 8\nu^{7} - 30\nu^{6} + 36\nu^{5} - 18\nu^{4} + 81\nu^{3} - 324\nu^{2} + 297\nu - 162 ) / 243 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -4\nu^{9} + 5\nu^{8} - 7\nu^{7} + 15\nu^{6} - 45\nu^{5} + 117\nu^{4} + 162\nu^{2} - 108\nu + 567 ) / 243 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 4\nu^{9} + 4\nu^{8} - 11\nu^{7} - 6\nu^{6} + 45\nu^{5} - 36\nu^{4} - 81\nu^{3} - 162\nu^{2} + 351\nu - 324 ) / 243 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{9} - \nu^{8} + 2\nu^{7} + 4\nu^{6} + 3\nu^{5} - 27\nu^{4} + 18\nu^{3} - 27\nu - 135 ) / 54 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -19\nu^{9} - \nu^{8} - 4\nu^{7} + 6\nu^{6} - 153\nu^{5} + 333\nu^{4} + 162\nu^{3} + 162\nu^{2} - 513\nu + 2025 ) / 486 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{8} + 2\beta_{6} - \beta_{4} - 2\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{9} - 2\beta_{8} - 3\beta_{7} + 2\beta_{6} - \beta_{4} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{8} + 2\beta_{6} + 3\beta_{5} - \beta_{4} + 4\beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{8} + 2\beta_{6} + 3\beta_{5} + 8\beta_{4} + 9\beta_{3} - 5\beta_{2} - 9\beta _1 - 18 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -8\beta_{8} - 7\beta_{6} + 3\beta_{5} + 8\beta_{4} + 9\beta_{3} + 4\beta_{2} + 18\beta _1 + 36 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 18 \beta_{9} + 28 \beta_{8} + 18 \beta_{7} + 11 \beta_{6} - 6 \beta_{5} + 8 \beta_{4} + 9 \beta_{3} - 23 \beta_{2} + 18 \beta _1 + 36 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 18 \beta_{9} + 10 \beta_{8} - 45 \beta_{7} + 29 \beta_{6} + 12 \beta_{5} + 8 \beta_{4} + 9 \beta_{3} - 50 \beta_{2} + 18 \beta _1 + 36 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 54 \beta_{9} - 35 \beta_{8} - 27 \beta_{7} + 74 \beta_{6} + 30 \beta_{5} - 46 \beta_{4} - 72 \beta_{3} + 58 \beta_{2} + 99 \beta _1 + 117 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 90 \beta_{9} + 55 \beta_{8} - 9 \beta_{7} + 65 \beta_{6} + 48 \beta_{5} + 89 \beta_{4} + 90 \beta_{3} - 23 \beta_{2} - 306 \beta _1 - 288 ) / 3 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3456\mathbb{Z}\right)^\times\).

\(n\) \(2053\) \(2431\) \(2945\)
\(\chi(n)\) \(1\) \(1\) \(-1 - \beta_{1}\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1153.1
−1.13593 1.30754i
1.72806 + 0.117480i
−1.41743 + 0.995434i
0.756905 + 1.55791i
1.06839 1.36328i
−1.13593 + 1.30754i
1.72806 0.117480i
−1.41743 0.995434i
0.756905 1.55791i
1.06839 + 1.36328i
0 0 0 −1.59327 2.75962i 0 −0.607060 + 1.05146i 0 0 0
1153.2 0 0 0 −0.705463 1.22190i 0 1.17123 2.02864i 0 0 0
1153.3 0 0 0 −0.115851 0.200661i 0 −0.230793 + 0.399745i 0 0 0
1153.4 0 0 0 1.07447 + 1.86104i 0 0.153174 0.265305i 0 0 0
1153.5 0 0 0 1.34011 + 2.32114i 0 −2.48656 + 4.30684i 0 0 0
2305.1 0 0 0 −1.59327 + 2.75962i 0 −0.607060 1.05146i 0 0 0
2305.2 0 0 0 −0.705463 + 1.22190i 0 1.17123 + 2.02864i 0 0 0
2305.3 0 0 0 −0.115851 + 0.200661i 0 −0.230793 0.399745i 0 0 0
2305.4 0 0 0 1.07447 1.86104i 0 0.153174 + 0.265305i 0 0 0
2305.5 0 0 0 1.34011 2.32114i 0 −2.48656 4.30684i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2305.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3456.2.i.e 10
3.b odd 2 1 1152.2.i.e 10
4.b odd 2 1 3456.2.i.h 10
8.b even 2 1 3456.2.i.f 10
8.d odd 2 1 3456.2.i.g 10
9.c even 3 1 inner 3456.2.i.e 10
9.d odd 6 1 1152.2.i.e 10
12.b even 2 1 1152.2.i.h yes 10
24.f even 2 1 1152.2.i.f yes 10
24.h odd 2 1 1152.2.i.g yes 10
36.f odd 6 1 3456.2.i.h 10
36.h even 6 1 1152.2.i.h yes 10
72.j odd 6 1 1152.2.i.g yes 10
72.l even 6 1 1152.2.i.f yes 10
72.n even 6 1 3456.2.i.f 10
72.p odd 6 1 3456.2.i.g 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.2.i.e 10 3.b odd 2 1
1152.2.i.e 10 9.d odd 6 1
1152.2.i.f yes 10 24.f even 2 1
1152.2.i.f yes 10 72.l even 6 1
1152.2.i.g yes 10 24.h odd 2 1
1152.2.i.g yes 10 72.j odd 6 1
1152.2.i.h yes 10 12.b even 2 1
1152.2.i.h yes 10 36.h even 6 1
3456.2.i.e 10 1.a even 1 1 trivial
3456.2.i.e 10 9.c even 3 1 inner
3456.2.i.f 10 8.b even 2 1
3456.2.i.f 10 72.n even 6 1
3456.2.i.g 10 8.d odd 2 1
3456.2.i.g 10 72.p odd 6 1
3456.2.i.h 10 4.b odd 2 1
3456.2.i.h 10 36.f odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3456, [\chi])\):

\( T_{5}^{10} + 12T_{5}^{8} - 4T_{5}^{7} + 117T_{5}^{6} - 18T_{5}^{5} + 328T_{5}^{4} + 198T_{5}^{3} + 717T_{5}^{2} + 162T_{5} + 36 \) Copy content Toggle raw display
\( T_{7}^{10} + 4T_{7}^{9} + 24T_{7}^{8} + 129T_{7}^{6} + 138T_{7}^{5} + 240T_{7}^{4} + 48T_{7}^{3} + 33T_{7}^{2} - 2T_{7} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + 12 T^{8} - 4 T^{7} + 117 T^{6} + \cdots + 36 \) Copy content Toggle raw display
$7$ \( T^{10} + 4 T^{9} + 24 T^{8} + 129 T^{6} + \cdots + 4 \) Copy content Toggle raw display
$11$ \( T^{10} + T^{9} + 21 T^{8} + 80 T^{7} + \cdots + 961 \) Copy content Toggle raw display
$13$ \( T^{10} - 6 T^{9} + 36 T^{8} + \cdots + 2304 \) Copy content Toggle raw display
$17$ \( (T^{5} - 3 T^{4} - 24 T^{3} + 80 T^{2} + \cdots - 108)^{2} \) Copy content Toggle raw display
$19$ \( (T^{5} + 9 T^{4} - 104 T^{2} - 48 T + 144)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + 4 T^{9} + 48 T^{8} + \cdots + 1024 \) Copy content Toggle raw display
$29$ \( T^{10} - 4 T^{9} + 96 T^{8} + \cdots + 34596 \) Copy content Toggle raw display
$31$ \( T^{10} + 8 T^{9} + 132 T^{8} + \cdots + 22810176 \) Copy content Toggle raw display
$37$ \( (T^{5} + 10 T^{4} - 68 T^{3} - 796 T^{2} + \cdots + 344)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} - 5 T^{9} + 171 T^{8} + \cdots + 319587129 \) Copy content Toggle raw display
$43$ \( T^{10} - 13 T^{9} + 219 T^{8} + \cdots + 720801 \) Copy content Toggle raw display
$47$ \( T^{10} - 6 T^{9} + 120 T^{8} + \cdots + 12194064 \) Copy content Toggle raw display
$53$ \( (T^{5} - 144 T^{3} + 132 T^{2} + \cdots - 9648)^{2} \) Copy content Toggle raw display
$59$ \( T^{10} + 13 T^{9} + \cdots + 2802325969 \) Copy content Toggle raw display
$61$ \( T^{10} - 10 T^{9} + \cdots + 2249415184 \) Copy content Toggle raw display
$67$ \( T^{10} - 17 T^{9} + 249 T^{8} + \cdots + 63001 \) Copy content Toggle raw display
$71$ \( (T^{5} + 4 T^{4} - 164 T^{3} - 1080 T^{2} + \cdots + 14472)^{2} \) Copy content Toggle raw display
$73$ \( (T^{5} + 17 T^{4} + 76 T^{3} - 140 T^{2} + \cdots - 2972)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + 6 T^{9} + \cdots + 6986619396 \) Copy content Toggle raw display
$83$ \( T^{10} - 12 T^{9} + 162 T^{8} + \cdots + 8088336 \) Copy content Toggle raw display
$89$ \( (T^{5} + 22 T^{4} - 80 T^{3} - 3388 T^{2} + \cdots + 27464)^{2} \) Copy content Toggle raw display
$97$ \( T^{10} - 27 T^{9} + \cdots + 10710387081 \) Copy content Toggle raw display
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