Properties

Label 3456.2.i.k
Level $3456$
Weight $2$
Character orbit 3456.i
Analytic conductor $27.596$
Analytic rank $0$
Dimension $12$
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: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Defining polynomial: \( x^{12} - 2 x^{11} + 3 x^{10} - 8 x^{9} + 22 x^{8} - 42 x^{7} + 51 x^{6} - 126 x^{5} + 198 x^{4} - 216 x^{3} + 243 x^{2} - 486 x + 729 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{4} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{5} + (\beta_{10} + \beta_{7} - 1) q^{7}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{5} + (\beta_{10} + \beta_{7} - 1) q^{7} + ( - \beta_{8} - \beta_{7} + 1) q^{11} + ( - \beta_{10} - \beta_{9} + 2 \beta_{7} - \beta_{3} - \beta_1) q^{13} + ( - \beta_{5} - \beta_{4} - \beta_{3} - 1) q^{17} + (\beta_{5} - \beta_{2} + \beta_1) q^{19} + ( - \beta_{11} - \beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} + \beta_{4}) q^{23} + (\beta_{11} - \beta_{9} + \beta_{8} + 3 \beta_{7} + \beta_{6} + \beta_{2} - 3) q^{25} + (\beta_{10} - \beta_{9}) q^{29} + ( - \beta_{11} + \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4}) q^{31} + ( - \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{35} + (\beta_{3} - \beta_{2} + \beta_1) q^{37} + ( - \beta_{10} - \beta_{9} + \beta_{7} + \beta_{6} - \beta_{3} - \beta_1) q^{41} + (\beta_{11} - \beta_{10} - \beta_{7} - \beta_{6} - \beta_{2} + 1) q^{43} + ( - \beta_{10} + 3 \beta_{7} + 2 \beta_{6} + 2 \beta_{2} - 3) q^{47} + (\beta_{11} - \beta_{10} + \beta_{8} - 3 \beta_{7} + \beta_{5} - \beta_{4} - \beta_{3}) q^{49} + ( - \beta_{3} - \beta_{2} + \beta_1 - 2) q^{53} + (\beta_{5} + \beta_{4} + 2 \beta_{3} + 3 \beta_{2} + 1) q^{55} + (\beta_{11} - \beta_{10} - 2 \beta_{9} + \beta_{7} - \beta_{6} - \beta_{4} - \beta_{3} - 2 \beta_1) q^{59} + (\beta_{10} - \beta_{9} - 2 \beta_{7} + 2) q^{61} + ( - \beta_{11} - \beta_{10} - 2 \beta_{9} - \beta_{8} - 2 \beta_{7} - 4 \beta_{6} - 4 \beta_{2} + \cdots + 2) q^{65}+ \cdots + (\beta_{10} - \beta_{9} - 2 \beta_{8} + \beta_{7} - 3 \beta_{6} - 3 \beta_{2} - 1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{5} - 6 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{5} - 6 q^{7} + 4 q^{11} + 10 q^{13} - 4 q^{17} - 4 q^{19} + 8 q^{23} - 14 q^{25} + 2 q^{29} - 8 q^{31} + 8 q^{35} + 2 q^{41} + 2 q^{43} - 14 q^{47} - 18 q^{49} - 24 q^{53} + 16 q^{55} + 6 q^{59} + 14 q^{61} + 8 q^{65} - 4 q^{67} - 28 q^{71} + 60 q^{73} - 2 q^{77} - 16 q^{79} + 24 q^{83} + 16 q^{85} + 48 q^{89} + 52 q^{91} - 20 q^{95} - 14 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 3 x^{10} - 8 x^{9} + 22 x^{8} - 42 x^{7} + 51 x^{6} - 126 x^{5} + 198 x^{4} - 216 x^{3} + 243 x^{2} - 486 x + 729 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{11} + \nu^{10} - 3 \nu^{9} - 26 \nu^{8} + 25 \nu^{7} - 3 \nu^{6} + 141 \nu^{5} - 270 \nu^{4} + 117 \nu^{3} - 27 \nu^{2} + 1701 \nu - 972 ) / 486 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{11} - 2 \nu^{10} + 3 \nu^{9} - 8 \nu^{8} + 13 \nu^{7} - 24 \nu^{6} + 51 \nu^{5} - 108 \nu^{4} + 81 \nu^{3} - 54 \nu^{2} + 135 \nu - 162 ) / 162 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 2 \nu^{11} - \nu^{10} + 20 \nu^{8} - 34 \nu^{7} + 63 \nu^{6} - 240 \nu^{5} + 252 \nu^{4} - 630 \nu^{3} + 567 \nu^{2} - 1620 \nu + 1458 ) / 486 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 7 \nu^{11} + 5 \nu^{10} - 3 \nu^{9} + 56 \nu^{8} - 109 \nu^{7} + 123 \nu^{6} - 195 \nu^{5} + 720 \nu^{4} - 549 \nu^{3} + 621 \nu^{2} - 1377 \nu + 2430 ) / 486 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( \nu^{11} - 2\nu^{10} - 2\nu^{8} + 4\nu^{7} - 15\nu^{5} + 18\nu^{4} + 9\nu^{3} + 162\nu^{2} - 216\nu + 81 ) / 81 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11 \nu^{11} - 2 \nu^{10} - 12 \nu^{9} + 16 \nu^{8} - 131 \nu^{7} + 42 \nu^{6} - 120 \nu^{5} + 810 \nu^{4} - 315 \nu^{3} + 540 \nu^{2} + 5346 ) / 486 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 11 \nu^{11} + 4 \nu^{10} - 33 \nu^{9} + 52 \nu^{8} - 179 \nu^{7} + 192 \nu^{6} - 327 \nu^{5} + 954 \nu^{4} - 693 \nu^{3} + 1404 \nu^{2} - 567 \nu + 4374 ) / 486 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 17 \nu^{11} + 13 \nu^{10} - 45 \nu^{9} + 64 \nu^{8} - 287 \nu^{7} + 243 \nu^{6} - 453 \nu^{5} + 1314 \nu^{4} - 1233 \nu^{3} + 2025 \nu^{2} - 81 \nu + 7290 ) / 486 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 31 \nu^{11} - \nu^{10} - 66 \nu^{9} + 95 \nu^{8} - 421 \nu^{7} + 357 \nu^{6} - 708 \nu^{5} + 2313 \nu^{4} - 1089 \nu^{3} + 3375 \nu^{2} - 1458 \nu + 12393 ) / 486 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 43 \nu^{11} - 17 \nu^{10} + 117 \nu^{9} - 173 \nu^{8} + 745 \nu^{7} - 729 \nu^{6} + 1311 \nu^{5} - 3735 \nu^{4} + 2925 \nu^{3} - 5589 \nu^{2} + 2025 \nu - 19197 ) / 486 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 64 \nu^{11} - 20 \nu^{10} + 165 \nu^{9} - 215 \nu^{8} + 1030 \nu^{7} - 960 \nu^{6} + 1617 \nu^{5} - 5229 \nu^{4} + 3816 \nu^{3} - 7506 \nu^{2} + 2349 \nu - 27945 ) / 486 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{11} - \beta_{10} + \beta_{7} + \beta_{6} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{10} + \beta_{9} + 2\beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{11} - 3\beta_{10} - \beta_{8} - 4\beta_{7} - \beta_{6} + \beta_{5} - 4\beta_{3} - \beta_{2} - 2\beta _1 + 7 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{9} + \beta_{7} + 4\beta_{6} + 3\beta_{5} + \beta_{4} - 2\beta_{3} - 5\beta_{2} + \beta _1 - 7 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 6 \beta_{11} + 8 \beta_{10} - 4 \beta_{9} + \beta_{8} + 7 \beta_{7} - 2 \beta_{6} + 2 \beta_{5} + 5 \beta_{4} - 3 \beta_{3} + 6 \beta_{2} - \beta _1 + 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 4 \beta_{11} - 6 \beta_{10} - 20 \beta_{8} - 14 \beta_{7} - 8 \beta_{6} - 4 \beta_{5} + 6 \beta_{4} + \beta_{3} - 5 \beta_{2} + 11 \beta _1 + 35 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 6 \beta_{11} + 30 \beta_{10} + 14 \beta_{9} + 9 \beta_{8} + 23 \beta_{7} + 2 \beta_{6} + 3 \beta_{5} + 5 \beta_{4} + 23 \beta_{3} - 34 \beta_{2} + 29 \beta _1 + 46 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 30 \beta_{11} - 8 \beta_{10} - 2 \beta_{9} + 14 \beta_{8} + 113 \beta_{7} + 14 \beta_{6} - 5 \beta_{5} + 13 \beta_{4} - 6 \beta_{3} + 45 \beta_{2} - 5 \beta _1 + 72 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 37 \beta_{11} - 69 \beta_{10} + 30 \beta_{9} - 13 \beta_{8} - 160 \beta_{7} + 5 \beta_{6} - 2 \beta_{5} + 63 \beta_{4} + 5 \beta_{3} - 4 \beta_{2} + 58 \beta _1 + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 48 \beta_{11} + 45 \beta_{10} - 5 \beta_{9} + 126 \beta_{8} - 257 \beta_{7} - 83 \beta_{6} + 63 \beta_{5} + 37 \beta_{4} - 95 \beta_{3} - 167 \beta_{2} - 44 \beta _1 + 263 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 105 \beta_{11} - 277 \beta_{10} - 256 \beta_{9} - 56 \beta_{8} + 121 \beta_{7} + 139 \beta_{6} + 152 \beta_{5} - 25 \beta_{4} - 267 \beta_{3} + 81 \beta_{2} - 175 \beta _1 + 351 ) / 3 \) Copy content Toggle raw display

Character values

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

\(n\) \(2053\) \(2431\) \(2945\)
\(\chi(n)\) \(1\) \(1\) \(-\beta_{7}\)

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
−0.433633 + 1.67689i
1.73202 0.0102491i
1.19051 1.25805i
−1.28252 + 1.16410i
0.952418 + 1.44669i
−1.15879 1.28733i
−0.433633 1.67689i
1.73202 + 0.0102491i
1.19051 + 1.25805i
−1.28252 1.16410i
0.952418 1.44669i
−1.15879 + 1.28733i
0 0 0 −2.22043 3.84590i 0 −1.45488 + 2.51992i 0 0 0
1153.2 0 0 0 −0.551563 0.955334i 0 1.62490 2.81442i 0 0 0
1153.3 0 0 0 −0.268104 0.464369i 0 −2.35014 + 4.07056i 0 0 0
1153.4 0 0 0 1.05471 + 1.82681i 0 1.43914 2.49267i 0 0 0
1153.5 0 0 0 1.24278 + 2.15256i 0 −0.909142 + 1.57468i 0 0 0
1153.6 0 0 0 1.74260 + 3.01828i 0 −1.34988 + 2.33807i 0 0 0
2305.1 0 0 0 −2.22043 + 3.84590i 0 −1.45488 2.51992i 0 0 0
2305.2 0 0 0 −0.551563 + 0.955334i 0 1.62490 + 2.81442i 0 0 0
2305.3 0 0 0 −0.268104 + 0.464369i 0 −2.35014 4.07056i 0 0 0
2305.4 0 0 0 1.05471 1.82681i 0 1.43914 + 2.49267i 0 0 0
2305.5 0 0 0 1.24278 2.15256i 0 −0.909142 1.57468i 0 0 0
2305.6 0 0 0 1.74260 3.01828i 0 −1.34988 2.33807i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 2305.6
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.k 12
3.b odd 2 1 1152.2.i.i 12
4.b odd 2 1 3456.2.i.l 12
8.b even 2 1 3456.2.i.i 12
8.d odd 2 1 3456.2.i.j 12
9.c even 3 1 inner 3456.2.i.k 12
9.d odd 6 1 1152.2.i.i 12
12.b even 2 1 1152.2.i.k yes 12
24.f even 2 1 1152.2.i.j yes 12
24.h odd 2 1 1152.2.i.l yes 12
36.f odd 6 1 3456.2.i.l 12
36.h even 6 1 1152.2.i.k yes 12
72.j odd 6 1 1152.2.i.l yes 12
72.l even 6 1 1152.2.i.j yes 12
72.n even 6 1 3456.2.i.i 12
72.p odd 6 1 3456.2.i.j 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1152.2.i.i 12 3.b odd 2 1
1152.2.i.i 12 9.d odd 6 1
1152.2.i.j yes 12 24.f even 2 1
1152.2.i.j yes 12 72.l even 6 1
1152.2.i.k yes 12 12.b even 2 1
1152.2.i.k yes 12 36.h even 6 1
1152.2.i.l yes 12 24.h odd 2 1
1152.2.i.l yes 12 72.j odd 6 1
3456.2.i.i 12 8.b even 2 1
3456.2.i.i 12 72.n even 6 1
3456.2.i.j 12 8.d odd 2 1
3456.2.i.j 12 72.p odd 6 1
3456.2.i.k 12 1.a even 1 1 trivial
3456.2.i.k 12 9.c even 3 1 inner
3456.2.i.l 12 4.b odd 2 1
3456.2.i.l 12 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 \(S_{2}^{\mathrm{new}}(3456, [\chi])\):

\( T_{5}^{12} - 2 T_{5}^{11} + 24 T_{5}^{10} - 60 T_{5}^{9} + 465 T_{5}^{8} - 948 T_{5}^{7} + 2928 T_{5}^{6} - 1674 T_{5}^{5} + 4665 T_{5}^{4} + 1720 T_{5}^{3} + 9424 T_{5}^{2} + 4224 T_{5} + 2304 \) Copy content Toggle raw display
\( T_{7}^{12} + 6 T_{7}^{11} + 48 T_{7}^{10} + 152 T_{7}^{9} + 861 T_{7}^{8} + 2400 T_{7}^{7} + 10164 T_{7}^{6} + 21192 T_{7}^{5} + 67353 T_{7}^{4} + 117452 T_{7}^{3} + 294516 T_{7}^{2} + 324048 T_{7} + 394384 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} - 2 T^{11} + 24 T^{10} + \cdots + 2304 \) Copy content Toggle raw display
$7$ \( T^{12} + 6 T^{11} + 48 T^{10} + \cdots + 394384 \) Copy content Toggle raw display
$11$ \( T^{12} - 4 T^{11} + 47 T^{10} + \cdots + 229441 \) Copy content Toggle raw display
$13$ \( T^{12} - 10 T^{11} + 108 T^{10} + \cdots + 6533136 \) Copy content Toggle raw display
$17$ \( (T^{6} + 2 T^{5} - 83 T^{4} - 176 T^{3} + \cdots + 1812)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 2 T^{5} - 65 T^{4} - 80 T^{3} + \cdots - 3408)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} - 8 T^{11} + 128 T^{10} + \cdots + 204304 \) Copy content Toggle raw display
$29$ \( T^{12} - 2 T^{11} + 88 T^{10} + \cdots + 229704336 \) Copy content Toggle raw display
$31$ \( T^{12} + 8 T^{11} + \cdots + 1021953024 \) Copy content Toggle raw display
$37$ \( (T^{6} - 60 T^{4} + 68 T^{3} + 876 T^{2} + \cdots - 128)^{2} \) Copy content Toggle raw display
$41$ \( T^{12} - 2 T^{11} + 85 T^{10} + \cdots + 2259009 \) Copy content Toggle raw display
$43$ \( T^{12} - 2 T^{11} + 103 T^{10} + \cdots + 16621929 \) Copy content Toggle raw display
$47$ \( T^{12} + 14 T^{11} + 216 T^{10} + \cdots + 2178576 \) Copy content Toggle raw display
$53$ \( (T^{6} + 12 T^{5} - 24 T^{4} - 516 T^{3} + \cdots + 1728)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} - 6 T^{11} + \cdots + 4100737369 \) Copy content Toggle raw display
$61$ \( T^{12} - 14 T^{11} + 200 T^{10} + \cdots + 60715264 \) Copy content Toggle raw display
$67$ \( T^{12} + 4 T^{11} + \cdots + 3020711521 \) Copy content Toggle raw display
$71$ \( (T^{6} + 14 T^{5} - 72 T^{4} - 1608 T^{3} + \cdots + 1728)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 30 T^{5} + 225 T^{4} + \cdots + 39892)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + 16 T^{11} + \cdots + 674337024 \) Copy content Toggle raw display
$83$ \( T^{12} - 24 T^{11} + \cdots + 734843664 \) Copy content Toggle raw display
$89$ \( (T^{6} - 24 T^{5} + 156 T^{4} + 68 T^{3} + \cdots - 2864)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + 14 T^{11} + \cdots + 78140934369 \) Copy content Toggle raw display
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