Properties

Label 3344.2.a.bb
Level $3344$
Weight $2$
Character orbit 3344.a
Self dual yes
Analytic conductor $26.702$
Analytic rank $0$
Dimension $9$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3344 = 2^{4} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3344.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.7019744359\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
Defining polynomial: \( x^{9} - x^{8} - 22x^{7} + 22x^{6} + 152x^{5} - 136x^{4} - 341x^{3} + 169x^{2} + 196x + 32 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 1672)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{8}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{3} + (\beta_{7} + 1) q^{5} - \beta_{8} q^{7} + (\beta_{2} + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{3} + (\beta_{7} + 1) q^{5} - \beta_{8} q^{7} + (\beta_{2} + 2) q^{9} - q^{11} + \beta_{5} q^{13} + (\beta_{5} - \beta_{3} - \beta_1 - 1) q^{15} + ( - \beta_{4} - \beta_1 + 1) q^{17} + q^{19} + (\beta_{7} + \beta_{6} + \beta_{4} - \beta_{3} + \beta_1) q^{21} + (\beta_{3} - \beta_1 - 1) q^{23} + (\beta_{7} - \beta_{6} - \beta_{5} - \beta_{2} + 4) q^{25} + ( - \beta_{5} - \beta_{4} - 2 \beta_1 + 1) q^{27} + ( - \beta_{7} - \beta_{6} + \beta_{5} + \beta_{3} + 1) q^{29} + (\beta_{8} + \beta_{5} + \beta_{4} + \beta_{2} - \beta_1) q^{31} + \beta_1 q^{33} + ( - 2 \beta_{8} - \beta_{7} - \beta_{4} + \beta_{2} - \beta_1 + 1) q^{35} + ( - \beta_{8} + \beta_{2} + 3) q^{37} + (2 \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} - \beta_{2} + \beta_1 - 1) q^{39} + (2 \beta_{6} - \beta_{5} + \beta_{4} + \beta_1 - 1) q^{41} + ( - \beta_{6} - \beta_{5} + \beta_{3} + \beta_1 + 2) q^{43} + ( - 2 \beta_{8} + 2 \beta_{7} - 2 \beta_{5} - \beta_{3} - \beta_{2} + \beta_1 + 4) q^{45} + (2 \beta_{7} - 2) q^{47} + (\beta_{7} + \beta_{3} - \beta_1 + 3) q^{49} + (2 \beta_{8} + \beta_{4} + 2 \beta_{2} - \beta_1 + 3) q^{51} + ( - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - 2 \beta_1 + 3) q^{53} + ( - \beta_{7} - 1) q^{55} - \beta_1 q^{57} + ( - \beta_{7} + \beta_{6} - \beta_{5} + \beta_{4} + \beta_{3} + 2) q^{59} + (\beta_{8} - \beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} + \beta_{2} - \beta_1 + 2) q^{61} + ( - \beta_{8} + 2 \beta_{7} - \beta_{6} + \beta_{5} - \beta_{4} - \beta_{3} - 2 \beta_{2} - 1) q^{63} + ( - 2 \beta_{7} - 2 \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 2) q^{65} + (2 \beta_{8} + \beta_{7} - \beta_{6} - \beta_{3} - 2 \beta_{2} - \beta_1 - 1) q^{67} + (2 \beta_{8} - 3 \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} + 2 \beta_1 + 2) q^{69} + (\beta_{8} + \beta_{7} - \beta_{6} - \beta_{5} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 1) q^{71} + (\beta_{7} + \beta_{6} - \beta_{5} + 2) q^{73} + ( - \beta_{7} - \beta_{6} + 2 \beta_{5} - \beta_{4} - 2 \beta_{3} - 3 \beta_1 - 1) q^{75} + \beta_{8} q^{77} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{4} + \beta_1 - 1) q^{79} + (2 \beta_{8} - 2 \beta_{7} - \beta_{6} + \beta_{5} + \beta_{2} - 2 \beta_1 + 3) q^{81} + (\beta_{7} - \beta_{4} - \beta_{3} + 2 \beta_1 - 1) q^{83} + ( - 2 \beta_{8} - \beta_{7} - 3 \beta_{6} + \beta_{5} - \beta_{4} - 2 \beta_{2} - \beta_1 + 3) q^{85} + (2 \beta_{8} + 2 \beta_{6} - 2 \beta_{5} + \beta_{4} + \beta_{3} - \beta_{2} - 3) q^{87} + (2 \beta_{8} + 2 \beta_{5} + \beta_{4} - \beta_{3}) q^{89} + ( - 2 \beta_{8} + 2 \beta_{6} - 2 \beta_{5} + \beta_{4} + \beta_{3} + 2 \beta_{2} + \beta_1 + 4) q^{91} + ( - 2 \beta_{8} + \beta_{7} - 2 \beta_{5} - 2 \beta_{4} + \beta_{3} - \beta_{2} - 3 \beta_1 + 7) q^{93} + (\beta_{7} + 1) q^{95} + (2 \beta_{7} + \beta_{4} - \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{97} + ( - \beta_{2} - 2) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - q^{3} + 6 q^{5} - 3 q^{7} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - q^{3} + 6 q^{5} - 3 q^{7} + 18 q^{9} - 9 q^{11} + q^{13} - 6 q^{15} + 7 q^{17} + 9 q^{19} + 3 q^{21} - 13 q^{23} + 31 q^{25} + 5 q^{27} + 9 q^{29} + 4 q^{31} + q^{33} + 4 q^{35} + 24 q^{37} - 13 q^{39} - 6 q^{41} + 14 q^{43} + 26 q^{45} - 24 q^{47} + 20 q^{49} + 33 q^{51} + 19 q^{53} - 6 q^{55} - q^{57} + 19 q^{59} + 28 q^{61} - 16 q^{63} + 16 q^{65} - 5 q^{67} + 35 q^{69} - 16 q^{71} + 15 q^{73} - 3 q^{75} + 3 q^{77} - 2 q^{79} + 37 q^{81} - 8 q^{83} + 20 q^{85} - 23 q^{87} + 12 q^{89} + 29 q^{91} + 44 q^{93} + 6 q^{95} - 4 q^{97} - 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{9} - x^{8} - 22x^{7} + 22x^{6} + 152x^{5} - 136x^{4} - 341x^{3} + 169x^{2} + 196x + 32 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 5 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 28 \nu^{8} + 66 \nu^{7} + 685 \nu^{6} - 1506 \nu^{5} - 5225 \nu^{4} + 9908 \nu^{3} + 12355 \nu^{2} - 15117 \nu - 6181 ) / 555 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 17\nu^{8} - 48\nu^{7} - 404\nu^{6} + 954\nu^{5} + 2986\nu^{4} - 5302\nu^{3} - 6863\nu^{2} + 5868\nu + 3170 ) / 222 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -17\nu^{8} + 48\nu^{7} + 404\nu^{6} - 954\nu^{5} - 2986\nu^{4} + 5524\nu^{3} + 6863\nu^{2} - 7644\nu - 2948 ) / 222 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 107 \nu^{8} + 54 \nu^{7} + 2360 \nu^{6} - 1434 \nu^{5} - 16300 \nu^{4} + 10192 \nu^{3} + 34865 \nu^{2} - 14538 \nu - 10994 ) / 1110 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 109 \nu^{8} + 138 \nu^{7} + 2290 \nu^{6} - 3048 \nu^{5} - 14810 \nu^{4} + 19304 \nu^{3} + 29365 \nu^{2} - 28026 \nu - 13378 ) / 1110 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -8\nu^{8} + 3\nu^{7} + 164\nu^{6} - 92\nu^{5} - 996\nu^{4} + 706\nu^{3} + 1606\nu^{2} - 1005\nu - 360 ) / 74 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{5} + \beta_{4} + 8\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{8} - 2\beta_{7} - \beta_{6} + \beta_{5} + 10\beta_{2} - 2\beta _1 + 39 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{7} + \beta_{6} + 14\beta_{5} + 12\beta_{4} - 3\beta_{3} + 71\beta _1 - 21 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 30\beta_{8} - 36\beta_{7} - 11\beta_{6} + 17\beta_{5} + \beta_{3} + 99\beta_{2} - 33\beta _1 + 335 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -2\beta_{8} - 12\beta_{7} + 12\beta_{6} + 162\beta_{5} + 122\beta_{4} - 56\beta_{3} - 6\beta_{2} + 655\beta _1 - 280 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 356 \beta_{8} - 482 \beta_{7} - 108 \beta_{6} + 212 \beta_{5} - 4 \beta_{4} + 34 \beta_{3} + 983 \beta_{2} - 418 \beta _1 + 3019 \) Copy content Toggle raw display

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
3.12163
2.61392
2.34973
1.21469
−0.223788
−0.438092
−1.48312
−2.93149
−3.22348
0 −3.12163 0 −1.30304 0 −0.855540 0 6.74459 0
1.2 0 −2.61392 0 4.05097 0 −3.36136 0 3.83255 0
1.3 0 −2.34973 0 3.03417 0 3.94780 0 2.52125 0
1.4 0 −1.21469 0 −1.18050 0 −1.93448 0 −1.52452 0
1.5 0 0.223788 0 −4.30360 0 0.878402 0 −2.94992 0
1.6 0 0.438092 0 3.19099 0 −3.98768 0 −2.80808 0
1.7 0 1.48312 0 0.694979 0 3.89404 0 −0.800348 0
1.8 0 2.93149 0 3.77651 0 2.25533 0 5.59362 0
1.9 0 3.22348 0 −1.96048 0 −3.83651 0 7.39085 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.9
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(1\)
\(11\) \(1\)
\(19\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3344.2.a.bb 9
4.b odd 2 1 1672.2.a.k 9
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1672.2.a.k 9 4.b odd 2 1
3344.2.a.bb 9 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(3344))\):

\( T_{3}^{9} + T_{3}^{8} - 22T_{3}^{7} - 22T_{3}^{6} + 152T_{3}^{5} + 136T_{3}^{4} - 341T_{3}^{3} - 169T_{3}^{2} + 196T_{3} - 32 \) Copy content Toggle raw display
\( T_{5}^{9} - 6T_{5}^{8} - 20T_{5}^{7} + 162T_{5}^{6} + 16T_{5}^{5} - 1080T_{5}^{4} + 201T_{5}^{3} + 2654T_{5}^{2} + 316T_{5} - 1336 \) Copy content Toggle raw display
\( T_{7}^{9} + 3T_{7}^{8} - 37T_{7}^{7} - 109T_{7}^{6} + 433T_{7}^{5} + 1235T_{7}^{4} - 1614T_{7}^{3} - 4294T_{7}^{2} + 1044T_{7} + 2592 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{9} \) Copy content Toggle raw display
$3$ \( T^{9} + T^{8} - 22 T^{7} - 22 T^{6} + \cdots - 32 \) Copy content Toggle raw display
$5$ \( T^{9} - 6 T^{8} - 20 T^{7} + \cdots - 1336 \) Copy content Toggle raw display
$7$ \( T^{9} + 3 T^{8} - 37 T^{7} + \cdots + 2592 \) Copy content Toggle raw display
$11$ \( (T + 1)^{9} \) Copy content Toggle raw display
$13$ \( T^{9} - T^{8} - 73 T^{7} + 91 T^{6} + \cdots - 27592 \) Copy content Toggle raw display
$17$ \( T^{9} - 7 T^{8} - 96 T^{7} + \cdots - 752000 \) Copy content Toggle raw display
$19$ \( (T - 1)^{9} \) Copy content Toggle raw display
$23$ \( T^{9} + 13 T^{8} - 53 T^{7} + \cdots + 983040 \) Copy content Toggle raw display
$29$ \( T^{9} - 9 T^{8} - 155 T^{7} + \cdots + 7577384 \) Copy content Toggle raw display
$31$ \( T^{9} - 4 T^{8} - 180 T^{7} + \cdots + 494096 \) Copy content Toggle raw display
$37$ \( T^{9} - 24 T^{8} + 157 T^{7} + \cdots + 190720 \) Copy content Toggle raw display
$41$ \( T^{9} + 6 T^{8} - 301 T^{7} + \cdots - 4555776 \) Copy content Toggle raw display
$43$ \( T^{9} - 14 T^{8} - 133 T^{7} + \cdots - 1550336 \) Copy content Toggle raw display
$47$ \( T^{9} + 24 T^{8} + 112 T^{7} + \cdots + 786432 \) Copy content Toggle raw display
$53$ \( T^{9} - 19 T^{8} - 170 T^{7} + \cdots + 16384 \) Copy content Toggle raw display
$59$ \( T^{9} - 19 T^{8} - 191 T^{7} + \cdots - 10710016 \) Copy content Toggle raw display
$61$ \( T^{9} - 28 T^{8} + 78 T^{7} + \cdots + 21155840 \) Copy content Toggle raw display
$67$ \( T^{9} + 5 T^{8} - 468 T^{7} + \cdots - 34237440 \) Copy content Toggle raw display
$71$ \( T^{9} + 16 T^{8} - 218 T^{7} + \cdots + 22501488 \) Copy content Toggle raw display
$73$ \( T^{9} - 15 T^{8} - 40 T^{7} + \cdots + 38400 \) Copy content Toggle raw display
$79$ \( T^{9} + 2 T^{8} - 250 T^{7} + \cdots - 1196032 \) Copy content Toggle raw display
$83$ \( T^{9} + 8 T^{8} - 283 T^{7} + \cdots - 14477312 \) Copy content Toggle raw display
$89$ \( T^{9} - 12 T^{8} - 343 T^{7} + \cdots + 307200 \) Copy content Toggle raw display
$97$ \( T^{9} + 4 T^{8} - 643 T^{7} + \cdots - 113837312 \) Copy content Toggle raw display
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