Properties

Label 3330.2.e.c
Level $3330$
Weight $2$
Character orbit 3330.e
Analytic conductor $26.590$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3330.e (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(26.5901838731\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
Defining polynomial: \(x^{10} + 19 x^{8} + 103 x^{6} + 210 x^{4} + 140 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 370)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - q^{2} + q^{4} + \beta_{5} q^{5} -\beta_{2} q^{7} - q^{8} +O(q^{10})\) \( q - q^{2} + q^{4} + \beta_{5} q^{5} -\beta_{2} q^{7} - q^{8} -\beta_{5} q^{10} -\beta_{7} q^{11} + ( \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{13} + \beta_{2} q^{14} + q^{16} + ( 2 - \beta_{8} ) q^{17} + ( \beta_{4} + \beta_{6} - \beta_{9} ) q^{19} + \beta_{5} q^{20} + \beta_{7} q^{22} + ( \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{23} + ( 1 - \beta_{3} - \beta_{4} + \beta_{9} ) q^{25} + ( -\beta_{4} + \beta_{6} + \beta_{7} + \beta_{8} ) q^{26} -\beta_{2} q^{28} + ( -\beta_{2} - \beta_{3} - \beta_{5} - \beta_{9} ) q^{29} + ( \beta_{1} + \beta_{2} - \beta_{9} ) q^{31} - q^{32} + ( -2 + \beta_{8} ) q^{34} + ( -\beta_{1} + \beta_{4} + \beta_{6} - \beta_{7} ) q^{35} + ( 1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} - \beta_{8} ) q^{37} + ( -\beta_{4} - \beta_{6} + \beta_{9} ) q^{38} -\beta_{5} q^{40} + ( \beta_{4} - \beta_{6} - \beta_{7} ) q^{41} + ( 2 - \beta_{3} + \beta_{5} + \beta_{7} - 2 \beta_{8} ) q^{43} -\beta_{7} q^{44} + ( -\beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} ) q^{46} + ( -2 \beta_{1} - 2 \beta_{3} + \beta_{4} - 2 \beta_{5} + \beta_{6} - \beta_{9} ) q^{47} + ( -1 + 2 \beta_{7} + \beta_{8} ) q^{49} + ( -1 + \beta_{3} + \beta_{4} - \beta_{9} ) q^{50} + ( \beta_{4} - \beta_{6} - \beta_{7} - \beta_{8} ) q^{52} + ( -\beta_{2} + \beta_{4} + \beta_{6} + \beta_{9} ) q^{53} + ( \beta_{2} - \beta_{3} + \beta_{6} + 2 \beta_{7} ) q^{55} + \beta_{2} q^{56} + ( \beta_{2} + \beta_{3} + \beta_{5} + \beta_{9} ) q^{58} + ( -3 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} ) q^{59} + ( -2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{9} ) q^{61} + ( -\beta_{1} - \beta_{2} + \beta_{9} ) q^{62} + q^{64} + ( -1 - 3 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} + 2 \beta_{8} + \beta_{9} ) q^{65} + ( \beta_{1} - 2 \beta_{2} + \beta_{4} + \beta_{6} - 2 \beta_{9} ) q^{67} + ( 2 - \beta_{8} ) q^{68} + ( \beta_{1} - \beta_{4} - \beta_{6} + \beta_{7} ) q^{70} + ( 2 \beta_{3} + 2 \beta_{4} - 2 \beta_{5} - 2 \beta_{6} - 2 \beta_{7} ) q^{71} + ( -2 \beta_{1} + 2 \beta_{2} - \beta_{3} - \beta_{5} ) q^{73} + ( -1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} - \beta_{6} + \beta_{8} ) q^{74} + ( \beta_{4} + \beta_{6} - \beta_{9} ) q^{76} + ( 2 \beta_{1} - 3 \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{9} ) q^{77} + ( \beta_{1} - 2 \beta_{2} + 2 \beta_{3} + 2 \beta_{5} ) q^{79} + \beta_{5} q^{80} + ( -\beta_{4} + \beta_{6} + \beta_{7} ) q^{82} + ( -3 \beta_{1} - 2 \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + \beta_{6} ) q^{83} + ( -2 - 2 \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{9} ) q^{85} + ( -2 + \beta_{3} - \beta_{5} - \beta_{7} + 2 \beta_{8} ) q^{86} + \beta_{7} q^{88} + ( 2 \beta_{2} - 2 \beta_{9} ) q^{89} + ( -\beta_{1} - \beta_{3} - 2 \beta_{4} - \beta_{5} - 2 \beta_{6} + \beta_{9} ) q^{91} + ( \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{92} + ( 2 \beta_{1} + 2 \beta_{3} - \beta_{4} + 2 \beta_{5} - \beta_{6} + \beta_{9} ) q^{94} + ( -\beta_{1} - 2 \beta_{2} - 3 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} ) q^{95} + ( -\beta_{3} + \beta_{5} - \beta_{7} + 2 \beta_{8} ) q^{97} + ( 1 - 2 \beta_{7} - \beta_{8} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10q - 10q^{2} + 10q^{4} - 3q^{5} - 10q^{8} + O(q^{10}) \) \( 10q - 10q^{2} + 10q^{4} - 3q^{5} - 10q^{8} + 3q^{10} + 2q^{13} + 10q^{16} + 18q^{17} - 3q^{20} + 10q^{23} + 5q^{25} - 2q^{26} - 10q^{32} - 18q^{34} + 8q^{37} + 3q^{40} + 4q^{41} + 10q^{43} - 10q^{46} - 8q^{49} - 5q^{50} + 2q^{52} - 5q^{55} + 10q^{64} - 2q^{65} + 18q^{68} + 20q^{71} - 8q^{74} - 3q^{80} - 4q^{82} - 28q^{85} - 10q^{86} + 10q^{92} - 2q^{95} - 2q^{97} + 8q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{10} + 19 x^{8} + 103 x^{6} + 210 x^{4} + 140 x^{2} + 16\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( -3 \nu^{9} - 45 \nu^{7} - 121 \nu^{5} - 26 \nu^{3} + 12 \nu \)\()/8\)
\(\beta_{3}\)\(=\)\((\)\( \nu^{9} + 6 \nu^{8} + 15 \nu^{7} + 94 \nu^{6} + 39 \nu^{5} + 302 \nu^{4} - 10 \nu^{3} + 212 \nu^{2} - 44 \nu + 16 \)\()/8\)
\(\beta_{4}\)\(=\)\((\)\( 3 \nu^{9} + 2 \nu^{8} + 49 \nu^{7} + 34 \nu^{6} + 181 \nu^{5} + 142 \nu^{4} + 186 \nu^{3} + 196 \nu^{2} + 20 \nu + 64 \)\()/8\)
\(\beta_{5}\)\(=\)\((\)\( \nu^{9} - 6 \nu^{8} + 15 \nu^{7} - 94 \nu^{6} + 39 \nu^{5} - 302 \nu^{4} - 10 \nu^{3} - 212 \nu^{2} - 44 \nu - 16 \)\()/8\)
\(\beta_{6}\)\(=\)\((\)\( 3 \nu^{9} - 2 \nu^{8} + 49 \nu^{7} - 34 \nu^{6} + 181 \nu^{5} - 142 \nu^{4} + 186 \nu^{3} - 196 \nu^{2} + 20 \nu - 64 \)\()/8\)
\(\beta_{7}\)\(=\)\((\)\( 3 \nu^{8} + 49 \nu^{6} + 181 \nu^{4} + 186 \nu^{2} + 28 \)\()/4\)
\(\beta_{8}\)\(=\)\((\)\( 3 \nu^{8} + 49 \nu^{6} + 181 \nu^{4} + 190 \nu^{2} + 44 \)\()/4\)
\(\beta_{9}\)\(=\)\((\)\( -\nu^{9} - 17 \nu^{7} - 71 \nu^{5} - 100 \nu^{3} - 46 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{8} - \beta_{7} - 4\)
\(\nu^{3}\)\(=\)\(-\beta_{9} - \beta_{6} + \beta_{5} - \beta_{4} + \beta_{3} - 7 \beta_{1}\)
\(\nu^{4}\)\(=\)\(-14 \beta_{8} + 10 \beta_{7} - 3 \beta_{6} - \beta_{5} + 3 \beta_{4} + \beta_{3} + 32\)
\(\nu^{5}\)\(=\)\(14 \beta_{9} + 14 \beta_{6} - 17 \beta_{5} + 14 \beta_{4} - 17 \beta_{3} - 2 \beta_{2} + 68 \beta_{1}\)
\(\nu^{6}\)\(=\)\(170 \beta_{8} - 108 \beta_{7} + 45 \beta_{6} + 16 \beta_{5} - 45 \beta_{4} - 16 \beta_{3} - 330\)
\(\nu^{7}\)\(=\)\(-170 \beta_{9} - 169 \beta_{6} + 215 \beta_{5} - 169 \beta_{4} + 215 \beta_{3} + 32 \beta_{2} - 748 \beta_{1}\)
\(\nu^{8}\)\(=\)\(-1994 \beta_{8} + 1224 \beta_{7} - 554 \beta_{6} - 201 \beta_{5} + 554 \beta_{4} + 201 \beta_{3} + 3698\)
\(\nu^{9}\)\(=\)\(1994 \beta_{9} + 1979 \beta_{6} - 2548 \beta_{5} + 1979 \beta_{4} - 2548 \beta_{3} - 402 \beta_{2} + 8542 \beta_{1}\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3330\mathbb{Z}\right)^\times\).

\(n\) \(371\) \(631\) \(667\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
739.1
0.987983i
0.987983i
1.76216i
1.76216i
3.40359i
3.40359i
0.377861i
0.377861i
1.78647i
1.78647i
−1.00000 0 1.00000 −1.85396 1.25013i 0 4.78937i −1.00000 0 1.85396 + 1.25013i
739.2 −1.00000 0 1.00000 −1.85396 + 1.25013i 0 4.78937i −1.00000 0 1.85396 1.25013i
739.3 −1.00000 0 1.00000 −1.62868 1.53213i 0 1.22131i −1.00000 0 1.62868 + 1.53213i
739.4 −1.00000 0 1.00000 −1.62868 + 1.53213i 0 1.22131i −1.00000 0 1.62868 1.53213i
739.5 −1.00000 0 1.00000 −1.28269 1.83159i 0 2.06225i −1.00000 0 1.28269 + 1.83159i
739.6 −1.00000 0 1.00000 −1.28269 + 1.83159i 0 2.06225i −1.00000 0 1.28269 1.83159i
739.7 −1.00000 0 1.00000 1.04797 1.97529i 0 0.631751i −1.00000 0 −1.04797 + 1.97529i
739.8 −1.00000 0 1.00000 1.04797 + 1.97529i 0 0.631751i −1.00000 0 −1.04797 1.97529i
739.9 −1.00000 0 1.00000 2.21736 0.288618i 0 3.14934i −1.00000 0 −2.21736 + 0.288618i
739.10 −1.00000 0 1.00000 2.21736 + 0.288618i 0 3.14934i −1.00000 0 −2.21736 0.288618i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 739.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
185.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3330.2.e.c 10
3.b odd 2 1 370.2.c.b yes 10
5.b even 2 1 3330.2.e.d 10
15.d odd 2 1 370.2.c.a 10
15.e even 4 2 1850.2.d.i 20
37.b even 2 1 3330.2.e.d 10
111.d odd 2 1 370.2.c.a 10
185.d even 2 1 inner 3330.2.e.c 10
555.b odd 2 1 370.2.c.b yes 10
555.n even 4 2 1850.2.d.i 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
370.2.c.a 10 15.d odd 2 1
370.2.c.a 10 111.d odd 2 1
370.2.c.b yes 10 3.b odd 2 1
370.2.c.b yes 10 555.b odd 2 1
1850.2.d.i 20 15.e even 4 2
1850.2.d.i 20 555.n even 4 2
3330.2.e.c 10 1.a even 1 1 trivial
3330.2.e.c 10 185.d even 2 1 inner
3330.2.e.d 10 5.b even 2 1
3330.2.e.d 10 37.b even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(3330, [\chi])\):

\( T_{7}^{10} + 39 T_{7}^{8} + 438 T_{7}^{6} + 1684 T_{7}^{4} + 2048 T_{7}^{2} + 576 \)
\( T_{13}^{5} - T_{13}^{4} - 39 T_{13}^{3} + 100 T_{13}^{2} + 160 T_{13} - 488 \)
\( T_{17}^{5} - 9 T_{17}^{4} + 4 T_{17}^{3} + 108 T_{17}^{2} - 112 T_{17} - 144 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T )^{10} \)
$3$ \( T^{10} \)
$5$ \( 3125 + 1875 T + 250 T^{2} - 400 T^{3} - 95 T^{4} - 22 T^{5} - 19 T^{6} - 16 T^{7} + 2 T^{8} + 3 T^{9} + T^{10} \)
$7$ \( 576 + 2048 T^{2} + 1684 T^{4} + 438 T^{6} + 39 T^{8} + T^{10} \)
$11$ \( ( 48 + 16 T - 51 T^{2} - 28 T^{3} + T^{5} )^{2} \)
$13$ \( ( -488 + 160 T + 100 T^{2} - 39 T^{3} - T^{4} + T^{5} )^{2} \)
$17$ \( ( -144 - 112 T + 108 T^{2} + 4 T^{3} - 9 T^{4} + T^{5} )^{2} \)
$19$ \( 9216 + 374336 T^{2} + 72240 T^{4} + 4668 T^{6} + 118 T^{8} + T^{10} \)
$23$ \( ( -768 + 256 T + 308 T^{2} - 63 T^{3} - 5 T^{4} + T^{5} )^{2} \)
$29$ \( 2262016 + 1251872 T^{2} + 194729 T^{4} + 10258 T^{6} + 182 T^{8} + T^{10} \)
$31$ \( 60516 + 346130 T^{2} + 68053 T^{4} + 4502 T^{6} + 116 T^{8} + T^{10} \)
$37$ \( 69343957 - 14993288 T + 4913341 T^{2} - 788544 T^{3} + 195434 T^{4} - 25584 T^{5} + 5282 T^{6} - 576 T^{7} + 97 T^{8} - 8 T^{9} + T^{10} \)
$41$ \( ( 36 - 8 T - 89 T^{2} - 44 T^{3} - 2 T^{4} + T^{5} )^{2} \)
$43$ \( ( 2624 + 1856 T + 76 T^{2} - 88 T^{3} - 5 T^{4} + T^{5} )^{2} \)
$47$ \( 4596736 + 10494016 T^{2} + 992816 T^{4} + 29148 T^{6} + 310 T^{8} + T^{10} \)
$53$ \( 39337984 + 10468352 T^{2} + 826000 T^{4} + 22964 T^{6} + 257 T^{8} + T^{10} \)
$59$ \( 21827584 + 7068992 T^{2} + 590448 T^{4} + 18844 T^{6} + 238 T^{8} + T^{10} \)
$61$ \( 82944 + 89792 T^{2} + 28169 T^{4} + 3402 T^{6} + 150 T^{8} + T^{10} \)
$67$ \( 559417104 + 70672748 T^{2} + 2839394 T^{4} + 48727 T^{6} + 367 T^{8} + T^{10} \)
$71$ \( ( -4608 - 2816 T + 1896 T^{2} - 164 T^{3} - 10 T^{4} + T^{5} )^{2} \)
$73$ \( 589824 + 785408 T^{2} + 189152 T^{4} + 10965 T^{6} + 189 T^{8} + T^{10} \)
$79$ \( 186486336 + 29601332 T^{2} + 1472598 T^{4} + 31019 T^{6} + 291 T^{8} + T^{10} \)
$83$ \( 1024 + 23736128 T^{2} + 2459520 T^{4} + 62212 T^{6} + 466 T^{8} + T^{10} \)
$89$ \( 1230045184 + 117901312 T^{2} + 3949632 T^{4} + 59888 T^{6} + 412 T^{8} + T^{10} \)
$97$ \( ( -1168 + 2824 T + 248 T^{2} - 178 T^{3} + T^{4} + T^{5} )^{2} \)
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