Properties

Label 1350.2.e.k
Level 1350
Weight 2
Character orbit 1350.e
Analytic conductor 10.780
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1350.e (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.7798042729\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: no (minimal twist has level 450)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 + \beta_{1} ) q^{2} -\beta_{1} q^{4} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{7} + q^{8} +O(q^{10})\) \( q + ( -1 + \beta_{1} ) q^{2} -\beta_{1} q^{4} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{7} + q^{8} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{11} + ( -2 \beta_{1} + \beta_{2} ) q^{13} + ( -2 \beta_{1} + \beta_{2} ) q^{14} + ( -1 + \beta_{1} ) q^{16} -2 \beta_{3} q^{17} + ( 5 + \beta_{3} ) q^{19} -2 \beta_{2} q^{22} -\beta_{2} q^{23} + ( 2 - \beta_{3} ) q^{26} + ( 2 - \beta_{3} ) q^{28} + ( \beta_{2} - \beta_{3} ) q^{29} + ( -2 \beta_{1} - \beta_{2} ) q^{31} -\beta_{1} q^{32} + ( -2 \beta_{2} + 2 \beta_{3} ) q^{34} + ( -4 - 3 \beta_{3} ) q^{37} + ( -5 + 5 \beta_{1} + \beta_{2} - \beta_{3} ) q^{38} + 9 \beta_{1} q^{41} + ( -5 + 5 \beta_{1} - \beta_{2} + \beta_{3} ) q^{43} + 2 \beta_{3} q^{44} + \beta_{3} q^{46} + ( -6 + 6 \beta_{1} + 2 \beta_{2} - 2 \beta_{3} ) q^{47} + ( -3 \beta_{1} + 4 \beta_{2} ) q^{49} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{52} + ( 6 - \beta_{3} ) q^{53} + ( -2 + 2 \beta_{1} - \beta_{2} + \beta_{3} ) q^{56} -\beta_{2} q^{58} + ( -3 \beta_{1} - \beta_{2} ) q^{59} + ( -8 + 8 \beta_{1} ) q^{61} + ( 2 + \beta_{3} ) q^{62} + q^{64} + ( 7 \beta_{1} - 3 \beta_{2} ) q^{67} + 2 \beta_{2} q^{68} + ( -6 - 3 \beta_{3} ) q^{71} - q^{73} + ( 4 - 4 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} ) q^{74} + ( -5 \beta_{1} - \beta_{2} ) q^{76} + ( 12 \beta_{1} - 4 \beta_{2} ) q^{77} + ( -2 + 2 \beta_{1} + 6 \beta_{2} - 6 \beta_{3} ) q^{79} -9 q^{82} + ( -3 + 3 \beta_{1} + \beta_{2} - \beta_{3} ) q^{83} + ( -5 \beta_{1} + \beta_{2} ) q^{86} + ( 2 \beta_{2} - 2 \beta_{3} ) q^{88} + 9 q^{89} + ( 10 - 4 \beta_{3} ) q^{91} + ( \beta_{2} - \beta_{3} ) q^{92} + ( -6 \beta_{1} - 2 \beta_{2} ) q^{94} + ( 1 - \beta_{1} + 4 \beta_{2} - 4 \beta_{3} ) q^{97} + ( 3 - 4 \beta_{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 2q^{2} - 2q^{4} - 4q^{7} + 4q^{8} + O(q^{10}) \) \( 4q - 2q^{2} - 2q^{4} - 4q^{7} + 4q^{8} - 4q^{13} - 4q^{14} - 2q^{16} + 20q^{19} + 8q^{26} + 8q^{28} - 4q^{31} - 2q^{32} - 16q^{37} - 10q^{38} + 18q^{41} - 10q^{43} - 12q^{47} - 6q^{49} - 4q^{52} + 24q^{53} - 4q^{56} - 6q^{59} - 16q^{61} + 8q^{62} + 4q^{64} + 14q^{67} - 24q^{71} - 4q^{73} + 8q^{74} - 10q^{76} + 24q^{77} - 4q^{79} - 36q^{82} - 6q^{83} - 10q^{86} + 36q^{89} + 40q^{91} - 12q^{94} + 2q^{97} + 12q^{98} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - 2 x^{2} + 4\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} \)\(/2\)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 2 \nu \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 4 \nu \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{3} + \beta_{2}\)\()/3\)
\(\nu^{2}\)\(=\)\(2 \beta_{1}\)
\(\nu^{3}\)\(=\)\((\)\(-2 \beta_{3} + 4 \beta_{2}\)\()/3\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1350\mathbb{Z}\right)^\times\).

\(n\) \(1001\) \(1027\)
\(\chi(n)\) \(-\beta_{1}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
−1.22474 0.707107i
1.22474 + 0.707107i
−1.22474 + 0.707107i
1.22474 0.707107i
−0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 −2.22474 + 3.85337i 1.00000 0 0
451.2 −0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 0.224745 0.389270i 1.00000 0 0
901.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 −2.22474 3.85337i 1.00000 0 0
901.2 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 0.224745 + 0.389270i 1.00000 0 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
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 1350.2.e.k 4
3.b odd 2 1 450.2.e.m yes 4
5.b even 2 1 1350.2.e.n 4
5.c odd 4 2 1350.2.j.g 8
9.c even 3 1 inner 1350.2.e.k 4
9.c even 3 1 4050.2.a.by 2
9.d odd 6 1 450.2.e.m yes 4
9.d odd 6 1 4050.2.a.br 2
15.d odd 2 1 450.2.e.l 4
15.e even 4 2 450.2.j.f 8
45.h odd 6 1 450.2.e.l 4
45.h odd 6 1 4050.2.a.bu 2
45.j even 6 1 1350.2.e.n 4
45.j even 6 1 4050.2.a.bl 2
45.k odd 12 2 1350.2.j.g 8
45.k odd 12 2 4050.2.c.w 4
45.l even 12 2 450.2.j.f 8
45.l even 12 2 4050.2.c.y 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.2.e.l 4 15.d odd 2 1
450.2.e.l 4 45.h odd 6 1
450.2.e.m yes 4 3.b odd 2 1
450.2.e.m yes 4 9.d odd 6 1
450.2.j.f 8 15.e even 4 2
450.2.j.f 8 45.l even 12 2
1350.2.e.k 4 1.a even 1 1 trivial
1350.2.e.k 4 9.c even 3 1 inner
1350.2.e.n 4 5.b even 2 1
1350.2.e.n 4 45.j even 6 1
1350.2.j.g 8 5.c odd 4 2
1350.2.j.g 8 45.k odd 12 2
4050.2.a.bl 2 45.j even 6 1
4050.2.a.br 2 9.d odd 6 1
4050.2.a.bu 2 45.h odd 6 1
4050.2.a.by 2 9.c even 3 1
4050.2.c.w 4 45.k odd 12 2
4050.2.c.y 4 45.l even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1350, [\chi])\):

\( T_{7}^{4} + 4 T_{7}^{3} + 18 T_{7}^{2} - 8 T_{7} + 4 \)
\( T_{11}^{4} + 24 T_{11}^{2} + 576 \)
\( T_{17}^{2} - 24 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 + T + T^{2} )^{2} \)
$3$ \( \)
$5$ \( \)
$7$ \( 1 + 4 T + 4 T^{2} - 8 T^{3} - 17 T^{4} - 56 T^{5} + 196 T^{6} + 1372 T^{7} + 2401 T^{8} \)
$11$ \( 1 + 2 T^{2} - 117 T^{4} + 242 T^{6} + 14641 T^{8} \)
$13$ \( 1 + 4 T - 8 T^{2} - 8 T^{3} + 199 T^{4} - 104 T^{5} - 1352 T^{6} + 8788 T^{7} + 28561 T^{8} \)
$17$ \( ( 1 + 10 T^{2} + 289 T^{4} )^{2} \)
$19$ \( ( 1 - 10 T + 57 T^{2} - 190 T^{3} + 361 T^{4} )^{2} \)
$23$ \( 1 - 40 T^{2} + 1071 T^{4} - 21160 T^{6} + 279841 T^{8} \)
$29$ \( 1 - 52 T^{2} + 1863 T^{4} - 43732 T^{6} + 707281 T^{8} \)
$31$ \( 1 + 4 T - 44 T^{2} - 8 T^{3} + 2143 T^{4} - 248 T^{5} - 42284 T^{6} + 119164 T^{7} + 923521 T^{8} \)
$37$ \( ( 1 + 8 T + 36 T^{2} + 296 T^{3} + 1369 T^{4} )^{2} \)
$41$ \( ( 1 - 9 T + 40 T^{2} - 369 T^{3} + 1681 T^{4} )^{2} \)
$43$ \( 1 + 10 T - 5 T^{2} + 190 T^{3} + 4876 T^{4} + 8170 T^{5} - 9245 T^{6} + 795070 T^{7} + 3418801 T^{8} \)
$47$ \( 1 + 12 T + 38 T^{2} + 144 T^{3} + 2259 T^{4} + 6768 T^{5} + 83942 T^{6} + 1245876 T^{7} + 4879681 T^{8} \)
$53$ \( ( 1 - 12 T + 136 T^{2} - 636 T^{3} + 2809 T^{4} )^{2} \)
$59$ \( 1 + 6 T - 85 T^{2} + 18 T^{3} + 9036 T^{4} + 1062 T^{5} - 295885 T^{6} + 1232274 T^{7} + 12117361 T^{8} \)
$61$ \( ( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( ( 1 - 14 T + 67 T^{2} )^{2}( 1 + 14 T + 129 T^{2} + 938 T^{3} + 4489 T^{4} ) \)
$71$ \( ( 1 + 12 T + 124 T^{2} + 852 T^{3} + 5041 T^{4} )^{2} \)
$73$ \( ( 1 + T + 73 T^{2} )^{4} \)
$79$ \( 1 + 4 T + 70 T^{2} - 848 T^{3} - 4589 T^{4} - 66992 T^{5} + 436870 T^{6} + 1972156 T^{7} + 38950081 T^{8} \)
$83$ \( 1 + 6 T - 133 T^{2} + 18 T^{3} + 18684 T^{4} + 1494 T^{5} - 916237 T^{6} + 3430722 T^{7} + 47458321 T^{8} \)
$89$ \( ( 1 - 9 T + 89 T^{2} )^{4} \)
$97$ \( 1 - 2 T - 95 T^{2} + 190 T^{3} + 4 T^{4} + 18430 T^{5} - 893855 T^{6} - 1825346 T^{7} + 88529281 T^{8} \)
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