Properties

Label 1350.2.q.a
Level 1350
Weight 2
Character orbit 1350.q
Analytic conductor 10.780
Analytic rank 0
Dimension 8
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 1350 = 2 \cdot 3^{3} \cdot 5^{2} \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 1350.q (of order \(12\), degree \(4\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(10.7798042729\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 450)
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{24}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{24}^{3} - \zeta_{24}^{7} ) q^{2} + ( \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{4} + ( \zeta_{24}^{3} - 3 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{7} + ( \zeta_{24} - \zeta_{24}^{5} ) q^{8} +O(q^{10})\) \( q + ( \zeta_{24}^{3} - \zeta_{24}^{7} ) q^{2} + ( \zeta_{24}^{2} - \zeta_{24}^{6} ) q^{4} + ( \zeta_{24}^{3} - 3 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{7} + ( \zeta_{24} - \zeta_{24}^{5} ) q^{8} + ( -2 - 2 \zeta_{24}^{4} ) q^{11} + ( -\zeta_{24} + 2 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{13} + ( \zeta_{24}^{2} - 3 \zeta_{24}^{4} + \zeta_{24}^{6} ) q^{14} + ( 1 - \zeta_{24}^{4} ) q^{16} + ( 3 - 6 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{19} + ( -4 \zeta_{24}^{3} + 2 \zeta_{24}^{7} ) q^{22} + ( -3 \zeta_{24} - 6 \zeta_{24}^{3} + 3 \zeta_{24}^{7} ) q^{23} + ( -1 + 2 \zeta_{24}^{4} - 3 \zeta_{24}^{6} ) q^{26} + ( \zeta_{24} - 3 \zeta_{24}^{3} + \zeta_{24}^{5} ) q^{28} + ( 9 - \zeta_{24}^{2} - 9 \zeta_{24}^{4} + 2 \zeta_{24}^{6} ) q^{29} + ( 3 \zeta_{24}^{2} + 5 \zeta_{24}^{4} + 3 \zeta_{24}^{6} ) q^{31} -\zeta_{24}^{7} q^{32} + ( -3 \zeta_{24} + 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{37} + ( -3 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{38} + ( 2 - \zeta_{24}^{4} ) q^{41} + ( 4 \zeta_{24} + 3 \zeta_{24}^{3} - 2 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{43} + ( -4 \zeta_{24}^{2} + 2 \zeta_{24}^{6} ) q^{44} + ( -3 - 6 \zeta_{24}^{2} + 3 \zeta_{24}^{6} ) q^{46} + ( -6 \zeta_{24}^{3} + 6 \zeta_{24}^{7} ) q^{47} + ( 12 - 5 \zeta_{24}^{2} - 6 \zeta_{24}^{4} + 5 \zeta_{24}^{6} ) q^{49} + ( \zeta_{24}^{3} - 3 \zeta_{24}^{5} + \zeta_{24}^{7} ) q^{52} + ( 3 \zeta_{24} - 3 \zeta_{24}^{3} - 3 \zeta_{24}^{5} + 6 \zeta_{24}^{7} ) q^{53} + ( 1 - 3 \zeta_{24}^{2} + \zeta_{24}^{4} ) q^{56} + ( -\zeta_{24} + 2 \zeta_{24}^{5} - 9 \zeta_{24}^{7} ) q^{58} + ( -2 \zeta_{24}^{2} - 9 \zeta_{24}^{4} - 2 \zeta_{24}^{6} ) q^{59} + ( 4 - 4 \zeta_{24}^{4} ) q^{61} + ( 3 \zeta_{24} + 5 \zeta_{24}^{3} + 3 \zeta_{24}^{5} ) q^{62} -\zeta_{24}^{6} q^{64} + ( -9 \zeta_{24} - 4 \zeta_{24}^{3} + 2 \zeta_{24}^{7} ) q^{67} + ( -1 + 2 \zeta_{24}^{4} + 9 \zeta_{24}^{6} ) q^{71} + ( -3 \zeta_{24} - 3 \zeta_{24}^{5} ) q^{73} + ( -3 + 3 \zeta_{24}^{2} + 3 \zeta_{24}^{4} - 6 \zeta_{24}^{6} ) q^{74} + ( -3 \zeta_{24}^{2} - 2 \zeta_{24}^{4} - 3 \zeta_{24}^{6} ) q^{76} + ( -6 \zeta_{24} + 12 \zeta_{24}^{5} - 6 \zeta_{24}^{7} ) q^{77} + 10 \zeta_{24}^{2} q^{79} + ( \zeta_{24}^{3} - 2 \zeta_{24}^{7} ) q^{82} + ( -3 \zeta_{24}^{3} + 12 \zeta_{24}^{5} - 3 \zeta_{24}^{7} ) q^{83} + ( 4 + 3 \zeta_{24}^{2} - 2 \zeta_{24}^{4} - 3 \zeta_{24}^{6} ) q^{86} + ( -4 \zeta_{24} + 2 \zeta_{24}^{5} ) q^{88} + ( 10 \zeta_{24}^{2} - 5 \zeta_{24}^{6} ) q^{89} + ( -12 + 12 \zeta_{24}^{2} - 6 \zeta_{24}^{6} ) q^{91} + ( -6 \zeta_{24} - 3 \zeta_{24}^{3} + 3 \zeta_{24}^{5} + 3 \zeta_{24}^{7} ) q^{92} + ( -6 \zeta_{24}^{2} + 6 \zeta_{24}^{6} ) q^{94} + ( 5 \zeta_{24}^{3} + 6 \zeta_{24}^{5} + 5 \zeta_{24}^{7} ) q^{97} + ( -5 \zeta_{24} + 6 \zeta_{24}^{3} + 5 \zeta_{24}^{5} - 12 \zeta_{24}^{7} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 24q^{11} - 12q^{14} + 4q^{16} + 36q^{29} + 20q^{31} + 12q^{41} - 24q^{46} + 72q^{49} + 12q^{56} - 36q^{59} + 16q^{61} - 12q^{74} - 8q^{76} + 24q^{86} - 96q^{91} + O(q^{100}) \)

Character values

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

\(n\) \(1001\) \(1027\)
\(\chi(n)\) \(\zeta_{24}^{4}\) \(-\zeta_{24}^{6}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
143.1
−0.965926 0.258819i
0.965926 + 0.258819i
−0.965926 + 0.258819i
0.965926 0.258819i
−0.258819 + 0.965926i
0.258819 0.965926i
−0.258819 0.965926i
0.258819 + 0.965926i
−0.965926 + 0.258819i 0 0.866025 0.500000i 0 0 0.328169 + 1.22474i −0.707107 + 0.707107i 0 0
143.2 0.965926 0.258819i 0 0.866025 0.500000i 0 0 −0.328169 1.22474i 0.707107 0.707107i 0 0
557.1 −0.965926 0.258819i 0 0.866025 + 0.500000i 0 0 0.328169 1.22474i −0.707107 0.707107i 0 0
557.2 0.965926 + 0.258819i 0 0.866025 + 0.500000i 0 0 −0.328169 + 1.22474i 0.707107 + 0.707107i 0 0
1007.1 −0.258819 0.965926i 0 −0.866025 + 0.500000i 0 0 4.57081 1.22474i 0.707107 + 0.707107i 0 0
1007.2 0.258819 + 0.965926i 0 −0.866025 + 0.500000i 0 0 −4.57081 + 1.22474i −0.707107 0.707107i 0 0
1043.1 −0.258819 + 0.965926i 0 −0.866025 0.500000i 0 0 4.57081 + 1.22474i 0.707107 0.707107i 0 0
1043.2 0.258819 0.965926i 0 −0.866025 0.500000i 0 0 −4.57081 1.22474i −0.707107 + 0.707107i 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1043.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
45.l even 12 2 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1350.2.q.a 8
3.b odd 2 1 450.2.p.e yes 8
5.b even 2 1 inner 1350.2.q.a 8
5.c odd 4 2 1350.2.q.d 8
9.c even 3 1 450.2.p.c 8
9.d odd 6 1 1350.2.q.d 8
15.d odd 2 1 450.2.p.e yes 8
15.e even 4 2 450.2.p.c 8
45.h odd 6 1 1350.2.q.d 8
45.j even 6 1 450.2.p.c 8
45.k odd 12 2 450.2.p.e yes 8
45.l even 12 2 inner 1350.2.q.a 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
450.2.p.c 8 9.c even 3 1
450.2.p.c 8 15.e even 4 2
450.2.p.c 8 45.j even 6 1
450.2.p.e yes 8 3.b odd 2 1
450.2.p.e yes 8 15.d odd 2 1
450.2.p.e yes 8 45.k odd 12 2
1350.2.q.a 8 1.a even 1 1 trivial
1350.2.q.a 8 5.b even 2 1 inner
1350.2.q.a 8 45.l even 12 2 inner
1350.2.q.d 8 5.c odd 4 2
1350.2.q.d 8 9.d odd 6 1
1350.2.q.d 8 45.h odd 6 1

Hecke kernels

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

\( T_{7}^{8} - 36 T_{7}^{6} + 396 T_{7}^{4} + 1296 T_{7}^{2} + 1296 \)
\( T_{11}^{2} + 6 T_{11} + 12 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{4} + T^{8} \)
$3$ \( \)
$5$ \( \)
$7$ \( 1 - 36 T^{2} + 634 T^{4} - 7272 T^{6} + 59571 T^{8} - 356328 T^{10} + 1522234 T^{12} - 4235364 T^{14} + 5764801 T^{16} \)
$11$ \( ( 1 + 6 T + 23 T^{2} + 66 T^{3} + 121 T^{4} )^{4} \)
$13$ \( 1 - 36 T^{2} + 682 T^{4} - 9000 T^{6} + 106947 T^{8} - 1521000 T^{10} + 19478602 T^{12} - 173765124 T^{14} + 815730721 T^{16} \)
$17$ \( ( 1 + 289 T^{4} )^{4} \)
$19$ \( ( 1 - 14 T^{2} + 339 T^{4} - 5054 T^{6} + 130321 T^{8} )^{2} \)
$23$ \( 1 - 108 T^{2} + 5818 T^{4} - 208440 T^{6} + 5501811 T^{8} - 110264760 T^{10} + 1628114938 T^{12} - 15987876012 T^{14} + 78310985281 T^{16} \)
$29$ \( ( 1 - 18 T + 188 T^{2} - 1404 T^{3} + 8259 T^{4} - 40716 T^{5} + 158108 T^{6} - 439002 T^{7} + 707281 T^{8} )^{2} \)
$31$ \( ( 1 - 10 T + 40 T^{2} + 20 T^{3} - 461 T^{4} + 620 T^{5} + 38440 T^{6} - 297910 T^{7} + 923521 T^{8} )^{2} \)
$37$ \( 1 - 644 T^{4} - 1762266 T^{8} - 1206959684 T^{12} + 3512479453921 T^{16} \)
$41$ \( ( 1 - 3 T + 44 T^{2} - 123 T^{3} + 1681 T^{4} )^{4} \)
$43$ \( 1 - 72 T^{2} + 1633 T^{4} + 6840 T^{6} - 214704 T^{8} + 12647160 T^{10} + 5582902033 T^{12} - 455138139528 T^{14} + 11688200277601 T^{16} \)
$47$ \( 1 + 1054 T^{4} - 3768765 T^{8} + 5143183774 T^{12} + 23811286661761 T^{16} \)
$53$ \( 1 + 508 T^{4} - 3205722 T^{8} + 4008364348 T^{12} + 62259690411361 T^{16} \)
$59$ \( ( 1 + 18 T + 137 T^{2} + 1242 T^{3} + 12372 T^{4} + 73278 T^{5} + 476897 T^{6} + 3696822 T^{7} + 12117361 T^{8} )^{2} \)
$61$ \( ( 1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4} )^{4} \)
$67$ \( 1 - 216 T^{2} + 26737 T^{4} - 2415960 T^{6} + 174766032 T^{8} - 10845244440 T^{10} + 538780522177 T^{12} - 19539010548504 T^{14} + 406067677556641 T^{16} \)
$71$ \( ( 1 - 116 T^{2} + 12474 T^{4} - 584756 T^{6} + 25411681 T^{8} )^{2} \)
$73$ \( ( 1 + 3503 T^{4} + 28398241 T^{8} )^{2} \)
$79$ \( ( 1 + 58 T^{2} - 2877 T^{4} + 361978 T^{6} + 38950081 T^{8} )^{2} \)
$83$ \( 1 - 432 T^{2} + 91513 T^{4} - 12659760 T^{6} + 1239875616 T^{8} - 87213086640 T^{10} + 4343053329673 T^{12} - 141238241295408 T^{14} + 2252292232139041 T^{16} \)
$89$ \( ( 1 + 103 T^{2} + 7921 T^{4} )^{4} \)
$97$ \( 1 + 360 T^{2} + 65929 T^{4} + 8182440 T^{6} + 834546960 T^{8} + 76988577960 T^{10} + 5836646967049 T^{12} + 299869921774440 T^{14} + 7837433594376961 T^{16} \)
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