Properties

Label 350.2.j.d
Level 350
Weight 2
Character orbit 350.j
Analytic conductor 2.795
Analytic rank 0
Dimension 4
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 350.j (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.79476407074\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{3} + \zeta_{12}^{2} q^{4} + 2 q^{6} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{3} q^{8} + ( 1 - \zeta_{12}^{2} ) q^{9} +O(q^{10})\) \( q + \zeta_{12} q^{2} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{3} + \zeta_{12}^{2} q^{4} + 2 q^{6} + ( 2 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{3} q^{8} + ( 1 - \zeta_{12}^{2} ) q^{9} -3 \zeta_{12}^{2} q^{11} + 2 \zeta_{12} q^{12} + \zeta_{12}^{3} q^{13} + ( 3 - \zeta_{12}^{2} ) q^{14} + ( -1 + \zeta_{12}^{2} ) q^{16} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{17} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{18} + ( -1 + \zeta_{12}^{2} ) q^{19} + ( 4 - 6 \zeta_{12}^{2} ) q^{21} -3 \zeta_{12}^{3} q^{22} + 9 \zeta_{12} q^{23} + 2 \zeta_{12}^{2} q^{24} + ( -1 + \zeta_{12}^{2} ) q^{26} + 4 \zeta_{12}^{3} q^{27} + ( 3 \zeta_{12} - \zeta_{12}^{3} ) q^{28} -6 q^{29} -8 \zeta_{12}^{2} q^{31} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{32} -6 \zeta_{12} q^{33} -6 q^{34} + q^{36} + 7 \zeta_{12} q^{37} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{38} + 2 \zeta_{12}^{2} q^{39} + 3 q^{41} + ( 4 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{42} -2 \zeta_{12}^{3} q^{43} + ( 3 - 3 \zeta_{12}^{2} ) q^{44} + 9 \zeta_{12}^{2} q^{46} -9 \zeta_{12} q^{47} + 2 \zeta_{12}^{3} q^{48} + ( 3 - 8 \zeta_{12}^{2} ) q^{49} + ( -12 + 12 \zeta_{12}^{2} ) q^{51} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{52} + ( -9 \zeta_{12} + 9 \zeta_{12}^{3} ) q^{53} + ( -4 + 4 \zeta_{12}^{2} ) q^{54} + ( 1 + 2 \zeta_{12}^{2} ) q^{56} + 2 \zeta_{12}^{3} q^{57} -6 \zeta_{12} q^{58} + ( -8 + 8 \zeta_{12}^{2} ) q^{61} -8 \zeta_{12}^{3} q^{62} + ( -\zeta_{12} - 2 \zeta_{12}^{3} ) q^{63} - q^{64} -6 \zeta_{12}^{2} q^{66} + ( 8 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{67} -6 \zeta_{12} q^{68} + 18 q^{69} + \zeta_{12} q^{72} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{73} + 7 \zeta_{12}^{2} q^{74} - q^{76} + ( -9 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{77} + 2 \zeta_{12}^{3} q^{78} + ( -10 + 10 \zeta_{12}^{2} ) q^{79} + 11 \zeta_{12}^{2} q^{81} + 3 \zeta_{12} q^{82} + ( 6 - 2 \zeta_{12}^{2} ) q^{84} + ( 2 - 2 \zeta_{12}^{2} ) q^{86} + ( -12 \zeta_{12} + 12 \zeta_{12}^{3} ) q^{87} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{88} + ( 6 - 6 \zeta_{12}^{2} ) q^{89} + ( 1 + 2 \zeta_{12}^{2} ) q^{91} + 9 \zeta_{12}^{3} q^{92} -16 \zeta_{12} q^{93} -9 \zeta_{12}^{2} q^{94} + ( -2 + 2 \zeta_{12}^{2} ) q^{96} -10 \zeta_{12}^{3} q^{97} + ( 3 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{98} -3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} + 8q^{6} + 2q^{9} + O(q^{10}) \) \( 4q + 2q^{4} + 8q^{6} + 2q^{9} - 6q^{11} + 10q^{14} - 2q^{16} - 2q^{19} + 4q^{21} + 4q^{24} - 2q^{26} - 24q^{29} - 16q^{31} - 24q^{34} + 4q^{36} + 4q^{39} + 12q^{41} + 6q^{44} + 18q^{46} - 4q^{49} - 24q^{51} - 8q^{54} + 8q^{56} - 16q^{61} - 4q^{64} - 12q^{66} + 72q^{69} + 14q^{74} - 4q^{76} - 20q^{79} + 22q^{81} + 20q^{84} + 4q^{86} + 12q^{89} + 8q^{91} - 18q^{94} - 4q^{96} - 12q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/350\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1 + \zeta_{12}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i −1.73205 + 1.00000i 0.500000 + 0.866025i 0 2.00000 −1.73205 + 2.00000i 1.00000i 0.500000 0.866025i 0
149.2 0.866025 + 0.500000i 1.73205 1.00000i 0.500000 + 0.866025i 0 2.00000 1.73205 2.00000i 1.00000i 0.500000 0.866025i 0
249.1 −0.866025 + 0.500000i −1.73205 1.00000i 0.500000 0.866025i 0 2.00000 −1.73205 2.00000i 1.00000i 0.500000 + 0.866025i 0
249.2 0.866025 0.500000i 1.73205 + 1.00000i 0.500000 0.866025i 0 2.00000 1.73205 + 2.00000i 1.00000i 0.500000 + 0.866025i 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
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.2.j.d 4
5.b even 2 1 inner 350.2.j.d 4
5.c odd 4 1 70.2.e.d 2
5.c odd 4 1 350.2.e.b 2
7.c even 3 1 inner 350.2.j.d 4
7.c even 3 1 2450.2.c.q 2
7.d odd 6 1 2450.2.c.e 2
15.e even 4 1 630.2.k.d 2
20.e even 4 1 560.2.q.b 2
35.f even 4 1 490.2.e.g 2
35.i odd 6 1 2450.2.c.e 2
35.j even 6 1 inner 350.2.j.d 4
35.j even 6 1 2450.2.c.q 2
35.k even 12 1 490.2.a.d 1
35.k even 12 1 490.2.e.g 2
35.k even 12 1 2450.2.a.v 1
35.l odd 12 1 70.2.e.d 2
35.l odd 12 1 350.2.e.b 2
35.l odd 12 1 490.2.a.a 1
35.l odd 12 1 2450.2.a.bf 1
105.w odd 12 1 4410.2.a.bg 1
105.x even 12 1 630.2.k.d 2
105.x even 12 1 4410.2.a.x 1
140.w even 12 1 560.2.q.b 2
140.w even 12 1 3920.2.a.bh 1
140.x odd 12 1 3920.2.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.d 2 5.c odd 4 1
70.2.e.d 2 35.l odd 12 1
350.2.e.b 2 5.c odd 4 1
350.2.e.b 2 35.l odd 12 1
350.2.j.d 4 1.a even 1 1 trivial
350.2.j.d 4 5.b even 2 1 inner
350.2.j.d 4 7.c even 3 1 inner
350.2.j.d 4 35.j even 6 1 inner
490.2.a.a 1 35.l odd 12 1
490.2.a.d 1 35.k even 12 1
490.2.e.g 2 35.f even 4 1
490.2.e.g 2 35.k even 12 1
560.2.q.b 2 20.e even 4 1
560.2.q.b 2 140.w even 12 1
630.2.k.d 2 15.e even 4 1
630.2.k.d 2 105.x even 12 1
2450.2.a.v 1 35.k even 12 1
2450.2.a.bf 1 35.l odd 12 1
2450.2.c.e 2 7.d odd 6 1
2450.2.c.e 2 35.i odd 6 1
2450.2.c.q 2 7.c even 3 1
2450.2.c.q 2 35.j even 6 1
3920.2.a.e 1 140.x odd 12 1
3920.2.a.bh 1 140.w even 12 1
4410.2.a.x 1 105.x even 12 1
4410.2.a.bg 1 105.w odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(350, [\chi])\):

\( T_{3}^{4} - 4 T_{3}^{2} + 16 \)
\( T_{11}^{2} + 3 T_{11} + 9 \)
\( T_{13}^{2} + 1 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( 1 + 2 T^{2} - 5 T^{4} + 18 T^{6} + 81 T^{8} \)
$5$ 1
$7$ \( 1 + 2 T^{2} + 49 T^{4} \)
$11$ \( ( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} )^{2} \)
$13$ \( ( 1 - 25 T^{2} + 169 T^{4} )^{2} \)
$17$ \( 1 - 2 T^{2} - 285 T^{4} - 578 T^{6} + 83521 T^{8} \)
$19$ \( ( 1 - 7 T + 19 T^{2} )^{2}( 1 + 8 T + 19 T^{2} )^{2} \)
$23$ \( 1 - 35 T^{2} + 696 T^{4} - 18515 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 + 6 T + 29 T^{2} )^{4} \)
$31$ \( ( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} )^{2} \)
$37$ \( 1 + 25 T^{2} - 744 T^{4} + 34225 T^{6} + 1874161 T^{8} \)
$41$ \( ( 1 - 3 T + 41 T^{2} )^{4} \)
$43$ \( ( 1 - 82 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( 1 + 13 T^{2} - 2040 T^{4} + 28717 T^{6} + 4879681 T^{8} \)
$53$ \( 1 + 25 T^{2} - 2184 T^{4} + 70225 T^{6} + 7890481 T^{8} \)
$59$ \( ( 1 - 59 T^{2} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 + 8 T + 3 T^{2} + 488 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( 1 + 70 T^{2} + 411 T^{4} + 314230 T^{6} + 20151121 T^{8} \)
$71$ \( ( 1 + 71 T^{2} )^{4} \)
$73$ \( 1 + 130 T^{2} + 11571 T^{4} + 692770 T^{6} + 28398241 T^{8} \)
$79$ \( ( 1 + 10 T + 21 T^{2} + 790 T^{3} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 - 83 T^{2} )^{4} \)
$89$ \( ( 1 - 6 T - 53 T^{2} - 534 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 94 T^{2} + 9409 T^{4} )^{2} \)
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