Properties

Label 350.2.j.f
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.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{3} + \zeta_{12}^{2} q^{4} + 3 q^{6} + ( -3 \zeta_{12} + \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{3} q^{8} + ( 6 - 6 \zeta_{12}^{2} ) q^{9} +O(q^{10})\) \( q + \zeta_{12} q^{2} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{3} + \zeta_{12}^{2} q^{4} + 3 q^{6} + ( -3 \zeta_{12} + \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{3} q^{8} + ( 6 - 6 \zeta_{12}^{2} ) q^{9} + 2 \zeta_{12}^{2} q^{11} + 3 \zeta_{12} q^{12} + ( -1 - 2 \zeta_{12}^{2} ) q^{14} + ( -1 + \zeta_{12}^{2} ) q^{16} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{17} + ( 6 \zeta_{12} - 6 \zeta_{12}^{3} ) q^{18} + ( -6 + 6 \zeta_{12}^{2} ) q^{19} + ( -9 + 3 \zeta_{12}^{2} ) q^{21} + 2 \zeta_{12}^{3} q^{22} -3 \zeta_{12} q^{23} + 3 \zeta_{12}^{2} q^{24} -9 \zeta_{12}^{3} q^{27} + ( -\zeta_{12} - 2 \zeta_{12}^{3} ) q^{28} -9 q^{29} + 4 \zeta_{12}^{2} q^{31} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{32} + 6 \zeta_{12} q^{33} + 4 q^{34} + 6 q^{36} -4 \zeta_{12} q^{37} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{38} -7 q^{41} + ( -9 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{42} -5 \zeta_{12}^{3} q^{43} + ( -2 + 2 \zeta_{12}^{2} ) q^{44} -3 \zeta_{12}^{2} q^{46} + 8 \zeta_{12} q^{47} + 3 \zeta_{12}^{3} q^{48} + ( 5 + 3 \zeta_{12}^{2} ) q^{49} + ( 12 - 12 \zeta_{12}^{2} ) q^{51} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{53} + ( 9 - 9 \zeta_{12}^{2} ) q^{54} + ( 2 - 3 \zeta_{12}^{2} ) q^{56} + 18 \zeta_{12}^{3} q^{57} -9 \zeta_{12} q^{58} + 10 \zeta_{12}^{2} q^{59} + ( -1 + \zeta_{12}^{2} ) q^{61} + 4 \zeta_{12}^{3} q^{62} + ( -12 \zeta_{12} + 18 \zeta_{12}^{3} ) q^{63} - q^{64} + 6 \zeta_{12}^{2} q^{66} + ( 9 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{67} + 4 \zeta_{12} q^{68} -9 q^{69} + 2 q^{71} + 6 \zeta_{12} q^{72} + ( -4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{73} -4 \zeta_{12}^{2} q^{74} -6 q^{76} + ( -2 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{77} + ( 10 - 10 \zeta_{12}^{2} ) q^{79} -9 \zeta_{12}^{2} q^{81} -7 \zeta_{12} q^{82} -7 \zeta_{12}^{3} q^{83} + ( -3 - 6 \zeta_{12}^{2} ) q^{84} + ( 5 - 5 \zeta_{12}^{2} ) q^{86} + ( -27 \zeta_{12} + 27 \zeta_{12}^{3} ) q^{87} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{88} + ( 1 - \zeta_{12}^{2} ) q^{89} -3 \zeta_{12}^{3} q^{92} + 12 \zeta_{12} q^{93} + 8 \zeta_{12}^{2} q^{94} + ( -3 + 3 \zeta_{12}^{2} ) q^{96} -14 \zeta_{12}^{3} q^{97} + ( 5 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{98} + 12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} + 12q^{6} + 12q^{9} + O(q^{10}) \) \( 4q + 2q^{4} + 12q^{6} + 12q^{9} + 4q^{11} - 8q^{14} - 2q^{16} - 12q^{19} - 30q^{21} + 6q^{24} - 36q^{29} + 8q^{31} + 16q^{34} + 24q^{36} - 28q^{41} - 4q^{44} - 6q^{46} + 26q^{49} + 24q^{51} + 18q^{54} + 2q^{56} + 20q^{59} - 2q^{61} - 4q^{64} + 12q^{66} - 36q^{69} + 8q^{71} - 8q^{74} - 24q^{76} + 20q^{79} - 18q^{81} - 24q^{84} + 10q^{86} + 2q^{89} + 16q^{94} - 6q^{96} + 48q^{99} + O(q^{100}) \)

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 −2.59808 + 1.50000i 0.500000 + 0.866025i 0 3.00000 2.59808 + 0.500000i 1.00000i 3.00000 5.19615i 0
149.2 0.866025 + 0.500000i 2.59808 1.50000i 0.500000 + 0.866025i 0 3.00000 −2.59808 0.500000i 1.00000i 3.00000 5.19615i 0
249.1 −0.866025 + 0.500000i −2.59808 1.50000i 0.500000 0.866025i 0 3.00000 2.59808 0.500000i 1.00000i 3.00000 + 5.19615i 0
249.2 0.866025 0.500000i 2.59808 + 1.50000i 0.500000 0.866025i 0 3.00000 −2.59808 + 0.500000i 1.00000i 3.00000 + 5.19615i 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.f 4
5.b even 2 1 inner 350.2.j.f 4
5.c odd 4 1 70.2.e.a 2
5.c odd 4 1 350.2.e.l 2
7.c even 3 1 inner 350.2.j.f 4
7.c even 3 1 2450.2.c.s 2
7.d odd 6 1 2450.2.c.a 2
15.e even 4 1 630.2.k.f 2
20.e even 4 1 560.2.q.i 2
35.f even 4 1 490.2.e.f 2
35.i odd 6 1 2450.2.c.a 2
35.j even 6 1 inner 350.2.j.f 4
35.j even 6 1 2450.2.c.s 2
35.k even 12 1 490.2.a.e 1
35.k even 12 1 490.2.e.f 2
35.k even 12 1 2450.2.a.q 1
35.l odd 12 1 70.2.e.a 2
35.l odd 12 1 350.2.e.l 2
35.l odd 12 1 490.2.a.k 1
35.l odd 12 1 2450.2.a.b 1
105.w odd 12 1 4410.2.a.h 1
105.x even 12 1 630.2.k.f 2
105.x even 12 1 4410.2.a.r 1
140.w even 12 1 560.2.q.i 2
140.w even 12 1 3920.2.a.b 1
140.x odd 12 1 3920.2.a.bk 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.a 2 5.c odd 4 1
70.2.e.a 2 35.l odd 12 1
350.2.e.l 2 5.c odd 4 1
350.2.e.l 2 35.l odd 12 1
350.2.j.f 4 1.a even 1 1 trivial
350.2.j.f 4 5.b even 2 1 inner
350.2.j.f 4 7.c even 3 1 inner
350.2.j.f 4 35.j even 6 1 inner
490.2.a.e 1 35.k even 12 1
490.2.a.k 1 35.l odd 12 1
490.2.e.f 2 35.f even 4 1
490.2.e.f 2 35.k even 12 1
560.2.q.i 2 20.e even 4 1
560.2.q.i 2 140.w even 12 1
630.2.k.f 2 15.e even 4 1
630.2.k.f 2 105.x even 12 1
2450.2.a.b 1 35.l odd 12 1
2450.2.a.q 1 35.k even 12 1
2450.2.c.a 2 7.d odd 6 1
2450.2.c.a 2 35.i odd 6 1
2450.2.c.s 2 7.c even 3 1
2450.2.c.s 2 35.j even 6 1
3920.2.a.b 1 140.w even 12 1
3920.2.a.bk 1 140.x odd 12 1
4410.2.a.h 1 105.w odd 12 1
4410.2.a.r 1 105.x even 12 1

Hecke kernels

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

\( T_{3}^{4} - 9 T_{3}^{2} + 81 \)
\( T_{11}^{2} - 2 T_{11} + 4 \)
\( T_{13} \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( ( 1 - 3 T^{2} )^{2}( 1 + 3 T^{2} + 9 T^{4} ) \)
$5$ \( \)
$7$ \( 1 - 13 T^{2} + 49 T^{4} \)
$11$ \( ( 1 - 2 T - 7 T^{2} - 22 T^{3} + 121 T^{4} )^{2} \)
$13$ \( ( 1 - 13 T^{2} )^{4} \)
$17$ \( 1 + 18 T^{2} + 35 T^{4} + 5202 T^{6} + 83521 T^{8} \)
$19$ \( ( 1 + 6 T + 17 T^{2} + 114 T^{3} + 361 T^{4} )^{2} \)
$23$ \( 1 + 37 T^{2} + 840 T^{4} + 19573 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 + 9 T + 29 T^{2} )^{4} \)
$31$ \( ( 1 - 11 T + 31 T^{2} )^{2}( 1 + 7 T + 31 T^{2} )^{2} \)
$37$ \( 1 + 58 T^{2} + 1995 T^{4} + 79402 T^{6} + 1874161 T^{8} \)
$41$ \( ( 1 + 7 T + 41 T^{2} )^{4} \)
$43$ \( ( 1 - 61 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( 1 + 30 T^{2} - 1309 T^{4} + 66270 T^{6} + 4879681 T^{8} \)
$53$ \( 1 + 102 T^{2} + 7595 T^{4} + 286518 T^{6} + 7890481 T^{8} \)
$59$ \( ( 1 - 10 T + 41 T^{2} - 590 T^{3} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 - 13 T + 61 T^{2} )^{2}( 1 + 14 T + 61 T^{2} )^{2} \)
$67$ \( 1 + 53 T^{2} - 1680 T^{4} + 237917 T^{6} + 20151121 T^{8} \)
$71$ \( ( 1 - 2 T + 71 T^{2} )^{4} \)
$73$ \( 1 + 130 T^{2} + 11571 T^{4} + 692770 T^{6} + 28398241 T^{8} \)
$79$ \( ( 1 - 10 T + 21 T^{2} - 790 T^{3} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 - 117 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - T - 88 T^{2} - 89 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 + 2 T^{2} + 9409 T^{4} )^{2} \)
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