Properties

Label 350.2.j.a
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: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
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.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + ( -2 \zeta_{12} + 2 \zeta_{12}^{3} ) q^{3} + \zeta_{12}^{2} q^{4} -2 q^{6} + ( 2 \zeta_{12} + \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{3} q^{8} + ( 1 - \zeta_{12}^{2} ) q^{9} +O(q^{10})\) \( 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} + \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} + 5 \zeta_{12}^{3} q^{13} + ( -1 + 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 - 2 \zeta_{12}^{2} ) q^{21} -3 \zeta_{12}^{3} q^{22} -3 \zeta_{12} q^{23} -2 \zeta_{12}^{2} q^{24} + ( -5 + 5 \zeta_{12}^{2} ) q^{26} -4 \zeta_{12}^{3} q^{27} + ( -\zeta_{12} + 3 \zeta_{12}^{3} ) q^{28} + 6 q^{29} + 4 \zeta_{12}^{2} q^{31} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{32} + 6 \zeta_{12} q^{33} -6 q^{34} + q^{36} + 11 \zeta_{12} q^{37} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{38} -10 \zeta_{12}^{2} q^{39} + 3 q^{41} + ( -4 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{42} -10 \zeta_{12}^{3} q^{43} + ( 3 - 3 \zeta_{12}^{2} ) q^{44} -3 \zeta_{12}^{2} q^{46} + 3 \zeta_{12} q^{47} -2 \zeta_{12}^{3} q^{48} + ( -5 + 8 \zeta_{12}^{2} ) q^{49} + ( 12 - 12 \zeta_{12}^{2} ) q^{51} + ( -5 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{52} + ( 3 \zeta_{12} - 3 \zeta_{12}^{3} ) q^{53} + ( 4 - 4 \zeta_{12}^{2} ) q^{54} + ( -3 + 2 \zeta_{12}^{2} ) q^{56} -2 \zeta_{12}^{3} q^{57} + 6 \zeta_{12} q^{58} + ( 4 - 4 \zeta_{12}^{2} ) q^{61} + 4 \zeta_{12}^{3} q^{62} + ( 3 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{63} - q^{64} + 6 \zeta_{12}^{2} q^{66} + ( 4 \zeta_{12} - 4 \zeta_{12}^{3} ) q^{67} -6 \zeta_{12} q^{68} + 6 q^{69} + 12 q^{71} + \zeta_{12} q^{72} + ( -4 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{73} + 11 \zeta_{12}^{2} q^{74} - q^{76} + ( 3 \zeta_{12} - 9 \zeta_{12}^{3} ) q^{77} -10 \zeta_{12}^{3} q^{78} + ( -10 + 10 \zeta_{12}^{2} ) q^{79} + 11 \zeta_{12}^{2} q^{81} + 3 \zeta_{12} q^{82} -12 \zeta_{12}^{3} q^{83} + ( 2 - 6 \zeta_{12}^{2} ) q^{84} + ( 10 - 10 \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} + ( -15 + 10 \zeta_{12}^{2} ) q^{91} -3 \zeta_{12}^{3} q^{92} -8 \zeta_{12} q^{93} + 3 \zeta_{12}^{2} q^{94} + ( 2 - 2 \zeta_{12}^{2} ) q^{96} -14 \zeta_{12}^{3} q^{97} + ( -5 \zeta_{12} + 8 \zeta_{12}^{3} ) q^{98} -3 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} - 8q^{6} + 2q^{9} + O(q^{10}) \) \( 4q + 2q^{4} - 8q^{6} + 2q^{9} - 6q^{11} + 2q^{14} - 2q^{16} - 2q^{19} - 20q^{21} - 4q^{24} - 10q^{26} + 24q^{29} + 8q^{31} - 24q^{34} + 4q^{36} - 20q^{39} + 12q^{41} + 6q^{44} - 6q^{46} - 4q^{49} + 24q^{51} + 8q^{54} - 8q^{56} + 8q^{61} - 4q^{64} + 12q^{66} + 24q^{69} + 48q^{71} + 22q^{74} - 4q^{76} - 20q^{79} + 22q^{81} - 4q^{84} + 20q^{86} + 12q^{89} - 40q^{91} + 6q^{94} + 4q^{96} - 12q^{99} + O(q^{100}) \)

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.a 4
5.b even 2 1 inner 350.2.j.a 4
5.c odd 4 1 70.2.e.b 2
5.c odd 4 1 350.2.e.h 2
7.c even 3 1 inner 350.2.j.a 4
7.c even 3 1 2450.2.c.f 2
7.d odd 6 1 2450.2.c.p 2
15.e even 4 1 630.2.k.e 2
20.e even 4 1 560.2.q.d 2
35.f even 4 1 490.2.e.a 2
35.i odd 6 1 2450.2.c.p 2
35.j even 6 1 inner 350.2.j.a 4
35.j even 6 1 2450.2.c.f 2
35.k even 12 1 490.2.a.j 1
35.k even 12 1 490.2.e.a 2
35.k even 12 1 2450.2.a.f 1
35.l odd 12 1 70.2.e.b 2
35.l odd 12 1 350.2.e.h 2
35.l odd 12 1 490.2.a.g 1
35.l odd 12 1 2450.2.a.p 1
105.w odd 12 1 4410.2.a.c 1
105.x even 12 1 630.2.k.e 2
105.x even 12 1 4410.2.a.m 1
140.w even 12 1 560.2.q.d 2
140.w even 12 1 3920.2.a.be 1
140.x odd 12 1 3920.2.a.g 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.b 2 5.c odd 4 1
70.2.e.b 2 35.l odd 12 1
350.2.e.h 2 5.c odd 4 1
350.2.e.h 2 35.l odd 12 1
350.2.j.a 4 1.a even 1 1 trivial
350.2.j.a 4 5.b even 2 1 inner
350.2.j.a 4 7.c even 3 1 inner
350.2.j.a 4 35.j even 6 1 inner
490.2.a.g 1 35.l odd 12 1
490.2.a.j 1 35.k even 12 1
490.2.e.a 2 35.f even 4 1
490.2.e.a 2 35.k even 12 1
560.2.q.d 2 20.e even 4 1
560.2.q.d 2 140.w even 12 1
630.2.k.e 2 15.e even 4 1
630.2.k.e 2 105.x even 12 1
2450.2.a.f 1 35.k even 12 1
2450.2.a.p 1 35.l odd 12 1
2450.2.c.f 2 7.c even 3 1
2450.2.c.f 2 35.j even 6 1
2450.2.c.p 2 7.d odd 6 1
2450.2.c.p 2 35.i odd 6 1
3920.2.a.g 1 140.x odd 12 1
3920.2.a.be 1 140.w even 12 1
4410.2.a.c 1 105.w odd 12 1
4410.2.a.m 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 \(S_{2}^{\mathrm{new}}(350, [\chi])\):

\( T_{3}^{4} - 4 T_{3}^{2} + 16 \)
\( T_{11}^{2} + 3 T_{11} + 9 \)
\( T_{13}^{2} + 25 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( 16 - 4 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( 49 + 2 T^{2} + T^{4} \)
$11$ \( ( 9 + 3 T + T^{2} )^{2} \)
$13$ \( ( 25 + T^{2} )^{2} \)
$17$ \( 1296 - 36 T^{2} + T^{4} \)
$19$ \( ( 1 + T + T^{2} )^{2} \)
$23$ \( 81 - 9 T^{2} + T^{4} \)
$29$ \( ( -6 + T )^{4} \)
$31$ \( ( 16 - 4 T + T^{2} )^{2} \)
$37$ \( 14641 - 121 T^{2} + T^{4} \)
$41$ \( ( -3 + T )^{4} \)
$43$ \( ( 100 + T^{2} )^{2} \)
$47$ \( 81 - 9 T^{2} + T^{4} \)
$53$ \( 81 - 9 T^{2} + T^{4} \)
$59$ \( T^{4} \)
$61$ \( ( 16 - 4 T + T^{2} )^{2} \)
$67$ \( 256 - 16 T^{2} + T^{4} \)
$71$ \( ( -12 + T )^{4} \)
$73$ \( 256 - 16 T^{2} + T^{4} \)
$79$ \( ( 100 + 10 T + T^{2} )^{2} \)
$83$ \( ( 144 + T^{2} )^{2} \)
$89$ \( ( 36 - 6 T + T^{2} )^{2} \)
$97$ \( ( 196 + T^{2} )^{2} \)
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