Properties

Label 350.2.c.b
Level 350
Weight 2
Character orbit 350.c
Analytic conductor 2.795
Analytic rank 0
Dimension 2
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) = \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 350.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

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

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{2} - q^{4} -i q^{7} -i q^{8} + 3 q^{9} +O(q^{10})\) \( q + i q^{2} - q^{4} -i q^{7} -i q^{8} + 3 q^{9} + 4 q^{11} + 6 i q^{13} + q^{14} + q^{16} + 2 i q^{17} + 3 i q^{18} + 4 i q^{22} -6 q^{26} + i q^{28} -6 q^{29} + 8 q^{31} + i q^{32} -2 q^{34} -3 q^{36} -10 i q^{37} + 2 q^{41} -4 i q^{43} -4 q^{44} + 8 i q^{47} - q^{49} -6 i q^{52} + 2 i q^{53} - q^{56} -6 i q^{58} + 8 q^{59} -14 q^{61} + 8 i q^{62} -3 i q^{63} - q^{64} -12 i q^{67} -2 i q^{68} -16 q^{71} -3 i q^{72} -2 i q^{73} + 10 q^{74} -4 i q^{77} + 8 q^{79} + 9 q^{81} + 2 i q^{82} -8 i q^{83} + 4 q^{86} -4 i q^{88} -10 q^{89} + 6 q^{91} -8 q^{94} + 2 i q^{97} -i q^{98} + 12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 6q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 6q^{9} + 8q^{11} + 2q^{14} + 2q^{16} - 12q^{26} - 12q^{29} + 16q^{31} - 4q^{34} - 6q^{36} + 4q^{41} - 8q^{44} - 2q^{49} - 2q^{56} + 16q^{59} - 28q^{61} - 2q^{64} - 32q^{71} + 20q^{74} + 16q^{79} + 18q^{81} + 8q^{86} - 20q^{89} + 12q^{91} - 16q^{94} + 24q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
1.00000i
1.00000i
1.00000i 0 −1.00000 0 0 1.00000i 1.00000i 3.00000 0
99.2 1.00000i 0 −1.00000 0 0 1.00000i 1.00000i 3.00000 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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.2.c.b 2
3.b odd 2 1 3150.2.g.c 2
4.b odd 2 1 2800.2.g.n 2
5.b even 2 1 inner 350.2.c.b 2
5.c odd 4 1 70.2.a.a 1
5.c odd 4 1 350.2.a.b 1
7.b odd 2 1 2450.2.c.k 2
15.d odd 2 1 3150.2.g.c 2
15.e even 4 1 630.2.a.d 1
15.e even 4 1 3150.2.a.bj 1
20.d odd 2 1 2800.2.g.n 2
20.e even 4 1 560.2.a.d 1
20.e even 4 1 2800.2.a.m 1
35.c odd 2 1 2450.2.c.k 2
35.f even 4 1 490.2.a.h 1
35.f even 4 1 2450.2.a.l 1
35.k even 12 2 490.2.e.c 2
35.l odd 12 2 490.2.e.d 2
40.i odd 4 1 2240.2.a.n 1
40.k even 4 1 2240.2.a.q 1
55.e even 4 1 8470.2.a.j 1
60.l odd 4 1 5040.2.a.bm 1
105.k odd 4 1 4410.2.a.b 1
140.j odd 4 1 3920.2.a.t 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.a.a 1 5.c odd 4 1
350.2.a.b 1 5.c odd 4 1
350.2.c.b 2 1.a even 1 1 trivial
350.2.c.b 2 5.b even 2 1 inner
490.2.a.h 1 35.f even 4 1
490.2.e.c 2 35.k even 12 2
490.2.e.d 2 35.l odd 12 2
560.2.a.d 1 20.e even 4 1
630.2.a.d 1 15.e even 4 1
2240.2.a.n 1 40.i odd 4 1
2240.2.a.q 1 40.k even 4 1
2450.2.a.l 1 35.f even 4 1
2450.2.c.k 2 7.b odd 2 1
2450.2.c.k 2 35.c odd 2 1
2800.2.a.m 1 20.e even 4 1
2800.2.g.n 2 4.b odd 2 1
2800.2.g.n 2 20.d odd 2 1
3150.2.a.bj 1 15.e even 4 1
3150.2.g.c 2 3.b odd 2 1
3150.2.g.c 2 15.d odd 2 1
3920.2.a.t 1 140.j odd 4 1
4410.2.a.b 1 105.k odd 4 1
5040.2.a.bm 1 60.l odd 4 1
8470.2.a.j 1 55.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3} \) acting on \(S_{2}^{\mathrm{new}}(350, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( ( 1 - 3 T^{2} )^{2} \)
$5$ 1
$7$ \( 1 + T^{2} \)
$11$ \( ( 1 - 4 T + 11 T^{2} )^{2} \)
$13$ \( ( 1 - 4 T + 13 T^{2} )( 1 + 4 T + 13 T^{2} ) \)
$17$ \( ( 1 - 8 T + 17 T^{2} )( 1 + 8 T + 17 T^{2} ) \)
$19$ \( ( 1 + 19 T^{2} )^{2} \)
$23$ \( ( 1 - 23 T^{2} )^{2} \)
$29$ \( ( 1 + 6 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 - 8 T + 31 T^{2} )^{2} \)
$37$ \( 1 + 26 T^{2} + 1369 T^{4} \)
$41$ \( ( 1 - 2 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 70 T^{2} + 1849 T^{4} \)
$47$ \( 1 - 30 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 102 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 - 8 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 + 14 T + 61 T^{2} )^{2} \)
$67$ \( 1 + 10 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 + 16 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 142 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 - 8 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 102 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 + 10 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 190 T^{2} + 9409 T^{4} \)
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