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 $2$

Related objects

Downloads

Learn more

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: \(\Q(\sqrt{-1}) \)
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
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.

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

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 \(S_{2}^{\mathrm{new}}(350, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 1 \) Copy content Toggle raw display
$11$ \( (T - 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 36 \) Copy content Toggle raw display
$17$ \( T^{2} + 4 \) Copy content Toggle raw display
$19$ \( T^{2} \) Copy content Toggle raw display
$23$ \( T^{2} \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( (T - 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 100 \) Copy content Toggle raw display
$41$ \( (T - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( T^{2} + 64 \) Copy content Toggle raw display
$53$ \( T^{2} + 4 \) Copy content Toggle raw display
$59$ \( (T - 8)^{2} \) Copy content Toggle raw display
$61$ \( (T + 14)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 144 \) Copy content Toggle raw display
$71$ \( (T + 16)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 4 \) Copy content Toggle raw display
$79$ \( (T - 8)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 64 \) Copy content Toggle raw display
$89$ \( (T + 10)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 4 \) Copy content Toggle raw display
show more
show less