Properties

Label 350.2.c.d
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 14)
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.

\(f(q)\) \(=\) \( q + i q^{2} -2 i q^{3} - q^{4} + 2 q^{6} -i q^{7} -i q^{8} - q^{9} +O(q^{10})\) \( q + i q^{2} -2 i q^{3} - q^{4} + 2 q^{6} -i q^{7} -i q^{8} - q^{9} + 2 i q^{12} -4 i q^{13} + q^{14} + q^{16} -6 i q^{17} -i q^{18} -2 q^{19} -2 q^{21} -2 q^{24} + 4 q^{26} -4 i q^{27} + i q^{28} + 6 q^{29} -4 q^{31} + i q^{32} + 6 q^{34} + q^{36} -2 i q^{37} -2 i q^{38} -8 q^{39} + 6 q^{41} -2 i q^{42} + 8 i q^{43} + 12 i q^{47} -2 i q^{48} - q^{49} -12 q^{51} + 4 i q^{52} + 6 i q^{53} + 4 q^{54} - q^{56} + 4 i q^{57} + 6 i q^{58} + 6 q^{59} + 8 q^{61} -4 i q^{62} + i q^{63} - q^{64} + 4 i q^{67} + 6 i q^{68} + i q^{72} + 2 i q^{73} + 2 q^{74} + 2 q^{76} -8 i q^{78} -8 q^{79} -11 q^{81} + 6 i q^{82} -6 i q^{83} + 2 q^{84} -8 q^{86} -12 i q^{87} + 6 q^{89} -4 q^{91} + 8 i q^{93} -12 q^{94} + 2 q^{96} + 10 i q^{97} -i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{4} + 4q^{6} - 2q^{9} + O(q^{10}) \) \( 2q - 2q^{4} + 4q^{6} - 2q^{9} + 2q^{14} + 2q^{16} - 4q^{19} - 4q^{21} - 4q^{24} + 8q^{26} + 12q^{29} - 8q^{31} + 12q^{34} + 2q^{36} - 16q^{39} + 12q^{41} - 2q^{49} - 24q^{51} + 8q^{54} - 2q^{56} + 12q^{59} + 16q^{61} - 2q^{64} + 4q^{74} + 4q^{76} - 16q^{79} - 22q^{81} + 4q^{84} - 16q^{86} + 12q^{89} - 8q^{91} - 24q^{94} + 4q^{96} + O(q^{100}) \)

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 2.00000i −1.00000 0 2.00000 1.00000i 1.00000i −1.00000 0
99.2 1.00000i 2.00000i −1.00000 0 2.00000 1.00000i 1.00000i −1.00000 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.d 2
3.b odd 2 1 3150.2.g.j 2
4.b odd 2 1 2800.2.g.h 2
5.b even 2 1 inner 350.2.c.d 2
5.c odd 4 1 14.2.a.a 1
5.c odd 4 1 350.2.a.f 1
7.b odd 2 1 2450.2.c.c 2
15.d odd 2 1 3150.2.g.j 2
15.e even 4 1 126.2.a.b 1
15.e even 4 1 3150.2.a.i 1
20.d odd 2 1 2800.2.g.h 2
20.e even 4 1 112.2.a.c 1
20.e even 4 1 2800.2.a.g 1
35.c odd 2 1 2450.2.c.c 2
35.f even 4 1 98.2.a.a 1
35.f even 4 1 2450.2.a.t 1
35.k even 12 2 98.2.c.a 2
35.l odd 12 2 98.2.c.b 2
40.i odd 4 1 448.2.a.g 1
40.k even 4 1 448.2.a.a 1
45.k odd 12 2 1134.2.f.l 2
45.l even 12 2 1134.2.f.f 2
55.e even 4 1 1694.2.a.e 1
60.l odd 4 1 1008.2.a.h 1
65.f even 4 1 2366.2.d.b 2
65.h odd 4 1 2366.2.a.j 1
65.k even 4 1 2366.2.d.b 2
80.i odd 4 1 1792.2.b.c 2
80.j even 4 1 1792.2.b.g 2
80.s even 4 1 1792.2.b.g 2
80.t odd 4 1 1792.2.b.c 2
85.g odd 4 1 4046.2.a.f 1
95.g even 4 1 5054.2.a.c 1
105.k odd 4 1 882.2.a.i 1
105.w odd 12 2 882.2.g.d 2
105.x even 12 2 882.2.g.c 2
115.e even 4 1 7406.2.a.a 1
120.q odd 4 1 4032.2.a.r 1
120.w even 4 1 4032.2.a.w 1
140.j odd 4 1 784.2.a.b 1
140.w even 12 2 784.2.i.c 2
140.x odd 12 2 784.2.i.i 2
280.s even 4 1 3136.2.a.e 1
280.y odd 4 1 3136.2.a.z 1
420.w even 4 1 7056.2.a.bd 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
14.2.a.a 1 5.c odd 4 1
98.2.a.a 1 35.f even 4 1
98.2.c.a 2 35.k even 12 2
98.2.c.b 2 35.l odd 12 2
112.2.a.c 1 20.e even 4 1
126.2.a.b 1 15.e even 4 1
350.2.a.f 1 5.c odd 4 1
350.2.c.d 2 1.a even 1 1 trivial
350.2.c.d 2 5.b even 2 1 inner
448.2.a.a 1 40.k even 4 1
448.2.a.g 1 40.i odd 4 1
784.2.a.b 1 140.j odd 4 1
784.2.i.c 2 140.w even 12 2
784.2.i.i 2 140.x odd 12 2
882.2.a.i 1 105.k odd 4 1
882.2.g.c 2 105.x even 12 2
882.2.g.d 2 105.w odd 12 2
1008.2.a.h 1 60.l odd 4 1
1134.2.f.f 2 45.l even 12 2
1134.2.f.l 2 45.k odd 12 2
1694.2.a.e 1 55.e even 4 1
1792.2.b.c 2 80.i odd 4 1
1792.2.b.c 2 80.t odd 4 1
1792.2.b.g 2 80.j even 4 1
1792.2.b.g 2 80.s even 4 1
2366.2.a.j 1 65.h odd 4 1
2366.2.d.b 2 65.f even 4 1
2366.2.d.b 2 65.k even 4 1
2450.2.a.t 1 35.f even 4 1
2450.2.c.c 2 7.b odd 2 1
2450.2.c.c 2 35.c odd 2 1
2800.2.a.g 1 20.e even 4 1
2800.2.g.h 2 4.b odd 2 1
2800.2.g.h 2 20.d odd 2 1
3136.2.a.e 1 280.s even 4 1
3136.2.a.z 1 280.y odd 4 1
3150.2.a.i 1 15.e even 4 1
3150.2.g.j 2 3.b odd 2 1
3150.2.g.j 2 15.d odd 2 1
4032.2.a.r 1 120.q odd 4 1
4032.2.a.w 1 120.w even 4 1
4046.2.a.f 1 85.g odd 4 1
5054.2.a.c 1 95.g even 4 1
7056.2.a.bd 1 420.w even 4 1
7406.2.a.a 1 115.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(350, [\chi])\).

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{2} \)
$3$ \( 1 - 2 T^{2} + 9 T^{4} \)
$5$ \( \)
$7$ \( 1 + T^{2} \)
$11$ \( ( 1 + 11 T^{2} )^{2} \)
$13$ \( ( 1 - 6 T + 13 T^{2} )( 1 + 6 T + 13 T^{2} ) \)
$17$ \( 1 + 2 T^{2} + 289 T^{4} \)
$19$ \( ( 1 + 2 T + 19 T^{2} )^{2} \)
$23$ \( ( 1 - 23 T^{2} )^{2} \)
$29$ \( ( 1 - 6 T + 29 T^{2} )^{2} \)
$31$ \( ( 1 + 4 T + 31 T^{2} )^{2} \)
$37$ \( ( 1 - 12 T + 37 T^{2} )( 1 + 12 T + 37 T^{2} ) \)
$41$ \( ( 1 - 6 T + 41 T^{2} )^{2} \)
$43$ \( 1 - 22 T^{2} + 1849 T^{4} \)
$47$ \( 1 + 50 T^{2} + 2209 T^{4} \)
$53$ \( 1 - 70 T^{2} + 2809 T^{4} \)
$59$ \( ( 1 - 6 T + 59 T^{2} )^{2} \)
$61$ \( ( 1 - 8 T + 61 T^{2} )^{2} \)
$67$ \( 1 - 118 T^{2} + 4489 T^{4} \)
$71$ \( ( 1 + 71 T^{2} )^{2} \)
$73$ \( 1 - 142 T^{2} + 5329 T^{4} \)
$79$ \( ( 1 + 8 T + 79 T^{2} )^{2} \)
$83$ \( 1 - 130 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 - 6 T + 89 T^{2} )^{2} \)
$97$ \( 1 - 94 T^{2} + 9409 T^{4} \)
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