Properties

Label 350.2.a.h
Level 350
Weight 2
Character orbit 350.a
Self dual yes
Analytic conductor 2.795
Analytic rank 0
Dimension 2
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 350.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(2.79476407074\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(\beta = \sqrt{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + q^{2} + \beta q^{3} + q^{4} + \beta q^{6} + q^{7} + q^{8} + 3 q^{9} +O(q^{10})\) \( q + q^{2} + \beta q^{3} + q^{4} + \beta q^{6} + q^{7} + q^{8} + 3 q^{9} -2 \beta q^{11} + \beta q^{12} + ( -2 - \beta ) q^{13} + q^{14} + q^{16} -2 q^{17} + 3 q^{18} + ( 4 - \beta ) q^{19} + \beta q^{21} -2 \beta q^{22} + ( 2 - 2 \beta ) q^{23} + \beta q^{24} + ( -2 - \beta ) q^{26} + q^{28} + ( 2 + 2 \beta ) q^{29} + ( 4 + 2 \beta ) q^{31} + q^{32} -12 q^{33} -2 q^{34} + 3 q^{36} -2 q^{37} + ( 4 - \beta ) q^{38} + ( -6 - 2 \beta ) q^{39} + ( -6 + 2 \beta ) q^{41} + \beta q^{42} + ( -4 + 2 \beta ) q^{43} -2 \beta q^{44} + ( 2 - 2 \beta ) q^{46} + ( -4 - 2 \beta ) q^{47} + \beta q^{48} + q^{49} -2 \beta q^{51} + ( -2 - \beta ) q^{52} + ( 6 + 2 \beta ) q^{53} + q^{56} + ( -6 + 4 \beta ) q^{57} + ( 2 + 2 \beta ) q^{58} + ( -4 + \beta ) q^{59} + ( 6 - \beta ) q^{61} + ( 4 + 2 \beta ) q^{62} + 3 q^{63} + q^{64} -12 q^{66} + 8 q^{67} -2 q^{68} + ( -12 + 2 \beta ) q^{69} + ( -6 + 2 \beta ) q^{71} + 3 q^{72} + ( 2 - 2 \beta ) q^{73} -2 q^{74} + ( 4 - \beta ) q^{76} -2 \beta q^{77} + ( -6 - 2 \beta ) q^{78} + ( 2 + 2 \beta ) q^{79} -9 q^{81} + ( -6 + 2 \beta ) q^{82} + \beta q^{83} + \beta q^{84} + ( -4 + 2 \beta ) q^{86} + ( 12 + 2 \beta ) q^{87} -2 \beta q^{88} -10 q^{89} + ( -2 - \beta ) q^{91} + ( 2 - 2 \beta ) q^{92} + ( 12 + 4 \beta ) q^{93} + ( -4 - 2 \beta ) q^{94} + \beta q^{96} + ( -6 - 4 \beta ) q^{97} + q^{98} -6 \beta q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{2} + 2q^{4} + 2q^{7} + 2q^{8} + 6q^{9} + O(q^{10}) \) \( 2q + 2q^{2} + 2q^{4} + 2q^{7} + 2q^{8} + 6q^{9} - 4q^{13} + 2q^{14} + 2q^{16} - 4q^{17} + 6q^{18} + 8q^{19} + 4q^{23} - 4q^{26} + 2q^{28} + 4q^{29} + 8q^{31} + 2q^{32} - 24q^{33} - 4q^{34} + 6q^{36} - 4q^{37} + 8q^{38} - 12q^{39} - 12q^{41} - 8q^{43} + 4q^{46} - 8q^{47} + 2q^{49} - 4q^{52} + 12q^{53} + 2q^{56} - 12q^{57} + 4q^{58} - 8q^{59} + 12q^{61} + 8q^{62} + 6q^{63} + 2q^{64} - 24q^{66} + 16q^{67} - 4q^{68} - 24q^{69} - 12q^{71} + 6q^{72} + 4q^{73} - 4q^{74} + 8q^{76} - 12q^{78} + 4q^{79} - 18q^{81} - 12q^{82} - 8q^{86} + 24q^{87} - 20q^{89} - 4q^{91} + 4q^{92} + 24q^{93} - 8q^{94} - 12q^{97} + 2q^{98} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.44949
2.44949
1.00000 −2.44949 1.00000 0 −2.44949 1.00000 1.00000 3.00000 0
1.2 1.00000 2.44949 1.00000 0 2.44949 1.00000 1.00000 3.00000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.2.a.h 2
3.b odd 2 1 3150.2.a.bs 2
4.b odd 2 1 2800.2.a.bl 2
5.b even 2 1 350.2.a.g 2
5.c odd 4 2 70.2.c.a 4
7.b odd 2 1 2450.2.a.bq 2
15.d odd 2 1 3150.2.a.bt 2
15.e even 4 2 630.2.g.g 4
20.d odd 2 1 2800.2.a.bm 2
20.e even 4 2 560.2.g.e 4
35.c odd 2 1 2450.2.a.bl 2
35.f even 4 2 490.2.c.e 4
35.k even 12 4 490.2.i.f 8
35.l odd 12 4 490.2.i.c 8
40.i odd 4 2 2240.2.g.j 4
40.k even 4 2 2240.2.g.i 4
60.l odd 4 2 5040.2.t.t 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.c.a 4 5.c odd 4 2
350.2.a.g 2 5.b even 2 1
350.2.a.h 2 1.a even 1 1 trivial
490.2.c.e 4 35.f even 4 2
490.2.i.c 8 35.l odd 12 4
490.2.i.f 8 35.k even 12 4
560.2.g.e 4 20.e even 4 2
630.2.g.g 4 15.e even 4 2
2240.2.g.i 4 40.k even 4 2
2240.2.g.j 4 40.i odd 4 2
2450.2.a.bl 2 35.c odd 2 1
2450.2.a.bq 2 7.b odd 2 1
2800.2.a.bl 2 4.b odd 2 1
2800.2.a.bm 2 20.d odd 2 1
3150.2.a.bs 2 3.b odd 2 1
3150.2.a.bt 2 15.d odd 2 1
5040.2.t.t 4 60.l odd 4 2

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)
\(7\) \(-1\)

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(350))\):

\( T_{3}^{2} - 6 \)
\( T_{13}^{2} + 4 T_{13} - 2 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T )^{2} \)
$3$ \( 1 + 9 T^{4} \)
$5$ \( \)
$7$ \( ( 1 - T )^{2} \)
$11$ \( 1 - 2 T^{2} + 121 T^{4} \)
$13$ \( 1 + 4 T + 24 T^{2} + 52 T^{3} + 169 T^{4} \)
$17$ \( ( 1 + 2 T + 17 T^{2} )^{2} \)
$19$ \( 1 - 8 T + 48 T^{2} - 152 T^{3} + 361 T^{4} \)
$23$ \( 1 - 4 T + 26 T^{2} - 92 T^{3} + 529 T^{4} \)
$29$ \( 1 - 4 T + 38 T^{2} - 116 T^{3} + 841 T^{4} \)
$31$ \( 1 - 8 T + 54 T^{2} - 248 T^{3} + 961 T^{4} \)
$37$ \( ( 1 + 2 T + 37 T^{2} )^{2} \)
$41$ \( 1 + 12 T + 94 T^{2} + 492 T^{3} + 1681 T^{4} \)
$43$ \( 1 + 8 T + 78 T^{2} + 344 T^{3} + 1849 T^{4} \)
$47$ \( 1 + 8 T + 86 T^{2} + 376 T^{3} + 2209 T^{4} \)
$53$ \( 1 - 12 T + 118 T^{2} - 636 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 8 T + 128 T^{2} + 472 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 12 T + 152 T^{2} - 732 T^{3} + 3721 T^{4} \)
$67$ \( ( 1 - 8 T + 67 T^{2} )^{2} \)
$71$ \( 1 + 12 T + 154 T^{2} + 852 T^{3} + 5041 T^{4} \)
$73$ \( 1 - 4 T + 126 T^{2} - 292 T^{3} + 5329 T^{4} \)
$79$ \( 1 - 4 T + 138 T^{2} - 316 T^{3} + 6241 T^{4} \)
$83$ \( 1 + 160 T^{2} + 6889 T^{4} \)
$89$ \( ( 1 + 10 T + 89 T^{2} )^{2} \)
$97$ \( 1 + 12 T + 134 T^{2} + 1164 T^{3} + 9409 T^{4} \)
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