Properties

Label 350.2.e.c
Level 350
Weight 2
Character orbit 350.e
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.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.79476407074\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 70)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} + ( -3 + 2 \zeta_{6} ) q^{7} + q^{8} + 3 \zeta_{6} q^{9} +O(q^{10})\) \( q -\zeta_{6} q^{2} + ( -1 + \zeta_{6} ) q^{4} + ( -3 + 2 \zeta_{6} ) q^{7} + q^{8} + 3 \zeta_{6} q^{9} + ( -3 + 3 \zeta_{6} ) q^{11} -5 q^{13} + ( 2 + \zeta_{6} ) q^{14} -\zeta_{6} q^{16} + ( 2 - 2 \zeta_{6} ) q^{17} + ( 3 - 3 \zeta_{6} ) q^{18} + 5 \zeta_{6} q^{19} + 3 q^{22} + 7 \zeta_{6} q^{23} + 5 \zeta_{6} q^{26} + ( 1 - 3 \zeta_{6} ) q^{28} -4 q^{29} + ( 2 - 2 \zeta_{6} ) q^{31} + ( -1 + \zeta_{6} ) q^{32} -2 q^{34} -3 q^{36} -\zeta_{6} q^{37} + ( 5 - 5 \zeta_{6} ) q^{38} + 3 q^{41} + 2 q^{43} -3 \zeta_{6} q^{44} + ( 7 - 7 \zeta_{6} ) q^{46} + 7 \zeta_{6} q^{47} + ( 5 - 8 \zeta_{6} ) q^{49} + ( 5 - 5 \zeta_{6} ) q^{52} + ( -9 + 9 \zeta_{6} ) q^{53} + ( -3 + 2 \zeta_{6} ) q^{56} + 4 \zeta_{6} q^{58} + ( 4 - 4 \zeta_{6} ) q^{59} -6 \zeta_{6} q^{61} -2 q^{62} + ( -6 - 3 \zeta_{6} ) q^{63} + q^{64} + ( -2 + 2 \zeta_{6} ) q^{67} + 2 \zeta_{6} q^{68} -6 q^{71} + 3 \zeta_{6} q^{72} + ( 16 - 16 \zeta_{6} ) q^{73} + ( -1 + \zeta_{6} ) q^{74} -5 q^{76} + ( 3 - 9 \zeta_{6} ) q^{77} -14 \zeta_{6} q^{79} + ( -9 + 9 \zeta_{6} ) q^{81} -3 \zeta_{6} q^{82} -6 q^{83} -2 \zeta_{6} q^{86} + ( -3 + 3 \zeta_{6} ) q^{88} -2 \zeta_{6} q^{89} + ( 15 - 10 \zeta_{6} ) q^{91} -7 q^{92} + ( 7 - 7 \zeta_{6} ) q^{94} -12 q^{97} + ( -8 + 3 \zeta_{6} ) q^{98} -9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} - q^{4} - 4q^{7} + 2q^{8} + 3q^{9} + O(q^{10}) \) \( 2q - q^{2} - q^{4} - 4q^{7} + 2q^{8} + 3q^{9} - 3q^{11} - 10q^{13} + 5q^{14} - q^{16} + 2q^{17} + 3q^{18} + 5q^{19} + 6q^{22} + 7q^{23} + 5q^{26} - q^{28} - 8q^{29} + 2q^{31} - q^{32} - 4q^{34} - 6q^{36} - q^{37} + 5q^{38} + 6q^{41} + 4q^{43} - 3q^{44} + 7q^{46} + 7q^{47} + 2q^{49} + 5q^{52} - 9q^{53} - 4q^{56} + 4q^{58} + 4q^{59} - 6q^{61} - 4q^{62} - 15q^{63} + 2q^{64} - 2q^{67} + 2q^{68} - 12q^{71} + 3q^{72} + 16q^{73} - q^{74} - 10q^{76} - 3q^{77} - 14q^{79} - 9q^{81} - 3q^{82} - 12q^{83} - 2q^{86} - 3q^{88} - 2q^{89} + 20q^{91} - 14q^{92} + 7q^{94} - 24q^{97} - 13q^{98} - 18q^{99} + O(q^{100}) \)

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1
0.500000 0.866025i
0.500000 + 0.866025i
−0.500000 + 0.866025i 0 −0.500000 0.866025i 0 0 −2.00000 1.73205i 1.00000 1.50000 2.59808i 0
151.1 −0.500000 0.866025i 0 −0.500000 + 0.866025i 0 0 −2.00000 + 1.73205i 1.00000 1.50000 + 2.59808i 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
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.2.e.c 2
5.b even 2 1 350.2.e.j 2
5.c odd 4 2 70.2.i.b 4
7.c even 3 1 inner 350.2.e.c 2
7.c even 3 1 2450.2.a.ba 1
7.d odd 6 1 2450.2.a.bb 1
15.e even 4 2 630.2.u.a 4
20.e even 4 2 560.2.bw.d 4
35.f even 4 2 490.2.i.a 4
35.i odd 6 1 2450.2.a.j 1
35.j even 6 1 350.2.e.j 2
35.j even 6 1 2450.2.a.k 1
35.k even 12 2 490.2.c.d 2
35.k even 12 2 490.2.i.a 4
35.l odd 12 2 70.2.i.b 4
35.l odd 12 2 490.2.c.a 2
105.x even 12 2 630.2.u.a 4
140.w even 12 2 560.2.bw.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.i.b 4 5.c odd 4 2
70.2.i.b 4 35.l odd 12 2
350.2.e.c 2 1.a even 1 1 trivial
350.2.e.c 2 7.c even 3 1 inner
350.2.e.j 2 5.b even 2 1
350.2.e.j 2 35.j even 6 1
490.2.c.a 2 35.l odd 12 2
490.2.c.d 2 35.k even 12 2
490.2.i.a 4 35.f even 4 2
490.2.i.a 4 35.k even 12 2
560.2.bw.d 4 20.e even 4 2
560.2.bw.d 4 140.w even 12 2
630.2.u.a 4 15.e even 4 2
630.2.u.a 4 105.x even 12 2
2450.2.a.j 1 35.i odd 6 1
2450.2.a.k 1 35.j even 6 1
2450.2.a.ba 1 7.c even 3 1
2450.2.a.bb 1 7.d odd 6 1

Hecke kernels

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

\( T_{3} \)
\( T_{11}^{2} + 3 T_{11} + 9 \)
\( T_{13} + 5 \)
\( T_{17}^{2} - 2 T_{17} + 4 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( ( 1 - 3 T + 3 T^{2} )( 1 + 3 T + 3 T^{2} ) \)
$5$ \( \)
$7$ \( 1 + 4 T + 7 T^{2} \)
$11$ \( 1 + 3 T - 2 T^{2} + 33 T^{3} + 121 T^{4} \)
$13$ \( ( 1 + 5 T + 13 T^{2} )^{2} \)
$17$ \( 1 - 2 T - 13 T^{2} - 34 T^{3} + 289 T^{4} \)
$19$ \( 1 - 5 T + 6 T^{2} - 95 T^{3} + 361 T^{4} \)
$23$ \( 1 - 7 T + 26 T^{2} - 161 T^{3} + 529 T^{4} \)
$29$ \( ( 1 + 4 T + 29 T^{2} )^{2} \)
$31$ \( 1 - 2 T - 27 T^{2} - 62 T^{3} + 961 T^{4} \)
$37$ \( ( 1 - 10 T + 37 T^{2} )( 1 + 11 T + 37 T^{2} ) \)
$41$ \( ( 1 - 3 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 - 2 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 7 T + 2 T^{2} - 329 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 9 T + 28 T^{2} + 477 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 4 T - 43 T^{2} - 236 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 6 T - 25 T^{2} + 366 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 6 T + 71 T^{2} )^{2} \)
$73$ \( 1 - 16 T + 183 T^{2} - 1168 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 14 T + 117 T^{2} + 1106 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 + 6 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 2 T - 85 T^{2} + 178 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 + 12 T + 97 T^{2} )^{2} \)
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