Properties

Label 350.2.e.k
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: yes
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.

\(f(q)\) \(=\) \( q + \zeta_{6} q^{2} + ( 2 - 2 \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} + 2 q^{6} + ( 2 + \zeta_{6} ) q^{7} - q^{8} -\zeta_{6} q^{9} +O(q^{10})\) \( q + \zeta_{6} q^{2} + ( 2 - 2 \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} + 2 q^{6} + ( 2 + \zeta_{6} ) q^{7} - q^{8} -\zeta_{6} q^{9} + 2 \zeta_{6} q^{12} + 2 q^{13} + ( -1 + 3 \zeta_{6} ) q^{14} -\zeta_{6} q^{16} + ( 3 - 3 \zeta_{6} ) q^{17} + ( 1 - \zeta_{6} ) q^{18} -8 \zeta_{6} q^{19} + ( 6 - 4 \zeta_{6} ) q^{21} + 9 \zeta_{6} q^{23} + ( -2 + 2 \zeta_{6} ) q^{24} + 2 \zeta_{6} q^{26} + 4 q^{27} + ( -3 + 2 \zeta_{6} ) q^{28} -6 q^{29} + ( -5 + 5 \zeta_{6} ) q^{31} + ( 1 - \zeta_{6} ) q^{32} + 3 q^{34} + q^{36} -8 \zeta_{6} q^{37} + ( 8 - 8 \zeta_{6} ) q^{38} + ( 4 - 4 \zeta_{6} ) q^{39} -3 q^{41} + ( 4 + 2 \zeta_{6} ) q^{42} -10 q^{43} + ( -9 + 9 \zeta_{6} ) q^{46} + 3 \zeta_{6} q^{47} -2 q^{48} + ( 3 + 5 \zeta_{6} ) q^{49} -6 \zeta_{6} q^{51} + ( -2 + 2 \zeta_{6} ) q^{52} + ( -6 + 6 \zeta_{6} ) q^{53} + 4 \zeta_{6} q^{54} + ( -2 - \zeta_{6} ) q^{56} -16 q^{57} -6 \zeta_{6} q^{58} + ( -12 + 12 \zeta_{6} ) q^{59} + 4 \zeta_{6} q^{61} -5 q^{62} + ( 1 - 3 \zeta_{6} ) q^{63} + q^{64} + ( -2 + 2 \zeta_{6} ) q^{67} + 3 \zeta_{6} q^{68} + 18 q^{69} -9 q^{71} + \zeta_{6} q^{72} + ( 10 - 10 \zeta_{6} ) q^{73} + ( 8 - 8 \zeta_{6} ) q^{74} + 8 q^{76} + 4 q^{78} -5 \zeta_{6} q^{79} + ( 11 - 11 \zeta_{6} ) q^{81} -3 \zeta_{6} q^{82} -6 q^{83} + ( -2 + 6 \zeta_{6} ) q^{84} -10 \zeta_{6} q^{86} + ( -12 + 12 \zeta_{6} ) q^{87} -3 \zeta_{6} q^{89} + ( 4 + 2 \zeta_{6} ) q^{91} -9 q^{92} + 10 \zeta_{6} q^{93} + ( -3 + 3 \zeta_{6} ) q^{94} -2 \zeta_{6} q^{96} + 5 q^{97} + ( -5 + 8 \zeta_{6} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + q^{2} + 2q^{3} - q^{4} + 4q^{6} + 5q^{7} - 2q^{8} - q^{9} + O(q^{10}) \) \( 2q + q^{2} + 2q^{3} - q^{4} + 4q^{6} + 5q^{7} - 2q^{8} - q^{9} + 2q^{12} + 4q^{13} + q^{14} - q^{16} + 3q^{17} + q^{18} - 8q^{19} + 8q^{21} + 9q^{23} - 2q^{24} + 2q^{26} + 8q^{27} - 4q^{28} - 12q^{29} - 5q^{31} + q^{32} + 6q^{34} + 2q^{36} - 8q^{37} + 8q^{38} + 4q^{39} - 6q^{41} + 10q^{42} - 20q^{43} - 9q^{46} + 3q^{47} - 4q^{48} + 11q^{49} - 6q^{51} - 2q^{52} - 6q^{53} + 4q^{54} - 5q^{56} - 32q^{57} - 6q^{58} - 12q^{59} + 4q^{61} - 10q^{62} - q^{63} + 2q^{64} - 2q^{67} + 3q^{68} + 36q^{69} - 18q^{71} + q^{72} + 10q^{73} + 8q^{74} + 16q^{76} + 8q^{78} - 5q^{79} + 11q^{81} - 3q^{82} - 12q^{83} + 2q^{84} - 10q^{86} - 12q^{87} - 3q^{89} + 10q^{91} - 18q^{92} + 10q^{93} - 3q^{94} - 2q^{96} + 10q^{97} - 2q^{98} + O(q^{100}) \)

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 1.00000 + 1.73205i −0.500000 0.866025i 0 2.00000 2.50000 0.866025i −1.00000 −0.500000 + 0.866025i 0
151.1 0.500000 + 0.866025i 1.00000 1.73205i −0.500000 + 0.866025i 0 2.00000 2.50000 + 0.866025i −1.00000 −0.500000 0.866025i 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.k yes 2
5.b even 2 1 350.2.e.a 2
5.c odd 4 2 350.2.j.e 4
7.c even 3 1 inner 350.2.e.k yes 2
7.c even 3 1 2450.2.a.e 1
7.d odd 6 1 2450.2.a.o 1
35.i odd 6 1 2450.2.a.u 1
35.j even 6 1 350.2.e.a 2
35.j even 6 1 2450.2.a.be 1
35.k even 12 2 2450.2.c.d 2
35.l odd 12 2 350.2.j.e 4
35.l odd 12 2 2450.2.c.o 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.e.a 2 5.b even 2 1
350.2.e.a 2 35.j even 6 1
350.2.e.k yes 2 1.a even 1 1 trivial
350.2.e.k yes 2 7.c even 3 1 inner
350.2.j.e 4 5.c odd 4 2
350.2.j.e 4 35.l odd 12 2
2450.2.a.e 1 7.c even 3 1
2450.2.a.o 1 7.d odd 6 1
2450.2.a.u 1 35.i odd 6 1
2450.2.a.be 1 35.j even 6 1
2450.2.c.d 2 35.k even 12 2
2450.2.c.o 2 35.l odd 12 2

Hecke kernels

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

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

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T + T^{2} \)
$3$ \( 1 - 2 T + T^{2} - 6 T^{3} + 9 T^{4} \)
$5$ \( \)
$7$ \( 1 - 5 T + 7 T^{2} \)
$11$ \( 1 - 11 T^{2} + 121 T^{4} \)
$13$ \( ( 1 - 2 T + 13 T^{2} )^{2} \)
$17$ \( 1 - 3 T - 8 T^{2} - 51 T^{3} + 289 T^{4} \)
$19$ \( ( 1 + T + 19 T^{2} )( 1 + 7 T + 19 T^{2} ) \)
$23$ \( 1 - 9 T + 58 T^{2} - 207 T^{3} + 529 T^{4} \)
$29$ \( ( 1 + 6 T + 29 T^{2} )^{2} \)
$31$ \( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} \)
$37$ \( 1 + 8 T + 27 T^{2} + 296 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 + 3 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 + 10 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 3 T - 38 T^{2} - 141 T^{3} + 2209 T^{4} \)
$53$ \( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \)
$59$ \( 1 + 12 T + 85 T^{2} + 708 T^{3} + 3481 T^{4} \)
$61$ \( 1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4} \)
$67$ \( 1 + 2 T - 63 T^{2} + 134 T^{3} + 4489 T^{4} \)
$71$ \( ( 1 + 9 T + 71 T^{2} )^{2} \)
$73$ \( ( 1 - 17 T + 73 T^{2} )( 1 + 7 T + 73 T^{2} ) \)
$79$ \( 1 + 5 T - 54 T^{2} + 395 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 + 6 T + 83 T^{2} )^{2} \)
$89$ \( 1 + 3 T - 80 T^{2} + 267 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 - 5 T + 97 T^{2} )^{2} \)
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