Properties

Label 350.2.e.e
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.

\(f(q)\) \(=\) \( q -\zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} - q^{6} + ( -1 + 3 \zeta_{6} ) q^{7} + q^{8} + 2 \zeta_{6} q^{9} +O(q^{10})\) \( q -\zeta_{6} q^{2} + ( 1 - \zeta_{6} ) q^{3} + ( -1 + \zeta_{6} ) q^{4} - q^{6} + ( -1 + 3 \zeta_{6} ) q^{7} + q^{8} + 2 \zeta_{6} q^{9} + ( 6 - 6 \zeta_{6} ) q^{11} + \zeta_{6} q^{12} + 4 q^{13} + ( 3 - 2 \zeta_{6} ) q^{14} -\zeta_{6} q^{16} + ( 2 - 2 \zeta_{6} ) q^{18} -2 \zeta_{6} q^{19} + ( 2 + \zeta_{6} ) q^{21} -6 q^{22} -3 \zeta_{6} q^{23} + ( 1 - \zeta_{6} ) q^{24} -4 \zeta_{6} q^{26} + 5 q^{27} + ( -2 - \zeta_{6} ) q^{28} -3 q^{29} + ( -8 + 8 \zeta_{6} ) q^{31} + ( -1 + \zeta_{6} ) q^{32} -6 \zeta_{6} q^{33} -2 q^{36} -4 \zeta_{6} q^{37} + ( -2 + 2 \zeta_{6} ) q^{38} + ( 4 - 4 \zeta_{6} ) q^{39} + 9 q^{41} + ( 1 - 3 \zeta_{6} ) q^{42} + 7 q^{43} + 6 \zeta_{6} q^{44} + ( -3 + 3 \zeta_{6} ) q^{46} - q^{48} + ( -8 + 3 \zeta_{6} ) q^{49} + ( -4 + 4 \zeta_{6} ) q^{52} + ( -6 + 6 \zeta_{6} ) q^{53} -5 \zeta_{6} q^{54} + ( -1 + 3 \zeta_{6} ) q^{56} -2 q^{57} + 3 \zeta_{6} q^{58} + ( 6 - 6 \zeta_{6} ) q^{59} -5 \zeta_{6} q^{61} + 8 q^{62} + ( -6 + 4 \zeta_{6} ) q^{63} + q^{64} + ( -6 + 6 \zeta_{6} ) q^{66} + ( 5 - 5 \zeta_{6} ) q^{67} -3 q^{69} -6 q^{71} + 2 \zeta_{6} q^{72} + ( -16 + 16 \zeta_{6} ) q^{73} + ( -4 + 4 \zeta_{6} ) q^{74} + 2 q^{76} + ( 12 + 6 \zeta_{6} ) q^{77} -4 q^{78} -2 \zeta_{6} q^{79} + ( -1 + \zeta_{6} ) q^{81} -9 \zeta_{6} q^{82} -3 q^{83} + ( -3 + 2 \zeta_{6} ) q^{84} -7 \zeta_{6} q^{86} + ( -3 + 3 \zeta_{6} ) q^{87} + ( 6 - 6 \zeta_{6} ) q^{88} + 15 \zeta_{6} q^{89} + ( -4 + 12 \zeta_{6} ) q^{91} + 3 q^{92} + 8 \zeta_{6} q^{93} + \zeta_{6} q^{96} -14 q^{97} + ( 3 + 5 \zeta_{6} ) q^{98} + 12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - q^{2} + q^{3} - q^{4} - 2q^{6} + q^{7} + 2q^{8} + 2q^{9} + O(q^{10}) \) \( 2q - q^{2} + q^{3} - q^{4} - 2q^{6} + q^{7} + 2q^{8} + 2q^{9} + 6q^{11} + q^{12} + 8q^{13} + 4q^{14} - q^{16} + 2q^{18} - 2q^{19} + 5q^{21} - 12q^{22} - 3q^{23} + q^{24} - 4q^{26} + 10q^{27} - 5q^{28} - 6q^{29} - 8q^{31} - q^{32} - 6q^{33} - 4q^{36} - 4q^{37} - 2q^{38} + 4q^{39} + 18q^{41} - q^{42} + 14q^{43} + 6q^{44} - 3q^{46} - 2q^{48} - 13q^{49} - 4q^{52} - 6q^{53} - 5q^{54} + q^{56} - 4q^{57} + 3q^{58} + 6q^{59} - 5q^{61} + 16q^{62} - 8q^{63} + 2q^{64} - 6q^{66} + 5q^{67} - 6q^{69} - 12q^{71} + 2q^{72} - 16q^{73} - 4q^{74} + 4q^{76} + 30q^{77} - 8q^{78} - 2q^{79} - q^{81} - 9q^{82} - 6q^{83} - 4q^{84} - 7q^{86} - 3q^{87} + 6q^{88} + 15q^{89} + 4q^{91} + 6q^{92} + 8q^{93} + q^{96} - 28q^{97} + 11q^{98} + 24q^{99} + O(q^{100}) \)

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 0.500000 + 0.866025i −0.500000 0.866025i 0 −1.00000 0.500000 2.59808i 1.00000 1.00000 1.73205i 0
151.1 −0.500000 0.866025i 0.500000 0.866025i −0.500000 + 0.866025i 0 −1.00000 0.500000 + 2.59808i 1.00000 1.00000 + 1.73205i 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.e 2
5.b even 2 1 70.2.e.c 2
5.c odd 4 2 350.2.j.b 4
7.c even 3 1 inner 350.2.e.e 2
7.c even 3 1 2450.2.a.w 1
7.d odd 6 1 2450.2.a.bc 1
15.d odd 2 1 630.2.k.b 2
20.d odd 2 1 560.2.q.g 2
35.c odd 2 1 490.2.e.h 2
35.i odd 6 1 490.2.a.b 1
35.i odd 6 1 490.2.e.h 2
35.j even 6 1 70.2.e.c 2
35.j even 6 1 490.2.a.c 1
35.k even 12 2 2450.2.c.l 2
35.l odd 12 2 350.2.j.b 4
35.l odd 12 2 2450.2.c.g 2
105.o odd 6 1 630.2.k.b 2
105.o odd 6 1 4410.2.a.bm 1
105.p even 6 1 4410.2.a.bd 1
140.p odd 6 1 560.2.q.g 2
140.p odd 6 1 3920.2.a.p 1
140.s even 6 1 3920.2.a.bc 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
70.2.e.c 2 5.b even 2 1
70.2.e.c 2 35.j even 6 1
350.2.e.e 2 1.a even 1 1 trivial
350.2.e.e 2 7.c even 3 1 inner
350.2.j.b 4 5.c odd 4 2
350.2.j.b 4 35.l odd 12 2
490.2.a.b 1 35.i odd 6 1
490.2.a.c 1 35.j even 6 1
490.2.e.h 2 35.c odd 2 1
490.2.e.h 2 35.i odd 6 1
560.2.q.g 2 20.d odd 2 1
560.2.q.g 2 140.p odd 6 1
630.2.k.b 2 15.d odd 2 1
630.2.k.b 2 105.o odd 6 1
2450.2.a.w 1 7.c even 3 1
2450.2.a.bc 1 7.d odd 6 1
2450.2.c.g 2 35.l odd 12 2
2450.2.c.l 2 35.k even 12 2
3920.2.a.p 1 140.p odd 6 1
3920.2.a.bc 1 140.s even 6 1
4410.2.a.bd 1 105.p even 6 1
4410.2.a.bm 1 105.o odd 6 1

Hecke kernels

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

\( T_{3}^{2} - T_{3} + 1 \)
\( T_{11}^{2} - 6 T_{11} + 36 \)
\( T_{13} - 4 \)
\( T_{17} \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 + T + T^{2} \)
$3$ \( 1 - T - 2 T^{2} - 3 T^{3} + 9 T^{4} \)
$5$ \( \)
$7$ \( 1 - T + 7 T^{2} \)
$11$ \( 1 - 6 T + 25 T^{2} - 66 T^{3} + 121 T^{4} \)
$13$ \( ( 1 - 4 T + 13 T^{2} )^{2} \)
$17$ \( 1 - 17 T^{2} + 289 T^{4} \)
$19$ \( 1 + 2 T - 15 T^{2} + 38 T^{3} + 361 T^{4} \)
$23$ \( 1 + 3 T - 14 T^{2} + 69 T^{3} + 529 T^{4} \)
$29$ \( ( 1 + 3 T + 29 T^{2} )^{2} \)
$31$ \( 1 + 8 T + 33 T^{2} + 248 T^{3} + 961 T^{4} \)
$37$ \( 1 + 4 T - 21 T^{2} + 148 T^{3} + 1369 T^{4} \)
$41$ \( ( 1 - 9 T + 41 T^{2} )^{2} \)
$43$ \( ( 1 - 7 T + 43 T^{2} )^{2} \)
$47$ \( 1 - 47 T^{2} + 2209 T^{4} \)
$53$ \( 1 + 6 T - 17 T^{2} + 318 T^{3} + 2809 T^{4} \)
$59$ \( 1 - 6 T - 23 T^{2} - 354 T^{3} + 3481 T^{4} \)
$61$ \( 1 + 5 T - 36 T^{2} + 305 T^{3} + 3721 T^{4} \)
$67$ \( ( 1 - 16 T + 67 T^{2} )( 1 + 11 T + 67 T^{2} ) \)
$71$ \( ( 1 + 6 T + 71 T^{2} )^{2} \)
$73$ \( 1 + 16 T + 183 T^{2} + 1168 T^{3} + 5329 T^{4} \)
$79$ \( 1 + 2 T - 75 T^{2} + 158 T^{3} + 6241 T^{4} \)
$83$ \( ( 1 + 3 T + 83 T^{2} )^{2} \)
$89$ \( 1 - 15 T + 136 T^{2} - 1335 T^{3} + 7921 T^{4} \)
$97$ \( ( 1 + 14 T + 97 T^{2} )^{2} \)
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