Properties

Label 350.2.e.j
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: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
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.

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

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.j 2
5.b even 2 1 350.2.e.c 2
5.c odd 4 2 70.2.i.b 4
7.c even 3 1 inner 350.2.e.j 2
7.c even 3 1 2450.2.a.k 1
7.d odd 6 1 2450.2.a.j 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.bb 1
35.j even 6 1 350.2.e.c 2
35.j even 6 1 2450.2.a.ba 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 5.b even 2 1
350.2.e.c 2 35.j even 6 1
350.2.e.j 2 1.a even 1 1 trivial
350.2.e.j 2 7.c even 3 1 inner
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 7.d odd 6 1
2450.2.a.k 1 7.c even 3 1
2450.2.a.ba 1 35.j even 6 1
2450.2.a.bb 1 35.i odd 6 1

Hecke kernels

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

\( T_{3} \) Copy content Toggle raw display
\( T_{11}^{2} + 3T_{11} + 9 \) Copy content Toggle raw display
\( T_{13} - 5 \) Copy content Toggle raw display
\( T_{17}^{2} + 2T_{17} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 4T + 7 \) Copy content Toggle raw display
$11$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$13$ \( (T - 5)^{2} \) Copy content Toggle raw display
$17$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$19$ \( T^{2} - 5T + 25 \) Copy content Toggle raw display
$23$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$29$ \( (T + 4)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$37$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$41$ \( (T - 3)^{2} \) Copy content Toggle raw display
$43$ \( (T + 2)^{2} \) Copy content Toggle raw display
$47$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$53$ \( T^{2} - 9T + 81 \) Copy content Toggle raw display
$59$ \( T^{2} - 4T + 16 \) Copy content Toggle raw display
$61$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$67$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$71$ \( (T + 6)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 16T + 256 \) Copy content Toggle raw display
$79$ \( T^{2} + 14T + 196 \) Copy content Toggle raw display
$83$ \( (T - 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$97$ \( (T - 12)^{2} \) Copy content Toggle raw display
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