Properties

Label 350.2.j.e
Level 350
Weight 2
Character orbit 350.j
Analytic conductor 2.795
Analytic rank 0
Dimension 4
CM no
Inner twists 4

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) = \( 350 = 2 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 350.j (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.79476407074\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{12}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \zeta_{12} q^{2} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{3} + \zeta_{12}^{2} q^{4} + 2 q^{6} + ( \zeta_{12} + 2 \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{3} q^{8} + ( 1 - \zeta_{12}^{2} ) q^{9} +O(q^{10})\) \( q + \zeta_{12} q^{2} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{3} + \zeta_{12}^{2} q^{4} + 2 q^{6} + ( \zeta_{12} + 2 \zeta_{12}^{3} ) q^{7} + \zeta_{12}^{3} q^{8} + ( 1 - \zeta_{12}^{2} ) q^{9} + 2 \zeta_{12} q^{12} -2 \zeta_{12}^{3} q^{13} + ( -2 + 3 \zeta_{12}^{2} ) q^{14} + ( -1 + \zeta_{12}^{2} ) q^{16} + ( -3 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{17} + ( \zeta_{12} - \zeta_{12}^{3} ) q^{18} + ( 8 - 8 \zeta_{12}^{2} ) q^{19} + ( 2 + 4 \zeta_{12}^{2} ) q^{21} -9 \zeta_{12} q^{23} + 2 \zeta_{12}^{2} q^{24} + ( 2 - 2 \zeta_{12}^{2} ) q^{26} + 4 \zeta_{12}^{3} q^{27} + ( -2 \zeta_{12} + 3 \zeta_{12}^{3} ) q^{28} + 6 q^{29} -5 \zeta_{12}^{2} q^{31} + ( -\zeta_{12} + \zeta_{12}^{3} ) q^{32} -3 q^{34} + q^{36} -8 \zeta_{12} q^{37} + ( 8 \zeta_{12} - 8 \zeta_{12}^{3} ) q^{38} -4 \zeta_{12}^{2} q^{39} -3 q^{41} + ( 2 \zeta_{12} + 4 \zeta_{12}^{3} ) q^{42} + 10 \zeta_{12}^{3} q^{43} -9 \zeta_{12}^{2} q^{46} + 3 \zeta_{12} q^{47} + 2 \zeta_{12}^{3} q^{48} + ( -8 + 5 \zeta_{12}^{2} ) q^{49} + ( -6 + 6 \zeta_{12}^{2} ) q^{51} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{52} + ( -6 \zeta_{12} + 6 \zeta_{12}^{3} ) q^{53} + ( -4 + 4 \zeta_{12}^{2} ) q^{54} + ( -3 + \zeta_{12}^{2} ) q^{56} -16 \zeta_{12}^{3} q^{57} + 6 \zeta_{12} q^{58} + 12 \zeta_{12}^{2} q^{59} + ( 4 - 4 \zeta_{12}^{2} ) q^{61} -5 \zeta_{12}^{3} q^{62} + ( 3 \zeta_{12} - \zeta_{12}^{3} ) q^{63} - q^{64} + ( 2 \zeta_{12} - 2 \zeta_{12}^{3} ) q^{67} -3 \zeta_{12} q^{68} -18 q^{69} -9 q^{71} + \zeta_{12} q^{72} + ( 10 \zeta_{12} - 10 \zeta_{12}^{3} ) q^{73} -8 \zeta_{12}^{2} q^{74} + 8 q^{76} -4 \zeta_{12}^{3} q^{78} + ( 5 - 5 \zeta_{12}^{2} ) q^{79} + 11 \zeta_{12}^{2} q^{81} -3 \zeta_{12} q^{82} + 6 \zeta_{12}^{3} q^{83} + ( -4 + 6 \zeta_{12}^{2} ) q^{84} + ( -10 + 10 \zeta_{12}^{2} ) q^{86} + ( 12 \zeta_{12} - 12 \zeta_{12}^{3} ) q^{87} + ( 3 - 3 \zeta_{12}^{2} ) q^{89} + ( 6 - 2 \zeta_{12}^{2} ) q^{91} -9 \zeta_{12}^{3} q^{92} -10 \zeta_{12} q^{93} + 3 \zeta_{12}^{2} q^{94} + ( -2 + 2 \zeta_{12}^{2} ) q^{96} + 5 \zeta_{12}^{3} q^{97} + ( -8 \zeta_{12} + 5 \zeta_{12}^{3} ) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} + 8q^{6} + 2q^{9} + O(q^{10}) \) \( 4q + 2q^{4} + 8q^{6} + 2q^{9} - 2q^{14} - 2q^{16} + 16q^{19} + 16q^{21} + 4q^{24} + 4q^{26} + 24q^{29} - 10q^{31} - 12q^{34} + 4q^{36} - 8q^{39} - 12q^{41} - 18q^{46} - 22q^{49} - 12q^{51} - 8q^{54} - 10q^{56} + 24q^{59} + 8q^{61} - 4q^{64} - 72q^{69} - 36q^{71} - 16q^{74} + 32q^{76} + 10q^{79} + 22q^{81} - 4q^{84} - 20q^{86} + 6q^{89} + 20q^{91} + 6q^{94} - 4q^{96} + O(q^{100}) \)

Character values

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

\(n\) \(101\) \(127\)
\(\chi(n)\) \(-1 + \zeta_{12}^{2}\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
149.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i −1.73205 + 1.00000i 0.500000 + 0.866025i 0 2.00000 −0.866025 2.50000i 1.00000i 0.500000 0.866025i 0
149.2 0.866025 + 0.500000i 1.73205 1.00000i 0.500000 + 0.866025i 0 2.00000 0.866025 + 2.50000i 1.00000i 0.500000 0.866025i 0
249.1 −0.866025 + 0.500000i −1.73205 1.00000i 0.500000 0.866025i 0 2.00000 −0.866025 + 2.50000i 1.00000i 0.500000 + 0.866025i 0
249.2 0.866025 0.500000i 1.73205 + 1.00000i 0.500000 0.866025i 0 2.00000 0.866025 2.50000i 1.00000i 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
5.b even 2 1 inner
7.c even 3 1 inner
35.j even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 350.2.j.e 4
5.b even 2 1 inner 350.2.j.e 4
5.c odd 4 1 350.2.e.a 2
5.c odd 4 1 350.2.e.k yes 2
7.c even 3 1 inner 350.2.j.e 4
7.c even 3 1 2450.2.c.o 2
7.d odd 6 1 2450.2.c.d 2
35.i odd 6 1 2450.2.c.d 2
35.j even 6 1 inner 350.2.j.e 4
35.j even 6 1 2450.2.c.o 2
35.k even 12 1 2450.2.a.o 1
35.k even 12 1 2450.2.a.u 1
35.l odd 12 1 350.2.e.a 2
35.l odd 12 1 350.2.e.k yes 2
35.l odd 12 1 2450.2.a.e 1
35.l odd 12 1 2450.2.a.be 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
350.2.e.a 2 5.c odd 4 1
350.2.e.a 2 35.l odd 12 1
350.2.e.k yes 2 5.c odd 4 1
350.2.e.k yes 2 35.l odd 12 1
350.2.j.e 4 1.a even 1 1 trivial
350.2.j.e 4 5.b even 2 1 inner
350.2.j.e 4 7.c even 3 1 inner
350.2.j.e 4 35.j even 6 1 inner
2450.2.a.e 1 35.l odd 12 1
2450.2.a.o 1 35.k even 12 1
2450.2.a.u 1 35.k even 12 1
2450.2.a.be 1 35.l odd 12 1
2450.2.c.d 2 7.d odd 6 1
2450.2.c.d 2 35.i odd 6 1
2450.2.c.o 2 7.c even 3 1
2450.2.c.o 2 35.j even 6 1

Hecke kernels

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

\( T_{3}^{4} - 4 T_{3}^{2} + 16 \)
\( T_{11} \)
\( T_{13}^{2} + 4 \)

Hecke Characteristic Polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( 1 + 2 T^{2} - 5 T^{4} + 18 T^{6} + 81 T^{8} \)
$5$ \( \)
$7$ \( 1 + 11 T^{2} + 49 T^{4} \)
$11$ \( ( 1 - 11 T^{2} + 121 T^{4} )^{2} \)
$13$ \( ( 1 - 22 T^{2} + 169 T^{4} )^{2} \)
$17$ \( 1 + 25 T^{2} + 336 T^{4} + 7225 T^{6} + 83521 T^{8} \)
$19$ \( ( 1 - 7 T + 19 T^{2} )^{2}( 1 - T + 19 T^{2} )^{2} \)
$23$ \( 1 - 35 T^{2} + 696 T^{4} - 18515 T^{6} + 279841 T^{8} \)
$29$ \( ( 1 - 6 T + 29 T^{2} )^{4} \)
$31$ \( ( 1 + 5 T - 6 T^{2} + 155 T^{3} + 961 T^{4} )^{2} \)
$37$ \( 1 + 10 T^{2} - 1269 T^{4} + 13690 T^{6} + 1874161 T^{8} \)
$41$ \( ( 1 + 3 T + 41 T^{2} )^{4} \)
$43$ \( ( 1 + 14 T^{2} + 1849 T^{4} )^{2} \)
$47$ \( 1 + 85 T^{2} + 5016 T^{4} + 187765 T^{6} + 4879681 T^{8} \)
$53$ \( 1 + 70 T^{2} + 2091 T^{4} + 196630 T^{6} + 7890481 T^{8} \)
$59$ \( ( 1 - 12 T + 85 T^{2} - 708 T^{3} + 3481 T^{4} )^{2} \)
$61$ \( ( 1 - 4 T - 45 T^{2} - 244 T^{3} + 3721 T^{4} )^{2} \)
$67$ \( 1 + 130 T^{2} + 12411 T^{4} + 583570 T^{6} + 20151121 T^{8} \)
$71$ \( ( 1 + 9 T + 71 T^{2} )^{4} \)
$73$ \( ( 1 - 97 T^{2} + 5329 T^{4} )( 1 + 143 T^{2} + 5329 T^{4} ) \)
$79$ \( ( 1 - 5 T - 54 T^{2} - 395 T^{3} + 6241 T^{4} )^{2} \)
$83$ \( ( 1 - 130 T^{2} + 6889 T^{4} )^{2} \)
$89$ \( ( 1 - 3 T - 80 T^{2} - 267 T^{3} + 7921 T^{4} )^{2} \)
$97$ \( ( 1 - 169 T^{2} + 9409 T^{4} )^{2} \)
show more
show less