Properties

Label 950.2.l.e
Level $950$
Weight $2$
Character orbit 950.l
Analytic conductor $7.586$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.l (of order \(9\), degree \(6\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{18})\)
Defining polynomial: \(x^{6} - x^{3} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{18}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( \zeta_{18} - \zeta_{18}^{4} ) q^{2} + ( 1 + \zeta_{18} - \zeta_{18}^{3} + \zeta_{18}^{5} ) q^{3} -\zeta_{18}^{5} q^{4} + ( \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{6} + ( 1 + \zeta_{18} - \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{7} -\zeta_{18}^{3} q^{8} + ( -2 + \zeta_{18} + \zeta_{18}^{3} - 2 \zeta_{18}^{4} ) q^{9} +O(q^{10})\) \( q + ( \zeta_{18} - \zeta_{18}^{4} ) q^{2} + ( 1 + \zeta_{18} - \zeta_{18}^{3} + \zeta_{18}^{5} ) q^{3} -\zeta_{18}^{5} q^{4} + ( \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{6} + ( 1 + \zeta_{18} - \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{7} -\zeta_{18}^{3} q^{8} + ( -2 + \zeta_{18} + \zeta_{18}^{3} - 2 \zeta_{18}^{4} ) q^{9} + \zeta_{18}^{3} q^{11} + ( 1 + \zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{3} ) q^{12} + ( 1 + \zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{5} ) q^{13} + ( -\zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{14} -\zeta_{18} q^{16} + ( 2 \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{3} - 2 \zeta_{18}^{4} ) q^{17} + ( -\zeta_{18} - \zeta_{18}^{2} + 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{18} + ( -1 + 4 \zeta_{18} + 2 \zeta_{18}^{2} - \zeta_{18}^{3} - 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{19} + ( 1 + 2 \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} - 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{21} + \zeta_{18} q^{22} + ( 1 - \zeta_{18}^{3} - \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{23} + ( -1 + \zeta_{18}^{2} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{24} + ( -1 + \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{26} + ( -3 \zeta_{18}^{2} - 3 \zeta_{18}^{4} ) q^{27} + ( -\zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{3} ) q^{28} + ( -4 + 2 \zeta_{18} + 2 \zeta_{18}^{3} - 4 \zeta_{18}^{4} ) q^{29} + ( 1 + 2 \zeta_{18} - 2 \zeta_{18}^{2} - \zeta_{18}^{3} ) q^{31} + ( -\zeta_{18}^{2} + \zeta_{18}^{5} ) q^{32} + ( 1 - \zeta_{18}^{2} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{33} + ( -1 + \zeta_{18} - 2 \zeta_{18}^{5} ) q^{34} + ( -1 + \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{5} ) q^{36} + ( -2 + 3 \zeta_{18} + 3 \zeta_{18}^{2} - 3 \zeta_{18}^{4} ) q^{37} + ( 2 - 2 \zeta_{18} + 2 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{38} + ( -2 + 2 \zeta_{18} + 2 \zeta_{18}^{2} - 3 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{39} + ( -3 + 2 \zeta_{18} - 4 \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{5} ) q^{41} + ( 1 - \zeta_{18}^{3} - \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{42} + ( 1 - \zeta_{18}^{2} + 4 \zeta_{18}^{3} + 3 \zeta_{18}^{4} + 5 \zeta_{18}^{5} ) q^{43} + ( \zeta_{18}^{2} - \zeta_{18}^{5} ) q^{44} + ( -\zeta_{18}^{2} + 3 \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{46} + ( -2 + \zeta_{18} + 3 \zeta_{18}^{2} + \zeta_{18}^{3} - 2 \zeta_{18}^{4} ) q^{47} + ( 1 - \zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{48} + ( -\zeta_{18} - 2 \zeta_{18}^{2} + 4 \zeta_{18}^{3} - 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{49} + ( 1 + \zeta_{18} + \zeta_{18}^{3} - 2 \zeta_{18}^{4} ) q^{51} + ( 1 - \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{52} + ( -1 - 4 \zeta_{18} + 5 \zeta_{18}^{3} + 5 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{53} + ( -3 - 3 \zeta_{18}^{2} ) q^{54} + ( -1 - \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{5} ) q^{56} + ( -2 + \zeta_{18} + 5 \zeta_{18}^{2} + 5 \zeta_{18}^{3} - 3 \zeta_{18}^{4} - 6 \zeta_{18}^{5} ) q^{57} + ( -2 \zeta_{18} - 2 \zeta_{18}^{2} + 4 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{58} + ( 4 + 3 \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{3} - 3 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{59} + ( -1 + 4 \zeta_{18} - 3 \zeta_{18}^{3} - 3 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{61} + ( -2 + 2 \zeta_{18}^{2} - \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{62} + ( -2 - 2 \zeta_{18} + \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{63} + ( -1 + \zeta_{18}^{3} ) q^{64} + ( -1 + \zeta_{18} + \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{66} + ( 1 + \zeta_{18}^{2} + \zeta_{18}^{4} ) q^{67} + ( -\zeta_{18} + \zeta_{18}^{2} - 2 \zeta_{18}^{3} + \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{68} + ( -2 - 3 \zeta_{18} + 4 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{69} + ( 4 - 4 \zeta_{18}^{2} - 6 \zeta_{18}^{3} + 4 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{71} + ( 1 - 2 \zeta_{18} + \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{72} + ( 4 - \zeta_{18} + 3 \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{5} ) q^{73} + ( 3 - 2 \zeta_{18} + 2 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{74} + ( 2 - \zeta_{18}^{2} - 4 \zeta_{18}^{3} - 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{76} + ( 1 + \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{5} ) q^{77} + ( 2 - 2 \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{3} + 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{78} + ( 3 + 6 \zeta_{18} + \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{5} ) q^{79} + ( -3 + 3 \zeta_{18}^{2} - 3 \zeta_{18}^{4} - 3 \zeta_{18}^{5} ) q^{81} + ( -4 - 4 \zeta_{18} + 2 \zeta_{18}^{2} + \zeta_{18}^{3} + 3 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{82} + ( -6 - 9 \zeta_{18} + 8 \zeta_{18}^{2} + 6 \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{83} + ( -\zeta_{18}^{2} - 2 \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{84} + ( -1 + 5 \zeta_{18} + 3 \zeta_{18}^{2} + 5 \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{86} + ( -6 \zeta_{18} + 6 \zeta_{18}^{3} - 6 \zeta_{18}^{5} ) q^{87} + ( 1 - \zeta_{18}^{3} ) q^{88} + ( -4 - 4 \zeta_{18} + 4 \zeta_{18}^{2} + 8 \zeta_{18}^{3} - 4 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{89} + ( \zeta_{18} - \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{91} + ( -1 + 3 \zeta_{18} - \zeta_{18}^{2} ) q^{92} + ( -2 + 5 \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} - 5 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{93} + ( 3 - \zeta_{18} - \zeta_{18}^{2} + 2 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{94} + ( -1 - \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{96} + ( -8 + \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} + 8 \zeta_{18}^{5} ) q^{97} + ( -2 + 4 \zeta_{18} - 3 \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{5} ) q^{98} + ( -1 + 2 \zeta_{18} - \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 3q^{3} + 3q^{6} + 3q^{7} - 3q^{8} - 9q^{9} + O(q^{10}) \) \( 6q + 3q^{3} + 3q^{6} + 3q^{7} - 3q^{8} - 9q^{9} + 3q^{11} + 3q^{12} + 3q^{13} - 3q^{14} + 3q^{17} - 9q^{19} + 3q^{21} + 3q^{23} - 6q^{24} - 3q^{26} - 3q^{28} - 18q^{29} + 3q^{31} + 6q^{33} - 6q^{34} - 9q^{36} - 12q^{37} + 6q^{38} - 12q^{39} - 21q^{41} + 3q^{42} + 18q^{43} + 9q^{46} - 9q^{47} + 3q^{48} + 12q^{49} + 9q^{51} + 3q^{52} + 9q^{53} - 18q^{54} - 6q^{56} + 3q^{57} + 27q^{59} - 15q^{61} - 12q^{62} - 9q^{63} - 3q^{64} - 3q^{66} + 6q^{67} - 6q^{68} - 6q^{69} + 6q^{71} + 9q^{72} + 21q^{73} + 18q^{74} + 6q^{77} + 15q^{78} + 12q^{79} - 18q^{81} - 21q^{82} - 18q^{83} - 6q^{84} + 9q^{86} + 18q^{87} + 3q^{88} - 3q^{91} - 6q^{92} - 15q^{93} + 18q^{94} - 6q^{96} - 45q^{97} - 15q^{98} - 9q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/950\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(1\) \(-\zeta_{18}^{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
101.1
0.939693 + 0.342020i
−0.173648 0.984808i
0.939693 0.342020i
−0.766044 + 0.642788i
−0.173648 + 0.984808i
−0.766044 0.642788i
0.766044 0.642788i 1.26604 + 0.460802i 0.173648 0.984808i 0 1.26604 0.460802i 1.43969 2.49362i −0.500000 0.866025i −0.907604 0.761570i 0
251.1 −0.939693 0.342020i −0.439693 2.49362i 0.766044 + 0.642788i 0 −0.439693 + 2.49362i 0.326352 0.565258i −0.500000 0.866025i −3.20574 + 1.16679i 0
301.1 0.766044 + 0.642788i 1.26604 0.460802i 0.173648 + 0.984808i 0 1.26604 + 0.460802i 1.43969 + 2.49362i −0.500000 + 0.866025i −0.907604 + 0.761570i 0
351.1 0.173648 + 0.984808i 0.673648 0.565258i −0.939693 + 0.342020i 0 0.673648 + 0.565258i −0.266044 + 0.460802i −0.500000 0.866025i −0.386659 + 2.19285i 0
651.1 −0.939693 + 0.342020i −0.439693 + 2.49362i 0.766044 0.642788i 0 −0.439693 2.49362i 0.326352 + 0.565258i −0.500000 + 0.866025i −3.20574 1.16679i 0
701.1 0.173648 0.984808i 0.673648 + 0.565258i −0.939693 0.342020i 0 0.673648 0.565258i −0.266044 0.460802i −0.500000 + 0.866025i −0.386659 2.19285i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 701.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.e even 9 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.l.e yes 6
5.b even 2 1 950.2.l.b 6
5.c odd 4 2 950.2.u.d 12
19.e even 9 1 inner 950.2.l.e yes 6
95.p even 18 1 950.2.l.b 6
95.q odd 36 2 950.2.u.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.l.b 6 5.b even 2 1
950.2.l.b 6 95.p even 18 1
950.2.l.e yes 6 1.a even 1 1 trivial
950.2.l.e yes 6 19.e even 9 1 inner
950.2.u.d 12 5.c odd 4 2
950.2.u.d 12 95.q odd 36 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 3 T_{3}^{5} + 9 T_{3}^{4} - 24 T_{3}^{3} + 36 T_{3}^{2} - 27 T_{3} + 9 \) acting on \(S_{2}^{\mathrm{new}}(950, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 + T^{3} + T^{6} \)
$3$ \( 9 - 27 T + 36 T^{2} - 24 T^{3} + 9 T^{4} - 3 T^{5} + T^{6} \)
$5$ \( T^{6} \)
$7$ \( 1 + 3 T^{2} - 2 T^{3} + 9 T^{4} - 3 T^{5} + T^{6} \)
$11$ \( ( 1 - T + T^{2} )^{3} \)
$13$ \( 1 - 6 T + 15 T^{2} - 19 T^{3} + 12 T^{4} - 3 T^{5} + T^{6} \)
$17$ \( 9 + 27 T + 9 T^{2} - 24 T^{3} + 18 T^{4} - 3 T^{5} + T^{6} \)
$19$ \( 6859 + 3249 T - 179 T^{3} + 9 T^{5} + T^{6} \)
$23$ \( 289 + 102 T + 219 T^{2} + T^{3} - 12 T^{4} - 3 T^{5} + T^{6} \)
$29$ \( 5184 + 5184 T + 2592 T^{2} + 720 T^{3} + 144 T^{4} + 18 T^{5} + T^{6} \)
$31$ \( 361 - 171 T + 138 T^{2} - 11 T^{3} + 18 T^{4} - 3 T^{5} + T^{6} \)
$37$ \( ( -19 - 15 T + 6 T^{2} + T^{3} )^{2} \)
$41$ \( 25281 + 17172 T + 5976 T^{2} + 1374 T^{3} + 216 T^{4} + 21 T^{5} + T^{6} \)
$43$ \( 29241 - 26163 T + 10530 T^{2} - 2196 T^{3} + 252 T^{4} - 18 T^{5} + T^{6} \)
$47$ \( 81 - 81 T + 162 T^{2} - 72 T^{3} + 9 T^{4} + 9 T^{5} + T^{6} \)
$53$ \( 32041 - 27387 T + 9891 T^{2} - 1792 T^{3} + 162 T^{4} - 9 T^{5} + T^{6} \)
$59$ \( 110889 - 59940 T + 16281 T^{2} - 2925 T^{3} + 360 T^{4} - 27 T^{5} + T^{6} \)
$61$ \( 25281 + 15741 T + 4545 T^{2} + 840 T^{3} + 126 T^{4} + 15 T^{5} + T^{6} \)
$67$ \( 9 - 27 T + 36 T^{2} - 30 T^{3} + 18 T^{4} - 6 T^{5} + T^{6} \)
$71$ \( 166464 + 44064 T + 6192 T^{2} + 672 T^{3} - 6 T^{5} + T^{6} \)
$73$ \( 361 - 456 T + 1581 T^{2} - 847 T^{3} + 192 T^{4} - 21 T^{5} + T^{6} \)
$79$ \( 2809 + 5565 T + 4548 T^{2} - 899 T^{3} + 87 T^{4} - 12 T^{5} + T^{6} \)
$83$ \( 4068289 + 223887 T + 48627 T^{2} + 2036 T^{3} + 435 T^{4} + 18 T^{5} + T^{6} \)
$89$ \( 4096 + 9216 T + 4608 T^{2} - 1792 T^{3} + 288 T^{4} + T^{6} \)
$97$ \( 289 - 918 T + 28071 T^{2} + 7381 T^{3} + 828 T^{4} + 45 T^{5} + T^{6} \)
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