Properties

Label 950.2.l.c
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: no (minimal twist has level 190)
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.

\(f(q)\) \(=\) \( q + ( -\zeta_{18} + \zeta_{18}^{4} ) q^{2} + ( -\zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{5} ) q^{3} -\zeta_{18}^{5} q^{4} + ( 1 + \zeta_{18} - \zeta_{18}^{3} ) q^{6} + ( -2 \zeta_{18} + 2 \zeta_{18}^{2} ) q^{7} + \zeta_{18}^{3} q^{8} + ( -1 + \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{9} +O(q^{10})\) \( q + ( -\zeta_{18} + \zeta_{18}^{4} ) q^{2} + ( -\zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{5} ) q^{3} -\zeta_{18}^{5} q^{4} + ( 1 + \zeta_{18} - \zeta_{18}^{3} ) q^{6} + ( -2 \zeta_{18} + 2 \zeta_{18}^{2} ) q^{7} + \zeta_{18}^{3} q^{8} + ( -1 + \zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{9} + ( -\zeta_{18}^{2} + 4 \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{11} + ( -\zeta_{18}^{2} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{12} + ( 2 + 2 \zeta_{18} - 2 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{5} ) q^{13} + ( -2 + 2 \zeta_{18}^{2} - 2 \zeta_{18}^{5} ) q^{14} -\zeta_{18} q^{16} + ( \zeta_{18} - 2 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - \zeta_{18}^{4} ) q^{17} + ( 1 - \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{18} + ( 2 - 4 \zeta_{18} - 2 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{19} + ( 2 - 2 \zeta_{18} + 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{21} + ( 1 - 4 \zeta_{18} + \zeta_{18}^{2} ) q^{22} + ( -2 \zeta_{18} + 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{23} + ( 1 - \zeta_{18}^{2} - \zeta_{18}^{3} ) q^{24} + ( 2 - 2 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{26} + ( 3 \zeta_{18} + \zeta_{18}^{3} + 3 \zeta_{18}^{5} ) q^{27} + ( -2 + 2 \zeta_{18} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{4} ) q^{28} + ( -4 + 4 \zeta_{18} - 2 \zeta_{18}^{2} + 4 \zeta_{18}^{3} - 4 \zeta_{18}^{4} ) q^{29} + ( 4 - 2 \zeta_{18} - 2 \zeta_{18}^{2} - 4 \zeta_{18}^{3} + 4 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{31} + ( \zeta_{18}^{2} - \zeta_{18}^{5} ) q^{32} + ( 4 - 4 \zeta_{18}^{2} - 3 \zeta_{18}^{3} + \zeta_{18}^{4} + \zeta_{18}^{5} ) q^{33} + ( 2 - 2 \zeta_{18} + \zeta_{18}^{5} ) q^{34} + ( -\zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{36} + 4 q^{37} + ( 2 + 2 \zeta_{18}^{2} - \zeta_{18}^{3} + 2 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{38} + ( -4 + 2 \zeta_{18} + 2 \zeta_{18}^{2} - 4 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{39} + ( 5 + 4 \zeta_{18}^{2} - \zeta_{18}^{3} + \zeta_{18}^{5} ) q^{41} + ( -2 \zeta_{18} + 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{42} + ( -6 + 6 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{43} + ( -1 - \zeta_{18} + 4 \zeta_{18}^{2} + \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{44} + ( -2 \zeta_{18} + 4 \zeta_{18}^{3} - 2 \zeta_{18}^{5} ) q^{46} + ( 6 - 4 \zeta_{18} + 4 \zeta_{18}^{2} - 4 \zeta_{18}^{3} + 6 \zeta_{18}^{4} ) q^{47} + ( 1 + \zeta_{18}^{4} ) q^{48} + ( 4 \zeta_{18}^{2} - \zeta_{18}^{3} + 4 \zeta_{18}^{4} ) q^{49} + ( 1 + \zeta_{18} - 2 \zeta_{18}^{2} - \zeta_{18}^{3} + 2 \zeta_{18}^{5} ) q^{51} + ( 2 - 2 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{4} ) q^{52} + ( 2 - 2 \zeta_{18} - 2 \zeta_{18}^{5} ) q^{53} + ( -\zeta_{18} - 3 \zeta_{18}^{2} - 3 \zeta_{18}^{3} + 3 \zeta_{18}^{5} ) q^{54} + ( -2 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{56} + ( 2 + 4 \zeta_{18} + \zeta_{18}^{2} - 2 \zeta_{18}^{3} + \zeta_{18}^{4} + 3 \zeta_{18}^{5} ) q^{57} + ( 2 - 4 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{58} + ( 1 + 8 \zeta_{18} + \zeta_{18}^{2} - \zeta_{18}^{3} - 8 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{59} + ( -2 + 2 \zeta_{18} + 6 \zeta_{18}^{5} ) q^{61} + ( 2 - 2 \zeta_{18}^{2} - 4 \zeta_{18}^{3} + 4 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{62} + ( 2 + 2 \zeta_{18} - 4 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 4 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{63} + ( -1 + \zeta_{18}^{3} ) q^{64} + ( 4 - \zeta_{18} - \zeta_{18}^{2} - \zeta_{18}^{3} + 4 \zeta_{18}^{4} ) q^{66} + ( 1 + 7 \zeta_{18} - 2 \zeta_{18}^{2} + 7 \zeta_{18}^{3} + \zeta_{18}^{4} ) q^{67} + ( -2 \zeta_{18} + 2 \zeta_{18}^{2} - \zeta_{18}^{3} + 2 \zeta_{18}^{4} - 2 \zeta_{18}^{5} ) q^{68} + ( 4 + 2 \zeta_{18} - 6 \zeta_{18}^{2} - 4 \zeta_{18}^{3} + 4 \zeta_{18}^{4} + 4 \zeta_{18}^{5} ) q^{69} + ( 2 - 2 \zeta_{18}^{2} + 4 \zeta_{18}^{3} + 4 \zeta_{18}^{4} + 6 \zeta_{18}^{5} ) q^{71} + ( -1 + \zeta_{18} - \zeta_{18}^{5} ) q^{72} + ( -1 - 2 \zeta_{18} - 3 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{5} ) q^{73} + ( -4 \zeta_{18} + 4 \zeta_{18}^{4} ) q^{74} + ( -2 - \zeta_{18} - 2 \zeta_{18}^{2} + 4 \zeta_{18}^{3} + 2 \zeta_{18}^{4} ) q^{76} + ( 2 - 10 \zeta_{18}^{4} + 10 \zeta_{18}^{5} ) q^{77} + ( -2 + 4 \zeta_{18} + 2 \zeta_{18}^{2} - 2 \zeta_{18}^{3} - 4 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{78} + ( 4 - 2 \zeta_{18} + 2 \zeta_{18}^{2} - 2 \zeta_{18}^{3} + 2 \zeta_{18}^{5} ) q^{79} + ( -5 + 5 \zeta_{18}^{2} - \zeta_{18}^{3} - 3 \zeta_{18}^{4} - 6 \zeta_{18}^{5} ) q^{81} + ( -4 - 4 \zeta_{18} - \zeta_{18}^{3} + 5 \zeta_{18}^{4} ) q^{82} + ( -2 - 9 \zeta_{18} + \zeta_{18}^{2} + 2 \zeta_{18}^{3} + 8 \zeta_{18}^{4} + 8 \zeta_{18}^{5} ) q^{83} + ( -2 \zeta_{18} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{5} ) q^{84} + ( -6 + 4 \zeta_{18} + \zeta_{18}^{2} + 4 \zeta_{18}^{3} - 6 \zeta_{18}^{4} ) q^{86} + ( -2 \zeta_{18} + 4 \zeta_{18}^{3} - 2 \zeta_{18}^{5} ) q^{87} + ( -4 + \zeta_{18} + 4 \zeta_{18}^{3} - \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{88} + ( 8 + 8 \zeta_{18} + \zeta_{18}^{2} - 8 \zeta_{18}^{4} - \zeta_{18}^{5} ) q^{89} + ( 4 - 8 \zeta_{18} + 4 \zeta_{18}^{3} + 4 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{91} + ( -4 \zeta_{18} + 2 \zeta_{18}^{2} + 2 \zeta_{18}^{3} - 2 \zeta_{18}^{5} ) q^{92} + ( -2 + 6 \zeta_{18} + 4 \zeta_{18}^{2} - 4 \zeta_{18}^{3} - 6 \zeta_{18}^{4} + 2 \zeta_{18}^{5} ) q^{93} + ( -4 - 2 \zeta_{18} - 2 \zeta_{18}^{2} + 6 \zeta_{18}^{4} - 4 \zeta_{18}^{5} ) q^{94} + ( -\zeta_{18} - \zeta_{18}^{2} + \zeta_{18}^{4} ) q^{96} + ( -6 + 4 \zeta_{18} - 3 \zeta_{18}^{2} + 3 \zeta_{18}^{3} - 4 \zeta_{18}^{4} + 6 \zeta_{18}^{5} ) q^{97} + ( -4 + \zeta_{18} - 4 \zeta_{18}^{2} ) q^{98} + ( -6 + 5 \zeta_{18} + \zeta_{18}^{3} + \zeta_{18}^{4} - 5 \zeta_{18}^{5} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 3q^{3} + 3q^{6} + 3q^{8} - 3q^{9} + O(q^{10}) \) \( 6q - 3q^{3} + 3q^{6} + 3q^{8} - 3q^{9} + 12q^{11} + 6q^{13} - 12q^{14} + 6q^{17} + 6q^{18} + 6q^{19} + 12q^{21} + 6q^{22} + 6q^{23} + 3q^{24} + 6q^{26} + 3q^{27} - 6q^{28} - 12q^{29} + 12q^{31} + 15q^{33} + 12q^{34} - 3q^{36} + 24q^{37} + 9q^{38} - 24q^{39} + 27q^{41} + 6q^{42} - 30q^{43} - 6q^{44} + 12q^{46} + 24q^{47} + 6q^{48} - 3q^{49} + 3q^{51} + 6q^{52} + 12q^{53} - 9q^{54} + 6q^{57} + 12q^{58} + 3q^{59} - 12q^{61} + 18q^{63} - 3q^{64} + 21q^{66} + 27q^{67} - 3q^{68} + 12q^{69} + 24q^{71} - 6q^{72} - 12q^{73} + 12q^{77} - 18q^{78} + 18q^{79} - 33q^{81} - 27q^{82} - 6q^{83} + 6q^{84} - 24q^{86} + 12q^{87} - 12q^{88} + 48q^{89} + 36q^{91} + 6q^{92} - 24q^{93} - 24q^{94} - 27q^{97} - 24q^{98} - 33q^{99} + O(q^{100}) \)

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 −1.43969 0.524005i 0.173648 0.984808i 0 1.43969 0.524005i −0.347296 + 0.601535i 0.500000 + 0.866025i −0.500000 0.419550i 0
251.1 0.939693 + 0.342020i −0.326352 1.85083i 0.766044 + 0.642788i 0 0.326352 1.85083i −1.53209 + 2.65366i 0.500000 + 0.866025i −0.500000 + 0.181985i 0
301.1 −0.766044 0.642788i −1.43969 + 0.524005i 0.173648 + 0.984808i 0 1.43969 + 0.524005i −0.347296 0.601535i 0.500000 0.866025i −0.500000 + 0.419550i 0
351.1 −0.173648 0.984808i 0.266044 0.223238i −0.939693 + 0.342020i 0 −0.266044 0.223238i 1.87939 3.25519i 0.500000 + 0.866025i −0.500000 + 2.83564i 0
651.1 0.939693 0.342020i −0.326352 + 1.85083i 0.766044 0.642788i 0 0.326352 + 1.85083i −1.53209 2.65366i 0.500000 0.866025i −0.500000 0.181985i 0
701.1 −0.173648 + 0.984808i 0.266044 + 0.223238i −0.939693 0.342020i 0 −0.266044 + 0.223238i 1.87939 + 3.25519i 0.500000 0.866025i −0.500000 2.83564i 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.c 6
5.b even 2 1 190.2.k.a 6
5.c odd 4 2 950.2.u.c 12
19.e even 9 1 inner 950.2.l.c 6
95.o odd 18 1 3610.2.a.w 3
95.p even 18 1 190.2.k.a 6
95.p even 18 1 3610.2.a.x 3
95.q odd 36 2 950.2.u.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
190.2.k.a 6 5.b even 2 1
190.2.k.a 6 95.p even 18 1
950.2.l.c 6 1.a even 1 1 trivial
950.2.l.c 6 19.e even 9 1 inner
950.2.u.c 12 5.c odd 4 2
950.2.u.c 12 95.q odd 36 2
3610.2.a.w 3 95.o odd 18 1
3610.2.a.x 3 95.p even 18 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 3 T_{3}^{5} + 6 T_{3}^{4} + 8 T_{3}^{3} + 3 T_{3}^{2} - 3 T_{3} + 1 \) acting on \(S_{2}^{\mathrm{new}}(950, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{3} + T^{6} \)
$3$ \( 1 - 3 T + 3 T^{2} + 8 T^{3} + 6 T^{4} + 3 T^{5} + T^{6} \)
$5$ \( T^{6} \)
$7$ \( 64 + 96 T + 144 T^{2} + 16 T^{3} + 12 T^{4} + T^{6} \)
$11$ \( 2601 - 2295 T + 1413 T^{2} - 438 T^{3} + 99 T^{4} - 12 T^{5} + T^{6} \)
$13$ \( 64 - 192 T + 240 T^{2} - 152 T^{3} + 48 T^{4} - 6 T^{5} + T^{6} \)
$17$ \( 9 - 54 T + 144 T^{2} - 111 T^{3} + 36 T^{4} - 6 T^{5} + T^{6} \)
$19$ \( 6859 - 2166 T - 228 T^{2} + 169 T^{3} - 12 T^{4} - 6 T^{5} + T^{6} \)
$23$ \( 576 + 576 T^{2} + 240 T^{3} - 6 T^{5} + T^{6} \)
$29$ \( 576 + 1728 T + 2304 T^{2} + 888 T^{3} + 144 T^{4} + 12 T^{5} + T^{6} \)
$31$ \( 23104 + 1824 T + 1968 T^{2} - 448 T^{3} + 132 T^{4} - 12 T^{5} + T^{6} \)
$37$ \( ( -4 + T )^{6} \)
$41$ \( 81 + 162 T + 3888 T^{2} - 1692 T^{3} + 306 T^{4} - 27 T^{5} + T^{6} \)
$43$ \( 2809 - 318 T + 5766 T^{2} + 2395 T^{3} + 390 T^{4} + 30 T^{5} + T^{6} \)
$47$ \( 166464 - 102816 T + 28224 T^{2} - 4296 T^{3} + 396 T^{4} - 24 T^{5} + T^{6} \)
$53$ \( 576 - 864 T + 576 T^{2} - 240 T^{3} + 72 T^{4} - 12 T^{5} + T^{6} \)
$59$ \( 239121 + 8802 T + 1980 T^{2} + 624 T^{3} - 18 T^{4} - 3 T^{5} + T^{6} \)
$61$ \( 18496 - 11424 T + 1056 T^{2} + 224 T^{3} + 96 T^{4} + 12 T^{5} + T^{6} \)
$67$ \( 982081 - 383517 T + 60957 T^{2} - 5624 T^{3} + 450 T^{4} - 27 T^{5} + T^{6} \)
$71$ \( 166464 - 102816 T + 28224 T^{2} - 4296 T^{3} + 396 T^{4} - 24 T^{5} + T^{6} \)
$73$ \( 289 - 255 T + 132 T^{2} + T^{3} + 39 T^{4} + 12 T^{5} + T^{6} \)
$79$ \( 64 + 432 T^{2} - 424 T^{3} + 144 T^{4} - 18 T^{5} + T^{6} \)
$83$ \( 25281 + 32913 T + 43803 T^{2} - 924 T^{3} + 243 T^{4} + 6 T^{5} + T^{6} \)
$89$ \( 103041 - 46224 T + 40536 T^{2} - 8961 T^{3} + 936 T^{4} - 48 T^{5} + T^{6} \)
$97$ \( 253009 + 95067 T + 19197 T^{2} + 2656 T^{3} + 324 T^{4} + 27 T^{5} + T^{6} \)
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