Properties

Label 950.2.e.m
Level $950$
Weight $2$
Character orbit 950.e
Analytic conductor $7.586$
Analytic rank $0$
Dimension $8$
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.e (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.39075800976.1
Defining polynomial: \(x^{8} - x^{7} + 12 x^{6} - 13 x^{5} + 125 x^{4} - 116 x^{3} + 232 x^{2} + 96 x + 64\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{5} q^{2} -\beta_{1} q^{3} + ( -1 - \beta_{5} ) q^{4} + ( -\beta_{1} + \beta_{4} ) q^{6} + ( -\beta_{2} + \beta_{3} - \beta_{6} - \beta_{7} ) q^{7} - q^{8} + ( -2 + \beta_{1} - \beta_{2} - 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{9} +O(q^{10})\) \( q -\beta_{5} q^{2} -\beta_{1} q^{3} + ( -1 - \beta_{5} ) q^{4} + ( -\beta_{1} + \beta_{4} ) q^{6} + ( -\beta_{2} + \beta_{3} - \beta_{6} - \beta_{7} ) q^{7} - q^{8} + ( -2 + \beta_{1} - \beta_{2} - 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{9} + ( 2 - \beta_{2} - \beta_{4} - \beta_{7} ) q^{11} + \beta_{4} q^{12} + ( -3 - \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} ) q^{13} + ( 1 + 2 \beta_{5} - \beta_{6} - \beta_{7} ) q^{14} + \beta_{5} q^{16} + ( \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + 2 \beta_{5} + \beta_{7} ) q^{17} + ( -2 + \beta_{3} + \beta_{4} - \beta_{6} ) q^{18} + ( 2 \beta_{1} - \beta_{3} - \beta_{4} + \beta_{5} ) q^{19} + ( -2 + 3 \beta_{1} + 2 \beta_{6} + 2 \beta_{7} ) q^{21} + ( 1 + \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} ) q^{22} + ( -1 - \beta_{2} + 2 \beta_{3} - \beta_{4} - 3 \beta_{5} + \beta_{6} - \beta_{7} ) q^{23} + \beta_{1} q^{24} + ( -4 + \beta_{2} - \beta_{3} + \beta_{4} + \beta_{6} + \beta_{7} ) q^{26} + ( -5 + 2 \beta_{2} - \beta_{3} + 4 \beta_{4} + \beta_{6} + 2 \beta_{7} ) q^{27} + ( 1 + \beta_{2} - \beta_{3} + 2 \beta_{5} ) q^{28} + ( 4 - \beta_{1} + \beta_{3} + 3 \beta_{5} + \beta_{6} - \beta_{7} ) q^{29} + ( 5 - \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{6} - \beta_{7} ) q^{31} + ( 1 + \beta_{5} ) q^{32} + ( -2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - \beta_{5} + \beta_{7} ) q^{33} + ( 2 + \beta_{1} - \beta_{2} + 2 \beta_{5} - \beta_{6} + \beta_{7} ) q^{34} + ( -\beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + 2 \beta_{5} - \beta_{7} ) q^{36} + ( 1 - \beta_{2} + 2 \beta_{4} - \beta_{7} ) q^{37} + ( 2 \beta_{1} - \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{7} ) q^{38} + ( -1 - 2 \beta_{2} + 3 \beta_{3} + 2 \beta_{4} - 3 \beta_{6} - 2 \beta_{7} ) q^{39} + ( 2 + 3 \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + \beta_{5} - 2 \beta_{6} ) q^{41} + ( 2 + 3 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} ) q^{42} + ( -2 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 5 \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{43} + ( -1 + \beta_{1} - \beta_{3} - \beta_{6} + \beta_{7} ) q^{44} + ( 1 - 2 \beta_{2} + \beta_{3} - 2 \beta_{4} - \beta_{6} - 2 \beta_{7} ) q^{46} + ( -3 - \beta_{2} + \beta_{3} - 4 \beta_{5} ) q^{47} + ( \beta_{1} - \beta_{4} ) q^{48} + ( 4 - \beta_{3} + 2 \beta_{4} + \beta_{6} ) q^{49} + ( -3 - \beta_{1} + \beta_{2} - 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{51} + ( -1 + \beta_{1} + 2 \beta_{5} + \beta_{6} + \beta_{7} ) q^{52} + ( 3 + \beta_{2} - \beta_{3} + 4 \beta_{5} ) q^{53} + ( -2 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} + \beta_{7} ) q^{54} + ( \beta_{2} - \beta_{3} + \beta_{6} + \beta_{7} ) q^{56} + ( 10 - 2 \beta_{1} - \beta_{4} + 6 \beta_{5} - 2 \beta_{7} ) q^{57} + ( 5 - \beta_{2} - \beta_{4} - \beta_{7} ) q^{58} + ( 1 + 2 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} ) q^{59} + ( -1 + 3 \beta_{1} - \beta_{2} - 2 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{61} + ( 1 - \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 3 \beta_{5} - \beta_{6} - 2 \beta_{7} ) q^{62} + ( 8 - 5 \beta_{1} + 4 \beta_{2} - \beta_{3} + 2 \beta_{4} + 9 \beta_{5} + 3 \beta_{6} - 3 \beta_{7} ) q^{63} + q^{64} + ( -1 - 2 \beta_{1} - \beta_{2} + 3 \beta_{4} - \beta_{5} - \beta_{6} + \beta_{7} ) q^{66} + ( -2 + \beta_{2} - \beta_{3} - \beta_{5} ) q^{67} + ( 2 + \beta_{3} + \beta_{4} - \beta_{6} ) q^{68} + ( 2 + 2 \beta_{2} - 4 \beta_{3} - \beta_{4} + 4 \beta_{6} + 2 \beta_{7} ) q^{69} + ( -4 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - 5 \beta_{5} - \beta_{7} ) q^{71} + ( 2 - \beta_{1} + \beta_{2} + 2 \beta_{5} + \beta_{6} - \beta_{7} ) q^{72} + ( -2 - \beta_{1} - 4 \beta_{5} + 2 \beta_{6} + 2 \beta_{7} ) q^{73} + ( 1 + 4 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{6} ) q^{74} + ( -\beta_{4} + \beta_{7} ) q^{76} + ( 4 - \beta_{2} + 3 \beta_{3} + 3 \beta_{4} - 3 \beta_{6} - \beta_{7} ) q^{77} + ( 2 + \beta_{2} + \beta_{3} + \beta_{4} + 5 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} ) q^{78} + ( 1 + \beta_{1} - 2 \beta_{2} - 2 \beta_{3} - 2 \beta_{4} + 6 \beta_{5} - \beta_{6} + \beta_{7} ) q^{79} + ( 2 + 2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + 10 \beta_{5} - 2 \beta_{6} - 3 \beta_{7} ) q^{81} + ( -1 + 3 \beta_{1} - 2 \beta_{3} - \beta_{4} + \beta_{5} - 2 \beta_{6} + 2 \beta_{7} ) q^{82} + ( 7 + \beta_{3} - \beta_{4} - \beta_{6} ) q^{83} + ( 4 - 2 \beta_{2} + 2 \beta_{3} - 3 \beta_{4} - 2 \beta_{6} - 2 \beta_{7} ) q^{84} + ( -3 - \beta_{1} - \beta_{2} + 2 \beta_{3} - 5 \beta_{5} + \beta_{6} - \beta_{7} ) q^{86} + ( 1 - \beta_{3} - 7 \beta_{4} + \beta_{6} ) q^{87} + ( -2 + \beta_{2} + \beta_{4} + \beta_{7} ) q^{88} + ( 5 + 2 \beta_{1} - 2 \beta_{2} + 2 \beta_{3} - 2 \beta_{4} + 3 \beta_{5} ) q^{89} + ( -5 + 7 \beta_{1} + \beta_{2} - 2 \beta_{3} - 6 \beta_{4} - 3 \beta_{5} - \beta_{6} + \beta_{7} ) q^{91} + ( 2 - \beta_{2} - \beta_{3} - \beta_{4} + 3 \beta_{5} - 2 \beta_{6} - \beta_{7} ) q^{92} + ( -4 + \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} + \beta_{5} + 4 \beta_{6} + 3 \beta_{7} ) q^{93} + ( -2 - \beta_{2} + \beta_{3} - \beta_{6} - \beta_{7} ) q^{94} -\beta_{4} q^{96} + ( -\beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - 4 \beta_{5} + \beta_{7} ) q^{97} + ( 4 \beta_{1} - \beta_{2} - \beta_{3} - \beta_{4} - 4 \beta_{5} + \beta_{7} ) q^{98} + ( -15 - 4 \beta_{1} - 2 \beta_{2} + \beta_{3} + 5 \beta_{4} - 16 \beta_{5} - \beta_{6} + \beta_{7} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{2} - q^{3} - 4q^{4} + q^{6} - 12q^{7} - 8q^{8} - 11q^{9} + O(q^{10}) \) \( 8q + 4q^{2} - q^{3} - 4q^{4} + q^{6} - 12q^{7} - 8q^{8} - 11q^{9} + 10q^{11} + 2q^{12} - 9q^{13} - 6q^{14} - 4q^{16} - 5q^{17} - 22q^{18} - q^{21} + 5q^{22} - 6q^{23} + q^{24} - 18q^{26} - 16q^{27} + 6q^{28} + 17q^{29} + 22q^{31} + 4q^{32} + 4q^{33} + 5q^{34} - 11q^{36} + 8q^{37} - 36q^{39} + 7q^{41} + q^{42} + 13q^{43} - 5q^{44} - 12q^{46} - 14q^{47} - q^{48} + 44q^{49} - 9q^{51} - 9q^{52} + 14q^{53} - 8q^{54} + 12q^{56} + 48q^{57} + 34q^{58} + 14q^{59} - 9q^{61} + 11q^{62} + 45q^{63} + 8q^{64} - 4q^{66} - 6q^{67} + 10q^{68} + 54q^{69} + 14q^{71} + 11q^{72} + 11q^{73} + 4q^{74} + 10q^{77} - 18q^{78} - 17q^{79} - 36q^{81} - 7q^{82} + 46q^{83} + 2q^{84} - 13q^{86} + 2q^{87} - 10q^{88} + 14q^{89} - 25q^{91} - 6q^{92} - 13q^{93} - 28q^{94} - 2q^{96} + 17q^{97} + 22q^{98} - 60q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - x^{7} + 12 x^{6} - 13 x^{5} + 125 x^{4} - 116 x^{3} + 232 x^{2} + 96 x + 64\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( 3 \nu^{7} + 403 \nu^{6} - 150 \nu^{5} + 4817 \nu^{4} - 2663 \nu^{3} + 45286 \nu^{2} - 24992 \nu + 58672 \)\()/15888\)
\(\beta_{3}\)\(=\)\((\)\( -93 \nu^{7} + 85 \nu^{6} - 2301 \nu^{5} + 2933 \nu^{4} - 20057 \nu^{3} + 31681 \nu^{2} - 116300 \nu + 82432 \)\()/39720\)
\(\beta_{4}\)\(=\)\((\)\( -154 \nu^{7} - 55 \nu^{6} - 1568 \nu^{5} - 126 \nu^{4} - 19421 \nu^{3} - 4592 \nu^{2} - 2240 \nu - 17824 \)\()/39720\)
\(\beta_{5}\)\(=\)\((\)\( 557 \nu^{7} - 865 \nu^{6} + 6574 \nu^{5} - 10377 \nu^{4} + 69373 \nu^{3} - 103454 \nu^{2} + 120040 \nu - 30448 \)\()/79440\)
\(\beta_{6}\)\(=\)\((\)\( -456 \nu^{7} + 310 \nu^{6} - 5997 \nu^{5} + 2636 \nu^{4} - 61934 \nu^{3} + 20857 \nu^{2} - 121580 \nu - 124136 \)\()/39720\)
\(\beta_{7}\)\(=\)\((\)\( 944 \nu^{7} - 845 \nu^{6} + 10063 \nu^{5} - 11264 \nu^{4} + 104841 \nu^{3} - 84843 \nu^{2} + 76320 \nu + 145024 \)\()/39720\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{7} - \beta_{6} - 5 \beta_{5} - \beta_{2} + \beta_{1} - 5\)
\(\nu^{3}\)\(=\)\(-2 \beta_{7} - \beta_{6} - 10 \beta_{4} + \beta_{3} - 2 \beta_{2} + 5\)
\(\nu^{4}\)\(=\)\(-12 \beta_{7} - 2 \beta_{6} + 46 \beta_{5} + 10 \beta_{4} + 10 \beta_{3} + 10 \beta_{2} - 7 \beta_{1} + 2\)
\(\nu^{5}\)\(=\)\(13 \beta_{7} - 13 \beta_{6} - 25 \beta_{5} + 88 \beta_{4} - 24 \beta_{3} + 11 \beta_{2} - 75 \beta_{1} - 49\)
\(\nu^{6}\)\(=\)\(22 \beta_{7} + 123 \beta_{6} - 154 \beta_{4} - 123 \beta_{3} + 22 \beta_{2} + 409\)
\(\nu^{7}\)\(=\)\(92 \beta_{7} + 246 \beta_{6} + 366 \beta_{5} + 154 \beta_{4} + 154 \beta_{3} + 154 \beta_{2} + 725 \beta_{1} - 246\)

Character values

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

\(n\) \(77\) \(401\)
\(\chi(n)\) \(1\) \(-1 - \beta_{5}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
201.1
1.51772 2.62877i
0.851703 1.47519i
−0.236942 + 0.410396i
−1.63248 + 2.82754i
1.51772 + 2.62877i
0.851703 + 1.47519i
−0.236942 0.410396i
−1.63248 2.82754i
0.500000 0.866025i −1.51772 + 2.62877i −0.500000 0.866025i 0 1.51772 + 2.62877i −4.93155 −1.00000 −3.10694 5.38138i 0
201.2 0.500000 0.866025i −0.851703 + 1.47519i −0.500000 0.866025i 0 0.851703 + 1.47519i 3.74324 −1.00000 0.0492032 + 0.0852224i 0
201.3 0.500000 0.866025i 0.236942 0.410396i −0.500000 0.866025i 0 −0.236942 0.410396i −2.19155 −1.00000 1.38772 + 2.40360i 0
201.4 0.500000 0.866025i 1.63248 2.82754i −0.500000 0.866025i 0 −1.63248 2.82754i −2.62013 −1.00000 −3.82998 6.63372i 0
501.1 0.500000 + 0.866025i −1.51772 2.62877i −0.500000 + 0.866025i 0 1.51772 2.62877i −4.93155 −1.00000 −3.10694 + 5.38138i 0
501.2 0.500000 + 0.866025i −0.851703 1.47519i −0.500000 + 0.866025i 0 0.851703 1.47519i 3.74324 −1.00000 0.0492032 0.0852224i 0
501.3 0.500000 + 0.866025i 0.236942 + 0.410396i −0.500000 + 0.866025i 0 −0.236942 + 0.410396i −2.19155 −1.00000 1.38772 2.40360i 0
501.4 0.500000 + 0.866025i 1.63248 + 2.82754i −0.500000 + 0.866025i 0 −1.63248 + 2.82754i −2.62013 −1.00000 −3.82998 + 6.63372i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 501.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.e.m yes 8
5.b even 2 1 950.2.e.l 8
5.c odd 4 2 950.2.j.i 16
19.c even 3 1 inner 950.2.e.m yes 8
95.i even 6 1 950.2.e.l 8
95.m odd 12 2 950.2.j.i 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.e.l 8 5.b even 2 1
950.2.e.l 8 95.i even 6 1
950.2.e.m yes 8 1.a even 1 1 trivial
950.2.e.m yes 8 19.c even 3 1 inner
950.2.j.i 16 5.c odd 4 2
950.2.j.i 16 95.m odd 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(950, [\chi])\):

\(T_{3}^{8} + \cdots\)
\( T_{7}^{4} + 6 T_{7}^{3} - 7 T_{7}^{2} - 82 T_{7} - 106 \)
\( T_{11}^{4} - 5 T_{11}^{3} - 20 T_{11}^{2} + 123 T_{11} - 117 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T + T^{2} )^{4} \)
$3$ \( 64 - 96 T + 232 T^{2} + 116 T^{3} + 125 T^{4} + 13 T^{5} + 12 T^{6} + T^{7} + T^{8} \)
$5$ \( T^{8} \)
$7$ \( ( -106 - 82 T - 7 T^{2} + 6 T^{3} + T^{4} )^{2} \)
$11$ \( ( -117 + 123 T - 20 T^{2} - 5 T^{3} + T^{4} )^{2} \)
$13$ \( 64516 + 36576 T + 20990 T^{2} + 4428 T^{3} + 1551 T^{4} + 297 T^{5} + 80 T^{6} + 9 T^{7} + T^{8} \)
$17$ \( 50625 - 16875 T + 11925 T^{2} - 150 T^{3} + 934 T^{4} + 10 T^{5} + 53 T^{6} + 5 T^{7} + T^{8} \)
$19$ \( 130321 + 12635 T^{2} + 684 T^{4} + 35 T^{6} + T^{8} \)
$23$ \( ( 1296 - 108 T + 45 T^{2} + 3 T^{3} + T^{4} )^{2} \)
$29$ \( 202500 + 35550 T^{2} - 15300 T^{3} + 6691 T^{4} - 1343 T^{5} + 210 T^{6} - 17 T^{7} + T^{8} \)
$31$ \( ( 1118 + 360 T - 41 T^{2} - 11 T^{3} + T^{4} )^{2} \)
$37$ \( ( 3096 + 228 T - 110 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$41$ \( 5948721 - 2538999 T + 839781 T^{2} - 138246 T^{3} + 19726 T^{4} - 1382 T^{5} + 149 T^{6} - 7 T^{7} + T^{8} \)
$43$ \( 29584 - 51600 T + 90172 T^{2} - 4172 T^{3} + 4073 T^{4} - 613 T^{5} + 168 T^{6} - 13 T^{7} + T^{8} \)
$47$ \( 54756 + 1404 T + 12438 T^{2} + 6234 T^{3} + 3127 T^{4} + 754 T^{5} + 143 T^{6} + 14 T^{7} + T^{8} \)
$53$ \( 54756 - 1404 T + 12438 T^{2} - 6234 T^{3} + 3127 T^{4} - 754 T^{5} + 143 T^{6} - 14 T^{7} + T^{8} \)
$59$ \( 363609 - 97686 T + 49158 T^{2} - 10728 T^{3} + 4315 T^{4} - 856 T^{5} + 158 T^{6} - 14 T^{7} + T^{8} \)
$61$ \( 256 + 2048 T + 15792 T^{2} + 5024 T^{3} + 2537 T^{4} - 77 T^{5} + 118 T^{6} + 9 T^{7} + T^{8} \)
$67$ \( 16 + 56 T + 168 T^{2} + 146 T^{3} + 137 T^{4} - 14 T^{5} + 43 T^{6} + 6 T^{7} + T^{8} \)
$71$ \( 2862864 - 1664928 T + 895500 T^{2} - 89688 T^{3} + 17317 T^{4} - 1366 T^{5} + 239 T^{6} - 14 T^{7} + T^{8} \)
$73$ \( 6718464 - 2426112 T + 738720 T^{2} - 106632 T^{3} + 15697 T^{4} - 1289 T^{5} + 174 T^{6} - 11 T^{7} + T^{8} \)
$79$ \( 7022500 + 2703000 T + 963550 T^{2} + 119680 T^{3} + 20831 T^{4} + 1547 T^{5} + 318 T^{6} + 17 T^{7} + T^{8} \)
$83$ \( ( -1872 + 144 T + 131 T^{2} - 23 T^{3} + T^{4} )^{2} \)
$89$ \( 8555625 - 3246750 T + 1103400 T^{2} - 130740 T^{3} + 20401 T^{4} - 1604 T^{5} + 240 T^{6} - 14 T^{7} + T^{8} \)
$97$ \( 81 - 459 T + 2691 T^{2} + 204 T^{3} + 976 T^{4} - 272 T^{5} + 279 T^{6} - 17 T^{7} + T^{8} \)
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