Properties

Label 1950.2.z.p
Level $1950$
Weight $2$
Character orbit 1950.z
Analytic conductor $15.571$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1950 = 2 \cdot 3 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1950.z (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.5708283941\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.3317760000.2
Defining polynomial: \(x^{8} - 25 x^{4} + 625\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{1} q^{2} + \beta_{1} q^{3} + \beta_{2} q^{4} + \beta_{2} q^{6} + ( 2 \beta_{1} - 2 \beta_{3} + \beta_{5} ) q^{7} + \beta_{3} q^{8} + \beta_{2} q^{9} +O(q^{10})\) \( q + \beta_{1} q^{2} + \beta_{1} q^{3} + \beta_{2} q^{4} + \beta_{2} q^{6} + ( 2 \beta_{1} - 2 \beta_{3} + \beta_{5} ) q^{7} + \beta_{3} q^{8} + \beta_{2} q^{9} + ( -1 + \beta_{2} ) q^{11} + \beta_{3} q^{12} + ( -\beta_{1} - \beta_{3} + \beta_{5} ) q^{13} + ( 2 + \beta_{7} ) q^{14} + ( -1 + \beta_{2} ) q^{16} + ( 2 \beta_{1} - 2 \beta_{3} + \beta_{5} ) q^{17} + \beta_{3} q^{18} + ( 2 \beta_{2} - \beta_{4} ) q^{19} + ( 2 + \beta_{7} ) q^{21} + ( -\beta_{1} + \beta_{3} ) q^{22} + ( -\beta_{1} - \beta_{6} ) q^{23} + ( -1 + \beta_{2} ) q^{24} + ( 1 - 2 \beta_{2} + \beta_{7} ) q^{26} + \beta_{3} q^{27} + ( 2 \beta_{1} + \beta_{6} ) q^{28} + ( -2 + 2 \beta_{2} - 2 \beta_{4} + 2 \beta_{7} ) q^{29} + ( -4 - \beta_{7} ) q^{31} + ( -\beta_{1} + \beta_{3} ) q^{32} + ( -\beta_{1} + \beta_{3} ) q^{33} + ( 2 + \beta_{7} ) q^{34} + ( -1 + \beta_{2} ) q^{36} + ( -3 \beta_{1} - \beta_{6} ) q^{37} + ( 2 \beta_{3} + \beta_{5} - \beta_{6} ) q^{38} + ( 1 - 2 \beta_{2} + \beta_{7} ) q^{39} + ( -2 + 2 \beta_{2} + \beta_{4} - \beta_{7} ) q^{41} + ( 2 \beta_{1} + \beta_{6} ) q^{42} + \beta_{5} q^{43} - q^{44} + ( -\beta_{2} - \beta_{4} ) q^{46} + 6 \beta_{3} q^{47} + ( -\beta_{1} + \beta_{3} ) q^{48} + ( 7 - 7 \beta_{2} - 4 \beta_{4} + 4 \beta_{7} ) q^{49} + ( 2 + \beta_{7} ) q^{51} + ( \beta_{1} - 2 \beta_{3} + \beta_{6} ) q^{52} + ( 2 \beta_{3} - 3 \beta_{5} + 3 \beta_{6} ) q^{53} + ( -1 + \beta_{2} ) q^{54} + ( 2 \beta_{2} + \beta_{4} ) q^{56} + ( 2 \beta_{3} + \beta_{5} - \beta_{6} ) q^{57} + ( -2 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} ) q^{58} + 10 \beta_{2} q^{59} + ( -3 \beta_{2} - \beta_{4} ) q^{61} + ( -4 \beta_{1} - \beta_{6} ) q^{62} + ( 2 \beta_{1} + \beta_{6} ) q^{63} - q^{64} - q^{66} + ( 2 \beta_{1} + \beta_{6} ) q^{68} + ( -\beta_{2} - \beta_{4} ) q^{69} + ( -\beta_{2} + 3 \beta_{4} ) q^{71} + ( -\beta_{1} + \beta_{3} ) q^{72} + ( 3 \beta_{3} - 2 \beta_{5} + 2 \beta_{6} ) q^{73} + ( -3 \beta_{2} - \beta_{4} ) q^{74} + ( -2 + 2 \beta_{2} - \beta_{4} + \beta_{7} ) q^{76} + ( 2 \beta_{3} - \beta_{5} + \beta_{6} ) q^{77} + ( \beta_{1} - 2 \beta_{3} + \beta_{6} ) q^{78} + ( 4 - 3 \beta_{7} ) q^{79} + ( -1 + \beta_{2} ) q^{81} + ( -2 \beta_{1} + 2 \beta_{3} - \beta_{5} ) q^{82} -9 \beta_{3} q^{83} + ( 2 \beta_{2} + \beta_{4} ) q^{84} + \beta_{7} q^{86} + ( -2 \beta_{1} + 2 \beta_{3} + 2 \beta_{5} ) q^{87} -\beta_{1} q^{88} + ( -2 + 2 \beta_{2} - 3 \beta_{4} + 3 \beta_{7} ) q^{89} + ( 8 - 12 \beta_{2} - 3 \beta_{4} + \beta_{7} ) q^{91} + ( -\beta_{3} + \beta_{5} - \beta_{6} ) q^{92} + ( -4 \beta_{1} - \beta_{6} ) q^{93} + ( -6 + 6 \beta_{2} ) q^{94} - q^{96} + ( -\beta_{1} + \beta_{3} - 2 \beta_{5} ) q^{97} + ( 7 \beta_{1} - 7 \beta_{3} + 4 \beta_{5} ) q^{98} - q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + 4q^{4} + 4q^{6} + 4q^{9} + O(q^{10}) \) \( 8q + 4q^{4} + 4q^{6} + 4q^{9} - 4q^{11} + 16q^{14} - 4q^{16} + 8q^{19} + 16q^{21} - 4q^{24} - 8q^{29} - 32q^{31} + 16q^{34} - 4q^{36} - 8q^{41} - 8q^{44} - 4q^{46} + 28q^{49} + 16q^{51} - 4q^{54} + 8q^{56} + 40q^{59} - 12q^{61} - 8q^{64} - 8q^{66} - 4q^{69} - 4q^{71} - 12q^{74} - 8q^{76} + 32q^{79} - 4q^{81} + 8q^{84} - 8q^{89} + 16q^{91} - 24q^{94} - 8q^{96} - 8q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{8} - 25 x^{4} + 625\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu^{2} \)\(/5\)
\(\beta_{2}\)\(=\)\( \nu^{4} \)\(/25\)
\(\beta_{3}\)\(=\)\( \nu^{6} \)\(/125\)
\(\beta_{4}\)\(=\)\((\)\( \nu^{7} + 125 \nu \)\()/125\)
\(\beta_{5}\)\(=\)\((\)\( -\nu^{7} + 125 \nu \)\()/125\)
\(\beta_{6}\)\(=\)\((\)\( -\nu^{7} + 5 \nu^{5} + 25 \nu^{3} \)\()/125\)
\(\beta_{7}\)\(=\)\((\)\( -\nu^{5} + 5 \nu^{3} + 25 \nu \)\()/25\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{5} + \beta_{4}\)\()/2\)
\(\nu^{2}\)\(=\)\(5 \beta_{1}\)
\(\nu^{3}\)\(=\)\((\)\(5 \beta_{7} + 5 \beta_{6} - 5 \beta_{5}\)\()/2\)
\(\nu^{4}\)\(=\)\(25 \beta_{2}\)
\(\nu^{5}\)\(=\)\((\)\(-25 \beta_{7} + 25 \beta_{6} + 25 \beta_{4}\)\()/2\)
\(\nu^{6}\)\(=\)\(125 \beta_{3}\)
\(\nu^{7}\)\(=\)\((\)\(-125 \beta_{5} + 125 \beta_{4}\)\()/2\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1950\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(1301\) \(1327\)
\(\chi(n)\) \(-\beta_{2}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1699.1
−0.578737 + 2.15988i
0.578737 2.15988i
−2.15988 0.578737i
2.15988 + 0.578737i
−0.578737 2.15988i
0.578737 + 2.15988i
−2.15988 + 0.578737i
2.15988 0.578737i
−0.866025 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i 0 0.500000 + 0.866025i −4.47066 + 2.58114i 1.00000i 0.500000 + 0.866025i 0
1699.2 −0.866025 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i 0 0.500000 + 0.866025i 1.00656 0.581139i 1.00000i 0.500000 + 0.866025i 0
1699.3 0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i 0 0.500000 + 0.866025i −1.00656 + 0.581139i 1.00000i 0.500000 + 0.866025i 0
1699.4 0.866025 + 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i 0 0.500000 + 0.866025i 4.47066 2.58114i 1.00000i 0.500000 + 0.866025i 0
1849.1 −0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i 0 0.500000 0.866025i −4.47066 2.58114i 1.00000i 0.500000 0.866025i 0
1849.2 −0.866025 + 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i 0 0.500000 0.866025i 1.00656 + 0.581139i 1.00000i 0.500000 0.866025i 0
1849.3 0.866025 0.500000i 0.866025 0.500000i 0.500000 0.866025i 0 0.500000 0.866025i −1.00656 0.581139i 1.00000i 0.500000 0.866025i 0
1849.4 0.866025 0.500000i 0.866025 0.500000i 0.500000 0.866025i 0 0.500000 0.866025i 4.47066 + 2.58114i 1.00000i 0.500000 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1849.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
13.c even 3 1 inner
65.n even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1950.2.z.p 8
5.b even 2 1 inner 1950.2.z.p 8
5.c odd 4 1 1950.2.i.bc 4
5.c odd 4 1 1950.2.i.bd yes 4
13.c even 3 1 inner 1950.2.z.p 8
65.n even 6 1 inner 1950.2.z.p 8
65.q odd 12 1 1950.2.i.bc 4
65.q odd 12 1 1950.2.i.bd yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1950.2.i.bc 4 5.c odd 4 1
1950.2.i.bc 4 65.q odd 12 1
1950.2.i.bd yes 4 5.c odd 4 1
1950.2.i.bd yes 4 65.q odd 12 1
1950.2.z.p 8 1.a even 1 1 trivial
1950.2.z.p 8 5.b even 2 1 inner
1950.2.z.p 8 13.c even 3 1 inner
1950.2.z.p 8 65.n even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1950, [\chi])\):

\( T_{7}^{8} - 28 T_{7}^{6} + 748 T_{7}^{4} - 1008 T_{7}^{2} + 1296 \)
\( T_{11}^{2} + T_{11} + 1 \)
\( T_{17}^{8} - 28 T_{17}^{6} + 748 T_{17}^{4} - 1008 T_{17}^{2} + 1296 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$3$ \( ( 1 - T^{2} + T^{4} )^{2} \)
$5$ \( T^{8} \)
$7$ \( 1296 - 1008 T^{2} + 748 T^{4} - 28 T^{6} + T^{8} \)
$11$ \( ( 1 + T + T^{2} )^{4} \)
$13$ \( 28561 - 2366 T^{2} + 27 T^{4} - 14 T^{6} + T^{8} \)
$17$ \( 1296 - 1008 T^{2} + 748 T^{4} - 28 T^{6} + T^{8} \)
$19$ \( ( 36 + 24 T + 22 T^{2} - 4 T^{3} + T^{4} )^{2} \)
$23$ \( 6561 - 1782 T^{2} + 403 T^{4} - 22 T^{6} + T^{8} \)
$29$ \( ( 1296 - 144 T + 52 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$31$ \( ( 6 + 8 T + T^{2} )^{4} \)
$37$ \( 1 - 38 T^{2} + 1443 T^{4} - 38 T^{6} + T^{8} \)
$41$ \( ( 36 - 24 T + 22 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$43$ \( ( 100 - 10 T^{2} + T^{4} )^{2} \)
$47$ \( ( 36 + T^{2} )^{4} \)
$53$ \( ( 7396 + 188 T^{2} + T^{4} )^{2} \)
$59$ \( ( 100 - 10 T + T^{2} )^{4} \)
$61$ \( ( 1 - 6 T + 37 T^{2} + 6 T^{3} + T^{4} )^{2} \)
$67$ \( T^{8} \)
$71$ \( ( 7921 - 178 T + 93 T^{2} + 2 T^{3} + T^{4} )^{2} \)
$73$ \( ( 961 + 98 T^{2} + T^{4} )^{2} \)
$79$ \( ( -74 - 8 T + T^{2} )^{4} \)
$83$ \( ( 81 + T^{2} )^{4} \)
$89$ \( ( 7396 - 344 T + 102 T^{2} + 4 T^{3} + T^{4} )^{2} \)
$97$ \( 2313441 - 124722 T^{2} + 5203 T^{4} - 82 T^{6} + T^{8} \)
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