Properties

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

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(15.5708283941\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 78)
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.

\(f(q)\) \(=\) \( q + (\zeta_{12}^{2} - 1) q^{2} + \zeta_{12} q^{3} - \zeta_{12}^{2} q^{4} + (\zeta_{12}^{3} - \zeta_{12}) q^{6} + (\zeta_{12}^{3} - 3 \zeta_{12}^{2} + \zeta_{12}) q^{7} + q^{8} + \zeta_{12}^{2} q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{12}^{2} - 1) q^{2} + \zeta_{12} q^{3} - \zeta_{12}^{2} q^{4} + (\zeta_{12}^{3} - \zeta_{12}) q^{6} + (\zeta_{12}^{3} - 3 \zeta_{12}^{2} + \zeta_{12}) q^{7} + q^{8} + \zeta_{12}^{2} q^{9} + (\zeta_{12}^{2} - 3 \zeta_{12} + 1) q^{11} - \zeta_{12}^{3} q^{12} + (3 \zeta_{12}^{2} - 2 \zeta_{12} - 3) q^{13} + (\zeta_{12}^{3} - 2 \zeta_{12} + 3) q^{14} + (\zeta_{12}^{2} - 1) q^{16} + ( - 3 \zeta_{12}^{2} + 6) q^{17} - q^{18} + ( - 3 \zeta_{12}^{3} - \zeta_{12}^{2} + 3 \zeta_{12} + 2) q^{19} + ( - 3 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{21} + ( - 3 \zeta_{12}^{3} + \zeta_{12}^{2} + 3 \zeta_{12} - 2) q^{22} + ( - 3 \zeta_{12}^{2} - 3 \zeta_{12} - 3) q^{23} + \zeta_{12} q^{24} + ( - 2 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 2 \zeta_{12}) q^{26} + \zeta_{12}^{3} q^{27} + ( - 2 \zeta_{12}^{3} + 3 \zeta_{12}^{2} + \zeta_{12} - 3) q^{28} + (3 \zeta_{12}^{2} - 3) q^{29} + ( - 6 \zeta_{12}^{3} - 4 \zeta_{12}^{2} + 2) q^{31} - \zeta_{12}^{2} q^{32} + (\zeta_{12}^{3} - 3 \zeta_{12}^{2} + \zeta_{12}) q^{33} + (6 \zeta_{12}^{2} - 3) q^{34} + ( - \zeta_{12}^{2} + 1) q^{36} + (3 \zeta_{12}^{2} - 3) q^{37} + (3 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{38} + (3 \zeta_{12}^{3} - 2 \zeta_{12}^{2} - 3 \zeta_{12}) q^{39} + (2 \zeta_{12}^{2} + 3 \zeta_{12} + 2) q^{41} + ( - \zeta_{12}^{2} + 3 \zeta_{12} - 1) q^{42} + (\zeta_{12}^{3} - 3 \zeta_{12}^{2} - \zeta_{12} + 6) q^{43} + (3 \zeta_{12}^{3} - 2 \zeta_{12}^{2} + 1) q^{44} + ( - 3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + 3 \zeta_{12} + 6) q^{46} + (\zeta_{12}^{3} - 2 \zeta_{12} - 3) q^{47} + (\zeta_{12}^{3} - \zeta_{12}) q^{48} + ( - 12 \zeta_{12}^{3} + 5 \zeta_{12}^{2} + 6 \zeta_{12} - 5) q^{49} + ( - 3 \zeta_{12}^{3} + 6 \zeta_{12}) q^{51} + (2 \zeta_{12}^{3} + 3) q^{52} + 3 \zeta_{12}^{3} q^{53} - \zeta_{12} q^{54} + (\zeta_{12}^{3} - 3 \zeta_{12}^{2} + \zeta_{12}) q^{56} + ( - \zeta_{12}^{3} + 2 \zeta_{12} + 3) q^{57} - 3 \zeta_{12}^{2} q^{58} + ( - 8 \zeta_{12}^{2} + 16) q^{59} + ( - 3 \zeta_{12}^{3} - 10 \zeta_{12}^{2} - 3 \zeta_{12}) q^{61} + (2 \zeta_{12}^{2} + 6 \zeta_{12} + 2) q^{62} + (2 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - \zeta_{12} + 3) q^{63} + q^{64} + (\zeta_{12}^{3} - 2 \zeta_{12} + 3) q^{66} + (2 \zeta_{12}^{3} - 9 \zeta_{12}^{2} - \zeta_{12} + 9) q^{67} + ( - 3 \zeta_{12}^{2} - 3) q^{68} + ( - 3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 3 \zeta_{12}) q^{69} + (3 \zeta_{12}^{3} - 3 \zeta_{12}^{2} - 3 \zeta_{12} + 6) q^{71} + \zeta_{12}^{2} q^{72} + (7 \zeta_{12}^{3} - 14 \zeta_{12}) q^{73} - 3 \zeta_{12}^{2} q^{74} + ( - \zeta_{12}^{2} - 3 \zeta_{12} - 1) q^{76} + (12 \zeta_{12}^{3} - 12 \zeta_{12}^{2} + 6) q^{77} + ( - 3 \zeta_{12}^{3} + 2) q^{78} + (6 \zeta_{12}^{3} - 12 \zeta_{12} + 2) q^{79} + (\zeta_{12}^{2} - 1) q^{81} + (3 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 3 \zeta_{12} - 4) q^{82} + ( - 5 \zeta_{12}^{3} + 10 \zeta_{12} - 3) q^{83} + (3 \zeta_{12}^{3} - \zeta_{12}^{2} - 3 \zeta_{12} + 2) q^{84} + ( - \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 3) q^{86} + (3 \zeta_{12}^{3} - 3 \zeta_{12}) q^{87} + (\zeta_{12}^{2} - 3 \zeta_{12} + 1) q^{88} + (2 \zeta_{12}^{2} + 6 \zeta_{12} + 2) q^{89} + (9 \zeta_{12}^{3} - 4 \zeta_{12}^{2} - 6 \zeta_{12} + 11) q^{91} + (3 \zeta_{12}^{3} + 6 \zeta_{12}^{2} - 3) q^{92} + ( - 4 \zeta_{12}^{3} - 6 \zeta_{12}^{2} + 2 \zeta_{12} + 6) q^{93} + ( - 2 \zeta_{12}^{3} - 3 \zeta_{12}^{2} + \zeta_{12} + 3) q^{94} - \zeta_{12}^{3} q^{96} + 6 \zeta_{12}^{2} q^{97} + (6 \zeta_{12}^{3} - 5 \zeta_{12}^{2} + 6 \zeta_{12}) q^{98} + ( - 3 \zeta_{12}^{3} + 2 \zeta_{12}^{2} - 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - 2 q^{4} - 6 q^{7} + 4 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - 2 q^{4} - 6 q^{7} + 4 q^{8} + 2 q^{9} + 6 q^{11} - 6 q^{13} + 12 q^{14} - 2 q^{16} + 18 q^{17} - 4 q^{18} + 6 q^{19} - 6 q^{22} - 18 q^{23} - 6 q^{26} - 6 q^{28} - 6 q^{29} - 2 q^{32} - 6 q^{33} + 2 q^{36} - 6 q^{37} - 4 q^{39} + 12 q^{41} - 6 q^{42} + 18 q^{43} + 18 q^{46} - 12 q^{47} - 10 q^{49} + 12 q^{52} - 6 q^{56} + 12 q^{57} - 6 q^{58} + 48 q^{59} - 20 q^{61} + 12 q^{62} + 6 q^{63} + 4 q^{64} + 12 q^{66} + 18 q^{67} - 18 q^{68} - 6 q^{69} + 18 q^{71} + 2 q^{72} - 6 q^{74} - 6 q^{76} + 8 q^{78} + 8 q^{79} - 2 q^{81} - 12 q^{82} - 12 q^{83} + 6 q^{84} + 6 q^{88} + 12 q^{89} + 36 q^{91} + 12 q^{93} + 6 q^{94} + 12 q^{97} - 10 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(301\) \(1301\) \(1327\)
\(\chi(n)\) \(\zeta_{12}^{2}\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 0.500000i
0.866025 + 0.500000i
−0.500000 0.866025i −0.866025 + 0.500000i −0.500000 + 0.866025i 0 0.866025 + 0.500000i −2.36603 + 4.09808i 1.00000 0.500000 0.866025i 0
49.2 −0.500000 0.866025i 0.866025 0.500000i −0.500000 + 0.866025i 0 −0.866025 0.500000i −0.633975 + 1.09808i 1.00000 0.500000 0.866025i 0
199.1 −0.500000 + 0.866025i −0.866025 0.500000i −0.500000 0.866025i 0 0.866025 0.500000i −2.36603 4.09808i 1.00000 0.500000 + 0.866025i 0
199.2 −0.500000 + 0.866025i 0.866025 + 0.500000i −0.500000 0.866025i 0 −0.866025 + 0.500000i −0.633975 1.09808i 1.00000 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
65.l 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.y.a 4
5.b even 2 1 1950.2.y.h 4
5.c odd 4 1 78.2.i.b 4
5.c odd 4 1 1950.2.bc.c 4
13.e even 6 1 1950.2.y.h 4
15.e even 4 1 234.2.l.a 4
20.e even 4 1 624.2.bv.d 4
60.l odd 4 1 1872.2.by.k 4
65.f even 4 1 1014.2.e.j 4
65.h odd 4 1 1014.2.i.f 4
65.k even 4 1 1014.2.e.h 4
65.l even 6 1 inner 1950.2.y.a 4
65.o even 12 1 1014.2.a.j 2
65.o even 12 1 1014.2.e.h 4
65.q odd 12 1 1014.2.b.d 4
65.q odd 12 1 1014.2.i.f 4
65.r odd 12 1 78.2.i.b 4
65.r odd 12 1 1014.2.b.d 4
65.r odd 12 1 1950.2.bc.c 4
65.t even 12 1 1014.2.a.h 2
65.t even 12 1 1014.2.e.j 4
195.bc odd 12 1 3042.2.a.v 2
195.bf even 12 1 234.2.l.a 4
195.bf even 12 1 3042.2.b.l 4
195.bl even 12 1 3042.2.b.l 4
195.bn odd 12 1 3042.2.a.s 2
260.be odd 12 1 8112.2.a.bx 2
260.bg even 12 1 624.2.bv.d 4
260.bl odd 12 1 8112.2.a.bq 2
780.cw odd 12 1 1872.2.by.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.i.b 4 5.c odd 4 1
78.2.i.b 4 65.r odd 12 1
234.2.l.a 4 15.e even 4 1
234.2.l.a 4 195.bf even 12 1
624.2.bv.d 4 20.e even 4 1
624.2.bv.d 4 260.bg even 12 1
1014.2.a.h 2 65.t even 12 1
1014.2.a.j 2 65.o even 12 1
1014.2.b.d 4 65.q odd 12 1
1014.2.b.d 4 65.r odd 12 1
1014.2.e.h 4 65.k even 4 1
1014.2.e.h 4 65.o even 12 1
1014.2.e.j 4 65.f even 4 1
1014.2.e.j 4 65.t even 12 1
1014.2.i.f 4 65.h odd 4 1
1014.2.i.f 4 65.q odd 12 1
1872.2.by.k 4 60.l odd 4 1
1872.2.by.k 4 780.cw odd 12 1
1950.2.y.a 4 1.a even 1 1 trivial
1950.2.y.a 4 65.l even 6 1 inner
1950.2.y.h 4 5.b even 2 1
1950.2.y.h 4 13.e even 6 1
1950.2.bc.c 4 5.c odd 4 1
1950.2.bc.c 4 65.r odd 12 1
3042.2.a.s 2 195.bn odd 12 1
3042.2.a.v 2 195.bc odd 12 1
3042.2.b.l 4 195.bf even 12 1
3042.2.b.l 4 195.bl even 12 1
8112.2.a.bq 2 260.bl odd 12 1
8112.2.a.bx 2 260.be odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} + 6T_{7}^{3} + 30T_{7}^{2} + 36T_{7} + 36 \) acting on \(S_{2}^{\mathrm{new}}(1950, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 6 T^{3} + 30 T^{2} + 36 T + 36 \) Copy content Toggle raw display
$11$ \( T^{4} - 6 T^{3} + 6 T^{2} + 36 T + 36 \) Copy content Toggle raw display
$13$ \( T^{4} + 6 T^{3} + 23 T^{2} + 78 T + 169 \) Copy content Toggle raw display
$17$ \( (T^{2} - 9 T + 27)^{2} \) Copy content Toggle raw display
$19$ \( T^{4} - 6 T^{3} + 6 T^{2} + 36 T + 36 \) Copy content Toggle raw display
$23$ \( T^{4} + 18 T^{3} + 126 T^{2} + \cdots + 324 \) Copy content Toggle raw display
$29$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 96T^{2} + 576 \) Copy content Toggle raw display
$37$ \( (T^{2} + 3 T + 9)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} - 12 T^{3} + 51 T^{2} - 36 T + 9 \) Copy content Toggle raw display
$43$ \( T^{4} - 18 T^{3} + 134 T^{2} + \cdots + 676 \) Copy content Toggle raw display
$47$ \( (T^{2} + 6 T + 6)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 9)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 24 T + 192)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 20 T^{3} + 327 T^{2} + \cdots + 5329 \) Copy content Toggle raw display
$67$ \( T^{4} - 18 T^{3} + 246 T^{2} + \cdots + 6084 \) Copy content Toggle raw display
$71$ \( T^{4} - 18 T^{3} + 126 T^{2} + \cdots + 324 \) Copy content Toggle raw display
$73$ \( (T^{2} - 147)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 4 T - 104)^{2} \) Copy content Toggle raw display
$83$ \( (T^{2} + 6 T - 66)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} - 12 T^{3} + 24 T^{2} + \cdots + 576 \) Copy content Toggle raw display
$97$ \( (T^{2} - 6 T + 36)^{2} \) Copy content Toggle raw display
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