Properties

Label 1950.2.z.b
Level $1950$
Weight $2$
Character orbit 1950.z
Analytic conductor $15.571$
Analytic rank $0$
Dimension $4$
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: \(4\)
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: \(\Z[a_1, a_2]\)
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.

\(f(q)\) \(=\) \( q + (\zeta_{12}^{3} - \zeta_{12}) q^{2} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{3} + ( - \zeta_{12}^{2} + 1) q^{4} + (\zeta_{12}^{2} - 1) q^{6} + 2 \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} + ( - \zeta_{12}^{2} + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{12}^{3} - \zeta_{12}) q^{2} + ( - \zeta_{12}^{3} + \zeta_{12}) q^{3} + ( - \zeta_{12}^{2} + 1) q^{4} + (\zeta_{12}^{2} - 1) q^{6} + 2 \zeta_{12} q^{7} + \zeta_{12}^{3} q^{8} + ( - \zeta_{12}^{2} + 1) q^{9} - 2 \zeta_{12}^{2} q^{11} - \zeta_{12}^{3} q^{12} + ( - \zeta_{12}^{3} - 3 \zeta_{12}) q^{13} - 2 q^{14} - \zeta_{12}^{2} q^{16} - 5 \zeta_{12} q^{17} + \zeta_{12}^{3} q^{18} + (2 \zeta_{12}^{2} - 2) q^{19} + 2 q^{21} + 2 \zeta_{12} q^{22} + (6 \zeta_{12}^{3} - 6 \zeta_{12}) q^{23} + \zeta_{12}^{2} q^{24} + (\zeta_{12}^{2} + 3) q^{26} - \zeta_{12}^{3} q^{27} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{28} - 9 \zeta_{12}^{2} q^{29} - 4 q^{31} + \zeta_{12} q^{32} - 2 \zeta_{12} q^{33} + 5 q^{34} - \zeta_{12}^{2} q^{36} + (11 \zeta_{12}^{3} - 11 \zeta_{12}) q^{37} - 2 \zeta_{12}^{3} q^{38} + ( - \zeta_{12}^{2} - 3) q^{39} - 5 \zeta_{12}^{2} q^{41} + (2 \zeta_{12}^{3} - 2 \zeta_{12}) q^{42} + 10 \zeta_{12} q^{43} - 2 q^{44} + ( - 6 \zeta_{12}^{2} + 6) q^{46} + 2 \zeta_{12}^{3} q^{47} - \zeta_{12} q^{48} - 3 \zeta_{12}^{2} q^{49} - 5 q^{51} + (3 \zeta_{12}^{3} - 4 \zeta_{12}) q^{52} + \zeta_{12}^{3} q^{53} + \zeta_{12}^{2} q^{54} + (2 \zeta_{12}^{2} - 2) q^{56} + 2 \zeta_{12}^{3} q^{57} + 9 \zeta_{12} q^{58} + (8 \zeta_{12}^{2} - 8) q^{59} + ( - 11 \zeta_{12}^{2} + 11) q^{61} + ( - 4 \zeta_{12}^{3} + 4 \zeta_{12}) q^{62} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{63} - q^{64} + 2 q^{66} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{67} + (5 \zeta_{12}^{3} - 5 \zeta_{12}) q^{68} + (6 \zeta_{12}^{2} - 6) q^{69} + ( - 14 \zeta_{12}^{2} + 14) q^{71} + \zeta_{12} q^{72} + 13 \zeta_{12}^{3} q^{73} + ( - 11 \zeta_{12}^{2} + 11) q^{74} + 2 \zeta_{12}^{2} q^{76} - 4 \zeta_{12}^{3} q^{77} + ( - 3 \zeta_{12}^{3} + 4 \zeta_{12}) q^{78} + 4 q^{79} - \zeta_{12}^{2} q^{81} + 5 \zeta_{12} q^{82} - 6 \zeta_{12}^{3} q^{83} + ( - 2 \zeta_{12}^{2} + 2) q^{84} - 10 q^{86} - 9 \zeta_{12} q^{87} + ( - 2 \zeta_{12}^{3} + 2 \zeta_{12}) q^{88} + 2 \zeta_{12}^{2} q^{89} + ( - 8 \zeta_{12}^{2} + 2) q^{91} + 6 \zeta_{12}^{3} q^{92} + (4 \zeta_{12}^{3} - 4 \zeta_{12}) q^{93} - 2 \zeta_{12}^{2} q^{94} + q^{96} + 2 \zeta_{12} q^{97} + 3 \zeta_{12} q^{98} - 2 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{4} - 2 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{4} - 2 q^{6} + 2 q^{9} - 4 q^{11} - 8 q^{14} - 2 q^{16} - 4 q^{19} + 8 q^{21} + 2 q^{24} + 14 q^{26} - 18 q^{29} - 16 q^{31} + 20 q^{34} - 2 q^{36} - 14 q^{39} - 10 q^{41} - 8 q^{44} + 12 q^{46} - 6 q^{49} - 20 q^{51} + 2 q^{54} - 4 q^{56} - 16 q^{59} + 22 q^{61} - 4 q^{64} + 8 q^{66} - 12 q^{69} + 28 q^{71} + 22 q^{74} + 4 q^{76} + 16 q^{79} - 2 q^{81} + 4 q^{84} - 40 q^{86} + 4 q^{89} - 8 q^{91} - 4 q^{94} + 4 q^{96} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

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

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1699.1
0.866025 0.500000i
−0.866025 + 0.500000i
0.866025 + 0.500000i
−0.866025 0.500000i
−0.866025 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i 0 −0.500000 0.866025i 1.73205 1.00000i 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 −1.73205 + 1.00000i 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 1.73205 + 1.00000i 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 −1.73205 1.00000i 1.00000i 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
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.b 4
5.b even 2 1 inner 1950.2.z.b 4
5.c odd 4 1 78.2.e.b 2
5.c odd 4 1 1950.2.i.b 2
13.c even 3 1 inner 1950.2.z.b 4
15.e even 4 1 234.2.h.b 2
20.e even 4 1 624.2.q.b 2
60.l odd 4 1 1872.2.t.i 2
65.f even 4 1 1014.2.i.e 4
65.h odd 4 1 1014.2.e.d 2
65.k even 4 1 1014.2.i.e 4
65.n even 6 1 inner 1950.2.z.b 4
65.o even 12 1 1014.2.b.a 2
65.o even 12 1 1014.2.i.e 4
65.q odd 12 1 78.2.e.b 2
65.q odd 12 1 1014.2.a.a 1
65.q odd 12 1 1950.2.i.b 2
65.r odd 12 1 1014.2.a.e 1
65.r odd 12 1 1014.2.e.d 2
65.t even 12 1 1014.2.b.a 2
65.t even 12 1 1014.2.i.e 4
195.bc odd 12 1 3042.2.b.d 2
195.bf even 12 1 3042.2.a.d 1
195.bl even 12 1 234.2.h.b 2
195.bl even 12 1 3042.2.a.m 1
195.bn odd 12 1 3042.2.b.d 2
260.bg even 12 1 8112.2.a.bb 1
260.bj even 12 1 624.2.q.b 2
260.bj even 12 1 8112.2.a.x 1
780.cj odd 12 1 1872.2.t.i 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.e.b 2 5.c odd 4 1
78.2.e.b 2 65.q odd 12 1
234.2.h.b 2 15.e even 4 1
234.2.h.b 2 195.bl even 12 1
624.2.q.b 2 20.e even 4 1
624.2.q.b 2 260.bj even 12 1
1014.2.a.a 1 65.q odd 12 1
1014.2.a.e 1 65.r odd 12 1
1014.2.b.a 2 65.o even 12 1
1014.2.b.a 2 65.t even 12 1
1014.2.e.d 2 65.h odd 4 1
1014.2.e.d 2 65.r odd 12 1
1014.2.i.e 4 65.f even 4 1
1014.2.i.e 4 65.k even 4 1
1014.2.i.e 4 65.o even 12 1
1014.2.i.e 4 65.t even 12 1
1872.2.t.i 2 60.l odd 4 1
1872.2.t.i 2 780.cj odd 12 1
1950.2.i.b 2 5.c odd 4 1
1950.2.i.b 2 65.q odd 12 1
1950.2.z.b 4 1.a even 1 1 trivial
1950.2.z.b 4 5.b even 2 1 inner
1950.2.z.b 4 13.c even 3 1 inner
1950.2.z.b 4 65.n even 6 1 inner
3042.2.a.d 1 195.bf even 12 1
3042.2.a.m 1 195.bl even 12 1
3042.2.b.d 2 195.bc odd 12 1
3042.2.b.d 2 195.bn odd 12 1
8112.2.a.x 1 260.bj even 12 1
8112.2.a.bb 1 260.bg even 12 1

Hecke kernels

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

\( T_{7}^{4} - 4T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{11}^{2} + 2T_{11} + 4 \) Copy content Toggle raw display
\( T_{17}^{4} - 25T_{17}^{2} + 625 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$11$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - T^{2} + 169 \) Copy content Toggle raw display
$17$ \( T^{4} - 25T^{2} + 625 \) Copy content Toggle raw display
$19$ \( (T^{2} + 2 T + 4)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} - 36T^{2} + 1296 \) Copy content Toggle raw display
$29$ \( (T^{2} + 9 T + 81)^{2} \) Copy content Toggle raw display
$31$ \( (T + 4)^{4} \) Copy content Toggle raw display
$37$ \( T^{4} - 121 T^{2} + 14641 \) Copy content Toggle raw display
$41$ \( (T^{2} + 5 T + 25)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 100 T^{2} + 10000 \) Copy content Toggle raw display
$47$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} + 8 T + 64)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 11 T + 121)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
$71$ \( (T^{2} - 14 T + 196)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 169)^{2} \) Copy content Toggle raw display
$79$ \( (T - 4)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 2 T + 4)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} - 4T^{2} + 16 \) Copy content Toggle raw display
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