Properties

Label 1950.2.i.m
Level $1950$
Weight $2$
Character orbit 1950.i
Analytic conductor $15.571$
Analytic rank $0$
Dimension $2$
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.i (of order \(3\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.5708283941\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-3}) \)
Defining polynomial: \( x^{2} - x + 1 \) 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: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a primitive root of unity \(\zeta_{6}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(301\) \(1301\) \(1327\)
\(\chi(n)\) \(-\zeta_{6}\) \(1\) \(1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
451.1
0.500000 + 0.866025i
0.500000 0.866025i
−0.500000 + 0.866025i 0.500000 0.866025i −0.500000 0.866025i 0 0.500000 + 0.866025i 1.00000 + 1.73205i 1.00000 −0.500000 0.866025i 0
601.1 −0.500000 0.866025i 0.500000 + 0.866025i −0.500000 + 0.866025i 0 0.500000 0.866025i 1.00000 1.73205i 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
13.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1950.2.i.m 2
5.b even 2 1 78.2.e.a 2
5.c odd 4 2 1950.2.z.g 4
13.c even 3 1 inner 1950.2.i.m 2
15.d odd 2 1 234.2.h.a 2
20.d odd 2 1 624.2.q.g 2
60.h even 2 1 1872.2.t.c 2
65.d even 2 1 1014.2.e.a 2
65.g odd 4 2 1014.2.i.b 4
65.l even 6 1 1014.2.a.f 1
65.l even 6 1 1014.2.e.a 2
65.n even 6 1 78.2.e.a 2
65.n even 6 1 1014.2.a.c 1
65.q odd 12 2 1950.2.z.g 4
65.s odd 12 2 1014.2.b.c 2
65.s odd 12 2 1014.2.i.b 4
195.x odd 6 1 234.2.h.a 2
195.x odd 6 1 3042.2.a.i 1
195.y odd 6 1 3042.2.a.h 1
195.bh even 12 2 3042.2.b.h 2
260.v odd 6 1 624.2.q.g 2
260.v odd 6 1 8112.2.a.m 1
260.w odd 6 1 8112.2.a.c 1
780.br even 6 1 1872.2.t.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.e.a 2 5.b even 2 1
78.2.e.a 2 65.n even 6 1
234.2.h.a 2 15.d odd 2 1
234.2.h.a 2 195.x odd 6 1
624.2.q.g 2 20.d odd 2 1
624.2.q.g 2 260.v odd 6 1
1014.2.a.c 1 65.n even 6 1
1014.2.a.f 1 65.l even 6 1
1014.2.b.c 2 65.s odd 12 2
1014.2.e.a 2 65.d even 2 1
1014.2.e.a 2 65.l even 6 1
1014.2.i.b 4 65.g odd 4 2
1014.2.i.b 4 65.s odd 12 2
1872.2.t.c 2 60.h even 2 1
1872.2.t.c 2 780.br even 6 1
1950.2.i.m 2 1.a even 1 1 trivial
1950.2.i.m 2 13.c even 3 1 inner
1950.2.z.g 4 5.c odd 4 2
1950.2.z.g 4 65.q odd 12 2
3042.2.a.h 1 195.y odd 6 1
3042.2.a.i 1 195.x odd 6 1
3042.2.b.h 2 195.bh even 12 2
8112.2.a.c 1 260.w odd 6 1
8112.2.a.m 1 260.v odd 6 1

Hecke kernels

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

\( T_{7}^{2} - 2T_{7} + 4 \) Copy content Toggle raw display
\( T_{11}^{2} + 6T_{11} + 36 \) Copy content Toggle raw display
\( T_{17}^{2} + 3T_{17} + 9 \) Copy content Toggle raw display
\( T_{19}^{2} + 2T_{19} + 4 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + T + 1 \) Copy content Toggle raw display
$3$ \( T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} - 2T + 4 \) Copy content Toggle raw display
$11$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$13$ \( T^{2} - 7T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$19$ \( T^{2} + 2T + 4 \) Copy content Toggle raw display
$23$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$29$ \( T^{2} + 3T + 9 \) Copy content Toggle raw display
$31$ \( (T + 4)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 7T + 49 \) Copy content Toggle raw display
$41$ \( T^{2} - 3T + 9 \) Copy content Toggle raw display
$43$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$47$ \( (T + 6)^{2} \) Copy content Toggle raw display
$53$ \( (T + 3)^{2} \) Copy content Toggle raw display
$59$ \( T^{2} \) Copy content Toggle raw display
$61$ \( T^{2} - 7T + 49 \) Copy content Toggle raw display
$67$ \( T^{2} + 10T + 100 \) Copy content Toggle raw display
$71$ \( T^{2} + 6T + 36 \) Copy content Toggle raw display
$73$ \( (T - 13)^{2} \) Copy content Toggle raw display
$79$ \( (T + 4)^{2} \) Copy content Toggle raw display
$83$ \( (T - 6)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 18T + 324 \) Copy content Toggle raw display
$97$ \( T^{2} - 14T + 196 \) Copy content Toggle raw display
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