Properties

Label 1950.2.f.g
Level $1950$
Weight $2$
Character orbit 1950.f
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.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

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

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

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

\(n\) \(301\) \(1301\) \(1327\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
649.1
1.00000i
1.00000i
1.00000 1.00000i 1.00000 0 1.00000i −2.00000 1.00000 −1.00000 0
649.2 1.00000 1.00000i 1.00000 0 1.00000i −2.00000 1.00000 −1.00000 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.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1950.2.f.g 2
5.b even 2 1 1950.2.f.d 2
5.c odd 4 1 78.2.b.a 2
5.c odd 4 1 1950.2.b.c 2
13.b even 2 1 1950.2.f.d 2
15.e even 4 1 234.2.b.a 2
20.e even 4 1 624.2.c.a 2
35.f even 4 1 3822.2.c.d 2
40.i odd 4 1 2496.2.c.f 2
40.k even 4 1 2496.2.c.m 2
60.l odd 4 1 1872.2.c.b 2
65.d even 2 1 inner 1950.2.f.g 2
65.f even 4 1 1014.2.a.b 1
65.h odd 4 1 78.2.b.a 2
65.h odd 4 1 1950.2.b.c 2
65.k even 4 1 1014.2.a.g 1
65.o even 12 2 1014.2.e.b 2
65.q odd 12 2 1014.2.i.c 4
65.r odd 12 2 1014.2.i.c 4
65.t even 12 2 1014.2.e.e 2
195.j odd 4 1 3042.2.a.c 1
195.s even 4 1 234.2.b.a 2
195.u odd 4 1 3042.2.a.n 1
260.l odd 4 1 8112.2.a.g 1
260.p even 4 1 624.2.c.a 2
260.s odd 4 1 8112.2.a.j 1
455.s even 4 1 3822.2.c.d 2
520.bc even 4 1 2496.2.c.m 2
520.bg odd 4 1 2496.2.c.f 2
780.w odd 4 1 1872.2.c.b 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
78.2.b.a 2 5.c odd 4 1
78.2.b.a 2 65.h odd 4 1
234.2.b.a 2 15.e even 4 1
234.2.b.a 2 195.s even 4 1
624.2.c.a 2 20.e even 4 1
624.2.c.a 2 260.p even 4 1
1014.2.a.b 1 65.f even 4 1
1014.2.a.g 1 65.k even 4 1
1014.2.e.b 2 65.o even 12 2
1014.2.e.e 2 65.t even 12 2
1014.2.i.c 4 65.q odd 12 2
1014.2.i.c 4 65.r odd 12 2
1872.2.c.b 2 60.l odd 4 1
1872.2.c.b 2 780.w odd 4 1
1950.2.b.c 2 5.c odd 4 1
1950.2.b.c 2 65.h odd 4 1
1950.2.f.d 2 5.b even 2 1
1950.2.f.d 2 13.b even 2 1
1950.2.f.g 2 1.a even 1 1 trivial
1950.2.f.g 2 65.d even 2 1 inner
2496.2.c.f 2 40.i odd 4 1
2496.2.c.f 2 520.bg odd 4 1
2496.2.c.m 2 40.k even 4 1
2496.2.c.m 2 520.bc even 4 1
3042.2.a.c 1 195.j odd 4 1
3042.2.a.n 1 195.u odd 4 1
3822.2.c.d 2 35.f even 4 1
3822.2.c.d 2 455.s even 4 1
8112.2.a.g 1 260.l odd 4 1
8112.2.a.j 1 260.s odd 4 1

Hecke kernels

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

\( T_{7} + 2 \) Copy content Toggle raw display
\( T_{11} \) Copy content Toggle raw display
\( T_{19}^{2} + 36 \) Copy content Toggle raw display
\( T_{37} - 8 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( (T + 2)^{2} \) Copy content Toggle raw display
$11$ \( T^{2} \) Copy content Toggle raw display
$13$ \( T^{2} - 4T + 13 \) Copy content Toggle raw display
$17$ \( T^{2} + 4 \) Copy content Toggle raw display
$19$ \( T^{2} + 36 \) Copy content Toggle raw display
$23$ \( T^{2} + 16 \) Copy content Toggle raw display
$29$ \( (T - 10)^{2} \) Copy content Toggle raw display
$31$ \( T^{2} + 100 \) Copy content Toggle raw display
$37$ \( (T - 8)^{2} \) Copy content Toggle raw display
$41$ \( T^{2} + 100 \) Copy content Toggle raw display
$43$ \( T^{2} + 16 \) Copy content Toggle raw display
$47$ \( (T + 12)^{2} \) Copy content Toggle raw display
$53$ \( T^{2} + 36 \) Copy content Toggle raw display
$59$ \( T^{2} + 16 \) Copy content Toggle raw display
$61$ \( (T - 2)^{2} \) Copy content Toggle raw display
$67$ \( (T + 2)^{2} \) Copy content Toggle raw display
$71$ \( T^{2} \) Copy content Toggle raw display
$73$ \( (T - 4)^{2} \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( (T - 4)^{2} \) Copy content Toggle raw display
$89$ \( T^{2} + 36 \) Copy content Toggle raw display
$97$ \( (T + 12)^{2} \) Copy content Toggle raw display
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