Properties

Label 75.2.b.a
Level $75$
Weight $2$
Character orbit 75.b
Analytic conductor $0.599$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 75.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.598878015160\)
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: yes
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.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(-1\)

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
1.00000i
1.00000i
2.00000i 1.00000i −2.00000 0 −2.00000 3.00000i 0 −1.00000 0
49.2 2.00000i 1.00000i −2.00000 0 −2.00000 3.00000i 0 −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
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.2.b.a 2
3.b odd 2 1 225.2.b.a 2
4.b odd 2 1 1200.2.f.d 2
5.b even 2 1 inner 75.2.b.a 2
5.c odd 4 1 75.2.a.a 1
5.c odd 4 1 75.2.a.c yes 1
8.b even 2 1 4800.2.f.l 2
8.d odd 2 1 4800.2.f.y 2
12.b even 2 1 3600.2.f.p 2
15.d odd 2 1 225.2.b.a 2
15.e even 4 1 225.2.a.a 1
15.e even 4 1 225.2.a.e 1
20.d odd 2 1 1200.2.f.d 2
20.e even 4 1 1200.2.a.c 1
20.e even 4 1 1200.2.a.p 1
35.f even 4 1 3675.2.a.b 1
35.f even 4 1 3675.2.a.q 1
40.e odd 2 1 4800.2.f.y 2
40.f even 2 1 4800.2.f.l 2
40.i odd 4 1 4800.2.a.bb 1
40.i odd 4 1 4800.2.a.bq 1
40.k even 4 1 4800.2.a.be 1
40.k even 4 1 4800.2.a.br 1
55.e even 4 1 9075.2.a.a 1
55.e even 4 1 9075.2.a.s 1
60.h even 2 1 3600.2.f.p 2
60.l odd 4 1 3600.2.a.j 1
60.l odd 4 1 3600.2.a.bk 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.2.a.a 1 5.c odd 4 1
75.2.a.c yes 1 5.c odd 4 1
75.2.b.a 2 1.a even 1 1 trivial
75.2.b.a 2 5.b even 2 1 inner
225.2.a.a 1 15.e even 4 1
225.2.a.e 1 15.e even 4 1
225.2.b.a 2 3.b odd 2 1
225.2.b.a 2 15.d odd 2 1
1200.2.a.c 1 20.e even 4 1
1200.2.a.p 1 20.e even 4 1
1200.2.f.d 2 4.b odd 2 1
1200.2.f.d 2 20.d odd 2 1
3600.2.a.j 1 60.l odd 4 1
3600.2.a.bk 1 60.l odd 4 1
3600.2.f.p 2 12.b even 2 1
3600.2.f.p 2 60.h even 2 1
3675.2.a.b 1 35.f even 4 1
3675.2.a.q 1 35.f even 4 1
4800.2.a.bb 1 40.i odd 4 1
4800.2.a.be 1 40.k even 4 1
4800.2.a.bq 1 40.i odd 4 1
4800.2.a.br 1 40.k even 4 1
4800.2.f.l 2 8.b even 2 1
4800.2.f.l 2 40.f even 2 1
4800.2.f.y 2 8.d odd 2 1
4800.2.f.y 2 40.e odd 2 1
9075.2.a.a 1 55.e even 4 1
9075.2.a.s 1 55.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} + 4 \) acting on \(S_{2}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 9 \) Copy content Toggle raw display
$11$ \( (T - 2)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1 \) Copy content Toggle raw display
$17$ \( T^{2} + 4 \) Copy content Toggle raw display
$19$ \( (T - 5)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 36 \) Copy content Toggle raw display
$29$ \( (T + 10)^{2} \) Copy content Toggle raw display
$31$ \( (T + 3)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4 \) Copy content Toggle raw display
$41$ \( (T + 8)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 1 \) Copy content Toggle raw display
$47$ \( T^{2} + 4 \) Copy content Toggle raw display
$53$ \( T^{2} + 16 \) Copy content Toggle raw display
$59$ \( (T - 10)^{2} \) Copy content Toggle raw display
$61$ \( (T - 7)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 9 \) Copy content Toggle raw display
$71$ \( (T + 8)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 196 \) Copy content Toggle raw display
$79$ \( T^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 36 \) Copy content Toggle raw display
$89$ \( T^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 289 \) Copy content Toggle raw display
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