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: \(x^{2} + 1\)
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.

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

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 \(S_{2}^{\mathrm{new}}(75, [\chi])\).

Hecke characteristic polynomials

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