Properties

Label 75.3.c.f
Level $75$
Weight $3$
Character orbit 75.c
Analytic conductor $2.044$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 75.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.04360198270\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-11}) \)
Defining polynomial: \(x^{2} - x + 3\)
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 \(\beta = \frac{1}{2}(1 + \sqrt{-11})\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( 1 - 2 \beta ) q^{2} + ( 3 - \beta ) q^{3} -7 q^{4} + ( -3 - 5 \beta ) q^{6} + ( -3 + 6 \beta ) q^{8} + ( 6 - 5 \beta ) q^{9} +O(q^{10})\) \( q + ( 1 - 2 \beta ) q^{2} + ( 3 - \beta ) q^{3} -7 q^{4} + ( -3 - 5 \beta ) q^{6} + ( -3 + 6 \beta ) q^{8} + ( 6 - 5 \beta ) q^{9} + ( -5 + 10 \beta ) q^{11} + ( -21 + 7 \beta ) q^{12} + 10 q^{13} + 5 q^{16} + ( 1 - 2 \beta ) q^{17} + ( -24 - 7 \beta ) q^{18} + 7 q^{19} + 55 q^{22} + ( 6 - 12 \beta ) q^{23} + ( 9 + 15 \beta ) q^{24} + ( 10 - 20 \beta ) q^{26} + ( 3 - 16 \beta ) q^{27} + ( -10 + 20 \beta ) q^{29} + 42 q^{31} + ( -7 + 14 \beta ) q^{32} + ( 15 + 25 \beta ) q^{33} -11 q^{34} + ( -42 + 35 \beta ) q^{36} -40 q^{37} + ( 7 - 14 \beta ) q^{38} + ( 30 - 10 \beta ) q^{39} + ( 5 - 10 \beta ) q^{41} -50 q^{43} + ( 35 - 70 \beta ) q^{44} -66 q^{46} + ( -14 + 28 \beta ) q^{47} + ( 15 - 5 \beta ) q^{48} -49 q^{49} + ( -3 - 5 \beta ) q^{51} -70 q^{52} + ( -14 + 28 \beta ) q^{53} + ( -93 + 10 \beta ) q^{54} + ( 21 - 7 \beta ) q^{57} + 110 q^{58} + ( -20 + 40 \beta ) q^{59} -8 q^{61} + ( 42 - 84 \beta ) q^{62} + 97 q^{64} + ( 165 - 55 \beta ) q^{66} + 45 q^{67} + ( -7 + 14 \beta ) q^{68} + ( -18 - 30 \beta ) q^{69} + ( -10 + 20 \beta ) q^{71} + ( 72 + 21 \beta ) q^{72} -35 q^{73} + ( -40 + 80 \beta ) q^{74} -49 q^{76} + ( -30 - 50 \beta ) q^{78} + 12 q^{79} + ( -39 - 35 \beta ) q^{81} -55 q^{82} + ( 21 - 42 \beta ) q^{83} + ( -50 + 100 \beta ) q^{86} + ( 30 + 50 \beta ) q^{87} -165 q^{88} + ( 45 - 90 \beta ) q^{89} + ( -42 + 84 \beta ) q^{92} + ( 126 - 42 \beta ) q^{93} + 154 q^{94} + ( 21 + 35 \beta ) q^{96} -70 q^{97} + ( -49 + 98 \beta ) q^{98} + ( 120 + 35 \beta ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 5q^{3} - 14q^{4} - 11q^{6} + 7q^{9} + O(q^{10}) \) \( 2q + 5q^{3} - 14q^{4} - 11q^{6} + 7q^{9} - 35q^{12} + 20q^{13} + 10q^{16} - 55q^{18} + 14q^{19} + 110q^{22} + 33q^{24} - 10q^{27} + 84q^{31} + 55q^{33} - 22q^{34} - 49q^{36} - 80q^{37} + 50q^{39} - 100q^{43} - 132q^{46} + 25q^{48} - 98q^{49} - 11q^{51} - 140q^{52} - 176q^{54} + 35q^{57} + 220q^{58} - 16q^{61} + 194q^{64} + 275q^{66} + 90q^{67} - 66q^{69} + 165q^{72} - 70q^{73} - 98q^{76} - 110q^{78} + 24q^{79} - 113q^{81} - 110q^{82} + 110q^{87} - 330q^{88} + 210q^{93} + 308q^{94} + 77q^{96} - 140q^{97} + 275q^{99} + O(q^{100}) \)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
26.1
0.500000 + 1.65831i
0.500000 1.65831i
3.31662i 2.50000 1.65831i −7.00000 0 −5.50000 8.29156i 0 9.94987i 3.50000 8.29156i 0
26.2 3.31662i 2.50000 + 1.65831i −7.00000 0 −5.50000 + 8.29156i 0 9.94987i 3.50000 + 8.29156i 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
3.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.3.c.f yes 2
3.b odd 2 1 inner 75.3.c.f yes 2
4.b odd 2 1 1200.3.l.f 2
5.b even 2 1 75.3.c.c 2
5.c odd 4 2 75.3.d.c 4
12.b even 2 1 1200.3.l.f 2
15.d odd 2 1 75.3.c.c 2
15.e even 4 2 75.3.d.c 4
20.d odd 2 1 1200.3.l.s 2
20.e even 4 2 1200.3.c.d 4
60.h even 2 1 1200.3.l.s 2
60.l odd 4 2 1200.3.c.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.3.c.c 2 5.b even 2 1
75.3.c.c 2 15.d odd 2 1
75.3.c.f yes 2 1.a even 1 1 trivial
75.3.c.f yes 2 3.b odd 2 1 inner
75.3.d.c 4 5.c odd 4 2
75.3.d.c 4 15.e even 4 2
1200.3.c.d 4 20.e even 4 2
1200.3.c.d 4 60.l odd 4 2
1200.3.l.f 2 4.b odd 2 1
1200.3.l.f 2 12.b even 2 1
1200.3.l.s 2 20.d odd 2 1
1200.3.l.s 2 60.h even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(75, [\chi])\):

\( T_{2}^{2} + 11 \)
\( T_{7} \)
\( T_{13} - 10 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 11 + T^{2} \)
$3$ \( 9 - 5 T + T^{2} \)
$5$ \( T^{2} \)
$7$ \( T^{2} \)
$11$ \( 275 + T^{2} \)
$13$ \( ( -10 + T )^{2} \)
$17$ \( 11 + T^{2} \)
$19$ \( ( -7 + T )^{2} \)
$23$ \( 396 + T^{2} \)
$29$ \( 1100 + T^{2} \)
$31$ \( ( -42 + T )^{2} \)
$37$ \( ( 40 + T )^{2} \)
$41$ \( 275 + T^{2} \)
$43$ \( ( 50 + T )^{2} \)
$47$ \( 2156 + T^{2} \)
$53$ \( 2156 + T^{2} \)
$59$ \( 4400 + T^{2} \)
$61$ \( ( 8 + T )^{2} \)
$67$ \( ( -45 + T )^{2} \)
$71$ \( 1100 + T^{2} \)
$73$ \( ( 35 + T )^{2} \)
$79$ \( ( -12 + T )^{2} \)
$83$ \( 4851 + T^{2} \)
$89$ \( 22275 + T^{2} \)
$97$ \( ( 70 + T )^{2} \)
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