Properties

Label 75.3.d.b
Level $75$
Weight $3$
Character orbit 75.d
Analytic conductor $2.044$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(2.04360198270\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
Defining polynomial: \(x^{4} + 3 x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q -\beta_{3} q^{2} + ( \beta_{1} - \beta_{3} ) q^{3} + q^{4} + ( 5 - \beta_{2} ) q^{6} -3 \beta_{1} q^{7} + 3 \beta_{3} q^{8} + ( 1 - 2 \beta_{2} ) q^{9} +O(q^{10})\) \( q -\beta_{3} q^{2} + ( \beta_{1} - \beta_{3} ) q^{3} + q^{4} + ( 5 - \beta_{2} ) q^{6} -3 \beta_{1} q^{7} + 3 \beta_{3} q^{8} + ( 1 - 2 \beta_{2} ) q^{9} -\beta_{2} q^{11} + ( \beta_{1} - \beta_{3} ) q^{12} -8 \beta_{1} q^{13} + 3 \beta_{2} q^{14} -19 q^{16} + 2 \beta_{3} q^{17} + ( 10 \beta_{1} - \beta_{3} ) q^{18} + 2 q^{19} + ( 12 + 3 \beta_{2} ) q^{21} + 5 \beta_{1} q^{22} -6 \beta_{3} q^{23} + ( -15 + 3 \beta_{2} ) q^{24} + 8 \beta_{2} q^{26} + ( 11 \beta_{1} + 7 \beta_{3} ) q^{27} -3 \beta_{1} q^{28} -7 \beta_{2} q^{29} -18 q^{31} + 7 \beta_{3} q^{32} + ( 5 \beta_{1} + 4 \beta_{3} ) q^{33} -10 q^{34} + ( 1 - 2 \beta_{2} ) q^{36} -8 \beta_{1} q^{37} -2 \beta_{3} q^{38} + ( 32 + 8 \beta_{2} ) q^{39} -14 \beta_{2} q^{41} + ( -15 \beta_{1} - 12 \beta_{3} ) q^{42} -8 \beta_{1} q^{43} -\beta_{2} q^{44} + 30 q^{46} -22 \beta_{3} q^{47} + ( -19 \beta_{1} + 19 \beta_{3} ) q^{48} + 13 q^{49} + ( -10 + 2 \beta_{2} ) q^{51} -8 \beta_{1} q^{52} -2 \beta_{3} q^{53} + ( -35 - 11 \beta_{2} ) q^{54} -9 \beta_{2} q^{56} + ( 2 \beta_{1} - 2 \beta_{3} ) q^{57} + 35 \beta_{1} q^{58} + \beta_{2} q^{59} + 82 q^{61} + 18 \beta_{3} q^{62} + ( -3 \beta_{1} - 24 \beta_{3} ) q^{63} + 41 q^{64} + ( -20 - 5 \beta_{2} ) q^{66} + 12 \beta_{1} q^{67} + 2 \beta_{3} q^{68} + ( 30 - 6 \beta_{2} ) q^{69} + 28 \beta_{2} q^{71} + ( -30 \beta_{1} + 3 \beta_{3} ) q^{72} + 37 \beta_{1} q^{73} + 8 \beta_{2} q^{74} + 2 q^{76} -12 \beta_{3} q^{77} + ( -40 \beta_{1} - 32 \beta_{3} ) q^{78} -138 q^{79} + ( -79 - 4 \beta_{2} ) q^{81} + 70 \beta_{1} q^{82} + 42 \beta_{3} q^{83} + ( 12 + 3 \beta_{2} ) q^{84} + 8 \beta_{2} q^{86} + ( 35 \beta_{1} + 28 \beta_{3} ) q^{87} -15 \beta_{1} q^{88} + 24 \beta_{2} q^{89} -96 q^{91} -6 \beta_{3} q^{92} + ( -18 \beta_{1} + 18 \beta_{3} ) q^{93} + 110 q^{94} + ( -35 + 7 \beta_{2} ) q^{96} -83 \beta_{1} q^{97} -13 \beta_{3} q^{98} + ( -40 - \beta_{2} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 4q^{4} + 20q^{6} + 4q^{9} + O(q^{10}) \) \( 4q + 4q^{4} + 20q^{6} + 4q^{9} - 76q^{16} + 8q^{19} + 48q^{21} - 60q^{24} - 72q^{31} - 40q^{34} + 4q^{36} + 128q^{39} + 120q^{46} + 52q^{49} - 40q^{51} - 140q^{54} + 328q^{61} + 164q^{64} - 80q^{66} + 120q^{69} + 8q^{76} - 552q^{79} - 316q^{81} + 48q^{84} - 384q^{91} + 440q^{94} - 140q^{96} - 160q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} + 3 x^{2} + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( 2 \nu^{3} + 4 \nu \)
\(\beta_{2}\)\(=\)\( 2 \nu^{3} + 8 \nu \)
\(\beta_{3}\)\(=\)\( 2 \nu^{2} + 3 \)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\((\)\(\beta_{2} - \beta_{1}\)\()/4\)
\(\nu^{2}\)\(=\)\((\)\(\beta_{3} - 3\)\()/2\)
\(\nu^{3}\)\(=\)\((\)\(-\beta_{2} + 2 \beta_{1}\)\()/2\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
74.1
0.618034i
0.618034i
1.61803i
1.61803i
−2.23607 −2.23607 2.00000i 1.00000 0 5.00000 + 4.47214i 6.00000i 6.70820 1.00000 + 8.94427i 0
74.2 −2.23607 −2.23607 + 2.00000i 1.00000 0 5.00000 4.47214i 6.00000i 6.70820 1.00000 8.94427i 0
74.3 2.23607 2.23607 2.00000i 1.00000 0 5.00000 4.47214i 6.00000i −6.70820 1.00000 8.94427i 0
74.4 2.23607 2.23607 + 2.00000i 1.00000 0 5.00000 + 4.47214i 6.00000i −6.70820 1.00000 + 8.94427i 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
5.b even 2 1 inner
15.d 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.d.b 4
3.b odd 2 1 inner 75.3.d.b 4
4.b odd 2 1 1200.3.c.f 4
5.b even 2 1 inner 75.3.d.b 4
5.c odd 4 1 15.3.c.a 2
5.c odd 4 1 75.3.c.e 2
12.b even 2 1 1200.3.c.f 4
15.d odd 2 1 inner 75.3.d.b 4
15.e even 4 1 15.3.c.a 2
15.e even 4 1 75.3.c.e 2
20.d odd 2 1 1200.3.c.f 4
20.e even 4 1 240.3.l.b 2
20.e even 4 1 1200.3.l.g 2
40.i odd 4 1 960.3.l.c 2
40.k even 4 1 960.3.l.b 2
45.k odd 12 2 405.3.i.b 4
45.l even 12 2 405.3.i.b 4
60.h even 2 1 1200.3.c.f 4
60.l odd 4 1 240.3.l.b 2
60.l odd 4 1 1200.3.l.g 2
120.q odd 4 1 960.3.l.b 2
120.w even 4 1 960.3.l.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.3.c.a 2 5.c odd 4 1
15.3.c.a 2 15.e even 4 1
75.3.c.e 2 5.c odd 4 1
75.3.c.e 2 15.e even 4 1
75.3.d.b 4 1.a even 1 1 trivial
75.3.d.b 4 3.b odd 2 1 inner
75.3.d.b 4 5.b even 2 1 inner
75.3.d.b 4 15.d odd 2 1 inner
240.3.l.b 2 20.e even 4 1
240.3.l.b 2 60.l odd 4 1
405.3.i.b 4 45.k odd 12 2
405.3.i.b 4 45.l even 12 2
960.3.l.b 2 40.k even 4 1
960.3.l.b 2 120.q odd 4 1
960.3.l.c 2 40.i odd 4 1
960.3.l.c 2 120.w even 4 1
1200.3.c.f 4 4.b odd 2 1
1200.3.c.f 4 12.b even 2 1
1200.3.c.f 4 20.d odd 2 1
1200.3.c.f 4 60.h even 2 1
1200.3.l.g 2 20.e even 4 1
1200.3.l.g 2 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - 5 \) acting on \(S_{3}^{\mathrm{new}}(75, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( ( -5 + T^{2} )^{2} \)
$3$ \( 81 - 2 T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( ( 36 + T^{2} )^{2} \)
$11$ \( ( 20 + T^{2} )^{2} \)
$13$ \( ( 256 + T^{2} )^{2} \)
$17$ \( ( -20 + T^{2} )^{2} \)
$19$ \( ( -2 + T )^{4} \)
$23$ \( ( -180 + T^{2} )^{2} \)
$29$ \( ( 980 + T^{2} )^{2} \)
$31$ \( ( 18 + T )^{4} \)
$37$ \( ( 256 + T^{2} )^{2} \)
$41$ \( ( 3920 + T^{2} )^{2} \)
$43$ \( ( 256 + T^{2} )^{2} \)
$47$ \( ( -2420 + T^{2} )^{2} \)
$53$ \( ( -20 + T^{2} )^{2} \)
$59$ \( ( 20 + T^{2} )^{2} \)
$61$ \( ( -82 + T )^{4} \)
$67$ \( ( 576 + T^{2} )^{2} \)
$71$ \( ( 15680 + T^{2} )^{2} \)
$73$ \( ( 5476 + T^{2} )^{2} \)
$79$ \( ( 138 + T )^{4} \)
$83$ \( ( -8820 + T^{2} )^{2} \)
$89$ \( ( 11520 + T^{2} )^{2} \)
$97$ \( ( 27556 + T^{2} )^{2} \)
show more
show less