Properties

Label 75.8.a.a
Level $75$
Weight $8$
Character orbit 75.a
Self dual yes
Analytic conductor $23.429$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [75,8,Mod(1,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 75.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(23.4288769113\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \( q - 6 q^{2} + 27 q^{3} - 92 q^{4} - 162 q^{6} + 64 q^{7} + 1320 q^{8} + 729 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - 6 q^{2} + 27 q^{3} - 92 q^{4} - 162 q^{6} + 64 q^{7} + 1320 q^{8} + 729 q^{9} - 948 q^{11} - 2484 q^{12} + 5098 q^{13} - 384 q^{14} + 3856 q^{16} - 28386 q^{17} - 4374 q^{18} - 8620 q^{19} + 1728 q^{21} + 5688 q^{22} + 15288 q^{23} + 35640 q^{24} - 30588 q^{26} + 19683 q^{27} - 5888 q^{28} + 36510 q^{29} - 276808 q^{31} - 192096 q^{32} - 25596 q^{33} + 170316 q^{34} - 67068 q^{36} - 268526 q^{37} + 51720 q^{38} + 137646 q^{39} - 629718 q^{41} - 10368 q^{42} - 685772 q^{43} + 87216 q^{44} - 91728 q^{46} - 583296 q^{47} + 104112 q^{48} - 819447 q^{49} - 766422 q^{51} - 469016 q^{52} + 428058 q^{53} - 118098 q^{54} + 84480 q^{56} - 232740 q^{57} - 219060 q^{58} + 1306380 q^{59} + 300662 q^{61} + 1660848 q^{62} + 46656 q^{63} + 659008 q^{64} + 153576 q^{66} + 507244 q^{67} + 2611512 q^{68} + 412776 q^{69} + 5560632 q^{71} + 962280 q^{72} - 1369082 q^{73} + 1611156 q^{74} + 793040 q^{76} - 60672 q^{77} - 825876 q^{78} - 6913720 q^{79} + 531441 q^{81} + 3778308 q^{82} + 4376748 q^{83} - 158976 q^{84} + 4114632 q^{86} + 985770 q^{87} - 1251360 q^{88} - 8528310 q^{89} + 326272 q^{91} - 1406496 q^{92} - 7473816 q^{93} + 3499776 q^{94} - 5186592 q^{96} + 8826814 q^{97} + 4916682 q^{98} - 691092 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
0
−6.00000 27.0000 −92.0000 0 −162.000 64.0000 1320.00 729.000 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(5\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.8.a.a 1
3.b odd 2 1 225.8.a.i 1
5.b even 2 1 3.8.a.a 1
5.c odd 4 2 75.8.b.c 2
15.d odd 2 1 9.8.a.a 1
15.e even 4 2 225.8.b.f 2
20.d odd 2 1 48.8.a.g 1
35.c odd 2 1 147.8.a.b 1
35.i odd 6 2 147.8.e.a 2
35.j even 6 2 147.8.e.b 2
40.e odd 2 1 192.8.a.a 1
40.f even 2 1 192.8.a.i 1
45.h odd 6 2 81.8.c.c 2
45.j even 6 2 81.8.c.a 2
55.d odd 2 1 363.8.a.b 1
60.h even 2 1 144.8.a.b 1
65.d even 2 1 507.8.a.a 1
105.g even 2 1 441.8.a.a 1
120.i odd 2 1 576.8.a.w 1
120.m even 2 1 576.8.a.x 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.8.a.a 1 5.b even 2 1
9.8.a.a 1 15.d odd 2 1
48.8.a.g 1 20.d odd 2 1
75.8.a.a 1 1.a even 1 1 trivial
75.8.b.c 2 5.c odd 4 2
81.8.c.a 2 45.j even 6 2
81.8.c.c 2 45.h odd 6 2
144.8.a.b 1 60.h even 2 1
147.8.a.b 1 35.c odd 2 1
147.8.e.a 2 35.i odd 6 2
147.8.e.b 2 35.j even 6 2
192.8.a.a 1 40.e odd 2 1
192.8.a.i 1 40.f even 2 1
225.8.a.i 1 3.b odd 2 1
225.8.b.f 2 15.e even 4 2
363.8.a.b 1 55.d odd 2 1
441.8.a.a 1 105.g even 2 1
507.8.a.a 1 65.d even 2 1
576.8.a.w 1 120.i odd 2 1
576.8.a.x 1 120.m even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2} + 6 \) acting on \(S_{8}^{\mathrm{new}}(\Gamma_0(75))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T + 6 \) Copy content Toggle raw display
$3$ \( T - 27 \) Copy content Toggle raw display
$5$ \( T \) Copy content Toggle raw display
$7$ \( T - 64 \) Copy content Toggle raw display
$11$ \( T + 948 \) Copy content Toggle raw display
$13$ \( T - 5098 \) Copy content Toggle raw display
$17$ \( T + 28386 \) Copy content Toggle raw display
$19$ \( T + 8620 \) Copy content Toggle raw display
$23$ \( T - 15288 \) Copy content Toggle raw display
$29$ \( T - 36510 \) Copy content Toggle raw display
$31$ \( T + 276808 \) Copy content Toggle raw display
$37$ \( T + 268526 \) Copy content Toggle raw display
$41$ \( T + 629718 \) Copy content Toggle raw display
$43$ \( T + 685772 \) Copy content Toggle raw display
$47$ \( T + 583296 \) Copy content Toggle raw display
$53$ \( T - 428058 \) Copy content Toggle raw display
$59$ \( T - 1306380 \) Copy content Toggle raw display
$61$ \( T - 300662 \) Copy content Toggle raw display
$67$ \( T - 507244 \) Copy content Toggle raw display
$71$ \( T - 5560632 \) Copy content Toggle raw display
$73$ \( T + 1369082 \) Copy content Toggle raw display
$79$ \( T + 6913720 \) Copy content Toggle raw display
$83$ \( T - 4376748 \) Copy content Toggle raw display
$89$ \( T + 8528310 \) Copy content Toggle raw display
$97$ \( T - 8826814 \) Copy content Toggle raw display
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