Properties

Label 75.3.f.c
Level $75$
Weight $3$
Character orbit 75.f
Analytic conductor $2.044$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [75,3,Mod(7,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.7");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 75.f (of order \(4\), degree \(2\), minimal)

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: no
Analytic conductor: \(2.04360198270\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 15)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

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.

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1 + 1) q^{2} + \beta_{3} q^{3} + (2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{4} + (\beta_{3} - \beta_1 - 3) q^{6} + ( - \beta_{2} + 2 \beta_1 - 1) q^{7} + (\beta_{3} + 3 \beta_{2} - 3) q^{8} - 3 \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + \beta_1 + 1) q^{2} + \beta_{3} q^{3} + (2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{4} + (\beta_{3} - \beta_1 - 3) q^{6} + ( - \beta_{2} + 2 \beta_1 - 1) q^{7} + (\beta_{3} + 3 \beta_{2} - 3) q^{8} - 3 \beta_{2} q^{9} + (3 \beta_{3} - 3 \beta_1 + 4) q^{11} + ( - 6 \beta_{2} - \beta_1 - 6) q^{12} + ( - 2 \beta_{3} - 8 \beta_{2} + 8) q^{13} + (\beta_{3} + 4 \beta_{2} + \beta_1) q^{14} + ( - 4 \beta_{3} + 4 \beta_1 - 5) q^{16} + (10 \beta_{2} - 6 \beta_1 + 10) q^{17} + ( - 3 \beta_{3} - 3 \beta_{2} + 3) q^{18} + ( - 6 \beta_{3} - 6 \beta_{2} - 6 \beta_1) q^{19} + ( - \beta_{3} + \beta_1 - 6) q^{21} + ( - 5 \beta_{2} - 2 \beta_1 - 5) q^{22} + (2 \beta_{3} + 14 \beta_{2} - 14) q^{23} + ( - 3 \beta_{3} - 3 \beta_{2} - 3 \beta_1) q^{24} + ( - 10 \beta_{3} + 10 \beta_1 + 22) q^{26} + 3 \beta_1 q^{27} + ( - 2 \beta_{3} + 11 \beta_{2} - 11) q^{28} + ( - 7 \beta_{3} + 18 \beta_{2} - 7 \beta_1) q^{29} + (6 \beta_{3} - 6 \beta_1 - 4) q^{31} + (19 \beta_{2} + 7 \beta_1 + 19) q^{32} + (4 \beta_{3} - 9 \beta_{2} + 9) q^{33} + (4 \beta_{3} + 2 \beta_{2} + 4 \beta_1) q^{34} + ( - 6 \beta_{3} + 6 \beta_1 + 3) q^{36} + ( - 16 \beta_{2} - 18 \beta_1 - 16) q^{37} + ( - 18 \beta_{3} - 24 \beta_{2} + 24) q^{38} + (8 \beta_{3} + 6 \beta_{2} + 8 \beta_1) q^{39} + (6 \beta_{3} - 6 \beta_1 - 14) q^{41} + ( - 3 \beta_{2} - 4 \beta_1 - 3) q^{42} + (20 \beta_{3} - 2 \beta_{2} + 2) q^{43} + (5 \beta_{3} - 32 \beta_{2} + 5 \beta_1) q^{44} + (16 \beta_{3} - 16 \beta_1 - 34) q^{46} + ( - 32 \beta_{2} + 10 \beta_1 - 32) q^{47} + ( - 5 \beta_{3} + 12 \beta_{2} - 12) q^{48} + ( - 4 \beta_{3} - 35 \beta_{2} - 4 \beta_1) q^{49} + (10 \beta_{3} - 10 \beta_1 + 18) q^{51} + (20 \beta_{2} + 34 \beta_1 + 20) q^{52} + (12 \beta_{3} + 14 \beta_{2} - 14) q^{53} + (3 \beta_{3} + 9 \beta_{2} + 3 \beta_1) q^{54} + (5 \beta_{3} - 5 \beta_1) q^{56} + (18 \beta_{2} + 6 \beta_1 + 18) q^{57} + (4 \beta_{3} - 3 \beta_{2} + 3) q^{58} + (31 \beta_{3} + 36 \beta_{2} + 31 \beta_1) q^{59} + ( - 18 \beta_{3} + 18 \beta_1 + 50) q^{61} + ( - 22 \beta_{2} - 16 \beta_1 - 22) q^{62} + ( - 6 \beta_{3} + 3 \beta_{2} - 3) q^{63} + (10 \beta_{3} + 79 \beta_{2} + 10 \beta_1) q^{64} + ( - 5 \beta_{3} + 5 \beta_1 + 6) q^{66} + (50 \beta_{2} + 4 \beta_1 + 50) q^{67} + (34 \beta_{3} - 26 \beta_{2} + 26) q^{68} + ( - 14 \beta_{3} - 6 \beta_{2} - 14 \beta_1) q^{69} - 68 q^{71} + (9 \beta_{2} + 3 \beta_1 + 9) q^{72} + ( - 48 \beta_{3} + 19 \beta_{2} - 19) q^{73} + ( - 34 \beta_{3} - 86 \beta_{2} - 34 \beta_1) q^{74} + ( - 18 \beta_{3} + 18 \beta_1 + 78) q^{76} + ( - 22 \beta_{2} + 14 \beta_1 - 22) q^{77} + (22 \beta_{3} + 30 \beta_{2} - 30) q^{78} + (10 \beta_{3} + 10 \beta_1) q^{79} - 9 q^{81} + ( - 32 \beta_{2} - 26 \beta_1 - 32) q^{82} + ( - 14 \beta_{3} - 4 \beta_{2} + 4) q^{83} + ( - 11 \beta_{3} + 6 \beta_{2} - 11 \beta_1) q^{84} + (18 \beta_{3} - 18 \beta_1 - 56) q^{86} + (21 \beta_{2} - 18 \beta_1 + 21) q^{87} + ( - 14 \beta_{3} + 3 \beta_{2} - 3) q^{88} + ( - 36 \beta_{3} - 6 \beta_{2} - 36 \beta_1) q^{89} + ( - 14 \beta_{3} + 14 \beta_1 - 4) q^{91} + ( - 26 \beta_{2} - 58 \beta_1 - 26) q^{92} + ( - 4 \beta_{3} - 18 \beta_{2} + 18) q^{93} + ( - 22 \beta_{3} - 34 \beta_{2} - 22 \beta_1) q^{94} + (19 \beta_{3} - 19 \beta_1 - 21) q^{96} + (5 \beta_{2} - 16 \beta_1 + 5) q^{97} + ( - 43 \beta_{3} - 47 \beta_{2} + 47) q^{98} + (9 \beta_{3} - 12 \beta_{2} + 9 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} - 12 q^{6} - 4 q^{7} - 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} - 12 q^{6} - 4 q^{7} - 12 q^{8} + 16 q^{11} - 24 q^{12} + 32 q^{13} - 20 q^{16} + 40 q^{17} + 12 q^{18} - 24 q^{21} - 20 q^{22} - 56 q^{23} + 88 q^{26} - 44 q^{28} - 16 q^{31} + 76 q^{32} + 36 q^{33} + 12 q^{36} - 64 q^{37} + 96 q^{38} - 56 q^{41} - 12 q^{42} + 8 q^{43} - 136 q^{46} - 128 q^{47} - 48 q^{48} + 72 q^{51} + 80 q^{52} - 56 q^{53} + 72 q^{57} + 12 q^{58} + 200 q^{61} - 88 q^{62} - 12 q^{63} + 24 q^{66} + 200 q^{67} + 104 q^{68} - 272 q^{71} + 36 q^{72} - 76 q^{73} + 312 q^{76} - 88 q^{77} - 120 q^{78} - 36 q^{81} - 128 q^{82} + 16 q^{83} - 224 q^{86} + 84 q^{87} - 12 q^{88} - 16 q^{91} - 104 q^{92} + 72 q^{93} - 84 q^{96} + 20 q^{97} + 188 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{2} ) / 3 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} ) / 3 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( 3\beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 3\beta_{3} \) Copy content Toggle raw display

Character values

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

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(\beta_{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−1.22474 1.22474i
1.22474 + 1.22474i
−1.22474 + 1.22474i
1.22474 1.22474i
−0.224745 0.224745i 1.22474 1.22474i 3.89898i 0 −0.550510 −3.44949 3.44949i −1.77526 + 1.77526i 3.00000i 0
7.2 2.22474 + 2.22474i −1.22474 + 1.22474i 5.89898i 0 −5.44949 1.44949 + 1.44949i −4.22474 + 4.22474i 3.00000i 0
43.1 −0.224745 + 0.224745i 1.22474 + 1.22474i 3.89898i 0 −0.550510 −3.44949 + 3.44949i −1.77526 1.77526i 3.00000i 0
43.2 2.22474 2.22474i −1.22474 1.22474i 5.89898i 0 −5.44949 1.44949 1.44949i −4.22474 4.22474i 3.00000i 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.c odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.3.f.c 4
3.b odd 2 1 225.3.g.a 4
4.b odd 2 1 1200.3.bg.k 4
5.b even 2 1 15.3.f.a 4
5.c odd 4 1 15.3.f.a 4
5.c odd 4 1 inner 75.3.f.c 4
15.d odd 2 1 45.3.g.b 4
15.e even 4 1 45.3.g.b 4
15.e even 4 1 225.3.g.a 4
20.d odd 2 1 240.3.bg.a 4
20.e even 4 1 240.3.bg.a 4
20.e even 4 1 1200.3.bg.k 4
40.e odd 2 1 960.3.bg.h 4
40.f even 2 1 960.3.bg.i 4
40.i odd 4 1 960.3.bg.i 4
40.k even 4 1 960.3.bg.h 4
45.h odd 6 2 405.3.l.f 8
45.j even 6 2 405.3.l.h 8
45.k odd 12 2 405.3.l.h 8
45.l even 12 2 405.3.l.f 8
60.h even 2 1 720.3.bh.k 4
60.l odd 4 1 720.3.bh.k 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.3.f.a 4 5.b even 2 1
15.3.f.a 4 5.c odd 4 1
45.3.g.b 4 15.d odd 2 1
45.3.g.b 4 15.e even 4 1
75.3.f.c 4 1.a even 1 1 trivial
75.3.f.c 4 5.c odd 4 1 inner
225.3.g.a 4 3.b odd 2 1
225.3.g.a 4 15.e even 4 1
240.3.bg.a 4 20.d odd 2 1
240.3.bg.a 4 20.e even 4 1
405.3.l.f 8 45.h odd 6 2
405.3.l.f 8 45.l even 12 2
405.3.l.h 8 45.j even 6 2
405.3.l.h 8 45.k odd 12 2
720.3.bh.k 4 60.h even 2 1
720.3.bh.k 4 60.l odd 4 1
960.3.bg.h 4 40.e odd 2 1
960.3.bg.h 4 40.k even 4 1
960.3.bg.i 4 40.f even 2 1
960.3.bg.i 4 40.i odd 4 1
1200.3.bg.k 4 4.b odd 2 1
1200.3.bg.k 4 20.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} - 4T_{2}^{3} + 8T_{2}^{2} + 4T_{2} + 1 \) acting on \(S_{3}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - 4 T^{3} + 8 T^{2} + 4 T + 1 \) Copy content Toggle raw display
$3$ \( T^{4} + 9 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 4 T^{3} + 8 T^{2} - 40 T + 100 \) Copy content Toggle raw display
$11$ \( (T^{2} - 8 T - 38)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 32 T^{3} + 512 T^{2} + \cdots + 13456 \) Copy content Toggle raw display
$17$ \( T^{4} - 40 T^{3} + 800 T^{2} + \cdots + 8464 \) Copy content Toggle raw display
$19$ \( T^{4} + 504 T^{2} + 32400 \) Copy content Toggle raw display
$23$ \( T^{4} + 56 T^{3} + 1568 T^{2} + \cdots + 144400 \) Copy content Toggle raw display
$29$ \( T^{4} + 1236T^{2} + 900 \) Copy content Toggle raw display
$31$ \( (T^{2} + 8 T - 200)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 64 T^{3} + 2048 T^{2} + \cdots + 211600 \) Copy content Toggle raw display
$41$ \( (T^{2} + 28 T - 20)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} - 8 T^{3} + 32 T^{2} + \cdots + 1420864 \) Copy content Toggle raw display
$47$ \( T^{4} + 128 T^{3} + 8192 T^{2} + \cdots + 3055504 \) Copy content Toggle raw display
$53$ \( T^{4} + 56 T^{3} + 1568 T^{2} + \cdots + 1600 \) Copy content Toggle raw display
$59$ \( T^{4} + 14124 T^{2} + \cdots + 19980900 \) Copy content Toggle raw display
$61$ \( (T^{2} - 100 T + 556)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} - 200 T^{3} + \cdots + 24522304 \) Copy content Toggle raw display
$71$ \( (T + 68)^{4} \) Copy content Toggle raw display
$73$ \( T^{4} + 76 T^{3} + 2888 T^{2} + \cdots + 38316100 \) Copy content Toggle raw display
$79$ \( (T^{2} + 600)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} - 16 T^{3} + 128 T^{2} + \cdots + 309136 \) Copy content Toggle raw display
$89$ \( T^{4} + 15624 T^{2} + \cdots + 59907600 \) Copy content Toggle raw display
$97$ \( T^{4} - 20 T^{3} + 200 T^{2} + \cdots + 515524 \) Copy content Toggle raw display
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