Properties

Label 75.22.b.d
Level $75$
Weight $22$
Character orbit 75.b
Analytic conductor $209.608$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [75,22,Mod(49,75)]
 
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, 1]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.49");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 75.b (of order \(2\), degree \(1\), not 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: \(209.608008215\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{649})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 325x^{2} + 26244 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 3)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{2} + 333 \beta_1) q^{2} - 59049 \beta_1 q^{3} + ( - 666 \beta_{3} - 589618) q^{4} + (59049 \beta_{3} + 19663317) q^{6} + ( - 182016 \beta_{2} + 339948056 \beta_1) q^{7} + (1285756 \beta_{2} - 1213527924 \beta_1) q^{8} - 3486784401 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + 333 \beta_1) q^{2} - 59049 \beta_1 q^{3} + ( - 666 \beta_{3} - 589618) q^{4} + (59049 \beta_{3} + 19663317) q^{6} + ( - 182016 \beta_{2} + 339948056 \beta_1) q^{7} + (1285756 \beta_{2} - 1213527924 \beta_1) q^{8} - 3486784401 q^{9} + (31263232 \beta_{3} + 109934561484) q^{11} + (39326634 \beta_{2} + 34816353282 \beta_1) q^{12} + (475236864 \beta_{2} + 24234454978 \beta_1) q^{13} + ( - 279336728 \beta_{3} + 355648853448) q^{14} + ( - 611332056 \beta_{3} - 4144368220280) q^{16} + (1677304320 \beta_{2} - 5666764520718 \beta_1) q^{17} + ( - 3486784401 \beta_{2} - 1161099205533 \beta_1) q^{18} + (22384332288 \beta_{3} - 5980292505812) q^{19} + ( - 10747862784 \beta_{3} + 20073592758744) q^{21} + (120345217740 \beta_{2} + 117138574281564 \beta_1) q^{22} + (73613782528 \beta_{2} + 73254195031752 \beta_1) q^{23} + (75922606044 \beta_{3} - 71657610384276) q^{24} + ( - 182488330690 \beta_{3} - 12\!\cdots\!58) q^{26}+ \cdots + ( - 10\!\cdots\!32 \beta_{3} - 38\!\cdots\!84) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2358472 q^{4} + 78653268 q^{6} - 13947137604 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2358472 q^{4} + 78653268 q^{6} - 13947137604 q^{9} + 439738245936 q^{11} + 1422595413792 q^{14} - 16577472881120 q^{16} - 23921170023248 q^{19} + 80294371034976 q^{21} - 286630441537104 q^{24} - 49\!\cdots\!32 q^{26}+ \cdots - 15\!\cdots\!36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 325x^{2} + 26244 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + 163\nu ) / 162 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7\nu^{3} + 3409\nu ) / 18 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( 126\nu^{2} + 20475 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - 63\beta_1 ) / 126 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{3} - 20475 ) / 126 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -163\beta_{2} + 30681\beta_1 ) / 126 \) Copy content Toggle raw display

Character values

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

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(-1\)

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.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
12.2377i
13.2377i
13.2377i
12.2377i
1937.96i 59049.0i −1.65852e6 0 1.14434e8 4.78205e7i 8.50053e8i −3.48678e9 0
49.2 1271.96i 59049.0i 479282. 0 −7.51077e7 6.32076e8i 3.27711e9i −3.48678e9 0
49.3 1271.96i 59049.0i 479282. 0 −7.51077e7 6.32076e8i 3.27711e9i −3.48678e9 0
49.4 1937.96i 59049.0i −1.65852e6 0 1.14434e8 4.78205e7i 8.50053e8i −3.48678e9 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 75.22.b.d 4
5.b even 2 1 inner 75.22.b.d 4
5.c odd 4 1 3.22.a.c 2
5.c odd 4 1 75.22.a.d 2
15.e even 4 1 9.22.a.e 2
20.e even 4 1 48.22.a.g 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3.22.a.c 2 5.c odd 4 1
9.22.a.e 2 15.e even 4 1
48.22.a.g 2 20.e even 4 1
75.22.a.d 2 5.c odd 4 1
75.22.b.d 4 1.a even 1 1 trivial
75.22.b.d 4 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 5373540T_{2}^{2} + 6076185560064 \) acting on \(S_{22}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + \cdots + 6076185560064 \) Copy content Toggle raw display
$3$ \( (T^{2} + 3486784401)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + \cdots + 91\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{2} + \cdots + 95\!\cdots\!12)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + \cdots + 33\!\cdots\!64 \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 61\!\cdots\!76 \) Copy content Toggle raw display
$19$ \( (T^{2} + \cdots - 12\!\cdots\!20)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{2} + \cdots - 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + \cdots + 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{2} + \cdots - 32\!\cdots\!80)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 41\!\cdots\!36 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 30\!\cdots\!56 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{2} + \cdots - 83\!\cdots\!20)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + \cdots - 74\!\cdots\!84)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 14\!\cdots\!16 \) Copy content Toggle raw display
$71$ \( (T^{2} + \cdots + 73\!\cdots\!84)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{2} + \cdots + 69\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 11\!\cdots\!44 \) Copy content Toggle raw display
$89$ \( (T^{2} + \cdots - 17\!\cdots\!80)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 75\!\cdots\!76 \) Copy content Toggle raw display
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