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Properties

Label 75.22.b.h
Level $75$
Weight $22$
Character orbit 75.b
Analytic conductor $209.608$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

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: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 7024333x^{6} + 16978215612516x^{4} + 16436248843740390400x^{2} + 5387455445730918400000000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{8}\cdot 5^{4}\cdot 7^{4} \)
Twist minimal: no (minimal twist has level 15)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (224 \beta_{5} + \beta_1) q^{2} - 59049 \beta_{5} q^{3} + (\beta_{3} - 294 \beta_{2} + 290854) q^{4} + (59049 \beta_{2} + 13226976) q^{6} + (11 \beta_{7} + 908 \beta_{6} + \cdots - 325419 \beta_1) q^{7}+ \cdots - 3486784401 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (224 \beta_{5} + \beta_1) q^{2} - 59049 \beta_{5} q^{3} + (\beta_{3} - 294 \beta_{2} + 290854) q^{4} + (59049 \beta_{2} + 13226976) q^{6} + (11 \beta_{7} + 908 \beta_{6} + \cdots - 325419 \beta_1) q^{7}+ \cdots + ( - 6506339692266 \beta_{4} + \cdots - 27\!\cdots\!22) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2326246 q^{4} + 105933906 q^{6} - 27894275208 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2326246 q^{4} + 105933906 q^{6} - 27894275208 q^{9} + 62983660512 q^{11} + 4466616253344 q^{14} - 22649362623550 q^{16} + 48540706601504 q^{19} + 27703134067392 q^{21} + 9076496073642 q^{24} + 949503245742756 q^{26} - 94\!\cdots\!92 q^{29}+ \cdots - 21\!\cdots\!12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 7024333x^{6} + 16978215612516x^{4} + 16436248843740390400x^{2} + 5387455445730918400000000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} - 356496147\nu^{4} - 1262087507347644\nu^{2} - 835611438336273920000 ) / 130872716198550720 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -77\nu^{6} + 27450203319\nu^{4} + 162617096165043948\nu^{2} + 179256308809908735593920 ) / 65436358099275360 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 418333 \nu^{6} + 2338948492779 \nu^{4} + \cdots + 18\!\cdots\!60 ) / 33\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 180004157 \nu^{7} + \cdots - 14\!\cdots\!00 \nu ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 158634902419577 \nu^{7} + \cdots + 83\!\cdots\!00 \nu ) / 75\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 88\!\cdots\!03 \nu^{7} + \cdots + 22\!\cdots\!00 \nu ) / 16\!\cdots\!00 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 154\beta_{2} - 1756122 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -22\beta_{7} + 711\beta_{6} + 270553984\beta_{5} - 2190262\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 22\beta_{4} - 3511455\beta_{3} - 902203782\beta_{2} + 3846432146268 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 77267674\beta_{7} - 2857159051\beta_{6} - 1584770384001636\beta_{5} + 5426554397630\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 7842915234\beta_{4} + 10267329483759\beta_{3} + 3602020238259942\beta_{2} - 9529959400602927172 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 220297092996090 \beta_{7} + \cdots - 14\!\cdots\!98 \beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

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
− 1492.88i
− 1711.97i
− 1069.36i
− 849.272i
849.272i
1069.36i
1711.97i
1492.88i
− 1716.88i 59049.0i −850541. 0 1.01380e8 − 6.24477e8i − 2.14029e9i −3.48678e9 0
49.2 − 1487.97i − 59049.0i −116903. 0 −8.78631e7 1.26833e9i − 2.94655e9i −3.48678e9 0
49.3 − 1293.36i 59049.0i 424379. 0 7.63714e7 1.27960e9i − 3.26124e9i −3.48678e9 0
49.4 − 625.272i − 59049.0i 1.70619e6 0 −3.69217e7 − 3.78633e8i − 2.37812e9i −3.48678e9 0
49.5 625.272i 59049.0i 1.70619e6 0 −3.69217e7 3.78633e8i 2.37812e9i −3.48678e9 0
49.6 1293.36i − 59049.0i 424379. 0 7.63714e7 − 1.27960e9i 3.26124e9i −3.48678e9 0
49.7 1487.97i 59049.0i −116903. 0 −8.78631e7 − 1.26833e9i 2.94655e9i −3.48678e9 0
49.8 1716.88i − 59049.0i −850541. 0 1.01380e8 6.24477e8i 2.14029e9i −3.48678e9 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.8
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.h 8
5.b even 2 1 inner 75.22.b.h 8
5.c odd 4 1 15.22.a.e ✓ 4
5.c odd 4 1 75.22.a.h 4
15.e even 4 1 45.22.a.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.22.a.e ✓ 4 5.c odd 4 1
45.22.a.g 4 15.e even 4 1
75.22.a.h 4 5.c odd 4 1
75.22.b.h 8 1.a even 1 1 trivial
75.22.b.h 8 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{8} + 7225485T_{2}^{6} + 17832839942628T_{2}^{4} + 16844437212865623040T_{2}^{2} + 4268203620075659666128896 \) acting on \(S_{22}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} + \cdots + 42\!\cdots\!96 \) Copy content Toggle raw display
$3$ \( (T^{2} + 3486784401)^{4} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 28\!\cdots\!64)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 72\!\cdots\!56 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 97\!\cdots\!36 \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 34\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 95\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots - 51\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 25\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 70\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 28\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 45\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 97\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 27\!\cdots\!76)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 78\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 17\!\cdots\!44)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 98\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 63\!\cdots\!36 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 85\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 28\!\cdots\!76 \) Copy content Toggle raw display
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