Properties

Label 75.22.b.g
Level $75$
Weight $22$
Character orbit 75.b
Analytic conductor $209.608$
Analytic rank $0$
Dimension $6$
CM no
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: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 250653x^{4} + 15711440196x^{2} + 5841695641600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{4}\cdot 5^{2} \)
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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{4} + 767 \beta_1) q^{2} + 59049 \beta_1 q^{3} + (16 \beta_{3} + 1324 \beta_{2} - 421908) q^{4} + (59049 \beta_{2} - 45290583) q^{6} + (6045 \beta_{5} - 207601 \beta_{4} - 155291491 \beta_1) q^{7} + (36800 \beta_{5} + 16176 \beta_{4} - 1279953456 \beta_1) q^{8} - 3486784401 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} + 767 \beta_1) q^{2} + 59049 \beta_1 q^{3} + (16 \beta_{3} + 1324 \beta_{2} - 421908) q^{4} + (59049 \beta_{2} - 45290583) q^{6} + (6045 \beta_{5} - 207601 \beta_{4} - 155291491 \beta_1) q^{7} + (36800 \beta_{5} + 16176 \beta_{4} - 1279953456 \beta_1) q^{8} - 3486784401 q^{9} + (696322 \beta_{3} - 2618198 \beta_{2} + \cdots - 55777904786) q^{11}+ \cdots + ( - 24\!\cdots\!22 \beta_{3} + \cdots + 19\!\cdots\!86) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2528800 q^{4} - 271625400 q^{6} - 20920706406 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2528800 q^{4} - 271625400 q^{6} - 20920706406 q^{9} - 334672665112 q^{11} + 3137717805312 q^{14} + 510535909888 q^{16} - 7875401481656 q^{19} + 54994326249456 q^{21} + 453481740093312 q^{24} + 59\!\cdots\!12 q^{26}+ \cdots + 11\!\cdots\!12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 250653x^{4} + 15711440196x^{2} + 5841695641600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{5} + 2166307\nu^{3} + 287198905724\nu ) / 5538787712640 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 41\nu^{4} + 4374529\nu^{2} - 63678988555 ) / 40103595 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -99\nu^{4} - 13171251\nu^{2} - 64137673740 ) / 891191 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 3419\nu^{5} + 708836239\nu^{3} + 36140757832172\nu ) / 426060593280 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -11111\nu^{5} - 2305342507\nu^{3} - 106595286708572\nu ) / 30771042848 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{5} + 45\beta_{4} + 135\beta_1 ) / 360 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -41\beta_{3} - 4455\beta_{2} - 10024635 ) / 120 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -125449\beta_{5} - 5626305\beta_{4} + 823112685\beta_1 ) / 360 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 4374529\beta_{3} + 592706295\beta_{2} + 1255964271915 ) / 120 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 15437858881\beta_{5} + 735646851945\beta_{4} - 172076952973365\beta_1 ) / 360 \) 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
19.3401i
362.802i
344.462i
344.462i
362.802i
19.3401i
2395.52i 59049.0i −3.64136e6 0 −1.41453e8 8.95100e8i 3.69919e9i −3.48678e9 0
49.2 999.805i 59049.0i 1.09754e6 0 5.90375e7 9.82313e7i 3.19407e9i −3.48678e9 0
49.3 904.285i 59049.0i 1.27942e6 0 −5.33971e7 5.27665e8i 3.05338e9i −3.48678e9 0
49.4 904.285i 59049.0i 1.27942e6 0 −5.33971e7 5.27665e8i 3.05338e9i −3.48678e9 0
49.5 999.805i 59049.0i 1.09754e6 0 5.90375e7 9.82313e7i 3.19407e9i −3.48678e9 0
49.6 2395.52i 59049.0i −3.64136e6 0 −1.41453e8 8.95100e8i 3.69919e9i −3.48678e9 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.6
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.g 6
5.b even 2 1 inner 75.22.b.g 6
5.c odd 4 1 15.22.a.b 3
5.c odd 4 1 75.22.a.g 3
15.e even 4 1 45.22.a.e 3
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
15.22.a.b 3 5.c odd 4 1
45.22.a.e 3 15.e even 4 1
75.22.a.g 3 5.c odd 4 1
75.22.b.g 6 1.a even 1 1 trivial
75.22.b.g 6 5.b even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} + 7555856T_{2}^{4} + 11246247952384T_{2}^{2} + 4690729282886959104 \) acting on \(S_{22}^{\mathrm{new}}(75, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{6} + 7555856 T^{4} + \cdots + 46\!\cdots\!04 \) Copy content Toggle raw display
$3$ \( (T^{2} + 3486784401)^{3} \) Copy content Toggle raw display
$5$ \( T^{6} \) Copy content Toggle raw display
$7$ \( T^{6} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{3} + 167336332556 T^{2} + \cdots - 24\!\cdots\!32)^{2} \) Copy content Toggle raw display
$13$ \( T^{6} + \cdots + 26\!\cdots\!16 \) Copy content Toggle raw display
$17$ \( T^{6} + \cdots + 17\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( (T^{3} + 3937700740828 T^{2} + \cdots + 10\!\cdots\!20)^{2} \) Copy content Toggle raw display
$23$ \( T^{6} + \cdots + 94\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{3} - 930273612785494 T^{2} + \cdots + 56\!\cdots\!40)^{2} \) Copy content Toggle raw display
$31$ \( (T^{3} + \cdots - 49\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{6} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{3} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{6} + \cdots + 13\!\cdots\!56 \) Copy content Toggle raw display
$47$ \( T^{6} + \cdots + 16\!\cdots\!84 \) Copy content Toggle raw display
$53$ \( T^{6} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{3} + \cdots - 34\!\cdots\!60)^{2} \) Copy content Toggle raw display
$61$ \( (T^{3} + \cdots - 87\!\cdots\!28)^{2} \) Copy content Toggle raw display
$67$ \( T^{6} + \cdots + 55\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( (T^{3} + \cdots - 51\!\cdots\!88)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{3} + \cdots + 82\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{6} + \cdots + 59\!\cdots\!96 \) Copy content Toggle raw display
$89$ \( (T^{3} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{6} + \cdots + 73\!\cdots\!24 \) Copy content Toggle raw display
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