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Properties

Label 30.5.b.a
Level $30$
Weight $5$
Character orbit 30.b
Analytic conductor $3.101$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [30,5,Mod(29,30)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(30, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("30.29");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 30 = 2 \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 30.b (of order \(2\), degree \(1\), 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: \(3.10109889252\)
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: \( x^{8} + 80x^{6} + 2612x^{4} + 38240x^{2} + 256036 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{2} \)
Twist minimal: yes
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{2} + \beta_1) q^{3} + 8 q^{4} + ( - \beta_{5} + \beta_{2} + 2 \beta_1) q^{5} + ( - \beta_{4} + 4) q^{6} + (\beta_{7} - \beta_{6} + \cdots + 2 \beta_1) q^{7}+ \cdots + (\beta_{7} + \beta_{6} - \beta_{5} + \cdots - 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{2} + \beta_1) q^{3} + 8 q^{4} + ( - \beta_{5} + \beta_{2} + 2 \beta_1) q^{5} + ( - \beta_{4} + 4) q^{6} + (\beta_{7} - \beta_{6} + \cdots + 2 \beta_1) q^{7}+ \cdots + (17 \beta_{7} + 17 \beta_{6} + \cdots - 2768) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 64 q^{4} + 32 q^{6} - 56 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 64 q^{4} + 32 q^{6} - 56 q^{9} + 160 q^{10} - 760 q^{15} + 512 q^{16} - 816 q^{19} - 1024 q^{21} + 256 q^{24} + 840 q^{25} - 1600 q^{30} + 6224 q^{31} - 3392 q^{34} - 448 q^{36} + 10560 q^{39} + 1280 q^{40} - 6080 q^{45} - 2368 q^{46} - 21240 q^{49} - 3376 q^{51} + 5920 q^{54} + 5440 q^{55} - 6080 q^{60} - 368 q^{61} + 4096 q^{64} + 13696 q^{66} + 17296 q^{69} + 14080 q^{70} + 5440 q^{75} - 6528 q^{76} - 1584 q^{79} - 33784 q^{81} - 8192 q^{84} - 3440 q^{85} - 14560 q^{90} + 30720 q^{91} + 6080 q^{94} + 2048 q^{96} - 22144 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 80x^{6} + 2612x^{4} + 38240x^{2} + 256036 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -16\nu^{7} - 1027\nu^{5} - 27624\nu^{3} - 224750\nu ) / 132572 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 9\nu^{7} + 46\nu^{6} + 766\nu^{5} + 2576\nu^{4} + 20058\nu^{3} + 70380\nu^{2} + 173500\nu + 648416 ) / 48208 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 178 \nu^{7} + 759 \nu^{6} - 23854 \nu^{5} + 42504 \nu^{4} - 870748 \nu^{3} + 1161270 \nu^{2} + \cdots + 10698864 ) / 795432 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 16\nu^{7} + 253\nu^{6} + 1027\nu^{5} + 14168\nu^{4} + 27624\nu^{3} + 254518\nu^{2} + 489894\nu + 914848 ) / 66286 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 608 \nu^{7} + 759 \nu^{6} - 39026 \nu^{5} + 42504 \nu^{4} - 784568 \nu^{3} + 1161270 \nu^{2} + \cdots + 10698864 ) / 795432 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1139 \nu^{7} - 12650 \nu^{6} + 42038 \nu^{5} - 973544 \nu^{4} + 160190 \nu^{3} - 24657380 \nu^{2} + \cdots - 206419664 ) / 1590864 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2117 \nu^{7} + 506 \nu^{6} + 117242 \nu^{5} + 293480 \nu^{4} + 2146994 \nu^{3} + 12440516 \nu^{2} + \cdots + 162506960 ) / 1590864 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} + 2\beta_{5} + 2\beta_{4} - 2\beta_{3} + 12\beta_1 ) / 24 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{7} + \beta_{6} + 6\beta_{5} - 4\beta_{4} + 2\beta_{3} + 16\beta_{2} - 8\beta _1 - 240 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -13\beta_{7} - 13\beta_{6} + 4\beta_{5} - 26\beta_{4} + 20\beta_{3} - 24\beta_{2} - 372\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( \beta_{7} - 17\beta_{6} - 58\beta_{5} + 32\beta_{4} - 26\beta_{3} - 186\beta_{2} + 102\beta _1 + 882 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 187\beta_{7} + 187\beta_{6} - 858\beta_{5} + 374\beta_{4} - 742\beta_{3} + 1600\beta_{2} + 13204\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 877 \beta_{7} + 1139 \beta_{6} + 2954 \beta_{5} - 524 \beta_{4} + 2430 \beta_{3} + 12784 \beta_{2} + \cdots + 240 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1709 \beta_{7} + 1709 \beta_{6} + 17060 \beta_{5} + 3418 \beta_{4} + 13572 \beta_{3} + \cdots - 194492 \beta_1 ) / 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/30\mathbb{Z}\right)^\times\).

\(n\) \(7\) \(11\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
29.1
−1.41421 − 3.69234i
−1.41421 + 3.69234i
−1.41421 − 5.51059i
−1.41421 + 5.51059i
1.41421 + 5.51059i
1.41421 − 5.51059i
1.41421 + 3.69234i
1.41421 − 3.69234i
−2.82843 −7.33029 − 5.22176i 8.00000 10.6772 + 22.6053i 20.7332 + 14.7694i 65.2135i −22.6274 26.4664 + 76.5541i −30.1996 − 63.9373i
29.2 −2.82843 −7.33029 + 5.22176i 8.00000 10.6772 − 22.6053i 20.7332 − 14.7694i − 65.2135i −22.6274 26.4664 − 76.5541i −30.1996 + 63.9373i
29.3 −2.82843 4.50187 − 7.79315i 8.00000 −24.8193 + 3.00033i −12.7332 + 22.0424i − 76.5454i −22.6274 −40.4664 − 70.1674i 70.1996 − 8.48622i
29.4 −2.82843 4.50187 + 7.79315i 8.00000 −24.8193 − 3.00033i −12.7332 − 22.0424i 76.5454i −22.6274 −40.4664 + 70.1674i 70.1996 + 8.48622i
29.5 2.82843 −4.50187 − 7.79315i 8.00000 24.8193 − 3.00033i −12.7332 − 22.0424i − 76.5454i 22.6274 −40.4664 + 70.1674i 70.1996 − 8.48622i
29.6 2.82843 −4.50187 + 7.79315i 8.00000 24.8193 + 3.00033i −12.7332 + 22.0424i 76.5454i 22.6274 −40.4664 − 70.1674i 70.1996 + 8.48622i
29.7 2.82843 7.33029 − 5.22176i 8.00000 −10.6772 − 22.6053i 20.7332 − 14.7694i 65.2135i 22.6274 26.4664 − 76.5541i −30.1996 − 63.9373i
29.8 2.82843 7.33029 + 5.22176i 8.00000 −10.6772 + 22.6053i 20.7332 + 14.7694i − 65.2135i 22.6274 26.4664 + 76.5541i −30.1996 + 63.9373i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 30.5.b.a ✓ 8
3.b odd 2 1 inner 30.5.b.a ✓ 8
4.b odd 2 1 240.5.c.d 8
5.b even 2 1 inner 30.5.b.a ✓ 8
5.c odd 4 2 150.5.d.e 8
12.b even 2 1 240.5.c.d 8
15.d odd 2 1 inner 30.5.b.a ✓ 8
15.e even 4 2 150.5.d.e 8
20.d odd 2 1 240.5.c.d 8
60.h even 2 1 240.5.c.d 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.5.b.a ✓ 8 1.a even 1 1 trivial
30.5.b.a ✓ 8 3.b odd 2 1 inner
30.5.b.a ✓ 8 5.b even 2 1 inner
30.5.b.a ✓ 8 15.d odd 2 1 inner
150.5.d.e 8 5.c odd 4 2
150.5.d.e 8 15.e even 4 2
240.5.c.d 8 4.b odd 2 1
240.5.c.d 8 12.b even 2 1
240.5.c.d 8 20.d odd 2 1
240.5.c.d 8 60.h even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{5}^{\mathrm{new}}(30, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 8)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} + 28 T^{6} + \cdots + 43046721 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 152587890625 \) Copy content Toggle raw display
$7$ \( (T^{4} + 10112 T^{2} + 24918016)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 17824 T^{2} + 30031744)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 79200 T^{2} + 1341360000)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 47464 T^{2} + 449948944)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 204 T - 150876)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 326824 T^{2} + 20024514064)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 1516320 T^{2} + 39114057600)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 1556 T - 402716)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 851552 T^{2} + 181093238656)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots + 6152820619264)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 2953568 T^{2} + 170627671936)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 207400 T^{2} + 1656490000)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 109209438910224)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 18587629012864)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 92 T - 643004)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + \cdots + 423153629435776)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots + 354933728409600)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + \cdots + 5998752102400)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 396 T - 5766876)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + \cdots + 29588682444304)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots + 11\!\cdots\!64)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots + 427910510411776)^{2} \) Copy content Toggle raw display
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