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Properties

Label 30.11.b.a
Level $30$
Weight $11$
Character orbit 30.b
Analytic conductor $19.061$
Analytic rank $0$
Dimension $20$
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] = [30,11,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, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("30.29");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 30 = 2 \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
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: \(19.0607175802\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} + 486208 x^{18} + 102177590160 x^{16} + \cdots + 19\!\cdots\!56 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{68}\cdot 3^{32}\cdot 5^{12} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + ( - \beta_{3} - \beta_{2}) q^{3} + 512 q^{4} + ( - \beta_{4} - 2 \beta_{3} + 2 \beta_{2}) q^{5} + (\beta_1 + 304) q^{6} + (\beta_{10} - \beta_{3}) q^{7} - 512 \beta_{2} q^{8} + ( - \beta_{5} + 2 \beta_{4} + \cdots - 3268) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + ( - \beta_{3} - \beta_{2}) q^{3} + 512 q^{4} + ( - \beta_{4} - 2 \beta_{3} + 2 \beta_{2}) q^{5} + (\beta_1 + 304) q^{6} + (\beta_{10} - \beta_{3}) q^{7} - 512 \beta_{2} q^{8} + ( - \beta_{5} + 2 \beta_{4} + \cdots - 3268) q^{9}+ \cdots + (64298 \beta_{17} - 19625 \beta_{16} + \cdots - 1978563692) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 10240 q^{4} + 6080 q^{6} - 65360 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 10240 q^{4} + 6080 q^{6} - 65360 q^{9} - 33536 q^{10} + 2083928 q^{15} + 5242880 q^{16} + 3209040 q^{19} - 654460 q^{21} + 3112960 q^{24} - 7480596 q^{25} + 34614848 q^{30} + 86022560 q^{31} + 81633280 q^{34} - 33464320 q^{36} + 93388800 q^{39} - 17170432 q^{40} + 440775292 q^{45} + 696625280 q^{46} - 1574325300 q^{49} - 2571307840 q^{51} + 681294400 q^{54} - 2158399088 q^{55} + 1066971136 q^{60} + 1300067680 q^{61} + 2684354560 q^{64} - 308783360 q^{66} - 2771050340 q^{69} - 443470976 q^{70} + 2946710128 q^{75} + 1643028480 q^{76} + 9709493760 q^{79} + 7324584140 q^{81} - 335083520 q^{84} + 10789531168 q^{85} - 5101384192 q^{90} - 20875353600 q^{91} + 7266454400 q^{94} + 1593835520 q^{96} - 39571273840 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} + 486208 x^{18} + 102177590160 x^{16} + \cdots + 19\!\cdots\!56 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 13\!\cdots\!67 \nu^{18} + \cdots - 50\!\cdots\!12 ) / 26\!\cdots\!61 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 25\!\cdots\!07 \nu^{19} + \cdots - 93\!\cdots\!12 \nu ) / 15\!\cdots\!46 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 12\!\cdots\!71 \nu^{19} + \cdots + 48\!\cdots\!84 \nu ) / 29\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 31\!\cdots\!79 \nu^{19} + \cdots + 28\!\cdots\!84 ) / 43\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 31\!\cdots\!99 \nu^{19} + \cdots - 12\!\cdots\!76 ) / 21\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 12\!\cdots\!51 \nu^{19} + \cdots + 10\!\cdots\!16 ) / 21\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 27\!\cdots\!49 \nu^{19} + \cdots + 28\!\cdots\!84 ) / 43\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 37\!\cdots\!33 \nu^{19} + \cdots + 24\!\cdots\!52 ) / 43\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 10\!\cdots\!73 \nu^{19} + \cdots + 73\!\cdots\!72 ) / 43\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 28\!\cdots\!11 \nu^{19} + \cdots - 10\!\cdots\!00 \nu ) / 87\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 76\!\cdots\!84 \nu^{19} + \cdots - 70\!\cdots\!00 ) / 15\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 78\!\cdots\!11 \nu^{19} + \cdots + 38\!\cdots\!72 ) / 95\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 48\!\cdots\!35 \nu^{19} + \cdots + 22\!\cdots\!12 ) / 43\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 71\!\cdots\!55 \nu^{19} + \cdots + 32\!\cdots\!36 ) / 62\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 25\!\cdots\!52 \nu^{19} + \cdots - 67\!\cdots\!76 ) / 21\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( - 52\!\cdots\!33 \nu^{19} + \cdots - 41\!\cdots\!64 ) / 43\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 26\!\cdots\!82 \nu^{19} + \cdots - 30\!\cdots\!76 ) / 21\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 56\!\cdots\!63 \nu^{19} + \cdots - 14\!\cdots\!76 ) / 43\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 46\!\cdots\!55 \nu^{19} + \cdots + 10\!\cdots\!72 ) / 24\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 10 \beta_{18} - 2 \beta_{17} + 7 \beta_{16} - 17 \beta_{15} + 3 \beta_{14} - 7 \beta_{13} + \cdots - 10 ) / 21600 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 531 \beta_{17} + 1526 \beta_{16} + 4054 \beta_{15} - 551 \beta_{14} - 1446 \beta_{13} + \cdots - 1050209280 ) / 21600 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 1440 \beta_{19} + 799690 \beta_{18} + 264014 \beta_{17} - 511759 \beta_{16} + 1618349 \beta_{15} + \cdots + 861340 ) / 21600 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 69993604 \beta_{17} - 66859829 \beta_{16} - 224687976 \beta_{15} + 125699 \beta_{14} + \cdots + 34606482059520 ) / 10800 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 410205600 \beta_{19} - 150143549270 \beta_{18} - 46718062039 \beta_{17} + 88471092689 \beta_{16} + \cdots - 150298405520 ) / 43200 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 17111193355801 \beta_{17} + 12365729421306 \beta_{16} + 44211715228304 \beta_{15} + \cdots - 51\!\cdots\!80 ) / 21600 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 48621110096160 \beta_{19} + \cdots + 13\!\cdots\!80 ) / 43200 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 88\!\cdots\!18 \beta_{17} + \cdots + 20\!\cdots\!60 ) / 1080 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 27\!\cdots\!80 \beta_{19} + \cdots - 62\!\cdots\!70 ) / 21600 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 17\!\cdots\!11 \beta_{17} + \cdots - 36\!\cdots\!80 ) / 21600 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 30\!\cdots\!20 \beta_{19} + \cdots + 58\!\cdots\!60 ) / 21600 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 84\!\cdots\!52 \beta_{17} + \cdots + 16\!\cdots\!60 ) / 10800 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 65\!\cdots\!40 \beta_{19} + \cdots - 10\!\cdots\!60 ) / 43200 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 16\!\cdots\!17 \beta_{17} + \cdots - 29\!\cdots\!60 ) / 21600 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 70\!\cdots\!20 \beta_{19} + \cdots + 10\!\cdots\!00 ) / 43200 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 38\!\cdots\!34 \beta_{17} + \cdots + 68\!\cdots\!20 ) / 540 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 37\!\cdots\!00 \beta_{19} + \cdots - 48\!\cdots\!90 ) / 21600 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( - 14\!\cdots\!43 \beta_{17} + \cdots - 25\!\cdots\!40 ) / 21600 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 39\!\cdots\!40 \beta_{19} + \cdots + 45\!\cdots\!00 ) / 21600 \) 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 + 307.050i
1.41421 − 307.050i
1.41421 − 194.561i
1.41421 + 194.561i
1.41421 − 231.440i
1.41421 + 231.440i
1.41421 − 230.332i
1.41421 + 230.332i
1.41421 − 66.0507i
1.41421 + 66.0507i
−1.41421 − 66.0507i
−1.41421 + 66.0507i
−1.41421 − 230.332i
−1.41421 + 230.332i
−1.41421 − 231.440i
−1.41421 + 231.440i
−1.41421 − 194.561i
−1.41421 + 194.561i
−1.41421 + 307.050i
−1.41421 − 307.050i
−22.6274 −233.559 − 67.0765i 512.000 −2581.60 − 1760.95i 5284.83 + 1517.77i − 22553.8i −11585.2 50050.5 + 31332.6i 58415.0 + 39845.8i
29.2 −22.6274 −233.559 + 67.0765i 512.000 −2581.60 + 1760.95i 5284.83 − 1517.77i 22553.8i −11585.2 50050.5 − 31332.6i 58415.0 − 39845.8i
29.3 −22.6274 −161.610 − 181.469i 512.000 3033.71 − 749.807i 3656.83 + 4106.18i − 5243.70i −11585.2 −6813.12 + 58654.6i −68645.1 + 16966.2i
29.4 −22.6274 −161.610 + 181.469i 512.000 3033.71 + 749.807i 3656.83 − 4106.18i 5243.70i −11585.2 −6813.12 − 58654.6i −68645.1 − 16966.2i
29.5 −22.6274 13.7033 − 242.613i 512.000 −1831.56 − 2532.00i −310.070 + 5489.71i 26803.8i −11585.2 −58673.4 − 6649.20i 41443.4 + 57292.6i
29.6 −22.6274 13.7033 + 242.613i 512.000 −1831.56 + 2532.00i −310.070 − 5489.71i − 26803.8i −11585.2 −58673.4 + 6649.20i 41443.4 − 57292.6i
29.7 −22.6274 89.2939 − 225.999i 512.000 −292.293 + 3111.30i −2020.49 + 5113.78i − 9956.97i −11585.2 −43102.2 − 40360.7i 6613.84 − 70400.7i
29.8 −22.6274 89.2939 + 225.999i 512.000 −292.293 − 3111.30i −2020.49 − 5113.78i 9956.97i −11585.2 −43102.2 + 40360.7i 6613.84 + 70400.7i
29.9 −22.6274 224.997 − 91.7898i 512.000 2042.27 − 2365.33i −5091.10 + 2076.97i − 21265.0i −11585.2 42198.3 − 41304.9i −46211.2 + 53521.2i
29.10 −22.6274 224.997 + 91.7898i 512.000 2042.27 + 2365.33i −5091.10 − 2076.97i 21265.0i −11585.2 42198.3 + 41304.9i −46211.2 − 53521.2i
29.11 22.6274 −224.997 − 91.7898i 512.000 −2042.27 + 2365.33i −5091.10 − 2076.97i − 21265.0i 11585.2 42198.3 + 41304.9i −46211.2 + 53521.2i
29.12 22.6274 −224.997 + 91.7898i 512.000 −2042.27 − 2365.33i −5091.10 + 2076.97i 21265.0i 11585.2 42198.3 − 41304.9i −46211.2 − 53521.2i
29.13 22.6274 −89.2939 − 225.999i 512.000 292.293 − 3111.30i −2020.49 − 5113.78i − 9956.97i 11585.2 −43102.2 + 40360.7i 6613.84 − 70400.7i
29.14 22.6274 −89.2939 + 225.999i 512.000 292.293 + 3111.30i −2020.49 + 5113.78i 9956.97i 11585.2 −43102.2 − 40360.7i 6613.84 + 70400.7i
29.15 22.6274 −13.7033 − 242.613i 512.000 1831.56 + 2532.00i −310.070 − 5489.71i 26803.8i 11585.2 −58673.4 + 6649.20i 41443.4 + 57292.6i
29.16 22.6274 −13.7033 + 242.613i 512.000 1831.56 − 2532.00i −310.070 + 5489.71i − 26803.8i 11585.2 −58673.4 − 6649.20i 41443.4 − 57292.6i
29.17 22.6274 161.610 − 181.469i 512.000 −3033.71 + 749.807i 3656.83 − 4106.18i − 5243.70i 11585.2 −6813.12 − 58654.6i −68645.1 + 16966.2i
29.18 22.6274 161.610 + 181.469i 512.000 −3033.71 − 749.807i 3656.83 + 4106.18i 5243.70i 11585.2 −6813.12 + 58654.6i −68645.1 − 16966.2i
29.19 22.6274 233.559 − 67.0765i 512.000 2581.60 + 1760.95i 5284.83 − 1517.77i − 22553.8i 11585.2 50050.5 − 31332.6i 58415.0 + 39845.8i
29.20 22.6274 233.559 + 67.0765i 512.000 2581.60 − 1760.95i 5284.83 + 1517.77i 22553.8i 11585.2 50050.5 + 31332.6i 58415.0 − 39845.8i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.20
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 30.11.b.a ✓ 20
3.b odd 2 1 inner 30.11.b.a ✓ 20
5.b even 2 1 inner 30.11.b.a ✓ 20
5.c odd 4 2 150.11.d.e 20
15.d odd 2 1 inner 30.11.b.a ✓ 20
15.e even 4 2 150.11.d.e 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.11.b.a ✓ 20 1.a even 1 1 trivial
30.11.b.a ✓ 20 3.b odd 2 1 inner
30.11.b.a ✓ 20 5.b even 2 1 inner
30.11.b.a ✓ 20 15.d odd 2 1 inner
150.11.d.e 20 5.c odd 4 2
150.11.d.e 20 15.e even 4 2

Hecke kernels

This newform subspace is the entire newspace \(S_{11}^{\mathrm{new}}(30, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 512)^{10} \) Copy content Toggle raw display
$3$ \( T^{20} + \cdots + 51\!\cdots\!01 \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 78\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots + 45\!\cdots\!24)^{2} \) Copy content Toggle raw display
$11$ \( (T^{10} + \cdots + 25\!\cdots\!32)^{2} \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 70\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{10} + \cdots - 12\!\cdots\!68)^{2} \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots - 10\!\cdots\!32)^{4} \) Copy content Toggle raw display
$23$ \( (T^{10} + \cdots - 11\!\cdots\!68)^{2} \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{5} + \cdots + 20\!\cdots\!32)^{4} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 20\!\cdots\!24)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots + 35\!\cdots\!32)^{2} \) Copy content Toggle raw display
$43$ \( (T^{10} + \cdots + 21\!\cdots\!76)^{2} \) Copy content Toggle raw display
$47$ \( (T^{10} + \cdots - 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{10} + \cdots - 20\!\cdots\!68)^{2} \) Copy content Toggle raw display
$59$ \( (T^{10} + \cdots + 38\!\cdots\!32)^{2} \) Copy content Toggle raw display
$61$ \( (T^{5} + \cdots - 79\!\cdots\!24)^{4} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 17\!\cdots\!24)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{5} + \cdots - 22\!\cdots\!68)^{4} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 27\!\cdots\!32)^{2} \) Copy content Toggle raw display
$89$ \( (T^{10} + \cdots + 54\!\cdots\!32)^{2} \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 97\!\cdots\!76)^{2} \) Copy content Toggle raw display
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