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Properties

Label 30.9.b.a
Level $30$
Weight $9$
Character orbit 30.b
Analytic conductor $12.221$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [30,9,Mod(29,30)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("30.29"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(30, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 30 = 2 \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 30.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.2213583018\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 11030 x^{14} + 49274731 x^{12} + 114127354194 x^{10} + 145952808215673 x^{8} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{40}\cdot 3^{20}\cdot 5^{10} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} - \beta_1 q^{3} + 128 q^{4} + ( - \beta_{6} - \beta_{2} + \beta_1) q^{5} + (\beta_{7} - 16) q^{6} + ( - \beta_{5} + \beta_{2} + 6 \beta_1) q^{7} - 128 \beta_{2} q^{8} + (\beta_{13} - \beta_{12} - \beta_{6} + \cdots + 1370) q^{9}+ \cdots + ( - 1122 \beta_{14} + 2930 \beta_{13} + \cdots + 47159485) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2048 q^{4} - 256 q^{6} + 21928 q^{9} + 1280 q^{10} - 62920 q^{15} + 262144 q^{16} + 96048 q^{19} + 667112 q^{21} - 32768 q^{24} + 1292160 q^{25} - 1269760 q^{30} - 5127152 q^{31} + 2971136 q^{34}+ \cdots + 754526432 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 11030 x^{14} + 49274731 x^{12} + 114127354194 x^{10} + 145952808215673 x^{8} + \cdots + 37\!\cdots\!00 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 13\!\cdots\!77 \nu^{15} + \cdots + 26\!\cdots\!00 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 21\!\cdots\!33 \nu^{15} + \cdots - 29\!\cdots\!70 \nu ) / 26\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 54\!\cdots\!69 \nu^{15} + \cdots - 19\!\cdots\!00 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 79\!\cdots\!87 \nu^{15} + \cdots - 79\!\cdots\!00 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 58\!\cdots\!03 \nu^{15} + \cdots + 86\!\cdots\!00 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 93\!\cdots\!71 \nu^{15} + \cdots + 13\!\cdots\!00 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 14\!\cdots\!14 \nu^{15} + \cdots - 15\!\cdots\!00 ) / 29\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 18\!\cdots\!71 \nu^{15} + \cdots + 45\!\cdots\!00 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 12\!\cdots\!83 \nu^{15} + \cdots - 28\!\cdots\!00 ) / 15\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 39\!\cdots\!61 \nu^{15} + \cdots + 20\!\cdots\!00 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 22\!\cdots\!71 \nu^{15} + \cdots + 38\!\cdots\!00 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 23\!\cdots\!97 \nu^{15} + \cdots + 27\!\cdots\!00 ) / 23\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 44\!\cdots\!53 \nu^{15} + \cdots + 15\!\cdots\!00 ) / 29\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 56\!\cdots\!41 \nu^{15} + \cdots - 29\!\cdots\!00 ) / 47\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 46\!\cdots\!07 \nu^{15} + \cdots + 37\!\cdots\!00 ) / 47\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 15 \beta_{14} + 70 \beta_{13} + 10 \beta_{12} - 143 \beta_{11} + 10 \beta_{8} + 162 \beta_{7} + \cdots + 5 ) / 21600 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 325 \beta_{14} - 1160 \beta_{13} + 1670 \beta_{12} + 1345 \beta_{11} + 40 \beta_{10} + \cdots - 29780165 ) / 21600 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 2610 \beta_{15} + 17555 \beta_{14} - 25965 \beta_{13} - 46720 \beta_{12} + 158817 \beta_{11} + \cdots - 23360 ) / 10800 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 928355 \beta_{14} + 3140840 \beta_{13} - 4424970 \beta_{12} - 3496615 \beta_{11} + \cdots + 62398670115 ) / 21600 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 23625900 \beta_{15} - 82694735 \beta_{14} + 66700150 \beta_{13} + 317212250 \beta_{12} + \cdots + 158606125 ) / 21600 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 263615125 \beta_{14} - 777082320 \beta_{13} + 1026934390 \beta_{12} + 763319265 \beta_{11} + \cdots - 14525740546645 ) / 2160 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 82255558560 \beta_{15} + 189492946145 \beta_{14} - 128453194470 \beta_{13} - 943150706110 \beta_{12} + \cdots - 471575353055 ) / 21600 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 7410566959425 \beta_{14} + 19095735893480 \beta_{13} - 23370337868110 \beta_{12} + \cdots + 34\!\cdots\!45 ) / 21600 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 130873528680030 \beta_{15} - 216982881668095 \beta_{14} + 149485022628150 \beta_{13} + \cdots + 667401784705525 ) / 10800 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 20\!\cdots\!85 \beta_{14} + \cdots - 85\!\cdots\!65 ) / 21600 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 79\!\cdots\!80 \beta_{15} + \cdots - 36\!\cdots\!65 ) / 21600 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 13\!\cdots\!72 \beta_{14} + \cdots + 53\!\cdots\!56 ) / 540 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 23\!\cdots\!40 \beta_{15} + \cdots + 99\!\cdots\!25 ) / 21600 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 14\!\cdots\!05 \beta_{14} + \cdots - 52\!\cdots\!45 ) / 21600 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 34\!\cdots\!30 \beta_{15} + \cdots - 13\!\cdots\!50 ) / 10800 \) 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.

Copy content comment:embeddings in the coefficient field
 
Copy content 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 + 45.0810i
1.41421 − 45.0810i
1.41421 + 12.8953i
1.41421 − 12.8953i
1.41421 − 26.0340i
1.41421 + 26.0340i
1.41421 + 51.4455i
1.41421 − 51.4455i
−1.41421 − 51.4455i
−1.41421 + 51.4455i
−1.41421 + 26.0340i
−1.41421 − 26.0340i
−1.41421 − 12.8953i
−1.41421 + 12.8953i
−1.41421 − 45.0810i
−1.41421 + 45.0810i
−11.3137 −80.8113 − 5.52629i 128.000 391.715 − 487.016i 914.275 + 62.5228i 4348.51i −1448.15 6499.92 + 893.173i −4431.75 + 5509.96i
29.2 −11.3137 −80.8113 + 5.52629i 128.000 391.715 + 487.016i 914.275 − 62.5228i − 4348.51i −1448.15 6499.92 − 893.173i −4431.75 − 5509.96i
29.3 −11.3137 −38.6282 − 71.1959i 128.000 −447.247 + 436.572i 437.028 + 805.490i 730.421i −1448.15 −3576.72 + 5500.34i 5060.02 − 4939.25i
29.4 −11.3137 −38.6282 + 71.1959i 128.000 −447.247 − 436.572i 437.028 − 805.490i − 730.421i −1448.15 −3576.72 − 5500.34i 5060.02 + 4939.25i
29.5 −11.3137 59.7326 − 54.7084i 128.000 556.269 + 284.937i −675.798 + 618.955i 2732.68i −1448.15 574.975 − 6535.76i −6293.47 − 3223.70i
29.6 −11.3137 59.7326 + 54.7084i 128.000 556.269 − 284.937i −675.798 − 618.955i − 2732.68i −1448.15 574.975 + 6535.76i −6293.47 + 3223.70i
29.7 −11.3137 65.3637 − 47.8392i 128.000 −529.022 − 332.808i −739.506 + 541.238i − 1228.22i −1448.15 1983.83 − 6253.89i 5985.20 + 3765.29i
29.8 −11.3137 65.3637 + 47.8392i 128.000 −529.022 + 332.808i −739.506 − 541.238i 1228.22i −1448.15 1983.83 + 6253.89i 5985.20 − 3765.29i
29.9 11.3137 −65.3637 − 47.8392i 128.000 529.022 + 332.808i −739.506 − 541.238i − 1228.22i 1448.15 1983.83 + 6253.89i 5985.20 + 3765.29i
29.10 11.3137 −65.3637 + 47.8392i 128.000 529.022 − 332.808i −739.506 + 541.238i 1228.22i 1448.15 1983.83 − 6253.89i 5985.20 − 3765.29i
29.11 11.3137 −59.7326 − 54.7084i 128.000 −556.269 − 284.937i −675.798 − 618.955i 2732.68i 1448.15 574.975 + 6535.76i −6293.47 − 3223.70i
29.12 11.3137 −59.7326 + 54.7084i 128.000 −556.269 + 284.937i −675.798 + 618.955i − 2732.68i 1448.15 574.975 − 6535.76i −6293.47 + 3223.70i
29.13 11.3137 38.6282 − 71.1959i 128.000 447.247 − 436.572i 437.028 − 805.490i 730.421i 1448.15 −3576.72 − 5500.34i 5060.02 − 4939.25i
29.14 11.3137 38.6282 + 71.1959i 128.000 447.247 + 436.572i 437.028 + 805.490i − 730.421i 1448.15 −3576.72 + 5500.34i 5060.02 + 4939.25i
29.15 11.3137 80.8113 − 5.52629i 128.000 −391.715 + 487.016i 914.275 − 62.5228i 4348.51i 1448.15 6499.92 − 893.173i −4431.75 + 5509.96i
29.16 11.3137 80.8113 + 5.52629i 128.000 −391.715 − 487.016i 914.275 + 62.5228i − 4348.51i 1448.15 6499.92 + 893.173i −4431.75 − 5509.96i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 29.16
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 30.9.b.a ✓ 16
3.b odd 2 1 inner 30.9.b.a ✓ 16
4.b odd 2 1 240.9.c.e 16
5.b even 2 1 inner 30.9.b.a ✓ 16
5.c odd 4 2 150.9.d.e 16
12.b even 2 1 240.9.c.e 16
15.d odd 2 1 inner 30.9.b.a ✓ 16
15.e even 4 2 150.9.d.e 16
20.d odd 2 1 240.9.c.e 16
60.h even 2 1 240.9.c.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
30.9.b.a ✓ 16 1.a even 1 1 trivial
30.9.b.a ✓ 16 3.b odd 2 1 inner
30.9.b.a ✓ 16 5.b even 2 1 inner
30.9.b.a ✓ 16 15.d odd 2 1 inner
150.9.d.e 16 5.c odd 4 2
150.9.d.e 16 15.e even 4 2
240.9.c.e 16 4.b odd 2 1
240.9.c.e 16 12.b even 2 1
240.9.c.e 16 20.d odd 2 1
240.9.c.e 16 60.h even 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{9}^{\mathrm{new}}(30, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 128)^{8} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 34\!\cdots\!81 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 54\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( (T^{8} + \cdots + 11\!\cdots\!16)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 31\!\cdots\!96)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 13\!\cdots\!16)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 45\!\cdots\!76)^{4} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 82\!\cdots\!76)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots - 15\!\cdots\!44)^{4} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 16\!\cdots\!56)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 26\!\cdots\!16)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 11\!\cdots\!16)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 15\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 67\!\cdots\!76)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 17\!\cdots\!16)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots + 94\!\cdots\!76)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 98\!\cdots\!36)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 85\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 84\!\cdots\!84)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 24\!\cdots\!96)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 34\!\cdots\!16)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 84\!\cdots\!96)^{2} \) Copy content Toggle raw display
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