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Properties

Label 130.2.p.b
Level $130$
Weight $2$
Character orbit 130.p
Analytic conductor $1.038$
Analytic rank $0$
Dimension $16$
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] = [130,2,Mod(7,130)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(130, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([3, 11]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("130.7");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 130 = 2 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 130.p (of order \(12\), degree \(4\), 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: \(1.03805522628\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{13} - 48 x^{12} + 16 x^{11} + 8 x^{10} + 80 x^{9} + 2208 x^{8} + 760 x^{7} + 192 x^{6} + \cdots + 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{5} + \beta_{3}) q^{2} + ( - \beta_{4} + \beta_1) q^{3} + (\beta_{8} + 1) q^{4} + ( - \beta_{8} + \beta_{6} - 1) q^{5} + (\beta_{11} - \beta_{2}) q^{6} + ( - \beta_{15} + \beta_{12} + \beta_{10}) q^{7} + \beta_{3} q^{8} + (\beta_{10} - \beta_{9} + \beta_{5}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{5} + \beta_{3}) q^{2} + ( - \beta_{4} + \beta_1) q^{3} + (\beta_{8} + 1) q^{4} + ( - \beta_{8} + \beta_{6} - 1) q^{5} + (\beta_{11} - \beta_{2}) q^{6} + ( - \beta_{15} + \beta_{12} + \beta_{10}) q^{7} + \beta_{3} q^{8} + (\beta_{10} - \beta_{9} + \beta_{5}) q^{9} + ( - \beta_{10} - \beta_{3}) q^{10} + ( - \beta_{12} - \beta_{7} + \cdots - \beta_1) q^{11}+ \cdots + ( - \beta_{9} - 4 \beta_{8} + \beta_{6} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{4} - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{4} - 2 q^{5} + 6 q^{11} + 2 q^{13} + 6 q^{15} - 8 q^{16} - 16 q^{17} - 16 q^{18} + 8 q^{20} - 6 q^{22} - 6 q^{23} - 14 q^{25} - 6 q^{26} - 12 q^{27} - 6 q^{29} + 6 q^{30} - 6 q^{33} - 14 q^{34} - 20 q^{37} + 6 q^{38} - 6 q^{39} - 44 q^{41} + 6 q^{42} + 6 q^{44} - 6 q^{46} + 52 q^{47} - 2 q^{49} + 10 q^{52} - 24 q^{53} + 6 q^{54} + 64 q^{55} + 6 q^{56} + 8 q^{58} - 46 q^{59} + 6 q^{61} + 12 q^{62} + 90 q^{63} - 16 q^{64} - 22 q^{65} + 52 q^{66} + 12 q^{67} - 2 q^{68} + 58 q^{69} - 32 q^{70} + 6 q^{71} - 8 q^{72} + 24 q^{74} + 44 q^{75} - 6 q^{76} + 58 q^{77} + 38 q^{78} + 10 q^{80} - 24 q^{81} - 44 q^{82} - 64 q^{83} + 6 q^{84} + 40 q^{85} - 44 q^{87} - 24 q^{89} - 18 q^{90} + 38 q^{91} - 26 q^{93} - 6 q^{95} - 6 q^{97} - 26 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{13} - 48 x^{12} + 16 x^{11} + 8 x^{10} + 80 x^{9} + 2208 x^{8} + 760 x^{7} + 192 x^{6} + \cdots + 4096 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 590814673 \nu^{15} - 4146468380 \nu^{14} - 928390744 \nu^{13} + 4089687260 \nu^{12} + \cdots - 16628864598016 ) / 104859315460224 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 238350271 \nu^{15} + 6776112136 \nu^{14} + 920962592 \nu^{13} + 1417547900 \nu^{12} + \cdots - 1698553709056 ) / 24672780108288 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 42599 \nu^{15} + 44468 \nu^{14} - 981520 \nu^{13} + 201412 \nu^{12} + 1784920 \nu^{11} + \cdots + 953416704 ) / 3150915456 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 12187170977 \nu^{15} + 112830647620 \nu^{14} - 929509456 \nu^{13} - 44571751012 \nu^{12} + \cdots - 17112277705216 ) / 419437261840896 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 1265139977 \nu^{15} - 1178805226 \nu^{14} + 5432135497 \nu^{13} + 22313466160 \nu^{12} + \cdots - 38434909738112 ) / 26214828865056 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 1638690067 \nu^{15} + 2114100244 \nu^{14} - 13423970953 \nu^{13} + 22733755088 \nu^{12} + \cdots - 4170710020480 ) / 26214828865056 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 931071 \nu^{15} + 170396 \nu^{14} - 177872 \nu^{13} + 201796 \nu^{12} - 45497056 \nu^{11} + \cdots - 12832193536 ) / 12603661824 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 9852329051 \nu^{15} + 28705092392 \nu^{14} + 47979745276 \nu^{13} + \cdots - 27683541191168 ) / 104859315460224 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 5879837650 \nu^{15} - 12865387787 \nu^{14} + 11969268317 \nu^{13} + 23302404986 \nu^{12} + \cdots - 4094491643392 ) / 26214828865056 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 28207661905 \nu^{15} - 232377364 \nu^{14} + 1044233224 \nu^{13} - 118518889004 \nu^{12} + \cdots - 12479663080448 ) / 104859315460224 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 38410768049 \nu^{15} + 34463647156 \nu^{14} - 14862048280 \nu^{13} - 159419819020 \nu^{12} + \cdots + 6777896482304 ) / 104859315460224 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 38453963605 \nu^{15} + 1001037668 \nu^{14} + 27774368104 \nu^{13} - 190478280212 \nu^{12} + \cdots + 165237149091328 ) / 104859315460224 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 38686340969 \nu^{15} - 43195556 \nu^{14} + 33462609488 \nu^{13} - 197381780260 \nu^{12} + \cdots - 138661529586688 ) / 104859315460224 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 10245718021 \nu^{15} - 12016563769 \nu^{14} - 7887648452 \nu^{13} - 43717796048 \nu^{12} + \cdots + 7461651612160 ) / 26214828865056 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{15} - \beta_{12} - \beta_{10} + \beta_{9} - 4\beta_{5} + 4\beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{12} - \beta_{10} - \beta_{9} - \beta_{7} + \beta_{6} - 6\beta_{2} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -8\beta_{13} + \beta_{11} + 24\beta_{8} - 8\beta_{7} - 8\beta_{6} - \beta_{2} + \beta _1 + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 9 \beta_{15} - 9 \beta_{14} - 8 \beta_{13} - \beta_{12} - 9 \beta_{10} + \beta_{9} + 4 \beta_{8} + \cdots + 40 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 56\beta_{15} - 58\beta_{12} - 2\beta_{10} - 2\beta_{9} - 14\beta_{4} + 160\beta_{3} - 14\beta_{2} \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 56 \beta_{15} - 56 \beta_{14} - 72 \beta_{13} - 56 \beta_{12} + 274 \beta_{11} + 16 \beta_{10} + \cdots + 56 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 418 \beta_{14} - 32 \beta_{13} + 144 \beta_{11} + 1096 \beta_{8} - 386 \beta_{7} - 418 \beta_{6} + \cdots + 144 \beta_1 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 176 \beta_{15} - 176 \beta_{14} + 176 \beta_{13} - 562 \beta_{12} - 386 \beta_{10} - 386 \beta_{9} + \cdots - 576 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 352 \beta_{15} - 352 \beta_{12} + 1314 \beta_{11} + 2672 \beta_{10} - 3024 \beta_{9} + \cdots - 1314 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 4338 \beta_{15} - 4338 \beta_{14} - 2672 \beta_{13} + 1666 \beta_{12} + 13296 \beta_{11} + \cdots - 5256 \beta_{3} \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( -18640\beta_{14} + 18640\beta_{13} + 3332\beta_{7} - 3332\beta_{6} - 11260\beta_{4} + 11260\beta_{2} - 53184 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 18640 \beta_{15} + 18640 \beta_{14} + 33232 \beta_{13} - 18640 \beta_{12} + 14592 \beta_{10} + \cdots - 45040 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 160260 \beta_{15} + 131076 \beta_{12} + 92864 \beta_{11} + 160260 \beta_{10} - 131076 \beta_{9} + \cdots - 92864 \beta_1 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( - 122048 \beta_{15} - 122048 \beta_{14} + 122048 \beta_{13} + 253124 \beta_{12} + 131076 \beta_{10} + \cdots - 371456 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/130\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(41\)
\(\chi(n)\) \(\beta_{3}\) \(-\beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1
−2.38987 − 0.640364i
−1.18141 − 0.316559i
0.935952 + 0.250788i
2.63533 + 0.706135i
−0.706135 − 2.63533i
−0.250788 − 0.935952i
0.316559 + 1.18141i
0.640364 + 2.38987i
−2.38987 + 0.640364i
−1.18141 + 0.316559i
0.935952 − 0.250788i
2.63533 − 0.706135i
−0.706135 + 2.63533i
−0.250788 + 0.935952i
0.316559 − 1.18141i
0.640364 − 2.38987i
0.866025 + 0.500000i −0.640364 − 2.38987i 0.500000 + 0.866025i −0.606329 − 2.15229i 0.640364 − 2.38987i −0.551051 − 0.954448i 1.00000i −2.70334 + 1.56078i 0.551051 − 2.16710i
7.2 0.866025 + 0.500000i −0.316559 − 1.18141i 0.500000 + 0.866025i 1.44768 + 1.70418i 0.316559 − 1.18141i −0.401632 − 0.695647i 1.00000i 1.30255 − 0.752028i 0.401632 + 2.19970i
7.3 0.866025 + 0.500000i 0.250788 + 0.935952i 0.500000 + 0.866025i −2.23384 − 0.0997335i −0.250788 + 0.935952i 1.88470 + 3.26439i 1.00000i 1.78496 − 1.03055i −1.88470 − 1.20329i
7.4 0.866025 + 0.500000i 0.706135 + 2.63533i 0.500000 + 0.866025i 0.892496 − 2.05023i −0.706135 + 2.63533i −1.79804 − 3.11430i 1.00000i −3.84827 + 2.22180i 1.79804 − 1.32930i
37.1 −0.866025 + 0.500000i −2.63533 − 0.706135i 0.500000 − 0.866025i 0.892496 + 2.05023i 2.63533 − 0.706135i 1.79804 − 3.11430i 1.00000i 3.84827 + 2.22180i −1.79804 − 1.32930i
37.2 −0.866025 + 0.500000i −0.935952 − 0.250788i 0.500000 − 0.866025i −2.23384 + 0.0997335i 0.935952 − 0.250788i −1.88470 + 3.26439i 1.00000i −1.78496 − 1.03055i 1.88470 − 1.20329i
37.3 −0.866025 + 0.500000i 1.18141 + 0.316559i 0.500000 − 0.866025i 1.44768 − 1.70418i −1.18141 + 0.316559i 0.401632 − 0.695647i 1.00000i −1.30255 − 0.752028i −0.401632 + 2.19970i
37.4 −0.866025 + 0.500000i 2.38987 + 0.640364i 0.500000 − 0.866025i −0.606329 + 2.15229i −2.38987 + 0.640364i 0.551051 − 0.954448i 1.00000i 2.70334 + 1.56078i −0.551051 − 2.16710i
93.1 0.866025 − 0.500000i −0.640364 + 2.38987i 0.500000 − 0.866025i −0.606329 + 2.15229i 0.640364 + 2.38987i −0.551051 + 0.954448i − 1.00000i −2.70334 − 1.56078i 0.551051 + 2.16710i
93.2 0.866025 − 0.500000i −0.316559 + 1.18141i 0.500000 − 0.866025i 1.44768 − 1.70418i 0.316559 + 1.18141i −0.401632 + 0.695647i − 1.00000i 1.30255 + 0.752028i 0.401632 − 2.19970i
93.3 0.866025 − 0.500000i 0.250788 − 0.935952i 0.500000 − 0.866025i −2.23384 + 0.0997335i −0.250788 − 0.935952i 1.88470 − 3.26439i − 1.00000i 1.78496 + 1.03055i −1.88470 + 1.20329i
93.4 0.866025 − 0.500000i 0.706135 − 2.63533i 0.500000 − 0.866025i 0.892496 + 2.05023i −0.706135 − 2.63533i −1.79804 + 3.11430i − 1.00000i −3.84827 − 2.22180i 1.79804 + 1.32930i
123.1 −0.866025 − 0.500000i −2.63533 + 0.706135i 0.500000 + 0.866025i 0.892496 − 2.05023i 2.63533 + 0.706135i 1.79804 + 3.11430i − 1.00000i 3.84827 − 2.22180i −1.79804 + 1.32930i
123.2 −0.866025 − 0.500000i −0.935952 + 0.250788i 0.500000 + 0.866025i −2.23384 − 0.0997335i 0.935952 + 0.250788i −1.88470 − 3.26439i − 1.00000i −1.78496 + 1.03055i 1.88470 + 1.20329i
123.3 −0.866025 − 0.500000i 1.18141 − 0.316559i 0.500000 + 0.866025i 1.44768 + 1.70418i −1.18141 − 0.316559i 0.401632 + 0.695647i − 1.00000i −1.30255 + 0.752028i −0.401632 − 2.19970i
123.4 −0.866025 − 0.500000i 2.38987 − 0.640364i 0.500000 + 0.866025i −0.606329 − 2.15229i −2.38987 − 0.640364i 0.551051 + 0.954448i − 1.00000i 2.70334 − 1.56078i −0.551051 + 2.16710i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 7.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.t even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 130.2.p.b ✓ 16
5.b even 2 1 650.2.t.g 16
5.c odd 4 1 130.2.s.b yes 16
5.c odd 4 1 650.2.w.g 16
13.f odd 12 1 130.2.s.b yes 16
65.o even 12 1 650.2.t.g 16
65.s odd 12 1 650.2.w.g 16
65.t even 12 1 inner 130.2.p.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.2.p.b ✓ 16 1.a even 1 1 trivial
130.2.p.b ✓ 16 65.t even 12 1 inner
130.2.s.b yes 16 5.c odd 4 1
130.2.s.b yes 16 13.f odd 12 1
650.2.t.g 16 5.b even 2 1
650.2.t.g 16 65.o even 12 1
650.2.w.g 16 5.c odd 4 1
650.2.w.g 16 65.s odd 12 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 4 T_{3}^{13} - 48 T_{3}^{12} - 16 T_{3}^{11} + 8 T_{3}^{10} - 80 T_{3}^{9} + 2208 T_{3}^{8} + \cdots + 4096 \) acting on \(S_{2}^{\mathrm{new}}(130, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - T^{2} + 1)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} + 4 T^{13} + \cdots + 4096 \) Copy content Toggle raw display
$5$ \( (T^{8} + T^{7} + 4 T^{6} + \cdots + 625)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} + 29 T^{14} + \cdots + 20736 \) Copy content Toggle raw display
$11$ \( T^{16} - 6 T^{15} + \cdots + 2408704 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 815730721 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 15513698916 \) Copy content Toggle raw display
$19$ \( T^{16} + 3 T^{14} + \cdots + 589824 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 6996987904 \) Copy content Toggle raw display
$29$ \( T^{16} + 6 T^{15} + \cdots + 2143296 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 345518244864 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 856089081 \) Copy content Toggle raw display
$41$ \( T^{16} + 44 T^{15} + \cdots + 8596624 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 9764601856 \) Copy content Toggle raw display
$47$ \( (T^{8} - 26 T^{7} + \cdots + 75232)^{2} \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 7629498409 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 2479944646656 \) Copy content Toggle raw display
$61$ \( T^{16} - 6 T^{15} + \cdots + 1633284 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 123184152576 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 5473632256 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 903408428484 \) Copy content Toggle raw display
$79$ \( T^{16} + 268 T^{14} + \cdots + 14017536 \) Copy content Toggle raw display
$83$ \( (T^{8} + 32 T^{7} + \cdots - 5965248)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 2057050114564 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 478697992372224 \) Copy content Toggle raw display
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