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Properties

Label 130.3.h.d
Level $130$
Weight $3$
Character orbit 130.h
Analytic conductor $3.542$
Analytic rank $0$
Dimension $12$
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] = [130,3,Mod(77,130)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(130, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([1, 2]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("130.77");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 130 = 2 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 130.h (of order \(4\), degree \(2\), 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: \(3.54224343668\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + 8 x^{10} + 4 x^{9} + 660 x^{8} - 2696 x^{7} + 5512 x^{6} + 5536 x^{5} + 39140 x^{4} + \cdots + 8836 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{3} + 1) q^{2} + \beta_{2} q^{3} - 2 \beta_{3} q^{4} + ( - \beta_{11} + \beta_{6} - \beta_{3} - 1) q^{5} + (\beta_{2} - \beta_1) q^{6} + (\beta_{10} + \beta_{7} - \beta_{3} + \cdots + 1) q^{7}+ \cdots + ( - \beta_{11} - \beta_{10} + \cdots - \beta_1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{3} + 1) q^{2} + \beta_{2} q^{3} - 2 \beta_{3} q^{4} + ( - \beta_{11} + \beta_{6} - \beta_{3} - 1) q^{5} + (\beta_{2} - \beta_1) q^{6} + (\beta_{10} + \beta_{7} - \beta_{3} + \cdots + 1) q^{7}+ \cdots + ( - 10 \beta_{11} + 10 \beta_{10} + \cdots + 74) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{2} + 4 q^{3} - 8 q^{5} + 12 q^{7} - 24 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{2} + 4 q^{3} - 8 q^{5} + 12 q^{7} - 24 q^{8} - 16 q^{10} - 8 q^{12} + 12 q^{13} + 76 q^{15} - 48 q^{16} - 24 q^{17} - 36 q^{18} - 56 q^{19} - 16 q^{20} - 40 q^{22} + 16 q^{23} - 16 q^{24} + 56 q^{25} + 32 q^{26} + 28 q^{27} - 24 q^{28} + 64 q^{30} - 48 q^{32} + 136 q^{33} - 48 q^{34} + 44 q^{35} - 72 q^{36} + 40 q^{37} - 56 q^{38} + 56 q^{39} + 64 q^{42} + 56 q^{43} - 80 q^{44} - 112 q^{45} - 60 q^{47} - 16 q^{48} + 164 q^{50} + 232 q^{51} + 40 q^{52} - 284 q^{53} + 56 q^{54} + 60 q^{55} - 48 q^{56} - 24 q^{57} + 192 q^{58} + 96 q^{59} - 24 q^{60} - 104 q^{61} - 32 q^{62} - 116 q^{63} - 500 q^{65} + 272 q^{66} - 28 q^{67} - 48 q^{68} + 32 q^{70} - 72 q^{72} + 8 q^{73} + 92 q^{75} - 20 q^{77} + 152 q^{78} + 32 q^{80} + 84 q^{81} + 160 q^{82} - 52 q^{83} + 128 q^{84} - 48 q^{85} - 364 q^{87} - 80 q^{88} - 704 q^{89} + 16 q^{90} + 56 q^{91} - 32 q^{92} - 200 q^{93} - 16 q^{95} + 128 q^{97} - 324 q^{98} + 960 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 4 x^{11} + 8 x^{10} + 4 x^{9} + 660 x^{8} - 2696 x^{7} + 5512 x^{6} + 5536 x^{5} + 39140 x^{4} + \cdots + 8836 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 148518488522 \nu^{11} + 441209072689 \nu^{10} - 797127544238 \nu^{9} + \cdots + 48\!\cdots\!76 ) / 10\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 26056342183127 \nu^{11} - 97244999771974 \nu^{10} + 187713911048633 \nu^{9} + \cdots - 67\!\cdots\!16 ) / 50\!\cdots\!50 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 554817133538717 \nu^{11} + \cdots + 37\!\cdots\!76 ) / 87\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 15562341520078 \nu^{11} - 42995406887016 \nu^{10} + 61536163584827 \nu^{9} + \cdots - 73\!\cdots\!04 ) / 18\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 88572978349109 \nu^{11} + 238371416990858 \nu^{10} - 290319141451711 \nu^{9} + \cdots - 21\!\cdots\!28 ) / 92\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 41\!\cdots\!84 \nu^{11} + \cdots + 18\!\cdots\!72 ) / 43\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 68\!\cdots\!58 \nu^{11} + \cdots - 19\!\cdots\!64 ) / 43\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 72\!\cdots\!77 \nu^{11} + \cdots - 17\!\cdots\!16 ) / 43\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 20\!\cdots\!62 \nu^{11} + \cdots + 11\!\cdots\!76 ) / 87\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 17\!\cdots\!97 \nu^{11} + \cdots + 11\!\cdots\!76 ) / 43\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + \beta_{10} + \beta_{9} - \beta_{8} + 12\beta_{3} - \beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + 2\beta_{7} + \beta_{6} - 3\beta_{3} - 20\beta_{2} + 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -19\beta_{11} + 19\beta_{10} - 3\beta_{7} + 18\beta_{6} - 24\beta_{5} + 3\beta_{4} - 16\beta_{2} - 16\beta _1 - 238 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 16\beta_{11} - 13\beta_{9} - 9\beta_{8} + 13\beta_{6} + 9\beta_{5} - 60\beta_{4} + 132\beta_{3} - 414\beta _1 + 132 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 394 \beta_{11} - 394 \beta_{10} - 322 \beta_{9} + 550 \beta_{8} + 100 \beta_{7} + 100 \beta_{4} + \cdots - 190 \beta_1 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 748 \beta_{10} - 34 \beta_{9} - 404 \beta_{8} - 1528 \beta_{7} - 34 \beta_{6} - 404 \beta_{5} + \cdots - 4806 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 8552 \beta_{11} - 8552 \beta_{10} + 2774 \beta_{7} - 6002 \beta_{6} + 12638 \beta_{5} - 2774 \beta_{4} + \cdots + 116892 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 25336 \beta_{11} - 4436 \beta_{9} + 13586 \beta_{8} + 4436 \beta_{6} - 13586 \beta_{5} + \cdots - 152250 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 191522 \beta_{11} + 191522 \beta_{10} + 116880 \beta_{9} - 292916 \beta_{8} - 73782 \beta_{7} + \cdots - 69416 \beta_1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 753136 \beta_{10} - 200142 \beta_{9} + 408502 \beta_{8} + 909432 \beta_{7} - 200142 \beta_{6} + \cdots + 4442332 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(27\) \(41\)
\(\chi(n)\) \(\beta_{3}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
77.1
−3.52851 − 3.52851i
−1.82030 − 1.82030i
0.204138 + 0.204138i
0.927990 + 0.927990i
3.03848 + 3.03848i
3.17820 + 3.17820i
−3.52851 + 3.52851i
−1.82030 + 1.82030i
0.204138 − 0.204138i
0.927990 − 0.927990i
3.03848 − 3.03848i
3.17820 − 3.17820i
1.00000 − 1.00000i −3.52851 + 3.52851i − 2.00000i −3.04175 − 3.96835i 7.05702i 8.24305 − 8.24305i −2.00000 − 2.00000i − 15.9008i −7.01009 − 0.926602i
77.2 1.00000 − 1.00000i −1.82030 + 1.82030i − 2.00000i −4.88125 − 1.08324i 3.64059i −9.03540 + 9.03540i −2.00000 − 2.00000i 2.37304i −5.96449 + 3.79800i
77.3 1.00000 − 1.00000i 0.204138 − 0.204138i − 2.00000i 2.02919 + 4.56972i − 0.408276i 8.16873 − 8.16873i −2.00000 − 2.00000i 8.91666i 6.59892 + 2.54053i
77.4 1.00000 − 1.00000i 0.927990 − 0.927990i − 2.00000i 2.10915 − 4.53338i − 1.85598i 0.198121 − 0.198121i −2.00000 − 2.00000i 7.27767i −2.42423 − 6.64252i
77.5 1.00000 − 1.00000i 3.03848 − 3.03848i − 2.00000i −4.97244 − 0.524263i − 6.07695i 1.49385 − 1.49385i −2.00000 − 2.00000i − 9.46469i −5.49670 + 4.44818i
77.6 1.00000 − 1.00000i 3.17820 − 3.17820i − 2.00000i 4.75709 + 1.53951i − 6.35640i −3.06835 + 3.06835i −2.00000 − 2.00000i − 11.2019i 6.29660 − 3.21759i
103.1 1.00000 + 1.00000i −3.52851 − 3.52851i 2.00000i −3.04175 + 3.96835i − 7.05702i 8.24305 + 8.24305i −2.00000 + 2.00000i 15.9008i −7.01009 + 0.926602i
103.2 1.00000 + 1.00000i −1.82030 − 1.82030i 2.00000i −4.88125 + 1.08324i − 3.64059i −9.03540 − 9.03540i −2.00000 + 2.00000i − 2.37304i −5.96449 − 3.79800i
103.3 1.00000 + 1.00000i 0.204138 + 0.204138i 2.00000i 2.02919 − 4.56972i 0.408276i 8.16873 + 8.16873i −2.00000 + 2.00000i − 8.91666i 6.59892 − 2.54053i
103.4 1.00000 + 1.00000i 0.927990 + 0.927990i 2.00000i 2.10915 + 4.53338i 1.85598i 0.198121 + 0.198121i −2.00000 + 2.00000i − 7.27767i −2.42423 + 6.64252i
103.5 1.00000 + 1.00000i 3.03848 + 3.03848i 2.00000i −4.97244 + 0.524263i 6.07695i 1.49385 + 1.49385i −2.00000 + 2.00000i 9.46469i −5.49670 − 4.44818i
103.6 1.00000 + 1.00000i 3.17820 + 3.17820i 2.00000i 4.75709 − 1.53951i 6.35640i −3.06835 − 3.06835i −2.00000 + 2.00000i 11.2019i 6.29660 + 3.21759i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 77.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.h odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 130.3.h.d yes 12
5.b even 2 1 650.3.h.c 12
5.c odd 4 1 130.3.h.c ✓ 12
5.c odd 4 1 650.3.h.e 12
13.b even 2 1 130.3.h.c ✓ 12
65.d even 2 1 650.3.h.e 12
65.h odd 4 1 inner 130.3.h.d yes 12
65.h odd 4 1 650.3.h.c 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.3.h.c ✓ 12 5.c odd 4 1
130.3.h.c ✓ 12 13.b even 2 1
130.3.h.d yes 12 1.a even 1 1 trivial
130.3.h.d yes 12 65.h odd 4 1 inner
650.3.h.c 12 5.b even 2 1
650.3.h.c 12 65.h odd 4 1
650.3.h.e 12 5.c odd 4 1
650.3.h.e 12 65.d even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(130, [\chi])\):

\( T_{3}^{12} - 4 T_{3}^{11} + 8 T_{3}^{10} + 4 T_{3}^{9} + 660 T_{3}^{8} - 2696 T_{3}^{7} + 5512 T_{3}^{6} + \cdots + 8836 \) Copy content Toggle raw display
\( T_{7}^{12} - 12 T_{7}^{11} + 72 T_{7}^{10} + 168 T_{7}^{9} + 27048 T_{7}^{8} - 326800 T_{7}^{7} + \cdots + 19536400 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 2)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} - 4 T^{11} + \cdots + 8836 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 244140625 \) Copy content Toggle raw display
$7$ \( T^{12} - 12 T^{11} + \cdots + 19536400 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 75048602500 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 23298085122481 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 58192242451456 \) Copy content Toggle raw display
$19$ \( (T^{6} + 28 T^{5} + \cdots - 153470)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 1933351202500 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 592077174708496 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 580839498774544 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{6} - 48 T^{5} + \cdots + 52652722)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 52 T^{5} + \cdots - 520990780)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 54\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 86\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 55\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( (T^{6} + 352 T^{5} + \cdots - 723056964128)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 58\!\cdots\!24 \) Copy content Toggle raw display
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