Properties

Label 130.3.f.a
Level $130$
Weight $3$
Character orbit 130.f
Analytic conductor $3.542$
Analytic rank $0$
Dimension $14$
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(99,130)]
 
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([2, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("130.99");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 130 = 2 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 130.f (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: \(14\)
Relative dimension: \(7\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} + 84x^{12} + 2668x^{10} + 40556x^{8} + 303080x^{6} + 1000960x^{4} + 1045476x^{2} + 193600 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{2} \)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} - 1) q^{2} + \beta_1 q^{3} + 2 \beta_{4} q^{4} + \beta_{8} q^{5} + (\beta_{2} - \beta_1) q^{6} - \beta_{12} q^{7} + ( - 2 \beta_{4} + 2) q^{8} + (\beta_{3} - \beta_{2} - 3) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} - 1) q^{2} + \beta_1 q^{3} + 2 \beta_{4} q^{4} + \beta_{8} q^{5} + (\beta_{2} - \beta_1) q^{6} - \beta_{12} q^{7} + ( - 2 \beta_{4} + 2) q^{8} + (\beta_{3} - \beta_{2} - 3) q^{9} + ( - \beta_{11} - \beta_{8}) q^{10} + ( - \beta_{6} + \beta_{4} + 1) q^{11} - 2 \beta_{2} q^{12} + (\beta_{12} - \beta_{11} + \beta_{9} - \beta_{8} - \beta_{7} - \beta_{6} - \beta_{2} + \beta_1) q^{13} + (\beta_{13} + \beta_{12}) q^{14} + (\beta_{13} - \beta_{11} - 2 \beta_{10} + \beta_{9} + \beta_{7} - \beta_{5} + 4 \beta_{4} + \beta_{3}) q^{15} - 4 q^{16} + (2 \beta_{11} + 2 \beta_{10} + \beta_{8} - \beta_{7} - \beta_{6} + \beta_{5} - 2 \beta_{2} - 6) q^{17} + (\beta_{9} + 3 \beta_{4} - \beta_{3} + \beta_{2} + \beta_1 + 3) q^{18} + ( - \beta_{11} - \beta_{7} + \beta_{5} - 4 \beta_{4} + \beta_{2} - \beta_1 + 4) q^{19} + 2 \beta_{11} q^{20} + (2 \beta_{13} - \beta_{11} + \beta_{9} + \beta_{7} - 2 \beta_{6} + \beta_{4} - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{21} + (\beta_{6} + \beta_{5} - 2 \beta_{4}) q^{22} + ( - \beta_{13} - \beta_{12} - \beta_{11} - \beta_{10} + 2 \beta_{6} - 2 \beta_{5} + 2 \beta_{3} + \cdots - 2) q^{23}+ \cdots + (5 \beta_{10} - 2 \beta_{9} - 5 \beta_{8} - 3 \beta_{6} + 20 \beta_{4} + 2 \beta_{3} + \cdots + 20) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q - 14 q^{2} - 4 q^{5} + 28 q^{8} - 42 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 14 q - 14 q^{2} - 4 q^{5} + 28 q^{8} - 42 q^{9} + 2 q^{10} + 8 q^{11} - 8 q^{13} + 4 q^{15} - 56 q^{16} - 96 q^{17} + 42 q^{18} + 44 q^{19} + 4 q^{20} + 4 q^{21} - 8 q^{23} + 46 q^{25} + 10 q^{26} + 72 q^{29} + 40 q^{30} - 64 q^{31} + 56 q^{32} - 56 q^{33} + 96 q^{34} + 124 q^{35} - 98 q^{37} - 88 q^{38} - 144 q^{39} - 12 q^{40} + 138 q^{41} + 120 q^{43} - 16 q^{44} - 80 q^{45} + 8 q^{46} + 40 q^{47} - 18 q^{50} - 4 q^{52} - 120 q^{54} - 76 q^{55} + 264 q^{57} - 72 q^{58} - 20 q^{59} - 88 q^{60} - 180 q^{61} + 128 q^{62} + 160 q^{63} - 2 q^{65} + 112 q^{66} - 8 q^{67} - 104 q^{70} - 100 q^{71} - 84 q^{72} + 54 q^{73} + 196 q^{74} - 520 q^{75} + 88 q^{76} + 336 q^{77} + 24 q^{78} + 344 q^{79} + 16 q^{80} + 38 q^{81} - 276 q^{82} - 296 q^{83} - 8 q^{84} + 284 q^{85} - 120 q^{86} + 32 q^{88} - 398 q^{89} - 6 q^{90} + 48 q^{91} - 120 q^{93} - 80 q^{94} + 476 q^{95} + 162 q^{97} - 130 q^{98} + 292 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} + 84x^{12} + 2668x^{10} + 40556x^{8} + 303080x^{6} + 1000960x^{4} + 1045476x^{2} + 193600 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 4669 \nu^{12} - 346476 \nu^{10} - 9112322 \nu^{8} - 103760778 \nu^{6} - 488446468 \nu^{4} - 690311148 \nu^{2} - 48986080 ) / 180068312 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 4669 \nu^{12} - 346476 \nu^{10} - 9112322 \nu^{8} - 103760778 \nu^{6} - 488446468 \nu^{4} - 510242836 \nu^{2} + 2111833664 ) / 180068312 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 27833 \nu^{13} - 1824382 \nu^{11} - 36146084 \nu^{9} - 126439728 \nu^{7} + 2978059940 \nu^{5} + 25869391800 \nu^{3} + \cdots + 46835492772 \nu ) / 19807514320 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 11986499 \nu^{13} - 169174104 \nu^{12} + 682799468 \nu^{11} - 15606716422 \nu^{10} + 85649480770 \nu^{9} + \cdots + 11599785775600 ) / 12498541535920 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11986499 \nu^{13} + 169174104 \nu^{12} + 682799468 \nu^{11} + 15606716422 \nu^{10} + 85649480770 \nu^{9} + \cdots - 11599785775600 ) / 12498541535920 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 72278873 \nu^{13} - 495832062 \nu^{12} - 6855618434 \nu^{11} - 38633004036 \nu^{10} - 244843021960 \nu^{9} + \cdots - 101689195713600 ) / 12498541535920 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 72278873 \nu^{13} + 495832062 \nu^{12} - 6855618434 \nu^{11} + 38633004036 \nu^{10} - 244843021960 \nu^{9} + \cdots + 101689195713600 ) / 12498541535920 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 89797 \nu^{13} - 8109888 \nu^{11} - 284301206 \nu^{9} - 4948204422 \nu^{7} - 44732915380 \nu^{5} - 193183463940 \nu^{3} + \cdots - 293610948192 \nu ) / 9903757160 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 168852424 \nu^{13} - 25605129 \nu^{12} - 13976177177 \nu^{11} - 2271622562 \nu^{10} - 433967819980 \nu^{9} + \cdots + 16994772218600 ) / 6249270767960 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 168852424 \nu^{13} - 25605129 \nu^{12} + 13976177177 \nu^{11} - 2271622562 \nu^{10} + 433967819980 \nu^{9} - 69913931065 \nu^{8} + \cdots + 16994772218600 ) / 6249270767960 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 200996947 \nu^{13} - 143545314 \nu^{12} - 16529243204 \nu^{11} - 9278768697 \nu^{10} - 509099620484 \nu^{9} + \cdots + 41334214997680 ) / 6249270767960 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 200996947 \nu^{13} - 143545314 \nu^{12} + 16529243204 \nu^{11} - 9278768697 \nu^{10} + 509099620484 \nu^{9} - 185362278750 \nu^{8} + \cdots + 41334214997680 ) / 6249270767960 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - \beta_{2} - 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{13} - \beta_{12} - 2\beta_{11} + 2\beta_{10} - 2\beta_{9} - \beta_{8} - \beta_{7} - 10\beta_{4} - 20\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{13} + \beta_{12} - 2 \beta_{11} - 2 \beta_{10} + 6 \beta_{8} - 6 \beta_{7} - 5 \beta_{6} + 5 \beta_{5} - 30 \beta_{3} + 42 \beta_{2} + 254 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 33 \beta_{13} + 33 \beta_{12} + 74 \beta_{11} - 74 \beta_{10} + 96 \beta_{9} + 24 \beta_{8} + 24 \beta_{7} - 16 \beta_{6} - 16 \beta_{5} + 540 \beta_{4} + 478 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 4 \beta_{13} - 4 \beta_{12} + 115 \beta_{11} + 115 \beta_{10} - 273 \beta_{8} + 273 \beta_{7} + 196 \beta_{6} - 196 \beta_{5} + 868 \beta_{3} - 1528 \beta_{2} - 6362 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 914 \beta_{13} - 914 \beta_{12} - 2324 \beta_{11} + 2324 \beta_{10} - 3556 \beta_{9} - 596 \beta_{8} - 596 \beta_{7} + 772 \beta_{6} + 772 \beta_{5} - 20736 \beta_{4} - 12798 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 772 \beta_{13} - 772 \beta_{12} - 4234 \beta_{11} - 4234 \beta_{10} + 9890 \beta_{8} - 9890 \beta_{7} - 6292 \beta_{6} + 6292 \beta_{5} - 25566 \beta_{3} + 51966 \beta_{2} + 174604 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 24658 \beta_{13} + 24658 \beta_{12} + 70776 \beta_{11} - 70776 \beta_{10} + 119908 \beta_{9} + 17234 \beta_{8} + 17234 \beta_{7} - 28252 \beta_{6} - 28252 \beta_{5} + 716044 \beta_{4} + \cdots + 366236 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 45302 \beta_{13} + 45302 \beta_{12} + 135388 \beta_{11} + 135388 \beta_{10} - 331420 \beta_{8} + 331420 \beta_{7} + 193830 \beta_{6} - 193830 \beta_{5} + 767984 \beta_{3} - 1702808 \beta_{2} + \cdots - 5053060 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 675178 \beta_{13} - 675178 \beta_{12} - 2151304 \beta_{11} + 2151304 \beta_{10} - 3882672 \beta_{9} - 544712 \beta_{8} - 544712 \beta_{7} + 945072 \beta_{6} + 945072 \beta_{5} + \cdots - 10878172 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 1870796 \beta_{13} - 1870796 \beta_{12} - 4129946 \beta_{11} - 4129946 \beta_{10} + 10731254 \beta_{8} - 10731254 \beta_{7} - 5936516 \beta_{6} + 5936516 \beta_{5} + \cdots + 150784228 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( - 18986992 \beta_{13} + 18986992 \beta_{12} + 65713672 \beta_{11} - 65713672 \beta_{10} + 123344480 \beta_{9} + 17669072 \beta_{8} + 17669072 \beta_{7} + \cdots + 329791252 \beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(27\) \(41\)
\(\chi(n)\) \(-1\) \(-\beta_{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
5.57826i
3.39370i
1.18066i
0.483741i
2.42206i
3.84840i
4.36592i
4.36592i
3.84840i
2.42206i
0.483741i
1.18066i
3.39370i
5.57826i
−1.00000 + 1.00000i 5.57826i 2.00000i 4.38321 2.40571i 5.57826 + 5.57826i 1.29593 + 1.29593i 2.00000 + 2.00000i −22.1170 −1.97750 + 6.78893i
99.2 −1.00000 + 1.00000i 3.39370i 2.00000i −3.57158 + 3.49911i 3.39370 + 3.39370i −4.87525 4.87525i 2.00000 + 2.00000i −2.51723 0.0724711 7.07070i
99.3 −1.00000 + 1.00000i 1.18066i 2.00000i −4.99448 0.234938i 1.18066 + 1.18066i 5.55697 + 5.55697i 2.00000 + 2.00000i 7.60603 5.22942 4.75954i
99.4 −1.00000 + 1.00000i 0.483741i 2.00000i 4.45445 + 2.27109i 0.483741 + 0.483741i −4.16889 4.16889i 2.00000 + 2.00000i 8.76599 −6.72554 + 2.18336i
99.5 −1.00000 + 1.00000i 2.42206i 2.00000i 1.68330 4.70813i −2.42206 2.42206i 7.75310 + 7.75310i 2.00000 + 2.00000i 3.13365 3.02484 + 6.39143i
99.6 −1.00000 + 1.00000i 3.84840i 2.00000i 0.429852 + 4.98149i −3.84840 3.84840i 2.44952 + 2.44952i 2.00000 + 2.00000i −5.81016 −5.41134 4.55164i
99.7 −1.00000 + 1.00000i 4.36592i 2.00000i −4.38475 2.40291i −4.36592 4.36592i −8.01138 8.01138i 2.00000 + 2.00000i −10.0613 6.78766 1.98184i
109.1 −1.00000 1.00000i 4.36592i 2.00000i −4.38475 + 2.40291i −4.36592 + 4.36592i −8.01138 + 8.01138i 2.00000 2.00000i −10.0613 6.78766 + 1.98184i
109.2 −1.00000 1.00000i 3.84840i 2.00000i 0.429852 4.98149i −3.84840 + 3.84840i 2.44952 2.44952i 2.00000 2.00000i −5.81016 −5.41134 + 4.55164i
109.3 −1.00000 1.00000i 2.42206i 2.00000i 1.68330 + 4.70813i −2.42206 + 2.42206i 7.75310 7.75310i 2.00000 2.00000i 3.13365 3.02484 6.39143i
109.4 −1.00000 1.00000i 0.483741i 2.00000i 4.45445 2.27109i 0.483741 0.483741i −4.16889 + 4.16889i 2.00000 2.00000i 8.76599 −6.72554 2.18336i
109.5 −1.00000 1.00000i 1.18066i 2.00000i −4.99448 + 0.234938i 1.18066 1.18066i 5.55697 5.55697i 2.00000 2.00000i 7.60603 5.22942 + 4.75954i
109.6 −1.00000 1.00000i 3.39370i 2.00000i −3.57158 3.49911i 3.39370 3.39370i −4.87525 + 4.87525i 2.00000 2.00000i −2.51723 0.0724711 + 7.07070i
109.7 −1.00000 1.00000i 5.57826i 2.00000i 4.38321 + 2.40571i 5.57826 5.57826i 1.29593 1.29593i 2.00000 2.00000i −22.1170 −1.97750 6.78893i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
65.g 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.f.a 14
5.b even 2 1 130.3.f.b yes 14
5.c odd 4 1 650.3.k.l 14
5.c odd 4 1 650.3.k.m 14
13.d odd 4 1 130.3.f.b yes 14
65.f even 4 1 650.3.k.m 14
65.g odd 4 1 inner 130.3.f.a 14
65.k even 4 1 650.3.k.l 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.3.f.a 14 1.a even 1 1 trivial
130.3.f.a 14 65.g odd 4 1 inner
130.3.f.b yes 14 5.b even 2 1
130.3.f.b yes 14 13.d odd 4 1
650.3.k.l 14 5.c odd 4 1
650.3.k.l 14 65.k even 4 1
650.3.k.m 14 5.c odd 4 1
650.3.k.m 14 65.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{14} - 64 T_{7}^{11} + 19184 T_{7}^{10} - 5776 T_{7}^{9} + 2048 T_{7}^{8} - 410992 T_{7}^{7} + 58999264 T_{7}^{6} + 1511872 T_{7}^{5} + 3695744 T_{7}^{4} - 2804394944 T_{7}^{3} + \cdots + 63476270208 \) acting on \(S_{3}^{\mathrm{new}}(130, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 2)^{7} \) Copy content Toggle raw display
$3$ \( T^{14} + 84 T^{12} + 2668 T^{10} + \cdots + 193600 \) Copy content Toggle raw display
$5$ \( T^{14} + 4 T^{13} + \cdots + 6103515625 \) Copy content Toggle raw display
$7$ \( T^{14} - 64 T^{11} + \cdots + 63476270208 \) Copy content Toggle raw display
$11$ \( T^{14} - 8 T^{13} + \cdots + 35146428192 \) Copy content Toggle raw display
$13$ \( T^{14} + 8 T^{13} + \cdots + 39\!\cdots\!89 \) Copy content Toggle raw display
$17$ \( (T^{7} + 48 T^{6} - 30 T^{5} + \cdots - 210654400)^{2} \) Copy content Toggle raw display
$19$ \( T^{14} - 44 T^{13} + \cdots + 35\!\cdots\!52 \) Copy content Toggle raw display
$23$ \( (T^{7} + 4 T^{6} - 2380 T^{5} + \cdots + 389696200)^{2} \) Copy content Toggle raw display
$29$ \( (T^{7} - 36 T^{6} - 3016 T^{5} + \cdots + 6769663296)^{2} \) Copy content Toggle raw display
$31$ \( T^{14} + 64 T^{13} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{14} + 98 T^{13} + \cdots + 16\!\cdots\!88 \) Copy content Toggle raw display
$41$ \( T^{14} - 138 T^{13} + \cdots + 50\!\cdots\!28 \) Copy content Toggle raw display
$43$ \( (T^{7} - 60 T^{6} - 3992 T^{5} + \cdots - 3768790720)^{2} \) Copy content Toggle raw display
$47$ \( T^{14} - 40 T^{13} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{14} + 15648 T^{12} + \cdots + 65\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{14} + 20 T^{13} + \cdots + 18\!\cdots\!08 \) Copy content Toggle raw display
$61$ \( (T^{7} + 90 T^{6} + \cdots - 398456455688)^{2} \) Copy content Toggle raw display
$67$ \( T^{14} + 8 T^{13} + \cdots + 17\!\cdots\!72 \) Copy content Toggle raw display
$71$ \( T^{14} + 100 T^{13} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{14} - 54 T^{13} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{7} - 172 T^{6} + \cdots + 2603398327360)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + 296 T^{13} + \cdots + 16\!\cdots\!92 \) Copy content Toggle raw display
$89$ \( T^{14} + 398 T^{13} + \cdots + 70\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{14} - 162 T^{13} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
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