Properties

Label 130.3.k.b
Level $130$
Weight $3$
Character orbit 130.k
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(21,130)]
 
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([0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("130.21");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 130 = 2 \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 130.k (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: \( x^{12} + 92x^{10} + 3284x^{8} + 58196x^{6} + 540184x^{4} + 2488032x^{2} + 4435236 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{6} + 1) q^{2} + \beta_{3} q^{3} + 2 \beta_{6} q^{4} - \beta_{7} q^{5} + (\beta_{3} + \beta_1) q^{6} + ( - \beta_{8} + \beta_{6} + \beta_{4} - 1) q^{7} + (2 \beta_{6} - 2) q^{8} + ( - \beta_{8} + \beta_{7} - \beta_{2} + 6) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{6} + 1) q^{2} + \beta_{3} q^{3} + 2 \beta_{6} q^{4} - \beta_{7} q^{5} + (\beta_{3} + \beta_1) q^{6} + ( - \beta_{8} + \beta_{6} + \beta_{4} - 1) q^{7} + (2 \beta_{6} - 2) q^{8} + ( - \beta_{8} + \beta_{7} - \beta_{2} + 6) q^{9} + ( - \beta_{8} - \beta_{7}) q^{10} + (\beta_{11} + \beta_{10} - \beta_{8} + \beta_{6} - \beta_{4} + \beta_{3} - \beta_{2} - \beta_1 - 1) q^{11} + 2 \beta_1 q^{12} + (\beta_{10} + \beta_{8} + 2 \beta_{7} - 4 \beta_{6} + \beta_{3} + \beta_{2} - 1) q^{13} + ( - \beta_{8} + \beta_{7} + \beta_{5} + \beta_{4} - 2) q^{14} + (\beta_{9} + 2 \beta_{6} + 2) q^{15} - 4 q^{16} + (2 \beta_{11} - \beta_{10} + \beta_{9} + 2 \beta_{8} + 2 \beta_{7} - 2 \beta_{6} + \beta_{5} - \beta_{4}) q^{17} + ( - \beta_{11} + 2 \beta_{7} + 6 \beta_{6} - \beta_{2} + 6) q^{18} + ( - \beta_{11} - \beta_{9} - 11 \beta_{7} - 6 \beta_{6} - \beta_{5} - 2 \beta_{3} - \beta_{2} + \cdots - 6) q^{19}+ \cdots + (5 \beta_{11} + 9 \beta_{10} - 9 \beta_{8} - 42 \beta_{6} - \beta_{4} - 2 \beta_{3} + \cdots + 42) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{2} - 8 q^{7} - 24 q^{8} + 76 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{2} - 8 q^{7} - 24 q^{8} + 76 q^{9} - 16 q^{11} - 20 q^{13} - 16 q^{14} + 20 q^{15} - 48 q^{16} + 76 q^{18} - 68 q^{19} - 44 q^{21} - 32 q^{22} + 24 q^{26} - 96 q^{27} - 16 q^{28} + 216 q^{29} + 64 q^{31} - 48 q^{32} + 52 q^{33} + 32 q^{34} - 40 q^{35} - 28 q^{37} + 140 q^{39} - 88 q^{42} - 32 q^{44} - 80 q^{45} + 16 q^{46} - 104 q^{47} - 60 q^{50} + 88 q^{52} - 424 q^{53} - 96 q^{54} - 80 q^{55} - 100 q^{57} + 216 q^{58} - 108 q^{59} - 40 q^{60} + 88 q^{61} + 312 q^{63} + 100 q^{65} + 104 q^{66} + 240 q^{67} + 64 q^{68} - 40 q^{70} + 12 q^{71} - 152 q^{72} - 164 q^{73} - 56 q^{74} + 136 q^{76} + 120 q^{78} + 392 q^{79} - 204 q^{81} + 536 q^{83} - 88 q^{84} + 60 q^{85} + 160 q^{86} - 672 q^{87} - 364 q^{89} - 244 q^{91} + 32 q^{92} - 236 q^{93} - 208 q^{94} + 192 q^{97} + 484 q^{98} + 484 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 92x^{10} + 3284x^{8} + 58196x^{6} + 540184x^{4} + 2488032x^{2} + 4435236 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{10} + 83\nu^{8} + 2582\nu^{6} + 38018\nu^{4} + 274522\nu^{2} + 808344 ) / 8280 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -4\nu^{10} - 326\nu^{8} - 9713\nu^{6} - 131246\nu^{4} - 803284\nu^{2} - 1779570 ) / 7176 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 97 \nu^{11} - 1404 \nu^{10} + 10706 \nu^{9} - 132327 \nu^{8} + 422714 \nu^{7} - 4699188 \nu^{6} + 7215926 \nu^{5} - 77322492 \nu^{4} + 49700224 \nu^{3} + \cdots - 1512668196 ) / 5812560 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 97 \nu^{11} - 1404 \nu^{10} - 10706 \nu^{9} - 132327 \nu^{8} - 422714 \nu^{7} - 4699188 \nu^{6} - 7215926 \nu^{5} - 77322492 \nu^{4} - 49700224 \nu^{3} + \cdots - 1512668196 ) / 5812560 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -65\nu^{11} - 5332\nu^{9} - 160648\nu^{7} - 2209234\nu^{5} - 13850108\nu^{3} - 31590072\nu ) / 1162512 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 418 \nu^{11} + 351 \nu^{10} + 33434 \nu^{9} + 29133 \nu^{8} + 972896 \nu^{7} + 906282 \nu^{6} + 12739364 \nu^{5} + 13344318 \nu^{4} + 74294836 \nu^{3} + \cdots + 240134544 ) / 5812560 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 418 \nu^{11} - 351 \nu^{10} + 33434 \nu^{9} - 29133 \nu^{8} + 972896 \nu^{7} - 906282 \nu^{6} + 12739364 \nu^{5} - 13344318 \nu^{4} + 74294836 \nu^{3} + \cdots - 240134544 ) / 5812560 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 299 \nu^{11} - 5022 \nu^{10} + 24187 \nu^{9} - 399816 \nu^{8} + 700198 \nu^{7} - 11586564 \nu^{6} + 8748022 \nu^{5} - 151502076 \nu^{4} + 45050138 \nu^{3} + \cdots - 1865553768 ) / 5812560 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 299 \nu^{11} - 5022 \nu^{10} - 24187 \nu^{9} - 399816 \nu^{8} - 700198 \nu^{7} - 11586564 \nu^{6} - 8748022 \nu^{5} - 151502076 \nu^{4} + \cdots - 1865553768 ) / 5812560 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -799\nu^{11} - 68972\nu^{9} - 2235278\nu^{7} - 33904562\nu^{5} - 239508388\nu^{3} - 625263876\nu ) / 5812560 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - \beta_{7} + \beta_{2} - 15 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{11} - 2\beta_{10} + 2\beta_{9} + 8\beta_{6} + \beta_{5} - \beta_{4} - 18\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{10} + 3\beta_{9} - 46\beta_{8} + 46\beta_{7} + 2\beta_{5} + 2\beta_{4} - 8\beta_{3} - 32\beta_{2} + 306 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 90 \beta_{11} + 80 \beta_{10} - 80 \beta_{9} - 49 \beta_{8} - 49 \beta_{7} - 476 \beta_{6} - 31 \beta_{5} + 31 \beta_{4} + 394 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 170 \beta_{10} - 170 \beta_{9} + 1532 \beta_{8} - 1532 \beta_{7} - 108 \beta_{5} - 108 \beta_{4} + 488 \beta_{3} + 920 \beta_{2} - 7550 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 3316 \beta_{11} - 2560 \beta_{10} + 2560 \beta_{9} + 2822 \beta_{8} + 2822 \beta_{7} + 19600 \beta_{6} + 874 \beta_{5} - 874 \beta_{4} - 9846 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 7012 \beta_{10} + 7012 \beta_{9} - 46498 \beta_{8} + 46498 \beta_{7} + 4128 \beta_{5} + 4128 \beta_{4} - 21056 \beta_{3} - 26842 \beta_{2} + 206458 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 113532 \beta_{11} + 77468 \beta_{10} - 77468 \beta_{9} - 117980 \beta_{8} - 117980 \beta_{7} - 709856 \beta_{6} - 25598 \beta_{5} + 25598 \beta_{4} + 267192 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 257110 \beta_{10} - 257110 \beta_{9} + 1378016 \beta_{8} - 1378016 \beta_{7} - 139804 \beta_{5} - 139804 \beta_{4} + 791776 \beta_{3} + 802780 \beta_{2} - 5965936 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 3750380 \beta_{11} - 2320600 \beta_{10} + 2320600 \beta_{9} + 4368818 \beta_{8} + 4368818 \beta_{7} + 24244360 \beta_{6} + 780282 \beta_{5} - 780282 \beta_{4} + \cdots - 7625444 \beta_1 \) Copy content Toggle raw display

Character values

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

\(n\) \(27\) \(41\)
\(\chi(n)\) \(1\) \(\beta_{6}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1
5.60976i
3.38353i
2.25366i
2.78364i
3.76217i
4.70114i
5.60976i
3.38353i
2.25366i
2.78364i
3.76217i
4.70114i
1.00000 + 1.00000i −5.60976 2.00000i −1.58114 1.58114i −5.60976 5.60976i 7.31442 7.31442i −2.00000 + 2.00000i 22.4694 3.16228i
21.2 1.00000 + 1.00000i −3.38353 2.00000i 1.58114 + 1.58114i −3.38353 3.38353i −2.57599 + 2.57599i −2.00000 + 2.00000i 2.44830 3.16228i
21.3 1.00000 + 1.00000i −2.25366 2.00000i −1.58114 1.58114i −2.25366 2.25366i −8.02207 + 8.02207i −2.00000 + 2.00000i −3.92101 3.16228i
21.4 1.00000 + 1.00000i 2.78364 2.00000i 1.58114 + 1.58114i 2.78364 + 2.78364i 6.97332 6.97332i −2.00000 + 2.00000i −1.25135 3.16228i
21.5 1.00000 + 1.00000i 3.76217 2.00000i 1.58114 + 1.58114i 3.76217 + 3.76217i −9.55961 + 9.55961i −2.00000 + 2.00000i 5.15394 3.16228i
21.6 1.00000 + 1.00000i 4.70114 2.00000i −1.58114 1.58114i 4.70114 + 4.70114i 1.86993 1.86993i −2.00000 + 2.00000i 13.1007 3.16228i
31.1 1.00000 1.00000i −5.60976 2.00000i −1.58114 + 1.58114i −5.60976 + 5.60976i 7.31442 + 7.31442i −2.00000 2.00000i 22.4694 3.16228i
31.2 1.00000 1.00000i −3.38353 2.00000i 1.58114 1.58114i −3.38353 + 3.38353i −2.57599 2.57599i −2.00000 2.00000i 2.44830 3.16228i
31.3 1.00000 1.00000i −2.25366 2.00000i −1.58114 + 1.58114i −2.25366 + 2.25366i −8.02207 8.02207i −2.00000 2.00000i −3.92101 3.16228i
31.4 1.00000 1.00000i 2.78364 2.00000i 1.58114 1.58114i 2.78364 2.78364i 6.97332 + 6.97332i −2.00000 2.00000i −1.25135 3.16228i
31.5 1.00000 1.00000i 3.76217 2.00000i 1.58114 1.58114i 3.76217 3.76217i −9.55961 9.55961i −2.00000 2.00000i 5.15394 3.16228i
31.6 1.00000 1.00000i 4.70114 2.00000i −1.58114 + 1.58114i 4.70114 4.70114i 1.86993 + 1.86993i −2.00000 2.00000i 13.1007 3.16228i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 21.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
13.d 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.k.b 12
3.b odd 2 1 1170.3.r.a 12
5.b even 2 1 650.3.k.j 12
5.c odd 4 1 650.3.f.k 12
5.c odd 4 1 650.3.f.n 12
13.d odd 4 1 inner 130.3.k.b 12
39.f even 4 1 1170.3.r.a 12
65.f even 4 1 650.3.f.k 12
65.g odd 4 1 650.3.k.j 12
65.k even 4 1 650.3.f.n 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.3.k.b 12 1.a even 1 1 trivial
130.3.k.b 12 13.d odd 4 1 inner
650.3.f.k 12 5.c odd 4 1
650.3.f.k 12 65.f even 4 1
650.3.f.n 12 5.c odd 4 1
650.3.f.n 12 65.k even 4 1
650.3.k.j 12 5.b even 2 1
650.3.k.j 12 65.g odd 4 1
1170.3.r.a 12 3.b odd 2 1
1170.3.r.a 12 39.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 46T_{3}^{4} + 16T_{3}^{3} + 584T_{3}^{2} - 168T_{3} - 2106 \) acting on \(S_{3}^{\mathrm{new}}(130, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 2)^{6} \) Copy content Toggle raw display
$3$ \( (T^{6} - 46 T^{4} + 16 T^{3} + 584 T^{2} + \cdots - 2106)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 25)^{3} \) Copy content Toggle raw display
$7$ \( T^{12} + 8 T^{11} + \cdots + 22720135824 \) Copy content Toggle raw display
$11$ \( T^{12} + 16 T^{11} + \cdots + 1025119800324 \) Copy content Toggle raw display
$13$ \( T^{12} + 20 T^{11} + \cdots + 23298085122481 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 258806497951296 \) Copy content Toggle raw display
$19$ \( T^{12} + 68 T^{11} + \cdots + 14\!\cdots\!44 \) Copy content Toggle raw display
$23$ \( T^{12} + 5432 T^{10} + \cdots + 88\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( (T^{6} - 108 T^{5} + 3656 T^{4} + \cdots + 23146884)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} - 64 T^{11} + \cdots + 9750768154884 \) Copy content Toggle raw display
$37$ \( T^{12} + 28 T^{11} + \cdots + 33\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( T^{12} + 96848 T^{9} + \cdots + 12\!\cdots\!04 \) Copy content Toggle raw display
$43$ \( T^{12} + 13392 T^{10} + \cdots + 16\!\cdots\!96 \) Copy content Toggle raw display
$47$ \( T^{12} + 104 T^{11} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{6} + 212 T^{5} + 14764 T^{4} + \cdots - 113961024)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + 108 T^{11} + \cdots + 89\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( (T^{6} - 44 T^{5} - 8764 T^{4} + \cdots + 137156004)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} - 240 T^{11} + \cdots + 94\!\cdots\!24 \) Copy content Toggle raw display
$71$ \( T^{12} - 12 T^{11} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( T^{12} + 164 T^{11} + \cdots + 80988840384400 \) Copy content Toggle raw display
$79$ \( (T^{6} - 196 T^{5} + \cdots - 54334041840)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} - 536 T^{11} + \cdots + 23\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{12} + 364 T^{11} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{12} - 192 T^{11} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
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