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Properties

Label 130.3.k.a
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: \( x^{12} + 76x^{10} + 1956x^{8} + 19924x^{6} + 77560x^{4} + 85248x^{2} + 2916 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{4} - 1) q^{2} - \beta_{2} q^{3} + 2 \beta_{4} q^{4} - \beta_{6} q^{5} + (\beta_{2} - \beta_1) q^{6} + (\beta_{8} + \beta_{5}) q^{7} + ( - 2 \beta_{4} + 2) q^{8} + ( - \beta_{11} - \beta_{10} + \beta_{9} + \cdots + 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} - 1) q^{2} - \beta_{2} q^{3} + 2 \beta_{4} q^{4} - \beta_{6} q^{5} + (\beta_{2} - \beta_1) q^{6} + (\beta_{8} + \beta_{5}) q^{7} + ( - 2 \beta_{4} + 2) q^{8} + ( - \beta_{11} - \beta_{10} + \beta_{9} + \cdots + 4) q^{9}+ \cdots + (5 \beta_{10} + 4 \beta_{8} - 3 \beta_{7} + \cdots - 5) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} + 24 q^{8} + 44 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{2} + 24 q^{8} + 44 q^{9} + 8 q^{11} - 4 q^{13} + 20 q^{15} - 48 q^{16} - 44 q^{18} + 36 q^{19} + 52 q^{21} - 16 q^{22} + 8 q^{26} + 144 q^{27} - 8 q^{29} - 136 q^{31} + 48 q^{32} - 60 q^{33} - 40 q^{35} - 44 q^{37} - 172 q^{39} + 32 q^{41} - 104 q^{42} + 16 q^{44} - 96 q^{46} - 16 q^{47} + 60 q^{50} - 8 q^{52} + 24 q^{53} - 144 q^{54} - 212 q^{57} + 8 q^{58} + 124 q^{59} - 40 q^{60} + 24 q^{61} - 80 q^{63} - 60 q^{65} + 120 q^{66} + 136 q^{67} + 40 q^{70} + 84 q^{71} + 88 q^{72} + 12 q^{73} + 88 q^{74} - 72 q^{76} + 312 q^{78} - 168 q^{79} + 596 q^{81} + 160 q^{83} + 104 q^{84} + 60 q^{85} + 224 q^{86} - 64 q^{87} - 44 q^{89} - 404 q^{91} + 192 q^{92} + 4 q^{93} + 32 q^{94} - 192 q^{97} + 60 q^{98} - 52 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 76x^{10} + 1956x^{8} + 19924x^{6} + 77560x^{4} + 85248x^{2} + 2916 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{10} + 62\nu^{8} + 1088\nu^{6} + 3626\nu^{4} - 13712\nu^{2} + 4320 ) / 25584 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 381\nu^{10} + 27886\nu^{8} + 666104\nu^{6} + 5594338\nu^{4} + 10074960\nu^{2} - 14932512 ) / 1867632 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -40\nu^{11} - 3013\nu^{9} - 76566\nu^{7} - 767584\nu^{5} - 3004498\nu^{3} - 3780144\nu ) / 690768 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 7226 \nu^{11} - 4797 \nu^{10} - 545018 \nu^{9} - 350181 \nu^{8} - 13818696 \nu^{7} + \cdots - 86605038 ) / 50426064 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 7226 \nu^{11} - 4797 \nu^{10} + 545018 \nu^{9} - 350181 \nu^{8} + 13818696 \nu^{7} + \cdots - 86605038 ) / 50426064 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -545\nu^{11} - 42851\nu^{9} - 1166468\nu^{7} - 13107342\nu^{5} - 59511202\nu^{3} - 83385324\nu ) / 1867632 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 19079 \nu^{11} - 1422 \nu^{10} - 1462190 \nu^{9} - 116946 \nu^{8} - 38169996 \nu^{7} + \cdots - 316556910 ) / 50426064 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 19079 \nu^{11} + 1422 \nu^{10} - 1462190 \nu^{9} + 116946 \nu^{8} - 38169996 \nu^{7} + \cdots + 316556910 ) / 50426064 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 29716 \nu^{11} + 5580 \nu^{10} + 2252269 \nu^{9} + 432306 \nu^{8} + 57681021 \nu^{7} + \cdots + 438480054 ) / 50426064 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 29716 \nu^{11} + 5580 \nu^{10} - 2252269 \nu^{9} + 432306 \nu^{8} - 57681021 \nu^{7} + \cdots + 438480054 ) / 50426064 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{11} + \beta_{10} - \beta_{9} + \beta_{8} - \beta_{3} + \beta_{2} - 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 2\beta_{11} - 2\beta_{10} + \beta_{9} + \beta_{8} - 2\beta_{7} + 6\beta_{6} - 6\beta_{5} - 14\beta_{4} - 20\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 35 \beta_{11} - 35 \beta_{10} + 35 \beta_{9} - 35 \beta_{8} - 12 \beta_{6} - 12 \beta_{5} + \cdots + 326 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 94 \beta_{11} + 94 \beta_{10} - 11 \beta_{9} - 11 \beta_{8} + 58 \beta_{7} - 243 \beta_{6} + \cdots + 528 \beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 1122 \beta_{11} + 1122 \beta_{10} - 1050 \beta_{9} + 1050 \beta_{8} + 600 \beta_{6} + 600 \beta_{5} + \cdots - 9342 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 3564 \beta_{11} - 3564 \beta_{10} - 342 \beta_{9} - 342 \beta_{8} - 1452 \beta_{7} + 7986 \beta_{6} + \cdots - 15418 \beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 35206 \beta_{11} - 35206 \beta_{10} + 30958 \beta_{9} - 30958 \beta_{8} - 23544 \beta_{6} + \cdots + 279622 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 126428 \beta_{11} + 126428 \beta_{10} + 29306 \beta_{9} + 29306 \beta_{8} + 33692 \beta_{7} + \cdots + 467684 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 1102658 \beta_{11} + 1102658 \beta_{10} - 917618 \beta_{9} + 917618 \beta_{8} + 850440 \beta_{6} + \cdots - 8537120 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 4354756 \beta_{11} - 4354756 \beta_{10} - 1416862 \beta_{9} - 1416862 \beta_{8} - 721276 \beta_{7} + \cdots - 14440296 \beta_1 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
21.1
− 5.08458i
− 2.34685i
− 1.30814i
− 0.187967i
3.23045i
5.69710i
5.08458i
2.34685i
1.30814i
0.187967i
− 3.23045i
− 5.69710i
−1.00000 − 1.00000i −5.08458 2.00000i −1.58114 − 1.58114i 5.08458 + 5.08458i −4.62330 + 4.62330i 2.00000 − 2.00000i 16.8530 3.16228i
21.2 −1.00000 − 1.00000i −2.34685 2.00000i 1.58114 + 1.58114i 2.34685 + 2.34685i 4.49093 − 4.49093i 2.00000 − 2.00000i −3.49227 − 3.16228i
21.3 −1.00000 − 1.00000i −1.30814 2.00000i −1.58114 − 1.58114i 1.30814 + 1.30814i 2.97945 − 2.97945i 2.00000 − 2.00000i −7.28877 3.16228i
21.4 −1.00000 − 1.00000i −0.187967 2.00000i 1.58114 + 1.58114i 0.187967 + 0.187967i −7.64727 + 7.64727i 2.00000 − 2.00000i −8.96467 − 3.16228i
21.5 −1.00000 − 1.00000i 3.23045 2.00000i −1.58114 − 1.58114i −3.23045 − 3.23045i 4.80613 − 4.80613i 2.00000 − 2.00000i 1.43578 3.16228i
21.6 −1.00000 − 1.00000i 5.69710 2.00000i 1.58114 + 1.58114i −5.69710 − 5.69710i −0.00593756 + 0.00593756i 2.00000 − 2.00000i 23.4569 − 3.16228i
31.1 −1.00000 + 1.00000i −5.08458 − 2.00000i −1.58114 + 1.58114i 5.08458 − 5.08458i −4.62330 − 4.62330i 2.00000 + 2.00000i 16.8530 − 3.16228i
31.2 −1.00000 + 1.00000i −2.34685 − 2.00000i 1.58114 − 1.58114i 2.34685 − 2.34685i 4.49093 + 4.49093i 2.00000 + 2.00000i −3.49227 3.16228i
31.3 −1.00000 + 1.00000i −1.30814 − 2.00000i −1.58114 + 1.58114i 1.30814 − 1.30814i 2.97945 + 2.97945i 2.00000 + 2.00000i −7.28877 − 3.16228i
31.4 −1.00000 + 1.00000i −0.187967 − 2.00000i 1.58114 − 1.58114i 0.187967 − 0.187967i −7.64727 − 7.64727i 2.00000 + 2.00000i −8.96467 3.16228i
31.5 −1.00000 + 1.00000i 3.23045 − 2.00000i −1.58114 + 1.58114i −3.23045 + 3.23045i 4.80613 + 4.80613i 2.00000 + 2.00000i 1.43578 − 3.16228i
31.6 −1.00000 + 1.00000i 5.69710 − 2.00000i 1.58114 − 1.58114i −5.69710 + 5.69710i −0.00593756 − 0.00593756i 2.00000 + 2.00000i 23.4569 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.a ✓ 12
3.b odd 2 1 1170.3.r.b 12
5.b even 2 1 650.3.k.k 12
5.c odd 4 1 650.3.f.l 12
5.c odd 4 1 650.3.f.m 12
13.d odd 4 1 inner 130.3.k.a ✓ 12
39.f even 4 1 1170.3.r.b 12
65.f even 4 1 650.3.f.m 12
65.g odd 4 1 650.3.k.k 12
65.k even 4 1 650.3.f.l 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
130.3.k.a ✓ 12 1.a even 1 1 trivial
130.3.k.a ✓ 12 13.d odd 4 1 inner
650.3.f.l 12 5.c odd 4 1
650.3.f.l 12 65.k even 4 1
650.3.f.m 12 5.c odd 4 1
650.3.f.m 12 65.f even 4 1
650.3.k.k 12 5.b even 2 1
650.3.k.k 12 65.g odd 4 1
1170.3.r.b 12 3.b odd 2 1
1170.3.r.b 12 39.f even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} - 38T_{3}^{4} - 24T_{3}^{3} + 256T_{3}^{2} + 336T_{3} + 54 \) 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
$3$ \( (T^{6} - 38 T^{4} + \cdots + 54)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + 25)^{3} \) Copy content Toggle raw display
$7$ \( T^{12} - 424 T^{9} + \cdots + 11664 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 231339836484 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 23298085122481 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 77722561281600 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 172186884 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 76889744100 \) Copy content Toggle raw display
$29$ \( (T^{6} + 4 T^{5} + \cdots - 5430204)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 99\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 50\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 10\!\cdots\!84 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 40\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( (T^{6} - 12 T^{5} + \cdots - 67585859136)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 16\!\cdots\!84 \) Copy content Toggle raw display
$61$ \( (T^{6} - 12 T^{5} + \cdots - 71111196)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 30\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 76\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{6} + 84 T^{5} + \cdots + 9524927376)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 22\!\cdots\!64 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 17\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 26\!\cdots\!44 \) Copy content Toggle raw display
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