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Properties

Label 2366.4.a.bg
Level $2366$
Weight $4$
Character orbit 2366.a
Self dual yes
Analytic conductor $139.599$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2366,4,Mod(1,2366)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2366, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2366.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2366 = 2 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2366.a (trivial)

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: yes
Analytic conductor: \(139.598519074\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} - 219 x^{10} + 1022 x^{9} + 17084 x^{8} - 65540 x^{7} - 566763 x^{6} + 1871300 x^{5} + \cdots + 543166 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2\cdot 13 \)
Twist minimal: no (minimal twist has level 182)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + 2 q^{2} + (\beta_{5} + \beta_1) q^{3} + 4 q^{4} + ( - \beta_{5} - \beta_{2} + 2) q^{5} + (2 \beta_{5} + 2 \beta_1) q^{6} - 7 q^{7} + 8 q^{8} + (\beta_{7} - 2 \beta_{5} - \beta_{3} + \cdots + 13) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + (\beta_{5} + \beta_1) q^{3} + 4 q^{4} + ( - \beta_{5} - \beta_{2} + 2) q^{5} + (2 \beta_{5} + 2 \beta_1) q^{6} - 7 q^{7} + 8 q^{8} + (\beta_{7} - 2 \beta_{5} - \beta_{3} + \cdots + 13) q^{9}+ \cdots + (9 \beta_{11} - 11 \beta_{10} + \cdots + 258) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 24 q^{2} + 6 q^{3} + 48 q^{4} + 28 q^{5} + 12 q^{6} - 84 q^{7} + 96 q^{8} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 24 q^{2} + 6 q^{3} + 48 q^{4} + 28 q^{5} + 12 q^{6} - 84 q^{7} + 96 q^{8} + 162 q^{9} + 56 q^{10} + 6 q^{11} + 24 q^{12} - 168 q^{14} - 138 q^{15} + 192 q^{16} - 56 q^{17} + 324 q^{18} + 158 q^{19} + 112 q^{20} - 42 q^{21} + 12 q^{22} + 414 q^{23} + 48 q^{24} + 478 q^{25} + 390 q^{27} - 336 q^{28} + 222 q^{29} - 276 q^{30} - 200 q^{31} + 384 q^{32} + 844 q^{33} - 112 q^{34} - 196 q^{35} + 648 q^{36} - 560 q^{37} + 316 q^{38} + 224 q^{40} - 66 q^{41} - 84 q^{42} + 484 q^{43} + 24 q^{44} + 542 q^{45} + 828 q^{46} + 618 q^{47} + 96 q^{48} + 588 q^{49} + 956 q^{50} + 992 q^{51} + 504 q^{53} + 780 q^{54} + 2584 q^{55} - 672 q^{56} - 1164 q^{57} + 444 q^{58} + 1460 q^{59} - 552 q^{60} - 2 q^{61} - 400 q^{62} - 1134 q^{63} + 768 q^{64} + 1688 q^{66} + 334 q^{67} - 224 q^{68} + 4660 q^{69} - 392 q^{70} - 196 q^{71} + 1296 q^{72} + 490 q^{73} - 1120 q^{74} - 338 q^{75} + 632 q^{76} - 42 q^{77} + 2942 q^{79} + 448 q^{80} + 2824 q^{81} - 132 q^{82} + 236 q^{83} - 168 q^{84} + 1352 q^{85} + 968 q^{86} + 1456 q^{87} + 48 q^{88} + 3566 q^{89} + 1084 q^{90} + 1656 q^{92} + 1884 q^{93} + 1236 q^{94} + 5754 q^{95} + 192 q^{96} + 3032 q^{97} + 1176 q^{98} + 3670 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} - 219 x^{10} + 1022 x^{9} + 17084 x^{8} - 65540 x^{7} - 566763 x^{6} + 1871300 x^{5} + \cdots + 543166 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 22\!\cdots\!67 \nu^{11} + \cdots - 11\!\cdots\!06 ) / 12\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1904663209127 \nu^{11} + 19462036530079 \nu^{10} - 459272103322540 \nu^{9} + \cdots - 66\!\cdots\!14 ) / 10\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 74\!\cdots\!57 \nu^{11} + \cdots - 19\!\cdots\!34 ) / 32\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 5468269259185 \nu^{11} + 23532019877671 \nu^{10} + \cdots - 82\!\cdots\!62 ) / 20\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 11\!\cdots\!47 \nu^{11} + \cdots + 43\!\cdots\!10 ) / 12\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 12841201727497 \nu^{11} + 27602003225263 \nu^{10} + \cdots - 49\!\cdots\!02 ) / 10\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 19\!\cdots\!73 \nu^{11} + \cdots - 26\!\cdots\!30 ) / 12\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 51\!\cdots\!71 \nu^{11} + \cdots - 26\!\cdots\!50 ) / 21\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 82\!\cdots\!39 \nu^{11} + \cdots + 39\!\cdots\!42 ) / 28\!\cdots\!36 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 60165898412177 \nu^{11} - 377355513926759 \nu^{10} + \cdots + 11\!\cdots\!10 ) / 10\!\cdots\!28 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - 4\beta_{5} + \beta_{3} + \beta _1 + 39 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -2\beta_{11} + \beta_{10} - \beta_{8} + 6\beta_{7} - 55\beta_{5} + 3\beta_{4} + 8\beta_{3} + 4\beta_{2} + 66\beta _1 + 58 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 18 \beta_{11} + 8 \beta_{10} + 3 \beta_{9} - 11 \beta_{8} + 118 \beta_{7} + 8 \beta_{6} - 765 \beta_{5} + \cdots + 2862 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( - 339 \beta_{11} + 126 \beta_{10} + 20 \beta_{9} - 196 \beta_{8} + 1077 \beta_{7} + 29 \beta_{6} + \cdots + 13308 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 3449 \beta_{11} + 1397 \beta_{10} + 504 \beta_{9} - 2115 \beta_{8} + 15302 \beta_{7} + 1195 \beta_{6} + \cdots + 279225 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 48896 \beta_{11} + 17122 \beta_{10} + 4699 \beta_{9} - 29639 \beta_{8} + 163586 \beta_{7} + \cdots + 2200883 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 534641 \beta_{11} + 206489 \beta_{10} + 72181 \beta_{9} - 328466 \beta_{8} + 2082948 \beta_{7} + \cdots + 33152717 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 6856641 \beta_{11} + 2405107 \beta_{10} + 783153 \beta_{9} - 4222423 \beta_{8} + 23686031 \beta_{7} + \cdots + 330901544 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 78042015 \beta_{11} + 29307359 \beta_{10} + 10139075 \beta_{9} - 48051925 \beta_{8} + \cdots + 4348380149 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 959434466 \beta_{11} + 340455429 \beta_{10} + 117051090 \beta_{9} - 593509978 \beta_{8} + \cdots + 47962198087 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−7.32981
−6.15503
−8.14920
−6.44172
−0.129358
1.61813
−0.254418
1.28463
6.59050
6.39016
6.70645
11.8697
2.00000 −9.06186 4.00000 19.3198 −18.1237 −7.00000 8.00000 55.1173 38.6396
1.2 2.00000 −7.88708 4.00000 −7.14146 −15.7742 −7.00000 8.00000 35.2060 −14.2829
1.3 2.00000 −6.41715 4.00000 16.3154 −12.8343 −7.00000 8.00000 14.1798 32.6307
1.4 2.00000 −4.70967 4.00000 −13.2105 −9.41934 −7.00000 8.00000 −4.81899 −26.4211
1.5 2.00000 −1.86141 4.00000 5.44751 −3.72282 −7.00000 8.00000 −23.5352 10.8950
1.6 2.00000 −0.113922 4.00000 −7.70020 −0.227844 −7.00000 8.00000 −26.9870 −15.4004
1.7 2.00000 1.47763 4.00000 −5.54905 2.95527 −7.00000 8.00000 −24.8166 −11.0981
1.8 2.00000 3.01668 4.00000 12.8832 6.03336 −7.00000 8.00000 −17.8997 25.7665
1.9 2.00000 4.85845 4.00000 13.8421 9.71690 −7.00000 8.00000 −3.39547 27.6843
1.10 2.00000 8.12221 4.00000 16.6961 16.2444 −7.00000 8.00000 38.9702 33.3921
1.11 2.00000 8.43850 4.00000 −18.3312 16.8770 −7.00000 8.00000 44.2083 −36.6625
1.12 2.00000 10.1376 4.00000 −4.57164 20.2752 −7.00000 8.00000 75.7713 −9.14327
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(7\) \(1\)
\(13\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2366.4.a.bg 12
13.b even 2 1 2366.4.a.bd 12
13.f odd 12 2 182.4.m.b ✓ 24
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
182.4.m.b ✓ 24 13.f odd 12 2
2366.4.a.bd 12 13.b even 2 1
2366.4.a.bg 12 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(2366))\):

\( T_{3}^{12} - 6 T_{3}^{11} - 225 T_{3}^{10} + 1184 T_{3}^{9} + 18272 T_{3}^{8} - 81416 T_{3}^{7} + \cdots + 6892600 \) Copy content Toggle raw display
\( T_{5}^{12} - 28 T_{5}^{11} - 597 T_{5}^{10} + 20164 T_{5}^{9} + 109349 T_{5}^{8} - 5078942 T_{5}^{7} + \cdots + 1727154468325 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{12} \) Copy content Toggle raw display
$3$ \( T^{12} - 6 T^{11} + \cdots + 6892600 \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 1727154468325 \) Copy content Toggle raw display
$7$ \( (T + 7)^{12} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots - 19\!\cdots\!92 \) Copy content Toggle raw display
$13$ \( T^{12} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots - 66\!\cdots\!68 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots - 13\!\cdots\!88 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots - 46\!\cdots\!68 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots - 36\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots - 56\!\cdots\!44 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 73\!\cdots\!00 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 48\!\cdots\!88 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 48\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots - 50\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 18\!\cdots\!92 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 26\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots - 40\!\cdots\!64 \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots - 20\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots - 27\!\cdots\!08 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots - 41\!\cdots\!92 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 86\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 89\!\cdots\!96 \) Copy content Toggle raw display
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