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Properties

Label 8.15.d.b
Level $8$
Weight $15$
Character orbit 8.d
Analytic conductor $9.946$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8,15,Mod(3,8)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 15, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8.3");
 
S:= CuspForms(chi, 15);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 15 \)
Character orbit: \([\chi]\) \(=\) 8.d (of order \(2\), degree \(1\), 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: \(9.94631745215\)
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: \( x^{12} - x^{11} + 4349 x^{10} - 33891 x^{9} + 12151288 x^{8} - 474141530 x^{7} + 82897017850 x^{6} + \cdots + 37\!\cdots\!50 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{66}\cdot 3^{6}\cdot 5^{2}\cdot 7 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_1 + 18) q^{2} + (\beta_{3} + 3 \beta_1 - 252) q^{3} + (\beta_{2} + 21 \beta_1 - 2573) q^{4} + (\beta_{6} - \beta_{3} + 75 \beta_1 - 13) q^{5} + ( - \beta_{7} + 40 \beta_{3} + \cdots + 43283) q^{6}+ \cdots + ( - 2 \beta_{11} + \beta_{10} + \cdots + 1097776) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_1 + 18) q^{2} + (\beta_{3} + 3 \beta_1 - 252) q^{3} + (\beta_{2} + 21 \beta_1 - 2573) q^{4} + (\beta_{6} - \beta_{3} + 75 \beta_1 - 13) q^{5} + ( - \beta_{7} + 40 \beta_{3} + \cdots + 43283) q^{6}+ \cdots + ( - 13202336 \beta_{11} + \cdots - 18616970820572) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 218 q^{2} - 3024 q^{3} - 30828 q^{4} + 518556 q^{6} - 1097608 q^{8} + 13188036 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 218 q^{2} - 3024 q^{3} - 30828 q^{4} + 518556 q^{6} - 1097608 q^{8} + 13188036 q^{9} - 14533440 q^{10} - 28256720 q^{11} + 34920024 q^{12} + 191568384 q^{14} - 185822448 q^{16} + 270339544 q^{17} + 1420811358 q^{18} - 2481505872 q^{19} - 1679371200 q^{20} + 3042383484 q^{22} + 7581335184 q^{24} - 15857276820 q^{25} - 2773507776 q^{26} - 16574868000 q^{27} + 25329333120 q^{28} + 42207767040 q^{30} + 38309251808 q^{32} - 136227597840 q^{33} + 350437044 q^{34} + 149949623040 q^{35} - 150590403492 q^{36} + 102789916636 q^{38} - 66999085440 q^{40} + 264287409880 q^{41} - 110343609600 q^{42} + 32253127344 q^{43} - 585547356392 q^{44} + 864780977664 q^{46} - 2387663418144 q^{48} - 646589230644 q^{49} - 388785556630 q^{50} + 4755867895776 q^{51} + 798005307840 q^{52} + 1305053764344 q^{54} - 1050155264256 q^{56} - 7479401742480 q^{57} + 389204742720 q^{58} + 1223083947184 q^{59} + 4350689397120 q^{60} + 9957296947200 q^{62} - 16809671099328 q^{64} - 8069319822720 q^{65} - 6067132925784 q^{66} - 9309378171216 q^{67} + 32301846360616 q^{68} + 35197935521280 q^{70} - 43695386222808 q^{72} + 3619334364696 q^{73} - 55499920147776 q^{74} + 9079078926000 q^{75} + 33532610502360 q^{76} + 92515055193600 q^{78} - 86826189154560 q^{80} + 56467107312444 q^{81} - 146233962574956 q^{82} - 18774355695824 q^{83} + 186893160787200 q^{84} + 96253393476220 q^{86} - 166888683024624 q^{88} + 54781416936088 q^{89} - 488020221650880 q^{90} + 36699395136768 q^{91} + 413167093560960 q^{92} + 496016398930944 q^{94} - 616114307580864 q^{96} + 73839238696536 q^{97} - 523654870565638 q^{98} - 223606851712368 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - x^{11} + 4349 x^{10} - 33891 x^{9} + 12151288 x^{8} - 474141530 x^{7} + 82897017850 x^{6} + \cdots + 37\!\cdots\!50 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( 2\nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( 4\nu^{2} + 30\nu + 2897 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 636923 \nu^{11} - 78759886 \nu^{10} - 245335731 \nu^{9} - 64007142854 \nu^{8} + \cdots - 82\!\cdots\!70 ) / 22\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 5206759 \nu^{11} + 414020870 \nu^{10} - 7749618561 \nu^{9} + 469354871518 \nu^{8} + \cdots + 59\!\cdots\!22 ) / 22\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 360140747 \nu^{11} + 10850839058 \nu^{10} + 582416384253 \nu^{9} - 21486353930342 \nu^{8} + \cdots - 40\!\cdots\!50 ) / 11\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 376164913 \nu^{11} + 12640903702 \nu^{10} + 313658894247 \nu^{9} - 19280426207698 \nu^{8} + \cdots + 82\!\cdots\!90 ) / 11\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 21282279 \nu^{11} + 537238778 \nu^{10} + 9930623105 \nu^{9} - 1438088847582 \nu^{8} + \cdots + 39\!\cdots\!90 ) / 28\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 820762567 \nu^{11} + 105005693498 \nu^{10} + 1459283923873 \nu^{9} + 30272617601378 \nu^{8} + \cdots + 95\!\cdots\!10 ) / 56\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 622670923 \nu^{11} - 41902712082 \nu^{10} - 799090716797 \nu^{9} + 1364985661798 \nu^{8} + \cdots - 34\!\cdots\!90 ) / 28\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 3005874881 \nu^{11} + 197184062346 \nu^{10} - 915675936919 \nu^{9} + \cdots + 23\!\cdots\!10 ) / 11\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 94370893 \nu^{11} - 10467446018 \nu^{10} - 118995218133 \nu^{9} - 16384430595658 \nu^{8} + \cdots - 10\!\cdots\!30 ) / 35\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{2} - 15\beta _1 - 2897 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{9} + \beta_{8} + \beta_{6} - 3\beta_{4} + 124\beta_{3} - 33\beta_{2} - 2688\beta _1 + 59612 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - \beta_{11} - 25 \beta_{10} - 13 \beta_{9} - 52 \beta_{8} - 158 \beta_{7} + 618 \beta_{6} + \cdots - 7126878 ) / 8 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 148 \beta_{11} - 300 \beta_{10} - 498 \beta_{9} + 142 \beta_{8} + 3464 \beta_{7} - 14374 \beta_{6} + \cdots + 278006249 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 61372 \beta_{11} + 50148 \beta_{10} + 63727 \beta_{9} - 13405 \beta_{8} + 370328 \beta_{7} + \cdots - 113439771901 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 8045658 \beta_{11} + 2413130 \beta_{10} - 8887157 \beta_{9} + 1964433 \beta_{8} - 80674228 \beta_{7} + \cdots - 344011645228 ) / 8 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 326554601 \beta_{11} - 409692737 \beta_{10} + 146557429 \beta_{9} + 388951174 \beta_{8} + \cdots - 362165003016430 ) / 8 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 9853827781 \beta_{11} + 6711836611 \beta_{10} - 7664756327 \beta_{9} - 32926070282 \beta_{8} + \cdots + 53\!\cdots\!54 ) / 4 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 1038882617494 \beta_{11} + 171139828006 \beta_{10} - 245567624140 \beta_{9} + 2070000797374 \beta_{8} + \cdots - 24\!\cdots\!13 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 146144280031564 \beta_{11} - 17558620143468 \beta_{10} + 9708632887341 \beta_{9} + \cdots - 23\!\cdots\!04 ) / 8 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(7\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
3.1
−64.5010 − 31.8690i
−64.5010 + 31.8690i
−37.2836 − 57.4111i
−37.2836 + 57.4111i
4.85446 − 62.4824i
4.85446 + 62.4824i
5.50725 − 62.3341i
5.50725 + 62.3341i
41.6607 − 39.1087i
41.6607 + 39.1087i
50.2622 − 24.1659i
50.2622 + 24.1659i
−111.002 − 63.7381i −2044.69 8258.92 + 14150.1i 54892.6i 226964. + 130324.i 432198.i −14856.1 − 2.09710e6i −602230. 3.49875e6 − 6.09319e6i
3.2 −111.002 + 63.7381i −2044.69 8258.92 − 14150.1i − 54892.6i 226964. − 130324.i − 432198.i −14856.1 + 2.09710e6i −602230. 3.49875e6 + 6.09319e6i
3.3 −56.5672 − 114.822i 1525.47 −9984.31 + 12990.3i − 71626.9i −86291.8 − 175158.i − 647418.i 2.05637e6 + 411594.i −2.45590e6 −8.22436e6 + 4.05173e6i
3.4 −56.5672 + 114.822i 1525.47 −9984.31 − 12990.3i 71626.9i −86291.8 + 175158.i 647418.i 2.05637e6 − 411594.i −2.45590e6 −8.22436e6 − 4.05173e6i
3.5 27.7089 − 124.965i 563.873 −14848.4 − 6925.28i 134561.i 15624.3 − 70464.4i 255103.i −1.27685e6 + 1.66364e6i −4.46502e6 1.68154e7 + 3.72854e6i
3.6 27.7089 + 124.965i 563.873 −14848.4 + 6925.28i − 134561.i 15624.3 + 70464.4i − 255103.i −1.27685e6 − 1.66364e6i −4.46502e6 1.68154e7 − 3.72854e6i
3.7 29.0145 − 124.668i −3879.84 −14700.3 − 7234.37i − 109885.i −112571. + 483692.i 642807.i −1.32842e6 + 1.62276e6i 1.02702e7 −1.36992e7 − 3.18827e6i
3.8 29.0145 + 124.668i −3879.84 −14700.3 + 7234.37i 109885.i −112571. − 483692.i − 642807.i −1.32842e6 − 1.62276e6i 1.02702e7 −1.36992e7 + 3.18827e6i
3.9 101.321 − 78.2175i 3476.15 4148.05 − 15850.2i − 78301.9i 352208. − 271895.i 1.22276e6i −819477. − 1.93042e6i 7.30063e6 −6.12457e6 − 7.93365e6i
3.10 101.321 + 78.2175i 3476.15 4148.05 + 15850.2i 78301.9i 352208. + 271895.i − 1.22276e6i −819477. + 1.93042e6i 7.30063e6 −6.12457e6 + 7.93365e6i
3.11 118.524 − 48.3317i −1152.97 11712.1 − 11457.0i 9668.61i −136655. + 55725.2i − 1.34658e6i 834433. − 1.92400e6i −3.45362e6 467301. + 1.14597e6i
3.12 118.524 + 48.3317i −1152.97 11712.1 + 11457.0i − 9668.61i −136655. − 55725.2i 1.34658e6i 834433. + 1.92400e6i −3.45362e6 467301. − 1.14597e6i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 8.15.d.b ✓ 12
3.b odd 2 1 72.15.b.b 12
4.b odd 2 1 32.15.d.b 12
8.b even 2 1 32.15.d.b 12
8.d odd 2 1 inner 8.15.d.b ✓ 12
24.f even 2 1 72.15.b.b 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.15.d.b ✓ 12 1.a even 1 1 trivial
8.15.d.b ✓ 12 8.d odd 2 1 inner
32.15.d.b 12 4.b odd 2 1
32.15.d.b 12 8.b even 2 1
72.15.b.b 12 3.b odd 2 1
72.15.b.b 12 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{6} + 1512 T_{3}^{5} - 16502844 T_{3}^{4} - 18520805952 T_{3}^{3} + 47859959296944 T_{3}^{2} + \cdots - 27\!\cdots\!00 \) acting on \(S_{15}^{\mathrm{new}}(8, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + \cdots + 19\!\cdots\!16 \) Copy content Toggle raw display
$3$ \( (T^{6} + \cdots - 27\!\cdots\!00)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 57\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{6} + \cdots + 12\!\cdots\!04)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{6} + \cdots - 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + \cdots - 20\!\cdots\!64)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 80\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 23\!\cdots\!64)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + \cdots - 50\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 31\!\cdots\!44)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{6} + \cdots - 33\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots - 85\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 13\!\cdots\!96)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
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