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Properties

Label 8.13.d.b
Level $8$
Weight $13$
Character orbit 8.d
Analytic conductor $7.312$
Analytic rank $0$
Dimension $10$
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] = [8,13,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, 13, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8.3");
 
S:= CuspForms(chi, 13);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8 = 2^{3} \)
Weight: \( k \) \(=\) \( 13 \)
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: \(7.31195053821\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 5 x^{9} + 468 x^{8} + 1496 x^{7} + 710096 x^{6} + 29155008 x^{5} + 143571008 x^{4} + \cdots + 824967906703360 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{45}\cdot 3^{3}\cdot 23 \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_1 - 11) q^{2} + ( - \beta_{2} + 3 \beta_1 - 66) q^{3} + ( - \beta_{3} + \beta_{2} + 12 \beta_1 - 244) q^{4} + ( - \beta_{5} + \beta_{2} + 48 \beta_1) q^{5} + ( - \beta_{9} - \beta_{4} + \cdots - 13081) q^{6}+ \cdots + (4 \beta_{9} - 4 \beta_{7} + \cdots + 169275) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_1 - 11) q^{2} + ( - \beta_{2} + 3 \beta_1 - 66) q^{3} + ( - \beta_{3} + \beta_{2} + 12 \beta_1 - 244) q^{4} + ( - \beta_{5} + \beta_{2} + 48 \beta_1) q^{5} + ( - \beta_{9} - \beta_{4} + \cdots - 13081) q^{6}+ \cdots + ( - 3014528 \beta_{9} + \cdots - 71728800198) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 110 q^{2} - 660 q^{3} - 2444 q^{4} - 130788 q^{6} + 27160 q^{8} + 1692798 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 110 q^{2} - 660 q^{3} - 2444 q^{4} - 130788 q^{6} + 27160 q^{8} + 1692798 q^{9} + 1873200 q^{10} - 4591444 q^{11} - 5427720 q^{12} + 12728736 q^{14} - 38294000 q^{16} + 42876500 q^{17} + 8229270 q^{18} + 36062828 q^{19} + 17615520 q^{20} - 227004580 q^{22} + 169102992 q^{24} - 706946390 q^{25} + 151295184 q^{26} + 1382414040 q^{27} - 132075840 q^{28} + 456064800 q^{30} - 2651204000 q^{32} - 478018200 q^{33} + 4454663012 q^{34} - 2838470400 q^{35} - 4074739428 q^{36} + 1644178460 q^{38} + 8180322240 q^{40} - 12339248044 q^{41} + 14692585920 q^{42} + 25081495340 q^{43} - 27950589832 q^{44} - 16813594656 q^{46} + 13310114400 q^{48} - 39532486838 q^{49} + 24888425650 q^{50} + 36757299288 q^{51} - 36172521120 q^{52} - 74007907272 q^{54} + 154450364544 q^{56} + 80408630760 q^{57} + 193270394640 q^{58} - 109448026708 q^{59} - 347360715840 q^{60} - 299237961600 q^{62} + 276213192256 q^{64} + 96485235840 q^{65} + 486280823688 q^{66} - 49860896020 q^{67} - 296812951960 q^{68} - 799057954560 q^{70} + 1077572984520 q^{72} - 58835592940 q^{73} + 742739480496 q^{74} - 251792743380 q^{75} - 699737494024 q^{76} - 1795838526240 q^{78} + 1981932232320 q^{80} + 259515975474 q^{81} + 1654109754980 q^{82} - 146236977940 q^{83} - 3144240693120 q^{84} - 2261898070564 q^{86} + 2170357811600 q^{88} + 913341514388 q^{89} + 3983485096080 q^{90} + 361546645248 q^{91} - 2649411172800 q^{92} - 2517413216064 q^{94} + 2836588548672 q^{96} - 1474226441260 q^{97} + 2377890492370 q^{98} - 717160631484 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 5 x^{9} + 468 x^{8} + 1496 x^{7} + 710096 x^{6} + 29155008 x^{5} + 143571008 x^{4} + \cdots + 824967906703360 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{9} + \nu^{8} + 474 \nu^{7} + 4340 \nu^{6} + 736136 \nu^{5} + 33571824 \nu^{4} + \cdots + 44108019403776 ) / 549755813888 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 4493 \nu^{9} - 30093 \nu^{8} + 3196142 \nu^{7} - 128916068 \nu^{6} + \cdots - 14\!\cdots\!40 ) / 23639499997184 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 8707 \nu^{9} + 53757 \nu^{8} + 1815154 \nu^{7} - 101763804 \nu^{6} + \cdots - 21\!\cdots\!56 ) / 23639499997184 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1615 \nu^{9} + 11215 \nu^{8} - 1231674 \nu^{7} + 48040268 \nu^{6} + 1229377784 \nu^{5} + \cdots + 51\!\cdots\!48 ) / 1477468749824 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 27321 \nu^{9} + 1186119 \nu^{8} - 6815370 \nu^{7} - 965949524 \nu^{6} + \cdots + 64\!\cdots\!84 ) / 23639499997184 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 136211 \nu^{9} + 361491 \nu^{8} - 76989650 \nu^{7} + 3208778268 \nu^{6} + \cdots + 41\!\cdots\!20 ) / 23639499997184 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 38493 \nu^{9} - 13015459 \nu^{8} + 358357042 \nu^{7} - 2234490716 \nu^{6} + \cdots - 10\!\cdots\!00 ) / 23639499997184 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 26341 \nu^{9} - 1612827 \nu^{8} + 35745538 \nu^{7} - 95398332 \nu^{6} + \cdots - 55\!\cdots\!56 ) / 2954937499648 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 123429 \nu^{9} - 4683739 \nu^{8} + 61948546 \nu^{7} + 2245161796 \nu^{6} + \cdots - 14\!\cdots\!24 ) / 11819749998592 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + 6\beta_{2} + 26\beta _1 + 256 ) / 512 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 8\beta_{6} + 3\beta_{4} + 40\beta_{3} + 186\beta_{2} + 390\beta _1 - 46656 ) / 512 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 32 \beta_{9} + 64 \beta_{8} - 96 \beta_{7} + 72 \beta_{6} - 128 \beta_{5} - 193 \beta_{4} + \cdots - 583616 ) / 512 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 4320 \beta_{9} + 1856 \beta_{8} - 1120 \beta_{7} + 552 \beta_{6} - 24192 \beta_{5} + \cdots - 126891712 ) / 512 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 193888 \beta_{9} + 83008 \beta_{8} + 44832 \beta_{7} - 11416 \beta_{6} - 342144 \beta_{5} + \cdots - 7937852096 ) / 512 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 2658400 \beta_{9} + 65088 \beta_{8} + 2715680 \beta_{7} - 4321240 \beta_{6} - 3255424 \beta_{5} + \cdots + 2085743424 ) / 512 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 35934880 \beta_{9} - 57652160 \beta_{8} + 98783008 \beta_{7} - 216557976 \beta_{6} + \cdots - 4460137581248 ) / 512 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 4710587296 \beta_{9} - 4144748992 \beta_{8} + 2120907808 \beta_{7} - 3951195992 \beta_{6} + \cdots - 451877316724928 ) / 512 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 266511889568 \beta_{9} - 114305216448 \beta_{8} - 23011923680 \beta_{7} - 151881717016 \beta_{6} + \cdots - 86\!\cdots\!80 ) / 512 \) 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
−25.4137 + 6.09761i
−25.4137 − 6.09761i
−11.5862 + 26.7344i
−11.5862 − 26.7344i
−10.5406 + 27.3936i
−10.5406 − 27.3936i
17.4695 + 29.8739i
17.4695 − 29.8739i
32.5710 + 17.8321i
32.5710 − 17.8321i
−62.8274 − 12.1952i 24.0930 3798.55 + 1532.39i − 25133.2i −1513.70 − 293.819i 190438.i −219965. − 142600.i −530861. −306505. + 1.57906e6i
3.2 −62.8274 + 12.1952i 24.0930 3798.55 − 1532.39i 25133.2i −1513.70 + 293.819i − 190438.i −219965. + 142600.i −530861. −306505. − 1.57906e6i
3.3 −35.1724 − 53.4687i 1272.20 −1621.80 + 3761.25i 19079.0i −44746.4 − 68023.0i 136156.i 258152. − 45576.2i 1.08706e6 1.02013e6 − 671054.i
3.4 −35.1724 + 53.4687i 1272.20 −1621.80 − 3761.25i − 19079.0i −44746.4 + 68023.0i − 136156.i 258152. + 45576.2i 1.08706e6 1.02013e6 + 671054.i
3.5 −33.0812 − 54.7872i −875.418 −1907.27 + 3624.85i 2272.99i 28959.9 + 47961.7i − 92079.6i 261690. − 15420.3i 234916. 124531. − 75193.1i
3.6 −33.0812 + 54.7872i −875.418 −1907.27 − 3624.85i − 2272.99i 28959.9 − 47961.7i 92079.6i 261690. + 15420.3i 234916. 124531. + 75193.1i
3.7 22.9390 − 59.7478i 271.191 −3043.60 − 2741.11i − 11036.2i 6220.87 − 16203.1i − 58770.2i −233593. + 118970.i −457896. −659388. − 253159.i
3.8 22.9390 + 59.7478i 271.191 −3043.60 + 2741.11i 11036.2i 6220.87 + 16203.1i 58770.2i −233593. − 118970.i −457896. −659388. + 253159.i
3.9 53.1419 − 35.6642i −1022.07 1552.12 − 3790.53i 21249.1i −54314.6 + 36451.3i 149114.i −52703.6 − 256791.i 513182. 757833. + 1.12922e6i
3.10 53.1419 + 35.6642i −1022.07 1552.12 + 3790.53i − 21249.1i −54314.6 − 36451.3i − 149114.i −52703.6 + 256791.i 513182. 757833. − 1.12922e6i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 3.10
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 8.13.d.b ✓ 10
3.b odd 2 1 72.13.b.b 10
4.b odd 2 1 32.13.d.b 10
8.b even 2 1 32.13.d.b 10
8.d odd 2 1 inner 8.13.d.b ✓ 10
24.f even 2 1 72.13.b.b 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
8.13.d.b ✓ 10 1.a even 1 1 trivial
8.13.d.b ✓ 10 8.d odd 2 1 inner
32.13.d.b 10 4.b odd 2 1
32.13.d.b 10 8.b even 2 1
72.13.b.b 10 3.b odd 2 1
72.13.b.b 10 24.f even 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{5} + 330T_{3}^{4} - 1697352T_{3}^{3} - 685590480T_{3}^{2} + 326191796496T_{3} - 7437355394400 \) acting on \(S_{13}^{\mathrm{new}}(8, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$3$ \( (T^{5} + \cdots - 7437355394400)^{2} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 65\!\cdots\!00 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 43\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{5} + \cdots - 78\!\cdots\!52)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{5} + \cdots - 38\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{5} + \cdots - 43\!\cdots\!88)^{2} \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 82\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 86\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots + 24\!\cdots\!68)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + \cdots - 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 35\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{5} + \cdots - 45\!\cdots\!08)^{2} \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( (T^{5} + \cdots + 46\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( T^{10} + \cdots + 53\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{5} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 42\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( (T^{5} + \cdots + 82\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{5} + \cdots - 11\!\cdots\!08)^{2} \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots + 30\!\cdots\!00)^{2} \) Copy content Toggle raw display
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