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Properties

Label 10.16.b.a
Level $10$
Weight $16$
Character orbit 10.b
Analytic conductor $14.269$
Analytic rank $0$
Dimension $8$
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] = [10,16,Mod(9,10)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(10, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 16, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("10.9");
 
S:= CuspForms(chi, 16);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 10 = 2 \cdot 5 \)
Weight: \( k \) \(=\) \( 16 \)
Character orbit: \([\chi]\) \(=\) 10.b (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: \(14.2693505100\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 4 x^{7} + 8 x^{6} + 6172534 x^{5} + 23752924445 x^{4} + 1095295465934 x^{3} + \cdots + 59\!\cdots\!64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{38}\cdot 3^{2}\cdot 5^{11} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{2} q^{3} - 16384 q^{4} + ( - \beta_{4} + 34 \beta_1 + 31425) q^{5} + ( - \beta_{5} - 6656) q^{6} + ( - \beta_{6} - 7 \beta_{4} + \cdots - 3769 \beta_1) q^{7}+ \cdots + ( - \beta_{7} + 3 \beta_{5} + \cdots - 5436397) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{2} q^{3} - 16384 q^{4} + ( - \beta_{4} + 34 \beta_1 + 31425) q^{5} + ( - \beta_{5} - 6656) q^{6} + ( - \beta_{6} - 7 \beta_{4} + \cdots - 3769 \beta_1) q^{7}+ \cdots + ( - 54225083 \beta_{7} + \cdots + 203280853669156) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 131072 q^{4} + 251400 q^{5} - 53248 q^{6} - 43491176 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 131072 q^{4} + 251400 q^{5} - 53248 q^{6} - 43491176 q^{9} + 4403200 q^{10} + 95435616 q^{11} - 499347456 q^{14} + 5448800 q^{15} + 2147483648 q^{16} + 6479216160 q^{19} - 4118937600 q^{20} - 14760325504 q^{21} + 872415232 q^{24} - 2241855000 q^{25} + 66288525312 q^{26} - 244549636560 q^{29} - 32701542400 q^{30} + 522311705216 q^{31} + 322563211264 q^{34} - 1829607146400 q^{35} + 712559427584 q^{36} - 2307595824192 q^{39} - 72142028800 q^{40} + 6699117519216 q^{41} - 1563617132544 q^{44} - 9090807477800 q^{45} + 12178733699072 q^{46} - 15809163185544 q^{49} - 13485542400000 q^{50} + 40555579650176 q^{51} - 7241111674880 q^{54} - 39746288199200 q^{55} + 8181308719104 q^{56} + 3791808509280 q^{59} - 89273139200 q^{60} + 57800629300816 q^{61} - 35184372088832 q^{64} + 58028394892800 q^{65} - 82398766186496 q^{66} + 59060996328448 q^{69} + 60817223987200 q^{70} - 245426235422784 q^{71} + 53331092987904 q^{74} + 226448486200000 q^{75} - 106155477565440 q^{76} + 624094605411840 q^{79} + 67484673638400 q^{80} - 13\!\cdots\!52 q^{81}+ \cdots + 16\!\cdots\!48 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4 x^{7} + 8 x^{6} + 6172534 x^{5} + 23752924445 x^{4} + 1095295465934 x^{3} + \cdots + 59\!\cdots\!64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 64\!\cdots\!88 \nu^{7} + \cdots + 16\!\cdots\!36 ) / 22\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 14\!\cdots\!93 \nu^{7} + \cdots + 29\!\cdots\!46 ) / 22\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 17\!\cdots\!97 \nu^{7} + \cdots - 13\!\cdots\!16 ) / 22\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 13\!\cdots\!61 \nu^{7} + \cdots + 28\!\cdots\!92 ) / 22\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 99\!\cdots\!32 \nu^{7} + \cdots - 18\!\cdots\!84 ) / 11\!\cdots\!95 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 32\!\cdots\!86 \nu^{7} + \cdots + 19\!\cdots\!42 ) / 43\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 18\!\cdots\!64 \nu^{7} + \cdots - 42\!\cdots\!32 ) / 11\!\cdots\!85 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{5} - 128\beta_{2} - 62\beta _1 + 1280 ) / 2560 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 54\beta_{6} - 472\beta_{4} - 62\beta_{3} - 7290\beta_{2} - 2475629\beta_1 ) / 3200 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 8389 \beta_{7} + 34614 \beta_{6} + 309201 \beta_{5} + 205568 \beta_{4} - 22038 \beta_{3} + \cdots - 14814107200 ) / 6400 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 3148390 \beta_{7} + 19906715 \beta_{5} + 111200480 \beta_{4} - 22240096 \beta_{3} + \cdots - 38044183380800 ) / 3200 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1560843715 \beta_{7} - 7181039070 \beta_{6} + 42362136545 \beta_{5} + 48712219680 \beta_{4} + \cdots - 47\!\cdots\!00 ) / 6400 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 227391930747 \beta_{6} + 1857732780786 \beta_{4} + 235111397709 \beta_{3} + \cdots + 50\!\cdots\!88 \beta_1 ) / 400 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 225416540685531 \beta_{7} + \cdots + 10\!\cdots\!00 ) / 6400 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/10\mathbb{Z}\right)^\times\).

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
9.1
−249.000 + 249.000i
−182.037 + 182.037i
149.707 − 149.707i
283.331 − 283.331i
283.331 + 283.331i
149.707 + 149.707i
−182.037 − 182.037i
−249.000 − 249.000i
− 128.000i − 5042.00i −16384.0 79524.6 − 155542.i −645376. − 3.81841e6i 2.09715e6i −1.10729e7 −1.99094e7 − 1.01791e7i
9.2 − 128.000i − 3702.75i −16384.0 44593.5 + 168905.i −473951. 2.91723e6i 2.09715e6i 638585. 2.16199e7 − 5.70797e6i
9.3 − 128.000i 2932.14i −16384.0 −160703. + 68498.7i 375313. − 1.80018e6i 2.09715e6i 5.75148e6 8.76783e6 + 2.05700e7i
9.4 − 128.000i 5604.61i −16384.0 162285. − 64661.7i 717390. 750784.i 2.09715e6i −1.70628e7 −8.27669e6 − 2.07725e7i
9.5 128.000i − 5604.61i −16384.0 162285. + 64661.7i 717390. − 750784.i − 2.09715e6i −1.70628e7 −8.27669e6 + 2.07725e7i
9.6 128.000i − 2932.14i −16384.0 −160703. − 68498.7i 375313. 1.80018e6i − 2.09715e6i 5.75148e6 8.76783e6 − 2.05700e7i
9.7 128.000i 3702.75i −16384.0 44593.5 − 168905.i −473951. − 2.91723e6i − 2.09715e6i 638585. 2.16199e7 + 5.70797e6i
9.8 128.000i 5042.00i −16384.0 79524.6 + 155542.i −645376. 3.81841e6i − 2.09715e6i −1.10729e7 −1.99094e7 + 1.01791e7i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 9.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 10.16.b.a ✓ 8
3.b odd 2 1 90.16.c.c 8
4.b odd 2 1 80.16.c.c 8
5.b even 2 1 inner 10.16.b.a ✓ 8
5.c odd 4 1 50.16.a.j 4
5.c odd 4 1 50.16.a.k 4
15.d odd 2 1 90.16.c.c 8
20.d odd 2 1 80.16.c.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
10.16.b.a ✓ 8 1.a even 1 1 trivial
10.16.b.a ✓ 8 5.b even 2 1 inner
50.16.a.j 4 5.c odd 4 1
50.16.a.k 4 5.c odd 4 1
80.16.c.c 8 4.b odd 2 1
80.16.c.c 8 20.d odd 2 1
90.16.c.c 8 3.b odd 2 1
90.16.c.c 8 15.d odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{16}^{\mathrm{new}}(10, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 16384)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} + \cdots + 94\!\cdots\!56 \) Copy content Toggle raw display
$5$ \( T^{8} + \cdots + 86\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 22\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( (T^{4} + \cdots + 12\!\cdots\!16)^{2} \) Copy content Toggle raw display
$13$ \( T^{8} + \cdots + 24\!\cdots\!96 \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 72\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( (T^{4} + \cdots + 17\!\cdots\!00)^{2} \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + \cdots - 12\!\cdots\!84)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$41$ \( (T^{4} + \cdots - 28\!\cdots\!84)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + \cdots + 92\!\cdots\!16 \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 71\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 79\!\cdots\!56 \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots + 56\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + \cdots - 11\!\cdots\!84)^{2} \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 21\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{4} + \cdots - 35\!\cdots\!84)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 18\!\cdots\!36 \) Copy content Toggle raw display
$79$ \( (T^{4} + \cdots - 30\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$89$ \( (T^{4} + \cdots - 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + \cdots + 27\!\cdots\!76 \) Copy content Toggle raw display
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