↑

Properties

Label 320.9.b.d
Level $320$
Weight $9$
Character orbit 320.b
Analytic conductor $130.361$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [320,9,Mod(191,320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(320, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.191");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 320.b (of order \(2\), degree \(1\), not 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: \(130.361155220\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 5 x^{15} + 26 x^{14} - 834 x^{13} + 4390 x^{12} - 61783 x^{11} + 466168 x^{10} + \cdots + 206161212459445 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{120}\cdot 5^{16} \)
Twist minimal: no (minimal twist has level 20)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{2} q^{5} + (\beta_{3} + 3 \beta_1) q^{7} + (\beta_{4} - \beta_{2} - 2425) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{2} q^{5} + (\beta_{3} + 3 \beta_1) q^{7} + (\beta_{4} - \beta_{2} - 2425) q^{9} + (\beta_{5} - \beta_{3} + 38 \beta_1) q^{11} + (\beta_{8} + \beta_{4} + 8 \beta_{2} - 3212) q^{13} + ( - \beta_{9} - 2 \beta_{3} + 11 \beta_1) q^{15} + ( - \beta_{6} + \beta_{4} + \cdots + 1722) q^{17}+ \cdots + (364 \beta_{15} - 68 \beta_{14} + \cdots - 236015 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 38800 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 38800 q^{9} - 51392 q^{13} + 27552 q^{17} - 414496 q^{21} + 1250000 q^{25} - 2764896 q^{29} - 5521600 q^{33} - 9009472 q^{37} - 8576448 q^{41} - 1580000 q^{45} - 32803600 q^{49} - 2452032 q^{53} + 11957760 q^{57} - 8371712 q^{61} + 9060000 q^{65} - 7527264 q^{69} + 61907232 q^{73} + 156997440 q^{77} + 140586672 q^{81} + 106960000 q^{85} + 106647456 q^{89} - 105563840 q^{93} + 171851232 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 5 x^{15} + 26 x^{14} - 834 x^{13} + 4390 x^{12} - 61783 x^{11} + 466168 x^{10} + \cdots + 206161212459445 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 47\!\cdots\!97 \nu^{15} + \cdots + 17\!\cdots\!55 ) / 21\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 17\!\cdots\!00 \nu^{15} + \cdots - 45\!\cdots\!25 ) / 27\!\cdots\!13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 30\!\cdots\!38 \nu^{15} + \cdots + 14\!\cdots\!05 ) / 36\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 71\!\cdots\!23 \nu^{15} + \cdots + 13\!\cdots\!65 ) / 34\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 34\!\cdots\!80 \nu^{15} + \cdots - 65\!\cdots\!65 ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 38\!\cdots\!43 \nu^{15} + \cdots - 83\!\cdots\!15 ) / 85\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 17\!\cdots\!22 \nu^{15} + \cdots - 15\!\cdots\!85 ) / 34\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 90\!\cdots\!94 \nu^{15} + \cdots + 78\!\cdots\!05 ) / 17\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 62\!\cdots\!93 \nu^{15} + \cdots + 15\!\cdots\!10 ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 26\!\cdots\!35 \nu^{15} + \cdots + 67\!\cdots\!19 ) / 34\!\cdots\!44 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 29\!\cdots\!67 \nu^{15} + \cdots + 47\!\cdots\!25 ) / 21\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 33\!\cdots\!07 \nu^{15} + \cdots - 25\!\cdots\!85 ) / 21\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 28\!\cdots\!67 \nu^{15} + \cdots + 27\!\cdots\!95 ) / 17\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 42\!\cdots\!27 \nu^{15} + \cdots - 67\!\cdots\!75 ) / 10\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 62\!\cdots\!98 \nu^{15} + \cdots - 10\!\cdots\!45 ) / 10\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 125 \beta_{13} + 250 \beta_{11} + 125 \beta_{6} - 500 \beta_{5} + 1250 \beta_{4} + \cdots + 1280000 ) / 4096000 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 460 \beta_{15} + 80 \beta_{14} - 340 \beta_{13} - 150 \beta_{12} - 455 \beta_{11} + 540 \beta_{10} + \cdots - 3456000 ) / 2048000 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 1540 \beta_{15} + 1400 \beta_{14} + 935 \beta_{13} - 3750 \beta_{12} - 275 \beta_{11} + \cdots + 286336000 ) / 2048000 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 13360 \beta_{15} + 4080 \beta_{14} - 32445 \beta_{13} - 83400 \beta_{12} + 12170 \beta_{11} + \cdots - 384768000 ) / 4096000 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 17160 \beta_{15} + 11900 \beta_{14} - 20100 \beta_{13} - 28500 \beta_{12} - 7925 \beta_{11} + \cdots + 2544256000 ) / 204800 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1319360 \beta_{15} + 2760640 \beta_{14} + 473665 \beta_{13} - 3125200 \beta_{12} + \cdots + 135429888000 ) / 4096000 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 21049060 \beta_{15} + 7236280 \beta_{14} + 8062375 \beta_{13} - 27158650 \beta_{12} + \cdots - 1261794944000 ) / 2048000 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 85121340 \beta_{15} + 57226320 \beta_{14} + 39905640 \beta_{13} - 303103350 \beta_{12} + \cdots - 13971868544000 ) / 2048000 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 280783120 \beta_{15} + 1853537200 \beta_{14} + 2257133305 \beta_{13} - 4204407000 \beta_{12} + \cdots - 274986937600000 ) / 4096000 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 922199700 \beta_{15} + 560713520 \beta_{14} + 1433960570 \beta_{13} - 1340047350 \beta_{12} + \cdots - 26097613337600 ) / 204800 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 85757091760 \beta_{15} + 32461878640 \beta_{14} + 203917244755 \beta_{13} - 119262460200 \beta_{12} + \cdots - 27\!\cdots\!00 ) / 4096000 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 474556965580 \beta_{15} + 419403046760 \beta_{14} + 1343043438720 \beta_{13} + \cdots - 20\!\cdots\!00 ) / 2048000 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 680228635220 \beta_{15} - 52589800760 \beta_{14} + 16759213985315 \beta_{13} + \cdots - 11\!\cdots\!00 ) / 2048000 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 64454182180640 \beta_{15} - 57661875059840 \beta_{14} + 251033604861615 \beta_{13} + \cdots - 29\!\cdots\!00 ) / 4096000 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 20769865367400 \beta_{15} - 11914738204920 \beta_{14} + 102849505676185 \beta_{13} + \cdots - 11\!\cdots\!00 ) / 204800 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/320\mathbb{Z}\right)^\times\).

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
191.1
−5.64565 + 3.35245i
−6.29294 − 5.63875i
3.05707 − 7.10588i
9.29581 − 2.24761i
−2.93811 + 7.65619i
2.32463 − 7.96873i
−4.16577 − 5.52698i
6.86496 + 2.82928i
6.86496 − 2.82928i
−4.16577 + 5.52698i
2.32463 + 7.96873i
−2.93811 − 7.65619i
9.29581 + 2.24761i
3.05707 + 7.10588i
−6.29294 + 5.63875i
−5.64565 − 3.35245i
0 − 150.211i 0 −279.508 0 − 2626.96i 0 −16002.3 0
191.2 0 − 137.297i 0 279.508 0 3940.57i 0 −12289.4 0
191.3 0 − 110.171i 0 279.508 0 − 3540.70i 0 −5576.56 0
191.4 0 − 98.1237i 0 −279.508 0 820.952i 0 −3067.27 0
191.5 0 − 75.7492i 0 279.508 0 210.345i 0 823.060 0
191.6 0 − 39.9624i 0 −279.508 0 2633.20i 0 4964.01 0
191.7 0 − 27.2434i 0 −279.508 0 − 3325.58i 0 5818.80 0
191.8 0 − 25.1248i 0 279.508 0 − 2973.76i 0 5929.74 0
191.9 0 25.1248i 0 279.508 0 2973.76i 0 5929.74 0
191.10 0 27.2434i 0 −279.508 0 3325.58i 0 5818.80 0
191.11 0 39.9624i 0 −279.508 0 − 2633.20i 0 4964.01 0
191.12 0 75.7492i 0 279.508 0 − 210.345i 0 823.060 0
191.13 0 98.1237i 0 −279.508 0 − 820.952i 0 −3067.27 0
191.14 0 110.171i 0 279.508 0 3540.70i 0 −5576.56 0
191.15 0 137.297i 0 279.508 0 − 3940.57i 0 −12289.4 0
191.16 0 150.211i 0 −279.508 0 2626.96i 0 −16002.3 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 191.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.9.b.d 16
4.b odd 2 1 inner 320.9.b.d 16
8.b even 2 1 20.9.b.a ✓ 16
8.d odd 2 1 20.9.b.a ✓ 16
24.f even 2 1 180.9.c.a 16
24.h odd 2 1 180.9.c.a 16
40.e odd 2 1 100.9.b.d 16
40.f even 2 1 100.9.b.d 16
40.i odd 4 2 100.9.d.c 32
40.k even 4 2 100.9.d.c 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
20.9.b.a ✓ 16 8.b even 2 1
20.9.b.a ✓ 16 8.d odd 2 1
100.9.b.d 16 40.e odd 2 1
100.9.b.d 16 40.f even 2 1
100.9.d.c 32 40.i odd 4 2
100.9.d.c 32 40.k even 4 2
180.9.c.a 16 24.f even 2 1
180.9.c.a 16 24.h odd 2 1
320.9.b.d 16 1.a even 1 1 trivial
320.9.b.d 16 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 71888 T_{3}^{14} + 2013496736 T_{3}^{12} + 27929868057600 T_{3}^{10} + \cdots + 21\!\cdots\!00 \) acting on \(S_{9}^{\mathrm{new}}(320, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$5$ \( (T^{2} - 78125)^{8} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 69\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 36\!\cdots\!00)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 92\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots - 61\!\cdots\!04)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 81\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots - 37\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots - 56\!\cdots\!64)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 26\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots - 18\!\cdots\!00)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 60\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 83\!\cdots\!76)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 75\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 26\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 10\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 24\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots - 36\!\cdots\!24)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 40\!\cdots\!00)^{2} \) Copy content Toggle raw display
show more
show less