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Properties

Label 320.9.b.c
Level $320$
Weight $9$
Character orbit 320.b
Analytic conductor $130.361$
Analytic rank $0$
Dimension $16$
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] = [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: \( x^{16} - 4 x^{15} - 176 x^{14} - 2098 x^{13} - 763929 x^{12} - 13756924 x^{11} - 353821633 x^{10} + \cdots + 15\!\cdots\!49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{96}\cdot 5^{16} \)
Twist minimal: no (minimal twist has level 160)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{2} q^{5} + ( - \beta_{6} - \beta_{3} - 6 \beta_1) q^{7} + ( - \beta_{7} - 8 \beta_{2} - 3929) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{3} + \beta_{2} q^{5} + ( - \beta_{6} - \beta_{3} - 6 \beta_1) q^{7} + ( - \beta_{7} - 8 \beta_{2} - 3929) q^{9} + ( - \beta_{12} + \beta_{6} + \cdots - 23 \beta_1) q^{11}+ \cdots + ( - 2694 \beta_{14} + \cdots + 55475 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 62864 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 62864 q^{9} + 156608 q^{13} - 200800 q^{17} + 948832 q^{21} + 1250000 q^{25} - 365280 q^{29} + 3551296 q^{33} + 6036928 q^{37} + 16234304 q^{41} - 9420000 q^{45} - 12229136 q^{49} - 3165120 q^{53} - 59394048 q^{57} - 44720384 q^{61} + 22180000 q^{65} + 52459552 q^{69} - 85566944 q^{73} + 105324480 q^{77} + 116911792 q^{81} - 480000 q^{85} + 97761952 q^{89} - 81664704 q^{93} + 505237216 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} - 176 x^{14} - 2098 x^{13} - 763929 x^{12} - 13756924 x^{11} - 353821633 x^{10} + \cdots + 15\!\cdots\!49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 33\!\cdots\!88 \nu^{15} + \cdots + 33\!\cdots\!11 ) / 11\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 20\!\cdots\!00 \nu^{15} + \cdots - 50\!\cdots\!93 ) / 18\!\cdots\!91 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 60\!\cdots\!36 \nu^{15} + \cdots - 16\!\cdots\!16 ) / 51\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 13\!\cdots\!74 \nu^{15} + \cdots - 27\!\cdots\!41 ) / 34\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 33\!\cdots\!30 \nu^{15} + \cdots - 18\!\cdots\!47 ) / 34\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 67\!\cdots\!84 \nu^{15} + \cdots - 20\!\cdots\!25 ) / 34\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 24\!\cdots\!00 \nu^{15} + \cdots + 77\!\cdots\!22 ) / 57\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 24\!\cdots\!60 \nu^{15} + \cdots - 43\!\cdots\!43 ) / 57\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 29\!\cdots\!16 \nu^{15} + \cdots + 20\!\cdots\!45 ) / 63\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 54\!\cdots\!44 \nu^{15} + \cdots + 38\!\cdots\!77 ) / 57\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 39\!\cdots\!36 \nu^{15} + \cdots - 33\!\cdots\!29 ) / 34\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 50\!\cdots\!38 \nu^{15} + \cdots + 65\!\cdots\!04 ) / 34\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 28\!\cdots\!68 \nu^{15} + \cdots + 21\!\cdots\!51 ) / 19\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 10\!\cdots\!34 \nu^{15} + \cdots - 62\!\cdots\!92 ) / 34\!\cdots\!25 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 64\!\cdots\!16 \nu^{15} + \cdots - 75\!\cdots\!32 ) / 19\!\cdots\!75 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 100 \beta_{15} - 20 \beta_{14} - 10 \beta_{13} + 110 \beta_{12} - 190 \beta_{11} - 10 \beta_{10} + \cdots + 64000 ) / 256000 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 610 \beta_{15} + 660 \beta_{14} - 685 \beta_{13} + 370 \beta_{12} - 1730 \beta_{11} + \cdots + 5888000 ) / 256000 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 35680 \beta_{15} + 11820 \beta_{14} + 35140 \beta_{13} - 37260 \beta_{12} + 46040 \beta_{11} + \cdots + 135520000 ) / 256000 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 1901460 \beta_{15} - 843860 \beta_{14} - 135170 \beta_{13} + 637230 \beta_{12} + 2595330 \beta_{11} + \cdots + 50604096000 ) / 256000 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 25726410 \beta_{15} - 2066520 \beta_{14} + 4580275 \beta_{13} - 20071940 \beta_{12} + \cdots + 555226521600 ) / 102400 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 665336820 \beta_{15} - 99893300 \beta_{14} + 24953920 \beta_{13} + 851598775 \beta_{12} + \cdots + 27044120464000 ) / 128000 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 111697966670 \beta_{15} + 34086891120 \beta_{14} - 11714195595 \beta_{13} - 16938092160 \beta_{12} + \cdots + 12\!\cdots\!00 ) / 512000 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1718390918540 \beta_{15} - 237284451980 \beta_{14} + 203775202250 \beta_{13} + \cdots + 33\!\cdots\!00 ) / 256000 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 57246312307040 \beta_{15} - 3953017173500 \beta_{14} + 14780789145590 \beta_{13} + \cdots + 14\!\cdots\!00 ) / 256000 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 396801520271950 \beta_{15} - 107480329816560 \beta_{14} + 15733985998845 \beta_{13} + \cdots + 84\!\cdots\!00 ) / 51200 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 57\!\cdots\!50 \beta_{15} + 642846775628320 \beta_{14} - 667521188035415 \beta_{13} + \cdots + 14\!\cdots\!00 ) / 256000 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 18\!\cdots\!40 \beta_{15} + \cdots + 38\!\cdots\!00 ) / 256000 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 12\!\cdots\!90 \beta_{15} + \cdots + 27\!\cdots\!00 ) / 512000 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 21\!\cdots\!40 \beta_{15} + \cdots + 42\!\cdots\!00 ) / 256000 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 25\!\cdots\!70 \beta_{15} + \cdots + 63\!\cdots\!00 ) / 102400 \) 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
32.4041 − 0.317814i
−27.0447 − 4.73664i
−10.0457 − 6.67711i
7.55504 − 13.0092i
5.60709 + 24.6919i
−13.6065 − 20.5689i
9.24706 − 23.7517i
−2.11640 + 15.1238i
−2.11640 − 15.1238i
9.24706 + 23.7517i
−13.6065 + 20.5689i
5.60709 − 24.6919i
7.55504 + 13.0092i
−10.0457 + 6.67711i
−27.0447 + 4.73664i
32.4041 + 0.317814i
0 − 151.451i 0 279.508 0 − 1297.85i 0 −16376.4 0
191.2 0 − 142.480i 0 279.508 0 − 911.678i 0 −13739.6 0
191.3 0 − 116.799i 0 −279.508 0 4072.51i 0 −7080.89 0
191.4 0 − 111.031i 0 −279.508 0 − 2421.94i 0 −5766.78 0
191.5 0 − 85.8157i 0 −279.508 0 2682.27i 0 −803.335 0
191.6 0 − 76.0383i 0 279.508 0 4406.09i 0 779.176 0
191.7 0 − 36.9561i 0 279.508 0 808.337i 0 5195.25 0
191.8 0 − 14.1592i 0 −279.508 0 − 68.2554i 0 6360.52 0
191.9 0 14.1592i 0 −279.508 0 68.2554i 0 6360.52 0
191.10 0 36.9561i 0 279.508 0 − 808.337i 0 5195.25 0
191.11 0 76.0383i 0 279.508 0 − 4406.09i 0 779.176 0
191.12 0 85.8157i 0 −279.508 0 − 2682.27i 0 −803.335 0
191.13 0 111.031i 0 −279.508 0 2421.94i 0 −5766.78 0
191.14 0 116.799i 0 −279.508 0 − 4072.51i 0 −7080.89 0
191.15 0 142.480i 0 279.508 0 911.678i 0 −13739.6 0
191.16 0 151.451i 0 279.508 0 1297.85i 0 −16376.4 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.c 16
4.b odd 2 1 inner 320.9.b.c 16
8.b even 2 1 160.9.b.a ✓ 16
8.d odd 2 1 160.9.b.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
160.9.b.a ✓ 16 8.b even 2 1
160.9.b.a ✓ 16 8.d odd 2 1
320.9.b.c 16 1.a even 1 1 trivial
320.9.b.c 16 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} + 83920 T_{3}^{14} + 2838343456 T_{3}^{12} + 49670226186240 T_{3}^{10} + \cdots + 91\!\cdots\!36 \) 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
$3$ \( T^{16} + \cdots + 91\!\cdots\!36 \) Copy content Toggle raw display
$5$ \( (T^{2} - 78125)^{8} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 57\!\cdots\!16 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots - 14\!\cdots\!56)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots - 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 22\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots - 17\!\cdots\!80)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 20\!\cdots\!00 \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 49\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots - 68\!\cdots\!00)^{2} \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 67\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 61\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots - 32\!\cdots\!80)^{2} \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 36\!\cdots\!96 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 11\!\cdots\!16)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 19\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 53\!\cdots\!36 \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots - 30\!\cdots\!64)^{2} \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 95\!\cdots\!96 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 39\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 98\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots - 69\!\cdots\!84)^{2} \) Copy content Toggle raw display
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