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Properties

Label 320.5.m.b
Level $320$
Weight $5$
Character orbit 320.m
Analytic conductor $33.078$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [320,5,Mod(33,320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(320, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 3]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.33");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 320.m (of order \(4\), degree \(2\), 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: \(33.0783881868\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 37 x^{14} - 5307 x^{12} + 38350 x^{10} + 13148446 x^{8} + 403724206 x^{6} + 4702609725 x^{4} + \cdots + 36739305625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{4}\cdot 5^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{4} + \beta_{3} + 1) q^{3} + \beta_{10} q^{5} + ( - \beta_{13} - \beta_{11}) q^{7} + ( - \beta_{9} - 2 \beta_{5} + \cdots + 34 \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{4} + \beta_{3} + 1) q^{3} + \beta_{10} q^{5} + ( - \beta_{13} - \beta_{11}) q^{7} + ( - \beta_{9} - 2 \beta_{5} + \cdots + 34 \beta_{3}) q^{9}+ \cdots + (830 \beta_{5} + 830 \beta_{4} + \cdots + 5311) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 20 q^{3}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 20 q^{3} + 240 q^{17} - 592 q^{19} - 2040 q^{25} - 2680 q^{27} + 4800 q^{33} + 5220 q^{35} - 5352 q^{41} + 4420 q^{43} + 5560 q^{57} + 4128 q^{59} + 11640 q^{65} - 19020 q^{67} - 12840 q^{73} - 4340 q^{75} - 30536 q^{81} - 36900 q^{83} + 13560 q^{97} + 92472 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 37 x^{14} - 5307 x^{12} + 38350 x^{10} + 13148446 x^{8} + 403724206 x^{6} + 4702609725 x^{4} + \cdots + 36739305625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 28\!\cdots\!83 \nu^{14} + \cdots + 70\!\cdots\!25 ) / 60\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 24\!\cdots\!49 \nu^{14} + \cdots + 13\!\cdots\!75 ) / 20\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 321835006514 \nu^{15} + 14185543083213 \nu^{13} + \cdots + 98\!\cdots\!11 \nu ) / 13\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 10\!\cdots\!29 \nu^{15} + \cdots - 41\!\cdots\!00 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 10\!\cdots\!29 \nu^{15} + \cdots - 41\!\cdots\!00 ) / 53\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 54\!\cdots\!03 \nu^{15} + \cdots - 70\!\cdots\!66 \nu ) / 80\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 99\!\cdots\!97 \nu^{15} + \cdots + 10\!\cdots\!50 ) / 96\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 54\!\cdots\!56 \nu^{15} + \cdots - 42\!\cdots\!00 ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 22\!\cdots\!16 \nu^{15} + \cdots + 12\!\cdots\!83 \nu ) / 16\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 91\!\cdots\!51 \nu^{15} + \cdots + 82\!\cdots\!25 ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 64\!\cdots\!29 \nu^{15} + \cdots - 94\!\cdots\!00 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 25\!\cdots\!07 \nu^{15} + \cdots - 90\!\cdots\!25 ) / 96\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 86\!\cdots\!64 \nu^{15} + \cdots - 10\!\cdots\!75 ) / 28\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 52\!\cdots\!23 \nu^{15} + \cdots - 10\!\cdots\!25 ) / 48\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 65\!\cdots\!99 \nu^{15} + \cdots + 49\!\cdots\!00 ) / 48\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 4 \beta_{15} + 3 \beta_{14} + 10 \beta_{13} - 7 \beta_{11} + 19 \beta_{10} - 10 \beta_{8} + \cdots + 150 \beta_{3} ) / 300 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 9 \beta_{15} + 17 \beta_{14} - 35 \beta_{13} - 5 \beta_{12} + 32 \beta_{11} + 41 \beta_{10} + \cdots + 1265 ) / 300 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 152 \beta_{15} + 159 \beta_{14} + 320 \beta_{13} - 656 \beta_{11} + 707 \beta_{10} + \cdots - 12825 \beta_{3} ) / 150 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 478 \beta_{15} + 264 \beta_{14} - 1045 \beta_{13} + 840 \beta_{12} + 2169 \beta_{11} + 1247 \beta_{10} + \cdots + 147815 ) / 100 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 14439 \beta_{15} + 16008 \beta_{14} + 46080 \beta_{13} - 40407 \beta_{11} + \cdots - 2133475 \beta_{3} ) / 300 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 36201 \beta_{15} + 40513 \beta_{14} - 93490 \beta_{13} + 33005 \beta_{12} + 170773 \beta_{11} + \cdots + 4809385 ) / 75 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1029564 \beta_{15} + 1013193 \beta_{14} + 2358150 \beta_{13} - 3786927 \beta_{11} + \cdots - 229360900 \beta_{3} ) / 300 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 4404223 \beta_{15} + 3978349 \beta_{14} - 11005245 \beta_{13} + 5570115 \beta_{12} + 20452604 \beta_{11} + \cdots + 355783665 ) / 100 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 17770502 \beta_{15} + 19712739 \beta_{14} + 52071380 \beta_{13} - 57286916 \beta_{11} + \cdots - 9726684675 \beta_{3} ) / 150 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 1094261424 \beta_{15} + 1074517112 \beta_{14} - 2745249935 \beta_{13} + 1229889220 \beta_{12} + \cdots + 32852550215 ) / 300 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 552712489 \beta_{15} + 491147388 \beta_{14} + 1097566240 \beta_{13} + \cdots - 1536496852275 \beta_{3} ) / 300 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 6921260342 \beta_{15} + 6598961446 \beta_{14} - 17380657180 \beta_{13} + 8157600510 \beta_{12} + \cdots - 104194664315 ) / 25 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 161749176036 \beta_{15} - 165887416917 \beta_{14} - 413119724070 \beta_{13} + \cdots - 113091583665850 \beta_{3} ) / 300 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 5852823186969 \beta_{15} + 5639363250697 \beta_{14} - 14682972977035 \beta_{13} + \cdots - 355618976355935 ) / 300 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 12645930440988 \beta_{15} - 13187707246581 \beta_{14} - 33127829679600 \beta_{13} + \cdots - 37\!\cdots\!25 \beta_{3} ) / 150 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(191\) \(257\) \(261\)
\(\chi(n)\) \(1\) \(\beta_{3}\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
33.1
0.961975 − 5.59259i
−0.961975 − 5.59259i
−0.743878 − 1.74174i
0.743878 − 1.74174i
−8.58423 + 1.19677i
8.58423 + 1.19677i
0.763089 + 4.63756i
−0.763089 + 4.63756i
0.961975 + 5.59259i
−0.961975 + 5.59259i
−0.743878 + 1.74174i
0.743878 + 1.74174i
−8.58423 − 1.19677i
8.58423 − 1.19677i
0.763089 − 4.63756i
−0.763089 − 4.63756i
0 −9.18519 + 9.18519i 0 −8.23041 − 23.6064i 0 −18.0198 + 18.0198i 0 − 87.7354i 0
33.2 0 −9.18519 + 9.18519i 0 8.23041 + 23.6064i 0 18.0198 − 18.0198i 0 − 87.7354i 0
33.3 0 −1.48347 + 1.48347i 0 −19.7461 + 15.3327i 0 −31.7886 + 31.7886i 0 76.5986i 0
33.4 0 −1.48347 + 1.48347i 0 19.7461 − 15.3327i 0 31.7886 − 31.7886i 0 76.5986i 0
33.5 0 4.39354 − 4.39354i 0 −23.1660 − 9.39876i 0 30.5174 − 30.5174i 0 42.3936i 0
33.6 0 4.39354 − 4.39354i 0 23.1660 + 9.39876i 0 −30.5174 + 30.5174i 0 42.3936i 0
33.7 0 11.2751 − 11.2751i 0 −0.830035 − 24.9862i 0 −61.3878 + 61.3878i 0 − 173.257i 0
33.8 0 11.2751 − 11.2751i 0 0.830035 + 24.9862i 0 61.3878 − 61.3878i 0 − 173.257i 0
97.1 0 −9.18519 − 9.18519i 0 −8.23041 + 23.6064i 0 −18.0198 − 18.0198i 0 87.7354i 0
97.2 0 −9.18519 − 9.18519i 0 8.23041 − 23.6064i 0 18.0198 + 18.0198i 0 87.7354i 0
97.3 0 −1.48347 − 1.48347i 0 −19.7461 − 15.3327i 0 −31.7886 − 31.7886i 0 − 76.5986i 0
97.4 0 −1.48347 − 1.48347i 0 19.7461 + 15.3327i 0 31.7886 + 31.7886i 0 − 76.5986i 0
97.5 0 4.39354 + 4.39354i 0 −23.1660 + 9.39876i 0 30.5174 + 30.5174i 0 − 42.3936i 0
97.6 0 4.39354 + 4.39354i 0 23.1660 − 9.39876i 0 −30.5174 − 30.5174i 0 − 42.3936i 0
97.7 0 11.2751 + 11.2751i 0 −0.830035 + 24.9862i 0 −61.3878 − 61.3878i 0 173.257i 0
97.8 0 11.2751 + 11.2751i 0 0.830035 − 24.9862i 0 61.3878 + 61.3878i 0 173.257i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 33.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 inner
20.e even 4 1 inner
40.i odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.5.m.b yes 16
4.b odd 2 1 320.5.m.a ✓ 16
5.c odd 4 1 320.5.m.a ✓ 16
8.b even 2 1 320.5.m.a ✓ 16
8.d odd 2 1 inner 320.5.m.b yes 16
20.e even 4 1 inner 320.5.m.b yes 16
40.i odd 4 1 inner 320.5.m.b yes 16
40.k even 4 1 320.5.m.a ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.5.m.a ✓ 16 4.b odd 2 1
320.5.m.a ✓ 16 5.c odd 4 1
320.5.m.a ✓ 16 8.b even 2 1
320.5.m.a ✓ 16 40.k even 4 1
320.5.m.b yes 16 1.a even 1 1 trivial
320.5.m.b yes 16 8.d odd 2 1 inner
320.5.m.b yes 16 20.e even 4 1 inner
320.5.m.b yes 16 40.i odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 10T_{3}^{7} + 50T_{3}^{6} + 820T_{3}^{5} + 37864T_{3}^{4} - 235080T_{3}^{3} + 793800T_{3}^{2} + 3402000T_{3} + 7290000 \) acting on \(S_{5}^{\mathrm{new}}(320, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} - 10 T^{7} + \cdots + 7290000)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 23\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 20\!\cdots\!56)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 36\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 29\!\cdots\!00)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 148 T^{3} + \cdots - 278890736)^{4} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 37\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 51\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 10\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{4} + 1338 T^{3} + \cdots - 408089526896)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 49\!\cdots\!00)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 15\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{4} + \cdots - 18204334631504)^{4} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 92\!\cdots\!00)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 28\!\cdots\!00)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 23\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 46\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 33\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 10\!\cdots\!36)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 63\!\cdots\!00)^{2} \) Copy content Toggle raw display
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