Properties

Label 320.6.d.a
Level $320$
Weight $6$
Character orbit 320.d
Analytic conductor $51.323$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [320,6,Mod(161,320)]
 
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([0, 1, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("320.161");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 320 = 2^{6} \cdot 5 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 320.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: \(51.3228223402\)
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: \( x^{8} - 4x^{7} - 384x^{6} + 506x^{5} + 49869x^{4} + 29654x^{3} - 2235516x^{2} - 1528906x + 34180205 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{12} \)
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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{3} + 25 \beta_1 q^{5} + ( - \beta_{6} - 3 \beta_{2} - 40) q^{7} + ( - \beta_{6} + \beta_{3} - 5 \beta_{2} - 149) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{3} + 25 \beta_1 q^{5} + ( - \beta_{6} - 3 \beta_{2} - 40) q^{7} + ( - \beta_{6} + \beta_{3} - 5 \beta_{2} - 149) q^{9} + ( - 3 \beta_{7} + 2 \beta_{5} - 92 \beta_1) q^{11} + (\beta_{7} - 5 \beta_{5} + 3 \beta_{4} + 220 \beta_1) q^{13} - 25 \beta_{2} q^{15} + ( - 5 \beta_{6} - 5 \beta_{3} - 3 \beta_{2} - 300) q^{17} + (\beta_{7} - 14 \beta_{5} - 30 \beta_{4} + 204 \beta_1) q^{19} + ( - 10 \beta_{7} + 20 \beta_{5} + 120 \beta_{4} - 1090 \beta_1) q^{21} + (11 \beta_{6} + 10 \beta_{3} - 117 \beta_{2} + 220) q^{23} - 625 q^{25} + (30 \beta_{5} + 58 \beta_{4} - 1980 \beta_1) q^{27} + (20 \beta_{7} - 10 \beta_{5} + 210 \beta_{4} - 1480 \beta_1) q^{29} + ( - 40 \beta_{3} - 150 \beta_{2} + 3880) q^{31} + (55 \beta_{6} + 35 \beta_{3} + 21 \beta_{2} + 470) q^{33} + (25 \beta_{7} + 75 \beta_{4} - 1000 \beta_1) q^{35} + (4 \beta_{7} - 40 \beta_{5} + 312 \beta_{4} - 1230 \beta_1) q^{37} + ( - 50 \beta_{6} + 20 \beta_{3} - 450 \beta_{2} + 560) q^{39} + ( - 47 \beta_{6} - 33 \beta_{3} + 165 \beta_{2} + 2698) q^{41} + (50 \beta_{7} - 70 \beta_{5} - 135 \beta_{4} + 7020 \beta_1) q^{43} + (25 \beta_{7} + 25 \beta_{5} + 125 \beta_{4} - 3725 \beta_1) q^{45} + (43 \beta_{6} + 80 \beta_{3} - 321 \beta_{2} + 5960) q^{47} + (35 \beta_{6} - 95 \beta_{3} + 195 \beta_{2} + 10353) q^{49} + ( - 102 \beta_{7} + 48 \beta_{5} + 330 \beta_{4} - 216 \beta_1) q^{51} + ( - 153 \beta_{7} + 45 \beta_{5} + 141 \beta_{4} - 10020 \beta_1) q^{53} + ( - 75 \beta_{6} - 50 \beta_{3} + 2300) q^{55} + ( - 155 \beta_{6} + 125 \beta_{3} - 1157 \beta_{2} - 13330) q^{57} + ( - 135 \beta_{7} + 30 \beta_{5} - 570 \beta_{4} + 13524 \beta_1) q^{59} + ( - 30 \beta_{7} - 50 \beta_{5} - 1290 \beta_{4} - 17350 \beta_1) q^{61} + (167 \beta_{6} - 110 \beta_{3} + 1551 \beta_{2} + 40300) q^{63} + (25 \beta_{6} + 125 \beta_{3} + 75 \beta_{2} - 5500) q^{65} + (110 \beta_{7} - 10 \beta_{5} - 723 \beta_{4} - 18300 \beta_1) q^{67} + (340 \beta_{7} + 10 \beta_{5} + 270 \beta_{4} - 47870 \beta_1) q^{69} + (180 \beta_{6} + 180 \beta_{3} + 1290 \beta_{2} + 30840) q^{71} + ( - 5 \beta_{6} - 205 \beta_{3} + 141 \beta_{2} + 5800) q^{73} + 625 \beta_{4} q^{75} + ( - 103 \beta_{7} + 155 \beta_{5} - 2709 \beta_{4} + \cdots - 49990 \beta_1) q^{77}+ \cdots + (245 \beta_{7} - 190 \beta_{5} - 1980 \beta_{4} - 22564 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 320 q^{7} - 1192 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 320 q^{7} - 1192 q^{9} - 2400 q^{17} + 1760 q^{23} - 5000 q^{25} + 31040 q^{31} + 3760 q^{33} + 4480 q^{39} + 21584 q^{41} + 47680 q^{47} + 82824 q^{49} + 18400 q^{55} - 106640 q^{57} + 322400 q^{63} - 44000 q^{65} + 246720 q^{71} + 46400 q^{73} + 325760 q^{79} - 82328 q^{81} + 636320 q^{87} - 78192 q^{89} - 40800 q^{95} + 24960 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 4x^{7} - 384x^{6} + 506x^{5} + 49869x^{4} + 29654x^{3} - 2235516x^{2} - 1528906x + 34180205 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 1351312 \nu^{7} - 13653287 \nu^{6} - 394224929 \nu^{5} + 2862879190 \nu^{4} + 38223976863 \nu^{3} - 163085600918 \nu^{2} + \cdots + 3699160158415 ) / 281446960630 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1351312 \nu^{7} + 13653287 \nu^{6} + 394224929 \nu^{5} - 2862879190 \nu^{4} - 38223976863 \nu^{3} + 163085600918 \nu^{2} + \cdots - 3980607119045 ) / 281446960630 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 46680604 \nu^{7} - 595364489 \nu^{6} - 11237259467 \nu^{5} + 129626329330 \nu^{4} + 680074862701 \nu^{3} + \cdots + 233594355624595 ) / 844340881890 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 324498 \nu^{7} - 4782019 \nu^{6} - 86408283 \nu^{5} + 1112583304 \nu^{4} + 8082523469 \nu^{3} - 76894837308 \nu^{2} + \cdots + 1741708282831 ) / 5117217466 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 8770602 \nu^{7} + 204376019 \nu^{6} + 1922738643 \nu^{5} - 61548799060 \nu^{4} - 143197714893 \nu^{3} + 5939121203576 \nu^{2} + \cdots - 148559382301075 ) / 76758261990 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 87017312 \nu^{7} - 1189115032 \nu^{6} - 22832683396 \nu^{5} + 269861003750 \nu^{4} + 1960333630208 \nu^{3} + \cdots + 269387125903700 ) / 422170440945 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 109345770 \nu^{7} - 502754801 \nu^{6} - 43714650957 \nu^{5} + 138978838510 \nu^{4} + 5044385010495 \nu^{3} + \cdots + 305925739736305 ) / 422170440945 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} + \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} - 2\beta_{4} - \beta_{3} + 7\beta_{2} + 2\beta _1 + 392 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -3\beta_{7} + 3\beta_{6} - 3\beta_{5} - 21\beta_{4} - 33\beta_{3} + 559\beta_{2} + 1178\beta _1 + 3154 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -3\beta_{7} + 196\beta_{6} - 33\beta_{5} - 561\beta_{4} - 226\beta_{3} + 2182\beta_{2} + 3156\beta _1 + 56086 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 985 \beta_{7} + 1635 \beta_{6} - 1135 \beta_{5} - 10945 \beta_{4} - 7515 \beta_{3} + 95369 \beta_{2} + 282394 \beta _1 + 915374 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 2460 \beta_{7} + 33721 \beta_{6} - 11355 \beta_{5} - 144457 \beta_{4} - 47416 \beta_{3} + 512212 \beta_{2} + 1380952 \beta _1 + 9668402 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 119749 \beta_{7} + 242214 \beta_{6} - 167944 \beta_{5} - 1811908 \beta_{4} - 750429 \beta_{3} + 8864437 \beta_{2} + 34334284 \beta _1 + 105600322 ) / 4 \) Copy content Toggle raw display

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
161.1
14.2856 0.500000i
−10.3099 + 0.500000i
−7.33903 + 0.500000i
5.36332 0.500000i
5.36332 + 0.500000i
−7.33903 0.500000i
−10.3099 0.500000i
14.2856 + 0.500000i
0 27.5713i 0 25.0000i 0 −220.359 0 −517.174 0
161.2 0 21.6198i 0 25.0000i 0 −169.466 0 −224.417 0
161.3 0 15.6781i 0 25.0000i 0 164.681 0 −2.80186 0
161.4 0 9.72664i 0 25.0000i 0 65.1432 0 148.393 0
161.5 0 9.72664i 0 25.0000i 0 65.1432 0 148.393 0
161.6 0 15.6781i 0 25.0000i 0 164.681 0 −2.80186 0
161.7 0 21.6198i 0 25.0000i 0 −169.466 0 −224.417 0
161.8 0 27.5713i 0 25.0000i 0 −220.359 0 −517.174 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.8
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 320.6.d.a 8
4.b odd 2 1 320.6.d.b yes 8
8.b even 2 1 inner 320.6.d.a 8
8.d odd 2 1 320.6.d.b yes 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.6.d.a 8 1.a even 1 1 trivial
320.6.d.a 8 8.b even 2 1 inner
320.6.d.b yes 8 4.b odd 2 1
320.6.d.b yes 8 8.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{6}^{\mathrm{new}}(320, [\chi])\):

\( T_{3}^{8} + 1568T_{3}^{6} + 796456T_{3}^{4} + 149500800T_{3}^{2} + 8262810000 \) Copy content Toggle raw display
\( T_{7}^{4} + 160T_{7}^{3} - 41520T_{7}^{2} - 4400400T_{7} + 400612500 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{8} \) Copy content Toggle raw display
$3$ \( T^{8} + 1568 T^{6} + \cdots + 8262810000 \) Copy content Toggle raw display
$5$ \( (T^{2} + 625)^{4} \) Copy content Toggle raw display
$7$ \( (T^{4} + 160 T^{3} - 41520 T^{2} + \cdots + 400612500)^{2} \) Copy content Toggle raw display
$11$ \( T^{8} + 826480 T^{6} + \cdots + 30\!\cdots\!96 \) Copy content Toggle raw display
$13$ \( T^{8} + 1797040 T^{6} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( (T^{4} + 1200 T^{3} + \cdots - 184493516400)^{2} \) Copy content Toggle raw display
$19$ \( T^{8} + 15180240 T^{6} + \cdots + 34\!\cdots\!56 \) Copy content Toggle raw display
$23$ \( (T^{4} - 880 T^{3} + \cdots - 2621307487500)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 106326400 T^{6} + \cdots + 16\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{4} - 15520 T^{3} + \cdots - 20\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{8} + 253620240 T^{6} + \cdots + 41\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{4} - 10792 T^{3} + \cdots - 27\!\cdots\!36)^{2} \) Copy content Toggle raw display
$43$ \( T^{8} + 633919200 T^{6} + \cdots + 17\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( (T^{4} - 23840 T^{3} + \cdots + 85\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( T^{8} + 2361270960 T^{6} + \cdots + 85\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{8} + 2677354704 T^{6} + \cdots + 22\!\cdots\!76 \) Copy content Toggle raw display
$61$ \( T^{8} + 4174546000 T^{6} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$67$ \( T^{8} + 3263149152 T^{6} + \cdots + 46\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{4} - 123360 T^{3} + \cdots - 46\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} - 23200 T^{3} + \cdots + 12\!\cdots\!00)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} - 162880 T^{3} + \cdots - 19\!\cdots\!00)^{2} \) Copy content Toggle raw display
$83$ \( T^{8} + 1663499008 T^{6} + \cdots + 11\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{4} + 39096 T^{3} + \cdots + 35\!\cdots\!04)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} - 12480 T^{3} + \cdots - 31\!\cdots\!00)^{2} \) Copy content Toggle raw display
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