Properties

Label 320.6.d.c
Level $320$
Weight $6$
Character orbit 320.d
Analytic conductor $51.323$
Analytic rank $0$
Dimension $12$
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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 343 x^{10} - 696 x^{9} + 44406 x^{8} + 179640 x^{7} - 2401691 x^{6} - 15554592 x^{5} + 26901210 x^{4} + 434775816 x^{3} + 1271335685 x^{2} + \cdots + 653157349 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{30} \)
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + 3 \beta_1) q^{3} - 25 \beta_1 q^{5} + (\beta_{11} - \beta_{3} + \beta_{2} - 22) q^{7} + ( - \beta_{11} - \beta_{8} - \beta_{6} + 12 \beta_{3} - 3 \beta_{2} - 36) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + 3 \beta_1) q^{3} - 25 \beta_1 q^{5} + (\beta_{11} - \beta_{3} + \beta_{2} - 22) q^{7} + ( - \beta_{11} - \beta_{8} - \beta_{6} + 12 \beta_{3} - 3 \beta_{2} - 36) q^{9} + ( - \beta_{10} + 2 \beta_{9} + 2 \beta_{7} - 2 \beta_{5} - 8 \beta_{4} - 63 \beta_1) q^{11} + ( - \beta_{10} - 3 \beta_{9} + \beta_{7} + 22 \beta_{5} - 4 \beta_{4} + 147 \beta_1) q^{13} + ( - 25 \beta_{3} + 75) q^{15} + ( - 5 \beta_{11} - 3 \beta_{8} + 5 \beta_{6} - 20 \beta_{3} + 2 \beta_{2} + \cdots + 201) q^{17}+ \cdots + ( - 119 \beta_{10} + 26 \beta_{9} + 156 \beta_{7} - 3442 \beta_{5} + \cdots + 14133 \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 268 q^{7} - 428 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 268 q^{7} - 428 q^{9} + 900 q^{15} + 2400 q^{17} - 8108 q^{23} - 7500 q^{25} - 7976 q^{31} - 20776 q^{33} - 40984 q^{39} - 56408 q^{41} - 21172 q^{47} - 5540 q^{49} - 18400 q^{55} + 39992 q^{57} - 179516 q^{63} + 44000 q^{65} - 367704 q^{71} + 58736 q^{73} - 26192 q^{79} + 411692 q^{81} - 183200 q^{87} + 87672 q^{89} - 121000 q^{95} - 172336 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 343 x^{10} - 696 x^{9} + 44406 x^{8} + 179640 x^{7} - 2401691 x^{6} - 15554592 x^{5} + 26901210 x^{4} + 434775816 x^{3} + 1271335685 x^{2} + \cdots + 653157349 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 11\!\cdots\!59 \nu^{11} + \cdots - 21\!\cdots\!01 ) / 19\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 36815915 \nu^{11} - 1614964016 \nu^{10} - 8316126004 \nu^{9} + 393167792559 \nu^{8} + 1368899772199 \nu^{7} + \cdots + 43\!\cdots\!49 ) / 484488895993706 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 35\!\cdots\!86 \nu^{11} + \cdots + 58\!\cdots\!42 ) / 36\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 10\!\cdots\!99 \nu^{11} + \cdots - 25\!\cdots\!63 ) / 36\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 74273403 \nu^{11} + 389497022 \nu^{10} + 22748159420 \nu^{9} - 69843390937 \nu^{8} - 2729723240483 \nu^{7} + \cdots - 14\!\cdots\!63 ) / 143817507231506 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 13\!\cdots\!27 \nu^{11} + \cdots - 98\!\cdots\!33 ) / 95\!\cdots\!78 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 54\!\cdots\!65 \nu^{11} + \cdots - 51\!\cdots\!31 ) / 36\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 34\!\cdots\!02 \nu^{11} + \cdots - 53\!\cdots\!30 ) / 18\!\cdots\!02 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 44\!\cdots\!17 \nu^{11} + \cdots - 28\!\cdots\!75 ) / 19\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 12\!\cdots\!93 \nu^{11} + \cdots - 16\!\cdots\!59 ) / 36\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 78\!\cdots\!91 \nu^{11} + \cdots + 71\!\cdots\!65 ) / 18\!\cdots\!02 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( -2\beta_{3} + \beta_{2} - 2\beta_1 ) / 4 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{6} + \beta_{5} + 2\beta_{4} - 6\beta_{3} + 3\beta_{2} + 229 ) / 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 3\beta_{9} - 2\beta_{8} + \beta_{6} + 9\beta_{5} + 18\beta_{4} - 338\beta_{3} + 149\beta_{2} - 689\beta _1 + 1393 ) / 8 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2 \beta_{11} - 2 \beta_{10} + \beta_{9} + 86 \beta_{6} + 150 \beta_{5} + 340 \beta_{4} - 856 \beta_{3} + 500 \beta_{2} - 1393 \beta _1 + 19254 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 92 \beta_{11} + 435 \beta_{9} - 236 \beta_{8} + 10 \beta_{7} + 423 \beta_{6} + 2515 \beta_{5} + 4310 \beta_{4} - 30494 \beta_{3} + 13393 \beta_{2} - 97419 \beta _1 + 197149 ) / 8 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 337 \beta_{11} - 359 \beta_{10} + 637 \beta_{9} - 348 \beta_{8} - 138 \beta_{7} + 7422 \beta_{6} + 20465 \beta_{5} + 46592 \beta_{4} - 100358 \beta_{3} + 58831 \beta_{2} - 299206 \beta _1 + 1730389 ) / 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 6986 \beta_{11} - 1218 \beta_{10} + 26740 \beta_{9} - 12629 \beta_{8} + 1197 \beta_{7} + 32355 \beta_{6} + 210315 \beta_{5} + 358806 \beta_{4} - 1416094 \beta_{3} + 649064 \beta_{2} + \cdots + 11481466 ) / 4 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 25936 \beta_{11} - 52196 \beta_{10} + 132395 \beta_{9} - 69564 \beta_{8} - 28588 \beta_{7} + 661484 \beta_{6} + 2692110 \beta_{5} + 5882600 \beta_{4} - 10862912 \beta_{3} + \cdots + 160121062 ) / 4 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 1550544 \beta_{11} - 640692 \beta_{10} + 6276069 \beta_{9} - 2721428 \beta_{8} + 247830 \beta_{7} + 7884109 \beta_{6} + 59538957 \beta_{5} + 102090018 \beta_{4} + \cdots + 2470443355 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 798437 \beta_{11} - 7197563 \beta_{10} + 20707699 \beta_{9} - 10001826 \beta_{8} - 4091736 \beta_{7} + 60493781 \beta_{6} + 343512064 \beta_{5} + 712125342 \beta_{4} + \cdots + 15038933230 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 156982508 \beta_{11} - 115919892 \beta_{10} + 723555305 \beta_{9} - 294490040 \beta_{8} + 11081026 \beta_{7} + 869353185 \beta_{6} + 7752531457 \beta_{5} + \cdots + 253907779519 ) / 8 \) 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
10.0457 + 0.500000i
−8.35048 0.500000i
10.2435 + 0.500000i
−1.02698 0.500000i
−7.49748 0.500000i
−3.41426 + 0.500000i
−3.41426 0.500000i
−7.49748 + 0.500000i
−1.02698 + 0.500000i
10.2435 0.500000i
−8.35048 + 0.500000i
10.0457 0.500000i
0 30.0196i 0 25.0000i 0 45.4594 0 −658.179 0
161.2 0 20.6292i 0 25.0000i 0 93.5254 0 −182.562 0
161.3 0 16.5588i 0 25.0000i 0 −55.4690 0 −31.1927 0
161.4 0 5.98217i 0 25.0000i 0 −86.0039 0 207.214 0
161.5 0 5.06676i 0 25.0000i 0 −250.333 0 217.328 0
161.6 0 3.09968i 0 25.0000i 0 118.821 0 233.392 0
161.7 0 3.09968i 0 25.0000i 0 118.821 0 233.392 0
161.8 0 5.06676i 0 25.0000i 0 −250.333 0 217.328 0
161.9 0 5.98217i 0 25.0000i 0 −86.0039 0 207.214 0
161.10 0 16.5588i 0 25.0000i 0 −55.4690 0 −31.1927 0
161.11 0 20.6292i 0 25.0000i 0 93.5254 0 −182.562 0
161.12 0 30.0196i 0 25.0000i 0 45.4594 0 −658.179 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 161.12
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.c 12
4.b odd 2 1 320.6.d.d yes 12
8.b even 2 1 inner 320.6.d.c 12
8.d odd 2 1 320.6.d.d yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
320.6.d.c 12 1.a even 1 1 trivial
320.6.d.c 12 8.b even 2 1 inner
320.6.d.d yes 12 4.b odd 2 1
320.6.d.d yes 12 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 \(S_{6}^{\mathrm{new}}(320, [\chi])\):

\( T_{3}^{12} + 1672 T_{3}^{10} + 862572 T_{3}^{8} + 160687168 T_{3}^{6} + 8614933552 T_{3}^{4} + 165296582784 T_{3}^{2} + 928201218624 \) Copy content Toggle raw display
\( T_{7}^{6} + 134T_{7}^{5} - 40058T_{7}^{4} - 1534896T_{7}^{3} + 328685460T_{7}^{2} + 4498079256T_{7} - 603298353096 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 1672 T^{10} + \cdots + 928201218624 \) Copy content Toggle raw display
$5$ \( (T^{2} + 625)^{6} \) Copy content Toggle raw display
$7$ \( (T^{6} + 134 T^{5} + \cdots - 603298353096)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + 930440 T^{10} + \cdots + 74\!\cdots\!24 \) Copy content Toggle raw display
$13$ \( T^{12} + 2201672 T^{10} + \cdots + 42\!\cdots\!84 \) Copy content Toggle raw display
$17$ \( (T^{6} - 1200 T^{5} + \cdots - 80\!\cdots\!52)^{2} \) Copy content Toggle raw display
$19$ \( T^{12} + 9819352 T^{10} + \cdots + 39\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( (T^{6} + 4054 T^{5} + \cdots - 54\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( T^{12} + 127367552 T^{10} + \cdots + 22\!\cdots\!00 \) Copy content Toggle raw display
$31$ \( (T^{6} + 3988 T^{5} + \cdots + 23\!\cdots\!56)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + 534056664 T^{10} + \cdots + 18\!\cdots\!96 \) Copy content Toggle raw display
$41$ \( (T^{6} + 28204 T^{5} + \cdots + 10\!\cdots\!24)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + 909883048 T^{10} + \cdots + 67\!\cdots\!24 \) Copy content Toggle raw display
$47$ \( (T^{6} + 10586 T^{5} + \cdots - 43\!\cdots\!92)^{2} \) Copy content Toggle raw display
$53$ \( T^{12} + 2546234376 T^{10} + \cdots + 56\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{12} + 5293822488 T^{10} + \cdots + 18\!\cdots\!56 \) Copy content Toggle raw display
$61$ \( T^{12} + 859187576 T^{10} + \cdots + 94\!\cdots\!24 \) Copy content Toggle raw display
$67$ \( T^{12} + 8268753448 T^{10} + \cdots + 70\!\cdots\!84 \) Copy content Toggle raw display
$71$ \( (T^{6} + 183852 T^{5} + \cdots - 49\!\cdots\!08)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} - 29368 T^{5} + \cdots - 70\!\cdots\!12)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 13096 T^{5} + \cdots + 30\!\cdots\!96)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + 31827334904 T^{10} + \cdots + 12\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{6} - 43836 T^{5} + \cdots + 10\!\cdots\!48)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + 86168 T^{5} + \cdots - 11\!\cdots\!32)^{2} \) Copy content Toggle raw display
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