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Properties

Label 640.3.e.i
Level $640$
Weight $3$
Character orbit 640.e
Analytic conductor $17.439$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [640,3,Mod(319,640)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(640, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("640.319");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 640 = 2^{7} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 640.e (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: \(17.4387369191\)
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: \( x^{16} - 16x^{14} + 104x^{12} - 208x^{10} - 352x^{8} + 2312x^{6} + 2497x^{4} - 9072x^{2} + 5184 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{38} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{9} q^{3} + (\beta_{8} - \beta_{7} + \beta_{6}) q^{5} + \beta_{2} q^{7} + \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{9} q^{3} + (\beta_{8} - \beta_{7} + \beta_{6}) q^{5} + \beta_{2} q^{7} + \beta_{3} q^{9} - \beta_{10} q^{11} - \beta_{4} q^{13} + (\beta_{12} + \beta_{5}) q^{15} + (\beta_{13} + \beta_{11}) q^{17} - \beta_1 q^{19} + (2 \beta_{8} - 3 \beta_{6}) q^{21} + (3 \beta_{5} - 2 \beta_{2}) q^{23} + (\beta_{13} - 3 \beta_{3} + 12) q^{25} + ( - \beta_{14} + 7 \beta_{9}) q^{27} + ( - \beta_{8} + 6 \beta_{6}) q^{29} + \beta_{15} q^{31} + ( - \beta_{13} + \beta_{11}) q^{33} + ( - \beta_{9} - \beta_1) q^{35} + (\beta_{8} - 2 \beta_{7} + \beta_{6}) q^{37} + (\beta_{15} + 2 \beta_{12} + \cdots - \beta_{2}) q^{39}+ \cdots + ( - 4 \beta_{10} - \beta_1) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 192 q^{25} + 48 q^{41} - 448 q^{49} + 112 q^{65} - 1024 q^{81} + 576 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 16x^{14} + 104x^{12} - 208x^{10} - 352x^{8} + 2312x^{6} + 2497x^{4} - 9072x^{2} + 5184 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 107 \nu^{15} - 20616 \nu^{13} + 405096 \nu^{11} - 3942096 \nu^{9} + 18971856 \nu^{7} + \cdots + 70276680 \nu ) / 26605800 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 166169 \nu^{15} - 1635736 \nu^{13} - 3327256 \nu^{11} + 136167536 \nu^{9} + \cdots - 4014734328 \nu ) / 5108313600 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 64 \nu^{14} + 1124 \nu^{12} - 8128 \nu^{10} + 23738 \nu^{8} - 208 \nu^{6} - 201082 \nu^{4} + \cdots + 903879 ) / 255825 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 1823 \nu^{14} + 59944 \nu^{12} - 639704 \nu^{10} + 2768704 \nu^{8} - 750944 \nu^{6} + \cdots + 94261320 ) / 6139800 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 2594467 \nu^{15} - 42206984 \nu^{13} + 280245880 \nu^{11} - 606467120 \nu^{9} + \cdots - 34903083816 \nu ) / 5108313600 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 787403 \nu^{14} + 11791048 \nu^{12} - 68476856 \nu^{10} + 73856176 \nu^{8} + \cdots + 4210501608 ) / 425692800 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 6037667 \nu^{14} + 89980168 \nu^{12} - 528626552 \nu^{10} + 674731312 \nu^{8} + \cdots + 19602293544 ) / 2554156800 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 21737 \nu^{14} + 330520 \nu^{12} - 1997672 \nu^{10} + 2991760 \nu^{8} + 9181408 \nu^{6} + \cdots + 107743608 ) / 8513856 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 35629 \nu^{15} + 540104 \nu^{13} - 3257128 \nu^{11} + 4797488 \nu^{9} + 15627392 \nu^{7} + \cdots + 161468424 \nu ) / 24559200 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 42167 \nu^{15} + 731912 \nu^{13} - 5129704 \nu^{11} + 12247184 \nu^{9} + 15658256 \nu^{7} + \cdots + 640887192 \nu ) / 26605800 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 28668 \nu^{14} - 436304 \nu^{12} + 2667564 \nu^{10} - 4284464 \nu^{8} - 10698096 \nu^{6} + \cdots - 145167120 ) / 3325725 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 4062209 \nu^{15} + 61852504 \nu^{13} - 376881448 \nu^{11} + 586977808 \nu^{9} + \cdots + 16024732344 \nu ) / 851385600 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 148156 \nu^{14} + 2217968 \nu^{12} - 13185388 \nu^{10} + 17821688 \nu^{8} + 70179632 \nu^{6} + \cdots + 776858040 ) / 9977175 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 67447 \nu^{15} - 1010648 \nu^{13} + 5957112 \nu^{11} - 7593872 \nu^{9} - 33149568 \nu^{7} + \cdots - 341155224 \nu ) / 8186400 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 539347 \nu^{15} - 8084360 \nu^{13} + 47970808 \nu^{11} - 63783728 \nu^{9} - 258622112 \nu^{7} + \cdots - 2755349928 \nu ) / 16372800 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{14} - 2\beta_{12} - \beta_{10} - \beta_{9} - 9\beta_{5} + 7\beta_{2} + \beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{11} + \beta_{8} + 10\beta_{7} - 5\beta_{6} + \beta_{4} + 32 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{15} + 2\beta_{14} - 12\beta_{12} - 3\beta_{10} + 74\beta_{9} - 10\beta_{5} + 30\beta_{2} + 7\beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta_{13} + 17\beta_{11} + 56\beta_{8} + 16\beta_{7} - 40\beta_{6} + 4\beta_{4} - 24\beta_{3} + 96 ) / 16 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 6 \beta_{15} + 41 \beta_{14} - 122 \beta_{12} + 17 \beta_{10} + 521 \beta_{9} + 111 \beta_{5} + \cdots + 7 \beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 60\beta_{13} + 140\beta_{11} + 409\beta_{8} - 122\beta_{7} - 239\beta_{6} + 7\beta_{4} - 544 ) / 16 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 86 \beta_{15} + 449 \beta_{14} - 726 \beta_{12} + 189 \beta_{10} + 3137 \beta_{9} + 1073 \beta_{5} + \cdots - 197 \beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 405 \beta_{13} + 695 \beta_{11} + 2624 \beta_{8} - 2240 \beta_{7} - 928 \beta_{6} - 260 \beta_{4} + \cdots - 9216 ) / 16 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 513 \beta_{15} + 2414 \beta_{14} - 2780 \beta_{12} + 1049 \beta_{10} + 11702 \beta_{9} + \cdots - 3125 \beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 1344 \beta_{13} + 1574 \beta_{11} + 12121 \beta_{8} - 20630 \beta_{7} + 4315 \beta_{6} - 3551 \beta_{4} + \cdots - 82768 ) / 16 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 632 \beta_{15} + 1715 \beta_{14} + 3114 \beta_{12} + 4071 \beta_{10} - 16237 \beta_{9} + \cdots - 26663 \beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 7437 \beta_{13} - 17743 \beta_{11} + 21608 \beta_{8} - 137264 \beta_{7} + 108632 \beta_{6} + \cdots - 549132 ) / 16 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 21696 \beta_{15} - 112735 \beta_{14} + 183778 \beta_{12} + 11045 \beta_{10} - 792895 \beta_{9} + \cdots - 158357 \beta_1 ) / 32 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 160704 \beta_{13} - 297394 \beta_{11} - 358793 \beta_{8} - 620378 \beta_{7} + 1116589 \beta_{6} + \cdots - 2461504 ) / 16 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 287263 \beta_{15} - 1432546 \beta_{14} + 2073228 \beta_{12} - 5421 \beta_{10} - 8866474 \beta_{9} + \cdots - 505559 \beta_1 ) / 32 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/640\mathbb{Z}\right)^\times\).

\(n\) \(257\) \(261\) \(511\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
319.1
−2.58263 − 0.771341i
2.58263 − 0.771341i
−0.332843 + 1.47845i
0.332843 + 1.47845i
1.94211 − 0.837921i
−1.94211 − 0.837921i
0.973371 + 0.130814i
−0.973371 + 0.130814i
−1.94211 + 0.837921i
1.94211 + 0.837921i
−0.973371 − 0.130814i
0.973371 − 0.130814i
2.58263 + 0.771341i
−2.58263 + 0.771341i
0.332843 − 1.47845i
−0.332843 − 1.47845i
0 − 3.62258i 0 −4.96837 − 0.561553i 0 −6.45101 0 −4.12311 0
319.2 0 − 3.62258i 0 −4.96837 + 0.561553i 0 6.45101 0 −4.12311 0
319.3 0 − 3.62258i 0 4.96837 − 0.561553i 0 −6.45101 0 −4.12311 0
319.4 0 − 3.62258i 0 4.96837 + 0.561553i 0 6.45101 0 −4.12311 0
319.5 0 − 2.20837i 0 −3.50932 − 3.56155i 0 −0.620058 0 4.12311 0
319.6 0 − 2.20837i 0 −3.50932 + 3.56155i 0 0.620058 0 4.12311 0
319.7 0 − 2.20837i 0 3.50932 − 3.56155i 0 −0.620058 0 4.12311 0
319.8 0 − 2.20837i 0 3.50932 + 3.56155i 0 0.620058 0 4.12311 0
319.9 0 2.20837i 0 −3.50932 − 3.56155i 0 0.620058 0 4.12311 0
319.10 0 2.20837i 0 −3.50932 + 3.56155i 0 −0.620058 0 4.12311 0
319.11 0 2.20837i 0 3.50932 − 3.56155i 0 0.620058 0 4.12311 0
319.12 0 2.20837i 0 3.50932 + 3.56155i 0 −0.620058 0 4.12311 0
319.13 0 3.62258i 0 −4.96837 − 0.561553i 0 6.45101 0 −4.12311 0
319.14 0 3.62258i 0 −4.96837 + 0.561553i 0 −6.45101 0 −4.12311 0
319.15 0 3.62258i 0 4.96837 − 0.561553i 0 6.45101 0 −4.12311 0
319.16 0 3.62258i 0 4.96837 + 0.561553i 0 −6.45101 0 −4.12311 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 319.16
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 640.3.e.i ✓ 16
4.b odd 2 1 inner 640.3.e.i ✓ 16
5.b even 2 1 inner 640.3.e.i ✓ 16
8.b even 2 1 inner 640.3.e.i ✓ 16
8.d odd 2 1 inner 640.3.e.i ✓ 16
16.e even 4 1 1280.3.h.h 8
16.e even 4 1 1280.3.h.l 8
16.f odd 4 1 1280.3.h.h 8
16.f odd 4 1 1280.3.h.l 8
20.d odd 2 1 inner 640.3.e.i ✓ 16
40.e odd 2 1 inner 640.3.e.i ✓ 16
40.f even 2 1 inner 640.3.e.i ✓ 16
80.k odd 4 1 1280.3.h.h 8
80.k odd 4 1 1280.3.h.l 8
80.q even 4 1 1280.3.h.h 8
80.q even 4 1 1280.3.h.l 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.3.e.i ✓ 16 1.a even 1 1 trivial
640.3.e.i ✓ 16 4.b odd 2 1 inner
640.3.e.i ✓ 16 5.b even 2 1 inner
640.3.e.i ✓ 16 8.b even 2 1 inner
640.3.e.i ✓ 16 8.d odd 2 1 inner
640.3.e.i ✓ 16 20.d odd 2 1 inner
640.3.e.i ✓ 16 40.e odd 2 1 inner
640.3.e.i ✓ 16 40.f even 2 1 inner
1280.3.h.h 8 16.e even 4 1
1280.3.h.h 8 16.f odd 4 1
1280.3.h.h 8 80.k odd 4 1
1280.3.h.h 8 80.q even 4 1
1280.3.h.l 8 16.e even 4 1
1280.3.h.l 8 16.f odd 4 1
1280.3.h.l 8 80.k odd 4 1
1280.3.h.l 8 80.q even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(640, [\chi])\):

\( T_{3}^{4} + 18T_{3}^{2} + 64 \) Copy content Toggle raw display
\( T_{7}^{4} - 42T_{7}^{2} + 16 \) Copy content Toggle raw display
\( T_{13}^{4} - 564T_{13}^{2} + 77824 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{4} + 18 T^{2} + 64)^{4} \) Copy content Toggle raw display
$5$ \( (T^{8} - 48 T^{6} + \cdots + 390625)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 42 T^{2} + 16)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} - 328 T^{2} + 4864)^{4} \) Copy content Toggle raw display
$13$ \( (T^{4} - 564 T^{2} + 77824)^{4} \) Copy content Toggle raw display
$17$ \( (T^{4} + 768 T^{2} + 77824)^{4} \) Copy content Toggle raw display
$19$ \( (T^{4} - 1032 T^{2} + 4864)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 426 T^{2} + 44944)^{4} \) Copy content Toggle raw display
$29$ \( (T^{4} + 1424 T^{2} + 262144)^{4} \) Copy content Toggle raw display
$31$ \( (T^{4} + 3456 T^{2} + 1245184)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} - 148 T^{2} + 4864)^{4} \) Copy content Toggle raw display
$41$ \( (T^{2} - 6 T - 2048)^{8} \) Copy content Toggle raw display
$43$ \( (T^{4} + 6258 T^{2} + 9684544)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 3402 T^{2} + 1373584)^{4} \) Copy content Toggle raw display
$53$ \( (T^{4} - 1268 T^{2} + 19456)^{4} \) Copy content Toggle raw display
$59$ \( (T^{4} - 5576 T^{2} + 1405696)^{4} \) Copy content Toggle raw display
$61$ \( (T^{4} + 12276 T^{2} + 37650496)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 914 T^{2} + 141376)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 13824 T^{2} + 25214976)^{4} \) Copy content Toggle raw display
$73$ \( (T^{4} + 17856 T^{2} + 79691776)^{4} \) Copy content Toggle raw display
$79$ \( (T^{4} + 9088 T^{2} + 4980736)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} + 12242 T^{2} + 8248384)^{4} \) Copy content Toggle raw display
$89$ \( (T^{2} - 72 T + 1228)^{8} \) Copy content Toggle raw display
$97$ \( (T^{4} + 33024 T^{2} + 270905344)^{4} \) Copy content Toggle raw display
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