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Properties

Label 640.4.c.a
Level $640$
Weight $4$
Character orbit 640.c
Analytic conductor $37.761$
Analytic rank $0$
Dimension $18$
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [640,4,Mod(129,640)]
 
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([0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("640.129");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 640 = 2^{7} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 640.c (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: \(37.7612224037\)
Analytic rank: \(0\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 6 x^{17} + 18 x^{16} + 54 x^{15} + 2104 x^{14} - 10372 x^{13} + 25818 x^{12} + 35384 x^{11} + \cdots + 44745800 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{54}\cdot 5 \)
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 \(1,\beta_1,\ldots,\beta_{17}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{3} + \beta_{8} q^{5} - \beta_{4} q^{7} + ( - \beta_{2} - 9) q^{9} + ( - \beta_{8} - \beta_{7} - 2) q^{11} + (\beta_{13} - 2 \beta_1) q^{13} - \beta_{3} q^{15} - \beta_{10} q^{17} + (\beta_{16} + \beta_{8} + \beta_{2} - 4) q^{19}+ \cdots + (12 \beta_{17} - 11 \beta_{16} + \cdots + 204) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 2 q^{5} - 162 q^{9} - 28 q^{11} + 4 q^{15} - 68 q^{19} - 136 q^{21} - 22 q^{25} - 340 q^{29} + 336 q^{31} - 236 q^{35} + 1000 q^{39} + 236 q^{41} + 90 q^{45} - 882 q^{49} - 48 q^{51} - 1532 q^{55}+ \cdots + 3628 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{18} - 6 x^{17} + 18 x^{16} + 54 x^{15} + 2104 x^{14} - 10372 x^{13} + 25818 x^{12} + 35384 x^{11} + \cdots + 44745800 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 82\!\cdots\!34 \nu^{17} + \cdots - 83\!\cdots\!00 ) / 93\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 60\!\cdots\!53 \nu^{17} + \cdots - 16\!\cdots\!00 ) / 46\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 13\!\cdots\!97 \nu^{17} + \cdots - 19\!\cdots\!00 ) / 11\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 10\!\cdots\!26 \nu^{17} + \cdots + 28\!\cdots\!00 ) / 51\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 28\!\cdots\!41 \nu^{17} + \cdots + 19\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 17\!\cdots\!82 \nu^{17} + \cdots - 21\!\cdots\!00 ) / 55\!\cdots\!75 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 37\!\cdots\!71 \nu^{17} + \cdots + 21\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 42\!\cdots\!21 \nu^{17} + \cdots + 22\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 34\!\cdots\!21 \nu^{17} + \cdots + 56\!\cdots\!00 ) / 54\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 26\!\cdots\!86 \nu^{17} + \cdots + 53\!\cdots\!00 ) / 36\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 80\!\cdots\!19 \nu^{17} + \cdots - 28\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 10\!\cdots\!79 \nu^{17} + \cdots + 70\!\cdots\!20 ) / 86\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 13\!\cdots\!77 \nu^{17} + \cdots + 16\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 11\!\cdots\!56 \nu^{17} + \cdots - 14\!\cdots\!00 ) / 54\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 23\!\cdots\!23 \nu^{17} + \cdots - 20\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 35\!\cdots\!11 \nu^{17} + \cdots - 60\!\cdots\!00 ) / 10\!\cdots\!00 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 29\!\cdots\!39 \nu^{17} + \cdots - 32\!\cdots\!00 ) / 54\!\cdots\!00 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{8} + \beta_{5} + \beta_{2} + 16\beta _1 + 11 ) / 32 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{15} - \beta_{14} - 2 \beta_{11} - 2 \beta_{10} + 2 \beta_{9} - 2 \beta_{8} + \beta_{7} + \cdots + 10 \beta_1 ) / 16 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 2 \beta_{16} - 5 \beta_{15} - 3 \beta_{14} - 8 \beta_{13} - 4 \beta_{11} - 4 \beta_{10} + 8 \beta_{9} + \cdots - 246 ) / 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 8 \beta_{17} + 12 \beta_{16} - 5 \beta_{15} + 5 \beta_{14} + 2 \beta_{12} + 4 \beta_{9} - 22 \beta_{8} + \cdots - 2312 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 62 \beta_{17} - 26 \beta_{16} + 183 \beta_{15} + 321 \beta_{14} + 480 \beta_{13} + 84 \beta_{12} + \cdots - 16298 ) / 16 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 1124 \beta_{15} + 1124 \beta_{14} + 1424 \beta_{13} + 876 \beta_{12} + 1518 \beta_{11} + 1806 \beta_{10} + \cdots + 876 ) / 8 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 5492 \beta_{17} - 2060 \beta_{16} + 17118 \beta_{15} + 9538 \beta_{14} + 24272 \beta_{13} + \cdots + 951259 ) / 16 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 14866 \beta_{17} - 18230 \beta_{16} + 7737 \beta_{15} - 7737 \beta_{14} - 7552 \beta_{12} + \cdots + 1899284 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 180540 \beta_{17} - 116338 \beta_{16} - 236029 \beta_{15} - 439659 \beta_{14} - 594752 \beta_{13} + \cdots + 25707103 ) / 8 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 1538741 \beta_{15} - 1538741 \beta_{14} - 2302420 \beta_{13} - 1073390 \beta_{12} - 1617443 \beta_{11} + \cdots - 1073390 ) / 4 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 10561438 \beta_{17} + 8182354 \beta_{16} - 22385531 \beta_{15} - 11521621 \beta_{14} + \cdots - 1372378156 ) / 8 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 18473300 \beta_{17} + 20085300 \beta_{16} - 8993054 \beta_{15} + 8993054 \beta_{14} + \cdots - 2063291616 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 582105976 \beta_{17} + 493058080 \beta_{16} + 563059896 \beta_{15} + 1136519752 \beta_{14} + \cdots - 70604794551 ) / 8 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 4072396237 \beta_{15} + 4072396237 \beta_{14} + 6393952496 \beta_{13} + 2513151592 \beta_{12} + \cdots + 2513151592 ) / 4 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 15509240116 \beta_{17} - 13802623738 \beta_{16} + 28806584327 \beta_{15} + 13841587969 \beta_{14} + \cdots + 1837427259512 ) / 4 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( 45054755800 \beta_{17} - 45943889832 \beta_{16} + 21465239701 \beta_{15} - 21465239701 \beta_{14} + \cdots + 4957460113458 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( 809973824974 \beta_{17} - 742329462014 \beta_{16} - 685075177395 \beta_{15} - 1458692844885 \beta_{14} + \cdots + 92902775935068 ) / 4 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
129.1
5.02516 − 5.02516i
−4.06961 − 4.06961i
3.60477 − 3.60477i
−3.29321 − 3.29321i
−2.73089 − 2.73089i
2.50594 − 2.50594i
1.08207 − 1.08207i
0.603102 − 0.603102i
0.272660 − 0.272660i
0.272660 + 0.272660i
0.603102 + 0.603102i
1.08207 + 1.08207i
2.50594 + 2.50594i
−2.73089 + 2.73089i
−3.29321 + 3.29321i
3.60477 + 3.60477i
−4.06961 + 4.06961i
5.02516 + 5.02516i
0 − 10.0503i 0 7.89047 + 7.92089i 0 − 1.07469i 0 −74.0089 0
129.2 0 − 8.13921i 0 −8.94136 − 6.71209i 0 − 29.4240i 0 −39.2468 0
129.3 0 − 7.20955i 0 −9.78943 − 5.40066i 0 25.2040i 0 −24.9776 0
129.4 0 − 6.58643i 0 −3.76329 + 10.5279i 0 12.8578i 0 −16.3810 0
129.5 0 − 5.46177i 0 10.7429 − 3.09696i 0 3.45757i 0 −2.83093 0
129.6 0 − 5.01189i 0 2.25874 − 10.9498i 0 − 24.9246i 0 1.88099 0
129.7 0 − 2.16413i 0 1.75536 + 11.0417i 0 − 5.03970i 0 22.3165 0
129.8 0 − 1.20620i 0 8.92439 − 6.73463i 0 32.5795i 0 25.5451 0
129.9 0 − 0.545319i 0 −10.0777 + 4.84141i 0 − 11.8531i 0 26.7026 0
129.10 0 0.545319i 0 −10.0777 − 4.84141i 0 11.8531i 0 26.7026 0
129.11 0 1.20620i 0 8.92439 + 6.73463i 0 − 32.5795i 0 25.5451 0
129.12 0 2.16413i 0 1.75536 − 11.0417i 0 5.03970i 0 22.3165 0
129.13 0 5.01189i 0 2.25874 + 10.9498i 0 24.9246i 0 1.88099 0
129.14 0 5.46177i 0 10.7429 + 3.09696i 0 − 3.45757i 0 −2.83093 0
129.15 0 6.58643i 0 −3.76329 − 10.5279i 0 − 12.8578i 0 −16.3810 0
129.16 0 7.20955i 0 −9.78943 + 5.40066i 0 − 25.2040i 0 −24.9776 0
129.17 0 8.13921i 0 −8.94136 + 6.71209i 0 29.4240i 0 −39.2468 0
129.18 0 10.0503i 0 7.89047 − 7.92089i 0 1.07469i 0 −74.0089 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 129.18
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 640.4.c.a ✓ 18
4.b odd 2 1 640.4.c.b yes 18
5.b even 2 1 inner 640.4.c.a ✓ 18
8.b even 2 1 640.4.c.d yes 18
8.d odd 2 1 640.4.c.c yes 18
20.d odd 2 1 640.4.c.b yes 18
40.e odd 2 1 640.4.c.c yes 18
40.f even 2 1 640.4.c.d yes 18
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
640.4.c.a ✓ 18 1.a even 1 1 trivial
640.4.c.a ✓ 18 5.b even 2 1 inner
640.4.c.b yes 18 4.b odd 2 1
640.4.c.b yes 18 20.d odd 2 1
640.4.c.c yes 18 8.d odd 2 1
640.4.c.c yes 18 40.e odd 2 1
640.4.c.d yes 18 8.b even 2 1
640.4.c.d yes 18 40.f even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(640, [\chi])\):

\( T_{11}^{9} + 14 T_{11}^{8} - 6352 T_{11}^{7} - 114752 T_{11}^{6} + 11124448 T_{11}^{5} + \cdots + 1653672113664 \) Copy content Toggle raw display
\( T_{29}^{9} + 170 T_{29}^{8} - 111984 T_{29}^{7} - 19236832 T_{29}^{6} + 3430140896 T_{29}^{5} + \cdots + 83\!\cdots\!20 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{18} \) Copy content Toggle raw display
$3$ \( T^{18} + \cdots + 22909849600 \) Copy content Toggle raw display
$5$ \( T^{18} + \cdots + 74\!\cdots\!25 \) Copy content Toggle raw display
$7$ \( T^{18} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$11$ \( (T^{9} + \cdots + 1653672113664)^{2} \) Copy content Toggle raw display
$13$ \( T^{18} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$17$ \( T^{18} + \cdots + 71\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{9} + \cdots + 60\!\cdots\!80)^{2} \) Copy content Toggle raw display
$23$ \( T^{18} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
$29$ \( (T^{9} + \cdots + 83\!\cdots\!20)^{2} \) Copy content Toggle raw display
$31$ \( (T^{9} + \cdots + 76\!\cdots\!00)^{2} \) Copy content Toggle raw display
$37$ \( T^{18} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{9} + \cdots - 40\!\cdots\!28)^{2} \) Copy content Toggle raw display
$43$ \( T^{18} + \cdots + 23\!\cdots\!00 \) Copy content Toggle raw display
$47$ \( T^{18} + \cdots + 88\!\cdots\!00 \) Copy content Toggle raw display
$53$ \( T^{18} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{9} + \cdots - 16\!\cdots\!00)^{2} \) Copy content Toggle raw display
$61$ \( (T^{9} + \cdots - 52\!\cdots\!60)^{2} \) Copy content Toggle raw display
$67$ \( T^{18} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{9} + \cdots + 13\!\cdots\!60)^{2} \) Copy content Toggle raw display
$73$ \( T^{18} + \cdots + 40\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{9} + \cdots + 76\!\cdots\!80)^{2} \) Copy content Toggle raw display
$83$ \( T^{18} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{9} + \cdots + 16\!\cdots\!84)^{2} \) Copy content Toggle raw display
$97$ \( T^{18} + \cdots + 15\!\cdots\!64 \) Copy content Toggle raw display
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