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Properties

Label 644.2.d.a
Level $644$
Weight $2$
Character orbit 644.d
Analytic conductor $5.142$
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] = [644,2,Mod(321,644)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(644, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("644.321");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 644 = 2^{2} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 644.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: \(5.14236589017\)
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: \( x^{16} - 2x^{14} + 68x^{12} - 46x^{10} + 4950x^{8} - 2254x^{6} + 163268x^{4} - 235298x^{2} + 5764801 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
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.

\(f(q)\) \(=\) \( q - \beta_{7} q^{3} + \beta_{2} q^{5} + \beta_{8} q^{7} + ( - \beta_{4} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{7} q^{3} + \beta_{2} q^{5} + \beta_{8} q^{7} + ( - \beta_{4} - 1) q^{9} + ( - \beta_{15} + \beta_{14} + \cdots + \beta_1) q^{11}+ \cdots + ( - \beta_{15} + 3 \beta_{14} - \beta_{12}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 8 q^{9} - 16 q^{23} + 8 q^{25} - 4 q^{35} + 16 q^{39} + 4 q^{49} - 8 q^{71} - 28 q^{77} + 32 q^{81} + 16 q^{85} - 8 q^{93} + 56 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2x^{14} + 68x^{12} - 46x^{10} + 4950x^{8} - 2254x^{6} + 163268x^{4} - 235298x^{2} + 5764801 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{15} - 849 \nu^{13} - 4069 \nu^{11} - 62787 \nu^{9} - 201333 \nu^{7} - 3304819 \nu^{5} + \cdots - 107279081 \nu ) / 45177216 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 22 \nu^{14} - 495 \nu^{12} + 4384 \nu^{10} + 72405 \nu^{8} + 242430 \nu^{6} + 3396043 \nu^{4} + \cdots + 197532671 ) / 56471520 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 19 \nu^{14} + 60 \nu^{12} + 3497 \nu^{10} + 988 \nu^{8} + 135161 \nu^{6} + 331828 \nu^{4} + \cdots - 470596 ) / 7529536 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1087 \nu^{14} + 4575 \nu^{12} - 37901 \nu^{10} + 249285 \nu^{8} - 950805 \nu^{6} + \cdots + 846955151 ) / 225886080 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 19 \nu^{14} - 452 \nu^{12} - 2713 \nu^{10} - 27644 \nu^{8} - 117129 \nu^{6} - 1331036 \nu^{4} + \cdots - 35294700 ) / 3764768 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2889 \nu^{14} - 935 \nu^{12} - 89387 \nu^{10} + 312675 \nu^{8} - 4683075 \nu^{6} + \cdots + 1410964457 ) / 451772160 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{15} - 2\nu^{13} + 68\nu^{11} - 46\nu^{9} + 4950\nu^{7} - 2254\nu^{5} + 163268\nu^{3} - 235298\nu ) / 823543 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 769 \nu^{15} + 20697 \nu^{13} + 137485 \nu^{11} + 1705539 \nu^{9} + 3410709 \nu^{7} + \cdots + 3093815753 \nu ) / 632481024 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 841 \nu^{14} + 1699 \nu^{12} + 52827 \nu^{10} - 166527 \nu^{8} + 2170659 \nu^{6} + \cdots - 603892317 ) / 90354432 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1207 \nu^{15} - 11871 \nu^{13} + 2549 \nu^{11} - 619365 \nu^{9} + 774525 \nu^{7} + \cdots - 1242255791 \nu ) / 632481024 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 1559 \nu^{15} - 6535 \nu^{13} - 113117 \nu^{11} + 291675 \nu^{9} - 4951245 \nu^{7} + \cdots + 1086253217 \nu ) / 790601280 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 1153 \nu^{14} - 1277 \nu^{12} + 21123 \nu^{10} - 225567 \nu^{8} + 2728395 \nu^{6} + \cdots - 773306877 ) / 90354432 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 1251 \nu^{15} + 885 \nu^{13} - 79433 \nu^{11} + 295735 \nu^{9} - 4190065 \nu^{7} + \cdots + 738482773 \nu ) / 527067520 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 4507 \nu^{15} + 14555 \nu^{13} + 218521 \nu^{11} + 182865 \nu^{9} + 11979225 \nu^{7} + \cdots - 1142254141 \nu ) / 790601280 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{10} + \beta_{7} - \beta_{4} + \beta_{3} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{15} + 4\beta_{14} - \beta_{12} - \beta_{11} + \beta_{9} + 4\beta_{8} + 4\beta_{2} + \beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( -3\beta_{13} - \beta_{10} - 16\beta_{7} + 2\beta_{6} + 10\beta_{5} + 2\beta_{4} + 4\beta_{3} - 15 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -7\beta_{15} + 28\beta_{14} - 29\beta_{12} + 17\beta_{11} - \beta_{9} + 13\beta_{8} - 12\beta_{2} + 4\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 51\beta_{13} - 32\beta_{10} + 43\beta_{7} + 44\beta_{5} + 51\beta_{4} - 13\beta_{3} - 28 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -68\beta_{15} - 72\beta_{14} + 8\beta_{12} - 100\beta_{11} - 140\beta_{9} + 213\beta_{8} - 248\beta_{2} - 48\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 72\beta_{13} - 184\beta_{10} + 296\beta_{7} - 136\beta_{6} - 468\beta_{5} - 80\beta_{4} + 128\beta_{3} - 1511 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 484 \beta_{15} - 856 \beta_{14} + 1928 \beta_{12} + 12 \beta_{11} + 324 \beta_{9} - 512 \beta_{8} + \cdots - 2059 \beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( - 1600 \beta_{13} - 395 \beta_{10} - 1011 \beta_{7} + 16 \beta_{6} - 2292 \beta_{5} + 2443 \beta_{4} + \cdots + 348 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 2767 \beta_{15} - 8468 \beta_{14} - 2333 \beta_{12} + 6039 \beta_{11} + 6201 \beta_{9} + \cdots - 1651 \beta_1 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( 1897 \beta_{13} + 10251 \beta_{10} + 18064 \beta_{7} - 5422 \beta_{6} + 3774 \beta_{5} - 9086 \beta_{4} + \cdots + 49153 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( 12137 \beta_{15} - 24396 \beta_{14} - 10913 \beta_{12} - 62935 \beta_{11} - 25673 \beta_{9} + \cdots + 20916 \beta_1 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( - 25657 \beta_{13} + 31776 \beta_{10} - 176041 \beta_{7} - 13680 \beta_{6} - 53384 \beta_{5} + \cdots + 345224 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( 392248 \beta_{15} + 254096 \beta_{14} + 283808 \beta_{12} + 160616 \beta_{11} + 69368 \beta_{9} + \cdots + 378032 \beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/644\mathbb{Z}\right)^\times\).

\(n\) \(185\) \(281\) \(323\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
321.1
0.986926 + 2.45479i
−0.986926 − 2.45479i
2.33151 − 1.25062i
−2.33151 + 1.25062i
−2.38903 − 1.13690i
2.38903 + 1.13690i
1.54356 + 2.14882i
−1.54356 − 2.14882i
1.54356 − 2.14882i
−1.54356 + 2.14882i
−2.38903 + 1.13690i
2.38903 − 1.13690i
2.33151 + 1.25062i
−2.33151 − 1.25062i
0.986926 − 2.45479i
−0.986926 + 2.45479i
0 − 3.02295i 0 −1.77749 0 −0.986926 + 2.45479i 0 −6.13825 0
321.2 0 − 3.02295i 0 1.77749 0 0.986926 − 2.45479i 0 −6.13825 0
321.3 0 − 1.90081i 0 −0.529536 0 −2.33151 − 1.25062i 0 −0.613080 0
321.4 0 − 1.90081i 0 0.529536 0 2.33151 + 1.25062i 0 −0.613080 0
321.5 0 − 1.09458i 0 −3.38856 0 2.38903 − 1.13690i 0 1.80189 0
321.6 0 − 1.09458i 0 3.38856 0 −2.38903 + 1.13690i 0 1.80189 0
321.7 0 − 0.224852i 0 −2.66041 0 −1.54356 + 2.14882i 0 2.94944 0
321.8 0 − 0.224852i 0 2.66041 0 1.54356 − 2.14882i 0 2.94944 0
321.9 0 0.224852i 0 −2.66041 0 −1.54356 − 2.14882i 0 2.94944 0
321.10 0 0.224852i 0 2.66041 0 1.54356 + 2.14882i 0 2.94944 0
321.11 0 1.09458i 0 −3.38856 0 2.38903 + 1.13690i 0 1.80189 0
321.12 0 1.09458i 0 3.38856 0 −2.38903 − 1.13690i 0 1.80189 0
321.13 0 1.90081i 0 −0.529536 0 −2.33151 + 1.25062i 0 −0.613080 0
321.14 0 1.90081i 0 0.529536 0 2.33151 − 1.25062i 0 −0.613080 0
321.15 0 3.02295i 0 −1.77749 0 −0.986926 − 2.45479i 0 −6.13825 0
321.16 0 3.02295i 0 1.77749 0 0.986926 + 2.45479i 0 −6.13825 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 321.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
23.b odd 2 1 inner
161.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 644.2.d.a ✓ 16
3.b odd 2 1 5796.2.k.b 16
4.b odd 2 1 2576.2.f.g 16
7.b odd 2 1 inner 644.2.d.a ✓ 16
21.c even 2 1 5796.2.k.b 16
23.b odd 2 1 inner 644.2.d.a ✓ 16
28.d even 2 1 2576.2.f.g 16
69.c even 2 1 5796.2.k.b 16
92.b even 2 1 2576.2.f.g 16
161.c even 2 1 inner 644.2.d.a ✓ 16
483.c odd 2 1 5796.2.k.b 16
644.h odd 2 1 2576.2.f.g 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
644.2.d.a ✓ 16 1.a even 1 1 trivial
644.2.d.a ✓ 16 7.b odd 2 1 inner
644.2.d.a ✓ 16 23.b odd 2 1 inner
644.2.d.a ✓ 16 161.c even 2 1 inner
2576.2.f.g 16 4.b odd 2 1
2576.2.f.g 16 28.d even 2 1
2576.2.f.g 16 92.b even 2 1
2576.2.f.g 16 644.h odd 2 1
5796.2.k.b 16 3.b odd 2 1
5796.2.k.b 16 21.c even 2 1
5796.2.k.b 16 69.c even 2 1
5796.2.k.b 16 483.c odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(644, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} + 14 T^{6} + 49 T^{4} + \cdots + 2)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} - 22 T^{6} + \cdots + 72)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} - 2 T^{14} + \cdots + 5764801 \) Copy content Toggle raw display
$11$ \( (T^{8} + 54 T^{6} + \cdots + 144)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 40 T^{6} + \cdots + 2592)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} - 60 T^{6} + \cdots + 648)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} - 78 T^{6} + \cdots + 28800)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + 8 T^{7} + \cdots + 279841)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} - 93 T^{2} + \cdots + 1854)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + 114 T^{6} + \cdots + 450)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 84 T^{6} + \cdots + 9216)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + 264 T^{6} + \cdots + 12781568)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 176 T^{6} + \cdots + 36864)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + 258 T^{6} + \cdots + 55778)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 310 T^{6} + \cdots + 34105600)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + 154 T^{6} + \cdots + 1620000)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 102 T^{6} + \cdots + 307328)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + 412 T^{6} + \cdots + 36144144)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 2 T^{3} + \cdots + 1080)^{4} \) Copy content Toggle raw display
$73$ \( (T^{8} + 240 T^{6} + \cdots + 968832)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + 502 T^{6} + \cdots + 114318864)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} - 250 T^{6} + \cdots + 28800)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} - 482 T^{6} + \cdots + 165888)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} - 342 T^{6} + \cdots + 19208)^{2} \) Copy content Toggle raw display
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