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Properties

Label 500.3.d.c
Level $500$
Weight $3$
Character orbit 500.d
Analytic conductor $13.624$
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] = [500,3,Mod(499,500)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(500, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("500.499");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 500 = 2^{2} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 500.d (of order \(2\), degree \(1\), not 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: \(13.6240132180\)
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: \( x^{16} + 12x^{14} + 83x^{12} + 444x^{10} + 2000x^{8} + 7104x^{6} + 21248x^{4} + 49152x^{2} + 65536 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{8} \)
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.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} + \beta_{4} q^{3} + (\beta_{3} - 2) q^{4} + ( - \beta_{8} - \beta_{3} - 1) q^{6} + ( - \beta_{12} + \beta_{9} + \cdots + \beta_1) q^{7}+ \cdots + ( - \beta_{13} - \beta_{10} + \cdots + \beta_{3}) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} + \beta_{4} q^{3} + (\beta_{3} - 2) q^{4} + ( - \beta_{8} - \beta_{3} - 1) q^{6} + ( - \beta_{12} + \beta_{9} + \cdots + \beta_1) q^{7}+ \cdots + ( - 5 \beta_{13} + 17 \beta_{10} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 24 q^{4} - 30 q^{6} + 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 24 q^{4} - 30 q^{6} + 16 q^{9} - 50 q^{14} - 44 q^{16} - 120 q^{21} + 70 q^{24} - 188 q^{26} - 48 q^{29} - 248 q^{34} - 34 q^{36} + 192 q^{41} + 180 q^{44} - 330 q^{46} - 144 q^{49} - 20 q^{54} + 50 q^{56} - 488 q^{61} - 144 q^{64} - 240 q^{66} - 40 q^{69} - 388 q^{74} + 760 q^{76} - 344 q^{81} + 840 q^{84} - 330 q^{86} + 152 q^{89} - 530 q^{94} + 250 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 12x^{14} + 83x^{12} + 444x^{10} + 2000x^{8} + 7104x^{6} + 21248x^{4} + 49152x^{2} + 65536 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{15} + 12\nu^{13} + 83\nu^{11} + 444\nu^{9} + 2000\nu^{7} + 7104\nu^{5} + 21248\nu^{3} + 49152\nu ) / 16384 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 7\nu^{14} + 36\nu^{12} + 133\nu^{10} + 660\nu^{8} + 1264\nu^{6} - 1728\nu^{4} + 37120\nu^{2} + 12288 ) / 24576 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{14} - 12\nu^{12} - 83\nu^{10} - 444\nu^{8} - 2000\nu^{6} - 7104\nu^{4} - 21248\nu^{2} - 40960 ) / 4096 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 13 \nu^{15} - 268 \nu^{13} - 1399 \nu^{11} - 7900 \nu^{9} - 31696 \nu^{7} - 106432 \nu^{5} + \cdots - 741376 \nu ) / 98304 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 13\nu^{14} + 108\nu^{12} + 631\nu^{10} + 3324\nu^{8} + 13264\nu^{6} + 40896\nu^{4} + 115456\nu^{2} + 184320 ) / 24576 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 19 \nu^{15} - 340 \nu^{13} - 1897 \nu^{11} - 10564 \nu^{9} - 43696 \nu^{7} - 149056 \nu^{5} + \cdots - 839680 \nu ) / 98304 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 15 \nu^{14} - 196 \nu^{12} - 1053 \nu^{10} - 5428 \nu^{8} - 23664 \nu^{6} - 68416 \nu^{4} + \cdots - 421888 ) / 24576 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 19 \nu^{14} + 340 \nu^{12} + 1897 \nu^{10} + 10564 \nu^{8} + 43696 \nu^{6} + 149056 \nu^{4} + \cdots + 962560 ) / 24576 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3\nu^{15} + 16\nu^{13} + 105\nu^{11} + 568\nu^{9} + 2016\nu^{7} + 6784\nu^{5} + 18432\nu^{3} + 13312\nu ) / 12288 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 25 \nu^{14} - 188 \nu^{12} - 1243 \nu^{10} - 5900 \nu^{8} - 22288 \nu^{6} - 68288 \nu^{4} + \cdots - 241664 ) / 24576 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 11 \nu^{15} - 116 \nu^{13} - 721 \nu^{11} - 3556 \nu^{9} - 14896 \nu^{7} - 46144 \nu^{5} + \cdots - 184320 \nu ) / 32768 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 45 \nu^{15} + 332 \nu^{13} + 2007 \nu^{11} + 9884 \nu^{9} + 36816 \nu^{7} + 107456 \nu^{5} + \cdots + 364544 \nu ) / 98304 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( -7\nu^{14} - 48\nu^{12} - 277\nu^{10} - 1272\nu^{8} - 5056\nu^{6} - 14976\nu^{4} - 40960\nu^{2} - 49152 ) / 3072 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( -5\nu^{15} - 46\nu^{13} - 263\nu^{11} - 1378\nu^{9} - 5624\nu^{7} - 17056\nu^{5} - 48512\nu^{3} - 85504\nu ) / 6144 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 125 \nu^{15} - 1036 \nu^{13} - 5831 \nu^{11} - 28252 \nu^{9} - 112592 \nu^{7} + \cdots - 1478656 \nu ) / 98304 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{6} - \beta_{4} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{5} - \beta_{3} + \beta_{2} - 3 ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2\beta_{12} + 2\beta_{11} - 2\beta_{9} - 3\beta_{6} + 3\beta_{4} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -2\beta_{13} + 2\beta_{10} + 2\beta_{8} + 4\beta_{7} - 3\beta_{5} - \beta_{3} - \beta_{2} - 9 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -4\beta_{15} + 6\beta_{14} - 10\beta_{12} - 2\beta_{11} + 12\beta_{9} - 3\beta_{6} - \beta_{4} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 4\beta_{13} + 4\beta_{10} - 12\beta_{8} - 20\beta_{7} + 23\beta_{5} - 5\beta_{3} - 11\beta_{2} + 13 ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 8\beta_{15} - 40\beta_{14} - 38\beta_{12} - 22\beta_{11} - 18\beta_{9} + 17\beta_{6} + 23\beta_{4} - 27\beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 38\beta_{13} - 22\beta_{10} + 10\beta_{8} + 36\beta_{7} + 177\beta_{5} + 75\beta_{3} + 11\beta_{2} + 35 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 76 \beta_{15} - 18 \beta_{14} + 110 \beta_{12} + 22 \beta_{11} + 140 \beta_{9} + 49 \beta_{6} + \cdots - 123 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( -28\beta_{13} - 316\beta_{10} - 12\beta_{8} - 100\beta_{7} - 829\beta_{5} - 89\beta_{3} - 71\beta_{2} + 561 ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 56 \beta_{15} + 616 \beta_{14} + 1122 \beta_{12} - 142 \beta_{11} - 874 \beta_{9} - 107 \beta_{6} + \cdots + 1465 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 594 \beta_{13} + 866 \beta_{10} + 642 \beta_{8} + 116 \beta_{7} - 1851 \beta_{5} + 535 \beta_{3} + \cdots - 5281 ) / 2 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 1188 \beta_{15} - 138 \beta_{14} - 1978 \beta_{12} + 1486 \beta_{11} - 884 \beta_{9} + \cdots - 5255 \beta_1 ) / 2 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 1212 \beta_{13} + 3396 \beta_{10} - 1356 \beta_{8} + 2508 \beta_{7} + 8991 \beta_{5} - 2173 \beta_{3} + \cdots + 21029 ) / 2 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 2424 \beta_{15} - 4104 \beta_{14} - 13686 \beta_{12} - 102 \beta_{11} + 14238 \beta_{9} + \cdots + 5341 \beta_1 ) / 2 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/500\mathbb{Z}\right)^\times\).

\(n\) \(251\) \(377\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
499.1
−1.65148 + 1.12811i
−1.65148 − 1.12811i
−1.32013 + 1.50242i
−1.32013 − 1.50242i
−0.575882 + 1.91530i
−0.575882 − 1.91530i
−0.445249 + 1.94981i
−0.445249 − 1.94981i
0.445249 + 1.94981i
0.445249 − 1.94981i
0.575882 + 1.91530i
0.575882 − 1.91530i
1.32013 + 1.50242i
1.32013 − 1.50242i
1.65148 + 1.12811i
1.65148 − 1.12811i
−1.65148 − 1.12811i 3.99031 1.45475 + 3.72608i 0 −6.58991 − 4.50149i 5.08174 1.80093 − 7.79466i 6.92259 0
499.2 −1.65148 + 1.12811i 3.99031 1.45475 − 3.72608i 0 −6.58991 + 4.50149i 5.08174 1.80093 + 7.79466i 6.92259 0
499.3 −1.32013 − 1.50242i 0.0620601 −0.514527 + 3.96677i 0 −0.0819273 − 0.0932403i 5.11803 6.63899 − 4.46361i −8.99615 0
499.4 −1.32013 + 1.50242i 0.0620601 −0.514527 − 3.96677i 0 −0.0819273 + 0.0932403i 5.11803 6.63899 + 4.46361i −8.99615 0
499.5 −0.575882 − 1.91530i −2.01926 −3.33672 + 2.20597i 0 1.16286 + 3.86748i 3.07482 6.14664 + 5.12043i −4.92259 0
499.6 −0.575882 + 1.91530i −2.01926 −3.33672 − 2.20597i 0 1.16286 − 3.86748i 3.07482 6.14664 − 5.12043i −4.92259 0
499.7 −0.445249 − 1.94981i 4.47171 −3.60351 + 1.73630i 0 −1.99102 − 8.71897i −9.92608 4.98991 + 6.25306i 10.9961 0
499.8 −0.445249 + 1.94981i 4.47171 −3.60351 − 1.73630i 0 −1.99102 + 8.71897i −9.92608 4.98991 − 6.25306i 10.9961 0
499.9 0.445249 − 1.94981i −4.47171 −3.60351 − 1.73630i 0 −1.99102 + 8.71897i 9.92608 −4.98991 + 6.25306i 10.9961 0
499.10 0.445249 + 1.94981i −4.47171 −3.60351 + 1.73630i 0 −1.99102 − 8.71897i 9.92608 −4.98991 − 6.25306i 10.9961 0
499.11 0.575882 − 1.91530i 2.01926 −3.33672 − 2.20597i 0 1.16286 − 3.86748i −3.07482 −6.14664 + 5.12043i −4.92259 0
499.12 0.575882 + 1.91530i 2.01926 −3.33672 + 2.20597i 0 1.16286 + 3.86748i −3.07482 −6.14664 − 5.12043i −4.92259 0
499.13 1.32013 − 1.50242i −0.0620601 −0.514527 − 3.96677i 0 −0.0819273 + 0.0932403i −5.11803 −6.63899 − 4.46361i −8.99615 0
499.14 1.32013 + 1.50242i −0.0620601 −0.514527 + 3.96677i 0 −0.0819273 − 0.0932403i −5.11803 −6.63899 + 4.46361i −8.99615 0
499.15 1.65148 − 1.12811i −3.99031 1.45475 − 3.72608i 0 −6.58991 + 4.50149i −5.08174 −1.80093 − 7.79466i 6.92259 0
499.16 1.65148 + 1.12811i −3.99031 1.45475 + 3.72608i 0 −6.58991 − 4.50149i −5.08174 −1.80093 + 7.79466i 6.92259 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 499.16
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
20.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 500.3.d.c 16
4.b odd 2 1 inner 500.3.d.c 16
5.b even 2 1 inner 500.3.d.c 16
5.c odd 4 2 500.3.b.b ✓ 16
20.d odd 2 1 inner 500.3.d.c 16
20.e even 4 2 500.3.b.b ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
500.3.b.b ✓ 16 5.c odd 4 2
500.3.b.b ✓ 16 20.e even 4 2
500.3.d.c 16 1.a even 1 1 trivial
500.3.d.c 16 4.b odd 2 1 inner
500.3.d.c 16 5.b even 2 1 inner
500.3.d.c 16 20.d odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 40T_{3}^{6} + 465T_{3}^{4} - 1300T_{3}^{2} + 5 \) acting on \(S_{3}^{\mathrm{new}}(500, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 12 T^{14} + \cdots + 65536 \) Copy content Toggle raw display
$3$ \( (T^{8} - 40 T^{6} + 465 T^{4} + \cdots + 5)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} - 160 T^{6} + \cdots + 630125)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 520 T^{6} + \cdots + 99278080)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + 904 T^{6} + \cdots + 275845376)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + 1204 T^{6} + \cdots + 275845376)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + 1980 T^{6} + \cdots + 690023680)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} - 2280 T^{6} + \cdots + 25205)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 12 T^{3} + \cdots + 341521)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + 3640 T^{6} + \cdots + 1379226880)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + 7444 T^{6} + \cdots + 187522509056)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} - 48 T^{3} + \cdots + 122641)^{4} \) Copy content Toggle raw display
$43$ \( (T^{8} - 5200 T^{6} + \cdots + 67285800125)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} - 6920 T^{6} + \cdots + 42228969005)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + 18384 T^{6} + \cdots + 10754195456)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 72821900135680)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 122 T^{3} + \cdots - 1984859)^{4} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 10\!\cdots\!80)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 1279949620480)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 227925793126656)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 7038931559680)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 46\!\cdots\!05)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 19 T - 2441)^{8} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 10\!\cdots\!56)^{2} \) Copy content Toggle raw display
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