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Properties

Label 1000.2.q.b
Level $1000$
Weight $2$
Character orbit 1000.q
Analytic conductor $7.985$
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] = [1000,2,Mod(49,1000)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1000, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1000.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1000 = 2^{3} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1000.q (of order \(10\), degree \(4\), 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: \(7.98504020213\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{10})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + x^{14} - 4x^{12} - 49x^{10} + 11x^{8} + 395x^{6} + 900x^{4} + 1125x^{2} + 625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: no (minimal twist has level 200)
Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

$q$-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.

\(f(q)\) \(=\) \( q + (\beta_{10} + \beta_{9}) q^{3} + (\beta_{14} + \beta_{11} + \cdots - \beta_{5}) q^{7}+ \cdots + ( - \beta_{6} + \beta_{3} - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{10} + \beta_{9}) q^{3} + (\beta_{14} + \beta_{11} + \cdots - \beta_{5}) q^{7}+ \cdots + ( - 2 \beta_{8} - 3 \beta_{4} + \cdots + 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{9} - 4 q^{11} - 6 q^{19} - 6 q^{21} - 12 q^{29} - 20 q^{31} + 2 q^{39} - 8 q^{41} + 8 q^{49} - 40 q^{51} - 58 q^{59} + 30 q^{61} + 24 q^{69} - 32 q^{71} - 46 q^{79} + 64 q^{81} + 14 q^{89} + 58 q^{91} + 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + x^{14} - 4x^{12} - 49x^{10} + 11x^{8} + 395x^{6} + 900x^{4} + 1125x^{2} + 625 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 948392 \nu^{14} - 2081693 \nu^{12} - 1547103 \nu^{10} - 35207443 \nu^{8} + 136185777 \nu^{6} + \cdots + 181387875 ) / 171780125 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 122987 \nu^{14} + 97172 \nu^{12} - 701513 \nu^{10} - 5799603 \nu^{8} + 3932417 \nu^{6} + \cdots + 52412250 ) / 15616375 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 1379032 \nu^{14} - 312743 \nu^{12} - 4990978 \nu^{10} - 61900293 \nu^{8} + 94590252 \nu^{6} + \cdots + 563878625 ) / 171780125 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 281280 \nu^{14} + 69219 \nu^{12} + 939009 \nu^{10} + 12381889 \nu^{8} - 18275316 \nu^{6} + \cdots - 167096475 ) / 34356025 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1157122 \nu^{15} + 1428203 \nu^{13} + 8678488 \nu^{11} + 43919978 \nu^{9} + \cdots + 1705295750 \nu ) / 858900625 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 441862 \nu^{14} + 108648 \nu^{12} + 1544473 \nu^{10} + 20140738 \nu^{8} - 29602957 \nu^{6} + \cdots - 215410900 ) / 34356025 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 2817591 \nu^{14} + 2379174 \nu^{12} + 8884104 \nu^{10} + 118381374 \nu^{8} - 262226761 \nu^{6} + \cdots - 860602375 ) / 171780125 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 626638 \nu^{14} + 144621 \nu^{12} + 2783831 \nu^{10} + 27217946 \nu^{8} - 41997039 \nu^{6} + \cdots - 222107450 ) / 34356025 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2293 \nu^{15} + 486 \nu^{13} - 12284 \nu^{11} - 101834 \nu^{9} + 117086 \nu^{7} + 906528 \nu^{5} + \cdots + 955700 \nu ) / 633875 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3183877 \nu^{15} - 4006568 \nu^{13} - 6543803 \nu^{11} - 134015793 \nu^{9} + \cdots + 1245786250 \nu ) / 858900625 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 8105189 \nu^{15} + 777904 \nu^{13} - 26062291 \nu^{11} - 376639746 \nu^{9} + \cdots + 5975622625 \nu ) / 858900625 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 8189122 \nu^{15} + 3158678 \nu^{13} + 32153713 \nu^{11} + 353467203 \nu^{9} + \cdots - 3331016750 \nu ) / 858900625 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 8211971 \nu^{15} - 5580699 \nu^{13} - 25580004 \nu^{11} - 355005349 \nu^{9} + \cdots + 3295922875 \nu ) / 858900625 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 2000435 \nu^{15} + 181037 \nu^{13} + 8319942 \nu^{11} + 89497307 \nu^{9} + \cdots - 651093200 \nu ) / 171780125 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 14757833 \nu^{15} + 273347 \nu^{13} + 57022912 \nu^{11} + 679004122 \nu^{9} + \cdots - 6958879375 \nu ) / 858900625 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{15} + 2\beta_{14} - \beta_{13} - \beta_{12} - \beta_{11} + 2\beta_{10} + 3\beta_{9} + 2\beta_{5} ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{8} + 2\beta_{7} + 3\beta_{6} + 2\beta_{4} + 11\beta_{3} - 4\beta_{2} - \beta _1 + 4 ) / 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{15} - 3\beta_{14} - 16\beta_{13} - 6\beta_{12} - \beta_{11} + 17\beta_{10} - 2\beta_{9} + 2\beta_{5} ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( -7\beta_{8} - 13\beta_{7} + 3\beta_{6} + 7\beta_{4} + 6\beta_{3} - 19\beta_{2} - 26\beta _1 + 14 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -16\beta_{15} + 22\beta_{14} - 6\beta_{13} - 51\beta_{12} - 6\beta_{11} - 43\beta_{10} - 67\beta_{9} - 8\beta_{5} ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 23\beta_{8} - 23\beta_{7} + 63\beta_{6} + 52\beta_{4} + 156\beta_{3} + 11\beta_{2} - 11\beta _1 + 144 ) / 5 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 31 \beta_{15} - 53 \beta_{14} + 9 \beta_{13} - 31 \beta_{12} - 106 \beta_{11} - 33 \beta_{10} + \cdots + 137 \beta_{5} ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -57\beta_{8} - 18\beta_{7} + 363\beta_{6} - 208\beta_{4} + 381\beta_{3} - 114\beta_{2} - 96\beta _1 + 39 ) / 5 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 209 \beta_{15} - 418 \beta_{14} - 271 \beta_{13} - 551 \beta_{12} - 76 \beta_{11} - 133 \beta_{10} + \cdots + 342 \beta_{5} ) / 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 208\beta_{8} - 688\beta_{7} + 208\beta_{6} - 493\beta_{4} - 149\beta_{3} - 344\beta_{2} - 896\beta _1 - 606 ) / 5 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 136 \beta_{15} + 77 \beta_{14} + 1584 \beta_{13} - 1661 \beta_{12} - 136 \beta_{11} + \cdots + 77 \beta_{5} ) / 5 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 2208 \beta_{8} - 683 \beta_{7} + 2888 \beta_{6} - 683 \beta_{4} + 3956 \beta_{3} + 2891 \beta_{2} + \cdots + 1784 ) / 5 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 1299 \beta_{15} - 5453 \beta_{14} + 5559 \beta_{13} + 2144 \beta_{12} - 3376 \beta_{11} + \cdots + 6752 \beta_{5} ) / 5 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 393 \beta_{8} + 1012 \beta_{7} + 10778 \beta_{6} - 18118 \beta_{4} - 619 \beta_{3} + 1631 \beta_{2} + \cdots - 18511 ) / 5 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 17084 \beta_{15} - 22378 \beta_{14} + 5294 \beta_{13} - 8776 \beta_{12} + 5294 \beta_{11} + \cdots + 8542 \beta_{5} ) / 5 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1000\mathbb{Z}\right)^\times\).

\(n\) \(377\) \(501\) \(751\)
\(\chi(n)\) \(1 + \beta_{3} + \beta_{4} + \beta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
−0.644389 − 0.983224i
0.0566033 + 1.17421i
−0.0566033 − 1.17421i
0.644389 + 0.983224i
0.0566033 − 1.17421i
−0.644389 + 0.983224i
0.644389 − 0.983224i
−0.0566033 + 1.17421i
0.917186 + 1.66637i
−1.86824 − 0.357358i
1.86824 + 0.357358i
−0.917186 − 1.66637i
−1.86824 + 0.357358i
0.917186 − 1.66637i
−0.917186 + 1.66637i
1.86824 − 0.357358i
0 −0.587785 + 0.190983i 0 0 0 − 0.833366i 0 −2.11803 + 1.53884i 0
49.2 0 −0.587785 + 0.190983i 0 0 0 1.83337i 0 −2.11803 + 1.53884i 0
49.3 0 0.587785 − 0.190983i 0 0 0 − 1.83337i 0 −2.11803 + 1.53884i 0
49.4 0 0.587785 − 0.190983i 0 0 0 0.833366i 0 −2.11803 + 1.53884i 0
449.1 0 −0.587785 − 0.190983i 0 0 0 − 1.83337i 0 −2.11803 − 1.53884i 0
449.2 0 −0.587785 − 0.190983i 0 0 0 0.833366i 0 −2.11803 − 1.53884i 0
449.3 0 0.587785 + 0.190983i 0 0 0 − 0.833366i 0 −2.11803 − 1.53884i 0
449.4 0 0.587785 + 0.190983i 0 0 0 1.83337i 0 −2.11803 − 1.53884i 0
649.1 0 −0.951057 + 1.30902i 0 0 0 − 2.77447i 0 0.118034 + 0.363271i 0
649.2 0 −0.951057 + 1.30902i 0 0 0 3.77447i 0 0.118034 + 0.363271i 0
649.3 0 0.951057 − 1.30902i 0 0 0 − 3.77447i 0 0.118034 + 0.363271i 0
649.4 0 0.951057 − 1.30902i 0 0 0 2.77447i 0 0.118034 + 0.363271i 0
849.1 0 −0.951057 − 1.30902i 0 0 0 − 3.77447i 0 0.118034 − 0.363271i 0
849.2 0 −0.951057 − 1.30902i 0 0 0 2.77447i 0 0.118034 − 0.363271i 0
849.3 0 0.951057 + 1.30902i 0 0 0 − 2.77447i 0 0.118034 − 0.363271i 0
849.4 0 0.951057 + 1.30902i 0 0 0 3.77447i 0 0.118034 − 0.363271i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 49.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
25.d even 5 1 inner
25.e even 10 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1000.2.q.b 16
5.b even 2 1 inner 1000.2.q.b 16
5.c odd 4 1 200.2.m.b ✓ 8
5.c odd 4 1 1000.2.m.b 8
20.e even 4 1 400.2.u.e 8
25.d even 5 1 inner 1000.2.q.b 16
25.e even 10 1 inner 1000.2.q.b 16
25.f odd 20 1 200.2.m.b ✓ 8
25.f odd 20 1 1000.2.m.b 8
25.f odd 20 1 5000.2.a.f 4
25.f odd 20 1 5000.2.a.i 4
100.l even 20 1 400.2.u.e 8
100.l even 20 1 10000.2.a.q 4
100.l even 20 1 10000.2.a.z 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
200.2.m.b ✓ 8 5.c odd 4 1
200.2.m.b ✓ 8 25.f odd 20 1
400.2.u.e 8 20.e even 4 1
400.2.u.e 8 100.l even 20 1
1000.2.m.b 8 5.c odd 4 1
1000.2.m.b 8 25.f odd 20 1
1000.2.q.b 16 1.a even 1 1 trivial
1000.2.q.b 16 5.b even 2 1 inner
1000.2.q.b 16 25.d even 5 1 inner
1000.2.q.b 16 25.e even 10 1 inner
5000.2.a.f 4 25.f odd 20 1
5000.2.a.i 4 25.f odd 20 1
10000.2.a.q 4 100.l even 20 1
10000.2.a.z 4 100.l even 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} + T_{3}^{6} + 6T_{3}^{4} - 4T_{3}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(1000, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} + T^{6} + 6 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} + 26 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 2 T^{7} + \cdots + 3481)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} - 92 T^{14} + \cdots + 38950081 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 577200625 \) Copy content Toggle raw display
$19$ \( (T^{8} + 3 T^{7} + \cdots + 2401)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} - 8 T^{14} + \cdots + 5764801 \) Copy content Toggle raw display
$29$ \( (T^{8} + 6 T^{7} + \cdots + 116281)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 10 T^{7} + \cdots + 15625)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 20544834434881 \) Copy content Toggle raw display
$41$ \( (T^{8} + 4 T^{7} + \cdots + 7921)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 394 T^{6} + \cdots + 25563136)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 9354951841 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 245635219456 \) Copy content Toggle raw display
$59$ \( (T^{8} + 29 T^{7} + \cdots + 1936)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 15 T^{7} + \cdots + 21025)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + 48 T^{14} + \cdots + 923521 \) Copy content Toggle raw display
$71$ \( (T^{8} + 16 T^{7} + \cdots + 15768841)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} - 83 T^{14} + \cdots + 3748096 \) Copy content Toggle raw display
$79$ \( (T^{8} + 23 T^{7} + \cdots + 9759376)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 68701569626881 \) Copy content Toggle raw display
$89$ \( (T^{8} - 7 T^{7} + \cdots + 80656)^{2} \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 18753758574721 \) Copy content Toggle raw display
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