↑

Properties

Label 250.2.e.c
Level $250$
Weight $2$
Character orbit 250.e
Analytic conductor $1.996$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [250,2,Mod(49,250)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(250, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([7]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("250.49");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 250 = 2 \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 250.e (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: \(1.99626005053\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: no (minimal twist has level 50)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{13} + \beta_{12} + \cdots + \beta_{9}) q^{2}+ \cdots + ( - \beta_{8} - \beta_{7} - 3 \beta_{6} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{13} + \beta_{12} + \cdots + \beta_{9}) q^{2}+ \cdots + ( - 2 \beta_{8} - \beta_{4} - \beta_{3} + \cdots + 6) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 4 q^{4} - 6 q^{6} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 4 q^{4} - 6 q^{6} + 2 q^{9} + 2 q^{11} + 2 q^{14} - 4 q^{16} - 40 q^{19} - 38 q^{21} - 4 q^{24} + 44 q^{26} + 30 q^{29} - 18 q^{31} + 2 q^{34} - 2 q^{36} + 24 q^{39} - 18 q^{41} - 2 q^{44} + 14 q^{46} + 8 q^{49} + 52 q^{51} + 50 q^{54} - 2 q^{56} - 20 q^{59} + 12 q^{61} + 4 q^{64} - 52 q^{66} - 86 q^{69} - 18 q^{71} - 48 q^{74} - 20 q^{76} + 20 q^{79} - 34 q^{81} - 52 q^{84} - 46 q^{86} + 30 q^{89} + 2 q^{91} + 2 q^{94} - 6 q^{96} + 84 q^{99}+O(q^{100}) \) 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 \(\left(\mathbb{Z}/250\mathbb{Z}\right)^\times\).

\(n\) \(127\)
\(\chi(n)\) \(-\beta_{3}\)

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
0.917186 + 1.66637i
−1.86824 − 0.357358i
1.86824 + 0.357358i
−0.917186 − 1.66637i
0.644389 + 0.983224i
−0.0566033 − 1.17421i
0.0566033 + 1.17421i
−0.644389 − 0.983224i
0.644389 − 0.983224i
−0.0566033 + 1.17421i
0.0566033 − 1.17421i
−0.644389 + 0.983224i
0.917186 − 1.66637i
−1.86824 + 0.357358i
1.86824 − 0.357358i
−0.917186 + 1.66637i
−0.587785 + 0.809017i −1.63079 + 0.529876i −0.309017 − 0.951057i 0 0.529876 − 1.63079i 2.77447i 0.951057 + 0.309017i −0.0483405 + 0.0351215i 0
49.2 −0.587785 + 0.809017i 2.21858 − 0.720859i −0.309017 − 0.951057i 0 −0.720859 + 2.21858i − 3.77447i 0.951057 + 0.309017i 1.97539 − 1.43521i 0
49.3 0.587785 − 0.809017i −2.21858 + 0.720859i −0.309017 − 0.951057i 0 −0.720859 + 2.21858i 3.77447i −0.951057 − 0.309017i 1.97539 − 1.43521i 0
49.4 0.587785 − 0.809017i 1.63079 − 0.529876i −0.309017 − 0.951057i 0 0.529876 − 1.63079i − 2.77447i −0.951057 − 0.309017i −0.0483405 + 0.0351215i 0
99.1 −0.951057 + 0.309017i −0.792578 − 1.09089i 0.809017 − 0.587785i 0 1.09089 + 0.792578i − 0.833366i −0.587785 + 0.809017i 0.365190 − 1.12394i 0
99.2 −0.951057 + 0.309017i 1.74363 + 2.39991i 0.809017 − 0.587785i 0 −2.39991 − 1.74363i 1.83337i −0.587785 + 0.809017i −1.79224 + 5.51595i 0
99.3 0.951057 − 0.309017i −1.74363 − 2.39991i 0.809017 − 0.587785i 0 −2.39991 − 1.74363i − 1.83337i 0.587785 − 0.809017i −1.79224 + 5.51595i 0
99.4 0.951057 − 0.309017i 0.792578 + 1.09089i 0.809017 − 0.587785i 0 1.09089 + 0.792578i 0.833366i 0.587785 − 0.809017i 0.365190 − 1.12394i 0
149.1 −0.951057 − 0.309017i −0.792578 + 1.09089i 0.809017 + 0.587785i 0 1.09089 − 0.792578i 0.833366i −0.587785 − 0.809017i 0.365190 + 1.12394i 0
149.2 −0.951057 − 0.309017i 1.74363 − 2.39991i 0.809017 + 0.587785i 0 −2.39991 + 1.74363i − 1.83337i −0.587785 − 0.809017i −1.79224 − 5.51595i 0
149.3 0.951057 + 0.309017i −1.74363 + 2.39991i 0.809017 + 0.587785i 0 −2.39991 + 1.74363i 1.83337i 0.587785 + 0.809017i −1.79224 − 5.51595i 0
149.4 0.951057 + 0.309017i 0.792578 − 1.09089i 0.809017 + 0.587785i 0 1.09089 − 0.792578i − 0.833366i 0.587785 + 0.809017i 0.365190 + 1.12394i 0
199.1 −0.587785 − 0.809017i −1.63079 − 0.529876i −0.309017 + 0.951057i 0 0.529876 + 1.63079i − 2.77447i 0.951057 − 0.309017i −0.0483405 − 0.0351215i 0
199.2 −0.587785 − 0.809017i 2.21858 + 0.720859i −0.309017 + 0.951057i 0 −0.720859 − 2.21858i 3.77447i 0.951057 − 0.309017i 1.97539 + 1.43521i 0
199.3 0.587785 + 0.809017i −2.21858 − 0.720859i −0.309017 + 0.951057i 0 −0.720859 − 2.21858i − 3.77447i −0.951057 + 0.309017i 1.97539 + 1.43521i 0
199.4 0.587785 + 0.809017i 1.63079 + 0.529876i −0.309017 + 0.951057i 0 0.529876 + 1.63079i 2.77447i −0.951057 + 0.309017i −0.0483405 − 0.0351215i 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 250.2.e.c 16
5.b even 2 1 inner 250.2.e.c 16
5.c odd 4 1 50.2.d.b ✓ 8
5.c odd 4 1 250.2.d.d 8
15.e even 4 1 450.2.h.e 8
20.e even 4 1 400.2.u.d 8
25.d even 5 1 inner 250.2.e.c 16
25.d even 5 1 1250.2.b.e 8
25.e even 10 1 inner 250.2.e.c 16
25.e even 10 1 1250.2.b.e 8
25.f odd 20 1 50.2.d.b ✓ 8
25.f odd 20 1 250.2.d.d 8
25.f odd 20 1 1250.2.a.f 4
25.f odd 20 1 1250.2.a.l 4
75.l even 20 1 450.2.h.e 8
100.l even 20 1 400.2.u.d 8
100.l even 20 1 10000.2.a.t 4
100.l even 20 1 10000.2.a.x 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.2.d.b ✓ 8 5.c odd 4 1
50.2.d.b ✓ 8 25.f odd 20 1
250.2.d.d 8 5.c odd 4 1
250.2.d.d 8 25.f odd 20 1
250.2.e.c 16 1.a even 1 1 trivial
250.2.e.c 16 5.b even 2 1 inner
250.2.e.c 16 25.d even 5 1 inner
250.2.e.c 16 25.e even 10 1 inner
400.2.u.d 8 20.e even 4 1
400.2.u.d 8 100.l even 20 1
450.2.h.e 8 15.e even 4 1
450.2.h.e 8 75.l even 20 1
1250.2.a.f 4 25.f odd 20 1
1250.2.a.l 4 25.f odd 20 1
1250.2.b.e 8 25.d even 5 1
1250.2.b.e 8 25.e even 10 1
10000.2.a.t 4 100.l even 20 1
10000.2.a.x 4 100.l even 20 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 7T_{3}^{14} + 78T_{3}^{12} - 764T_{3}^{10} + 4625T_{3}^{8} - 12224T_{3}^{6} + 19968T_{3}^{4} - 28672T_{3}^{2} + 65536 \) acting on \(S_{2}^{\mathrm{new}}(250, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{8} - T^{6} + T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{16} - 7 T^{14} + \cdots + 65536 \) 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
$11$ \( (T^{8} - T^{7} - 3 T^{6} + \cdots + 256)^{2} \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots + 1568239201 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 141158161 \) Copy content Toggle raw display
$19$ \( (T^{8} + 20 T^{7} + \cdots + 6400)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} - 117 T^{14} + \cdots + 65536 \) Copy content Toggle raw display
$29$ \( (T^{8} - 15 T^{7} + \cdots + 25)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + 9 T^{7} + \cdots + 1597696)^{2} \) Copy content Toggle raw display
$37$ \( T^{16} - 58 T^{14} + \cdots + 25411681 \) Copy content Toggle raw display
$41$ \( (T^{8} + 9 T^{7} + \cdots + 7921)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + 114 T^{6} + \cdots + 30976)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} - 33 T^{14} + \cdots + 65536 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 4294967296 \) Copy content Toggle raw display
$59$ \( (T^{8} + 10 T^{7} + \cdots + 102400)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} - 6 T^{7} + \cdots + 2920681)^{2} \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 794123370496 \) Copy content Toggle raw display
$71$ \( (T^{8} + 9 T^{7} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 1380756603136 \) Copy content Toggle raw display
$79$ \( (T^{8} - 10 T^{7} + \cdots + 102400)^{2} \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 180227832610816 \) Copy content Toggle raw display
$89$ \( (T^{8} - 15 T^{7} + \cdots + 9610000)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + 76 T^{6} + 15126 T^{4} + \cdots + 1)^{2} \) Copy content Toggle raw display
show more
show less