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Properties

Label 1250.4.a.n
Level $1250$
Weight $4$
Character orbit 1250.a
Self dual yes
Analytic conductor $73.752$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1250,4,Mod(1,1250)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1250, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1250.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1250 = 2 \cdot 5^{4} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1250.a (trivial)

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: yes
Analytic conductor: \(73.7523875072\)
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: \( x^{16} - 8 x^{15} - 125 x^{14} + 990 x^{13} + 6166 x^{12} - 47880 x^{11} - 151199 x^{10} + \cdots - 45086320 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 5^{15} \)
Twist minimal: no (minimal twist has level 50)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-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.

\(f(q)\) \(=\) \( q + 2 q^{2} + ( - \beta_{3} + 1) q^{3} + 4 q^{4} + ( - 2 \beta_{3} + 2) q^{6} + (\beta_1 + 3) q^{7} + 8 q^{8} + (\beta_{5} - \beta_{3} - \beta_{2} + 14) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 q^{2} + ( - \beta_{3} + 1) q^{3} + 4 q^{4} + ( - 2 \beta_{3} + 2) q^{6} + (\beta_1 + 3) q^{7} + 8 q^{8} + (\beta_{5} - \beta_{3} - \beta_{2} + 14) q^{9} + (\beta_{11} - \beta_{3} + \beta_{2} + 4) q^{11} + ( - 4 \beta_{3} + 4) q^{12} + ( - \beta_{14} - \beta_{6} - \beta_{2} + 13) q^{13} + (2 \beta_1 + 6) q^{14} + 16 q^{16} + (\beta_{14} + \beta_{12} + \beta_{11} + \cdots + 13) q^{17}+ \cdots + ( - 13 \beta_{14} + 13 \beta_{13} + \cdots + 29) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 32 q^{2} + 12 q^{3} + 64 q^{4} + 24 q^{6} + 56 q^{7} + 128 q^{8} + 212 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 32 q^{2} + 12 q^{3} + 64 q^{4} + 24 q^{6} + 56 q^{7} + 128 q^{8} + 212 q^{9} + 72 q^{11} + 48 q^{12} + 202 q^{13} + 112 q^{14} + 256 q^{16} + 216 q^{17} + 424 q^{18} + 100 q^{19} + 192 q^{21} + 144 q^{22} + 292 q^{23} + 96 q^{24} + 404 q^{26} + 570 q^{27} + 224 q^{28} + 400 q^{29} + 102 q^{31} + 512 q^{32} + 664 q^{33} + 432 q^{34} + 848 q^{36} + 646 q^{37} + 200 q^{38} + 104 q^{39} + 532 q^{41} + 384 q^{42} + 902 q^{43} + 288 q^{44} + 584 q^{46} + 776 q^{47} + 192 q^{48} + 1038 q^{49} + 442 q^{51} + 808 q^{52} + 632 q^{53} + 1140 q^{54} + 448 q^{56} + 1400 q^{57} + 800 q^{58} + 1000 q^{59} + 662 q^{61} + 204 q^{62} + 932 q^{63} + 1024 q^{64} + 1328 q^{66} + 1326 q^{67} + 864 q^{68} + 1854 q^{69} + 1292 q^{71} + 1696 q^{72} + 2272 q^{73} + 1292 q^{74} + 400 q^{76} + 2582 q^{77} + 208 q^{78} + 320 q^{79} + 2956 q^{81} + 1064 q^{82} + 2842 q^{83} + 768 q^{84} + 1804 q^{86} + 2920 q^{87} + 576 q^{88} + 2780 q^{89} + 812 q^{91} + 1168 q^{92} + 2824 q^{93} + 1552 q^{94} + 384 q^{96} + 3796 q^{97} + 2076 q^{98} + 1054 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 8 x^{15} - 125 x^{14} + 990 x^{13} + 6166 x^{12} - 47880 x^{11} - 151199 x^{10} + \cdots - 45086320 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 10\!\cdots\!52 \nu^{15} + \cdots - 24\!\cdots\!80 ) / 56\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 17\!\cdots\!55 \nu^{15} + \cdots + 67\!\cdots\!12 ) / 57\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 11\!\cdots\!34 \nu^{15} + \cdots - 58\!\cdots\!40 ) / 27\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 49\!\cdots\!15 \nu^{15} + \cdots + 89\!\cdots\!40 ) / 11\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 24\!\cdots\!30 \nu^{15} + \cdots - 30\!\cdots\!80 ) / 27\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 16\!\cdots\!00 \nu^{15} + \cdots + 11\!\cdots\!40 ) / 11\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 18\!\cdots\!23 \nu^{15} + \cdots + 19\!\cdots\!20 ) / 11\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 20\!\cdots\!93 \nu^{15} + \cdots - 84\!\cdots\!40 ) / 11\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 34\!\cdots\!77 \nu^{15} + \cdots + 17\!\cdots\!80 ) / 11\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 35\!\cdots\!46 \nu^{15} + \cdots - 18\!\cdots\!40 ) / 56\!\cdots\!40 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 22\!\cdots\!63 \nu^{15} + \cdots - 13\!\cdots\!60 ) / 28\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 11\!\cdots\!69 \nu^{15} + \cdots + 63\!\cdots\!60 ) / 11\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 68\!\cdots\!14 \nu^{15} + \cdots + 43\!\cdots\!40 ) / 62\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 33\!\cdots\!28 \nu^{15} + \cdots - 17\!\cdots\!20 ) / 28\!\cdots\!20 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 96\!\cdots\!12 \nu^{15} + \cdots - 56\!\cdots\!20 ) / 56\!\cdots\!40 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2\beta_{15} - \beta_{14} + \beta_{13} - 2\beta_{10} + \beta_{8} + \beta_{7} - \beta_{4} + \beta_{3} - \beta_{2} + 14 ) / 25 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -\beta_{13} - 2\beta_{9} + 3\beta_{7} + 5\beta_{6} + 10\beta_{5} - 14\beta_{3} + 25\beta_{2} + 480 ) / 25 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 42 \beta_{15} - 41 \beta_{14} + 20 \beta_{13} - 10 \beta_{12} + 25 \beta_{11} - 62 \beta_{10} + \cdots + 853 ) / 25 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 6 \beta_{15} - 17 \beta_{14} - 4 \beta_{13} - 10 \beta_{12} - \beta_{11} - 9 \beta_{10} - 60 \beta_{9} + \cdots + 3307 ) / 5 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 1397 \beta_{15} - 2041 \beta_{14} + 514 \beta_{13} - 660 \beta_{12} + 1090 \beta_{11} - 2277 \beta_{10} + \cdots + 52964 ) / 25 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 3245 \beta_{15} - 7660 \beta_{14} - 204 \beta_{13} - 3025 \beta_{12} + 700 \beta_{11} - 4390 \beta_{10} + \cdots + 702760 ) / 25 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 61672 \beta_{15} - 106431 \beta_{14} + 14896 \beta_{13} - 31895 \beta_{12} + 41795 \beta_{11} + \cdots + 3221970 ) / 25 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 261440 \beta_{15} - 544420 \beta_{14} + 13787 \beta_{13} - 149080 \beta_{12} + 63280 \beta_{11} + \cdots + 33960410 ) / 25 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 3224772 \beta_{15} - 5822681 \beta_{14} + 507978 \beta_{13} - 1395110 \beta_{12} + 1567065 \beta_{11} + \cdots + 194254031 ) / 25 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 18584110 \beta_{15} - 35645675 \beta_{14} + 1635028 \beta_{13} - 6802820 \beta_{12} + 3629765 \beta_{11} + \cdots + 1787868565 ) / 25 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 184485067 \beta_{15} - 331185701 \beta_{14} + 22243900 \beta_{13} - 58400540 \beta_{12} + \cdots + 11688551212 ) / 25 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 1241341835 \beta_{15} - 2252024620 \beta_{14} + 130370544 \beta_{13} - 295653145 \beta_{12} + \cdots + 99687703960 ) / 25 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 11032868732 \beta_{15} - 19368771371 \beta_{14} + 1239554622 \beta_{13} - 2371370905 \beta_{12} + \cdots + 705067008718 ) / 25 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 16050094008 \beta_{15} - 27999480800 \beta_{14} + 1879634153 \beta_{13} - 2455860046 \beta_{12} + \cdots + 1154335918310 ) / 5 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 673699868032 \beta_{15} - 1154262608961 \beta_{14} + 78867026844 \beta_{13} - 92921614590 \beta_{12} + \cdots + 42716589320789 ) / 25 \) Copy content Toggle raw display

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
4.06650
−3.09423
6.57044
−5.86465
4.25130
0.828229
0.685901
−0.735805
1.18615
−2.44669
3.41748
−3.75075
−5.52002
−5.42784
5.97064
7.86333
2.00000 −9.35299 4.00000 0 −18.7060 −11.0064 8.00000 60.4783 0
1.2 2.00000 −7.50360 4.00000 0 −15.0072 17.5556 8.00000 29.3040 0
1.3 2.00000 −7.10598 4.00000 0 −14.2120 23.3487 8.00000 23.4949 0
1.4 2.00000 −6.27628 4.00000 0 −12.5526 9.80956 8.00000 12.3916 0
1.5 2.00000 −4.37967 4.00000 0 −8.75934 −31.2051 8.00000 −7.81848 0
1.6 2.00000 −3.19342 4.00000 0 −6.38684 2.29951 8.00000 −16.8021 0
1.7 2.00000 −0.313373 4.00000 0 −0.626745 −8.77601 8.00000 −26.9018 0
1.8 2.00000 −0.246957 4.00000 0 −0.493914 −19.9138 8.00000 −26.9390 0
1.9 2.00000 2.01244 4.00000 0 4.02487 29.3318 8.00000 −22.9501 0
1.10 2.00000 3.03585 4.00000 0 6.07171 −1.93842 8.00000 −17.7836 0
1.11 2.00000 4.88240 4.00000 0 9.76480 32.3828 8.00000 −3.16219 0
1.12 2.00000 5.02730 4.00000 0 10.0546 −11.4048 8.00000 −1.72622 0
1.13 2.00000 7.10721 4.00000 0 14.2144 25.2037 8.00000 23.5124 0
1.14 2.00000 8.70632 4.00000 0 17.4126 24.3678 8.00000 48.8001 0
1.15 2.00000 9.73880 4.00000 0 19.4776 −26.8849 8.00000 67.8443 0
1.16 2.00000 9.86194 4.00000 0 19.7239 2.82989 8.00000 70.2578 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(2\) \(-1\)
\(5\) \(-1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1250.4.a.n 16
5.b even 2 1 1250.4.a.m 16
25.d even 5 2 250.4.d.c 32
25.e even 10 2 250.4.d.d 32
25.f odd 20 2 50.4.e.a ✓ 32
25.f odd 20 2 250.4.e.b 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
50.4.e.a ✓ 32 25.f odd 20 2
250.4.d.c 32 25.d even 5 2
250.4.d.d 32 25.e even 10 2
250.4.e.b 32 25.f odd 20 2
1250.4.a.m 16 5.b even 2 1
1250.4.a.n 16 1.a even 1 1 trivial

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{16} - 12 T_{3}^{15} - 250 T_{3}^{14} + 3170 T_{3}^{13} + 23530 T_{3}^{12} - 325606 T_{3}^{11} + \cdots + 3019248976 \) acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1250))\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T - 2)^{16} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 3019248976 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots - 54\!\cdots\!84 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots - 70\!\cdots\!44 \) Copy content Toggle raw display
$13$ \( T^{16} + \cdots - 62\!\cdots\!19 \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 69\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots - 93\!\cdots\!00 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots - 10\!\cdots\!64 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots - 12\!\cdots\!75 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots - 10\!\cdots\!44 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 13\!\cdots\!41 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 99\!\cdots\!16 \) Copy content Toggle raw display
$43$ \( T^{16} + \cdots + 82\!\cdots\!76 \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 30\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots - 81\!\cdots\!04 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 74\!\cdots\!81 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 13\!\cdots\!36 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 73\!\cdots\!16 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 55\!\cdots\!96 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 59\!\cdots\!00 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 15\!\cdots\!56 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 27\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 86\!\cdots\!81 \) Copy content Toggle raw display
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