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Properties

Label 625.2.d.n
Level $625$
Weight $2$
Character orbit 625.d
Analytic conductor $4.991$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [625,2,Mod(126,625)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(625, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("625.126");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 625 = 5^{4} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 625.d (of order \(5\), 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: \(4.99065012633\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 25x^{14} + 239x^{12} + 1165x^{10} + 3166x^{8} + 4820x^{6} + 3809x^{4} + 1205x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 5^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{12} - \beta_{8} + \cdots - \beta_{5}) q^{2}+ \cdots + (\beta_{15} - \beta_{13} + \beta_{12} + \cdots + 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{12} - \beta_{8} + \cdots - \beta_{5}) q^{2}+ \cdots + ( - 6 \beta_{14} - 2 \beta_{13} + \cdots - 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 5 q^{3} - 8 q^{4} - 3 q^{6} + 20 q^{7} - 10 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 5 q^{3} - 8 q^{4} - 3 q^{6} + 20 q^{7} - 10 q^{8} + 3 q^{9} + 2 q^{11} - 25 q^{12} - 5 q^{13} + 9 q^{14} - 14 q^{16} + 10 q^{17} - 10 q^{18} + 7 q^{21} + 40 q^{22} - 15 q^{23} + 10 q^{24} + 22 q^{26} - 20 q^{27} - 30 q^{28} - 10 q^{29} + 17 q^{31} + 60 q^{32} - 5 q^{33} - q^{34} - 4 q^{36} + 15 q^{37} + 15 q^{38} - 9 q^{39} + 12 q^{41} + 45 q^{42} + 49 q^{44} - 33 q^{46} - 25 q^{47} + 20 q^{48} - 8 q^{49} - 28 q^{51} - 20 q^{52} - 30 q^{54} - 35 q^{56} - 20 q^{57} - 5 q^{58} + 20 q^{59} - 23 q^{61} - 15 q^{62} - 10 q^{63} - 28 q^{64} - 26 q^{66} + 80 q^{68} + 6 q^{69} + 22 q^{71} - 5 q^{72} - 40 q^{73} - 36 q^{74} - 20 q^{76} + 40 q^{77} + 25 q^{78} + 75 q^{79} + 11 q^{81} - 90 q^{82} - 25 q^{83} - 31 q^{84} + 17 q^{86} + 20 q^{87} + 5 q^{89} + 22 q^{91} - 60 q^{92} - 80 q^{93} - 51 q^{94} - 28 q^{96} - 40 q^{97} - 15 q^{98} - 44 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 25x^{14} + 239x^{12} + 1165x^{10} + 3166x^{8} + 4820x^{6} + 3809x^{4} + 1205x^{2} + 1 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 29\nu^{14} + 619\nu^{12} + 4515\nu^{10} + 13949\nu^{8} + 15463\nu^{6} - 4230\nu^{4} - 12623\nu^{2} - 1474 ) / 801 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -19\nu^{14} - 470\nu^{12} - 4422\nu^{10} - 20887\nu^{8} - 52142\nu^{6} - 63150\nu^{4} - 25961\nu^{2} + 1601 ) / 801 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 29 \nu^{15} - 29 \nu^{14} + 886 \nu^{13} - 619 \nu^{12} + 10389 \nu^{11} - 4515 \nu^{10} + 59873 \nu^{9} + \cdots + 673 ) / 1602 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 22 \nu^{15} + \nu^{14} + 488 \nu^{13} + 6 \nu^{12} + 3907 \nu^{11} - 142 \nu^{10} + 15210 \nu^{9} + \cdots + 431 ) / 534 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 22 \nu^{15} + \nu^{14} - 488 \nu^{13} + 6 \nu^{12} - 3907 \nu^{11} - 142 \nu^{10} - 15210 \nu^{9} + \cdots + 431 ) / 534 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 32 \nu^{15} - 16 \nu^{14} + 1082 \nu^{13} - 541 \nu^{12} + 13968 \nu^{11} - 6717 \nu^{10} + \cdots - 1556 ) / 1602 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 32 \nu^{15} + 16 \nu^{14} + 1082 \nu^{13} + 541 \nu^{12} + 13968 \nu^{11} + 6717 \nu^{10} + \cdots + 1556 ) / 1602 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 23 \nu^{15} - 119 \nu^{14} - 316 \nu^{13} - 2583 \nu^{12} + 240 \nu^{11} - 19770 \nu^{10} + \cdots + 3891 ) / 1602 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 72 \nu^{15} + \nu^{14} + 1678 \nu^{13} + 6 \nu^{12} + 14429 \nu^{11} - 142 \nu^{10} + 60887 \nu^{9} + \cdots + 164 ) / 534 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 35 \nu^{15} - 250 \nu^{14} + 833 \nu^{13} - 5683 \nu^{12} + 7668 \nu^{11} - 46914 \nu^{10} + \cdots - 950 ) / 1602 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 52 \nu^{15} - 221 \nu^{14} - 1113 \nu^{13} - 5064 \nu^{12} - 8280 \nu^{11} - 42399 \nu^{10} + \cdots - 21 ) / 1602 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 135 \nu^{14} + 3124 \nu^{12} + 26487 \nu^{10} + 108723 \nu^{8} + 230851 \nu^{6} + 241140 \nu^{4} + \cdots - 2335 ) / 801 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 35 \nu^{15} + 250 \nu^{14} + 833 \nu^{13} + 5683 \nu^{12} + 7668 \nu^{11} + 46914 \nu^{10} + \cdots + 950 ) / 1602 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 14 \nu^{15} - 282 \nu^{14} - 351 \nu^{13} - 6320 \nu^{12} - 3441 \nu^{11} - 51003 \nu^{10} + \cdots + 1634 ) / 1602 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 187 \nu^{15} + 484 \nu^{14} - 4415 \nu^{13} + 10914 \nu^{12} - 38772 \nu^{11} + 88980 \nu^{10} + \cdots + 2391 ) / 1602 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 4 \beta_{15} - \beta_{14} + 3 \beta_{13} + \beta_{12} - 3 \beta_{11} - \beta_{10} - 4 \beta_{9} + \cdots + 3 \beta_1 ) / 5 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{14} + \beta_{13} + \beta_{11} - \beta_{10} + \beta_{4} - 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 26 \beta_{15} + \beta_{14} - 12 \beta_{13} - 4 \beta_{12} + 25 \beta_{11} + 14 \beta_{10} + 34 \beta_{9} + \cdots + 3 ) / 5 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( - 11 \beta_{14} - 9 \beta_{13} - 2 \beta_{12} - 11 \beta_{11} + 9 \beta_{10} - \beta_{7} + \beta_{6} + \cdots + 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 226 \beta_{15} + 6 \beta_{14} + 67 \beta_{13} + 34 \beta_{12} - 232 \beta_{11} - 159 \beta_{10} + \cdots - 55 ) / 5 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 109 \beta_{14} + 84 \beta_{13} + 21 \beta_{12} + 109 \beta_{11} - 84 \beta_{10} + 19 \beta_{7} + \cdots - 155 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2184 \beta_{15} - 111 \beta_{14} - 503 \beta_{13} - 346 \beta_{12} + 2295 \beta_{11} + 1681 \beta_{10} + \cdots + 677 ) / 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 1082 \beta_{14} - 820 \beta_{13} - 194 \beta_{12} - 1082 \beta_{11} + 820 \beta_{10} - 237 \beta_{7} + \cdots + 1509 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 21824 \beta_{15} + 1359 \beta_{14} + 4493 \beta_{13} + 3576 \beta_{12} - 23183 \beta_{11} + \cdots - 7385 ) / 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 10857 \beta_{14} + 8194 \beta_{13} + 1830 \beta_{12} + 10857 \beta_{11} - 8194 \beta_{10} + 2603 \beta_{7} + \cdots - 15185 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 220506 \beta_{15} - 14974 \beta_{14} - 43387 \beta_{13} - 36754 \beta_{12} + 235480 \beta_{11} + \cdots + 77143 ) / 5 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 109708 \beta_{14} - 82728 \beta_{13} - 17847 \beta_{12} - 109708 \beta_{11} + 82728 \beta_{10} + \cdots + 154077 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 2236846 \beta_{15} + 157931 \beta_{14} + 432317 \beta_{13} + 375854 \beta_{12} - 2394777 \beta_{11} + \cdots - 792740 ) / 5 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( 1112529 \beta_{14} + 838847 \beta_{13} + 177854 \beta_{12} + 1112529 \beta_{11} - 838847 \beta_{10} + \cdots - 1566411 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 22725724 \beta_{15} - 1632836 \beta_{14} - 4361108 \beta_{13} - 3832591 \beta_{12} + 24358560 \beta_{11} + \cdots + 8095157 ) / 5 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/625\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1 + \beta_{4} + \beta_{5} - \beta_{9}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
126.1
0.991969i
3.18910i
− 1.20005i
− 1.80544i
1.63097i
− 1.51514i
− 2.04679i
0.0288455i
− 1.63097i
1.51514i
2.04679i
− 0.0288455i
− 0.991969i
− 3.18910i
1.20005i
1.80544i
−0.520202 − 1.60102i −0.574677 + 0.417528i −0.674615 + 0.490137i 0 0.967418 + 0.702870i 4.59110 −1.58816 − 1.15387i −0.771126 + 2.37328i 0
126.2 0.154814 + 0.476469i −2.50250 + 1.81817i 1.41498 − 1.02804i 0 −1.25372 − 0.910884i 0.0237879 1.51951 + 1.10399i 2.02970 − 6.24676i 0
126.3 0.718805 + 2.21225i 1.86261 − 1.35327i −2.75935 + 2.00479i 0 4.33262 + 3.14783i 3.59425 −2.65482 − 1.92884i 0.710939 − 2.18805i 0
126.4 0.764617 + 2.35325i −1.71249 + 1.24419i −3.33511 + 2.42310i 0 −4.23730 − 3.07858i −0.973070 −4.24866 − 3.08683i 0.457541 − 1.40817i 0
251.1 −2.15604 + 1.56645i 0.234569 − 0.721930i 1.57669 − 4.85257i 0 0.625130 + 1.92395i −2.04213 2.55484 + 7.86300i 1.96089 + 1.42467i 0
251.2 −0.855434 + 0.621509i −0.212419 + 0.653760i −0.272540 + 0.838792i 0 −0.224607 − 0.691269i 1.01199 −0.941671 − 2.89816i 2.04477 + 1.48561i 0
251.3 0.264347 − 0.192059i −0.530081 + 1.63142i −0.585041 + 1.80057i 0 0.173205 + 0.533069i 3.42409 0.393106 + 1.20986i 0.0465016 + 0.0337854i 0
251.4 1.62909 − 1.18361i 0.934982 − 2.87758i 0.634989 − 1.95429i 0 −1.88274 − 5.79449i 0.369971 −0.0341417 − 0.105077i −4.97921 − 3.61761i 0
376.1 −2.15604 − 1.56645i 0.234569 + 0.721930i 1.57669 + 4.85257i 0 0.625130 − 1.92395i −2.04213 2.55484 − 7.86300i 1.96089 − 1.42467i 0
376.2 −0.855434 − 0.621509i −0.212419 − 0.653760i −0.272540 − 0.838792i 0 −0.224607 + 0.691269i 1.01199 −0.941671 + 2.89816i 2.04477 − 1.48561i 0
376.3 0.264347 + 0.192059i −0.530081 − 1.63142i −0.585041 − 1.80057i 0 0.173205 − 0.533069i 3.42409 0.393106 − 1.20986i 0.0465016 − 0.0337854i 0
376.4 1.62909 + 1.18361i 0.934982 + 2.87758i 0.634989 + 1.95429i 0 −1.88274 + 5.79449i 0.369971 −0.0341417 + 0.105077i −4.97921 + 3.61761i 0
501.1 −0.520202 + 1.60102i −0.574677 − 0.417528i −0.674615 − 0.490137i 0 0.967418 − 0.702870i 4.59110 −1.58816 + 1.15387i −0.771126 − 2.37328i 0
501.2 0.154814 − 0.476469i −2.50250 − 1.81817i 1.41498 + 1.02804i 0 −1.25372 + 0.910884i 0.0237879 1.51951 − 1.10399i 2.02970 + 6.24676i 0
501.3 0.718805 − 2.21225i 1.86261 + 1.35327i −2.75935 − 2.00479i 0 4.33262 − 3.14783i 3.59425 −2.65482 + 1.92884i 0.710939 + 2.18805i 0
501.4 0.764617 − 2.35325i −1.71249 − 1.24419i −3.33511 − 2.42310i 0 −4.23730 + 3.07858i −0.973070 −4.24866 + 3.08683i 0.457541 + 1.40817i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 126.4
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 625.2.d.n 16
5.b even 2 1 625.2.d.p 16
5.c odd 4 2 625.2.e.k 32
25.d even 5 1 625.2.a.g yes 8
25.d even 5 2 625.2.d.m 16
25.d even 5 1 inner 625.2.d.n 16
25.e even 10 1 625.2.a.e ✓ 8
25.e even 10 1 625.2.d.p 16
25.e even 10 2 625.2.d.q 16
25.f odd 20 2 625.2.b.d 16
25.f odd 20 4 625.2.e.j 32
25.f odd 20 2 625.2.e.k 32
75.h odd 10 1 5625.2.a.be 8
75.j odd 10 1 5625.2.a.s 8
100.h odd 10 1 10000.2.a.bn 8
100.j odd 10 1 10000.2.a.be 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
625.2.a.e ✓ 8 25.e even 10 1
625.2.a.g yes 8 25.d even 5 1
625.2.b.d 16 25.f odd 20 2
625.2.d.m 16 25.d even 5 2
625.2.d.n 16 1.a even 1 1 trivial
625.2.d.n 16 25.d even 5 1 inner
625.2.d.p 16 5.b even 2 1
625.2.d.p 16 25.e even 10 1
625.2.d.q 16 25.e even 10 2
625.2.e.j 32 25.f odd 20 4
625.2.e.k 32 5.c odd 4 2
625.2.e.k 32 25.f odd 20 2
5625.2.a.s 8 75.j odd 10 1
5625.2.a.be 8 75.h odd 10 1
10000.2.a.be 8 100.j odd 10 1
10000.2.a.bn 8 100.h odd 10 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(625, [\chi])\):

\( T_{2}^{16} + 8 T_{2}^{14} + 10 T_{2}^{13} + 48 T_{2}^{12} - 20 T_{2}^{11} + 336 T_{2}^{10} - 185 T_{2}^{9} + \cdots + 81 \) Copy content Toggle raw display
\( T_{3}^{16} + 5 T_{3}^{15} + 17 T_{3}^{14} + 50 T_{3}^{13} + 168 T_{3}^{12} + 295 T_{3}^{11} + 529 T_{3}^{10} + \cdots + 841 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} + 8 T^{14} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{16} + 5 T^{15} + \cdots + 841 \) Copy content Toggle raw display
$5$ \( T^{16} \) Copy content Toggle raw display
$7$ \( (T^{8} - 10 T^{7} + 24 T^{6} + \cdots + 1)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} - 2 T^{15} + \cdots + 5861241 \) Copy content Toggle raw display
$13$ \( T^{16} + 5 T^{15} + \cdots + 130321 \) Copy content Toggle raw display
$17$ \( T^{16} - 10 T^{15} + \cdots + 2595321 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 110775625 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 2124195921 \) Copy content Toggle raw display
$29$ \( T^{16} + \cdots + 3717950625 \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 576048001 \) Copy content Toggle raw display
$37$ \( T^{16} - 15 T^{15} + \cdots + 6561 \) Copy content Toggle raw display
$41$ \( T^{16} + \cdots + 237782041641 \) Copy content Toggle raw display
$43$ \( (T^{8} - 99 T^{6} + \cdots - 1949)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 3244555521 \) Copy content Toggle raw display
$53$ \( T^{16} + 17 T^{14} + \cdots + 3606201 \) Copy content Toggle raw display
$59$ \( T^{16} - 20 T^{15} + \cdots + 50625 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 10718253841 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 3049560197401 \) Copy content Toggle raw display
$71$ \( T^{16} + \cdots + 280529001 \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 56212142281 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 62262725625 \) Copy content Toggle raw display
$83$ \( T^{16} + \cdots + 150377514648201 \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 3419818025625 \) Copy content Toggle raw display
$97$ \( T^{16} + \cdots + 945602601241 \) Copy content Toggle raw display
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