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Properties

Label 625.2.d.q
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.

\(f(q)\) \(=\) \( q + ( - \beta_{7} - \beta_{5} + \beta_{3} + 1) q^{2} + ( - \beta_{15} + \beta_{5}) q^{3} + (\beta_{12} - \beta_{10} + \cdots - \beta_1) q^{4}+ \cdots + (2 \beta_{14} + \beta_{12} + \beta_{10} + \cdots - 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{7} - \beta_{5} + \beta_{3} + 1) q^{2} + ( - \beta_{15} + \beta_{5}) q^{3} + (\beta_{12} - \beta_{10} + \cdots - \beta_1) q^{4}+ \cdots + ( - 4 \beta_{14} + 2 \beta_{13} + \cdots - 3) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 5 q^{2} - 3 q^{4} + 7 q^{6} - 20 q^{7} + 5 q^{8} - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 5 q^{2} - 3 q^{4} + 7 q^{6} - 20 q^{7} + 5 q^{8} - 12 q^{9} - 3 q^{11} - 15 q^{12} + 5 q^{13} - q^{14} + q^{16} + 25 q^{17} + 10 q^{18} + 10 q^{19} + 7 q^{21} + 35 q^{22} + 15 q^{23} + 10 q^{24} + 22 q^{26} - 35 q^{28} - 8 q^{31} - 60 q^{32} - 6 q^{34} + q^{36} + 5 q^{37} + 35 q^{38} + q^{39} - 8 q^{41} + 10 q^{42} - 31 q^{44} + 42 q^{46} + 5 q^{47} + 25 q^{48} - 8 q^{49} - 28 q^{51} - 15 q^{52} + 10 q^{53} + 50 q^{54} + 35 q^{56} + 20 q^{57} - 35 q^{58} - 15 q^{59} + 17 q^{61} - 5 q^{62} - 10 q^{63} + 37 q^{64} + 44 q^{66} + 10 q^{67} - 80 q^{68} - 9 q^{69} - 13 q^{71} - 20 q^{72} - 40 q^{73} - 36 q^{74} - 20 q^{76} + 45 q^{77} - 5 q^{78} - 55 q^{79} - 19 q^{81} + 90 q^{82} + 15 q^{83} + 59 q^{84} + 7 q^{86} + 60 q^{87} - 40 q^{88} - 28 q^{91} - 45 q^{92} + 80 q^{93} + 4 q^{94} - 43 q^{96} - 40 q^{97} - 45 q^{98} - 44 q^{99}+O(q^{100}) \) 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 \(\left(\mathbb{Z}/625\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-\beta_{5}\)

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
− 2.04679i
0.0288455i
1.63097i
− 1.51514i
1.80544i
1.20005i
− 3.18910i
− 0.991969i
− 1.80544i
− 1.20005i
3.18910i
0.991969i
2.04679i
− 0.0288455i
− 1.63097i
1.51514i
−0.823534 − 2.53458i 0.614111 − 0.446178i −4.12784 + 2.99905i 0 −1.63661 − 1.18907i 2.04213 6.68866 + 4.85960i −0.748993 + 2.30516i 0
126.2 −0.326747 − 1.00562i −0.556121 + 0.404046i 0.713519 − 0.518402i 0 0.588029 + 0.427228i −1.01199 −2.46533 − 1.79116i −0.781033 + 2.40377i 0
126.3 0.100972 + 0.310759i −1.38777 + 1.00827i 1.53166 − 1.11281i 0 −0.453455 − 0.329455i −3.42409 1.02917 + 0.747732i −0.0177620 + 0.0546659i 0
126.4 0.622258 + 1.91511i 2.44781 − 1.77844i −1.66242 + 1.20782i 0 4.92909 + 3.58119i −0.369971 −0.0893841 − 0.0649413i 1.90189 − 5.85342i 0
251.1 −1.36191 + 0.989484i −0.219507 + 0.675574i 0.257680 − 0.793058i 0 −0.369521 − 1.13727i −4.59110 −0.606623 − 1.86699i 2.01883 + 1.46677i 0
251.2 0.405309 − 0.294474i −0.955869 + 2.94186i −0.540474 + 1.66341i 0 0.478880 + 1.47384i −0.0237879 0.580400 + 1.78629i −5.31382 − 3.86071i 0
251.3 1.88186 − 1.36725i 0.711454 − 2.18963i 1.05398 − 3.24381i 0 −1.65491 − 5.09330i −3.59425 −1.01405 − 3.12093i −1.86126 − 1.35229i 0
251.4 2.00179 − 1.45439i −0.654112 + 2.01315i 1.27390 − 3.92066i 0 1.61850 + 4.98124i 0.973070 −1.62284 − 4.99460i −1.19786 − 0.870294i 0
376.1 −1.36191 − 0.989484i −0.219507 − 0.675574i 0.257680 + 0.793058i 0 −0.369521 + 1.13727i −4.59110 −0.606623 + 1.86699i 2.01883 − 1.46677i 0
376.2 0.405309 + 0.294474i −0.955869 − 2.94186i −0.540474 − 1.66341i 0 0.478880 − 1.47384i −0.0237879 0.580400 − 1.78629i −5.31382 + 3.86071i 0
376.3 1.88186 + 1.36725i 0.711454 + 2.18963i 1.05398 + 3.24381i 0 −1.65491 + 5.09330i −3.59425 −1.01405 + 3.12093i −1.86126 + 1.35229i 0
376.4 2.00179 + 1.45439i −0.654112 − 2.01315i 1.27390 + 3.92066i 0 1.61850 − 4.98124i 0.973070 −1.62284 + 4.99460i −1.19786 + 0.870294i 0
501.1 −0.823534 + 2.53458i 0.614111 + 0.446178i −4.12784 − 2.99905i 0 −1.63661 + 1.18907i 2.04213 6.68866 − 4.85960i −0.748993 − 2.30516i 0
501.2 −0.326747 + 1.00562i −0.556121 − 0.404046i 0.713519 + 0.518402i 0 0.588029 − 0.427228i −1.01199 −2.46533 + 1.79116i −0.781033 − 2.40377i 0
501.3 0.100972 − 0.310759i −1.38777 − 1.00827i 1.53166 + 1.11281i 0 −0.453455 + 0.329455i −3.42409 1.02917 − 0.747732i −0.0177620 − 0.0546659i 0
501.4 0.622258 − 1.91511i 2.44781 + 1.77844i −1.66242 − 1.20782i 0 4.92909 − 3.58119i −0.369971 −0.0893841 + 0.0649413i 1.90189 + 5.85342i 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.q 16
5.b even 2 1 625.2.d.m 16
5.c odd 4 2 625.2.e.j 32
25.d even 5 1 625.2.a.e ✓ 8
25.d even 5 2 625.2.d.p 16
25.d even 5 1 inner 625.2.d.q 16
25.e even 10 1 625.2.a.g yes 8
25.e even 10 1 625.2.d.m 16
25.e even 10 2 625.2.d.n 16
25.f odd 20 2 625.2.b.d 16
25.f odd 20 2 625.2.e.j 32
25.f odd 20 4 625.2.e.k 32
75.h odd 10 1 5625.2.a.s 8
75.j odd 10 1 5625.2.a.be 8
100.h odd 10 1 10000.2.a.be 8
100.j odd 10 1 10000.2.a.bn 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
625.2.a.e ✓ 8 25.d even 5 1
625.2.a.g yes 8 25.e even 10 1
625.2.b.d 16 25.f odd 20 2
625.2.d.m 16 5.b even 2 1
625.2.d.m 16 25.e even 10 1
625.2.d.n 16 25.e even 10 2
625.2.d.p 16 25.d even 5 2
625.2.d.q 16 1.a even 1 1 trivial
625.2.d.q 16 25.d even 5 1 inner
625.2.e.j 32 5.c odd 4 2
625.2.e.j 32 25.f odd 20 2
625.2.e.k 32 25.f odd 20 4
5625.2.a.s 8 75.h odd 10 1
5625.2.a.be 8 75.j odd 10 1
10000.2.a.be 8 100.h odd 10 1
10000.2.a.bn 8 100.j odd 10 1

Hecke kernels

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

\( T_{2}^{16} - 5 T_{2}^{15} + 18 T_{2}^{14} - 50 T_{2}^{13} + 138 T_{2}^{12} - 245 T_{2}^{11} + 386 T_{2}^{10} + \cdots + 81 \) Copy content Toggle raw display
\( T_{3}^{16} + 12 T_{3}^{14} - 10 T_{3}^{13} + 88 T_{3}^{12} + 80 T_{3}^{11} + 629 T_{3}^{10} + 1130 T_{3}^{9} + \cdots + 841 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} - 5 T^{15} + \cdots + 81 \) Copy content Toggle raw display
$3$ \( T^{16} + 12 T^{14} + \cdots + 841 \) 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
$11$ \( T^{16} + 3 T^{15} + \cdots + 5861241 \) Copy content Toggle raw display
$13$ \( T^{16} - 5 T^{15} + \cdots + 130321 \) Copy content Toggle raw display
$17$ \( T^{16} - 25 T^{15} + \cdots + 2595321 \) 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
$37$ \( T^{16} - 5 T^{15} + \cdots + 6561 \) 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
$53$ \( T^{16} - 10 T^{15} + \cdots + 3606201 \) Copy content Toggle raw display
$59$ \( T^{16} + 15 T^{15} + \cdots + 50625 \) 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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