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Properties

Label 725.2.q.a
Level $725$
Weight $2$
Character orbit 725.q
Analytic conductor $5.789$
Analytic rank $0$
Dimension $12$
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] = [725,2,Mod(51,725)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(725, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 13]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("725.51");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 725 = 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 725.q (of order \(14\), degree \(6\), 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: \(5.78915414654\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{14})\)
Coefficient field: 12.0.7877952219361.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3x^{11} + 13x^{9} - 18x^{8} - 14x^{7} + 57x^{6} - 28x^{5} - 72x^{4} + 104x^{3} - 96x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 29)
Sato-Tate group: $\mathrm{SU}(2)[C_{14}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \beta_{10} - \beta_{9} - \beta_{7} + \cdots + 1) q^{2}+ \cdots + (\beta_{11} + \beta_{10} + 2 \beta_{9} + \cdots - 1) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{10} - \beta_{9} - \beta_{7} + \cdots + 1) q^{2}+ \cdots + (4 \beta_{11} - 3 \beta_{10} + \cdots - 8) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 7 q^{2} + 7 q^{3} - q^{4} - 3 q^{6} + 11 q^{7} - 14 q^{8} - 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 7 q^{2} + 7 q^{3} - q^{4} - 3 q^{6} + 11 q^{7} - 14 q^{8} - 3 q^{9} + 7 q^{11} - 9 q^{13} - 7 q^{14} + 9 q^{16} - 42 q^{18} - 7 q^{19} - 7 q^{21} + 4 q^{22} + 5 q^{23} - 25 q^{24} - 21 q^{26} + 7 q^{27} - 12 q^{28} - 15 q^{29} - 21 q^{31} + 17 q^{33} - 13 q^{34} - 40 q^{36} - 7 q^{37} - 28 q^{38} + 21 q^{39} - 50 q^{42} - 7 q^{43} + 42 q^{44} + 7 q^{47} + 14 q^{48} + 13 q^{49} + 20 q^{51} + 6 q^{52} + 10 q^{53} - 38 q^{54} - 21 q^{56} + 14 q^{57} + 57 q^{58} + 44 q^{59} - 7 q^{61} - 37 q^{62} + 13 q^{63} - 26 q^{64} + 21 q^{66} + 37 q^{67} - 14 q^{68} + 21 q^{69} - 21 q^{71} - 35 q^{72} - 14 q^{73} + 7 q^{76} + 7 q^{77} - 17 q^{78} + 49 q^{79} + q^{81} - 22 q^{82} - 5 q^{83} + 21 q^{84} - 44 q^{86} - 15 q^{87} + 66 q^{88} + 7 q^{89} - 3 q^{91} + 6 q^{92} - 19 q^{93} + 66 q^{94} + 30 q^{96} - 14 q^{97} + 42 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 3x^{11} + 13x^{9} - 18x^{8} - 14x^{7} + 57x^{6} - 28x^{5} - 72x^{4} + 104x^{3} - 96x + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{11} + 7 \nu^{10} - 10 \nu^{9} - 7 \nu^{8} + 40 \nu^{7} - 14 \nu^{6} - 83 \nu^{5} + 102 \nu^{4} + \cdots + 96 ) / 128 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - \nu^{11} + 5 \nu^{10} - 10 \nu^{9} + 7 \nu^{8} + 4 \nu^{7} - 58 \nu^{6} + 59 \nu^{5} + 38 \nu^{4} + \cdots - 288 ) / 128 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - \nu^{11} + 5 \nu^{10} - 10 \nu^{9} + 7 \nu^{8} + 36 \nu^{7} - 58 \nu^{6} - 5 \nu^{5} + 134 \nu^{4} + \cdots - 32 ) / 128 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( \nu^{11} - 3 \nu^{10} + 4 \nu^{9} + \nu^{8} - 18 \nu^{7} + 22 \nu^{6} + 17 \nu^{5} - 52 \nu^{4} + \cdots + 64 ) / 64 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - \nu^{11} + 5 \nu^{10} - 10 \nu^{9} - 25 \nu^{8} + 68 \nu^{7} + 6 \nu^{6} - 165 \nu^{5} + 134 \nu^{4} + \cdots + 224 ) / 128 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3 \nu^{11} + \nu^{10} - 18 \nu^{9} + 11 \nu^{8} + 36 \nu^{7} - 50 \nu^{6} - 49 \nu^{5} + 94 \nu^{4} + \cdots + 96 ) / 128 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{10} + 3\nu^{9} - 9\nu^{7} + 10\nu^{6} + 6\nu^{5} - 29\nu^{4} + 16\nu^{3} + 20\nu^{2} - 40\nu + 16 ) / 16 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - \nu^{11} + 5 \nu^{10} - 10 \nu^{9} - 9 \nu^{8} + 36 \nu^{7} - 26 \nu^{6} - 53 \nu^{5} + 86 \nu^{4} + \cdots - 32 ) / 64 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{11} - 5\nu^{9} + 5\nu^{8} + 9\nu^{7} - 16\nu^{6} - 5\nu^{5} + 31\nu^{4} - 12\nu^{2} + 8\nu + 16 ) / 32 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 3 \nu^{11} - 11 \nu^{10} - 14 \nu^{9} + 59 \nu^{8} - 40 \nu^{7} - 122 \nu^{6} + 199 \nu^{5} + \cdots - 352 ) / 128 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 15 \nu^{11} + 31 \nu^{10} + 30 \nu^{9} - 151 \nu^{8} + 80 \nu^{7} + 290 \nu^{6} - 435 \nu^{5} + \cdots + 480 ) / 128 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( -\beta_{8} + \beta_{5} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{9} - \beta_{6} - \beta_{4} - \beta_{3} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + \beta_{7} - \beta_{6} + \beta_{5} + \beta_{4} + \beta_{3} + \beta_{2} - 1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{10} + \beta_{9} + 2\beta_{8} - 3\beta_{6} - \beta_{5} - \beta_{3} - 2\beta_{2} + 2\beta _1 + 1 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -\beta_{10} + \beta_{8} - \beta_{7} - \beta_{5} + 5\beta_{4} + 4\beta_{3} - 2\beta _1 - 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 2 \beta_{11} + 2 \beta_{10} + 2 \beta_{9} - 3 \beta_{8} - 3 \beta_{7} + 2 \beta_{6} - 2 \beta_{5} + \cdots - 4 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -5\beta_{10} + \beta_{9} - 6\beta_{8} - 5\beta_{7} + 5\beta_{6} + 5\beta_{3} + \beta_{2} - 10\beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 4 \beta_{11} + 4 \beta_{10} + 10 \beta_{9} - 11 \beta_{8} + 4 \beta_{7} + 4 \beta_{6} + \beta_{5} + \cdots - 10 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( - 6 \beta_{11} - 10 \beta_{10} - 3 \beta_{9} - 4 \beta_{8} - 11 \beta_{6} + 4 \beta_{5} - 15 \beta_{4} + \cdots + 5 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 2\beta_{11} + 9\beta_{9} + 9\beta_{7} - 7\beta_{6} - 9\beta_{5} + \beta_{4} + 9\beta_{3} + 5\beta_{2} + 18\beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( - 18 \beta_{11} - 29 \beta_{10} - 29 \beta_{9} - 8 \beta_{8} - 28 \beta_{7} - 7 \beta_{6} + \beta_{5} + \cdots + 11 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/725\mathbb{Z}\right)^\times\).

\(n\) \(176\) \(552\)
\(\chi(n)\) \(\beta_{11}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
51.1
1.38491 + 0.286410i
0.639551 − 1.26134i
−1.41140 − 0.0891373i
1.23295 − 0.692694i
1.38491 − 0.286410i
0.639551 + 1.26134i
0.911180 − 1.08155i
−1.25719 + 0.647667i
0.911180 + 1.08155i
−1.25719 − 0.647667i
−1.41140 + 0.0891373i
1.23295 + 0.692694i
0.536089 − 0.122359i −0.855966 + 1.77743i −1.52952 + 0.736577i 0 −0.241390 + 1.05760i 4.03077 + 1.94112i −1.58965 + 1.26771i −0.556117 − 0.697349i 0
51.2 2.21089 − 0.504621i 1.23248 − 2.55926i 2.83146 − 1.36356i 0 1.43341 − 6.28018i −1.40728 − 0.677709i 2.02596 − 1.61565i −3.16036 − 3.96297i 0
151.1 −1.21127 − 0.965958i 2.86109 + 0.653024i 0.0890656 + 0.390222i 0 −2.83476 − 3.55468i 0.759522 − 3.32768i −1.07536 + 2.23300i 5.05647 + 2.43507i 0
151.2 0.909335 + 0.725171i −0.960118 − 0.219141i −0.144024 − 0.631009i 0 −0.714155 − 0.895521i 0.339509 − 1.48749i 1.33591 − 2.77404i −1.82910 − 0.880850i 0
526.1 0.536089 + 0.122359i −0.855966 − 1.77743i −1.52952 − 0.736577i 0 −0.241390 − 1.05760i 4.03077 − 1.94112i −1.58965 − 1.26771i −0.556117 + 0.697349i 0
526.2 2.21089 + 0.504621i 1.23248 + 2.55926i 2.83146 + 1.36356i 0 1.43341 + 6.28018i −1.40728 + 0.677709i 2.02596 + 1.61565i −3.16036 + 3.96297i 0
651.1 −0.0741982 + 0.154074i 0.879032 + 0.701005i 1.22875 + 1.54080i 0 −0.173229 + 0.0834229i 1.82432 − 2.28763i −0.662012 + 0.151100i −0.386273 − 1.69237i 0
651.2 1.12916 − 2.34472i 0.343489 + 0.273923i −2.97573 − 3.73144i 0 1.03013 − 0.496082i −0.0468435 + 0.0587399i −7.03485 + 1.60566i −0.624612 − 2.73660i 0
676.1 −0.0741982 − 0.154074i 0.879032 − 0.701005i 1.22875 − 1.54080i 0 −0.173229 − 0.0834229i 1.82432 + 2.28763i −0.662012 − 0.151100i −0.386273 + 1.69237i 0
676.2 1.12916 + 2.34472i 0.343489 − 0.273923i −2.97573 + 3.73144i 0 1.03013 + 0.496082i −0.0468435 − 0.0587399i −7.03485 − 1.60566i −0.624612 + 2.73660i 0
701.1 −1.21127 + 0.965958i 2.86109 − 0.653024i 0.0890656 − 0.390222i 0 −2.83476 + 3.55468i 0.759522 + 3.32768i −1.07536 − 2.23300i 5.05647 − 2.43507i 0
701.2 0.909335 − 0.725171i −0.960118 + 0.219141i −0.144024 + 0.631009i 0 −0.714155 + 0.895521i 0.339509 + 1.48749i 1.33591 + 2.77404i −1.82910 + 0.880850i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 51.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
29.e even 14 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 725.2.q.a 12
5.b even 2 1 29.2.e.a ✓ 12
5.c odd 4 2 725.2.p.a 24
15.d odd 2 1 261.2.o.a 12
20.d odd 2 1 464.2.y.d 12
29.e even 14 1 inner 725.2.q.a 12
145.d even 2 1 841.2.e.i 12
145.f odd 4 2 841.2.d.m 24
145.l even 14 1 29.2.e.a ✓ 12
145.l even 14 1 841.2.b.e 12
145.l even 14 1 841.2.e.a 12
145.l even 14 1 841.2.e.e 12
145.l even 14 1 841.2.e.f 12
145.l even 14 1 841.2.e.h 12
145.n even 14 1 841.2.b.e 12
145.n even 14 1 841.2.e.a 12
145.n even 14 1 841.2.e.e 12
145.n even 14 1 841.2.e.f 12
145.n even 14 1 841.2.e.h 12
145.n even 14 1 841.2.e.i 12
145.q odd 28 2 725.2.p.a 24
145.s odd 28 2 841.2.a.k 12
145.s odd 28 4 841.2.d.k 24
145.s odd 28 4 841.2.d.l 24
145.s odd 28 2 841.2.d.m 24
435.bb odd 14 1 261.2.o.a 12
435.bk even 28 2 7569.2.a.bp 12
580.y odd 14 1 464.2.y.d 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.2.e.a ✓ 12 5.b even 2 1
29.2.e.a ✓ 12 145.l even 14 1
261.2.o.a 12 15.d odd 2 1
261.2.o.a 12 435.bb odd 14 1
464.2.y.d 12 20.d odd 2 1
464.2.y.d 12 580.y odd 14 1
725.2.p.a 24 5.c odd 4 2
725.2.p.a 24 145.q odd 28 2
725.2.q.a 12 1.a even 1 1 trivial
725.2.q.a 12 29.e even 14 1 inner
841.2.a.k 12 145.s odd 28 2
841.2.b.e 12 145.l even 14 1
841.2.b.e 12 145.n even 14 1
841.2.d.k 24 145.s odd 28 4
841.2.d.l 24 145.s odd 28 4
841.2.d.m 24 145.f odd 4 2
841.2.d.m 24 145.s odd 28 2
841.2.e.a 12 145.l even 14 1
841.2.e.a 12 145.n even 14 1
841.2.e.e 12 145.l even 14 1
841.2.e.e 12 145.n even 14 1
841.2.e.f 12 145.l even 14 1
841.2.e.f 12 145.n even 14 1
841.2.e.h 12 145.l even 14 1
841.2.e.h 12 145.n even 14 1
841.2.e.i 12 145.d even 2 1
841.2.e.i 12 145.n even 14 1
7569.2.a.bp 12 435.bk even 28 2

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{12} - 7 T_{2}^{11} + 23 T_{2}^{10} - 42 T_{2}^{9} + 32 T_{2}^{8} - 7 T_{2}^{7} + 92 T_{2}^{6} + \cdots + 1 \) acting on \(S_{2}^{\mathrm{new}}(725, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 7 T^{11} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{12} - 7 T^{11} + \cdots + 64 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} - 11 T^{11} + \cdots + 64 \) Copy content Toggle raw display
$11$ \( T^{12} - 7 T^{11} + \cdots + 10816 \) Copy content Toggle raw display
$13$ \( T^{12} + 9 T^{11} + \cdots + 841 \) Copy content Toggle raw display
$17$ \( T^{12} + 71 T^{10} + \cdots + 53824 \) Copy content Toggle raw display
$19$ \( T^{12} + 7 T^{11} + \cdots + 3136 \) Copy content Toggle raw display
$23$ \( T^{12} - 5 T^{11} + \cdots + 64 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 594823321 \) Copy content Toggle raw display
$31$ \( T^{12} + 21 T^{11} + \cdots + 817216 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 110103049 \) Copy content Toggle raw display
$41$ \( T^{12} + 99 T^{10} + \cdots + 107584 \) Copy content Toggle raw display
$43$ \( T^{12} + 7 T^{11} + \cdots + 24364096 \) Copy content Toggle raw display
$47$ \( T^{12} - 7 T^{11} + \cdots + 11343424 \) Copy content Toggle raw display
$53$ \( T^{12} - 10 T^{11} + \cdots + 9409 \) Copy content Toggle raw display
$59$ \( (T^{6} - 22 T^{5} + \cdots + 1856)^{2} \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 325694209 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 415833664 \) Copy content Toggle raw display
$71$ \( T^{12} + \cdots + 2671649344 \) Copy content Toggle raw display
$73$ \( T^{12} + 14 T^{11} + \cdots + 625681 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 30056463424 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 419758144 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 203946961 \) Copy content Toggle raw display
$97$ \( T^{12} + 14 T^{11} + \cdots + 1697809 \) Copy content Toggle raw display
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