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Properties

Label 725.2.d.c
Level $725$
Weight $2$
Character orbit 725.d
Analytic conductor $5.789$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [725,2,Mod(724,725)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(725, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("725.724");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 725 = 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 725.d (of order \(2\), degree \(1\), 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: \(5.78915414654\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 34x^{8} + 193x^{4} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{11} \)
Twist minimal: no (minimal twist has level 145)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

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

\(f(q)\) \(=\) \( q + \beta_{7} q^{2} + ( - \beta_{7} + \beta_{5}) q^{3} + (\beta_{4} + 1) q^{4} + ( - \beta_{4} - 2) q^{6} - \beta_{3} q^{7} + (\beta_{10} + 2 \beta_{7} - \beta_{5}) q^{8} + (\beta_1 + 2) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{7} q^{2} + ( - \beta_{7} + \beta_{5}) q^{3} + (\beta_{4} + 1) q^{4} + ( - \beta_{4} - 2) q^{6} - \beta_{3} q^{7} + (\beta_{10} + 2 \beta_{7} - \beta_{5}) q^{8} + (\beta_1 + 2) q^{9} - \beta_{2} q^{11} + ( - \beta_{10} - 3 \beta_{7} - \beta_{5}) q^{12} + \beta_{9} q^{13} + ( - \beta_{11} + \beta_{8} - 2 \beta_{2}) q^{14} + (2 \beta_{4} + \beta_1 + 4) q^{16} + (\beta_{10} - \beta_{7} - \beta_{5}) q^{17} + (\beta_{10} + \beta_{7}) q^{18} + (\beta_{11} + \beta_{2}) q^{19} + (\beta_{11} - 2 \beta_{8}) q^{21} + (\beta_{6} - \beta_{3}) q^{22} + ( - \beta_{9} + \beta_{6} - \beta_{3}) q^{23} + ( - 3 \beta_{4} - \beta_1 - 7) q^{24} + (\beta_{11} + \beta_{8} - \beta_{2}) q^{26} + 4 \beta_{5} q^{27} + ( - \beta_{9} + 5 \beta_{6} - 3 \beta_{3}) q^{28} + ( - \beta_{11} + \beta_{4} + \beta_{2} + 1) q^{29} + (\beta_{11} + 2 \beta_{8} + \beta_{2}) q^{31} + (\beta_{10} + 5 \beta_{7}) q^{32} + (\beta_{9} - 3 \beta_{6}) q^{33} + (\beta_{4} + \beta_1 - 3) q^{34} + (3 \beta_{4} - \beta_1) q^{36} + (\beta_{10} - 3 \beta_{7} + \beta_{5}) q^{37} + (\beta_{9} - 2 \beta_{6} + 3 \beta_{3}) q^{38} + ( - 2 \beta_{8} - 2 \beta_{2}) q^{39} + (\beta_{11} - 2 \beta_{8}) q^{41} + (\beta_{9} - 5 \beta_{6} + 4 \beta_{3}) q^{42} + (2 \beta_{10} - \beta_{7} + \beta_{5}) q^{43} + ( - \beta_{11} + 2 \beta_{8} - \beta_{2}) q^{44} + ( - 2 \beta_{11} + \beta_{8} - 2 \beta_{2}) q^{46} + ( - \beta_{7} + \beta_{5}) q^{47} + ( - 2 \beta_{10} - 9 \beta_{7} + 5 \beta_{5}) q^{48} + ( - 2 \beta_{4} - \beta_1 - 2) q^{49} + ( - \beta_1 - 1) q^{51} + ( - \beta_{9} + 2 \beta_{6}) q^{52} + ( - \beta_{9} - 2 \beta_{6} + 2 \beta_{3}) q^{53} + 4 q^{54} + ( - 2 \beta_{11} + 5 \beta_{8} - 6 \beta_{2}) q^{56} + ( - \beta_{9} + 3 \beta_{6} - 2 \beta_{3}) q^{57} + (\beta_{10} - \beta_{9} + \cdots - \beta_{3}) q^{58}+ \cdots + ( - 2 \beta_{8} - 5 \beta_{2}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{4} - 24 q^{6} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{4} - 24 q^{6} + 28 q^{9} + 52 q^{16} - 88 q^{24} + 12 q^{29} - 32 q^{34} - 4 q^{36} - 28 q^{49} - 16 q^{51} + 48 q^{54} + 32 q^{59} + 92 q^{64} - 112 q^{71} - 80 q^{74} + 76 q^{81} + 8 q^{86} - 32 q^{91} - 24 q^{94} - 120 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 34x^{8} + 193x^{4} + 4 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{8} - 49\nu^{4} - 279 ) / 59 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 4\nu^{11} + 137\nu^{7} + 821\nu^{3} + 118\nu ) / 118 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -7\nu^{10} - 225\nu^{6} - 1068\nu^{2} ) / 118 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -5\nu^{8} - 127\nu^{4} - 156 ) / 118 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -12\nu^{11} + \nu^{9} - 411\nu^{7} + 49\nu^{5} - 2345\nu^{3} + 338\nu ) / 236 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -4\nu^{10} - 137\nu^{6} - 821\nu^{2} ) / 59 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -20\nu^{11} + \nu^{9} - 685\nu^{7} + 49\nu^{5} - 3987\nu^{3} + 574\nu ) / 236 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -16\nu^{11} - \nu^{9} - 548\nu^{7} - 49\nu^{5} - 3166\nu^{3} - 456\nu ) / 118 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -12\nu^{10} - 411\nu^{6} - 2345\nu^{2} ) / 59 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 39\nu^{11} - 7\nu^{9} + 1321\nu^{7} - 225\nu^{5} + 7400\nu^{3} - 1068\nu ) / 118 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -39\nu^{11} - 7\nu^{9} - 1321\nu^{7} - 225\nu^{5} - 7400\nu^{3} - 1068\nu ) / 118 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{7} - \beta_{5} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{9} - 3\beta_{6} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{8} - 3\beta_{7} + 5\beta_{5} + 4\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2\beta_{4} - 5\beta _1 - 21 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{11} + \beta_{10} - 7\beta_{8} - 11\beta_{7} + 25\beta_{5} - 18\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -25\beta_{9} + 61\beta_{6} + 16\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 8\beta_{11} - 8\beta_{10} - 41\beta_{8} + 45\beta_{7} - 127\beta_{5} - 86\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( -98\beta_{4} + 127\beta _1 + 471 ) / 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -49\beta_{11} - 49\beta_{10} + 225\beta_{8} + 201\beta_{7} - 651\beta_{5} + 426\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 651\beta_{9} - 1503\beta_{6} - 548\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( -274\beta_{11} + 274\beta_{10} + 1199\beta_{8} - 955\beta_{7} + 3353\beta_{5} + 2154\beta_{2} ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(176\) \(552\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
724.1
−1.15723 − 1.15723i
−1.15723 + 1.15723i
−0.268540 + 0.268540i
−0.268540 − 0.268540i
1.60895 − 1.60895i
1.60895 + 1.60895i
−1.60895 + 1.60895i
−1.60895 − 1.60895i
0.268540 − 0.268540i
0.268540 + 0.268540i
1.15723 + 1.15723i
1.15723 − 1.15723i
−2.68667 2.31446 5.21819 0 −6.21819 − 4.21819i −8.64620 2.35673 0
724.2 −2.68667 2.31446 5.21819 0 −6.21819 4.21819i −8.64620 2.35673 0
724.3 −1.30397 0.537080 −0.299664 0 −0.700336 − 1.29966i 2.99869 −2.71155 0
724.4 −1.30397 0.537080 −0.299664 0 −0.700336 1.29966i 2.99869 −2.71155 0
724.5 −0.285442 −3.21789 −1.91852 0 0.918523 − 2.91852i 1.11851 7.35482 0
724.6 −0.285442 −3.21789 −1.91852 0 0.918523 2.91852i 1.11851 7.35482 0
724.7 0.285442 3.21789 −1.91852 0 0.918523 − 2.91852i −1.11851 7.35482 0
724.8 0.285442 3.21789 −1.91852 0 0.918523 2.91852i −1.11851 7.35482 0
724.9 1.30397 −0.537080 −0.299664 0 −0.700336 − 1.29966i −2.99869 −2.71155 0
724.10 1.30397 −0.537080 −0.299664 0 −0.700336 1.29966i −2.99869 −2.71155 0
724.11 2.68667 −2.31446 5.21819 0 −6.21819 − 4.21819i 8.64620 2.35673 0
724.12 2.68667 −2.31446 5.21819 0 −6.21819 4.21819i 8.64620 2.35673 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 724.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
29.b even 2 1 inner
145.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 725.2.d.c 12
5.b even 2 1 inner 725.2.d.c 12
5.c odd 4 1 145.2.c.b ✓ 6
5.c odd 4 1 725.2.c.e 6
15.e even 4 1 1305.2.d.b 6
20.e even 4 1 2320.2.g.i 6
29.b even 2 1 inner 725.2.d.c 12
145.d even 2 1 inner 725.2.d.c 12
145.e even 4 1 4205.2.a.m 6
145.h odd 4 1 145.2.c.b ✓ 6
145.h odd 4 1 725.2.c.e 6
145.j even 4 1 4205.2.a.m 6
435.p even 4 1 1305.2.d.b 6
580.o even 4 1 2320.2.g.i 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
145.2.c.b ✓ 6 5.c odd 4 1
145.2.c.b ✓ 6 145.h odd 4 1
725.2.c.e 6 5.c odd 4 1
725.2.c.e 6 145.h odd 4 1
725.2.d.c 12 1.a even 1 1 trivial
725.2.d.c 12 5.b even 2 1 inner
725.2.d.c 12 29.b even 2 1 inner
725.2.d.c 12 145.d even 2 1 inner
1305.2.d.b 6 15.e even 4 1
1305.2.d.b 6 435.p even 4 1
2320.2.g.i 6 20.e even 4 1
2320.2.g.i 6 580.o even 4 1
4205.2.a.m 6 145.e even 4 1
4205.2.a.m 6 145.j even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{6} - 9T_{2}^{4} + 13T_{2}^{2} - 1 \) acting on \(S_{2}^{\mathrm{new}}(725, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{6} - 9 T^{4} + 13 T^{2} - 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{6} - 16 T^{4} + \cdots - 16)^{2} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( (T^{6} + 28 T^{4} + \cdots + 256)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 16 T^{4} + \cdots + 16)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 52 T^{4} + \cdots + 256)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 52 T^{4} + \cdots - 64)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 56 T^{4} + \cdots + 1296)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + 68 T^{4} + \cdots + 9216)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} - 6 T^{5} + \cdots + 24389)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 152 T^{4} + \cdots + 144)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} - 100 T^{4} + \cdots - 20736)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 184 T^{4} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} - 176 T^{4} + \cdots - 121104)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 16 T^{4} + \cdots - 16)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 260 T^{4} + \cdots + 331776)^{2} \) Copy content Toggle raw display
$59$ \( (T^{3} - 8 T^{2} - 4 T + 48)^{4} \) Copy content Toggle raw display
$61$ \( (T^{6} + 104 T^{4} + \cdots + 2304)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} + 84 T^{4} + \cdots + 64)^{2} \) Copy content Toggle raw display
$71$ \( (T^{3} + 28 T^{2} + \cdots + 576)^{4} \) Copy content Toggle raw display
$73$ \( (T^{6} - 252 T^{4} + \cdots - 419904)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 176 T^{4} + \cdots + 121104)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 76 T^{4} + \cdots + 576)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 160 T^{4} + \cdots + 65536)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} - 28 T^{4} + \cdots - 576)^{2} \) Copy content Toggle raw display
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