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Properties

Label 725.2.b.b
Level $725$
Weight $2$
Character orbit 725.b
Analytic conductor $5.789$
Analytic rank $0$
Dimension $4$
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(349,725)]
 
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, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("725.349");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 725 = 5^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 725.b (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: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 29)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{2} + \beta_1) q^{2} + (\beta_{2} + \beta_1) q^{3} + ( - 2 \beta_{3} - 1) q^{4} + ( - 2 \beta_{3} - 3) q^{6} + 2 \beta_{2} q^{7} + ( - \beta_{2} - 3 \beta_1) q^{8} - 2 \beta_{3} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{2} + \beta_1) q^{2} + (\beta_{2} + \beta_1) q^{3} + ( - 2 \beta_{3} - 1) q^{4} + ( - 2 \beta_{3} - 3) q^{6} + 2 \beta_{2} q^{7} + ( - \beta_{2} - 3 \beta_1) q^{8} - 2 \beta_{3} q^{9} + ( - \beta_{3} + 1) q^{11} + ( - 3 \beta_{2} - 5 \beta_1) q^{12} + ( - 2 \beta_{2} - \beta_1) q^{13} + ( - 2 \beta_{3} - 4) q^{14} + 3 q^{16} + ( - 2 \beta_{2} + 2 \beta_1) q^{17} + ( - 2 \beta_{2} - 4 \beta_1) q^{18} - 6 q^{19} + ( - 2 \beta_{3} - 4) q^{21} - \beta_1 q^{22} + (4 \beta_{2} - 2 \beta_1) q^{23} + (4 \beta_{3} + 5) q^{24} + (3 \beta_{3} + 5) q^{26} + (\beta_{2} - \beta_1) q^{27} + ( - 2 \beta_{2} - 8 \beta_1) q^{28} - q^{29} + (5 \beta_{3} + 3) q^{31} + (\beta_{2} - 3 \beta_1) q^{32} - \beta_1 q^{33} + 2 q^{34} + (2 \beta_{3} + 8) q^{36} + 4 \beta_1 q^{37} + ( - 6 \beta_{2} - 6 \beta_1) q^{38} + (3 \beta_{3} + 5) q^{39} + ( - 6 \beta_{3} + 4) q^{41} + ( - 6 \beta_{2} - 8 \beta_1) q^{42} + ( - \beta_{2} + 5 \beta_1) q^{43} + ( - \beta_{3} + 3) q^{44} + ( - 2 \beta_{3} - 6) q^{46} + (3 \beta_{2} - \beta_1) q^{47} + (3 \beta_{2} + 3 \beta_1) q^{48} - q^{49} + 2 q^{51} + (4 \beta_{2} + 9 \beta_1) q^{52} + (6 \beta_{2} + \beta_1) q^{53} - q^{54} + (6 \beta_{3} + 4) q^{56} + ( - 6 \beta_{2} - 6 \beta_1) q^{57} + ( - \beta_{2} - \beta_1) q^{58} + (4 \beta_{3} - 2) q^{59} + ( - 2 \beta_{3} - 2) q^{61} + (8 \beta_{2} + 13 \beta_1) q^{62} - 8 \beta_1 q^{63} + (2 \beta_{3} + 7) q^{64} + (\beta_{3} + 1) q^{66} - 4 \beta_{2} q^{67} + ( - 2 \beta_{2} + 6 \beta_1) q^{68} + ( - 2 \beta_{3} - 6) q^{69} + ( - 2 \beta_{3} - 6) q^{71} + (6 \beta_{2} + 4 \beta_1) q^{72} + 4 \beta_1 q^{73} + ( - 4 \beta_{3} - 4) q^{74} + (12 \beta_{3} + 6) q^{76} + (2 \beta_{2} - 4 \beta_1) q^{77} + (8 \beta_{2} + 11 \beta_1) q^{78} + (\beta_{3} + 1) q^{79} + ( - 6 \beta_{3} - 1) q^{81} + ( - 2 \beta_{2} - 8 \beta_1) q^{82} + (4 \beta_{2} + 2 \beta_1) q^{83} + (10 \beta_{3} + 12) q^{84} + ( - 4 \beta_{3} - 3) q^{86} + ( - \beta_{2} - \beta_1) q^{87} + (2 \beta_{2} - \beta_1) q^{88} + (6 \beta_{3} + 4) q^{89} + (2 \beta_{3} + 8) q^{91} - 14 \beta_1 q^{92} + (8 \beta_{2} + 13 \beta_1) q^{93} + ( - 2 \beta_{3} - 5) q^{94} + (2 \beta_{3} + 1) q^{96} + ( - 6 \beta_{2} + 4 \beta_1) q^{97} + ( - \beta_{2} - \beta_1) q^{98} + ( - 2 \beta_{3} + 4) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 12 q^{6}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 12 q^{6} + 4 q^{11} - 16 q^{14} + 12 q^{16} - 24 q^{19} - 16 q^{21} + 20 q^{24} + 20 q^{26} - 4 q^{29} + 12 q^{31} + 8 q^{34} + 32 q^{36} + 20 q^{39} + 16 q^{41} + 12 q^{44} - 24 q^{46} - 4 q^{49} + 8 q^{51} - 4 q^{54} + 16 q^{56} - 8 q^{59} - 8 q^{61} + 28 q^{64} + 4 q^{66} - 24 q^{69} - 24 q^{71} - 16 q^{74} + 24 q^{76} + 4 q^{79} - 4 q^{81} + 48 q^{84} - 12 q^{86} + 16 q^{89} + 32 q^{91} - 20 q^{94} + 4 q^{96} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{8}^{2} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -\zeta_{8}^{3} + \zeta_{8} \) Copy content Toggle raw display

Display \(\zeta_{8}^j\) in terms of \(\beta_i\)

\(\zeta_{8}\)\(=\) \( ( \beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} + \beta_{2} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{8}^j\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
349.1
0.707107 − 0.707107i
−0.707107 − 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
− 2.41421i − 2.41421i −3.82843 0 −5.82843 − 2.82843i 4.41421i −2.82843 0
349.2 − 0.414214i − 0.414214i 1.82843 0 −0.171573 − 2.82843i − 1.58579i 2.82843 0
349.3 0.414214i 0.414214i 1.82843 0 −0.171573 2.82843i 1.58579i 2.82843 0
349.4 2.41421i 2.41421i −3.82843 0 −5.82843 2.82843i − 4.41421i −2.82843 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b 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.b.b 4
5.b even 2 1 inner 725.2.b.b 4
5.c odd 4 1 29.2.a.a ✓ 2
5.c odd 4 1 725.2.a.b 2
15.e even 4 1 261.2.a.d 2
15.e even 4 1 6525.2.a.o 2
20.e even 4 1 464.2.a.h 2
35.f even 4 1 1421.2.a.j 2
40.i odd 4 1 1856.2.a.r 2
40.k even 4 1 1856.2.a.w 2
55.e even 4 1 3509.2.a.j 2
60.l odd 4 1 4176.2.a.bq 2
65.h odd 4 1 4901.2.a.g 2
85.g odd 4 1 8381.2.a.e 2
145.e even 4 1 841.2.b.a 4
145.h odd 4 1 841.2.a.d 2
145.j even 4 1 841.2.b.a 4
145.o even 28 6 841.2.e.k 24
145.p odd 28 6 841.2.d.j 12
145.q odd 28 6 841.2.d.f 12
145.t even 28 6 841.2.e.k 24
435.p even 4 1 7569.2.a.c 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
29.2.a.a ✓ 2 5.c odd 4 1
261.2.a.d 2 15.e even 4 1
464.2.a.h 2 20.e even 4 1
725.2.a.b 2 5.c odd 4 1
725.2.b.b 4 1.a even 1 1 trivial
725.2.b.b 4 5.b even 2 1 inner
841.2.a.d 2 145.h odd 4 1
841.2.b.a 4 145.e even 4 1
841.2.b.a 4 145.j even 4 1
841.2.d.f 12 145.q odd 28 6
841.2.d.j 12 145.p odd 28 6
841.2.e.k 24 145.o even 28 6
841.2.e.k 24 145.t even 28 6
1421.2.a.j 2 35.f even 4 1
1856.2.a.r 2 40.i odd 4 1
1856.2.a.w 2 40.k even 4 1
3509.2.a.j 2 55.e even 4 1
4176.2.a.bq 2 60.l odd 4 1
4901.2.a.g 2 65.h odd 4 1
6525.2.a.o 2 15.e even 4 1
7569.2.a.c 2 435.p even 4 1
8381.2.a.e 2 85.g odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(725, [\chi])\):

\( T_{2}^{4} + 6T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{3}^{4} + 6T_{3}^{2} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 6T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{4} + 6T^{2} + 1 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} - 2 T - 1)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 18T^{2} + 49 \) Copy content Toggle raw display
$17$ \( T^{4} + 24T^{2} + 16 \) Copy content Toggle raw display
$19$ \( (T + 6)^{4} \) Copy content Toggle raw display
$23$ \( T^{4} + 72T^{2} + 784 \) Copy content Toggle raw display
$29$ \( (T + 1)^{4} \) Copy content Toggle raw display
$31$ \( (T^{2} - 6 T - 41)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$41$ \( (T^{2} - 8 T - 56)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 54T^{2} + 529 \) Copy content Toggle raw display
$47$ \( T^{4} + 38T^{2} + 289 \) Copy content Toggle raw display
$53$ \( T^{4} + 146T^{2} + 5041 \) Copy content Toggle raw display
$59$ \( (T^{2} + 4 T - 28)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 4 T - 4)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 32)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} + 12 T + 28)^{2} \) Copy content Toggle raw display
$73$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 2 T - 1)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 72T^{2} + 784 \) Copy content Toggle raw display
$89$ \( (T^{2} - 8 T - 56)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 176T^{2} + 3136 \) Copy content Toggle raw display
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