Properties

Label 825.3.b.a
Level $825$
Weight $3$
Character orbit 825.b
Analytic conductor $22.480$
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] = [825,3,Mod(76,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.76");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 825.b (of order \(2\), degree \(1\), 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: \(22.4796218097\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.0.39744.5
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} + 12x^{2} + 4x + 22 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 33)
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.

\(f(q)\) \(=\) \( q - \beta_{3} q^{2} + \beta_1 q^{3} + ( - 2 \beta_1 - 5) q^{4} + ( - \beta_{3} + \beta_{2}) q^{6} + \beta_{2} q^{7} + (3 \beta_{3} - 2 \beta_{2}) q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{3} q^{2} + \beta_1 q^{3} + ( - 2 \beta_1 - 5) q^{4} + ( - \beta_{3} + \beta_{2}) q^{6} + \beta_{2} q^{7} + (3 \beta_{3} - 2 \beta_{2}) q^{8} + 3 q^{9} + (\beta_{3} - 2 \beta_{2} + \beta_1 + 5) q^{11} + ( - 5 \beta_1 - 6) q^{12} + (6 \beta_{3} - \beta_{2}) q^{13} + ( - 7 \beta_1 + 3) q^{14} + (12 \beta_1 + 1) q^{16} + (2 \beta_{3} - 2 \beta_{2}) q^{17} - 3 \beta_{3} q^{18} + (2 \beta_{3} + 4 \beta_{2}) q^{19} + ( - 2 \beta_{3} - \beta_{2}) q^{21} + ( - 6 \beta_{3} + \beta_{2} + 16 \beta_1 + 3) q^{22} + ( - 5 \beta_1 + 23) q^{23} + (7 \beta_{3} - \beta_{2}) q^{24} + (19 \beta_1 + 51) q^{26} + 3 \beta_1 q^{27} + (4 \beta_{3} - 3 \beta_{2}) q^{28} + (8 \beta_{3} - 4 \beta_{2}) q^{29} + ( - 18 \beta_1 + 20) q^{31} + ( - \beta_{3} + 4 \beta_{2}) q^{32} + (5 \beta_{3} + \beta_{2} + 5 \beta_1 + 3) q^{33} + (18 \beta_1 + 12) q^{34} + ( - 6 \beta_1 - 15) q^{36} + ( - 2 \beta_1 + 16) q^{37} + ( - 24 \beta_1 + 30) q^{38} + (8 \beta_{3} - 5 \beta_{2}) q^{39} + ( - 4 \beta_{3} - 6 \beta_{2}) q^{41} + (3 \beta_1 - 21) q^{42} + ( - 2 \beta_{3} - 12 \beta_{2}) q^{43} + ( - 15 \beta_{3} + 8 \beta_{2} - 15 \beta_1 - 31) q^{44} + ( - 18 \beta_{3} - 5 \beta_{2}) q^{46} + (3 \beta_1 - 25) q^{47} + (\beta_1 + 36) q^{48} + (10 \beta_1 + 25) q^{49} + 6 \beta_{3} q^{51} + ( - 46 \beta_{3} + 15 \beta_{2}) q^{52} + (11 \beta_1 - 7) q^{53} + ( - 3 \beta_{3} + 3 \beta_{2}) q^{54} + (\beta_1 + 39) q^{56} + ( - 6 \beta_{3} - 6 \beta_{2}) q^{57} + (44 \beta_1 + 60) q^{58} + ( - 42 \beta_1 + 10) q^{59} + ( - 2 \beta_{3} - 11 \beta_{2}) q^{61} + ( - 2 \beta_{3} - 18 \beta_{2}) q^{62} + 3 \beta_{2} q^{63} + (18 \beta_1 + 7) q^{64} + ( - 8 \beta_{3} + 5 \beta_{2} + 3 \beta_1 + 48) q^{66} + 34 q^{67} + ( - 22 \beta_{3} + 10 \beta_{2}) q^{68} + (23 \beta_1 - 15) q^{69} + (\beta_1 - 71) q^{71} + (9 \beta_{3} - 6 \beta_{2}) q^{72} + ( - 4 \beta_{3} + 10 \beta_{2}) q^{73} + ( - 14 \beta_{3} - 2 \beta_{2}) q^{74} + (2 \beta_{3} - 8 \beta_{2}) q^{76} + ( - 2 \beta_{3} + 4 \beta_{2} - 13 \beta_1 + 45) q^{77} + (51 \beta_1 + 57) q^{78} + ( - 24 \beta_{3} + 5 \beta_{2}) q^{79} + 9 q^{81} + (34 \beta_1 - 54) q^{82} + ( - 16 \beta_{3} + 6 \beta_{2}) q^{83} + (10 \beta_{3} - \beta_{2}) q^{84} + (80 \beta_1 - 54) q^{86} + (16 \beta_{3} - 4 \beta_{2}) q^{87} + (22 \beta_{3} - 11 \beta_{2} - 22 \beta_1 - 99) q^{88} + (18 \beta_1 + 76) q^{89} + (32 \beta_1 + 6) q^{91} + ( - 21 \beta_1 - 85) q^{92} + (20 \beta_1 - 54) q^{93} + (22 \beta_{3} + 3 \beta_{2}) q^{94} + ( - 9 \beta_{3} - 3 \beta_{2}) q^{96} + (42 \beta_1 + 94) q^{97} + ( - 35 \beta_{3} + 10 \beta_{2}) q^{98} + (3 \beta_{3} - 6 \beta_{2} + 3 \beta_1 + 15) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 20 q^{4} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 20 q^{4} + 12 q^{9} + 20 q^{11} - 24 q^{12} + 12 q^{14} + 4 q^{16} + 12 q^{22} + 92 q^{23} + 204 q^{26} + 80 q^{31} + 12 q^{33} + 48 q^{34} - 60 q^{36} + 64 q^{37} + 120 q^{38} - 84 q^{42} - 124 q^{44} - 100 q^{47} + 144 q^{48} + 100 q^{49} - 28 q^{53} + 156 q^{56} + 240 q^{58} + 40 q^{59} + 28 q^{64} + 192 q^{66} + 136 q^{67} - 60 q^{69} - 284 q^{71} + 180 q^{77} + 228 q^{78} + 36 q^{81} - 216 q^{82} - 216 q^{86} - 396 q^{88} + 304 q^{89} + 24 q^{91} - 340 q^{92} - 216 q^{93} + 376 q^{97} + 60 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - 2x^{3} + 12x^{2} + 4x + 22 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{3} + \nu^{2} + 2\nu + 23 ) / 13 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + \nu^{2} + 28\nu + 10 ) / 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -3\nu^{3} + 10\nu^{2} - 32\nu + 9 ) / 13 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{2} - \beta _1 + 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + \beta_{2} + 2\beta _1 - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( -\beta_{3} - 2\beta_{2} + 12\beta _1 - 19 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
76.1
−0.366025 1.29224i
1.36603 + 3.21405i
1.36603 3.21405i
−0.366025 + 1.29224i
3.53045i 1.73205 −8.46410 0 6.11492i 2.58447i 15.7603i 3.00000 0
76.2 2.35285i −1.73205 −1.53590 0 4.07525i 6.42810i 5.79766i 3.00000 0
76.3 2.35285i −1.73205 −1.53590 0 4.07525i 6.42810i 5.79766i 3.00000 0
76.4 3.53045i 1.73205 −8.46410 0 6.11492i 2.58447i 15.7603i 3.00000 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
11.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 825.3.b.a 4
5.b even 2 1 33.3.c.a 4
5.c odd 4 2 825.3.h.a 8
11.b odd 2 1 inner 825.3.b.a 4
15.d odd 2 1 99.3.c.b 4
20.d odd 2 1 528.3.j.c 4
40.e odd 2 1 2112.3.j.d 4
40.f even 2 1 2112.3.j.a 4
55.d odd 2 1 33.3.c.a 4
55.e even 4 2 825.3.h.a 8
55.h odd 10 4 363.3.g.e 16
55.j even 10 4 363.3.g.e 16
60.h even 2 1 1584.3.j.f 4
165.d even 2 1 99.3.c.b 4
220.g even 2 1 528.3.j.c 4
440.c even 2 1 2112.3.j.d 4
440.o odd 2 1 2112.3.j.a 4
660.g odd 2 1 1584.3.j.f 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.3.c.a 4 5.b even 2 1
33.3.c.a 4 55.d odd 2 1
99.3.c.b 4 15.d odd 2 1
99.3.c.b 4 165.d even 2 1
363.3.g.e 16 55.h odd 10 4
363.3.g.e 16 55.j even 10 4
528.3.j.c 4 20.d odd 2 1
528.3.j.c 4 220.g even 2 1
825.3.b.a 4 1.a even 1 1 trivial
825.3.b.a 4 11.b odd 2 1 inner
825.3.h.a 8 5.c odd 4 2
825.3.h.a 8 55.e even 4 2
1584.3.j.f 4 60.h even 2 1
1584.3.j.f 4 660.g odd 2 1
2112.3.j.a 4 40.f even 2 1
2112.3.j.a 4 440.o odd 2 1
2112.3.j.d 4 40.e odd 2 1
2112.3.j.d 4 440.c even 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(825, [\chi])\):

\( T_{2}^{4} + 18T_{2}^{2} + 69 \) Copy content Toggle raw display
\( T_{23}^{2} - 46T_{23} + 454 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 18T^{2} + 69 \) Copy content Toggle raw display
$3$ \( (T^{2} - 3)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 48T^{2} + 276 \) Copy content Toggle raw display
$11$ \( T^{4} - 20 T^{3} + 330 T^{2} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{4} + 624 T^{2} + 33396 \) Copy content Toggle raw display
$17$ \( T^{4} + 216T^{2} + 9936 \) Copy content Toggle raw display
$19$ \( T^{4} + 936T^{2} + 9936 \) Copy content Toggle raw display
$23$ \( (T^{2} - 46 T + 454)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} + 1536 T^{2} + 70656 \) Copy content Toggle raw display
$31$ \( (T^{2} - 40 T - 572)^{2} \) Copy content Toggle raw display
$37$ \( (T^{2} - 32 T + 244)^{2} \) Copy content Toggle raw display
$41$ \( T^{4} + 2304 T^{2} + 4416 \) Copy content Toggle raw display
$43$ \( T^{4} + 7272 T^{2} + \cdots + 3843024 \) Copy content Toggle raw display
$47$ \( (T^{2} + 50 T + 598)^{2} \) Copy content Toggle raw display
$53$ \( (T^{2} + 14 T - 314)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 20 T - 5192)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 6144 T^{2} + \cdots + 2596884 \) Copy content Toggle raw display
$67$ \( (T - 34)^{4} \) Copy content Toggle raw display
$71$ \( (T^{2} + 142 T + 5038)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 4608 T^{2} + \cdots + 4809024 \) Copy content Toggle raw display
$79$ \( T^{4} + 10128 T^{2} + \cdots + 5643924 \) Copy content Toggle raw display
$83$ \( T^{4} + 5184 T^{2} + 4416 \) Copy content Toggle raw display
$89$ \( (T^{2} - 152 T + 4804)^{2} \) Copy content Toggle raw display
$97$ \( (T^{2} - 188 T + 3544)^{2} \) Copy content Toggle raw display
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