Properties

Label 825.4.c.f
Level $825$
Weight $4$
Character orbit 825.c
Analytic conductor $48.677$
Analytic rank $0$
Dimension $2$
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] = [825,4,Mod(199,825)]
 
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, 1, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.199");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.c (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: \(48.6765757547\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-1}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
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 \(i = \sqrt{-1}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + i q^{2} - 3 i q^{3} + 7 q^{4} + 3 q^{6} + 26 i q^{7} + 15 i q^{8} - 9 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + i q^{2} - 3 i q^{3} + 7 q^{4} + 3 q^{6} + 26 i q^{7} + 15 i q^{8} - 9 q^{9} + 11 q^{11} - 21 i q^{12} - 32 i q^{13} - 26 q^{14} + 41 q^{16} - 74 i q^{17} - 9 i q^{18} + 60 q^{19} + 78 q^{21} + 11 i q^{22} - 182 i q^{23} + 45 q^{24} + 32 q^{26} + 27 i q^{27} + 182 i q^{28} + 90 q^{29} - 8 q^{31} + 161 i q^{32} - 33 i q^{33} + 74 q^{34} - 63 q^{36} + 66 i q^{37} + 60 i q^{38} - 96 q^{39} + 422 q^{41} + 78 i q^{42} + 408 i q^{43} + 77 q^{44} + 182 q^{46} + 506 i q^{47} - 123 i q^{48} - 333 q^{49} - 222 q^{51} - 224 i q^{52} + 348 i q^{53} - 27 q^{54} - 390 q^{56} - 180 i q^{57} + 90 i q^{58} + 200 q^{59} + 132 q^{61} - 8 i q^{62} - 234 i q^{63} + 167 q^{64} + 33 q^{66} + 1036 i q^{67} - 518 i q^{68} - 546 q^{69} + 762 q^{71} - 135 i q^{72} - 542 i q^{73} - 66 q^{74} + 420 q^{76} + 286 i q^{77} - 96 i q^{78} + 550 q^{79} + 81 q^{81} + 422 i q^{82} - 132 i q^{83} + 546 q^{84} - 408 q^{86} - 270 i q^{87} + 165 i q^{88} - 570 q^{89} + 832 q^{91} - 1274 i q^{92} + 24 i q^{93} - 506 q^{94} + 483 q^{96} - 14 i q^{97} - 333 i q^{98} - 99 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 14 q^{4} + 6 q^{6} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 14 q^{4} + 6 q^{6} - 18 q^{9} + 22 q^{11} - 52 q^{14} + 82 q^{16} + 120 q^{19} + 156 q^{21} + 90 q^{24} + 64 q^{26} + 180 q^{29} - 16 q^{31} + 148 q^{34} - 126 q^{36} - 192 q^{39} + 844 q^{41} + 154 q^{44} + 364 q^{46} - 666 q^{49} - 444 q^{51} - 54 q^{54} - 780 q^{56} + 400 q^{59} + 264 q^{61} + 334 q^{64} + 66 q^{66} - 1092 q^{69} + 1524 q^{71} - 132 q^{74} + 840 q^{76} + 1100 q^{79} + 162 q^{81} + 1092 q^{84} - 816 q^{86} - 1140 q^{89} + 1664 q^{91} - 1012 q^{94} + 966 q^{96} - 198 q^{99}+O(q^{100}) \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
1.00000i
1.00000i
1.00000i 3.00000i 7.00000 0 3.00000 26.0000i 15.0000i −9.00000 0
199.2 1.00000i 3.00000i 7.00000 0 3.00000 26.0000i 15.0000i −9.00000 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
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 825.4.c.f 2
5.b even 2 1 inner 825.4.c.f 2
5.c odd 4 1 33.4.a.b 1
5.c odd 4 1 825.4.a.f 1
15.e even 4 1 99.4.a.a 1
15.e even 4 1 2475.4.a.e 1
20.e even 4 1 528.4.a.h 1
35.f even 4 1 1617.4.a.d 1
40.i odd 4 1 2112.4.a.u 1
40.k even 4 1 2112.4.a.h 1
55.e even 4 1 363.4.a.d 1
60.l odd 4 1 1584.4.a.l 1
165.l odd 4 1 1089.4.a.e 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.a.b 1 5.c odd 4 1
99.4.a.a 1 15.e even 4 1
363.4.a.d 1 55.e even 4 1
528.4.a.h 1 20.e even 4 1
825.4.a.f 1 5.c odd 4 1
825.4.c.f 2 1.a even 1 1 trivial
825.4.c.f 2 5.b even 2 1 inner
1089.4.a.e 1 165.l odd 4 1
1584.4.a.l 1 60.l odd 4 1
1617.4.a.d 1 35.f even 4 1
2112.4.a.h 1 40.k even 4 1
2112.4.a.u 1 40.i odd 4 1
2475.4.a.e 1 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(825, [\chi])\):

\( T_{2}^{2} + 1 \) Copy content Toggle raw display
\( T_{7}^{2} + 676 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 1 \) Copy content Toggle raw display
$3$ \( T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 676 \) Copy content Toggle raw display
$11$ \( (T - 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1024 \) Copy content Toggle raw display
$17$ \( T^{2} + 5476 \) Copy content Toggle raw display
$19$ \( (T - 60)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 33124 \) Copy content Toggle raw display
$29$ \( (T - 90)^{2} \) Copy content Toggle raw display
$31$ \( (T + 8)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 4356 \) Copy content Toggle raw display
$41$ \( (T - 422)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 166464 \) Copy content Toggle raw display
$47$ \( T^{2} + 256036 \) Copy content Toggle raw display
$53$ \( T^{2} + 121104 \) Copy content Toggle raw display
$59$ \( (T - 200)^{2} \) Copy content Toggle raw display
$61$ \( (T - 132)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 1073296 \) Copy content Toggle raw display
$71$ \( (T - 762)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 293764 \) Copy content Toggle raw display
$79$ \( (T - 550)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 17424 \) Copy content Toggle raw display
$89$ \( (T + 570)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 196 \) Copy content Toggle raw display
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