Properties

Label 825.4.c.a
Level $825$
Weight $4$
Character orbit 825.c
Analytic conductor $48.677$
Analytic rank $1$
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: \(1\)
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: \(\Z[a_1, a_2, a_3]\)
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.

\(f(q)\) \(=\) \( q + 5 i q^{2} + 3 i q^{3} - 17 q^{4} - 15 q^{6} + 32 i q^{7} - 45 i q^{8} - 9 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + 5 i q^{2} + 3 i q^{3} - 17 q^{4} - 15 q^{6} + 32 i q^{7} - 45 i q^{8} - 9 q^{9} - 11 q^{11} - 51 i q^{12} - 38 i q^{13} - 160 q^{14} + 89 q^{16} + 2 i q^{17} - 45 i q^{18} - 72 q^{19} - 96 q^{21} - 55 i q^{22} + 68 i q^{23} + 135 q^{24} + 190 q^{26} - 27 i q^{27} - 544 i q^{28} + 54 q^{29} - 152 q^{31} + 85 i q^{32} - 33 i q^{33} - 10 q^{34} + 153 q^{36} - 174 i q^{37} - 360 i q^{38} + 114 q^{39} + 94 q^{41} - 480 i q^{42} - 528 i q^{43} + 187 q^{44} - 340 q^{46} + 340 i q^{47} + 267 i q^{48} - 681 q^{49} - 6 q^{51} + 646 i q^{52} - 438 i q^{53} + 135 q^{54} + 1440 q^{56} - 216 i q^{57} + 270 i q^{58} - 20 q^{59} + 570 q^{61} - 760 i q^{62} - 288 i q^{63} + 287 q^{64} + 165 q^{66} + 460 i q^{67} - 34 i q^{68} - 204 q^{69} - 1092 q^{71} + 405 i q^{72} + 562 i q^{73} + 870 q^{74} + 1224 q^{76} - 352 i q^{77} + 570 i q^{78} + 16 q^{79} + 81 q^{81} + 470 i q^{82} + 372 i q^{83} + 1632 q^{84} + 2640 q^{86} + 162 i q^{87} + 495 i q^{88} + 966 q^{89} + 1216 q^{91} - 1156 i q^{92} - 456 i q^{93} - 1700 q^{94} - 255 q^{96} + 526 i q^{97} - 3405 i q^{98} + 99 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 34 q^{4} - 30 q^{6} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 34 q^{4} - 30 q^{6} - 18 q^{9} - 22 q^{11} - 320 q^{14} + 178 q^{16} - 144 q^{19} - 192 q^{21} + 270 q^{24} + 380 q^{26} + 108 q^{29} - 304 q^{31} - 20 q^{34} + 306 q^{36} + 228 q^{39} + 188 q^{41} + 374 q^{44} - 680 q^{46} - 1362 q^{49} - 12 q^{51} + 270 q^{54} + 2880 q^{56} - 40 q^{59} + 1140 q^{61} + 574 q^{64} + 330 q^{66} - 408 q^{69} - 2184 q^{71} + 1740 q^{74} + 2448 q^{76} + 32 q^{79} + 162 q^{81} + 3264 q^{84} + 5280 q^{86} + 1932 q^{89} + 2432 q^{91} - 3400 q^{94} - 510 q^{96} + 198 q^{99}+O(q^{100}) \) 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
5.00000i 3.00000i −17.0000 0 −15.0000 32.0000i 45.0000i −9.00000 0
199.2 5.00000i 3.00000i −17.0000 0 −15.0000 32.0000i 45.0000i −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.a 2
5.b even 2 1 inner 825.4.c.a 2
5.c odd 4 1 33.4.a.a 1
5.c odd 4 1 825.4.a.i 1
15.e even 4 1 99.4.a.b 1
15.e even 4 1 2475.4.a.b 1
20.e even 4 1 528.4.a.a 1
35.f even 4 1 1617.4.a.a 1
40.i odd 4 1 2112.4.a.l 1
40.k even 4 1 2112.4.a.y 1
55.e even 4 1 363.4.a.h 1
60.l odd 4 1 1584.4.a.t 1
165.l odd 4 1 1089.4.a.a 1
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
33.4.a.a 1 5.c odd 4 1
99.4.a.b 1 15.e even 4 1
363.4.a.h 1 55.e even 4 1
528.4.a.a 1 20.e even 4 1
825.4.a.i 1 5.c odd 4 1
825.4.c.a 2 1.a even 1 1 trivial
825.4.c.a 2 5.b even 2 1 inner
1089.4.a.a 1 165.l odd 4 1
1584.4.a.t 1 60.l odd 4 1
1617.4.a.a 1 35.f even 4 1
2112.4.a.l 1 40.i odd 4 1
2112.4.a.y 1 40.k even 4 1
2475.4.a.b 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 \(S_{4}^{\mathrm{new}}(825, [\chi])\):

\( T_{2}^{2} + 25 \) Copy content Toggle raw display
\( T_{7}^{2} + 1024 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 25 \) Copy content Toggle raw display
$3$ \( T^{2} + 9 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 1024 \) Copy content Toggle raw display
$11$ \( (T + 11)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 1444 \) Copy content Toggle raw display
$17$ \( T^{2} + 4 \) Copy content Toggle raw display
$19$ \( (T + 72)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 4624 \) Copy content Toggle raw display
$29$ \( (T - 54)^{2} \) Copy content Toggle raw display
$31$ \( (T + 152)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 30276 \) Copy content Toggle raw display
$41$ \( (T - 94)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 278784 \) Copy content Toggle raw display
$47$ \( T^{2} + 115600 \) Copy content Toggle raw display
$53$ \( T^{2} + 191844 \) Copy content Toggle raw display
$59$ \( (T + 20)^{2} \) Copy content Toggle raw display
$61$ \( (T - 570)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 211600 \) Copy content Toggle raw display
$71$ \( (T + 1092)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 315844 \) Copy content Toggle raw display
$79$ \( (T - 16)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} + 138384 \) Copy content Toggle raw display
$89$ \( (T - 966)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 276676 \) Copy content Toggle raw display
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