Properties

Label 650.4.b.f
Level $650$
Weight $4$
Character orbit 650.b
Analytic conductor $38.351$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [650,4,Mod(599,650)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(650, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("650.599");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 650 = 2 \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 650.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: \(38.3512415037\)
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: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
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.

\(f(q)\) \(=\) \( q - 2 i q^{2} + 4 i q^{3} - 4 q^{4} + 8 q^{6} - 20 i q^{7} + 8 i q^{8} + 11 q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - 2 i q^{2} + 4 i q^{3} - 4 q^{4} + 8 q^{6} - 20 i q^{7} + 8 i q^{8} + 11 q^{9} - 48 q^{11} - 16 i q^{12} + 13 i q^{13} - 40 q^{14} + 16 q^{16} - 66 i q^{17} - 22 i q^{18} + 16 q^{19} + 80 q^{21} + 96 i q^{22} + 168 i q^{23} - 32 q^{24} + 26 q^{26} + 152 i q^{27} + 80 i q^{28} - 6 q^{29} + 20 q^{31} - 32 i q^{32} - 192 i q^{33} - 132 q^{34} - 44 q^{36} - 254 i q^{37} - 32 i q^{38} - 52 q^{39} - 390 q^{41} - 160 i q^{42} - 124 i q^{43} + 192 q^{44} + 336 q^{46} + 468 i q^{47} + 64 i q^{48} - 57 q^{49} + 264 q^{51} - 52 i q^{52} + 558 i q^{53} + 304 q^{54} + 160 q^{56} + 64 i q^{57} + 12 i q^{58} + 96 q^{59} - 826 q^{61} - 40 i q^{62} - 220 i q^{63} - 64 q^{64} - 384 q^{66} + 160 i q^{67} + 264 i q^{68} - 672 q^{69} - 420 q^{71} + 88 i q^{72} + 362 i q^{73} - 508 q^{74} - 64 q^{76} + 960 i q^{77} + 104 i q^{78} - 776 q^{79} - 311 q^{81} + 780 i q^{82} - 320 q^{84} - 248 q^{86} - 24 i q^{87} - 384 i q^{88} - 1626 q^{89} + 260 q^{91} - 672 i q^{92} + 80 i q^{93} + 936 q^{94} + 128 q^{96} + 1294 i q^{97} + 114 i q^{98} - 528 q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4} + 16 q^{6} + 22 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{4} + 16 q^{6} + 22 q^{9} - 96 q^{11} - 80 q^{14} + 32 q^{16} + 32 q^{19} + 160 q^{21} - 64 q^{24} + 52 q^{26} - 12 q^{29} + 40 q^{31} - 264 q^{34} - 88 q^{36} - 104 q^{39} - 780 q^{41} + 384 q^{44} + 672 q^{46} - 114 q^{49} + 528 q^{51} + 608 q^{54} + 320 q^{56} + 192 q^{59} - 1652 q^{61} - 128 q^{64} - 768 q^{66} - 1344 q^{69} - 840 q^{71} - 1016 q^{74} - 128 q^{76} - 1552 q^{79} - 622 q^{81} - 640 q^{84} - 496 q^{86} - 3252 q^{89} + 520 q^{91} + 1872 q^{94} + 256 q^{96} - 1056 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/650\mathbb{Z}\right)^\times\).

\(n\) \(27\) \(301\)
\(\chi(n)\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
599.1
1.00000i
1.00000i
2.00000i 4.00000i −4.00000 0 8.00000 20.0000i 8.00000i 11.0000 0
599.2 2.00000i 4.00000i −4.00000 0 8.00000 20.0000i 8.00000i 11.0000 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 650.4.b.f 2
5.b even 2 1 inner 650.4.b.f 2
5.c odd 4 1 26.4.a.c 1
5.c odd 4 1 650.4.a.b 1
15.e even 4 1 234.4.a.e 1
20.e even 4 1 208.4.a.b 1
35.f even 4 1 1274.4.a.d 1
40.i odd 4 1 832.4.a.d 1
40.k even 4 1 832.4.a.o 1
60.l odd 4 1 1872.4.a.q 1
65.f even 4 1 338.4.b.d 2
65.h odd 4 1 338.4.a.c 1
65.k even 4 1 338.4.b.d 2
65.o even 12 2 338.4.e.a 4
65.q odd 12 2 338.4.c.a 2
65.r odd 12 2 338.4.c.e 2
65.t even 12 2 338.4.e.a 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
26.4.a.c 1 5.c odd 4 1
208.4.a.b 1 20.e even 4 1
234.4.a.e 1 15.e even 4 1
338.4.a.c 1 65.h odd 4 1
338.4.b.d 2 65.f even 4 1
338.4.b.d 2 65.k even 4 1
338.4.c.a 2 65.q odd 12 2
338.4.c.e 2 65.r odd 12 2
338.4.e.a 4 65.o even 12 2
338.4.e.a 4 65.t even 12 2
650.4.a.b 1 5.c odd 4 1
650.4.b.f 2 1.a even 1 1 trivial
650.4.b.f 2 5.b even 2 1 inner
832.4.a.d 1 40.i odd 4 1
832.4.a.o 1 40.k even 4 1
1274.4.a.d 1 35.f even 4 1
1872.4.a.q 1 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(650, [\chi])\):

\( T_{3}^{2} + 16 \) Copy content Toggle raw display
\( T_{7}^{2} + 400 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{2} + 4 \) Copy content Toggle raw display
$3$ \( T^{2} + 16 \) Copy content Toggle raw display
$5$ \( T^{2} \) Copy content Toggle raw display
$7$ \( T^{2} + 400 \) Copy content Toggle raw display
$11$ \( (T + 48)^{2} \) Copy content Toggle raw display
$13$ \( T^{2} + 169 \) Copy content Toggle raw display
$17$ \( T^{2} + 4356 \) Copy content Toggle raw display
$19$ \( (T - 16)^{2} \) Copy content Toggle raw display
$23$ \( T^{2} + 28224 \) Copy content Toggle raw display
$29$ \( (T + 6)^{2} \) Copy content Toggle raw display
$31$ \( (T - 20)^{2} \) Copy content Toggle raw display
$37$ \( T^{2} + 64516 \) Copy content Toggle raw display
$41$ \( (T + 390)^{2} \) Copy content Toggle raw display
$43$ \( T^{2} + 15376 \) Copy content Toggle raw display
$47$ \( T^{2} + 219024 \) Copy content Toggle raw display
$53$ \( T^{2} + 311364 \) Copy content Toggle raw display
$59$ \( (T - 96)^{2} \) Copy content Toggle raw display
$61$ \( (T + 826)^{2} \) Copy content Toggle raw display
$67$ \( T^{2} + 25600 \) Copy content Toggle raw display
$71$ \( (T + 420)^{2} \) Copy content Toggle raw display
$73$ \( T^{2} + 131044 \) Copy content Toggle raw display
$79$ \( (T + 776)^{2} \) Copy content Toggle raw display
$83$ \( T^{2} \) Copy content Toggle raw display
$89$ \( (T + 1626)^{2} \) Copy content Toggle raw display
$97$ \( T^{2} + 1674436 \) Copy content Toggle raw display
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