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Properties

Label 1300.4.c.d
Level $1300$
Weight $4$
Character orbit 1300.c
Analytic conductor $76.702$
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] = [1300,4,Mod(1249,1300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1300, 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("1300.1249");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1300 = 2^{2} \cdot 5^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1300.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: \(76.7024830075\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{217})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 109x^{2} + 2916 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 52)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{2} + \beta_1) q^{3} + ( - 13 \beta_{2} + \beta_1) q^{7} + (3 \beta_{3} - 31) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{2} + \beta_1) q^{3} + ( - 13 \beta_{2} + \beta_1) q^{7} + (3 \beta_{3} - 31) q^{9} + 2 \beta_{3} q^{11} + 13 \beta_{2} q^{13} + (\beta_{2} + \beta_1) q^{17} + (10 \beta_{3} + 16) q^{19} + (15 \beta_{3} - 82) q^{21} - 24 \beta_1 q^{23} + (163 \beta_{2} - 7 \beta_1) q^{27} + ( - 24 \beta_{3} + 42) q^{29} + (12 \beta_{3} - 208) q^{31} + (106 \beta_{2} - 2 \beta_1) q^{33} + ( - 55 \beta_{2} - 35 \beta_1) q^{37} + ( - 13 \beta_{3} + 26) q^{39} + (14 \beta_{3} + 246) q^{41} + ( - 191 \beta_{2} - \beta_1) q^{43} + ( - 233 \beta_{2} - 11 \beta_1) q^{47} + (27 \beta_{3} + 93) q^{49} + (\beta_{3} - 54) q^{51} + (236 \beta_{2} - 22 \beta_1) q^{53} + (514 \beta_{2} + 6 \beta_1) q^{57} + ( - 22 \beta_{3} - 120) q^{59} + (74 \beta_{3} - 166) q^{61} + (526 \beta_{2} - 70 \beta_1) q^{63} + (310 \beta_{2} - 18 \beta_1) q^{67} + ( - 48 \beta_{3} + 1344) q^{69} + (61 \beta_{3} + 666) q^{71} + ( - 94 \beta_{2} - 72 \beta_1) q^{73} + (82 \beta_{2} - 26 \beta_1) q^{77} + (72 \beta_{3} + 16) q^{79} + ( - 96 \beta_{3} - 119) q^{81} + ( - 708 \beta_{2} - 24 \beta_1) q^{83} + ( - 1314 \beta_{2} + 66 \beta_1) q^{87} + ( - 80 \beta_{3} - 138) q^{89} + ( - 13 \beta_{3} + 182) q^{91} + (844 \beta_{2} - 220 \beta_1) q^{93} + ( - 238 \beta_{2} - 164 \beta_1) q^{97} + ( - 56 \beta_{3} + 324) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 118 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 118 q^{9} + 4 q^{11} + 84 q^{19} - 298 q^{21} + 120 q^{29} - 808 q^{31} + 78 q^{39} + 1012 q^{41} + 426 q^{49} - 214 q^{51} - 524 q^{59} - 516 q^{61} + 5280 q^{69} + 2786 q^{71} + 208 q^{79} - 668 q^{81} - 712 q^{89} + 702 q^{91} + 1184 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 109x^{2} + 2916 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 55\nu ) / 54 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 55 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 55 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 54\beta_{2} - 55\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1300\mathbb{Z}\right)^\times\).

\(n\) \(301\) \(651\) \(677\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1249.1
− 7.86546i
− 6.86546i
6.86546i
7.86546i
0 − 8.86546i 0 0 0 − 20.8655i 0 −51.5964 0
1249.2 0 − 5.86546i 0 0 0 6.13454i 0 −7.40362 0
1249.3 0 5.86546i 0 0 0 − 6.13454i 0 −7.40362 0
1249.4 0 8.86546i 0 0 0 20.8655i 0 −51.5964 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 1300.4.c.d 4
5.b even 2 1 inner 1300.4.c.d 4
5.c odd 4 1 52.4.a.b ✓ 2
5.c odd 4 1 1300.4.a.g 2
15.e even 4 1 468.4.a.e 2
20.e even 4 1 208.4.a.k 2
40.i odd 4 1 832.4.a.x 2
40.k even 4 1 832.4.a.t 2
60.l odd 4 1 1872.4.a.u 2
65.f even 4 1 676.4.d.b 4
65.h odd 4 1 676.4.a.c 2
65.k even 4 1 676.4.d.b 4
65.o even 12 2 676.4.h.f 8
65.q odd 12 2 676.4.e.e 4
65.r odd 12 2 676.4.e.d 4
65.t even 12 2 676.4.h.f 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
52.4.a.b ✓ 2 5.c odd 4 1
208.4.a.k 2 20.e even 4 1
468.4.a.e 2 15.e even 4 1
676.4.a.c 2 65.h odd 4 1
676.4.d.b 4 65.f even 4 1
676.4.d.b 4 65.k even 4 1
676.4.e.d 4 65.r odd 12 2
676.4.e.e 4 65.q odd 12 2
676.4.h.f 8 65.o even 12 2
676.4.h.f 8 65.t even 12 2
832.4.a.t 2 40.k even 4 1
832.4.a.x 2 40.i odd 4 1
1300.4.a.g 2 5.c odd 4 1
1300.4.c.d 4 1.a even 1 1 trivial
1300.4.c.d 4 5.b even 2 1 inner
1872.4.a.u 2 60.l odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{4} + 113T_{3}^{2} + 2704 \) acting on \(S_{4}^{\mathrm{new}}(1300, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} \) Copy content Toggle raw display
$3$ \( T^{4} + 113T^{2} + 2704 \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} + 473 T^{2} + 16384 \) Copy content Toggle raw display
$11$ \( (T^{2} - 2 T - 216)^{2} \) Copy content Toggle raw display
$13$ \( (T^{2} + 169)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + 109T^{2} + 2916 \) Copy content Toggle raw display
$19$ \( (T^{2} - 42 T - 4984)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 62784 T^{2} + 967458816 \) Copy content Toggle raw display
$29$ \( (T^{2} - 60 T - 30348)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 404 T + 32992)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 4231502500 \) Copy content Toggle raw display
$41$ \( (T^{2} - 506 T + 53376)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + \cdots + 1313047696 \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 2042316864 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 1207701504 \) Copy content Toggle raw display
$59$ \( (T^{2} + 262 T - 9096)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 258 T - 280432)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 7086945856 \) Copy content Toggle raw display
$71$ \( (T^{2} - 1393 T + 283248)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 77210625424 \) Copy content Toggle raw display
$79$ \( (T^{2} - 104 T - 278528)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + \cdots + 205361236224 \) Copy content Toggle raw display
$89$ \( (T^{2} + 356 T - 315516)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + \cdots + 2058570691984 \) Copy content Toggle raw display
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