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Properties

Label 1275.2.b.d
Level $1275$
Weight $2$
Character orbit 1275.b
Analytic conductor $10.181$
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] = [1275,2,Mod(1174,1275)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1275, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1275.1174");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1275 = 3 \cdot 5^{2} \cdot 17 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1275.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: \(10.1809262577\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) 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 51)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{2} q^{3} + (\beta_{3} - 3) q^{4} + ( - \beta_{3} + 1) q^{6} + (4 \beta_{2} - \beta_1) q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{2} q^{3} + (\beta_{3} - 3) q^{4} + ( - \beta_{3} + 1) q^{6} + (4 \beta_{2} - \beta_1) q^{8} - q^{9} - \beta_{3} q^{11} + ( - 2 \beta_{2} + \beta_1) q^{12} + ( - 3 \beta_{2} - \beta_1) q^{13} + ( - 3 \beta_{3} + 3) q^{16} + \beta_{2} q^{17} - \beta_1 q^{18} - 3 \beta_{3} q^{19} - 4 \beta_{2} q^{22} + (5 \beta_{2} + \beta_1) q^{23} + (\beta_{3} - 5) q^{24} + (2 \beta_{3} + 2) q^{26} - \beta_{2} q^{27} + ( - 4 \beta_{3} + 2) q^{29} - 2 \beta_{3} q^{31} + ( - 4 \beta_{2} + \beta_1) q^{32} + ( - \beta_{2} - \beta_1) q^{33} + ( - \beta_{3} + 1) q^{34} + ( - \beta_{3} + 3) q^{36} + 2 \beta_1 q^{37} - 12 \beta_{2} q^{38} + (\beta_{3} + 2) q^{39} + (\beta_{3} - 2) q^{41} + (3 \beta_{2} + 3 \beta_1) q^{43} + (2 \beta_{3} - 4) q^{44} - 4 \beta_{3} q^{46} + ( - 6 \beta_{2} + 2 \beta_1) q^{47} - 3 \beta_1 q^{48} + 7 q^{49} - q^{51} + 2 \beta_{2} q^{52} + ( - 2 \beta_{2} + 4 \beta_1) q^{53} + (\beta_{3} - 1) q^{54} + ( - 3 \beta_{2} - 3 \beta_1) q^{57} + ( - 16 \beta_{2} + 2 \beta_1) q^{58} + (2 \beta_{3} - 4) q^{59} + ( - 2 \beta_{3} + 6) q^{61} - 8 \beta_{2} q^{62} + ( - \beta_{3} - 3) q^{64} + 4 q^{66} + 4 \beta_{2} q^{67} + ( - 2 \beta_{2} + \beta_1) q^{68} + ( - \beta_{3} - 4) q^{69} + 4 \beta_{3} q^{71} + ( - 4 \beta_{2} + \beta_1) q^{72} + (2 \beta_{2} - 4 \beta_1) q^{73} + (2 \beta_{3} - 10) q^{74} + (6 \beta_{3} - 12) q^{76} + (4 \beta_{2} + 2 \beta_1) q^{78} - 6 \beta_{3} q^{79} + q^{81} + (4 \beta_{2} - 2 \beta_1) q^{82} + (6 \beta_{2} + 2 \beta_1) q^{83} - 12 q^{86} + ( - 2 \beta_{2} - 4 \beta_1) q^{87} - 4 \beta_1 q^{88} + ( - 2 \beta_{3} - 2) q^{89} + ( - 6 \beta_{2} + 2 \beta_1) q^{92} + ( - 2 \beta_{2} - 2 \beta_1) q^{93} + (8 \beta_{3} - 16) q^{94} + ( - \beta_{3} + 5) q^{96} + ( - 8 \beta_{2} - 2 \beta_1) q^{97} + 7 \beta_1 q^{98} + \beta_{3} q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{4} + 2 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 10 q^{4} + 2 q^{6} - 4 q^{9} - 2 q^{11} + 6 q^{16} - 6 q^{19} - 18 q^{24} + 12 q^{26} - 4 q^{31} + 2 q^{34} + 10 q^{36} + 10 q^{39} - 6 q^{41} - 12 q^{44} - 8 q^{46} + 28 q^{49} - 4 q^{51} - 2 q^{54} - 12 q^{59} + 20 q^{61} - 14 q^{64} + 16 q^{66} - 18 q^{69} + 8 q^{71} - 36 q^{74} - 36 q^{76} - 12 q^{79} + 4 q^{81} - 48 q^{86} - 12 q^{89} - 48 q^{94} + 18 q^{96} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} + 9x^{2} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{3} + 5\nu ) / 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{2} + 5 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} - 5 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{2} - 5\beta_1 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1275\mathbb{Z}\right)^\times\).

\(n\) \(52\) \(751\) \(851\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1174.1
− 2.56155i
− 1.56155i
1.56155i
2.56155i
− 2.56155i 1.00000i −4.56155 0 2.56155 0 6.56155i −1.00000 0
1174.2 − 1.56155i − 1.00000i −0.438447 0 −1.56155 0 − 2.43845i −1.00000 0
1174.3 1.56155i 1.00000i −0.438447 0 −1.56155 0 2.43845i −1.00000 0
1174.4 2.56155i − 1.00000i −4.56155 0 2.56155 0 − 6.56155i −1.00000 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 1275.2.b.d 4
5.b even 2 1 inner 1275.2.b.d 4
5.c odd 4 1 51.2.a.b ✓ 2
5.c odd 4 1 1275.2.a.n 2
15.e even 4 1 153.2.a.e 2
15.e even 4 1 3825.2.a.s 2
20.e even 4 1 816.2.a.m 2
35.f even 4 1 2499.2.a.o 2
40.i odd 4 1 3264.2.a.bl 2
40.k even 4 1 3264.2.a.bg 2
55.e even 4 1 6171.2.a.p 2
60.l odd 4 1 2448.2.a.v 2
65.h odd 4 1 8619.2.a.q 2
85.f odd 4 1 867.2.d.c 4
85.g odd 4 1 867.2.a.f 2
85.i odd 4 1 867.2.d.c 4
85.k odd 8 2 867.2.e.f 8
85.n odd 8 2 867.2.e.f 8
85.o even 16 4 867.2.h.j 16
85.r even 16 4 867.2.h.j 16
105.k odd 4 1 7497.2.a.v 2
120.q odd 4 1 9792.2.a.cz 2
120.w even 4 1 9792.2.a.cy 2
255.o even 4 1 2601.2.a.t 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
51.2.a.b ✓ 2 5.c odd 4 1
153.2.a.e 2 15.e even 4 1
816.2.a.m 2 20.e even 4 1
867.2.a.f 2 85.g odd 4 1
867.2.d.c 4 85.f odd 4 1
867.2.d.c 4 85.i odd 4 1
867.2.e.f 8 85.k odd 8 2
867.2.e.f 8 85.n odd 8 2
867.2.h.j 16 85.o even 16 4
867.2.h.j 16 85.r even 16 4
1275.2.a.n 2 5.c odd 4 1
1275.2.b.d 4 1.a even 1 1 trivial
1275.2.b.d 4 5.b even 2 1 inner
2448.2.a.v 2 60.l odd 4 1
2499.2.a.o 2 35.f even 4 1
2601.2.a.t 2 255.o even 4 1
3264.2.a.bg 2 40.k even 4 1
3264.2.a.bl 2 40.i odd 4 1
3825.2.a.s 2 15.e even 4 1
6171.2.a.p 2 55.e even 4 1
7497.2.a.v 2 105.k odd 4 1
8619.2.a.q 2 65.h odd 4 1
9792.2.a.cy 2 120.w even 4 1
9792.2.a.cz 2 120.q odd 4 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(1275, [\chi])\):

\( T_{2}^{4} + 9T_{2}^{2} + 16 \) Copy content Toggle raw display
\( T_{7} \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 9T^{2} + 16 \) Copy content Toggle raw display
$3$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( T^{4} \) Copy content Toggle raw display
$11$ \( (T^{2} + T - 4)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} + 21T^{2} + 4 \) Copy content Toggle raw display
$17$ \( (T^{2} + 1)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} + 3 T - 36)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 49T^{2} + 256 \) Copy content Toggle raw display
$29$ \( (T^{2} - 68)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} + 2 T - 16)^{2} \) Copy content Toggle raw display
$37$ \( T^{4} + 36T^{2} + 256 \) Copy content Toggle raw display
$41$ \( (T^{2} + 3 T - 2)^{2} \) Copy content Toggle raw display
$43$ \( T^{4} + 81T^{2} + 1296 \) Copy content Toggle raw display
$47$ \( T^{4} + 132T^{2} + 1024 \) Copy content Toggle raw display
$53$ \( T^{4} + 168T^{2} + 2704 \) Copy content Toggle raw display
$59$ \( (T^{2} + 6 T - 8)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 10 T + 8)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} + 16)^{2} \) Copy content Toggle raw display
$71$ \( (T^{2} - 4 T - 64)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 168T^{2} + 2704 \) Copy content Toggle raw display
$79$ \( (T^{2} + 6 T - 144)^{2} \) Copy content Toggle raw display
$83$ \( T^{4} + 84T^{2} + 64 \) Copy content Toggle raw display
$89$ \( (T^{2} + 6 T - 8)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 132T^{2} + 1024 \) Copy content Toggle raw display
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