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Properties

Label 275.2.b.d
Level $275$
Weight $2$
Character orbit 275.b
Analytic conductor $2.196$
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] = [275,2,Mod(199,275)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(275, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("275.199");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 275.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: \(2.19588605559\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 55)
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.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + ( - \beta_{2} + \beta_1) q^{3} + ( - \beta_{3} - 1) q^{4} + ( - \beta_{3} - 4) q^{6} + ( - \beta_{2} - \beta_1) q^{7} + ( - \beta_{2} - 2 \beta_1) q^{8} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + ( - \beta_{2} + \beta_1) q^{3} + ( - \beta_{3} - 1) q^{4} + ( - \beta_{3} - 4) q^{6} + ( - \beta_{2} - \beta_1) q^{7} + ( - \beta_{2} - 2 \beta_1) q^{8} - 5 q^{9} + q^{11} + ( - 3 \beta_{2} - 5 \beta_1) q^{12} + (3 \beta_{2} + \beta_1) q^{13} + (\beta_{3} + 2) q^{14} + 3 q^{16} + (\beta_{2} + 3 \beta_1) q^{17} - 5 \beta_1 q^{18} + 2 \beta_{3} q^{21} + \beta_1 q^{22} + ( - \beta_{2} + \beta_1) q^{23} + (3 \beta_{3} + 4) q^{24} - \beta_{3} q^{26} + (2 \beta_{2} - 2 \beta_1) q^{27} + ( - \beta_{2} + 3 \beta_1) q^{28} + (2 \beta_{3} - 2) q^{29} + ( - 2 \beta_{2} - \beta_1) q^{32} + ( - \beta_{2} + \beta_1) q^{33} + ( - 3 \beta_{3} - 8) q^{34} + (5 \beta_{3} + 5) q^{36} + (\beta_{2} - 3 \beta_1) q^{37} + ( - 4 \beta_{3} + 8) q^{39} + 6 q^{41} + (2 \beta_{2} + 6 \beta_1) q^{42} + (3 \beta_{2} + 3 \beta_1) q^{43} + ( - \beta_{3} - 1) q^{44} + ( - \beta_{3} - 4) q^{46} + ( - \beta_{2} + \beta_1) q^{47} + ( - 3 \beta_{2} + 3 \beta_1) q^{48} + 3 q^{49} + ( - 4 \beta_{3} - 8) q^{51} + (5 \beta_{2} - \beta_1) q^{52} + ( - \beta_{2} - 5 \beta_1) q^{53} + (2 \beta_{3} + 8) q^{54} + ( - \beta_{3} - 6) q^{56} + (2 \beta_{2} + 4 \beta_1) q^{58} + ( - 2 \beta_{3} + 4) q^{59} + ( - 4 \beta_{3} + 2) q^{61} + (5 \beta_{2} + 5 \beta_1) q^{63} + (\beta_{3} + 7) q^{64} + ( - \beta_{3} - 4) q^{66} + ( - \beta_{2} + 5 \beta_1) q^{67} + ( - \beta_{2} - 11 \beta_1) q^{68} - 8 q^{69} + 4 \beta_{3} q^{71} + (5 \beta_{2} + 10 \beta_1) q^{72} + (3 \beta_{2} + \beta_1) q^{73} + (3 \beta_{3} + 10) q^{74} + ( - \beta_{2} - \beta_1) q^{77} + ( - 4 \beta_{2} - 4 \beta_1) q^{78} - 4 q^{79} + q^{81} + 6 \beta_1 q^{82} + (3 \beta_{2} + 3 \beta_1) q^{83} + ( - 2 \beta_{3} - 16) q^{84} + ( - 3 \beta_{3} - 6) q^{86} + (10 \beta_{2} + 6 \beta_1) q^{87} + ( - \beta_{2} - 2 \beta_1) q^{88} + (4 \beta_{3} + 2) q^{89} + ( - 2 \beta_{3} + 8) q^{91} + ( - 3 \beta_{2} - 5 \beta_1) q^{92} + ( - \beta_{3} - 4) q^{94} + (3 \beta_{3} - 4) q^{96} + ( - 3 \beta_{2} + \beta_1) q^{97} + 3 \beta_1 q^{98} - 5 q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} - 16 q^{6} - 20 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{4} - 16 q^{6} - 20 q^{9} + 4 q^{11} + 8 q^{14} + 12 q^{16} + 16 q^{24} - 8 q^{29} - 32 q^{34} + 20 q^{36} + 32 q^{39} + 24 q^{41} - 4 q^{44} - 16 q^{46} + 12 q^{49} - 32 q^{51} + 32 q^{54} - 24 q^{56} + 16 q^{59} + 8 q^{61} + 28 q^{64} - 16 q^{66} - 32 q^{69} + 40 q^{74} - 16 q^{79} + 4 q^{81} - 64 q^{84} - 24 q^{86} + 8 q^{89} + 32 q^{91} - 16 q^{94} - 16 q^{96} - 20 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring

\(\beta_{1}\)\(=\) \( \zeta_{8}^{3} + \zeta_{8}^{2} + \zeta_{8} \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( -\zeta_{8}^{3} + \zeta_{8}^{2} - \zeta_{8} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( -2\zeta_{8}^{3} + 2\zeta_{8} \) Copy content Toggle raw display

Display \(\zeta_{8}^j\) in terms of \(\beta_i\)

\(\zeta_{8}\)\(=\) \( ( \beta_{3} - \beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display
\(\zeta_{8}^{2}\)\(=\) \( ( \beta_{2} + \beta_1 ) / 2 \) Copy content Toggle raw display
\(\zeta_{8}^{3}\)\(=\) \( ( -\beta_{3} - \beta_{2} + \beta_1 ) / 4 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\zeta_{8}^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/275\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(177\)
\(\chi(n)\) \(1\) \(-1\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
199.1
0.707107 − 0.707107i
−0.707107 − 0.707107i
−0.707107 + 0.707107i
0.707107 + 0.707107i
− 2.41421i − 2.82843i −3.82843 0 −6.82843 2.00000i 4.41421i −5.00000 0
199.2 − 0.414214i − 2.82843i 1.82843 0 −1.17157 − 2.00000i − 1.58579i −5.00000 0
199.3 0.414214i 2.82843i 1.82843 0 −1.17157 2.00000i 1.58579i −5.00000 0
199.4 2.41421i 2.82843i −3.82843 0 −6.82843 − 2.00000i − 4.41421i −5.00000 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 275.2.b.d 4
3.b odd 2 1 2475.2.c.l 4
4.b odd 2 1 4400.2.b.q 4
5.b even 2 1 inner 275.2.b.d 4
5.c odd 4 1 55.2.a.b ✓ 2
5.c odd 4 1 275.2.a.c 2
15.d odd 2 1 2475.2.c.l 4
15.e even 4 1 495.2.a.b 2
15.e even 4 1 2475.2.a.x 2
20.d odd 2 1 4400.2.b.q 4
20.e even 4 1 880.2.a.m 2
20.e even 4 1 4400.2.a.bn 2
35.f even 4 1 2695.2.a.f 2
40.i odd 4 1 3520.2.a.bn 2
40.k even 4 1 3520.2.a.bo 2
55.e even 4 1 605.2.a.d 2
55.e even 4 1 3025.2.a.o 2
55.k odd 20 4 605.2.g.f 8
55.l even 20 4 605.2.g.l 8
60.l odd 4 1 7920.2.a.ch 2
65.h odd 4 1 9295.2.a.g 2
165.l odd 4 1 5445.2.a.y 2
220.i odd 4 1 9680.2.a.bn 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
55.2.a.b ✓ 2 5.c odd 4 1
275.2.a.c 2 5.c odd 4 1
275.2.b.d 4 1.a even 1 1 trivial
275.2.b.d 4 5.b even 2 1 inner
495.2.a.b 2 15.e even 4 1
605.2.a.d 2 55.e even 4 1
605.2.g.f 8 55.k odd 20 4
605.2.g.l 8 55.l even 20 4
880.2.a.m 2 20.e even 4 1
2475.2.a.x 2 15.e even 4 1
2475.2.c.l 4 3.b odd 2 1
2475.2.c.l 4 15.d odd 2 1
2695.2.a.f 2 35.f even 4 1
3025.2.a.o 2 55.e even 4 1
3520.2.a.bn 2 40.i odd 4 1
3520.2.a.bo 2 40.k even 4 1
4400.2.a.bn 2 20.e even 4 1
4400.2.b.q 4 4.b odd 2 1
4400.2.b.q 4 20.d odd 2 1
5445.2.a.y 2 165.l odd 4 1
7920.2.a.ch 2 60.l odd 4 1
9295.2.a.g 2 65.h odd 4 1
9680.2.a.bn 2 220.i odd 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{4} + 6T_{2}^{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(275, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} + 6T^{2} + 1 \) Copy content Toggle raw display
$3$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} \) Copy content Toggle raw display
$7$ \( (T^{2} + 4)^{2} \) Copy content Toggle raw display
$11$ \( (T - 1)^{4} \) Copy content Toggle raw display
$13$ \( T^{4} + 48T^{2} + 64 \) Copy content Toggle raw display
$17$ \( T^{4} + 48T^{2} + 64 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 4 T - 28)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} \) Copy content Toggle raw display
$37$ \( T^{4} + 72T^{2} + 784 \) Copy content Toggle raw display
$41$ \( (T - 6)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 8)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} + 136T^{2} + 16 \) Copy content Toggle raw display
$59$ \( (T^{2} - 8 T - 16)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} - 4 T - 124)^{2} \) Copy content Toggle raw display
$67$ \( T^{4} + 176T^{2} + 3136 \) Copy content Toggle raw display
$71$ \( (T^{2} - 128)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + 48T^{2} + 64 \) Copy content Toggle raw display
$79$ \( (T + 4)^{4} \) Copy content Toggle raw display
$83$ \( (T^{2} + 36)^{2} \) Copy content Toggle raw display
$89$ \( (T^{2} - 4 T - 124)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 72T^{2} + 784 \) Copy content Toggle raw display
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