Properties

Label 275.2.l.b
Level $275$
Weight $2$
Character orbit 275.l
Analytic conductor $2.196$
Analytic rank $1$
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(31,275)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(275, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([4, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("275.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 275 = 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 275.l (of order \(5\), degree \(4\), 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: \(1\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 primitive root of unity \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\zeta_{10}^{3} - \zeta_{10}^{2} - 1) q^{2} + ( - \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 1) q^{3} + ( - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{4} + ( - 2 \zeta_{10}^{2} + \zeta_{10} - 2) q^{5} + (\zeta_{10}^{2} - \zeta_{10}) q^{6} - 2 \zeta_{10}^{3} q^{7} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 1) q^{8} + (\zeta_{10}^{3} + 3 \zeta_{10} - 3) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + (\zeta_{10}^{3} - \zeta_{10}^{2} - 1) q^{2} + ( - \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 1) q^{3} + ( - \zeta_{10}^{3} + \zeta_{10}^{2}) q^{4} + ( - 2 \zeta_{10}^{2} + \zeta_{10} - 2) q^{5} + (\zeta_{10}^{2} - \zeta_{10}) q^{6} - 2 \zeta_{10}^{3} q^{7} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 1) q^{8} + (\zeta_{10}^{3} + 3 \zeta_{10} - 3) q^{9} + (\zeta_{10}^{2} + 2 \zeta_{10} + 1) q^{10} + ( - 2 \zeta_{10}^{3} - \zeta_{10} - 2) q^{11} + (\zeta_{10}^{3} - 3 \zeta_{10}^{2} + 3 \zeta_{10} - 1) q^{12} + (5 \zeta_{10}^{3} + 2 \zeta_{10} - 2) q^{13} + (2 \zeta_{10}^{3} + 2 \zeta_{10} - 2) q^{14} + (3 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 1) q^{15} + (3 \zeta_{10}^{3} - 3 \zeta_{10}^{2} - 3) q^{16} + ( - \zeta_{10}^{3} + 1) q^{17} + ( - 4 \zeta_{10}^{3} - \zeta_{10} + 1) q^{18} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 6) q^{19} + (\zeta_{10}^{2} - 3 \zeta_{10} + 1) q^{20} + (2 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 2 \zeta_{10}) q^{21} + (3 \zeta_{10}^{2} + 2 \zeta_{10} + 1) q^{22} + ( - 2 \zeta_{10}^{2} - 2) q^{23} + (\zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} - 1) q^{24} + 5 \zeta_{10}^{2} q^{25} + ( - 7 \zeta_{10}^{3} - 5 \zeta_{10} + 5) q^{26} + (2 \zeta_{10}^{3} - \zeta_{10}^{2} + 2 \zeta_{10}) q^{27} + ( - 2 \zeta_{10} + 2) q^{28} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - 7) q^{29} + (\zeta_{10}^{3} - \zeta_{10}^{2} + 2) q^{30} + ( - 3 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 3 \zeta_{10}) q^{31} + (\zeta_{10}^{3} - \zeta_{10}^{2} + 4) q^{32} + (3 \zeta_{10}^{3} - 7 \zeta_{10}^{2} + 6 \zeta_{10} - 3) q^{33} + (2 \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 2) q^{34} + (2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} - 2) q^{35} + (3 \zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{36} + (5 \zeta_{10}^{3} - 3 \zeta_{10}^{2} + 5 \zeta_{10}) q^{37} + ( - 6 \zeta_{10}^{3} + 6 \zeta_{10}^{2} + 8) q^{38} + ( - \zeta_{10}^{3} + 4 \zeta_{10}^{2} - \zeta_{10}) q^{39} - 5 \zeta_{10} q^{40} + ( - 2 \zeta_{10}^{2} - 6 \zeta_{10} - 2) q^{41} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10}) q^{42} + (6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} - 2) q^{43} + (2 \zeta_{10}^{3} - 3 \zeta_{10}^{2} - \zeta_{10} + 1) q^{44} + ( - 7 \zeta_{10}^{3} + 8 \zeta_{10}^{2} - 8 \zeta_{10} + 7) q^{45} + (2 \zeta_{10}^{2} + 2 \zeta_{10} + 2) q^{46} + ( - 6 \zeta_{10}^{3} + 10 \zeta_{10}^{2} - 10 \zeta_{10} + 6) q^{47} + (3 \zeta_{10}^{2} - 3 \zeta_{10}) q^{48} + 3 \zeta_{10} q^{49} + ( - 5 \zeta_{10}^{3} - 5 \zeta_{10}) q^{50} + ( - \zeta_{10} + 1) q^{51} + (2 \zeta_{10}^{3} + 3 \zeta_{10} - 3) q^{52} + ( - 2 \zeta_{10}^{2} + 5 \zeta_{10} - 2) q^{53} + ( - \zeta_{10}^{3} - 2 \zeta_{10}^{2} - \zeta_{10}) q^{54} + (4 \zeta_{10}^{3} + 5 \zeta_{10}^{2} - 2 \zeta_{10} + 2) q^{55} + ( - 2 \zeta_{10}^{3} - 4 \zeta_{10} + 4) q^{56} + (4 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 6 \zeta_{10} - 4) q^{57} + ( - 7 \zeta_{10}^{3} + 7 \zeta_{10}^{2} + 6) q^{58} + (4 \zeta_{10}^{3} + 3 \zeta_{10}^{2} + 4 \zeta_{10}) q^{59} + ( - 4 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 3) q^{60} - 3 \zeta_{10}^{2} q^{61} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 3 \zeta_{10}) q^{62} + (6 \zeta_{10}^{2} - 4 \zeta_{10} + 6) q^{63} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 3) q^{64} + ( - 9 \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 9) q^{65} + (\zeta_{10}^{3} - 3 \zeta_{10}^{2} + 4 \zeta_{10}) q^{66} - 7 \zeta_{10}^{3} q^{67} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{68} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2}) q^{69} + ( - 6 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} + 6) q^{70} + ( - 7 \zeta_{10}^{3} - 5 \zeta_{10} + 5) q^{71} + (7 \zeta_{10}^{3} - \zeta_{10} + 1) q^{72} + ( - 11 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 11) q^{73} + ( - 2 \zeta_{10}^{3} - 5 \zeta_{10}^{2} - 2 \zeta_{10}) q^{74} + ( - 5 \zeta_{10}^{2} + 10 \zeta_{10} - 5) q^{75} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} - 2) q^{76} + (6 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 2 \zeta_{10} - 2) q^{77} + ( - 3 \zeta_{10}^{3} + \zeta_{10}^{2} - 3 \zeta_{10}) q^{78} + (4 \zeta_{10}^{2} - 12 \zeta_{10} + 4) q^{79} + (3 \zeta_{10}^{2} + 6 \zeta_{10} + 3) q^{80} + ( - 6 \zeta_{10}^{2} + 2 \zeta_{10} - 6) q^{81} + (8 \zeta_{10}^{2} + 2 \zeta_{10} + 8) q^{82} + (2 \zeta_{10}^{3} - 10 \zeta_{10}^{2} + 10 \zeta_{10} - 2) q^{83} + ( - 4 \zeta_{10}^{3} + 6 \zeta_{10}^{2} - 4 \zeta_{10}) q^{84} + (\zeta_{10}^{3} - \zeta_{10}^{2} - 3) q^{85} + ( - 2 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 8) q^{86} + (8 \zeta_{10}^{3} - 17 \zeta_{10}^{2} + 17 \zeta_{10} - 8) q^{87} + (2 \zeta_{10}^{3} - 6 \zeta_{10}^{2} - 3 \zeta_{10}) q^{88} + ( - 5 \zeta_{10}^{2} + 10 \zeta_{10} - 5) q^{89} + (6 \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} - 6) q^{90} + (4 \zeta_{10}^{2} + 6 \zeta_{10} + 4) q^{91} - 2 \zeta_{10} q^{92} + ( - 9 \zeta_{10}^{2} + 15 \zeta_{10} - 9) q^{93} + (2 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 4 \zeta_{10} - 2) q^{94} + (10 \zeta_{10}^{2} + 10) q^{95} + ( - 5 \zeta_{10}^{3} + 11 \zeta_{10}^{2} - 11 \zeta_{10} + 5) q^{96} + (4 \zeta_{10}^{2} - 9 \zeta_{10} + 4) q^{97} + ( - 3 \zeta_{10}^{2} - 3) q^{98} + ( - 3 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 8 \zeta_{10} + 13) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{2} - q^{3} - 2 q^{4} - 5 q^{5} - 2 q^{6} - 2 q^{7} - 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{2} - q^{3} - 2 q^{4} - 5 q^{5} - 2 q^{6} - 2 q^{7} - 8 q^{9} + 5 q^{10} - 11 q^{11} + 3 q^{12} - q^{13} - 4 q^{14} + 10 q^{15} - 6 q^{16} + 3 q^{17} - q^{18} - 20 q^{19} + 8 q^{21} + 3 q^{22} - 6 q^{23} + 5 q^{24} - 5 q^{25} + 8 q^{26} + 5 q^{27} + 6 q^{28} - 30 q^{29} + 10 q^{30} - 12 q^{31} + 18 q^{32} + 4 q^{33} - 4 q^{34} - 10 q^{35} + 9 q^{36} + 13 q^{37} + 20 q^{38} - 6 q^{39} - 5 q^{40} - 12 q^{41} + 6 q^{42} + 4 q^{43} + 8 q^{44} + 5 q^{45} + 8 q^{46} - 2 q^{47} - 6 q^{48} + 3 q^{49} - 10 q^{50} + 3 q^{51} - 7 q^{52} - q^{53} + 5 q^{55} + 10 q^{56} + 10 q^{58} + 5 q^{59} - 20 q^{60} + 3 q^{61} - 9 q^{62} + 14 q^{63} + 8 q^{64} + 25 q^{65} + 8 q^{66} - 7 q^{67} + q^{68} + 4 q^{69} + 10 q^{70} + 8 q^{71} + 10 q^{72} + 29 q^{73} + q^{74} - 5 q^{75} - 2 q^{77} - 7 q^{78} + 15 q^{80} - 16 q^{81} + 26 q^{82} + 14 q^{83} - 14 q^{84} - 10 q^{85} + 28 q^{86} + 10 q^{87} + 5 q^{88} - 5 q^{89} - 20 q^{90} + 18 q^{91} - 2 q^{92} - 12 q^{93} - 14 q^{94} + 30 q^{95} - 7 q^{96} + 3 q^{97} - 9 q^{98} + 37 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(177\)
\(\chi(n)\) \(-\zeta_{10}^{3}\) \(\zeta_{10}^{2}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
−0.309017 0.951057i
−0.309017 + 0.951057i
0.809017 + 0.587785i
0.809017 0.587785i
0.618034 −0.809017 + 2.48990i −1.61803 −0.690983 2.12663i −0.500000 + 1.53884i −1.61803 1.17557i −2.23607 −3.11803 2.26538i −0.427051 1.31433i
71.1 0.618034 −0.809017 2.48990i −1.61803 −0.690983 + 2.12663i −0.500000 1.53884i −1.61803 + 1.17557i −2.23607 −3.11803 + 2.26538i −0.427051 + 1.31433i
91.1 −1.61803 0.309017 0.224514i 0.618034 −1.80902 1.31433i −0.500000 + 0.363271i 0.618034 1.90211i 2.23607 −0.881966 + 2.71441i 2.92705 + 2.12663i
136.1 −1.61803 0.309017 + 0.224514i 0.618034 −1.80902 + 1.31433i −0.500000 0.363271i 0.618034 + 1.90211i 2.23607 −0.881966 2.71441i 2.92705 2.12663i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
275.l even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 275.2.l.b yes 4
11.c even 5 1 275.2.k.b 4
25.d even 5 1 275.2.k.b 4
275.l even 5 1 inner 275.2.l.b yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
275.2.k.b 4 11.c even 5 1
275.2.k.b 4 25.d even 5 1
275.2.l.b yes 4 1.a even 1 1 trivial
275.2.l.b yes 4 275.l even 5 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(275, [\chi])\):

\( T_{2}^{2} + T_{2} - 1 \) Copy content Toggle raw display
\( T_{3}^{4} + T_{3}^{3} + 6T_{3}^{2} - 4T_{3} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T - 1)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + T^{3} + 6 T^{2} - 4 T + 1 \) Copy content Toggle raw display
$5$ \( T^{4} + 5 T^{3} + 15 T^{2} + 25 T + 25 \) Copy content Toggle raw display
$7$ \( T^{4} + 2 T^{3} + 4 T^{2} + 8 T + 16 \) Copy content Toggle raw display
$11$ \( T^{4} + 11 T^{3} + 51 T^{2} + \cdots + 121 \) Copy content Toggle raw display
$13$ \( T^{4} + T^{3} + 31 T^{2} - 99 T + 121 \) Copy content Toggle raw display
$17$ \( T^{4} - 3 T^{3} + 4 T^{2} - 2 T + 1 \) Copy content Toggle raw display
$19$ \( (T^{2} + 10 T + 20)^{2} \) Copy content Toggle raw display
$23$ \( T^{4} + 6 T^{3} + 16 T^{2} + 16 T + 16 \) Copy content Toggle raw display
$29$ \( (T^{2} + 15 T + 55)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + 12 T^{3} + 54 T^{2} - 27 T + 81 \) Copy content Toggle raw display
$37$ \( T^{4} - 13 T^{3} + 94 T^{2} + \cdots + 961 \) Copy content Toggle raw display
$41$ \( T^{4} + 12 T^{3} + 64 T^{2} + \cdots + 1936 \) Copy content Toggle raw display
$43$ \( (T^{2} - 2 T - 44)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 2 T^{3} + 124 T^{2} + \cdots + 1936 \) Copy content Toggle raw display
$53$ \( T^{4} + T^{3} + 31 T^{2} - 99 T + 121 \) Copy content Toggle raw display
$59$ \( T^{4} - 5 T^{3} + 85 T^{2} + 75 T + 25 \) Copy content Toggle raw display
$61$ \( T^{4} - 3 T^{3} + 9 T^{2} - 27 T + 81 \) Copy content Toggle raw display
$67$ \( T^{4} + 7 T^{3} + 49 T^{2} + \cdots + 2401 \) Copy content Toggle raw display
$71$ \( T^{4} - 8 T^{3} + 114 T^{2} + \cdots + 121 \) Copy content Toggle raw display
$73$ \( T^{4} - 29 T^{3} + 346 T^{2} + \cdots + 3481 \) Copy content Toggle raw display
$79$ \( T^{4} + 160 T^{2} + 1600 T + 6400 \) Copy content Toggle raw display
$83$ \( T^{4} - 14 T^{3} + 276 T^{2} + \cdots + 1936 \) Copy content Toggle raw display
$89$ \( T^{4} + 5 T^{3} + 150 T^{2} + \cdots + 625 \) Copy content Toggle raw display
$97$ \( T^{4} - 3 T^{3} + 109 T^{2} + \cdots + 841 \) Copy content Toggle raw display
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