Properties

Label 115.2.g.a
Level $115$
Weight $2$
Character orbit 115.g
Analytic conductor $0.918$
Analytic rank $0$
Dimension $10$
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] = [115,2,Mod(6,115)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(115, base_ring=CyclotomicField(22))
 
chi = DirichletCharacter(H, H._module([0, 18]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("115.6");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 115.g (of order \(11\), degree \(10\), 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: \(0.918279623245\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\Q(\zeta_{22})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{11}]$

$q$-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 \(\zeta_{22}\). We also show the integral \(q\)-expansion of the trace form.

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/115\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(1\) \(\zeta_{22}^{4}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1
0.959493 0.281733i
−0.841254 0.540641i
−0.415415 0.909632i
−0.415415 + 0.909632i
−0.841254 + 0.540641i
0.654861 + 0.755750i
0.142315 0.989821i
0.142315 + 0.989821i
0.959493 + 0.281733i
0.654861 0.755750i
0.925839 2.02730i 0.959493 + 0.281733i −1.94306 2.24241i −0.841254 + 0.540641i 1.45949 1.68434i −0.0125459 + 0.0872586i −2.06815 + 0.607265i −1.68251 1.08128i 0.317178 + 2.20602i
16.1 1.57028 1.81219i −0.841254 + 0.540641i −0.533654 3.71165i −0.415415 0.909632i −0.341254 + 2.37347i 1.31718 0.386758i −3.52977 2.26844i −0.830830 + 1.81926i −2.30075 0.675560i
26.1 −0.0125459 0.0872586i −0.415415 + 0.909632i 1.91153 0.561276i 0.654861 0.755750i 0.0845850 + 0.0248364i −1.30075 + 0.835939i −0.146201 0.320135i 1.30972 + 1.51150i −0.0741615 0.0476607i
31.1 −0.0125459 + 0.0872586i −0.415415 0.909632i 1.91153 + 0.561276i 0.654861 + 0.755750i 0.0845850 0.0248364i −1.30075 0.835939i −0.146201 + 0.320135i 1.30972 1.51150i −0.0741615 + 0.0476607i
36.1 1.57028 + 1.81219i −0.841254 0.540641i −0.533654 + 3.71165i −0.415415 + 0.909632i −0.341254 2.37347i 1.31718 + 0.386758i −3.52977 + 2.26844i −0.830830 1.81926i −2.30075 + 0.675560i
41.1 1.31718 + 0.386758i 0.654861 0.755750i −0.0971309 0.0624222i 0.142315 0.989821i 1.15486 0.742184i 0.925839 + 2.02730i −1.90176 2.19475i 0.284630 + 1.97964i 0.570276 1.24873i
71.1 −1.30075 0.835939i 0.142315 + 0.989821i 0.162317 + 0.355426i 0.959493 + 0.281733i 0.642315 1.40647i 1.57028 1.81219i −0.354114 + 2.46292i 1.91899 0.563465i −1.01255 1.16854i
81.1 −1.30075 + 0.835939i 0.142315 0.989821i 0.162317 0.355426i 0.959493 0.281733i 0.642315 + 1.40647i 1.57028 + 1.81219i −0.354114 2.46292i 1.91899 + 0.563465i −1.01255 + 1.16854i
96.1 0.925839 + 2.02730i 0.959493 0.281733i −1.94306 + 2.24241i −0.841254 0.540641i 1.45949 + 1.68434i −0.0125459 0.0872586i −2.06815 0.607265i −1.68251 + 1.08128i 0.317178 2.20602i
101.1 1.31718 0.386758i 0.654861 + 0.755750i −0.0971309 + 0.0624222i 0.142315 + 0.989821i 1.15486 + 0.742184i 0.925839 2.02730i −1.90176 + 2.19475i 0.284630 1.97964i 0.570276 + 1.24873i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 6.1
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.c even 11 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 115.2.g.a 10
5.b even 2 1 575.2.k.a 10
5.c odd 4 2 575.2.p.a 20
23.c even 11 1 inner 115.2.g.a 10
23.c even 11 1 2645.2.a.n 5
23.d odd 22 1 2645.2.a.o 5
115.j even 22 1 575.2.k.a 10
115.k odd 44 2 575.2.p.a 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
115.2.g.a 10 1.a even 1 1 trivial
115.2.g.a 10 23.c even 11 1 inner
575.2.k.a 10 5.b even 2 1
575.2.k.a 10 115.j even 22 1
575.2.p.a 20 5.c odd 4 2
575.2.p.a 20 115.k odd 44 2
2645.2.a.n 5 23.c even 11 1
2645.2.a.o 5 23.d odd 22 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{10} - 5T_{2}^{9} + 14T_{2}^{8} - 15T_{2}^{7} - 2T_{2}^{6} + 21T_{2}^{5} + 38T_{2}^{4} - 157T_{2}^{3} + 125T_{2}^{2} + 2T_{2} + 1 \) acting on \(S_{2}^{\mathrm{new}}(115, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{10} - 5 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$3$ \( T^{10} - T^{9} + \cdots + 1 \) Copy content Toggle raw display
$5$ \( T^{10} - T^{9} + \cdots + 1 \) Copy content Toggle raw display
$7$ \( T^{10} - 5 T^{9} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( T^{10} - 8 T^{9} + \cdots + 7921 \) Copy content Toggle raw display
$13$ \( T^{10} + 11 T^{8} + \cdots + 121 \) Copy content Toggle raw display
$17$ \( T^{10} + 23 T^{9} + \cdots + 466489 \) Copy content Toggle raw display
$19$ \( T^{10} + 13 T^{9} + \cdots + 11881 \) Copy content Toggle raw display
$23$ \( T^{10} + 21 T^{9} + \cdots + 6436343 \) Copy content Toggle raw display
$29$ \( T^{10} + \cdots + 127893481 \) Copy content Toggle raw display
$31$ \( T^{10} + 20 T^{9} + \cdots + 7921 \) Copy content Toggle raw display
$37$ \( T^{10} - 13 T^{9} + \cdots + 11485321 \) Copy content Toggle raw display
$41$ \( T^{10} + \cdots + 193293409 \) Copy content Toggle raw display
$43$ \( T^{10} - 15 T^{9} + \cdots + 4280761 \) Copy content Toggle raw display
$47$ \( (T^{5} + 26 T^{4} + \cdots - 9811)^{2} \) Copy content Toggle raw display
$53$ \( T^{10} - 6 T^{9} + \cdots + 1408969 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 437688241 \) Copy content Toggle raw display
$61$ \( T^{10} + 3 T^{9} + \cdots + 436921 \) Copy content Toggle raw display
$67$ \( T^{10} - 20 T^{9} + \cdots + 25593481 \) Copy content Toggle raw display
$71$ \( T^{10} + 22 T^{9} + \cdots + 1437601 \) Copy content Toggle raw display
$73$ \( T^{10} - 43 T^{9} + \cdots + 78623689 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 246772681 \) Copy content Toggle raw display
$83$ \( T^{10} + 17 T^{9} + \cdots + 10029889 \) Copy content Toggle raw display
$89$ \( T^{10} - 57 T^{9} + \cdots + 43256929 \) Copy content Toggle raw display
$97$ \( T^{10} - 52 T^{9} + \cdots + 62047129 \) Copy content Toggle raw display
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