Properties

Label 215.2.e.b
Level $215$
Weight $2$
Character orbit 215.e
Analytic conductor $1.717$
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] = [215,2,Mod(6,215)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(215, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("215.6");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 215 = 5 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 215.e (of order \(3\), degree \(2\), 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: \(1.71678364346\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{13})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 4x^{2} + 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{3} + (\beta_{3} + 1) q^{4} + \beta_{2} q^{5} + ( - 2 \beta_{3} + 5 \beta_{2} - 2 \beta_1 + 2) q^{6} + ( - \beta_{2} - 1) q^{7} + 3 q^{8} + ( - 4 \beta_{2} + 3 \beta_1 - 4) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{3} + (\beta_{3} + 1) q^{4} + \beta_{2} q^{5} + ( - 2 \beta_{3} + 5 \beta_{2} - 2 \beta_1 + 2) q^{6} + ( - \beta_{2} - 1) q^{7} + 3 q^{8} + ( - 4 \beta_{2} + 3 \beta_1 - 4) q^{9} + ( - \beta_{3} + \beta_{2} - \beta_1 + 1) q^{10} + (3 \beta_{3} - 2) q^{11} + ( - 3 \beta_{3} + 7 \beta_{2} - 3 \beta_1 + 3) q^{12} + (4 \beta_{2} + \beta_1 + 4) q^{13} + ( - \beta_{2} + \beta_1 - 1) q^{14} + ( - 2 \beta_{2} + \beta_1 - 2) q^{15} + (\beta_{3} - 2) q^{16} + (3 \beta_{2} + \beta_1 + 3) q^{17} + ( - 13 \beta_{2} + 4 \beta_1 - 13) q^{18} + ( - \beta_{3} + 2 \beta_{2} - \beta_1 + 1) q^{20} + (\beta_{3} + 1) q^{21} + (\beta_{3} + 9) q^{22} + (3 \beta_{3} - 3 \beta_{2} + 3 \beta_1 - 3) q^{23} + ( - 3 \beta_{3} + 6 \beta_{2} - 3 \beta_1 + 3) q^{24} + ( - \beta_{2} - 1) q^{25} + (\beta_{2} - 4 \beta_1 + 1) q^{26} + (4 \beta_{3} + 7) q^{27} + ( - 2 \beta_{2} + \beta_1 - 2) q^{28} + ( - \beta_{2} + 2 \beta_1 - 1) q^{29} + ( - 5 \beta_{2} + 2 \beta_1 - 5) q^{30} + (2 \beta_{3} - 4 \beta_{2} + 2 \beta_1 - 2) q^{31} + ( - \beta_{3} - 3) q^{32} + ( - 4 \beta_{3} + 11 \beta_{2} - 4 \beta_1 + 4) q^{33} - 3 \beta_1 q^{34} + q^{35} + ( - 17 \beta_{2} + 7 \beta_1 - 17) q^{36} + ( - 3 \beta_{3} + 5 \beta_{2} - 3 \beta_1 + 3) q^{37} + ( - 3 \beta_{3} - 2) q^{39} + 3 \beta_{2} q^{40} - 7 q^{41} + (2 \beta_{3} + 3) q^{42} + ( - \beta_{2} + 6) q^{43} + (4 \beta_{3} + 7) q^{44} + (3 \beta_{3} + 1) q^{45} + (3 \beta_{3} - 12 \beta_{2} + 3 \beta_1 - 3) q^{46} + ( - \beta_{3} - 8) q^{47} + \beta_{2} q^{48} - 6 \beta_{2} q^{49} + ( - \beta_{2} + \beta_1 - 1) q^{50} + ( - 2 \beta_{3} - 1) q^{51} + (5 \beta_{2} - 3 \beta_1 + 5) q^{52} + ( - 5 \beta_{3} - 5 \beta_1 + 5) q^{53} + (11 \beta_{3} + 12) q^{54} + ( - 3 \beta_{3} + \beta_{2} - 3 \beta_1 + 3) q^{55} + ( - 3 \beta_{2} - 3) q^{56} + ( - 7 \beta_{2} + \beta_1 - 7) q^{58} + (\beta_{3} - 7) q^{59} + ( - 7 \beta_{2} + 3 \beta_1 - 7) q^{60} + ( - 11 \beta_{2} - \beta_1 - 11) q^{61} + (4 \beta_{3} - 10 \beta_{2} + 4 \beta_1 - 4) q^{62} + ( - 3 \beta_{3} + 4 \beta_{2} - 3 \beta_1 + 3) q^{63} + ( - 6 \beta_{3} + 1) q^{64} + (\beta_{3} - 5) q^{65} + ( - 11 \beta_{3} + 23 \beta_{2} - 11 \beta_1 + 11) q^{66} + ( - 3 \beta_{3} - 3 \beta_1 + 3) q^{67} + (3 \beta_{2} - 2 \beta_1 + 3) q^{68} + (15 \beta_{2} - 6 \beta_1 + 15) q^{69} + \beta_{3} q^{70} + (8 \beta_{2} + 3 \beta_1 + 8) q^{71} + ( - 12 \beta_{2} + 9 \beta_1 - 12) q^{72} + (2 \beta_{2} + 3 \beta_1 + 2) q^{73} + ( - 5 \beta_{3} + 14 \beta_{2} - 5 \beta_1 + 5) q^{74} + (\beta_{3} + 1) q^{75} + ( - \beta_{2} + 3 \beta_1 - 1) q^{77} + ( - 5 \beta_{3} - 9) q^{78} + (4 \beta_{2} + \beta_1 + 4) q^{79} + ( - \beta_{3} - \beta_{2} - \beta_1 + 1) q^{80} + ( - 6 \beta_{3} + 22 \beta_{2} - 6 \beta_1 + 6) q^{81} - 7 \beta_{3} q^{82} + (3 \beta_{3} - 6 \beta_{2} + 3 \beta_1 - 3) q^{83} + (3 \beta_{3} + 4) q^{84} + (\beta_{3} - 4) q^{85} + (7 \beta_{3} - \beta_{2} + \beta_1 - 1) q^{86} + (3 \beta_{3} + 5) q^{87} + (9 \beta_{3} - 6) q^{88} + (\beta_{3} + 8 \beta_{2} + \beta_1 - 1) q^{89} + (4 \beta_{3} + 9) q^{90} + ( - \beta_{3} - 4 \beta_{2} - \beta_1 + 1) q^{91} + (6 \beta_{3} - 15 \beta_{2} + 6 \beta_1 - 6) q^{92} + (14 \beta_{2} - 6 \beta_1 + 14) q^{93} + ( - 9 \beta_{3} - 3) q^{94} + (5 \beta_{3} - 11 \beta_{2} + 5 \beta_1 - 5) q^{96} + 6 q^{97} + (6 \beta_{3} - 6 \beta_{2} + 6 \beta_1 - 6) q^{98} + ( - 31 \beta_{2} + 6 \beta_1 - 31) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 2 q^{2} - 3 q^{3} + 6 q^{4} - 2 q^{5} - 8 q^{6} - 2 q^{7} + 12 q^{8} - 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 2 q^{2} - 3 q^{3} + 6 q^{4} - 2 q^{5} - 8 q^{6} - 2 q^{7} + 12 q^{8} - 5 q^{9} - q^{10} - 2 q^{11} - 11 q^{12} + 9 q^{13} - q^{14} - 3 q^{15} - 6 q^{16} + 7 q^{17} - 22 q^{18} - 3 q^{20} + 6 q^{21} + 38 q^{22} + 3 q^{23} - 9 q^{24} - 2 q^{25} - 2 q^{26} + 36 q^{27} - 3 q^{28} - 8 q^{30} + 6 q^{31} - 14 q^{32} - 18 q^{33} - 3 q^{34} + 4 q^{35} - 27 q^{36} - 7 q^{37} - 14 q^{39} - 6 q^{40} - 28 q^{41} + 16 q^{42} + 26 q^{43} + 36 q^{44} + 10 q^{45} + 21 q^{46} - 34 q^{47} - 2 q^{48} + 12 q^{49} - q^{50} - 8 q^{51} + 7 q^{52} + 5 q^{53} + 70 q^{54} + q^{55} - 6 q^{56} - 13 q^{58} - 26 q^{59} - 11 q^{60} - 23 q^{61} + 16 q^{62} - 5 q^{63} - 8 q^{64} - 18 q^{65} - 35 q^{66} + 3 q^{67} + 4 q^{68} + 24 q^{69} + 2 q^{70} + 19 q^{71} - 15 q^{72} + 7 q^{73} - 23 q^{74} + 6 q^{75} + q^{77} - 46 q^{78} + 9 q^{79} + 3 q^{80} - 38 q^{81} - 14 q^{82} + 9 q^{83} + 22 q^{84} - 14 q^{85} + 13 q^{86} + 26 q^{87} - 6 q^{88} - 17 q^{89} + 44 q^{90} + 9 q^{91} + 24 q^{92} + 22 q^{93} - 30 q^{94} + 17 q^{96} + 24 q^{97} + 6 q^{98} - 56 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 4x^{2} + 3x + 9 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( -\nu^{3} + 4\nu^{2} - 4\nu - 3 ) / 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{3} + 7 ) / 4 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{3} + 3\beta_{2} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 4\beta_{3} - 7 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/215\mathbb{Z}\right)^\times\).

\(n\) \(46\) \(87\)
\(\chi(n)\) \(-1 - \beta_{2}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1
1.15139 + 1.99426i
−0.651388 1.12824i
1.15139 1.99426i
−0.651388 + 1.12824i
−1.30278 0.151388 0.262211i −0.302776 −0.500000 + 0.866025i −0.197224 + 0.341603i −0.500000 0.866025i 3.00000 1.45416 + 2.51868i 0.651388 1.12824i
6.2 2.30278 −1.65139 + 2.86029i 3.30278 −0.500000 + 0.866025i −3.80278 + 6.58660i −0.500000 0.866025i 3.00000 −3.95416 6.84881i −1.15139 + 1.99426i
36.1 −1.30278 0.151388 + 0.262211i −0.302776 −0.500000 0.866025i −0.197224 0.341603i −0.500000 + 0.866025i 3.00000 1.45416 2.51868i 0.651388 + 1.12824i
36.2 2.30278 −1.65139 2.86029i 3.30278 −0.500000 0.866025i −3.80278 6.58660i −0.500000 + 0.866025i 3.00000 −3.95416 + 6.84881i −1.15139 1.99426i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 215.2.e.b 4
43.c even 3 1 inner 215.2.e.b 4
43.c even 3 1 9245.2.a.e 2
43.d odd 6 1 9245.2.a.d 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
215.2.e.b 4 1.a even 1 1 trivial
215.2.e.b 4 43.c even 3 1 inner
9245.2.a.d 2 43.d odd 6 1
9245.2.a.e 2 43.c even 3 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{2}^{2} - T_{2} - 3 \) acting on \(S_{2}^{\mathrm{new}}(215, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T - 3)^{2} \) Copy content Toggle raw display
$3$ \( T^{4} + 3 T^{3} + 10 T^{2} - 3 T + 1 \) Copy content Toggle raw display
$5$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$7$ \( (T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$11$ \( (T^{2} + T - 29)^{2} \) Copy content Toggle raw display
$13$ \( T^{4} - 9 T^{3} + 64 T^{2} - 153 T + 289 \) Copy content Toggle raw display
$17$ \( T^{4} - 7 T^{3} + 40 T^{2} - 63 T + 81 \) Copy content Toggle raw display
$19$ \( T^{4} \) Copy content Toggle raw display
$23$ \( T^{4} - 3 T^{3} + 36 T^{2} + 81 T + 729 \) Copy content Toggle raw display
$29$ \( T^{4} + 13T^{2} + 169 \) Copy content Toggle raw display
$31$ \( T^{4} - 6 T^{3} + 40 T^{2} + 24 T + 16 \) Copy content Toggle raw display
$37$ \( T^{4} + 7 T^{3} + 66 T^{2} - 119 T + 289 \) Copy content Toggle raw display
$41$ \( (T + 7)^{4} \) Copy content Toggle raw display
$43$ \( (T^{2} - 13 T + 43)^{2} \) Copy content Toggle raw display
$47$ \( (T^{2} + 17 T + 69)^{2} \) Copy content Toggle raw display
$53$ \( T^{4} - 5 T^{3} + 100 T^{2} + \cdots + 5625 \) Copy content Toggle raw display
$59$ \( (T^{2} + 13 T + 39)^{2} \) Copy content Toggle raw display
$61$ \( T^{4} + 23 T^{3} + 400 T^{2} + \cdots + 16641 \) Copy content Toggle raw display
$67$ \( T^{4} - 3 T^{3} + 36 T^{2} + 81 T + 729 \) Copy content Toggle raw display
$71$ \( T^{4} - 19 T^{3} + 300 T^{2} + \cdots + 3721 \) Copy content Toggle raw display
$73$ \( T^{4} - 7 T^{3} + 66 T^{2} + 119 T + 289 \) Copy content Toggle raw display
$79$ \( T^{4} - 9 T^{3} + 64 T^{2} - 153 T + 289 \) Copy content Toggle raw display
$83$ \( T^{4} - 9 T^{3} + 90 T^{2} + 81 T + 81 \) Copy content Toggle raw display
$89$ \( T^{4} + 17 T^{3} + 220 T^{2} + \cdots + 4761 \) Copy content Toggle raw display
$97$ \( (T - 6)^{4} \) Copy content Toggle raw display
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