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Properties

Label 950.2.u.d
Level $950$
Weight $2$
Character orbit 950.u
Analytic conductor $7.586$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [950,2,Mod(99,950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(950, base_ring=CyclotomicField(18))
 
chi = DirichletCharacter(H, H._module([9, 2]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("950.99");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 950 = 2 \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 950.u (of order \(18\), degree \(6\), 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: \(7.58578819202\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{18})\)
Coefficient field: \(\Q(\zeta_{36})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{6} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

$q$-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 \(\zeta_{36}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \zeta_{36}^{5} q^{2} + (\zeta_{36}^{11} + \zeta_{36}^{3} - \zeta_{36}) q^{3} + \zeta_{36}^{10} q^{4} + ( - \zeta_{36}^{10} + \cdots + \zeta_{36}^{4}) q^{6} + \cdots + (2 \zeta_{36}^{8} - \zeta_{36}^{6} + \cdots + 2) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q - \zeta_{36}^{5} q^{2} + (\zeta_{36}^{11} + \zeta_{36}^{3} - \zeta_{36}) q^{3} + \zeta_{36}^{10} q^{4} + ( - \zeta_{36}^{10} + \cdots + \zeta_{36}^{4}) q^{6} + \cdots + (\zeta_{36}^{8} + \zeta_{36}^{6} + \cdots + 1) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 6 q^{6} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 6 q^{6} + 18 q^{9} + 6 q^{11} + 6 q^{14} + 18 q^{19} + 6 q^{21} + 12 q^{24} - 6 q^{26} + 36 q^{29} + 6 q^{31} + 12 q^{34} - 18 q^{36} + 24 q^{39} - 42 q^{41} + 18 q^{46} - 24 q^{49} + 18 q^{51} + 36 q^{54} - 12 q^{56} - 54 q^{59} - 30 q^{61} + 6 q^{64} - 6 q^{66} + 12 q^{69} + 12 q^{71} - 36 q^{74} - 24 q^{79} - 36 q^{81} + 12 q^{84} + 18 q^{86} - 6 q^{91} - 36 q^{94} - 12 q^{96} + 18 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/950\mathbb{Z}\right)^\times\).

\(n\) \(77\) \(401\)
\(\chi(n)\) \(-1\) \(-\zeta_{36}^{10}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
99.1
−0.642788 + 0.766044i
0.642788 − 0.766044i
0.984808 − 0.173648i
−0.984808 + 0.173648i
0.342020 + 0.939693i
−0.342020 − 0.939693i
−0.642788 − 0.766044i
0.642788 + 0.766044i
0.342020 − 0.939693i
−0.342020 + 0.939693i
0.984808 + 0.173648i
−0.984808 − 0.173648i
−0.342020 + 0.939693i 2.49362 − 0.439693i −0.766044 − 0.642788i 0 −0.439693 + 2.49362i −0.565258 − 0.326352i 0.866025 − 0.500000i 3.20574 − 1.16679i 0
99.2 0.342020 − 0.939693i −2.49362 + 0.439693i −0.766044 − 0.642788i 0 −0.439693 + 2.49362i 0.565258 + 0.326352i −0.866025 + 0.500000i 3.20574 − 1.16679i 0
149.1 −0.642788 + 0.766044i −0.460802 − 1.26604i −0.173648 − 0.984808i 0 1.26604 + 0.460802i −2.49362 + 1.43969i 0.866025 + 0.500000i 0.907604 − 0.761570i 0
149.2 0.642788 − 0.766044i 0.460802 + 1.26604i −0.173648 − 0.984808i 0 1.26604 + 0.460802i 2.49362 − 1.43969i −0.866025 − 0.500000i 0.907604 − 0.761570i 0
199.1 −0.984808 + 0.173648i −0.565258 − 0.673648i 0.939693 − 0.342020i 0 0.673648 + 0.565258i −0.460802 − 0.266044i −0.866025 + 0.500000i 0.386659 − 2.19285i 0
199.2 0.984808 − 0.173648i 0.565258 + 0.673648i 0.939693 − 0.342020i 0 0.673648 + 0.565258i 0.460802 + 0.266044i 0.866025 − 0.500000i 0.386659 − 2.19285i 0
499.1 −0.342020 − 0.939693i 2.49362 + 0.439693i −0.766044 + 0.642788i 0 −0.439693 − 2.49362i −0.565258 + 0.326352i 0.866025 + 0.500000i 3.20574 + 1.16679i 0
499.2 0.342020 + 0.939693i −2.49362 − 0.439693i −0.766044 + 0.642788i 0 −0.439693 − 2.49362i 0.565258 − 0.326352i −0.866025 − 0.500000i 3.20574 + 1.16679i 0
549.1 −0.984808 − 0.173648i −0.565258 + 0.673648i 0.939693 + 0.342020i 0 0.673648 − 0.565258i −0.460802 + 0.266044i −0.866025 − 0.500000i 0.386659 + 2.19285i 0
549.2 0.984808 + 0.173648i 0.565258 − 0.673648i 0.939693 + 0.342020i 0 0.673648 − 0.565258i 0.460802 − 0.266044i 0.866025 + 0.500000i 0.386659 + 2.19285i 0
899.1 −0.642788 − 0.766044i −0.460802 + 1.26604i −0.173648 + 0.984808i 0 1.26604 − 0.460802i −2.49362 − 1.43969i 0.866025 − 0.500000i 0.907604 + 0.761570i 0
899.2 0.642788 + 0.766044i 0.460802 − 1.26604i −0.173648 + 0.984808i 0 1.26604 − 0.460802i 2.49362 + 1.43969i −0.866025 + 0.500000i 0.907604 + 0.761570i 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 99.2
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
19.e even 9 1 inner
95.p even 18 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 950.2.u.d 12
5.b even 2 1 inner 950.2.u.d 12
5.c odd 4 1 950.2.l.b ✓ 6
5.c odd 4 1 950.2.l.e yes 6
19.e even 9 1 inner 950.2.u.d 12
95.p even 18 1 inner 950.2.u.d 12
95.q odd 36 1 950.2.l.b ✓ 6
95.q odd 36 1 950.2.l.e yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
950.2.l.b ✓ 6 5.c odd 4 1
950.2.l.b ✓ 6 95.q odd 36 1
950.2.l.e yes 6 5.c odd 4 1
950.2.l.e yes 6 95.q odd 36 1
950.2.u.d 12 1.a even 1 1 trivial
950.2.u.d 12 5.b even 2 1 inner
950.2.u.d 12 19.e even 9 1 inner
950.2.u.d 12 95.p even 18 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(950, [\chi])\):

\( T_{3}^{12} - 9T_{3}^{10} + 9T_{3}^{8} + 72T_{3}^{6} + 162T_{3}^{4} + 81T_{3}^{2} + 81 \) Copy content Toggle raw display
\( T_{7}^{12} - 9T_{7}^{10} + 75T_{7}^{8} - 52T_{7}^{6} + 27T_{7}^{4} - 6T_{7}^{2} + 1 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - T^{6} + 1 \) Copy content Toggle raw display
$3$ \( T^{12} - 9 T^{10} + \cdots + 81 \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} - 9 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$11$ \( (T^{2} - T + 1)^{6} \) Copy content Toggle raw display
$13$ \( T^{12} - 15 T^{10} + \cdots + 1 \) Copy content Toggle raw display
$17$ \( T^{12} - 27 T^{10} + \cdots + 81 \) Copy content Toggle raw display
$19$ \( (T^{6} - 9 T^{5} + \cdots + 6859)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 33 T^{10} + \cdots + 83521 \) Copy content Toggle raw display
$29$ \( (T^{6} - 18 T^{5} + \cdots + 5184)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} - 3 T^{5} + \cdots + 361)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 66 T^{4} + \cdots + 361)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 21 T^{5} + \cdots + 25281)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 855036081 \) Copy content Toggle raw display
$47$ \( T^{12} + 63 T^{10} + \cdots + 6561 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 1026625681 \) Copy content Toggle raw display
$59$ \( (T^{6} + 27 T^{5} + \cdots + 110889)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + 15 T^{5} + \cdots + 25281)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + 36 T^{8} + \cdots + 81 \) Copy content Toggle raw display
$71$ \( (T^{6} - 6 T^{5} + \cdots + 166464)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + 57 T^{10} + \cdots + 130321 \) Copy content Toggle raw display
$79$ \( (T^{6} + 12 T^{5} + \cdots + 2809)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 16550975387521 \) Copy content Toggle raw display
$89$ \( (T^{6} + 288 T^{4} + \cdots + 4096)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + 369 T^{10} + \cdots + 83521 \) Copy content Toggle raw display
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