Properties

Label 468.2.c.c
Level $468$
Weight $2$
Character orbit 468.c
Analytic conductor $3.737$
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] = [468,2,Mod(287,468)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(468, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 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("468.287");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 468 = 2^{2} \cdot 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 468.c (of order \(2\), degree \(1\), 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: \(3.73699881460\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: 12.0.1279179096064000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} + 5x^{8} - 4x^{6} + 20x^{4} + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + \beta_{2} q^{4} + (\beta_{10} + \beta_{3}) q^{5} + ( - \beta_{7} + \beta_{6}) q^{7} + \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_1 q^{2} + \beta_{2} q^{4} + (\beta_{10} + \beta_{3}) q^{5} + ( - \beta_{7} + \beta_{6}) q^{7} + \beta_{3} q^{8} + (\beta_{11} + \beta_{8} + \beta_{7}) q^{10} + ( - \beta_{5} - \beta_{3} + \beta_1) q^{11} - q^{13} + ( - \beta_{10} - \beta_{5} - 2 \beta_{4} - \beta_{3}) q^{14} + (\beta_{11} + \beta_{7} - 2) q^{16} + ( - \beta_{9} + \beta_{4} - 2 \beta_1) q^{17} + (\beta_{11} + \beta_{8} + 2 \beta_{2}) q^{19} + (\beta_{10} + 2 \beta_{9} - \beta_{5} + 2 \beta_{4} - \beta_{3}) q^{20} + ( - \beta_{11} + \beta_{7} + \beta_{2} + 2) q^{22} + (\beta_{9} - 2 \beta_{5} - 2 \beta_{3}) q^{23} + (\beta_{8} - 2 \beta_{7} - 2 \beta_{6} - \beta_{2} - 3) q^{25} - \beta_1 q^{26} + ( - \beta_{11} - \beta_{8} + \beta_{7} + 2 \beta_{6}) q^{28} - 3 \beta_{4} q^{29} + ( - \beta_{11} - 2 \beta_{7} + 2 \beta_{6} + \beta_{2}) q^{31} + (\beta_{10} - \beta_{5} + 2 \beta_{4} - 2 \beta_1) q^{32} + ( - \beta_{6} - 2 \beta_{2} + 4) q^{34} + ( - 2 \beta_{10} - 3 \beta_{9} + 2 \beta_{3} + 2 \beta_1) q^{35} + ( - \beta_{8} - 2 \beta_{7} - 2 \beta_{6} + \beta_{2} + 2) q^{37} + (\beta_{10} + 2 \beta_{9} - \beta_{5} + \beta_{3}) q^{38} + ( - \beta_{11} + \beta_{8} + \beta_{7} - 2 \beta_{6} - 4) q^{40} + ( - \beta_{10} + 2 \beta_{4} - \beta_{3}) q^{41} + ( - 2 \beta_{11} - \beta_{8} - \beta_{2}) q^{43} + ( - \beta_{10} + \beta_{5} + 2 \beta_{4} + \beta_{3} + 2 \beta_1) q^{44} + ( - 2 \beta_{11} + 2 \beta_{7}) q^{46} + ( - 2 \beta_{9} + 3 \beta_{5} + 3 \beta_{3} + \beta_1) q^{47} + (\beta_{8} - \beta_{2} - 1) q^{49} + (2 \beta_{10} + 2 \beta_{9} + 2 \beta_{5} - 4 \beta_{4} - 3 \beta_1) q^{50} - \beta_{2} q^{52} + ( - \beta_{9} + \beta_{4} - 2 \beta_1) q^{53} + (4 \beta_{11} + \beta_{8} - \beta_{2}) q^{55} + ( - 3 \beta_{10} - 2 \beta_{9} - \beta_{5} + 2 \beta_{4} - \beta_{3}) q^{56} + 3 \beta_{6} q^{58} + ( - \beta_{9} + \beta_{5} + \beta_{3} + \beta_1) q^{59} + ( - \beta_{8} + \beta_{2} + 8) q^{61} + ( - 3 \beta_{10} - \beta_{5} - 4 \beta_{4} - \beta_{3}) q^{62} + (\beta_{8} + 2 \beta_{7} - 2 \beta_{6} - 2 \beta_{2} + 2) q^{64} + ( - \beta_{10} - \beta_{3}) q^{65} + ( - 2 \beta_{11} - \beta_{8} - \beta_{7} + \beta_{6} - \beta_{2}) q^{67} + (\beta_{10} + \beta_{5} - \beta_{3} + 4 \beta_1) q^{68} + (2 \beta_{11} - 2 \beta_{8} + 2 \beta_{7} + 2 \beta_{2} + 4) q^{70} + ( - 2 \beta_{10} - \beta_{9} - 3 \beta_{5} - \beta_{3} + \beta_1) q^{71} + (\beta_{8} - 2 \beta_{7} - 2 \beta_{6} - \beta_{2} - 2) q^{73} + (2 \beta_{10} - 2 \beta_{9} + 2 \beta_{5} - 4 \beta_{4} + 4 \beta_{3} + 2 \beta_1) q^{74} + (\beta_{11} + \beta_{8} + 3 \beta_{7} - 8) q^{76} + ( - \beta_{9} - 4 \beta_{4} - 2 \beta_1) q^{77} + ( - 2 \beta_{11} - 2 \beta_{8} - 4 \beta_{2}) q^{79} + (\beta_{10} + 2 \beta_{9} + 3 \beta_{5} + 2 \beta_{4} + \beta_{3} - 4 \beta_1) q^{80} + ( - \beta_{11} - \beta_{8} - \beta_{7} - 2 \beta_{6}) q^{82} + (4 \beta_{10} + 3 \beta_{9} + \beta_{5} - 3 \beta_{3} + \beta_1) q^{83} + ( - \beta_{8} - 2 \beta_{7} - 2 \beta_{6} + \beta_{2}) q^{85} + ( - 2 \beta_{10} - 2 \beta_{9} + 2 \beta_{5}) q^{86} + (\beta_{11} - \beta_{8} - \beta_{7} - 2 \beta_{6} + 2 \beta_{2} - 4) q^{88} + ( - \beta_{10} + 4 \beta_{4} - \beta_{3}) q^{89} + (\beta_{7} - \beta_{6}) q^{91} + ( - 2 \beta_{10} + 2 \beta_{5} + 4 \beta_{4}) q^{92} + (3 \beta_{11} - 3 \beta_{7} + \beta_{2} + 2) q^{94} + (2 \beta_{10} + 5 \beta_{9} - 2 \beta_{3} - 6 \beta_1) q^{95} + ( - \beta_{8} + 2 \beta_{7} + 2 \beta_{6} + \beta_{2} - 6) q^{97} + (2 \beta_{9} - 2 \beta_{3} - \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 4 q^{10} - 12 q^{13} - 20 q^{16} + 28 q^{22} - 52 q^{25} + 12 q^{28} + 44 q^{34} + 8 q^{37} - 52 q^{40} + 8 q^{46} - 12 q^{49} + 12 q^{58} + 96 q^{61} + 24 q^{64} + 56 q^{70} - 40 q^{73} - 84 q^{76} - 12 q^{82} - 16 q^{85} - 60 q^{88} + 12 q^{94} - 56 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} + 5x^{8} - 4x^{6} + 20x^{4} + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{9} + 2\nu^{7} - \nu^{5} + 6\nu^{3} - 8\nu ) / 16 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{11} + 3\nu^{7} - 12\nu^{5} - 12\nu^{3} - 48\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( \nu^{10} - 2\nu^{8} + \nu^{6} - 6\nu^{4} + 8\nu^{2} ) / 16 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( -\nu^{8} + 2\nu^{6} - \nu^{4} + 6\nu^{2} - 8 ) / 8 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( \nu^{10} + 5\nu^{6} - 4\nu^{4} + 12\nu^{2} ) / 8 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( \nu^{11} + 5\nu^{7} - 4\nu^{5} + 20\nu^{3} ) / 16 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( -\nu^{11} + 4\nu^{9} - 5\nu^{7} + 24\nu^{5} - 36\nu^{3} + 48\nu ) / 32 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( \nu^{8} - 2\nu^{6} + 9\nu^{4} - 6\nu^{2} + 24 ) / 8 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} + \beta_{7} - 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{10} - \beta_{5} + 2\beta_{4} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{8} + 2\beta_{7} - 2\beta_{6} - 2\beta_{2} + 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 2\beta_{10} + 2\beta_{9} + 2\beta_{5} + 4\beta_{4} - \beta_{3} + 2\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -\beta_{11} + 2\beta_{8} - 5\beta_{7} - 4\beta_{6} + 2\beta_{2} - 2 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 3\beta_{10} + 4\beta_{9} + 5\beta_{5} - 10\beta_{4} + 4\beta_{3} - 2\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 4\beta_{11} + 3\beta_{8} - 6\beta_{7} + 10\beta_{6} - 2\beta_{2} - 18 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( -6\beta_{10} + 6\beta_{9} - 14\beta_{5} - 12\beta_{4} - 15\beta_{3} - 18\beta_1 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/468\mathbb{Z}\right)^\times\).

\(n\) \(145\) \(209\) \(235\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
287.1
−1.31293 0.525570i
−1.31293 + 0.525570i
−0.921588 1.07270i
−0.921588 + 1.07270i
−0.653376 1.25423i
−0.653376 + 1.25423i
0.653376 1.25423i
0.653376 + 1.25423i
0.921588 1.07270i
0.921588 + 1.07270i
1.31293 0.525570i
1.31293 + 0.525570i
−1.31293 0.525570i 0 1.44755 + 1.38007i 4.09430i 0 3.71352i −1.17521 2.57272i 0 −2.15184 + 5.37551i
287.2 −1.31293 + 0.525570i 0 1.44755 1.38007i 4.09430i 0 3.71352i −1.17521 + 2.57272i 0 −2.15184 5.37551i
287.3 −0.921588 1.07270i 0 −0.301352 + 1.97717i 0.852353i 0 2.60664i 2.39862 1.49887i 0 −0.914316 + 0.785518i
287.4 −0.921588 + 1.07270i 0 −0.301352 1.97717i 0.852353i 0 2.60664i 2.39862 + 1.49887i 0 −0.914316 0.785518i
287.5 −0.653376 1.25423i 0 −1.14620 + 1.63897i 3.24195i 0 1.84803i 2.80455 + 0.366740i 0 4.06615 2.11821i
287.6 −0.653376 + 1.25423i 0 −1.14620 1.63897i 3.24195i 0 1.84803i 2.80455 0.366740i 0 4.06615 + 2.11821i
287.7 0.653376 1.25423i 0 −1.14620 1.63897i 3.24195i 0 1.84803i −2.80455 + 0.366740i 0 4.06615 + 2.11821i
287.8 0.653376 + 1.25423i 0 −1.14620 + 1.63897i 3.24195i 0 1.84803i −2.80455 0.366740i 0 4.06615 2.11821i
287.9 0.921588 1.07270i 0 −0.301352 1.97717i 0.852353i 0 2.60664i −2.39862 1.49887i 0 −0.914316 0.785518i
287.10 0.921588 + 1.07270i 0 −0.301352 + 1.97717i 0.852353i 0 2.60664i −2.39862 + 1.49887i 0 −0.914316 + 0.785518i
287.11 1.31293 0.525570i 0 1.44755 1.38007i 4.09430i 0 3.71352i 1.17521 2.57272i 0 −2.15184 5.37551i
287.12 1.31293 + 0.525570i 0 1.44755 + 1.38007i 4.09430i 0 3.71352i 1.17521 + 2.57272i 0 −2.15184 + 5.37551i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 287.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
4.b odd 2 1 inner
12.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 468.2.c.c 12
3.b odd 2 1 inner 468.2.c.c 12
4.b odd 2 1 inner 468.2.c.c 12
8.b even 2 1 7488.2.d.m 12
8.d odd 2 1 7488.2.d.m 12
12.b even 2 1 inner 468.2.c.c 12
24.f even 2 1 7488.2.d.m 12
24.h odd 2 1 7488.2.d.m 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
468.2.c.c 12 1.a even 1 1 trivial
468.2.c.c 12 3.b odd 2 1 inner
468.2.c.c 12 4.b odd 2 1 inner
468.2.c.c 12 12.b even 2 1 inner
7488.2.d.m 12 8.b even 2 1
7488.2.d.m 12 8.d odd 2 1
7488.2.d.m 12 24.f even 2 1
7488.2.d.m 12 24.h odd 2 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 28T_{5}^{4} + 196T_{5}^{2} + 128 \) acting on \(S_{2}^{\mathrm{new}}(468, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} + 5 T^{8} - 4 T^{6} + 20 T^{4} + \cdots + 64 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} + 28 T^{4} + 196 T^{2} + 128)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 24 T^{4} + 164 T^{2} + 320)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} - 38 T^{4} + 156 T^{2} - 40)^{2} \) Copy content Toggle raw display
$13$ \( (T + 1)^{12} \) Copy content Toggle raw display
$17$ \( (T^{6} + 46 T^{4} + 444 T^{2} + 200)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 76 T^{4} + 1444 T^{2} + 80)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} - 88 T^{4} + 656 T^{2} - 640)^{2} \) Copy content Toggle raw display
$29$ \( (T^{2} + 18)^{6} \) Copy content Toggle raw display
$31$ \( (T^{6} + 124 T^{4} + 4164 T^{2} + \cdots + 23120)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 2 T^{2} - 80 T + 320)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} + 52 T^{4} + 580 T^{2} + 800)^{2} \) Copy content Toggle raw display
$43$ \( (T^{6} + 104 T^{4} + 3344 T^{2} + \cdots + 32000)^{2} \) Copy content Toggle raw display
$47$ \( (T^{6} - 198 T^{4} + 8156 T^{2} + \cdots - 88360)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 46 T^{4} + 444 T^{2} + 200)^{2} \) Copy content Toggle raw display
$59$ \( (T^{2} - 10)^{6} \) Copy content Toggle raw display
$61$ \( (T^{3} - 24 T^{2} + 164 T - 320)^{4} \) Copy content Toggle raw display
$67$ \( (T^{6} + 136 T^{4} + 2404 T^{2} + \cdots + 5120)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 222 T^{4} + 14636 T^{2} + \cdots - 289000)^{2} \) Copy content Toggle raw display
$73$ \( (T^{3} + 10 T^{2} - 32 T - 256)^{4} \) Copy content Toggle raw display
$79$ \( (T^{6} + 304 T^{4} + 23104 T^{2} + \cdots + 5120)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} - 702 T^{4} + 162476 T^{2} + \cdots - 12409960)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + 124 T^{4} + 3652 T^{2} + \cdots + 8192)^{2} \) Copy content Toggle raw display
$97$ \( (T^{3} + 14 T^{2} - 128)^{4} \) Copy content Toggle raw display
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