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Properties

Label 3150.3.c.c
Level $3150$
Weight $3$
Character orbit 3150.c
Analytic conductor $85.831$
Analytic rank $0$
Dimension $8$
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] = [3150,3,Mod(449,3150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3150, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3150.449");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 3150.c (of order \(2\), degree \(1\), 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: \(85.8312832735\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.157351936.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{6} q^{2} + 2 q^{4} - \beta_1 q^{7} + 2 \beta_{6} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{6} q^{2} + 2 q^{4} - \beta_1 q^{7} + 2 \beta_{6} q^{8} + (7 \beta_{7} - 9 \beta_{3}) q^{11} + (3 \beta_{2} - \beta_1) q^{13} + ( - 2 \beta_{7} + \beta_{3}) q^{14} + 4 q^{16} + ( - 19 \beta_{6} - 4 \beta_{5}) q^{17} + (5 \beta_{4} + 15) q^{19} + ( - 11 \beta_{2} + 7 \beta_1) q^{22} + ( - 7 \beta_{6} - 13 \beta_{5}) q^{23} + ( - 2 \beta_{7} + 4 \beta_{3}) q^{26} - 2 \beta_1 q^{28} + (7 \beta_{7} - 15 \beta_{3}) q^{29} + (8 \beta_{4} + 4) q^{31} + 4 \beta_{6} q^{32} + ( - 4 \beta_{4} - 34) q^{34} + (28 \beta_{2} + 5 \beta_1) q^{37} + (20 \beta_{6} + 10 \beta_{5}) q^{38} + (18 \beta_{7} - 6 \beta_{3}) q^{41} + (55 \beta_{2} + 6 \beta_1) q^{43} + (14 \beta_{7} - 18 \beta_{3}) q^{44} + ( - 13 \beta_{4} - 1) q^{46} + (27 \beta_{6} + 30 \beta_{5}) q^{47} - 7 q^{49} + (6 \beta_{2} - 2 \beta_1) q^{52} + (37 \beta_{6} + 28 \beta_{5}) q^{53} + ( - 4 \beta_{7} + 2 \beta_{3}) q^{56} + ( - 23 \beta_{2} + 7 \beta_1) q^{58} + (36 \beta_{7} - 33 \beta_{3}) q^{59} + (24 \beta_{4} - 22) q^{61} + (12 \beta_{6} + 16 \beta_{5}) q^{62} + 8 q^{64} + ( - 38 \beta_{2} - 7 \beta_1) q^{67} + ( - 38 \beta_{6} - 8 \beta_{5}) q^{68} + ( - 43 \beta_{7} + 6 \beta_{3}) q^{71} + ( - 21 \beta_{2} + 35 \beta_1) q^{73} + (10 \beta_{7} + 23 \beta_{3}) q^{74} + (10 \beta_{4} + 30) q^{76} + (19 \beta_{6} - 11 \beta_{5}) q^{77} + ( - 19 \beta_{4} - 36) q^{79} + (6 \beta_{2} + 18 \beta_1) q^{82} + ( - 96 \beta_{6} + 6 \beta_{5}) q^{83} + (12 \beta_{7} + 49 \beta_{3}) q^{86} + ( - 22 \beta_{2} + 14 \beta_1) q^{88} + (34 \beta_{7} + 39 \beta_{3}) q^{89} + (3 \beta_{4} - 7) q^{91} + ( - 14 \beta_{6} - 26 \beta_{5}) q^{92} + (30 \beta_{4} + 24) q^{94} + (38 \beta_{2} + 36 \beta_1) q^{97} - 7 \beta_{6} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 16 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 16 q^{4} + 32 q^{16} + 120 q^{19} + 32 q^{31} - 272 q^{34} - 8 q^{46} - 56 q^{49} - 176 q^{61} + 64 q^{64} + 240 q^{76} - 288 q^{79} - 56 q^{91} + 192 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + x^{4} + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 2\nu^{4} + 1 ) / 3 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{6} + 5\nu^{2} ) / 12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -\nu^{7} + 4\nu^{5} + 7\nu^{3} - 4\nu ) / 24 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -\nu^{6} + 3\nu^{2} ) / 4 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -\nu^{7} + 2\nu^{5} - 5\nu^{3} + 10\nu ) / 12 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -\nu^{7} - 4\nu^{5} + 7\nu^{3} + 4\nu ) / 24 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} + 2\nu^{5} + 5\nu^{3} + 10\nu ) / 12 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( ( \beta_{7} + \beta_{6} + \beta_{5} - \beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{4} + 3\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( \beta_{7} + 2\beta_{6} - \beta_{5} + 2\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 3\beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( \beta_{7} - 5\beta_{6} + \beta_{5} + 5\beta_{3} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -5\beta_{4} + 9\beta_{2} ) / 2 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 7\beta_{7} - 10\beta_{6} - 7\beta_{5} - 10\beta_{3} ) / 2 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3150\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(451\) \(2801\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
0.581861 − 1.28897i
−1.28897 − 0.581861i
−1.28897 + 0.581861i
0.581861 + 1.28897i
1.28897 + 0.581861i
−0.581861 + 1.28897i
−0.581861 − 1.28897i
1.28897 − 0.581861i
−1.41421 0 2.00000 0 0 − 2.64575i −2.82843 0 0
449.2 −1.41421 0 2.00000 0 0 − 2.64575i −2.82843 0 0
449.3 −1.41421 0 2.00000 0 0 2.64575i −2.82843 0 0
449.4 −1.41421 0 2.00000 0 0 2.64575i −2.82843 0 0
449.5 1.41421 0 2.00000 0 0 − 2.64575i 2.82843 0 0
449.6 1.41421 0 2.00000 0 0 − 2.64575i 2.82843 0 0
449.7 1.41421 0 2.00000 0 0 2.64575i 2.82843 0 0
449.8 1.41421 0 2.00000 0 0 2.64575i 2.82843 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
5.b even 2 1 inner
15.d odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3150.3.c.c 8
3.b odd 2 1 inner 3150.3.c.c 8
5.b even 2 1 inner 3150.3.c.c 8
5.c odd 4 1 3150.3.e.a ✓ 4
5.c odd 4 1 3150.3.e.d yes 4
15.d odd 2 1 inner 3150.3.c.c 8
15.e even 4 1 3150.3.e.a ✓ 4
15.e even 4 1 3150.3.e.d yes 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3150.3.c.c 8 1.a even 1 1 trivial
3150.3.c.c 8 3.b odd 2 1 inner
3150.3.c.c 8 5.b even 2 1 inner
3150.3.c.c 8 15.d odd 2 1 inner
3150.3.e.a ✓ 4 5.c odd 4 1
3150.3.e.a ✓ 4 15.e even 4 1
3150.3.e.d yes 4 5.c odd 4 1
3150.3.e.d yes 4 15.e even 4 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{11}^{4} + 464T_{11}^{2} + 12321 \) acting on \(S_{3}^{\mathrm{new}}(3150, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} \) Copy content Toggle raw display
$7$ \( (T^{2} + 7)^{4} \) Copy content Toggle raw display
$11$ \( (T^{4} + 464 T^{2} + 12321)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 32 T^{2} + 4)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 1268 T^{2} + 272484)^{2} \) Copy content Toggle raw display
$19$ \( (T^{2} - 30 T + 50)^{4} \) Copy content Toggle raw display
$23$ \( (T^{4} - 1184 T^{2} + 349281)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 872 T^{2} + 8649)^{2} \) Copy content Toggle raw display
$31$ \( (T^{2} - 8 T - 432)^{4} \) Copy content Toggle raw display
$37$ \( (T^{4} + 1918 T^{2} + 370881)^{2} \) Copy content Toggle raw display
$41$ \( (T^{4} + 2304 T^{2} + 1245456)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 6554 T^{2} + 7689529)^{2} \) Copy content Toggle raw display
$47$ \( (T^{4} - 6876 T^{2} + 8191044)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} - 7604 T^{2} + 2842596)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} + 9972 T^{2} + 16695396)^{2} \) Copy content Toggle raw display
$61$ \( (T^{2} + 44 T - 3548)^{4} \) Copy content Toggle raw display
$67$ \( (T^{4} + 3574 T^{2} + 1212201)^{2} \) Copy content Toggle raw display
$71$ \( (T^{4} + 13904 T^{2} + 35892081)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 18032 T^{2} + 66161956)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} + 72 T - 1231)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 39456 T^{2} + 379314576)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} + 20636 T^{2} + 4955076)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 21032 T^{2} + 58186384)^{2} \) Copy content Toggle raw display
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