Properties

Label 330.2.m.c
Level $330$
Weight $2$
Character orbit 330.m
Analytic conductor $2.635$
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] = [330,2,Mod(31,330)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(330, base_ring=CyclotomicField(10))
 
chi = DirichletCharacter(H, H._module([0, 0, 6]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("330.31");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 330 = 2 \cdot 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 330.m (of order \(5\), degree \(4\), 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: \(2.63506326670\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{10})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-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 \(\zeta_{10}\). We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{2} - \zeta_{10}^{2} q^{3} - \zeta_{10}^{3} q^{4} - \zeta_{10} q^{5} - \zeta_{10} q^{6} + ( - 2 \zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{7} - \zeta_{10}^{2} q^{8} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{9} +O(q^{10}) \) Copy content Toggle raw display \( q + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{2} - \zeta_{10}^{2} q^{3} - \zeta_{10}^{3} q^{4} - \zeta_{10} q^{5} - \zeta_{10} q^{6} + ( - 2 \zeta_{10}^{3} - 2 \zeta_{10} + 2) q^{7} - \zeta_{10}^{2} q^{8} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{9} - q^{10} + (3 \zeta_{10}^{3} - \zeta_{10}^{2} - \zeta_{10} - 1) q^{11} - q^{12} + ( - \zeta_{10}^{2} + \zeta_{10}) q^{13} + ( - 2 \zeta_{10}^{3} - 2 \zeta_{10}) q^{14} + \zeta_{10}^{3} q^{15} - \zeta_{10} q^{16} + (3 \zeta_{10}^{2} - 2 \zeta_{10} + 3) q^{17} + \zeta_{10}^{3} q^{18} + ( - 2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 2 \zeta_{10}) q^{19} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{20} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 2) q^{21} + (\zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2) q^{22} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} + 3) q^{23} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{24} + \zeta_{10}^{2} q^{25} + ( - \zeta_{10} + 1) q^{26} + \zeta_{10} q^{27} + ( - 2 \zeta_{10}^{2} - 2) q^{28} + ( - 2 \zeta_{10}^{3} - 5 \zeta_{10} + 5) q^{29} + \zeta_{10}^{2} q^{30} + ( - 2 \zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} + 2) q^{31} - q^{32} + (2 \zeta_{10}^{3} + \zeta_{10} + 2) q^{33} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2} + 1) q^{34} + (2 \zeta_{10}^{3} - 2) q^{35} + \zeta_{10}^{2} q^{36} + ( - \zeta_{10}^{3} + 3 \zeta_{10} - 3) q^{37} + ( - 2 \zeta_{10}^{2} - 2 \zeta_{10} - 2) q^{38} + ( - \zeta_{10}^{2} + \zeta_{10} - 1) q^{39} + \zeta_{10}^{3} q^{40} + (6 \zeta_{10}^{3} + 2 \zeta_{10}^{2} + 6 \zeta_{10}) q^{41} + (2 \zeta_{10}^{3} - 2) q^{42} + ( - 3 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 1) q^{43} + (2 \zeta_{10}^{3} - \zeta_{10}^{2} + 4 \zeta_{10} - 2) q^{44} + q^{45} + ( - 3 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 2 \zeta_{10} + 3) q^{46} + ( - \zeta_{10}^{3} + 13 \zeta_{10}^{2} - \zeta_{10}) q^{47} + \zeta_{10}^{3} q^{48} + ( - 4 \zeta_{10}^{2} + 3 \zeta_{10} - 4) q^{49} + \zeta_{10} q^{50} + ( - \zeta_{10}^{3} - 3 \zeta_{10} + 3) q^{51} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10}) q^{52} + (2 \zeta_{10}^{3} - 6 \zeta_{10}^{2} + 6 \zeta_{10} - 2) q^{53} + q^{54} + ( - 2 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 2 \zeta_{10} + 3) q^{55} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 2) q^{56} + (4 \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} - 4) q^{57} + ( - 5 \zeta_{10}^{3} + 3 \zeta_{10}^{2} - 5 \zeta_{10}) q^{58} + ( - 5 \zeta_{10}^{3} - 3 \zeta_{10} + 3) q^{59} + \zeta_{10} q^{60} + (4 \zeta_{10}^{2} + 4 \zeta_{10} + 4) q^{61} + ( - 2 \zeta_{10}^{3} - 3 \zeta_{10} + 3) q^{62} + (2 \zeta_{10}^{3} + 2 \zeta_{10}) q^{63} + (\zeta_{10}^{3} - \zeta_{10}^{2} + \zeta_{10} - 1) q^{64} + (\zeta_{10}^{3} - \zeta_{10}^{2}) q^{65} + ( - 2 \zeta_{10}^{3} + 4 \zeta_{10}^{2} - 2 \zeta_{10} + 3) q^{66} + ( - 7 \zeta_{10}^{3} + 7 \zeta_{10}^{2} + 2) q^{67} + ( - \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2 \zeta_{10} + 1) q^{68} + ( - \zeta_{10}^{3} - 2 \zeta_{10}^{2} - \zeta_{10}) q^{69} + (2 \zeta_{10}^{3} + 2 \zeta_{10} - 2) q^{70} + ( - 12 \zeta_{10}^{2} + 8 \zeta_{10} - 12) q^{71} + \zeta_{10} q^{72} + ( - 2 \zeta_{10}^{3} - 4 \zeta_{10} + 4) q^{73} + (3 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 3 \zeta_{10}) q^{74} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{75} + (2 \zeta_{10}^{3} - 2 \zeta_{10}^{2} - 4) q^{76} + (6 \zeta_{10}^{3} + 4 \zeta_{10}^{2} + 2 \zeta_{10}) q^{77} + (\zeta_{10}^{3} - \zeta_{10}^{2}) q^{78} + ( - 5 \zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} + 5) q^{79} + \zeta_{10}^{2} q^{80} - \zeta_{10}^{3} q^{81} + (6 \zeta_{10}^{2} + 2 \zeta_{10} + 6) q^{82} + (4 \zeta_{10}^{2} + 4 \zeta_{10} + 4) q^{83} + (2 \zeta_{10}^{3} + 2 \zeta_{10} - 2) q^{84} + ( - 3 \zeta_{10}^{3} + 2 \zeta_{10}^{2} - 3 \zeta_{10}) q^{85} + (\zeta_{10}^{3} - 4 \zeta_{10}^{2} + 4 \zeta_{10} - 1) q^{86} + (5 \zeta_{10}^{3} - 5 \zeta_{10}^{2} - 2) q^{87} + (2 \zeta_{10}^{3} + \zeta_{10} + 2) q^{88} + (6 \zeta_{10}^{3} - 6 \zeta_{10}^{2} - 4) q^{89} + ( - \zeta_{10}^{3} + \zeta_{10}^{2} - \zeta_{10} + 1) q^{90} - 2 \zeta_{10}^{2} q^{91} + ( - 3 \zeta_{10}^{3} - \zeta_{10} + 1) q^{92} + ( - 3 \zeta_{10}^{2} + \zeta_{10} - 3) q^{93} + ( - \zeta_{10}^{2} + 13 \zeta_{10} - 1) q^{94} + (4 \zeta_{10}^{3} + 2 \zeta_{10} - 2) q^{95} + \zeta_{10}^{2} q^{96} + (14 \zeta_{10}^{3} - 8 \zeta_{10}^{2} + 8 \zeta_{10} - 14) q^{97} + (4 \zeta_{10}^{3} - 4 \zeta_{10}^{2} - 1) q^{98} + ( - \zeta_{10}^{3} - 2 \zeta_{10}^{2} + 2) q^{99} +O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + q^{2} + q^{3} - q^{4} - q^{5} - q^{6} + 4 q^{7} + q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + q^{2} + q^{3} - q^{4} - q^{5} - q^{6} + 4 q^{7} + q^{8} - q^{9} - 4 q^{10} - q^{11} - 4 q^{12} + 2 q^{13} - 4 q^{14} + q^{15} - q^{16} + 7 q^{17} + q^{18} - 2 q^{19} - q^{20} - 4 q^{21} - 9 q^{22} + 10 q^{23} - q^{24} - q^{25} + 3 q^{26} + q^{27} - 6 q^{28} + 13 q^{29} - q^{30} + 8 q^{31} - 4 q^{32} + 11 q^{33} - 2 q^{34} - 6 q^{35} - q^{36} - 10 q^{37} - 8 q^{38} - 2 q^{39} + q^{40} + 10 q^{41} - 6 q^{42} - 10 q^{43} - q^{44} + 4 q^{45} + 5 q^{46} - 15 q^{47} + q^{48} - 9 q^{49} + q^{50} + 8 q^{51} - 3 q^{52} + 6 q^{53} + 4 q^{54} + 4 q^{55} - 4 q^{56} - 8 q^{57} - 13 q^{58} + 4 q^{59} + q^{60} + 16 q^{61} + 7 q^{62} + 4 q^{63} - q^{64} + 2 q^{65} + 4 q^{66} - 6 q^{67} + 7 q^{68} - 4 q^{70} - 28 q^{71} + q^{72} + 10 q^{73} + 10 q^{74} + q^{75} - 12 q^{76} + 4 q^{77} + 2 q^{78} + 23 q^{79} - q^{80} - q^{81} + 20 q^{82} + 16 q^{83} - 4 q^{84} - 8 q^{85} + 5 q^{86} + 2 q^{87} + 11 q^{88} - 4 q^{89} + q^{90} + 2 q^{91} - 8 q^{93} + 10 q^{94} - 2 q^{95} - q^{96} - 26 q^{97} + 4 q^{98} + 9 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/330\mathbb{Z}\right)^\times\).

\(n\) \(67\) \(211\) \(221\)
\(\chi(n)\) \(1\) \(-\zeta_{10}^{3}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
31.1
−0.309017 0.951057i
0.809017 + 0.587785i
−0.309017 + 0.951057i
0.809017 0.587785i
−0.309017 + 0.951057i 0.809017 0.587785i −0.809017 0.587785i 0.309017 + 0.951057i 0.309017 + 0.951057i 1.00000 + 0.726543i 0.809017 0.587785i 0.309017 0.951057i −1.00000
91.1 0.809017 0.587785i −0.309017 0.951057i 0.309017 0.951057i −0.809017 0.587785i −0.809017 0.587785i 1.00000 3.07768i −0.309017 0.951057i −0.809017 + 0.587785i −1.00000
181.1 −0.309017 0.951057i 0.809017 + 0.587785i −0.809017 + 0.587785i 0.309017 0.951057i 0.309017 0.951057i 1.00000 0.726543i 0.809017 + 0.587785i 0.309017 + 0.951057i −1.00000
301.1 0.809017 + 0.587785i −0.309017 + 0.951057i 0.309017 + 0.951057i −0.809017 + 0.587785i −0.809017 + 0.587785i 1.00000 + 3.07768i −0.309017 + 0.951057i −0.809017 0.587785i −1.00000
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
11.c even 5 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 330.2.m.c 4
3.b odd 2 1 990.2.n.d 4
11.c even 5 1 inner 330.2.m.c 4
11.c even 5 1 3630.2.a.be 2
11.d odd 10 1 3630.2.a.bm 2
33.h odd 10 1 990.2.n.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.m.c 4 1.a even 1 1 trivial
330.2.m.c 4 11.c even 5 1 inner
990.2.n.d 4 3.b odd 2 1
990.2.n.d 4 33.h odd 10 1
3630.2.a.be 2 11.c even 5 1
3630.2.a.bm 2 11.d odd 10 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{4} - 4T_{7}^{3} + 16T_{7}^{2} - 24T_{7} + 16 \) acting on \(S_{2}^{\mathrm{new}}(330, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{4} - T^{3} + T^{2} - T + 1 \) Copy content Toggle raw display
$3$ \( T^{4} - T^{3} + T^{2} - T + 1 \) Copy content Toggle raw display
$5$ \( T^{4} + T^{3} + T^{2} + T + 1 \) Copy content Toggle raw display
$7$ \( T^{4} - 4 T^{3} + 16 T^{2} - 24 T + 16 \) Copy content Toggle raw display
$11$ \( T^{4} + T^{3} - 9 T^{2} + 11 T + 121 \) Copy content Toggle raw display
$13$ \( T^{4} - 2 T^{3} + 4 T^{2} - 3 T + 1 \) Copy content Toggle raw display
$17$ \( T^{4} - 7 T^{3} + 34 T^{2} - 88 T + 121 \) Copy content Toggle raw display
$19$ \( T^{4} + 2 T^{3} + 24 T^{2} - 32 T + 16 \) Copy content Toggle raw display
$23$ \( (T^{2} - 5 T + 5)^{2} \) Copy content Toggle raw display
$29$ \( T^{4} - 13 T^{3} + 94 T^{2} + \cdots + 961 \) Copy content Toggle raw display
$31$ \( T^{4} - 8 T^{3} + 34 T^{2} - 77 T + 121 \) Copy content Toggle raw display
$37$ \( T^{4} + 10 T^{3} + 40 T^{2} + 25 T + 25 \) Copy content Toggle raw display
$41$ \( T^{4} - 10 T^{3} + 160 T^{2} + \cdots + 400 \) Copy content Toggle raw display
$43$ \( (T^{2} + 5 T - 5)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + 15 T^{3} + 160 T^{2} + \cdots + 24025 \) Copy content Toggle raw display
$53$ \( T^{4} - 6 T^{3} + 76 T^{2} - 56 T + 16 \) Copy content Toggle raw display
$59$ \( T^{4} - 4 T^{3} + 46 T^{2} + 11 T + 1 \) Copy content Toggle raw display
$61$ \( T^{4} - 16 T^{3} + 96 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$67$ \( (T^{2} + 3 T - 59)^{2} \) Copy content Toggle raw display
$71$ \( T^{4} + 28 T^{3} + 544 T^{2} + \cdots + 30976 \) Copy content Toggle raw display
$73$ \( T^{4} - 10 T^{3} + 60 T^{2} + \cdots + 400 \) Copy content Toggle raw display
$79$ \( T^{4} - 23 T^{3} + 304 T^{2} + \cdots + 10201 \) Copy content Toggle raw display
$83$ \( T^{4} - 16 T^{3} + 96 T^{2} + \cdots + 256 \) Copy content Toggle raw display
$89$ \( (T^{2} + 2 T - 44)^{2} \) Copy content Toggle raw display
$97$ \( T^{4} + 26 T^{3} + 256 T^{2} + \cdots + 5776 \) Copy content Toggle raw display
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