Properties

Label 330.2.m.e
Level $330$
Weight $2$
Character orbit 330.m
Analytic conductor $2.635$
Analytic rank $0$
Dimension $8$
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: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{5})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - 3x^{7} + 14x^{6} - 12x^{5} + 121x^{4} + 120x^{3} + 1400x^{2} + 3000x + 10000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
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 basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + \beta_{4} q^{3} - \beta_{3} q^{4} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{5} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{6} + ( - \beta_{7} + \beta_{4} - \beta_{3} - \beta_{2}) q^{7} + \beta_{4} q^{8} - \beta_{2} q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{2} q^{2} + \beta_{4} q^{3} - \beta_{3} q^{4} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{5} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{6} + ( - \beta_{7} + \beta_{4} - \beta_{3} - \beta_{2}) q^{7} + \beta_{4} q^{8} - \beta_{2} q^{9} + q^{10} + ( - \beta_{7} + \beta_{6} + \beta_{5} + \beta_{2} + \beta_1 - 1) q^{11} + q^{12} + ( - \beta_{7} + \beta_{5} - \beta_{3} - 2 \beta_{2} + 1) q^{13} + (\beta_{2} + \beta_1 - 1) q^{14} - \beta_{3} q^{15} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{16} + ( - \beta_{7} + \beta_{6} + 2 \beta_{4} - \beta_{3} - \beta_{2} + 1) q^{17} - \beta_{3} q^{18} + ( - \beta_{6} - \beta_{5} + 3 \beta_{4} - \beta_{3} - 2 \beta_{2} + 2) q^{19} - \beta_{2} q^{20} + (\beta_{5} - \beta_{4} + \beta_{3}) q^{21} + ( - \beta_{5} + \beta_{4}) q^{22} + ( - \beta_{7} + \beta_{5} - 3 \beta_{4} + 3 \beta_{3} + \beta_1) q^{23} - \beta_{2} q^{24} + \beta_{4} q^{25} + (\beta_{6} + \beta_{4} - \beta_{3} + \beta_1 - 1) q^{26} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{27} + (\beta_{7} - \beta_{6} - \beta_{5} - \beta_1 + 1) q^{28} + \beta_{7} q^{29} + \beta_{4} q^{30} + ( - \beta_{6} + 3 \beta_{3} - \beta_{2} - 3) q^{31} + q^{32} + (\beta_{7} + 1) q^{33} + ( - \beta_{7} + \beta_1 - 2) q^{34} + \beta_{6} q^{35} + \beta_{4} q^{36} + ( - 3 \beta_{4} - 4 \beta_{3} + 3 \beta_{2}) q^{37} + (\beta_{7} - \beta_{6} - 3 \beta_{4} + \beta_{3} + \beta_{2} - 2) q^{38} + (\beta_{5} - \beta_{4} + 2 \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{39} - \beta_{3} q^{40} + (2 \beta_{6} + 2 \beta_{5} + 2 \beta_{3} + \beta_{2} + \beta_1 - 1) q^{41} + \beta_{6} q^{42} + ( - \beta_{7} + 6 \beta_{4} - 6 \beta_{3} + \beta_1 + 2) q^{43} + ( - \beta_{6} - \beta_{4}) q^{44} + q^{45} + (\beta_{7} - \beta_{5} - 3 \beta_{3} - 3 \beta_{2} + 3) q^{46} + ( - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - 2 \beta_{2} - 3 \beta_1 + 2) q^{47} - \beta_{3} q^{48} + (\beta_{7} - \beta_{6} - 4 \beta_{4} + 4 \beta_{3} + 4 \beta_{2} - 3) q^{49} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{50} + (\beta_{7} - \beta_{6} + \beta_{3} - \beta_{2} - \beta_1 + 1) q^{51} + ( - \beta_{6} - \beta_{5} + \beta_{4} - \beta_{3} - \beta_1) q^{52} + (\beta_{6} + 6 \beta_{3} + 2 \beta_{2} - 6) q^{53} + q^{54} + ( - \beta_{2} - \beta_1) q^{55} + (\beta_{5} - \beta_{4} + \beta_{3}) q^{56} + ( - \beta_{7} + \beta_{6} + \beta_{5} + 2 \beta_{3} - \beta_{2} - 2) q^{57} - \beta_1 q^{58} + (2 \beta_{7} - 2 \beta_{6} - \beta_{4} - 4 \beta_{3} - \beta_{2} - 2 \beta_1 + 2) q^{59} + ( - \beta_{4} + \beta_{3} + \beta_{2} - 1) q^{60} + (2 \beta_{7} - 2 \beta_{6} - 2 \beta_{5} + 4 \beta_{4} - 4 \beta_{3} - 4 \beta_{2} + \cdots + 6) q^{61}+ \cdots + ( - \beta_{5} + \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 2 q^{2} - 2 q^{3} - 2 q^{4} - 2 q^{5} - 2 q^{6} - 3 q^{7} - 2 q^{8} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 2 q^{2} - 2 q^{3} - 2 q^{4} - 2 q^{5} - 2 q^{6} - 3 q^{7} - 2 q^{8} - 2 q^{9} + 8 q^{10} + 8 q^{12} + 3 q^{13} - 3 q^{14} - 2 q^{15} - 2 q^{16} + 5 q^{17} - 2 q^{18} + 4 q^{19} - 2 q^{20} + 2 q^{21} + 16 q^{23} - 2 q^{24} - 2 q^{25} - 7 q^{26} - 2 q^{27} + 2 q^{28} - 3 q^{29} - 2 q^{30} - 22 q^{31} + 8 q^{32} + 5 q^{33} - 10 q^{34} + 2 q^{35} - 2 q^{36} + 4 q^{37} - 11 q^{38} + 3 q^{39} - 2 q^{40} + q^{41} + 2 q^{42} - 2 q^{43} + 8 q^{45} + 11 q^{46} - q^{47} - 2 q^{48} - 5 q^{49} - 2 q^{50} - 7 q^{52} - 30 q^{53} + 8 q^{54} - 5 q^{55} + 2 q^{56} - 11 q^{57} - 3 q^{58} - 8 q^{59} - 2 q^{60} + 12 q^{61} + 13 q^{62} - 3 q^{63} - 2 q^{64} + 8 q^{65} - 5 q^{66} + 58 q^{67} + 5 q^{68} - 19 q^{69} - 3 q^{70} - 2 q^{71} - 2 q^{72} + 12 q^{73} + 4 q^{74} - 2 q^{75} + 14 q^{76} + 22 q^{77} + 8 q^{78} + 41 q^{79} - 2 q^{80} - 2 q^{81} - 4 q^{82} - 8 q^{83} - 3 q^{84} - 27 q^{86} + 2 q^{87} + 5 q^{88} + 10 q^{89} - 2 q^{90} - 47 q^{91} - 19 q^{92} - 22 q^{93} - q^{94} + 4 q^{95} - 2 q^{96} - 40 q^{97} + 30 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - 3x^{7} + 14x^{6} - 12x^{5} + 121x^{4} + 120x^{3} + 1400x^{2} + 3000x + 10000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{7} - 13\nu^{6} + 144\nu^{5} - 1452\nu^{4} + 14641\nu^{3} - 146290\nu^{2} + 133300\nu + 1000 ) / 1331000 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 41\nu^{7} - 423\nu^{6} + 3374\nu^{5} - 18392\nu^{4} + 29161\nu^{3} - 7180\nu^{2} + 9300\nu + 52000 ) / 1331000 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -289\nu^{7} + 1557\nu^{6} - 3116\nu^{5} - 3872\nu^{4} + 3751\nu^{3} - 72190\nu^{2} - 79800\nu - 630000 ) / 1331000 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 3\nu^{7} - 28\nu^{6} + 179\nu^{5} - 242\nu^{4} + 121\nu^{3} + 481\nu^{2} + 710\nu + 4100 ) / 13310 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 324\nu^{7} + 683\nu^{6} - 5429\nu^{5} + 29282\nu^{4} - 40656\nu^{3} + 239135\nu^{2} + 47200\nu + 2084000 ) / 665500 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( \nu^{7} - 2\nu^{6} + 12\nu^{5} + 121\nu^{3} + 241\nu^{2} + 1641\nu + 3310 ) / 1331 \) Copy content Toggle raw display
\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - \beta_{6} - \beta_{5} - \beta_{3} - 11\beta_{2} + 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( 12\beta_{7} - 12\beta_{6} - 11\beta_{5} - 22\beta_{3} - 22\beta_{2} - 11\beta _1 + 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 23\beta_{7} - 22\beta_{6} + 10\beta_{4} - 142\beta_{3} - 32\beta_{2} - 22\beta _1 + 22 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 33\beta_{7} + 132\beta_{5} + 220\beta_{4} - 220\beta_{3} - 33\beta _1 - 10 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 385\beta_{6} + 385\beta_{5} + 1320\beta_{4} + 385\beta_{3} + 715\beta_{2} - 43\beta _1 - 715 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( -758\beta_{7} + 2463\beta_{6} + 758\beta_{5} + 6313\beta_{3} + 6743\beta_{2} - 6313 \) Copy content Toggle raw display

Character values

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

\(n\) \(67\) \(211\) \(221\)
\(\chi(n)\) \(1\) \(-\beta_{3}\) \(1\)

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
−1.98622 + 1.44308i
3.29524 2.39413i
−0.886361 2.72794i
1.07734 + 3.31572i
−1.98622 1.44308i
3.29524 + 2.39413i
−0.886361 + 2.72794i
1.07734 3.31572i
0.309017 0.951057i −0.809017 + 0.587785i −0.809017 0.587785i 0.309017 + 0.951057i 0.309017 + 0.951057i −3.29524 2.39413i −0.809017 + 0.587785i 0.309017 0.951057i 1.00000
31.2 0.309017 0.951057i −0.809017 + 0.587785i −0.809017 0.587785i 0.309017 + 0.951057i 0.309017 + 0.951057i 1.98622 + 1.44308i −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 −1.07734 + 3.31572i 0.309017 + 0.951057i −0.809017 + 0.587785i 1.00000
91.2 −0.809017 + 0.587785i 0.309017 + 0.951057i 0.309017 0.951057i −0.809017 0.587785i −0.809017 0.587785i 0.886361 2.72794i 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 −3.29524 + 2.39413i −0.809017 0.587785i 0.309017 + 0.951057i 1.00000
181.2 0.309017 + 0.951057i −0.809017 0.587785i −0.809017 + 0.587785i 0.309017 0.951057i 0.309017 0.951057i 1.98622 1.44308i −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 −1.07734 3.31572i 0.309017 0.951057i −0.809017 0.587785i 1.00000
301.2 −0.809017 0.587785i 0.309017 0.951057i 0.309017 + 0.951057i −0.809017 + 0.587785i −0.809017 + 0.587785i 0.886361 + 2.72794i 0.309017 0.951057i −0.809017 0.587785i 1.00000
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 31.2
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.e 8
3.b odd 2 1 990.2.n.k 8
11.c even 5 1 inner 330.2.m.e 8
11.c even 5 1 3630.2.a.bt 4
11.d odd 10 1 3630.2.a.br 4
33.h odd 10 1 990.2.n.k 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
330.2.m.e 8 1.a even 1 1 trivial
330.2.m.e 8 11.c even 5 1 inner
990.2.n.k 8 3.b odd 2 1
990.2.n.k 8 33.h odd 10 1
3630.2.a.br 4 11.d odd 10 1
3630.2.a.bt 4 11.c even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{7}^{8} + 3T_{7}^{7} + 14T_{7}^{6} + 12T_{7}^{5} + 121T_{7}^{4} - 120T_{7}^{3} + 1400T_{7}^{2} - 3000T_{7} + 10000 \) acting on \(S_{2}^{\mathrm{new}}(330, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$3$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$5$ \( (T^{4} + T^{3} + T^{2} + T + 1)^{2} \) Copy content Toggle raw display
$7$ \( T^{8} + 3 T^{7} + 14 T^{6} + \cdots + 10000 \) Copy content Toggle raw display
$11$ \( T^{8} + 11 T^{6} - 50 T^{5} + \cdots + 14641 \) Copy content Toggle raw display
$13$ \( T^{8} - 3 T^{7} + 2 T^{6} + 45 T^{5} + \cdots + 121 \) Copy content Toggle raw display
$17$ \( T^{8} - 5 T^{7} + 54 T^{6} - 280 T^{5} + \cdots + 256 \) Copy content Toggle raw display
$19$ \( T^{8} - 4 T^{7} + 16 T^{6} + \cdots + 48400 \) Copy content Toggle raw display
$23$ \( (T^{4} - 8 T^{3} - 29 T^{2} + 330 T - 605)^{2} \) Copy content Toggle raw display
$29$ \( T^{8} + 3 T^{7} + 14 T^{6} + \cdots + 10000 \) Copy content Toggle raw display
$31$ \( T^{8} + 22 T^{7} + 234 T^{6} + \cdots + 48400 \) Copy content Toggle raw display
$37$ \( (T^{4} - 2 T^{3} + 64 T^{2} + 247 T + 361)^{2} \) Copy content Toggle raw display
$41$ \( T^{8} - T^{7} + 52 T^{6} + \cdots + 6739216 \) Copy content Toggle raw display
$43$ \( (T^{4} + T^{3} - 107 T^{2} + 242 T + 404)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + T^{7} + 202 T^{6} + \cdots + 7027801 \) Copy content Toggle raw display
$53$ \( T^{8} + 30 T^{7} + 466 T^{6} + \cdots + 355216 \) Copy content Toggle raw display
$59$ \( T^{8} + 8 T^{7} + 4 T^{6} - 448 T^{5} + \cdots + 3025 \) Copy content Toggle raw display
$61$ \( T^{8} - 12 T^{7} + 112 T^{6} + \cdots + 495616 \) Copy content Toggle raw display
$67$ \( (T^{4} - 29 T^{3} + 83 T^{2} + 3662 T - 26476)^{2} \) Copy content Toggle raw display
$71$ \( T^{8} + 2 T^{7} + 184 T^{6} + \cdots + 2560000 \) Copy content Toggle raw display
$73$ \( (T^{4} - 6 T^{3} + 36 T^{2} - 216 T + 1296)^{2} \) Copy content Toggle raw display
$79$ \( T^{8} - 41 T^{7} + 1006 T^{6} + \cdots + 31584400 \) Copy content Toggle raw display
$83$ \( (T^{4} + 4 T^{3} + 96 T^{2} - 256 T + 256)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 5 T^{3} - 145 T^{2} + 350 T + 500)^{2} \) Copy content Toggle raw display
$97$ \( T^{8} + 40 T^{7} + 744 T^{6} + \cdots + 26050816 \) Copy content Toggle raw display
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