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Properties

Label 378.4.g.a
Level $378$
Weight $4$
Character orbit 378.g
Analytic conductor $22.303$
Analytic rank $0$
Dimension $6$
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] = [378,4,Mod(109,378)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(378, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("378.109");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 378.g (of order \(3\), degree \(2\), 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: \(22.3027219822\)
Analytic rank: \(0\)
Dimension: \(6\)
Relative dimension: \(3\) over \(\Q(\zeta_{3})\)
Coefficient field: 6.0.11184604443.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 43x^{4} - 210x^{3} + 1849x^{2} - 4515x + 11025 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{5}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - 2 \beta_1 q^{2} + (4 \beta_1 - 4) q^{4} + ( - \beta_{5} + \beta_{3} - 2 \beta_1) q^{5} + ( - 2 \beta_{5} - \beta_{2} - 2 \beta_1 + 3) q^{7} + 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - 2 \beta_1 q^{2} + (4 \beta_1 - 4) q^{4} + ( - \beta_{5} + \beta_{3} - 2 \beta_1) q^{5} + ( - 2 \beta_{5} - \beta_{2} - 2 \beta_1 + 3) q^{7} + 8 q^{8} + (2 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} + 4 \beta_1 - 4) q^{10} + ( - \beta_{5} - 3 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} - 11 \beta_1 + 11) q^{11} + ( - \beta_{5} + \beta_{4} + \beta_{2} + 3) q^{13} + ( - 2 \beta_{5} + 6 \beta_{2} - 2 \beta_1 - 4) q^{14} - 16 \beta_1 q^{16} + (\beta_{5} - 5 \beta_{4} + 6 \beta_{3} - 6 \beta_{2} - 14 \beta_1 + 14) q^{17} + (4 \beta_{5} + 5 \beta_{4} - 4 \beta_{3} - 5 \beta_{2} + 22 \beta_1) q^{19} + (4 \beta_{5} - 4 \beta_{4} - 4 \beta_{2} + 8) q^{20} + ( - 4 \beta_{5} + 4 \beta_{4} + 2 \beta_{3} + 6 \beta_{2} - 22) q^{22} + ( - 8 \beta_{5} - 9 \beta_{4} + 8 \beta_{3} + 9 \beta_{2} - \beta_1) q^{23} + (5 \beta_{5} + 9 \beta_{4} - 4 \beta_{3} + 4 \beta_{2} - 40 \beta_1 + 40) q^{25} + (2 \beta_{5} - 2 \beta_{3} - 6 \beta_1) q^{26} + (12 \beta_{5} - 8 \beta_{2} + 12 \beta_1 - 4) q^{28} + (20 \beta_{5} - 20 \beta_{4} + 11 \beta_{3} - 9 \beta_{2} - 61) q^{29} + (22 \beta_{5} + 7 \beta_{4} + 15 \beta_{3} - 15 \beta_{2} + 39 \beta_1 - 39) q^{31} + (32 \beta_1 - 32) q^{32} + ( - 12 \beta_{5} + 12 \beta_{4} - 2 \beta_{3} + 10 \beta_{2} - 28) q^{34} + ( - 7 \beta_{5} + 14 \beta_{4} + 7 \beta_{3} + 7 \beta_{2} + 77 \beta_1 - 126) q^{35} + ( - 9 \beta_{5} + 15 \beta_{4} + 9 \beta_{3} - 15 \beta_{2} + 130 \beta_1) q^{37} + ( - 10 \beta_{5} - 8 \beta_{4} - 2 \beta_{3} + 2 \beta_{2} - 44 \beta_1 + 44) q^{38} + ( - 8 \beta_{5} + 8 \beta_{3} - 16 \beta_1) q^{40} + (3 \beta_{5} - 3 \beta_{4} + 15 \beta_{3} + 12 \beta_{2} + 63) q^{41} + ( - 38 \beta_{5} + 38 \beta_{4} - \beta_{3} + 37 \beta_{2} - 133) q^{43} + (12 \beta_{5} + 4 \beta_{4} - 12 \beta_{3} - 4 \beta_{2} + 44 \beta_1) q^{44} + (18 \beta_{5} + 16 \beta_{4} + 2 \beta_{3} - 2 \beta_{2} + 2 \beta_1 - 2) q^{46} + ( - 7 \beta_{5} - 14 \beta_{4} + 7 \beta_{3} + 14 \beta_{2} + 54 \beta_1) q^{47} + ( - 16 \beta_{5} + 21 \beta_{4} + 35 \beta_{3} + 6 \beta_{2} - 37 \beta_1 + 66) q^{49} + (8 \beta_{5} - 8 \beta_{4} - 10 \beta_{3} - 18 \beta_{2} - 80) q^{50} + ( - 4 \beta_{4} + 4 \beta_{3} - 4 \beta_{2} + 12 \beta_1 - 12) q^{52} + (22 \beta_{5} + 41 \beta_{4} - 19 \beta_{3} + 19 \beta_{2} - 111 \beta_1 + 111) q^{53} + ( - 22 \beta_{5} + 22 \beta_{4} + 11 \beta_{3} + 33 \beta_{2} - 226) q^{55} + ( - 16 \beta_{5} - 8 \beta_{2} - 16 \beta_1 + 24) q^{56} + ( - 18 \beta_{5} + 22 \beta_{4} + 18 \beta_{3} - 22 \beta_{2} + 122 \beta_1) q^{58} + (7 \beta_{5} - 60 \beta_{4} + 67 \beta_{3} - 67 \beta_{2} - 160 \beta_1 + 160) q^{59} + (16 \beta_{5} + 42 \beta_{4} - 16 \beta_{3} - 42 \beta_{2} - 123 \beta_1) q^{61} + ( - 30 \beta_{5} + 30 \beta_{4} - 44 \beta_{3} - 14 \beta_{2} + 78) q^{62} + 64 q^{64} + (4 \beta_{5} + 5 \beta_{4} - 4 \beta_{3} - 5 \beta_{2} + 75 \beta_1) q^{65} + (51 \beta_{5} + 46 \beta_{4} + 5 \beta_{3} - 5 \beta_{2} + 43 \beta_1 - 43) q^{67} + (20 \beta_{5} - 4 \beta_{4} - 20 \beta_{3} + 4 \beta_{2} + 56 \beta_1) q^{68} + (14 \beta_{5} + 14 \beta_{4} - 42 \beta_{3} + 98 \beta_1 + 154) q^{70} + ( - 61 \beta_{5} + 61 \beta_{4} - 27 \beta_{3} + 34 \beta_{2} - 134) q^{71} + (7 \beta_{5} + 27 \beta_{4} - 20 \beta_{3} + 20 \beta_{2} - 113 \beta_1 + 113) q^{73} + ( - 30 \beta_{5} + 18 \beta_{4} - 48 \beta_{3} + 48 \beta_{2} - 260 \beta_1 + 260) q^{74} + (4 \beta_{5} - 4 \beta_{4} + 20 \beta_{3} + 16 \beta_{2} - 88) q^{76} + ( - 35 \beta_{5} + 49 \beta_{3} + 14 \beta_{2} - 280 \beta_1 - 77) q^{77} + (76 \beta_{5} + 27 \beta_{4} - 76 \beta_{3} - 27 \beta_{2} + 489 \beta_1) q^{79} + (16 \beta_{4} - 16 \beta_{3} + 16 \beta_{2} + 32 \beta_1 - 32) q^{80} + (24 \beta_{5} + 30 \beta_{4} - 24 \beta_{3} - 30 \beta_{2} - 126 \beta_1) q^{82} + ( - 53 \beta_{5} + 53 \beta_{4} + 28 \beta_{3} + 81 \beta_{2} - 62) q^{83} + ( - 19 \beta_{5} + 19 \beta_{4} + 29 \beta_{3} + 48 \beta_{2} - 472) q^{85} + (74 \beta_{5} - 2 \beta_{4} - 74 \beta_{3} + 2 \beta_{2} + 266 \beta_1) q^{86} + ( - 8 \beta_{5} - 24 \beta_{4} + 16 \beta_{3} - 16 \beta_{2} - 88 \beta_1 + 88) q^{88} + (42 \beta_{5} + 84 \beta_{4} - 42 \beta_{3} - 84 \beta_{2} + 453 \beta_1) q^{89} + ( - 10 \beta_{5} + 21 \beta_{4} - 14 \beta_{3} + 2 \beta_{2} + 116 \beta_1 - 34) q^{91} + ( - 4 \beta_{5} + 4 \beta_{4} - 36 \beta_{3} - 32 \beta_{2} + 4) q^{92} + (28 \beta_{5} + 14 \beta_{4} + 14 \beta_{3} - 14 \beta_{2} - 108 \beta_1 + 108) q^{94} + ( - 55 \beta_{4} + 55 \beta_{3} - 55 \beta_{2} - 173 \beta_1 + 173) q^{95} + ( - 6 \beta_{5} + 6 \beta_{4} - 38 \beta_{3} - 32 \beta_{2} + 220) q^{97} + (12 \beta_{5} + 70 \beta_{4} - 112 \beta_{3} + 20 \beta_{2} - 58 \beta_1 - 74) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 6 q^{2} - 12 q^{4} - 5 q^{5} + 8 q^{7} + 48 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 6 q^{2} - 12 q^{4} - 5 q^{5} + 8 q^{7} + 48 q^{8} - 10 q^{10} + 29 q^{11} + 20 q^{13} - 20 q^{14} - 48 q^{16} + 38 q^{17} + 57 q^{19} + 40 q^{20} - 116 q^{22} + 14 q^{23} + 134 q^{25} - 20 q^{26} + 8 q^{28} - 362 q^{29} - 88 q^{31} - 96 q^{32} - 152 q^{34} - 490 q^{35} + 384 q^{37} + 114 q^{38} - 40 q^{40} + 432 q^{41} - 726 q^{43} + 116 q^{44} + 28 q^{46} + 183 q^{47} + 372 q^{49} - 536 q^{50} - 40 q^{52} + 396 q^{53} - 1268 q^{55} + 64 q^{56} + 362 q^{58} + 427 q^{59} - 427 q^{61} + 352 q^{62} + 384 q^{64} + 216 q^{65} - 32 q^{67} + 152 q^{68} + 1162 q^{70} - 790 q^{71} + 373 q^{73} + 768 q^{74} - 456 q^{76} - 1211 q^{77} + 1364 q^{79} - 80 q^{80} - 432 q^{82} - 154 q^{83} - 2678 q^{85} + 726 q^{86} + 232 q^{88} + 1233 q^{89} + 131 q^{91} - 112 q^{92} + 366 q^{94} + 464 q^{95} + 1180 q^{97} - 720 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{6} + 43x^{4} - 210x^{3} + 1849x^{2} - 4515x + 11025 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( \nu^{5} + 43\nu^{3} - 105\nu^{2} + 1849\nu ) / 4515 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( \nu^{4} + 4\nu^{3} + 43\nu^{2} - 62\nu + 827 ) / 43 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( \nu^{4} + 3\nu^{3} + 43\nu^{2} - 148\nu + 932 ) / 43 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -29\nu^{5} - 827\nu^{3} + 7560\nu^{2} - 40076\nu + 86835 ) / 4515 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -29\nu^{5} - 932\nu^{3} + 7560\nu^{2} - 35561\nu + 97860 ) / 4515 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{5} - \beta_{4} - \beta_{3} + \beta_{2} ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 5\beta_{5} - 2\beta_{4} + 7\beta_{3} - 7\beta_{2} + 87\beta _1 - 87 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -86\beta_{5} + 86\beta_{4} - 43\beta_{3} + 43\beta_{2} + 315 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 191\beta_{5} - 320\beta_{4} - 191\beta_{3} + 320\beta_{2} - 3741\beta_1 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 2374\beta_{5} - 2059\beta_{4} + 4433\beta_{3} - 4433\beta_{2} + 22680\beta _1 - 22680 ) / 3 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/378\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\)
\(\chi(n)\) \(1\) \(-\beta_{1}\)

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 \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
2.16527 + 3.75036i
1.60692 + 2.78326i
−3.77219 − 6.53362i
2.16527 − 3.75036i
1.60692 − 2.78326i
−3.77219 + 6.53362i
−1.00000 − 1.73205i 0 −2.00000 + 3.46410i −6.70319 − 11.6103i 0 5.32531 − 17.7381i 8.00000 0 −13.4064 + 23.2205i
109.2 −1.00000 − 1.73205i 0 −2.00000 + 3.46410i 0.301070 + 0.521469i 0 −17.6665 + 5.55825i 8.00000 0 0.602141 − 1.04294i
109.3 −1.00000 − 1.73205i 0 −2.00000 + 3.46410i 3.90212 + 6.75867i 0 16.3412 + 8.71578i 8.00000 0 7.80425 − 13.5173i
163.1 −1.00000 + 1.73205i 0 −2.00000 − 3.46410i −6.70319 + 11.6103i 0 5.32531 + 17.7381i 8.00000 0 −13.4064 − 23.2205i
163.2 −1.00000 + 1.73205i 0 −2.00000 − 3.46410i 0.301070 − 0.521469i 0 −17.6665 − 5.55825i 8.00000 0 0.602141 + 1.04294i
163.3 −1.00000 + 1.73205i 0 −2.00000 − 3.46410i 3.90212 − 6.75867i 0 16.3412 − 8.71578i 8.00000 0 7.80425 + 13.5173i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.3
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.4.g.a ✓ 6
3.b odd 2 1 378.4.g.b yes 6
7.c even 3 1 inner 378.4.g.a ✓ 6
21.h odd 6 1 378.4.g.b yes 6
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.4.g.a ✓ 6 1.a even 1 1 trivial
378.4.g.a ✓ 6 7.c even 3 1 inner
378.4.g.b yes 6 3.b odd 2 1
378.4.g.b yes 6 21.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} + 5T_{5}^{5} + 133T_{5}^{4} - 666T_{5}^{3} + 11349T_{5}^{2} - 6804T_{5} + 3969 \) acting on \(S_{4}^{\mathrm{new}}(378, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 2 T + 4)^{3} \) Copy content Toggle raw display
$3$ \( T^{6} \) Copy content Toggle raw display
$5$ \( T^{6} + 5 T^{5} + 133 T^{4} + \cdots + 3969 \) Copy content Toggle raw display
$7$ \( T^{6} - 8 T^{5} - 154 T^{4} + \cdots + 40353607 \) Copy content Toggle raw display
$11$ \( T^{6} - 29 T^{5} + 1429 T^{4} + \cdots + 4862025 \) Copy content Toggle raw display
$13$ \( (T^{3} - 10 T^{2} - 83 T + 603)^{2} \) Copy content Toggle raw display
$17$ \( T^{6} - 38 T^{5} + \cdots + 2114252361 \) Copy content Toggle raw display
$19$ \( T^{6} - 57 T^{5} + \cdots + 19329618961 \) Copy content Toggle raw display
$23$ \( T^{6} - 14 T^{5} + \cdots + 148635694089 \) Copy content Toggle raw display
$29$ \( (T^{3} + 181 T^{2} - 28782 T + 862407)^{2} \) Copy content Toggle raw display
$31$ \( T^{6} + 88 T^{5} + \cdots + 27799667506521 \) Copy content Toggle raw display
$37$ \( T^{6} - 384 T^{5} + \cdots + 1957974001 \) Copy content Toggle raw display
$41$ \( (T^{3} - 216 T^{2} - 16605 T - 164025)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + 363 T^{2} - 119652 T - 6611597)^{2} \) Copy content Toggle raw display
$47$ \( T^{6} - 183 T^{5} + \cdots + 2568361041 \) Copy content Toggle raw display
$53$ \( T^{6} - 396 T^{5} + \cdots + 13\!\cdots\!61 \) Copy content Toggle raw display
$59$ \( T^{6} - 427 T^{5} + \cdots + 15\!\cdots\!69 \) Copy content Toggle raw display
$61$ \( T^{6} + 427 T^{5} + \cdots + 1548419366025 \) Copy content Toggle raw display
$67$ \( T^{6} + 32 T^{5} + \cdots + 70\!\cdots\!25 \) Copy content Toggle raw display
$71$ \( (T^{3} + 395 T^{2} - 301020 T - 115887969)^{2} \) Copy content Toggle raw display
$73$ \( T^{6} + \cdots + 202747156022761 \) Copy content Toggle raw display
$79$ \( T^{6} - 1364 T^{5} + \cdots + 21738803675121 \) Copy content Toggle raw display
$83$ \( (T^{3} + 77 T^{2} - 630648 T + 172613511)^{2} \) Copy content Toggle raw display
$89$ \( T^{6} - 1233 T^{5} + \cdots + 37\!\cdots\!69 \) Copy content Toggle raw display
$97$ \( (T^{3} - 590 T^{2} - 95128 T + 67884200)^{2} \) Copy content Toggle raw display
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