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Properties

Label 378.4.g.f
Level $378$
Weight $4$
Character orbit 378.g
Analytic conductor $22.303$
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] = [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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + 18x^{6} + 9x^{5} + 283x^{4} - 48x^{3} + 186x^{2} + 40x + 100 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{3} \)
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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + 2 \beta_1 q^{2} + (4 \beta_1 - 4) q^{4} + ( - \beta_{2} - \beta_1) q^{5} + ( - \beta_{4} - 2 \beta_1 + 2) q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + 2 \beta_1 q^{2} + (4 \beta_1 - 4) q^{4} + ( - \beta_{2} - \beta_1) q^{5} + ( - \beta_{4} - 2 \beta_1 + 2) q^{7} - 8 q^{8} + ( - 2 \beta_{7} - 2 \beta_{2} - 2 \beta_1) q^{10} + (\beta_{6} - \beta_{5} - 8 \beta_1 + 8) q^{11} + (2 \beta_{7} + 2 \beta_{6} + \cdots + \beta_1) q^{13}+ \cdots + ( - 28 \beta_{7} + 14 \beta_{6} + \cdots + 102) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 8 q^{2} - 16 q^{4} - 2 q^{5} + 6 q^{7} - 64 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{2} - 16 q^{4} - 2 q^{5} + 6 q^{7} - 64 q^{8} + 4 q^{10} + 32 q^{11} - 4 q^{13} + 36 q^{14} - 64 q^{16} - 58 q^{17} + 70 q^{19} + 16 q^{20} + 128 q^{22} + 86 q^{23} - 156 q^{25} - 4 q^{26} + 48 q^{28} - 212 q^{29} - 64 q^{31} + 128 q^{32} - 232 q^{34} + 8 q^{35} - 146 q^{37} - 140 q^{38} + 16 q^{40} - 780 q^{41} + 880 q^{43} + 128 q^{44} - 172 q^{46} - 306 q^{47} + 50 q^{49} - 624 q^{50} + 8 q^{52} - 90 q^{53} - 64 q^{55} - 48 q^{56} - 212 q^{58} + 148 q^{59} - 364 q^{61} - 256 q^{62} + 512 q^{64} + 1296 q^{65} - 954 q^{67} - 232 q^{68} + 20 q^{70} - 1360 q^{71} - 54 q^{73} + 292 q^{74} - 560 q^{76} + 2224 q^{77} - 226 q^{79} - 32 q^{80} - 780 q^{82} - 3136 q^{83} + 3920 q^{85} + 880 q^{86} - 256 q^{88} - 1458 q^{89} + 3836 q^{91} - 688 q^{92} + 612 q^{94} + 1310 q^{95} - 4344 q^{97} + 776 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} - x^{7} + 18x^{6} + 9x^{5} + 283x^{4} - 48x^{3} + 186x^{2} + 40x + 100 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 5251 \nu^{7} + 17661 \nu^{6} - 103443 \nu^{5} + 165901 \nu^{4} - 1314483 \nu^{3} + \cdots + 2144550 ) / 2252390 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 5533 \nu^{7} + 45633 \nu^{6} - 267279 \nu^{5} + 771473 \nu^{4} - 3396399 \nu^{3} + \cdots + 5541150 ) / 2252390 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 25642 \nu^{7} - 123347 \nu^{6} + 400691 \nu^{5} - 1182462 \nu^{4} + 3388681 \nu^{3} + \cdots + 20095080 ) / 2252390 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 26206 \nu^{7} + 179291 \nu^{6} - 728363 \nu^{5} + 2393606 \nu^{4} - 7552513 \nu^{3} + \cdots - 13301880 ) / 2252390 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 16807 \nu^{7} - 3492 \nu^{6} + 296256 \nu^{5} + 417823 \nu^{4} + 4902586 \nu^{3} + 3475204 \nu^{2} + \cdots + 5873470 ) / 321770 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 42509 \nu^{7} + 56798 \nu^{6} - 783152 \nu^{5} - 84149 \nu^{4} - 11885520 \nu^{3} + \cdots + 5545466 ) / 450478 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 47855 \nu^{7} + 35959 \nu^{6} - 821980 \nu^{5} - 661525 \nu^{4} - 13230433 \nu^{3} + \cdots - 3549092 ) / 450478 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{4} + \beta_{3} - 2\beta_{2} + 2\beta_1 ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -2\beta_{7} - \beta_{6} - 3\beta_{5} - 4\beta_{4} + 4\beta_{3} - 2\beta_{2} + 106\beta _1 - 104 ) / 12 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -14\beta_{7} + 11\beta_{6} - 9\beta_{5} - 11\beta_{4} - 9\beta_{3} + 2\beta _1 - 68 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 24\beta_{6} + 24\beta_{5} + \beta_{4} - 23\beta_{3} + 18\beta_{2} - 570\beta _1 - 24 ) / 4 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 502\beta_{7} - 297\beta_{6} + 421\beta_{5} + 124\beta_{4} - 124\beta_{3} + 502\beta_{2} - 3174\beta _1 + 3552 ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 322\beta_{7} - 369\beta_{6} + 46\beta_{5} + 369\beta_{4} + 46\beta_{3} - 323\beta _1 + 8316 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -3072\beta_{6} - 3072\beta_{5} + 5065\beta_{4} + 8137\beta_{3} - 9326\beta_{2} + 74654\beta _1 + 3072 ) / 12 \) 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
−0.338925 − 0.587036i
−1.84763 − 3.20018i
2.24123 + 3.88192i
0.445324 + 0.771324i
−0.338925 + 0.587036i
−1.84763 + 3.20018i
2.24123 − 3.88192i
0.445324 − 0.771324i
1.00000 + 1.73205i 0 −2.00000 + 3.46410i −8.05411 − 13.9501i 0 6.77345 − 17.2372i −8.00000 0 16.1082 − 27.9003i
109.2 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −5.04834 − 8.74398i 0 −5.48778 + 17.6885i −8.00000 0 10.0967 − 17.4880i
109.3 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 5.59791 + 9.69587i 0 −16.7643 − 7.87127i −8.00000 0 −11.1958 + 19.3917i
109.4 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 6.50453 + 11.2662i 0 18.4787 − 1.24034i −8.00000 0 −13.0091 + 22.5324i
163.1 1.00000 − 1.73205i 0 −2.00000 − 3.46410i −8.05411 + 13.9501i 0 6.77345 + 17.2372i −8.00000 0 16.1082 + 27.9003i
163.2 1.00000 − 1.73205i 0 −2.00000 − 3.46410i −5.04834 + 8.74398i 0 −5.48778 − 17.6885i −8.00000 0 10.0967 + 17.4880i
163.3 1.00000 − 1.73205i 0 −2.00000 − 3.46410i 5.59791 − 9.69587i 0 −16.7643 + 7.87127i −8.00000 0 −11.1958 − 19.3917i
163.4 1.00000 − 1.73205i 0 −2.00000 − 3.46410i 6.50453 − 11.2662i 0 18.4787 + 1.24034i −8.00000 0 −13.0091 − 22.5324i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.4
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.f yes 8
3.b odd 2 1 378.4.g.c ✓ 8
7.c even 3 1 inner 378.4.g.f yes 8
21.h odd 6 1 378.4.g.c ✓ 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.4.g.c ✓ 8 3.b odd 2 1
378.4.g.c ✓ 8 21.h odd 6 1
378.4.g.f yes 8 1.a even 1 1 trivial
378.4.g.f yes 8 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{8} + 2 T_{5}^{7} + 330 T_{5}^{6} - 412 T_{5}^{5} + 82828 T_{5}^{4} - 55632 T_{5}^{3} + \cdots + 561121344 \) acting on \(S_{4}^{\mathrm{new}}(378, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 4)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} \) Copy content Toggle raw display
$5$ \( T^{8} + 2 T^{7} + \cdots + 561121344 \) Copy content Toggle raw display
$7$ \( T^{8} + \cdots + 13841287201 \) Copy content Toggle raw display
$11$ \( T^{8} + \cdots + 53616328704 \) Copy content Toggle raw display
$13$ \( (T^{4} + 2 T^{3} + \cdots - 513576)^{2} \) Copy content Toggle raw display
$17$ \( T^{8} + \cdots + 1132436505600 \) Copy content Toggle raw display
$19$ \( T^{8} + \cdots + 543448598016256 \) Copy content Toggle raw display
$23$ \( T^{8} + \cdots + 2210504256 \) Copy content Toggle raw display
$29$ \( (T^{4} + 106 T^{3} + \cdots - 175957920)^{2} \) Copy content Toggle raw display
$31$ \( T^{8} + \cdots + 139065597225 \) Copy content Toggle raw display
$37$ \( T^{8} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display
$41$ \( (T^{4} + 390 T^{3} + \cdots + 2132840160)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 440 T^{3} + \cdots - 49652003)^{2} \) Copy content Toggle raw display
$47$ \( T^{8} + \cdots + 73\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{8} + \cdots + 17\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{8} + \cdots + 22\!\cdots\!16 \) Copy content Toggle raw display
$61$ \( T^{8} + \cdots + 80\!\cdots\!25 \) Copy content Toggle raw display
$67$ \( T^{8} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{4} + 680 T^{3} + \cdots - 5339649600)^{2} \) Copy content Toggle raw display
$73$ \( T^{8} + \cdots + 38\!\cdots\!56 \) Copy content Toggle raw display
$79$ \( T^{8} + \cdots + 20\!\cdots\!56 \) Copy content Toggle raw display
$83$ \( (T^{4} + 1568 T^{3} + \cdots - 310761735840)^{2} \) Copy content Toggle raw display
$89$ \( T^{8} + \cdots + 29\!\cdots\!00 \) Copy content Toggle raw display
$97$ \( (T^{4} + \cdots - 1160560203227)^{2} \) Copy content Toggle raw display
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