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Properties

Label 378.4.g.h
Level $378$
Weight $4$
Character orbit 378.g
Analytic conductor $22.303$
Analytic rank $0$
Dimension $10$
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: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 5 x^{9} + 181 x^{8} - 694 x^{7} + 11513 x^{6} - 32131 x^{5} + 301447 x^{4} - 550142 x^{3} + \cdots + 6694527 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{8}\cdot 7 \)
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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (2 \beta_{2} + 2) q^{2} + 4 \beta_{2} q^{4} - \beta_{5} q^{5} + (\beta_{3} + 3 \beta_{2}) q^{7} - 8 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (2 \beta_{2} + 2) q^{2} + 4 \beta_{2} q^{4} - \beta_{5} q^{5} + (\beta_{3} + 3 \beta_{2}) q^{7} - 8 q^{8} + ( - 2 \beta_{5} + 2 \beta_1) q^{10} + (\beta_{8} + \beta_{4} + \cdots - 3 \beta_{2}) q^{11}+ \cdots + (6 \beta_{9} + 8 \beta_{7} + \cdots - 182) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 10 q^{2} - 20 q^{4} - 2 q^{5} - 13 q^{7} - 80 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 10 q^{2} - 20 q^{4} - 2 q^{5} - 13 q^{7} - 80 q^{8} + 4 q^{10} + 14 q^{11} + 120 q^{13} - 52 q^{14} - 80 q^{16} + 8 q^{17} - 31 q^{19} + 16 q^{20} + 56 q^{22} + 152 q^{23} - 277 q^{25} + 120 q^{26} - 52 q^{28} - 266 q^{29} - 69 q^{31} + 160 q^{32} + 32 q^{34} - 7 q^{35} - 184 q^{37} + 62 q^{38} + 16 q^{40} + 744 q^{41} - 238 q^{43} + 56 q^{44} - 304 q^{46} - 117 q^{47} - 815 q^{49} - 1108 q^{50} - 240 q^{52} + 429 q^{53} + 248 q^{55} + 104 q^{56} - 266 q^{58} - 65 q^{59} + 735 q^{61} - 276 q^{62} + 640 q^{64} + 528 q^{65} - 728 q^{67} + 32 q^{68} - 760 q^{70} - 2314 q^{71} - 185 q^{73} + 368 q^{74} + 248 q^{76} - 2843 q^{77} - 3030 q^{79} - 32 q^{80} + 744 q^{82} + 2708 q^{83} + 3566 q^{85} - 238 q^{86} - 112 q^{88} + 2421 q^{89} - 4575 q^{91} - 1216 q^{92} + 234 q^{94} + 4364 q^{95} + 2638 q^{97} - 794 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} - 5 x^{9} + 181 x^{8} - 694 x^{7} + 11513 x^{6} - 32131 x^{5} + 301447 x^{4} - 550142 x^{3} + \cdots + 6694527 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 29 \nu^{8} - 116 \nu^{7} + 3436 \nu^{6} - 9902 \nu^{5} + 116611 \nu^{4} - 216854 \nu^{3} + \cdots - 73857 ) / 118944 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 1946 \nu^{9} + 8757 \nu^{8} - 234020 \nu^{7} + 778204 \nu^{6} - 9084392 \nu^{5} + \cdots - 295466325 ) / 1805213088 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 8025 \nu^{9} - 3229 \nu^{8} + 745180 \nu^{7} + 1551946 \nu^{6} + 10387381 \nu^{5} + \cdots + 4751955789 ) / 601737696 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 24101 \nu^{9} - 996309 \nu^{8} + 2429900 \nu^{7} - 115770406 \nu^{6} - 34063591 \nu^{5} + \cdots - 179047981875 ) / 1805213088 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 8615 \nu^{9} + 34588 \nu^{8} + 1136476 \nu^{7} + 3867634 \nu^{6} + 53315821 \nu^{5} + \cdots - 4847728788 ) / 601737696 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 3196 \nu^{9} + 91857 \nu^{8} + 90680 \nu^{7} + 11405672 \nu^{6} - 10307458 \nu^{5} + \cdots + 4617205971 ) / 200579232 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 50893 \nu^{9} + 1569456 \nu^{8} - 157052 \nu^{7} + 216206038 \nu^{6} - 384640793 \nu^{5} + \cdots + 475543146288 ) / 1805213088 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 50893 \nu^{9} - 449085 \nu^{8} + 7917112 \nu^{7} - 49634294 \nu^{6} + 384620629 \nu^{5} + \cdots - 54950782551 ) / 1805213088 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 27861 \nu^{9} + 158258 \nu^{8} - 3830496 \nu^{7} + 16813206 \nu^{6} - 165458907 \nu^{5} + \cdots - 1262012010 ) / 200579232 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{9} + 6\beta_{8} + 5\beta_{6} - 15\beta_{5} - 12\beta_{3} + 12\beta_{2} + 3\beta _1 + 39 ) / 63 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -13\beta_{9} - 48\beta_{8} + 23\beta_{6} - 27\beta_{5} - 30\beta_{3} + 30\beta_{2} - 45\beta _1 - 2076 ) / 63 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 74 \beta_{9} - 374 \beta_{8} + 28 \beta_{7} - 202 \beta_{6} + 956 \beta_{5} + 56 \beta_{4} + \cdots - 4545 ) / 63 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 761 \beta_{9} + 2252 \beta_{8} + 140 \beta_{7} - 1663 \beta_{6} + 2539 \beta_{5} + 112 \beta_{4} + \cdots + 94977 ) / 63 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 3805 \beta_{9} + 21416 \beta_{8} - 2044 \beta_{7} + 8105 \beta_{6} - 53225 \beta_{5} - 4508 \beta_{4} + \cdots + 300078 ) / 63 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 5548 \beta_{9} - 12320 \beta_{8} - 1544 \beta_{7} + 14444 \beta_{6} - 27364 \beta_{5} - 1972 \beta_{4} + \cdots - 673611 ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 195383 \beta_{9} - 1166670 \beta_{8} + 124236 \beta_{7} - 301135 \beta_{6} + 2906469 \beta_{5} + \cdots - 18895929 ) / 63 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 1971719 \beta_{9} + 2749020 \beta_{8} + 726012 \beta_{7} - 5845033 \beta_{6} + 13532877 \beta_{5} + \cdots + 238591776 ) / 63 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 10604842 \beta_{9} + 61868314 \beta_{8} - 7095116 \beta_{7} + 9264578 \beta_{6} - 155542936 \beta_{5} + \cdots + 1169800371 ) / 63 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(29\) \(325\)
\(\chi(n)\) \(1\) \(-1 - \beta_{2}\)

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
0.500000 − 7.61408i
0.500000 + 6.59007i
0.500000 − 3.38997i
0.500000 + 7.27226i
0.500000 − 1.99225i
0.500000 + 7.61408i
0.500000 − 6.59007i
0.500000 + 3.38997i
0.500000 − 7.27226i
0.500000 + 1.99225i
1.00000 + 1.73205i 0 −2.00000 + 3.46410i −10.8847 − 18.8528i 0 −18.1808 + 3.52981i −8.00000 0 21.7693 − 37.7056i
109.2 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −3.33613 − 5.77834i 0 14.5345 − 11.4781i −8.00000 0 6.67226 − 11.5567i
109.3 1.00000 + 1.73205i 0 −2.00000 + 3.46410i −0.334902 − 0.580067i 0 6.55596 + 17.3211i −8.00000 0 0.669803 − 1.16013i
109.4 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 5.38076 + 9.31975i 0 −8.21950 − 16.5964i −8.00000 0 −10.7615 + 18.6395i
109.5 1.00000 + 1.73205i 0 −2.00000 + 3.46410i 8.17493 + 14.1594i 0 −1.19020 + 18.4820i −8.00000 0 −16.3499 + 28.3188i
163.1 1.00000 − 1.73205i 0 −2.00000 − 3.46410i −10.8847 + 18.8528i 0 −18.1808 − 3.52981i −8.00000 0 21.7693 + 37.7056i
163.2 1.00000 − 1.73205i 0 −2.00000 − 3.46410i −3.33613 + 5.77834i 0 14.5345 + 11.4781i −8.00000 0 6.67226 + 11.5567i
163.3 1.00000 − 1.73205i 0 −2.00000 − 3.46410i −0.334902 + 0.580067i 0 6.55596 − 17.3211i −8.00000 0 0.669803 + 1.16013i
163.4 1.00000 − 1.73205i 0 −2.00000 − 3.46410i 5.38076 − 9.31975i 0 −8.21950 + 16.5964i −8.00000 0 −10.7615 − 18.6395i
163.5 1.00000 − 1.73205i 0 −2.00000 − 3.46410i 8.17493 − 14.1594i 0 −1.19020 − 18.4820i −8.00000 0 −16.3499 − 28.3188i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.5
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.h yes 10
3.b odd 2 1 378.4.g.g ✓ 10
7.c even 3 1 inner 378.4.g.h yes 10
21.h odd 6 1 378.4.g.g ✓ 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.4.g.g ✓ 10 3.b odd 2 1
378.4.g.g ✓ 10 21.h odd 6 1
378.4.g.h yes 10 1.a even 1 1 trivial
378.4.g.h yes 10 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{10} + 2 T_{5}^{9} + 453 T_{5}^{8} - 2428 T_{5}^{7} + 173800 T_{5}^{6} - 431451 T_{5}^{5} + \cdots + 293025924 \) acting on \(S_{4}^{\mathrm{new}}(378, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 2 T + 4)^{5} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + \cdots + 293025924 \) Copy content Toggle raw display
$7$ \( T^{10} + \cdots + 4747561509943 \) Copy content Toggle raw display
$11$ \( T^{10} + \cdots + 573171864056064 \) Copy content Toggle raw display
$13$ \( (T^{5} - 60 T^{4} + \cdots - 251504424)^{2} \) Copy content Toggle raw display
$17$ \( T^{10} + \cdots + 66\!\cdots\!96 \) Copy content Toggle raw display
$19$ \( T^{10} + \cdots + 13\!\cdots\!04 \) Copy content Toggle raw display
$23$ \( T^{10} + \cdots + 14\!\cdots\!84 \) Copy content Toggle raw display
$29$ \( (T^{5} + 133 T^{4} + \cdots + 8523490437)^{2} \) Copy content Toggle raw display
$31$ \( T^{10} + \cdots + 32\!\cdots\!84 \) Copy content Toggle raw display
$37$ \( T^{10} + \cdots + 24\!\cdots\!24 \) Copy content Toggle raw display
$41$ \( (T^{5} + \cdots - 1260804686436)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + 119 T^{4} + \cdots + 454373888128)^{2} \) Copy content Toggle raw display
$47$ \( T^{10} + \cdots + 12\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{10} + \cdots + 38\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{10} + \cdots + 98\!\cdots\!29 \) Copy content Toggle raw display
$61$ \( T^{10} + \cdots + 12\!\cdots\!84 \) Copy content Toggle raw display
$67$ \( T^{10} + \cdots + 27\!\cdots\!56 \) Copy content Toggle raw display
$71$ \( (T^{5} + \cdots + 81092003351688)^{2} \) Copy content Toggle raw display
$73$ \( T^{10} + \cdots + 35\!\cdots\!01 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 90\!\cdots\!61 \) Copy content Toggle raw display
$83$ \( (T^{5} + \cdots + 6912513777708)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + \cdots + 81\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( (T^{5} + \cdots - 13\!\cdots\!48)^{2} \) Copy content Toggle raw display
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