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Properties

Label 378.3.n.c
Level $378$
Weight $3$
Character orbit 378.n
Analytic conductor $10.300$
Analytic rank $0$
Dimension $12$
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] = [378,3,Mod(271,378)]
 
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, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("378.271");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 378.n (of order \(6\), 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: \(10.2997539928\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 13 x^{10} - 10 x^{9} + 89 x^{8} - 70 x^{7} + 314 x^{6} - 138 x^{5} + 673 x^{4} + \cdots + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3^{6}\cdot 7 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + (\beta_{7} + \beta_{3}) q^{2} + 2 \beta_{2} q^{4} + ( - \beta_{3} - \beta_1 - 1) q^{5} + (\beta_{11} + \beta_{10} + \cdots - \beta_1) q^{7}+ \cdots - 2 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} + \beta_{3}) q^{2} + 2 \beta_{2} q^{4} + ( - \beta_{3} - \beta_1 - 1) q^{5} + (\beta_{11} + \beta_{10} + \cdots - \beta_1) q^{7}+ \cdots + (6 \beta_{11} + 2 \beta_{10} + \cdots + 42) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{4} - 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{4} - 6 q^{5} - 12 q^{10} + 6 q^{11} - 12 q^{14} - 24 q^{16} + 24 q^{17} - 30 q^{19} + 96 q^{23} + 36 q^{25} - 48 q^{26} + 24 q^{28} - 108 q^{29} + 36 q^{31} + 108 q^{35} - 12 q^{37} + 96 q^{38} + 24 q^{40} - 156 q^{43} + 12 q^{44} + 48 q^{46} - 42 q^{47} - 108 q^{49} - 336 q^{50} + 48 q^{52} - 60 q^{53} + 24 q^{56} + 12 q^{58} + 66 q^{59} + 186 q^{61} + 96 q^{64} + 180 q^{65} - 156 q^{67} - 48 q^{68} - 72 q^{70} - 372 q^{71} - 366 q^{73} + 24 q^{74} + 534 q^{77} + 84 q^{79} + 24 q^{80} - 192 q^{82} + 276 q^{85} + 12 q^{86} + 306 q^{89} + 390 q^{91} - 384 q^{92} + 24 q^{94} - 420 q^{95} + 216 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 13 x^{10} - 10 x^{9} + 89 x^{8} - 70 x^{7} + 314 x^{6} - 138 x^{5} + 673 x^{4} + \cdots + 49 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 14482023 \nu^{11} + 1108530048 \nu^{10} - 5430958 \nu^{9} + 10719869112 \nu^{8} + \cdots + 143166932020 ) / 13107836357 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 71898534 \nu^{11} + 150039156 \nu^{10} - 924828780 \nu^{9} + 765827019 \nu^{8} + \cdots - 10624742541 ) / 13107836357 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 12964 \nu^{11} + 36004 \nu^{10} + 111523 \nu^{9} + 494982 \nu^{8} + 1243935 \nu^{7} + \cdots - 1258677 ) / 1478939 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 126019412 \nu^{11} - 1739516115 \nu^{10} + 1693046408 \nu^{9} - 15748888799 \nu^{8} + \cdots - 70870073876 ) / 13107836357 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 28907597 \nu^{11} - 158525426 \nu^{10} - 308220881 \nu^{9} - 1787607711 \nu^{8} + \cdots - 1765718010 ) / 1191621487 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 30372251 \nu^{11} - 55436888 \nu^{10} + 196314506 \nu^{9} - 150385285 \nu^{8} + \cdots - 5981168536 ) / 1191621487 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 339116963 \nu^{11} - 858033488 \nu^{10} + 3848478922 \nu^{9} - 4850321008 \nu^{8} + \cdots + 4244563617 ) / 13107836357 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 440979046 \nu^{11} - 2907702596 \nu^{10} + 7430185959 \nu^{9} - 25313969181 \nu^{8} + \cdots - 35048072661 ) / 13107836357 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 515153246 \nu^{11} - 628751536 \nu^{10} - 3150655989 \nu^{9} - 12094955534 \nu^{8} + \cdots + 109295678856 ) / 13107836357 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 786254350 \nu^{11} + 217358764 \nu^{10} - 7477881970 \nu^{9} - 6683844172 \nu^{8} + \cdots - 40297188715 ) / 13107836357 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1665037430 \nu^{11} - 1776367118 \nu^{10} + 17563724837 \nu^{9} + 386406512 \nu^{8} + \cdots + 110051637780 ) / 13107836357 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2 \beta_{11} + 5 \beta_{10} - \beta_{9} - 2 \beta_{8} + 8 \beta_{7} - 3 \beta_{6} + \beta_{5} + \cdots + 3 ) / 21 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} + 6 \beta_{10} + 3 \beta_{9} - \beta_{8} + 32 \beta_{7} + 2 \beta_{6} - 3 \beta_{5} + \cdots + 5 ) / 21 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 9 \beta_{11} - 26 \beta_{10} - 6 \beta_{9} + 9 \beta_{8} - \beta_{7} - 4 \beta_{6} + 6 \beta_{5} + \cdots - 87 ) / 21 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 24 \beta_{11} - 94 \beta_{10} - 12 \beta_{9} - 10 \beta_{8} - 331 \beta_{7} - 50 \beta_{6} - 2 \beta_{5} + \cdots - 503 ) / 21 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 11 \beta_{11} - 179 \beta_{10} + 19 \beta_{9} - 4 \beta_{8} - 747 \beta_{7} - 62 \beta_{6} - 19 \beta_{5} + \cdots + 34 ) / 21 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 22 \beta_{11} + 56 \beta_{10} - 8 \beta_{9} + 22 \beta_{8} + 43 \beta_{7} + 29 \beta_{6} + 28 \beta_{5} + \cdots + 518 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 138 \beta_{11} + 3344 \beta_{10} + 69 \beta_{9} - 9 \beta_{8} + 7897 \beta_{7} + 1411 \beta_{6} + \cdots + 8886 ) / 21 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 405 \beta_{11} + 5102 \beta_{10} + 1137 \beta_{9} - 771 \beta_{8} + 24168 \beta_{7} + 2739 \beta_{6} + \cdots + 1503 ) / 21 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1503 \beta_{11} - 14306 \beta_{10} + 365 \beta_{9} - 1503 \beta_{8} - 7435 \beta_{7} - 6297 \beta_{6} + \cdots - 84808 ) / 21 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 12554 \beta_{11} - 85963 \beta_{10} - 6277 \beta_{9} - 2740 \beta_{8} - 244288 \beta_{7} + \cdots - 288910 ) / 21 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 7886 \beta_{11} - 146150 \beta_{10} - 18006 \beta_{9} + 12946 \beta_{8} - 638503 \beta_{7} + \cdots - 23066 ) / 21 \) 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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
271.1
−0.125028 + 0.216554i
−0.852454 + 1.47649i
0.770375 − 1.33433i
1.52125 − 2.63488i
0.791791 − 1.37142i
−1.10593 + 1.91553i
−0.125028 − 0.216554i
−0.852454 − 1.47649i
0.770375 + 1.33433i
1.52125 + 2.63488i
0.791791 + 1.37142i
−1.10593 − 1.91553i
−0.707107 + 1.22474i 0 −1.00000 − 1.73205i −6.42174 − 3.70759i 0 −3.10640 + 6.27298i 2.82843 0 9.08171 − 5.24333i
271.2 −0.707107 + 1.22474i 0 −1.00000 − 1.73205i −1.39053 − 0.802823i 0 0.744740 − 6.96027i 2.82843 0 1.96651 − 1.13536i
271.3 −0.707107 + 1.22474i 0 −1.00000 − 1.73205i 8.43359 + 4.86914i 0 4.48298 + 5.37614i 2.82843 0 −11.9269 + 6.88600i
271.4 0.707107 − 1.22474i 0 −1.00000 − 1.73205i −4.89915 − 2.82852i 0 −5.20889 − 4.67627i −2.82843 0 −6.92844 + 4.00014i
271.5 0.707107 − 1.22474i 0 −1.00000 − 1.73205i 0.200623 + 0.115830i 0 6.91963 + 1.05770i −2.82843 0 0.283724 − 0.163808i
271.6 0.707107 − 1.22474i 0 −1.00000 − 1.73205i 1.07720 + 0.621924i 0 −3.83206 + 5.85793i −2.82843 0 1.52340 − 0.879534i
325.1 −0.707107 − 1.22474i 0 −1.00000 + 1.73205i −6.42174 + 3.70759i 0 −3.10640 − 6.27298i 2.82843 0 9.08171 + 5.24333i
325.2 −0.707107 − 1.22474i 0 −1.00000 + 1.73205i −1.39053 + 0.802823i 0 0.744740 + 6.96027i 2.82843 0 1.96651 + 1.13536i
325.3 −0.707107 − 1.22474i 0 −1.00000 + 1.73205i 8.43359 − 4.86914i 0 4.48298 − 5.37614i 2.82843 0 −11.9269 − 6.88600i
325.4 0.707107 + 1.22474i 0 −1.00000 + 1.73205i −4.89915 + 2.82852i 0 −5.20889 + 4.67627i −2.82843 0 −6.92844 − 4.00014i
325.5 0.707107 + 1.22474i 0 −1.00000 + 1.73205i 0.200623 − 0.115830i 0 6.91963 − 1.05770i −2.82843 0 0.283724 + 0.163808i
325.6 0.707107 + 1.22474i 0 −1.00000 + 1.73205i 1.07720 − 0.621924i 0 −3.83206 − 5.85793i −2.82843 0 1.52340 + 0.879534i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 271.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.d odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.3.n.c ✓ 12
3.b odd 2 1 378.3.n.e yes 12
7.d odd 6 1 inner 378.3.n.c ✓ 12
21.g even 6 1 378.3.n.e yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.3.n.c ✓ 12 1.a even 1 1 trivial
378.3.n.c ✓ 12 7.d odd 6 1 inner
378.3.n.e yes 12 3.b odd 2 1
378.3.n.e yes 12 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} + 6 T_{5}^{11} - 75 T_{5}^{10} - 522 T_{5}^{9} + 5967 T_{5}^{8} + 62802 T_{5}^{7} + \cdots + 35721 \) acting on \(S_{3}^{\mathrm{new}}(378, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} + 2 T^{2} + 4)^{3} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} + 6 T^{11} + \cdots + 35721 \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 13841287201 \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 1275989841 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 424711586601 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 22772566187721 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 32\!\cdots\!41 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 9694249646481 \) Copy content Toggle raw display
$29$ \( (T^{6} + 54 T^{5} + \cdots + 233834769)^{2} \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 34\!\cdots\!29 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 24\!\cdots\!29 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 98\!\cdots\!69 \) Copy content Toggle raw display
$43$ \( (T^{6} + 78 T^{5} + \cdots - 40196423)^{2} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 96\!\cdots\!29 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 38640931089489 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 62\!\cdots\!61 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 11\!\cdots\!09 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 10\!\cdots\!21 \) Copy content Toggle raw display
$71$ \( (T^{6} + 186 T^{5} + \cdots + 4059941697)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 33\!\cdots\!89 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 11\!\cdots\!29 \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 27\!\cdots\!89 \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 10\!\cdots\!21 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 85\!\cdots\!04 \) Copy content Toggle raw display
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