↑

Properties

Label 378.3.n.d
Level $378$
Weight $3$
Character orbit 378.n
Analytic conductor $10.300$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

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: \( x^{12} - 2 x^{11} + 19 x^{10} - 22 x^{9} + 252 x^{8} - 310 x^{7} + 983 x^{6} + 78 x^{5} + 935 x^{4} + \cdots + 196 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{6} \)
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.

\(f(q)\) \(=\) \( q + ( - \beta_{4} + \beta_{3}) q^{2} + 2 \beta_1 q^{4} + \beta_{2} q^{5} + ( - \beta_{10} + \beta_1) q^{7} - 2 \beta_{3} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - \beta_{4} + \beta_{3}) q^{2} + 2 \beta_1 q^{4} + \beta_{2} q^{5} + ( - \beta_{10} + \beta_1) q^{7} - 2 \beta_{3} q^{8} + ( - \beta_{11} - \beta_{7} + \beta_{6} + \cdots + 1) q^{10}+ \cdots + (3 \beta_{9} + \beta_{8} + \cdots + 12 \beta_{2}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{4} - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{4} - 12 q^{7} + 12 q^{10} - 24 q^{16} + 120 q^{19} + 24 q^{22} + 30 q^{25} - 12 q^{28} - 54 q^{31} - 90 q^{37} - 24 q^{40} + 84 q^{43} - 72 q^{46} - 96 q^{49} - 84 q^{52} - 24 q^{58} + 66 q^{61} + 96 q^{64} + 342 q^{67} - 372 q^{70} - 12 q^{73} - 378 q^{79} + 120 q^{82} + 792 q^{85} - 24 q^{88} - 708 q^{91} - 672 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 19 x^{10} - 22 x^{9} + 252 x^{8} - 310 x^{7} + 983 x^{6} + 78 x^{5} + 935 x^{4} + \cdots + 196 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 153570350 \nu^{11} - 334714188 \nu^{10} + 2916338636 \nu^{9} - 3822714777 \nu^{8} + \cdots - 28640130834 ) / 58080930754 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 6689343307 \nu^{11} + 86680686235 \nu^{10} - 133526450686 \nu^{9} + 1869411253248 \nu^{8} + \cdots + 46661947109540 ) / 2497480022422 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 423076 \nu^{11} - 628257 \nu^{10} + 7184563 \nu^{9} - 4150840 \nu^{8} + 93668678 \nu^{7} + \cdots + 236511156 ) / 34043702 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 15719468020 \nu^{11} + 51926290541 \nu^{10} - 346624990200 \nu^{9} + 746461140650 \nu^{8} + \cdots + 4928105132468 ) / 1248740011211 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 39772771025 \nu^{11} - 205421012701 \nu^{10} + 1075765669368 \nu^{9} - 3355960706176 \nu^{8} + \cdots - 20716976230132 ) / 2497480022422 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 41739393418 \nu^{11} - 218391365092 \nu^{10} + 1170548048848 \nu^{9} - 3692037476673 \nu^{8} + \cdots - 46478989711658 ) / 2497480022422 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 59858347418 \nu^{11} - 6593603131 \nu^{10} - 785808882764 \nu^{9} - 1306926153531 \nu^{8} + \cdots - 72446323652318 ) / 2497480022422 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 82714356426 \nu^{11} - 230944972903 \nu^{10} + 1724847163959 \nu^{9} - 3092019708828 \nu^{8} + \cdots - 5275847353140 ) / 2497480022422 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 106006420157 \nu^{11} + 145895456574 \nu^{10} - 1795932734447 \nu^{9} + \cdots - 39101828892672 ) / 2497480022422 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 167594281762 \nu^{11} + 516301536581 \nu^{10} - 3557946979024 \nu^{9} + \cdots + 26315894979164 ) / 2497480022422 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 186941995090 \nu^{11} + 354952388017 \nu^{10} - 3398553032272 \nu^{9} + \cdots - 60985148629524 ) / 2497480022422 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{11} - \beta_{7} - \beta_{6} - 6\beta_{4} - 3\beta _1 - 1 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - \beta_{11} - \beta_{10} + 4 \beta_{9} - 2 \beta_{7} + 2 \beta_{6} - 4 \beta_{5} - 4 \beta_{4} + \cdots - 34 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - \beta_{11} + 6 \beta_{10} - 3 \beta_{8} + 5 \beta_{7} + 6 \beta_{6} + 3 \beta_{4} + 39 \beta_{3} + \cdots - 5 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 19 \beta_{11} + 36 \beta_{10} + 8 \beta_{8} + 19 \beta_{7} - 17 \beta_{6} + 56 \beta_{5} + 48 \beta_{4} + \cdots - 17 ) / 6 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 119 \beta_{11} - 119 \beta_{10} - 10 \beta_{9} + 200 \beta_{8} + 54 \beta_{7} - 54 \beta_{6} + \cdots + 262 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 273 \beta_{11} - 157 \beta_{10} - 380 \beta_{9} + 82 \beta_{8} + 116 \beta_{7} - 157 \beta_{6} + \cdots + 2740 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 2463 \beta_{11} - 956 \beta_{10} - 1470 \beta_{8} - 2463 \beta_{7} - 1507 \beta_{6} - 294 \beta_{5} + \cdots - 1507 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 2949 \beta_{11} - 2949 \beta_{10} + 10384 \beta_{9} - 5440 \beta_{8} - 7866 \beta_{7} + 7866 \beta_{6} + \cdots - 71406 ) / 6 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 7637 \beta_{11} + 17419 \beta_{10} + 3144 \beta_{9} - 10485 \beta_{8} + 9782 \beta_{7} + 17419 \beta_{6} + \cdots - 26390 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 75269 \beta_{11} + 111616 \beta_{10} + 42532 \beta_{8} + 75269 \beta_{7} - 36347 \beta_{6} + \cdots - 36347 ) / 6 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 256791 \beta_{11} - 256791 \beta_{10} - 117678 \beta_{9} + 593648 \beta_{8} + 235602 \beta_{7} + \cdots + 1197990 ) / 6 \) Copy content Toggle raw display

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

Character values

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

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

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
271.1
1.09354 + 1.89407i
−0.375017 − 0.649549i
1.90280 + 3.29574i
0.488584 + 0.846253i
−1.78923 − 3.09904i
−0.320674 − 0.555424i
1.09354 − 1.89407i
−0.375017 + 0.649549i
1.90280 − 3.29574i
0.488584 − 0.846253i
−1.78923 + 3.09904i
−0.320674 + 0.555424i
−0.707107 + 1.22474i 0 −1.00000 − 1.73205i −7.21817 − 4.16741i 0 −4.94472 − 4.95477i 2.82843 0 10.2080 − 5.89361i
271.2 −0.707107 + 1.22474i 0 −1.00000 − 1.73205i 1.45092 + 0.837688i 0 −3.22042 + 6.21522i 2.82843 0 −2.05191 + 1.18467i
271.3 −0.707107 + 1.22474i 0 −1.00000 − 1.73205i 3.64593 + 2.10498i 0 5.16514 − 4.72455i 2.82843 0 −5.15613 + 2.97689i
271.4 0.707107 − 1.22474i 0 −1.00000 − 1.73205i −3.64593 − 2.10498i 0 5.16514 − 4.72455i −2.82843 0 −5.15613 + 2.97689i
271.5 0.707107 − 1.22474i 0 −1.00000 − 1.73205i −1.45092 − 0.837688i 0 −3.22042 + 6.21522i −2.82843 0 −2.05191 + 1.18467i
271.6 0.707107 − 1.22474i 0 −1.00000 − 1.73205i 7.21817 + 4.16741i 0 −4.94472 − 4.95477i −2.82843 0 10.2080 − 5.89361i
325.1 −0.707107 − 1.22474i 0 −1.00000 + 1.73205i −7.21817 + 4.16741i 0 −4.94472 + 4.95477i 2.82843 0 10.2080 + 5.89361i
325.2 −0.707107 − 1.22474i 0 −1.00000 + 1.73205i 1.45092 − 0.837688i 0 −3.22042 − 6.21522i 2.82843 0 −2.05191 − 1.18467i
325.3 −0.707107 − 1.22474i 0 −1.00000 + 1.73205i 3.64593 − 2.10498i 0 5.16514 + 4.72455i 2.82843 0 −5.15613 − 2.97689i
325.4 0.707107 + 1.22474i 0 −1.00000 + 1.73205i −3.64593 + 2.10498i 0 5.16514 + 4.72455i −2.82843 0 −5.15613 − 2.97689i
325.5 0.707107 + 1.22474i 0 −1.00000 + 1.73205i −1.45092 + 0.837688i 0 −3.22042 − 6.21522i −2.82843 0 −2.05191 − 1.18467i
325.6 0.707107 + 1.22474i 0 −1.00000 + 1.73205i 7.21817 − 4.16741i 0 −4.94472 + 4.95477i −2.82843 0 10.2080 + 5.89361i
\(n\): 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
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 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.d ✓ 12
3.b odd 2 1 inner 378.3.n.d ✓ 12
7.d odd 6 1 inner 378.3.n.d ✓ 12
21.g even 6 1 inner 378.3.n.d ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.3.n.d ✓ 12 1.a even 1 1 trivial
378.3.n.d ✓ 12 3.b odd 2 1 inner
378.3.n.d ✓ 12 7.d odd 6 1 inner
378.3.n.d ✓ 12 21.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{12} - 90T_{5}^{10} + 6624T_{5}^{8} - 125928T_{5}^{6} + 1867536T_{5}^{4} - 5101056T_{5}^{2} + 11943936 \) 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
$5$ \( T^{12} - 90 T^{10} + \cdots + 11943936 \) Copy content Toggle raw display
$7$ \( (T^{6} + 6 T^{5} + \cdots + 117649)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 8707129344 \) Copy content Toggle raw display
$13$ \( (T^{6} + 375 T^{4} + \cdots + 1939248)^{2} \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 21\!\cdots\!16 \) Copy content Toggle raw display
$19$ \( (T^{6} - 60 T^{5} + \cdots + 90816012)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + 1440 T^{10} + \cdots + 34012224 \) Copy content Toggle raw display
$29$ \( (T^{6} - 4824 T^{4} + \cdots - 802722312)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + 27 T^{5} + \cdots + 44861067)^{2} \) Copy content Toggle raw display
$37$ \( (T^{6} + 45 T^{5} + \cdots + 38416)^{2} \) Copy content Toggle raw display
$41$ \( (T^{6} + 2232 T^{4} + \cdots + 245913624)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} - 21 T^{2} + \cdots + 72161)^{4} \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 44\!\cdots\!16 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 14\!\cdots\!24 \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 339186773016576 \) Copy content Toggle raw display
$61$ \( (T^{6} - 33 T^{5} + \cdots + 50835029787)^{2} \) Copy content Toggle raw display
$67$ \( (T^{6} - 171 T^{5} + \cdots + 11847016336)^{2} \) Copy content Toggle raw display
$71$ \( (T^{6} - 2736 T^{4} + \cdots - 1492992)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + 6 T^{5} + \cdots + 36243262188)^{2} \) Copy content Toggle raw display
$79$ \( (T^{6} + 189 T^{5} + \cdots + 46478185744)^{2} \) Copy content Toggle raw display
$83$ \( (T^{6} + 7614 T^{4} + \cdots + 9835668864)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 48\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( (T^{6} + 18993 T^{4} + \cdots + 8621131347)^{2} \) Copy content Toggle raw display
show more
show less