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Properties

Label 378.4.d.d
Level $378$
Weight $4$
Character orbit 378.d
Analytic conductor $22.303$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [378,4,Mod(377,378)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(378, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("378.377");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 378.d (of order \(2\), degree \(1\), 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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 2 x^{11} + 2 x^{10} - 8 x^{9} - 28 x^{8} + 88 x^{7} - 88 x^{6} + 212 x^{5} + 1316 x^{4} + \cdots + 2500 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{12} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{4} q^{2} - 4 q^{4} - \beta_{6} q^{5} + ( - \beta_{2} + \beta_1 - 2) q^{7} + 4 \beta_{4} q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{4} q^{2} - 4 q^{4} - \beta_{6} q^{5} + ( - \beta_{2} + \beta_1 - 2) q^{7} + 4 \beta_{4} q^{8} + ( - \beta_{8} + \beta_{2}) q^{10} + (\beta_{9} + \beta_{7} - 3 \beta_{4}) q^{11} + (\beta_{11} + 4 \beta_{2}) q^{13} + ( - 2 \beta_{9} - \beta_{6} + \cdots + 2 \beta_{4}) q^{14}+ \cdots + ( - 4 \beta_{10} + \cdots - 121 \beta_{4}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 48 q^{4} - 24 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 48 q^{4} - 24 q^{7} + 192 q^{16} - 168 q^{22} - 696 q^{25} + 96 q^{28} + 228 q^{37} - 360 q^{43} - 264 q^{46} + 1356 q^{49} + 1776 q^{58} - 768 q^{64} - 4920 q^{67} + 840 q^{70} + 2040 q^{79} - 1392 q^{85} + 672 q^{88} + 5736 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 2 x^{11} + 2 x^{10} - 8 x^{9} - 28 x^{8} + 88 x^{7} - 88 x^{6} + 212 x^{5} + 1316 x^{4} + \cdots + 2500 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 79269 \nu^{11} - 124377 \nu^{10} + 121608 \nu^{9} + 2560740 \nu^{8} - 3856218 \nu^{7} + \cdots + 3295612852 ) / 170288734 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10046903 \nu^{11} - 11901791 \nu^{10} - 17266449 \nu^{9} - 107079119 \nu^{8} + \cdots - 8127001750 ) / 21286091750 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 85914 \nu^{11} - 39204 \nu^{10} - 885834 \nu^{9} + 2821833 \nu^{8} + 2758419 \nu^{7} + \cdots + 1229892006 ) / 85144367 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 162061 \nu^{11} - 60187 \nu^{10} - 16073 \nu^{9} - 1008093 \nu^{8} - 6298713 \nu^{7} + \cdots - 247562750 ) / 138221375 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 46532148 \nu^{11} + 56216121 \nu^{10} + 37898829 \nu^{9} + 281154159 \nu^{8} + \cdots + 73436577000 ) / 21286091750 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 807878 \nu^{11} + 1446581 \nu^{10} - 656631 \nu^{9} + 5166999 \nu^{8} + 23340034 \nu^{7} + \cdots + 1157034500 ) / 276442750 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 68331526 \nu^{11} - 22153342 \nu^{10} + 124007932 \nu^{9} - 528756888 \nu^{8} + \cdots - 137347454000 ) / 21286091750 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 601893 \nu^{11} + 750321 \nu^{10} - 1068381 \nu^{9} + 4199139 \nu^{8} + 20735424 \nu^{7} + \cdots + 1464891750 ) / 58000250 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 255020291 \nu^{11} - 94173347 \nu^{10} - 3495088 \nu^{9} - 1603625958 \nu^{8} + \cdots - 395060576500 ) / 21286091750 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 8405811 \nu^{11} + 17572497 \nu^{10} - 6308472 \nu^{9} + 53944638 \nu^{8} + \cdots + 12261939000 ) / 608174050 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 16366077 \nu^{11} + 20347569 \nu^{10} - 28524234 \nu^{9} + 125034246 \nu^{8} + \cdots + 39040572000 ) / 608174050 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -3\beta_{8} - 2\beta_{7} + 9\beta_{6} - 3\beta_{5} - 7\beta_{4} + \beta_{3} + 9\beta_{2} + 12 ) / 72 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -3\beta_{10} - 3\beta_{9} + 15\beta_{6} + 5\beta_{5} + 42\beta_{4} ) / 36 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -\beta_{9} + 2\beta_{7} + 6\beta_{4} + \beta_{3} - \beta _1 + 10 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 24\beta_{11} - 63\beta_{8} + \beta_{3} + 9\beta_{2} - 24\beta _1 + 504 ) / 36 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 27 \beta_{11} - 27 \beta_{10} - 27 \beta_{9} - 67 \beta_{8} + 38 \beta_{7} + 186 \beta_{6} - 52 \beta_{5} + \cdots - 264 ) / 36 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( -84\beta_{9} + 14\beta_{7} + 820\beta_{4} ) / 9 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 210 \beta_{11} + 210 \beta_{10} - 210 \beta_{9} - 509 \beta_{8} + 246 \beta_{7} - 1317 \beta_{6} + \cdots + 2316 ) / 36 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 579\beta_{11} - 1440\beta_{8} - 70\beta_{3} + 630\beta_{2} + 579\beta _1 - 10752 ) / 18 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( -789\beta_{9} + 808\beta_{7} + 5189\beta_{4} - 404\beta_{3} + 789\beta _1 - 9570 ) / 9 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 4002\beta_{10} - 4002\beta_{9} + 1226\beta_{7} - 17637\beta_{6} - 2201\beta_{5} + 36649\beta_{4} ) / 18 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 5841 \beta_{11} + 5841 \beta_{10} + 5841 \beta_{9} - 14191 \beta_{8} - 5374 \beta_{7} - 33378 \beta_{6} + \cdots - 75732 ) / 18 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(29\) \(325\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
377.1
2.57860 + 0.690933i
1.07862 + 0.289017i
0.613924 + 2.29120i
−2.29120 − 0.613924i
−0.289017 − 1.07862i
−0.690933 − 2.57860i
2.57860 − 0.690933i
1.07862 − 0.289017i
0.613924 − 2.29120i
−2.29120 + 0.613924i
−0.289017 + 1.07862i
−0.690933 + 2.57860i
− 2.00000i 0 −4.00000 −11.7352 0 −16.7593 − 7.88197i 8.00000i 0 23.4704i
377.2 − 2.00000i 0 −4.00000 −7.93144 0 18.5182 − 0.274414i 8.00000i 0 15.8629i
377.3 − 2.00000i 0 −4.00000 −0.614090 0 −7.75891 − 16.8166i 8.00000i 0 1.22818i
377.4 − 2.00000i 0 −4.00000 0.614090 0 −7.75891 + 16.8166i 8.00000i 0 − 1.22818i
377.5 − 2.00000i 0 −4.00000 7.93144 0 18.5182 + 0.274414i 8.00000i 0 − 15.8629i
377.6 − 2.00000i 0 −4.00000 11.7352 0 −16.7593 + 7.88197i 8.00000i 0 − 23.4704i
377.7 2.00000i 0 −4.00000 −11.7352 0 −16.7593 + 7.88197i − 8.00000i 0 − 23.4704i
377.8 2.00000i 0 −4.00000 −7.93144 0 18.5182 + 0.274414i − 8.00000i 0 − 15.8629i
377.9 2.00000i 0 −4.00000 −0.614090 0 −7.75891 + 16.8166i − 8.00000i 0 − 1.22818i
377.10 2.00000i 0 −4.00000 0.614090 0 −7.75891 − 16.8166i − 8.00000i 0 1.22818i
377.11 2.00000i 0 −4.00000 7.93144 0 18.5182 − 0.274414i − 8.00000i 0 15.8629i
377.12 2.00000i 0 −4.00000 11.7352 0 −16.7593 − 7.88197i − 8.00000i 0 23.4704i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 377.12
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 378.4.d.d ✓ 12
3.b odd 2 1 inner 378.4.d.d ✓ 12
7.b odd 2 1 inner 378.4.d.d ✓ 12
21.c even 2 1 inner 378.4.d.d ✓ 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.4.d.d ✓ 12 1.a even 1 1 trivial
378.4.d.d ✓ 12 3.b odd 2 1 inner
378.4.d.d ✓ 12 7.b odd 2 1 inner
378.4.d.d ✓ 12 21.c even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{6} - 201T_{5}^{4} + 8739T_{5}^{2} - 3267 \) acting on \(S_{4}^{\mathrm{new}}(378, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{6} \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( (T^{6} - 201 T^{4} + \cdots - 3267)^{2} \) Copy content Toggle raw display
$7$ \( (T^{6} + 12 T^{5} + \cdots + 40353607)^{2} \) Copy content Toggle raw display
$11$ \( (T^{6} + 2907 T^{4} + \cdots + 43758225)^{2} \) Copy content Toggle raw display
$13$ \( (T^{6} + 8004 T^{4} + \cdots + 15252781248)^{2} \) Copy content Toggle raw display
$17$ \( (T^{6} - 2640 T^{4} + \cdots - 31027968)^{2} \) Copy content Toggle raw display
$19$ \( (T^{6} + 33105 T^{4} + \cdots + 32978680227)^{2} \) Copy content Toggle raw display
$23$ \( (T^{6} + \cdots + 3014324464761)^{2} \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots + 69332001109056)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 5942113171875)^{2} \) Copy content Toggle raw display
$37$ \( (T^{3} - 57 T^{2} + \cdots - 2625211)^{4} \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots - 13\!\cdots\!27)^{2} \) Copy content Toggle raw display
$43$ \( (T^{3} + 90 T^{2} + \cdots - 1617784)^{4} \) Copy content Toggle raw display
$47$ \( (T^{6} + \cdots - 14\!\cdots\!00)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + \cdots + 194221015142400)^{2} \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots - 35\!\cdots\!48)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots + 18750480003072)^{2} \) Copy content Toggle raw display
$67$ \( (T^{3} + 1230 T^{2} + \cdots + 15933496)^{4} \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 55\!\cdots\!41)^{2} \) Copy content Toggle raw display
$73$ \( (T^{6} + \cdots + 26\!\cdots\!92)^{2} \) Copy content Toggle raw display
$79$ \( (T^{3} - 510 T^{2} + \cdots + 7801192)^{4} \) Copy content Toggle raw display
$83$ \( (T^{6} + \cdots - 39\!\cdots\!00)^{2} \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots - 20\!\cdots\!27)^{2} \) Copy content Toggle raw display
$97$ \( (T^{6} + \cdots + 34\!\cdots\!48)^{2} \) Copy content Toggle raw display
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