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Properties

Label 378.4.k.d
Level $378$
Weight $4$
Character orbit 378.k
Analytic conductor $22.303$
Analytic rank $0$
Dimension $20$
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(215,378)]
 
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([3, 5]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("378.215");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 378.k (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: \(22.3027219822\)
Analytic rank: \(0\)
Dimension: \(20\)
Relative dimension: \(10\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - 2 x^{19} + 2 x^{18} - 32 x^{17} - 669 x^{16} + 1752 x^{15} - 1654 x^{14} + 13878 x^{13} + \cdots + 2458624 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{18}\cdot 7^{2} \)
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 \(1,\beta_1,\ldots,\beta_{19}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_1 q^{2} - 4 \beta_{5} q^{4} + \beta_{6} q^{5} + (\beta_{12} + \beta_{5} + 1) q^{7} + (4 \beta_{8} - 4 \beta_1) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_1 q^{2} - 4 \beta_{5} q^{4} + \beta_{6} q^{5} + (\beta_{12} + \beta_{5} + 1) q^{7} + (4 \beta_{8} - 4 \beta_1) q^{8} + ( - \beta_{5} - \beta_{3} - 2) q^{10} + (\beta_{15} - \beta_{14} + \cdots + \beta_{2}) q^{11}+ \cdots + ( - 4 \beta_{19} - 13 \beta_{15} + \cdots - 136 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 40 q^{4} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 40 q^{4} + 4 q^{7} - 24 q^{10} - 160 q^{16} + 408 q^{19} + 240 q^{22} - 646 q^{25} + 56 q^{28} - 102 q^{31} + 194 q^{37} - 96 q^{40} - 2332 q^{43} - 624 q^{46} + 2840 q^{49} - 648 q^{52} + 96 q^{58} + 1878 q^{61} - 1280 q^{64} - 386 q^{67} + 3672 q^{70} + 1788 q^{73} + 814 q^{79} - 672 q^{82} - 4560 q^{85} + 480 q^{88} + 2724 q^{91} - 1536 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 2 x^{19} + 2 x^{18} - 32 x^{17} - 669 x^{16} + 1752 x^{15} - 1654 x^{14} + 13878 x^{13} + \cdots + 2458624 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 32\!\cdots\!04 \nu^{19} + \cdots + 14\!\cdots\!52 ) / 64\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 12\!\cdots\!47 \nu^{19} + \cdots - 53\!\cdots\!32 ) / 12\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 69\!\cdots\!65 \nu^{19} + \cdots + 45\!\cdots\!76 ) / 49\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 83\!\cdots\!91 \nu^{19} + \cdots - 80\!\cdots\!56 ) / 49\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 30\!\cdots\!55 \nu^{19} + \cdots + 93\!\cdots\!68 ) / 11\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 42\!\cdots\!53 \nu^{19} + \cdots + 11\!\cdots\!56 ) / 12\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 87\!\cdots\!49 \nu^{19} + \cdots + 37\!\cdots\!68 ) / 24\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 67\!\cdots\!51 \nu^{19} + \cdots + 20\!\cdots\!56 ) / 12\!\cdots\!96 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 13\!\cdots\!55 \nu^{19} + \cdots - 32\!\cdots\!88 ) / 24\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 23\!\cdots\!89 \nu^{19} + \cdots - 59\!\cdots\!32 ) / 12\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 21\!\cdots\!13 \nu^{19} + \cdots + 10\!\cdots\!68 ) / 84\!\cdots\!56 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 22\!\cdots\!69 \nu^{19} + \cdots + 15\!\cdots\!20 ) / 49\!\cdots\!48 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 36\!\cdots\!93 \nu^{19} + \cdots - 46\!\cdots\!16 ) / 70\!\cdots\!64 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 78\!\cdots\!98 \nu^{19} + \cdots - 31\!\cdots\!40 ) / 12\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 10\!\cdots\!63 \nu^{19} + \cdots + 35\!\cdots\!48 ) / 12\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 29\!\cdots\!05 \nu^{19} + \cdots - 24\!\cdots\!56 ) / 35\!\cdots\!32 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( 27\!\cdots\!88 \nu^{19} + \cdots + 25\!\cdots\!64 ) / 30\!\cdots\!28 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 10\!\cdots\!05 \nu^{19} + \cdots - 11\!\cdots\!16 ) / 12\!\cdots\!12 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( 17\!\cdots\!91 \nu^{19} + \cdots - 52\!\cdots\!76 ) / 12\!\cdots\!12 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 2 \beta_{16} + 10 \beta_{15} + 10 \beta_{14} + 8 \beta_{13} + 4 \beta_{12} + 2 \beta_{10} + \cdots + 2 ) / 252 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2\beta_{19} + 2\beta_{15} + 2\beta_{11} + \beta_{9} + \beta_{8} + \beta_{6} + 2\beta_{2} + 108\beta_1 ) / 18 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 14 \beta_{19} + 14 \beta_{18} - 14 \beta_{17} + 190 \beta_{16} - 30 \beta_{15} - 58 \beta_{14} + \cdots + 1329 ) / 252 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 29 \beta_{18} - 58 \beta_{17} + 5 \beta_{16} + 4 \beta_{13} - 10 \beta_{12} - 21 \beta_{7} + \cdots + 24 ) / 9 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 602 \beta_{19} - 1204 \beta_{18} - 602 \beta_{17} - 6464 \beta_{16} + 5664 \beta_{15} + 5062 \beta_{14} + \cdots - 47163 ) / 252 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 528 \beta_{19} + 111 \beta_{15} + 88 \beta_{14} + 1056 \beta_{11} + 130 \beta_{10} + \cdots + 22599 \beta_1 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 43148 \beta_{19} - 21574 \beta_{18} - 43148 \beta_{17} - 35982 \beta_{16} - 179546 \beta_{15} + \cdots + 57556 ) / 252 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 43122 \beta_{18} - 21561 \beta_{17} - 4555 \beta_{16} + 2995 \beta_{13} - 25645 \beta_{12} + \cdots - 1902020 ) / 9 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 722078 \beta_{19} - 722078 \beta_{18} + 722078 \beta_{17} - 3698402 \beta_{16} - 549292 \beta_{15} + \cdots - 60042905 ) / 252 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 2354832 \beta_{19} - 1512607 \beta_{15} - 401626 \beta_{14} + 1177416 \beta_{11} + \cdots - 970809 \beta_1 ) / 18 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 7751086 \beta_{19} - 15502172 \beta_{18} - 7751086 \beta_{17} - 41757948 \beta_{16} + \cdots - 635917939 ) / 84 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 16132663 \beta_{18} + 16132663 \beta_{17} - 8808194 \beta_{16} - 7510669 \beta_{13} + \cdots - 1432484346 ) / 9 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 1459546564 \beta_{19} + 729773282 \beta_{18} + 1459546564 \beta_{17} + 650046826 \beta_{16} + \cdots - 1379820108 ) / 252 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 887335304 \beta_{19} - 1452278678 \beta_{15} - 564943374 \beta_{14} - 887335304 \beta_{11} + \cdots - 38202741684 \beta_1 ) / 18 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 22482218038 \beta_{19} - 22482218038 \beta_{18} + 22482218038 \beta_{17} - 75352700158 \beta_{16} + \cdots - 1917929890939 ) / 252 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( 12242668145 \beta_{18} + 24485336290 \beta_{17} - 334765091 \beta_{16} - 8839689094 \beta_{13} + \cdots - 11907903054 ) / 9 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 683043635582 \beta_{19} + 1366087271164 \beta_{18} + 683043635582 \beta_{17} + \cdots + 57396999531141 ) / 252 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( ( 677799636936 \beta_{19} - 64931379499 \beta_{15} - 2215604392 \beta_{14} - 1355599273872 \beta_{11} + \cdots - 28771586877555 \beta_1 ) / 18 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 41060113896340 \beta_{19} + 20530056948170 \beta_{18} + 41060113896340 \beta_{17} + \cdots - 32759579342940 ) / 252 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(29\) \(325\)
\(\chi(n)\) \(-1\) \(1 + \beta_{5}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
215.1
1.94968 + 0.522414i
−4.83258 − 1.29489i
0.489109 + 0.131056i
5.16003 + 1.38263i
−1.40021 − 0.375184i
0.375184 − 1.40021i
−1.38263 + 5.16003i
−0.131056 + 0.489109i
1.29489 − 4.83258i
−0.522414 + 1.94968i
1.94968 − 0.522414i
−4.83258 + 1.29489i
0.489109 − 0.131056i
5.16003 − 1.38263i
−1.40021 + 0.375184i
0.375184 + 1.40021i
−1.38263 − 5.16003i
−0.131056 − 0.489109i
1.29489 + 4.83258i
−0.522414 − 1.94968i
−1.73205 − 1.00000i 0 2.00000 + 3.46410i −9.83277 + 17.0309i 0 9.86964 + 15.6713i − 8.00000i 0 34.0617 − 19.6655i
215.2 −1.73205 − 1.00000i 0 2.00000 + 3.46410i −3.43124 + 5.94308i 0 −18.5152 + 0.434813i − 8.00000i 0 11.8862 − 6.86248i
215.3 −1.73205 − 1.00000i 0 2.00000 + 3.46410i 0.523601 − 0.906903i 0 9.66492 − 15.7984i − 8.00000i 0 −1.81381 + 1.04720i
215.4 −1.73205 − 1.00000i 0 2.00000 + 3.46410i 3.80525 − 6.59089i 0 18.4140 − 1.98070i − 8.00000i 0 −13.1818 + 7.61050i
215.5 −1.73205 − 1.00000i 0 2.00000 + 3.46410i 10.6672 − 18.4761i 0 −18.4334 − 1.79115i − 8.00000i 0 −36.9523 + 21.3344i
215.6 1.73205 + 1.00000i 0 2.00000 + 3.46410i −10.6672 + 18.4761i 0 −18.4334 − 1.79115i 8.00000i 0 −36.9523 + 21.3344i
215.7 1.73205 + 1.00000i 0 2.00000 + 3.46410i −3.80525 + 6.59089i 0 18.4140 − 1.98070i 8.00000i 0 −13.1818 + 7.61050i
215.8 1.73205 + 1.00000i 0 2.00000 + 3.46410i −0.523601 + 0.906903i 0 9.66492 − 15.7984i 8.00000i 0 −1.81381 + 1.04720i
215.9 1.73205 + 1.00000i 0 2.00000 + 3.46410i 3.43124 − 5.94308i 0 −18.5152 + 0.434813i 8.00000i 0 11.8862 − 6.86248i
215.10 1.73205 + 1.00000i 0 2.00000 + 3.46410i 9.83277 − 17.0309i 0 9.86964 + 15.6713i 8.00000i 0 34.0617 − 19.6655i
269.1 −1.73205 + 1.00000i 0 2.00000 − 3.46410i −9.83277 − 17.0309i 0 9.86964 − 15.6713i 8.00000i 0 34.0617 + 19.6655i
269.2 −1.73205 + 1.00000i 0 2.00000 − 3.46410i −3.43124 − 5.94308i 0 −18.5152 − 0.434813i 8.00000i 0 11.8862 + 6.86248i
269.3 −1.73205 + 1.00000i 0 2.00000 − 3.46410i 0.523601 + 0.906903i 0 9.66492 + 15.7984i 8.00000i 0 −1.81381 − 1.04720i
269.4 −1.73205 + 1.00000i 0 2.00000 − 3.46410i 3.80525 + 6.59089i 0 18.4140 + 1.98070i 8.00000i 0 −13.1818 − 7.61050i
269.5 −1.73205 + 1.00000i 0 2.00000 − 3.46410i 10.6672 + 18.4761i 0 −18.4334 + 1.79115i 8.00000i 0 −36.9523 − 21.3344i
269.6 1.73205 − 1.00000i 0 2.00000 − 3.46410i −10.6672 − 18.4761i 0 −18.4334 + 1.79115i − 8.00000i 0 −36.9523 − 21.3344i
269.7 1.73205 − 1.00000i 0 2.00000 − 3.46410i −3.80525 − 6.59089i 0 18.4140 + 1.98070i − 8.00000i 0 −13.1818 − 7.61050i
269.8 1.73205 − 1.00000i 0 2.00000 − 3.46410i −0.523601 − 0.906903i 0 9.66492 + 15.7984i − 8.00000i 0 −1.81381 − 1.04720i
269.9 1.73205 − 1.00000i 0 2.00000 − 3.46410i 3.43124 + 5.94308i 0 −18.5152 − 0.434813i − 8.00000i 0 11.8862 + 6.86248i
269.10 1.73205 − 1.00000i 0 2.00000 − 3.46410i 9.83277 + 17.0309i 0 9.86964 − 15.6713i − 8.00000i 0 34.0617 + 19.6655i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 215.10
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 378.4.k.d ✓ 20
3.b odd 2 1 inner 378.4.k.d ✓ 20
7.d odd 6 1 inner 378.4.k.d ✓ 20
21.g even 6 1 inner 378.4.k.d ✓ 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
378.4.k.d ✓ 20 1.a even 1 1 trivial
378.4.k.d ✓ 20 3.b odd 2 1 inner
378.4.k.d ✓ 20 7.d odd 6 1 inner
378.4.k.d ✓ 20 21.g even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{20} + 948 T_{5}^{18} + 630504 T_{5}^{16} + 212105088 T_{5}^{14} + 51449923440 T_{5}^{12} + \cdots + 27\!\cdots\!44 \) acting on \(S_{4}^{\mathrm{new}}(378, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 4 T^{2} + 16)^{5} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} + \cdots + 27\!\cdots\!44 \) Copy content Toggle raw display
$7$ \( (T^{10} + \cdots + 4747561509943)^{2} \) Copy content Toggle raw display
$11$ \( T^{20} + \cdots + 34\!\cdots\!00 \) Copy content Toggle raw display
$13$ \( (T^{10} + \cdots + 63\!\cdots\!72)^{2} \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 60\!\cdots\!04 \) Copy content Toggle raw display
$19$ \( (T^{10} + \cdots + 31277618111232)^{2} \) Copy content Toggle raw display
$23$ \( T^{20} + \cdots + 11\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{10} + \cdots + 49\!\cdots\!00)^{2} \) Copy content Toggle raw display
$31$ \( (T^{10} + \cdots + 98\!\cdots\!75)^{2} \) Copy content Toggle raw display
$37$ \( (T^{10} + \cdots + 13\!\cdots\!00)^{2} \) Copy content Toggle raw display
$41$ \( (T^{10} + \cdots - 43\!\cdots\!72)^{2} \) Copy content Toggle raw display
$43$ \( (T^{5} + 583 T^{4} + \cdots + 642834729047)^{4} \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 15\!\cdots\!44 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 95\!\cdots\!76 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 64\!\cdots\!24 \) Copy content Toggle raw display
$61$ \( (T^{10} + \cdots + 13\!\cdots\!63)^{2} \) Copy content Toggle raw display
$67$ \( (T^{10} + \cdots + 16\!\cdots\!36)^{2} \) Copy content Toggle raw display
$71$ \( (T^{10} + \cdots + 76\!\cdots\!36)^{2} \) Copy content Toggle raw display
$73$ \( (T^{10} + \cdots + 67\!\cdots\!28)^{2} \) Copy content Toggle raw display
$79$ \( (T^{10} + \cdots + 52\!\cdots\!96)^{2} \) Copy content Toggle raw display
$83$ \( (T^{10} + \cdots - 24\!\cdots\!48)^{2} \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 30\!\cdots\!64 \) Copy content Toggle raw display
$97$ \( (T^{10} + \cdots + 81\!\cdots\!03)^{2} \) Copy content Toggle raw display
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