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Properties

Label 2394.2.cq.f
Level $2394$
Weight $2$
Character orbit 2394.cq
Analytic conductor $19.116$
Analytic rank $0$
Dimension $20$
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] = [2394,2,Mod(449,2394)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2394, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2394.449");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2394 = 2 \cdot 3^{2} \cdot 7 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2394.cq (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: \(19.1161862439\)
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: \( x^{20} - 4 x^{19} + 34 x^{18} - 88 x^{17} + 560 x^{16} - 1260 x^{15} + 5896 x^{14} - 9944 x^{13} + \cdots + 1444 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{5} \)
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.

\(f(q)\) \(=\) \( q + (\beta_{7} + 1) q^{2} + \beta_{7} q^{4} + \beta_{2} q^{5} - q^{7} - q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q + (\beta_{7} + 1) q^{2} + \beta_{7} q^{4} + \beta_{2} q^{5} - q^{7} - q^{8} + \beta_{3} q^{10} + (\beta_{15} + \beta_{14}) q^{11} + \beta_{16} q^{13} + ( - \beta_{7} - 1) q^{14} + ( - \beta_{7} - 1) q^{16} + (\beta_{19} + \beta_{13} + \beta_{2}) q^{17} + (\beta_{19} + \beta_{16} + \beta_{10} + \cdots + 1) q^{19}+ \cdots + (\beta_{7} + 1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q + 10 q^{2} - 10 q^{4} - 6 q^{5} - 20 q^{7} - 20 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 20 q + 10 q^{2} - 10 q^{4} - 6 q^{5} - 20 q^{7} - 20 q^{8} - 6 q^{10} - 10 q^{14} - 10 q^{16} + 6 q^{17} + 2 q^{19} + 6 q^{22} + 8 q^{25} + 10 q^{28} + 8 q^{29} + 10 q^{32} + 6 q^{34} + 6 q^{35} - 8 q^{38} + 6 q^{40} - 18 q^{41} - 8 q^{43} + 6 q^{44} + 12 q^{47} + 20 q^{49} + 16 q^{50} - 12 q^{53} + 18 q^{55} + 20 q^{56} + 16 q^{58} + 12 q^{59} - 4 q^{61} - 12 q^{62} + 20 q^{64} - 8 q^{65} - 24 q^{67} + 6 q^{70} - 4 q^{71} + 18 q^{74} - 10 q^{76} + 24 q^{79} + 6 q^{80} + 18 q^{82} + 30 q^{85} + 8 q^{86} - 2 q^{89} + 32 q^{95} + 24 q^{97} + 10 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{20} - 4 x^{19} + 34 x^{18} - 88 x^{17} + 560 x^{16} - 1260 x^{15} + 5896 x^{14} - 9944 x^{13} + \cdots + 1444 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 12\!\cdots\!09 \nu^{19} + \cdots - 64\!\cdots\!97 ) / 24\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 39\!\cdots\!00 \nu^{19} + \cdots + 13\!\cdots\!54 ) / 27\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 45\!\cdots\!17 \nu^{19} + \cdots + 66\!\cdots\!72 ) / 27\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 62\!\cdots\!50 \nu^{19} + \cdots + 63\!\cdots\!10 ) / 27\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 68\!\cdots\!27 \nu^{19} + \cdots - 14\!\cdots\!34 ) / 27\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 34\!\cdots\!07 \nu^{19} + \cdots + 47\!\cdots\!58 ) / 91\!\cdots\!22 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 20\!\cdots\!88 \nu^{19} + \cdots - 47\!\cdots\!76 ) / 46\!\cdots\!98 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 15\!\cdots\!95 \nu^{19} + \cdots - 78\!\cdots\!26 ) / 27\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 77\!\cdots\!83 \nu^{19} + \cdots + 18\!\cdots\!80 ) / 13\!\cdots\!33 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 27\!\cdots\!00 \nu^{19} + \cdots + 18\!\cdots\!44 ) / 45\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 34\!\cdots\!02 \nu^{19} + \cdots + 21\!\cdots\!90 ) / 45\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 42\!\cdots\!31 \nu^{19} + \cdots + 12\!\cdots\!20 ) / 45\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 87\!\cdots\!21 \nu^{19} + \cdots + 78\!\cdots\!06 ) / 91\!\cdots\!22 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 93\!\cdots\!46 \nu^{19} + \cdots - 20\!\cdots\!64 ) / 91\!\cdots\!22 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 98\!\cdots\!80 \nu^{19} + \cdots + 42\!\cdots\!38 ) / 91\!\cdots\!22 \) Copy content Toggle raw display
\(\beta_{16}\)\(=\) \( ( 50\!\cdots\!48 \nu^{19} + \cdots - 10\!\cdots\!84 ) / 45\!\cdots\!11 \) Copy content Toggle raw display
\(\beta_{17}\)\(=\) \( ( - 40\!\cdots\!88 \nu^{19} + \cdots - 60\!\cdots\!72 ) / 27\!\cdots\!66 \) Copy content Toggle raw display
\(\beta_{18}\)\(=\) \( ( 20\!\cdots\!95 \nu^{19} + \cdots + 36\!\cdots\!20 ) / 13\!\cdots\!33 \) Copy content Toggle raw display
\(\beta_{19}\)\(=\) \( ( - 18\!\cdots\!21 \nu^{19} + \cdots - 41\!\cdots\!10 ) / 91\!\cdots\!22 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2 \beta_{19} + 2 \beta_{18} - \beta_{17} - 3 \beta_{14} + \beta_{13} - \beta_{12} + \beta_{11} + \cdots + 3 ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{19} - 3 \beta_{17} + 3 \beta_{16} + 2 \beta_{15} + 4 \beta_{14} - 2 \beta_{13} + 3 \beta_{10} + \cdots + 3 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 16 \beta_{19} - 19 \beta_{18} - 19 \beta_{17} + 33 \beta_{16} - 4 \beta_{15} + 37 \beta_{14} - 28 \beta_{13} + \cdots - 21 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 17 \beta_{19} - 36 \beta_{18} - 6 \beta_{17} - 5 \beta_{15} - 7 \beta_{14} + 11 \beta_{13} + \cdots - 159 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 318 \beta_{19} + 29 \beta_{18} + 224 \beta_{17} - 369 \beta_{16} + 118 \beta_{15} - 52 \beta_{14} + \cdots - 108 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 55 \beta_{19} + 170 \beta_{18} + 170 \beta_{17} - 146 \beta_{16} - 22 \beta_{15} - 154 \beta_{14} + \cdots + 386 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 1640 \beta_{19} + 2576 \beta_{18} + 503 \beta_{17} + 162 \beta_{16} - 702 \beta_{15} - 4107 \beta_{14} + \cdots + 3879 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 4721 \beta_{19} - 1062 \beta_{18} - 4887 \beta_{17} + 5463 \beta_{16} + 137 \beta_{15} + 4036 \beta_{14} + \cdots + 4455 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 17746 \beta_{19} - 36355 \beta_{18} - 36355 \beta_{17} + 42087 \beta_{16} - 8434 \beta_{15} + \cdots - 29775 ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 34295 \beta_{19} - 56895 \beta_{18} - 11604 \beta_{17} - 10110 \beta_{16} + 8731 \beta_{15} + \cdots - 170280 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 391020 \beta_{19} + 72737 \beta_{18} + 352004 \beta_{17} - 500319 \beta_{16} + 199468 \beta_{15} + \cdots - 238254 ) / 6 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( - 78762 \beta_{19} + 262146 \beta_{18} + 262146 \beta_{17} - 196159 \beta_{16} - 10912 \beta_{15} + \cdots + 427175 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 2328020 \beta_{19} + 4147904 \beta_{18} + 799691 \beta_{17} + 659922 \beta_{16} - 1318248 \beta_{15} + \cdots + 7491051 ) / 6 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 7383683 \beta_{19} - 1297578 \beta_{18} - 7742124 \beta_{17} + 8379312 \beta_{16} - 1718641 \beta_{15} + \cdots + 6342870 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 21861520 \beta_{19} - 57638545 \beta_{18} - 57638545 \beta_{17} + 53994189 \beta_{16} - 11020306 \beta_{15} + \cdots - 55917429 ) / 6 \) Copy content Toggle raw display
\(\nu^{16}\)\(=\) \( ( - 54952244 \beta_{19} - 90530697 \beta_{18} - 13662546 \beta_{17} - 22583865 \beta_{16} + \cdots - 224220816 ) / 3 \) Copy content Toggle raw display
\(\nu^{17}\)\(=\) \( ( - 556467066 \beta_{19} + 92606975 \beta_{18} + 579755798 \beta_{17} - 714284499 \beta_{16} + \cdots - 418942218 ) / 6 \) Copy content Toggle raw display
\(\nu^{18}\)\(=\) \( - 112816817 \beta_{19} + 401612823 \beta_{18} + 401612823 \beta_{17} - 279824657 \beta_{16} + \cdots + 560009949 \) Copy content Toggle raw display
\(\nu^{19}\)\(=\) \( ( 3743835260 \beta_{19} + 6865199276 \beta_{18} + 991457759 \beta_{17} + 1544917308 \beta_{16} + \cdots + 13085601483 ) / 6 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2394\mathbb{Z}\right)^\times\).

\(n\) \(533\) \(1009\) \(1711\)
\(\chi(n)\) \(-1\) \(-\beta_{7}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
449.1
−1.54953 − 2.68386i
−0.466524 − 0.808043i
−0.596339 − 1.03289i
0.632902 + 1.09622i
1.58789 + 2.75031i
−0.0330345 − 0.0572175i
1.14441 + 1.98217i
1.72236 + 2.98321i
−1.38912 − 2.40603i
0.946994 + 1.64024i
−1.54953 + 2.68386i
−0.466524 + 0.808043i
−0.596339 + 1.03289i
0.632902 − 1.09622i
1.58789 − 2.75031i
−0.0330345 + 0.0572175i
1.14441 − 1.98217i
1.72236 − 2.98321i
−1.38912 + 2.40603i
0.946994 − 1.64024i
0.500000 + 0.866025i 0 −0.500000 + 0.866025i −3.64144 + 2.10239i 0 −1.00000 −1.00000 0 −3.64144 − 2.10239i
449.2 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −2.68523 + 1.55032i 0 −1.00000 −1.00000 0 −2.68523 − 1.55032i
449.3 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −2.24323 + 1.29513i 0 −1.00000 −1.00000 0 −2.24323 − 1.29513i
449.4 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −1.58290 + 0.913888i 0 −1.00000 −1.00000 0 −1.58290 − 0.913888i
449.5 0.500000 + 0.866025i 0 −0.500000 + 0.866025i −0.383660 + 0.221506i 0 −1.00000 −1.00000 0 −0.383660 − 0.221506i
449.6 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.337433 − 0.194817i 0 −1.00000 −1.00000 0 0.337433 + 0.194817i
449.7 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 0.943190 − 0.544551i 0 −1.00000 −1.00000 0 0.943190 + 0.544551i
449.8 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 1.17752 − 0.679840i 0 −1.00000 −1.00000 0 1.17752 + 0.679840i
449.9 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 2.36607 − 1.36605i 0 −1.00000 −1.00000 0 2.36607 + 1.36605i
449.10 0.500000 + 0.866025i 0 −0.500000 + 0.866025i 2.71225 − 1.56592i 0 −1.00000 −1.00000 0 2.71225 + 1.56592i
1205.1 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −3.64144 − 2.10239i 0 −1.00000 −1.00000 0 −3.64144 + 2.10239i
1205.2 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −2.68523 − 1.55032i 0 −1.00000 −1.00000 0 −2.68523 + 1.55032i
1205.3 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −2.24323 − 1.29513i 0 −1.00000 −1.00000 0 −2.24323 + 1.29513i
1205.4 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −1.58290 − 0.913888i 0 −1.00000 −1.00000 0 −1.58290 + 0.913888i
1205.5 0.500000 − 0.866025i 0 −0.500000 − 0.866025i −0.383660 − 0.221506i 0 −1.00000 −1.00000 0 −0.383660 + 0.221506i
1205.6 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 0.337433 + 0.194817i 0 −1.00000 −1.00000 0 0.337433 − 0.194817i
1205.7 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 0.943190 + 0.544551i 0 −1.00000 −1.00000 0 0.943190 − 0.544551i
1205.8 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 1.17752 + 0.679840i 0 −1.00000 −1.00000 0 1.17752 − 0.679840i
1205.9 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 2.36607 + 1.36605i 0 −1.00000 −1.00000 0 2.36607 − 1.36605i
1205.10 0.500000 − 0.866025i 0 −0.500000 − 0.866025i 2.71225 + 1.56592i 0 −1.00000 −1.00000 0 2.71225 − 1.56592i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 449.10
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
57.f even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2394.2.cq.f yes 20
3.b odd 2 1 2394.2.cq.e ✓ 20
19.d odd 6 1 2394.2.cq.e ✓ 20
57.f even 6 1 inner 2394.2.cq.f yes 20
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2394.2.cq.e ✓ 20 3.b odd 2 1
2394.2.cq.e ✓ 20 19.d odd 6 1
2394.2.cq.f yes 20 1.a even 1 1 trivial
2394.2.cq.f yes 20 57.f even 6 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{20} + 6 T_{5}^{19} - 11 T_{5}^{18} - 138 T_{5}^{17} + 107 T_{5}^{16} + 2028 T_{5}^{15} + \cdots + 18225 \) acting on \(S_{2}^{\mathrm{new}}(2394, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - T + 1)^{10} \) Copy content Toggle raw display
$3$ \( T^{20} \) Copy content Toggle raw display
$5$ \( T^{20} + 6 T^{19} + \cdots + 18225 \) Copy content Toggle raw display
$7$ \( (T + 1)^{20} \) Copy content Toggle raw display
$11$ \( T^{20} + 118 T^{18} + \cdots + 41990400 \) Copy content Toggle raw display
$13$ \( T^{20} + \cdots + 1520064144 \) Copy content Toggle raw display
$17$ \( T^{20} + \cdots + 10430945424 \) Copy content Toggle raw display
$19$ \( T^{20} + \cdots + 6131066257801 \) Copy content Toggle raw display
$23$ \( T^{20} - 80 T^{18} + \cdots + 1296 \) Copy content Toggle raw display
$29$ \( T^{20} + \cdots + 167961600 \) Copy content Toggle raw display
$31$ \( T^{20} + \cdots + 28095123456 \) Copy content Toggle raw display
$37$ \( T^{20} + \cdots + 13404082176 \) Copy content Toggle raw display
$41$ \( T^{20} + \cdots + 413744890118400 \) Copy content Toggle raw display
$43$ \( T^{20} + \cdots + 64745067930624 \) Copy content Toggle raw display
$47$ \( T^{20} + \cdots + 186064547904 \) Copy content Toggle raw display
$53$ \( T^{20} + \cdots + 1179510336 \) Copy content Toggle raw display
$59$ \( T^{20} + \cdots + 55943518611600 \) Copy content Toggle raw display
$61$ \( T^{20} + \cdots + 48579686464 \) Copy content Toggle raw display
$67$ \( T^{20} + \cdots + 196061604840000 \) Copy content Toggle raw display
$71$ \( T^{20} + \cdots + 67\!\cdots\!44 \) Copy content Toggle raw display
$73$ \( T^{20} + \cdots + 7368401670400 \) Copy content Toggle raw display
$79$ \( T^{20} + \cdots + 43\!\cdots\!24 \) Copy content Toggle raw display
$83$ \( T^{20} + \cdots + 1717525575936 \) Copy content Toggle raw display
$89$ \( T^{20} + \cdots + 103511479824 \) Copy content Toggle raw display
$97$ \( T^{20} + \cdots + 90\!\cdots\!00 \) Copy content Toggle raw display
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