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Properties

Label 2394.2.o.w
Level $2394$
Weight $2$
Character orbit 2394.o
Analytic conductor $19.116$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2394,2,Mod(505,2394)]
 
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([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2394.505");
 
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.o (of order \(3\), 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: \(10\)
Relative dimension: \(5\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 10x^{8} + 81x^{6} - 8x^{5} + 190x^{4} - 160x^{3} + 361x^{2} - 152x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{9}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{6} q^{2} + (\beta_{6} - 1) q^{4} + ( - \beta_{8} - \beta_{6}) q^{5} - q^{7} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{6} q^{2} + (\beta_{6} - 1) q^{4} + ( - \beta_{8} - \beta_{6}) q^{5} - q^{7} + q^{8} + (\beta_{8} + \beta_{6} - \beta_{3} - 1) q^{10} + ( - \beta_{2} + \beta_1) q^{11} + (\beta_{9} - \beta_{8} + \beta_{6} + \cdots - 1) q^{13}+ \cdots - \beta_{6} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 5 q^{2} - 5 q^{4} - 3 q^{5} - 10 q^{7} + 10 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 5 q^{2} - 5 q^{4} - 3 q^{5} - 10 q^{7} + 10 q^{8} - 3 q^{10} - 6 q^{11} - 6 q^{13} + 5 q^{14} - 5 q^{16} - 5 q^{17} - 5 q^{19} + 6 q^{20} + 3 q^{22} - 4 q^{23} - 12 q^{25} + 12 q^{26} + 5 q^{28} - 6 q^{29} - 5 q^{32} - 5 q^{34} + 3 q^{35} + 14 q^{37} + 4 q^{38} - 3 q^{40} - 11 q^{41} + 4 q^{43} + 3 q^{44} + 8 q^{46} + 2 q^{47} + 10 q^{49} + 24 q^{50} - 6 q^{52} - 2 q^{53} + 5 q^{55} - 10 q^{56} + 12 q^{58} - 14 q^{59} + 12 q^{61} + 10 q^{64} - 12 q^{65} - 14 q^{67} + 10 q^{68} + 3 q^{70} + 12 q^{71} - 2 q^{73} - 7 q^{74} + q^{76} + 6 q^{77} - 12 q^{79} - 3 q^{80} - 11 q^{82} + 8 q^{83} + 21 q^{85} + 4 q^{86} - 6 q^{88} - 3 q^{89} + 6 q^{91} - 4 q^{92} - 4 q^{94} + 52 q^{95} - 24 q^{97} - 5 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{10} + 10x^{8} + 81x^{6} - 8x^{5} + 190x^{4} - 160x^{3} + 361x^{2} - 152x + 64 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 46321 \nu^{9} - 4361222 \nu^{8} - 1836304 \nu^{7} - 44473234 \nu^{6} - 18363040 \nu^{5} + \cdots - 879284821 ) / 137765889 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 137083 \nu^{9} - 1164073 \nu^{8} - 490136 \nu^{7} - 14130854 \nu^{6} - 4901360 \nu^{5} + \cdots + 42225022 ) / 137765889 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 191347 \nu^{9} + 2044951 \nu^{8} + 861032 \nu^{7} + 21309377 \nu^{6} + 8610320 \nu^{5} + \cdots + 514329926 ) / 137765889 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 784225 \nu^{9} + 7046840 \nu^{8} - 11534582 \nu^{7} + 56452024 \nu^{6} - 69423857 \nu^{5} + \cdots + 47213056 ) / 551063556 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 409274 \nu^{9} + 2424799 \nu^{8} + 1020968 \nu^{7} + 23905649 \nu^{6} + 10209680 \nu^{5} + \cdots + 112332638 ) / 137765889 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 2392841 \nu^{9} + 228808 \nu^{8} - 24073114 \nu^{7} - 60928 \nu^{6} - 194809177 \nu^{5} + \cdots + 375288152 ) / 367375704 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 10266365 \nu^{9} - 9884480 \nu^{8} - 105673594 \nu^{7} - 88277380 \nu^{6} - 827126125 \nu^{5} + \cdots - 35595520 ) / 551063556 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 13024489 \nu^{9} - 1017700 \nu^{8} - 130943558 \nu^{7} - 294176 \nu^{6} - 1079825765 \nu^{5} + \cdots + 2035615768 ) / 551063556 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 55661575 \nu^{9} - 2587768 \nu^{8} - 537651470 \nu^{7} + 7984960 \nu^{6} - 4320309551 \nu^{5} + \cdots + 6943417000 ) / 1102127112 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( -\beta_{9} + \beta_{8} + \beta_{7} + \beta_{6} - \beta_{3} - \beta _1 - 1 ) / 2 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{8} - 4\beta_{6} + \beta_{4} + \beta_{2} \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 5\beta_{5} + 9\beta_{3} + 7\beta _1 + 7 ) / 2 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 2\beta_{9} - 9\beta_{8} - \beta_{7} + 24\beta_{6} - 9\beta_{4} + 9\beta_{3} + 2\beta _1 - 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 51\beta_{9} - 71\beta_{8} - 31\beta_{7} - 35\beta_{6} - 31\beta_{5} ) / 2 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( -14\beta_{5} - 75\beta_{3} - 71\beta_{2} - 24\beta _1 + 160 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( -377\beta_{9} + 555\beta_{8} + 215\beta_{7} + 153\beta_{6} + 16\beta_{4} - 555\beta_{3} - 377\beta _1 - 153 ) / 2 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( -230\beta_{9} + 615\beta_{8} + 141\beta_{7} - 1116\beta_{6} + 141\beta_{5} + 539\beta_{4} + 539\beta_{2} \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1577\beta_{5} + 4345\beta_{3} + 304\beta_{2} + 2833\beta _1 + 481 ) / 2 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(533\) \(1009\) \(1711\)
\(\chi(n)\) \(1\) \(-1 + \beta_{6}\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
505.1
0.238404 + 0.412928i
−1.29481 − 2.24267i
0.600301 + 1.03975i
−0.955644 − 1.65522i
1.41175 + 2.44521i
0.238404 − 0.412928i
−1.29481 + 2.24267i
0.600301 − 1.03975i
−0.955644 + 1.65522i
1.41175 − 2.44521i
−0.500000 + 0.866025i 0 −0.500000 − 0.866025i −2.05460 + 3.55867i 0 −1.00000 1.00000 0 −2.05460 − 3.55867i
505.2 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −1.20236 + 2.08254i 0 −1.00000 1.00000 0 −1.20236 − 2.08254i
505.3 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i −0.662663 + 1.14777i 0 −1.00000 1.00000 0 −0.662663 − 1.14777i
505.4 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 0.880752 − 1.52551i 0 −1.00000 1.00000 0 0.880752 + 1.52551i
505.5 −0.500000 + 0.866025i 0 −0.500000 − 0.866025i 1.53887 − 2.66540i 0 −1.00000 1.00000 0 1.53887 + 2.66540i
1261.1 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −2.05460 − 3.55867i 0 −1.00000 1.00000 0 −2.05460 + 3.55867i
1261.2 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −1.20236 − 2.08254i 0 −1.00000 1.00000 0 −1.20236 + 2.08254i
1261.3 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i −0.662663 − 1.14777i 0 −1.00000 1.00000 0 −0.662663 + 1.14777i
1261.4 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 0.880752 + 1.52551i 0 −1.00000 1.00000 0 0.880752 − 1.52551i
1261.5 −0.500000 − 0.866025i 0 −0.500000 + 0.866025i 1.53887 + 2.66540i 0 −1.00000 1.00000 0 1.53887 − 2.66540i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 505.5
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
19.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 2394.2.o.w ✓ 10
3.b odd 2 1 2394.2.o.x yes 10
19.c even 3 1 inner 2394.2.o.w ✓ 10
57.h odd 6 1 2394.2.o.x yes 10
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
2394.2.o.w ✓ 10 1.a even 1 1 trivial
2394.2.o.w ✓ 10 19.c even 3 1 inner
2394.2.o.x yes 10 3.b odd 2 1
2394.2.o.x yes 10 57.h odd 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(2394, [\chi])\):

\( T_{5}^{10} + 3 T_{5}^{9} + 23 T_{5}^{8} + 26 T_{5}^{7} + 261 T_{5}^{6} + 325 T_{5}^{5} + 1461 T_{5}^{4} + \cdots + 5041 \) Copy content Toggle raw display
\( T_{11}^{5} + 3T_{11}^{4} - 24T_{11}^{3} - 56T_{11}^{2} - 16T_{11} + 16 \) Copy content Toggle raw display
\( T_{13}^{10} + 6 T_{13}^{9} + 54 T_{13}^{8} + 116 T_{13}^{7} + 1011 T_{13}^{6} + 2422 T_{13}^{5} + \cdots + 51076 \) Copy content Toggle raw display
\( T_{17}^{10} + 5 T_{17}^{9} + 37 T_{17}^{8} + 44 T_{17}^{7} + 352 T_{17}^{6} + 156 T_{17}^{5} + \cdots + 2704 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + T + 1)^{5} \) Copy content Toggle raw display
$3$ \( T^{10} \) Copy content Toggle raw display
$5$ \( T^{10} + 3 T^{9} + \cdots + 5041 \) Copy content Toggle raw display
$7$ \( (T + 1)^{10} \) Copy content Toggle raw display
$11$ \( (T^{5} + 3 T^{4} - 24 T^{3} + \cdots + 16)^{2} \) Copy content Toggle raw display
$13$ \( T^{10} + 6 T^{9} + \cdots + 51076 \) Copy content Toggle raw display
$17$ \( T^{10} + 5 T^{9} + \cdots + 2704 \) Copy content Toggle raw display
$19$ \( T^{10} + 5 T^{9} + \cdots + 2476099 \) Copy content Toggle raw display
$23$ \( T^{10} + 4 T^{9} + \cdots + 215296 \) Copy content Toggle raw display
$29$ \( T^{10} + 6 T^{9} + \cdots + 1024 \) Copy content Toggle raw display
$31$ \( (T^{5} - 128 T^{3} + \cdots + 4864)^{2} \) Copy content Toggle raw display
$37$ \( (T^{5} - 7 T^{4} + \cdots + 512)^{2} \) Copy content Toggle raw display
$41$ \( T^{10} + 11 T^{9} + \cdots + 20736 \) Copy content Toggle raw display
$43$ \( T^{10} - 4 T^{9} + \cdots + 16384 \) Copy content Toggle raw display
$47$ \( T^{10} - 2 T^{9} + \cdots + 817216 \) Copy content Toggle raw display
$53$ \( T^{10} + 2 T^{9} + \cdots + 23104 \) Copy content Toggle raw display
$59$ \( T^{10} + 14 T^{9} + \cdots + 18198756 \) Copy content Toggle raw display
$61$ \( T^{10} - 12 T^{9} + \cdots + 14684224 \) Copy content Toggle raw display
$67$ \( T^{10} + 14 T^{9} + \cdots + 3182656 \) Copy content Toggle raw display
$71$ \( T^{10} - 12 T^{9} + \cdots + 5053504 \) Copy content Toggle raw display
$73$ \( T^{10} + 2 T^{9} + \cdots + 15936064 \) Copy content Toggle raw display
$79$ \( T^{10} + \cdots + 959512576 \) Copy content Toggle raw display
$83$ \( (T^{5} - 4 T^{4} + \cdots - 122968)^{2} \) Copy content Toggle raw display
$89$ \( T^{10} + 3 T^{9} + \cdots + 156816 \) Copy content Toggle raw display
$97$ \( T^{10} + 24 T^{9} + \cdots + 262144 \) Copy content Toggle raw display
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