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Properties

Label 592.3.k.e
Level $592$
Weight $3$
Character orbit 592.k
Analytic conductor $16.131$
Analytic rank $0$
Dimension $12$
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] = [592,3,Mod(401,592)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(592, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("592.401");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 592 = 2^{4} \cdot 37 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 592.k (of order \(4\), degree \(2\), not 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: \(16.1308316501\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} + 18 x^{10} + 8 x^{9} + 42 x^{8} - 268 x^{7} + 884 x^{6} + 704 x^{5} + 761 x^{4} + \cdots + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 37)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_{5} q^{3} + (\beta_{11} - \beta_{6} - 1) q^{5} + (\beta_{11} - \beta_{10} + \cdots + \beta_1) q^{7}+ \cdots + (\beta_{11} - \beta_{10} - \beta_{4} + \cdots - 5) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{5} q^{3} + (\beta_{11} - \beta_{6} - 1) q^{5} + (\beta_{11} - \beta_{10} + \cdots + \beta_1) q^{7}+ \cdots + (14 \beta_{11} + 14 \beta_{10} + \cdots - 12) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 6 q^{5} + 4 q^{7} - 60 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 6 q^{5} + 4 q^{7} - 60 q^{9} + 14 q^{13} + 2 q^{15} + 2 q^{17} - 14 q^{19} - 56 q^{23} + 60 q^{29} - 72 q^{31} + 56 q^{33} + 154 q^{35} - 66 q^{37} + 46 q^{39} - 70 q^{43} + 232 q^{45} + 384 q^{47} + 144 q^{49} + 126 q^{51} - 56 q^{53} - 70 q^{55} - 94 q^{57} - 184 q^{59} + 132 q^{61} + 400 q^{63} + 368 q^{69} - 68 q^{71} - 116 q^{75} + 2 q^{79} - 76 q^{81} - 108 q^{83} + 420 q^{87} + 278 q^{89} + 450 q^{91} + 584 q^{93} - 244 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 6 x^{11} + 18 x^{10} + 8 x^{9} + 42 x^{8} - 268 x^{7} + 884 x^{6} + 704 x^{5} + 761 x^{4} + \cdots + 16 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 1147429625 \nu^{11} - 7950477538 \nu^{10} + 30130260222 \nu^{9} - 30812713531 \nu^{8} + \cdots - 31702472561346 ) / 6472908901282 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 2383585029 \nu^{11} + 13065354770 \nu^{10} - 37789653290 \nu^{9} - 26728073300 \nu^{8} + \cdots - 7096843485882 ) / 6472908901282 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 22666832839 \nu^{11} - 126631722778 \nu^{10} + 385593203213 \nu^{9} + 105672415293 \nu^{8} + \cdots + 9609041030712 ) / 25891635605128 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 43439014063 \nu^{11} + 245969084305 \nu^{10} - 725700954999 \nu^{9} + \cdots - 9336516192680 ) / 25891635605128 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 77991823075 \nu^{11} - 472718108508 \nu^{10} + 1429983524890 \nu^{9} + \cdots + 12075070737780 ) / 12945817802564 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 385191945317 \nu^{11} - 2337459833000 \nu^{10} + 7074338317870 \nu^{9} + \cdots + 59127484519700 ) / 12945817802564 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 389618604143 \nu^{11} + 2372313842020 \nu^{10} - 7214583139376 \nu^{9} + \cdots + 4240742854992 ) / 12945817802564 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 428273927397 \nu^{11} + 2566965018248 \nu^{10} - 7709610342782 \nu^{9} + \cdots - 133686977444264 ) / 12945817802564 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 925851580180 \nu^{11} - 5667654113167 \nu^{10} + 17364795575951 \nu^{9} + \cdots + 16031325091000 ) / 25891635605128 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 1017662160901 \nu^{11} - 6083306132567 \nu^{10} + 18191287173440 \nu^{9} + \cdots + 314224969405808 ) / 25891635605128 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{7} - 5\beta_{6} - \beta_{3} + \beta _1 - 1 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{9} + 2\beta_{7} - 5\beta_{6} - 11\beta_{3} - 2\beta_{2} - 16 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( 3\beta_{9} - 3\beta_{8} + 4\beta_{4} - 23\beta_{3} - 17\beta_{2} - 23\beta _1 - 78 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( -4\beta_{10} - 27\beta_{8} - 46\beta_{7} + 119\beta_{6} + 12\beta_{5} + 12\beta_{4} - 46\beta_{2} - 167\beta _1 - 115 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( - 12 \beta_{11} - 12 \beta_{10} - 85 \beta_{9} - 85 \beta_{8} - 313 \beta_{7} + 863 \beta_{6} + \cdots + 467 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( - 112 \beta_{11} - 595 \beta_{9} - 938 \beta_{7} + 2399 \beta_{6} + 352 \beta_{5} - 352 \beta_{4} + \cdots + 5366 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( - 352 \beta_{11} + 352 \beta_{10} - 1885 \beta_{9} + 1885 \beta_{8} - 2492 \beta_{4} + 8875 \beta_{3} + \cdots + 24174 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 2492 \beta_{10} + 12295 \beta_{8} + 18678 \beta_{7} - 47219 \beta_{6} - 7892 \beta_{5} - 7892 \beta_{4} + \cdots + 44727 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 7892 \beta_{11} + 7892 \beta_{10} + 38865 \beta_{9} + 38865 \beta_{8} + 118201 \beta_{7} - 299551 \beta_{6} + \cdots - 181675 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 51672 \beta_{11} + 247603 \beta_{9} + 369714 \beta_{7} - 928479 \beta_{6} - 163352 \beta_{5} + \cdots - 2022342 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/592\mathbb{Z}\right)^\times\).

\(n\) \(113\) \(149\) \(223\)
\(\chi(n)\) \(-\beta_{6}\) \(1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
401.1
−1.15759 − 1.15759i
2.02961 + 2.02961i
−0.0496173 − 0.0496173i
0.781387 + 0.781387i
3.14278 + 3.14278i
−1.74658 − 1.74658i
−1.74658 + 1.74658i
3.14278 − 3.14278i
0.781387 − 0.781387i
−0.0496173 + 0.0496173i
2.02961 − 2.02961i
−1.15759 + 1.15759i
0 − 5.58276i 0 −2.97944 + 2.97944i 0 −8.27885 0 −22.1672 0
401.2 0 − 3.18930i 0 −0.460662 + 0.460662i 0 2.31735 0 −1.17160 0
401.3 0 − 0.377912i 0 5.26886 − 5.26886i 0 5.54265 0 8.85718 0
401.4 0 3.27551i 0 −3.73585 + 3.73585i 0 −11.2506 0 −1.72900 0
401.5 0 3.31038i 0 −1.68576 + 1.68576i 0 11.3827 0 −1.95861 0
401.6 0 4.56407i 0 0.592850 − 0.592850i 0 2.28677 0 −11.8308 0
561.1 0 − 4.56407i 0 0.592850 + 0.592850i 0 2.28677 0 −11.8308 0
561.2 0 − 3.31038i 0 −1.68576 − 1.68576i 0 11.3827 0 −1.95861 0
561.3 0 − 3.27551i 0 −3.73585 − 3.73585i 0 −11.2506 0 −1.72900 0
561.4 0 0.377912i 0 5.26886 + 5.26886i 0 5.54265 0 8.85718 0
561.5 0 3.18930i 0 −0.460662 − 0.460662i 0 2.31735 0 −1.17160 0
561.6 0 5.58276i 0 −2.97944 − 2.97944i 0 −8.27885 0 −22.1672 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 401.6
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
37.d odd 4 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 592.3.k.e 12
4.b odd 2 1 37.3.d.a ✓ 12
12.b even 2 1 333.3.i.a 12
37.d odd 4 1 inner 592.3.k.e 12
148.g even 4 1 37.3.d.a ✓ 12
444.j odd 4 1 333.3.i.a 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
37.3.d.a ✓ 12 4.b odd 2 1
37.3.d.a ✓ 12 148.g even 4 1
333.3.i.a 12 12.b even 2 1
333.3.i.a 12 444.j odd 4 1
592.3.k.e 12 1.a even 1 1 trivial
592.3.k.e 12 37.d odd 4 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(592, [\chi])\):

\( T_{3}^{12} + 84T_{3}^{10} + 2656T_{3}^{8} + 39842T_{3}^{6} + 287376T_{3}^{4} + 816676T_{3}^{2} + 110889 \) Copy content Toggle raw display
\( T_{5}^{12} + 6 T_{5}^{11} + 18 T_{5}^{10} + 120 T_{5}^{9} + 2663 T_{5}^{8} + 19142 T_{5}^{7} + \cdots + 46656 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} \) Copy content Toggle raw display
$3$ \( T^{12} + 84 T^{10} + \cdots + 110889 \) Copy content Toggle raw display
$5$ \( T^{12} + 6 T^{11} + \cdots + 46656 \) Copy content Toggle raw display
$7$ \( (T^{6} - 2 T^{5} + \cdots + 31140)^{2} \) Copy content Toggle raw display
$11$ \( T^{12} + \cdots + 1467889749225 \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 1245652423744 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 286601604793344 \) Copy content Toggle raw display
$19$ \( T^{12} + \cdots + 57846022144 \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 926935411836996 \) Copy content Toggle raw display
$29$ \( T^{12} + \cdots + 109768966596 \) Copy content Toggle raw display
$31$ \( T^{12} + \cdots + 56851117441024 \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 65\!\cdots\!81 \) Copy content Toggle raw display
$41$ \( T^{12} + \cdots + 11954142225 \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 800194650194944 \) Copy content Toggle raw display
$47$ \( (T^{6} - 192 T^{5} + \cdots + 189858180)^{2} \) Copy content Toggle raw display
$53$ \( (T^{6} + 28 T^{5} + \cdots + 744687180)^{2} \) Copy content Toggle raw display
$59$ \( T^{12} + \cdots + 45\!\cdots\!00 \) Copy content Toggle raw display
$61$ \( T^{12} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 26\!\cdots\!44 \) Copy content Toggle raw display
$71$ \( (T^{6} + 34 T^{5} + \cdots - 165884527056)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 83\!\cdots\!41 \) Copy content Toggle raw display
$79$ \( T^{12} + \cdots + 11\!\cdots\!16 \) Copy content Toggle raw display
$83$ \( (T^{6} + 54 T^{5} + \cdots - 359355430320)^{2} \) Copy content Toggle raw display
$89$ \( T^{12} + \cdots + 51\!\cdots\!04 \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 42\!\cdots\!76 \) Copy content Toggle raw display
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