Properties

Label 42.6.e.d
Level $42$
Weight $6$
Character orbit 42.e
Analytic conductor $6.736$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [42,6,Mod(25,42)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(42, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 4]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("42.25");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 42.e (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: \(6.73612043215\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{505})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} + 127x^{2} + 126x + 15876 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 7 \)
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 \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( - 4 \beta_1 + 4) q^{2} - 9 \beta_1 q^{3} - 16 \beta_1 q^{4} + (3 \beta_{3} - 2 \beta_{2} + 10 \beta_1 - 9) q^{5} - 36 q^{6} + ( - 2 \beta_{3} + 3 \beta_{2} + \cdots - 111) q^{7}+ \cdots + (81 \beta_1 - 81) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + ( - 4 \beta_1 + 4) q^{2} - 9 \beta_1 q^{3} - 16 \beta_1 q^{4} + (3 \beta_{3} - 2 \beta_{2} + 10 \beta_1 - 9) q^{5} - 36 q^{6} + ( - 2 \beta_{3} + 3 \beta_{2} + \cdots - 111) q^{7}+ \cdots + (1134 \beta_{3} + 567 \beta_{2} + \cdots + 5022) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 8 q^{2} - 18 q^{3} - 32 q^{4} - 17 q^{5} - 144 q^{6} - 408 q^{7} - 256 q^{8} - 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 8 q^{2} - 18 q^{3} - 32 q^{4} - 17 q^{5} - 144 q^{6} - 408 q^{7} - 256 q^{8} - 162 q^{9} + 68 q^{10} - 145 q^{11} - 288 q^{12} - 1430 q^{13} - 612 q^{14} + 306 q^{15} - 512 q^{16} - 1372 q^{17} + 648 q^{18} - 1081 q^{19} + 544 q^{20} + 2295 q^{21} - 1160 q^{22} + 4508 q^{23} + 1152 q^{24} - 6267 q^{25} - 2860 q^{26} + 2916 q^{27} + 4080 q^{28} + 15730 q^{29} + 612 q^{30} - 8816 q^{31} + 2048 q^{32} - 1305 q^{33} - 10976 q^{34} + 24278 q^{35} + 5184 q^{36} + 14573 q^{37} + 4324 q^{38} + 6435 q^{39} + 1088 q^{40} + 14700 q^{41} + 14688 q^{42} - 11842 q^{43} - 2320 q^{44} - 1377 q^{45} - 18032 q^{46} - 44808 q^{47} + 9216 q^{48} + 17014 q^{49} - 50136 q^{50} - 12348 q^{51} + 11440 q^{52} - 9417 q^{53} + 5832 q^{54} - 170750 q^{55} + 26112 q^{56} + 19458 q^{57} + 31460 q^{58} + 5077 q^{59} - 2448 q^{60} - 42368 q^{61} - 70528 q^{62} + 12393 q^{63} + 16384 q^{64} + 18450 q^{65} + 5220 q^{66} - 30501 q^{67} - 21952 q^{68} - 81144 q^{69} + 69100 q^{70} + 183488 q^{71} + 10368 q^{72} + 85665 q^{73} - 58292 q^{74} - 56403 q^{75} + 34592 q^{76} + 154585 q^{77} + 51480 q^{78} - 94646 q^{79} - 4352 q^{80} - 13122 q^{81} + 29400 q^{82} + 67682 q^{83} + 22032 q^{84} + 518224 q^{85} - 23684 q^{86} - 70785 q^{87} + 9280 q^{88} - 27558 q^{89} - 11016 q^{90} + 149395 q^{91} - 144256 q^{92} - 79344 q^{93} + 179232 q^{94} - 343246 q^{95} + 18432 q^{96} - 93342 q^{97} + 16652 q^{98} + 23490 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{4} - x^{3} + 127x^{2} + 126x + 15876 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( -\nu^{3} + 127\nu^{2} - 127\nu + 15876 ) / 16002 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 125\nu^{3} + 127\nu^{2} - 32131\nu + 47754 ) / 16002 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 379\nu^{3} - 127\nu^{2} + 16129\nu + 63756 ) / 16002 \) Copy content Toggle raw display
\(\nu\)\(=\) \( ( \beta_{3} - 3\beta_{2} + 4\beta _1 + 1 ) / 7 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 3\beta_{3} - 2\beta_{2} + 887\beta _1 - 886 ) / 7 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 254\beta_{3} + 127\beta_{2} + 127\beta _1 - 1517 ) / 7 \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/42\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(31\)
\(\chi(n)\) \(1\) \(-1 + \beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
25.1
5.86805 10.1638i
−5.36805 + 9.29774i
5.86805 + 10.1638i
−5.36805 9.29774i
2.00000 + 3.46410i −4.50000 + 7.79423i −8.00000 + 13.8564i −43.5764 75.4765i −36.0000 −113.236 + 63.1236i −64.0000 −40.5000 70.1481i 174.305 301.906i
25.2 2.00000 + 3.46410i −4.50000 + 7.79423i −8.00000 + 13.8564i 35.0764 + 60.7540i −36.0000 −90.7639 92.5684i −64.0000 −40.5000 70.1481i −140.305 + 243.016i
37.1 2.00000 3.46410i −4.50000 7.79423i −8.00000 13.8564i −43.5764 + 75.4765i −36.0000 −113.236 63.1236i −64.0000 −40.5000 + 70.1481i 174.305 + 301.906i
37.2 2.00000 3.46410i −4.50000 7.79423i −8.00000 13.8564i 35.0764 60.7540i −36.0000 −90.7639 + 92.5684i −64.0000 −40.5000 + 70.1481i −140.305 243.016i
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

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

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 42.6.e.d 4
3.b odd 2 1 126.6.g.g 4
4.b odd 2 1 336.6.q.h 4
7.b odd 2 1 294.6.e.y 4
7.c even 3 1 inner 42.6.e.d 4
7.c even 3 1 294.6.a.p 2
7.d odd 6 1 294.6.a.o 2
7.d odd 6 1 294.6.e.y 4
21.g even 6 1 882.6.a.bs 2
21.h odd 6 1 126.6.g.g 4
21.h odd 6 1 882.6.a.bm 2
28.g odd 6 1 336.6.q.h 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.6.e.d 4 1.a even 1 1 trivial
42.6.e.d 4 7.c even 3 1 inner
126.6.g.g 4 3.b odd 2 1
126.6.g.g 4 21.h odd 6 1
294.6.a.o 2 7.d odd 6 1
294.6.a.p 2 7.c even 3 1
294.6.e.y 4 7.b odd 2 1
294.6.e.y 4 7.d odd 6 1
336.6.q.h 4 4.b odd 2 1
336.6.q.h 4 28.g odd 6 1
882.6.a.bm 2 21.h odd 6 1
882.6.a.bs 2 21.g even 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{4} + 17T_{5}^{3} + 6403T_{5}^{2} - 103938T_{5} + 37380996 \) acting on \(S_{6}^{\mathrm{new}}(42, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} - 4 T + 16)^{2} \) Copy content Toggle raw display
$3$ \( (T^{2} + 9 T + 81)^{2} \) Copy content Toggle raw display
$5$ \( T^{4} + 17 T^{3} + \cdots + 37380996 \) Copy content Toggle raw display
$7$ \( T^{4} + 408 T^{3} + \cdots + 282475249 \) Copy content Toggle raw display
$11$ \( T^{4} + \cdots + 88726536900 \) Copy content Toggle raw display
$13$ \( (T^{2} + 715 T + 121620)^{2} \) Copy content Toggle raw display
$17$ \( T^{4} + \cdots + 4015631241216 \) Copy content Toggle raw display
$19$ \( T^{4} + \cdots + 17788453428496 \) Copy content Toggle raw display
$23$ \( T^{4} + \cdots + 24815700919296 \) Copy content Toggle raw display
$29$ \( (T^{2} - 7865 T - 6069780)^{2} \) Copy content Toggle raw display
$31$ \( T^{4} + \cdots + 331894030125681 \) Copy content Toggle raw display
$37$ \( T^{4} + \cdots + 156377425969216 \) Copy content Toggle raw display
$41$ \( (T^{2} - 7350 T - 13441680)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 5921 T - 15788666)^{2} \) Copy content Toggle raw display
$47$ \( T^{4} + \cdots + 24\!\cdots\!96 \) Copy content Toggle raw display
$53$ \( T^{4} + \cdots + 92983900409856 \) Copy content Toggle raw display
$59$ \( T^{4} + \cdots + 76\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( T^{4} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$67$ \( T^{4} + \cdots + 15\!\cdots\!76 \) Copy content Toggle raw display
$71$ \( (T^{2} - 91744 T + 1804825884)^{2} \) Copy content Toggle raw display
$73$ \( T^{4} + \cdots + 66\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( T^{4} + \cdots + 47\!\cdots\!81 \) Copy content Toggle raw display
$83$ \( (T^{2} - 33841 T + 280358334)^{2} \) Copy content Toggle raw display
$89$ \( T^{4} + \cdots + 6584027556096 \) Copy content Toggle raw display
$97$ \( (T^{2} + 46671 T - 4062781666)^{2} \) Copy content Toggle raw display
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