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Properties

Label 42.4.d.a
Level $42$
Weight $4$
Character orbit 42.d
Analytic conductor $2.478$
Analytic rank $0$
Dimension $8$
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] = [42,4,Mod(41,42)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(42, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("42.41");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 42.d (of order \(2\), degree \(1\), 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: \(2.47808022024\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.116876510171136.13
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 15x^{6} + 455x^{4} + 5097x^{2} + 21904 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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 \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{3} q^{2} + \beta_{5} q^{3} - 4 q^{4} + (\beta_{7} - 2 \beta_{6} + 2 \beta_{5}) q^{5} + (\beta_{7} - \beta_{2}) q^{6} + ( - \beta_{2} - \beta_1) q^{7} - 4 \beta_{3} q^{8} + ( - \beta_{4} + 4 \beta_{3} + \beta_1 - 7) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{3} q^{2} + \beta_{5} q^{3} - 4 q^{4} + (\beta_{7} - 2 \beta_{6} + 2 \beta_{5}) q^{5} + (\beta_{7} - \beta_{2}) q^{6} + ( - \beta_{2} - \beta_1) q^{7} - 4 \beta_{3} q^{8} + ( - \beta_{4} + 4 \beta_{3} + \beta_1 - 7) q^{9} + ( - 2 \beta_{6} - 2 \beta_{5} - 4 \beta_{2}) q^{10} + (\beta_{6} + \beta_{5} + \cdots + \beta_{2}) q^{11}+ \cdots + ( - 15 \beta_{6} - 15 \beta_{5} + \cdots + 504) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 32 q^{4} + 4 q^{7} - 60 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 32 q^{4} + 4 q^{7} - 60 q^{9} + 252 q^{15} + 128 q^{16} - 120 q^{18} - 168 q^{21} - 240 q^{22} + 320 q^{25} - 16 q^{28} + 312 q^{30} + 240 q^{36} - 592 q^{37} - 804 q^{39} - 216 q^{42} - 1696 q^{43} + 1344 q^{46} + 2192 q^{49} + 504 q^{51} + 2532 q^{57} - 1488 q^{58} - 1008 q^{60} - 2496 q^{63} - 512 q^{64} + 496 q^{67} - 192 q^{70} + 480 q^{72} + 1080 q^{78} + 2824 q^{79} + 672 q^{84} - 3936 q^{85} + 960 q^{88} - 264 q^{91} - 1512 q^{93} + 4032 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{8} + 15x^{6} + 455x^{4} + 5097x^{2} + 21904 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 5\nu^{6} + 497\nu^{4} + 4214\nu^{2} + 96213 ) / 4879 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 117\nu^{6} + 896\nu^{4} + 42987\nu^{2} + 261728 ) / 19516 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( -12\nu^{7} - 217\nu^{5} - 3980\nu^{3} - 56243\nu ) / 51578 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 696 \nu^{7} - 1147 \nu^{6} + 12586 \nu^{5} - 5698 \nu^{4} + 385574 \nu^{3} - 208495 \nu^{2} + \cdots - 1238908 ) / 722092 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 1411 \nu^{7} - 2035 \nu^{6} - 10472 \nu^{5} - 21756 \nu^{4} - 553945 \nu^{3} - 1173529 \nu^{2} + \cdots - 7206120 ) / 1444184 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 1411 \nu^{7} - 2035 \nu^{6} + 10472 \nu^{5} - 21756 \nu^{4} + 553945 \nu^{3} - 1173529 \nu^{2} + \cdots - 7206120 ) / 1444184 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 59\nu^{7} + 182\nu^{5} + 18557\nu^{3} + 96890\nu ) / 42476 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 3\beta_{7} - 2\beta_{6} + 4\beta_{5} + 2\beta_{4} + \beta_{3} + \beta_{2} ) / 12 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -11\beta_{6} - 11\beta_{5} - 5\beta_{2} - \beta _1 - 23 ) / 6 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( -63\beta_{7} + 70\beta_{6} - 56\beta_{5} + 14\beta_{4} + 211\beta_{3} + 7\beta_{2} ) / 12 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 19\beta_{6} + 19\beta_{5} + 5\beta_{2} + 23\beta _1 - 331 ) / 2 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( -1113\beta_{7} + 326\beta_{6} - 748\beta_{5} - 422\beta_{4} - 3883\beta_{3} - 211\beta_{2} ) / 12 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 3605\beta_{6} + 3605\beta_{5} + 2723\beta_{2} - 161\beta _1 + 2633 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 26961\beta_{7} - 19738\beta_{6} + 13352\beta_{5} - 6386\beta_{4} - 56029\beta_{3} - 3193\beta_{2} ) / 12 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(29\) \(31\)
\(\chi(n)\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
41.1
−0.696949 + 2.67617i
−3.04373 − 3.17617i
3.04373 − 3.17617i
0.696949 + 2.67617i
−0.696949 − 2.67617i
−3.04373 + 3.17617i
3.04373 + 3.17617i
0.696949 − 2.67617i
− 2.00000i −4.30448 + 2.91058i −4.00000 −11.3967 5.82116 + 8.60895i −17.0570 + 7.21506i 8.00000i 10.0570 − 25.0570i 22.7935i
41.2 − 2.00000i −0.985634 − 5.10182i −4.00000 −14.1462 −10.2036 + 1.97127i 18.0570 − 4.11618i 8.00000i −25.0570 + 10.0570i 28.2923i
41.3 − 2.00000i 0.985634 + 5.10182i −4.00000 14.1462 10.2036 − 1.97127i 18.0570 + 4.11618i 8.00000i −25.0570 + 10.0570i − 28.2923i
41.4 − 2.00000i 4.30448 − 2.91058i −4.00000 11.3967 −5.82116 − 8.60895i −17.0570 − 7.21506i 8.00000i 10.0570 − 25.0570i − 22.7935i
41.5 2.00000i −4.30448 − 2.91058i −4.00000 −11.3967 5.82116 − 8.60895i −17.0570 − 7.21506i − 8.00000i 10.0570 + 25.0570i − 22.7935i
41.6 2.00000i −0.985634 + 5.10182i −4.00000 −14.1462 −10.2036 − 1.97127i 18.0570 + 4.11618i − 8.00000i −25.0570 − 10.0570i − 28.2923i
41.7 2.00000i 0.985634 − 5.10182i −4.00000 14.1462 10.2036 + 1.97127i 18.0570 − 4.11618i − 8.00000i −25.0570 − 10.0570i 28.2923i
41.8 2.00000i 4.30448 + 2.91058i −4.00000 11.3967 −5.82116 + 8.60895i −17.0570 + 7.21506i − 8.00000i 10.0570 + 25.0570i 22.7935i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 41.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.b odd 2 1 inner
21.c even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 42.4.d.a ✓ 8
3.b odd 2 1 inner 42.4.d.a ✓ 8
4.b odd 2 1 336.4.k.c 8
7.b odd 2 1 inner 42.4.d.a ✓ 8
7.c even 3 2 294.4.f.b 16
7.d odd 6 2 294.4.f.b 16
12.b even 2 1 336.4.k.c 8
21.c even 2 1 inner 42.4.d.a ✓ 8
21.g even 6 2 294.4.f.b 16
21.h odd 6 2 294.4.f.b 16
28.d even 2 1 336.4.k.c 8
84.h odd 2 1 336.4.k.c 8
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.d.a ✓ 8 1.a even 1 1 trivial
42.4.d.a ✓ 8 3.b odd 2 1 inner
42.4.d.a ✓ 8 7.b odd 2 1 inner
42.4.d.a ✓ 8 21.c even 2 1 inner
294.4.f.b 16 7.c even 3 2
294.4.f.b 16 7.d odd 6 2
294.4.f.b 16 21.g even 6 2
294.4.f.b 16 21.h odd 6 2
336.4.k.c 8 4.b odd 2 1
336.4.k.c 8 12.b even 2 1
336.4.k.c 8 28.d even 2 1
336.4.k.c 8 84.h odd 2 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(42, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{2} + 4)^{4} \) Copy content Toggle raw display
$3$ \( T^{8} + 30 T^{6} + \cdots + 531441 \) Copy content Toggle raw display
$5$ \( (T^{4} - 330 T^{2} + 25992)^{2} \) Copy content Toggle raw display
$7$ \( (T^{4} - 2 T^{3} + \cdots + 117649)^{2} \) Copy content Toggle raw display
$11$ \( (T^{4} + 2916 T^{2} + 1016064)^{2} \) Copy content Toggle raw display
$13$ \( (T^{4} + 2958 T^{2} + 1002528)^{2} \) Copy content Toggle raw display
$17$ \( (T^{4} - 16116 T^{2} + 40069152)^{2} \) Copy content Toggle raw display
$19$ \( (T^{4} + 18762 T^{2} + 43022088)^{2} \) Copy content Toggle raw display
$23$ \( (T^{4} + 23976 T^{2} + 4511376)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + 19764 T^{2} + 54997056)^{2} \) Copy content Toggle raw display
$31$ \( (T^{4} + 112860 T^{2} + 1170505728)^{2} \) Copy content Toggle raw display
$37$ \( (T + 74)^{8} \) Copy content Toggle raw display
$41$ \( (T^{4} - 94740 T^{2} + 10071072)^{2} \) Copy content Toggle raw display
$43$ \( (T^{2} + 424 T - 33968)^{4} \) Copy content Toggle raw display
$47$ \( (T^{4} - 255480 T^{2} + 9930350592)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + 209268 T^{2} + 9046292544)^{2} \) Copy content Toggle raw display
$59$ \( (T^{4} - 580830 T^{2} + 4021968672)^{2} \) Copy content Toggle raw display
$61$ \( (T^{4} + 183678 T^{2} + 8432329248)^{2} \) Copy content Toggle raw display
$67$ \( (T^{2} - 124 T - 1088)^{4} \) Copy content Toggle raw display
$71$ \( (T^{4} + 607716 T^{2} + 12745506816)^{2} \) Copy content Toggle raw display
$73$ \( (T^{4} + 1099680 T^{2} + 148979386368)^{2} \) Copy content Toggle raw display
$79$ \( (T^{2} - 706 T + 24736)^{4} \) Copy content Toggle raw display
$83$ \( (T^{4} - 1502862 T^{2} + 422927723808)^{2} \) Copy content Toggle raw display
$89$ \( (T^{4} - 392796 T^{2} + 35407798272)^{2} \) Copy content Toggle raw display
$97$ \( (T^{4} + 1338360 T^{2} + 94643822592)^{2} \) Copy content Toggle raw display
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