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Properties

Label 126.4.k.a
Level $126$
Weight $4$
Character orbit 126.k
Analytic conductor $7.434$
Analytic rank $0$
Dimension $16$
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] = [126,4,Mod(17,126)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(126, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("126.17");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 126.k (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: \(7.43424066072\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 26x^{14} + 451x^{12} - 4330x^{10} + 30081x^{8} - 130232x^{6} + 401200x^{4} - 595840x^{2} + 614656 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{12}\cdot 7^{2} \)
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 \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{12} q^{2} + ( - 4 \beta_1 + 4) q^{4} + (\beta_{15} - \beta_{14} + \cdots - 2 \beta_{4}) q^{5}+ \cdots + ( - 4 \beta_{12} + 4 \beta_{4}) q^{8}+O(q^{10}) \) Copy content Toggle raw display \( q - \beta_{12} q^{2} + ( - 4 \beta_1 + 4) q^{4} + (\beta_{15} - \beta_{14} + \cdots - 2 \beta_{4}) q^{5}+ \cdots + ( - 26 \beta_{15} + \cdots + 32 \beta_{3}) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 32 q^{4} - 64 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 32 q^{4} - 64 q^{7} + 72 q^{10} - 128 q^{16} + 684 q^{19} - 48 q^{22} - 212 q^{25} + 16 q^{28} - 1248 q^{31} + 1252 q^{37} + 288 q^{40} - 1112 q^{43} + 672 q^{46} - 1088 q^{49} - 336 q^{52} + 552 q^{58} + 3264 q^{61} - 1024 q^{64} + 68 q^{67} - 888 q^{70} - 3132 q^{73} + 2744 q^{79} - 864 q^{82} - 672 q^{85} - 96 q^{88} - 5748 q^{91} - 2160 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 26x^{14} + 451x^{12} - 4330x^{10} + 30081x^{8} - 130232x^{6} + 401200x^{4} - 595840x^{2} + 614656 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 142677179 \nu^{14} + 3449734804 \nu^{12} - 58606224029 \nu^{10} + 522506995056 \nu^{8} + \cdots + 67829831095936 ) / 42688070765536 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 268395985 \nu^{14} + 3082276126 \nu^{12} - 35491743131 \nu^{10} - 313799736514 \nu^{8} + \cdots - 508725620940208 ) / 21344035382768 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 742856039 \nu^{15} + 26162302198 \nu^{13} - 135143792323 \nu^{11} + \cdots + 18\!\cdots\!40 \nu ) / 341504566124288 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 4843621507 \nu^{15} + 20981759882 \nu^{13} + 279055868863 \nu^{11} + \cdots - 40\!\cdots\!92 \nu ) / 11\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 306102998 \nu^{15} - 154321871 \nu^{13} - 45181905369 \nu^{11} + 1381985194213 \nu^{9} + \cdots - 708753261192760 \nu ) / 74704123839688 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 32388 \nu^{15} - 519129 \nu^{13} + 6905658 \nu^{11} - 11173011 \nu^{9} - 146158518 \nu^{7} + \cdots + 30203812464 \nu ) / 3440822336 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 1696491632 \nu^{14} - 43710871023 \nu^{12} + 751023188358 \nu^{10} - 7152478495349 \nu^{8} + \cdots - 573488655962320 ) / 21344035382768 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 13640578801 \nu^{14} - 205507917882 \nu^{12} + 2374377814083 \nu^{10} + \cdots + 48\!\cdots\!36 ) / 170752283062144 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 3267 \nu^{14} - 37902 \nu^{12} + 492633 \nu^{10} + 624450 \nu^{8} - 1471341 \nu^{6} + \cdots - 126923328 ) / 37158464 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 16719559391 \nu^{14} + 543918658634 \nu^{12} - 9567740081797 \nu^{10} + \cdots + 47\!\cdots\!72 ) / 170752283062144 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 10775336067 \nu^{14} + 300711423882 \nu^{12} - 4761110826945 \nu^{10} + \cdots + 18\!\cdots\!44 ) / 85376141531072 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 441308025 \nu^{15} + 10288095186 \nu^{13} - 161810981131 \nu^{11} + \cdots - 10865608428608 \nu ) / 24393183294592 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 24034842351 \nu^{15} - 614713688662 \nu^{13} + 11461071464877 \nu^{11} + \cdots - 66\!\cdots\!36 \nu ) / 11\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 66561205569 \nu^{15} - 1803932697690 \nu^{13} + 31311073580403 \nu^{11} + \cdots - 24\!\cdots\!92 \nu ) / 23\!\cdots\!16 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 20335707403 \nu^{15} - 409133424622 \nu^{13} + 6641293590097 \nu^{11} + \cdots - 45\!\cdots\!40 \nu ) / 341504566124288 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 6\beta_{15} - 4\beta_{13} + 13\beta_{12} - 12\beta_{5} - 4\beta_{4} - 4\beta_{3} ) / 42 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -7\beta_{11} + 18\beta_{10} - 7\beta_{9} - 6\beta_{8} + 6\beta_{7} - 9\beta_{2} - 414\beta _1 + 405 ) / 63 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 6 \beta_{15} + 112 \beta_{14} + 192 \beta_{13} + 531 \beta_{12} + 56 \beta_{6} - 96 \beta_{5} + \cdots - 186 \beta_{3} ) / 126 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 35\beta_{11} + 30\beta_{10} - 70\beta_{9} + 60\beta_{8} + 59\beta_{7} - \beta_{2} - 1201\beta _1 - 60 ) / 21 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( - 2196 \beta_{15} + 1120 \beta_{14} + 2808 \beta_{13} - 936 \beta_{12} - 1120 \beta_{6} + \cdots + 162 \beta_{3} ) / 126 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 2702 \beta_{11} - 2364 \beta_{10} - 1351 \beta_{9} + 3546 \beta_{8} - 249 \beta_{7} + 2862 \beta_{2} + \cdots - 36537 ) / 63 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 23370 \beta_{15} - 16520 \beta_{14} + 10176 \beta_{13} - 100419 \beta_{12} - 33040 \beta_{6} + \cdots + 26388 \beta_{3} ) / 126 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 5607 \beta_{11} - 15042 \beta_{10} + 5607 \beta_{9} + 5014 \beta_{8} - 13274 \beta_{7} + 11651 \beta_{2} + \cdots - 125127 ) / 21 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 14054 \beta_{15} - 146720 \beta_{14} - 78352 \beta_{13} - 378345 \beta_{12} - 73360 \beta_{6} + \cdots + 92406 \beta_{3} ) / 42 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 207683 \beta_{11} - 188838 \beta_{10} + 415366 \beta_{9} - 377676 \beta_{8} - 448317 \beta_{7} + \cdots + 377676 ) / 63 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 3894420 \beta_{15} - 2812376 \beta_{14} - 4206672 \beta_{13} + 1402224 \beta_{12} + \cdots - 544986 \beta_{3} ) / 126 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 5118946 \beta_{11} + 4706196 \beta_{10} + 2559473 \beta_{9} - 7059294 \beta_{8} + 932733 \beta_{7} + \cdots + 56875641 ) / 63 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 40877778 \beta_{15} + 35280896 \beta_{14} - 17013960 \beta_{13} + 193160979 \beta_{12} + \cdots - 47727636 \beta_{3} ) / 126 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 10514133 \beta_{11} + 29197230 \beta_{10} - 10514133 \beta_{9} - 9732410 \beta_{8} + \cdots + 211338591 ) / 21 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 85151466 \beta_{15} + 877655632 \beta_{14} + 416411712 \beta_{13} + 2182771107 \beta_{12} + \cdots - 501563178 \beta_{3} ) / 126 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\)
\(\chi(n)\) \(-1\) \(\beta_{1}\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
17.1
2.03494 − 1.17487i
−2.17636 + 1.25652i
−1.16892 + 0.674875i
3.04238 − 1.75652i
−3.04238 + 1.75652i
1.16892 − 0.674875i
2.17636 − 1.25652i
−2.03494 + 1.17487i
2.03494 + 1.17487i
−2.17636 − 1.25652i
−1.16892 − 0.674875i
3.04238 + 1.75652i
−3.04238 − 1.75652i
1.16892 + 0.674875i
2.17636 + 1.25652i
−2.03494 − 1.17487i
−1.73205 + 1.00000i 0 2.00000 − 3.46410i −9.88140 − 17.1151i 0 −6.68600 + 17.2713i 8.00000i 0 34.2302 + 19.7628i
17.2 −1.73205 + 1.00000i 0 2.00000 − 3.46410i −3.62059 − 6.27105i 0 12.4914 + 13.6735i 8.00000i 0 12.5421 + 7.24119i
17.3 −1.73205 + 1.00000i 0 2.00000 − 3.46410i 2.38434 + 4.12981i 0 −3.43532 − 18.1989i 8.00000i 0 −8.25961 − 4.76869i
17.4 −1.73205 + 1.00000i 0 2.00000 − 3.46410i 5.92150 + 10.2563i 0 −18.3701 − 2.35362i 8.00000i 0 −20.5127 − 11.8430i
17.5 1.73205 − 1.00000i 0 2.00000 − 3.46410i −5.92150 − 10.2563i 0 −18.3701 − 2.35362i − 8.00000i 0 −20.5127 − 11.8430i
17.6 1.73205 − 1.00000i 0 2.00000 − 3.46410i −2.38434 − 4.12981i 0 −3.43532 − 18.1989i − 8.00000i 0 −8.25961 − 4.76869i
17.7 1.73205 − 1.00000i 0 2.00000 − 3.46410i 3.62059 + 6.27105i 0 12.4914 + 13.6735i − 8.00000i 0 12.5421 + 7.24119i
17.8 1.73205 − 1.00000i 0 2.00000 − 3.46410i 9.88140 + 17.1151i 0 −6.68600 + 17.2713i − 8.00000i 0 34.2302 + 19.7628i
89.1 −1.73205 − 1.00000i 0 2.00000 + 3.46410i −9.88140 + 17.1151i 0 −6.68600 − 17.2713i − 8.00000i 0 34.2302 − 19.7628i
89.2 −1.73205 − 1.00000i 0 2.00000 + 3.46410i −3.62059 + 6.27105i 0 12.4914 − 13.6735i − 8.00000i 0 12.5421 − 7.24119i
89.3 −1.73205 − 1.00000i 0 2.00000 + 3.46410i 2.38434 − 4.12981i 0 −3.43532 + 18.1989i − 8.00000i 0 −8.25961 + 4.76869i
89.4 −1.73205 − 1.00000i 0 2.00000 + 3.46410i 5.92150 − 10.2563i 0 −18.3701 + 2.35362i − 8.00000i 0 −20.5127 + 11.8430i
89.5 1.73205 + 1.00000i 0 2.00000 + 3.46410i −5.92150 + 10.2563i 0 −18.3701 + 2.35362i 8.00000i 0 −20.5127 + 11.8430i
89.6 1.73205 + 1.00000i 0 2.00000 + 3.46410i −2.38434 + 4.12981i 0 −3.43532 + 18.1989i 8.00000i 0 −8.25961 + 4.76869i
89.7 1.73205 + 1.00000i 0 2.00000 + 3.46410i 3.62059 − 6.27105i 0 12.4914 − 13.6735i 8.00000i 0 12.5421 − 7.24119i
89.8 1.73205 + 1.00000i 0 2.00000 + 3.46410i 9.88140 − 17.1151i 0 −6.68600 − 17.2713i 8.00000i 0 34.2302 − 19.7628i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 17.8
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
7.d odd 6 1 inner
21.g even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 126.4.k.a ✓ 16
3.b odd 2 1 inner 126.4.k.a ✓ 16
4.b odd 2 1 1008.4.bt.c 16
7.b odd 2 1 882.4.k.a 16
7.c even 3 1 882.4.d.c 16
7.c even 3 1 882.4.k.a 16
7.d odd 6 1 inner 126.4.k.a ✓ 16
7.d odd 6 1 882.4.d.c 16
12.b even 2 1 1008.4.bt.c 16
21.c even 2 1 882.4.k.a 16
21.g even 6 1 inner 126.4.k.a ✓ 16
21.g even 6 1 882.4.d.c 16
21.h odd 6 1 882.4.d.c 16
21.h odd 6 1 882.4.k.a 16
28.f even 6 1 1008.4.bt.c 16
84.j odd 6 1 1008.4.bt.c 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
126.4.k.a ✓ 16 1.a even 1 1 trivial
126.4.k.a ✓ 16 3.b odd 2 1 inner
126.4.k.a ✓ 16 7.d odd 6 1 inner
126.4.k.a ✓ 16 21.g even 6 1 inner
882.4.d.c 16 7.c even 3 1
882.4.d.c 16 7.d odd 6 1
882.4.d.c 16 21.g even 6 1
882.4.d.c 16 21.h odd 6 1
882.4.k.a 16 7.b odd 2 1
882.4.k.a 16 7.c even 3 1
882.4.k.a 16 21.c even 2 1
882.4.k.a 16 21.h odd 6 1
1008.4.bt.c 16 4.b odd 2 1
1008.4.bt.c 16 12.b even 2 1
1008.4.bt.c 16 28.f even 6 1
1008.4.bt.c 16 84.j odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{4}^{\mathrm{new}}(126, [\chi])\).

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 4 T^{2} + 16)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 42\!\cdots\!76 \) Copy content Toggle raw display
$7$ \( (T^{8} + 32 T^{7} + \cdots + 13841287201)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 51\!\cdots\!36 \) Copy content Toggle raw display
$13$ \( (T^{8} + 12498 T^{6} + \cdots + 138656437956)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 55\!\cdots\!56 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 95\!\cdots\!84)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 84\!\cdots\!76 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 11\!\cdots\!16)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 35\!\cdots\!41)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 38\!\cdots\!64)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 399206154260736)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 278 T^{3} + \cdots - 863420834)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 35\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 38\!\cdots\!36 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 57\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 80\!\cdots\!36)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 34\!\cdots\!36)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 21\!\cdots\!36)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 11\!\cdots\!64)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 39\!\cdots\!81)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 37\!\cdots\!36)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 33\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 17\!\cdots\!44)^{2} \) Copy content Toggle raw display
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