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Properties

Label 42.4.f.a
Level $42$
Weight $4$
Character orbit 42.f
Analytic conductor $2.478$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [42,4,Mod(5,42)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("42.5"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(42, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([3, 5])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 42 = 2 \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 42.f (of order \(6\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(2.47808022024\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} - x^{14} - 2 x^{13} + 9 x^{12} - 24 x^{11} + 714 x^{10} - 1940 x^{9} - 2834 x^{8} + \cdots + 43046721 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{8} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content 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.

\(f(q)\) \(=\) \( q - \beta_{2} q^{2} + \beta_1 q^{3} + (4 \beta_{5} + 4) q^{4} + ( - \beta_{15} - \beta_{11} + \cdots - \beta_{2}) q^{5} + (\beta_{5} - \beta_{4}) q^{6} + ( - \beta_{14} + \beta_{8} - \beta_{6} + \cdots + 5) q^{7}+ \cdots + (6 \beta_{15} - 48 \beta_{14} + \cdots - 297) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 32 q^{4} + 80 q^{7} + 18 q^{9} - 36 q^{10} - 128 q^{16} - 48 q^{18} - 342 q^{19} - 450 q^{21} + 24 q^{22} - 48 q^{24} - 194 q^{25} + 88 q^{28} + 360 q^{30} + 804 q^{31} + 1332 q^{33} + 144 q^{36}+ \cdots - 4284 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 2 x^{15} - x^{14} - 2 x^{13} + 9 x^{12} - 24 x^{11} + 714 x^{10} - 1940 x^{9} - 2834 x^{8} + \cdots + 43046721 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 11783 \nu^{15} - 129119 \nu^{14} - 517471 \nu^{13} + 606928 \nu^{12} - 14603832 \nu^{11} + \cdots + 756665695800 ) / 802467406944 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 99716 \nu^{15} - 525581 \nu^{14} - 4176229 \nu^{13} + 461611 \nu^{12} + \cdots - 74485176237 ) / 4413570738192 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 212507 \nu^{15} - 5626388 \nu^{14} - 25621768 \nu^{13} - 99499205 \nu^{12} + \cdots - 21889683312741 ) / 8827141476384 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 2357 \nu^{15} - 76480 \nu^{14} + 402562 \nu^{13} + 93215 \nu^{12} - 52902 \nu^{11} + \cdots + 475470165321 ) / 90072872208 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 1025 \nu^{15} - 7522 \nu^{14} - 1448 \nu^{13} + 4529 \nu^{12} + 66042 \nu^{11} + \cdots - 60620943429 ) / 30024290736 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 38623 \nu^{15} + 1198056 \nu^{14} - 5469812 \nu^{13} + 5048117 \nu^{12} + \cdots - 9719194424139 ) / 980793497376 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 286331 \nu^{15} - 1172447 \nu^{14} - 3463231 \nu^{13} - 17257904 \nu^{12} + \cdots + 545449784760 ) / 4413570738192 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 80557 \nu^{15} - 475105 \nu^{14} + 29131 \nu^{13} + 26780 \nu^{12} + 12598524 \nu^{11} + \cdots + 476938536804 ) / 326931165792 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 370944 \nu^{15} - 1248091 \nu^{14} - 538303 \nu^{13} - 14904299 \nu^{12} + \cdots + 5989636082637 ) / 980793497376 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( - 193901 \nu^{15} - 416618 \nu^{14} - 1981174 \nu^{13} + 5790835 \nu^{12} + \cdots + 1369512296739 ) / 326931165792 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( 5816395 \nu^{15} + 4462603 \nu^{14} - 58985857 \nu^{13} - 70253912 \nu^{12} + \cdots - 185904439092000 ) / 8827141476384 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( - 3030617 \nu^{15} - 6794630 \nu^{14} + 2799530 \nu^{13} - 11951165 \nu^{12} + \cdots + 2314040260275 ) / 2942380492128 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 1111135 \nu^{15} - 1280964 \nu^{14} + 6759296 \nu^{13} + 6088015 \nu^{12} + \cdots - 890478819513 ) / 980793497376 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( - 28607 \nu^{15} + 29539 \nu^{14} + 231701 \nu^{13} + 96310 \nu^{12} - 379746 \nu^{11} + \cdots + 576366896376 ) / 22518218052 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 4503491 \nu^{15} - 4051820 \nu^{14} + 29323772 \nu^{13} + 95485939 \nu^{12} + \cdots + 55258803118467 ) / 2942380492128 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{13} - \beta_{12} + \beta_{9} + \beta_{8} + \beta_{6} - 2\beta_{5} ) / 6 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{15} - \beta_{13} + \beta_{11} - \beta_{10} + \beta_{9} - \beta_{8} + \beta_{6} + 2 \beta_{5} + \cdots + 2 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 2 \beta_{15} + 2 \beta_{12} + 4 \beta_{11} - 6 \beta_{10} + 2 \beta_{9} - 2 \beta_{8} + 27 \beta_{7} + \cdots + 8 ) / 6 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - \beta_{15} + \beta_{14} - \beta_{13} - 2 \beta_{12} + \beta_{11} + 7 \beta_{10} - \beta_{9} + \cdots + 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 32 \beta_{15} - 13 \beta_{14} + 4 \beta_{13} - 12 \beta_{12} + 16 \beta_{11} + 4 \beta_{10} + 8 \beta_{9} + \cdots + 123 ) / 6 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 56 \beta_{15} - 16 \beta_{14} + 12 \beta_{13} + 66 \beta_{12} - 112 \beta_{11} + 28 \beta_{10} + \cdots - 715 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 368 \beta_{15} - 704 \beta_{14} - 9 \beta_{13} + 261 \beta_{12} - 368 \beta_{11} - 372 \beta_{10} + \cdots - 704 ) / 6 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 642 \beta_{15} + 682 \beta_{14} + 259 \beta_{13} - 262 \beta_{12} - 321 \beta_{11} + 259 \beta_{10} + \cdots + 10092 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 370 \beta_{15} - 1520 \beta_{14} + 2312 \beta_{13} + 1554 \beta_{12} - 740 \beta_{11} + 814 \beta_{10} + \cdots + 80552 ) / 6 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( 4273 \beta_{15} - 2997 \beta_{14} + 7897 \beta_{13} - 8840 \beta_{12} - 4273 \beta_{11} - 6513 \beta_{10} + \cdots - 2997 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 40896 \beta_{15} + 10137 \beta_{14} - 39840 \beta_{13} + 16424 \beta_{12} + 20448 \beta_{11} + \cdots + 208345 ) / 6 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 31536 \beta_{15} - 20440 \beta_{14} + 13868 \beta_{13} - 14848 \beta_{12} + 63072 \beta_{11} + \cdots + 622547 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 40992 \beta_{15} + 224312 \beta_{14} + 210401 \beta_{13} - 313729 \beta_{12} - 40992 \beta_{11} + \cdots + 224312 ) / 6 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 432322 \beta_{15} - 100872 \beta_{14} - 339137 \beta_{13} + 382692 \beta_{12} + 216161 \beta_{11} + \cdots - 4241030 ) / 3 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 2937310 \beta_{15} - 3314576 \beta_{14} - 576456 \beta_{13} + 1478274 \beta_{12} - 5874620 \beta_{11} + \cdots - 16054616 ) / 6 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(29\) \(31\)
\(\chi(n)\) \(-1\) \(-\beta_{5}\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
5.1
2.41164 − 1.78437i
2.30541 + 1.91966i
−1.62928 − 2.51902i
−2.58777 + 1.51770i
−2.81518 − 1.03671i
−0.0204843 + 2.99993i
0.339489 − 2.98073i
2.99617 + 0.151487i
2.41164 + 1.78437i
2.30541 − 1.91966i
−1.62928 + 2.51902i
−2.58777 − 1.51770i
−2.81518 + 1.03671i
−0.0204843 − 2.99993i
0.339489 + 2.98073i
2.99617 − 0.151487i
−1.73205 − 1.00000i −4.17709 − 3.09062i 2.00000 + 3.46410i −4.27911 + 7.41164i 4.14431 + 9.53020i −6.41772 + 17.3728i − 8.00000i 7.89612 + 25.8196i 14.8233 − 8.55823i
5.2 −1.73205 − 1.00000i −3.99309 + 3.32495i 2.00000 + 3.46410i 9.90442 − 17.1550i 10.2412 − 1.76589i 18.4277 − 1.84901i − 8.00000i 4.88947 − 26.5536i −34.3099 + 19.8088i
5.3 −1.73205 − 1.00000i 2.82199 − 4.36307i 2.00000 + 3.46410i 2.24534 − 3.88904i −9.25090 + 4.73506i −9.71288 − 15.7690i − 8.00000i −11.0727 − 24.6251i −7.77808 + 4.49068i
5.4 −1.73205 − 1.00000i 4.48216 + 2.62874i 2.00000 + 3.46410i −5.27257 + 9.13236i −5.13458 − 9.03527i 17.7029 + 5.44135i − 8.00000i 13.1794 + 23.5649i 18.2647 − 10.5451i
5.5 1.73205 + 1.00000i −4.87603 + 1.79564i 2.00000 + 3.46410i −9.90442 + 17.1550i −10.2412 − 1.76589i 18.4277 − 1.84901i 8.00000i 20.5514 − 17.5112i −34.3099 + 19.8088i
5.6 1.73205 + 1.00000i −0.0354799 − 5.19603i 2.00000 + 3.46410i 5.27257 − 9.13236i 5.13458 − 9.03527i 17.7029 + 5.44135i 8.00000i −26.9975 + 0.368709i 18.2647 − 10.5451i
5.7 1.73205 + 1.00000i 0.588012 + 5.16277i 2.00000 + 3.46410i 4.27911 − 7.41164i −4.14431 + 9.53020i −6.41772 + 17.3728i 8.00000i −26.3085 + 6.07155i 14.8233 − 8.55823i
5.8 1.73205 + 1.00000i 5.18952 − 0.262384i 2.00000 + 3.46410i −2.24534 + 3.88904i 9.25090 + 4.73506i −9.71288 − 15.7690i 8.00000i 26.8623 − 2.72329i −7.77808 + 4.49068i
17.1 −1.73205 + 1.00000i −4.17709 + 3.09062i 2.00000 − 3.46410i −4.27911 − 7.41164i 4.14431 − 9.53020i −6.41772 − 17.3728i 8.00000i 7.89612 − 25.8196i 14.8233 + 8.55823i
17.2 −1.73205 + 1.00000i −3.99309 − 3.32495i 2.00000 − 3.46410i 9.90442 + 17.1550i 10.2412 + 1.76589i 18.4277 + 1.84901i 8.00000i 4.88947 + 26.5536i −34.3099 − 19.8088i
17.3 −1.73205 + 1.00000i 2.82199 + 4.36307i 2.00000 − 3.46410i 2.24534 + 3.88904i −9.25090 − 4.73506i −9.71288 + 15.7690i 8.00000i −11.0727 + 24.6251i −7.77808 − 4.49068i
17.4 −1.73205 + 1.00000i 4.48216 − 2.62874i 2.00000 − 3.46410i −5.27257 − 9.13236i −5.13458 + 9.03527i 17.7029 − 5.44135i 8.00000i 13.1794 − 23.5649i 18.2647 + 10.5451i
17.5 1.73205 − 1.00000i −4.87603 − 1.79564i 2.00000 − 3.46410i −9.90442 − 17.1550i −10.2412 + 1.76589i 18.4277 + 1.84901i − 8.00000i 20.5514 + 17.5112i −34.3099 − 19.8088i
17.6 1.73205 − 1.00000i −0.0354799 + 5.19603i 2.00000 − 3.46410i 5.27257 + 9.13236i 5.13458 + 9.03527i 17.7029 − 5.44135i − 8.00000i −26.9975 − 0.368709i 18.2647 + 10.5451i
17.7 1.73205 − 1.00000i 0.588012 − 5.16277i 2.00000 − 3.46410i 4.27911 + 7.41164i −4.14431 − 9.53020i −6.41772 − 17.3728i − 8.00000i −26.3085 − 6.07155i 14.8233 + 8.55823i
17.8 1.73205 − 1.00000i 5.18952 + 0.262384i 2.00000 − 3.46410i −2.24534 − 3.88904i 9.25090 − 4.73506i −9.71288 + 15.7690i − 8.00000i 26.8623 + 2.72329i −7.77808 − 4.49068i
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 5.8
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 42.4.f.a ✓ 16
3.b odd 2 1 inner 42.4.f.a ✓ 16
4.b odd 2 1 336.4.bc.e 16
7.b odd 2 1 294.4.f.a 16
7.c even 3 1 294.4.d.a 16
7.c even 3 1 294.4.f.a 16
7.d odd 6 1 inner 42.4.f.a ✓ 16
7.d odd 6 1 294.4.d.a 16
12.b even 2 1 336.4.bc.e 16
21.c even 2 1 294.4.f.a 16
21.g even 6 1 inner 42.4.f.a ✓ 16
21.g even 6 1 294.4.d.a 16
21.h odd 6 1 294.4.d.a 16
21.h odd 6 1 294.4.f.a 16
28.f even 6 1 336.4.bc.e 16
84.j odd 6 1 336.4.bc.e 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
42.4.f.a ✓ 16 1.a even 1 1 trivial
42.4.f.a ✓ 16 3.b odd 2 1 inner
42.4.f.a ✓ 16 7.d odd 6 1 inner
42.4.f.a ✓ 16 21.g even 6 1 inner
294.4.d.a 16 7.c even 3 1
294.4.d.a 16 7.d odd 6 1
294.4.d.a 16 21.g even 6 1
294.4.d.a 16 21.h odd 6 1
294.4.f.a 16 7.b odd 2 1
294.4.f.a 16 7.c even 3 1
294.4.f.a 16 21.c even 2 1
294.4.f.a 16 21.h odd 6 1
336.4.bc.e 16 4.b odd 2 1
336.4.bc.e 16 12.b even 2 1
336.4.bc.e 16 28.f even 6 1
336.4.bc.e 16 84.j odd 6 1

Hecke kernels

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

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( (T^{4} - 4 T^{2} + 16)^{4} \) Copy content Toggle raw display
$3$ \( T^{16} + \cdots + 282429536481 \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 41\!\cdots\!56 \) Copy content Toggle raw display
$7$ \( (T^{8} - 40 T^{7} + \cdots + 13841287201)^{2} \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 24\!\cdots\!16 \) Copy content Toggle raw display
$13$ \( (T^{8} + 4551 T^{6} + \cdots + 16498888704)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 87\!\cdots\!00 \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 69773043356676)^{2} \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 40\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 20\!\cdots\!96)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 81\!\cdots\!01)^{2} \) Copy content Toggle raw display
$37$ \( (T^{8} + \cdots + 81\!\cdots\!56)^{2} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 15\!\cdots\!76)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} - 433 T^{3} + \cdots + 966156928)^{4} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 20\!\cdots\!76 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 65\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 11\!\cdots\!36 \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 25\!\cdots\!44)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 30\!\cdots\!04)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 18\!\cdots\!16)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 13\!\cdots\!64)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 11\!\cdots\!61)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 16\!\cdots\!56)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 18\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 10\!\cdots\!36)^{2} \) Copy content Toggle raw display
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