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Properties

Label 756.4.k.h
Level $756$
Weight $4$
Character orbit 756.k
Analytic conductor $44.605$
Analytic rank $0$
Dimension $16$
Inner twists $2$

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [756,4,Mod(109,756)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(756, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 4, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("756.109"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 756.k (of order \(3\), degree \(2\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,7,0,-11] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(44.6054439643\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(\zeta_{3})\)
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: \( x^{16} + 198 x^{14} + 15603 x^{12} + 623882 x^{10} + 13296387 x^{8} + 143321550 x^{6} + 644261986 x^{4} + \cdots + 9753129 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{16} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-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.

\(f(q)\) \(=\) \( q + ( - \beta_{9} + \beta_{6} + 1) q^{5} + ( - \beta_{8} - \beta_{6} - 1) q^{7} + (\beta_{12} + 5 \beta_{6} - \beta_{2}) q^{11} + ( - \beta_{3} - 3) q^{13} + ( - \beta_{15} + \beta_{10} + \cdots - 4 \beta_{6}) q^{17}+ \cdots + (23 \beta_{11} + 23 \beta_{10} + \cdots - 67) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 7 q^{5} - 11 q^{7} - 43 q^{11} - 52 q^{13} + 38 q^{17} + 32 q^{19} - 118 q^{23} - 143 q^{25} + 160 q^{29} + 5 q^{31} + 161 q^{35} - 136 q^{37} - 216 q^{41} + 800 q^{43} - 78 q^{47} + 253 q^{49} + 105 q^{53}+ \cdots - 838 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} + 198 x^{14} + 15603 x^{12} + 623882 x^{10} + 13296387 x^{8} + 143321550 x^{6} + 644261986 x^{4} + \cdots + 9753129 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( 185453027 \nu^{14} + 27907229611 \nu^{12} + 1421923819922 \nu^{10} + 24053854230680 \nu^{8} + \cdots + 12\!\cdots\!69 ) / 30\!\cdots\!08 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 11657671505 \nu^{14} + 1758560464243 \nu^{12} + 89813203127036 \nu^{10} + \cdots - 12\!\cdots\!89 ) / 39\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 472599229 \nu^{14} - 85179069068 \nu^{12} - 5993715493588 \nu^{10} + \cdots - 83\!\cdots\!15 ) / 15\!\cdots\!04 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 1633568945 \nu^{14} + 295505272273 \nu^{12} + 20821761658082 \nu^{10} + \cdots + 27\!\cdots\!31 ) / 19\!\cdots\!24 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( - 5560560679 \nu^{14} - 933065805551 \nu^{12} - 58835089543072 \nu^{10} + \cdots + 30\!\cdots\!63 ) / 57\!\cdots\!72 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 48251363 \nu^{15} - 9516374031 \nu^{13} - 746065202406 \nu^{11} - 29634673049896 \nu^{9} + \cdots - 19\!\cdots\!96 ) / 38\!\cdots\!92 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 2266065251721 \nu^{15} - 2613754913423 \nu^{14} + 437932003859853 \nu^{13} + \cdots + 48\!\cdots\!83 ) / 27\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 2266065251721 \nu^{15} - 2613754913423 \nu^{14} - 437932003859853 \nu^{13} + \cdots + 48\!\cdots\!83 ) / 27\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( 2358957514041 \nu^{15} - 836578604797 \nu^{14} + 456271543108767 \nu^{13} + \cdots - 57\!\cdots\!59 ) / 27\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 4175931932781 \nu^{15} - 2105861859949 \nu^{14} + 847059176771661 \nu^{13} + \cdots + 32\!\cdots\!01 ) / 27\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 4175931932781 \nu^{15} - 2105861859949 \nu^{14} - 847059176771661 \nu^{13} + \cdots + 32\!\cdots\!01 ) / 27\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 5211752701899 \nu^{15} + 4045212012235 \nu^{14} + \cdots - 45\!\cdots\!83 ) / 27\!\cdots\!76 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 2005879751955 \nu^{15} - 1929514555613 \nu^{14} - 416045959758795 \nu^{13} + \cdots + 10\!\cdots\!61 ) / 39\!\cdots\!68 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 8008663928157 \nu^{15} - 2131895122019 \nu^{14} + \cdots - 37\!\cdots\!65 ) / 13\!\cdots\!88 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( 13380131425987 \nu^{15} + 3967938967405 \nu^{14} + \cdots + 66\!\cdots\!99 ) / 92\!\cdots\!92 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( 2 \beta_{14} - 2 \beta_{13} - 2 \beta_{12} + 4 \beta_{11} - 4 \beta_{10} + 4 \beta_{9} + 8 \beta_{8} + \cdots - 15 ) / 54 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( \beta_{11} + \beta_{10} + 12\beta_{8} + 12\beta_{7} - \beta_{5} - 2\beta_{3} + 2\beta_{2} + 4\beta _1 - 450 ) / 18 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 86 \beta_{14} + 98 \beta_{13} + 146 \beta_{12} - 166 \beta_{11} + 166 \beta_{10} + 140 \beta_{9} + \cdots + 1401 ) / 54 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 13 \beta_{11} + 13 \beta_{10} - 660 \beta_{8} - 660 \beta_{7} + 98 \beta_{5} + 54 \beta_{4} + \cdots + 18183 ) / 18 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 432 \beta_{15} + 3866 \beta_{14} - 5312 \beta_{13} - 8306 \beta_{12} + 9169 \beta_{11} - 9169 \beta_{10} + \cdots - 76281 ) / 54 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 963 \beta_{11} - 963 \beta_{10} + 11914 \beta_{8} + 11914 \beta_{7} - 2227 \beta_{5} - 1488 \beta_{4} + \cdots - 276800 ) / 6 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 23760 \beta_{15} - 83530 \beta_{14} + 147376 \beta_{13} + 231310 \beta_{12} - 268409 \beta_{11} + \cdots + 2107938 ) / 27 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 228368 \beta_{11} + 228368 \beta_{10} - 1959942 \beta_{8} - 1959942 \beta_{7} + 420361 \beta_{5} + \cdots + 40755357 ) / 18 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 3764880 \beta_{15} + 6951416 \beta_{14} - 16460378 \beta_{13} - 25621616 \beta_{12} + 31892503 \beta_{11} + \cdots - 243513672 ) / 54 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 14467264 \beta_{11} - 14467264 \beta_{10} + 108485688 \beta_{8} + 108485688 \beta_{7} + \cdots - 2097180597 ) / 18 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 260915688 \beta_{15} - 277264922 \beta_{14} + 923346488 \beta_{13} + 1417635494 \beta_{12} + \cdots + 14453032857 ) / 54 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 277439287 \beta_{11} + 277439287 \beta_{10} - 2014647730 \beta_{8} - 2014647730 \beta_{7} + \cdots + 37133049029 ) / 6 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 16888977792 \beta_{15} + 10371691352 \beta_{14} - 52019447762 \beta_{13} - 78575511836 \beta_{12} + \cdots - 866909089176 ) / 54 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 22814776186 \beta_{11} - 22814776186 \beta_{10} + 169236148452 \beta_{8} + 169236148452 \beta_{7} + \cdots - 3023328077307 ) / 9 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 1051844722272 \beta_{15} - 343013179694 \beta_{14} + 2942605212902 \beta_{13} + \cdots + 52045147860513 ) / 54 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/756\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(325\) \(379\)
\(\chi(n)\) \(1\) \(-1 - \beta_{6}\) \(1\)

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
109.1
2.87126i
4.58725i
− 7.59982i
− 5.69902i
− 0.121095i
6.82033i
1.20013i
− 5.52313i
− 2.87126i
− 4.58725i
7.59982i
5.69902i
0.121095i
− 6.82033i
− 1.20013i
5.52313i
0 0 0 −9.20902 − 15.9505i 0 −14.4823 + 11.5440i 0 0 0
109.2 0 0 0 −6.30299 − 10.9171i 0 1.85564 − 18.4271i 0 0 0
109.3 0 0 0 −4.37589 − 7.57927i 0 18.4405 + 1.71677i 0 0 0
109.4 0 0 0 1.79893 + 3.11584i 0 −2.37215 + 18.3677i 0 0 0
109.5 0 0 0 2.65607 + 4.60045i 0 −17.9579 − 4.52938i 0 0 0
109.6 0 0 0 3.53383 + 6.12078i 0 8.87541 − 16.2551i 0 0 0
109.7 0 0 0 7.09811 + 12.2943i 0 −15.3427 − 10.3731i 0 0 0
109.8 0 0 0 8.30096 + 14.3777i 0 15.4834 + 10.1619i 0 0 0
541.1 0 0 0 −9.20902 + 15.9505i 0 −14.4823 − 11.5440i 0 0 0
541.2 0 0 0 −6.30299 + 10.9171i 0 1.85564 + 18.4271i 0 0 0
541.3 0 0 0 −4.37589 + 7.57927i 0 18.4405 − 1.71677i 0 0 0
541.4 0 0 0 1.79893 − 3.11584i 0 −2.37215 − 18.3677i 0 0 0
541.5 0 0 0 2.65607 − 4.60045i 0 −17.9579 + 4.52938i 0 0 0
541.6 0 0 0 3.53383 − 6.12078i 0 8.87541 + 16.2551i 0 0 0
541.7 0 0 0 7.09811 − 12.2943i 0 −15.3427 + 10.3731i 0 0 0
541.8 0 0 0 8.30096 − 14.3777i 0 15.4834 − 10.1619i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 109.8
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 756.4.k.h yes 16
3.b odd 2 1 756.4.k.f ✓ 16
7.c even 3 1 inner 756.4.k.h yes 16
21.h odd 6 1 756.4.k.f ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.4.k.f ✓ 16 3.b odd 2 1
756.4.k.f ✓ 16 21.h odd 6 1
756.4.k.h yes 16 1.a even 1 1 trivial
756.4.k.h yes 16 7.c even 3 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(756, [\chi])\):

\( T_{5}^{16} - 7 T_{5}^{15} + 596 T_{5}^{14} - 4681 T_{5}^{13} + 248330 T_{5}^{12} - 1887607 T_{5}^{11} + \cdots + 41\!\cdots\!04 \) Copy content Toggle raw display
\( T_{19}^{16} - 32 T_{19}^{15} + 28953 T_{19}^{14} - 1681168 T_{19}^{13} + 618545783 T_{19}^{12} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( T^{16} \) Copy content Toggle raw display
$5$ \( T^{16} + \cdots + 41\!\cdots\!04 \) Copy content Toggle raw display
$7$ \( T^{16} + \cdots + 19\!\cdots\!01 \) Copy content Toggle raw display
$11$ \( T^{16} + \cdots + 16\!\cdots\!24 \) Copy content Toggle raw display
$13$ \( (T^{8} + 26 T^{7} + \cdots - 277849442808)^{2} \) Copy content Toggle raw display
$17$ \( T^{16} + \cdots + 26\!\cdots\!64 \) Copy content Toggle raw display
$19$ \( T^{16} + \cdots + 10\!\cdots\!16 \) Copy content Toggle raw display
$23$ \( T^{16} + \cdots + 35\!\cdots\!96 \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 80\!\cdots\!01)^{2} \) Copy content Toggle raw display
$31$ \( T^{16} + \cdots + 16\!\cdots\!64 \) Copy content Toggle raw display
$37$ \( T^{16} + \cdots + 98\!\cdots\!16 \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 50\!\cdots\!72)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots - 79\!\cdots\!92)^{2} \) Copy content Toggle raw display
$47$ \( T^{16} + \cdots + 14\!\cdots\!36 \) Copy content Toggle raw display
$53$ \( T^{16} + \cdots + 15\!\cdots\!96 \) Copy content Toggle raw display
$59$ \( T^{16} + \cdots + 19\!\cdots\!69 \) Copy content Toggle raw display
$61$ \( T^{16} + \cdots + 19\!\cdots\!44 \) Copy content Toggle raw display
$67$ \( T^{16} + \cdots + 21\!\cdots\!04 \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots - 10\!\cdots\!48)^{2} \) Copy content Toggle raw display
$73$ \( T^{16} + \cdots + 14\!\cdots\!61 \) Copy content Toggle raw display
$79$ \( T^{16} + \cdots + 10\!\cdots\!09 \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 10\!\cdots\!32)^{2} \) Copy content Toggle raw display
$89$ \( T^{16} + \cdots + 19\!\cdots\!36 \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots - 41\!\cdots\!12)^{2} \) Copy content Toggle raw display
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