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Properties

Label 756.2.i.b
Level $756$
Weight $2$
Character orbit 756.i
Analytic conductor $6.037$
Analytic rank $0$
Dimension $14$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [756,2,Mod(37,756)] 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, 2, 2])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("756.37"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 756.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.03669039281\)
Analytic rank: \(0\)
Dimension: \(14\)
Relative dimension: \(7\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{14} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{14} - 5x^{12} - 3x^{11} + 7x^{10} + 30x^{9} - 117x^{7} + 270x^{5} + 189x^{4} - 243x^{3} - 1215x^{2} + 2187 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{7} \)
Twist minimal: no (minimal twist has level 252)
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 \(1,\beta_1,\ldots,\beta_{13}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q - \beta_{3} q^{5} + (\beta_{12} + \beta_{5}) q^{7} + \beta_{13} q^{11} - \beta_{11} q^{13} + (\beta_{12} + \beta_{9} - \beta_{6} + \cdots - \beta_1) q^{17} + ( - \beta_{13} - \beta_{7} - \beta_1 + 1) q^{19}+ \cdots + ( - \beta_{12} + \beta_{10} + \cdots - 2 \beta_1) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 14 q + 2 q^{5} + 6 q^{7} - 2 q^{11} + 2 q^{13} - 2 q^{17} + 7 q^{19} - 11 q^{23} - 9 q^{25} - q^{29} + 2 q^{31} + 19 q^{35} + 10 q^{37} + 33 q^{41} + 7 q^{43} - 6 q^{47} - 4 q^{49} + 15 q^{53} - 28 q^{55}+ \cdots - 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{14} - 5x^{12} - 3x^{11} + 7x^{10} + 30x^{9} - 117x^{7} + 270x^{5} + 189x^{4} - 243x^{3} - 1215x^{2} + 2187 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 35 \nu^{13} + 72 \nu^{12} + 157 \nu^{11} + 312 \nu^{10} - 290 \nu^{9} - 1383 \nu^{8} + \cdots - 3645 ) / 43011 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 10 \nu^{13} - 45 \nu^{12} + 196 \nu^{11} + 249 \nu^{10} + 136 \nu^{9} + 84 \nu^{8} - 2274 \nu^{7} + \cdots + 140940 ) / 14337 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 26 \nu^{13} + 7 \nu^{12} - 70 \nu^{11} - 266 \nu^{10} - 301 \nu^{9} + 955 \nu^{8} + 846 \nu^{7} + \cdots - 30618 ) / 14337 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( 25 \nu^{13} + 294 \nu^{12} + 109 \nu^{11} - 924 \nu^{10} - 1715 \nu^{9} - 351 \nu^{8} + \cdots - 218700 ) / 43011 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 56 \nu^{13} - 441 \nu^{12} + 62 \nu^{11} + 1227 \nu^{10} + 761 \nu^{9} - 165 \nu^{8} - 8622 \nu^{7} + \cdots + 293058 ) / 43011 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 44 \nu^{13} + 78 \nu^{12} + 145 \nu^{11} - 150 \nu^{10} - 392 \nu^{9} - 387 \nu^{8} + \cdots - 48114 ) / 14337 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 73 \nu^{13} - 360 \nu^{12} - 40 \nu^{11} + 804 \nu^{10} + 1703 \nu^{9} - 417 \nu^{8} + \cdots + 239841 ) / 43011 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 73 \nu^{13} + 69 \nu^{12} - 28 \nu^{11} + 36 \nu^{10} - 373 \nu^{9} - 1446 \nu^{8} + \cdots - 121257 ) / 14337 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 142 \nu^{13} - 351 \nu^{12} + 386 \nu^{11} + 1398 \nu^{10} + 869 \nu^{9} - 2478 \nu^{8} + \cdots + 187353 ) / 43011 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 49 \nu^{13} + 125 \nu^{12} - 77 \nu^{11} - 313 \nu^{10} - 368 \nu^{9} + 140 \nu^{8} + 2352 \nu^{7} + \cdots - 68283 ) / 14337 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 281 \nu^{13} - 6 \nu^{12} + 208 \nu^{11} + 657 \nu^{10} - 500 \nu^{9} - 6555 \nu^{8} + \cdots - 334611 ) / 43011 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 179 \nu^{13} + 471 \nu^{12} - 769 \nu^{11} - 2352 \nu^{10} - 763 \nu^{9} + 5184 \nu^{8} + \cdots - 324405 ) / 43011 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( 443 \nu^{13} + 546 \nu^{12} - 1099 \nu^{11} - 2736 \nu^{10} - 2092 \nu^{9} + 8283 \nu^{8} + \cdots - 414072 ) / 43011 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{12} - \beta_{10} + \beta_{5} + \beta_{4} - \beta_{3} + \beta _1 - 1 ) / 3 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( 2 \beta_{13} - 2 \beta_{12} + 2 \beta_{11} - \beta_{10} - \beta_{9} - \beta_{8} + \beta_{7} - 2 \beta_{5} + \cdots + 4 ) / 3 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 4\beta_{12} + 2\beta_{10} + 3\beta_{9} + 3\beta_{7} + 3\beta_{6} + 4\beta_{5} + \beta_{4} - \beta_{3} + 4\beta _1 - 4 ) / 3 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( 2 \beta_{13} + 2 \beta_{11} - 3 \beta_{10} - \beta_{9} - 4 \beta_{7} + 2 \beta_{6} + 2 \beta_{5} + \cdots + 9 ) / 3 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 3 \beta_{13} + 3 \beta_{12} + 3 \beta_{11} - 3 \beta_{10} + 3 \beta_{9} + 4 \beta_{8} + \beta_{7} + \cdots - 11 ) / 3 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 8 \beta_{13} - 19 \beta_{12} - 7 \beta_{11} + 7 \beta_{10} + 5 \beta_{9} + \beta_{8} + 9 \beta_{7} + \cdots + 23 ) / 3 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 21 \beta_{13} + 32 \beta_{12} - 12 \beta_{11} + 7 \beta_{10} + 39 \beta_{9} + 2 \beta_{8} - 34 \beta_{7} + \cdots + 18 ) / 3 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 17 \beta_{13} - 26 \beta_{12} - 22 \beta_{11} - 85 \beta_{10} - 13 \beta_{9} - \beta_{8} - 32 \beta_{7} + \cdots - 50 ) / 3 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( - 9 \beta_{13} + \beta_{12} + 18 \beta_{11} + 86 \beta_{10} + 75 \beta_{9} - 69 \beta_{8} + 42 \beta_{7} + \cdots - 115 ) / 3 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 16 \beta_{13} + 72 \beta_{12} - 79 \beta_{11} - 48 \beta_{10} + 62 \beta_{9} + 9 \beta_{8} + \cdots - 306 ) / 3 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( - 150 \beta_{13} + 57 \beta_{12} + 12 \beta_{11} + 105 \beta_{10} + 66 \beta_{9} - 122 \beta_{8} + \cdots - 650 ) / 3 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( 134 \beta_{13} - 280 \beta_{12} - 97 \beta_{11} - 56 \beta_{10} - 328 \beta_{9} + 199 \beta_{8} + \cdots - 598 ) / 3 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( - 363 \beta_{13} - 247 \beta_{12} - 516 \beta_{11} + 1393 \beta_{10} + 426 \beta_{9} - 196 \beta_{8} + \cdots - 1089 ) / 3 \) Copy content Toggle raw display

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

Character values

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

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

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.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
37.1
−1.58203 + 0.705117i
−0.473632 − 1.66604i
1.13119 + 1.31165i
1.64515 − 0.541745i
−1.73040 − 0.0755709i
1.68442 + 0.403398i
−0.674693 + 1.59524i
−1.58203 − 0.705117i
−0.473632 + 1.66604i
1.13119 − 1.31165i
1.64515 + 0.541745i
−1.73040 + 0.0755709i
1.68442 − 0.403398i
−0.674693 − 1.59524i
0 0 0 −1.26013 − 2.18261i 0 0.527655 + 2.59260i 0 0 0
37.2 0 0 0 −0.951504 − 1.64805i 0 2.11495 − 1.58965i 0 0 0
37.3 0 0 0 −0.764702 − 1.32450i 0 −1.91978 + 1.82056i 0 0 0
37.4 0 0 0 −0.381918 − 0.661502i 0 2.62892 + 0.297968i 0 0 0
37.5 0 0 0 0.483929 + 0.838189i 0 −1.52054 − 2.16517i 0 0 0
37.6 0 0 0 1.80173 + 3.12069i 0 −1.02133 − 2.44067i 0 0 0
37.7 0 0 0 2.07260 + 3.58985i 0 2.19013 + 1.48437i 0 0 0
613.1 0 0 0 −1.26013 + 2.18261i 0 0.527655 − 2.59260i 0 0 0
613.2 0 0 0 −0.951504 + 1.64805i 0 2.11495 + 1.58965i 0 0 0
613.3 0 0 0 −0.764702 + 1.32450i 0 −1.91978 − 1.82056i 0 0 0
613.4 0 0 0 −0.381918 + 0.661502i 0 2.62892 − 0.297968i 0 0 0
613.5 0 0 0 0.483929 − 0.838189i 0 −1.52054 + 2.16517i 0 0 0
613.6 0 0 0 1.80173 − 3.12069i 0 −1.02133 + 2.44067i 0 0 0
613.7 0 0 0 2.07260 − 3.58985i 0 2.19013 − 1.48437i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 37.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.h even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 756.2.i.b 14
3.b odd 2 1 252.2.i.b ✓ 14
4.b odd 2 1 3024.2.q.j 14
7.b odd 2 1 5292.2.i.i 14
7.c even 3 1 756.2.l.b 14
7.c even 3 1 5292.2.j.h 14
7.d odd 6 1 5292.2.j.g 14
7.d odd 6 1 5292.2.l.i 14
9.c even 3 1 756.2.l.b 14
9.c even 3 1 2268.2.k.f 14
9.d odd 6 1 252.2.l.b yes 14
9.d odd 6 1 2268.2.k.e 14
12.b even 2 1 1008.2.q.j 14
21.c even 2 1 1764.2.i.i 14
21.g even 6 1 1764.2.j.h 14
21.g even 6 1 1764.2.l.i 14
21.h odd 6 1 252.2.l.b yes 14
21.h odd 6 1 1764.2.j.g 14
28.g odd 6 1 3024.2.t.j 14
36.f odd 6 1 3024.2.t.j 14
36.h even 6 1 1008.2.t.j 14
63.g even 3 1 2268.2.k.f 14
63.g even 3 1 5292.2.j.h 14
63.h even 3 1 inner 756.2.i.b 14
63.i even 6 1 1764.2.i.i 14
63.j odd 6 1 252.2.i.b ✓ 14
63.k odd 6 1 5292.2.j.g 14
63.l odd 6 1 5292.2.l.i 14
63.n odd 6 1 1764.2.j.g 14
63.n odd 6 1 2268.2.k.e 14
63.o even 6 1 1764.2.l.i 14
63.s even 6 1 1764.2.j.h 14
63.t odd 6 1 5292.2.i.i 14
84.n even 6 1 1008.2.t.j 14
252.u odd 6 1 3024.2.q.j 14
252.bb even 6 1 1008.2.q.j 14
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
252.2.i.b ✓ 14 3.b odd 2 1
252.2.i.b ✓ 14 63.j odd 6 1
252.2.l.b yes 14 9.d odd 6 1
252.2.l.b yes 14 21.h odd 6 1
756.2.i.b 14 1.a even 1 1 trivial
756.2.i.b 14 63.h even 3 1 inner
756.2.l.b 14 7.c even 3 1
756.2.l.b 14 9.c even 3 1
1008.2.q.j 14 12.b even 2 1
1008.2.q.j 14 252.bb even 6 1
1008.2.t.j 14 36.h even 6 1
1008.2.t.j 14 84.n even 6 1
1764.2.i.i 14 21.c even 2 1
1764.2.i.i 14 63.i even 6 1
1764.2.j.g 14 21.h odd 6 1
1764.2.j.g 14 63.n odd 6 1
1764.2.j.h 14 21.g even 6 1
1764.2.j.h 14 63.s even 6 1
1764.2.l.i 14 21.g even 6 1
1764.2.l.i 14 63.o even 6 1
2268.2.k.e 14 9.d odd 6 1
2268.2.k.e 14 63.n odd 6 1
2268.2.k.f 14 9.c even 3 1
2268.2.k.f 14 63.g even 3 1
3024.2.q.j 14 4.b odd 2 1
3024.2.q.j 14 252.u odd 6 1
3024.2.t.j 14 28.g odd 6 1
3024.2.t.j 14 36.f odd 6 1
5292.2.i.i 14 7.b odd 2 1
5292.2.i.i 14 63.t odd 6 1
5292.2.j.g 14 7.d odd 6 1
5292.2.j.g 14 63.k odd 6 1
5292.2.j.h 14 7.c even 3 1
5292.2.j.h 14 63.g even 3 1
5292.2.l.i 14 7.d odd 6 1
5292.2.l.i 14 63.l odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{5}^{14} - 2 T_{5}^{13} + 24 T_{5}^{12} + 16 T_{5}^{11} + 295 T_{5}^{10} + 357 T_{5}^{9} + 2670 T_{5}^{8} + \cdots + 6561 \) acting on \(S_{2}^{\mathrm{new}}(756, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{14} \) Copy content Toggle raw display
$3$ \( T^{14} \) Copy content Toggle raw display
$5$ \( T^{14} - 2 T^{13} + \cdots + 6561 \) Copy content Toggle raw display
$7$ \( T^{14} - 6 T^{13} + \cdots + 823543 \) Copy content Toggle raw display
$11$ \( T^{14} + 2 T^{13} + \cdots + 6561 \) Copy content Toggle raw display
$13$ \( T^{14} + \cdots + 150626529 \) Copy content Toggle raw display
$17$ \( T^{14} + 2 T^{13} + \cdots + 6561 \) Copy content Toggle raw display
$19$ \( T^{14} - 7 T^{13} + \cdots + 4084441 \) Copy content Toggle raw display
$23$ \( T^{14} + \cdots + 105822369 \) Copy content Toggle raw display
$29$ \( T^{14} + \cdots + 145660761 \) Copy content Toggle raw display
$31$ \( (T^{7} - T^{6} + \cdots - 117504)^{2} \) Copy content Toggle raw display
$37$ \( T^{14} + \cdots + 1566893056 \) Copy content Toggle raw display
$41$ \( T^{14} + \cdots + 1108290681 \) Copy content Toggle raw display
$43$ \( T^{14} - 7 T^{13} + \cdots + 4084441 \) Copy content Toggle raw display
$47$ \( (T^{7} + 3 T^{6} + \cdots + 11664)^{2} \) Copy content Toggle raw display
$53$ \( T^{14} + \cdots + 952401321 \) Copy content Toggle raw display
$59$ \( (T^{7} + 14 T^{6} + \cdots - 26244)^{2} \) Copy content Toggle raw display
$61$ \( (T^{7} - 10 T^{6} + \cdots - 12192)^{2} \) Copy content Toggle raw display
$67$ \( (T^{7} + 6 T^{6} + \cdots + 10816)^{2} \) Copy content Toggle raw display
$71$ \( (T^{7} + T^{6} - 116 T^{5} + \cdots + 972)^{2} \) Copy content Toggle raw display
$73$ \( T^{14} + \cdots + 2748590329 \) Copy content Toggle raw display
$79$ \( (T^{7} - 10 T^{6} + \cdots - 233232)^{2} \) Copy content Toggle raw display
$83$ \( T^{14} + \cdots + 901054679121 \) Copy content Toggle raw display
$89$ \( T^{14} + \cdots + 16524331209 \) Copy content Toggle raw display
$97$ \( T^{14} + \cdots + 767677849 \) Copy content Toggle raw display
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