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Properties

Label 756.4.f.e
Level $756$
Weight $4$
Character orbit 756.f
Analytic conductor $44.605$
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] = [756,4,Mod(377,756)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(756, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) 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("756.377"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Level: \( N \) \(=\) \( 756 = 2^{2} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 756.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0,0,0,0,14] 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: \(16\)
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: \( x^{16} - 4 x^{15} - 36 x^{14} - 780 x^{13} + 17052 x^{12} - 100012 x^{11} + 336938 x^{10} + \cdots + 1976222698452 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{8}\cdot 3^{26} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-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.

\(f(q)\) \(=\) \( q - \beta_1 q^{5} + ( - \beta_{6} + 1) q^{7} - \beta_{2} q^{11} + \beta_{10} q^{13} + ( - \beta_{7} + 2 \beta_1) q^{17} + (\beta_{12} + \beta_{6} - \beta_{3}) q^{19} - \beta_{15} q^{23} + ( - \beta_{13} - \beta_{6} - \beta_{3} + 25) q^{25}+ \cdots + (\beta_{12} - 3 \beta_{11} + \cdots - 7 \beta_{3}) q^{97}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 14 q^{7} + 388 q^{25} - 344 q^{37} - 404 q^{43} - 14 q^{49} + 1772 q^{67} - 1456 q^{79} - 4704 q^{85} + 2436 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 4 x^{15} - 36 x^{14} - 780 x^{13} + 17052 x^{12} - 100012 x^{11} + 336938 x^{10} + \cdots + 1976222698452 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 50\!\cdots\!71 \nu^{15} + \cdots - 32\!\cdots\!44 ) / 20\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( 53\!\cdots\!48 \nu^{15} + \cdots - 25\!\cdots\!28 ) / 16\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( - 26\!\cdots\!83 \nu^{15} + \cdots - 18\!\cdots\!32 ) / 27\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 43\!\cdots\!45 \nu^{15} + \cdots + 15\!\cdots\!88 ) / 41\!\cdots\!34 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 46\!\cdots\!67 \nu^{15} + \cdots + 73\!\cdots\!08 ) / 41\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( 35\!\cdots\!67 \nu^{15} + \cdots - 25\!\cdots\!32 ) / 27\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 21\!\cdots\!37 \nu^{15} + \cdots + 91\!\cdots\!56 ) / 13\!\cdots\!84 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( - 79\!\cdots\!71 \nu^{15} + \cdots - 64\!\cdots\!64 ) / 20\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 17\!\cdots\!37 \nu^{15} + \cdots - 63\!\cdots\!68 ) / 41\!\cdots\!52 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 14\!\cdots\!13 \nu^{15} + \cdots - 14\!\cdots\!08 ) / 27\!\cdots\!80 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 48\!\cdots\!27 \nu^{15} + \cdots + 22\!\cdots\!12 ) / 92\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 66\!\cdots\!09 \nu^{15} + \cdots - 65\!\cdots\!04 ) / 92\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 18\!\cdots\!30 \nu^{15} + \cdots - 38\!\cdots\!07 ) / 23\!\cdots\!63 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 18\!\cdots\!18 \nu^{15} + \cdots + 22\!\cdots\!52 ) / 20\!\cdots\!60 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 22\!\cdots\!58 \nu^{15} + \cdots + 11\!\cdots\!16 ) / 16\!\cdots\!36 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( \beta_{13} - \beta_{12} - \beta_{11} + 4\beta_{10} - 13\beta_{6} + 2\beta_{5} - \beta_{4} - 3\beta_{3} - 2\beta _1 + 15 ) / 54 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( - 2 \beta_{15} + 3 \beta_{13} + 27 \beta_{12} - 3 \beta_{11} - 42 \beta_{10} + 4 \beta_{9} + \cdots + 303 ) / 54 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( 75 \beta_{15} + 90 \beta_{14} + 35 \beta_{13} - 187 \beta_{12} - 73 \beta_{11} + 310 \beta_{10} + \cdots + 9447 ) / 54 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 348 \beta_{15} - 456 \beta_{14} - 187 \beta_{13} - 177 \beta_{12} - 243 \beta_{11} - 702 \beta_{10} + \cdots - 168087 ) / 54 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 185 \beta_{15} - 1470 \beta_{14} + 2484 \beta_{13} + 14106 \beta_{12} + 762 \beta_{11} - 34692 \beta_{10} + \cdots + 1331532 ) / 54 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( 86484 \beta_{15} + 93744 \beta_{14} - 6156 \beta_{13} - 324166 \beta_{12} - 63514 \beta_{11} + \cdots - 3752616 ) / 54 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( - 1305248 \beta_{15} - 1747704 \beta_{14} - 157811 \beta_{13} + 2903445 \beta_{12} + 313827 \beta_{11} + \cdots - 77630403 ) / 54 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( 12359960 \beta_{15} + 15314064 \beta_{14} + 2758717 \beta_{13} - 17857295 \beta_{12} + \cdots + 1989716457 ) / 54 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 1495143 \beta_{15} + 2463102 \beta_{14} - 47807879 \beta_{13} - 76332053 \beta_{12} + \cdots - 25499756847 ) / 54 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 1419400766 \beta_{15} - 1783812348 \beta_{14} + 231150197 \beta_{13} + 3777900555 \beta_{12} + \cdots + 183177655401 ) / 54 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 26259684693 \beta_{15} + 33168326202 \beta_{14} - 1194390100 \beta_{13} - 58549095474 \beta_{12} + \cdots - 331044395364 ) / 54 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 252581362464 \beta_{15} - 320058627984 \beta_{14} - 46856809454 \beta_{13} + 559432013600 \beta_{12} + \cdots - 25268536822242 ) / 54 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 859039949404 \beta_{15} + 1075874127432 \beta_{14} + 707210227551 \beta_{13} - 2090969425657 \beta_{12} + \cdots + 460509115389951 ) / 54 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( 17973487813366 \beta_{15} + 22691684473956 \beta_{14} - 8815903979085 \beta_{13} + \cdots - 52\!\cdots\!05 ) / 54 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( - 436691238430245 \beta_{15} - 550831794415794 \beta_{14} + 44009007275475 \beta_{13} + \cdots + 28\!\cdots\!11 ) / 54 \) Copy content Toggle raw display

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

Character values

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

\(n\) \(29\) \(325\) \(379\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
377.1
−3.65554 + 1.50294i
−3.65554 − 1.50294i
5.48402 − 5.91781i
5.48402 + 5.91781i
−8.90572 − 7.58540i
−8.90572 + 7.58540i
8.07724 + 2.16386i
8.07724 − 2.16386i
4.61314 + 2.16386i
4.61314 − 2.16386i
−5.44161 − 7.58540i
−5.44161 + 7.58540i
2.01992 − 5.91781i
2.01992 + 5.91781i
−0.191442 + 1.50294i
−0.191442 − 1.50294i
0 0 0 −19.1271 0 8.29412 − 16.5592i 0 0 0
377.2 0 0 0 −19.1271 0 8.29412 + 16.5592i 0 0 0
377.3 0 0 0 −13.6625 0 −3.07615 − 18.2630i 0 0 0
377.4 0 0 0 −13.6625 0 −3.07615 + 18.2630i 0 0 0
377.5 0 0 0 −6.67001 0 −18.2446 − 3.18348i 0 0 0
377.6 0 0 0 −6.67001 0 −18.2446 + 3.18348i 0 0 0
377.7 0 0 0 −0.0103268 0 16.5266 − 8.35886i 0 0 0
377.8 0 0 0 −0.0103268 0 16.5266 + 8.35886i 0 0 0
377.9 0 0 0 0.0103268 0 16.5266 − 8.35886i 0 0 0
377.10 0 0 0 0.0103268 0 16.5266 + 8.35886i 0 0 0
377.11 0 0 0 6.67001 0 −18.2446 − 3.18348i 0 0 0
377.12 0 0 0 6.67001 0 −18.2446 + 3.18348i 0 0 0
377.13 0 0 0 13.6625 0 −3.07615 − 18.2630i 0 0 0
377.14 0 0 0 13.6625 0 −3.07615 + 18.2630i 0 0 0
377.15 0 0 0 19.1271 0 8.29412 − 16.5592i 0 0 0
377.16 0 0 0 19.1271 0 8.29412 + 16.5592i 0 0 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 377.16
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 756.4.f.e ✓ 16
3.b odd 2 1 inner 756.4.f.e ✓ 16
7.b odd 2 1 inner 756.4.f.e ✓ 16
21.c even 2 1 inner 756.4.f.e ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
756.4.f.e ✓ 16 1.a even 1 1 trivial
756.4.f.e ✓ 16 3.b odd 2 1 inner
756.4.f.e ✓ 16 7.b odd 2 1 inner
756.4.f.e ✓ 16 21.c even 2 1 inner

Hecke kernels

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

\( T_{5}^{8} - 597T_{5}^{6} + 92871T_{5}^{4} - 3038175T_{5}^{2} + 324 \) Copy content Toggle raw display
\( T_{13}^{8} + 12039T_{13}^{6} + 40934880T_{13}^{4} + 36747462780T_{13}^{2} + 6774514512192 \) 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
$5$ \( (T^{8} - 597 T^{6} + \cdots + 324)^{2} \) Copy content Toggle raw display
$7$ \( (T^{8} - 7 T^{7} + \cdots + 13841287201)^{2} \) Copy content Toggle raw display
$11$ \( (T^{8} + 4977 T^{6} + \cdots + 18492257232)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 6774514512192)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 291127268252304)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 196092353150208)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 24\!\cdots\!00)^{2} \) Copy content Toggle raw display
$29$ \( (T^{8} + \cdots + 27\!\cdots\!68)^{2} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 58307319287028)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + 86 T^{3} + \cdots + 106819876)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 430755413603904)^{2} \) Copy content Toggle raw display
$43$ \( (T^{4} + 101 T^{3} + \cdots - 254922206)^{4} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 15\!\cdots\!04)^{2} \) Copy content Toggle raw display
$53$ \( (T^{8} + \cdots + 11\!\cdots\!88)^{2} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 51\!\cdots\!56)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 31\!\cdots\!48)^{2} \) Copy content Toggle raw display
$67$ \( (T^{4} - 443 T^{3} + \cdots + 35842596664)^{4} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 65\!\cdots\!88)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 94\!\cdots\!52)^{2} \) Copy content Toggle raw display
$79$ \( (T^{4} + 364 T^{3} + \cdots + 2048222656)^{4} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 56\!\cdots\!36)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 27\!\cdots\!24)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 21\!\cdots\!28)^{2} \) Copy content Toggle raw display
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