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Properties

Label 784.6.f.e
Level $784$
Weight $6$
Character orbit 784.f
Analytic conductor $125.741$
Analytic rank $0$
Dimension $16$
Inner twists $4$

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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] = [784,6,Mod(783,784)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(784, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 0, 1])) N = Newforms(chi, 6, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("784.783"); S:= CuspForms(chi, 6); N := Newforms(S);
 
Level: \( N \) \(=\) \( 784 = 2^{4} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 784.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [16,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(125.740914733\)
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: \( x^{16} - 232 x^{14} + 23092 x^{12} - 1258560 x^{10} + 40996255 x^{8} - 775625040 x^{6} + \cdots + 20552376321 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{34}\cdot 7^{18} \)
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.

\(f(q)\) \(=\) \( q + \beta_{9} q^{3} + ( - \beta_{13} + \beta_{10}) q^{5} + (\beta_{3} + \beta_{2} + \beta_1 + 107) q^{9} + ( - \beta_{6} - \beta_{5}) q^{11} + (3 \beta_{13} + 4 \beta_{11} + \cdots - 6 \beta_{8}) q^{13}+ \cdots + ( - 99 \beta_{7} - 13 \beta_{6} + \cdots - 51 \beta_{4}) q^{99}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 1712 q^{9} - 496 q^{25} - 17024 q^{29} + 85664 q^{53} - 95424 q^{57} + 233344 q^{65} - 511216 q^{81} + 697952 q^{85} - 857024 q^{93}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{16} - 232 x^{14} + 23092 x^{12} - 1258560 x^{10} + 40996255 x^{8} - 775625040 x^{6} + \cdots + 20552376321 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( ( - 702301302265822 \nu^{14} + \cdots + 29\!\cdots\!30 ) / 41\!\cdots\!71 \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( ( - 17\!\cdots\!16 \nu^{14} + \cdots - 52\!\cdots\!04 ) / 53\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 28\!\cdots\!88 \nu^{14} + \cdots + 29\!\cdots\!00 ) / 17\!\cdots\!29 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( - 62\!\cdots\!28 \nu^{14} + \cdots - 69\!\cdots\!58 ) / 17\!\cdots\!29 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( 22\!\cdots\!40 \nu^{14} + \cdots + 22\!\cdots\!54 ) / 53\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( - 15\!\cdots\!96 \nu^{14} + \cdots + 29\!\cdots\!34 ) / 53\!\cdots\!87 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( - 63\!\cdots\!20 \nu^{14} + \cdots + 21\!\cdots\!96 ) / 17\!\cdots\!29 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( 75\!\cdots\!99 \nu^{15} + \cdots - 89\!\cdots\!17 \nu ) / 11\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( - 18\!\cdots\!72 \nu^{15} + \cdots + 62\!\cdots\!27 \nu ) / 28\!\cdots\!41 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 86\!\cdots\!15 \nu^{15} + \cdots - 95\!\cdots\!66 \nu ) / 11\!\cdots\!89 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( - 10\!\cdots\!49 \nu^{15} + \cdots + 23\!\cdots\!46 \nu ) / 25\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{12}\)\(=\) \( ( 19\!\cdots\!68 \nu^{15} + \cdots + 17\!\cdots\!32 \nu ) / 28\!\cdots\!41 \) Copy content Toggle raw display
\(\beta_{13}\)\(=\) \( ( - 29\!\cdots\!38 \nu^{15} + \cdots + 95\!\cdots\!74 \nu ) / 25\!\cdots\!69 \) Copy content Toggle raw display
\(\beta_{14}\)\(=\) \( ( 42\!\cdots\!06 \nu^{15} + \cdots - 33\!\cdots\!09 \nu ) / 16\!\cdots\!73 \) Copy content Toggle raw display
\(\beta_{15}\)\(=\) \( ( - 13\!\cdots\!91 \nu^{15} + \cdots + 39\!\cdots\!60 \nu ) / 28\!\cdots\!41 \) Copy content Toggle raw display

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

\(\nu\)\(=\) \( ( - 10 \beta_{15} - 7 \beta_{14} + 98 \beta_{13} - 28 \beta_{12} - 392 \beta_{11} + \cdots + 84 \beta_{8} ) / 10976 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( ( -49\beta_{7} + 49\beta_{6} - 53\beta_{5} - 104\beta_{4} - 539\beta_{3} - 392\beta_{2} - 1358\beta _1 + 159152 ) / 5488 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( ( - 30 \beta_{15} - 217 \beta_{14} + 8869 \beta_{13} - 623 \beta_{12} - 23471 \beta_{11} + \cdots + 11914 \beta_{8} ) / 5488 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( ( - 7301 \beta_{7} + 9506 \beta_{6} + 1601 \beta_{5} - 6835 \beta_{4} - 28567 \beta_{3} + \cdots + 5241040 ) / 5488 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( ( 17706 \beta_{15} - 19593 \beta_{14} + 1352302 \beta_{13} - 46032 \beta_{12} + \cdots + 3128636 \beta_{8} ) / 10976 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( ( - 352702 \beta_{7} + 481278 \beta_{6} + 208338 \beta_{5} - 176660 \beta_{4} - 546889 \beta_{3} + \cdots + 65449888 ) / 2744 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( ( 141250 \beta_{15} - 33229 \beta_{14} + 13404328 \beta_{13} - 122038 \beta_{12} + \cdots + 38985440 \beta_{8} ) / 1568 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( ( - 13222944 \beta_{7} + 18361231 \beta_{6} + 9147967 \beta_{5} - 4953297 \beta_{4} - 2212595 \beta_{3} + \cdots - 712182562 ) / 1372 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( ( 15435946 \beta_{15} + 36949367 \beta_{14} + 3064535117 \beta_{13} + 44772665 \beta_{12} + \cdots + 9367489866 \beta_{8} ) / 5488 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( ( - 3264987453 \beta_{7} + 4568317089 \beta_{6} + 2267970567 \beta_{5} - 1184810988 \beta_{4} + \cdots - 951721831088 ) / 5488 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( ( 998983630 \beta_{15} + 11038184269 \beta_{14} + 348904730104 \beta_{13} + \cdots + 1046236200464 \beta_{8} ) / 10976 \) Copy content Toggle raw display
\(\nu^{12}\)\(=\) \( ( - 80495034641 \beta_{7} + 112986199494 \beta_{6} + 53894529501 \beta_{5} - 31040666155 \beta_{4} + \cdots - 55403494991152 ) / 2744 \) Copy content Toggle raw display
\(\nu^{13}\)\(=\) \( ( 79714548098 \beta_{15} + 1069613856003 \beta_{14} + 14929442801252 \beta_{13} + \cdots + 43322365364040 \beta_{8} ) / 10976 \) Copy content Toggle raw display
\(\nu^{14}\)\(=\) \( ( - 704495745653 \beta_{7} + 988127384559 \beta_{6} + 451223419251 \beta_{5} - 293820800566 \beta_{4} + \cdots - 13\!\cdots\!12 ) / 784 \) Copy content Toggle raw display
\(\nu^{15}\)\(=\) \( ( 3856327960356 \beta_{15} + 41786566017462 \beta_{14} + 99222248301561 \beta_{13} + \cdots + 271079956281442 \beta_{8} ) / 5488 \) Copy content Toggle raw display

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

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/784\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(687\) \(689\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
783.1
−5.38401 − 2.60313i
−5.38401 + 2.60313i
−1.27628 − 0.597902i
−1.27628 + 0.597902i
−5.40349 − 0.215219i
−5.40349 + 0.215219i
8.13547 + 1.67925i
8.13547 − 1.67925i
−8.13547 + 1.67925i
−8.13547 − 1.67925i
5.40349 − 0.215219i
5.40349 + 0.215219i
1.27628 − 0.597902i
1.27628 + 0.597902i
5.38401 − 2.60313i
5.38401 + 2.60313i
0 −24.0806 0 − 1.38632i 0 0 0 336.874 0
783.2 0 −24.0806 0 1.38632i 0 0 0 336.874 0
783.3 0 −22.3313 0 − 90.0418i 0 0 0 255.687 0
783.4 0 −22.3313 0 90.0418i 0 0 0 255.687 0
783.5 0 −16.1465 0 − 48.6032i 0 0 0 17.7104 0
783.6 0 −16.1465 0 48.6032i 0 0 0 17.7104 0
783.7 0 −7.79289 0 − 46.3928i 0 0 0 −182.271 0
783.8 0 −7.79289 0 46.3928i 0 0 0 −182.271 0
783.9 0 7.79289 0 − 46.3928i 0 0 0 −182.271 0
783.10 0 7.79289 0 46.3928i 0 0 0 −182.271 0
783.11 0 16.1465 0 − 48.6032i 0 0 0 17.7104 0
783.12 0 16.1465 0 48.6032i 0 0 0 17.7104 0
783.13 0 22.3313 0 − 90.0418i 0 0 0 255.687 0
783.14 0 22.3313 0 90.0418i 0 0 0 255.687 0
783.15 0 24.0806 0 − 1.38632i 0 0 0 336.874 0
783.16 0 24.0806 0 1.38632i 0 0 0 336.874 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 783.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
4.b odd 2 1 inner
7.b odd 2 1 inner
28.d even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 784.6.f.e ✓ 16
4.b odd 2 1 inner 784.6.f.e ✓ 16
7.b odd 2 1 inner 784.6.f.e ✓ 16
28.d even 2 1 inner 784.6.f.e ✓ 16
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
784.6.f.e ✓ 16 1.a even 1 1 trivial
784.6.f.e ✓ 16 4.b odd 2 1 inner
784.6.f.e ✓ 16 7.b odd 2 1 inner
784.6.f.e ✓ 16 28.d even 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator \( T_{3}^{8} - 1400T_{3}^{6} + 651700T_{3}^{4} - 110028912T_{3}^{2} + 4578428484 \) acting on \(S_{6}^{\mathrm{new}}(784, [\chi])\). Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{16} \) Copy content Toggle raw display
$3$ \( (T^{8} - 1400 T^{6} + \cdots + 4578428484)^{2} \) Copy content Toggle raw display
$5$ \( (T^{8} + 12624 T^{6} + \cdots + 79221983296)^{2} \) Copy content Toggle raw display
$7$ \( T^{16} \) Copy content Toggle raw display
$11$ \( (T^{8} + \cdots + 37\!\cdots\!64)^{2} \) Copy content Toggle raw display
$13$ \( (T^{8} + \cdots + 15\!\cdots\!44)^{2} \) Copy content Toggle raw display
$17$ \( (T^{8} + \cdots + 50\!\cdots\!16)^{2} \) Copy content Toggle raw display
$19$ \( (T^{8} + \cdots + 62\!\cdots\!56)^{2} \) Copy content Toggle raw display
$23$ \( (T^{8} + \cdots + 17\!\cdots\!84)^{2} \) Copy content Toggle raw display
$29$ \( (T^{4} + \cdots + 346802178400832)^{4} \) Copy content Toggle raw display
$31$ \( (T^{8} + \cdots + 26\!\cdots\!44)^{2} \) Copy content Toggle raw display
$37$ \( (T^{4} + \cdots + 354129424811008)^{4} \) Copy content Toggle raw display
$41$ \( (T^{8} + \cdots + 20\!\cdots\!84)^{2} \) Copy content Toggle raw display
$43$ \( (T^{8} + \cdots + 39\!\cdots\!76)^{2} \) Copy content Toggle raw display
$47$ \( (T^{8} + \cdots + 50\!\cdots\!76)^{2} \) Copy content Toggle raw display
$53$ \( (T^{4} + \cdots + 32\!\cdots\!04)^{4} \) Copy content Toggle raw display
$59$ \( (T^{8} + \cdots + 10\!\cdots\!04)^{2} \) Copy content Toggle raw display
$61$ \( (T^{8} + \cdots + 86\!\cdots\!56)^{2} \) Copy content Toggle raw display
$67$ \( (T^{8} + \cdots + 60\!\cdots\!04)^{2} \) Copy content Toggle raw display
$71$ \( (T^{8} + \cdots + 24\!\cdots\!16)^{2} \) Copy content Toggle raw display
$73$ \( (T^{8} + \cdots + 50\!\cdots\!64)^{2} \) Copy content Toggle raw display
$79$ \( (T^{8} + \cdots + 72\!\cdots\!04)^{2} \) Copy content Toggle raw display
$83$ \( (T^{8} + \cdots + 36\!\cdots\!76)^{2} \) Copy content Toggle raw display
$89$ \( (T^{8} + \cdots + 54\!\cdots\!36)^{2} \) Copy content Toggle raw display
$97$ \( (T^{8} + \cdots + 57\!\cdots\!44)^{2} \) Copy content Toggle raw display
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