Properties

Label 688.2.i.e
Level 688
Weight 2
Character orbit 688.i
Analytic conductor 5.494
Analytic rank 0
Dimension 4
CM no
Inner twists 2

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Newspace parameters

Level: \( N \) \(=\) \( 688 = 2^{4} \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 688.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.49370765906\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{5})\)
Defining polynomial: \(x^{4} - x^{3} + 2 x^{2} + x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 43)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\beta_2,\beta_3\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + ( -1 - \beta_{1} - \beta_{3} ) q^{3} + ( 2 - 2 \beta_{1} + 2 \beta_{3} ) q^{5} + ( 2 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{7} + ( 3 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{9} +O(q^{10})\) \( q + ( -1 - \beta_{1} - \beta_{3} ) q^{3} + ( 2 - 2 \beta_{1} + 2 \beta_{3} ) q^{5} + ( 2 \beta_{1} + 2 \beta_{2} + \beta_{3} ) q^{7} + ( 3 \beta_{1} + 3 \beta_{2} - \beta_{3} ) q^{9} + ( 2 - \beta_{2} ) q^{11} + ( -\beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{13} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{15} + ( 5 \beta_{1} + 5 \beta_{2} - 2 \beta_{3} ) q^{17} + ( 2 - 2 \beta_{1} + 2 \beta_{3} ) q^{19} + ( 3 - 5 \beta_{2} ) q^{21} + ( 6 - \beta_{1} + 6 \beta_{3} ) q^{23} + ( -4 \beta_{1} - 4 \beta_{2} + 3 \beta_{3} ) q^{25} + ( -1 - 2 \beta_{2} ) q^{27} + 3 \beta_{3} q^{29} + ( -3 - 4 \beta_{1} - 3 \beta_{3} ) q^{33} + ( 2 - 2 \beta_{2} ) q^{35} + 3 \beta_{1} q^{37} + ( 2 - \beta_{2} ) q^{39} + ( 3 - 4 \beta_{2} ) q^{41} + ( 7 + \beta_{3} ) q^{43} + ( 8 + 2 \beta_{2} ) q^{45} + ( -6 - 3 \beta_{2} ) q^{47} + ( 2 - 8 \beta_{1} + 2 \beta_{3} ) q^{49} + ( 3 - 8 \beta_{2} ) q^{51} + ( 3 - \beta_{1} + 3 \beta_{3} ) q^{53} + ( 2 - 4 \beta_{1} + 2 \beta_{3} ) q^{55} + ( 2 \beta_{1} + 2 \beta_{2} ) q^{57} + ( 3 + 5 \beta_{2} ) q^{59} + ( 3 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} ) q^{61} + ( -5 - 7 \beta_{1} - 5 \beta_{3} ) q^{63} + ( -8 - 6 \beta_{2} ) q^{65} + ( 1 - 3 \beta_{1} + \beta_{3} ) q^{67} + ( -4 \beta_{1} - 4 \beta_{2} - 5 \beta_{3} ) q^{69} + ( 11 \beta_{1} + 11 \beta_{2} - 7 \beta_{3} ) q^{71} + ( 3 \beta_{1} + 3 \beta_{2} - 3 \beta_{3} ) q^{73} + ( -1 + 5 \beta_{2} ) q^{75} + ( 7 \beta_{1} + 7 \beta_{2} + 4 \beta_{3} ) q^{77} + ( -\beta_{1} - \beta_{2} + 3 \beta_{3} ) q^{79} + ( -4 + 6 \beta_{1} - 4 \beta_{3} ) q^{81} + ( -5 + 13 \beta_{1} - 5 \beta_{3} ) q^{83} + ( 14 + 4 \beta_{2} ) q^{85} + ( 3 - 3 \beta_{2} ) q^{87} + ( -3 + 3 \beta_{1} - 3 \beta_{3} ) q^{89} + ( -1 - 3 \beta_{1} - \beta_{3} ) q^{91} + ( -4 \beta_{1} - 4 \beta_{2} + 8 \beta_{3} ) q^{95} + ( -6 + 2 \beta_{2} ) q^{97} + ( 8 \beta_{1} + 8 \beta_{2} + \beta_{3} ) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 3q^{3} + 2q^{5} - 4q^{7} - q^{9} + O(q^{10}) \) \( 4q - 3q^{3} + 2q^{5} - 4q^{7} - q^{9} + 10q^{11} - 5q^{13} - 2q^{15} - q^{17} + 2q^{19} + 22q^{21} + 11q^{23} - 2q^{25} - 6q^{29} - 10q^{33} + 12q^{35} + 3q^{37} + 10q^{39} + 20q^{41} + 26q^{43} + 28q^{45} - 18q^{47} - 4q^{49} + 28q^{51} + 5q^{53} - 2q^{57} + 2q^{59} + q^{61} - 17q^{63} - 20q^{65} - q^{67} + 14q^{69} + 3q^{71} + 3q^{73} - 14q^{75} - 15q^{77} - 5q^{79} - 2q^{81} + 3q^{83} + 48q^{85} + 18q^{87} - 3q^{89} - 5q^{91} - 12q^{95} - 28q^{97} - 10q^{99} + O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{4} - x^{3} + 2 x^{2} + x + 1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\((\)\( \nu^{3} + 1 \)\()/2\)
\(\beta_{3}\)\(=\)\((\)\( -\nu^{3} + 2 \nu^{2} - 2 \nu - 1 \)\()/2\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{3} + \beta_{2} + \beta_{1}\)
\(\nu^{3}\)\(=\)\(2 \beta_{2} - 1\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/688\mathbb{Z}\right)^\times\).

\(n\) \(431\) \(433\) \(517\)
\(\chi(n)\) \(1\) \(\beta_{3}\) \(1\)

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.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
49.1
0.809017 1.40126i
−0.309017 + 0.535233i
0.809017 + 1.40126i
−0.309017 0.535233i
0 −1.30902 + 2.26728i 0 −0.618034 + 1.07047i 0 −2.11803 3.66854i 0 −1.92705 3.33775i 0
49.2 0 −0.190983 + 0.330792i 0 1.61803 2.80252i 0 0.118034 + 0.204441i 0 1.42705 + 2.47172i 0
337.1 0 −1.30902 2.26728i 0 −0.618034 1.07047i 0 −2.11803 + 3.66854i 0 −1.92705 + 3.33775i 0
337.2 0 −0.190983 0.330792i 0 1.61803 + 2.80252i 0 0.118034 0.204441i 0 1.42705 2.47172i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
43.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 688.2.i.e 4
4.b odd 2 1 43.2.c.b 4
12.b even 2 1 387.2.h.d 4
43.c even 3 1 inner 688.2.i.e 4
172.f even 6 1 1849.2.a.h 2
172.g odd 6 1 43.2.c.b 4
172.g odd 6 1 1849.2.a.e 2
516.p even 6 1 387.2.h.d 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
43.2.c.b 4 4.b odd 2 1
43.2.c.b 4 172.g odd 6 1
387.2.h.d 4 12.b even 2 1
387.2.h.d 4 516.p even 6 1
688.2.i.e 4 1.a even 1 1 trivial
688.2.i.e 4 43.c even 3 1 inner
1849.2.a.e 2 172.g odd 6 1
1849.2.a.h 2 172.f even 6 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(688, [\chi])\):

\( T_{3}^{4} + 3 T_{3}^{3} + 8 T_{3}^{2} + 3 T_{3} + 1 \)
\( T_{5}^{4} - 2 T_{5}^{3} + 8 T_{5}^{2} + 8 T_{5} + 16 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ 1
$3$ \( 1 + 3 T + 2 T^{2} + 3 T^{3} + 13 T^{4} + 9 T^{5} + 18 T^{6} + 81 T^{7} + 81 T^{8} \)
$5$ \( 1 - 2 T - 2 T^{2} + 8 T^{3} - 9 T^{4} + 40 T^{5} - 50 T^{6} - 250 T^{7} + 625 T^{8} \)
$7$ \( 1 + 4 T + 3 T^{2} - 4 T^{3} + 8 T^{4} - 28 T^{5} + 147 T^{6} + 1372 T^{7} + 2401 T^{8} \)
$11$ \( ( 1 - 5 T + 27 T^{2} - 55 T^{3} + 121 T^{4} )^{2} \)
$13$ \( 1 + 5 T - 6 T^{2} + 25 T^{3} + 467 T^{4} + 325 T^{5} - 1014 T^{6} + 10985 T^{7} + 28561 T^{8} \)
$17$ \( 1 + T - 2 T^{2} - 31 T^{3} - 297 T^{4} - 527 T^{5} - 578 T^{6} + 4913 T^{7} + 83521 T^{8} \)
$19$ \( 1 - 2 T - 30 T^{2} + 8 T^{3} + 719 T^{4} + 152 T^{5} - 10830 T^{6} - 13718 T^{7} + 130321 T^{8} \)
$23$ \( 1 - 11 T + 46 T^{2} - 319 T^{3} + 2313 T^{4} - 7337 T^{5} + 24334 T^{6} - 133837 T^{7} + 279841 T^{8} \)
$29$ \( ( 1 + 3 T - 20 T^{2} + 87 T^{3} + 841 T^{4} )^{2} \)
$31$ \( ( 1 - 31 T^{2} + 961 T^{4} )^{2} \)
$37$ \( 1 - 3 T - 56 T^{2} + 27 T^{3} + 2523 T^{4} + 999 T^{5} - 76664 T^{6} - 151959 T^{7} + 1874161 T^{8} \)
$41$ \( ( 1 - 10 T + 87 T^{2} - 410 T^{3} + 1681 T^{4} )^{2} \)
$43$ \( ( 1 - 13 T + 43 T^{2} )^{2} \)
$47$ \( ( 1 + 9 T + 103 T^{2} + 423 T^{3} + 2209 T^{4} )^{2} \)
$53$ \( 1 - 5 T - 86 T^{2} - 25 T^{3} + 8187 T^{4} - 1325 T^{5} - 241574 T^{6} - 744385 T^{7} + 7890481 T^{8} \)
$59$ \( ( 1 - T + 87 T^{2} - 59 T^{3} + 3481 T^{4} )^{2} \)
$61$ \( 1 - T - 110 T^{2} + 11 T^{3} + 8539 T^{4} + 671 T^{5} - 409310 T^{6} - 226981 T^{7} + 13845841 T^{8} \)
$67$ \( 1 + T - 122 T^{2} - 11 T^{3} + 10573 T^{4} - 737 T^{5} - 547658 T^{6} + 300763 T^{7} + 20151121 T^{8} \)
$71$ \( 1 - 3 T + 16 T^{2} + 447 T^{3} - 5631 T^{4} + 31737 T^{5} + 80656 T^{6} - 1073733 T^{7} + 25411681 T^{8} \)
$73$ \( 1 - 3 T - 128 T^{2} + 27 T^{3} + 12783 T^{4} + 1971 T^{5} - 682112 T^{6} - 1167051 T^{7} + 28398241 T^{8} \)
$79$ \( 1 + 5 T - 138 T^{2} + 25 T^{3} + 18353 T^{4} + 1975 T^{5} - 861258 T^{6} + 2465195 T^{7} + 38950081 T^{8} \)
$83$ \( 1 - 3 T + 52 T^{2} + 627 T^{3} - 5787 T^{4} + 52041 T^{5} + 358228 T^{6} - 1715361 T^{7} + 47458321 T^{8} \)
$89$ \( 1 + 3 T - 160 T^{2} - 27 T^{3} + 19839 T^{4} - 2403 T^{5} - 1267360 T^{6} + 2114907 T^{7} + 62742241 T^{8} \)
$97$ \( ( 1 + 14 T + 238 T^{2} + 1358 T^{3} + 9409 T^{4} )^{2} \)
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