Properties

Label 637.2.q.j
Level $637$
Weight $2$
Character orbit 637.q
Analytic conductor $5.086$
Analytic rank $0$
Dimension $32$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 637 = 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 637.q (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(5.08647060876\)
Analytic rank: \(0\)
Dimension: \(32\)
Relative dimension: \(16\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.

\(\operatorname{Tr}(f)(q) = \) \( 32q + 20q^{4} - 28q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 32q + 20q^{4} - 28q^{9} - 12q^{15} - 28q^{16} + 8q^{22} + 24q^{23} - 40q^{25} - 24q^{29} + 24q^{30} - 60q^{32} + 92q^{36} - 32q^{39} + 12q^{43} - 24q^{46} + 12q^{50} + 72q^{53} - 132q^{58} + 32q^{64} + 48q^{67} - 48q^{71} + 72q^{72} + 24q^{74} - 156q^{78} + 96q^{79} - 64q^{81} + 12q^{85} + 56q^{88} + 168q^{92} - 48q^{93} + 84q^{95} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
491.1 −2.08022 + 1.20101i −0.888571 1.53905i 1.88487 3.26470i 0.706642i 3.69684 + 2.13437i 0 4.25098i −0.0791164 + 0.137034i 0.848687 + 1.46997i
491.2 −2.08022 + 1.20101i 0.888571 + 1.53905i 1.88487 3.26470i 0.706642i −3.69684 2.13437i 0 4.25098i −0.0791164 + 0.137034i −0.848687 1.46997i
491.3 −2.04719 + 1.18194i −1.49347 2.58676i 1.79399 3.10727i 3.20263i 6.11481 + 3.53039i 0 3.75379i −2.96088 + 5.12839i 3.78533 + 6.55639i
491.4 −2.04719 + 1.18194i 1.49347 + 2.58676i 1.79399 3.10727i 3.20263i −6.11481 3.53039i 0 3.75379i −2.96088 + 5.12839i −3.78533 6.55639i
491.5 −1.12902 + 0.651838i −0.134969 0.233774i −0.150215 + 0.260179i 1.56818i 0.304765 + 0.175956i 0 2.99901i 1.46357 2.53497i −1.02220 1.77050i
491.6 −1.12902 + 0.651838i 0.134969 + 0.233774i −0.150215 + 0.260179i 1.56818i −0.304765 0.175956i 0 2.99901i 1.46357 2.53497i 1.02220 + 1.77050i
491.7 −0.250157 + 0.144428i −0.969921 1.67995i −0.958281 + 1.65979i 4.29339i 0.485264 + 0.280167i 0 1.13132i −0.381493 + 0.660765i −0.620085 1.07402i
491.8 −0.250157 + 0.144428i 0.969921 + 1.67995i −0.958281 + 1.65979i 4.29339i −0.485264 0.280167i 0 1.13132i −0.381493 + 0.660765i 0.620085 + 1.07402i
491.9 0.489742 0.282753i −1.54556 2.67698i −0.840102 + 1.45510i 1.80514i −1.51385 0.874021i 0 2.08118i −3.27749 + 5.67679i 0.510408 + 0.884053i
491.10 0.489742 0.282753i 1.54556 + 2.67698i −0.840102 + 1.45510i 1.80514i 1.51385 + 0.874021i 0 2.08118i −3.27749 + 5.67679i −0.510408 0.884053i
491.11 0.900699 0.520019i −0.384681 0.666288i −0.459161 + 0.795291i 1.67669i −0.692964 0.400083i 0 3.03516i 1.20404 2.08546i −0.871910 1.51019i
491.12 0.900699 0.520019i 0.384681 + 0.666288i −0.459161 + 0.795291i 1.67669i 0.692964 + 0.400083i 0 3.03516i 1.20404 2.08546i 0.871910 + 1.51019i
491.13 1.81104 1.04560i −0.663994 1.15007i 1.18657 2.05519i 3.48900i −2.40503 1.38855i 0 0.780297i 0.618224 1.07080i −3.64811 6.31871i
491.14 1.81104 1.04560i 0.663994 + 1.15007i 1.18657 2.05519i 3.48900i 2.40503 + 1.38855i 0 0.780297i 0.618224 1.07080i 3.64811 + 6.31871i
491.15 2.30510 1.33085i −1.59481 2.76230i 2.54233 4.40345i 0.329389i −7.35241 4.24492i 0 8.21047i −3.58685 + 6.21261i −0.438368 0.759275i
491.16 2.30510 1.33085i 1.59481 + 2.76230i 2.54233 4.40345i 0.329389i 7.35241 + 4.24492i 0 8.21047i −3.58685 + 6.21261i 0.438368 + 0.759275i
589.1 −2.08022 1.20101i −0.888571 + 1.53905i 1.88487 + 3.26470i 0.706642i 3.69684 2.13437i 0 4.25098i −0.0791164 0.137034i 0.848687 1.46997i
589.2 −2.08022 1.20101i 0.888571 1.53905i 1.88487 + 3.26470i 0.706642i −3.69684 + 2.13437i 0 4.25098i −0.0791164 0.137034i −0.848687 + 1.46997i
589.3 −2.04719 1.18194i −1.49347 + 2.58676i 1.79399 + 3.10727i 3.20263i 6.11481 3.53039i 0 3.75379i −2.96088 5.12839i 3.78533 6.55639i
589.4 −2.04719 1.18194i 1.49347 2.58676i 1.79399 + 3.10727i 3.20263i −6.11481 + 3.53039i 0 3.75379i −2.96088 5.12839i −3.78533 + 6.55639i
See all 32 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 589.16
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
13.e even 6 1 inner
91.t odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 637.2.q.j 32
7.b odd 2 1 inner 637.2.q.j 32
7.c even 3 1 637.2.k.j 32
7.c even 3 1 637.2.u.j 32
7.d odd 6 1 637.2.k.j 32
7.d odd 6 1 637.2.u.j 32
13.e even 6 1 inner 637.2.q.j 32
13.f odd 12 2 8281.2.a.cx 32
91.k even 6 1 637.2.u.j 32
91.l odd 6 1 637.2.u.j 32
91.p odd 6 1 637.2.k.j 32
91.t odd 6 1 inner 637.2.q.j 32
91.u even 6 1 637.2.k.j 32
91.bc even 12 2 8281.2.a.cx 32
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
637.2.k.j 32 7.c even 3 1
637.2.k.j 32 7.d odd 6 1
637.2.k.j 32 91.p odd 6 1
637.2.k.j 32 91.u even 6 1
637.2.q.j 32 1.a even 1 1 trivial
637.2.q.j 32 7.b odd 2 1 inner
637.2.q.j 32 13.e even 6 1 inner
637.2.q.j 32 91.t odd 6 1 inner
637.2.u.j 32 7.c even 3 1
637.2.u.j 32 7.d odd 6 1
637.2.u.j 32 91.k even 6 1
637.2.u.j 32 91.l odd 6 1
8281.2.a.cx 32 13.f odd 12 2
8281.2.a.cx 32 91.bc even 12 2

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(637, [\chi])\):

\(T_{2}^{16} - \cdots\)
\(T_{3}^{32} + \cdots\)