Properties

Label 315.2.bq.a
Level 315
Weight 2
Character orbit 315.bq
Analytic conductor 2.515
Analytic rank 0
Dimension 88
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 315 = 3^{2} \cdot 5 \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 315.bq (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(2.51528766367\)
Analytic rank: \(0\)
Dimension: \(88\)
Relative dimension: \(44\) over \(\Q(\zeta_{6})\)
Coefficient ring index: multiple of None
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.

\(\operatorname{Tr}(f)(q) = \) \( 88q + 76q^{4} - 3q^{5} - 12q^{6} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 88q + 76q^{4} - 3q^{5} - 12q^{6} - 6q^{10} - 12q^{11} - 12q^{14} - 6q^{15} + 52q^{16} - 12q^{19} - 6q^{20} - 12q^{21} - 48q^{24} + q^{25} - 12q^{26} + 6q^{29} + 33q^{30} - 12q^{34} - 36q^{36} + 12q^{39} - 30q^{40} + 6q^{41} - 84q^{44} + 21q^{45} - 18q^{46} - 8q^{49} + 30q^{50} - 24q^{51} - 60q^{54} - 78q^{56} - 12q^{59} - 45q^{60} - 8q^{64} - 24q^{66} + 60q^{69} + 15q^{70} - 30q^{74} - 48q^{75} - 48q^{76} - 16q^{79} - 69q^{80} + 36q^{81} - 90q^{84} - 7q^{85} - 12q^{86} + 72q^{89} + 33q^{90} + 20q^{91} - 60q^{96} + 30q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
164.1 −2.70830 0.577858 + 1.63281i 5.33489 −0.422616 + 2.19577i −1.56501 4.42215i −1.01783 + 2.44214i −9.03188 −2.33216 + 1.88707i 1.14457 5.94680i
164.2 −2.57600 −1.19095 1.25763i 4.63577 2.13393 + 0.668088i 3.06790 + 3.23965i 2.40367 + 1.10561i −6.78974 −0.163255 + 2.99555i −5.49700 1.72099i
164.3 −2.53172 1.70470 0.306605i 4.40959 −1.09270 1.95090i −4.31581 + 0.776236i 2.57916 + 0.589845i −6.10041 2.81199 1.04534i 2.76640 + 4.93913i
164.4 −2.46032 0.957408 1.44339i 4.05317 −0.494408 + 2.18072i −2.35553 + 3.55119i −0.227160 2.63598i −5.05144 −1.16674 2.76382i 1.21640 5.36528i
164.5 −2.29795 0.238210 + 1.71559i 3.28056 −0.0571900 2.23534i −0.547395 3.94234i −0.338914 2.62395i −2.94267 −2.88651 + 0.817344i 0.131420 + 5.13669i
164.6 −2.16166 −1.44589 + 0.953629i 2.67276 −2.00646 + 0.986977i 3.12551 2.06142i 2.53153 0.769008i −1.45428 1.18118 2.75768i 4.33727 2.13351i
164.7 −2.06518 1.70672 + 0.295145i 2.26496 2.20004 0.399802i −3.52468 0.609527i −2.20598 + 1.46070i −0.547180 2.82578 + 1.00746i −4.54346 + 0.825662i
164.8 −2.04520 −0.124265 1.72759i 2.18286 −1.93784 1.11569i 0.254147 + 3.53327i −0.907139 + 2.48538i −0.373988 −2.96912 + 0.429357i 3.96329 + 2.28181i
164.9 −2.00210 −1.55371 + 0.765499i 2.00841 2.11856 + 0.715328i 3.11068 1.53261i −2.15805 1.53063i −0.0168431 1.82802 2.37873i −4.24158 1.43216i
164.10 −1.70249 −0.695317 1.58636i 0.898480 1.25685 1.84941i 1.18377 + 2.70076i −0.756494 2.53529i 1.87533 −2.03307 + 2.20604i −2.13977 + 3.14861i
164.11 −1.50968 1.73201 0.0112588i 0.279131 −1.98951 + 1.02071i −2.61479 + 0.0169972i −2.58044 0.584218i 2.59796 2.99975 0.0390010i 3.00352 1.54095i
164.12 −1.37324 −1.73021 + 0.0799234i −0.114213 0.214801 2.22573i 2.37599 0.109754i 1.26290 + 2.32489i 2.90332 2.98722 0.276568i −0.294973 + 3.05646i
164.13 −1.33100 1.20337 + 1.24576i −0.228439 0.821920 + 2.07953i −1.60168 1.65810i 1.38620 2.25354i 2.96605 −0.103821 + 2.99820i −1.09398 2.76786i
164.14 −1.20693 −0.344453 + 1.69745i −0.543308 −2.01662 0.966042i 0.415733 2.04872i −2.56388 + 0.653073i 3.06961 −2.76270 1.16939i 2.43393 + 1.16595i
164.15 −1.20290 1.27624 1.17099i −0.553033 2.10508 0.754091i −1.53519 + 1.40858i 2.51709 0.815010i 3.07104 0.257567 2.98892i −2.53219 + 0.907096i
164.16 −0.932214 −0.704225 + 1.58242i −1.13098 1.20128 + 1.88598i 0.656489 1.47516i 0.405582 + 2.61448i 2.91874 −2.00813 2.22877i −1.11985 1.75814i
164.17 −0.827410 −1.03098 1.39179i −1.31539 −1.03969 + 1.97966i 0.853047 + 1.15158i 2.63782 + 0.204740i 2.74319 −0.874143 + 2.86982i 0.860253 1.63799i
164.18 −0.586009 −1.67705 0.433007i −1.65659 −2.03395 0.928995i 0.982768 + 0.253746i −1.03863 2.43336i 2.14280 2.62501 + 1.45235i 1.19192 + 0.544399i
164.19 −0.504719 −0.161348 1.72452i −1.74526 1.16790 + 1.90683i 0.0814357 + 0.870398i −2.52898 0.777347i 1.89030 −2.94793 + 0.556497i −0.589461 0.962416i
164.20 −0.487080 1.51865 0.832881i −1.76275 −1.77632 + 1.35819i −0.739706 + 0.405680i 1.01806 + 2.44204i 1.83276 1.61262 2.52972i 0.865213 0.661546i
See all 88 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 194.44
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
63.i even 6 1 inner
315.bq even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.bq.a yes 88
3.b odd 2 1 945.2.bq.a 88
5.b even 2 1 inner 315.2.bq.a yes 88
7.d odd 6 1 315.2.u.a 88
9.c even 3 1 945.2.u.a 88
9.d odd 6 1 315.2.u.a 88
15.d odd 2 1 945.2.bq.a 88
21.g even 6 1 945.2.u.a 88
35.i odd 6 1 315.2.u.a 88
45.h odd 6 1 315.2.u.a 88
45.j even 6 1 945.2.u.a 88
63.i even 6 1 inner 315.2.bq.a yes 88
63.t odd 6 1 945.2.bq.a 88
105.p even 6 1 945.2.u.a 88
315.q odd 6 1 945.2.bq.a 88
315.bq even 6 1 inner 315.2.bq.a yes 88
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
315.2.u.a 88 7.d odd 6 1
315.2.u.a 88 9.d odd 6 1
315.2.u.a 88 35.i odd 6 1
315.2.u.a 88 45.h odd 6 1
315.2.bq.a yes 88 1.a even 1 1 trivial
315.2.bq.a yes 88 5.b even 2 1 inner
315.2.bq.a yes 88 63.i even 6 1 inner
315.2.bq.a yes 88 315.bq even 6 1 inner
945.2.u.a 88 9.c even 3 1
945.2.u.a 88 21.g even 6 1
945.2.u.a 88 45.j even 6 1
945.2.u.a 88 105.p even 6 1
945.2.bq.a 88 3.b odd 2 1
945.2.bq.a 88 15.d odd 2 1
945.2.bq.a 88 63.t odd 6 1
945.2.bq.a 88 315.q odd 6 1

Hecke kernels

This newform subspace is the entire newspace \(S_{2}^{\mathrm{new}}(315, [\chi])\).

Hecke Characteristic Polynomials

There are no characteristic polynomials of Hecke operators in the database