# Properties

 Label 605.2.j.k Level $605$ Weight $2$ Character orbit 605.j Analytic conductor $4.831$ Analytic rank $0$ Dimension $48$ CM no Inner twists $16$

# Learn more about

## Newspace parameters

 Level: $$N$$ $$=$$ $$605 = 5 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 605.j (of order $$10$$, degree $$4$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.83094932229$$ Analytic rank: $$0$$ Dimension: $$48$$ Relative dimension: $$12$$ over $$\Q(\zeta_{10})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$48q + 20q^{4} + 6q^{5} + 20q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$48q + 20q^{4} + 6q^{5} + 20q^{9} + 32q^{14} - 20q^{15} - 36q^{16} - 26q^{20} + 10q^{25} - 20q^{26} - 8q^{31} + 48q^{34} - 92q^{36} - 72q^{45} + 4q^{49} + 192q^{56} + 32q^{59} + 92q^{60} - 28q^{64} + 16q^{69} + 12q^{70} - 112q^{71} - 36q^{75} + 106q^{80} + 20q^{81} + 56q^{86} - 432q^{89} - 72q^{91} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
9.1 −2.48013 0.805844i −1.25775 1.73114i 3.88364 + 2.82163i 0.0494116 2.23552i 1.72435 + 5.30700i −0.581248 + 0.800020i −4.29253 5.90817i −0.487865 + 1.50149i −1.92403 + 5.50457i
9.2 −2.48013 0.805844i 1.25775 + 1.73114i 3.88364 + 2.82163i −1.35398 1.77953i −1.72435 5.30700i −0.581248 + 0.800020i −4.29253 5.90817i −0.487865 + 1.50149i 1.92403 + 5.50457i
9.3 −1.92380 0.625081i −1.71325 2.35809i 1.69226 + 1.22950i 2.16467 + 0.560538i 1.82196 + 5.60741i −1.88754 + 2.59798i −0.109082 0.150138i −1.69829 + 5.22681i −3.81402 2.43146i
9.4 −1.92380 0.625081i 1.71325 + 2.35809i 1.69226 + 1.22950i −1.42178 + 1.72585i −1.82196 5.60741i −1.88754 + 2.59798i −0.109082 0.150138i −1.69829 + 5.22681i 3.81402 2.43146i
9.5 −0.312280 0.101466i −0.565450 0.778275i −1.53081 1.11220i −2.23364 + 0.104146i 0.0976102 + 0.300413i 1.92358 2.64758i 0.751190 + 1.03393i 0.641073 1.97302i 0.708089 + 0.194116i
9.6 −0.312280 0.101466i 0.565450 + 0.778275i −1.53081 1.11220i 1.86827 1.22865i −0.0976102 0.300413i 1.92358 2.64758i 0.751190 + 1.03393i 0.641073 1.97302i −0.708089 + 0.194116i
9.7 0.312280 + 0.101466i −0.565450 0.778275i −1.53081 1.11220i −2.23364 + 0.104146i −0.0976102 0.300413i −1.92358 + 2.64758i −0.751190 1.03393i 0.641073 1.97302i −0.708089 0.194116i
9.8 0.312280 + 0.101466i 0.565450 + 0.778275i −1.53081 1.11220i 1.86827 1.22865i 0.0976102 + 0.300413i −1.92358 + 2.64758i −0.751190 1.03393i 0.641073 1.97302i 0.708089 0.194116i
9.9 1.92380 + 0.625081i −1.71325 2.35809i 1.69226 + 1.22950i 2.16467 + 0.560538i −1.82196 5.60741i 1.88754 2.59798i 0.109082 + 0.150138i −1.69829 + 5.22681i 3.81402 + 2.43146i
9.10 1.92380 + 0.625081i 1.71325 + 2.35809i 1.69226 + 1.22950i −1.42178 + 1.72585i 1.82196 + 5.60741i 1.88754 2.59798i 0.109082 + 0.150138i −1.69829 + 5.22681i −3.81402 + 2.43146i
9.11 2.48013 + 0.805844i −1.25775 1.73114i 3.88364 + 2.82163i 0.0494116 2.23552i −1.72435 5.30700i 0.581248 0.800020i 4.29253 + 5.90817i −0.487865 + 1.50149i 1.92403 5.50457i
9.12 2.48013 + 0.805844i 1.25775 + 1.73114i 3.88364 + 2.82163i −1.35398 1.77953i 1.72435 + 5.30700i 0.581248 0.800020i 4.29253 + 5.90817i −0.487865 + 1.50149i −1.92403 5.50457i
124.1 −1.53281 2.10973i −2.03508 + 0.661236i −1.48342 + 4.56549i 1.27403 + 1.83762i 4.51440 + 3.27991i 0.940480 + 0.305580i 6.94547 2.25672i 1.27725 0.927974i 1.92403 5.50457i
124.2 −1.53281 2.10973i 2.03508 0.661236i −1.48342 + 4.56549i 2.14138 + 0.643821i −4.51440 3.27991i 0.940480 + 0.305580i 6.94547 2.25672i 1.27725 0.927974i −1.92403 5.50457i
124.3 −1.18898 1.63648i −2.77210 + 0.900709i −0.646385 + 1.98937i −2.08073 + 0.818876i 4.76995 + 3.46557i 3.05411 + 0.992339i 0.176498 0.0573478i 4.44619 3.23035i 3.81402 + 2.43146i
124.4 −1.18898 1.63648i 2.77210 0.900709i −0.646385 + 1.98937i 0.135816 2.23194i −4.76995 3.46557i 3.05411 + 0.992339i 0.176498 0.0573478i 4.44619 3.23035i −3.81402 + 2.43146i
124.5 −0.193000 0.265641i −0.914917 + 0.297274i 0.584718 1.79958i 1.74584 1.39716i 0.255547 + 0.185666i −3.11241 1.01128i −1.21545 + 0.394924i −1.67835 + 1.21939i −0.708089 0.194116i
124.6 −0.193000 0.265641i 0.914917 0.297274i 0.584718 1.79958i −0.789281 + 2.09214i −0.255547 0.185666i −3.11241 1.01128i −1.21545 + 0.394924i −1.67835 + 1.21939i 0.708089 0.194116i
124.7 0.193000 + 0.265641i −0.914917 + 0.297274i 0.584718 1.79958i 1.74584 1.39716i −0.255547 0.185666i 3.11241 + 1.01128i 1.21545 0.394924i −1.67835 + 1.21939i 0.708089 + 0.194116i
124.8 0.193000 + 0.265641i 0.914917 0.297274i 0.584718 1.79958i −0.789281 + 2.09214i 0.255547 + 0.185666i 3.11241 + 1.01128i 1.21545 0.394924i −1.67835 + 1.21939i −0.708089 + 0.194116i
See all 48 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 444.12 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner
11.b odd 2 1 inner
11.c even 5 3 inner
11.d odd 10 3 inner
55.d odd 2 1 inner
55.h odd 10 3 inner
55.j even 10 3 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 605.2.j.k 48
5.b even 2 1 inner 605.2.j.k 48
11.b odd 2 1 inner 605.2.j.k 48
11.c even 5 1 605.2.b.h 12
11.c even 5 3 inner 605.2.j.k 48
11.d odd 10 1 605.2.b.h 12
11.d odd 10 3 inner 605.2.j.k 48
55.d odd 2 1 inner 605.2.j.k 48
55.h odd 10 1 605.2.b.h 12
55.h odd 10 3 inner 605.2.j.k 48
55.j even 10 1 605.2.b.h 12
55.j even 10 3 inner 605.2.j.k 48
55.k odd 20 2 3025.2.a.bo 12
55.l even 20 2 3025.2.a.bo 12

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
605.2.b.h 12 11.c even 5 1
605.2.b.h 12 11.d odd 10 1
605.2.b.h 12 55.h odd 10 1
605.2.b.h 12 55.j even 10 1
605.2.j.k 48 1.a even 1 1 trivial
605.2.j.k 48 5.b even 2 1 inner
605.2.j.k 48 11.b odd 2 1 inner
605.2.j.k 48 11.c even 5 3 inner
605.2.j.k 48 11.d odd 10 3 inner
605.2.j.k 48 55.d odd 2 1 inner
605.2.j.k 48 55.h odd 10 3 inner
605.2.j.k 48 55.j even 10 3 inner
3025.2.a.bo 12 55.k odd 20 2
3025.2.a.bo 12 55.l even 20 2

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(605, [\chi])$$:

 $$T_{2}^{24} - \cdots$$ $$17\!\cdots\!96$$$$T_{19}^{8} +$$$$63\!\cdots\!04$$$$T_{19}^{6} +$$$$22\!\cdots\!72$$$$T_{19}^{4} +$$$$78\!\cdots\!04$$$$T_{19}^{2} +$$$$22\!\cdots\!36$$">$$T_{19}^{24} + \cdots$$