# Properties

 Label 900.3.u.d Level $900$ Weight $3$ Character orbit 900.u Analytic conductor $24.523$ Analytic rank $0$ Dimension $32$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$900 = 2^{2} \cdot 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$3$$ Character orbit: $$[\chi]$$ $$=$$ 900.u (of order $$6$$, degree $$2$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$24.5232237924$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$32q - 28q^{9} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$32q - 28q^{9} - 4q^{19} + 2q^{21} - 18q^{29} + 16q^{31} - 38q^{39} + 108q^{41} + 90q^{49} + 180q^{51} - 18q^{59} - 110q^{61} + 294q^{69} - 22q^{79} - 260q^{81} - 268q^{91} - 504q^{99} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
149.1 0 −2.83603 0.978225i 0 0 0 4.69840 2.71262i 0 7.08615 + 5.54856i 0
149.2 0 −2.70777 + 1.29149i 0 0 0 3.00081 1.73252i 0 5.66408 6.99415i 0
149.3 0 −2.56994 1.54771i 0 0 0 −1.25581 + 0.725042i 0 4.20921 + 7.95503i 0
149.4 0 −2.37034 + 1.83888i 0 0 0 −7.15437 + 4.13058i 0 2.23701 8.71756i 0
149.5 0 −1.60876 + 2.53217i 0 0 0 8.13249 4.69530i 0 −3.82376 8.14732i 0
149.6 0 −1.28947 2.70874i 0 0 0 −9.36178 + 5.40503i 0 −5.67451 + 6.98569i 0
149.7 0 −0.708005 + 2.91526i 0 0 0 −8.59526 + 4.96248i 0 −7.99746 4.12804i 0
149.8 0 −0.386835 2.97496i 0 0 0 4.03103 2.32731i 0 −8.70072 + 2.30163i 0
149.9 0 0.386835 + 2.97496i 0 0 0 −4.03103 + 2.32731i 0 −8.70072 + 2.30163i 0
149.10 0 0.708005 2.91526i 0 0 0 8.59526 4.96248i 0 −7.99746 4.12804i 0
149.11 0 1.28947 + 2.70874i 0 0 0 9.36178 5.40503i 0 −5.67451 + 6.98569i 0
149.12 0 1.60876 2.53217i 0 0 0 −8.13249 + 4.69530i 0 −3.82376 8.14732i 0
149.13 0 2.37034 1.83888i 0 0 0 7.15437 4.13058i 0 2.23701 8.71756i 0
149.14 0 2.56994 + 1.54771i 0 0 0 1.25581 0.725042i 0 4.20921 + 7.95503i 0
149.15 0 2.70777 1.29149i 0 0 0 −3.00081 + 1.73252i 0 5.66408 6.99415i 0
149.16 0 2.83603 + 0.978225i 0 0 0 −4.69840 + 2.71262i 0 7.08615 + 5.54856i 0
749.1 0 −2.83603 + 0.978225i 0 0 0 4.69840 + 2.71262i 0 7.08615 5.54856i 0
749.2 0 −2.70777 1.29149i 0 0 0 3.00081 + 1.73252i 0 5.66408 + 6.99415i 0
749.3 0 −2.56994 + 1.54771i 0 0 0 −1.25581 0.725042i 0 4.20921 7.95503i 0
749.4 0 −2.37034 1.83888i 0 0 0 −7.15437 4.13058i 0 2.23701 + 8.71756i 0
See all 32 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 749.16 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
9.d odd 6 1 inner
45.h odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 900.3.u.d 32
3.b odd 2 1 2700.3.u.d 32
5.b even 2 1 inner 900.3.u.d 32
5.c odd 4 1 900.3.p.d 16
5.c odd 4 1 900.3.p.e yes 16
9.c even 3 1 2700.3.u.d 32
9.d odd 6 1 inner 900.3.u.d 32
15.d odd 2 1 2700.3.u.d 32
15.e even 4 1 2700.3.p.d 16
15.e even 4 1 2700.3.p.e 16
45.h odd 6 1 inner 900.3.u.d 32
45.j even 6 1 2700.3.u.d 32
45.k odd 12 1 2700.3.p.d 16
45.k odd 12 1 2700.3.p.e 16
45.l even 12 1 900.3.p.d 16
45.l even 12 1 900.3.p.e yes 16

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
900.3.p.d 16 5.c odd 4 1
900.3.p.d 16 45.l even 12 1
900.3.p.e yes 16 5.c odd 4 1
900.3.p.e yes 16 45.l even 12 1
900.3.u.d 32 1.a even 1 1 trivial
900.3.u.d 32 5.b even 2 1 inner
900.3.u.d 32 9.d odd 6 1 inner
900.3.u.d 32 45.h odd 6 1 inner
2700.3.p.d 16 15.e even 4 1
2700.3.p.d 16 45.k odd 12 1
2700.3.p.e 16 15.e even 4 1
2700.3.p.e 16 45.k odd 12 1
2700.3.u.d 32 3.b odd 2 1
2700.3.u.d 32 9.c even 3 1
2700.3.u.d 32 15.d odd 2 1
2700.3.u.d 32 45.j even 6 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$18\!\cdots\!38$$$$T_{7}^{20} -$$$$10\!\cdots\!88$$$$T_{7}^{18} +$$$$45\!\cdots\!13$$$$T_{7}^{16} -$$$$14\!\cdots\!88$$$$T_{7}^{14} +$$$$37\!\cdots\!08$$$$T_{7}^{12} -$$$$68\!\cdots\!68$$$$T_{7}^{10} +$$$$90\!\cdots\!68$$$$T_{7}^{8} -$$$$77\!\cdots\!68$$$$T_{7}^{6} +$$$$43\!\cdots\!08$$$$T_{7}^{4} -$$$$85\!\cdots\!52$$$$T_{7}^{2} +$$$$12\!\cdots\!76$$">$$T_{7}^{32} - \cdots$$ acting on $$S_{3}^{\mathrm{new}}(900, [\chi])$$.