# Properties

 Label 3234.2.e.d Level $3234$ Weight $2$ Character orbit 3234.e Analytic conductor $25.824$ Analytic rank $0$ Dimension $24$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3234.e (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$25.8236200137$$ Analytic rank: $$0$$ Dimension: $$24$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 24 q^{4} + 24 q^{6} - 24 q^{9}+O(q^{10})$$ 24 * q - 24 * q^4 + 24 * q^6 - 24 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q - 24 q^{4} + 24 q^{6} - 24 q^{9} + 24 q^{16} - 16 q^{17} + 32 q^{19} - 8 q^{22} - 24 q^{24} - 8 q^{25} + 8 q^{33} + 24 q^{36} + 16 q^{37} + 16 q^{41} - 24 q^{54} - 16 q^{55} + 16 q^{62} - 24 q^{64} - 64 q^{67} + 16 q^{68} + 64 q^{71} - 32 q^{76} + 24 q^{81} + 16 q^{83} + 8 q^{88} - 16 q^{93} - 64 q^{94} + 24 q^{96}+O(q^{100})$$ 24 * q - 24 * q^4 + 24 * q^6 - 24 * q^9 + 24 * q^16 - 16 * q^17 + 32 * q^19 - 8 * q^22 - 24 * q^24 - 8 * q^25 + 8 * q^33 + 24 * q^36 + 16 * q^37 + 16 * q^41 - 24 * q^54 - 16 * q^55 + 16 * q^62 - 24 * q^64 - 64 * q^67 + 16 * q^68 + 64 * q^71 - 32 * q^76 + 24 * q^81 + 16 * q^83 + 8 * q^88 - 16 * q^93 - 64 * q^94 + 24 * q^96

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
2155.1 1.00000i 1.00000i −1.00000 4.05345i 1.00000 0 1.00000i −1.00000 −4.05345
2155.2 1.00000i 1.00000i −1.00000 3.45711i 1.00000 0 1.00000i −1.00000 −3.45711
2155.3 1.00000i 1.00000i −1.00000 2.66749i 1.00000 0 1.00000i −1.00000 −2.66749
2155.4 1.00000i 1.00000i −1.00000 0.516505i 1.00000 0 1.00000i −1.00000 −0.516505
2155.5 1.00000i 1.00000i −1.00000 0.357203i 1.00000 0 1.00000i −1.00000 −0.357203
2155.6 1.00000i 1.00000i −1.00000 0.313884i 1.00000 0 1.00000i −1.00000 −0.313884
2155.7 1.00000i 1.00000i −1.00000 0.500309i 1.00000 0 1.00000i −1.00000 0.500309
2155.8 1.00000i 1.00000i −1.00000 0.942563i 1.00000 0 1.00000i −1.00000 0.942563
2155.9 1.00000i 1.00000i −1.00000 1.58222i 1.00000 0 1.00000i −1.00000 1.58222
2155.10 1.00000i 1.00000i −1.00000 1.98215i 1.00000 0 1.00000i −1.00000 1.98215
2155.11 1.00000i 1.00000i −1.00000 2.84462i 1.00000 0 1.00000i −1.00000 2.84462
2155.12 1.00000i 1.00000i −1.00000 3.51377i 1.00000 0 1.00000i −1.00000 3.51377
2155.13 1.00000i 1.00000i −1.00000 3.51377i 1.00000 0 1.00000i −1.00000 3.51377
2155.14 1.00000i 1.00000i −1.00000 2.84462i 1.00000 0 1.00000i −1.00000 2.84462
2155.15 1.00000i 1.00000i −1.00000 1.98215i 1.00000 0 1.00000i −1.00000 1.98215
2155.16 1.00000i 1.00000i −1.00000 1.58222i 1.00000 0 1.00000i −1.00000 1.58222
2155.17 1.00000i 1.00000i −1.00000 0.942563i 1.00000 0 1.00000i −1.00000 0.942563
2155.18 1.00000i 1.00000i −1.00000 0.500309i 1.00000 0 1.00000i −1.00000 0.500309
2155.19 1.00000i 1.00000i −1.00000 0.313884i 1.00000 0 1.00000i −1.00000 −0.313884
2155.20 1.00000i 1.00000i −1.00000 0.357203i 1.00000 0 1.00000i −1.00000 −0.357203
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 2155.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
77.b even 2 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 3234.2.e.d yes 24
7.b odd 2 1 3234.2.e.c 24
11.b odd 2 1 3234.2.e.c 24
77.b even 2 1 inner 3234.2.e.d yes 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
3234.2.e.c 24 7.b odd 2 1
3234.2.e.c 24 11.b odd 2 1
3234.2.e.d yes 24 1.a even 1 1 trivial
3234.2.e.d yes 24 77.b even 2 1 inner

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(3234, [\chi])$$:

 $$T_{5}^{24} + 64 T_{5}^{22} + 1696 T_{5}^{20} + 24160 T_{5}^{18} + 201024 T_{5}^{16} + 995200 T_{5}^{14} + 2870592 T_{5}^{12} + 4574208 T_{5}^{10} + 3727360 T_{5}^{8} + 1498112 T_{5}^{6} + 301056 T_{5}^{4} + \cdots + 1024$$ T5^24 + 64*T5^22 + 1696*T5^20 + 24160*T5^18 + 201024*T5^16 + 995200*T5^14 + 2870592*T5^12 + 4574208*T5^10 + 3727360*T5^8 + 1498112*T5^6 + 301056*T5^4 + 28672*T5^2 + 1024 $$T_{13}^{12} - 76 T_{13}^{10} - 48 T_{13}^{9} + 2134 T_{13}^{8} + 2432 T_{13}^{7} - 26512 T_{13}^{6} - 40000 T_{13}^{5} + 138220 T_{13}^{4} + 247808 T_{13}^{3} - 211312 T_{13}^{2} - 510912 T_{13} - 207224$$ T13^12 - 76*T13^10 - 48*T13^9 + 2134*T13^8 + 2432*T13^7 - 26512*T13^6 - 40000*T13^5 + 138220*T13^4 + 247808*T13^3 - 211312*T13^2 - 510912*T13 - 207224