# Properties

 Label 230.4.e.a Level $230$ Weight $4$ Character orbit 230.e Analytic conductor $13.570$ Analytic rank $0$ Dimension $72$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$230 = 2 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 230.e (of order $$4$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.5704393013$$ Analytic rank: $$0$$ Dimension: $$72$$ Relative dimension: $$36$$ over $$\Q(i)$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$72q - 16q^{3} - 16q^{6} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$72q - 16q^{3} - 16q^{6} - 64q^{12} + 192q^{13} - 1152q^{16} + 32q^{18} + 276q^{23} + 880q^{25} + 304q^{26} + 728q^{27} + 608q^{31} + 688q^{35} + 2816q^{36} - 2208q^{41} - 256q^{46} + 144q^{47} + 256q^{48} + 272q^{50} + 768q^{52} - 2360q^{55} - 1376q^{62} - 1104q^{70} + 4528q^{71} + 128q^{72} - 1296q^{73} - 2568q^{75} - 752q^{77} + 6000q^{78} - 11560q^{81} + 80q^{82} - 904q^{85} + 6760q^{87} + 1104q^{92} - 5288q^{93} + 9264q^{95} + 256q^{96} - 448q^{98} + O(q^{100})$$

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
137.1 −1.41421 1.41421i −7.12210 + 7.12210i 4.00000i −1.63222 11.0606i 20.1443 −16.1147 + 16.1147i 5.65685 5.65685i 74.4485i −13.3337 + 17.9503i
137.2 −1.41421 1.41421i −7.12210 + 7.12210i 4.00000i 1.63222 + 11.0606i 20.1443 16.1147 16.1147i 5.65685 5.65685i 74.4485i 13.3337 17.9503i
137.3 −1.41421 1.41421i −4.47764 + 4.47764i 4.00000i −11.1722 + 0.427507i 12.6647 −6.97055 + 6.97055i 5.65685 5.65685i 13.0984i 16.4044 + 15.1952i
137.4 −1.41421 1.41421i −4.47764 + 4.47764i 4.00000i 11.1722 0.427507i 12.6647 6.97055 6.97055i 5.65685 5.65685i 13.0984i −16.4044 15.1952i
137.5 −1.41421 1.41421i −3.58924 + 3.58924i 4.00000i −11.1634 0.614616i 10.1519 16.2976 16.2976i 5.65685 5.65685i 1.23476i 14.9183 + 16.6567i
137.6 −1.41421 1.41421i −3.58924 + 3.58924i 4.00000i 11.1634 + 0.614616i 10.1519 −16.2976 + 16.2976i 5.65685 5.65685i 1.23476i −14.9183 16.6567i
137.7 −1.41421 1.41421i −2.02411 + 2.02411i 4.00000i −2.14096 + 10.9734i 5.72506 −1.46247 + 1.46247i 5.65685 5.65685i 18.8059i 18.5466 12.4910i
137.8 −1.41421 1.41421i −2.02411 + 2.02411i 4.00000i 2.14096 10.9734i 5.72506 1.46247 1.46247i 5.65685 5.65685i 18.8059i −18.5466 + 12.4910i
137.9 −1.41421 1.41421i 0.422683 0.422683i 4.00000i −11.1421 + 0.923475i −1.19553 −23.7633 + 23.7633i 5.65685 5.65685i 26.6427i 17.0634 + 14.4514i
137.10 −1.41421 1.41421i 0.422683 0.422683i 4.00000i 11.1421 0.923475i −1.19553 23.7633 23.7633i 5.65685 5.65685i 26.6427i −17.0634 14.4514i
137.11 −1.41421 1.41421i 0.933603 0.933603i 4.00000i −4.48554 10.2411i −2.64063 −3.80706 + 3.80706i 5.65685 5.65685i 25.2568i −8.13958 + 20.8266i
137.12 −1.41421 1.41421i 0.933603 0.933603i 4.00000i 4.48554 + 10.2411i −2.64063 3.80706 3.80706i 5.65685 5.65685i 25.2568i 8.13958 20.8266i
137.13 −1.41421 1.41421i 3.25915 3.25915i 4.00000i −7.62223 + 8.17934i −9.21827 22.5290 22.5290i 5.65685 5.65685i 5.75586i 22.3468 0.787870i
137.14 −1.41421 1.41421i 3.25915 3.25915i 4.00000i 7.62223 8.17934i −9.21827 −22.5290 + 22.5290i 5.65685 5.65685i 5.75586i −22.3468 + 0.787870i
137.15 −1.41421 1.41421i 5.03526 5.03526i 4.00000i −6.80528 8.87064i −14.2419 11.1481 11.1481i 5.65685 5.65685i 23.7076i −2.92086 + 22.1691i
137.16 −1.41421 1.41421i 5.03526 5.03526i 4.00000i 6.80528 + 8.87064i −14.2419 −11.1481 + 11.1481i 5.65685 5.65685i 23.7076i 2.92086 22.1691i
137.17 −1.41421 1.41421i 6.26949 6.26949i 4.00000i −8.82273 + 6.86728i −17.7328 −11.5218 + 11.5218i 5.65685 5.65685i 51.6130i 22.1890 + 2.76543i
137.18 −1.41421 1.41421i 6.26949 6.26949i 4.00000i 8.82273 6.86728i −17.7328 11.5218 11.5218i 5.65685 5.65685i 51.6130i −22.1890 2.76543i
137.19 1.41421 + 1.41421i −6.86623 + 6.86623i 4.00000i −11.1776 + 0.245509i −19.4206 21.1410 21.1410i −5.65685 + 5.65685i 67.2901i −16.1548 15.4604i
137.20 1.41421 + 1.41421i −6.86623 + 6.86623i 4.00000i 11.1776 0.245509i −19.4206 −21.1410 + 21.1410i −5.65685 + 5.65685i 67.2901i 16.1548 + 15.4604i
See all 72 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 183.36 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.c odd 4 1 inner
23.b odd 2 1 inner
115.e even 4 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 230.4.e.a 72
5.c odd 4 1 inner 230.4.e.a 72
23.b odd 2 1 inner 230.4.e.a 72
115.e even 4 1 inner 230.4.e.a 72

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
230.4.e.a 72 1.a even 1 1 trivial
230.4.e.a 72 5.c odd 4 1 inner
230.4.e.a 72 23.b odd 2 1 inner
230.4.e.a 72 115.e even 4 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{4}^{\mathrm{new}}(230, [\chi])$$.