# Properties

 Label 138.7.b.a Level $138$ Weight $7$ Character orbit 138.b Analytic conductor $31.747$ Analytic rank $0$ Dimension $24$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$138 = 2 \cdot 3 \cdot 23$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 138.b (of order $$2$$, degree $$1$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$31.7474635395$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 768 q^{4} + 5832 q^{9}+O(q^{10})$$ 24 * q + 768 * q^4 + 5832 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$24 q + 768 q^{4} + 5832 q^{9} - 768 q^{13} + 24576 q^{16} - 44104 q^{23} - 119448 q^{25} - 53888 q^{26} + 3456 q^{29} + 50976 q^{31} + 149008 q^{35} + 186624 q^{36} + 11664 q^{39} - 3920 q^{41} - 150720 q^{46} + 441088 q^{47} - 32472 q^{49} + 8320 q^{50} - 24576 q^{52} + 826176 q^{55} - 307200 q^{58} - 1210160 q^{59} + 783744 q^{62} + 786432 q^{64} + 361584 q^{69} - 2480064 q^{70} + 1531264 q^{71} + 593472 q^{73} + 23328 q^{75} + 1068784 q^{77} + 171072 q^{78} + 1417176 q^{81} + 1454592 q^{82} - 1318272 q^{85} + 697248 q^{87} - 1411328 q^{92} - 983664 q^{93} + 1115712 q^{94} + 4047632 q^{95} - 2409344 q^{98}+O(q^{100})$$ 24 * q + 768 * q^4 + 5832 * q^9 - 768 * q^13 + 24576 * q^16 - 44104 * q^23 - 119448 * q^25 - 53888 * q^26 + 3456 * q^29 + 50976 * q^31 + 149008 * q^35 + 186624 * q^36 + 11664 * q^39 - 3920 * q^41 - 150720 * q^46 + 441088 * q^47 - 32472 * q^49 + 8320 * q^50 - 24576 * q^52 + 826176 * q^55 - 307200 * q^58 - 1210160 * q^59 + 783744 * q^62 + 786432 * q^64 + 361584 * q^69 - 2480064 * q^70 + 1531264 * q^71 + 593472 * q^73 + 23328 * q^75 + 1068784 * q^77 + 171072 * q^78 + 1417176 * q^81 + 1454592 * q^82 - 1318272 * q^85 + 697248 * q^87 - 1411328 * q^92 - 983664 * q^93 + 1115712 * q^94 + 4047632 * q^95 - 2409344 * q^98

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
91.1 −5.65685 −15.5885 32.0000 191.970i 88.1816 321.404i −181.019 243.000 1085.94i
91.2 −5.65685 −15.5885 32.0000 138.652i 88.1816 40.7846i −181.019 243.000 784.336i
91.3 −5.65685 −15.5885 32.0000 33.5442i 88.1816 442.137i −181.019 243.000 189.755i
91.4 −5.65685 −15.5885 32.0000 33.5442i 88.1816 442.137i −181.019 243.000 189.755i
91.5 −5.65685 −15.5885 32.0000 138.652i 88.1816 40.7846i −181.019 243.000 784.336i
91.6 −5.65685 −15.5885 32.0000 191.970i 88.1816 321.404i −181.019 243.000 1085.94i
91.7 −5.65685 15.5885 32.0000 203.893i −88.1816 33.7438i −181.019 243.000 1153.39i
91.8 −5.65685 15.5885 32.0000 155.973i −88.1816 550.532i −181.019 243.000 882.319i
91.9 −5.65685 15.5885 32.0000 29.6262i −88.1816 53.4742i −181.019 243.000 167.591i
91.10 −5.65685 15.5885 32.0000 29.6262i −88.1816 53.4742i −181.019 243.000 167.591i
91.11 −5.65685 15.5885 32.0000 155.973i −88.1816 550.532i −181.019 243.000 882.319i
91.12 −5.65685 15.5885 32.0000 203.893i −88.1816 33.7438i −181.019 243.000 1153.39i
91.13 5.65685 −15.5885 32.0000 223.235i −88.1816 652.987i 181.019 243.000 1262.81i
91.14 5.65685 −15.5885 32.0000 124.369i −88.1816 215.829i 181.019 243.000 703.535i
91.15 5.65685 −15.5885 32.0000 38.5076i −88.1816 40.4171i 181.019 243.000 217.832i
91.16 5.65685 −15.5885 32.0000 38.5076i −88.1816 40.4171i 181.019 243.000 217.832i
91.17 5.65685 −15.5885 32.0000 124.369i −88.1816 215.829i 181.019 243.000 703.535i
91.18 5.65685 −15.5885 32.0000 223.235i −88.1816 652.987i 181.019 243.000 1262.81i
91.19 5.65685 15.5885 32.0000 227.787i 88.1816 132.014i 181.019 243.000 1288.56i
91.20 5.65685 15.5885 32.0000 56.1927i 88.1816 504.037i 181.019 243.000 317.874i
See all 24 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 91.24 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

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

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 138.7.b.a 24
3.b odd 2 1 414.7.b.c 24
23.b odd 2 1 inner 138.7.b.a 24
69.c even 2 1 414.7.b.c 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
138.7.b.a 24 1.a even 1 1 trivial
138.7.b.a 24 23.b odd 2 1 inner
414.7.b.c 24 3.b odd 2 1
414.7.b.c 24 69.c even 2 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{7}^{\mathrm{new}}(138, [\chi])$$.