# Properties

 Label 72.7.m.a Level $72$ Weight $7$ Character orbit 72.m Analytic conductor $16.564$ Analytic rank $0$ Dimension $36$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$72 = 2^{3} \cdot 3^{2}$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 72.m (of order $$6$$, degree $$2$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$16.5638940206$$ Analytic rank: $$0$$ Dimension: $$36$$ Relative dimension: $$18$$ 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.

 $$\operatorname{Tr}(f)(q) =$$ $$36 q + 10 q^{3} + 74 q^{9}+O(q^{10})$$ 36 * q + 10 * q^3 + 74 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$36 q + 10 q^{3} + 74 q^{9} + 1350 q^{11} + 7912 q^{15} + 9540 q^{19} + 3828 q^{21} + 30888 q^{23} + 56250 q^{25} + 11392 q^{27} + 38556 q^{29} + 27720 q^{31} + 33514 q^{33} + 134068 q^{39} + 179226 q^{41} + 15930 q^{43} - 185620 q^{45} + 187596 q^{47} - 198774 q^{49} - 158098 q^{51} - 197064 q^{55} - 244990 q^{57} - 408618 q^{59} + 17136 q^{61} - 417048 q^{63} - 125712 q^{65} + 27090 q^{67} - 848504 q^{69} - 534060 q^{73} - 1405714 q^{75} + 48168 q^{77} + 172620 q^{79} + 349010 q^{81} + 1801980 q^{83} - 791568 q^{85} + 28500 q^{87} + 538560 q^{91} - 1116448 q^{93} + 1832652 q^{95} + 770706 q^{97} - 614260 q^{99}+O(q^{100})$$ 36 * q + 10 * q^3 + 74 * q^9 + 1350 * q^11 + 7912 * q^15 + 9540 * q^19 + 3828 * q^21 + 30888 * q^23 + 56250 * q^25 + 11392 * q^27 + 38556 * q^29 + 27720 * q^31 + 33514 * q^33 + 134068 * q^39 + 179226 * q^41 + 15930 * q^43 - 185620 * q^45 + 187596 * q^47 - 198774 * q^49 - 158098 * q^51 - 197064 * q^55 - 244990 * q^57 - 408618 * q^59 + 17136 * q^61 - 417048 * q^63 - 125712 * q^65 + 27090 * q^67 - 848504 * q^69 - 534060 * q^73 - 1405714 * q^75 + 48168 * q^77 + 172620 * q^79 + 349010 * q^81 + 1801980 * q^83 - 791568 * q^85 + 28500 * q^87 + 538560 * q^91 - 1116448 * q^93 + 1832652 * q^95 + 770706 * q^97 - 614260 * q^99

## 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
41.1 0 −26.9340 1.88607i 0 43.3462 + 25.0260i 0 47.2533 + 81.8451i 0 721.885 + 101.599i 0
41.2 0 −26.5258 + 5.03801i 0 −212.009 122.404i 0 −198.515 343.838i 0 678.237 267.274i 0
41.3 0 −26.4412 + 5.46471i 0 174.849 + 100.949i 0 −193.914 335.869i 0 669.274 288.987i 0
41.4 0 −19.7087 18.4545i 0 −11.9109 6.87674i 0 174.223 + 301.763i 0 47.8620 + 727.427i 0
41.5 0 −14.5191 + 22.7639i 0 26.1779 + 15.1138i 0 301.407 + 522.052i 0 −307.391 661.023i 0
41.6 0 −13.2266 + 23.5384i 0 −180.740 104.350i 0 173.716 + 300.885i 0 −379.115 622.666i 0
41.7 0 −13.0560 + 23.6335i 0 49.0773 + 28.3348i 0 −226.487 392.287i 0 −388.084 617.116i 0
41.8 0 −11.5131 24.4223i 0 −113.268 65.3953i 0 −143.592 248.709i 0 −463.898 + 562.352i 0
41.9 0 0.709327 26.9907i 0 157.631 + 91.0083i 0 199.766 + 346.005i 0 −727.994 38.2905i 0
41.10 0 1.83005 26.9379i 0 86.3751 + 49.8687i 0 −207.661 359.679i 0 −722.302 98.5953i 0
41.11 0 4.90324 + 26.5511i 0 −9.93230 5.73442i 0 −128.008 221.717i 0 −680.917 + 260.372i 0
41.12 0 12.6322 + 23.8627i 0 152.846 + 88.2455i 0 152.534 + 264.197i 0 −409.853 + 602.878i 0
41.13 0 16.4370 21.4202i 0 −124.675 71.9811i 0 53.2895 + 92.3001i 0 −188.649 704.168i 0
41.14 0 22.0600 15.5677i 0 −91.6877 52.9359i 0 238.433 + 412.978i 0 244.291 686.850i 0
41.15 0 22.2209 + 15.3372i 0 −93.2385 53.8313i 0 −32.4834 56.2629i 0 258.541 + 681.614i 0
41.16 0 24.3041 11.7607i 0 76.0344 + 43.8985i 0 −287.890 498.640i 0 452.374 571.663i 0
41.17 0 24.8361 + 10.5910i 0 −87.4624 50.4964i 0 −21.1751 36.6764i 0 504.663 + 526.077i 0
41.18 0 26.9914 0.680128i 0 158.588 + 91.5607i 0 99.1040 + 171.653i 0 728.075 36.7153i 0
65.1 0 −26.9340 + 1.88607i 0 43.3462 25.0260i 0 47.2533 81.8451i 0 721.885 101.599i 0
65.2 0 −26.5258 5.03801i 0 −212.009 + 122.404i 0 −198.515 + 343.838i 0 678.237 + 267.274i 0
See all 36 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 65.18 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
9.d odd 6 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 72.7.m.a 36
3.b odd 2 1 216.7.m.a 36
4.b odd 2 1 144.7.q.d 36
9.c even 3 1 216.7.m.a 36
9.c even 3 1 648.7.e.c 36
9.d odd 6 1 inner 72.7.m.a 36
9.d odd 6 1 648.7.e.c 36
12.b even 2 1 432.7.q.d 36
36.f odd 6 1 432.7.q.d 36
36.h even 6 1 144.7.q.d 36

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.7.m.a 36 1.a even 1 1 trivial
72.7.m.a 36 9.d odd 6 1 inner
144.7.q.d 36 4.b odd 2 1
144.7.q.d 36 36.h even 6 1
216.7.m.a 36 3.b odd 2 1
216.7.m.a 36 9.c even 3 1
432.7.q.d 36 12.b even 2 1
432.7.q.d 36 36.f odd 6 1
648.7.e.c 36 9.c even 3 1
648.7.e.c 36 9.d odd 6 1

## Hecke kernels

This newform subspace is the entire newspace $$S_{7}^{\mathrm{new}}(72, [\chi])$$.