Properties

 Label 180.7.c.b Level $180$ Weight $7$ Character orbit 180.c Analytic conductor $41.410$ Analytic rank $0$ Dimension $24$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$180 = 2^{2} \cdot 3^{2} \cdot 5$$ Weight: $$k$$ $$=$$ $$7$$ Character orbit: $$[\chi]$$ $$=$$ 180.c (of order $$2$$, degree $$1$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$41.4097350516$$ Analytic rank: $$0$$ Dimension: $$24$$ Twist minimal: no (minimal twist has level 60) 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.

 $$\operatorname{Tr}(f)(q) =$$ $$24q - 20q^{2} - 246q^{4} + 340q^{8} + O(q^{10})$$ $$\operatorname{Tr}(f)(q) =$$ $$24q - 20q^{2} - 246q^{4} + 340q^{8} - 750q^{10} + 5040q^{13} + 2596q^{14} + 4194q^{16} - 7000q^{20} + 45780q^{22} + 75000q^{25} - 75852q^{26} + 54300q^{28} - 132800q^{29} + 10700q^{32} - 173484q^{34} - 69840q^{37} - 215800q^{38} - 14250q^{40} + 70448q^{41} + 395668q^{44} - 158760q^{46} - 642984q^{49} - 62500q^{50} - 210240q^{52} + 644320q^{53} + 917708q^{56} - 1345020q^{58} - 222864q^{61} - 1948520q^{62} + 935922q^{64} - 266000q^{65} - 572680q^{68} + 220500q^{70} + 771120q^{73} + 589164q^{74} - 191544q^{76} - 1383840q^{77} + 946000q^{80} + 2672520q^{82} - 372000q^{85} - 1781528q^{86} + 956940q^{88} + 1566224q^{89} + 3040560q^{92} - 3788352q^{94} - 1666800q^{97} + 2709660q^{98} + O(q^{100})$$

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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
91.1 −7.99899 0.126876i 0 63.9678 + 2.02976i 55.9017 0 335.195i −511.421 24.3521i 0 −447.157 7.09260i
91.2 −7.99899 + 0.126876i 0 63.9678 2.02976i 55.9017 0 335.195i −511.421 + 24.3521i 0 −447.157 + 7.09260i
91.3 −7.41350 3.00666i 0 45.9200 + 44.5798i −55.9017 0 453.408i −206.391 468.558i 0 414.427 + 168.078i
91.4 −7.41350 + 3.00666i 0 45.9200 44.5798i −55.9017 0 453.408i −206.391 + 468.558i 0 414.427 168.078i
91.5 −5.40467 5.89827i 0 −5.57916 + 63.7564i 55.9017 0 125.427i 406.206 311.674i 0 −302.130 329.723i
91.6 −5.40467 + 5.89827i 0 −5.57916 63.7564i 55.9017 0 125.427i 406.206 + 311.674i 0 −302.130 + 329.723i
91.7 −5.21792 6.06410i 0 −9.54653 + 63.2840i −55.9017 0 234.050i 433.573 272.320i 0 291.691 + 338.993i
91.8 −5.21792 + 6.06410i 0 −9.54653 63.2840i −55.9017 0 234.050i 433.573 + 272.320i 0 291.691 338.993i
91.9 −3.43879 7.22321i 0 −40.3494 + 49.6782i −55.9017 0 472.024i 497.589 + 120.619i 0 192.234 + 403.790i
91.10 −3.43879 + 7.22321i 0 −40.3494 49.6782i −55.9017 0 472.024i 497.589 120.619i 0 192.234 403.790i
91.11 −2.59351 7.56794i 0 −50.5474 + 39.2551i 55.9017 0 671.100i 428.176 + 280.731i 0 −144.982 423.061i
91.12 −2.59351 + 7.56794i 0 −50.5474 39.2551i 55.9017 0 671.100i 428.176 280.731i 0 −144.982 + 423.061i
91.13 −0.331992 7.99311i 0 −63.7796 + 5.30729i 55.9017 0 552.003i 63.5960 + 508.035i 0 −18.5589 446.828i
91.14 −0.331992 + 7.99311i 0 −63.7796 5.30729i 55.9017 0 552.003i 63.5960 508.035i 0 −18.5589 + 446.828i
91.15 1.36400 7.88286i 0 −60.2790 21.5045i 55.9017 0 86.8984i −251.737 + 445.839i 0 76.2500 440.665i
91.16 1.36400 + 7.88286i 0 −60.2790 + 21.5045i 55.9017 0 86.8984i −251.737 445.839i 0 76.2500 + 440.665i
91.17 3.08457 7.38143i 0 −44.9709 45.5370i −55.9017 0 200.003i −474.844 + 191.487i 0 −172.433 + 412.634i
91.18 3.08457 + 7.38143i 0 −44.9709 + 45.5370i −55.9017 0 200.003i −474.844 191.487i 0 −172.433 412.634i
91.19 3.53904 7.17462i 0 −38.9505 50.7825i −55.9017 0 99.9522i −502.192 + 99.7338i 0 −197.838 + 401.074i
91.20 3.53904 + 7.17462i 0 −38.9505 + 50.7825i −55.9017 0 99.9522i −502.192 99.7338i 0 −197.838 401.074i
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
4.b odd 2 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 180.7.c.b 24
3.b odd 2 1 60.7.c.a 24
4.b odd 2 1 inner 180.7.c.b 24
12.b even 2 1 60.7.c.a 24

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
60.7.c.a 24 3.b odd 2 1
60.7.c.a 24 12.b even 2 1
180.7.c.b 24 1.a even 1 1 trivial
180.7.c.b 24 4.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$12\!\cdots\!80$$$$T_{7}^{20} +$$$$51\!\cdots\!48$$$$T_{7}^{18} +$$$$12\!\cdots\!20$$$$T_{7}^{16} +$$$$20\!\cdots\!40$$$$T_{7}^{14} +$$$$20\!\cdots\!84$$$$T_{7}^{12} +$$$$13\!\cdots\!20$$$$T_{7}^{10} +$$$$53\!\cdots\!00$$$$T_{7}^{8} +$$$$12\!\cdots\!92$$$$T_{7}^{6} +$$$$17\!\cdots\!00$$$$T_{7}^{4} +$$$$11\!\cdots\!80$$$$T_{7}^{2} +$$$$30\!\cdots\!16$$">$$T_{7}^{24} + \cdots$$ acting on $$S_{7}^{\mathrm{new}}(180, [\chi])$$.