Properties

 Label 75.4.g.b Level $75$ Weight $4$ Character orbit 75.g Analytic conductor $4.425$ Analytic rank $0$ Dimension $28$ CM no Inner twists $2$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 75.g (of order $$5$$, degree $$4$$, minimal)

Newform invariants

 Self dual: no Analytic conductor: $$4.42514325043$$ Analytic rank: $$0$$ Dimension: $$28$$ Relative dimension: $$7$$ over $$\Q(\zeta_{5})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{5}]$

$q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$28 q - 21 q^{3} - 30 q^{4} - 15 q^{5} - 54 q^{7} - 63 q^{8} - 63 q^{9}+O(q^{10})$$ 28 * q - 21 * q^3 - 30 * q^4 - 15 * q^5 - 54 * q^7 - 63 * q^8 - 63 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$28 q - 21 q^{3} - 30 q^{4} - 15 q^{5} - 54 q^{7} - 63 q^{8} - 63 q^{9} + 165 q^{10} + 19 q^{11} - 60 q^{12} + 4 q^{13} - 24 q^{14} + 45 q^{15} - 66 q^{16} + 208 q^{17} + 42 q^{19} + 295 q^{20} + 3 q^{21} - 89 q^{22} + 32 q^{23} + 126 q^{24} + 95 q^{25} + 206 q^{26} - 189 q^{27} - 482 q^{28} - 716 q^{29} - 645 q^{30} + 637 q^{31} - 844 q^{32} + 42 q^{33} - 90 q^{34} + 430 q^{35} - 180 q^{36} + 216 q^{37} + 2314 q^{38} + 12 q^{39} - 500 q^{40} - 38 q^{41} + 933 q^{42} - 1392 q^{43} + 603 q^{44} + 270 q^{45} + 1622 q^{46} - 536 q^{47} - 198 q^{48} + 162 q^{49} - 2265 q^{50} - 876 q^{51} - 1922 q^{52} + 1672 q^{53} - 1000 q^{55} + 3000 q^{56} - 1104 q^{57} - 827 q^{58} + 973 q^{59} + 1365 q^{60} - 2712 q^{61} + 1057 q^{62} + 234 q^{63} + 4439 q^{64} - 4360 q^{65} + 1098 q^{66} + 2768 q^{67} - 1370 q^{68} + 396 q^{69} + 3230 q^{70} - 1074 q^{71} - 567 q^{72} - 1018 q^{73} - 1414 q^{74} + 765 q^{75} - 11408 q^{76} + 1607 q^{77} + 168 q^{78} - 1820 q^{79} - 1290 q^{80} - 567 q^{81} + 1772 q^{82} + 4045 q^{83} + 774 q^{84} + 1850 q^{85} - 3986 q^{86} + 1392 q^{87} + 2407 q^{88} + 4542 q^{89} - 180 q^{90} + 4412 q^{91} - 1089 q^{92} - 5334 q^{93} + 5137 q^{94} - 720 q^{95} + 1623 q^{96} - 5977 q^{97} - 10689 q^{98} - 594 q^{99}+O(q^{100})$$ 28 * q - 21 * q^3 - 30 * q^4 - 15 * q^5 - 54 * q^7 - 63 * q^8 - 63 * q^9 + 165 * q^10 + 19 * q^11 - 60 * q^12 + 4 * q^13 - 24 * q^14 + 45 * q^15 - 66 * q^16 + 208 * q^17 + 42 * q^19 + 295 * q^20 + 3 * q^21 - 89 * q^22 + 32 * q^23 + 126 * q^24 + 95 * q^25 + 206 * q^26 - 189 * q^27 - 482 * q^28 - 716 * q^29 - 645 * q^30 + 637 * q^31 - 844 * q^32 + 42 * q^33 - 90 * q^34 + 430 * q^35 - 180 * q^36 + 216 * q^37 + 2314 * q^38 + 12 * q^39 - 500 * q^40 - 38 * q^41 + 933 * q^42 - 1392 * q^43 + 603 * q^44 + 270 * q^45 + 1622 * q^46 - 536 * q^47 - 198 * q^48 + 162 * q^49 - 2265 * q^50 - 876 * q^51 - 1922 * q^52 + 1672 * q^53 - 1000 * q^55 + 3000 * q^56 - 1104 * q^57 - 827 * q^58 + 973 * q^59 + 1365 * q^60 - 2712 * q^61 + 1057 * q^62 + 234 * q^63 + 4439 * q^64 - 4360 * q^65 + 1098 * q^66 + 2768 * q^67 - 1370 * q^68 + 396 * q^69 + 3230 * q^70 - 1074 * q^71 - 567 * q^72 - 1018 * q^73 - 1414 * q^74 + 765 * q^75 - 11408 * q^76 + 1607 * q^77 + 168 * q^78 - 1820 * q^79 - 1290 * q^80 - 567 * q^81 + 1772 * q^82 + 4045 * q^83 + 774 * q^84 + 1850 * q^85 - 3986 * q^86 + 1392 * q^87 + 2407 * q^88 + 4542 * q^89 - 180 * q^90 + 4412 * q^91 - 1089 * q^92 - 5334 * q^93 + 5137 * q^94 - 720 * q^95 + 1623 * q^96 - 5977 * q^97 - 10689 * q^98 - 594 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
16.1 −3.76530 2.73565i 0.927051 + 2.85317i 4.22155 + 12.9926i 5.34090 9.82216i 4.31465 13.2791i −26.0445 8.14206 25.0587i −7.28115 + 5.29007i −46.9800 + 22.3725i
16.2 −3.16070 2.29638i 0.927051 + 2.85317i 2.24451 + 6.90789i −11.0852 1.45535i 3.62184 11.1469i 22.0918 −0.889297 + 2.73697i −7.28115 + 5.29007i 31.6950 + 30.0558i
16.3 −1.08389 0.787491i 0.927051 + 2.85317i −1.91746 5.90135i −3.83217 + 10.5031i 1.24203 3.82256i −12.2101 −5.88101 + 18.0999i −7.28115 + 5.29007i 12.4247 8.36635i
16.4 0.109191 + 0.0793317i 0.927051 + 2.85317i −2.46651 7.59113i −6.22327 9.28822i −0.125121 + 0.385084i −17.3099 0.666555 2.05145i −7.28115 + 5.29007i 0.0573272 1.50789i
16.5 0.772797 + 0.561470i 0.927051 + 2.85317i −2.19017 6.74065i 11.0970 1.36218i −0.885547 + 2.72543i 12.4836 4.45357 13.7067i −7.28115 + 5.29007i 9.34059 + 5.17797i
16.6 3.25026 + 2.36146i 0.927051 + 2.85317i 2.51561 + 7.74226i −6.00716 + 9.42943i −3.72447 + 11.4627i 1.75849 −0.174670 + 0.537580i −7.28115 + 5.29007i −41.7920 + 16.4625i
16.7 3.87763 + 2.81726i 0.927051 + 2.85317i 4.62691 + 14.2402i 7.51887 8.27445i −4.44337 + 13.6753i 0.140520 −10.3279 + 31.7859i −7.28115 + 5.29007i 52.4667 10.9026i
31.1 −1.53022 4.70953i −2.42705 + 1.76336i −13.3660 + 9.71093i −9.83672 + 5.31403i 12.0185 + 8.73195i 28.2766 34.1374 + 24.8023i 2.78115 8.55951i 40.0789 + 38.1947i
31.2 −0.907834 2.79403i −2.42705 + 1.76336i −0.510282 + 0.370741i 10.7234 + 3.16365i 7.13022 + 5.18041i 18.9115 −17.5148 12.7253i 2.78115 8.55951i −0.895733 32.8335i
31.3 −0.722966 2.22506i −2.42705 + 1.76336i 2.04392 1.48499i −1.87995 + 11.0212i 5.67825 + 4.12549i −32.9322 −19.9239 14.4756i 2.78115 8.55951i 25.8819 3.78491i
31.4 −0.177563 0.546483i −2.42705 + 1.76336i 6.20502 4.50821i −9.45304 5.96993i 1.39460 + 1.01324i −2.67744 −7.28438 5.29241i 2.78115 8.55951i −1.58395 + 6.22597i
31.5 0.536885 + 1.65236i −2.42705 + 1.76336i 4.03008 2.92802i 8.44668 7.32487i −4.21675 3.06365i 7.66213 18.2465 + 13.2569i 2.78115 8.55951i 16.6382 + 10.0244i
31.6 1.31310 + 4.04131i −2.42705 + 1.76336i −8.13578 + 5.91099i −8.20823 7.59111i −10.3132 7.49299i −28.2853 −7.06929 5.13614i 2.78115 8.55951i 19.8998 43.1398i
31.7 1.48860 + 4.58143i −2.42705 + 1.76336i −12.3014 + 8.93752i 5.89885 + 9.49755i −11.6916 8.49444i 1.13492 −28.0809 20.4020i 2.78115 8.55951i −34.7314 + 41.1632i
46.1 −1.53022 + 4.70953i −2.42705 1.76336i −13.3660 9.71093i −9.83672 5.31403i 12.0185 8.73195i 28.2766 34.1374 24.8023i 2.78115 + 8.55951i 40.0789 38.1947i
46.2 −0.907834 + 2.79403i −2.42705 1.76336i −0.510282 0.370741i 10.7234 3.16365i 7.13022 5.18041i 18.9115 −17.5148 + 12.7253i 2.78115 + 8.55951i −0.895733 + 32.8335i
46.3 −0.722966 + 2.22506i −2.42705 1.76336i 2.04392 + 1.48499i −1.87995 11.0212i 5.67825 4.12549i −32.9322 −19.9239 + 14.4756i 2.78115 + 8.55951i 25.8819 + 3.78491i
46.4 −0.177563 + 0.546483i −2.42705 1.76336i 6.20502 + 4.50821i −9.45304 + 5.96993i 1.39460 1.01324i −2.67744 −7.28438 + 5.29241i 2.78115 + 8.55951i −1.58395 6.22597i
46.5 0.536885 1.65236i −2.42705 1.76336i 4.03008 + 2.92802i 8.44668 + 7.32487i −4.21675 + 3.06365i 7.66213 18.2465 13.2569i 2.78115 + 8.55951i 16.6382 10.0244i
46.6 1.31310 4.04131i −2.42705 1.76336i −8.13578 5.91099i −8.20823 + 7.59111i −10.3132 + 7.49299i −28.2853 −7.06929 + 5.13614i 2.78115 + 8.55951i 19.8998 + 43.1398i
See all 28 embeddings
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 61.7 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
25.d even 5 1 inner

Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.4.g.b 28
3.b odd 2 1 225.4.h.a 28
25.d even 5 1 inner 75.4.g.b 28
25.d even 5 1 1875.4.a.g 14
25.e even 10 1 1875.4.a.f 14
75.j odd 10 1 225.4.h.a 28

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.4.g.b 28 1.a even 1 1 trivial
75.4.g.b 28 25.d even 5 1 inner
225.4.h.a 28 3.b odd 2 1
225.4.h.a 28 75.j odd 10 1
1875.4.a.f 14 25.e even 10 1
1875.4.a.g 14 25.d even 5 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{28} + 43 T_{2}^{26} + 21 T_{2}^{25} + 1285 T_{2}^{24} + 803 T_{2}^{23} + 33580 T_{2}^{22} + 26209 T_{2}^{21} + 852818 T_{2}^{20} + 655690 T_{2}^{19} + 13441499 T_{2}^{18} + 14783120 T_{2}^{17} + \cdots + 1769380096$$ acting on $$S_{4}^{\mathrm{new}}(75, [\chi])$$.