# Properties

 Label 75.9.j.a Level $75$ Weight $9$ Character orbit 75.j Analytic conductor $30.553$ Analytic rank $0$ Dimension $312$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 75.j (of order $$10$$, degree $$4$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$30.5533957546$$ Analytic rank: $$0$$ Dimension: $$312$$ Relative dimension: $$78$$ over $$\Q(\zeta_{10})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{10}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$312 q - 73 q^{3} + 9722 q^{4} - 515 q^{6} - 176 q^{7} + 7907 q^{9}+O(q^{10})$$ 312 * q - 73 * q^3 + 9722 * q^4 - 515 * q^6 - 176 * q^7 + 7907 * q^9 $$\operatorname{Tr}(f)(q) =$$ $$312 q - 73 q^{3} + 9722 q^{4} - 515 q^{6} - 176 q^{7} + 7907 q^{9} - 1340 q^{10} - 41153 q^{12} - 47466 q^{13} - 183555 q^{15} - 1375138 q^{16} - 124530 q^{18} - 215676 q^{19} - 322116 q^{21} + 506210 q^{22} - 885600 q^{24} - 2285000 q^{25} + 1668122 q^{27} + 795754 q^{28} + 511395 q^{30} + 137814 q^{31} + 185030 q^{33} - 3354230 q^{34} - 1217933 q^{36} + 8147494 q^{37} + 6148789 q^{39} + 12667680 q^{40} - 20632975 q^{42} + 6373244 q^{43} + 42801695 q^{45} - 5724210 q^{46} + 45688462 q^{48} + 215489176 q^{49} + 4033340 q^{51} + 50335444 q^{52} - 11570780 q^{54} + 1252690 q^{55} + 127383454 q^{57} + 910270 q^{58} + 42215760 q^{60} - 25290306 q^{61} - 103896031 q^{63} + 154389042 q^{64} - 29877340 q^{66} - 47397296 q^{67} + 41856605 q^{69} - 258904800 q^{70} - 239805005 q^{72} - 140624106 q^{73} - 35216050 q^{75} - 212861216 q^{76} + 88431575 q^{78} - 106524466 q^{79} - 142401613 q^{81} - 440177940 q^{82} - 80599906 q^{84} + 255505030 q^{85} - 214355530 q^{87} - 142625130 q^{88} - 76416350 q^{90} - 49394062 q^{91} - 843269096 q^{93} + 501646740 q^{94} - 115465550 q^{96} + 129442874 q^{97} + 136982720 q^{99}+O(q^{100})$$ 312 * q - 73 * q^3 + 9722 * q^4 - 515 * q^6 - 176 * q^7 + 7907 * q^9 - 1340 * q^10 - 41153 * q^12 - 47466 * q^13 - 183555 * q^15 - 1375138 * q^16 - 124530 * q^18 - 215676 * q^19 - 322116 * q^21 + 506210 * q^22 - 885600 * q^24 - 2285000 * q^25 + 1668122 * q^27 + 795754 * q^28 + 511395 * q^30 + 137814 * q^31 + 185030 * q^33 - 3354230 * q^34 - 1217933 * q^36 + 8147494 * q^37 + 6148789 * q^39 + 12667680 * q^40 - 20632975 * q^42 + 6373244 * q^43 + 42801695 * q^45 - 5724210 * q^46 + 45688462 * q^48 + 215489176 * q^49 + 4033340 * q^51 + 50335444 * q^52 - 11570780 * q^54 + 1252690 * q^55 + 127383454 * q^57 + 910270 * q^58 + 42215760 * q^60 - 25290306 * q^61 - 103896031 * q^63 + 154389042 * q^64 - 29877340 * q^66 - 47397296 * q^67 + 41856605 * q^69 - 258904800 * q^70 - 239805005 * q^72 - 140624106 * q^73 - 35216050 * q^75 - 212861216 * q^76 + 88431575 * q^78 - 106524466 * q^79 - 142401613 * q^81 - 440177940 * q^82 - 80599906 * q^84 + 255505030 * q^85 - 214355530 * q^87 - 142625130 * q^88 - 76416350 * q^90 - 49394062 * q^91 - 843269096 * q^93 + 501646740 * q^94 - 115465550 * q^96 + 129442874 * q^97 + 136982720 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
11.1 −18.2305 25.0921i −80.9595 2.56149i −218.155 + 671.412i −34.5911 + 624.042i 1411.66 + 2078.14i −3717.22 13272.8 4312.60i 6547.88 + 414.754i 16289.1 10508.6i
11.2 −17.9702 24.7339i −3.85988 + 80.9080i −209.728 + 645.475i 609.095 + 140.101i 2070.53 1358.46i 3815.59 12290.4 3993.39i −6531.20 624.590i −7480.33 17582.9i
11.3 −17.6326 24.2691i 12.6048 + 80.0132i −198.976 + 612.384i −559.573 278.393i 1719.60 1716.75i −2015.68 11066.8 3595.82i −6243.24 + 2017.11i 3110.35 + 18489.2i
11.4 −17.3034 23.8161i 79.9396 + 13.0637i −188.690 + 580.728i 518.937 348.324i −1072.10 2129.89i −3638.55 9928.28 3225.89i 6219.68 + 2088.62i −17275.1 6331.84i
11.5 −17.0444 23.4596i −29.5485 75.4181i −180.733 + 556.239i −601.366 170.248i −1265.64 + 1978.65i 1578.44 9069.54 2946.87i −4814.77 + 4456.98i 6255.98 + 17009.6i
11.6 −16.9218 23.2908i 40.8105 69.9679i −177.008 + 544.774i 356.232 + 513.541i −2320.20 + 233.469i 1515.19 8674.23 2818.43i −3230.00 5710.85i 5932.71 16987.0i
11.7 −16.6151 22.8688i 76.4502 26.7649i −167.810 + 516.465i −553.809 + 289.691i −1882.31 1303.62i −219.465 7716.82 2507.35i 5128.28 4092.37i 15826.5 + 7851.68i
11.8 −16.3948 22.5655i −80.8710 4.56896i −161.305 + 496.445i 295.172 550.907i 1222.76 + 1899.80i 1770.00 7056.09 2292.66i 6519.25 + 738.994i −17270.8 + 2371.32i
11.9 −15.8525 21.8191i −5.79551 80.7924i −145.663 + 448.304i 426.206 457.136i −1670.94 + 1407.21i −1651.43 5524.34 1794.97i −6493.82 + 936.466i −16730.7 2052.68i
11.10 −14.9990 20.6444i 74.9382 + 30.7451i −122.112 + 375.821i −233.811 579.619i −489.288 2008.20i 4241.10 3377.31 1097.36i 4670.48 + 4607.96i −8458.94 + 13520.6i
11.11 −14.0772 19.3756i −55.8311 + 58.6847i −98.1381 + 302.038i −216.170 586.426i 1923.00 + 255.644i −1564.18 1402.67 455.754i −326.786 6552.86i −8319.29 + 12443.7i
11.12 −13.8207 19.0226i 34.2136 + 73.4196i −91.7383 + 282.341i −323.695 + 534.646i 923.774 1665.54i −716.299 913.979 296.970i −4219.86 + 5023.89i 14644.1 1231.69i
11.13 −13.7113 18.8720i −65.6121 + 47.4980i −89.0441 + 274.050i −427.105 + 456.297i 1796.01 + 586.972i 3153.01 713.322 231.773i 2048.89 6232.88i 14467.4 + 1803.89i
11.14 −13.0609 17.9768i −58.5411 55.9816i −73.4700 + 226.118i 333.636 + 528.500i −241.770 + 1783.55i 2630.47 −385.592 + 125.287i 293.121 + 6554.45i 5143.16 12900.4i
11.15 −13.0487 17.9600i 76.2368 + 27.3669i −73.1849 + 225.240i 326.265 + 533.081i −503.283 1726.32i 791.787 −404.710 + 131.498i 5063.11 + 4172.73i 5316.81 12815.8i
11.16 −12.6564 17.4201i −41.7102 + 69.4353i −64.1661 + 197.483i 571.801 + 252.327i 1737.47 152.208i −2610.02 −990.225 + 321.744i −3081.52 5792.32i −2841.41 13154.4i
11.17 −11.9719 16.4779i 55.6718 58.8358i −49.0862 + 151.072i −314.756 539.957i −1635.98 212.979i −1258.15 −1881.96 + 611.485i −362.291 6550.99i −5129.12 + 11650.8i
11.18 −11.3839 15.6686i −62.4373 51.6002i −36.8032 + 113.269i 619.683 + 81.3505i −97.7219 + 1565.72i −1394.12 −2521.68 + 819.342i 1235.84 + 6443.56i −5779.76 10635.6i
11.19 −11.3177 15.5774i −6.75329 80.7180i −35.4584 + 109.130i −344.132 + 521.726i −1180.95 + 1018.74i −4430.09 −2586.70 + 840.469i −6469.79 + 1090.22i 12021.9 544.023i
11.20 −10.5050 14.4588i −77.0617 24.9499i −19.5955 + 60.3088i −616.818 100.797i 448.784 + 1376.32i −818.953 −3273.49 + 1063.62i 5316.01 + 3845.36i 5022.24 + 9977.35i
See next 80 embeddings (of 312 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 71.78 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
3.b odd 2 1 inner
25.d even 5 1 inner
75.j odd 10 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.9.j.a 312
3.b odd 2 1 inner 75.9.j.a 312
25.d even 5 1 inner 75.9.j.a 312
75.j odd 10 1 inner 75.9.j.a 312

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.9.j.a 312 1.a even 1 1 trivial
75.9.j.a 312 3.b odd 2 1 inner
75.9.j.a 312 25.d even 5 1 inner
75.9.j.a 312 75.j odd 10 1 inner

## Hecke kernels

This newform subspace is the entire newspace $$S_{9}^{\mathrm{new}}(75, [\chi])$$.