# Properties

 Label 75.9.k.a Level $75$ Weight $9$ Character orbit 75.k Analytic conductor $30.553$ Analytic rank $0$ Dimension $320$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$75 = 3 \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$9$$ Character orbit: $$[\chi]$$ $$=$$ 75.k (of order $$20$$, degree $$8$$, minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$30.5533957546$$ Analytic rank: $$0$$ Dimension: $$320$$ Relative dimension: $$40$$ over $$\Q(\zeta_{20})$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{20}]$

## $q$-expansion

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

 $$\operatorname{Tr}(f)(q) =$$ $$320 q + 444 q^{5} - 4540 q^{7} - 17460 q^{8}+O(q^{10})$$ 320 * q + 444 * q^5 - 4540 * q^7 - 17460 * q^8 $$\operatorname{Tr}(f)(q) =$$ $$320 q + 444 q^{5} - 4540 q^{7} - 17460 q^{8} + 25496 q^{10} - 22680 q^{12} - 133420 q^{13} + 59616 q^{15} + 1310720 q^{16} + 168000 q^{17} - 718900 q^{19} - 641844 q^{20} + 2257100 q^{22} + 1089120 q^{23} - 1459756 q^{25} + 1827840 q^{26} + 3260740 q^{28} + 3913800 q^{29} + 3585384 q^{30} - 641460 q^{32} - 1458000 q^{33} - 4121600 q^{34} + 853140 q^{35} - 22394880 q^{36} - 14553460 q^{37} + 17491260 q^{38} + 3203652 q^{40} + 8749440 q^{41} + 9643860 q^{42} - 23946520 q^{43} + 4277700 q^{44} - 2204496 q^{45} + 26491200 q^{47} - 3661200 q^{48} - 47100924 q^{50} - 169691020 q^{52} - 97028940 q^{53} + 27431764 q^{55} - 4714200 q^{57} + 179119320 q^{58} + 95587200 q^{59} + 38705364 q^{60} + 36738240 q^{61} - 296094660 q^{62} - 9928980 q^{63} - 59695100 q^{64} + 24018972 q^{65} - 88558160 q^{67} + 197811840 q^{68} + 220501420 q^{70} + 38185020 q^{72} + 110964400 q^{73} + 51345576 q^{75} - 97175880 q^{77} - 131709240 q^{78} - 138506200 q^{79} - 339741204 q^{80} + 382637520 q^{81} + 217169400 q^{82} + 862340400 q^{83} + 743029200 q^{84} + 619431268 q^{85} - 335756340 q^{87} - 1380992420 q^{88} - 1141686000 q^{89} - 265834224 q^{90} - 407198400 q^{92} + 402028920 q^{93} + 290042200 q^{94} + 1436174256 q^{95} - 165325860 q^{96} + 1057772920 q^{97} + 1077594720 q^{98}+O(q^{100})$$ 320 * q + 444 * q^5 - 4540 * q^7 - 17460 * q^8 + 25496 * q^10 - 22680 * q^12 - 133420 * q^13 + 59616 * q^15 + 1310720 * q^16 + 168000 * q^17 - 718900 * q^19 - 641844 * q^20 + 2257100 * q^22 + 1089120 * q^23 - 1459756 * q^25 + 1827840 * q^26 + 3260740 * q^28 + 3913800 * q^29 + 3585384 * q^30 - 641460 * q^32 - 1458000 * q^33 - 4121600 * q^34 + 853140 * q^35 - 22394880 * q^36 - 14553460 * q^37 + 17491260 * q^38 + 3203652 * q^40 + 8749440 * q^41 + 9643860 * q^42 - 23946520 * q^43 + 4277700 * q^44 - 2204496 * q^45 + 26491200 * q^47 - 3661200 * q^48 - 47100924 * q^50 - 169691020 * q^52 - 97028940 * q^53 + 27431764 * q^55 - 4714200 * q^57 + 179119320 * q^58 + 95587200 * q^59 + 38705364 * q^60 + 36738240 * q^61 - 296094660 * q^62 - 9928980 * q^63 - 59695100 * q^64 + 24018972 * q^65 - 88558160 * q^67 + 197811840 * q^68 + 220501420 * q^70 + 38185020 * q^72 + 110964400 * q^73 + 51345576 * q^75 - 97175880 * q^77 - 131709240 * q^78 - 138506200 * q^79 - 339741204 * q^80 + 382637520 * q^81 + 217169400 * q^82 + 862340400 * q^83 + 743029200 * q^84 + 619431268 * q^85 - 335756340 * q^87 - 1380992420 * q^88 - 1141686000 * q^89 - 265834224 * q^90 - 407198400 * q^92 + 402028920 * q^93 + 290042200 * q^94 + 1436174256 * q^95 - 165325860 * q^96 + 1057772920 * q^97 + 1077594720 * 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 $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
13.1 −31.1680 + 4.93653i −21.2310 + 41.6683i 703.605 228.615i 559.718 + 278.101i 456.033 1403.52i 1699.82 1699.82i −13603.4 + 6931.29i −1285.49 1769.32i −18818.2 5904.80i
13.2 −29.6660 + 4.69863i 21.2310 41.6683i 614.522 199.670i −583.809 223.142i −434.056 + 1335.89i 2556.62 2556.62i −10441.1 + 5320.03i −1285.49 1769.32i 18367.7 + 3876.61i
13.3 −28.6989 + 4.54547i −21.2310 + 41.6683i 559.498 181.792i −53.0876 622.741i 419.907 1292.34i −1163.90 + 1163.90i −8602.90 + 4383.40i −1285.49 1769.32i 4354.21 + 17630.7i
13.4 −27.5831 + 4.36873i 21.2310 41.6683i 498.269 161.897i 111.872 + 614.906i −403.580 + 1242.09i −1146.98 + 1146.98i −6666.44 + 3396.72i −1285.49 1769.32i −5772.14 16472.3i
13.5 −26.6845 + 4.22641i 21.2310 41.6683i 450.730 146.451i 365.591 506.921i −390.433 + 1201.63i 516.394 516.394i −5246.00 + 2672.97i −1285.49 1769.32i −7613.15 + 15072.1i
13.6 −24.6881 + 3.91021i −21.2310 + 41.6683i 350.742 113.963i −624.780 16.5691i 361.222 1111.73i 1512.89 1512.89i −2512.05 + 1279.95i −1285.49 1769.32i 15489.4 2033.96i
13.7 −22.6671 + 3.59011i −21.2310 + 41.6683i 257.437 83.6462i −191.814 + 594.838i 331.652 1020.72i −1074.42 + 1074.42i −300.281 + 153.001i −1285.49 1769.32i 2212.32 14171.9i
13.8 −21.3788 + 3.38607i −21.2310 + 41.6683i 202.117 65.6718i 611.912 + 127.235i 312.802 962.707i −2654.08 + 2654.08i 838.588 427.282i −1285.49 1769.32i −13512.8 648.155i
13.9 −20.1080 + 3.18480i 21.2310 41.6683i 150.719 48.9717i −314.176 540.295i −294.209 + 905.483i −3324.52 + 3324.52i 1769.06 901.383i −1285.49 1769.32i 8038.19 + 9863.67i
13.10 −19.1290 + 3.02974i 21.2310 41.6683i 113.269 36.8032i −568.921 + 258.756i −279.885 + 861.396i 339.914 339.914i 2362.46 1203.73i −1285.49 1769.32i 10098.9 6673.41i
13.11 −16.6143 + 2.63144i 21.2310 41.6683i 25.6393 8.33071i 450.950 + 432.746i −243.091 + 748.156i 2407.36 2407.36i 3432.86 1749.13i −1285.49 1769.32i −8630.96 6003.11i
13.12 −14.4958 + 2.29591i −21.2310 + 41.6683i −38.6136 + 12.5463i 555.364 286.698i 212.094 652.759i 2517.93 2517.93i 3878.60 1976.25i −1285.49 1769.32i −7392.21 + 5430.98i
13.13 −13.4052 + 2.12318i 21.2310 41.6683i −68.2788 + 22.1851i 619.380 83.6279i −196.137 + 603.649i −1056.12 + 1056.12i 3964.00 2019.76i −1285.49 1769.32i −8125.36 + 2436.10i
13.14 −11.6504 + 1.84524i −21.2310 + 41.6683i −111.144 + 36.1127i −559.413 278.715i 170.462 524.628i 285.116 285.116i 3918.78 1996.72i −1285.49 1769.32i 7031.68 + 2214.88i
13.15 −9.28548 + 1.47068i −21.2310 + 41.6683i −159.413 + 51.7965i −136.341 609.948i 135.860 418.134i −1847.56 + 1847.56i 3548.45 1808.03i −1285.49 1769.32i 2163.02 + 5463.14i
13.16 −9.06505 + 1.43576i 21.2310 41.6683i −163.357 + 53.0778i −123.101 612.757i −132.635 + 408.208i 2061.95 2061.95i 3498.12 1782.38i −1285.49 1769.32i 1995.69 + 5377.93i
13.17 −6.59120 + 1.04394i −21.2310 + 41.6683i −201.116 + 65.3467i −166.950 + 602.290i 96.4387 296.808i 872.910 872.910i 2779.56 1416.26i −1285.49 1769.32i 471.642 4144.10i
13.18 −5.04321 + 0.798766i −21.2310 + 41.6683i −218.674 + 71.0517i 357.372 + 512.748i 73.7894 227.101i 267.664 267.664i 2210.75 1126.43i −1285.49 1769.32i −2211.87 2300.44i
13.19 −2.33258 + 0.369444i 21.2310 41.6683i −238.166 + 77.3848i 379.135 496.872i −34.1290 + 105.038i −309.158 + 309.158i 1065.64 542.970i −1285.49 1769.32i −700.795 + 1299.06i
13.20 −0.0204195 + 0.00323412i 21.2310 41.6683i −243.470 + 79.1082i 217.278 + 586.017i −0.298766 + 0.919507i −2757.81 + 2757.81i 9.43137 4.80552i −1285.49 1769.32i −6.33194 11.2634i
See next 80 embeddings (of 320 total)
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 73.40 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.f odd 20 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 75.9.k.a 320
25.f odd 20 1 inner 75.9.k.a 320

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
75.9.k.a 320 1.a even 1 1 trivial
75.9.k.a 320 25.f odd 20 1 inner

## Hecke kernels

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