Properties

Label 69.7.f.a
Level $69$
Weight $7$
Character orbit 69.f
Analytic conductor $15.874$
Analytic rank $0$
Dimension $240$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 69 = 3 \cdot 23 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 69.f (of order \(22\), degree \(10\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(15.8737317698\)
Analytic rank: \(0\)
Dimension: \(240\)
Relative dimension: \(24\) over \(\Q(\zeta_{22})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{22}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 240 q + 20 q^{2} - 816 q^{4} + 324 q^{6} + 940 q^{8} - 5832 q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 240 q + 20 q^{2} - 816 q^{4} + 324 q^{6} + 940 q^{8} - 5832 q^{9} - 384 q^{13} + 33816 q^{16} + 22880 q^{17} + 4860 q^{18} - 31680 q^{19} - 157696 q^{20} + 36488 q^{23} + 39204 q^{24} + 200664 q^{25} + 99032 q^{26} - 202752 q^{28} - 181712 q^{29} - 128496 q^{31} + 439300 q^{32} - 400026 q^{34} + 368744 q^{35} + 42282 q^{36} + 446688 q^{37} + 550550 q^{38} - 52488 q^{39} - 156750 q^{40} - 606080 q^{41} - 1274130 q^{42} - 429000 q^{43} - 867482 q^{44} + 309288 q^{46} + 1111152 q^{47} + 837864 q^{48} + 722136 q^{49} + 1786926 q^{50} + 463320 q^{51} - 812550 q^{52} - 367840 q^{53} - 354294 q^{54} - 398088 q^{55} - 3696550 q^{56} - 1240272 q^{57} + 1903134 q^{58} + 545800 q^{59} + 2450250 q^{60} - 495264 q^{61} + 488776 q^{62} - 2699808 q^{64} - 290400 q^{67} + 23328 q^{69} - 392616 q^{70} + 255392 q^{71} + 228420 q^{72} + 1346880 q^{73} - 5387470 q^{74} + 365472 q^{75} + 2123022 q^{76} + 6134240 q^{77} + 171072 q^{78} + 4025472 q^{79} + 10626462 q^{80} - 1417176 q^{81} + 2025492 q^{82} - 2474560 q^{83} - 8866152 q^{85} - 15634300 q^{86} - 611712 q^{87} - 10636560 q^{88} - 5901720 q^{89} + 6566394 q^{92} - 2484432 q^{93} + 13931490 q^{94} + 8183072 q^{95} - 4682124 q^{96} + 12319560 q^{97} + 14139580 q^{98} + 2993760 q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
7.1 −10.0393 + 11.5859i −13.1138 + 8.42776i −24.3387 169.280i −28.9992 + 13.2435i 34.0100 236.544i −177.630 604.953i 1380.21 + 887.009i 100.946 221.041i 137.692 468.937i
7.2 −9.70976 + 11.2057i 13.1138 8.42776i −22.1792 154.260i 28.9455 13.2190i −32.8937 + 228.781i 60.2734 + 205.272i 1145.64 + 736.258i 100.946 221.041i −132.927 + 452.706i
7.3 −7.55085 + 8.71415i −13.1138 + 8.42776i −9.81285 68.2498i −14.6117 + 6.67294i 25.5800 177.913i 9.29392 + 31.6522i 48.0315 + 30.8680i 100.946 221.041i 52.1818 177.715i
7.4 −7.46772 + 8.61821i 13.1138 8.42776i −9.39853 65.3682i −12.8791 + 5.88169i −25.2984 + 175.954i −31.2375 106.385i 19.5745 + 12.5798i 100.946 221.041i 45.4879 154.918i
7.5 −7.37873 + 8.51551i −13.1138 + 8.42776i −8.96008 62.3187i 186.661 85.2451i 24.9969 173.857i 65.3761 + 222.650i −9.86246 6.33822i 100.946 221.041i −651.414 + 2218.51i
7.6 −5.37670 + 6.20505i −13.1138 + 8.42776i −0.485518 3.37685i −45.2350 + 20.6582i 18.2146 126.686i −13.8057 47.0180i −418.489 268.946i 100.946 221.041i 115.031 391.758i
7.7 −4.49904 + 5.19217i 13.1138 8.42776i 2.39089 + 16.6290i 160.721 73.3986i −15.2414 + 106.006i −169.038 575.691i −466.992 300.117i 100.946 221.041i −341.990 + 1164.71i
7.8 −4.02561 + 4.64580i 13.1138 8.42776i 3.73022 + 25.9443i −145.258 + 66.3370i −13.6375 + 94.8511i 94.3193 + 321.222i −466.519 299.813i 100.946 221.041i 276.562 941.884i
7.9 −3.87564 + 4.47273i 13.1138 8.42776i 4.12344 + 28.6791i 175.461 80.1304i −13.1295 + 91.3177i 148.103 + 504.394i −462.896 297.485i 100.946 221.041i −321.623 + 1095.35i
7.10 −2.99454 + 3.45588i −13.1138 + 8.42776i 6.13230 + 42.6511i −203.315 + 92.8510i 10.1446 70.5571i −106.030 361.104i −411.960 264.751i 100.946 221.041i 287.953 980.679i
7.11 −0.826670 + 0.954028i −13.1138 + 8.42776i 8.88136 + 61.7712i 46.0457 21.0284i 2.80051 19.4779i 134.637 + 458.531i −134.239 86.2703i 100.946 221.041i −18.0029 + 61.3124i
7.12 0.0241383 0.0278571i 13.1138 8.42776i 9.10796 + 63.3472i −89.8556 + 41.0357i 0.0817732 0.568745i −160.665 547.174i 3.96908 + 2.55077i 100.946 221.041i −1.02583 + 3.49365i
7.13 0.203656 0.235032i −13.1138 + 8.42776i 9.09439 + 63.2528i 171.608 78.3705i −0.689926 + 4.79854i −84.9865 289.438i 33.4624 + 21.5050i 100.946 221.041i 16.5294 56.2939i
7.14 0.439258 0.506931i −13.1138 + 8.42776i 9.04412 + 62.9032i −167.785 + 76.6249i −1.48807 + 10.3498i 132.687 + 451.891i 71.9745 + 46.2552i 100.946 221.041i −34.8575 + 118.714i
7.15 0.991787 1.14458i 13.1138 8.42776i 8.78172 + 61.0782i −45.4301 + 20.7472i 3.35987 23.3684i 28.4560 + 96.9124i 160.160 + 102.928i 100.946 221.041i −21.3100 + 72.5754i
7.16 3.76160 4.34111i 13.1138 8.42776i 4.41249 + 30.6896i 172.538 78.7957i 12.7431 88.6305i 43.8846 + 149.457i 459.089 + 295.039i 100.946 221.041i 306.959 1045.41i
7.17 4.04842 4.67213i −13.1138 + 8.42776i 3.66909 + 25.5191i −6.23883 + 2.84918i −13.7148 + 95.3887i −83.3001 283.694i 466.929 + 300.077i 100.946 221.041i −11.9457 + 40.6833i
7.18 5.80964 6.70469i 13.1138 8.42776i −2.09271 14.5551i −138.264 + 63.1430i 19.6813 136.886i 130.352 + 443.939i 367.902 + 236.436i 100.946 221.041i −379.910 + 1293.85i
7.19 6.08888 7.02694i −13.1138 + 8.42776i −3.19530 22.2238i −77.8070 + 35.5333i −20.6273 + 143.466i −42.4272 144.494i 324.984 + 208.854i 100.946 221.041i −224.067 + 763.103i
7.20 6.34640 7.32413i 13.1138 8.42776i −4.25801 29.6151i 79.9255 36.5007i 21.4997 149.533i −34.3232 116.894i 277.849 + 178.563i 100.946 221.041i 239.903 817.033i
See next 80 embeddings (of 240 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 67.24
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
23.d odd 22 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 69.7.f.a 240
23.d odd 22 1 inner 69.7.f.a 240
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
69.7.f.a 240 1.a even 1 1 trivial
69.7.f.a 240 23.d odd 22 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{7}^{\mathrm{new}}(69, [\chi])\).