Properties

Label 91.10.u.a
Level $91$
Weight $10$
Character orbit 91.u
Analytic conductor $46.868$
Analytic rank $0$
Dimension $164$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 91.u (of order \(6\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(46.8682610909\)
Analytic rank: \(0\)
Dimension: \(164\)
Relative dimension: \(82\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 164 q - 3 q^{2} + 322 q^{3} + 20481 q^{4} - 6 q^{6} + 1023 q^{7} + 1021818 q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 164 q - 3 q^{2} + 322 q^{3} + 20481 q^{4} - 6 q^{6} + 1023 q^{7} + 1021818 q^{9} - 82050 q^{10} + 43064 q^{12} - 232462 q^{13} + 82708 q^{14} + 338445 q^{15} - 4712643 q^{16} + 638795 q^{17} - 1176873 q^{18} + 571062 q^{20} + 1367523 q^{21} - 1417332 q^{22} + 1074033 q^{23} + 29660776 q^{25} - 4350684 q^{26} + 7350574 q^{27} - 5807904 q^{28} + 996812 q^{29} + 18354032 q^{30} + 21474090 q^{31} + 786429 q^{32} - 14921172 q^{35} + 131881331 q^{36} + 6476901 q^{37} - 8950274 q^{38} + 27439160 q^{39} - 7901342 q^{40} + 24649770 q^{41} + 107058858 q^{42} + 15892214 q^{43} + 1176612 q^{44} + 20834727 q^{45} + 277543272 q^{46} + 69222114 q^{47} - 15455476 q^{48} - 69917499 q^{49} + 309166224 q^{50} + 52823362 q^{51} - 95118408 q^{52} - 155327528 q^{53} + 82174923 q^{54} - 77925569 q^{55} - 282686919 q^{56} - 216056427 q^{59} + 290928939 q^{60} + 937523074 q^{61} + 238770500 q^{62} - 521903604 q^{63} - 2035250696 q^{64} + 418729590 q^{65} - 154929237 q^{66} - 598026797 q^{68} + 127053056 q^{69} + 1180478502 q^{70} - 269829840 q^{71} + 35353098 q^{73} - 460031457 q^{74} + 803288072 q^{75} + 1140965370 q^{76} - 1032378781 q^{77} - 835494667 q^{78} - 683323974 q^{79} + 3448679004 q^{81} + 649989068 q^{82} + 2491042464 q^{84} - 196261605 q^{85} - 572477022 q^{86} - 1849967933 q^{87} - 74806442 q^{88} - 207815607 q^{89} - 9086462948 q^{90} - 1222123653 q^{91} - 2683122798 q^{92} + 389990277 q^{93} + 5123642528 q^{94} + 198982451 q^{95} - 392112 q^{96} + 6155159124 q^{97} + 1935772011 q^{98} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
30.1 −38.4423 + 22.1947i −211.181 729.208 1263.02i 1212.41 + 699.984i 8118.27 4687.09i −6075.53 + 1855.15i 42010.8i 24914.3 −62143.7
30.2 −37.6617 + 21.7440i 84.7460 689.605 1194.43i 1069.03 + 617.207i −3191.68 + 1842.72i 664.570 6317.59i 37713.2i −12501.1 −53682.2
30.3 −37.1580 + 21.4532i 7.72989 664.476 1150.91i −1575.27 909.485i −287.227 + 165.831i 1841.00 + 6079.83i 35052.4i −19623.2 78045.3
30.4 −35.5005 + 20.4962i 132.067 584.192 1011.85i 1239.03 + 715.356i −4688.46 + 2706.88i −3482.51 + 5312.79i 26906.8i −2241.26 −58648.4
30.5 −35.1232 + 20.2784i −146.367 566.427 981.080i −436.798 252.185i 5140.89 2968.10i 5917.60 + 2309.88i 25179.8i 1740.41 20455.7
30.6 −35.0393 + 20.2299i 269.124 562.501 974.280i −63.4069 36.6080i −9429.90 + 5444.36i 5725.53 + 2751.71i 24802.0i 52744.6 2962.31
30.7 −34.8194 + 20.1030i 195.742 552.260 956.543i −1750.16 1010.46i −6815.62 + 3935.00i −6019.15 2030.62i 23822.8i 18631.9 81252.8
30.8 −32.7341 + 18.8991i −212.836 458.349 793.883i −1677.20 968.331i 6967.00 4022.40i 454.420 6336.17i 15296.8i 25616.1 73202.1
30.9 −32.3735 + 18.6908i −146.497 442.694 766.768i 1496.61 + 864.068i 4742.60 2738.14i 2994.62 5602.31i 13957.8i 1778.26 −64600.5
30.10 −32.2435 + 18.6158i −94.9975 437.095 757.071i −1072.58 619.257i 3063.05 1768.45i −6352.31 41.5992i 13484.9i −10658.5 46111.8
30.11 −32.1895 + 18.5846i 70.4411 434.776 753.054i −714.561 412.552i −2267.46 + 1309.12i 5001.70 3916.20i 13289.9i −14721.1 30668.5
30.12 −29.1216 + 16.8134i −81.0362 309.379 535.861i 1658.21 + 957.369i 2359.91 1362.49i 4828.80 + 4127.50i 3589.95i −13116.1 −64386.4
30.13 −28.4816 + 16.4439i 94.5061 284.803 493.293i 1116.08 + 644.368i −2691.69 + 1554.05i −1759.05 + 6104.04i 1894.52i −10751.6 −42383.6
30.14 −28.2880 + 16.3321i 21.2897 277.473 480.598i 611.718 + 353.175i −602.244 + 347.706i −6100.90 1769.93i 1402.82i −19229.7 −23072.3
30.15 −27.3854 + 15.8110i 258.288 243.973 422.574i 2120.68 + 1224.37i −7073.31 + 4083.78i −4845.78 4107.55i 760.607i 47029.5 −77434.2
30.16 −26.6838 + 15.4059i −254.883 218.682 378.768i 455.574 + 263.026i 6801.24 3926.70i −1765.12 + 6102.29i 2299.67i 45282.4 −16208.6
30.17 −26.3184 + 15.1950i 183.597 205.773 356.410i −1465.14 845.901i −4831.98 + 2789.75i −2061.39 6008.69i 3052.77i 14024.8 51413.7
30.18 −23.8716 + 13.7823i 180.122 123.902 214.605i 97.1768 + 56.1050i −4299.81 + 2482.50i 6336.56 449.013i 7282.43i 12761.0 −3093.02
30.19 −23.7092 + 13.6885i −252.262 118.751 205.683i −1023.39 590.853i 5980.94 3453.10i 5001.22 + 3916.81i 7514.94i 43953.3 32351.6
30.20 −22.2527 + 12.8476i 176.414 74.1218 128.383i −1048.20 605.176i −3925.69 + 2266.50i −1476.29 + 6178.53i 9346.80i 11439.0 31100.2
See next 80 embeddings (of 164 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 88.82
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
91.u even 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.10.u.a yes 164
7.c even 3 1 91.10.k.a 164
13.e even 6 1 91.10.k.a 164
91.u even 6 1 inner 91.10.u.a yes 164
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.10.k.a 164 7.c even 3 1
91.10.k.a 164 13.e even 6 1
91.10.u.a yes 164 1.a even 1 1 trivial
91.10.u.a yes 164 91.u even 6 1 inner

Hecke kernels

This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(91, [\chi])\).