Properties

Label 91.10.bc.a
Level $91$
Weight $10$
Character orbit 91.bc
Analytic conductor $46.868$
Analytic rank $0$
Dimension $328$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 91.bc (of order \(12\), degree \(4\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(46.8682610909\)
Analytic rank: \(0\)
Dimension: \(328\)
Relative dimension: \(82\) over \(\Q(\zeta_{12})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{12}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 328 q - 8 q^{2} - 12 q^{4} - 916 q^{7} - 2056 q^{8} + 1023512 q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 328 q - 8 q^{2} - 12 q^{4} - 916 q^{7} - 2056 q^{8} + 1023512 q^{9} - 83032 q^{11} + 21448 q^{14} - 108452 q^{15} + 9961468 q^{16} - 733972 q^{18} - 2737398 q^{21} + 2436220 q^{22} - 12 q^{23} - 15223908 q^{28} - 668252 q^{29} + 13271412 q^{30} - 14102516 q^{32} - 1621814 q^{35} + 4973904 q^{36} + 61121584 q^{37} + 69333140 q^{39} + 95547212 q^{42} - 352770276 q^{43} - 64700376 q^{44} - 218239228 q^{46} + 107128758 q^{49} - 564307244 q^{50} - 200988768 q^{53} - 414599232 q^{56} + 697378756 q^{57} + 543458848 q^{58} + 212624012 q^{60} + 840037568 q^{63} - 1107622952 q^{65} + 358974440 q^{67} + 709259864 q^{70} - 1649468064 q^{71} + 2246747752 q^{72} + 615361080 q^{74} + 3791118716 q^{78} - 1806303888 q^{79} - 6487448396 q^{81} + 7421566672 q^{84} - 1238748100 q^{85} - 2371677312 q^{86} - 9232155156 q^{88} - 5018797796 q^{91} - 3132502872 q^{92} + 1279633044 q^{93} + 4801302204 q^{95} + 8438875060 q^{98} + 8386949632 q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
6.1 −11.3834 42.4834i −216.879 125.215i −1231.85 + 711.210i −460.761 460.761i −2850.75 + 10639.1i 6338.10 426.780i 28314.1 + 28314.1i 21516.3 + 37267.3i −14329.7 + 24819.7i
6.2 −11.3834 42.4834i 216.879 + 125.215i −1231.85 + 711.210i 460.761 + 460.761i 2850.75 10639.1i −5275.56 3538.65i 28314.1 + 28314.1i 21516.3 + 37267.3i 14329.7 24819.7i
6.3 −11.2984 42.1661i −30.5561 17.6416i −1206.92 + 696.818i −1738.47 1738.47i −398.643 + 1487.76i −3004.11 + 5597.23i 27214.1 + 27214.1i −9219.05 15967.9i −53662.8 + 92946.6i
6.4 −11.2984 42.1661i 30.5561 + 17.6416i −1206.92 + 696.818i 1738.47 + 1738.47i 398.643 1487.76i −196.979 + 6349.39i 27214.1 + 27214.1i −9219.05 15967.9i 53662.8 92946.6i
6.5 −10.4056 38.8341i −174.119 100.528i −956.405 + 552.181i 1035.54 + 1035.54i −2092.09 + 7807.80i −6203.64 1366.90i 16840.0 + 16840.0i 10370.1 + 17961.6i 29438.9 50989.7i
6.6 −10.4056 38.8341i 174.119 + 100.528i −956.405 + 552.181i −1035.54 1035.54i 2092.09 7807.80i 6055.96 + 1918.05i 16840.0 + 16840.0i 10370.1 + 17961.6i −29438.9 + 50989.7i
6.7 −10.3743 38.7173i −28.8864 16.6776i −947.997 + 547.326i −258.041 258.041i −346.035 + 1291.42i 3584.60 5244.45i 16514.1 + 16514.1i −9285.22 16082.5i −7313.67 + 12667.6i
6.8 −10.3743 38.7173i 28.8864 + 16.6776i −947.997 + 547.326i 258.041 + 258.041i 346.035 1291.42i −482.125 6334.13i 16514.1 + 16514.1i −9285.22 16082.5i 7313.67 12667.6i
6.9 −9.34057 34.8595i −52.8126 30.4914i −684.533 + 395.215i 0.948714 + 0.948714i −569.614 + 2125.83i −6165.54 + 1529.63i 7105.25 + 7105.25i −7982.05 13825.3i 24.2102 41.9332i
6.10 −9.34057 34.8595i 52.8126 + 30.4914i −684.533 + 395.215i −0.948714 0.948714i 569.614 2125.83i 4574.69 + 4407.47i 7105.25 + 7105.25i −7982.05 13825.3i −24.2102 + 41.9332i
6.11 −8.78129 32.7722i −159.314 91.9801i −553.502 + 319.564i 74.9438 + 74.9438i −1615.41 + 6028.78i 1498.81 + 6173.10i 3049.93 + 3049.93i 7079.17 + 12261.5i 1797.97 3114.18i
6.12 −8.78129 32.7722i 159.314 + 91.9801i −553.502 + 319.564i −74.9438 74.9438i 1615.41 6028.78i −4384.56 + 4596.65i 3049.93 + 3049.93i 7079.17 + 12261.5i −1797.97 + 3114.18i
6.13 −8.71884 32.5391i −158.391 91.4473i −539.372 + 311.407i −1215.97 1215.97i −1594.63 + 5951.23i −4870.43 4078.29i 2639.61 + 2639.61i 6883.72 + 11922.9i −28964.8 + 50168.5i
6.14 −8.71884 32.5391i 158.391 + 91.4473i −539.372 + 311.407i 1215.97 + 1215.97i 1594.63 5951.23i 6257.07 1096.69i 2639.61 + 2639.61i 6883.72 + 11922.9i 28964.8 50168.5i
6.15 −8.37802 31.2672i −114.682 66.2115i −464.042 + 267.915i 1420.94 + 1420.94i −1109.44 + 4140.50i 6248.40 + 1145.02i 545.436 + 545.436i −1073.58 1859.49i 32524.2 56333.6i
6.16 −8.37802 31.2672i 114.682 + 66.2115i −464.042 + 267.915i −1420.94 1420.94i 1109.44 4140.50i −5983.79 2132.59i 545.436 + 545.436i −1073.58 1859.49i −32524.2 + 56333.6i
6.17 −7.52760 28.0934i −122.597 70.7812i −289.168 + 166.951i −1762.96 1762.96i −1065.63 + 3976.97i 5140.90 3731.58i −3662.72 3662.72i 178.466 + 309.112i −36256.6 + 62798.3i
6.18 −7.52760 28.0934i 122.597 + 70.7812i −289.168 + 166.951i 1762.96 + 1762.96i 1065.63 3976.97i −2586.36 5802.10i −3662.72 3662.72i 178.466 + 309.112i 36256.6 62798.3i
6.19 −6.75438 25.2077i −153.838 88.8186i −146.400 + 84.5241i 1293.94 + 1293.94i −1199.83 + 4477.82i 376.502 6341.28i −6328.59 6328.59i 5935.98 + 10281.4i 23877.4 41356.9i
6.20 −6.75438 25.2077i 153.838 + 88.8186i −146.400 + 84.5241i −1293.94 1293.94i 1199.83 4477.82i 2844.58 5679.96i −6328.59 6328.59i 5935.98 + 10281.4i −23877.4 + 41356.9i
See next 80 embeddings (of 328 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 76.82
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.b odd 2 1 inner
13.f odd 12 1 inner
91.bc even 12 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 91.10.bc.a 328
7.b odd 2 1 inner 91.10.bc.a 328
13.f odd 12 1 inner 91.10.bc.a 328
91.bc even 12 1 inner 91.10.bc.a 328
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
91.10.bc.a 328 1.a even 1 1 trivial
91.10.bc.a 328 7.b odd 2 1 inner
91.10.bc.a 328 13.f odd 12 1 inner
91.10.bc.a 328 91.bc even 12 1 inner

Hecke kernels

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