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Properties

Label 83.13.b.c
Level $83$
Weight $13$
Character orbit 83.b
Analytic conductor $75.861$
Analytic rank $0$
Dimension $80$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 83 \)
Weight: \( k \) \(=\) \( 13 \)
Character orbit: \([\chi]\) \(=\) 83.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(75.8614868339\)
Analytic rank: \(0\)
Dimension: \(80\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 80 q + 918 q^{3} - 181150 q^{4} + 103918 q^{7} + 14902142 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 80 q + 918 q^{3} - 181150 q^{4} + 103918 q^{7} + 14902142 q^{9} + 1177482 q^{10} + 510406 q^{11} - 9260402 q^{12} + 203791850 q^{16} + 5571718 q^{17} + 46565862 q^{21} + 804389464 q^{23} - 4263713272 q^{25} + 18061794 q^{26} - 2326165338 q^{27} - 204811652 q^{28} + 1486960270 q^{29} - 2621683648 q^{30} - 665743010 q^{31} + 135224502 q^{33} - 54936709824 q^{36} - 280627202 q^{37} + 6646145310 q^{38} - 7119722058 q^{40} + 51318109072 q^{41} - 11512674650 q^{44} + 85368259738 q^{48} + 148785395094 q^{49} - 100143389562 q^{51} - 63584241050 q^{59} - 29216180978 q^{61} - 332932206620 q^{63} - 323596534090 q^{64} + 362112989184 q^{65} - 86115426752 q^{68} + 272383417100 q^{69} + 105630718656 q^{70} + 785418808326 q^{75} - 663355117738 q^{77} + 1483841884620 q^{78} + 2430778545148 q^{81} + 837315119192 q^{83} - 3013574788354 q^{84} + 452180651958 q^{86} - 682263689498 q^{87} - 499714512022 q^{90} + 997428187414 q^{92} - 1487992716298 q^{93} + 3169817690580 q^{94} + 1542762610848 q^{95} + 2021347267420 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
82.1 − 125.545i 1153.33 −11665.6 − 3006.48i − 144796.i 213778. 950326.i 798738. −377449.
82.2 − 124.655i −510.562 −11442.8 12605.4i 63644.0i 14598.4 915813.i −270767. 1.57132e6
82.3 − 118.839i 844.819 −10026.8 − 13361.5i − 100398.i −171650. 704815.i 182277. −1.58787e6
82.4 − 115.485i −823.731 −9240.80 − 17633.0i 95128.7i −95956.6 594147.i 147092. −2.03635e6
82.5 − 112.239i 1121.74 −8501.62 14550.4i − 125903.i −118949. 494483.i 726860. 1.63312e6
82.6 − 112.222i −1315.59 −8497.84 − 11772.7i 147639.i 180491. 493984.i 1.19935e6 −1.32116e6
82.7 − 111.294i 126.234 −8290.25 − 22804.9i − 14049.0i 79992.9 466793.i −515506. −2.53804e6
82.8 − 108.230i 117.748 −7617.70 16840.2i − 12743.8i −114224. 381152.i −517576. 1.82261e6
82.9 − 107.274i 399.023 −7411.71 1290.82i − 42804.8i 30813.1 355690.i −372222. 138472.
82.10 − 106.238i −1381.89 −7190.55 23413.1i 146810.i −193774. 328759.i 1.37818e6 2.48737e6
82.11 − 102.696i −81.5431 −6450.37 18424.7i 8374.11i 214939. 241783.i −524792. 1.89213e6
82.12 − 97.6509i 955.647 −5439.71 28739.5i − 93319.9i 19697.5 131214.i 381821. 2.80644e6
82.13 − 95.5397i −812.758 −5031.83 − 16377.5i 77650.6i −194728. 89409.2i 129134. −1.56470e6
82.14 − 91.3709i −1031.91 −4252.64 26144.5i 94286.4i 118004. 14312.3i 533393. 2.38885e6
82.15 − 88.7339i 1350.47 −3777.70 − 29230.4i − 119832.i −67443.9 − 28244.1i 1.29232e6 −2.59373e6
82.16 − 87.7892i −807.149 −3610.94 − 1433.36i 70858.9i 30455.0 − 42583.4i 120049. −125833.
82.17 − 87.6467i 305.164 −3585.94 − 19977.0i − 26746.6i 69148.7 − 44704.8i −438316. −1.75092e6
82.18 − 87.1338i 1220.66 −3496.31 751.299i − 106361.i 90841.6 − 52253.7i 958575. 65463.6
82.19 − 82.5668i −867.031 −2721.28 8790.22i 71588.0i −11496.7 − 113506.i 220301. 725781.
82.20 − 75.4904i 74.7419 −1602.80 3300.31i − 5642.29i −221798. − 188213.i −525855. 249142.
See all 80 embeddings
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 82.80
Significant digits:
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Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
83.b odd 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 83.13.b.c ✓ 80
83.b odd 2 1 inner 83.13.b.c ✓ 80
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
83.13.b.c ✓ 80 1.a even 1 1 trivial
83.13.b.c ✓ 80 83.b odd 2 1 inner

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{13}^{\mathrm{new}}(83, [\chi])\):

\( T_{2}^{80} + 254415 T_{2}^{78} + 31084966710 T_{2}^{76} + \cdots + 33\!\cdots\!00 \) Copy content Toggle raw display
\( T_{3}^{40} - 459 T_{3}^{39} - 14249015 T_{3}^{38} + 6797605355 T_{3}^{37} + 92487915212076 T_{3}^{36} + \cdots + 49\!\cdots\!00 \) Copy content Toggle raw display