Properties

Label 983.6.a.a
Level $983$
Weight $6$
Character orbit 983.a
Self dual yes
Analytic conductor $157.657$
Analytic rank $1$
Dimension $191$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 983 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 983.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(157.657294876\)
Analytic rank: \(1\)
Dimension: \(191\)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

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

\(\operatorname{Tr}(f)(q) = \) \( 191 q - 37 q^{2} - 74 q^{3} + 2821 q^{4} - 247 q^{5} - 335 q^{6} - 1569 q^{7} - 1761 q^{8} + 13251 q^{9}+O(q^{10}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q) = \) \( 191 q - 37 q^{2} - 74 q^{3} + 2821 q^{4} - 247 q^{5} - 335 q^{6} - 1569 q^{7} - 1761 q^{8} + 13251 q^{9} - 2867 q^{10} - 1878 q^{11} - 3400 q^{12} - 6854 q^{13} - 1601 q^{14} - 3461 q^{15} + 38457 q^{16} - 10730 q^{17} - 17249 q^{18} - 5817 q^{19} - 9988 q^{20} - 15999 q^{21} - 20287 q^{22} - 15625 q^{23} - 19747 q^{24} + 67082 q^{25} - 9868 q^{26} - 22892 q^{27} - 72720 q^{28} - 17960 q^{29} - 27464 q^{30} - 25604 q^{31} - 68869 q^{32} - 60654 q^{33} - 42876 q^{34} - 30018 q^{35} + 172922 q^{36} - 114862 q^{37} + 4404 q^{38} - 73500 q^{39} - 137154 q^{40} - 90896 q^{41} - 10652 q^{42} - 121447 q^{43} - 57962 q^{44} - 109019 q^{45} - 136262 q^{46} - 86994 q^{47} - 133347 q^{48} + 278242 q^{49} - 93911 q^{50} - 66966 q^{51} - 284241 q^{52} - 122112 q^{53} - 130806 q^{54} - 134904 q^{55} - 100292 q^{56} - 423426 q^{57} - 307669 q^{58} - 85704 q^{59} - 238277 q^{60} - 206736 q^{61} - 190602 q^{62} - 387623 q^{63} + 411903 q^{64} - 244408 q^{65} - 113963 q^{66} - 337002 q^{67} - 388031 q^{68} - 165342 q^{69} - 183925 q^{70} - 174806 q^{71} - 753621 q^{72} - 1009738 q^{73} - 204958 q^{74} - 282676 q^{75} - 326869 q^{76} - 332288 q^{77} - 591801 q^{78} - 488092 q^{79} - 259068 q^{80} + 385959 q^{81} - 523996 q^{82} - 315720 q^{83} - 750486 q^{84} - 1001755 q^{85} - 287709 q^{86} - 316995 q^{87} - 836923 q^{88} - 298065 q^{89} - 751039 q^{90} - 521459 q^{91} - 640932 q^{92} - 554391 q^{93} - 623481 q^{94} - 491883 q^{95} - 767843 q^{96} - 1468693 q^{97} - 714146 q^{98} - 842507 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −11.1976 −23.7621 93.3852 76.4799 266.078 −19.5877 −687.365 321.638 −856.388
1.2 −11.1262 29.5188 91.7918 67.1507 −328.432 −250.979 −665.254 628.360 −747.131
1.3 −10.9287 4.75838 87.4366 −74.3663 −52.0029 −185.895 −605.851 −220.358 812.728
1.4 −10.8272 14.3547 85.2276 −26.4676 −155.420 140.784 −576.304 −36.9436 286.569
1.5 −10.8057 19.0440 84.7636 −41.1837 −205.784 −3.45765 −570.149 119.674 445.020
1.6 −10.7746 −26.7262 84.0930 −46.3632 287.965 −164.750 −561.284 471.287 499.547
1.7 −10.6536 −21.7329 81.4987 −24.1347 231.533 102.660 −527.338 229.319 257.121
1.8 −10.6191 17.1432 80.7650 60.4272 −182.045 43.6018 −517.839 50.8902 −641.681
1.9 −10.4383 −14.1728 76.9585 25.9677 147.941 174.030 −469.291 −42.1308 −271.059
1.10 −10.4121 −2.05023 76.4119 54.9016 21.3472 184.115 −462.422 −238.797 −571.641
1.11 −10.1648 24.7155 71.3234 −57.1368 −251.229 −211.752 −399.716 367.858 580.785
1.12 −10.0336 −29.0942 68.6734 −75.3888 291.920 17.3499 −367.966 603.471 756.422
1.13 −9.88271 −13.0845 65.6680 −13.4569 129.310 −107.789 −332.731 −71.7966 132.990
1.14 −9.83897 −20.7623 64.8053 50.1722 204.279 3.16220 −322.770 188.071 −493.643
1.15 −9.80318 16.6427 64.1023 2.99274 −163.152 −200.824 −314.705 33.9806 −29.3383
1.16 −9.78124 27.6828 63.6727 21.4120 −270.772 51.6520 −309.799 523.335 −209.436
1.17 −9.77505 10.8192 63.5517 −59.8667 −105.759 −143.755 −308.419 −125.944 585.200
1.18 −9.77050 −4.83034 63.4627 −33.1528 47.1948 39.6159 −307.406 −219.668 323.919
1.19 −9.71106 −11.8693 62.3048 93.6186 115.264 163.882 −294.291 −102.119 −909.136
1.20 −9.64751 −11.8379 61.0744 −72.0672 114.206 −61.6914 −280.496 −102.865 695.269
See next 80 embeddings (of 191 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.191
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(983\) \(1\)

Inner twists

This newform does not admit any (nontrivial) inner twists.

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 983.6.a.a 191
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
983.6.a.a 191 1.a even 1 1 trivial