Properties

Label 1997.4.a.a
Level $1997$
Weight $4$
Character orbit 1997.a
Self dual yes
Analytic conductor $117.827$
Analytic rank $1$
Dimension $239$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1997 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1997.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(117.826814281\)
Analytic rank: \(1\)
Dimension: \(239\)
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.

\(\operatorname{Tr}(f)(q) = \) \( 239q - 16q^{2} - 106q^{3} + 872q^{4} - 85q^{5} - 111q^{6} - 352q^{7} - 210q^{8} + 1961q^{9} + O(q^{10}) \)
\(\operatorname{Tr}(f)(q) = \) \( 239q - 16q^{2} - 106q^{3} + 872q^{4} - 85q^{5} - 111q^{6} - 352q^{7} - 210q^{8} + 1961q^{9} - 273q^{10} - 294q^{11} - 864q^{12} - 797q^{13} - 220q^{14} - 580q^{15} + 2816q^{16} - 439q^{17} - 536q^{18} - 1704q^{19} - 933q^{20} - 596q^{21} - 1046q^{22} - 829q^{23} - 1237q^{24} + 4364q^{25} - 818q^{26} - 3670q^{27} - 3690q^{28} - 316q^{29} - 888q^{30} - 2595q^{31} - 1881q^{32} - 2066q^{33} - 2605q^{34} - 2450q^{35} + 5863q^{36} - 1912q^{37} - 1709q^{38} - 914q^{39} - 3582q^{40} - 1064q^{41} - 3228q^{42} - 5184q^{43} - 2656q^{44} - 3967q^{45} - 2521q^{46} - 4909q^{47} - 7461q^{48} + 7193q^{49} - 1906q^{50} - 3240q^{51} - 9614q^{52} - 2722q^{53} - 3754q^{54} - 6018q^{55} - 2347q^{56} - 2032q^{57} - 6709q^{58} - 6318q^{59} - 5821q^{60} - 2990q^{61} - 2117q^{62} - 8738q^{63} + 6866q^{64} - 1738q^{65} - 3080q^{66} - 14729q^{67} - 3897q^{68} - 2080q^{69} - 7445q^{70} - 3240q^{71} - 8263q^{72} - 8828q^{73} - 3103q^{74} - 12716q^{75} - 14843q^{76} - 3818q^{77} - 8029q^{78} - 4794q^{79} - 10336q^{80} + 11899q^{81} - 13447q^{82} - 11434q^{83} - 7957q^{84} - 8188q^{85} - 5196q^{86} - 11266q^{87} - 11861q^{88} - 4845q^{89} - 7759q^{90} - 12734q^{91} - 8644q^{92} - 10130q^{93} - 6909q^{94} - 3686q^{95} - 11958q^{96} - 16108q^{97} - 6845q^{98} - 12372q^{99} + O(q^{100}) \)

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 \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1 −5.59440 −8.36152 23.2974 −4.51522 46.7777 −9.81959 −85.5797 42.9151 25.2599
1.2 −5.52780 3.43977 22.5565 −10.5187 −19.0144 −34.6164 −80.4656 −15.1679 58.1450
1.3 −5.51428 7.20768 22.4073 9.62817 −39.7451 −3.48165 −79.4456 24.9506 −53.0924
1.4 −5.50614 −8.56073 22.3176 13.6270 47.1366 0.910634 −78.8349 46.2860 −75.0320
1.5 −5.50179 8.24154 22.2697 −8.43995 −45.3433 −6.56247 −78.5092 40.9229 46.4349
1.6 −5.43887 0.991773 21.5813 −3.96164 −5.39413 −25.9935 −73.8672 −26.0164 21.5469
1.7 −5.32569 −8.11542 20.3630 −6.55697 43.2203 14.6874 −65.8417 38.8601 34.9204
1.8 −5.28431 4.93207 19.9239 14.3183 −26.0626 −11.8358 −63.0097 −2.67464 −75.6624
1.9 −5.28041 −3.44790 19.8827 16.8405 18.2063 27.1027 −62.7455 −15.1120 −88.9245
1.10 −5.27417 −3.32423 19.8169 −16.6866 17.5325 −23.2918 −62.3242 −15.9495 88.0082
1.11 −5.23446 3.65257 19.3996 −6.07203 −19.1192 18.8571 −59.6706 −13.6587 31.7838
1.12 −5.19350 0.362833 18.9725 7.86550 −1.88438 18.4887 −56.9857 −26.8684 −40.8495
1.13 −5.12919 −8.38676 18.3086 −1.93936 43.0173 −30.3660 −52.8747 43.3378 9.94735
1.14 −5.08140 3.08170 17.8206 −3.39990 −15.6594 5.43909 −49.9023 −17.5031 17.2762
1.15 −5.05732 −7.84414 17.5765 13.2678 39.6703 9.68917 −48.4315 34.5305 −67.0996
1.16 −5.02812 −2.19694 17.2820 −12.8877 11.0465 −2.11320 −46.6709 −22.1734 64.8011
1.17 −4.91538 −4.80472 16.1610 1.32248 23.6170 24.0734 −40.1143 −3.91469 −6.50051
1.18 −4.88824 6.13395 15.8948 2.37453 −29.9842 −2.66042 −38.5919 10.6254 −11.6073
1.19 −4.85005 −9.72090 15.5230 −21.5228 47.1469 −0.473007 −36.4868 67.4960 104.387
1.20 −4.81468 −1.36926 15.1811 −10.0416 6.59255 −5.82889 −34.5748 −25.1251 48.3471
See next 80 embeddings (of 239 total)
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.239
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(1997\) \(-1\)

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 1997.4.a.a 239
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1997.4.a.a 239 1.a even 1 1 trivial