LMFDB - Embedded newform 1143.3.b.a.890.80

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1143,3,Mod(890,1143)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1143, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1143.890");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1143 = 3^{2} \cdot 127 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1143.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$31.1444942164$
Analytic rank:	$0$
Dimension:	$84$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			890.80
Character	$\chi$	$=$	1143.890
Dual form			1143.3.b.a.890.5

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+3.63886i q^{2} -9.24130 q^{4} +3.49336i q^{5} -9.62828 q^{7} -19.0724i q^{8} +O(q^{10})$	\(q+3.63886i q^{2} -9.24130 q^{4} +3.49336i q^{5} -9.62828 q^{7} -19.0724i q^{8} -12.7118 q^{10} -19.0515i q^{11} +20.1962 q^{13} -35.0360i q^{14} +32.4365 q^{16} +0.819801i q^{17} +7.28753 q^{19} -32.2832i q^{20} +69.3258 q^{22} -0.446991i q^{23} +12.7965 q^{25} +73.4911i q^{26} +88.9779 q^{28} +43.3692i q^{29} -12.5475 q^{31} +41.7423i q^{32} -2.98314 q^{34} -33.6350i q^{35} -31.1753 q^{37} +26.5183i q^{38} +66.6266 q^{40} +17.5224i q^{41} +21.4646 q^{43} +176.061i q^{44} +1.62654 q^{46} -61.7080i q^{47} +43.7038 q^{49} +46.5645i q^{50} -186.639 q^{52} -65.6615i q^{53} +66.5537 q^{55} +183.634i q^{56} -157.814 q^{58} -15.3487i q^{59} +59.7978 q^{61} -45.6585i q^{62} -22.1485 q^{64} +70.5525i q^{65} -51.8217 q^{67} -7.57603i q^{68} +122.393 q^{70} -11.6672i q^{71} +123.785 q^{73} -113.443i q^{74} -67.3463 q^{76} +183.433i q^{77} +52.3561 q^{79} +113.312i q^{80} -63.7615 q^{82} +92.4955i q^{83} -2.86386 q^{85} +78.1067i q^{86} -363.357 q^{88} -1.56409i q^{89} -194.455 q^{91} +4.13078i q^{92} +224.547 q^{94} +25.4580i q^{95} -40.8492 q^{97} +159.032i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 84 q - 160 q^{4}+O(q^{10}) $ 84 * q - 160 * q^4	$ 84 q - 160 q^{4} - 48 q^{10} + 16 q^{13} + 360 q^{16} + 64 q^{19} - 8 q^{22} - 388 q^{25} - 120 q^{28} - 160 q^{31} + 192 q^{34} - 152 q^{37} + 208 q^{40} - 24 q^{43} + 56 q^{46} + 564 q^{49} - 80 q^{52} + 136 q^{55} - 136 q^{58} + 168 q^{61} - 736 q^{64} + 168 q^{67} - 608 q^{70} + 80 q^{73} - 32 q^{76} - 168 q^{79} + 528 q^{82} + 288 q^{85} - 392 q^{88} + 176 q^{91} + 176 q^{94} - 120 q^{97}+O(q^{100}) $ 84 * q - 160 * q^4 - 48 * q^10 + 16 * q^13 + 360 * q^16 + 64 * q^19 - 8 * q^22 - 388 * q^25 - 120 * q^28 - 160 * q^31 + 192 * q^34 - 152 * q^37 + 208 * q^40 - 24 * q^43 + 56 * q^46 + 564 * q^49 - 80 * q^52 + 136 * q^55 - 136 * q^58 + 168 * q^61 - 736 * q^64 + 168 * q^67 - 608 * q^70 + 80 * q^73 - 32 * q^76 - 168 * q^79 + 528 * q^82 + 288 * q^85 - 392 * q^88 + 176 * q^91 + 176 * q^94 - 120 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1143\mathbb{Z}\right)^\times$.

$n$	$128$	$892$
$\chi(n)$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	3.63886i	1.81943i	0.415233	+	0.909715i	$0.363700\pi$
$2$	3.63886i	1.81943i	−0.415233	+	0.909715i	$0.636300\pi$
$3$	0	0
$3$	0	0
$4$	−9.24130	−2.31033
$5$	3.49336i	0.698671i	0.936998	+	0.349336i	$0.113593\pi$
$5$	3.49336i	0.698671i	−0.936998	+	0.349336i	$0.886407\pi$
$6$	0	0
$7$	−9.62828	−1.37547	−0.687734	−	0.725962i	$-0.741395\pi$
$7$	−9.62828	−1.37547	−0.687734	+	0.725962i	$0.741395\pi$
$8$	− 19.0724i	− 2.38405i
$9$	0	0
$10$	−12.7118	−1.27118
$11$	− 19.0515i	− 1.73196i	−0.500083	−	0.865978i	$-0.666697\pi$
$11$	− 19.0515i	− 1.73196i	0.500083	−	0.865978i	$-0.333303\pi$
$12$	0	0
$13$	20.1962	1.55355	0.776776	−	0.629777i	$-0.216854\pi$
$13$	20.1962	1.55355	0.776776	+	0.629777i	$0.216854\pi$
$14$	− 35.0360i	− 2.50257i
$15$	0	0
$16$	32.4365	2.02728
$17$	0.819801i	0.0482236i	0.999709	+	0.0241118i	$0.00767577\pi$
$17$	0.819801i	0.0482236i	−0.999709	+	0.0241118i	$0.992324\pi$
$18$	0	0
$19$	7.28753	0.383554	0.191777	−	0.981438i	$-0.438575\pi$
$19$	7.28753	0.383554	0.191777	+	0.981438i	$0.438575\pi$
$20$	− 32.2832i	− 1.61416i
$21$	0	0
$22$	69.3258	3.15117
$23$	− 0.446991i	− 0.0194344i	−0.999953	−	0.00971719i	$-0.996907\pi$
$23$	− 0.446991i	− 0.0194344i	0.999953	−	0.00971719i	$-0.00309313\pi$
$24$	0	0
$25$	12.7965	0.511858
$26$	73.4911i	2.82658i
$27$	0	0
$28$	88.9779	3.17778
$29$	43.3692i	1.49549i	0.663986	+	0.747745i	$0.268863\pi$
$29$	43.3692i	1.49549i	−0.663986	+	0.747745i	$0.731137\pi$
$30$	0	0
$31$	−12.5475	−0.404757	−0.202379	−	0.979307i	$-0.564867\pi$
$31$	−12.5475	−0.404757	−0.202379	+	0.979307i	$0.564867\pi$
$32$	41.7423i	1.30445i
$33$	0	0
$34$	−2.98314	−0.0877395
$35$	− 33.6350i	− 0.961001i
$36$	0	0
$37$	−31.1753	−0.842576	−0.421288	−	0.906927i	$-0.638422\pi$
$37$	−31.1753	−0.842576	−0.421288	+	0.906927i	$0.638422\pi$
$38$	26.5183i	0.697850i
$39$	0	0
$40$	66.6266	1.66566
$41$	17.5224i	0.427375i	0.976902	+	0.213688i	$0.0685474\pi$
$41$	17.5224i	0.427375i	−0.976902	+	0.213688i	$0.931453\pi$
$42$	0	0
$43$	21.4646	0.499177	0.249588	−	0.968352i	$-0.419705\pi$
$43$	21.4646	0.499177	0.249588	+	0.968352i	$0.419705\pi$
$44$	176.061i	4.00138i
$45$	0	0
$46$	1.62654	0.0353595
$47$	− 61.7080i	− 1.31294i	−0.754354	−	0.656468i	$-0.772050\pi$
$47$	− 61.7080i	− 1.31294i	0.754354	−	0.656468i	$-0.227950\pi$
$48$	0	0
$49$	43.7038	0.891915
$50$	46.5645i	0.931290i
$51$	0	0
$52$	−186.639	−3.58921
$53$	− 65.6615i	− 1.23890i	−0.785038	−	0.619448i	$-0.787356\pi$
$53$	− 65.6615i	− 1.23890i	0.785038	−	0.619448i	$-0.212644\pi$
$54$	0	0
$55$	66.5537	1.21007
$56$	183.634i	3.27918i
$57$	0	0
$58$	−157.814	−2.72094
$59$	− 15.3487i	− 0.260147i	−0.991504	−	0.130073i	$-0.958479\pi$
$59$	− 15.3487i	− 0.260147i	0.991504	−	0.130073i	$-0.0415213\pi$
$60$	0	0
$61$	59.7978	0.980292	0.490146	−	0.871640i	$-0.336943\pi$
$61$	59.7978	0.980292	0.490146	+	0.871640i	$0.336943\pi$
$62$	− 45.6585i	− 0.736427i
$63$	0	0
$64$	−22.1485	−0.346070
$65$	70.5525i	1.08542i
$66$	0	0
$67$	−51.8217	−0.773458	−0.386729	−	0.922193i	$-0.626395\pi$
$67$	−51.8217	−0.773458	−0.386729	+	0.922193i	$0.626395\pi$
$68$	− 7.57603i	− 0.111412i
$69$	0	0
$70$	122.393	1.74847
$71$	− 11.6672i	− 0.164327i	−0.996619	−	0.0821635i	$-0.973817\pi$
$71$	− 11.6672i	− 0.164327i	0.996619	−	0.0821635i	$-0.0261830\pi$
$72$	0	0
$73$	123.785	1.69568	0.847842	−	0.530248i	$-0.177901\pi$
$73$	123.785	1.69568	0.847842	+	0.530248i	$0.177901\pi$
$74$	− 113.443i	− 1.53301i
$75$	0	0
$76$	−67.3463	−0.886136
$77$	183.433i	2.38225i
$78$	0	0
$79$	52.3561	0.662736	0.331368	−	0.943502i	$-0.392490\pi$
$79$	52.3561	0.662736	0.331368	+	0.943502i	$0.392490\pi$
$80$	113.312i	1.41640i
$81$	0	0
$82$	−63.7615	−0.777579
$83$	92.4955i	1.11440i	0.830377	+	0.557202i	$0.188125\pi$
$83$	92.4955i	1.11440i	−0.830377	+	0.557202i	$0.811875\pi$
$84$	0	0
$85$	−2.86386	−0.0336925
$86$	78.1067i	0.908217i
$87$	0	0
$88$	−363.357	−4.12906
$89$	− 1.56409i	− 0.0175740i	−0.999961	−	0.00878700i	$-0.997203\pi$
$89$	− 1.56409i	− 0.0175740i	0.999961	−	0.00878700i	$-0.00279703\pi$
$90$	0	0
$91$	−194.455	−2.13686
$92$	4.13078i	0.0448998i
$93$	0	0
$94$	224.547	2.38880
$95$	25.4580i	0.267978i
$96$	0	0
$97$	−40.8492	−0.421125	−0.210563	−	0.977580i	$-0.567530\pi$
$97$	−40.8492	−0.421125	−0.210563	+	0.977580i	$0.567530\pi$
$98$	159.032i	1.62278i
$99$	0	0
$100$	−118.256	−1.18256
$101$	− 95.6674i	− 0.947202i	−0.880739	−	0.473601i	$-0.842954\pi$
$101$	− 95.6674i	− 0.947202i	0.880739	−	0.473601i	$-0.157046\pi$
$102$	0	0
$103$	−20.6832	−0.200808	−0.100404	−	0.994947i	$-0.532013\pi$
$103$	−20.6832	−0.200808	−0.100404	+	0.994947i	$0.532013\pi$
$104$	− 385.189i	− 3.70374i
$105$	0	0
$106$	238.933	2.25409
$107$	− 102.522i	− 0.958152i	−0.877773	−	0.479076i	$-0.840972\pi$
$107$	− 102.522i	− 0.958152i	0.877773	−	0.479076i	$-0.159028\pi$
$108$	0	0
$109$	100.438	0.921450	0.460725	−	0.887543i	$-0.347589\pi$
$109$	100.438	0.921450	0.460725	+	0.887543i	$0.347589\pi$
$110$	242.180i	2.20163i
$111$	0	0
$112$	−312.307	−2.78846
$113$	117.964i	1.04393i	0.852967	+	0.521964i	$0.174801\pi$
$113$	117.964i	1.04393i	−0.852967	+	0.521964i	$0.825199\pi$
$114$	0	0
$115$	1.56150	0.0135782
$116$	− 400.788i	− 3.45507i
$117$	0	0
$118$	55.8517	0.473319
$119$	− 7.89328i	− 0.0663301i
$120$	0	0
$121$	−241.960	−1.99967
$122$	217.596i	1.78357i
$123$	0	0
$124$	115.955	0.935121
$125$	132.037i	1.05629i
$126$	0	0
$127$	−11.2694	−0.0887357
$127$	−11.2694	−0.0887357
$128$	86.3740i	0.674797i
$129$	0	0
$130$	−256.731	−1.97485
$131$	180.115i	1.37493i	0.726219	+	0.687463i	$0.241276\pi$
$131$	180.115i	1.37493i	−0.726219	+	0.687463i	$0.758724\pi$
$132$	0	0
$133$	−70.1664	−0.527567
$134$	− 188.572i	− 1.40725i
$135$	0	0
$136$	15.6355	0.114967
$137$	179.632i	1.31118i	0.755116	+	0.655591i	$0.227581\pi$
$137$	179.632i	1.31118i	−0.755116	+	0.655591i	$0.772419\pi$
$138$	0	0
$139$	238.348	1.71474	0.857369	−	0.514703i	$-0.172098\pi$
$139$	238.348	1.71474	0.857369	+	0.514703i	$0.172098\pi$
$140$	310.831i	2.22022i
$141$	0	0
$142$	42.4554	0.298981
$143$	− 384.768i	− 2.69068i
$144$	0	0
$145$	−151.504	−1.04486
$146$	450.436i	3.08518i
$147$	0	0
$148$	288.100	1.94662
$149$	152.663i	1.02458i	0.858812	+	0.512292i	$0.171203\pi$
$149$	152.663i	1.02458i	−0.858812	+	0.512292i	$0.828797\pi$
$150$	0	0
$151$	298.056	1.97388	0.986939	−	0.161092i	$-0.0515015\pi$
$151$	298.056	1.97388	0.986939	+	0.161092i	$0.0515015\pi$
$152$	− 138.991i	− 0.914411i
$153$	0	0
$154$	−667.488	−4.33434
$155$	− 43.8328i	− 0.282792i
$156$	0	0
$157$	90.2523	0.574855	0.287428	−	0.957802i	$-0.407200\pi$
$157$	90.2523	0.574855	0.287428	+	0.957802i	$0.407200\pi$
$158$	190.517i	1.20580i
$159$	0	0
$160$	−145.821	−0.911379
$161$	4.30375i	0.0267314i
$162$	0	0
$163$	307.712	1.88780	0.943901	−	0.330229i	$-0.107126\pi$
$163$	307.712	1.88780	0.943901	+	0.330229i	$0.107126\pi$
$164$	− 161.930i	− 0.987376i
$165$	0	0
$166$	−336.578	−2.02758
$167$	− 40.6463i	− 0.243391i	−0.992567	−	0.121696i	$-0.961167\pi$
$167$	− 40.6463i	− 0.243391i	0.992567	−	0.121696i	$-0.0388331\pi$
$168$	0	0
$169$	238.886	1.41352
$170$	− 10.4212i	− 0.0613011i
$171$	0	0
$172$	−198.361	−1.15326
$173$	234.964i	1.35818i	0.734057	+	0.679088i	$0.237624\pi$
$173$	234.964i	1.35818i	−0.734057	+	0.679088i	$0.762376\pi$
$174$	0	0
$175$	−123.208	−0.704045
$176$	− 617.963i	− 3.51116i
$177$	0	0
$178$	5.69149	0.0319747
$179$	− 254.540i	− 1.42201i	−0.703186	−	0.711006i	$-0.748240\pi$
$179$	− 254.540i	− 1.42201i	0.703186	−	0.711006i	$-0.251760\pi$
$180$	0	0
$181$	−160.679	−0.887731	−0.443865	−	0.896094i	$-0.646393\pi$
$181$	−160.679	−0.887731	−0.443865	+	0.896094i	$0.646393\pi$
$182$	− 707.593i	− 3.88787i
$183$	0	0
$184$	−8.52517	−0.0463325
$185$	− 108.906i	− 0.588684i
$186$	0	0
$187$	15.6184	0.0835211
$188$	570.262i	3.03331i
$189$	0	0
$190$	−92.6379	−0.487568
$191$	− 168.426i	− 0.881814i	−0.897553	−	0.440907i	$-0.854657\pi$
$191$	− 168.426i	− 0.881814i	0.897553	−	0.440907i	$-0.145343\pi$
$192$	0	0
$193$	325.539	1.68673	0.843364	−	0.537343i	$-0.180572\pi$
$193$	325.539	1.68673	0.843364	+	0.537343i	$0.180572\pi$
$194$	− 148.644i	− 0.766208i
$195$	0	0
$196$	−403.880	−2.06061
$197$	− 154.639i	− 0.784968i	−0.919759	−	0.392484i	$-0.871616\pi$
$197$	− 154.639i	− 0.784968i	0.919759	−	0.392484i	$-0.128384\pi$
$198$	0	0
$199$	−310.045	−1.55801	−0.779007	−	0.627016i	$-0.784276\pi$
$199$	−310.045	−1.55801	−0.779007	+	0.627016i	$0.784276\pi$
$200$	− 244.059i	− 1.22029i
$201$	0	0
$202$	348.120	1.72337
$203$	− 417.571i	− 2.05700i
$204$	0	0
$205$	−61.2119	−0.298595
$206$	− 75.2632i	− 0.365355i
$207$	0	0
$208$	655.092	3.14948
$209$	− 138.838i	− 0.664299i
$210$	0	0
$211$	212.697	1.00804	0.504021	−	0.863691i	$-0.331853\pi$
$211$	212.697	1.00804	0.504021	+	0.863691i	$0.331853\pi$
$212$	606.798i	2.86225i
$213$	0	0
$214$	373.064	1.74329
$215$	74.9835i	0.348761i
$216$	0	0
$217$	120.811	0.556731
$218$	365.480i	1.67651i
$219$	0	0
$220$	−615.043	−2.79565
$221$	16.5569i	0.0749179i
$222$	0	0
$223$	−259.955	−1.16572	−0.582859	−	0.812573i	$-0.698066\pi$
$223$	−259.955	−1.16572	−0.582859	+	0.812573i	$0.698066\pi$
$224$	− 401.906i	− 1.79423i
$225$	0	0
$226$	−429.254	−1.89936
$227$	213.321i	0.939742i	0.882735	+	0.469871i	$0.155700\pi$
$227$	213.321i	0.939742i	−0.882735	+	0.469871i	$0.844300\pi$
$228$	0	0
$229$	−72.3915	−0.316120	−0.158060	−	0.987430i	$-0.550524\pi$
$229$	−72.3915	−0.316120	−0.158060	+	0.987430i	$0.550524\pi$
$230$	5.68208i	0.0247047i
$231$	0	0
$232$	827.153	3.56532
$233$	− 254.852i	− 1.09378i	−0.837203	−	0.546892i	$-0.815811\pi$
$233$	− 254.852i	− 1.09378i	0.837203	−	0.546892i	$-0.184189\pi$
$234$	0	0
$235$	215.568	0.917311
$236$	141.842i	0.601024i
$237$	0	0
$238$	28.7225	0.120683
$239$	− 99.5547i	− 0.416547i	−0.978071	−	0.208273i	$-0.933216\pi$
$239$	− 99.5547i	− 0.416547i	0.978071	−	0.208273i	$-0.0667844\pi$
$240$	0	0
$241$	280.958	1.16580	0.582900	−	0.812544i	$-0.301918\pi$
$241$	280.958	1.16580	0.582900	+	0.812544i	$0.301918\pi$
$242$	− 880.458i	− 3.63826i
$243$	0	0
$244$	−552.610	−2.26479
$245$	152.673i	0.623155i
$246$	0	0
$247$	147.180	0.595872
$248$	239.310i	0.964960i
$249$	0	0
$250$	−480.462	−1.92185
$251$	− 394.884i	− 1.57324i	−0.617436	−	0.786621i	$-0.711828\pi$
$251$	− 394.884i	− 1.57324i	0.617436	−	0.786621i	$-0.288172\pi$
$252$	0	0
$253$	−8.51585	−0.0336595
$254$	− 41.0079i	− 0.161448i
$255$	0	0
$256$	−402.897	−1.57381
$257$	21.2709i	0.0827660i	0.999143	+	0.0413830i	$0.0131764\pi$
$257$	21.2709i	0.0827660i	−0.999143	+	0.0413830i	$0.986824\pi$
$258$	0	0
$259$	300.165	1.15894
$260$	− 651.997i	− 2.50768i
$261$	0	0
$262$	−655.415	−2.50158
$263$	380.580i	1.44707i	0.690287	+	0.723536i	$0.257484\pi$
$263$	380.580i	1.44707i	−0.690287	+	0.723536i	$0.742516\pi$
$264$	0	0
$265$	229.379	0.865581
$266$	− 255.326i	− 0.959872i
$267$	0	0
$268$	478.900	1.78694
$269$	− 184.288i	− 0.685087i	−0.939502	−	0.342543i	$-0.888712\pi$
$269$	− 184.288i	− 0.685087i	0.939502	−	0.342543i	$-0.111288\pi$
$270$	0	0
$271$	−509.810	−1.88122	−0.940608	−	0.339494i	$-0.889744\pi$
$271$	−509.810	−1.88122	−0.940608	+	0.339494i	$0.889744\pi$
$272$	26.5915i	0.0977627i
$273$	0	0
$274$	−653.656	−2.38561
$275$	− 243.792i	− 0.886515i
$276$	0	0
$277$	78.0363	0.281720	0.140860	−	0.990030i	$-0.455013\pi$
$277$	78.0363	0.281720	0.140860	+	0.990030i	$0.455013\pi$
$278$	867.317i	3.11984i
$279$	0	0
$280$	−641.500	−2.29107
$281$	143.781i	0.511676i	0.966720	+	0.255838i	$0.0823514\pi$
$281$	143.781i	0.511676i	−0.966720	+	0.255838i	$0.917649\pi$
$282$	0	0
$283$	275.281	0.972724	0.486362	−	0.873757i	$-0.338324\pi$
$283$	275.281	0.972724	0.486362	+	0.873757i	$0.338324\pi$
$284$	107.820i	0.379649i
$285$	0	0
$286$	1400.12	4.89551
$287$	− 168.710i	− 0.587841i
$288$	0	0
$289$	288.328	0.997674
$290$	− 551.302i	− 1.90104i
$291$	0	0
$292$	−1143.93	−3.91758
$293$	− 267.650i	− 0.913480i	−0.889600	−	0.456740i	$-0.849017\pi$
$293$	− 267.650i	− 0.913480i	0.889600	−	0.456740i	$-0.150983\pi$
$294$	0	0
$295$	53.6184	0.181757
$296$	594.587i	2.00874i
$297$	0	0
$298$	−555.519	−1.86416
$299$	− 9.02751i	− 0.0301923i
$300$	0	0
$301$	−206.667	−0.686602
$302$	1084.58i	3.59133i
$303$	0	0
$304$	236.382	0.777572
$305$	208.895i	0.684902i
$306$	0	0
$307$	151.145	0.492329	0.246164	−	0.969228i	$-0.420830\pi$
$307$	151.145	0.492329	0.246164	+	0.969228i	$0.420830\pi$
$308$	− 1695.16i	− 5.50377i
$309$	0	0
$310$	159.501	0.514521
$311$	493.265i	1.58606i	0.609182	+	0.793030i	$0.291498\pi$
$311$	493.265i	1.58606i	−0.609182	+	0.793030i	$0.708502\pi$
$312$	0	0
$313$	99.1298	0.316708	0.158354	−	0.987382i	$-0.449381\pi$
$313$	99.1298	0.316708	0.158354	+	0.987382i	$0.449381\pi$
$314$	328.415i	1.04591i
$315$	0	0
$316$	−483.839	−1.53114
$317$	− 195.074i	− 0.615375i	−0.951487	−	0.307687i	$-0.900445\pi$
$317$	− 195.074i	− 0.615375i	0.951487	−	0.307687i	$-0.0995551\pi$
$318$	0	0
$319$	826.249	2.59012
$320$	− 77.3725i	− 0.241789i
$321$	0	0
$322$	−15.6608	−0.0486359
$323$	5.97433i	0.0184964i
$324$	0	0
$325$	258.439	0.795198
$326$	1119.72i	3.43472i
$327$	0	0
$328$	334.193	1.01888
$329$	594.142i	1.80590i
$330$	0	0
$331$	116.566	0.352163	0.176082	−	0.984376i	$-0.443658\pi$
$331$	116.566	0.352163	0.176082	+	0.984376i	$0.443658\pi$
$332$	− 854.778i	− 2.57463i
$333$	0	0
$334$	147.906	0.442833
$335$	− 181.032i	− 0.540393i
$336$	0	0
$337$	−193.848	−0.575216	−0.287608	−	0.957748i	$-0.592860\pi$
$337$	−193.848	−0.575216	−0.287608	+	0.957748i	$0.592860\pi$
$338$	869.271i	2.57181i
$339$	0	0
$340$	26.4658	0.0778405
$341$	239.048i	0.701021i
$342$	0	0
$343$	50.9930	0.148668
$344$	− 409.381i	− 1.19006i
$345$	0	0
$346$	−855.003	−2.47111
$347$	− 365.092i	− 1.05214i	−0.850442	−	0.526069i	$-0.823665\pi$
$347$	− 365.092i	− 1.05214i	0.850442	−	0.526069i	$-0.176335\pi$
$348$	0	0
$349$	619.739	1.77576	0.887879	−	0.460078i	$-0.152179\pi$
$349$	619.739	1.77576	0.887879	+	0.460078i	$0.152179\pi$
$350$	− 448.336i	− 1.28096i
$351$	0	0
$352$	795.253	2.25924
$353$	553.745i	1.56868i	0.620330	+	0.784341i	$0.286999\pi$
$353$	553.745i	1.56868i	−0.620330	+	0.784341i	$0.713001\pi$
$354$	0	0
$355$	40.7577	0.114811
$356$	14.4542i	0.0406017i
$357$	0	0
$358$	926.235	2.58725
$359$	97.3629i	0.271206i	0.990763	+	0.135603i	$0.0432972\pi$
$359$	97.3629i	0.271206i	−0.990763	+	0.135603i	$0.956703\pi$
$360$	0	0
$361$	−307.892	−0.852886
$362$	− 584.689i	− 1.61516i
$363$	0	0
$364$	1797.01	4.93685
$365$	432.425i	1.18473i
$366$	0	0
$367$	−491.906	−1.34034	−0.670172	−	0.742206i	$-0.733780\pi$
$367$	−491.906	−1.34034	−0.670172	+	0.742206i	$0.733780\pi$
$368$	− 14.4988i	− 0.0393989i
$369$	0	0
$370$	396.295	1.07107
$371$	632.208i	1.70406i
$372$	0	0
$373$	379.909	1.01852	0.509261	−	0.860612i	$-0.329919\pi$
$373$	379.909	1.01852	0.509261	+	0.860612i	$0.329919\pi$
$374$	56.8333i	0.151961i
$375$	0	0
$376$	−1176.92	−3.13010
$377$	875.892i	2.32332i
$378$	0	0
$379$	646.240	1.70512	0.852559	−	0.522631i	$-0.175049\pi$
$379$	646.240	1.70512	0.852559	+	0.522631i	$0.175049\pi$
$380$	− 235.265i	− 0.619118i
$381$	0	0
$382$	612.880	1.60440
$383$	363.219i	0.948354i	0.880430	+	0.474177i	$0.157254\pi$
$383$	363.219i	0.948354i	−0.880430	+	0.474177i	$0.842746\pi$
$384$	0	0
$385$	−640.798	−1.66441
$386$	1184.59i	3.06888i
$387$	0	0
$388$	377.499	0.972937
$389$	− 662.872i	− 1.70404i	−0.523509	−	0.852020i	$-0.675377\pi$
$389$	− 662.872i	− 1.70404i	0.523509	−	0.852020i	$-0.324623\pi$
$390$	0	0
$391$	0.366444	0.000937196 0
$392$	− 833.535i	− 2.12637i
$393$	0	0
$394$	562.708	1.42819
$395$	182.899i	0.463035i
$396$	0	0
$397$	24.6407	0.0620674	0.0310337	−	0.999518i	$-0.490120\pi$
$397$	24.6407	0.0620674	0.0310337	+	0.999518i	$0.490120\pi$
$398$	− 1128.21i	− 2.83470i
$399$	0	0
$400$	415.072	1.03768
$401$	− 273.425i	− 0.681858i	−0.940089	−	0.340929i	$-0.889258\pi$
$401$	− 273.425i	− 0.681858i	0.940089	−	0.340929i	$-0.110742\pi$
$402$	0	0
$403$	−253.411	−0.628811
$404$	884.092i	2.18835i
$405$	0	0
$406$	1519.48	3.74257
$407$	593.936i	1.45930i
$408$	0	0
$409$	601.798	1.47139	0.735695	−	0.677313i	$-0.236856\pi$
$409$	601.798	1.47139	0.735695	+	0.677313i	$0.236856\pi$
$410$	− 222.742i	− 0.543272i
$411$	0	0
$412$	191.140	0.463931
$413$	147.781i	0.357824i
$414$	0	0
$415$	−323.120	−0.778602
$416$	843.034i	2.02653i
$417$	0	0
$418$	505.214	1.20865
$419$	− 242.412i	− 0.578550i	−0.957246	−	0.289275i	$-0.906586\pi$
$419$	− 242.412i	− 0.578550i	0.957246	−	0.289275i	$-0.0934142\pi$
$420$	0	0
$421$	76.6518	0.182071	0.0910354	−	0.995848i	$-0.470982\pi$
$421$	76.6518	0.182071	0.0910354	+	0.995848i	$0.470982\pi$
$422$	773.975i	1.83406i
$423$	0	0
$424$	−1252.32	−2.95359
$425$	10.4906i	0.0246836i
$426$	0	0
$427$	−575.750	−1.34836
$428$	947.440i	2.21364i
$429$	0	0
$430$	−272.855	−0.634545
$431$	− 818.889i	− 1.89997i	−0.312290	−	0.949987i	$-0.601096\pi$
$431$	− 818.889i	− 1.89997i	0.312290	−	0.949987i	$-0.398904\pi$
$432$	0	0
$433$	15.8770	0.0366674	0.0183337	−	0.999832i	$-0.494164\pi$
$433$	15.8770	0.0366674	0.0183337	+	0.999832i	$0.494164\pi$
$434$	439.613i	1.01293i
$435$	0	0
$436$	−928.178	−2.12885
$437$	− 3.25746i	− 0.00745414i
$438$	0	0
$439$	−617.332	−1.40622	−0.703111	−	0.711080i	$-0.748206\pi$
$439$	−617.332	−1.40622	−0.703111	+	0.711080i	$0.748206\pi$
$440$	− 1269.34i	− 2.88486i
$441$	0	0
$442$	−60.2481	−0.136308
$443$	− 548.280i	− 1.23765i	−0.785528	−	0.618826i	$-0.787608\pi$
$443$	− 548.280i	− 1.23765i	0.785528	−	0.618826i	$-0.212392\pi$
$444$	0	0
$445$	5.46391	0.0122785
$446$	− 945.941i	− 2.12094i
$447$	0	0
$448$	213.252	0.476008
$449$	209.223i	0.465976i	0.972480	+	0.232988i	$0.0748503\pi$
$449$	209.223i	0.465976i	−0.972480	+	0.232988i	$0.925150\pi$
$450$	0	0
$451$	333.828	0.740195
$452$	− 1090.14i	− 2.41181i
$453$	0	0
$454$	−776.247	−1.70980
$455$	− 679.299i	− 1.49296i
$456$	0	0
$457$	−505.510	−1.10615	−0.553074	−	0.833132i	$-0.686545\pi$
$457$	−505.510	−1.10615	−0.553074	+	0.833132i	$0.686545\pi$
$458$	− 263.422i	− 0.575158i
$459$	0	0
$460$	−14.4303	−0.0313702
$461$	− 839.959i	− 1.82204i	−0.412366	−	0.911018i	$-0.635298\pi$
$461$	− 839.959i	− 1.82204i	0.412366	−	0.911018i	$-0.364702\pi$
$462$	0	0
$463$	−287.564	−0.621089	−0.310545	−	0.950559i	$-0.600511\pi$
$463$	−287.564	−0.621089	−0.310545	+	0.950559i	$0.600511\pi$
$464$	1406.74i	3.03178i
$465$	0	0
$466$	927.369	1.99006
$467$	729.628i	1.56237i	0.624298	+	0.781187i	$0.285385\pi$
$467$	729.628i	1.56237i	−0.624298	+	0.781187i	$0.714615\pi$
$468$	0	0
$469$	498.954	1.06387
$470$	784.422i	1.66898i
$471$	0	0
$472$	−292.735	−0.620202
$473$	− 408.933i	− 0.864552i
$474$	0	0
$475$	93.2546	0.196325
$476$	72.9442i	0.153244i
$477$	0	0
$478$	362.266	0.757878
$479$	202.711i	0.423196i	0.977357	+	0.211598i	$0.0678667\pi$
$479$	202.711i	0.423196i	−0.977357	+	0.211598i	$0.932133\pi$
$480$	0	0
$481$	−629.622	−1.30899
$482$	1022.37i	2.12109i
$483$	0	0
$484$	2236.02	4.61989
$485$	− 142.701i	− 0.294228i
$486$	0	0
$487$	706.928	1.45160	0.725799	−	0.687907i	$-0.241470\pi$
$487$	706.928	1.45160	0.725799	+	0.687907i	$0.241470\pi$
$488$	− 1140.49i	− 2.33706i
$489$	0	0
$490$	−555.556	−1.13379
$491$	− 246.901i	− 0.502854i	−0.967876	−	0.251427i	$-0.919100\pi$
$491$	− 246.901i	− 0.502854i	0.967876	−	0.251427i	$-0.0808999\pi$
$492$	0	0
$493$	−35.5541	−0.0721179
$494$	535.569i	1.08415i
$495$	0	0
$496$	−406.996	−0.820556
$497$	112.335i	0.226027i
$498$	0	0
$499$	480.346	0.962618	0.481309	−	0.876551i	$-0.340162\pi$
$499$	480.346	0.962618	0.481309	+	0.876551i	$0.340162\pi$
$500$	− 1220.19i	− 2.44038i
$501$	0	0
$502$	1436.93	2.86240
$503$	− 955.396i	− 1.89940i	−0.313167	−	0.949698i	$-0.601390\pi$
$503$	− 955.396i	− 1.89940i	0.313167	−	0.949698i	$-0.398610\pi$
$504$	0	0
$505$	334.201	0.661783
$506$	− 30.9880i	− 0.0612411i
$507$	0	0
$508$	104.144	0.205008
$509$	− 666.159i	− 1.30876i	−0.756166	−	0.654380i	$-0.772930\pi$
$509$	− 666.159i	− 1.30876i	0.756166	−	0.654380i	$-0.227070\pi$
$510$	0	0
$511$	−1191.84	−2.33236
$512$	− 1120.59i	− 2.18865i
$513$	0	0
$514$	−77.4017	−0.150587
$515$	− 72.2537i	− 0.140299i
$516$	0	0
$517$	−1175.63	−2.27395
$518$	1092.26i	2.10860i
$519$	0	0
$520$	1345.60	2.58770
$521$	− 742.086i	− 1.42435i	−0.702002	−	0.712175i	$-0.747710\pi$
$521$	− 742.086i	− 1.42435i	0.702002	−	0.712175i	$-0.252290\pi$
$522$	0	0
$523$	−171.187	−0.327317	−0.163659	−	0.986517i	$-0.552330\pi$
$523$	−171.187	−0.327317	−0.163659	+	0.986517i	$0.552330\pi$
$524$	− 1664.50i	− 3.17653i
$525$	0	0
$526$	−1384.88	−2.63285
$527$	− 10.2864i	− 0.0195189i
$528$	0	0
$529$	528.800	0.999622
$530$	834.678i	1.57486i
$531$	0	0
$532$	648.429	1.21885
$533$	353.885i	0.663949i
$534$	0	0
$535$	358.147	0.669434
$536$	988.362i	1.84396i
$537$	0	0
$538$	670.599	1.24647
$539$	− 832.624i	− 1.54476i
$540$	0	0
$541$	−853.931	−1.57843	−0.789215	−	0.614117i	$-0.789512\pi$
$541$	−853.931	−1.57843	−0.789215	+	0.614117i	$0.789512\pi$
$542$	− 1855.13i	− 3.42274i
$543$	0	0
$544$	−34.2204	−0.0629051
$545$	350.866i	0.643791i
$546$	0	0
$547$	304.847	0.557306	0.278653	−	0.960392i	$-0.410112\pi$
$547$	304.847	0.557306	0.278653	+	0.960392i	$0.410112\pi$
$548$	− 1660.03i	− 3.02926i
$549$	0	0
$550$	887.124	1.61295
$551$	316.055i	0.573602i
$552$	0	0
$553$	−504.100	−0.911573
$554$	283.963i	0.512569i
$555$	0	0
$556$	−2202.65	−3.96160
$557$	376.762i	0.676413i	0.941072	+	0.338207i	$0.109820\pi$
$557$	376.762i	0.676413i	−0.941072	+	0.338207i	$0.890180\pi$
$558$	0	0
$559$	433.503	0.775497
$560$	− 1091.00i	− 1.94822i
$561$	0	0
$562$	−523.199	−0.930959
$563$	343.827i	0.610705i	0.952239	+	0.305352i	$0.0987742\pi$
$563$	343.827i	0.610705i	−0.952239	+	0.305352i	$0.901226\pi$
$564$	0	0
$565$	−412.090	−0.729363
$566$	1001.71i	1.76980i
$567$	0	0
$568$	−222.521	−0.391763
$569$	601.227i	1.05664i	0.849046	+	0.528319i	$0.177177\pi$
$569$	601.227i	1.05664i	−0.849046	+	0.528319i	$0.822823\pi$
$570$	0	0
$571$	595.727	1.04330	0.521652	−	0.853158i	$-0.325316\pi$
$571$	595.727	1.04330	0.521652	+	0.853158i	$0.325316\pi$
$572$	3555.75i	6.21635i
$573$	0	0
$574$	613.914	1.06954
$575$	− 5.71990i	− 0.00994765i
$576$	0	0
$577$	68.3783	0.118507	0.0592533	−	0.998243i	$-0.481128\pi$
$577$	68.3783	0.118507	0.0592533	+	0.998243i	$0.481128\pi$
$578$	1049.18i	1.81520i
$579$	0	0
$580$	1400.10	2.41396
$581$	− 890.572i	− 1.53283i
$582$	0	0
$583$	−1250.95	−2.14571
$584$	− 2360.87i	− 4.04259i
$585$	0	0
$586$	973.939	1.66201
$587$	651.244i	1.10944i	0.832036	+	0.554722i	$0.187175\pi$
$587$	651.244i	1.10944i	−0.832036	+	0.554722i	$0.812825\pi$
$588$	0	0
$589$	−91.4401	−0.155246
$590$	195.110i	0.330695i
$591$	0	0
$592$	−1011.22	−1.70814
$593$	12.2860i	0.0207183i	0.999946	+	0.0103592i	$0.00329748\pi$
$593$	12.2860i	0.0207183i	−0.999946	+	0.0103592i	$0.996703\pi$
$594$	0	0
$595$	27.5740	0.0463429
$596$	− 1410.80i	− 2.36712i
$597$	0	0
$598$	32.8498	0.0549328
$599$	10.2076i	0.0170410i	0.999964	+	0.00852051i	$0.00271220\pi$
$599$	10.2076i	0.0170410i	−0.999964	+	0.00852051i	$0.997288\pi$
$600$	0	0
$601$	−370.191	−0.615959	−0.307980	−	0.951393i	$-0.599653\pi$
$601$	−370.191	−0.615959	−0.307980	+	0.951393i	$0.599653\pi$
$602$	− 752.033i	− 1.24922i
$603$	0	0
$604$	−2754.42	−4.56030
$605$	− 845.252i	− 1.39711i
$606$	0	0
$607$	166.851	0.274878	0.137439	−	0.990510i	$-0.456113\pi$
$607$	166.851	0.274878	0.137439	+	0.990510i	$0.456113\pi$
$608$	304.198i	0.500326i
$609$	0	0
$610$	−760.140	−1.24613
$611$	− 1246.27i	− 2.03971i
$612$	0	0
$613$	6.09767	0.00994726	0.00497363	−	0.999988i	$-0.498417\pi$
$613$	6.09767	0.00994726	0.00497363	+	0.999988i	$0.498417\pi$
$614$	549.995i	0.895758i
$615$	0	0
$616$	3498.51	5.67939
$617$	− 304.059i	− 0.492803i	−0.969168	−	0.246401i	$-0.920752\pi$
$617$	− 304.059i	− 0.492803i	0.969168	−	0.246401i	$-0.0792482\pi$
$618$	0	0
$619$	−446.526	−0.721367	−0.360683	−	0.932688i	$-0.617457\pi$
$619$	−446.526	−0.721367	−0.360683	+	0.932688i	$0.617457\pi$
$620$	405.072i	0.653342i
$621$	0	0
$622$	−1794.92	−2.88573
$623$	15.0595i	0.0241725i
$624$	0	0
$625$	−141.339	−0.226143
$626$	360.719i	0.576229i
$627$	0	0
$628$	−834.049	−1.32810
$629$	− 25.5576i	− 0.0406320i
$630$	0	0
$631$	786.982	1.24720	0.623599	−	0.781745i	$-0.285670\pi$
$631$	786.982	1.24720	0.623599	+	0.781745i	$0.285670\pi$
$632$	− 998.556i	− 1.57999i
$633$	0	0
$634$	709.846	1.11963
$635$	− 39.3681i	− 0.0619971i
$636$	0	0
$637$	882.650	1.38564
$638$	3006.60i	4.71254i
$639$	0	0
$640$	−301.735	−0.471461
$641$	− 596.401i	− 0.930423i	−0.885199	−	0.465212i	$-0.845978\pi$
$641$	− 596.401i	− 0.930423i	0.885199	−	0.465212i	$-0.154022\pi$
$642$	0	0
$643$	773.061	1.20227	0.601136	−	0.799146i	$-0.294715\pi$
$643$	773.061	1.20227	0.601136	+	0.799146i	$0.294715\pi$
$644$	− 39.7723i	− 0.0617582i
$645$	0	0
$646$	−21.7397	−0.0336529
$647$	112.273i	0.173529i	0.996229	+	0.0867644i	$0.0276527\pi$
$647$	112.273i	0.173529i	−0.996229	+	0.0867644i	$0.972347\pi$
$648$	0	0
$649$	−292.415	−0.450563
$650$	940.425i	1.44681i
$651$	0	0
$652$	−2843.66	−4.36144
$653$	− 1069.49i	− 1.63781i	−0.573929	−	0.818905i	$-0.694581\pi$
$653$	− 1069.49i	− 1.63781i	0.573929	−	0.818905i	$-0.305419\pi$
$654$	0	0
$655$	−629.207	−0.960622
$656$	568.364i	0.866409i
$657$	0	0
$658$	−2162.00	−3.28571
$659$	830.211i	1.25980i	0.776675	+	0.629902i	$0.216905\pi$
$659$	830.211i	1.25980i	−0.776675	+	0.629902i	$0.783095\pi$
$660$	0	0
$661$	−696.017	−1.05298	−0.526488	−	0.850183i	$-0.676492\pi$
$661$	−696.017	−1.05298	−0.526488	+	0.850183i	$0.676492\pi$
$662$	424.168i	0.640737i
$663$	0	0
$664$	1764.11	2.65679
$665$	− 245.116i	− 0.368596i
$666$	0	0
$667$	19.3856	0.0290639
$668$	375.625i	0.562313i
$669$	0	0
$670$	658.749	0.983207
$671$	− 1139.24i	− 1.69782i
$672$	0	0
$673$	−122.055	−0.181359	−0.0906796	−	0.995880i	$-0.528904\pi$
$673$	−122.055	−0.181359	−0.0906796	+	0.995880i	$0.528904\pi$
$674$	− 705.386i	− 1.04657i
$675$	0	0
$676$	−2207.61	−3.26570
$677$	− 10.1764i	− 0.0150317i	−0.999972	−	0.00751584i	$-0.997608\pi$
$677$	− 10.1764i	− 0.0150317i	0.999972	−	0.00751584i	$-0.00239239\pi$
$678$	0	0
$679$	393.307	0.579245
$680$	54.6206i	0.0803243i
$681$	0	0
$682$	−869.863	−1.27546
$683$	356.039i	0.521287i	0.965435	+	0.260643i	$0.0839347\pi$
$683$	356.039i	0.521287i	−0.965435	+	0.260643i	$0.916065\pi$
$684$	0	0
$685$	−627.519	−0.916086
$686$	185.557i	0.270491i
$687$	0	0
$688$	696.236	1.01197
$689$	− 1326.11i	− 1.92469i
$690$	0	0
$691$	707.950	1.02453	0.512265	−	0.858828i	$-0.328807\pi$
$691$	707.950	1.02453	0.512265	+	0.858828i	$0.328807\pi$
$692$	− 2171.38i	− 3.13783i
$693$	0	0
$694$	1328.52	1.91429
$695$	832.636i	1.19804i
$696$	0	0
$697$	−14.3649	−0.0206096
$698$	2255.14i	3.23087i
$699$	0	0
$700$	1138.60	1.62657
$701$	− 578.940i	− 0.825877i	−0.910759	−	0.412939i	$-0.864502\pi$
$701$	− 578.940i	− 0.825877i	0.910759	−	0.412939i	$-0.135498\pi$
$702$	0	0
$703$	−227.191	−0.323174
$704$	421.962i	0.599377i
$705$	0	0
$706$	−2015.00	−2.85411
$707$	921.113i	1.30285i
$708$	0	0
$709$	−115.048	−0.162269	−0.0811343	−	0.996703i	$-0.525854\pi$
$709$	−115.048	−0.162269	−0.0811343	+	0.996703i	$0.525854\pi$
$710$	148.312i	0.208890i
$711$	0	0
$712$	−29.8308	−0.0418972
$713$	5.60861i	0.00786621i
$714$	0	0
$715$	1344.13	1.87990
$716$	2352.28i	3.28531i
$717$	0	0
$718$	−354.290	−0.493440
$719$	− 626.313i	− 0.871089i	−0.900167	−	0.435545i	$-0.856556\pi$
$719$	− 626.313i	− 0.871089i	0.900167	−	0.435545i	$-0.143444\pi$
$720$	0	0
$721$	199.143	0.276205
$722$	− 1120.38i	− 1.55177i
$723$	0	0
$724$	1484.89	2.05095
$725$	554.972i	0.765479i
$726$	0	0
$727$	−773.461	−1.06391	−0.531954	−	0.846773i	$-0.678542\pi$
$727$	−773.461	−1.06391	−0.531954	+	0.846773i	$0.678542\pi$
$728$	3708.71i	5.09438i
$729$	0	0
$730$	−1573.53	−2.15553
$731$	17.5967i	0.0240721i
$732$	0	0
$733$	148.562	0.202677	0.101339	−	0.994852i	$-0.467687\pi$
$733$	148.562	0.202677	0.101339	+	0.994852i	$0.467687\pi$
$734$	− 1789.98i	− 2.43866i
$735$	0	0
$736$	18.6584	0.0253511
$737$	987.281i	1.33959i
$738$	0	0
$739$	−1320.96	−1.78749	−0.893745	−	0.448575i	$-0.851932\pi$
$739$	−1320.96	−1.78749	−0.893745	+	0.448575i	$0.851932\pi$
$740$	1006.44i	1.36005i
$741$	0	0
$742$	−2300.51	−3.10042
$743$	674.905i	0.908352i	0.890912	+	0.454176i	$0.150066\pi$
$743$	674.905i	0.908352i	−0.890912	+	0.454176i	$0.849934\pi$
$744$	0	0
$745$	−533.306	−0.715847
$746$	1382.44i	1.85313i
$747$	0	0
$748$	−144.335	−0.192961
$749$	987.114i	1.31791i
$750$	0	0
$751$	−128.638	−0.171289	−0.0856446	−	0.996326i	$-0.527295\pi$
$751$	−128.638	−0.171289	−0.0856446	+	0.996326i	$0.527295\pi$
$752$	− 2001.59i	− 2.66169i
$753$	0	0
$754$	−3187.25	−4.22712
$755$	1041.22i	1.37909i
$756$	0	0
$757$	−877.612	−1.15933	−0.579665	−	0.814855i	$-0.696816\pi$
$757$	−877.612	−1.15933	−0.579665	+	0.814855i	$0.696816\pi$
$758$	2351.58i	3.10234i
$759$	0	0
$760$	485.543	0.638873
$761$	379.587i	0.498800i	0.968401	+	0.249400i	$0.0802334\pi$
$761$	379.587i	0.498800i	−0.968401	+	0.249400i	$0.919767\pi$
$762$	0	0
$763$	−967.046	−1.26743
$764$	1556.48i	2.03728i
$765$	0	0
$766$	−1321.70	−1.72546
$767$	− 309.984i	− 0.404152i
$768$	0	0
$769$	−37.2784	−0.0484764	−0.0242382	−	0.999706i	$-0.507716\pi$
$769$	−37.2784	−0.0484764	−0.0242382	+	0.999706i	$0.507716\pi$
$770$	− 2331.77i	− 3.02828i
$771$	0	0
$772$	−3008.40	−3.89689
$773$	− 90.6078i	− 0.117216i	−0.998281	−	0.0586079i	$-0.981334\pi$
$773$	− 90.6078i	− 0.117216i	0.998281	−	0.0586079i	$-0.0186662\pi$
$774$	0	0
$775$	−160.563	−0.207178
$776$	779.090i	1.00398i
$777$	0	0
$778$	2412.10	3.10038
$779$	127.695i	0.163922i
$780$	0	0
$781$	−222.278	−0.284607
$782$	1.33344i	0.00170516i
$783$	0	0
$784$	1417.60	1.80816
$785$	315.283i	0.401635i
$786$	0	0
$787$	−605.048	−0.768804	−0.384402	−	0.923166i	$-0.625592\pi$
$787$	−605.048	−0.768804	−0.384402	+	0.923166i	$0.625592\pi$
$788$	1429.06i	1.81353i
$789$	0	0
$790$	−665.543	−0.842459
$791$	− 1135.79i	− 1.43589i
$792$	0	0
$793$	1207.69	1.52294
$794$	89.6642i	0.112927i
$795$	0	0
$796$	2865.22	3.59952
$797$	91.0829i	0.114282i	0.998366	+	0.0571411i	$0.0181985\pi$
$797$	91.0829i	0.114282i	−0.998366	+	0.0571411i	$0.981802\pi$
$798$	0	0
$799$	50.5883	0.0633145
$800$	534.153i	0.667692i
$801$	0	0
$802$	994.956	1.24059
$803$	− 2358.29i	− 2.93685i
$804$	0	0
$805$	−15.0346	−0.0186765
$806$	− 922.127i	− 1.14408i
$807$	0	0
$808$	−1824.60	−2.25817
$809$	− 936.735i	− 1.15789i	−0.815366	−	0.578947i	$-0.803464\pi$
$809$	− 936.735i	− 1.15789i	0.815366	−	0.578947i	$-0.196536\pi$
$810$	0	0
$811$	−275.363	−0.339536	−0.169768	−	0.985484i	$-0.554302\pi$
$811$	−275.363	−0.339536	−0.169768	+	0.985484i	$0.554302\pi$
$812$	3858.90i	4.75234i
$813$	0	0
$814$	−2161.25	−2.65510
$815$	1074.95i	1.31895i
$816$	0	0
$817$	156.424	0.191461
$818$	2189.86i	2.67709i
$819$	0	0
$820$	565.678	0.689851
$821$	243.447i	0.296525i	0.988948	+	0.148263i	$0.0473681\pi$
$821$	243.447i	0.296525i	−0.988948	+	0.148263i	$0.952632\pi$
$822$	0	0
$823$	−554.519	−0.673777	−0.336889	−	0.941545i	$-0.609375\pi$
$823$	−554.519	−0.673777	−0.336889	+	0.941545i	$0.609375\pi$
$824$	394.477i	0.478734i
$825$	0	0
$826$	−537.756	−0.651036
$827$	− 1064.18i	− 1.28680i	−0.765530	−	0.643400i	$-0.777523\pi$
$827$	− 1064.18i	− 1.28680i	0.765530	−	0.643400i	$-0.222477\pi$
$828$	0	0
$829$	1090.98	1.31602	0.658012	−	0.753007i	$-0.271397\pi$
$829$	1090.98	1.31602	0.658012	+	0.753007i	$0.271397\pi$
$830$	− 1175.79i	− 1.41661i
$831$	0	0
$832$	−447.314	−0.537637
$833$	35.8285i	0.0430113i
$834$	0	0
$835$	141.992	0.170050
$836$	1283.05i	1.53475i
$837$	0	0
$838$	882.105	1.05263
$839$	− 978.978i	− 1.16684i	−0.812171	−	0.583419i	$-0.801714\pi$
$839$	− 978.978i	− 1.16684i	0.812171	−	0.583419i	$-0.198286\pi$
$840$	0	0
$841$	−1039.89	−1.23649
$842$	278.925i	0.331265i
$843$	0	0
$844$	−1965.60	−2.32891
$845$	834.513i	0.987589i
$846$	0	0
$847$	2329.66	2.75048
$848$	− 2129.83i	− 2.51159i
$849$	0	0
$850$	−38.1736	−0.0449102
$851$	13.9351i	0.0163749i
$852$	0	0
$853$	301.741	0.353741	0.176870	−	0.984234i	$-0.443403\pi$
$853$	301.741	0.353741	0.176870	+	0.984234i	$0.443403\pi$
$854$	− 2095.08i	− 2.45325i
$855$	0	0
$856$	−1955.34	−2.28428
$857$	− 1467.38i	− 1.71222i	−0.516790	−	0.856112i	$-0.672873\pi$
$857$	− 1467.38i	− 1.71222i	0.516790	−	0.856112i	$-0.327127\pi$
$858$	0	0
$859$	12.3456	0.0143720	0.00718600	−	0.999974i	$-0.497713\pi$
$859$	12.3456	0.0143720	0.00718600	+	0.999974i	$0.497713\pi$
$860$	− 692.945i	− 0.805750i
$861$	0	0
$862$	2979.82	3.45687
$863$	− 10.8046i	− 0.0125198i	−0.999980	−	0.00625990i	$-0.998007\pi$
$863$	− 10.8046i	− 0.0125198i	0.999980	−	0.00625990i	$-0.00199260\pi$
$864$	0	0
$865$	−820.815	−0.948919
$866$	57.7742i	0.0667138i
$867$	0	0
$868$	−1116.45	−1.28623
$869$	− 997.463i	− 1.14783i
$870$	0	0
$871$	−1046.60	−1.20161
$872$	− 1915.59i	− 2.19678i
$873$	0	0
$874$	11.8534	0.0135623
$875$	− 1271.28i	− 1.45290i
$876$	0	0
$877$	366.912	0.418372	0.209186	−	0.977876i	$-0.432919\pi$
$877$	366.912	0.418372	0.209186	+	0.977876i	$0.432919\pi$
$878$	− 2246.38i	− 2.55852i
$879$	0	0
$880$	2158.77	2.45314
$881$	308.365i	0.350017i	0.984567	+	0.175008i	$0.0559952\pi$
$881$	308.365i	0.350017i	−0.984567	+	0.175008i	$0.944005\pi$
$882$	0	0
$883$	−497.722	−0.563672	−0.281836	−	0.959463i	$-0.590943\pi$
$883$	−497.722	−0.563672	−0.281836	+	0.959463i	$0.590943\pi$
$884$	− 153.007i	− 0.173085i
$885$	0	0
$886$	1995.11	2.25182
$887$	1001.96i	1.12961i	0.825225	+	0.564805i	$0.191048\pi$
$887$	1001.96i	1.12961i	−0.825225	+	0.564805i	$0.808952\pi$
$888$	0	0
$889$	108.505	0.122053
$890$	19.8824i	0.0223398i
$891$	0	0
$892$	2402.32	2.69319
$893$	− 449.699i	− 0.503582i
$894$	0	0
$895$	889.199	0.993519
$896$	− 831.633i	− 0.928162i
$897$	0	0
$898$	−761.334	−0.847811
$899$	− 544.174i	− 0.605310i
$900$	0	0
$901$	53.8294	0.0597440
$902$	1214.75i	1.34673i
$903$	0	0
$904$	2249.85	2.48877
$905$	− 561.310i	− 0.620232i
$906$	0	0
$907$	968.351	1.06764	0.533821	−	0.845598i	$-0.320756\pi$
$907$	968.351	1.06764	0.533821	+	0.845598i	$0.320756\pi$
$908$	− 1971.37i	− 2.17111i
$909$	0	0
$910$	2471.87	2.71635
$911$	620.552i	0.681176i	0.940213	+	0.340588i	$0.110626\pi$
$911$	620.552i	0.681176i	−0.940213	+	0.340588i	$0.889374\pi$
$912$	0	0
$913$	1762.18	1.93010
$914$	− 1839.48i	− 2.01256i
$915$	0	0
$916$	668.992	0.730340
$917$	− 1734.20i	− 1.89117i
$918$	0	0
$919$	153.844	0.167403	0.0837016	−	0.996491i	$-0.473326\pi$
$919$	153.844	0.167403	0.0837016	+	0.996491i	$0.473326\pi$
$920$	− 29.7815i	− 0.0323712i
$921$	0	0
$922$	3056.49	3.31507
$923$	− 235.633i	− 0.255290i
$924$	0	0
$925$	−398.933	−0.431279
$926$	− 1046.41i	− 1.13003i
$927$	0	0
$928$	−1810.33	−1.95079
$929$	1440.34i	1.55042i	0.631705	+	0.775209i	$0.282356\pi$
$929$	1440.34i	1.55042i	−0.631705	+	0.775209i	$0.717644\pi$
$930$	0	0
$931$	318.493	0.342098
$932$	2355.16i	2.52700i
$933$	0	0
$934$	−2655.01	−2.84263
$935$	54.5608i	0.0583538i
$936$	0	0
$937$	−1174.61	−1.25359	−0.626795	−	0.779184i	$-0.715634\pi$
$937$	−1174.61	−1.25359	−0.626795	+	0.779184i	$0.715634\pi$
$938$	1815.62i	1.93563i
$939$	0	0
$940$	−1992.13	−2.11929
$941$	1621.47i	1.72313i	0.507644	+	0.861567i	$0.330516\pi$
$941$	1621.47i	1.72313i	−0.507644	+	0.861567i	$0.669484\pi$
$942$	0	0
$943$	7.83234	0.00830577
$944$	− 497.857i	− 0.527390i
$945$	0	0
$946$	1488.05	1.57299
$947$	642.612i	0.678576i	0.940682	+	0.339288i	$0.110186\pi$
$947$	642.612i	0.678576i	−0.940682	+	0.339288i	$0.889814\pi$
$948$	0	0
$949$	2499.98	2.63433
$950$	339.340i	0.357200i
$951$	0	0
$952$	−150.543	−0.158134
$953$	447.223i	0.469279i	0.972082	+	0.234639i	$0.0753909\pi$
$953$	447.223i	0.469279i	−0.972082	+	0.234639i	$0.924609\pi$
$954$	0	0
$955$	588.374	0.616098
$956$	920.015i	0.962359i
$957$	0	0
$958$	−737.636	−0.769975
$959$	− 1729.55i	− 1.80349i
$960$	0	0
$961$	−803.561	−0.836172
$962$	− 2291.11i	− 2.38161i
$963$	0	0
$964$	−2596.42	−2.69338
$965$	1137.22i	1.17847i
$966$	0	0
$967$	−1084.47	−1.12148	−0.560742	−	0.827991i	$-0.689484\pi$
$967$	−1084.47	−1.12148	−0.560742	+	0.827991i	$0.689484\pi$
$968$	4614.75i	4.76730i
$969$	0	0
$970$	519.268	0.535328
$971$	− 202.150i	− 0.208187i	−0.994567	−	0.104094i	$-0.966806\pi$
$971$	− 202.150i	− 0.208187i	0.994567	−	0.104094i	$-0.0331942\pi$
$972$	0	0
$973$	−2294.89	−2.35857
$974$	2572.41i	2.64108i
$975$	0	0
$976$	1939.63	1.98733
$977$	− 443.613i	− 0.454057i	−0.973888	−	0.227028i	$-0.927099\pi$
$977$	− 443.613i	− 0.454057i	0.973888	−	0.227028i	$-0.0729010\pi$
$978$	0	0
$979$	−29.7982	−0.0304374
$980$	− 1410.90i	− 1.43969i
$981$	0	0
$982$	898.440	0.914908
$983$	− 350.462i	− 0.356522i	−0.983983	−	0.178261i	$-0.942953\pi$
$983$	− 350.462i	− 0.356522i	0.983983	−	0.178261i	$-0.0570472\pi$
$984$	0	0
$985$	540.208	0.548435
$986$	− 129.377i	− 0.131213i
$987$	0	0
$988$	−1360.14	−1.37666
$989$	− 9.59448i	− 0.00970119i
$990$	0	0
$991$	−889.003	−0.897076	−0.448538	−	0.893764i	$-0.648055\pi$
$991$	−889.003	−0.897076	−0.448538	+	0.893764i	$0.648055\pi$
$992$	− 523.760i	− 0.527984i
$993$	0	0
$994$	−408.772	−0.411240
$995$	− 1083.10i	− 1.08854i
$996$	0	0
$997$	1156.35	1.15983	0.579913	−	0.814678i	$-0.303086\pi$
$997$	1156.35	1.15983	0.579913	+	0.814678i	$0.303086\pi$
$998$	1747.91i	1.75142i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1143.3.b.a.890.80	yes	84
3.2	odd	2	inner	1143.3.b.a.890.5	✓	84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1143.3.b.a.890.5	✓	84	3.2	odd	2	inner
1143.3.b.a.890.80	yes	84	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1143 = 3^{2} \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1143.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.63886i q^{2} -9.24130 q^{4} +3.49336i q^{5} -9.62828 q^{7} -19.0724i q^{8} +O(q^{10})\)	\(q+3.63886i q^{2} -9.24130 q^{4} +3.49336i q^{5} -9.62828 q^{7} -19.0724i q^{8} -12.7118 q^{10} -19.0515i q^{11} +20.1962 q^{13} -35.0360i q^{14} +32.4365 q^{16} +0.819801i q^{17} +7.28753 q^{19} -32.2832i q^{20} +69.3258 q^{22} -0.446991i q^{23} +12.7965 q^{25} +73.4911i q^{26} +88.9779 q^{28} +43.3692i q^{29} -12.5475 q^{31} +41.7423i q^{32} -2.98314 q^{34} -33.6350i q^{35} -31.1753 q^{37} +26.5183i q^{38} +66.6266 q^{40} +17.5224i q^{41} +21.4646 q^{43} +176.061i q^{44} +1.62654 q^{46} -61.7080i q^{47} +43.7038 q^{49} +46.5645i q^{50} -186.639 q^{52} -65.6615i q^{53} +66.5537 q^{55} +183.634i q^{56} -157.814 q^{58} -15.3487i q^{59} +59.7978 q^{61} -45.6585i q^{62} -22.1485 q^{64} +70.5525i q^{65} -51.8217 q^{67} -7.57603i q^{68} +122.393 q^{70} -11.6672i q^{71} +123.785 q^{73} -113.443i q^{74} -67.3463 q^{76} +183.433i q^{77} +52.3561 q^{79} +113.312i q^{80} -63.7615 q^{82} +92.4955i q^{83} -2.86386 q^{85} +78.1067i q^{86} -363.357 q^{88} -1.56409i q^{89} -194.455 q^{91} +4.13078i q^{92} +224.547 q^{94} +25.4580i q^{95} -40.8492 q^{97} +159.032i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 84 q - 160 q^{4}+O(q^{10}) \) 84 * q - 160 * q^4	\( 84 q - 160 q^{4} - 48 q^{10} + 16 q^{13} + 360 q^{16} + 64 q^{19} - 8 q^{22} - 388 q^{25} - 120 q^{28} - 160 q^{31} + 192 q^{34} - 152 q^{37} + 208 q^{40} - 24 q^{43} + 56 q^{46} + 564 q^{49} - 80 q^{52} + 136 q^{55} - 136 q^{58} + 168 q^{61} - 736 q^{64} + 168 q^{67} - 608 q^{70} + 80 q^{73} - 32 q^{76} - 168 q^{79} + 528 q^{82} + 288 q^{85} - 392 q^{88} + 176 q^{91} + 176 q^{94} - 120 q^{97}+O(q^{100}) \) 84 * q - 160 * q^4 - 48 * q^10 + 16 * q^13 + 360 * q^16 + 64 * q^19 - 8 * q^22 - 388 * q^25 - 120 * q^28 - 160 * q^31 + 192 * q^34 - 152 * q^37 + 208 * q^40 - 24 * q^43 + 56 * q^46 + 564 * q^49 - 80 * q^52 + 136 * q^55 - 136 * q^58 + 168 * q^61 - 736 * q^64 + 168 * q^67 - 608 * q^70 + 80 * q^73 - 32 * q^76 - 168 * q^79 + 528 * q^82 + 288 * q^85 - 392 * q^88 + 176 * q^91 + 176 * q^94 - 120 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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