LMFDB - Embedded newform 1143.3.b.a.890.67

Newspace parameters

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

Newform invariants

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			890.67
Character	$\chi$	$=$	1143.890
Dual form			1143.3.b.a.890.18

$q$-expansion

$f(q)$	$=$	$q+2.51986i q^{2} -2.34969 q^{4} -1.17863i q^{5} +3.92062 q^{7} +4.15855i q^{8} +O(q^{10})$	\(q+2.51986i q^{2} -2.34969 q^{4} -1.17863i q^{5} +3.92062 q^{7} +4.15855i q^{8} +2.96998 q^{10} -0.116151i q^{11} -19.8712 q^{13} +9.87941i q^{14} -19.8777 q^{16} +4.09340i q^{17} +0.981142 q^{19} +2.76941i q^{20} +0.292685 q^{22} +3.96962i q^{23} +23.6108 q^{25} -50.0726i q^{26} -9.21224 q^{28} +1.24741i q^{29} -1.50131 q^{31} -33.4548i q^{32} -10.3148 q^{34} -4.62096i q^{35} -62.1179 q^{37} +2.47234i q^{38} +4.90139 q^{40} +55.0004i q^{41} -48.4928 q^{43} +0.272919i q^{44} -10.0029 q^{46} +61.7325i q^{47} -33.6287 q^{49} +59.4960i q^{50} +46.6911 q^{52} +27.1346i q^{53} -0.136899 q^{55} +16.3041i q^{56} -3.14329 q^{58} -58.0100i q^{59} +7.12498 q^{61} -3.78310i q^{62} +4.79062 q^{64} +23.4207i q^{65} -103.496 q^{67} -9.61821i q^{68} +11.6442 q^{70} -31.8315i q^{71} -47.2798 q^{73} -156.528i q^{74} -2.30538 q^{76} -0.455385i q^{77} -40.9794 q^{79} +23.4284i q^{80} -138.593 q^{82} -93.8255i q^{83} +4.82460 q^{85} -122.195i q^{86} +0.483021 q^{88} +92.6845i q^{89} -77.9074 q^{91} -9.32737i q^{92} -155.557 q^{94} -1.15640i q^{95} +156.948 q^{97} -84.7396i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 84q - 160q^{4} + O(q^{10}) $	$ 84q - 160q^{4} - 48q^{10} + 16q^{13} + 360q^{16} + 64q^{19} - 8q^{22} - 388q^{25} - 120q^{28} - 160q^{31} + 192q^{34} - 152q^{37} + 208q^{40} - 24q^{43} + 56q^{46} + 564q^{49} - 80q^{52} + 136q^{55} - 136q^{58} + 168q^{61} - 736q^{64} + 168q^{67} - 608q^{70} + 80q^{73} - 32q^{76} - 168q^{79} + 528q^{82} + 288q^{85} - 392q^{88} + 176q^{91} + 176q^{94} - 120q^{97} + O(q^{100}) $

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 $

$2$	2.51986i	1.25993i	0.776624	+	0.629965i	$0.216931\pi$
$2$	2.51986i	1.25993i	−0.776624	+	0.629965i	$0.783069\pi$
$3$	0	0
$3$	0	0
$4$	−2.34969	−0.587422
$5$	− 1.17863i	− 0.235726i	−0.993030	−	0.117863i	$-0.962396\pi$
$5$	− 1.17863i	− 0.235726i	0.993030	−	0.117863i	$-0.0376043\pi$
$6$	0	0
$7$	3.92062	0.560089	0.280044	−	0.959987i	$-0.409651\pi$
$7$	3.92062	0.560089	0.280044	+	0.959987i	$0.409651\pi$
$8$	4.15855i	0.519819i
$9$	0	0
$10$	2.96998	0.296998
$11$	− 0.116151i	− 0.0105592i	−0.999986	−	0.00527960i	$-0.998319\pi$
$11$	− 0.116151i	− 0.0105592i	0.999986	−	0.00527960i	$-0.00168056\pi$
$12$	0	0
$13$	−19.8712	−1.52855	−0.764276	−	0.644889i	$-0.776904\pi$
$13$	−19.8712	−1.52855	−0.764276	+	0.644889i	$0.776904\pi$
$14$	9.87941i	0.705672i
$15$	0	0
$16$	−19.8777	−1.24236
$17$	4.09340i	0.240788i	0.992726	+	0.120394i	$0.0384158\pi$
$17$	4.09340i	0.240788i	−0.992726	+	0.120394i	$0.961584\pi$
$18$	0	0
$19$	0.981142	0.0516391	0.0258195	−	0.999667i	$-0.491780\pi$
$19$	0.981142	0.0516391	0.0258195	+	0.999667i	$0.491780\pi$
$20$	2.76941i	0.138471i
$21$	0	0
$22$	0.292685	0.0133038
$23$	3.96962i	0.172592i	0.996270	+	0.0862961i	$0.0275031\pi$
$23$	3.96962i	0.172592i	−0.996270	+	0.0862961i	$0.972497\pi$
$24$	0	0
$25$	23.6108	0.944433
$26$	− 50.0726i	− 1.92587i
$27$	0	0
$28$	−9.21224	−0.329009
$29$	1.24741i	0.0430140i	0.999769	+	0.0215070i	$0.00684642\pi$
$29$	1.24741i	0.0430140i	−0.999769	+	0.0215070i	$0.993154\pi$
$30$	0	0
$31$	−1.50131	−0.0484294	−0.0242147	−	0.999707i	$-0.507709\pi$
$31$	−1.50131	−0.0484294	−0.0242147	+	0.999707i	$0.507709\pi$
$32$	− 33.4548i	− 1.04546i
$33$	0	0
$34$	−10.3148	−0.303376
$35$	− 4.62096i	− 0.132027i
$36$	0	0
$37$	−62.1179	−1.67886	−0.839430	−	0.543467i	$-0.817111\pi$
$37$	−62.1179	−1.67886	−0.839430	+	0.543467i	$0.817111\pi$
$38$	2.47234i	0.0650616i
$39$	0	0
$40$	4.90139	0.122535
$41$	55.0004i	1.34147i	0.741695	+	0.670737i	$0.234022\pi$
$41$	55.0004i	1.34147i	−0.741695	+	0.670737i	$0.765978\pi$
$42$	0	0
$43$	−48.4928	−1.12774	−0.563870	−	0.825863i	$-0.690688\pi$
$43$	−48.4928	−1.12774	−0.563870	+	0.825863i	$0.690688\pi$
$44$	0.272919i	0.00620271i
$45$	0	0
$46$	−10.0029	−0.217454
$47$	61.7325i	1.31346i	0.754127	+	0.656729i	$0.228060\pi$
$47$	61.7325i	1.31346i	−0.754127	+	0.656729i	$0.771940\pi$
$48$	0	0
$49$	−33.6287	−0.686301
$50$	59.4960i	1.18992i
$51$	0	0
$52$	46.6911	0.897906
$53$	27.1346i	0.511974i	0.966680	+	0.255987i	$0.0824004\pi$
$53$	27.1346i	0.511974i	−0.966680	+	0.255987i	$0.917600\pi$
$54$	0	0
$55$	−0.136899	−0.00248907
$56$	16.3041i	0.291145i
$57$	0	0
$58$	−3.14329	−0.0541946
$59$	− 58.0100i	− 0.983221i	−0.870815	−	0.491611i	$-0.836408\pi$
$59$	− 58.0100i	− 0.983221i	0.870815	−	0.491611i	$-0.163592\pi$
$60$	0	0
$61$	7.12498	0.116803	0.0584015	−	0.998293i	$-0.481400\pi$
$61$	7.12498	0.116803	0.0584015	+	0.998293i	$0.481400\pi$
$62$	− 3.78310i	− 0.0610177i
$63$	0	0
$64$	4.79062	0.0748535
$65$	23.4207i	0.360319i
$66$	0	0
$67$	−103.496	−1.54472	−0.772358	−	0.635188i	$-0.780923\pi$
$67$	−103.496	−1.54472	−0.772358	+	0.635188i	$0.780923\pi$
$68$	− 9.61821i	− 0.141444i
$69$	0	0
$70$	11.6442	0.166345
$71$	− 31.8315i	− 0.448331i	−0.974551	−	0.224166i	$-0.928034\pi$
$71$	− 31.8315i	− 0.448331i	0.974551	−	0.224166i	$-0.0719657\pi$
$72$	0	0
$73$	−47.2798	−0.647668	−0.323834	−	0.946114i	$-0.604972\pi$
$73$	−47.2798	−0.647668	−0.323834	+	0.946114i	$0.604972\pi$
$74$	− 156.528i	− 2.11525i
$75$	0	0
$76$	−2.30538	−0.0303339
$77$	− 0.455385i	− 0.00591409i
$78$	0	0
$79$	−40.9794	−0.518726	−0.259363	−	0.965780i	$-0.583513\pi$
$79$	−40.9794	−0.518726	−0.259363	+	0.965780i	$0.583513\pi$
$80$	23.4284i	0.292856i
$81$	0	0
$82$	−138.593	−1.69016
$83$	− 93.8255i	− 1.13043i	−0.824945	−	0.565214i	$-0.808794\pi$
$83$	− 93.8255i	− 1.13043i	0.824945	−	0.565214i	$-0.191206\pi$
$84$	0	0
$85$	4.82460	0.0567600
$86$	− 122.195i	− 1.42087i
$87$	0	0
$88$	0.483021	0.00548887
$89$	92.6845i	1.04140i	0.853740	+	0.520700i	$0.174329\pi$
$89$	92.6845i	1.04140i	−0.853740	+	0.520700i	$0.825671\pi$
$90$	0	0
$91$	−77.9074	−0.856125
$92$	− 9.32737i	− 0.101385i
$93$	0	0
$94$	−155.557	−1.65486
$95$	− 1.15640i	− 0.0121727i
$96$	0	0
$97$	156.948	1.61802	0.809008	−	0.587797i	$-0.200005\pi$
$97$	156.948	1.61802	0.809008	+	0.587797i	$0.200005\pi$
$98$	− 84.7396i	− 0.864690i
$99$	0	0
$100$	−55.4781	−0.554781
$101$	− 88.0834i	− 0.872113i	−0.899919	−	0.436057i	$-0.856375\pi$
$101$	− 88.0834i	− 0.872113i	0.899919	−	0.436057i	$-0.143625\pi$
$102$	0	0
$103$	−24.6377	−0.239201	−0.119600	−	0.992822i	$-0.538161\pi$
$103$	−24.6377	−0.239201	−0.119600	+	0.992822i	$0.538161\pi$
$104$	− 82.6353i	− 0.794570i
$105$	0	0
$106$	−68.3754	−0.645051
$107$	53.5903i	0.500844i	0.968137	+	0.250422i	$0.0805693\pi$
$107$	53.5903i	0.500844i	−0.968137	+	0.250422i	$0.919431\pi$
$108$	0	0
$109$	−110.122	−1.01029	−0.505147	−	0.863033i	$-0.668562\pi$
$109$	−110.122	−1.01029	−0.505147	+	0.863033i	$0.668562\pi$
$110$	− 0.344966i	− 0.00313606i
$111$	0	0
$112$	−77.9330	−0.695830
$113$	− 117.149i	− 1.03672i	−0.855164	−	0.518358i	$-0.826544\pi$
$113$	− 117.149i	− 1.03672i	0.855164	−	0.518358i	$-0.173456\pi$
$114$	0	0
$115$	4.67871	0.0406844
$116$	− 2.93102i	− 0.0252674i
$117$	0	0
$118$	146.177	1.23879
$119$	16.0487i	0.134863i
$120$	0	0
$121$	120.987	0.999889
$122$	17.9539i	0.147163i
$123$	0	0
$124$	3.52762	0.0284485
$125$	− 57.2941i	− 0.458353i
$126$	0	0
$127$	−11.2694	−0.0887357
$127$	−11.2694	−0.0887357
$128$	− 121.748i	− 0.951154i
$129$	0	0
$130$	−59.0170	−0.453977
$131$	213.224i	1.62766i	0.581100	+	0.813832i	$0.302622\pi$
$131$	213.224i	1.62766i	−0.581100	+	0.813832i	$0.697378\pi$
$132$	0	0
$133$	3.84669	0.0289225
$134$	− 260.795i	− 1.94623i
$135$	0	0
$136$	−17.0226	−0.125166
$137$	92.6280i	0.676117i	0.941125	+	0.338058i	$0.109770\pi$
$137$	92.6280i	0.676117i	−0.941125	+	0.338058i	$0.890230\pi$
$138$	0	0
$139$	−46.4344	−0.334060	−0.167030	−	0.985952i	$-0.553418\pi$
$139$	−46.4344	−0.334060	−0.167030	+	0.985952i	$0.553418\pi$
$140$	10.8578i	0.0775558i
$141$	0	0
$142$	80.2109	0.564866
$143$	2.30806i	0.0161403i
$144$	0	0
$145$	1.47023	0.0101395
$146$	− 119.138i	− 0.816016i
$147$	0	0
$148$	145.958	0.986200
$149$	139.438i	0.935824i	0.883775	+	0.467912i	$0.154994\pi$
$149$	139.438i	0.935824i	−0.883775	+	0.467912i	$0.845006\pi$
$150$	0	0
$151$	281.925	1.86705	0.933526	−	0.358510i	$-0.116715\pi$
$151$	281.925	1.86705	0.933526	+	0.358510i	$0.116715\pi$
$152$	4.08013i	0.0268430i
$153$	0	0
$154$	1.14751	0.00745134
$155$	1.76949i	0.0114161i
$156$	0	0
$157$	−89.4645	−0.569837	−0.284919	−	0.958552i	$-0.591967\pi$
$157$	−89.4645	−0.569837	−0.284919	+	0.958552i	$0.591967\pi$
$158$	− 103.262i	− 0.653559i
$159$	0	0
$160$	−39.4308	−0.246443
$161$	15.5634i	0.0966669i
$162$	0	0
$163$	−89.8376	−0.551151	−0.275575	−	0.961279i	$-0.588868\pi$
$163$	−89.8376	−0.551151	−0.275575	+	0.961279i	$0.588868\pi$
$164$	− 129.234i	− 0.788012i
$165$	0	0
$166$	236.427	1.42426
$167$	− 216.719i	− 1.29772i	−0.760908	−	0.648860i	$-0.775246\pi$
$167$	− 216.719i	− 1.29772i	0.760908	−	0.648860i	$-0.224754\pi$
$168$	0	0
$169$	225.864	1.33647
$170$	12.1573i	0.0715135i
$171$	0	0
$172$	113.943	0.662460
$173$	147.874i	0.854763i	0.904071	+	0.427382i	$0.140564\pi$
$173$	147.874i	0.854763i	−0.904071	+	0.427382i	$0.859436\pi$
$174$	0	0
$175$	92.5691	0.528967
$176$	2.30882i	0.0131183i
$177$	0	0
$178$	−233.552	−1.31209
$179$	18.6836i	0.104378i	0.998637	+	0.0521888i	$0.0166198\pi$
$179$	18.6836i	0.104378i	−0.998637	+	0.0521888i	$0.983380\pi$
$180$	0	0
$181$	−5.48027	−0.0302777	−0.0151389	−	0.999885i	$-0.504819\pi$
$181$	−5.48027	−0.0302777	−0.0151389	+	0.999885i	$0.504819\pi$
$182$	− 196.316i	− 1.07866i
$183$	0	0
$184$	−16.5079	−0.0897167
$185$	73.2139i	0.395751i
$186$	0	0
$187$	0.475453	0.00254253
$188$	− 145.052i	− 0.771554i
$189$	0	0
$190$	2.91397	0.0153367
$191$	− 230.705i	− 1.20788i	−0.797029	−	0.603941i	$-0.793596\pi$
$191$	− 230.705i	− 1.20788i	0.797029	−	0.603941i	$-0.206404\pi$
$192$	0	0
$193$	−232.264	−1.20344	−0.601721	−	0.798706i	$-0.705518\pi$
$193$	−232.264	−1.20344	−0.601721	+	0.798706i	$0.705518\pi$
$194$	395.486i	2.03859i
$195$	0	0
$196$	79.0171	0.403148
$197$	46.5025i	0.236053i	0.993010	+	0.118027i	$0.0376568\pi$
$197$	46.5025i	0.236053i	−0.993010	+	0.118027i	$0.962343\pi$
$198$	0	0
$199$	230.998	1.16079	0.580397	−	0.814334i	$-0.302897\pi$
$199$	230.998	1.16079	0.580397	+	0.814334i	$0.302897\pi$
$200$	98.1868i	0.490934i
$201$	0	0
$202$	221.958	1.09880
$203$	4.89061i	0.0240917i
$204$	0	0
$205$	64.8251	0.316220
$206$	− 62.0835i	− 0.301376i
$207$	0	0
$208$	394.994	1.89901
$209$	− 0.113961i	0 0.000545267i
$210$	0	0
$211$	−132.024	−0.625704	−0.312852	−	0.949802i	$-0.601284\pi$
$211$	−132.024	−0.625704	−0.312852	+	0.949802i	$0.601284\pi$
$212$	− 63.7579i	− 0.300745i
$213$	0	0
$214$	−135.040	−0.631028
$215$	57.1550i	0.265837i
$216$	0	0
$217$	−5.88608	−0.0271248
$218$	− 277.492i	− 1.27290i
$219$	0	0
$220$	0.321670	0.00146214
$221$	− 81.3407i	− 0.368057i
$222$	0	0
$223$	−268.297	−1.20312	−0.601562	−	0.798826i	$-0.705455\pi$
$223$	−268.297	−1.20312	−0.601562	+	0.798826i	$0.705455\pi$
$224$	− 131.164i	− 0.585553i
$225$	0	0
$226$	295.199	1.30619
$227$	− 104.979i	− 0.462461i	−0.972899	−	0.231231i	$-0.925725\pi$
$227$	− 104.979i	− 0.462461i	0.972899	−	0.231231i	$-0.0742752\pi$
$228$	0	0
$229$	−146.937	−0.641647	−0.320823	−	0.947139i	$-0.603960\pi$
$229$	−146.937	−0.641647	−0.320823	+	0.947139i	$0.603960\pi$
$230$	11.7897i	0.0512595i
$231$	0	0
$232$	−5.18740	−0.0223595
$233$	− 96.9595i	− 0.416135i	−0.978114	−	0.208068i	$-0.933283\pi$
$233$	− 96.9595i	− 0.416135i	0.978114	−	0.208068i	$-0.0667174\pi$
$234$	0	0
$235$	72.7597	0.309616
$236$	136.306i	0.577566i
$237$	0	0
$238$	−40.4404	−0.169918
$239$	198.906i	0.832244i	0.909309	+	0.416122i	$0.136611\pi$
$239$	198.906i	0.832244i	−0.909309	+	0.416122i	$0.863389\pi$
$240$	0	0
$241$	264.284	1.09661	0.548307	−	0.836277i	$-0.315273\pi$
$241$	264.284	1.09661	0.548307	+	0.836277i	$0.315273\pi$
$242$	304.869i	1.25979i
$243$	0	0
$244$	−16.7415	−0.0686126
$245$	39.6358i	0.161779i
$246$	0	0
$247$	−19.4965	−0.0789330
$248$	− 6.24328i	− 0.0251745i
$249$	0	0
$250$	144.373	0.577492
$251$	31.6442i	0.126073i	0.998011	+	0.0630363i	$0.0200784\pi$
$251$	31.6442i	0.126073i	−0.998011	+	0.0630363i	$0.979922\pi$
$252$	0	0
$253$	0.461076	0.00182244
$254$	− 28.3974i	− 0.111801i
$255$	0	0
$256$	325.949	1.27324
$257$	345.962i	1.34615i	0.739572	+	0.673077i	$0.235028\pi$
$257$	345.962i	1.34615i	−0.739572	+	0.673077i	$0.764972\pi$
$258$	0	0
$259$	−243.541	−0.940311
$260$	− 55.0315i	− 0.211659i
$261$	0	0
$262$	−537.294	−2.05074
$263$	263.188i	1.00071i	0.865819	+	0.500357i	$0.166798\pi$
$263$	263.188i	1.00071i	−0.865819	+	0.500357i	$0.833202\pi$
$264$	0	0
$265$	31.9816	0.120685
$266$	9.69311i	0.0364403i
$267$	0	0
$268$	243.183	0.907400
$269$	− 412.009i	− 1.53163i	−0.643059	−	0.765816i	$-0.722335\pi$
$269$	− 412.009i	− 1.53163i	0.643059	−	0.765816i	$-0.277665\pi$
$270$	0	0
$271$	−339.138	−1.25143	−0.625716	−	0.780051i	$-0.715193\pi$
$271$	−339.138	−1.25143	−0.625716	+	0.780051i	$0.715193\pi$
$272$	− 81.3674i	− 0.299145i
$273$	0	0
$274$	−233.410	−0.851860
$275$	− 2.74243i	− 0.00997246i
$276$	0	0
$277$	345.943	1.24889	0.624447	−	0.781068i	$-0.285325\pi$
$277$	345.943	1.24889	0.624447	+	0.781068i	$0.285325\pi$
$278$	− 117.008i	− 0.420892i
$279$	0	0
$280$	19.2165	0.0686303
$281$	531.371i	1.89100i	0.325621	+	0.945500i	$0.394427\pi$
$281$	531.371i	1.89100i	−0.325621	+	0.945500i	$0.605573\pi$
$282$	0	0
$283$	168.014	0.593689	0.296844	−	0.954926i	$-0.404066\pi$
$283$	168.014	0.593689	0.296844	+	0.954926i	$0.404066\pi$
$284$	74.7942i	0.263360i
$285$	0	0
$286$	−5.81599	−0.0203356
$287$	215.636i	0.751344i
$288$	0	0
$289$	272.244	0.942021
$290$	3.70477i	0.0127751i
$291$	0	0
$292$	111.093	0.380455
$293$	81.2495i	0.277302i	0.990341	+	0.138651i	$0.0442766\pi$
$293$	81.2495i	0.277302i	−0.990341	+	0.138651i	$0.955723\pi$
$294$	0	0
$295$	−68.3723	−0.231770
$296$	− 258.320i	− 0.872703i
$297$	0	0
$298$	−351.363	−1.17907
$299$	− 78.8810i	− 0.263816i
$300$	0	0
$301$	−190.122	−0.631635
$302$	710.411i	2.35235i
$303$	0	0
$304$	−19.5029	−0.0641542
$305$	− 8.39770i	− 0.0275334i
$306$	0	0
$307$	65.7639	0.214215	0.107107	−	0.994247i	$-0.465841\pi$
$307$	65.7639	0.214215	0.107107	+	0.994247i	$0.465841\pi$
$308$	1.07001i	0.00347407i
$309$	0	0
$310$	−4.45886	−0.0143834
$311$	321.306i	1.03314i	0.856245	+	0.516569i	$0.172791\pi$
$311$	321.306i	1.03314i	−0.856245	+	0.516569i	$0.827209\pi$
$312$	0	0
$313$	−175.188	−0.559705	−0.279852	−	0.960043i	$-0.590286\pi$
$313$	−175.188	−0.559705	−0.279852	+	0.960043i	$0.590286\pi$
$314$	− 225.438i	− 0.717955i
$315$	0	0
$316$	96.2888	0.304711
$317$	− 335.093i	− 1.05708i	−0.848909	−	0.528538i	$-0.822740\pi$
$317$	− 335.093i	− 1.05708i	0.848909	−	0.528538i	$-0.177260\pi$
$318$	0	0
$319$	0.144888	0.000454194 0
$320$	− 5.64636i	− 0.0176449i
$321$	0	0
$322$	−39.2175	−0.121794
$323$	4.01621i	0.0124341i
$324$	0	0
$325$	−469.175	−1.44362
$326$	− 226.378i	− 0.694411i
$327$	0	0
$328$	−228.722	−0.697323
$329$	242.030i	0.735653i
$330$	0	0
$331$	133.280	0.402659	0.201330	−	0.979524i	$-0.435474\pi$
$331$	133.280	0.402659	0.201330	+	0.979524i	$0.435474\pi$
$332$	220.461i	0.664038i
$333$	0	0
$334$	546.102	1.63504
$335$	121.983i	0.364129i
$336$	0	0
$337$	303.240	0.899822	0.449911	−	0.893073i	$-0.351456\pi$
$337$	303.240	0.899822	0.449911	+	0.893073i	$0.351456\pi$
$338$	569.145i	1.68386i
$339$	0	0
$340$	−11.3363	−0.0333421
$341$	0.174379i	0 0.000511376i
$342$	0	0
$343$	−323.956	−0.944478
$344$	− 201.660i	− 0.586221i
$345$	0	0
$346$	−372.622	−1.07694
$347$	− 139.489i	− 0.401985i	−0.979593	−	0.200992i	$-0.935583\pi$
$347$	− 139.489i	− 0.401985i	0.979593	−	0.200992i	$-0.0644166\pi$
$348$	0	0
$349$	412.489	1.18192	0.590958	−	0.806702i	$-0.298750\pi$
$349$	412.489	1.18192	0.590958	+	0.806702i	$0.298750\pi$
$350$	233.261i	0.666461i
$351$	0	0
$352$	−3.88582	−0.0110393
$353$	− 4.08080i	− 0.0115603i	−0.999983	−	0.00578016i	$-0.998160\pi$
$353$	− 4.08080i	− 0.0115603i	0.999983	−	0.00578016i	$-0.00183989\pi$
$354$	0	0
$355$	−37.5175	−0.105683
$356$	− 217.780i	− 0.611741i
$357$	0	0
$358$	−47.0800	−0.131508
$359$	30.4463i	0.0848087i	0.999101	+	0.0424044i	$0.0135018\pi$
$359$	30.4463i	0.0848087i	−0.999101	+	0.0424044i	$0.986498\pi$
$360$	0	0
$361$	−360.037	−0.997333
$362$	− 13.8095i	− 0.0381478i
$363$	0	0
$364$	183.058	0.502907
$365$	55.7253i	0.152672i
$366$	0	0
$367$	599.427	1.63332	0.816658	−	0.577122i	$-0.195824\pi$
$367$	599.427	1.63332	0.816658	+	0.577122i	$0.195824\pi$
$368$	− 78.9070i	− 0.214421i
$369$	0	0
$370$	−184.489	−0.498618
$371$	106.385i	0.286751i
$372$	0	0
$373$	96.8953	0.259773	0.129886	−	0.991529i	$-0.458539\pi$
$373$	96.8953	0.259773	0.129886	+	0.991529i	$0.458539\pi$
$374$	1.19807i	0.00320341i
$375$	0	0
$376$	−256.718	−0.682760
$377$	− 24.7874i	− 0.0657492i
$378$	0	0
$379$	678.905	1.79131	0.895653	−	0.444753i	$-0.146709\pi$
$379$	678.905	1.79131	0.895653	+	0.444753i	$0.146709\pi$
$380$	2.71719i	0.00715049i
$381$	0	0
$382$	581.345	1.52185
$383$	− 24.6072i	− 0.0642486i	−0.999484	−	0.0321243i	$-0.989773\pi$
$383$	− 24.6072i	− 0.0642486i	0.999484	−	0.0321243i	$-0.0102272\pi$
$384$	0	0
$385$	−0.536730	−0.00139410
$386$	− 585.273i	− 1.51625i
$387$	0	0
$388$	−368.778	−0.950459
$389$	105.592i	0.271444i	0.990747	+	0.135722i	$0.0433353\pi$
$389$	105.592i	0.271444i	−0.990747	+	0.135722i	$0.956665\pi$
$390$	0	0
$391$	−16.2492	−0.0415582
$392$	− 139.847i	− 0.356752i
$393$	0	0
$394$	−117.180	−0.297410
$395$	48.2995i	0.122277i
$396$	0	0
$397$	−656.200	−1.65290	−0.826449	−	0.563012i	$-0.809643\pi$
$397$	−656.200	−1.65290	−0.826449	+	0.563012i	$0.809643\pi$
$398$	582.083i	1.46252i
$399$	0	0
$400$	−469.330	−1.17332
$401$	458.161i	1.14255i	0.820760	+	0.571273i	$0.193550\pi$
$401$	458.161i	1.14255i	−0.820760	+	0.571273i	$0.806450\pi$
$402$	0	0
$403$	29.8329	0.0740269
$404$	206.969i	0.512299i
$405$	0	0
$406$	−12.3236	−0.0303538
$407$	7.21506i	0.0177274i
$408$	0	0
$409$	303.807	0.742803	0.371402	−	0.928472i	$-0.378877\pi$
$409$	303.807	0.742803	0.371402	+	0.928472i	$0.378877\pi$
$410$	163.350i	0.398415i
$411$	0	0
$412$	57.8909	0.140512
$413$	− 227.435i	− 0.550691i
$414$	0	0
$415$	−110.585	−0.266471
$416$	664.787i	1.59805i
$417$	0	0
$418$	0.287165	0.000686998 0
$419$	743.246i	1.77386i	0.461906	+	0.886929i	$0.347166\pi$
$419$	743.246i	1.77386i	−0.461906	+	0.886929i	$0.652834\pi$
$420$	0	0
$421$	−487.274	−1.15742	−0.578710	−	0.815533i	$-0.696444\pi$
$421$	−487.274	−1.15742	−0.578710	+	0.815533i	$0.696444\pi$
$422$	− 332.681i	− 0.788343i
$423$	0	0
$424$	−112.841	−0.266134
$425$	96.6486i	0.227408i
$426$	0	0
$427$	27.9343	0.0654200
$428$	− 125.921i	− 0.294207i
$429$	0	0
$430$	−144.023	−0.334936
$431$	834.575i	1.93637i	0.250237	+	0.968185i	$0.419491\pi$
$431$	834.575i	1.93637i	−0.250237	+	0.968185i	$0.580509\pi$
$432$	0	0
$433$	−454.402	−1.04943	−0.524713	−	0.851279i	$-0.675828\pi$
$433$	−454.402	−1.04943	−0.524713	+	0.851279i	$0.675828\pi$
$434$	− 14.8321i	− 0.0341753i
$435$	0	0
$436$	258.753	0.593469
$437$	3.89476i	0.00891250i
$438$	0	0
$439$	−69.1547	−0.157528	−0.0787639	−	0.996893i	$-0.525097\pi$
$439$	−69.1547	−0.157528	−0.0787639	+	0.996893i	$0.525097\pi$
$440$	− 0.569302i	− 0.00129387i
$441$	0	0
$442$	204.967	0.463726
$443$	550.971i	1.24373i	0.783126	+	0.621863i	$0.213624\pi$
$443$	550.971i	1.24373i	−0.783126	+	0.621863i	$0.786376\pi$
$444$	0	0
$445$	109.241	0.245485
$446$	− 676.070i	− 1.51585i
$447$	0	0
$448$	18.7822	0.0419246
$449$	660.419i	1.47087i	0.677597	+	0.735433i	$0.263021\pi$
$449$	660.419i	1.47087i	−0.677597	+	0.735433i	$0.736979\pi$
$450$	0	0
$451$	6.38837	0.0141649
$452$	275.263i	0.608990i
$453$	0	0
$454$	264.532	0.582669
$455$	91.8239i	0.201811i
$456$	0	0
$457$	332.594	0.727777	0.363888	−	0.931443i	$-0.381449\pi$
$457$	332.594	0.727777	0.363888	+	0.931443i	$0.381449\pi$
$458$	− 370.261i	− 0.808429i
$459$	0	0
$460$	−10.9935	−0.0238989
$461$	− 129.547i	− 0.281013i	−0.990080	−	0.140506i	$-0.955127\pi$
$461$	− 129.547i	− 0.281013i	0.990080	−	0.140506i	$-0.0448730\pi$
$462$	0	0
$463$	26.7590	0.0577949	0.0288975	−	0.999582i	$-0.490800\pi$
$463$	26.7590	0.0577949	0.0288975	+	0.999582i	$0.490800\pi$
$464$	− 24.7956i	− 0.0534388i
$465$	0	0
$466$	244.324	0.524301
$467$	300.066i	0.642540i	0.946988	+	0.321270i	$0.104110\pi$
$467$	300.066i	0.642540i	−0.946988	+	0.321270i	$0.895890\pi$
$468$	0	0
$469$	−405.768	−0.865178
$470$	183.344i	0.390094i
$471$	0	0
$472$	241.238	0.511097
$473$	5.63250i	0.0119080i
$474$	0	0
$475$	23.1656	0.0487697
$476$	− 37.7094i	− 0.0792214i
$477$	0	0
$478$	−501.216	−1.04857
$479$	− 34.2241i	− 0.0714491i	−0.999362	−	0.0357246i	$-0.988626\pi$
$479$	− 34.2241i	− 0.0714491i	0.999362	−	0.0357246i	$-0.0113739\pi$
$480$	0	0
$481$	1234.36	2.56623
$482$	665.958i	1.38166i
$483$	0	0
$484$	−284.281	−0.587357
$485$	− 184.983i	− 0.381408i
$486$	0	0
$487$	241.028	0.494924	0.247462	−	0.968898i	$-0.420404\pi$
$487$	241.028	0.494924	0.247462	+	0.968898i	$0.420404\pi$
$488$	29.6296i	0.0607163i
$489$	0	0
$490$	−99.8766	−0.203830
$491$	221.985i	0.452107i	0.974115	+	0.226054i	$0.0725825\pi$
$491$	221.985i	0.452107i	−0.974115	+	0.226054i	$0.927418\pi$
$492$	0	0
$493$	−5.10613	−0.0103573
$494$	− 49.1283i	− 0.0994501i
$495$	0	0
$496$	29.8427	0.0601667
$497$	− 124.799i	− 0.251105i
$498$	0	0
$499$	618.851	1.24018	0.620091	−	0.784530i	$-0.287096\pi$
$499$	618.851	1.24018	0.620091	+	0.784530i	$0.287096\pi$
$500$	134.623i	0.269247i
$501$	0	0
$502$	−79.7389	−0.158843
$503$	172.036i	0.342020i	0.985269	+	0.171010i	$0.0547030\pi$
$503$	172.036i	0.342020i	−0.985269	+	0.171010i	$0.945297\pi$
$504$	0	0
$505$	−103.818	−0.205580
$506$	1.16185i	0.00229614i
$507$	0	0
$508$	26.4797	0.0521253
$509$	592.690i	1.16442i	0.813038	+	0.582211i	$0.197812\pi$
$509$	592.690i	1.16442i	−0.813038	+	0.582211i	$0.802188\pi$
$510$	0	0
$511$	−185.366	−0.362751
$512$	334.356i	0.653039i
$513$	0	0
$514$	−871.774	−1.69606
$515$	29.0387i	0.0563858i
$516$	0	0
$517$	7.17030	0.0138691
$518$	− 613.688i	− 1.18473i
$519$	0	0
$520$	−97.3963	−0.187301
$521$	− 622.280i	− 1.19439i	−0.802094	−	0.597197i	$-0.796281\pi$
$521$	− 622.280i	− 1.19439i	0.802094	−	0.597197i	$-0.203719\pi$
$522$	0	0
$523$	−357.150	−0.682887	−0.341443	−	0.939902i	$-0.610916\pi$
$523$	−357.150	−0.682887	−0.341443	+	0.939902i	$0.610916\pi$
$524$	− 501.010i	− 0.956126i
$525$	0	0
$526$	−663.195	−1.26083
$527$	− 6.14547i	− 0.0116612i
$528$	0	0
$529$	513.242	0.970212
$530$	80.5892i	0.152055i
$531$	0	0
$532$	−9.03852	−0.0169897
$533$	− 1092.92i	− 2.05051i
$534$	0	0
$535$	63.1630	0.118062
$536$	− 430.393i	− 0.802972i
$537$	0	0
$538$	1038.21	1.92975
$539$	3.90602i	0.00724678i
$540$	0	0
$541$	−277.130	−0.512255	−0.256128	−	0.966643i	$-0.582447\pi$
$541$	−277.130	−0.512255	−0.256128	+	0.966643i	$0.582447\pi$
$542$	− 854.580i	− 1.57672i
$543$	0	0
$544$	136.944	0.251735
$545$	129.793i	0.238152i
$546$	0	0
$547$	629.328	1.15051	0.575254	−	0.817975i	$-0.304903\pi$
$547$	629.328	1.15051	0.575254	+	0.817975i	$0.304903\pi$
$548$	− 217.647i	− 0.397166i
$549$	0	0
$550$	6.91053	0.0125646
$551$	1.22388i	0.00222120i
$552$	0	0
$553$	−160.665	−0.290533
$554$	871.729i	1.57352i
$555$	0	0
$556$	109.106	0.196234
$557$	− 92.7731i	− 0.166559i	−0.996526	−	0.0832793i	$-0.973461\pi$
$557$	− 92.7731i	− 0.166559i	0.996526	−	0.0832793i	$-0.0265394\pi$
$558$	0	0
$559$	963.610	1.72381
$560$	91.8541i	0.164025i
$561$	0	0
$562$	−1338.98	−2.38253
$563$	1037.13i	1.84214i	0.389396	+	0.921071i	$0.372684\pi$
$563$	1037.13i	1.84214i	−0.389396	+	0.921071i	$0.627316\pi$
$564$	0	0
$565$	−138.075	−0.244380
$566$	423.371i	0.748006i
$567$	0	0
$568$	132.373	0.233051
$569$	192.618i	0.338521i	0.985571	+	0.169260i	$0.0541379\pi$
$569$	192.618i	0.338521i	−0.985571	+	0.169260i	$0.945862\pi$
$570$	0	0
$571$	447.153	0.783105	0.391552	−	0.920156i	$-0.371938\pi$
$571$	447.153	0.783105	0.391552	+	0.920156i	$0.371938\pi$
$572$	− 5.42323i	− 0.00948117i
$573$	0	0
$574$	−543.372	−0.946641
$575$	93.7260i	0.163002i
$576$	0	0
$577$	−345.436	−0.598675	−0.299338	−	0.954147i	$-0.596766\pi$
$577$	−345.436	−0.598675	−0.299338	+	0.954147i	$0.596766\pi$
$578$	686.017i	1.18688i
$579$	0	0
$580$	−3.45458	−0.00595617
$581$	− 367.854i	− 0.633140i
$582$	0	0
$583$	3.15172	0.00540603
$584$	− 196.615i	− 0.336670i
$585$	0	0
$586$	−204.737	−0.349381
$587$	459.636i	0.783026i	0.920173	+	0.391513i	$0.128048\pi$
$587$	459.636i	0.783026i	−0.920173	+	0.391513i	$0.871952\pi$
$588$	0	0
$589$	−1.47300	−0.00250085
$590$	− 172.289i	− 0.292014i
$591$	0	0
$592$	1234.76	2.08575
$593$	− 782.439i	− 1.31946i	−0.751504	−	0.659729i	$-0.770671\pi$
$593$	− 782.439i	− 1.31946i	0.751504	−	0.659729i	$-0.229329\pi$
$594$	0	0
$595$	18.9154	0.0317906
$596$	− 327.635i	− 0.549724i
$597$	0	0
$598$	198.769	0.332390
$599$	− 162.105i	− 0.270626i	−0.990803	−	0.135313i	$-0.956796\pi$
$599$	− 162.105i	− 0.270626i	0.990803	−	0.135313i	$-0.0432040\pi$
$600$	0	0
$601$	−410.144	−0.682436	−0.341218	−	0.939984i	$-0.610839\pi$
$601$	−410.144	−0.682436	−0.341218	+	0.939984i	$0.610839\pi$
$602$	− 479.081i	− 0.795815i
$603$	0	0
$604$	−662.436	−1.09675
$605$	− 142.598i	− 0.235699i
$606$	0	0
$607$	−564.201	−0.929490	−0.464745	−	0.885444i	$-0.653854\pi$
$607$	−564.201	−0.929490	−0.464745	+	0.885444i	$0.653854\pi$
$608$	− 32.8240i	− 0.0539868i
$609$	0	0
$610$	21.1610	0.0346902
$611$	− 1226.70i	− 2.00769i
$612$	0	0
$613$	842.376	1.37419	0.687093	−	0.726569i	$-0.258886\pi$
$613$	842.376	1.37419	0.687093	+	0.726569i	$0.258886\pi$
$614$	165.716i	0.269896i
$615$	0	0
$616$	1.89374	0.00307425
$617$	− 298.792i	− 0.484266i	−0.970243	−	0.242133i	$-0.922153\pi$
$617$	− 298.792i	− 0.484266i	0.970243	−	0.242133i	$-0.0778470\pi$
$618$	0	0
$619$	402.914	0.650910	0.325455	−	0.945557i	$-0.394482\pi$
$619$	402.914	0.650910	0.325455	+	0.945557i	$0.394482\pi$
$620$	− 4.15775i	− 0.00670605i
$621$	0	0
$622$	−809.646	−1.30168
$623$	363.381i	0.583276i
$624$	0	0
$625$	522.742	0.836388
$626$	− 441.448i	− 0.705189i
$627$	0	0
$628$	210.214	0.334735
$629$	− 254.273i	− 0.404250i
$630$	0	0
$631$	617.724	0.978960	0.489480	−	0.872015i	$-0.337187\pi$
$631$	617.724	0.978960	0.489480	+	0.872015i	$0.337187\pi$
$632$	− 170.415i	− 0.269644i
$633$	0	0
$634$	844.388	1.33184
$635$	13.2825i	0.0209173i
$636$	0	0
$637$	668.243	1.04905
$638$	0.365097i	0 0.000572252i
$639$	0	0
$640$	−143.495	−0.224211
$641$	− 871.699i	− 1.35991i	−0.733256	−	0.679953i	$-0.762000\pi$
$641$	− 871.699i	− 1.35991i	0.733256	−	0.679953i	$-0.238000\pi$
$642$	0	0
$643$	−915.020	−1.42305	−0.711524	−	0.702661i	$-0.751995\pi$
$643$	−915.020	−1.42305	−0.711524	+	0.702661i	$0.751995\pi$
$644$	− 36.5691i	− 0.0567843i
$645$	0	0
$646$	−10.1203	−0.0156661
$647$	99.8398i	0.154312i	0.997019	+	0.0771559i	$0.0245839\pi$
$647$	99.8398i	0.154312i	−0.997019	+	0.0771559i	$0.975416\pi$
$648$	0	0
$649$	−6.73794	−0.0103820
$650$	− 1182.26i	− 1.81885i
$651$	0	0
$652$	211.090	0.323758
$653$	− 322.455i	− 0.493805i	−0.969040	−	0.246903i	$-0.920587\pi$
$653$	− 322.455i	− 0.493805i	0.969040	−	0.246903i	$-0.0794128\pi$
$654$	0	0
$655$	251.312	0.383682
$656$	− 1093.28i	− 1.66659i
$657$	0	0
$658$	−609.881	−0.926871
$659$	− 1044.59i	− 1.58512i	−0.609796	−	0.792558i	$-0.708749\pi$
$659$	− 1044.59i	− 1.58512i	0.609796	−	0.792558i	$-0.291251\pi$
$660$	0	0
$661$	−557.410	−0.843282	−0.421641	−	0.906763i	$-0.638546\pi$
$661$	−557.410	−0.843282	−0.421641	+	0.906763i	$0.638546\pi$
$662$	335.848i	0.507323i
$663$	0	0
$664$	390.178	0.587617
$665$	− 4.53382i	− 0.00681777i
$666$	0	0
$667$	−4.95173	−0.00742388
$668$	509.223i	0.762310i
$669$	0	0
$670$	−307.381	−0.458777
$671$	− 0.827575i	− 0.00123335i
$672$	0	0
$673$	−1035.16	−1.53812	−0.769062	−	0.639174i	$-0.779276\pi$
$673$	−1035.16	−1.53812	−0.769062	+	0.639174i	$0.779276\pi$
$674$	764.122i	1.13371i
$675$	0	0
$676$	−530.710	−0.785074
$677$	623.131i	0.920430i	0.887807	+	0.460215i	$0.152228\pi$
$677$	623.131i	0.920430i	−0.887807	+	0.460215i	$0.847772\pi$
$678$	0	0
$679$	615.332	0.906233
$680$	20.0633i	0.0295049i
$681$	0	0
$682$	−0.439411	−0.000644298 0
$683$	103.181i	0.151070i	0.997143	+	0.0755351i	$0.0240665\pi$
$683$	103.181i	0.151070i	−0.997143	+	0.0755351i	$0.975934\pi$
$684$	0	0
$685$	109.174	0.159378
$686$	− 816.323i	− 1.18998i
$687$	0	0
$688$	963.927	1.40106
$689$	− 539.197i	− 0.782579i
$690$	0	0
$691$	326.699	0.472791	0.236396	−	0.971657i	$-0.424034\pi$
$691$	326.699	0.472791	0.236396	+	0.971657i	$0.424034\pi$
$692$	− 347.458i	− 0.502107i
$693$	0	0
$694$	351.492	0.506472
$695$	54.7289i	0.0787466i
$696$	0	0
$697$	−225.139	−0.323011
$698$	1039.41i	1.48913i
$699$	0	0
$700$	−217.509	−0.310727
$701$	− 882.429i	− 1.25881i	−0.777076	−	0.629407i	$-0.783298\pi$
$701$	− 882.429i	− 1.25881i	0.777076	−	0.629407i	$-0.216702\pi$
$702$	0	0
$703$	−60.9465	−0.0866948
$704$	− 0.556436i	0 0.000790393i
$705$	0	0
$706$	10.2830	0.0145652
$707$	− 345.342i	− 0.488461i
$708$	0	0
$709$	29.3380	0.0413794	0.0206897	−	0.999786i	$-0.493414\pi$
$709$	29.3380	0.0413794	0.0206897	+	0.999786i	$0.493414\pi$
$710$	− 94.5389i	− 0.133153i
$711$	0	0
$712$	−385.433	−0.541339
$713$	− 5.95964i	− 0.00835854i
$714$	0	0
$715$	2.72035	0.00380468
$716$	− 43.9006i	− 0.0613138i
$717$	0	0
$718$	−76.7205	−0.106853
$719$	− 112.619i	− 0.156633i	−0.996929	−	0.0783163i	$-0.975046\pi$
$719$	− 112.619i	− 0.156633i	0.996929	−	0.0783163i	$-0.0249544\pi$
$720$	0	0
$721$	−96.5950	−0.133974
$722$	− 907.243i	− 1.25657i
$723$	0	0
$724$	12.8769	0.0177858
$725$	29.4523i	0.0406239i
$726$	0	0
$727$	60.0874	0.0826511	0.0413256	−	0.999146i	$-0.486842\pi$
$727$	60.0874	0.0826511	0.0413256	+	0.999146i	$0.486842\pi$
$728$	− 323.982i	− 0.445030i
$729$	0	0
$730$	−140.420	−0.192356
$731$	− 198.500i	− 0.271546i
$732$	0	0
$733$	497.573	0.678817	0.339409	−	0.940639i	$-0.389773\pi$
$733$	497.573	0.678817	0.339409	+	0.940639i	$0.389773\pi$
$734$	1510.47i	2.05786i
$735$	0	0
$736$	132.803	0.180439
$737$	12.0212i	0.0163110i
$738$	0	0
$739$	216.118	0.292447	0.146223	−	0.989252i	$-0.453288\pi$
$739$	216.118	0.292447	0.146223	+	0.989252i	$0.453288\pi$
$740$	− 172.030i	− 0.232473i
$741$	0	0
$742$	−268.074	−0.361286
$743$	− 1121.65i	− 1.50962i	−0.655944	−	0.754810i	$-0.727729\pi$
$743$	− 1121.65i	− 1.50962i	0.655944	−	0.754810i	$-0.272271\pi$
$744$	0	0
$745$	164.345	0.220598
$746$	244.162i	0.327296i
$747$	0	0
$748$	−1.11717	−0.00149354
$749$	210.107i	0.280517i
$750$	0	0
$751$	235.599	0.313714	0.156857	−	0.987621i	$-0.449864\pi$
$751$	235.599	0.313714	0.156857	+	0.987621i	$0.449864\pi$
$752$	− 1227.10i	− 1.63178i
$753$	0	0
$754$	62.4609	0.0828393
$755$	− 332.285i	− 0.440112i
$756$	0	0
$757$	−2.57579	−0.00340263	−0.00170132	−	0.999999i	$-0.500542\pi$
$757$	−2.57579	−0.00340263	−0.00170132	+	0.999999i	$0.500542\pi$
$758$	1710.75i	2.25692i
$759$	0	0
$760$	4.80896	0.00632758
$761$	− 311.007i	− 0.408682i	−0.978900	−	0.204341i	$-0.934495\pi$
$761$	− 311.007i	− 0.408682i	0.978900	−	0.204341i	$-0.0655052\pi$
$762$	0	0
$763$	−431.747	−0.565854
$764$	542.086i	0.709537i
$765$	0	0
$766$	62.0067	0.0809486
$767$	1152.73i	1.50291i
$768$	0	0
$769$	−794.978	−1.03378	−0.516891	−	0.856051i	$-0.672911\pi$
$769$	−794.978	−1.03378	−0.516891	+	0.856051i	$0.672911\pi$
$770$	− 1.35248i	− 0.00175647i
$771$	0	0
$772$	545.749	0.706929
$773$	319.749i	0.413647i	0.978378	+	0.206823i	$0.0663125\pi$
$773$	319.749i	0.413647i	−0.978378	+	0.206823i	$0.933687\pi$
$774$	0	0
$775$	−35.4472	−0.0457384
$776$	652.675i	0.841075i
$777$	0	0
$778$	−266.076	−0.342000
$779$	53.9633i	0.0692725i
$780$	0	0
$781$	−3.69727	−0.00473402
$782$	− 40.9458i	− 0.0523603i
$783$	0	0
$784$	668.462	0.852631
$785$	105.445i	0.134325i
$786$	0	0
$787$	548.874	0.697426	0.348713	−	0.937230i	$-0.386619\pi$
$787$	548.874	0.697426	0.348713	+	0.937230i	$0.386619\pi$
$788$	− 109.266i	− 0.138663i
$789$	0	0
$790$	−121.708	−0.154061
$791$	− 459.296i	− 0.580653i
$792$	0	0
$793$	−141.582	−0.178539
$794$	− 1653.53i	− 2.08253i
$795$	0	0
$796$	−542.774	−0.681876
$797$	− 362.036i	− 0.454249i	−0.973866	−	0.227124i	$-0.927068\pi$
$797$	− 362.036i	− 0.454249i	0.973866	−	0.227124i	$-0.0729324\pi$
$798$	0	0
$799$	−252.696	−0.316265
$800$	− 789.897i	− 0.987371i
$801$	0	0
$802$	−1154.50	−1.43953
$803$	5.49160i	0.00683885i
$804$	0	0
$805$	18.3434	0.0227869
$806$	75.1746i	0.0932687i
$807$	0	0
$808$	366.299	0.453341
$809$	− 1305.60i	− 1.61385i	−0.590655	−	0.806924i	$-0.701131\pi$
$809$	− 1305.60i	− 1.61385i	0.590655	−	0.806924i	$-0.298869\pi$
$810$	0	0
$811$	6.45607	0.00796063	0.00398032	−	0.999992i	$-0.498733\pi$
$811$	6.45607	0.00796063	0.00398032	+	0.999992i	$0.498733\pi$
$812$	− 11.4914i	− 0.0141520i
$813$	0	0
$814$	−18.1809	−0.0223353
$815$	105.885i	0.129920i
$816$	0	0
$817$	−47.5784	−0.0582355
$818$	765.550i	0.935880i
$819$	0	0
$820$	−152.319	−0.185755
$821$	− 240.575i	− 0.293027i	−0.989209	−	0.146513i	$-0.953195\pi$
$821$	− 240.575i	− 0.293027i	0.989209	−	0.146513i	$-0.0468051\pi$
$822$	0	0
$823$	−226.340	−0.275018	−0.137509	−	0.990501i	$-0.543910\pi$
$823$	−226.340	−0.275018	−0.137509	+	0.990501i	$0.543910\pi$
$824$	− 102.457i	− 0.124341i
$825$	0	0
$826$	573.105	0.693832
$827$	− 1086.04i	− 1.31323i	−0.754228	−	0.656613i	$-0.771988\pi$
$827$	− 1086.04i	− 1.31323i	0.754228	−	0.656613i	$-0.228012\pi$
$828$	0	0
$829$	458.391	0.552945	0.276473	−	0.961022i	$-0.410835\pi$
$829$	458.391	0.552945	0.276473	+	0.961022i	$0.410835\pi$
$830$	− 278.660i	− 0.335734i
$831$	0	0
$832$	−95.1953	−0.114417
$833$	− 137.656i	− 0.165253i
$834$	0	0
$835$	−255.432	−0.305906
$836$	0.267773i	0 0.000320302i
$837$	0	0
$838$	−1872.88	−2.23494
$839$	770.491i	0.918344i	0.888347	+	0.459172i	$0.151854\pi$
$839$	770.491i	0.918344i	−0.888347	+	0.459172i	$0.848146\pi$
$840$	0	0
$841$	839.444	0.998150
$842$	− 1227.86i	− 1.45827i
$843$	0	0
$844$	310.214	0.367553
$845$	− 266.210i	− 0.315041i
$846$	0	0
$847$	474.342	0.560026
$848$	− 539.374i	− 0.636055i
$849$	0	0
$850$	−243.541	−0.286519
$851$	− 246.584i	− 0.289758i
$852$	0	0
$853$	−142.876	−0.167498	−0.0837489	−	0.996487i	$-0.526689\pi$
$853$	−142.876	−0.167498	−0.0837489	+	0.996487i	$0.526689\pi$
$854$	70.3906i	0.0824246i
$855$	0	0
$856$	−222.858	−0.260348
$857$	905.026i	1.05604i	0.849232	+	0.528020i	$0.177065\pi$
$857$	905.026i	1.05604i	−0.849232	+	0.528020i	$0.822935\pi$
$858$	0	0
$859$	143.642	0.167220	0.0836101	−	0.996499i	$-0.473355\pi$
$859$	143.642	0.167220	0.0836101	+	0.996499i	$0.473355\pi$
$860$	− 134.297i	− 0.156159i
$861$	0	0
$862$	−2103.01	−2.43969
$863$	− 915.923i	− 1.06132i	−0.847583	−	0.530662i	$-0.821943\pi$
$863$	− 915.923i	− 1.06132i	0.847583	−	0.530662i	$-0.178057\pi$
$864$	0	0
$865$	174.289	0.201490
$866$	− 1145.03i	− 1.32220i
$867$	0	0
$868$	13.8305	0.0159337
$869$	4.75980i	0.00547733i
$870$	0	0
$871$	2056.59	2.36118
$872$	− 457.948i	− 0.525170i
$873$	0	0
$874$	−9.81425	−0.0112291
$875$	− 224.629i	− 0.256718i
$876$	0	0
$877$	−172.726	−0.196951	−0.0984753	−	0.995139i	$-0.531397\pi$
$877$	−172.726	−0.196951	−0.0984753	+	0.995139i	$0.531397\pi$
$878$	− 174.260i	− 0.198474i
$879$	0	0
$880$	2.72124	0.00309232
$881$	383.373i	0.435156i	0.976043	+	0.217578i	$0.0698157\pi$
$881$	383.373i	0.435156i	−0.976043	+	0.217578i	$0.930184\pi$
$882$	0	0
$883$	−1187.34	−1.34467	−0.672334	−	0.740248i	$-0.734708\pi$
$883$	−1187.34	−1.34467	−0.672334	+	0.740248i	$0.734708\pi$
$884$	191.125i	0.216205i
$885$	0	0
$886$	−1388.37	−1.56701
$887$	1142.28i	1.28781i	0.765107	+	0.643903i	$0.222686\pi$
$887$	1142.28i	1.28781i	−0.765107	+	0.643903i	$0.777314\pi$
$888$	0	0
$889$	−44.1832	−0.0496998
$890$	275.271i	0.309293i
$891$	0	0
$892$	630.414	0.706742
$893$	60.5684i	0.0678257i
$894$	0	0
$895$	22.0210	0.0246045
$896$	− 477.327i	− 0.532731i
$897$	0	0
$898$	−1664.16	−1.85319
$899$	− 1.87275i	− 0.00208314i
$900$	0	0
$901$	−111.073	−0.123277
$902$	16.0978i	0.0178468i
$903$	0	0
$904$	487.169	0.538904
$905$	6.45920i	0.00713724i
$906$	0	0
$907$	−412.022	−0.454269	−0.227135	−	0.973863i	$-0.572936\pi$
$907$	−412.022	−0.454269	−0.227135	+	0.973863i	$0.572936\pi$
$908$	246.667i	0.271660i
$909$	0	0
$910$	−231.383	−0.254267
$911$	− 1002.94i	− 1.10093i	−0.834859	−	0.550464i	$-0.814451\pi$
$911$	− 1002.94i	− 1.10093i	0.834859	−	0.550464i	$-0.185549\pi$
$912$	0	0
$913$	−10.8979	−0.0119364
$914$	838.090i	0.916947i
$915$	0	0
$916$	345.256	0.376918
$917$	835.970i	0.911636i
$918$	0	0
$919$	−1339.72	−1.45781	−0.728903	−	0.684616i	$-0.759970\pi$
$919$	−1339.72	−1.45781	−0.728903	+	0.684616i	$0.759970\pi$
$920$	19.4566i	0.0211485i
$921$	0	0
$922$	326.440	0.354056
$923$	632.530i	0.685298i
$924$	0	0
$925$	−1466.65	−1.58557
$926$	67.4290i	0.0728175i
$927$	0	0
$928$	41.7318	0.0449696
$929$	− 355.644i	− 0.382824i	−0.981510	−	0.191412i	$-0.938693\pi$
$929$	− 355.644i	− 0.382824i	0.981510	−	0.191412i	$-0.0613067\pi$
$930$	0	0
$931$	−32.9946	−0.0354399
$932$	227.825i	0.244447i
$933$	0	0
$934$	−756.125	−0.809556
$935$	− 0.560383i	0 0.000599340i
$936$	0	0
$937$	548.887	0.585792	0.292896	−	0.956144i	$-0.405381\pi$
$937$	548.887	0.585792	0.292896	+	0.956144i	$0.405381\pi$
$938$	− 1022.48i	− 1.09006i
$939$	0	0
$940$	−170.963	−0.181875
$941$	648.277i	0.688924i	0.938800	+	0.344462i	$0.111939\pi$
$941$	648.277i	0.688924i	−0.938800	+	0.344462i	$0.888061\pi$
$942$	0	0
$943$	−218.331	−0.231528
$944$	1153.11i	1.22151i
$945$	0	0
$946$	−14.1931	−0.0150033
$947$	− 284.695i	− 0.300628i	−0.988638	−	0.150314i	$-0.951971\pi$
$947$	− 284.695i	− 0.300628i	0.988638	−	0.150314i	$-0.0480285\pi$
$948$	0	0
$949$	939.504	0.989994
$950$	58.3740i	0.0614463i
$951$	0	0
$952$	−66.7392	−0.0701042
$953$	159.812i	0.167694i	0.996479	+	0.0838468i	$0.0267206\pi$
$953$	159.812i	0.167694i	−0.996479	+	0.0838468i	$0.973279\pi$
$954$	0	0
$955$	−271.916	−0.284729
$956$	− 467.368i	− 0.488879i
$957$	0	0
$958$	86.2400	0.0900208
$959$	363.159i	0.378686i
$960$	0	0
$961$	−958.746	−0.997655
$962$	3110.40i	3.23326i
$963$	0	0
$964$	−620.985	−0.644176
$965$	273.753i	0.283682i
$966$	0	0
$967$	487.083	0.503705	0.251852	−	0.967766i	$-0.418960\pi$
$967$	487.083	0.503705	0.251852	+	0.967766i	$0.418960\pi$
$968$	503.128i	0.519761i
$969$	0	0
$970$	466.131	0.480547
$971$	126.842i	0.130630i	0.997865	+	0.0653150i	$0.0208052\pi$
$971$	126.842i	0.130630i	−0.997865	+	0.0653150i	$0.979195\pi$
$972$	0	0
$973$	−182.052	−0.187103
$974$	607.356i	0.623569i
$975$	0	0
$976$	−141.628	−0.145111
$977$	− 1131.60i	− 1.15824i	−0.815243	−	0.579119i	$-0.803396\pi$
$977$	− 1131.60i	− 1.15824i	0.815243	−	0.579119i	$-0.196604\pi$
$978$	0	0
$979$	10.7654	0.0109963
$980$	− 93.1318i	− 0.0950324i
$981$	0	0
$982$	−559.370	−0.569623
$983$	− 120.256i	− 0.122336i	−0.998127	−	0.0611679i	$-0.980517\pi$
$983$	− 120.256i	− 0.122336i	0.998127	−	0.0611679i	$-0.0194825\pi$
$984$	0	0
$985$	54.8091	0.0556438
$986$	− 12.8667i	− 0.0130494i
$987$	0	0
$988$	45.8106	0.0463670
$989$	− 192.498i	− 0.194639i
$990$	0	0
$991$	1347.36	1.35959	0.679796	−	0.733401i	$-0.262068\pi$
$991$	1347.36	1.35959	0.679796	+	0.733401i	$0.262068\pi$
$992$	50.2262i	0.0506312i
$993$	0	0
$994$	314.477	0.316375
$995$	− 272.261i	− 0.273629i
$996$	0	0
$997$	−275.154	−0.275982	−0.137991	−	0.990433i	$-0.544065\pi$
$997$	−275.154	−0.275982	−0.137991	+	0.990433i	$0.544065\pi$
$998$	1559.42i	1.56254i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1143.3.b.a.890.67	yes	84
3.2	odd	2	inner	1143.3.b.a.890.18	✓	84

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1143.3.b.a.890.18	✓	84	3.2	odd	2	inner
1143.3.b.a.890.67	yes	84	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Degree 1	Degree 2
Degree 3	Degree 4
$\zeta$ zeros

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)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Properties

Related objects

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