LMFDB - Embedded newform 810.3.d.c.161.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [810,3,Mod(161,810)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(810, 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("810.161");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 810 = 2 \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	810.d (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:	$22.0709014132$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 12x^{14} + 138x^{12} - 1040x^{10} + 5541x^{8} - 26220x^{6} + 99328x^{4} - 202728x^{2} + 181476 $ x^16 - 12x^14 + 138x^12 - 1040x^10 + 5541x^8 - 26220x^6 + 99328x^4 - 202728*x^2 + 181476
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12}\cdot 3^{14} $
Twist minimal:	no (minimal twist has level 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.1
Root			$-2.11536 - 0.410927i$ of defining polynomial
Character	$\chi$	$=$	810.161
Dual form			810.3.d.c.161.13

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.41421i q^{2} -2.00000 q^{4} -2.23607i q^{5} -6.63197 q^{7} +2.82843i q^{8} -3.16228 q^{10} -21.0354i q^{11} -9.78248 q^{13} +9.37902i q^{14} +4.00000 q^{16} +12.9017i q^{17} +16.1821 q^{19} +4.47214i q^{20} -29.7486 q^{22} -16.9365i q^{23} -5.00000 q^{25} +13.8345i q^{26} +13.2639 q^{28} +50.0591i q^{29} -10.0642 q^{31} -5.65685i q^{32} +18.2457 q^{34} +14.8295i q^{35} +2.54425 q^{37} -22.8850i q^{38} +6.32456 q^{40} +53.7648i q^{41} -74.8349 q^{43} +42.0709i q^{44} -23.9518 q^{46} +47.2003i q^{47} -5.01700 q^{49} +7.07107i q^{50} +19.5650 q^{52} +1.08813i q^{53} -47.0367 q^{55} -18.7580i q^{56} +70.7943 q^{58} -9.63952i q^{59} -44.9145 q^{61} +14.2330i q^{62} -8.00000 q^{64} +21.8743i q^{65} -3.34135 q^{67} -25.8033i q^{68} +20.9721 q^{70} +38.4670i q^{71} -88.9201 q^{73} -3.59811i q^{74} -32.3642 q^{76} +139.506i q^{77} +117.647 q^{79} -8.94427i q^{80} +76.0349 q^{82} +147.123i q^{83} +28.8490 q^{85} +105.833i q^{86} +59.4972 q^{88} -145.331i q^{89} +64.8771 q^{91} +33.8729i q^{92} +66.7513 q^{94} -36.1843i q^{95} -47.0534 q^{97} +7.09511i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 32 q^{4} + 8 q^{7} - 40 q^{13} + 64 q^{16} + 80 q^{19} - 48 q^{22} - 80 q^{25} - 16 q^{28} + 32 q^{31} + 96 q^{34} - 88 q^{37} - 184 q^{43} + 24 q^{46} + 168 q^{49} + 80 q^{52} + 152 q^{61} - 128 q^{64}+ \cdots + 32 q^{97}+O(q^{100}) $ 16 * q - 32 * q^4 + 8 * q^7 - 40 * q^13 + 64 * q^16 + 80 * q^19 - 48 * q^22 - 80 * q^25 - 16 * q^28 + 32 * q^31 + 96 * q^34 - 88 * q^37 - 184 * q^43 + 24 * q^46 + 168 * q^49 + 80 * q^52 + 152 * q^61 - 128 * q^64 - 112 * q^67 - 120 * q^70 + 416 * q^73 - 160 * q^76 - 160 * q^79 - 192 * q^82 + 120 * q^85 + 96 * q^88 - 656 * q^91 + 168 * q^94 + 32 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/810\mathbb{Z}\right)^\times$.

$n$	$487$	$731$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 1.41421i	− 0.707107i
$2$	− 1.41421i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−2.00000	−0.500000
$5$	− 2.23607i	− 0.447214i
$5$	− 2.23607i	− 0.447214i
$6$	0	0
$7$	−6.63197	−0.947424	−0.473712	−	0.880680i	$-0.657086\pi$
$7$	−6.63197	−0.947424	−0.473712	+	0.880680i	$0.657086\pi$
$8$	2.82843i	0.353553i
$9$	0	0
$10$	−3.16228	−0.316228
$11$	− 21.0354i	− 1.91231i	−0.292857	−	0.956156i	$-0.594606\pi$
$11$	− 21.0354i	− 1.91231i	0.292857	−	0.956156i	$-0.405394\pi$
$12$	0	0
$13$	−9.78248	−0.752498	−0.376249	−	0.926519i	$-0.622786\pi$
$13$	−9.78248	−0.752498	−0.376249	+	0.926519i	$0.622786\pi$
$14$	9.37902i	0.669930i
$15$	0	0
$16$	4.00000	0.250000
$17$	12.9017i	0.758921i	0.925208	+	0.379460i	$0.123890\pi$
$17$	12.9017i	0.758921i	−0.925208	+	0.379460i	$0.876110\pi$
$18$	0	0
$19$	16.1821	0.851691	0.425845	−	0.904796i	$-0.359977\pi$
$19$	16.1821	0.851691	0.425845	+	0.904796i	$0.359977\pi$
$20$	4.47214i	0.223607i
$21$	0	0
$22$	−29.7486	−1.35221
$23$	− 16.9365i	− 0.736368i	−0.929753	−	0.368184i	$-0.879980\pi$
$23$	− 16.9365i	− 0.736368i	0.929753	−	0.368184i	$-0.120020\pi$
$24$	0	0
$25$	−5.00000	−0.200000
$26$	13.8345i	0.532097i
$27$	0	0
$28$	13.2639	0.473712
$29$	50.0591i	1.72618i	0.505053	+	0.863089i	$0.331473\pi$
$29$	50.0591i	1.72618i	−0.505053	+	0.863089i	$0.668527\pi$
$30$	0	0
$31$	−10.0642	−0.324653	−0.162326	−	0.986737i	$-0.551900\pi$
$31$	−10.0642	−0.324653	−0.162326	+	0.986737i	$0.551900\pi$
$32$	− 5.65685i	− 0.176777i
$33$	0	0
$34$	18.2457	0.536638
$35$	14.8295i	0.423701i
$36$	0	0
$37$	2.54425	0.0687635	0.0343817	−	0.999409i	$-0.489054\pi$
$37$	2.54425	0.0687635	0.0343817	+	0.999409i	$0.489054\pi$
$38$	− 22.8850i	− 0.602236i
$39$	0	0
$40$	6.32456	0.158114
$41$	53.7648i	1.31134i	0.755049	+	0.655669i	$0.227613\pi$
$41$	53.7648i	1.31134i	−0.755049	+	0.655669i	$0.772387\pi$
$42$	0	0
$43$	−74.8349	−1.74035	−0.870173	−	0.492746i	$-0.835993\pi$
$43$	−74.8349	−1.74035	−0.870173	+	0.492746i	$0.835993\pi$
$44$	42.0709i	0.956156i
$45$	0	0
$46$	−23.9518	−0.520690
$47$	47.2003i	1.00426i	0.864792	+	0.502131i	$0.167450\pi$
$47$	47.2003i	1.00426i	−0.864792	+	0.502131i	$0.832550\pi$
$48$	0	0
$49$	−5.01700	−0.102388
$50$	7.07107i	0.141421i
$51$	0	0
$52$	19.5650	0.376249
$53$	1.08813i	0.0205307i	0.999947	+	0.0102654i	$0.00326762\pi$
$53$	1.08813i	0.0205307i	−0.999947	+	0.0102654i	$0.996732\pi$
$54$	0	0
$55$	−47.0367	−0.855212
$56$	− 18.7580i	− 0.334965i
$57$	0	0
$58$	70.7943	1.22059
$59$	− 9.63952i	− 0.163382i	−0.996658	−	0.0816908i	$-0.973968\pi$
$59$	− 9.63952i	− 0.163382i	0.996658	−	0.0816908i	$-0.0260320\pi$
$60$	0	0
$61$	−44.9145	−0.736303	−0.368152	−	0.929766i	$-0.620009\pi$
$61$	−44.9145	−0.736303	−0.368152	+	0.929766i	$0.620009\pi$
$62$	14.2330i	0.229564i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	21.8743i	0.336527i
$66$	0	0
$67$	−3.34135	−0.0498708	−0.0249354	−	0.999689i	$-0.507938\pi$
$67$	−3.34135	−0.0498708	−0.0249354	+	0.999689i	$0.507938\pi$
$68$	− 25.8033i	− 0.379460i
$69$	0	0
$70$	20.9721	0.299602
$71$	38.4670i	0.541788i	0.962609	+	0.270894i	$0.0873193\pi$
$71$	38.4670i	0.541788i	−0.962609	+	0.270894i	$0.912681\pi$
$72$	0	0
$73$	−88.9201	−1.21808	−0.609042	−	0.793138i	$-0.708446\pi$
$73$	−88.9201	−1.21808	−0.609042	+	0.793138i	$0.708446\pi$
$74$	− 3.59811i	− 0.0486231i
$75$	0	0
$76$	−32.3642	−0.425845
$77$	139.506i	1.81177i
$78$	0	0
$79$	117.647	1.48920	0.744602	−	0.667509i	$-0.232639\pi$
$79$	117.647	1.48920	0.744602	+	0.667509i	$0.232639\pi$
$80$	− 8.94427i	− 0.111803i
$81$	0	0
$82$	76.0349	0.927255
$83$	147.123i	1.77256i	0.463146	+	0.886282i	$0.346721\pi$
$83$	147.123i	1.77256i	−0.463146	+	0.886282i	$0.653279\pi$
$84$	0	0
$85$	28.8490	0.339400
$86$	105.833i	1.23061i
$87$	0	0
$88$	59.4972	0.676105
$89$	− 145.331i	− 1.63293i	−0.577396	−	0.816464i	$-0.695931\pi$
$89$	− 145.331i	− 1.63293i	0.577396	−	0.816464i	$-0.304069\pi$
$90$	0	0
$91$	64.8771	0.712935
$92$	33.8729i	0.368184i
$93$	0	0
$94$	66.7513	0.710120
$95$	− 36.1843i	− 0.380888i
$96$	0	0
$97$	−47.0534	−0.485086	−0.242543	−	0.970141i	$-0.577982\pi$
$97$	−47.0534	−0.485086	−0.242543	+	0.970141i	$0.577982\pi$
$98$	7.09511i	0.0723991i
$99$	0	0
$100$	10.0000	0.100000
$101$	− 22.3550i	− 0.221337i	−0.993857	−	0.110668i	$-0.964701\pi$
$101$	− 22.3550i	− 0.221337i	0.993857	−	0.110668i	$-0.0352991\pi$
$102$	0	0
$103$	123.899	1.20290	0.601452	−	0.798909i	$-0.294589\pi$
$103$	123.899	1.20290	0.601452	+	0.798909i	$0.294589\pi$
$104$	− 27.6690i	− 0.266048i
$105$	0	0
$106$	1.53884	0.0145174
$107$	47.3690i	0.442701i	0.975194	+	0.221351i	$0.0710465\pi$
$107$	47.3690i	0.442701i	−0.975194	+	0.221351i	$0.928954\pi$
$108$	0	0
$109$	−26.8745	−0.246555	−0.123278	−	0.992372i	$-0.539341\pi$
$109$	−26.8745	−0.246555	−0.123278	+	0.992372i	$0.539341\pi$
$110$	66.5199i	0.604726i
$111$	0	0
$112$	−26.5279	−0.236856
$113$	18.8099i	0.166459i	0.996530	+	0.0832297i	$0.0265235\pi$
$113$	18.8099i	0.166459i	−0.996530	+	0.0832297i	$0.973476\pi$
$114$	0	0
$115$	−37.8711	−0.329314
$116$	− 100.118i	− 0.863089i
$117$	0	0
$118$	−13.6323	−0.115528
$119$	− 85.5634i	− 0.719020i
$120$	0	0
$121$	−321.490	−2.65694
$122$	63.5187i	0.520645i
$123$	0	0
$124$	20.1285	0.162326
$125$	11.1803i	0.0894427i
$126$	0	0
$127$	−128.820	−1.01433	−0.507166	−	0.861849i	$-0.669307\pi$
$127$	−128.820	−1.01433	−0.507166	+	0.861849i	$0.669307\pi$
$128$	11.3137i	0.0883883i
$129$	0	0
$130$	30.9349	0.237961
$131$	30.8116i	0.235203i	0.993061	+	0.117601i	$0.0375205\pi$
$131$	30.8116i	0.235203i	−0.993061	+	0.117601i	$0.962480\pi$
$132$	0	0
$133$	−107.319	−0.806912
$134$	4.72538i	0.0352640i
$135$	0	0
$136$	−36.4914	−0.268319
$137$	− 106.545i	− 0.777697i	−0.921302	−	0.388849i	$-0.872873\pi$
$137$	− 106.545i	− 0.777697i	0.921302	−	0.388849i	$-0.127127\pi$
$138$	0	0
$139$	114.237	0.821845	0.410923	−	0.911670i	$-0.365207\pi$
$139$	114.237	0.821845	0.410923	+	0.911670i	$0.365207\pi$
$140$	− 29.6591i	− 0.211850i
$141$	0	0
$142$	54.4005	0.383102
$143$	205.779i	1.43901i
$144$	0	0
$145$	111.936	0.771970
$146$	125.752i	0.861315i
$147$	0	0
$148$	−5.08850	−0.0343817
$149$	− 120.884i	− 0.811301i	−0.914028	−	0.405651i	$-0.867045\pi$
$149$	− 120.884i	− 0.811301i	0.914028	−	0.405651i	$-0.132955\pi$
$150$	0	0
$151$	126.988	0.840978	0.420489	−	0.907298i	$-0.361859\pi$
$151$	126.988	0.840978	0.420489	+	0.907298i	$0.361859\pi$
$152$	45.7700i	0.301118i
$153$	0	0
$154$	197.292	1.28112
$155$	22.5043i	0.145189i
$156$	0	0
$157$	104.213	0.663776	0.331888	−	0.943319i	$-0.392314\pi$
$157$	104.213	0.663776	0.331888	+	0.943319i	$0.392314\pi$
$158$	− 166.378i	− 1.05303i
$159$	0	0
$160$	−12.6491	−0.0790569
$161$	112.322i	0.697652i
$162$	0	0
$163$	−188.533	−1.15664	−0.578321	−	0.815809i	$-0.696292\pi$
$163$	−188.533	−1.15664	−0.578321	+	0.815809i	$0.696292\pi$
$164$	− 107.530i	− 0.655669i
$165$	0	0
$166$	208.063	1.25339
$167$	67.9163i	0.406684i	0.979108	+	0.203342i	$0.0651804\pi$
$167$	67.9163i	0.406684i	−0.979108	+	0.203342i	$0.934820\pi$
$168$	0	0
$169$	−73.3031	−0.433746
$170$	− 40.7986i	− 0.239992i
$171$	0	0
$172$	149.670	0.870173
$173$	122.393i	0.707472i	0.935345	+	0.353736i	$0.115089\pi$
$173$	122.393i	0.707472i	−0.935345	+	0.353736i	$0.884911\pi$
$174$	0	0
$175$	33.1598	0.189485
$176$	− 84.1418i	− 0.478078i
$177$	0	0
$178$	−205.529	−1.15466
$179$	78.4259i	0.438134i	0.975710	+	0.219067i	$0.0703013\pi$
$179$	78.4259i	0.438134i	−0.975710	+	0.219067i	$0.929699\pi$
$180$	0	0
$181$	219.562	1.21305	0.606525	−	0.795064i	$-0.292563\pi$
$181$	219.562	1.21305	0.606525	+	0.795064i	$0.292563\pi$
$182$	− 91.7500i	− 0.504121i
$183$	0	0
$184$	47.9035	0.260345
$185$	− 5.68911i	− 0.0307520i
$186$	0	0
$187$	271.392	1.45129
$188$	− 94.4006i	− 0.502131i
$189$	0	0
$190$	−51.1724	−0.269328
$191$	− 322.982i	− 1.69100i	−0.533973	−	0.845501i	$-0.679302\pi$
$191$	− 322.982i	− 1.69100i	0.533973	−	0.845501i	$-0.320698\pi$
$192$	0	0
$193$	−172.784	−0.895252	−0.447626	−	0.894221i	$-0.647730\pi$
$193$	−172.784	−0.895252	−0.447626	+	0.894221i	$0.647730\pi$
$194$	66.5435i	0.343008i
$195$	0	0
$196$	10.0340	0.0511939
$197$	− 71.7321i	− 0.364123i	−0.983287	−	0.182061i	$-0.941723\pi$
$197$	− 71.7321i	− 0.364123i	0.983287	−	0.182061i	$-0.0582769\pi$
$198$	0	0
$199$	133.225	0.669472	0.334736	−	0.942312i	$-0.391353\pi$
$199$	133.225	0.669472	0.334736	+	0.942312i	$0.391353\pi$
$200$	− 14.1421i	− 0.0707107i
$201$	0	0
$202$	−31.6148	−0.156509
$203$	− 331.991i	− 1.63542i
$204$	0	0
$205$	120.222	0.586448
$206$	− 175.220i	− 0.850581i
$207$	0	0
$208$	−39.1299	−0.188125
$209$	− 340.398i	− 1.62870i
$210$	0	0
$211$	−376.327	−1.78354	−0.891771	−	0.452486i	$-0.850537\pi$
$211$	−376.327	−1.78354	−0.891771	+	0.452486i	$0.850537\pi$
$212$	− 2.17626i	− 0.0102654i
$213$	0	0
$214$	66.9899	0.313037
$215$	167.336i	0.778307i
$216$	0	0
$217$	66.7457	0.307584
$218$	38.0063i	0.174341i
$219$	0	0
$220$	94.0733	0.427606
$221$	− 126.210i	− 0.571087i
$222$	0	0
$223$	59.1770	0.265368	0.132684	−	0.991158i	$-0.457641\pi$
$223$	59.1770	0.265368	0.132684	+	0.991158i	$0.457641\pi$
$224$	37.5161i	0.167482i
$225$	0	0
$226$	26.6012	0.117705
$227$	149.839i	0.660082i	0.943967	+	0.330041i	$0.107063\pi$
$227$	149.839i	0.660082i	−0.943967	+	0.330041i	$0.892937\pi$
$228$	0	0
$229$	−170.864	−0.746130	−0.373065	−	0.927805i	$-0.621693\pi$
$229$	−170.864	−0.746130	−0.373065	+	0.927805i	$0.621693\pi$
$230$	53.5578i	0.232860i
$231$	0	0
$232$	−141.589	−0.610296
$233$	− 448.983i	− 1.92697i	−0.267770	−	0.963483i	$-0.586287\pi$
$233$	− 448.983i	− 1.92697i	0.267770	−	0.963483i	$-0.413713\pi$
$234$	0	0
$235$	105.543	0.449119
$236$	19.2790i	0.0816908i
$237$	0	0
$238$	−121.005	−0.508424
$239$	231.916i	0.970360i	0.874414	+	0.485180i	$0.161246\pi$
$239$	231.916i	0.970360i	−0.874414	+	0.485180i	$0.838754\pi$
$240$	0	0
$241$	−397.341	−1.64872	−0.824359	−	0.566067i	$-0.808464\pi$
$241$	−397.341	−1.64872	−0.824359	+	0.566067i	$0.808464\pi$
$242$	454.655i	1.87874i
$243$	0	0
$244$	89.8290	0.368152
$245$	11.2184i	0.0457892i
$246$	0	0
$247$	−158.301	−0.640896
$248$	− 28.4660i	− 0.114782i
$249$	0	0
$250$	15.8114	0.0632456
$251$	− 193.179i	− 0.769636i	−0.922992	−	0.384818i	$-0.874264\pi$
$251$	− 193.179i	− 0.769636i	0.922992	−	0.384818i	$-0.125736\pi$
$252$	0	0
$253$	−356.266	−1.40816
$254$	182.179i	0.717240i
$255$	0	0
$256$	16.0000	0.0625000
$257$	− 120.866i	− 0.470297i	−0.971959	−	0.235148i	$-0.924442\pi$
$257$	− 120.866i	− 0.470297i	0.971959	−	0.235148i	$-0.0755576\pi$
$258$	0	0
$259$	−16.8734	−0.0651482
$260$	− 43.7486i	− 0.168264i
$261$	0	0
$262$	43.5741	0.166313
$263$	− 301.036i	− 1.14462i	−0.820036	−	0.572312i	$-0.806047\pi$
$263$	− 301.036i	− 1.14462i	0.820036	−	0.572312i	$-0.193953\pi$
$264$	0	0
$265$	2.43313	0.00918161
$266$	151.772i	0.570573i
$267$	0	0
$268$	6.68269	0.0249354
$269$	− 128.065i	− 0.476077i	−0.971256	−	0.238038i	$-0.923496\pi$
$269$	− 128.065i	− 0.476077i	0.971256	−	0.238038i	$-0.0765044\pi$
$270$	0	0
$271$	−512.831	−1.89236	−0.946182	−	0.323635i	$-0.895095\pi$
$271$	−512.831	−1.89236	−0.946182	+	0.323635i	$0.895095\pi$
$272$	51.6066i	0.189730i
$273$	0	0
$274$	−150.677	−0.549915
$275$	105.177i	0.382463i
$276$	0	0
$277$	−277.522	−1.00188	−0.500942	−	0.865481i	$-0.667013\pi$
$277$	−277.522	−1.00188	−0.500942	+	0.865481i	$0.667013\pi$
$278$	− 161.555i	− 0.581133i
$279$	0	0
$280$	−41.9442	−0.149801
$281$	− 103.195i	− 0.367243i	−0.982997	−	0.183622i	$-0.941218\pi$
$281$	− 103.195i	− 0.367243i	0.982997	−	0.183622i	$-0.0587821\pi$
$282$	0	0
$283$	−164.068	−0.579746	−0.289873	−	0.957065i	$-0.593613\pi$
$283$	−164.068	−0.579746	−0.289873	+	0.957065i	$0.593613\pi$
$284$	− 76.9339i	− 0.270894i
$285$	0	0
$286$	291.015	1.01754
$287$	− 356.567i	− 1.24239i
$288$	0	0
$289$	122.547	0.424039
$290$	− 158.301i	− 0.545865i
$291$	0	0
$292$	177.840	0.609042
$293$	292.987i	0.999957i	0.866038	+	0.499979i	$0.166659\pi$
$293$	292.987i	0.999957i	−0.866038	+	0.499979i	$0.833341\pi$
$294$	0	0
$295$	−21.5546	−0.0730665
$296$	7.19622i	0.0243116i
$297$	0	0
$298$	−170.956	−0.573677
$299$	165.680i	0.554115i
$300$	0	0
$301$	496.303	1.64885
$302$	− 179.588i	− 0.594661i
$303$	0	0
$304$	64.7285	0.212923
$305$	100.432i	0.329285i
$306$	0	0
$307$	−467.606	−1.52315	−0.761573	−	0.648079i	$-0.775573\pi$
$307$	−467.606	−1.52315	−0.761573	+	0.648079i	$0.775573\pi$
$308$	− 279.013i	− 0.905885i
$309$	0	0
$310$	31.8259	0.102664
$311$	− 537.805i	− 1.72928i	−0.502395	−	0.864638i	$-0.667548\pi$
$311$	− 537.805i	− 1.72928i	0.502395	−	0.864638i	$-0.332452\pi$
$312$	0	0
$313$	−266.258	−0.850664	−0.425332	−	0.905037i	$-0.639843\pi$
$313$	−266.258	−0.850664	−0.425332	+	0.905037i	$0.639843\pi$
$314$	− 147.379i	− 0.469361i
$315$	0	0
$316$	−235.294	−0.744602
$317$	476.875i	1.50434i	0.658971	+	0.752168i	$0.270992\pi$
$317$	476.875i	1.50434i	−0.658971	+	0.752168i	$0.729008\pi$
$318$	0	0
$319$	1053.02	3.30099
$320$	17.8885i	0.0559017i
$321$	0	0
$322$	158.847	0.493315
$323$	208.776i	0.646366i
$324$	0	0
$325$	48.9124	0.150500
$326$	266.625i	0.817869i
$327$	0	0
$328$	−152.070	−0.463628
$329$	− 313.031i	− 0.951461i
$330$	0	0
$331$	35.3440	0.106780	0.0533898	−	0.998574i	$-0.482997\pi$
$331$	35.3440	0.106780	0.0533898	+	0.998574i	$0.482997\pi$
$332$	− 294.246i	− 0.886282i
$333$	0	0
$334$	96.0482	0.287569
$335$	7.47148i	0.0223029i
$336$	0	0
$337$	−248.493	−0.737367	−0.368684	−	0.929555i	$-0.620191\pi$
$337$	−248.493	−0.737367	−0.368684	+	0.929555i	$0.620191\pi$
$338$	103.666i	0.306705i
$339$	0	0
$340$	−57.6980	−0.169700
$341$	211.706i	0.620838i
$342$	0	0
$343$	358.239	1.04443
$344$	− 211.665i	− 0.615305i
$345$	0	0
$346$	173.089	0.500258
$347$	2.69290i	0.00776053i	0.999992	+	0.00388026i	$0.00123513\pi$
$347$	2.69290i	0.00776053i	−0.999992	+	0.00388026i	$0.998765\pi$
$348$	0	0
$349$	392.385	1.12431	0.562156	−	0.827031i	$-0.309972\pi$
$349$	392.385	1.12431	0.562156	+	0.827031i	$0.309972\pi$
$350$	− 46.8951i	− 0.133986i
$351$	0	0
$352$	−118.994	−0.338052
$353$	19.7825i	0.0560410i	0.999607	+	0.0280205i	$0.00892037\pi$
$353$	19.7825i	0.0560410i	−0.999607	+	0.0280205i	$0.991080\pi$
$354$	0	0
$355$	86.0147	0.242295
$356$	290.661i	0.816464i
$357$	0	0
$358$	110.911	0.309807
$359$	− 415.325i	− 1.15689i	−0.815720	−	0.578447i	$-0.803659\pi$
$359$	− 415.325i	− 1.15689i	0.815720	−	0.578447i	$-0.196341\pi$
$360$	0	0
$361$	−99.1389	−0.274623
$362$	− 310.508i	− 0.857757i
$363$	0	0
$364$	−129.754	−0.356467
$365$	198.831i	0.544744i
$366$	0	0
$367$	−533.387	−1.45337	−0.726685	−	0.686970i	$-0.758940\pi$
$367$	−533.387	−1.45337	−0.726685	+	0.686970i	$0.758940\pi$
$368$	− 67.7458i	− 0.184092i
$369$	0	0
$370$	−8.04562	−0.0217449
$371$	− 7.21643i	− 0.0194513i
$372$	0	0
$373$	−215.580	−0.577961	−0.288981	−	0.957335i	$-0.593316\pi$
$373$	−215.580	−0.577961	−0.288981	+	0.957335i	$0.593316\pi$
$374$	− 383.806i	− 1.02622i
$375$	0	0
$376$	−133.503	−0.355060
$377$	− 489.702i	− 1.29895i
$378$	0	0
$379$	176.846	0.466612	0.233306	−	0.972403i	$-0.425046\pi$
$379$	176.846	0.466612	0.233306	+	0.972403i	$0.425046\pi$
$380$	72.3686i	0.190444i
$381$	0	0
$382$	−456.765	−1.19572
$383$	− 17.2027i	− 0.0449156i	−0.999748	−	0.0224578i	$-0.992851\pi$
$383$	− 17.2027i	− 0.0449156i	0.999748	−	0.0224578i	$-0.00714914\pi$
$384$	0	0
$385$	311.946	0.810249
$386$	244.353i	0.633039i
$387$	0	0
$388$	94.1068	0.242543
$389$	344.533i	0.885689i	0.896598	+	0.442845i	$0.146031\pi$
$389$	344.533i	0.885689i	−0.896598	+	0.442845i	$0.853969\pi$
$390$	0	0
$391$	218.508	0.558845
$392$	− 14.1902i	− 0.0361996i
$393$	0	0
$394$	−101.445	−0.257474
$395$	− 263.067i	− 0.665992i
$396$	0	0
$397$	68.9955	0.173792	0.0868960	−	0.996217i	$-0.472305\pi$
$397$	68.9955	0.173792	0.0868960	+	0.996217i	$0.472305\pi$
$398$	− 188.408i	− 0.473388i
$399$	0	0
$400$	−20.0000	−0.0500000
$401$	178.188i	0.444358i	0.975006	+	0.222179i	$0.0713169\pi$
$401$	178.188i	0.444358i	−0.975006	+	0.222179i	$0.928683\pi$
$402$	0	0
$403$	98.4532	0.244301
$404$	44.7100i	0.110668i
$405$	0	0
$406$	−469.506	−1.15642
$407$	− 53.5194i	− 0.131497i
$408$	0	0
$409$	11.2906	0.0276054	0.0138027	−	0.999905i	$-0.495606\pi$
$409$	11.2906	0.0276054	0.0138027	+	0.999905i	$0.495606\pi$
$410$	− 170.019i	− 0.414681i
$411$	0	0
$412$	−247.798	−0.601452
$413$	63.9290i	0.154792i
$414$	0	0
$415$	328.977	0.792715
$416$	55.3381i	0.133024i
$417$	0	0
$418$	−481.396	−1.15166
$419$	65.0792i	0.155320i	0.996980	+	0.0776601i	$0.0247449\pi$
$419$	65.0792i	0.155320i	−0.996980	+	0.0776601i	$0.975255\pi$
$420$	0	0
$421$	−230.088	−0.546527	−0.273263	−	0.961939i	$-0.588103\pi$
$421$	−230.088	−0.546527	−0.273263	+	0.961939i	$0.588103\pi$
$422$	532.207i	1.26116i
$423$	0	0
$424$	−3.07769	−0.00725870
$425$	− 64.5083i	− 0.151784i
$426$	0	0
$427$	297.872	0.697591
$428$	− 94.7380i	− 0.221351i
$429$	0	0
$430$	236.649	0.550346
$431$	− 67.4877i	− 0.156584i	−0.996930	−	0.0782920i	$-0.975053\pi$
$431$	− 67.4877i	− 0.156584i	0.996930	−	0.0782920i	$-0.0249467\pi$
$432$	0	0
$433$	132.060	0.304989	0.152495	−	0.988304i	$-0.451269\pi$
$433$	132.060	0.304989	0.152495	+	0.988304i	$0.451269\pi$
$434$	− 94.3927i	− 0.217495i
$435$	0	0
$436$	53.7491	0.123278
$437$	− 274.068i	− 0.627157i
$438$	0	0
$439$	689.023	1.56953	0.784764	−	0.619795i	$-0.212784\pi$
$439$	689.023	1.56953	0.784764	+	0.619795i	$0.212784\pi$
$440$	− 133.040i	− 0.302363i
$441$	0	0
$442$	−178.488	−0.403819
$443$	68.0825i	0.153685i	0.997043	+	0.0768426i	$0.0244839\pi$
$443$	68.0825i	0.153685i	−0.997043	+	0.0768426i	$0.975516\pi$
$444$	0	0
$445$	−324.969	−0.730268
$446$	− 83.6889i	− 0.187643i
$447$	0	0
$448$	53.0557	0.118428
$449$	685.554i	1.52685i	0.645898	+	0.763423i	$0.276483\pi$
$449$	685.554i	1.52685i	−0.645898	+	0.763423i	$0.723517\pi$
$450$	0	0
$451$	1130.97	2.50769
$452$	− 37.6198i	− 0.0832297i
$453$	0	0
$454$	211.904	0.466748
$455$	− 145.070i	− 0.318834i
$456$	0	0
$457$	519.396	1.13653	0.568266	−	0.822845i	$-0.307614\pi$
$457$	519.396	1.13653	0.568266	+	0.822845i	$0.307614\pi$
$458$	241.638i	0.527593i
$459$	0	0
$460$	75.7421	0.164657
$461$	89.5330i	0.194215i	0.995274	+	0.0971074i	$0.0309590\pi$
$461$	89.5330i	0.194215i	−0.995274	+	0.0971074i	$0.969041\pi$
$462$	0	0
$463$	−509.961	−1.10143	−0.550714	−	0.834694i	$-0.685645\pi$
$463$	−509.961	−1.10143	−0.550714	+	0.834694i	$0.685645\pi$
$464$	200.237i	0.431544i
$465$	0	0
$466$	−634.958	−1.36257
$467$	539.803i	1.15590i	0.816074	+	0.577948i	$0.196146\pi$
$467$	539.803i	1.15590i	−0.816074	+	0.577948i	$0.803854\pi$
$468$	0	0
$469$	22.1597	0.0472488
$470$	− 149.260i	− 0.317575i
$471$	0	0
$472$	27.2647	0.0577641
$473$	1574.19i	3.32809i
$474$	0	0
$475$	−80.9106	−0.170338
$476$	171.127i	0.359510i
$477$	0	0
$478$	327.979	0.686148
$479$	108.803i	0.227146i	0.993530	+	0.113573i	$0.0362296\pi$
$479$	108.803i	0.227146i	−0.993530	+	0.113573i	$0.963770\pi$
$480$	0	0
$481$	−24.8891	−0.0517444
$482$	561.925i	1.16582i
$483$	0	0
$484$	642.979	1.32847
$485$	105.215i	0.216937i
$486$	0	0
$487$	342.524	0.703334	0.351667	−	0.936125i	$-0.385615\pi$
$487$	342.524	0.703334	0.351667	+	0.936125i	$0.385615\pi$
$488$	− 127.037i	− 0.260323i
$489$	0	0
$490$	15.8652	0.0323779
$491$	− 497.280i	− 1.01279i	−0.862302	−	0.506395i	$-0.830978\pi$
$491$	− 497.280i	− 1.01279i	0.862302	−	0.506395i	$-0.169022\pi$
$492$	0	0
$493$	−645.846	−1.31003
$494$	223.872i	0.453182i
$495$	0	0
$496$	−40.2569	−0.0811632
$497$	− 255.112i	− 0.513303i
$498$	0	0
$499$	155.033	0.310687	0.155343	−	0.987861i	$-0.450352\pi$
$499$	155.033	0.310687	0.155343	+	0.987861i	$0.450352\pi$
$500$	− 22.3607i	− 0.0447214i
$501$	0	0
$502$	−273.196	−0.544215
$503$	580.438i	1.15395i	0.816761	+	0.576976i	$0.195767\pi$
$503$	580.438i	1.15395i	−0.816761	+	0.576976i	$0.804233\pi$
$504$	0	0
$505$	−49.9873	−0.0989848
$506$	503.836i	0.995723i
$507$	0	0
$508$	257.640	0.507166
$509$	− 182.684i	− 0.358907i	−0.983766	−	0.179454i	$-0.942567\pi$
$509$	− 182.684i	− 0.358907i	0.983766	−	0.179454i	$-0.0574330\pi$
$510$	0	0
$511$	589.715	1.15404
$512$	− 22.6274i	− 0.0441942i
$513$	0	0
$514$	−170.931	−0.332550
$515$	− 277.047i	− 0.537955i
$516$	0	0
$517$	992.879	1.92046
$518$	23.8626i	0.0460667i
$519$	0	0
$520$	−61.8698	−0.118980
$521$	47.7611i	0.0916720i	0.998949	+	0.0458360i	$0.0145952\pi$
$521$	47.7611i	0.0916720i	−0.998949	+	0.0458360i	$0.985405\pi$
$522$	0	0
$523$	266.851	0.510230	0.255115	−	0.966911i	$-0.417887\pi$
$523$	266.851	0.510230	0.255115	+	0.966911i	$0.417887\pi$
$524$	− 61.6231i	− 0.117601i
$525$	0	0
$526$	−425.729	−0.809371
$527$	− 129.845i	− 0.246386i
$528$	0	0
$529$	242.157	0.457763
$530$	− 3.44096i	− 0.00649238i
$531$	0	0
$532$	214.639	0.403456
$533$	− 525.953i	− 0.986779i
$534$	0	0
$535$	105.920	0.197982
$536$	− 9.45076i	− 0.0176320i
$537$	0	0
$538$	−181.111	−0.336637
$539$	105.535i	0.195798i
$540$	0	0
$541$	−707.266	−1.30733	−0.653666	−	0.756783i	$-0.726770\pi$
$541$	−707.266	−1.30733	−0.653666	+	0.756783i	$0.726770\pi$
$542$	725.252i	1.33810i
$543$	0	0
$544$	72.9828	0.134160
$545$	60.0933i	0.110263i
$546$	0	0
$547$	206.328	0.377200	0.188600	−	0.982054i	$-0.439605\pi$
$547$	206.328	0.377200	0.188600	+	0.982054i	$0.439605\pi$
$548$	213.089i	0.388849i
$549$	0	0
$550$	148.743	0.270442
$551$	810.063i	1.47017i
$552$	0	0
$553$	−780.232	−1.41091
$554$	392.475i	0.708439i
$555$	0	0
$556$	−228.473	−0.410923
$557$	643.753i	1.15575i	0.816125	+	0.577876i	$0.196118\pi$
$557$	643.753i	1.15575i	−0.816125	+	0.577876i	$0.803882\pi$
$558$	0	0
$559$	732.071	1.30961
$560$	59.3181i	0.105925i
$561$	0	0
$562$	−145.940	−0.259680
$563$	723.644i	1.28534i	0.766145	+	0.642668i	$0.222172\pi$
$563$	723.644i	1.28534i	−0.766145	+	0.642668i	$0.777828\pi$
$564$	0	0
$565$	42.0602	0.0744429
$566$	232.027i	0.409942i
$567$	0	0
$568$	−108.801	−0.191551
$569$	380.132i	0.668071i	0.942561	+	0.334035i	$0.108411\pi$
$569$	380.132i	0.668071i	−0.942561	+	0.334035i	$0.891589\pi$
$570$	0	0
$571$	41.4584	0.0726066	0.0363033	−	0.999341i	$-0.488442\pi$
$571$	41.4584	0.0726066	0.0363033	+	0.999341i	$0.488442\pi$
$572$	− 411.557i	− 0.719506i
$573$	0	0
$574$	−504.261	−0.878504
$575$	84.6823i	0.147274i
$576$	0	0
$577$	−601.252	−1.04203	−0.521016	−	0.853547i	$-0.674447\pi$
$577$	−601.252	−1.04203	−0.521016	+	0.853547i	$0.674447\pi$
$578$	− 173.308i	− 0.299841i
$579$	0	0
$580$	−223.871	−0.385985
$581$	− 975.714i	− 1.67937i
$582$	0	0
$583$	22.8892	0.0392611
$584$	− 251.504i	− 0.430658i
$585$	0	0
$586$	414.347	0.707076
$587$	− 727.976i	− 1.24016i	−0.784537	−	0.620082i	$-0.787099\pi$
$587$	− 727.976i	− 1.24016i	0.784537	−	0.620082i	$-0.212901\pi$
$588$	0	0
$589$	−162.861	−0.276504
$590$	30.4828i	0.0516658i
$591$	0	0
$592$	10.1770	0.0171909
$593$	418.998i	0.706573i	0.935515	+	0.353286i	$0.114936\pi$
$593$	418.998i	0.706573i	−0.935515	+	0.353286i	$0.885064\pi$
$594$	0	0
$595$	−191.326	−0.321555
$596$	241.768i	0.405651i
$597$	0	0
$598$	234.308	0.391819
$599$	878.072i	1.46590i	0.680284	+	0.732948i	$0.261856\pi$
$599$	878.072i	1.46590i	−0.680284	+	0.732948i	$0.738144\pi$
$600$	0	0
$601$	−532.458	−0.885954	−0.442977	−	0.896533i	$-0.646078\pi$
$601$	−532.458	−0.885954	−0.442977	+	0.896533i	$0.646078\pi$
$602$	− 701.878i	− 1.16591i
$603$	0	0
$604$	−253.975	−0.420489
$605$	718.873i	1.18822i
$606$	0	0
$607$	94.5285	0.155731	0.0778653	−	0.996964i	$-0.475190\pi$
$607$	94.5285	0.155731	0.0778653	+	0.996964i	$0.475190\pi$
$608$	− 91.5399i	− 0.150559i
$609$	0	0
$610$	142.032	0.232840
$611$	− 461.736i	− 0.755705i
$612$	0	0
$613$	−816.657	−1.33223	−0.666115	−	0.745849i	$-0.732044\pi$
$613$	−816.657	−1.33223	−0.666115	+	0.745849i	$0.732044\pi$
$614$	661.294i	1.07703i
$615$	0	0
$616$	−394.584	−0.640558
$617$	− 189.639i	− 0.307357i	−0.988121	−	0.153678i	$-0.950888\pi$
$617$	− 189.639i	− 0.307357i	0.988121	−	0.153678i	$-0.0491119\pi$
$618$	0	0
$619$	−95.4624	−0.154220	−0.0771101	−	0.997023i	$-0.524569\pi$
$619$	−95.4624	−0.154220	−0.0771101	+	0.997023i	$0.524569\pi$
$620$	− 45.0086i	− 0.0725946i
$621$	0	0
$622$	−760.571	−1.22278
$623$	963.828i	1.54708i
$624$	0	0
$625$	25.0000	0.0400000
$626$	376.546i	0.601510i
$627$	0	0
$628$	−208.426	−0.331888
$629$	32.8250i	0.0521861i
$630$	0	0
$631$	−192.912	−0.305724	−0.152862	−	0.988248i	$-0.548849\pi$
$631$	−192.912	−0.305724	−0.152862	+	0.988248i	$0.548849\pi$
$632$	332.756i	0.526513i
$633$	0	0
$634$	674.403	1.06373
$635$	288.050i	0.453623i
$636$	0	0
$637$	49.0787	0.0770467
$638$	− 1489.19i	− 2.33415i
$639$	0	0
$640$	25.2982	0.0395285
$641$	− 319.897i	− 0.499060i	−0.968367	−	0.249530i	$-0.919724\pi$
$641$	− 319.897i	− 0.499060i	0.968367	−	0.249530i	$-0.0802761\pi$
$642$	0	0
$643$	152.348	0.236934	0.118467	−	0.992958i	$-0.462202\pi$
$643$	152.348	0.236934	0.118467	+	0.992958i	$0.462202\pi$
$644$	− 224.644i	− 0.348826i
$645$	0	0
$646$	295.254	0.457050
$647$	− 632.862i	− 0.978148i	−0.872242	−	0.489074i	$-0.837335\pi$
$647$	− 632.862i	− 0.978148i	0.872242	−	0.489074i	$-0.162665\pi$
$648$	0	0
$649$	−202.772	−0.312437
$650$	− 69.1726i	− 0.106419i
$651$	0	0
$652$	377.065	0.578321
$653$	− 817.873i	− 1.25249i	−0.779628	−	0.626243i	$-0.784592\pi$
$653$	− 817.873i	− 1.25249i	0.779628	−	0.626243i	$-0.215408\pi$
$654$	0	0
$655$	68.8967	0.105186
$656$	215.059i	0.327834i
$657$	0	0
$658$	−442.692	−0.672785
$659$	425.477i	0.645640i	0.946460	+	0.322820i	$0.104631\pi$
$659$	425.477i	0.645640i	−0.946460	+	0.322820i	$0.895369\pi$
$660$	0	0
$661$	285.972	0.432635	0.216318	−	0.976323i	$-0.430595\pi$
$661$	285.972	0.432635	0.216318	+	0.976323i	$0.430595\pi$
$662$	− 49.9840i	− 0.0755046i
$663$	0	0
$664$	−416.126	−0.626696
$665$	239.973i	0.360862i
$666$	0	0
$667$	847.824	1.27110
$668$	− 135.833i	− 0.203342i
$669$	0	0
$670$	10.5663	0.0157705
$671$	944.796i	1.40804i
$672$	0	0
$673$	1197.42	1.77922	0.889612	−	0.456718i	$-0.150975\pi$
$673$	1197.42	1.77922	0.889612	+	0.456718i	$0.150975\pi$
$674$	351.422i	0.521398i
$675$	0	0
$676$	146.606	0.216873
$677$	− 905.553i	− 1.33760i	−0.743444	−	0.668799i	$-0.766809\pi$
$677$	− 905.553i	− 1.33760i	0.743444	−	0.668799i	$-0.233191\pi$
$678$	0	0
$679$	312.056	0.459582
$680$	81.5972i	0.119996i
$681$	0	0
$682$	299.397	0.438998
$683$	− 31.1970i	− 0.0456764i	−0.999739	−	0.0228382i	$-0.992730\pi$
$683$	− 31.1970i	− 0.0456764i	0.999739	−	0.0228382i	$-0.00727026\pi$
$684$	0	0
$685$	−238.241	−0.347797
$686$	− 506.626i	− 0.738523i
$687$	0	0
$688$	−299.340	−0.435087
$689$	− 10.6446i	− 0.0154493i
$690$	0	0
$691$	−203.064	−0.293869	−0.146935	−	0.989146i	$-0.546941\pi$
$691$	−203.064	−0.293869	−0.146935	+	0.989146i	$0.546941\pi$
$692$	− 244.785i	− 0.353736i
$693$	0	0
$694$	3.80834	0.00548752
$695$	− 255.441i	− 0.367540i
$696$	0	0
$697$	−693.655	−0.995201
$698$	− 554.916i	− 0.795008i
$699$	0	0
$700$	−66.3197	−0.0947424
$701$	− 633.699i	− 0.903993i	−0.892020	−	0.451997i	$-0.850712\pi$
$701$	− 633.699i	− 0.903993i	0.892020	−	0.451997i	$-0.149288\pi$
$702$	0	0
$703$	41.1714	0.0585652
$704$	168.284i	0.239039i
$705$	0	0
$706$	27.9767	0.0396270
$707$	148.258i	0.209700i
$708$	0	0
$709$	−149.380	−0.210691	−0.105346	−	0.994436i	$-0.533595\pi$
$709$	−149.380	−0.210691	−0.105346	+	0.994436i	$0.533595\pi$
$710$	− 121.643i	− 0.171328i
$711$	0	0
$712$	411.057	0.577328
$713$	170.452i	0.239064i
$714$	0	0
$715$	460.135	0.643546
$716$	− 156.852i	− 0.219067i
$717$	0	0
$718$	−587.359	−0.818048
$719$	257.590i	0.358262i	0.983825	+	0.179131i	$0.0573285\pi$
$719$	257.590i	0.358262i	−0.983825	+	0.179131i	$0.942671\pi$
$720$	0	0
$721$	−821.694	−1.13966
$722$	140.204i	0.194188i
$723$	0	0
$724$	−439.124	−0.606525
$725$	− 250.296i	− 0.345235i
$726$	0	0
$727$	607.381	0.835463	0.417731	−	0.908571i	$-0.362825\pi$
$727$	607.381	0.835463	0.417731	+	0.908571i	$0.362825\pi$
$728$	183.500i	0.252061i
$729$	0	0
$730$	281.190	0.385192
$731$	− 965.494i	− 1.32079i
$732$	0	0
$733$	887.285	1.21048	0.605242	−	0.796041i	$-0.293076\pi$
$733$	887.285	1.21048	0.605242	+	0.796041i	$0.293076\pi$
$734$	754.323i	1.02769i
$735$	0	0
$736$	−95.8070	−0.130173
$737$	70.2867i	0.0953686i
$738$	0	0
$739$	−30.1955	−0.0408599	−0.0204300	−	0.999791i	$-0.506504\pi$
$739$	−30.1955	−0.0408599	−0.0204300	+	0.999791i	$0.506504\pi$
$740$	11.3782i	0.0153760i
$741$	0	0
$742$	−10.2056	−0.0137541
$743$	− 570.016i	− 0.767181i	−0.923503	−	0.383591i	$-0.874687\pi$
$743$	− 570.016i	− 0.767181i	0.923503	−	0.383591i	$-0.125313\pi$
$744$	0	0
$745$	−270.305	−0.362825
$746$	304.876i	0.408680i
$747$	0	0
$748$	−542.784	−0.725647
$749$	− 314.150i	− 0.419426i
$750$	0	0
$751$	427.494	0.569233	0.284617	−	0.958641i	$-0.408134\pi$
$751$	427.494	0.569233	0.284617	+	0.958641i	$0.408134\pi$
$752$	188.801i	0.251065i
$753$	0	0
$754$	−692.544	−0.918493
$755$	− 283.953i	− 0.376097i
$756$	0	0
$757$	−587.791	−0.776474	−0.388237	−	0.921560i	$-0.626916\pi$
$757$	−587.791	−0.776474	−0.388237	+	0.921560i	$0.626916\pi$
$758$	− 250.098i	− 0.329945i
$759$	0	0
$760$	102.345	0.134664
$761$	1311.43i	1.72330i	0.507502	+	0.861651i	$0.330569\pi$
$761$	1311.43i	1.72330i	−0.507502	+	0.861651i	$0.669431\pi$
$762$	0	0
$763$	178.231	0.233592
$764$	645.963i	0.845501i
$765$	0	0
$766$	−24.3283	−0.0317601
$767$	94.2984i	0.122944i
$768$	0	0
$769$	−1191.52	−1.54945	−0.774724	−	0.632300i	$-0.782111\pi$
$769$	−1191.52	−1.54945	−0.774724	+	0.632300i	$0.782111\pi$
$770$	− 441.158i	− 0.572932i
$771$	0	0
$772$	345.567	0.447626
$773$	204.848i	0.265003i	0.991183	+	0.132502i	$0.0423010\pi$
$773$	204.848i	0.265003i	−0.991183	+	0.132502i	$0.957699\pi$
$774$	0	0
$775$	50.3212	0.0649306
$776$	− 133.087i	− 0.171504i
$777$	0	0
$778$	487.244	0.626277
$779$	870.029i	1.11685i
$780$	0	0
$781$	809.169	1.03607
$782$	− 309.017i	− 0.395163i
$783$	0	0
$784$	−20.0680	−0.0255970
$785$	− 233.027i	− 0.296850i
$786$	0	0
$787$	133.227	0.169285	0.0846426	−	0.996411i	$-0.473025\pi$
$787$	133.227	0.169285	0.0846426	+	0.996411i	$0.473025\pi$
$788$	143.464i	0.182061i
$789$	0	0
$790$	−372.033	−0.470928
$791$	− 124.747i	− 0.157708i
$792$	0	0
$793$	439.375	0.554067
$794$	− 97.5743i	− 0.122890i
$795$	0	0
$796$	−266.450	−0.334736
$797$	− 608.921i	− 0.764016i	−0.924159	−	0.382008i	$-0.875233\pi$
$797$	− 608.921i	− 0.764016i	0.924159	−	0.382008i	$-0.124767\pi$
$798$	0	0
$799$	−608.962	−0.762155
$800$	28.2843i	0.0353553i
$801$	0	0
$802$	251.995	0.314208
$803$	1870.47i	2.32936i
$804$	0	0
$805$	251.160	0.312000
$806$	− 139.234i	− 0.172747i
$807$	0	0
$808$	63.2295	0.0782544
$809$	− 164.342i	− 0.203142i	−0.994828	−	0.101571i	$-0.967613\pi$
$809$	− 164.342i	− 0.203142i	0.994828	−	0.101571i	$-0.0323869\pi$
$810$	0	0
$811$	−1026.77	−1.26606	−0.633028	−	0.774129i	$-0.718188\pi$
$811$	−1026.77	−1.26606	−0.633028	+	0.774129i	$0.718188\pi$
$812$	663.981i	0.817711i
$813$	0	0
$814$	−75.6879	−0.0929826
$815$	421.572i	0.517266i
$816$	0	0
$817$	−1210.99	−1.48224
$818$	− 15.9673i	− 0.0195200i
$819$	0	0
$820$	−240.444	−0.293224
$821$	302.693i	0.368688i	0.982862	+	0.184344i	$0.0590161\pi$
$821$	302.693i	0.368688i	−0.982862	+	0.184344i	$0.940984\pi$
$822$	0	0
$823$	−697.119	−0.847046	−0.423523	−	0.905885i	$-0.639207\pi$
$823$	−697.119	−0.847046	−0.423523	+	0.905885i	$0.639207\pi$
$824$	350.439i	0.425291i
$825$	0	0
$826$	90.4092	0.109454
$827$	− 454.516i	− 0.549596i	−0.961502	−	0.274798i	$-0.911389\pi$
$827$	− 454.516i	− 0.549596i	0.961502	−	0.274798i	$-0.0886110\pi$
$828$	0	0
$829$	591.488	0.713496	0.356748	−	0.934201i	$-0.383885\pi$
$829$	591.488	0.713496	0.356748	+	0.934201i	$0.383885\pi$
$830$	− 465.243i	− 0.560534i
$831$	0	0
$832$	78.2598	0.0940623
$833$	− 64.7276i	− 0.0777043i
$834$	0	0
$835$	151.865	0.181875
$836$	680.796i	0.814349i
$837$	0	0
$838$	92.0359	0.109828
$839$	− 1549.73i	− 1.84712i	−0.383457	−	0.923559i	$-0.625267\pi$
$839$	− 1549.73i	− 1.84712i	0.383457	−	0.923559i	$-0.374733\pi$
$840$	0	0
$841$	−1664.92	−1.97969
$842$	325.393i	0.386453i
$843$	0	0
$844$	752.655	0.891771
$845$	163.911i	0.193977i
$846$	0	0
$847$	2132.11	2.51725
$848$	4.35251i	0.00513268i
$849$	0	0
$850$	−91.2285	−0.107328
$851$	− 43.0906i	− 0.0506352i
$852$	0	0
$853$	−1641.55	−1.92444	−0.962219	−	0.272276i	$-0.912224\pi$
$853$	−1641.55	−1.92444	−0.962219	+	0.272276i	$0.912224\pi$
$854$	− 421.254i	− 0.493272i
$855$	0	0
$856$	−133.980	−0.156518
$857$	318.643i	0.371812i	0.982568	+	0.185906i	$0.0595219\pi$
$857$	318.643i	0.371812i	−0.982568	+	0.185906i	$0.940478\pi$
$858$	0	0
$859$	−628.332	−0.731470	−0.365735	−	0.930719i	$-0.619182\pi$
$859$	−628.332	−0.731470	−0.365735	+	0.930719i	$0.619182\pi$
$860$	− 334.672i	− 0.389153i
$861$	0	0
$862$	−95.4421	−0.110722
$863$	183.620i	0.212770i	0.994325	+	0.106385i	$0.0339276\pi$
$863$	183.620i	0.212770i	−0.994325	+	0.106385i	$0.966072\pi$
$864$	0	0
$865$	273.678	0.316391
$866$	− 186.761i	− 0.215660i
$867$	0	0
$868$	−133.491	−0.153792
$869$	− 2474.76i	− 2.84782i
$870$	0	0
$871$	32.6866	0.0375277
$872$	− 76.0126i	− 0.0871705i
$873$	0	0
$874$	−387.590	−0.443467
$875$	− 74.1477i	− 0.0847402i
$876$	0	0
$877$	−1445.29	−1.64800	−0.823999	−	0.566592i	$-0.808262\pi$
$877$	−1445.29	−1.64800	−0.823999	+	0.566592i	$0.808262\pi$
$878$	− 974.425i	− 1.10982i
$879$	0	0
$880$	−188.147	−0.213803
$881$	− 608.977i	− 0.691234i	−0.938376	−	0.345617i	$-0.887670\pi$
$881$	− 608.977i	− 0.691234i	0.938376	−	0.345617i	$-0.112330\pi$
$882$	0	0
$883$	−120.519	−0.136488	−0.0682440	−	0.997669i	$-0.521740\pi$
$883$	−120.519	−0.136488	−0.0682440	+	0.997669i	$0.521740\pi$
$884$	252.420i	0.285543i
$885$	0	0
$886$	96.2832	0.108672
$887$	− 1607.42i	− 1.81220i	−0.423064	−	0.906100i	$-0.639046\pi$
$887$	− 1607.42i	− 1.81220i	0.423064	−	0.906100i	$-0.360954\pi$
$888$	0	0
$889$	854.330	0.961002
$890$	459.576i	0.516377i
$891$	0	0
$892$	−118.354	−0.132684
$893$	763.801i	0.855320i
$894$	0	0
$895$	175.366	0.195939
$896$	− 75.0322i	− 0.0837412i
$897$	0	0
$898$	969.520	1.07964
$899$	− 503.807i	− 0.560408i
$900$	0	0
$901$	−14.0386	−0.0155812
$902$	− 1599.43i	− 1.77320i
$903$	0	0
$904$	−53.2025	−0.0588523
$905$	− 490.956i	− 0.542493i
$906$	0	0
$907$	647.989	0.714431	0.357215	−	0.934022i	$-0.383726\pi$
$907$	647.989	0.714431	0.357215	+	0.934022i	$0.383726\pi$
$908$	− 299.677i	− 0.330041i
$909$	0	0
$910$	−205.159	−0.225450
$911$	− 200.538i	− 0.220130i	−0.993924	−	0.110065i	$-0.964894\pi$
$911$	− 200.538i	− 0.220130i	0.993924	−	0.110065i	$-0.0351059\pi$
$912$	0	0
$913$	3094.79	3.38970
$914$	− 734.536i	− 0.803650i
$915$	0	0
$916$	341.727	0.373065
$917$	− 204.341i	− 0.222837i
$918$	0	0
$919$	−619.581	−0.674191	−0.337095	−	0.941471i	$-0.609444\pi$
$919$	−619.581	−0.674191	−0.337095	+	0.941471i	$0.609444\pi$
$920$	− 107.116i	− 0.116430i
$921$	0	0
$922$	126.619	0.137331
$923$	− 376.302i	− 0.407695i
$924$	0	0
$925$	−12.7212	−0.0137527
$926$	721.194i	0.778828i
$927$	0	0
$928$	283.177	0.305148
$929$	− 710.823i	− 0.765149i	−0.923925	−	0.382574i	$-0.875038\pi$
$929$	− 710.823i	− 0.765149i	0.923925	−	0.382574i	$-0.124962\pi$
$930$	0	0
$931$	−81.1857	−0.0872027
$932$	897.966i	0.963483i
$933$	0	0
$934$	763.397	0.817342
$935$	− 606.851i	− 0.649039i
$936$	0	0
$937$	435.674	0.464967	0.232483	−	0.972600i	$-0.425315\pi$
$937$	435.674	0.464967	0.232483	+	0.972600i	$0.425315\pi$
$938$	− 31.3386i	− 0.0334100i
$939$	0	0
$940$	−211.086	−0.224560
$941$	− 126.099i	− 0.134005i	−0.997753	−	0.0670027i	$-0.978656\pi$
$941$	− 126.099i	− 0.134005i	0.997753	−	0.0670027i	$-0.0213436\pi$
$942$	0	0
$943$	910.585	0.965626
$944$	− 38.5581i	− 0.0408454i
$945$	0	0
$946$	2226.23	2.35331
$947$	− 567.372i	− 0.599126i	−0.954077	−	0.299563i	$-0.903159\pi$
$947$	− 567.372i	− 0.599126i	0.954077	−	0.299563i	$-0.0968408\pi$
$948$	0	0
$949$	869.859	0.916606
$950$	114.425i	0.120447i
$951$	0	0
$952$	242.010	0.254212
$953$	873.128i	0.916189i	0.888903	+	0.458095i	$0.151468\pi$
$953$	873.128i	0.916189i	−0.888903	+	0.458095i	$0.848532\pi$
$954$	0	0
$955$	−722.209	−0.756239
$956$	− 463.832i	− 0.485180i
$957$	0	0
$958$	153.871	0.160616
$959$	706.600i	0.736809i
$960$	0	0
$961$	−859.711	−0.894601
$962$	35.1985i	0.0365888i
$963$	0	0
$964$	794.682	0.824359
$965$	386.356i	0.400369i
$966$	0	0
$967$	−840.748	−0.869439	−0.434720	−	0.900566i	$-0.643153\pi$
$967$	−840.748	−0.869439	−0.434720	+	0.900566i	$0.643153\pi$
$968$	− 909.310i	− 0.939370i
$969$	0	0
$970$	148.796	0.153398
$971$	258.099i	0.265807i	0.991129	+	0.132903i	$0.0424300\pi$
$971$	258.099i	0.265807i	−0.991129	+	0.132903i	$0.957570\pi$
$972$	0	0
$973$	−757.613	−0.778636
$974$	− 484.402i	− 0.497333i
$975$	0	0
$976$	−179.658	−0.184076
$977$	1151.99i	1.17911i	0.807728	+	0.589555i	$0.200697\pi$
$977$	1151.99i	1.17911i	−0.807728	+	0.589555i	$0.799303\pi$
$978$	0	0
$979$	−3057.09	−3.12267
$980$	− 22.4367i	− 0.0228946i
$981$	0	0
$982$	−703.260	−0.716151
$983$	− 639.379i	− 0.650436i	−0.945639	−	0.325218i	$-0.894562\pi$
$983$	− 639.379i	− 0.650436i	0.945639	−	0.325218i	$-0.105438\pi$
$984$	0	0
$985$	−160.398	−0.162841
$986$	913.364i	0.926333i
$987$	0	0
$988$	316.602	0.320448
$989$	1267.44i	1.28153i
$990$	0	0
$991$	1875.09	1.89212	0.946060	−	0.323992i	$-0.105025\pi$
$991$	1875.09	1.89212	0.946060	+	0.323992i	$0.105025\pi$
$992$	56.9319i	0.0573910i
$993$	0	0
$994$	−360.782	−0.362960
$995$	− 297.900i	− 0.299397i
$996$	0	0
$997$	−945.395	−0.948239	−0.474120	−	0.880460i	$-0.657234\pi$
$997$	−945.395	−0.948239	−0.474120	+	0.880460i	$0.657234\pi$
$998$	− 219.249i	− 0.219689i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	810.3.d.c.161.1		16
3.2	odd	2	inner	810.3.d.c.161.13		16
9.2	odd	6		270.3.h.a.71.6		16
9.4	even	3		270.3.h.a.251.6		16
9.5	odd	6		90.3.h.a.11.3	✓	16
9.7	even	3		90.3.h.a.41.3	yes	16
36.7	odd	6		720.3.bs.d.401.5		16
36.11	even	6		2160.3.bs.d.881.1		16
36.23	even	6		720.3.bs.d.641.5		16
36.31	odd	6		2160.3.bs.d.1601.1		16
45.2	even	12		1350.3.k.b.449.14		32
45.4	even	6		1350.3.i.g.251.2		16
45.7	odd	12		450.3.k.c.149.3		32
45.13	odd	12		1350.3.k.b.899.14		32
45.14	odd	6		450.3.i.g.101.6		16
45.22	odd	12		1350.3.k.b.899.3		32
45.23	even	12		450.3.k.c.299.3		32
45.29	odd	6		1350.3.i.g.1151.2		16
45.32	even	12		450.3.k.c.299.14		32
45.34	even	6		450.3.i.g.401.6		16
45.38	even	12		1350.3.k.b.449.3		32
45.43	odd	12		450.3.k.c.149.14		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.3.h.a.11.3	✓	16	9.5	odd	6
90.3.h.a.41.3	yes	16	9.7	even	3
270.3.h.a.71.6		16	9.2	odd	6
270.3.h.a.251.6		16	9.4	even	3
450.3.i.g.101.6		16	45.14	odd	6
450.3.i.g.401.6		16	45.34	even	6
450.3.k.c.149.3		32	45.7	odd	12
450.3.k.c.149.14		32	45.43	odd	12
450.3.k.c.299.3		32	45.23	even	12
450.3.k.c.299.14		32	45.32	even	12
720.3.bs.d.401.5		16	36.7	odd	6
720.3.bs.d.641.5		16	36.23	even	6
810.3.d.c.161.1		16	1.1	even	1	trivial
810.3.d.c.161.13		16	3.2	odd	2	inner
1350.3.i.g.251.2		16	45.4	even	6
1350.3.i.g.1151.2		16	45.29	odd	6
1350.3.k.b.449.3		32	45.38	even	12
1350.3.k.b.449.14		32	45.2	even	12
1350.3.k.b.899.3		32	45.22	odd	12
1350.3.k.b.899.14		32	45.13	odd	12
2160.3.bs.d.881.1		16	36.11	even	6
2160.3.bs.d.1601.1		16	36.31	odd	6

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:	\( N \)	\(=\)	\( 810 = 2 \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	810.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421i q^{2} -2.00000 q^{4} -2.23607i q^{5} -6.63197 q^{7} +2.82843i q^{8} -3.16228 q^{10} -21.0354i q^{11} -9.78248 q^{13} +9.37902i q^{14} +4.00000 q^{16} +12.9017i q^{17} +16.1821 q^{19} +4.47214i q^{20} -29.7486 q^{22} -16.9365i q^{23} -5.00000 q^{25} +13.8345i q^{26} +13.2639 q^{28} +50.0591i q^{29} -10.0642 q^{31} -5.65685i q^{32} +18.2457 q^{34} +14.8295i q^{35} +2.54425 q^{37} -22.8850i q^{38} +6.32456 q^{40} +53.7648i q^{41} -74.8349 q^{43} +42.0709i q^{44} -23.9518 q^{46} +47.2003i q^{47} -5.01700 q^{49} +7.07107i q^{50} +19.5650 q^{52} +1.08813i q^{53} -47.0367 q^{55} -18.7580i q^{56} +70.7943 q^{58} -9.63952i q^{59} -44.9145 q^{61} +14.2330i q^{62} -8.00000 q^{64} +21.8743i q^{65} -3.34135 q^{67} -25.8033i q^{68} +20.9721 q^{70} +38.4670i q^{71} -88.9201 q^{73} -3.59811i q^{74} -32.3642 q^{76} +139.506i q^{77} +117.647 q^{79} -8.94427i q^{80} +76.0349 q^{82} +147.123i q^{83} +28.8490 q^{85} +105.833i q^{86} +59.4972 q^{88} -145.331i q^{89} +64.8771 q^{91} +33.8729i q^{92} +66.7513 q^{94} -36.1843i q^{95} -47.0534 q^{97} +7.09511i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 32 q^{4} + 8 q^{7} - 40 q^{13} + 64 q^{16} + 80 q^{19} - 48 q^{22} - 80 q^{25} - 16 q^{28} + 32 q^{31} + 96 q^{34} - 88 q^{37} - 184 q^{43} + 24 q^{46} + 168 q^{49} + 80 q^{52} + 152 q^{61} - 128 q^{64}+ \cdots + 32 q^{97}+O(q^{100}) \) 16 * q - 32 * q^4 + 8 * q^7 - 40 * q^13 + 64 * q^16 + 80 * q^19 - 48 * q^22 - 80 * q^25 - 16 * q^28 + 32 * q^31 + 96 * q^34 - 88 * q^37 - 184 * q^43 + 24 * q^46 + 168 * q^49 + 80 * q^52 + 152 * q^61 - 128 * q^64 - 112 * q^67 - 120 * q^70 + 416 * q^73 - 160 * q^76 - 160 * q^79 - 192 * q^82 + 120 * q^85 + 96 * q^88 - 656 * q^91 + 168 * q^94 + 32 * q^97

Introduction

L-functions

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