LMFDB - Embedded newform 810.5.d.c.161.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [810,5,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, 5, names="a")

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

chi := DirichletCharacter("810.161");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 810 = 2 \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 5 $
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:	$83.7296700979$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 90)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.5
Character	$\chi$	$=$	810.161
Dual form			810.5.d.c.161.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{2} -8.00000 q^{4} -11.1803i q^{5} +90.5214 q^{7} +22.6274i q^{8} +O(q^{10})$	\(q-2.82843i q^{2} -8.00000 q^{4} -11.1803i q^{5} +90.5214 q^{7} +22.6274i q^{8} -31.6228 q^{10} +40.6295i q^{11} +65.0232 q^{13} -256.033i q^{14} +64.0000 q^{16} -555.313i q^{17} +178.769 q^{19} +89.4427i q^{20} +114.918 q^{22} +160.270i q^{23} -125.000 q^{25} -183.914i q^{26} -724.171 q^{28} +1158.72i q^{29} +419.600 q^{31} -181.019i q^{32} -1570.66 q^{34} -1012.06i q^{35} -1113.34 q^{37} -505.634i q^{38} +252.982 q^{40} -1185.85i q^{41} +3662.94 q^{43} -325.036i q^{44} +453.311 q^{46} -979.717i q^{47} +5793.13 q^{49} +353.553i q^{50} -520.186 q^{52} +2078.98i q^{53} +454.252 q^{55} +2048.27i q^{56} +3277.35 q^{58} +4609.19i q^{59} +1617.33 q^{61} -1186.81i q^{62} -512.000 q^{64} -726.982i q^{65} +1135.51 q^{67} +4442.50i q^{68} -2862.54 q^{70} -8247.59i q^{71} -3506.51 q^{73} +3149.00i q^{74} -1430.15 q^{76} +3677.84i q^{77} +10164.2 q^{79} -715.542i q^{80} -3354.09 q^{82} +1652.66i q^{83} -6208.59 q^{85} -10360.4i q^{86} -919.341 q^{88} +7258.01i q^{89} +5886.00 q^{91} -1282.16i q^{92} -2771.06 q^{94} -1998.70i q^{95} -8340.59 q^{97} -16385.4i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 256 q^{4} - 104 q^{7}+O(q^{10}) $ 32 * q - 256 * q^4 - 104 * q^7	$ 32 q - 256 q^{4} - 104 q^{7} + 40 q^{13} + 2048 q^{16} - 200 q^{19} + 1344 q^{22} - 4000 q^{25} + 832 q^{28} - 1472 q^{31} - 384 q^{34} - 4136 q^{37} - 272 q^{43} - 2112 q^{46} + 22296 q^{49} - 320 q^{52} - 12344 q^{61} - 16384 q^{64} + 40936 q^{67} + 4800 q^{70} - 41432 q^{73} + 1600 q^{76} - 14048 q^{79} - 17664 q^{82} - 17400 q^{85} - 10752 q^{88} + 69392 q^{91} + 1344 q^{94} + 36664 q^{97}+O(q^{100}) $ 32 * q - 256 * q^4 - 104 * q^7 + 40 * q^13 + 2048 * q^16 - 200 * q^19 + 1344 * q^22 - 4000 * q^25 + 832 * q^28 - 1472 * q^31 - 384 * q^34 - 4136 * q^37 - 272 * q^43 - 2112 * q^46 + 22296 * q^49 - 320 * q^52 - 12344 * q^61 - 16384 * q^64 + 40936 * q^67 + 4800 * q^70 - 41432 * q^73 + 1600 * q^76 - 14048 * q^79 - 17664 * q^82 - 17400 * q^85 - 10752 * q^88 + 69392 * q^91 + 1344 * q^94 + 36664 * q^97

Display 100 coefficients 10 coefficients

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 $

$2$	− 2.82843i	− 0.707107i
$2$	− 2.82843i	− 0.707107i
$3$	0	0
$3$	0	0
$4$	−8.00000	−0.500000
$5$	− 11.1803i	− 0.447214i
$5$	− 11.1803i	− 0.447214i
$6$	0	0
$7$	90.5214	1.84738	0.923688	−	0.383145i	$-0.125159\pi$
$7$	90.5214	1.84738	0.923688	+	0.383145i	$0.125159\pi$
$8$	22.6274i	0.353553i
$9$	0	0
$10$	−31.6228	−0.316228
$11$	40.6295i	0.335781i	0.985806	+	0.167890i	$0.0536955\pi$
$11$	40.6295i	0.335781i	−0.985806	+	0.167890i	$0.946305\pi$
$12$	0	0
$13$	65.0232	0.384753	0.192376	−	0.981321i	$-0.438381\pi$
$13$	65.0232	0.384753	0.192376	+	0.981321i	$0.438381\pi$
$14$	− 256.033i	− 1.30629i
$15$	0	0
$16$	64.0000	0.250000
$17$	− 555.313i	− 1.92150i	−0.277419	−	0.960749i	$-0.589479\pi$
$17$	− 555.313i	− 1.92150i	0.277419	−	0.960749i	$-0.410521\pi$
$18$	0	0
$19$	178.769	0.495204	0.247602	−	0.968862i	$-0.420357\pi$
$19$	178.769	0.495204	0.247602	+	0.968862i	$0.420357\pi$
$20$	89.4427i	0.223607i
$21$	0	0
$22$	114.918	0.237433
$23$	160.270i	0.302967i	0.988460	+	0.151484i	$0.0484051\pi$
$23$	160.270i	0.302967i	−0.988460	+	0.151484i	$0.951595\pi$
$24$	0	0
$25$	−125.000	−0.200000
$26$	− 183.914i	− 0.272061i
$27$	0	0
$28$	−724.171	−0.923688
$29$	1158.72i	1.37779i	0.724863	+	0.688893i	$0.241903\pi$
$29$	1158.72i	1.37779i	−0.724863	+	0.688893i	$0.758097\pi$
$30$	0	0
$31$	419.600	0.436629	0.218314	−	0.975878i	$-0.429944\pi$
$31$	419.600	0.436629	0.218314	+	0.975878i	$0.429944\pi$
$32$	− 181.019i	− 0.176777i
$33$	0	0
$34$	−1570.66	−1.35870
$35$	− 1012.06i	− 0.826172i
$36$	0	0
$37$	−1113.34	−0.813250	−0.406625	−	0.913595i	$-0.633294\pi$
$37$	−1113.34	−0.813250	−0.406625	+	0.913595i	$0.633294\pi$
$38$	− 505.634i	− 0.350162i
$39$	0	0
$40$	252.982	0.158114
$41$	− 1185.85i	− 0.705443i	−0.935728	−	0.352721i	$-0.885256\pi$
$41$	− 1185.85i	− 0.705443i	0.935728	−	0.352721i	$-0.114744\pi$
$42$	0	0
$43$	3662.94	1.98104	0.990519	−	0.137375i	$-0.0438665\pi$
$43$	3662.94	1.98104	0.990519	+	0.137375i	$0.0438665\pi$
$44$	− 325.036i	− 0.167890i
$45$	0	0
$46$	453.311	0.214230
$47$	− 979.717i	− 0.443512i	−0.975102	−	0.221756i	$-0.928821\pi$
$47$	− 979.717i	− 0.443512i	0.975102	−	0.221756i	$-0.0711788\pi$
$48$	0	0
$49$	5793.13	2.41280
$50$	353.553i	0.141421i
$51$	0	0
$52$	−520.186	−0.192376
$53$	2078.98i	0.740115i	0.929009	+	0.370057i	$0.120662\pi$
$53$	2078.98i	0.740115i	−0.929009	+	0.370057i	$0.879338\pi$
$54$	0	0
$55$	454.252	0.150166
$56$	2048.27i	0.653146i
$57$	0	0
$58$	3277.35	0.974242
$59$	4609.19i	1.32410i	0.749460	+	0.662050i	$0.230313\pi$
$59$	4609.19i	1.32410i	−0.749460	+	0.662050i	$0.769687\pi$
$60$	0	0
$61$	1617.33	0.434649	0.217325	−	0.976099i	$-0.430267\pi$
$61$	1617.33	0.434649	0.217325	+	0.976099i	$0.430267\pi$
$62$	− 1186.81i	− 0.308743i
$63$	0	0
$64$	−512.000	−0.125000
$65$	− 726.982i	− 0.172067i
$66$	0	0
$67$	1135.51	0.252954	0.126477	−	0.991970i	$-0.459633\pi$
$67$	1135.51	0.252954	0.126477	+	0.991970i	$0.459633\pi$
$68$	4442.50i	0.960749i
$69$	0	0
$70$	−2862.54	−0.584192
$71$	− 8247.59i	− 1.63610i	−0.575145	−	0.818051i	$-0.695055\pi$
$71$	− 8247.59i	− 1.63610i	0.575145	−	0.818051i	$-0.304945\pi$
$72$	0	0
$73$	−3506.51	−0.658004	−0.329002	−	0.944329i	$-0.606712\pi$
$73$	−3506.51	−0.658004	−0.329002	+	0.944329i	$0.606712\pi$
$74$	3149.00i	0.575055i
$75$	0	0
$76$	−1430.15	−0.247602
$77$	3677.84i	0.620314i
$78$	0	0
$79$	10164.2	1.62861	0.814306	−	0.580436i	$-0.197118\pi$
$79$	10164.2	1.62861	0.814306	+	0.580436i	$0.197118\pi$
$80$	− 715.542i	− 0.111803i
$81$	0	0
$82$	−3354.09	−0.498823
$83$	1652.66i	0.239898i	0.992780	+	0.119949i	$0.0382731\pi$
$83$	1652.66i	0.239898i	−0.992780	+	0.119949i	$0.961727\pi$
$84$	0	0
$85$	−6208.59	−0.859320
$86$	− 10360.4i	− 1.40081i
$87$	0	0
$88$	−919.341	−0.118716
$89$	7258.01i	0.916299i	0.888875	+	0.458150i	$0.151488\pi$
$89$	7258.01i	0.916299i	−0.888875	+	0.458150i	$0.848512\pi$
$90$	0	0
$91$	5886.00	0.710783
$92$	− 1282.16i	− 0.151484i
$93$	0	0
$94$	−2771.06	−0.313610
$95$	− 1998.70i	− 0.221462i
$96$	0	0
$97$	−8340.59	−0.886448	−0.443224	−	0.896411i	$-0.646165\pi$
$97$	−8340.59	−0.886448	−0.443224	+	0.896411i	$0.646165\pi$
$98$	− 16385.4i	− 1.70611i
$99$	0	0
$100$	1000.00	0.100000
$101$	5270.41i	0.516656i	0.966057	+	0.258328i	$0.0831715\pi$
$101$	5270.41i	0.516656i	−0.966057	+	0.258328i	$0.916829\pi$
$102$	0	0
$103$	−7836.33	−0.738649	−0.369325	−	0.929300i	$-0.620411\pi$
$103$	−7836.33	−0.738649	−0.369325	+	0.929300i	$0.620411\pi$
$104$	1471.31i	0.136031i
$105$	0	0
$106$	5880.25	0.523340
$107$	− 4051.14i	− 0.353842i	−0.984225	−	0.176921i	$-0.943386\pi$
$107$	− 4051.14i	− 0.353842i	0.984225	−	0.176921i	$-0.0566138\pi$
$108$	0	0
$109$	8665.79	0.729382	0.364691	−	0.931129i	$-0.381175\pi$
$109$	8665.79	0.729382	0.364691	+	0.931129i	$0.381175\pi$
$110$	− 1284.82i	− 0.106183i
$111$	0	0
$112$	5793.37	0.461844
$113$	− 11086.4i	− 0.868231i	−0.900857	−	0.434116i	$-0.857061\pi$
$113$	− 11086.4i	− 0.868231i	0.900857	−	0.434116i	$-0.142939\pi$
$114$	0	0
$115$	1791.87	0.135491
$116$	− 9269.74i	− 0.688893i
$117$	0	0
$118$	13036.8	0.936280
$119$	− 50267.7i	− 3.54973i
$120$	0	0
$121$	12990.2	0.887251
$122$	− 4574.50i	− 0.307343i
$123$	0	0
$124$	−3356.80	−0.218314
$125$	1397.54i	0.0894427i
$126$	0	0
$127$	−24871.6	−1.54204	−0.771021	−	0.636810i	$-0.780254\pi$
$127$	−24871.6	−1.54204	−0.771021	+	0.636810i	$0.780254\pi$
$128$	1448.15i	0.0883883i
$129$	0	0
$130$	−2056.22	−0.121670
$131$	− 20594.1i	− 1.20005i	−0.799981	−	0.600026i	$-0.795157\pi$
$131$	− 20594.1i	− 1.20005i	0.799981	−	0.600026i	$-0.204843\pi$
$132$	0	0
$133$	16182.4	0.914829
$134$	− 3211.71i	− 0.178866i
$135$	0	0
$136$	12565.3	0.679352
$137$	− 20190.3i	− 1.07573i	−0.843032	−	0.537863i	$-0.819232\pi$
$137$	− 20190.3i	− 1.07573i	0.843032	−	0.537863i	$-0.180768\pi$
$138$	0	0
$139$	20162.3	1.04354	0.521772	−	0.853085i	$-0.325271\pi$
$139$	20162.3	1.04354	0.521772	+	0.853085i	$0.325271\pi$
$140$	8096.48i	0.413086i
$141$	0	0
$142$	−23327.7	−1.15690
$143$	2641.86i	0.129193i
$144$	0	0
$145$	12954.9	0.616165
$146$	9917.90i	0.465279i
$147$	0	0
$148$	8906.71	0.406625
$149$	− 30922.8i	− 1.39286i	−0.717626	−	0.696429i	$-0.754771\pi$
$149$	− 30922.8i	− 1.39286i	0.717626	−	0.696429i	$-0.245229\pi$
$150$	0	0
$151$	10583.8	0.464180	0.232090	−	0.972694i	$-0.425444\pi$
$151$	10583.8	0.464180	0.232090	+	0.972694i	$0.425444\pi$
$152$	4045.08i	0.175081i
$153$	0	0
$154$	10402.5	0.438628
$155$	− 4691.27i	− 0.195266i
$156$	0	0
$157$	−42648.0	−1.73021	−0.865106	−	0.501589i	$-0.832749\pi$
$157$	−42648.0	−1.73021	−0.865106	+	0.501589i	$0.832749\pi$
$158$	− 28748.6i	− 1.15160i
$159$	0	0
$160$	−2023.86	−0.0790569
$161$	14507.8i	0.559695i
$162$	0	0
$163$	25755.6	0.969386	0.484693	−	0.874684i	$-0.338931\pi$
$163$	25755.6	0.969386	0.484693	+	0.874684i	$0.338931\pi$
$164$	9486.80i	0.352721i
$165$	0	0
$166$	4674.42	0.169634
$167$	− 43027.8i	− 1.54282i	−0.636337	−	0.771411i	$-0.719551\pi$
$167$	− 43027.8i	− 1.54282i	0.636337	−	0.771411i	$-0.280449\pi$
$168$	0	0
$169$	−24333.0	−0.851965
$170$	17560.5i	0.607631i
$171$	0	0
$172$	−29303.5	−0.990519
$173$	− 30415.1i	− 1.01624i	−0.861286	−	0.508121i	$-0.830340\pi$
$173$	− 30415.1i	− 1.01624i	0.861286	−	0.508121i	$-0.169660\pi$
$174$	0	0
$175$	−11315.2	−0.369475
$176$	2600.29i	0.0839452i
$177$	0	0
$178$	20528.7	0.647921
$179$	− 5692.33i	− 0.177658i	−0.996047	−	0.0888289i	$-0.971688\pi$
$179$	− 5692.33i	− 0.177658i	0.996047	−	0.0888289i	$-0.0283124\pi$
$180$	0	0
$181$	−28534.7	−0.870997	−0.435498	−	0.900190i	$-0.643428\pi$
$181$	−28534.7	−0.870997	−0.435498	+	0.900190i	$0.643428\pi$
$182$	− 16648.1i	− 0.502600i
$183$	0	0
$184$	−3626.49	−0.107115
$185$	12447.5i	0.363696i
$186$	0	0
$187$	22562.1	0.645202
$188$	7837.74i	0.221756i
$189$	0	0
$190$	−5653.16	−0.156597
$191$	14904.9i	0.408565i	0.978912	+	0.204283i	$0.0654861\pi$
$191$	14904.9i	0.408565i	−0.978912	+	0.204283i	$0.934514\pi$
$192$	0	0
$193$	−38164.7	−1.02458	−0.512291	−	0.858812i	$-0.671203\pi$
$193$	−38164.7	−1.02458	−0.512291	+	0.858812i	$0.671203\pi$
$194$	23590.8i	0.626814i
$195$	0	0
$196$	−46345.0	−1.20640
$197$	− 7450.14i	− 0.191969i	−0.995383	−	0.0959846i	$-0.969400\pi$
$197$	− 7450.14i	− 0.191969i	0.995383	−	0.0959846i	$-0.0306000\pi$
$198$	0	0
$199$	59827.1	1.51075	0.755374	−	0.655294i	$-0.227455\pi$
$199$	59827.1	1.51075	0.755374	+	0.655294i	$0.227455\pi$
$200$	− 2828.43i	− 0.0707107i
$201$	0	0
$202$	14907.0	0.365331
$203$	104889.i	2.54529i
$204$	0	0
$205$	−13258.2	−0.315484
$206$	22164.5i	0.522304i
$207$	0	0
$208$	4161.49	0.0961882
$209$	7263.28i	0.166280i
$210$	0	0
$211$	35732.2	0.802593	0.401296	−	0.915948i	$-0.368560\pi$
$211$	35732.2	0.802593	0.401296	+	0.915948i	$0.368560\pi$
$212$	− 16631.9i	− 0.370057i
$213$	0	0
$214$	−11458.4	−0.250204
$215$	− 40952.9i	− 0.885947i
$216$	0	0
$217$	37982.8	0.806618
$218$	− 24510.5i	− 0.515751i
$219$	0	0
$220$	−3634.01	−0.0750829
$221$	− 36108.3i	− 0.739302i
$222$	0	0
$223$	60051.5	1.20758	0.603788	−	0.797145i	$-0.293657\pi$
$223$	60051.5	1.20758	0.603788	+	0.797145i	$0.293657\pi$
$224$	− 16386.1i	− 0.326573i
$225$	0	0
$226$	−31357.2	−0.613932
$227$	57930.8i	1.12424i	0.827057	+	0.562118i	$0.190013\pi$
$227$	57930.8i	1.12424i	−0.827057	+	0.562118i	$0.809987\pi$
$228$	0	0
$229$	−24473.1	−0.466679	−0.233340	−	0.972395i	$-0.574965\pi$
$229$	−24473.1	−0.466679	−0.233340	+	0.972395i	$0.574965\pi$
$230$	− 5068.17i	− 0.0958067i
$231$	0	0
$232$	−26218.8	−0.487121
$233$	3870.02i	0.0712855i	0.999365	+	0.0356427i	$0.0113478\pi$
$233$	3870.02i	0.0712855i	−0.999365	+	0.0356427i	$0.988652\pi$
$234$	0	0
$235$	−10953.6	−0.198344
$236$	− 36873.5i	− 0.662050i
$237$	0	0
$238$	−142179.	−2.51004
$239$	9548.11i	0.167156i	0.996501	+	0.0835780i	$0.0266348\pi$
$239$	9548.11i	0.167156i	−0.996501	+	0.0835780i	$0.973365\pi$
$240$	0	0
$241$	62255.1	1.07187	0.535933	−	0.844260i	$-0.319960\pi$
$241$	62255.1	1.07187	0.535933	+	0.844260i	$0.319960\pi$
$242$	− 36742.0i	− 0.627381i
$243$	0	0
$244$	−12938.6	−0.217325
$245$	− 64769.1i	− 1.07904i
$246$	0	0
$247$	11624.1	0.190531
$248$	9494.47i	0.154372i
$249$	0	0
$250$	3952.85	0.0632456
$251$	− 28870.9i	− 0.458261i	−0.973396	−	0.229130i	$-0.926412\pi$
$251$	− 28870.9i	− 0.458261i	0.973396	−	0.229130i	$-0.0735882\pi$
$252$	0	0
$253$	−6511.68	−0.101731
$254$	70347.5i	1.09039i
$255$	0	0
$256$	4096.00	0.0625000
$257$	− 109344.i	− 1.65550i	−0.561094	−	0.827752i	$-0.689619\pi$
$257$	− 109344.i	− 1.65550i	0.561094	−	0.827752i	$-0.310381\pi$
$258$	0	0
$259$	−100781.	−1.50238
$260$	5815.86i	0.0860334i
$261$	0	0
$262$	−58248.9	−0.848565
$263$	− 73987.3i	− 1.06966i	−0.844960	−	0.534830i	$-0.820376\pi$
$263$	− 73987.3i	− 1.06966i	0.844960	−	0.534830i	$-0.179624\pi$
$264$	0	0
$265$	23243.7	0.330989
$266$	− 45770.7i	− 0.646881i
$267$	0	0
$268$	−9084.09	−0.126477
$269$	− 50470.3i	− 0.697480i	−0.937220	−	0.348740i	$-0.886610\pi$
$269$	− 50470.3i	− 0.697480i	0.937220	−	0.348740i	$-0.113390\pi$
$270$	0	0
$271$	−26280.6	−0.357847	−0.178923	−	0.983863i	$-0.557261\pi$
$271$	−26280.6	−0.357847	−0.178923	+	0.983863i	$0.557261\pi$
$272$	− 35540.0i	− 0.480375i
$273$	0	0
$274$	−57106.8	−0.760653
$275$	− 5078.69i	− 0.0671562i
$276$	0	0
$277$	−51200.1	−0.667285	−0.333642	−	0.942700i	$-0.608278\pi$
$277$	−51200.1	−0.667285	−0.333642	+	0.942700i	$0.608278\pi$
$278$	− 57027.6i	− 0.737897i
$279$	0	0
$280$	22900.3	0.292096
$281$	84346.9i	1.06821i	0.845418	+	0.534105i	$0.179351\pi$
$281$	84346.9i	1.06821i	−0.845418	+	0.534105i	$0.820649\pi$
$282$	0	0
$283$	71274.9	0.889946	0.444973	−	0.895544i	$-0.353213\pi$
$283$	71274.9	0.889946	0.444973	+	0.895544i	$0.353213\pi$
$284$	65980.7i	0.818051i
$285$	0	0
$286$	7472.31	0.0913530
$287$	− 107345.i	− 1.30322i
$288$	0	0
$289$	−224852.	−2.69216
$290$	− 36641.9i	− 0.435694i
$291$	0	0
$292$	28052.0	0.329002
$293$	60371.4i	0.703227i	0.936145	+	0.351614i	$0.114367\pi$
$293$	60371.4i	0.703227i	−0.936145	+	0.351614i	$0.885633\pi$
$294$	0	0
$295$	51532.3	0.592155
$296$	− 25192.0i	− 0.287527i
$297$	0	0
$298$	−87463.0	−0.984899
$299$	10421.3i	0.116568i
$300$	0	0
$301$	331575.	3.65972
$302$	− 29935.4i	− 0.328225i
$303$	0	0
$304$	11441.2	0.123801
$305$	− 18082.3i	− 0.194381i
$306$	0	0
$307$	99288.9	1.05347	0.526737	−	0.850028i	$-0.323415\pi$
$307$	99288.9	1.05347	0.526737	+	0.850028i	$0.323415\pi$
$308$	− 29422.7i	− 0.310157i
$309$	0	0
$310$	−13268.9	−0.138074
$311$	− 21215.7i	− 0.219350i	−0.993968	−	0.109675i	$-0.965019\pi$
$311$	− 21215.7i	− 0.219350i	0.993968	−	0.109675i	$-0.0349810\pi$
$312$	0	0
$313$	−62595.9	−0.638935	−0.319468	−	0.947597i	$-0.603504\pi$
$313$	−62595.9	−0.638935	−0.319468	+	0.947597i	$0.603504\pi$
$314$	120627.i	1.22344i
$315$	0	0
$316$	−81313.3	−0.814306
$317$	42687.5i	0.424798i	0.977183	+	0.212399i	$0.0681276\pi$
$317$	42687.5i	0.424798i	−0.977183	+	0.212399i	$0.931872\pi$
$318$	0	0
$319$	−47078.1	−0.462634
$320$	5724.33i	0.0559017i
$321$	0	0
$322$	41034.4	0.395764
$323$	− 99272.6i	− 0.951534i
$324$	0	0
$325$	−8127.91	−0.0769506
$326$	− 72847.9i	− 0.685459i
$327$	0	0
$328$	26832.7	0.249412
$329$	− 88685.4i	− 0.819333i
$330$	0	0
$331$	−196707.	−1.79541	−0.897704	−	0.440599i	$-0.854766\pi$
$331$	−196707.	−1.79541	−0.897704	+	0.440599i	$0.854766\pi$
$332$	− 13221.3i	− 0.119949i
$333$	0	0
$334$	−121701.	−1.09094
$335$	− 12695.4i	− 0.113125i
$336$	0	0
$337$	122679.	1.08022	0.540108	−	0.841596i	$-0.318384\pi$
$337$	122679.	1.08022	0.540108	+	0.841596i	$0.318384\pi$
$338$	68824.1i	0.602430i
$339$	0	0
$340$	49668.7	0.429660
$341$	17048.1i	0.146612i
$342$	0	0
$343$	307060.	2.60997
$344$	82882.9i	0.700403i
$345$	0	0
$346$	−86026.9	−0.718592
$347$	26649.5i	0.221324i	0.993858	+	0.110662i	$0.0352972\pi$
$347$	26649.5i	0.221324i	−0.993858	+	0.110662i	$0.964703\pi$
$348$	0	0
$349$	121314.	0.996002	0.498001	−	0.867176i	$-0.334068\pi$
$349$	121314.	0.996002	0.498001	+	0.867176i	$0.334068\pi$
$350$	32004.2i	0.261258i
$351$	0	0
$352$	7354.72	0.0593582
$353$	127818.i	1.02575i	0.858464	+	0.512874i	$0.171419\pi$
$353$	127818.i	1.02575i	−0.858464	+	0.512874i	$0.828581\pi$
$354$	0	0
$355$	−92210.9	−0.731687
$356$	− 58064.1i	− 0.458150i
$357$	0	0
$358$	−16100.3	−0.125623
$359$	113214.i	0.878440i	0.898379	+	0.439220i	$0.144745\pi$
$359$	113214.i	0.878440i	−0.898379	+	0.439220i	$0.855255\pi$
$360$	0	0
$361$	−98362.7	−0.754773
$362$	80708.4i	0.615888i
$363$	0	0
$364$	−47088.0	−0.355392
$365$	39203.9i	0.294269i
$366$	0	0
$367$	130460.	0.968601	0.484300	−	0.874902i	$-0.339074\pi$
$367$	130460.	0.968601	0.484300	+	0.874902i	$0.339074\pi$
$368$	10257.3i	0.0757419i
$369$	0	0
$370$	35206.9	0.257172
$371$	188192.i	1.36727i
$372$	0	0
$373$	−164423.	−1.18180	−0.590900	−	0.806745i	$-0.701227\pi$
$373$	−164423.	−1.18180	−0.590900	+	0.806745i	$0.701227\pi$
$374$	− 63815.2i	− 0.456227i
$375$	0	0
$376$	22168.5	0.156805
$377$	75343.6i	0.530107i
$378$	0	0
$379$	−13086.7	−0.0911073	−0.0455536	−	0.998962i	$-0.514505\pi$
$379$	−13086.7	−0.0911073	−0.0455536	+	0.998962i	$0.514505\pi$
$380$	15989.6i	0.110731i
$381$	0	0
$382$	42157.3	0.288899
$383$	85239.3i	0.581088i	0.956862	+	0.290544i	$0.0938363\pi$
$383$	85239.3i	0.581088i	−0.956862	+	0.290544i	$0.906164\pi$
$384$	0	0
$385$	41119.5	0.277413
$386$	107946.i	0.724489i
$387$	0	0
$388$	66724.7	0.443224
$389$	108168.i	0.714823i	0.933947	+	0.357411i	$0.116341\pi$
$389$	108168.i	0.714823i	−0.933947	+	0.357411i	$0.883659\pi$
$390$	0	0
$391$	88999.9	0.582151
$392$	131084.i	0.853053i
$393$	0	0
$394$	−21072.2	−0.135743
$395$	− 113639.i	− 0.728337i
$396$	0	0
$397$	117602.	0.746164	0.373082	−	0.927798i	$-0.378301\pi$
$397$	117602.	0.746164	0.373082	+	0.927798i	$0.378301\pi$
$398$	− 169217.i	− 1.06826i
$399$	0	0
$400$	−8000.00	−0.0500000
$401$	147488.i	0.917206i	0.888641	+	0.458603i	$0.151650\pi$
$401$	147488.i	0.917206i	−0.888641	+	0.458603i	$0.848350\pi$
$402$	0	0
$403$	27283.8	0.167994
$404$	− 42163.2i	− 0.258328i
$405$	0	0
$406$	296670.	1.79979
$407$	− 45234.4i	− 0.273074i
$408$	0	0
$409$	−104282.	−0.623397	−0.311699	−	0.950181i	$-0.600898\pi$
$409$	−104282.	−0.623397	−0.311699	+	0.950181i	$0.600898\pi$
$410$	37499.9i	0.223081i
$411$	0	0
$412$	62690.6	0.369325
$413$	417230.i	2.44611i
$414$	0	0
$415$	18477.3	0.107286
$416$	− 11770.5i	− 0.0680154i
$417$	0	0
$418$	20543.7	0.117578
$419$	97880.5i	0.557530i	0.960359	+	0.278765i	$0.0899250\pi$
$419$	97880.5i	0.557530i	−0.960359	+	0.278765i	$0.910075\pi$
$420$	0	0
$421$	246370.	1.39003	0.695015	−	0.718996i	$-0.255398\pi$
$421$	246370.	1.39003	0.695015	+	0.718996i	$0.255398\pi$
$422$	− 101066.i	− 0.567519i
$423$	0	0
$424$	−47042.0	−0.261670
$425$	69414.1i	0.384300i
$426$	0	0
$427$	146403.	0.802961
$428$	32409.1i	0.176921i
$429$	0	0
$430$	−115832.	−0.626459
$431$	− 111607.i	− 0.600809i	−0.953812	−	0.300404i	$-0.902878\pi$
$431$	− 111607.i	− 0.600809i	0.953812	−	0.300404i	$-0.0971216\pi$
$432$	0	0
$433$	144498.	0.770700	0.385350	−	0.922770i	$-0.374081\pi$
$433$	144498.	0.770700	0.385350	+	0.922770i	$0.374081\pi$
$434$	− 107432.i	− 0.570365i
$435$	0	0
$436$	−69326.3	−0.364691
$437$	28651.2i	0.150031i
$438$	0	0
$439$	−205699.	−1.06734	−0.533670	−	0.845693i	$-0.679188\pi$
$439$	−205699.	−1.06734	−0.533670	+	0.845693i	$0.679188\pi$
$440$	10278.5i	0.0530916i
$441$	0	0
$442$	−102130.	−0.522766
$443$	− 293250.i	− 1.49428i	−0.664669	−	0.747138i	$-0.731427\pi$
$443$	− 293250.i	− 1.49428i	0.664669	−	0.747138i	$-0.268573\pi$
$444$	0	0
$445$	81147.0	0.409782
$446$	− 169851.i	− 0.853885i
$447$	0	0
$448$	−46347.0	−0.230922
$449$	43704.9i	0.216789i	0.994108	+	0.108395i	$0.0345710\pi$
$449$	43704.9i	0.216789i	−0.994108	+	0.108395i	$0.965429\pi$
$450$	0	0
$451$	48180.5	0.236874
$452$	88691.6i	0.434116i
$453$	0	0
$454$	163853.	0.794955
$455$	− 65807.4i	− 0.317872i
$456$	0	0
$457$	−18082.9	−0.0865834	−0.0432917	−	0.999062i	$-0.513784\pi$
$457$	−18082.9	−0.0865834	−0.0432917	+	0.999062i	$0.513784\pi$
$458$	69220.4i	0.329992i
$459$	0	0
$460$	−14335.0	−0.0677456
$461$	78440.3i	0.369094i	0.982824	+	0.184547i	$0.0590818\pi$
$461$	78440.3i	0.369094i	−0.982824	+	0.184547i	$0.940918\pi$
$462$	0	0
$463$	−351459.	−1.63950	−0.819752	−	0.572718i	$-0.805889\pi$
$463$	−351459.	−1.63950	−0.819752	+	0.572718i	$0.805889\pi$
$464$	74157.9i	0.344446i
$465$	0	0
$466$	10946.1	0.0504065
$467$	345695.i	1.58511i	0.609800	+	0.792555i	$0.291250\pi$
$467$	345695.i	1.58511i	−0.609800	+	0.792555i	$0.708750\pi$
$468$	0	0
$469$	102788.	0.467301
$470$	30981.4i	0.140251i
$471$	0	0
$472$	−104294.	−0.468140
$473$	148823.i	0.665195i
$474$	0	0
$475$	−22346.1	−0.0990409
$476$	402142.i	1.77486i
$477$	0	0
$478$	27006.1	0.118197
$479$	355914.i	1.55122i	0.631211	+	0.775611i	$0.282558\pi$
$479$	355914.i	1.55122i	−0.631211	+	0.775611i	$0.717442\pi$
$480$	0	0
$481$	−72392.9	−0.312900
$482$	− 176084.i	− 0.757924i
$483$	0	0
$484$	−103922.	−0.443626
$485$	93250.7i	0.396432i
$486$	0	0
$487$	220779.	0.930892	0.465446	−	0.885076i	$-0.345894\pi$
$487$	220779.	0.930892	0.465446	+	0.885076i	$0.345894\pi$
$488$	36596.0i	0.153672i
$489$	0	0
$490$	−183195.	−0.762994
$491$	26225.6i	0.108784i	0.998520	+	0.0543918i	$0.0173220\pi$
$491$	26225.6i	0.108784i	−0.998520	+	0.0543918i	$0.982678\pi$
$492$	0	0
$493$	643451.	2.64741
$494$	− 32878.0i	− 0.134726i
$495$	0	0
$496$	26854.4	0.109157
$497$	− 746584.i	− 3.02250i
$498$	0	0
$499$	−116452.	−0.467678	−0.233839	−	0.972275i	$-0.575129\pi$
$499$	−116452.	−0.467678	−0.233839	+	0.972275i	$0.575129\pi$
$500$	− 11180.3i	− 0.0447214i
$501$	0	0
$502$	−81659.2	−0.324039
$503$	− 90136.7i	− 0.356259i	−0.984007	−	0.178130i	$-0.942995\pi$
$503$	− 90136.7i	− 0.356259i	0.984007	−	0.178130i	$-0.0570046\pi$
$504$	0	0
$505$	58924.9	0.231056
$506$	18417.8i	0.0719345i
$507$	0	0
$508$	198973.	0.771021
$509$	32250.2i	0.124479i	0.998061	+	0.0622396i	$0.0198243\pi$
$509$	32250.2i	0.124479i	−0.998061	+	0.0622396i	$0.980176\pi$
$510$	0	0
$511$	−317414.	−1.21558
$512$	− 11585.2i	− 0.0441942i
$513$	0	0
$514$	−309273.	−1.17062
$515$	87612.8i	0.330334i
$516$	0	0
$517$	39805.4	0.148923
$518$	285052.i	1.06234i
$519$	0	0
$520$	16449.7	0.0608348
$521$	18369.3i	0.0676734i	0.999427	+	0.0338367i	$0.0107726\pi$
$521$	18369.3i	0.0676734i	−0.999427	+	0.0338367i	$0.989227\pi$
$522$	0	0
$523$	−56173.8	−0.205367	−0.102683	−	0.994714i	$-0.532743\pi$
$523$	−56173.8	−0.205367	−0.102683	+	0.994714i	$0.532743\pi$
$524$	164753.i	0.600026i
$525$	0	0
$526$	−209268.	−0.756364
$527$	− 233010.i	− 0.838982i
$528$	0	0
$529$	254155.	0.908211
$530$	− 65743.2i	− 0.234045i
$531$	0	0
$532$	−129459.	−0.457414
$533$	− 77107.8i	− 0.271421i
$534$	0	0
$535$	−45293.1	−0.158243
$536$	25693.7i	0.0894328i
$537$	0	0
$538$	−142752.	−0.493193
$539$	235372.i	0.810172i
$540$	0	0
$541$	190027.	0.649264	0.324632	−	0.945840i	$-0.394760\pi$
$541$	190027.	0.649264	0.324632	+	0.945840i	$0.394760\pi$
$542$	74332.8i	0.253036i
$543$	0	0
$544$	−100522.	−0.339676
$545$	− 96886.4i	− 0.326189i
$546$	0	0
$547$	−152961.	−0.511217	−0.255608	−	0.966780i	$-0.582276\pi$
$547$	−152961.	−0.511217	−0.255608	+	0.966780i	$0.582276\pi$
$548$	161522.i	0.537863i
$549$	0	0
$550$	−14364.7	−0.0474866
$551$	207143.i	0.682285i
$552$	0	0
$553$	920075.	3.00866
$554$	144816.i	0.471842i
$555$	0	0
$556$	−161298.	−0.521772
$557$	151549.i	0.488476i	0.969715	+	0.244238i	$0.0785378\pi$
$557$	151549.i	0.488476i	−0.969715	+	0.244238i	$0.921462\pi$
$558$	0	0
$559$	238176.	0.762210
$560$	− 64771.9i	− 0.206543i
$561$	0	0
$562$	238569.	0.755338
$563$	104672.i	0.330228i	0.986275	+	0.165114i	$0.0527992\pi$
$563$	104672.i	0.330228i	−0.986275	+	0.165114i	$0.947201\pi$
$564$	0	0
$565$	−123950.	−0.388285
$566$	− 201596.i	− 0.629287i
$567$	0	0
$568$	186622.	0.578450
$569$	49628.8i	0.153288i	0.997059	+	0.0766441i	$0.0244205\pi$
$569$	49628.8i	0.153288i	−0.997059	+	0.0766441i	$0.975579\pi$
$570$	0	0
$571$	−111744.	−0.342731	−0.171365	−	0.985208i	$-0.554818\pi$
$571$	−111744.	−0.342731	−0.171365	+	0.985208i	$0.554818\pi$
$572$	− 21134.9i	− 0.0645964i
$573$	0	0
$574$	−303617.	−0.921515
$575$	− 20033.7i	− 0.0605935i
$576$	0	0
$577$	−481502.	−1.44626	−0.723130	−	0.690712i	$-0.757297\pi$
$577$	−481502.	−1.44626	−0.723130	+	0.690712i	$0.757297\pi$
$578$	635976.i	1.90364i
$579$	0	0
$580$	−103639.	−0.308082
$581$	149601.i	0.443182i
$582$	0	0
$583$	−84468.0	−0.248516
$584$	− 79343.2i	− 0.232640i
$585$	0	0
$586$	170756.	0.497257
$587$	74046.3i	0.214895i	0.994211	+	0.107448i	$0.0342678\pi$
$587$	74046.3i	0.214895i	−0.994211	+	0.107448i	$0.965732\pi$
$588$	0	0
$589$	75011.4	0.216220
$590$	− 145755.i	− 0.418717i
$591$	0	0
$592$	−71253.7	−0.203313
$593$	− 10200.9i	− 0.0290088i	−0.999895	−	0.0145044i	$-0.995383\pi$
$593$	− 10200.9i	− 0.0290088i	0.999895	−	0.0145044i	$-0.00461705\pi$
$594$	0	0
$595$	−562010.	−1.58749
$596$	247383.i	0.696429i
$597$	0	0
$598$	29475.8	0.0824257
$599$	564030.i	1.57198i	0.618236	+	0.785992i	$0.287847\pi$
$599$	564030.i	1.57198i	−0.618236	+	0.785992i	$0.712153\pi$
$600$	0	0
$601$	606277.	1.67850	0.839252	−	0.543743i	$-0.182994\pi$
$601$	606277.	1.67850	0.839252	+	0.543743i	$0.182994\pi$
$602$	− 937834.i	− 2.58781i
$603$	0	0
$604$	−84670.1	−0.232090
$605$	− 145235.i	− 0.396791i
$606$	0	0
$607$	−244655.	−0.664014	−0.332007	−	0.943277i	$-0.607726\pi$
$607$	−244655.	−0.664014	−0.332007	+	0.943277i	$0.607726\pi$
$608$	− 32360.6i	− 0.0875406i
$609$	0	0
$610$	−51144.5	−0.137448
$611$	− 63704.4i	− 0.170642i
$612$	0	0
$613$	241760.	0.643374	0.321687	−	0.946846i	$-0.395750\pi$
$613$	241760.	0.643374	0.321687	+	0.946846i	$0.395750\pi$
$614$	− 280831.i	− 0.744918i
$615$	0	0
$616$	−83220.0	−0.219314
$617$	− 311420.i	− 0.818044i	−0.912525	−	0.409022i	$-0.865870\pi$
$617$	− 311420.i	− 0.818044i	0.912525	−	0.409022i	$-0.134130\pi$
$618$	0	0
$619$	157384.	0.410752	0.205376	−	0.978683i	$-0.434158\pi$
$619$	157384.	0.410752	0.205376	+	0.978683i	$0.434158\pi$
$620$	37530.2i	0.0976332i
$621$	0	0
$622$	−60007.1	−0.155104
$623$	657005.i	1.69275i
$624$	0	0
$625$	15625.0	0.0400000
$626$	177048.i	0.451796i
$627$	0	0
$628$	341184.	0.865106
$629$	618252.i	1.56266i
$630$	0	0
$631$	−385317.	−0.967741	−0.483871	−	0.875140i	$-0.660769\pi$
$631$	−385317.	−0.967741	−0.483871	+	0.875140i	$0.660769\pi$
$632$	229989.i	0.575801i
$633$	0	0
$634$	120738.	0.300377
$635$	278073.i	0.689622i
$636$	0	0
$637$	376688.	0.928331
$638$	133157.i	0.327132i
$639$	0	0
$640$	16190.9	0.0395285
$641$	658038.i	1.60153i	0.598978	+	0.800765i	$0.295574\pi$
$641$	658038.i	1.60153i	−0.598978	+	0.800765i	$0.704426\pi$
$642$	0	0
$643$	278663.	0.673997	0.336999	−	0.941505i	$-0.390588\pi$
$643$	278663.	0.673997	0.336999	+	0.941505i	$0.390588\pi$
$644$	− 116063.i	− 0.279847i
$645$	0	0
$646$	−280785.	−0.672836
$647$	− 89120.0i	− 0.212896i	−0.994318	−	0.106448i	$-0.966052\pi$
$647$	− 89120.0i	− 0.212896i	0.994318	−	0.106448i	$-0.0339477\pi$
$648$	0	0
$649$	−187269.	−0.444607
$650$	22989.2i	0.0544123i
$651$	0	0
$652$	−206045.	−0.484693
$653$	− 797771.i	− 1.87091i	−0.353452	−	0.935453i	$-0.614992\pi$
$653$	− 797771.i	− 1.87091i	0.353452	−	0.935453i	$-0.385008\pi$
$654$	0	0
$655$	−230249.	−0.536679
$656$	− 75894.4i	− 0.176361i
$657$	0	0
$658$	−250840.	−0.579356
$659$	− 263402.i	− 0.606525i	−0.952907	−	0.303263i	$-0.901924\pi$
$659$	− 263402.i	− 0.606525i	0.952907	−	0.303263i	$-0.0980759\pi$
$660$	0	0
$661$	142384.	0.325881	0.162941	−	0.986636i	$-0.447902\pi$
$661$	142384.	0.325881	0.162941	+	0.986636i	$0.447902\pi$
$662$	556371.i	1.26955i
$663$	0	0
$664$	−37395.4	−0.0848168
$665$	− 180925.i	− 0.409124i
$666$	0	0
$667$	−185707.	−0.417424
$668$	344222.i	0.771411i
$669$	0	0
$670$	−35908.0	−0.0799911
$671$	65711.3i	0.145947i
$672$	0	0
$673$	−544960.	−1.20319	−0.601596	−	0.798801i	$-0.705468\pi$
$673$	−544960.	−1.20319	−0.601596	+	0.798801i	$0.705468\pi$
$674$	− 346988.i	− 0.763827i
$675$	0	0
$676$	194664.	0.425983
$677$	− 155312.i	− 0.338865i	−0.985542	−	0.169433i	$-0.945806\pi$
$677$	− 155312.i	− 0.338865i	0.985542	−	0.169433i	$-0.0541936\pi$
$678$	0	0
$679$	−755002.	−1.63760
$680$	− 140484.i	− 0.303816i
$681$	0	0
$682$	48219.4	0.103670
$683$	− 704750.i	− 1.51075i	−0.655291	−	0.755376i	$-0.727454\pi$
$683$	− 704750.i	− 1.51075i	0.655291	−	0.755376i	$-0.272546\pi$
$684$	0	0
$685$	−225734.	−0.481079
$686$	− 868498.i	− 1.84553i
$687$	0	0
$688$	234428.	0.495260
$689$	135182.i	0.284761i
$690$	0	0
$691$	262488.	0.549734	0.274867	−	0.961482i	$-0.411366\pi$
$691$	262488.	0.549734	0.274867	+	0.961482i	$0.411366\pi$
$692$	243321.i	0.508121i
$693$	0	0
$694$	75376.0	0.156500
$695$	− 225421.i	− 0.466687i
$696$	0	0
$697$	−658518.	−1.35551
$698$	− 343128.i	− 0.704280i
$699$	0	0
$700$	90521.4	0.184738
$701$	− 471234.i	− 0.958959i	−0.877553	−	0.479480i	$-0.840825\pi$
$701$	− 471234.i	− 0.958959i	0.877553	−	0.479480i	$-0.159175\pi$
$702$	0	0
$703$	−199030.	−0.402725
$704$	− 20802.3i	− 0.0419726i
$705$	0	0
$706$	361523.	0.725314
$707$	477085.i	0.954458i
$708$	0	0
$709$	−760243.	−1.51238	−0.756188	−	0.654354i	$-0.772941\pi$
$709$	−760243.	−1.51238	−0.756188	+	0.654354i	$0.772941\pi$
$710$	260812.i	0.517381i
$711$	0	0
$712$	−164230.	−0.323961
$713$	67249.2i	0.132284i
$714$	0	0
$715$	29536.9	0.0577767
$716$	45538.7i	0.0888289i
$717$	0	0
$718$	320218.	0.621151
$719$	− 53596.6i	− 0.103676i	−0.998656	−	0.0518381i	$-0.983492\pi$
$719$	− 53596.6i	− 0.103676i	0.998656	−	0.0518381i	$-0.0165080\pi$
$720$	0	0
$721$	−709356.	−1.36456
$722$	278212.i	0.533705i
$723$	0	0
$724$	228278.	0.435498
$725$	− 144840.i	− 0.275557i
$726$	0	0
$727$	−726798.	−1.37513	−0.687567	−	0.726121i	$-0.741321\pi$
$727$	−726798.	−1.37513	−0.687567	+	0.726121i	$0.741321\pi$
$728$	133185.i	0.251300i
$729$	0	0
$730$	110885.	0.208079
$731$	− 2.03408e6i	− 3.80656i
$732$	0	0
$733$	715960.	1.33254	0.666271	−	0.745710i	$-0.267889\pi$
$733$	715960.	1.33254	0.666271	+	0.745710i	$0.267889\pi$
$734$	− 368996.i	− 0.684904i
$735$	0	0
$736$	29011.9	0.0535576
$737$	46135.2i	0.0849372i
$738$	0	0
$739$	−127972.	−0.234330	−0.117165	−	0.993112i	$-0.537381\pi$
$739$	−127972.	−0.234330	−0.117165	+	0.993112i	$0.537381\pi$
$740$	− 99580.1i	− 0.181848i
$741$	0	0
$742$	532289.	0.966806
$743$	281585.i	0.510072i	0.966931	+	0.255036i	$0.0820874\pi$
$743$	281585.i	0.510072i	−0.966931	+	0.255036i	$0.917913\pi$
$744$	0	0
$745$	−345728.	−0.622905
$746$	465057.i	0.835659i
$747$	0	0
$748$	−180497.	−0.322601
$749$	− 366715.i	− 0.653680i
$750$	0	0
$751$	297953.	0.528285	0.264142	−	0.964484i	$-0.414911\pi$
$751$	297953.	0.528285	0.264142	+	0.964484i	$0.414911\pi$
$752$	− 62701.9i	− 0.110878i
$753$	0	0
$754$	213104.	0.374842
$755$	− 118330.i	− 0.207588i
$756$	0	0
$757$	853881.	1.49007	0.745033	−	0.667028i	$-0.232434\pi$
$757$	853881.	1.49007	0.745033	+	0.667028i	$0.232434\pi$
$758$	37014.9i	0.0644226i
$759$	0	0
$760$	45225.3	0.0782987
$761$	622340.i	1.07463i	0.843382	+	0.537314i	$0.180561\pi$
$761$	622340.i	1.07463i	−0.843382	+	0.537314i	$0.819439\pi$
$762$	0	0
$763$	784439.	1.34744
$764$	− 119239.i	− 0.204283i
$765$	0	0
$766$	241093.	0.410891
$767$	299704.i	0.509451i
$768$	0	0
$769$	−357409.	−0.604384	−0.302192	−	0.953247i	$-0.597718\pi$
$769$	−357409.	−0.604384	−0.302192	+	0.953247i	$0.597718\pi$
$770$	− 116304.i	− 0.196160i
$771$	0	0
$772$	305317.	0.512291
$773$	− 390495.i	− 0.653516i	−0.945108	−	0.326758i	$-0.894044\pi$
$773$	− 390495.i	− 0.653516i	0.945108	−	0.326758i	$-0.105956\pi$
$774$	0	0
$775$	−52450.0	−0.0873258
$776$	− 188726.i	− 0.313407i
$777$	0	0
$778$	305944.	0.505456
$779$	− 211993.i	− 0.349338i
$780$	0	0
$781$	335095.	0.549372
$782$	− 251730.i	− 0.411643i
$783$	0	0
$784$	370760.	0.603200
$785$	476819.i	0.773775i
$786$	0	0
$787$	−1.14723e6	−1.85226	−0.926129	−	0.377208i	$-0.876884\pi$
$787$	−1.14723e6	−1.85226	−0.926129	+	0.377208i	$0.876884\pi$
$788$	59601.1i	0.0959846i
$789$	0	0
$790$	−321419.	−0.515012
$791$	− 1.00356e6i	− 1.60395i
$792$	0	0
$793$	105164.	0.167233
$794$	− 332629.i	− 0.527617i
$795$	0	0
$796$	−478617.	−0.755374
$797$	− 142458.i	− 0.224269i	−0.993693	−	0.112135i	$-0.964231\pi$
$797$	− 142458.i	− 0.224269i	0.993693	−	0.112135i	$-0.0357688\pi$
$798$	0	0
$799$	−544050.	−0.852207
$800$	22627.4i	0.0353553i
$801$	0	0
$802$	417158.	0.648563
$803$	− 142468.i	− 0.220945i
$804$	0	0
$805$	162203.	0.250303
$806$	− 77170.2i	− 0.118790i
$807$	0	0
$808$	−119256.	−0.182665
$809$	− 677403.i	− 1.03502i	−0.855676	−	0.517512i	$-0.826858\pi$
$809$	− 677403.i	− 1.03502i	0.855676	−	0.517512i	$-0.173142\pi$
$810$	0	0
$811$	354053.	0.538303	0.269152	−	0.963098i	$-0.413257\pi$
$811$	354053.	0.538303	0.269152	+	0.963098i	$0.413257\pi$
$812$	− 839110.i	− 1.27264i
$813$	0	0
$814$	−127942.	−0.193092
$815$	− 287957.i	− 0.433523i
$816$	0	0
$817$	654819.	0.981019
$818$	294955.i	0.440808i
$819$	0	0
$820$	106066.	0.157742
$821$	32507.4i	0.0482276i	0.999709	+	0.0241138i	$0.00767640\pi$
$821$	32507.4i	0.0482276i	−0.999709	+	0.0241138i	$0.992324\pi$
$822$	0	0
$823$	97518.7	0.143975	0.0719877	−	0.997406i	$-0.477066\pi$
$823$	97518.7	0.143975	0.0719877	+	0.997406i	$0.477066\pi$
$824$	− 177316.i	− 0.261152i
$825$	0	0
$826$	1.18011e6	1.72966
$827$	880714.i	1.28773i	0.765140	+	0.643864i	$0.222670\pi$
$827$	880714.i	1.28773i	−0.765140	+	0.643864i	$0.777330\pi$
$828$	0	0
$829$	−564895.	−0.821975	−0.410987	−	0.911641i	$-0.634816\pi$
$829$	−564895.	−0.821975	−0.410987	+	0.911641i	$0.634816\pi$
$830$	− 52261.7i	− 0.0758625i
$831$	0	0
$832$	−33291.9	−0.0480941
$833$	− 3.21700e6i	− 4.63619i
$834$	0	0
$835$	−481065.	−0.689971
$836$	− 58106.3i	− 0.0831401i
$837$	0	0
$838$	276848.	0.394233
$839$	733064.i	1.04140i	0.853739	+	0.520701i	$0.174329\pi$
$839$	733064.i	1.04140i	−0.853739	+	0.520701i	$0.825671\pi$
$840$	0	0
$841$	−635346.	−0.898294
$842$	− 696840.i	− 0.982899i
$843$	0	0
$844$	−285858.	−0.401296
$845$	272051.i	0.381010i
$846$	0	0
$847$	1.17590e6	1.63909
$848$	133055.i	0.185029i
$849$	0	0
$850$	196333.	0.271741
$851$	− 178435.i	− 0.246388i
$852$	0	0
$853$	387698.	0.532838	0.266419	−	0.963857i	$-0.414159\pi$
$853$	387698.	0.532838	0.266419	+	0.963857i	$0.414159\pi$
$854$	− 414090.i	− 0.567779i
$855$	0	0
$856$	91666.8	0.125102
$857$	782691.i	1.06568i	0.846214	+	0.532842i	$0.178876\pi$
$857$	782691.i	1.06568i	−0.846214	+	0.532842i	$0.821124\pi$
$858$	0	0
$859$	−1.03753e6	−1.40609	−0.703045	−	0.711145i	$-0.748177\pi$
$859$	−1.03753e6	−1.40609	−0.703045	+	0.711145i	$0.748177\pi$
$860$	327623.i	0.442974i
$861$	0	0
$862$	−315672.	−0.424836
$863$	325737.i	0.437367i	0.975796	+	0.218683i	$0.0701762\pi$
$863$	325737.i	0.437367i	−0.975796	+	0.218683i	$0.929824\pi$
$864$	0	0
$865$	−340051.	−0.454477
$866$	− 408702.i	− 0.544967i
$867$	0	0
$868$	−303863.	−0.403309
$869$	412965.i	0.546857i
$870$	0	0
$871$	73834.6	0.0973249
$872$	196084.i	0.257875i
$873$	0	0
$874$	81037.9	0.106088
$875$	126508.i	0.165234i
$876$	0	0
$877$	352135.	0.457836	0.228918	−	0.973446i	$-0.426481\pi$
$877$	352135.	0.457836	0.228918	+	0.973446i	$0.426481\pi$
$878$	581804.i	0.754724i
$879$	0	0
$880$	29072.1	0.0375415
$881$	1.26422e6i	1.62882i	0.580291	+	0.814409i	$0.302939\pi$
$881$	1.26422e6i	1.62882i	−0.580291	+	0.814409i	$0.697061\pi$
$882$	0	0
$883$	−1.24132e6	−1.59208	−0.796038	−	0.605246i	$-0.793075\pi$
$883$	−1.24132e6	−1.59208	−0.796038	+	0.605246i	$0.793075\pi$
$884$	288866.i	0.369651i
$885$	0	0
$886$	−829437.	−1.05661
$887$	902225.i	1.14675i	0.819294	+	0.573373i	$0.194366\pi$
$887$	902225.i	1.14675i	−0.819294	+	0.573373i	$0.805634\pi$
$888$	0	0
$889$	−2.25141e6	−2.84873
$890$	− 229518.i	− 0.289759i
$891$	0	0
$892$	−480412.	−0.603788
$893$	− 175143.i	− 0.219629i
$894$	0	0
$895$	−63642.2	−0.0794510
$896$	131089.i	0.163287i
$897$	0	0
$898$	123616.	0.153293
$899$	486198.i	0.601581i
$900$	0	0
$901$	1.15449e6	1.42213
$902$	− 136275.i	− 0.167495i
$903$	0	0
$904$	250858.	0.306966
$905$	319028.i	0.389522i
$906$	0	0
$907$	−475859.	−0.578447	−0.289223	−	0.957262i	$-0.593397\pi$
$907$	−475859.	−0.578447	−0.289223	+	0.957262i	$0.593397\pi$
$908$	− 463446.i	− 0.562118i
$909$	0	0
$910$	−186132.	−0.224769
$911$	889172.i	1.07139i	0.844410	+	0.535697i	$0.179951\pi$
$911$	889172.i	1.07139i	−0.844410	+	0.535697i	$0.820049\pi$
$912$	0	0
$913$	−67146.7	−0.0805532
$914$	51146.0i	0.0612237i
$915$	0	0
$916$	195785.	0.233340
$917$	− 1.86421e6i	− 2.21695i
$918$	0	0
$919$	−856608.	−1.01426	−0.507132	−	0.861868i	$-0.669294\pi$
$919$	−856608.	−1.01426	−0.507132	+	0.861868i	$0.669294\pi$
$920$	40545.4i	0.0479034i
$921$	0	0
$922$	221863.	0.260989
$923$	− 536285.i	− 0.629495i
$924$	0	0
$925$	139167.	0.162650
$926$	994076.i	1.15930i
$927$	0	0
$928$	209750.	0.243560
$929$	− 1.53053e6i	− 1.77341i	−0.462335	−	0.886705i	$-0.652988\pi$
$929$	− 1.53053e6i	− 1.77341i	0.462335	−	0.886705i	$-0.347012\pi$
$930$	0	0
$931$	1.03563e6	1.19483
$932$	− 30960.1i	− 0.0356427i
$933$	0	0
$934$	977774.	1.12084
$935$	− 252252.i	− 0.288543i
$936$	0	0
$937$	465485.	0.530183	0.265092	−	0.964223i	$-0.414598\pi$
$937$	465485.	0.530183	0.265092	+	0.964223i	$0.414598\pi$
$938$	− 290729.i	− 0.330432i
$939$	0	0
$940$	87628.5	0.0991722
$941$	1.51732e6i	1.71355i	0.515691	+	0.856775i	$0.327535\pi$
$941$	1.51732e6i	1.71355i	−0.515691	+	0.856775i	$0.672465\pi$
$942$	0	0
$943$	190056.	0.213726
$944$	294988.i	0.331025i
$945$	0	0
$946$	420936.	0.470364
$947$	− 1.13070e6i	− 1.26081i	−0.776267	−	0.630404i	$-0.782889\pi$
$947$	− 1.13070e6i	− 1.26081i	0.776267	−	0.630404i	$-0.217111\pi$
$948$	0	0
$949$	−228004.	−0.253169
$950$	63204.3i	0.0700325i
$951$	0	0
$952$	1.13743e6	1.25502
$953$	975410.i	1.07399i	0.843584	+	0.536997i	$0.180441\pi$
$953$	975410.i	1.07399i	−0.843584	+	0.536997i	$0.819559\pi$
$954$	0	0
$955$	166641.	0.182716
$956$	− 76384.9i	− 0.0835780i
$957$	0	0
$958$	1.00668e6	1.09688
$959$	− 1.82765e6i	− 1.98727i
$960$	0	0
$961$	−747457.	−0.809355
$962$	204758.i	0.221254i
$963$	0	0
$964$	−498040.	−0.535933
$965$	426694.i	0.458207i
$966$	0	0
$967$	780376.	0.834547	0.417274	−	0.908781i	$-0.362986\pi$
$967$	780376.	0.834547	0.417274	+	0.908781i	$0.362986\pi$
$968$	293936.i	0.313691i
$969$	0	0
$970$	263753.	0.280320
$971$	305121.i	0.323619i	0.986822	+	0.161809i	$0.0517329\pi$
$971$	305121.i	0.323619i	−0.986822	+	0.161809i	$0.948267\pi$
$972$	0	0
$973$	1.82512e6	1.92782
$974$	− 624456.i	− 0.658240i
$975$	0	0
$976$	103509.	0.108662
$977$	− 861787.i	− 0.902840i	−0.892312	−	0.451420i	$-0.850918\pi$
$977$	− 861787.i	− 0.902840i	0.892312	−	0.451420i	$-0.149082\pi$
$978$	0	0
$979$	−294889.	−0.307676
$980$	518153.i	0.539518i
$981$	0	0
$982$	74177.3	0.0769216
$983$	− 252426.i	− 0.261233i	−0.991433	−	0.130616i	$-0.958304\pi$
$983$	− 252426.i	− 0.261233i	0.991433	−	0.130616i	$-0.0416956\pi$
$984$	0	0
$985$	−83295.1	−0.0858513
$986$	− 1.81995e6i	− 1.87200i
$987$	0	0
$988$	−92993.0	−0.0952657
$989$	587058.i	0.600190i
$990$	0	0
$991$	60269.6	0.0613693	0.0306847	−	0.999529i	$-0.490231\pi$
$991$	60269.6	0.0613693	0.0306847	+	0.999529i	$0.490231\pi$
$992$	− 75955.8i	− 0.0771858i
$993$	0	0
$994$	−2.11166e6	−2.13723
$995$	− 668888.i	− 0.675627i
$996$	0	0
$997$	1.39782e6	1.40624	0.703122	−	0.711069i	$-0.251789\pi$
$997$	1.39782e6	1.40624	0.703122	+	0.711069i	$0.251789\pi$
$998$	329377.i	0.330699i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	810.5.d.c.161.5		32
3.2	odd	2	inner	810.5.d.c.161.6		32
9.2	odd	6		270.5.h.a.71.9		32
9.4	even	3		270.5.h.a.251.9		32
9.5	odd	6		90.5.h.a.11.5	✓	32
9.7	even	3		90.5.h.a.41.5	yes	32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.5.h.a.11.5	✓	32	9.5	odd	6
90.5.h.a.41.5	yes	32	9.7	even	3
270.5.h.a.71.9		32	9.2	odd	6
270.5.h.a.251.9		32	9.4	even	3
810.5.d.c.161.5		32	1.1	even	1	trivial
810.5.d.c.161.6		32	3.2	odd	2	inner

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:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	810.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.82843i q^{2} -8.00000 q^{4} -11.1803i q^{5} +90.5214 q^{7} +22.6274i q^{8} +O(q^{10})\)	\(q-2.82843i q^{2} -8.00000 q^{4} -11.1803i q^{5} +90.5214 q^{7} +22.6274i q^{8} -31.6228 q^{10} +40.6295i q^{11} +65.0232 q^{13} -256.033i q^{14} +64.0000 q^{16} -555.313i q^{17} +178.769 q^{19} +89.4427i q^{20} +114.918 q^{22} +160.270i q^{23} -125.000 q^{25} -183.914i q^{26} -724.171 q^{28} +1158.72i q^{29} +419.600 q^{31} -181.019i q^{32} -1570.66 q^{34} -1012.06i q^{35} -1113.34 q^{37} -505.634i q^{38} +252.982 q^{40} -1185.85i q^{41} +3662.94 q^{43} -325.036i q^{44} +453.311 q^{46} -979.717i q^{47} +5793.13 q^{49} +353.553i q^{50} -520.186 q^{52} +2078.98i q^{53} +454.252 q^{55} +2048.27i q^{56} +3277.35 q^{58} +4609.19i q^{59} +1617.33 q^{61} -1186.81i q^{62} -512.000 q^{64} -726.982i q^{65} +1135.51 q^{67} +4442.50i q^{68} -2862.54 q^{70} -8247.59i q^{71} -3506.51 q^{73} +3149.00i q^{74} -1430.15 q^{76} +3677.84i q^{77} +10164.2 q^{79} -715.542i q^{80} -3354.09 q^{82} +1652.66i q^{83} -6208.59 q^{85} -10360.4i q^{86} -919.341 q^{88} +7258.01i q^{89} +5886.00 q^{91} -1282.16i q^{92} -2771.06 q^{94} -1998.70i q^{95} -8340.59 q^{97} -16385.4i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 256 q^{4} - 104 q^{7}+O(q^{10}) \) 32 * q - 256 * q^4 - 104 * q^7	\( 32 q - 256 q^{4} - 104 q^{7} + 40 q^{13} + 2048 q^{16} - 200 q^{19} + 1344 q^{22} - 4000 q^{25} + 832 q^{28} - 1472 q^{31} - 384 q^{34} - 4136 q^{37} - 272 q^{43} - 2112 q^{46} + 22296 q^{49} - 320 q^{52} - 12344 q^{61} - 16384 q^{64} + 40936 q^{67} + 4800 q^{70} - 41432 q^{73} + 1600 q^{76} - 14048 q^{79} - 17664 q^{82} - 17400 q^{85} - 10752 q^{88} + 69392 q^{91} + 1344 q^{94} + 36664 q^{97}+O(q^{100}) \) 32 * q - 256 * q^4 - 104 * q^7 + 40 * q^13 + 2048 * q^16 - 200 * q^19 + 1344 * q^22 - 4000 * q^25 + 832 * q^28 - 1472 * q^31 - 384 * q^34 - 4136 * q^37 - 272 * q^43 - 2112 * q^46 + 22296 * q^49 - 320 * q^52 - 12344 * q^61 - 16384 * q^64 + 40936 * q^67 + 4800 * q^70 - 41432 * q^73 + 1600 * q^76 - 14048 * q^79 - 17664 * q^82 - 17400 * q^85 - 10752 * q^88 + 69392 * q^91 + 1344 * q^94 + 36664 * q^97

Introduction

L-functions

Modular forms

Varieties

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Database

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