LMFDB - Embedded newform 810.5.d.c.161.7

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			161.7
Character	$\chi$	$=$	810.161
Dual form			810.5.d.c.161.8

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.82843i q^{2} -8.00000 q^{4} -11.1803i q^{5} +49.1951 q^{7} +22.6274i q^{8} +O(q^{10})$	\(q-2.82843i q^{2} -8.00000 q^{4} -11.1803i q^{5} +49.1951 q^{7} +22.6274i q^{8} -31.6228 q^{10} -96.2002i q^{11} -113.983 q^{13} -139.145i q^{14} +64.0000 q^{16} +351.453i q^{17} -253.282 q^{19} +89.4427i q^{20} -272.095 q^{22} -313.171i q^{23} -125.000 q^{25} +322.392i q^{26} -393.561 q^{28} +1127.36i q^{29} +340.531 q^{31} -181.019i q^{32} +994.058 q^{34} -550.018i q^{35} +75.9627 q^{37} +716.390i q^{38} +252.982 q^{40} -1144.61i q^{41} -2591.02 q^{43} +769.601i q^{44} -885.781 q^{46} -3558.29i q^{47} +19.1598 q^{49} +353.553i q^{50} +911.861 q^{52} +2926.47i q^{53} -1075.55 q^{55} +1113.16i q^{56} +3188.66 q^{58} -130.797i q^{59} -5629.54 q^{61} -963.168i q^{62} -512.000 q^{64} +1274.36i q^{65} +2670.08 q^{67} -2811.62i q^{68} -1555.69 q^{70} +2279.92i q^{71} -8045.40 q^{73} -214.855i q^{74} +2026.26 q^{76} -4732.58i q^{77} -8420.23 q^{79} -715.542i q^{80} -3237.45 q^{82} -12679.3i q^{83} +3929.36 q^{85} +7328.50i q^{86} +2176.76 q^{88} +10724.8i q^{89} -5607.39 q^{91} +2505.37i q^{92} -10064.4 q^{94} +2831.78i q^{95} +11723.3 q^{97} -54.1920i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	49.1951	1.00398	0.501991	−	0.864873i	$-0.332601\pi$
$7$	49.1951	1.00398	0.501991	+	0.864873i	$0.332601\pi$
$8$	22.6274i	0.353553i
$9$	0	0
$10$	−31.6228	−0.316228
$11$	− 96.2002i	− 0.795043i	−0.917593	−	0.397521i	$-0.869870\pi$
$11$	− 96.2002i	− 0.795043i	0.917593	−	0.397521i	$-0.130130\pi$
$12$	0	0
$13$	−113.983	−0.674453	−0.337227	−	0.941423i	$-0.609489\pi$
$13$	−113.983	−0.674453	−0.337227	+	0.941423i	$0.609489\pi$
$14$	− 139.145i	− 0.709922i
$15$	0	0
$16$	64.0000	0.250000
$17$	351.453i	1.21610i	0.793899	+	0.608050i	$0.208048\pi$
$17$	351.453i	1.21610i	−0.793899	+	0.608050i	$0.791952\pi$
$18$	0	0
$19$	−253.282	−0.701613	−0.350806	−	0.936448i	$-0.614092\pi$
$19$	−253.282	−0.701613	−0.350806	+	0.936448i	$0.614092\pi$
$20$	89.4427i	0.223607i
$21$	0	0
$22$	−272.095	−0.562180
$23$	− 313.171i	− 0.592006i	−0.955187	−	0.296003i	$-0.904346\pi$
$23$	− 313.171i	− 0.592006i	0.955187	−	0.296003i	$-0.0956538\pi$
$24$	0	0
$25$	−125.000	−0.200000
$26$	322.392i	0.476911i
$27$	0	0
$28$	−393.561	−0.501991
$29$	1127.36i	1.34050i	0.742135	+	0.670251i	$0.233813\pi$
$29$	1127.36i	1.34050i	−0.742135	+	0.670251i	$0.766187\pi$
$30$	0	0
$31$	340.531	0.354351	0.177175	−	0.984179i	$-0.443304\pi$
$31$	340.531	0.354351	0.177175	+	0.984179i	$0.443304\pi$
$32$	− 181.019i	− 0.176777i
$33$	0	0
$34$	994.058	0.859912
$35$	− 550.018i	− 0.448994i
$36$	0	0
$37$	75.9627	0.0554878	0.0277439	−	0.999615i	$-0.491168\pi$
$37$	75.9627	0.0554878	0.0277439	+	0.999615i	$0.491168\pi$
$38$	716.390i	0.496115i
$39$	0	0
$40$	252.982	0.158114
$41$	− 1144.61i	− 0.680910i	−0.940261	−	0.340455i	$-0.889419\pi$
$41$	− 1144.61i	− 0.680910i	0.940261	−	0.340455i	$-0.110581\pi$
$42$	0	0
$43$	−2591.02	−1.40131	−0.700653	−	0.713502i	$-0.747108\pi$
$43$	−2591.02	−1.40131	−0.700653	+	0.713502i	$0.747108\pi$
$44$	769.601i	0.397521i
$45$	0	0
$46$	−885.781	−0.418611
$47$	− 3558.29i	− 1.61081i	−0.592722	−	0.805407i	$-0.701947\pi$
$47$	− 3558.29i	− 1.61081i	0.592722	−	0.805407i	$-0.298053\pi$
$48$	0	0
$49$	19.1598	0.00797991
$50$	353.553i	0.141421i
$51$	0	0
$52$	911.861	0.337227
$53$	2926.47i	1.04182i	0.853612	+	0.520910i	$0.174407\pi$
$53$	2926.47i	1.04182i	−0.853612	+	0.520910i	$0.825593\pi$
$54$	0	0
$55$	−1075.55	−0.355554
$56$	1113.16i	0.354961i
$57$	0	0
$58$	3188.66	0.947877
$59$	− 130.797i	− 0.0375747i	−0.999824	−	0.0187873i	$-0.994019\pi$
$59$	− 130.797i	− 0.0375747i	0.999824	−	0.0187873i	$-0.00598055\pi$
$60$	0	0
$61$	−5629.54	−1.51291	−0.756456	−	0.654045i	$-0.773071\pi$
$61$	−5629.54	−1.51291	−0.756456	+	0.654045i	$0.773071\pi$
$62$	− 963.168i	− 0.250564i
$63$	0	0
$64$	−512.000	−0.125000
$65$	1274.36i	0.301625i
$66$	0	0
$67$	2670.08	0.594806	0.297403	−	0.954752i	$-0.403879\pi$
$67$	2670.08	0.594806	0.297403	+	0.954752i	$0.403879\pi$
$68$	− 2811.62i	− 0.608050i
$69$	0	0
$70$	−1555.69	−0.317487
$71$	2279.92i	0.452275i	0.974095	+	0.226137i	$0.0726098\pi$
$71$	2279.92i	0.452275i	−0.974095	+	0.226137i	$0.927390\pi$
$72$	0	0
$73$	−8045.40	−1.50974	−0.754869	−	0.655875i	$-0.772300\pi$
$73$	−8045.40	−1.50974	−0.754869	+	0.655875i	$0.772300\pi$
$74$	− 214.855i	− 0.0392358i
$75$	0	0
$76$	2026.26	0.350806
$77$	− 4732.58i	− 0.798209i
$78$	0	0
$79$	−8420.23	−1.34918	−0.674589	−	0.738193i	$-0.735679\pi$
$79$	−8420.23	−1.34918	−0.674589	+	0.738193i	$0.735679\pi$
$80$	− 715.542i	− 0.111803i
$81$	0	0
$82$	−3237.45	−0.481476
$83$	− 12679.3i	− 1.84052i	−0.391313	−	0.920258i	$-0.627979\pi$
$83$	− 12679.3i	− 1.84052i	0.391313	−	0.920258i	$-0.372021\pi$
$84$	0	0
$85$	3929.36	0.543856
$86$	7328.50i	0.990874i
$87$	0	0
$88$	2176.76	0.281090
$89$	10724.8i	1.35397i	0.735997	+	0.676985i	$0.236714\pi$
$89$	10724.8i	1.35397i	−0.735997	+	0.676985i	$0.763286\pi$
$90$	0	0
$91$	−5607.39	−0.677139
$92$	2505.37i	0.296003i
$93$	0	0
$94$	−10064.4	−1.13902
$95$	2831.78i	0.313771i
$96$	0	0
$97$	11723.3	1.24596	0.622982	−	0.782237i	$-0.285921\pi$
$97$	11723.3	1.24596	0.622982	+	0.782237i	$0.285921\pi$
$98$	− 54.1920i	− 0.00564265i
$99$	0	0
$100$	1000.00	0.100000
$101$	12192.5i	1.19522i	0.801786	+	0.597612i	$0.203884\pi$
$101$	12192.5i	1.19522i	−0.801786	+	0.597612i	$0.796116\pi$
$102$	0	0
$103$	−8197.62	−0.772704	−0.386352	−	0.922351i	$-0.626265\pi$
$103$	−8197.62	−0.772704	−0.386352	+	0.922351i	$0.626265\pi$
$104$	− 2579.13i	− 0.238455i
$105$	0	0
$106$	8277.31	0.736678
$107$	21177.8i	1.84975i	0.380270	+	0.924876i	$0.375831\pi$
$107$	21177.8i	1.84975i	−0.380270	+	0.924876i	$0.624169\pi$
$108$	0	0
$109$	14536.2	1.22348	0.611740	−	0.791059i	$-0.290470\pi$
$109$	14536.2	1.22348	0.611740	+	0.791059i	$0.290470\pi$
$110$	3042.12i	0.251415i
$111$	0	0
$112$	3148.49	0.250996
$113$	8890.90i	0.696288i	0.937441	+	0.348144i	$0.113188\pi$
$113$	8890.90i	0.696288i	−0.937441	+	0.348144i	$0.886812\pi$
$114$	0	0
$115$	−3501.36	−0.264753
$116$	− 9018.89i	− 0.670251i
$117$	0	0
$118$	−369.951	−0.0265693
$119$	17289.8i	1.22094i
$120$	0	0
$121$	5386.53	0.367907
$122$	15922.7i	1.06979i
$123$	0	0
$124$	−2724.25	−0.177175
$125$	1397.54i	0.0894427i
$126$	0	0
$127$	13893.3	0.861385	0.430692	−	0.902499i	$-0.358269\pi$
$127$	13893.3	0.861385	0.430692	+	0.902499i	$0.358269\pi$
$128$	1448.15i	0.0883883i
$129$	0	0
$130$	3604.45	0.213281
$131$	3299.12i	0.192245i	0.995370	+	0.0961225i	$0.0306440\pi$
$131$	3299.12i	0.192245i	−0.995370	+	0.0961225i	$0.969356\pi$
$132$	0	0
$133$	−12460.2	−0.704407
$134$	− 7552.14i	− 0.420591i
$135$	0	0
$136$	−7952.47	−0.429956
$137$	24473.9i	1.30395i	0.758239	+	0.651976i	$0.226060\pi$
$137$	24473.9i	1.30395i	−0.758239	+	0.651976i	$0.773940\pi$
$138$	0	0
$139$	−17136.1	−0.886914	−0.443457	−	0.896296i	$-0.646248\pi$
$139$	−17136.1	−0.886914	−0.443457	+	0.896296i	$0.646248\pi$
$140$	4400.15i	0.224497i
$141$	0	0
$142$	6448.58	0.319806
$143$	10965.1i	0.536219i
$144$	0	0
$145$	12604.3	0.599490
$146$	22755.8i	1.06755i
$147$	0	0
$148$	−607.702	−0.0277439
$149$	17433.2i	0.785243i	0.919700	+	0.392622i	$0.128432\pi$
$149$	17433.2i	0.785243i	−0.919700	+	0.392622i	$0.871568\pi$
$150$	0	0
$151$	−5464.62	−0.239666	−0.119833	−	0.992794i	$-0.538236\pi$
$151$	−5464.62	−0.239666	−0.119833	+	0.992794i	$0.538236\pi$
$152$	− 5731.12i	− 0.248058i
$153$	0	0
$154$	−13385.8	−0.564419
$155$	− 3807.25i	− 0.158471i
$156$	0	0
$157$	−42494.7	−1.72399	−0.861996	−	0.506915i	$-0.830786\pi$
$157$	−42494.7	−1.72399	−0.861996	+	0.506915i	$0.830786\pi$
$158$	23816.0i	0.954013i
$159$	0	0
$160$	−2023.86	−0.0790569
$161$	− 15406.5i	− 0.594363i
$162$	0	0
$163$	−12868.0	−0.484323	−0.242162	−	0.970236i	$-0.577856\pi$
$163$	−12868.0	−0.484323	−0.242162	+	0.970236i	$0.577856\pi$
$164$	9156.88i	0.340455i
$165$	0	0
$166$	−35862.5	−1.30144
$167$	33203.2i	1.19055i	0.803522	+	0.595275i	$0.202957\pi$
$167$	33203.2i	1.19055i	−0.803522	+	0.595275i	$0.797043\pi$
$168$	0	0
$169$	−15569.0	−0.545113
$170$	− 11113.9i	− 0.384564i
$171$	0	0
$172$	20728.1	0.700653
$173$	− 27039.6i	− 0.903457i	−0.892156	−	0.451728i	$-0.850808\pi$
$173$	− 27039.6i	− 0.903457i	0.892156	−	0.451728i	$-0.149192\pi$
$174$	0	0
$175$	−6149.39	−0.200796
$176$	− 6156.81i	− 0.198761i
$177$	0	0
$178$	30334.3	0.957401
$179$	− 50554.7i	− 1.57781i	−0.614513	−	0.788906i	$-0.710648\pi$
$179$	− 50554.7i	− 1.57781i	0.614513	−	0.788906i	$-0.289352\pi$
$180$	0	0
$181$	−4462.41	−0.136211	−0.0681054	−	0.997678i	$-0.521695\pi$
$181$	−4462.41	−0.136211	−0.0681054	+	0.997678i	$0.521695\pi$
$182$	15860.1i	0.478810i
$183$	0	0
$184$	7086.25	0.209306
$185$	− 849.289i	− 0.0248149i
$186$	0	0
$187$	33809.8	0.966851
$188$	28466.3i	0.805407i
$189$	0	0
$190$	8009.49	0.221869
$191$	4764.36i	0.130598i	0.997866	+	0.0652992i	$0.0208002\pi$
$191$	4764.36i	0.130598i	−0.997866	+	0.0652992i	$0.979200\pi$
$192$	0	0
$193$	62069.6	1.66634	0.833172	−	0.553015i	$-0.186523\pi$
$193$	62069.6	1.66634	0.833172	+	0.553015i	$0.186523\pi$
$194$	− 33158.4i	− 0.881029i
$195$	0	0
$196$	−153.278	−0.00398996
$197$	30470.8i	0.785148i	0.919720	+	0.392574i	$0.128415\pi$
$197$	30470.8i	0.785148i	−0.919720	+	0.392574i	$0.871585\pi$
$198$	0	0
$199$	26636.9	0.672632	0.336316	−	0.941749i	$-0.390819\pi$
$199$	26636.9	0.672632	0.336316	+	0.941749i	$0.390819\pi$
$200$	− 2828.43i	− 0.0707107i
$201$	0	0
$202$	34485.5	0.845150
$203$	55460.7i	1.34584i
$204$	0	0
$205$	−12797.1	−0.304512
$206$	23186.4i	0.546384i
$207$	0	0
$208$	−7294.89	−0.168613
$209$	24365.8i	0.557812i
$210$	0	0
$211$	−78677.0	−1.76719	−0.883594	−	0.468253i	$-0.844884\pi$
$211$	−78677.0	−1.76719	−0.883594	+	0.468253i	$0.844884\pi$
$212$	− 23411.8i	− 0.520910i
$213$	0	0
$214$	59899.9	1.30797
$215$	28968.4i	0.626684i
$216$	0	0
$217$	16752.5	0.355762
$218$	− 41114.5i	− 0.865132i
$219$	0	0
$220$	8604.40	0.177777
$221$	− 40059.5i	− 0.820202i
$222$	0	0
$223$	−85807.7	−1.72551	−0.862753	−	0.505626i	$-0.831262\pi$
$223$	−85807.7	−1.72551	−0.862753	+	0.505626i	$0.831262\pi$
$224$	− 8905.27i	− 0.177481i
$225$	0	0
$226$	25147.3	0.492350
$227$	37252.2i	0.722936i	0.932384	+	0.361468i	$0.117724\pi$
$227$	37252.2i	0.722936i	−0.932384	+	0.361468i	$0.882276\pi$
$228$	0	0
$229$	21434.6	0.408737	0.204369	−	0.978894i	$-0.434486\pi$
$229$	21434.6	0.408737	0.204369	+	0.978894i	$0.434486\pi$
$230$	9903.34i	0.187209i
$231$	0	0
$232$	−25509.3	−0.473939
$233$	31750.4i	0.584840i	0.956290	+	0.292420i	$0.0944605\pi$
$233$	31750.4i	0.584840i	−0.956290	+	0.292420i	$0.905540\pi$
$234$	0	0
$235$	−39782.9	−0.720378
$236$	1046.38i	0.0187873i
$237$	0	0
$238$	48902.8	0.863336
$239$	61678.6i	1.07979i	0.841733	+	0.539894i	$0.181536\pi$
$239$	61678.6i	1.07979i	−0.841733	+	0.539894i	$0.818464\pi$
$240$	0	0
$241$	70047.7	1.20604	0.603018	−	0.797728i	$-0.293965\pi$
$241$	70047.7	1.20604	0.603018	+	0.797728i	$0.293965\pi$
$242$	− 15235.4i	− 0.260150i
$243$	0	0
$244$	45036.3	0.756456
$245$	− 214.213i	− 0.00356872i
$246$	0	0
$247$	28869.8	0.473205
$248$	7705.34i	0.125282i
$249$	0	0
$250$	3952.85	0.0632456
$251$	91245.1i	1.44831i	0.689636	+	0.724156i	$0.257771\pi$
$251$	91245.1i	1.44831i	−0.689636	+	0.724156i	$0.742229\pi$
$252$	0	0
$253$	−30127.1	−0.470670
$254$	− 39296.1i	− 0.609091i
$255$	0	0
$256$	4096.00	0.0625000
$257$	− 52399.4i	− 0.793341i	−0.917961	−	0.396671i	$-0.870165\pi$
$257$	− 52399.4i	− 0.793341i	0.917961	−	0.396671i	$-0.129835\pi$
$258$	0	0
$259$	3737.00	0.0557087
$260$	− 10194.9i	− 0.150812i
$261$	0	0
$262$	9331.31	0.135938
$263$	− 30655.0i	− 0.443189i	−0.975139	−	0.221595i	$-0.928874\pi$
$263$	− 30655.0i	− 0.443189i	0.975139	−	0.221595i	$-0.0711262\pi$
$264$	0	0
$265$	32719.0	0.465916
$266$	35242.9i	0.498091i
$267$	0	0
$268$	−21360.7	−0.297403
$269$	7299.68i	0.100879i	0.998727	+	0.0504393i	$0.0160621\pi$
$269$	7299.68i	0.100879i	−0.998727	+	0.0504393i	$0.983938\pi$
$270$	0	0
$271$	−48800.5	−0.664486	−0.332243	−	0.943194i	$-0.607805\pi$
$271$	−48800.5	−0.664486	−0.332243	+	0.943194i	$0.607805\pi$
$272$	22493.0i	0.304025i
$273$	0	0
$274$	69222.6	0.922034
$275$	12025.0i	0.159009i
$276$	0	0
$277$	18302.9	0.238539	0.119270	−	0.992862i	$-0.461945\pi$
$277$	18302.9	0.238539	0.119270	+	0.992862i	$0.461945\pi$
$278$	48468.1i	0.627143i
$279$	0	0
$280$	12445.5	0.158743
$281$	47262.4i	0.598554i	0.954166	+	0.299277i	$0.0967454\pi$
$281$	47262.4i	0.598554i	−0.954166	+	0.299277i	$0.903255\pi$
$282$	0	0
$283$	116677.	1.45684	0.728420	−	0.685131i	$-0.240255\pi$
$283$	116677.	1.45684	0.728420	+	0.685131i	$0.240255\pi$
$284$	− 18239.3i	− 0.226137i
$285$	0	0
$286$	31014.1	0.379164
$287$	− 56309.2i	− 0.683622i
$288$	0	0
$289$	−39998.0	−0.478897
$290$	− 35650.3i	− 0.423904i
$291$	0	0
$292$	64363.2	0.754869
$293$	24486.2i	0.285224i	0.989779	+	0.142612i	$0.0455501\pi$
$293$	24486.2i	0.285224i	−0.989779	+	0.142612i	$0.954450\pi$
$294$	0	0
$295$	−1462.36	−0.0168039
$296$	1718.84i	0.0196179i
$297$	0	0
$298$	49308.5	0.555251
$299$	35696.1i	0.399280i
$300$	0	0
$301$	−127465.	−1.40689
$302$	15456.3i	0.169469i
$303$	0	0
$304$	−16210.1	−0.175403
$305$	62940.2i	0.676594i
$306$	0	0
$307$	−103102.	−1.09394	−0.546968	−	0.837154i	$-0.684218\pi$
$307$	−103102.	−1.09394	−0.546968	+	0.837154i	$0.684218\pi$
$308$	37860.6i	0.399104i
$309$	0	0
$310$	−10768.5	−0.112056
$311$	− 88795.7i	− 0.918060i	−0.888421	−	0.459030i	$-0.848197\pi$
$311$	− 88795.7i	− 0.918060i	0.888421	−	0.459030i	$-0.151803\pi$
$312$	0	0
$313$	−69209.6	−0.706444	−0.353222	−	0.935540i	$-0.614914\pi$
$313$	−69209.6	−0.706444	−0.353222	+	0.935540i	$0.614914\pi$
$314$	120193.i	1.21905i
$315$	0	0
$316$	67361.8	0.674589
$317$	126817.i	1.26200i	0.775783	+	0.630999i	$0.217355\pi$
$317$	126817.i	1.26200i	−0.775783	+	0.630999i	$0.782645\pi$
$318$	0	0
$319$	108452.	1.06576
$320$	5724.33i	0.0559017i
$321$	0	0
$322$	−43576.1	−0.420278
$323$	− 89016.7i	− 0.853231i
$324$	0	0
$325$	14247.8	0.134891
$326$	36396.2i	0.342468i
$327$	0	0
$328$	25899.6	0.240738
$329$	− 175050.i	− 1.61723i
$330$	0	0
$331$	76584.8	0.699015	0.349508	−	0.936934i	$-0.386349\pi$
$331$	76584.8	0.699015	0.349508	+	0.936934i	$0.386349\pi$
$332$	101434.i	0.920258i
$333$	0	0
$334$	93912.9	0.841845
$335$	− 29852.5i	− 0.266005i
$336$	0	0
$337$	−141729.	−1.24795	−0.623977	−	0.781443i	$-0.714484\pi$
$337$	−141729.	−1.24795	−0.623977	+	0.781443i	$0.714484\pi$
$338$	44035.7i	0.385453i
$339$	0	0
$340$	−31434.9	−0.271928
$341$	− 32759.2i	− 0.281724i
$342$	0	0
$343$	−117175.	−0.995970
$344$	− 58628.0i	− 0.495437i
$345$	0	0
$346$	−76479.4	−0.638840
$347$	118472.i	0.983916i	0.870619	+	0.491958i	$0.163719\pi$
$347$	118472.i	0.983916i	−0.870619	+	0.491958i	$0.836281\pi$
$348$	0	0
$349$	85526.1	0.702179	0.351089	−	0.936342i	$-0.385811\pi$
$349$	85526.1	0.702179	0.351089	+	0.936342i	$0.385811\pi$
$350$	17393.1i	0.141984i
$351$	0	0
$352$	−17414.1	−0.140545
$353$	− 149927.i	− 1.20318i	−0.798805	−	0.601590i	$-0.794534\pi$
$353$	− 149927.i	− 1.20318i	0.798805	−	0.601590i	$-0.205466\pi$
$354$	0	0
$355$	25490.2	0.202263
$356$	− 85798.4i	− 0.676985i
$357$	0	0
$358$	−142990.	−1.11568
$359$	− 61273.8i	− 0.475429i	−0.971335	−	0.237715i	$-0.923602\pi$
$359$	− 61273.8i	− 0.475429i	0.971335	−	0.237715i	$-0.0763983\pi$
$360$	0	0
$361$	−66169.1	−0.507739
$362$	12621.6i	0.0963156i
$363$	0	0
$364$	44859.1	0.338570
$365$	89950.3i	0.675176i
$366$	0	0
$367$	8152.67	0.0605296	0.0302648	−	0.999542i	$-0.490365\pi$
$367$	8152.67	0.0605296	0.0302648	+	0.999542i	$0.490365\pi$
$368$	− 20042.9i	− 0.148001i
$369$	0	0
$370$	−2402.15	−0.0175468
$371$	143968.i	1.04597i
$372$	0	0
$373$	−224482.	−1.61348	−0.806741	−	0.590905i	$-0.798771\pi$
$373$	−224482.	−1.61348	−0.806741	+	0.590905i	$0.798771\pi$
$374$	− 95628.6i	− 0.683667i
$375$	0	0
$376$	80514.9	0.569509
$377$	− 128500.i	− 0.904106i
$378$	0	0
$379$	−20153.1	−0.140302	−0.0701509	−	0.997536i	$-0.522348\pi$
$379$	−20153.1	−0.140302	−0.0701509	+	0.997536i	$0.522348\pi$
$380$	− 22654.3i	− 0.156885i
$381$	0	0
$382$	13475.7	0.0923471
$383$	− 63325.6i	− 0.431700i	−0.976427	−	0.215850i	$-0.930748\pi$
$383$	− 63325.6i	− 0.431700i	0.976427	−	0.215850i	$-0.0692522\pi$
$384$	0	0
$385$	−52911.8	−0.356970
$386$	− 175559.i	− 1.17828i
$387$	0	0
$388$	−93786.1	−0.622982
$389$	− 194066.i	− 1.28248i	−0.767342	−	0.641239i	$-0.778421\pi$
$389$	− 194066.i	− 1.28248i	0.767342	−	0.641239i	$-0.221579\pi$
$390$	0	0
$391$	110065.	0.719937
$392$	433.536i	0.00282132i
$393$	0	0
$394$	86184.5	0.555184
$395$	94141.0i	0.603371i
$396$	0	0
$397$	66142.9	0.419665	0.209832	−	0.977737i	$-0.432708\pi$
$397$	66142.9	0.419665	0.209832	+	0.977737i	$0.432708\pi$
$398$	− 75340.5i	− 0.475623i
$399$	0	0
$400$	−8000.00	−0.0500000
$401$	− 290480.i	− 1.80645i	−0.429163	−	0.903227i	$-0.641191\pi$
$401$	− 290480.i	− 1.80645i	0.429163	−	0.903227i	$-0.358809\pi$
$402$	0	0
$403$	−38814.6	−0.238993
$404$	− 97539.8i	− 0.597612i
$405$	0	0
$406$	156866.	0.951652
$407$	− 7307.63i	− 0.0441151i
$408$	0	0
$409$	−223523.	−1.33622	−0.668108	−	0.744064i	$-0.732896\pi$
$409$	−223523.	−1.33622	−0.668108	+	0.744064i	$0.732896\pi$
$410$	36195.7i	0.215323i
$411$	0	0
$412$	65580.9	0.386352
$413$	− 6434.60i	− 0.0377243i
$414$	0	0
$415$	−141759.	−0.823104
$416$	20633.1i	0.119228i
$417$	0	0
$418$	68916.9	0.394433
$419$	111993.i	0.637914i	0.947769	+	0.318957i	$0.103332\pi$
$419$	111993.i	0.637914i	−0.947769	+	0.318957i	$0.896668\pi$
$420$	0	0
$421$	−300299.	−1.69430	−0.847148	−	0.531358i	$-0.821682\pi$
$421$	−300299.	−1.69430	−0.847148	+	0.531358i	$0.821682\pi$
$422$	222532.i	1.24959i
$423$	0	0
$424$	−66218.5	−0.368339
$425$	− 43931.6i	− 0.243220i
$426$	0	0
$427$	−276946.	−1.51894
$428$	− 169422.i	− 0.924876i
$429$	0	0
$430$	81935.1	0.443132
$431$	− 306017.i	− 1.64737i	−0.567048	−	0.823685i	$-0.691915\pi$
$431$	− 306017.i	− 1.64737i	0.567048	−	0.823685i	$-0.308085\pi$
$432$	0	0
$433$	253079.	1.34984	0.674918	−	0.737893i	$-0.264179\pi$
$433$	253079.	1.34984	0.674918	+	0.737893i	$0.264179\pi$
$434$	− 47383.1i	− 0.251562i
$435$	0	0
$436$	−116289.	−0.611740
$437$	79320.6i	0.415359i
$438$	0	0
$439$	269111.	1.39638	0.698189	−	0.715914i	$-0.253990\pi$
$439$	269111.	1.39638	0.698189	+	0.715914i	$0.253990\pi$
$440$	− 24336.9i	− 0.125707i
$441$	0	0
$442$	−113305.	−0.579971
$443$	− 202671.i	− 1.03272i	−0.856371	−	0.516361i	$-0.827286\pi$
$443$	− 202671.i	− 1.03272i	0.856371	−	0.516361i	$-0.172714\pi$
$444$	0	0
$445$	119907.	0.605514
$446$	242701.i	1.22012i
$447$	0	0
$448$	−25187.9	−0.125498
$449$	− 76904.0i	− 0.381466i	−0.981642	−	0.190733i	$-0.938914\pi$
$449$	− 76904.0i	− 0.381466i	0.981642	−	0.190733i	$-0.0610865\pi$
$450$	0	0
$451$	−110112.	−0.541353
$452$	− 71127.2i	− 0.348144i
$453$	0	0
$454$	105365.	0.511193
$455$	62692.5i	0.302826i
$456$	0	0
$457$	−220998.	−1.05817	−0.529085	−	0.848569i	$-0.677465\pi$
$457$	−220998.	−1.05817	−0.529085	+	0.848569i	$0.677465\pi$
$458$	− 60626.2i	− 0.289021i
$459$	0	0
$460$	28010.9	0.132376
$461$	− 43395.3i	− 0.204193i	−0.994775	−	0.102096i	$-0.967445\pi$
$461$	− 43395.3i	− 0.204193i	0.994775	−	0.102096i	$-0.0325550\pi$
$462$	0	0
$463$	77526.9	0.361652	0.180826	−	0.983515i	$-0.442123\pi$
$463$	77526.9	0.361652	0.180826	+	0.983515i	$0.442123\pi$
$464$	72151.1i	0.335125i
$465$	0	0
$466$	89803.6	0.413544
$467$	− 144596.i	− 0.663015i	−0.943453	−	0.331507i	$-0.892443\pi$
$467$	− 144596.i	− 0.663015i	0.943453	−	0.331507i	$-0.107557\pi$
$468$	0	0
$469$	131355.	0.597175
$470$	112523.i	0.509384i
$471$	0	0
$472$	2959.61	0.0132847
$473$	249256.i	1.11410i
$474$	0	0
$475$	31660.3	0.140323
$476$	− 138318.i	− 0.610471i
$477$	0	0
$478$	174453.	0.763525
$479$	− 136132.i	− 0.593322i	−0.954983	−	0.296661i	$-0.904127\pi$
$479$	− 136132.i	− 0.593322i	0.954983	−	0.296661i	$-0.0958731\pi$
$480$	0	0
$481$	−8658.43	−0.0374239
$482$	− 198125.i	− 0.852796i
$483$	0	0
$484$	−43092.2	−0.183954
$485$	− 131070.i	− 0.557212i
$486$	0	0
$487$	−42708.6	−0.180077	−0.0900384	−	0.995938i	$-0.528699\pi$
$487$	−42708.6	−0.180077	−0.0900384	+	0.995938i	$0.528699\pi$
$488$	− 127382.i	− 0.534895i
$489$	0	0
$490$	−605.885	−0.00252347
$491$	154556.i	0.641094i	0.947233	+	0.320547i	$0.103867\pi$
$491$	154556.i	0.641094i	−0.947233	+	0.320547i	$0.896133\pi$
$492$	0	0
$493$	−396214.	−1.63018
$494$	− 81656.1i	− 0.334607i
$495$	0	0
$496$	21794.0	0.0885877
$497$	112161.i	0.454076i
$498$	0	0
$499$	51859.2	0.208269	0.104134	−	0.994563i	$-0.466793\pi$
$499$	51859.2	0.208269	0.104134	+	0.994563i	$0.466793\pi$
$500$	− 11180.3i	− 0.0447214i
$501$	0	0
$502$	258080.	1.02411
$503$	− 334798.i	− 1.32327i	−0.749828	−	0.661633i	$-0.769864\pi$
$503$	− 334798.i	− 1.32327i	0.749828	−	0.661633i	$-0.230136\pi$
$504$	0	0
$505$	136316.	0.534520
$506$	85212.3i	0.332814i
$507$	0	0
$508$	−111146.	−0.430692
$509$	− 173371.i	− 0.669178i	−0.942364	−	0.334589i	$-0.891403\pi$
$509$	− 173371.i	− 0.669178i	0.942364	−	0.334589i	$-0.108597\pi$
$510$	0	0
$511$	−395794.	−1.51575
$512$	− 11585.2i	− 0.0441942i
$513$	0	0
$514$	−148208.	−0.560977
$515$	91652.1i	0.345564i
$516$	0	0
$517$	−342308.	−1.28067
$518$	− 10569.8i	− 0.0393920i
$519$	0	0
$520$	−28835.6	−0.106640
$521$	200698.i	0.739379i	0.929155	+	0.369690i	$0.120536\pi$
$521$	200698.i	0.739379i	−0.929155	+	0.369690i	$0.879464\pi$
$522$	0	0
$523$	−303311.	−1.10888	−0.554441	−	0.832223i	$-0.687068\pi$
$523$	−303311.	−1.10888	−0.554441	+	0.832223i	$0.687068\pi$
$524$	− 26392.9i	− 0.0961225i
$525$	0	0
$526$	−86705.4	−0.313382
$527$	119681.i	0.430926i
$528$	0	0
$529$	181765.	0.649529
$530$	− 92543.2i	− 0.329452i
$531$	0	0
$532$	99682.0	0.352203
$533$	130466.i	0.459242i
$534$	0	0
$535$	236775.	0.827234
$536$	60417.1i	0.210296i
$537$	0	0
$538$	20646.6	0.0713320
$539$	− 1843.17i	− 0.00634437i
$540$	0	0
$541$	−127347.	−0.435104	−0.217552	−	0.976049i	$-0.569807\pi$
$541$	−127347.	−0.435104	−0.217552	+	0.976049i	$0.569807\pi$
$542$	138029.i	0.469863i
$543$	0	0
$544$	63619.7	0.214978
$545$	− 162519.i	− 0.547157i
$546$	0	0
$547$	37960.8	0.126870	0.0634352	−	0.997986i	$-0.479794\pi$
$547$	37960.8	0.126870	0.0634352	+	0.997986i	$0.479794\pi$
$548$	− 195791.i	− 0.651976i
$549$	0	0
$550$	34011.9	0.112436
$551$	− 285541.i	− 0.940513i
$552$	0	0
$553$	−414234.	−1.35455
$554$	− 51768.4i	− 0.168673i
$555$	0	0
$556$	137088.	0.443457
$557$	309520.i	0.997651i	0.866703	+	0.498825i	$0.166235\pi$
$557$	309520.i	0.997651i	−0.866703	+	0.498825i	$0.833765\pi$
$558$	0	0
$559$	295331.	0.945116
$560$	− 35201.2i	− 0.112249i
$561$	0	0
$562$	133678.	0.423241
$563$	− 581536.i	− 1.83468i	−0.398108	−	0.917338i	$-0.630333\pi$
$563$	− 581536.i	− 1.83468i	0.398108	−	0.917338i	$-0.369667\pi$
$564$	0	0
$565$	99403.3	0.311389
$566$	− 330012.i	− 1.03014i
$567$	0	0
$568$	−51588.6	−0.159903
$569$	− 259660.i	− 0.802013i	−0.916075	−	0.401006i	$-0.868661\pi$
$569$	− 259660.i	− 0.802013i	0.916075	−	0.401006i	$-0.131339\pi$
$570$	0	0
$571$	13162.2	0.0403697	0.0201848	−	0.999796i	$-0.493575\pi$
$571$	13162.2	0.0403697	0.0201848	+	0.999796i	$0.493575\pi$
$572$	− 87721.2i	− 0.268110i
$573$	0	0
$574$	−159267.	−0.483393
$575$	39146.4i	0.118401i
$576$	0	0
$577$	323202.	0.970784	0.485392	−	0.874297i	$-0.338677\pi$
$577$	323202.	0.970784	0.485392	+	0.874297i	$0.338677\pi$
$578$	113131.i	0.338631i
$579$	0	0
$580$	−100834.	−0.299745
$581$	− 623760.i	− 1.84784i
$582$	0	0
$583$	281527.	0.828291
$584$	− 182047.i	− 0.533773i
$585$	0	0
$586$	69257.4	0.201684
$587$	− 292720.i	− 0.849525i	−0.905305	−	0.424763i	$-0.860358\pi$
$587$	− 292720.i	− 0.849525i	0.905305	−	0.424763i	$-0.139642\pi$
$588$	0	0
$589$	−86250.5	−0.248617
$590$	4136.18i	0.0118822i
$591$	0	0
$592$	4861.61	0.0138719
$593$	328987.i	0.935554i	0.883847	+	0.467777i	$0.154945\pi$
$593$	328987.i	0.935554i	−0.883847	+	0.467777i	$0.845055\pi$
$594$	0	0
$595$	193305.	0.546022
$596$	− 139465.i	− 0.392622i
$597$	0	0
$598$	100964.	0.282334
$599$	− 250539.i	− 0.698266i	−0.937073	−	0.349133i	$-0.886476\pi$
$599$	− 250539.i	− 0.698266i	0.937073	−	0.349133i	$-0.113524\pi$
$600$	0	0
$601$	363868.	1.00738	0.503692	−	0.863883i	$-0.331975\pi$
$601$	363868.	1.00738	0.503692	+	0.863883i	$0.331975\pi$
$602$	360527.i	0.994819i
$603$	0	0
$604$	43717.0	0.119833
$605$	− 60223.2i	− 0.164533i
$606$	0	0
$607$	319447.	0.867005	0.433503	−	0.901152i	$-0.357277\pi$
$607$	319447.	0.867005	0.433503	+	0.901152i	$0.357277\pi$
$608$	45849.0i	0.124029i
$609$	0	0
$610$	178022.	0.478424
$611$	405583.i	1.08642i
$612$	0	0
$613$	−604584.	−1.60892	−0.804462	−	0.594003i	$-0.797547\pi$
$613$	−604584.	−1.60892	−0.804462	+	0.594003i	$0.797547\pi$
$614$	291618.i	0.773529i
$615$	0	0
$616$	107086.	0.282209
$617$	− 26.7317i	0 7.02193e-5i	−1.00000		3.51097e-5i	$-0.999989\pi$
$617$	− 26.7317i	0 7.02193e-5i	1.00000		3.51097e-5i	$-1.11758e-5\pi$
$618$	0	0
$619$	115824.	0.302285	0.151143	−	0.988512i	$-0.451705\pi$
$619$	115824.	0.302285	0.151143	+	0.988512i	$0.451705\pi$
$620$	30458.0i	0.0792353i
$621$	0	0
$622$	−251152.	−0.649167
$623$	527608.i	1.35936i
$624$	0	0
$625$	15625.0	0.0400000
$626$	195754.i	0.499531i
$627$	0	0
$628$	339957.	0.861996
$629$	26697.3i	0.0674786i
$630$	0	0
$631$	−284115.	−0.713568	−0.356784	−	0.934187i	$-0.616127\pi$
$631$	−284115.	−0.713568	−0.356784	+	0.934187i	$0.616127\pi$
$632$	− 190528.i	− 0.477007i
$633$	0	0
$634$	358693.	0.892368
$635$	− 155331.i	− 0.385223i
$636$	0	0
$637$	−2183.88	−0.00538208
$638$	− 306750.i	− 0.753603i
$639$	0	0
$640$	16190.9	0.0395285
$641$	− 87886.0i	− 0.213896i	−0.994265	−	0.106948i	$-0.965892\pi$
$641$	− 87886.0i	− 0.213896i	0.994265	−	0.106948i	$-0.0341079\pi$
$642$	0	0
$643$	−70340.7	−0.170132	−0.0850658	−	0.996375i	$-0.527110\pi$
$643$	−70340.7	−0.170132	−0.0850658	+	0.996375i	$0.527110\pi$
$644$	123252.i	0.297181i
$645$	0	0
$646$	−251777.	−0.603325
$647$	− 251015.i	− 0.599640i	−0.953996	−	0.299820i	$-0.903073\pi$
$647$	− 251015.i	− 0.599640i	0.953996	−	0.299820i	$-0.0969266\pi$
$648$	0	0
$649$	−12582.7	−0.0298735
$650$	− 40298.9i	− 0.0953821i
$651$	0	0
$652$	102944.	0.242162
$653$	23838.5i	0.0559052i	0.999609	+	0.0279526i	$0.00889874\pi$
$653$	23838.5i	0.0559052i	−0.999609	+	0.0279526i	$0.991101\pi$
$654$	0	0
$655$	36885.2	0.0859746
$656$	− 73255.0i	− 0.170228i
$657$	0	0
$658$	−495117.	−1.14355
$659$	− 374084.i	− 0.861386i	−0.902498	−	0.430693i	$-0.858269\pi$
$659$	− 374084.i	− 0.861386i	0.902498	−	0.430693i	$-0.141731\pi$
$660$	0	0
$661$	829840.	1.89929	0.949645	−	0.313327i	$-0.101443\pi$
$661$	829840.	1.89929	0.949645	+	0.313327i	$0.101443\pi$
$662$	− 216615.i	− 0.494278i
$663$	0	0
$664$	286900.	0.650721
$665$	139310.i	0.315020i
$666$	0	0
$667$	353057.	0.793584
$668$	− 265626.i	− 0.595275i
$669$	0	0
$670$	−84435.5	−0.188094
$671$	541563.i	1.20283i
$672$	0	0
$673$	−476337.	−1.05168	−0.525841	−	0.850583i	$-0.676249\pi$
$673$	−476337.	−1.05168	−0.525841	+	0.850583i	$0.676249\pi$
$674$	400870.i	0.882436i
$675$	0	0
$676$	124552.	0.272556
$677$	151782.i	0.331164i	0.986196	+	0.165582i	$0.0529503\pi$
$677$	151782.i	0.331164i	−0.986196	+	0.165582i	$0.947050\pi$
$678$	0	0
$679$	576728.	1.25092
$680$	88911.3i	0.192282i
$681$	0	0
$682$	−92656.9	−0.199209
$683$	711647.i	1.52554i	0.646670	+	0.762770i	$0.276161\pi$
$683$	711647.i	1.52554i	−0.646670	+	0.762770i	$0.723839\pi$
$684$	0	0
$685$	273626.	0.583145
$686$	331421.i	0.704257i
$687$	0	0
$688$	−165825.	−0.350327
$689$	− 333567.i	− 0.702659i
$690$	0	0
$691$	−348157.	−0.729153	−0.364577	−	0.931173i	$-0.618786\pi$
$691$	−348157.	−0.729153	−0.364577	+	0.931173i	$0.618786\pi$
$692$	216316.i	0.451728i
$693$	0	0
$694$	335090.	0.695733
$695$	191587.i	0.396640i
$696$	0	0
$697$	402276.	0.828054
$698$	− 241904.i	− 0.496515i
$699$	0	0
$700$	49195.1	0.100398
$701$	702624.i	1.42984i	0.699207	+	0.714920i	$0.253537\pi$
$701$	702624.i	1.42984i	−0.699207	+	0.714920i	$0.746463\pi$
$702$	0	0
$703$	−19240.0	−0.0389309
$704$	49254.5i	0.0993803i
$705$	0	0
$706$	−424057.	−0.850776
$707$	599810.i	1.19998i
$708$	0	0
$709$	−855514.	−1.70190	−0.850951	−	0.525244i	$-0.823974\pi$
$709$	−855514.	−1.70190	−0.850951	+	0.525244i	$0.823974\pi$
$710$	− 72097.3i	− 0.143022i
$711$	0	0
$712$	−242674.	−0.478701
$713$	− 106644.i	− 0.209778i
$714$	0	0
$715$	122594.	0.239805
$716$	404438.i	0.788906i
$717$	0	0
$718$	−173308.	−0.336179
$719$	− 628845.i	− 1.21643i	−0.793774	−	0.608213i	$-0.791886\pi$
$719$	− 628845.i	− 1.21643i	0.793774	−	0.608213i	$-0.208114\pi$
$720$	0	0
$721$	−403283.	−0.775781
$722$	187154.i	0.359026i
$723$	0	0
$724$	35699.2	0.0681054
$725$	− 140920.i	− 0.268100i
$726$	0	0
$727$	−528399.	−0.999754	−0.499877	−	0.866096i	$-0.666621\pi$
$727$	−528399.	−0.999754	−0.499877	+	0.866096i	$0.666621\pi$
$728$	− 126881.i	− 0.239405i
$729$	0	0
$730$	254418.	0.477421
$731$	− 910620.i	− 1.70413i
$732$	0	0
$733$	−897234.	−1.66993	−0.834964	−	0.550304i	$-0.814512\pi$
$733$	−897234.	−1.66993	−0.834964	+	0.550304i	$0.814512\pi$
$734$	− 23059.2i	− 0.0428009i
$735$	0	0
$736$	−56690.0	−0.104653
$737$	− 256863.i	− 0.472896i
$738$	0	0
$739$	592410.	1.08476	0.542380	−	0.840133i	$-0.317523\pi$
$739$	592410.	1.08476	0.542380	+	0.840133i	$0.317523\pi$
$740$	6794.31i	0.0124074i
$741$	0	0
$742$	407203.	0.739611
$743$	409669.i	0.742088i	0.928615	+	0.371044i	$0.121000\pi$
$743$	409669.i	0.742088i	−0.928615	+	0.371044i	$0.879000\pi$
$744$	0	0
$745$	194909.	0.351171
$746$	634931.i	1.14090i
$747$	0	0
$748$	−270478.	−0.483425
$749$	1.04184e6i	1.85712i
$750$	0	0
$751$	878893.	1.55832	0.779159	−	0.626826i	$-0.215646\pi$
$751$	878893.	1.55832	0.779159	+	0.626826i	$0.215646\pi$
$752$	− 227730.i	− 0.402703i
$753$	0	0
$754$	−363452.	−0.639299
$755$	61096.4i	0.107182i
$756$	0	0
$757$	−497041.	−0.867362	−0.433681	−	0.901066i	$-0.642786\pi$
$757$	−497041.	−0.867362	−0.433681	+	0.901066i	$0.642786\pi$
$758$	57001.6i	0.0992084i
$759$	0	0
$760$	−64075.9	−0.110935
$761$	− 73857.1i	− 0.127533i	−0.997965	−	0.0637665i	$-0.979689\pi$
$761$	− 73857.1i	− 0.127533i	0.997965	−	0.0637665i	$-0.0203113\pi$
$762$	0	0
$763$	715109.	1.22835
$764$	− 38114.9i	− 0.0652992i
$765$	0	0
$766$	−179112.	−0.305258
$767$	14908.6i	0.0253424i
$768$	0	0
$769$	540707.	0.914343	0.457172	−	0.889379i	$-0.348863\pi$
$769$	540707.	0.914343	0.457172	+	0.889379i	$0.348863\pi$
$770$	149657.i	0.252416i
$771$	0	0
$772$	−496557.	−0.833172
$773$	− 23921.3i	− 0.0400337i	−0.999800	−	0.0200169i	$-0.993628\pi$
$773$	− 23921.3i	− 0.0400337i	0.999800	−	0.0200169i	$-0.00637199\pi$
$774$	0	0
$775$	−42566.4	−0.0708702
$776$	265267.i	0.440514i
$777$	0	0
$778$	−548901.	−0.906848
$779$	289909.i	0.477735i
$780$	0	0
$781$	219328.	0.359578
$782$	− 311310.i	− 0.509073i
$783$	0	0
$784$	1226.23	0.00199498
$785$	475105.i	0.770993i
$786$	0	0
$787$	236097.	0.381190	0.190595	−	0.981669i	$-0.438958\pi$
$787$	236097.	0.381190	0.190595	+	0.981669i	$0.438958\pi$
$788$	− 243767.i	− 0.392574i
$789$	0	0
$790$	266271.	0.426648
$791$	437389.i	0.699060i
$792$	0	0
$793$	641670.	1.02039
$794$	− 187080.i	− 0.296748i
$795$	0	0
$796$	−213095.	−0.336316
$797$	1.02477e6i	1.61329i	0.591039	+	0.806643i	$0.298718\pi$
$797$	1.02477e6i	1.61329i	−0.591039	+	0.806643i	$0.701282\pi$
$798$	0	0
$799$	1.25057e6	1.95891
$800$	22627.4i	0.0353553i
$801$	0	0
$802$	−821601.	−1.27736
$803$	773968.i	1.20031i
$804$	0	0
$805$	−172250.	−0.265807
$806$	109784.i	0.168994i
$807$	0	0
$808$	−275884.	−0.422575
$809$	− 880763.i	− 1.34574i	−0.739759	−	0.672871i	$-0.765061\pi$
$809$	− 880763.i	− 1.34574i	0.739759	−	0.672871i	$-0.234939\pi$
$810$	0	0
$811$	946858.	1.43960	0.719802	−	0.694179i	$-0.244232\pi$
$811$	946858.	1.43960	0.719802	+	0.694179i	$0.244232\pi$
$812$	− 443685.i	− 0.672919i
$813$	0	0
$814$	−20669.1	−0.0311941
$815$	143868.i	0.216596i
$816$	0	0
$817$	656258.	0.983175
$818$	632220.i	0.944847i
$819$	0	0
$820$	102377.	0.152256
$821$	− 890064.i	− 1.32049i	−0.751050	−	0.660245i	$-0.770452\pi$
$821$	− 890064.i	− 1.32049i	0.751050	−	0.660245i	$-0.229548\pi$
$822$	0	0
$823$	−543081.	−0.801798	−0.400899	−	0.916122i	$-0.631302\pi$
$823$	−543081.	−0.801798	−0.400899	+	0.916122i	$0.631302\pi$
$824$	− 185491.i	− 0.273192i
$825$	0	0
$826$	−18199.8	−0.0266751
$827$	− 177655.i	− 0.259756i	−0.991530	−	0.129878i	$-0.958541\pi$
$827$	− 177655.i	− 0.259756i	0.991530	−	0.129878i	$-0.0414586\pi$
$828$	0	0
$829$	743143.	1.08134	0.540672	−	0.841234i	$-0.318170\pi$
$829$	743143.	1.08134	0.540672	+	0.841234i	$0.318170\pi$
$830$	400955.i	0.582022i
$831$	0	0
$832$	58359.1	0.0843067
$833$	6733.75i	0.00970436i
$834$	0	0
$835$	371223.	0.532430
$836$	− 194926.i	− 0.278906i
$837$	0	0
$838$	316763.	0.451073
$839$	− 207462.i	− 0.294724i	−0.989083	−	0.147362i	$-0.952922\pi$
$839$	− 207462.i	− 0.294724i	0.989083	−	0.147362i	$-0.0470782\pi$
$840$	0	0
$841$	−563663.	−0.796943
$842$	849373.i	1.19805i
$843$	0	0
$844$	629416.	0.883594
$845$	174066.i	0.243782i
$846$	0	0
$847$	264991.	0.369372
$848$	187294.i	0.260455i
$849$	0	0
$850$	−124257.	−0.171982
$851$	− 23789.3i	− 0.0328491i
$852$	0	0
$853$	327894.	0.450646	0.225323	−	0.974284i	$-0.427656\pi$
$853$	327894.	0.450646	0.225323	+	0.974284i	$0.427656\pi$
$854$	783322.i	1.07405i
$855$	0	0
$856$	−479199.	−0.653986
$857$	871170.i	1.18615i	0.805145	+	0.593077i	$0.202087\pi$
$857$	871170.i	1.18615i	−0.805145	+	0.593077i	$0.797913\pi$
$858$	0	0
$859$	−339809.	−0.460520	−0.230260	−	0.973129i	$-0.573958\pi$
$859$	−339809.	−0.460520	−0.230260	+	0.973129i	$0.573958\pi$
$860$	− 231748.i	− 0.313342i
$861$	0	0
$862$	−865547.	−1.16487
$863$	− 379236.i	− 0.509199i	−0.967047	−	0.254600i	$-0.918056\pi$
$863$	− 379236.i	− 0.509199i	0.967047	−	0.254600i	$-0.0819437\pi$
$864$	0	0
$865$	−302311.	−0.404038
$866$	− 715816.i	− 0.954478i
$867$	0	0
$868$	−134020.	−0.177881
$869$	810027.i	1.07265i
$870$	0	0
$871$	−304343.	−0.401169
$872$	328916.i	0.432566i
$873$	0	0
$874$	224353.	0.293703
$875$	68752.3i	0.0897989i
$876$	0	0
$877$	95986.1	0.124798	0.0623992	−	0.998051i	$-0.480125\pi$
$877$	95986.1	0.124798	0.0623992	+	0.998051i	$0.480125\pi$
$878$	− 761162.i	− 0.987388i
$879$	0	0
$880$	−68835.2	−0.0888885
$881$	− 97012.4i	− 0.124990i	−0.998045	−	0.0624950i	$-0.980094\pi$
$881$	− 97012.4i	− 0.124990i	0.998045	−	0.0624950i	$-0.0199058\pi$
$882$	0	0
$883$	−845225.	−1.08405	−0.542027	−	0.840361i	$-0.682343\pi$
$883$	−845225.	−1.08405	−0.542027	+	0.840361i	$0.682343\pi$
$884$	320476.i	0.410101i
$885$	0	0
$886$	−573239.	−0.730245
$887$	− 744606.i	− 0.946409i	−0.880953	−	0.473205i	$-0.843097\pi$
$887$	− 744606.i	− 0.946409i	0.880953	−	0.473205i	$-0.156903\pi$
$888$	0	0
$889$	683481.	0.864815
$890$	− 339148.i	− 0.428163i
$891$	0	0
$892$	686461.	0.862753
$893$	901251.i	1.13017i
$894$	0	0
$895$	−565219.	−0.705619
$896$	71242.1i	0.0887403i
$897$	0	0
$898$	−217517.	−0.269737
$899$	383902.i	0.475008i
$900$	0	0
$901$	−1.02852e6	−1.26696
$902$	311443.i	0.382794i
$903$	0	0
$904$	−201178.	−0.246175
$905$	49891.2i	0.0609154i
$906$	0	0
$907$	736427.	0.895190	0.447595	−	0.894236i	$-0.352281\pi$
$907$	736427.	0.895190	0.447595	+	0.894236i	$0.352281\pi$
$908$	− 298018.i	− 0.361468i
$909$	0	0
$910$	177321.	0.214130
$911$	347020.i	0.418136i	0.977901	+	0.209068i	$0.0670430\pi$
$911$	347020.i	0.418136i	−0.977901	+	0.209068i	$0.932957\pi$
$912$	0	0
$913$	−1.21975e6	−1.46329
$914$	625076.i	0.748239i
$915$	0	0
$916$	−171477.	−0.204369
$917$	162300.i	0.193010i
$918$	0	0
$919$	1.51291e6	1.79136	0.895678	−	0.444703i	$-0.146691\pi$
$919$	1.51291e6	1.79136	0.895678	+	0.444703i	$0.146691\pi$
$920$	− 79226.7i	− 0.0936043i
$921$	0	0
$922$	−122740.	−0.144386
$923$	− 259871.i	− 0.305038i
$924$	0	0
$925$	−9495.34	−0.0110976
$926$	− 219279.i	− 0.255726i
$927$	0	0
$928$	204074.	0.236969
$929$	1.31258e6i	1.52088i	0.649408	+	0.760440i	$0.275017\pi$
$929$	1.31258e6i	1.52088i	−0.649408	+	0.760440i	$0.724983\pi$
$930$	0	0
$931$	−4852.83	−0.00559881
$932$	− 254003.i	− 0.292420i
$933$	0	0
$934$	−408980.	−0.468822
$935$	− 378005.i	− 0.432389i
$936$	0	0
$937$	−837419.	−0.953814	−0.476907	−	0.878954i	$-0.658242\pi$
$937$	−837419.	−0.953814	−0.476907	+	0.878954i	$0.658242\pi$
$938$	− 371528.i	− 0.422266i
$939$	0	0
$940$	318263.	0.360189
$941$	39183.5i	0.0442511i	0.999755	+	0.0221256i	$0.00704336\pi$
$941$	39183.5i	0.0442511i	−0.999755	+	0.0221256i	$0.992957\pi$
$942$	0	0
$943$	−358459.	−0.403103
$944$	− 8371.04i	− 0.00939367i
$945$	0	0
$946$	705003.	0.787787
$947$	− 940742.i	− 1.04899i	−0.851414	−	0.524494i	$-0.824254\pi$
$947$	− 940742.i	− 1.04899i	0.851414	−	0.524494i	$-0.175746\pi$
$948$	0	0
$949$	917036.	1.01825
$950$	− 89548.8i	− 0.0992230i
$951$	0	0
$952$	−391222.	−0.431668
$953$	− 984970.i	− 1.08452i	−0.840211	−	0.542259i	$-0.817569\pi$
$953$	− 984970.i	− 1.08452i	0.840211	−	0.542259i	$-0.182431\pi$
$954$	0	0
$955$	53267.2	0.0584054
$956$	− 493429.i	− 0.539894i
$957$	0	0
$958$	−385041.	−0.419542
$959$	1.20400e6i	1.30915i
$960$	0	0
$961$	−807560.	−0.874435
$962$	24489.7i	0.0264627i
$963$	0	0
$964$	−560382.	−0.603018
$965$	− 693959.i	− 0.745211i
$966$	0	0
$967$	−1.70961e6	−1.82829	−0.914143	−	0.405392i	$-0.867135\pi$
$967$	−1.70961e6	−1.82829	−0.914143	+	0.405392i	$0.867135\pi$
$968$	121883.i	0.130075i
$969$	0	0
$970$	−370722.	−0.394008
$971$	− 717106.i	− 0.760580i	−0.924867	−	0.380290i	$-0.875824\pi$
$971$	− 717106.i	− 0.760580i	0.924867	−	0.380290i	$-0.124176\pi$
$972$	0	0
$973$	−843010.	−0.890445
$974$	120798.i	0.127333i
$975$	0	0
$976$	−360291.	−0.378228
$977$	− 289225.i	− 0.303003i	−0.988457	−	0.151501i	$-0.951589\pi$
$977$	− 289225.i	− 0.303003i	0.988457	−	0.151501i	$-0.0484108\pi$
$978$	0	0
$979$	1.03173e6	1.07646
$980$	1713.70i	0.00178436i
$981$	0	0
$982$	437149.	0.453322
$983$	1.62065e6i	1.67718i	0.544760	+	0.838592i	$0.316621\pi$
$983$	1.62065e6i	1.67718i	−0.544760	+	0.838592i	$0.683379\pi$
$984$	0	0
$985$	340674.	0.351129
$986$	1.12066e6i	1.15271i
$987$	0	0
$988$	−230958.	−0.236603
$989$	811431.i	0.829582i
$990$	0	0
$991$	−722454.	−0.735636	−0.367818	−	0.929898i	$-0.619895\pi$
$991$	−722454.	−0.735636	−0.367818	+	0.929898i	$0.619895\pi$
$992$	− 61642.7i	− 0.0626410i
$993$	0	0
$994$	317239.	0.321080
$995$	− 297810.i	− 0.300810i
$996$	0	0
$997$	−602932.	−0.606566	−0.303283	−	0.952900i	$-0.598083\pi$
$997$	−602932.	−0.606566	−0.303283	+	0.952900i	$0.598083\pi$
$998$	− 146680.i	− 0.147268i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	810.5.d.c.161.7		32
3.2	odd	2	inner	810.5.d.c.161.8		32
9.2	odd	6		90.5.h.a.41.12	yes	32
9.4	even	3		90.5.h.a.11.12	✓	32
9.5	odd	6		270.5.h.a.251.3		32
9.7	even	3		270.5.h.a.71.3		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
90.5.h.a.11.12	✓	32	9.4	even	3
90.5.h.a.41.12	yes	32	9.2	odd	6
270.5.h.a.71.3		32	9.7	even	3
270.5.h.a.251.3		32	9.5	odd	6
810.5.d.c.161.7		32	1.1	even	1	trivial
810.5.d.c.161.8		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)

\(f(q)\)	\(=\)	\(q-2.82843i q^{2} -8.00000 q^{4} -11.1803i q^{5} +49.1951 q^{7} +22.6274i q^{8} +O(q^{10})\)	\(q-2.82843i q^{2} -8.00000 q^{4} -11.1803i q^{5} +49.1951 q^{7} +22.6274i q^{8} -31.6228 q^{10} -96.2002i q^{11} -113.983 q^{13} -139.145i q^{14} +64.0000 q^{16} +351.453i q^{17} -253.282 q^{19} +89.4427i q^{20} -272.095 q^{22} -313.171i q^{23} -125.000 q^{25} +322.392i q^{26} -393.561 q^{28} +1127.36i q^{29} +340.531 q^{31} -181.019i q^{32} +994.058 q^{34} -550.018i q^{35} +75.9627 q^{37} +716.390i q^{38} +252.982 q^{40} -1144.61i q^{41} -2591.02 q^{43} +769.601i q^{44} -885.781 q^{46} -3558.29i q^{47} +19.1598 q^{49} +353.553i q^{50} +911.861 q^{52} +2926.47i q^{53} -1075.55 q^{55} +1113.16i q^{56} +3188.66 q^{58} -130.797i q^{59} -5629.54 q^{61} -963.168i q^{62} -512.000 q^{64} +1274.36i q^{65} +2670.08 q^{67} -2811.62i q^{68} -1555.69 q^{70} +2279.92i q^{71} -8045.40 q^{73} -214.855i q^{74} +2026.26 q^{76} -4732.58i q^{77} -8420.23 q^{79} -715.542i q^{80} -3237.45 q^{82} -12679.3i q^{83} +3929.36 q^{85} +7328.50i q^{86} +2176.76 q^{88} +10724.8i q^{89} -5607.39 q^{91} +2505.37i q^{92} -10064.4 q^{94} +2831.78i q^{95} +11723.3 q^{97} -54.1920i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 256 q^{4} - 104 q^{7}+O(q^{10}) \) 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

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Database

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