LMFDB - Embedded newform 810.3.d.b.161.5

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:	$8$
Coefficient field:	8.0.3317760000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 8x^{6} + 13x^{4} + 12x^{2} + 36 $ x^8 - 8x^6 + 13x^4 + 12*x^2 + 36
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4}\cdot 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			161.5
Root			$-2.15988 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	810.161
Dual form			810.3.d.b.161.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421i q^{2} -2.00000 q^{4} -2.23607i q^{5} -2.45930 q^{7} -2.82843i q^{8} +O(q^{10})$	\(q+1.41421i q^{2} -2.00000 q^{4} -2.23607i q^{5} -2.45930 q^{7} -2.82843i q^{8} +3.16228 q^{10} -21.2757i q^{11} +7.81697 q^{13} -3.47797i q^{14} +4.00000 q^{16} +26.2143i q^{17} -36.1829 q^{19} +4.47214i q^{20} +30.0884 q^{22} +36.2450i q^{23} -5.00000 q^{25} +11.0549i q^{26} +4.91860 q^{28} +40.1949i q^{29} -10.6208 q^{31} +5.65685i q^{32} -37.0726 q^{34} +5.49916i q^{35} +0.543363 q^{37} -51.1704i q^{38} -6.32456 q^{40} -10.1819i q^{41} -13.2474 q^{43} +42.5514i q^{44} -51.2582 q^{46} -57.0347i q^{47} -42.9519 q^{49} -7.07107i q^{50} -15.6339 q^{52} +73.2739i q^{53} -47.5740 q^{55} +6.95594i q^{56} -56.8442 q^{58} -23.2311i q^{59} +30.6988 q^{61} -15.0201i q^{62} -8.00000 q^{64} -17.4793i q^{65} -71.0750 q^{67} -52.4286i q^{68} -7.77698 q^{70} +59.6959i q^{71} -38.5521 q^{73} +0.768431i q^{74} +72.3659 q^{76} +52.3233i q^{77} -25.5763 q^{79} -8.94427i q^{80} +14.3994 q^{82} +30.4861i q^{83} +58.6170 q^{85} -18.7346i q^{86} -60.1768 q^{88} +140.382i q^{89} -19.2243 q^{91} -72.4901i q^{92} +80.6592 q^{94} +80.9075i q^{95} -58.0756 q^{97} -60.7431i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 16 q^{4} - 8 q^{7}+O(q^{10}) $ 8 * q - 16 * q^4 - 8 * q^7	$ 8 q - 16 q^{4} - 8 q^{7} + 88 q^{13} + 32 q^{16} - 80 q^{19} + 24 q^{22} - 40 q^{25} + 16 q^{28} - 152 q^{31} + 24 q^{34} + 136 q^{37} - 32 q^{43} - 48 q^{49} - 176 q^{52} - 120 q^{55} - 168 q^{58} + 232 q^{61} - 64 q^{64} + 184 q^{67} + 120 q^{70} + 280 q^{73} + 160 q^{76} + 184 q^{79} - 24 q^{82} - 48 q^{88} + 32 q^{91} + 312 q^{94} - 512 q^{97}+O(q^{100}) $ 8 * q - 16 * q^4 - 8 * q^7 + 88 * q^13 + 32 * q^16 - 80 * q^19 + 24 * q^22 - 40 * q^25 + 16 * q^28 - 152 * q^31 + 24 * q^34 + 136 * q^37 - 32 * q^43 - 48 * q^49 - 176 * q^52 - 120 * q^55 - 168 * q^58 + 232 * q^61 - 64 * q^64 + 184 * q^67 + 120 * q^70 + 280 * q^73 + 160 * q^76 + 184 * q^79 - 24 * q^82 - 48 * q^88 + 32 * q^91 + 312 * q^94 - 512 * 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 $

$2$	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
$7$	−2.45930	−0.351328	−0.175664	−	0.984450i	$-0.556207\pi$
$7$	−2.45930	−0.351328	−0.175664	+	0.984450i	$0.556207\pi$
$8$	− 2.82843i	− 0.353553i
$9$	0	0
$10$	3.16228	0.316228
$11$	− 21.2757i	− 1.93416i	−0.254479	−	0.967078i	$-0.581904\pi$
$11$	− 21.2757i	− 1.93416i	0.254479	−	0.967078i	$-0.418096\pi$
$12$	0	0
$13$	7.81697	0.601306	0.300653	−	0.953734i	$-0.402795\pi$
$13$	7.81697	0.601306	0.300653	+	0.953734i	$0.402795\pi$
$14$	− 3.47797i	− 0.248427i
$15$	0	0
$16$	4.00000	0.250000
$17$	26.2143i	1.54202i	0.636824	+	0.771009i	$0.280248\pi$
$17$	26.2143i	1.54202i	−0.636824	+	0.771009i	$0.719752\pi$
$18$	0	0
$19$	−36.1829	−1.90437	−0.952183	−	0.305529i	$-0.901167\pi$
$19$	−36.1829	−1.90437	−0.952183	+	0.305529i	$0.901167\pi$
$20$	4.47214i	0.223607i
$21$	0	0
$22$	30.0884	1.36766
$23$	36.2450i	1.57587i	0.615757	+	0.787936i	$0.288850\pi$
$23$	36.2450i	1.57587i	−0.615757	+	0.787936i	$0.711150\pi$
$24$	0	0
$25$	−5.00000	−0.200000
$26$	11.0549i	0.425187i
$27$	0	0
$28$	4.91860	0.175664
$29$	40.1949i	1.38603i	0.720922	+	0.693016i	$0.243718\pi$
$29$	40.1949i	1.38603i	−0.720922	+	0.693016i	$0.756282\pi$
$30$	0	0
$31$	−10.6208	−0.342607	−0.171304	−	0.985218i	$-0.554798\pi$
$31$	−10.6208	−0.342607	−0.171304	+	0.985218i	$0.554798\pi$
$32$	5.65685i	0.176777i
$33$	0	0
$34$	−37.0726	−1.09037
$35$	5.49916i	0.157119i
$36$	0	0
$37$	0.543363	0.0146855	0.00734274	−	0.999973i	$-0.497663\pi$
$37$	0.543363	0.0146855	0.00734274	+	0.999973i	$0.497663\pi$
$38$	− 51.1704i	− 1.34659i
$39$	0	0
$40$	−6.32456	−0.158114
$41$	− 10.1819i	− 0.248339i	−0.992261	−	0.124170i	$-0.960373\pi$
$41$	− 10.1819i	− 0.248339i	0.992261	−	0.124170i	$-0.0396267\pi$
$42$	0	0
$43$	−13.2474	−0.308078	−0.154039	−	0.988065i	$-0.549228\pi$
$43$	−13.2474	−0.308078	−0.154039	+	0.988065i	$0.549228\pi$
$44$	42.5514i	0.967078i
$45$	0	0
$46$	−51.2582	−1.11431
$47$	− 57.0347i	− 1.21350i	−0.794891	−	0.606752i	$-0.792472\pi$
$47$	− 57.0347i	− 1.21350i	0.794891	−	0.606752i	$-0.207528\pi$
$48$	0	0
$49$	−42.9519	−0.876568
$50$	− 7.07107i	− 0.141421i
$51$	0	0
$52$	−15.6339	−0.300653
$53$	73.2739i	1.38253i	0.722603	+	0.691264i	$0.242946\pi$
$53$	73.2739i	1.38253i	−0.722603	+	0.691264i	$0.757054\pi$
$54$	0	0
$55$	−47.5740	−0.864981
$56$	6.95594i	0.124213i
$57$	0	0
$58$	−56.8442	−0.980072
$59$	− 23.2311i	− 0.393747i	−0.980429	−	0.196873i	$-0.936921\pi$
$59$	− 23.2311i	− 0.393747i	0.980429	−	0.196873i	$-0.0630788\pi$
$60$	0	0
$61$	30.6988	0.503259	0.251629	−	0.967824i	$-0.419034\pi$
$61$	30.6988	0.503259	0.251629	+	0.967824i	$0.419034\pi$
$62$	− 15.0201i	− 0.242260i
$63$	0	0
$64$	−8.00000	−0.125000
$65$	− 17.4793i	− 0.268912i
$66$	0	0
$67$	−71.0750	−1.06082	−0.530411	−	0.847741i	$-0.677962\pi$
$67$	−71.0750	−1.06082	−0.530411	+	0.847741i	$0.677962\pi$
$68$	− 52.4286i	− 0.771009i
$69$	0	0
$70$	−7.77698	−0.111100
$71$	59.6959i	0.840787i	0.907342	+	0.420394i	$0.138108\pi$
$71$	59.6959i	0.840787i	−0.907342	+	0.420394i	$0.861892\pi$
$72$	0	0
$73$	−38.5521	−0.528110	−0.264055	−	0.964508i	$-0.585060\pi$
$73$	−38.5521	−0.528110	−0.264055	+	0.964508i	$0.585060\pi$
$74$	0.768431i	0.0103842i
$75$	0	0
$76$	72.3659	0.952183
$77$	52.3233i	0.679524i
$78$	0	0
$79$	−25.5763	−0.323751	−0.161875	−	0.986811i	$-0.551754\pi$
$79$	−25.5763	−0.323751	−0.161875	+	0.986811i	$0.551754\pi$
$80$	− 8.94427i	− 0.111803i
$81$	0	0
$82$	14.3994	0.175602
$83$	30.4861i	0.367302i	0.982991	+	0.183651i	$0.0587917\pi$
$83$	30.4861i	0.367302i	−0.982991	+	0.183651i	$0.941208\pi$
$84$	0	0
$85$	58.6170	0.689611
$86$	− 18.7346i	− 0.217844i
$87$	0	0
$88$	−60.1768	−0.683828
$89$	140.382i	1.57733i	0.614825	+	0.788664i	$0.289227\pi$
$89$	140.382i	1.57733i	−0.614825	+	0.788664i	$0.710773\pi$
$90$	0	0
$91$	−19.2243	−0.211256
$92$	− 72.4901i	− 0.787936i
$93$	0	0
$94$	80.6592	0.858077
$95$	80.9075i	0.851658i
$96$	0	0
$97$	−58.0756	−0.598717	−0.299359	−	0.954141i	$-0.596773\pi$
$97$	−58.0756	−0.598717	−0.299359	+	0.954141i	$0.596773\pi$
$98$	− 60.7431i	− 0.619827i
$99$	0	0
$100$	10.0000	0.100000
$101$	− 91.3875i	− 0.904827i	−0.891808	−	0.452414i	$-0.850563\pi$
$101$	− 91.3875i	− 0.904827i	0.891808	−	0.452414i	$-0.149437\pi$
$102$	0	0
$103$	−152.430	−1.47990	−0.739951	−	0.672661i	$-0.765151\pi$
$103$	−152.430	−1.47990	−0.739951	+	0.672661i	$0.765151\pi$
$104$	− 22.1097i	− 0.212594i
$105$	0	0
$106$	−103.625	−0.977594
$107$	85.6101i	0.800094i	0.916495	+	0.400047i	$0.131006\pi$
$107$	85.6101i	0.800094i	−0.916495	+	0.400047i	$0.868994\pi$
$108$	0	0
$109$	−67.6745	−0.620867	−0.310433	−	0.950595i	$-0.600474\pi$
$109$	−67.6745	−0.620867	−0.310433	+	0.950595i	$0.600474\pi$
$110$	− 67.2797i	− 0.611634i
$111$	0	0
$112$	−9.83719	−0.0878321
$113$	− 193.012i	− 1.70807i	−0.520217	−	0.854034i	$-0.674149\pi$
$113$	− 193.012i	− 1.70807i	0.520217	−	0.854034i	$-0.325851\pi$
$114$	0	0
$115$	81.0464	0.704751
$116$	− 80.3898i	− 0.693016i
$117$	0	0
$118$	32.8537	0.278421
$119$	− 64.4688i	− 0.541755i
$120$	0	0
$121$	−331.656	−2.74096
$122$	43.4146i	0.355858i
$123$	0	0
$124$	21.2416	0.171304
$125$	11.1803i	0.0894427i
$126$	0	0
$127$	163.277	1.28565	0.642823	−	0.766015i	$-0.277763\pi$
$127$	163.277	1.28565	0.642823	+	0.766015i	$0.277763\pi$
$128$	− 11.3137i	− 0.0883883i
$129$	0	0
$130$	24.7194	0.190149
$131$	− 132.081i	− 1.00825i	−0.863630	−	0.504126i	$-0.831815\pi$
$131$	− 132.081i	− 1.00825i	0.863630	−	0.504126i	$-0.168185\pi$
$132$	0	0
$133$	88.9846	0.669057
$134$	− 100.515i	− 0.750114i
$135$	0	0
$136$	74.1453	0.545186
$137$	− 60.6956i	− 0.443034i	−0.975157	−	0.221517i	$-0.928899\pi$
$137$	− 60.6956i	− 0.443034i	0.975157	−	0.221517i	$-0.0711008\pi$
$138$	0	0
$139$	67.4480	0.485237	0.242619	−	0.970122i	$-0.421994\pi$
$139$	67.4480	0.485237	0.242619	+	0.970122i	$0.421994\pi$
$140$	− 10.9983i	− 0.0785594i
$141$	0	0
$142$	−84.4227	−0.594526
$143$	− 166.312i	− 1.16302i
$144$	0	0
$145$	89.8786	0.619852
$146$	− 54.5208i	− 0.373430i
$147$	0	0
$148$	−1.08673	−0.00734274
$149$	273.791i	1.83752i	0.394811	+	0.918762i	$0.370810\pi$
$149$	273.791i	1.83752i	−0.394811	+	0.918762i	$0.629190\pi$
$150$	0	0
$151$	86.8563	0.575207	0.287604	−	0.957750i	$-0.407141\pi$
$151$	86.8563	0.575207	0.287604	+	0.957750i	$0.407141\pi$
$152$	102.341i	0.673295i
$153$	0	0
$154$	−73.9964	−0.480496
$155$	23.7489i	0.153219i
$156$	0	0
$157$	140.952	0.897783	0.448891	−	0.893586i	$-0.351819\pi$
$157$	140.952	0.897783	0.448891	+	0.893586i	$0.351819\pi$
$158$	− 36.1704i	− 0.228926i
$159$	0	0
$160$	12.6491	0.0790569
$161$	− 89.1374i	− 0.553648i
$162$	0	0
$163$	144.246	0.884943	0.442471	−	0.896783i	$-0.354102\pi$
$163$	144.246	0.884943	0.442471	+	0.896783i	$0.354102\pi$
$164$	20.3638i	0.124170i
$165$	0	0
$166$	−43.1139	−0.259722
$167$	176.263i	1.05547i	0.849410	+	0.527734i	$0.176958\pi$
$167$	176.263i	1.05547i	−0.849410	+	0.527734i	$0.823042\pi$
$168$	0	0
$169$	−107.895	−0.638432
$170$	82.8969i	0.487629i
$171$	0	0
$172$	26.4947	0.154039
$173$	34.9133i	0.201811i	0.994896	+	0.100905i	$0.0321740\pi$
$173$	34.9133i	0.201811i	−0.994896	+	0.100905i	$0.967826\pi$
$174$	0	0
$175$	12.2965	0.0702657
$176$	− 85.1029i	− 0.483539i
$177$	0	0
$178$	−198.530	−1.11534
$179$	45.3633i	0.253426i	0.991939	+	0.126713i	$0.0404427\pi$
$179$	45.3633i	0.253426i	−0.991939	+	0.126713i	$0.959557\pi$
$180$	0	0
$181$	−7.95905	−0.0439727	−0.0219863	−	0.999758i	$-0.506999\pi$
$181$	−7.95905	−0.0439727	−0.0219863	+	0.999758i	$0.506999\pi$
$182$	− 27.1872i	− 0.149380i
$183$	0	0
$184$	102.516	0.557155
$185$	− 1.21500i	− 0.00656754i
$186$	0	0
$187$	557.728	2.98250
$188$	114.069i	0.606752i
$189$	0	0
$190$	−114.421	−0.602213
$191$	112.541i	0.589222i	0.955617	+	0.294611i	$0.0951901\pi$
$191$	112.541i	0.589222i	−0.955617	+	0.294611i	$0.904810\pi$
$192$	0	0
$193$	−280.991	−1.45591	−0.727955	−	0.685625i	$-0.759529\pi$
$193$	−280.991	−1.45591	−0.727955	+	0.685625i	$0.759529\pi$
$194$	− 82.1313i	− 0.423357i
$195$	0	0
$196$	85.9037	0.438284
$197$	− 106.289i	− 0.539540i	−0.962925	−	0.269770i	$-0.913052\pi$
$197$	− 106.289i	− 0.539540i	0.962925	−	0.269770i	$-0.0869476\pi$
$198$	0	0
$199$	−335.479	−1.68583	−0.842913	−	0.538050i	$-0.819161\pi$
$199$	−335.479	−1.68583	−0.842913	+	0.538050i	$0.819161\pi$
$200$	14.1421i	0.0707107i
$201$	0	0
$202$	129.241	0.639809
$203$	− 98.8513i	− 0.486952i
$204$	0	0
$205$	−22.7675	−0.111061
$206$	− 215.568i	− 1.04645i
$207$	0	0
$208$	31.2679	0.150326
$209$	769.818i	3.68334i
$210$	0	0
$211$	277.461	1.31498	0.657490	−	0.753463i	$-0.271618\pi$
$211$	277.461	1.31498	0.657490	+	0.753463i	$0.271618\pi$
$212$	− 146.548i	− 0.691264i
$213$	0	0
$214$	−121.071	−0.565752
$215$	29.6220i	0.137777i
$216$	0	0
$217$	26.1198	0.120368
$218$	− 95.7062i	− 0.439019i
$219$	0	0
$220$	95.1479	0.432491
$221$	204.916i	0.927224i
$222$	0	0
$223$	135.390	0.607128	0.303564	−	0.952811i	$-0.401823\pi$
$223$	135.390	0.607128	0.303564	+	0.952811i	$0.401823\pi$
$224$	− 13.9119i	− 0.0621067i
$225$	0	0
$226$	272.960	1.20779
$227$	− 73.8325i	− 0.325253i	−0.986688	−	0.162627i	$-0.948003\pi$
$227$	− 73.8325i	− 0.325253i	0.986688	−	0.162627i	$-0.0519966\pi$
$228$	0	0
$229$	155.095	0.677270	0.338635	−	0.940918i	$-0.390035\pi$
$229$	155.095	0.677270	0.338635	+	0.940918i	$0.390035\pi$
$230$	114.617i	0.498334i
$231$	0	0
$232$	113.688	0.490036
$233$	− 252.085i	− 1.08191i	−0.841051	−	0.540955i	$-0.818063\pi$
$233$	− 252.085i	− 1.08191i	0.841051	−	0.540955i	$-0.181937\pi$
$234$	0	0
$235$	−127.533	−0.542695
$236$	46.4621i	0.196873i
$237$	0	0
$238$	91.1726	0.383078
$239$	− 324.896i	− 1.35940i	−0.733492	−	0.679698i	$-0.762111\pi$
$239$	− 324.896i	− 1.35940i	0.733492	−	0.679698i	$-0.237889\pi$
$240$	0	0
$241$	−132.451	−0.549590	−0.274795	−	0.961503i	$-0.588610\pi$
$241$	−132.451	−0.549590	−0.274795	+	0.961503i	$0.588610\pi$
$242$	− 469.033i	− 1.93815i
$243$	0	0
$244$	−61.3975	−0.251629
$245$	96.0433i	0.392013i
$246$	0	0
$247$	−282.841	−1.14511
$248$	30.0402i	0.121130i
$249$	0	0
$250$	−15.8114	−0.0632456
$251$	80.8492i	0.322108i	0.986946	+	0.161054i	$0.0514894\pi$
$251$	80.8492i	0.322108i	−0.986946	+	0.161054i	$0.948511\pi$
$252$	0	0
$253$	771.139	3.04798
$254$	230.909i	0.909089i
$255$	0	0
$256$	16.0000	0.0625000
$257$	170.256i	0.662477i	0.943547	+	0.331238i	$0.107466\pi$
$257$	170.256i	0.662477i	−0.943547	+	0.331238i	$0.892534\pi$
$258$	0	0
$259$	−1.33629	−0.00515942
$260$	34.9586i	0.134456i
$261$	0	0
$262$	186.791	0.712942
$263$	− 69.1449i	− 0.262908i	−0.991322	−	0.131454i	$-0.958035\pi$
$263$	− 69.1449i	− 0.262908i	0.991322	−	0.131454i	$-0.0419646\pi$
$264$	0	0
$265$	163.846	0.618285
$266$	125.843i	0.473095i
$267$	0	0
$268$	142.150	0.530411
$269$	54.7154i	0.203403i	0.994815	+	0.101702i	$0.0324287\pi$
$269$	54.7154i	0.203403i	−0.994815	+	0.101702i	$0.967571\pi$
$270$	0	0
$271$	−129.945	−0.479501	−0.239751	−	0.970834i	$-0.577066\pi$
$271$	−129.945	−0.479501	−0.239751	+	0.970834i	$0.577066\pi$
$272$	104.857i	0.385505i
$273$	0	0
$274$	85.8366	0.313272
$275$	106.379i	0.386831i
$276$	0	0
$277$	−367.857	−1.32801	−0.664003	−	0.747730i	$-0.731144\pi$
$277$	−367.857	−1.32801	−0.664003	+	0.747730i	$0.731144\pi$
$278$	95.3859i	0.343115i
$279$	0	0
$280$	15.5540	0.0555499
$281$	33.0465i	0.117603i	0.998270	+	0.0588016i	$0.0187279\pi$
$281$	33.0465i	0.117603i	−0.998270	+	0.0588016i	$0.981272\pi$
$282$	0	0
$283$	38.9174	0.137517	0.0687586	−	0.997633i	$-0.478096\pi$
$283$	38.9174	0.137517	0.0687586	+	0.997633i	$0.478096\pi$
$284$	− 119.392i	− 0.420394i
$285$	0	0
$286$	235.200	0.822379
$287$	25.0404i	0.0872486i
$288$	0	0
$289$	−398.190	−1.37782
$290$	127.107i	0.438302i
$291$	0	0
$292$	77.1041	0.264055
$293$	171.454i	0.585168i	0.956240	+	0.292584i	$0.0945150\pi$
$293$	171.454i	0.585168i	−0.956240	+	0.292584i	$0.905485\pi$
$294$	0	0
$295$	−51.9462	−0.176089
$296$	− 1.53686i	− 0.00519210i
$297$	0	0
$298$	−387.199	−1.29933
$299$	283.326i	0.947580i
$300$	0	0
$301$	32.5792	0.108237
$302$	122.833i	0.406733i
$303$	0	0
$304$	−144.732	−0.476091
$305$	− 68.6445i	− 0.225064i
$306$	0	0
$307$	−299.803	−0.976558	−0.488279	−	0.872688i	$-0.662375\pi$
$307$	−299.803	−0.976558	−0.488279	+	0.872688i	$0.662375\pi$
$308$	− 104.647i	− 0.339762i
$309$	0	0
$310$	−33.5860	−0.108342
$311$	− 58.9911i	− 0.189682i	−0.995492	−	0.0948410i	$-0.969766\pi$
$311$	− 58.9911i	− 0.189682i	0.995492	−	0.0948410i	$-0.0302343\pi$
$312$	0	0
$313$	302.553	0.966622	0.483311	−	0.875449i	$-0.339434\pi$
$313$	302.553	0.966622	0.483311	+	0.875449i	$0.339434\pi$
$314$	199.336i	0.634828i
$315$	0	0
$316$	51.1526	0.161875
$317$	− 204.288i	− 0.644441i	−0.946665	−	0.322220i	$-0.895571\pi$
$317$	− 204.288i	− 0.644441i	0.946665	−	0.322220i	$-0.104429\pi$
$318$	0	0
$319$	855.176	2.68080
$320$	17.8885i	0.0559017i
$321$	0	0
$322$	126.059	0.391488
$323$	− 948.511i	− 2.93657i
$324$	0	0
$325$	−39.0849	−0.120261
$326$	203.994i	0.625749i
$327$	0	0
$328$	−28.7988	−0.0878012
$329$	140.265i	0.426338i
$330$	0	0
$331$	−93.9806	−0.283929	−0.141965	−	0.989872i	$-0.545342\pi$
$331$	−93.9806	−0.283929	−0.141965	+	0.989872i	$0.545342\pi$
$332$	− 60.9722i	− 0.183651i
$333$	0	0
$334$	−249.274	−0.746328
$335$	158.929i	0.474414i
$336$	0	0
$337$	576.177	1.70972	0.854862	−	0.518856i	$-0.173642\pi$
$337$	576.177	1.70972	0.854862	+	0.518856i	$0.173642\pi$
$338$	− 152.587i	− 0.451439i
$339$	0	0
$340$	−117.234	−0.344806
$341$	225.966i	0.662656i
$342$	0	0
$343$	226.137	0.659292
$344$	37.4692i	0.108922i
$345$	0	0
$346$	−49.3748	−0.142702
$347$	108.964i	0.314016i	0.987597	+	0.157008i	$0.0501848\pi$
$347$	108.964i	0.314016i	−0.987597	+	0.157008i	$0.949815\pi$
$348$	0	0
$349$	513.437	1.47117	0.735584	−	0.677434i	$-0.236908\pi$
$349$	513.437	1.47117	0.735584	+	0.677434i	$0.236908\pi$
$350$	17.3899i	0.0496853i
$351$	0	0
$352$	120.354	0.341914
$353$	375.870i	1.06479i	0.846497	+	0.532394i	$0.178707\pi$
$353$	375.870i	1.06479i	−0.846497	+	0.532394i	$0.821293\pi$
$354$	0	0
$355$	133.484	0.376011
$356$	− 280.764i	− 0.788664i
$357$	0	0
$358$	−64.1533	−0.179199
$359$	13.4600i	0.0374930i	0.999824	+	0.0187465i	$0.00596755\pi$
$359$	13.4600i	0.0374930i	−0.999824	+	0.0187465i	$0.994032\pi$
$360$	0	0
$361$	948.205	2.62661
$362$	− 11.2558i	− 0.0310934i
$363$	0	0
$364$	38.4485	0.105628
$365$	86.2050i	0.236178i
$366$	0	0
$367$	121.679	0.331552	0.165776	−	0.986163i	$-0.446987\pi$
$367$	121.679	0.331552	0.165776	+	0.986163i	$0.446987\pi$
$368$	144.980i	0.393968i
$369$	0	0
$370$	1.71826	0.00464395
$371$	− 180.202i	− 0.485721i
$372$	0	0
$373$	−449.280	−1.20450	−0.602252	−	0.798306i	$-0.705730\pi$
$373$	−449.280	−1.20450	−0.602252	+	0.798306i	$0.705730\pi$
$374$	788.747i	2.10895i
$375$	0	0
$376$	−161.318	−0.429038
$377$	314.203i	0.833429i
$378$	0	0
$379$	−458.988	−1.21105	−0.605525	−	0.795826i	$-0.707037\pi$
$379$	−458.988	−1.21105	−0.605525	+	0.795826i	$0.707037\pi$
$380$	− 161.815i	− 0.425829i
$381$	0	0
$382$	−159.158	−0.416643
$383$	− 650.092i	− 1.69737i	−0.528900	−	0.848684i	$-0.677395\pi$
$383$	− 650.092i	− 1.69737i	0.528900	−	0.848684i	$-0.322605\pi$
$384$	0	0
$385$	116.999	0.303892
$386$	− 397.381i	− 1.02948i
$387$	0	0
$388$	116.151	0.299359
$389$	− 301.276i	− 0.774489i	−0.921977	−	0.387244i	$-0.873427\pi$
$389$	− 301.276i	− 0.774489i	0.921977	−	0.387244i	$-0.126573\pi$
$390$	0	0
$391$	−950.139	−2.43002
$392$	121.486i	0.309914i
$393$	0	0
$394$	150.316	0.381512
$395$	57.1904i	0.144786i
$396$	0	0
$397$	324.620	0.817682	0.408841	−	0.912606i	$-0.365933\pi$
$397$	324.620	0.817682	0.408841	+	0.912606i	$0.365933\pi$
$398$	− 474.440i	− 1.19206i
$399$	0	0
$400$	−20.0000	−0.0500000
$401$	− 232.195i	− 0.579039i	−0.957172	−	0.289520i	$-0.906504\pi$
$401$	− 232.195i	− 0.579039i	0.957172	−	0.289520i	$-0.0934955\pi$
$402$	0	0
$403$	−83.0226	−0.206012
$404$	182.775i	0.452414i
$405$	0	0
$406$	139.797	0.344327
$407$	− 11.5604i	− 0.0284040i
$408$	0	0
$409$	174.664	0.427051	0.213526	−	0.976937i	$-0.431505\pi$
$409$	174.664	0.427051	0.213526	+	0.976937i	$0.431505\pi$
$410$	− 32.1980i	− 0.0785318i
$411$	0	0
$412$	304.860	0.739951
$413$	57.1321i	0.138334i
$414$	0	0
$415$	68.1690	0.164263
$416$	44.2195i	0.106297i
$417$	0	0
$418$	−1088.69	−2.60452
$419$	712.265i	1.69992i	0.526850	+	0.849958i	$0.323373\pi$
$419$	712.265i	1.69992i	−0.526850	+	0.849958i	$0.676627\pi$
$420$	0	0
$421$	−370.284	−0.879535	−0.439768	−	0.898112i	$-0.644939\pi$
$421$	−370.284	−0.879535	−0.439768	+	0.898112i	$0.644939\pi$
$422$	392.389i	0.929831i
$423$	0	0
$424$	207.250	0.488797
$425$	− 131.072i	− 0.308404i
$426$	0	0
$427$	−75.4974	−0.176809
$428$	− 171.220i	− 0.400047i
$429$	0	0
$430$	−41.8918	−0.0974229
$431$	− 339.199i	− 0.787004i	−0.919324	−	0.393502i	$-0.871263\pi$
$431$	− 339.199i	− 0.787004i	0.919324	−	0.393502i	$-0.128737\pi$
$432$	0	0
$433$	−163.324	−0.377191	−0.188596	−	0.982055i	$-0.560394\pi$
$433$	−163.324	−0.377191	−0.188596	+	0.982055i	$0.560394\pi$
$434$	36.9389i	0.0851127i
$435$	0	0
$436$	135.349	0.310433
$437$	− 1311.45i	− 3.00104i
$438$	0	0
$439$	−457.739	−1.04269	−0.521343	−	0.853347i	$-0.674569\pi$
$439$	−457.739	−1.04269	−0.521343	+	0.853347i	$0.674569\pi$
$440$	134.559i	0.305817i
$441$	0	0
$442$	−289.796	−0.655646
$443$	− 263.959i	− 0.595845i	−0.954590	−	0.297922i	$-0.903706\pi$
$443$	− 263.959i	− 0.595845i	0.954590	−	0.297922i	$-0.0962936\pi$
$444$	0	0
$445$	313.904	0.705402
$446$	191.470i	0.429305i
$447$	0	0
$448$	19.6744	0.0439160
$449$	502.704i	1.11961i	0.828625	+	0.559804i	$0.189123\pi$
$449$	502.704i	1.11961i	−0.828625	+	0.559804i	$0.810877\pi$
$450$	0	0
$451$	−216.628	−0.480327
$452$	386.023i	0.854034i
$453$	0	0
$454$	104.415	0.229989
$455$	42.9868i	0.0944764i
$456$	0	0
$457$	−370.634	−0.811016	−0.405508	−	0.914092i	$-0.632905\pi$
$457$	−370.634	−0.811016	−0.405508	+	0.914092i	$0.632905\pi$
$458$	219.337i	0.478902i
$459$	0	0
$460$	−162.093	−0.352376
$461$	− 589.279i	− 1.27826i	−0.769098	−	0.639131i	$-0.779294\pi$
$461$	− 589.279i	− 1.27826i	0.769098	−	0.639131i	$-0.220706\pi$
$462$	0	0
$463$	82.5881	0.178376	0.0891880	−	0.996015i	$-0.471573\pi$
$463$	82.5881	0.178376	0.0891880	+	0.996015i	$0.471573\pi$
$464$	160.780i	0.346508i
$465$	0	0
$466$	356.502	0.765026
$467$	− 723.629i	− 1.54953i	−0.632251	−	0.774764i	$-0.717869\pi$
$467$	− 723.629i	− 1.54953i	0.632251	−	0.774764i	$-0.282131\pi$
$468$	0	0
$469$	174.795	0.372697
$470$	− 180.360i	− 0.383744i
$471$	0	0
$472$	−65.7074	−0.139211
$473$	281.847i	0.595871i
$474$	0	0
$475$	180.915	0.380873
$476$	128.938i	0.270877i
$477$	0	0
$478$	459.472	0.961238
$479$	− 670.262i	− 1.39929i	−0.714489	−	0.699647i	$-0.753341\pi$
$479$	− 670.262i	− 1.39929i	0.714489	−	0.699647i	$-0.246659\pi$
$480$	0	0
$481$	4.24745	0.00883046
$482$	− 187.314i	− 0.388619i
$483$	0	0
$484$	663.313	1.37048
$485$	129.861i	0.267755i
$486$	0	0
$487$	−488.875	−1.00385	−0.501925	−	0.864911i	$-0.667375\pi$
$487$	−488.875	−1.00385	−0.501925	+	0.864911i	$0.667375\pi$
$488$	− 86.8292i	− 0.177929i
$489$	0	0
$490$	−135.826	−0.277195
$491$	− 338.979i	− 0.690385i	−0.938532	−	0.345192i	$-0.887814\pi$
$491$	− 338.979i	− 0.690385i	0.938532	−	0.345192i	$-0.112186\pi$
$492$	0	0
$493$	−1053.68	−2.13729
$494$	− 399.998i	− 0.809712i
$495$	0	0
$496$	−42.4833	−0.0856518
$497$	− 146.810i	− 0.295392i
$498$	0	0
$499$	232.499	0.465929	0.232965	−	0.972485i	$-0.425157\pi$
$499$	232.499	0.465929	0.232965	+	0.972485i	$0.425157\pi$
$500$	− 22.3607i	− 0.0447214i
$501$	0	0
$502$	−114.338	−0.227765
$503$	21.1469i	0.0420416i	0.999779	+	0.0210208i	$0.00669162\pi$
$503$	21.1469i	0.0420416i	−0.999779	+	0.0210208i	$0.993308\pi$
$504$	0	0
$505$	−204.349	−0.404651
$506$	1090.56i	2.15525i
$507$	0	0
$508$	−326.554	−0.642823
$509$	354.789i	0.697032i	0.937303	+	0.348516i	$0.113314\pi$
$509$	354.789i	0.697032i	−0.937303	+	0.348516i	$0.886686\pi$
$510$	0	0
$511$	94.8110	0.185540
$512$	22.6274i	0.0441942i
$513$	0	0
$514$	−240.779	−0.468442
$515$	340.843i	0.661832i
$516$	0	0
$517$	−1213.45	−2.34711
$518$	− 1.88980i	− 0.00364826i
$519$	0	0
$520$	−49.4389	−0.0950747
$521$	854.414i	1.63995i	0.572399	+	0.819975i	$0.306013\pi$
$521$	854.414i	1.63995i	−0.572399	+	0.819975i	$0.693987\pi$
$522$	0	0
$523$	20.4554	0.0391116	0.0195558	−	0.999809i	$-0.493775\pi$
$523$	20.4554	0.0391116	0.0195558	+	0.999809i	$0.493775\pi$
$524$	264.162i	0.504126i
$525$	0	0
$526$	97.7856	0.185904
$527$	− 278.417i	− 0.528306i
$528$	0	0
$529$	−784.703	−1.48337
$530$	231.713i	0.437193i
$531$	0	0
$532$	−177.969	−0.334529
$533$	− 79.5917i	− 0.149328i
$534$	0	0
$535$	191.430	0.357813
$536$	201.031i	0.375057i
$537$	0	0
$538$	−77.3793	−0.143828
$539$	913.832i	1.69542i
$540$	0	0
$541$	−649.938	−1.20136	−0.600682	−	0.799488i	$-0.705104\pi$
$541$	−649.938	−1.20136	−0.600682	+	0.799488i	$0.705104\pi$
$542$	− 183.770i	− 0.339059i
$543$	0	0
$544$	−148.291	−0.272593
$545$	151.325i	0.277660i
$546$	0	0
$547$	−720.199	−1.31663	−0.658317	−	0.752741i	$-0.728731\pi$
$547$	−720.199	−1.31663	−0.658317	+	0.752741i	$0.728731\pi$
$548$	121.391i	0.221517i
$549$	0	0
$550$	−150.442	−0.273531
$551$	− 1454.37i	− 2.63951i
$552$	0	0
$553$	62.8998	0.113743
$554$	− 520.229i	− 0.939042i
$555$	0	0
$556$	−134.896	−0.242619
$557$	13.3409i	0.0239514i	0.999928	+	0.0119757i	$0.00381207\pi$
$557$	13.3409i	0.0239514i	−0.999928	+	0.0119757i	$0.996188\pi$
$558$	0	0
$559$	−103.554	−0.185249
$560$	21.9966i	0.0392797i
$561$	0	0
$562$	−46.7348	−0.0831580
$563$	− 743.654i	− 1.32088i	−0.750880	−	0.660439i	$-0.770370\pi$
$563$	− 743.654i	− 1.32088i	0.750880	−	0.660439i	$-0.229630\pi$
$564$	0	0
$565$	−431.587	−0.763871
$566$	55.0375i	0.0972393i
$567$	0	0
$568$	168.845	0.297263
$569$	− 271.569i	− 0.477274i	−0.971109	−	0.238637i	$-0.923299\pi$
$569$	− 271.569i	− 0.477274i	0.971109	−	0.238637i	$-0.0767007\pi$
$570$	0	0
$571$	−669.626	−1.17273	−0.586363	−	0.810049i	$-0.699441\pi$
$571$	−669.626	−1.17273	−0.586363	+	0.810049i	$0.699441\pi$
$572$	332.623i	0.581509i
$573$	0	0
$574$	−35.4124	−0.0616941
$575$	− 181.225i	− 0.315174i
$576$	0	0
$577$	845.698	1.46568	0.732841	−	0.680400i	$-0.238194\pi$
$577$	845.698	1.46568	0.732841	+	0.680400i	$0.238194\pi$
$578$	− 563.126i	− 0.974266i
$579$	0	0
$580$	−179.757	−0.309926
$581$	− 74.9744i	− 0.129044i
$582$	0	0
$583$	1558.96	2.67402
$584$	109.042i	0.186715i
$585$	0	0
$586$	−242.473	−0.413776
$587$	− 557.579i	− 0.949878i	−0.880019	−	0.474939i	$-0.842470\pi$
$587$	− 557.579i	− 0.949878i	0.880019	−	0.474939i	$-0.157530\pi$
$588$	0	0
$589$	384.292	0.652449
$590$	− 73.4631i	− 0.124514i
$591$	0	0
$592$	2.17345	0.00367137
$593$	− 901.058i	− 1.51949i	−0.650221	−	0.759745i	$-0.725324\pi$
$593$	− 901.058i	− 1.51949i	0.650221	−	0.759745i	$-0.274676\pi$
$594$	0	0
$595$	−144.157	−0.242280
$596$	− 547.582i	− 0.918762i
$597$	0	0
$598$	−400.684	−0.670040
$599$	− 708.507i	− 1.18282i	−0.806372	−	0.591408i	$-0.798572\pi$
$599$	− 708.507i	− 1.18282i	0.806372	−	0.591408i	$-0.201428\pi$
$600$	0	0
$601$	−859.849	−1.43070	−0.715349	−	0.698767i	$-0.753732\pi$
$601$	−859.849	−1.43070	−0.715349	+	0.698767i	$0.753732\pi$
$602$	46.0740i	0.0765348i
$603$	0	0
$604$	−173.713	−0.287604
$605$	741.606i	1.22580i
$606$	0	0
$607$	−310.076	−0.510834	−0.255417	−	0.966831i	$-0.582213\pi$
$607$	−310.076	−0.510834	−0.255417	+	0.966831i	$0.582213\pi$
$608$	− 204.682i	− 0.336647i
$609$	0	0
$610$	97.0780	0.159144
$611$	− 445.839i	− 0.729687i
$612$	0	0
$613$	1162.31	1.89609	0.948047	−	0.318132i	$-0.103055\pi$
$613$	1162.31	1.89609	0.948047	+	0.318132i	$0.103055\pi$
$614$	− 423.986i	− 0.690531i
$615$	0	0
$616$	147.993	0.240248
$617$	812.672i	1.31713i	0.752522	+	0.658567i	$0.228837\pi$
$617$	812.672i	1.31713i	−0.752522	+	0.658567i	$0.771163\pi$
$618$	0	0
$619$	417.890	0.675106	0.337553	−	0.941307i	$-0.390401\pi$
$619$	417.890	0.675106	0.337553	+	0.941307i	$0.390401\pi$
$620$	− 47.4977i	− 0.0766093i
$621$	0	0
$622$	83.4260	0.134125
$623$	− 345.241i	− 0.554160i
$624$	0	0
$625$	25.0000	0.0400000
$626$	427.874i	0.683505i
$627$	0	0
$628$	−281.904	−0.448891
$629$	14.2439i	0.0226453i
$630$	0	0
$631$	−436.728	−0.692121	−0.346061	−	0.938212i	$-0.612481\pi$
$631$	−436.728	−0.692121	−0.346061	+	0.938212i	$0.612481\pi$
$632$	72.3407i	0.114463i
$633$	0	0
$634$	288.906	0.455688
$635$	− 365.099i	− 0.574959i
$636$	0	0
$637$	−335.753	−0.527085
$638$	1209.40i	1.89561i
$639$	0	0
$640$	−25.2982	−0.0395285
$641$	− 336.366i	− 0.524751i	−0.964966	−	0.262376i	$-0.915494\pi$
$641$	− 336.366i	− 0.524751i	0.964966	−	0.262376i	$-0.0845059\pi$
$642$	0	0
$643$	1102.03	1.71389	0.856943	−	0.515410i	$-0.172361\pi$
$643$	1102.03	1.71389	0.856943	+	0.515410i	$0.172361\pi$
$644$	178.275i	0.276824i
$645$	0	0
$646$	1341.40	2.07647
$647$	840.423i	1.29895i	0.760381	+	0.649477i	$0.225012\pi$
$647$	840.423i	1.29895i	−0.760381	+	0.649477i	$0.774988\pi$
$648$	0	0
$649$	−494.258	−0.761568
$650$	− 55.2743i	− 0.0850374i
$651$	0	0
$652$	−288.491	−0.442471
$653$	1100.43i	1.68520i	0.538541	+	0.842599i	$0.318976\pi$
$653$	1100.43i	1.68520i	−0.538541	+	0.842599i	$0.681024\pi$
$654$	0	0
$655$	−295.342	−0.450904
$656$	− 40.7277i	− 0.0620848i
$657$	0	0
$658$	−198.365	−0.301467
$659$	645.046i	0.978826i	0.872052	+	0.489413i	$0.162789\pi$
$659$	645.046i	0.978826i	−0.872052	+	0.489413i	$0.837211\pi$
$660$	0	0
$661$	565.238	0.855126	0.427563	−	0.903985i	$-0.359372\pi$
$661$	565.238	0.855126	0.427563	+	0.903985i	$0.359372\pi$
$662$	− 132.909i	− 0.200768i
$663$	0	0
$664$	86.2277	0.129861
$665$	− 198.976i	− 0.299212i
$666$	0	0
$667$	−1456.87	−2.18421
$668$	− 352.526i	− 0.527734i
$669$	0	0
$670$	−224.759	−0.335461
$671$	− 653.138i	− 0.973381i
$672$	0	0
$673$	108.798	0.161661	0.0808303	−	0.996728i	$-0.474243\pi$
$673$	108.798	0.161661	0.0808303	+	0.996728i	$0.474243\pi$
$674$	814.837i	1.20896i
$675$	0	0
$676$	215.790	0.319216
$677$	1279.20i	1.88952i	0.327768	+	0.944758i	$0.393703\pi$
$677$	1279.20i	1.88952i	−0.327768	+	0.944758i	$0.606297\pi$
$678$	0	0
$679$	142.825	0.210346
$680$	− 165.794i	− 0.243814i
$681$	0	0
$682$	−319.564	−0.468568
$683$	853.097i	1.24904i	0.781008	+	0.624522i	$0.214706\pi$
$683$	853.097i	1.24904i	−0.781008	+	0.624522i	$0.785294\pi$
$684$	0	0
$685$	−135.720	−0.198131
$686$	319.806i	0.466190i
$687$	0	0
$688$	−52.9894	−0.0770195
$689$	572.780i	0.831321i
$690$	0	0
$691$	−168.457	−0.243788	−0.121894	−	0.992543i	$-0.538897\pi$
$691$	−168.457	−0.243788	−0.121894	+	0.992543i	$0.538897\pi$
$692$	− 69.8266i	− 0.100905i
$693$	0	0
$694$	−154.098	−0.222043
$695$	− 150.818i	− 0.217005i
$696$	0	0
$697$	266.912	0.382944
$698$	726.110i	1.04027i
$699$	0	0
$700$	−24.5930	−0.0351328
$701$	310.614i	0.443101i	0.975149	+	0.221551i	$0.0711118\pi$
$701$	310.614i	0.443101i	−0.975149	+	0.221551i	$0.928888\pi$
$702$	0	0
$703$	−19.6605	−0.0279665
$704$	170.206i	0.241770i
$705$	0	0
$706$	−531.560	−0.752918
$707$	224.749i	0.317891i
$708$	0	0
$709$	−1103.78	−1.55681	−0.778404	−	0.627764i	$-0.783970\pi$
$709$	−1103.78	−1.55681	−0.778404	+	0.627764i	$0.783970\pi$
$710$	188.775i	0.265880i
$711$	0	0
$712$	397.061	0.557669
$713$	− 384.952i	− 0.539905i
$714$	0	0
$715$	−371.884	−0.520118
$716$	− 90.7265i	− 0.126713i
$717$	0	0
$718$	−19.0353	−0.0265116
$719$	− 542.890i	− 0.755062i	−0.925997	−	0.377531i	$-0.876773\pi$
$719$	− 542.890i	− 0.755062i	0.925997	−	0.377531i	$-0.123227\pi$
$720$	0	0
$721$	374.870	0.519931
$722$	1340.96i	1.85729i
$723$	0	0
$724$	15.9181	0.0219863
$725$	− 200.975i	− 0.277206i
$726$	0	0
$727$	−337.995	−0.464918	−0.232459	−	0.972606i	$-0.574677\pi$
$727$	−337.995	−0.464918	−0.232459	+	0.972606i	$0.574677\pi$
$728$	54.3744i	0.0746901i
$729$	0	0
$730$	−121.912	−0.167003
$731$	− 347.270i	− 0.475062i
$732$	0	0
$733$	765.050	1.04372	0.521862	−	0.853030i	$-0.325237\pi$
$733$	765.050	1.04372	0.521862	+	0.853030i	$0.325237\pi$
$734$	172.081i	0.234442i
$735$	0	0
$736$	−205.033	−0.278577
$737$	1512.17i	2.05179i
$738$	0	0
$739$	119.322	0.161464	0.0807321	−	0.996736i	$-0.474274\pi$
$739$	119.322	0.161464	0.0807321	+	0.996736i	$0.474274\pi$
$740$	2.42999i	0.00328377i
$741$	0	0
$742$	254.845	0.343457
$743$	499.379i	0.672111i	0.941842	+	0.336056i	$0.109093\pi$
$743$	499.379i	0.672111i	−0.941842	+	0.336056i	$0.890907\pi$
$744$	0	0
$745$	612.216	0.821766
$746$	− 635.377i	− 0.851712i
$747$	0	0
$748$	−1115.46	−1.49125
$749$	− 210.541i	− 0.281096i
$750$	0	0
$751$	383.510	0.510666	0.255333	−	0.966853i	$-0.417815\pi$
$751$	383.510	0.510666	0.255333	+	0.966853i	$0.417815\pi$
$752$	− 228.139i	− 0.303376i
$753$	0	0
$754$	−444.350	−0.589323
$755$	− 194.217i	− 0.257240i
$756$	0	0
$757$	−742.273	−0.980545	−0.490273	−	0.871569i	$-0.663103\pi$
$757$	−742.273	−0.980545	−0.490273	+	0.871569i	$0.663103\pi$
$758$	− 649.107i	− 0.856342i
$759$	0	0
$760$	228.841	0.301107
$761$	997.222i	1.31041i	0.755451	+	0.655205i	$0.227418\pi$
$761$	997.222i	1.31041i	−0.755451	+	0.655205i	$0.772582\pi$
$762$	0	0
$763$	166.432	0.218128
$764$	− 225.083i	− 0.294611i
$765$	0	0
$766$	919.369	1.20022
$767$	− 181.597i	− 0.236762i
$768$	0	0
$769$	−212.696	−0.276588	−0.138294	−	0.990391i	$-0.544162\pi$
$769$	−212.696	−0.276588	−0.138294	+	0.990391i	$0.544162\pi$
$770$	165.461i	0.214884i
$771$	0	0
$772$	561.981	0.727955
$773$	− 1227.49i	− 1.58795i	−0.607950	−	0.793975i	$-0.708008\pi$
$773$	− 1227.49i	− 1.58795i	0.607950	−	0.793975i	$-0.291992\pi$
$774$	0	0
$775$	53.1041	0.0685214
$776$	164.263i	0.211679i
$777$	0	0
$778$	426.069	0.547646
$779$	368.412i	0.472929i
$780$	0	0
$781$	1270.07	1.62621
$782$	− 1343.70i	− 1.71829i
$783$	0	0
$784$	−171.807	−0.219142
$785$	− 315.178i	− 0.401501i
$786$	0	0
$787$	943.383	1.19871	0.599354	−	0.800484i	$-0.295424\pi$
$787$	943.383	1.19871	0.599354	+	0.800484i	$0.295424\pi$
$788$	212.579i	0.269770i
$789$	0	0
$790$	−80.8794	−0.102379
$791$	474.673i	0.600093i
$792$	0	0
$793$	239.971	0.302612
$794$	459.082i	0.578188i
$795$	0	0
$796$	670.959	0.842913
$797$	− 1067.60i	− 1.33952i	−0.742579	−	0.669759i	$-0.766398\pi$
$797$	− 1067.60i	− 1.33952i	0.742579	−	0.669759i	$-0.233602\pi$
$798$	0	0
$799$	1495.12	1.87125
$800$	− 28.2843i	− 0.0353553i
$801$	0	0
$802$	328.373	0.409442
$803$	820.223i	1.02145i
$804$	0	0
$805$	−199.317	−0.247599
$806$	− 117.412i	− 0.145672i
$807$	0	0
$808$	−258.483	−0.319905
$809$	− 663.893i	− 0.820634i	−0.911943	−	0.410317i	$-0.865418\pi$
$809$	− 663.893i	− 0.820634i	0.911943	−	0.410317i	$-0.134582\pi$
$810$	0	0
$811$	−146.090	−0.180135	−0.0900677	−	0.995936i	$-0.528708\pi$
$811$	−146.090	−0.180135	−0.0900677	+	0.995936i	$0.528708\pi$
$812$	197.703i	0.243476i
$813$	0	0
$814$	16.3489	0.0200847
$815$	− 322.543i	− 0.395758i
$816$	0	0
$817$	479.329	0.586693
$818$	247.012i	0.301971i
$819$	0	0
$820$	45.5349	0.0555304
$821$	1141.50i	1.39038i	0.718825	+	0.695191i	$0.244680\pi$
$821$	1141.50i	1.39038i	−0.718825	+	0.695191i	$0.755320\pi$
$822$	0	0
$823$	−1312.84	−1.59519	−0.797595	−	0.603194i	$-0.793895\pi$
$823$	−1312.84	−1.59519	−0.797595	+	0.603194i	$0.793895\pi$
$824$	431.137i	0.523224i
$825$	0	0
$826$	−80.7970	−0.0978172
$827$	− 661.135i	− 0.799437i	−0.916638	−	0.399719i	$-0.869108\pi$
$827$	− 661.135i	− 0.799437i	0.916638	−	0.399719i	$-0.130892\pi$
$828$	0	0
$829$	877.228	1.05818	0.529088	−	0.848567i	$-0.322534\pi$
$829$	877.228	1.05818	0.529088	+	0.848567i	$0.322534\pi$
$830$	96.4055i	0.116151i
$831$	0	0
$832$	−62.5358	−0.0751632
$833$	− 1125.95i	− 1.35168i
$834$	0	0
$835$	394.136	0.472019
$836$	− 1539.64i	− 1.84167i
$837$	0	0
$838$	−1007.29	−1.20202
$839$	− 609.457i	− 0.726409i	−0.931709	−	0.363205i	$-0.881683\pi$
$839$	− 609.457i	− 0.726409i	0.931709	−	0.363205i	$-0.118317\pi$
$840$	0	0
$841$	−774.632	−0.921084
$842$	− 523.661i	− 0.621925i
$843$	0	0
$844$	−554.922	−0.657490
$845$	241.260i	0.285515i
$846$	0	0
$847$	815.642	0.962977
$848$	293.096i	0.345632i
$849$	0	0
$850$	185.363	0.218074
$851$	19.6942i	0.0231424i
$852$	0	0
$853$	−391.451	−0.458911	−0.229456	−	0.973319i	$-0.573695\pi$
$853$	−391.451	−0.458911	−0.229456	+	0.973319i	$0.573695\pi$
$854$	− 106.769i	− 0.125023i
$855$	0	0
$856$	242.142	0.282876
$857$	− 207.971i	− 0.242673i	−0.992611	−	0.121337i	$-0.961282\pi$
$857$	− 207.971i	− 0.242673i	0.992611	−	0.121337i	$-0.0387181\pi$
$858$	0	0
$859$	399.379	0.464935	0.232467	−	0.972604i	$-0.425320\pi$
$859$	399.379	0.464935	0.232467	+	0.972604i	$0.425320\pi$
$860$	− 59.2440i	− 0.0688884i
$861$	0	0
$862$	479.699	0.556496
$863$	− 703.747i	− 0.815466i	−0.913101	−	0.407733i	$-0.866319\pi$
$863$	− 703.747i	− 0.815466i	0.913101	−	0.407733i	$-0.133681\pi$
$864$	0	0
$865$	78.0685	0.0902526
$866$	− 230.975i	− 0.266715i
$867$	0	0
$868$	−52.2395	−0.0601838
$869$	544.155i	0.626185i
$870$	0	0
$871$	−555.592	−0.637878
$872$	191.412i	0.219510i
$873$	0	0
$874$	1854.67	2.12205
$875$	− 27.4958i	− 0.0314238i
$876$	0	0
$877$	−963.822	−1.09900	−0.549499	−	0.835494i	$-0.685182\pi$
$877$	−963.822	−1.09900	−0.549499	+	0.835494i	$0.685182\pi$
$878$	− 647.341i	− 0.737291i
$879$	0	0
$880$	−190.296	−0.216245
$881$	438.268i	0.497467i	0.968572	+	0.248733i	$0.0800143\pi$
$881$	438.268i	0.497467i	−0.968572	+	0.248733i	$0.919986\pi$
$882$	0	0
$883$	563.995	0.638726	0.319363	−	0.947632i	$-0.396531\pi$
$883$	563.995	0.638726	0.319363	+	0.947632i	$0.396531\pi$
$884$	− 409.833i	− 0.463612i
$885$	0	0
$886$	373.295	0.421326
$887$	− 584.041i	− 0.658446i	−0.944252	−	0.329223i	$-0.893213\pi$
$887$	− 584.041i	− 0.658446i	0.944252	−	0.329223i	$-0.106787\pi$
$888$	0	0
$889$	−401.547	−0.451684
$890$	443.927i	0.498795i
$891$	0	0
$892$	−270.779	−0.303564
$893$	2063.68i	2.31095i
$894$	0	0
$895$	101.435	0.113336
$896$	27.8238i	0.0310533i
$897$	0	0
$898$	−710.930	−0.791682
$899$	− 426.903i	− 0.474864i
$900$	0	0
$901$	−1920.83	−2.13188
$902$	− 306.358i	− 0.339643i
$903$	0	0
$904$	−545.920	−0.603893
$905$	17.7970i	0.0196652i
$906$	0	0
$907$	−1084.50	−1.19570	−0.597851	−	0.801607i	$-0.703979\pi$
$907$	−1084.50	−1.19570	−0.597851	+	0.801607i	$0.703979\pi$
$908$	147.665i	0.162627i
$909$	0	0
$910$	−60.7925	−0.0668049
$911$	921.978i	1.01205i	0.862519	+	0.506025i	$0.168886\pi$
$911$	921.978i	1.01205i	−0.862519	+	0.506025i	$0.831114\pi$
$912$	0	0
$913$	648.614	0.710420
$914$	− 524.156i	− 0.573475i
$915$	0	0
$916$	−310.190	−0.338635
$917$	324.827i	0.354228i
$918$	0	0
$919$	537.501	0.584876	0.292438	−	0.956284i	$-0.405533\pi$
$919$	537.501	0.584876	0.292438	+	0.956284i	$0.405533\pi$
$920$	− 229.234i	− 0.249167i
$921$	0	0
$922$	833.366	0.903868
$923$	466.641i	0.505570i
$924$	0	0
$925$	−2.71681	−0.00293709
$926$	116.797i	0.126131i
$927$	0	0
$928$	−227.377	−0.245018
$929$	− 632.559i	− 0.680903i	−0.940262	−	0.340451i	$-0.889420\pi$
$929$	− 632.559i	− 0.680903i	0.940262	−	0.340451i	$-0.110580\pi$
$930$	0	0
$931$	1554.12	1.66931
$932$	504.170i	0.540955i
$933$	0	0
$934$	1023.37	1.09568
$935$	− 1247.12i	− 1.33382i
$936$	0	0
$937$	1028.22	1.09736	0.548679	−	0.836033i	$-0.315131\pi$
$937$	1028.22	1.09736	0.548679	+	0.836033i	$0.315131\pi$
$938$	247.197i	0.263536i
$939$	0	0
$940$	255.067	0.271348
$941$	− 570.099i	− 0.605843i	−0.953015	−	0.302922i	$-0.902038\pi$
$941$	− 570.099i	− 0.605843i	0.953015	−	0.302922i	$-0.0979621\pi$
$942$	0	0
$943$	369.044	0.391351
$944$	− 92.9242i	− 0.0984367i
$945$	0	0
$946$	−398.592	−0.421345
$947$	757.748i	0.800156i	0.916481	+	0.400078i	$0.131017\pi$
$947$	757.748i	0.800156i	−0.916481	+	0.400078i	$0.868983\pi$
$948$	0	0
$949$	−301.360	−0.317556
$950$	255.852i	0.269318i
$951$	0	0
$952$	−182.345	−0.191539
$953$	− 949.465i	− 0.996291i	−0.867093	−	0.498146i	$-0.834015\pi$
$953$	− 949.465i	− 0.996291i	0.867093	−	0.498146i	$-0.165985\pi$
$954$	0	0
$955$	251.650	0.263508
$956$	649.791i	0.679698i
$957$	0	0
$958$	947.894	0.989450
$959$	149.269i	0.155650i
$960$	0	0
$961$	−848.198	−0.882620
$962$	6.00680i	0.00624408i
$963$	0	0
$964$	264.902	0.274795
$965$	628.314i	0.651103i
$966$	0	0
$967$	144.077	0.148993	0.0744967	−	0.997221i	$-0.476265\pi$
$967$	144.077	0.148993	0.0744967	+	0.997221i	$0.476265\pi$
$968$	938.066i	0.969076i
$969$	0	0
$970$	−183.651	−0.189331
$971$	1539.16i	1.58513i	0.609789	+	0.792564i	$0.291254\pi$
$971$	1539.16i	1.58513i	−0.609789	+	0.792564i	$0.708746\pi$
$972$	0	0
$973$	−165.875	−0.170478
$974$	− 691.374i	− 0.709829i
$975$	0	0
$976$	122.795	0.125815
$977$	366.569i	0.375198i	0.982246	+	0.187599i	$0.0600706\pi$
$977$	366.569i	0.375198i	−0.982246	+	0.187599i	$0.939929\pi$
$978$	0	0
$979$	2986.73	3.05080
$980$	− 192.087i	− 0.196007i
$981$	0	0
$982$	479.389	0.488176
$983$	1525.80i	1.55219i	0.630618	+	0.776093i	$0.282801\pi$
$983$	1525.80i	1.55219i	−0.630618	+	0.776093i	$0.717199\pi$
$984$	0	0
$985$	−237.670	−0.241289
$986$	− 1490.13i	− 1.51129i
$987$	0	0
$988$	565.682	0.572553
$989$	− 480.151i	− 0.485492i
$990$	0	0
$991$	−192.145	−0.193890	−0.0969449	−	0.995290i	$-0.530907\pi$
$991$	−192.145	−0.193890	−0.0969449	+	0.995290i	$0.530907\pi$
$992$	− 60.0804i	− 0.0605649i
$993$	0	0
$994$	207.621	0.208874
$995$	750.155i	0.753924i
$996$	0	0
$997$	1449.61	1.45397	0.726986	−	0.686652i	$-0.240920\pi$
$997$	1449.61	1.45397	0.726986	+	0.686652i	$0.240920\pi$
$998$	328.803i	0.329462i
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	810.3.d.b.161.5	yes	8
3.2	odd	2	inner	810.3.d.b.161.3	✓	8
9.2	odd	6		810.3.h.c.701.2		16
9.4	even	3		810.3.h.c.431.2		16
9.5	odd	6		810.3.h.c.431.8		16
9.7	even	3		810.3.h.c.701.8		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
810.3.d.b.161.3	✓	8	3.2	odd	2	inner
810.3.d.b.161.5	yes	8	1.1	even	1	trivial
810.3.h.c.431.2		16	9.4	even	3
810.3.h.c.431.8		16	9.5	odd	6
810.3.h.c.701.2		16	9.2	odd	6
810.3.h.c.701.8		16	9.7	even	3

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)

\(f(q)\)	\(=\)	\(q+1.41421i q^{2} -2.00000 q^{4} -2.23607i q^{5} -2.45930 q^{7} -2.82843i q^{8} +O(q^{10})\)	\(q+1.41421i q^{2} -2.00000 q^{4} -2.23607i q^{5} -2.45930 q^{7} -2.82843i q^{8} +3.16228 q^{10} -21.2757i q^{11} +7.81697 q^{13} -3.47797i q^{14} +4.00000 q^{16} +26.2143i q^{17} -36.1829 q^{19} +4.47214i q^{20} +30.0884 q^{22} +36.2450i q^{23} -5.00000 q^{25} +11.0549i q^{26} +4.91860 q^{28} +40.1949i q^{29} -10.6208 q^{31} +5.65685i q^{32} -37.0726 q^{34} +5.49916i q^{35} +0.543363 q^{37} -51.1704i q^{38} -6.32456 q^{40} -10.1819i q^{41} -13.2474 q^{43} +42.5514i q^{44} -51.2582 q^{46} -57.0347i q^{47} -42.9519 q^{49} -7.07107i q^{50} -15.6339 q^{52} +73.2739i q^{53} -47.5740 q^{55} +6.95594i q^{56} -56.8442 q^{58} -23.2311i q^{59} +30.6988 q^{61} -15.0201i q^{62} -8.00000 q^{64} -17.4793i q^{65} -71.0750 q^{67} -52.4286i q^{68} -7.77698 q^{70} +59.6959i q^{71} -38.5521 q^{73} +0.768431i q^{74} +72.3659 q^{76} +52.3233i q^{77} -25.5763 q^{79} -8.94427i q^{80} +14.3994 q^{82} +30.4861i q^{83} +58.6170 q^{85} -18.7346i q^{86} -60.1768 q^{88} +140.382i q^{89} -19.2243 q^{91} -72.4901i q^{92} +80.6592 q^{94} +80.9075i q^{95} -58.0756 q^{97} -60.7431i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 16 q^{4} - 8 q^{7}+O(q^{10}) \) 8 * q - 16 * q^4 - 8 * q^7	\( 8 q - 16 q^{4} - 8 q^{7} + 88 q^{13} + 32 q^{16} - 80 q^{19} + 24 q^{22} - 40 q^{25} + 16 q^{28} - 152 q^{31} + 24 q^{34} + 136 q^{37} - 32 q^{43} - 48 q^{49} - 176 q^{52} - 120 q^{55} - 168 q^{58} + 232 q^{61} - 64 q^{64} + 184 q^{67} + 120 q^{70} + 280 q^{73} + 160 q^{76} + 184 q^{79} - 24 q^{82} - 48 q^{88} + 32 q^{91} + 312 q^{94} - 512 q^{97}+O(q^{100}) \) 8 * q - 16 * q^4 - 8 * q^7 + 88 * q^13 + 32 * q^16 - 80 * q^19 + 24 * q^22 - 40 * q^25 + 16 * q^28 - 152 * q^31 + 24 * q^34 + 136 * q^37 - 32 * q^43 - 48 * q^49 - 176 * q^52 - 120 * q^55 - 168 * q^58 + 232 * q^61 - 64 * q^64 + 184 * q^67 + 120 * q^70 + 280 * q^73 + 160 * q^76 + 184 * q^79 - 24 * q^82 - 48 * q^88 + 32 * q^91 + 312 * q^94 - 512 * q^97

Introduction

L-functions

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