LMFDB - Embedded newform 363.5.c.e.241.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [363,5,Mod(241,363)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(363, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("363.241");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 363 = 3 \cdot 11^{2} $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	363.c (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:	$37.5232965994$
Analytic rank:	$0$
Dimension:	$32$
Twist minimal:	no (minimal twist has level 33)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			241.7
Character	$\chi$	$=$	363.241
Dual form			363.5.c.e.241.26

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-4.72406i q^{2} +5.19615 q^{3} -6.31678 q^{4} +17.3409 q^{5} -24.5470i q^{6} +82.0604i q^{7} -45.7442i q^{8} +27.0000 q^{9} +O(q^{10})$	\(q-4.72406i q^{2} +5.19615 q^{3} -6.31678 q^{4} +17.3409 q^{5} -24.5470i q^{6} +82.0604i q^{7} -45.7442i q^{8} +27.0000 q^{9} -81.9194i q^{10} -32.8230 q^{12} +170.528i q^{13} +387.659 q^{14} +90.1058 q^{15} -317.167 q^{16} +330.492i q^{17} -127.550i q^{18} -270.776i q^{19} -109.538 q^{20} +426.399i q^{21} +414.069 q^{23} -237.694i q^{24} -324.294 q^{25} +805.585 q^{26} +140.296 q^{27} -518.358i q^{28} +1074.05i q^{29} -425.666i q^{30} +543.157 q^{31} +766.410i q^{32} +1561.27 q^{34} +1423.00i q^{35} -170.553 q^{36} +628.025 q^{37} -1279.16 q^{38} +886.089i q^{39} -793.244i q^{40} +245.867i q^{41} +2014.33 q^{42} +3521.17i q^{43} +468.204 q^{45} -1956.09i q^{46} +2360.53 q^{47} -1648.05 q^{48} -4332.92 q^{49} +1531.99i q^{50} +1717.29i q^{51} -1077.19i q^{52} +1023.27 q^{53} -662.768i q^{54} +3753.79 q^{56} -1406.99i q^{57} +5073.89 q^{58} +2169.34 q^{59} -569.179 q^{60} -3862.07i q^{61} -2565.91i q^{62} +2215.63i q^{63} -1454.10 q^{64} +2957.10i q^{65} -70.0820 q^{67} -2087.65i q^{68} +2151.57 q^{69} +6722.34 q^{70} -458.341 q^{71} -1235.09i q^{72} -6657.52i q^{73} -2966.83i q^{74} -1685.08 q^{75} +1710.43i q^{76} +4185.94 q^{78} +8184.42i q^{79} -5499.95 q^{80} +729.000 q^{81} +1161.49 q^{82} -317.535i q^{83} -2693.47i q^{84} +5731.02i q^{85} +16634.2 q^{86} +5580.93i q^{87} -5300.49 q^{89} -2211.82i q^{90} -13993.6 q^{91} -2615.58 q^{92} +2822.33 q^{93} -11151.3i q^{94} -4695.49i q^{95} +3982.38i q^{96} -5258.60 q^{97} +20469.0i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 32 q - 244 q^{4} + 36 q^{5} + 864 q^{9}+O(q^{10}) $ 32 * q - 244 * q^4 + 36 * q^5 + 864 * q^9	$ 32 q - 244 q^{4} + 36 q^{5} + 864 q^{9} + 360 q^{12} + 2220 q^{14} - 108 q^{15} + 3908 q^{16} + 468 q^{20} - 2196 q^{23} + 7280 q^{25} + 3564 q^{26} - 5872 q^{31} - 2320 q^{34} - 6588 q^{36} - 656 q^{37} - 2616 q^{38} - 1404 q^{42} + 972 q^{45} + 2640 q^{47} - 9936 q^{48} - 6988 q^{49} + 4560 q^{53} - 5604 q^{56} - 24644 q^{58} + 39612 q^{59} + 20592 q^{60} - 6232 q^{64} + 2796 q^{67} - 10476 q^{69} - 72692 q^{70} - 51828 q^{71} - 18072 q^{75} + 53640 q^{78} - 27624 q^{80} + 23328 q^{81} - 11548 q^{82} + 106284 q^{86} - 38748 q^{89} + 30672 q^{91} + 27000 q^{92} + 42624 q^{93} - 50544 q^{97}+O(q^{100}) $ 32 * q - 244 * q^4 + 36 * q^5 + 864 * q^9 + 360 * q^12 + 2220 * q^14 - 108 * q^15 + 3908 * q^16 + 468 * q^20 - 2196 * q^23 + 7280 * q^25 + 3564 * q^26 - 5872 * q^31 - 2320 * q^34 - 6588 * q^36 - 656 * q^37 - 2616 * q^38 - 1404 * q^42 + 972 * q^45 + 2640 * q^47 - 9936 * q^48 - 6988 * q^49 + 4560 * q^53 - 5604 * q^56 - 24644 * q^58 + 39612 * q^59 + 20592 * q^60 - 6232 * q^64 + 2796 * q^67 - 10476 * q^69 - 72692 * q^70 - 51828 * q^71 - 18072 * q^75 + 53640 * q^78 - 27624 * q^80 + 23328 * q^81 - 11548 * q^82 + 106284 * q^86 - 38748 * q^89 + 30672 * q^91 + 27000 * q^92 + 42624 * q^93 - 50544 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/363\mathbb{Z}\right)^\times$.

$n$	$122$	$244$
$\chi(n)$	$1$	$-1$

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 $

$2$	− 4.72406i	− 1.18102i	−0.807032	−	0.590508i	$-0.798927\pi$
$2$	− 4.72406i	− 1.18102i	0.807032	−	0.590508i	$-0.201073\pi$
$3$	5.19615	0.577350
$3$	5.19615	0.577350
$4$	−6.31678	−0.394799
$5$	17.3409	0.693635	0.346818	−	0.937933i	$-0.387262\pi$
$5$	17.3409	0.693635	0.346818	+	0.937933i	$0.387262\pi$
$6$	− 24.5470i	− 0.681860i
$7$	82.0604i	1.67470i	0.546665	+	0.837351i	$0.315897\pi$
$7$	82.0604i	1.67470i	−0.546665	+	0.837351i	$0.684103\pi$
$8$	− 45.7442i	− 0.714752i
$9$	27.0000	0.333333
$10$	− 81.9194i	− 0.819194i
$11$	0	0
$11$	0	0
$12$	−32.8230	−0.227937
$13$	170.528i	1.00904i	0.863400	+	0.504520i	$0.168331\pi$
$13$	170.528i	1.00904i	−0.863400	+	0.504520i	$0.831669\pi$
$14$	387.659	1.97785
$15$	90.1058	0.400470
$16$	−317.167	−1.23893
$17$	330.492i	1.14357i	0.820403	+	0.571786i	$0.193749\pi$
$17$	330.492i	1.14357i	−0.820403	+	0.571786i	$0.806251\pi$
$18$	− 127.550i	− 0.393672i
$19$	− 270.776i	− 0.750071i	−0.927010	−	0.375035i	$-0.877631\pi$
$19$	− 270.776i	− 0.750071i	0.927010	−	0.375035i	$-0.122369\pi$
$20$	−109.538	−0.273846
$21$	426.399i	0.966890i
$22$	0	0
$23$	414.069	0.782739	0.391370	−	0.920234i	$-0.372001\pi$
$23$	414.069	0.782739	0.391370	+	0.920234i	$0.372001\pi$
$24$	− 237.694i	− 0.412662i
$25$	−324.294	−0.518870
$26$	805.585	1.19169
$27$	140.296	0.192450
$28$	− 518.358i	− 0.661171i
$29$	1074.05i	1.27711i	0.769575	+	0.638556i	$0.220468\pi$
$29$	1074.05i	1.27711i	−0.769575	+	0.638556i	$0.779532\pi$
$30$	− 425.666i	− 0.472962i
$31$	543.157	0.565200	0.282600	−	0.959238i	$-0.408803\pi$
$31$	543.157	0.565200	0.282600	+	0.959238i	$0.408803\pi$
$32$	766.410i	0.748447i
$33$	0	0
$34$	1561.27	1.35058
$35$	1423.00i	1.16163i
$36$	−170.553	−0.131600
$37$	628.025	0.458747	0.229374	−	0.973338i	$-0.426332\pi$
$37$	628.025	0.458747	0.229374	+	0.973338i	$0.426332\pi$
$38$	−1279.16	−0.885846
$39$	886.089i	0.582570i
$40$	− 793.244i	− 0.495777i
$41$	245.867i	0.146262i	0.997322	+	0.0731312i	$0.0232992\pi$
$41$	245.867i	0.146262i	−0.997322	+	0.0731312i	$0.976701\pi$
$42$	2014.33	1.14191
$43$	3521.17i	1.90436i	0.305533	+	0.952181i	$0.401165\pi$
$43$	3521.17i	1.90436i	−0.305533	+	0.952181i	$0.598835\pi$
$44$	0	0
$45$	468.204	0.231212
$46$	− 1956.09i	− 0.924428i
$47$	2360.53	1.06859	0.534297	−	0.845297i	$-0.320576\pi$
$47$	2360.53	1.06859	0.534297	+	0.845297i	$0.320576\pi$
$48$	−1648.05	−0.715298
$49$	−4332.92	−1.80463
$50$	1531.99i	0.612794i
$51$	1717.29i	0.660241i
$52$	− 1077.19i	− 0.398368i
$53$	1023.27	0.364283	0.182142	−	0.983272i	$-0.441697\pi$
$53$	1023.27	0.364283	0.182142	+	0.983272i	$0.441697\pi$
$54$	− 662.768i	− 0.227287i
$55$	0	0
$56$	3753.79	1.19700
$57$	− 1406.99i	− 0.433054i
$58$	5073.89	1.50829
$59$	2169.34	0.623196	0.311598	−	0.950214i	$-0.399136\pi$
$59$	2169.34	0.623196	0.311598	+	0.950214i	$0.399136\pi$
$60$	−569.179	−0.158105
$61$	− 3862.07i	− 1.03791i	−0.854801	−	0.518956i	$-0.826321\pi$
$61$	− 3862.07i	− 1.03791i	0.854801	−	0.518956i	$-0.173679\pi$
$62$	− 2565.91i	− 0.667510i
$63$	2215.63i	0.558234i
$64$	−1454.10	−0.355005
$65$	2957.10i	0.699906i
$66$	0	0
$67$	−70.0820	−0.0156119	−0.00780597	−	0.999970i	$-0.502485\pi$
$67$	−70.0820	−0.0156119	−0.00780597	+	0.999970i	$0.502485\pi$
$68$	− 2087.65i	− 0.451480i
$69$	2151.57	0.451915
$70$	6722.34	1.37191
$71$	−458.341	−0.0909226	−0.0454613	−	0.998966i	$-0.514476\pi$
$71$	−458.341	−0.0909226	−0.0454613	+	0.998966i	$0.514476\pi$
$72$	− 1235.09i	− 0.238251i
$73$	− 6657.52i	− 1.24930i	−0.780905	−	0.624650i	$-0.785242\pi$
$73$	− 6657.52i	− 1.24930i	0.780905	−	0.624650i	$-0.214758\pi$
$74$	− 2966.83i	− 0.541788i
$75$	−1685.08	−0.299570
$76$	1710.43i	0.296127i
$77$	0	0
$78$	4185.94	0.688024
$79$	8184.42i	1.31140i	0.755023	+	0.655698i	$0.227626\pi$
$79$	8184.42i	1.31140i	−0.755023	+	0.655698i	$0.772374\pi$
$80$	−5499.95	−0.859367
$81$	729.000	0.111111
$82$	1161.49	0.172738
$83$	− 317.535i	− 0.0460931i	−0.999734	−	0.0230465i	$-0.992663\pi$
$83$	− 317.535i	− 0.0460931i	0.999734	−	0.0230465i	$-0.00733659\pi$
$84$	− 2693.47i	− 0.381727i
$85$	5731.02i	0.793221i
$86$	16634.2	2.24908
$87$	5580.93i	0.737341i
$88$	0	0
$89$	−5300.49	−0.669169	−0.334585	−	0.942366i	$-0.608596\pi$
$89$	−5300.49	−0.669169	−0.334585	+	0.942366i	$0.608596\pi$
$90$	− 2211.82i	− 0.273065i
$91$	−13993.6	−1.68984
$92$	−2615.58	−0.309024
$93$	2822.33	0.326318
$94$	− 11151.3i	− 1.26203i
$95$	− 4695.49i	− 0.520276i
$96$	3982.38i	0.432116i
$97$	−5258.60	−0.558890	−0.279445	−	0.960162i	$-0.590150\pi$
$97$	−5258.60	−0.558890	−0.279445	+	0.960162i	$0.590150\pi$
$98$	20469.0i	2.13130i
$99$	0	0
$100$	2048.49	0.204849
$101$	− 20315.3i	− 1.99150i	−0.0920853	−	0.995751i	$-0.529353\pi$
$101$	− 20315.3i	− 1.99150i	0.0920853	−	0.995751i	$-0.470647\pi$
$102$	8112.58	0.779755
$103$	19713.5	1.85819	0.929093	−	0.369845i	$-0.120589\pi$
$103$	19713.5	1.85819	0.929093	+	0.369845i	$0.120589\pi$
$104$	7800.65	0.721214
$105$	7394.12i	0.670669i
$106$	− 4834.00i	− 0.430224i
$107$	− 9239.08i	− 0.806977i	−0.914985	−	0.403488i	$-0.867798\pi$
$107$	− 9239.08i	− 0.806977i	0.914985	−	0.403488i	$-0.132202\pi$
$108$	−886.220	−0.0759791
$109$	− 14033.0i	− 1.18113i	−0.806990	−	0.590566i	$-0.798905\pi$
$109$	− 14033.0i	− 1.18113i	0.806990	−	0.590566i	$-0.201095\pi$
$110$	0	0
$111$	3263.31	0.264858
$112$	− 26026.8i	− 2.07484i
$113$	15846.6	1.24102	0.620511	−	0.784198i	$-0.286925\pi$
$113$	15846.6	1.24102	0.620511	+	0.784198i	$0.286925\pi$
$114$	−6646.72	−0.511443
$115$	7180.32	0.542935
$116$	− 6784.54i	− 0.504202i
$117$	4604.25i	0.336347i
$118$	− 10248.1i	− 0.736004i
$119$	−27120.3	−1.91514
$120$	− 4121.81i	− 0.286237i
$121$	0	0
$122$	−18244.7	−1.22579
$123$	1277.56i	0.0844447i
$124$	−3431.01	−0.223140
$125$	−16461.6	−1.05354
$126$	10466.8	0.659284
$127$	− 22718.9i	− 1.40857i	−0.709915	−	0.704287i	$-0.751267\pi$
$127$	− 22718.9i	− 1.40857i	0.709915	−	0.704287i	$-0.248733\pi$
$128$	19131.8i	1.16771i
$129$	18296.5i	1.09948i
$130$	13969.5	0.826600
$131$	− 10558.8i	− 0.615277i	−0.951503	−	0.307638i	$-0.900461\pi$
$131$	− 10558.8i	− 0.615277i	0.951503	−	0.307638i	$-0.0995387\pi$
$132$	0	0
$133$	22220.0	1.25615
$134$	331.072i	0.0184380i
$135$	2432.86	0.133490
$136$	15118.1	0.817370
$137$	−19436.9	−1.03559	−0.517793	−	0.855506i	$-0.673246\pi$
$137$	−19436.9	−1.03559	−0.517793	+	0.855506i	$0.673246\pi$
$138$	− 10164.1i	− 0.533719i
$139$	− 21119.9i	− 1.09311i	−0.837424	−	0.546554i	$-0.815939\pi$
$139$	− 21119.9i	− 1.09311i	0.837424	−	0.546554i	$-0.184061\pi$
$140$	− 8988.78i	− 0.458611i
$141$	12265.7	0.616953
$142$	2165.23i	0.107381i
$143$	0	0
$144$	−8563.50	−0.412978
$145$	18625.0i	0.885849i
$146$	−31450.6	−1.47544
$147$	−22514.5	−1.04190
$148$	−3967.09	−0.181113
$149$	18431.1i	0.830192i	0.909778	+	0.415096i	$0.136252\pi$
$149$	18431.1i	0.830192i	−0.909778	+	0.415096i	$0.863748\pi$
$150$	7960.43i	0.353797i
$151$	6078.73i	0.266599i	0.991076	+	0.133300i	$0.0425573\pi$
$151$	6078.73i	0.266599i	−0.991076	+	0.133300i	$0.957443\pi$
$152$	−12386.4	−0.536115
$153$	8923.29i	0.381190i
$154$	0	0
$155$	9418.82	0.392043
$156$	− 5597.23i	− 0.229998i
$157$	22970.8	0.931914	0.465957	−	0.884807i	$-0.345710\pi$
$157$	22970.8	0.931914	0.465957	+	0.884807i	$0.345710\pi$
$158$	38663.7	1.54878
$159$	5317.08	0.210319
$160$	13290.2i	0.519149i
$161$	33978.7i	1.31086i
$162$	− 3443.84i	− 0.131224i
$163$	−25261.4	−0.950785	−0.475393	−	0.879774i	$-0.657694\pi$
$163$	−25261.4	−0.950785	−0.475393	+	0.879774i	$0.657694\pi$
$164$	− 1553.09i	− 0.0577442i
$165$	0	0
$166$	−1500.06	−0.0544366
$167$	39852.1i	1.42895i	0.699660	+	0.714476i	$0.253335\pi$
$167$	39852.1i	1.42895i	−0.699660	+	0.714476i	$0.746665\pi$
$168$	19505.2	0.691087
$169$	−518.765	−0.0181634
$170$	27073.7	0.936807
$171$	− 7310.94i	− 0.250024i
$172$	− 22242.4i	− 0.751840i
$173$	29238.6i	0.976931i	0.872583	+	0.488465i	$0.162443\pi$
$173$	29238.6i	0.976931i	−0.872583	+	0.488465i	$0.837557\pi$
$174$	26364.7	0.870811
$175$	− 26611.7i	− 0.868954i
$176$	0	0
$177$	11272.2	0.359802
$178$	25039.8i	0.790299i
$179$	−12644.2	−0.394627	−0.197313	−	0.980340i	$-0.563222\pi$
$179$	−12644.2	−0.394627	−0.197313	+	0.980340i	$0.563222\pi$
$180$	−2957.54	−0.0912821
$181$	3733.32	0.113956	0.0569782	−	0.998375i	$-0.481853\pi$
$181$	3733.32	0.113956	0.0569782	+	0.998375i	$0.481853\pi$
$182$	66106.6i	1.99573i
$183$	− 20067.9i	− 0.599239i
$184$	− 18941.2i	− 0.559465i
$185$	10890.5	0.318203
$186$	− 13332.9i	− 0.385387i
$187$	0	0
$188$	−14910.9	−0.421880
$189$	11512.8i	0.322297i
$190$	−22181.8	−0.614454
$191$	−25025.4	−0.685984	−0.342992	−	0.939338i	$-0.611440\pi$
$191$	−25025.4	−0.685984	−0.342992	+	0.939338i	$0.611440\pi$
$192$	−7555.73	−0.204962
$193$	26401.1i	0.708773i	0.935099	+	0.354386i	$0.115310\pi$
$193$	26401.1i	0.708773i	−0.935099	+	0.354386i	$0.884690\pi$
$194$	24842.0i	0.660058i
$195$	15365.6i	0.404091i
$196$	27370.1	0.712465
$197$	40491.4i	1.04335i	0.853144	+	0.521676i	$0.174693\pi$
$197$	40491.4i	1.04335i	−0.853144	+	0.521676i	$0.825307\pi$
$198$	0	0
$199$	−57554.9	−1.45337	−0.726685	−	0.686971i	$-0.758940\pi$
$199$	−57554.9	−1.45337	−0.726685	+	0.686971i	$0.758940\pi$
$200$	14834.6i	0.370864i
$201$	−364.157	−0.00901356
$202$	−95970.8	−2.35200
$203$	−88137.1	−2.13878
$204$	− 10847.7i	− 0.260662i
$205$	4263.55i	0.101453i
$206$	− 93127.8i	− 2.19455i
$207$	11179.9	0.260913
$208$	− 54085.8i	− 1.25013i
$209$	0	0
$210$	34930.3	0.792071
$211$	36089.2i	0.810610i	0.914181	+	0.405305i	$0.132835\pi$
$211$	36089.2i	0.810610i	−0.914181	+	0.405305i	$0.867165\pi$
$212$	−6463.78	−0.143819
$213$	−2381.61	−0.0524942
$214$	−43646.0	−0.953053
$215$	61060.1i	1.32093i
$216$	− 6417.73i	− 0.137554i
$217$	44571.7i	0.946542i
$218$	−66292.9	−1.39494
$219$	− 34593.5i	− 0.721284i
$220$	0	0
$221$	−56358.1	−1.15391
$222$	− 15416.1i	− 0.312801i
$223$	5318.08	0.106941	0.0534706	−	0.998569i	$-0.482972\pi$
$223$	5318.08	0.106941	0.0534706	+	0.998569i	$0.482972\pi$
$224$	−62891.9	−1.25343
$225$	−8755.94	−0.172957
$226$	− 74860.4i	− 1.46567i
$227$	18829.2i	0.365410i	0.983168	+	0.182705i	$0.0584853\pi$
$227$	18829.2i	0.365410i	−0.983168	+	0.182705i	$0.941515\pi$
$228$	8887.66i	0.170969i
$229$	12234.6	0.233302	0.116651	−	0.993173i	$-0.462784\pi$
$229$	12234.6	0.233302	0.116651	+	0.993173i	$0.462784\pi$
$230$	− 33920.3i	− 0.641215i
$231$	0	0
$232$	49131.5	0.912819
$233$	34531.1i	0.636060i	0.948081	+	0.318030i	$0.103021\pi$
$233$	34531.1i	0.636060i	−0.948081	+	0.318030i	$0.896979\pi$
$234$	21750.8	0.397231
$235$	40933.6	0.741215
$236$	−13703.3	−0.246037
$237$	42527.5i	0.757135i
$238$	128118.i	2.26181i
$239$	− 61945.9i	− 1.08447i	−0.840227	−	0.542234i	$-0.817579\pi$
$239$	− 61945.9i	− 1.08447i	0.840227	−	0.542234i	$-0.182421\pi$
$240$	−28578.6	−0.496156
$241$	10177.0i	0.175221i	0.996155	+	0.0876107i	$0.0279232\pi$
$241$	10177.0i	0.175221i	−0.996155	+	0.0876107i	$0.972077\pi$
$242$	0	0
$243$	3788.00	0.0641500
$244$	24395.9i	0.409767i
$245$	−75136.5	−1.25175
$246$	6035.29	0.0997305
$247$	46174.8	0.756852
$248$	− 24846.3i	− 0.403978i
$249$	− 1649.96i	− 0.0266118i
$250$	77765.6i	1.24425i
$251$	4663.08	0.0740160	0.0370080	−	0.999315i	$-0.488217\pi$
$251$	4663.08	0.0740160	0.0370080	+	0.999315i	$0.488217\pi$
$252$	− 13995.7i	− 0.220390i
$253$	0	0
$254$	−107326.	−1.66355
$255$	29779.3i	0.457966i
$256$	67114.3	1.02408
$257$	51921.7	0.786109	0.393055	−	0.919515i	$-0.371418\pi$
$257$	51921.7	0.786109	0.393055	+	0.919515i	$0.371418\pi$
$258$	86433.9	1.29851
$259$	51536.0i	0.768265i
$260$	− 18679.4i	− 0.276322i
$261$	28999.4i	0.425704i
$262$	−49880.3	−0.726652
$263$	− 69680.9i	− 1.00740i	−0.863879	−	0.503700i	$-0.831972\pi$
$263$	− 69680.9i	− 1.00740i	0.863879	−	0.503700i	$-0.168028\pi$
$264$	0	0
$265$	17744.4	0.252680
$266$	− 104969.i	− 1.48353i
$267$	−27542.1	−0.386345
$268$	442.693	0.00616357
$269$	55972.2	0.773513	0.386757	−	0.922182i	$-0.373595\pi$
$269$	55972.2	0.773513	0.386757	+	0.922182i	$0.373595\pi$
$270$	− 11493.0i	− 0.157654i
$271$	− 84527.3i	− 1.15096i	−0.817818	−	0.575478i	$-0.804816\pi$
$271$	− 84527.3i	− 1.15096i	0.817818	−	0.575478i	$-0.195184\pi$
$272$	− 104821.i	− 1.41681i
$273$	−72712.8	−0.975632
$274$	91821.2i	1.22304i
$275$	0	0
$276$	−13591.0	−0.178415
$277$	− 64847.1i	− 0.845144i	−0.906329	−	0.422572i	$-0.861127\pi$
$277$	− 64847.1i	− 0.845144i	0.906329	−	0.422572i	$-0.138873\pi$
$278$	−99772.0	−1.29098
$279$	14665.2	0.188400
$280$	65093.9	0.830280
$281$	2907.14i	0.0368175i	0.999831	+	0.0184087i	$0.00586001\pi$
$281$	2907.14i	0.0368175i	−0.999831	+	0.0184087i	$0.994140\pi$
$282$	− 57943.7i	− 0.728632i
$283$	140908.i	1.75939i	0.475542	+	0.879693i	$0.342252\pi$
$283$	140908.i	1.75939i	−0.475542	+	0.879693i	$0.657748\pi$
$284$	2895.24	0.0358961
$285$	− 24398.5i	− 0.300381i
$286$	0	0
$287$	−20176.0	−0.244946
$288$	20693.1i	0.249482i
$289$	−25704.0	−0.307755
$290$	87985.6	1.04620
$291$	−27324.5	−0.322675
$292$	42054.1i	0.493222i
$293$	− 67312.2i	− 0.784076i	−0.919949	−	0.392038i	$-0.871770\pi$
$293$	− 67312.2i	− 0.784076i	0.919949	−	0.392038i	$-0.128230\pi$
$294$	106360.i	1.23050i
$295$	37618.3	0.432270
$296$	− 28728.5i	− 0.327891i
$297$	0	0
$298$	87069.6	0.980470
$299$	70610.3i	0.789816i
$300$	10644.3	0.118270
$301$	−288948.	−3.18924
$302$	28716.3	0.314858
$303$	− 105561.i	− 1.14979i
$304$	85881.0i	0.929287i
$305$	− 66971.7i	− 0.719933i
$306$	42154.2	0.450192
$307$	− 85320.9i	− 0.905272i	−0.891696	−	0.452636i	$-0.850484\pi$
$307$	− 85320.9i	− 0.905272i	0.891696	−	0.452636i	$-0.149516\pi$
$308$	0	0
$309$	102434.	1.07282
$310$	− 44495.1i	− 0.463009i
$311$	−30774.3	−0.318176	−0.159088	−	0.987264i	$-0.550855\pi$
$311$	−30774.3	−0.318176	−0.159088	+	0.987264i	$0.550855\pi$
$312$	40533.4	0.416393
$313$	113713.	1.16070	0.580352	−	0.814366i	$-0.302915\pi$
$313$	113713.	1.16070	0.580352	+	0.814366i	$0.302915\pi$
$314$	− 108515.i	− 1.10061i
$315$	38421.0i	0.387211i
$316$	− 51699.2i	− 0.517737i
$317$	49528.1	0.492871	0.246436	−	0.969159i	$-0.420741\pi$
$317$	49528.1	0.492871	0.246436	+	0.969159i	$0.420741\pi$
$318$	− 25118.2i	− 0.248390i
$319$	0	0
$320$	−25215.4	−0.246244
$321$	− 48007.7i	− 0.465908i
$322$	160518.	1.54814
$323$	89489.2	0.857760
$324$	−4604.93	−0.0438665
$325$	− 55301.2i	− 0.523561i
$326$	119337.i	1.12289i
$327$	− 72917.7i	− 0.681927i
$328$	11247.0	0.104541
$329$	193706.i	1.78958i
$330$	0	0
$331$	−148612.	−1.35644	−0.678218	−	0.734861i	$-0.737247\pi$
$331$	−148612.	−1.35644	−0.678218	+	0.734861i	$0.737247\pi$
$332$	2005.80i	0.0181975i
$333$	16956.7	0.152916
$334$	188264.	1.68762
$335$	−1215.28	−0.0108290
$336$	− 135239.i	− 1.19791i
$337$	− 110406.i	− 0.972153i	−0.873916	−	0.486077i	$-0.838428\pi$
$337$	− 110406.i	− 0.972153i	0.873916	−	0.486077i	$-0.161572\pi$
$338$	2450.68i	0.0214513i
$339$	82341.4	0.716505
$340$	− 36201.6i	− 0.313163i
$341$	0	0
$342$	−34537.4	−0.295282
$343$	− 158534.i	− 1.34752i
$344$	161073.	1.36115
$345$	37310.0	0.313464
$346$	138125.	1.15377
$347$	− 42009.8i	− 0.348892i	−0.984667	−	0.174446i	$-0.944187\pi$
$347$	− 42009.8i	− 0.348892i	0.984667	−	0.174446i	$-0.0558135\pi$
$348$	− 35253.5i	− 0.291101i
$349$	− 93812.2i	− 0.770209i	−0.922873	−	0.385104i	$-0.874165\pi$
$349$	− 93812.2i	− 0.770209i	0.922873	−	0.385104i	$-0.125835\pi$
$350$	−125715.	−1.02625
$351$	23924.4i	0.194190i
$352$	0	0
$353$	−43599.5	−0.349891	−0.174945	−	0.984578i	$-0.555975\pi$
$353$	−43599.5	−0.349891	−0.174945	+	0.984578i	$0.555975\pi$
$354$	− 53250.8i	− 0.424932i
$355$	−7948.03	−0.0630671
$356$	33482.0	0.264187
$357$	−140921.	−1.10571
$358$	59732.2i	0.466061i
$359$	− 196677.i	− 1.52603i	−0.646380	−	0.763016i	$-0.723718\pi$
$359$	− 196677.i	− 1.52603i	0.646380	−	0.763016i	$-0.276282\pi$
$360$	− 21417.6i	− 0.165259i
$361$	57001.6	0.437394
$362$	− 17636.5i	− 0.134584i
$363$	0	0
$364$	88394.5	0.667148
$365$	− 115447.i	− 0.866559i
$366$	−94802.1	−0.707711
$367$	−188987.	−1.40314	−0.701570	−	0.712601i	$-0.747517\pi$
$367$	−188987.	−1.40314	−0.701570	+	0.712601i	$0.747517\pi$
$368$	−131329.	−0.969761
$369$	6638.41i	0.0487542i
$370$	− 51447.4i	− 0.375803i
$371$	83970.1i	0.610066i
$372$	−17828.0	−0.128830
$373$	37487.8i	0.269447i	0.990883	+	0.134723i	$0.0430146\pi$
$373$	37487.8i	0.269447i	−0.990883	+	0.134723i	$0.956985\pi$
$374$	0	0
$375$	−85536.9	−0.608263
$376$	− 107980.i	− 0.763781i
$377$	−183156.	−1.28866
$378$	54387.0	0.380638
$379$	−186336.	−1.29723	−0.648617	−	0.761115i	$-0.724652\pi$
$379$	−186336.	−1.29723	−0.648617	+	0.761115i	$0.724652\pi$
$380$	29660.4i	0.205404i
$381$	− 118051.i	− 0.813241i
$382$	118221.i	0.810158i
$383$	−120052.	−0.818413	−0.409206	−	0.912442i	$-0.634194\pi$
$383$	−120052.	−0.818413	−0.409206	+	0.912442i	$0.634194\pi$
$384$	99411.8i	0.674180i
$385$	0	0
$386$	124720.	0.837072
$387$	95071.5i	0.634788i
$388$	33217.4	0.220649
$389$	126406.	0.835351	0.417676	−	0.908596i	$-0.362845\pi$
$389$	126406.	0.835351	0.417676	+	0.908596i	$0.362845\pi$
$390$	72587.9	0.477238
$391$	136847.i	0.895118i
$392$	198206.i	1.28986i
$393$	− 54864.9i	− 0.355230i
$394$	191284.	1.23221
$395$	141925.i	0.909630i
$396$	0	0
$397$	180759.	1.14688	0.573441	−	0.819247i	$-0.305608\pi$
$397$	180759.	1.14688	0.573441	+	0.819247i	$0.305608\pi$
$398$	271893.i	1.71645i
$399$	115458.	0.725236
$400$	102855.	0.642846
$401$	−262540.	−1.63270	−0.816351	−	0.577556i	$-0.804007\pi$
$401$	−262540.	−1.63270	−0.816351	+	0.577556i	$0.804007\pi$
$402$	1720.30i	0.0106452i
$403$	92623.5i	0.570310i
$404$	128327.i	0.786243i
$405$	12641.5	0.0770706
$406$	416365.i	2.52594i
$407$	0	0
$408$	78555.9	0.471909
$409$	3562.25i	0.0212950i	0.999943	+	0.0106475i	$0.00338927\pi$
$409$	3562.25i	0.0212950i	−0.999943	+	0.0106475i	$0.996611\pi$
$410$	20141.3	0.119817
$411$	−100997.	−0.597896
$412$	−124526.	−0.733610
$413$	178017.i	1.04367i
$414$	− 52814.4i	− 0.308143i
$415$	− 5506.34i	− 0.0319718i
$416$	−130694.	−0.755214
$417$	− 109742.i	− 0.631106i
$418$	0	0
$419$	287386.	1.63696	0.818480	−	0.574536i	$-0.194817\pi$
$419$	287386.	1.63696	0.818480	+	0.574536i	$0.194817\pi$
$420$	− 46707.1i	− 0.264779i
$421$	−65613.8	−0.370195	−0.185098	−	0.982720i	$-0.559260\pi$
$421$	−65613.8	−0.370195	−0.185098	+	0.982720i	$0.559260\pi$
$422$	170488.	0.957344
$423$	63734.2	0.356198
$424$	− 46808.7i	− 0.260372i
$425$	− 107177.i	− 0.593365i
$426$	11250.9i	0.0619965i
$427$	316923.	1.73820
$428$	58361.2i	0.318593i
$429$	0	0
$430$	288452.	1.56004
$431$	− 13257.5i	− 0.0713688i	−0.999363	−	0.0356844i	$-0.988639\pi$
$431$	− 13257.5i	− 0.0713688i	0.999363	−	0.0356844i	$-0.0113611\pi$
$432$	−44497.3	−0.238433
$433$	123397.	0.658156	0.329078	−	0.944303i	$-0.393262\pi$
$433$	123397.	0.658156	0.329078	+	0.944303i	$0.393262\pi$
$434$	210560.	1.11788
$435$	96778.3i	0.511445i
$436$	88643.5i	0.466309i
$437$	− 112120.i	− 0.587110i
$438$	−163422.	−0.851848
$439$	141451.i	0.733969i	0.930227	+	0.366984i	$0.119610\pi$
$439$	141451.i	0.733969i	−0.930227	+	0.366984i	$0.880390\pi$
$440$	0	0
$441$	−116989.	−0.601543
$442$	266239.i	1.36279i
$443$	167118.	0.851560	0.425780	−	0.904827i	$-0.360000\pi$
$443$	167118.	0.851560	0.425780	+	0.904827i	$0.360000\pi$
$444$	−20613.6	−0.104566
$445$	−91915.1	−0.464159
$446$	− 25123.0i	− 0.126299i
$447$	95770.7i	0.479311i
$448$	− 119324.i	− 0.594528i
$449$	365924.	1.81509	0.907545	−	0.419954i	$-0.137954\pi$
$449$	365924.	1.81509	0.907545	+	0.419954i	$0.137954\pi$
$450$	41363.6i	0.204265i
$451$	0	0
$452$	−100100.	−0.489954
$453$	31586.0i	0.153921i
$454$	88950.4	0.431555
$455$	−242661.	−1.17213
$456$	−64361.6	−0.309526
$457$	− 258654.i	− 1.23848i	−0.785203	−	0.619238i	$-0.787442\pi$
$457$	− 258654.i	− 1.23848i	0.785203	−	0.619238i	$-0.212558\pi$
$458$	− 57797.0i	− 0.275533i
$459$	46366.8i	0.220080i
$460$	−45356.5	−0.214350
$461$	41493.9i	0.195246i	0.995223	+	0.0976231i	$0.0311240\pi$
$461$	41493.9i	0.195246i	−0.995223	+	0.0976231i	$0.968876\pi$
$462$	0	0
$463$	−300615.	−1.40232	−0.701161	−	0.713003i	$-0.747335\pi$
$463$	−300615.	−1.40232	−0.701161	+	0.713003i	$0.747335\pi$
$464$	− 340653.i	− 1.58226i
$465$	48941.6	0.226346
$466$	163127.	0.751197
$467$	−194042.	−0.889738	−0.444869	−	0.895596i	$-0.646750\pi$
$467$	−194042.	−0.889738	−0.444869	+	0.895596i	$0.646750\pi$
$468$	− 29084.1i	− 0.132789i
$469$	− 5750.96i	− 0.0261454i
$470$	− 193373.i	− 0.875386i
$471$	119360.	0.538041
$472$	− 99234.8i	− 0.445431i
$473$	0	0
$474$	200903.	0.894188
$475$	87810.9i	0.389190i
$476$	171313.	0.756096
$477$	27628.3	0.121428
$478$	−292636.	−1.28077
$479$	− 195928.i	− 0.853938i	−0.904266	−	0.426969i	$-0.859581\pi$
$479$	− 195928.i	− 0.853938i	0.904266	−	0.426969i	$-0.140419\pi$
$480$	69058.0i	0.299731i
$481$	107096.i	0.462895i
$482$	48077.0	0.206939
$483$	176558.i	0.756823i
$484$	0	0
$485$	−91188.7	−0.387666
$486$	− 17894.7i	− 0.0757622i
$487$	92827.3	0.391397	0.195699	−	0.980664i	$-0.437303\pi$
$487$	92827.3	0.391397	0.195699	+	0.980664i	$0.437303\pi$
$488$	−176667.	−0.741851
$489$	−131262.	−0.548936
$490$	354950.i	1.47834i
$491$	− 125334.i	− 0.519884i	−0.965624	−	0.259942i	$-0.916296\pi$
$491$	− 125334.i	− 0.519884i	0.965624	−	0.259942i	$-0.0837035\pi$
$492$	− 8070.09i	− 0.0333386i
$493$	−354965.	−1.46047
$494$	− 218133.i	− 0.893855i
$495$	0	0
$496$	−172271.	−0.700245
$497$	− 37611.7i	− 0.152268i
$498$	−7794.52	−0.0314290
$499$	79119.9	0.317749	0.158875	−	0.987299i	$-0.449213\pi$
$499$	79119.9	0.317749	0.158875	+	0.987299i	$0.449213\pi$
$500$	103984.	0.415937
$501$	207077.i	0.825006i
$502$	− 22028.7i	− 0.0874141i
$503$	141888.i	0.560802i	0.959883	+	0.280401i	$0.0904674\pi$
$503$	141888.i	0.560802i	−0.959883	+	0.280401i	$0.909533\pi$
$504$	101352.	0.398999
$505$	− 352285.i	− 1.38138i
$506$	0	0
$507$	−2695.58	−0.0104867
$508$	143510.i	0.556104i
$509$	257010.	0.992005	0.496003	−	0.868321i	$-0.334801\pi$
$509$	257010.	0.992005	0.496003	+	0.868321i	$0.334801\pi$
$510$	140679.	0.540866
$511$	546319.	2.09221
$512$	− 10943.3i	− 0.0417453i
$513$	− 37988.8i	− 0.144351i
$514$	− 245282.i	− 0.928408i
$515$	341849.	1.28890
$516$	− 115575.i	− 0.434075i
$517$	0	0
$518$	243459.	0.907333
$519$	151928.i	0.564031i
$520$	135270.	0.500260
$521$	17559.0	0.0646879	0.0323440	−	0.999477i	$-0.489703\pi$
$521$	17559.0	0.0646879	0.0323440	+	0.999477i	$0.489703\pi$
$522$	136995.	0.502763
$523$	− 437021.i	− 1.59771i	−0.601522	−	0.798856i	$-0.705439\pi$
$523$	− 437021.i	− 1.59771i	0.601522	−	0.798856i	$-0.294561\pi$
$524$	66697.4i	0.242910i
$525$	− 138278.i	− 0.501691i
$526$	−329177.	−1.18976
$527$	179509.i	0.646347i
$528$	0	0
$529$	−108388.	−0.387319
$530$	− 83825.8i	− 0.298419i
$531$	58572.3	0.207732
$532$	−140359.	−0.495925
$533$	−41927.2	−0.147585
$534$	130111.i	0.456280i
$535$	− 160214.i	− 0.559747i
$536$	3205.84i	0.0111587i
$537$	−65701.4	−0.227838
$538$	− 264416.i	− 0.913531i
$539$	0	0
$540$	−15367.8	−0.0527017
$541$	− 116691.i	− 0.398697i	−0.979929	−	0.199349i	$-0.936117\pi$
$541$	− 116691.i	− 0.398697i	0.979929	−	0.199349i	$-0.0638826\pi$
$542$	−399312.	−1.35930
$543$	19398.9	0.0657927
$544$	−253292.	−0.855902
$545$	− 243345.i	− 0.819274i
$546$	343500.i	1.15224i
$547$	− 75756.8i	− 0.253190i	−0.991954	−	0.126595i	$-0.959595\pi$
$547$	− 75756.8i	− 0.253190i	0.991954	−	0.126595i	$-0.0404049\pi$
$548$	122779.	0.408848
$549$	− 104276.i	− 0.345971i
$550$	0	0
$551$	290827.	0.957924
$552$	− 98421.6i	− 0.323007i
$553$	−671617.	−2.19620
$554$	−306342.	−0.998129
$555$	56588.7	0.183715
$556$	133410.i	0.431558i
$557$	470751.i	1.51733i	0.651479	+	0.758667i	$0.274149\pi$
$557$	470751.i	1.51733i	−0.651479	+	0.758667i	$0.725851\pi$
$558$	− 69279.6i	− 0.222503i
$559$	−600457.	−1.92158
$560$	− 451328.i	− 1.43918i
$561$	0	0
$562$	13733.5	0.0434820
$563$	56944.3i	0.179653i	0.995957	+	0.0898263i	$0.0286312\pi$
$563$	56944.3i	0.179653i	−0.995957	+	0.0898263i	$0.971369\pi$
$564$	−77479.4	−0.243572
$565$	274794.	0.860817
$566$	665656.	2.07786
$567$	59822.1i	0.186078i
$568$	20966.4i	0.0649871i
$569$	405682.i	1.25303i	0.779409	+	0.626515i	$0.215519\pi$
$569$	405682.i	1.25303i	−0.779409	+	0.626515i	$0.784481\pi$
$570$	−115260.	−0.354755
$571$	196880.i	0.603851i	0.953331	+	0.301926i	$0.0976294\pi$
$571$	196880.i	0.603851i	−0.953331	+	0.301926i	$0.902371\pi$
$572$	0	0
$573$	−130036.	−0.396053
$574$	95312.6i	0.289285i
$575$	−134280.	−0.406140
$576$	−39260.7	−0.118335
$577$	253346.	0.760962	0.380481	−	0.924789i	$-0.375759\pi$
$577$	253346.	0.760962	0.380481	+	0.924789i	$0.375759\pi$
$578$	121427.i	0.363464i
$579$	137184.i	0.409210i
$580$	− 117650.i	− 0.349732i
$581$	26057.1	0.0771922
$582$	129083.i	0.381085i
$583$	0	0
$584$	−304543.	−0.892941
$585$	79841.8i	0.233302i
$586$	−317987.	−0.926007
$587$	−601990.	−1.74708	−0.873541	−	0.486750i	$-0.838182\pi$
$587$	−601990.	−1.74708	−0.873541	+	0.486750i	$0.838182\pi$
$588$	142219.	0.411342
$589$	− 147074.i	− 0.423940i
$590$	− 177711.i	− 0.510518i
$591$	210400.i	0.602379i
$592$	−199189.	−0.568357
$593$	165959.i	0.471945i	0.971760	+	0.235973i	$0.0758276\pi$
$593$	165959.i	0.471945i	−0.971760	+	0.235973i	$0.924172\pi$
$594$	0	0
$595$	−470290.	−1.32841
$596$	− 116425.i	− 0.327759i
$597$	−299064.	−0.839103
$598$	333568.	0.932785
$599$	421440.	1.17458	0.587290	−	0.809377i	$-0.300195\pi$
$599$	421440.	1.17458	0.587290	+	0.809377i	$0.300195\pi$
$600$	77082.6i	0.214118i
$601$	− 202063.i	− 0.559421i	−0.960084	−	0.279710i	$-0.909762\pi$
$601$	− 202063.i	− 0.559421i	0.960084	−	0.279710i	$-0.0902385\pi$
$602$	1.36501e6i	3.76655i
$603$	−1892.21	−0.00520398
$604$	− 38398.0i	− 0.105253i
$605$	0	0
$606$	−498679.	−1.35793
$607$	− 73350.5i	− 0.199079i	−0.995034	−	0.0995395i	$-0.968263\pi$
$607$	− 73350.5i	− 0.199079i	0.995034	−	0.0995395i	$-0.0317370\pi$
$608$	207525.	0.561388
$609$	−457974.	−1.23483
$610$	−316379.	−0.850252
$611$	402536.i	1.07826i
$612$	− 56366.4i	− 0.150493i
$613$	− 65690.1i	− 0.174815i	−0.996173	−	0.0874075i	$-0.972142\pi$
$613$	− 65690.1i	− 0.174815i	0.996173	−	0.0874075i	$-0.0278582\pi$
$614$	−403062.	−1.06914
$615$	22154.1i	0.0585738i
$616$	0	0
$617$	−300650.	−0.789752	−0.394876	−	0.918734i	$-0.629213\pi$
$617$	−300650.	−0.789752	−0.394876	+	0.918734i	$0.629213\pi$
$618$	− 483906.i	− 1.26702i
$619$	−378552.	−0.987972	−0.493986	−	0.869470i	$-0.664461\pi$
$619$	−378552.	−0.987972	−0.493986	+	0.869470i	$0.664461\pi$
$620$	−59496.6	−0.154778
$621$	58092.3	0.150638
$622$	145380.i	0.375771i
$623$	− 434960.i	− 1.12066i
$624$	− 281038.i	− 0.721765i
$625$	−82774.6	−0.211903
$626$	− 537187.i	− 1.37081i
$627$	0	0
$628$	−145101.	−0.367918
$629$	207557.i	0.524610i
$630$	181503.	0.457302
$631$	−229677.	−0.576845	−0.288423	−	0.957503i	$-0.593131\pi$
$631$	−229677.	−0.576845	−0.288423	+	0.957503i	$0.593131\pi$
$632$	374389.	0.937323
$633$	187525.i	0.468006i
$634$	− 233974.i	− 0.582089i
$635$	− 393966.i	− 0.977037i
$636$	−33586.8	−0.0830337
$637$	− 738883.i	− 1.82094i
$638$	0	0
$639$	−12375.2	−0.0303075
$640$	331762.i	0.809967i
$641$	556155.	1.35357	0.676784	−	0.736182i	$-0.263373\pi$
$641$	556155.	1.35357	0.676784	+	0.736182i	$0.263373\pi$
$642$	−226791.	−0.550245
$643$	−599886.	−1.45093	−0.725466	−	0.688259i	$-0.758375\pi$
$643$	−599886.	−1.45093	−0.725466	+	0.688259i	$0.758375\pi$
$644$	− 214636.i	− 0.517524i
$645$	317278.i	0.762641i
$646$	− 422753.i	− 1.01303i
$647$	−371278.	−0.886932	−0.443466	−	0.896291i	$-0.646251\pi$
$647$	−371278.	−0.886932	−0.443466	+	0.896291i	$0.646251\pi$
$648$	− 33347.5i	− 0.0794169i
$649$	0	0
$650$	−261246.	−0.618334
$651$	231602.i	0.546486i
$652$	159571.	0.375369
$653$	423861.	0.994023	0.497012	−	0.867744i	$-0.334431\pi$
$653$	423861.	0.994023	0.497012	+	0.867744i	$0.334431\pi$
$654$	−344468.	−0.805366
$655$	− 183098.i	− 0.426777i
$656$	− 77980.9i	− 0.181209i
$657$	− 179753.i	− 0.416434i
$658$	915078.	2.11352
$659$	− 620818.i	− 1.42953i	−0.699364	−	0.714765i	$-0.746534\pi$
$659$	− 620818.i	− 1.42953i	0.699364	−	0.714765i	$-0.253466\pi$
$660$	0	0
$661$	69379.3	0.158791	0.0793957	−	0.996843i	$-0.474701\pi$
$661$	69379.3	0.158791	0.0793957	+	0.996843i	$0.474701\pi$
$662$	702055.i	1.60197i
$663$	−292845.	−0.666210
$664$	−14525.4	−0.0329451
$665$	385314.	0.871307
$666$	− 80104.4i	− 0.180596i
$667$	444731.i	0.999646i
$668$	− 251737.i	− 0.564149i
$669$	27633.6	0.0617426
$670$	5741.08i	0.0127892i
$671$	0	0
$672$	−326796.	−0.723666
$673$	− 605741.i	− 1.33739i	−0.743538	−	0.668693i	$-0.766854\pi$
$673$	− 605741.i	− 1.33739i	0.743538	−	0.668693i	$-0.233146\pi$
$674$	−521567.	−1.14813
$675$	−45497.2	−0.0998567
$676$	3276.93	0.00717089
$677$	3316.52i	0.00723612i	0.999993	+	0.00361806i	$0.00115167\pi$
$677$	3316.52i	0.00723612i	−0.999993	+	0.00361806i	$0.998848\pi$
$678$	− 388986.i	− 0.846204i
$679$	− 431523.i	− 0.935975i
$680$	262161.	0.566957
$681$	97839.5i	0.210970i
$682$	0	0
$683$	780979.	1.67416	0.837082	−	0.547078i	$-0.184260\pi$
$683$	780979.	1.67416	0.837082	+	0.547078i	$0.184260\pi$
$684$	46181.6i	0.0987090i
$685$	−337053.	−0.718318
$686$	−748924.	−1.59144
$687$	63572.8	0.134697
$688$	− 1.11680e6i	− 2.35938i
$689$	174496.i	0.367577i
$690$	− 176255.i	− 0.370206i
$691$	−249922.	−0.523417	−0.261708	−	0.965147i	$-0.584286\pi$
$691$	−249922.	−0.523417	−0.261708	+	0.965147i	$0.584286\pi$
$692$	− 184694.i	− 0.385691i
$693$	0	0
$694$	−198457.	−0.412047
$695$	− 366238.i	− 0.758218i
$696$	255295.	0.527016
$697$	−81257.2	−0.167262
$698$	−443175.	−0.909629
$699$	179429.i	0.367229i
$700$	168100.i	0.343062i
$701$	282457.i	0.574799i	0.957811	+	0.287399i	$0.0927907\pi$
$701$	282457.i	0.574799i	−0.957811	+	0.287399i	$0.907209\pi$
$702$	113020.	0.229341
$703$	− 170054.i	− 0.344093i
$704$	0	0
$705$	212697.	0.427941
$706$	205967.i	0.413227i
$707$	1.66708e6	3.33517
$708$	−71204.3	−0.142049
$709$	714406.	1.42119	0.710595	−	0.703601i	$-0.248426\pi$
$709$	714406.	1.42119	0.710595	+	0.703601i	$0.248426\pi$
$710$	37547.0i	0.0744833i
$711$	220979.i	0.437132i
$712$	242466.i	0.478290i
$713$	224905.	0.442404
$714$	665721.i	1.30586i
$715$	0	0
$716$	79870.9	0.155798
$717$	− 321880.i	− 0.626118i
$718$	−929112.	−1.80227
$719$	705192.	1.36411	0.682055	−	0.731301i	$-0.261086\pi$
$719$	705192.	1.36411	0.682055	+	0.731301i	$0.261086\pi$
$720$	−148499.	−0.286456
$721$	1.61770e6i	3.11191i
$722$	− 269279.i	− 0.516569i
$723$	52881.4i	0.101164i
$724$	−23582.6	−0.0449898
$725$	− 348308.i	− 0.662655i
$726$	0	0
$727$	−348406.	−0.659200	−0.329600	−	0.944121i	$-0.606914\pi$
$727$	−348406.	−0.659200	−0.329600	+	0.944121i	$0.606914\pi$
$728$	640125.i	1.20782i
$729$	19683.0	0.0370370
$730$	−545380.	−1.02342
$731$	−1.16372e6	−2.17777
$732$	126765.i	0.236579i
$733$	− 507563.i	− 0.944673i	−0.881418	−	0.472337i	$-0.843411\pi$
$733$	− 507563.i	− 0.944673i	0.881418	−	0.472337i	$-0.156589\pi$
$734$	892789.i	1.65713i
$735$	−390421.	−0.722701
$736$	317347.i	0.585839i
$737$	0	0
$738$	31360.3	0.0575794
$739$	713204.i	1.30595i	0.757381	+	0.652973i	$0.226478\pi$
$739$	713204.i	1.30595i	−0.757381	+	0.652973i	$0.773522\pi$
$740$	−68792.9	−0.125626
$741$	239931.	0.436969
$742$	396680.	0.720498
$743$	− 933593.i	− 1.69114i	−0.533863	−	0.845571i	$-0.679260\pi$
$743$	− 933593.i	− 1.69114i	0.533863	−	0.845571i	$-0.320740\pi$
$744$	− 129105.i	− 0.233237i
$745$	319611.i	0.575850i
$746$	177095.	0.318221
$747$	− 8573.45i	− 0.0153644i
$748$	0	0
$749$	758163.	1.35145
$750$	404082.i	0.718368i
$751$	119751.	0.212324	0.106162	−	0.994349i	$-0.466144\pi$
$751$	119751.	0.212324	0.106162	+	0.994349i	$0.466144\pi$
$752$	−748680.	−1.32392
$753$	24230.1	0.0427332
$754$	865239.i	1.52193i
$755$	105411.i	0.184923i
$756$	− 72723.6i	− 0.127242i
$757$	−237232.	−0.413982	−0.206991	−	0.978343i	$-0.566367\pi$
$757$	−237232.	−0.413982	−0.206991	+	0.978343i	$0.566367\pi$
$758$	880262.i	1.53205i
$759$	0	0
$760$	−214791.	−0.371868
$761$	55190.5i	0.0953005i	0.998864	+	0.0476503i	$0.0151733\pi$
$761$	55190.5i	0.0953005i	−0.998864	+	0.0476503i	$0.984827\pi$
$762$	−557680.	−0.960451
$763$	1.15156e6	1.97804
$764$	158080.	0.270825
$765$	154738.i	0.264407i
$766$	567134.i	0.966559i
$767$	369934.i	0.628830i
$768$	348736.	0.591255
$769$	73815.8i	0.124824i	0.998050	+	0.0624118i	$0.0198792\pi$
$769$	73815.8i	0.124824i	−0.998050	+	0.0624118i	$0.980121\pi$
$770$	0	0
$771$	269793.	0.453860
$772$	− 166770.i	− 0.279823i
$773$	−370770.	−0.620506	−0.310253	−	0.950654i	$-0.600414\pi$
$773$	−370770.	−0.620506	−0.310253	+	0.950654i	$0.600414\pi$
$774$	449124.	0.749694
$775$	−176143.	−0.293266
$776$	240550.i	0.399468i
$777$	267789.i	0.443558i
$778$	− 597151.i	− 0.986563i
$779$	66574.8	0.109707
$780$	− 97060.9i	− 0.159535i
$781$	0	0
$782$	646472.	1.05715
$783$	150685.i	0.245780i
$784$	1.37426e6	2.23581
$785$	398333.	0.646408
$786$	−259186.	−0.419533
$787$	163047.i	0.263247i	0.991300	+	0.131623i	$0.0420189\pi$
$787$	163047.i	0.263247i	−0.991300	+	0.131623i	$0.957981\pi$
$788$	− 255775.i	− 0.411914i
$789$	− 362072.i	− 0.581623i
$790$	670463.	1.07429
$791$	1.30038e6i	2.07834i
$792$	0	0
$793$	658591.	1.04730
$794$	− 853916.i	− 1.35448i
$795$	92202.7	0.145885
$796$	363561.	0.573788
$797$	437542.	0.688816	0.344408	−	0.938820i	$-0.388080\pi$
$797$	437542.	0.688816	0.344408	+	0.938820i	$0.388080\pi$
$798$	− 545433.i	− 0.856516i
$799$	780135.i	1.22201i
$800$	− 248542.i	− 0.388347i
$801$	−143113.	−0.223056
$802$	1.24026e6i	1.92825i
$803$	0	0
$804$	2300.30	0.00355854
$805$	589220.i	0.909255i
$806$	437559.	0.673545
$807$	290840.	0.446588
$808$	−929307.	−1.42343
$809$	− 61091.6i	− 0.0933435i	−0.998910	−	0.0466718i	$-0.985139\pi$
$809$	− 61091.6i	− 0.0933435i	0.998910	−	0.0466718i	$-0.0148615\pi$
$810$	− 59719.2i	− 0.0910216i
$811$	206289.i	0.313641i	0.987627	+	0.156821i	$0.0501245\pi$
$811$	206289.i	0.313641i	−0.987627	+	0.156821i	$0.949876\pi$
$812$	556743.	0.844389
$813$	− 439217.i	− 0.664504i
$814$	0	0
$815$	−438055.	−0.659498
$816$	− 544666.i	− 0.817994i
$817$	953446.	1.42841
$818$	16828.3	0.0251498
$819$	−377827.	−0.563281
$820$	− 26931.9i	− 0.0400534i
$821$	1.33425e6i	1.97948i	0.142885	+	0.989739i	$0.454362\pi$
$821$	1.33425e6i	1.97948i	−0.142885	+	0.989739i	$0.545638\pi$
$822$	477117.i	0.706124i
$823$	−758169.	−1.11935	−0.559676	−	0.828712i	$-0.689074\pi$
$823$	−758169.	−1.11935	−0.559676	+	0.828712i	$0.689074\pi$
$824$	− 901777.i	− 1.32814i
$825$	0	0
$826$	840965.	1.23259
$827$	− 292627.i	− 0.427861i	−0.976849	−	0.213931i	$-0.931373\pi$
$827$	− 292627.i	− 0.427861i	0.976849	−	0.213931i	$-0.0686267\pi$
$828$	−70620.7	−0.103008
$829$	−511408.	−0.744146	−0.372073	−	0.928204i	$-0.621353\pi$
$829$	−511408.	−0.744146	−0.372073	+	0.928204i	$0.621353\pi$
$830$	−26012.3	−0.0377592
$831$	− 336955.i	− 0.487944i
$832$	− 247965.i	− 0.358214i
$833$	− 1.43199e6i	− 2.06372i
$834$	−518430.	−0.745347
$835$	691070.i	0.991172i
$836$	0	0
$837$	76202.9	0.108773
$838$	− 1.35763e6i	− 1.93328i
$839$	−627178.	−0.890977	−0.445489	−	0.895288i	$-0.646970\pi$
$839$	−627178.	−0.890977	−0.445489	+	0.895288i	$0.646970\pi$
$840$	338238.	0.479362
$841$	−446304.	−0.631014
$842$	309964.i	0.437206i
$843$	15106.0i	0.0212566i
$844$	− 227967.i	− 0.320028i
$845$	−8995.85	−0.0125988
$846$	− 301084.i	− 0.420676i
$847$	0	0
$848$	−324548.	−0.451322
$849$	732177.i	1.01578i
$850$	−506309.	−0.700774
$851$	260046.	0.359079
$852$	15044.1	0.0207246
$853$	25187.5i	0.0346168i	0.999850	+	0.0173084i	$0.00550970\pi$
$853$	25187.5i	0.0346168i	−0.999850	+	0.0173084i	$0.994490\pi$
$854$	− 1.49717e6i	− 2.05284i
$855$	− 126778.i	− 0.173425i
$856$	−422634.	−0.576789
$857$	287947.i	0.392058i	0.980598	+	0.196029i	$0.0628047\pi$
$857$	287947.i	0.392058i	−0.980598	+	0.196029i	$0.937195\pi$
$858$	0	0
$859$	633150.	0.858065	0.429032	−	0.903289i	$-0.358855\pi$
$859$	633150.	0.858065	0.429032	+	0.903289i	$0.358855\pi$
$860$	− 385703.i	− 0.521503i
$861$	−104837.	−0.141420
$862$	−62629.5	−0.0842877
$863$	133773.	0.179617	0.0898085	−	0.995959i	$-0.471375\pi$
$863$	133773.	0.179617	0.0898085	+	0.995959i	$0.471375\pi$
$864$	107524.i	0.144039i
$865$	507022.i	0.677633i
$866$	− 582935.i	− 0.777292i
$867$	−133562.	−0.177683
$868$	− 281550.i	− 0.373694i
$869$	0	0
$870$	457187.	0.604025
$871$	− 11950.9i	− 0.0157531i
$872$	−641929.	−0.844216
$873$	−141982.	−0.186297
$874$	−529661.	−0.693386
$875$	− 1.35085e6i	− 1.76437i
$876$	218520.i	0.284762i
$877$	− 236828.i	− 0.307917i	−0.988077	−	0.153958i	$-0.950798\pi$
$877$	− 236828.i	− 0.307917i	0.988077	−	0.153958i	$-0.0492021\pi$
$878$	668225.	0.866829
$879$	− 349764.i	− 0.452687i
$880$	0	0
$881$	772779.	0.995642	0.497821	−	0.867280i	$-0.334134\pi$
$881$	772779.	0.995642	0.497821	+	0.867280i	$0.334134\pi$
$882$	552662.i	0.710432i
$883$	913673.	1.17184	0.585922	−	0.810368i	$-0.300733\pi$
$883$	913673.	1.17184	0.585922	+	0.810368i	$0.300733\pi$
$884$	356002.	0.455562
$885$	195471.	0.249571
$886$	− 789475.i	− 1.00571i
$887$	− 368420.i	− 0.468270i	−0.972204	−	0.234135i	$-0.924774\pi$
$887$	− 368420.i	− 0.468270i	0.972204	−	0.234135i	$-0.0752258\pi$
$888$	− 149277.i	− 0.189308i
$889$	1.86432e6	2.35894
$890$	434213.i	0.548179i
$891$	0	0
$892$	−33593.1	−0.0422203
$893$	− 639173.i	− 0.801522i
$894$	452427.	0.566074
$895$	−219262.	−0.273727
$896$	−1.56997e6	−1.95557
$897$	366902.i	0.456000i
$898$	− 1.72865e6i	− 2.14365i
$899$	583379.i	0.721824i
$900$	55309.3	0.0682831
$901$	338183.i	0.416584i
$902$	0	0
$903$	−1.50142e6	−1.84131
$904$	− 724890.i	− 0.887024i
$905$	64739.1	0.0790441
$906$	149214.	0.181783
$907$	1.21472e6	1.47660	0.738298	−	0.674474i	$-0.235630\pi$
$907$	1.21472e6	1.47660	0.738298	+	0.674474i	$0.235630\pi$
$908$	− 118940.i	− 0.144263i
$909$	− 548513.i	− 0.663834i
$910$	1.14635e6i	1.38431i
$911$	415635.	0.500813	0.250407	−	0.968141i	$-0.419436\pi$
$911$	415635.	0.500813	0.250407	+	0.968141i	$0.419436\pi$
$912$	446251.i	0.536524i
$913$	0	0
$914$	−1.22190e6	−1.46266
$915$	− 347995.i	− 0.415653i
$916$	−77283.2	−0.0921073
$917$	866457.	1.03041
$918$	219040.	0.259918
$919$	− 1.35520e6i	− 1.60462i	−0.596908	−	0.802310i	$-0.703604\pi$
$919$	− 1.35520e6i	− 1.60462i	0.596908	−	0.802310i	$-0.296396\pi$
$920$	− 328458.i	− 0.388064i
$921$	− 443341.i	− 0.522659i
$922$	196020.	0.230589
$923$	− 78159.9i	− 0.0917446i
$924$	0	0
$925$	−203665.	−0.238030
$926$	1.42012e6i	1.65617i
$927$	532265.	0.619396
$928$	−823163.	−0.955850
$929$	1.21537e6	1.40824	0.704121	−	0.710080i	$-0.251341\pi$
$929$	1.21537e6	1.40824	0.704121	+	0.710080i	$0.251341\pi$
$930$	− 231203.i	− 0.267318i
$931$	1.17325e6i	1.35360i
$932$	− 218125.i	− 0.251116i
$933$	−159908.	−0.183699
$934$	916667.i	1.05079i
$935$	0	0
$936$	210618.	0.240405
$937$	− 422605.i	− 0.481344i	−0.970606	−	0.240672i	$-0.922632\pi$
$937$	− 422605.i	− 0.481344i	0.970606	−	0.240672i	$-0.0773678\pi$
$938$	−27167.9	−0.0308781
$939$	590870.	0.670132
$940$	−258568.	−0.292631
$941$	1.03567e6i	1.16962i	0.811172	+	0.584808i	$0.198830\pi$
$941$	1.03567e6i	1.16962i	−0.811172	+	0.584808i	$0.801170\pi$
$942$	− 563862.i	− 0.635435i
$943$	101806.i	0.114485i
$944$	−688044.	−0.772098
$945$	199641.i	0.223556i
$946$	0	0
$947$	181714.	0.202623	0.101312	−	0.994855i	$-0.467696\pi$
$947$	181714.	0.202623	0.101312	+	0.994855i	$0.467696\pi$
$948$	− 268637.i	− 0.298916i
$949$	1.13529e6	1.26060
$950$	414824.	0.459639
$951$	257356.	0.284559
$952$	1.24060e6i	1.36885i
$953$	− 1.21679e6i	− 1.33977i	−0.742467	−	0.669883i	$-0.766344\pi$
$953$	− 1.21679e6i	− 1.33977i	0.742467	−	0.669883i	$-0.233656\pi$
$954$	− 130518.i	− 0.143408i
$955$	−433962.	−0.475822
$956$	391299.i	0.428147i
$957$	0	0
$958$	−925578.	−1.00851
$959$	− 1.59500e6i	− 1.73430i
$960$	−131023.	−0.142169
$961$	−628501.	−0.680549
$962$	505927.	0.546686
$963$	− 249455.i	− 0.268992i
$964$	− 64286.1i	− 0.0691772i
$965$	457818.i	0.491630i
$966$	834073.	0.893820
$967$	− 1.10152e6i	− 1.17798i	−0.808140	−	0.588990i	$-0.799526\pi$
$967$	− 1.10152e6i	− 1.17798i	0.808140	−	0.588990i	$-0.200474\pi$
$968$	0	0
$969$	465000.	0.495228
$970$	430781.i	0.457840i
$971$	−1.21766e6	−1.29148	−0.645739	−	0.763558i	$-0.723451\pi$
$971$	−1.21766e6	−1.29148	−0.645739	+	0.763558i	$0.723451\pi$
$972$	−23927.9	−0.0253264
$973$	1.73311e6	1.83063
$974$	− 438522.i	− 0.462246i
$975$	− 287353.i	− 0.302278i
$976$	1.22492e6i	1.28590i
$977$	−719518.	−0.753794	−0.376897	−	0.926255i	$-0.623009\pi$
$977$	−719518.	−0.753794	−0.376897	+	0.926255i	$0.623009\pi$
$978$	620091.i	0.648302i
$979$	0	0
$980$	474621.	0.494191
$981$	− 378892.i	− 0.393710i
$982$	−592087.	−0.613992
$983$	1.77925e6	1.84132	0.920660	−	0.390366i	$-0.127651\pi$
$983$	1.77925e6	1.84132	0.920660	+	0.390366i	$0.127651\pi$
$984$	58441.1	0.0603570
$985$	702157.i	0.723705i
$986$	1.67688e6i	1.72484i
$987$	1.00652e6i	1.03321i
$988$	−291676.	−0.298804
$989$	1.45801e6i	1.49062i
$990$	0	0
$991$	51647.2	0.0525895	0.0262948	−	0.999654i	$-0.491629\pi$
$991$	51647.2	0.0525895	0.0262948	+	0.999654i	$0.491629\pi$
$992$	416281.i	0.423022i
$993$	−772213.	−0.783138
$994$	−177680.	−0.179831
$995$	−998052.	−1.00811
$996$	10422.4i	0.0105063i
$997$	776154.i	0.780832i	0.920638	+	0.390416i	$0.127669\pi$
$997$	776154.i	0.780832i	−0.920638	+	0.390416i	$0.872331\pi$
$998$	− 373767.i	− 0.375267i
$999$	88109.4	0.0882859

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	363.5.c.e.241.7		32
11.3	even	5		33.5.g.a.13.2	✓	32
11.7	odd	10		33.5.g.a.28.2	yes	32
11.10	odd	2	inner	363.5.c.e.241.26		32
33.14	odd	10		99.5.k.c.46.7		32
33.29	even	10		99.5.k.c.28.7		32

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.5.g.a.13.2	✓	32	11.3	even	5
33.5.g.a.28.2	yes	32	11.7	odd	10
99.5.k.c.28.7		32	33.29	even	10
99.5.k.c.46.7		32	33.14	odd	10
363.5.c.e.241.7		32	1.1	even	1	trivial
363.5.c.e.241.26		32	11.10	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:	\( N \)	\(=\)	\( 363 = 3 \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	363.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-4.72406i q^{2} +5.19615 q^{3} -6.31678 q^{4} +17.3409 q^{5} -24.5470i q^{6} +82.0604i q^{7} -45.7442i q^{8} +27.0000 q^{9} +O(q^{10})\)	\(q-4.72406i q^{2} +5.19615 q^{3} -6.31678 q^{4} +17.3409 q^{5} -24.5470i q^{6} +82.0604i q^{7} -45.7442i q^{8} +27.0000 q^{9} -81.9194i q^{10} -32.8230 q^{12} +170.528i q^{13} +387.659 q^{14} +90.1058 q^{15} -317.167 q^{16} +330.492i q^{17} -127.550i q^{18} -270.776i q^{19} -109.538 q^{20} +426.399i q^{21} +414.069 q^{23} -237.694i q^{24} -324.294 q^{25} +805.585 q^{26} +140.296 q^{27} -518.358i q^{28} +1074.05i q^{29} -425.666i q^{30} +543.157 q^{31} +766.410i q^{32} +1561.27 q^{34} +1423.00i q^{35} -170.553 q^{36} +628.025 q^{37} -1279.16 q^{38} +886.089i q^{39} -793.244i q^{40} +245.867i q^{41} +2014.33 q^{42} +3521.17i q^{43} +468.204 q^{45} -1956.09i q^{46} +2360.53 q^{47} -1648.05 q^{48} -4332.92 q^{49} +1531.99i q^{50} +1717.29i q^{51} -1077.19i q^{52} +1023.27 q^{53} -662.768i q^{54} +3753.79 q^{56} -1406.99i q^{57} +5073.89 q^{58} +2169.34 q^{59} -569.179 q^{60} -3862.07i q^{61} -2565.91i q^{62} +2215.63i q^{63} -1454.10 q^{64} +2957.10i q^{65} -70.0820 q^{67} -2087.65i q^{68} +2151.57 q^{69} +6722.34 q^{70} -458.341 q^{71} -1235.09i q^{72} -6657.52i q^{73} -2966.83i q^{74} -1685.08 q^{75} +1710.43i q^{76} +4185.94 q^{78} +8184.42i q^{79} -5499.95 q^{80} +729.000 q^{81} +1161.49 q^{82} -317.535i q^{83} -2693.47i q^{84} +5731.02i q^{85} +16634.2 q^{86} +5580.93i q^{87} -5300.49 q^{89} -2211.82i q^{90} -13993.6 q^{91} -2615.58 q^{92} +2822.33 q^{93} -11151.3i q^{94} -4695.49i q^{95} +3982.38i q^{96} -5258.60 q^{97} +20469.0i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 32 q - 244 q^{4} + 36 q^{5} + 864 q^{9}+O(q^{10}) \) 32 * q - 244 * q^4 + 36 * q^5 + 864 * q^9	\( 32 q - 244 q^{4} + 36 q^{5} + 864 q^{9} + 360 q^{12} + 2220 q^{14} - 108 q^{15} + 3908 q^{16} + 468 q^{20} - 2196 q^{23} + 7280 q^{25} + 3564 q^{26} - 5872 q^{31} - 2320 q^{34} - 6588 q^{36} - 656 q^{37} - 2616 q^{38} - 1404 q^{42} + 972 q^{45} + 2640 q^{47} - 9936 q^{48} - 6988 q^{49} + 4560 q^{53} - 5604 q^{56} - 24644 q^{58} + 39612 q^{59} + 20592 q^{60} - 6232 q^{64} + 2796 q^{67} - 10476 q^{69} - 72692 q^{70} - 51828 q^{71} - 18072 q^{75} + 53640 q^{78} - 27624 q^{80} + 23328 q^{81} - 11548 q^{82} + 106284 q^{86} - 38748 q^{89} + 30672 q^{91} + 27000 q^{92} + 42624 q^{93} - 50544 q^{97}+O(q^{100}) \) 32 * q - 244 * q^4 + 36 * q^5 + 864 * q^9 + 360 * q^12 + 2220 * q^14 - 108 * q^15 + 3908 * q^16 + 468 * q^20 - 2196 * q^23 + 7280 * q^25 + 3564 * q^26 - 5872 * q^31 - 2320 * q^34 - 6588 * q^36 - 656 * q^37 - 2616 * q^38 - 1404 * q^42 + 972 * q^45 + 2640 * q^47 - 9936 * q^48 - 6988 * q^49 + 4560 * q^53 - 5604 * q^56 - 24644 * q^58 + 39612 * q^59 + 20592 * q^60 - 6232 * q^64 + 2796 * q^67 - 10476 * q^69 - 72692 * q^70 - 51828 * q^71 - 18072 * q^75 + 53640 * q^78 - 27624 * q^80 + 23328 * q^81 - 11548 * q^82 + 106284 * q^86 - 38748 * q^89 + 30672 * q^91 + 27000 * q^92 + 42624 * q^93 - 50544 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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