LMFDB - Embedded newform 8003.2.a.b.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8003,2,Mod(1,8003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8003, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8003 = 53 \cdot 151 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8003.a (trivial)

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:	yes
Analytic conductor:	$63.9042767376$
Analytic rank:	$1$
Dimension:	$153$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	8003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.36991 q^{2} -1.12262 q^{3} +3.61648 q^{4} +0.983312 q^{5} +2.66051 q^{6} +2.71051 q^{7} -3.83091 q^{8} -1.73972 q^{9} +O(q^{10})$	\(q-2.36991 q^{2} -1.12262 q^{3} +3.61648 q^{4} +0.983312 q^{5} +2.66051 q^{6} +2.71051 q^{7} -3.83091 q^{8} -1.73972 q^{9} -2.33036 q^{10} -1.16701 q^{11} -4.05993 q^{12} +3.98883 q^{13} -6.42368 q^{14} -1.10389 q^{15} +1.84596 q^{16} -4.49166 q^{17} +4.12299 q^{18} -7.69190 q^{19} +3.55613 q^{20} -3.04288 q^{21} +2.76571 q^{22} -0.541286 q^{23} +4.30066 q^{24} -4.03310 q^{25} -9.45317 q^{26} +5.32091 q^{27} +9.80251 q^{28} +6.32794 q^{29} +2.61611 q^{30} -9.15254 q^{31} +3.28705 q^{32} +1.31011 q^{33} +10.6448 q^{34} +2.66528 q^{35} -6.29168 q^{36} +5.19039 q^{37} +18.2291 q^{38} -4.47794 q^{39} -3.76698 q^{40} +1.05534 q^{41} +7.21135 q^{42} +3.56306 q^{43} -4.22046 q^{44} -1.71069 q^{45} +1.28280 q^{46} +13.5615 q^{47} -2.07232 q^{48} +0.346880 q^{49} +9.55809 q^{50} +5.04242 q^{51} +14.4255 q^{52} -1.00000 q^{53} -12.6101 q^{54} -1.14753 q^{55} -10.3837 q^{56} +8.63508 q^{57} -14.9967 q^{58} -2.11908 q^{59} -3.99218 q^{60} -1.28622 q^{61} +21.6907 q^{62} -4.71554 q^{63} -11.4820 q^{64} +3.92226 q^{65} -3.10484 q^{66} +14.7669 q^{67} -16.2440 q^{68} +0.607658 q^{69} -6.31647 q^{70} -6.06876 q^{71} +6.66473 q^{72} +4.45096 q^{73} -12.3008 q^{74} +4.52764 q^{75} -27.8176 q^{76} -3.16319 q^{77} +10.6123 q^{78} +5.53740 q^{79} +1.81516 q^{80} -0.754189 q^{81} -2.50106 q^{82} +3.68374 q^{83} -11.0045 q^{84} -4.41670 q^{85} -8.44413 q^{86} -7.10388 q^{87} +4.47071 q^{88} +7.35347 q^{89} +4.05418 q^{90} +10.8118 q^{91} -1.95755 q^{92} +10.2748 q^{93} -32.1395 q^{94} -7.56353 q^{95} -3.69011 q^{96} -16.9296 q^{97} -0.822076 q^{98} +2.03027 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 153 q - 9 q^{2} - 17 q^{3} + 137 q^{4} - 31 q^{5} - 10 q^{6} - 17 q^{7} - 30 q^{8} + 136 q^{9}+O(q^{10}) $ 153 * q - 9 * q^2 - 17 * q^3 + 137 * q^4 - 31 * q^5 - 10 * q^6 - 17 * q^7 - 30 * q^8 + 136 * q^9	\( 153 q - 9 q^{2} - 17 q^{3} + 137 q^{4} - 31 q^{5} - 10 q^{6} - 17 q^{7} - 30 q^{8} + 136 q^{9} - 34 q^{10} - q^{11} - 60 q^{12} - 101 q^{13} - 16 q^{14} - 14 q^{15} + 97 q^{16} - 12 q^{17} - 45 q^{18} - 45 q^{19} - 52 q^{20} - 76 q^{21} - 46 q^{22} - 28 q^{23} - 30 q^{24} + 84 q^{25} - 22 q^{26} - 68 q^{27} - 64 q^{28} - 14 q^{29} - q^{30} - 70 q^{31} - 54 q^{32} - 85 q^{33} - 59 q^{34} - 16 q^{35} + 87 q^{36} - 167 q^{37} - 48 q^{38} - 28 q^{39} - 68 q^{40} - 38 q^{41} + 2 q^{42} - 71 q^{43} - 10 q^{44} - 151 q^{45} - 37 q^{46} - 37 q^{47} - 166 q^{48} + 74 q^{49} - 3 q^{50} - 11 q^{51} - 183 q^{52} - 153 q^{53} - 40 q^{54} - 88 q^{55} - 69 q^{56} - 26 q^{57} - 43 q^{58} - 34 q^{59} - 12 q^{60} - 90 q^{61} - 37 q^{62} - 36 q^{63} + 58 q^{64} - 19 q^{65} + 52 q^{66} - 86 q^{67} - 22 q^{68} - 81 q^{69} - 144 q^{70} - 50 q^{71} - 190 q^{72} - 171 q^{73} - 14 q^{74} - 69 q^{75} - 88 q^{76} - 72 q^{77} - 61 q^{78} - 13 q^{79} - 84 q^{80} + 117 q^{81} - 124 q^{82} - 72 q^{83} - 106 q^{84} - 193 q^{85} - 44 q^{86} - 65 q^{87} - 89 q^{88} - 10 q^{89} - 152 q^{90} - 67 q^{91} - 29 q^{92} - 129 q^{93} - 43 q^{94} - 29 q^{95} - 106 q^{96} - 177 q^{97} - 69 q^{98} - 11 q^{99}+O(q^{100}) \) 153 * q - 9 * q^2 - 17 * q^3 + 137 * q^4 - 31 * q^5 - 10 * q^6 - 17 * q^7 - 30 * q^8 + 136 * q^9 - 34 * q^10 - q^11 - 60 * q^12 - 101 * q^13 - 16 * q^14 - 14 * q^15 + 97 * q^16 - 12 * q^17 - 45 * q^18 - 45 * q^19 - 52 * q^20 - 76 * q^21 - 46 * q^22 - 28 * q^23 - 30 * q^24 + 84 * q^25 - 22 * q^26 - 68 * q^27 - 64 * q^28 - 14 * q^29 - q^30 - 70 * q^31 - 54 * q^32 - 85 * q^33 - 59 * q^34 - 16 * q^35 + 87 * q^36 - 167 * q^37 - 48 * q^38 - 28 * q^39 - 68 * q^40 - 38 * q^41 + 2 * q^42 - 71 * q^43 - 10 * q^44 - 151 * q^45 - 37 * q^46 - 37 * q^47 - 166 * q^48 + 74 * q^49 - 3 * q^50 - 11 * q^51 - 183 * q^52 - 153 * q^53 - 40 * q^54 - 88 * q^55 - 69 * q^56 - 26 * q^57 - 43 * q^58 - 34 * q^59 - 12 * q^60 - 90 * q^61 - 37 * q^62 - 36 * q^63 + 58 * q^64 - 19 * q^65 + 52 * q^66 - 86 * q^67 - 22 * q^68 - 81 * q^69 - 144 * q^70 - 50 * q^71 - 190 * q^72 - 171 * q^73 - 14 * q^74 - 69 * q^75 - 88 * q^76 - 72 * q^77 - 61 * q^78 - 13 * q^79 - 84 * q^80 + 117 * q^81 - 124 * q^82 - 72 * q^83 - 106 * q^84 - 193 * q^85 - 44 * q^86 - 65 * q^87 - 89 * q^88 - 10 * q^89 - 152 * q^90 - 67 * q^91 - 29 * q^92 - 129 * q^93 - 43 * q^94 - 29 * q^95 - 106 * q^96 - 177 * q^97 - 69 * q^98 - 11 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−2.36991	−1.67578	−0.837890	−	0.545839i	$-0.816211\pi$
$2$	−2.36991	−1.67578	−0.837890	+	0.545839i	$0.816211\pi$
$3$	−1.12262	−0.648145	−0.324073	−	0.946032i	$-0.605052\pi$
$3$	−1.12262	−0.648145	−0.324073	+	0.946032i	$0.605052\pi$
$4$	3.61648	1.80824
$5$	0.983312	0.439750	0.219875	−	0.975528i	$-0.429435\pi$
$5$	0.983312	0.439750	0.219875	+	0.975528i	$0.429435\pi$
$6$	2.66051	1.08615
$7$	2.71051	1.02448	0.512239	−	0.858843i	$-0.328816\pi$
$7$	2.71051	1.02448	0.512239	+	0.858843i	$0.328816\pi$
$8$	−3.83091	−1.35443
$9$	−1.73972	−0.579908
$10$	−2.33036	−0.736925
$11$	−1.16701	−0.351866	−0.175933	−	0.984402i	$-0.556294\pi$
$11$	−1.16701	−0.351866	−0.175933	+	0.984402i	$0.556294\pi$
$12$	−4.05993	−1.17200
$13$	3.98883	1.10630	0.553151	−	0.833081i	$-0.313425\pi$
$13$	3.98883	1.10630	0.553151	+	0.833081i	$0.313425\pi$
$14$	−6.42368	−1.71680
$15$	−1.10389	−0.285022
$16$	1.84596	0.461491
$17$	−4.49166	−1.08939	−0.544693	−	0.838635i	$-0.683354\pi$
$17$	−4.49166	−1.08939	−0.544693	+	0.838635i	$0.683354\pi$
$18$	4.12299	0.971798
$19$	−7.69190	−1.76464	−0.882321	−	0.470648i	$-0.844020\pi$
$19$	−7.69190	−1.76464	−0.882321	+	0.470648i	$0.844020\pi$
$20$	3.55613	0.795174
$21$	−3.04288	−0.664010
$22$	2.76571	0.589651
$23$	−0.541286	−0.112866	−0.0564329	−	0.998406i	$-0.517973\pi$
$23$	−0.541286	−0.112866	−0.0564329	+	0.998406i	$0.517973\pi$
$24$	4.30066	0.877868
$25$	−4.03310	−0.806620
$26$	−9.45317	−1.85392
$27$	5.32091	1.02401
$28$	9.80251	1.85250
$29$	6.32794	1.17507	0.587535	−	0.809199i	$-0.300099\pi$
$29$	6.32794	1.17507	0.587535	+	0.809199i	$0.300099\pi$
$30$	2.61611	0.477634
$31$	−9.15254	−1.64385	−0.821923	−	0.569599i	$-0.807099\pi$
$31$	−9.15254	−1.64385	−0.821923	+	0.569599i	$0.807099\pi$
$32$	3.28705	0.581075
$33$	1.31011	0.228060
$34$	10.6448	1.82557
$35$	2.66528	0.450514
$36$	−6.29168	−1.04861
$37$	5.19039	0.853294	0.426647	−	0.904418i	$-0.359695\pi$
$37$	5.19039	0.853294	0.426647	+	0.904418i	$0.359695\pi$
$38$	18.2291	2.95715
$39$	−4.47794	−0.717044
$40$	−3.76698	−0.595612
$41$	1.05534	0.164817	0.0824083	−	0.996599i	$-0.473739\pi$
$41$	1.05534	0.164817	0.0824083	+	0.996599i	$0.473739\pi$
$42$	7.21135	1.11274
$43$	3.56306	0.543361	0.271681	−	0.962387i	$-0.412421\pi$
$43$	3.56306	0.543361	0.271681	+	0.962387i	$0.412421\pi$
$44$	−4.22046	−0.636259
$45$	−1.71069	−0.255015
$46$	1.28280	0.189138
$47$	13.5615	1.97815	0.989073	−	0.147429i	$-0.0470998\pi$
$47$	13.5615	1.97815	0.989073	+	0.147429i	$0.0470998\pi$
$48$	−2.07232	−0.299113
$49$	0.346880	0.0495544
$50$	9.55809	1.35172
$51$	5.04242	0.706081
$52$	14.4255	2.00046
$53$	−1.00000	−0.137361
$53$	−1.00000	−0.137361
$54$	−12.6101	−1.71602
$55$	−1.14753	−0.154733
$56$	−10.3837	−1.38759
$57$	8.63508	1.14374
$58$	−14.9967	−1.96916
$59$	−2.11908	−0.275881	−0.137940	−	0.990441i	$-0.544048\pi$
$59$	−2.11908	−0.275881	−0.137940	+	0.990441i	$0.544048\pi$
$60$	−3.99218	−0.515388
$61$	−1.28622	−0.164683	−0.0823415	−	0.996604i	$-0.526240\pi$
$61$	−1.28622	−0.164683	−0.0823415	+	0.996604i	$0.526240\pi$
$62$	21.6907	2.75472
$63$	−4.71554	−0.594103
$64$	−11.4820	−1.43524
$65$	3.92226	0.486496
$66$	−3.10484	−0.382179
$67$	14.7669	1.80407	0.902034	−	0.431665i	$-0.142074\pi$
$67$	14.7669	1.80407	0.902034	+	0.431665i	$0.142074\pi$
$68$	−16.2440	−1.96987
$69$	0.607658	0.0731535
$70$	−6.31647	−0.754963
$71$	−6.06876	−0.720229	−0.360115	−	0.932908i	$-0.617262\pi$
$71$	−6.06876	−0.720229	−0.360115	+	0.932908i	$0.617262\pi$
$72$	6.66473	0.785446
$73$	4.45096	0.520945	0.260473	−	0.965481i	$-0.416122\pi$
$73$	4.45096	0.520945	0.260473	+	0.965481i	$0.416122\pi$
$74$	−12.3008	−1.42993
$75$	4.52764	0.522807
$76$	−27.8176	−3.19090
$77$	−3.16319	−0.360479
$78$	10.6123	1.20161
$79$	5.53740	0.623006	0.311503	−	0.950245i	$-0.399168\pi$
$79$	5.53740	0.623006	0.311503	+	0.950245i	$0.399168\pi$
$80$	1.81516	0.202941
$81$	−0.754189	−0.0837988
$82$	−2.50106	−0.276196
$83$	3.68374	0.404343	0.202171	−	0.979350i	$-0.435200\pi$
$83$	3.68374	0.404343	0.202171	+	0.979350i	$0.435200\pi$
$84$	−11.0045	−1.20069
$85$	−4.41670	−0.479058
$86$	−8.44413	−0.910554
$87$	−7.10388	−0.761615
$88$	4.47071	0.476579
$89$	7.35347	0.779466	0.389733	−	0.920928i	$-0.372567\pi$
$89$	7.35347	0.779466	0.389733	+	0.920928i	$0.372567\pi$
$90$	4.05418	0.427349
$91$	10.8118	1.13338
$92$	−1.95755	−0.204089
$93$	10.2748	1.06545
$94$	−32.1395	−3.31494
$95$	−7.56353	−0.776002
$96$	−3.69011	−0.376621
$97$	−16.9296	−1.71894	−0.859470	−	0.511187i	$-0.829206\pi$
$97$	−16.9296	−1.71894	−0.859470	+	0.511187i	$0.829206\pi$
$98$	−0.822076	−0.0830422
$99$	2.03027	0.204050
$100$	−14.5856	−1.45856
$101$	−2.24894	−0.223777	−0.111889	−	0.993721i	$-0.535690\pi$
$101$	−2.24894	−0.223777	−0.111889	+	0.993721i	$0.535690\pi$
$102$	−11.9501	−1.18324
$103$	0.513601	0.0506066	0.0253033	−	0.999680i	$-0.491945\pi$
$103$	0.513601	0.0506066	0.0253033	+	0.999680i	$0.491945\pi$
$104$	−15.2808	−1.49841
$105$	−2.99210	−0.291999
$106$	2.36991	0.230186
$107$	1.93001	0.186581	0.0932907	−	0.995639i	$-0.470261\pi$
$107$	1.93001	0.186581	0.0932907	+	0.995639i	$0.470261\pi$
$108$	19.2430	1.85165
$109$	9.12126	0.873658	0.436829	−	0.899544i	$-0.356101\pi$
$109$	9.12126	0.873658	0.436829	+	0.899544i	$0.356101\pi$
$110$	2.71955	0.259299
$111$	−5.82683	−0.553059
$112$	5.00351	0.472787
$113$	15.8955	1.49532	0.747661	−	0.664081i	$-0.231177\pi$
$113$	15.8955	1.49532	0.747661	+	0.664081i	$0.231177\pi$
$114$	−20.4644	−1.91666
$115$	−0.532253	−0.0496328
$116$	22.8849	2.12481
$117$	−6.93946	−0.641553
$118$	5.02204	0.462316
$119$	−12.1747	−1.11605
$120$	4.22889	0.386043
$121$	−9.63809	−0.876190
$122$	3.04822	0.275973
$123$	−1.18475	−0.106825
$124$	−33.1000	−2.97247
$125$	−8.88235	−0.794462
$126$	11.1754	0.995586
$127$	−13.2914	−1.17942	−0.589709	−	0.807616i	$-0.700758\pi$
$127$	−13.2914	−1.17942	−0.589709	+	0.807616i	$0.700758\pi$
$128$	20.6371	1.82408
$129$	−3.99996	−0.352177
$130$	−9.29541	−0.815261
$131$	10.1842	0.889796	0.444898	−	0.895581i	$-0.353240\pi$
$131$	10.1842	0.889796	0.444898	+	0.895581i	$0.353240\pi$
$132$	4.73798	0.412388
$133$	−20.8490	−1.80784
$134$	−34.9963	−3.02322
$135$	5.23211	0.450309
$136$	17.2071	1.47550
$137$	−13.7811	−1.17740	−0.588701	−	0.808351i	$-0.700360\pi$
$137$	−13.7811	−1.17740	−0.588701	+	0.808351i	$0.700360\pi$
$138$	−1.44010	−0.122589
$139$	−21.5588	−1.82859	−0.914297	−	0.405046i	$-0.867256\pi$
$139$	−21.5588	−1.82859	−0.914297	+	0.405046i	$0.867256\pi$
$140$	9.63893	0.814638
$141$	−15.2244	−1.28213
$142$	14.3824	1.20695
$143$	−4.65500	−0.389270
$144$	−3.21147	−0.267622
$145$	6.22234	0.516737
$146$	−10.5484	−0.872990
$147$	−0.389415	−0.0321184
$148$	18.7709	1.54296
$149$	9.36635	0.767321	0.383661	−	0.923474i	$-0.374663\pi$
$149$	9.36635	0.767321	0.383661	+	0.923474i	$0.374663\pi$
$150$	−10.7301	−0.876109
$151$	−1.00000	−0.0813788
$151$	−1.00000	−0.0813788
$152$	29.4670	2.39009
$153$	7.81424	0.631744
$154$	7.49648	0.604084
$155$	−8.99980	−0.722882
$156$	−16.1944	−1.29659
$157$	−9.54420	−0.761710	−0.380855	−	0.924635i	$-0.624370\pi$
$157$	−9.54420	−0.761710	−0.380855	+	0.924635i	$0.624370\pi$
$158$	−13.1231	−1.04402
$159$	1.12262	0.0890296
$160$	3.23220	0.255528
$161$	−1.46716	−0.115629
$162$	1.78736	0.140428
$163$	23.7545	1.86060	0.930300	−	0.366800i	$-0.119547\pi$
$163$	23.7545	1.86060	0.930300	+	0.366800i	$0.119547\pi$
$164$	3.81662	0.298028
$165$	1.28824	0.100290
$166$	−8.73013	−0.677589
$167$	10.3388	0.800039	0.400019	−	0.916507i	$-0.369003\pi$
$167$	10.3388	0.800039	0.400019	+	0.916507i	$0.369003\pi$
$168$	11.6570	0.899357
$169$	2.91074	0.223903
$170$	10.4672	0.802796
$171$	13.3818	1.02333
$172$	12.8857	0.982528
$173$	−10.0843	−0.766692	−0.383346	−	0.923605i	$-0.625228\pi$
$173$	−10.0843	−0.766692	−0.383346	+	0.923605i	$0.625228\pi$
$174$	16.8356	1.27630
$175$	−10.9318	−0.826364
$176$	−2.15426	−0.162383
$177$	2.37892	0.178811
$178$	−17.4271	−1.30621
$179$	−0.651822	−0.0487195	−0.0243597	−	0.999703i	$-0.507755\pi$
$179$	−0.651822	−0.0487195	−0.0243597	+	0.999703i	$0.507755\pi$
$180$	−6.18668	−0.461128
$181$	−8.09432	−0.601646	−0.300823	−	0.953680i	$-0.597261\pi$
$181$	−8.09432	−0.601646	−0.300823	+	0.953680i	$0.597261\pi$
$182$	−25.6229	−1.89930
$183$	1.44393	0.106738
$184$	2.07362	0.152869
$185$	5.10377	0.375236
$186$	−24.3504	−1.78546
$187$	5.24180	0.383319
$188$	49.0448	3.57696
$189$	14.4224	1.04907
$190$	17.9249	1.30041
$191$	5.73456	0.414939	0.207469	−	0.978242i	$-0.433477\pi$
$191$	5.73456	0.414939	0.207469	+	0.978242i	$0.433477\pi$
$192$	12.8899	0.930247
$193$	−14.9959	−1.07943	−0.539716	−	0.841847i	$-0.681468\pi$
$193$	−14.9959	−1.07943	−0.539716	+	0.841847i	$0.681468\pi$
$194$	40.1216	2.88056
$195$	−4.40321	−0.315320
$196$	1.25449	0.0896061
$197$	−8.24257	−0.587259	−0.293629	−	0.955919i	$-0.594863\pi$
$197$	−8.24257	−0.587259	−0.293629	+	0.955919i	$0.594863\pi$
$198$	−4.81157	−0.341943
$199$	−15.5799	−1.10443	−0.552216	−	0.833701i	$-0.686218\pi$
$199$	−15.5799	−1.10443	−0.552216	+	0.833701i	$0.686218\pi$
$200$	15.4504	1.09251
$201$	−16.5777	−1.16930
$202$	5.32978	0.375002
$203$	17.1520	1.20383
$204$	18.2358	1.27676
$205$	1.03773	0.0724781
$206$	−1.21719	−0.0848055
$207$	0.941688	0.0654518
$208$	7.36323	0.510548
$209$	8.97651	0.620918
$210$	7.09100	0.489326
$211$	−2.92004	−0.201024	−0.100512	−	0.994936i	$-0.532048\pi$
$211$	−2.92004	−0.201024	−0.100512	+	0.994936i	$0.532048\pi$
$212$	−3.61648	−0.248381
$213$	6.81291	0.466813
$214$	−4.57396	−0.312669
$215$	3.50360	0.238943
$216$	−20.3839	−1.38695
$217$	−24.8081	−1.68408
$218$	−21.6166	−1.46406
$219$	−4.99674	−0.337648
$220$	−4.15003	−0.279795
$221$	−17.9164	−1.20519
$222$	13.8091	0.926805
$223$	−2.61809	−0.175321	−0.0876603	−	0.996150i	$-0.527939\pi$
$223$	−2.61809	−0.175321	−0.0876603	+	0.996150i	$0.527939\pi$
$224$	8.90960	0.595298
$225$	7.01648	0.467765
$226$	−37.6709	−2.50583
$227$	−24.5090	−1.62672	−0.813360	−	0.581761i	$-0.802364\pi$
$227$	−24.5090	−1.62672	−0.813360	+	0.581761i	$0.802364\pi$
$228$	31.2286	2.06816
$229$	−3.84039	−0.253780	−0.126890	−	0.991917i	$-0.540500\pi$
$229$	−3.84039	−0.253780	−0.126890	+	0.991917i	$0.540500\pi$
$230$	1.26139	0.0831737
$231$	3.55106	0.233643
$232$	−24.2418	−1.59155
$233$	4.39610	0.287998	0.143999	−	0.989578i	$-0.454004\pi$
$233$	4.39610	0.287998	0.143999	+	0.989578i	$0.454004\pi$
$234$	16.4459	1.07510
$235$	13.3352	0.869890
$236$	−7.66362	−0.498859
$237$	−6.21639	−0.403798
$238$	28.8529	1.87026
$239$	−9.21626	−0.596150	−0.298075	−	0.954542i	$-0.596345\pi$
$239$	−9.21626	−0.596150	−0.298075	+	0.954542i	$0.596345\pi$
$240$	−2.03773	−0.131535
$241$	9.96847	0.642126	0.321063	−	0.947058i	$-0.395960\pi$
$241$	9.96847	0.642126	0.321063	+	0.947058i	$0.395960\pi$
$242$	22.8414	1.46830
$243$	−15.1161	−0.969696
$244$	−4.65157	−0.297786
$245$	0.341092	0.0217915
$246$	2.80775	0.179015
$247$	−30.6816	−1.95223
$248$	35.0626	2.22648
$249$	−4.13544	−0.262073
$250$	21.0504	1.33134
$251$	7.17902	0.453136	0.226568	−	0.973995i	$-0.427249\pi$
$251$	7.17902	0.453136	0.226568	+	0.973995i	$0.427249\pi$
$252$	−17.0537	−1.07428
$253$	0.631685	0.0397137
$254$	31.4994	1.97645
$255$	4.95827	0.310499
$256$	−25.9442	−1.62151
$257$	15.6262	0.974739	0.487369	−	0.873196i	$-0.337957\pi$
$257$	15.6262	0.974739	0.487369	+	0.873196i	$0.337957\pi$
$258$	9.47955	0.590171
$259$	14.0686	0.874181
$260$	14.1848	0.879702
$261$	−11.0089	−0.681432
$262$	−24.1356	−1.49110
$263$	−4.76886	−0.294061	−0.147030	−	0.989132i	$-0.546972\pi$
$263$	−4.76886	−0.294061	−0.147030	+	0.989132i	$0.546972\pi$
$264$	−5.01891	−0.308892
$265$	−0.983312	−0.0604043
$266$	49.4102	3.02954
$267$	−8.25515	−0.505207
$268$	53.4043	3.26219
$269$	1.74684	0.106507	0.0532533	−	0.998581i	$-0.483041\pi$
$269$	1.74684	0.106507	0.0532533	+	0.998581i	$0.483041\pi$
$270$	−12.3996	−0.754618
$271$	−21.3661	−1.29790	−0.648948	−	0.760833i	$-0.724791\pi$
$271$	−21.3661	−1.29790	−0.648948	+	0.760833i	$0.724791\pi$
$272$	−8.29143	−0.502742
$273$	−12.1375	−0.734595
$274$	32.6600	1.97307
$275$	4.70666	0.283822
$276$	2.19758	0.132279
$277$	−25.4291	−1.52789	−0.763944	−	0.645282i	$-0.776740\pi$
$277$	−25.4291	−1.52789	−0.763944	+	0.645282i	$0.776740\pi$
$278$	51.0924	3.06432
$279$	15.9229	0.953279
$280$	−10.2104	−0.610191
$281$	−25.1491	−1.50027	−0.750134	−	0.661286i	$-0.770011\pi$
$281$	−25.1491	−1.50027	−0.750134	+	0.661286i	$0.770011\pi$
$282$	36.0805	2.14856
$283$	−12.1371	−0.721473	−0.360737	−	0.932668i	$-0.617475\pi$
$283$	−12.1371	−0.721473	−0.360737	+	0.932668i	$0.617475\pi$
$284$	−21.9475	−1.30235
$285$	8.49097	0.502962
$286$	11.0319	0.652331
$287$	2.86052	0.168851
$288$	−5.71857	−0.336970
$289$	3.17498	0.186763
$290$	−14.7464	−0.865938
$291$	19.0055	1.11412
$292$	16.0968	0.941994
$293$	11.9117	0.695891	0.347945	−	0.937515i	$-0.386879\pi$
$293$	11.9117	0.695891	0.347945	+	0.937515i	$0.386879\pi$
$294$	0.922879	0.0538234
$295$	−2.08372	−0.121319
$296$	−19.8839	−1.15573
$297$	−6.20955	−0.360315
$298$	−22.1974	−1.28586
$299$	−2.15910	−0.124864
$300$	16.3741	0.945360
$301$	9.65772	0.556662
$302$	2.36991	0.136373
$303$	2.52470	0.145040
$304$	−14.1990	−0.814366
$305$	−1.26475	−0.0724194
$306$	−18.5191	−1.05866
$307$	2.40653	0.137348	0.0686739	−	0.997639i	$-0.478123\pi$
$307$	2.40653	0.137348	0.0686739	+	0.997639i	$0.478123\pi$
$308$	−11.4396	−0.651833
$309$	−0.576579	−0.0328004
$310$	21.3287	1.21139
$311$	1.75136	0.0993103	0.0496552	−	0.998766i	$-0.484188\pi$
$311$	1.75136	0.0993103	0.0496552	+	0.998766i	$0.484188\pi$
$312$	17.1546	0.971187
$313$	−33.2343	−1.87851	−0.939257	−	0.343215i	$-0.888484\pi$
$313$	−33.2343	−1.87851	−0.939257	+	0.343215i	$0.888484\pi$
$314$	22.6189	1.27646
$315$	−4.63685	−0.261257
$316$	20.0259	1.12654
$317$	−4.62324	−0.259667	−0.129833	−	0.991536i	$-0.541444\pi$
$317$	−4.62324	−0.259667	−0.129833	+	0.991536i	$0.541444\pi$
$318$	−2.66051	−0.149194
$319$	−7.38476	−0.413467
$320$	−11.2903	−0.631149
$321$	−2.16667	−0.120932
$322$	3.47704	0.193768
$323$	34.5494	1.92238
$324$	−2.72751	−0.151528
$325$	−16.0873	−0.892365
$326$	−56.2962	−3.11796
$327$	−10.2397	−0.566257
$328$	−4.04292	−0.223233
$329$	36.7586	2.02657
$330$	−3.05302	−0.168063
$331$	12.7582	0.701255	0.350628	−	0.936515i	$-0.385968\pi$
$331$	12.7582	0.701255	0.350628	+	0.936515i	$0.385968\pi$
$332$	13.3222	0.731148
$333$	−9.02984	−0.494832
$334$	−24.5020	−1.34069
$335$	14.5205	0.793339
$336$	−5.61704	−0.306435
$337$	14.2594	0.776758	0.388379	−	0.921500i	$-0.373035\pi$
$337$	14.2594	0.776758	0.388379	+	0.921500i	$0.373035\pi$
$338$	−6.89820	−0.375212
$339$	−17.8446	−0.969185
$340$	−15.9729	−0.866252
$341$	10.6811	0.578414
$342$	−31.7136	−1.71488
$343$	−18.0334	−0.973710
$344$	−13.6498	−0.735946
$345$	0.597518	0.0321693
$346$	23.8988	1.28481
$347$	29.2819	1.57193	0.785967	−	0.618268i	$-0.212165\pi$
$347$	29.2819	1.57193	0.785967	+	0.618268i	$0.212165\pi$
$348$	−25.6910	−1.37718
$349$	7.25835	0.388530	0.194265	−	0.980949i	$-0.437768\pi$
$349$	7.25835	0.388530	0.194265	+	0.980949i	$0.437768\pi$
$350$	25.9073	1.38480
$351$	21.2242	1.13286
$352$	−3.83602	−0.204461
$353$	−7.49718	−0.399034	−0.199517	−	0.979894i	$-0.563937\pi$
$353$	−7.49718	−0.399034	−0.199517	+	0.979894i	$0.563937\pi$
$354$	−5.63784	−0.299648
$355$	−5.96748	−0.316721
$356$	26.5937	1.40946
$357$	13.6676	0.723364
$358$	1.54476	0.0816431
$359$	28.0726	1.48161	0.740807	−	0.671718i	$-0.234443\pi$
$359$	28.0726	1.48161	0.740807	+	0.671718i	$0.234443\pi$
$360$	6.55351	0.345400
$361$	40.1653	2.11396
$362$	19.1828	1.00823
$363$	10.8199	0.567898
$364$	39.1005	2.04942
$365$	4.37668	0.229086
$366$	−3.42199	−0.178870
$367$	−13.4701	−0.703133	−0.351567	−	0.936163i	$-0.614351\pi$
$367$	−13.4701	−0.703133	−0.351567	+	0.936163i	$0.614351\pi$
$368$	−0.999194	−0.0520866
$369$	−1.83600	−0.0955784
$370$	−12.0955	−0.628814
$371$	−2.71051	−0.140723
$372$	37.1587	1.92659
$373$	30.9636	1.60324	0.801618	−	0.597836i	$-0.203973\pi$
$373$	30.9636	1.60324	0.801618	+	0.597836i	$0.203973\pi$
$374$	−12.4226	−0.642358
$375$	9.97151	0.514926
$376$	−51.9528	−2.67926
$377$	25.2411	1.29998
$378$	−34.1798	−1.75802
$379$	26.6297	1.36788	0.683938	−	0.729541i	$-0.260266\pi$
$379$	26.6297	1.36788	0.683938	+	0.729541i	$0.260266\pi$
$380$	−27.3533	−1.40320
$381$	14.9212	0.764434
$382$	−13.5904	−0.695346
$383$	−22.4532	−1.14731	−0.573653	−	0.819098i	$-0.694474\pi$
$383$	−22.4532	−1.14731	−0.573653	+	0.819098i	$0.694474\pi$
$384$	−23.1676	−1.18227
$385$	−3.11040	−0.158521
$386$	35.5390	1.80889
$387$	−6.19874	−0.315100
$388$	−61.2255	−3.10825
$389$	−28.9292	−1.46677	−0.733385	−	0.679813i	$-0.762061\pi$
$389$	−28.9292	−1.46677	−0.733385	+	0.679813i	$0.762061\pi$
$390$	10.4352	0.528407
$391$	2.43127	0.122955
$392$	−1.32887	−0.0671180
$393$	−11.4330	−0.576717
$394$	19.5342	0.984116
$395$	5.44499	0.273967
$396$	7.34244	0.368971
$397$	24.8995	1.24967	0.624836	−	0.780756i	$-0.285166\pi$
$397$	24.8995	1.24967	0.624836	+	0.780756i	$0.285166\pi$
$398$	36.9231	1.85079
$399$	23.4055	1.17174
$400$	−7.44495	−0.372248
$401$	11.8978	0.594147	0.297073	−	0.954855i	$-0.403989\pi$
$401$	11.8978	0.594147	0.297073	+	0.954855i	$0.403989\pi$
$402$	39.2876	1.95949
$403$	−36.5079	−1.81859
$404$	−8.13323	−0.404643
$405$	−0.741603	−0.0368505
$406$	−40.6486	−2.01736
$407$	−6.05723	−0.300246
$408$	−19.3171	−0.956338
$409$	−26.5680	−1.31370	−0.656852	−	0.754020i	$-0.728112\pi$
$409$	−26.5680	−1.31370	−0.656852	+	0.754020i	$0.728112\pi$
$410$	−2.45933	−0.121457
$411$	15.4710	0.763127
$412$	1.85743	0.0915089
$413$	−5.74380	−0.282634
$414$	−2.23172	−0.109683
$415$	3.62226	0.177810
$416$	13.1115	0.642844
$417$	24.2023	1.18519
$418$	−21.2735	−1.04052
$419$	−33.8122	−1.65183	−0.825917	−	0.563792i	$-0.809342\pi$
$419$	−33.8122	−1.65183	−0.825917	+	0.563792i	$0.809342\pi$
$420$	−10.8209	−0.528004
$421$	29.9688	1.46059	0.730295	−	0.683132i	$-0.239383\pi$
$421$	29.9688	1.46059	0.730295	+	0.683132i	$0.239383\pi$
$422$	6.92023	0.336871
$423$	−23.5932	−1.14714
$424$	3.83091	0.186046
$425$	18.1153	0.878721
$426$	−16.1460	−0.782276
$427$	−3.48630	−0.168714
$428$	6.97985	0.337384
$429$	5.22579	0.252304
$430$	−8.30321	−0.400417
$431$	−7.42622	−0.357708	−0.178854	−	0.983876i	$-0.557239\pi$
$431$	−7.42622	−0.357708	−0.178854	+	0.983876i	$0.557239\pi$
$432$	9.82221	0.472571
$433$	−18.0839	−0.869057	−0.434529	−	0.900658i	$-0.643085\pi$
$433$	−18.0839	−0.869057	−0.434529	+	0.900658i	$0.643085\pi$
$434$	58.7930	2.82215
$435$	−6.98532	−0.334921
$436$	32.9869	1.57978
$437$	4.16351	0.199168
$438$	11.8418	0.565824
$439$	22.7821	1.08733	0.543666	−	0.839302i	$-0.317036\pi$
$439$	22.7821	1.08733	0.543666	+	0.839302i	$0.317036\pi$
$440$	4.39610	0.209576
$441$	−0.603476	−0.0287370
$442$	42.4604	2.01963
$443$	21.6143	1.02693	0.513464	−	0.858111i	$-0.328362\pi$
$443$	21.6143	1.02693	0.513464	+	0.858111i	$0.328362\pi$
$444$	−21.0726	−1.00006
$445$	7.23075	0.342770
$446$	6.20465	0.293799
$447$	−10.5149	−0.497335
$448$	−31.1220	−1.47038
$449$	−27.2867	−1.28774	−0.643869	−	0.765135i	$-0.722672\pi$
$449$	−27.2867	−1.28774	−0.643869	+	0.765135i	$0.722672\pi$
$450$	−16.6284	−0.783872
$451$	−1.23159	−0.0579934
$452$	57.4857	2.70390
$453$	1.12262	0.0527453
$454$	58.0842	2.72603
$455$	10.6313	0.498405
$456$	−33.0802	−1.54912
$457$	−31.8099	−1.48801	−0.744003	−	0.668176i	$-0.767075\pi$
$457$	−31.8099	−1.48801	−0.744003	+	0.668176i	$0.767075\pi$
$458$	9.10138	0.425279
$459$	−23.8997	−1.11554
$460$	−1.92488	−0.0897480
$461$	−18.3939	−0.856687	−0.428344	−	0.903616i	$-0.640903\pi$
$461$	−18.3939	−0.856687	−0.428344	+	0.903616i	$0.640903\pi$
$462$	−8.41570	−0.391534
$463$	−5.82135	−0.270541	−0.135270	−	0.990809i	$-0.543190\pi$
$463$	−5.82135	−0.270541	−0.135270	+	0.990809i	$0.543190\pi$
$464$	11.6811	0.542284
$465$	10.1034	0.468532
$466$	−10.4184	−0.482622
$467$	−7.64472	−0.353755	−0.176878	−	0.984233i	$-0.556600\pi$
$467$	−7.64472	−0.353755	−0.176878	+	0.984233i	$0.556600\pi$
$468$	−25.0964	−1.16008
$469$	40.0260	1.84823
$470$	−31.6031	−1.45774
$471$	10.7145	0.493699
$472$	8.11802	0.373662
$473$	−4.15812	−0.191191
$474$	14.7323	0.676677
$475$	31.0222	1.42339
$476$	−44.0295	−2.01809
$477$	1.73972	0.0796565
$478$	21.8417	0.999017
$479$	−20.0354	−0.915439	−0.457719	−	0.889097i	$-0.651334\pi$
$479$	−20.0354	−0.915439	−0.457719	+	0.889097i	$0.651334\pi$
$480$	−3.62853	−0.165619
$481$	20.7036	0.944001
$482$	−23.6244	−1.07606
$483$	1.64707	0.0749441
$484$	−34.8560	−1.58436
$485$	−16.6471	−0.755904
$486$	35.8237	1.62500
$487$	−0.629141	−0.0285091	−0.0142545	−	0.999898i	$-0.504538\pi$
$487$	−0.629141	−0.0285091	−0.0142545	+	0.999898i	$0.504538\pi$
$488$	4.92738	0.223052
$489$	−26.6673	−1.20594
$490$	−0.808357	−0.0365178
$491$	−6.37157	−0.287545	−0.143772	−	0.989611i	$-0.545923\pi$
$491$	−6.37157	−0.287545	−0.143772	+	0.989611i	$0.545923\pi$
$492$	−4.28461	−0.193165
$493$	−28.4229	−1.28010
$494$	72.7128	3.27150
$495$	1.99639	0.0897311
$496$	−16.8953	−0.758620
$497$	−16.4495	−0.737859
$498$	9.80062	0.439176
$499$	21.4500	0.960235	0.480118	−	0.877204i	$-0.340594\pi$
$499$	21.4500	0.960235	0.480118	+	0.877204i	$0.340594\pi$
$500$	−32.1228	−1.43658
$501$	−11.6065	−0.518541
$502$	−17.0137	−0.759356
$503$	38.6546	1.72352	0.861761	−	0.507314i	$-0.169362\pi$
$503$	38.6546	1.72352	0.861761	+	0.507314i	$0.169362\pi$
$504$	18.0648	0.804672
$505$	−2.21140	−0.0984062
$506$	−1.49704	−0.0665515
$507$	−3.26766	−0.145122
$508$	−48.0679	−2.13267
$509$	−18.7062	−0.829137	−0.414568	−	0.910018i	$-0.636067\pi$
$509$	−18.7062	−0.829137	−0.414568	+	0.910018i	$0.636067\pi$
$510$	−11.7507	−0.520328
$511$	12.0644	0.533697
$512$	20.2112	0.893219
$513$	−40.9279	−1.80701
$514$	−37.0328	−1.63345
$515$	0.505030	0.0222543
$516$	−14.4658	−0.636820
$517$	−15.8264	−0.696043
$518$	−33.3414	−1.46494
$519$	11.3208	0.496927
$520$	−15.0258	−0.658926
$521$	−0.398041	−0.0174385	−0.00871925	−	0.999962i	$-0.502775\pi$
$521$	−0.398041	−0.0174385	−0.00871925	+	0.999962i	$0.502775\pi$
$522$	26.0900	1.14193
$523$	3.76150	0.164479	0.0822395	−	0.996613i	$-0.473793\pi$
$523$	3.76150	0.164479	0.0822395	+	0.996613i	$0.473793\pi$
$524$	36.8309	1.60896
$525$	12.2722	0.535604
$526$	11.3018	0.492781
$527$	41.1101	1.79078
$528$	2.41841	0.105248
$529$	−22.7070	−0.987261
$530$	2.33036	0.101224
$531$	3.68662	0.159986
$532$	−75.3999	−3.26900
$533$	4.20957	0.182337
$534$	19.5640	0.846616
$535$	1.89780	0.0820492
$536$	−56.5708	−2.44349
$537$	0.731748	0.0315773
$538$	−4.13985	−0.178482
$539$	−0.404813	−0.0174365
$540$	18.9218	0.814266
$541$	−25.9551	−1.11590	−0.557948	−	0.829876i	$-0.688411\pi$
$541$	−25.9551	−1.11590	−0.557948	+	0.829876i	$0.688411\pi$
$542$	50.6357	2.17499
$543$	9.08685	0.389954
$544$	−14.7643	−0.633015
$545$	8.96904	0.384192
$546$	28.7648	1.23102
$547$	−22.3179	−0.954245	−0.477123	−	0.878837i	$-0.658320\pi$
$547$	−22.3179	−0.954245	−0.477123	+	0.878837i	$0.658320\pi$
$548$	−49.8392	−2.12902
$549$	2.23766	0.0955010
$550$	−11.1544	−0.475624
$551$	−48.6739	−2.07358
$552$	−2.32789	−0.0990814
$553$	15.0092	0.638256
$554$	60.2648	2.56041
$555$	−5.72959	−0.243208
$556$	−77.9669	−3.30653
$557$	19.4876	0.825716	0.412858	−	0.910795i	$-0.364531\pi$
$557$	19.4876	0.825716	0.412858	+	0.910795i	$0.364531\pi$
$558$	−37.7359	−1.59749
$559$	14.2124	0.601122
$560$	4.92001	0.207908
$561$	−5.88455	−0.248446
$562$	59.6011	2.51412
$563$	−9.60392	−0.404757	−0.202379	−	0.979307i	$-0.564867\pi$
$563$	−9.60392	−0.404757	−0.202379	+	0.979307i	$0.564867\pi$
$564$	−55.0587	−2.31839
$565$	15.6302	0.657568
$566$	28.7638	1.20903
$567$	−2.04424	−0.0858500
$568$	23.2489	0.975502
$569$	−29.3422	−1.23009	−0.615045	−	0.788492i	$-0.710862\pi$
$569$	−29.3422	−1.23009	−0.615045	+	0.788492i	$0.710862\pi$
$570$	−20.1228	−0.842853
$571$	−40.7553	−1.70555	−0.852777	−	0.522274i	$-0.825084\pi$
$571$	−40.7553	−1.70555	−0.852777	+	0.522274i	$0.825084\pi$
$572$	−16.8347	−0.703894
$573$	−6.43774	−0.268940
$574$	−6.77917	−0.282957
$575$	2.18306	0.0910399
$576$	19.9754	0.832310
$577$	4.25234	0.177027	0.0885135	−	0.996075i	$-0.471788\pi$
$577$	4.25234	0.177027	0.0885135	+	0.996075i	$0.471788\pi$
$578$	−7.52442	−0.312974
$579$	16.8347	0.699628
$580$	22.5030	0.934384
$581$	9.98481	0.414240
$582$	−45.0413	−1.86702
$583$	1.16701	0.0483326
$584$	−17.0512	−0.705585
$585$	−6.82365	−0.282123
$586$	−28.2297	−1.16616
$587$	25.1946	1.03989	0.519947	−	0.854199i	$-0.325952\pi$
$587$	25.1946	1.03989	0.519947	+	0.854199i	$0.325952\pi$
$588$	−1.40831	−0.0580778
$589$	70.4004	2.90080
$590$	4.93823	0.203304
$591$	9.25327	0.380629
$592$	9.58127	0.393788
$593$	−35.7256	−1.46707	−0.733537	−	0.679649i	$-0.762132\pi$
$593$	−35.7256	−1.46707	−0.733537	+	0.679649i	$0.762132\pi$
$594$	14.7161	0.603808
$595$	−11.9715	−0.490784
$596$	33.8732	1.38750
$597$	17.4904	0.715832
$598$	5.11686	0.209244
$599$	−4.26301	−0.174182	−0.0870909	−	0.996200i	$-0.527757\pi$
$599$	−4.26301	−0.174182	−0.0870909	+	0.996200i	$0.527757\pi$
$600$	−17.3450	−0.708106
$601$	−32.8171	−1.33864	−0.669319	−	0.742975i	$-0.733414\pi$
$601$	−32.8171	−1.33864	−0.669319	+	0.742975i	$0.733414\pi$
$602$	−22.8879	−0.932843
$603$	−25.6904	−1.04619
$604$	−3.61648	−0.147152
$605$	−9.47725	−0.385305
$606$	−5.98332	−0.243056
$607$	−9.88402	−0.401180	−0.200590	−	0.979675i	$-0.564286\pi$
$607$	−9.88402	−0.401180	−0.200590	+	0.979675i	$0.564286\pi$
$608$	−25.2837	−1.02539
$609$	−19.2551	−0.780258
$610$	2.99735	0.121359
$611$	54.0944	2.18843
$612$	28.2600	1.14234
$613$	−12.8058	−0.517220	−0.258610	−	0.965982i	$-0.583264\pi$
$613$	−12.8058	−0.517220	−0.258610	+	0.965982i	$0.583264\pi$
$614$	−5.70326	−0.230165
$615$	−1.16498	−0.0469763
$616$	12.1179	0.488245
$617$	20.7771	0.836454	0.418227	−	0.908343i	$-0.362652\pi$
$617$	20.7771	0.836454	0.418227	+	0.908343i	$0.362652\pi$
$618$	1.36644	0.0549663
$619$	2.15920	0.0867857	0.0433929	−	0.999058i	$-0.486183\pi$
$619$	2.15920	0.0867857	0.0433929	+	0.999058i	$0.486183\pi$
$620$	−32.5476	−1.30714
$621$	−2.88013	−0.115576
$622$	−4.15056	−0.166422
$623$	19.9317	0.798545
$624$	−8.26611	−0.330909
$625$	11.4314	0.457255
$626$	78.7624	3.14798
$627$	−10.0772	−0.402445
$628$	−34.5164	−1.37735
$629$	−23.3134	−0.929568
$630$	10.9889	0.437809
$631$	−16.7489	−0.666761	−0.333381	−	0.942792i	$-0.608189\pi$
$631$	−16.7489	−0.666761	−0.333381	+	0.942792i	$0.608189\pi$
$632$	−21.2133	−0.843819
$633$	3.27809	0.130292
$634$	10.9567	0.435145
$635$	−13.0696	−0.518649
$636$	4.05993	0.160987
$637$	1.38365	0.0548221
$638$	17.5012	0.692880
$639$	10.5580	0.417667
$640$	20.2927	0.802140
$641$	−17.3573	−0.685571	−0.342785	−	0.939414i	$-0.611370\pi$
$641$	−17.3573	−0.685571	−0.342785	+	0.939414i	$0.611370\pi$
$642$	5.13482	0.202655
$643$	−1.78770	−0.0705002	−0.0352501	−	0.999379i	$-0.511223\pi$
$643$	−1.78770	−0.0705002	−0.0352501	+	0.999379i	$0.511223\pi$
$644$	−5.30596	−0.209084
$645$	−3.93321	−0.154870
$646$	−81.8789	−3.22148
$647$	−22.2521	−0.874821	−0.437411	−	0.899262i	$-0.644104\pi$
$647$	−22.2521	−0.874821	−0.437411	+	0.899262i	$0.644104\pi$
$648$	2.88923	0.113500
$649$	2.47299	0.0970732
$650$	38.1255	1.49541
$651$	27.8501	1.09153
$652$	85.9078	3.36441
$653$	19.4641	0.761688	0.380844	−	0.924639i	$-0.375633\pi$
$653$	19.4641	0.761688	0.380844	+	0.924639i	$0.375633\pi$
$654$	24.2672	0.948923
$655$	10.0142	0.391288
$656$	1.94812	0.0760613
$657$	−7.74344	−0.302100
$658$	−87.1145	−3.39608
$659$	6.04659	0.235542	0.117771	−	0.993041i	$-0.462425\pi$
$659$	6.04659	0.235542	0.117771	+	0.993041i	$0.462425\pi$
$660$	4.65891	0.181348
$661$	37.9723	1.47695	0.738475	−	0.674281i	$-0.235546\pi$
$661$	37.9723	1.47695	0.738475	+	0.674281i	$0.235546\pi$
$662$	−30.2359	−1.17515
$663$	20.1134	0.781138
$664$	−14.1121	−0.547655
$665$	−20.5010	−0.794996
$666$	21.3999	0.829230
$667$	−3.42523	−0.132625
$668$	37.3900	1.44666
$669$	2.93912	0.113633
$670$	−34.4123	−1.32946
$671$	1.50102	0.0579464
$672$	−10.0021	−0.385839
$673$	−13.0726	−0.503912	−0.251956	−	0.967739i	$-0.581074\pi$
$673$	−13.0726	−0.503912	−0.251956	+	0.967739i	$0.581074\pi$
$674$	−33.7935	−1.30168
$675$	−21.4598	−0.825986
$676$	10.5266	0.404871
$677$	5.91181	0.227209	0.113605	−	0.993526i	$-0.463760\pi$
$677$	5.91181	0.227209	0.113605	+	0.993526i	$0.463760\pi$
$678$	42.2901	1.62414
$679$	−45.8879	−1.76101
$680$	16.9200	0.648852
$681$	27.5143	1.05435
$682$	−25.3133	−0.969295
$683$	−29.7175	−1.13711	−0.568554	−	0.822646i	$-0.692497\pi$
$683$	−29.7175	−1.13711	−0.568554	+	0.822646i	$0.692497\pi$
$684$	48.3949	1.85043
$685$	−13.5511	−0.517762
$686$	42.7375	1.63172
$687$	4.31130	0.164486
$688$	6.57728	0.250756
$689$	−3.98883	−0.151962
$690$	−1.41606	−0.0539086
$691$	−17.1289	−0.651614	−0.325807	−	0.945436i	$-0.605636\pi$
$691$	−17.1289	−0.651614	−0.325807	+	0.945436i	$0.605636\pi$
$692$	−36.4695	−1.38636
$693$	5.50308	0.209045
$694$	−69.3955	−2.63422
$695$	−21.1990	−0.804124
$696$	27.2143	1.03156
$697$	−4.74023	−0.179549
$698$	−17.2016	−0.651091
$699$	−4.93515	−0.186665
$700$	−39.5345	−1.49426
$701$	−21.9772	−0.830066	−0.415033	−	0.909806i	$-0.636230\pi$
$701$	−21.9772	−0.830066	−0.415033	+	0.909806i	$0.636230\pi$
$702$	−50.2994	−1.89843
$703$	−39.9239	−1.50576
$704$	13.3995	0.505014
$705$	−14.9703	−0.563815
$706$	17.7676	0.668694
$707$	−6.09577	−0.229255
$708$	8.60333	0.323333
$709$	21.8867	0.821971	0.410985	−	0.911642i	$-0.365185\pi$
$709$	21.8867	0.821971	0.410985	+	0.911642i	$0.365185\pi$
$710$	14.1424	0.530755
$711$	−9.63354	−0.361286
$712$	−28.1705	−1.05573
$713$	4.95414	0.185534
$714$	−32.3909	−1.21220
$715$	−4.57731	−0.171182
$716$	−2.35730	−0.0880964
$717$	10.3464	0.386392
$718$	−66.5295	−2.48286
$719$	−42.2807	−1.57681	−0.788403	−	0.615159i	$-0.789092\pi$
$719$	−42.2807	−1.57681	−0.788403	+	0.615159i	$0.789092\pi$
$720$	−3.15787	−0.117687
$721$	1.39212	0.0518453
$722$	−95.1881	−3.54253
$723$	−11.1908	−0.416191
$724$	−29.2729	−1.08792
$725$	−25.5212	−0.947834
$726$	−25.6422	−0.951673
$727$	−21.4427	−0.795264	−0.397632	−	0.917545i	$-0.630168\pi$
$727$	−21.4427	−0.795264	−0.397632	+	0.917545i	$0.630168\pi$
$728$	−41.4189	−1.53509
$729$	19.2322	0.712302
$730$	−10.3723	−0.383898
$731$	−16.0040	−0.591931
$732$	5.22195	0.193009
$733$	−26.8627	−0.992196	−0.496098	−	0.868267i	$-0.665234\pi$
$733$	−26.8627	−0.992196	−0.496098	+	0.868267i	$0.665234\pi$
$734$	31.9229	1.17830
$735$	−0.382916	−0.0141241
$736$	−1.77924	−0.0655835
$737$	−17.2331	−0.634791
$738$	4.35116	0.160168
$739$	1.75000	0.0643748	0.0321874	−	0.999482i	$-0.489753\pi$
$739$	1.75000	0.0643748	0.0321874	+	0.999482i	$0.489753\pi$
$740$	18.4577	0.678517
$741$	34.4438	1.26533
$742$	6.42368	0.235821
$743$	3.76118	0.137984	0.0689921	−	0.997617i	$-0.478022\pi$
$743$	3.76118	0.137984	0.0689921	+	0.997617i	$0.478022\pi$
$744$	−39.3620	−1.44308
$745$	9.21004	0.337430
$746$	−73.3811	−2.68667
$747$	−6.40868	−0.234481
$748$	18.9569	0.693132
$749$	5.23133	0.191148
$750$	−23.6316	−0.862903
$751$	34.3160	1.25221	0.626105	−	0.779739i	$-0.284648\pi$
$751$	34.3160	1.25221	0.626105	+	0.779739i	$0.284648\pi$
$752$	25.0340	0.912896
$753$	−8.05932	−0.293698
$754$	−59.8191	−2.17848
$755$	−0.983312	−0.0357864
$756$	52.1583	1.89698
$757$	−24.5496	−0.892271	−0.446135	−	0.894966i	$-0.647200\pi$
$757$	−24.5496	−0.892271	−0.446135	+	0.894966i	$0.647200\pi$
$758$	−63.1100	−2.29226
$759$	−0.709143	−0.0257402
$760$	28.9752	1.05104
$761$	23.0877	0.836927	0.418464	−	0.908234i	$-0.362569\pi$
$761$	23.0877	0.836927	0.418464	+	0.908234i	$0.362569\pi$
$762$	−35.3618	−1.28102
$763$	24.7233	0.895043
$764$	20.7389	0.750308
$765$	7.68383	0.277810
$766$	53.2122	1.92263
$767$	−8.45265	−0.305208
$768$	29.1255	1.05098
$769$	−48.4011	−1.74539	−0.872693	−	0.488269i	$-0.837629\pi$
$769$	−48.4011	−1.74539	−0.872693	+	0.488269i	$0.837629\pi$
$770$	7.37138	0.265646
$771$	−17.5423	−0.631772
$772$	−54.2325	−1.95187
$773$	−1.22525	−0.0440690	−0.0220345	−	0.999757i	$-0.507014\pi$
$773$	−1.22525	−0.0440690	−0.0220345	+	0.999757i	$0.507014\pi$
$774$	14.6905	0.528038
$775$	36.9131	1.32596
$776$	64.8558	2.32819
$777$	−15.7937	−0.566596
$778$	68.5597	2.45798
$779$	−8.11757	−0.290842
$780$	−15.9241	−0.570175
$781$	7.08230	0.253424
$782$	−5.76189	−0.206045
$783$	33.6704	1.20328
$784$	0.640329	0.0228689
$785$	−9.38493	−0.334962
$786$	27.0951	0.966450
$787$	−7.61598	−0.271480	−0.135740	−	0.990744i	$-0.543341\pi$
$787$	−7.61598	−0.271480	−0.135740	+	0.990744i	$0.543341\pi$
$788$	−29.8091	−1.06190
$789$	5.35362	0.190594
$790$	−12.9041	−0.459109
$791$	43.0849	1.53192
$792$	−7.77780	−0.276372
$793$	−5.13049	−0.182189
$794$	−59.0097	−2.09418
$795$	1.10389	0.0391508
$796$	−56.3445	−1.99708
$797$	12.5111	0.443168	0.221584	−	0.975141i	$-0.428877\pi$
$797$	12.5111	0.443168	0.221584	+	0.975141i	$0.428877\pi$
$798$	−55.4689	−1.96358
$799$	−60.9135	−2.15497
$800$	−13.2570	−0.468706
$801$	−12.7930	−0.452019
$802$	−28.1967	−0.995659
$803$	−5.19431	−0.183303
$804$	−59.9527	−2.11437
$805$	−1.44268	−0.0508477
$806$	86.5205	3.04756
$807$	−1.96103	−0.0690317
$808$	8.61547	0.303091
$809$	−9.90149	−0.348118	−0.174059	−	0.984735i	$-0.555688\pi$
$809$	−9.90149	−0.348118	−0.174059	+	0.984735i	$0.555688\pi$
$810$	1.75753	0.0617534
$811$	21.3543	0.749852	0.374926	−	0.927055i	$-0.377668\pi$
$811$	21.3543	0.749852	0.374926	+	0.927055i	$0.377668\pi$
$812$	62.0297	2.17682
$813$	23.9860	0.841225
$814$	14.3551	0.503146
$815$	23.3581	0.818199
$816$	9.30813	0.325850
$817$	−27.4067	−0.958838
$818$	62.9638	2.20148
$819$	−18.8095	−0.657257
$820$	3.75293	0.131058
$821$	−7.50177	−0.261814	−0.130907	−	0.991395i	$-0.541789\pi$
$821$	−7.50177	−0.261814	−0.130907	+	0.991395i	$0.541789\pi$
$822$	−36.6648	−1.27883
$823$	7.58769	0.264490	0.132245	−	0.991217i	$-0.457781\pi$
$823$	7.58769	0.264490	0.132245	+	0.991217i	$0.457781\pi$
$824$	−1.96756	−0.0685432
$825$	−5.28379	−0.183958
$826$	13.6123	0.473632
$827$	5.16163	0.179487	0.0897437	−	0.995965i	$-0.471395\pi$
$827$	5.16163	0.179487	0.0897437	+	0.995965i	$0.471395\pi$
$828$	3.40559	0.118353
$829$	−40.0913	−1.39243	−0.696214	−	0.717834i	$-0.745133\pi$
$829$	−40.0913	−1.39243	−0.696214	+	0.717834i	$0.745133\pi$
$830$	−8.58444	−0.297970
$831$	28.5473	0.990293
$832$	−45.7995	−1.58781
$833$	−1.55807	−0.0539839
$834$	−57.3574	−1.98612
$835$	10.1662	0.351817
$836$	32.4634	1.12277
$837$	−48.6999	−1.68331
$838$	80.1319	2.76811
$839$	−7.91738	−0.273338	−0.136669	−	0.990617i	$-0.543640\pi$
$839$	−7.91738	−0.273338	−0.136669	+	0.990617i	$0.543640\pi$
$840$	11.4625	0.395492
$841$	11.0428	0.380788
$842$	−71.0234	−2.44763
$843$	28.2329	0.972391
$844$	−10.5603	−0.363499
$845$	2.86217	0.0984615
$846$	55.9139	1.92236
$847$	−26.1242	−0.897637
$848$	−1.84596	−0.0633906
$849$	13.6253	0.467619
$850$	−42.9316	−1.47254
$851$	−2.80948	−0.0963078
$852$	24.6388	0.844110
$853$	−42.3846	−1.45122	−0.725610	−	0.688106i	$-0.758442\pi$
$853$	−42.3846	−1.45122	−0.725610	+	0.688106i	$0.758442\pi$
$854$	8.26223	0.282728
$855$	13.1585	0.450010
$856$	−7.39371	−0.252712
$857$	45.8746	1.56705	0.783523	−	0.621363i	$-0.213421\pi$
$857$	45.8746	1.56705	0.783523	+	0.621363i	$0.213421\pi$
$858$	−12.3847	−0.422805
$859$	15.1368	0.516459	0.258230	−	0.966084i	$-0.416861\pi$
$859$	15.1368	0.516459	0.258230	+	0.966084i	$0.416861\pi$
$860$	12.6707	0.432067
$861$	−3.21127	−0.109440
$862$	17.5995	0.599441
$863$	7.19201	0.244819	0.122409	−	0.992480i	$-0.460938\pi$
$863$	7.19201	0.244819	0.122409	+	0.992480i	$0.460938\pi$
$864$	17.4901	0.595026
$865$	−9.91596	−0.337153
$866$	42.8573	1.45635
$867$	−3.56429	−0.121050
$868$	−89.7179	−3.04523
$869$	−6.46219	−0.219215
$870$	16.5546	0.561253
$871$	58.9027	1.99584
$872$	−34.9428	−1.18331
$873$	29.4528	0.996827
$874$	−9.86716	−0.333762
$875$	−24.0757	−0.813908
$876$	−18.0706	−0.610549
$877$	46.3350	1.56462	0.782310	−	0.622889i	$-0.214041\pi$
$877$	46.3350	1.56462	0.782310	+	0.622889i	$0.214041\pi$
$878$	−53.9916	−1.82213
$879$	−13.3723	−0.451038
$880$	−2.11830	−0.0714080
$881$	33.9098	1.14245	0.571226	−	0.820793i	$-0.306468\pi$
$881$	33.9098	1.14245	0.571226	+	0.820793i	$0.306468\pi$
$882$	1.43019	0.0481568
$883$	−10.1550	−0.341744	−0.170872	−	0.985293i	$-0.554658\pi$
$883$	−10.1550	−0.341744	−0.170872	+	0.985293i	$0.554658\pi$
$884$	−64.7944	−2.17927
$885$	2.33922	0.0786321
$886$	−51.2241	−1.72091
$887$	−51.0245	−1.71323	−0.856617	−	0.515952i	$-0.827438\pi$
$887$	−51.0245	−1.71323	−0.856617	+	0.515952i	$0.827438\pi$
$888$	22.3221	0.749080
$889$	−36.0264	−1.20829
$890$	−17.1362	−0.574408
$891$	0.880145	0.0294860
$892$	−9.46828	−0.317022
$893$	−104.313	−3.49072
$894$	24.9193	0.833425
$895$	−0.640944	−0.0214244
$896$	55.9371	1.86873
$897$	2.42384	0.0809298
$898$	64.6670	2.15797
$899$	−57.9168	−1.93163
$900$	25.3749	0.845832
$901$	4.49166	0.149639
$902$	2.91876	0.0971842
$903$	−10.8419	−0.360797
$904$	−60.8942	−2.02531
$905$	−7.95924	−0.264574
$906$	−2.66051	−0.0883895
$907$	3.85912	0.128140	0.0640700	−	0.997945i	$-0.479592\pi$
$907$	3.85912	0.128140	0.0640700	+	0.997945i	$0.479592\pi$
$908$	−88.6363	−2.94150
$909$	3.91253	0.129770
$910$	−25.1953	−0.835217
$911$	−28.0357	−0.928864	−0.464432	−	0.885609i	$-0.653742\pi$
$911$	−28.0357	−0.928864	−0.464432	+	0.885609i	$0.653742\pi$
$912$	15.9400	0.527827
$913$	−4.29895	−0.142275
$914$	75.3867	2.49357
$915$	1.41983	0.0469383
$916$	−13.8887	−0.458895
$917$	27.6043	0.911576
$918$	56.6402	1.86940
$919$	−50.1373	−1.65388	−0.826938	−	0.562293i	$-0.809919\pi$
$919$	−50.1373	−1.65388	−0.826938	+	0.562293i	$0.809919\pi$
$920$	2.03901	0.0672243
$921$	−2.70162	−0.0890213
$922$	43.5918	1.43562
$923$	−24.2072	−0.796791
$924$	12.8423	0.422482
$925$	−20.9333	−0.688284
$926$	13.7961	0.453367
$927$	−0.893524	−0.0293472
$928$	20.8003	0.682803
$929$	−8.05560	−0.264296	−0.132148	−	0.991230i	$-0.542187\pi$
$929$	−8.05560	−0.264296	−0.132148	+	0.991230i	$0.542187\pi$
$930$	−23.9441	−0.785157
$931$	−2.66817	−0.0874457
$932$	15.8984	0.520770
$933$	−1.96611	−0.0643675
$934$	18.1173	0.592816
$935$	5.15432	0.168564
$936$	26.5845	0.868940
$937$	32.2970	1.05510	0.527548	−	0.849525i	$-0.323111\pi$
$937$	32.2970	1.05510	0.527548	+	0.849525i	$0.323111\pi$
$938$	−94.8580	−3.09722
$939$	37.3095	1.21755
$940$	48.2263	1.57297
$941$	4.04283	0.131793	0.0658963	−	0.997826i	$-0.479009\pi$
$941$	4.04283	0.131793	0.0658963	+	0.997826i	$0.479009\pi$
$942$	−25.3924	−0.827331
$943$	−0.571241	−0.0186022
$944$	−3.91175	−0.127317
$945$	14.1817	0.461331
$946$	9.85438	0.320393
$947$	−24.8884	−0.808764	−0.404382	−	0.914590i	$-0.632513\pi$
$947$	−24.8884	−0.808764	−0.404382	+	0.914590i	$0.632513\pi$
$948$	−22.4815	−0.730164
$949$	17.7541	0.576323
$950$	−73.5198	−2.38530
$951$	5.19014	0.168302
$952$	46.6402	1.51162
$953$	−20.1417	−0.652452	−0.326226	−	0.945292i	$-0.605777\pi$
$953$	−20.1417	−0.652452	−0.326226	+	0.945292i	$0.605777\pi$
$954$	−4.12299	−0.133487
$955$	5.63886	0.182469
$956$	−33.3304	−1.07798
$957$	8.29028	0.267987
$958$	47.4820	1.53407
$959$	−37.3539	−1.20622
$960$	12.6748	0.409076
$961$	52.7691	1.70223
$962$	−49.0656	−1.58194
$963$	−3.35769	−0.108200
$964$	36.0508	1.16112
$965$	−14.7457	−0.474680
$966$	−3.90340	−0.125590
$967$	21.5490	0.692970	0.346485	−	0.938056i	$-0.387375\pi$
$967$	21.5490	0.692970	0.346485	+	0.938056i	$0.387375\pi$
$968$	36.9227	1.18674
$969$	−38.7858	−1.24598
$970$	39.4521	1.26673
$971$	−14.5288	−0.466252	−0.233126	−	0.972447i	$-0.574895\pi$
$971$	−14.5288	−0.466252	−0.233126	+	0.972447i	$0.574895\pi$
$972$	−54.6669	−1.75344
$973$	−58.4354	−1.87335
$974$	1.49101	0.0477750
$975$	18.0600	0.578382
$976$	−2.37431	−0.0759997
$977$	−54.7312	−1.75101	−0.875503	−	0.483212i	$-0.839470\pi$
$977$	−54.7312	−1.75101	−0.875503	+	0.483212i	$0.839470\pi$
$978$	63.1992	2.02089
$979$	−8.58156	−0.274268
$980$	1.23355	0.0394043
$981$	−15.8685	−0.506641
$982$	15.1000	0.481862
$983$	3.50474	0.111784	0.0558919	−	0.998437i	$-0.482200\pi$
$983$	3.50474	0.111784	0.0558919	+	0.998437i	$0.482200\pi$
$984$	4.53866	0.144687
$985$	−8.10501	−0.258247
$986$	67.3598	2.14517
$987$	−41.2659	−1.31351
$988$	−110.960	−3.53009
$989$	−1.92863	−0.0613270
$990$	−4.73127	−0.150370
$991$	−18.1278	−0.575847	−0.287924	−	0.957653i	$-0.592965\pi$
$991$	−18.1278	−0.575847	−0.287924	+	0.957653i	$0.592965\pi$
$992$	−30.0849	−0.955197
$993$	−14.3226	−0.454515
$994$	38.9837	1.23649
$995$	−15.3199	−0.485674
$996$	−14.9557	−0.473890
$997$	23.1100	0.731901	0.365951	−	0.930634i	$-0.380744\pi$
$997$	23.1100	0.731901	0.365951	+	0.930634i	$0.380744\pi$
$998$	−50.8347	−1.60914
$999$	27.6176	0.873782

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8003.2.a.b.1.16	✓	153

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8003.2.a.b.1.16	✓	153	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8003 = 53 \cdot 151 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8003.a (trivial)

\(f(q)\)	\(=\)	\(q-2.36991 q^{2} -1.12262 q^{3} +3.61648 q^{4} +0.983312 q^{5} +2.66051 q^{6} +2.71051 q^{7} -3.83091 q^{8} -1.73972 q^{9} +O(q^{10})\)	\(q-2.36991 q^{2} -1.12262 q^{3} +3.61648 q^{4} +0.983312 q^{5} +2.66051 q^{6} +2.71051 q^{7} -3.83091 q^{8} -1.73972 q^{9} -2.33036 q^{10} -1.16701 q^{11} -4.05993 q^{12} +3.98883 q^{13} -6.42368 q^{14} -1.10389 q^{15} +1.84596 q^{16} -4.49166 q^{17} +4.12299 q^{18} -7.69190 q^{19} +3.55613 q^{20} -3.04288 q^{21} +2.76571 q^{22} -0.541286 q^{23} +4.30066 q^{24} -4.03310 q^{25} -9.45317 q^{26} +5.32091 q^{27} +9.80251 q^{28} +6.32794 q^{29} +2.61611 q^{30} -9.15254 q^{31} +3.28705 q^{32} +1.31011 q^{33} +10.6448 q^{34} +2.66528 q^{35} -6.29168 q^{36} +5.19039 q^{37} +18.2291 q^{38} -4.47794 q^{39} -3.76698 q^{40} +1.05534 q^{41} +7.21135 q^{42} +3.56306 q^{43} -4.22046 q^{44} -1.71069 q^{45} +1.28280 q^{46} +13.5615 q^{47} -2.07232 q^{48} +0.346880 q^{49} +9.55809 q^{50} +5.04242 q^{51} +14.4255 q^{52} -1.00000 q^{53} -12.6101 q^{54} -1.14753 q^{55} -10.3837 q^{56} +8.63508 q^{57} -14.9967 q^{58} -2.11908 q^{59} -3.99218 q^{60} -1.28622 q^{61} +21.6907 q^{62} -4.71554 q^{63} -11.4820 q^{64} +3.92226 q^{65} -3.10484 q^{66} +14.7669 q^{67} -16.2440 q^{68} +0.607658 q^{69} -6.31647 q^{70} -6.06876 q^{71} +6.66473 q^{72} +4.45096 q^{73} -12.3008 q^{74} +4.52764 q^{75} -27.8176 q^{76} -3.16319 q^{77} +10.6123 q^{78} +5.53740 q^{79} +1.81516 q^{80} -0.754189 q^{81} -2.50106 q^{82} +3.68374 q^{83} -11.0045 q^{84} -4.41670 q^{85} -8.44413 q^{86} -7.10388 q^{87} +4.47071 q^{88} +7.35347 q^{89} +4.05418 q^{90} +10.8118 q^{91} -1.95755 q^{92} +10.2748 q^{93} -32.1395 q^{94} -7.56353 q^{95} -3.69011 q^{96} -16.9296 q^{97} -0.822076 q^{98} +2.03027 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 153 q - 9 q^{2} - 17 q^{3} + 137 q^{4} - 31 q^{5} - 10 q^{6} - 17 q^{7} - 30 q^{8} + 136 q^{9}+O(q^{10}) \) 153 * q - 9 * q^2 - 17 * q^3 + 137 * q^4 - 31 * q^5 - 10 * q^6 - 17 * q^7 - 30 * q^8 + 136 * q^9	\( 153 q - 9 q^{2} - 17 q^{3} + 137 q^{4} - 31 q^{5} - 10 q^{6} - 17 q^{7} - 30 q^{8} + 136 q^{9} - 34 q^{10} - q^{11} - 60 q^{12} - 101 q^{13} - 16 q^{14} - 14 q^{15} + 97 q^{16} - 12 q^{17} - 45 q^{18} - 45 q^{19} - 52 q^{20} - 76 q^{21} - 46 q^{22} - 28 q^{23} - 30 q^{24} + 84 q^{25} - 22 q^{26} - 68 q^{27} - 64 q^{28} - 14 q^{29} - q^{30} - 70 q^{31} - 54 q^{32} - 85 q^{33} - 59 q^{34} - 16 q^{35} + 87 q^{36} - 167 q^{37} - 48 q^{38} - 28 q^{39} - 68 q^{40} - 38 q^{41} + 2 q^{42} - 71 q^{43} - 10 q^{44} - 151 q^{45} - 37 q^{46} - 37 q^{47} - 166 q^{48} + 74 q^{49} - 3 q^{50} - 11 q^{51} - 183 q^{52} - 153 q^{53} - 40 q^{54} - 88 q^{55} - 69 q^{56} - 26 q^{57} - 43 q^{58} - 34 q^{59} - 12 q^{60} - 90 q^{61} - 37 q^{62} - 36 q^{63} + 58 q^{64} - 19 q^{65} + 52 q^{66} - 86 q^{67} - 22 q^{68} - 81 q^{69} - 144 q^{70} - 50 q^{71} - 190 q^{72} - 171 q^{73} - 14 q^{74} - 69 q^{75} - 88 q^{76} - 72 q^{77} - 61 q^{78} - 13 q^{79} - 84 q^{80} + 117 q^{81} - 124 q^{82} - 72 q^{83} - 106 q^{84} - 193 q^{85} - 44 q^{86} - 65 q^{87} - 89 q^{88} - 10 q^{89} - 152 q^{90} - 67 q^{91} - 29 q^{92} - 129 q^{93} - 43 q^{94} - 29 q^{95} - 106 q^{96} - 177 q^{97} - 69 q^{98} - 11 q^{99}+O(q^{100}) \) 153 * q - 9 * q^2 - 17 * q^3 + 137 * q^4 - 31 * q^5 - 10 * q^6 - 17 * q^7 - 30 * q^8 + 136 * q^9 - 34 * q^10 - q^11 - 60 * q^12 - 101 * q^13 - 16 * q^14 - 14 * q^15 + 97 * q^16 - 12 * q^17 - 45 * q^18 - 45 * q^19 - 52 * q^20 - 76 * q^21 - 46 * q^22 - 28 * q^23 - 30 * q^24 + 84 * q^25 - 22 * q^26 - 68 * q^27 - 64 * q^28 - 14 * q^29 - q^30 - 70 * q^31 - 54 * q^32 - 85 * q^33 - 59 * q^34 - 16 * q^35 + 87 * q^36 - 167 * q^37 - 48 * q^38 - 28 * q^39 - 68 * q^40 - 38 * q^41 + 2 * q^42 - 71 * q^43 - 10 * q^44 - 151 * q^45 - 37 * q^46 - 37 * q^47 - 166 * q^48 + 74 * q^49 - 3 * q^50 - 11 * q^51 - 183 * q^52 - 153 * q^53 - 40 * q^54 - 88 * q^55 - 69 * q^56 - 26 * q^57 - 43 * q^58 - 34 * q^59 - 12 * q^60 - 90 * q^61 - 37 * q^62 - 36 * q^63 + 58 * q^64 - 19 * q^65 + 52 * q^66 - 86 * q^67 - 22 * q^68 - 81 * q^69 - 144 * q^70 - 50 * q^71 - 190 * q^72 - 171 * q^73 - 14 * q^74 - 69 * q^75 - 88 * q^76 - 72 * q^77 - 61 * q^78 - 13 * q^79 - 84 * q^80 + 117 * q^81 - 124 * q^82 - 72 * q^83 - 106 * q^84 - 193 * q^85 - 44 * q^86 - 65 * q^87 - 89 * q^88 - 10 * q^89 - 152 * q^90 - 67 * q^91 - 29 * q^92 - 129 * q^93 - 43 * q^94 - 29 * q^95 - 106 * q^96 - 177 * q^97 - 69 * q^98 - 11 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more