LMFDB - Embedded newform 4003.2.a.b.1.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4003,2,Mod(1,4003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4003, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4003 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4003.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:	$31.9641159291$
Analytic rank:	$1$
Dimension:	$152$
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.15
Character	$\chi$	$=$	4003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.43441 q^{2} +1.77953 q^{3} +3.92633 q^{4} -0.360928 q^{5} -4.33209 q^{6} -1.53438 q^{7} -4.68946 q^{8} +0.166714 q^{9} +O(q^{10})$	\(q-2.43441 q^{2} +1.77953 q^{3} +3.92633 q^{4} -0.360928 q^{5} -4.33209 q^{6} -1.53438 q^{7} -4.68946 q^{8} +0.166714 q^{9} +0.878646 q^{10} -2.71903 q^{11} +6.98700 q^{12} +5.42134 q^{13} +3.73530 q^{14} -0.642281 q^{15} +3.56340 q^{16} +1.82952 q^{17} -0.405849 q^{18} -1.34221 q^{19} -1.41712 q^{20} -2.73047 q^{21} +6.61922 q^{22} +6.95468 q^{23} -8.34502 q^{24} -4.86973 q^{25} -13.1977 q^{26} -5.04191 q^{27} -6.02447 q^{28} -6.05028 q^{29} +1.56357 q^{30} +3.54754 q^{31} +0.704175 q^{32} -4.83859 q^{33} -4.45379 q^{34} +0.553801 q^{35} +0.654572 q^{36} -1.69847 q^{37} +3.26748 q^{38} +9.64741 q^{39} +1.69256 q^{40} -5.25484 q^{41} +6.64706 q^{42} -3.53428 q^{43} -10.6758 q^{44} -0.0601717 q^{45} -16.9305 q^{46} +0.427352 q^{47} +6.34116 q^{48} -4.64568 q^{49} +11.8549 q^{50} +3.25568 q^{51} +21.2860 q^{52} +2.88425 q^{53} +12.2740 q^{54} +0.981375 q^{55} +7.19541 q^{56} -2.38850 q^{57} +14.7288 q^{58} -13.7321 q^{59} -2.52181 q^{60} +8.87911 q^{61} -8.63616 q^{62} -0.255802 q^{63} -8.84104 q^{64} -1.95671 q^{65} +11.7791 q^{66} +9.16717 q^{67} +7.18330 q^{68} +12.3760 q^{69} -1.34817 q^{70} +5.39556 q^{71} -0.781797 q^{72} +11.6556 q^{73} +4.13476 q^{74} -8.66581 q^{75} -5.26995 q^{76} +4.17202 q^{77} -23.4857 q^{78} -2.01441 q^{79} -1.28613 q^{80} -9.47235 q^{81} +12.7924 q^{82} -6.98293 q^{83} -10.7207 q^{84} -0.660326 q^{85} +8.60386 q^{86} -10.7666 q^{87} +12.7508 q^{88} +1.05385 q^{89} +0.146482 q^{90} -8.31838 q^{91} +27.3064 q^{92} +6.31295 q^{93} -1.04035 q^{94} +0.484441 q^{95} +1.25310 q^{96} -0.00171825 q^{97} +11.3095 q^{98} -0.453299 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 152 q - 22 q^{2} - 18 q^{3} + 138 q^{4} - 59 q^{5} - 17 q^{6} - 19 q^{7} - 66 q^{8} + 106 q^{9}+O(q^{10}) $ 152 * q - 22 * q^2 - 18 * q^3 + 138 * q^4 - 59 * q^5 - 17 * q^6 - 19 * q^7 - 66 * q^8 + 106 * q^9	\( 152 q - 22 q^{2} - 18 q^{3} + 138 q^{4} - 59 q^{5} - 17 q^{6} - 19 q^{7} - 66 q^{8} + 106 q^{9} - 15 q^{10} - 40 q^{11} - 53 q^{12} - 59 q^{13} - 36 q^{14} - 40 q^{15} + 118 q^{16} - 93 q^{17} - 59 q^{18} - 16 q^{19} - 108 q^{20} - 62 q^{21} - 37 q^{22} - 107 q^{23} - 31 q^{24} + 101 q^{25} - 64 q^{26} - 63 q^{27} - 53 q^{28} - 124 q^{29} - 68 q^{30} - 15 q^{31} - 129 q^{32} - 49 q^{33} - 76 q^{35} + 45 q^{36} - 98 q^{37} - 125 q^{38} - 47 q^{39} - 7 q^{40} - 56 q^{41} - 84 q^{42} - 62 q^{43} - 114 q^{44} - 142 q^{45} - 3 q^{46} - 111 q^{47} - 92 q^{48} + 117 q^{49} - 64 q^{50} - 21 q^{51} - 85 q^{52} - 347 q^{53} + 3 q^{54} - 16 q^{55} - 73 q^{56} - 115 q^{57} - 29 q^{58} - 50 q^{59} - 54 q^{60} - 62 q^{61} - 55 q^{62} - 70 q^{63} + 64 q^{64} - 147 q^{65} + 34 q^{66} - 86 q^{67} - 174 q^{68} - 104 q^{69} - 7 q^{70} - 86 q^{71} - 139 q^{72} - 27 q^{73} - 52 q^{74} - 49 q^{75} - 11 q^{76} - 346 q^{77} - 59 q^{78} - 17 q^{79} - 149 q^{80} - 8 q^{81} - 31 q^{82} - 106 q^{83} - 51 q^{84} - 69 q^{85} - 85 q^{86} - 32 q^{87} - 113 q^{88} - 59 q^{89} + 10 q^{90} - 9 q^{91} - 314 q^{92} - 230 q^{93} + 7 q^{94} - 74 q^{95} - 54 q^{96} - 60 q^{97} - 77 q^{98} - 96 q^{99}+O(q^{100}) \) 152 * q - 22 * q^2 - 18 * q^3 + 138 * q^4 - 59 * q^5 - 17 * q^6 - 19 * q^7 - 66 * q^8 + 106 * q^9 - 15 * q^10 - 40 * q^11 - 53 * q^12 - 59 * q^13 - 36 * q^14 - 40 * q^15 + 118 * q^16 - 93 * q^17 - 59 * q^18 - 16 * q^19 - 108 * q^20 - 62 * q^21 - 37 * q^22 - 107 * q^23 - 31 * q^24 + 101 * q^25 - 64 * q^26 - 63 * q^27 - 53 * q^28 - 124 * q^29 - 68 * q^30 - 15 * q^31 - 129 * q^32 - 49 * q^33 - 76 * q^35 + 45 * q^36 - 98 * q^37 - 125 * q^38 - 47 * q^39 - 7 * q^40 - 56 * q^41 - 84 * q^42 - 62 * q^43 - 114 * q^44 - 142 * q^45 - 3 * q^46 - 111 * q^47 - 92 * q^48 + 117 * q^49 - 64 * q^50 - 21 * q^51 - 85 * q^52 - 347 * q^53 + 3 * q^54 - 16 * q^55 - 73 * q^56 - 115 * q^57 - 29 * q^58 - 50 * q^59 - 54 * q^60 - 62 * q^61 - 55 * q^62 - 70 * q^63 + 64 * q^64 - 147 * q^65 + 34 * q^66 - 86 * q^67 - 174 * q^68 - 104 * q^69 - 7 * q^70 - 86 * q^71 - 139 * q^72 - 27 * q^73 - 52 * q^74 - 49 * q^75 - 11 * q^76 - 346 * q^77 - 59 * q^78 - 17 * q^79 - 149 * q^80 - 8 * q^81 - 31 * q^82 - 106 * q^83 - 51 * q^84 - 69 * q^85 - 85 * q^86 - 32 * q^87 - 113 * q^88 - 59 * q^89 + 10 * q^90 - 9 * q^91 - 314 * q^92 - 230 * q^93 + 7 * q^94 - 74 * q^95 - 54 * q^96 - 60 * q^97 - 77 * q^98 - 96 * 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 $

$2$	−2.43441	−1.72138	−0.860692	−	0.509126i	$-0.829969\pi$
$2$	−2.43441	−1.72138	−0.860692	+	0.509126i	$0.829969\pi$
$3$	1.77953	1.02741	0.513705	−	0.857967i	$-0.328273\pi$
$3$	1.77953	1.02741	0.513705	+	0.857967i	$0.328273\pi$
$4$	3.92633	1.96316
$5$	−0.360928	−0.161412	−0.0807060	−	0.996738i	$-0.525717\pi$
$5$	−0.360928	−0.161412	−0.0807060	+	0.996738i	$0.525717\pi$
$6$	−4.33209	−1.76857
$7$	−1.53438	−0.579940	−0.289970	−	0.957036i	$-0.593645\pi$
$7$	−1.53438	−0.579940	−0.289970	+	0.957036i	$0.593645\pi$
$8$	−4.68946	−1.65798
$9$	0.166714	0.0555712
$10$	0.878646	0.277852
$11$	−2.71903	−0.819818	−0.409909	−	0.912126i	$-0.634440\pi$
$11$	−2.71903	−0.819818	−0.409909	+	0.912126i	$0.634440\pi$
$12$	6.98700	2.01697
$13$	5.42134	1.50361	0.751804	−	0.659386i	$-0.229184\pi$
$13$	5.42134	1.50361	0.751804	+	0.659386i	$0.229184\pi$
$14$	3.73530	0.998300
$15$	−0.642281	−0.165836
$16$	3.56340	0.890849
$17$	1.82952	0.443724	0.221862	−	0.975078i	$-0.428787\pi$
$17$	1.82952	0.443724	0.221862	+	0.975078i	$0.428787\pi$
$18$	−0.405849	−0.0956594
$19$	−1.34221	−0.307924	−0.153962	−	0.988077i	$-0.549203\pi$
$19$	−1.34221	−0.307924	−0.153962	+	0.988077i	$0.549203\pi$
$20$	−1.41712	−0.316878
$21$	−2.73047	−0.595837
$22$	6.61922	1.41122
$23$	6.95468	1.45015	0.725076	−	0.688669i	$-0.241805\pi$
$23$	6.95468	1.45015	0.725076	+	0.688669i	$0.241805\pi$
$24$	−8.34502	−1.70342
$25$	−4.86973	−0.973946
$26$	−13.1977	−2.58829
$27$	−5.04191	−0.970316
$28$	−6.02447	−1.13852
$29$	−6.05028	−1.12351	−0.561755	−	0.827304i	$-0.689874\pi$
$29$	−6.05028	−1.12351	−0.561755	+	0.827304i	$0.689874\pi$
$30$	1.56357	0.285468
$31$	3.54754	0.637157	0.318579	−	0.947896i	$-0.396794\pi$
$31$	3.54754	0.637157	0.318579	+	0.947896i	$0.396794\pi$
$32$	0.704175	0.124482
$33$	−4.83859	−0.842290
$34$	−4.45379	−0.763819
$35$	0.553801	0.0936094
$36$	0.654572	0.109095
$37$	−1.69847	−0.279227	−0.139613	−	0.990206i	$-0.544586\pi$
$37$	−1.69847	−0.279227	−0.139613	+	0.990206i	$0.544586\pi$
$38$	3.26748	0.530055
$39$	9.64741	1.54482
$40$	1.69256	0.267617
$41$	−5.25484	−0.820668	−0.410334	−	0.911935i	$-0.634588\pi$
$41$	−5.25484	−0.820668	−0.410334	+	0.911935i	$0.634588\pi$
$42$	6.64706	1.02566
$43$	−3.53428	−0.538972	−0.269486	−	0.963004i	$-0.586854\pi$
$43$	−3.53428	−0.538972	−0.269486	+	0.963004i	$0.586854\pi$
$44$	−10.6758	−1.60944
$45$	−0.0601717	−0.00896986
$46$	−16.9305	−2.49627
$47$	0.427352	0.0623357	0.0311679	−	0.999514i	$-0.490077\pi$
$47$	0.427352	0.0623357	0.0311679	+	0.999514i	$0.490077\pi$
$48$	6.34116	0.915267
$49$	−4.64568	−0.663669
$50$	11.8549	1.67654
$51$	3.25568	0.455886
$52$	21.2860	2.95183
$53$	2.88425	0.396182	0.198091	−	0.980184i	$-0.436526\pi$
$53$	2.88425	0.396182	0.198091	+	0.980184i	$0.436526\pi$
$54$	12.2740	1.67029
$55$	0.981375	0.132329
$56$	7.19541	0.961527
$57$	−2.38850	−0.316364
$58$	14.7288	1.93399
$59$	−13.7321	−1.78777	−0.893886	−	0.448295i	$-0.852032\pi$
$59$	−13.7321	−1.78777	−0.893886	+	0.448295i	$0.852032\pi$
$60$	−2.52181	−0.325564
$61$	8.87911	1.13685	0.568427	−	0.822734i	$-0.307552\pi$
$61$	8.87911	1.13685	0.568427	+	0.822734i	$0.307552\pi$
$62$	−8.63616	−1.09679
$63$	−0.255802	−0.0322280
$64$	−8.84104	−1.10513
$65$	−1.95671	−0.242701
$66$	11.7791	1.44990
$67$	9.16717	1.11995	0.559974	−	0.828510i	$-0.310811\pi$
$67$	9.16717	1.11995	0.559974	+	0.828510i	$0.310811\pi$
$68$	7.18330	0.871103
$69$	12.3760	1.48990
$70$	−1.34817	−0.161138
$71$	5.39556	0.640335	0.320168	−	0.947361i	$-0.396261\pi$
$71$	5.39556	0.640335	0.320168	+	0.947361i	$0.396261\pi$
$72$	−0.781797	−0.0921357
$73$	11.6556	1.36419	0.682093	−	0.731266i	$-0.261070\pi$
$73$	11.6556	1.36419	0.682093	+	0.731266i	$0.261070\pi$
$74$	4.13476	0.480656
$75$	−8.66581	−1.00064
$76$	−5.26995	−0.604505
$77$	4.17202	0.475446
$78$	−23.4857	−2.65923
$79$	−2.01441	−0.226639	−0.113320	−	0.993559i	$-0.536148\pi$
$79$	−2.01441	−0.226639	−0.113320	+	0.993559i	$0.536148\pi$
$80$	−1.28613	−0.143794
$81$	−9.47235	−1.05248
$82$	12.7924	1.41268
$83$	−6.98293	−0.766477	−0.383238	−	0.923650i	$-0.625191\pi$
$83$	−6.98293	−0.766477	−0.383238	+	0.923650i	$0.625191\pi$
$84$	−10.7207	−1.16972
$85$	−0.660326	−0.0716224
$86$	8.60386	0.927778
$87$	−10.7666	−1.15430
$88$	12.7508	1.35924
$89$	1.05385	0.111708	0.0558540	−	0.998439i	$-0.482212\pi$
$89$	1.05385	0.111708	0.0558540	+	0.998439i	$0.482212\pi$
$90$	0.146482	0.0154406
$91$	−8.31838	−0.872003
$92$	27.3064	2.84689
$93$	6.31295	0.654622
$94$	−1.04035	−0.107304
$95$	0.484441	0.0497026
$96$	1.25310	0.127894
$97$	−0.00171825	−0.000174462 0	−8.72309e−5	−	1.00000i	$-0.500028\pi$
$97$	−0.00171825	−0.000174462 0	−8.72309e−5		1.00000i	$0.500028\pi$
$98$	11.3095	1.14243
$99$	−0.453299	−0.0455583
$100$	−19.1202	−1.91202
$101$	−10.8815	−1.08275	−0.541373	−	0.840783i	$-0.682095\pi$
$101$	−10.8815	−1.08275	−0.541373	+	0.840783i	$0.682095\pi$
$102$	−7.92564	−0.784756
$103$	9.23290	0.909745	0.454872	−	0.890557i	$-0.349685\pi$
$103$	9.23290	0.909745	0.454872	+	0.890557i	$0.349685\pi$
$104$	−25.4232	−2.49295
$105$	0.985503	0.0961752
$106$	−7.02144	−0.681982
$107$	−18.9603	−1.83296	−0.916481	−	0.400078i	$-0.868983\pi$
$107$	−18.9603	−1.83296	−0.916481	+	0.400078i	$0.868983\pi$
$108$	−19.7962	−1.90489
$109$	2.88088	0.275938	0.137969	−	0.990437i	$-0.455943\pi$
$109$	2.88088	0.275938	0.137969	+	0.990437i	$0.455943\pi$
$110$	−2.38906	−0.227788
$111$	−3.02247	−0.286880
$112$	−5.46760	−0.516639
$113$	−10.4326	−0.981417	−0.490709	−	0.871324i	$-0.663262\pi$
$113$	−10.4326	−0.981417	−0.490709	+	0.871324i	$0.663262\pi$
$114$	5.81457	0.544584
$115$	−2.51014	−0.234072
$116$	−23.7554	−2.20563
$117$	0.903811	0.0835574
$118$	33.4296	3.07744
$119$	−2.80718	−0.257333
$120$	3.01196	0.274953
$121$	−3.60688	−0.327898
$122$	−21.6153	−1.95696
$123$	−9.35112	−0.843162
$124$	13.9288	1.25084
$125$	3.56227	0.318619
$126$	0.622725	0.0554768
$127$	1.76375	0.156507	0.0782537	−	0.996933i	$-0.475066\pi$
$127$	1.76375	0.156507	0.0782537	+	0.996933i	$0.475066\pi$
$128$	20.1143	1.77787
$129$	−6.28934	−0.553745
$130$	4.76344	0.417781
$131$	−11.3760	−0.993928	−0.496964	−	0.867771i	$-0.665552\pi$
$131$	−11.3760	−0.993928	−0.496964	+	0.867771i	$0.665552\pi$
$132$	−18.9979	−1.65355
$133$	2.05946	0.178578
$134$	−22.3166	−1.92786
$135$	1.81977	0.156621
$136$	−8.57947	−0.735683
$137$	−7.11757	−0.608095	−0.304047	−	0.952657i	$-0.598338\pi$
$137$	−7.11757	−0.608095	−0.304047	+	0.952657i	$0.598338\pi$
$138$	−30.1283	−2.56469
$139$	0.279750	0.0237281	0.0118640	−	0.999930i	$-0.496223\pi$
$139$	0.279750	0.0237281	0.0118640	+	0.999930i	$0.496223\pi$
$140$	2.17440	0.183771
$141$	0.760485	0.0640444
$142$	−13.1350	−1.10226
$143$	−14.7408	−1.23269
$144$	0.594067	0.0495056
$145$	2.18372	0.181348
$146$	−28.3745	−2.34829
$147$	−8.26712	−0.681860
$148$	−6.66875	−0.548168
$149$	0.431693	0.0353656	0.0176828	−	0.999844i	$-0.494371\pi$
$149$	0.431693	0.0353656	0.0176828	+	0.999844i	$0.494371\pi$
$150$	21.0961	1.72249
$151$	−0.974531	−0.0793062	−0.0396531	−	0.999214i	$-0.512625\pi$
$151$	−0.974531	−0.0793062	−0.0396531	+	0.999214i	$0.512625\pi$
$152$	6.29424	0.510530
$153$	0.305006	0.0246583
$154$	−10.1564	−0.818425
$155$	−1.28041	−0.102845
$156$	37.8789	3.03274
$157$	2.38774	0.190563	0.0952813	−	0.995450i	$-0.469625\pi$
$157$	2.38774	0.190563	0.0952813	+	0.995450i	$0.469625\pi$
$158$	4.90390	0.390133
$159$	5.13260	0.407042
$160$	−0.254157	−0.0200929
$161$	−10.6711	−0.841001
$162$	23.0595	1.81173
$163$	−1.77098	−0.138714	−0.0693571	−	0.997592i	$-0.522095\pi$
$163$	−1.77098	−0.138714	−0.0693571	+	0.997592i	$0.522095\pi$
$164$	−20.6322	−1.61111
$165$	1.74638	0.135956
$166$	16.9993	1.31940
$167$	16.4248	1.27099	0.635496	−	0.772104i	$-0.280795\pi$
$167$	16.4248	1.27099	0.635496	+	0.772104i	$0.280795\pi$
$168$	12.8044	0.987882
$169$	16.3909	1.26084
$170$	1.60750	0.123290
$171$	−0.223765	−0.0171117
$172$	−13.8767	−1.05809
$173$	−17.3309	−1.31765	−0.658823	−	0.752298i	$-0.728945\pi$
$173$	−17.3309	−1.31765	−0.658823	+	0.752298i	$0.728945\pi$
$174$	26.2104	1.98700
$175$	7.47201	0.564831
$176$	−9.68898	−0.730335
$177$	−24.4367	−1.83677
$178$	−2.56550	−0.192292
$179$	−13.4213	−1.00315	−0.501577	−	0.865113i	$-0.667247\pi$
$179$	−13.4213	−1.00315	−0.501577	+	0.865113i	$0.667247\pi$
$180$	−0.236254	−0.0176093
$181$	−11.0756	−0.823241	−0.411620	−	0.911355i	$-0.635037\pi$
$181$	−11.0756	−0.823241	−0.411620	+	0.911355i	$0.635037\pi$
$182$	20.2503	1.50105
$183$	15.8006	1.16801
$184$	−32.6137	−2.40432
$185$	0.613026	0.0450705
$186$	−15.3683	−1.12686
$187$	−4.97452	−0.363773
$188$	1.67793	0.122375
$189$	7.73619	0.562725
$190$	−1.17933	−0.0855574
$191$	−25.9102	−1.87480	−0.937398	−	0.348259i	$-0.886773\pi$
$191$	−25.9102	−1.87480	−0.937398	+	0.348259i	$0.886773\pi$
$192$	−15.7329	−1.13542
$193$	−7.76187	−0.558711	−0.279356	−	0.960188i	$-0.590121\pi$
$193$	−7.76187	−0.558711	−0.279356	+	0.960188i	$0.590121\pi$
$194$	0.00418291	0.000300316 0
$195$	−3.48203	−0.249353
$196$	−18.2405	−1.30289
$197$	−7.21440	−0.514005	−0.257002	−	0.966411i	$-0.582735\pi$
$197$	−7.21440	−0.514005	−0.257002	+	0.966411i	$0.582735\pi$
$198$	1.10351	0.0784234
$199$	−8.30703	−0.588870	−0.294435	−	0.955672i	$-0.595131\pi$
$199$	−8.30703	−0.588870	−0.294435	+	0.955672i	$0.595131\pi$
$200$	22.8364	1.61478
$201$	16.3132	1.15065
$202$	26.4899	1.86382
$203$	9.28342	0.651569
$204$	12.7829	0.894980
$205$	1.89662	0.132466
$206$	−22.4766	−1.56602
$207$	1.15944	0.0805867
$208$	19.3184	1.33949
$209$	3.64951	0.252442
$210$	−2.39911	−0.165554
$211$	2.41851	0.166497	0.0832485	−	0.996529i	$-0.473470\pi$
$211$	2.41851	0.166497	0.0832485	+	0.996529i	$0.473470\pi$
$212$	11.3245	0.777771
$213$	9.60154	0.657887
$214$	46.1571	3.15523
$215$	1.27562	0.0869966
$216$	23.6438	1.60876
$217$	−5.44327	−0.369513
$218$	−7.01322	−0.474995
$219$	20.7415	1.40158
$220$	3.85320	0.259783
$221$	9.91845	0.667187
$222$	7.35792	0.493831
$223$	1.47657	0.0988787	0.0494394	−	0.998777i	$-0.484257\pi$
$223$	1.47657	0.0988787	0.0494394	+	0.998777i	$0.484257\pi$
$224$	−1.08047	−0.0721920
$225$	−0.811851	−0.0541234
$226$	25.3972	1.68940
$227$	−6.29268	−0.417660	−0.208830	−	0.977952i	$-0.566965\pi$
$227$	−6.29268	−0.417660	−0.208830	+	0.977952i	$0.566965\pi$
$228$	−9.37802	−0.621075
$229$	10.3797	0.685908	0.342954	−	0.939352i	$-0.388572\pi$
$229$	10.3797	0.685908	0.342954	+	0.939352i	$0.388572\pi$
$230$	6.11070	0.402928
$231$	7.42422	0.488478
$232$	28.3726	1.86275
$233$	6.96882	0.456543	0.228271	−	0.973598i	$-0.426693\pi$
$233$	6.96882	0.456543	0.228271	+	0.973598i	$0.426693\pi$
$234$	−2.20024	−0.143834
$235$	−0.154244	−0.0100617
$236$	−53.9169	−3.50969
$237$	−3.58470	−0.232851
$238$	6.83380	0.442970
$239$	3.81790	0.246959	0.123480	−	0.992347i	$-0.460595\pi$
$239$	3.81790	0.246959	0.123480	+	0.992347i	$0.460595\pi$
$240$	−2.28870	−0.147735
$241$	3.58227	0.230754	0.115377	−	0.993322i	$-0.463192\pi$
$241$	3.58227	0.230754	0.115377	+	0.993322i	$0.463192\pi$
$242$	8.78060	0.564438
$243$	−1.73057	−0.111016
$244$	34.8623	2.23183
$245$	1.67676	0.107124
$246$	22.7644	1.45141
$247$	−7.27657	−0.462997
$248$	−16.6361	−1.05639
$249$	−12.4263	−0.787486
$250$	−8.67200	−0.548465
$251$	−8.55140	−0.539759	−0.269880	−	0.962894i	$-0.586984\pi$
$251$	−8.55140	−0.539759	−0.269880	+	0.962894i	$0.586984\pi$
$252$	−1.00436	−0.0632688
$253$	−18.9100	−1.18886
$254$	−4.29368	−0.269409
$255$	−1.17507	−0.0735856
$256$	−31.2843	−1.95527
$257$	6.56368	0.409431	0.204716	−	0.978821i	$-0.434373\pi$
$257$	6.56368	0.409431	0.204716	+	0.978821i	$0.434373\pi$
$258$	15.3108	0.953208
$259$	2.60609	0.161935
$260$	−7.68270	−0.476461
$261$	−1.00866	−0.0624348
$262$	27.6938	1.71093
$263$	29.7712	1.83577	0.917884	−	0.396849i	$-0.129896\pi$
$263$	29.7712	1.83577	0.917884	+	0.396849i	$0.129896\pi$
$264$	22.6904	1.39650
$265$	−1.04101	−0.0639486
$266$	−5.01355	−0.307401
$267$	1.87536	0.114770
$268$	35.9933	2.19864
$269$	−8.69549	−0.530173	−0.265087	−	0.964225i	$-0.585401\pi$
$269$	−8.69549	−0.530173	−0.265087	+	0.964225i	$0.585401\pi$
$270$	−4.43005	−0.269604
$271$	−2.99039	−0.181653	−0.0908266	−	0.995867i	$-0.528951\pi$
$271$	−2.99039	−0.181653	−0.0908266	+	0.995867i	$0.528951\pi$
$272$	6.51931	0.395291
$273$	−14.8028	−0.895905
$274$	17.3270	1.04676
$275$	13.2409	0.798459
$276$	48.5924	2.92492
$277$	−9.40388	−0.565025	−0.282512	−	0.959264i	$-0.591168\pi$
$277$	−9.40388	−0.565025	−0.282512	+	0.959264i	$0.591168\pi$
$278$	−0.681025	−0.0408451
$279$	0.591424	0.0354076
$280$	−2.59703	−0.155202
$281$	23.4111	1.39659	0.698294	−	0.715811i	$-0.253943\pi$
$281$	23.4111	1.39659	0.698294	+	0.715811i	$0.253943\pi$
$282$	−1.85133	−0.110245
$283$	−26.0157	−1.54647	−0.773235	−	0.634119i	$-0.781363\pi$
$283$	−26.0157	−1.54647	−0.773235	+	0.634119i	$0.781363\pi$
$284$	21.1847	1.25708
$285$	0.862076	0.0510650
$286$	35.8850	2.12193
$287$	8.06291	0.475938
$288$	0.117396	0.00691760
$289$	−13.6529	−0.803109
$290$	−5.31606	−0.312170
$291$	−0.00305767	−0.000179244 0
$292$	45.7637	2.67812
$293$	0.0159037	0.000929102 0	0.000464551	−	1.00000i	$-0.499852\pi$
$293$	0.0159037	0.000929102 0	0.000464551		1.00000i	$0.499852\pi$
$294$	20.1255	1.17374
$295$	4.95632	0.288568
$296$	7.96491	0.462951
$297$	13.7091	0.795483
$298$	−1.05091	−0.0608779
$299$	37.7037	2.18046
$300$	−34.0248	−1.96442
$301$	5.42292	0.312572
$302$	2.37240	0.136517
$303$	−19.3638	−1.11242
$304$	−4.78282	−0.274314
$305$	−3.20472	−0.183502
$306$	−0.742508	−0.0424464
$307$	3.89875	0.222514	0.111257	−	0.993792i	$-0.464512\pi$
$307$	3.89875	0.222514	0.111257	+	0.993792i	$0.464512\pi$
$308$	16.3807	0.933378
$309$	16.4302	0.934681
$310$	3.11703	0.177036
$311$	−6.30304	−0.357413	−0.178706	−	0.983902i	$-0.557191\pi$
$311$	−6.30304	−0.357413	−0.178706	+	0.983902i	$0.557191\pi$
$312$	−45.2412	−2.56128
$313$	−9.76144	−0.551749	−0.275874	−	0.961194i	$-0.588967\pi$
$313$	−9.76144	−0.551749	−0.275874	+	0.961194i	$0.588967\pi$
$314$	−5.81273	−0.328031
$315$	0.0923261	0.00520199
$316$	−7.90925	−0.444930
$317$	−9.23281	−0.518566	−0.259283	−	0.965801i	$-0.583486\pi$
$317$	−9.23281	−0.518566	−0.259283	+	0.965801i	$0.583486\pi$
$318$	−12.4948	−0.700675
$319$	16.4509	0.921074
$320$	3.19098	0.178381
$321$	−33.7404	−1.88320
$322$	25.9778	1.44769
$323$	−2.45560	−0.136633
$324$	−37.1915	−2.06620
$325$	−26.4005	−1.46443
$326$	4.31129	0.238780
$327$	5.12660	0.283501
$328$	24.6424	1.36065
$329$	−0.655720	−0.0361510
$330$	−4.25140	−0.234032
$331$	−9.86665	−0.542320	−0.271160	−	0.962534i	$-0.587407\pi$
$331$	−9.86665	−0.542320	−0.271160	+	0.962534i	$0.587407\pi$
$332$	−27.4173	−1.50472
$333$	−0.283158	−0.0155170
$334$	−39.9847	−2.18787
$335$	−3.30869	−0.180773
$336$	−9.72973	−0.530801
$337$	−29.9516	−1.63157	−0.815785	−	0.578356i	$-0.803695\pi$
$337$	−29.9516	−1.63157	−0.815785	+	0.578356i	$0.803695\pi$
$338$	−39.9021	−2.17039
$339$	−18.5651	−1.00832
$340$	−2.59266	−0.140607
$341$	−9.64588	−0.522353
$342$	0.544734	0.0294558
$343$	17.8689	0.964829
$344$	16.5739	0.893603
$345$	−4.46686	−0.240488
$346$	42.1905	2.26818
$347$	−17.2299	−0.924951	−0.462475	−	0.886632i	$-0.653039\pi$
$347$	−17.2299	−0.924951	−0.462475	+	0.886632i	$0.653039\pi$
$348$	−42.2734	−2.26609
$349$	−21.5405	−1.15304	−0.576519	−	0.817084i	$-0.695589\pi$
$349$	−21.5405	−1.15304	−0.576519	+	0.817084i	$0.695589\pi$
$350$	−18.1899	−0.972291
$351$	−27.3339	−1.45898
$352$	−1.91467	−0.102052
$353$	−28.0720	−1.49412	−0.747060	−	0.664757i	$-0.768535\pi$
$353$	−28.0720	−1.49412	−0.747060	+	0.664757i	$0.768535\pi$
$354$	59.4888	3.16179
$355$	−1.94741	−0.103358
$356$	4.13777	0.219301
$357$	−4.99544	−0.264387
$358$	32.6729	1.72681
$359$	14.6434	0.772851	0.386426	−	0.922321i	$-0.373710\pi$
$359$	14.6434	0.772851	0.386426	+	0.922321i	$0.373710\pi$
$360$	0.282173	0.0148718
$361$	−17.1985	−0.905183
$362$	26.9624	1.41711
$363$	−6.41853	−0.336885
$364$	−32.6607	−1.71189
$365$	−4.20684	−0.220196
$366$	−38.4651	−2.01060
$367$	19.7503	1.03096	0.515479	−	0.856902i	$-0.327614\pi$
$367$	19.7503	1.03096	0.515479	+	0.856902i	$0.327614\pi$
$368$	24.7823	1.29187
$369$	−0.876053	−0.0456055
$370$	−1.49235	−0.0775837
$371$	−4.42553	−0.229762
$372$	24.7867	1.28513
$373$	26.4196	1.36796	0.683978	−	0.729503i	$-0.260248\pi$
$373$	26.4196	1.36796	0.683978	+	0.729503i	$0.260248\pi$
$374$	12.1100	0.626193
$375$	6.33915	0.327352
$376$	−2.00405	−0.103351
$377$	−32.8006	−1.68932
$378$	−18.8330	−0.968666
$379$	−5.76350	−0.296051	−0.148026	−	0.988984i	$-0.547292\pi$
$379$	−5.76350	−0.296051	−0.148026	+	0.988984i	$0.547292\pi$
$380$	1.90208	0.0975744
$381$	3.13864	0.160797
$382$	63.0759	3.22725
$383$	−8.45821	−0.432194	−0.216097	−	0.976372i	$-0.569333\pi$
$383$	−8.45821	−0.432194	−0.216097	+	0.976372i	$0.569333\pi$
$384$	35.7940	1.82660
$385$	−1.50580	−0.0767427
$386$	18.8955	0.961757
$387$	−0.589212	−0.0299513
$388$	−0.00674641	−0.000342497 0
$389$	−33.0117	−1.67376	−0.836880	−	0.547387i	$-0.815623\pi$
$389$	−33.0117	−1.67376	−0.836880	+	0.547387i	$0.815623\pi$
$390$	8.47666	0.429232
$391$	12.7237	0.643467
$392$	21.7858	1.10035
$393$	−20.2439	−1.02117
$394$	17.5628	0.884800
$395$	0.727059	0.0365823
$396$	−1.77980	−0.0894384
$397$	13.4482	0.674944	0.337472	−	0.941336i	$-0.390428\pi$
$397$	13.4482	0.674944	0.337472	+	0.941336i	$0.390428\pi$
$398$	20.2227	1.01367
$399$	3.66486	0.183472
$400$	−17.3528	−0.867639
$401$	−7.22760	−0.360929	−0.180465	−	0.983581i	$-0.557760\pi$
$401$	−7.22760	−0.360929	−0.180465	+	0.983581i	$0.557760\pi$
$402$	−39.7130	−1.98070
$403$	19.2324	0.958036
$404$	−42.7242	−2.12561
$405$	3.41884	0.169883
$406$	−22.5996	−1.12160
$407$	4.61819	0.228915
$408$	−15.2674	−0.755848
$409$	17.1033	0.845704	0.422852	−	0.906199i	$-0.361029\pi$
$409$	17.1033	0.845704	0.422852	+	0.906199i	$0.361029\pi$
$410$	−4.61714	−0.228024
$411$	−12.6659	−0.624763
$412$	36.2514	1.78598
$413$	21.0703	1.03680
$414$	−2.82255	−0.138721
$415$	2.52034	0.123719
$416$	3.81757	0.187172
$417$	0.497822	0.0243785
$418$	−8.88438	−0.434549
$419$	15.8437	0.774015	0.387008	−	0.922077i	$-0.373509\pi$
$419$	15.8437	0.774015	0.387008	+	0.922077i	$0.373509\pi$
$420$	3.86941	0.188808
$421$	−12.9353	−0.630430	−0.315215	−	0.949020i	$-0.602077\pi$
$421$	−12.9353	−0.630430	−0.315215	+	0.949020i	$0.602077\pi$
$422$	−5.88763	−0.286605
$423$	0.0712455	0.00346407
$424$	−13.5256	−0.656861
$425$	−8.90927	−0.432163
$426$	−23.3740	−1.13248
$427$	−13.6239	−0.659307
$428$	−74.4444	−3.59841
$429$	−26.2316	−1.26647
$430$	−3.10538	−0.149755
$431$	7.16212	0.344987	0.172494	−	0.985011i	$-0.444818\pi$
$431$	7.16212	0.344987	0.172494	+	0.985011i	$0.444818\pi$
$432$	−17.9663	−0.864405
$433$	−18.9370	−0.910052	−0.455026	−	0.890478i	$-0.650370\pi$
$433$	−18.9370	−0.910052	−0.455026	+	0.890478i	$0.650370\pi$
$434$	13.2511	0.636075
$435$	3.88599	0.186319
$436$	11.3113	0.541712
$437$	−9.33464	−0.446536
$438$	−50.4931	−2.41265
$439$	35.9615	1.71635	0.858173	−	0.513360i	$-0.171599\pi$
$439$	35.9615	1.71635	0.858173	+	0.513360i	$0.171599\pi$
$440$	−4.60212	−0.219398
$441$	−0.774499	−0.0368809
$442$	−24.1455	−1.14849
$443$	−6.42771	−0.305390	−0.152695	−	0.988273i	$-0.548795\pi$
$443$	−6.42771	−0.305390	−0.152695	+	0.988273i	$0.548795\pi$
$444$	−11.8672	−0.563193
$445$	−0.380365	−0.0180310
$446$	−3.59458	−0.170208
$447$	0.768209	0.0363350
$448$	13.5655	0.640910
$449$	−13.8258	−0.652482	−0.326241	−	0.945287i	$-0.605782\pi$
$449$	−13.8258	−0.652482	−0.326241	+	0.945287i	$0.605782\pi$
$450$	1.97637	0.0931671
$451$	14.2881	0.672799
$452$	−40.9618	−1.92668
$453$	−1.73420	−0.0814800
$454$	15.3189	0.718953
$455$	3.00234	0.140752
$456$	11.2008	0.524524
$457$	29.4748	1.37878	0.689388	−	0.724393i	$-0.257880\pi$
$457$	29.4748	1.37878	0.689388	+	0.724393i	$0.257880\pi$
$458$	−25.2683	−1.18071
$459$	−9.22427	−0.430552
$460$	−9.85564	−0.459522
$461$	−24.8082	−1.15543	−0.577716	−	0.816238i	$-0.696056\pi$
$461$	−24.8082	−1.15543	−0.577716	+	0.816238i	$0.696056\pi$
$462$	−18.0736	−0.840858
$463$	5.79512	0.269322	0.134661	−	0.990892i	$-0.457005\pi$
$463$	5.79512	0.269322	0.134661	+	0.990892i	$0.457005\pi$
$464$	−21.5596	−1.00088
$465$	−2.27852	−0.105664
$466$	−16.9649	−0.785885
$467$	−4.45534	−0.206169	−0.103084	−	0.994673i	$-0.532871\pi$
$467$	−4.45534	−0.206169	−0.103084	+	0.994673i	$0.532871\pi$
$468$	3.54866	0.164037
$469$	−14.0659	−0.649503
$470$	0.375491	0.0173201
$471$	4.24905	0.195786
$472$	64.3963	2.96408
$473$	9.60980	0.441859
$474$	8.72661	0.400827
$475$	6.53620	0.299901
$476$	−11.0219	−0.505188
$477$	0.480844	0.0220163
$478$	−9.29431	−0.425112
$479$	27.5003	1.25652	0.628260	−	0.778003i	$-0.283767\pi$
$479$	27.5003	1.25652	0.628260	+	0.778003i	$0.283767\pi$
$480$	−0.452279	−0.0206436
$481$	−9.20798	−0.419848
$482$	−8.72069	−0.397217
$483$	−18.9895	−0.864053
$484$	−14.1618	−0.643717
$485$	0.000620165 0	2.81602e−5 0
$486$	4.21291	0.191101
$487$	−43.9550	−1.99179	−0.995895	−	0.0905110i	$-0.971150\pi$
$487$	−43.9550	−1.99179	−0.995895	+	0.0905110i	$0.971150\pi$
$488$	−41.6382	−1.88487
$489$	−3.15151	−0.142516
$490$	−4.08191	−0.184402
$491$	18.6143	0.840051	0.420026	−	0.907512i	$-0.362021\pi$
$491$	18.6143	0.840051	0.420026	+	0.907512i	$0.362021\pi$
$492$	−36.7156	−1.65527
$493$	−11.0691	−0.498528
$494$	17.7141	0.796996
$495$	0.163609	0.00735366
$496$	12.6413	0.567611
$497$	−8.27883	−0.371356
$498$	30.2507	1.35557
$499$	14.9742	0.670339	0.335170	−	0.942158i	$-0.391206\pi$
$499$	14.9742	0.670339	0.335170	+	0.942158i	$0.391206\pi$
$500$	13.9866	0.625501
$501$	29.2284	1.30583
$502$	20.8176	0.929133
$503$	−35.6500	−1.58955	−0.794777	−	0.606902i	$-0.792412\pi$
$503$	−35.6500	−1.58955	−0.794777	+	0.606902i	$0.792412\pi$
$504$	1.19957	0.0534332
$505$	3.92743	0.174768
$506$	46.0346	2.04649
$507$	29.1681	1.29540
$508$	6.92505	0.307250
$509$	14.4290	0.639554	0.319777	−	0.947493i	$-0.396392\pi$
$509$	14.4290	0.639554	0.319777	+	0.947493i	$0.396392\pi$
$510$	2.86059	0.126669
$511$	−17.8841	−0.791146
$512$	35.9301	1.58790
$513$	6.76730	0.298783
$514$	−15.9787	−0.704789
$515$	−3.33242	−0.146844
$516$	−24.6940	−1.08709
$517$	−1.16198	−0.0511040
$518$	−6.34429	−0.278752
$519$	−30.8408	−1.35376
$520$	9.17594	0.402392
$521$	−38.7147	−1.69612	−0.848062	−	0.529898i	$-0.822230\pi$
$521$	−38.7147	−1.69612	−0.848062	+	0.529898i	$0.822230\pi$
$522$	2.45550	0.107474
$523$	9.85415	0.430892	0.215446	−	0.976516i	$-0.430879\pi$
$523$	9.85415	0.430892	0.215446	+	0.976516i	$0.430879\pi$
$524$	−44.6660	−1.95124
$525$	13.2966	0.580313
$526$	−72.4750	−3.16006
$527$	6.49030	0.282722
$528$	−17.2418	−0.750353
$529$	25.3676	1.10294
$530$	2.53424	0.110080
$531$	−2.28933	−0.0993486
$532$	8.08610	0.350577
$533$	−28.4883	−1.23396
$534$	−4.56538	−0.197563
$535$	6.84331	0.295862
$536$	−42.9891	−1.85685
$537$	−23.8835	−1.03065
$538$	21.1683	0.912632
$539$	12.6318	0.544088
$540$	7.14500	0.307472
$541$	−19.9504	−0.857737	−0.428868	−	0.903367i	$-0.641088\pi$
$541$	−19.9504	−0.857737	−0.428868	+	0.903367i	$0.641088\pi$
$542$	7.27982	0.312695
$543$	−19.7093	−0.845806
$544$	1.28830	0.0552355
$545$	−1.03979	−0.0445397
$546$	36.0360	1.54220
$547$	−29.7185	−1.27067	−0.635337	−	0.772235i	$-0.719139\pi$
$547$	−29.7185	−1.27067	−0.635337	+	0.772235i	$0.719139\pi$
$548$	−27.9459	−1.19379
$549$	1.48027	0.0631763
$550$	−32.2338	−1.37445
$551$	8.12075	0.345955
$552$	−58.0370	−2.47022
$553$	3.09087	0.131437
$554$	22.8929	0.972624
$555$	1.09090	0.0463059
$556$	1.09839	0.0465821
$557$	−34.6065	−1.46632	−0.733161	−	0.680055i	$-0.761956\pi$
$557$	−34.6065	−1.46632	−0.733161	+	0.680055i	$0.761956\pi$
$558$	−1.43977	−0.0609501
$559$	−19.1605	−0.810403
$560$	1.97341	0.0833918
$561$	−8.85229	−0.373744
$562$	−56.9920	−2.40406
$563$	−8.96186	−0.377697	−0.188849	−	0.982006i	$-0.560476\pi$
$563$	−8.96186	−0.377697	−0.188849	+	0.982006i	$0.560476\pi$
$564$	2.98591	0.125730
$565$	3.76542	0.158413
$566$	63.3326	2.66207
$567$	14.5342	0.610377
$568$	−25.3023	−1.06166
$569$	23.1417	0.970151	0.485075	−	0.874472i	$-0.338792\pi$
$569$	23.1417	0.970151	0.485075	+	0.874472i	$0.338792\pi$
$570$	−2.09864	−0.0879025
$571$	−37.6945	−1.57746	−0.788732	−	0.614737i	$-0.789262\pi$
$571$	−37.6945	−1.57746	−0.788732	+	0.614737i	$0.789262\pi$
$572$	−57.8772	−2.41997
$573$	−46.1079	−1.92618
$574$	−19.6284	−0.819273
$575$	−33.8674	−1.41237
$576$	−1.47392	−0.0614134
$577$	−9.43782	−0.392902	−0.196451	−	0.980514i	$-0.562942\pi$
$577$	−9.43782	−0.392902	−0.196451	+	0.980514i	$0.562942\pi$
$578$	33.2366	1.38246
$579$	−13.8124	−0.574026
$580$	8.57400	0.356016
$581$	10.7145	0.444511
$582$	0.00744360	0.000308547 0
$583$	−7.84237	−0.324798
$584$	−54.6585	−2.26179
$585$	−0.326211	−0.0134872
$586$	−0.0387159	−0.00159934
$587$	4.20125	0.173404	0.0867020	−	0.996234i	$-0.472367\pi$
$587$	4.20125	0.173404	0.0867020	+	0.996234i	$0.472367\pi$
$588$	−32.4594	−1.33860
$589$	−4.76155	−0.196196
$590$	−12.0657	−0.496736
$591$	−12.8382	−0.528094
$592$	−6.05232	−0.248749
$593$	−14.2869	−0.586694	−0.293347	−	0.956006i	$-0.594769\pi$
$593$	−14.2869	−0.586694	−0.293347	+	0.956006i	$0.594769\pi$
$594$	−33.3735	−1.36933
$595$	1.01319	0.0415367
$596$	1.69497	0.0694286
$597$	−14.7826	−0.605010
$598$	−91.7860	−3.75341
$599$	−8.45517	−0.345469	−0.172734	−	0.984968i	$-0.555260\pi$
$599$	−8.45517	−0.345469	−0.172734	+	0.984968i	$0.555260\pi$
$600$	40.6380	1.65904
$601$	−10.3852	−0.423623	−0.211811	−	0.977311i	$-0.567936\pi$
$601$	−10.3852	−0.423623	−0.211811	+	0.977311i	$0.567936\pi$
$602$	−13.2016	−0.538056
$603$	1.52829	0.0622369
$604$	−3.82633	−0.155691
$605$	1.30182	0.0529267
$606$	47.1394	1.91491
$607$	9.68019	0.392907	0.196453	−	0.980513i	$-0.437058\pi$
$607$	9.68019	0.392907	0.196453	+	0.980513i	$0.437058\pi$
$608$	−0.945150	−0.0383309
$609$	16.5201	0.669428
$610$	7.80159	0.315877
$611$	2.31682	0.0937286
$612$	1.19755	0.0484082
$613$	−9.12829	−0.368688	−0.184344	−	0.982862i	$-0.559016\pi$
$613$	−9.12829	−0.368688	−0.184344	+	0.982862i	$0.559016\pi$
$614$	−9.49114	−0.383031
$615$	3.37509	0.136097
$616$	−19.5645	−0.788278
$617$	38.2907	1.54153	0.770763	−	0.637122i	$-0.219875\pi$
$617$	38.2907	1.54153	0.770763	+	0.637122i	$0.219875\pi$
$618$	−39.9977	−1.60894
$619$	16.1469	0.648997	0.324499	−	0.945886i	$-0.394804\pi$
$619$	16.1469	0.648997	0.324499	+	0.945886i	$0.394804\pi$
$620$	−5.02731	−0.201901
$621$	−35.0649	−1.40710
$622$	15.3442	0.615245
$623$	−1.61701	−0.0647840
$624$	34.3776	1.37620
$625$	23.0629	0.922517
$626$	23.7633	0.949772
$627$	6.49439	0.259361
$628$	9.37506	0.374106
$629$	−3.10738	−0.123900
$630$	−0.224759	−0.00895462
$631$	−9.21589	−0.366879	−0.183439	−	0.983031i	$-0.558723\pi$
$631$	−9.21589	−0.366879	−0.183439	+	0.983031i	$0.558723\pi$
$632$	9.44652	0.375762
$633$	4.30380	0.171061
$634$	22.4764	0.892652
$635$	−0.636587	−0.0252622
$636$	20.1523	0.799090
$637$	−25.1858	−0.997899
$638$	−40.0482	−1.58552
$639$	0.899514	0.0355842
$640$	−7.25983	−0.286970
$641$	9.68779	0.382645	0.191322	−	0.981527i	$-0.438722\pi$
$641$	9.68779	0.382645	0.191322	+	0.981527i	$0.438722\pi$
$642$	82.1377	3.24172
$643$	24.8826	0.981272	0.490636	−	0.871365i	$-0.336764\pi$
$643$	24.8826	0.981272	0.490636	+	0.871365i	$0.336764\pi$
$644$	−41.8983	−1.65102
$645$	2.27000	0.0893812
$646$	5.97792	0.235198
$647$	21.0575	0.827855	0.413928	−	0.910310i	$-0.364157\pi$
$647$	21.0575	0.827855	0.413928	+	0.910310i	$0.364157\pi$
$648$	44.4202	1.74499
$649$	37.3381	1.46565
$650$	64.2694	2.52085
$651$	−9.68644	−0.379642
$652$	−6.95346	−0.272319
$653$	25.7195	1.00648	0.503240	−	0.864146i	$-0.332141\pi$
$653$	25.7195	1.00648	0.503240	+	0.864146i	$0.332141\pi$
$654$	−12.4802	−0.488015
$655$	4.10593	0.160432
$656$	−18.7251	−0.731091
$657$	1.94315	0.0758094
$658$	1.59629	0.0622298
$659$	23.2097	0.904122	0.452061	−	0.891987i	$-0.350689\pi$
$659$	23.2097	0.904122	0.452061	+	0.891987i	$0.350689\pi$
$660$	6.85687	0.266903
$661$	−23.1703	−0.901222	−0.450611	−	0.892720i	$-0.648794\pi$
$661$	−23.1703	−0.901222	−0.450611	+	0.892720i	$0.648794\pi$
$662$	24.0194	0.933542
$663$	17.6501	0.685475
$664$	32.7462	1.27080
$665$	−0.743316	−0.0288246
$666$	0.689321	0.0267107
$667$	−42.0778	−1.62926
$668$	64.4893	2.49517
$669$	2.62760	0.101589
$670$	8.05470	0.311180
$671$	−24.1426	−0.932013
$672$	−1.92273	−0.0741708
$673$	29.2139	1.12611	0.563056	−	0.826419i	$-0.309626\pi$
$673$	29.2139	1.12611	0.563056	+	0.826419i	$0.309626\pi$
$674$	72.9144	2.80856
$675$	24.5527	0.945035
$676$	64.3561	2.47523
$677$	−20.5320	−0.789108	−0.394554	−	0.918873i	$-0.629101\pi$
$677$	−20.5320	−0.789108	−0.394554	+	0.918873i	$0.629101\pi$
$678$	45.1950	1.73570
$679$	0.00263644	0.000101177 0
$680$	3.09657	0.118748
$681$	−11.1980	−0.429108
$682$	23.4820	0.899171
$683$	−1.82937	−0.0699990	−0.0349995	−	0.999387i	$-0.511143\pi$
$683$	−1.82937	−0.0699990	−0.0349995	+	0.999387i	$0.511143\pi$
$684$	−0.878573	−0.0335931
$685$	2.56893	0.0981538
$686$	−43.5001	−1.66084
$687$	18.4709	0.704709
$688$	−12.5940	−0.480143
$689$	15.6365	0.595703
$690$	10.8742	0.413972
$691$	−37.7553	−1.43628	−0.718139	−	0.695900i	$-0.755006\pi$
$691$	−37.7553	−1.43628	−0.718139	+	0.695900i	$0.755006\pi$
$692$	−68.0469	−2.58676
$693$	0.695533	0.0264211
$694$	41.9446	1.59220
$695$	−0.100970	−0.00383000
$696$	50.4898	1.91381
$697$	−9.61383	−0.364150
$698$	52.4383	1.98482
$699$	12.4012	0.469056
$700$	29.3376	1.10886
$701$	22.9884	0.868260	0.434130	−	0.900850i	$-0.357056\pi$
$701$	22.9884	0.868260	0.434130	+	0.900850i	$0.357056\pi$
$702$	66.5418	2.51146
$703$	2.27970	0.0859806
$704$	24.0391	0.906006
$705$	−0.274480	−0.0103375
$706$	68.3385	2.57195
$707$	16.6963	0.627928
$708$	−95.9465	−3.60589
$709$	23.9602	0.899844	0.449922	−	0.893068i	$-0.351452\pi$
$709$	23.9602	0.899844	0.449922	+	0.893068i	$0.351452\pi$
$710$	4.74079	0.177919
$711$	−0.335830	−0.0125946
$712$	−4.94200	−0.185209
$713$	24.6720	0.923975
$714$	12.1609	0.455111
$715$	5.32037	0.198970
$716$	−52.6964	−1.96936
$717$	6.79405	0.253728
$718$	−35.6481	−1.33037
$719$	31.4605	1.17328	0.586639	−	0.809848i	$-0.300451\pi$
$719$	31.4605	1.17328	0.586639	+	0.809848i	$0.300451\pi$
$720$	−0.214416	−0.00799080
$721$	−14.1668	−0.527598
$722$	41.8681	1.55817
$723$	6.37474	0.237079
$724$	−43.4863	−1.61616
$725$	29.4633	1.09424
$726$	15.6253	0.579909
$727$	−1.16498	−0.0432068	−0.0216034	−	0.999767i	$-0.506877\pi$
$727$	−1.16498	−0.0432068	−0.0216034	+	0.999767i	$0.506877\pi$
$728$	39.0088	1.44576
$729$	25.3375	0.938424
$730$	10.2411	0.379042
$731$	−6.46603	−0.239155
$732$	62.0384	2.29300
$733$	−22.0307	−0.813721	−0.406860	−	0.913490i	$-0.633376\pi$
$733$	−22.0307	−0.813721	−0.406860	+	0.913490i	$0.633376\pi$
$734$	−48.0803	−1.77468
$735$	2.98384	0.110060
$736$	4.89731	0.180517
$737$	−24.9258	−0.918155
$738$	2.13267	0.0785046
$739$	−31.9974	−1.17704	−0.588521	−	0.808482i	$-0.700290\pi$
$739$	−31.9974	−1.17704	−0.588521	+	0.808482i	$0.700290\pi$
$740$	2.40694	0.0884809
$741$	−12.9489	−0.475688
$742$	10.7735	0.395509
$743$	−47.2497	−1.73342	−0.866711	−	0.498810i	$-0.833771\pi$
$743$	−47.2497	−1.73342	−0.866711	+	0.498810i	$0.833771\pi$
$744$	−29.6043	−1.08535
$745$	−0.155810	−0.00570844
$746$	−64.3160	−2.35478
$747$	−1.16415	−0.0425940
$748$	−19.5316	−0.714146
$749$	29.0923	1.06301
$750$	−15.4320	−0.563499
$751$	18.0746	0.659551	0.329775	−	0.944059i	$-0.393027\pi$
$751$	18.0746	0.659551	0.329775	+	0.944059i	$0.393027\pi$
$752$	1.52283	0.0555317
$753$	−15.2174	−0.554554
$754$	79.8500	2.90797
$755$	0.351736	0.0128010
$756$	30.3748	1.10472
$757$	42.5737	1.54737	0.773683	−	0.633573i	$-0.218412\pi$
$757$	42.5737	1.54737	0.773683	+	0.633573i	$0.218412\pi$
$758$	14.0307	0.509618
$759$	−33.6508	−1.22145
$760$	−2.27177	−0.0824058
$761$	−3.79319	−0.137503	−0.0687515	−	0.997634i	$-0.521902\pi$
$761$	−3.79319	−0.137503	−0.0687515	+	0.997634i	$0.521902\pi$
$762$	−7.64071	−0.276794
$763$	−4.42036	−0.160028
$764$	−101.732	−3.68053
$765$	−0.110085	−0.00398014
$766$	20.5907	0.743973
$767$	−74.4466	−2.68811
$768$	−55.6713	−2.00886
$769$	25.9928	0.937324	0.468662	−	0.883378i	$-0.344736\pi$
$769$	25.9928	0.937324	0.468662	+	0.883378i	$0.344736\pi$
$770$	3.66573	0.132104
$771$	11.6802	0.420654
$772$	−30.4756	−1.09684
$773$	−17.9650	−0.646154	−0.323077	−	0.946373i	$-0.604717\pi$
$773$	−17.9650	−0.646154	−0.323077	+	0.946373i	$0.604717\pi$
$774$	1.43438	0.0515578
$775$	−17.2756	−0.620557
$776$	0.00805766	0.000289253 0
$777$	4.63761	0.166373
$778$	80.3638	2.88118
$779$	7.05309	0.252703
$780$	−13.6716	−0.489521
$781$	−14.6707	−0.524959
$782$	−30.9747	−1.10765
$783$	30.5050	1.09016
$784$	−16.5544	−0.591229
$785$	−0.861804	−0.0307591
$786$	49.2819	1.75783
$787$	21.3822	0.762193	0.381096	−	0.924535i	$-0.375547\pi$
$787$	21.3822	0.762193	0.381096	+	0.924535i	$0.375547\pi$
$788$	−28.3261	−1.00908
$789$	52.9785	1.88609
$790$	−1.76996	−0.0629722
$791$	16.0076	0.569164
$792$	2.12573	0.0755346
$793$	48.1366	1.70938
$794$	−32.7383	−1.16184
$795$	−1.85250	−0.0657014
$796$	−32.6161	−1.15605
$797$	−11.1237	−0.394023	−0.197012	−	0.980401i	$-0.563124\pi$
$797$	−11.1237	−0.394023	−0.197012	+	0.980401i	$0.563124\pi$
$798$	−8.92175	−0.315826
$799$	0.781850	0.0276599
$800$	−3.42914	−0.121239
$801$	0.175691	0.00620775
$802$	17.5949	0.621298
$803$	−31.6919	−1.11838
$804$	64.0511	2.25891
$805$	3.85151	0.135748
$806$	−46.8195	−1.64915
$807$	−15.4739	−0.544705
$808$	51.0282	1.79517
$809$	−14.7939	−0.520127	−0.260063	−	0.965592i	$-0.583743\pi$
$809$	−14.7939	−0.520127	−0.260063	+	0.965592i	$0.583743\pi$
$810$	−8.32284	−0.292435
$811$	17.8191	0.625713	0.312857	−	0.949800i	$-0.398714\pi$
$811$	17.8191	0.625713	0.312857	+	0.949800i	$0.398714\pi$
$812$	36.4498	1.27914
$813$	−5.32148	−0.186632
$814$	−11.2425	−0.394051
$815$	0.639198	0.0223901
$816$	11.6013	0.406126
$817$	4.74374	0.165962
$818$	−41.6364	−1.45578
$819$	−1.38679	−0.0484583
$820$	7.44675	0.260052
$821$	3.65192	0.127453	0.0637265	−	0.997967i	$-0.479701\pi$
$821$	3.65192	0.127453	0.0637265	+	0.997967i	$0.479701\pi$
$822$	30.8339	1.07546
$823$	−41.6032	−1.45020	−0.725098	−	0.688646i	$-0.758206\pi$
$823$	−41.6032	−1.45020	−0.725098	+	0.688646i	$0.758206\pi$
$824$	−43.2973	−1.50833
$825$	23.5626	0.820345
$826$	−51.2936	−1.78473
$827$	6.24366	0.217113	0.108557	−	0.994090i	$-0.465377\pi$
$827$	6.24366	0.217113	0.108557	+	0.994090i	$0.465377\pi$
$828$	4.55234	0.158205
$829$	28.0990	0.975920	0.487960	−	0.872866i	$-0.337741\pi$
$829$	28.0990	0.975920	0.487960	+	0.872866i	$0.337741\pi$
$830$	−6.13553	−0.212967
$831$	−16.7345	−0.580512
$832$	−47.9303	−1.66168
$833$	−8.49937	−0.294486
$834$	−1.21190	−0.0419647
$835$	−5.92819	−0.205154
$836$	14.3292	0.495585
$837$	−17.8864	−0.618244
$838$	−38.5700	−1.33238
$839$	33.8225	1.16768	0.583842	−	0.811868i	$-0.301549\pi$
$839$	33.8225	1.16768	0.583842	+	0.811868i	$0.301549\pi$
$840$	−4.62148	−0.159456
$841$	7.60594	0.262274
$842$	31.4899	1.08521
$843$	41.6606	1.43487
$844$	9.49586	0.326861
$845$	−5.91595	−0.203515
$846$	−0.173440	−0.00596300
$847$	5.53431	0.190161
$848$	10.2777	0.352939
$849$	−46.2955	−1.58886
$850$	21.6888	0.743919
$851$	−11.8123	−0.404921
$852$	37.6988	1.29154
$853$	22.7291	0.778229	0.389114	−	0.921189i	$-0.372781\pi$
$853$	22.7291	0.778229	0.389114	+	0.921189i	$0.372781\pi$
$854$	33.1661	1.13492
$855$	0.0807630	0.00276204
$856$	88.9137	3.03901
$857$	33.6001	1.14776	0.573878	−	0.818941i	$-0.305438\pi$
$857$	33.6001	1.14776	0.573878	+	0.818941i	$0.305438\pi$
$858$	63.8584	2.18009
$859$	5.67293	0.193558	0.0967790	−	0.995306i	$-0.469146\pi$
$859$	5.67293	0.193558	0.0967790	+	0.995306i	$0.469146\pi$
$860$	5.00850	0.170789
$861$	14.3482	0.488984
$862$	−17.4355	−0.593855
$863$	25.2822	0.860616	0.430308	−	0.902682i	$-0.358405\pi$
$863$	25.2822	0.860616	0.430308	+	0.902682i	$0.358405\pi$
$864$	−3.55039	−0.120787
$865$	6.25522	0.212684
$866$	46.1002	1.56655
$867$	−24.2956	−0.825122
$868$	−21.3721	−0.725415
$869$	5.47725	0.185803
$870$	−9.46006	−0.320726
$871$	49.6984	1.68396
$872$	−13.5098	−0.457499
$873$	−0.000286456 0	−9.69505e−6 0
$874$	22.7243	0.768661
$875$	−5.46586	−0.184780
$876$	81.4378	2.75153
$877$	−38.7794	−1.30949	−0.654744	−	0.755851i	$-0.727224\pi$
$877$	−38.7794	−1.30949	−0.654744	+	0.755851i	$0.727224\pi$
$878$	−87.5447	−2.95449
$879$	0.0283010	0.000954568 0
$880$	3.49703	0.117885
$881$	−25.7886	−0.868838	−0.434419	−	0.900711i	$-0.643046\pi$
$881$	−25.7886	−0.868838	−0.434419	+	0.900711i	$0.643046\pi$
$882$	1.88544	0.0634862
$883$	−24.1823	−0.813799	−0.406899	−	0.913473i	$-0.633390\pi$
$883$	−24.1823	−0.813799	−0.406899	+	0.913473i	$0.633390\pi$
$884$	38.9431	1.30980
$885$	8.81990	0.296478
$886$	15.6477	0.525693
$887$	−7.03011	−0.236048	−0.118024	−	0.993011i	$-0.537656\pi$
$887$	−7.03011	−0.236048	−0.118024	+	0.993011i	$0.537656\pi$
$888$	14.1738	0.475640
$889$	−2.70626	−0.0907649
$890$	0.925962	0.0310383
$891$	25.7556	0.862845
$892$	5.79752	0.194115
$893$	−0.573596	−0.0191947
$894$	−1.87013	−0.0625465
$895$	4.84412	0.161921
$896$	−30.8630	−1.03106
$897$	67.0947	2.24023
$898$	33.6577	1.12317
$899$	−21.4636	−0.715852
$900$	−3.18759	−0.106253
$901$	5.27680	0.175796
$902$	−34.7829	−1.15815
$903$	9.65022	0.321139
$904$	48.9233	1.62717
$905$	3.99749	0.132881
$906$	4.22176	0.140258
$907$	5.32177	0.176707	0.0883533	−	0.996089i	$-0.471840\pi$
$907$	5.32177	0.176707	0.0883533	+	0.996089i	$0.471840\pi$
$908$	−24.7071	−0.819935
$909$	−1.81409	−0.0601695
$910$	−7.30891	−0.242288
$911$	40.6208	1.34583	0.672913	−	0.739722i	$-0.265043\pi$
$911$	40.6208	1.34583	0.672913	+	0.739722i	$0.265043\pi$
$912$	−8.51116	−0.281833
$913$	18.9868	0.628372
$914$	−71.7537	−2.37340
$915$	−5.70289	−0.188532
$916$	40.7540	1.34655
$917$	17.4551	0.576419
$918$	22.4556	0.741146
$919$	39.1305	1.29080	0.645398	−	0.763847i	$-0.276692\pi$
$919$	39.1305	1.29080	0.645398	+	0.763847i	$0.276692\pi$
$920$	11.7712	0.388086
$921$	6.93793	0.228613
$922$	60.3932	1.98894
$923$	29.2512	0.962814
$924$	29.1499	0.958962
$925$	8.27109	0.271952
$926$	−14.1077	−0.463607
$927$	1.53925	0.0505556
$928$	−4.26046	−0.139856
$929$	41.7517	1.36983	0.684914	−	0.728623i	$-0.259840\pi$
$929$	41.7517	1.36983	0.684914	+	0.728623i	$0.259840\pi$
$930$	5.54684	0.181888
$931$	6.23548	0.204360
$932$	27.3619	0.896268
$933$	−11.2164	−0.367209
$934$	10.8461	0.354896
$935$	1.79545	0.0587174
$936$	−4.23839	−0.138536
$937$	−14.1376	−0.461854	−0.230927	−	0.972971i	$-0.574176\pi$
$937$	−14.1376	−0.461854	−0.230927	+	0.972971i	$0.574176\pi$
$938$	34.2421	1.11805
$939$	−17.3707	−0.566872
$940$	−0.605611	−0.0197528
$941$	47.9322	1.56254	0.781272	−	0.624191i	$-0.214571\pi$
$941$	47.9322	1.56254	0.781272	+	0.624191i	$0.214571\pi$
$942$	−10.3439	−0.337023
$943$	−36.5457	−1.19009
$944$	−48.9330	−1.59263
$945$	−2.79221	−0.0908306
$946$	−23.3942	−0.760610
$947$	−10.5619	−0.343215	−0.171607	−	0.985165i	$-0.554896\pi$
$947$	−10.5619	−0.343215	−0.171607	+	0.985165i	$0.554896\pi$
$948$	−14.0747	−0.457125
$949$	63.1890	2.05120
$950$	−15.9118	−0.516245
$951$	−16.4300	−0.532780
$952$	13.1641	0.426653
$953$	37.2634	1.20708	0.603540	−	0.797333i	$-0.293756\pi$
$953$	37.2634	1.20708	0.603540	+	0.797333i	$0.293756\pi$
$954$	−1.17057	−0.0378986
$955$	9.35173	0.302615
$956$	14.9903	0.484821
$957$	29.2748	0.946320
$958$	−66.9469	−2.16296
$959$	10.9210	0.352659
$960$	5.67844	0.183271
$961$	−18.4149	−0.594030
$962$	22.4159	0.722719
$963$	−3.16094	−0.101860
$964$	14.0652	0.453008
$965$	2.80148	0.0901828
$966$	46.2282	1.48737
$967$	−24.1777	−0.777501	−0.388751	−	0.921343i	$-0.627093\pi$
$967$	−24.1777	−0.777501	−0.388751	+	0.921343i	$0.627093\pi$
$968$	16.9143	0.543646
$969$	−4.36980	−0.140378
$970$	−0.00150973	−4.84746e−5 0
$971$	56.3468	1.80825	0.904127	−	0.427264i	$-0.140522\pi$
$971$	56.3468	1.80825	0.904127	+	0.427264i	$0.140522\pi$
$972$	−6.79478	−0.217943
$973$	−0.429242	−0.0137609
$974$	107.004	3.42864
$975$	−46.9803	−1.50457
$976$	31.6398	1.01276
$977$	−49.4353	−1.58157	−0.790787	−	0.612091i	$-0.790329\pi$
$977$	−49.4353	−1.58157	−0.790787	+	0.612091i	$0.790329\pi$
$978$	7.67206	0.245325
$979$	−2.86545	−0.0915803
$980$	6.58351	0.210302
$981$	0.480282	0.0153342
$982$	−45.3147	−1.44605
$983$	−21.5557	−0.687519	−0.343760	−	0.939058i	$-0.611701\pi$
$983$	−21.5557	−0.687519	−0.343760	+	0.939058i	$0.611701\pi$
$984$	43.8517	1.39794
$985$	2.60388	0.0829666
$986$	26.9467	0.858158
$987$	−1.16687	−0.0371419
$988$	−28.5702	−0.908939
$989$	−24.5798	−0.781591
$990$	−0.398290	−0.0126585
$991$	−10.9082	−0.346509	−0.173255	−	0.984877i	$-0.555428\pi$
$991$	−10.9082	−0.346509	−0.173255	+	0.984877i	$0.555428\pi$
$992$	2.49809	0.0793145
$993$	−17.5580	−0.557185
$994$	20.1540	0.639247
$995$	2.99824	0.0950507
$996$	−48.7898	−1.54596
$997$	55.3657	1.75345	0.876725	−	0.480993i	$-0.159724\pi$
$997$	55.3657	1.75345	0.876725	+	0.480993i	$0.159724\pi$
$998$	−36.4534	−1.15391
$999$	8.56352	0.270938

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4003.2.a.b.1.15	✓	152

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4003.2.a.b.1.15	✓	152	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:	\( N \)	\(=\)	\( 4003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4003.a (trivial)

\(f(q)\)	\(=\)	\(q-2.43441 q^{2} +1.77953 q^{3} +3.92633 q^{4} -0.360928 q^{5} -4.33209 q^{6} -1.53438 q^{7} -4.68946 q^{8} +0.166714 q^{9} +O(q^{10})\)	\(q-2.43441 q^{2} +1.77953 q^{3} +3.92633 q^{4} -0.360928 q^{5} -4.33209 q^{6} -1.53438 q^{7} -4.68946 q^{8} +0.166714 q^{9} +0.878646 q^{10} -2.71903 q^{11} +6.98700 q^{12} +5.42134 q^{13} +3.73530 q^{14} -0.642281 q^{15} +3.56340 q^{16} +1.82952 q^{17} -0.405849 q^{18} -1.34221 q^{19} -1.41712 q^{20} -2.73047 q^{21} +6.61922 q^{22} +6.95468 q^{23} -8.34502 q^{24} -4.86973 q^{25} -13.1977 q^{26} -5.04191 q^{27} -6.02447 q^{28} -6.05028 q^{29} +1.56357 q^{30} +3.54754 q^{31} +0.704175 q^{32} -4.83859 q^{33} -4.45379 q^{34} +0.553801 q^{35} +0.654572 q^{36} -1.69847 q^{37} +3.26748 q^{38} +9.64741 q^{39} +1.69256 q^{40} -5.25484 q^{41} +6.64706 q^{42} -3.53428 q^{43} -10.6758 q^{44} -0.0601717 q^{45} -16.9305 q^{46} +0.427352 q^{47} +6.34116 q^{48} -4.64568 q^{49} +11.8549 q^{50} +3.25568 q^{51} +21.2860 q^{52} +2.88425 q^{53} +12.2740 q^{54} +0.981375 q^{55} +7.19541 q^{56} -2.38850 q^{57} +14.7288 q^{58} -13.7321 q^{59} -2.52181 q^{60} +8.87911 q^{61} -8.63616 q^{62} -0.255802 q^{63} -8.84104 q^{64} -1.95671 q^{65} +11.7791 q^{66} +9.16717 q^{67} +7.18330 q^{68} +12.3760 q^{69} -1.34817 q^{70} +5.39556 q^{71} -0.781797 q^{72} +11.6556 q^{73} +4.13476 q^{74} -8.66581 q^{75} -5.26995 q^{76} +4.17202 q^{77} -23.4857 q^{78} -2.01441 q^{79} -1.28613 q^{80} -9.47235 q^{81} +12.7924 q^{82} -6.98293 q^{83} -10.7207 q^{84} -0.660326 q^{85} +8.60386 q^{86} -10.7666 q^{87} +12.7508 q^{88} +1.05385 q^{89} +0.146482 q^{90} -8.31838 q^{91} +27.3064 q^{92} +6.31295 q^{93} -1.04035 q^{94} +0.484441 q^{95} +1.25310 q^{96} -0.00171825 q^{97} +11.3095 q^{98} -0.453299 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 152 q - 22 q^{2} - 18 q^{3} + 138 q^{4} - 59 q^{5} - 17 q^{6} - 19 q^{7} - 66 q^{8} + 106 q^{9}+O(q^{10}) \) 152 * q - 22 * q^2 - 18 * q^3 + 138 * q^4 - 59 * q^5 - 17 * q^6 - 19 * q^7 - 66 * q^8 + 106 * q^9	\( 152 q - 22 q^{2} - 18 q^{3} + 138 q^{4} - 59 q^{5} - 17 q^{6} - 19 q^{7} - 66 q^{8} + 106 q^{9} - 15 q^{10} - 40 q^{11} - 53 q^{12} - 59 q^{13} - 36 q^{14} - 40 q^{15} + 118 q^{16} - 93 q^{17} - 59 q^{18} - 16 q^{19} - 108 q^{20} - 62 q^{21} - 37 q^{22} - 107 q^{23} - 31 q^{24} + 101 q^{25} - 64 q^{26} - 63 q^{27} - 53 q^{28} - 124 q^{29} - 68 q^{30} - 15 q^{31} - 129 q^{32} - 49 q^{33} - 76 q^{35} + 45 q^{36} - 98 q^{37} - 125 q^{38} - 47 q^{39} - 7 q^{40} - 56 q^{41} - 84 q^{42} - 62 q^{43} - 114 q^{44} - 142 q^{45} - 3 q^{46} - 111 q^{47} - 92 q^{48} + 117 q^{49} - 64 q^{50} - 21 q^{51} - 85 q^{52} - 347 q^{53} + 3 q^{54} - 16 q^{55} - 73 q^{56} - 115 q^{57} - 29 q^{58} - 50 q^{59} - 54 q^{60} - 62 q^{61} - 55 q^{62} - 70 q^{63} + 64 q^{64} - 147 q^{65} + 34 q^{66} - 86 q^{67} - 174 q^{68} - 104 q^{69} - 7 q^{70} - 86 q^{71} - 139 q^{72} - 27 q^{73} - 52 q^{74} - 49 q^{75} - 11 q^{76} - 346 q^{77} - 59 q^{78} - 17 q^{79} - 149 q^{80} - 8 q^{81} - 31 q^{82} - 106 q^{83} - 51 q^{84} - 69 q^{85} - 85 q^{86} - 32 q^{87} - 113 q^{88} - 59 q^{89} + 10 q^{90} - 9 q^{91} - 314 q^{92} - 230 q^{93} + 7 q^{94} - 74 q^{95} - 54 q^{96} - 60 q^{97} - 77 q^{98} - 96 q^{99}+O(q^{100}) \) 152 * q - 22 * q^2 - 18 * q^3 + 138 * q^4 - 59 * q^5 - 17 * q^6 - 19 * q^7 - 66 * q^8 + 106 * q^9 - 15 * q^10 - 40 * q^11 - 53 * q^12 - 59 * q^13 - 36 * q^14 - 40 * q^15 + 118 * q^16 - 93 * q^17 - 59 * q^18 - 16 * q^19 - 108 * q^20 - 62 * q^21 - 37 * q^22 - 107 * q^23 - 31 * q^24 + 101 * q^25 - 64 * q^26 - 63 * q^27 - 53 * q^28 - 124 * q^29 - 68 * q^30 - 15 * q^31 - 129 * q^32 - 49 * q^33 - 76 * q^35 + 45 * q^36 - 98 * q^37 - 125 * q^38 - 47 * q^39 - 7 * q^40 - 56 * q^41 - 84 * q^42 - 62 * q^43 - 114 * q^44 - 142 * q^45 - 3 * q^46 - 111 * q^47 - 92 * q^48 + 117 * q^49 - 64 * q^50 - 21 * q^51 - 85 * q^52 - 347 * q^53 + 3 * q^54 - 16 * q^55 - 73 * q^56 - 115 * q^57 - 29 * q^58 - 50 * q^59 - 54 * q^60 - 62 * q^61 - 55 * q^62 - 70 * q^63 + 64 * q^64 - 147 * q^65 + 34 * q^66 - 86 * q^67 - 174 * q^68 - 104 * q^69 - 7 * q^70 - 86 * q^71 - 139 * q^72 - 27 * q^73 - 52 * q^74 - 49 * q^75 - 11 * q^76 - 346 * q^77 - 59 * q^78 - 17 * q^79 - 149 * q^80 - 8 * q^81 - 31 * q^82 - 106 * q^83 - 51 * q^84 - 69 * q^85 - 85 * q^86 - 32 * q^87 - 113 * q^88 - 59 * q^89 + 10 * q^90 - 9 * q^91 - 314 * q^92 - 230 * q^93 + 7 * q^94 - 74 * q^95 - 54 * q^96 - 60 * q^97 - 77 * q^98 - 96 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more