LMFDB - Embedded newform 4009.2.a.f.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4009,2,Mod(1,4009)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4009, 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("4009.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4009 = 19 \cdot 211 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4009.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:	$32.0120261703$
Analytic rank:	$0$
Dimension:	$83$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Character	$\chi$	$=$	4009.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.54651 q^{2} +2.15718 q^{3} +4.48471 q^{4} +0.595796 q^{5} -5.49327 q^{6} +3.08751 q^{7} -6.32733 q^{8} +1.65341 q^{9} +O(q^{10})$	\(q-2.54651 q^{2} +2.15718 q^{3} +4.48471 q^{4} +0.595796 q^{5} -5.49327 q^{6} +3.08751 q^{7} -6.32733 q^{8} +1.65341 q^{9} -1.51720 q^{10} +5.42807 q^{11} +9.67430 q^{12} -2.92262 q^{13} -7.86237 q^{14} +1.28524 q^{15} +7.14318 q^{16} -3.02394 q^{17} -4.21041 q^{18} -1.00000 q^{19} +2.67197 q^{20} +6.66030 q^{21} -13.8226 q^{22} +6.90252 q^{23} -13.6492 q^{24} -4.64503 q^{25} +7.44247 q^{26} -2.90484 q^{27} +13.8466 q^{28} -4.42575 q^{29} -3.27287 q^{30} +5.08904 q^{31} -5.53551 q^{32} +11.7093 q^{33} +7.70049 q^{34} +1.83953 q^{35} +7.41504 q^{36} -7.09124 q^{37} +2.54651 q^{38} -6.30460 q^{39} -3.76980 q^{40} +0.679209 q^{41} -16.9605 q^{42} +8.67076 q^{43} +24.3433 q^{44} +0.985092 q^{45} -17.5773 q^{46} -1.38156 q^{47} +15.4091 q^{48} +2.53272 q^{49} +11.8286 q^{50} -6.52317 q^{51} -13.1071 q^{52} +8.41072 q^{53} +7.39720 q^{54} +3.23402 q^{55} -19.5357 q^{56} -2.15718 q^{57} +11.2702 q^{58} +11.3963 q^{59} +5.76391 q^{60} +11.6490 q^{61} -12.9593 q^{62} +5.10491 q^{63} -0.190124 q^{64} -1.74128 q^{65} -29.8178 q^{66} +4.79925 q^{67} -13.5615 q^{68} +14.8900 q^{69} -4.68437 q^{70} -8.52052 q^{71} -10.4616 q^{72} +3.95527 q^{73} +18.0579 q^{74} -10.0201 q^{75} -4.48471 q^{76} +16.7592 q^{77} +16.0547 q^{78} +13.2687 q^{79} +4.25588 q^{80} -11.2265 q^{81} -1.72961 q^{82} -0.341171 q^{83} +29.8695 q^{84} -1.80165 q^{85} -22.0802 q^{86} -9.54711 q^{87} -34.3452 q^{88} -5.13429 q^{89} -2.50855 q^{90} -9.02362 q^{91} +30.9558 q^{92} +10.9779 q^{93} +3.51814 q^{94} -0.595796 q^{95} -11.9411 q^{96} -13.4076 q^{97} -6.44960 q^{98} +8.97480 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 83 q + 11 q^{2} + 95 q^{4} + 15 q^{5} + 23 q^{6} + 19 q^{7} + 30 q^{8} + 101 q^{9}+O(q^{10}) $ 83 * q + 11 * q^2 + 95 * q^4 + 15 * q^5 + 23 * q^6 + 19 * q^7 + 30 * q^8 + 101 * q^9	\( 83 q + 11 q^{2} + 95 q^{4} + 15 q^{5} + 23 q^{6} + 19 q^{7} + 30 q^{8} + 101 q^{9} + 9 q^{10} + 56 q^{11} - 2 q^{12} - 5 q^{13} + 6 q^{14} + 19 q^{15} + 123 q^{16} + 19 q^{17} + 40 q^{18} - 83 q^{19} + 49 q^{20} + 9 q^{21} + 18 q^{22} + 74 q^{23} + 38 q^{24} + 98 q^{25} + 28 q^{26} + 6 q^{27} + 50 q^{28} + 16 q^{29} + 56 q^{30} + 24 q^{31} + 81 q^{32} + 13 q^{33} + 9 q^{34} + 71 q^{35} + 156 q^{36} - 6 q^{37} - 11 q^{38} + 126 q^{39} + q^{40} - q^{42} + 34 q^{43} + 140 q^{44} + 42 q^{45} + 34 q^{46} + 53 q^{47} + 16 q^{48} + 118 q^{49} + 51 q^{50} + 57 q^{51} + 32 q^{52} + q^{53} + 53 q^{54} + 60 q^{55} - 2 q^{56} - 2 q^{58} + 44 q^{59} - 9 q^{60} + 21 q^{61} + 28 q^{62} + 83 q^{63} + 154 q^{64} + 44 q^{65} + 17 q^{66} + 5 q^{67} + 63 q^{68} - 36 q^{69} - 48 q^{70} + 193 q^{71} + 135 q^{72} + 54 q^{73} + 127 q^{74} + 5 q^{75} - 95 q^{76} + 54 q^{77} + 45 q^{78} + 54 q^{79} + 45 q^{80} + 147 q^{81} - 35 q^{82} + 84 q^{83} + 12 q^{84} + 28 q^{85} + 60 q^{86} + 51 q^{87} + 23 q^{88} - 24 q^{89} + 31 q^{90} + 28 q^{91} + 108 q^{92} + 39 q^{93} - 49 q^{94} - 15 q^{95} + 25 q^{96} - 22 q^{97} - 67 q^{98} + 132 q^{99}+O(q^{100}) \) 83 * q + 11 * q^2 + 95 * q^4 + 15 * q^5 + 23 * q^6 + 19 * q^7 + 30 * q^8 + 101 * q^9 + 9 * q^10 + 56 * q^11 - 2 * q^12 - 5 * q^13 + 6 * q^14 + 19 * q^15 + 123 * q^16 + 19 * q^17 + 40 * q^18 - 83 * q^19 + 49 * q^20 + 9 * q^21 + 18 * q^22 + 74 * q^23 + 38 * q^24 + 98 * q^25 + 28 * q^26 + 6 * q^27 + 50 * q^28 + 16 * q^29 + 56 * q^30 + 24 * q^31 + 81 * q^32 + 13 * q^33 + 9 * q^34 + 71 * q^35 + 156 * q^36 - 6 * q^37 - 11 * q^38 + 126 * q^39 + q^40 - q^42 + 34 * q^43 + 140 * q^44 + 42 * q^45 + 34 * q^46 + 53 * q^47 + 16 * q^48 + 118 * q^49 + 51 * q^50 + 57 * q^51 + 32 * q^52 + q^53 + 53 * q^54 + 60 * q^55 - 2 * q^56 - 2 * q^58 + 44 * q^59 - 9 * q^60 + 21 * q^61 + 28 * q^62 + 83 * q^63 + 154 * q^64 + 44 * q^65 + 17 * q^66 + 5 * q^67 + 63 * q^68 - 36 * q^69 - 48 * q^70 + 193 * q^71 + 135 * q^72 + 54 * q^73 + 127 * q^74 + 5 * q^75 - 95 * q^76 + 54 * q^77 + 45 * q^78 + 54 * q^79 + 45 * q^80 + 147 * q^81 - 35 * q^82 + 84 * q^83 + 12 * q^84 + 28 * q^85 + 60 * q^86 + 51 * q^87 + 23 * q^88 - 24 * q^89 + 31 * q^90 + 28 * q^91 + 108 * q^92 + 39 * q^93 - 49 * q^94 - 15 * q^95 + 25 * q^96 - 22 * q^97 - 67 * q^98 + 132 * 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 $

$2$	−2.54651	−1.80065	−0.900327	−	0.435215i	$-0.856673\pi$
$2$	−2.54651	−1.80065	−0.900327	+	0.435215i	$0.856673\pi$
$3$	2.15718	1.24545	0.622723	−	0.782442i	$-0.286026\pi$
$3$	2.15718	1.24545	0.622723	+	0.782442i	$0.286026\pi$
$4$	4.48471	2.24235
$5$	0.595796	0.266448	0.133224	−	0.991086i	$-0.457467\pi$
$5$	0.595796	0.266448	0.133224	+	0.991086i	$0.457467\pi$
$6$	−5.49327	−2.24262
$7$	3.08751	1.16697	0.583485	−	0.812124i	$-0.301689\pi$
$7$	3.08751	1.16697	0.583485	+	0.812124i	$0.301689\pi$
$8$	−6.32733	−2.23705
$9$	1.65341	0.551135
$10$	−1.51720	−0.479781
$11$	5.42807	1.63663	0.818313	−	0.574773i	$-0.194910\pi$
$11$	5.42807	1.63663	0.818313	+	0.574773i	$0.194910\pi$
$12$	9.67430	2.79273
$13$	−2.92262	−0.810589	−0.405294	−	0.914186i	$-0.632831\pi$
$13$	−2.92262	−0.810589	−0.405294	+	0.914186i	$0.632831\pi$
$14$	−7.86237	−2.10131
$15$	1.28524	0.331847
$16$	7.14318	1.78579
$17$	−3.02394	−0.733413	−0.366707	−	0.930337i	$-0.619515\pi$
$17$	−3.02394	−0.733413	−0.366707	+	0.930337i	$0.619515\pi$
$18$	−4.21041	−0.992403
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	2.67197	0.597471
$21$	6.66030	1.45340
$22$	−13.8226	−2.94699
$23$	6.90252	1.43928	0.719638	−	0.694350i	$-0.244308\pi$
$23$	6.90252	1.43928	0.719638	+	0.694350i	$0.244308\pi$
$24$	−13.6492	−2.78612
$25$	−4.64503	−0.929005
$26$	7.44247	1.45959
$27$	−2.90484	−0.559037
$28$	13.8466	2.61676
$29$	−4.42575	−0.821840	−0.410920	−	0.911671i	$-0.634793\pi$
$29$	−4.42575	−0.821840	−0.410920	+	0.911671i	$0.634793\pi$
$30$	−3.27287	−0.597541
$31$	5.08904	0.914018	0.457009	−	0.889462i	$-0.348921\pi$
$31$	5.08904	0.914018	0.457009	+	0.889462i	$0.348921\pi$
$32$	−5.53551	−0.978550
$33$	11.7093	2.03833
$34$	7.70049	1.32062
$35$	1.83953	0.310937
$36$	7.41504	1.23584
$37$	−7.09124	−1.16579	−0.582897	−	0.812546i	$-0.698081\pi$
$37$	−7.09124	−1.16579	−0.582897	+	0.812546i	$0.698081\pi$
$38$	2.54651	0.413098
$39$	−6.30460	−1.00954
$40$	−3.76980	−0.596057
$41$	0.679209	0.106075	0.0530373	−	0.998593i	$-0.483110\pi$
$41$	0.679209	0.106075	0.0530373	+	0.998593i	$0.483110\pi$
$42$	−16.9605	−2.61706
$43$	8.67076	1.32228	0.661139	−	0.750263i	$-0.270073\pi$
$43$	8.67076	1.32228	0.661139	+	0.750263i	$0.270073\pi$
$44$	24.3433	3.66989
$45$	0.985092	0.146849
$46$	−17.5773	−2.59164
$47$	−1.38156	−0.201521	−0.100760	−	0.994911i	$-0.532128\pi$
$47$	−1.38156	−0.201521	−0.100760	+	0.994911i	$0.532128\pi$
$48$	15.4091	2.22411
$49$	2.53272	0.361818
$50$	11.8286	1.67282
$51$	−6.52317	−0.913426
$52$	−13.1071	−1.81763
$53$	8.41072	1.15530	0.577651	−	0.816284i	$-0.303970\pi$
$53$	8.41072	1.15530	0.577651	+	0.816284i	$0.303970\pi$
$54$	7.39720	1.00663
$55$	3.23402	0.436076
$56$	−19.5357	−2.61057
$57$	−2.15718	−0.285725
$58$	11.2702	1.47985
$59$	11.3963	1.48367	0.741834	−	0.670584i	$-0.233956\pi$
$59$	11.3963	1.48367	0.741834	+	0.670584i	$0.233956\pi$
$60$	5.76391	0.744117
$61$	11.6490	1.49150	0.745752	−	0.666223i	$-0.232090\pi$
$61$	11.6490	1.49150	0.745752	+	0.666223i	$0.232090\pi$
$62$	−12.9593	−1.64583
$63$	5.10491	0.643158
$64$	−0.190124	−0.0237655
$65$	−1.74128	−0.215980
$66$	−29.8178	−3.67032
$67$	4.79925	0.586322	0.293161	−	0.956063i	$-0.405293\pi$
$67$	4.79925	0.586322	0.293161	+	0.956063i	$0.405293\pi$
$68$	−13.5615	−1.64457
$69$	14.8900	1.79254
$70$	−4.68437	−0.559889
$71$	−8.52052	−1.01120	−0.505600	−	0.862768i	$-0.668729\pi$
$71$	−8.52052	−1.01120	−0.505600	+	0.862768i	$0.668729\pi$
$72$	−10.4616	−1.23292
$73$	3.95527	0.462929	0.231464	−	0.972843i	$-0.425648\pi$
$73$	3.95527	0.462929	0.231464	+	0.972843i	$0.425648\pi$
$74$	18.0579	2.09919
$75$	−10.0201	−1.15703
$76$	−4.48471	−0.514431
$77$	16.7592	1.90989
$78$	16.0547	1.81784
$79$	13.2687	1.49285	0.746425	−	0.665470i	$-0.231769\pi$
$79$	13.2687	1.49285	0.746425	+	0.665470i	$0.231769\pi$
$80$	4.25588	0.475821
$81$	−11.2265	−1.24739
$82$	−1.72961	−0.191004
$83$	−0.341171	−0.0374484	−0.0187242	−	0.999825i	$-0.505960\pi$
$83$	−0.341171	−0.0374484	−0.0187242	+	0.999825i	$0.505960\pi$
$84$	29.8695	3.25903
$85$	−1.80165	−0.195416
$86$	−22.0802	−2.38097
$87$	−9.54711	−1.02356
$88$	−34.3452	−3.66121
$89$	−5.13429	−0.544234	−0.272117	−	0.962264i	$-0.587724\pi$
$89$	−5.13429	−0.544234	−0.272117	+	0.962264i	$0.587724\pi$
$90$	−2.50855	−0.264424
$91$	−9.02362	−0.945932
$92$	30.9558	3.22736
$93$	10.9779	1.13836
$94$	3.51814	0.362869
$95$	−0.595796	−0.0611274
$96$	−11.9411	−1.21873
$97$	−13.4076	−1.36134	−0.680668	−	0.732592i	$-0.738310\pi$
$97$	−13.4076	−1.36134	−0.680668	+	0.732592i	$0.738310\pi$
$98$	−6.44960	−0.651508
$99$	8.97480	0.902002
$100$	−20.8316	−2.08316
$101$	14.6388	1.45662	0.728308	−	0.685250i	$-0.240307\pi$
$101$	14.6388	1.45662	0.728308	+	0.685250i	$0.240307\pi$
$102$	16.6113	1.64476
$103$	−9.09449	−0.896106	−0.448053	−	0.894007i	$-0.647883\pi$
$103$	−9.09449	−0.896106	−0.448053	+	0.894007i	$0.647883\pi$
$104$	18.4924	1.81333
$105$	3.96818	0.387255
$106$	−21.4180	−2.08030
$107$	10.6556	1.03011	0.515056	−	0.857156i	$-0.327771\pi$
$107$	10.6556	1.03011	0.515056	+	0.857156i	$0.327771\pi$
$108$	−13.0274	−1.25356
$109$	15.8398	1.51718	0.758588	−	0.651570i	$-0.225890\pi$
$109$	15.8398	1.51718	0.758588	+	0.651570i	$0.225890\pi$
$110$	−8.23547	−0.785221
$111$	−15.2971	−1.45193
$112$	22.0546	2.08397
$113$	18.9777	1.78527	0.892634	−	0.450782i	$-0.148855\pi$
$113$	18.9777	1.78527	0.892634	+	0.450782i	$0.148855\pi$
$114$	5.49327	0.514491
$115$	4.11250	0.383492
$116$	−19.8482	−1.84286
$117$	−4.83227	−0.446744
$118$	−29.0207	−2.67157
$119$	−9.33645	−0.855871
$120$	−8.13211	−0.742357
$121$	18.4640	1.67854
$122$	−29.6643	−2.68568
$123$	1.46517	0.132110
$124$	22.8228	2.04955
$125$	−5.74647	−0.513980
$126$	−12.9997	−1.15810
$127$	−9.48838	−0.841958	−0.420979	−	0.907070i	$-0.638313\pi$
$127$	−9.48838	−0.841958	−0.420979	+	0.907070i	$0.638313\pi$
$128$	11.5552	1.02134
$129$	18.7043	1.64683
$130$	4.43420	0.388905
$131$	18.2243	1.59227	0.796134	−	0.605121i	$-0.206875\pi$
$131$	18.2243	1.59227	0.796134	+	0.605121i	$0.206875\pi$
$132$	52.5128	4.57065
$133$	−3.08751	−0.267721
$134$	−12.2213	−1.05576
$135$	−1.73069	−0.148954
$136$	19.1335	1.64068
$137$	9.08343	0.776050	0.388025	−	0.921649i	$-0.373157\pi$
$137$	9.08343	0.776050	0.388025	+	0.921649i	$0.373157\pi$
$138$	−37.9174	−3.22774
$139$	−0.0593213	−0.00503156	−0.00251578	−	0.999997i	$-0.500801\pi$
$139$	−0.0593213	−0.00503156	−0.00251578	+	0.999997i	$0.500801\pi$
$140$	8.24974	0.697230
$141$	−2.98026	−0.250983
$142$	21.6976	1.82082
$143$	−15.8642	−1.32663
$144$	11.8106	0.984214
$145$	−2.63684	−0.218978
$146$	−10.0721	−0.833574
$147$	5.46353	0.450624
$148$	−31.8021	−2.61412
$149$	13.5707	1.11175	0.555876	−	0.831265i	$-0.312383\pi$
$149$	13.5707	1.11175	0.555876	+	0.831265i	$0.312383\pi$
$150$	25.5164	2.08340
$151$	−0.800976	−0.0651825	−0.0325913	−	0.999469i	$-0.510376\pi$
$151$	−0.800976	−0.0651825	−0.0325913	+	0.999469i	$0.510376\pi$
$152$	6.32733	0.513214
$153$	−4.99980	−0.404210
$154$	−42.6775	−3.43905
$155$	3.03203	0.243538
$156$	−28.2743	−2.26375
$157$	1.25381	0.100065	0.0500324	−	0.998748i	$-0.484068\pi$
$157$	1.25381	0.100065	0.0500324	+	0.998748i	$0.484068\pi$
$158$	−33.7890	−2.68811
$159$	18.1434	1.43887
$160$	−3.29804	−0.260733
$161$	21.3116	1.67959
$162$	28.5883	2.24611
$163$	16.5465	1.29602	0.648012	−	0.761630i	$-0.275601\pi$
$163$	16.5465	1.29602	0.648012	+	0.761630i	$0.275601\pi$
$164$	3.04605	0.237857
$165$	6.97635	0.543108
$166$	0.868795	0.0674316
$167$	−9.14908	−0.707977	−0.353989	−	0.935250i	$-0.615175\pi$
$167$	−9.14908	−0.707977	−0.353989	+	0.935250i	$0.615175\pi$
$168$	−42.1419	−3.25132
$169$	−4.45830	−0.342946
$170$	4.58792	0.351877
$171$	−1.65341	−0.126439
$172$	38.8858	2.96501
$173$	16.8482	1.28095	0.640474	−	0.767980i	$-0.278738\pi$
$173$	16.8482	1.28095	0.640474	+	0.767980i	$0.278738\pi$
$174$	24.3118	1.84307
$175$	−14.3416	−1.08412
$176$	38.7737	2.92268
$177$	24.5837	1.84783
$178$	13.0745	0.979977
$179$	14.2246	1.06320	0.531600	−	0.846996i	$-0.321591\pi$
$179$	14.2246	1.06320	0.531600	+	0.846996i	$0.321591\pi$
$180$	4.41785	0.329287
$181$	−16.6315	−1.23621	−0.618106	−	0.786095i	$-0.712100\pi$
$181$	−16.6315	−1.23621	−0.618106	+	0.786095i	$0.712100\pi$
$182$	22.9787	1.70330
$183$	25.1290	1.85759
$184$	−43.6745	−3.21973
$185$	−4.22493	−0.310623
$186$	−27.9554	−2.04979
$187$	−16.4142	−1.20032
$188$	−6.19587	−0.451881
$189$	−8.96873	−0.652379
$190$	1.51720	0.110069
$191$	−10.8945	−0.788298	−0.394149	−	0.919046i	$-0.628961\pi$
$191$	−10.8945	−0.788298	−0.394149	+	0.919046i	$0.628961\pi$
$192$	−0.410130	−0.0295986
$193$	−22.9914	−1.65496	−0.827480	−	0.561495i	$-0.810226\pi$
$193$	−22.9914	−1.65496	−0.827480	+	0.561495i	$0.810226\pi$
$194$	34.1426	2.45129
$195$	−3.75626	−0.268991
$196$	11.3585	0.811323
$197$	8.18419	0.583099	0.291550	−	0.956556i	$-0.405829\pi$
$197$	8.18419	0.583099	0.291550	+	0.956556i	$0.405829\pi$
$198$	−22.8544	−1.62419
$199$	−26.8306	−1.90197	−0.950985	−	0.309238i	$-0.899926\pi$
$199$	−26.8306	−1.90197	−0.950985	+	0.309238i	$0.899926\pi$
$200$	29.3906	2.07823
$201$	10.3528	0.730232
$202$	−37.2779	−2.62286
$203$	−13.6645	−0.959063
$204$	−29.2545	−2.04822
$205$	0.404670	0.0282634
$206$	23.1592	1.61358
$207$	11.4127	0.793235
$208$	−20.8768	−1.44754
$209$	−5.42807	−0.375468
$210$	−10.1050	−0.697312
$211$	1.00000	0.0688428
$211$	1.00000	0.0688428
$212$	37.7196	2.59059
$213$	−18.3803	−1.25939
$214$	−27.1345	−1.85488
$215$	5.16600	0.352318
$216$	18.3799	1.25059
$217$	15.7125	1.06663
$218$	−40.3361	−2.73191
$219$	8.53220	0.576553
$220$	14.5036	0.977835
$221$	8.83782	0.594496
$222$	38.9541	2.61443
$223$	−18.7055	−1.25261	−0.626305	−	0.779578i	$-0.715433\pi$
$223$	−18.7055	−1.25261	−0.626305	+	0.779578i	$0.715433\pi$
$224$	−17.0910	−1.14194
$225$	−7.68011	−0.512007
$226$	−48.3268	−3.21465
$227$	−10.5728	−0.701740	−0.350870	−	0.936424i	$-0.614114\pi$
$227$	−10.5728	−0.701740	−0.350870	+	0.936424i	$0.614114\pi$
$228$	−9.67430	−0.640696
$229$	9.08903	0.600620	0.300310	−	0.953842i	$-0.402910\pi$
$229$	9.08903	0.600620	0.300310	+	0.953842i	$0.402910\pi$
$230$	−10.4725	−0.690537
$231$	36.1526	2.37867
$232$	28.0031	1.83850
$233$	2.97019	0.194584	0.0972919	−	0.995256i	$-0.468982\pi$
$233$	2.97019	0.194584	0.0972919	+	0.995256i	$0.468982\pi$
$234$	12.3054	0.804431
$235$	−0.823126	−0.0536948
$236$	51.1089	3.32691
$237$	28.6230	1.85926
$238$	23.7753	1.54113
$239$	11.4413	0.740076	0.370038	−	0.929017i	$-0.379345\pi$
$239$	11.4413	0.740076	0.370038	+	0.929017i	$0.379345\pi$
$240$	9.18067	0.592610
$241$	−26.9436	−1.73559	−0.867795	−	0.496923i	$-0.834463\pi$
$241$	−26.9436	−1.73559	−0.867795	+	0.496923i	$0.834463\pi$
$242$	−47.0186	−3.02247
$243$	−15.5029	−0.994514
$244$	52.2425	3.34448
$245$	1.50899	0.0964056
$246$	−3.73108	−0.237885
$247$	2.92262	0.185962
$248$	−32.2000	−2.04470
$249$	−0.735966	−0.0466399
$250$	14.6334	0.925499
$251$	−7.86974	−0.496734	−0.248367	−	0.968666i	$-0.579894\pi$
$251$	−7.86974	−0.496734	−0.248367	+	0.968666i	$0.579894\pi$
$252$	22.8940	1.44219
$253$	37.4674	2.35555
$254$	24.1623	1.51607
$255$	−3.88648	−0.243381
$256$	−29.0451	−1.81532
$257$	−13.2110	−0.824080	−0.412040	−	0.911166i	$-0.635184\pi$
$257$	−13.2110	−0.824080	−0.412040	+	0.911166i	$0.635184\pi$
$258$	−47.6308	−2.96536
$259$	−21.8943	−1.36045
$260$	−7.80915	−0.484303
$261$	−7.31755	−0.452945
$262$	−46.4084	−2.86712
$263$	−22.4733	−1.38577	−0.692883	−	0.721050i	$-0.743660\pi$
$263$	−22.4733	−1.38577	−0.692883	+	0.721050i	$0.743660\pi$
$264$	−74.0886	−4.55984
$265$	5.01107	0.307828
$266$	7.86237	0.482073
$267$	−11.0756	−0.677814
$268$	21.5232	1.31474
$269$	−18.0436	−1.10014	−0.550068	−	0.835120i	$-0.685398\pi$
$269$	−18.0436	−1.10014	−0.550068	+	0.835120i	$0.685398\pi$
$270$	4.40722	0.268215
$271$	−22.3442	−1.35731	−0.678657	−	0.734456i	$-0.737437\pi$
$271$	−22.3442	−1.35731	−0.678657	+	0.734456i	$0.737437\pi$
$272$	−21.6005	−1.30973
$273$	−19.4655	−1.17811
$274$	−23.1310	−1.39740
$275$	−25.2135	−1.52043
$276$	66.7771	4.01951
$277$	−17.7646	−1.06737	−0.533687	−	0.845682i	$-0.679194\pi$
$277$	−17.7646	−1.06737	−0.533687	+	0.845682i	$0.679194\pi$
$278$	0.151062	0.00906010
$279$	8.41424	0.503748
$280$	−11.6393	−0.695580
$281$	4.28036	0.255345	0.127673	−	0.991816i	$-0.459249\pi$
$281$	4.28036	0.255345	0.127673	+	0.991816i	$0.459249\pi$
$282$	7.58926	0.451934
$283$	−2.85245	−0.169561	−0.0847803	−	0.996400i	$-0.527019\pi$
$283$	−2.85245	−0.169561	−0.0847803	+	0.996400i	$0.527019\pi$
$284$	−38.2120	−2.26747
$285$	−1.28524	−0.0761308
$286$	40.3983	2.38880
$287$	2.09706	0.123786
$288$	−9.15245	−0.539313
$289$	−7.85579	−0.462105
$290$	6.71474	0.394303
$291$	−28.9225	−1.69547
$292$	17.7382	1.03805
$293$	17.5327	1.02427	0.512135	−	0.858905i	$-0.328855\pi$
$293$	17.5327	1.02427	0.512135	+	0.858905i	$0.328855\pi$
$294$	−13.9129	−0.811418
$295$	6.78985	0.395320
$296$	44.8686	2.60794
$297$	−15.7677	−0.914934
$298$	−34.5578	−2.00188
$299$	−20.1734	−1.16666
$300$	−44.9374	−2.59446
$301$	26.7711	1.54306
$302$	2.03969	0.117371
$303$	31.5785	1.81414
$304$	−7.14318	−0.409689
$305$	6.94044	0.397409
$306$	12.7320	0.727842
$307$	−20.6540	−1.17878	−0.589392	−	0.807847i	$-0.700633\pi$
$307$	−20.6540	−1.17878	−0.589392	+	0.807847i	$0.700633\pi$
$308$	75.1602	4.28265
$309$	−19.6184	−1.11605
$310$	−7.72109	−0.438528
$311$	18.3700	1.04167	0.520835	−	0.853657i	$-0.325621\pi$
$311$	18.3700	1.04167	0.520835	+	0.853657i	$0.325621\pi$
$312$	39.8913	2.25840
$313$	−34.0340	−1.92372	−0.961859	−	0.273545i	$-0.911804\pi$
$313$	−34.0340	−1.92372	−0.961859	+	0.273545i	$0.911804\pi$
$314$	−3.19284	−0.180182
$315$	3.04148	0.171368
$316$	59.5064	3.34750
$317$	−35.3811	−1.98720	−0.993601	−	0.112946i	$-0.963971\pi$
$317$	−35.3811	−1.98720	−0.993601	+	0.112946i	$0.963971\pi$
$318$	−46.2023	−2.59090
$319$	−24.0233	−1.34504
$320$	−0.113275	−0.00633226
$321$	22.9859	1.28295
$322$	−54.2702	−3.02436
$323$	3.02394	0.168257
$324$	−50.3474	−2.79708
$325$	13.5756	0.753041
$326$	−42.1358	−2.33369
$327$	34.1692	1.88956
$328$	−4.29758	−0.237294
$329$	−4.26557	−0.235168
$330$	−17.7653	−0.977950
$331$	30.5648	1.68000	0.839998	−	0.542590i	$-0.182556\pi$
$331$	30.5648	1.68000	0.839998	+	0.542590i	$0.182556\pi$
$332$	−1.53005	−0.0839725
$333$	−11.7247	−0.642510
$334$	23.2982	1.27482
$335$	2.85937	0.156224
$336$	47.5757	2.59547
$337$	20.2963	1.10561	0.552804	−	0.833312i	$-0.313558\pi$
$337$	20.2963	1.10561	0.552804	+	0.833312i	$0.313558\pi$
$338$	11.3531	0.617527
$339$	40.9381	2.22345
$340$	−8.07988	−0.438193
$341$	27.6237	1.49591
$342$	4.21041	0.227673
$343$	−13.7928	−0.744739
$344$	−54.8627	−2.95800
$345$	8.87137	0.477619
$346$	−42.9042	−2.30654
$347$	−1.54639	−0.0830148	−0.0415074	−	0.999138i	$-0.513216\pi$
$347$	−1.54639	−0.0830148	−0.0415074	+	0.999138i	$0.513216\pi$
$348$	−42.8160	−2.29518
$349$	3.33938	0.178753	0.0893763	−	0.995998i	$-0.471513\pi$
$349$	3.33938	0.178753	0.0893763	+	0.995998i	$0.471513\pi$
$350$	36.5209	1.95213
$351$	8.48974	0.453149
$352$	−30.0472	−1.60152
$353$	21.5291	1.14588	0.572938	−	0.819598i	$-0.305803\pi$
$353$	21.5291	1.14588	0.572938	+	0.819598i	$0.305803\pi$
$354$	−62.6027	−3.32730
$355$	−5.07649	−0.269432
$356$	−23.0258	−1.22037
$357$	−20.1404	−1.06594
$358$	−36.2232	−1.91445
$359$	13.1902	0.696154	0.348077	−	0.937466i	$-0.386835\pi$
$359$	13.1902	0.696154	0.348077	+	0.937466i	$0.386835\pi$
$360$	−6.23300	−0.328508
$361$	1.00000	0.0526316
$362$	42.3523	2.22599
$363$	39.8300	2.09053
$364$	−40.4683	−2.12111
$365$	2.35653	0.123346
$366$	−63.9912	−3.34487
$367$	34.5138	1.80161	0.900803	−	0.434227i	$-0.142978\pi$
$367$	34.5138	1.80161	0.900803	+	0.434227i	$0.142978\pi$
$368$	49.3060	2.57025
$369$	1.12301	0.0584614
$370$	10.7588	0.559325
$371$	25.9682	1.34820
$372$	49.2329	2.55261
$373$	15.6113	0.808322	0.404161	−	0.914688i	$-0.367563\pi$
$373$	15.6113	0.808322	0.404161	+	0.914688i	$0.367563\pi$
$374$	41.7988	2.16136
$375$	−12.3961	−0.640134
$376$	8.74156	0.450811
$377$	12.9348	0.666174
$378$	22.8389	1.17471
$379$	−8.17714	−0.420032	−0.210016	−	0.977698i	$-0.567352\pi$
$379$	−8.17714	−0.420032	−0.210016	+	0.977698i	$0.567352\pi$
$380$	−2.67197	−0.137069
$381$	−20.4681	−1.04861
$382$	27.7429	1.41945
$383$	−10.9965	−0.561897	−0.280948	−	0.959723i	$-0.590649\pi$
$383$	−10.9965	−0.561897	−0.280948	+	0.959723i	$0.590649\pi$
$384$	24.9265	1.27203
$385$	9.98508	0.508887
$386$	58.5479	2.98001
$387$	14.3363	0.728754
$388$	−60.1292	−3.05260
$389$	−9.14964	−0.463905	−0.231952	−	0.972727i	$-0.574511\pi$
$389$	−9.14964	−0.463905	−0.231952	+	0.972727i	$0.574511\pi$
$390$	9.56534	0.484360
$391$	−20.8728	−1.05558
$392$	−16.0254	−0.809403
$393$	39.3131	1.98308
$394$	−20.8411	−1.04996
$395$	7.90546	0.397767
$396$	40.2494	2.02261
$397$	−12.6895	−0.636869	−0.318434	−	0.947945i	$-0.603157\pi$
$397$	−12.6895	−0.636869	−0.318434	+	0.947945i	$0.603157\pi$
$398$	68.3243	3.42479
$399$	−6.66030	−0.333432
$400$	−33.1803	−1.65901
$401$	31.1175	1.55393	0.776966	−	0.629542i	$-0.216757\pi$
$401$	31.1175	1.55393	0.776966	+	0.629542i	$0.216757\pi$
$402$	−26.3636	−1.31490
$403$	−14.8733	−0.740893
$404$	65.6508	3.26625
$405$	−6.68868	−0.332363
$406$	34.7969	1.72694
$407$	−38.4918	−1.90797
$408$	41.2742	2.04338
$409$	27.1727	1.34360	0.671802	−	0.740730i	$-0.265520\pi$
$409$	27.1727	1.34360	0.671802	+	0.740730i	$0.265520\pi$
$410$	−1.03050	−0.0508925
$411$	19.5946	0.966528
$412$	−40.7861	−2.00939
$413$	35.1861	1.73139
$414$	−29.0625	−1.42834
$415$	−0.203268	−0.00997805
$416$	16.1782	0.793201
$417$	−0.127966	−0.00626654
$418$	13.8226	0.676087
$419$	−1.61581	−0.0789377	−0.0394689	−	0.999221i	$-0.512567\pi$
$419$	−1.61581	−0.0789377	−0.0394689	+	0.999221i	$0.512567\pi$
$420$	17.7961	0.868362
$421$	5.12894	0.249969	0.124985	−	0.992159i	$-0.460112\pi$
$421$	5.12894	0.249969	0.124985	+	0.992159i	$0.460112\pi$
$422$	−2.54651	−0.123962
$423$	−2.28427	−0.111065
$424$	−53.2174	−2.58446
$425$	14.0463	0.681345
$426$	46.8055	2.26773
$427$	35.9665	1.74054
$428$	47.7871	2.30988
$429$	−34.2218	−1.65225
$430$	−13.1553	−0.634403
$431$	−23.3889	−1.12660	−0.563302	−	0.826251i	$-0.690469\pi$
$431$	−23.3889	−1.12660	−0.563302	+	0.826251i	$0.690469\pi$
$432$	−20.7498	−0.998325
$433$	−25.2028	−1.21117	−0.605584	−	0.795781i	$-0.707061\pi$
$433$	−25.2028	−1.21117	−0.605584	+	0.795781i	$0.707061\pi$
$434$	−40.0119	−1.92063
$435$	−5.68813	−0.272725
$436$	71.0368	3.40204
$437$	−6.90252	−0.330192
$438$	−21.7273	−1.03817
$439$	26.0228	1.24200	0.620999	−	0.783811i	$-0.286727\pi$
$439$	26.0228	1.24200	0.620999	+	0.783811i	$0.286727\pi$
$440$	−20.4627	−0.975522
$441$	4.18762	0.199410
$442$	−22.5056	−1.07048
$443$	−28.4681	−1.35256	−0.676280	−	0.736645i	$-0.736409\pi$
$443$	−28.4681	−1.35256	−0.676280	+	0.736645i	$0.736409\pi$
$444$	−68.6028	−3.25574
$445$	−3.05899	−0.145010
$446$	47.6336	2.25552
$447$	29.2743	1.38463
$448$	−0.587009	−0.0277336
$449$	8.58841	0.405312	0.202656	−	0.979250i	$-0.435043\pi$
$449$	8.58841	0.405312	0.202656	+	0.979250i	$0.435043\pi$
$450$	19.5575	0.921948
$451$	3.68679	0.173604
$452$	85.1092	4.00320
$453$	−1.72785	−0.0811813
$454$	26.9237	1.26359
$455$	−5.37623	−0.252042
$456$	13.6492	0.639180
$457$	−21.1869	−0.991082	−0.495541	−	0.868584i	$-0.665030\pi$
$457$	−21.1869	−0.991082	−0.495541	+	0.868584i	$0.665030\pi$
$458$	−23.1453	−1.08151
$459$	8.78406	0.410005
$460$	18.4433	0.859925
$461$	−35.1751	−1.63827	−0.819133	−	0.573604i	$-0.805545\pi$
$461$	−35.1751	−1.63827	−0.819133	+	0.573604i	$0.805545\pi$
$462$	−92.0629	−4.28315
$463$	5.65659	0.262884	0.131442	−	0.991324i	$-0.458039\pi$
$463$	5.65659	0.262884	0.131442	+	0.991324i	$0.458039\pi$
$464$	−31.6139	−1.46764
$465$	6.54062	0.303314
$466$	−7.56362	−0.350378
$467$	18.3215	0.847819	0.423909	−	0.905705i	$-0.360657\pi$
$467$	18.3215	0.847819	0.423909	+	0.905705i	$0.360657\pi$
$468$	−21.6713	−1.00176
$469$	14.8177	0.684220
$470$	2.09610	0.0966857
$471$	2.70469	0.124625
$472$	−72.1079	−3.31903
$473$	47.0655	2.16407
$474$	−72.8887	−3.34789
$475$	4.64503	0.213128
$476$	−41.8712	−1.91916
$477$	13.9063	0.636727
$478$	−29.1354	−1.33262
$479$	−24.1397	−1.10297	−0.551486	−	0.834184i	$-0.685939\pi$
$479$	−24.1397	−1.10297	−0.551486	+	0.834184i	$0.685939\pi$
$480$	−7.11444	−0.324728
$481$	20.7250	0.944979
$482$	68.6121	3.12520
$483$	45.9729	2.09184
$484$	82.8054	3.76388
$485$	−7.98819	−0.362725
$486$	39.4784	1.79077
$487$	11.7587	0.532839	0.266419	−	0.963857i	$-0.414159\pi$
$487$	11.7587	0.532839	0.266419	+	0.963857i	$0.414159\pi$
$488$	−73.7072	−3.33657
$489$	35.6937	1.61413
$490$	−3.84265	−0.173593
$491$	−7.60319	−0.343127	−0.171564	−	0.985173i	$-0.554882\pi$
$491$	−7.60319	−0.343127	−0.171564	+	0.985173i	$0.554882\pi$
$492$	6.57087	0.296238
$493$	13.3832	0.602749
$494$	−7.44247	−0.334853
$495$	5.34715	0.240337
$496$	36.3519	1.63225
$497$	−26.3072	−1.18004
$498$	1.87414	0.0839824
$499$	−4.31612	−0.193216	−0.0966080	−	0.995323i	$-0.530799\pi$
$499$	−4.31612	−0.193216	−0.0966080	+	0.995323i	$0.530799\pi$
$500$	−25.7712	−1.15252
$501$	−19.7362	−0.881747
$502$	20.0404	0.894445
$503$	24.3414	1.08533	0.542665	−	0.839949i	$-0.317415\pi$
$503$	24.3414	1.08533	0.542665	+	0.839949i	$0.317415\pi$
$504$	−32.3004	−1.43877
$505$	8.72174	0.388113
$506$	−95.4110	−4.24154
$507$	−9.61733	−0.427121
$508$	−42.5526	−1.88797
$509$	0.857352	0.0380015	0.0190007	−	0.999819i	$-0.493952\pi$
$509$	0.857352	0.0380015	0.0190007	+	0.999819i	$0.493952\pi$
$510$	9.89695	0.438244
$511$	12.2119	0.540224
$512$	50.8533	2.24742
$513$	2.90484	0.128252
$514$	33.6420	1.48388
$515$	−5.41846	−0.238766
$516$	83.8835	3.69277
$517$	−7.49919	−0.329814
$518$	55.7540	2.44969
$519$	36.3446	1.59535
$520$	11.0177	0.483157
$521$	−25.6877	−1.12540	−0.562699	−	0.826662i	$-0.690237\pi$
$521$	−25.6877	−1.12540	−0.562699	+	0.826662i	$0.690237\pi$
$522$	18.6342	0.815597
$523$	−38.6953	−1.69203	−0.846013	−	0.533162i	$-0.821003\pi$
$523$	−38.6953	−1.69203	−0.846013	+	0.533162i	$0.821003\pi$
$524$	81.7308	3.57043
$525$	−30.9373	−1.35021
$526$	57.2286	2.49528
$527$	−15.3889	−0.670353
$528$	83.6416	3.64003
$529$	24.6448	1.07151
$530$	−12.7607	−0.554291
$531$	18.8426	0.817701
$532$	−13.8466	−0.600325
$533$	−1.98507	−0.0859829
$534$	28.2040	1.22051
$535$	6.34854	0.274471
$536$	−30.3664	−1.31163
$537$	30.6850	1.32416
$538$	45.9481	1.98096
$539$	13.7478	0.592160
$540$	−7.76165	−0.334008
$541$	17.7536	0.763289	0.381644	−	0.924309i	$-0.375358\pi$
$541$	17.7536	0.763289	0.381644	+	0.924309i	$0.375358\pi$
$542$	56.8997	2.44405
$543$	−35.8771	−1.53963
$544$	16.7391	0.717681
$545$	9.43728	0.404249
$546$	49.5691	2.12136
$547$	−28.4478	−1.21634	−0.608170	−	0.793807i	$-0.708096\pi$
$547$	−28.4478	−1.21634	−0.608170	+	0.793807i	$0.708096\pi$
$548$	40.7365	1.74018
$549$	19.2606	0.822021
$550$	64.2065	2.73777
$551$	4.42575	0.188543
$552$	−94.2136	−4.01000
$553$	40.9674	1.74211
$554$	45.2378	1.92197
$555$	−9.11392	−0.386865
$556$	−0.266038	−0.0112825
$557$	20.9661	0.888362	0.444181	−	0.895937i	$-0.353495\pi$
$557$	20.9661	0.888362	0.444181	+	0.895937i	$0.353495\pi$
$558$	−21.4269	−0.907075
$559$	−25.3413	−1.07182
$560$	13.1401	0.555269
$561$	−35.4082	−1.49494
$562$	−10.9000	−0.459788
$563$	−5.02358	−0.211718	−0.105859	−	0.994381i	$-0.533759\pi$
$563$	−5.02358	−0.211718	−0.105859	+	0.994381i	$0.533759\pi$
$564$	−13.3656	−0.562793
$565$	11.3068	0.475681
$566$	7.26379	0.305320
$567$	−34.6618	−1.45566
$568$	53.9121	2.26210
$569$	9.04801	0.379313	0.189656	−	0.981851i	$-0.439263\pi$
$569$	9.04801	0.379313	0.189656	+	0.981851i	$0.439263\pi$
$570$	3.27287	0.137085
$571$	14.9578	0.625965	0.312983	−	0.949759i	$-0.398672\pi$
$571$	14.9578	0.625965	0.312983	+	0.949759i	$0.398672\pi$
$572$	−71.1462	−2.97477
$573$	−23.5013	−0.981783
$574$	−5.34019	−0.222895
$575$	−32.0624	−1.33709
$576$	−0.314351	−0.0130980
$577$	−13.1931	−0.549237	−0.274618	−	0.961553i	$-0.588552\pi$
$577$	−13.1931	−0.549237	−0.274618	+	0.961553i	$0.588552\pi$
$578$	20.0048	0.832091
$579$	−49.5966	−2.06116
$580$	−11.8255	−0.491025
$581$	−1.05337	−0.0437011
$582$	73.6515	3.05295
$583$	45.6540	1.89080
$584$	−25.0263	−1.03559
$585$	−2.87905	−0.119034
$586$	−44.6471	−1.84436
$587$	−18.5238	−0.764558	−0.382279	−	0.924047i	$-0.624861\pi$
$587$	−18.5238	−0.764558	−0.382279	+	0.924047i	$0.624861\pi$
$588$	24.5023	1.01046
$589$	−5.08904	−0.209690
$590$	−17.2904	−0.711835
$591$	17.6547	0.726218
$592$	−50.6540	−2.08187
$593$	34.2973	1.40842	0.704211	−	0.709991i	$-0.251301\pi$
$593$	34.2973	1.40842	0.704211	+	0.709991i	$0.251301\pi$
$594$	40.1525	1.64748
$595$	−5.56262	−0.228045
$596$	60.8604	2.49294
$597$	−57.8783	−2.36880
$598$	51.3718	2.10075
$599$	−28.5857	−1.16798	−0.583989	−	0.811761i	$-0.698509\pi$
$599$	−28.5857	−1.16798	−0.583989	+	0.811761i	$0.698509\pi$
$600$	63.4007	2.58832
$601$	−23.7626	−0.969298	−0.484649	−	0.874709i	$-0.661053\pi$
$601$	−23.7626	−0.969298	−0.484649	+	0.874709i	$0.661053\pi$
$602$	−68.1727	−2.77851
$603$	7.93511	0.323143
$604$	−3.59214	−0.146162
$605$	11.0008	0.447244
$606$	−80.4149	−3.26663
$607$	21.6050	0.876920	0.438460	−	0.898751i	$-0.355524\pi$
$607$	21.6050	0.876920	0.438460	+	0.898751i	$0.355524\pi$
$608$	5.53551	0.224495
$609$	−29.4768	−1.19446
$610$	−17.6739	−0.715595
$611$	4.03776	0.163350
$612$	−22.4226	−0.906381
$613$	15.4592	0.624391	0.312195	−	0.950018i	$-0.398936\pi$
$613$	15.4592	0.624391	0.312195	+	0.950018i	$0.398936\pi$
$614$	52.5955	2.12258
$615$	0.872944	0.0352005
$616$	−106.041	−4.27252
$617$	−23.0157	−0.926579	−0.463289	−	0.886207i	$-0.653331\pi$
$617$	−23.0157	−0.926579	−0.463289	+	0.886207i	$0.653331\pi$
$618$	49.9584	2.00962
$619$	24.9073	1.00111	0.500554	−	0.865705i	$-0.333130\pi$
$619$	24.9073	1.00111	0.500554	+	0.865705i	$0.333130\pi$
$620$	13.5978	0.546099
$621$	−20.0507	−0.804608
$622$	−46.7795	−1.87569
$623$	−15.8522	−0.635105
$624$	−45.0349	−1.80284
$625$	19.8014	0.792057
$626$	86.6680	3.46395
$627$	−11.7093	−0.467624
$628$	5.62297	0.224381
$629$	21.4435	0.855008
$630$	−7.74516	−0.308575
$631$	41.0981	1.63609	0.818045	−	0.575154i	$-0.195058\pi$
$631$	41.0981	1.63609	0.818045	+	0.575154i	$0.195058\pi$
$632$	−83.9556	−3.33958
$633$	2.15718	0.0857400
$634$	90.0984	3.57826
$635$	−5.65314	−0.224338
$636$	81.3678	3.22644
$637$	−7.40218	−0.293285
$638$	61.1755	2.42196
$639$	−14.0879	−0.557308
$640$	6.88453	0.272135
$641$	3.14179	0.124093	0.0620466	−	0.998073i	$-0.480237\pi$
$641$	3.14179	0.124093	0.0620466	+	0.998073i	$0.480237\pi$
$642$	−58.5338	−2.31015
$643$	25.1320	0.991111	0.495555	−	0.868576i	$-0.334965\pi$
$643$	25.1320	0.991111	0.495555	+	0.868576i	$0.334965\pi$
$644$	95.5763	3.76624
$645$	11.1440	0.438794
$646$	−7.70049	−0.302972
$647$	12.9166	0.507802	0.253901	−	0.967230i	$-0.418286\pi$
$647$	12.9166	0.507802	0.253901	+	0.967230i	$0.418286\pi$
$648$	71.0335	2.79046
$649$	61.8597	2.42821
$650$	−34.5705	−1.35597
$651$	33.8945	1.32843
$652$	74.2063	2.90614
$653$	−22.8107	−0.892651	−0.446325	−	0.894871i	$-0.647268\pi$
$653$	−22.8107	−0.892651	−0.446325	+	0.894871i	$0.647268\pi$
$654$	−87.0121	−3.40244
$655$	10.8580	0.424257
$656$	4.85171	0.189427
$657$	6.53966	0.255136
$658$	10.8623	0.423457
$659$	−6.66998	−0.259826	−0.129913	−	0.991525i	$-0.541470\pi$
$659$	−6.66998	−0.259826	−0.129913	+	0.991525i	$0.541470\pi$
$660$	31.2869	1.21784
$661$	28.8746	1.12309	0.561546	−	0.827446i	$-0.310207\pi$
$661$	28.8746	1.12309	0.561546	+	0.827446i	$0.310207\pi$
$662$	−77.8336	−3.02509
$663$	19.0647	0.740413
$664$	2.15870	0.0837738
$665$	−1.83953	−0.0713338
$666$	29.8570	1.15694
$667$	−30.5488	−1.18285
$668$	−41.0310	−1.58754
$669$	−40.3509	−1.56006
$670$	−7.28142	−0.281306
$671$	63.2317	2.44103
$672$	−36.8682	−1.42222
$673$	27.2270	1.04952	0.524761	−	0.851250i	$-0.324155\pi$
$673$	27.2270	1.04952	0.524761	+	0.851250i	$0.324155\pi$
$674$	−51.6846	−1.99082
$675$	13.4931	0.519348
$676$	−19.9942	−0.769006
$677$	1.56444	0.0601261	0.0300631	−	0.999548i	$-0.490429\pi$
$677$	1.56444	0.0601261	0.0300631	+	0.999548i	$0.490429\pi$
$678$	−104.249	−4.00367
$679$	−41.3961	−1.58864
$680$	11.3996	0.437156
$681$	−22.8073	−0.873979
$682$	−70.3439	−2.69361
$683$	−20.9167	−0.800357	−0.400178	−	0.916437i	$-0.631052\pi$
$683$	−20.9167	−0.800357	−0.400178	+	0.916437i	$0.631052\pi$
$684$	−7.41504	−0.283521
$685$	5.41187	0.206777
$686$	35.1234	1.34102
$687$	19.6066	0.748039
$688$	61.9368	2.36132
$689$	−24.5813	−0.936474
$690$	−22.5910	−0.860026
$691$	−31.4031	−1.19463	−0.597315	−	0.802007i	$-0.703766\pi$
$691$	−31.4031	−1.19463	−0.597315	+	0.802007i	$0.703766\pi$
$692$	75.5594	2.87234
$693$	27.7098	1.05261
$694$	3.93790	0.149481
$695$	−0.0353434	−0.00134065
$696$	60.4077	2.28975
$697$	−2.05389	−0.0777965
$698$	−8.50375	−0.321872
$699$	6.40722	0.242343
$700$	−64.3177	−2.43098
$701$	−33.1991	−1.25391	−0.626957	−	0.779054i	$-0.715700\pi$
$701$	−33.1991	−1.25391	−0.626957	+	0.779054i	$0.715700\pi$
$702$	−21.6192	−0.815964
$703$	7.09124	0.267451
$704$	−1.03200	−0.0388951
$705$	−1.77563	−0.0668739
$706$	−54.8240	−2.06333
$707$	45.1975	1.69983
$708$	110.251	4.14348
$709$	−47.9844	−1.80209	−0.901046	−	0.433724i	$-0.857199\pi$
$709$	−47.9844	−1.80209	−0.901046	+	0.433724i	$0.857199\pi$
$710$	12.9273	0.485154
$711$	21.9386	0.822762
$712$	32.4864	1.21748
$713$	35.1272	1.31552
$714$	51.2876	1.91939
$715$	−9.45182	−0.353478
$716$	63.7933	2.38407
$717$	24.6809	0.921724
$718$	−33.5891	−1.25353
$719$	4.36301	0.162713	0.0813564	−	0.996685i	$-0.474075\pi$
$719$	4.36301	0.162713	0.0813564	+	0.996685i	$0.474075\pi$
$720$	7.03669	0.262242
$721$	−28.0793	−1.04573
$722$	−2.54651	−0.0947712
$723$	−58.1221	−2.16158
$724$	−74.5875	−2.77202
$725$	20.5577	0.763494
$726$	−101.427	−3.76433
$727$	6.89570	0.255747	0.127874	−	0.991790i	$-0.459185\pi$
$727$	6.89570	0.255747	0.127874	+	0.991790i	$0.459185\pi$
$728$	57.0954	2.11610
$729$	0.236855	0.00877239
$730$	−6.00093	−0.222104
$731$	−26.2199	−0.969776
$732$	112.696	4.16537
$733$	21.5833	0.797197	0.398598	−	0.917126i	$-0.369497\pi$
$733$	21.5833	0.797197	0.398598	+	0.917126i	$0.369497\pi$
$734$	−87.8897	−3.24407
$735$	3.25515	0.120068
$736$	−38.2090	−1.40840
$737$	26.0507	0.959589
$738$	−2.85975	−0.105269
$739$	−32.1421	−1.18237	−0.591183	−	0.806537i	$-0.701339\pi$
$739$	−32.1421	−1.18237	−0.591183	+	0.806537i	$0.701339\pi$
$740$	−18.9476	−0.696527
$741$	6.30460	0.231605
$742$	−66.1282	−2.42764
$743$	22.1613	0.813020	0.406510	−	0.913646i	$-0.366746\pi$
$743$	22.1613	0.813020	0.406510	+	0.913646i	$0.366746\pi$
$744$	−69.4611	−2.54657
$745$	8.08535	0.296224
$746$	−39.7543	−1.45551
$747$	−0.564094	−0.0206391
$748$	−73.6127	−2.69155
$749$	32.8992	1.20211
$750$	31.5669	1.15266
$751$	−14.3426	−0.523369	−0.261685	−	0.965153i	$-0.584278\pi$
$751$	−14.3426	−0.523369	−0.261685	+	0.965153i	$0.584278\pi$
$752$	−9.86870	−0.359875
$753$	−16.9764	−0.618655
$754$	−32.9385	−1.19955
$755$	−0.477218	−0.0173678
$756$	−40.2221	−1.46286
$757$	7.32146	0.266103	0.133052	−	0.991109i	$-0.457522\pi$
$757$	7.32146	0.266103	0.133052	+	0.991109i	$0.457522\pi$
$758$	20.8232	0.756331
$759$	80.8237	2.93372
$760$	3.76980	0.136745
$761$	−17.9552	−0.650875	−0.325437	−	0.945564i	$-0.605512\pi$
$761$	−17.9552	−0.650875	−0.325437	+	0.945564i	$0.605512\pi$
$762$	52.1222	1.88819
$763$	48.9055	1.77050
$764$	−48.8586	−1.76764
$765$	−2.97886	−0.107701
$766$	28.0028	1.01178
$767$	−33.3069	−1.20264
$768$	−62.6554	−2.26088
$769$	10.5215	0.379416	0.189708	−	0.981841i	$-0.439246\pi$
$769$	10.5215	0.379416	0.189708	+	0.981841i	$0.439246\pi$
$770$	−25.4271	−0.916329
$771$	−28.4985	−1.02635
$772$	−103.110	−3.71101
$773$	−11.8202	−0.425144	−0.212572	−	0.977145i	$-0.568184\pi$
$773$	−11.8202	−0.425144	−0.212572	+	0.977145i	$0.568184\pi$
$774$	−36.5075	−1.31223
$775$	−23.6387	−0.849128
$776$	84.8343	3.04537
$777$	−47.2298	−1.69436
$778$	23.2996	0.835332
$779$	−0.679209	−0.0243352
$780$	−16.8457	−0.603173
$781$	−46.2500	−1.65496
$782$	53.1528	1.90074
$783$	12.8561	0.459439
$784$	18.0917	0.646132
$785$	0.747015	0.0266621
$786$	−100.111	−3.57085
$787$	14.1571	0.504646	0.252323	−	0.967643i	$-0.418806\pi$
$787$	14.1571	0.504646	0.252323	+	0.967643i	$0.418806\pi$
$788$	36.7037	1.30751
$789$	−48.4790	−1.72590
$790$	−20.1313	−0.716240
$791$	58.5937	2.08335
$792$	−56.7865	−2.01782
$793$	−34.0457	−1.20900
$794$	32.3140	1.14678
$795$	10.8098	0.383383
$796$	−120.327	−4.26489
$797$	−41.0910	−1.45552	−0.727759	−	0.685833i	$-0.759438\pi$
$797$	−41.0910	−1.45552	−0.727759	+	0.685833i	$0.759438\pi$
$798$	16.9605	0.600396
$799$	4.17774	0.147798
$800$	25.7126	0.909078
$801$	−8.48907	−0.299947
$802$	−79.2409	−2.79809
$803$	21.4695	0.757641
$804$	46.4294	1.63744
$805$	12.6974	0.447524
$806$	37.8750	1.33409
$807$	−38.9232	−1.37016
$808$	−92.6245	−3.25852
$809$	6.09536	0.214301	0.107151	−	0.994243i	$-0.465827\pi$
$809$	6.09536	0.214301	0.107151	+	0.994243i	$0.465827\pi$
$810$	17.0328	0.598471
$811$	−38.2795	−1.34417	−0.672087	−	0.740472i	$-0.734602\pi$
$811$	−38.2795	−1.34417	−0.672087	+	0.740472i	$0.734602\pi$
$812$	−61.2814	−2.15056
$813$	−48.2003	−1.69046
$814$	98.0196	3.43559
$815$	9.85835	0.345323
$816$	−46.5962	−1.63119
$817$	−8.67076	−0.303351
$818$	−69.1956	−2.41937
$819$	−14.9197	−0.521336
$820$	1.81483	0.0633765
$821$	11.7725	0.410864	0.205432	−	0.978671i	$-0.434140\pi$
$821$	11.7725	0.410864	0.205432	+	0.978671i	$0.434140\pi$
$822$	−49.8977	−1.74038
$823$	−36.7622	−1.28145	−0.640725	−	0.767771i	$-0.721366\pi$
$823$	−36.7622	−1.28145	−0.640725	+	0.767771i	$0.721366\pi$
$824$	57.5438	2.00463
$825$	−54.3900	−1.89362
$826$	−89.6017	−3.11764
$827$	−23.1921	−0.806468	−0.403234	−	0.915097i	$-0.632114\pi$
$827$	−23.1921	−0.806468	−0.403234	+	0.915097i	$0.632114\pi$
$828$	51.1825	1.77871
$829$	51.6813	1.79497	0.897484	−	0.441048i	$-0.145393\pi$
$829$	51.6813	1.79497	0.897484	+	0.441048i	$0.145393\pi$
$830$	0.517625	0.0179670
$831$	−38.3214	−1.32936
$832$	0.555659	0.0192640
$833$	−7.65880	−0.265362
$834$	0.325867	0.0112839
$835$	−5.45099	−0.188639
$836$	−24.3433	−0.841931
$837$	−14.7828	−0.510970
$838$	4.11469	0.142139
$839$	45.6906	1.57741	0.788707	−	0.614770i	$-0.210751\pi$
$839$	45.6906	1.57741	0.788707	+	0.614770i	$0.210751\pi$
$840$	−25.1080	−0.866307
$841$	−9.41277	−0.324578
$842$	−13.0609	−0.450108
$843$	9.23349	0.318018
$844$	4.48471	0.154370
$845$	−2.65624	−0.0913773
$846$	5.81692	0.199990
$847$	57.0077	1.95881
$848$	60.0793	2.06313
$849$	−6.15324	−0.211179
$850$	−35.7690	−1.22687
$851$	−48.9475	−1.67790
$852$	−82.4301	−2.82401
$853$	23.4289	0.802191	0.401095	−	0.916036i	$-0.368630\pi$
$853$	23.4289	0.802191	0.401095	+	0.916036i	$0.368630\pi$
$854$	−91.5890	−3.13411
$855$	−0.985092	−0.0336894
$856$	−67.4212	−2.30441
$857$	−40.1205	−1.37049	−0.685245	−	0.728313i	$-0.740305\pi$
$857$	−40.1205	−1.37049	−0.685245	+	0.728313i	$0.740305\pi$
$858$	87.1462	2.97512
$859$	48.4058	1.65158	0.825792	−	0.563974i	$-0.190728\pi$
$859$	48.4058	1.65158	0.825792	+	0.563974i	$0.190728\pi$
$860$	23.1680	0.790022
$861$	4.52374	0.154169
$862$	59.5601	2.02862
$863$	−23.5912	−0.803055	−0.401528	−	0.915847i	$-0.631521\pi$
$863$	−23.5912	−0.803055	−0.401528	+	0.915847i	$0.631521\pi$
$864$	16.0798	0.547045
$865$	10.0381	0.341306
$866$	64.1791	2.18089
$867$	−16.9463	−0.575527
$868$	70.4658	2.39176
$869$	72.0237	2.44324
$870$	14.4849	0.491083
$871$	−14.0264	−0.475266
$872$	−100.223	−3.39400
$873$	−22.1682	−0.750280
$874$	17.5773	0.594562
$875$	−17.7423	−0.599799
$876$	38.2644	1.29283
$877$	7.05549	0.238247	0.119123	−	0.992879i	$-0.461992\pi$
$877$	7.05549	0.238247	0.119123	+	0.992879i	$0.461992\pi$
$878$	−66.2672	−2.23641
$879$	37.8211	1.27567
$880$	23.1012	0.778741
$881$	−50.6855	−1.70764	−0.853819	−	0.520570i	$-0.825720\pi$
$881$	−50.6855	−1.70764	−0.853819	+	0.520570i	$0.825720\pi$
$882$	−10.6638	−0.359069
$883$	−24.0937	−0.810819	−0.405409	−	0.914135i	$-0.632871\pi$
$883$	−24.0937	−0.810819	−0.405409	+	0.914135i	$0.632871\pi$
$884$	39.6350	1.33307
$885$	14.6469	0.492350
$886$	72.4942	2.43549
$887$	−55.1568	−1.85198	−0.925991	−	0.377545i	$-0.876768\pi$
$887$	−55.1568	−1.85198	−0.925991	+	0.377545i	$0.876768\pi$
$888$	96.7895	3.24804
$889$	−29.2955	−0.982539
$890$	7.78975	0.261113
$891$	−60.9381	−2.04150
$892$	−83.8885	−2.80879
$893$	1.38156	0.0462320
$894$	−74.5473	−2.49323
$895$	8.47498	0.283287
$896$	35.6767	1.19188
$897$	−43.5177	−1.45301
$898$	−21.8705	−0.729827
$899$	−22.5228	−0.751177
$900$	−34.4430	−1.14810
$901$	−25.4335	−0.847313
$902$	−9.38845	−0.312601
$903$	57.7499	1.92180
$904$	−120.078	−3.99373
$905$	−9.90900	−0.329386
$906$	4.39997	0.146179
$907$	48.1487	1.59875	0.799376	−	0.600831i	$-0.205164\pi$
$907$	48.1487	1.59875	0.799376	+	0.600831i	$0.205164\pi$
$908$	−47.4158	−1.57355
$909$	24.2039	0.802792
$910$	13.6906	0.453840
$911$	19.7151	0.653190	0.326595	−	0.945164i	$-0.394099\pi$
$911$	19.7151	0.653190	0.326595	+	0.945164i	$0.394099\pi$
$912$	−15.4091	−0.510246
$913$	−1.85190	−0.0612890
$914$	53.9527	1.78460
$915$	14.9717	0.494951
$916$	40.7616	1.34680
$917$	56.2678	1.85813
$918$	−22.3687	−0.738277
$919$	−26.8139	−0.884510	−0.442255	−	0.896889i	$-0.645821\pi$
$919$	−26.8139	−0.884510	−0.442255	+	0.896889i	$0.645821\pi$
$920$	−26.0211	−0.857890
$921$	−44.5542	−1.46811
$922$	89.5736	2.94995
$923$	24.9022	0.819667
$924$	162.134	5.33381
$925$	32.9390	1.08303
$926$	−14.4046	−0.473363
$927$	−15.0369	−0.493876
$928$	24.4988	0.804212
$929$	−32.2552	−1.05826	−0.529130	−	0.848541i	$-0.677482\pi$
$929$	−32.2552	−1.05826	−0.529130	+	0.848541i	$0.677482\pi$
$930$	−16.6557	−0.546163
$931$	−2.53272	−0.0830066
$932$	13.3204	0.436325
$933$	39.6274	1.29734
$934$	−46.6559	−1.52663
$935$	−9.77949	−0.319824
$936$	30.5754	0.999387
$937$	−35.5095	−1.16004	−0.580022	−	0.814601i	$-0.696956\pi$
$937$	−35.5095	−1.16004	−0.580022	+	0.814601i	$0.696956\pi$
$938$	−37.7335	−1.23204
$939$	−73.4174	−2.39589
$940$	−3.69148	−0.120403
$941$	6.09352	0.198643	0.0993215	−	0.995055i	$-0.468333\pi$
$941$	6.09352	0.198643	0.0993215	+	0.995055i	$0.468333\pi$
$942$	−6.88751	−0.224407
$943$	4.68826	0.152671
$944$	81.4056	2.64953
$945$	−5.34353	−0.173825
$946$	−119.853	−3.89675
$947$	−28.1442	−0.914564	−0.457282	−	0.889322i	$-0.651177\pi$
$947$	−28.1442	−0.914564	−0.457282	+	0.889322i	$0.651177\pi$
$948$	128.366	4.16913
$949$	−11.5597	−0.375245
$950$	−11.8286	−0.383771
$951$	−76.3233	−2.47495
$952$	59.0747	1.91462
$953$	20.3085	0.657856	0.328928	−	0.944355i	$-0.393313\pi$
$953$	20.3085	0.657856	0.328928	+	0.944355i	$0.393313\pi$
$954$	−35.4126	−1.14653
$955$	−6.49090	−0.210041
$956$	51.3109	1.65951
$957$	−51.8224	−1.67518
$958$	61.4721	1.98607
$959$	28.0452	0.905627
$960$	−0.244354	−0.00788649
$961$	−5.10168	−0.164570
$962$	−52.7764	−1.70158
$963$	17.6180	0.567731
$964$	−120.834	−3.89181
$965$	−13.6982	−0.440961
$966$	−117.070	−3.76668
$967$	15.1754	0.488009	0.244005	−	0.969774i	$-0.421539\pi$
$967$	15.1754	0.488009	0.244005	+	0.969774i	$0.421539\pi$
$968$	−116.828	−3.75498
$969$	6.52317	0.209554
$970$	20.3420	0.653142
$971$	6.48116	0.207990	0.103995	−	0.994578i	$-0.466837\pi$
$971$	6.48116	0.207990	0.103995	+	0.994578i	$0.466837\pi$
$972$	−69.5261	−2.23005
$973$	−0.183155	−0.00587168
$974$	−29.9437	−0.959458
$975$	29.2850	0.937872
$976$	83.2111	2.66352
$977$	17.1441	0.548487	0.274244	−	0.961660i	$-0.411573\pi$
$977$	17.1441	0.548487	0.274244	+	0.961660i	$0.411573\pi$
$978$	−90.8944	−2.90648
$979$	−27.8693	−0.890707
$980$	6.76736	0.216175
$981$	26.1896	0.836169
$982$	19.3616	0.617853
$983$	−10.4299	−0.332662	−0.166331	−	0.986070i	$-0.553192\pi$
$983$	−10.4299	−0.332662	−0.166331	+	0.986070i	$0.553192\pi$
$984$	−9.27063	−0.295537
$985$	4.87610	0.155366
$986$	−34.0804	−1.08534
$987$	−9.20158	−0.292890
$988$	13.1071	0.416992
$989$	59.8501	1.90312
$990$	−13.6166	−0.432763
$991$	0.647633	0.0205728	0.0102864	−	0.999947i	$-0.496726\pi$
$991$	0.647633	0.0205728	0.0102864	+	0.999947i	$0.496726\pi$
$992$	−28.1704	−0.894413
$993$	65.9337	2.09234
$994$	66.9915	2.12484
$995$	−15.9856	−0.506776
$996$	−3.30059	−0.104583
$997$	49.7951	1.57703	0.788513	−	0.615017i	$-0.210851\pi$
$997$	49.7951	1.57703	0.788513	+	0.615017i	$0.210851\pi$
$998$	10.9910	0.347915
$999$	20.5989	0.651721

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4009.2.a.f.1.5	✓	83

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4009.2.a.f.1.5	✓	83	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:	\( N \)	\(=\)	\( 4009 = 19 \cdot 211 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4009.a (trivial)

\(f(q)\)	\(=\)	\(q-2.54651 q^{2} +2.15718 q^{3} +4.48471 q^{4} +0.595796 q^{5} -5.49327 q^{6} +3.08751 q^{7} -6.32733 q^{8} +1.65341 q^{9} +O(q^{10})\)	\(q-2.54651 q^{2} +2.15718 q^{3} +4.48471 q^{4} +0.595796 q^{5} -5.49327 q^{6} +3.08751 q^{7} -6.32733 q^{8} +1.65341 q^{9} -1.51720 q^{10} +5.42807 q^{11} +9.67430 q^{12} -2.92262 q^{13} -7.86237 q^{14} +1.28524 q^{15} +7.14318 q^{16} -3.02394 q^{17} -4.21041 q^{18} -1.00000 q^{19} +2.67197 q^{20} +6.66030 q^{21} -13.8226 q^{22} +6.90252 q^{23} -13.6492 q^{24} -4.64503 q^{25} +7.44247 q^{26} -2.90484 q^{27} +13.8466 q^{28} -4.42575 q^{29} -3.27287 q^{30} +5.08904 q^{31} -5.53551 q^{32} +11.7093 q^{33} +7.70049 q^{34} +1.83953 q^{35} +7.41504 q^{36} -7.09124 q^{37} +2.54651 q^{38} -6.30460 q^{39} -3.76980 q^{40} +0.679209 q^{41} -16.9605 q^{42} +8.67076 q^{43} +24.3433 q^{44} +0.985092 q^{45} -17.5773 q^{46} -1.38156 q^{47} +15.4091 q^{48} +2.53272 q^{49} +11.8286 q^{50} -6.52317 q^{51} -13.1071 q^{52} +8.41072 q^{53} +7.39720 q^{54} +3.23402 q^{55} -19.5357 q^{56} -2.15718 q^{57} +11.2702 q^{58} +11.3963 q^{59} +5.76391 q^{60} +11.6490 q^{61} -12.9593 q^{62} +5.10491 q^{63} -0.190124 q^{64} -1.74128 q^{65} -29.8178 q^{66} +4.79925 q^{67} -13.5615 q^{68} +14.8900 q^{69} -4.68437 q^{70} -8.52052 q^{71} -10.4616 q^{72} +3.95527 q^{73} +18.0579 q^{74} -10.0201 q^{75} -4.48471 q^{76} +16.7592 q^{77} +16.0547 q^{78} +13.2687 q^{79} +4.25588 q^{80} -11.2265 q^{81} -1.72961 q^{82} -0.341171 q^{83} +29.8695 q^{84} -1.80165 q^{85} -22.0802 q^{86} -9.54711 q^{87} -34.3452 q^{88} -5.13429 q^{89} -2.50855 q^{90} -9.02362 q^{91} +30.9558 q^{92} +10.9779 q^{93} +3.51814 q^{94} -0.595796 q^{95} -11.9411 q^{96} -13.4076 q^{97} -6.44960 q^{98} +8.97480 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 83 q + 11 q^{2} + 95 q^{4} + 15 q^{5} + 23 q^{6} + 19 q^{7} + 30 q^{8} + 101 q^{9}+O(q^{10}) \) 83 * q + 11 * q^2 + 95 * q^4 + 15 * q^5 + 23 * q^6 + 19 * q^7 + 30 * q^8 + 101 * q^9	\( 83 q + 11 q^{2} + 95 q^{4} + 15 q^{5} + 23 q^{6} + 19 q^{7} + 30 q^{8} + 101 q^{9} + 9 q^{10} + 56 q^{11} - 2 q^{12} - 5 q^{13} + 6 q^{14} + 19 q^{15} + 123 q^{16} + 19 q^{17} + 40 q^{18} - 83 q^{19} + 49 q^{20} + 9 q^{21} + 18 q^{22} + 74 q^{23} + 38 q^{24} + 98 q^{25} + 28 q^{26} + 6 q^{27} + 50 q^{28} + 16 q^{29} + 56 q^{30} + 24 q^{31} + 81 q^{32} + 13 q^{33} + 9 q^{34} + 71 q^{35} + 156 q^{36} - 6 q^{37} - 11 q^{38} + 126 q^{39} + q^{40} - q^{42} + 34 q^{43} + 140 q^{44} + 42 q^{45} + 34 q^{46} + 53 q^{47} + 16 q^{48} + 118 q^{49} + 51 q^{50} + 57 q^{51} + 32 q^{52} + q^{53} + 53 q^{54} + 60 q^{55} - 2 q^{56} - 2 q^{58} + 44 q^{59} - 9 q^{60} + 21 q^{61} + 28 q^{62} + 83 q^{63} + 154 q^{64} + 44 q^{65} + 17 q^{66} + 5 q^{67} + 63 q^{68} - 36 q^{69} - 48 q^{70} + 193 q^{71} + 135 q^{72} + 54 q^{73} + 127 q^{74} + 5 q^{75} - 95 q^{76} + 54 q^{77} + 45 q^{78} + 54 q^{79} + 45 q^{80} + 147 q^{81} - 35 q^{82} + 84 q^{83} + 12 q^{84} + 28 q^{85} + 60 q^{86} + 51 q^{87} + 23 q^{88} - 24 q^{89} + 31 q^{90} + 28 q^{91} + 108 q^{92} + 39 q^{93} - 49 q^{94} - 15 q^{95} + 25 q^{96} - 22 q^{97} - 67 q^{98} + 132 q^{99}+O(q^{100}) \) 83 * q + 11 * q^2 + 95 * q^4 + 15 * q^5 + 23 * q^6 + 19 * q^7 + 30 * q^8 + 101 * q^9 + 9 * q^10 + 56 * q^11 - 2 * q^12 - 5 * q^13 + 6 * q^14 + 19 * q^15 + 123 * q^16 + 19 * q^17 + 40 * q^18 - 83 * q^19 + 49 * q^20 + 9 * q^21 + 18 * q^22 + 74 * q^23 + 38 * q^24 + 98 * q^25 + 28 * q^26 + 6 * q^27 + 50 * q^28 + 16 * q^29 + 56 * q^30 + 24 * q^31 + 81 * q^32 + 13 * q^33 + 9 * q^34 + 71 * q^35 + 156 * q^36 - 6 * q^37 - 11 * q^38 + 126 * q^39 + q^40 - q^42 + 34 * q^43 + 140 * q^44 + 42 * q^45 + 34 * q^46 + 53 * q^47 + 16 * q^48 + 118 * q^49 + 51 * q^50 + 57 * q^51 + 32 * q^52 + q^53 + 53 * q^54 + 60 * q^55 - 2 * q^56 - 2 * q^58 + 44 * q^59 - 9 * q^60 + 21 * q^61 + 28 * q^62 + 83 * q^63 + 154 * q^64 + 44 * q^65 + 17 * q^66 + 5 * q^67 + 63 * q^68 - 36 * q^69 - 48 * q^70 + 193 * q^71 + 135 * q^72 + 54 * q^73 + 127 * q^74 + 5 * q^75 - 95 * q^76 + 54 * q^77 + 45 * q^78 + 54 * q^79 + 45 * q^80 + 147 * q^81 - 35 * q^82 + 84 * q^83 + 12 * q^84 + 28 * q^85 + 60 * q^86 + 51 * q^87 + 23 * q^88 - 24 * q^89 + 31 * q^90 + 28 * q^91 + 108 * q^92 + 39 * q^93 - 49 * q^94 - 15 * q^95 + 25 * q^96 - 22 * q^97 - 67 * q^98 + 132 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more