LMFDB - Embedded newform 4006.2.a.h.1.16

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4006,2,Mod(1,4006)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4006, 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("4006.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4006 = 2 \cdot 2003 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4006.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:	$31.9880710497$
Analytic rank:	$0$
Dimension:	$42$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.16
Character	$\chi$	$=$	4006.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} -0.963522 q^{3} +1.00000 q^{4} +3.93503 q^{5} +0.963522 q^{6} -4.13063 q^{7} -1.00000 q^{8} -2.07162 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} -0.963522 q^{3} +1.00000 q^{4} +3.93503 q^{5} +0.963522 q^{6} -4.13063 q^{7} -1.00000 q^{8} -2.07162 q^{9} -3.93503 q^{10} -4.05582 q^{11} -0.963522 q^{12} -2.35505 q^{13} +4.13063 q^{14} -3.79149 q^{15} +1.00000 q^{16} +6.68608 q^{17} +2.07162 q^{18} -6.89261 q^{19} +3.93503 q^{20} +3.97996 q^{21} +4.05582 q^{22} -3.83032 q^{23} +0.963522 q^{24} +10.4844 q^{25} +2.35505 q^{26} +4.88662 q^{27} -4.13063 q^{28} -9.84209 q^{29} +3.79149 q^{30} +5.64450 q^{31} -1.00000 q^{32} +3.90788 q^{33} -6.68608 q^{34} -16.2541 q^{35} -2.07162 q^{36} -4.92565 q^{37} +6.89261 q^{38} +2.26914 q^{39} -3.93503 q^{40} +1.40661 q^{41} -3.97996 q^{42} -0.448274 q^{43} -4.05582 q^{44} -8.15190 q^{45} +3.83032 q^{46} -6.67505 q^{47} -0.963522 q^{48} +10.0621 q^{49} -10.4844 q^{50} -6.44219 q^{51} -2.35505 q^{52} +9.71273 q^{53} -4.88662 q^{54} -15.9598 q^{55} +4.13063 q^{56} +6.64118 q^{57} +9.84209 q^{58} +11.5727 q^{59} -3.79149 q^{60} -3.31787 q^{61} -5.64450 q^{62} +8.55712 q^{63} +1.00000 q^{64} -9.26718 q^{65} -3.90788 q^{66} +7.22963 q^{67} +6.68608 q^{68} +3.69060 q^{69} +16.2541 q^{70} +12.0779 q^{71} +2.07162 q^{72} +6.64791 q^{73} +4.92565 q^{74} -10.1020 q^{75} -6.89261 q^{76} +16.7531 q^{77} -2.26914 q^{78} +0.751208 q^{79} +3.93503 q^{80} +1.50650 q^{81} -1.40661 q^{82} +10.3804 q^{83} +3.97996 q^{84} +26.3099 q^{85} +0.448274 q^{86} +9.48308 q^{87} +4.05582 q^{88} +7.47305 q^{89} +8.15190 q^{90} +9.72784 q^{91} -3.83032 q^{92} -5.43860 q^{93} +6.67505 q^{94} -27.1226 q^{95} +0.963522 q^{96} +2.66017 q^{97} -10.0621 q^{98} +8.40214 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 42 q - 42 q^{2} + 42 q^{4} + 27 q^{5} - 10 q^{7} - 42 q^{8} + 56 q^{9}+O(q^{10}) $ 42 * q - 42 * q^2 + 42 * q^4 + 27 * q^5 - 10 * q^7 - 42 * q^8 + 56 * q^9	\( 42 q - 42 q^{2} + 42 q^{4} + 27 q^{5} - 10 q^{7} - 42 q^{8} + 56 q^{9} - 27 q^{10} + 23 q^{11} + 15 q^{13} + 10 q^{14} + 6 q^{15} + 42 q^{16} + 14 q^{17} - 56 q^{18} - 4 q^{19} + 27 q^{20} + 26 q^{21} - 23 q^{22} + 12 q^{23} + 45 q^{25} - 15 q^{26} - 3 q^{27} - 10 q^{28} + 41 q^{29} - 6 q^{30} + 18 q^{31} - 42 q^{32} + 25 q^{33} - 14 q^{34} + 8 q^{35} + 56 q^{36} + 33 q^{37} + 4 q^{38} + 10 q^{39} - 27 q^{40} + 84 q^{41} - 26 q^{42} - 36 q^{43} + 23 q^{44} + 66 q^{45} - 12 q^{46} + 28 q^{47} + 58 q^{49} - 45 q^{50} + 17 q^{51} + 15 q^{52} + 68 q^{53} + 3 q^{54} - 28 q^{55} + 10 q^{56} - 9 q^{57} - 41 q^{58} + 59 q^{59} + 6 q^{60} + 41 q^{61} - 18 q^{62} - 28 q^{63} + 42 q^{64} + 44 q^{65} - 25 q^{66} + 14 q^{68} + 67 q^{69} - 8 q^{70} + 69 q^{71} - 56 q^{72} - 27 q^{73} - 33 q^{74} + 14 q^{75} - 4 q^{76} + 43 q^{77} - 10 q^{78} - 19 q^{79} + 27 q^{80} + 74 q^{81} - 84 q^{82} + 20 q^{83} + 26 q^{84} + 16 q^{85} + 36 q^{86} - 28 q^{87} - 23 q^{88} + 123 q^{89} - 66 q^{90} - 9 q^{91} + 12 q^{92} + 48 q^{93} - 28 q^{94} + 28 q^{95} + 10 q^{97} - 58 q^{98} + 75 q^{99}+O(q^{100}) \) 42 * q - 42 * q^2 + 42 * q^4 + 27 * q^5 - 10 * q^7 - 42 * q^8 + 56 * q^9 - 27 * q^10 + 23 * q^11 + 15 * q^13 + 10 * q^14 + 6 * q^15 + 42 * q^16 + 14 * q^17 - 56 * q^18 - 4 * q^19 + 27 * q^20 + 26 * q^21 - 23 * q^22 + 12 * q^23 + 45 * q^25 - 15 * q^26 - 3 * q^27 - 10 * q^28 + 41 * q^29 - 6 * q^30 + 18 * q^31 - 42 * q^32 + 25 * q^33 - 14 * q^34 + 8 * q^35 + 56 * q^36 + 33 * q^37 + 4 * q^38 + 10 * q^39 - 27 * q^40 + 84 * q^41 - 26 * q^42 - 36 * q^43 + 23 * q^44 + 66 * q^45 - 12 * q^46 + 28 * q^47 + 58 * q^49 - 45 * q^50 + 17 * q^51 + 15 * q^52 + 68 * q^53 + 3 * q^54 - 28 * q^55 + 10 * q^56 - 9 * q^57 - 41 * q^58 + 59 * q^59 + 6 * q^60 + 41 * q^61 - 18 * q^62 - 28 * q^63 + 42 * q^64 + 44 * q^65 - 25 * q^66 + 14 * q^68 + 67 * q^69 - 8 * q^70 + 69 * q^71 - 56 * q^72 - 27 * q^73 - 33 * q^74 + 14 * q^75 - 4 * q^76 + 43 * q^77 - 10 * q^78 - 19 * q^79 + 27 * q^80 + 74 * q^81 - 84 * q^82 + 20 * q^83 + 26 * q^84 + 16 * q^85 + 36 * q^86 - 28 * q^87 - 23 * q^88 + 123 * q^89 - 66 * q^90 - 9 * q^91 + 12 * q^92 + 48 * q^93 - 28 * q^94 + 28 * q^95 + 10 * q^97 - 58 * q^98 + 75 * 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 $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	−0.963522	−0.556290	−0.278145	−	0.960539i	$-0.589720\pi$
$3$	−0.963522	−0.556290	−0.278145	+	0.960539i	$0.589720\pi$
$4$	1.00000	0.500000
$5$	3.93503	1.75980	0.879899	−	0.475161i	$-0.157610\pi$
$5$	3.93503	1.75980	0.879899	+	0.475161i	$0.157610\pi$
$6$	0.963522	0.393356
$7$	−4.13063	−1.56123	−0.780616	−	0.625011i	$-0.785094\pi$
$7$	−4.13063	−1.56123	−0.780616	+	0.625011i	$0.785094\pi$
$8$	−1.00000	−0.353553
$9$	−2.07162	−0.690542
$10$	−3.93503	−1.24436
$11$	−4.05582	−1.22288	−0.611438	−	0.791292i	$-0.709409\pi$
$11$	−4.05582	−1.22288	−0.611438	+	0.791292i	$0.709409\pi$
$12$	−0.963522	−0.278145
$13$	−2.35505	−0.653173	−0.326587	−	0.945167i	$-0.605898\pi$
$13$	−2.35505	−0.653173	−0.326587	+	0.945167i	$0.605898\pi$
$14$	4.13063	1.10396
$15$	−3.79149	−0.978958
$16$	1.00000	0.250000
$17$	6.68608	1.62161	0.810807	−	0.585314i	$-0.199029\pi$
$17$	6.68608	1.62161	0.810807	+	0.585314i	$0.199029\pi$
$18$	2.07162	0.488287
$19$	−6.89261	−1.58127	−0.790636	−	0.612286i	$-0.790250\pi$
$19$	−6.89261	−1.58127	−0.790636	+	0.612286i	$0.790250\pi$
$20$	3.93503	0.879899
$21$	3.97996	0.868497
$22$	4.05582	0.864704
$23$	−3.83032	−0.798677	−0.399338	−	0.916804i	$-0.630760\pi$
$23$	−3.83032	−0.798677	−0.399338	+	0.916804i	$0.630760\pi$
$24$	0.963522	0.196678
$25$	10.4844	2.09689
$26$	2.35505	0.461863
$27$	4.88662	0.940431
$28$	−4.13063	−0.780616
$29$	−9.84209	−1.82763	−0.913815	−	0.406130i	$-0.866878\pi$
$29$	−9.84209	−1.82763	−0.913815	+	0.406130i	$0.866878\pi$
$30$	3.79149	0.692228
$31$	5.64450	1.01378	0.506891	−	0.862010i	$-0.330795\pi$
$31$	5.64450	1.01378	0.506891	+	0.862010i	$0.330795\pi$
$32$	−1.00000	−0.176777
$33$	3.90788	0.680274
$34$	−6.68608	−1.14665
$35$	−16.2541	−2.74745
$36$	−2.07162	−0.345271
$37$	−4.92565	−0.809771	−0.404886	−	0.914367i	$-0.632689\pi$
$37$	−4.92565	−0.809771	−0.404886	+	0.914367i	$0.632689\pi$
$38$	6.89261	1.11813
$39$	2.26914	0.363354
$40$	−3.93503	−0.622182
$41$	1.40661	0.219676	0.109838	−	0.993950i	$-0.464967\pi$
$41$	1.40661	0.219676	0.109838	+	0.993950i	$0.464967\pi$
$42$	−3.97996	−0.614120
$43$	−0.448274	−0.0683611	−0.0341806	−	0.999416i	$-0.510882\pi$
$43$	−0.448274	−0.0683611	−0.0341806	+	0.999416i	$0.510882\pi$
$44$	−4.05582	−0.611438
$45$	−8.15190	−1.21521
$46$	3.83032	0.564750
$47$	−6.67505	−0.973656	−0.486828	−	0.873498i	$-0.661846\pi$
$47$	−6.67505	−0.973656	−0.486828	+	0.873498i	$0.661846\pi$
$48$	−0.963522	−0.139072
$49$	10.0621	1.43744
$50$	−10.4844	−1.48272
$51$	−6.44219	−0.902087
$52$	−2.35505	−0.326587
$53$	9.71273	1.33415	0.667073	−	0.744992i	$-0.267547\pi$
$53$	9.71273	1.33415	0.667073	+	0.744992i	$0.267547\pi$
$54$	−4.88662	−0.664985
$55$	−15.9598	−2.15202
$56$	4.13063	0.551979
$57$	6.64118	0.879646
$58$	9.84209	1.29233
$59$	11.5727	1.50664	0.753321	−	0.657653i	$-0.228451\pi$
$59$	11.5727	1.50664	0.753321	+	0.657653i	$0.228451\pi$
$60$	−3.79149	−0.489479
$61$	−3.31787	−0.424810	−0.212405	−	0.977182i	$-0.568130\pi$
$61$	−3.31787	−0.424810	−0.212405	+	0.977182i	$0.568130\pi$
$62$	−5.64450	−0.716852
$63$	8.55712	1.07810
$64$	1.00000	0.125000
$65$	−9.26718	−1.14945
$66$	−3.90788	−0.481026
$67$	7.22963	0.883240	0.441620	−	0.897202i	$-0.354404\pi$
$67$	7.22963	0.883240	0.441620	+	0.897202i	$0.354404\pi$
$68$	6.68608	0.810807
$69$	3.69060	0.444296
$70$	16.2541	1.94274
$71$	12.0779	1.43338	0.716692	−	0.697390i	$-0.245655\pi$
$71$	12.0779	1.43338	0.716692	+	0.697390i	$0.245655\pi$
$72$	2.07162	0.244143
$73$	6.64791	0.778079	0.389039	−	0.921221i	$-0.372807\pi$
$73$	6.64791	0.778079	0.389039	+	0.921221i	$0.372807\pi$
$74$	4.92565	0.572595
$75$	−10.1020	−1.16648
$76$	−6.89261	−0.790636
$77$	16.7531	1.90919
$78$	−2.26914	−0.256930
$79$	0.751208	0.0845175	0.0422587	−	0.999107i	$-0.486545\pi$
$79$	0.751208	0.0845175	0.0422587	+	0.999107i	$0.486545\pi$
$80$	3.93503	0.439949
$81$	1.50650	0.167389
$82$	−1.40661	−0.155334
$83$	10.3804	1.13939	0.569697	−	0.821855i	$-0.307061\pi$
$83$	10.3804	1.13939	0.569697	+	0.821855i	$0.307061\pi$
$84$	3.97996	0.434249
$85$	26.3099	2.85371
$86$	0.448274	0.0483386
$87$	9.48308	1.01669
$88$	4.05582	0.432352
$89$	7.47305	0.792142	0.396071	−	0.918220i	$-0.370373\pi$
$89$	7.47305	0.792142	0.396071	+	0.918220i	$0.370373\pi$
$90$	8.15190	0.859286
$91$	9.72784	1.01975
$92$	−3.83032	−0.399338
$93$	−5.43860	−0.563957
$94$	6.67505	0.688478
$95$	−27.1226	−2.78272
$96$	0.963522	0.0983391
$97$	2.66017	0.270099	0.135050	−	0.990839i	$-0.456881\pi$
$97$	2.66017	0.270099	0.135050	+	0.990839i	$0.456881\pi$
$98$	−10.0621	−1.01643
$99$	8.40214	0.844447
$100$	10.4844	1.04844
$101$	−2.68922	−0.267587	−0.133793	−	0.991009i	$-0.542716\pi$
$101$	−2.68922	−0.267587	−0.133793	+	0.991009i	$0.542716\pi$
$102$	6.44219	0.637872
$103$	9.53875	0.939881	0.469940	−	0.882698i	$-0.344275\pi$
$103$	9.53875	0.939881	0.469940	+	0.882698i	$0.344275\pi$
$104$	2.35505	0.230932
$105$	15.6612	1.52838
$106$	−9.71273	−0.943384
$107$	−15.3258	−1.48160	−0.740799	−	0.671727i	$-0.765553\pi$
$107$	−15.3258	−1.48160	−0.740799	+	0.671727i	$0.765553\pi$
$108$	4.88662	0.470216
$109$	13.3648	1.28012	0.640060	−	0.768325i	$-0.278910\pi$
$109$	13.3648	1.28012	0.640060	+	0.768325i	$0.278910\pi$
$110$	15.9598	1.52170
$111$	4.74597	0.450468
$112$	−4.13063	−0.390308
$113$	−19.9580	−1.87749	−0.938746	−	0.344610i	$-0.888011\pi$
$113$	−19.9580	−1.87749	−0.938746	+	0.344610i	$0.888011\pi$
$114$	−6.64118	−0.622004
$115$	−15.0724	−1.40551
$116$	−9.84209	−0.913815
$117$	4.87878	0.451043
$118$	−11.5727	−1.06536
$119$	−27.6177	−2.53171
$120$	3.79149	0.346114
$121$	5.44970	0.495427
$122$	3.31787	0.300386
$123$	−1.35530	−0.122203
$124$	5.64450	0.506891
$125$	21.5814	1.93030
$126$	−8.55712	−0.762328
$127$	−1.76242	−0.156389	−0.0781947	−	0.996938i	$-0.524916\pi$
$127$	−1.76242	−0.156389	−0.0781947	+	0.996938i	$0.524916\pi$
$128$	−1.00000	−0.0883883
$129$	0.431922	0.0380286
$130$	9.26718	0.812786
$131$	−12.6757	−1.10748	−0.553741	−	0.832689i	$-0.686800\pi$
$131$	−12.6757	−1.10748	−0.553741	+	0.832689i	$0.686800\pi$
$132$	3.90788	0.340137
$133$	28.4708	2.46873
$134$	−7.22963	−0.624545
$135$	19.2290	1.65497
$136$	−6.68608	−0.573327
$137$	−3.56996	−0.305002	−0.152501	−	0.988303i	$-0.548733\pi$
$137$	−3.56996	−0.305002	−0.152501	+	0.988303i	$0.548733\pi$
$138$	−3.69060	−0.314165
$139$	9.55379	0.810342	0.405171	−	0.914241i	$-0.367212\pi$
$139$	9.55379	0.810342	0.405171	+	0.914241i	$0.367212\pi$
$140$	−16.2541	−1.37373
$141$	6.43156	0.541635
$142$	−12.0779	−1.01356
$143$	9.55166	0.798750
$144$	−2.07162	−0.172635
$145$	−38.7289	−3.21626
$146$	−6.64791	−0.550185
$147$	−9.69507	−0.799636
$148$	−4.92565	−0.404886
$149$	18.8960	1.54802	0.774008	−	0.633175i	$-0.218249\pi$
$149$	18.8960	1.54802	0.774008	+	0.633175i	$0.218249\pi$
$150$	10.1020	0.824824
$151$	−3.78813	−0.308274	−0.154137	−	0.988050i	$-0.549260\pi$
$151$	−3.78813	−0.308274	−0.154137	+	0.988050i	$0.549260\pi$
$152$	6.89261	0.559064
$153$	−13.8511	−1.11979
$154$	−16.7531	−1.35000
$155$	22.2113	1.78405
$156$	2.26914	0.181677
$157$	19.1044	1.52470	0.762350	−	0.647165i	$-0.224046\pi$
$157$	19.1044	1.52470	0.762350	+	0.647165i	$0.224046\pi$
$158$	−0.751208	−0.0597629
$159$	−9.35843	−0.742172
$160$	−3.93503	−0.311091
$161$	15.8216	1.24692
$162$	−1.50650	−0.118362
$163$	−9.82853	−0.769830	−0.384915	−	0.922952i	$-0.625769\pi$
$163$	−9.82853	−0.769830	−0.384915	+	0.922952i	$0.625769\pi$
$164$	1.40661	0.109838
$165$	15.3776	1.19714
$166$	−10.3804	−0.805673
$167$	−18.8840	−1.46129	−0.730646	−	0.682757i	$-0.760781\pi$
$167$	−18.8840	−1.46129	−0.730646	+	0.682757i	$0.760781\pi$
$168$	−3.97996	−0.307060
$169$	−7.45374	−0.573365
$170$	−26.3099	−2.01788
$171$	14.2789	1.09193
$172$	−0.448274	−0.0341806
$173$	4.96019	0.377116	0.188558	−	0.982062i	$-0.439619\pi$
$173$	4.96019	0.377116	0.188558	+	0.982062i	$0.439619\pi$
$174$	−9.48308	−0.718910
$175$	−43.3073	−3.27373
$176$	−4.05582	−0.305719
$177$	−11.1506	−0.838130
$178$	−7.47305	−0.560129
$179$	−7.90431	−0.590796	−0.295398	−	0.955374i	$-0.595452\pi$
$179$	−7.90431	−0.590796	−0.295398	+	0.955374i	$0.595452\pi$
$180$	−8.15190	−0.607607
$181$	16.7982	1.24860	0.624300	−	0.781184i	$-0.285384\pi$
$181$	16.7982	1.24860	0.624300	+	0.781184i	$0.285384\pi$
$182$	−9.72784	−0.721075
$183$	3.19684	0.236317
$184$	3.83032	0.282375
$185$	−19.3825	−1.42503
$186$	5.43860	0.398778
$187$	−27.1176	−1.98303
$188$	−6.67505	−0.486828
$189$	−20.1848	−1.46823
$190$	27.1226	1.96768
$191$	−17.3989	−1.25894	−0.629469	−	0.777025i	$-0.716728\pi$
$191$	−17.3989	−1.25894	−0.629469	+	0.777025i	$0.716728\pi$
$192$	−0.963522	−0.0695362
$193$	21.7252	1.56382	0.781908	−	0.623394i	$-0.214247\pi$
$193$	21.7252	1.56382	0.781908	+	0.623394i	$0.214247\pi$
$194$	−2.66017	−0.190989
$195$	8.92914	0.639429
$196$	10.0621	0.718722
$197$	4.59268	0.327215	0.163608	−	0.986526i	$-0.447687\pi$
$197$	4.59268	0.327215	0.163608	+	0.986526i	$0.447687\pi$
$198$	−8.40214	−0.597114
$199$	22.0584	1.56368	0.781840	−	0.623479i	$-0.214281\pi$
$199$	22.0584	1.56368	0.781840	+	0.623479i	$0.214281\pi$
$200$	−10.4844	−0.741362
$201$	−6.96591	−0.491338
$202$	2.68922	0.189213
$203$	40.6541	2.85335
$204$	−6.44219	−0.451044
$205$	5.53505	0.386585
$206$	−9.53875	−0.664596
$207$	7.93498	0.551520
$208$	−2.35505	−0.163293
$209$	27.9552	1.93370
$210$	−15.6612	−1.08073
$211$	18.9004	1.30116	0.650579	−	0.759439i	$-0.274526\pi$
$211$	18.9004	1.30116	0.650579	+	0.759439i	$0.274526\pi$
$212$	9.71273	0.667073
$213$	−11.6373	−0.797377
$214$	15.3258	1.04765
$215$	−1.76397	−0.120302
$216$	−4.88662	−0.332493
$217$	−23.3153	−1.58275
$218$	−13.3648	−0.905181
$219$	−6.40541	−0.432837
$220$	−15.9598	−1.07601
$221$	−15.7461	−1.05919
$222$	−4.74597	−0.318529
$223$	−0.907796	−0.0607905	−0.0303953	−	0.999538i	$-0.509677\pi$
$223$	−0.907796	−0.0607905	−0.0303953	+	0.999538i	$0.509677\pi$
$224$	4.13063	0.275989
$225$	−21.7198	−1.44799
$226$	19.9580	1.32759
$227$	−13.6652	−0.906994	−0.453497	−	0.891258i	$-0.649824\pi$
$227$	−13.6652	−0.906994	−0.453497	+	0.891258i	$0.649824\pi$
$228$	6.64118	0.439823
$229$	−3.14278	−0.207681	−0.103840	−	0.994594i	$-0.533113\pi$
$229$	−3.14278	−0.207681	−0.103840	+	0.994594i	$0.533113\pi$
$230$	15.0724	0.993845
$231$	−16.1420	−1.06207
$232$	9.84209	0.646165
$233$	−2.38493	−0.156242	−0.0781211	−	0.996944i	$-0.524892\pi$
$233$	−2.38493	−0.156242	−0.0781211	+	0.996944i	$0.524892\pi$
$234$	−4.87878	−0.318936
$235$	−26.2665	−1.71344
$236$	11.5727	0.753321
$237$	−0.723806	−0.0470162
$238$	27.6177	1.79019
$239$	15.0769	0.975246	0.487623	−	0.873054i	$-0.337864\pi$
$239$	15.0769	0.975246	0.487623	+	0.873054i	$0.337864\pi$
$240$	−3.79149	−0.244739
$241$	−25.0351	−1.61266	−0.806328	−	0.591469i	$-0.798548\pi$
$241$	−25.0351	−1.61266	−0.806328	+	0.591469i	$0.798548\pi$
$242$	−5.44970	−0.350320
$243$	−16.1114	−1.03355
$244$	−3.31787	−0.212405
$245$	39.5947	2.52961
$246$	1.35530	0.0864108
$247$	16.2324	1.03285
$248$	−5.64450	−0.358426
$249$	−10.0017	−0.633833
$250$	−21.5814	−1.36493
$251$	24.9709	1.57615	0.788076	−	0.615578i	$-0.211077\pi$
$251$	24.9709	1.57615	0.788076	+	0.615578i	$0.211077\pi$
$252$	8.55712	0.539048
$253$	15.5351	0.976683
$254$	1.76242	0.110584
$255$	−25.3502	−1.58749
$256$	1.00000	0.0625000
$257$	−7.65976	−0.477803	−0.238901	−	0.971044i	$-0.576787\pi$
$257$	−7.65976	−0.477803	−0.238901	+	0.971044i	$0.576787\pi$
$258$	−0.431922	−0.0268903
$259$	20.3460	1.26424
$260$	−9.26718	−0.574726
$261$	20.3891	1.26205
$262$	12.6757	0.783107
$263$	24.7689	1.52731	0.763657	−	0.645623i	$-0.223402\pi$
$263$	24.7689	1.52731	0.763657	+	0.645623i	$0.223402\pi$
$264$	−3.90788	−0.240513
$265$	38.2199	2.34783
$266$	−28.4708	−1.74566
$267$	−7.20045	−0.440661
$268$	7.22963	0.441620
$269$	−2.22314	−0.135547	−0.0677736	−	0.997701i	$-0.521590\pi$
$269$	−2.22314	−0.135547	−0.0677736	+	0.997701i	$0.521590\pi$
$270$	−19.2290	−1.17024
$271$	−28.4759	−1.72979	−0.864895	−	0.501953i	$-0.832615\pi$
$271$	−28.4759	−1.72979	−0.864895	+	0.501953i	$0.832615\pi$
$272$	6.68608	0.405403
$273$	−9.37299	−0.567279
$274$	3.56996	0.215669
$275$	−42.5230	−2.56423
$276$	3.69060	0.222148
$277$	15.4471	0.928124	0.464062	−	0.885803i	$-0.346391\pi$
$277$	15.4471	0.928124	0.464062	+	0.885803i	$0.346391\pi$
$278$	−9.55379	−0.572998
$279$	−11.6933	−0.700059
$280$	16.2541	0.971371
$281$	−8.83749	−0.527200	−0.263600	−	0.964632i	$-0.584910\pi$
$281$	−8.83749	−0.527200	−0.263600	+	0.964632i	$0.584910\pi$
$282$	−6.43156	−0.382994
$283$	27.4740	1.63316	0.816580	−	0.577233i	$-0.195867\pi$
$283$	27.4740	1.63316	0.816580	+	0.577233i	$0.195867\pi$
$284$	12.0779	0.716692
$285$	26.1332	1.54800
$286$	−9.55166	−0.564802
$287$	−5.81019	−0.342965
$288$	2.07162	0.122072
$289$	27.7037	1.62963
$290$	38.7289	2.27424
$291$	−2.56313	−0.150254
$292$	6.64791	0.389039
$293$	6.58607	0.384762	0.192381	−	0.981320i	$-0.438379\pi$
$293$	6.58607	0.384762	0.192381	+	0.981320i	$0.438379\pi$
$294$	9.69507	0.565428
$295$	45.5390	2.65138
$296$	4.92565	0.286297
$297$	−19.8193	−1.15003
$298$	−18.8960	−1.09461
$299$	9.02059	0.521674
$300$	−10.1020	−0.583239
$301$	1.85165	0.106728
$302$	3.78813	0.217982
$303$	2.59112	0.148856
$304$	−6.89261	−0.395318
$305$	−13.0559	−0.747579
$306$	13.8511	0.791812
$307$	−7.62885	−0.435402	−0.217701	−	0.976016i	$-0.569856\pi$
$307$	−7.62885	−0.435402	−0.217701	+	0.976016i	$0.569856\pi$
$308$	16.7531	0.954597
$309$	−9.19080	−0.522846
$310$	−22.2113	−1.26151
$311$	−8.31502	−0.471502	−0.235751	−	0.971814i	$-0.575755\pi$
$311$	−8.31502	−0.471502	−0.235751	+	0.971814i	$0.575755\pi$
$312$	−2.26914	−0.128465
$313$	2.90961	0.164461	0.0822303	−	0.996613i	$-0.473796\pi$
$313$	2.90961	0.164461	0.0822303	+	0.996613i	$0.473796\pi$
$314$	−19.1044	−1.07813
$315$	33.6725	1.89723
$316$	0.751208	0.0422587
$317$	−21.6112	−1.21381	−0.606904	−	0.794776i	$-0.707589\pi$
$317$	−21.6112	−1.21381	−0.606904	+	0.794776i	$0.707589\pi$
$318$	9.35843	0.524795
$319$	39.9178	2.23497
$320$	3.93503	0.219975
$321$	14.7667	0.824198
$322$	−15.8216	−0.881705
$323$	−46.0845	−2.56421
$324$	1.50650	0.0836945
$325$	−24.6914	−1.36963
$326$	9.82853	0.544352
$327$	−12.8773	−0.712118
$328$	−1.40661	−0.0776671
$329$	27.5721	1.52010
$330$	−15.3776	−0.846509
$331$	−1.66370	−0.0914455	−0.0457227	−	0.998954i	$-0.514559\pi$
$331$	−1.66370	−0.0914455	−0.0457227	+	0.998954i	$0.514559\pi$
$332$	10.3804	0.569697
$333$	10.2041	0.559181
$334$	18.8840	1.03329
$335$	28.4488	1.55432
$336$	3.97996	0.217124
$337$	−29.5009	−1.60702	−0.803508	−	0.595294i	$-0.797036\pi$
$337$	−29.5009	−1.60702	−0.803508	+	0.595294i	$0.797036\pi$
$338$	7.45374	0.405430
$339$	19.2300	1.04443
$340$	26.3099	1.42686
$341$	−22.8931	−1.23973
$342$	−14.2789	−0.772114
$343$	−12.6484	−0.682952
$344$	0.448274	0.0241693
$345$	14.5226	0.781871
$346$	−4.96019	−0.266662
$347$	−0.432072	−0.0231948	−0.0115974	−	0.999933i	$-0.503692\pi$
$347$	−0.432072	−0.0231948	−0.0115974	+	0.999933i	$0.503692\pi$
$348$	9.48308	0.508346
$349$	−20.1616	−1.07923	−0.539614	−	0.841913i	$-0.681430\pi$
$349$	−20.1616	−1.07923	−0.539614	+	0.841913i	$0.681430\pi$
$350$	43.3073	2.31487
$351$	−11.5082	−0.614264
$352$	4.05582	0.216176
$353$	−9.61810	−0.511920	−0.255960	−	0.966687i	$-0.582391\pi$
$353$	−9.61810	−0.511920	−0.255960	+	0.966687i	$0.582391\pi$
$354$	11.1506	0.592647
$355$	47.5269	2.52247
$356$	7.47305	0.396071
$357$	26.6103	1.40837
$358$	7.90431	0.417756
$359$	8.62071	0.454984	0.227492	−	0.973780i	$-0.426948\pi$
$359$	8.62071	0.454984	0.227492	+	0.973780i	$0.426948\pi$
$360$	8.15190	0.429643
$361$	28.5081	1.50042
$362$	−16.7982	−0.882894
$363$	−5.25091	−0.275601
$364$	9.72784	0.509877
$365$	26.1597	1.36926
$366$	−3.19684	−0.167102
$367$	−15.7556	−0.822435	−0.411217	−	0.911537i	$-0.634896\pi$
$367$	−15.7556	−0.822435	−0.411217	+	0.911537i	$0.634896\pi$
$368$	−3.83032	−0.199669
$369$	−2.91397	−0.151695
$370$	19.3825	1.00765
$371$	−40.1197	−2.08291
$372$	−5.43860	−0.281978
$373$	−17.7522	−0.919174	−0.459587	−	0.888133i	$-0.652003\pi$
$373$	−17.7522	−0.919174	−0.459587	+	0.888133i	$0.652003\pi$
$374$	27.1176	1.40222
$375$	−20.7942	−1.07381
$376$	6.67505	0.344239
$377$	23.1786	1.19376
$378$	20.1848	1.03820
$379$	−19.6451	−1.00910	−0.504550	−	0.863382i	$-0.668342\pi$
$379$	−19.6451	−1.00910	−0.504550	+	0.863382i	$0.668342\pi$
$380$	−27.1226	−1.39136
$381$	1.69813	0.0869978
$382$	17.3989	0.890204
$383$	−3.12580	−0.159721	−0.0798605	−	0.996806i	$-0.525447\pi$
$383$	−3.12580	−0.159721	−0.0798605	+	0.996806i	$0.525447\pi$
$384$	0.963522	0.0491695
$385$	65.9239	3.35979
$386$	−21.7252	−1.10579
$387$	0.928655	0.0472062
$388$	2.66017	0.135050
$389$	−15.0042	−0.760745	−0.380372	−	0.924833i	$-0.624204\pi$
$389$	−15.0042	−0.760745	−0.380372	+	0.924833i	$0.624204\pi$
$390$	−8.92914	−0.452145
$391$	−25.6098	−1.29514
$392$	−10.0621	−0.508213
$393$	12.2133	0.616081
$394$	−4.59268	−0.231376
$395$	2.95602	0.148734
$396$	8.40214	0.422223
$397$	17.0532	0.855877	0.427938	−	0.903808i	$-0.359240\pi$
$397$	17.0532	0.855877	0.427938	+	0.903808i	$0.359240\pi$
$398$	−22.0584	−1.10569
$399$	−27.4323	−1.37333
$400$	10.4844	0.524222
$401$	36.5783	1.82663	0.913317	−	0.407249i	$-0.133512\pi$
$401$	36.5783	1.82663	0.913317	+	0.407249i	$0.133512\pi$
$402$	6.96591	0.347428
$403$	−13.2931	−0.662175
$404$	−2.68922	−0.133793
$405$	5.92812	0.294571
$406$	−40.6541	−2.01763
$407$	19.9775	0.990250
$408$	6.44219	0.318936
$409$	−10.6259	−0.525418	−0.262709	−	0.964875i	$-0.584616\pi$
$409$	−10.6259	−0.525418	−0.262709	+	0.964875i	$0.584616\pi$
$410$	−5.53505	−0.273357
$411$	3.43974	0.169670
$412$	9.53875	0.469940
$413$	−47.8027	−2.35222
$414$	−7.93498	−0.389983
$415$	40.8470	2.00510
$416$	2.35505	0.115466
$417$	−9.20529	−0.450785
$418$	−27.9552	−1.36733
$419$	27.4177	1.33944	0.669720	−	0.742613i	$-0.266414\pi$
$419$	27.4177	1.33944	0.669720	+	0.742613i	$0.266414\pi$
$420$	15.6612	0.764190
$421$	5.71707	0.278633	0.139316	−	0.990248i	$-0.455510\pi$
$421$	5.71707	0.278633	0.139316	+	0.990248i	$0.455510\pi$
$422$	−18.9004	−0.920057
$423$	13.8282	0.672350
$424$	−9.71273	−0.471692
$425$	70.0998	3.40034
$426$	11.6373	0.563831
$427$	13.7049	0.663226
$428$	−15.3258	−0.740799
$429$	−9.20324	−0.444337
$430$	1.76397	0.0850662
$431$	25.0136	1.20486	0.602432	−	0.798171i	$-0.294199\pi$
$431$	25.0136	1.20486	0.602432	+	0.798171i	$0.294199\pi$
$432$	4.88662	0.235108
$433$	−14.9838	−0.720077	−0.360038	−	0.932938i	$-0.617236\pi$
$433$	−14.9838	−0.720077	−0.360038	+	0.932938i	$0.617236\pi$
$434$	23.3153	1.11917
$435$	37.3162	1.78917
$436$	13.3648	0.640060
$437$	26.4009	1.26293
$438$	6.40541	0.306062
$439$	−11.9313	−0.569449	−0.284725	−	0.958609i	$-0.591902\pi$
$439$	−11.9313	−0.569449	−0.284725	+	0.958609i	$0.591902\pi$
$440$	15.9598	0.760852
$441$	−20.8449	−0.992615
$442$	15.7461	0.748963
$443$	3.75556	0.178432	0.0892161	−	0.996012i	$-0.471564\pi$
$443$	3.75556	0.178432	0.0892161	+	0.996012i	$0.471564\pi$
$444$	4.74597	0.225234
$445$	29.4067	1.39401
$446$	0.907796	0.0429854
$447$	−18.2067	−0.861146
$448$	−4.13063	−0.195154
$449$	33.3477	1.57378	0.786888	−	0.617096i	$-0.211691\pi$
$449$	33.3477	1.57378	0.786888	+	0.617096i	$0.211691\pi$
$450$	21.7198	1.02388
$451$	−5.70496	−0.268636
$452$	−19.9580	−0.938746
$453$	3.64995	0.171489
$454$	13.6652	0.641342
$455$	38.2793	1.79456
$456$	−6.64118	−0.311002
$457$	−23.9744	−1.12148	−0.560738	−	0.827993i	$-0.689483\pi$
$457$	−23.9744	−1.12148	−0.560738	+	0.827993i	$0.689483\pi$
$458$	3.14278	0.146852
$459$	32.6724	1.52502
$460$	−15.0724	−0.702755
$461$	34.2062	1.59314	0.796571	−	0.604546i	$-0.206645\pi$
$461$	34.2062	1.59314	0.796571	+	0.604546i	$0.206645\pi$
$462$	16.1420	0.750993
$463$	19.8775	0.923785	0.461893	−	0.886936i	$-0.347171\pi$
$463$	19.8775	0.923785	0.461893	+	0.886936i	$0.347171\pi$
$464$	−9.84209	−0.456908
$465$	−21.4010	−0.992450
$466$	2.38493	0.110480
$467$	9.69288	0.448533	0.224267	−	0.974528i	$-0.428001\pi$
$467$	9.69288	0.448533	0.224267	+	0.974528i	$0.428001\pi$
$468$	4.87878	0.225522
$469$	−29.8629	−1.37894
$470$	26.2665	1.21158
$471$	−18.4075	−0.848175
$472$	−11.5727	−0.532678
$473$	1.81812	0.0835972
$474$	0.723806	0.0332455
$475$	−72.2651	−3.31575
$476$	−27.6177	−1.26586
$477$	−20.1211	−0.921283
$478$	−15.0769	−0.689603
$479$	12.4782	0.570144	0.285072	−	0.958506i	$-0.407982\pi$
$479$	12.4782	0.570144	0.285072	+	0.958506i	$0.407982\pi$
$480$	3.79149	0.173057
$481$	11.6001	0.528921
$482$	25.0351	1.14032
$483$	−15.2445	−0.693649
$484$	5.44970	0.247713
$485$	10.4678	0.475320
$486$	16.1114	0.730829
$487$	−1.99248	−0.0902880	−0.0451440	−	0.998980i	$-0.514375\pi$
$487$	−1.99248	−0.0902880	−0.0451440	+	0.998980i	$0.514375\pi$
$488$	3.31787	0.150193
$489$	9.47001	0.428249
$490$	−39.5947	−1.78871
$491$	1.61635	0.0729447	0.0364724	−	0.999335i	$-0.488388\pi$
$491$	1.61635	0.0729447	0.0364724	+	0.999335i	$0.488388\pi$
$492$	−1.35530	−0.0611017
$493$	−65.8050	−2.96371
$494$	−16.2324	−0.730332
$495$	33.0627	1.48606
$496$	5.64450	0.253446
$497$	−49.8894	−2.23784
$498$	10.0017	0.448188
$499$	15.3895	0.688931	0.344465	−	0.938799i	$-0.388060\pi$
$499$	15.3895	0.688931	0.344465	+	0.938799i	$0.388060\pi$
$500$	21.5814	0.965150
$501$	18.1952	0.812902
$502$	−24.9709	−1.11451
$503$	11.4473	0.510409	0.255204	−	0.966887i	$-0.417857\pi$
$503$	11.4473	0.510409	0.255204	+	0.966887i	$0.417857\pi$
$504$	−8.55712	−0.381164
$505$	−10.5821	−0.470899
$506$	−15.5351	−0.690619
$507$	7.18185	0.318957
$508$	−1.76242	−0.0781947
$509$	27.0142	1.19738	0.598691	−	0.800980i	$-0.295688\pi$
$509$	27.0142	1.19738	0.598691	+	0.800980i	$0.295688\pi$
$510$	25.3502	1.12253
$511$	−27.4601	−1.21476
$512$	−1.00000	−0.0441942
$513$	−33.6816	−1.48708
$514$	7.65976	0.337858
$515$	37.5352	1.65400
$516$	0.431922	0.0190143
$517$	27.0728	1.19066
$518$	−20.3460	−0.893953
$519$	−4.77926	−0.209786
$520$	9.26718	0.406393
$521$	39.1767	1.71636	0.858181	−	0.513348i	$-0.171595\pi$
$521$	39.1767	1.71636	0.858181	+	0.513348i	$0.171595\pi$
$522$	−20.3891	−0.892408
$523$	25.0242	1.09423	0.547117	−	0.837056i	$-0.315725\pi$
$523$	25.0242	1.09423	0.547117	+	0.837056i	$0.315725\pi$
$524$	−12.6757	−0.553741
$525$	41.7276	1.82114
$526$	−24.7689	−1.07997
$527$	37.7396	1.64396
$528$	3.90788	0.170068
$529$	−8.32865	−0.362115
$530$	−38.2199	−1.66016
$531$	−23.9744	−1.04040
$532$	28.4708	1.23437
$533$	−3.31264	−0.143486
$534$	7.20045	0.311594
$535$	−60.3073	−2.60731
$536$	−7.22963	−0.312273
$537$	7.61598	0.328654
$538$	2.22314	0.0958464
$539$	−40.8101	−1.75782
$540$	19.2290	0.827484
$541$	−21.6025	−0.928763	−0.464381	−	0.885635i	$-0.653723\pi$
$541$	−21.6025	−0.928763	−0.464381	+	0.885635i	$0.653723\pi$
$542$	28.4759	1.22315
$543$	−16.1855	−0.694584
$544$	−6.68608	−0.286663
$545$	52.5910	2.25275
$546$	9.37299	0.401127
$547$	0.988323	0.0422576	0.0211288	−	0.999777i	$-0.493274\pi$
$547$	0.988323	0.0422576	0.0211288	+	0.999777i	$0.493274\pi$
$548$	−3.56996	−0.152501
$549$	6.87338	0.293349
$550$	42.5230	1.81319
$551$	67.8377	2.88998
$552$	−3.69060	−0.157082
$553$	−3.10296	−0.131951
$554$	−15.4471	−0.656283
$555$	18.6755	0.792732
$556$	9.55379	0.405171
$557$	30.7856	1.30443	0.652214	−	0.758035i	$-0.273840\pi$
$557$	30.7856	1.30443	0.652214	+	0.758035i	$0.273840\pi$
$558$	11.6933	0.495016
$559$	1.05571	0.0446517
$560$	−16.2541	−0.686863
$561$	26.1284	1.10314
$562$	8.83749	0.372787
$563$	3.90728	0.164672	0.0823360	−	0.996605i	$-0.473762\pi$
$563$	3.90728	0.164672	0.0823360	+	0.996605i	$0.473762\pi$
$564$	6.43156	0.270817
$565$	−78.5353	−3.30401
$566$	−27.4740	−1.15482
$567$	−6.22280	−0.261333
$568$	−12.0779	−0.506778
$569$	−5.64684	−0.236728	−0.118364	−	0.992970i	$-0.537765\pi$
$569$	−5.64684	−0.236728	−0.118364	+	0.992970i	$0.537765\pi$
$570$	−26.1332	−1.09460
$571$	28.9161	1.21010	0.605051	−	0.796186i	$-0.293153\pi$
$571$	28.9161	1.21010	0.605051	+	0.796186i	$0.293153\pi$
$572$	9.55166	0.399375
$573$	16.7642	0.700335
$574$	5.81019	0.242513
$575$	−40.1587	−1.67474
$576$	−2.07162	−0.0863177
$577$	−36.6975	−1.52774	−0.763869	−	0.645372i	$-0.776702\pi$
$577$	−36.6975	−1.52774	−0.763869	+	0.645372i	$0.776702\pi$
$578$	−27.7037	−1.15232
$579$	−20.9327	−0.869935
$580$	−38.7289	−1.60813
$581$	−42.8775	−1.77886
$582$	2.56313	0.106245
$583$	−39.3931	−1.63150
$584$	−6.64791	−0.275092
$585$	19.1981	0.793745
$586$	−6.58607	−0.272068
$587$	−24.7399	−1.02112	−0.510562	−	0.859841i	$-0.670563\pi$
$587$	−24.7399	−1.02112	−0.510562	+	0.859841i	$0.670563\pi$
$588$	−9.69507	−0.399818
$589$	−38.9053	−1.60307
$590$	−45.5390	−1.87481
$591$	−4.42515	−0.182026
$592$	−4.92565	−0.202443
$593$	5.63477	0.231392	0.115696	−	0.993285i	$-0.463090\pi$
$593$	5.63477	0.231392	0.115696	+	0.993285i	$0.463090\pi$
$594$	19.8193	0.813195
$595$	−108.677	−4.45530
$596$	18.8960	0.774008
$597$	−21.2538	−0.869860
$598$	−9.02059	−0.368879
$599$	21.1799	0.865388	0.432694	−	0.901541i	$-0.357563\pi$
$599$	21.1799	0.865388	0.432694	+	0.901541i	$0.357563\pi$
$600$	10.1020	0.412412
$601$	33.7342	1.37605	0.688023	−	0.725688i	$-0.258479\pi$
$601$	33.7342	1.37605	0.688023	+	0.725688i	$0.258479\pi$
$602$	−1.85165	−0.0754678
$603$	−14.9771	−0.609914
$604$	−3.78813	−0.154137
$605$	21.4447	0.871851
$606$	−2.59112	−0.105257
$607$	36.9253	1.49875	0.749376	−	0.662145i	$-0.230354\pi$
$607$	36.9253	1.49875	0.749376	+	0.662145i	$0.230354\pi$
$608$	6.89261	0.279532
$609$	−39.1711	−1.58729
$610$	13.0559	0.528618
$611$	15.7201	0.635966
$612$	−13.8511	−0.559896
$613$	24.1571	0.975694	0.487847	−	0.872929i	$-0.337782\pi$
$613$	24.1571	0.975694	0.487847	+	0.872929i	$0.337782\pi$
$614$	7.62885	0.307875
$615$	−5.33315	−0.215053
$616$	−16.7531	−0.675002
$617$	3.49122	0.140551	0.0702757	−	0.997528i	$-0.477612\pi$
$617$	3.49122	0.140551	0.0702757	+	0.997528i	$0.477612\pi$
$618$	9.19080	0.369708
$619$	13.7959	0.554503	0.277252	−	0.960797i	$-0.410576\pi$
$619$	13.7959	0.554503	0.277252	+	0.960797i	$0.410576\pi$
$620$	22.2113	0.892026
$621$	−18.7173	−0.751101
$622$	8.31502	0.333402
$623$	−30.8684	−1.23672
$624$	2.26914	0.0908384
$625$	32.5012	1.30005
$626$	−2.90961	−0.116291
$627$	−26.9355	−1.07570
$628$	19.1044	0.762350
$629$	−32.9333	−1.31314
$630$	−33.6725	−1.34154
$631$	−40.1074	−1.59665	−0.798325	−	0.602227i	$-0.794280\pi$
$631$	−40.1074	−1.59665	−0.798325	+	0.602227i	$0.794280\pi$
$632$	−0.751208	−0.0298814
$633$	−18.2110	−0.723821
$634$	21.6112	0.858291
$635$	−6.93516	−0.275214
$636$	−9.35843	−0.371086
$637$	−23.6968	−0.938900
$638$	−39.9178	−1.58036
$639$	−25.0209	−0.989811
$640$	−3.93503	−0.155546
$641$	3.99931	0.157963	0.0789817	−	0.996876i	$-0.474833\pi$
$641$	3.99931	0.157963	0.0789817	+	0.996876i	$0.474833\pi$
$642$	−14.7667	−0.582796
$643$	−46.9113	−1.85000	−0.925001	−	0.379966i	$-0.875936\pi$
$643$	−46.9113	−1.85000	−0.925001	+	0.379966i	$0.875936\pi$
$644$	15.8216	0.623460
$645$	1.69962	0.0669226
$646$	46.0845	1.81317
$647$	24.5144	0.963760	0.481880	−	0.876237i	$-0.339954\pi$
$647$	24.5144	0.963760	0.481880	+	0.876237i	$0.339954\pi$
$648$	−1.50650	−0.0591810
$649$	−46.9370	−1.84244
$650$	24.6914	0.968475
$651$	22.4649	0.880467
$652$	−9.82853	−0.384915
$653$	43.6951	1.70992	0.854960	−	0.518694i	$-0.173582\pi$
$653$	43.6951	1.70992	0.854960	+	0.518694i	$0.173582\pi$
$654$	12.8773	0.503543
$655$	−49.8792	−1.94894
$656$	1.40661	0.0549189
$657$	−13.7720	−0.537296
$658$	−27.5721	−1.07487
$659$	−19.4526	−0.757765	−0.378883	−	0.925445i	$-0.623692\pi$
$659$	−19.4526	−0.757765	−0.378883	+	0.925445i	$0.623692\pi$
$660$	15.3776	0.598572
$661$	−19.3679	−0.753325	−0.376663	−	0.926351i	$-0.622928\pi$
$661$	−19.3679	−0.753325	−0.376663	+	0.926351i	$0.622928\pi$
$662$	1.66370	0.0646617
$663$	15.1717	0.589219
$664$	−10.3804	−0.402836
$665$	112.033	4.34447
$666$	−10.2041	−0.395400
$667$	37.6984	1.45969
$668$	−18.8840	−0.730646
$669$	0.874682	0.0338172
$670$	−28.4488	−1.09907
$671$	13.4567	0.519490
$672$	−3.97996	−0.153530
$673$	2.54597	0.0981399	0.0490700	−	0.998795i	$-0.484374\pi$
$673$	2.54597	0.0981399	0.0490700	+	0.998795i	$0.484374\pi$
$674$	29.5009	1.13633
$675$	51.2335	1.97198
$676$	−7.45374	−0.286682
$677$	−44.9529	−1.72768	−0.863840	−	0.503767i	$-0.831947\pi$
$677$	−44.9529	−1.72768	−0.863840	+	0.503767i	$0.831947\pi$
$678$	−19.2300	−0.738523
$679$	−10.9882	−0.421688
$680$	−26.3099	−1.00894
$681$	13.1668	0.504552
$682$	22.8931	0.876622
$683$	28.0094	1.07175	0.535874	−	0.844298i	$-0.319982\pi$
$683$	28.0094	1.07175	0.535874	+	0.844298i	$0.319982\pi$
$684$	14.2789	0.545967
$685$	−14.0479	−0.536742
$686$	12.6484	0.482920
$687$	3.02814	0.115531
$688$	−0.448274	−0.0170903
$689$	−22.8740	−0.871429
$690$	−14.5226	−0.552866
$691$	18.9256	0.719962	0.359981	−	0.932960i	$-0.382783\pi$
$691$	18.9256	0.719962	0.359981	+	0.932960i	$0.382783\pi$
$692$	4.96019	0.188558
$693$	−34.7061	−1.31838
$694$	0.432072	0.0164012
$695$	37.5944	1.42604
$696$	−9.48308	−0.359455
$697$	9.40472	0.356229
$698$	20.1616	0.763129
$699$	2.29794	0.0869160
$700$	−43.3073	−1.63686
$701$	16.6016	0.627032	0.313516	−	0.949583i	$-0.398493\pi$
$701$	16.6016	0.627032	0.313516	+	0.949583i	$0.398493\pi$
$702$	11.5082	0.434351
$703$	33.9505	1.28047
$704$	−4.05582	−0.152860
$705$	25.3083	0.953167
$706$	9.61810	0.361982
$707$	11.1082	0.417765
$708$	−11.1506	−0.419065
$709$	41.7124	1.56654	0.783271	−	0.621681i	$-0.213550\pi$
$709$	41.7124	1.56654	0.783271	+	0.621681i	$0.213550\pi$
$710$	−47.5269	−1.78365
$711$	−1.55622	−0.0583628
$712$	−7.47305	−0.280064
$713$	−21.6202	−0.809684
$714$	−26.6103	−0.995866
$715$	37.5860	1.40564
$716$	−7.90431	−0.295398
$717$	−14.5270	−0.542519
$718$	−8.62071	−0.321722
$719$	−33.2677	−1.24068	−0.620339	−	0.784334i	$-0.713005\pi$
$719$	−33.2677	−1.24068	−0.620339	+	0.784334i	$0.713005\pi$
$720$	−8.15190	−0.303803
$721$	−39.4010	−1.46737
$722$	−28.5081	−1.06096
$723$	24.1219	0.897104
$724$	16.7982	0.624300
$725$	−103.189	−3.83234
$726$	5.25091	0.194879
$727$	−23.6941	−0.878765	−0.439382	−	0.898300i	$-0.644803\pi$
$727$	−23.6941	−0.878765	−0.439382	+	0.898300i	$0.644803\pi$
$728$	−9.72784	−0.360538
$729$	11.0042	0.407563
$730$	−26.1597	−0.968214
$731$	−2.99720	−0.110855
$732$	3.19684	0.118159
$733$	−29.7595	−1.09919	−0.549596	−	0.835431i	$-0.685218\pi$
$733$	−29.7595	−1.09919	−0.549596	+	0.835431i	$0.685218\pi$
$734$	15.7556	0.581549
$735$	−38.1504	−1.40720
$736$	3.83032	0.141187
$737$	−29.3221	−1.08009
$738$	2.91397	0.107265
$739$	45.3317	1.66755	0.833776	−	0.552102i	$-0.186174\pi$
$739$	45.3317	1.66755	0.833776	+	0.552102i	$0.186174\pi$
$740$	−19.3825	−0.712517
$741$	−15.6403	−0.574561
$742$	40.1197	1.47284
$743$	−14.8545	−0.544957	−0.272479	−	0.962162i	$-0.587843\pi$
$743$	−14.8545	−0.544957	−0.272479	+	0.962162i	$0.587843\pi$
$744$	5.43860	0.199389
$745$	74.3561	2.72420
$746$	17.7522	0.649954
$747$	−21.5042	−0.786799
$748$	−27.1176	−0.991516
$749$	63.3051	2.31312
$750$	20.7942	0.759296
$751$	14.0047	0.511040	0.255520	−	0.966804i	$-0.417753\pi$
$751$	14.0047	0.511040	0.255520	+	0.966804i	$0.417753\pi$
$752$	−6.67505	−0.243414
$753$	−24.0601	−0.876797
$754$	−23.1786	−0.844115
$755$	−14.9064	−0.542499
$756$	−20.1848	−0.734116
$757$	−0.537935	−0.0195516	−0.00977579	−	0.999952i	$-0.503112\pi$
$757$	−0.537935	−0.0195516	−0.00977579	+	0.999952i	$0.503112\pi$
$758$	19.6451	0.713542
$759$	−14.9684	−0.543319
$760$	27.1226	0.983840
$761$	−3.34848	−0.121382	−0.0606912	−	0.998157i	$-0.519330\pi$
$761$	−3.34848	−0.121382	−0.0606912	+	0.998157i	$0.519330\pi$
$762$	−1.69813	−0.0615167
$763$	−55.2052	−1.99856
$764$	−17.3989	−0.629469
$765$	−54.5043	−1.97061
$766$	3.12580	0.112940
$767$	−27.2544	−0.984098
$768$	−0.963522	−0.0347681
$769$	−30.7722	−1.10967	−0.554837	−	0.831959i	$-0.687219\pi$
$769$	−30.7722	−1.10967	−0.554837	+	0.831959i	$0.687219\pi$
$770$	−65.9239	−2.37573
$771$	7.38035	0.265797
$772$	21.7252	0.781908
$773$	−44.2158	−1.59033	−0.795165	−	0.606393i	$-0.792616\pi$
$773$	−44.2158	−1.59033	−0.795165	+	0.606393i	$0.792616\pi$
$774$	−0.928655	−0.0333798
$775$	59.1794	2.12579
$776$	−2.66017	−0.0954946
$777$	−19.6039	−0.703284
$778$	15.0042	0.537928
$779$	−9.69522	−0.347367
$780$	8.92914	0.319714
$781$	−48.9858	−1.75285
$782$	25.6098	0.915806
$783$	−48.0946	−1.71876
$784$	10.0621	0.359361
$785$	75.1764	2.68316
$786$	−12.2133	−0.435635
$787$	−40.6842	−1.45024	−0.725118	−	0.688625i	$-0.758215\pi$
$787$	−40.6842	−1.45024	−0.725118	+	0.688625i	$0.758215\pi$
$788$	4.59268	0.163608
$789$	−23.8653	−0.849629
$790$	−2.95602	−0.105171
$791$	82.4392	2.93120
$792$	−8.40214	−0.298557
$793$	7.81375	0.277474
$794$	−17.0532	−0.605196
$795$	−36.8257	−1.30607
$796$	22.0584	0.781840
$797$	7.74219	0.274242	0.137121	−	0.990554i	$-0.456215\pi$
$797$	7.74219	0.274242	0.137121	+	0.990554i	$0.456215\pi$
$798$	27.4323	0.971092
$799$	−44.6299	−1.57889
$800$	−10.4844	−0.370681
$801$	−15.4814	−0.547007
$802$	−36.5783	−1.29163
$803$	−26.9627	−0.951494
$804$	−6.96591	−0.245669
$805$	62.2586	2.19433
$806$	13.2931	0.468229
$807$	2.14205	0.0754036
$808$	2.68922	0.0946063
$809$	−27.9187	−0.981568	−0.490784	−	0.871281i	$-0.663290\pi$
$809$	−27.9187	−0.981568	−0.490784	+	0.871281i	$0.663290\pi$
$810$	−5.92812	−0.208293
$811$	51.4329	1.80605	0.903027	−	0.429585i	$-0.141340\pi$
$811$	51.4329	1.80605	0.903027	+	0.429585i	$0.141340\pi$
$812$	40.6541	1.42668
$813$	27.4372	0.962265
$814$	−19.9775	−0.700213
$815$	−38.6755	−1.35474
$816$	−6.44219	−0.225522
$817$	3.08978	0.108098
$818$	10.6259	0.371526
$819$	−20.1524	−0.704183
$820$	5.53505	0.193292
$821$	−0.109702	−0.00382864	−0.00191432	−	0.999998i	$-0.500609\pi$
$821$	−0.109702	−0.00382864	−0.00191432	+	0.999998i	$0.500609\pi$
$822$	−3.43974	−0.119975
$823$	42.9288	1.49641	0.748203	−	0.663470i	$-0.230917\pi$
$823$	42.9288	1.49641	0.748203	+	0.663470i	$0.230917\pi$
$824$	−9.53875	−0.332298
$825$	40.9719	1.42646
$826$	47.8027	1.66327
$827$	31.2031	1.08504	0.542519	−	0.840044i	$-0.317471\pi$
$827$	31.2031	1.08504	0.542519	+	0.840044i	$0.317471\pi$
$828$	7.93498	0.275760
$829$	20.0515	0.696416	0.348208	−	0.937417i	$-0.386790\pi$
$829$	20.0515	0.696416	0.348208	+	0.937417i	$0.386790\pi$
$830$	−40.8470	−1.41782
$831$	−14.8836	−0.516306
$832$	−2.35505	−0.0816467
$833$	67.2761	2.33098
$834$	9.20529	0.318753
$835$	−74.3092	−2.57158
$836$	27.9552	0.966851
$837$	27.5826	0.953392
$838$	−27.4177	−0.947127
$839$	29.6533	1.02375	0.511873	−	0.859061i	$-0.328952\pi$
$839$	29.6533	1.02375	0.511873	+	0.859061i	$0.328952\pi$
$840$	−15.6612	−0.540364
$841$	67.8668	2.34023
$842$	−5.71707	−0.197023
$843$	8.51512	0.293276
$844$	18.9004	0.650579
$845$	−29.3307	−1.00901
$846$	−13.8282	−0.475423
$847$	−22.5107	−0.773476
$848$	9.71273	0.333537
$849$	−26.4718	−0.908510
$850$	−70.0998	−2.40440
$851$	18.8668	0.646745
$852$	−11.6373	−0.398688
$853$	31.3113	1.07208	0.536039	−	0.844193i	$-0.319920\pi$
$853$	31.3113	1.07208	0.536039	+	0.844193i	$0.319920\pi$
$854$	−13.7049	−0.468972
$855$	56.1878	1.92158
$856$	15.3258	0.523824
$857$	−31.3497	−1.07088	−0.535442	−	0.844572i	$-0.679855\pi$
$857$	−31.3497	−1.07088	−0.535442	+	0.844572i	$0.679855\pi$
$858$	9.20324	0.314193
$859$	−37.4371	−1.27734	−0.638669	−	0.769482i	$-0.720515\pi$
$859$	−37.4371	−1.27734	−0.638669	+	0.769482i	$0.720515\pi$
$860$	−1.76397	−0.0601509
$861$	5.59825	0.190788
$862$	−25.0136	−0.851967
$863$	26.9784	0.918356	0.459178	−	0.888344i	$-0.348144\pi$
$863$	26.9784	0.918356	0.459178	+	0.888344i	$0.348144\pi$
$864$	−4.88662	−0.166246
$865$	19.5185	0.663649
$866$	14.9838	0.509171
$867$	−26.6931	−0.906546
$868$	−23.3153	−0.791374
$869$	−3.04677	−0.103354
$870$	−37.3162	−1.26514
$871$	−17.0261	−0.576909
$872$	−13.3648	−0.452591
$873$	−5.51088	−0.186515
$874$	−26.4009	−0.893024
$875$	−89.1448	−3.01364
$876$	−6.40541	−0.216419
$877$	0.171268	0.00578332	0.00289166	−	0.999996i	$-0.499080\pi$
$877$	0.171268	0.00578332	0.00289166	+	0.999996i	$0.499080\pi$
$878$	11.9313	0.402661
$879$	−6.34582	−0.214039
$880$	−15.9598	−0.538004
$881$	36.1238	1.21704	0.608522	−	0.793537i	$-0.291763\pi$
$881$	36.1238	1.21704	0.608522	+	0.793537i	$0.291763\pi$
$882$	20.8449	0.701885
$883$	10.3128	0.347052	0.173526	−	0.984829i	$-0.444484\pi$
$883$	10.3128	0.347052	0.173526	+	0.984829i	$0.444484\pi$
$884$	−15.7461	−0.529597
$885$	−43.8779	−1.47494
$886$	−3.75556	−0.126171
$887$	−42.7590	−1.43571	−0.717854	−	0.696194i	$-0.754876\pi$
$887$	−42.7590	−1.43571	−0.717854	+	0.696194i	$0.754876\pi$
$888$	−4.74597	−0.159264
$889$	7.27990	0.244160
$890$	−29.4067	−0.985713
$891$	−6.11010	−0.204696
$892$	−0.907796	−0.0303953
$893$	46.0085	1.53962
$894$	18.2067	0.608922
$895$	−31.1037	−1.03968
$896$	4.13063	0.137995
$897$	−8.69154	−0.290202
$898$	−33.3477	−1.11283
$899$	−55.5537	−1.85282
$900$	−21.7198	−0.723994
$901$	64.9401	2.16347
$902$	5.70496	0.189955
$903$	−1.78411	−0.0593715
$904$	19.9580	0.663794
$905$	66.1014	2.19728
$906$	−3.64995	−0.121261
$907$	−42.3372	−1.40578	−0.702892	−	0.711296i	$-0.748108\pi$
$907$	−42.3372	−1.40578	−0.702892	+	0.711296i	$0.748108\pi$
$908$	−13.6652	−0.453497
$909$	5.57105	0.184780
$910$	−38.2793	−1.26895
$911$	12.0015	0.397627	0.198813	−	0.980037i	$-0.436291\pi$
$911$	12.0015	0.397627	0.198813	+	0.980037i	$0.436291\pi$
$912$	6.64118	0.219912
$913$	−42.1009	−1.39334
$914$	23.9744	0.793004
$915$	12.5797	0.415871
$916$	−3.14278	−0.103840
$917$	52.3586	1.72903
$918$	−32.6724	−1.07835
$919$	−1.55744	−0.0513752	−0.0256876	−	0.999670i	$-0.508178\pi$
$919$	−1.55744	−0.0513752	−0.0256876	+	0.999670i	$0.508178\pi$
$920$	15.0724	0.496923
$921$	7.35057	0.242210
$922$	−34.2062	−1.12652
$923$	−28.4441	−0.936248
$924$	−16.1420	−0.531033
$925$	−51.6426	−1.69800
$926$	−19.8775	−0.653215
$927$	−19.7607	−0.649027
$928$	9.84209	0.323083
$929$	51.5731	1.69206	0.846030	−	0.533136i	$-0.178987\pi$
$929$	51.5731	1.69206	0.846030	+	0.533136i	$0.178987\pi$
$930$	21.4010	0.701768
$931$	−69.3542	−2.27299
$932$	−2.38493	−0.0781211
$933$	8.01171	0.262292
$934$	−9.69288	−0.317161
$935$	−106.708	−3.48974
$936$	−4.87878	−0.159468
$937$	−35.3650	−1.15532	−0.577661	−	0.816276i	$-0.696035\pi$
$937$	−35.3650	−1.15532	−0.577661	+	0.816276i	$0.696035\pi$
$938$	29.8629	0.975060
$939$	−2.80347	−0.0914878
$940$	−26.2665	−0.856718
$941$	−21.3120	−0.694753	−0.347376	−	0.937726i	$-0.612927\pi$
$941$	−21.3120	−0.694753	−0.347376	+	0.937726i	$0.612927\pi$
$942$	18.4075	0.599750
$943$	−5.38777	−0.175450
$944$	11.5727	0.376660
$945$	−79.4279	−2.58379
$946$	−1.81812	−0.0591122
$947$	−2.09288	−0.0680095	−0.0340047	−	0.999422i	$-0.510826\pi$
$947$	−2.09288	−0.0680095	−0.0340047	+	0.999422i	$0.510826\pi$
$948$	−0.723806	−0.0235081
$949$	−15.6562	−0.508220
$950$	72.2651	2.34459
$951$	20.8229	0.675229
$952$	27.6177	0.895096
$953$	−33.9879	−1.10098	−0.550489	−	0.834843i	$-0.685559\pi$
$953$	−33.9879	−1.10098	−0.550489	+	0.834843i	$0.685559\pi$
$954$	20.1211	0.651446
$955$	−68.4651	−2.21548
$956$	15.0769	0.487623
$957$	−38.4617	−1.24329
$958$	−12.4782	−0.403153
$959$	14.7462	0.476179
$960$	−3.79149	−0.122370
$961$	0.860388	0.0277545
$962$	−11.6001	−0.374004
$963$	31.7492	1.02311
$964$	−25.0351	−0.806328
$965$	85.4894	2.75200
$966$	15.2445	0.490484
$967$	−21.0298	−0.676273	−0.338136	−	0.941097i	$-0.609797\pi$
$967$	−21.0298	−0.676273	−0.338136	+	0.941097i	$0.609797\pi$
$968$	−5.44970	−0.175160
$969$	44.4035	1.42645
$970$	−10.4678	−0.336102
$971$	6.97180	0.223736	0.111868	−	0.993723i	$-0.464317\pi$
$971$	6.97180	0.223736	0.111868	+	0.993723i	$0.464317\pi$
$972$	−16.1114	−0.516774
$973$	−39.4632	−1.26513
$974$	1.99248	0.0638433
$975$	23.7907	0.761912
$976$	−3.31787	−0.106202
$977$	24.0859	0.770576	0.385288	−	0.922796i	$-0.374102\pi$
$977$	24.0859	0.770576	0.385288	+	0.922796i	$0.374102\pi$
$978$	−9.47001	−0.302818
$979$	−30.3094	−0.968692
$980$	39.5947	1.26481
$981$	−27.6869	−0.883976
$982$	−1.61635	−0.0515797
$983$	−10.6987	−0.341235	−0.170617	−	0.985337i	$-0.554576\pi$
$983$	−10.6987	−0.341235	−0.170617	+	0.985337i	$0.554576\pi$
$984$	1.35530	0.0432054
$985$	18.0723	0.575832
$986$	65.8050	2.09566
$987$	−26.5664	−0.845617
$988$	16.2324	0.516423
$989$	1.71703	0.0545984
$990$	−33.0627	−1.05080
$991$	25.5913	0.812935	0.406467	−	0.913665i	$-0.366761\pi$
$991$	25.5913	0.812935	0.406467	+	0.913665i	$0.366761\pi$
$992$	−5.64450	−0.179213
$993$	1.60302	0.0508702
$994$	49.8894	1.58239
$995$	86.8005	2.75176
$996$	−10.0017	−0.316917
$997$	−0.293787	−0.00930433	−0.00465216	−	0.999989i	$-0.501481\pi$
$997$	−0.293787	−0.00930433	−0.00465216	+	0.999989i	$0.501481\pi$
$998$	−15.3895	−0.487148
$999$	−24.0698	−0.761534

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4006.2.a.h.1.16	✓	42

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4006.2.a.h.1.16	✓	42	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:	\( N \)	\(=\)	\( 4006 = 2 \cdot 2003 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4006.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} -0.963522 q^{3} +1.00000 q^{4} +3.93503 q^{5} +0.963522 q^{6} -4.13063 q^{7} -1.00000 q^{8} -2.07162 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} -0.963522 q^{3} +1.00000 q^{4} +3.93503 q^{5} +0.963522 q^{6} -4.13063 q^{7} -1.00000 q^{8} -2.07162 q^{9} -3.93503 q^{10} -4.05582 q^{11} -0.963522 q^{12} -2.35505 q^{13} +4.13063 q^{14} -3.79149 q^{15} +1.00000 q^{16} +6.68608 q^{17} +2.07162 q^{18} -6.89261 q^{19} +3.93503 q^{20} +3.97996 q^{21} +4.05582 q^{22} -3.83032 q^{23} +0.963522 q^{24} +10.4844 q^{25} +2.35505 q^{26} +4.88662 q^{27} -4.13063 q^{28} -9.84209 q^{29} +3.79149 q^{30} +5.64450 q^{31} -1.00000 q^{32} +3.90788 q^{33} -6.68608 q^{34} -16.2541 q^{35} -2.07162 q^{36} -4.92565 q^{37} +6.89261 q^{38} +2.26914 q^{39} -3.93503 q^{40} +1.40661 q^{41} -3.97996 q^{42} -0.448274 q^{43} -4.05582 q^{44} -8.15190 q^{45} +3.83032 q^{46} -6.67505 q^{47} -0.963522 q^{48} +10.0621 q^{49} -10.4844 q^{50} -6.44219 q^{51} -2.35505 q^{52} +9.71273 q^{53} -4.88662 q^{54} -15.9598 q^{55} +4.13063 q^{56} +6.64118 q^{57} +9.84209 q^{58} +11.5727 q^{59} -3.79149 q^{60} -3.31787 q^{61} -5.64450 q^{62} +8.55712 q^{63} +1.00000 q^{64} -9.26718 q^{65} -3.90788 q^{66} +7.22963 q^{67} +6.68608 q^{68} +3.69060 q^{69} +16.2541 q^{70} +12.0779 q^{71} +2.07162 q^{72} +6.64791 q^{73} +4.92565 q^{74} -10.1020 q^{75} -6.89261 q^{76} +16.7531 q^{77} -2.26914 q^{78} +0.751208 q^{79} +3.93503 q^{80} +1.50650 q^{81} -1.40661 q^{82} +10.3804 q^{83} +3.97996 q^{84} +26.3099 q^{85} +0.448274 q^{86} +9.48308 q^{87} +4.05582 q^{88} +7.47305 q^{89} +8.15190 q^{90} +9.72784 q^{91} -3.83032 q^{92} -5.43860 q^{93} +6.67505 q^{94} -27.1226 q^{95} +0.963522 q^{96} +2.66017 q^{97} -10.0621 q^{98} +8.40214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 42 q - 42 q^{2} + 42 q^{4} + 27 q^{5} - 10 q^{7} - 42 q^{8} + 56 q^{9}+O(q^{10}) \) 42 * q - 42 * q^2 + 42 * q^4 + 27 * q^5 - 10 * q^7 - 42 * q^8 + 56 * q^9	\( 42 q - 42 q^{2} + 42 q^{4} + 27 q^{5} - 10 q^{7} - 42 q^{8} + 56 q^{9} - 27 q^{10} + 23 q^{11} + 15 q^{13} + 10 q^{14} + 6 q^{15} + 42 q^{16} + 14 q^{17} - 56 q^{18} - 4 q^{19} + 27 q^{20} + 26 q^{21} - 23 q^{22} + 12 q^{23} + 45 q^{25} - 15 q^{26} - 3 q^{27} - 10 q^{28} + 41 q^{29} - 6 q^{30} + 18 q^{31} - 42 q^{32} + 25 q^{33} - 14 q^{34} + 8 q^{35} + 56 q^{36} + 33 q^{37} + 4 q^{38} + 10 q^{39} - 27 q^{40} + 84 q^{41} - 26 q^{42} - 36 q^{43} + 23 q^{44} + 66 q^{45} - 12 q^{46} + 28 q^{47} + 58 q^{49} - 45 q^{50} + 17 q^{51} + 15 q^{52} + 68 q^{53} + 3 q^{54} - 28 q^{55} + 10 q^{56} - 9 q^{57} - 41 q^{58} + 59 q^{59} + 6 q^{60} + 41 q^{61} - 18 q^{62} - 28 q^{63} + 42 q^{64} + 44 q^{65} - 25 q^{66} + 14 q^{68} + 67 q^{69} - 8 q^{70} + 69 q^{71} - 56 q^{72} - 27 q^{73} - 33 q^{74} + 14 q^{75} - 4 q^{76} + 43 q^{77} - 10 q^{78} - 19 q^{79} + 27 q^{80} + 74 q^{81} - 84 q^{82} + 20 q^{83} + 26 q^{84} + 16 q^{85} + 36 q^{86} - 28 q^{87} - 23 q^{88} + 123 q^{89} - 66 q^{90} - 9 q^{91} + 12 q^{92} + 48 q^{93} - 28 q^{94} + 28 q^{95} + 10 q^{97} - 58 q^{98} + 75 q^{99}+O(q^{100}) \) 42 * q - 42 * q^2 + 42 * q^4 + 27 * q^5 - 10 * q^7 - 42 * q^8 + 56 * q^9 - 27 * q^10 + 23 * q^11 + 15 * q^13 + 10 * q^14 + 6 * q^15 + 42 * q^16 + 14 * q^17 - 56 * q^18 - 4 * q^19 + 27 * q^20 + 26 * q^21 - 23 * q^22 + 12 * q^23 + 45 * q^25 - 15 * q^26 - 3 * q^27 - 10 * q^28 + 41 * q^29 - 6 * q^30 + 18 * q^31 - 42 * q^32 + 25 * q^33 - 14 * q^34 + 8 * q^35 + 56 * q^36 + 33 * q^37 + 4 * q^38 + 10 * q^39 - 27 * q^40 + 84 * q^41 - 26 * q^42 - 36 * q^43 + 23 * q^44 + 66 * q^45 - 12 * q^46 + 28 * q^47 + 58 * q^49 - 45 * q^50 + 17 * q^51 + 15 * q^52 + 68 * q^53 + 3 * q^54 - 28 * q^55 + 10 * q^56 - 9 * q^57 - 41 * q^58 + 59 * q^59 + 6 * q^60 + 41 * q^61 - 18 * q^62 - 28 * q^63 + 42 * q^64 + 44 * q^65 - 25 * q^66 + 14 * q^68 + 67 * q^69 - 8 * q^70 + 69 * q^71 - 56 * q^72 - 27 * q^73 - 33 * q^74 + 14 * q^75 - 4 * q^76 + 43 * q^77 - 10 * q^78 - 19 * q^79 + 27 * q^80 + 74 * q^81 - 84 * q^82 + 20 * q^83 + 26 * q^84 + 16 * q^85 + 36 * q^86 - 28 * q^87 - 23 * q^88 + 123 * q^89 - 66 * q^90 - 9 * q^91 + 12 * q^92 + 48 * q^93 - 28 * q^94 + 28 * q^95 + 10 * q^97 - 58 * q^98 + 75 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more