LMFDB - Embedded newform 1339.2.a.f.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1339,2,Mod(1,1339)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1339, 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("1339.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1339 = 13 \cdot 103 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1339.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:	$10.6919688306$
Analytic rank:	$0$
Dimension:	$28$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Character	$\chi$	$=$	1339.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.779581 q^{2} -1.24905 q^{3} -1.39225 q^{4} +1.96235 q^{5} +0.973735 q^{6} +3.90578 q^{7} +2.64454 q^{8} -1.43988 q^{9} +O(q^{10})$	\(q-0.779581 q^{2} -1.24905 q^{3} -1.39225 q^{4} +1.96235 q^{5} +0.973735 q^{6} +3.90578 q^{7} +2.64454 q^{8} -1.43988 q^{9} -1.52981 q^{10} +5.25330 q^{11} +1.73899 q^{12} +1.00000 q^{13} -3.04487 q^{14} -2.45107 q^{15} +0.722875 q^{16} -0.637414 q^{17} +1.12250 q^{18} +6.18255 q^{19} -2.73208 q^{20} -4.87851 q^{21} -4.09537 q^{22} -2.06099 q^{23} -3.30315 q^{24} -1.14920 q^{25} -0.779581 q^{26} +5.54562 q^{27} -5.43783 q^{28} -2.34460 q^{29} +1.91080 q^{30} +0.549740 q^{31} -5.85261 q^{32} -6.56162 q^{33} +0.496916 q^{34} +7.66449 q^{35} +2.00467 q^{36} -7.82532 q^{37} -4.81980 q^{38} -1.24905 q^{39} +5.18950 q^{40} +7.09468 q^{41} +3.80319 q^{42} -5.70072 q^{43} -7.31392 q^{44} -2.82554 q^{45} +1.60671 q^{46} +4.06141 q^{47} -0.902907 q^{48} +8.25511 q^{49} +0.895892 q^{50} +0.796161 q^{51} -1.39225 q^{52} +7.88533 q^{53} -4.32326 q^{54} +10.3088 q^{55} +10.3290 q^{56} -7.72231 q^{57} +1.82781 q^{58} -10.1334 q^{59} +3.41250 q^{60} -9.82301 q^{61} -0.428567 q^{62} -5.62384 q^{63} +3.11684 q^{64} +1.96235 q^{65} +5.11532 q^{66} +9.54922 q^{67} +0.887442 q^{68} +2.57427 q^{69} -5.97509 q^{70} +1.65976 q^{71} -3.80781 q^{72} +8.52900 q^{73} +6.10047 q^{74} +1.43540 q^{75} -8.60768 q^{76} +20.5182 q^{77} +0.973735 q^{78} +15.8251 q^{79} +1.41853 q^{80} -2.60712 q^{81} -5.53088 q^{82} -9.50521 q^{83} +6.79212 q^{84} -1.25083 q^{85} +4.44418 q^{86} +2.92852 q^{87} +13.8925 q^{88} -12.3471 q^{89} +2.20274 q^{90} +3.90578 q^{91} +2.86942 q^{92} -0.686651 q^{93} -3.16620 q^{94} +12.1323 q^{95} +7.31020 q^{96} +12.1782 q^{97} -6.43553 q^{98} -7.56410 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 28 q + 6 q^{2} + 5 q^{3} + 32 q^{4} + 27 q^{5} + 9 q^{6} + 6 q^{7} + 21 q^{8} + 39 q^{9}+O(q^{10}) $ 28 * q + 6 * q^2 + 5 * q^3 + 32 * q^4 + 27 * q^5 + 9 * q^6 + 6 * q^7 + 21 * q^8 + 39 * q^9	\( 28 q + 6 q^{2} + 5 q^{3} + 32 q^{4} + 27 q^{5} + 9 q^{6} + 6 q^{7} + 21 q^{8} + 39 q^{9} + 5 q^{10} + 9 q^{11} + 15 q^{12} + 28 q^{13} + 20 q^{14} + 4 q^{15} + 32 q^{16} - 7 q^{17} + 10 q^{18} - 3 q^{19} + 53 q^{20} + 25 q^{21} - 14 q^{22} - 6 q^{23} + 10 q^{24} + 45 q^{25} + 6 q^{26} + 8 q^{27} - 12 q^{28} + 25 q^{29} - 43 q^{30} + q^{31} + 28 q^{32} + 17 q^{33} + 10 q^{34} + 11 q^{35} + 27 q^{36} + 18 q^{37} + 2 q^{38} + 5 q^{39} + 37 q^{40} + 56 q^{41} - 29 q^{42} - 30 q^{43} + 30 q^{44} + 38 q^{45} - 27 q^{46} + 40 q^{47} + 19 q^{48} + 36 q^{49} + 20 q^{50} - 18 q^{51} + 32 q^{52} + 51 q^{53} - 36 q^{54} - 5 q^{55} - 13 q^{56} + 31 q^{57} - 29 q^{58} + 29 q^{59} + 48 q^{60} + 16 q^{61} - 38 q^{62} - 23 q^{63} - 5 q^{64} + 27 q^{65} + 25 q^{66} - 2 q^{67} - 43 q^{68} + 38 q^{69} + 10 q^{70} + 34 q^{71} - 28 q^{72} + 14 q^{73} + 21 q^{74} + 19 q^{75} + 5 q^{76} + 34 q^{77} + 9 q^{78} + 3 q^{79} + 58 q^{80} + 4 q^{81} - 9 q^{82} + 38 q^{83} - 78 q^{84} - 27 q^{85} + 49 q^{86} + 8 q^{87} - 25 q^{88} + 125 q^{89} - 74 q^{90} + 6 q^{91} - 35 q^{92} + 36 q^{93} - q^{94} - 12 q^{95} + 26 q^{96} - 2 q^{97} + 18 q^{98} + 56 q^{99}+O(q^{100}) \) 28 * q + 6 * q^2 + 5 * q^3 + 32 * q^4 + 27 * q^5 + 9 * q^6 + 6 * q^7 + 21 * q^8 + 39 * q^9 + 5 * q^10 + 9 * q^11 + 15 * q^12 + 28 * q^13 + 20 * q^14 + 4 * q^15 + 32 * q^16 - 7 * q^17 + 10 * q^18 - 3 * q^19 + 53 * q^20 + 25 * q^21 - 14 * q^22 - 6 * q^23 + 10 * q^24 + 45 * q^25 + 6 * q^26 + 8 * q^27 - 12 * q^28 + 25 * q^29 - 43 * q^30 + q^31 + 28 * q^32 + 17 * q^33 + 10 * q^34 + 11 * q^35 + 27 * q^36 + 18 * q^37 + 2 * q^38 + 5 * q^39 + 37 * q^40 + 56 * q^41 - 29 * q^42 - 30 * q^43 + 30 * q^44 + 38 * q^45 - 27 * q^46 + 40 * q^47 + 19 * q^48 + 36 * q^49 + 20 * q^50 - 18 * q^51 + 32 * q^52 + 51 * q^53 - 36 * q^54 - 5 * q^55 - 13 * q^56 + 31 * q^57 - 29 * q^58 + 29 * q^59 + 48 * q^60 + 16 * q^61 - 38 * q^62 - 23 * q^63 - 5 * q^64 + 27 * q^65 + 25 * q^66 - 2 * q^67 - 43 * q^68 + 38 * q^69 + 10 * q^70 + 34 * q^71 - 28 * q^72 + 14 * q^73 + 21 * q^74 + 19 * q^75 + 5 * q^76 + 34 * q^77 + 9 * q^78 + 3 * q^79 + 58 * q^80 + 4 * q^81 - 9 * q^82 + 38 * q^83 - 78 * q^84 - 27 * q^85 + 49 * q^86 + 8 * q^87 - 25 * q^88 + 125 * q^89 - 74 * q^90 + 6 * q^91 - 35 * q^92 + 36 * q^93 - q^94 - 12 * q^95 + 26 * q^96 - 2 * q^97 + 18 * q^98 + 56 * 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 $

$2$	−0.779581	−0.551247	−0.275624	−	0.961266i	$-0.588884\pi$
$2$	−0.779581	−0.551247	−0.275624	+	0.961266i	$0.588884\pi$
$3$	−1.24905	−0.721139	−0.360569	−	0.932732i	$-0.617418\pi$
$3$	−1.24905	−0.721139	−0.360569	+	0.932732i	$0.617418\pi$
$4$	−1.39225	−0.696127
$5$	1.96235	0.877588	0.438794	−	0.898588i	$-0.355406\pi$
$5$	1.96235	0.877588	0.438794	+	0.898588i	$0.355406\pi$
$6$	0.973735	0.397526
$7$	3.90578	1.47625	0.738123	−	0.674666i	$-0.235712\pi$
$7$	3.90578	1.47625	0.738123	+	0.674666i	$0.235712\pi$
$8$	2.64454	0.934985
$9$	−1.43988	−0.479959
$10$	−1.52981	−0.483768
$11$	5.25330	1.58393	0.791964	−	0.610568i	$-0.209059\pi$
$11$	5.25330	1.58393	0.791964	+	0.610568i	$0.209059\pi$
$12$	1.73899	0.502004
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−3.04487	−0.813776
$15$	−2.45107	−0.632862
$16$	0.722875	0.180719
$17$	−0.637414	−0.154596	−0.0772978	−	0.997008i	$-0.524629\pi$
$17$	−0.637414	−0.154596	−0.0772978	+	0.997008i	$0.524629\pi$
$18$	1.12250	0.264576
$19$	6.18255	1.41837	0.709187	−	0.705020i	$-0.249062\pi$
$19$	6.18255	1.41837	0.709187	+	0.705020i	$0.249062\pi$
$20$	−2.73208	−0.610912
$21$	−4.87851	−1.06458
$22$	−4.09537	−0.873136
$23$	−2.06099	−0.429746	−0.214873	−	0.976642i	$-0.568934\pi$
$23$	−2.06099	−0.429746	−0.214873	+	0.976642i	$0.568934\pi$
$24$	−3.30315	−0.674254
$25$	−1.14920	−0.229839
$26$	−0.779581	−0.152888
$27$	5.54562	1.06726
$28$	−5.43783	−1.02765
$29$	−2.34460	−0.435382	−0.217691	−	0.976018i	$-0.569852\pi$
$29$	−2.34460	−0.435382	−0.217691	+	0.976018i	$0.569852\pi$
$30$	1.91080	0.348864
$31$	0.549740	0.0987362	0.0493681	−	0.998781i	$-0.484279\pi$
$31$	0.549740	0.0987362	0.0493681	+	0.998781i	$0.484279\pi$
$32$	−5.85261	−1.03461
$33$	−6.56162	−1.14223
$34$	0.496916	0.0852204
$35$	7.66449	1.29554
$36$	2.00467	0.334112
$37$	−7.82532	−1.28647	−0.643237	−	0.765667i	$-0.722409\pi$
$37$	−7.82532	−1.28647	−0.643237	+	0.765667i	$0.722409\pi$
$38$	−4.81980	−0.781875
$39$	−1.24905	−0.200008
$40$	5.18950	0.820531
$41$	7.09468	1.10800	0.554002	−	0.832516i	$-0.313100\pi$
$41$	7.09468	1.10800	0.554002	+	0.832516i	$0.313100\pi$
$42$	3.80319	0.586845
$43$	−5.70072	−0.869352	−0.434676	−	0.900587i	$-0.643137\pi$
$43$	−5.70072	−0.869352	−0.434676	+	0.900587i	$0.643137\pi$
$44$	−7.31392	−1.10261
$45$	−2.82554	−0.421206
$46$	1.60671	0.236896
$47$	4.06141	0.592418	0.296209	−	0.955123i	$-0.404278\pi$
$47$	4.06141	0.592418	0.296209	+	0.955123i	$0.404278\pi$
$48$	−0.902907	−0.130323
$49$	8.25511	1.17930
$50$	0.895892	0.126698
$51$	0.796161	0.111485
$52$	−1.39225	−0.193071
$53$	7.88533	1.08313	0.541567	−	0.840658i	$-0.317831\pi$
$53$	7.88533	1.08313	0.541567	+	0.840658i	$0.317831\pi$
$54$	−4.32326	−0.588322
$55$	10.3088	1.39004
$56$	10.3290	1.38027
$57$	−7.72231	−1.02284
$58$	1.82781	0.240003
$59$	−10.1334	−1.31925	−0.659627	−	0.751593i	$-0.729286\pi$
$59$	−10.1334	−1.31925	−0.659627	+	0.751593i	$0.729286\pi$
$60$	3.41250	0.440552
$61$	−9.82301	−1.25771	−0.628854	−	0.777524i	$-0.716476\pi$
$61$	−9.82301	−1.25771	−0.628854	+	0.777524i	$0.716476\pi$
$62$	−0.428567	−0.0544280
$63$	−5.62384	−0.708538
$64$	3.11684	0.389605
$65$	1.96235	0.243399
$66$	5.11532	0.629652
$67$	9.54922	1.16662	0.583311	−	0.812249i	$-0.301757\pi$
$67$	9.54922	1.16662	0.583311	+	0.812249i	$0.301757\pi$
$68$	0.887442	0.107618
$69$	2.57427	0.309906
$70$	−5.97509	−0.714160
$71$	1.65976	0.196977	0.0984885	−	0.995138i	$-0.468599\pi$
$71$	1.65976	0.196977	0.0984885	+	0.995138i	$0.468599\pi$
$72$	−3.80781	−0.448755
$73$	8.52900	0.998243	0.499122	−	0.866532i	$-0.333656\pi$
$73$	8.52900	0.998243	0.499122	+	0.866532i	$0.333656\pi$
$74$	6.10047	0.709165
$75$	1.43540	0.165746
$76$	−8.60768	−0.987368
$77$	20.5182	2.33827
$78$	0.973735	0.110254
$79$	15.8251	1.78046	0.890232	−	0.455507i	$-0.150542\pi$
$79$	15.8251	1.78046	0.890232	+	0.455507i	$0.150542\pi$
$80$	1.41853	0.158597
$81$	−2.60712	−0.289680
$82$	−5.53088	−0.610784
$83$	−9.50521	−1.04333	−0.521666	−	0.853150i	$-0.674689\pi$
$83$	−9.50521	−1.04333	−0.521666	+	0.853150i	$0.674689\pi$
$84$	6.79212	0.741081
$85$	−1.25083	−0.135671
$86$	4.44418	0.479228
$87$	2.92852	0.313971
$88$	13.8925	1.48095
$89$	−12.3471	−1.30879	−0.654394	−	0.756154i	$-0.727076\pi$
$89$	−12.3471	−1.30879	−0.654394	+	0.756154i	$0.727076\pi$
$90$	2.20274	0.232189
$91$	3.90578	0.409437
$92$	2.86942	0.299157
$93$	−0.686651	−0.0712024
$94$	−3.16620	−0.326568
$95$	12.1323	1.24475
$96$	7.31020	0.746094
$97$	12.1782	1.23651	0.618254	−	0.785979i	$-0.287840\pi$
$97$	12.1782	1.23651	0.618254	+	0.785979i	$0.287840\pi$
$98$	−6.43553	−0.650087
$99$	−7.56410	−0.760221
$100$	1.59997	0.159997
$101$	2.63678	0.262369	0.131185	−	0.991358i	$-0.458122\pi$
$101$	2.63678	0.262369	0.131185	+	0.991358i	$0.458122\pi$
$102$	−0.620672	−0.0614557
$103$	−1.00000	−0.0985329
$103$	−1.00000	−0.0985329
$104$	2.64454	0.259318
$105$	−9.57332	−0.934261
$106$	−6.14726	−0.597074
$107$	18.9097	1.82807	0.914035	−	0.405635i	$-0.132950\pi$
$107$	18.9097	1.82807	0.914035	+	0.405635i	$0.132950\pi$
$108$	−7.72091	−0.742945
$109$	13.0595	1.25087	0.625437	−	0.780275i	$-0.284921\pi$
$109$	13.0595	1.25087	0.625437	+	0.780275i	$0.284921\pi$
$110$	−8.03653	−0.766253
$111$	9.77420	0.927726
$112$	2.82339	0.266785
$113$	−7.14185	−0.671849	−0.335925	−	0.941889i	$-0.609049\pi$
$113$	−7.14185	−0.671849	−0.335925	+	0.941889i	$0.609049\pi$
$114$	6.02016	0.563840
$115$	−4.04437	−0.377140
$116$	3.26428	0.303081
$117$	−1.43988	−0.133117
$118$	7.89979	0.727235
$119$	−2.48960	−0.228221
$120$	−6.48193	−0.591717
$121$	16.5971	1.50883
$122$	7.65783	0.693307
$123$	−8.86160	−0.799024
$124$	−0.765377	−0.0687329
$125$	−12.0669	−1.07929
$126$	4.38424	0.390579
$127$	−10.6779	−0.947511	−0.473755	−	0.880657i	$-0.657102\pi$
$127$	−10.6779	−0.947511	−0.473755	+	0.880657i	$0.657102\pi$
$128$	9.27540	0.819837
$129$	7.12048	0.626923
$130$	−1.52981	−0.134173
$131$	10.3048	0.900339	0.450169	−	0.892943i	$-0.351364\pi$
$131$	10.3048	0.900339	0.450169	+	0.892943i	$0.351364\pi$
$132$	9.13544	0.795138
$133$	24.1477	2.09387
$134$	−7.44439	−0.643097
$135$	10.8824	0.936611
$136$	−1.68566	−0.144545
$137$	−1.30157	−0.111200	−0.0556002	−	0.998453i	$-0.517707\pi$
$137$	−1.30157	−0.111200	−0.0556002	+	0.998453i	$0.517707\pi$
$138$	−2.00686	−0.170835
$139$	−14.1925	−1.20380	−0.601898	−	0.798573i	$-0.705589\pi$
$139$	−14.1925	−1.20380	−0.601898	+	0.798573i	$0.705589\pi$
$140$	−10.6709	−0.901857
$141$	−5.07290	−0.427215
$142$	−1.29392	−0.108583
$143$	5.25330	0.439303
$144$	−1.04085	−0.0867377
$145$	−4.60092	−0.382086
$146$	−6.64904	−0.550279
$147$	−10.3110	−0.850440
$148$	10.8948	0.895549
$149$	15.9066	1.30312	0.651561	−	0.758596i	$-0.274114\pi$
$149$	15.9066	1.30312	0.651561	+	0.758596i	$0.274114\pi$
$150$	−1.11901	−0.0913670
$151$	−7.15507	−0.582271	−0.291136	−	0.956682i	$-0.594033\pi$
$151$	−7.15507	−0.582271	−0.291136	+	0.956682i	$0.594033\pi$
$152$	16.3500	1.32616
$153$	0.917798	0.0741996
$154$	−15.9956	−1.28896
$155$	1.07878	0.0866497
$156$	1.73899	0.139231
$157$	4.54275	0.362551	0.181275	−	0.983432i	$-0.441977\pi$
$157$	4.54275	0.362551	0.181275	+	0.983432i	$0.441977\pi$
$158$	−12.3370	−0.981476
$159$	−9.84916	−0.781089
$160$	−11.4849	−0.907957
$161$	−8.04976	−0.634410
$162$	2.03246	0.159685
$163$	−7.97741	−0.624839	−0.312419	−	0.949944i	$-0.601139\pi$
$163$	−7.97741	−0.624839	−0.312419	+	0.949944i	$0.601139\pi$
$164$	−9.87760	−0.771311
$165$	−12.8762	−1.00241
$166$	7.41008	0.575134
$167$	−1.00428	−0.0777134	−0.0388567	−	0.999245i	$-0.512372\pi$
$167$	−1.00428	−0.0777134	−0.0388567	+	0.999245i	$0.512372\pi$
$168$	−12.9014	−0.995364
$169$	1.00000	0.0769231
$170$	0.975121	0.0747884
$171$	−8.90212	−0.680762
$172$	7.93685	0.605179
$173$	6.96442	0.529495	0.264747	−	0.964318i	$-0.414711\pi$
$173$	6.96442	0.529495	0.264747	+	0.964318i	$0.414711\pi$
$174$	−2.28302	−0.173075
$175$	−4.48851	−0.339299
$176$	3.79748	0.286246
$177$	12.6571	0.951364
$178$	9.62556	0.721466
$179$	−13.6078	−1.01709	−0.508547	−	0.861034i	$-0.669817\pi$
$179$	−13.6078	−1.01709	−0.508547	+	0.861034i	$0.669817\pi$
$180$	3.93387	0.293213
$181$	−8.83198	−0.656476	−0.328238	−	0.944595i	$-0.606455\pi$
$181$	−8.83198	−0.656476	−0.328238	+	0.944595i	$0.606455\pi$
$182$	−3.04487	−0.225701
$183$	12.2694	0.906981
$184$	−5.45036	−0.401806
$185$	−15.3560	−1.12899
$186$	0.535301	0.0392501
$187$	−3.34852	−0.244868
$188$	−5.65451	−0.412398
$189$	21.6600	1.57553
$190$	−9.45812	−0.686164
$191$	17.1539	1.24121	0.620606	−	0.784123i	$-0.286887\pi$
$191$	17.1539	1.24121	0.620606	+	0.784123i	$0.286887\pi$
$192$	−3.89308	−0.280959
$193$	−13.6800	−0.984709	−0.492355	−	0.870395i	$-0.663864\pi$
$193$	−13.6800	−0.984709	−0.492355	+	0.870395i	$0.663864\pi$
$194$	−9.49388	−0.681621
$195$	−2.45107	−0.175524
$196$	−11.4932	−0.820943
$197$	6.22424	0.443459	0.221729	−	0.975108i	$-0.428830\pi$
$197$	6.22424	0.443459	0.221729	+	0.975108i	$0.428830\pi$
$198$	5.89683	0.419070
$199$	−9.11955	−0.646468	−0.323234	−	0.946319i	$-0.604770\pi$
$199$	−9.11955	−0.646468	−0.323234	+	0.946319i	$0.604770\pi$
$200$	−3.03909	−0.214896
$201$	−11.9274	−0.841297
$202$	−2.05558	−0.144630
$203$	−9.15750	−0.642730
$204$	−1.10846	−0.0776076
$205$	13.9222	0.972370
$206$	0.779581	0.0543160
$207$	2.96757	0.206260
$208$	0.722875	0.0501224
$209$	32.4788	2.24660
$210$	7.46318	0.515008
$211$	−6.77715	−0.466558	−0.233279	−	0.972410i	$-0.574946\pi$
$211$	−6.77715	−0.466558	−0.233279	+	0.972410i	$0.574946\pi$
$212$	−10.9784	−0.753998
$213$	−2.07312	−0.142048
$214$	−14.7416	−1.00772
$215$	−11.1868	−0.762933
$216$	14.6656	0.997868
$217$	2.14716	0.145759
$218$	−10.1809	−0.689540
$219$	−10.6531	−0.719872
$220$	−14.3524	−0.967641
$221$	−0.637414	−0.0428771
$222$	−7.61978	−0.511406
$223$	−11.1333	−0.745542	−0.372771	−	0.927923i	$-0.621592\pi$
$223$	−11.1333	−0.745542	−0.372771	+	0.927923i	$0.621592\pi$
$224$	−22.8590	−1.52733
$225$	1.65470	0.110314
$226$	5.56765	0.370355
$227$	−6.58746	−0.437225	−0.218612	−	0.975812i	$-0.570153\pi$
$227$	−6.58746	−0.437225	−0.218612	+	0.975812i	$0.570153\pi$
$228$	10.7514	0.712029
$229$	1.21239	0.0801167	0.0400584	−	0.999197i	$-0.487246\pi$
$229$	1.21239	0.0801167	0.0400584	+	0.999197i	$0.487246\pi$
$230$	3.15292	0.207897
$231$	−25.6282	−1.68621
$232$	−6.20039	−0.407075
$233$	−1.86431	−0.122135	−0.0610675	−	0.998134i	$-0.519451\pi$
$233$	−1.86431	−0.122135	−0.0610675	+	0.998134i	$0.519451\pi$
$234$	1.12250	0.0733802
$235$	7.96989	0.519898
$236$	14.1082	0.918367
$237$	−19.7663	−1.28396
$238$	1.94084	0.125806
$239$	−0.528308	−0.0341734	−0.0170867	−	0.999854i	$-0.505439\pi$
$239$	−0.528308	−0.0341734	−0.0170867	+	0.999854i	$0.505439\pi$
$240$	−1.77182	−0.114370
$241$	−4.86533	−0.313403	−0.156702	−	0.987646i	$-0.550086\pi$
$241$	−4.86533	−0.313403	−0.156702	+	0.987646i	$0.550086\pi$
$242$	−12.9388	−0.831737
$243$	−13.3805	−0.858356
$244$	13.6761	0.875523
$245$	16.1994	1.03494
$246$	6.90834	0.440460
$247$	6.18255	0.393386
$248$	1.45381	0.0923168
$249$	11.8725	0.752387
$250$	9.40709	0.594957
$251$	15.0421	0.949450	0.474725	−	0.880134i	$-0.342548\pi$
$251$	15.0421	0.949450	0.474725	+	0.880134i	$0.342548\pi$
$252$	7.82982	0.493232
$253$	−10.8270	−0.680686
$254$	8.32429	0.522312
$255$	1.56234	0.0978377
$256$	−13.4646	−0.841537
$257$	7.41396	0.462470	0.231235	−	0.972898i	$-0.425723\pi$
$257$	7.41396	0.462470	0.231235	+	0.972898i	$0.425723\pi$
$258$	−5.55099	−0.345590
$259$	−30.5640	−1.89915
$260$	−2.73208	−0.169437
$261$	3.37594	0.208965
$262$	−8.03346	−0.496309
$263$	3.90539	0.240817	0.120408	−	0.992724i	$-0.461580\pi$
$263$	3.90539	0.240817	0.120408	+	0.992724i	$0.461580\pi$
$264$	−17.3524	−1.06797
$265$	15.4738	0.950545
$266$	−18.8251	−1.15424
$267$	15.4221	0.943818
$268$	−13.2949	−0.812117
$269$	−16.1721	−0.986027	−0.493014	−	0.870022i	$-0.664105\pi$
$269$	−16.1721	−0.986027	−0.493014	+	0.870022i	$0.664105\pi$
$270$	−8.48374	−0.516304
$271$	0.690419	0.0419400	0.0209700	−	0.999780i	$-0.493325\pi$
$271$	0.690419	0.0419400	0.0209700	+	0.999780i	$0.493325\pi$
$272$	−0.460771	−0.0279383
$273$	−4.87851	−0.295261
$274$	1.01468	0.0612989
$275$	−6.03707	−0.364049
$276$	−3.58404	−0.215734
$277$	−8.15848	−0.490196	−0.245098	−	0.969498i	$-0.578820\pi$
$277$	−8.15848	−0.490196	−0.245098	+	0.969498i	$0.578820\pi$
$278$	11.0642	0.663589
$279$	−0.791558	−0.0473893
$280$	20.2690	1.21131
$281$	27.1417	1.61914	0.809568	−	0.587026i	$-0.199701\pi$
$281$	27.1417	1.61914	0.809568	+	0.587026i	$0.199701\pi$
$282$	3.95474	0.235501
$283$	−11.4605	−0.681258	−0.340629	−	0.940198i	$-0.610640\pi$
$283$	−11.4605	−0.681258	−0.340629	+	0.940198i	$0.610640\pi$
$284$	−2.31080	−0.137121
$285$	−15.1538	−0.897636
$286$	−4.09537	−0.242164
$287$	27.7103	1.63569
$288$	8.42705	0.496569
$289$	−16.5937	−0.976100
$290$	3.58679	0.210624
$291$	−15.2111	−0.891693
$292$	−11.8745	−0.694904
$293$	−5.34681	−0.312364	−0.156182	−	0.987728i	$-0.549919\pi$
$293$	−5.34681	−0.312364	−0.156182	+	0.987728i	$0.549919\pi$
$294$	8.03829	0.468803
$295$	−19.8852	−1.15776
$296$	−20.6943	−1.20283
$297$	29.1328	1.69046
$298$	−12.4005	−0.718342
$299$	−2.06099	−0.119190
$300$	−1.99844	−0.115380
$301$	−22.2658	−1.28338
$302$	5.57796	0.320975
$303$	−3.29346	−0.189204
$304$	4.46921	0.256327
$305$	−19.2761	−1.10375
$306$	−0.715498	−0.0409023
$307$	−4.00293	−0.228459	−0.114230	−	0.993454i	$-0.536440\pi$
$307$	−4.00293	−0.228459	−0.114230	+	0.993454i	$0.536440\pi$
$308$	−28.5665	−1.62773
$309$	1.24905	0.0710559
$310$	−0.840996	−0.0477654
$311$	24.6809	1.39953	0.699763	−	0.714375i	$-0.253289\pi$
$311$	24.6809	1.39953	0.699763	+	0.714375i	$0.253289\pi$
$312$	−3.30315	−0.187004
$313$	22.0126	1.24422	0.622112	−	0.782928i	$-0.286275\pi$
$313$	22.0126	1.24422	0.622112	+	0.782928i	$0.286275\pi$
$314$	−3.54144	−0.199855
$315$	−11.0359	−0.621804
$316$	−22.0326	−1.23943
$317$	26.7886	1.50460	0.752299	−	0.658821i	$-0.228945\pi$
$317$	26.7886	1.50460	0.752299	+	0.658821i	$0.228945\pi$
$318$	7.67822	0.430573
$319$	−12.3169	−0.689613
$320$	6.11631	0.341912
$321$	−23.6191	−1.31829
$322$	6.27544	0.349717
$323$	−3.94084	−0.219274
$324$	3.62977	0.201654
$325$	−1.14920	−0.0637460
$326$	6.21904	0.344441
$327$	−16.3120	−0.902053
$328$	18.7622	1.03597
$329$	15.8630	0.874554
$330$	10.0380	0.552575
$331$	14.2132	0.781228	0.390614	−	0.920555i	$-0.372263\pi$
$331$	14.2132	0.781228	0.390614	+	0.920555i	$0.372263\pi$
$332$	13.2337	0.726291
$333$	11.2675	0.617455
$334$	0.782917	0.0428393
$335$	18.7389	1.02381
$336$	−3.52655	−0.192389
$337$	6.46683	0.352271	0.176135	−	0.984366i	$-0.443640\pi$
$337$	6.46683	0.352271	0.176135	+	0.984366i	$0.443640\pi$
$338$	−0.779581	−0.0424036
$339$	8.92052	0.484496
$340$	1.74147	0.0944443
$341$	2.88794	0.156391
$342$	6.93992	0.375268
$343$	4.90220	0.264694
$344$	−15.0758	−0.812831
$345$	5.05162	0.271970
$346$	−5.42933	−0.291883
$347$	19.4162	1.04232	0.521159	−	0.853459i	$-0.325500\pi$
$347$	19.4162	1.04232	0.521159	+	0.853459i	$0.325500\pi$
$348$	−4.07724	−0.218563
$349$	18.7020	1.00110	0.500548	−	0.865709i	$-0.333132\pi$
$349$	18.7020	1.00110	0.500548	+	0.865709i	$0.333132\pi$
$350$	3.49916	0.187038
$351$	5.54562	0.296003
$352$	−30.7455	−1.63874
$353$	−14.9691	−0.796727	−0.398363	−	0.917228i	$-0.630422\pi$
$353$	−14.9691	−0.796727	−0.398363	+	0.917228i	$0.630422\pi$
$354$	−9.86722	−0.524437
$355$	3.25702	0.172865
$356$	17.1903	0.911082
$357$	3.10963	0.164579
$358$	10.6084	0.560671
$359$	33.9580	1.79224	0.896118	−	0.443816i	$-0.146376\pi$
$359$	33.9580	1.79224	0.896118	+	0.443816i	$0.146376\pi$
$360$	−7.47224	−0.393822
$361$	19.2239	1.01179
$362$	6.88525	0.361880
$363$	−20.7306	−1.08807
$364$	−5.43783	−0.285020
$365$	16.7368	0.876046
$366$	−9.56500	−0.499971
$367$	−9.75901	−0.509416	−0.254708	−	0.967018i	$-0.581979\pi$
$367$	−9.75901	−0.509416	−0.254708	+	0.967018i	$0.581979\pi$
$368$	−1.48984	−0.0776631
$369$	−10.2155	−0.531796
$370$	11.9712	0.622355
$371$	30.7984	1.59897
$372$	0.955993	0.0495659
$373$	−24.7826	−1.28319	−0.641597	−	0.767042i	$-0.721728\pi$
$373$	−24.7826	−1.28319	−0.641597	+	0.767042i	$0.721728\pi$
$374$	2.61045	0.134983
$375$	15.0721	0.778319
$376$	10.7405	0.553901
$377$	−2.34460	−0.120753
$378$	−16.8857	−0.868507
$379$	−28.7349	−1.47601	−0.738007	−	0.674793i	$-0.764233\pi$
$379$	−28.7349	−1.47601	−0.738007	+	0.674793i	$0.764233\pi$
$380$	−16.8912	−0.866502
$381$	13.3372	0.683286
$382$	−13.3728	−0.684214
$383$	5.67410	0.289933	0.144966	−	0.989437i	$-0.453693\pi$
$383$	5.67410	0.289933	0.144966	+	0.989437i	$0.453693\pi$
$384$	−11.5854	−0.591216
$385$	40.2638	2.05204
$386$	10.6647	0.542818
$387$	8.20834	0.417254
$388$	−16.9551	−0.860766
$389$	4.95258	0.251106	0.125553	−	0.992087i	$-0.459930\pi$
$389$	4.95258	0.251106	0.125553	+	0.992087i	$0.459930\pi$
$390$	1.91080	0.0967574
$391$	1.31370	0.0664368
$392$	21.8310	1.10263
$393$	−12.8713	−0.649269
$394$	−4.85230	−0.244455
$395$	31.0544	1.56251
$396$	10.5311	0.529210
$397$	−12.1030	−0.607432	−0.303716	−	0.952763i	$-0.598227\pi$
$397$	−12.1030	−0.607432	−0.303716	+	0.952763i	$0.598227\pi$
$398$	7.10943	0.356363
$399$	−30.1616	−1.50997
$400$	−0.830726	−0.0415363
$401$	8.52229	0.425583	0.212791	−	0.977098i	$-0.431744\pi$
$401$	8.52229	0.425583	0.212791	+	0.977098i	$0.431744\pi$
$402$	9.29841	0.463762
$403$	0.549740	0.0273845
$404$	−3.67106	−0.182642
$405$	−5.11607	−0.254220
$406$	7.13901	0.354303
$407$	−41.1087	−2.03768
$408$	2.10548	0.104237
$409$	−7.15729	−0.353905	−0.176953	−	0.984219i	$-0.556624\pi$
$409$	−7.15729	−0.353905	−0.176953	+	0.984219i	$0.556624\pi$
$410$	−10.8535	−0.536016
$411$	1.62572	0.0801909
$412$	1.39225	0.0685914
$413$	−39.5787	−1.94754
$414$	−2.31346	−0.113700
$415$	−18.6525	−0.915616
$416$	−5.85261	−0.286948
$417$	17.7272	0.868104
$418$	−25.3198	−1.23843
$419$	−11.6706	−0.570147	−0.285074	−	0.958506i	$-0.592018\pi$
$419$	−11.6706	−0.570147	−0.285074	+	0.958506i	$0.592018\pi$
$420$	13.3285	0.650364
$421$	−17.5870	−0.857137	−0.428569	−	0.903509i	$-0.640982\pi$
$421$	−17.5870	−0.857137	−0.428569	+	0.903509i	$0.640982\pi$
$422$	5.28334	0.257189
$423$	−5.84793	−0.284336
$424$	20.8530	1.01271
$425$	0.732514	0.0355322
$426$	1.61616	0.0783034
$427$	−38.3665	−1.85668
$428$	−26.3271	−1.27257
$429$	−6.56162	−0.316798
$430$	8.72101	0.420565
$431$	−28.1810	−1.35743	−0.678716	−	0.734401i	$-0.737463\pi$
$431$	−28.1810	−1.35743	−0.678716	+	0.734401i	$0.737463\pi$
$432$	4.00879	0.192873
$433$	−28.9345	−1.39050	−0.695252	−	0.718766i	$-0.744707\pi$
$433$	−28.9345	−1.39050	−0.695252	+	0.718766i	$0.744707\pi$
$434$	−1.67389	−0.0803491
$435$	5.74677	0.275537
$436$	−18.1821	−0.870766
$437$	−12.7422	−0.609540
$438$	8.30498	0.396827
$439$	22.3303	1.06577	0.532884	−	0.846188i	$-0.321108\pi$
$439$	22.3303	1.06577	0.532884	+	0.846188i	$0.321108\pi$
$440$	27.2620	1.29966
$441$	−11.8864	−0.566017
$442$	0.496916	0.0236359
$443$	−36.6684	−1.74217	−0.871083	−	0.491135i	$-0.836582\pi$
$443$	−36.6684	−1.74217	−0.871083	+	0.491135i	$0.836582\pi$
$444$	−13.6082	−0.645815
$445$	−24.2293	−1.14858
$446$	8.67932	0.410978
$447$	−19.8682	−0.939732
$448$	12.1737	0.575152
$449$	8.06378	0.380553	0.190277	−	0.981731i	$-0.439061\pi$
$449$	8.06378	0.380553	0.190277	+	0.981731i	$0.439061\pi$
$450$	−1.28998	−0.0608100
$451$	37.2705	1.75500
$452$	9.94327	0.467692
$453$	8.93703	0.419898
$454$	5.13546	0.241019
$455$	7.66449	0.359317
$456$	−20.4219	−0.956344
$457$	34.0372	1.59219	0.796097	−	0.605169i	$-0.206894\pi$
$457$	34.0372	1.59219	0.796097	+	0.605169i	$0.206894\pi$
$458$	−0.945154	−0.0441641
$459$	−3.53486	−0.164993
$460$	5.63079	0.262537
$461$	38.0976	1.77438	0.887191	−	0.461401i	$-0.152653\pi$
$461$	38.0976	1.77438	0.887191	+	0.461401i	$0.152653\pi$
$462$	19.9793	0.929521
$463$	−30.3586	−1.41089	−0.705443	−	0.708767i	$-0.749252\pi$
$463$	−30.3586	−1.41089	−0.705443	+	0.708767i	$0.749252\pi$
$464$	−1.69486	−0.0786817
$465$	−1.34745	−0.0624864
$466$	1.45338	0.0673266
$467$	−25.2438	−1.16815	−0.584073	−	0.811701i	$-0.698542\pi$
$467$	−25.2438	−1.16815	−0.584073	+	0.811701i	$0.698542\pi$
$468$	2.00467	0.0926661
$469$	37.2971	1.72222
$470$	−6.21318	−0.286593
$471$	−5.67411	−0.261449
$472$	−26.7981	−1.23348
$473$	−29.9476	−1.37699
$474$	15.4095	0.707780
$475$	−7.10497	−0.325998
$476$	3.46615	0.158871
$477$	−11.3539	−0.519860
$478$	0.411859	0.0188380
$479$	7.58839	0.346722	0.173361	−	0.984858i	$-0.444537\pi$
$479$	7.58839	0.346722	0.173361	+	0.984858i	$0.444537\pi$
$480$	14.3451	0.654763
$481$	−7.82532	−0.356804
$482$	3.79292	0.172763
$483$	10.0545	0.457498
$484$	−23.1074	−1.05034
$485$	23.8978	1.08514
$486$	10.4311	0.473166
$487$	−37.3602	−1.69295	−0.846477	−	0.532426i	$-0.821280\pi$
$487$	−37.3602	−1.69295	−0.846477	+	0.532426i	$0.821280\pi$
$488$	−25.9773	−1.17594
$489$	9.96417	0.450595
$490$	−12.6287	−0.570508
$491$	24.5747	1.10904	0.554519	−	0.832171i	$-0.312902\pi$
$491$	24.5747	1.10904	0.554519	+	0.832171i	$0.312902\pi$
$492$	12.3376	0.556222
$493$	1.49448	0.0673081
$494$	−4.81980	−0.216853
$495$	−14.8434	−0.667161
$496$	0.397393	0.0178435
$497$	6.48265	0.290786
$498$	−9.25555	−0.414751
$499$	−16.7213	−0.748548	−0.374274	−	0.927318i	$-0.622108\pi$
$499$	−16.7213	−0.748548	−0.374274	+	0.927318i	$0.622108\pi$
$500$	16.8001	0.751324
$501$	1.25439	0.0560421
$502$	−11.7265	−0.523382
$503$	−28.0808	−1.25206	−0.626031	−	0.779798i	$-0.715322\pi$
$503$	−28.0808	−1.25206	−0.626031	+	0.779798i	$0.715322\pi$
$504$	−14.8725	−0.662472
$505$	5.17427	0.230252
$506$	8.44051	0.375226
$507$	−1.24905	−0.0554722
$508$	14.8663	0.659587
$509$	4.24261	0.188050	0.0940252	−	0.995570i	$-0.470027\pi$
$509$	4.24261	0.188050	0.0940252	+	0.995570i	$0.470027\pi$
$510$	−1.21797	−0.0539328
$511$	33.3124	1.47365
$512$	−8.05405	−0.355942
$513$	34.2861	1.51377
$514$	−5.77978	−0.254935
$515$	−1.96235	−0.0864713
$516$	−9.91351	−0.436418
$517$	21.3358	0.938347
$518$	23.8271	1.04690
$519$	−8.69890	−0.381839
$520$	5.18950	0.227574
$521$	19.4450	0.851901	0.425950	−	0.904747i	$-0.359940\pi$
$521$	19.4450	0.851901	0.425950	+	0.904747i	$0.359940\pi$
$522$	−2.63182	−0.115192
$523$	27.5723	1.20565	0.602826	−	0.797873i	$-0.294041\pi$
$523$	27.5723	1.20565	0.602826	+	0.797873i	$0.294041\pi$
$524$	−14.3470	−0.626750
$525$	5.60637	0.244682
$526$	−3.04457	−0.132750
$527$	−0.350412	−0.0152642
$528$	−4.74323	−0.206423
$529$	−18.7523	−0.815319
$530$	−12.0630	−0.523985
$531$	14.5908	0.633188
$532$	−33.6197	−1.45760
$533$	7.09468	0.307305
$534$	−12.0228	−0.520277
$535$	37.1074	1.60429
$536$	25.2533	1.09077
$537$	16.9968	0.733466
$538$	12.6074	0.543545
$539$	43.3665	1.86793
$540$	−15.1511	−0.652000
$541$	24.8550	1.06860	0.534299	−	0.845296i	$-0.320576\pi$
$541$	24.8550	1.06860	0.534299	+	0.845296i	$0.320576\pi$
$542$	−0.538238	−0.0231193
$543$	11.0316	0.473410
$544$	3.73054	0.159945
$545$	25.6273	1.09775
$546$	3.80319	0.162762
$547$	3.34139	0.142868	0.0714339	−	0.997445i	$-0.477243\pi$
$547$	3.34139	0.142868	0.0714339	+	0.997445i	$0.477243\pi$
$548$	1.81211	0.0774095
$549$	14.1439	0.603648
$550$	4.70639	0.200681
$551$	−14.4956	−0.617534
$552$	6.80776	0.289758
$553$	61.8094	2.62840
$554$	6.36020	0.270219
$555$	19.1804	0.814161
$556$	19.7596	0.837995
$557$	−3.15315	−0.133603	−0.0668016	−	0.997766i	$-0.521279\pi$
$557$	−3.15315	−0.133603	−0.0668016	+	0.997766i	$0.521279\pi$
$558$	0.617084	0.0261232
$559$	−5.70072	−0.241115
$560$	5.54047	0.234128
$561$	4.18247	0.176584
$562$	−21.1591	−0.892544
$563$	26.9324	1.13507	0.567533	−	0.823350i	$-0.307898\pi$
$563$	26.9324	1.13507	0.567533	+	0.823350i	$0.307898\pi$
$564$	7.06276	0.297396
$565$	−14.0148	−0.589607
$566$	8.93441	0.375541
$567$	−10.1828	−0.427639
$568$	4.38929	0.184170
$569$	−25.5656	−1.07177	−0.535884	−	0.844292i	$-0.680021\pi$
$569$	−25.5656	−1.07177	−0.535884	+	0.844292i	$0.680021\pi$
$570$	11.8136	0.494819
$571$	−16.9446	−0.709108	−0.354554	−	0.935036i	$-0.615367\pi$
$571$	−16.9446	−0.709108	−0.354554	+	0.935036i	$0.615367\pi$
$572$	−7.31392	−0.305810
$573$	−21.4260	−0.895085
$574$	−21.6024	−0.901667
$575$	2.36848	0.0987725
$576$	−4.48786	−0.186994
$577$	−0.153495	−0.00639007	−0.00319503	−	0.999995i	$-0.501017\pi$
$577$	−0.153495	−0.00639007	−0.00319503	+	0.999995i	$0.501017\pi$
$578$	12.9361	0.538072
$579$	17.0870	0.710112
$580$	6.40565	0.265980
$581$	−37.1252	−1.54021
$582$	11.8583	0.491543
$583$	41.4240	1.71561
$584$	22.5552	0.933343
$585$	−2.82554	−0.116822
$586$	4.16827	0.172190
$587$	−32.6914	−1.34932	−0.674659	−	0.738129i	$-0.735709\pi$
$587$	−32.6914	−1.34932	−0.674659	+	0.738129i	$0.735709\pi$
$588$	14.3556	0.592014
$589$	3.39879	0.140045
$590$	15.5021	0.638212
$591$	−7.77438	−0.319795
$592$	−5.65673	−0.232490
$593$	24.1377	0.991215	0.495608	−	0.868547i	$-0.334945\pi$
$593$	24.1377	0.991215	0.495608	+	0.868547i	$0.334945\pi$
$594$	−22.7114	−0.931859
$595$	−4.88545	−0.200284
$596$	−22.1461	−0.907138
$597$	11.3908	0.466193
$598$	1.60671	0.0657031
$599$	20.3757	0.832529	0.416265	−	0.909243i	$-0.363339\pi$
$599$	20.3757	0.832529	0.416265	+	0.909243i	$0.363339\pi$
$600$	3.79598	0.154970
$601$	39.4917	1.61090	0.805449	−	0.592665i	$-0.201924\pi$
$601$	39.4917	1.61090	0.805449	+	0.592665i	$0.201924\pi$
$602$	17.3580	0.707458
$603$	−13.7497	−0.559931
$604$	9.96167	0.405334
$605$	32.5693	1.32413
$606$	2.56752	0.104298
$607$	25.5073	1.03531	0.517654	−	0.855590i	$-0.326805\pi$
$607$	25.5073	1.03531	0.517654	+	0.855590i	$0.326805\pi$
$608$	−36.1841	−1.46746
$609$	11.4382	0.463498
$610$	15.0273	0.608438
$611$	4.06141	0.164307
$612$	−1.27781	−0.0516523
$613$	−44.8991	−1.81346	−0.906729	−	0.421714i	$-0.861429\pi$
$613$	−44.8991	−1.81346	−0.906729	+	0.421714i	$0.861429\pi$
$614$	3.12061	0.125938
$615$	−17.3895	−0.701214
$616$	54.2612	2.18624
$617$	−28.3288	−1.14048	−0.570238	−	0.821480i	$-0.693149\pi$
$617$	−28.3288	−1.14048	−0.570238	+	0.821480i	$0.693149\pi$
$618$	−0.973735	−0.0391694
$619$	−21.5295	−0.865346	−0.432673	−	0.901551i	$-0.642429\pi$
$619$	−21.5295	−0.865346	−0.432673	+	0.901551i	$0.642429\pi$
$620$	−1.50193	−0.0603191
$621$	−11.4295	−0.458648
$622$	−19.2408	−0.771485
$623$	−48.2250	−1.93209
$624$	−0.902907	−0.0361452
$625$	−17.9334	−0.717334
$626$	−17.1606	−0.685875
$627$	−40.5676	−1.62011
$628$	−6.32465	−0.252381
$629$	4.98797	0.198883
$630$	8.60340	0.342768
$631$	−28.7988	−1.14646	−0.573231	−	0.819394i	$-0.694310\pi$
$631$	−28.7988	−1.14646	−0.573231	+	0.819394i	$0.694310\pi$
$632$	41.8501	1.66471
$633$	8.46499	0.336453
$634$	−20.8839	−0.829406
$635$	−20.9537	−0.831524
$636$	13.7125	0.543737
$637$	8.25511	0.327080
$638$	9.60201	0.380147
$639$	−2.38985	−0.0945409
$640$	18.2015	0.719479
$641$	−15.3084	−0.604646	−0.302323	−	0.953206i	$-0.597762\pi$
$641$	−15.3084	−0.604646	−0.302323	+	0.953206i	$0.597762\pi$
$642$	18.4130	0.726704
$643$	−10.4762	−0.413142	−0.206571	−	0.978432i	$-0.566230\pi$
$643$	−10.4762	−0.413142	−0.206571	+	0.978432i	$0.566230\pi$
$644$	11.2073	0.441630
$645$	13.9728	0.550180
$646$	3.07221	0.120874
$647$	−36.5387	−1.43648	−0.718242	−	0.695794i	$-0.755053\pi$
$647$	−36.5387	−1.43648	−0.718242	+	0.695794i	$0.755053\pi$
$648$	−6.89462	−0.270846
$649$	−53.2336	−2.08960
$650$	0.895892	0.0351398
$651$	−2.68191	−0.105112
$652$	11.1066	0.434967
$653$	30.4578	1.19190	0.595952	−	0.803020i	$-0.296775\pi$
$653$	30.4578	1.19190	0.595952	+	0.803020i	$0.296775\pi$
$654$	12.7165	0.497254
$655$	20.2217	0.790126
$656$	5.12857	0.200237
$657$	−12.2807	−0.479116
$658$	−12.3665	−0.482095
$659$	34.6954	1.35154	0.675771	−	0.737111i	$-0.263811\pi$
$659$	34.6954	1.35154	0.675771	+	0.737111i	$0.263811\pi$
$660$	17.9269	0.697803
$661$	−34.7796	−1.35277	−0.676386	−	0.736548i	$-0.736455\pi$
$661$	−34.7796	−1.35277	−0.676386	+	0.736548i	$0.736455\pi$
$662$	−11.0803	−0.430650
$663$	0.796161	0.0309203
$664$	−25.1369	−0.975500
$665$	47.3861	1.83755
$666$	−8.78393	−0.340370
$667$	4.83220	0.187103
$668$	1.39821	0.0540984
$669$	13.9060	0.537639
$670$	−14.6085	−0.564375
$671$	−51.6032	−1.99212
$672$	28.5520	1.10142
$673$	35.8975	1.38375	0.691873	−	0.722019i	$-0.256786\pi$
$673$	35.8975	1.38375	0.691873	+	0.722019i	$0.256786\pi$
$674$	−5.04142	−0.194188
$675$	−6.37301	−0.245297
$676$	−1.39225	−0.0535482
$677$	5.27414	0.202702	0.101351	−	0.994851i	$-0.467684\pi$
$677$	5.27414	0.202702	0.101351	+	0.994851i	$0.467684\pi$
$678$	−6.95427	−0.267077
$679$	47.5653	1.82539
$680$	−3.30786	−0.126851
$681$	8.22805	0.315300
$682$	−2.25139	−0.0862101
$683$	43.6948	1.67194	0.835968	−	0.548779i	$-0.184907\pi$
$683$	43.6948	1.67194	0.835968	+	0.548779i	$0.184907\pi$
$684$	12.3940	0.473897
$685$	−2.55413	−0.0975881
$686$	−3.82166	−0.145912
$687$	−1.51433	−0.0577753
$688$	−4.12091	−0.157108
$689$	7.88533	0.300407
$690$	−3.93814	−0.149923
$691$	41.8120	1.59060	0.795302	−	0.606213i	$-0.207312\pi$
$691$	41.8120	1.59060	0.795302	+	0.606213i	$0.207312\pi$
$692$	−9.69623	−0.368595
$693$	−29.5437	−1.12227
$694$	−15.1365	−0.574575
$695$	−27.8507	−1.05644
$696$	7.74458	0.293558
$697$	−4.52225	−0.171292
$698$	−14.5797	−0.551851
$699$	2.32861	0.0880763
$700$	6.24914	0.236195
$701$	−15.2390	−0.575570	−0.287785	−	0.957695i	$-0.592919\pi$
$701$	−15.2390	−0.575570	−0.287785	+	0.957695i	$0.592919\pi$
$702$	−4.32326	−0.163171
$703$	−48.3804	−1.82470
$704$	16.3737	0.617106
$705$	−9.95478	−0.374919
$706$	11.6697	0.439193
$707$	10.2987	0.387321
$708$	−17.6219	−0.662270
$709$	15.4927	0.581839	0.290920	−	0.956748i	$-0.406039\pi$
$709$	15.4927	0.581839	0.290920	+	0.956748i	$0.406039\pi$
$710$	−2.53911	−0.0952911
$711$	−22.7862	−0.854550
$712$	−32.6523	−1.22370
$713$	−1.13301	−0.0424314
$714$	−2.42421	−0.0907237
$715$	10.3088	0.385527
$716$	18.9455	0.708027
$717$	0.659882	0.0246437
$718$	−26.4730	−0.987965
$719$	−19.8518	−0.740346	−0.370173	−	0.928963i	$-0.620702\pi$
$719$	−19.8518	−0.740346	−0.370173	+	0.928963i	$0.620702\pi$
$720$	−2.04251	−0.0761200
$721$	−3.90578	−0.145459
$722$	−14.9866	−0.557744
$723$	6.07703	0.226007
$724$	12.2964	0.456990
$725$	2.69441	0.100068
$726$	16.1612	0.599798
$727$	−26.6490	−0.988357	−0.494179	−	0.869360i	$-0.664531\pi$
$727$	−26.6490	−0.988357	−0.494179	+	0.869360i	$0.664531\pi$
$728$	10.3290	0.382817
$729$	24.5342	0.908674
$730$	−13.0477	−0.482918
$731$	3.63372	0.134398
$732$	−17.0821	−0.631374
$733$	10.9453	0.404275	0.202138	−	0.979357i	$-0.435211\pi$
$733$	10.9453	0.404275	0.202138	+	0.979357i	$0.435211\pi$
$734$	7.60794	0.280814
$735$	−20.2338	−0.746336
$736$	12.0622	0.444617
$737$	50.1649	1.84785
$738$	7.96379	0.293151
$739$	25.3343	0.931937	0.465969	−	0.884801i	$-0.345706\pi$
$739$	25.3343	0.931937	0.465969	+	0.884801i	$0.345706\pi$
$740$	21.3794	0.785923
$741$	−7.72231	−0.283686
$742$	−24.0098	−0.881428
$743$	−19.2100	−0.704747	−0.352374	−	0.935859i	$-0.614625\pi$
$743$	−19.2100	−0.704747	−0.352374	+	0.935859i	$0.614625\pi$
$744$	−1.81588	−0.0665732
$745$	31.2143	1.14360
$746$	19.3201	0.707357
$747$	13.6863	0.500757
$748$	4.66199	0.170459
$749$	73.8571	2.69868
$750$	−11.7499	−0.429046
$751$	−47.2873	−1.72554	−0.862770	−	0.505597i	$-0.831272\pi$
$751$	−47.2873	−1.72554	−0.862770	+	0.505597i	$0.831272\pi$
$752$	2.93589	0.107061
$753$	−18.7883	−0.684685
$754$	1.82781	0.0665648
$755$	−14.0407	−0.510994
$756$	−30.1562	−1.09677
$757$	−9.71997	−0.353278	−0.176639	−	0.984276i	$-0.556523\pi$
$757$	−9.71997	−0.353278	−0.176639	+	0.984276i	$0.556523\pi$
$758$	22.4012	0.813649
$759$	13.5234	0.490869
$760$	32.0843	1.16382
$761$	−38.8196	−1.40721	−0.703604	−	0.710592i	$-0.748427\pi$
$761$	−38.8196	−1.40721	−0.703604	+	0.710592i	$0.748427\pi$
$762$	−10.3974	−0.376660
$763$	51.0075	1.84660
$764$	−23.8825	−0.864040
$765$	1.80104	0.0651167
$766$	−4.42342	−0.159825
$767$	−10.1334	−0.365895
$768$	16.8179	0.606865
$769$	41.5317	1.49767	0.748836	−	0.662755i	$-0.230613\pi$
$769$	41.5317	1.49767	0.748836	+	0.662755i	$0.230613\pi$
$770$	−31.3889	−1.13118
$771$	−9.26040	−0.333505
$772$	19.0460	0.685482
$773$	26.3734	0.948585	0.474293	−	0.880367i	$-0.342704\pi$
$773$	26.3734	0.948585	0.474293	+	0.880367i	$0.342704\pi$
$774$	−6.39907	−0.230010
$775$	−0.631759	−0.0226935
$776$	32.2057	1.15612
$777$	38.1759	1.36955
$778$	−3.86094	−0.138421
$779$	43.8632	1.57156
$780$	3.41250	0.122187
$781$	8.71919	0.311997
$782$	−1.02414	−0.0366231
$783$	−13.0023	−0.464664
$784$	5.96742	0.213122
$785$	8.91444	0.318170
$786$	10.0342	0.357908
$787$	−42.8665	−1.52803	−0.764014	−	0.645200i	$-0.776774\pi$
$787$	−42.8665	−1.52803	−0.764014	+	0.645200i	$0.776774\pi$
$788$	−8.66572	−0.308703
$789$	−4.87802	−0.173662
$790$	−24.2094	−0.861331
$791$	−27.8945	−0.991814
$792$	−20.0035	−0.710795
$793$	−9.82301	−0.348825
$794$	9.43527	0.334845
$795$	−19.3275	−0.685475
$796$	12.6967	0.450023
$797$	−10.4650	−0.370689	−0.185345	−	0.982674i	$-0.559340\pi$
$797$	−10.4650	−0.370689	−0.185345	+	0.982674i	$0.559340\pi$
$798$	23.5134	0.832367
$799$	−2.58880	−0.0915851
$800$	6.72581	0.237793
$801$	17.7783	0.628165
$802$	−6.64382	−0.234601
$803$	44.8053	1.58115
$804$	16.6060	0.585649
$805$	−15.7964	−0.556751
$806$	−0.428567	−0.0150956
$807$	20.1997	0.711062
$808$	6.97305	0.245311
$809$	12.7501	0.448270	0.224135	−	0.974558i	$-0.428044\pi$
$809$	12.7501	0.448270	0.224135	+	0.974558i	$0.428044\pi$
$810$	3.98839	0.140138
$811$	36.6465	1.28683	0.643416	−	0.765517i	$-0.277516\pi$
$811$	36.6465	1.28683	0.643416	+	0.765517i	$0.277516\pi$
$812$	12.7496	0.447422
$813$	−0.862367	−0.0302445
$814$	32.0476	1.12327
$815$	−15.6544	−0.548351
$816$	0.575525	0.0201474
$817$	−35.2450	−1.23307
$818$	5.57969	0.195089
$819$	−5.62384	−0.196513
$820$	−19.3833	−0.676893
$821$	−30.4988	−1.06442	−0.532208	−	0.846613i	$-0.678638\pi$
$821$	−30.4988	−1.06442	−0.532208	+	0.846613i	$0.678638\pi$
$822$	−1.26738	−0.0442050
$823$	31.4016	1.09459	0.547295	−	0.836940i	$-0.315658\pi$
$823$	31.4016	1.09459	0.547295	+	0.836940i	$0.315658\pi$
$824$	−2.64454	−0.0921268
$825$	7.54060	0.262530
$826$	30.8548	1.07358
$827$	−31.9229	−1.11007	−0.555033	−	0.831828i	$-0.687294\pi$
$827$	−31.9229	−1.11007	−0.555033	+	0.831828i	$0.687294\pi$
$828$	−4.13161	−0.143583
$829$	−7.36901	−0.255936	−0.127968	−	0.991778i	$-0.540846\pi$
$829$	−7.36901	−0.255936	−0.127968	+	0.991778i	$0.540846\pi$
$830$	14.5411	0.504730
$831$	10.1903	0.353499
$832$	3.11684	0.108057
$833$	−5.26192	−0.182315
$834$	−13.8198	−0.478540
$835$	−1.97074	−0.0682004
$836$	−45.2187	−1.56392
$837$	3.04865	0.105377
$838$	9.09819	0.314292
$839$	−53.0777	−1.83244	−0.916222	−	0.400671i	$-0.868777\pi$
$839$	−53.0777	−1.83244	−0.916222	+	0.400671i	$0.868777\pi$
$840$	−25.3170	−0.873520
$841$	−23.5028	−0.810443
$842$	13.7105	0.472494
$843$	−33.9013	−1.16762
$844$	9.43551	0.324784
$845$	1.96235	0.0675068
$846$	4.55894	0.156740
$847$	64.8247	2.22740
$848$	5.70011	0.195743
$849$	14.3148	0.491281
$850$	−0.571054	−0.0195870
$851$	16.1279	0.552857
$852$	2.88630	0.0988831
$853$	−22.8190	−0.781307	−0.390654	−	0.920538i	$-0.627751\pi$
$853$	−22.8190	−0.781307	−0.390654	+	0.920538i	$0.627751\pi$
$854$	29.9098	1.02349
$855$	−17.4690	−0.597428
$856$	50.0074	1.70922
$857$	−33.5698	−1.14672	−0.573362	−	0.819302i	$-0.694361\pi$
$857$	−33.5698	−1.14672	−0.573362	+	0.819302i	$0.694361\pi$
$858$	5.11532	0.174634
$859$	−6.19625	−0.211413	−0.105707	−	0.994397i	$-0.533710\pi$
$859$	−6.19625	−0.211413	−0.105707	+	0.994397i	$0.533710\pi$
$860$	15.5748	0.531098
$861$	−34.6115	−1.17956
$862$	21.9694	0.748281
$863$	−6.82954	−0.232480	−0.116240	−	0.993221i	$-0.537084\pi$
$863$	−6.82954	−0.232480	−0.116240	+	0.993221i	$0.537084\pi$
$864$	−32.4564	−1.10419
$865$	13.6666	0.464678
$866$	22.5568	0.766512
$867$	20.7263	0.703903
$868$	−2.98939	−0.101467
$869$	83.1340	2.82013
$870$	−4.48008	−0.151889
$871$	9.54922	0.323563
$872$	34.5363	1.16955
$873$	−17.5351	−0.593473
$874$	9.93355	0.336007
$875$	−47.1305	−1.59330
$876$	14.8319	0.501122
$877$	−18.3496	−0.619621	−0.309810	−	0.950798i	$-0.600266\pi$
$877$	−18.3496	−0.619621	−0.309810	+	0.950798i	$0.600266\pi$
$878$	−17.4083	−0.587502
$879$	6.67842	0.225258
$880$	7.45197	0.251206
$881$	−33.2710	−1.12093	−0.560465	−	0.828178i	$-0.689377\pi$
$881$	−33.2710	−1.12093	−0.560465	+	0.828178i	$0.689377\pi$
$882$	9.26638	0.312015
$883$	−33.1702	−1.11627	−0.558133	−	0.829752i	$-0.688482\pi$
$883$	−33.1702	−1.11627	−0.558133	+	0.829752i	$0.688482\pi$
$884$	0.887442	0.0298479
$885$	24.8376	0.834906
$886$	28.5860	0.960364
$887$	37.6508	1.26419	0.632095	−	0.774891i	$-0.282195\pi$
$887$	37.6508	1.26419	0.632095	+	0.774891i	$0.282195\pi$
$888$	25.8482	0.867410
$889$	−41.7055	−1.39876
$890$	18.8887	0.633150
$891$	−13.6960	−0.458832
$892$	15.5004	0.518991
$893$	25.1099	0.840270
$894$	15.4888	0.518024
$895$	−26.7032	−0.892590
$896$	36.2277	1.21028
$897$	2.57427	0.0859525
$898$	−6.28637	−0.209779
$899$	−1.28892	−0.0429879
$900$	−2.30377	−0.0767922
$901$	−5.02622	−0.167448
$902$	−29.0554	−0.967438
$903$	27.8110	0.925493
$904$	−18.8869	−0.628169
$905$	−17.3314	−0.576115
$906$	−6.96714	−0.231468
$907$	56.8033	1.88612	0.943061	−	0.332621i	$-0.107933\pi$
$907$	56.8033	1.88612	0.943061	+	0.332621i	$0.107933\pi$
$908$	9.17141	0.304364
$909$	−3.79664	−0.125926
$910$	−5.97509	−0.198072
$911$	3.88518	0.128722	0.0643609	−	0.997927i	$-0.479499\pi$
$911$	3.88518	0.128722	0.0643609	+	0.997927i	$0.479499\pi$
$912$	−5.58227	−0.184847
$913$	−49.9337	−1.65256
$914$	−26.5348	−0.877693
$915$	24.0768	0.795956
$916$	−1.68795	−0.0557714
$917$	40.2485	1.32912
$918$	2.75571	0.0909519
$919$	−25.0385	−0.825944	−0.412972	−	0.910744i	$-0.635509\pi$
$919$	−25.0385	−0.825944	−0.412972	+	0.910744i	$0.635509\pi$
$920$	−10.6955	−0.352620
$921$	4.99986	0.164751
$922$	−29.7002	−0.978124
$923$	1.65976	0.0546316
$924$	35.6810	1.17382
$925$	8.99283	0.295682
$926$	23.6670	0.777746
$927$	1.43988	0.0472918
$928$	13.7221	0.450448
$929$	31.9404	1.04793	0.523966	−	0.851739i	$-0.324452\pi$
$929$	31.9404	1.04793	0.523966	+	0.851739i	$0.324452\pi$
$930$	1.05045	0.0344455
$931$	51.0377	1.67269
$932$	2.59559	0.0850215
$933$	−30.8277	−1.00925
$934$	19.6796	0.643937
$935$	−6.57096	−0.214893
$936$	−3.80781	−0.124462
$937$	−27.2350	−0.889727	−0.444864	−	0.895598i	$-0.646748\pi$
$937$	−27.2350	−0.889727	−0.444864	+	0.895598i	$0.646748\pi$
$938$	−29.0761	−0.949370
$939$	−27.4948	−0.897258
$940$	−11.0961	−0.361915
$941$	51.6632	1.68417	0.842085	−	0.539344i	$-0.181328\pi$
$941$	51.6632	1.68417	0.842085	+	0.539344i	$0.181328\pi$
$942$	4.42343	0.144123
$943$	−14.6221	−0.476160
$944$	−7.32517	−0.238414
$945$	42.5044	1.38267
$946$	23.3466	0.759063
$947$	−25.2012	−0.818929	−0.409465	−	0.912326i	$-0.634284\pi$
$947$	−25.2012	−0.818929	−0.409465	+	0.912326i	$0.634284\pi$
$948$	27.5197	0.893800
$949$	8.52900	0.276863
$950$	5.53890	0.179706
$951$	−33.4603	−1.08502
$952$	−6.58383	−0.213383
$953$	−31.7437	−1.02828	−0.514140	−	0.857706i	$-0.671889\pi$
$953$	−31.7437	−1.02828	−0.514140	+	0.857706i	$0.671889\pi$
$954$	8.85130	0.286571
$955$	33.6619	1.08927
$956$	0.735538	0.0237890
$957$	15.3844	0.497307
$958$	−5.91577	−0.191130
$959$	−5.08363	−0.164159
$960$	−7.63957	−0.246566
$961$	−30.6978	−0.990251
$962$	6.10047	0.196687
$963$	−27.2277	−0.877399
$964$	6.77377	0.218168
$965$	−26.8449	−0.864169
$966$	−7.83833	−0.252194
$967$	−38.1765	−1.22767	−0.613836	−	0.789433i	$-0.710374\pi$
$967$	−38.1765	−1.22767	−0.613836	+	0.789433i	$0.710374\pi$
$968$	43.8917	1.41073
$969$	4.92231	0.158127
$970$	−18.6303	−0.598183
$971$	41.7737	1.34058	0.670290	−	0.742099i	$-0.266170\pi$
$971$	41.7737	1.34058	0.670290	+	0.742099i	$0.266170\pi$
$972$	18.6290	0.597525
$973$	−55.4330	−1.77710
$974$	29.1253	0.933235
$975$	1.43540	0.0459697
$976$	−7.10081	−0.227291
$977$	−42.2121	−1.35048	−0.675242	−	0.737597i	$-0.735961\pi$
$977$	−42.2121	−1.35048	−0.675242	+	0.737597i	$0.735961\pi$
$978$	−7.76788	−0.248389
$979$	−64.8629	−2.07303
$980$	−22.5537	−0.720450
$981$	−18.8041	−0.600368
$982$	−19.1579	−0.611354
$983$	−15.0720	−0.480721	−0.240360	−	0.970684i	$-0.577266\pi$
$983$	−15.0720	−0.480721	−0.240360	+	0.970684i	$0.577266\pi$
$984$	−23.4348	−0.747075
$985$	12.2141	0.389174
$986$	−1.16507	−0.0371034
$987$	−19.8136	−0.630675
$988$	−8.60768	−0.273847
$989$	11.7491	0.373600
$990$	11.5716	0.367770
$991$	−45.9929	−1.46101	−0.730506	−	0.682906i	$-0.760716\pi$
$991$	−45.9929	−1.46101	−0.730506	+	0.682906i	$0.760716\pi$
$992$	−3.21741	−0.102153
$993$	−17.7530	−0.563374
$994$	−5.05375	−0.160295
$995$	−17.8957	−0.567332
$996$	−16.5295	−0.523757
$997$	−41.5175	−1.31487	−0.657435	−	0.753511i	$-0.728359\pi$
$997$	−41.5175	−1.31487	−0.657435	+	0.753511i	$0.728359\pi$
$998$	13.0356	0.412635
$999$	−43.3963	−1.37300

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1339.2.a.f.1.10	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1339.2.a.f.1.10	✓	28	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:	\( N \)	\(=\)	\( 1339 = 13 \cdot 103 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1339.a (trivial)

\(f(q)\)	\(=\)	\(q-0.779581 q^{2} -1.24905 q^{3} -1.39225 q^{4} +1.96235 q^{5} +0.973735 q^{6} +3.90578 q^{7} +2.64454 q^{8} -1.43988 q^{9} +O(q^{10})\)	\(q-0.779581 q^{2} -1.24905 q^{3} -1.39225 q^{4} +1.96235 q^{5} +0.973735 q^{6} +3.90578 q^{7} +2.64454 q^{8} -1.43988 q^{9} -1.52981 q^{10} +5.25330 q^{11} +1.73899 q^{12} +1.00000 q^{13} -3.04487 q^{14} -2.45107 q^{15} +0.722875 q^{16} -0.637414 q^{17} +1.12250 q^{18} +6.18255 q^{19} -2.73208 q^{20} -4.87851 q^{21} -4.09537 q^{22} -2.06099 q^{23} -3.30315 q^{24} -1.14920 q^{25} -0.779581 q^{26} +5.54562 q^{27} -5.43783 q^{28} -2.34460 q^{29} +1.91080 q^{30} +0.549740 q^{31} -5.85261 q^{32} -6.56162 q^{33} +0.496916 q^{34} +7.66449 q^{35} +2.00467 q^{36} -7.82532 q^{37} -4.81980 q^{38} -1.24905 q^{39} +5.18950 q^{40} +7.09468 q^{41} +3.80319 q^{42} -5.70072 q^{43} -7.31392 q^{44} -2.82554 q^{45} +1.60671 q^{46} +4.06141 q^{47} -0.902907 q^{48} +8.25511 q^{49} +0.895892 q^{50} +0.796161 q^{51} -1.39225 q^{52} +7.88533 q^{53} -4.32326 q^{54} +10.3088 q^{55} +10.3290 q^{56} -7.72231 q^{57} +1.82781 q^{58} -10.1334 q^{59} +3.41250 q^{60} -9.82301 q^{61} -0.428567 q^{62} -5.62384 q^{63} +3.11684 q^{64} +1.96235 q^{65} +5.11532 q^{66} +9.54922 q^{67} +0.887442 q^{68} +2.57427 q^{69} -5.97509 q^{70} +1.65976 q^{71} -3.80781 q^{72} +8.52900 q^{73} +6.10047 q^{74} +1.43540 q^{75} -8.60768 q^{76} +20.5182 q^{77} +0.973735 q^{78} +15.8251 q^{79} +1.41853 q^{80} -2.60712 q^{81} -5.53088 q^{82} -9.50521 q^{83} +6.79212 q^{84} -1.25083 q^{85} +4.44418 q^{86} +2.92852 q^{87} +13.8925 q^{88} -12.3471 q^{89} +2.20274 q^{90} +3.90578 q^{91} +2.86942 q^{92} -0.686651 q^{93} -3.16620 q^{94} +12.1323 q^{95} +7.31020 q^{96} +12.1782 q^{97} -6.43553 q^{98} -7.56410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 28 q + 6 q^{2} + 5 q^{3} + 32 q^{4} + 27 q^{5} + 9 q^{6} + 6 q^{7} + 21 q^{8} + 39 q^{9}+O(q^{10}) \) 28 * q + 6 * q^2 + 5 * q^3 + 32 * q^4 + 27 * q^5 + 9 * q^6 + 6 * q^7 + 21 * q^8 + 39 * q^9	\( 28 q + 6 q^{2} + 5 q^{3} + 32 q^{4} + 27 q^{5} + 9 q^{6} + 6 q^{7} + 21 q^{8} + 39 q^{9} + 5 q^{10} + 9 q^{11} + 15 q^{12} + 28 q^{13} + 20 q^{14} + 4 q^{15} + 32 q^{16} - 7 q^{17} + 10 q^{18} - 3 q^{19} + 53 q^{20} + 25 q^{21} - 14 q^{22} - 6 q^{23} + 10 q^{24} + 45 q^{25} + 6 q^{26} + 8 q^{27} - 12 q^{28} + 25 q^{29} - 43 q^{30} + q^{31} + 28 q^{32} + 17 q^{33} + 10 q^{34} + 11 q^{35} + 27 q^{36} + 18 q^{37} + 2 q^{38} + 5 q^{39} + 37 q^{40} + 56 q^{41} - 29 q^{42} - 30 q^{43} + 30 q^{44} + 38 q^{45} - 27 q^{46} + 40 q^{47} + 19 q^{48} + 36 q^{49} + 20 q^{50} - 18 q^{51} + 32 q^{52} + 51 q^{53} - 36 q^{54} - 5 q^{55} - 13 q^{56} + 31 q^{57} - 29 q^{58} + 29 q^{59} + 48 q^{60} + 16 q^{61} - 38 q^{62} - 23 q^{63} - 5 q^{64} + 27 q^{65} + 25 q^{66} - 2 q^{67} - 43 q^{68} + 38 q^{69} + 10 q^{70} + 34 q^{71} - 28 q^{72} + 14 q^{73} + 21 q^{74} + 19 q^{75} + 5 q^{76} + 34 q^{77} + 9 q^{78} + 3 q^{79} + 58 q^{80} + 4 q^{81} - 9 q^{82} + 38 q^{83} - 78 q^{84} - 27 q^{85} + 49 q^{86} + 8 q^{87} - 25 q^{88} + 125 q^{89} - 74 q^{90} + 6 q^{91} - 35 q^{92} + 36 q^{93} - q^{94} - 12 q^{95} + 26 q^{96} - 2 q^{97} + 18 q^{98} + 56 q^{99}+O(q^{100}) \) 28 * q + 6 * q^2 + 5 * q^3 + 32 * q^4 + 27 * q^5 + 9 * q^6 + 6 * q^7 + 21 * q^8 + 39 * q^9 + 5 * q^10 + 9 * q^11 + 15 * q^12 + 28 * q^13 + 20 * q^14 + 4 * q^15 + 32 * q^16 - 7 * q^17 + 10 * q^18 - 3 * q^19 + 53 * q^20 + 25 * q^21 - 14 * q^22 - 6 * q^23 + 10 * q^24 + 45 * q^25 + 6 * q^26 + 8 * q^27 - 12 * q^28 + 25 * q^29 - 43 * q^30 + q^31 + 28 * q^32 + 17 * q^33 + 10 * q^34 + 11 * q^35 + 27 * q^36 + 18 * q^37 + 2 * q^38 + 5 * q^39 + 37 * q^40 + 56 * q^41 - 29 * q^42 - 30 * q^43 + 30 * q^44 + 38 * q^45 - 27 * q^46 + 40 * q^47 + 19 * q^48 + 36 * q^49 + 20 * q^50 - 18 * q^51 + 32 * q^52 + 51 * q^53 - 36 * q^54 - 5 * q^55 - 13 * q^56 + 31 * q^57 - 29 * q^58 + 29 * q^59 + 48 * q^60 + 16 * q^61 - 38 * q^62 - 23 * q^63 - 5 * q^64 + 27 * q^65 + 25 * q^66 - 2 * q^67 - 43 * q^68 + 38 * q^69 + 10 * q^70 + 34 * q^71 - 28 * q^72 + 14 * q^73 + 21 * q^74 + 19 * q^75 + 5 * q^76 + 34 * q^77 + 9 * q^78 + 3 * q^79 + 58 * q^80 + 4 * q^81 - 9 * q^82 + 38 * q^83 - 78 * q^84 - 27 * q^85 + 49 * q^86 + 8 * q^87 - 25 * q^88 + 125 * q^89 - 74 * q^90 + 6 * q^91 - 35 * q^92 + 36 * q^93 - q^94 - 12 * q^95 + 26 * q^96 - 2 * q^97 + 18 * q^98 + 56 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more