LMFDB - Embedded newform 845.2.a.j.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [845,2,Mod(1,845)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(845, 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("845.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 845 = 5 \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	845.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:	$6.74735897080$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{14})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 2x + 1 $ x^3 - x^2 - 2*x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.445042$ of defining polynomial
Character	$\chi$	$=$	845.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.445042 q^{2} -3.24698 q^{3} -1.80194 q^{4} -1.00000 q^{5} -1.44504 q^{6} +3.24698 q^{7} -1.69202 q^{8} +7.54288 q^{9} +O(q^{10})$	\(q+0.445042 q^{2} -3.24698 q^{3} -1.80194 q^{4} -1.00000 q^{5} -1.44504 q^{6} +3.24698 q^{7} -1.69202 q^{8} +7.54288 q^{9} -0.445042 q^{10} -0.692021 q^{11} +5.85086 q^{12} +1.44504 q^{14} +3.24698 q^{15} +2.85086 q^{16} +3.74094 q^{17} +3.35690 q^{18} -1.53319 q^{19} +1.80194 q^{20} -10.5429 q^{21} -0.307979 q^{22} -1.22521 q^{23} +5.49396 q^{24} +1.00000 q^{25} -14.7506 q^{27} -5.85086 q^{28} -6.07069 q^{29} +1.44504 q^{30} -8.45473 q^{31} +4.65279 q^{32} +2.24698 q^{33} +1.66487 q^{34} -3.24698 q^{35} -13.5918 q^{36} -1.89008 q^{37} -0.682333 q^{38} +1.69202 q^{40} -0.457123 q^{41} -4.69202 q^{42} -6.19806 q^{43} +1.24698 q^{44} -7.54288 q^{45} -0.545269 q^{46} +11.5429 q^{47} -9.25667 q^{48} +3.54288 q^{49} +0.445042 q^{50} -12.1468 q^{51} +0.801938 q^{53} -6.56465 q^{54} +0.692021 q^{55} -5.49396 q^{56} +4.97823 q^{57} -2.70171 q^{58} +6.60388 q^{59} -5.85086 q^{60} -4.19806 q^{61} -3.76271 q^{62} +24.4916 q^{63} -3.63102 q^{64} +1.00000 q^{66} -13.8116 q^{67} -6.74094 q^{68} +3.97823 q^{69} -1.44504 q^{70} -9.87263 q^{71} -12.7627 q^{72} +8.05429 q^{73} -0.841166 q^{74} -3.24698 q^{75} +2.76271 q^{76} -2.24698 q^{77} -16.5157 q^{79} -2.85086 q^{80} +25.2664 q^{81} -0.203439 q^{82} +6.17092 q^{83} +18.9976 q^{84} -3.74094 q^{85} -2.75840 q^{86} +19.7114 q^{87} +1.17092 q^{88} -10.5254 q^{89} -3.35690 q^{90} +2.20775 q^{92} +27.4523 q^{93} +5.13706 q^{94} +1.53319 q^{95} -15.1075 q^{96} -3.45473 q^{97} +1.57673 q^{98} -5.21983 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} - 5 q^{3} - q^{4} - 3 q^{5} - 4 q^{6} + 5 q^{7} + 4 q^{9}+O(q^{10}) $ 3 * q + q^2 - 5 * q^3 - q^4 - 3 * q^5 - 4 * q^6 + 5 * q^7 + 4 * q^9	$ 3 q + q^{2} - 5 q^{3} - q^{4} - 3 q^{5} - 4 q^{6} + 5 q^{7} + 4 q^{9} - q^{10} + 3 q^{11} + 4 q^{12} + 4 q^{14} + 5 q^{15} - 5 q^{16} - 3 q^{17} + 6 q^{18} - 8 q^{19} + q^{20} - 13 q^{21} - 6 q^{22} - 2 q^{23} + 7 q^{24} + 3 q^{25} - 8 q^{27} - 4 q^{28} - 6 q^{29} + 4 q^{30} - 3 q^{31} - 4 q^{32} + 2 q^{33} + 6 q^{34} - 5 q^{35} - 13 q^{36} - 5 q^{37} - 19 q^{38} - 20 q^{41} - 9 q^{42} - 23 q^{43} - q^{44} - 4 q^{45} - 24 q^{46} + 16 q^{47} - q^{48} - 8 q^{49} + q^{50} - 9 q^{51} - 2 q^{53} + 2 q^{54} - 3 q^{55} - 7 q^{56} + 18 q^{57} + 19 q^{58} + 11 q^{59} - 4 q^{60} - 17 q^{61} + 6 q^{62} + 23 q^{63} + 4 q^{64} + 3 q^{66} - 15 q^{67} - 6 q^{68} + 15 q^{69} - 4 q^{70} - 13 q^{71} - 21 q^{72} + 12 q^{73} - 11 q^{74} - 5 q^{75} - 9 q^{76} - 2 q^{77} - 37 q^{79} + 5 q^{80} + 27 q^{81} - 2 q^{82} + 29 q^{83} + 16 q^{84} + 3 q^{85} - 10 q^{86} + 10 q^{87} + 14 q^{88} + 3 q^{89} - 6 q^{90} - 11 q^{92} + 19 q^{93} + 10 q^{94} + 8 q^{95} - 5 q^{96} + 12 q^{97} + 2 q^{98} - 17 q^{99}+O(q^{100}) $ 3 * q + q^2 - 5 * q^3 - q^4 - 3 * q^5 - 4 * q^6 + 5 * q^7 + 4 * q^9 - q^10 + 3 * q^11 + 4 * q^12 + 4 * q^14 + 5 * q^15 - 5 * q^16 - 3 * q^17 + 6 * q^18 - 8 * q^19 + q^20 - 13 * q^21 - 6 * q^22 - 2 * q^23 + 7 * q^24 + 3 * q^25 - 8 * q^27 - 4 * q^28 - 6 * q^29 + 4 * q^30 - 3 * q^31 - 4 * q^32 + 2 * q^33 + 6 * q^34 - 5 * q^35 - 13 * q^36 - 5 * q^37 - 19 * q^38 - 20 * q^41 - 9 * q^42 - 23 * q^43 - q^44 - 4 * q^45 - 24 * q^46 + 16 * q^47 - q^48 - 8 * q^49 + q^50 - 9 * q^51 - 2 * q^53 + 2 * q^54 - 3 * q^55 - 7 * q^56 + 18 * q^57 + 19 * q^58 + 11 * q^59 - 4 * q^60 - 17 * q^61 + 6 * q^62 + 23 * q^63 + 4 * q^64 + 3 * q^66 - 15 * q^67 - 6 * q^68 + 15 * q^69 - 4 * q^70 - 13 * q^71 - 21 * q^72 + 12 * q^73 - 11 * q^74 - 5 * q^75 - 9 * q^76 - 2 * q^77 - 37 * q^79 + 5 * q^80 + 27 * q^81 - 2 * q^82 + 29 * q^83 + 16 * q^84 + 3 * q^85 - 10 * q^86 + 10 * q^87 + 14 * q^88 + 3 * q^89 - 6 * q^90 - 11 * q^92 + 19 * q^93 + 10 * q^94 + 8 * q^95 - 5 * q^96 + 12 * q^97 + 2 * q^98 - 17 * 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 $

$2$	0.445042	0.314692	0.157346	−	0.987544i	$-0.449706\pi$
$2$	0.445042	0.314692	0.157346	+	0.987544i	$0.449706\pi$
$3$	−3.24698	−1.87464	−0.937322	−	0.348464i	$-0.886703\pi$
$3$	−3.24698	−1.87464	−0.937322	+	0.348464i	$0.886703\pi$
$4$	−1.80194	−0.900969
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	−1.44504	−0.589936
$7$	3.24698	1.22724	0.613621	−	0.789600i	$-0.289712\pi$
$7$	3.24698	1.22724	0.613621	+	0.789600i	$0.289712\pi$
$8$	−1.69202	−0.598220
$9$	7.54288	2.51429
$10$	−0.445042	−0.140735
$11$	−0.692021	−0.208652	−0.104326	−	0.994543i	$-0.533269\pi$
$11$	−0.692021	−0.208652	−0.104326	+	0.994543i	$0.533269\pi$
$12$	5.85086	1.68900
$13$	0	0
$13$	0	0
$14$	1.44504	0.386204
$15$	3.24698	0.838367
$16$	2.85086	0.712714
$17$	3.74094	0.907311	0.453655	−	0.891177i	$-0.350120\pi$
$17$	3.74094	0.907311	0.453655	+	0.891177i	$0.350120\pi$
$18$	3.35690	0.791228
$19$	−1.53319	−0.351737	−0.175869	−	0.984414i	$-0.556273\pi$
$19$	−1.53319	−0.351737	−0.175869	+	0.984414i	$0.556273\pi$
$20$	1.80194	0.402926
$21$	−10.5429	−2.30064
$22$	−0.307979	−0.0656612
$23$	−1.22521	−0.255474	−0.127737	−	0.991808i	$-0.540771\pi$
$23$	−1.22521	−0.255474	−0.127737	+	0.991808i	$0.540771\pi$
$24$	5.49396	1.12145
$25$	1.00000	0.200000
$26$	0	0
$27$	−14.7506	−2.83876
$28$	−5.85086	−1.10571
$29$	−6.07069	−1.12730	−0.563649	−	0.826014i	$-0.690603\pi$
$29$	−6.07069	−1.12730	−0.563649	+	0.826014i	$0.690603\pi$
$30$	1.44504	0.263827
$31$	−8.45473	−1.51851	−0.759257	−	0.650791i	$-0.774438\pi$
$31$	−8.45473	−1.51851	−0.759257	+	0.650791i	$0.774438\pi$
$32$	4.65279	0.822505
$33$	2.24698	0.391149
$34$	1.66487	0.285524
$35$	−3.24698	−0.548840
$36$	−13.5918	−2.26530
$37$	−1.89008	−0.310728	−0.155364	−	0.987857i	$-0.549655\pi$
$37$	−1.89008	−0.310728	−0.155364	+	0.987857i	$0.549655\pi$
$38$	−0.682333	−0.110689
$39$	0	0
$40$	1.69202	0.267532
$41$	−0.457123	−0.0713907	−0.0356953	−	0.999363i	$-0.511365\pi$
$41$	−0.457123	−0.0713907	−0.0356953	+	0.999363i	$0.511365\pi$
$42$	−4.69202	−0.723995
$43$	−6.19806	−0.945196	−0.472598	−	0.881278i	$-0.656684\pi$
$43$	−6.19806	−0.945196	−0.472598	+	0.881278i	$0.656684\pi$
$44$	1.24698	0.187989
$45$	−7.54288	−1.12443
$46$	−0.545269	−0.0803956
$47$	11.5429	1.68370	0.841851	−	0.539710i	$-0.181466\pi$
$47$	11.5429	1.68370	0.841851	+	0.539710i	$0.181466\pi$
$48$	−9.25667	−1.33608
$49$	3.54288	0.506125
$50$	0.445042	0.0629384
$51$	−12.1468	−1.70089
$52$	0	0
$53$	0.801938	0.110155	0.0550773	−	0.998482i	$-0.482459\pi$
$53$	0.801938	0.110155	0.0550773	+	0.998482i	$0.482459\pi$
$54$	−6.56465	−0.893335
$55$	0.692021	0.0933122
$56$	−5.49396	−0.734161
$57$	4.97823	0.659383
$58$	−2.70171	−0.354752
$59$	6.60388	0.859751	0.429876	−	0.902888i	$-0.358557\pi$
$59$	6.60388	0.859751	0.429876	+	0.902888i	$0.358557\pi$
$60$	−5.85086	−0.755342
$61$	−4.19806	−0.537507	−0.268753	−	0.963209i	$-0.586612\pi$
$61$	−4.19806	−0.537507	−0.268753	+	0.963209i	$0.586612\pi$
$62$	−3.76271	−0.477865
$63$	24.4916	3.08565
$64$	−3.63102	−0.453878
$65$	0	0
$66$	1.00000	0.123091
$67$	−13.8116	−1.68736	−0.843679	−	0.536847i	$-0.819615\pi$
$67$	−13.8116	−1.68736	−0.843679	+	0.536847i	$0.819615\pi$
$68$	−6.74094	−0.817459
$69$	3.97823	0.478923
$70$	−1.44504	−0.172716
$71$	−9.87263	−1.17167	−0.585833	−	0.810432i	$-0.699232\pi$
$71$	−9.87263	−1.17167	−0.585833	+	0.810432i	$0.699232\pi$
$72$	−12.7627	−1.50410
$73$	8.05429	0.942684	0.471342	−	0.881951i	$-0.343770\pi$
$73$	8.05429	0.942684	0.471342	+	0.881951i	$0.343770\pi$
$74$	−0.841166	−0.0977836
$75$	−3.24698	−0.374929
$76$	2.76271	0.316904
$77$	−2.24698	−0.256067
$78$	0	0
$79$	−16.5157	−1.85816	−0.929082	−	0.369873i	$-0.879401\pi$
$79$	−16.5157	−1.85816	−0.929082	+	0.369873i	$0.879401\pi$
$80$	−2.85086	−0.318735
$81$	25.2664	2.80737
$82$	−0.203439	−0.0224661
$83$	6.17092	0.677346	0.338673	−	0.940904i	$-0.390022\pi$
$83$	6.17092	0.677346	0.338673	+	0.940904i	$0.390022\pi$
$84$	18.9976	2.07281
$85$	−3.74094	−0.405762
$86$	−2.75840	−0.297446
$87$	19.7114	2.11328
$88$	1.17092	0.124820
$89$	−10.5254	−1.11569	−0.557846	−	0.829944i	$-0.688372\pi$
$89$	−10.5254	−1.11569	−0.557846	+	0.829944i	$0.688372\pi$
$90$	−3.35690	−0.353848
$91$	0	0
$92$	2.20775	0.230174
$93$	27.4523	2.84667
$94$	5.13706	0.529848
$95$	1.53319	0.157302
$96$	−15.1075	−1.54191
$97$	−3.45473	−0.350775	−0.175387	−	0.984500i	$-0.556118\pi$
$97$	−3.45473	−0.350775	−0.175387	+	0.984500i	$0.556118\pi$
$98$	1.57673	0.159274
$99$	−5.21983	−0.524613
$100$	−1.80194	−0.180194
$101$	4.32304	0.430159	0.215079	−	0.976597i	$-0.430999\pi$
$101$	4.32304	0.430159	0.215079	+	0.976597i	$0.430999\pi$
$102$	−5.40581	−0.535255
$103$	−4.25906	−0.419658	−0.209829	−	0.977738i	$-0.567291\pi$
$103$	−4.25906	−0.419658	−0.209829	+	0.977738i	$0.567291\pi$
$104$	0	0
$105$	10.5429	1.02888
$106$	0.356896	0.0346648
$107$	−11.2174	−1.08443	−0.542215	−	0.840240i	$-0.682414\pi$
$107$	−11.2174	−1.08443	−0.542215	+	0.840240i	$0.682414\pi$
$108$	26.5797	2.55763
$109$	2.37196	0.227193	0.113596	−	0.993527i	$-0.463763\pi$
$109$	2.37196	0.227193	0.113596	+	0.993527i	$0.463763\pi$
$110$	0.307979	0.0293646
$111$	6.13706	0.582504
$112$	9.25667	0.874673
$113$	−1.72886	−0.162637	−0.0813186	−	0.996688i	$-0.525913\pi$
$113$	−1.72886	−0.162637	−0.0813186	+	0.996688i	$0.525913\pi$
$114$	2.21552	0.207503
$115$	1.22521	0.114251
$116$	10.9390	1.01566
$117$	0	0
$118$	2.93900	0.270557
$119$	12.1468	1.11349
$120$	−5.49396	−0.501528
$121$	−10.5211	−0.956464
$122$	−1.86831	−0.169149
$123$	1.48427	0.133832
$124$	15.2349	1.36813
$125$	−1.00000	−0.0894427
$126$	10.8998	0.971029
$127$	3.78986	0.336295	0.168148	−	0.985762i	$-0.446221\pi$
$127$	3.78986	0.336295	0.168148	+	0.985762i	$0.446221\pi$
$128$	−10.9215	−0.965337
$129$	20.1250	1.77191
$130$	0	0
$131$	5.48858	0.479540	0.239770	−	0.970830i	$-0.422928\pi$
$131$	5.48858	0.479540	0.239770	+	0.970830i	$0.422928\pi$
$132$	−4.04892	−0.352413
$133$	−4.97823	−0.431667
$134$	−6.14675	−0.530998
$135$	14.7506	1.26953
$136$	−6.32975	−0.542771
$137$	2.58211	0.220604	0.110302	−	0.993898i	$-0.464818\pi$
$137$	2.58211	0.220604	0.110302	+	0.993898i	$0.464818\pi$
$138$	1.77048	0.150713
$139$	−14.4330	−1.22419	−0.612094	−	0.790785i	$-0.709673\pi$
$139$	−14.4330	−1.22419	−0.612094	+	0.790785i	$0.709673\pi$
$140$	5.85086	0.494488
$141$	−37.4795	−3.15634
$142$	−4.39373	−0.368714
$143$	0	0
$144$	21.5036	1.79197
$145$	6.07069	0.504143
$146$	3.58450	0.296655
$147$	−11.5036	−0.948805
$148$	3.40581	0.279956
$149$	−0.572417	−0.0468942	−0.0234471	−	0.999725i	$-0.507464\pi$
$149$	−0.572417	−0.0468942	−0.0234471	+	0.999725i	$0.507464\pi$
$150$	−1.44504	−0.117987
$151$	1.23490	0.100495	0.0502473	−	0.998737i	$-0.483999\pi$
$151$	1.23490	0.100495	0.0502473	+	0.998737i	$0.483999\pi$
$152$	2.59419	0.210416
$153$	28.2174	2.28124
$154$	−1.00000	−0.0805823
$155$	8.45473	0.679100
$156$	0	0
$157$	11.5375	0.920793	0.460396	−	0.887713i	$-0.347707\pi$
$157$	11.5375	0.920793	0.460396	+	0.887713i	$0.347707\pi$
$158$	−7.35019	−0.584750
$159$	−2.60388	−0.206501
$160$	−4.65279	−0.367836
$161$	−3.97823	−0.313528
$162$	11.2446	0.883458
$163$	−7.28382	−0.570512	−0.285256	−	0.958451i	$-0.592079\pi$
$163$	−7.28382	−0.570512	−0.285256	+	0.958451i	$0.592079\pi$
$164$	0.823708	0.0643208
$165$	−2.24698	−0.174927
$166$	2.74632	0.213155
$167$	−14.4480	−1.11802	−0.559011	−	0.829160i	$-0.688819\pi$
$167$	−14.4480	−1.11802	−0.559011	+	0.829160i	$0.688819\pi$
$168$	17.8388	1.37629
$169$	0	0
$170$	−1.66487	−0.127690
$171$	−11.5646	−0.884371
$172$	11.1685	0.851592
$173$	1.81940	0.138326	0.0691631	−	0.997605i	$-0.477967\pi$
$173$	1.81940	0.138326	0.0691631	+	0.997605i	$0.477967\pi$
$174$	8.77240	0.665034
$175$	3.24698	0.245449
$176$	−1.97285	−0.148709
$177$	−21.4426	−1.61173
$178$	−4.68425	−0.351100
$179$	−18.2470	−1.36384	−0.681922	−	0.731425i	$-0.738856\pi$
$179$	−18.2470	−1.36384	−0.681922	+	0.731425i	$0.738856\pi$
$180$	13.5918	1.01307
$181$	−11.8605	−0.881587	−0.440794	−	0.897608i	$-0.645303\pi$
$181$	−11.8605	−0.881587	−0.440794	+	0.897608i	$0.645303\pi$
$182$	0	0
$183$	13.6310	1.00763
$184$	2.07308	0.152830
$185$	1.89008	0.138962
$186$	12.2174	0.895826
$187$	−2.58881	−0.189313
$188$	−20.7995	−1.51696
$189$	−47.8950	−3.48385
$190$	0.682333	0.0495016
$191$	8.56465	0.619716	0.309858	−	0.950783i	$-0.399718\pi$
$191$	8.56465	0.619716	0.309858	+	0.950783i	$0.399718\pi$
$192$	11.7899	0.850860
$193$	21.6993	1.56195	0.780976	−	0.624562i	$-0.214722\pi$
$193$	21.6993	1.56195	0.780976	+	0.624562i	$0.214722\pi$
$194$	−1.53750	−0.110386
$195$	0	0
$196$	−6.38404	−0.456003
$197$	21.8431	1.55626	0.778128	−	0.628106i	$-0.216170\pi$
$197$	21.8431	1.55626	0.778128	+	0.628106i	$0.216170\pi$
$198$	−2.32304	−0.165092
$199$	−3.67025	−0.260177	−0.130089	−	0.991502i	$-0.541526\pi$
$199$	−3.67025	−0.260177	−0.130089	+	0.991502i	$0.541526\pi$
$200$	−1.69202	−0.119644
$201$	44.8461	3.16320
$202$	1.92394	0.135368
$203$	−19.7114	−1.38347
$204$	21.8877	1.53244
$205$	0.457123	0.0319269
$206$	−1.89546	−0.132063
$207$	−9.24160	−0.642336
$208$	0	0
$209$	1.06100	0.0733908
$210$	4.69202	0.323780
$211$	−15.2862	−1.05235	−0.526173	−	0.850378i	$-0.676374\pi$
$211$	−15.2862	−1.05235	−0.526173	+	0.850378i	$0.676374\pi$
$212$	−1.44504	−0.0992459
$213$	32.0562	2.19646
$214$	−4.99223	−0.341262
$215$	6.19806	0.422704
$216$	24.9584	1.69820
$217$	−27.4523	−1.86359
$218$	1.05562	0.0714958
$219$	−26.1521	−1.76720
$220$	−1.24698	−0.0840713
$221$	0	0
$222$	2.73125	0.183310
$223$	−3.66786	−0.245618	−0.122809	−	0.992430i	$-0.539190\pi$
$223$	−3.66786	−0.245618	−0.122809	+	0.992430i	$0.539190\pi$
$224$	15.1075	1.00941
$225$	7.54288	0.502858
$226$	−0.769414	−0.0511806
$227$	18.5676	1.23238	0.616188	−	0.787599i	$-0.288676\pi$
$227$	18.5676	1.23238	0.616188	+	0.787599i	$0.288676\pi$
$228$	−8.97046	−0.594083
$229$	16.0978	1.06377	0.531887	−	0.846815i	$-0.321483\pi$
$229$	16.0978	1.06377	0.531887	+	0.846815i	$0.321483\pi$
$230$	0.545269	0.0359540
$231$	7.29590	0.480035
$232$	10.2717	0.674372
$233$	−20.3056	−1.33026	−0.665132	−	0.746726i	$-0.731625\pi$
$233$	−20.3056	−1.33026	−0.665132	+	0.746726i	$0.731625\pi$
$234$	0	0
$235$	−11.5429	−0.752974
$236$	−11.8998	−0.774609
$237$	53.6262	3.48340
$238$	5.40581	0.350407
$239$	−20.7168	−1.34006	−0.670028	−	0.742335i	$-0.733718\pi$
$239$	−20.7168	−1.34006	−0.670028	+	0.742335i	$0.733718\pi$
$240$	9.25667	0.597515
$241$	−25.8582	−1.66567	−0.832835	−	0.553521i	$-0.813284\pi$
$241$	−25.8582	−1.66567	−0.832835	+	0.553521i	$0.813284\pi$
$242$	−4.68233	−0.300992
$243$	−37.7875	−2.42407
$244$	7.56465	0.484277
$245$	−3.54288	−0.226346
$246$	0.660563	0.0421159
$247$	0	0
$248$	14.3056	0.908406
$249$	−20.0368	−1.26978
$250$	−0.445042	−0.0281469
$251$	−26.2989	−1.65997	−0.829985	−	0.557785i	$-0.811651\pi$
$251$	−26.2989	−1.65997	−0.829985	+	0.557785i	$0.811651\pi$
$252$	−44.1323	−2.78007
$253$	0.847871	0.0533052
$254$	1.68664	0.105829
$255$	12.1468	0.760659
$256$	2.40150	0.150094
$257$	−6.04354	−0.376986	−0.188493	−	0.982075i	$-0.560360\pi$
$257$	−6.04354	−0.376986	−0.188493	+	0.982075i	$0.560360\pi$
$258$	8.95646	0.557605
$259$	−6.13706	−0.381339
$260$	0	0
$261$	−45.7904	−2.83436
$262$	2.44265	0.150907
$263$	−16.5429	−1.02008	−0.510039	−	0.860151i	$-0.670369\pi$
$263$	−16.5429	−1.02008	−0.510039	+	0.860151i	$0.670369\pi$
$264$	−3.80194	−0.233993
$265$	−0.801938	−0.0492626
$266$	−2.21552	−0.135842
$267$	34.1758	2.09153
$268$	24.8877	1.52026
$269$	−12.0954	−0.737472	−0.368736	−	0.929534i	$-0.620209\pi$
$269$	−12.0954	−0.737472	−0.368736	+	0.929534i	$0.620209\pi$
$270$	6.56465	0.399512
$271$	13.2687	0.806019	0.403010	−	0.915196i	$-0.367964\pi$
$271$	13.2687	0.806019	0.403010	+	0.915196i	$0.367964\pi$
$272$	10.6649	0.646653
$273$	0	0
$274$	1.14914	0.0694224
$275$	−0.692021	−0.0417305
$276$	−7.16852	−0.431494
$277$	−8.48858	−0.510029	−0.255015	−	0.966937i	$-0.582080\pi$
$277$	−8.48858	−0.510029	−0.255015	+	0.966937i	$0.582080\pi$
$278$	−6.42327	−0.385242
$279$	−63.7730	−3.81799
$280$	5.49396	0.328327
$281$	26.7603	1.59639	0.798193	−	0.602401i	$-0.205789\pi$
$281$	26.7603	1.59639	0.798193	+	0.602401i	$0.205789\pi$
$282$	−16.6799	−0.993276
$283$	−11.6649	−0.693405	−0.346702	−	0.937975i	$-0.612699\pi$
$283$	−11.6649	−0.693405	−0.346702	+	0.937975i	$0.612699\pi$
$284$	17.7899	1.05563
$285$	−4.97823	−0.294885
$286$	0	0
$287$	−1.48427	−0.0876137
$288$	35.0954	2.06802
$289$	−3.00538	−0.176787
$290$	2.70171	0.158650
$291$	11.2174	0.657578
$292$	−14.5133	−0.849329
$293$	−10.6407	−0.621637	−0.310818	−	0.950469i	$-0.600603\pi$
$293$	−10.6407	−0.621637	−0.310818	+	0.950469i	$0.600603\pi$
$294$	−5.11960	−0.298581
$295$	−6.60388	−0.384492
$296$	3.19806	0.185884
$297$	10.2078	0.592314
$298$	−0.254749	−0.0147572
$299$	0	0
$300$	5.85086	0.337799
$301$	−20.1250	−1.15998
$302$	0.549581	0.0316249
$303$	−14.0368	−0.806395
$304$	−4.37090	−0.250688
$305$	4.19806	0.240380
$306$	12.5579	0.717890
$307$	−0.291585	−0.0166416	−0.00832082	−	0.999965i	$-0.502649\pi$
$307$	−0.291585	−0.0166416	−0.00832082	+	0.999965i	$0.502649\pi$
$308$	4.04892	0.230708
$309$	13.8291	0.786709
$310$	3.76271	0.213708
$311$	30.2707	1.71649	0.858246	−	0.513238i	$-0.171554\pi$
$311$	30.2707	1.71649	0.858246	+	0.513238i	$0.171554\pi$
$312$	0	0
$313$	8.50902	0.480959	0.240479	−	0.970654i	$-0.422695\pi$
$313$	8.50902	0.480959	0.240479	+	0.970654i	$0.422695\pi$
$314$	5.13467	0.289766
$315$	−24.4916	−1.37994
$316$	29.7603	1.67415
$317$	11.5550	0.648991	0.324496	−	0.945887i	$-0.394805\pi$
$317$	11.5550	0.648991	0.324496	+	0.945887i	$0.394805\pi$
$318$	−1.15883	−0.0649842
$319$	4.20105	0.235213
$320$	3.63102	0.202980
$321$	36.4228	2.03292
$322$	−1.77048	−0.0986649
$323$	−5.73556	−0.319135
$324$	−45.5284	−2.52936
$325$	0	0
$326$	−3.24160	−0.179536
$327$	−7.70171	−0.425906
$328$	0.773463	0.0427073
$329$	37.4795	2.06631
$330$	−1.00000	−0.0550482
$331$	27.4403	1.50825	0.754126	−	0.656729i	$-0.228061\pi$
$331$	27.4403	1.50825	0.754126	+	0.656729i	$0.228061\pi$
$332$	−11.1196	−0.610268
$333$	−14.2567	−0.781261
$334$	−6.42998	−0.351833
$335$	13.8116	0.754610
$336$	−30.0562	−1.63970
$337$	−34.7265	−1.89167	−0.945836	−	0.324646i	$-0.894755\pi$
$337$	−34.7265	−1.89167	−0.945836	+	0.324646i	$0.894755\pi$
$338$	0	0
$339$	5.61356	0.304887
$340$	6.74094	0.365579
$341$	5.85086	0.316842
$342$	−5.14675	−0.278304
$343$	−11.2252	−0.606104
$344$	10.4873	0.565435
$345$	−3.97823	−0.214181
$346$	0.809707	0.0435301
$347$	29.6262	1.59042	0.795210	−	0.606334i	$-0.207361\pi$
$347$	29.6262	1.59042	0.795210	+	0.606334i	$0.207361\pi$
$348$	−35.5187	−1.90400
$349$	15.9463	0.853586	0.426793	−	0.904349i	$-0.359643\pi$
$349$	15.9463	0.853586	0.426793	+	0.904349i	$0.359643\pi$
$350$	1.44504	0.0772407
$351$	0	0
$352$	−3.21983	−0.171618
$353$	−22.2828	−1.18599	−0.592996	−	0.805206i	$-0.702055\pi$
$353$	−22.2828	−1.18599	−0.592996	+	0.805206i	$0.702055\pi$
$354$	−9.54288	−0.507198
$355$	9.87263	0.523985
$356$	18.9661	1.00520
$357$	−39.4403	−2.08740
$358$	−8.12067	−0.429191
$359$	−34.6896	−1.83085	−0.915424	−	0.402490i	$-0.868145\pi$
$359$	−34.6896	−1.83085	−0.915424	+	0.402490i	$0.868145\pi$
$360$	12.7627	0.672654
$361$	−16.6493	−0.876281
$362$	−5.27844	−0.277429
$363$	34.1618	1.79303
$364$	0	0
$365$	−8.05429	−0.421581
$366$	6.06638	0.317095
$367$	9.71246	0.506986	0.253493	−	0.967337i	$-0.418420\pi$
$367$	9.71246	0.506986	0.253493	+	0.967337i	$0.418420\pi$
$368$	−3.49289	−0.182080
$369$	−3.44803	−0.179497
$370$	0.841166	0.0437302
$371$	2.60388	0.135186
$372$	−49.4674	−2.56477
$373$	15.0411	0.778801	0.389401	−	0.921069i	$-0.372682\pi$
$373$	15.0411	0.778801	0.389401	+	0.921069i	$0.372682\pi$
$374$	−1.15213	−0.0595752
$375$	3.24698	0.167673
$376$	−19.5308	−1.00722
$377$	0	0
$378$	−21.3153	−1.09634
$379$	13.1347	0.674683	0.337341	−	0.941382i	$-0.390472\pi$
$379$	13.1347	0.674683	0.337341	+	0.941382i	$0.390472\pi$
$380$	−2.76271	−0.141724
$381$	−12.3056	−0.630434
$382$	3.81163	0.195020
$383$	3.02177	0.154405	0.0772026	−	0.997015i	$-0.475401\pi$
$383$	3.02177	0.154405	0.0772026	+	0.997015i	$0.475401\pi$
$384$	35.4620	1.80966
$385$	2.24698	0.114517
$386$	9.65710	0.491534
$387$	−46.7512	−2.37650
$388$	6.22521	0.316037
$389$	31.4282	1.59347	0.796736	−	0.604328i	$-0.206558\pi$
$389$	31.4282	1.59347	0.796736	+	0.604328i	$0.206558\pi$
$390$	0	0
$391$	−4.58343	−0.231794
$392$	−5.99462	−0.302774
$393$	−17.8213	−0.898966
$394$	9.72109	0.489741
$395$	16.5157	0.830997
$396$	9.40581	0.472660
$397$	−23.0489	−1.15679	−0.578396	−	0.815756i	$-0.696321\pi$
$397$	−23.0489	−1.15679	−0.578396	+	0.815756i	$0.696321\pi$
$398$	−1.63342	−0.0818757
$399$	16.1642	0.809223
$400$	2.85086	0.142543
$401$	19.8103	0.989279	0.494640	−	0.869098i	$-0.335300\pi$
$401$	19.8103	0.989279	0.494640	+	0.869098i	$0.335300\pi$
$402$	19.9584	0.995433
$403$	0	0
$404$	−7.78986	−0.387560
$405$	−25.2664	−1.25550
$406$	−8.77240	−0.435367
$407$	1.30798	0.0648341
$408$	20.5526	1.01750
$409$	32.9705	1.63028	0.815142	−	0.579261i	$-0.196659\pi$
$409$	32.9705	1.63028	0.815142	+	0.579261i	$0.196659\pi$
$410$	0.203439	0.0100471
$411$	−8.38404	−0.413554
$412$	7.67456	0.378099
$413$	21.4426	1.05512
$414$	−4.11290	−0.202138
$415$	−6.17092	−0.302918
$416$	0	0
$417$	46.8635	2.29492
$418$	0.472189	0.0230955
$419$	21.6136	1.05589	0.527946	−	0.849278i	$-0.322962\pi$
$419$	21.6136	1.05589	0.527946	+	0.849278i	$0.322962\pi$
$420$	−18.9976	−0.926988
$421$	4.11769	0.200684	0.100342	−	0.994953i	$-0.468006\pi$
$421$	4.11769	0.200684	0.100342	+	0.994953i	$0.468006\pi$
$422$	−6.80300	−0.331165
$423$	87.0665	4.23332
$424$	−1.35690	−0.0658967
$425$	3.74094	0.181462
$426$	14.2664	0.691207
$427$	−13.6310	−0.659651
$428$	20.2131	0.977038
$429$	0	0
$430$	2.75840	0.133022
$431$	18.8616	0.908532	0.454266	−	0.890866i	$-0.349902\pi$
$431$	18.8616	0.908532	0.454266	+	0.890866i	$0.349902\pi$
$432$	−42.0519	−2.02322
$433$	3.31336	0.159230	0.0796148	−	0.996826i	$-0.474631\pi$
$433$	3.31336	0.159230	0.0796148	+	0.996826i	$0.474631\pi$
$434$	−12.2174	−0.586456
$435$	−19.7114	−0.945089
$436$	−4.27413	−0.204694
$437$	1.87848	0.0898597
$438$	−11.6388	−0.556123
$439$	−14.3894	−0.686770	−0.343385	−	0.939195i	$-0.611573\pi$
$439$	−14.3894	−0.686770	−0.343385	+	0.939195i	$0.611573\pi$
$440$	−1.17092	−0.0558212
$441$	26.7235	1.27255
$442$	0	0
$443$	12.4577	0.591884	0.295942	−	0.955206i	$-0.404367\pi$
$443$	12.4577	0.591884	0.295942	+	0.955206i	$0.404367\pi$
$444$	−11.0586	−0.524818
$445$	10.5254	0.498953
$446$	−1.63235	−0.0772940
$447$	1.85862	0.0879099
$448$	−11.7899	−0.557018
$449$	30.5478	1.44164	0.720819	−	0.693123i	$-0.243766\pi$
$449$	30.5478	1.44164	0.720819	+	0.693123i	$0.243766\pi$
$450$	3.35690	0.158246
$451$	0.316339	0.0148958
$452$	3.11529	0.146531
$453$	−4.00969	−0.188392
$454$	8.26337	0.387819
$455$	0	0
$456$	−8.42327	−0.394456
$457$	9.79225	0.458062	0.229031	−	0.973419i	$-0.426444\pi$
$457$	9.79225	0.458062	0.229031	+	0.973419i	$0.426444\pi$
$458$	7.16421	0.334762
$459$	−55.1812	−2.57564
$460$	−2.20775	−0.102937
$461$	−13.2620	−0.617675	−0.308838	−	0.951115i	$-0.599940\pi$
$461$	−13.2620	−0.617675	−0.308838	+	0.951115i	$0.599940\pi$
$462$	3.24698	0.151063
$463$	−17.2403	−0.801224	−0.400612	−	0.916248i	$-0.631202\pi$
$463$	−17.2403	−0.801224	−0.400612	+	0.916248i	$0.631202\pi$
$464$	−17.3067	−0.803441
$465$	−27.4523	−1.27307
$466$	−9.03684	−0.418623
$467$	−4.98254	−0.230565	−0.115282	−	0.993333i	$-0.536777\pi$
$467$	−4.98254	−0.230565	−0.115282	+	0.993333i	$0.536777\pi$
$468$	0	0
$469$	−44.8461	−2.07080
$470$	−5.13706	−0.236955
$471$	−37.4620	−1.72616
$472$	−11.1739	−0.514320
$473$	4.28919	0.197217
$474$	23.8659	1.09620
$475$	−1.53319	−0.0703475
$476$	−21.8877	−1.00322
$477$	6.04892	0.276961
$478$	−9.21983	−0.421705
$479$	−5.11529	−0.233724	−0.116862	−	0.993148i	$-0.537283\pi$
$479$	−5.11529	−0.233724	−0.116862	+	0.993148i	$0.537283\pi$
$480$	15.1075	0.689561
$481$	0	0
$482$	−11.5080	−0.524173
$483$	12.9172	0.587754
$484$	18.9584	0.861744
$485$	3.45473	0.156871
$486$	−16.8170	−0.762835
$487$	−6.33704	−0.287159	−0.143579	−	0.989639i	$-0.545861\pi$
$487$	−6.33704	−0.287159	−0.143579	+	0.989639i	$0.545861\pi$
$488$	7.10321	0.321547
$489$	23.6504	1.06951
$490$	−1.57673	−0.0712293
$491$	−31.3381	−1.41427	−0.707135	−	0.707079i	$-0.750012\pi$
$491$	−31.3381	−1.41427	−0.707135	+	0.707079i	$0.750012\pi$
$492$	−2.67456	−0.120579
$493$	−22.7101	−1.02281
$494$	0	0
$495$	5.21983	0.234614
$496$	−24.1032	−1.08227
$497$	−32.0562	−1.43792
$498$	−8.91723	−0.399591
$499$	−23.7047	−1.06117	−0.530584	−	0.847632i	$-0.678027\pi$
$499$	−23.7047	−1.06117	−0.530584	+	0.847632i	$0.678027\pi$
$500$	1.80194	0.0805851
$501$	46.9124	2.09589
$502$	−11.7041	−0.522380
$503$	15.7125	0.700584	0.350292	−	0.936641i	$-0.386082\pi$
$503$	15.7125	0.700584	0.350292	+	0.936641i	$0.386082\pi$
$504$	−41.4403	−1.84590
$505$	−4.32304	−0.192373
$506$	0.377338	0.0167747
$507$	0	0
$508$	−6.82908	−0.302992
$509$	36.3551	1.61141	0.805706	−	0.592316i	$-0.201786\pi$
$509$	36.3551	1.61141	0.805706	+	0.592316i	$0.201786\pi$
$510$	5.40581	0.239373
$511$	26.1521	1.15690
$512$	22.9119	1.01257
$513$	22.6155	0.998498
$514$	−2.68963	−0.118634
$515$	4.25906	0.187677
$516$	−36.2640	−1.59643
$517$	−7.98792	−0.351308
$518$	−2.73125	−0.120004
$519$	−5.90754	−0.259312
$520$	0	0
$521$	9.55735	0.418715	0.209358	−	0.977839i	$-0.432863\pi$
$521$	9.55735	0.418715	0.209358	+	0.977839i	$0.432863\pi$
$522$	−20.3787	−0.891950
$523$	3.57242	0.156211	0.0781054	−	0.996945i	$-0.475113\pi$
$523$	3.57242	0.156211	0.0781054	+	0.996945i	$0.475113\pi$
$524$	−9.89008	−0.432050
$525$	−10.5429	−0.460129
$526$	−7.36227	−0.321010
$527$	−31.6286	−1.37776
$528$	6.40581	0.278777
$529$	−21.4989	−0.934733
$530$	−0.356896	−0.0155026
$531$	49.8122	2.16167
$532$	8.97046	0.388919
$533$	0	0
$534$	15.2097	0.658187
$535$	11.2174	0.484972
$536$	23.3696	1.00941
$537$	59.2476	2.55672
$538$	−5.38298	−0.232077
$539$	−2.45175	−0.105604
$540$	−26.5797	−1.14381
$541$	−9.99330	−0.429645	−0.214823	−	0.976653i	$-0.568917\pi$
$541$	−9.99330	−0.429645	−0.214823	+	0.976653i	$0.568917\pi$
$542$	5.90515	0.253648
$543$	38.5109	1.65266
$544$	17.4058	0.746268
$545$	−2.37196	−0.101604
$546$	0	0
$547$	−17.9836	−0.768923	−0.384462	−	0.923141i	$-0.625613\pi$
$547$	−17.9836	−0.768923	−0.384462	+	0.923141i	$0.625613\pi$
$548$	−4.65279	−0.198757
$549$	−31.6655	−1.35145
$550$	−0.307979	−0.0131322
$551$	9.30750	0.396513
$552$	−6.73125	−0.286501
$553$	−53.6262	−2.28042
$554$	−3.77777	−0.160502
$555$	−6.13706	−0.260504
$556$	26.0073	1.10296
$557$	16.5670	0.701968	0.350984	−	0.936381i	$-0.385847\pi$
$557$	16.5670	0.701968	0.350984	+	0.936381i	$0.385847\pi$
$558$	−28.3817	−1.20149
$559$	0	0
$560$	−9.25667	−0.391166
$561$	8.40581	0.354894
$562$	11.9095	0.502370
$563$	31.7356	1.33749	0.668747	−	0.743490i	$-0.266831\pi$
$563$	31.7356	1.33749	0.668747	+	0.743490i	$0.266831\pi$
$564$	67.5357	2.84377
$565$	1.72886	0.0727336
$566$	−5.19136	−0.218209
$567$	82.0393	3.44533
$568$	16.7047	0.700913
$569$	13.9336	0.584128	0.292064	−	0.956399i	$-0.405658\pi$
$569$	13.9336	0.584128	0.292064	+	0.956399i	$0.405658\pi$
$570$	−2.21552	−0.0927979
$571$	−26.2959	−1.10045	−0.550225	−	0.835017i	$-0.685458\pi$
$571$	−26.2959	−1.10045	−0.550225	+	0.835017i	$0.685458\pi$
$572$	0	0
$573$	−27.8092	−1.16175
$574$	−0.660563	−0.0275713
$575$	−1.22521	−0.0510948
$576$	−27.3884	−1.14118
$577$	−20.0339	−0.834020	−0.417010	−	0.908902i	$-0.636922\pi$
$577$	−20.0339	−0.834020	−0.417010	+	0.908902i	$0.636922\pi$
$578$	−1.33752	−0.0556334
$579$	−70.4572	−2.92810
$580$	−10.9390	−0.454217
$581$	20.0368	0.831268
$582$	4.99223	0.206935
$583$	−0.554958	−0.0229840
$584$	−13.6280	−0.563932
$585$	0	0
$586$	−4.73556	−0.195624
$587$	−20.0103	−0.825913	−0.412956	−	0.910751i	$-0.635504\pi$
$587$	−20.0103	−0.825913	−0.412956	+	0.910751i	$0.635504\pi$
$588$	20.7289	0.854844
$589$	12.9627	0.534118
$590$	−2.93900	−0.120997
$591$	−70.9241	−2.91743
$592$	−5.38835	−0.221460
$593$	−36.7023	−1.50718	−0.753591	−	0.657343i	$-0.771680\pi$
$593$	−36.7023	−1.50718	−0.753591	+	0.657343i	$0.771680\pi$
$594$	4.54288	0.186396
$595$	−12.1468	−0.497968
$596$	1.03146	0.0422502
$597$	11.9172	0.487740
$598$	0	0
$599$	13.5386	0.553171	0.276585	−	0.960989i	$-0.410797\pi$
$599$	13.5386	0.553171	0.276585	+	0.960989i	$0.410797\pi$
$600$	5.49396	0.224290
$601$	42.4965	1.73347	0.866734	−	0.498771i	$-0.166215\pi$
$601$	42.4965	1.73347	0.866734	+	0.498771i	$0.166215\pi$
$602$	−8.95646	−0.365038
$603$	−104.179	−4.24251
$604$	−2.22521	−0.0905425
$605$	10.5211	0.427744
$606$	−6.24698	−0.253766
$607$	−12.4969	−0.507235	−0.253618	−	0.967305i	$-0.581620\pi$
$607$	−12.4969	−0.507235	−0.253618	+	0.967305i	$0.581620\pi$
$608$	−7.13361	−0.289306
$609$	64.0025	2.59351
$610$	1.86831	0.0756458
$611$	0	0
$612$	−50.8461	−2.05533
$613$	−30.0694	−1.21449	−0.607245	−	0.794515i	$-0.707725\pi$
$613$	−30.0694	−1.21449	−0.607245	+	0.794515i	$0.707725\pi$
$614$	−0.129768	−0.00523699
$615$	−1.48427	−0.0598516
$616$	3.80194	0.153184
$617$	28.1196	1.13205	0.566026	−	0.824387i	$-0.308480\pi$
$617$	28.1196	1.13205	0.566026	+	0.824387i	$0.308480\pi$
$618$	6.15452	0.247571
$619$	20.3424	0.817631	0.408815	−	0.912617i	$-0.365942\pi$
$619$	20.3424	0.817631	0.408815	+	0.912617i	$0.365942\pi$
$620$	−15.2349	−0.611848
$621$	18.0726	0.725229
$622$	13.4717	0.540167
$623$	−34.1758	−1.36923
$624$	0	0
$625$	1.00000	0.0400000
$626$	3.78687	0.151354
$627$	−3.44504	−0.137582
$628$	−20.7899	−0.829606
$629$	−7.07069	−0.281927
$630$	−10.8998	−0.434257
$631$	−3.66355	−0.145843	−0.0729217	−	0.997338i	$-0.523232\pi$
$631$	−3.66355	−0.145843	−0.0729217	+	0.997338i	$0.523232\pi$
$632$	27.9450	1.11159
$633$	49.6340	1.97277
$634$	5.14244	0.204232
$635$	−3.78986	−0.150396
$636$	4.69202	0.186051
$637$	0	0
$638$	1.86964	0.0740198
$639$	−74.4680	−2.94591
$640$	10.9215	0.431712
$641$	5.73556	0.226541	0.113271	−	0.993564i	$-0.463867\pi$
$641$	5.73556	0.226541	0.113271	+	0.993564i	$0.463867\pi$
$642$	16.2097	0.639745
$643$	−7.35929	−0.290222	−0.145111	−	0.989415i	$-0.546354\pi$
$643$	−7.35929	−0.290222	−0.145111	+	0.989415i	$0.546354\pi$
$644$	7.16852	0.282479
$645$	−20.1250	−0.792420
$646$	−2.55257	−0.100429
$647$	49.2664	1.93686	0.968430	−	0.249285i	$-0.0801956\pi$
$647$	49.2664	1.93686	0.968430	+	0.249285i	$0.0801956\pi$
$648$	−42.7512	−1.67943
$649$	−4.57002	−0.179389
$650$	0	0
$651$	89.1372	3.49356
$652$	13.1250	0.514014
$653$	8.50796	0.332942	0.166471	−	0.986046i	$-0.446763\pi$
$653$	8.50796	0.332942	0.166471	+	0.986046i	$0.446763\pi$
$654$	−3.42758	−0.134029
$655$	−5.48858	−0.214457
$656$	−1.30319	−0.0508811
$657$	60.7525	2.37018
$658$	16.6799	0.650252
$659$	−5.42221	−0.211219	−0.105610	−	0.994408i	$-0.533679\pi$
$659$	−5.42221	−0.211219	−0.105610	+	0.994408i	$0.533679\pi$
$660$	4.04892	0.157604
$661$	−16.2784	−0.633158	−0.316579	−	0.948566i	$-0.602534\pi$
$661$	−16.2784	−0.633158	−0.316579	+	0.948566i	$0.602534\pi$
$662$	12.2121	0.474635
$663$	0	0
$664$	−10.4413	−0.405202
$665$	4.97823	0.193047
$666$	−6.34481	−0.245857
$667$	7.43786	0.287995
$668$	26.0344	1.00730
$669$	11.9095	0.460446
$670$	6.14675	0.237470
$671$	2.90515	0.112152
$672$	−49.0538	−1.89229
$673$	−22.8866	−0.882215	−0.441107	−	0.897454i	$-0.645414\pi$
$673$	−22.8866	−0.882215	−0.441107	+	0.897454i	$0.645414\pi$
$674$	−15.4547	−0.595294
$675$	−14.7506	−0.567752
$676$	0	0
$677$	−36.9711	−1.42091	−0.710456	−	0.703741i	$-0.751511\pi$
$677$	−36.9711	−1.42091	−0.710456	+	0.703741i	$0.751511\pi$
$678$	2.49827	0.0959455
$679$	−11.2174	−0.430486
$680$	6.32975	0.242735
$681$	−60.2887	−2.31027
$682$	2.60388	0.0997075
$683$	−21.3177	−0.815698	−0.407849	−	0.913049i	$-0.633721\pi$
$683$	−21.3177	−0.815698	−0.407849	+	0.913049i	$0.633721\pi$
$684$	20.8388	0.796790
$685$	−2.58211	−0.0986572
$686$	−4.99569	−0.190736
$687$	−52.2693	−1.99420
$688$	−17.6698	−0.673654
$689$	0	0
$690$	−1.77048	−0.0674010
$691$	20.9554	0.797181	0.398590	−	0.917129i	$-0.369500\pi$
$691$	20.9554	0.797181	0.398590	+	0.917129i	$0.369500\pi$
$692$	−3.27844	−0.124628
$693$	−16.9487	−0.643827
$694$	13.1849	0.500493
$695$	14.4330	0.547473
$696$	−33.3521	−1.26421
$697$	−1.71007	−0.0647736
$698$	7.09677	0.268617
$699$	65.9318	2.49377
$700$	−5.85086	−0.221142
$701$	−36.9439	−1.39535	−0.697676	−	0.716413i	$-0.745782\pi$
$701$	−36.9439	−1.39535	−0.697676	+	0.716413i	$0.745782\pi$
$702$	0	0
$703$	2.89785	0.109295
$704$	2.51275	0.0947027
$705$	37.4795	1.41156
$706$	−9.91676	−0.373222
$707$	14.0368	0.527910
$708$	38.6383	1.45212
$709$	6.41358	0.240867	0.120434	−	0.992721i	$-0.461572\pi$
$709$	6.41358	0.240867	0.120434	+	0.992721i	$0.461572\pi$
$710$	4.39373	0.164894
$711$	−124.576	−4.67197
$712$	17.8092	0.667429
$713$	10.3588	0.387941
$714$	−17.5526	−0.656888
$715$	0	0
$716$	32.8799	1.22878
$717$	67.2669	2.51213
$718$	−15.4383	−0.576154
$719$	−13.7265	−0.511911	−0.255955	−	0.966689i	$-0.582390\pi$
$719$	−13.7265	−0.511911	−0.255955	+	0.966689i	$0.582390\pi$
$720$	−21.5036	−0.801394
$721$	−13.8291	−0.515022
$722$	−7.40965	−0.275759
$723$	83.9609	3.12254
$724$	21.3720	0.794283
$725$	−6.07069	−0.225460
$726$	15.2034	0.564253
$727$	33.1008	1.22764	0.613821	−	0.789445i	$-0.289632\pi$
$727$	33.1008	1.22764	0.613821	+	0.789445i	$0.289632\pi$
$728$	0	0
$729$	46.8961	1.73689
$730$	−3.58450	−0.132668
$731$	−23.1866	−0.857586
$732$	−24.5623	−0.907847
$733$	50.6902	1.87229	0.936143	−	0.351620i	$-0.114369\pi$
$733$	50.6902	1.87229	0.936143	+	0.351620i	$0.114369\pi$
$734$	4.32245	0.159545
$735$	11.5036	0.424318
$736$	−5.70065	−0.210129
$737$	9.55794	0.352071
$738$	−1.53452	−0.0564863
$739$	−24.5448	−0.902895	−0.451447	−	0.892298i	$-0.649092\pi$
$739$	−24.5448	−0.902895	−0.451447	+	0.892298i	$0.649092\pi$
$740$	−3.40581	−0.125200
$741$	0	0
$742$	1.15883	0.0425421
$743$	−8.90217	−0.326589	−0.163294	−	0.986577i	$-0.552212\pi$
$743$	−8.90217	−0.326589	−0.163294	+	0.986577i	$0.552212\pi$
$744$	−46.4499	−1.70294
$745$	0.572417	0.0209717
$746$	6.69394	0.245083
$747$	46.5465	1.70305
$748$	4.66487	0.170565
$749$	−36.4228	−1.33086
$750$	1.44504	0.0527655
$751$	−3.00059	−0.109493	−0.0547466	−	0.998500i	$-0.517435\pi$
$751$	−3.00059	−0.109493	−0.0547466	+	0.998500i	$0.517435\pi$
$752$	32.9071	1.20000
$753$	85.3919	3.11185
$754$	0	0
$755$	−1.23490	−0.0449425
$756$	86.3038	3.13884
$757$	−23.1691	−0.842096	−0.421048	−	0.907038i	$-0.638338\pi$
$757$	−23.1691	−0.842096	−0.421048	+	0.907038i	$0.638338\pi$
$758$	5.84548	0.212317
$759$	−2.75302	−0.0999283
$760$	−2.59419	−0.0941010
$761$	−13.6431	−0.494562	−0.247281	−	0.968944i	$-0.579537\pi$
$761$	−13.6431	−0.494562	−0.247281	+	0.968944i	$0.579537\pi$
$762$	−5.47650	−0.198393
$763$	7.70171	0.278821
$764$	−15.4330	−0.558345
$765$	−28.2174	−1.02020
$766$	1.34481	0.0485901
$767$	0	0
$768$	−7.79763	−0.281373
$769$	−31.8853	−1.14981	−0.574907	−	0.818219i	$-0.694962\pi$
$769$	−31.8853	−1.14981	−0.574907	+	0.818219i	$0.694962\pi$
$770$	1.00000	0.0360375
$771$	19.6233	0.706714
$772$	−39.1008	−1.40727
$773$	−18.4168	−0.662407	−0.331204	−	0.943559i	$-0.607455\pi$
$773$	−18.4168	−0.662407	−0.331204	+	0.943559i	$0.607455\pi$
$774$	−20.8062	−0.747865
$775$	−8.45473	−0.303703
$776$	5.84548	0.209840
$777$	19.9269	0.714874
$778$	13.9869	0.501453
$779$	0.700856	0.0251108
$780$	0	0
$781$	6.83207	0.244471
$782$	−2.03982	−0.0729438
$783$	89.5465	3.20013
$784$	10.1002	0.360722
$785$	−11.5375	−0.411791
$786$	−7.93123	−0.282898
$787$	31.5230	1.12367	0.561837	−	0.827248i	$-0.310095\pi$
$787$	31.5230	1.12367	0.561837	+	0.827248i	$0.310095\pi$
$788$	−39.3599	−1.40214
$789$	53.7144	1.91228
$790$	7.35019	0.261508
$791$	−5.61356	−0.199595
$792$	8.83207	0.313834
$793$	0	0
$794$	−10.2577	−0.364033
$795$	2.60388	0.0923499
$796$	6.61356	0.234412
$797$	41.3967	1.46635	0.733173	−	0.680042i	$-0.238038\pi$
$797$	41.3967	1.46635	0.733173	+	0.680042i	$0.238038\pi$
$798$	7.19375	0.254656
$799$	43.1812	1.52764
$800$	4.65279	0.164501
$801$	−79.3919	−2.80518
$802$	8.81641	0.311318
$803$	−5.57374	−0.196693
$804$	−80.8098	−2.84994
$805$	3.97823	0.140214
$806$	0	0
$807$	39.2737	1.38250
$808$	−7.31468	−0.257330
$809$	33.9748	1.19449	0.597245	−	0.802059i	$-0.296262\pi$
$809$	33.9748	1.19449	0.597245	+	0.802059i	$0.296262\pi$
$810$	−11.2446	−0.395095
$811$	42.4174	1.48948	0.744739	−	0.667356i	$-0.232574\pi$
$811$	42.4174	1.48948	0.744739	+	0.667356i	$0.232574\pi$
$812$	35.5187	1.24646
$813$	−43.0834	−1.51100
$814$	0.582105	0.0204028
$815$	7.28382	0.255141
$816$	−34.6286	−1.21224
$817$	9.50279	0.332461
$818$	14.6732	0.513038
$819$	0	0
$820$	−0.823708	−0.0287651
$821$	20.8823	0.728798	0.364399	−	0.931243i	$-0.381274\pi$
$821$	20.8823	0.728798	0.364399	+	0.931243i	$0.381274\pi$
$822$	−3.73125	−0.130142
$823$	−31.4196	−1.09522	−0.547608	−	0.836735i	$-0.684462\pi$
$823$	−31.4196	−1.09522	−0.547608	+	0.836735i	$0.684462\pi$
$824$	7.20642	0.251048
$825$	2.24698	0.0782298
$826$	9.54288	0.332039
$827$	30.8471	1.07266	0.536330	−	0.844008i	$-0.319810\pi$
$827$	30.8471	1.07266	0.536330	+	0.844008i	$0.319810\pi$
$828$	16.6528	0.578725
$829$	30.9004	1.07321	0.536607	−	0.843832i	$-0.319706\pi$
$829$	30.9004	1.07321	0.536607	+	0.843832i	$0.319706\pi$
$830$	−2.74632	−0.0953260
$831$	27.5623	0.956124
$832$	0	0
$833$	13.2537	0.459213
$834$	20.8562	0.722192
$835$	14.4480	0.499995
$836$	−1.91185	−0.0661229
$837$	124.713	4.31070
$838$	9.61894	0.332281
$839$	17.9782	0.620677	0.310339	−	0.950626i	$-0.399558\pi$
$839$	17.9782	0.620677	0.310339	+	0.950626i	$0.399558\pi$
$840$	−17.8388	−0.615496
$841$	7.85325	0.270802
$842$	1.83254	0.0631536
$843$	−86.8902	−2.99266
$844$	27.5448	0.948131
$845$	0	0
$846$	38.7482	1.33219
$847$	−34.1618	−1.17381
$848$	2.28621	0.0785087
$849$	37.8756	1.29989
$850$	1.66487	0.0571047
$851$	2.31575	0.0793828
$852$	−57.7633	−1.97894
$853$	4.04413	0.138468	0.0692342	−	0.997600i	$-0.477944\pi$
$853$	4.04413	0.138468	0.0692342	+	0.997600i	$0.477944\pi$
$854$	−6.06638	−0.207587
$855$	11.5646	0.395503
$856$	18.9801	0.648728
$857$	25.7888	0.880928	0.440464	−	0.897770i	$-0.354814\pi$
$857$	25.7888	0.880928	0.440464	+	0.897770i	$0.354814\pi$
$858$	0	0
$859$	−5.63043	−0.192108	−0.0960539	−	0.995376i	$-0.530622\pi$
$859$	−5.63043	−0.192108	−0.0960539	+	0.995376i	$0.530622\pi$
$860$	−11.1685	−0.380843
$861$	4.81940	0.164245
$862$	8.39421	0.285908
$863$	−20.6112	−0.701612	−0.350806	−	0.936448i	$-0.614092\pi$
$863$	−20.6112	−0.701612	−0.350806	+	0.936448i	$0.614092\pi$
$864$	−68.6316	−2.33489
$865$	−1.81940	−0.0618613
$866$	1.47458	0.0501083
$867$	9.75840	0.331413
$868$	49.4674	1.67903
$869$	11.4292	0.387710
$870$	−8.77240	−0.297412
$871$	0	0
$872$	−4.01341	−0.135911
$873$	−26.0586	−0.881950
$874$	0.836001	0.0282781
$875$	−3.24698	−0.109768
$876$	47.1245	1.59219
$877$	3.46383	0.116965	0.0584826	−	0.998288i	$-0.481374\pi$
$877$	3.46383	0.116965	0.0584826	+	0.998288i	$0.481374\pi$
$878$	−6.40389	−0.216121
$879$	34.5502	1.16535
$880$	1.97285	0.0665049
$881$	−28.2040	−0.950218	−0.475109	−	0.879927i	$-0.657591\pi$
$881$	−28.2040	−0.950218	−0.475109	+	0.879927i	$0.657591\pi$
$882$	11.8931	0.400460
$883$	36.4577	1.22690	0.613450	−	0.789734i	$-0.289781\pi$
$883$	36.4577	1.22690	0.613450	+	0.789734i	$0.289781\pi$
$884$	0	0
$885$	21.4426	0.720787
$886$	5.54420	0.186261
$887$	6.17331	0.207279	0.103640	−	0.994615i	$-0.466951\pi$
$887$	6.17331	0.207279	0.103640	+	0.994615i	$0.466951\pi$
$888$	−10.3840	−0.348466
$889$	12.3056	0.412716
$890$	4.68425	0.157016
$891$	−17.4849	−0.585765
$892$	6.60925	0.221294
$893$	−17.6974	−0.592221
$894$	0.827166	0.0276646
$895$	18.2470	0.609929
$896$	−35.4620	−1.18470
$897$	0	0
$898$	13.5950	0.453672
$899$	51.3260	1.71182
$900$	−13.5918	−0.453060
$901$	3.00000	0.0999445
$902$	0.140784	0.00468760
$903$	65.3454	2.17456
$904$	2.92526	0.0972928
$905$	11.8605	0.394258
$906$	−1.78448	−0.0592854
$907$	−15.1860	−0.504242	−0.252121	−	0.967696i	$-0.581128\pi$
$907$	−15.1860	−0.504242	−0.252121	+	0.967696i	$0.581128\pi$
$908$	−33.4577	−1.11033
$909$	32.6082	1.08155
$910$	0	0
$911$	37.0810	1.22855	0.614274	−	0.789093i	$-0.289449\pi$
$911$	37.0810	1.22855	0.614274	+	0.789093i	$0.289449\pi$
$912$	14.1922	0.469951
$913$	−4.27041	−0.141330
$914$	4.35796	0.144149
$915$	−13.6310	−0.450628
$916$	−29.0073	−0.958428
$917$	17.8213	0.588512
$918$	−24.5579	−0.810533
$919$	−30.5590	−1.00805	−0.504024	−	0.863689i	$-0.668148\pi$
$919$	−30.5590	−1.00805	−0.504024	+	0.863689i	$0.668148\pi$
$920$	−2.07308	−0.0683474
$921$	0.946771	0.0311972
$922$	−5.90217	−0.194377
$923$	0	0
$924$	−13.1468	−0.432496
$925$	−1.89008	−0.0621456
$926$	−7.67264	−0.252139
$927$	−32.1256	−1.05514
$928$	−28.2457	−0.927209
$929$	−27.2489	−0.894007	−0.447004	−	0.894532i	$-0.647509\pi$
$929$	−27.2489	−0.894007	−0.447004	+	0.894532i	$0.647509\pi$
$930$	−12.2174	−0.400626
$931$	−5.43190	−0.178023
$932$	36.5894	1.19853
$933$	−98.2882	−3.21781
$934$	−2.21744	−0.0725568
$935$	2.58881	0.0846631
$936$	0	0
$937$	57.8286	1.88918	0.944589	−	0.328255i	$-0.106460\pi$
$937$	57.8286	1.88918	0.944589	+	0.328255i	$0.106460\pi$
$938$	−19.9584	−0.651664
$939$	−27.6286	−0.901626
$940$	20.7995	0.678406
$941$	−40.1463	−1.30873	−0.654366	−	0.756178i	$-0.727064\pi$
$941$	−40.1463	−1.30873	−0.654366	+	0.756178i	$0.727064\pi$
$942$	−16.6722	−0.543209
$943$	0.560072	0.0182385
$944$	18.8267	0.612757
$945$	47.8950	1.55802
$946$	1.90887	0.0620627
$947$	−0.987918	−0.0321030	−0.0160515	−	0.999871i	$-0.505110\pi$
$947$	−0.987918	−0.0321030	−0.0160515	+	0.999871i	$0.505110\pi$
$948$	−96.6311	−3.13843
$949$	0	0
$950$	−0.682333	−0.0221378
$951$	−37.5187	−1.21663
$952$	−20.5526	−0.666112
$953$	−43.3351	−1.40376	−0.701881	−	0.712294i	$-0.747656\pi$
$953$	−43.3351	−1.40376	−0.701881	+	0.712294i	$0.747656\pi$
$954$	2.69202	0.0871574
$955$	−8.56465	−0.277145
$956$	37.3303	1.20735
$957$	−13.6407	−0.440942
$958$	−2.27652	−0.0735510
$959$	8.38404	0.270735
$960$	−11.7899	−0.380516
$961$	40.4825	1.30589
$962$	0	0
$963$	−84.6118	−2.72658
$964$	46.5948	1.50072
$965$	−21.6993	−0.698526
$966$	5.74871	0.184962
$967$	4.89977	0.157566	0.0787830	−	0.996892i	$-0.474897\pi$
$967$	4.89977	0.157566	0.0787830	+	0.996892i	$0.474897\pi$
$968$	17.8019	0.572176
$969$	18.6233	0.598265
$970$	1.53750	0.0493661
$971$	−37.5153	−1.20392	−0.601961	−	0.798526i	$-0.705614\pi$
$971$	−37.5153	−1.20392	−0.601961	+	0.798526i	$0.705614\pi$
$972$	68.0907	2.18401
$973$	−46.8635	−1.50238
$974$	−2.82025	−0.0903666
$975$	0	0
$976$	−11.9681	−0.383088
$977$	−59.7663	−1.91209	−0.956046	−	0.293215i	$-0.905275\pi$
$977$	−59.7663	−1.91209	−0.956046	+	0.293215i	$0.905275\pi$
$978$	10.5254	0.336566
$979$	7.28382	0.232792
$980$	6.38404	0.203931
$981$	17.8914	0.571229
$982$	−13.9468	−0.445059
$983$	39.2452	1.25173	0.625863	−	0.779933i	$-0.284747\pi$
$983$	39.2452	1.25173	0.625863	+	0.779933i	$0.284747\pi$
$984$	−2.51142	−0.0800611
$985$	−21.8431	−0.695979
$986$	−10.1069	−0.321870
$987$	−121.695	−3.87360
$988$	0	0
$989$	7.59392	0.241473
$990$	2.32304	0.0738312
$991$	−12.2577	−0.389380	−0.194690	−	0.980865i	$-0.562370\pi$
$991$	−12.2577	−0.389380	−0.194690	+	0.980865i	$0.562370\pi$
$992$	−39.3381	−1.24899
$993$	−89.0980	−2.82744
$994$	−14.2664	−0.452501
$995$	3.67025	0.116355
$996$	36.1051	1.14403
$997$	−24.5687	−0.778098	−0.389049	−	0.921217i	$-0.627196\pi$
$997$	−24.5687	−0.778098	−0.389049	+	0.921217i	$0.627196\pi$
$998$	−10.5496	−0.333941
$999$	27.8799	0.882082

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	845.2.a.j.1.2	yes	3
3.2	odd	2		7605.2.a.br.1.2		3
5.4	even	2		4225.2.a.bd.1.2		3
13.2	odd	12		845.2.m.i.316.4		12
13.3	even	3		845.2.e.j.191.2		6
13.4	even	6		845.2.e.l.146.2		6
13.5	odd	4		845.2.c.f.506.3		6
13.6	odd	12		845.2.m.i.361.3		12
13.7	odd	12		845.2.m.i.361.4		12
13.8	odd	4		845.2.c.f.506.4		6
13.9	even	3		845.2.e.j.146.2		6
13.10	even	6		845.2.e.l.191.2		6
13.11	odd	12		845.2.m.i.316.3		12
13.12	even	2		845.2.a.h.1.2	✓	3
39.38	odd	2		7605.2.a.by.1.2		3
65.64	even	2		4225.2.a.bf.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
845.2.a.h.1.2	✓	3	13.12	even	2
845.2.a.j.1.2	yes	3	1.1	even	1	trivial
845.2.c.f.506.3		6	13.5	odd	4
845.2.c.f.506.4		6	13.8	odd	4
845.2.e.j.146.2		6	13.9	even	3
845.2.e.j.191.2		6	13.3	even	3
845.2.e.l.146.2		6	13.4	even	6
845.2.e.l.191.2		6	13.10	even	6
845.2.m.i.316.3		12	13.11	odd	12
845.2.m.i.316.4		12	13.2	odd	12
845.2.m.i.361.3		12	13.6	odd	12
845.2.m.i.361.4		12	13.7	odd	12
4225.2.a.bd.1.2		3	5.4	even	2
4225.2.a.bf.1.2		3	65.64	even	2
7605.2.a.br.1.2		3	3.2	odd	2
7605.2.a.by.1.2		3	39.38	odd	2

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:	\( N \)	\(=\)	\( 845 = 5 \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	845.a (trivial)

\(f(q)\)	\(=\)	\(q+0.445042 q^{2} -3.24698 q^{3} -1.80194 q^{4} -1.00000 q^{5} -1.44504 q^{6} +3.24698 q^{7} -1.69202 q^{8} +7.54288 q^{9} +O(q^{10})\)	\(q+0.445042 q^{2} -3.24698 q^{3} -1.80194 q^{4} -1.00000 q^{5} -1.44504 q^{6} +3.24698 q^{7} -1.69202 q^{8} +7.54288 q^{9} -0.445042 q^{10} -0.692021 q^{11} +5.85086 q^{12} +1.44504 q^{14} +3.24698 q^{15} +2.85086 q^{16} +3.74094 q^{17} +3.35690 q^{18} -1.53319 q^{19} +1.80194 q^{20} -10.5429 q^{21} -0.307979 q^{22} -1.22521 q^{23} +5.49396 q^{24} +1.00000 q^{25} -14.7506 q^{27} -5.85086 q^{28} -6.07069 q^{29} +1.44504 q^{30} -8.45473 q^{31} +4.65279 q^{32} +2.24698 q^{33} +1.66487 q^{34} -3.24698 q^{35} -13.5918 q^{36} -1.89008 q^{37} -0.682333 q^{38} +1.69202 q^{40} -0.457123 q^{41} -4.69202 q^{42} -6.19806 q^{43} +1.24698 q^{44} -7.54288 q^{45} -0.545269 q^{46} +11.5429 q^{47} -9.25667 q^{48} +3.54288 q^{49} +0.445042 q^{50} -12.1468 q^{51} +0.801938 q^{53} -6.56465 q^{54} +0.692021 q^{55} -5.49396 q^{56} +4.97823 q^{57} -2.70171 q^{58} +6.60388 q^{59} -5.85086 q^{60} -4.19806 q^{61} -3.76271 q^{62} +24.4916 q^{63} -3.63102 q^{64} +1.00000 q^{66} -13.8116 q^{67} -6.74094 q^{68} +3.97823 q^{69} -1.44504 q^{70} -9.87263 q^{71} -12.7627 q^{72} +8.05429 q^{73} -0.841166 q^{74} -3.24698 q^{75} +2.76271 q^{76} -2.24698 q^{77} -16.5157 q^{79} -2.85086 q^{80} +25.2664 q^{81} -0.203439 q^{82} +6.17092 q^{83} +18.9976 q^{84} -3.74094 q^{85} -2.75840 q^{86} +19.7114 q^{87} +1.17092 q^{88} -10.5254 q^{89} -3.35690 q^{90} +2.20775 q^{92} +27.4523 q^{93} +5.13706 q^{94} +1.53319 q^{95} -15.1075 q^{96} -3.45473 q^{97} +1.57673 q^{98} -5.21983 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} - 5 q^{3} - q^{4} - 3 q^{5} - 4 q^{6} + 5 q^{7} + 4 q^{9}+O(q^{10}) \) 3 * q + q^2 - 5 * q^3 - q^4 - 3 * q^5 - 4 * q^6 + 5 * q^7 + 4 * q^9	\( 3 q + q^{2} - 5 q^{3} - q^{4} - 3 q^{5} - 4 q^{6} + 5 q^{7} + 4 q^{9} - q^{10} + 3 q^{11} + 4 q^{12} + 4 q^{14} + 5 q^{15} - 5 q^{16} - 3 q^{17} + 6 q^{18} - 8 q^{19} + q^{20} - 13 q^{21} - 6 q^{22} - 2 q^{23} + 7 q^{24} + 3 q^{25} - 8 q^{27} - 4 q^{28} - 6 q^{29} + 4 q^{30} - 3 q^{31} - 4 q^{32} + 2 q^{33} + 6 q^{34} - 5 q^{35} - 13 q^{36} - 5 q^{37} - 19 q^{38} - 20 q^{41} - 9 q^{42} - 23 q^{43} - q^{44} - 4 q^{45} - 24 q^{46} + 16 q^{47} - q^{48} - 8 q^{49} + q^{50} - 9 q^{51} - 2 q^{53} + 2 q^{54} - 3 q^{55} - 7 q^{56} + 18 q^{57} + 19 q^{58} + 11 q^{59} - 4 q^{60} - 17 q^{61} + 6 q^{62} + 23 q^{63} + 4 q^{64} + 3 q^{66} - 15 q^{67} - 6 q^{68} + 15 q^{69} - 4 q^{70} - 13 q^{71} - 21 q^{72} + 12 q^{73} - 11 q^{74} - 5 q^{75} - 9 q^{76} - 2 q^{77} - 37 q^{79} + 5 q^{80} + 27 q^{81} - 2 q^{82} + 29 q^{83} + 16 q^{84} + 3 q^{85} - 10 q^{86} + 10 q^{87} + 14 q^{88} + 3 q^{89} - 6 q^{90} - 11 q^{92} + 19 q^{93} + 10 q^{94} + 8 q^{95} - 5 q^{96} + 12 q^{97} + 2 q^{98} - 17 q^{99}+O(q^{100}) \) 3 * q + q^2 - 5 * q^3 - q^4 - 3 * q^5 - 4 * q^6 + 5 * q^7 + 4 * q^9 - q^10 + 3 * q^11 + 4 * q^12 + 4 * q^14 + 5 * q^15 - 5 * q^16 - 3 * q^17 + 6 * q^18 - 8 * q^19 + q^20 - 13 * q^21 - 6 * q^22 - 2 * q^23 + 7 * q^24 + 3 * q^25 - 8 * q^27 - 4 * q^28 - 6 * q^29 + 4 * q^30 - 3 * q^31 - 4 * q^32 + 2 * q^33 + 6 * q^34 - 5 * q^35 - 13 * q^36 - 5 * q^37 - 19 * q^38 - 20 * q^41 - 9 * q^42 - 23 * q^43 - q^44 - 4 * q^45 - 24 * q^46 + 16 * q^47 - q^48 - 8 * q^49 + q^50 - 9 * q^51 - 2 * q^53 + 2 * q^54 - 3 * q^55 - 7 * q^56 + 18 * q^57 + 19 * q^58 + 11 * q^59 - 4 * q^60 - 17 * q^61 + 6 * q^62 + 23 * q^63 + 4 * q^64 + 3 * q^66 - 15 * q^67 - 6 * q^68 + 15 * q^69 - 4 * q^70 - 13 * q^71 - 21 * q^72 + 12 * q^73 - 11 * q^74 - 5 * q^75 - 9 * q^76 - 2 * q^77 - 37 * q^79 + 5 * q^80 + 27 * q^81 - 2 * q^82 + 29 * q^83 + 16 * q^84 + 3 * q^85 - 10 * q^86 + 10 * q^87 + 14 * q^88 + 3 * q^89 - 6 * q^90 - 11 * q^92 + 19 * q^93 + 10 * q^94 + 8 * q^95 - 5 * q^96 + 12 * q^97 + 2 * q^98 - 17 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more