LMFDB - Embedded newform 1520.2.a.n.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1520,2,Mod(1,1520)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1520, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 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("1520.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1520 = 2^{4} \cdot 5 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1520.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:	$12.1372611072$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 190)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.56155$ of defining polynomial
Character	$\chi$	$=$	1520.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.56155 q^{3} +1.00000 q^{5} -1.56155 q^{7} -0.561553 q^{9} +O(q^{10})$	$q-1.56155 q^{3} +1.00000 q^{5} -1.56155 q^{7} -0.561553 q^{9} -4.00000 q^{11} -6.68466 q^{13} -1.56155 q^{15} +7.56155 q^{17} +1.00000 q^{19} +2.43845 q^{21} +4.68466 q^{23} +1.00000 q^{25} +5.56155 q^{27} +6.68466 q^{29} -3.12311 q^{31} +6.24621 q^{33} -1.56155 q^{35} -6.00000 q^{37} +10.4384 q^{39} -4.24621 q^{41} +11.1231 q^{43} -0.561553 q^{45} +10.2462 q^{47} -4.56155 q^{49} -11.8078 q^{51} -0.438447 q^{53} -4.00000 q^{55} -1.56155 q^{57} +1.56155 q^{59} +2.87689 q^{61} +0.876894 q^{63} -6.68466 q^{65} -1.56155 q^{67} -7.31534 q^{69} +6.24621 q^{71} +10.6847 q^{73} -1.56155 q^{75} +6.24621 q^{77} -3.12311 q^{79} -7.00000 q^{81} -11.1231 q^{83} +7.56155 q^{85} -10.4384 q^{87} +2.00000 q^{89} +10.4384 q^{91} +4.87689 q^{93} +1.00000 q^{95} +6.00000 q^{97} +2.24621 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{3} + 2 q^{5} + q^{7} + 3 q^{9}+O(q^{10}) $ 2 * q + q^3 + 2 * q^5 + q^7 + 3 * q^9	$ 2 q + q^{3} + 2 q^{5} + q^{7} + 3 q^{9} - 8 q^{11} - q^{13} + q^{15} + 11 q^{17} + 2 q^{19} + 9 q^{21} - 3 q^{23} + 2 q^{25} + 7 q^{27} + q^{29} + 2 q^{31} - 4 q^{33} + q^{35} - 12 q^{37} + 25 q^{39} + 8 q^{41} + 14 q^{43} + 3 q^{45} + 4 q^{47} - 5 q^{49} - 3 q^{51} - 5 q^{53} - 8 q^{55} + q^{57} - q^{59} + 14 q^{61} + 10 q^{63} - q^{65} + q^{67} - 27 q^{69} - 4 q^{71} + 9 q^{73} + q^{75} - 4 q^{77} + 2 q^{79} - 14 q^{81} - 14 q^{83} + 11 q^{85} - 25 q^{87} + 4 q^{89} + 25 q^{91} + 18 q^{93} + 2 q^{95} + 12 q^{97} - 12 q^{99}+O(q^{100}) $ 2 * q + q^3 + 2 * q^5 + q^7 + 3 * q^9 - 8 * q^11 - q^13 + q^15 + 11 * q^17 + 2 * q^19 + 9 * q^21 - 3 * q^23 + 2 * q^25 + 7 * q^27 + q^29 + 2 * q^31 - 4 * q^33 + q^35 - 12 * q^37 + 25 * q^39 + 8 * q^41 + 14 * q^43 + 3 * q^45 + 4 * q^47 - 5 * q^49 - 3 * q^51 - 5 * q^53 - 8 * q^55 + q^57 - q^59 + 14 * q^61 + 10 * q^63 - q^65 + q^67 - 27 * q^69 - 4 * q^71 + 9 * q^73 + q^75 - 4 * q^77 + 2 * q^79 - 14 * q^81 - 14 * q^83 + 11 * q^85 - 25 * q^87 + 4 * q^89 + 25 * q^91 + 18 * q^93 + 2 * q^95 + 12 * q^97 - 12 * 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	0
$2$	0	0
$3$	−1.56155	−0.901563	−0.450781	−	0.892634i	$-0.648855\pi$
$3$	−1.56155	−0.901563	−0.450781	+	0.892634i	$0.648855\pi$
$4$	0	0
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	−1.56155	−0.590211	−0.295106	−	0.955465i	$-0.595355\pi$
$7$	−1.56155	−0.590211	−0.295106	+	0.955465i	$0.595355\pi$
$8$	0	0
$9$	−0.561553	−0.187184
$10$	0	0
$11$	−4.00000	−1.20605	−0.603023	−	0.797724i	$-0.706037\pi$
$11$	−4.00000	−1.20605	−0.603023	+	0.797724i	$0.706037\pi$
$12$	0	0
$13$	−6.68466	−1.85399	−0.926995	−	0.375073i	$-0.877618\pi$
$13$	−6.68466	−1.85399	−0.926995	+	0.375073i	$0.877618\pi$
$14$	0	0
$15$	−1.56155	−0.403191
$16$	0	0
$17$	7.56155	1.83395	0.916973	−	0.398949i	$-0.130625\pi$
$17$	7.56155	1.83395	0.916973	+	0.398949i	$0.130625\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	2.43845	0.532113
$22$	0	0
$23$	4.68466	0.976819	0.488409	−	0.872615i	$-0.337577\pi$
$23$	4.68466	0.976819	0.488409	+	0.872615i	$0.337577\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	5.56155	1.07032
$28$	0	0
$29$	6.68466	1.24131	0.620655	−	0.784084i	$-0.286867\pi$
$29$	6.68466	1.24131	0.620655	+	0.784084i	$0.286867\pi$
$30$	0	0
$31$	−3.12311	−0.560926	−0.280463	−	0.959865i	$-0.590488\pi$
$31$	−3.12311	−0.560926	−0.280463	+	0.959865i	$0.590488\pi$
$32$	0	0
$33$	6.24621	1.08733
$34$	0	0
$35$	−1.56155	−0.263951
$36$	0	0
$37$	−6.00000	−0.986394	−0.493197	−	0.869918i	$-0.664172\pi$
$37$	−6.00000	−0.986394	−0.493197	+	0.869918i	$0.664172\pi$
$38$	0	0
$39$	10.4384	1.67149
$40$	0	0
$41$	−4.24621	−0.663147	−0.331573	−	0.943429i	$-0.607579\pi$
$41$	−4.24621	−0.663147	−0.331573	+	0.943429i	$0.607579\pi$
$42$	0	0
$43$	11.1231	1.69626	0.848129	−	0.529790i	$-0.177729\pi$
$43$	11.1231	1.69626	0.848129	+	0.529790i	$0.177729\pi$
$44$	0	0
$45$	−0.561553	−0.0837114
$46$	0	0
$47$	10.2462	1.49456	0.747282	−	0.664507i	$-0.231359\pi$
$47$	10.2462	1.49456	0.747282	+	0.664507i	$0.231359\pi$
$48$	0	0
$49$	−4.56155	−0.651650
$50$	0	0
$51$	−11.8078	−1.65342
$52$	0	0
$53$	−0.438447	−0.0602254	−0.0301127	−	0.999547i	$-0.509587\pi$
$53$	−0.438447	−0.0602254	−0.0301127	+	0.999547i	$0.509587\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	0	0
$57$	−1.56155	−0.206833
$58$	0	0
$59$	1.56155	0.203297	0.101648	−	0.994820i	$-0.467588\pi$
$59$	1.56155	0.203297	0.101648	+	0.994820i	$0.467588\pi$
$60$	0	0
$61$	2.87689	0.368349	0.184174	−	0.982894i	$-0.441039\pi$
$61$	2.87689	0.368349	0.184174	+	0.982894i	$0.441039\pi$
$62$	0	0
$63$	0.876894	0.110478
$64$	0	0
$65$	−6.68466	−0.829130
$66$	0	0
$67$	−1.56155	−0.190774	−0.0953870	−	0.995440i	$-0.530409\pi$
$67$	−1.56155	−0.190774	−0.0953870	+	0.995440i	$0.530409\pi$
$68$	0	0
$69$	−7.31534	−0.880664
$70$	0	0
$71$	6.24621	0.741289	0.370644	−	0.928775i	$-0.379137\pi$
$71$	6.24621	0.741289	0.370644	+	0.928775i	$0.379137\pi$
$72$	0	0
$73$	10.6847	1.25054	0.625272	−	0.780407i	$-0.284988\pi$
$73$	10.6847	1.25054	0.625272	+	0.780407i	$0.284988\pi$
$74$	0	0
$75$	−1.56155	−0.180313
$76$	0	0
$77$	6.24621	0.711822
$78$	0	0
$79$	−3.12311	−0.351377	−0.175688	−	0.984446i	$-0.556215\pi$
$79$	−3.12311	−0.351377	−0.175688	+	0.984446i	$0.556215\pi$
$80$	0	0
$81$	−7.00000	−0.777778
$82$	0	0
$83$	−11.1231	−1.22092	−0.610460	−	0.792047i	$-0.709015\pi$
$83$	−11.1231	−1.22092	−0.610460	+	0.792047i	$0.709015\pi$
$84$	0	0
$85$	7.56155	0.820166
$86$	0	0
$87$	−10.4384	−1.11912
$88$	0	0
$89$	2.00000	0.212000	0.106000	−	0.994366i	$-0.466196\pi$
$89$	2.00000	0.212000	0.106000	+	0.994366i	$0.466196\pi$
$90$	0	0
$91$	10.4384	1.09425
$92$	0	0
$93$	4.87689	0.505710
$94$	0	0
$95$	1.00000	0.102598
$96$	0	0
$97$	6.00000	0.609208	0.304604	−	0.952479i	$-0.401476\pi$
$97$	6.00000	0.609208	0.304604	+	0.952479i	$0.401476\pi$
$98$	0	0
$99$	2.24621	0.225753
$100$	0	0
$101$	7.36932	0.733274	0.366637	−	0.930364i	$-0.380509\pi$
$101$	7.36932	0.733274	0.366637	+	0.930364i	$0.380509\pi$
$102$	0	0
$103$	14.2462	1.40372	0.701860	−	0.712314i	$-0.252353\pi$
$103$	14.2462	1.40372	0.701860	+	0.712314i	$0.252353\pi$
$104$	0	0
$105$	2.43845	0.237968
$106$	0	0
$107$	9.56155	0.924350	0.462175	−	0.886789i	$-0.347069\pi$
$107$	9.56155	0.924350	0.462175	+	0.886789i	$0.347069\pi$
$108$	0	0
$109$	3.56155	0.341135	0.170567	−	0.985346i	$-0.445440\pi$
$109$	3.56155	0.341135	0.170567	+	0.985346i	$0.445440\pi$
$110$	0	0
$111$	9.36932	0.889296
$112$	0	0
$113$	17.1231	1.61081	0.805403	−	0.592727i	$-0.201949\pi$
$113$	17.1231	1.61081	0.805403	+	0.592727i	$0.201949\pi$
$114$	0	0
$115$	4.68466	0.436847
$116$	0	0
$117$	3.75379	0.347038
$118$	0	0
$119$	−11.8078	−1.08242
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	6.63068	0.597869
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−4.87689	−0.432754	−0.216377	−	0.976310i	$-0.569424\pi$
$127$	−4.87689	−0.432754	−0.216377	+	0.976310i	$0.569424\pi$
$128$	0	0
$129$	−17.3693	−1.52928
$130$	0	0
$131$	−16.4924	−1.44095	−0.720475	−	0.693481i	$-0.756076\pi$
$131$	−16.4924	−1.44095	−0.720475	+	0.693481i	$0.756076\pi$
$132$	0	0
$133$	−1.56155	−0.135404
$134$	0	0
$135$	5.56155	0.478662
$136$	0	0
$137$	5.80776	0.496191	0.248095	−	0.968736i	$-0.420195\pi$
$137$	5.80776	0.496191	0.248095	+	0.968736i	$0.420195\pi$
$138$	0	0
$139$	16.4924	1.39887	0.699435	−	0.714697i	$-0.253435\pi$
$139$	16.4924	1.39887	0.699435	+	0.714697i	$0.253435\pi$
$140$	0	0
$141$	−16.0000	−1.34744
$142$	0	0
$143$	26.7386	2.23600
$144$	0	0
$145$	6.68466	0.555131
$146$	0	0
$147$	7.12311	0.587504
$148$	0	0
$149$	−11.3693	−0.931411	−0.465705	−	0.884940i	$-0.654199\pi$
$149$	−11.3693	−0.931411	−0.465705	+	0.884940i	$0.654199\pi$
$150$	0	0
$151$	3.12311	0.254155	0.127077	−	0.991893i	$-0.459440\pi$
$151$	3.12311	0.254155	0.127077	+	0.991893i	$0.459440\pi$
$152$	0	0
$153$	−4.24621	−0.343286
$154$	0	0
$155$	−3.12311	−0.250854
$156$	0	0
$157$	−3.75379	−0.299585	−0.149792	−	0.988717i	$-0.547861\pi$
$157$	−3.75379	−0.299585	−0.149792	+	0.988717i	$0.547861\pi$
$158$	0	0
$159$	0.684658	0.0542969
$160$	0	0
$161$	−7.31534	−0.576530
$162$	0	0
$163$	−9.36932	−0.733862	−0.366931	−	0.930248i	$-0.619591\pi$
$163$	−9.36932	−0.733862	−0.366931	+	0.930248i	$0.619591\pi$
$164$	0	0
$165$	6.24621	0.486267
$166$	0	0
$167$	−17.3693	−1.34408	−0.672039	−	0.740516i	$-0.734581\pi$
$167$	−17.3693	−1.34408	−0.672039	+	0.740516i	$0.734581\pi$
$168$	0	0
$169$	31.6847	2.43728
$170$	0	0
$171$	−0.561553	−0.0429430
$172$	0	0
$173$	3.75379	0.285395	0.142698	−	0.989766i	$-0.454422\pi$
$173$	3.75379	0.285395	0.142698	+	0.989766i	$0.454422\pi$
$174$	0	0
$175$	−1.56155	−0.118042
$176$	0	0
$177$	−2.43845	−0.183285
$178$	0	0
$179$	5.75379	0.430058	0.215029	−	0.976608i	$-0.431015\pi$
$179$	5.75379	0.430058	0.215029	+	0.976608i	$0.431015\pi$
$180$	0	0
$181$	−18.0000	−1.33793	−0.668965	−	0.743294i	$-0.733262\pi$
$181$	−18.0000	−1.33793	−0.668965	+	0.743294i	$0.733262\pi$
$182$	0	0
$183$	−4.49242	−0.332089
$184$	0	0
$185$	−6.00000	−0.441129
$186$	0	0
$187$	−30.2462	−2.21182
$188$	0	0
$189$	−8.68466	−0.631716
$190$	0	0
$191$	−8.68466	−0.628400	−0.314200	−	0.949357i	$-0.601736\pi$
$191$	−8.68466	−0.628400	−0.314200	+	0.949357i	$0.601736\pi$
$192$	0	0
$193$	18.4924	1.33111	0.665557	−	0.746347i	$-0.268194\pi$
$193$	18.4924	1.33111	0.665557	+	0.746347i	$0.268194\pi$
$194$	0	0
$195$	10.4384	0.747513
$196$	0	0
$197$	−3.75379	−0.267446	−0.133723	−	0.991019i	$-0.542693\pi$
$197$	−3.75379	−0.267446	−0.133723	+	0.991019i	$0.542693\pi$
$198$	0	0
$199$	3.80776	0.269925	0.134963	−	0.990851i	$-0.456909\pi$
$199$	3.80776	0.269925	0.134963	+	0.990851i	$0.456909\pi$
$200$	0	0
$201$	2.43845	0.171995
$202$	0	0
$203$	−10.4384	−0.732635
$204$	0	0
$205$	−4.24621	−0.296568
$206$	0	0
$207$	−2.63068	−0.182845
$208$	0	0
$209$	−4.00000	−0.276686
$210$	0	0
$211$	−20.6847	−1.42399	−0.711995	−	0.702184i	$-0.752208\pi$
$211$	−20.6847	−1.42399	−0.711995	+	0.702184i	$0.752208\pi$
$212$	0	0
$213$	−9.75379	−0.668319
$214$	0	0
$215$	11.1231	0.758590
$216$	0	0
$217$	4.87689	0.331065
$218$	0	0
$219$	−16.6847	−1.12744
$220$	0	0
$221$	−50.5464	−3.40012
$222$	0	0
$223$	−1.36932	−0.0916962	−0.0458481	−	0.998948i	$-0.514599\pi$
$223$	−1.36932	−0.0916962	−0.0458481	+	0.998948i	$0.514599\pi$
$224$	0	0
$225$	−0.561553	−0.0374369
$226$	0	0
$227$	2.93087	0.194529	0.0972643	−	0.995259i	$-0.468991\pi$
$227$	2.93087	0.194529	0.0972643	+	0.995259i	$0.468991\pi$
$228$	0	0
$229$	18.4924	1.22201	0.611007	−	0.791625i	$-0.290765\pi$
$229$	18.4924	1.22201	0.611007	+	0.791625i	$0.290765\pi$
$230$	0	0
$231$	−9.75379	−0.641752
$232$	0	0
$233$	10.0000	0.655122	0.327561	−	0.944830i	$-0.393773\pi$
$233$	10.0000	0.655122	0.327561	+	0.944830i	$0.393773\pi$
$234$	0	0
$235$	10.2462	0.668389
$236$	0	0
$237$	4.87689	0.316788
$238$	0	0
$239$	5.56155	0.359747	0.179873	−	0.983690i	$-0.442431\pi$
$239$	5.56155	0.359747	0.179873	+	0.983690i	$0.442431\pi$
$240$	0	0
$241$	14.8769	0.958305	0.479153	−	0.877732i	$-0.340944\pi$
$241$	14.8769	0.958305	0.479153	+	0.877732i	$0.340944\pi$
$242$	0	0
$243$	−5.75379	−0.369106
$244$	0	0
$245$	−4.56155	−0.291427
$246$	0	0
$247$	−6.68466	−0.425335
$248$	0	0
$249$	17.3693	1.10074
$250$	0	0
$251$	10.2462	0.646735	0.323368	−	0.946273i	$-0.395185\pi$
$251$	10.2462	0.646735	0.323368	+	0.946273i	$0.395185\pi$
$252$	0	0
$253$	−18.7386	−1.17809
$254$	0	0
$255$	−11.8078	−0.739431
$256$	0	0
$257$	14.0000	0.873296	0.436648	−	0.899632i	$-0.356166\pi$
$257$	14.0000	0.873296	0.436648	+	0.899632i	$0.356166\pi$
$258$	0	0
$259$	9.36932	0.582181
$260$	0	0
$261$	−3.75379	−0.232354
$262$	0	0
$263$	5.75379	0.354794	0.177397	−	0.984139i	$-0.443232\pi$
$263$	5.75379	0.354794	0.177397	+	0.984139i	$0.443232\pi$
$264$	0	0
$265$	−0.438447	−0.0269336
$266$	0	0
$267$	−3.12311	−0.191131
$268$	0	0
$269$	−26.0000	−1.58525	−0.792624	−	0.609711i	$-0.791286\pi$
$269$	−26.0000	−1.58525	−0.792624	+	0.609711i	$0.791286\pi$
$270$	0	0
$271$	−6.93087	−0.421020	−0.210510	−	0.977592i	$-0.567513\pi$
$271$	−6.93087	−0.421020	−0.210510	+	0.977592i	$0.567513\pi$
$272$	0	0
$273$	−16.3002	−0.986532
$274$	0	0
$275$	−4.00000	−0.241209
$276$	0	0
$277$	9.12311	0.548154	0.274077	−	0.961708i	$-0.411628\pi$
$277$	9.12311	0.548154	0.274077	+	0.961708i	$0.411628\pi$
$278$	0	0
$279$	1.75379	0.104997
$280$	0	0
$281$	2.00000	0.119310	0.0596550	−	0.998219i	$-0.481000\pi$
$281$	2.00000	0.119310	0.0596550	+	0.998219i	$0.481000\pi$
$282$	0	0
$283$	12.8769	0.765452	0.382726	−	0.923862i	$-0.374985\pi$
$283$	12.8769	0.765452	0.382726	+	0.923862i	$0.374985\pi$
$284$	0	0
$285$	−1.56155	−0.0924984
$286$	0	0
$287$	6.63068	0.391397
$288$	0	0
$289$	40.1771	2.36336
$290$	0	0
$291$	−9.36932	−0.549239
$292$	0	0
$293$	23.1771	1.35402	0.677010	−	0.735974i	$-0.263275\pi$
$293$	23.1771	1.35402	0.677010	+	0.735974i	$0.263275\pi$
$294$	0	0
$295$	1.56155	0.0909171
$296$	0	0
$297$	−22.2462	−1.29086
$298$	0	0
$299$	−31.3153	−1.81101
$300$	0	0
$301$	−17.3693	−1.00115
$302$	0	0
$303$	−11.5076	−0.661093
$304$	0	0
$305$	2.87689	0.164730
$306$	0	0
$307$	0.492423	0.0281040	0.0140520	−	0.999901i	$-0.495527\pi$
$307$	0.492423	0.0281040	0.0140520	+	0.999901i	$0.495527\pi$
$308$	0	0
$309$	−22.2462	−1.26554
$310$	0	0
$311$	−8.68466	−0.492462	−0.246231	−	0.969211i	$-0.579192\pi$
$311$	−8.68466	−0.492462	−0.246231	+	0.969211i	$0.579192\pi$
$312$	0	0
$313$	−32.0540	−1.81180	−0.905899	−	0.423494i	$-0.860803\pi$
$313$	−32.0540	−1.81180	−0.905899	+	0.423494i	$0.860803\pi$
$314$	0	0
$315$	0.876894	0.0494074
$316$	0	0
$317$	−24.0540	−1.35101	−0.675503	−	0.737357i	$-0.736073\pi$
$317$	−24.0540	−1.35101	−0.675503	+	0.737357i	$0.736073\pi$
$318$	0	0
$319$	−26.7386	−1.49708
$320$	0	0
$321$	−14.9309	−0.833360
$322$	0	0
$323$	7.56155	0.420736
$324$	0	0
$325$	−6.68466	−0.370798
$326$	0	0
$327$	−5.56155	−0.307555
$328$	0	0
$329$	−16.0000	−0.882109
$330$	0	0
$331$	−1.56155	−0.0858307	−0.0429154	−	0.999079i	$-0.513665\pi$
$331$	−1.56155	−0.0858307	−0.0429154	+	0.999079i	$0.513665\pi$
$332$	0	0
$333$	3.36932	0.184637
$334$	0	0
$335$	−1.56155	−0.0853167
$336$	0	0
$337$	−26.0000	−1.41631	−0.708155	−	0.706057i	$-0.750472\pi$
$337$	−26.0000	−1.41631	−0.708155	+	0.706057i	$0.750472\pi$
$338$	0	0
$339$	−26.7386	−1.45224
$340$	0	0
$341$	12.4924	0.676503
$342$	0	0
$343$	18.0540	0.974823
$344$	0	0
$345$	−7.31534	−0.393845
$346$	0	0
$347$	33.3693	1.79136	0.895679	−	0.444700i	$-0.146690\pi$
$347$	33.3693	1.79136	0.895679	+	0.444700i	$0.146690\pi$
$348$	0	0
$349$	20.2462	1.08375	0.541877	−	0.840458i	$-0.317714\pi$
$349$	20.2462	1.08375	0.541877	+	0.840458i	$0.317714\pi$
$350$	0	0
$351$	−37.1771	−1.98437
$352$	0	0
$353$	24.9309	1.32694	0.663468	−	0.748205i	$-0.269084\pi$
$353$	24.9309	1.32694	0.663468	+	0.748205i	$0.269084\pi$
$354$	0	0
$355$	6.24621	0.331514
$356$	0	0
$357$	18.4384	0.975866
$358$	0	0
$359$	−5.56155	−0.293528	−0.146764	−	0.989172i	$-0.546886\pi$
$359$	−5.56155	−0.293528	−0.146764	+	0.989172i	$0.546886\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	−7.80776	−0.409801
$364$	0	0
$365$	10.6847	0.559261
$366$	0	0
$367$	10.2462	0.534848	0.267424	−	0.963579i	$-0.413828\pi$
$367$	10.2462	0.534848	0.267424	+	0.963579i	$0.413828\pi$
$368$	0	0
$369$	2.38447	0.124131
$370$	0	0
$371$	0.684658	0.0355457
$372$	0	0
$373$	−27.5616	−1.42708	−0.713542	−	0.700613i	$-0.752910\pi$
$373$	−27.5616	−1.42708	−0.713542	+	0.700613i	$0.752910\pi$
$374$	0	0
$375$	−1.56155	−0.0806382
$376$	0	0
$377$	−44.6847	−2.30138
$378$	0	0
$379$	−6.43845	−0.330721	−0.165360	−	0.986233i	$-0.552879\pi$
$379$	−6.43845	−0.330721	−0.165360	+	0.986233i	$0.552879\pi$
$380$	0	0
$381$	7.61553	0.390155
$382$	0	0
$383$	−30.2462	−1.54551	−0.772755	−	0.634705i	$-0.781122\pi$
$383$	−30.2462	−1.54551	−0.772755	+	0.634705i	$0.781122\pi$
$384$	0	0
$385$	6.24621	0.318336
$386$	0	0
$387$	−6.24621	−0.317513
$388$	0	0
$389$	1.12311	0.0569437	0.0284719	−	0.999595i	$-0.490936\pi$
$389$	1.12311	0.0569437	0.0284719	+	0.999595i	$0.490936\pi$
$390$	0	0
$391$	35.4233	1.79143
$392$	0	0
$393$	25.7538	1.29911
$394$	0	0
$395$	−3.12311	−0.157140
$396$	0	0
$397$	1.12311	0.0563671	0.0281835	−	0.999603i	$-0.491028\pi$
$397$	1.12311	0.0563671	0.0281835	+	0.999603i	$0.491028\pi$
$398$	0	0
$399$	2.43845	0.122075
$400$	0	0
$401$	−20.2462	−1.01105	−0.505524	−	0.862813i	$-0.668701\pi$
$401$	−20.2462	−1.01105	−0.505524	+	0.862813i	$0.668701\pi$
$402$	0	0
$403$	20.8769	1.03995
$404$	0	0
$405$	−7.00000	−0.347833
$406$	0	0
$407$	24.0000	1.18964
$408$	0	0
$409$	−24.7386	−1.22325	−0.611623	−	0.791149i	$-0.709483\pi$
$409$	−24.7386	−1.22325	−0.611623	+	0.791149i	$0.709483\pi$
$410$	0	0
$411$	−9.06913	−0.447347
$412$	0	0
$413$	−2.43845	−0.119988
$414$	0	0
$415$	−11.1231	−0.546012
$416$	0	0
$417$	−25.7538	−1.26117
$418$	0	0
$419$	33.8617	1.65425	0.827127	−	0.562015i	$-0.189974\pi$
$419$	33.8617	1.65425	0.827127	+	0.562015i	$0.189974\pi$
$420$	0	0
$421$	4.93087	0.240316	0.120158	−	0.992755i	$-0.461660\pi$
$421$	4.93087	0.240316	0.120158	+	0.992755i	$0.461660\pi$
$422$	0	0
$423$	−5.75379	−0.279759
$424$	0	0
$425$	7.56155	0.366789
$426$	0	0
$427$	−4.49242	−0.217404
$428$	0	0
$429$	−41.7538	−2.01589
$430$	0	0
$431$	16.0000	0.770693	0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000	0.770693	0.385346	+	0.922772i	$0.374082\pi$
$432$	0	0
$433$	39.3693	1.89197	0.945984	−	0.324212i	$-0.105099\pi$
$433$	39.3693	1.89197	0.945984	+	0.324212i	$0.105099\pi$
$434$	0	0
$435$	−10.4384	−0.500485
$436$	0	0
$437$	4.68466	0.224098
$438$	0	0
$439$	4.87689	0.232761	0.116381	−	0.993205i	$-0.462871\pi$
$439$	4.87689	0.232761	0.116381	+	0.993205i	$0.462871\pi$
$440$	0	0
$441$	2.56155	0.121979
$442$	0	0
$443$	14.2462	0.676858	0.338429	−	0.940992i	$-0.390105\pi$
$443$	14.2462	0.676858	0.338429	+	0.940992i	$0.390105\pi$
$444$	0	0
$445$	2.00000	0.0948091
$446$	0	0
$447$	17.7538	0.839725
$448$	0	0
$449$	20.7386	0.978717	0.489358	−	0.872083i	$-0.337231\pi$
$449$	20.7386	0.978717	0.489358	+	0.872083i	$0.337231\pi$
$450$	0	0
$451$	16.9848	0.799785
$452$	0	0
$453$	−4.87689	−0.229136
$454$	0	0
$455$	10.4384	0.489362
$456$	0	0
$457$	18.6847	0.874031	0.437016	−	0.899454i	$-0.356035\pi$
$457$	18.6847	0.874031	0.437016	+	0.899454i	$0.356035\pi$
$458$	0	0
$459$	42.0540	1.96291
$460$	0	0
$461$	20.2462	0.942960	0.471480	−	0.881877i	$-0.343720\pi$
$461$	20.2462	0.942960	0.471480	+	0.881877i	$0.343720\pi$
$462$	0	0
$463$	13.7538	0.639193	0.319596	−	0.947554i	$-0.396453\pi$
$463$	13.7538	0.639193	0.319596	+	0.947554i	$0.396453\pi$
$464$	0	0
$465$	4.87689	0.226161
$466$	0	0
$467$	−1.75379	−0.0811557	−0.0405778	−	0.999176i	$-0.512920\pi$
$467$	−1.75379	−0.0811557	−0.0405778	+	0.999176i	$0.512920\pi$
$468$	0	0
$469$	2.43845	0.112597
$470$	0	0
$471$	5.86174	0.270095
$472$	0	0
$473$	−44.4924	−2.04576
$474$	0	0
$475$	1.00000	0.0458831
$476$	0	0
$477$	0.246211	0.0112732
$478$	0	0
$479$	−32.0000	−1.46212	−0.731059	−	0.682315i	$-0.760973\pi$
$479$	−32.0000	−1.46212	−0.731059	+	0.682315i	$0.760973\pi$
$480$	0	0
$481$	40.1080	1.82877
$482$	0	0
$483$	11.4233	0.519778
$484$	0	0
$485$	6.00000	0.272446
$486$	0	0
$487$	−23.6155	−1.07012	−0.535061	−	0.844814i	$-0.679711\pi$
$487$	−23.6155	−1.07012	−0.535061	+	0.844814i	$0.679711\pi$
$488$	0	0
$489$	14.6307	0.661622
$490$	0	0
$491$	−7.12311	−0.321461	−0.160731	−	0.986998i	$-0.551385\pi$
$491$	−7.12311	−0.321461	−0.160731	+	0.986998i	$0.551385\pi$
$492$	0	0
$493$	50.5464	2.27650
$494$	0	0
$495$	2.24621	0.100960
$496$	0	0
$497$	−9.75379	−0.437517
$498$	0	0
$499$	32.1080	1.43735	0.718675	−	0.695347i	$-0.244749\pi$
$499$	32.1080	1.43735	0.718675	+	0.695347i	$0.244749\pi$
$500$	0	0
$501$	27.1231	1.21177
$502$	0	0
$503$	−30.0540	−1.34004	−0.670020	−	0.742343i	$-0.733715\pi$
$503$	−30.0540	−1.34004	−0.670020	+	0.742343i	$0.733715\pi$
$504$	0	0
$505$	7.36932	0.327930
$506$	0	0
$507$	−49.4773	−2.19736
$508$	0	0
$509$	−30.4924	−1.35155	−0.675776	−	0.737107i	$-0.736192\pi$
$509$	−30.4924	−1.35155	−0.675776	+	0.737107i	$0.736192\pi$
$510$	0	0
$511$	−16.6847	−0.738086
$512$	0	0
$513$	5.56155	0.245549
$514$	0	0
$515$	14.2462	0.627763
$516$	0	0
$517$	−40.9848	−1.80251
$518$	0	0
$519$	−5.86174	−0.257302
$520$	0	0
$521$	5.12311	0.224447	0.112224	−	0.993683i	$-0.464203\pi$
$521$	5.12311	0.224447	0.112224	+	0.993683i	$0.464203\pi$
$522$	0	0
$523$	19.3153	0.844601	0.422300	−	0.906456i	$-0.361223\pi$
$523$	19.3153	0.844601	0.422300	+	0.906456i	$0.361223\pi$
$524$	0	0
$525$	2.43845	0.106423
$526$	0	0
$527$	−23.6155	−1.02871
$528$	0	0
$529$	−1.05398	−0.0458250
$530$	0	0
$531$	−0.876894	−0.0380540
$532$	0	0
$533$	28.3845	1.22947
$534$	0	0
$535$	9.56155	0.413382
$536$	0	0
$537$	−8.98485	−0.387725
$538$	0	0
$539$	18.2462	0.785920
$540$	0	0
$541$	−41.6155	−1.78919	−0.894596	−	0.446877i	$-0.852536\pi$
$541$	−41.6155	−1.78919	−0.894596	+	0.446877i	$0.852536\pi$
$542$	0	0
$543$	28.1080	1.20623
$544$	0	0
$545$	3.56155	0.152560
$546$	0	0
$547$	16.4924	0.705165	0.352583	−	0.935781i	$-0.385304\pi$
$547$	16.4924	0.705165	0.352583	+	0.935781i	$0.385304\pi$
$548$	0	0
$549$	−1.61553	−0.0689491
$550$	0	0
$551$	6.68466	0.284776
$552$	0	0
$553$	4.87689	0.207387
$554$	0	0
$555$	9.36932	0.397705
$556$	0	0
$557$	−1.61553	−0.0684521	−0.0342261	−	0.999414i	$-0.510897\pi$
$557$	−1.61553	−0.0684521	−0.0342261	+	0.999414i	$0.510897\pi$
$558$	0	0
$559$	−74.3542	−3.14485
$560$	0	0
$561$	47.2311	1.99410
$562$	0	0
$563$	−24.4924	−1.03223	−0.516116	−	0.856519i	$-0.672623\pi$
$563$	−24.4924	−1.03223	−0.516116	+	0.856519i	$0.672623\pi$
$564$	0	0
$565$	17.1231	0.720374
$566$	0	0
$567$	10.9309	0.459053
$568$	0	0
$569$	5.12311	0.214772	0.107386	−	0.994217i	$-0.465752\pi$
$569$	5.12311	0.214772	0.107386	+	0.994217i	$0.465752\pi$
$570$	0	0
$571$	19.6155	0.820884	0.410442	−	0.911887i	$-0.365374\pi$
$571$	19.6155	0.820884	0.410442	+	0.911887i	$0.365374\pi$
$572$	0	0
$573$	13.5616	0.566542
$574$	0	0
$575$	4.68466	0.195364
$576$	0	0
$577$	−22.6847	−0.944375	−0.472187	−	0.881498i	$-0.656535\pi$
$577$	−22.6847	−0.944375	−0.472187	+	0.881498i	$0.656535\pi$
$578$	0	0
$579$	−28.8769	−1.20008
$580$	0	0
$581$	17.3693	0.720601
$582$	0	0
$583$	1.75379	0.0726345
$584$	0	0
$585$	3.75379	0.155200
$586$	0	0
$587$	−17.3693	−0.716908	−0.358454	−	0.933547i	$-0.616696\pi$
$587$	−17.3693	−0.716908	−0.358454	+	0.933547i	$0.616696\pi$
$588$	0	0
$589$	−3.12311	−0.128685
$590$	0	0
$591$	5.86174	0.241120
$592$	0	0
$593$	24.2462	0.995673	0.497836	−	0.867271i	$-0.334128\pi$
$593$	24.2462	0.995673	0.497836	+	0.867271i	$0.334128\pi$
$594$	0	0
$595$	−11.8078	−0.484071
$596$	0	0
$597$	−5.94602	−0.243355
$598$	0	0
$599$	−45.8617	−1.87386	−0.936930	−	0.349517i	$-0.886346\pi$
$599$	−45.8617	−1.87386	−0.936930	+	0.349517i	$0.886346\pi$
$600$	0	0
$601$	18.0000	0.734235	0.367118	−	0.930175i	$-0.380345\pi$
$601$	18.0000	0.734235	0.367118	+	0.930175i	$0.380345\pi$
$602$	0	0
$603$	0.876894	0.0357099
$604$	0	0
$605$	5.00000	0.203279
$606$	0	0
$607$	12.8769	0.522657	0.261329	−	0.965250i	$-0.415839\pi$
$607$	12.8769	0.522657	0.261329	+	0.965250i	$0.415839\pi$
$608$	0	0
$609$	16.3002	0.660517
$610$	0	0
$611$	−68.4924	−2.77091
$612$	0	0
$613$	−19.3693	−0.782319	−0.391160	−	0.920323i	$-0.627926\pi$
$613$	−19.3693	−0.782319	−0.391160	+	0.920323i	$0.627926\pi$
$614$	0	0
$615$	6.63068	0.267375
$616$	0	0
$617$	−4.24621	−0.170946	−0.0854730	−	0.996340i	$-0.527240\pi$
$617$	−4.24621	−0.170946	−0.0854730	+	0.996340i	$0.527240\pi$
$618$	0	0
$619$	36.0000	1.44696	0.723481	−	0.690344i	$-0.242541\pi$
$619$	36.0000	1.44696	0.723481	+	0.690344i	$0.242541\pi$
$620$	0	0
$621$	26.0540	1.04551
$622$	0	0
$623$	−3.12311	−0.125125
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	6.24621	0.249450
$628$	0	0
$629$	−45.3693	−1.80899
$630$	0	0
$631$	−12.4924	−0.497315	−0.248658	−	0.968591i	$-0.579989\pi$
$631$	−12.4924	−0.497315	−0.248658	+	0.968591i	$0.579989\pi$
$632$	0	0
$633$	32.3002	1.28382
$634$	0	0
$635$	−4.87689	−0.193534
$636$	0	0
$637$	30.4924	1.20815
$638$	0	0
$639$	−3.50758	−0.138758
$640$	0	0
$641$	−35.8617	−1.41645	−0.708227	−	0.705985i	$-0.750505\pi$
$641$	−35.8617	−1.41645	−0.708227	+	0.705985i	$0.750505\pi$
$642$	0	0
$643$	−7.61553	−0.300327	−0.150164	−	0.988661i	$-0.547980\pi$
$643$	−7.61553	−0.300327	−0.150164	+	0.988661i	$0.547980\pi$
$644$	0	0
$645$	−17.3693	−0.683916
$646$	0	0
$647$	−9.56155	−0.375903	−0.187952	−	0.982178i	$-0.560185\pi$
$647$	−9.56155	−0.375903	−0.187952	+	0.982178i	$0.560185\pi$
$648$	0	0
$649$	−6.24621	−0.245185
$650$	0	0
$651$	−7.61553	−0.298476
$652$	0	0
$653$	1.12311	0.0439505	0.0219753	−	0.999759i	$-0.493004\pi$
$653$	1.12311	0.0439505	0.0219753	+	0.999759i	$0.493004\pi$
$654$	0	0
$655$	−16.4924	−0.644412
$656$	0	0
$657$	−6.00000	−0.234082
$658$	0	0
$659$	42.9309	1.67235	0.836175	−	0.548463i	$-0.184787\pi$
$659$	42.9309	1.67235	0.836175	+	0.548463i	$0.184787\pi$
$660$	0	0
$661$	9.80776	0.381478	0.190739	−	0.981641i	$-0.438912\pi$
$661$	9.80776	0.381478	0.190739	+	0.981641i	$0.438912\pi$
$662$	0	0
$663$	78.9309	3.06542
$664$	0	0
$665$	−1.56155	−0.0605544
$666$	0	0
$667$	31.3153	1.21253
$668$	0	0
$669$	2.13826	0.0826699
$670$	0	0
$671$	−11.5076	−0.444245
$672$	0	0
$673$	−3.36932	−0.129878	−0.0649388	−	0.997889i	$-0.520685\pi$
$673$	−3.36932	−0.129878	−0.0649388	+	0.997889i	$0.520685\pi$
$674$	0	0
$675$	5.56155	0.214064
$676$	0	0
$677$	36.4384	1.40044	0.700222	−	0.713926i	$-0.253085\pi$
$677$	36.4384	1.40044	0.700222	+	0.713926i	$0.253085\pi$
$678$	0	0
$679$	−9.36932	−0.359561
$680$	0	0
$681$	−4.57671	−0.175380
$682$	0	0
$683$	−4.00000	−0.153056	−0.0765279	−	0.997067i	$-0.524383\pi$
$683$	−4.00000	−0.153056	−0.0765279	+	0.997067i	$0.524383\pi$
$684$	0	0
$685$	5.80776	0.221903
$686$	0	0
$687$	−28.8769	−1.10172
$688$	0	0
$689$	2.93087	0.111657
$690$	0	0
$691$	8.87689	0.337693	0.168846	−	0.985642i	$-0.445996\pi$
$691$	8.87689	0.337693	0.168846	+	0.985642i	$0.445996\pi$
$692$	0	0
$693$	−3.50758	−0.133242
$694$	0	0
$695$	16.4924	0.625593
$696$	0	0
$697$	−32.1080	−1.21618
$698$	0	0
$699$	−15.6155	−0.590634
$700$	0	0
$701$	−29.1231	−1.09996	−0.549982	−	0.835176i	$-0.685366\pi$
$701$	−29.1231	−1.09996	−0.549982	+	0.835176i	$0.685366\pi$
$702$	0	0
$703$	−6.00000	−0.226294
$704$	0	0
$705$	−16.0000	−0.602595
$706$	0	0
$707$	−11.5076	−0.432787
$708$	0	0
$709$	38.0000	1.42712	0.713560	−	0.700594i	$-0.247082\pi$
$709$	38.0000	1.42712	0.713560	+	0.700594i	$0.247082\pi$
$710$	0	0
$711$	1.75379	0.0657722
$712$	0	0
$713$	−14.6307	−0.547923
$714$	0	0
$715$	26.7386	0.999968
$716$	0	0
$717$	−8.68466	−0.324335
$718$	0	0
$719$	29.5616	1.10246	0.551230	−	0.834353i	$-0.314159\pi$
$719$	29.5616	1.10246	0.551230	+	0.834353i	$0.314159\pi$
$720$	0	0
$721$	−22.2462	−0.828492
$722$	0	0
$723$	−23.2311	−0.863972
$724$	0	0
$725$	6.68466	0.248262
$726$	0	0
$727$	36.6847	1.36056	0.680279	−	0.732953i	$-0.261858\pi$
$727$	36.6847	1.36056	0.680279	+	0.732953i	$0.261858\pi$
$728$	0	0
$729$	29.9848	1.11055
$730$	0	0
$731$	84.1080	3.11084
$732$	0	0
$733$	−45.1231	−1.66666	−0.833330	−	0.552776i	$-0.813569\pi$
$733$	−45.1231	−1.66666	−0.833330	+	0.552776i	$0.813569\pi$
$734$	0	0
$735$	7.12311	0.262740
$736$	0	0
$737$	6.24621	0.230082
$738$	0	0
$739$	−16.8769	−0.620827	−0.310413	−	0.950602i	$-0.600467\pi$
$739$	−16.8769	−0.620827	−0.310413	+	0.950602i	$0.600467\pi$
$740$	0	0
$741$	10.4384	0.383466
$742$	0	0
$743$	−27.1231	−0.995050	−0.497525	−	0.867450i	$-0.665758\pi$
$743$	−27.1231	−0.995050	−0.497525	+	0.867450i	$0.665758\pi$
$744$	0	0
$745$	−11.3693	−0.416540
$746$	0	0
$747$	6.24621	0.228537
$748$	0	0
$749$	−14.9309	−0.545562
$750$	0	0
$751$	43.1231	1.57358	0.786792	−	0.617218i	$-0.211740\pi$
$751$	43.1231	1.57358	0.786792	+	0.617218i	$0.211740\pi$
$752$	0	0
$753$	−16.0000	−0.583072
$754$	0	0
$755$	3.12311	0.113661
$756$	0	0
$757$	9.50758	0.345559	0.172779	−	0.984961i	$-0.444725\pi$
$757$	9.50758	0.345559	0.172779	+	0.984961i	$0.444725\pi$
$758$	0	0
$759$	29.2614	1.06212
$760$	0	0
$761$	−28.5464	−1.03481	−0.517403	−	0.855742i	$-0.673101\pi$
$761$	−28.5464	−1.03481	−0.517403	+	0.855742i	$0.673101\pi$
$762$	0	0
$763$	−5.56155	−0.201342
$764$	0	0
$765$	−4.24621	−0.153522
$766$	0	0
$767$	−10.4384	−0.376910
$768$	0	0
$769$	31.5616	1.13814	0.569069	−	0.822290i	$-0.307304\pi$
$769$	31.5616	1.13814	0.569069	+	0.822290i	$0.307304\pi$
$770$	0	0
$771$	−21.8617	−0.787331
$772$	0	0
$773$	−9.80776	−0.352761	−0.176380	−	0.984322i	$-0.556439\pi$
$773$	−9.80776	−0.352761	−0.176380	+	0.984322i	$0.556439\pi$
$774$	0	0
$775$	−3.12311	−0.112185
$776$	0	0
$777$	−14.6307	−0.524873
$778$	0	0
$779$	−4.24621	−0.152136
$780$	0	0
$781$	−24.9848	−0.894028
$782$	0	0
$783$	37.1771	1.32860
$784$	0	0
$785$	−3.75379	−0.133978
$786$	0	0
$787$	−31.8078	−1.13382	−0.566912	−	0.823778i	$-0.691862\pi$
$787$	−31.8078	−1.13382	−0.566912	+	0.823778i	$0.691862\pi$
$788$	0	0
$789$	−8.98485	−0.319869
$790$	0	0
$791$	−26.7386	−0.950716
$792$	0	0
$793$	−19.2311	−0.682915
$794$	0	0
$795$	0.684658	0.0242823
$796$	0	0
$797$	42.3002	1.49835	0.749175	−	0.662372i	$-0.230450\pi$
$797$	42.3002	1.49835	0.749175	+	0.662372i	$0.230450\pi$
$798$	0	0
$799$	77.4773	2.74095
$800$	0	0
$801$	−1.12311	−0.0396830
$802$	0	0
$803$	−42.7386	−1.50821
$804$	0	0
$805$	−7.31534	−0.257832
$806$	0	0
$807$	40.6004	1.42920
$808$	0	0
$809$	−8.43845	−0.296680	−0.148340	−	0.988936i	$-0.547393\pi$
$809$	−8.43845	−0.296680	−0.148340	+	0.988936i	$0.547393\pi$
$810$	0	0
$811$	0.192236	0.00675032	0.00337516	−	0.999994i	$-0.498926\pi$
$811$	0.192236	0.00675032	0.00337516	+	0.999994i	$0.498926\pi$
$812$	0	0
$813$	10.8229	0.379576
$814$	0	0
$815$	−9.36932	−0.328193
$816$	0	0
$817$	11.1231	0.389148
$818$	0	0
$819$	−5.86174	−0.204826
$820$	0	0
$821$	7.36932	0.257191	0.128595	−	0.991697i	$-0.458953\pi$
$821$	7.36932	0.257191	0.128595	+	0.991697i	$0.458953\pi$
$822$	0	0
$823$	33.5616	1.16988	0.584941	−	0.811076i	$-0.301118\pi$
$823$	33.5616	1.16988	0.584941	+	0.811076i	$0.301118\pi$
$824$	0	0
$825$	6.24621	0.217465
$826$	0	0
$827$	6.43845	0.223887	0.111943	−	0.993715i	$-0.464292\pi$
$827$	6.43845	0.223887	0.111943	+	0.993715i	$0.464292\pi$
$828$	0	0
$829$	16.0540	0.557578	0.278789	−	0.960352i	$-0.410067\pi$
$829$	16.0540	0.557578	0.278789	+	0.960352i	$0.410067\pi$
$830$	0	0
$831$	−14.2462	−0.494196
$832$	0	0
$833$	−34.4924	−1.19509
$834$	0	0
$835$	−17.3693	−0.601090
$836$	0	0
$837$	−17.3693	−0.600371
$838$	0	0
$839$	12.4924	0.431286	0.215643	−	0.976472i	$-0.430815\pi$
$839$	12.4924	0.431286	0.215643	+	0.976472i	$0.430815\pi$
$840$	0	0
$841$	15.6847	0.540850
$842$	0	0
$843$	−3.12311	−0.107565
$844$	0	0
$845$	31.6847	1.08999
$846$	0	0
$847$	−7.80776	−0.268278
$848$	0	0
$849$	−20.1080	−0.690103
$850$	0	0
$851$	−28.1080	−0.963528
$852$	0	0
$853$	24.7386	0.847035	0.423517	−	0.905888i	$-0.360795\pi$
$853$	24.7386	0.847035	0.423517	+	0.905888i	$0.360795\pi$
$854$	0	0
$855$	−0.561553	−0.0192047
$856$	0	0
$857$	39.3693	1.34483	0.672415	−	0.740174i	$-0.265257\pi$
$857$	39.3693	1.34483	0.672415	+	0.740174i	$0.265257\pi$
$858$	0	0
$859$	−12.9848	−0.443037	−0.221519	−	0.975156i	$-0.571101\pi$
$859$	−12.9848	−0.443037	−0.221519	+	0.975156i	$0.571101\pi$
$860$	0	0
$861$	−10.3542	−0.352869
$862$	0	0
$863$	14.2462	0.484947	0.242473	−	0.970158i	$-0.422041\pi$
$863$	14.2462	0.484947	0.242473	+	0.970158i	$0.422041\pi$
$864$	0	0
$865$	3.75379	0.127633
$866$	0	0
$867$	−62.7386	−2.13072
$868$	0	0
$869$	12.4924	0.423776
$870$	0	0
$871$	10.4384	0.353693
$872$	0	0
$873$	−3.36932	−0.114034
$874$	0	0
$875$	−1.56155	−0.0527901
$876$	0	0
$877$	24.9309	0.841856	0.420928	−	0.907094i	$-0.361705\pi$
$877$	24.9309	0.841856	0.420928	+	0.907094i	$0.361705\pi$
$878$	0	0
$879$	−36.1922	−1.22073
$880$	0	0
$881$	−22.9848	−0.774379	−0.387190	−	0.922000i	$-0.626554\pi$
$881$	−22.9848	−0.774379	−0.387190	+	0.922000i	$0.626554\pi$
$882$	0	0
$883$	47.6155	1.60239	0.801195	−	0.598403i	$-0.204198\pi$
$883$	47.6155	1.60239	0.801195	+	0.598403i	$0.204198\pi$
$884$	0	0
$885$	−2.43845	−0.0819675
$886$	0	0
$887$	4.49242	0.150841	0.0754204	−	0.997152i	$-0.475970\pi$
$887$	4.49242	0.150841	0.0754204	+	0.997152i	$0.475970\pi$
$888$	0	0
$889$	7.61553	0.255417
$890$	0	0
$891$	28.0000	0.938035
$892$	0	0
$893$	10.2462	0.342876
$894$	0	0
$895$	5.75379	0.192328
$896$	0	0
$897$	48.9006	1.63274
$898$	0	0
$899$	−20.8769	−0.696283
$900$	0	0
$901$	−3.31534	−0.110450
$902$	0	0
$903$	27.1231	0.902600
$904$	0	0
$905$	−18.0000	−0.598340
$906$	0	0
$907$	−25.1771	−0.835991	−0.417996	−	0.908449i	$-0.637267\pi$
$907$	−25.1771	−0.835991	−0.417996	+	0.908449i	$0.637267\pi$
$908$	0	0
$909$	−4.13826	−0.137257
$910$	0	0
$911$	28.4924	0.943996	0.471998	−	0.881600i	$-0.343533\pi$
$911$	28.4924	0.943996	0.471998	+	0.881600i	$0.343533\pi$
$912$	0	0
$913$	44.4924	1.47248
$914$	0	0
$915$	−4.49242	−0.148515
$916$	0	0
$917$	25.7538	0.850465
$918$	0	0
$919$	30.9309	1.02032	0.510158	−	0.860081i	$-0.329587\pi$
$919$	30.9309	1.02032	0.510158	+	0.860081i	$0.329587\pi$
$920$	0	0
$921$	−0.768944	−0.0253376
$922$	0	0
$923$	−41.7538	−1.37434
$924$	0	0
$925$	−6.00000	−0.197279
$926$	0	0
$927$	−8.00000	−0.262754
$928$	0	0
$929$	34.3002	1.12535	0.562676	−	0.826677i	$-0.309772\pi$
$929$	34.3002	1.12535	0.562676	+	0.826677i	$0.309772\pi$
$930$	0	0
$931$	−4.56155	−0.149499
$932$	0	0
$933$	13.5616	0.443985
$934$	0	0
$935$	−30.2462	−0.989157
$936$	0	0
$937$	36.4384	1.19039	0.595196	−	0.803580i	$-0.297074\pi$
$937$	36.4384	1.19039	0.595196	+	0.803580i	$0.297074\pi$
$938$	0	0
$939$	50.0540	1.63345
$940$	0	0
$941$	34.1922	1.11464	0.557318	−	0.830299i	$-0.311831\pi$
$941$	34.1922	1.11464	0.557318	+	0.830299i	$0.311831\pi$
$942$	0	0
$943$	−19.8920	−0.647774
$944$	0	0
$945$	−8.68466	−0.282512
$946$	0	0
$947$	17.7538	0.576921	0.288460	−	0.957492i	$-0.406857\pi$
$947$	17.7538	0.576921	0.288460	+	0.957492i	$0.406857\pi$
$948$	0	0
$949$	−71.4233	−2.31850
$950$	0	0
$951$	37.5616	1.21802
$952$	0	0
$953$	−30.1080	−0.975292	−0.487646	−	0.873041i	$-0.662144\pi$
$953$	−30.1080	−0.975292	−0.487646	+	0.873041i	$0.662144\pi$
$954$	0	0
$955$	−8.68466	−0.281029
$956$	0	0
$957$	41.7538	1.34971
$958$	0	0
$959$	−9.06913	−0.292857
$960$	0	0
$961$	−21.2462	−0.685362
$962$	0	0
$963$	−5.36932	−0.173024
$964$	0	0
$965$	18.4924	0.595292
$966$	0	0
$967$	48.4924	1.55941	0.779706	−	0.626146i	$-0.215369\pi$
$967$	48.4924	1.55941	0.779706	+	0.626146i	$0.215369\pi$
$968$	0	0
$969$	−11.8078	−0.379320
$970$	0	0
$971$	23.5076	0.754394	0.377197	−	0.926133i	$-0.376888\pi$
$971$	23.5076	0.754394	0.377197	+	0.926133i	$0.376888\pi$
$972$	0	0
$973$	−25.7538	−0.825629
$974$	0	0
$975$	10.4384	0.334298
$976$	0	0
$977$	−20.7386	−0.663488	−0.331744	−	0.943370i	$-0.607637\pi$
$977$	−20.7386	−0.663488	−0.331744	+	0.943370i	$0.607637\pi$
$978$	0	0
$979$	−8.00000	−0.255681
$980$	0	0
$981$	−2.00000	−0.0638551
$982$	0	0
$983$	−27.1231	−0.865093	−0.432546	−	0.901612i	$-0.642385\pi$
$983$	−27.1231	−0.865093	−0.432546	+	0.901612i	$0.642385\pi$
$984$	0	0
$985$	−3.75379	−0.119606
$986$	0	0
$987$	24.9848	0.795276
$988$	0	0
$989$	52.1080	1.65694
$990$	0	0
$991$	−11.1231	−0.353337	−0.176669	−	0.984270i	$-0.556532\pi$
$991$	−11.1231	−0.353337	−0.176669	+	0.984270i	$0.556532\pi$
$992$	0	0
$993$	2.43845	0.0773818
$994$	0	0
$995$	3.80776	0.120714
$996$	0	0
$997$	32.7386	1.03684	0.518421	−	0.855125i	$-0.326520\pi$
$997$	32.7386	1.03684	0.518421	+	0.855125i	$0.326520\pi$
$998$	0	0
$999$	−33.3693	−1.05576

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1520.2.a.n.1.1		2
4.3	odd	2		190.2.a.d.1.2	✓	2
5.4	even	2		7600.2.a.y.1.2		2
8.3	odd	2		6080.2.a.bh.1.1		2
8.5	even	2		6080.2.a.bb.1.2		2
12.11	even	2		1710.2.a.w.1.2		2
20.3	even	4		950.2.b.f.799.4		4
20.7	even	4		950.2.b.f.799.1		4
20.19	odd	2		950.2.a.h.1.1		2
28.27	even	2		9310.2.a.bc.1.1		2
60.59	even	2		8550.2.a.br.1.1		2
76.75	even	2		3610.2.a.t.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
190.2.a.d.1.2	✓	2	4.3	odd	2
950.2.a.h.1.1		2	20.19	odd	2
950.2.b.f.799.1		4	20.7	even	4
950.2.b.f.799.4		4	20.3	even	4
1520.2.a.n.1.1		2	1.1	even	1	trivial
1710.2.a.w.1.2		2	12.11	even	2
3610.2.a.t.1.1		2	76.75	even	2
6080.2.a.bb.1.2		2	8.5	even	2
6080.2.a.bh.1.1		2	8.3	odd	2
7600.2.a.y.1.2		2	5.4	even	2
8550.2.a.br.1.1		2	60.59	even	2
9310.2.a.bc.1.1		2	28.27	even	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:	\( N \)	\(=\)	\( 1520 = 2^{4} \cdot 5 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1520.a (trivial)

\(f(q)\)	\(=\)	\(q-1.56155 q^{3} +1.00000 q^{5} -1.56155 q^{7} -0.561553 q^{9} +O(q^{10})\)	\(q-1.56155 q^{3} +1.00000 q^{5} -1.56155 q^{7} -0.561553 q^{9} -4.00000 q^{11} -6.68466 q^{13} -1.56155 q^{15} +7.56155 q^{17} +1.00000 q^{19} +2.43845 q^{21} +4.68466 q^{23} +1.00000 q^{25} +5.56155 q^{27} +6.68466 q^{29} -3.12311 q^{31} +6.24621 q^{33} -1.56155 q^{35} -6.00000 q^{37} +10.4384 q^{39} -4.24621 q^{41} +11.1231 q^{43} -0.561553 q^{45} +10.2462 q^{47} -4.56155 q^{49} -11.8078 q^{51} -0.438447 q^{53} -4.00000 q^{55} -1.56155 q^{57} +1.56155 q^{59} +2.87689 q^{61} +0.876894 q^{63} -6.68466 q^{65} -1.56155 q^{67} -7.31534 q^{69} +6.24621 q^{71} +10.6847 q^{73} -1.56155 q^{75} +6.24621 q^{77} -3.12311 q^{79} -7.00000 q^{81} -11.1231 q^{83} +7.56155 q^{85} -10.4384 q^{87} +2.00000 q^{89} +10.4384 q^{91} +4.87689 q^{93} +1.00000 q^{95} +6.00000 q^{97} +2.24621 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{3} + 2 q^{5} + q^{7} + 3 q^{9}+O(q^{10}) \) 2 * q + q^3 + 2 * q^5 + q^7 + 3 * q^9	\( 2 q + q^{3} + 2 q^{5} + q^{7} + 3 q^{9} - 8 q^{11} - q^{13} + q^{15} + 11 q^{17} + 2 q^{19} + 9 q^{21} - 3 q^{23} + 2 q^{25} + 7 q^{27} + q^{29} + 2 q^{31} - 4 q^{33} + q^{35} - 12 q^{37} + 25 q^{39} + 8 q^{41} + 14 q^{43} + 3 q^{45} + 4 q^{47} - 5 q^{49} - 3 q^{51} - 5 q^{53} - 8 q^{55} + q^{57} - q^{59} + 14 q^{61} + 10 q^{63} - q^{65} + q^{67} - 27 q^{69} - 4 q^{71} + 9 q^{73} + q^{75} - 4 q^{77} + 2 q^{79} - 14 q^{81} - 14 q^{83} + 11 q^{85} - 25 q^{87} + 4 q^{89} + 25 q^{91} + 18 q^{93} + 2 q^{95} + 12 q^{97} - 12 q^{99}+O(q^{100}) \) 2 * q + q^3 + 2 * q^5 + q^7 + 3 * q^9 - 8 * q^11 - q^13 + q^15 + 11 * q^17 + 2 * q^19 + 9 * q^21 - 3 * q^23 + 2 * q^25 + 7 * q^27 + q^29 + 2 * q^31 - 4 * q^33 + q^35 - 12 * q^37 + 25 * q^39 + 8 * q^41 + 14 * q^43 + 3 * q^45 + 4 * q^47 - 5 * q^49 - 3 * q^51 - 5 * q^53 - 8 * q^55 + q^57 - q^59 + 14 * q^61 + 10 * q^63 - q^65 + q^67 - 27 * q^69 - 4 * q^71 + 9 * q^73 + q^75 - 4 * q^77 + 2 * q^79 - 14 * q^81 - 14 * q^83 + 11 * q^85 - 25 * q^87 + 4 * q^89 + 25 * q^91 + 18 * q^93 + 2 * q^95 + 12 * q^97 - 12 * q^99

Introduction

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