LMFDB - Embedded newform 1755.2.a.o.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1755,2,Mod(1,1755)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1755, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1755.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1755 = 3^{3} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1755.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:	$14.0137455547$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.12357.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + 3x + 3 $ x^4 - x^3 - 5x^2 + 3x + 3
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			1.3
Root			$-0.563729$ of defining polynomial
Character	$\chi$	$=$	1755.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.563729 q^{2} -1.68221 q^{4} -1.00000 q^{5} +3.19425 q^{7} -2.07577 q^{8} +O(q^{10})$	\(q+0.563729 q^{2} -1.68221 q^{4} -1.00000 q^{5} +3.19425 q^{7} -2.07577 q^{8} -0.563729 q^{10} -2.11848 q^{11} +1.00000 q^{13} +1.80069 q^{14} +2.19425 q^{16} -4.75798 q^{17} -1.52102 q^{19} +1.68221 q^{20} -1.19425 q^{22} +4.44019 q^{23} +1.00000 q^{25} +0.563729 q^{26} -5.37340 q^{28} -5.63950 q^{29} +1.68221 q^{31} +5.38850 q^{32} -2.68221 q^{34} -3.19425 q^{35} -4.20323 q^{37} -0.857441 q^{38} +2.07577 q^{40} +0.245939 q^{41} -12.2739 q^{43} +3.56373 q^{44} +2.50306 q^{46} -9.53459 q^{47} +3.20323 q^{49} +0.563729 q^{50} -1.68221 q^{52} -0.293712 q^{53} +2.11848 q^{55} -6.63052 q^{56} -3.17915 q^{58} -9.47324 q^{59} +5.55867 q^{61} +0.948310 q^{62} -1.35085 q^{64} -1.00000 q^{65} +0.463882 q^{67} +8.00392 q^{68} -1.80069 q^{70} +5.84272 q^{71} -4.03765 q^{73} -2.36948 q^{74} +2.55867 q^{76} -6.76695 q^{77} -10.5346 q^{79} -2.19425 q^{80} +0.138643 q^{82} -13.4351 q^{83} +4.75798 q^{85} -6.91917 q^{86} +4.39747 q^{88} -7.97984 q^{89} +3.19425 q^{91} -7.46933 q^{92} -5.37492 q^{94} +1.52102 q^{95} +11.5346 q^{97} +1.80575 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{2} + 3 q^{4} - 4 q^{5} + q^{7} - 3 q^{8}+O(q^{10}) $ 4 * q - q^2 + 3 * q^4 - 4 * q^5 + q^7 - 3 * q^8	$ 4 q - q^{2} + 3 q^{4} - 4 q^{5} + q^{7} - 3 q^{8} + q^{10} - 2 q^{11} + 4 q^{13} - 9 q^{14} - 3 q^{16} - 4 q^{17} - 4 q^{19} - 3 q^{20} + 7 q^{22} - 7 q^{23} + 4 q^{25} - q^{26} - 2 q^{28} - 14 q^{29} - 3 q^{31} - 2 q^{32} - q^{34} - q^{35} - 5 q^{37} - 14 q^{38} + 3 q^{40} - 12 q^{41} - 4 q^{43} + 11 q^{44} + 8 q^{46} - 11 q^{47} + q^{49} - q^{50} + 3 q^{52} - 15 q^{53} + 2 q^{55} - 18 q^{56} - 5 q^{58} - 9 q^{59} - 9 q^{61} + 5 q^{62} - 11 q^{64} - 4 q^{65} + 8 q^{67} + 4 q^{68} + 9 q^{70} + 3 q^{71} + 13 q^{73} - 18 q^{74} - 21 q^{76} - 12 q^{77} - 15 q^{79} + 3 q^{80} + 18 q^{82} - q^{83} + 4 q^{85} - 5 q^{86} - 6 q^{88} - 8 q^{89} + q^{91} - 29 q^{92} - 23 q^{94} + 4 q^{95} + 19 q^{97} + 19 q^{98}+O(q^{100}) $ 4 * q - q^2 + 3 * q^4 - 4 * q^5 + q^7 - 3 * q^8 + q^10 - 2 * q^11 + 4 * q^13 - 9 * q^14 - 3 * q^16 - 4 * q^17 - 4 * q^19 - 3 * q^20 + 7 * q^22 - 7 * q^23 + 4 * q^25 - q^26 - 2 * q^28 - 14 * q^29 - 3 * q^31 - 2 * q^32 - q^34 - q^35 - 5 * q^37 - 14 * q^38 + 3 * q^40 - 12 * q^41 - 4 * q^43 + 11 * q^44 + 8 * q^46 - 11 * q^47 + q^49 - q^50 + 3 * q^52 - 15 * q^53 + 2 * q^55 - 18 * q^56 - 5 * q^58 - 9 * q^59 - 9 * q^61 + 5 * q^62 - 11 * q^64 - 4 * q^65 + 8 * q^67 + 4 * q^68 + 9 * q^70 + 3 * q^71 + 13 * q^73 - 18 * q^74 - 21 * q^76 - 12 * q^77 - 15 * q^79 + 3 * q^80 + 18 * q^82 - q^83 + 4 * q^85 - 5 * q^86 - 6 * q^88 - 8 * q^89 + q^91 - 29 * q^92 - 23 * q^94 + 4 * q^95 + 19 * q^97 + 19 * q^98

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 $

$2$	0.563729	0.398617	0.199308	−	0.979937i	$-0.436130\pi$
$2$	0.563729	0.398617	0.199308	+	0.979937i	$0.436130\pi$
$3$	0	0
$3$	0	0
$4$	−1.68221	−0.841105
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	3.19425	1.20731	0.603656	−	0.797245i	$-0.293710\pi$
$7$	3.19425	1.20731	0.603656	+	0.797245i	$0.293710\pi$
$8$	−2.07577	−0.733895
$9$	0	0
$10$	−0.563729	−0.178267
$11$	−2.11848	−0.638746	−0.319373	−	0.947629i	$-0.603472\pi$
$11$	−2.11848	−0.638746	−0.319373	+	0.947629i	$0.603472\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	1.80069	0.481255
$15$	0	0
$16$	2.19425	0.548562
$17$	−4.75798	−1.15398	−0.576990	−	0.816752i	$-0.695773\pi$
$17$	−4.75798	−1.15398	−0.576990	+	0.816752i	$0.695773\pi$
$18$	0	0
$19$	−1.52102	−0.348945	−0.174473	−	0.984662i	$-0.555822\pi$
$19$	−1.52102	−0.348945	−0.174473	+	0.984662i	$0.555822\pi$
$20$	1.68221	0.376154
$21$	0	0
$22$	−1.19425	−0.254615
$23$	4.44019	0.925843	0.462922	−	0.886399i	$-0.346801\pi$
$23$	4.44019	0.925843	0.462922	+	0.886399i	$0.346801\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0.563729	0.110556
$27$	0	0
$28$	−5.37340	−1.01548
$29$	−5.63950	−1.04723	−0.523614	−	0.851956i	$-0.675417\pi$
$29$	−5.63950	−1.04723	−0.523614	+	0.851956i	$0.675417\pi$
$30$	0	0
$31$	1.68221	0.302134	0.151067	−	0.988524i	$-0.451729\pi$
$31$	1.68221	0.302134	0.151067	+	0.988524i	$0.451729\pi$
$32$	5.38850	0.952561
$33$	0	0
$34$	−2.68221	−0.459995
$35$	−3.19425	−0.539927
$36$	0	0
$37$	−4.20323	−0.691006	−0.345503	−	0.938418i	$-0.612292\pi$
$37$	−4.20323	−0.691006	−0.345503	+	0.938418i	$0.612292\pi$
$38$	−0.857441	−0.139095
$39$	0	0
$40$	2.07577	0.328208
$41$	0.245939	0.0384092	0.0192046	−	0.999816i	$-0.493887\pi$
$41$	0.245939	0.0384092	0.0192046	+	0.999816i	$0.493887\pi$
$42$	0	0
$43$	−12.2739	−1.87176	−0.935879	−	0.352323i	$-0.885392\pi$
$43$	−12.2739	−1.87176	−0.935879	+	0.352323i	$0.885392\pi$
$44$	3.56373	0.537252
$45$	0	0
$46$	2.50306	0.369056
$47$	−9.53459	−1.39076	−0.695381	−	0.718641i	$-0.744765\pi$
$47$	−9.53459	−1.39076	−0.695381	+	0.718641i	$0.744765\pi$
$48$	0	0
$49$	3.20323	0.457604
$50$	0.563729	0.0797233
$51$	0	0
$52$	−1.68221	−0.233281
$53$	−0.293712	−0.0403444	−0.0201722	−	0.999797i	$-0.506421\pi$
$53$	−0.293712	−0.0403444	−0.0201722	+	0.999797i	$0.506421\pi$
$54$	0	0
$55$	2.11848	0.285656
$56$	−6.63052	−0.886041
$57$	0	0
$58$	−3.17915	−0.417443
$59$	−9.47324	−1.23331	−0.616656	−	0.787233i	$-0.711513\pi$
$59$	−9.47324	−1.23331	−0.616656	+	0.787233i	$0.711513\pi$
$60$	0	0
$61$	5.55867	0.711715	0.355857	−	0.934540i	$-0.384189\pi$
$61$	5.55867	0.711715	0.355857	+	0.934540i	$0.384189\pi$
$62$	0.948310	0.120436
$63$	0	0
$64$	−1.35085	−0.168856
$65$	−1.00000	−0.124035
$66$	0	0
$67$	0.463882	0.0566723	0.0283361	−	0.999598i	$-0.490979\pi$
$67$	0.463882	0.0566723	0.0283361	+	0.999598i	$0.490979\pi$
$68$	8.00392	0.970617
$69$	0	0
$70$	−1.80069	−0.215224
$71$	5.84272	0.693404	0.346702	−	0.937975i	$-0.387302\pi$
$71$	5.84272	0.693404	0.346702	+	0.937975i	$0.387302\pi$
$72$	0	0
$73$	−4.03765	−0.472571	−0.236286	−	0.971684i	$-0.575930\pi$
$73$	−4.03765	−0.472571	−0.236286	+	0.971684i	$0.575930\pi$
$74$	−2.36948	−0.275446
$75$	0	0
$76$	2.55867	0.293499
$77$	−6.76695	−0.771166
$78$	0	0
$79$	−10.5346	−1.18523	−0.592617	−	0.805484i	$-0.701905\pi$
$79$	−10.5346	−1.18523	−0.592617	+	0.805484i	$0.701905\pi$
$80$	−2.19425	−0.245324
$81$	0	0
$82$	0.138643	0.0153105
$83$	−13.4351	−1.47470	−0.737348	−	0.675513i	$-0.763922\pi$
$83$	−13.4351	−1.47470	−0.737348	+	0.675513i	$0.763922\pi$
$84$	0	0
$85$	4.75798	0.516075
$86$	−6.91917	−0.746113
$87$	0	0
$88$	4.39747	0.468772
$89$	−7.97984	−0.845861	−0.422931	−	0.906162i	$-0.638999\pi$
$89$	−7.97984	−0.845861	−0.422931	+	0.906162i	$0.638999\pi$
$90$	0	0
$91$	3.19425	0.334848
$92$	−7.46933	−0.778731
$93$	0	0
$94$	−5.37492	−0.554381
$95$	1.52102	0.156053
$96$	0	0
$97$	11.5346	1.17116	0.585580	−	0.810615i	$-0.300867\pi$
$97$	11.5346	1.17116	0.585580	+	0.810615i	$0.300867\pi$
$98$	1.80575	0.182408
$99$	0	0
$100$	−1.68221	−0.168221
$101$	1.64341	0.163526	0.0817629	−	0.996652i	$-0.473945\pi$
$101$	1.64341	0.163526	0.0817629	+	0.996652i	$0.473945\pi$
$102$	0	0
$103$	5.72424	0.564026	0.282013	−	0.959411i	$-0.408998\pi$
$103$	5.72424	0.564026	0.282013	+	0.959411i	$0.408998\pi$
$104$	−2.07577	−0.203546
$105$	0	0
$106$	−0.165574	−0.0160820
$107$	−8.54901	−0.826464	−0.413232	−	0.910626i	$-0.635600\pi$
$107$	−8.54901	−0.826464	−0.413232	+	0.910626i	$0.635600\pi$
$108$	0	0
$109$	0.739344	0.0708163	0.0354081	−	0.999373i	$-0.488727\pi$
$109$	0.739344	0.0708163	0.0354081	+	0.999373i	$0.488727\pi$
$110$	1.19425	0.113867
$111$	0	0
$112$	7.00898	0.662286
$113$	−2.54577	−0.239486	−0.119743	−	0.992805i	$-0.538207\pi$
$113$	−2.54577	−0.239486	−0.119743	+	0.992805i	$0.538207\pi$
$114$	0	0
$115$	−4.44019	−0.414050
$116$	9.48682	0.880829
$117$	0	0
$118$	−5.34034	−0.491618
$119$	−15.1982	−1.39321
$120$	0	0
$121$	−6.51204	−0.592004
$122$	3.13358	0.283701
$123$	0	0
$124$	−2.82983	−0.254126
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−8.39747	−0.745155	−0.372578	−	0.928001i	$-0.621526\pi$
$127$	−8.39747	−0.745155	−0.372578	+	0.928001i	$0.621526\pi$
$128$	−11.5385	−1.01987
$129$	0	0
$130$	−0.563729	−0.0494423
$131$	−1.30881	−0.114352	−0.0571758	−	0.998364i	$-0.518210\pi$
$131$	−1.30881	−0.114352	−0.0571758	+	0.998364i	$0.518210\pi$
$132$	0	0
$133$	−4.85850	−0.421286
$134$	0.261504	0.0225905
$135$	0	0
$136$	9.87646	0.846899
$137$	−7.70982	−0.658694	−0.329347	−	0.944209i	$-0.606829\pi$
$137$	−7.70982	−0.658694	−0.329347	+	0.944209i	$0.606829\pi$
$138$	0	0
$139$	1.41543	0.120055	0.0600276	−	0.998197i	$-0.480881\pi$
$139$	1.41543	0.120055	0.0600276	+	0.998197i	$0.480881\pi$
$140$	5.37340	0.454135
$141$	0	0
$142$	3.29371	0.276402
$143$	−2.11848	−0.177156
$144$	0	0
$145$	5.63950	0.468335
$146$	−2.27614	−0.188375
$147$	0	0
$148$	7.07071	0.581209
$149$	−6.44916	−0.528336	−0.264168	−	0.964477i	$-0.585097\pi$
$149$	−6.44916	−0.528336	−0.264168	+	0.964477i	$0.585097\pi$
$150$	0	0
$151$	22.5417	1.83442	0.917210	−	0.398403i	$-0.130435\pi$
$151$	22.5417	1.83442	0.917210	+	0.398403i	$0.130435\pi$
$152$	3.15728	0.256089
$153$	0	0
$154$	−3.81473	−0.307400
$155$	−1.68221	−0.135118
$156$	0	0
$157$	18.9831	1.51501	0.757507	−	0.652827i	$-0.226417\pi$
$157$	18.9831	1.51501	0.757507	+	0.652827i	$0.226417\pi$
$158$	−5.93865	−0.472454
$159$	0	0
$160$	−5.38850	−0.425998
$161$	14.1831	1.11778
$162$	0	0
$163$	−12.8085	−1.00324	−0.501620	−	0.865088i	$-0.667263\pi$
$163$	−12.8085	−1.00324	−0.501620	+	0.865088i	$0.667263\pi$
$164$	−0.413720	−0.0323061
$165$	0	0
$166$	−7.57377	−0.587839
$167$	−8.26887	−0.639865	−0.319932	−	0.947440i	$-0.603660\pi$
$167$	−8.26887	−0.639865	−0.319932	+	0.947440i	$0.603660\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	2.68221	0.205716
$171$	0	0
$172$	20.6473	1.57434
$173$	−21.1314	−1.60659	−0.803294	−	0.595582i	$-0.796921\pi$
$173$	−21.1314	−1.60659	−0.803294	+	0.595582i	$0.796921\pi$
$174$	0	0
$175$	3.19425	0.241463
$176$	−4.64847	−0.350392
$177$	0	0
$178$	−4.49847	−0.337174
$179$	9.95682	0.744208	0.372104	−	0.928191i	$-0.378636\pi$
$179$	9.95682	0.744208	0.372104	+	0.928191i	$0.378636\pi$
$180$	0	0
$181$	−12.2980	−0.914104	−0.457052	−	0.889440i	$-0.651095\pi$
$181$	−12.2980	−0.914104	−0.457052	+	0.889440i	$0.651095\pi$
$182$	1.80069	0.133476
$183$	0	0
$184$	−9.21680	−0.679471
$185$	4.20323	0.309027
$186$	0	0
$187$	10.0797	0.737100
$188$	16.0392	1.16978
$189$	0	0
$190$	0.857441	0.0622053
$191$	14.1543	1.02417	0.512085	−	0.858935i	$-0.328873\pi$
$191$	14.1543	1.02417	0.512085	+	0.858935i	$0.328873\pi$
$192$	0	0
$193$	−5.68833	−0.409455	−0.204728	−	0.978819i	$-0.565631\pi$
$193$	−5.68833	−0.409455	−0.204728	+	0.978819i	$0.565631\pi$
$194$	6.50238	0.466844
$195$	0	0
$196$	−5.38850	−0.384893
$197$	14.3543	1.02270	0.511351	−	0.859372i	$-0.329145\pi$
$197$	14.3543	1.02270	0.511351	+	0.859372i	$0.329145\pi$
$198$	0	0
$199$	0.207823	0.0147322	0.00736608	−	0.999973i	$-0.497655\pi$
$199$	0.207823	0.0147322	0.00736608	+	0.999973i	$0.497655\pi$
$200$	−2.07577	−0.146779
$201$	0	0
$202$	0.926440	0.0651841
$203$	−18.0140	−1.26433
$204$	0	0
$205$	−0.245939	−0.0171771
$206$	3.22692	0.224830
$207$	0	0
$208$	2.19425	0.152144
$209$	3.22224	0.222887
$210$	0	0
$211$	−9.67936	−0.666354	−0.333177	−	0.942864i	$-0.608121\pi$
$211$	−9.67936	−0.666354	−0.333177	+	0.942864i	$0.608121\pi$
$212$	0.494085	0.0339339
$213$	0	0
$214$	−4.81932	−0.329442
$215$	12.2739	0.837075
$216$	0	0
$217$	5.37340	0.364770
$218$	0.416789	0.0282285
$219$	0	0
$220$	−3.56373	−0.240267
$221$	−4.75798	−0.320056
$222$	0	0
$223$	11.3690	0.761326	0.380663	−	0.924714i	$-0.375696\pi$
$223$	11.3690	0.761326	0.380663	+	0.924714i	$0.375696\pi$
$224$	17.2122	1.15004
$225$	0	0
$226$	−1.43513	−0.0954632
$227$	15.5536	1.03233	0.516165	−	0.856489i	$-0.327359\pi$
$227$	15.5536	1.03233	0.516165	+	0.856489i	$0.327359\pi$
$228$	0	0
$229$	8.94104	0.590841	0.295420	−	0.955367i	$-0.404540\pi$
$229$	8.94104	0.590841	0.295420	+	0.955367i	$0.404540\pi$
$230$	−2.50306	−0.165047
$231$	0	0
$232$	11.7063	0.768555
$233$	11.2875	0.739469	0.369735	−	0.929137i	$-0.379449\pi$
$233$	11.2875	0.739469	0.369735	+	0.929137i	$0.379449\pi$
$234$	0	0
$235$	9.53459	0.621968
$236$	15.9360	1.03734
$237$	0	0
$238$	−8.56765	−0.555358
$239$	26.3357	1.70351	0.851756	−	0.523938i	$-0.175538\pi$
$239$	26.3357	1.70351	0.851756	+	0.523938i	$0.175538\pi$
$240$	0	0
$241$	−13.7619	−0.886482	−0.443241	−	0.896403i	$-0.646171\pi$
$241$	−13.7619	−0.886482	−0.443241	+	0.896403i	$0.646171\pi$
$242$	−3.67102	−0.235982
$243$	0	0
$244$	−9.35085	−0.598627
$245$	−3.20323	−0.204647
$246$	0	0
$247$	−1.52102	−0.0967799
$248$	−3.49188	−0.221734
$249$	0	0
$250$	−0.563729	−0.0356533
$251$	−23.0365	−1.45405	−0.727026	−	0.686610i	$-0.759098\pi$
$251$	−23.0365	−1.45405	−0.727026	+	0.686610i	$0.759098\pi$
$252$	0	0
$253$	−9.40645	−0.591379
$254$	−4.73390	−0.297031
$255$	0	0
$256$	−3.80290	−0.237681
$257$	28.1177	1.75394	0.876968	−	0.480550i	$-0.159563\pi$
$257$	28.1177	1.75394	0.876968	+	0.480550i	$0.159563\pi$
$258$	0	0
$259$	−13.4261	−0.834260
$260$	1.68221	0.104326
$261$	0	0
$262$	−0.737816	−0.0455824
$263$	31.9219	1.96839	0.984196	−	0.177084i	$-0.0566665\pi$
$263$	31.9219	1.96839	0.984196	+	0.177084i	$0.0566665\pi$
$264$	0	0
$265$	0.293712	0.0180426
$266$	−2.73888	−0.167931
$267$	0	0
$268$	−0.780348	−0.0476673
$269$	21.2840	1.29771	0.648853	−	0.760913i	$-0.275249\pi$
$269$	21.2840	1.29771	0.648853	+	0.760913i	$0.275249\pi$
$270$	0	0
$271$	14.4923	0.880342	0.440171	−	0.897914i	$-0.354918\pi$
$271$	14.4923	0.880342	0.440171	+	0.897914i	$0.354918\pi$
$272$	−10.4402	−0.633029
$273$	0	0
$274$	−4.34625	−0.262567
$275$	−2.11848	−0.127749
$276$	0	0
$277$	25.8297	1.55196	0.775980	−	0.630758i	$-0.217256\pi$
$277$	25.8297	1.55196	0.775980	+	0.630758i	$0.217256\pi$
$278$	0.797918	0.0478560
$279$	0	0
$280$	6.63052	0.396249
$281$	−19.0528	−1.13659	−0.568296	−	0.822824i	$-0.692397\pi$
$281$	−19.0528	−1.13659	−0.568296	+	0.822824i	$0.692397\pi$
$282$	0	0
$283$	16.2452	0.965676	0.482838	−	0.875710i	$-0.339606\pi$
$283$	16.2452	0.965676	0.482838	+	0.875710i	$0.339606\pi$
$284$	−9.82869	−0.583225
$285$	0	0
$286$	−1.19425	−0.0706174
$287$	0.785589	0.0463719
$288$	0	0
$289$	5.63835	0.331668
$290$	3.17915	0.186686
$291$	0	0
$292$	6.79218	0.397482
$293$	24.3138	1.42042	0.710212	−	0.703987i	$-0.248599\pi$
$293$	24.3138	1.42042	0.710212	+	0.703987i	$0.248599\pi$
$294$	0	0
$295$	9.47324	0.551554
$296$	8.72492	0.507126
$297$	0	0
$298$	−3.63558	−0.210604
$299$	4.44019	0.256783
$300$	0	0
$301$	−39.2060	−2.25980
$302$	12.7074	0.731231
$303$	0	0
$304$	−3.33749	−0.191418
$305$	−5.55867	−0.318288
$306$	0	0
$307$	28.8067	1.64409	0.822043	−	0.569426i	$-0.192835\pi$
$307$	28.8067	1.64409	0.822043	+	0.569426i	$0.192835\pi$
$308$	11.3834	0.648632
$309$	0	0
$310$	−0.948310	−0.0538604
$311$	3.12814	0.177380	0.0886902	−	0.996059i	$-0.471732\pi$
$311$	3.12814	0.177380	0.0886902	+	0.996059i	$0.471732\pi$
$312$	0	0
$313$	−4.15507	−0.234858	−0.117429	−	0.993081i	$-0.537465\pi$
$313$	−4.15507	−0.234858	−0.117429	+	0.993081i	$0.537465\pi$
$314$	10.7013	0.603910
$315$	0	0
$316$	17.7214	0.996906
$317$	−16.3928	−0.920711	−0.460356	−	0.887735i	$-0.652278\pi$
$317$	−16.3928	−0.920711	−0.460356	+	0.887735i	$0.652278\pi$
$318$	0	0
$319$	11.9472	0.668913
$320$	1.35085	0.0755146
$321$	0	0
$322$	7.99540	0.445566
$323$	7.23696	0.402675
$324$	0	0
$325$	1.00000	0.0554700
$326$	−7.22054	−0.399908
$327$	0	0
$328$	−0.510512	−0.0281883
$329$	−30.4559	−1.67909
$330$	0	0
$331$	−13.5992	−0.747478	−0.373739	−	0.927534i	$-0.621924\pi$
$331$	−13.5992	−0.747478	−0.373739	+	0.927534i	$0.621924\pi$
$332$	22.6007	1.24037
$333$	0	0
$334$	−4.66140	−0.255061
$335$	−0.463882	−0.0253446
$336$	0	0
$337$	18.5120	1.00842	0.504208	−	0.863583i	$-0.331785\pi$
$337$	18.5120	1.00842	0.504208	+	0.863583i	$0.331785\pi$
$338$	0.563729	0.0306628
$339$	0	0
$340$	−8.00392	−0.434073
$341$	−3.56373	−0.192987
$342$	0	0
$343$	−12.1278	−0.654842
$344$	25.4778	1.37367
$345$	0	0
$346$	−11.9124	−0.640413
$347$	−30.1055	−1.61615	−0.808073	−	0.589082i	$-0.799489\pi$
$347$	−30.1055	−1.61615	−0.808073	+	0.589082i	$0.799489\pi$
$348$	0	0
$349$	−27.5794	−1.47629	−0.738146	−	0.674641i	$-0.764298\pi$
$349$	−27.5794	−1.47629	−0.738146	+	0.674641i	$0.764298\pi$
$350$	1.80069	0.0962510
$351$	0	0
$352$	−11.4154	−0.608444
$353$	−15.5534	−0.827824	−0.413912	−	0.910317i	$-0.635838\pi$
$353$	−15.5534	−0.827824	−0.413912	+	0.910317i	$0.635838\pi$
$354$	0	0
$355$	−5.84272	−0.310100
$356$	13.4238	0.711458
$357$	0	0
$358$	5.61295	0.296654
$359$	−33.4182	−1.76375	−0.881873	−	0.471488i	$-0.843717\pi$
$359$	−33.4182	−1.76375	−0.881873	+	0.471488i	$0.843717\pi$
$360$	0	0
$361$	−16.6865	−0.878237
$362$	−6.93274	−0.364377
$363$	0	0
$364$	−5.37340	−0.281643
$365$	4.03765	0.211340
$366$	0	0
$367$	−5.07683	−0.265008	−0.132504	−	0.991182i	$-0.542302\pi$
$367$	−5.07683	−0.265008	−0.132504	+	0.991182i	$0.542302\pi$
$368$	9.74288	0.507883
$369$	0	0
$370$	2.36948	0.123183
$371$	−0.938189	−0.0487083
$372$	0	0
$373$	12.4621	0.645261	0.322630	−	0.946525i	$-0.395433\pi$
$373$	12.4621	0.645261	0.322630	+	0.946525i	$0.395433\pi$
$374$	5.68221	0.293820
$375$	0	0
$376$	19.7916	1.02067
$377$	−5.63950	−0.290449
$378$	0	0
$379$	−15.8990	−0.816677	−0.408339	−	0.912831i	$-0.633892\pi$
$379$	−15.8990	−0.816677	−0.408339	+	0.912831i	$0.633892\pi$
$380$	−2.55867	−0.131257
$381$	0	0
$382$	7.97919	0.408251
$383$	−10.4114	−0.531999	−0.266000	−	0.963973i	$-0.585702\pi$
$383$	−10.4114	−0.531999	−0.266000	+	0.963973i	$0.585702\pi$
$384$	0	0
$385$	6.76695	0.344876
$386$	−3.20668	−0.163216
$387$	0	0
$388$	−19.4036	−0.985068
$389$	−26.9743	−1.36765	−0.683826	−	0.729645i	$-0.739685\pi$
$389$	−26.9743	−1.36765	−0.683826	+	0.729645i	$0.739685\pi$
$390$	0	0
$391$	−21.1263	−1.06840
$392$	−6.64915	−0.335833
$393$	0	0
$394$	8.09193	0.407666
$395$	10.5346	0.530053
$396$	0	0
$397$	−26.7647	−1.34328	−0.671640	−	0.740878i	$-0.734410\pi$
$397$	−26.7647	−1.34328	−0.671640	+	0.740878i	$0.734410\pi$
$398$	0.117156	0.00587248
$399$	0	0
$400$	2.19425	0.109712
$401$	33.2413	1.65999	0.829995	−	0.557771i	$-0.188343\pi$
$401$	33.2413	1.65999	0.829995	+	0.557771i	$0.188343\pi$
$402$	0	0
$403$	1.68221	0.0837968
$404$	−2.76457	−0.137542
$405$	0	0
$406$	−10.1550	−0.503984
$407$	8.90445	0.441377
$408$	0	0
$409$	−13.6502	−0.674958	−0.337479	−	0.941333i	$-0.609574\pi$
$409$	−13.6502	−0.674958	−0.337479	+	0.941333i	$0.609574\pi$
$410$	−0.138643	−0.00684708
$411$	0	0
$412$	−9.62938	−0.474405
$413$	−30.2599	−1.48899
$414$	0	0
$415$	13.4351	0.659504
$416$	5.38850	0.264193
$417$	0	0
$418$	1.81647	0.0888465
$419$	32.6721	1.59614	0.798068	−	0.602567i	$-0.205856\pi$
$419$	32.6721	1.59614	0.798068	+	0.602567i	$0.205856\pi$
$420$	0	0
$421$	−4.74832	−0.231419	−0.115709	−	0.993283i	$-0.536914\pi$
$421$	−4.74832	−0.231419	−0.115709	+	0.993283i	$0.536914\pi$
$422$	−5.45653	−0.265620
$423$	0	0
$424$	0.609678	0.0296086
$425$	−4.75798	−0.230796
$426$	0	0
$427$	17.7558	0.859262
$428$	14.3812	0.695143
$429$	0	0
$430$	6.91917	0.333672
$431$	20.9927	1.01118	0.505592	−	0.862773i	$-0.331274\pi$
$431$	20.9927	1.01118	0.505592	+	0.862773i	$0.331274\pi$
$432$	0	0
$433$	−9.77087	−0.469558	−0.234779	−	0.972049i	$-0.575437\pi$
$433$	−9.77087	−0.469558	−0.234779	+	0.972049i	$0.575437\pi$
$434$	3.02914	0.145403
$435$	0	0
$436$	−1.24373	−0.0595639
$437$	−6.75360	−0.323068
$438$	0	0
$439$	−28.5973	−1.36488	−0.682438	−	0.730943i	$-0.739081\pi$
$439$	−28.5973	−1.36488	−0.682438	+	0.730943i	$0.739081\pi$
$440$	−4.39747	−0.209641
$441$	0	0
$442$	−2.68221	−0.127580
$443$	−11.9259	−0.566619	−0.283309	−	0.959029i	$-0.591432\pi$
$443$	−11.9259	−0.566619	−0.283309	+	0.959029i	$0.591432\pi$
$444$	0	0
$445$	7.97984	0.378281
$446$	6.40904	0.303477
$447$	0	0
$448$	−4.31494	−0.203862
$449$	9.45444	0.446183	0.223091	−	0.974798i	$-0.428385\pi$
$449$	9.45444	0.446183	0.223091	+	0.974798i	$0.428385\pi$
$450$	0	0
$451$	−0.521016	−0.0245337
$452$	4.28253	0.201433
$453$	0	0
$454$	8.76802	0.411503
$455$	−3.19425	−0.149749
$456$	0	0
$457$	27.7260	1.29697	0.648483	−	0.761229i	$-0.275404\pi$
$457$	27.7260	1.29697	0.648483	+	0.761229i	$0.275404\pi$
$458$	5.04032	0.235519
$459$	0	0
$460$	7.46933	0.348259
$461$	11.7927	0.549243	0.274621	−	0.961552i	$-0.411447\pi$
$461$	11.7927	0.549243	0.274621	+	0.961552i	$0.411447\pi$
$462$	0	0
$463$	−12.5117	−0.581470	−0.290735	−	0.956804i	$-0.593900\pi$
$463$	−12.5117	−0.581470	−0.290735	+	0.956804i	$0.593900\pi$
$464$	−12.3745	−0.574470
$465$	0	0
$466$	6.36309	0.294765
$467$	−31.8387	−1.47332	−0.736660	−	0.676263i	$-0.763598\pi$
$467$	−31.8387	−1.47332	−0.736660	+	0.676263i	$0.763598\pi$
$468$	0	0
$469$	1.48176	0.0684211
$470$	5.37492	0.247927
$471$	0	0
$472$	19.6643	0.905121
$473$	26.0021	1.19558
$474$	0	0
$475$	−1.52102	−0.0697890
$476$	25.5665	1.17184
$477$	0	0
$478$	14.8462	0.679048
$479$	19.2234	0.878339	0.439170	−	0.898404i	$-0.355273\pi$
$479$	19.2234	0.878339	0.439170	+	0.898404i	$0.355273\pi$
$480$	0	0
$481$	−4.20323	−0.191651
$482$	−7.75798	−0.353366
$483$	0	0
$484$	10.9546	0.497937
$485$	−11.5346	−0.523759
$486$	0	0
$487$	−26.7270	−1.21112	−0.605558	−	0.795801i	$-0.707050\pi$
$487$	−26.7270	−1.21112	−0.605558	+	0.795801i	$0.707050\pi$
$488$	−11.5385	−0.522324
$489$	0	0
$490$	−1.80575	−0.0815755
$491$	16.7205	0.754587	0.377294	−	0.926094i	$-0.376855\pi$
$491$	16.7205	0.754587	0.377294	+	0.926094i	$0.376855\pi$
$492$	0	0
$493$	26.8326	1.20848
$494$	−0.857441	−0.0385781
$495$	0	0
$496$	3.69119	0.165739
$497$	18.6631	0.837155
$498$	0	0
$499$	−23.0674	−1.03264	−0.516318	−	0.856397i	$-0.672698\pi$
$499$	−23.0674	−1.03264	−0.516318	+	0.856397i	$0.672698\pi$
$500$	1.68221	0.0752307
$501$	0	0
$502$	−12.9863	−0.579609
$503$	19.6890	0.877888	0.438944	−	0.898514i	$-0.355353\pi$
$503$	19.6890	0.877888	0.438944	+	0.898514i	$0.355353\pi$
$504$	0	0
$505$	−1.64341	−0.0731309
$506$	−5.30269	−0.235733
$507$	0	0
$508$	14.1263	0.626754
$509$	−3.63052	−0.160920	−0.0804600	−	0.996758i	$-0.525639\pi$
$509$	−3.63052	−0.160920	−0.0804600	+	0.996758i	$0.525639\pi$
$510$	0	0
$511$	−12.8973	−0.570541
$512$	20.9332	0.925126
$513$	0	0
$514$	15.8508	0.699148
$515$	−5.72424	−0.252240
$516$	0	0
$517$	20.1988	0.888344
$518$	−7.56871	−0.332550
$519$	0	0
$520$	2.07577	0.0910285
$521$	−23.3511	−1.02303	−0.511514	−	0.859275i	$-0.670915\pi$
$521$	−23.3511	−1.02303	−0.511514	+	0.859275i	$0.670915\pi$
$522$	0	0
$523$	30.2675	1.32351	0.661753	−	0.749722i	$-0.269813\pi$
$523$	30.2675	1.32351	0.661753	+	0.749722i	$0.269813\pi$
$524$	2.20170	0.0961816
$525$	0	0
$526$	17.9953	0.784633
$527$	−8.00392	−0.348656
$528$	0	0
$529$	−3.28473	−0.142815
$530$	0.165574	0.00719207
$531$	0	0
$532$	8.17302	0.354346
$533$	0.245939	0.0106528
$534$	0	0
$535$	8.54901	0.369606
$536$	−0.962912	−0.0415915
$537$	0	0
$538$	11.9984	0.517287
$539$	−6.78597	−0.292293
$540$	0	0
$541$	32.6983	1.40581	0.702906	−	0.711283i	$-0.251886\pi$
$541$	32.6983	1.40581	0.702906	+	0.711283i	$0.251886\pi$
$542$	8.16971	0.350919
$543$	0	0
$544$	−25.6384	−1.09924
$545$	−0.739344	−0.0316700
$546$	0	0
$547$	−36.1604	−1.54611	−0.773053	−	0.634341i	$-0.781272\pi$
$547$	−36.1604	−1.54611	−0.773053	+	0.634341i	$0.781272\pi$
$548$	12.9695	0.554031
$549$	0	0
$550$	−1.19425	−0.0509229
$551$	8.57777	0.365425
$552$	0	0
$553$	−33.6501	−1.43095
$554$	14.5610	0.618637
$555$	0	0
$556$	−2.38105	−0.100979
$557$	33.6971	1.42779	0.713896	−	0.700251i	$-0.246929\pi$
$557$	33.6971	1.42779	0.713896	+	0.700251i	$0.246929\pi$
$558$	0	0
$559$	−12.2739	−0.519132
$560$	−7.00898	−0.296183
$561$	0	0
$562$	−10.7406	−0.453064
$563$	41.0379	1.72954	0.864771	−	0.502166i	$-0.167463\pi$
$563$	41.0379	1.72954	0.864771	+	0.502166i	$0.167463\pi$
$564$	0	0
$565$	2.54577	0.107102
$566$	9.15788	0.384934
$567$	0	0
$568$	−12.1281	−0.508885
$569$	−35.0474	−1.46926	−0.734632	−	0.678466i	$-0.762645\pi$
$569$	−35.0474	−1.46926	−0.734632	+	0.678466i	$0.762645\pi$
$570$	0	0
$571$	−32.6307	−1.36555	−0.682776	−	0.730628i	$-0.739228\pi$
$571$	−32.6307	−1.36555	−0.682776	+	0.730628i	$0.739228\pi$
$572$	3.56373	0.149007
$573$	0	0
$574$	0.442859	0.0184846
$575$	4.44019	0.185169
$576$	0	0
$577$	−2.19607	−0.0914237	−0.0457119	−	0.998955i	$-0.514556\pi$
$577$	−2.19607	−0.0914237	−0.0457119	+	0.998955i	$0.514556\pi$
$578$	3.17850	0.132208
$579$	0	0
$580$	−9.48682	−0.393919
$581$	−42.9151	−1.78042
$582$	0	0
$583$	0.622223	0.0257698
$584$	8.38123	0.346818
$585$	0	0
$586$	13.7064	0.566205
$587$	29.2531	1.20740	0.603702	−	0.797210i	$-0.293692\pi$
$587$	29.2531	1.20740	0.603702	+	0.797210i	$0.293692\pi$
$588$	0	0
$589$	−2.55867	−0.105428
$590$	5.34034	0.219858
$591$	0	0
$592$	−9.22292	−0.379060
$593$	−27.0512	−1.11086	−0.555430	−	0.831563i	$-0.687446\pi$
$593$	−27.0512	−1.11086	−0.555430	+	0.831563i	$0.687446\pi$
$594$	0	0
$595$	15.1982	0.623064
$596$	10.8488	0.444386
$597$	0	0
$598$	2.50306	0.102358
$599$	10.2137	0.417318	0.208659	−	0.977988i	$-0.433090\pi$
$599$	10.2137	0.417318	0.208659	+	0.977988i	$0.433090\pi$
$600$	0	0
$601$	−29.7665	−1.21420	−0.607100	−	0.794625i	$-0.707667\pi$
$601$	−29.7665	−1.21420	−0.607100	+	0.794625i	$0.707667\pi$
$602$	−22.1016	−0.900792
$603$	0	0
$604$	−37.9199	−1.54294
$605$	6.51204	0.264752
$606$	0	0
$607$	18.7181	0.759745	0.379872	−	0.925039i	$-0.375968\pi$
$607$	18.7181	0.759745	0.379872	+	0.925039i	$0.375968\pi$
$608$	−8.19599	−0.332391
$609$	0	0
$610$	−3.13358	−0.126875
$611$	−9.53459	−0.385728
$612$	0	0
$613$	45.1349	1.82298	0.911491	−	0.411320i	$-0.134932\pi$
$613$	45.1349	1.82298	0.911491	+	0.411320i	$0.134932\pi$
$614$	16.2392	0.655360
$615$	0	0
$616$	14.0466	0.565955
$617$	15.8127	0.636597	0.318298	−	0.947991i	$-0.396889\pi$
$617$	15.8127	0.636597	0.318298	+	0.947991i	$0.396889\pi$
$618$	0	0
$619$	−28.4143	−1.14207	−0.571034	−	0.820926i	$-0.693458\pi$
$619$	−28.4143	−1.14207	−0.571034	+	0.820926i	$0.693458\pi$
$620$	2.82983	0.113649
$621$	0	0
$622$	1.76342	0.0707068
$623$	−25.4896	−1.02122
$624$	0	0
$625$	1.00000	0.0400000
$626$	−2.34233	−0.0936184
$627$	0	0
$628$	−31.9335	−1.27429
$629$	19.9989	0.797407
$630$	0	0
$631$	30.9426	1.23180	0.615902	−	0.787823i	$-0.288792\pi$
$631$	30.9426	1.23180	0.615902	+	0.787823i	$0.288792\pi$
$632$	21.8674	0.869837
$633$	0	0
$634$	−9.24109	−0.367011
$635$	8.39747	0.333244
$636$	0	0
$637$	3.20323	0.126916
$638$	6.73496	0.266640
$639$	0	0
$640$	11.5385	0.456100
$641$	−37.5106	−1.48158	−0.740790	−	0.671737i	$-0.765549\pi$
$641$	−37.5106	−1.48158	−0.740790	+	0.671737i	$0.765549\pi$
$642$	0	0
$643$	−12.8355	−0.506181	−0.253090	−	0.967443i	$-0.581447\pi$
$643$	−12.8355	−0.506181	−0.253090	+	0.967443i	$0.581447\pi$
$644$	−23.8589	−0.940172
$645$	0	0
$646$	4.07968	0.160513
$647$	1.14341	0.0449520	0.0224760	−	0.999747i	$-0.492845\pi$
$647$	1.14341	0.0449520	0.0224760	+	0.999747i	$0.492845\pi$
$648$	0	0
$649$	20.0689	0.787773
$650$	0.563729	0.0221113
$651$	0	0
$652$	21.5466	0.843831
$653$	24.3241	0.951875	0.475938	−	0.879479i	$-0.342109\pi$
$653$	24.3241	0.951875	0.475938	+	0.879479i	$0.342109\pi$
$654$	0	0
$655$	1.30881	0.0511396
$656$	0.539651	0.0210698
$657$	0	0
$658$	−17.1688	−0.669311
$659$	−4.78206	−0.186282	−0.0931412	−	0.995653i	$-0.529691\pi$
$659$	−4.78206	−0.186282	−0.0931412	+	0.995653i	$0.529691\pi$
$660$	0	0
$661$	25.6681	0.998372	0.499186	−	0.866495i	$-0.333632\pi$
$661$	25.6681	0.998372	0.499186	+	0.866495i	$0.333632\pi$
$662$	−7.66625	−0.297957
$663$	0	0
$664$	27.8882	1.08227
$665$	4.85850	0.188405
$666$	0	0
$667$	−25.0404	−0.969569
$668$	13.9100	0.538193
$669$	0	0
$670$	−0.261504	−0.0101028
$671$	−11.7759	−0.454605
$672$	0	0
$673$	47.3779	1.82628	0.913142	−	0.407641i	$-0.133649\pi$
$673$	47.3779	1.82628	0.913142	+	0.407641i	$0.133649\pi$
$674$	10.4358	0.401971
$675$	0	0
$676$	−1.68221	−0.0647004
$677$	−38.9982	−1.49882	−0.749411	−	0.662105i	$-0.769663\pi$
$677$	−38.9982	−1.49882	−0.749411	+	0.662105i	$0.769663\pi$
$678$	0	0
$679$	36.8444	1.41396
$680$	−9.87646	−0.378745
$681$	0	0
$682$	−2.00898	−0.0769277
$683$	−9.90274	−0.378918	−0.189459	−	0.981889i	$-0.560673\pi$
$683$	−9.90274	−0.378918	−0.189459	+	0.981889i	$0.560673\pi$
$684$	0	0
$685$	7.70982	0.294577
$686$	−6.83681	−0.261031
$687$	0	0
$688$	−26.9321	−1.02678
$689$	−0.293712	−0.0111895
$690$	0	0
$691$	−21.8721	−0.832053	−0.416027	−	0.909352i	$-0.636578\pi$
$691$	−21.8721	−0.832053	−0.416027	+	0.909352i	$0.636578\pi$
$692$	35.5474	1.35131
$693$	0	0
$694$	−16.9713	−0.644223
$695$	−1.41543	−0.0536903
$696$	0	0
$697$	−1.17017	−0.0443234
$698$	−15.5473	−0.588474
$699$	0	0
$700$	−5.37340	−0.203095
$701$	−3.48519	−0.131634	−0.0658169	−	0.997832i	$-0.520965\pi$
$701$	−3.48519	−0.131634	−0.0658169	+	0.997832i	$0.520965\pi$
$702$	0	0
$703$	6.39317	0.241123
$704$	2.86174	0.107856
$705$	0	0
$706$	−8.76790	−0.329984
$707$	5.24947	0.197427
$708$	0	0
$709$	−40.9256	−1.53699	−0.768497	−	0.639853i	$-0.778995\pi$
$709$	−40.9256	−1.53699	−0.768497	+	0.639853i	$0.778995\pi$
$710$	−3.29371	−0.123611
$711$	0	0
$712$	16.5643	0.620773
$713$	7.46933	0.279728
$714$	0	0
$715$	2.11848	0.0792267
$716$	−16.7495	−0.625957
$717$	0	0
$718$	−18.8388	−0.703058
$719$	−25.3044	−0.943695	−0.471847	−	0.881680i	$-0.656413\pi$
$719$	−25.3044	−0.943695	−0.471847	+	0.881680i	$0.656413\pi$
$720$	0	0
$721$	18.2847	0.680956
$722$	−9.40667	−0.350080
$723$	0	0
$724$	20.6878	0.768857
$725$	−5.63950	−0.209446
$726$	0	0
$727$	18.7375	0.694936	0.347468	−	0.937692i	$-0.387042\pi$
$727$	18.7375	0.694936	0.347468	+	0.937692i	$0.387042\pi$
$728$	−6.63052	−0.245743
$729$	0	0
$730$	2.27614	0.0842438
$731$	58.3991	2.15997
$732$	0	0
$733$	−13.0751	−0.482939	−0.241470	−	0.970408i	$-0.577629\pi$
$733$	−13.0751	−0.482939	−0.241470	+	0.970408i	$0.577629\pi$
$734$	−2.86196	−0.105637
$735$	0	0
$736$	23.9259	0.881922
$737$	−0.982726	−0.0361992
$738$	0	0
$739$	−2.36004	−0.0868154	−0.0434077	−	0.999057i	$-0.513821\pi$
$739$	−2.36004	−0.0868154	−0.0434077	+	0.999057i	$0.513821\pi$
$740$	−7.07071	−0.259924
$741$	0	0
$742$	−0.528884	−0.0194160
$743$	18.2018	0.667759	0.333879	−	0.942616i	$-0.391642\pi$
$743$	18.2018	0.667759	0.333879	+	0.942616i	$0.391642\pi$
$744$	0	0
$745$	6.44916	0.236279
$746$	7.02522	0.257212
$747$	0	0
$748$	−16.9561	−0.619978
$749$	−27.3077	−0.997801
$750$	0	0
$751$	2.03583	0.0742884	0.0371442	−	0.999310i	$-0.488174\pi$
$751$	2.03583	0.0742884	0.0371442	+	0.999310i	$0.488174\pi$
$752$	−20.9213	−0.762920
$753$	0	0
$754$	−3.17915	−0.115778
$755$	−22.5417	−0.820378
$756$	0	0
$757$	−12.2334	−0.444632	−0.222316	−	0.974975i	$-0.571362\pi$
$757$	−12.2334	−0.444632	−0.222316	+	0.974975i	$0.571362\pi$
$758$	−8.96273	−0.325541
$759$	0	0
$760$	−3.15728	−0.114526
$761$	−4.41714	−0.160121	−0.0800606	−	0.996790i	$-0.525511\pi$
$761$	−4.41714	−0.160121	−0.0800606	+	0.996790i	$0.525511\pi$
$762$	0	0
$763$	2.36165	0.0854974
$764$	−23.8105	−0.861434
$765$	0	0
$766$	−5.86923	−0.212064
$767$	−9.47324	−0.342059
$768$	0	0
$769$	−6.93177	−0.249966	−0.124983	−	0.992159i	$-0.539888\pi$
$769$	−6.93177	−0.249966	−0.124983	+	0.992159i	$0.539888\pi$
$770$	3.81473	0.137473
$771$	0	0
$772$	9.56897	0.344395
$773$	−45.2451	−1.62735	−0.813677	−	0.581317i	$-0.802538\pi$
$773$	−45.2451	−1.62735	−0.813677	+	0.581317i	$0.802538\pi$
$774$	0	0
$775$	1.68221	0.0604268
$776$	−23.9431	−0.859508
$777$	0	0
$778$	−15.2062	−0.545169
$779$	−0.374077	−0.0134027
$780$	0	0
$781$	−12.3777	−0.442909
$782$	−11.9095	−0.425883
$783$	0	0
$784$	7.02867	0.251024
$785$	−18.9831	−0.677535
$786$	0	0
$787$	−2.73913	−0.0976393	−0.0488197	−	0.998808i	$-0.515546\pi$
$787$	−2.73913	−0.0976393	−0.0488197	+	0.998808i	$0.515546\pi$
$788$	−24.1469	−0.860199
$789$	0	0
$790$	5.93865	0.211288
$791$	−8.13184	−0.289135
$792$	0	0
$793$	5.55867	0.197394
$794$	−15.0880	−0.535454
$795$	0	0
$796$	−0.349601	−0.0123913
$797$	−25.6780	−0.909561	−0.454781	−	0.890603i	$-0.650282\pi$
$797$	−25.6780	−0.909561	−0.454781	+	0.890603i	$0.650282\pi$
$798$	0	0
$799$	45.3654	1.60491
$800$	5.38850	0.190512
$801$	0	0
$802$	18.7391	0.661699
$803$	8.55369	0.301853
$804$	0	0
$805$	−14.1831	−0.499887
$806$	0.948310	0.0334028
$807$	0	0
$808$	−3.41135	−0.120011
$809$	−38.4964	−1.35346	−0.676731	−	0.736231i	$-0.736604\pi$
$809$	−38.4964	−1.35346	−0.676731	+	0.736231i	$0.736604\pi$
$810$	0	0
$811$	41.4184	1.45440	0.727199	−	0.686427i	$-0.240822\pi$
$811$	41.4184	1.45440	0.727199	+	0.686427i	$0.240822\pi$
$812$	30.3033	1.06344
$813$	0	0
$814$	5.01970	0.175940
$815$	12.8085	0.448663
$816$	0	0
$817$	18.6689	0.653140
$818$	−7.69500	−0.269049
$819$	0	0
$820$	0.413720	0.0144477
$821$	−30.6953	−1.07127	−0.535637	−	0.844448i	$-0.679928\pi$
$821$	−30.6953	−1.07127	−0.535637	+	0.844448i	$0.679928\pi$
$822$	0	0
$823$	−22.1775	−0.773060	−0.386530	−	0.922277i	$-0.626327\pi$
$823$	−22.1775	−0.773060	−0.386530	+	0.922277i	$0.626327\pi$
$824$	−11.8822	−0.413936
$825$	0	0
$826$	−17.0584	−0.593537
$827$	22.1424	0.769968	0.384984	−	0.922923i	$-0.374207\pi$
$827$	22.1424	0.769968	0.384984	+	0.922923i	$0.374207\pi$
$828$	0	0
$829$	−50.0945	−1.73985	−0.869927	−	0.493181i	$-0.835834\pi$
$829$	−50.0945	−1.73985	−0.869927	+	0.493181i	$0.835834\pi$
$830$	7.57377	0.262889
$831$	0	0
$832$	−1.35085	−0.0468321
$833$	−15.2409	−0.528065
$834$	0	0
$835$	8.26887	0.286156
$836$	−5.42049	−0.187472
$837$	0	0
$838$	18.4182	0.636246
$839$	22.0611	0.761632	0.380816	−	0.924651i	$-0.375643\pi$
$839$	22.0611	0.761632	0.380816	+	0.924651i	$0.375643\pi$
$840$	0	0
$841$	2.80393	0.0966871
$842$	−2.67677	−0.0922474
$843$	0	0
$844$	16.2827	0.560474
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−20.8011	−0.714733
$848$	−0.644477	−0.0221314
$849$	0	0
$850$	−2.68221	−0.0919990
$851$	−18.6631	−0.639763
$852$	0	0
$853$	−0.911555	−0.0312110	−0.0156055	−	0.999878i	$-0.504968\pi$
$853$	−0.911555	−0.0312110	−0.0156055	+	0.999878i	$0.504968\pi$
$854$	10.0094	0.342516
$855$	0	0
$856$	17.7458	0.606538
$857$	−12.3643	−0.422358	−0.211179	−	0.977447i	$-0.567730\pi$
$857$	−12.3643	−0.422358	−0.211179	+	0.977447i	$0.567730\pi$
$858$	0	0
$859$	23.4875	0.801382	0.400691	−	0.916213i	$-0.368770\pi$
$859$	23.4875	0.801382	0.400691	+	0.916213i	$0.368770\pi$
$860$	−20.6473	−0.704068
$861$	0	0
$862$	11.8342	0.403075
$863$	48.9471	1.66618	0.833089	−	0.553139i	$-0.186570\pi$
$863$	48.9471	1.66618	0.833089	+	0.553139i	$0.186570\pi$
$864$	0	0
$865$	21.1314	0.718488
$866$	−5.50812	−0.187174
$867$	0	0
$868$	−9.03918	−0.306810
$869$	22.3173	0.757063
$870$	0	0
$871$	0.463882	0.0157181
$872$	−1.53471	−0.0519717
$873$	0	0
$874$	−3.80720	−0.128780
$875$	−3.19425	−0.107985
$876$	0	0
$877$	16.2682	0.549339	0.274670	−	0.961539i	$-0.411432\pi$
$877$	16.2682	0.549339	0.274670	+	0.961539i	$0.411432\pi$
$878$	−16.1212	−0.544063
$879$	0	0
$880$	4.64847	0.156700
$881$	−47.7399	−1.60840	−0.804199	−	0.594360i	$-0.797405\pi$
$881$	−47.7399	−1.60840	−0.804199	+	0.594360i	$0.797405\pi$
$882$	0	0
$883$	−14.3582	−0.483192	−0.241596	−	0.970377i	$-0.577671\pi$
$883$	−14.3582	−0.483192	−0.241596	+	0.970377i	$0.577671\pi$
$884$	8.00392	0.269201
$885$	0	0
$886$	−6.72300	−0.225864
$887$	31.4266	1.05520	0.527601	−	0.849492i	$-0.323092\pi$
$887$	31.4266	1.05520	0.527601	+	0.849492i	$0.323092\pi$
$888$	0	0
$889$	−26.8236	−0.899635
$890$	4.49847	0.150789
$891$	0	0
$892$	−19.1251	−0.640355
$893$	14.5023	0.485300
$894$	0	0
$895$	−9.95682	−0.332820
$896$	−36.8569	−1.23130
$897$	0	0
$898$	5.32974	0.177856
$899$	−9.48682	−0.316403
$900$	0	0
$901$	1.39747	0.0465566
$902$	−0.293712	−0.00977954
$903$	0	0
$904$	5.28444	0.175758
$905$	12.2980	0.408800
$906$	0	0
$907$	−14.8568	−0.493311	−0.246655	−	0.969103i	$-0.579332\pi$
$907$	−14.8568	−0.493311	−0.246655	+	0.969103i	$0.579332\pi$
$908$	−26.1644	−0.868297
$909$	0	0
$910$	−1.80069	−0.0596923
$911$	43.8879	1.45407	0.727036	−	0.686600i	$-0.240897\pi$
$911$	43.8879	1.45407	0.727036	+	0.686600i	$0.240897\pi$
$912$	0	0
$913$	28.4621	0.941957
$914$	15.6299	0.516992
$915$	0	0
$916$	−15.0407	−0.496959
$917$	−4.18068	−0.138058
$918$	0	0
$919$	−43.5714	−1.43729	−0.718645	−	0.695377i	$-0.755237\pi$
$919$	−43.5714	−1.43729	−0.718645	+	0.695377i	$0.755237\pi$
$920$	9.21680	0.303869
$921$	0	0
$922$	6.64791	0.218937
$923$	5.84272	0.192316
$924$	0	0
$925$	−4.20323	−0.138201
$926$	−7.05323	−0.231784
$927$	0	0
$928$	−30.3884	−0.997549
$929$	34.2867	1.12491	0.562456	−	0.826827i	$-0.309857\pi$
$929$	34.2867	1.12491	0.562456	+	0.826827i	$0.309857\pi$
$930$	0	0
$931$	−4.87216	−0.159679
$932$	−18.9880	−0.621971
$933$	0	0
$934$	−17.9484	−0.587290
$935$	−10.0797	−0.329641
$936$	0	0
$937$	−25.4336	−0.830879	−0.415440	−	0.909621i	$-0.636372\pi$
$937$	−25.4336	−0.830879	−0.415440	+	0.909621i	$0.636372\pi$
$938$	0.835309	0.0272738
$939$	0	0
$940$	−16.0392	−0.523140
$941$	−12.2014	−0.397754	−0.198877	−	0.980024i	$-0.563729\pi$
$941$	−12.2014	−0.397754	−0.198877	+	0.980024i	$0.563729\pi$
$942$	0	0
$943$	1.09201	0.0355609
$944$	−20.7867	−0.676548
$945$	0	0
$946$	14.6581	0.476577
$947$	25.4266	0.826254	0.413127	−	0.910673i	$-0.364437\pi$
$947$	25.4266	0.826254	0.413127	+	0.910673i	$0.364437\pi$
$948$	0	0
$949$	−4.03765	−0.131068
$950$	−0.857441	−0.0278191
$951$	0	0
$952$	31.5479	1.02247
$953$	−43.1195	−1.39678	−0.698390	−	0.715718i	$-0.746100\pi$
$953$	−43.1195	−1.39678	−0.698390	+	0.715718i	$0.746100\pi$
$954$	0	0
$955$	−14.1543	−0.458023
$956$	−44.3021	−1.43283
$957$	0	0
$958$	10.8368	0.350120
$959$	−24.6271	−0.795250
$960$	0	0
$961$	−28.1702	−0.908715
$962$	−2.36948	−0.0763951
$963$	0	0
$964$	23.1504	0.745624
$965$	5.68833	0.183114
$966$	0	0
$967$	−49.4289	−1.58953	−0.794763	−	0.606919i	$-0.792405\pi$
$967$	−49.4289	−1.58953	−0.794763	+	0.606919i	$0.792405\pi$
$968$	13.5175	0.434468
$969$	0	0
$970$	−6.50238	−0.208779
$971$	−29.6355	−0.951047	−0.475524	−	0.879703i	$-0.657741\pi$
$971$	−29.6355	−0.951047	−0.475524	+	0.879703i	$0.657741\pi$
$972$	0	0
$973$	4.52123	0.144944
$974$	−15.0668	−0.482771
$975$	0	0
$976$	12.1971	0.390420
$977$	−3.87629	−0.124014	−0.0620068	−	0.998076i	$-0.519750\pi$
$977$	−3.87629	−0.124014	−0.0620068	+	0.998076i	$0.519750\pi$
$978$	0	0
$979$	16.9051	0.540290
$980$	5.38850	0.172129
$981$	0	0
$982$	9.42585	0.300791
$983$	4.09530	0.130620	0.0653099	−	0.997865i	$-0.479196\pi$
$983$	4.09530	0.130620	0.0653099	+	0.997865i	$0.479196\pi$
$984$	0	0
$985$	−14.3543	−0.457366
$986$	15.1263	0.481720
$987$	0	0
$988$	2.55867	0.0814021
$989$	−54.4986	−1.73295
$990$	0	0
$991$	18.0556	0.573555	0.286778	−	0.957997i	$-0.407416\pi$
$991$	18.0556	0.573555	0.286778	+	0.957997i	$0.407416\pi$
$992$	9.06458	0.287801
$993$	0	0
$994$	10.5209	0.333704
$995$	−0.207823	−0.00658842
$996$	0	0
$997$	18.4349	0.583839	0.291920	−	0.956443i	$-0.405706\pi$
$997$	18.4349	0.583839	0.291920	+	0.956443i	$0.405706\pi$
$998$	−13.0037	−0.411626
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1755.2.a.o.1.3	✓	4
3.2	odd	2		1755.2.a.q.1.2	yes	4
5.4	even	2		8775.2.a.br.1.2		4
15.14	odd	2		8775.2.a.bj.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1755.2.a.o.1.3	✓	4	1.1	even	1	trivial
1755.2.a.q.1.2	yes	4	3.2	odd	2
8775.2.a.bj.1.3		4	15.14	odd	2
8775.2.a.br.1.2		4	5.4	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:	\( N \)	\(=\)	\( 1755 = 3^{3} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1755.a (trivial)

\(f(q)\)	\(=\)	\(q+0.563729 q^{2} -1.68221 q^{4} -1.00000 q^{5} +3.19425 q^{7} -2.07577 q^{8} +O(q^{10})\)	\(q+0.563729 q^{2} -1.68221 q^{4} -1.00000 q^{5} +3.19425 q^{7} -2.07577 q^{8} -0.563729 q^{10} -2.11848 q^{11} +1.00000 q^{13} +1.80069 q^{14} +2.19425 q^{16} -4.75798 q^{17} -1.52102 q^{19} +1.68221 q^{20} -1.19425 q^{22} +4.44019 q^{23} +1.00000 q^{25} +0.563729 q^{26} -5.37340 q^{28} -5.63950 q^{29} +1.68221 q^{31} +5.38850 q^{32} -2.68221 q^{34} -3.19425 q^{35} -4.20323 q^{37} -0.857441 q^{38} +2.07577 q^{40} +0.245939 q^{41} -12.2739 q^{43} +3.56373 q^{44} +2.50306 q^{46} -9.53459 q^{47} +3.20323 q^{49} +0.563729 q^{50} -1.68221 q^{52} -0.293712 q^{53} +2.11848 q^{55} -6.63052 q^{56} -3.17915 q^{58} -9.47324 q^{59} +5.55867 q^{61} +0.948310 q^{62} -1.35085 q^{64} -1.00000 q^{65} +0.463882 q^{67} +8.00392 q^{68} -1.80069 q^{70} +5.84272 q^{71} -4.03765 q^{73} -2.36948 q^{74} +2.55867 q^{76} -6.76695 q^{77} -10.5346 q^{79} -2.19425 q^{80} +0.138643 q^{82} -13.4351 q^{83} +4.75798 q^{85} -6.91917 q^{86} +4.39747 q^{88} -7.97984 q^{89} +3.19425 q^{91} -7.46933 q^{92} -5.37492 q^{94} +1.52102 q^{95} +11.5346 q^{97} +1.80575 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{2} + 3 q^{4} - 4 q^{5} + q^{7} - 3 q^{8}+O(q^{10}) \) 4 * q - q^2 + 3 * q^4 - 4 * q^5 + q^7 - 3 * q^8	\( 4 q - q^{2} + 3 q^{4} - 4 q^{5} + q^{7} - 3 q^{8} + q^{10} - 2 q^{11} + 4 q^{13} - 9 q^{14} - 3 q^{16} - 4 q^{17} - 4 q^{19} - 3 q^{20} + 7 q^{22} - 7 q^{23} + 4 q^{25} - q^{26} - 2 q^{28} - 14 q^{29} - 3 q^{31} - 2 q^{32} - q^{34} - q^{35} - 5 q^{37} - 14 q^{38} + 3 q^{40} - 12 q^{41} - 4 q^{43} + 11 q^{44} + 8 q^{46} - 11 q^{47} + q^{49} - q^{50} + 3 q^{52} - 15 q^{53} + 2 q^{55} - 18 q^{56} - 5 q^{58} - 9 q^{59} - 9 q^{61} + 5 q^{62} - 11 q^{64} - 4 q^{65} + 8 q^{67} + 4 q^{68} + 9 q^{70} + 3 q^{71} + 13 q^{73} - 18 q^{74} - 21 q^{76} - 12 q^{77} - 15 q^{79} + 3 q^{80} + 18 q^{82} - q^{83} + 4 q^{85} - 5 q^{86} - 6 q^{88} - 8 q^{89} + q^{91} - 29 q^{92} - 23 q^{94} + 4 q^{95} + 19 q^{97} + 19 q^{98}+O(q^{100}) \) 4 * q - q^2 + 3 * q^4 - 4 * q^5 + q^7 - 3 * q^8 + q^10 - 2 * q^11 + 4 * q^13 - 9 * q^14 - 3 * q^16 - 4 * q^17 - 4 * q^19 - 3 * q^20 + 7 * q^22 - 7 * q^23 + 4 * q^25 - q^26 - 2 * q^28 - 14 * q^29 - 3 * q^31 - 2 * q^32 - q^34 - q^35 - 5 * q^37 - 14 * q^38 + 3 * q^40 - 12 * q^41 - 4 * q^43 + 11 * q^44 + 8 * q^46 - 11 * q^47 + q^49 - q^50 + 3 * q^52 - 15 * q^53 + 2 * q^55 - 18 * q^56 - 5 * q^58 - 9 * q^59 - 9 * q^61 + 5 * q^62 - 11 * q^64 - 4 * q^65 + 8 * q^67 + 4 * q^68 + 9 * q^70 + 3 * q^71 + 13 * q^73 - 18 * q^74 - 21 * q^76 - 12 * q^77 - 15 * q^79 + 3 * q^80 + 18 * q^82 - q^83 + 4 * q^85 - 5 * q^86 - 6 * q^88 - 8 * q^89 + q^91 - 29 * q^92 - 23 * q^94 + 4 * q^95 + 19 * q^97 + 19 * q^98

Introduction

L-functions

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