LMFDB - Embedded newform 1755.2.a.u.1.4

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:	$0$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - x^{6} - 12x^{5} + 9x^{4} + 39x^{3} - 16x^{2} - 30x + 12 $ x^7 - x^6 - 12x^5 + 9x^4 + 39x^3 - 16x^2 - 30*x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.401763$ 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)

$f(q)$	$=$	$q-0.401763 q^{2} -1.83859 q^{4} -1.00000 q^{5} +2.55731 q^{7} +1.54220 q^{8} +O(q^{10})$	\(q-0.401763 q^{2} -1.83859 q^{4} -1.00000 q^{5} +2.55731 q^{7} +1.54220 q^{8} +0.401763 q^{10} -6.34809 q^{11} +1.00000 q^{13} -1.02743 q^{14} +3.05757 q^{16} +1.45193 q^{17} -2.89616 q^{19} +1.83859 q^{20} +2.55043 q^{22} +1.87972 q^{23} +1.00000 q^{25} -0.401763 q^{26} -4.70184 q^{28} +3.57786 q^{29} -5.67050 q^{31} -4.31283 q^{32} -0.583334 q^{34} -2.55731 q^{35} +6.18373 q^{37} +1.16357 q^{38} -1.54220 q^{40} -3.33062 q^{41} +9.46580 q^{43} +11.6715 q^{44} -0.755201 q^{46} -2.77434 q^{47} -0.460148 q^{49} -0.401763 q^{50} -1.83859 q^{52} +9.34202 q^{53} +6.34809 q^{55} +3.94390 q^{56} -1.43745 q^{58} -11.1512 q^{59} +7.83229 q^{61} +2.27820 q^{62} -4.38241 q^{64} -1.00000 q^{65} +9.22216 q^{67} -2.66951 q^{68} +1.02743 q^{70} -13.2535 q^{71} +14.8993 q^{73} -2.48439 q^{74} +5.32484 q^{76} -16.2341 q^{77} +4.01232 q^{79} -3.05757 q^{80} +1.33812 q^{82} +10.5733 q^{83} -1.45193 q^{85} -3.80301 q^{86} -9.79005 q^{88} +8.56408 q^{89} +2.55731 q^{91} -3.45602 q^{92} +1.11463 q^{94} +2.89616 q^{95} -7.57479 q^{97} +0.184871 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - q^{2} + 11 q^{4} - 7 q^{5} + 2 q^{7} - 6 q^{8}+O(q^{10}) $ 7 * q - q^2 + 11 * q^4 - 7 * q^5 + 2 * q^7 - 6 * q^8	$ 7 q - q^{2} + 11 q^{4} - 7 q^{5} + 2 q^{7} - 6 q^{8} + q^{10} - q^{11} + 7 q^{13} + 12 q^{14} + 23 q^{16} - 11 q^{17} + 2 q^{19} - 11 q^{20} + 16 q^{22} + q^{23} + 7 q^{25} - q^{26} + 10 q^{28} + 4 q^{29} - 13 q^{32} + q^{34} - 2 q^{35} + 23 q^{37} + 15 q^{38} + 6 q^{40} - 2 q^{41} + 8 q^{43} - 10 q^{44} + 37 q^{46} - 2 q^{47} + 43 q^{49} - q^{50} + 11 q^{52} - 10 q^{53} + q^{55} + 68 q^{56} - 26 q^{58} + 13 q^{59} + 21 q^{61} - 9 q^{62} + 46 q^{64} - 7 q^{65} + 21 q^{67} - 53 q^{68} - 12 q^{70} + 10 q^{71} + 13 q^{73} + 68 q^{74} - 41 q^{76} - 6 q^{77} + 8 q^{79} - 23 q^{80} - 26 q^{82} + 4 q^{83} + 11 q^{85} + 12 q^{86} + 44 q^{88} + 27 q^{89} + 2 q^{91} + 9 q^{92} - 24 q^{94} - 2 q^{95} - 15 q^{97} - 7 q^{98}+O(q^{100}) $ 7 * q - q^2 + 11 * q^4 - 7 * q^5 + 2 * q^7 - 6 * q^8 + q^10 - q^11 + 7 * q^13 + 12 * q^14 + 23 * q^16 - 11 * q^17 + 2 * q^19 - 11 * q^20 + 16 * q^22 + q^23 + 7 * q^25 - q^26 + 10 * q^28 + 4 * q^29 - 13 * q^32 + q^34 - 2 * q^35 + 23 * q^37 + 15 * q^38 + 6 * q^40 - 2 * q^41 + 8 * q^43 - 10 * q^44 + 37 * q^46 - 2 * q^47 + 43 * q^49 - q^50 + 11 * q^52 - 10 * q^53 + q^55 + 68 * q^56 - 26 * q^58 + 13 * q^59 + 21 * q^61 - 9 * q^62 + 46 * q^64 - 7 * q^65 + 21 * q^67 - 53 * q^68 - 12 * q^70 + 10 * q^71 + 13 * q^73 + 68 * q^74 - 41 * q^76 - 6 * q^77 + 8 * q^79 - 23 * q^80 - 26 * q^82 + 4 * q^83 + 11 * q^85 + 12 * q^86 + 44 * q^88 + 27 * q^89 + 2 * q^91 + 9 * q^92 - 24 * q^94 - 2 * q^95 - 15 * q^97 - 7 * 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 $

$2$	−0.401763	−0.284089	−0.142045	−	0.989860i	$-0.545368\pi$
$2$	−0.401763	−0.284089	−0.142045	+	0.989860i	$0.545368\pi$
$3$	0	0
$3$	0	0
$4$	−1.83859	−0.919293
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	2.55731	0.966574	0.483287	−	0.875462i	$-0.339443\pi$
$7$	2.55731	0.966574	0.483287	+	0.875462i	$0.339443\pi$
$8$	1.54220	0.545251
$9$	0	0
$10$	0.401763	0.127049
$11$	−6.34809	−1.91402	−0.957011	−	0.290052i	$-0.906328\pi$
$11$	−6.34809	−1.91402	−0.957011	+	0.290052i	$0.906328\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−1.02743	−0.274593
$15$	0	0
$16$	3.05757	0.764393
$17$	1.45193	0.352146	0.176073	−	0.984377i	$-0.443661\pi$
$17$	1.45193	0.352146	0.176073	+	0.984377i	$0.443661\pi$
$18$	0	0
$19$	−2.89616	−0.664424	−0.332212	−	0.943205i	$-0.607795\pi$
$19$	−2.89616	−0.664424	−0.332212	+	0.943205i	$0.607795\pi$
$20$	1.83859	0.411120
$21$	0	0
$22$	2.55043	0.543753
$23$	1.87972	0.391948	0.195974	−	0.980609i	$-0.437213\pi$
$23$	1.87972	0.391948	0.195974	+	0.980609i	$0.437213\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	−0.401763	−0.0787922
$27$	0	0
$28$	−4.70184	−0.888565
$29$	3.57786	0.664393	0.332196	−	0.943210i	$-0.392210\pi$
$29$	3.57786	0.664393	0.332196	+	0.943210i	$0.392210\pi$
$30$	0	0
$31$	−5.67050	−1.01845	−0.509226	−	0.860633i	$-0.670068\pi$
$31$	−5.67050	−1.01845	−0.509226	+	0.860633i	$0.670068\pi$
$32$	−4.31283	−0.762407
$33$	0	0
$34$	−0.583334	−0.100041
$35$	−2.55731	−0.432265
$36$	0	0
$37$	6.18373	1.01660	0.508299	−	0.861181i	$-0.330274\pi$
$37$	6.18373	1.01660	0.508299	+	0.861181i	$0.330274\pi$
$38$	1.16357	0.188756
$39$	0	0
$40$	−1.54220	−0.243844
$41$	−3.33062	−0.520155	−0.260078	−	0.965588i	$-0.583748\pi$
$41$	−3.33062	−0.520155	−0.260078	+	0.965588i	$0.583748\pi$
$42$	0	0
$43$	9.46580	1.44352	0.721760	−	0.692143i	$-0.243333\pi$
$43$	9.46580	1.44352	0.721760	+	0.692143i	$0.243333\pi$
$44$	11.6715	1.75955
$45$	0	0
$46$	−0.755201	−0.111348
$47$	−2.77434	−0.404679	−0.202339	−	0.979315i	$-0.564854\pi$
$47$	−2.77434	−0.404679	−0.202339	+	0.979315i	$0.564854\pi$
$48$	0	0
$49$	−0.460148	−0.0657355
$50$	−0.401763	−0.0568179
$51$	0	0
$52$	−1.83859	−0.254966
$53$	9.34202	1.28323	0.641613	−	0.767029i	$-0.278266\pi$
$53$	9.34202	1.28323	0.641613	+	0.767029i	$0.278266\pi$
$54$	0	0
$55$	6.34809	0.855977
$56$	3.94390	0.527025
$57$	0	0
$58$	−1.43745	−0.188747
$59$	−11.1512	−1.45176	−0.725881	−	0.687820i	$-0.758568\pi$
$59$	−11.1512	−1.45176	−0.725881	+	0.687820i	$0.758568\pi$
$60$	0	0
$61$	7.83229	1.00282	0.501411	−	0.865209i	$-0.332814\pi$
$61$	7.83229	1.00282	0.501411	+	0.865209i	$0.332814\pi$
$62$	2.27820	0.289331
$63$	0	0
$64$	−4.38241	−0.547801
$65$	−1.00000	−0.124035
$66$	0	0
$67$	9.22216	1.12667	0.563333	−	0.826230i	$-0.309519\pi$
$67$	9.22216	1.12667	0.563333	+	0.826230i	$0.309519\pi$
$68$	−2.66951	−0.323725
$69$	0	0
$70$	1.02743	0.122802
$71$	−13.2535	−1.57290	−0.786450	−	0.617653i	$-0.788083\pi$
$71$	−13.2535	−1.57290	−0.786450	+	0.617653i	$0.788083\pi$
$72$	0	0
$73$	14.8993	1.74384	0.871918	−	0.489653i	$-0.162876\pi$
$73$	14.8993	1.74384	0.871918	+	0.489653i	$0.162876\pi$
$74$	−2.48439	−0.288805
$75$	0	0
$76$	5.32484	0.610801
$77$	−16.2341	−1.85004
$78$	0	0
$79$	4.01232	0.451422	0.225711	−	0.974194i	$-0.427530\pi$
$79$	4.01232	0.451422	0.225711	+	0.974194i	$0.427530\pi$
$80$	−3.05757	−0.341847
$81$	0	0
$82$	1.33812	0.147771
$83$	10.5733	1.16057	0.580287	−	0.814412i	$-0.302940\pi$
$83$	10.5733	1.16057	0.580287	+	0.814412i	$0.302940\pi$
$84$	0	0
$85$	−1.45193	−0.157484
$86$	−3.80301	−0.410089
$87$	0	0
$88$	−9.79005	−1.04362
$89$	8.56408	0.907791	0.453896	−	0.891055i	$-0.350034\pi$
$89$	8.56408	0.907791	0.453896	+	0.891055i	$0.350034\pi$
$90$	0	0
$91$	2.55731	0.268079
$92$	−3.45602	−0.360315
$93$	0	0
$94$	1.11463	0.114965
$95$	2.89616	0.297140
$96$	0	0
$97$	−7.57479	−0.769103	−0.384552	−	0.923103i	$-0.625644\pi$
$97$	−7.57479	−0.769103	−0.384552	+	0.923103i	$0.625644\pi$
$98$	0.184871	0.0186748
$99$	0	0
$100$	−1.83859	−0.183859
$101$	7.91815	0.787886	0.393943	−	0.919135i	$-0.371111\pi$
$101$	7.91815	0.787886	0.393943	+	0.919135i	$0.371111\pi$
$102$	0	0
$103$	10.4515	1.02982	0.514909	−	0.857245i	$-0.327826\pi$
$103$	10.4515	1.02982	0.514909	+	0.857245i	$0.327826\pi$
$104$	1.54220	0.151225
$105$	0	0
$106$	−3.75328	−0.364551
$107$	12.5181	1.21017	0.605086	−	0.796160i	$-0.293139\pi$
$107$	12.5181	1.21017	0.605086	+	0.796160i	$0.293139\pi$
$108$	0	0
$109$	19.2366	1.84254	0.921268	−	0.388929i	$-0.127155\pi$
$109$	19.2366	1.84254	0.921268	+	0.388929i	$0.127155\pi$
$110$	−2.55043	−0.243174
$111$	0	0
$112$	7.81917	0.738842
$113$	−1.59765	−0.150294	−0.0751471	−	0.997172i	$-0.523943\pi$
$113$	−1.59765	−0.150294	−0.0751471	+	0.997172i	$0.523943\pi$
$114$	0	0
$115$	−1.87972	−0.175285
$116$	−6.57821	−0.610772
$117$	0	0
$118$	4.48014	0.412430
$119$	3.71305	0.340375
$120$	0	0
$121$	29.2983	2.66348
$122$	−3.14673	−0.284891
$123$	0	0
$124$	10.4257	0.936255
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−7.20238	−0.639108	−0.319554	−	0.947568i	$-0.603533\pi$
$127$	−7.20238	−0.639108	−0.319554	+	0.947568i	$0.603533\pi$
$128$	10.3863	0.918032
$129$	0	0
$130$	0.401763	0.0352370
$131$	13.6813	1.19534	0.597670	−	0.801743i	$-0.296093\pi$
$131$	13.6813	1.19534	0.597670	+	0.801743i	$0.296093\pi$
$132$	0	0
$133$	−7.40639	−0.642215
$134$	−3.70513	−0.320074
$135$	0	0
$136$	2.23918	0.192008
$137$	13.5621	1.15869	0.579345	−	0.815082i	$-0.303308\pi$
$137$	13.5621	1.15869	0.579345	+	0.815082i	$0.303308\pi$
$138$	0	0
$139$	−10.7003	−0.907587	−0.453794	−	0.891107i	$-0.649930\pi$
$139$	−10.7003	−0.907587	−0.453794	+	0.891107i	$0.649930\pi$
$140$	4.70184	0.397378
$141$	0	0
$142$	5.32477	0.446845
$143$	−6.34809	−0.530854
$144$	0	0
$145$	−3.57786	−0.297125
$146$	−5.98600	−0.495405
$147$	0	0
$148$	−11.3693	−0.934552
$149$	5.46795	0.447952	0.223976	−	0.974595i	$-0.428096\pi$
$149$	5.46795	0.447952	0.223976	+	0.974595i	$0.428096\pi$
$150$	0	0
$151$	19.9767	1.62568	0.812841	−	0.582486i	$-0.197920\pi$
$151$	19.9767	1.62568	0.812841	+	0.582486i	$0.197920\pi$
$152$	−4.46646	−0.362278
$153$	0	0
$154$	6.52225	0.525578
$155$	5.67050	0.455465
$156$	0	0
$157$	−11.3585	−0.906503	−0.453252	−	0.891383i	$-0.649736\pi$
$157$	−11.3585	−0.906503	−0.453252	+	0.891383i	$0.649736\pi$
$158$	−1.61200	−0.128244
$159$	0	0
$160$	4.31283	0.340959
$161$	4.80703	0.378847
$162$	0	0
$163$	5.57479	0.436651	0.218326	−	0.975876i	$-0.429941\pi$
$163$	5.57479	0.436651	0.218326	+	0.975876i	$0.429941\pi$
$164$	6.12363	0.478175
$165$	0	0
$166$	−4.24798	−0.329707
$167$	−15.4470	−1.19532	−0.597662	−	0.801749i	$-0.703903\pi$
$167$	−15.4470	−1.19532	−0.597662	+	0.801749i	$0.703903\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0.583334	0.0447396
$171$	0	0
$172$	−17.4037	−1.32702
$173$	5.92586	0.450535	0.225267	−	0.974297i	$-0.427674\pi$
$173$	5.92586	0.450535	0.225267	+	0.974297i	$0.427674\pi$
$174$	0	0
$175$	2.55731	0.193315
$176$	−19.4098	−1.46307
$177$	0	0
$178$	−3.44073	−0.257894
$179$	18.1579	1.35718	0.678592	−	0.734516i	$-0.262591\pi$
$179$	18.1579	1.35718	0.678592	+	0.734516i	$0.262591\pi$
$180$	0	0
$181$	−25.4502	−1.89170	−0.945848	−	0.324610i	$-0.894767\pi$
$181$	−25.4502	−1.89170	−0.945848	+	0.324610i	$0.894767\pi$
$182$	−1.02743	−0.0761585
$183$	0	0
$184$	2.89890	0.213710
$185$	−6.18373	−0.454637
$186$	0	0
$187$	−9.21701	−0.674015
$188$	5.10086	0.372018
$189$	0	0
$190$	−1.16357	−0.0844142
$191$	0.717570	0.0519215	0.0259608	−	0.999663i	$-0.491736\pi$
$191$	0.717570	0.0519215	0.0259608	+	0.999663i	$0.491736\pi$
$192$	0	0
$193$	11.8483	0.852860	0.426430	−	0.904521i	$-0.359771\pi$
$193$	11.8483	0.852860	0.426430	+	0.904521i	$0.359771\pi$
$194$	3.04327	0.218494
$195$	0	0
$196$	0.846022	0.0604302
$197$	−25.4644	−1.81427	−0.907133	−	0.420844i	$-0.861734\pi$
$197$	−25.4644	−1.81427	−0.907133	+	0.420844i	$0.861734\pi$
$198$	0	0
$199$	6.04813	0.428740	0.214370	−	0.976752i	$-0.431230\pi$
$199$	6.04813	0.428740	0.214370	+	0.976752i	$0.431230\pi$
$200$	1.54220	0.109050
$201$	0	0
$202$	−3.18122	−0.223830
$203$	9.14972	0.642184
$204$	0	0
$205$	3.33062	0.232620
$206$	−4.19903	−0.292560
$207$	0	0
$208$	3.05757	0.212005
$209$	18.3851	1.27172
$210$	0	0
$211$	−12.8922	−0.887539	−0.443769	−	0.896141i	$-0.646359\pi$
$211$	−12.8922	−0.887539	−0.443769	+	0.896141i	$0.646359\pi$
$212$	−17.1761	−1.17966
$213$	0	0
$214$	−5.02931	−0.343797
$215$	−9.46580	−0.645562
$216$	0	0
$217$	−14.5012	−0.984408
$218$	−7.72857	−0.523445
$219$	0	0
$220$	−11.6715	−0.786894
$221$	1.45193	0.0976676
$222$	0	0
$223$	3.27541	0.219337	0.109669	−	0.993968i	$-0.465021\pi$
$223$	3.27541	0.219337	0.109669	+	0.993968i	$0.465021\pi$
$224$	−11.0292	−0.736922
$225$	0	0
$226$	0.641876	0.0426970
$227$	−9.58823	−0.636393	−0.318197	−	0.948025i	$-0.603077\pi$
$227$	−9.58823	−0.636393	−0.318197	+	0.948025i	$0.603077\pi$
$228$	0	0
$229$	4.08956	0.270245	0.135123	−	0.990829i	$-0.456857\pi$
$229$	4.08956	0.270245	0.135123	+	0.990829i	$0.456857\pi$
$230$	0.755201	0.0497965
$231$	0	0
$232$	5.51779	0.362261
$233$	7.89420	0.517166	0.258583	−	0.965989i	$-0.416744\pi$
$233$	7.89420	0.517166	0.258583	+	0.965989i	$0.416744\pi$
$234$	0	0
$235$	2.77434	0.180978
$236$	20.5024	1.33459
$237$	0	0
$238$	−1.49177	−0.0966969
$239$	−13.9244	−0.900696	−0.450348	−	0.892853i	$-0.648700\pi$
$239$	−13.9244	−0.900696	−0.450348	+	0.892853i	$0.648700\pi$
$240$	0	0
$241$	−18.6778	−1.20314	−0.601571	−	0.798819i	$-0.705458\pi$
$241$	−18.6778	−1.20314	−0.601571	+	0.798819i	$0.705458\pi$
$242$	−11.7710	−0.756667
$243$	0	0
$244$	−14.4003	−0.921888
$245$	0.460148	0.0293978
$246$	0	0
$247$	−2.89616	−0.184278
$248$	−8.74505	−0.555312
$249$	0	0
$250$	0.401763	0.0254097
$251$	5.68993	0.359145	0.179573	−	0.983745i	$-0.442528\pi$
$251$	5.68993	0.359145	0.179573	+	0.983745i	$0.442528\pi$
$252$	0	0
$253$	−11.9326	−0.750197
$254$	2.89365	0.181564
$255$	0	0
$256$	4.59197	0.286998
$257$	−24.9229	−1.55465	−0.777323	−	0.629101i	$-0.783423\pi$
$257$	−24.9229	−1.55465	−0.777323	+	0.629101i	$0.783423\pi$
$258$	0	0
$259$	15.8137	0.982617
$260$	1.83859	0.114024
$261$	0	0
$262$	−5.49664	−0.339583
$263$	−5.61614	−0.346306	−0.173153	−	0.984895i	$-0.555396\pi$
$263$	−5.61614	−0.346306	−0.173153	+	0.984895i	$0.555396\pi$
$264$	0	0
$265$	−9.34202	−0.573876
$266$	2.97561	0.182447
$267$	0	0
$268$	−16.9557	−1.03574
$269$	3.18019	0.193900	0.0969498	−	0.995289i	$-0.469091\pi$
$269$	3.18019	0.193900	0.0969498	+	0.995289i	$0.469091\pi$
$270$	0	0
$271$	−29.2960	−1.77961	−0.889804	−	0.456343i	$-0.849159\pi$
$271$	−29.2960	−1.77961	−0.889804	+	0.456343i	$0.849159\pi$
$272$	4.43939	0.269178
$273$	0	0
$274$	−5.44876	−0.329172
$275$	−6.34809	−0.382804
$276$	0	0
$277$	−11.6813	−0.701860	−0.350930	−	0.936402i	$-0.614135\pi$
$277$	−11.6813	−0.701860	−0.350930	+	0.936402i	$0.614135\pi$
$278$	4.29898	0.257836
$279$	0	0
$280$	−3.94390	−0.235693
$281$	12.2203	0.729005	0.364502	−	0.931202i	$-0.381239\pi$
$281$	12.2203	0.729005	0.364502	+	0.931202i	$0.381239\pi$
$282$	0	0
$283$	−10.3849	−0.617319	−0.308660	−	0.951173i	$-0.599880\pi$
$283$	−10.3849	−0.617319	−0.308660	+	0.951173i	$0.599880\pi$
$284$	24.3677	1.44596
$285$	0	0
$286$	2.55043	0.150810
$287$	−8.51743	−0.502768
$288$	0	0
$289$	−14.8919	−0.875993
$290$	1.43745	0.0844102
$291$	0	0
$292$	−27.3937	−1.60310
$293$	22.2008	1.29698	0.648492	−	0.761222i	$-0.275400\pi$
$293$	22.2008	1.29698	0.648492	+	0.761222i	$0.275400\pi$
$294$	0	0
$295$	11.1512	0.649248
$296$	9.53656	0.554301
$297$	0	0
$298$	−2.19682	−0.127258
$299$	1.87972	0.108707
$300$	0	0
$301$	24.2070	1.39527
$302$	−8.02591	−0.461839
$303$	0	0
$304$	−8.85522	−0.507881
$305$	−7.83229	−0.448476
$306$	0	0
$307$	32.6887	1.86564	0.932820	−	0.360342i	$-0.117340\pi$
$307$	32.6887	1.86564	0.932820	+	0.360342i	$0.117340\pi$
$308$	29.8477	1.70073
$309$	0	0
$310$	−2.27820	−0.129393
$311$	12.2155	0.692677	0.346338	−	0.938110i	$-0.387425\pi$
$311$	12.2155	0.692677	0.346338	+	0.938110i	$0.387425\pi$
$312$	0	0
$313$	−1.87637	−0.106059	−0.0530295	−	0.998593i	$-0.516888\pi$
$313$	−1.87637	−0.106059	−0.0530295	+	0.998593i	$0.516888\pi$
$314$	4.56341	0.257528
$315$	0	0
$316$	−7.37700	−0.414989
$317$	−5.86970	−0.329675	−0.164838	−	0.986321i	$-0.552710\pi$
$317$	−5.86970	−0.329675	−0.164838	+	0.986321i	$0.552710\pi$
$318$	0	0
$319$	−22.7126	−1.27166
$320$	4.38241	0.244984
$321$	0	0
$322$	−1.93129	−0.107626
$323$	−4.20503	−0.233974
$324$	0	0
$325$	1.00000	0.0554700
$326$	−2.23974	−0.124048
$327$	0	0
$328$	−5.13649	−0.283615
$329$	−7.09485	−0.391152
$330$	0	0
$331$	24.7130	1.35835	0.679175	−	0.733976i	$-0.262338\pi$
$331$	24.7130	1.35835	0.679175	+	0.733976i	$0.262338\pi$
$332$	−19.4400	−1.06691
$333$	0	0
$334$	6.20603	0.339579
$335$	−9.22216	−0.503861
$336$	0	0
$337$	25.3897	1.38307	0.691533	−	0.722345i	$-0.256936\pi$
$337$	25.3897	1.38307	0.691533	+	0.722345i	$0.256936\pi$
$338$	−0.401763	−0.0218530
$339$	0	0
$340$	2.66951	0.144774
$341$	35.9968	1.94934
$342$	0	0
$343$	−19.0779	−1.03011
$344$	14.5982	0.787081
$345$	0	0
$346$	−2.38079	−0.127992
$347$	25.2781	1.35700	0.678501	−	0.734599i	$-0.262630\pi$
$347$	25.2781	1.35700	0.678501	+	0.734599i	$0.262630\pi$
$348$	0	0
$349$	12.3375	0.660411	0.330205	−	0.943909i	$-0.392882\pi$
$349$	12.3375	0.660411	0.330205	+	0.943909i	$0.392882\pi$
$350$	−1.02743	−0.0549187
$351$	0	0
$352$	27.3782	1.45926
$353$	−17.9266	−0.954135	−0.477068	−	0.878867i	$-0.658300\pi$
$353$	−17.9266	−0.954135	−0.477068	+	0.878867i	$0.658300\pi$
$354$	0	0
$355$	13.2535	0.703423
$356$	−15.7458	−0.834526
$357$	0	0
$358$	−7.29516	−0.385561
$359$	8.69865	0.459097	0.229549	−	0.973297i	$-0.426275\pi$
$359$	8.69865	0.459097	0.229549	+	0.973297i	$0.426275\pi$
$360$	0	0
$361$	−10.6123	−0.558540
$362$	10.2249	0.537411
$363$	0	0
$364$	−4.70184	−0.246443
$365$	−14.8993	−0.779867
$366$	0	0
$367$	−25.7787	−1.34564	−0.672818	−	0.739808i	$-0.734916\pi$
$367$	−25.7787	−1.34564	−0.672818	+	0.739808i	$0.734916\pi$
$368$	5.74737	0.299602
$369$	0	0
$370$	2.48439	0.129157
$371$	23.8905	1.24033
$372$	0	0
$373$	11.8760	0.614914	0.307457	−	0.951562i	$-0.400522\pi$
$373$	11.8760	0.614914	0.307457	+	0.951562i	$0.400522\pi$
$374$	3.70306	0.191480
$375$	0	0
$376$	−4.27859	−0.220651
$377$	3.57786	0.184269
$378$	0	0
$379$	−28.1024	−1.44352	−0.721762	−	0.692141i	$-0.756668\pi$
$379$	−28.1024	−1.44352	−0.721762	+	0.692141i	$0.756668\pi$
$380$	−5.32484	−0.273158
$381$	0	0
$382$	−0.288293	−0.0147504
$383$	15.3113	0.782371	0.391185	−	0.920312i	$-0.372065\pi$
$383$	15.3113	0.782371	0.391185	+	0.920312i	$0.372065\pi$
$384$	0	0
$385$	16.2341	0.827364
$386$	−4.76021	−0.242289
$387$	0	0
$388$	13.9269	0.707031
$389$	−7.83076	−0.397035	−0.198518	−	0.980097i	$-0.563613\pi$
$389$	−7.83076	−0.397035	−0.198518	+	0.980097i	$0.563613\pi$
$390$	0	0
$391$	2.72923	0.138023
$392$	−0.709642	−0.0358423
$393$	0	0
$394$	10.2307	0.515414
$395$	−4.01232	−0.201882
$396$	0	0
$397$	31.6620	1.58907	0.794536	−	0.607217i	$-0.207714\pi$
$397$	31.6620	1.58907	0.794536	+	0.607217i	$0.207714\pi$
$398$	−2.42991	−0.121801
$399$	0	0
$400$	3.05757	0.152879
$401$	−15.4799	−0.773031	−0.386515	−	0.922283i	$-0.626321\pi$
$401$	−15.4799	−0.773031	−0.386515	+	0.922283i	$0.626321\pi$
$402$	0	0
$403$	−5.67050	−0.282468
$404$	−14.5582	−0.724298
$405$	0	0
$406$	−3.67602	−0.182438
$407$	−39.2549	−1.94579
$408$	0	0
$409$	−8.35785	−0.413269	−0.206634	−	0.978418i	$-0.566251\pi$
$409$	−8.35785	−0.413269	−0.206634	+	0.978418i	$0.566251\pi$
$410$	−1.33812	−0.0660850
$411$	0	0
$412$	−19.2160	−0.946705
$413$	−28.5171	−1.40323
$414$	0	0
$415$	−10.5733	−0.519024
$416$	−4.31283	−0.211454
$417$	0	0
$418$	−7.38645	−0.361283
$419$	22.2214	1.08559	0.542793	−	0.839867i	$-0.317367\pi$
$419$	22.2214	1.08559	0.542793	+	0.839867i	$0.317367\pi$
$420$	0	0
$421$	−17.1089	−0.833837	−0.416918	−	0.908944i	$-0.636890\pi$
$421$	−17.1089	−0.833837	−0.416918	+	0.908944i	$0.636890\pi$
$422$	5.17963	0.252140
$423$	0	0
$424$	14.4073	0.699680
$425$	1.45193	0.0704291
$426$	0	0
$427$	20.0296	0.969301
$428$	−23.0156	−1.11250
$429$	0	0
$430$	3.80301	0.183397
$431$	26.7827	1.29008	0.645039	−	0.764149i	$-0.276841\pi$
$431$	26.7827	1.29008	0.645039	+	0.764149i	$0.276841\pi$
$432$	0	0
$433$	8.26459	0.397171	0.198586	−	0.980084i	$-0.436365\pi$
$433$	8.26459	0.397171	0.198586	+	0.980084i	$0.436365\pi$
$434$	5.82606	0.279660
$435$	0	0
$436$	−35.3682	−1.69383
$437$	−5.44396	−0.260420
$438$	0	0
$439$	−19.5600	−0.933546	−0.466773	−	0.884377i	$-0.654584\pi$
$439$	−19.5600	−0.933546	−0.466773	+	0.884377i	$0.654584\pi$
$440$	9.79005	0.466722
$441$	0	0
$442$	−0.583334	−0.0277463
$443$	−7.39824	−0.351501	−0.175751	−	0.984435i	$-0.556235\pi$
$443$	−7.39824	−0.351501	−0.175751	+	0.984435i	$0.556235\pi$
$444$	0	0
$445$	−8.56408	−0.405977
$446$	−1.31594	−0.0623114
$447$	0	0
$448$	−11.2072	−0.529490
$449$	3.84532	0.181472	0.0907359	−	0.995875i	$-0.471078\pi$
$449$	3.84532	0.181472	0.0907359	+	0.995875i	$0.471078\pi$
$450$	0	0
$451$	21.1431	0.995588
$452$	2.93741	0.138164
$453$	0	0
$454$	3.85220	0.180793
$455$	−2.55731	−0.119889
$456$	0	0
$457$	−21.7310	−1.01653	−0.508266	−	0.861200i	$-0.669713\pi$
$457$	−21.7310	−1.01653	−0.508266	+	0.861200i	$0.669713\pi$
$458$	−1.64303	−0.0767739
$459$	0	0
$460$	3.45602	0.161138
$461$	−18.2548	−0.850210	−0.425105	−	0.905144i	$-0.639763\pi$
$461$	−18.2548	−0.850210	−0.425105	+	0.905144i	$0.639763\pi$
$462$	0	0
$463$	−0.479576	−0.0222878	−0.0111439	−	0.999938i	$-0.503547\pi$
$463$	−0.479576	−0.0222878	−0.0111439	+	0.999938i	$0.503547\pi$
$464$	10.9396	0.507857
$465$	0	0
$466$	−3.17160	−0.146921
$467$	1.42017	0.0657178	0.0328589	−	0.999460i	$-0.489539\pi$
$467$	1.42017	0.0657178	0.0328589	+	0.999460i	$0.489539\pi$
$468$	0	0
$469$	23.5840	1.08901
$470$	−1.11463	−0.0514139
$471$	0	0
$472$	−17.1974	−0.791575
$473$	−60.0898	−2.76293
$474$	0	0
$475$	−2.89616	−0.132885
$476$	−6.82676	−0.312904
$477$	0	0
$478$	5.59432	0.255878
$479$	23.1704	1.05868	0.529341	−	0.848409i	$-0.322439\pi$
$479$	23.1704	1.05868	0.529341	+	0.848409i	$0.322439\pi$
$480$	0	0
$481$	6.18373	0.281954
$482$	7.50405	0.341800
$483$	0	0
$484$	−53.8674	−2.44852
$485$	7.57479	0.343953
$486$	0	0
$487$	−4.31669	−0.195608	−0.0978040	−	0.995206i	$-0.531182\pi$
$487$	−4.31669	−0.195608	−0.0978040	+	0.995206i	$0.531182\pi$
$488$	12.0790	0.546790
$489$	0	0
$490$	−0.184871	−0.00835160
$491$	−28.2374	−1.27434	−0.637168	−	0.770725i	$-0.719894\pi$
$491$	−28.2374	−1.27434	−0.637168	+	0.770725i	$0.719894\pi$
$492$	0	0
$493$	5.19482	0.233963
$494$	1.16357	0.0523515
$495$	0	0
$496$	−17.3380	−0.778497
$497$	−33.8934	−1.52032
$498$	0	0
$499$	−32.1148	−1.43766	−0.718828	−	0.695188i	$-0.755321\pi$
$499$	−32.1148	−1.43766	−0.718828	+	0.695188i	$0.755321\pi$
$500$	1.83859	0.0822241
$501$	0	0
$502$	−2.28601	−0.102029
$503$	−8.03586	−0.358301	−0.179151	−	0.983822i	$-0.557335\pi$
$503$	−8.03586	−0.358301	−0.179151	+	0.983822i	$0.557335\pi$
$504$	0	0
$505$	−7.91815	−0.352353
$506$	4.79409	0.213123
$507$	0	0
$508$	13.2422	0.587527
$509$	−5.50467	−0.243990	−0.121995	−	0.992531i	$-0.538929\pi$
$509$	−5.50467	−0.243990	−0.121995	+	0.992531i	$0.538929\pi$
$510$	0	0
$511$	38.1023	1.68555
$512$	−22.6176	−0.999565
$513$	0	0
$514$	10.0131	0.441659
$515$	−10.4515	−0.460549
$516$	0	0
$517$	17.6118	0.774564
$518$	−6.35337	−0.279151
$519$	0	0
$520$	−1.54220	−0.0676301
$521$	22.5845	0.989446	0.494723	−	0.869051i	$-0.335269\pi$
$521$	22.5845	0.989446	0.494723	+	0.869051i	$0.335269\pi$
$522$	0	0
$523$	−2.48216	−0.108537	−0.0542687	−	0.998526i	$-0.517283\pi$
$523$	−2.48216	−0.108537	−0.0542687	+	0.998526i	$0.517283\pi$
$524$	−25.1542	−1.09887
$525$	0	0
$526$	2.25636	0.0983819
$527$	−8.23319	−0.358643
$528$	0	0
$529$	−19.4667	−0.846377
$530$	3.75328	0.163032
$531$	0	0
$532$	13.6173	0.590384
$533$	−3.33062	−0.144265
$534$	0	0
$535$	−12.5181	−0.541205
$536$	14.2224	0.614316
$537$	0	0
$538$	−1.27768	−0.0550848
$539$	2.92106	0.125819
$540$	0	0
$541$	13.9422	0.599421	0.299711	−	0.954030i	$-0.403110\pi$
$541$	13.9422	0.599421	0.299711	+	0.954030i	$0.403110\pi$
$542$	11.7701	0.505568
$543$	0	0
$544$	−6.26194	−0.268478
$545$	−19.2366	−0.824007
$546$	0	0
$547$	−5.77751	−0.247029	−0.123514	−	0.992343i	$-0.539416\pi$
$547$	−5.77751	−0.247029	−0.123514	+	0.992343i	$0.539416\pi$
$548$	−24.9351	−1.06518
$549$	0	0
$550$	2.55043	0.108751
$551$	−10.3621	−0.441439
$552$	0	0
$553$	10.2608	0.436332
$554$	4.69311	0.199391
$555$	0	0
$556$	19.6734	0.834339
$557$	7.91155	0.335223	0.167611	−	0.985853i	$-0.446395\pi$
$557$	7.91155	0.335223	0.167611	+	0.985853i	$0.446395\pi$
$558$	0	0
$559$	9.46580	0.400361
$560$	−7.81917	−0.330420
$561$	0	0
$562$	−4.90969	−0.207103
$563$	33.0552	1.39311	0.696556	−	0.717502i	$-0.254715\pi$
$563$	33.0552	1.39311	0.696556	+	0.717502i	$0.254715\pi$
$564$	0	0
$565$	1.59765	0.0672136
$566$	4.17228	0.175374
$567$	0	0
$568$	−20.4396	−0.857626
$569$	14.5702	0.610814	0.305407	−	0.952222i	$-0.401208\pi$
$569$	14.5702	0.610814	0.305407	+	0.952222i	$0.401208\pi$
$570$	0	0
$571$	−2.65302	−0.111026	−0.0555128	−	0.998458i	$-0.517679\pi$
$571$	−2.65302	−0.111026	−0.0555128	+	0.998458i	$0.517679\pi$
$572$	11.6715	0.488011
$573$	0	0
$574$	3.42199	0.142831
$575$	1.87972	0.0783896
$576$	0	0
$577$	−2.92116	−0.121610	−0.0608048	−	0.998150i	$-0.519367\pi$
$577$	−2.92116	−0.121610	−0.0608048	+	0.998150i	$0.519367\pi$
$578$	5.98301	0.248860
$579$	0	0
$580$	6.57821	0.273145
$581$	27.0393	1.12178
$582$	0	0
$583$	−59.3040	−2.45612
$584$	22.9778	0.950828
$585$	0	0
$586$	−8.91946	−0.368459
$587$	28.7270	1.18569	0.592844	−	0.805317i	$-0.298005\pi$
$587$	28.7270	1.18569	0.592844	+	0.805317i	$0.298005\pi$
$588$	0	0
$589$	16.4227	0.676684
$590$	−4.48014	−0.184444
$591$	0	0
$592$	18.9072	0.777081
$593$	18.9449	0.777973	0.388986	−	0.921244i	$-0.372825\pi$
$593$	18.9449	0.777973	0.388986	+	0.921244i	$0.372825\pi$
$594$	0	0
$595$	−3.71305	−0.152220
$596$	−10.0533	−0.411799
$597$	0	0
$598$	−0.755201	−0.0308825
$599$	−7.21695	−0.294877	−0.147438	−	0.989071i	$-0.547103\pi$
$599$	−7.21695	−0.294877	−0.147438	+	0.989071i	$0.547103\pi$
$600$	0	0
$601$	21.0580	0.858976	0.429488	−	0.903073i	$-0.358694\pi$
$601$	21.0580	0.858976	0.429488	+	0.903073i	$0.358694\pi$
$602$	−9.72548	−0.396381
$603$	0	0
$604$	−36.7289	−1.49448
$605$	−29.2983	−1.19114
$606$	0	0
$607$	−3.31489	−0.134547	−0.0672737	−	0.997735i	$-0.521430\pi$
$607$	−3.31489	−0.134547	−0.0672737	+	0.997735i	$0.521430\pi$
$608$	12.4906	0.506562
$609$	0	0
$610$	3.14673	0.127407
$611$	−2.77434	−0.112238
$612$	0	0
$613$	−1.43291	−0.0578748	−0.0289374	−	0.999581i	$-0.509212\pi$
$613$	−1.43291	−0.0578748	−0.0289374	+	0.999581i	$0.509212\pi$
$614$	−13.1331	−0.530009
$615$	0	0
$616$	−25.0362	−1.00874
$617$	15.6104	0.628452	0.314226	−	0.949348i	$-0.398255\pi$
$617$	15.6104	0.628452	0.314226	+	0.949348i	$0.398255\pi$
$618$	0	0
$619$	−4.82323	−0.193862	−0.0969311	−	0.995291i	$-0.530903\pi$
$619$	−4.82323	−0.193862	−0.0969311	+	0.995291i	$0.530903\pi$
$620$	−10.4257	−0.418706
$621$	0	0
$622$	−4.90773	−0.196782
$623$	21.9010	0.877447
$624$	0	0
$625$	1.00000	0.0400000
$626$	0.753858	0.0301302
$627$	0	0
$628$	20.8835	0.833342
$629$	8.97836	0.357991
$630$	0	0
$631$	−39.1264	−1.55760	−0.778798	−	0.627275i	$-0.784170\pi$
$631$	−39.1264	−1.55760	−0.778798	+	0.627275i	$0.784170\pi$
$632$	6.18782	0.246138
$633$	0	0
$634$	2.35823	0.0936572
$635$	7.20238	0.285818
$636$	0	0
$637$	−0.460148	−0.0182317
$638$	9.12509	0.361266
$639$	0	0
$640$	−10.3863	−0.410556
$641$	−21.9856	−0.868377	−0.434189	−	0.900822i	$-0.642965\pi$
$641$	−21.9856	−0.868377	−0.434189	+	0.900822i	$0.642965\pi$
$642$	0	0
$643$	0.753630	0.0297203	0.0148601	−	0.999890i	$-0.495270\pi$
$643$	0.753630	0.0297203	0.0148601	+	0.999890i	$0.495270\pi$
$644$	−8.83813	−0.348271
$645$	0	0
$646$	1.68943	0.0664696
$647$	21.2493	0.835396	0.417698	−	0.908586i	$-0.362837\pi$
$647$	21.2493	0.835396	0.417698	+	0.908586i	$0.362837\pi$
$648$	0	0
$649$	70.7888	2.77870
$650$	−0.401763	−0.0157584
$651$	0	0
$652$	−10.2497	−0.401410
$653$	28.9191	1.13169	0.565846	−	0.824511i	$-0.308550\pi$
$653$	28.9191	1.13169	0.565846	+	0.824511i	$0.308550\pi$
$654$	0	0
$655$	−13.6813	−0.534572
$656$	−10.1836	−0.397603
$657$	0	0
$658$	2.85045	0.111122
$659$	26.1949	1.02041	0.510204	−	0.860054i	$-0.329570\pi$
$659$	26.1949	1.02041	0.510204	+	0.860054i	$0.329570\pi$
$660$	0	0
$661$	−22.2331	−0.864769	−0.432384	−	0.901689i	$-0.642328\pi$
$661$	−22.2331	−0.864769	−0.432384	+	0.901689i	$0.642328\pi$
$662$	−9.92878	−0.385893
$663$	0	0
$664$	16.3062	0.632804
$665$	7.40639	0.287207
$666$	0	0
$667$	6.72537	0.260407
$668$	28.4006	1.09885
$669$	0	0
$670$	3.70513	0.143141
$671$	−49.7201	−1.91942
$672$	0	0
$673$	17.6506	0.680381	0.340190	−	0.940357i	$-0.389508\pi$
$673$	17.6506	0.680381	0.340190	+	0.940357i	$0.389508\pi$
$674$	−10.2007	−0.392914
$675$	0	0
$676$	−1.83859	−0.0707149
$677$	−0.199260	−0.00765820	−0.00382910	−	0.999993i	$-0.501219\pi$
$677$	−0.199260	−0.00765820	−0.00382910	+	0.999993i	$0.501219\pi$
$678$	0	0
$679$	−19.3711	−0.743395
$680$	−2.23918	−0.0858685
$681$	0	0
$682$	−14.4622	−0.553786
$683$	−15.8831	−0.607751	−0.303876	−	0.952712i	$-0.598281\pi$
$683$	−15.8831	−0.607751	−0.303876	+	0.952712i	$0.598281\pi$
$684$	0	0
$685$	−13.5621	−0.518182
$686$	7.66481	0.292644
$687$	0	0
$688$	28.9424	1.10342
$689$	9.34202	0.355903
$690$	0	0
$691$	−23.4153	−0.890760	−0.445380	−	0.895342i	$-0.646931\pi$
$691$	−23.4153	−0.890760	−0.445380	+	0.895342i	$0.646931\pi$
$692$	−10.8952	−0.414174
$693$	0	0
$694$	−10.1558	−0.385510
$695$	10.7003	0.405885
$696$	0	0
$697$	−4.83584	−0.183170
$698$	−4.95675	−0.187616
$699$	0	0
$700$	−4.70184	−0.177713
$701$	42.8646	1.61897	0.809486	−	0.587139i	$-0.199746\pi$
$701$	42.8646	1.61897	0.809486	+	0.587139i	$0.199746\pi$
$702$	0	0
$703$	−17.9091	−0.675453
$704$	27.8199	1.04850
$705$	0	0
$706$	7.20224	0.271060
$707$	20.2492	0.761550
$708$	0	0
$709$	31.5465	1.18476	0.592378	−	0.805660i	$-0.298189\pi$
$709$	31.5465	1.18476	0.592378	+	0.805660i	$0.298189\pi$
$710$	−5.32477	−0.199835
$711$	0	0
$712$	13.2076	0.494974
$713$	−10.6589	−0.399180
$714$	0	0
$715$	6.34809	0.237405
$716$	−33.3848	−1.24765
$717$	0	0
$718$	−3.49480	−0.130425
$719$	19.5446	0.728889	0.364445	−	0.931225i	$-0.381259\pi$
$719$	19.5446	0.728889	0.364445	+	0.931225i	$0.381259\pi$
$720$	0	0
$721$	26.7278	0.995395
$722$	4.26362	0.158675
$723$	0	0
$724$	46.7923	1.73902
$725$	3.57786	0.132879
$726$	0	0
$727$	−0.0765012	−0.00283727	−0.00141864	−	0.999999i	$-0.500452\pi$
$727$	−0.0765012	−0.00283727	−0.00141864	+	0.999999i	$0.500452\pi$
$728$	3.94390	0.146170
$729$	0	0
$730$	5.98600	0.221552
$731$	13.7437	0.508329
$732$	0	0
$733$	46.0394	1.70050	0.850252	−	0.526376i	$-0.176450\pi$
$733$	46.0394	1.70050	0.850252	+	0.526376i	$0.176450\pi$
$734$	10.3569	0.382281
$735$	0	0
$736$	−8.10689	−0.298824
$737$	−58.5431	−2.15646
$738$	0	0
$739$	−12.6860	−0.466661	−0.233330	−	0.972398i	$-0.574962\pi$
$739$	−12.6860	−0.466661	−0.233330	+	0.972398i	$0.574962\pi$
$740$	11.3693	0.417944
$741$	0	0
$742$	−9.59831	−0.352365
$743$	−1.47694	−0.0541836	−0.0270918	−	0.999633i	$-0.508625\pi$
$743$	−1.47694	−0.0541836	−0.0270918	+	0.999633i	$0.508625\pi$
$744$	0	0
$745$	−5.46795	−0.200330
$746$	−4.77132	−0.174691
$747$	0	0
$748$	16.9463	0.619617
$749$	32.0127	1.16972
$750$	0	0
$751$	−0.0924292	−0.00337279	−0.00168639	−	0.999999i	$-0.500537\pi$
$751$	−0.0924292	−0.00337279	−0.00168639	+	0.999999i	$0.500537\pi$
$752$	−8.48274	−0.309334
$753$	0	0
$754$	−1.43745	−0.0523490
$755$	−19.9767	−0.727027
$756$	0	0
$757$	−39.8770	−1.44935	−0.724676	−	0.689090i	$-0.758011\pi$
$757$	−39.8770	−1.44935	−0.724676	+	0.689090i	$0.758011\pi$
$758$	11.2905	0.410090
$759$	0	0
$760$	4.46646	0.162016
$761$	−19.6346	−0.711752	−0.355876	−	0.934533i	$-0.615817\pi$
$761$	−19.6346	−0.711752	−0.355876	+	0.934533i	$0.615817\pi$
$762$	0	0
$763$	49.1941	1.78095
$764$	−1.31931	−0.0477311
$765$	0	0
$766$	−6.15151	−0.222263
$767$	−11.1512	−0.402646
$768$	0	0
$769$	9.83452	0.354642	0.177321	−	0.984153i	$-0.443257\pi$
$769$	9.83452	0.354642	0.177321	+	0.984153i	$0.443257\pi$
$770$	−6.52225	−0.235046
$771$	0	0
$772$	−21.7841	−0.784028
$773$	−19.2327	−0.691752	−0.345876	−	0.938280i	$-0.612418\pi$
$773$	−19.2327	−0.691752	−0.345876	+	0.938280i	$0.612418\pi$
$774$	0	0
$775$	−5.67050	−0.203690
$776$	−11.6819	−0.419354
$777$	0	0
$778$	3.14611	0.112794
$779$	9.64600	0.345604
$780$	0	0
$781$	84.1344	3.01057
$782$	−1.09650	−0.0392108
$783$	0	0
$784$	−1.40694	−0.0502477
$785$	11.3585	0.405401
$786$	0	0
$787$	22.7379	0.810518	0.405259	−	0.914202i	$-0.367181\pi$
$787$	22.7379	0.810518	0.405259	+	0.914202i	$0.367181\pi$
$788$	46.8186	1.66784
$789$	0	0
$790$	1.61200	0.0573525
$791$	−4.08569	−0.145270
$792$	0	0
$793$	7.83229	0.278133
$794$	−12.7206	−0.451439
$795$	0	0
$796$	−11.1200	−0.394138
$797$	−16.2745	−0.576471	−0.288235	−	0.957560i	$-0.593069\pi$
$797$	−16.2745	−0.576471	−0.288235	+	0.957560i	$0.593069\pi$
$798$	0	0
$799$	−4.02816	−0.142506
$800$	−4.31283	−0.152481
$801$	0	0
$802$	6.21926	0.219610
$803$	−94.5824	−3.33774
$804$	0	0
$805$	−4.80703	−0.169425
$806$	2.27820	0.0802461
$807$	0	0
$808$	12.2114	0.429595
$809$	−6.77042	−0.238035	−0.119018	−	0.992892i	$-0.537974\pi$
$809$	−6.77042	−0.238035	−0.119018	+	0.992892i	$0.537974\pi$
$810$	0	0
$811$	−26.1657	−0.918803	−0.459402	−	0.888229i	$-0.651936\pi$
$811$	−26.1657	−0.918803	−0.459402	+	0.888229i	$0.651936\pi$
$812$	−16.8226	−0.590356
$813$	0	0
$814$	15.7712	0.552779
$815$	−5.57479	−0.195276
$816$	0	0
$817$	−27.4144	−0.959110
$818$	3.35787	0.117405
$819$	0	0
$820$	−6.12363	−0.213846
$821$	22.3233	0.779090	0.389545	−	0.921008i	$-0.372632\pi$
$821$	22.3233	0.779090	0.389545	+	0.921008i	$0.372632\pi$
$822$	0	0
$823$	−7.78703	−0.271439	−0.135719	−	0.990747i	$-0.543335\pi$
$823$	−7.78703	−0.271439	−0.135719	+	0.990747i	$0.543335\pi$
$824$	16.1183	0.561509
$825$	0	0
$826$	11.4571	0.398644
$827$	48.8603	1.69904	0.849520	−	0.527557i	$-0.176892\pi$
$827$	48.8603	1.69904	0.849520	+	0.527557i	$0.176892\pi$
$828$	0	0
$829$	28.1116	0.976358	0.488179	−	0.872744i	$-0.337661\pi$
$829$	28.1116	0.976358	0.488179	+	0.872744i	$0.337661\pi$
$830$	4.24798	0.147449
$831$	0	0
$832$	−4.38241	−0.151933
$833$	−0.668105	−0.0231485
$834$	0	0
$835$	15.4470	0.534565
$836$	−33.8026	−1.16909
$837$	0	0
$838$	−8.92774	−0.308404
$839$	−18.6588	−0.644173	−0.322086	−	0.946710i	$-0.604384\pi$
$839$	−18.6588	−0.644173	−0.322086	+	0.946710i	$0.604384\pi$
$840$	0	0
$841$	−16.1989	−0.558582
$842$	6.87373	0.236884
$843$	0	0
$844$	23.7035	0.815908
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	74.9249	2.57445
$848$	28.5639	0.980889
$849$	0	0
$850$	−0.583334	−0.0200082
$851$	11.6237	0.398454
$852$	0	0
$853$	20.2842	0.694519	0.347259	−	0.937769i	$-0.387112\pi$
$853$	20.2842	0.694519	0.347259	+	0.937769i	$0.387112\pi$
$854$	−8.04717	−0.275368
$855$	0	0
$856$	19.3055	0.659847
$857$	−37.1742	−1.26985	−0.634924	−	0.772575i	$-0.718968\pi$
$857$	−37.1742	−1.26985	−0.634924	+	0.772575i	$0.718968\pi$
$858$	0	0
$859$	41.9086	1.42990	0.714951	−	0.699174i	$-0.246449\pi$
$859$	41.9086	1.42990	0.714951	+	0.699174i	$0.246449\pi$
$860$	17.4037	0.593461
$861$	0	0
$862$	−10.7603	−0.366498
$863$	2.75327	0.0937225	0.0468613	−	0.998901i	$-0.485078\pi$
$863$	2.75327	0.0937225	0.0468613	+	0.998901i	$0.485078\pi$
$864$	0	0
$865$	−5.92586	−0.201485
$866$	−3.32041	−0.112832
$867$	0	0
$868$	26.6618	0.904960
$869$	−25.4706	−0.864031
$870$	0	0
$871$	9.22216	0.312481
$872$	29.6668	1.00464
$873$	0	0
$874$	2.18718	0.0739826
$875$	−2.55731	−0.0864530
$876$	0	0
$877$	2.75319	0.0929686	0.0464843	−	0.998919i	$-0.485198\pi$
$877$	2.75319	0.0929686	0.0464843	+	0.998919i	$0.485198\pi$
$878$	7.85847	0.265211
$879$	0	0
$880$	19.4098	0.654303
$881$	50.8561	1.71339	0.856693	−	0.515827i	$-0.172515\pi$
$881$	50.8561	1.71339	0.856693	+	0.515827i	$0.172515\pi$
$882$	0	0
$883$	16.4161	0.552444	0.276222	−	0.961094i	$-0.410917\pi$
$883$	16.4161	0.552444	0.276222	+	0.961094i	$0.410917\pi$
$884$	−2.66951	−0.0897852
$885$	0	0
$886$	2.97234	0.0998578
$887$	32.2894	1.08417	0.542086	−	0.840323i	$-0.317635\pi$
$887$	32.2894	1.08417	0.542086	+	0.840323i	$0.317635\pi$
$888$	0	0
$889$	−18.4187	−0.617745
$890$	3.44073	0.115334
$891$	0	0
$892$	−6.02212	−0.201635
$893$	8.03492	0.268878
$894$	0	0
$895$	−18.1579	−0.606951
$896$	26.5611	0.887345
$897$	0	0
$898$	−1.54491	−0.0515542
$899$	−20.2883	−0.676652
$900$	0	0
$901$	13.5640	0.451882
$902$	−8.49451	−0.282836
$903$	0	0
$904$	−2.46390	−0.0819480
$905$	25.4502	0.845992
$906$	0	0
$907$	12.6641	0.420504	0.210252	−	0.977647i	$-0.432572\pi$
$907$	12.6641	0.420504	0.210252	+	0.977647i	$0.432572\pi$
$908$	17.6288	0.585032
$909$	0	0
$910$	1.02743	0.0340591
$911$	−46.8797	−1.55319	−0.776597	−	0.629997i	$-0.783056\pi$
$911$	−46.8797	−1.55319	−0.776597	+	0.629997i	$0.783056\pi$
$912$	0	0
$913$	−67.1205	−2.22136
$914$	8.73070	0.288786
$915$	0	0
$916$	−7.51900	−0.248435
$917$	34.9873	1.15538
$918$	0	0
$919$	23.5810	0.777867	0.388933	−	0.921266i	$-0.372844\pi$
$919$	23.5810	0.777867	0.388933	+	0.921266i	$0.372844\pi$
$920$	−2.89890	−0.0955741
$921$	0	0
$922$	7.33410	0.241536
$923$	−13.2535	−0.436244
$924$	0	0
$925$	6.18373	0.203320
$926$	0.192676	0.00633172
$927$	0	0
$928$	−15.4307	−0.506538
$929$	−53.0861	−1.74170	−0.870850	−	0.491549i	$-0.836431\pi$
$929$	−53.0861	−1.74170	−0.870850	+	0.491549i	$0.836431\pi$
$930$	0	0
$931$	1.33266	0.0436762
$932$	−14.5142	−0.475427
$933$	0	0
$934$	−0.570574	−0.0186697
$935$	9.21701	0.301429
$936$	0	0
$937$	21.6222	0.706367	0.353184	−	0.935554i	$-0.385099\pi$
$937$	21.6222	0.706367	0.353184	+	0.935554i	$0.385099\pi$
$938$	−9.47517	−0.309375
$939$	0	0
$940$	−5.10086	−0.166372
$941$	42.1438	1.37385	0.686924	−	0.726729i	$-0.258960\pi$
$941$	42.1438	1.37385	0.686924	+	0.726729i	$0.258960\pi$
$942$	0	0
$943$	−6.26062	−0.203874
$944$	−34.0956	−1.10972
$945$	0	0
$946$	24.1418	0.784919
$947$	−59.1465	−1.92200	−0.961002	−	0.276541i	$-0.910812\pi$
$947$	−59.1465	−1.92200	−0.961002	+	0.276541i	$0.910812\pi$
$948$	0	0
$949$	14.8993	0.483653
$950$	1.16357	0.0377512
$951$	0	0
$952$	5.72628	0.185590
$953$	35.4266	1.14758	0.573790	−	0.819002i	$-0.305472\pi$
$953$	35.4266	1.14758	0.573790	+	0.819002i	$0.305472\pi$
$954$	0	0
$955$	−0.717570	−0.0232200
$956$	25.6012	0.828004
$957$	0	0
$958$	−9.30901	−0.300760
$959$	34.6826	1.11996
$960$	0	0
$961$	1.15453	0.0372430
$962$	−2.48439	−0.0801001
$963$	0	0
$964$	34.3407	1.10604
$965$	−11.8483	−0.381411
$966$	0	0
$967$	−16.5235	−0.531359	−0.265680	−	0.964061i	$-0.585596\pi$
$967$	−16.5235	−0.531359	−0.265680	+	0.964061i	$0.585596\pi$
$968$	45.1839	1.45227
$969$	0	0
$970$	−3.04327	−0.0977135
$971$	−36.0915	−1.15823	−0.579115	−	0.815246i	$-0.696602\pi$
$971$	−36.0915	−1.15823	−0.579115	+	0.815246i	$0.696602\pi$
$972$	0	0
$973$	−27.3640	−0.877250
$974$	1.73429	0.0555702
$975$	0	0
$976$	23.9478	0.766550
$977$	−40.7560	−1.30390	−0.651950	−	0.758262i	$-0.726049\pi$
$977$	−40.7560	−1.30390	−0.651950	+	0.758262i	$0.726049\pi$
$978$	0	0
$979$	−54.3656	−1.73753
$980$	−0.846022	−0.0270252
$981$	0	0
$982$	11.3447	0.362025
$983$	−41.9976	−1.33952	−0.669758	−	0.742579i	$-0.733602\pi$
$983$	−41.9976	−1.33952	−0.669758	+	0.742579i	$0.733602\pi$
$984$	0	0
$985$	25.4644	0.811365
$986$	−2.08709	−0.0664664
$987$	0	0
$988$	5.32484	0.169406
$989$	17.7930	0.565785
$990$	0	0
$991$	−23.7070	−0.753078	−0.376539	−	0.926401i	$-0.622886\pi$
$991$	−23.7070	−0.753078	−0.376539	+	0.926401i	$0.622886\pi$
$992$	24.4559	0.776474
$993$	0	0
$994$	13.6171	0.431908
$995$	−6.04813	−0.191738
$996$	0	0
$997$	8.77413	0.277879	0.138940	−	0.990301i	$-0.455631\pi$
$997$	8.77413	0.277879	0.138940	+	0.990301i	$0.455631\pi$
$998$	12.9026	0.408423
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1755.2.a.u.1.4	✓	7
3.2	odd	2		1755.2.a.v.1.4	yes	7
5.4	even	2		8775.2.a.bx.1.4		7
15.14	odd	2		8775.2.a.bw.1.4		7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1755.2.a.u.1.4	✓	7	1.1	even	1	trivial
1755.2.a.v.1.4	yes	7	3.2	odd	2
8775.2.a.bw.1.4		7	15.14	odd	2
8775.2.a.bx.1.4		7	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)

\(f(q)\)	\(=\)	\(q-0.401763 q^{2} -1.83859 q^{4} -1.00000 q^{5} +2.55731 q^{7} +1.54220 q^{8} +O(q^{10})\)	\(q-0.401763 q^{2} -1.83859 q^{4} -1.00000 q^{5} +2.55731 q^{7} +1.54220 q^{8} +0.401763 q^{10} -6.34809 q^{11} +1.00000 q^{13} -1.02743 q^{14} +3.05757 q^{16} +1.45193 q^{17} -2.89616 q^{19} +1.83859 q^{20} +2.55043 q^{22} +1.87972 q^{23} +1.00000 q^{25} -0.401763 q^{26} -4.70184 q^{28} +3.57786 q^{29} -5.67050 q^{31} -4.31283 q^{32} -0.583334 q^{34} -2.55731 q^{35} +6.18373 q^{37} +1.16357 q^{38} -1.54220 q^{40} -3.33062 q^{41} +9.46580 q^{43} +11.6715 q^{44} -0.755201 q^{46} -2.77434 q^{47} -0.460148 q^{49} -0.401763 q^{50} -1.83859 q^{52} +9.34202 q^{53} +6.34809 q^{55} +3.94390 q^{56} -1.43745 q^{58} -11.1512 q^{59} +7.83229 q^{61} +2.27820 q^{62} -4.38241 q^{64} -1.00000 q^{65} +9.22216 q^{67} -2.66951 q^{68} +1.02743 q^{70} -13.2535 q^{71} +14.8993 q^{73} -2.48439 q^{74} +5.32484 q^{76} -16.2341 q^{77} +4.01232 q^{79} -3.05757 q^{80} +1.33812 q^{82} +10.5733 q^{83} -1.45193 q^{85} -3.80301 q^{86} -9.79005 q^{88} +8.56408 q^{89} +2.55731 q^{91} -3.45602 q^{92} +1.11463 q^{94} +2.89616 q^{95} -7.57479 q^{97} +0.184871 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - q^{2} + 11 q^{4} - 7 q^{5} + 2 q^{7} - 6 q^{8}+O(q^{10}) \) 7 * q - q^2 + 11 * q^4 - 7 * q^5 + 2 * q^7 - 6 * q^8	\( 7 q - q^{2} + 11 q^{4} - 7 q^{5} + 2 q^{7} - 6 q^{8} + q^{10} - q^{11} + 7 q^{13} + 12 q^{14} + 23 q^{16} - 11 q^{17} + 2 q^{19} - 11 q^{20} + 16 q^{22} + q^{23} + 7 q^{25} - q^{26} + 10 q^{28} + 4 q^{29} - 13 q^{32} + q^{34} - 2 q^{35} + 23 q^{37} + 15 q^{38} + 6 q^{40} - 2 q^{41} + 8 q^{43} - 10 q^{44} + 37 q^{46} - 2 q^{47} + 43 q^{49} - q^{50} + 11 q^{52} - 10 q^{53} + q^{55} + 68 q^{56} - 26 q^{58} + 13 q^{59} + 21 q^{61} - 9 q^{62} + 46 q^{64} - 7 q^{65} + 21 q^{67} - 53 q^{68} - 12 q^{70} + 10 q^{71} + 13 q^{73} + 68 q^{74} - 41 q^{76} - 6 q^{77} + 8 q^{79} - 23 q^{80} - 26 q^{82} + 4 q^{83} + 11 q^{85} + 12 q^{86} + 44 q^{88} + 27 q^{89} + 2 q^{91} + 9 q^{92} - 24 q^{94} - 2 q^{95} - 15 q^{97} - 7 q^{98}+O(q^{100}) \) 7 * q - q^2 + 11 * q^4 - 7 * q^5 + 2 * q^7 - 6 * q^8 + q^10 - q^11 + 7 * q^13 + 12 * q^14 + 23 * q^16 - 11 * q^17 + 2 * q^19 - 11 * q^20 + 16 * q^22 + q^23 + 7 * q^25 - q^26 + 10 * q^28 + 4 * q^29 - 13 * q^32 + q^34 - 2 * q^35 + 23 * q^37 + 15 * q^38 + 6 * q^40 - 2 * q^41 + 8 * q^43 - 10 * q^44 + 37 * q^46 - 2 * q^47 + 43 * q^49 - q^50 + 11 * q^52 - 10 * q^53 + q^55 + 68 * q^56 - 26 * q^58 + 13 * q^59 + 21 * q^61 - 9 * q^62 + 46 * q^64 - 7 * q^65 + 21 * q^67 - 53 * q^68 - 12 * q^70 + 10 * q^71 + 13 * q^73 + 68 * q^74 - 41 * q^76 - 6 * q^77 + 8 * q^79 - 23 * q^80 - 26 * q^82 + 4 * q^83 + 11 * q^85 + 12 * q^86 + 44 * q^88 + 27 * q^89 + 2 * q^91 + 9 * q^92 - 24 * q^94 - 2 * q^95 - 15 * q^97 - 7 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more