LMFDB - Embedded newform 1755.2.a.s.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:	$0$
Dimension:	$4$
Coefficient field:	4.4.1957.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 4x^{2} - x + 1 $ x^4 - 4*x^2 - x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.76401$ 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)

$f(q)$	$=$	$q+1.56689 q^{2} +0.455140 q^{4} -1.00000 q^{5} +4.83690 q^{7} -2.42062 q^{8} +O(q^{10})$	\(q+1.56689 q^{2} +0.455140 q^{4} -1.00000 q^{5} +4.83690 q^{7} -2.42062 q^{8} -1.56689 q^{10} +3.41628 q^{11} +1.00000 q^{13} +7.57889 q^{14} -4.70313 q^{16} +4.74199 q^{17} -6.35974 q^{19} -0.455140 q^{20} +5.35293 q^{22} +1.93665 q^{23} +1.00000 q^{25} +1.56689 q^{26} +2.20147 q^{28} +8.29204 q^{29} -10.0849 q^{31} -2.52803 q^{32} +7.43016 q^{34} -4.83690 q^{35} -2.80287 q^{37} -9.96500 q^{38} +2.42062 q^{40} +0.148727 q^{41} +6.32276 q^{43} +1.55489 q^{44} +3.03452 q^{46} +10.1942 q^{47} +16.3956 q^{49} +1.56689 q^{50} +0.455140 q^{52} +13.1626 q^{53} -3.41628 q^{55} -11.7083 q^{56} +12.9927 q^{58} +1.52037 q^{59} +3.96360 q^{61} -15.8019 q^{62} +5.44511 q^{64} -1.00000 q^{65} +10.9098 q^{67} +2.15827 q^{68} -7.57889 q^{70} -7.36247 q^{71} -10.6637 q^{73} -4.39179 q^{74} -2.89457 q^{76} +16.5242 q^{77} +1.63838 q^{79} +4.70313 q^{80} +0.233038 q^{82} -9.81241 q^{83} -4.74199 q^{85} +9.90706 q^{86} -8.26953 q^{88} +3.90648 q^{89} +4.83690 q^{91} +0.881447 q^{92} +15.9731 q^{94} +6.35974 q^{95} -6.56935 q^{97} +25.6902 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 3 q^{2} + 3 q^{4} - 4 q^{5} - 3 q^{7} + 3 q^{8}+O(q^{10}) $ 4 * q + 3 * q^2 + 3 * q^4 - 4 * q^5 - 3 * q^7 + 3 * q^8	$ 4 q + 3 q^{2} + 3 q^{4} - 4 q^{5} - 3 q^{7} + 3 q^{8} - 3 q^{10} + 4 q^{11} + 4 q^{13} + 3 q^{14} - 3 q^{16} + 14 q^{17} - 4 q^{19} - 3 q^{20} + 9 q^{22} + 5 q^{23} + 4 q^{25} + 3 q^{26} + 2 q^{28} + 12 q^{29} - q^{31} + 4 q^{32} - q^{34} + 3 q^{35} - 15 q^{37} + 14 q^{38} - 3 q^{40} + 4 q^{41} - 4 q^{43} + 27 q^{44} + 26 q^{46} + 3 q^{47} + 21 q^{49} + 3 q^{50} + 3 q^{52} + 35 q^{53} - 4 q^{55} - 16 q^{56} + 21 q^{58} + 13 q^{59} - q^{61} + 13 q^{62} + q^{64} - 4 q^{65} + 6 q^{67} - 6 q^{68} - 3 q^{70} + q^{71} - 25 q^{73} - 16 q^{74} + 33 q^{76} + 30 q^{77} + 25 q^{79} + 3 q^{80} - 26 q^{82} - 25 q^{83} - 14 q^{85} + 11 q^{86} + 26 q^{88} - 4 q^{89} - 3 q^{91} + 17 q^{92} + 29 q^{94} + 4 q^{95} - 17 q^{97} + 31 q^{98}+O(q^{100}) $ 4 * q + 3 * q^2 + 3 * q^4 - 4 * q^5 - 3 * q^7 + 3 * q^8 - 3 * q^10 + 4 * q^11 + 4 * q^13 + 3 * q^14 - 3 * q^16 + 14 * q^17 - 4 * q^19 - 3 * q^20 + 9 * q^22 + 5 * q^23 + 4 * q^25 + 3 * q^26 + 2 * q^28 + 12 * q^29 - q^31 + 4 * q^32 - q^34 + 3 * q^35 - 15 * q^37 + 14 * q^38 - 3 * q^40 + 4 * q^41 - 4 * q^43 + 27 * q^44 + 26 * q^46 + 3 * q^47 + 21 * q^49 + 3 * q^50 + 3 * q^52 + 35 * q^53 - 4 * q^55 - 16 * q^56 + 21 * q^58 + 13 * q^59 - q^61 + 13 * q^62 + q^64 - 4 * q^65 + 6 * q^67 - 6 * q^68 - 3 * q^70 + q^71 - 25 * q^73 - 16 * q^74 + 33 * q^76 + 30 * q^77 + 25 * q^79 + 3 * q^80 - 26 * q^82 - 25 * q^83 - 14 * q^85 + 11 * q^86 + 26 * q^88 - 4 * q^89 - 3 * q^91 + 17 * q^92 + 29 * q^94 + 4 * q^95 - 17 * q^97 + 31 * 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 $

$2$	1.56689	1.10796	0.553979	−	0.832531i	$-0.313109\pi$
$2$	1.56689	1.10796	0.553979	+	0.832531i	$0.313109\pi$
$3$	0	0
$3$	0	0
$4$	0.455140	0.227570
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	4.83690	1.82818	0.914089	−	0.405513i	$-0.132907\pi$
$7$	4.83690	1.82818	0.914089	+	0.405513i	$0.132907\pi$
$8$	−2.42062	−0.855820
$9$	0	0
$10$	−1.56689	−0.495494
$11$	3.41628	1.03005	0.515024	−	0.857176i	$-0.327783\pi$
$11$	3.41628	1.03005	0.515024	+	0.857176i	$0.327783\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	7.57889	2.02554
$15$	0	0
$16$	−4.70313	−1.17578
$17$	4.74199	1.15010	0.575050	−	0.818118i	$-0.304982\pi$
$17$	4.74199	1.15010	0.575050	+	0.818118i	$0.304982\pi$
$18$	0	0
$19$	−6.35974	−1.45902	−0.729512	−	0.683968i	$-0.760253\pi$
$19$	−6.35974	−1.45902	−0.729512	+	0.683968i	$0.760253\pi$
$20$	−0.455140	−0.101772
$21$	0	0
$22$	5.35293	1.14125
$23$	1.93665	0.403820	0.201910	−	0.979404i	$-0.435285\pi$
$23$	1.93665	0.403820	0.201910	+	0.979404i	$0.435285\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	1.56689	0.307292
$27$	0	0
$28$	2.20147	0.416038
$29$	8.29204	1.53979	0.769897	−	0.638168i	$-0.220308\pi$
$29$	8.29204	1.53979	0.769897	+	0.638168i	$0.220308\pi$
$30$	0	0
$31$	−10.0849	−1.81130	−0.905650	−	0.424026i	$-0.860617\pi$
$31$	−10.0849	−1.81130	−0.905650	+	0.424026i	$0.860617\pi$
$32$	−2.52803	−0.446897
$33$	0	0
$34$	7.43016	1.27426
$35$	−4.83690	−0.817586
$36$	0	0
$37$	−2.80287	−0.460790	−0.230395	−	0.973097i	$-0.574002\pi$
$37$	−2.80287	−0.460790	−0.230395	+	0.973097i	$0.574002\pi$
$38$	−9.96500	−1.61654
$39$	0	0
$40$	2.42062	0.382734
$41$	0.148727	0.0232272	0.0116136	−	0.999933i	$-0.496303\pi$
$41$	0.148727	0.0232272	0.0116136	+	0.999933i	$0.496303\pi$
$42$	0	0
$43$	6.32276	0.964212	0.482106	−	0.876113i	$-0.339872\pi$
$43$	6.32276	0.964212	0.482106	+	0.876113i	$0.339872\pi$
$44$	1.55489	0.234408
$45$	0	0
$46$	3.03452	0.447415
$47$	10.1942	1.48697	0.743487	−	0.668750i	$-0.233171\pi$
$47$	10.1942	1.48697	0.743487	+	0.668750i	$0.233171\pi$
$48$	0	0
$49$	16.3956	2.34224
$50$	1.56689	0.221592
$51$	0	0
$52$	0.455140	0.0631166
$53$	13.1626	1.80802	0.904012	−	0.427508i	$-0.140608\pi$
$53$	13.1626	1.80802	0.904012	+	0.427508i	$0.140608\pi$
$54$	0	0
$55$	−3.41628	−0.460651
$56$	−11.7083	−1.56459
$57$	0	0
$58$	12.9927	1.70603
$59$	1.52037	0.197935	0.0989676	−	0.995091i	$-0.468446\pi$
$59$	1.52037	0.197935	0.0989676	+	0.995091i	$0.468446\pi$
$60$	0	0
$61$	3.96360	0.507487	0.253744	−	0.967271i	$-0.418338\pi$
$61$	3.96360	0.507487	0.253744	+	0.967271i	$0.418338\pi$
$62$	−15.8019	−2.00684
$63$	0	0
$64$	5.44511	0.680639
$65$	−1.00000	−0.124035
$66$	0	0
$67$	10.9098	1.33284	0.666422	−	0.745575i	$-0.267825\pi$
$67$	10.9098	1.33284	0.666422	+	0.745575i	$0.267825\pi$
$68$	2.15827	0.261728
$69$	0	0
$70$	−7.57889	−0.905851
$71$	−7.36247	−0.873765	−0.436882	−	0.899519i	$-0.643917\pi$
$71$	−7.36247	−0.873765	−0.436882	+	0.899519i	$0.643917\pi$
$72$	0	0
$73$	−10.6637	−1.24809	−0.624045	−	0.781388i	$-0.714512\pi$
$73$	−10.6637	−1.24809	−0.624045	+	0.781388i	$0.714512\pi$
$74$	−4.39179	−0.510535
$75$	0	0
$76$	−2.89457	−0.332030
$77$	16.5242	1.88311
$78$	0	0
$79$	1.63838	0.184332	0.0921662	−	0.995744i	$-0.470621\pi$
$79$	1.63838	0.184332	0.0921662	+	0.995744i	$0.470621\pi$
$80$	4.70313	0.525826
$81$	0	0
$82$	0.233038	0.0257348
$83$	−9.81241	−1.07705	−0.538526	−	0.842609i	$-0.681019\pi$
$83$	−9.81241	−1.07705	−0.538526	+	0.842609i	$0.681019\pi$
$84$	0	0
$85$	−4.74199	−0.514341
$86$	9.90706	1.06831
$87$	0	0
$88$	−8.26953	−0.881535
$89$	3.90648	0.414086	0.207043	−	0.978332i	$-0.433616\pi$
$89$	3.90648	0.414086	0.207043	+	0.978332i	$0.433616\pi$
$90$	0	0
$91$	4.83690	0.507045
$92$	0.881447	0.0918972
$93$	0	0
$94$	15.9731	1.64750
$95$	6.35974	0.652495
$96$	0	0
$97$	−6.56935	−0.667016	−0.333508	−	0.942747i	$-0.608232\pi$
$97$	−6.56935	−0.667016	−0.333508	+	0.942747i	$0.608232\pi$
$98$	25.6902	2.59510
$99$	0	0
$100$	0.455140	0.0455140
$101$	−6.89731	−0.686308	−0.343154	−	0.939279i	$-0.611495\pi$
$101$	−6.89731	−0.686308	−0.343154	+	0.939279i	$0.611495\pi$
$102$	0	0
$103$	6.44217	0.634766	0.317383	−	0.948298i	$-0.397196\pi$
$103$	6.44217	0.634766	0.317383	+	0.948298i	$0.397196\pi$
$104$	−2.42062	−0.237362
$105$	0	0
$106$	20.6243	2.00321
$107$	15.2239	1.47175	0.735873	−	0.677119i	$-0.236772\pi$
$107$	15.2239	1.47175	0.735873	+	0.677119i	$0.236772\pi$
$108$	0	0
$109$	−17.3745	−1.66417	−0.832086	−	0.554646i	$-0.812854\pi$
$109$	−17.3745	−1.66417	−0.832086	+	0.554646i	$0.812854\pi$
$110$	−5.35293	−0.510382
$111$	0	0
$112$	−22.7486	−2.14954
$113$	−10.9922	−1.03406	−0.517031	−	0.855967i	$-0.672963\pi$
$113$	−10.9922	−1.03406	−0.517031	+	0.855967i	$0.672963\pi$
$114$	0	0
$115$	−1.93665	−0.180594
$116$	3.77404	0.350411
$117$	0	0
$118$	2.38225	0.219304
$119$	22.9365	2.10259
$120$	0	0
$121$	0.670977	0.0609979
$122$	6.21052	0.562274
$123$	0	0
$124$	−4.59004	−0.412198
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−12.9160	−1.14611	−0.573056	−	0.819516i	$-0.694242\pi$
$127$	−12.9160	−1.14611	−0.573056	+	0.819516i	$0.694242\pi$
$128$	13.5879	1.20102
$129$	0	0
$130$	−1.56689	−0.137425
$131$	−7.28144	−0.636182	−0.318091	−	0.948060i	$-0.603042\pi$
$131$	−7.28144	−0.636182	−0.318091	+	0.948060i	$0.603042\pi$
$132$	0	0
$133$	−30.7614	−2.66736
$134$	17.0944	1.47673
$135$	0	0
$136$	−11.4786	−0.984279
$137$	−7.67295	−0.655545	−0.327772	−	0.944757i	$-0.606298\pi$
$137$	−7.67295	−0.655545	−0.327772	+	0.944757i	$0.606298\pi$
$138$	0	0
$139$	−8.30159	−0.704131	−0.352066	−	0.935975i	$-0.614521\pi$
$139$	−8.30159	−0.704131	−0.352066	+	0.935975i	$0.614521\pi$
$140$	−2.20147	−0.186058
$141$	0	0
$142$	−11.5362	−0.968094
$143$	3.41628	0.285684
$144$	0	0
$145$	−8.29204	−0.688617
$146$	−16.7088	−1.38283
$147$	0	0
$148$	−1.27570	−0.104862
$149$	−23.5347	−1.92804	−0.964020	−	0.265831i	$-0.914354\pi$
$149$	−23.5347	−1.92804	−0.964020	+	0.265831i	$0.914354\pi$
$150$	0	0
$151$	−9.02777	−0.734669	−0.367335	−	0.930089i	$-0.619730\pi$
$151$	−9.02777	−0.734669	−0.367335	+	0.930089i	$0.619730\pi$
$152$	15.3945	1.24866
$153$	0	0
$154$	25.8916	2.08641
$155$	10.0849	0.810038
$156$	0	0
$157$	5.83359	0.465571	0.232785	−	0.972528i	$-0.425216\pi$
$157$	5.83359	0.465571	0.232785	+	0.972528i	$0.425216\pi$
$158$	2.56716	0.204233
$159$	0	0
$160$	2.52803	0.199858
$161$	9.36740	0.738254
$162$	0	0
$163$	2.61255	0.204631	0.102315	−	0.994752i	$-0.467375\pi$
$163$	2.61255	0.204631	0.102315	+	0.994752i	$0.467375\pi$
$164$	0.0676915	0.00528581
$165$	0	0
$166$	−15.3750	−1.19333
$167$	−17.2138	−1.33205	−0.666024	−	0.745931i	$-0.732005\pi$
$167$	−17.2138	−1.33205	−0.666024	+	0.745931i	$0.732005\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−7.43016	−0.569868
$171$	0	0
$172$	2.87774	0.219426
$173$	−14.1687	−1.07722	−0.538612	−	0.842554i	$-0.681051\pi$
$173$	−14.1687	−1.07722	−0.538612	+	0.842554i	$0.681051\pi$
$174$	0	0
$175$	4.83690	0.365636
$176$	−16.0672	−1.21111
$177$	0	0
$178$	6.12101	0.458789
$179$	8.11175	0.606301	0.303150	−	0.952943i	$-0.401962\pi$
$179$	8.11175	0.606301	0.303150	+	0.952943i	$0.401962\pi$
$180$	0	0
$181$	16.8557	1.25288	0.626438	−	0.779472i	$-0.284512\pi$
$181$	16.8557	1.25288	0.626438	+	0.779472i	$0.284512\pi$
$182$	7.57889	0.561785
$183$	0	0
$184$	−4.68790	−0.345597
$185$	2.80287	0.206071
$186$	0	0
$187$	16.2000	1.18466
$188$	4.63978	0.338391
$189$	0	0
$190$	9.96500	0.722937
$191$	−3.73330	−0.270132	−0.135066	−	0.990837i	$-0.543125\pi$
$191$	−3.73330	−0.270132	−0.135066	+	0.990837i	$0.543125\pi$
$192$	0	0
$193$	14.7010	1.05820	0.529102	−	0.848558i	$-0.322529\pi$
$193$	14.7010	1.05820	0.529102	+	0.848558i	$0.322529\pi$
$194$	−10.2934	−0.739026
$195$	0	0
$196$	7.46231	0.533022
$197$	21.3606	1.52188	0.760940	−	0.648823i	$-0.224738\pi$
$197$	21.3606	1.52188	0.760940	+	0.648823i	$0.224738\pi$
$198$	0	0
$199$	−3.22435	−0.228568	−0.114284	−	0.993448i	$-0.536457\pi$
$199$	−3.22435	−0.228568	−0.114284	+	0.993448i	$0.536457\pi$
$200$	−2.42062	−0.171164
$201$	0	0
$202$	−10.8073	−0.760400
$203$	40.1078	2.81502
$204$	0	0
$205$	−0.148727	−0.0103875
$206$	10.0942	0.703293
$207$	0	0
$208$	−4.70313	−0.326103
$209$	−21.7266	−1.50286
$210$	0	0
$211$	−21.1156	−1.45366	−0.726829	−	0.686818i	$-0.759007\pi$
$211$	−21.1156	−1.45366	−0.726829	+	0.686818i	$0.759007\pi$
$212$	5.99083	0.411452
$213$	0	0
$214$	23.8541	1.63063
$215$	−6.32276	−0.431209
$216$	0	0
$217$	−48.7797	−3.31138
$218$	−27.2239	−1.84383
$219$	0	0
$220$	−1.55489	−0.104830
$221$	4.74199	0.318981
$222$	0	0
$223$	1.45675	0.0975509	0.0487755	−	0.998810i	$-0.484468\pi$
$223$	1.45675	0.0975509	0.0487755	+	0.998810i	$0.484468\pi$
$224$	−12.2278	−0.817007
$225$	0	0
$226$	−17.2236	−1.14570
$227$	7.25270	0.481379	0.240689	−	0.970602i	$-0.422627\pi$
$227$	7.25270	0.481379	0.240689	+	0.970602i	$0.422627\pi$
$228$	0	0
$229$	−16.5178	−1.09153	−0.545763	−	0.837939i	$-0.683760\pi$
$229$	−16.5178	−1.09153	−0.545763	+	0.837939i	$0.683760\pi$
$230$	−3.03452	−0.200090
$231$	0	0
$232$	−20.0719	−1.31779
$233$	−15.1975	−0.995621	−0.497811	−	0.867286i	$-0.665863\pi$
$233$	−15.1975	−0.995621	−0.497811	+	0.867286i	$0.665863\pi$
$234$	0	0
$235$	−10.1942	−0.664995
$236$	0.691981	0.0450441
$237$	0	0
$238$	35.9390	2.32958
$239$	−11.2855	−0.729999	−0.364999	−	0.931008i	$-0.618931\pi$
$239$	−11.2855	−0.729999	−0.364999	+	0.931008i	$0.618931\pi$
$240$	0	0
$241$	19.6018	1.26266	0.631332	−	0.775512i	$-0.282508\pi$
$241$	19.6018	1.26266	0.631332	+	0.775512i	$0.282508\pi$
$242$	1.05135	0.0675831
$243$	0	0
$244$	1.80399	0.115489
$245$	−16.3956	−1.04748
$246$	0	0
$247$	−6.35974	−0.404660
$248$	24.4117	1.55015
$249$	0	0
$250$	−1.56689	−0.0990987
$251$	−11.8277	−0.746560	−0.373280	−	0.927719i	$-0.621767\pi$
$251$	−11.8277	−0.746560	−0.373280	+	0.927719i	$0.621767\pi$
$252$	0	0
$253$	6.61614	0.415953
$254$	−20.2380	−1.26984
$255$	0	0
$256$	10.4006	0.650036
$257$	28.3217	1.76666	0.883331	−	0.468751i	$-0.155296\pi$
$257$	28.3217	1.76666	0.883331	+	0.468751i	$0.155296\pi$
$258$	0	0
$259$	−13.5572	−0.842405
$260$	−0.455140	−0.0282266
$261$	0	0
$262$	−11.4092	−0.704863
$263$	−1.23787	−0.0763301	−0.0381650	−	0.999271i	$-0.512151\pi$
$263$	−1.23787	−0.0763301	−0.0381650	+	0.999271i	$0.512151\pi$
$264$	0	0
$265$	−13.1626	−0.808573
$266$	−48.1997	−2.95532
$267$	0	0
$268$	4.96548	0.303315
$269$	11.1928	0.682436	0.341218	−	0.939984i	$-0.389161\pi$
$269$	11.1928	0.682436	0.341218	+	0.939984i	$0.389161\pi$
$270$	0	0
$271$	12.1776	0.739734	0.369867	−	0.929085i	$-0.379403\pi$
$271$	12.1776	0.739734	0.369867	+	0.929085i	$0.379403\pi$
$272$	−22.3022	−1.35227
$273$	0	0
$274$	−12.0227	−0.726316
$275$	3.41628	0.206010
$276$	0	0
$277$	−14.7097	−0.883822	−0.441911	−	0.897059i	$-0.645699\pi$
$277$	−14.7097	−0.883822	−0.441911	+	0.897059i	$0.645699\pi$
$278$	−13.0077	−0.780148
$279$	0	0
$280$	11.7083	0.699706
$281$	−16.9141	−1.00901	−0.504506	−	0.863408i	$-0.668326\pi$
$281$	−16.9141	−1.00901	−0.504506	+	0.863408i	$0.668326\pi$
$282$	0	0
$283$	−1.14058	−0.0678005	−0.0339003	−	0.999425i	$-0.510793\pi$
$283$	−1.14058	−0.0678005	−0.0339003	+	0.999425i	$0.510793\pi$
$284$	−3.35096	−0.198843
$285$	0	0
$286$	5.35293	0.316526
$287$	0.719377	0.0424635
$288$	0	0
$289$	5.48643	0.322731
$290$	−12.9927	−0.762958
$291$	0	0
$292$	−4.85347	−0.284028
$293$	−4.77356	−0.278874	−0.139437	−	0.990231i	$-0.544529\pi$
$293$	−4.77356	−0.278874	−0.139437	+	0.990231i	$0.544529\pi$
$294$	0	0
$295$	−1.52037	−0.0885193
$296$	6.78470	0.394353
$297$	0	0
$298$	−36.8763	−2.13619
$299$	1.93665	0.111999
$300$	0	0
$301$	30.5826	1.76275
$302$	−14.1455	−0.813982
$303$	0	0
$304$	29.9107	1.71549
$305$	−3.96360	−0.226955
$306$	0	0
$307$	−8.14958	−0.465121	−0.232561	−	0.972582i	$-0.574710\pi$
$307$	−8.14958	−0.465121	−0.232561	+	0.972582i	$0.574710\pi$
$308$	7.52084	0.428539
$309$	0	0
$310$	15.8019	0.897488
$311$	−16.4214	−0.931174	−0.465587	−	0.885002i	$-0.654157\pi$
$311$	−16.4214	−0.931174	−0.465587	+	0.885002i	$0.654157\pi$
$312$	0	0
$313$	−27.4550	−1.55185	−0.775925	−	0.630825i	$-0.782716\pi$
$313$	−27.4550	−1.55185	−0.775925	+	0.630825i	$0.782716\pi$
$314$	9.14058	0.515833
$315$	0	0
$316$	0.745694	0.0419485
$317$	8.60714	0.483425	0.241713	−	0.970348i	$-0.422291\pi$
$317$	8.60714	0.483425	0.241713	+	0.970348i	$0.422291\pi$
$318$	0	0
$319$	28.3280	1.58606
$320$	−5.44511	−0.304391
$321$	0	0
$322$	14.6777	0.817954
$323$	−30.1578	−1.67802
$324$	0	0
$325$	1.00000	0.0554700
$326$	4.09358	0.226722
$327$	0	0
$328$	−0.360011	−0.0198783
$329$	49.3083	2.71845
$330$	0	0
$331$	7.11797	0.391239	0.195620	−	0.980680i	$-0.437328\pi$
$331$	7.11797	0.391239	0.195620	+	0.980680i	$0.437328\pi$
$332$	−4.46602	−0.245105
$333$	0	0
$334$	−26.9722	−1.47585
$335$	−10.9098	−0.596066
$336$	0	0
$337$	−29.1170	−1.58610	−0.793052	−	0.609154i	$-0.791509\pi$
$337$	−29.1170	−1.58610	−0.793052	+	0.609154i	$0.791509\pi$
$338$	1.56689	0.0852275
$339$	0	0
$340$	−2.15827	−0.117048
$341$	−34.4528	−1.86573
$342$	0	0
$343$	45.4459	2.45385
$344$	−15.3050	−0.825191
$345$	0	0
$346$	−22.2007	−1.19352
$347$	5.54684	0.297770	0.148885	−	0.988855i	$-0.452432\pi$
$347$	5.54684	0.297770	0.148885	+	0.988855i	$0.452432\pi$
$348$	0	0
$349$	−7.98354	−0.427349	−0.213675	−	0.976905i	$-0.568543\pi$
$349$	−7.98354	−0.427349	−0.213675	+	0.976905i	$0.568543\pi$
$350$	7.57889	0.405109
$351$	0	0
$352$	−8.63646	−0.460325
$353$	−33.0263	−1.75781	−0.878905	−	0.476997i	$-0.841725\pi$
$353$	−33.0263	−1.75781	−0.878905	+	0.476997i	$0.841725\pi$
$354$	0	0
$355$	7.36247	0.390759
$356$	1.77799	0.0942335
$357$	0	0
$358$	12.7102	0.671755
$359$	6.03318	0.318419	0.159209	−	0.987245i	$-0.449105\pi$
$359$	6.03318	0.318419	0.159209	+	0.987245i	$0.449105\pi$
$360$	0	0
$361$	21.4462	1.12875
$362$	26.4110	1.38813
$363$	0	0
$364$	2.20147	0.115388
$365$	10.6637	0.558163
$366$	0	0
$367$	0.523202	0.0273109	0.0136555	−	0.999907i	$-0.495653\pi$
$367$	0.523202	0.0273109	0.0136555	+	0.999907i	$0.495653\pi$
$368$	−9.10832	−0.474804
$369$	0	0
$370$	4.39179	0.228318
$371$	63.6663	3.30539
$372$	0	0
$373$	17.0579	0.883227	0.441613	−	0.897205i	$-0.354406\pi$
$373$	17.0579	0.883227	0.441613	+	0.897205i	$0.354406\pi$
$374$	25.3835	1.31255
$375$	0	0
$376$	−24.6763	−1.27258
$377$	8.29204	0.427062
$378$	0	0
$379$	10.6042	0.544701	0.272350	−	0.962198i	$-0.412199\pi$
$379$	10.6042	0.544701	0.272350	+	0.962198i	$0.412199\pi$
$380$	2.89457	0.148488
$381$	0	0
$382$	−5.84967	−0.299295
$383$	−1.33309	−0.0681177	−0.0340589	−	0.999420i	$-0.510843\pi$
$383$	−1.33309	−0.0681177	−0.0340589	+	0.999420i	$0.510843\pi$
$384$	0	0
$385$	−16.5242	−0.842153
$386$	23.0349	1.17244
$387$	0	0
$388$	−2.98997	−0.151793
$389$	36.5988	1.85563	0.927817	−	0.373035i	$-0.121683\pi$
$389$	36.5988	1.85563	0.927817	+	0.373035i	$0.121683\pi$
$390$	0	0
$391$	9.18357	0.464433
$392$	−39.6877	−2.00453
$393$	0	0
$394$	33.4697	1.68618
$395$	−1.63838	−0.0824360
$396$	0	0
$397$	−2.15119	−0.107965	−0.0539825	−	0.998542i	$-0.517192\pi$
$397$	−2.15119	−0.107965	−0.0539825	+	0.998542i	$0.517192\pi$
$398$	−5.05220	−0.253244
$399$	0	0
$400$	−4.70313	−0.235156
$401$	3.76750	0.188140	0.0940700	−	0.995566i	$-0.470012\pi$
$401$	3.76750	0.188140	0.0940700	+	0.995566i	$0.470012\pi$
$402$	0	0
$403$	−10.0849	−0.502364
$404$	−3.13924	−0.156183
$405$	0	0
$406$	62.8445	3.11892
$407$	−9.57540	−0.474635
$408$	0	0
$409$	−5.69964	−0.281829	−0.140915	−	0.990022i	$-0.545004\pi$
$409$	−5.69964	−0.281829	−0.140915	+	0.990022i	$0.545004\pi$
$410$	−0.233038	−0.0115089
$411$	0	0
$412$	2.93209	0.144454
$413$	7.35388	0.361861
$414$	0	0
$415$	9.81241	0.481672
$416$	−2.52803	−0.123947
$417$	0	0
$418$	−34.0432	−1.66511
$419$	9.64401	0.471141	0.235570	−	0.971857i	$-0.424304\pi$
$419$	9.64401	0.471141	0.235570	+	0.971857i	$0.424304\pi$
$420$	0	0
$421$	24.4662	1.19241	0.596204	−	0.802833i	$-0.296675\pi$
$421$	24.4662	1.19241	0.596204	+	0.802833i	$0.296675\pi$
$422$	−33.0858	−1.61059
$423$	0	0
$424$	−31.8617	−1.54734
$425$	4.74199	0.230020
$426$	0	0
$427$	19.1716	0.927777
$428$	6.92899	0.334925
$429$	0	0
$430$	−9.90706	−0.477761
$431$	−9.29290	−0.447623	−0.223812	−	0.974632i	$-0.571850\pi$
$431$	−9.29290	−0.447623	−0.223812	+	0.974632i	$0.571850\pi$
$432$	0	0
$433$	38.5109	1.85071	0.925357	−	0.379096i	$-0.123765\pi$
$433$	38.5109	1.85071	0.925357	+	0.379096i	$0.123765\pi$
$434$	−76.4323	−3.66887
$435$	0	0
$436$	−7.90782	−0.378716
$437$	−12.3166	−0.589182
$438$	0	0
$439$	−0.00895992	−0.000427634 0	−0.000213817	−	1.00000i	$-0.500068\pi$
$439$	−0.00895992	−0.000427634 0	−0.000213817		1.00000i	$0.500068\pi$
$440$	8.26953	0.394234
$441$	0	0
$442$	7.43016	0.353417
$443$	14.1123	0.670497	0.335249	−	0.942130i	$-0.391180\pi$
$443$	14.1123	0.670497	0.335249	+	0.942130i	$0.391180\pi$
$444$	0	0
$445$	−3.90648	−0.185185
$446$	2.28256	0.108082
$447$	0	0
$448$	26.3375	1.24433
$449$	−14.9778	−0.706846	−0.353423	−	0.935464i	$-0.614982\pi$
$449$	−14.9778	−0.706846	−0.353423	+	0.935464i	$0.614982\pi$
$450$	0	0
$451$	0.508092	0.0239251
$452$	−5.00300	−0.235321
$453$	0	0
$454$	11.3642	0.533347
$455$	−4.83690	−0.226758
$456$	0	0
$457$	−7.83711	−0.366605	−0.183302	−	0.983057i	$-0.558679\pi$
$457$	−7.83711	−0.366605	−0.183302	+	0.983057i	$0.558679\pi$
$458$	−25.8815	−1.20937
$459$	0	0
$460$	−0.881447	−0.0410977
$461$	13.5482	0.631004	0.315502	−	0.948925i	$-0.397827\pi$
$461$	13.5482	0.631004	0.315502	+	0.948925i	$0.397827\pi$
$462$	0	0
$463$	−11.4521	−0.532224	−0.266112	−	0.963942i	$-0.585739\pi$
$463$	−11.4521	−0.532224	−0.266112	+	0.963942i	$0.585739\pi$
$464$	−38.9985	−1.81046
$465$	0	0
$466$	−23.8128	−1.10311
$467$	−8.33471	−0.385684	−0.192842	−	0.981230i	$-0.561771\pi$
$467$	−8.33471	−0.385684	−0.192842	+	0.981230i	$0.561771\pi$
$468$	0	0
$469$	52.7696	2.43668
$470$	−15.9731	−0.736786
$471$	0	0
$472$	−3.68024	−0.169397
$473$	21.6003	0.993184
$474$	0	0
$475$	−6.35974	−0.291805
$476$	10.4393	0.478486
$477$	0	0
$478$	−17.6831	−0.808808
$479$	−24.5659	−1.12245	−0.561223	−	0.827665i	$-0.689669\pi$
$479$	−24.5659	−1.12245	−0.561223	+	0.827665i	$0.689669\pi$
$480$	0	0
$481$	−2.80287	−0.127800
$482$	30.7139	1.39898
$483$	0	0
$484$	0.305389	0.0138813
$485$	6.56935	0.298299
$486$	0	0
$487$	4.94212	0.223949	0.111975	−	0.993711i	$-0.464282\pi$
$487$	4.94212	0.223949	0.111975	+	0.993711i	$0.464282\pi$
$488$	−9.59439	−0.434318
$489$	0	0
$490$	−25.6902	−1.16056
$491$	2.01769	0.0910569	0.0455285	−	0.998963i	$-0.485503\pi$
$491$	2.01769	0.0910569	0.0455285	+	0.998963i	$0.485503\pi$
$492$	0	0
$493$	39.3208	1.77092
$494$	−9.96500	−0.448346
$495$	0	0
$496$	47.4305	2.12969
$497$	−35.6116	−1.59740
$498$	0	0
$499$	25.0134	1.11975	0.559877	−	0.828576i	$-0.310848\pi$
$499$	25.0134	1.11975	0.559877	+	0.828576i	$0.310848\pi$
$500$	−0.455140	−0.0203545
$501$	0	0
$502$	−18.5327	−0.827157
$503$	−8.34999	−0.372307	−0.186154	−	0.982521i	$-0.559602\pi$
$503$	−8.34999	−0.372307	−0.186154	+	0.982521i	$0.559602\pi$
$504$	0	0
$505$	6.89731	0.306926
$506$	10.3668	0.460859
$507$	0	0
$508$	−5.87860	−0.260821
$509$	43.9765	1.94922	0.974612	−	0.223899i	$-0.0718785\pi$
$509$	43.9765	1.94922	0.974612	+	0.223899i	$0.0718785\pi$
$510$	0	0
$511$	−51.5792	−2.28173
$512$	−10.8794	−0.480804
$513$	0	0
$514$	44.3770	1.95739
$515$	−6.44217	−0.283876
$516$	0	0
$517$	34.8262	1.53165
$518$	−21.2427	−0.933350
$519$	0	0
$520$	2.42062	0.106151
$521$	−6.19038	−0.271205	−0.135603	−	0.990763i	$-0.543297\pi$
$521$	−6.19038	−0.271205	−0.135603	+	0.990763i	$0.543297\pi$
$522$	0	0
$523$	−35.3509	−1.54579	−0.772894	−	0.634535i	$-0.781192\pi$
$523$	−35.3509	−1.54579	−0.772894	+	0.634535i	$0.781192\pi$
$524$	−3.31407	−0.144776
$525$	0	0
$526$	−1.93960	−0.0845705
$527$	−47.8224	−2.08318
$528$	0	0
$529$	−19.2494	−0.836930
$530$	−20.6243	−0.895864
$531$	0	0
$532$	−14.0008	−0.607010
$533$	0.148727	0.00644207
$534$	0	0
$535$	−15.2239	−0.658185
$536$	−26.4085	−1.14067
$537$	0	0
$538$	17.5378	0.756110
$539$	56.0121	2.41261
$540$	0	0
$541$	−27.6090	−1.18700	−0.593501	−	0.804833i	$-0.702255\pi$
$541$	−27.6090	−1.18700	−0.593501	+	0.804833i	$0.702255\pi$
$542$	19.0809	0.819594
$543$	0	0
$544$	−11.9879	−0.513976
$545$	17.3745	0.744241
$546$	0	0
$547$	−6.32893	−0.270605	−0.135303	−	0.990804i	$-0.543201\pi$
$547$	−6.32893	−0.270605	−0.135303	+	0.990804i	$0.543201\pi$
$548$	−3.49227	−0.149182
$549$	0	0
$550$	5.35293	0.228250
$551$	−52.7352	−2.24660
$552$	0	0
$553$	7.92470	0.336993
$554$	−23.0485	−0.979237
$555$	0	0
$556$	−3.77838	−0.160239
$557$	−1.36156	−0.0576913	−0.0288456	−	0.999584i	$-0.509183\pi$
$557$	−1.36156	−0.0576913	−0.0288456	+	0.999584i	$0.509183\pi$
$558$	0	0
$559$	6.32276	0.267424
$560$	22.7486	0.961303
$561$	0	0
$562$	−26.5026	−1.11794
$563$	1.72023	0.0724992	0.0362496	−	0.999343i	$-0.488459\pi$
$563$	1.72023	0.0724992	0.0362496	+	0.999343i	$0.488459\pi$
$564$	0	0
$565$	10.9922	0.462446
$566$	−1.78716	−0.0751201
$567$	0	0
$568$	17.8218	0.747785
$569$	8.82818	0.370096	0.185048	−	0.982729i	$-0.440756\pi$
$569$	8.82818	0.370096	0.185048	+	0.982729i	$0.440756\pi$
$570$	0	0
$571$	−2.52782	−0.105786	−0.0528930	−	0.998600i	$-0.516844\pi$
$571$	−2.52782	−0.105786	−0.0528930	+	0.998600i	$0.516844\pi$
$572$	1.55489	0.0650131
$573$	0	0
$574$	1.12718	0.0470477
$575$	1.93665	0.0807639
$576$	0	0
$577$	9.01613	0.375347	0.187673	−	0.982232i	$-0.439905\pi$
$577$	9.01613	0.375347	0.187673	+	0.982232i	$0.439905\pi$
$578$	8.59663	0.357573
$579$	0	0
$580$	−3.77404	−0.156709
$581$	−47.4617	−1.96904
$582$	0	0
$583$	44.9672	1.86235
$584$	25.8128	1.06814
$585$	0	0
$586$	−7.47963	−0.308981
$587$	−41.1125	−1.69690	−0.848448	−	0.529278i	$-0.822463\pi$
$587$	−41.1125	−1.69690	−0.848448	+	0.529278i	$0.822463\pi$
$588$	0	0
$589$	64.1373	2.64273
$590$	−2.38225	−0.0980757
$591$	0	0
$592$	13.1823	0.541788
$593$	20.1507	0.827490	0.413745	−	0.910393i	$-0.364220\pi$
$593$	20.1507	0.827490	0.413745	+	0.910393i	$0.364220\pi$
$594$	0	0
$595$	−22.9365	−0.940306
$596$	−10.7116	−0.438764
$597$	0	0
$598$	3.03452	0.124091
$599$	−24.5125	−1.00155	−0.500776	−	0.865577i	$-0.666952\pi$
$599$	−24.5125	−1.00155	−0.500776	+	0.865577i	$0.666952\pi$
$600$	0	0
$601$	−16.9536	−0.691553	−0.345777	−	0.938317i	$-0.612385\pi$
$601$	−16.9536	−0.691553	−0.345777	+	0.938317i	$0.612385\pi$
$602$	47.9195	1.95305
$603$	0	0
$604$	−4.10890	−0.167189
$605$	−0.670977	−0.0272791
$606$	0	0
$607$	−23.8044	−0.966189	−0.483095	−	0.875568i	$-0.660487\pi$
$607$	−23.8044	−0.966189	−0.483095	+	0.875568i	$0.660487\pi$
$608$	16.0776	0.652033
$609$	0	0
$610$	−6.21052	−0.251457
$611$	10.1942	0.412412
$612$	0	0
$613$	−15.7008	−0.634148	−0.317074	−	0.948401i	$-0.602700\pi$
$613$	−15.7008	−0.634148	−0.317074	+	0.948401i	$0.602700\pi$
$614$	−12.7695	−0.515335
$615$	0	0
$616$	−39.9989	−1.61160
$617$	30.3640	1.22241	0.611204	−	0.791473i	$-0.290685\pi$
$617$	30.3640	1.22241	0.611204	+	0.791473i	$0.290685\pi$
$618$	0	0
$619$	13.8970	0.558569	0.279284	−	0.960208i	$-0.409903\pi$
$619$	13.8970	0.558569	0.279284	+	0.960208i	$0.409903\pi$
$620$	4.59004	0.184340
$621$	0	0
$622$	−25.7306	−1.03170
$623$	18.8953	0.757023
$624$	0	0
$625$	1.00000	0.0400000
$626$	−43.0190	−1.71938
$627$	0	0
$628$	2.65510	0.105950
$629$	−13.2912	−0.529954
$630$	0	0
$631$	−5.03981	−0.200632	−0.100316	−	0.994956i	$-0.531985\pi$
$631$	−5.03981	−0.200632	−0.100316	+	0.994956i	$0.531985\pi$
$632$	−3.96591	−0.157755
$633$	0	0
$634$	13.4864	0.535615
$635$	12.9160	0.512557
$636$	0	0
$637$	16.3956	0.649619
$638$	44.3868	1.75729
$639$	0	0
$640$	−13.5879	−0.537111
$641$	29.5688	1.16790	0.583948	−	0.811791i	$-0.301507\pi$
$641$	29.5688	1.16790	0.583948	+	0.811791i	$0.301507\pi$
$642$	0	0
$643$	−12.5905	−0.496522	−0.248261	−	0.968693i	$-0.579859\pi$
$643$	−12.5905	−0.496522	−0.248261	+	0.968693i	$0.579859\pi$
$644$	4.26348	0.168005
$645$	0	0
$646$	−47.2539	−1.85918
$647$	−36.1489	−1.42116	−0.710579	−	0.703617i	$-0.751567\pi$
$647$	−36.1489	−1.42116	−0.710579	+	0.703617i	$0.751567\pi$
$648$	0	0
$649$	5.19401	0.203883
$650$	1.56689	0.0614584
$651$	0	0
$652$	1.18908	0.0465678
$653$	23.1267	0.905018	0.452509	−	0.891760i	$-0.350529\pi$
$653$	23.1267	0.905018	0.452509	+	0.891760i	$0.350529\pi$
$654$	0	0
$655$	7.28144	0.284509
$656$	−0.699481	−0.0273101
$657$	0	0
$658$	77.2606	3.01193
$659$	45.9662	1.79059	0.895295	−	0.445474i	$-0.146965\pi$
$659$	45.9662	1.79059	0.895295	+	0.445474i	$0.146965\pi$
$660$	0	0
$661$	−36.6317	−1.42481	−0.712405	−	0.701769i	$-0.752394\pi$
$661$	−36.6317	−1.42481	−0.712405	+	0.701769i	$0.752394\pi$
$662$	11.1531	0.433476
$663$	0	0
$664$	23.7522	0.921763
$665$	30.7614	1.19288
$666$	0	0
$667$	16.0588	0.621799
$668$	−7.83471	−0.303134
$669$	0	0
$670$	−17.0944	−0.660416
$671$	13.5408	0.522736
$672$	0	0
$673$	−15.0358	−0.579587	−0.289794	−	0.957089i	$-0.593587\pi$
$673$	−15.0358	−0.579587	−0.289794	+	0.957089i	$0.593587\pi$
$674$	−45.6231	−1.75734
$675$	0	0
$676$	0.455140	0.0175054
$677$	6.67563	0.256565	0.128283	−	0.991738i	$-0.459054\pi$
$677$	6.67563	0.256565	0.128283	+	0.991738i	$0.459054\pi$
$678$	0	0
$679$	−31.7753	−1.21942
$680$	11.4786	0.440183
$681$	0	0
$682$	−53.9837	−2.06714
$683$	−31.7287	−1.21407	−0.607033	−	0.794677i	$-0.707640\pi$
$683$	−31.7287	−1.21407	−0.607033	+	0.794677i	$0.707640\pi$
$684$	0	0
$685$	7.67295	0.293168
$686$	71.2086	2.71876
$687$	0	0
$688$	−29.7367	−1.13370
$689$	13.1626	0.501456
$690$	0	0
$691$	−1.19616	−0.0455039	−0.0227520	−	0.999741i	$-0.507243\pi$
$691$	−1.19616	−0.0455039	−0.0227520	+	0.999741i	$0.507243\pi$
$692$	−6.44873	−0.245144
$693$	0	0
$694$	8.69127	0.329916
$695$	8.30159	0.314897
$696$	0	0
$697$	0.705260	0.0267136
$698$	−12.5093	−0.473485
$699$	0	0
$700$	2.20147	0.0832077
$701$	43.3421	1.63701	0.818504	−	0.574501i	$-0.194804\pi$
$701$	43.3421	1.63701	0.818504	+	0.574501i	$0.194804\pi$
$702$	0	0
$703$	17.8255	0.672303
$704$	18.6020	0.701091
$705$	0	0
$706$	−51.7485	−1.94758
$707$	−33.3616	−1.25469
$708$	0	0
$709$	−32.4902	−1.22019	−0.610097	−	0.792327i	$-0.708870\pi$
$709$	−32.4902	−1.22019	−0.610097	+	0.792327i	$0.708870\pi$
$710$	11.5362	0.432945
$711$	0	0
$712$	−9.45611	−0.354383
$713$	−19.5309	−0.731439
$714$	0	0
$715$	−3.41628	−0.127762
$716$	3.69198	0.137976
$717$	0	0
$718$	9.45331	0.352795
$719$	−38.0930	−1.42063	−0.710314	−	0.703885i	$-0.751447\pi$
$719$	−38.0930	−1.42063	−0.710314	+	0.703885i	$0.751447\pi$
$720$	0	0
$721$	31.1601	1.16046
$722$	33.6039	1.25061
$723$	0	0
$724$	7.67171	0.285117
$725$	8.29204	0.307959
$726$	0	0
$727$	12.9930	0.481883	0.240942	−	0.970540i	$-0.422544\pi$
$727$	12.9930	0.481883	0.240942	+	0.970540i	$0.422544\pi$
$728$	−11.7083	−0.433939
$729$	0	0
$730$	16.7088	0.618421
$731$	29.9824	1.10894
$732$	0	0
$733$	−50.3138	−1.85838	−0.929191	−	0.369599i	$-0.879495\pi$
$733$	−50.3138	−1.85838	−0.929191	+	0.369599i	$0.879495\pi$
$734$	0.819800	0.0302594
$735$	0	0
$736$	−4.89591	−0.180466
$737$	37.2709	1.37289
$738$	0	0
$739$	−25.2096	−0.927351	−0.463675	−	0.886005i	$-0.653470\pi$
$739$	−25.2096	−0.927351	−0.463675	+	0.886005i	$0.653470\pi$
$740$	1.27570	0.0468957
$741$	0	0
$742$	99.7580	3.66223
$743$	−7.98612	−0.292982	−0.146491	−	0.989212i	$-0.546798\pi$
$743$	−7.98612	−0.292982	−0.146491	+	0.989212i	$0.546798\pi$
$744$	0	0
$745$	23.5347	0.862245
$746$	26.7279	0.978578
$747$	0	0
$748$	7.37325	0.269593
$749$	73.6364	2.69062
$750$	0	0
$751$	47.4494	1.73145	0.865727	−	0.500517i	$-0.166857\pi$
$751$	47.4494	1.73145	0.865727	+	0.500517i	$0.166857\pi$
$752$	−47.9445	−1.74836
$753$	0	0
$754$	12.9927	0.473167
$755$	9.02777	0.328554
$756$	0	0
$757$	−35.6046	−1.29407	−0.647035	−	0.762460i	$-0.723991\pi$
$757$	−35.6046	−1.29407	−0.647035	+	0.762460i	$0.723991\pi$
$758$	16.6156	0.603506
$759$	0	0
$760$	−15.3945	−0.558418
$761$	25.3272	0.918110	0.459055	−	0.888408i	$-0.348188\pi$
$761$	25.3272	0.918110	0.459055	+	0.888408i	$0.348188\pi$
$762$	0	0
$763$	−84.0387	−3.04240
$764$	−1.69917	−0.0614740
$765$	0	0
$766$	−2.08880	−0.0754715
$767$	1.52037	0.0548974
$768$	0	0
$769$	−2.75046	−0.0991842	−0.0495921	−	0.998770i	$-0.515792\pi$
$769$	−2.75046	−0.0991842	−0.0495921	+	0.998770i	$0.515792\pi$
$770$	−25.8916	−0.933069
$771$	0	0
$772$	6.69103	0.240815
$773$	37.4021	1.34526	0.672630	−	0.739979i	$-0.265165\pi$
$773$	37.4021	1.34526	0.672630	+	0.739979i	$0.265165\pi$
$774$	0	0
$775$	−10.0849	−0.362260
$776$	15.9019	0.570846
$777$	0	0
$778$	57.3463	2.05596
$779$	−0.945862	−0.0338890
$780$	0	0
$781$	−25.1523	−0.900019
$782$	14.3896	0.514572
$783$	0	0
$784$	−77.1108	−2.75396
$785$	−5.83359	−0.208210
$786$	0	0
$787$	15.0544	0.536633	0.268316	−	0.963331i	$-0.413533\pi$
$787$	15.0544	0.536633	0.268316	+	0.963331i	$0.413533\pi$
$788$	9.72206	0.346334
$789$	0	0
$790$	−2.56716	−0.0913356
$791$	−53.1683	−1.89045
$792$	0	0
$793$	3.96360	0.140752
$794$	−3.37067	−0.119621
$795$	0	0
$796$	−1.46753	−0.0520153
$797$	11.0024	0.389725	0.194863	−	0.980831i	$-0.437574\pi$
$797$	11.0024	0.389725	0.194863	+	0.980831i	$0.437574\pi$
$798$	0	0
$799$	48.3407	1.71017
$800$	−2.52803	−0.0893794
$801$	0	0
$802$	5.90325	0.208451
$803$	−36.4302	−1.28559
$804$	0	0
$805$	−9.36740	−0.330157
$806$	−15.8019	−0.556598
$807$	0	0
$808$	16.6958	0.587356
$809$	−36.1824	−1.27210	−0.636052	−	0.771646i	$-0.719434\pi$
$809$	−36.1824	−1.27210	−0.636052	+	0.771646i	$0.719434\pi$
$810$	0	0
$811$	0.222006	0.00779569	0.00389785	−	0.999992i	$-0.498759\pi$
$811$	0.222006	0.00779569	0.00389785	+	0.999992i	$0.498759\pi$
$812$	18.2547	0.640614
$813$	0	0
$814$	−15.0036	−0.525876
$815$	−2.61255	−0.0915137
$816$	0	0
$817$	−40.2111	−1.40681
$818$	−8.93070	−0.312255
$819$	0	0
$820$	−0.0676915	−0.00236389
$821$	2.00595	0.0700081	0.0350040	−	0.999387i	$-0.488856\pi$
$821$	2.00595	0.0700081	0.0350040	+	0.999387i	$0.488856\pi$
$822$	0	0
$823$	−24.3282	−0.848027	−0.424013	−	0.905656i	$-0.639379\pi$
$823$	−24.3282	−0.848027	−0.424013	+	0.905656i	$0.639379\pi$
$824$	−15.5941	−0.543245
$825$	0	0
$826$	11.5227	0.400927
$827$	18.2410	0.634302	0.317151	−	0.948375i	$-0.397274\pi$
$827$	18.2410	0.634302	0.317151	+	0.948375i	$0.397274\pi$
$828$	0	0
$829$	−32.5007	−1.12880	−0.564398	−	0.825503i	$-0.690892\pi$
$829$	−32.5007	−1.12880	−0.564398	+	0.825503i	$0.690892\pi$
$830$	15.3750	0.533673
$831$	0	0
$832$	5.44511	0.188775
$833$	77.7479	2.69381
$834$	0	0
$835$	17.2138	0.595710
$836$	−9.88867	−0.342007
$837$	0	0
$838$	15.1111	0.522004
$839$	16.2098	0.559624	0.279812	−	0.960055i	$-0.409728\pi$
$839$	16.2098	0.559624	0.279812	+	0.960055i	$0.409728\pi$
$840$	0	0
$841$	39.7580	1.37097
$842$	38.3358	1.32114
$843$	0	0
$844$	−9.61056	−0.330809
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	3.24545	0.111515
$848$	−61.9054	−2.12584
$849$	0	0
$850$	7.43016	0.254853
$851$	−5.42819	−0.186076
$852$	0	0
$853$	39.1882	1.34178	0.670889	−	0.741558i	$-0.265913\pi$
$853$	39.1882	1.34178	0.670889	+	0.741558i	$0.265913\pi$
$854$	30.0397	1.02794
$855$	0	0
$856$	−36.8513	−1.25955
$857$	29.6558	1.01302	0.506512	−	0.862233i	$-0.330935\pi$
$857$	29.6558	1.01302	0.506512	+	0.862233i	$0.330935\pi$
$858$	0	0
$859$	46.8105	1.59715	0.798577	−	0.601893i	$-0.205587\pi$
$859$	46.8105	1.59715	0.798577	+	0.601893i	$0.205587\pi$
$860$	−2.87774	−0.0981301
$861$	0	0
$862$	−14.5609	−0.495948
$863$	−11.2611	−0.383333	−0.191667	−	0.981460i	$-0.561389\pi$
$863$	−11.2611	−0.383333	−0.191667	+	0.981460i	$0.561389\pi$
$864$	0	0
$865$	14.1687	0.481749
$866$	60.3423	2.05051
$867$	0	0
$868$	−22.2016	−0.753571
$869$	5.59718	0.189871
$870$	0	0
$871$	10.9098	0.369664
$872$	42.0571	1.42423
$873$	0	0
$874$	−19.2987	−0.652789
$875$	−4.83690	−0.163517
$876$	0	0
$877$	49.4996	1.67148	0.835742	−	0.549123i	$-0.185038\pi$
$877$	49.4996	1.67148	0.835742	+	0.549123i	$0.185038\pi$
$878$	−0.0140392	−0.000473800 0
$879$	0	0
$880$	16.0672	0.541625
$881$	17.5156	0.590115	0.295058	−	0.955479i	$-0.404661\pi$
$881$	17.5156	0.590115	0.295058	+	0.955479i	$0.404661\pi$
$882$	0	0
$883$	−57.9666	−1.95073	−0.975366	−	0.220594i	$-0.929200\pi$
$883$	−57.9666	−1.95073	−0.975366	+	0.220594i	$0.929200\pi$
$884$	2.15827	0.0725904
$885$	0	0
$886$	22.1124	0.742882
$887$	−10.1828	−0.341906	−0.170953	−	0.985279i	$-0.554685\pi$
$887$	−10.1828	−0.341906	−0.170953	+	0.985279i	$0.554685\pi$
$888$	0	0
$889$	−62.4735	−2.09530
$890$	−6.12101	−0.205177
$891$	0	0
$892$	0.663023	0.0221997
$893$	−64.8323	−2.16953
$894$	0	0
$895$	−8.11175	−0.271146
$896$	65.7236	2.19567
$897$	0	0
$898$	−23.4685	−0.783156
$899$	−83.6244	−2.78903
$900$	0	0
$901$	62.4169	2.07941
$902$	0.796124	0.0265080
$903$	0	0
$904$	26.6080	0.884970
$905$	−16.8557	−0.560303
$906$	0	0
$907$	29.7508	0.987858	0.493929	−	0.869502i	$-0.335560\pi$
$907$	29.7508	0.987858	0.493929	+	0.869502i	$0.335560\pi$
$908$	3.30099	0.109547
$909$	0	0
$910$	−7.57889	−0.251238
$911$	2.68914	0.0890951	0.0445476	−	0.999007i	$-0.485815\pi$
$911$	2.68914	0.0890951	0.0445476	+	0.999007i	$0.485815\pi$
$912$	0	0
$913$	−33.5220	−1.10942
$914$	−12.2799	−0.406183
$915$	0	0
$916$	−7.51791	−0.248399
$917$	−35.2196	−1.16305
$918$	0	0
$919$	30.8708	1.01833	0.509166	−	0.860668i	$-0.329954\pi$
$919$	30.8708	1.01833	0.509166	+	0.860668i	$0.329954\pi$
$920$	4.68790	0.154556
$921$	0	0
$922$	21.2286	0.699126
$923$	−7.36247	−0.242339
$924$	0	0
$925$	−2.80287	−0.0921579
$926$	−17.9442	−0.589682
$927$	0	0
$928$	−20.9625	−0.688129
$929$	−24.1475	−0.792252	−0.396126	−	0.918196i	$-0.629646\pi$
$929$	−24.1475	−0.792252	−0.396126	+	0.918196i	$0.629646\pi$
$930$	0	0
$931$	−104.272	−3.41738
$932$	−6.91699	−0.226573
$933$	0	0
$934$	−13.0596	−0.427322
$935$	−16.2000	−0.529795
$936$	0	0
$937$	−36.8676	−1.20441	−0.602206	−	0.798341i	$-0.705711\pi$
$937$	−36.8676	−1.20441	−0.602206	+	0.798341i	$0.705711\pi$
$938$	82.6841	2.69973
$939$	0	0
$940$	−4.63978	−0.151333
$941$	23.9878	0.781979	0.390989	−	0.920395i	$-0.372133\pi$
$941$	23.9878	0.781979	0.390989	+	0.920395i	$0.372133\pi$
$942$	0	0
$943$	0.288032	0.00937960
$944$	−7.15049	−0.232729
$945$	0	0
$946$	33.8453	1.10041
$947$	−12.9995	−0.422427	−0.211214	−	0.977440i	$-0.567742\pi$
$947$	−12.9995	−0.422427	−0.211214	+	0.977440i	$0.567742\pi$
$948$	0	0
$949$	−10.6637	−0.346158
$950$	−9.96500	−0.323307
$951$	0	0
$952$	−55.5207	−1.79944
$953$	3.55744	0.115237	0.0576184	−	0.998339i	$-0.481649\pi$
$953$	3.55744	0.115237	0.0576184	+	0.998339i	$0.481649\pi$
$954$	0	0
$955$	3.73330	0.120807
$956$	−5.13648	−0.166126
$957$	0	0
$958$	−38.4921	−1.24362
$959$	−37.1133	−1.19845
$960$	0	0
$961$	70.7051	2.28081
$962$	−4.39179	−0.141597
$963$	0	0
$964$	8.92158	0.287345
$965$	−14.7010	−0.473243
$966$	0	0
$967$	24.6421	0.792438	0.396219	−	0.918156i	$-0.370322\pi$
$967$	24.6421	0.792438	0.396219	+	0.918156i	$0.370322\pi$
$968$	−1.62418	−0.0522032
$969$	0	0
$970$	10.2934	0.330502
$971$	51.0325	1.63771	0.818855	−	0.574000i	$-0.194609\pi$
$971$	51.0325	1.63771	0.818855	+	0.574000i	$0.194609\pi$
$972$	0	0
$973$	−40.1540	−1.28728
$974$	7.74376	0.248126
$975$	0	0
$976$	−18.6413	−0.596694
$977$	−12.3716	−0.395802	−0.197901	−	0.980222i	$-0.563412\pi$
$977$	−12.3716	−0.395802	−0.197901	+	0.980222i	$0.563412\pi$
$978$	0	0
$979$	13.3456	0.426528
$980$	−7.46231	−0.238375
$981$	0	0
$982$	3.16149	0.100887
$983$	−30.0214	−0.957533	−0.478767	−	0.877942i	$-0.658916\pi$
$983$	−30.0214	−0.957533	−0.478767	+	0.877942i	$0.658916\pi$
$984$	0	0
$985$	−21.3606	−0.680605
$986$	61.6113	1.96210
$987$	0	0
$988$	−2.89457	−0.0920885
$989$	12.2450	0.389368
$990$	0	0
$991$	22.5077	0.714981	0.357490	−	0.933917i	$-0.383632\pi$
$991$	22.5077	0.714981	0.357490	+	0.933917i	$0.383632\pi$
$992$	25.4949	0.809464
$993$	0	0
$994$	−55.7994	−1.76985
$995$	3.22435	0.102219
$996$	0	0
$997$	5.10413	0.161649	0.0808247	−	0.996728i	$-0.474245\pi$
$997$	5.10413	0.161649	0.0808247	+	0.996728i	$0.474245\pi$
$998$	39.1932	1.24064
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1755.2.a.s.1.3	yes	4
3.2	odd	2		1755.2.a.m.1.2	✓	4
5.4	even	2		8775.2.a.bh.1.2		4
15.14	odd	2		8775.2.a.bt.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1755.2.a.m.1.2	✓	4	3.2	odd	2
1755.2.a.s.1.3	yes	4	1.1	even	1	trivial
8775.2.a.bh.1.2		4	5.4	even	2
8775.2.a.bt.1.3		4	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1755 = 3^{3} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1755.a (trivial)

\(f(q)\)	\(=\)	\(q+1.56689 q^{2} +0.455140 q^{4} -1.00000 q^{5} +4.83690 q^{7} -2.42062 q^{8} +O(q^{10})\)	\(q+1.56689 q^{2} +0.455140 q^{4} -1.00000 q^{5} +4.83690 q^{7} -2.42062 q^{8} -1.56689 q^{10} +3.41628 q^{11} +1.00000 q^{13} +7.57889 q^{14} -4.70313 q^{16} +4.74199 q^{17} -6.35974 q^{19} -0.455140 q^{20} +5.35293 q^{22} +1.93665 q^{23} +1.00000 q^{25} +1.56689 q^{26} +2.20147 q^{28} +8.29204 q^{29} -10.0849 q^{31} -2.52803 q^{32} +7.43016 q^{34} -4.83690 q^{35} -2.80287 q^{37} -9.96500 q^{38} +2.42062 q^{40} +0.148727 q^{41} +6.32276 q^{43} +1.55489 q^{44} +3.03452 q^{46} +10.1942 q^{47} +16.3956 q^{49} +1.56689 q^{50} +0.455140 q^{52} +13.1626 q^{53} -3.41628 q^{55} -11.7083 q^{56} +12.9927 q^{58} +1.52037 q^{59} +3.96360 q^{61} -15.8019 q^{62} +5.44511 q^{64} -1.00000 q^{65} +10.9098 q^{67} +2.15827 q^{68} -7.57889 q^{70} -7.36247 q^{71} -10.6637 q^{73} -4.39179 q^{74} -2.89457 q^{76} +16.5242 q^{77} +1.63838 q^{79} +4.70313 q^{80} +0.233038 q^{82} -9.81241 q^{83} -4.74199 q^{85} +9.90706 q^{86} -8.26953 q^{88} +3.90648 q^{89} +4.83690 q^{91} +0.881447 q^{92} +15.9731 q^{94} +6.35974 q^{95} -6.56935 q^{97} +25.6902 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{2} + 3 q^{4} - 4 q^{5} - 3 q^{7} + 3 q^{8}+O(q^{10}) \) 4 * q + 3 * q^2 + 3 * q^4 - 4 * q^5 - 3 * q^7 + 3 * q^8	\( 4 q + 3 q^{2} + 3 q^{4} - 4 q^{5} - 3 q^{7} + 3 q^{8} - 3 q^{10} + 4 q^{11} + 4 q^{13} + 3 q^{14} - 3 q^{16} + 14 q^{17} - 4 q^{19} - 3 q^{20} + 9 q^{22} + 5 q^{23} + 4 q^{25} + 3 q^{26} + 2 q^{28} + 12 q^{29} - q^{31} + 4 q^{32} - q^{34} + 3 q^{35} - 15 q^{37} + 14 q^{38} - 3 q^{40} + 4 q^{41} - 4 q^{43} + 27 q^{44} + 26 q^{46} + 3 q^{47} + 21 q^{49} + 3 q^{50} + 3 q^{52} + 35 q^{53} - 4 q^{55} - 16 q^{56} + 21 q^{58} + 13 q^{59} - q^{61} + 13 q^{62} + q^{64} - 4 q^{65} + 6 q^{67} - 6 q^{68} - 3 q^{70} + q^{71} - 25 q^{73} - 16 q^{74} + 33 q^{76} + 30 q^{77} + 25 q^{79} + 3 q^{80} - 26 q^{82} - 25 q^{83} - 14 q^{85} + 11 q^{86} + 26 q^{88} - 4 q^{89} - 3 q^{91} + 17 q^{92} + 29 q^{94} + 4 q^{95} - 17 q^{97} + 31 q^{98}+O(q^{100}) \) 4 * q + 3 * q^2 + 3 * q^4 - 4 * q^5 - 3 * q^7 + 3 * q^8 - 3 * q^10 + 4 * q^11 + 4 * q^13 + 3 * q^14 - 3 * q^16 + 14 * q^17 - 4 * q^19 - 3 * q^20 + 9 * q^22 + 5 * q^23 + 4 * q^25 + 3 * q^26 + 2 * q^28 + 12 * q^29 - q^31 + 4 * q^32 - q^34 + 3 * q^35 - 15 * q^37 + 14 * q^38 - 3 * q^40 + 4 * q^41 - 4 * q^43 + 27 * q^44 + 26 * q^46 + 3 * q^47 + 21 * q^49 + 3 * q^50 + 3 * q^52 + 35 * q^53 - 4 * q^55 - 16 * q^56 + 21 * q^58 + 13 * q^59 - q^61 + 13 * q^62 + q^64 - 4 * q^65 + 6 * q^67 - 6 * q^68 - 3 * q^70 + q^71 - 25 * q^73 - 16 * q^74 + 33 * q^76 + 30 * q^77 + 25 * q^79 + 3 * q^80 - 26 * q^82 - 25 * q^83 - 14 * q^85 + 11 * q^86 + 26 * q^88 - 4 * q^89 - 3 * q^91 + 17 * q^92 + 29 * q^94 + 4 * q^95 - 17 * q^97 + 31 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more