LMFDB - Embedded newform 1755.2.a.n.1.2

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:	4.4.3981.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 4x^{2} + 2x + 1 $ x^4 - x^3 - 4x^2 + 2x + 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.318459$ 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)

$f(q)$	$=$	$q-1.31846 q^{2} -0.261666 q^{4} +1.00000 q^{5} +0.241539 q^{7} +2.98191 q^{8} +O(q^{10})$	\(q-1.31846 q^{2} -0.261666 q^{4} +1.00000 q^{5} +0.241539 q^{7} +2.98191 q^{8} -1.31846 q^{10} +1.21704 q^{11} -1.00000 q^{13} -0.318459 q^{14} -3.40820 q^{16} -6.84024 q^{17} +1.07692 q^{19} -0.261666 q^{20} -1.60462 q^{22} -5.30474 q^{23} +1.00000 q^{25} +1.31846 q^{26} -0.0632025 q^{28} -8.66345 q^{29} +3.90499 q^{31} -1.47026 q^{32} +9.01858 q^{34} +0.241539 q^{35} +9.38215 q^{37} -1.41987 q^{38} +2.98191 q^{40} +3.70653 q^{41} -0.0769202 q^{43} -0.318459 q^{44} +6.99408 q^{46} -1.91230 q^{47} -6.94166 q^{49} -1.31846 q^{50} +0.261666 q^{52} +4.02499 q^{53} +1.21704 q^{55} +0.720248 q^{56} +11.4224 q^{58} -4.40179 q^{59} -1.61103 q^{61} -5.14857 q^{62} +8.75487 q^{64} -1.00000 q^{65} -12.8947 q^{67} +1.78986 q^{68} -0.318459 q^{70} -0.00436851 q^{71} +10.4361 q^{73} -12.3700 q^{74} -0.281794 q^{76} +0.293963 q^{77} -5.63537 q^{79} -3.40820 q^{80} -4.88691 q^{82} -13.0652 q^{83} -6.84024 q^{85} +0.101416 q^{86} +3.62912 q^{88} -5.50320 q^{89} -0.241539 q^{91} +1.38807 q^{92} +2.52129 q^{94} +1.07692 q^{95} -0.954884 q^{97} +9.15229 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 3 q^{2} + 3 q^{4} + 4 q^{5} - q^{7} - 9 q^{8}+O(q^{10}) $ 4 * q - 3 * q^2 + 3 * q^4 + 4 * q^5 - q^7 - 9 * q^8	$ 4 q - 3 q^{2} + 3 q^{4} + 4 q^{5} - q^{7} - 9 q^{8} - 3 q^{10} - 6 q^{11} - 4 q^{13} + q^{14} + 13 q^{16} - 2 q^{17} + 4 q^{19} + 3 q^{20} - 9 q^{22} - 9 q^{23} + 4 q^{25} + 3 q^{26} + 10 q^{28} - 16 q^{29} - 5 q^{31} - 26 q^{32} + 19 q^{34} - q^{35} - 13 q^{37} - 12 q^{38} - 9 q^{40} - 12 q^{41} + q^{44} + 2 q^{46} - 9 q^{47} - 11 q^{49} - 3 q^{50} - 3 q^{52} - 13 q^{53} - 6 q^{55} - 14 q^{56} - 9 q^{58} - 3 q^{59} + 3 q^{61} + 25 q^{62} + 45 q^{64} - 4 q^{65} + 6 q^{67} - 32 q^{68} + q^{70} - 11 q^{71} - 3 q^{73} - 4 q^{74} + 5 q^{76} - 10 q^{77} - 3 q^{79} + 13 q^{80} + 22 q^{82} - 19 q^{83} - 2 q^{85} + 9 q^{86} + 26 q^{88} - 16 q^{89} + q^{91} - 19 q^{92} + 25 q^{94} + 4 q^{95} - 35 q^{97} + 21 q^{98}+O(q^{100}) $ 4 * q - 3 * q^2 + 3 * q^4 + 4 * q^5 - q^7 - 9 * q^8 - 3 * q^10 - 6 * q^11 - 4 * q^13 + q^14 + 13 * q^16 - 2 * q^17 + 4 * q^19 + 3 * q^20 - 9 * q^22 - 9 * q^23 + 4 * q^25 + 3 * q^26 + 10 * q^28 - 16 * q^29 - 5 * q^31 - 26 * q^32 + 19 * q^34 - q^35 - 13 * q^37 - 12 * q^38 - 9 * q^40 - 12 * q^41 + q^44 + 2 * q^46 - 9 * q^47 - 11 * q^49 - 3 * q^50 - 3 * q^52 - 13 * q^53 - 6 * q^55 - 14 * q^56 - 9 * q^58 - 3 * q^59 + 3 * q^61 + 25 * q^62 + 45 * q^64 - 4 * q^65 + 6 * q^67 - 32 * q^68 + q^70 - 11 * q^71 - 3 * q^73 - 4 * q^74 + 5 * q^76 - 10 * q^77 - 3 * q^79 + 13 * q^80 + 22 * q^82 - 19 * q^83 - 2 * q^85 + 9 * q^86 + 26 * q^88 - 16 * q^89 + q^91 - 19 * q^92 + 25 * q^94 + 4 * q^95 - 35 * q^97 + 21 * 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 $

$2$	−1.31846	−0.932291	−0.466146	−	0.884708i	$-0.654358\pi$
$2$	−1.31846	−0.932291	−0.466146	+	0.884708i	$0.654358\pi$
$3$	0	0
$3$	0	0
$4$	−0.261666	−0.130833
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	0.241539	0.0912931	0.0456465	−	0.998958i	$-0.485465\pi$
$7$	0.241539	0.0912931	0.0456465	+	0.998958i	$0.485465\pi$
$8$	2.98191	1.05427
$9$	0	0
$10$	−1.31846	−0.416933
$11$	1.21704	0.366952	0.183476	−	0.983024i	$-0.441265\pi$
$11$	1.21704	0.366952	0.183476	+	0.983024i	$0.441265\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−0.318459	−0.0851117
$15$	0	0
$16$	−3.40820	−0.852050
$17$	−6.84024	−1.65900	−0.829501	−	0.558505i	$-0.811375\pi$
$17$	−6.84024	−1.65900	−0.829501	+	0.558505i	$0.811375\pi$
$18$	0	0
$19$	1.07692	0.247062	0.123531	−	0.992341i	$-0.460578\pi$
$19$	1.07692	0.247062	0.123531	+	0.992341i	$0.460578\pi$
$20$	−0.261666	−0.0585103
$21$	0	0
$22$	−1.60462	−0.342106
$23$	−5.30474	−1.10612	−0.553058	−	0.833143i	$-0.686539\pi$
$23$	−5.30474	−1.10612	−0.553058	+	0.833143i	$0.686539\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	1.31846	0.258571
$27$	0	0
$28$	−0.0632025	−0.0119442
$29$	−8.66345	−1.60876	−0.804382	−	0.594113i	$-0.797503\pi$
$29$	−8.66345	−1.60876	−0.804382	+	0.594113i	$0.797503\pi$
$30$	0	0
$31$	3.90499	0.701358	0.350679	−	0.936496i	$-0.385951\pi$
$31$	3.90499	0.701358	0.350679	+	0.936496i	$0.385951\pi$
$32$	−1.47026	−0.259907
$33$	0	0
$34$	9.01858	1.54667
$35$	0.241539	0.0408275
$36$	0	0
$37$	9.38215	1.54242	0.771208	−	0.636583i	$-0.219653\pi$
$37$	9.38215	1.54242	0.771208	+	0.636583i	$0.219653\pi$
$38$	−1.41987	−0.230334
$39$	0	0
$40$	2.98191	0.471482
$41$	3.70653	0.578863	0.289431	−	0.957199i	$-0.406534\pi$
$41$	3.70653	0.578863	0.289431	+	0.957199i	$0.406534\pi$
$42$	0	0
$43$	−0.0769202	−0.0117302	−0.00586511	−	0.999983i	$-0.501867\pi$
$43$	−0.0769202	−0.0117302	−0.00586511	+	0.999983i	$0.501867\pi$
$44$	−0.318459	−0.0480095
$45$	0	0
$46$	6.99408	1.03122
$47$	−1.91230	−0.278938	−0.139469	−	0.990226i	$-0.544540\pi$
$47$	−1.91230	−0.278938	−0.139469	+	0.990226i	$0.544540\pi$
$48$	0	0
$49$	−6.94166	−0.991666
$50$	−1.31846	−0.186458
$51$	0	0
$52$	0.261666	0.0362866
$53$	4.02499	0.552875	0.276437	−	0.961032i	$-0.410846\pi$
$53$	4.02499	0.552875	0.276437	+	0.961032i	$0.410846\pi$
$54$	0	0
$55$	1.21704	0.164106
$56$	0.720248	0.0962471
$57$	0	0
$58$	11.4224	1.49984
$59$	−4.40179	−0.573064	−0.286532	−	0.958071i	$-0.592502\pi$
$59$	−4.40179	−0.573064	−0.286532	+	0.958071i	$0.592502\pi$
$60$	0	0
$61$	−1.61103	−0.206271	−0.103136	−	0.994667i	$-0.532888\pi$
$61$	−1.61103	−0.206271	−0.103136	+	0.994667i	$0.532888\pi$
$62$	−5.14857	−0.653870
$63$	0	0
$64$	8.75487	1.09436
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−12.8947	−1.57534	−0.787670	−	0.616098i	$-0.788713\pi$
$67$	−12.8947	−1.57534	−0.787670	+	0.616098i	$0.788713\pi$
$68$	1.78986	0.217052
$69$	0	0
$70$	−0.318459	−0.0380631
$71$	−0.00436851	−0.000518447 0	−0.000259223	−	1.00000i	$-0.500083\pi$
$71$	−0.00436851	−0.000518447 0	−0.000259223		1.00000i	$0.500083\pi$
$72$	0	0
$73$	10.4361	1.22146	0.610728	−	0.791840i	$-0.290877\pi$
$73$	10.4361	1.22146	0.610728	+	0.791840i	$0.290877\pi$
$74$	−12.3700	−1.43798
$75$	0	0
$76$	−0.281794	−0.0323239
$77$	0.293963	0.0335002
$78$	0	0
$79$	−5.63537	−0.634029	−0.317014	−	0.948421i	$-0.602680\pi$
$79$	−5.63537	−0.634029	−0.317014	+	0.948421i	$0.602680\pi$
$80$	−3.40820	−0.381048
$81$	0	0
$82$	−4.88691	−0.539669
$83$	−13.0652	−1.43410	−0.717048	−	0.697023i	$-0.754507\pi$
$83$	−13.0652	−1.43410	−0.717048	+	0.697023i	$0.754507\pi$
$84$	0	0
$85$	−6.84024	−0.741929
$86$	0.101416	0.0109360
$87$	0	0
$88$	3.62912	0.386865
$89$	−5.50320	−0.583339	−0.291669	−	0.956519i	$-0.594211\pi$
$89$	−5.50320	−0.583339	−0.291669	+	0.956519i	$0.594211\pi$
$90$	0	0
$91$	−0.241539	−0.0253201
$92$	1.38807	0.144716
$93$	0	0
$94$	2.52129	0.260051
$95$	1.07692	0.110490
$96$	0	0
$97$	−0.954884	−0.0969537	−0.0484769	−	0.998824i	$-0.515437\pi$
$97$	−0.954884	−0.0969537	−0.0484769	+	0.998824i	$0.515437\pi$
$98$	9.15229	0.924521
$99$	0	0
$100$	−0.261666	−0.0261666
$101$	4.66767	0.464450	0.232225	−	0.972662i	$-0.425399\pi$
$101$	4.66767	0.464450	0.232225	+	0.972662i	$0.425399\pi$
$102$	0	0
$103$	−18.0132	−1.77489	−0.887444	−	0.460915i	$-0.847521\pi$
$103$	−18.0132	−1.77489	−0.887444	+	0.460915i	$0.847521\pi$
$104$	−2.98191	−0.292401
$105$	0	0
$106$	−5.30678	−0.515440
$107$	−4.89422	−0.473142	−0.236571	−	0.971614i	$-0.576024\pi$
$107$	−4.89422	−0.473142	−0.236571	+	0.971614i	$0.576024\pi$
$108$	0	0
$109$	13.0818	1.25301	0.626504	−	0.779418i	$-0.284485\pi$
$109$	13.0818	1.25301	0.626504	+	0.779418i	$0.284485\pi$
$110$	−1.60462	−0.152995
$111$	0	0
$112$	−0.823212	−0.0777862
$113$	−2.60511	−0.245069	−0.122534	−	0.992464i	$-0.539102\pi$
$113$	−2.60511	−0.245069	−0.122534	+	0.992464i	$0.539102\pi$
$114$	0	0
$115$	−5.30474	−0.494670
$116$	2.26693	0.210479
$117$	0	0
$118$	5.80358	0.534263
$119$	−1.65218	−0.151455
$120$	0	0
$121$	−9.51881	−0.865346
$122$	2.12408	0.192305
$123$	0	0
$124$	−1.02180	−0.0917608
$125$	1.00000	0.0894427
$126$	0	0
$127$	−14.5101	−1.28756	−0.643782	−	0.765209i	$-0.722636\pi$
$127$	−14.5101	−1.28756	−0.643782	+	0.765209i	$0.722636\pi$
$128$	−8.60242	−0.760354
$129$	0	0
$130$	1.31846	0.115636
$131$	−7.09460	−0.619858	−0.309929	−	0.950760i	$-0.600305\pi$
$131$	−7.09460	−0.619858	−0.309929	+	0.950760i	$0.600305\pi$
$132$	0	0
$133$	0.260118	0.0225551
$134$	17.0011	1.46868
$135$	0	0
$136$	−20.3970	−1.74903
$137$	−21.3297	−1.82232	−0.911161	−	0.412051i	$-0.864813\pi$
$137$	−21.3297	−1.82232	−0.911161	+	0.412051i	$0.864813\pi$
$138$	0	0
$139$	20.9103	1.77359	0.886796	−	0.462161i	$-0.152926\pi$
$139$	20.9103	1.77359	0.886796	+	0.462161i	$0.152926\pi$
$140$	−0.0632025	−0.00534159
$141$	0	0
$142$	0.00575970	0.000483343 0
$143$	−1.21704	−0.101774
$144$	0	0
$145$	−8.66345	−0.719461
$146$	−13.7596	−1.13875
$147$	0	0
$148$	−2.45499	−0.201799
$149$	−12.6692	−1.03790	−0.518951	−	0.854804i	$-0.673677\pi$
$149$	−12.6692	−1.03790	−0.518951	+	0.854804i	$0.673677\pi$
$150$	0	0
$151$	16.9236	1.37722	0.688610	−	0.725132i	$-0.258221\pi$
$151$	16.9236	1.37722	0.688610	+	0.725132i	$0.258221\pi$
$152$	3.21128	0.260469
$153$	0	0
$154$	−0.387578	−0.0312319
$155$	3.90499	0.313657
$156$	0	0
$157$	−19.4018	−1.54843	−0.774216	−	0.632922i	$-0.781855\pi$
$157$	−19.4018	−1.54843	−0.774216	+	0.632922i	$0.781855\pi$
$158$	7.43000	0.591099
$159$	0	0
$160$	−1.47026	−0.116234
$161$	−1.28130	−0.100981
$162$	0	0
$163$	3.05103	0.238975	0.119488	−	0.992836i	$-0.461875\pi$
$163$	3.05103	0.238975	0.119488	+	0.992836i	$0.461875\pi$
$164$	−0.969873	−0.0757344
$165$	0	0
$166$	17.2260	1.33700
$167$	5.44691	0.421494	0.210747	−	0.977541i	$-0.432410\pi$
$167$	5.44691	0.421494	0.210747	+	0.977541i	$0.432410\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	9.01858	0.691693
$171$	0	0
$172$	0.0201274	0.00153470
$173$	−9.44780	−0.718303	−0.359152	−	0.933279i	$-0.616934\pi$
$173$	−9.44780	−0.718303	−0.359152	+	0.933279i	$0.616934\pi$
$174$	0	0
$175$	0.241539	0.0182586
$176$	−4.14792	−0.312662
$177$	0	0
$178$	7.25575	0.543841
$179$	13.7042	1.02430	0.512150	−	0.858896i	$-0.328849\pi$
$179$	13.7042	1.02430	0.512150	+	0.858896i	$0.328849\pi$
$180$	0	0
$181$	2.47716	0.184126	0.0920630	−	0.995753i	$-0.470654\pi$
$181$	2.47716	0.184126	0.0920630	+	0.995753i	$0.470654\pi$
$182$	0.318459	0.0236057
$183$	0	0
$184$	−15.8183	−1.16614
$185$	9.38215	0.689790
$186$	0	0
$187$	−8.32487	−0.608775
$188$	0.500385	0.0364943
$189$	0	0
$190$	−1.41987	−0.103009
$191$	14.5948	1.05604	0.528022	−	0.849230i	$-0.322934\pi$
$191$	14.5948	1.05604	0.528022	+	0.849230i	$0.322934\pi$
$192$	0	0
$193$	−11.6281	−0.837006	−0.418503	−	0.908215i	$-0.637445\pi$
$193$	−11.6281	−0.837006	−0.418503	+	0.908215i	$0.637445\pi$
$194$	1.25897	0.0903891
$195$	0	0
$196$	1.81640	0.129743
$197$	−9.67717	−0.689470	−0.344735	−	0.938700i	$-0.612031\pi$
$197$	−9.67717	−0.689470	−0.344735	+	0.938700i	$0.612031\pi$
$198$	0	0
$199$	−12.1915	−0.864232	−0.432116	−	0.901818i	$-0.642233\pi$
$199$	−12.1915	−0.864232	−0.432116	+	0.901818i	$0.642233\pi$
$200$	2.98191	0.210853
$201$	0	0
$202$	−6.15413	−0.433003
$203$	−2.09256	−0.146869
$204$	0	0
$205$	3.70653	0.258875
$206$	23.7496	1.65471
$207$	0	0
$208$	3.40820	0.236316
$209$	1.31066	0.0906601
$210$	0	0
$211$	3.00437	0.206829	0.103415	−	0.994638i	$-0.467023\pi$
$211$	3.00437	0.206829	0.103415	+	0.994638i	$0.467023\pi$
$212$	−1.05320	−0.0723343
$213$	0	0
$214$	6.45282	0.441106
$215$	−0.0769202	−0.00524591
$216$	0	0
$217$	0.943207	0.0640291
$218$	−17.2478	−1.16817
$219$	0	0
$220$	−0.318459	−0.0214705
$221$	6.84024	0.460125
$222$	0	0
$223$	1.02499	0.0686383	0.0343192	−	0.999411i	$-0.489074\pi$
$223$	1.02499	0.0686383	0.0343192	+	0.999411i	$0.489074\pi$
$224$	−0.355124	−0.0237277
$225$	0	0
$226$	3.43474	0.228475
$227$	−24.3323	−1.61499	−0.807495	−	0.589875i	$-0.799177\pi$
$227$	−24.3323	−1.61499	−0.807495	+	0.589875i	$0.799177\pi$
$228$	0	0
$229$	−8.48794	−0.560899	−0.280449	−	0.959869i	$-0.590483\pi$
$229$	−8.48794	−0.560899	−0.280449	+	0.959869i	$0.590483\pi$
$230$	6.99408	0.461176
$231$	0	0
$232$	−25.8337	−1.69606
$233$	3.65361	0.239356	0.119678	−	0.992813i	$-0.461814\pi$
$233$	3.65361	0.239356	0.119678	+	0.992813i	$0.461814\pi$
$234$	0	0
$235$	−1.91230	−0.124745
$236$	1.15180	0.0749757
$237$	0	0
$238$	2.17834	0.141201
$239$	−1.67562	−0.108387	−0.0541936	−	0.998530i	$-0.517259\pi$
$239$	−1.67562	−0.108387	−0.0541936	+	0.998530i	$0.517259\pi$
$240$	0	0
$241$	−10.4483	−0.673034	−0.336517	−	0.941677i	$-0.609249\pi$
$241$	−10.4483	−0.673034	−0.336517	+	0.941677i	$0.609249\pi$
$242$	12.5502	0.806755
$243$	0	0
$244$	0.421552	0.0269871
$245$	−6.94166	−0.443486
$246$	0	0
$247$	−1.07692	−0.0685228
$248$	11.6444	0.739417
$249$	0	0
$250$	−1.31846	−0.0833867
$251$	12.8898	0.813600	0.406800	−	0.913517i	$-0.366645\pi$
$251$	12.8898	0.813600	0.406800	+	0.913517i	$0.366645\pi$
$252$	0	0
$253$	−6.45610	−0.405891
$254$	19.1310	1.20038
$255$	0	0
$256$	−6.16780	−0.385488
$257$	−9.75663	−0.608602	−0.304301	−	0.952576i	$-0.598423\pi$
$257$	−9.75663	−0.608602	−0.304301	+	0.952576i	$0.598423\pi$
$258$	0	0
$259$	2.26615	0.140812
$260$	0.261666	0.0162278
$261$	0	0
$262$	9.35394	0.577888
$263$	−8.74192	−0.539050	−0.269525	−	0.962993i	$-0.586867\pi$
$263$	−8.74192	−0.539050	−0.269525	+	0.962993i	$0.586867\pi$
$264$	0	0
$265$	4.02499	0.247253
$266$	−0.342955	−0.0210279
$267$	0	0
$268$	3.37411	0.206106
$269$	−20.7309	−1.26398	−0.631992	−	0.774975i	$-0.717762\pi$
$269$	−20.7309	−1.26398	−0.631992	+	0.774975i	$0.717762\pi$
$270$	0	0
$271$	−13.7590	−0.835797	−0.417898	−	0.908494i	$-0.637233\pi$
$271$	−13.7590	−0.835797	−0.417898	+	0.908494i	$0.637233\pi$
$272$	23.3129	1.41355
$273$	0	0
$274$	28.1224	1.69893
$275$	1.21704	0.0733904
$276$	0	0
$277$	−31.2755	−1.87916	−0.939580	−	0.342328i	$-0.888785\pi$
$277$	−31.2755	−1.87916	−0.939580	+	0.342328i	$0.888785\pi$
$278$	−27.5694	−1.65350
$279$	0	0
$280$	0.720248	0.0430430
$281$	23.0598	1.37563	0.687817	−	0.725884i	$-0.258569\pi$
$281$	23.0598	1.37563	0.687817	+	0.725884i	$0.258569\pi$
$282$	0	0
$283$	18.8153	1.11846	0.559228	−	0.829014i	$-0.311098\pi$
$283$	18.8153	1.11846	0.559228	+	0.829014i	$0.311098\pi$
$284$	0.00114309	6.78299e−5 0
$285$	0	0
$286$	1.60462	0.0948832
$287$	0.895271	0.0528462
$288$	0	0
$289$	29.7889	1.75229
$290$	11.4224	0.670747
$291$	0	0
$292$	−2.73078	−0.159807
$293$	31.1142	1.81771	0.908855	−	0.417113i	$-0.136958\pi$
$293$	31.1142	1.81771	0.908855	+	0.417113i	$0.136958\pi$
$294$	0	0
$295$	−4.40179	−0.256282
$296$	27.9768	1.62612
$297$	0	0
$298$	16.7038	0.967627
$299$	5.30474	0.306781
$300$	0	0
$301$	−0.0185792	−0.00107089
$302$	−22.3130	−1.28397
$303$	0	0
$304$	−3.67036	−0.210509
$305$	−1.61103	−0.0922473
$306$	0	0
$307$	18.5316	1.05766	0.528828	−	0.848729i	$-0.322632\pi$
$307$	18.5316	1.05766	0.528828	+	0.848729i	$0.322632\pi$
$308$	−0.0769202	−0.00438293
$309$	0	0
$310$	−5.14857	−0.292419
$311$	1.01669	0.0576515	0.0288257	−	0.999584i	$-0.490823\pi$
$311$	1.01669	0.0576515	0.0288257	+	0.999584i	$0.490823\pi$
$312$	0	0
$313$	−21.2598	−1.20168	−0.600838	−	0.799371i	$-0.705166\pi$
$313$	−21.2598	−1.20168	−0.600838	+	0.799371i	$0.705166\pi$
$314$	25.5805	1.44359
$315$	0	0
$316$	1.47459	0.0829519
$317$	−9.89168	−0.555572	−0.277786	−	0.960643i	$-0.589601\pi$
$317$	−9.89168	−0.555572	−0.277786	+	0.960643i	$0.589601\pi$
$318$	0	0
$319$	−10.5438	−0.590339
$320$	8.75487	0.489412
$321$	0	0
$322$	1.68934	0.0941434
$323$	−7.36640	−0.409877
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	−4.02266	−0.222795
$327$	0	0
$328$	11.0526	0.610275
$329$	−0.461895	−0.0254651
$330$	0	0
$331$	3.64423	0.200305	0.100152	−	0.994972i	$-0.468067\pi$
$331$	3.64423	0.200305	0.100152	+	0.994972i	$0.468067\pi$
$332$	3.41873	0.187627
$333$	0	0
$334$	−7.18152	−0.392955
$335$	−12.8947	−0.704513
$336$	0	0
$337$	−1.64423	−0.0895667	−0.0447833	−	0.998997i	$-0.514260\pi$
$337$	−1.64423	−0.0895667	−0.0447833	+	0.998997i	$0.514260\pi$
$338$	−1.31846	−0.0717147
$339$	0	0
$340$	1.78986	0.0970688
$341$	4.75254	0.257365
$342$	0	0
$343$	−3.36745	−0.181825
$344$	−0.229369	−0.0123668
$345$	0	0
$346$	12.4565	0.669668
$347$	26.1074	1.40152	0.700760	−	0.713397i	$-0.252844\pi$
$347$	26.1074	1.40152	0.700760	+	0.713397i	$0.252844\pi$
$348$	0	0
$349$	17.4874	0.936082	0.468041	−	0.883707i	$-0.344960\pi$
$349$	17.4874	0.936082	0.468041	+	0.883707i	$0.344960\pi$
$350$	−0.318459	−0.0170223
$351$	0	0
$352$	−1.78937	−0.0953736
$353$	−18.4316	−0.981017	−0.490509	−	0.871436i	$-0.663189\pi$
$353$	−18.4316	−0.981017	−0.490509	+	0.871436i	$0.663189\pi$
$354$	0	0
$355$	−0.00436851	−0.000231856 0
$356$	1.44000	0.0763200
$357$	0	0
$358$	−18.0684	−0.954946
$359$	12.4606	0.657647	0.328823	−	0.944391i	$-0.393348\pi$
$359$	12.4606	0.657647	0.328823	+	0.944391i	$0.393348\pi$
$360$	0	0
$361$	−17.8402	−0.938960
$362$	−3.26603	−0.171659
$363$	0	0
$364$	0.0632025	0.00331271
$365$	10.4361	0.546252
$366$	0	0
$367$	11.8891	0.620605	0.310302	−	0.950638i	$-0.399570\pi$
$367$	11.8891	0.620605	0.310302	+	0.950638i	$0.399570\pi$
$368$	18.0796	0.942465
$369$	0	0
$370$	−12.3700	−0.643085
$371$	0.972191	0.0504736
$372$	0	0
$373$	21.9902	1.13861	0.569305	−	0.822126i	$-0.307212\pi$
$373$	21.9902	1.13861	0.569305	+	0.822126i	$0.307212\pi$
$374$	10.9760	0.567555
$375$	0	0
$376$	−5.70232	−0.294075
$377$	8.66345	0.446191
$378$	0	0
$379$	24.9751	1.28288	0.641442	−	0.767171i	$-0.278336\pi$
$379$	24.9751	1.28288	0.641442	+	0.767171i	$0.278336\pi$
$380$	−0.281794	−0.0144557
$381$	0	0
$382$	−19.2427	−0.984541
$383$	35.6980	1.82408	0.912042	−	0.410098i	$-0.134505\pi$
$383$	35.6980	1.82408	0.912042	+	0.410098i	$0.134505\pi$
$384$	0	0
$385$	0.293963	0.0149817
$386$	15.3311	0.780334
$387$	0	0
$388$	0.249861	0.0126848
$389$	−32.2023	−1.63272	−0.816360	−	0.577543i	$-0.804011\pi$
$389$	−32.2023	−1.63272	−0.816360	+	0.577543i	$0.804011\pi$
$390$	0	0
$391$	36.2857	1.83505
$392$	−20.6994	−1.04548
$393$	0	0
$394$	12.7590	0.642787
$395$	−5.63537	−0.283546
$396$	0	0
$397$	−11.0485	−0.554511	−0.277255	−	0.960796i	$-0.589425\pi$
$397$	−11.0485	−0.554511	−0.277255	+	0.960796i	$0.589425\pi$
$398$	16.0740	0.805716
$399$	0	0
$400$	−3.40820	−0.170410
$401$	−25.3798	−1.26741	−0.633704	−	0.773576i	$-0.718466\pi$
$401$	−25.3798	−1.26741	−0.633704	+	0.773576i	$0.718466\pi$
$402$	0	0
$403$	−3.90499	−0.194522
$404$	−1.22137	−0.0607654
$405$	0	0
$406$	2.75895	0.136925
$407$	11.4185	0.565993
$408$	0	0
$409$	−4.22125	−0.208728	−0.104364	−	0.994539i	$-0.533281\pi$
$409$	−4.22125	−0.208728	−0.104364	+	0.994539i	$0.533281\pi$
$410$	−4.88691	−0.241347
$411$	0	0
$412$	4.71343	0.232214
$413$	−1.06320	−0.0523168
$414$	0	0
$415$	−13.0652	−0.641348
$416$	1.47026	0.0720853
$417$	0	0
$418$	−1.72805	−0.0845216
$419$	−13.1481	−0.642326	−0.321163	−	0.947024i	$-0.604074\pi$
$419$	−13.1481	−0.642326	−0.321163	+	0.947024i	$0.604074\pi$
$420$	0	0
$421$	34.5250	1.68264	0.841322	−	0.540534i	$-0.181778\pi$
$421$	34.5250	1.68264	0.841322	+	0.540534i	$0.181778\pi$
$422$	−3.96114	−0.192825
$423$	0	0
$424$	12.0022	0.582877
$425$	−6.84024	−0.331801
$426$	0	0
$427$	−0.389126	−0.0188311
$428$	1.28065	0.0619026
$429$	0	0
$430$	0.101416	0.00489072
$431$	−7.06088	−0.340110	−0.170055	−	0.985435i	$-0.554395\pi$
$431$	−7.06088	−0.340110	−0.170055	+	0.985435i	$0.554395\pi$
$432$	0	0
$433$	−26.3518	−1.26639	−0.633193	−	0.773994i	$-0.718256\pi$
$433$	−26.3518	−1.26639	−0.633193	+	0.773994i	$0.718256\pi$
$434$	−1.24358	−0.0596937
$435$	0	0
$436$	−3.42306	−0.163935
$437$	−5.71278	−0.273279
$438$	0	0
$439$	15.6761	0.748180	0.374090	−	0.927392i	$-0.377955\pi$
$439$	15.6761	0.748180	0.374090	+	0.927392i	$0.377955\pi$
$440$	3.62912	0.173011
$441$	0	0
$442$	−9.01858	−0.428970
$443$	39.3213	1.86821	0.934106	−	0.356995i	$-0.116199\pi$
$443$	39.3213	1.86821	0.934106	+	0.356995i	$0.116199\pi$
$444$	0	0
$445$	−5.50320	−0.260877
$446$	−1.35141	−0.0639909
$447$	0	0
$448$	2.11464	0.0999074
$449$	−2.32527	−0.109736	−0.0548682	−	0.998494i	$-0.517474\pi$
$449$	−2.32527	−0.109736	−0.0548682	+	0.998494i	$0.517474\pi$
$450$	0	0
$451$	4.51101	0.212415
$452$	0.681670	0.0320631
$453$	0	0
$454$	32.0811	1.50564
$455$	−0.241539	−0.0113235
$456$	0	0
$457$	12.1955	0.570481	0.285240	−	0.958456i	$-0.407927\pi$
$457$	12.1955	0.570481	0.285240	+	0.958456i	$0.407927\pi$
$458$	11.1910	0.522921
$459$	0	0
$460$	1.38807	0.0647192
$461$	−5.67529	−0.264325	−0.132162	−	0.991228i	$-0.542192\pi$
$461$	−5.67529	−0.264325	−0.132162	+	0.991228i	$0.542192\pi$
$462$	0	0
$463$	15.9250	0.740096	0.370048	−	0.929013i	$-0.379341\pi$
$463$	15.9250	0.740096	0.370048	+	0.929013i	$0.379341\pi$
$464$	29.5268	1.37075
$465$	0	0
$466$	−4.81714	−0.223150
$467$	30.3938	1.40646	0.703228	−	0.710965i	$-0.251741\pi$
$467$	30.3938	1.40646	0.703228	+	0.710965i	$0.251741\pi$
$468$	0	0
$469$	−3.11457	−0.143818
$470$	2.52129	0.116298
$471$	0	0
$472$	−13.1258	−0.604162
$473$	−0.0936151	−0.00430443
$474$	0	0
$475$	1.07692	0.0494125
$476$	0.432321	0.0198154
$477$	0	0
$478$	2.20924	0.101048
$479$	−0.493856	−0.0225648	−0.0112824	−	0.999936i	$-0.503591\pi$
$479$	−0.493856	−0.0225648	−0.0112824	+	0.999936i	$0.503591\pi$
$480$	0	0
$481$	−9.38215	−0.427789
$482$	13.7756	0.627464
$483$	0	0
$484$	2.49075	0.113216
$485$	−0.954884	−0.0433590
$486$	0	0
$487$	−12.8437	−0.582002	−0.291001	−	0.956723i	$-0.593988\pi$
$487$	−12.8437	−0.582002	−0.291001	+	0.956723i	$0.593988\pi$
$488$	−4.80395	−0.217465
$489$	0	0
$490$	9.15229	0.413458
$491$	4.77034	0.215283	0.107641	−	0.994190i	$-0.465670\pi$
$491$	4.77034	0.215283	0.107641	+	0.994190i	$0.465670\pi$
$492$	0	0
$493$	59.2601	2.66894
$494$	1.41987	0.0638832
$495$	0	0
$496$	−13.3090	−0.597591
$497$	−0.00105516	−4.73306e−5 0
$498$	0	0
$499$	−17.6837	−0.791633	−0.395816	−	0.918330i	$-0.629538\pi$
$499$	−17.6837	−0.791633	−0.395816	+	0.918330i	$0.629538\pi$
$500$	−0.261666	−0.0117021
$501$	0	0
$502$	−16.9947	−0.758512
$503$	−19.6134	−0.874519	−0.437259	−	0.899335i	$-0.644051\pi$
$503$	−19.6134	−0.874519	−0.437259	+	0.899335i	$0.644051\pi$
$504$	0	0
$505$	4.66767	0.207708
$506$	8.51210	0.378409
$507$	0	0
$508$	3.79680	0.168456
$509$	−14.0620	−0.623287	−0.311643	−	0.950199i	$-0.600879\pi$
$509$	−14.0620	−0.623287	−0.311643	+	0.950199i	$0.600879\pi$
$510$	0	0
$511$	2.52073	0.111510
$512$	25.3368	1.11974
$513$	0	0
$514$	12.8637	0.567394
$515$	−18.0132	−0.793754
$516$	0	0
$517$	−2.32735	−0.102357
$518$	−2.98783	−0.131278
$519$	0	0
$520$	−2.98191	−0.130766
$521$	4.05605	0.177699	0.0888494	−	0.996045i	$-0.471681\pi$
$521$	4.05605	0.177699	0.0888494	+	0.996045i	$0.471681\pi$
$522$	0	0
$523$	22.3971	0.979359	0.489679	−	0.871903i	$-0.337114\pi$
$523$	22.3971	0.979359	0.489679	+	0.871903i	$0.337114\pi$
$524$	1.85642	0.0810979
$525$	0	0
$526$	11.5259	0.502552
$527$	−26.7111	−1.16355
$528$	0	0
$529$	5.14028	0.223490
$530$	−5.30678	−0.230512
$531$	0	0
$532$	−0.0680641	−0.00295095
$533$	−3.70653	−0.160548
$534$	0	0
$535$	−4.89422	−0.211595
$536$	−38.4509	−1.66083
$537$	0	0
$538$	27.3328	1.17840
$539$	−8.44830	−0.363894
$540$	0	0
$541$	19.1448	0.823100	0.411550	−	0.911387i	$-0.364987\pi$
$541$	19.1448	0.823100	0.411550	+	0.911387i	$0.364987\pi$
$542$	18.1406	0.779206
$543$	0	0
$544$	10.0569	0.431187
$545$	13.0818	0.560362
$546$	0	0
$547$	−17.4082	−0.744321	−0.372160	−	0.928168i	$-0.621383\pi$
$547$	−17.4082	−0.744321	−0.372160	+	0.928168i	$0.621383\pi$
$548$	5.58127	0.238420
$549$	0	0
$550$	−1.60462	−0.0684213
$551$	−9.32985	−0.397465
$552$	0	0
$553$	−1.36116	−0.0578824
$554$	41.2354	1.75193
$555$	0	0
$556$	−5.47153	−0.232045
$557$	−2.70106	−0.114447	−0.0572237	−	0.998361i	$-0.518225\pi$
$557$	−2.70106	−0.114447	−0.0572237	+	0.998361i	$0.518225\pi$
$558$	0	0
$559$	0.0769202	0.00325338
$560$	−0.823212	−0.0347871
$561$	0	0
$562$	−30.4034	−1.28249
$563$	40.5401	1.70856	0.854280	−	0.519813i	$-0.173998\pi$
$563$	40.5401	1.70856	0.854280	+	0.519813i	$0.173998\pi$
$564$	0	0
$565$	−2.60511	−0.109598
$566$	−24.8073	−1.04273
$567$	0	0
$568$	−0.0130265	−0.000546580 0
$569$	14.9124	0.625160	0.312580	−	0.949891i	$-0.398807\pi$
$569$	14.9124	0.625160	0.312580	+	0.949891i	$0.398807\pi$
$570$	0	0
$571$	8.66859	0.362769	0.181385	−	0.983412i	$-0.441942\pi$
$571$	8.66859	0.362769	0.181385	+	0.983412i	$0.441942\pi$
$572$	0.318459	0.0133154
$573$	0	0
$574$	−1.18038	−0.0492680
$575$	−5.30474	−0.221223
$576$	0	0
$577$	−21.5229	−0.896011	−0.448006	−	0.894031i	$-0.647866\pi$
$577$	−21.5229	−0.896011	−0.448006	+	0.894031i	$0.647866\pi$
$578$	−39.2755	−1.63364
$579$	0	0
$580$	2.26693	0.0941293
$581$	−3.15576	−0.130923
$582$	0	0
$583$	4.89858	0.202879
$584$	31.1196	1.28774
$585$	0	0
$586$	−41.0227	−1.69463
$587$	48.3034	1.99369	0.996847	−	0.0793434i	$-0.0252824\pi$
$587$	48.3034	1.99369	0.996847	+	0.0793434i	$0.0252824\pi$
$588$	0	0
$589$	4.20537	0.173279
$590$	5.80358	0.238930
$591$	0	0
$592$	−31.9762	−1.31422
$593$	30.6791	1.25984	0.629921	−	0.776659i	$-0.283087\pi$
$593$	30.6791	1.25984	0.629921	+	0.776659i	$0.283087\pi$
$594$	0	0
$595$	−1.65218	−0.0677329
$596$	3.31510	0.135792
$597$	0	0
$598$	−6.99408	−0.286009
$599$	0.311308	0.0127197	0.00635985	−	0.999980i	$-0.497976\pi$
$599$	0.311308	0.0127197	0.00635985	+	0.999980i	$0.497976\pi$
$600$	0	0
$601$	21.8574	0.891582	0.445791	−	0.895137i	$-0.352923\pi$
$601$	21.8574	0.891582	0.445791	+	0.895137i	$0.352923\pi$
$602$	0.0244959	0.000998379 0
$603$	0	0
$604$	−4.42833	−0.180186
$605$	−9.51881	−0.386995
$606$	0	0
$607$	25.2380	1.02438	0.512189	−	0.858873i	$-0.328835\pi$
$607$	25.2380	1.02438	0.512189	+	0.858873i	$0.328835\pi$
$608$	−1.58335	−0.0642133
$609$	0	0
$610$	2.12408	0.0860014
$611$	1.91230	0.0773634
$612$	0	0
$613$	−13.5527	−0.547388	−0.273694	−	0.961817i	$-0.588246\pi$
$613$	−13.5527	−0.547388	−0.273694	+	0.961817i	$0.588246\pi$
$614$	−24.4332	−0.986043
$615$	0	0
$616$	0.876572	0.0353181
$617$	−19.4101	−0.781421	−0.390710	−	0.920514i	$-0.627771\pi$
$617$	−19.4101	−0.781421	−0.390710	+	0.920514i	$0.627771\pi$
$618$	0	0
$619$	37.6240	1.51224	0.756119	−	0.654434i	$-0.227093\pi$
$619$	37.6240	1.51224	0.756119	+	0.654434i	$0.227093\pi$
$620$	−1.02180	−0.0410367
$621$	0	0
$622$	−1.34047	−0.0537480
$623$	−1.32924	−0.0532548
$624$	0	0
$625$	1.00000	0.0400000
$626$	28.0302	1.12031
$627$	0	0
$628$	5.07679	0.202586
$629$	−64.1762	−2.55887
$630$	0	0
$631$	−41.8162	−1.66468	−0.832338	−	0.554269i	$-0.812998\pi$
$631$	−41.8162	−1.66468	−0.832338	+	0.554269i	$0.812998\pi$
$632$	−16.8042	−0.668435
$633$	0	0
$634$	13.0418	0.517955
$635$	−14.5101	−0.575816
$636$	0	0
$637$	6.94166	0.275039
$638$	13.9016	0.550368
$639$	0	0
$640$	−8.60242	−0.340041
$641$	37.7025	1.48916	0.744580	−	0.667534i	$-0.232650\pi$
$641$	37.7025	1.48916	0.744580	+	0.667534i	$0.232650\pi$
$642$	0	0
$643$	−49.1886	−1.93981	−0.969904	−	0.243489i	$-0.921708\pi$
$643$	−49.1886	−1.93981	−0.969904	+	0.243489i	$0.921708\pi$
$644$	0.335273	0.0132116
$645$	0	0
$646$	9.71229	0.382125
$647$	8.03026	0.315702	0.157851	−	0.987463i	$-0.449543\pi$
$647$	8.03026	0.315702	0.157851	+	0.987463i	$0.449543\pi$
$648$	0	0
$649$	−5.35717	−0.210287
$650$	1.31846	0.0517142
$651$	0	0
$652$	−0.798352	−0.0312659
$653$	−31.7528	−1.24258	−0.621291	−	0.783580i	$-0.713392\pi$
$653$	−31.7528	−1.24258	−0.621291	+	0.783580i	$0.713392\pi$
$654$	0	0
$655$	−7.09460	−0.277209
$656$	−12.6326	−0.493220
$657$	0	0
$658$	0.608989	0.0237409
$659$	−7.95890	−0.310035	−0.155017	−	0.987912i	$-0.549543\pi$
$659$	−7.95890	−0.310035	−0.155017	+	0.987912i	$0.549543\pi$
$660$	0	0
$661$	36.8485	1.43324	0.716621	−	0.697463i	$-0.245688\pi$
$661$	36.8485	1.43324	0.716621	+	0.697463i	$0.245688\pi$
$662$	−4.80476	−0.186742
$663$	0	0
$664$	−38.9594	−1.51192
$665$	0.260118	0.0100869
$666$	0	0
$667$	45.9574	1.77948
$668$	−1.42527	−0.0551454
$669$	0	0
$670$	17.0011	0.656811
$671$	−1.96069	−0.0756917
$672$	0	0
$673$	−39.5829	−1.52581	−0.762904	−	0.646512i	$-0.776227\pi$
$673$	−39.5829	−1.52581	−0.762904	+	0.646512i	$0.776227\pi$
$674$	2.16784	0.0835022
$675$	0	0
$676$	−0.261666	−0.0100641
$677$	−9.59372	−0.368717	−0.184358	−	0.982859i	$-0.559021\pi$
$677$	−9.59372	−0.368717	−0.184358	+	0.982859i	$0.559021\pi$
$678$	0	0
$679$	−0.230641	−0.00885120
$680$	−20.3970	−0.782190
$681$	0	0
$682$	−6.26603	−0.239939
$683$	−40.9033	−1.56512	−0.782561	−	0.622574i	$-0.786087\pi$
$683$	−40.9033	−1.56512	−0.782561	+	0.622574i	$0.786087\pi$
$684$	0	0
$685$	−21.3297	−0.814967
$686$	4.43985	0.169514
$687$	0	0
$688$	0.262159	0.00999472
$689$	−4.02499	−0.153340
$690$	0	0
$691$	−32.2975	−1.22866	−0.614328	−	0.789051i	$-0.710573\pi$
$691$	−32.2975	−1.22866	−0.614328	+	0.789051i	$0.710573\pi$
$692$	2.47217	0.0939778
$693$	0	0
$694$	−34.4216	−1.30662
$695$	20.9103	0.793175
$696$	0	0
$697$	−25.3536	−0.960335
$698$	−23.0565	−0.872701
$699$	0	0
$700$	−0.0632025	−0.00238883
$701$	−29.3316	−1.10784	−0.553920	−	0.832570i	$-0.686869\pi$
$701$	−29.3316	−1.10784	−0.553920	+	0.832570i	$0.686869\pi$
$702$	0	0
$703$	10.1038	0.381073
$704$	10.6551	0.401577
$705$	0	0
$706$	24.3014	0.914594
$707$	1.12742	0.0424011
$708$	0	0
$709$	5.47038	0.205444	0.102722	−	0.994710i	$-0.467245\pi$
$709$	5.47038	0.205444	0.102722	+	0.994710i	$0.467245\pi$
$710$	0.00575970	0.000216158 0
$711$	0	0
$712$	−16.4101	−0.614994
$713$	−20.7150	−0.775782
$714$	0	0
$715$	−1.21704	−0.0455148
$716$	−3.58593	−0.134012
$717$	0	0
$718$	−16.4288	−0.613118
$719$	−8.08819	−0.301639	−0.150819	−	0.988561i	$-0.548191\pi$
$719$	−8.08819	−0.301639	−0.150819	+	0.988561i	$0.548191\pi$
$720$	0	0
$721$	−4.35087	−0.162035
$722$	23.5216	0.875384
$723$	0	0
$724$	−0.648189	−0.0240898
$725$	−8.66345	−0.321753
$726$	0	0
$727$	25.7473	0.954914	0.477457	−	0.878655i	$-0.341559\pi$
$727$	25.7473	0.954914	0.477457	+	0.878655i	$0.341559\pi$
$728$	−0.720248	−0.0266942
$729$	0	0
$730$	−13.7596	−0.509266
$731$	0.526153	0.0194605
$732$	0	0
$733$	−9.27290	−0.342502	−0.171251	−	0.985227i	$-0.554781\pi$
$733$	−9.27290	−0.342502	−0.171251	+	0.985227i	$0.554781\pi$
$734$	−15.6753	−0.578584
$735$	0	0
$736$	7.79934	0.287487
$737$	−15.6934	−0.578074
$738$	0	0
$739$	−25.0241	−0.920525	−0.460262	−	0.887783i	$-0.652245\pi$
$739$	−25.0241	−0.920525	−0.460262	+	0.887783i	$0.652245\pi$
$740$	−2.45499	−0.0902473
$741$	0	0
$742$	−1.28179	−0.0470561
$743$	50.3455	1.84700	0.923499	−	0.383600i	$-0.125316\pi$
$743$	50.3455	1.84700	0.923499	+	0.383600i	$0.125316\pi$
$744$	0	0
$745$	−12.6692	−0.464164
$746$	−28.9932	−1.06152
$747$	0	0
$748$	2.17834	0.0796479
$749$	−1.18214	−0.0431946
$750$	0	0
$751$	21.0360	0.767616	0.383808	−	0.923413i	$-0.374612\pi$
$751$	21.0360	0.767616	0.383808	+	0.923413i	$0.374612\pi$
$752$	6.51750	0.237669
$753$	0	0
$754$	−11.4224	−0.415980
$755$	16.9236	0.615912
$756$	0	0
$757$	37.6577	1.36869	0.684347	−	0.729156i	$-0.260087\pi$
$757$	37.6577	1.36869	0.684347	+	0.729156i	$0.260087\pi$
$758$	−32.9286	−1.19602
$759$	0	0
$760$	3.21128	0.116485
$761$	16.0578	0.582094	0.291047	−	0.956709i	$-0.405996\pi$
$761$	16.0578	0.582094	0.291047	+	0.956709i	$0.405996\pi$
$762$	0	0
$763$	3.15976	0.114391
$764$	−3.81897	−0.138166
$765$	0	0
$766$	−47.0664	−1.70058
$767$	4.40179	0.158939
$768$	0	0
$769$	20.9601	0.755840	0.377920	−	0.925838i	$-0.376639\pi$
$769$	20.9601	0.755840	0.377920	+	0.925838i	$0.376639\pi$
$770$	−0.387578	−0.0139673
$771$	0	0
$772$	3.04267	0.109508
$773$	2.59765	0.0934309	0.0467155	−	0.998908i	$-0.485125\pi$
$773$	2.59765	0.0934309	0.0467155	+	0.998908i	$0.485125\pi$
$774$	0	0
$775$	3.90499	0.140272
$776$	−2.84738	−0.102215
$777$	0	0
$778$	42.4574	1.52217
$779$	3.99164	0.143015
$780$	0	0
$781$	−0.00531666	−0.000190245 0
$782$	−47.8412	−1.71080
$783$	0	0
$784$	23.6586	0.844948
$785$	−19.4018	−0.692480
$786$	0	0
$787$	−24.1065	−0.859303	−0.429652	−	0.902995i	$-0.641364\pi$
$787$	−24.1065	−0.859303	−0.429652	+	0.902995i	$0.641364\pi$
$788$	2.53219	0.0902055
$789$	0	0
$790$	7.43000	0.264348
$791$	−0.629236	−0.0223731
$792$	0	0
$793$	1.61103	0.0572094
$794$	14.5671	0.516966
$795$	0	0
$796$	3.19010	0.113070
$797$	48.3333	1.71205	0.856026	−	0.516933i	$-0.172926\pi$
$797$	48.3333	1.71205	0.856026	+	0.516933i	$0.172926\pi$
$798$	0	0
$799$	13.0806	0.462759
$800$	−1.47026	−0.0519815
$801$	0	0
$802$	33.4623	1.18159
$803$	12.7012	0.448216
$804$	0	0
$805$	−1.28130	−0.0451599
$806$	5.14857	0.181351
$807$	0	0
$808$	13.9186	0.489654
$809$	−25.5864	−0.899569	−0.449785	−	0.893137i	$-0.648499\pi$
$809$	−25.5864	−0.899569	−0.449785	+	0.893137i	$0.648499\pi$
$810$	0	0
$811$	−13.2844	−0.466477	−0.233239	−	0.972420i	$-0.574932\pi$
$811$	−13.2844	−0.466477	−0.233239	+	0.972420i	$0.574932\pi$
$812$	0.547552	0.0192153
$813$	0	0
$814$	−15.0548	−0.527671
$815$	3.05103	0.106873
$816$	0	0
$817$	−0.0828369	−0.00289810
$818$	5.56555	0.194595
$819$	0	0
$820$	−0.969873	−0.0338694
$821$	−41.4866	−1.44789	−0.723946	−	0.689857i	$-0.757674\pi$
$821$	−41.4866	−1.44789	−0.723946	+	0.689857i	$0.757674\pi$
$822$	0	0
$823$	−17.3421	−0.604508	−0.302254	−	0.953227i	$-0.597739\pi$
$823$	−17.3421	−0.604508	−0.302254	+	0.953227i	$0.597739\pi$
$824$	−53.7137	−1.87120
$825$	0	0
$826$	1.40179	0.0487745
$827$	−9.67661	−0.336489	−0.168244	−	0.985745i	$-0.553810\pi$
$827$	−9.67661	−0.336489	−0.168244	+	0.985745i	$0.553810\pi$
$828$	0	0
$829$	20.0799	0.697404	0.348702	−	0.937234i	$-0.386623\pi$
$829$	20.0799	0.697404	0.348702	+	0.937234i	$0.386623\pi$
$830$	17.2260	0.597923
$831$	0	0
$832$	−8.75487	−0.303521
$833$	47.4826	1.64518
$834$	0	0
$835$	5.44691	0.188498
$836$	−0.342955	−0.0118613
$837$	0	0
$838$	17.3352	0.598835
$839$	39.1466	1.35149	0.675745	−	0.737135i	$-0.263822\pi$
$839$	39.1466	1.35149	0.675745	+	0.737135i	$0.263822\pi$
$840$	0	0
$841$	46.0555	1.58812
$842$	−45.5198	−1.56871
$843$	0	0
$844$	−0.786142	−0.0270601
$845$	1.00000	0.0344010
$846$	0	0
$847$	−2.29916	−0.0790001
$848$	−13.7180	−0.471077
$849$	0	0
$850$	9.01858	0.309335
$851$	−49.7699	−1.70609
$852$	0	0
$853$	−10.9623	−0.375343	−0.187671	−	0.982232i	$-0.560094\pi$
$853$	−10.9623	−0.375343	−0.187671	+	0.982232i	$0.560094\pi$
$854$	0.513047	0.0175561
$855$	0	0
$856$	−14.5941	−0.498817
$857$	31.5421	1.07746	0.538729	−	0.842479i	$-0.318905\pi$
$857$	31.5421	1.07746	0.538729	+	0.842479i	$0.318905\pi$
$858$	0	0
$859$	−25.3261	−0.864117	−0.432058	−	0.901846i	$-0.642213\pi$
$859$	−25.3261	−0.864117	−0.432058	+	0.901846i	$0.642213\pi$
$860$	0.0201274	0.000686339 0
$861$	0	0
$862$	9.30947	0.317082
$863$	−15.6214	−0.531760	−0.265880	−	0.964006i	$-0.585662\pi$
$863$	−15.6214	−0.531760	−0.265880	+	0.964006i	$0.585662\pi$
$864$	0	0
$865$	−9.44780	−0.321235
$866$	34.7437	1.18064
$867$	0	0
$868$	−0.246805	−0.00837712
$869$	−6.85849	−0.232658
$870$	0	0
$871$	12.8947	0.436921
$872$	39.0087	1.32100
$873$	0	0
$874$	7.53207	0.254776
$875$	0.241539	0.00816550
$876$	0	0
$877$	−41.5153	−1.40187	−0.700936	−	0.713224i	$-0.747234\pi$
$877$	−41.5153	−1.40187	−0.700936	+	0.713224i	$0.747234\pi$
$878$	−20.6683	−0.697522
$879$	0	0
$880$	−4.14792	−0.139826
$881$	−23.5672	−0.794000	−0.397000	−	0.917819i	$-0.629949\pi$
$881$	−23.5672	−0.794000	−0.397000	+	0.917819i	$0.629949\pi$
$882$	0	0
$883$	−35.5716	−1.19708	−0.598541	−	0.801093i	$-0.704252\pi$
$883$	−35.5716	−1.19708	−0.598541	+	0.801093i	$0.704252\pi$
$884$	−1.78986	−0.0601995
$885$	0	0
$886$	−51.8436	−1.74172
$887$	−37.2591	−1.25104	−0.625519	−	0.780209i	$-0.715113\pi$
$887$	−37.2591	−1.25104	−0.625519	+	0.780209i	$0.715113\pi$
$888$	0	0
$889$	−3.50475	−0.117546
$890$	7.25575	0.243213
$891$	0	0
$892$	−0.268205	−0.00898016
$893$	−2.05940	−0.0689151
$894$	0	0
$895$	13.7042	0.458081
$896$	−2.07782	−0.0694150
$897$	0	0
$898$	3.06578	0.102306
$899$	−33.8307	−1.12832
$900$	0	0
$901$	−27.5319	−0.917221
$902$	−5.94758	−0.198033
$903$	0	0
$904$	−7.76823	−0.258367
$905$	2.47716	0.0823436
$906$	0	0
$907$	−6.82445	−0.226602	−0.113301	−	0.993561i	$-0.536142\pi$
$907$	−6.82445	−0.226602	−0.113301	+	0.993561i	$0.536142\pi$
$908$	6.36693	0.211294
$909$	0	0
$910$	0.318459	0.0105568
$911$	−4.91223	−0.162750	−0.0813748	−	0.996684i	$-0.525931\pi$
$911$	−4.91223	−0.162750	−0.0813748	+	0.996684i	$0.525931\pi$
$912$	0	0
$913$	−15.9010	−0.526245
$914$	−16.0792	−0.531854
$915$	0	0
$916$	2.22101	0.0733841
$917$	−1.71362	−0.0565888
$918$	0	0
$919$	28.9941	0.956428	0.478214	−	0.878243i	$-0.341284\pi$
$919$	28.9941	0.956428	0.478214	+	0.878243i	$0.341284\pi$
$920$	−15.8183	−0.521513
$921$	0	0
$922$	7.48263	0.246427
$923$	0.00436851	0.000143791 0
$924$	0	0
$925$	9.38215	0.308483
$926$	−20.9964	−0.689985
$927$	0	0
$928$	12.7375	0.418129
$929$	32.3716	1.06208	0.531039	−	0.847348i	$-0.321802\pi$
$929$	32.3716	1.06208	0.531039	+	0.847348i	$0.321802\pi$
$930$	0	0
$931$	−7.47561	−0.245003
$932$	−0.956027	−0.0313157
$933$	0	0
$934$	−40.0729	−1.31123
$935$	−8.32487	−0.272252
$936$	0	0
$937$	−15.4479	−0.504661	−0.252330	−	0.967641i	$-0.581197\pi$
$937$	−15.4479	−0.504661	−0.252330	+	0.967641i	$0.581197\pi$
$938$	4.10643	0.134080
$939$	0	0
$940$	0.500385	0.0163207
$941$	−18.4446	−0.601276	−0.300638	−	0.953738i	$-0.597200\pi$
$941$	−18.4446	−0.601276	−0.300638	+	0.953738i	$0.597200\pi$
$942$	0	0
$943$	−19.6622	−0.640289
$944$	15.0022	0.488279
$945$	0	0
$946$	0.123428	0.00401298
$947$	−15.0824	−0.490112	−0.245056	−	0.969509i	$-0.578806\pi$
$947$	−15.0824	−0.490112	−0.245056	+	0.969509i	$0.578806\pi$
$948$	0	0
$949$	−10.4361	−0.338771
$950$	−1.41987	−0.0460668
$951$	0	0
$952$	−4.92667	−0.159674
$953$	−44.4136	−1.43870	−0.719349	−	0.694649i	$-0.755560\pi$
$953$	−44.4136	−1.43870	−0.719349	+	0.694649i	$0.755560\pi$
$954$	0	0
$955$	14.5948	0.472278
$956$	0.438454	0.0141806
$957$	0	0
$958$	0.651128	0.0210370
$959$	−5.15196	−0.166365
$960$	0	0
$961$	−15.7510	−0.508098
$962$	12.3700	0.398824
$963$	0	0
$964$	2.73397	0.0880551
$965$	−11.6281	−0.374321
$966$	0	0
$967$	−42.9440	−1.38099	−0.690494	−	0.723339i	$-0.742607\pi$
$967$	−42.9440	−1.38099	−0.690494	+	0.723339i	$0.742607\pi$
$968$	−28.3843	−0.912305
$969$	0	0
$970$	1.25897	0.0404232
$971$	1.94775	0.0625064	0.0312532	−	0.999511i	$-0.490050\pi$
$971$	1.94775	0.0625064	0.0312532	+	0.999511i	$0.490050\pi$
$972$	0	0
$973$	5.05066	0.161917
$974$	16.9339	0.542596
$975$	0	0
$976$	5.49071	0.175753
$977$	13.2810	0.424898	0.212449	−	0.977172i	$-0.431856\pi$
$977$	13.2810	0.424898	0.212449	+	0.977172i	$0.431856\pi$
$978$	0	0
$979$	−6.69764	−0.214057
$980$	1.81640	0.0580227
$981$	0	0
$982$	−6.28950	−0.200706
$983$	59.2861	1.89093	0.945466	−	0.325720i	$-0.105607\pi$
$983$	59.2861	1.89093	0.945466	+	0.325720i	$0.105607\pi$
$984$	0	0
$985$	−9.67717	−0.308340
$986$	−78.1321	−2.48823
$987$	0	0
$988$	0.281794	0.00896505
$989$	0.408042	0.0129750
$990$	0	0
$991$	−9.02028	−0.286539	−0.143269	−	0.989684i	$-0.545762\pi$
$991$	−9.02028	−0.286539	−0.143269	+	0.989684i	$0.545762\pi$
$992$	−5.74135	−0.182288
$993$	0	0
$994$	0.00139119	4.41259e−5 0
$995$	−12.1915	−0.386496
$996$	0	0
$997$	−38.1707	−1.20888	−0.604439	−	0.796652i	$-0.706603\pi$
$997$	−38.1707	−1.20888	−0.604439	+	0.796652i	$0.706603\pi$
$998$	23.3153	0.738032
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1755.2.a.n.1.2	✓	4
3.2	odd	2		1755.2.a.t.1.3	yes	4
5.4	even	2		8775.2.a.bs.1.3		4
15.14	odd	2		8775.2.a.bg.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1755.2.a.n.1.2	✓	4	1.1	even	1	trivial
1755.2.a.t.1.3	yes	4	3.2	odd	2
8775.2.a.bg.1.2		4	15.14	odd	2
8775.2.a.bs.1.3		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)

\(f(q)\)	\(=\)	\(q-1.31846 q^{2} -0.261666 q^{4} +1.00000 q^{5} +0.241539 q^{7} +2.98191 q^{8} +O(q^{10})\)	\(q-1.31846 q^{2} -0.261666 q^{4} +1.00000 q^{5} +0.241539 q^{7} +2.98191 q^{8} -1.31846 q^{10} +1.21704 q^{11} -1.00000 q^{13} -0.318459 q^{14} -3.40820 q^{16} -6.84024 q^{17} +1.07692 q^{19} -0.261666 q^{20} -1.60462 q^{22} -5.30474 q^{23} +1.00000 q^{25} +1.31846 q^{26} -0.0632025 q^{28} -8.66345 q^{29} +3.90499 q^{31} -1.47026 q^{32} +9.01858 q^{34} +0.241539 q^{35} +9.38215 q^{37} -1.41987 q^{38} +2.98191 q^{40} +3.70653 q^{41} -0.0769202 q^{43} -0.318459 q^{44} +6.99408 q^{46} -1.91230 q^{47} -6.94166 q^{49} -1.31846 q^{50} +0.261666 q^{52} +4.02499 q^{53} +1.21704 q^{55} +0.720248 q^{56} +11.4224 q^{58} -4.40179 q^{59} -1.61103 q^{61} -5.14857 q^{62} +8.75487 q^{64} -1.00000 q^{65} -12.8947 q^{67} +1.78986 q^{68} -0.318459 q^{70} -0.00436851 q^{71} +10.4361 q^{73} -12.3700 q^{74} -0.281794 q^{76} +0.293963 q^{77} -5.63537 q^{79} -3.40820 q^{80} -4.88691 q^{82} -13.0652 q^{83} -6.84024 q^{85} +0.101416 q^{86} +3.62912 q^{88} -5.50320 q^{89} -0.241539 q^{91} +1.38807 q^{92} +2.52129 q^{94} +1.07692 q^{95} -0.954884 q^{97} +9.15229 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 3 q^{2} + 3 q^{4} + 4 q^{5} - q^{7} - 9 q^{8}+O(q^{10}) \) 4 * q - 3 * q^2 + 3 * q^4 + 4 * q^5 - q^7 - 9 * q^8	\( 4 q - 3 q^{2} + 3 q^{4} + 4 q^{5} - q^{7} - 9 q^{8} - 3 q^{10} - 6 q^{11} - 4 q^{13} + q^{14} + 13 q^{16} - 2 q^{17} + 4 q^{19} + 3 q^{20} - 9 q^{22} - 9 q^{23} + 4 q^{25} + 3 q^{26} + 10 q^{28} - 16 q^{29} - 5 q^{31} - 26 q^{32} + 19 q^{34} - q^{35} - 13 q^{37} - 12 q^{38} - 9 q^{40} - 12 q^{41} + q^{44} + 2 q^{46} - 9 q^{47} - 11 q^{49} - 3 q^{50} - 3 q^{52} - 13 q^{53} - 6 q^{55} - 14 q^{56} - 9 q^{58} - 3 q^{59} + 3 q^{61} + 25 q^{62} + 45 q^{64} - 4 q^{65} + 6 q^{67} - 32 q^{68} + q^{70} - 11 q^{71} - 3 q^{73} - 4 q^{74} + 5 q^{76} - 10 q^{77} - 3 q^{79} + 13 q^{80} + 22 q^{82} - 19 q^{83} - 2 q^{85} + 9 q^{86} + 26 q^{88} - 16 q^{89} + q^{91} - 19 q^{92} + 25 q^{94} + 4 q^{95} - 35 q^{97} + 21 q^{98}+O(q^{100}) \) 4 * q - 3 * q^2 + 3 * q^4 + 4 * q^5 - q^7 - 9 * q^8 - 3 * q^10 - 6 * q^11 - 4 * q^13 + q^14 + 13 * q^16 - 2 * q^17 + 4 * q^19 + 3 * q^20 - 9 * q^22 - 9 * q^23 + 4 * q^25 + 3 * q^26 + 10 * q^28 - 16 * q^29 - 5 * q^31 - 26 * q^32 + 19 * q^34 - q^35 - 13 * q^37 - 12 * q^38 - 9 * q^40 - 12 * q^41 + q^44 + 2 * q^46 - 9 * q^47 - 11 * q^49 - 3 * q^50 - 3 * q^52 - 13 * q^53 - 6 * q^55 - 14 * q^56 - 9 * q^58 - 3 * q^59 + 3 * q^61 + 25 * q^62 + 45 * q^64 - 4 * q^65 + 6 * q^67 - 32 * q^68 + q^70 - 11 * q^71 - 3 * q^73 - 4 * q^74 + 5 * q^76 - 10 * q^77 - 3 * q^79 + 13 * q^80 + 22 * q^82 - 19 * q^83 - 2 * q^85 + 9 * q^86 + 26 * q^88 - 16 * q^89 + q^91 - 19 * q^92 + 25 * q^94 + 4 * q^95 - 35 * q^97 + 21 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more