LMFDB - Embedded newform 1755.2.a.t.1.1

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:	4.4.3981.1
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			1.1
Root			$2.28400$ 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)

$f(q)$	$=$	$q-1.28400 q^{2} -0.351349 q^{4} -1.00000 q^{5} +1.77882 q^{7} +3.01913 q^{8} +O(q^{10})$	\(q-1.28400 q^{2} -0.351349 q^{4} -1.00000 q^{5} +1.77882 q^{7} +3.01913 q^{8} +1.28400 q^{10} +6.50064 q^{11} -1.00000 q^{13} -2.28400 q^{14} -3.17385 q^{16} -1.38084 q^{17} -3.06282 q^{19} +0.351349 q^{20} -8.34681 q^{22} +7.40380 q^{23} +1.00000 q^{25} +1.28400 q^{26} -0.624987 q^{28} +5.26487 q^{29} +2.04369 q^{31} -1.96303 q^{32} +1.77299 q^{34} -1.77882 q^{35} -5.90515 q^{37} +3.93265 q^{38} -3.01913 q^{40} -2.31732 q^{41} +4.06282 q^{43} -2.28400 q^{44} -9.50647 q^{46} -7.90445 q^{47} -3.83581 q^{49} -1.28400 q^{50} +0.351349 q^{52} -0.0333269 q^{53} -6.50064 q^{55} +5.37048 q^{56} -6.76008 q^{58} +0.913521 q^{59} -11.6071 q^{61} -2.62409 q^{62} +8.86824 q^{64} +1.00000 q^{65} +16.1697 q^{67} +0.485157 q^{68} +2.28400 q^{70} +10.7069 q^{71} -12.4479 q^{73} +7.58219 q^{74} +1.07612 q^{76} +11.5635 q^{77} +5.36753 q^{79} +3.17385 q^{80} +2.97544 q^{82} +6.17839 q^{83} +1.38084 q^{85} -5.21665 q^{86} +19.6263 q^{88} +7.13017 q^{89} -1.77882 q^{91} -2.60132 q^{92} +10.1493 q^{94} +3.06282 q^{95} -6.39414 q^{97} +4.92517 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.28400	−0.907924	−0.453962	−	0.891021i	$-0.649990\pi$
$2$	−1.28400	−0.907924	−0.453962	+	0.891021i	$0.649990\pi$
$3$	0	0
$3$	0	0
$4$	−0.351349	−0.175675
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	1.77882	0.672330	0.336165	−	0.941803i	$-0.390870\pi$
$7$	1.77882	0.672330	0.336165	+	0.941803i	$0.390870\pi$
$8$	3.01913	1.06742
$9$	0	0
$10$	1.28400	0.406036
$11$	6.50064	1.96002	0.980009	−	0.198953i	$-0.0637542\pi$
$11$	6.50064	1.96002	0.980009	+	0.198953i	$0.0637542\pi$
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	−2.28400	−0.610424
$15$	0	0
$16$	−3.17385	−0.793464
$17$	−1.38084	−0.334902	−0.167451	−	0.985880i	$-0.553554\pi$
$17$	−1.38084	−0.334902	−0.167451	+	0.985880i	$0.553554\pi$
$18$	0	0
$19$	−3.06282	−0.702658	−0.351329	−	0.936252i	$-0.614270\pi$
$19$	−3.06282	−0.702658	−0.351329	+	0.936252i	$0.614270\pi$
$20$	0.351349	0.0785641
$21$	0	0
$22$	−8.34681	−1.77955
$23$	7.40380	1.54380	0.771900	−	0.635744i	$-0.219307\pi$
$23$	7.40380	1.54380	0.771900	+	0.635744i	$0.219307\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	1.28400	0.251813
$27$	0	0
$28$	−0.624987	−0.118111
$29$	5.26487	0.977662	0.488831	−	0.872379i	$-0.337423\pi$
$29$	5.26487	0.977662	0.488831	+	0.872379i	$0.337423\pi$
$30$	0	0
$31$	2.04369	0.367057	0.183529	−	0.983014i	$-0.441248\pi$
$31$	2.04369	0.367057	0.183529	+	0.983014i	$0.441248\pi$
$32$	−1.96303	−0.347018
$33$	0	0
$34$	1.77299	0.304066
$35$	−1.77882	−0.300675
$36$	0	0
$37$	−5.90515	−0.970800	−0.485400	−	0.874292i	$-0.661326\pi$
$37$	−5.90515	−0.970800	−0.485400	+	0.874292i	$0.661326\pi$
$38$	3.93265	0.637960
$39$	0	0
$40$	−3.01913	−0.477366
$41$	−2.31732	−0.361905	−0.180953	−	0.983492i	$-0.557918\pi$
$41$	−2.31732	−0.361905	−0.180953	+	0.983492i	$0.557918\pi$
$42$	0	0
$43$	4.06282	0.619574	0.309787	−	0.950806i	$-0.399742\pi$
$43$	4.06282	0.619574	0.309787	+	0.950806i	$0.399742\pi$
$44$	−2.28400	−0.344326
$45$	0	0
$46$	−9.50647	−1.40165
$47$	−7.90445	−1.15298	−0.576491	−	0.817103i	$-0.695578\pi$
$47$	−7.90445	−1.15298	−0.576491	+	0.817103i	$0.695578\pi$
$48$	0	0
$49$	−3.83581	−0.547973
$50$	−1.28400	−0.181585
$51$	0	0
$52$	0.351349	0.0487234
$53$	−0.0333269	−0.00457780	−0.00228890	−	0.999997i	$-0.500729\pi$
$53$	−0.0333269	−0.00457780	−0.00228890	+	0.999997i	$0.500729\pi$
$54$	0	0
$55$	−6.50064	−0.876547
$56$	5.37048	0.717660
$57$	0	0
$58$	−6.76008	−0.887642
$59$	0.913521	0.118930	0.0594651	−	0.998230i	$-0.481060\pi$
$59$	0.913521	0.118930	0.0594651	+	0.998230i	$0.481060\pi$
$60$	0	0
$61$	−11.6071	−1.48614	−0.743071	−	0.669212i	$-0.766632\pi$
$61$	−11.6071	−1.48614	−0.743071	+	0.669212i	$0.766632\pi$
$62$	−2.62409	−0.333260
$63$	0	0
$64$	8.86824	1.10853
$65$	1.00000	0.124035
$66$	0	0
$67$	16.1697	1.97544	0.987722	−	0.156220i	$-0.0499309\pi$
$67$	16.1697	1.97544	0.987722	+	0.156220i	$0.0499309\pi$
$68$	0.485157	0.0588339
$69$	0	0
$70$	2.28400	0.272990
$71$	10.7069	1.27068	0.635339	−	0.772233i	$-0.280860\pi$
$71$	10.7069	1.27068	0.635339	+	0.772233i	$0.280860\pi$
$72$	0	0
$73$	−12.4479	−1.45691	−0.728457	−	0.685091i	$-0.759762\pi$
$73$	−12.4479	−1.45691	−0.728457	+	0.685091i	$0.759762\pi$
$74$	7.58219	0.881412
$75$	0	0
$76$	1.07612	0.123439
$77$	11.5635	1.31778
$78$	0	0
$79$	5.36753	0.603895	0.301947	−	0.953325i	$-0.402363\pi$
$79$	5.36753	0.603895	0.301947	+	0.953325i	$0.402363\pi$
$80$	3.17385	0.354848
$81$	0	0
$82$	2.97544	0.328582
$83$	6.17839	0.678166	0.339083	−	0.940756i	$-0.389883\pi$
$83$	6.17839	0.678166	0.339083	+	0.940756i	$0.389883\pi$
$84$	0	0
$85$	1.38084	0.149773
$86$	−5.21665	−0.562525
$87$	0	0
$88$	19.6263	2.09217
$89$	7.13017	0.755796	0.377898	−	0.925847i	$-0.376647\pi$
$89$	7.13017	0.755796	0.377898	+	0.925847i	$0.376647\pi$
$90$	0	0
$91$	−1.77882	−0.186471
$92$	−2.60132	−0.271207
$93$	0	0
$94$	10.1493	1.04682
$95$	3.06282	0.314238
$96$	0	0
$97$	−6.39414	−0.649227	−0.324613	−	0.945847i	$-0.605234\pi$
$97$	−6.39414	−0.649227	−0.324613	+	0.945847i	$0.605234\pi$
$98$	4.92517	0.497517
$99$	0	0
$100$	−0.351349	−0.0351349
$101$	18.2823	1.81916	0.909581	−	0.415528i	$-0.136403\pi$
$101$	18.2823	1.81916	0.909581	+	0.415528i	$0.136403\pi$
$102$	0	0
$103$	18.9796	1.87012	0.935058	−	0.354494i	$-0.115347\pi$
$103$	18.9796	1.87012	0.935058	+	0.354494i	$0.115347\pi$
$104$	−3.01913	−0.296050
$105$	0	0
$106$	0.0427917	0.00415629
$107$	−10.9236	−1.05602	−0.528011	−	0.849238i	$-0.677062\pi$
$107$	−10.9236	−1.05602	−0.528011	+	0.849238i	$0.677062\pi$
$108$	0	0
$109$	6.39798	0.612815	0.306408	−	0.951900i	$-0.400873\pi$
$109$	6.39798	0.612815	0.306408	+	0.951900i	$0.400873\pi$
$110$	8.34681	0.795837
$111$	0	0
$112$	−5.64571	−0.533469
$113$	−3.89932	−0.366817	−0.183409	−	0.983037i	$-0.558713\pi$
$113$	−3.89932	−0.366817	−0.183409	+	0.983037i	$0.558713\pi$
$114$	0	0
$115$	−7.40380	−0.690408
$116$	−1.84981	−0.171750
$117$	0	0
$118$	−1.17296	−0.107980
$119$	−2.45626	−0.225165
$120$	0	0
$121$	31.2584	2.84167
$122$	14.9036	1.34930
$123$	0	0
$124$	−0.718049	−0.0644827
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−6.14437	−0.545224	−0.272612	−	0.962124i	$-0.587888\pi$
$127$	−6.14437	−0.545224	−0.272612	+	0.962124i	$0.587888\pi$
$128$	−7.46073	−0.659442
$129$	0	0
$130$	−1.28400	−0.112614
$131$	6.91865	0.604485	0.302243	−	0.953231i	$-0.402265\pi$
$131$	6.91865	0.604485	0.302243	+	0.953231i	$0.402265\pi$
$132$	0	0
$133$	−5.44819	−0.472418
$134$	−20.7619	−1.79355
$135$	0	0
$136$	−4.16893	−0.357482
$137$	19.4371	1.66063	0.830313	−	0.557297i	$-0.188162\pi$
$137$	19.4371	1.66063	0.830313	+	0.557297i	$0.188162\pi$
$138$	0	0
$139$	3.25835	0.276369	0.138185	−	0.990406i	$-0.455873\pi$
$139$	3.25835	0.276369	0.138185	+	0.990406i	$0.455873\pi$
$140$	0.624987	0.0528210
$141$	0	0
$142$	−13.7477	−1.15368
$143$	−6.50064	−0.543611
$144$	0	0
$145$	−5.26487	−0.437224
$146$	15.9831	1.32277
$147$	0	0
$148$	2.07477	0.170545
$149$	−4.48280	−0.367246	−0.183623	−	0.982997i	$-0.558783\pi$
$149$	−4.48280	−0.367246	−0.183623	+	0.982997i	$0.558783\pi$
$150$	0	0
$151$	7.81668	0.636112	0.318056	−	0.948072i	$-0.396970\pi$
$151$	7.81668	0.636112	0.318056	+	0.948072i	$0.396970\pi$
$152$	−9.24703	−0.750033
$153$	0	0
$154$	−14.8475	−1.19644
$155$	−2.04369	−0.164153
$156$	0	0
$157$	−15.9135	−1.27004	−0.635019	−	0.772497i	$-0.719008\pi$
$157$	−15.9135	−1.27004	−0.635019	+	0.772497i	$0.719008\pi$
$158$	−6.89190	−0.548290
$159$	0	0
$160$	1.96303	0.155191
$161$	13.1700	1.03794
$162$	0	0
$163$	14.1123	1.10536	0.552681	−	0.833393i	$-0.313605\pi$
$163$	14.1123	1.10536	0.552681	+	0.833393i	$0.313605\pi$
$164$	0.814191	0.0635776
$165$	0	0
$166$	−7.93304	−0.615723
$167$	3.48062	0.269338	0.134669	−	0.990891i	$-0.457003\pi$
$167$	3.48062	0.269338	0.134669	+	0.990891i	$0.457003\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−1.77299	−0.135982
$171$	0	0
$172$	−1.42747	−0.108843
$173$	−10.6891	−0.812676	−0.406338	−	0.913723i	$-0.633195\pi$
$173$	−10.6891	−0.812676	−0.406338	+	0.913723i	$0.633195\pi$
$174$	0	0
$175$	1.77882	0.134466
$176$	−20.6321	−1.55520
$177$	0	0
$178$	−9.15512	−0.686205
$179$	5.83620	0.436218	0.218109	−	0.975924i	$-0.430011\pi$
$179$	5.83620	0.436218	0.218109	+	0.975924i	$0.430011\pi$
$180$	0	0
$181$	−10.9488	−0.813820	−0.406910	−	0.913468i	$-0.633394\pi$
$181$	−10.9488	−0.813820	−0.406910	+	0.913468i	$0.633394\pi$
$182$	2.28400	0.169301
$183$	0	0
$184$	22.3530	1.64789
$185$	5.90515	0.434155
$186$	0	0
$187$	−8.97634	−0.656415
$188$	2.77722	0.202550
$189$	0	0
$190$	−3.93265	−0.285304
$191$	19.1127	1.38295	0.691474	−	0.722401i	$-0.256962\pi$
$191$	19.1127	1.38295	0.691474	+	0.722401i	$0.256962\pi$
$192$	0	0
$193$	−8.58060	−0.617645	−0.308823	−	0.951120i	$-0.599935\pi$
$193$	−8.58060	−0.617645	−0.308823	+	0.951120i	$0.599935\pi$
$194$	8.21006	0.589448
$195$	0	0
$196$	1.34771	0.0962649
$197$	1.57707	0.112361	0.0561807	−	0.998421i	$-0.482108\pi$
$197$	1.57707	0.112361	0.0561807	+	0.998421i	$0.482108\pi$
$198$	0	0
$199$	23.8258	1.68896	0.844482	−	0.535583i	$-0.179908\pi$
$199$	23.8258	1.68896	0.844482	+	0.535583i	$0.179908\pi$
$200$	3.01913	0.213485
$201$	0	0
$202$	−23.4745	−1.65166
$203$	9.36524	0.657311
$204$	0	0
$205$	2.31732	0.161849
$206$	−24.3698	−1.69792
$207$	0	0
$208$	3.17385	0.220067
$209$	−19.9103	−1.37722
$210$	0	0
$211$	13.7069	0.943624	0.471812	−	0.881699i	$-0.343600\pi$
$211$	13.7069	0.943624	0.471812	+	0.881699i	$0.343600\pi$
$212$	0.0117094	0.000804204 0
$213$	0	0
$214$	14.0258	0.958787
$215$	−4.06282	−0.277082
$216$	0	0
$217$	3.63535	0.246783
$218$	−8.21499	−0.556389
$219$	0	0
$220$	2.28400	0.153987
$221$	1.38084	0.0928852
$222$	0	0
$223$	−2.96667	−0.198663	−0.0993316	−	0.995054i	$-0.531670\pi$
$223$	−2.96667	−0.198663	−0.0993316	+	0.995054i	$0.531670\pi$
$224$	−3.49188	−0.233311
$225$	0	0
$226$	5.00672	0.333042
$227$	1.74440	0.115780	0.0578900	−	0.998323i	$-0.481563\pi$
$227$	1.74440	0.115780	0.0578900	+	0.998323i	$0.481563\pi$
$228$	0	0
$229$	−9.01843	−0.595955	−0.297977	−	0.954573i	$-0.596312\pi$
$229$	−9.01843	−0.595955	−0.297977	+	0.954573i	$0.596312\pi$
$230$	9.50647	0.626838
$231$	0	0
$232$	15.8953	1.04358
$233$	−28.9057	−1.89368	−0.946839	−	0.321708i	$-0.895743\pi$
$233$	−28.9057	−1.89368	−0.946839	+	0.321708i	$0.895743\pi$
$234$	0	0
$235$	7.90445	0.515629
$236$	−0.320965	−0.0208930
$237$	0	0
$238$	3.15383	0.204432
$239$	−12.2225	−0.790606	−0.395303	−	0.918551i	$-0.629360\pi$
$239$	−12.2225	−0.790606	−0.395303	+	0.918551i	$0.629360\pi$
$240$	0	0
$241$	22.9352	1.47739	0.738694	−	0.674041i	$-0.235443\pi$
$241$	22.9352	1.47739	0.738694	+	0.674041i	$0.235443\pi$
$242$	−40.1357	−2.58002
$243$	0	0
$244$	4.07816	0.261078
$245$	3.83581	0.245061
$246$	0	0
$247$	3.06282	0.194882
$248$	6.17015	0.391805
$249$	0	0
$250$	1.28400	0.0812072
$251$	13.6305	0.860350	0.430175	−	0.902746i	$-0.358452\pi$
$251$	13.6305	0.860350	0.430175	+	0.902746i	$0.358452\pi$
$252$	0	0
$253$	48.1295	3.02588
$254$	7.88935	0.495022
$255$	0	0
$256$	−8.15691	−0.509807
$257$	−23.1785	−1.44583	−0.722917	−	0.690935i	$-0.757199\pi$
$257$	−23.1785	−1.44583	−0.722917	+	0.690935i	$0.757199\pi$
$258$	0	0
$259$	−10.5042	−0.652698
$260$	−0.351349	−0.0217898
$261$	0	0
$262$	−8.88353	−0.548826
$263$	7.00159	0.431737	0.215868	−	0.976422i	$-0.430742\pi$
$263$	7.00159	0.431737	0.215868	+	0.976422i	$0.430742\pi$
$264$	0	0
$265$	0.0333269	0.00204725
$266$	6.99546	0.428919
$267$	0	0
$268$	−5.68122	−0.347036
$269$	−1.65736	−0.101051	−0.0505254	−	0.998723i	$-0.516090\pi$
$269$	−1.65736	−0.101051	−0.0505254	+	0.998723i	$0.516090\pi$
$270$	0	0
$271$	1.02495	0.0622614	0.0311307	−	0.999515i	$-0.490089\pi$
$271$	1.02495	0.0622614	0.0311307	+	0.999515i	$0.490089\pi$
$272$	4.38258	0.265733
$273$	0	0
$274$	−24.9572	−1.50772
$275$	6.50064	0.392004
$276$	0	0
$277$	−11.3797	−0.683743	−0.341871	−	0.939747i	$-0.611061\pi$
$277$	−11.3797	−0.683743	−0.341871	+	0.939747i	$0.611061\pi$
$278$	−4.18371	−0.250922
$279$	0	0
$280$	−5.37048	−0.320947
$281$	13.5742	0.809770	0.404885	−	0.914368i	$-0.367312\pi$
$281$	13.5742	0.809770	0.404885	+	0.914368i	$0.367312\pi$
$282$	0	0
$283$	−0.697965	−0.0414897	−0.0207448	−	0.999785i	$-0.506604\pi$
$283$	−0.697965	−0.0414897	−0.0207448	+	0.999785i	$0.506604\pi$
$284$	−3.76187	−0.223226
$285$	0	0
$286$	8.34681	0.493557
$287$	−4.12210	−0.243320
$288$	0	0
$289$	−15.0933	−0.887840
$290$	6.76008	0.396966
$291$	0	0
$292$	4.37356	0.255943
$293$	2.80063	0.163614	0.0818072	−	0.996648i	$-0.473931\pi$
$293$	2.80063	0.163614	0.0818072	+	0.996648i	$0.473931\pi$
$294$	0	0
$295$	−0.913521	−0.0531872
$296$	−17.8284	−1.03625
$297$	0	0
$298$	5.75591	0.333431
$299$	−7.40380	−0.428173
$300$	0	0
$301$	7.22701	0.416558
$302$	−10.0366	−0.577542
$303$	0	0
$304$	9.72093	0.557534
$305$	11.6071	0.664623
$306$	0	0
$307$	−15.7377	−0.898198	−0.449099	−	0.893482i	$-0.648255\pi$
$307$	−15.7377	−0.898198	−0.449099	+	0.893482i	$0.648255\pi$
$308$	−4.06282	−0.231500
$309$	0	0
$310$	2.62409	0.149038
$311$	−31.4737	−1.78471	−0.892356	−	0.451333i	$-0.850949\pi$
$311$	−31.4737	−1.78471	−0.892356	+	0.451333i	$0.850949\pi$
$312$	0	0
$313$	10.0483	0.567964	0.283982	−	0.958830i	$-0.408344\pi$
$313$	10.0483	0.567964	0.283982	+	0.958830i	$0.408344\pi$
$314$	20.4329	1.15310
$315$	0	0
$316$	−1.88588	−0.106089
$317$	14.7692	0.829518	0.414759	−	0.909931i	$-0.363866\pi$
$317$	14.7692	0.829518	0.414759	+	0.909931i	$0.363866\pi$
$318$	0	0
$319$	34.2250	1.91623
$320$	−8.86824	−0.495750
$321$	0	0
$322$	−16.9103	−0.942373
$323$	4.22925	0.235322
$324$	0	0
$325$	−1.00000	−0.0554700
$326$	−18.1202	−1.00358
$327$	0	0
$328$	−6.99630	−0.386306
$329$	−14.0606	−0.775184
$330$	0	0
$331$	−9.51613	−0.523054	−0.261527	−	0.965196i	$-0.584226\pi$
$331$	−9.51613	−0.523054	−0.261527	+	0.965196i	$0.584226\pi$
$332$	−2.17077	−0.119137
$333$	0	0
$334$	−4.46911	−0.244539
$335$	−16.1697	−0.883446
$336$	0	0
$337$	11.5161	0.627324	0.313662	−	0.949535i	$-0.398444\pi$
$337$	11.5161	0.627324	0.313662	+	0.949535i	$0.398444\pi$
$338$	−1.28400	−0.0698403
$339$	0	0
$340$	−0.485157	−0.0263113
$341$	13.2853	0.719439
$342$	0	0
$343$	−19.2749	−1.04075
$344$	12.2662	0.661347
$345$	0	0
$346$	13.7248	0.737848
$347$	−32.4393	−1.74143	−0.870717	−	0.491785i	$-0.836345\pi$
$347$	−32.4393	−1.74143	−0.870717	+	0.491785i	$0.836345\pi$
$348$	0	0
$349$	31.2646	1.67355	0.836777	−	0.547544i	$-0.184437\pi$
$349$	31.2646	1.67355	0.836777	+	0.547544i	$0.184437\pi$
$350$	−2.28400	−0.122085
$351$	0	0
$352$	−12.7610	−0.680162
$353$	8.40764	0.447494	0.223747	−	0.974647i	$-0.428171\pi$
$353$	8.40764	0.447494	0.223747	+	0.974647i	$0.428171\pi$
$354$	0	0
$355$	−10.7069	−0.568265
$356$	−2.50518	−0.132774
$357$	0	0
$358$	−7.49367	−0.396053
$359$	1.16842	0.0616670	0.0308335	−	0.999525i	$-0.490184\pi$
$359$	1.16842	0.0616670	0.0308335	+	0.999525i	$0.490184\pi$
$360$	0	0
$361$	−9.61916	−0.506272
$362$	14.0583	0.738887
$363$	0	0
$364$	0.624987	0.0327582
$365$	12.4479	0.651552
$366$	0	0
$367$	−8.07606	−0.421567	−0.210783	−	0.977533i	$-0.567601\pi$
$367$	−8.07606	−0.421567	−0.210783	+	0.977533i	$0.567601\pi$
$368$	−23.4986	−1.22495
$369$	0	0
$370$	−7.58219	−0.394180
$371$	−0.0592825	−0.00307779
$372$	0	0
$373$	−18.4595	−0.955798	−0.477899	−	0.878415i	$-0.658602\pi$
$373$	−18.4595	−0.955798	−0.477899	+	0.878415i	$0.658602\pi$
$374$	11.5256	0.595974
$375$	0	0
$376$	−23.8645	−1.23072
$377$	−5.26487	−0.271155
$378$	0	0
$379$	13.6829	0.702842	0.351421	−	0.936218i	$-0.385699\pi$
$379$	13.6829	0.702842	0.351421	+	0.936218i	$0.385699\pi$
$380$	−1.07612	−0.0552037
$381$	0	0
$382$	−24.5407	−1.25561
$383$	7.03436	0.359439	0.179719	−	0.983718i	$-0.442481\pi$
$383$	7.03436	0.359439	0.179719	+	0.983718i	$0.442481\pi$
$384$	0	0
$385$	−11.5635	−0.589328
$386$	11.0175	0.560775
$387$	0	0
$388$	2.24658	0.114053
$389$	10.1415	0.514194	0.257097	−	0.966386i	$-0.417234\pi$
$389$	10.1415	0.514194	0.257097	+	0.966386i	$0.417234\pi$
$390$	0	0
$391$	−10.2235	−0.517022
$392$	−11.5808	−0.584918
$393$	0	0
$394$	−2.02495	−0.102016
$395$	−5.36753	−0.270070
$396$	0	0
$397$	26.2953	1.31973	0.659863	−	0.751386i	$-0.270614\pi$
$397$	26.2953	1.31973	0.659863	+	0.751386i	$0.270614\pi$
$398$	−30.5923	−1.53345
$399$	0	0
$400$	−3.17385	−0.158693
$401$	−8.05867	−0.402431	−0.201215	−	0.979547i	$-0.564489\pi$
$401$	−8.05867	−0.402431	−0.201215	+	0.979547i	$0.564489\pi$
$402$	0	0
$403$	−2.04369	−0.101803
$404$	−6.42349	−0.319581
$405$	0	0
$406$	−12.0250	−0.596788
$407$	−38.3873	−1.90279
$408$	0	0
$409$	23.0479	1.13964	0.569822	−	0.821768i	$-0.307012\pi$
$409$	23.0479	1.13964	0.569822	+	0.821768i	$0.307012\pi$
$410$	−2.97544	−0.146947
$411$	0	0
$412$	−6.66848	−0.328532
$413$	1.62499	0.0799604
$414$	0	0
$415$	−6.17839	−0.303285
$416$	1.96303	0.0962456
$417$	0	0
$418$	25.5647	1.25041
$419$	18.6220	0.909746	0.454873	−	0.890556i	$-0.349685\pi$
$419$	18.6220	0.909746	0.454873	+	0.890556i	$0.349685\pi$
$420$	0	0
$421$	25.2184	1.22907	0.614536	−	0.788889i	$-0.289343\pi$
$421$	25.2184	1.22907	0.614536	+	0.788889i	$0.289343\pi$
$422$	−17.5997	−0.856738
$423$	0	0
$424$	−0.100618	−0.00488645
$425$	−1.38084	−0.0669805
$426$	0	0
$427$	−20.6470	−0.999178
$428$	3.83799	0.185516
$429$	0	0
$430$	5.21665	0.251569
$431$	−10.5285	−0.507142	−0.253571	−	0.967317i	$-0.581605\pi$
$431$	−10.5285	−0.507142	−0.253571	+	0.967317i	$0.581605\pi$
$432$	0	0
$433$	−39.1255	−1.88025	−0.940126	−	0.340826i	$-0.889293\pi$
$433$	−39.1255	−1.88025	−0.940126	+	0.340826i	$0.889293\pi$
$434$	−4.66778	−0.224061
$435$	0	0
$436$	−2.24793	−0.107656
$437$	−22.6765	−1.08476
$438$	0	0
$439$	−11.4686	−0.547367	−0.273683	−	0.961820i	$-0.588242\pi$
$439$	−11.4686	−0.547367	−0.273683	+	0.961820i	$0.588242\pi$
$440$	−19.6263	−0.935646
$441$	0	0
$442$	−1.77299	−0.0843327
$443$	27.4770	1.30547	0.652736	−	0.757585i	$-0.273621\pi$
$443$	27.4770	1.30547	0.652736	+	0.757585i	$0.273621\pi$
$444$	0	0
$445$	−7.13017	−0.338002
$446$	3.80920	0.180371
$447$	0	0
$448$	15.7750	0.745298
$449$	5.01400	0.236625	0.118313	−	0.992976i	$-0.462251\pi$
$449$	5.01400	0.236625	0.118313	+	0.992976i	$0.462251\pi$
$450$	0	0
$451$	−15.0641	−0.709341
$452$	1.37002	0.0644405
$453$	0	0
$454$	−2.23981	−0.105119
$455$	1.77882	0.0833923
$456$	0	0
$457$	−23.4352	−1.09625	−0.548126	−	0.836396i	$-0.684658\pi$
$457$	−23.4352	−1.09625	−0.548126	+	0.836396i	$0.684658\pi$
$458$	11.5796	0.541081
$459$	0	0
$460$	2.60132	0.121287
$461$	35.2778	1.64305	0.821526	−	0.570172i	$-0.193123\pi$
$461$	35.2778	1.64305	0.821526	+	0.570172i	$0.193123\pi$
$462$	0	0
$463$	−17.6379	−0.819704	−0.409852	−	0.912152i	$-0.634420\pi$
$463$	−17.6379	−0.819704	−0.409852	+	0.912152i	$0.634420\pi$
$464$	−16.7099	−0.775739
$465$	0	0
$466$	37.1149	1.71931
$467$	27.6843	1.28108	0.640538	−	0.767926i	$-0.278711\pi$
$467$	27.6843	1.28108	0.640538	+	0.767926i	$0.278711\pi$
$468$	0	0
$469$	28.7630	1.32815
$470$	−10.1493	−0.468152
$471$	0	0
$472$	2.75804	0.126949
$473$	26.4109	1.21438
$474$	0	0
$475$	−3.06282	−0.140532
$476$	0.863005	0.0395558
$477$	0	0
$478$	15.6936	0.717810
$479$	17.5249	0.800733	0.400367	−	0.916355i	$-0.368883\pi$
$479$	17.5249	0.800733	0.400367	+	0.916355i	$0.368883\pi$
$480$	0	0
$481$	5.90515	0.269251
$482$	−29.4488	−1.34136
$483$	0	0
$484$	−10.9826	−0.499210
$485$	6.39414	0.290343
$486$	0	0
$487$	27.2820	1.23627	0.618134	−	0.786073i	$-0.287889\pi$
$487$	27.2820	1.23627	0.618134	+	0.786073i	$0.287889\pi$
$488$	−35.0435	−1.58634
$489$	0	0
$490$	−4.92517	−0.222496
$491$	32.8663	1.48323	0.741617	−	0.670823i	$-0.234059\pi$
$491$	32.8663	1.48323	0.741617	+	0.670823i	$0.234059\pi$
$492$	0	0
$493$	−7.26993	−0.327421
$494$	−3.93265	−0.176938
$495$	0	0
$496$	−6.48637	−0.291246
$497$	19.0457	0.854315
$498$	0	0
$499$	−43.0915	−1.92904	−0.964521	−	0.264005i	$-0.914956\pi$
$499$	−43.0915	−1.92904	−0.964521	+	0.264005i	$0.914956\pi$
$500$	0.351349	0.0157128
$501$	0	0
$502$	−17.5015	−0.781132
$503$	−21.3397	−0.951491	−0.475746	−	0.879583i	$-0.657822\pi$
$503$	−21.3397	−0.951491	−0.475746	+	0.879583i	$0.657822\pi$
$504$	0	0
$505$	−18.2823	−0.813554
$506$	−61.7982	−2.74726
$507$	0	0
$508$	2.15882	0.0957821
$509$	−34.6748	−1.53693	−0.768467	−	0.639889i	$-0.778980\pi$
$509$	−34.6748	−1.53693	−0.768467	+	0.639889i	$0.778980\pi$
$510$	0	0
$511$	−22.1425	−0.979527
$512$	25.3949	1.12231
$513$	0	0
$514$	29.7611	1.31271
$515$	−18.9796	−0.836342
$516$	0	0
$517$	−51.3840	−2.25987
$518$	13.4873	0.592600
$519$	0	0
$520$	3.01913	0.132398
$521$	−42.8273	−1.87630	−0.938148	−	0.346234i	$-0.887460\pi$
$521$	−42.8273	−1.87630	−0.938148	+	0.346234i	$0.887460\pi$
$522$	0	0
$523$	1.51490	0.0662421	0.0331211	−	0.999451i	$-0.489455\pi$
$523$	1.51490	0.0662421	0.0331211	+	0.999451i	$0.489455\pi$
$524$	−2.43086	−0.106193
$525$	0	0
$526$	−8.99003	−0.391984
$527$	−2.82200	−0.122928
$528$	0	0
$529$	31.8163	1.38332
$530$	−0.0427917	−0.00185875
$531$	0	0
$532$	1.91422	0.0829919
$533$	2.31732	0.100374
$534$	0	0
$535$	10.9236	0.472267
$536$	48.8184	2.10863
$537$	0	0
$538$	2.12804	0.0917464
$539$	−24.9352	−1.07404
$540$	0	0
$541$	−17.2312	−0.740826	−0.370413	−	0.928867i	$-0.620784\pi$
$541$	−17.2312	−0.740826	−0.370413	+	0.928867i	$0.620784\pi$
$542$	−1.31604	−0.0565286
$543$	0	0
$544$	2.71063	0.116217
$545$	−6.39798	−0.274059
$546$	0	0
$547$	−17.1739	−0.734301	−0.367150	−	0.930162i	$-0.619667\pi$
$547$	−17.1739	−0.734301	−0.367150	+	0.930162i	$0.619667\pi$
$548$	−6.82923	−0.291730
$549$	0	0
$550$	−8.34681	−0.355909
$551$	−16.1253	−0.686962
$552$	0	0
$553$	9.54786	0.406017
$554$	14.6116	0.620786
$555$	0	0
$556$	−1.14482	−0.0485511
$557$	40.6651	1.72304	0.861518	−	0.507727i	$-0.169514\pi$
$557$	40.6651	1.72304	0.861518	+	0.507727i	$0.169514\pi$
$558$	0	0
$559$	−4.06282	−0.171839
$560$	5.64571	0.238575
$561$	0	0
$562$	−17.4293	−0.735209
$563$	−2.07603	−0.0874943	−0.0437472	−	0.999043i	$-0.513930\pi$
$563$	−2.07603	−0.0874943	−0.0437472	+	0.999043i	$0.513930\pi$
$564$	0	0
$565$	3.89932	0.164046
$566$	0.896185	0.0376695
$567$	0	0
$568$	32.3256	1.35635
$569$	10.1882	0.427114	0.213557	−	0.976931i	$-0.431495\pi$
$569$	10.1882	0.427114	0.213557	+	0.976931i	$0.431495\pi$
$570$	0	0
$571$	9.41735	0.394104	0.197052	−	0.980393i	$-0.436863\pi$
$571$	9.41735	0.394104	0.197052	+	0.980393i	$0.436863\pi$
$572$	2.28400	0.0954987
$573$	0	0
$574$	5.29276	0.220916
$575$	7.40380	0.308760
$576$	0	0
$577$	−19.6650	−0.818666	−0.409333	−	0.912385i	$-0.634239\pi$
$577$	−19.6650	−0.818666	−0.409333	+	0.912385i	$0.634239\pi$
$578$	19.3797	0.806091
$579$	0	0
$580$	1.84981	0.0768092
$581$	10.9902	0.455952
$582$	0	0
$583$	−0.216646	−0.00897257
$584$	−37.5818	−1.55514
$585$	0	0
$586$	−3.59600	−0.148549
$587$	−5.75801	−0.237658	−0.118829	−	0.992915i	$-0.537914\pi$
$587$	−5.75801	−0.237658	−0.118829	+	0.992915i	$0.537914\pi$
$588$	0	0
$589$	−6.25944	−0.257916
$590$	1.17296	0.0482899
$591$	0	0
$592$	18.7421	0.770295
$593$	30.4075	1.24869	0.624343	−	0.781151i	$-0.285367\pi$
$593$	30.4075	1.24869	0.624343	+	0.781151i	$0.285367\pi$
$594$	0	0
$595$	2.45626	0.100697
$596$	1.57503	0.0645158
$597$	0	0
$598$	9.50647	0.388748
$599$	−35.9183	−1.46758	−0.733791	−	0.679376i	$-0.762251\pi$
$599$	−35.9183	−1.46758	−0.733791	+	0.679376i	$0.762251\pi$
$600$	0	0
$601$	−22.9698	−0.936959	−0.468480	−	0.883474i	$-0.655198\pi$
$601$	−22.9698	−0.936959	−0.468480	+	0.883474i	$0.655198\pi$
$602$	−9.27946	−0.378203
$603$	0	0
$604$	−2.74639	−0.111749
$605$	−31.2584	−1.27083
$606$	0	0
$607$	−40.6745	−1.65093	−0.825464	−	0.564455i	$-0.809086\pi$
$607$	−40.6745	−1.65093	−0.825464	+	0.564455i	$0.809086\pi$
$608$	6.01241	0.243835
$609$	0	0
$610$	−14.9036	−0.603427
$611$	7.90445	0.319780
$612$	0	0
$613$	−20.4430	−0.825683	−0.412842	−	0.910803i	$-0.635464\pi$
$613$	−20.4430	−0.825683	−0.412842	+	0.910803i	$0.635464\pi$
$614$	20.2072	0.815495
$615$	0	0
$616$	34.9116	1.40663
$617$	−18.5269	−0.745864	−0.372932	−	0.927859i	$-0.621648\pi$
$617$	−18.5269	−0.745864	−0.372932	+	0.927859i	$0.621648\pi$
$618$	0	0
$619$	−19.6266	−0.788860	−0.394430	−	0.918926i	$-0.629058\pi$
$619$	−19.6266	−0.788860	−0.394430	+	0.918926i	$0.629058\pi$
$620$	0.718049	0.0288375
$621$	0	0
$622$	40.4122	1.62038
$623$	12.6833	0.508144
$624$	0	0
$625$	1.00000	0.0400000
$626$	−12.9020	−0.515668
$627$	0	0
$628$	5.59121	0.223113
$629$	8.15405	0.325123
$630$	0	0
$631$	34.4444	1.37121	0.685605	−	0.727974i	$-0.259538\pi$
$631$	34.4444	1.37121	0.685605	+	0.727974i	$0.259538\pi$
$632$	16.2053	0.644611
$633$	0	0
$634$	−18.9636	−0.753139
$635$	6.14437	0.243832
$636$	0	0
$637$	3.83581	0.151980
$638$	−43.9449	−1.73979
$639$	0	0
$640$	7.46073	0.294911
$641$	17.8899	0.706608	0.353304	−	0.935509i	$-0.385058\pi$
$641$	17.8899	0.706608	0.353304	+	0.935509i	$0.385058\pi$
$642$	0	0
$643$	−31.8293	−1.25522	−0.627612	−	0.778526i	$-0.715968\pi$
$643$	−31.8293	−1.25522	−0.627612	+	0.778526i	$0.715968\pi$
$644$	−4.62728	−0.182340
$645$	0	0
$646$	−5.43035	−0.213654
$647$	−3.53179	−0.138849	−0.0694244	−	0.997587i	$-0.522116\pi$
$647$	−3.53179	−0.138849	−0.0694244	+	0.997587i	$0.522116\pi$
$648$	0	0
$649$	5.93847	0.233105
$650$	1.28400	0.0503625
$651$	0	0
$652$	−4.95836	−0.194184
$653$	28.6851	1.12253	0.561267	−	0.827635i	$-0.310314\pi$
$653$	28.6851	1.12253	0.561267	+	0.827635i	$0.310314\pi$
$654$	0	0
$655$	−6.91865	−0.270334
$656$	7.35485	0.287159
$657$	0	0
$658$	18.0537	0.703808
$659$	−47.0370	−1.83230	−0.916151	−	0.400834i	$-0.868720\pi$
$659$	−47.0370	−1.83230	−0.916151	+	0.400834i	$0.868720\pi$
$660$	0	0
$661$	−5.82124	−0.226420	−0.113210	−	0.993571i	$-0.536113\pi$
$661$	−5.82124	−0.226420	−0.113210	+	0.993571i	$0.536113\pi$
$662$	12.2187	0.474893
$663$	0	0
$664$	18.6534	0.723890
$665$	5.44819	0.211272
$666$	0	0
$667$	38.9801	1.50931
$668$	−1.22291	−0.0473160
$669$	0	0
$670$	20.7619	0.802101
$671$	−75.4539	−2.91287
$672$	0	0
$673$	3.56320	0.137351	0.0686757	−	0.997639i	$-0.478123\pi$
$673$	3.56320	0.137351	0.0686757	+	0.997639i	$0.478123\pi$
$674$	−14.7867	−0.569562
$675$	0	0
$676$	−0.351349	−0.0135134
$677$	25.9420	0.997032	0.498516	−	0.866880i	$-0.333878\pi$
$677$	25.9420	0.997032	0.498516	+	0.866880i	$0.333878\pi$
$678$	0	0
$679$	−11.3740	−0.436494
$680$	4.16893	0.159871
$681$	0	0
$682$	−17.0583	−0.653195
$683$	37.8982	1.45013	0.725067	−	0.688679i	$-0.241809\pi$
$683$	37.8982	1.45013	0.725067	+	0.688679i	$0.241809\pi$
$684$	0	0
$685$	−19.4371	−0.742655
$686$	24.7490	0.944920
$687$	0	0
$688$	−12.8948	−0.491609
$689$	0.0333269	0.00126965
$690$	0	0
$691$	−27.0681	−1.02972	−0.514860	−	0.857274i	$-0.672156\pi$
$691$	−27.0681	−1.02972	−0.514860	+	0.857274i	$0.672156\pi$
$692$	3.75561	0.142767
$693$	0	0
$694$	41.6520	1.58109
$695$	−3.25835	−0.123596
$696$	0	0
$697$	3.19985	0.121203
$698$	−40.1436	−1.51946
$699$	0	0
$700$	−0.624987	−0.0236223
$701$	−10.2636	−0.387651	−0.193826	−	0.981036i	$-0.562090\pi$
$701$	−10.2636	−0.387651	−0.193826	+	0.981036i	$0.562090\pi$
$702$	0	0
$703$	18.0864	0.682140
$704$	57.6493	2.17274
$705$	0	0
$706$	−10.7954	−0.406290
$707$	32.5210	1.22308
$708$	0	0
$709$	−21.5255	−0.808407	−0.404203	−	0.914669i	$-0.632451\pi$
$709$	−21.5255	−0.808407	−0.404203	+	0.914669i	$0.632451\pi$
$710$	13.7477	0.515941
$711$	0	0
$712$	21.5269	0.806754
$713$	15.1311	0.566663
$714$	0	0
$715$	6.50064	0.243110
$716$	−2.05055	−0.0766325
$717$	0	0
$718$	−1.50025	−0.0559889
$719$	4.65831	0.173726	0.0868629	−	0.996220i	$-0.472316\pi$
$719$	4.65831	0.173726	0.0868629	+	0.996220i	$0.472316\pi$
$720$	0	0
$721$	33.7613	1.25734
$722$	12.3510	0.459656
$723$	0	0
$724$	3.84687	0.142968
$725$	5.26487	0.195532
$726$	0	0
$727$	8.21625	0.304724	0.152362	−	0.988325i	$-0.451312\pi$
$727$	8.21625	0.304724	0.152362	+	0.988325i	$0.451312\pi$
$728$	−5.37048	−0.199043
$729$	0	0
$730$	−15.9831	−0.591559
$731$	−5.61009	−0.207497
$732$	0	0
$733$	43.7441	1.61573	0.807863	−	0.589370i	$-0.200624\pi$
$733$	43.7441	1.61573	0.807863	+	0.589370i	$0.200624\pi$
$734$	10.3696	0.382750
$735$	0	0
$736$	−14.5339	−0.535727
$737$	105.114	3.87191
$738$	0	0
$739$	21.5748	0.793642	0.396821	−	0.917896i	$-0.370113\pi$
$739$	21.5748	0.793642	0.396821	+	0.917896i	$0.370113\pi$
$740$	−2.07477	−0.0762701
$741$	0	0
$742$	0.0761186	0.00279440
$743$	32.7977	1.20323	0.601615	−	0.798786i	$-0.294524\pi$
$743$	32.7977	1.20323	0.601615	+	0.798786i	$0.294524\pi$
$744$	0	0
$745$	4.48280	0.164237
$746$	23.7020	0.867792
$747$	0	0
$748$	3.15383	0.115315
$749$	−19.4310	−0.709995
$750$	0	0
$751$	37.6923	1.37541	0.687705	−	0.725990i	$-0.258618\pi$
$751$	37.6923	1.37541	0.687705	+	0.725990i	$0.258618\pi$
$752$	25.0876	0.914850
$753$	0	0
$754$	6.76008	0.246188
$755$	−7.81668	−0.284478
$756$	0	0
$757$	−21.0879	−0.766451	−0.383226	−	0.923655i	$-0.625187\pi$
$757$	−21.0879	−0.766451	−0.383226	+	0.923655i	$0.625187\pi$
$758$	−17.5688	−0.638127
$759$	0	0
$760$	9.24703	0.335425
$761$	13.1276	0.475876	0.237938	−	0.971280i	$-0.423529\pi$
$761$	13.1276	0.475876	0.237938	+	0.971280i	$0.423529\pi$
$762$	0	0
$763$	11.3808	0.412014
$764$	−6.71524	−0.242949
$765$	0	0
$766$	−9.03210	−0.326343
$767$	−0.913521	−0.0329853
$768$	0	0
$769$	−19.6453	−0.708429	−0.354215	−	0.935164i	$-0.615252\pi$
$769$	−19.6453	−0.708429	−0.354215	+	0.935164i	$0.615252\pi$
$770$	14.8475	0.535065
$771$	0	0
$772$	3.01479	0.108505
$773$	26.2053	0.942540	0.471270	−	0.881989i	$-0.343796\pi$
$773$	26.2053	0.942540	0.471270	+	0.881989i	$0.343796\pi$
$774$	0	0
$775$	2.04369	0.0734114
$776$	−19.3047	−0.692999
$777$	0	0
$778$	−13.0216	−0.466849
$779$	7.09754	0.254296
$780$	0	0
$781$	69.6019	2.49055
$782$	13.1269	0.469417
$783$	0	0
$784$	12.1743	0.434796
$785$	15.9135	0.567978
$786$	0	0
$787$	12.1688	0.433771	0.216886	−	0.976197i	$-0.430410\pi$
$787$	12.1688	0.433771	0.216886	+	0.976197i	$0.430410\pi$
$788$	−0.554102	−0.0197391
$789$	0	0
$790$	6.89190	0.245203
$791$	−6.93618	−0.246622
$792$	0	0
$793$	11.6071	0.412182
$794$	−33.7632	−1.19821
$795$	0	0
$796$	−8.37117	−0.296708
$797$	0.747869	0.0264909	0.0132454	−	0.999912i	$-0.495784\pi$
$797$	0.747869	0.0264909	0.0132454	+	0.999912i	$0.495784\pi$
$798$	0	0
$799$	10.9148	0.386137
$800$	−1.96303	−0.0694037
$801$	0	0
$802$	10.3473	0.365376
$803$	−80.9193	−2.85558
$804$	0	0
$805$	−13.1700	−0.464182
$806$	2.62409	0.0924296
$807$	0	0
$808$	55.1967	1.94181
$809$	−3.92646	−0.138047	−0.0690235	−	0.997615i	$-0.521988\pi$
$809$	−3.92646	−0.138047	−0.0690235	+	0.997615i	$0.521988\pi$
$810$	0	0
$811$	−45.0478	−1.58184	−0.790920	−	0.611919i	$-0.790398\pi$
$811$	−45.0478	−1.58184	−0.790920	+	0.611919i	$0.790398\pi$
$812$	−3.29047	−0.115473
$813$	0	0
$814$	49.2892	1.72758
$815$	−14.1123	−0.494333
$816$	0	0
$817$	−12.4437	−0.435348
$818$	−29.5934	−1.03471
$819$	0	0
$820$	−0.814191	−0.0284328
$821$	−2.62736	−0.0916957	−0.0458478	−	0.998948i	$-0.514599\pi$
$821$	−2.62736	−0.0916957	−0.0458478	+	0.998948i	$0.514599\pi$
$822$	0	0
$823$	33.8965	1.18156	0.590778	−	0.806834i	$-0.298821\pi$
$823$	33.8965	1.18156	0.590778	+	0.806834i	$0.298821\pi$
$824$	57.3019	1.99621
$825$	0	0
$826$	−2.08648	−0.0725979
$827$	−30.7147	−1.06806	−0.534028	−	0.845467i	$-0.679322\pi$
$827$	−30.7147	−1.06806	−0.534028	+	0.845467i	$0.679322\pi$
$828$	0	0
$829$	51.0987	1.77473	0.887366	−	0.461066i	$-0.152533\pi$
$829$	51.0987	1.77473	0.887366	+	0.461066i	$0.152533\pi$
$830$	7.93304	0.275360
$831$	0	0
$832$	−8.86824	−0.307451
$833$	5.29663	0.183517
$834$	0	0
$835$	−3.48062	−0.120452
$836$	6.99546	0.241943
$837$	0	0
$838$	−23.9107	−0.825980
$839$	30.2779	1.04531	0.522654	−	0.852545i	$-0.324942\pi$
$839$	30.2779	1.04531	0.522654	+	0.852545i	$0.324942\pi$
$840$	0	0
$841$	−1.28114	−0.0441773
$842$	−32.3804	−1.11590
$843$	0	0
$844$	−4.81592	−0.165771
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	55.6029	1.91054
$848$	0.105775	0.00363232
$849$	0	0
$850$	1.77299	0.0608132
$851$	−43.7205	−1.49872
$852$	0	0
$853$	−46.9788	−1.60852	−0.804262	−	0.594275i	$-0.797439\pi$
$853$	−46.9788	−1.60852	−0.804262	+	0.594275i	$0.797439\pi$
$854$	26.5107	0.907177
$855$	0	0
$856$	−32.9797	−1.12722
$857$	14.3706	0.490889	0.245445	−	0.969411i	$-0.421066\pi$
$857$	14.3706	0.490889	0.245445	+	0.969411i	$0.421066\pi$
$858$	0	0
$859$	−25.0842	−0.855861	−0.427931	−	0.903812i	$-0.640757\pi$
$859$	−25.0842	−0.855861	−0.427931	+	0.903812i	$0.640757\pi$
$860$	1.42747	0.0486763
$861$	0	0
$862$	13.5186	0.460446
$863$	29.2581	0.995957	0.497978	−	0.867189i	$-0.334076\pi$
$863$	29.2581	0.995957	0.497978	+	0.867189i	$0.334076\pi$
$864$	0	0
$865$	10.6891	0.363440
$866$	50.2371	1.70713
$867$	0	0
$868$	−1.27728	−0.0433536
$869$	34.8924	1.18364
$870$	0	0
$871$	−16.1697	−0.547890
$872$	19.3163	0.654133
$873$	0	0
$874$	29.1166	0.984882
$875$	−1.77882	−0.0601350
$876$	0	0
$877$	12.8951	0.435437	0.217719	−	0.976012i	$-0.430139\pi$
$877$	12.8951	0.435437	0.217719	+	0.976012i	$0.430139\pi$
$878$	14.7257	0.496967
$879$	0	0
$880$	20.6321	0.695508
$881$	−30.9913	−1.04412	−0.522061	−	0.852908i	$-0.674837\pi$
$881$	−30.9913	−1.04412	−0.522061	+	0.852908i	$0.674837\pi$
$882$	0	0
$883$	−45.5323	−1.53228	−0.766141	−	0.642672i	$-0.777826\pi$
$883$	−45.5323	−1.53228	−0.766141	+	0.642672i	$0.777826\pi$
$884$	−0.485157	−0.0163176
$885$	0	0
$886$	−35.2804	−1.18527
$887$	47.4143	1.59202	0.796008	−	0.605286i	$-0.206941\pi$
$887$	47.4143	1.59202	0.796008	+	0.605286i	$0.206941\pi$
$888$	0	0
$889$	−10.9297	−0.366571
$890$	9.15512	0.306880
$891$	0	0
$892$	1.04234	0.0349001
$893$	24.2099	0.810152
$894$	0	0
$895$	−5.83620	−0.195083
$896$	−13.2713	−0.443363
$897$	0	0
$898$	−6.43797	−0.214838
$899$	10.7598	0.358858
$900$	0	0
$901$	0.0460190	0.00153312
$902$	19.3423	0.644027
$903$	0	0
$904$	−11.7725	−0.391549
$905$	10.9488	0.363952
$906$	0	0
$907$	43.0630	1.42988	0.714942	−	0.699183i	$-0.246453\pi$
$907$	43.0630	1.42988	0.714942	+	0.699183i	$0.246453\pi$
$908$	−0.612894	−0.0203396
$909$	0	0
$910$	−2.28400	−0.0757138
$911$	−50.4424	−1.67123	−0.835615	−	0.549315i	$-0.814889\pi$
$911$	−50.4424	−1.67123	−0.835615	+	0.549315i	$0.814889\pi$
$912$	0	0
$913$	40.1635	1.32922
$914$	30.0907	0.995312
$915$	0	0
$916$	3.16862	0.104694
$917$	12.3070	0.406413
$918$	0	0
$919$	−56.6069	−1.86729	−0.933644	−	0.358202i	$-0.883390\pi$
$919$	−56.6069	−1.86729	−0.933644	+	0.358202i	$0.883390\pi$
$920$	−22.3530	−0.736958
$921$	0	0
$922$	−45.2966	−1.49177
$923$	−10.7069	−0.352423
$924$	0	0
$925$	−5.90515	−0.194160
$926$	22.6471	0.744228
$927$	0	0
$928$	−10.3351	−0.339267
$929$	16.8102	0.551526	0.275763	−	0.961226i	$-0.411070\pi$
$929$	16.8102	0.551526	0.275763	+	0.961226i	$0.411070\pi$
$930$	0	0
$931$	11.7484	0.385037
$932$	10.1560	0.332671
$933$	0	0
$934$	−35.5466	−1.16312
$935$	8.97634	0.293558
$936$	0	0
$937$	19.9729	0.652486	0.326243	−	0.945286i	$-0.394217\pi$
$937$	19.9729	0.652486	0.326243	+	0.945286i	$0.394217\pi$
$938$	−36.9316	−1.20586
$939$	0	0
$940$	−2.77722	−0.0905830
$941$	−8.63415	−0.281465	−0.140733	−	0.990048i	$-0.544946\pi$
$941$	−8.63415	−0.281465	−0.140733	+	0.990048i	$0.544946\pi$
$942$	0	0
$943$	−17.1570	−0.558709
$944$	−2.89938	−0.0943668
$945$	0	0
$946$	−33.9116	−1.10256
$947$	−28.4106	−0.923221	−0.461610	−	0.887083i	$-0.652728\pi$
$947$	−28.4106	−0.923221	−0.461610	+	0.887083i	$0.652728\pi$
$948$	0	0
$949$	12.4479	0.404075
$950$	3.93265	0.127592
$951$	0	0
$952$	−7.41576	−0.240346
$953$	−40.7119	−1.31879	−0.659394	−	0.751798i	$-0.729187\pi$
$953$	−40.7119	−1.31879	−0.659394	+	0.751798i	$0.729187\pi$
$954$	0	0
$955$	−19.1127	−0.618473
$956$	4.29436	0.138889
$957$	0	0
$958$	−22.5019	−0.727004
$959$	34.5751	1.11649
$960$	0	0
$961$	−26.8233	−0.865269
$962$	−7.58219	−0.244460
$963$	0	0
$964$	−8.05828	−0.259540
$965$	8.58060	0.276219
$966$	0	0
$967$	11.1111	0.357308	0.178654	−	0.983912i	$-0.442826\pi$
$967$	11.1111	0.357308	0.178654	+	0.983912i	$0.442826\pi$
$968$	94.3730	3.03326
$969$	0	0
$970$	−8.21006	−0.263609
$971$	58.4121	1.87453	0.937267	−	0.348611i	$-0.113347\pi$
$971$	58.4121	1.87453	0.937267	+	0.348611i	$0.113347\pi$
$972$	0	0
$973$	5.79601	0.185811
$974$	−35.0301	−1.12244
$975$	0	0
$976$	36.8394	1.17920
$977$	−18.4783	−0.591174	−0.295587	−	0.955316i	$-0.595515\pi$
$977$	−18.4783	−0.591174	−0.295587	+	0.955316i	$0.595515\pi$
$978$	0	0
$979$	46.3507	1.48137
$980$	−1.34771	−0.0430510
$981$	0	0
$982$	−42.2002	−1.34666
$983$	−23.0929	−0.736548	−0.368274	−	0.929717i	$-0.620051\pi$
$983$	−23.0929	−0.736548	−0.368274	+	0.929717i	$0.620051\pi$
$984$	0	0
$985$	−1.57707	−0.0502496
$986$	9.33458	0.297274
$987$	0	0
$988$	−1.07612	−0.0342359
$989$	30.0803	0.956498
$990$	0	0
$991$	−37.8267	−1.20160	−0.600802	−	0.799398i	$-0.705152\pi$
$991$	−37.8267	−1.20160	−0.600802	+	0.799398i	$0.705152\pi$
$992$	−4.01183	−0.127376
$993$	0	0
$994$	−24.4546	−0.775653
$995$	−23.8258	−0.755328
$996$	0	0
$997$	24.0361	0.761231	0.380615	−	0.924733i	$-0.375712\pi$
$997$	24.0361	0.761231	0.380615	+	0.924733i	$0.375712\pi$
$998$	55.3295	1.75142
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1755.2.a.t.1.1	yes	4
3.2	odd	2		1755.2.a.n.1.4	✓	4
5.4	even	2		8775.2.a.bg.1.4		4
15.14	odd	2		8775.2.a.bs.1.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1755.2.a.n.1.4	✓	4	3.2	odd	2
1755.2.a.t.1.1	yes	4	1.1	even	1	trivial
8775.2.a.bg.1.4		4	5.4	even	2
8775.2.a.bs.1.1		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)

\(f(q)\)	\(=\)	\(q-1.28400 q^{2} -0.351349 q^{4} -1.00000 q^{5} +1.77882 q^{7} +3.01913 q^{8} +O(q^{10})\)	\(q-1.28400 q^{2} -0.351349 q^{4} -1.00000 q^{5} +1.77882 q^{7} +3.01913 q^{8} +1.28400 q^{10} +6.50064 q^{11} -1.00000 q^{13} -2.28400 q^{14} -3.17385 q^{16} -1.38084 q^{17} -3.06282 q^{19} +0.351349 q^{20} -8.34681 q^{22} +7.40380 q^{23} +1.00000 q^{25} +1.28400 q^{26} -0.624987 q^{28} +5.26487 q^{29} +2.04369 q^{31} -1.96303 q^{32} +1.77299 q^{34} -1.77882 q^{35} -5.90515 q^{37} +3.93265 q^{38} -3.01913 q^{40} -2.31732 q^{41} +4.06282 q^{43} -2.28400 q^{44} -9.50647 q^{46} -7.90445 q^{47} -3.83581 q^{49} -1.28400 q^{50} +0.351349 q^{52} -0.0333269 q^{53} -6.50064 q^{55} +5.37048 q^{56} -6.76008 q^{58} +0.913521 q^{59} -11.6071 q^{61} -2.62409 q^{62} +8.86824 q^{64} +1.00000 q^{65} +16.1697 q^{67} +0.485157 q^{68} +2.28400 q^{70} +10.7069 q^{71} -12.4479 q^{73} +7.58219 q^{74} +1.07612 q^{76} +11.5635 q^{77} +5.36753 q^{79} +3.17385 q^{80} +2.97544 q^{82} +6.17839 q^{83} +1.38084 q^{85} -5.21665 q^{86} +19.6263 q^{88} +7.13017 q^{89} -1.77882 q^{91} -2.60132 q^{92} +10.1493 q^{94} +3.06282 q^{95} -6.39414 q^{97} +4.92517 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{2} + 3 q^{4} - 4 q^{5} - q^{7} + 9 q^{8}+O(q^{10}) \) 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

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