LMFDB - Embedded newform 1755.2.a.s.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1755.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$14.0137455547$
Analytic rank:	$0$
Dimension:	$4$
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			1.4
Root			$-0.693822$ 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)

$f(q)$	$=$	$q+2.44129 q^{2} +3.95990 q^{4} -1.00000 q^{5} -1.87843 q^{7} +4.78469 q^{8} +O(q^{10})$	\(q+2.44129 q^{2} +3.95990 q^{4} -1.00000 q^{5} -1.87843 q^{7} +4.78469 q^{8} -2.44129 q^{10} +3.90626 q^{11} +1.00000 q^{13} -4.58580 q^{14} +3.76102 q^{16} -0.707370 q^{17} +8.23953 q^{19} -3.95990 q^{20} +9.53631 q^{22} +5.63005 q^{23} +1.00000 q^{25} +2.44129 q^{26} -7.43841 q^{28} +5.08147 q^{29} +8.59935 q^{31} -0.387645 q^{32} -1.72690 q^{34} +1.87843 q^{35} -4.74747 q^{37} +20.1151 q^{38} -4.78469 q^{40} -2.85891 q^{41} -7.89923 q^{43} +15.4684 q^{44} +13.7446 q^{46} -0.0330729 q^{47} -3.47149 q^{49} +2.44129 q^{50} +3.95990 q^{52} +0.507941 q^{53} -3.90626 q^{55} -8.98772 q^{56} +12.4053 q^{58} +4.72379 q^{59} -10.1385 q^{61} +20.9935 q^{62} -8.46839 q^{64} -1.00000 q^{65} -1.45069 q^{67} -2.80111 q^{68} +4.58580 q^{70} -9.59410 q^{71} -13.6642 q^{73} -11.5900 q^{74} +32.6277 q^{76} -7.33764 q^{77} +12.8456 q^{79} -3.76102 q^{80} -6.97943 q^{82} -9.80526 q^{83} +0.707370 q^{85} -19.2843 q^{86} +18.6902 q^{88} -10.8055 q^{89} -1.87843 q^{91} +22.2944 q^{92} -0.0807406 q^{94} -8.23953 q^{95} +3.64360 q^{97} -8.47491 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	2.44129	1.72625	0.863127	−	0.504987i	$-0.168503\pi$
$2$	2.44129	1.72625	0.863127	+	0.504987i	$0.168503\pi$
$3$	0	0
$3$	0	0
$4$	3.95990	1.97995
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	−1.87843	−0.709981	−0.354991	−	0.934870i	$-0.615516\pi$
$7$	−1.87843	−0.709981	−0.354991	+	0.934870i	$0.615516\pi$
$8$	4.78469	1.69164
$9$	0	0
$10$	−2.44129	−0.772004
$11$	3.90626	1.17778	0.588890	−	0.808213i	$-0.299565\pi$
$11$	3.90626	1.17778	0.588890	+	0.808213i	$0.299565\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−4.58580	−1.22561
$15$	0	0
$16$	3.76102	0.940254
$17$	−0.707370	−0.171562	−0.0857812	−	0.996314i	$-0.527339\pi$
$17$	−0.707370	−0.171562	−0.0857812	+	0.996314i	$0.527339\pi$
$18$	0	0
$19$	8.23953	1.89028	0.945139	−	0.326670i	$-0.105926\pi$
$19$	8.23953	1.89028	0.945139	+	0.326670i	$0.105926\pi$
$20$	−3.95990	−0.885461
$21$	0	0
$22$	9.53631	2.03315
$23$	5.63005	1.17395	0.586973	−	0.809606i	$-0.300319\pi$
$23$	5.63005	1.17395	0.586973	+	0.809606i	$0.300319\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	2.44129	0.478777
$27$	0	0
$28$	−7.43841	−1.40573
$29$	5.08147	0.943605	0.471802	−	0.881704i	$-0.343604\pi$
$29$	5.08147	0.943605	0.471802	+	0.881704i	$0.343604\pi$
$30$	0	0
$31$	8.59935	1.54449	0.772245	−	0.635325i	$-0.219134\pi$
$31$	8.59935	1.54449	0.772245	+	0.635325i	$0.219134\pi$
$32$	−0.387645	−0.0685266
$33$	0	0
$34$	−1.72690	−0.296160
$35$	1.87843	0.317513
$36$	0	0
$37$	−4.74747	−0.780479	−0.390240	−	0.920713i	$-0.627608\pi$
$37$	−4.74747	−0.780479	−0.390240	+	0.920713i	$0.627608\pi$
$38$	20.1151	3.26310
$39$	0	0
$40$	−4.78469	−0.756526
$41$	−2.85891	−0.446486	−0.223243	−	0.974763i	$-0.571664\pi$
$41$	−2.85891	−0.446486	−0.223243	+	0.974763i	$0.571664\pi$
$42$	0	0
$43$	−7.89923	−1.20462	−0.602311	−	0.798262i	$-0.705753\pi$
$43$	−7.89923	−1.20462	−0.602311	+	0.798262i	$0.705753\pi$
$44$	15.4684	2.33195
$45$	0	0
$46$	13.7446	2.02653
$47$	−0.0330729	−0.00482418	−0.00241209	−	0.999997i	$-0.500768\pi$
$47$	−0.0330729	−0.00482418	−0.00241209	+	0.999997i	$0.500768\pi$
$48$	0	0
$49$	−3.47149	−0.495927
$50$	2.44129	0.345251
$51$	0	0
$52$	3.95990	0.549140
$53$	0.507941	0.0697711	0.0348855	−	0.999391i	$-0.488893\pi$
$53$	0.507941	0.0697711	0.0348855	+	0.999391i	$0.488893\pi$
$54$	0	0
$55$	−3.90626	−0.526719
$56$	−8.98772	−1.20103
$57$	0	0
$58$	12.4053	1.62890
$59$	4.72379	0.614986	0.307493	−	0.951550i	$-0.400510\pi$
$59$	4.72379	0.614986	0.307493	+	0.951550i	$0.400510\pi$
$60$	0	0
$61$	−10.1385	−1.29811	−0.649053	−	0.760743i	$-0.724835\pi$
$61$	−10.1385	−1.29811	−0.649053	+	0.760743i	$0.724835\pi$
$62$	20.9935	2.66618
$63$	0	0
$64$	−8.46839	−1.05855
$65$	−1.00000	−0.124035
$66$	0	0
$67$	−1.45069	−0.177230	−0.0886150	−	0.996066i	$-0.528244\pi$
$67$	−1.45069	−0.177230	−0.0886150	+	0.996066i	$0.528244\pi$
$68$	−2.80111	−0.339685
$69$	0	0
$70$	4.58580	0.548108
$71$	−9.59410	−1.13861	−0.569305	−	0.822126i	$-0.692788\pi$
$71$	−9.59410	−1.13861	−0.569305	+	0.822126i	$0.692788\pi$
$72$	0	0
$73$	−13.6642	−1.59927	−0.799635	−	0.600486i	$-0.794974\pi$
$73$	−13.6642	−1.59927	−0.799635	+	0.600486i	$0.794974\pi$
$74$	−11.5900	−1.34730
$75$	0	0
$76$	32.6277	3.74266
$77$	−7.33764	−0.836202
$78$	0	0
$79$	12.8456	1.44524	0.722621	−	0.691245i	$-0.242937\pi$
$79$	12.8456	1.44524	0.722621	+	0.691245i	$0.242937\pi$
$80$	−3.76102	−0.420494
$81$	0	0
$82$	−6.97943	−0.770749
$83$	−9.80526	−1.07627	−0.538134	−	0.842860i	$-0.680870\pi$
$83$	−9.80526	−1.07627	−0.538134	+	0.842860i	$0.680870\pi$
$84$	0	0
$85$	0.707370	0.0767250
$86$	−19.2843	−2.07948
$87$	0	0
$88$	18.6902	1.99238
$89$	−10.8055	−1.14538	−0.572690	−	0.819772i	$-0.694100\pi$
$89$	−10.8055	−1.14538	−0.572690	+	0.819772i	$0.694100\pi$
$90$	0	0
$91$	−1.87843	−0.196913
$92$	22.2944	2.32436
$93$	0	0
$94$	−0.0807406	−0.00832776
$95$	−8.23953	−0.845358
$96$	0	0
$97$	3.64360	0.369951	0.184976	−	0.982743i	$-0.440779\pi$
$97$	3.64360	0.369951	0.184976	+	0.982743i	$0.440779\pi$
$98$	−8.47491	−0.856095
$99$	0	0
$100$	3.95990	0.395990
$101$	11.7941	1.17356	0.586778	−	0.809748i	$-0.300396\pi$
$101$	11.7941	1.17356	0.586778	+	0.809748i	$0.300396\pi$
$102$	0	0
$103$	−15.7540	−1.55229	−0.776143	−	0.630556i	$-0.782827\pi$
$103$	−15.7540	−1.55229	−0.776143	+	0.630556i	$0.782827\pi$
$104$	4.78469	0.469177
$105$	0	0
$106$	1.24003	0.120443
$107$	4.03197	0.389785	0.194893	−	0.980825i	$-0.437564\pi$
$107$	4.03197	0.389785	0.194893	+	0.980825i	$0.437564\pi$
$108$	0	0
$109$	−6.56700	−0.629005	−0.314502	−	0.949257i	$-0.601838\pi$
$109$	−6.56700	−0.629005	−0.314502	+	0.949257i	$0.601838\pi$
$110$	−9.53631	−0.909251
$111$	0	0
$112$	−7.06482	−0.667563
$113$	8.96515	0.843371	0.421685	−	0.906742i	$-0.361439\pi$
$113$	8.96515	0.843371	0.421685	+	0.906742i	$0.361439\pi$
$114$	0	0
$115$	−5.63005	−0.525005
$116$	20.1221	1.86829
$117$	0	0
$118$	11.5322	1.06162
$119$	1.32875	0.121806
$120$	0	0
$121$	4.25883	0.387166
$122$	−24.7511	−2.24086
$123$	0	0
$124$	34.0526	3.05801
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	3.74769	0.332554	0.166277	−	0.986079i	$-0.446825\pi$
$127$	3.74769	0.332554	0.166277	+	0.986079i	$0.446825\pi$
$128$	−19.8985	−1.75880
$129$	0	0
$130$	−2.44129	−0.214115
$131$	−1.13201	−0.0989044	−0.0494522	−	0.998776i	$-0.515748\pi$
$131$	−1.13201	−0.0989044	−0.0494522	+	0.998776i	$0.515748\pi$
$132$	0	0
$133$	−15.4774	−1.34206
$134$	−3.54156	−0.305944
$135$	0	0
$136$	−3.38454	−0.290222
$137$	19.1966	1.64007	0.820036	−	0.572312i	$-0.193953\pi$
$137$	19.1966	1.64007	0.820036	+	0.572312i	$0.193953\pi$
$138$	0	0
$139$	−3.13926	−0.266269	−0.133134	−	0.991098i	$-0.542504\pi$
$139$	−3.13926	−0.266269	−0.133134	+	0.991098i	$0.542504\pi$
$140$	7.43841	0.628661
$141$	0	0
$142$	−23.4220	−1.96553
$143$	3.90626	0.326657
$144$	0	0
$145$	−5.08147	−0.421993
$146$	−33.3582	−2.76075
$147$	0	0
$148$	−18.7995	−1.54531
$149$	22.6895	1.85880	0.929398	−	0.369080i	$-0.120327\pi$
$149$	22.6895	1.85880	0.929398	+	0.369080i	$0.120327\pi$
$150$	0	0
$151$	10.2663	0.835460	0.417730	−	0.908571i	$-0.362826\pi$
$151$	10.2663	0.835460	0.417730	+	0.908571i	$0.362826\pi$
$152$	39.4236	3.19767
$153$	0	0
$154$	−17.9133	−1.44350
$155$	−8.59935	−0.690716
$156$	0	0
$157$	−3.23323	−0.258040	−0.129020	−	0.991642i	$-0.541183\pi$
$157$	−3.23323	−0.258040	−0.129020	+	0.991642i	$0.541183\pi$
$158$	31.3598	2.49485
$159$	0	0
$160$	0.387645	0.0306460
$161$	−10.5757	−0.833480
$162$	0	0
$163$	−12.2809	−0.961914	−0.480957	−	0.876744i	$-0.659711\pi$
$163$	−12.2809	−0.961914	−0.480957	+	0.876744i	$0.659711\pi$
$164$	−11.3210	−0.884021
$165$	0	0
$166$	−23.9375	−1.85791
$167$	11.3963	0.881874	0.440937	−	0.897538i	$-0.354646\pi$
$167$	11.3963	0.881874	0.440937	+	0.897538i	$0.354646\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	1.72690	0.132447
$171$	0	0
$172$	−31.2802	−2.38509
$173$	−20.6964	−1.57352	−0.786758	−	0.617262i	$-0.788242\pi$
$173$	−20.6964	−1.57352	−0.786758	+	0.617262i	$0.788242\pi$
$174$	0	0
$175$	−1.87843	−0.141996
$176$	14.6915	1.10741
$177$	0	0
$178$	−26.3793	−1.97721
$179$	5.48139	0.409698	0.204849	−	0.978794i	$-0.434330\pi$
$179$	5.48139	0.409698	0.204849	+	0.978794i	$0.434330\pi$
$180$	0	0
$181$	−21.6814	−1.61156	−0.805781	−	0.592213i	$-0.798254\pi$
$181$	−21.6814	−1.61156	−0.805781	+	0.592213i	$0.798254\pi$
$182$	−4.58580	−0.339922
$183$	0	0
$184$	26.9380	1.98590
$185$	4.74747	0.349041
$186$	0	0
$187$	−2.76317	−0.202063
$188$	−0.130965	−0.00955164
$189$	0	0
$190$	−20.1151	−1.45930
$191$	−13.6745	−0.989453	−0.494727	−	0.869049i	$-0.664732\pi$
$191$	−13.6745	−0.989453	−0.494727	+	0.869049i	$0.664732\pi$
$192$	0	0
$193$	11.3931	0.820091	0.410045	−	0.912065i	$-0.365513\pi$
$193$	11.3931	0.820091	0.410045	+	0.912065i	$0.365513\pi$
$194$	8.89508	0.638630
$195$	0	0
$196$	−13.7467	−0.981910
$197$	20.2002	1.43920	0.719601	−	0.694388i	$-0.244325\pi$
$197$	20.2002	1.43920	0.719601	+	0.694388i	$0.244325\pi$
$198$	0	0
$199$	−11.4025	−0.808299	−0.404150	−	0.914693i	$-0.632432\pi$
$199$	−11.4025	−0.808299	−0.404150	+	0.914693i	$0.632432\pi$
$200$	4.78469	0.338329
$201$	0	0
$202$	28.7928	2.02585
$203$	−9.54520	−0.669942
$204$	0	0
$205$	2.85891	0.199675
$206$	−38.4601	−2.67964
$207$	0	0
$208$	3.76102	0.260780
$209$	32.1857	2.22633
$210$	0	0
$211$	8.58005	0.590675	0.295337	−	0.955393i	$-0.404568\pi$
$211$	8.58005	0.590675	0.295337	+	0.955393i	$0.404568\pi$
$212$	2.01140	0.138143
$213$	0	0
$214$	9.84321	0.672868
$215$	7.89923	0.538723
$216$	0	0
$217$	−16.1533	−1.09656
$218$	−16.0320	−1.08582
$219$	0	0
$220$	−15.4684	−1.04288
$221$	−0.707370	−0.0475828
$222$	0	0
$223$	−19.5647	−1.31015	−0.655074	−	0.755565i	$-0.727362\pi$
$223$	−19.5647	−1.31015	−0.655074	+	0.755565i	$0.727362\pi$
$224$	0.728165	0.0486526
$225$	0	0
$226$	21.8865	1.45587
$227$	−18.3427	−1.21745	−0.608723	−	0.793383i	$-0.708318\pi$
$227$	−18.3427	−1.21745	−0.608723	+	0.793383i	$0.708318\pi$
$228$	0	0
$229$	−5.50738	−0.363938	−0.181969	−	0.983304i	$-0.558247\pi$
$229$	−5.50738	−0.363938	−0.181969	+	0.983304i	$0.558247\pi$
$230$	−13.7446	−0.906291
$231$	0	0
$232$	24.3132	1.59624
$233$	−7.32172	−0.479662	−0.239831	−	0.970815i	$-0.577092\pi$
$233$	−7.32172	−0.479662	−0.239831	+	0.970815i	$0.577092\pi$
$234$	0	0
$235$	0.0330729	0.00215744
$236$	18.7058	1.21764
$237$	0	0
$238$	3.24386	0.210268
$239$	30.4774	1.97142	0.985711	−	0.168445i	$-0.0538745\pi$
$239$	30.4774	1.97142	0.985711	+	0.168445i	$0.0538745\pi$
$240$	0	0
$241$	−23.5317	−1.51581	−0.757904	−	0.652367i	$-0.773776\pi$
$241$	−23.5317	−1.51581	−0.757904	+	0.652367i	$0.773776\pi$
$242$	10.3970	0.668347
$243$	0	0
$244$	−40.1476	−2.57019
$245$	3.47149	0.221785
$246$	0	0
$247$	8.23953	0.524269
$248$	41.1452	2.61272
$249$	0	0
$250$	−2.44129	−0.154401
$251$	−1.13296	−0.0715119	−0.0357560	−	0.999361i	$-0.511384\pi$
$251$	−1.13296	−0.0715119	−0.0357560	+	0.999361i	$0.511384\pi$
$252$	0	0
$253$	21.9924	1.38265
$254$	9.14920	0.574072
$255$	0	0
$256$	−31.6413	−1.97758
$257$	24.1465	1.50622	0.753109	−	0.657896i	$-0.228553\pi$
$257$	24.1465	1.50622	0.753109	+	0.657896i	$0.228553\pi$
$258$	0	0
$259$	8.91781	0.554125
$260$	−3.95990	−0.245583
$261$	0	0
$262$	−2.76357	−0.170734
$263$	−5.70012	−0.351485	−0.175742	−	0.984436i	$-0.556233\pi$
$263$	−5.70012	−0.351485	−0.175742	+	0.984436i	$0.556233\pi$
$264$	0	0
$265$	−0.507941	−0.0312026
$266$	−37.7849	−2.31674
$267$	0	0
$268$	−5.74459	−0.350907
$269$	16.9435	1.03306	0.516531	−	0.856268i	$-0.327223\pi$
$269$	16.9435	1.03306	0.516531	+	0.856268i	$0.327223\pi$
$270$	0	0
$271$	−5.23355	−0.317915	−0.158958	−	0.987285i	$-0.550813\pi$
$271$	−5.23355	−0.317915	−0.158958	+	0.987285i	$0.550813\pi$
$272$	−2.66043	−0.161312
$273$	0	0
$274$	46.8644	2.83118
$275$	3.90626	0.235556
$276$	0	0
$277$	3.98883	0.239665	0.119833	−	0.992794i	$-0.461764\pi$
$277$	3.98883	0.239665	0.119833	+	0.992794i	$0.461764\pi$
$278$	−7.66385	−0.459647
$279$	0	0
$280$	8.98772	0.537119
$281$	3.14163	0.187414	0.0937071	−	0.995600i	$-0.470128\pi$
$281$	3.14163	0.187414	0.0937071	+	0.995600i	$0.470128\pi$
$282$	0	0
$283$	15.8933	0.944756	0.472378	−	0.881396i	$-0.343396\pi$
$283$	15.8933	0.944756	0.472378	+	0.881396i	$0.343396\pi$
$284$	−37.9917	−2.25439
$285$	0	0
$286$	9.53631	0.563894
$287$	5.37027	0.316997
$288$	0	0
$289$	−16.4996	−0.970566
$290$	−12.4053	−0.728467
$291$	0	0
$292$	−54.1088	−3.16648
$293$	−1.75162	−0.102331	−0.0511653	−	0.998690i	$-0.516294\pi$
$293$	−1.75162	−0.102331	−0.0511653	+	0.998690i	$0.516294\pi$
$294$	0	0
$295$	−4.72379	−0.275030
$296$	−22.7152	−1.32029
$297$	0	0
$298$	55.3916	3.20875
$299$	5.63005	0.325594
$300$	0	0
$301$	14.8382	0.855259
$302$	25.0630	1.44222
$303$	0	0
$304$	30.9890	1.77734
$305$	10.1385	0.580531
$306$	0	0
$307$	−18.5808	−1.06046	−0.530230	−	0.847854i	$-0.677895\pi$
$307$	−18.5808	−1.06046	−0.530230	+	0.847854i	$0.677895\pi$
$308$	−29.0563	−1.65564
$309$	0	0
$310$	−20.9935	−1.19235
$311$	−7.71759	−0.437625	−0.218812	−	0.975767i	$-0.570218\pi$
$311$	−7.71759	−0.437625	−0.218812	+	0.975767i	$0.570218\pi$
$312$	0	0
$313$	−5.65860	−0.319843	−0.159921	−	0.987130i	$-0.551124\pi$
$313$	−5.65860	−0.319843	−0.159921	+	0.987130i	$0.551124\pi$
$314$	−7.89325	−0.445442
$315$	0	0
$316$	50.8672	2.86151
$317$	−3.48161	−0.195547	−0.0977734	−	0.995209i	$-0.531172\pi$
$317$	−3.48161	−0.195547	−0.0977734	+	0.995209i	$0.531172\pi$
$318$	0	0
$319$	19.8495	1.11136
$320$	8.46839	0.473397
$321$	0	0
$322$	−25.8183	−1.43880
$323$	−5.82839	−0.324300
$324$	0	0
$325$	1.00000	0.0554700
$326$	−29.9812	−1.66051
$327$	0	0
$328$	−13.6790	−0.755296
$329$	0.0621253	0.00342508
$330$	0	0
$331$	0.184389	0.0101349	0.00506747	−	0.999987i	$-0.498387\pi$
$331$	0.184389	0.0101349	0.00506747	+	0.999987i	$0.498387\pi$
$332$	−38.8279	−2.13096
$333$	0	0
$334$	27.8217	1.52234
$335$	1.45069	0.0792597
$336$	0	0
$337$	16.5566	0.901895	0.450948	−	0.892550i	$-0.351086\pi$
$337$	16.5566	0.901895	0.450948	+	0.892550i	$0.351086\pi$
$338$	2.44129	0.132789
$339$	0	0
$340$	2.80111	0.151912
$341$	33.5913	1.81907
$342$	0	0
$343$	19.6700	1.06208
$344$	−37.7954	−2.03779
$345$	0	0
$346$	−50.5258	−2.71629
$347$	−28.4153	−1.52541	−0.762706	−	0.646745i	$-0.776130\pi$
$347$	−28.4153	−1.52541	−0.762706	+	0.646745i	$0.776130\pi$
$348$	0	0
$349$	−3.41674	−0.182894	−0.0914468	−	0.995810i	$-0.529149\pi$
$349$	−3.41674	−0.182894	−0.0914468	+	0.995810i	$0.529149\pi$
$350$	−4.58580	−0.245122
$351$	0	0
$352$	−1.51424	−0.0807093
$353$	−4.40894	−0.234664	−0.117332	−	0.993093i	$-0.537434\pi$
$353$	−4.40894	−0.234664	−0.117332	+	0.993093i	$0.537434\pi$
$354$	0	0
$355$	9.59410	0.509202
$356$	−42.7887	−2.26779
$357$	0	0
$358$	13.3817	0.707243
$359$	−16.0656	−0.847909	−0.423955	−	0.905683i	$-0.639358\pi$
$359$	−16.0656	−0.847909	−0.423955	+	0.905683i	$0.639358\pi$
$360$	0	0
$361$	48.8898	2.57315
$362$	−52.9305	−2.78197
$363$	0	0
$364$	−7.43841	−0.389879
$365$	13.6642	0.715215
$366$	0	0
$367$	−13.2919	−0.693832	−0.346916	−	0.937896i	$-0.612771\pi$
$367$	−13.2919	−0.693832	−0.346916	+	0.937896i	$0.612771\pi$
$368$	21.1747	1.10381
$369$	0	0
$370$	11.5900	0.602533
$371$	−0.954134	−0.0495361
$372$	0	0
$373$	16.1692	0.837211	0.418606	−	0.908168i	$-0.362519\pi$
$373$	16.1692	0.837211	0.418606	+	0.908168i	$0.362519\pi$
$374$	−6.74569	−0.348812
$375$	0	0
$376$	−0.158244	−0.00816079
$377$	5.08147	0.261709
$378$	0	0
$379$	−9.76722	−0.501708	−0.250854	−	0.968025i	$-0.580711\pi$
$379$	−9.76722	−0.501708	−0.250854	+	0.968025i	$0.580711\pi$
$380$	−32.6277	−1.67377
$381$	0	0
$382$	−33.3835	−1.70805
$383$	37.8683	1.93498	0.967490	−	0.252911i	$-0.0813879\pi$
$383$	37.8683	1.93498	0.967490	+	0.252911i	$0.0813879\pi$
$384$	0	0
$385$	7.33764	0.373961
$386$	27.8138	1.41568
$387$	0	0
$388$	14.4283	0.732485
$389$	33.9695	1.72232	0.861160	−	0.508334i	$-0.169738\pi$
$389$	33.9695	1.72232	0.861160	+	0.508334i	$0.169738\pi$
$390$	0	0
$391$	−3.98253	−0.201405
$392$	−16.6100	−0.838931
$393$	0	0
$394$	49.3145	2.48443
$395$	−12.8456	−0.646331
$396$	0	0
$397$	11.9438	0.599442	0.299721	−	0.954027i	$-0.403106\pi$
$397$	11.9438	0.599442	0.299721	+	0.954027i	$0.403106\pi$
$398$	−27.8367	−1.39533
$399$	0	0
$400$	3.76102	0.188051
$401$	−18.4368	−0.920690	−0.460345	−	0.887740i	$-0.652274\pi$
$401$	−18.4368	−0.920690	−0.460345	+	0.887740i	$0.652274\pi$
$402$	0	0
$403$	8.59935	0.428364
$404$	46.7034	2.32358
$405$	0	0
$406$	−23.3026	−1.15649
$407$	−18.5448	−0.919233
$408$	0	0
$409$	−18.3696	−0.908319	−0.454159	−	0.890920i	$-0.650060\pi$
$409$	−18.3696	−0.908319	−0.454159	+	0.890920i	$0.650060\pi$
$410$	6.97943	0.344689
$411$	0	0
$412$	−62.3842	−3.07345
$413$	−8.87334	−0.436628
$414$	0	0
$415$	9.80526	0.481321
$416$	−0.387645	−0.0190059
$417$	0	0
$418$	78.5747	3.84321
$419$	−28.1232	−1.37391	−0.686954	−	0.726700i	$-0.741053\pi$
$419$	−28.1232	−1.37391	−0.686954	+	0.726700i	$0.741053\pi$
$420$	0	0
$421$	−23.8082	−1.16034	−0.580170	−	0.814496i	$-0.697014\pi$
$421$	−23.8082	−1.16034	−0.580170	+	0.814496i	$0.697014\pi$
$422$	20.9464	1.01965
$423$	0	0
$424$	2.43034	0.118028
$425$	−0.707370	−0.0343125
$426$	0	0
$427$	19.0446	0.921631
$428$	15.9662	0.771755
$429$	0	0
$430$	19.2843	0.929972
$431$	−19.5212	−0.940301	−0.470150	−	0.882586i	$-0.655800\pi$
$431$	−19.5212	−0.940301	−0.470150	+	0.882586i	$0.655800\pi$
$432$	0	0
$433$	20.9819	1.00833	0.504163	−	0.863608i	$-0.331801\pi$
$433$	20.9819	1.00833	0.504163	+	0.863608i	$0.331801\pi$
$434$	−39.4349	−1.89294
$435$	0	0
$436$	−26.0047	−1.24540
$437$	46.3890	2.21908
$438$	0	0
$439$	−12.5366	−0.598341	−0.299170	−	0.954200i	$-0.596710\pi$
$439$	−12.5366	−0.598341	−0.299170	+	0.954200i	$0.596710\pi$
$440$	−18.6902	−0.891021
$441$	0	0
$442$	−1.72690	−0.0821400
$443$	−2.99744	−0.142413	−0.0712064	−	0.997462i	$-0.522685\pi$
$443$	−2.99744	−0.142413	−0.0712064	+	0.997462i	$0.522685\pi$
$444$	0	0
$445$	10.8055	0.512229
$446$	−47.7631	−2.26165
$447$	0	0
$448$	15.9073	0.751549
$449$	−34.0842	−1.60853	−0.804267	−	0.594268i	$-0.797442\pi$
$449$	−34.0842	−1.60853	−0.804267	+	0.594268i	$0.797442\pi$
$450$	0	0
$451$	−11.1676	−0.525863
$452$	35.5011	1.66983
$453$	0	0
$454$	−44.7798	−2.10162
$455$	1.87843	0.0880623
$456$	0	0
$457$	7.42646	0.347395	0.173697	−	0.984799i	$-0.444429\pi$
$457$	7.42646	0.347395	0.173697	+	0.984799i	$0.444429\pi$
$458$	−13.4451	−0.628249
$459$	0	0
$460$	−22.2944	−1.03948
$461$	−36.3918	−1.69494	−0.847468	−	0.530846i	$-0.821874\pi$
$461$	−36.3918	−1.69494	−0.847468	+	0.530846i	$0.821874\pi$
$462$	0	0
$463$	−40.5236	−1.88329	−0.941646	−	0.336604i	$-0.890722\pi$
$463$	−40.5236	−1.88329	−0.941646	+	0.336604i	$0.890722\pi$
$464$	19.1115	0.887228
$465$	0	0
$466$	−17.8745	−0.828018
$467$	−29.8604	−1.38177	−0.690887	−	0.722962i	$-0.742780\pi$
$467$	−29.8604	−1.38177	−0.690887	+	0.722962i	$0.742780\pi$
$468$	0	0
$469$	2.72503	0.125830
$470$	0.0807406	0.00372429
$471$	0	0
$472$	22.6019	1.04034
$473$	−30.8564	−1.41878
$474$	0	0
$475$	8.23953	0.378055
$476$	5.26171	0.241170
$477$	0	0
$478$	74.4043	3.40317
$479$	13.2997	0.607679	0.303839	−	0.952723i	$-0.401731\pi$
$479$	13.2997	0.607679	0.303839	+	0.952723i	$0.401731\pi$
$480$	0	0
$481$	−4.74747	−0.216466
$482$	−57.4476	−2.61667
$483$	0	0
$484$	16.8645	0.766570
$485$	−3.64360	−0.165447
$486$	0	0
$487$	42.2973	1.91667	0.958337	−	0.285640i	$-0.0922063\pi$
$487$	42.2973	1.91667	0.958337	+	0.285640i	$0.0922063\pi$
$488$	−48.5097	−2.19593
$489$	0	0
$490$	8.47491	0.382857
$491$	14.0921	0.635969	0.317985	−	0.948096i	$-0.396994\pi$
$491$	14.0921	0.635969	0.317985	+	0.948096i	$0.396994\pi$
$492$	0	0
$493$	−3.59448	−0.161887
$494$	20.1151	0.905020
$495$	0	0
$496$	32.3423	1.45221
$497$	18.0219	0.808392
$498$	0	0
$499$	−4.00364	−0.179228	−0.0896139	−	0.995977i	$-0.528563\pi$
$499$	−4.00364	−0.179228	−0.0896139	+	0.995977i	$0.528563\pi$
$500$	−3.95990	−0.177092
$501$	0	0
$502$	−2.76589	−0.123448
$503$	−4.25070	−0.189529	−0.0947647	−	0.995500i	$-0.530210\pi$
$503$	−4.25070	−0.189529	−0.0947647	+	0.995500i	$0.530210\pi$
$504$	0	0
$505$	−11.7941	−0.524830
$506$	53.6899	2.38681
$507$	0	0
$508$	14.8405	0.658440
$509$	−18.5127	−0.820560	−0.410280	−	0.911960i	$-0.634569\pi$
$509$	−18.5127	−0.820560	−0.410280	+	0.911960i	$0.634569\pi$
$510$	0	0
$511$	25.6672	1.13545
$512$	−37.4485	−1.65501
$513$	0	0
$514$	58.9487	2.60011
$515$	15.7540	0.694204
$516$	0	0
$517$	−0.129191	−0.00568182
$518$	21.7710	0.956561
$519$	0	0
$520$	−4.78469	−0.209823
$521$	25.7584	1.12849	0.564247	−	0.825606i	$-0.309167\pi$
$521$	25.7584	1.12849	0.564247	+	0.825606i	$0.309167\pi$
$522$	0	0
$523$	−10.8411	−0.474046	−0.237023	−	0.971504i	$-0.576172\pi$
$523$	−10.8411	−0.474046	−0.237023	+	0.971504i	$0.576172\pi$
$524$	−4.48266	−0.195826
$525$	0	0
$526$	−13.9157	−0.606751
$527$	−6.08292	−0.264976
$528$	0	0
$529$	8.69747	0.378151
$530$	−1.24003	−0.0538635
$531$	0	0
$532$	−61.2890	−2.65722
$533$	−2.85891	−0.123833
$534$	0	0
$535$	−4.03197	−0.174317
$536$	−6.94110	−0.299810
$537$	0	0
$538$	41.3640	1.78333
$539$	−13.5605	−0.584093
$540$	0	0
$541$	−8.54843	−0.367526	−0.183763	−	0.982971i	$-0.558828\pi$
$541$	−8.54843	−0.367526	−0.183763	+	0.982971i	$0.558828\pi$
$542$	−12.7766	−0.548803
$543$	0	0
$544$	0.274208	0.0117566
$545$	6.56700	0.281300
$546$	0	0
$547$	−36.5905	−1.56450	−0.782248	−	0.622967i	$-0.785927\pi$
$547$	−36.5905	−1.56450	−0.782248	+	0.622967i	$0.785927\pi$
$548$	76.0164	3.24726
$549$	0	0
$550$	9.53631	0.406629
$551$	41.8689	1.78367
$552$	0	0
$553$	−24.1296	−1.02609
$554$	9.73789	0.413723
$555$	0	0
$556$	−12.4312	−0.527199
$557$	−38.9411	−1.64999	−0.824995	−	0.565141i	$-0.808822\pi$
$557$	−38.9411	−1.64999	−0.824995	+	0.565141i	$0.808822\pi$
$558$	0	0
$559$	−7.89923	−0.334102
$560$	7.06482	0.298543
$561$	0	0
$562$	7.66964	0.323524
$563$	19.8100	0.834890	0.417445	−	0.908702i	$-0.362926\pi$
$563$	19.8100	0.834890	0.417445	+	0.908702i	$0.362926\pi$
$564$	0	0
$565$	−8.96515	−0.377167
$566$	38.8000	1.63089
$567$	0	0
$568$	−45.9048	−1.92612
$569$	2.56606	0.107575	0.0537873	−	0.998552i	$-0.482871\pi$
$569$	2.56606	0.107575	0.0537873	+	0.998552i	$0.482871\pi$
$570$	0	0
$571$	−8.93567	−0.373946	−0.186973	−	0.982365i	$-0.559868\pi$
$571$	−8.93567	−0.373946	−0.186973	+	0.982365i	$0.559868\pi$
$572$	15.4684	0.646766
$573$	0	0
$574$	13.1104	0.547217
$575$	5.63005	0.234789
$576$	0	0
$577$	−3.17002	−0.131970	−0.0659848	−	0.997821i	$-0.521019\pi$
$577$	−3.17002	−0.131970	−0.0659848	+	0.997821i	$0.521019\pi$
$578$	−40.2804	−1.67544
$579$	0	0
$580$	−20.1221	−0.835525
$581$	18.4185	0.764130
$582$	0	0
$583$	1.98415	0.0821750
$584$	−65.3788	−2.70539
$585$	0	0
$586$	−4.27621	−0.176648
$587$	22.1670	0.914928	0.457464	−	0.889228i	$-0.348758\pi$
$587$	22.1670	0.914928	0.457464	+	0.889228i	$0.348758\pi$
$588$	0	0
$589$	70.8546	2.91951
$590$	−11.5322	−0.474771
$591$	0	0
$592$	−17.8553	−0.733848
$593$	−13.3143	−0.546752	−0.273376	−	0.961907i	$-0.588140\pi$
$593$	−13.3143	−0.546752	−0.273376	+	0.961907i	$0.588140\pi$
$594$	0	0
$595$	−1.32875	−0.0544733
$596$	89.8481	3.68032
$597$	0	0
$598$	13.7446	0.562058
$599$	17.5427	0.716774	0.358387	−	0.933573i	$-0.383327\pi$
$599$	17.5427	0.716774	0.358387	+	0.933573i	$0.383327\pi$
$600$	0	0
$601$	−21.8997	−0.893308	−0.446654	−	0.894707i	$-0.647385\pi$
$601$	−21.8997	−0.893308	−0.446654	+	0.894707i	$0.647385\pi$
$602$	36.2243	1.47639
$603$	0	0
$604$	40.6535	1.65417
$605$	−4.25883	−0.173146
$606$	0	0
$607$	24.0784	0.977312	0.488656	−	0.872476i	$-0.337487\pi$
$607$	24.0784	0.977312	0.488656	+	0.872476i	$0.337487\pi$
$608$	−3.19401	−0.129534
$609$	0	0
$610$	24.7511	1.00214
$611$	−0.0330729	−0.00133799
$612$	0	0
$613$	15.5255	0.627067	0.313534	−	0.949577i	$-0.398487\pi$
$613$	15.5255	0.627067	0.313534	+	0.949577i	$0.398487\pi$
$614$	−45.3611	−1.83062
$615$	0	0
$616$	−35.1083	−1.41456
$617$	−2.29438	−0.0923681	−0.0461840	−	0.998933i	$-0.514706\pi$
$617$	−2.29438	−0.0923681	−0.0461840	+	0.998933i	$0.514706\pi$
$618$	0	0
$619$	−32.7126	−1.31483	−0.657415	−	0.753528i	$-0.728350\pi$
$619$	−32.7126	−1.31483	−0.657415	+	0.753528i	$0.728350\pi$
$620$	−34.0526	−1.36758
$621$	0	0
$622$	−18.8409	−0.755451
$623$	20.2974	0.813198
$624$	0	0
$625$	1.00000	0.0400000
$626$	−13.8143	−0.552130
$627$	0	0
$628$	−12.8033	−0.510906
$629$	3.35822	0.133901
$630$	0	0
$631$	12.3560	0.491884	0.245942	−	0.969284i	$-0.420903\pi$
$631$	12.3560	0.491884	0.245942	+	0.969284i	$0.420903\pi$
$632$	61.4621	2.44483
$633$	0	0
$634$	−8.49963	−0.337563
$635$	−3.74769	−0.148723
$636$	0	0
$637$	−3.47149	−0.137545
$638$	48.4584	1.91849
$639$	0	0
$640$	19.8985	0.786557
$641$	33.8352	1.33641	0.668206	−	0.743977i	$-0.267063\pi$
$641$	33.8352	1.33641	0.668206	+	0.743977i	$0.267063\pi$
$642$	0	0
$643$	6.68209	0.263516	0.131758	−	0.991282i	$-0.457938\pi$
$643$	6.68209	0.263516	0.131758	+	0.991282i	$0.457938\pi$
$644$	−41.8786	−1.65025
$645$	0	0
$646$	−14.2288	−0.559825
$647$	49.4950	1.94585	0.972924	−	0.231126i	$-0.0742410\pi$
$647$	49.4950	1.94585	0.972924	+	0.231126i	$0.0742410\pi$
$648$	0	0
$649$	18.4523	0.724318
$650$	2.44129	0.0957553
$651$	0	0
$652$	−48.6311	−1.90454
$653$	15.7399	0.615950	0.307975	−	0.951394i	$-0.400349\pi$
$653$	15.7399	0.615950	0.307975	+	0.951394i	$0.400349\pi$
$654$	0	0
$655$	1.13201	0.0442314
$656$	−10.7524	−0.419811
$657$	0	0
$658$	0.151666	0.00591255
$659$	23.3938	0.911293	0.455646	−	0.890161i	$-0.349408\pi$
$659$	23.3938	0.911293	0.455646	+	0.890161i	$0.349408\pi$
$660$	0	0
$661$	16.0146	0.622897	0.311449	−	0.950263i	$-0.399186\pi$
$661$	16.0146	0.622897	0.311449	+	0.950263i	$0.399186\pi$
$662$	0.450147	0.0174955
$663$	0	0
$664$	−46.9151	−1.82066
$665$	15.4774	0.600188
$666$	0	0
$667$	28.6089	1.10774
$668$	45.1283	1.74607
$669$	0	0
$670$	3.54156	0.136822
$671$	−39.6037	−1.52888
$672$	0	0
$673$	15.5333	0.598763	0.299382	−	0.954133i	$-0.403220\pi$
$673$	15.5333	0.598763	0.299382	+	0.954133i	$0.403220\pi$
$674$	40.4195	1.55690
$675$	0	0
$676$	3.95990	0.152304
$677$	−39.8295	−1.53077	−0.765385	−	0.643572i	$-0.777452\pi$
$677$	−39.8295	−1.53077	−0.765385	+	0.643572i	$0.777452\pi$
$678$	0	0
$679$	−6.84426	−0.262658
$680$	3.38454	0.129791
$681$	0	0
$682$	82.0060	3.14017
$683$	−42.9761	−1.64443	−0.822217	−	0.569174i	$-0.807263\pi$
$683$	−42.9761	−1.64443	−0.822217	+	0.569174i	$0.807263\pi$
$684$	0	0
$685$	−19.1966	−0.733463
$686$	48.0202	1.83342
$687$	0	0
$688$	−29.7091	−1.13265
$689$	0.507941	0.0193510
$690$	0	0
$691$	39.4885	1.50221	0.751106	−	0.660182i	$-0.229521\pi$
$691$	39.4885	1.50221	0.751106	+	0.660182i	$0.229521\pi$
$692$	−81.9556	−3.11548
$693$	0	0
$694$	−69.3700	−2.63325
$695$	3.13926	0.119079
$696$	0	0
$697$	2.02231	0.0766003
$698$	−8.34125	−0.315721
$699$	0	0
$700$	−7.43841	−0.281146
$701$	−6.01575	−0.227212	−0.113606	−	0.993526i	$-0.536240\pi$
$701$	−6.01575	−0.227212	−0.113606	+	0.993526i	$0.536240\pi$
$702$	0	0
$703$	−39.1169	−1.47532
$704$	−33.0797	−1.24674
$705$	0	0
$706$	−10.7635	−0.405090
$707$	−22.1544	−0.833203
$708$	0	0
$709$	41.8624	1.57217	0.786087	−	0.618116i	$-0.212104\pi$
$709$	41.8624	1.57217	0.786087	+	0.618116i	$0.212104\pi$
$710$	23.4220	0.879012
$711$	0	0
$712$	−51.7009	−1.93757
$713$	48.4148	1.81315
$714$	0	0
$715$	−3.90626	−0.146086
$716$	21.7058	0.811182
$717$	0	0
$718$	−39.2208	−1.46371
$719$	40.6530	1.51610	0.758051	−	0.652195i	$-0.226152\pi$
$719$	40.6530	1.51610	0.758051	+	0.652195i	$0.226152\pi$
$720$	0	0
$721$	29.5928	1.10209
$722$	119.354	4.44191
$723$	0	0
$724$	−85.8561	−3.19082
$725$	5.08147	0.188721
$726$	0	0
$727$	40.3239	1.49553	0.747765	−	0.663964i	$-0.231127\pi$
$727$	40.3239	1.49553	0.747765	+	0.663964i	$0.231127\pi$
$728$	−8.98772	−0.333107
$729$	0	0
$730$	33.3582	1.23464
$731$	5.58768	0.206668
$732$	0	0
$733$	14.0573	0.519220	0.259610	−	0.965714i	$-0.416406\pi$
$733$	14.0573	0.519220	0.259610	+	0.965714i	$0.416406\pi$
$734$	−32.4494	−1.19773
$735$	0	0
$736$	−2.18246	−0.0804466
$737$	−5.66677	−0.208738
$738$	0	0
$739$	−29.5959	−1.08870	−0.544352	−	0.838857i	$-0.683224\pi$
$739$	−29.5959	−1.08870	−0.544352	+	0.838857i	$0.683224\pi$
$740$	18.7995	0.691084
$741$	0	0
$742$	−2.32932	−0.0855119
$743$	−17.6332	−0.646898	−0.323449	−	0.946246i	$-0.604842\pi$
$743$	−17.6332	−0.646898	−0.323449	+	0.946246i	$0.604842\pi$
$744$	0	0
$745$	−22.6895	−0.831279
$746$	39.4738	1.44524
$747$	0	0
$748$	−10.9419	−0.400074
$749$	−7.57379	−0.276740
$750$	0	0
$751$	37.0059	1.35036	0.675182	−	0.737651i	$-0.264065\pi$
$751$	37.0059	1.35036	0.675182	+	0.737651i	$0.264065\pi$
$752$	−0.124388	−0.00453595
$753$	0	0
$754$	12.4053	0.451776
$755$	−10.2663	−0.373629
$756$	0	0
$757$	−9.30197	−0.338086	−0.169043	−	0.985609i	$-0.554068\pi$
$757$	−9.30197	−0.338086	−0.169043	+	0.985609i	$0.554068\pi$
$758$	−23.8446	−0.866075
$759$	0	0
$760$	−39.4236	−1.43004
$761$	−30.4395	−1.10343	−0.551715	−	0.834032i	$-0.686027\pi$
$761$	−30.4395	−1.10343	−0.551715	+	0.834032i	$0.686027\pi$
$762$	0	0
$763$	12.3357	0.446582
$764$	−54.1497	−1.95907
$765$	0	0
$766$	92.4475	3.34026
$767$	4.72379	0.170566
$768$	0	0
$769$	9.54124	0.344066	0.172033	−	0.985091i	$-0.444966\pi$
$769$	9.54124	0.344066	0.172033	+	0.985091i	$0.444966\pi$
$770$	17.9133	0.645551
$771$	0	0
$772$	45.1154	1.62374
$773$	30.7861	1.10730	0.553650	−	0.832749i	$-0.313235\pi$
$773$	30.7861	1.10730	0.553650	+	0.832749i	$0.313235\pi$
$774$	0	0
$775$	8.59935	0.308898
$776$	17.4335	0.625826
$777$	0	0
$778$	82.9294	2.97316
$779$	−23.5561	−0.843983
$780$	0	0
$781$	−37.4770	−1.34103
$782$	−9.72251	−0.347676
$783$	0	0
$784$	−13.0563	−0.466297
$785$	3.23323	0.115399
$786$	0	0
$787$	−49.9534	−1.78065	−0.890323	−	0.455329i	$-0.849522\pi$
$787$	−49.9534	−1.78065	−0.890323	+	0.455329i	$0.849522\pi$
$788$	79.9906	2.84955
$789$	0	0
$790$	−31.3598	−1.11573
$791$	−16.8404	−0.598777
$792$	0	0
$793$	−10.1385	−0.360030
$794$	29.1583	1.03479
$795$	0	0
$796$	−45.1526	−1.60039
$797$	48.7018	1.72511	0.862554	−	0.505965i	$-0.168864\pi$
$797$	48.7018	1.72511	0.862554	+	0.505965i	$0.168864\pi$
$798$	0	0
$799$	0.0233948	0.000827648 0
$800$	−0.387645	−0.0137053
$801$	0	0
$802$	−45.0096	−1.58934
$803$	−53.3757	−1.88359
$804$	0	0
$805$	10.5757	0.372744
$806$	20.9935	0.739465
$807$	0	0
$808$	56.4310	1.98524
$809$	28.7046	1.00920	0.504601	−	0.863353i	$-0.331640\pi$
$809$	28.7046	1.00920	0.504601	+	0.863353i	$0.331640\pi$
$810$	0	0
$811$	44.7887	1.57274	0.786371	−	0.617754i	$-0.211957\pi$
$811$	44.7887	1.57274	0.786371	+	0.617754i	$0.211957\pi$
$812$	−37.7981	−1.32645
$813$	0	0
$814$	−45.2733	−1.58683
$815$	12.2809	0.430181
$816$	0	0
$817$	−65.0859	−2.27707
$818$	−44.8456	−1.56799
$819$	0	0
$820$	11.3210	0.395346
$821$	−30.2155	−1.05453	−0.527264	−	0.849701i	$-0.676782\pi$
$821$	−30.2155	−1.05453	−0.527264	+	0.849701i	$0.676782\pi$
$822$	0	0
$823$	−7.30149	−0.254514	−0.127257	−	0.991870i	$-0.540617\pi$
$823$	−7.30149	−0.254514	−0.127257	+	0.991870i	$0.540617\pi$
$824$	−75.3779	−2.62592
$825$	0	0
$826$	−21.6624	−0.753731
$827$	−36.7129	−1.27663	−0.638317	−	0.769774i	$-0.720369\pi$
$827$	−36.7129	−1.27663	−0.638317	+	0.769774i	$0.720369\pi$
$828$	0	0
$829$	5.06359	0.175866	0.0879329	−	0.996126i	$-0.471974\pi$
$829$	5.06359	0.175866	0.0879329	+	0.996126i	$0.471974\pi$
$830$	23.9375	0.830883
$831$	0	0
$832$	−8.46839	−0.293588
$833$	2.45562	0.0850823
$834$	0	0
$835$	−11.3963	−0.394386
$836$	127.452	4.40803
$837$	0	0
$838$	−68.6569	−2.37171
$839$	23.2131	0.801407	0.400703	−	0.916208i	$-0.368766\pi$
$839$	23.2131	0.801407	0.400703	+	0.916208i	$0.368766\pi$
$840$	0	0
$841$	−3.17869	−0.109610
$842$	−58.1227	−2.00304
$843$	0	0
$844$	33.9761	1.16951
$845$	−1.00000	−0.0344010
$846$	0	0
$847$	−7.99993	−0.274881
$848$	1.91037	0.0656025
$849$	0	0
$850$	−1.72690	−0.0592320
$851$	−26.7285	−0.916241
$852$	0	0
$853$	−12.9056	−0.441879	−0.220939	−	0.975288i	$-0.570912\pi$
$853$	−12.9056	−0.441879	−0.220939	+	0.975288i	$0.570912\pi$
$854$	46.4933	1.59097
$855$	0	0
$856$	19.2917	0.659377
$857$	−23.7675	−0.811883	−0.405942	−	0.913899i	$-0.633056\pi$
$857$	−23.7675	−0.811883	−0.405942	+	0.913899i	$0.633056\pi$
$858$	0	0
$859$	54.5766	1.86213	0.931064	−	0.364856i	$-0.118882\pi$
$859$	54.5766	1.86213	0.931064	+	0.364856i	$0.118882\pi$
$860$	31.2802	1.06664
$861$	0	0
$862$	−47.6568	−1.62320
$863$	−27.6102	−0.939862	−0.469931	−	0.882703i	$-0.655721\pi$
$863$	−27.6102	−0.939862	−0.469931	+	0.882703i	$0.655721\pi$
$864$	0	0
$865$	20.6964	0.703697
$866$	51.2229	1.74063
$867$	0	0
$868$	−63.9655	−2.17113
$869$	50.1781	1.70218
$870$	0	0
$871$	−1.45069	−0.0491548
$872$	−31.4211	−1.06405
$873$	0	0
$874$	113.249	3.83070
$875$	1.87843	0.0635027
$876$	0	0
$877$	7.04475	0.237884	0.118942	−	0.992901i	$-0.462050\pi$
$877$	7.04475	0.237884	0.118942	+	0.992901i	$0.462050\pi$
$878$	−30.6056	−1.03289
$879$	0	0
$880$	−14.6915	−0.495250
$881$	−39.7425	−1.33896	−0.669479	−	0.742831i	$-0.733483\pi$
$881$	−39.7425	−1.33896	−0.669479	+	0.742831i	$0.733483\pi$
$882$	0	0
$883$	−29.4630	−0.991508	−0.495754	−	0.868463i	$-0.665108\pi$
$883$	−29.4630	−0.991508	−0.495754	+	0.868463i	$0.665108\pi$
$884$	−2.80111	−0.0942117
$885$	0	0
$886$	−7.31763	−0.245841
$887$	−34.9817	−1.17457	−0.587286	−	0.809380i	$-0.699804\pi$
$887$	−34.9817	−1.17457	−0.587286	+	0.809380i	$0.699804\pi$
$888$	0	0
$889$	−7.03979	−0.236107
$890$	26.3793	0.884237
$891$	0	0
$892$	−77.4742	−2.59403
$893$	−0.272505	−0.00911904
$894$	0	0
$895$	−5.48139	−0.183223
$896$	37.3780	1.24871
$897$	0	0
$898$	−83.2095	−2.77674
$899$	43.6973	1.45739
$900$	0	0
$901$	−0.359302	−0.0119701
$902$	−27.2634	−0.907773
$903$	0	0
$904$	42.8955	1.42668
$905$	21.6814	0.720713
$906$	0	0
$907$	−24.9386	−0.828072	−0.414036	−	0.910260i	$-0.635881\pi$
$907$	−24.9386	−0.828072	−0.414036	+	0.910260i	$0.635881\pi$
$908$	−72.6352	−2.41048
$909$	0	0
$910$	4.58580	0.152018
$911$	−47.5318	−1.57480	−0.787399	−	0.616443i	$-0.788573\pi$
$911$	−47.5318	−1.57480	−0.787399	+	0.616443i	$0.788573\pi$
$912$	0	0
$913$	−38.3019	−1.26761
$914$	18.1301	0.599692
$915$	0	0
$916$	−21.8087	−0.720579
$917$	2.12641	0.0702203
$918$	0	0
$919$	38.9811	1.28587	0.642934	−	0.765922i	$-0.277717\pi$
$919$	38.9811	1.28587	0.642934	+	0.765922i	$0.277717\pi$
$920$	−26.9380	−0.888121
$921$	0	0
$922$	−88.8430	−2.92589
$923$	−9.59410	−0.315794
$924$	0	0
$925$	−4.74747	−0.156096
$926$	−98.9300	−3.25104
$927$	0	0
$928$	−1.96980	−0.0646620
$929$	−24.7975	−0.813578	−0.406789	−	0.913522i	$-0.633352\pi$
$929$	−24.7975	−0.813578	−0.406789	+	0.913522i	$0.633352\pi$
$930$	0	0
$931$	−28.6034	−0.937439
$932$	−28.9933	−0.949707
$933$	0	0
$934$	−72.8979	−2.38529
$935$	2.76317	0.0903652
$936$	0	0
$937$	40.0098	1.30706	0.653531	−	0.756900i	$-0.273287\pi$
$937$	40.0098	1.30706	0.653531	+	0.756900i	$0.273287\pi$
$938$	6.65258	0.217214
$939$	0	0
$940$	0.130965	0.00427162
$941$	−13.5755	−0.442549	−0.221275	−	0.975212i	$-0.571022\pi$
$941$	−13.5755	−0.442549	−0.221275	+	0.975212i	$0.571022\pi$
$942$	0	0
$943$	−16.0958	−0.524151
$944$	17.7663	0.578243
$945$	0	0
$946$	−75.3295	−2.44917
$947$	6.37049	0.207013	0.103507	−	0.994629i	$-0.466994\pi$
$947$	6.37049	0.207013	0.103507	+	0.994629i	$0.466994\pi$
$948$	0	0
$949$	−13.6642	−0.443558
$950$	20.1151	0.652619
$951$	0	0
$952$	6.35764	0.206052
$953$	−27.4658	−0.889705	−0.444853	−	0.895604i	$-0.646744\pi$
$953$	−27.4658	−0.889705	−0.444853	+	0.895604i	$0.646744\pi$
$954$	0	0
$955$	13.6745	0.442497
$956$	120.688	3.90332
$957$	0	0
$958$	32.4684	1.04901
$959$	−36.0595	−1.16442
$960$	0	0
$961$	42.9488	1.38545
$962$	−11.5900	−0.373675
$963$	0	0
$964$	−93.1830	−3.00122
$965$	−11.3931	−0.366756
$966$	0	0
$967$	42.6335	1.37100	0.685500	−	0.728073i	$-0.259584\pi$
$967$	42.6335	1.37100	0.685500	+	0.728073i	$0.259584\pi$
$968$	20.3772	0.654947
$969$	0	0
$970$	−8.89508	−0.285604
$971$	18.1119	0.581240	0.290620	−	0.956839i	$-0.406138\pi$
$971$	18.1119	0.581240	0.290620	+	0.956839i	$0.406138\pi$
$972$	0	0
$973$	5.89689	0.189046
$974$	103.260	3.30866
$975$	0	0
$976$	−38.1312	−1.22055
$977$	−8.21880	−0.262943	−0.131471	−	0.991320i	$-0.541970\pi$
$977$	−8.21880	−0.262943	−0.131471	+	0.991320i	$0.541970\pi$
$978$	0	0
$979$	−42.2090	−1.34900
$980$	13.7467	0.439124
$981$	0	0
$982$	34.4030	1.09784
$983$	−38.5161	−1.22847	−0.614237	−	0.789122i	$-0.710536\pi$
$983$	−38.5161	−1.22847	−0.614237	+	0.789122i	$0.710536\pi$
$984$	0	0
$985$	−20.2002	−0.643630
$986$	−8.77516	−0.279458
$987$	0	0
$988$	32.6277	1.03803
$989$	−44.4731	−1.41416
$990$	0	0
$991$	−42.3874	−1.34648	−0.673241	−	0.739423i	$-0.735098\pi$
$991$	−42.3874	−1.34648	−0.673241	+	0.739423i	$0.735098\pi$
$992$	−3.33349	−0.105839
$993$	0	0
$994$	43.9967	1.39549
$995$	11.4025	0.361482
$996$	0	0
$997$	22.7549	0.720656	0.360328	−	0.932826i	$-0.382665\pi$
$997$	22.7549	0.720656	0.360328	+	0.932826i	$0.382665\pi$
$998$	−9.77406	−0.309392
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1755.2.a.s.1.4	yes	4
3.2	odd	2		1755.2.a.m.1.1	✓	4
5.4	even	2		8775.2.a.bh.1.1		4
15.14	odd	2		8775.2.a.bt.1.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1755.2.a.m.1.1	✓	4	3.2	odd	2
1755.2.a.s.1.4	yes	4	1.1	even	1	trivial
8775.2.a.bh.1.1		4	5.4	even	2
8775.2.a.bt.1.4		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)

\(f(q)\)	\(=\)	\(q+2.44129 q^{2} +3.95990 q^{4} -1.00000 q^{5} -1.87843 q^{7} +4.78469 q^{8} +O(q^{10})\)	\(q+2.44129 q^{2} +3.95990 q^{4} -1.00000 q^{5} -1.87843 q^{7} +4.78469 q^{8} -2.44129 q^{10} +3.90626 q^{11} +1.00000 q^{13} -4.58580 q^{14} +3.76102 q^{16} -0.707370 q^{17} +8.23953 q^{19} -3.95990 q^{20} +9.53631 q^{22} +5.63005 q^{23} +1.00000 q^{25} +2.44129 q^{26} -7.43841 q^{28} +5.08147 q^{29} +8.59935 q^{31} -0.387645 q^{32} -1.72690 q^{34} +1.87843 q^{35} -4.74747 q^{37} +20.1151 q^{38} -4.78469 q^{40} -2.85891 q^{41} -7.89923 q^{43} +15.4684 q^{44} +13.7446 q^{46} -0.0330729 q^{47} -3.47149 q^{49} +2.44129 q^{50} +3.95990 q^{52} +0.507941 q^{53} -3.90626 q^{55} -8.98772 q^{56} +12.4053 q^{58} +4.72379 q^{59} -10.1385 q^{61} +20.9935 q^{62} -8.46839 q^{64} -1.00000 q^{65} -1.45069 q^{67} -2.80111 q^{68} +4.58580 q^{70} -9.59410 q^{71} -13.6642 q^{73} -11.5900 q^{74} +32.6277 q^{76} -7.33764 q^{77} +12.8456 q^{79} -3.76102 q^{80} -6.97943 q^{82} -9.80526 q^{83} +0.707370 q^{85} -19.2843 q^{86} +18.6902 q^{88} -10.8055 q^{89} -1.87843 q^{91} +22.2944 q^{92} -0.0807406 q^{94} -8.23953 q^{95} +3.64360 q^{97} -8.47491 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 3 q^{2} + 3 q^{4} - 4 q^{5} - 3 q^{7} + 3 q^{8}+O(q^{10}) \) 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

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Database

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