LMFDB - Embedded newform 752.4.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [752,4,Mod(1,752)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(752, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("752.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 752 = 2^{4} \cdot 47 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	752.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:	$44.3694363243$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.1101.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 9x + 12 $ x^3 - x^2 - 9*x + 12
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 47)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.11903$ of defining polynomial
Character	$\chi$	$=$	752.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.72833 q^{3} -9.01945 q^{5} +11.3182 q^{7} -24.0129 q^{9} +O(q^{10})$	$q-1.72833 q^{3} -9.01945 q^{5} +11.3182 q^{7} -24.0129 q^{9} +40.6469 q^{11} -12.3983 q^{13} +15.5886 q^{15} +59.6717 q^{17} -26.4089 q^{19} -19.5615 q^{21} -107.097 q^{23} -43.6495 q^{25} +88.1670 q^{27} +173.657 q^{29} +332.301 q^{31} -70.2512 q^{33} -102.084 q^{35} -172.339 q^{37} +21.4283 q^{39} -178.190 q^{41} -63.2928 q^{43} +216.583 q^{45} -47.0000 q^{47} -214.899 q^{49} -103.132 q^{51} -402.256 q^{53} -366.613 q^{55} +45.6432 q^{57} -305.280 q^{59} -86.2464 q^{61} -271.782 q^{63} +111.826 q^{65} +681.333 q^{67} +185.098 q^{69} -726.348 q^{71} -79.8711 q^{73} +75.4406 q^{75} +460.049 q^{77} -279.707 q^{79} +495.966 q^{81} -556.598 q^{83} -538.206 q^{85} -300.135 q^{87} -342.069 q^{89} -140.326 q^{91} -574.325 q^{93} +238.194 q^{95} -1637.35 q^{97} -976.050 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 5 q^{3} - 6 q^{5} + 45 q^{7} - 42 q^{9}+O(q^{10}) $ 3 * q + 5 * q^3 - 6 * q^5 + 45 * q^7 - 42 * q^9	$ 3 q + 5 q^{3} - 6 q^{5} + 45 q^{7} - 42 q^{9} - 2 q^{11} - 80 q^{13} - 14 q^{15} - 39 q^{17} + 24 q^{19} + 24 q^{21} - 120 q^{23} - 171 q^{25} - 64 q^{27} - 184 q^{29} + 4 q^{31} - 208 q^{33} + 156 q^{35} - 589 q^{37} - 60 q^{39} - 92 q^{41} + 250 q^{43} - 78 q^{45} - 141 q^{47} + 30 q^{49} - 317 q^{51} + 459 q^{53} - 448 q^{55} + 216 q^{57} - 579 q^{59} + 267 q^{61} - 1044 q^{63} - 424 q^{65} + 540 q^{67} + 642 q^{69} - 749 q^{71} - 1924 q^{73} - 473 q^{75} - 288 q^{77} - 805 q^{79} + 291 q^{81} - 712 q^{83} - 1038 q^{85} - 1216 q^{87} + 835 q^{89} - 2040 q^{91} - 1500 q^{93} + 312 q^{95} - 2243 q^{97} - 554 q^{99}+O(q^{100}) $ 3 * q + 5 * q^3 - 6 * q^5 + 45 * q^7 - 42 * q^9 - 2 * q^11 - 80 * q^13 - 14 * q^15 - 39 * q^17 + 24 * q^19 + 24 * q^21 - 120 * q^23 - 171 * q^25 - 64 * q^27 - 184 * q^29 + 4 * q^31 - 208 * q^33 + 156 * q^35 - 589 * q^37 - 60 * q^39 - 92 * q^41 + 250 * q^43 - 78 * q^45 - 141 * q^47 + 30 * q^49 - 317 * q^51 + 459 * q^53 - 448 * q^55 + 216 * q^57 - 579 * q^59 + 267 * q^61 - 1044 * q^63 - 424 * q^65 + 540 * q^67 + 642 * q^69 - 749 * q^71 - 1924 * q^73 - 473 * q^75 - 288 * q^77 - 805 * q^79 + 291 * q^81 - 712 * q^83 - 1038 * q^85 - 1216 * q^87 + 835 * q^89 - 2040 * q^91 - 1500 * q^93 + 312 * q^95 - 2243 * q^97 - 554 * q^99

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−1.72833	−0.332617	−0.166308	−	0.986074i	$-0.553185\pi$
$3$	−1.72833	−0.332617	−0.166308	+	0.986074i	$0.553185\pi$
$4$	0	0
$5$	−9.01945	−0.806724	−0.403362	−	0.915040i	$-0.632159\pi$
$5$	−9.01945	−0.806724	−0.403362	+	0.915040i	$0.632159\pi$
$6$	0	0
$7$	11.3182	0.611124	0.305562	−	0.952172i	$-0.401156\pi$
$7$	11.3182	0.611124	0.305562	+	0.952172i	$0.401156\pi$
$8$	0	0
$9$	−24.0129	−0.889366
$10$	0	0
$11$	40.6469	1.11414	0.557069	−	0.830467i	$-0.311926\pi$
$11$	40.6469	1.11414	0.557069	+	0.830467i	$0.311926\pi$
$12$	0	0
$13$	−12.3983	−0.264513	−0.132257	−	0.991216i	$-0.542222\pi$
$13$	−12.3983	−0.264513	−0.132257	+	0.991216i	$0.542222\pi$
$14$	0	0
$15$	15.5886	0.268330
$16$	0	0
$17$	59.6717	0.851324	0.425662	−	0.904882i	$-0.360041\pi$
$17$	59.6717	0.851324	0.425662	+	0.904882i	$0.360041\pi$
$18$	0	0
$19$	−26.4089	−0.318874	−0.159437	−	0.987208i	$-0.550968\pi$
$19$	−26.4089	−0.318874	−0.159437	+	0.987208i	$0.550968\pi$
$20$	0	0
$21$	−19.5615	−0.203270
$22$	0	0
$23$	−107.097	−0.970923	−0.485461	−	0.874258i	$-0.661348\pi$
$23$	−107.097	−0.970923	−0.485461	+	0.874258i	$0.661348\pi$
$24$	0	0
$25$	−43.6495	−0.349196
$26$	0	0
$27$	88.1670	0.628435
$28$	0	0
$29$	173.657	1.11197	0.555987	−	0.831191i	$-0.312341\pi$
$29$	173.657	1.11197	0.555987	+	0.831191i	$0.312341\pi$
$30$	0	0
$31$	332.301	1.92526	0.962629	−	0.270823i	$-0.0872957\pi$
$31$	332.301	1.92526	0.962629	+	0.270823i	$0.0872957\pi$
$32$	0	0
$33$	−70.2512	−0.370581
$34$	0	0
$35$	−102.084	−0.493009
$36$	0	0
$37$	−172.339	−0.765739	−0.382869	−	0.923802i	$-0.625064\pi$
$37$	−172.339	−0.765739	−0.382869	+	0.923802i	$0.625064\pi$
$38$	0	0
$39$	21.4283	0.0879815
$40$	0	0
$41$	−178.190	−0.678746	−0.339373	−	0.940652i	$-0.610215\pi$
$41$	−178.190	−0.678746	−0.339373	+	0.940652i	$0.610215\pi$
$42$	0	0
$43$	−63.2928	−0.224467	−0.112233	−	0.993682i	$-0.535800\pi$
$43$	−63.2928	−0.224467	−0.112233	+	0.993682i	$0.535800\pi$
$44$	0	0
$45$	216.583	0.717473
$46$	0	0
$47$	−47.0000	−0.145865
$47$	−47.0000	−0.145865
$48$	0	0
$49$	−214.899	−0.626527
$50$	0	0
$51$	−103.132	−0.283165
$52$	0	0
$53$	−402.256	−1.04253	−0.521266	−	0.853395i	$-0.674540\pi$
$53$	−402.256	−1.04253	−0.521266	+	0.853395i	$0.674540\pi$
$54$	0	0
$55$	−366.613	−0.898801
$56$	0	0
$57$	45.6432	0.106063
$58$	0	0
$59$	−305.280	−0.673628	−0.336814	−	0.941571i	$-0.609349\pi$
$59$	−305.280	−0.673628	−0.336814	+	0.941571i	$0.609349\pi$
$60$	0	0
$61$	−86.2464	−0.181028	−0.0905141	−	0.995895i	$-0.528851\pi$
$61$	−86.2464	−0.181028	−0.0905141	+	0.995895i	$0.528851\pi$
$62$	0	0
$63$	−271.782	−0.543513
$64$	0	0
$65$	111.826	0.213389
$66$	0	0
$67$	681.333	1.24236	0.621179	−	0.783668i	$-0.286654\pi$
$67$	681.333	1.24236	0.621179	+	0.783668i	$0.286654\pi$
$68$	0	0
$69$	185.098	0.322945
$70$	0	0
$71$	−726.348	−1.21411	−0.607054	−	0.794661i	$-0.707649\pi$
$71$	−726.348	−1.21411	−0.607054	+	0.794661i	$0.707649\pi$
$72$	0	0
$73$	−79.8711	−0.128058	−0.0640288	−	0.997948i	$-0.520395\pi$
$73$	−79.8711	−0.128058	−0.0640288	+	0.997948i	$0.520395\pi$
$74$	0	0
$75$	75.4406	0.116148
$76$	0	0
$77$	460.049	0.680876
$78$	0	0
$79$	−279.707	−0.398348	−0.199174	−	0.979964i	$-0.563826\pi$
$79$	−279.707	−0.398348	−0.199174	+	0.979964i	$0.563826\pi$
$80$	0	0
$81$	495.966	0.680338
$82$	0	0
$83$	−556.598	−0.736080	−0.368040	−	0.929810i	$-0.619971\pi$
$83$	−556.598	−0.736080	−0.368040	+	0.929810i	$0.619971\pi$
$84$	0	0
$85$	−538.206	−0.686784
$86$	0	0
$87$	−300.135	−0.369861
$88$	0	0
$89$	−342.069	−0.407408	−0.203704	−	0.979033i	$-0.565298\pi$
$89$	−342.069	−0.407408	−0.203704	+	0.979033i	$0.565298\pi$
$90$	0	0
$91$	−140.326	−0.161650
$92$	0	0
$93$	−574.325	−0.640373
$94$	0	0
$95$	238.194	0.257244
$96$	0	0
$97$	−1637.35	−1.71390	−0.856949	−	0.515402i	$-0.827643\pi$
$97$	−1637.35	−1.71390	−0.856949	+	0.515402i	$0.827643\pi$
$98$	0	0
$99$	−976.050	−0.990876
$100$	0	0
$101$	−1453.29	−1.43176	−0.715879	−	0.698224i	$-0.753974\pi$
$101$	−1453.29	−1.43176	−0.715879	+	0.698224i	$0.753974\pi$
$102$	0	0
$103$	−1363.10	−1.30398	−0.651992	−	0.758226i	$-0.726067\pi$
$103$	−1363.10	−1.30398	−0.651992	+	0.758226i	$0.726067\pi$
$104$	0	0
$105$	176.434	0.163983
$106$	0	0
$107$	−274.746	−0.248231	−0.124116	−	0.992268i	$-0.539609\pi$
$107$	−274.746	−0.248231	−0.124116	+	0.992268i	$0.539609\pi$
$108$	0	0
$109$	1448.56	1.27291	0.636454	−	0.771315i	$-0.280401\pi$
$109$	1448.56	1.27291	0.636454	+	0.771315i	$0.280401\pi$
$110$	0	0
$111$	297.858	0.254698
$112$	0	0
$113$	−1591.30	−1.32475	−0.662376	−	0.749171i	$-0.730452\pi$
$113$	−1591.30	−1.32475	−0.662376	+	0.749171i	$0.730452\pi$
$114$	0	0
$115$	965.954	0.783267
$116$	0	0
$117$	297.719	0.235249
$118$	0	0
$119$	675.374	0.520264
$120$	0	0
$121$	321.172	0.241301
$122$	0	0
$123$	307.970	0.225762
$124$	0	0
$125$	1521.13	1.08843
$126$	0	0
$127$	−747.668	−0.522400	−0.261200	−	0.965285i	$-0.584118\pi$
$127$	−747.668	−0.522400	−0.261200	+	0.965285i	$0.584118\pi$
$128$	0	0
$129$	109.391	0.0746614
$130$	0	0
$131$	624.528	0.416529	0.208264	−	0.978073i	$-0.433219\pi$
$131$	624.528	0.416529	0.208264	+	0.978073i	$0.433219\pi$
$132$	0	0
$133$	−298.900	−0.194872
$134$	0	0
$135$	−795.218	−0.506974
$136$	0	0
$137$	−336.418	−0.209796	−0.104898	−	0.994483i	$-0.533452\pi$
$137$	−336.418	−0.209796	−0.104898	+	0.994483i	$0.533452\pi$
$138$	0	0
$139$	−1940.28	−1.18397	−0.591987	−	0.805948i	$-0.701656\pi$
$139$	−1940.28	−1.18397	−0.591987	+	0.805948i	$0.701656\pi$
$140$	0	0
$141$	81.2314	0.0485171
$142$	0	0
$143$	−503.953	−0.294704
$144$	0	0
$145$	−1566.29	−0.897056
$146$	0	0
$147$	371.416	0.208394
$148$	0	0
$149$	2810.71	1.54538	0.772692	−	0.634781i	$-0.218910\pi$
$149$	2810.71	1.54538	0.772692	+	0.634781i	$0.218910\pi$
$150$	0	0
$151$	−1710.82	−0.922014	−0.461007	−	0.887396i	$-0.652512\pi$
$151$	−1710.82	−0.922014	−0.461007	+	0.887396i	$0.652512\pi$
$152$	0	0
$153$	−1432.89	−0.757138
$154$	0	0
$155$	−2997.17	−1.55315
$156$	0	0
$157$	−1378.36	−0.700669	−0.350334	−	0.936625i	$-0.613932\pi$
$157$	−1378.36	−0.700669	−0.350334	+	0.936625i	$0.613932\pi$
$158$	0	0
$159$	695.231	0.346763
$160$	0	0
$161$	−1212.14	−0.593354
$162$	0	0
$163$	3255.46	1.56434	0.782170	−	0.623066i	$-0.214113\pi$
$163$	3255.46	1.56434	0.782170	+	0.623066i	$0.214113\pi$
$164$	0	0
$165$	633.627	0.298956
$166$	0	0
$167$	2337.08	1.08293	0.541463	−	0.840725i	$-0.317871\pi$
$167$	2337.08	1.08293	0.541463	+	0.840725i	$0.317871\pi$
$168$	0	0
$169$	−2043.28	−0.930033
$170$	0	0
$171$	634.153	0.283596
$172$	0	0
$173$	−3269.43	−1.43682	−0.718412	−	0.695618i	$-0.755130\pi$
$173$	−3269.43	−1.43682	−0.718412	+	0.695618i	$0.755130\pi$
$174$	0	0
$175$	−494.033	−0.213402
$176$	0	0
$177$	527.624	0.224060
$178$	0	0
$179$	2203.96	0.920290	0.460145	−	0.887844i	$-0.347797\pi$
$179$	2203.96	0.920290	0.460145	+	0.887844i	$0.347797\pi$
$180$	0	0
$181$	1522.63	0.625282	0.312641	−	0.949871i	$-0.398786\pi$
$181$	1522.63	0.625282	0.312641	+	0.949871i	$0.398786\pi$
$182$	0	0
$183$	149.062	0.0602130
$184$	0	0
$185$	1554.40	0.617740
$186$	0	0
$187$	2425.47	0.948491
$188$	0	0
$189$	997.889	0.384052
$190$	0	0
$191$	2019.27	0.764968	0.382484	−	0.923962i	$-0.375069\pi$
$191$	2019.27	0.764968	0.382484	+	0.923962i	$0.375069\pi$
$192$	0	0
$193$	−4447.95	−1.65891	−0.829456	−	0.558571i	$-0.811350\pi$
$193$	−4447.95	−1.65891	−0.829456	+	0.558571i	$0.811350\pi$
$194$	0	0
$195$	−193.272	−0.0709768
$196$	0	0
$197$	1790.04	0.647388	0.323694	−	0.946162i	$-0.395075\pi$
$197$	1790.04	0.647388	0.323694	+	0.946162i	$0.395075\pi$
$198$	0	0
$199$	−3481.42	−1.24016	−0.620079	−	0.784540i	$-0.712899\pi$
$199$	−3481.42	−1.24016	−0.620079	+	0.784540i	$0.712899\pi$
$200$	0	0
$201$	−1177.57	−0.413229
$202$	0	0
$203$	1965.48	0.679554
$204$	0	0
$205$	1607.18	0.547561
$206$	0	0
$207$	2571.70	0.863506
$208$	0	0
$209$	−1073.44	−0.355270
$210$	0	0
$211$	−5216.72	−1.70206	−0.851028	−	0.525120i	$-0.824021\pi$
$211$	−5216.72	−1.70206	−0.851028	+	0.525120i	$0.824021\pi$
$212$	0	0
$213$	1255.37	0.403833
$214$	0	0
$215$	570.867	0.181083
$216$	0	0
$217$	3761.04	1.17657
$218$	0	0
$219$	138.043	0.0425941
$220$	0	0
$221$	−739.827	−0.225186
$222$	0	0
$223$	−5075.44	−1.52411	−0.762055	−	0.647512i	$-0.775810\pi$
$223$	−5075.44	−1.52411	−0.762055	+	0.647512i	$0.775810\pi$
$224$	0	0
$225$	1048.15	0.310563
$226$	0	0
$227$	−1752.66	−0.512460	−0.256230	−	0.966616i	$-0.582480\pi$
$227$	−1752.66	−0.512460	−0.256230	+	0.966616i	$0.582480\pi$
$228$	0	0
$229$	−4676.77	−1.34956	−0.674781	−	0.738018i	$-0.735762\pi$
$229$	−4676.77	−1.34956	−0.674781	+	0.738018i	$0.735762\pi$
$230$	0	0
$231$	−795.115	−0.226471
$232$	0	0
$233$	4261.89	1.19831	0.599154	−	0.800634i	$-0.295504\pi$
$233$	4261.89	1.19831	0.599154	+	0.800634i	$0.295504\pi$
$234$	0	0
$235$	423.914	0.117673
$236$	0	0
$237$	483.426	0.132497
$238$	0	0
$239$	3227.38	0.873479	0.436740	−	0.899588i	$-0.356133\pi$
$239$	3227.38	0.873479	0.436740	+	0.899588i	$0.356133\pi$
$240$	0	0
$241$	−1310.37	−0.350242	−0.175121	−	0.984547i	$-0.556032\pi$
$241$	−1310.37	−0.350242	−0.175121	+	0.984547i	$0.556032\pi$
$242$	0	0
$243$	−3237.70	−0.854727
$244$	0	0
$245$	1938.27	0.505435
$246$	0	0
$247$	327.425	0.0843464
$248$	0	0
$249$	961.984	0.244832
$250$	0	0
$251$	−4906.61	−1.23387	−0.616937	−	0.787013i	$-0.711627\pi$
$251$	−4906.61	−1.23387	−0.616937	+	0.787013i	$0.711627\pi$
$252$	0	0
$253$	−4353.15	−1.08174
$254$	0	0
$255$	930.196	0.228436
$256$	0	0
$257$	2103.21	0.510485	0.255243	−	0.966877i	$-0.417845\pi$
$257$	2103.21	0.510485	0.255243	+	0.966877i	$0.417845\pi$
$258$	0	0
$259$	−1950.56	−0.467962
$260$	0	0
$261$	−4170.00	−0.988951
$262$	0	0
$263$	6993.90	1.63978	0.819890	−	0.572521i	$-0.194035\pi$
$263$	6993.90	1.63978	0.819890	+	0.572521i	$0.194035\pi$
$264$	0	0
$265$	3628.13	0.841035
$266$	0	0
$267$	591.208	0.135511
$268$	0	0
$269$	6466.63	1.46571	0.732857	−	0.680382i	$-0.238186\pi$
$269$	6466.63	1.46571	0.732857	+	0.680382i	$0.238186\pi$
$270$	0	0
$271$	8154.65	1.82790	0.913948	−	0.405831i	$-0.133018\pi$
$271$	8154.65	1.82790	0.913948	+	0.405831i	$0.133018\pi$
$272$	0	0
$273$	242.530	0.0537676
$274$	0	0
$275$	−1774.22	−0.389052
$276$	0	0
$277$	3722.71	0.807495	0.403748	−	0.914870i	$-0.367707\pi$
$277$	3722.71	0.807495	0.403748	+	0.914870i	$0.367707\pi$
$278$	0	0
$279$	−7979.50	−1.71226
$280$	0	0
$281$	−6046.33	−1.28361	−0.641804	−	0.766869i	$-0.721814\pi$
$281$	−6046.33	−1.28361	−0.641804	+	0.766869i	$0.721814\pi$
$282$	0	0
$283$	−268.000	−0.0562931	−0.0281466	−	0.999604i	$-0.508961\pi$
$283$	−268.000	−0.0562931	−0.0281466	+	0.999604i	$0.508961\pi$
$284$	0	0
$285$	−411.676	−0.0855635
$286$	0	0
$287$	−2016.78	−0.414798
$288$	0	0
$289$	−1352.29	−0.275248
$290$	0	0
$291$	2829.88	0.570071
$292$	0	0
$293$	2731.03	0.544535	0.272267	−	0.962222i	$-0.412226\pi$
$293$	2731.03	0.544535	0.272267	+	0.962222i	$0.412226\pi$
$294$	0	0
$295$	2753.46	0.543432
$296$	0	0
$297$	3583.72	0.700163
$298$	0	0
$299$	1327.82	0.256822
$300$	0	0
$301$	−716.360	−0.137177
$302$	0	0
$303$	2511.76	0.476227
$304$	0	0
$305$	777.896	0.146040
$306$	0	0
$307$	7009.57	1.30312	0.651559	−	0.758598i	$-0.274115\pi$
$307$	7009.57	1.30312	0.651559	+	0.758598i	$0.274115\pi$
$308$	0	0
$309$	2355.89	0.433727
$310$	0	0
$311$	867.418	0.158157	0.0790784	−	0.996868i	$-0.474802\pi$
$311$	867.418	0.158157	0.0790784	+	0.996868i	$0.474802\pi$
$312$	0	0
$313$	5724.10	1.03369	0.516845	−	0.856079i	$-0.327106\pi$
$313$	5724.10	1.03369	0.516845	+	0.856079i	$0.327106\pi$
$314$	0	0
$315$	2451.33	0.438465
$316$	0	0
$317$	5821.24	1.03140	0.515700	−	0.856769i	$-0.327532\pi$
$317$	5821.24	1.03140	0.515700	+	0.856769i	$0.327532\pi$
$318$	0	0
$319$	7058.61	1.23889
$320$	0	0
$321$	474.852	0.0825659
$322$	0	0
$323$	−1575.86	−0.271465
$324$	0	0
$325$	541.179	0.0923669
$326$	0	0
$327$	−2503.59	−0.423390
$328$	0	0
$329$	−531.954	−0.0891416
$330$	0	0
$331$	2300.44	0.382005	0.191002	−	0.981590i	$-0.438826\pi$
$331$	2300.44	0.382005	0.191002	+	0.981590i	$0.438826\pi$
$332$	0	0
$333$	4138.35	0.681022
$334$	0	0
$335$	−6145.25	−1.00224
$336$	0	0
$337$	−6399.32	−1.03440	−0.517201	−	0.855864i	$-0.673026\pi$
$337$	−6399.32	−1.03440	−0.517201	+	0.855864i	$0.673026\pi$
$338$	0	0
$339$	2750.29	0.440635
$340$	0	0
$341$	13507.0	2.14500
$342$	0	0
$343$	−6314.40	−0.994010
$344$	0	0
$345$	−1669.48	−0.260528
$346$	0	0
$347$	9188.96	1.42158	0.710791	−	0.703403i	$-0.248337\pi$
$347$	9188.96	1.42158	0.710791	+	0.703403i	$0.248337\pi$
$348$	0	0
$349$	6327.24	0.970457	0.485229	−	0.874387i	$-0.338736\pi$
$349$	6327.24	0.970457	0.485229	+	0.874387i	$0.338736\pi$
$350$	0	0
$351$	−1093.12	−0.166229
$352$	0	0
$353$	8212.76	1.23830	0.619152	−	0.785271i	$-0.287477\pi$
$353$	8212.76	1.23830	0.619152	+	0.785271i	$0.287477\pi$
$354$	0	0
$355$	6551.26	0.979450
$356$	0	0
$357$	−1167.27	−0.173049
$358$	0	0
$359$	5614.24	0.825371	0.412685	−	0.910874i	$-0.364591\pi$
$359$	5614.24	0.825371	0.412685	+	0.910874i	$0.364591\pi$
$360$	0	0
$361$	−6161.57	−0.898319
$362$	0	0
$363$	−555.091	−0.0802609
$364$	0	0
$365$	720.394	0.103307
$366$	0	0
$367$	3282.91	0.466938	0.233469	−	0.972364i	$-0.424992\pi$
$367$	3282.91	0.466938	0.233469	+	0.972364i	$0.424992\pi$
$368$	0	0
$369$	4278.85	0.603654
$370$	0	0
$371$	−4552.81	−0.637116
$372$	0	0
$373$	−9749.67	−1.35340	−0.676701	−	0.736258i	$-0.736591\pi$
$373$	−9749.67	−1.35340	−0.676701	+	0.736258i	$0.736591\pi$
$374$	0	0
$375$	−2629.00	−0.362030
$376$	0	0
$377$	−2153.05	−0.294131
$378$	0	0
$379$	7875.26	1.06735	0.533674	−	0.845690i	$-0.320811\pi$
$379$	7875.26	1.06735	0.533674	+	0.845690i	$0.320811\pi$
$380$	0	0
$381$	1292.21	0.173759
$382$	0	0
$383$	1599.65	0.213416	0.106708	−	0.994290i	$-0.465969\pi$
$383$	1599.65	0.213416	0.106708	+	0.994290i	$0.465969\pi$
$384$	0	0
$385$	−4149.39	−0.549279
$386$	0	0
$387$	1519.84	0.199633
$388$	0	0
$389$	−5101.31	−0.664901	−0.332451	−	0.943121i	$-0.607876\pi$
$389$	−5101.31	−0.664901	−0.332451	+	0.943121i	$0.607876\pi$
$390$	0	0
$391$	−6390.64	−0.826569
$392$	0	0
$393$	−1079.39	−0.138544
$394$	0	0
$395$	2522.81	0.321357
$396$	0	0
$397$	−8835.01	−1.11692	−0.558459	−	0.829532i	$-0.688607\pi$
$397$	−8835.01	−1.11692	−0.558459	+	0.829532i	$0.688607\pi$
$398$	0	0
$399$	516.597	0.0648176
$400$	0	0
$401$	−15444.1	−1.92330	−0.961649	−	0.274284i	$-0.911559\pi$
$401$	−15444.1	−1.92330	−0.961649	+	0.274284i	$0.911559\pi$
$402$	0	0
$403$	−4119.97	−0.509256
$404$	0	0
$405$	−4473.35	−0.548845
$406$	0	0
$407$	−7005.05	−0.853138
$408$	0	0
$409$	−13309.7	−1.60910	−0.804552	−	0.593882i	$-0.797595\pi$
$409$	−13309.7	−1.60910	−0.804552	+	0.593882i	$0.797595\pi$
$410$	0	0
$411$	581.440	0.0697818
$412$	0	0
$413$	−3455.21	−0.411671
$414$	0	0
$415$	5020.21	0.593813
$416$	0	0
$417$	3353.44	0.393809
$418$	0	0
$419$	2074.44	0.241869	0.120934	−	0.992661i	$-0.461411\pi$
$419$	2074.44	0.241869	0.120934	+	0.992661i	$0.461411\pi$
$420$	0	0
$421$	14903.2	1.72526	0.862631	−	0.505833i	$-0.168815\pi$
$421$	14903.2	1.72526	0.862631	+	0.505833i	$0.168815\pi$
$422$	0	0
$423$	1128.61	0.129727
$424$	0	0
$425$	−2604.64	−0.297279
$426$	0	0
$427$	−976.152	−0.110631
$428$	0	0
$429$	870.995	0.0980234
$430$	0	0
$431$	−7434.74	−0.830902	−0.415451	−	0.909616i	$-0.636376\pi$
$431$	−7434.74	−0.830902	−0.415451	+	0.909616i	$0.636376\pi$
$432$	0	0
$433$	4385.47	0.486725	0.243363	−	0.969935i	$-0.421749\pi$
$433$	4385.47	0.486725	0.243363	+	0.969935i	$0.421749\pi$
$434$	0	0
$435$	2707.06	0.298376
$436$	0	0
$437$	2828.30	0.309602
$438$	0	0
$439$	10300.7	1.11988	0.559939	−	0.828534i	$-0.310825\pi$
$439$	10300.7	1.11988	0.559939	+	0.828534i	$0.310825\pi$
$440$	0	0
$441$	5160.34	0.557212
$442$	0	0
$443$	−5700.48	−0.611372	−0.305686	−	0.952132i	$-0.598886\pi$
$443$	−5700.48	−0.611372	−0.305686	+	0.952132i	$0.598886\pi$
$444$	0	0
$445$	3085.28	0.328666
$446$	0	0
$447$	−4857.83	−0.514021
$448$	0	0
$449$	−8946.30	−0.940317	−0.470158	−	0.882582i	$-0.655803\pi$
$449$	−8946.30	−0.940317	−0.470158	+	0.882582i	$0.655803\pi$
$450$	0	0
$451$	−7242.87	−0.756216
$452$	0	0
$453$	2956.85	0.306677
$454$	0	0
$455$	1265.66	0.130407
$456$	0	0
$457$	−16498.3	−1.68875	−0.844374	−	0.535755i	$-0.820027\pi$
$457$	−16498.3	−1.68875	−0.844374	+	0.535755i	$0.820027\pi$
$458$	0	0
$459$	5261.07	0.535001
$460$	0	0
$461$	9016.08	0.910891	0.455445	−	0.890264i	$-0.349480\pi$
$461$	9016.08	0.910891	0.455445	+	0.890264i	$0.349480\pi$
$462$	0	0
$463$	−3241.41	−0.325359	−0.162679	−	0.986679i	$-0.552014\pi$
$463$	−3241.41	−0.325359	−0.162679	+	0.986679i	$0.552014\pi$
$464$	0	0
$465$	5180.10	0.516605
$466$	0	0
$467$	−12186.1	−1.20751	−0.603755	−	0.797170i	$-0.706329\pi$
$467$	−12186.1	−1.20751	−0.603755	+	0.797170i	$0.706329\pi$
$468$	0	0
$469$	7711.44	0.759235
$470$	0	0
$471$	2382.25	0.233054
$472$	0	0
$473$	−2572.66	−0.250087
$474$	0	0
$475$	1152.73	0.111350
$476$	0	0
$477$	9659.33	0.927192
$478$	0	0
$479$	9110.87	0.869074	0.434537	−	0.900654i	$-0.356912\pi$
$479$	9110.87	0.869074	0.434537	+	0.900654i	$0.356912\pi$
$480$	0	0
$481$	2136.71	0.202548
$482$	0	0
$483$	2094.97	0.197360
$484$	0	0
$485$	14768.0	1.38264
$486$	0	0
$487$	−535.218	−0.0498009	−0.0249004	−	0.999690i	$-0.507927\pi$
$487$	−535.218	−0.0498009	−0.0249004	+	0.999690i	$0.507927\pi$
$488$	0	0
$489$	−5626.50	−0.520325
$490$	0	0
$491$	−10810.8	−0.993658	−0.496829	−	0.867848i	$-0.665502\pi$
$491$	−10810.8	−0.993658	−0.496829	+	0.867848i	$0.665502\pi$
$492$	0	0
$493$	10362.4	0.946649
$494$	0	0
$495$	8803.43	0.799364
$496$	0	0
$497$	−8220.93	−0.741970
$498$	0	0
$499$	−16736.4	−1.50146	−0.750728	−	0.660612i	$-0.770297\pi$
$499$	−16736.4	−1.50146	−0.750728	+	0.660612i	$0.770297\pi$
$500$	0	0
$501$	−4039.24	−0.360199
$502$	0	0
$503$	−6612.65	−0.586170	−0.293085	−	0.956086i	$-0.594682\pi$
$503$	−6612.65	−0.586170	−0.293085	+	0.956086i	$0.594682\pi$
$504$	0	0
$505$	13107.9	1.15503
$506$	0	0
$507$	3531.46	0.309345
$508$	0	0
$509$	14953.5	1.30216	0.651081	−	0.759008i	$-0.274316\pi$
$509$	14953.5	1.30216	0.651081	+	0.759008i	$0.274316\pi$
$510$	0	0
$511$	−903.995	−0.0782591
$512$	0	0
$513$	−2328.39	−0.200392
$514$	0	0
$515$	12294.4	1.05196
$516$	0	0
$517$	−1910.41	−0.162514
$518$	0	0
$519$	5650.65	0.477911
$520$	0	0
$521$	822.169	0.0691361	0.0345680	−	0.999402i	$-0.488994\pi$
$521$	822.169	0.0691361	0.0345680	+	0.999402i	$0.488994\pi$
$522$	0	0
$523$	−97.2265	−0.00812890	−0.00406445	−	0.999992i	$-0.501294\pi$
$523$	−97.2265	−0.00812890	−0.00406445	+	0.999992i	$0.501294\pi$
$524$	0	0
$525$	853.850	0.0709811
$526$	0	0
$527$	19828.9	1.63902
$528$	0	0
$529$	−697.284	−0.0573095
$530$	0	0
$531$	7330.65	0.599102
$532$	0	0
$533$	2209.25	0.179537
$534$	0	0
$535$	2478.06	0.200254
$536$	0	0
$537$	−3809.17	−0.306104
$538$	0	0
$539$	−8734.98	−0.698037
$540$	0	0
$541$	−3856.31	−0.306462	−0.153231	−	0.988190i	$-0.548968\pi$
$541$	−3856.31	−0.306462	−0.153231	+	0.988190i	$0.548968\pi$
$542$	0	0
$543$	−2631.60	−0.207979
$544$	0	0
$545$	−13065.2	−1.02689
$546$	0	0
$547$	4147.40	0.324186	0.162093	−	0.986775i	$-0.448175\pi$
$547$	4147.40	0.324186	0.162093	+	0.986775i	$0.448175\pi$
$548$	0	0
$549$	2071.03	0.161000
$550$	0	0
$551$	−4586.07	−0.354580
$552$	0	0
$553$	−3165.78	−0.243440
$554$	0	0
$555$	−2686.52	−0.205471
$556$	0	0
$557$	85.2280	0.00648335	0.00324168	−	0.999995i	$-0.498968\pi$
$557$	85.2280	0.00648335	0.00324168	+	0.999995i	$0.498968\pi$
$558$	0	0
$559$	784.724	0.0593744
$560$	0	0
$561$	−4192.00	−0.315484
$562$	0	0
$563$	−12824.3	−0.960003	−0.480001	−	0.877268i	$-0.659364\pi$
$563$	−12824.3	−0.960003	−0.480001	+	0.877268i	$0.659364\pi$
$564$	0	0
$565$	14352.7	1.06871
$566$	0	0
$567$	5613.44	0.415771
$568$	0	0
$569$	−12096.8	−0.891253	−0.445626	−	0.895219i	$-0.647019\pi$
$569$	−12096.8	−0.891253	−0.445626	+	0.895219i	$0.647019\pi$
$570$	0	0
$571$	−5688.49	−0.416910	−0.208455	−	0.978032i	$-0.566844\pi$
$571$	−5688.49	−0.416910	−0.208455	+	0.978032i	$0.566844\pi$
$572$	0	0
$573$	−3489.95	−0.254441
$574$	0	0
$575$	4674.72	0.339042
$576$	0	0
$577$	−9878.78	−0.712754	−0.356377	−	0.934342i	$-0.615988\pi$
$577$	−9878.78	−0.712754	−0.356377	+	0.934342i	$0.615988\pi$
$578$	0	0
$579$	7687.51	0.551782
$580$	0	0
$581$	−6299.68	−0.449836
$582$	0	0
$583$	−16350.5	−1.16152
$584$	0	0
$585$	−2685.26	−0.189781
$586$	0	0
$587$	17342.9	1.21945	0.609726	−	0.792613i	$-0.291280\pi$
$587$	17342.9	1.21945	0.609726	+	0.792613i	$0.291280\pi$
$588$	0	0
$589$	−8775.69	−0.613915
$590$	0	0
$591$	−3093.78	−0.215332
$592$	0	0
$593$	698.313	0.0483580	0.0241790	−	0.999708i	$-0.492303\pi$
$593$	698.313	0.0483580	0.0241790	+	0.999708i	$0.492303\pi$
$594$	0	0
$595$	−6091.51	−0.419710
$596$	0	0
$597$	6017.03	0.412497
$598$	0	0
$599$	−9549.46	−0.651386	−0.325693	−	0.945476i	$-0.605598\pi$
$599$	−9549.46	−0.651386	−0.325693	+	0.945476i	$0.605598\pi$
$600$	0	0
$601$	27369.5	1.85761	0.928805	−	0.370568i	$-0.120837\pi$
$601$	27369.5	1.85761	0.928805	+	0.370568i	$0.120837\pi$
$602$	0	0
$603$	−16360.8	−1.10491
$604$	0	0
$605$	−2896.80	−0.194664
$606$	0	0
$607$	−18179.8	−1.21564	−0.607821	−	0.794074i	$-0.707956\pi$
$607$	−18179.8	−1.21564	−0.607821	+	0.794074i	$0.707956\pi$
$608$	0	0
$609$	−3396.99	−0.226031
$610$	0	0
$611$	582.720	0.0385832
$612$	0	0
$613$	−3148.69	−0.207462	−0.103731	−	0.994605i	$-0.533078\pi$
$613$	−3148.69	−0.207462	−0.103731	+	0.994605i	$0.533078\pi$
$614$	0	0
$615$	−2777.72	−0.182128
$616$	0	0
$617$	−24002.0	−1.56610	−0.783050	−	0.621959i	$-0.786337\pi$
$617$	−24002.0	−1.56610	−0.783050	+	0.621959i	$0.786337\pi$
$618$	0	0
$619$	−17997.2	−1.16861	−0.584304	−	0.811535i	$-0.698633\pi$
$619$	−17997.2	−1.16861	−0.584304	+	0.811535i	$0.698633\pi$
$620$	0	0
$621$	−9442.40	−0.610162
$622$	0	0
$623$	−3871.60	−0.248977
$624$	0	0
$625$	−8263.54	−0.528866
$626$	0	0
$627$	1855.25	0.118169
$628$	0	0
$629$	−10283.7	−0.651892
$630$	0	0
$631$	−5508.64	−0.347536	−0.173768	−	0.984787i	$-0.555594\pi$
$631$	−5508.64	−0.347536	−0.173768	+	0.984787i	$0.555594\pi$
$632$	0	0
$633$	9016.20	0.566133
$634$	0	0
$635$	6743.55	0.421433
$636$	0	0
$637$	2664.38	0.165725
$638$	0	0
$639$	17441.7	1.07979
$640$	0	0
$641$	11605.9	0.715142	0.357571	−	0.933886i	$-0.383605\pi$
$641$	11605.9	0.715142	0.357571	+	0.933886i	$0.383605\pi$
$642$	0	0
$643$	−7998.88	−0.490583	−0.245292	−	0.969449i	$-0.578884\pi$
$643$	−7998.88	−0.490583	−0.245292	+	0.969449i	$0.578884\pi$
$644$	0	0
$645$	−986.645	−0.0602312
$646$	0	0
$647$	5421.64	0.329438	0.164719	−	0.986341i	$-0.447328\pi$
$647$	5421.64	0.329438	0.164719	+	0.986341i	$0.447328\pi$
$648$	0	0
$649$	−12408.7	−0.750514
$650$	0	0
$651$	−6500.31	−0.391348
$652$	0	0
$653$	−21419.1	−1.28361	−0.641804	−	0.766869i	$-0.721814\pi$
$653$	−21419.1	−1.28361	−0.641804	+	0.766869i	$0.721814\pi$
$654$	0	0
$655$	−5632.90	−0.336024
$656$	0	0
$657$	1917.94	0.113890
$658$	0	0
$659$	11435.7	0.675982	0.337991	−	0.941149i	$-0.390253\pi$
$659$	11435.7	0.675982	0.337991	+	0.941149i	$0.390253\pi$
$660$	0	0
$661$	27123.2	1.59602	0.798011	−	0.602643i	$-0.205886\pi$
$661$	27123.2	1.59602	0.798011	+	0.602643i	$0.205886\pi$
$662$	0	0
$663$	1278.66	0.0749007
$664$	0	0
$665$	2695.92	0.157208
$666$	0	0
$667$	−18598.1	−1.07964
$668$	0	0
$669$	8772.02	0.506945
$670$	0	0
$671$	−3505.65	−0.201690
$672$	0	0
$673$	−11097.2	−0.635612	−0.317806	−	0.948156i	$-0.602946\pi$
$673$	−11097.2	−0.635612	−0.317806	+	0.948156i	$0.602946\pi$
$674$	0	0
$675$	−3848.44	−0.219447
$676$	0	0
$677$	−1647.83	−0.0935468	−0.0467734	−	0.998906i	$-0.514894\pi$
$677$	−1647.83	−0.0935468	−0.0467734	+	0.998906i	$0.514894\pi$
$678$	0	0
$679$	−18531.9	−1.04740
$680$	0	0
$681$	3029.18	0.170453
$682$	0	0
$683$	−9256.41	−0.518575	−0.259287	−	0.965800i	$-0.583488\pi$
$683$	−9256.41	−0.518575	−0.259287	+	0.965800i	$0.583488\pi$
$684$	0	0
$685$	3034.30	0.169248
$686$	0	0
$687$	8082.99	0.448887
$688$	0	0
$689$	4987.29	0.275763
$690$	0	0
$691$	2404.88	0.132397	0.0661983	−	0.997806i	$-0.478913\pi$
$691$	2404.88	0.132397	0.0661983	+	0.997806i	$0.478913\pi$
$692$	0	0
$693$	−11047.1	−0.605548
$694$	0	0
$695$	17500.3	0.955140
$696$	0	0
$697$	−10632.9	−0.577833
$698$	0	0
$699$	−7365.94	−0.398577
$700$	0	0
$701$	1158.57	0.0624232	0.0312116	−	0.999513i	$-0.490063\pi$
$701$	1158.57	0.0624232	0.0312116	+	0.999513i	$0.490063\pi$
$702$	0	0
$703$	4551.28	0.244174
$704$	0	0
$705$	−732.663	−0.0391400
$706$	0	0
$707$	−16448.6	−0.874982
$708$	0	0
$709$	−26868.0	−1.42320	−0.711600	−	0.702585i	$-0.752029\pi$
$709$	−26868.0	−1.42320	−0.711600	+	0.702585i	$0.752029\pi$
$710$	0	0
$711$	6716.58	0.354278
$712$	0	0
$713$	−35588.4	−1.86928
$714$	0	0
$715$	4545.38	0.237745
$716$	0	0
$717$	−5577.96	−0.290534
$718$	0	0
$719$	−12313.1	−0.638666	−0.319333	−	0.947643i	$-0.603459\pi$
$719$	−12313.1	−0.638666	−0.319333	+	0.947643i	$0.603459\pi$
$720$	0	0
$721$	−15427.8	−0.796896
$722$	0	0
$723$	2264.75	0.116496
$724$	0	0
$725$	−7580.02	−0.388296
$726$	0	0
$727$	−4104.02	−0.209367	−0.104683	−	0.994506i	$-0.533383\pi$
$727$	−4104.02	−0.209367	−0.104683	+	0.994506i	$0.533383\pi$
$728$	0	0
$729$	−7795.29	−0.396042
$730$	0	0
$731$	−3776.79	−0.191094
$732$	0	0
$733$	36113.9	1.81978	0.909888	−	0.414854i	$-0.136167\pi$
$733$	36113.9	1.81978	0.909888	+	0.414854i	$0.136167\pi$
$734$	0	0
$735$	−3349.97	−0.168116
$736$	0	0
$737$	27694.1	1.38416
$738$	0	0
$739$	4185.77	0.208357	0.104179	−	0.994559i	$-0.466779\pi$
$739$	4185.77	0.208357	0.104179	+	0.994559i	$0.466779\pi$
$740$	0	0
$741$	−565.898	−0.0280550
$742$	0	0
$743$	24708.4	1.22001	0.610003	−	0.792399i	$-0.291168\pi$
$743$	24708.4	1.22001	0.610003	+	0.792399i	$0.291168\pi$
$744$	0	0
$745$	−25351.1	−1.24670
$746$	0	0
$747$	13365.5	0.654644
$748$	0	0
$749$	−3109.63	−0.151700
$750$	0	0
$751$	−24154.5	−1.17365	−0.586824	−	0.809714i	$-0.699622\pi$
$751$	−24154.5	−1.17365	−0.586824	+	0.809714i	$0.699622\pi$
$752$	0	0
$753$	8480.22	0.410407
$754$	0	0
$755$	15430.6	0.743811
$756$	0	0
$757$	−29215.5	−1.40272	−0.701358	−	0.712809i	$-0.747423\pi$
$757$	−29215.5	−1.40272	−0.701358	+	0.712809i	$0.747423\pi$
$758$	0	0
$759$	7523.67	0.359805
$760$	0	0
$761$	29582.7	1.40916	0.704581	−	0.709624i	$-0.251135\pi$
$761$	29582.7	1.40916	0.704581	+	0.709624i	$0.251135\pi$
$762$	0	0
$763$	16395.1	0.777905
$764$	0	0
$765$	12923.9	0.610802
$766$	0	0
$767$	3784.95	0.178183
$768$	0	0
$769$	−30737.1	−1.44136	−0.720682	−	0.693265i	$-0.756171\pi$
$769$	−30737.1	−1.44136	−0.720682	+	0.693265i	$0.756171\pi$
$770$	0	0
$771$	−3635.04	−0.169796
$772$	0	0
$773$	−13581.6	−0.631947	−0.315973	−	0.948768i	$-0.602331\pi$
$773$	−13581.6	−0.631947	−0.315973	+	0.948768i	$0.602331\pi$
$774$	0	0
$775$	−14504.8	−0.672292
$776$	0	0
$777$	3371.21	0.155652
$778$	0	0
$779$	4705.79	0.216435
$780$	0	0
$781$	−29523.8	−1.35268
$782$	0	0
$783$	15310.8	0.698803
$784$	0	0
$785$	12432.0	0.565246
$786$	0	0
$787$	37723.8	1.70865	0.854325	−	0.519739i	$-0.173971\pi$
$787$	37723.8	1.70865	0.854325	+	0.519739i	$0.173971\pi$
$788$	0	0
$789$	−12087.7	−0.545418
$790$	0	0
$791$	−18010.6	−0.809588
$792$	0	0
$793$	1069.31	0.0478843
$794$	0	0
$795$	−6270.60	−0.279742
$796$	0	0
$797$	−6349.61	−0.282202	−0.141101	−	0.989995i	$-0.545064\pi$
$797$	−6349.61	−0.282202	−0.141101	+	0.989995i	$0.545064\pi$
$798$	0	0
$799$	−2804.57	−0.124178
$800$	0	0
$801$	8214.07	0.362334
$802$	0	0
$803$	−3246.51	−0.142674
$804$	0	0
$805$	10932.8	0.478673
$806$	0	0
$807$	−11176.4	−0.487521
$808$	0	0
$809$	32315.3	1.40438	0.702191	−	0.711988i	$-0.252205\pi$
$809$	32315.3	1.40438	0.702191	+	0.711988i	$0.252205\pi$
$810$	0	0
$811$	27000.2	1.16906	0.584528	−	0.811373i	$-0.301280\pi$
$811$	27000.2	1.16906	0.584528	+	0.811373i	$0.301280\pi$
$812$	0	0
$813$	−14093.9	−0.607989
$814$	0	0
$815$	−29362.5	−1.26199
$816$	0	0
$817$	1671.49	0.0715766
$818$	0	0
$819$	3369.64	0.143766
$820$	0	0
$821$	−8734.23	−0.371287	−0.185644	−	0.982617i	$-0.559437\pi$
$821$	−8734.23	−0.371287	−0.185644	+	0.982617i	$0.559437\pi$
$822$	0	0
$823$	−37165.8	−1.57414	−0.787070	−	0.616864i	$-0.788403\pi$
$823$	−37165.8	−1.57414	−0.787070	+	0.616864i	$0.788403\pi$
$824$	0	0
$825$	3066.43	0.129405
$826$	0	0
$827$	−44504.9	−1.87133	−0.935664	−	0.352892i	$-0.885198\pi$
$827$	−44504.9	−1.87133	−0.935664	+	0.352892i	$0.885198\pi$
$828$	0	0
$829$	8435.36	0.353404	0.176702	−	0.984264i	$-0.443457\pi$
$829$	8435.36	0.353404	0.176702	+	0.984264i	$0.443457\pi$
$830$	0	0
$831$	−6434.07	−0.268586
$832$	0	0
$833$	−12823.4	−0.533378
$834$	0	0
$835$	−21079.2	−0.873623
$836$	0	0
$837$	29298.0	1.20990
$838$	0	0
$839$	15830.0	0.651384	0.325692	−	0.945476i	$-0.394403\pi$
$839$	15830.0	0.651384	0.325692	+	0.945476i	$0.394403\pi$
$840$	0	0
$841$	5767.62	0.236484
$842$	0	0
$843$	10450.0	0.426950
$844$	0	0
$845$	18429.3	0.750280
$846$	0	0
$847$	3635.08	0.147465
$848$	0	0
$849$	463.192	0.0187240
$850$	0	0
$851$	18456.9	0.743473
$852$	0	0
$853$	−25517.9	−1.02429	−0.512144	−	0.858900i	$-0.671148\pi$
$853$	−25517.9	−1.02429	−0.512144	+	0.858900i	$0.671148\pi$
$854$	0	0
$855$	−5719.71	−0.228784
$856$	0	0
$857$	1353.59	0.0539530	0.0269765	−	0.999636i	$-0.491412\pi$
$857$	1353.59	0.0539530	0.0269765	+	0.999636i	$0.491412\pi$
$858$	0	0
$859$	−44829.9	−1.78065	−0.890325	−	0.455326i	$-0.849523\pi$
$859$	−44829.9	−1.78065	−0.890325	+	0.455326i	$0.849523\pi$
$860$	0	0
$861$	3485.66	0.137969
$862$	0	0
$863$	22467.5	0.886216	0.443108	−	0.896468i	$-0.353876\pi$
$863$	22467.5	0.886216	0.443108	+	0.896468i	$0.353876\pi$
$864$	0	0
$865$	29488.5	1.15912
$866$	0	0
$867$	2337.21	0.0915521
$868$	0	0
$869$	−11369.2	−0.443815
$870$	0	0
$871$	−8447.37	−0.328620
$872$	0	0
$873$	39317.6	1.52428
$874$	0	0
$875$	17216.4	0.665165
$876$	0	0
$877$	408.941	0.0157457	0.00787283	−	0.999969i	$-0.497494\pi$
$877$	408.941	0.0157457	0.00787283	+	0.999969i	$0.497494\pi$
$878$	0	0
$879$	−4720.12	−0.181121
$880$	0	0
$881$	−20458.5	−0.782366	−0.391183	−	0.920313i	$-0.627934\pi$
$881$	−20458.5	−0.782366	−0.391183	+	0.920313i	$0.627934\pi$
$882$	0	0
$883$	−3746.69	−0.142793	−0.0713965	−	0.997448i	$-0.522746\pi$
$883$	−3746.69	−0.142793	−0.0713965	+	0.997448i	$0.522746\pi$
$884$	0	0
$885$	−4758.88	−0.180755
$886$	0	0
$887$	−5289.38	−0.200225	−0.100113	−	0.994976i	$-0.531920\pi$
$887$	−5289.38	−0.200225	−0.100113	+	0.994976i	$0.531920\pi$
$888$	0	0
$889$	−8462.24	−0.319251
$890$	0	0
$891$	20159.5	0.757990
$892$	0	0
$893$	1241.22	0.0465126
$894$	0	0
$895$	−19878.5	−0.742420
$896$	0	0
$897$	−2294.90	−0.0854232
$898$	0	0
$899$	57706.3	2.14084
$900$	0	0
$901$	−24003.3	−0.887531
$902$	0	0
$903$	1238.10	0.0456274
$904$	0	0
$905$	−13733.3	−0.504430
$906$	0	0
$907$	10720.8	0.392480	0.196240	−	0.980556i	$-0.437127\pi$
$907$	10720.8	0.392480	0.196240	+	0.980556i	$0.437127\pi$
$908$	0	0
$909$	34897.6	1.27336
$910$	0	0
$911$	−3405.66	−0.123858	−0.0619290	−	0.998081i	$-0.519725\pi$
$911$	−3405.66	−0.123858	−0.0619290	+	0.998081i	$0.519725\pi$
$912$	0	0
$913$	−22624.0	−0.820094
$914$	0	0
$915$	−1344.46	−0.0485753
$916$	0	0
$917$	7068.52	0.254551
$918$	0	0
$919$	−3184.34	−0.114300	−0.0571500	−	0.998366i	$-0.518201\pi$
$919$	−3184.34	−0.114300	−0.0571500	+	0.998366i	$0.518201\pi$
$920$	0	0
$921$	−12114.8	−0.433439
$922$	0	0
$923$	9005.48	0.321147
$924$	0	0
$925$	7522.51	0.267393
$926$	0	0
$927$	32732.0	1.15972
$928$	0	0
$929$	24847.1	0.877510	0.438755	−	0.898607i	$-0.355420\pi$
$929$	24847.1	0.877510	0.438755	+	0.898607i	$0.355420\pi$
$930$	0	0
$931$	5675.24	0.199783
$932$	0	0
$933$	−1499.18	−0.0526056
$934$	0	0
$935$	−21876.4	−0.765171
$936$	0	0
$937$	16072.8	0.560381	0.280191	−	0.959944i	$-0.409602\pi$
$937$	16072.8	0.560381	0.280191	+	0.959944i	$0.409602\pi$
$938$	0	0
$939$	−9893.12	−0.343823
$940$	0	0
$941$	49359.1	1.70995	0.854974	−	0.518672i	$-0.173573\pi$
$941$	49359.1	1.70995	0.854974	+	0.518672i	$0.173573\pi$
$942$	0	0
$943$	19083.6	0.659010
$944$	0	0
$945$	−9000.41	−0.309824
$946$	0	0
$947$	−19624.5	−0.673400	−0.336700	−	0.941612i	$-0.609311\pi$
$947$	−19624.5	−0.673400	−0.336700	+	0.941612i	$0.609311\pi$
$948$	0	0
$949$	990.266	0.0338729
$950$	0	0
$951$	−10061.0	−0.343061
$952$	0	0
$953$	39172.6	1.33151	0.665753	−	0.746172i	$-0.268110\pi$
$953$	39172.6	1.33151	0.665753	+	0.746172i	$0.268110\pi$
$954$	0	0
$955$	−18212.7	−0.617118
$956$	0	0
$957$	−12199.6	−0.412076
$958$	0	0
$959$	−3807.64	−0.128212
$960$	0	0
$961$	80632.9	2.70662
$962$	0	0
$963$	6597.46	0.220768
$964$	0	0
$965$	40118.0	1.33829
$966$	0	0
$967$	50809.7	1.68969	0.844845	−	0.535010i	$-0.179692\pi$
$967$	50809.7	1.68969	0.844845	+	0.535010i	$0.179692\pi$
$968$	0	0
$969$	2723.60	0.0902939
$970$	0	0
$971$	49995.3	1.65235	0.826173	−	0.563417i	$-0.190514\pi$
$971$	49995.3	1.65235	0.826173	+	0.563417i	$0.190514\pi$
$972$	0	0
$973$	−21960.4	−0.723555
$974$	0	0
$975$	−935.335	−0.0307228
$976$	0	0
$977$	−6507.03	−0.213079	−0.106540	−	0.994308i	$-0.533977\pi$
$977$	−6507.03	−0.213079	−0.106540	+	0.994308i	$0.533977\pi$
$978$	0	0
$979$	−13904.1	−0.453908
$980$	0	0
$981$	−34784.1	−1.13208
$982$	0	0
$983$	37044.4	1.20197	0.600983	−	0.799262i	$-0.294776\pi$
$983$	37044.4	1.20197	0.600983	+	0.799262i	$0.294776\pi$
$984$	0	0
$985$	−16145.2	−0.522263
$986$	0	0
$987$	919.391	0.0296500
$988$	0	0
$989$	6778.46	0.217940
$990$	0	0
$991$	17372.8	0.556877	0.278439	−	0.960454i	$-0.410183\pi$
$991$	17372.8	0.556877	0.278439	+	0.960454i	$0.410183\pi$
$992$	0	0
$993$	−3975.91	−0.127061
$994$	0	0
$995$	31400.5	1.00046
$996$	0	0
$997$	−45015.4	−1.42994	−0.714971	−	0.699154i	$-0.753560\pi$
$997$	−45015.4	−1.42994	−0.714971	+	0.699154i	$0.753560\pi$
$998$	0	0
$999$	−15194.6	−0.481217

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	752.4.a.c.1.1		3
4.3	odd	2		47.4.a.a.1.2	✓	3
12.11	even	2		423.4.a.b.1.2		3
20.19	odd	2		1175.4.a.a.1.2		3
28.27	even	2		2303.4.a.a.1.2		3
188.187	even	2		2209.4.a.a.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
47.4.a.a.1.2	✓	3	4.3	odd	2
423.4.a.b.1.2		3	12.11	even	2
752.4.a.c.1.1		3	1.1	even	1	trivial
1175.4.a.a.1.2		3	20.19	odd	2
2209.4.a.a.1.2		3	188.187	even	2
2303.4.a.a.1.2		3	28.27	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:	\( N \)	\(=\)	\( 752 = 2^{4} \cdot 47 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	752.a (trivial)

\(f(q)\)	\(=\)	\(q-1.72833 q^{3} -9.01945 q^{5} +11.3182 q^{7} -24.0129 q^{9} +O(q^{10})\)	\(q-1.72833 q^{3} -9.01945 q^{5} +11.3182 q^{7} -24.0129 q^{9} +40.6469 q^{11} -12.3983 q^{13} +15.5886 q^{15} +59.6717 q^{17} -26.4089 q^{19} -19.5615 q^{21} -107.097 q^{23} -43.6495 q^{25} +88.1670 q^{27} +173.657 q^{29} +332.301 q^{31} -70.2512 q^{33} -102.084 q^{35} -172.339 q^{37} +21.4283 q^{39} -178.190 q^{41} -63.2928 q^{43} +216.583 q^{45} -47.0000 q^{47} -214.899 q^{49} -103.132 q^{51} -402.256 q^{53} -366.613 q^{55} +45.6432 q^{57} -305.280 q^{59} -86.2464 q^{61} -271.782 q^{63} +111.826 q^{65} +681.333 q^{67} +185.098 q^{69} -726.348 q^{71} -79.8711 q^{73} +75.4406 q^{75} +460.049 q^{77} -279.707 q^{79} +495.966 q^{81} -556.598 q^{83} -538.206 q^{85} -300.135 q^{87} -342.069 q^{89} -140.326 q^{91} -574.325 q^{93} +238.194 q^{95} -1637.35 q^{97} -976.050 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 5 q^{3} - 6 q^{5} + 45 q^{7} - 42 q^{9}+O(q^{10}) \) 3 * q + 5 * q^3 - 6 * q^5 + 45 * q^7 - 42 * q^9	\( 3 q + 5 q^{3} - 6 q^{5} + 45 q^{7} - 42 q^{9} - 2 q^{11} - 80 q^{13} - 14 q^{15} - 39 q^{17} + 24 q^{19} + 24 q^{21} - 120 q^{23} - 171 q^{25} - 64 q^{27} - 184 q^{29} + 4 q^{31} - 208 q^{33} + 156 q^{35} - 589 q^{37} - 60 q^{39} - 92 q^{41} + 250 q^{43} - 78 q^{45} - 141 q^{47} + 30 q^{49} - 317 q^{51} + 459 q^{53} - 448 q^{55} + 216 q^{57} - 579 q^{59} + 267 q^{61} - 1044 q^{63} - 424 q^{65} + 540 q^{67} + 642 q^{69} - 749 q^{71} - 1924 q^{73} - 473 q^{75} - 288 q^{77} - 805 q^{79} + 291 q^{81} - 712 q^{83} - 1038 q^{85} - 1216 q^{87} + 835 q^{89} - 2040 q^{91} - 1500 q^{93} + 312 q^{95} - 2243 q^{97} - 554 q^{99}+O(q^{100}) \) 3 * q + 5 * q^3 - 6 * q^5 + 45 * q^7 - 42 * q^9 - 2 * q^11 - 80 * q^13 - 14 * q^15 - 39 * q^17 + 24 * q^19 + 24 * q^21 - 120 * q^23 - 171 * q^25 - 64 * q^27 - 184 * q^29 + 4 * q^31 - 208 * q^33 + 156 * q^35 - 589 * q^37 - 60 * q^39 - 92 * q^41 + 250 * q^43 - 78 * q^45 - 141 * q^47 + 30 * q^49 - 317 * q^51 + 459 * q^53 - 448 * q^55 + 216 * q^57 - 579 * q^59 + 267 * q^61 - 1044 * q^63 - 424 * q^65 + 540 * q^67 + 642 * q^69 - 749 * q^71 - 1924 * q^73 - 473 * q^75 - 288 * q^77 - 805 * q^79 + 291 * q^81 - 712 * q^83 - 1038 * q^85 - 1216 * q^87 + 835 * q^89 - 2040 * q^91 - 1500 * q^93 + 312 * q^95 - 2243 * q^97 - 554 * q^99

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