LMFDB - Embedded newform 153.4.a.h.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [153,4,Mod(1,153)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(153, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("153.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 153 = 3^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	153.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:	$9.02729223088$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.1506848.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 17x^{2} + 18 $ x^4 - 17*x^2 + 18
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.98315$ of defining polynomial
Character	$\chi$	$=$	153.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.47680 q^{2} +4.08814 q^{4} -7.60718 q^{5} +16.3925 q^{7} +13.6008 q^{8} +O(q^{10})$	\(q-3.47680 q^{2} +4.08814 q^{4} -7.60718 q^{5} +16.3925 q^{7} +13.6008 q^{8} +26.4486 q^{10} -42.4328 q^{11} +62.2726 q^{13} -56.9936 q^{14} -79.9922 q^{16} +17.0000 q^{17} -52.0963 q^{19} -31.0992 q^{20} +147.530 q^{22} -67.8856 q^{23} -67.1308 q^{25} -216.509 q^{26} +67.0150 q^{28} -91.7954 q^{29} -177.082 q^{31} +169.311 q^{32} -59.1056 q^{34} -124.701 q^{35} -258.046 q^{37} +181.128 q^{38} -103.463 q^{40} -262.159 q^{41} -331.883 q^{43} -173.471 q^{44} +236.025 q^{46} +49.0592 q^{47} -74.2848 q^{49} +233.400 q^{50} +254.579 q^{52} +462.443 q^{53} +322.794 q^{55} +222.951 q^{56} +319.154 q^{58} +351.772 q^{59} +41.6873 q^{61} +615.679 q^{62} +51.2773 q^{64} -473.719 q^{65} +845.771 q^{67} +69.4984 q^{68} +433.560 q^{70} +420.922 q^{71} -402.152 q^{73} +897.175 q^{74} -212.977 q^{76} -695.581 q^{77} -736.775 q^{79} +608.515 q^{80} +911.473 q^{82} -1241.26 q^{83} -129.322 q^{85} +1153.89 q^{86} -577.118 q^{88} -463.658 q^{89} +1020.81 q^{91} -277.526 q^{92} -170.569 q^{94} +396.306 q^{95} -1236.59 q^{97} +258.273 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{2} + 26 q^{4} - 22 q^{5} - 24 q^{7} - 102 q^{8}+O(q^{10}) $ 4 * q - 4 * q^2 + 26 * q^4 - 22 * q^5 - 24 * q^7 - 102 * q^8	$ 4 q - 4 q^{2} + 26 q^{4} - 22 q^{5} - 24 q^{7} - 102 q^{8} - 2 q^{10} - 50 q^{11} + 26 q^{13} - 80 q^{14} + 138 q^{16} + 68 q^{17} + 34 q^{19} - 312 q^{20} - 254 q^{22} - 382 q^{23} + 138 q^{25} + 22 q^{26} + 52 q^{28} - 540 q^{29} - 356 q^{31} - 730 q^{32} - 68 q^{34} - 304 q^{35} - 404 q^{37} - 298 q^{38} + 332 q^{40} + 114 q^{41} + 570 q^{43} + 1368 q^{44} - 290 q^{46} - 496 q^{47} - 224 q^{49} + 1862 q^{50} - 1012 q^{52} - 92 q^{53} - 482 q^{55} + 1428 q^{56} + 1324 q^{58} + 48 q^{59} - 1036 q^{61} + 2564 q^{62} + 2898 q^{64} + 342 q^{65} + 812 q^{67} + 442 q^{68} + 152 q^{70} - 1044 q^{71} - 1212 q^{73} + 1444 q^{74} + 2268 q^{76} + 564 q^{77} + 488 q^{79} + 1000 q^{80} - 938 q^{82} - 1708 q^{83} - 374 q^{85} + 2446 q^{86} - 3868 q^{88} + 8 q^{89} + 716 q^{91} - 1356 q^{92} - 1224 q^{94} - 1010 q^{95} - 76 q^{97} + 1472 q^{98}+O(q^{100}) $ 4 * q - 4 * q^2 + 26 * q^4 - 22 * q^5 - 24 * q^7 - 102 * q^8 - 2 * q^10 - 50 * q^11 + 26 * q^13 - 80 * q^14 + 138 * q^16 + 68 * q^17 + 34 * q^19 - 312 * q^20 - 254 * q^22 - 382 * q^23 + 138 * q^25 + 22 * q^26 + 52 * q^28 - 540 * q^29 - 356 * q^31 - 730 * q^32 - 68 * q^34 - 304 * q^35 - 404 * q^37 - 298 * q^38 + 332 * q^40 + 114 * q^41 + 570 * q^43 + 1368 * q^44 - 290 * q^46 - 496 * q^47 - 224 * q^49 + 1862 * q^50 - 1012 * q^52 - 92 * q^53 - 482 * q^55 + 1428 * q^56 + 1324 * q^58 + 48 * q^59 - 1036 * q^61 + 2564 * q^62 + 2898 * q^64 + 342 * q^65 + 812 * q^67 + 442 * q^68 + 152 * q^70 - 1044 * q^71 - 1212 * q^73 + 1444 * q^74 + 2268 * q^76 + 564 * q^77 + 488 * q^79 + 1000 * q^80 - 938 * q^82 - 1708 * q^83 - 374 * q^85 + 2446 * q^86 - 3868 * q^88 + 8 * q^89 + 716 * q^91 - 1356 * q^92 - 1224 * q^94 - 1010 * q^95 - 76 * q^97 + 1472 * 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 $

$2$	−3.47680	−1.22923	−0.614617	−	0.788825i	$-0.710690\pi$
$2$	−3.47680	−1.22923	−0.614617	+	0.788825i	$0.710690\pi$
$3$	0	0
$3$	0	0
$4$	4.08814	0.511018
$5$	−7.60718	−0.680407	−0.340203	−	0.940352i	$-0.610496\pi$
$5$	−7.60718	−0.680407	−0.340203	+	0.940352i	$0.610496\pi$
$6$	0	0
$7$	16.3925	0.885114	0.442557	−	0.896740i	$-0.354071\pi$
$7$	16.3925	0.885114	0.442557	+	0.896740i	$0.354071\pi$
$8$	13.6008	0.601074
$9$	0	0
$10$	26.4486	0.836380
$11$	−42.4328	−1.16309	−0.581544	−	0.813515i	$-0.697551\pi$
$11$	−42.4328	−1.16309	−0.581544	+	0.813515i	$0.697551\pi$
$12$	0	0
$13$	62.2726	1.32856	0.664281	−	0.747483i	$-0.268738\pi$
$13$	62.2726	1.32856	0.664281	+	0.747483i	$0.268738\pi$
$14$	−56.9936	−1.08801
$15$	0	0
$16$	−79.9922	−1.24988
$17$	17.0000	0.242536
$17$	17.0000	0.242536
$18$	0	0
$19$	−52.0963	−0.629037	−0.314519	−	0.949251i	$-0.601843\pi$
$19$	−52.0963	−0.629037	−0.314519	+	0.949251i	$0.601843\pi$
$20$	−31.0992	−0.347700
$21$	0	0
$22$	147.530	1.42971
$23$	−67.8856	−0.615440	−0.307720	−	0.951477i	$-0.599566\pi$
$23$	−67.8856	−0.615440	−0.307720	+	0.951477i	$0.599566\pi$
$24$	0	0
$25$	−67.1308	−0.537046
$26$	−216.509	−1.63311
$27$	0	0
$28$	67.0150	0.452309
$29$	−91.7954	−0.587792	−0.293896	−	0.955837i	$-0.594952\pi$
$29$	−91.7954	−0.587792	−0.293896	+	0.955837i	$0.594952\pi$
$30$	0	0
$31$	−177.082	−1.02596	−0.512982	−	0.858399i	$-0.671459\pi$
$31$	−177.082	−1.02596	−0.512982	+	0.858399i	$0.671459\pi$
$32$	169.311	0.935320
$33$	0	0
$34$	−59.1056	−0.298133
$35$	−124.701	−0.602237
$36$	0	0
$37$	−258.046	−1.14655	−0.573277	−	0.819361i	$-0.694328\pi$
$37$	−258.046	−1.14655	−0.573277	+	0.819361i	$0.694328\pi$
$38$	181.128	0.773235
$39$	0	0
$40$	−103.463	−0.408975
$41$	−262.159	−0.998592	−0.499296	−	0.866431i	$-0.666408\pi$
$41$	−262.159	−0.998592	−0.499296	+	0.866431i	$0.666408\pi$
$42$	0	0
$43$	−331.883	−1.17701	−0.588507	−	0.808492i	$-0.700284\pi$
$43$	−331.883	−1.17701	−0.588507	+	0.808492i	$0.700284\pi$
$44$	−173.471	−0.594358
$45$	0	0
$46$	236.025	0.756520
$47$	49.0592	0.152256	0.0761278	−	0.997098i	$-0.475744\pi$
$47$	49.0592	0.152256	0.0761278	+	0.997098i	$0.475744\pi$
$48$	0	0
$49$	−74.2848	−0.216574
$50$	233.400	0.660156
$51$	0	0
$52$	254.579	0.678919
$53$	462.443	1.19852	0.599259	−	0.800556i	$-0.295462\pi$
$53$	462.443	1.19852	0.599259	+	0.800556i	$0.295462\pi$
$54$	0	0
$55$	322.794	0.791373
$56$	222.951	0.532019
$57$	0	0
$58$	319.154	0.722535
$59$	351.772	0.776217	0.388109	−	0.921614i	$-0.373129\pi$
$59$	351.772	0.776217	0.388109	+	0.921614i	$0.373129\pi$
$60$	0	0
$61$	41.6873	0.0875003	0.0437501	−	0.999043i	$-0.486069\pi$
$61$	41.6873	0.0875003	0.0437501	+	0.999043i	$0.486069\pi$
$62$	615.679	1.26115
$63$	0	0
$64$	51.2773	0.100151
$65$	−473.719	−0.903963
$66$	0	0
$67$	845.771	1.54220	0.771100	−	0.636715i	$-0.219707\pi$
$67$	845.771	1.54220	0.771100	+	0.636715i	$0.219707\pi$
$68$	69.4984	0.123940
$69$	0	0
$70$	433.560	0.740291
$71$	420.922	0.703580	0.351790	−	0.936079i	$-0.385573\pi$
$71$	420.922	0.703580	0.351790	+	0.936079i	$0.385573\pi$
$72$	0	0
$73$	−402.152	−0.644771	−0.322386	−	0.946608i	$-0.604485\pi$
$73$	−402.152	−0.644771	−0.322386	+	0.946608i	$0.604485\pi$
$74$	897.175	1.40939
$75$	0	0
$76$	−212.977	−0.321449
$77$	−695.581	−1.02947
$78$	0	0
$79$	−736.775	−1.04929	−0.524643	−	0.851322i	$-0.675801\pi$
$79$	−736.775	−1.04929	−0.524643	+	0.851322i	$0.675801\pi$
$80$	608.515	0.850426
$81$	0	0
$82$	911.473	1.22750
$83$	−1241.26	−1.64152	−0.820760	−	0.571273i	$-0.806450\pi$
$83$	−1241.26	−1.64152	−0.820760	+	0.571273i	$0.806450\pi$
$84$	0	0
$85$	−129.322	−0.165023
$86$	1153.89	1.44683
$87$	0	0
$88$	−577.118	−0.699102
$89$	−463.658	−0.552221	−0.276110	−	0.961126i	$-0.589046\pi$
$89$	−463.658	−0.552221	−0.276110	+	0.961126i	$0.589046\pi$
$90$	0	0
$91$	1020.81	1.17593
$92$	−277.526	−0.314501
$93$	0	0
$94$	−170.569	−0.187158
$95$	396.306	0.428001
$96$	0	0
$97$	−1236.59	−1.29440	−0.647200	−	0.762321i	$-0.724060\pi$
$97$	−1236.59	−1.29440	−0.647200	+	0.762321i	$0.724060\pi$
$98$	258.273	0.266220
$99$	0	0
$100$	−274.440	−0.274440
$101$	−812.390	−0.800355	−0.400178	−	0.916438i	$-0.631052\pi$
$101$	−812.390	−0.800355	−0.400178	+	0.916438i	$0.631052\pi$
$102$	0	0
$103$	−762.574	−0.729501	−0.364751	−	0.931105i	$-0.618846\pi$
$103$	−762.574	−0.729501	−0.364751	+	0.931105i	$0.618846\pi$
$104$	846.954	0.798564
$105$	0	0
$106$	−1607.82	−1.47326
$107$	−954.481	−0.862366	−0.431183	−	0.902265i	$-0.641904\pi$
$107$	−954.481	−0.862366	−0.431183	+	0.902265i	$0.641904\pi$
$108$	0	0
$109$	1383.24	1.21551	0.607754	−	0.794125i	$-0.292071\pi$
$109$	1383.24	1.21551	0.607754	+	0.794125i	$0.292071\pi$
$110$	−1122.29	−0.972783
$111$	0	0
$112$	−1311.28	−1.10628
$113$	1763.42	1.46804	0.734021	−	0.679127i	$-0.237641\pi$
$113$	1763.42	1.46804	0.734021	+	0.679127i	$0.237641\pi$
$114$	0	0
$115$	516.418	0.418750
$116$	−375.273	−0.300372
$117$	0	0
$118$	−1223.04	−0.954153
$119$	278.673	0.214672
$120$	0	0
$121$	469.542	0.352774
$122$	−144.939	−0.107558
$123$	0	0
$124$	−723.937	−0.524286
$125$	1461.57	1.04582
$126$	0	0
$127$	2136.72	1.49294	0.746469	−	0.665420i	$-0.231747\pi$
$127$	2136.72	1.49294	0.746469	+	0.665420i	$0.231747\pi$
$128$	−1532.77	−1.05843
$129$	0	0
$130$	1647.03	1.11118
$131$	−2681.59	−1.78849	−0.894244	−	0.447580i	$-0.852286\pi$
$131$	−2681.59	−1.78849	−0.894244	+	0.447580i	$0.852286\pi$
$132$	0	0
$133$	−853.990	−0.556770
$134$	−2940.58	−1.89572
$135$	0	0
$136$	231.213	0.145782
$137$	−2571.30	−1.60351	−0.801755	−	0.597653i	$-0.796100\pi$
$137$	−2571.30	−1.60351	−0.801755	+	0.597653i	$0.796100\pi$
$138$	0	0
$139$	−483.513	−0.295043	−0.147522	−	0.989059i	$-0.547130\pi$
$139$	−483.513	−0.295043	−0.147522	+	0.989059i	$0.547130\pi$
$140$	−509.795	−0.307754
$141$	0	0
$142$	−1463.46	−0.864865
$143$	−2642.40	−1.54523
$144$	0	0
$145$	698.304	0.399938
$146$	1398.20	0.792575
$147$	0	0
$148$	−1054.93	−0.585910
$149$	3194.69	1.75650	0.878252	−	0.478198i	$-0.158710\pi$
$149$	3194.69	1.75650	0.878252	+	0.478198i	$0.158710\pi$
$150$	0	0
$151$	1038.38	0.559617	0.279809	−	0.960056i	$-0.409729\pi$
$151$	1038.38	0.559617	0.279809	+	0.960056i	$0.409729\pi$
$152$	−708.549	−0.378098
$153$	0	0
$154$	2418.40	1.26545
$155$	1347.10	0.698073
$156$	0	0
$157$	658.734	0.334858	0.167429	−	0.985884i	$-0.446454\pi$
$157$	658.734	0.334858	0.167429	+	0.985884i	$0.446454\pi$
$158$	2561.62	1.28982
$159$	0	0
$160$	−1287.98	−0.636398
$161$	−1112.82	−0.544734
$162$	0	0
$163$	−603.959	−0.290219	−0.145110	−	0.989416i	$-0.546353\pi$
$163$	−603.959	−0.290219	−0.145110	+	0.989416i	$0.546353\pi$
$164$	−1071.74	−0.510298
$165$	0	0
$166$	4315.62	2.01781
$167$	−1748.74	−0.810311	−0.405155	−	0.914248i	$-0.632783\pi$
$167$	−1748.74	−0.810311	−0.405155	+	0.914248i	$0.632783\pi$
$168$	0	0
$169$	1680.87	0.765077
$170$	449.627	0.202852
$171$	0	0
$172$	−1356.78	−0.601475
$173$	1821.21	0.800372	0.400186	−	0.916434i	$-0.368946\pi$
$173$	1821.21	0.800372	0.400186	+	0.916434i	$0.368946\pi$
$174$	0	0
$175$	−1100.44	−0.475347
$176$	3394.29	1.45372
$177$	0	0
$178$	1612.05	0.678809
$179$	−3269.37	−1.36516	−0.682581	−	0.730810i	$-0.739143\pi$
$179$	−3269.37	−1.36516	−0.682581	+	0.730810i	$0.739143\pi$
$180$	0	0
$181$	4270.64	1.75378	0.876890	−	0.480691i	$-0.159614\pi$
$181$	4270.64	1.75378	0.876890	+	0.480691i	$0.159614\pi$
$182$	−3549.14	−1.44549
$183$	0	0
$184$	−923.295	−0.369925
$185$	1963.00	0.780124
$186$	0	0
$187$	−721.357	−0.282090
$188$	200.561	0.0778053
$189$	0	0
$190$	−1377.88	−0.526114
$191$	−328.693	−0.124520	−0.0622602	−	0.998060i	$-0.519831\pi$
$191$	−328.693	−0.124520	−0.0622602	+	0.998060i	$0.519831\pi$
$192$	0	0
$193$	−3214.93	−1.19904	−0.599522	−	0.800358i	$-0.704643\pi$
$193$	−3214.93	−1.19904	−0.599522	+	0.800358i	$0.704643\pi$
$194$	4299.38	1.59112
$195$	0	0
$196$	−303.687	−0.110673
$197$	3436.44	1.24282	0.621411	−	0.783484i	$-0.286560\pi$
$197$	3436.44	1.24282	0.621411	+	0.783484i	$0.286560\pi$
$198$	0	0
$199$	−3558.85	−1.26774	−0.633869	−	0.773440i	$-0.718534\pi$
$199$	−3558.85	−1.26774	−0.633869	+	0.773440i	$0.718534\pi$
$200$	−913.029	−0.322805
$201$	0	0
$202$	2824.52	0.983824
$203$	−1504.76	−0.520263
$204$	0	0
$205$	1994.29	0.679449
$206$	2651.32	0.896728
$207$	0	0
$208$	−4981.32	−1.66054
$209$	2210.59	0.731626
$210$	0	0
$211$	984.360	0.321166	0.160583	−	0.987022i	$-0.448662\pi$
$211$	984.360	0.321166	0.160583	+	0.987022i	$0.448662\pi$
$212$	1890.53	0.612463
$213$	0	0
$214$	3318.54	1.06005
$215$	2524.69	0.800849
$216$	0	0
$217$	−2902.83	−0.908095
$218$	−4809.25	−1.49414
$219$	0	0
$220$	1319.63	0.404406
$221$	1058.63	0.322224
$222$	0	0
$223$	−1077.90	−0.323684	−0.161842	−	0.986817i	$-0.551744\pi$
$223$	−1077.90	−0.323684	−0.161842	+	0.986817i	$0.551744\pi$
$224$	2775.44	0.827864
$225$	0	0
$226$	−6131.07	−1.80457
$227$	6639.72	1.94138	0.970692	−	0.240329i	$-0.0772552\pi$
$227$	6639.72	1.94138	0.970692	+	0.240329i	$0.0772552\pi$
$228$	0	0
$229$	−5261.85	−1.51840	−0.759198	−	0.650860i	$-0.774409\pi$
$229$	−5261.85	−1.51840	−0.759198	+	0.650860i	$0.774409\pi$
$230$	−1795.48	−0.514742
$231$	0	0
$232$	−1248.49	−0.353307
$233$	−1463.68	−0.411541	−0.205771	−	0.978600i	$-0.565970\pi$
$233$	−1463.68	−0.411541	−0.205771	+	0.978600i	$0.565970\pi$
$234$	0	0
$235$	−373.202	−0.103596
$236$	1438.09	0.396661
$237$	0	0
$238$	−968.891	−0.263882
$239$	−2244.08	−0.607354	−0.303677	−	0.952775i	$-0.598214\pi$
$239$	−2244.08	−0.607354	−0.303677	+	0.952775i	$0.598214\pi$
$240$	0	0
$241$	4728.95	1.26398	0.631989	−	0.774977i	$-0.282239\pi$
$241$	4728.95	1.26398	0.631989	+	0.774977i	$0.282239\pi$
$242$	−1632.50	−0.433642
$243$	0	0
$244$	170.424	0.0447142
$245$	565.098	0.147358
$246$	0	0
$247$	−3244.17	−0.835715
$248$	−2408.45	−0.616681
$249$	0	0
$250$	−5081.60	−1.28555
$251$	4629.50	1.16419	0.582094	−	0.813121i	$-0.302234\pi$
$251$	4629.50	1.16419	0.582094	+	0.813121i	$0.302234\pi$
$252$	0	0
$253$	2880.57	0.715811
$254$	−7428.95	−1.83517
$255$	0	0
$256$	4918.91	1.20091
$257$	7620.67	1.84967	0.924833	−	0.380373i	$-0.124204\pi$
$257$	7620.67	1.84967	0.924833	+	0.380373i	$0.124204\pi$
$258$	0	0
$259$	−4230.03	−1.01483
$260$	−1936.63	−0.461941
$261$	0	0
$262$	9323.37	2.19847
$263$	−4030.59	−0.945008	−0.472504	−	0.881329i	$-0.656650\pi$
$263$	−4030.59	−0.945008	−0.472504	+	0.881329i	$0.656650\pi$
$264$	0	0
$265$	−3517.89	−0.815479
$266$	2969.15	0.684400
$267$	0	0
$268$	3457.63	0.788091
$269$	−356.893	−0.0808928	−0.0404464	−	0.999182i	$-0.512878\pi$
$269$	−356.893	−0.0808928	−0.0404464	+	0.999182i	$0.512878\pi$
$270$	0	0
$271$	3.71432	0.000832578 0	0.000416289	−	1.00000i	$-0.499867\pi$
$271$	3.71432	0.000832578 0	0.000416289		1.00000i	$0.499867\pi$
$272$	−1359.87	−0.303140
$273$	0	0
$274$	8939.89	1.97109
$275$	2848.55	0.624632
$276$	0	0
$277$	4624.45	1.00309	0.501546	−	0.865131i	$-0.332765\pi$
$277$	4624.45	1.00309	0.501546	+	0.865131i	$0.332765\pi$
$278$	1681.08	0.362677
$279$	0	0
$280$	−1696.03	−0.361989
$281$	479.468	0.101789	0.0508945	−	0.998704i	$-0.483793\pi$
$281$	479.468	0.101789	0.0508945	+	0.998704i	$0.483793\pi$
$282$	0	0
$283$	2620.73	0.550482	0.275241	−	0.961375i	$-0.411242\pi$
$283$	2620.73	0.550482	0.275241	+	0.961375i	$0.411242\pi$
$284$	1720.79	0.359542
$285$	0	0
$286$	9187.10	1.89946
$287$	−4297.44	−0.883868
$288$	0	0
$289$	289.000	0.0588235
$290$	−2427.86	−0.491618
$291$	0	0
$292$	−1644.05	−0.329489
$293$	1166.08	0.232502	0.116251	−	0.993220i	$-0.462912\pi$
$293$	1166.08	0.232502	0.116251	+	0.993220i	$0.462912\pi$
$294$	0	0
$295$	−2675.99	−0.528144
$296$	−3509.62	−0.689164
$297$	0	0
$298$	−11107.3	−2.15916
$299$	−4227.41	−0.817650
$300$	0	0
$301$	−5440.40	−1.04179
$302$	−3610.24	−0.687901
$303$	0	0
$304$	4167.30	0.786220
$305$	−317.123	−0.0595358
$306$	0	0
$307$	7729.44	1.43695	0.718473	−	0.695555i	$-0.244841\pi$
$307$	7729.44	1.43695	0.718473	+	0.695555i	$0.244841\pi$
$308$	−2843.63	−0.526075
$309$	0	0
$310$	−4683.59	−0.858096
$311$	4549.54	0.829521	0.414760	−	0.909931i	$-0.363865\pi$
$311$	4549.54	0.829521	0.414760	+	0.909931i	$0.363865\pi$
$312$	0	0
$313$	−2687.97	−0.485410	−0.242705	−	0.970100i	$-0.578035\pi$
$313$	−2687.97	−0.485410	−0.242705	+	0.970100i	$0.578035\pi$
$314$	−2290.29	−0.411619
$315$	0	0
$316$	−3012.04	−0.536204
$317$	−5617.46	−0.995294	−0.497647	−	0.867380i	$-0.665802\pi$
$317$	−5617.46	−0.995294	−0.497647	+	0.867380i	$0.665802\pi$
$318$	0	0
$319$	3895.14	0.683654
$320$	−390.076	−0.0681435
$321$	0	0
$322$	3869.04	0.669606
$323$	−885.637	−0.152564
$324$	0	0
$325$	−4180.41	−0.713499
$326$	2099.85	0.356748
$327$	0	0
$328$	−3565.55	−0.600228
$329$	804.204	0.134764
$330$	0	0
$331$	8535.90	1.41745	0.708724	−	0.705485i	$-0.249271\pi$
$331$	8535.90	1.41745	0.708724	+	0.705485i	$0.249271\pi$
$332$	−5074.45	−0.838846
$333$	0	0
$334$	6080.04	0.996062
$335$	−6433.93	−1.04932
$336$	0	0
$337$	9073.71	1.46670	0.733348	−	0.679854i	$-0.237957\pi$
$337$	9073.71	1.46670	0.733348	+	0.679854i	$0.237957\pi$
$338$	−5844.06	−0.940459
$339$	0	0
$340$	−528.687	−0.0843296
$341$	7514.09	1.19329
$342$	0	0
$343$	−6840.36	−1.07681
$344$	−4513.85	−0.707473
$345$	0	0
$346$	−6332.00	−0.983845
$347$	−9087.35	−1.40586	−0.702932	−	0.711257i	$-0.748126\pi$
$347$	−9087.35	−1.40586	−0.702932	+	0.711257i	$0.748126\pi$
$348$	0	0
$349$	−8381.26	−1.28550	−0.642749	−	0.766077i	$-0.722206\pi$
$349$	−8381.26	−1.28550	−0.642749	+	0.766077i	$0.722206\pi$
$350$	3826.02	0.584313
$351$	0	0
$352$	−7184.34	−1.08786
$353$	7887.40	1.18925	0.594623	−	0.804005i	$-0.297301\pi$
$353$	7887.40	1.18925	0.594623	+	0.804005i	$0.297301\pi$
$354$	0	0
$355$	−3202.03	−0.478721
$356$	−1895.50	−0.282194
$357$	0	0
$358$	11366.9	1.67811
$359$	−8093.44	−1.18985	−0.594924	−	0.803782i	$-0.702818\pi$
$359$	−8093.44	−1.18985	−0.594924	+	0.803782i	$0.702818\pi$
$360$	0	0
$361$	−4144.98	−0.604312
$362$	−14848.2	−2.15581
$363$	0	0
$364$	4173.20	0.600920
$365$	3059.24	0.438707
$366$	0	0
$367$	871.023	0.123888	0.0619442	−	0.998080i	$-0.480270\pi$
$367$	871.023	0.123888	0.0619442	+	0.998080i	$0.480270\pi$
$368$	5430.32	0.769225
$369$	0	0
$370$	−6824.97	−0.958955
$371$	7580.61	1.06082
$372$	0	0
$373$	−11778.3	−1.63500	−0.817501	−	0.575927i	$-0.804641\pi$
$373$	−11778.3	−1.63500	−0.817501	+	0.575927i	$0.804641\pi$
$374$	2508.02	0.346755
$375$	0	0
$376$	667.242	0.0915169
$377$	−5716.34	−0.780919
$378$	0	0
$379$	1344.02	0.182157	0.0910787	−	0.995844i	$-0.470969\pi$
$379$	1344.02	0.182157	0.0910787	+	0.995844i	$0.470969\pi$
$380$	1620.15	0.218716
$381$	0	0
$382$	1142.80	0.153065
$383$	−874.156	−0.116625	−0.0583124	−	0.998298i	$-0.518572\pi$
$383$	−874.156	−0.116625	−0.0583124	+	0.998298i	$0.518572\pi$
$384$	0	0
$385$	5291.41	0.700455
$386$	11177.7	1.47391
$387$	0	0
$388$	−5055.35	−0.661461
$389$	4783.41	0.623467	0.311734	−	0.950170i	$-0.399090\pi$
$389$	4783.41	0.623467	0.311734	+	0.950170i	$0.399090\pi$
$390$	0	0
$391$	−1154.05	−0.149266
$392$	−1010.33	−0.130177
$393$	0	0
$394$	−11947.8	−1.52772
$395$	5604.78	0.713942
$396$	0	0
$397$	−5829.06	−0.736907	−0.368454	−	0.929646i	$-0.620113\pi$
$397$	−5829.06	−0.736907	−0.368454	+	0.929646i	$0.620113\pi$
$398$	12373.4	1.55835
$399$	0	0
$400$	5369.94	0.671243
$401$	3158.93	0.393390	0.196695	−	0.980465i	$-0.436979\pi$
$401$	3158.93	0.393390	0.196695	+	0.980465i	$0.436979\pi$
$402$	0	0
$403$	−11027.4	−1.36306
$404$	−3321.17	−0.408995
$405$	0	0
$406$	5231.75	0.639525
$407$	10949.6	1.33354
$408$	0	0
$409$	11171.5	1.35059	0.675297	−	0.737546i	$-0.264016\pi$
$409$	11171.5	1.35059	0.675297	+	0.737546i	$0.264016\pi$
$410$	−6933.74	−0.835202
$411$	0	0
$412$	−3117.51	−0.372788
$413$	5766.43	0.687040
$414$	0	0
$415$	9442.50	1.11690
$416$	10543.4	1.24263
$417$	0	0
$418$	−7685.78	−0.899340
$419$	−5329.40	−0.621380	−0.310690	−	0.950511i	$-0.600560\pi$
$419$	−5329.40	−0.621380	−0.310690	+	0.950511i	$0.600560\pi$
$420$	0	0
$421$	13072.6	1.51335	0.756673	−	0.653793i	$-0.226823\pi$
$421$	13072.6	1.51335	0.756673	+	0.653793i	$0.226823\pi$
$422$	−3422.42	−0.394789
$423$	0	0
$424$	6289.57	0.720398
$425$	−1141.22	−0.130253
$426$	0	0
$427$	683.361	0.0774477
$428$	−3902.05	−0.440684
$429$	0	0
$430$	−8777.84	−0.984431
$431$	−15823.8	−1.76846	−0.884231	−	0.467049i	$-0.845317\pi$
$431$	−15823.8	−1.76846	−0.884231	+	0.467049i	$0.845317\pi$
$432$	0	0
$433$	−874.552	−0.0970630	−0.0485315	−	0.998822i	$-0.515454\pi$
$433$	−874.552	−0.0970630	−0.0485315	+	0.998822i	$0.515454\pi$
$434$	10092.5	1.11626
$435$	0	0
$436$	5654.88	0.621146
$437$	3536.59	0.387135
$438$	0	0
$439$	−2029.21	−0.220612	−0.110306	−	0.993898i	$-0.535183\pi$
$439$	−2029.21	−0.220612	−0.110306	+	0.993898i	$0.535183\pi$
$440$	4390.24	0.475674
$441$	0	0
$442$	−3680.66	−0.396088
$443$	−2473.26	−0.265255	−0.132628	−	0.991166i	$-0.542341\pi$
$443$	−2473.26	−0.265255	−0.132628	+	0.991166i	$0.542341\pi$
$444$	0	0
$445$	3527.13	0.375735
$446$	3747.64	0.397884
$447$	0	0
$448$	840.565	0.0886451
$449$	−7824.08	−0.822363	−0.411182	−	0.911553i	$-0.634884\pi$
$449$	−7824.08	−0.822363	−0.411182	+	0.911553i	$0.634884\pi$
$450$	0	0
$451$	11124.1	1.16145
$452$	7209.12	0.750195
$453$	0	0
$454$	−23085.0	−2.38642
$455$	−7765.45	−0.800110
$456$	0	0
$457$	14341.7	1.46801	0.734003	−	0.679147i	$-0.237650\pi$
$457$	14341.7	1.46801	0.734003	+	0.679147i	$0.237650\pi$
$458$	18294.4	1.86646
$459$	0	0
$460$	2111.19	0.213988
$461$	14599.6	1.47499	0.737494	−	0.675353i	$-0.236009\pi$
$461$	14599.6	1.47499	0.737494	+	0.675353i	$0.236009\pi$
$462$	0	0
$463$	6007.86	0.603043	0.301522	−	0.953459i	$-0.402505\pi$
$463$	6007.86	0.603043	0.301522	+	0.953459i	$0.402505\pi$
$464$	7342.92	0.734669
$465$	0	0
$466$	5088.94	0.505881
$467$	−11725.0	−1.16182	−0.580908	−	0.813969i	$-0.697303\pi$
$467$	−11725.0	−1.16182	−0.580908	+	0.813969i	$0.697303\pi$
$468$	0	0
$469$	13864.3	1.36502
$470$	1297.55	0.127344
$471$	0	0
$472$	4784.36	0.466564
$473$	14082.7	1.36897
$474$	0	0
$475$	3497.27	0.337822
$476$	1139.25	0.109701
$477$	0	0
$478$	7802.23	0.746581
$479$	−15696.1	−1.49723	−0.748615	−	0.663005i	$-0.769281\pi$
$479$	−15696.1	−1.49723	−0.748615	+	0.663005i	$0.769281\pi$
$480$	0	0
$481$	−16069.2	−1.52327
$482$	−16441.6	−1.55373
$483$	0	0
$484$	1919.55	0.180274
$485$	9406.97	0.880718
$486$	0	0
$487$	5507.75	0.512484	0.256242	−	0.966613i	$-0.417516\pi$
$487$	5507.75	0.512484	0.256242	+	0.966613i	$0.417516\pi$
$488$	566.979	0.0525941
$489$	0	0
$490$	−1964.73	−0.181138
$491$	4604.09	0.423177	0.211588	−	0.977359i	$-0.432136\pi$
$491$	4604.09	0.423177	0.211588	+	0.977359i	$0.432136\pi$
$492$	0	0
$493$	−1560.52	−0.142561
$494$	11279.3	1.02729
$495$	0	0
$496$	14165.2	1.28233
$497$	6899.97	0.622749
$498$	0	0
$499$	7561.06	0.678316	0.339158	−	0.940729i	$-0.389858\pi$
$499$	7561.06	0.678316	0.339158	+	0.940729i	$0.389858\pi$
$500$	5975.12	0.534431
$501$	0	0
$502$	−16095.8	−1.43106
$503$	−16739.3	−1.48384	−0.741919	−	0.670490i	$-0.766084\pi$
$503$	−16739.3	−1.48384	−0.741919	+	0.670490i	$0.766084\pi$
$504$	0	0
$505$	6180.00	0.544567
$506$	−10015.2	−0.879900
$507$	0	0
$508$	8735.21	0.762918
$509$	17.0393	0.00148380	0.000741899	−	1.00000i	$-0.499764\pi$
$509$	17.0393	0.00148380	0.000741899		1.00000i	$0.499764\pi$
$510$	0	0
$511$	−6592.29	−0.570696
$512$	−4839.93	−0.417767
$513$	0	0
$514$	−26495.5	−2.27367
$515$	5801.04	0.496358
$516$	0	0
$517$	−2081.72	−0.177087
$518$	14707.0	1.24747
$519$	0	0
$520$	−6442.93	−0.543349
$521$	2876.59	0.241892	0.120946	−	0.992659i	$-0.461407\pi$
$521$	2876.59	0.241892	0.120946	+	0.992659i	$0.461407\pi$
$522$	0	0
$523$	13441.9	1.12385	0.561923	−	0.827190i	$-0.310062\pi$
$523$	13441.9	1.12385	0.561923	+	0.827190i	$0.310062\pi$
$524$	−10962.7	−0.913949
$525$	0	0
$526$	14013.6	1.16164
$527$	−3010.40	−0.248833
$528$	0	0
$529$	−7558.55	−0.621234
$530$	12231.0	1.00242
$531$	0	0
$532$	−3491.23	−0.284519
$533$	−16325.3	−1.32669
$534$	0	0
$535$	7260.91	0.586760
$536$	11503.1	0.926976
$537$	0	0
$538$	1240.85	0.0994362
$539$	3152.11	0.251894
$540$	0	0
$541$	−12958.0	−1.02978	−0.514888	−	0.857258i	$-0.672166\pi$
$541$	−12958.0	−1.02978	−0.514888	+	0.857258i	$0.672166\pi$
$542$	−12.9139	−0.00102343
$543$	0	0
$544$	2878.29	0.226848
$545$	−10522.6	−0.827040
$546$	0	0
$547$	−21637.6	−1.69133	−0.845664	−	0.533716i	$-0.820795\pi$
$547$	−21637.6	−1.69133	−0.845664	+	0.533716i	$0.820795\pi$
$548$	−10511.8	−0.819422
$549$	0	0
$550$	−9903.83	−0.767820
$551$	4782.20	0.369743
$552$	0	0
$553$	−12077.6	−0.928738
$554$	−16078.3	−1.23303
$555$	0	0
$556$	−1976.67	−0.150772
$557$	−21252.5	−1.61669	−0.808345	−	0.588709i	$-0.799636\pi$
$557$	−21252.5	−1.61669	−0.808345	+	0.588709i	$0.799636\pi$
$558$	0	0
$559$	−20667.2	−1.56374
$560$	9975.11	0.752724
$561$	0	0
$562$	−1667.02	−0.125122
$563$	−21952.4	−1.64331	−0.821654	−	0.569987i	$-0.806948\pi$
$563$	−21952.4	−1.64331	−0.821654	+	0.569987i	$0.806948\pi$
$564$	0	0
$565$	−13414.7	−0.998866
$566$	−9111.76	−0.676671
$567$	0	0
$568$	5724.85	0.422904
$569$	−5820.12	−0.428809	−0.214404	−	0.976745i	$-0.568781\pi$
$569$	−5820.12	−0.428809	−0.214404	+	0.976745i	$0.568781\pi$
$570$	0	0
$571$	−17406.4	−1.27572	−0.637858	−	0.770154i	$-0.720179\pi$
$571$	−17406.4	−1.27572	−0.637858	+	0.770154i	$0.720179\pi$
$572$	−10802.5	−0.789642
$573$	0	0
$574$	14941.4	1.08648
$575$	4557.21	0.330520
$576$	0	0
$577$	14297.2	1.03155	0.515773	−	0.856726i	$-0.327505\pi$
$577$	14297.2	1.03155	0.515773	+	0.856726i	$0.327505\pi$
$578$	−1004.80	−0.0723079
$579$	0	0
$580$	2854.77	0.204375
$581$	−20347.4	−1.45293
$582$	0	0
$583$	−19622.7	−1.39398
$584$	−5469.57	−0.387555
$585$	0	0
$586$	−4054.23	−0.285800
$587$	13376.1	0.940532	0.470266	−	0.882525i	$-0.344158\pi$
$587$	13376.1	0.940532	0.470266	+	0.882525i	$0.344158\pi$
$588$	0	0
$589$	9225.33	0.645370
$590$	9303.89	0.649212
$591$	0	0
$592$	20641.7	1.43305
$593$	16928.6	1.17230	0.586151	−	0.810202i	$-0.300643\pi$
$593$	16928.6	1.17230	0.586151	+	0.810202i	$0.300643\pi$
$594$	0	0
$595$	−2119.92	−0.146064
$596$	13060.3	0.897604
$597$	0	0
$598$	14697.9	1.00508
$599$	−1428.93	−0.0974697	−0.0487348	−	0.998812i	$-0.515519\pi$
$599$	−1428.93	−0.0974697	−0.0487348	+	0.998812i	$0.515519\pi$
$600$	0	0
$601$	−11646.1	−0.790438	−0.395219	−	0.918587i	$-0.629331\pi$
$601$	−11646.1	−0.790438	−0.395219	+	0.918587i	$0.629331\pi$
$602$	18915.2	1.28061
$603$	0	0
$604$	4245.05	0.285974
$605$	−3571.89	−0.240030
$606$	0	0
$607$	8424.88	0.563353	0.281677	−	0.959509i	$-0.409109\pi$
$607$	8424.88	0.563353	0.281677	+	0.959509i	$0.409109\pi$
$608$	−8820.48	−0.588351
$609$	0	0
$610$	1102.57	0.0731834
$611$	3055.04	0.202281
$612$	0	0
$613$	6734.36	0.443717	0.221858	−	0.975079i	$-0.428788\pi$
$613$	6734.36	0.443717	0.221858	+	0.975079i	$0.428788\pi$
$614$	−26873.7	−1.76634
$615$	0	0
$616$	−9460.43	−0.618785
$617$	−23683.5	−1.54532	−0.772661	−	0.634819i	$-0.781075\pi$
$617$	−23683.5	−1.54532	−0.772661	+	0.634819i	$0.781075\pi$
$618$	0	0
$619$	6795.00	0.441218	0.220609	−	0.975362i	$-0.429196\pi$
$619$	6795.00	0.441218	0.220609	+	0.975362i	$0.429196\pi$
$620$	5507.12	0.356728
$621$	0	0
$622$	−15817.9	−1.01968
$623$	−7600.53	−0.488778
$624$	0	0
$625$	−2727.11	−0.174535
$626$	9345.54	0.596682
$627$	0	0
$628$	2693.00	0.171118
$629$	−4386.79	−0.278080
$630$	0	0
$631$	19712.0	1.24361	0.621807	−	0.783170i	$-0.286399\pi$
$631$	19712.0	1.24361	0.621807	+	0.783170i	$0.286399\pi$
$632$	−10020.7	−0.630699
$633$	0	0
$634$	19530.8	1.22345
$635$	−16254.4	−1.01581
$636$	0	0
$637$	−4625.91	−0.287732
$638$	−13542.6	−0.840372
$639$	0	0
$640$	11660.1	0.720162
$641$	25070.0	1.54479	0.772393	−	0.635146i	$-0.219060\pi$
$641$	25070.0	1.54479	0.772393	+	0.635146i	$0.219060\pi$
$642$	0	0
$643$	−13096.3	−0.803214	−0.401607	−	0.915812i	$-0.631548\pi$
$643$	−13096.3	−0.803214	−0.401607	+	0.915812i	$0.631548\pi$
$644$	−4549.35	−0.278369
$645$	0	0
$646$	3079.18	0.187537
$647$	−13315.1	−0.809074	−0.404537	−	0.914522i	$-0.632567\pi$
$647$	−13315.1	−0.809074	−0.404537	+	0.914522i	$0.632567\pi$
$648$	0	0
$649$	−14926.7	−0.902809
$650$	14534.4	0.877058
$651$	0	0
$652$	−2469.07	−0.148307
$653$	−19517.3	−1.16964	−0.584818	−	0.811165i	$-0.698834\pi$
$653$	−19517.3	−1.16964	−0.584818	+	0.811165i	$0.698834\pi$
$654$	0	0
$655$	20399.4	1.21690
$656$	20970.6	1.24812
$657$	0	0
$658$	−2796.06	−0.165656
$659$	16684.0	0.986216	0.493108	−	0.869968i	$-0.335861\pi$
$659$	16684.0	0.986216	0.493108	+	0.869968i	$0.335861\pi$
$660$	0	0
$661$	10911.1	0.642048	0.321024	−	0.947071i	$-0.395973\pi$
$661$	10911.1	0.642048	0.321024	+	0.947071i	$0.395973\pi$
$662$	−29677.6	−1.74238
$663$	0	0
$664$	−16882.1	−0.986675
$665$	6496.46	0.378830
$666$	0	0
$667$	6231.59	0.361751
$668$	−7149.11	−0.414083
$669$	0	0
$670$	22369.5	1.28986
$671$	−1768.91	−0.101771
$672$	0	0
$673$	12539.1	0.718198	0.359099	−	0.933299i	$-0.383084\pi$
$673$	12539.1	0.718198	0.359099	+	0.933299i	$0.383084\pi$
$674$	−31547.5	−1.80291
$675$	0	0
$676$	6871.65	0.390968
$677$	31055.0	1.76299	0.881493	−	0.472198i	$-0.156539\pi$
$677$	31055.0	1.76299	0.881493	+	0.472198i	$0.156539\pi$
$678$	0	0
$679$	−20270.8	−1.14569
$680$	−1758.88	−0.0991910
$681$	0	0
$682$	−26125.0	−1.46683
$683$	27622.0	1.54748	0.773739	−	0.633504i	$-0.218384\pi$
$683$	27622.0	1.54748	0.773739	+	0.633504i	$0.218384\pi$
$684$	0	0
$685$	19560.3	1.09104
$686$	23782.6	1.32365
$687$	0	0
$688$	26548.0	1.47112
$689$	28797.5	1.59230
$690$	0	0
$691$	−17085.2	−0.940594	−0.470297	−	0.882508i	$-0.655853\pi$
$691$	−17085.2	−0.940594	−0.470297	+	0.882508i	$0.655853\pi$
$692$	7445.38	0.409004
$693$	0	0
$694$	31594.9	1.72814
$695$	3678.17	0.200749
$696$	0	0
$697$	−4456.69	−0.242194
$698$	29140.0	1.58018
$699$	0	0
$700$	−4498.77	−0.242911
$701$	−1835.02	−0.0988700	−0.0494350	−	0.998777i	$-0.515742\pi$
$701$	−1835.02	−0.0988700	−0.0494350	+	0.998777i	$0.515742\pi$
$702$	0	0
$703$	13443.3	0.721226
$704$	−2175.84	−0.116484
$705$	0	0
$706$	−27422.9	−1.46186
$707$	−13317.1	−0.708405
$708$	0	0
$709$	−11565.0	−0.612600	−0.306300	−	0.951935i	$-0.599091\pi$
$709$	−11565.0	−0.612600	−0.306300	+	0.951935i	$0.599091\pi$
$710$	11132.8	0.588460
$711$	0	0
$712$	−6306.10	−0.331926
$713$	12021.3	0.631420
$714$	0	0
$715$	20101.2	1.05139
$716$	−13365.6	−0.697622
$717$	0	0
$718$	28139.3	1.46260
$719$	23764.7	1.23265	0.616325	−	0.787492i	$-0.288621\pi$
$719$	23764.7	1.23265	0.616325	+	0.787492i	$0.288621\pi$
$720$	0	0
$721$	−12500.5	−0.645691
$722$	14411.3	0.742841
$723$	0	0
$724$	17459.0	0.896213
$725$	6162.30	0.315672
$726$	0	0
$727$	−17712.6	−0.903612	−0.451806	−	0.892116i	$-0.649220\pi$
$727$	−17712.6	−0.903612	−0.451806	+	0.892116i	$0.649220\pi$
$728$	13883.7	0.706820
$729$	0	0
$730$	−10636.4	−0.539274
$731$	−5642.00	−0.285468
$732$	0	0
$733$	381.951	0.0192465	0.00962324	−	0.999954i	$-0.496937\pi$
$733$	381.951	0.0192465	0.00962324	+	0.999954i	$0.496937\pi$
$734$	−3028.37	−0.152288
$735$	0	0
$736$	−11493.8	−0.575633
$737$	−35888.4	−1.79371
$738$	0	0
$739$	−5941.21	−0.295739	−0.147869	−	0.989007i	$-0.547242\pi$
$739$	−5941.21	−0.295739	−0.147869	+	0.989007i	$0.547242\pi$
$740$	8025.04	0.398657
$741$	0	0
$742$	−26356.3	−1.30400
$743$	−28444.3	−1.40447	−0.702235	−	0.711946i	$-0.747814\pi$
$743$	−28444.3	−1.40447	−0.702235	+	0.711946i	$0.747814\pi$
$744$	0	0
$745$	−24302.6	−1.19514
$746$	40950.7	2.00980
$747$	0	0
$748$	−2949.01	−0.144153
$749$	−15646.4	−0.763292
$750$	0	0
$751$	−12002.7	−0.583201	−0.291600	−	0.956540i	$-0.594188\pi$
$751$	−12002.7	−0.583201	−0.291600	+	0.956540i	$0.594188\pi$
$752$	−3924.35	−0.190301
$753$	0	0
$754$	19874.6	0.959932
$755$	−7899.15	−0.380768
$756$	0	0
$757$	24817.2	1.19154	0.595770	−	0.803155i	$-0.296847\pi$
$757$	24817.2	1.19154	0.595770	+	0.803155i	$0.296847\pi$
$758$	−4672.89	−0.223914
$759$	0	0
$760$	5390.06	0.257261
$761$	23691.4	1.12853	0.564267	−	0.825592i	$-0.309159\pi$
$761$	23691.4	1.12853	0.564267	+	0.825592i	$0.309159\pi$
$762$	0	0
$763$	22674.8	1.07586
$764$	−1343.74	−0.0636322
$765$	0	0
$766$	3039.27	0.143359
$767$	21905.7	1.03125
$768$	0	0
$769$	18595.9	0.872020	0.436010	−	0.899942i	$-0.356391\pi$
$769$	18595.9	0.872020	0.436010	+	0.899942i	$0.356391\pi$
$770$	−18397.2	−0.861024
$771$	0	0
$772$	−13143.1	−0.612733
$773$	−18268.7	−0.850040	−0.425020	−	0.905184i	$-0.639733\pi$
$773$	−18268.7	−0.850040	−0.425020	+	0.905184i	$0.639733\pi$
$774$	0	0
$775$	11887.7	0.550991
$776$	−16818.6	−0.778030
$777$	0	0
$778$	−16631.0	−0.766387
$779$	13657.5	0.628152
$780$	0	0
$781$	−17860.9	−0.818326
$782$	4012.42	0.183483
$783$	0	0
$784$	5942.21	0.270691
$785$	−5011.11	−0.227840
$786$	0	0
$787$	−8828.69	−0.399884	−0.199942	−	0.979808i	$-0.564075\pi$
$787$	−8828.69	−0.399884	−0.199942	+	0.979808i	$0.564075\pi$
$788$	14048.6	0.635104
$789$	0	0
$790$	−19486.7	−0.877602
$791$	28907.0	1.29938
$792$	0	0
$793$	2595.98	0.116250
$794$	20266.5	0.905832
$795$	0	0
$796$	−14549.1	−0.647837
$797$	−2785.92	−0.123817	−0.0619086	−	0.998082i	$-0.519719\pi$
$797$	−2785.92	−0.123817	−0.0619086	+	0.998082i	$0.519719\pi$
$798$	0	0
$799$	834.006	0.0369274
$800$	−11366.0	−0.502310
$801$	0	0
$802$	−10983.0	−0.483568
$803$	17064.4	0.749926
$804$	0	0
$805$	8465.40	0.370641
$806$	38340.0	1.67552
$807$	0	0
$808$	−11049.1	−0.481073
$809$	5946.07	0.258409	0.129204	−	0.991618i	$-0.458758\pi$
$809$	5946.07	0.258409	0.129204	+	0.991618i	$0.458758\pi$
$810$	0	0
$811$	−36744.2	−1.59095	−0.795477	−	0.605983i	$-0.792780\pi$
$811$	−36744.2	−1.59095	−0.795477	+	0.605983i	$0.792780\pi$
$812$	−6151.67	−0.265864
$813$	0	0
$814$	−38069.6	−1.63924
$815$	4594.43	0.197467
$816$	0	0
$817$	17289.9	0.740386
$818$	−38840.9	−1.66020
$819$	0	0
$820$	8152.93	0.347210
$821$	3372.76	0.143374	0.0716870	−	0.997427i	$-0.477162\pi$
$821$	3372.76	0.143374	0.0716870	+	0.997427i	$0.477162\pi$
$822$	0	0
$823$	−30830.8	−1.30583	−0.652913	−	0.757433i	$-0.726453\pi$
$823$	−30830.8	−1.30583	−0.652913	+	0.757433i	$0.726453\pi$
$824$	−10371.6	−0.438484
$825$	0	0
$826$	−20048.7	−0.844534
$827$	18590.5	0.781685	0.390842	−	0.920458i	$-0.372184\pi$
$827$	18590.5	0.781685	0.390842	+	0.920458i	$0.372184\pi$
$828$	0	0
$829$	−18532.2	−0.776417	−0.388209	−	0.921572i	$-0.626906\pi$
$829$	−18532.2	−0.776417	−0.388209	+	0.921572i	$0.626906\pi$
$830$	−32829.7	−1.37293
$831$	0	0
$832$	3193.17	0.133057
$833$	−1262.84	−0.0525268
$834$	0	0
$835$	13303.0	0.551341
$836$	9037.21	0.373874
$837$	0	0
$838$	18529.3	0.763822
$839$	31147.1	1.28166	0.640832	−	0.767681i	$-0.278589\pi$
$839$	31147.1	1.28166	0.640832	+	0.767681i	$0.278589\pi$
$840$	0	0
$841$	−15962.6	−0.654500
$842$	−45450.8	−1.86026
$843$	0	0
$844$	4024.20	0.164122
$845$	−12786.7	−0.520564
$846$	0	0
$847$	7696.98	0.312245
$848$	−36991.8	−1.49800
$849$	0	0
$850$	3967.81	0.160111
$851$	17517.6	0.705636
$852$	0	0
$853$	27617.8	1.10858	0.554288	−	0.832325i	$-0.312991\pi$
$853$	27617.8	1.10858	0.554288	+	0.832325i	$0.312991\pi$
$854$	−2375.91	−0.0952014
$855$	0	0
$856$	−12981.7	−0.518346
$857$	7374.42	0.293939	0.146969	−	0.989141i	$-0.453048\pi$
$857$	7374.42	0.293939	0.146969	+	0.989141i	$0.453048\pi$
$858$	0	0
$859$	−18875.2	−0.749726	−0.374863	−	0.927080i	$-0.622310\pi$
$859$	−18875.2	−0.749726	−0.374863	+	0.927080i	$0.622310\pi$
$860$	10321.3	0.409248
$861$	0	0
$862$	55016.3	2.17386
$863$	−30812.9	−1.21539	−0.607696	−	0.794170i	$-0.707906\pi$
$863$	−30812.9	−1.21539	−0.607696	+	0.794170i	$0.707906\pi$
$864$	0	0
$865$	−13854.3	−0.544579
$866$	3040.64	0.119313
$867$	0	0
$868$	−11867.2	−0.464053
$869$	31263.4	1.22041
$870$	0	0
$871$	52668.3	2.04891
$872$	18813.1	0.730610
$873$	0	0
$874$	−12296.0	−0.475879
$875$	23958.9	0.925667
$876$	0	0
$877$	34791.0	1.33958	0.669789	−	0.742552i	$-0.266385\pi$
$877$	34791.0	1.33958	0.669789	+	0.742552i	$0.266385\pi$
$878$	7055.15	0.271184
$879$	0	0
$880$	−25821.0	−0.989120
$881$	−39390.8	−1.50637	−0.753184	−	0.657810i	$-0.771483\pi$
$881$	−39390.8	−1.50637	−0.753184	+	0.657810i	$0.771483\pi$
$882$	0	0
$883$	−3249.30	−0.123837	−0.0619183	−	0.998081i	$-0.519722\pi$
$883$	−3249.30	−0.123837	−0.0619183	+	0.998081i	$0.519722\pi$
$884$	4327.84	0.164662
$885$	0	0
$886$	8599.03	0.326061
$887$	13724.9	0.519547	0.259774	−	0.965670i	$-0.416352\pi$
$887$	13724.9	0.519547	0.259774	+	0.965670i	$0.416352\pi$
$888$	0	0
$889$	35026.3	1.32142
$890$	−12263.1	−0.461866
$891$	0	0
$892$	−4406.61	−0.165408
$893$	−2555.80	−0.0957745
$894$	0	0
$895$	24870.7	0.928866
$896$	−25126.0	−0.936830
$897$	0	0
$898$	27202.8	1.01088
$899$	16255.3	0.603054
$900$	0	0
$901$	7861.53	0.290683
$902$	−38676.3	−1.42770
$903$	0	0
$904$	23983.9	0.882402
$905$	−32487.5	−1.19328
$906$	0	0
$907$	−21248.3	−0.777880	−0.388940	−	0.921263i	$-0.627159\pi$
$907$	−21248.3	−0.777880	−0.388940	+	0.921263i	$0.627159\pi$
$908$	27144.1	0.992081
$909$	0	0
$910$	26998.9	0.983523
$911$	−33293.3	−1.21082	−0.605410	−	0.795914i	$-0.706991\pi$
$911$	−33293.3	−1.21082	−0.605410	+	0.795914i	$0.706991\pi$
$912$	0	0
$913$	52670.2	1.90923
$914$	−49863.4	−1.80452
$915$	0	0
$916$	−21511.2	−0.775927
$917$	−43958.1	−1.58302
$918$	0	0
$919$	−23233.4	−0.833950	−0.416975	−	0.908918i	$-0.636910\pi$
$919$	−23233.4	−0.833950	−0.416975	+	0.908918i	$0.636910\pi$
$920$	7023.67	0.251700
$921$	0	0
$922$	−50759.8	−1.81311
$923$	26211.9	0.934750
$924$	0	0
$925$	17322.8	0.615753
$926$	−20888.1	−0.741282
$927$	0	0
$928$	−15542.0	−0.549774
$929$	16773.7	0.592386	0.296193	−	0.955128i	$-0.404283\pi$
$929$	16773.7	0.592386	0.296193	+	0.955128i	$0.404283\pi$
$930$	0	0
$931$	3869.96	0.136233
$932$	−5983.75	−0.210305
$933$	0	0
$934$	40765.5	1.42815
$935$	5487.50	0.191936
$936$	0	0
$937$	−25023.9	−0.872459	−0.436229	−	0.899835i	$-0.643686\pi$
$937$	−25023.9	−0.872459	−0.436229	+	0.899835i	$0.643686\pi$
$938$	−48203.5	−1.67793
$939$	0	0
$940$	−1525.70	−0.0529393
$941$	14494.2	0.502122	0.251061	−	0.967971i	$-0.419220\pi$
$941$	14494.2	0.502122	0.251061	+	0.967971i	$0.419220\pi$
$942$	0	0
$943$	17796.8	0.614574
$944$	−28139.0	−0.970177
$945$	0	0
$946$	−48962.7	−1.68279
$947$	−4203.16	−0.144228	−0.0721142	−	0.997396i	$-0.522975\pi$
$947$	−4203.16	−0.144228	−0.0721142	+	0.997396i	$0.522975\pi$
$948$	0	0
$949$	−25043.0	−0.856619
$950$	−12159.3	−0.415263
$951$	0	0
$952$	3790.16	0.129034
$953$	−9623.53	−0.327111	−0.163555	−	0.986534i	$-0.552296\pi$
$953$	−9623.53	−0.327111	−0.163555	+	0.986534i	$0.552296\pi$
$954$	0	0
$955$	2500.43	0.0847246
$956$	−9174.13	−0.310369
$957$	0	0
$958$	54572.2	1.84045
$959$	−42150.1	−1.41929
$960$	0	0
$961$	1567.11	0.0526034
$962$	55869.4	1.87246
$963$	0	0
$964$	19332.6	0.645915
$965$	24456.5	0.815838
$966$	0	0
$967$	46366.1	1.54192	0.770959	−	0.636885i	$-0.219777\pi$
$967$	46366.1	1.54192	0.770959	+	0.636885i	$0.219777\pi$
$968$	6386.12	0.212043
$969$	0	0
$970$	−32706.1	−1.08261
$971$	43540.7	1.43902	0.719509	−	0.694483i	$-0.244367\pi$
$971$	43540.7	1.43902	0.719509	+	0.694483i	$0.244367\pi$
$972$	0	0
$973$	−7926.00	−0.261147
$974$	−19149.3	−0.629963
$975$	0	0
$976$	−3334.66	−0.109365
$977$	−10563.4	−0.345910	−0.172955	−	0.984930i	$-0.555332\pi$
$977$	−10563.4	−0.345910	−0.172955	+	0.984930i	$0.555332\pi$
$978$	0	0
$979$	19674.3	0.642281
$980$	2310.20	0.0753027
$981$	0	0
$982$	−16007.5	−0.520183
$983$	−32907.4	−1.06774	−0.533868	−	0.845568i	$-0.679262\pi$
$983$	−32907.4	−1.06774	−0.533868	+	0.845568i	$0.679262\pi$
$984$	0	0
$985$	−26141.6	−0.845625
$986$	5425.62	0.175240
$987$	0	0
$988$	−13262.6	−0.427065
$989$	22530.0	0.724382
$990$	0	0
$991$	33988.3	1.08948	0.544740	−	0.838605i	$-0.316628\pi$
$991$	33988.3	1.08948	0.544740	+	0.838605i	$0.316628\pi$
$992$	−29982.0	−0.959605
$993$	0	0
$994$	−23989.8	−0.765504
$995$	27072.8	0.862578
$996$	0	0
$997$	−17809.2	−0.565720	−0.282860	−	0.959161i	$-0.591283\pi$
$997$	−17809.2	−0.565720	−0.282860	+	0.959161i	$0.591283\pi$
$998$	−26288.3	−0.833809
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	153.4.a.h.1.2	✓	4
3.2	odd	2		153.4.a.i.1.3	yes	4
4.3	odd	2		2448.4.a.bo.1.2		4
12.11	even	2		2448.4.a.bs.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
153.4.a.h.1.2	✓	4	1.1	even	1	trivial
153.4.a.i.1.3	yes	4	3.2	odd	2
2448.4.a.bo.1.2		4	4.3	odd	2
2448.4.a.bs.1.3		4	12.11	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:	\( N \)	\(=\)	\( 153 = 3^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	153.a (trivial)

\(f(q)\)	\(=\)	\(q-3.47680 q^{2} +4.08814 q^{4} -7.60718 q^{5} +16.3925 q^{7} +13.6008 q^{8} +O(q^{10})\)	\(q-3.47680 q^{2} +4.08814 q^{4} -7.60718 q^{5} +16.3925 q^{7} +13.6008 q^{8} +26.4486 q^{10} -42.4328 q^{11} +62.2726 q^{13} -56.9936 q^{14} -79.9922 q^{16} +17.0000 q^{17} -52.0963 q^{19} -31.0992 q^{20} +147.530 q^{22} -67.8856 q^{23} -67.1308 q^{25} -216.509 q^{26} +67.0150 q^{28} -91.7954 q^{29} -177.082 q^{31} +169.311 q^{32} -59.1056 q^{34} -124.701 q^{35} -258.046 q^{37} +181.128 q^{38} -103.463 q^{40} -262.159 q^{41} -331.883 q^{43} -173.471 q^{44} +236.025 q^{46} +49.0592 q^{47} -74.2848 q^{49} +233.400 q^{50} +254.579 q^{52} +462.443 q^{53} +322.794 q^{55} +222.951 q^{56} +319.154 q^{58} +351.772 q^{59} +41.6873 q^{61} +615.679 q^{62} +51.2773 q^{64} -473.719 q^{65} +845.771 q^{67} +69.4984 q^{68} +433.560 q^{70} +420.922 q^{71} -402.152 q^{73} +897.175 q^{74} -212.977 q^{76} -695.581 q^{77} -736.775 q^{79} +608.515 q^{80} +911.473 q^{82} -1241.26 q^{83} -129.322 q^{85} +1153.89 q^{86} -577.118 q^{88} -463.658 q^{89} +1020.81 q^{91} -277.526 q^{92} -170.569 q^{94} +396.306 q^{95} -1236.59 q^{97} +258.273 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{2} + 26 q^{4} - 22 q^{5} - 24 q^{7} - 102 q^{8}+O(q^{10}) \) 4 * q - 4 * q^2 + 26 * q^4 - 22 * q^5 - 24 * q^7 - 102 * q^8	\( 4 q - 4 q^{2} + 26 q^{4} - 22 q^{5} - 24 q^{7} - 102 q^{8} - 2 q^{10} - 50 q^{11} + 26 q^{13} - 80 q^{14} + 138 q^{16} + 68 q^{17} + 34 q^{19} - 312 q^{20} - 254 q^{22} - 382 q^{23} + 138 q^{25} + 22 q^{26} + 52 q^{28} - 540 q^{29} - 356 q^{31} - 730 q^{32} - 68 q^{34} - 304 q^{35} - 404 q^{37} - 298 q^{38} + 332 q^{40} + 114 q^{41} + 570 q^{43} + 1368 q^{44} - 290 q^{46} - 496 q^{47} - 224 q^{49} + 1862 q^{50} - 1012 q^{52} - 92 q^{53} - 482 q^{55} + 1428 q^{56} + 1324 q^{58} + 48 q^{59} - 1036 q^{61} + 2564 q^{62} + 2898 q^{64} + 342 q^{65} + 812 q^{67} + 442 q^{68} + 152 q^{70} - 1044 q^{71} - 1212 q^{73} + 1444 q^{74} + 2268 q^{76} + 564 q^{77} + 488 q^{79} + 1000 q^{80} - 938 q^{82} - 1708 q^{83} - 374 q^{85} + 2446 q^{86} - 3868 q^{88} + 8 q^{89} + 716 q^{91} - 1356 q^{92} - 1224 q^{94} - 1010 q^{95} - 76 q^{97} + 1472 q^{98}+O(q^{100}) \) 4 * q - 4 * q^2 + 26 * q^4 - 22 * q^5 - 24 * q^7 - 102 * q^8 - 2 * q^10 - 50 * q^11 + 26 * q^13 - 80 * q^14 + 138 * q^16 + 68 * q^17 + 34 * q^19 - 312 * q^20 - 254 * q^22 - 382 * q^23 + 138 * q^25 + 22 * q^26 + 52 * q^28 - 540 * q^29 - 356 * q^31 - 730 * q^32 - 68 * q^34 - 304 * q^35 - 404 * q^37 - 298 * q^38 + 332 * q^40 + 114 * q^41 + 570 * q^43 + 1368 * q^44 - 290 * q^46 - 496 * q^47 - 224 * q^49 + 1862 * q^50 - 1012 * q^52 - 92 * q^53 - 482 * q^55 + 1428 * q^56 + 1324 * q^58 + 48 * q^59 - 1036 * q^61 + 2564 * q^62 + 2898 * q^64 + 342 * q^65 + 812 * q^67 + 442 * q^68 + 152 * q^70 - 1044 * q^71 - 1212 * q^73 + 1444 * q^74 + 2268 * q^76 + 564 * q^77 + 488 * q^79 + 1000 * q^80 - 938 * q^82 - 1708 * q^83 - 374 * q^85 + 2446 * q^86 - 3868 * q^88 + 8 * q^89 + 716 * q^91 - 1356 * q^92 - 1224 * q^94 - 1010 * q^95 - 76 * q^97 + 1472 * q^98

Introduction

L-functions

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