LMFDB - Embedded newform 416.4.a.g.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [416,4,Mod(1,416)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(416, 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("416.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 416 = 2^{5} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	416.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:	$24.5447945624$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.1847677.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 18x^{2} + 19x - 4 $ x^4 - 2x^3 - 18x^2 + 19*x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-3.91115$ of defining polynomial
Character	$\chi$	$=$	416.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.83719 q^{3} +6.20824 q^{5} -19.4818 q^{7} -23.6247 q^{9} +45.5850 q^{11} -13.0000 q^{13} +11.4057 q^{15} -47.0412 q^{17} +10.2958 q^{19} -35.7918 q^{21} -155.854 q^{23} -86.4577 q^{25} -93.0073 q^{27} -69.1670 q^{29} +5.16685 q^{31} +83.7484 q^{33} -120.948 q^{35} -71.2928 q^{37} -23.8835 q^{39} -77.5835 q^{41} +296.629 q^{43} -146.668 q^{45} +296.666 q^{47} +36.5401 q^{49} -86.4237 q^{51} -601.913 q^{53} +283.003 q^{55} +18.9154 q^{57} -272.093 q^{59} -641.080 q^{61} +460.252 q^{63} -80.7072 q^{65} +1021.97 q^{67} -286.334 q^{69} -981.934 q^{71} -875.657 q^{73} -158.839 q^{75} -888.078 q^{77} -649.979 q^{79} +466.996 q^{81} -812.303 q^{83} -292.043 q^{85} -127.073 q^{87} +97.1670 q^{89} +253.263 q^{91} +9.49249 q^{93} +63.9191 q^{95} +1193.65 q^{97} -1076.93 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 14 q^{5} + 22 q^{9} - 52 q^{13} + 6 q^{17} - 182 q^{21} - 74 q^{25} - 432 q^{29} - 364 q^{33} - 790 q^{37} - 388 q^{41} - 1208 q^{45} - 514 q^{49} - 932 q^{53} - 468 q^{57} - 1244 q^{61} + 182 q^{65}+ \cdots - 1128 q^{97}+O(q^{100}) $ 4 * q - 14 * q^5 + 22 * q^9 - 52 * q^13 + 6 * q^17 - 182 * q^21 - 74 * q^25 - 432 * q^29 - 364 * q^33 - 790 * q^37 - 388 * q^41 - 1208 * q^45 - 514 * q^49 - 932 * q^53 - 468 * q^57 - 1244 * q^61 + 182 * q^65 - 1456 * q^69 + 536 * q^73 - 1300 * q^77 + 4 * q^81 - 1906 * q^85 + 544 * q^89 - 3224 * q^93 - 1128 * q^97

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
$3$	1.83719	0.353567	0.176784	−	0.984250i	$-0.443431\pi$
$3$	1.83719	0.353567	0.176784	+	0.984250i	$0.443431\pi$
$4$	0	0
$5$	6.20824	0.555282	0.277641	−	0.960685i	$-0.410447\pi$
$5$	6.20824	0.555282	0.277641	+	0.960685i	$0.410447\pi$
$6$	0	0
$7$	−19.4818	−1.05192	−0.525959	−	0.850510i	$-0.676293\pi$
$7$	−19.4818	−1.05192	−0.525959	+	0.850510i	$0.676293\pi$
$8$	0	0
$9$	−23.6247	−0.874990
$10$	0	0
$11$	45.5850	1.24949	0.624746	−	0.780828i	$-0.285203\pi$
$11$	45.5850	1.24949	0.624746	+	0.780828i	$0.285203\pi$
$12$	0	0
$13$	−13.0000	−0.277350
$13$	−13.0000	−0.277350
$14$	0	0
$15$	11.4057	0.196330
$16$	0	0
$17$	−47.0412	−0.671128	−0.335564	−	0.942017i	$-0.608927\pi$
$17$	−47.0412	−0.671128	−0.335564	+	0.942017i	$0.608927\pi$
$18$	0	0
$19$	10.2958	0.124317	0.0621586	−	0.998066i	$-0.480202\pi$
$19$	10.2958	0.124317	0.0621586	+	0.998066i	$0.480202\pi$
$20$	0	0
$21$	−35.7918	−0.371924
$22$	0	0
$23$	−155.854	−1.41295	−0.706475	−	0.707738i	$-0.749716\pi$
$23$	−155.854	−1.41295	−0.706475	+	0.707738i	$0.749716\pi$
$24$	0	0
$25$	−86.4577	−0.691662
$26$	0	0
$27$	−93.0073	−0.662935
$28$	0	0
$29$	−69.1670	−0.442896	−0.221448	−	0.975172i	$-0.571078\pi$
$29$	−69.1670	−0.442896	−0.221448	+	0.975172i	$0.571078\pi$
$30$	0	0
$31$	5.16685	0.0299353	0.0149676	−	0.999888i	$-0.495235\pi$
$31$	5.16685	0.0299353	0.0149676	+	0.999888i	$0.495235\pi$
$32$	0	0
$33$	83.7484	0.441780
$34$	0	0
$35$	−120.948	−0.584111
$36$	0	0
$37$	−71.2928	−0.316769	−0.158385	−	0.987377i	$-0.550629\pi$
$37$	−71.2928	−0.316769	−0.158385	+	0.987377i	$0.550629\pi$
$38$	0	0
$39$	−23.8835	−0.0980620
$40$	0	0
$41$	−77.5835	−0.295525	−0.147762	−	0.989023i	$-0.547207\pi$
$41$	−77.5835	−0.295525	−0.147762	+	0.989023i	$0.547207\pi$
$42$	0	0
$43$	296.629	1.05199	0.525993	−	0.850489i	$-0.323694\pi$
$43$	296.629	1.05199	0.525993	+	0.850489i	$0.323694\pi$
$44$	0	0
$45$	−146.668	−0.485866
$46$	0	0
$47$	296.666	0.920707	0.460354	−	0.887736i	$-0.347723\pi$
$47$	296.666	0.920707	0.460354	+	0.887736i	$0.347723\pi$
$48$	0	0
$49$	36.5401	0.106531
$50$	0	0
$51$	−86.4237	−0.237289
$52$	0	0
$53$	−601.913	−1.55998	−0.779992	−	0.625790i	$-0.784777\pi$
$53$	−601.913	−1.55998	−0.779992	+	0.625790i	$0.784777\pi$
$54$	0	0
$55$	283.003	0.693820
$56$	0	0
$57$	18.9154	0.0439545
$58$	0	0
$59$	−272.093	−0.600399	−0.300200	−	0.953876i	$-0.597053\pi$
$59$	−272.093	−0.600399	−0.300200	+	0.953876i	$0.597053\pi$
$60$	0	0
$61$	−641.080	−1.34561	−0.672803	−	0.739822i	$-0.734910\pi$
$61$	−641.080	−1.34561	−0.672803	+	0.739822i	$0.734910\pi$
$62$	0	0
$63$	460.252	0.920418
$64$	0	0
$65$	−80.7072	−0.154008
$66$	0	0
$67$	1021.97	1.86349	0.931743	−	0.363119i	$-0.118288\pi$
$67$	1021.97	1.86349	0.931743	+	0.363119i	$0.118288\pi$
$68$	0	0
$69$	−286.334	−0.499573
$70$	0	0
$71$	−981.934	−1.64133	−0.820663	−	0.571412i	$-0.806396\pi$
$71$	−981.934	−1.64133	−0.820663	+	0.571412i	$0.806396\pi$
$72$	0	0
$73$	−875.657	−1.40394	−0.701972	−	0.712204i	$-0.747697\pi$
$73$	−875.657	−1.40394	−0.701972	+	0.712204i	$0.747697\pi$
$74$	0	0
$75$	−158.839	−0.244549
$76$	0	0
$77$	−888.078	−1.31436
$78$	0	0
$79$	−649.979	−0.925675	−0.462838	−	0.886443i	$-0.653169\pi$
$79$	−649.979	−0.925675	−0.462838	+	0.886443i	$0.653169\pi$
$80$	0	0
$81$	466.996	0.640598
$82$	0	0
$83$	−812.303	−1.07424	−0.537120	−	0.843506i	$-0.680488\pi$
$83$	−812.303	−1.07424	−0.537120	+	0.843506i	$0.680488\pi$
$84$	0	0
$85$	−292.043	−0.372665
$86$	0	0
$87$	−127.073	−0.156594
$88$	0	0
$89$	97.1670	0.115727	0.0578634	−	0.998325i	$-0.481571\pi$
$89$	97.1670	0.115727	0.0578634	+	0.998325i	$0.481571\pi$
$90$	0	0
$91$	253.263	0.291750
$92$	0	0
$93$	9.49249	0.0105841
$94$	0	0
$95$	63.9191	0.0690311
$96$	0	0
$97$	1193.65	1.24945	0.624727	−	0.780843i	$-0.285210\pi$
$97$	1193.65	1.24945	0.624727	+	0.780843i	$0.285210\pi$
$98$	0	0
$99$	−1076.93	−1.09329
$100$	0	0
$101$	−596.412	−0.587577	−0.293788	−	0.955871i	$-0.594916\pi$
$101$	−596.412	−0.587577	−0.293788	+	0.955871i	$0.594916\pi$
$102$	0	0
$103$	1193.86	1.14209	0.571043	−	0.820920i	$-0.306539\pi$
$103$	1193.86	1.14209	0.571043	+	0.820920i	$0.306539\pi$
$104$	0	0
$105$	−222.204	−0.206523
$106$	0	0
$107$	1787.96	1.61541	0.807704	−	0.589588i	$-0.200710\pi$
$107$	1787.96	1.61541	0.807704	+	0.589588i	$0.200710\pi$
$108$	0	0
$109$	3.03264	0.00266490	0.00133245	−	0.999999i	$-0.499576\pi$
$109$	3.03264	0.00266490	0.00133245	+	0.999999i	$0.499576\pi$
$110$	0	0
$111$	−130.978	−0.111999
$112$	0	0
$113$	513.505	0.427491	0.213746	−	0.976889i	$-0.431434\pi$
$113$	513.505	0.427491	0.213746	+	0.976889i	$0.431434\pi$
$114$	0	0
$115$	−967.582	−0.784587
$116$	0	0
$117$	307.122	0.242679
$118$	0	0
$119$	916.447	0.705971
$120$	0	0
$121$	746.996	0.561229
$122$	0	0
$123$	−142.536	−0.104488
$124$	0	0
$125$	−1312.78	−0.939350
$126$	0	0
$127$	−207.447	−0.144945	−0.0724723	−	0.997370i	$-0.523089\pi$
$127$	−207.447	−0.144945	−0.0724723	+	0.997370i	$0.523089\pi$
$128$	0	0
$129$	544.963	0.371948
$130$	0	0
$131$	82.2113	0.0548308	0.0274154	−	0.999624i	$-0.491272\pi$
$131$	82.2113	0.0548308	0.0274154	+	0.999624i	$0.491272\pi$
$132$	0	0
$133$	−200.581	−0.130772
$134$	0	0
$135$	−577.412	−0.368116
$136$	0	0
$137$	2096.91	1.30767	0.653835	−	0.756637i	$-0.273159\pi$
$137$	2096.91	1.30767	0.653835	+	0.756637i	$0.273159\pi$
$138$	0	0
$139$	67.3623	0.0411050	0.0205525	−	0.999789i	$-0.493457\pi$
$139$	67.3623	0.0411050	0.0205525	+	0.999789i	$0.493457\pi$
$140$	0	0
$141$	545.033	0.325532
$142$	0	0
$143$	−592.605	−0.346547
$144$	0	0
$145$	−429.406	−0.245932
$146$	0	0
$147$	67.1312	0.0376659
$148$	0	0
$149$	−1041.49	−0.572631	−0.286316	−	0.958135i	$-0.592431\pi$
$149$	−1041.49	−0.572631	−0.286316	+	0.958135i	$0.592431\pi$
$150$	0	0
$151$	−696.787	−0.375522	−0.187761	−	0.982215i	$-0.560123\pi$
$151$	−696.787	−0.375522	−0.187761	+	0.982215i	$0.560123\pi$
$152$	0	0
$153$	1111.34	0.587230
$154$	0	0
$155$	32.0771	0.0166225
$156$	0	0
$157$	−269.011	−0.136748	−0.0683739	−	0.997660i	$-0.521781\pi$
$157$	−269.011	−0.136748	−0.0683739	+	0.997660i	$0.521781\pi$
$158$	0	0
$159$	−1105.83	−0.551560
$160$	0	0
$161$	3036.32	1.48631
$162$	0	0
$163$	−2031.20	−0.976047	−0.488023	−	0.872831i	$-0.662282\pi$
$163$	−2031.20	−0.976047	−0.488023	+	0.872831i	$0.662282\pi$
$164$	0	0
$165$	519.930	0.245312
$166$	0	0
$167$	2210.33	1.02419	0.512096	−	0.858928i	$-0.328869\pi$
$167$	2210.33	1.02419	0.512096	+	0.858928i	$0.328869\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	−243.236	−0.108776
$172$	0	0
$173$	1677.24	0.737098	0.368549	−	0.929608i	$-0.379855\pi$
$173$	1677.24	0.737098	0.368549	+	0.929608i	$0.379855\pi$
$174$	0	0
$175$	1684.35	0.727571
$176$	0	0
$177$	−499.888	−0.212282
$178$	0	0
$179$	2444.60	1.02077	0.510386	−	0.859945i	$-0.329503\pi$
$179$	2444.60	1.02077	0.510386	+	0.859945i	$0.329503\pi$
$180$	0	0
$181$	843.249	0.346289	0.173144	−	0.984896i	$-0.444607\pi$
$181$	843.249	0.346289	0.173144	+	0.984896i	$0.444607\pi$
$182$	0	0
$183$	−1177.79	−0.475762
$184$	0	0
$185$	−442.603	−0.175896
$186$	0	0
$187$	−2144.38	−0.838568
$188$	0	0
$189$	1811.95	0.697354
$190$	0	0
$191$	−2742.53	−1.03897	−0.519484	−	0.854480i	$-0.673876\pi$
$191$	−2742.53	−1.03897	−0.519484	+	0.854480i	$0.673876\pi$
$192$	0	0
$193$	801.584	0.298960	0.149480	−	0.988765i	$-0.452240\pi$
$193$	801.584	0.298960	0.149480	+	0.988765i	$0.452240\pi$
$194$	0	0
$195$	−148.274	−0.0544521
$196$	0	0
$197$	2867.94	1.03722	0.518609	−	0.855012i	$-0.326450\pi$
$197$	2867.94	1.03722	0.518609	+	0.855012i	$0.326450\pi$
$198$	0	0
$199$	4101.21	1.46094	0.730469	−	0.682945i	$-0.239301\pi$
$199$	4101.21	1.46094	0.730469	+	0.682945i	$0.239301\pi$
$200$	0	0
$201$	1877.55	0.658868
$202$	0	0
$203$	1347.50	0.465891
$204$	0	0
$205$	−481.657	−0.164100
$206$	0	0
$207$	3682.02	1.23632
$208$	0	0
$209$	469.336	0.155333
$210$	0	0
$211$	−186.162	−0.0607390	−0.0303695	−	0.999539i	$-0.509668\pi$
$211$	−186.162	−0.0607390	−0.0303695	+	0.999539i	$0.509668\pi$
$212$	0	0
$213$	−1804.00	−0.580320
$214$	0	0
$215$	1841.54	0.584150
$216$	0	0
$217$	−100.660	−0.0314895
$218$	0	0
$219$	−1608.75	−0.496389
$220$	0	0
$221$	611.536	0.186137
$222$	0	0
$223$	−6463.02	−1.94079	−0.970394	−	0.241527i	$-0.922352\pi$
$223$	−6463.02	−1.94079	−0.970394	+	0.241527i	$0.922352\pi$
$224$	0	0
$225$	2042.54	0.605197
$226$	0	0
$227$	4199.99	1.22803	0.614016	−	0.789294i	$-0.289553\pi$
$227$	4199.99	1.22803	0.614016	+	0.789294i	$0.289553\pi$
$228$	0	0
$229$	2068.59	0.596928	0.298464	−	0.954421i	$-0.403526\pi$
$229$	2068.59	0.596928	0.298464	+	0.954421i	$0.403526\pi$
$230$	0	0
$231$	−1631.57	−0.464716
$232$	0	0
$233$	1584.10	0.445398	0.222699	−	0.974887i	$-0.428513\pi$
$233$	1584.10	0.445398	0.222699	+	0.974887i	$0.428513\pi$
$234$	0	0
$235$	1841.78	0.511252
$236$	0	0
$237$	−1194.13	−0.327289
$238$	0	0
$239$	305.470	0.0826745	0.0413372	−	0.999145i	$-0.486838\pi$
$239$	305.470	0.0826745	0.0413372	+	0.999145i	$0.486838\pi$
$240$	0	0
$241$	1503.33	0.401817	0.200908	−	0.979610i	$-0.435611\pi$
$241$	1503.33	0.401817	0.200908	+	0.979610i	$0.435611\pi$
$242$	0	0
$243$	3369.16	0.889430
$244$	0	0
$245$	226.850	0.0591548
$246$	0	0
$247$	−133.846	−0.0344794
$248$	0	0
$249$	−1492.36	−0.379816
$250$	0	0
$251$	−5154.98	−1.29633	−0.648166	−	0.761499i	$-0.724464\pi$
$251$	−5154.98	−1.29633	−0.648166	+	0.761499i	$0.724464\pi$
$252$	0	0
$253$	−7104.63	−1.76547
$254$	0	0
$255$	−536.539	−0.131762
$256$	0	0
$257$	6005.25	1.45758	0.728789	−	0.684739i	$-0.240084\pi$
$257$	6005.25	1.45758	0.728789	+	0.684739i	$0.240084\pi$
$258$	0	0
$259$	1388.91	0.333215
$260$	0	0
$261$	1634.05	0.387530
$262$	0	0
$263$	−3760.32	−0.881640	−0.440820	−	0.897595i	$-0.645312\pi$
$263$	−3760.32	−0.881640	−0.440820	+	0.897595i	$0.645312\pi$
$264$	0	0
$265$	−3736.82	−0.866231
$266$	0	0
$267$	178.514	0.0409172
$268$	0	0
$269$	1359.08	0.308048	0.154024	−	0.988067i	$-0.450777\pi$
$269$	1359.08	0.308048	0.154024	+	0.988067i	$0.450777\pi$
$270$	0	0
$271$	1790.90	0.401438	0.200719	−	0.979649i	$-0.435672\pi$
$271$	1790.90	0.401438	0.200719	+	0.979649i	$0.435672\pi$
$272$	0	0
$273$	465.293	0.103153
$274$	0	0
$275$	−3941.18	−0.864225
$276$	0	0
$277$	6981.03	1.51426	0.757129	−	0.653266i	$-0.226601\pi$
$277$	6981.03	1.51426	0.757129	+	0.653266i	$0.226601\pi$
$278$	0	0
$279$	−122.065	−0.0261931
$280$	0	0
$281$	−1082.51	−0.229812	−0.114906	−	0.993376i	$-0.536657\pi$
$281$	−1082.51	−0.229812	−0.114906	+	0.993376i	$0.536657\pi$
$282$	0	0
$283$	4439.85	0.932585	0.466292	−	0.884631i	$-0.345589\pi$
$283$	4439.85	0.932585	0.466292	+	0.884631i	$0.345589\pi$
$284$	0	0
$285$	117.432	0.0244072
$286$	0	0
$287$	1511.47	0.310868
$288$	0	0
$289$	−2700.12	−0.549588
$290$	0	0
$291$	2192.97	0.441767
$292$	0	0
$293$	3570.20	0.711854	0.355927	−	0.934514i	$-0.384165\pi$
$293$	3570.20	0.711854	0.355927	+	0.934514i	$0.384165\pi$
$294$	0	0
$295$	−1689.22	−0.333391
$296$	0	0
$297$	−4239.74	−0.828332
$298$	0	0
$299$	2026.11	0.391882
$300$	0	0
$301$	−5778.85	−1.10660
$302$	0	0
$303$	−1095.72	−0.207748
$304$	0	0
$305$	−3979.98	−0.747191
$306$	0	0
$307$	−5636.05	−1.04777	−0.523887	−	0.851788i	$-0.675519\pi$
$307$	−5636.05	−1.04777	−0.523887	+	0.851788i	$0.675519\pi$
$308$	0	0
$309$	2193.35	0.403804
$310$	0	0
$311$	−9042.60	−1.64874	−0.824371	−	0.566049i	$-0.808471\pi$
$311$	−9042.60	−1.64874	−0.824371	+	0.566049i	$0.808471\pi$
$312$	0	0
$313$	−10852.3	−1.95976	−0.979882	−	0.199577i	$-0.936043\pi$
$313$	−10852.3	−1.95976	−0.979882	+	0.199577i	$0.936043\pi$
$314$	0	0
$315$	2857.36	0.511092
$316$	0	0
$317$	−3736.13	−0.661963	−0.330981	−	0.943637i	$-0.607380\pi$
$317$	−3736.13	−0.661963	−0.330981	+	0.943637i	$0.607380\pi$
$318$	0	0
$319$	−3152.98	−0.553395
$320$	0	0
$321$	3284.82	0.571156
$322$	0	0
$323$	−484.329	−0.0834327
$324$	0	0
$325$	1123.95	0.191832
$326$	0	0
$327$	5.57153	0.000942221 0
$328$	0	0
$329$	−5779.59	−0.968508
$330$	0	0
$331$	−6755.12	−1.12174	−0.560868	−	0.827905i	$-0.689533\pi$
$331$	−6755.12	−1.12174	−0.560868	+	0.827905i	$0.689533\pi$
$332$	0	0
$333$	1684.27	0.277170
$334$	0	0
$335$	6344.64	1.03476
$336$	0	0
$337$	−10073.2	−1.62826	−0.814130	−	0.580682i	$-0.802786\pi$
$337$	−10073.2	−1.62826	−0.814130	+	0.580682i	$0.802786\pi$
$338$	0	0
$339$	943.407	0.151147
$340$	0	0
$341$	235.531	0.0374039
$342$	0	0
$343$	5970.39	0.939856
$344$	0	0
$345$	−1777.63	−0.277404
$346$	0	0
$347$	−7102.09	−1.09873	−0.549366	−	0.835582i	$-0.685131\pi$
$347$	−7102.09	−1.09873	−0.549366	+	0.835582i	$0.685131\pi$
$348$	0	0
$349$	−5599.78	−0.858880	−0.429440	−	0.903095i	$-0.641289\pi$
$349$	−5599.78	−0.858880	−0.429440	+	0.903095i	$0.641289\pi$
$350$	0	0
$351$	1209.09	0.183865
$352$	0	0
$353$	−7062.56	−1.06488	−0.532439	−	0.846468i	$-0.678724\pi$
$353$	−7062.56	−1.06488	−0.532439	+	0.846468i	$0.678724\pi$
$354$	0	0
$355$	−6096.09	−0.911399
$356$	0	0
$357$	1683.69	0.249608
$358$	0	0
$359$	10391.4	1.52769	0.763844	−	0.645401i	$-0.223310\pi$
$359$	10391.4	1.52769	0.763844	+	0.645401i	$0.223310\pi$
$360$	0	0
$361$	−6753.00	−0.984545
$362$	0	0
$363$	1372.37	0.198432
$364$	0	0
$365$	−5436.29	−0.779585
$366$	0	0
$367$	9368.95	1.33258	0.666288	−	0.745695i	$-0.267882\pi$
$367$	9368.95	1.33258	0.666288	+	0.745695i	$0.267882\pi$
$368$	0	0
$369$	1832.89	0.258581
$370$	0	0
$371$	11726.3	1.64097
$372$	0	0
$373$	6303.38	0.875004	0.437502	−	0.899217i	$-0.355863\pi$
$373$	6303.38	0.875004	0.437502	+	0.899217i	$0.355863\pi$
$374$	0	0
$375$	−2411.83	−0.332123
$376$	0	0
$377$	899.171	0.122837
$378$	0	0
$379$	3980.44	0.539476	0.269738	−	0.962934i	$-0.413063\pi$
$379$	3980.44	0.539476	0.269738	+	0.962934i	$0.413063\pi$
$380$	0	0
$381$	−381.120	−0.0512477
$382$	0	0
$383$	2315.19	0.308879	0.154439	−	0.988002i	$-0.450643\pi$
$383$	2315.19	0.308879	0.154439	+	0.988002i	$0.450643\pi$
$384$	0	0
$385$	−5513.41	−0.729842
$386$	0	0
$387$	−7007.77	−0.920478
$388$	0	0
$389$	10739.7	1.39981	0.699905	−	0.714236i	$-0.253226\pi$
$389$	10739.7	1.39981	0.699905	+	0.714236i	$0.253226\pi$
$390$	0	0
$391$	7331.58	0.948271
$392$	0	0
$393$	151.038	0.0193864
$394$	0	0
$395$	−4035.23	−0.514011
$396$	0	0
$397$	−610.651	−0.0771982	−0.0385991	−	0.999255i	$-0.512290\pi$
$397$	−610.651	−0.0771982	−0.0385991	+	0.999255i	$0.512290\pi$
$398$	0	0
$399$	−368.506	−0.0462365
$400$	0	0
$401$	5952.35	0.741263	0.370631	−	0.928780i	$-0.379141\pi$
$401$	5952.35	0.741263	0.370631	+	0.928780i	$0.379141\pi$
$402$	0	0
$403$	−67.1691	−0.00830255
$404$	0	0
$405$	2899.22	0.355712
$406$	0	0
$407$	−3249.89	−0.395801
$408$	0	0
$409$	−15612.1	−1.88745	−0.943725	−	0.330730i	$-0.892705\pi$
$409$	−15612.1	−1.88745	−0.943725	+	0.330730i	$0.892705\pi$
$410$	0	0
$411$	3852.42	0.462350
$412$	0	0
$413$	5300.87	0.631571
$414$	0	0
$415$	−5042.98	−0.596506
$416$	0	0
$417$	123.757	0.0145334
$418$	0	0
$419$	−4971.26	−0.579623	−0.289811	−	0.957084i	$-0.593593\pi$
$419$	−4971.26	−0.579623	−0.289811	+	0.957084i	$0.593593\pi$
$420$	0	0
$421$	−1307.30	−0.151340	−0.0756698	−	0.997133i	$-0.524109\pi$
$421$	−1307.30	−0.151340	−0.0756698	+	0.997133i	$0.524109\pi$
$422$	0	0
$423$	−7008.66	−0.805610
$424$	0	0
$425$	4067.08	0.464193
$426$	0	0
$427$	12489.4	1.41547
$428$	0	0
$429$	−1088.73	−0.122528
$430$	0	0
$431$	−9146.52	−1.02221	−0.511105	−	0.859518i	$-0.670764\pi$
$431$	−9146.52	−1.02221	−0.511105	+	0.859518i	$0.670764\pi$
$432$	0	0
$433$	−3246.44	−0.360310	−0.180155	−	0.983638i	$-0.557660\pi$
$433$	−3246.44	−0.360310	−0.180155	+	0.983638i	$0.557660\pi$
$434$	0	0
$435$	−788.900	−0.0869537
$436$	0	0
$437$	−1604.65	−0.175654
$438$	0	0
$439$	−7918.55	−0.860892	−0.430446	−	0.902616i	$-0.641644\pi$
$439$	−7918.55	−0.860892	−0.430446	+	0.902616i	$0.641644\pi$
$440$	0	0
$441$	−863.251	−0.0932136
$442$	0	0
$443$	5069.07	0.543654	0.271827	−	0.962346i	$-0.412372\pi$
$443$	5069.07	0.543654	0.271827	+	0.962346i	$0.412372\pi$
$444$	0	0
$445$	603.237	0.0642610
$446$	0	0
$447$	−1913.41	−0.202464
$448$	0	0
$449$	8055.03	0.846638	0.423319	−	0.905981i	$-0.360865\pi$
$449$	8055.03	0.846638	0.423319	+	0.905981i	$0.360865\pi$
$450$	0	0
$451$	−3536.65	−0.369255
$452$	0	0
$453$	−1280.13	−0.132772
$454$	0	0
$455$	1572.32	0.162003
$456$	0	0
$457$	−16991.3	−1.73921	−0.869607	−	0.493745i	$-0.835627\pi$
$457$	−16991.3	−1.73921	−0.869607	+	0.493745i	$0.835627\pi$
$458$	0	0
$459$	4375.18	0.444914
$460$	0	0
$461$	−12875.9	−1.30085	−0.650424	−	0.759571i	$-0.725409\pi$
$461$	−12875.9	−1.30085	−0.650424	+	0.759571i	$0.725409\pi$
$462$	0	0
$463$	−135.611	−0.0136121	−0.00680604	−	0.999977i	$-0.502166\pi$
$463$	−135.611	−0.0136121	−0.00680604	+	0.999977i	$0.502166\pi$
$464$	0	0
$465$	58.9317	0.00587719
$466$	0	0
$467$	−9132.78	−0.904957	−0.452478	−	0.891775i	$-0.649460\pi$
$467$	−9132.78	−0.904957	−0.452478	+	0.891775i	$0.649460\pi$
$468$	0	0
$469$	−19909.8	−1.96023
$470$	0	0
$471$	−494.224	−0.0483495
$472$	0	0
$473$	13521.8	1.31445
$474$	0	0
$475$	−890.155	−0.0859855
$476$	0	0
$477$	14220.0	1.36497
$478$	0	0
$479$	2785.43	0.265698	0.132849	−	0.991136i	$-0.457587\pi$
$479$	2785.43	0.265698	0.132849	+	0.991136i	$0.457587\pi$
$480$	0	0
$481$	926.807	0.0878560
$482$	0	0
$483$	5578.30	0.525510
$484$	0	0
$485$	7410.49	0.693800
$486$	0	0
$487$	−10570.6	−0.983570	−0.491785	−	0.870717i	$-0.663655\pi$
$487$	−10570.6	−0.983570	−0.491785	+	0.870717i	$0.663655\pi$
$488$	0	0
$489$	−3731.70	−0.345098
$490$	0	0
$491$	−10962.1	−1.00756	−0.503781	−	0.863832i	$-0.668058\pi$
$491$	−10962.1	−1.00756	−0.503781	+	0.863832i	$0.668058\pi$
$492$	0	0
$493$	3253.70	0.297240
$494$	0	0
$495$	−6685.87	−0.607086
$496$	0	0
$497$	19129.8	1.72654
$498$	0	0
$499$	10631.0	0.953730	0.476865	−	0.878977i	$-0.341773\pi$
$499$	10631.0	0.953730	0.476865	+	0.878977i	$0.341773\pi$
$500$	0	0
$501$	4060.79	0.362121
$502$	0	0
$503$	8406.08	0.745146	0.372573	−	0.928003i	$-0.378476\pi$
$503$	8406.08	0.745146	0.372573	+	0.928003i	$0.378476\pi$
$504$	0	0
$505$	−3702.67	−0.326271
$506$	0	0
$507$	310.485	0.0271975
$508$	0	0
$509$	−7896.27	−0.687615	−0.343808	−	0.939040i	$-0.611717\pi$
$509$	−7896.27	−0.687615	−0.343808	+	0.939040i	$0.611717\pi$
$510$	0	0
$511$	17059.4	1.47683
$512$	0	0
$513$	−957.588	−0.0824143
$514$	0	0
$515$	7411.79	0.634180
$516$	0	0
$517$	13523.5	1.15042
$518$	0	0
$519$	3081.40	0.260614
$520$	0	0
$521$	−18913.1	−1.59040	−0.795200	−	0.606347i	$-0.792634\pi$
$521$	−18913.1	−1.59040	−0.795200	+	0.606347i	$0.792634\pi$
$522$	0	0
$523$	−2606.45	−0.217920	−0.108960	−	0.994046i	$-0.534752\pi$
$523$	−2606.45	−0.217920	−0.108960	+	0.994046i	$0.534752\pi$
$524$	0	0
$525$	3094.47	0.257245
$526$	0	0
$527$	−243.055	−0.0200904
$528$	0	0
$529$	12123.6	0.996430
$530$	0	0
$531$	6428.14	0.525343
$532$	0	0
$533$	1008.59	0.0819638
$534$	0	0
$535$	11100.1	0.897008
$536$	0	0
$537$	4491.20	0.360912
$538$	0	0
$539$	1665.68	0.133110
$540$	0	0
$541$	−15288.3	−1.21497	−0.607483	−	0.794333i	$-0.707821\pi$
$541$	−15288.3	−1.21497	−0.607483	+	0.794333i	$0.707821\pi$
$542$	0	0
$543$	1549.21	0.122436
$544$	0	0
$545$	18.8273	0.00147977
$546$	0	0
$547$	−18025.8	−1.40901	−0.704506	−	0.709698i	$-0.748831\pi$
$547$	−18025.8	−1.40901	−0.704506	+	0.709698i	$0.748831\pi$
$548$	0	0
$549$	15145.3	1.17739
$550$	0	0
$551$	−712.133	−0.0550596
$552$	0	0
$553$	12662.8	0.973734
$554$	0	0
$555$	−813.146	−0.0621913
$556$	0	0
$557$	11863.5	0.902463	0.451231	−	0.892407i	$-0.350985\pi$
$557$	11863.5	0.902463	0.451231	+	0.892407i	$0.350985\pi$
$558$	0	0
$559$	−3856.17	−0.291769
$560$	0	0
$561$	−3939.63	−0.296490
$562$	0	0
$563$	−5548.38	−0.415340	−0.207670	−	0.978199i	$-0.566588\pi$
$563$	−5548.38	−0.415340	−0.207670	+	0.978199i	$0.566588\pi$
$564$	0	0
$565$	3187.97	0.237378
$566$	0	0
$567$	−9097.91	−0.673856
$568$	0	0
$569$	10337.3	0.761618	0.380809	−	0.924654i	$-0.375646\pi$
$569$	10337.3	0.761618	0.380809	+	0.924654i	$0.375646\pi$
$570$	0	0
$571$	17630.0	1.29210	0.646052	−	0.763294i	$-0.276419\pi$
$571$	17630.0	1.29210	0.646052	+	0.763294i	$0.276419\pi$
$572$	0	0
$573$	−5038.56	−0.367345
$574$	0	0
$575$	13474.8	0.977284
$576$	0	0
$577$	7765.21	0.560260	0.280130	−	0.959962i	$-0.409622\pi$
$577$	7765.21	0.560260	0.280130	+	0.959962i	$0.409622\pi$
$578$	0	0
$579$	1472.66	0.105702
$580$	0	0
$581$	15825.1	1.13001
$582$	0	0
$583$	−27438.2	−1.94919
$584$	0	0
$585$	1906.69	0.134755
$586$	0	0
$587$	−20090.6	−1.41265	−0.706327	−	0.707886i	$-0.749649\pi$
$587$	−20090.6	−1.41265	−0.706327	+	0.707886i	$0.749649\pi$
$588$	0	0
$589$	53.1971	0.00372147
$590$	0	0
$591$	5268.94	0.366726
$592$	0	0
$593$	18437.8	1.27681	0.638405	−	0.769701i	$-0.279595\pi$
$593$	18437.8	1.27681	0.638405	+	0.769701i	$0.279595\pi$
$594$	0	0
$595$	5689.53	0.392013
$596$	0	0
$597$	7534.70	0.516540
$598$	0	0
$599$	1743.10	0.118900	0.0594501	−	0.998231i	$-0.481065\pi$
$599$	1743.10	0.118900	0.0594501	+	0.998231i	$0.481065\pi$
$600$	0	0
$601$	10234.2	0.694609	0.347304	−	0.937752i	$-0.387097\pi$
$601$	10234.2	0.694609	0.347304	+	0.937752i	$0.387097\pi$
$602$	0	0
$603$	−24143.8	−1.63053
$604$	0	0
$605$	4637.53	0.311640
$606$	0	0
$607$	−15838.6	−1.05909	−0.529546	−	0.848281i	$-0.677638\pi$
$607$	−15838.6	−1.05909	−0.529546	+	0.848281i	$0.677638\pi$
$608$	0	0
$609$	2475.61	0.164724
$610$	0	0
$611$	−3856.66	−0.255358
$612$	0	0
$613$	−10048.6	−0.662087	−0.331043	−	0.943616i	$-0.607401\pi$
$613$	−10048.6	−0.662087	−0.331043	+	0.943616i	$0.607401\pi$
$614$	0	0
$615$	−884.896	−0.0580203
$616$	0	0
$617$	−8515.00	−0.555593	−0.277797	−	0.960640i	$-0.589604\pi$
$617$	−8515.00	−0.555593	−0.277797	+	0.960640i	$0.589604\pi$
$618$	0	0
$619$	−4715.30	−0.306177	−0.153089	−	0.988212i	$-0.548922\pi$
$619$	−4715.30	−0.306177	−0.153089	+	0.988212i	$0.548922\pi$
$620$	0	0
$621$	14495.6	0.936695
$622$	0	0
$623$	−1892.99	−0.121735
$624$	0	0
$625$	2657.15	0.170058
$626$	0	0
$627$	862.260	0.0549208
$628$	0	0
$629$	3353.70	0.212593
$630$	0	0
$631$	1150.78	0.0726020	0.0363010	−	0.999341i	$-0.488442\pi$
$631$	1150.78	0.0726020	0.0363010	+	0.999341i	$0.488442\pi$
$632$	0	0
$633$	−342.015	−0.0214753
$634$	0	0
$635$	−1287.88	−0.0804851
$636$	0	0
$637$	−475.022	−0.0295464
$638$	0	0
$639$	23197.9	1.43614
$640$	0	0
$641$	14095.0	0.868518	0.434259	−	0.900788i	$-0.357010\pi$
$641$	14095.0	0.868518	0.434259	+	0.900788i	$0.357010\pi$
$642$	0	0
$643$	8245.60	0.505715	0.252858	−	0.967504i	$-0.418630\pi$
$643$	8245.60	0.505715	0.252858	+	0.967504i	$0.418630\pi$
$644$	0	0
$645$	3383.26	0.206536
$646$	0	0
$647$	25082.1	1.52408	0.762040	−	0.647530i	$-0.224198\pi$
$647$	25082.1	1.52408	0.762040	+	0.647530i	$0.224198\pi$
$648$	0	0
$649$	−12403.4	−0.750194
$650$	0	0
$651$	−184.931	−0.0111336
$652$	0	0
$653$	−14425.3	−0.864479	−0.432240	−	0.901759i	$-0.642277\pi$
$653$	−14425.3	−0.864479	−0.432240	+	0.901759i	$0.642277\pi$
$654$	0	0
$655$	510.388	0.0304466
$656$	0	0
$657$	20687.2	1.22844
$658$	0	0
$659$	32874.8	1.94328	0.971639	−	0.236469i	$-0.0759903\pi$
$659$	32874.8	1.94328	0.971639	+	0.236469i	$0.0759903\pi$
$660$	0	0
$661$	−16168.3	−0.951400	−0.475700	−	0.879607i	$-0.657805\pi$
$661$	−16168.3	−0.951400	−0.475700	+	0.879607i	$0.657805\pi$
$662$	0	0
$663$	1123.51	0.0658121
$664$	0	0
$665$	−1245.26	−0.0726151
$666$	0	0
$667$	10780.0	0.625791
$668$	0	0
$669$	−11873.8	−0.686199
$670$	0	0
$671$	−29223.7	−1.68132
$672$	0	0
$673$	16782.5	0.961247	0.480623	−	0.876927i	$-0.340410\pi$
$673$	16782.5	0.961247	0.480623	+	0.876927i	$0.340410\pi$
$674$	0	0
$675$	8041.20	0.458527
$676$	0	0
$677$	−34811.9	−1.97626	−0.988131	−	0.153612i	$-0.950909\pi$
$677$	−34811.9	−1.97626	−0.988131	+	0.153612i	$0.950909\pi$
$678$	0	0
$679$	−23254.5	−1.31432
$680$	0	0
$681$	7716.18	0.434192
$682$	0	0
$683$	−5547.22	−0.310774	−0.155387	−	0.987854i	$-0.549662\pi$
$683$	−5547.22	−0.310774	−0.155387	+	0.987854i	$0.549662\pi$
$684$	0	0
$685$	13018.1	0.726126
$686$	0	0
$687$	3800.40	0.211054
$688$	0	0
$689$	7824.87	0.432662
$690$	0	0
$691$	20003.7	1.10127	0.550636	−	0.834746i	$-0.314385\pi$
$691$	20003.7	1.10127	0.550636	+	0.834746i	$0.314385\pi$
$692$	0	0
$693$	20980.6	1.15005
$694$	0	0
$695$	418.202	0.0228249
$696$	0	0
$697$	3649.62	0.198335
$698$	0	0
$699$	2910.29	0.157478
$700$	0	0
$701$	−3289.73	−0.177249	−0.0886244	−	0.996065i	$-0.528247\pi$
$701$	−3289.73	−0.177249	−0.0886244	+	0.996065i	$0.528247\pi$
$702$	0	0
$703$	−734.019	−0.0393799
$704$	0	0
$705$	3383.70	0.180762
$706$	0	0
$707$	11619.2	0.618082
$708$	0	0
$709$	5892.45	0.312123	0.156062	−	0.987747i	$-0.450120\pi$
$709$	5892.45	0.312123	0.156062	+	0.987747i	$0.450120\pi$
$710$	0	0
$711$	15355.6	0.809957
$712$	0	0
$713$	−805.276	−0.0422971
$714$	0	0
$715$	−3679.04	−0.192431
$716$	0	0
$717$	561.206	0.0292310
$718$	0	0
$719$	−28981.2	−1.50322	−0.751612	−	0.659606i	$-0.770723\pi$
$719$	−28981.2	−1.50322	−0.751612	+	0.659606i	$0.770723\pi$
$720$	0	0
$721$	−23258.6	−1.20138
$722$	0	0
$723$	2761.90	0.142069
$724$	0	0
$725$	5980.02	0.306334
$726$	0	0
$727$	26044.2	1.32865	0.664323	−	0.747446i	$-0.268720\pi$
$727$	26044.2	1.32865	0.664323	+	0.747446i	$0.268720\pi$
$728$	0	0
$729$	−6419.10	−0.326124
$730$	0	0
$731$	−13953.8	−0.706017
$732$	0	0
$733$	−34919.4	−1.75959	−0.879793	−	0.475357i	$-0.842319\pi$
$733$	−34919.4	−1.75959	−0.879793	+	0.475357i	$0.842319\pi$
$734$	0	0
$735$	416.767	0.0209152
$736$	0	0
$737$	46586.5	2.32841
$738$	0	0
$739$	−18907.6	−0.941171	−0.470586	−	0.882354i	$-0.655957\pi$
$739$	−18907.6	−0.941171	−0.470586	+	0.882354i	$0.655957\pi$
$740$	0	0
$741$	−245.900	−0.0121908
$742$	0	0
$743$	13621.5	0.672579	0.336289	−	0.941759i	$-0.390828\pi$
$743$	13621.5	0.672579	0.336289	+	0.941759i	$0.390828\pi$
$744$	0	0
$745$	−6465.81	−0.317972
$746$	0	0
$747$	19190.4	0.939949
$748$	0	0
$749$	−34832.7	−1.69928
$750$	0	0
$751$	7690.73	0.373687	0.186843	−	0.982390i	$-0.440174\pi$
$751$	7690.73	0.373687	0.186843	+	0.982390i	$0.440174\pi$
$752$	0	0
$753$	−9470.68	−0.458341
$754$	0	0
$755$	−4325.83	−0.208520
$756$	0	0
$757$	11273.7	0.541279	0.270640	−	0.962681i	$-0.412765\pi$
$757$	11273.7	0.541279	0.270640	+	0.962681i	$0.412765\pi$
$758$	0	0
$759$	−13052.5	−0.624213
$760$	0	0
$761$	39972.3	1.90406	0.952032	−	0.305998i	$-0.0989899\pi$
$761$	39972.3	1.90406	0.952032	+	0.305998i	$0.0989899\pi$
$762$	0	0
$763$	−59.0812	−0.00280325
$764$	0	0
$765$	6899.45	0.326078
$766$	0	0
$767$	3537.22	0.166521
$768$	0	0
$769$	−33967.8	−1.59286	−0.796429	−	0.604731i	$-0.793280\pi$
$769$	−33967.8	−1.59286	−0.796429	+	0.604731i	$0.793280\pi$
$770$	0	0
$771$	11032.8	0.515352
$772$	0	0
$773$	−29974.6	−1.39471	−0.697356	−	0.716725i	$-0.745640\pi$
$773$	−29974.6	−1.39471	−0.697356	+	0.716725i	$0.745640\pi$
$774$	0	0
$775$	−446.714	−0.0207051
$776$	0	0
$777$	2551.70	0.117814
$778$	0	0
$779$	−798.787	−0.0367388
$780$	0	0
$781$	−44761.5	−2.05082
$782$	0	0
$783$	6433.04	0.293612
$784$	0	0
$785$	−1670.08	−0.0759336
$786$	0	0
$787$	−36015.1	−1.63126	−0.815628	−	0.578577i	$-0.803608\pi$
$787$	−36015.1	−1.63126	−0.815628	+	0.578577i	$0.803608\pi$
$788$	0	0
$789$	−6908.43	−0.311719
$790$	0	0
$791$	−10004.0	−0.449686
$792$	0	0
$793$	8334.04	0.373204
$794$	0	0
$795$	−6865.26	−0.306271
$796$	0	0
$797$	13889.5	0.617302	0.308651	−	0.951175i	$-0.400122\pi$
$797$	13889.5	0.617302	0.308651	+	0.951175i	$0.400122\pi$
$798$	0	0
$799$	−13955.5	−0.617912
$800$	0	0
$801$	−2295.54	−0.101260
$802$	0	0
$803$	−39916.9	−1.75422
$804$	0	0
$805$	18850.2	0.825321
$806$	0	0
$807$	2496.90	0.108916
$808$	0	0
$809$	−45197.9	−1.96424	−0.982122	−	0.188247i	$-0.939719\pi$
$809$	−45197.9	−1.96424	−0.982122	+	0.188247i	$0.939719\pi$
$810$	0	0
$811$	−15933.4	−0.689884	−0.344942	−	0.938624i	$-0.612101\pi$
$811$	−15933.4	−0.689884	−0.344942	+	0.938624i	$0.612101\pi$
$812$	0	0
$813$	3290.23	0.141935
$814$	0	0
$815$	−12610.2	−0.541982
$816$	0	0
$817$	3054.04	0.130780
$818$	0	0
$819$	−5983.28	−0.255278
$820$	0	0
$821$	1301.72	0.0553355	0.0276678	−	0.999617i	$-0.491192\pi$
$821$	1301.72	0.0553355	0.0276678	+	0.999617i	$0.491192\pi$
$822$	0	0
$823$	21002.5	0.889553	0.444776	−	0.895642i	$-0.353283\pi$
$823$	21002.5	0.889553	0.444776	+	0.895642i	$0.353283\pi$
$824$	0	0
$825$	−7240.69	−0.305562
$826$	0	0
$827$	4721.35	0.198522	0.0992608	−	0.995061i	$-0.468352\pi$
$827$	4721.35	0.198522	0.0992608	+	0.995061i	$0.468352\pi$
$828$	0	0
$829$	−19341.0	−0.810300	−0.405150	−	0.914250i	$-0.632781\pi$
$829$	−19341.0	−0.810300	−0.405150	+	0.914250i	$0.632781\pi$
$830$	0	0
$831$	12825.5	0.535392
$832$	0	0
$833$	−1718.89	−0.0714959
$834$	0	0
$835$	13722.2	0.568716
$836$	0	0
$837$	−480.555	−0.0198452
$838$	0	0
$839$	13489.1	0.555059	0.277530	−	0.960717i	$-0.410484\pi$
$839$	13489.1	0.555059	0.277530	+	0.960717i	$0.410484\pi$
$840$	0	0
$841$	−19604.9	−0.803843
$842$	0	0
$843$	−1988.78	−0.0812541
$844$	0	0
$845$	1049.19	0.0427140
$846$	0	0
$847$	−14552.8	−0.590367
$848$	0	0
$849$	8156.85	0.329732
$850$	0	0
$851$	11111.3	0.447580
$852$	0	0
$853$	−25868.7	−1.03837	−0.519184	−	0.854663i	$-0.673764\pi$
$853$	−25868.7	−1.03837	−0.519184	+	0.854663i	$0.673764\pi$
$854$	0	0
$855$	−1510.07	−0.0604016
$856$	0	0
$857$	18974.3	0.756301	0.378150	−	0.925744i	$-0.376560\pi$
$857$	18974.3	0.756301	0.378150	+	0.925744i	$0.376560\pi$
$858$	0	0
$859$	−1053.08	−0.0418286	−0.0209143	−	0.999781i	$-0.506658\pi$
$859$	−1053.08	−0.0418286	−0.0209143	+	0.999781i	$0.506658\pi$
$860$	0	0
$861$	2776.85	0.109913
$862$	0	0
$863$	35966.7	1.41868	0.709340	−	0.704867i	$-0.248993\pi$
$863$	35966.7	1.41868	0.709340	+	0.704867i	$0.248993\pi$
$864$	0	0
$865$	10412.7	0.409297
$866$	0	0
$867$	−4960.64	−0.194316
$868$	0	0
$869$	−29629.3	−1.15662
$870$	0	0
$871$	−13285.6	−0.516838
$872$	0	0
$873$	−28199.7	−1.09326
$874$	0	0
$875$	25575.3	0.988119
$876$	0	0
$877$	−42712.4	−1.64458	−0.822289	−	0.569070i	$-0.807303\pi$
$877$	−42712.4	−1.64458	−0.822289	+	0.569070i	$0.807303\pi$
$878$	0	0
$879$	6559.14	0.251688
$880$	0	0
$881$	29849.6	1.14150	0.570748	−	0.821125i	$-0.306653\pi$
$881$	29849.6	1.14150	0.570748	+	0.821125i	$0.306653\pi$
$882$	0	0
$883$	−51744.8	−1.97208	−0.986042	−	0.166495i	$-0.946755\pi$
$883$	−51744.8	−1.97208	−0.986042	+	0.166495i	$0.946755\pi$
$884$	0	0
$885$	−3103.42	−0.117876
$886$	0	0
$887$	−30718.2	−1.16282	−0.581408	−	0.813612i	$-0.697498\pi$
$887$	−30718.2	−1.16282	−0.581408	+	0.813612i	$0.697498\pi$
$888$	0	0
$889$	4041.44	0.152470
$890$	0	0
$891$	21288.0	0.800421
$892$	0	0
$893$	3054.43	0.114460
$894$	0	0
$895$	15176.7	0.566817
$896$	0	0
$897$	3722.34	0.138557
$898$	0	0
$899$	−357.376	−0.0132582
$900$	0	0
$901$	28314.7	1.04695
$902$	0	0
$903$	−10616.9	−0.391259
$904$	0	0
$905$	5235.10	0.192288
$906$	0	0
$907$	34123.7	1.24924	0.624620	−	0.780929i	$-0.285254\pi$
$907$	34123.7	1.24924	0.624620	+	0.780929i	$0.285254\pi$
$908$	0	0
$909$	14090.1	0.514124
$910$	0	0
$911$	4694.50	0.170731	0.0853654	−	0.996350i	$-0.472794\pi$
$911$	4694.50	0.170731	0.0853654	+	0.996350i	$0.472794\pi$
$912$	0	0
$913$	−37028.9	−1.34225
$914$	0	0
$915$	−7311.99	−0.264182
$916$	0	0
$917$	−1601.62	−0.0576775
$918$	0	0
$919$	12086.3	0.433831	0.216915	−	0.976190i	$-0.430400\pi$
$919$	12086.3	0.433831	0.216915	+	0.976190i	$0.430400\pi$
$920$	0	0
$921$	−10354.5	−0.370459
$922$	0	0
$923$	12765.1	0.455222
$924$	0	0
$925$	6163.81	0.219097
$926$	0	0
$927$	−28204.7	−0.999313
$928$	0	0
$929$	−16524.3	−0.583579	−0.291790	−	0.956483i	$-0.594251\pi$
$929$	−16524.3	−0.583579	−0.291790	+	0.956483i	$0.594251\pi$
$930$	0	0
$931$	376.211	0.0132436
$932$	0	0
$933$	−16613.0	−0.582942
$934$	0	0
$935$	−13312.8	−0.465642
$936$	0	0
$937$	8736.94	0.304614	0.152307	−	0.988333i	$-0.451330\pi$
$937$	8736.94	0.304614	0.152307	+	0.988333i	$0.451330\pi$
$938$	0	0
$939$	−19937.7	−0.692909
$940$	0	0
$941$	−8823.88	−0.305686	−0.152843	−	0.988251i	$-0.548843\pi$
$941$	−8823.88	−0.305686	−0.152843	+	0.988251i	$0.548843\pi$
$942$	0	0
$943$	12091.7	0.417562
$944$	0	0
$945$	11249.0	0.387228
$946$	0	0
$947$	46241.6	1.58675	0.793373	−	0.608736i	$-0.208323\pi$
$947$	46241.6	1.58675	0.793373	+	0.608736i	$0.208323\pi$
$948$	0	0
$949$	11383.5	0.389384
$950$	0	0
$951$	−6863.99	−0.234049
$952$	0	0
$953$	14160.2	0.481314	0.240657	−	0.970610i	$-0.422637\pi$
$953$	14160.2	0.481314	0.240657	+	0.970610i	$0.422637\pi$
$954$	0	0
$955$	−17026.3	−0.576920
$956$	0	0
$957$	−5792.63	−0.195663
$958$	0	0
$959$	−40851.5	−1.37556
$960$	0	0
$961$	−29764.3	−0.999104
$962$	0	0
$963$	−42240.1	−1.41347
$964$	0	0
$965$	4976.43	0.166007
$966$	0	0
$967$	20496.1	0.681604	0.340802	−	0.940135i	$-0.389301\pi$
$967$	20496.1	0.681604	0.340802	+	0.940135i	$0.389301\pi$
$968$	0	0
$969$	−889.804	−0.0294991
$970$	0	0
$971$	16603.1	0.548732	0.274366	−	0.961625i	$-0.411532\pi$
$971$	16603.1	0.548732	0.274366	+	0.961625i	$0.411532\pi$
$972$	0	0
$973$	−1312.34	−0.0432391
$974$	0	0
$975$	2064.91	0.0678257
$976$	0	0
$977$	−41331.9	−1.35345	−0.676727	−	0.736234i	$-0.736602\pi$
$977$	−41331.9	−1.35345	−0.676727	+	0.736234i	$0.736602\pi$
$978$	0	0
$979$	4429.36	0.144600
$980$	0	0
$981$	−71.6452	−0.00233176
$982$	0	0
$983$	39233.1	1.27298	0.636492	−	0.771283i	$-0.280385\pi$
$983$	39233.1	1.27298	0.636492	+	0.771283i	$0.280385\pi$
$984$	0	0
$985$	17804.8	0.575949
$986$	0	0
$987$	−10618.2	−0.342433
$988$	0	0
$989$	−46230.8	−1.48641
$990$	0	0
$991$	45762.9	1.46691	0.733454	−	0.679739i	$-0.237907\pi$
$991$	45762.9	1.46691	0.733454	+	0.679739i	$0.237907\pi$
$992$	0	0
$993$	−12410.4	−0.396609
$994$	0	0
$995$	25461.3	0.811233
$996$	0	0
$997$	20826.9	0.661579	0.330790	−	0.943705i	$-0.392685\pi$
$997$	20826.9	0.661579	0.330790	+	0.943705i	$0.392685\pi$
$998$	0	0
$999$	6630.75	0.209998

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	416.4.a.g.1.3	yes	4
4.3	odd	2	inner	416.4.a.g.1.2	✓	4
8.3	odd	2		832.4.a.be.1.3		4
8.5	even	2		832.4.a.be.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
416.4.a.g.1.2	✓	4	4.3	odd	2	inner
416.4.a.g.1.3	yes	4	1.1	even	1	trivial
832.4.a.be.1.2		4	8.5	even	2
832.4.a.be.1.3		4	8.3	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:	\( N \)	\(=\)	\( 416 = 2^{5} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	416.a (trivial)

\(f(q)\)	\(=\)	\(q+1.83719 q^{3} +6.20824 q^{5} -19.4818 q^{7} -23.6247 q^{9} +45.5850 q^{11} -13.0000 q^{13} +11.4057 q^{15} -47.0412 q^{17} +10.2958 q^{19} -35.7918 q^{21} -155.854 q^{23} -86.4577 q^{25} -93.0073 q^{27} -69.1670 q^{29} +5.16685 q^{31} +83.7484 q^{33} -120.948 q^{35} -71.2928 q^{37} -23.8835 q^{39} -77.5835 q^{41} +296.629 q^{43} -146.668 q^{45} +296.666 q^{47} +36.5401 q^{49} -86.4237 q^{51} -601.913 q^{53} +283.003 q^{55} +18.9154 q^{57} -272.093 q^{59} -641.080 q^{61} +460.252 q^{63} -80.7072 q^{65} +1021.97 q^{67} -286.334 q^{69} -981.934 q^{71} -875.657 q^{73} -158.839 q^{75} -888.078 q^{77} -649.979 q^{79} +466.996 q^{81} -812.303 q^{83} -292.043 q^{85} -127.073 q^{87} +97.1670 q^{89} +253.263 q^{91} +9.49249 q^{93} +63.9191 q^{95} +1193.65 q^{97} -1076.93 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 14 q^{5} + 22 q^{9} - 52 q^{13} + 6 q^{17} - 182 q^{21} - 74 q^{25} - 432 q^{29} - 364 q^{33} - 790 q^{37} - 388 q^{41} - 1208 q^{45} - 514 q^{49} - 932 q^{53} - 468 q^{57} - 1244 q^{61} + 182 q^{65}+ \cdots - 1128 q^{97}+O(q^{100}) \) 4 * q - 14 * q^5 + 22 * q^9 - 52 * q^13 + 6 * q^17 - 182 * q^21 - 74 * q^25 - 432 * q^29 - 364 * q^33 - 790 * q^37 - 388 * q^41 - 1208 * q^45 - 514 * q^49 - 932 * q^53 - 468 * q^57 - 1244 * q^61 + 182 * q^65 - 1456 * q^69 + 536 * q^73 - 1300 * q^77 + 4 * q^81 - 1906 * q^85 + 544 * q^89 - 3224 * q^93 - 1128 * q^97

Introduction

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