LMFDB - Embedded newform 832.4.a.be.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [832,4,Mod(1,832)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("832.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(832, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 832 = 2^{6} \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	832.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,14,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$49.0895891248$
Analytic rank:	$0$
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:	no (minimal twist has level 416)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$4.91115$ of defining polynomial
Character	$\chi$	$=$	832.1

$q$-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	$-0.556569\pi$
$3$	−1.83719	−0.353567	−0.176784	+	0.984250i	$0.556569\pi$
$4$	0	0
$5$	−6.20824	−0.555282	−0.277641	−	0.960685i	$-0.589553\pi$
$5$	−6.20824	−0.555282	−0.277641	+	0.960685i	$0.589553\pi$
$6$	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	$-0.714797\pi$
$11$	−45.5850	−1.24949	−0.624746	+	0.780828i	$0.714797\pi$
$12$	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	$-0.519798\pi$
$19$	−10.2958	−0.124317	−0.0621586	+	0.998066i	$0.519798\pi$
$20$	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	$-0.428922\pi$
$29$	69.1670	0.442896	0.221448	+	0.975172i	$0.428922\pi$
$30$	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	$-0.449371\pi$
$37$	71.2928	0.316769	0.158385	+	0.987377i	$0.449371\pi$
$38$	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	$-0.676306\pi$
$43$	−296.629	−1.05199	−0.525993	+	0.850489i	$0.676306\pi$
$44$	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	$-0.215223\pi$
$53$	601.913	1.55998	0.779992	+	0.625790i	$0.215223\pi$
$54$	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	$-0.402947\pi$
$59$	272.093	0.600399	0.300200	+	0.953876i	$0.402947\pi$
$60$	0	0
$61$	641.080	1.34561	0.672803	−	0.739822i	$-0.265090\pi$
$61$	641.080	1.34561	0.672803	+	0.739822i	$0.265090\pi$
$62$	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	$-0.881712\pi$
$67$	−1021.97	−1.86349	−0.931743	+	0.363119i	$0.881712\pi$
$68$	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	$-0.319512\pi$
$83$	812.303	1.07424	0.537120	+	0.843506i	$0.319512\pi$
$84$	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	$-0.405084\pi$
$101$	596.412	0.587577	0.293788	+	0.955871i	$0.405084\pi$
$102$	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	$-0.799290\pi$
$107$	−1787.96	−1.61541	−0.807704	+	0.589588i	$0.799290\pi$
$108$	0	0
$109$	−3.03264	−0.00266490	−0.00133245	−	0.999999i	$-0.500424\pi$
$109$	−3.03264	−0.00266490	−0.00133245	+	0.999999i	$0.500424\pi$
$110$	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	$-0.508728\pi$
$131$	−82.2113	−0.0548308	−0.0274154	+	0.999624i	$0.508728\pi$
$132$	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	$-0.506543\pi$
$139$	−67.3623	−0.0411050	−0.0205525	+	0.999789i	$0.506543\pi$
$140$	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	$-0.407569\pi$
$149$	1041.49	0.572631	0.286316	+	0.958135i	$0.407569\pi$
$150$	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	$-0.478219\pi$
$157$	269.011	0.136748	0.0683739	+	0.997660i	$0.478219\pi$
$158$	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	$-0.337718\pi$
$163$	2031.20	0.976047	0.488023	+	0.872831i	$0.337718\pi$
$164$	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	$-0.620145\pi$
$173$	−1677.24	−0.737098	−0.368549	+	0.929608i	$0.620145\pi$
$174$	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	$-0.670497\pi$
$179$	−2444.60	−1.02077	−0.510386	+	0.859945i	$0.670497\pi$
$180$	0	0
$181$	−843.249	−0.346289	−0.173144	−	0.984896i	$-0.555393\pi$
$181$	−843.249	−0.346289	−0.173144	+	0.984896i	$0.555393\pi$
$182$	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	$-0.673550\pi$
$197$	−2867.94	−1.03722	−0.518609	+	0.855012i	$0.673550\pi$
$198$	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	$-0.490332\pi$
$211$	186.162	0.0607390	0.0303695	+	0.999539i	$0.490332\pi$
$212$	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	$-0.710447\pi$
$227$	−4199.99	−1.22803	−0.614016	+	0.789294i	$0.710447\pi$
$228$	0	0
$229$	−2068.59	−0.596928	−0.298464	−	0.954421i	$-0.596474\pi$
$229$	−2068.59	−0.596928	−0.298464	+	0.954421i	$0.596474\pi$
$230$	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	$-0.275536\pi$
$251$	5154.98	1.29633	0.648166	+	0.761499i	$0.275536\pi$
$252$	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	$-0.549223\pi$
$269$	−1359.08	−0.308048	−0.154024	+	0.988067i	$0.549223\pi$
$270$	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	$-0.773399\pi$
$277$	−6981.03	−1.51426	−0.757129	+	0.653266i	$0.773399\pi$
$278$	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	$-0.654411\pi$
$283$	−4439.85	−0.932585	−0.466292	+	0.884631i	$0.654411\pi$
$284$	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	$-0.615835\pi$
$293$	−3570.20	−0.711854	−0.355927	+	0.934514i	$0.615835\pi$
$294$	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	$-0.324481\pi$
$307$	5636.05	1.04777	0.523887	+	0.851788i	$0.324481\pi$
$308$	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	$-0.392620\pi$
$317$	3736.13	0.661963	0.330981	+	0.943637i	$0.392620\pi$
$318$	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	$-0.310467\pi$
$331$	6755.12	1.12174	0.560868	+	0.827905i	$0.310467\pi$
$332$	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	$-0.314869\pi$
$347$	7102.09	1.09873	0.549366	+	0.835582i	$0.314869\pi$
$348$	0	0
$349$	5599.78	0.858880	0.429440	−	0.903095i	$-0.358711\pi$
$349$	5599.78	0.858880	0.429440	+	0.903095i	$0.358711\pi$
$350$	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	$-0.644137\pi$
$373$	−6303.38	−0.875004	−0.437502	+	0.899217i	$0.644137\pi$
$374$	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	$-0.586937\pi$
$379$	−3980.44	−0.539476	−0.269738	+	0.962934i	$0.586937\pi$
$380$	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	$-0.746774\pi$
$389$	−10739.7	−1.39981	−0.699905	+	0.714236i	$0.746774\pi$
$390$	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	$-0.487710\pi$
$397$	610.651	0.0771982	0.0385991	+	0.999255i	$0.487710\pi$
$398$	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	$-0.406407\pi$
$419$	4971.26	0.579623	0.289811	+	0.957084i	$0.406407\pi$
$420$	0	0
$421$	1307.30	0.151340	0.0756698	−	0.997133i	$-0.475891\pi$
$421$	1307.30	0.151340	0.0756698	+	0.997133i	$0.475891\pi$
$422$	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	$-0.587628\pi$
$443$	−5069.07	−0.543654	−0.271827	+	0.962346i	$0.587628\pi$
$444$	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	$-0.274591\pi$
$461$	12875.9	1.30085	0.650424	+	0.759571i	$0.274591\pi$
$462$	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	$-0.350540\pi$
$467$	9132.78	0.904957	0.452478	+	0.891775i	$0.350540\pi$
$468$	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	$-0.331942\pi$
$491$	10962.1	1.00756	0.503781	+	0.863832i	$0.331942\pi$
$492$	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	$-0.658227\pi$
$499$	−10631.0	−0.953730	−0.476865	+	0.878977i	$0.658227\pi$
$500$	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	$-0.388283\pi$
$509$	7896.27	0.687615	0.343808	+	0.939040i	$0.388283\pi$
$510$	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	$-0.465248\pi$
$523$	2606.45	0.217920	0.108960	+	0.994046i	$0.465248\pi$
$524$	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	$-0.292179\pi$
$541$	15288.3	1.21497	0.607483	+	0.794333i	$0.292179\pi$
$542$	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	$-0.251169\pi$
$547$	18025.8	1.40901	0.704506	+	0.709698i	$0.251169\pi$
$548$	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	$-0.649015\pi$
$557$	−11863.5	−0.902463	−0.451231	+	0.892407i	$0.649015\pi$
$558$	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	$-0.433412\pi$
$563$	5548.38	0.415340	0.207670	+	0.978199i	$0.433412\pi$
$564$	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	$-0.723581\pi$
$571$	−17630.0	−1.29210	−0.646052	+	0.763294i	$0.723581\pi$
$572$	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	$-0.250351\pi$
$587$	20090.6	1.41265	0.706327	+	0.707886i	$0.250351\pi$
$588$	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	$-0.392599\pi$
$613$	10048.6	0.662087	0.331043	+	0.943616i	$0.392599\pi$
$614$	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	$-0.451078\pi$
$619$	4715.30	0.306177	0.153089	+	0.988212i	$0.451078\pi$
$620$	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	$-0.581370\pi$
$643$	−8245.60	−0.505715	−0.252858	+	0.967504i	$0.581370\pi$
$644$	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	$-0.357723\pi$
$653$	14425.3	0.864479	0.432240	+	0.901759i	$0.357723\pi$
$654$	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	$-0.924010\pi$
$659$	−32874.8	−1.94328	−0.971639	+	0.236469i	$0.924010\pi$
$660$	0	0
$661$	16168.3	0.951400	0.475700	−	0.879607i	$-0.342195\pi$
$661$	16168.3	0.951400	0.475700	+	0.879607i	$0.342195\pi$
$662$	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	$-0.0490905\pi$
$677$	34811.9	1.97626	0.988131	+	0.153612i	$0.0490905\pi$
$678$	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	$-0.450338\pi$
$683$	5547.22	0.310774	0.155387	+	0.987854i	$0.450338\pi$
$684$	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	$-0.685615\pi$
$691$	−20003.7	−1.10127	−0.550636	+	0.834746i	$0.685615\pi$
$692$	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	$-0.471753\pi$
$701$	3289.73	0.177249	0.0886244	+	0.996065i	$0.471753\pi$
$702$	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	$-0.549880\pi$
$709$	−5892.45	−0.312123	−0.156062	+	0.987747i	$0.549880\pi$
$710$	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	$-0.157681\pi$
$733$	34919.4	1.75959	0.879793	+	0.475357i	$0.157681\pi$
$734$	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	$-0.344043\pi$
$739$	18907.6	0.941171	0.470586	+	0.882354i	$0.344043\pi$
$740$	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	$-0.587235\pi$
$757$	−11273.7	−0.541279	−0.270640	+	0.962681i	$0.587235\pi$
$758$	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	$-0.254360\pi$
$773$	29974.6	1.39471	0.697356	+	0.716725i	$0.254360\pi$
$774$	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	$-0.196392\pi$
$787$	36015.1	1.63126	0.815628	+	0.578577i	$0.196392\pi$
$788$	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	$-0.599878\pi$
$797$	−13889.5	−0.617302	−0.308651	+	0.951175i	$0.599878\pi$
$798$	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	$-0.387899\pi$
$811$	15933.4	0.689884	0.344942	+	0.938624i	$0.387899\pi$
$812$	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	$-0.508808\pi$
$821$	−1301.72	−0.0553355	−0.0276678	+	0.999617i	$0.508808\pi$
$822$	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	$-0.531648\pi$
$827$	−4721.35	−0.198522	−0.0992608	+	0.995061i	$0.531648\pi$
$828$	0	0
$829$	19341.0	0.810300	0.405150	−	0.914250i	$-0.367219\pi$
$829$	19341.0	0.810300	0.405150	+	0.914250i	$0.367219\pi$
$830$	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	$-0.326236\pi$
$853$	25868.7	1.03837	0.519184	+	0.854663i	$0.326236\pi$
$854$	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	$-0.493342\pi$
$859$	1053.08	0.0418286	0.0209143	+	0.999781i	$0.493342\pi$
$860$	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	$-0.192697\pi$
$877$	42712.4	1.64458	0.822289	+	0.569070i	$0.192697\pi$
$878$	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	$-0.0532451\pi$
$883$	51744.8	1.97208	0.986042	+	0.166495i	$0.0532451\pi$
$884$	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	$-0.714746\pi$
$907$	−34123.7	−1.24924	−0.624620	+	0.780929i	$0.714746\pi$
$908$	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	$-0.451157\pi$
$941$	8823.88	0.305686	0.152843	+	0.988251i	$0.451157\pi$
$942$	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	$-0.791677\pi$
$947$	−46241.6	−1.58675	−0.793373	+	0.608736i	$0.791677\pi$
$948$	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	$-0.588468\pi$
$971$	−16603.1	−0.548732	−0.274366	+	0.961625i	$0.588468\pi$
$972$	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	$-0.607315\pi$
$997$	−20826.9	−0.661579	−0.330790	+	0.943705i	$0.607315\pi$
$998$	0	0
$999$	6630.75	0.209998

Twists

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

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

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 832 = 2^{6} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	832.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

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more