LMFDB - Embedded newform 4368.2.a.bl.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(1,4368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4368.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4368.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:	$34.8786556029$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 1092)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.61803$ of defining polynomial
Character	$\chi$	$=$	4368.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.23607 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.23607 q^{11} +1.00000 q^{13} +3.23607 q^{15} +6.47214 q^{17} -2.47214 q^{19} -1.00000 q^{21} +4.47214 q^{23} +5.47214 q^{25} +1.00000 q^{27} -0.472136 q^{29} +1.23607 q^{33} -3.23607 q^{35} -4.47214 q^{37} +1.00000 q^{39} +0.763932 q^{41} -4.00000 q^{43} +3.23607 q^{45} +5.70820 q^{47} +1.00000 q^{49} +6.47214 q^{51} +10.0000 q^{53} +4.00000 q^{55} -2.47214 q^{57} -4.76393 q^{59} -2.94427 q^{61} -1.00000 q^{63} +3.23607 q^{65} -6.47214 q^{67} +4.47214 q^{69} -7.70820 q^{71} -4.47214 q^{73} +5.47214 q^{75} -1.23607 q^{77} +1.52786 q^{79} +1.00000 q^{81} -7.23607 q^{83} +20.9443 q^{85} -0.472136 q^{87} +12.1803 q^{89} -1.00000 q^{91} -8.00000 q^{95} -10.9443 q^{97} +1.23607 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} - 2 q^{11} + 2 q^{13} + 2 q^{15} + 4 q^{17} + 4 q^{19} - 2 q^{21} + 2 q^{25} + 2 q^{27} + 8 q^{29} - 2 q^{33} - 2 q^{35} + 2 q^{39} + 6 q^{41} - 8 q^{43} + 2 q^{45}+ \cdots - 2 q^{99}+O(q^{100}) $ 2 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9 - 2 * q^11 + 2 * q^13 + 2 * q^15 + 4 * q^17 + 4 * q^19 - 2 * q^21 + 2 * q^25 + 2 * q^27 + 8 * q^29 - 2 * q^33 - 2 * q^35 + 2 * q^39 + 6 * q^41 - 8 * q^43 + 2 * q^45 - 2 * q^47 + 2 * q^49 + 4 * q^51 + 20 * q^53 + 8 * q^55 + 4 * q^57 - 14 * q^59 + 12 * q^61 - 2 * q^63 + 2 * q^65 - 4 * q^67 - 2 * q^71 + 2 * q^75 + 2 * q^77 + 12 * q^79 + 2 * q^81 - 10 * q^83 + 24 * q^85 + 8 * q^87 + 2 * q^89 - 2 * q^91 - 16 * q^95 - 4 * q^97 - 2 * q^99

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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	3.23607	1.44721	0.723607	−	0.690212i	$-0.242483\pi$
$5$	3.23607	1.44721	0.723607	+	0.690212i	$0.242483\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	1.23607	0.372689	0.186344	−	0.982485i	$-0.440336\pi$
$11$	1.23607	0.372689	0.186344	+	0.982485i	$0.440336\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	0	0
$15$	3.23607	0.835549
$16$	0	0
$17$	6.47214	1.56972	0.784862	−	0.619671i	$-0.212734\pi$
$17$	6.47214	1.56972	0.784862	+	0.619671i	$0.212734\pi$
$18$	0	0
$19$	−2.47214	−0.567147	−0.283573	−	0.958951i	$-0.591520\pi$
$19$	−2.47214	−0.567147	−0.283573	+	0.958951i	$0.591520\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	4.47214	0.932505	0.466252	−	0.884652i	$-0.345604\pi$
$23$	4.47214	0.932505	0.466252	+	0.884652i	$0.345604\pi$
$24$	0	0
$25$	5.47214	1.09443
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	−0.472136	−0.0876734	−0.0438367	−	0.999039i	$-0.513958\pi$
$29$	−0.472136	−0.0876734	−0.0438367	+	0.999039i	$0.513958\pi$
$30$	0	0
$31$	0	0		−	1.00000i	$-0.5\pi$
$31$	0	0			1.00000i	$0.5\pi$
$32$	0	0
$33$	1.23607	0.215172
$34$	0	0
$35$	−3.23607	−0.546995
$36$	0	0
$37$	−4.47214	−0.735215	−0.367607	−	0.929981i	$-0.619823\pi$
$37$	−4.47214	−0.735215	−0.367607	+	0.929981i	$0.619823\pi$
$38$	0	0
$39$	1.00000	0.160128
$40$	0	0
$41$	0.763932	0.119306	0.0596531	−	0.998219i	$-0.481001\pi$
$41$	0.763932	0.119306	0.0596531	+	0.998219i	$0.481001\pi$
$42$	0	0
$43$	−4.00000	−0.609994	−0.304997	−	0.952353i	$-0.598656\pi$
$43$	−4.00000	−0.609994	−0.304997	+	0.952353i	$0.598656\pi$
$44$	0	0
$45$	3.23607	0.482405
$46$	0	0
$47$	5.70820	0.832627	0.416314	−	0.909221i	$-0.363322\pi$
$47$	5.70820	0.832627	0.416314	+	0.909221i	$0.363322\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	6.47214	0.906280
$52$	0	0
$53$	10.0000	1.37361	0.686803	−	0.726844i	$-0.259014\pi$
$53$	10.0000	1.37361	0.686803	+	0.726844i	$0.259014\pi$
$54$	0	0
$55$	4.00000	0.539360
$56$	0	0
$57$	−2.47214	−0.327442
$58$	0	0
$59$	−4.76393	−0.620211	−0.310106	−	0.950702i	$-0.600364\pi$
$59$	−4.76393	−0.620211	−0.310106	+	0.950702i	$0.600364\pi$
$60$	0	0
$61$	−2.94427	−0.376975	−0.188488	−	0.982076i	$-0.560359\pi$
$61$	−2.94427	−0.376975	−0.188488	+	0.982076i	$0.560359\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	3.23607	0.401385
$66$	0	0
$67$	−6.47214	−0.790697	−0.395349	−	0.918531i	$-0.629376\pi$
$67$	−6.47214	−0.790697	−0.395349	+	0.918531i	$0.629376\pi$
$68$	0	0
$69$	4.47214	0.538382
$70$	0	0
$71$	−7.70820	−0.914796	−0.457398	−	0.889262i	$-0.651218\pi$
$71$	−7.70820	−0.914796	−0.457398	+	0.889262i	$0.651218\pi$
$72$	0	0
$73$	−4.47214	−0.523424	−0.261712	−	0.965146i	$-0.584287\pi$
$73$	−4.47214	−0.523424	−0.261712	+	0.965146i	$0.584287\pi$
$74$	0	0
$75$	5.47214	0.631868
$76$	0	0
$77$	−1.23607	−0.140863
$78$	0	0
$79$	1.52786	0.171898	0.0859491	−	0.996300i	$-0.472608\pi$
$79$	1.52786	0.171898	0.0859491	+	0.996300i	$0.472608\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−7.23607	−0.794262	−0.397131	−	0.917762i	$-0.629994\pi$
$83$	−7.23607	−0.794262	−0.397131	+	0.917762i	$0.629994\pi$
$84$	0	0
$85$	20.9443	2.27173
$86$	0	0
$87$	−0.472136	−0.0506183
$88$	0	0
$89$	12.1803	1.29111	0.645557	−	0.763712i	$-0.276625\pi$
$89$	12.1803	1.29111	0.645557	+	0.763712i	$0.276625\pi$
$90$	0	0
$91$	−1.00000	−0.104828
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−8.00000	−0.820783
$96$	0	0
$97$	−10.9443	−1.11122	−0.555611	−	0.831442i	$-0.687516\pi$
$97$	−10.9443	−1.11122	−0.555611	+	0.831442i	$0.687516\pi$
$98$	0	0
$99$	1.23607	0.124230
$100$	0	0
$101$	8.94427	0.889988	0.444994	−	0.895533i	$-0.353206\pi$
$101$	8.94427	0.889988	0.444994	+	0.895533i	$0.353206\pi$
$102$	0	0
$103$	−0.944272	−0.0930419	−0.0465209	−	0.998917i	$-0.514813\pi$
$103$	−0.944272	−0.0930419	−0.0465209	+	0.998917i	$0.514813\pi$
$104$	0	0
$105$	−3.23607	−0.315808
$106$	0	0
$107$	12.4721	1.20573	0.602863	−	0.797844i	$-0.294026\pi$
$107$	12.4721	1.20573	0.602863	+	0.797844i	$0.294026\pi$
$108$	0	0
$109$	12.4721	1.19461	0.597307	−	0.802013i	$-0.296237\pi$
$109$	12.4721	1.19461	0.597307	+	0.802013i	$0.296237\pi$
$110$	0	0
$111$	−4.47214	−0.424476
$112$	0	0
$113$	11.5279	1.08445	0.542225	−	0.840233i	$-0.317582\pi$
$113$	11.5279	1.08445	0.542225	+	0.840233i	$0.317582\pi$
$114$	0	0
$115$	14.4721	1.34953
$116$	0	0
$117$	1.00000	0.0924500
$118$	0	0
$119$	−6.47214	−0.593300
$120$	0	0
$121$	−9.47214	−0.861103
$122$	0	0
$123$	0.763932	0.0688814
$124$	0	0
$125$	1.52786	0.136656
$126$	0	0
$127$	−6.47214	−0.574309	−0.287155	−	0.957884i	$-0.592709\pi$
$127$	−6.47214	−0.574309	−0.287155	+	0.957884i	$0.592709\pi$
$128$	0	0
$129$	−4.00000	−0.352180
$130$	0	0
$131$	10.4721	0.914955	0.457477	−	0.889221i	$-0.348753\pi$
$131$	10.4721	0.914955	0.457477	+	0.889221i	$0.348753\pi$
$132$	0	0
$133$	2.47214	0.214361
$134$	0	0
$135$	3.23607	0.278516
$136$	0	0
$137$	10.7639	0.919625	0.459812	−	0.888016i	$-0.347917\pi$
$137$	10.7639	0.919625	0.459812	+	0.888016i	$0.347917\pi$
$138$	0	0
$139$	17.8885	1.51729	0.758643	−	0.651506i	$-0.225863\pi$
$139$	17.8885	1.51729	0.758643	+	0.651506i	$0.225863\pi$
$140$	0	0
$141$	5.70820	0.480717
$142$	0	0
$143$	1.23607	0.103365
$144$	0	0
$145$	−1.52786	−0.126882
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−17.2361	−1.41203	−0.706017	−	0.708195i	$-0.749510\pi$
$149$	−17.2361	−1.41203	−0.706017	+	0.708195i	$0.749510\pi$
$150$	0	0
$151$	20.9443	1.70442	0.852210	−	0.523199i	$-0.175262\pi$
$151$	20.9443	1.70442	0.852210	+	0.523199i	$0.175262\pi$
$152$	0	0
$153$	6.47214	0.523241
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−18.0000	−1.43656	−0.718278	−	0.695756i	$-0.755069\pi$
$157$	−18.0000	−1.43656	−0.718278	+	0.695756i	$0.755069\pi$
$158$	0	0
$159$	10.0000	0.793052
$160$	0	0
$161$	−4.47214	−0.352454
$162$	0	0
$163$	−6.47214	−0.506937	−0.253468	−	0.967344i	$-0.581571\pi$
$163$	−6.47214	−0.506937	−0.253468	+	0.967344i	$0.581571\pi$
$164$	0	0
$165$	4.00000	0.311400
$166$	0	0
$167$	−4.18034	−0.323484	−0.161742	−	0.986833i	$-0.551711\pi$
$167$	−4.18034	−0.323484	−0.161742	+	0.986833i	$0.551711\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	0	0
$171$	−2.47214	−0.189049
$172$	0	0
$173$	12.0000	0.912343	0.456172	−	0.889892i	$-0.349220\pi$
$173$	12.0000	0.912343	0.456172	+	0.889892i	$0.349220\pi$
$174$	0	0
$175$	−5.47214	−0.413655
$176$	0	0
$177$	−4.76393	−0.358079
$178$	0	0
$179$	−10.9443	−0.818013	−0.409007	−	0.912531i	$-0.634125\pi$
$179$	−10.9443	−0.818013	−0.409007	+	0.912531i	$0.634125\pi$
$180$	0	0
$181$	−15.8885	−1.18099	−0.590493	−	0.807043i	$-0.701067\pi$
$181$	−15.8885	−1.18099	−0.590493	+	0.807043i	$0.701067\pi$
$182$	0	0
$183$	−2.94427	−0.217647
$184$	0	0
$185$	−14.4721	−1.06401
$186$	0	0
$187$	8.00000	0.585018
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−10.9443	−0.791900	−0.395950	−	0.918272i	$-0.629585\pi$
$191$	−10.9443	−0.791900	−0.395950	+	0.918272i	$0.629585\pi$
$192$	0	0
$193$	−26.3607	−1.89748	−0.948742	−	0.316053i	$-0.897642\pi$
$193$	−26.3607	−1.89748	−0.948742	+	0.316053i	$0.897642\pi$
$194$	0	0
$195$	3.23607	0.231740
$196$	0	0
$197$	5.23607	0.373054	0.186527	−	0.982450i	$-0.440277\pi$
$197$	5.23607	0.373054	0.186527	+	0.982450i	$0.440277\pi$
$198$	0	0
$199$	16.0000	1.13421	0.567105	−	0.823646i	$-0.308063\pi$
$199$	16.0000	1.13421	0.567105	+	0.823646i	$0.308063\pi$
$200$	0	0
$201$	−6.47214	−0.456509
$202$	0	0
$203$	0.472136	0.0331374
$204$	0	0
$205$	2.47214	0.172661
$206$	0	0
$207$	4.47214	0.310835
$208$	0	0
$209$	−3.05573	−0.211369
$210$	0	0
$211$	24.3607	1.67706	0.838529	−	0.544857i	$-0.183416\pi$
$211$	24.3607	1.67706	0.838529	+	0.544857i	$0.183416\pi$
$212$	0	0
$213$	−7.70820	−0.528157
$214$	0	0
$215$	−12.9443	−0.882792
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−4.47214	−0.302199
$220$	0	0
$221$	6.47214	0.435363
$222$	0	0
$223$	1.52786	0.102313	0.0511567	−	0.998691i	$-0.483709\pi$
$223$	1.52786	0.102313	0.0511567	+	0.998691i	$0.483709\pi$
$224$	0	0
$225$	5.47214	0.364809
$226$	0	0
$227$	11.2361	0.745764	0.372882	−	0.927879i	$-0.378370\pi$
$227$	11.2361	0.745764	0.372882	+	0.927879i	$0.378370\pi$
$228$	0	0
$229$	1.41641	0.0935989	0.0467994	−	0.998904i	$-0.485098\pi$
$229$	1.41641	0.0935989	0.0467994	+	0.998904i	$0.485098\pi$
$230$	0	0
$231$	−1.23607	−0.0813273
$232$	0	0
$233$	−21.4164	−1.40304	−0.701518	−	0.712652i	$-0.747494\pi$
$233$	−21.4164	−1.40304	−0.701518	+	0.712652i	$0.747494\pi$
$234$	0	0
$235$	18.4721	1.20499
$236$	0	0
$237$	1.52786	0.0992454
$238$	0	0
$239$	19.1246	1.23707	0.618534	−	0.785758i	$-0.287727\pi$
$239$	19.1246	1.23707	0.618534	+	0.785758i	$0.287727\pi$
$240$	0	0
$241$	−7.52786	−0.484912	−0.242456	−	0.970162i	$-0.577953\pi$
$241$	−7.52786	−0.484912	−0.242456	+	0.970162i	$0.577953\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	3.23607	0.206745
$246$	0	0
$247$	−2.47214	−0.157298
$248$	0	0
$249$	−7.23607	−0.458567
$250$	0	0
$251$	−14.4721	−0.913473	−0.456737	−	0.889602i	$-0.650982\pi$
$251$	−14.4721	−0.913473	−0.456737	+	0.889602i	$0.650982\pi$
$252$	0	0
$253$	5.52786	0.347534
$254$	0	0
$255$	20.9443	1.31158
$256$	0	0
$257$	−26.8328	−1.67379	−0.836893	−	0.547367i	$-0.815630\pi$
$257$	−26.8328	−1.67379	−0.836893	+	0.547367i	$0.815630\pi$
$258$	0	0
$259$	4.47214	0.277885
$260$	0	0
$261$	−0.472136	−0.0292245
$262$	0	0
$263$	−8.47214	−0.522414	−0.261207	−	0.965283i	$-0.584121\pi$
$263$	−8.47214	−0.522414	−0.261207	+	0.965283i	$0.584121\pi$
$264$	0	0
$265$	32.3607	1.98790
$266$	0	0
$267$	12.1803	0.745425
$268$	0	0
$269$	−12.9443	−0.789226	−0.394613	−	0.918847i	$-0.629121\pi$
$269$	−12.9443	−0.789226	−0.394613	+	0.918847i	$0.629121\pi$
$270$	0	0
$271$	−3.41641	−0.207532	−0.103766	−	0.994602i	$-0.533089\pi$
$271$	−3.41641	−0.207532	−0.103766	+	0.994602i	$0.533089\pi$
$272$	0	0
$273$	−1.00000	−0.0605228
$274$	0	0
$275$	6.76393	0.407880
$276$	0	0
$277$	19.8885	1.19499	0.597493	−	0.801874i	$-0.296163\pi$
$277$	19.8885	1.19499	0.597493	+	0.801874i	$0.296163\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	12.6525	0.754783	0.377392	−	0.926054i	$-0.376821\pi$
$281$	12.6525	0.754783	0.377392	+	0.926054i	$0.376821\pi$
$282$	0	0
$283$	−4.00000	−0.237775	−0.118888	−	0.992908i	$-0.537933\pi$
$283$	−4.00000	−0.237775	−0.118888	+	0.992908i	$0.537933\pi$
$284$	0	0
$285$	−8.00000	−0.473879
$286$	0	0
$287$	−0.763932	−0.0450935
$288$	0	0
$289$	24.8885	1.46403
$290$	0	0
$291$	−10.9443	−0.641565
$292$	0	0
$293$	0.763932	0.0446294	0.0223147	−	0.999751i	$-0.492896\pi$
$293$	0.763932	0.0446294	0.0223147	+	0.999751i	$0.492896\pi$
$294$	0	0
$295$	−15.4164	−0.897578
$296$	0	0
$297$	1.23607	0.0717239
$298$	0	0
$299$	4.47214	0.258630
$300$	0	0
$301$	4.00000	0.230556
$302$	0	0
$303$	8.94427	0.513835
$304$	0	0
$305$	−9.52786	−0.545564
$306$	0	0
$307$	−29.8885	−1.70583	−0.852915	−	0.522050i	$-0.825167\pi$
$307$	−29.8885	−1.70583	−0.852915	+	0.522050i	$0.825167\pi$
$308$	0	0
$309$	−0.944272	−0.0537178
$310$	0	0
$311$	10.4721	0.593820	0.296910	−	0.954905i	$-0.404044\pi$
$311$	10.4721	0.593820	0.296910	+	0.954905i	$0.404044\pi$
$312$	0	0
$313$	−32.8328	−1.85582	−0.927910	−	0.372804i	$-0.878396\pi$
$313$	−32.8328	−1.85582	−0.927910	+	0.372804i	$0.878396\pi$
$314$	0	0
$315$	−3.23607	−0.182332
$316$	0	0
$317$	−5.81966	−0.326865	−0.163432	−	0.986555i	$-0.552257\pi$
$317$	−5.81966	−0.326865	−0.163432	+	0.986555i	$0.552257\pi$
$318$	0	0
$319$	−0.583592	−0.0326749
$320$	0	0
$321$	12.4721	0.696127
$322$	0	0
$323$	−16.0000	−0.890264
$324$	0	0
$325$	5.47214	0.303539
$326$	0	0
$327$	12.4721	0.689711
$328$	0	0
$329$	−5.70820	−0.314703
$330$	0	0
$331$	−18.8328	−1.03515	−0.517573	−	0.855639i	$-0.673164\pi$
$331$	−18.8328	−1.03515	−0.517573	+	0.855639i	$0.673164\pi$
$332$	0	0
$333$	−4.47214	−0.245072
$334$	0	0
$335$	−20.9443	−1.14431
$336$	0	0
$337$	22.3607	1.21806	0.609032	−	0.793146i	$-0.291558\pi$
$337$	22.3607	1.21806	0.609032	+	0.793146i	$0.291558\pi$
$338$	0	0
$339$	11.5279	0.626108
$340$	0	0
$341$	0	0
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	14.4721	0.779154
$346$	0	0
$347$	−22.0000	−1.18102	−0.590511	−	0.807030i	$-0.701074\pi$
$347$	−22.0000	−1.18102	−0.590511	+	0.807030i	$0.701074\pi$
$348$	0	0
$349$	−25.4164	−1.36051	−0.680255	−	0.732976i	$-0.738131\pi$
$349$	−25.4164	−1.36051	−0.680255	+	0.732976i	$0.738131\pi$
$350$	0	0
$351$	1.00000	0.0533761
$352$	0	0
$353$	14.6525	0.779873	0.389936	−	0.920842i	$-0.372497\pi$
$353$	14.6525	0.779873	0.389936	+	0.920842i	$0.372497\pi$
$354$	0	0
$355$	−24.9443	−1.32390
$356$	0	0
$357$	−6.47214	−0.342542
$358$	0	0
$359$	18.1803	0.959522	0.479761	−	0.877399i	$-0.340723\pi$
$359$	18.1803	0.959522	0.479761	+	0.877399i	$0.340723\pi$
$360$	0	0
$361$	−12.8885	−0.678344
$362$	0	0
$363$	−9.47214	−0.497158
$364$	0	0
$365$	−14.4721	−0.757506
$366$	0	0
$367$	7.05573	0.368306	0.184153	−	0.982898i	$-0.441046\pi$
$367$	7.05573	0.368306	0.184153	+	0.982898i	$0.441046\pi$
$368$	0	0
$369$	0.763932	0.0397687
$370$	0	0
$371$	−10.0000	−0.519174
$372$	0	0
$373$	27.5279	1.42534	0.712669	−	0.701500i	$-0.247486\pi$
$373$	27.5279	1.42534	0.712669	+	0.701500i	$0.247486\pi$
$374$	0	0
$375$	1.52786	0.0788986
$376$	0	0
$377$	−0.472136	−0.0243162
$378$	0	0
$379$	21.8885	1.12434	0.562169	−	0.827022i	$-0.309967\pi$
$379$	21.8885	1.12434	0.562169	+	0.827022i	$0.309967\pi$
$380$	0	0
$381$	−6.47214	−0.331578
$382$	0	0
$383$	24.1803	1.23556	0.617779	−	0.786352i	$-0.288033\pi$
$383$	24.1803	1.23556	0.617779	+	0.786352i	$0.288033\pi$
$384$	0	0
$385$	−4.00000	−0.203859
$386$	0	0
$387$	−4.00000	−0.203331
$388$	0	0
$389$	14.0000	0.709828	0.354914	−	0.934899i	$-0.384510\pi$
$389$	14.0000	0.709828	0.354914	+	0.934899i	$0.384510\pi$
$390$	0	0
$391$	28.9443	1.46377
$392$	0	0
$393$	10.4721	0.528249
$394$	0	0
$395$	4.94427	0.248773
$396$	0	0
$397$	0.111456	0.00559383	0.00279691	−	0.999996i	$-0.499110\pi$
$397$	0.111456	0.00559383	0.00279691	+	0.999996i	$0.499110\pi$
$398$	0	0
$399$	2.47214	0.123762
$400$	0	0
$401$	−14.7639	−0.737276	−0.368638	−	0.929573i	$-0.620176\pi$
$401$	−14.7639	−0.737276	−0.368638	+	0.929573i	$0.620176\pi$
$402$	0	0
$403$	0	0
$404$	0	0
$405$	3.23607	0.160802
$406$	0	0
$407$	−5.52786	−0.274006
$408$	0	0
$409$	−34.9443	−1.72788	−0.863942	−	0.503592i	$-0.832011\pi$
$409$	−34.9443	−1.72788	−0.863942	+	0.503592i	$0.832011\pi$
$410$	0	0
$411$	10.7639	0.530946
$412$	0	0
$413$	4.76393	0.234418
$414$	0	0
$415$	−23.4164	−1.14947
$416$	0	0
$417$	17.8885	0.876006
$418$	0	0
$419$	−3.41641	−0.166902	−0.0834512	−	0.996512i	$-0.526594\pi$
$419$	−3.41641	−0.166902	−0.0834512	+	0.996512i	$0.526594\pi$
$420$	0	0
$421$	35.8885	1.74910	0.874550	−	0.484935i	$-0.161157\pi$
$421$	35.8885	1.74910	0.874550	+	0.484935i	$0.161157\pi$
$422$	0	0
$423$	5.70820	0.277542
$424$	0	0
$425$	35.4164	1.71795
$426$	0	0
$427$	2.94427	0.142483
$428$	0	0
$429$	1.23607	0.0596779
$430$	0	0
$431$	11.7082	0.563964	0.281982	−	0.959420i	$-0.409008\pi$
$431$	11.7082	0.563964	0.281982	+	0.959420i	$0.409008\pi$
$432$	0	0
$433$	22.0000	1.05725	0.528626	−	0.848855i	$-0.322707\pi$
$433$	22.0000	1.05725	0.528626	+	0.848855i	$0.322707\pi$
$434$	0	0
$435$	−1.52786	−0.0732555
$436$	0	0
$437$	−11.0557	−0.528867
$438$	0	0
$439$	32.9443	1.57234	0.786172	−	0.618008i	$-0.212060\pi$
$439$	32.9443	1.57234	0.786172	+	0.618008i	$0.212060\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−9.41641	−0.447387	−0.223694	−	0.974660i	$-0.571811\pi$
$443$	−9.41641	−0.447387	−0.223694	+	0.974660i	$0.571811\pi$
$444$	0	0
$445$	39.4164	1.86852
$446$	0	0
$447$	−17.2361	−0.815238
$448$	0	0
$449$	7.12461	0.336231	0.168116	−	0.985767i	$-0.446232\pi$
$449$	7.12461	0.336231	0.168116	+	0.985767i	$0.446232\pi$
$450$	0	0
$451$	0.944272	0.0444640
$452$	0	0
$453$	20.9443	0.984048
$454$	0	0
$455$	−3.23607	−0.151709
$456$	0	0
$457$	−12.4721	−0.583422	−0.291711	−	0.956507i	$-0.594225\pi$
$457$	−12.4721	−0.583422	−0.291711	+	0.956507i	$0.594225\pi$
$458$	0	0
$459$	6.47214	0.302093
$460$	0	0
$461$	−29.7082	−1.38365	−0.691825	−	0.722066i	$-0.743193\pi$
$461$	−29.7082	−1.38365	−0.691825	+	0.722066i	$0.743193\pi$
$462$	0	0
$463$	−15.4164	−0.716461	−0.358231	−	0.933633i	$-0.616620\pi$
$463$	−15.4164	−0.716461	−0.358231	+	0.933633i	$0.616620\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6.47214	−0.299495	−0.149747	−	0.988724i	$-0.547846\pi$
$467$	−6.47214	−0.299495	−0.149747	+	0.988724i	$0.547846\pi$
$468$	0	0
$469$	6.47214	0.298855
$470$	0	0
$471$	−18.0000	−0.829396
$472$	0	0
$473$	−4.94427	−0.227338
$474$	0	0
$475$	−13.5279	−0.620701
$476$	0	0
$477$	10.0000	0.457869
$478$	0	0
$479$	30.0689	1.37388	0.686941	−	0.726713i	$-0.258953\pi$
$479$	30.0689	1.37388	0.686941	+	0.726713i	$0.258953\pi$
$480$	0	0
$481$	−4.47214	−0.203912
$482$	0	0
$483$	−4.47214	−0.203489
$484$	0	0
$485$	−35.4164	−1.60818
$486$	0	0
$487$	22.8328	1.03465	0.517327	−	0.855788i	$-0.326927\pi$
$487$	22.8328	1.03465	0.517327	+	0.855788i	$0.326927\pi$
$488$	0	0
$489$	−6.47214	−0.292680
$490$	0	0
$491$	−7.52786	−0.339728	−0.169864	−	0.985468i	$-0.554333\pi$
$491$	−7.52786	−0.339728	−0.169864	+	0.985468i	$0.554333\pi$
$492$	0	0
$493$	−3.05573	−0.137623
$494$	0	0
$495$	4.00000	0.179787
$496$	0	0
$497$	7.70820	0.345760
$498$	0	0
$499$	13.8885	0.621737	0.310868	−	0.950453i	$-0.399380\pi$
$499$	13.8885	0.621737	0.310868	+	0.950453i	$0.399380\pi$
$500$	0	0
$501$	−4.18034	−0.186764
$502$	0	0
$503$	−1.52786	−0.0681241	−0.0340620	−	0.999420i	$-0.510844\pi$
$503$	−1.52786	−0.0681241	−0.0340620	+	0.999420i	$0.510844\pi$
$504$	0	0
$505$	28.9443	1.28800
$506$	0	0
$507$	1.00000	0.0444116
$508$	0	0
$509$	−27.5967	−1.22320	−0.611602	−	0.791165i	$-0.709475\pi$
$509$	−27.5967	−1.22320	−0.611602	+	0.791165i	$0.709475\pi$
$510$	0	0
$511$	4.47214	0.197836
$512$	0	0
$513$	−2.47214	−0.109147
$514$	0	0
$515$	−3.05573	−0.134651
$516$	0	0
$517$	7.05573	0.310311
$518$	0	0
$519$	12.0000	0.526742
$520$	0	0
$521$	−3.05573	−0.133874	−0.0669369	−	0.997757i	$-0.521323\pi$
$521$	−3.05573	−0.133874	−0.0669369	+	0.997757i	$0.521323\pi$
$522$	0	0
$523$	1.88854	0.0825803	0.0412901	−	0.999147i	$-0.486853\pi$
$523$	1.88854	0.0825803	0.0412901	+	0.999147i	$0.486853\pi$
$524$	0	0
$525$	−5.47214	−0.238824
$526$	0	0
$527$	0	0
$528$	0	0
$529$	−3.00000	−0.130435
$530$	0	0
$531$	−4.76393	−0.206737
$532$	0	0
$533$	0.763932	0.0330896
$534$	0	0
$535$	40.3607	1.74494
$536$	0	0
$537$	−10.9443	−0.472280
$538$	0	0
$539$	1.23607	0.0532412
$540$	0	0
$541$	−10.9443	−0.470531	−0.235266	−	0.971931i	$-0.575596\pi$
$541$	−10.9443	−0.470531	−0.235266	+	0.971931i	$0.575596\pi$
$542$	0	0
$543$	−15.8885	−0.681843
$544$	0	0
$545$	40.3607	1.72886
$546$	0	0
$547$	36.9443	1.57962	0.789812	−	0.613350i	$-0.210178\pi$
$547$	36.9443	1.57962	0.789812	+	0.613350i	$0.210178\pi$
$548$	0	0
$549$	−2.94427	−0.125658
$550$	0	0
$551$	1.16718	0.0497237
$552$	0	0
$553$	−1.52786	−0.0649714
$554$	0	0
$555$	−14.4721	−0.614308
$556$	0	0
$557$	−33.2361	−1.40826	−0.704129	−	0.710072i	$-0.748662\pi$
$557$	−33.2361	−1.40826	−0.704129	+	0.710072i	$0.748662\pi$
$558$	0	0
$559$	−4.00000	−0.169182
$560$	0	0
$561$	8.00000	0.337760
$562$	0	0
$563$	−36.9443	−1.55702	−0.778508	−	0.627635i	$-0.784023\pi$
$563$	−36.9443	−1.55702	−0.778508	+	0.627635i	$0.784023\pi$
$564$	0	0
$565$	37.3050	1.56943
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	9.41641	0.394756	0.197378	−	0.980327i	$-0.436757\pi$
$569$	9.41641	0.394756	0.197378	+	0.980327i	$0.436757\pi$
$570$	0	0
$571$	−44.3607	−1.85644	−0.928218	−	0.372036i	$-0.878660\pi$
$571$	−44.3607	−1.85644	−0.928218	+	0.372036i	$0.878660\pi$
$572$	0	0
$573$	−10.9443	−0.457204
$574$	0	0
$575$	24.4721	1.02056
$576$	0	0
$577$	1.05573	0.0439505	0.0219753	−	0.999759i	$-0.493004\pi$
$577$	1.05573	0.0439505	0.0219753	+	0.999759i	$0.493004\pi$
$578$	0	0
$579$	−26.3607	−1.09551
$580$	0	0
$581$	7.23607	0.300203
$582$	0	0
$583$	12.3607	0.511927
$584$	0	0
$585$	3.23607	0.133795
$586$	0	0
$587$	27.5967	1.13904	0.569520	−	0.821978i	$-0.307129\pi$
$587$	27.5967	1.13904	0.569520	+	0.821978i	$0.307129\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	5.23607	0.215383
$592$	0	0
$593$	40.1803	1.65001	0.825004	−	0.565126i	$-0.191173\pi$
$593$	40.1803	1.65001	0.825004	+	0.565126i	$0.191173\pi$
$594$	0	0
$595$	−20.9443	−0.858631
$596$	0	0
$597$	16.0000	0.654836
$598$	0	0
$599$	−35.3050	−1.44252	−0.721261	−	0.692664i	$-0.756437\pi$
$599$	−35.3050	−1.44252	−0.721261	+	0.692664i	$0.756437\pi$
$600$	0	0
$601$	26.9443	1.09908	0.549540	−	0.835467i	$-0.314803\pi$
$601$	26.9443	1.09908	0.549540	+	0.835467i	$0.314803\pi$
$602$	0	0
$603$	−6.47214	−0.263566
$604$	0	0
$605$	−30.6525	−1.24620
$606$	0	0
$607$	−13.8885	−0.563719	−0.281859	−	0.959456i	$-0.590951\pi$
$607$	−13.8885	−0.563719	−0.281859	+	0.959456i	$0.590951\pi$
$608$	0	0
$609$	0.472136	0.0191319
$610$	0	0
$611$	5.70820	0.230929
$612$	0	0
$613$	41.7771	1.68736	0.843680	−	0.536846i	$-0.180384\pi$
$613$	41.7771	1.68736	0.843680	+	0.536846i	$0.180384\pi$
$614$	0	0
$615$	2.47214	0.0996861
$616$	0	0
$617$	−45.2361	−1.82114	−0.910568	−	0.413360i	$-0.864355\pi$
$617$	−45.2361	−1.82114	−0.910568	+	0.413360i	$0.864355\pi$
$618$	0	0
$619$	−25.3050	−1.01709	−0.508546	−	0.861035i	$-0.669817\pi$
$619$	−25.3050	−1.01709	−0.508546	+	0.861035i	$0.669817\pi$
$620$	0	0
$621$	4.47214	0.179461
$622$	0	0
$623$	−12.1803	−0.487995
$624$	0	0
$625$	−22.4164	−0.896656
$626$	0	0
$627$	−3.05573	−0.122034
$628$	0	0
$629$	−28.9443	−1.15408
$630$	0	0
$631$	−5.52786	−0.220061	−0.110030	−	0.993928i	$-0.535095\pi$
$631$	−5.52786	−0.220061	−0.110030	+	0.993928i	$0.535095\pi$
$632$	0	0
$633$	24.3607	0.968250
$634$	0	0
$635$	−20.9443	−0.831148
$636$	0	0
$637$	1.00000	0.0396214
$638$	0	0
$639$	−7.70820	−0.304932
$640$	0	0
$641$	−29.7771	−1.17612	−0.588062	−	0.808816i	$-0.700109\pi$
$641$	−29.7771	−1.17612	−0.588062	+	0.808816i	$0.700109\pi$
$642$	0	0
$643$	−42.8328	−1.68916	−0.844581	−	0.535428i	$-0.820150\pi$
$643$	−42.8328	−1.68916	−0.844581	+	0.535428i	$0.820150\pi$
$644$	0	0
$645$	−12.9443	−0.509680
$646$	0	0
$647$	−42.8328	−1.68393	−0.841966	−	0.539531i	$-0.818602\pi$
$647$	−42.8328	−1.68393	−0.841966	+	0.539531i	$0.818602\pi$
$648$	0	0
$649$	−5.88854	−0.231146
$650$	0	0
$651$	0	0
$652$	0	0
$653$	6.36068	0.248913	0.124456	−	0.992225i	$-0.460281\pi$
$653$	6.36068	0.248913	0.124456	+	0.992225i	$0.460281\pi$
$654$	0	0
$655$	33.8885	1.32413
$656$	0	0
$657$	−4.47214	−0.174475
$658$	0	0
$659$	−35.8885	−1.39802	−0.699010	−	0.715112i	$-0.746376\pi$
$659$	−35.8885	−1.39802	−0.699010	+	0.715112i	$0.746376\pi$
$660$	0	0
$661$	13.4164	0.521838	0.260919	−	0.965361i	$-0.415974\pi$
$661$	13.4164	0.521838	0.260919	+	0.965361i	$0.415974\pi$
$662$	0	0
$663$	6.47214	0.251357
$664$	0	0
$665$	8.00000	0.310227
$666$	0	0
$667$	−2.11146	−0.0817559
$668$	0	0
$669$	1.52786	0.0590706
$670$	0	0
$671$	−3.63932	−0.140494
$672$	0	0
$673$	4.11146	0.158485	0.0792425	−	0.996855i	$-0.474750\pi$
$673$	4.11146	0.158485	0.0792425	+	0.996855i	$0.474750\pi$
$674$	0	0
$675$	5.47214	0.210623
$676$	0	0
$677$	31.4164	1.20743	0.603715	−	0.797200i	$-0.293686\pi$
$677$	31.4164	1.20743	0.603715	+	0.797200i	$0.293686\pi$
$678$	0	0
$679$	10.9443	0.420003
$680$	0	0
$681$	11.2361	0.430567
$682$	0	0
$683$	−42.7639	−1.63632	−0.818158	−	0.574993i	$-0.805005\pi$
$683$	−42.7639	−1.63632	−0.818158	+	0.574993i	$0.805005\pi$
$684$	0	0
$685$	34.8328	1.33089
$686$	0	0
$687$	1.41641	0.0540393
$688$	0	0
$689$	10.0000	0.380970
$690$	0	0
$691$	52.3607	1.99189	0.995947	−	0.0899413i	$-0.0286680\pi$
$691$	52.3607	1.99189	0.995947	+	0.0899413i	$0.0286680\pi$
$692$	0	0
$693$	−1.23607	−0.0469543
$694$	0	0
$695$	57.8885	2.19584
$696$	0	0
$697$	4.94427	0.187278
$698$	0	0
$699$	−21.4164	−0.810043
$700$	0	0
$701$	−37.7771	−1.42682	−0.713410	−	0.700746i	$-0.752850\pi$
$701$	−37.7771	−1.42682	−0.713410	+	0.700746i	$0.752850\pi$
$702$	0	0
$703$	11.0557	0.416975
$704$	0	0
$705$	18.4721	0.695701
$706$	0	0
$707$	−8.94427	−0.336384
$708$	0	0
$709$	−26.3607	−0.989996	−0.494998	−	0.868894i	$-0.664831\pi$
$709$	−26.3607	−0.989996	−0.494998	+	0.868894i	$0.664831\pi$
$710$	0	0
$711$	1.52786	0.0572994
$712$	0	0
$713$	0	0
$714$	0	0
$715$	4.00000	0.149592
$716$	0	0
$717$	19.1246	0.714222
$718$	0	0
$719$	−53.3050	−1.98794	−0.993970	−	0.109656i	$-0.965025\pi$
$719$	−53.3050	−1.98794	−0.993970	+	0.109656i	$0.965025\pi$
$720$	0	0
$721$	0.944272	0.0351665
$722$	0	0
$723$	−7.52786	−0.279964
$724$	0	0
$725$	−2.58359	−0.0959522
$726$	0	0
$727$	22.8328	0.846822	0.423411	−	0.905938i	$-0.360833\pi$
$727$	22.8328	0.846822	0.423411	+	0.905938i	$0.360833\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−25.8885	−0.957522
$732$	0	0
$733$	7.52786	0.278048	0.139024	−	0.990289i	$-0.455603\pi$
$733$	7.52786	0.278048	0.139024	+	0.990289i	$0.455603\pi$
$734$	0	0
$735$	3.23607	0.119364
$736$	0	0
$737$	−8.00000	−0.294684
$738$	0	0
$739$	4.58359	0.168610	0.0843051	−	0.996440i	$-0.473133\pi$
$739$	4.58359	0.168610	0.0843051	+	0.996440i	$0.473133\pi$
$740$	0	0
$741$	−2.47214	−0.0908162
$742$	0	0
$743$	11.3475	0.416300	0.208150	−	0.978097i	$-0.433256\pi$
$743$	11.3475	0.416300	0.208150	+	0.978097i	$0.433256\pi$
$744$	0	0
$745$	−55.7771	−2.04351
$746$	0	0
$747$	−7.23607	−0.264754
$748$	0	0
$749$	−12.4721	−0.455722
$750$	0	0
$751$	−36.9443	−1.34812	−0.674058	−	0.738679i	$-0.735450\pi$
$751$	−36.9443	−1.34812	−0.674058	+	0.738679i	$0.735450\pi$
$752$	0	0
$753$	−14.4721	−0.527394
$754$	0	0
$755$	67.7771	2.46666
$756$	0	0
$757$	14.5836	0.530050	0.265025	−	0.964242i	$-0.414620\pi$
$757$	14.5836	0.530050	0.265025	+	0.964242i	$0.414620\pi$
$758$	0	0
$759$	5.52786	0.200649
$760$	0	0
$761$	3.81966	0.138463	0.0692313	−	0.997601i	$-0.477945\pi$
$761$	3.81966	0.138463	0.0692313	+	0.997601i	$0.477945\pi$
$762$	0	0
$763$	−12.4721	−0.451522
$764$	0	0
$765$	20.9443	0.757242
$766$	0	0
$767$	−4.76393	−0.172016
$768$	0	0
$769$	−16.4721	−0.594000	−0.297000	−	0.954877i	$-0.595986\pi$
$769$	−16.4721	−0.594000	−0.297000	+	0.954877i	$0.595986\pi$
$770$	0	0
$771$	−26.8328	−0.966360
$772$	0	0
$773$	−33.1246	−1.19141	−0.595705	−	0.803204i	$-0.703127\pi$
$773$	−33.1246	−1.19141	−0.595705	+	0.803204i	$0.703127\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	4.47214	0.160437
$778$	0	0
$779$	−1.88854	−0.0676641
$780$	0	0
$781$	−9.52786	−0.340934
$782$	0	0
$783$	−0.472136	−0.0168728
$784$	0	0
$785$	−58.2492	−2.07900
$786$	0	0
$787$	−52.3607	−1.86646	−0.933228	−	0.359284i	$-0.883021\pi$
$787$	−52.3607	−1.86646	−0.933228	+	0.359284i	$0.883021\pi$
$788$	0	0
$789$	−8.47214	−0.301616
$790$	0	0
$791$	−11.5279	−0.409884
$792$	0	0
$793$	−2.94427	−0.104554
$794$	0	0
$795$	32.3607	1.14772
$796$	0	0
$797$	−21.5279	−0.762556	−0.381278	−	0.924460i	$-0.624516\pi$
$797$	−21.5279	−0.762556	−0.381278	+	0.924460i	$0.624516\pi$
$798$	0	0
$799$	36.9443	1.30699
$800$	0	0
$801$	12.1803	0.430371
$802$	0	0
$803$	−5.52786	−0.195074
$804$	0	0
$805$	−14.4721	−0.510076
$806$	0	0
$807$	−12.9443	−0.455660
$808$	0	0
$809$	3.52786	0.124033	0.0620165	−	0.998075i	$-0.480247\pi$
$809$	3.52786	0.124033	0.0620165	+	0.998075i	$0.480247\pi$
$810$	0	0
$811$	−36.0000	−1.26413	−0.632065	−	0.774915i	$-0.717793\pi$
$811$	−36.0000	−1.26413	−0.632065	+	0.774915i	$0.717793\pi$
$812$	0	0
$813$	−3.41641	−0.119819
$814$	0	0
$815$	−20.9443	−0.733646
$816$	0	0
$817$	9.88854	0.345956
$818$	0	0
$819$	−1.00000	−0.0349428
$820$	0	0
$821$	−15.3475	−0.535632	−0.267816	−	0.963470i	$-0.586302\pi$
$821$	−15.3475	−0.535632	−0.267816	+	0.963470i	$0.586302\pi$
$822$	0	0
$823$	23.7771	0.828817	0.414409	−	0.910091i	$-0.363988\pi$
$823$	23.7771	0.828817	0.414409	+	0.910091i	$0.363988\pi$
$824$	0	0
$825$	6.76393	0.235490
$826$	0	0
$827$	−47.4853	−1.65123	−0.825613	−	0.564237i	$-0.809170\pi$
$827$	−47.4853	−1.65123	−0.825613	+	0.564237i	$0.809170\pi$
$828$	0	0
$829$	6.00000	0.208389	0.104194	−	0.994557i	$-0.466774\pi$
$829$	6.00000	0.208389	0.104194	+	0.994557i	$0.466774\pi$
$830$	0	0
$831$	19.8885	0.689926
$832$	0	0
$833$	6.47214	0.224246
$834$	0	0
$835$	−13.5279	−0.468151
$836$	0	0
$837$	0	0
$838$	0	0
$839$	52.1803	1.80147	0.900733	−	0.434374i	$-0.143030\pi$
$839$	52.1803	1.80147	0.900733	+	0.434374i	$0.143030\pi$
$840$	0	0
$841$	−28.7771	−0.992313
$842$	0	0
$843$	12.6525	0.435774
$844$	0	0
$845$	3.23607	0.111324
$846$	0	0
$847$	9.47214	0.325466
$848$	0	0
$849$	−4.00000	−0.137280
$850$	0	0
$851$	−20.0000	−0.685591
$852$	0	0
$853$	32.8328	1.12417	0.562087	−	0.827078i	$-0.309999\pi$
$853$	32.8328	1.12417	0.562087	+	0.827078i	$0.309999\pi$
$854$	0	0
$855$	−8.00000	−0.273594
$856$	0	0
$857$	23.0557	0.787569	0.393784	−	0.919203i	$-0.371166\pi$
$857$	23.0557	0.787569	0.393784	+	0.919203i	$0.371166\pi$
$858$	0	0
$859$	45.8885	1.56570	0.782848	−	0.622212i	$-0.213766\pi$
$859$	45.8885	1.56570	0.782848	+	0.622212i	$0.213766\pi$
$860$	0	0
$861$	−0.763932	−0.0260347
$862$	0	0
$863$	29.8197	1.01507	0.507537	−	0.861630i	$-0.330556\pi$
$863$	29.8197	1.01507	0.507537	+	0.861630i	$0.330556\pi$
$864$	0	0
$865$	38.8328	1.32036
$866$	0	0
$867$	24.8885	0.845259
$868$	0	0
$869$	1.88854	0.0640645
$870$	0	0
$871$	−6.47214	−0.219300
$872$	0	0
$873$	−10.9443	−0.370407
$874$	0	0
$875$	−1.52786	−0.0516512
$876$	0	0
$877$	−27.8885	−0.941729	−0.470865	−	0.882205i	$-0.656058\pi$
$877$	−27.8885	−0.941729	−0.470865	+	0.882205i	$0.656058\pi$
$878$	0	0
$879$	0.763932	0.0257668
$880$	0	0
$881$	−22.8328	−0.769257	−0.384629	−	0.923071i	$-0.625670\pi$
$881$	−22.8328	−0.769257	−0.384629	+	0.923071i	$0.625670\pi$
$882$	0	0
$883$	−11.4164	−0.384193	−0.192096	−	0.981376i	$-0.561529\pi$
$883$	−11.4164	−0.384193	−0.192096	+	0.981376i	$0.561529\pi$
$884$	0	0
$885$	−15.4164	−0.518217
$886$	0	0
$887$	58.8328	1.97541	0.987706	−	0.156321i	$-0.0499634\pi$
$887$	58.8328	1.97541	0.987706	+	0.156321i	$0.0499634\pi$
$888$	0	0
$889$	6.47214	0.217068
$890$	0	0
$891$	1.23607	0.0414098
$892$	0	0
$893$	−14.1115	−0.472222
$894$	0	0
$895$	−35.4164	−1.18384
$896$	0	0
$897$	4.47214	0.149320
$898$	0	0
$899$	0	0
$900$	0	0
$901$	64.7214	2.15618
$902$	0	0
$903$	4.00000	0.133112
$904$	0	0
$905$	−51.4164	−1.70914
$906$	0	0
$907$	−16.3607	−0.543247	−0.271624	−	0.962404i	$-0.587561\pi$
$907$	−16.3607	−0.543247	−0.271624	+	0.962404i	$0.587561\pi$
$908$	0	0
$909$	8.94427	0.296663
$910$	0	0
$911$	3.88854	0.128833	0.0644166	−	0.997923i	$-0.479481\pi$
$911$	3.88854	0.128833	0.0644166	+	0.997923i	$0.479481\pi$
$912$	0	0
$913$	−8.94427	−0.296012
$914$	0	0
$915$	−9.52786	−0.314981
$916$	0	0
$917$	−10.4721	−0.345820
$918$	0	0
$919$	−10.4721	−0.345444	−0.172722	−	0.984971i	$-0.555256\pi$
$919$	−10.4721	−0.345444	−0.172722	+	0.984971i	$0.555256\pi$
$920$	0	0
$921$	−29.8885	−0.984861
$922$	0	0
$923$	−7.70820	−0.253719
$924$	0	0
$925$	−24.4721	−0.804639
$926$	0	0
$927$	−0.944272	−0.0310140
$928$	0	0
$929$	−13.3475	−0.437918	−0.218959	−	0.975734i	$-0.570266\pi$
$929$	−13.3475	−0.437918	−0.218959	+	0.975734i	$0.570266\pi$
$930$	0	0
$931$	−2.47214	−0.0810210
$932$	0	0
$933$	10.4721	0.342842
$934$	0	0
$935$	25.8885	0.846646
$936$	0	0
$937$	−5.05573	−0.165163	−0.0825817	−	0.996584i	$-0.526317\pi$
$937$	−5.05573	−0.165163	−0.0825817	+	0.996584i	$0.526317\pi$
$938$	0	0
$939$	−32.8328	−1.07146
$940$	0	0
$941$	16.5410	0.539222	0.269611	−	0.962969i	$-0.413105\pi$
$941$	16.5410	0.539222	0.269611	+	0.962969i	$0.413105\pi$
$942$	0	0
$943$	3.41641	0.111254
$944$	0	0
$945$	−3.23607	−0.105269
$946$	0	0
$947$	41.5967	1.35171	0.675856	−	0.737033i	$-0.263774\pi$
$947$	41.5967	1.35171	0.675856	+	0.737033i	$0.263774\pi$
$948$	0	0
$949$	−4.47214	−0.145172
$950$	0	0
$951$	−5.81966	−0.188715
$952$	0	0
$953$	57.1935	1.85268	0.926340	−	0.376690i	$-0.122938\pi$
$953$	57.1935	1.85268	0.926340	+	0.376690i	$0.122938\pi$
$954$	0	0
$955$	−35.4164	−1.14605
$956$	0	0
$957$	−0.583592	−0.0188649
$958$	0	0
$959$	−10.7639	−0.347585
$960$	0	0
$961$	−31.0000	−1.00000
$962$	0	0
$963$	12.4721	0.401909
$964$	0	0
$965$	−85.3050	−2.74606
$966$	0	0
$967$	−4.36068	−0.140230	−0.0701150	−	0.997539i	$-0.522337\pi$
$967$	−4.36068	−0.140230	−0.0701150	+	0.997539i	$0.522337\pi$
$968$	0	0
$969$	−16.0000	−0.513994
$970$	0	0
$971$	5.88854	0.188972	0.0944862	−	0.995526i	$-0.469879\pi$
$971$	5.88854	0.188972	0.0944862	+	0.995526i	$0.469879\pi$
$972$	0	0
$973$	−17.8885	−0.573480
$974$	0	0
$975$	5.47214	0.175249
$976$	0	0
$977$	−14.7639	−0.472340	−0.236170	−	0.971712i	$-0.575892\pi$
$977$	−14.7639	−0.472340	−0.236170	+	0.971712i	$0.575892\pi$
$978$	0	0
$979$	15.0557	0.481183
$980$	0	0
$981$	12.4721	0.398205
$982$	0	0
$983$	−18.8754	−0.602031	−0.301016	−	0.953619i	$-0.597326\pi$
$983$	−18.8754	−0.602031	−0.301016	+	0.953619i	$0.597326\pi$
$984$	0	0
$985$	16.9443	0.539889
$986$	0	0
$987$	−5.70820	−0.181694
$988$	0	0
$989$	−17.8885	−0.568823
$990$	0	0
$991$	−41.8885	−1.33063	−0.665317	−	0.746561i	$-0.731703\pi$
$991$	−41.8885	−1.33063	−0.665317	+	0.746561i	$0.731703\pi$
$992$	0	0
$993$	−18.8328	−0.597641
$994$	0	0
$995$	51.7771	1.64144
$996$	0	0
$997$	−53.7771	−1.70314	−0.851569	−	0.524243i	$-0.824348\pi$
$997$	−53.7771	−1.70314	−0.851569	+	0.524243i	$0.824348\pi$
$998$	0	0
$999$	−4.47214	−0.141492

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.a.bl.1.2		2
4.3	odd	2		1092.2.a.f.1.2	✓	2
12.11	even	2		3276.2.a.l.1.1		2
28.27	even	2		7644.2.a.p.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1092.2.a.f.1.2	✓	2	4.3	odd	2
3276.2.a.l.1.1		2	12.11	even	2
4368.2.a.bl.1.2		2	1.1	even	1	trivial
7644.2.a.p.1.1		2	28.27	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4368.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.23607 q^{5} -1.00000 q^{7} +1.00000 q^{9} +1.23607 q^{11} +1.00000 q^{13} +3.23607 q^{15} +6.47214 q^{17} -2.47214 q^{19} -1.00000 q^{21} +4.47214 q^{23} +5.47214 q^{25} +1.00000 q^{27} -0.472136 q^{29} +1.23607 q^{33} -3.23607 q^{35} -4.47214 q^{37} +1.00000 q^{39} +0.763932 q^{41} -4.00000 q^{43} +3.23607 q^{45} +5.70820 q^{47} +1.00000 q^{49} +6.47214 q^{51} +10.0000 q^{53} +4.00000 q^{55} -2.47214 q^{57} -4.76393 q^{59} -2.94427 q^{61} -1.00000 q^{63} +3.23607 q^{65} -6.47214 q^{67} +4.47214 q^{69} -7.70820 q^{71} -4.47214 q^{73} +5.47214 q^{75} -1.23607 q^{77} +1.52786 q^{79} +1.00000 q^{81} -7.23607 q^{83} +20.9443 q^{85} -0.472136 q^{87} +12.1803 q^{89} -1.00000 q^{91} -8.00000 q^{95} -10.9443 q^{97} +1.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{3} + 2 q^{5} - 2 q^{7} + 2 q^{9} - 2 q^{11} + 2 q^{13} + 2 q^{15} + 4 q^{17} + 4 q^{19} - 2 q^{21} + 2 q^{25} + 2 q^{27} + 8 q^{29} - 2 q^{33} - 2 q^{35} + 2 q^{39} + 6 q^{41} - 8 q^{43} + 2 q^{45}+ \cdots - 2 q^{99}+O(q^{100}) \) 2 * q + 2 * q^3 + 2 * q^5 - 2 * q^7 + 2 * q^9 - 2 * q^11 + 2 * q^13 + 2 * q^15 + 4 * q^17 + 4 * q^19 - 2 * q^21 + 2 * q^25 + 2 * q^27 + 8 * q^29 - 2 * q^33 - 2 * q^35 + 2 * q^39 + 6 * q^41 - 8 * q^43 + 2 * q^45 - 2 * q^47 + 2 * q^49 + 4 * q^51 + 20 * q^53 + 8 * q^55 + 4 * q^57 - 14 * q^59 + 12 * q^61 - 2 * q^63 + 2 * q^65 - 4 * q^67 - 2 * q^71 + 2 * q^75 + 2 * q^77 + 12 * q^79 + 2 * q^81 - 10 * q^83 + 24 * q^85 + 8 * q^87 + 2 * q^89 - 2 * q^91 - 16 * q^95 - 4 * q^97 - 2 * q^99

Introduction

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