LMFDB - Embedded newform 4600.2.a.ba.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4600,2,Mod(1,4600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4600, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4600.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4600 = 2^{3} \cdot 5^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4600.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:	$36.7311849298$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.15529.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 6x^{2} - x + 2 $ x^4 - x^3 - 6*x^2 - x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$3.02228$ of defining polynomial
Character	$\chi$	$=$	4600.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.751385 q^{3} -0.661751 q^{7} -2.43542 q^{9} +O(q^{10})$	$q-0.751385 q^{3} -0.661751 q^{7} -2.43542 q^{9} +5.38282 q^{11} +3.45017 q^{13} +6.04457 q^{17} +1.38282 q^{19} +0.497230 q^{21} +1.00000 q^{23} +4.08409 q^{27} -3.34579 q^{29} +0.863303 q^{31} -4.04457 q^{33} +3.82073 q^{37} -2.59240 q^{39} -1.91037 q^{41} -5.88005 q^{43} -4.11192 q^{47} -6.56209 q^{49} -4.54180 q^{51} +3.00554 q^{53} -1.03903 q^{57} -14.1094 q^{59} -8.26841 q^{61} +1.61164 q^{63} -2.49723 q^{67} -0.751385 q^{69} +6.65372 q^{71} +12.7434 q^{73} -3.56209 q^{77} +8.64568 q^{79} +4.23753 q^{81} +5.06486 q^{83} +2.51397 q^{87} +2.22384 q^{89} -2.28315 q^{91} -0.648673 q^{93} +14.7211 q^{97} -13.1094 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - q^{7} + 6 q^{9}+O(q^{10}) $ 4 * q - q^7 + 6 * q^9	$ 4 q - q^{7} + 6 q^{9} + q^{11} + 3 q^{13} + 2 q^{17} - 15 q^{19} + 8 q^{21} + 4 q^{23} + 27 q^{27} + q^{29} - 12 q^{31} + 6 q^{33} + 18 q^{37} - 3 q^{39} - 9 q^{41} - 9 q^{43} - 4 q^{47} - 3 q^{49} - 2 q^{51} + 6 q^{57} - 5 q^{59} + 14 q^{61} + 39 q^{63} - 16 q^{67} - 2 q^{71} + 21 q^{73} + 9 q^{77} - 21 q^{79} + 44 q^{81} - 9 q^{83} + 17 q^{87} - 16 q^{89} + 33 q^{91} - 29 q^{93} + 40 q^{97} - q^{99}+O(q^{100}) $ 4 * q - q^7 + 6 * q^9 + q^11 + 3 * q^13 + 2 * q^17 - 15 * q^19 + 8 * q^21 + 4 * q^23 + 27 * q^27 + q^29 - 12 * q^31 + 6 * q^33 + 18 * q^37 - 3 * q^39 - 9 * q^41 - 9 * q^43 - 4 * q^47 - 3 * q^49 - 2 * q^51 + 6 * q^57 - 5 * q^59 + 14 * q^61 + 39 * q^63 - 16 * q^67 - 2 * q^71 + 21 * q^73 + 9 * q^77 - 21 * q^79 + 44 * q^81 - 9 * q^83 + 17 * q^87 - 16 * q^89 + 33 * q^91 - 29 * q^93 + 40 * q^97 - q^99

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	0	0
$2$	0	0
$3$	−0.751385	−0.433812	−0.216906	−	0.976192i	$-0.569597\pi$
$3$	−0.751385	−0.433812	−0.216906	+	0.976192i	$0.569597\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−0.661751	−0.250118	−0.125059	−	0.992149i	$-0.539912\pi$
$7$	−0.661751	−0.250118	−0.125059	+	0.992149i	$0.539912\pi$
$8$	0	0
$9$	−2.43542	−0.811807
$10$	0	0
$11$	5.38282	1.62298	0.811490	−	0.584366i	$-0.198657\pi$
$11$	5.38282	1.62298	0.811490	+	0.584366i	$0.198657\pi$
$12$	0	0
$13$	3.45017	0.956904	0.478452	−	0.878114i	$-0.341198\pi$
$13$	3.45017	0.956904	0.478452	+	0.878114i	$0.341198\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	6.04457	1.46602	0.733012	−	0.680216i	$-0.238114\pi$
$17$	6.04457	1.46602	0.733012	+	0.680216i	$0.238114\pi$
$18$	0	0
$19$	1.38282	0.317240	0.158620	−	0.987340i	$-0.449296\pi$
$19$	1.38282	0.317240	0.158620	+	0.987340i	$0.449296\pi$
$20$	0	0
$21$	0.497230	0.108504
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	0	0
$26$	0	0
$27$	4.08409	0.785984
$28$	0	0
$29$	−3.34579	−0.621297	−0.310648	−	0.950525i	$-0.600546\pi$
$29$	−3.34579	−0.621297	−0.310648	+	0.950525i	$0.600546\pi$
$30$	0	0
$31$	0.863303	0.155054	0.0775269	−	0.996990i	$-0.475298\pi$
$31$	0.863303	0.155054	0.0775269	+	0.996990i	$0.475298\pi$
$32$	0	0
$33$	−4.04457	−0.704069
$34$	0	0
$35$	0	0
$36$	0	0
$37$	3.82073	0.628124	0.314062	−	0.949402i	$-0.398310\pi$
$37$	3.82073	0.628124	0.314062	+	0.949402i	$0.398310\pi$
$38$	0	0
$39$	−2.59240	−0.415117
$40$	0	0
$41$	−1.91037	−0.298349	−0.149175	−	0.988811i	$-0.547662\pi$
$41$	−1.91037	−0.298349	−0.149175	+	0.988811i	$0.547662\pi$
$42$	0	0
$43$	−5.88005	−0.896699	−0.448349	−	0.893858i	$-0.647988\pi$
$43$	−5.88005	−0.896699	−0.448349	+	0.893858i	$0.647988\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.11192	−0.599785	−0.299892	−	0.953973i	$-0.596951\pi$
$47$	−4.11192	−0.599785	−0.299892	+	0.953973i	$0.596951\pi$
$48$	0	0
$49$	−6.56209	−0.937441
$50$	0	0
$51$	−4.54180	−0.635979
$52$	0	0
$53$	3.00554	0.412843	0.206421	−	0.978463i	$-0.433818\pi$
$53$	3.00554	0.412843	0.206421	+	0.978463i	$0.433818\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−1.03903	−0.137623
$58$	0	0
$59$	−14.1094	−1.83689	−0.918445	−	0.395548i	$-0.870555\pi$
$59$	−14.1094	−1.83689	−0.918445	+	0.395548i	$0.870555\pi$
$60$	0	0
$61$	−8.26841	−1.05866	−0.529330	−	0.848416i	$-0.677557\pi$
$61$	−8.26841	−1.05866	−0.529330	+	0.848416i	$0.677557\pi$
$62$	0	0
$63$	1.61164	0.203048
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−2.49723	−0.305085	−0.152543	−	0.988297i	$-0.548746\pi$
$67$	−2.49723	−0.305085	−0.152543	+	0.988297i	$0.548746\pi$
$68$	0	0
$69$	−0.751385	−0.0904561
$70$	0	0
$71$	6.65372	0.789651	0.394825	−	0.918756i	$-0.370805\pi$
$71$	6.65372	0.789651	0.394825	+	0.918756i	$0.370805\pi$
$72$	0	0
$73$	12.7434	1.49150	0.745748	−	0.666228i	$-0.232092\pi$
$73$	12.7434	1.49150	0.745748	+	0.666228i	$0.232092\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−3.56209	−0.405937
$78$	0	0
$79$	8.64568	0.972715	0.486358	−	0.873760i	$-0.338325\pi$
$79$	8.64568	0.972715	0.486358	+	0.873760i	$0.338325\pi$
$80$	0	0
$81$	4.23753	0.470837
$82$	0	0
$83$	5.06486	0.555940	0.277970	−	0.960590i	$-0.410338\pi$
$83$	5.06486	0.555940	0.277970	+	0.960590i	$0.410338\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	2.51397	0.269526
$88$	0	0
$89$	2.22384	0.235726	0.117863	−	0.993030i	$-0.462396\pi$
$89$	2.22384	0.235726	0.117863	+	0.993030i	$0.462396\pi$
$90$	0	0
$91$	−2.28315	−0.239339
$92$	0	0
$93$	−0.648673	−0.0672643
$94$	0	0
$95$	0	0
$96$	0	0
$97$	14.7211	1.49470	0.747349	−	0.664432i	$-0.231326\pi$
$97$	14.7211	1.49470	0.747349	+	0.664432i	$0.231326\pi$
$98$	0	0
$99$	−13.1094	−1.31755
$100$	0	0
$101$	−2.88005	−0.286575	−0.143288	−	0.989681i	$-0.545767\pi$
$101$	−2.88005	−0.286575	−0.143288	+	0.989681i	$0.545767\pi$
$102$	0	0
$103$	6.70632	0.660793	0.330397	−	0.943842i	$-0.392817\pi$
$103$	6.70632	0.660793	0.330397	+	0.943842i	$0.392817\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	13.4126	1.28470	0.642349	−	0.766412i	$-0.277960\pi$
$109$	13.4126	1.28470	0.642349	+	0.766412i	$0.277960\pi$
$110$	0	0
$111$	−2.87084	−0.272488
$112$	0	0
$113$	−3.03903	−0.285888	−0.142944	−	0.989731i	$-0.545657\pi$
$113$	−3.03903	−0.285888	−0.142944	+	0.989731i	$0.545657\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	−8.40261	−0.776821
$118$	0	0
$119$	−4.00000	−0.366679
$120$	0	0
$121$	17.9747	1.63407
$122$	0	0
$123$	1.43542	0.129428
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−0.863303	−0.0766058	−0.0383029	−	0.999266i	$-0.512195\pi$
$127$	−0.863303	−0.0766058	−0.0383029	+	0.999266i	$0.512195\pi$
$128$	0	0
$129$	4.41818	0.388999
$130$	0	0
$131$	−1.83498	−0.160323	−0.0801616	−	0.996782i	$-0.525544\pi$
$131$	−1.83498	−0.160323	−0.0801616	+	0.996782i	$0.525544\pi$
$132$	0	0
$133$	−0.915081	−0.0793476
$134$	0	0
$135$	0	0
$136$	0	0
$137$	5.32350	0.454817	0.227409	−	0.973799i	$-0.426975\pi$
$137$	5.32350	0.454817	0.227409	+	0.973799i	$0.426975\pi$
$138$	0	0
$139$	−7.88808	−0.669058	−0.334529	−	0.942385i	$-0.608577\pi$
$139$	−7.88808	−0.669058	−0.334529	+	0.942385i	$0.608577\pi$
$140$	0	0
$141$	3.08963	0.260194
$142$	0	0
$143$	18.5716	1.55304
$144$	0	0
$145$	0	0
$146$	0	0
$147$	4.93065	0.406673
$148$	0	0
$149$	7.57683	0.620718	0.310359	−	0.950619i	$-0.399551\pi$
$149$	7.57683	0.620718	0.310359	+	0.950619i	$0.399551\pi$
$150$	0	0
$151$	9.30072	0.756882	0.378441	−	0.925625i	$-0.376460\pi$
$151$	9.30072	0.756882	0.378441	+	0.925625i	$0.376460\pi$
$152$	0	0
$153$	−14.7211	−1.19013
$154$	0	0
$155$	0	0
$156$	0	0
$157$	12.4477	0.993432	0.496716	−	0.867913i	$-0.334539\pi$
$157$	12.4477	0.993432	0.496716	+	0.867913i	$0.334539\pi$
$158$	0	0
$159$	−2.25832	−0.179096
$160$	0	0
$161$	−0.661751	−0.0521533
$162$	0	0
$163$	11.2553	0.881585	0.440793	−	0.897609i	$-0.354697\pi$
$163$	11.2553	0.881585	0.440793	+	0.897609i	$0.354697\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−0.985253	−0.0762412	−0.0381206	−	0.999273i	$-0.512137\pi$
$167$	−0.985253	−0.0762412	−0.0381206	+	0.999273i	$0.512137\pi$
$168$	0	0
$169$	−1.09635	−0.0843343
$170$	0	0
$171$	−3.36774	−0.257538
$172$	0	0
$173$	23.8253	1.81140	0.905701	−	0.423917i	$-0.139345\pi$
$173$	23.8253	1.81140	0.905701	+	0.423917i	$0.139345\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	10.6016	0.796866
$178$	0	0
$179$	−9.82827	−0.734599	−0.367300	−	0.930103i	$-0.619718\pi$
$179$	−9.82827	−0.734599	−0.367300	+	0.930103i	$0.619718\pi$
$180$	0	0
$181$	14.4972	1.07757	0.538785	−	0.842443i	$-0.318883\pi$
$181$	14.4972	1.07757	0.538785	+	0.842443i	$0.318883\pi$
$182$	0	0
$183$	6.21276	0.459260
$184$	0	0
$185$	0	0
$186$	0	0
$187$	32.5368	2.37933
$188$	0	0
$189$	−2.70265	−0.196589
$190$	0	0
$191$	26.0432	1.88442	0.942212	−	0.335018i	$-0.108743\pi$
$191$	26.0432	1.88442	0.942212	+	0.335018i	$0.108743\pi$
$192$	0	0
$193$	−22.3768	−1.61072	−0.805358	−	0.592789i	$-0.798027\pi$
$193$	−22.3768	−1.61072	−0.805358	+	0.592789i	$0.798027\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	20.7946	1.48156	0.740778	−	0.671750i	$-0.234457\pi$
$197$	20.7946	1.48156	0.740778	+	0.671750i	$0.234457\pi$
$198$	0	0
$199$	−17.2927	−1.22585	−0.612923	−	0.790143i	$-0.710006\pi$
$199$	−17.2927	−1.22585	−0.612923	+	0.790143i	$0.710006\pi$
$200$	0	0
$201$	1.87638	0.132350
$202$	0	0
$203$	2.21408	0.155398
$204$	0	0
$205$	0	0
$206$	0	0
$207$	−2.43542	−0.169273
$208$	0	0
$209$	7.44345	0.514875
$210$	0	0
$211$	−23.8289	−1.64045	−0.820226	−	0.572039i	$-0.806152\pi$
$211$	−23.8289	−1.64045	−0.820226	+	0.572039i	$0.806152\pi$
$212$	0	0
$213$	−4.99950	−0.342560
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−0.571292	−0.0387818
$218$	0	0
$219$	−9.57516	−0.647030
$220$	0	0
$221$	20.8548	1.40284
$222$	0	0
$223$	−0.656211	−0.0439431	−0.0219716	−	0.999759i	$-0.506994\pi$
$223$	−0.656211	−0.0439431	−0.0219716	+	0.999759i	$0.506994\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−5.76009	−0.382311	−0.191155	−	0.981560i	$-0.561223\pi$
$227$	−5.76009	−0.382311	−0.191155	+	0.981560i	$0.561223\pi$
$228$	0	0
$229$	11.1848	0.739113	0.369556	−	0.929208i	$-0.379510\pi$
$229$	11.1848	0.739113	0.369556	+	0.929208i	$0.379510\pi$
$230$	0	0
$231$	2.67650	0.176101
$232$	0	0
$233$	−7.28597	−0.477320	−0.238660	−	0.971103i	$-0.576708\pi$
$233$	−7.28597	−0.477320	−0.238660	+	0.971103i	$0.576708\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	−6.49624	−0.421976
$238$	0	0
$239$	−26.2532	−1.69818	−0.849088	−	0.528252i	$-0.822848\pi$
$239$	−26.2532	−1.69818	−0.849088	+	0.528252i	$0.822848\pi$
$240$	0	0
$241$	14.8708	0.957915	0.478958	−	0.877838i	$-0.341015\pi$
$241$	14.8708	0.957915	0.478958	+	0.877838i	$0.341015\pi$
$242$	0	0
$243$	−15.4363	−0.990239
$244$	0	0
$245$	0	0
$246$	0	0
$247$	4.77095	0.303568
$248$	0	0
$249$	−3.80566	−0.241174
$250$	0	0
$251$	27.5424	1.73846	0.869229	−	0.494410i	$-0.164616\pi$
$251$	27.5424	1.73846	0.869229	+	0.494410i	$0.164616\pi$
$252$	0	0
$253$	5.38282	0.338415
$254$	0	0
$255$	0	0
$256$	0	0
$257$	26.0994	1.62804	0.814018	−	0.580840i	$-0.197276\pi$
$257$	26.0994	1.62804	0.814018	+	0.580840i	$0.197276\pi$
$258$	0	0
$259$	−2.52837	−0.157105
$260$	0	0
$261$	8.14840	0.504373
$262$	0	0
$263$	−10.9600	−0.675821	−0.337911	−	0.941178i	$-0.609720\pi$
$263$	−10.9600	−0.675821	−0.337911	+	0.941178i	$0.609720\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−1.67096	−0.102261
$268$	0	0
$269$	−16.6229	−1.01352	−0.506758	−	0.862088i	$-0.669156\pi$
$269$	−16.6229	−1.01352	−0.506758	+	0.862088i	$0.669156\pi$
$270$	0	0
$271$	−28.8751	−1.75403	−0.877017	−	0.480458i	$-0.840470\pi$
$271$	−28.8751	−1.75403	−0.877017	+	0.480458i	$0.840470\pi$
$272$	0	0
$273$	1.71553	0.103828
$274$	0	0
$275$	0	0
$276$	0	0
$277$	26.4472	1.58906	0.794528	−	0.607227i	$-0.207718\pi$
$277$	26.4472	1.58906	0.794528	+	0.607227i	$0.207718\pi$
$278$	0	0
$279$	−2.10251	−0.125874
$280$	0	0
$281$	−10.3625	−0.618177	−0.309088	−	0.951033i	$-0.600024\pi$
$281$	−10.3625	−0.618177	−0.309088	+	0.951033i	$0.600024\pi$
$282$	0	0
$283$	0.709986	0.0422043	0.0211021	−	0.999777i	$-0.493282\pi$
$283$	0.709986	0.0422043	0.0211021	+	0.999777i	$0.493282\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1.26419	0.0746226
$288$	0	0
$289$	19.5368	1.14922
$290$	0	0
$291$	−11.0612	−0.648418
$292$	0	0
$293$	−6.49224	−0.379281	−0.189640	−	0.981854i	$-0.560732\pi$
$293$	−6.49224	−0.379281	−0.189640	+	0.981854i	$0.560732\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	21.9839	1.27564
$298$	0	0
$299$	3.45017	0.199528
$300$	0	0
$301$	3.89113	0.224281
$302$	0	0
$303$	2.16402	0.124320
$304$	0	0
$305$	0	0
$306$	0	0
$307$	0	0		−	1.00000i	$-0.5\pi$
$307$	0	0			1.00000i	$0.5\pi$
$308$	0	0
$309$	−5.03903	−0.286660
$310$	0	0
$311$	−14.8673	−0.843047	−0.421524	−	0.906817i	$-0.638505\pi$
$311$	−14.8673	−0.843047	−0.421524	+	0.906817i	$0.638505\pi$
$312$	0	0
$313$	14.2523	0.805590	0.402795	−	0.915290i	$-0.368039\pi$
$313$	14.2523	0.805590	0.402795	+	0.915290i	$0.368039\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−14.3388	−0.805347	−0.402674	−	0.915344i	$-0.631919\pi$
$317$	−14.3388	−0.805347	−0.402674	+	0.915344i	$0.631919\pi$
$318$	0	0
$319$	−18.0098	−1.00835
$320$	0	0
$321$	0	0
$322$	0	0
$323$	8.35854	0.465081
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−10.0781	−0.557318
$328$	0	0
$329$	2.72107	0.150017
$330$	0	0
$331$	22.4931	1.23633	0.618165	−	0.786048i	$-0.287876\pi$
$331$	22.4931	1.23633	0.618165	+	0.786048i	$0.287876\pi$
$332$	0	0
$333$	−9.30509	−0.509916
$334$	0	0
$335$	0	0
$336$	0	0
$337$	10.8152	0.589141	0.294571	−	0.955630i	$-0.404823\pi$
$337$	10.8152	0.589141	0.294571	+	0.955630i	$0.404823\pi$
$338$	0	0
$339$	2.28348	0.124022
$340$	0	0
$341$	4.64700	0.251649
$342$	0	0
$343$	8.97473	0.484590
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−13.5516	−0.727486	−0.363743	−	0.931499i	$-0.618501\pi$
$347$	−13.5516	−0.727486	−0.363743	+	0.931499i	$0.618501\pi$
$348$	0	0
$349$	0.102160	0.00546848	0.00273424	−	0.999996i	$-0.499130\pi$
$349$	0.102160	0.00546848	0.00273424	+	0.999996i	$0.499130\pi$
$350$	0	0
$351$	14.0908	0.752112
$352$	0	0
$353$	1.46269	0.0778513	0.0389256	−	0.999242i	$-0.487606\pi$
$353$	1.46269	0.0778513	0.0389256	+	0.999242i	$0.487606\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	3.00554	0.159070
$358$	0	0
$359$	23.2726	1.22828	0.614141	−	0.789196i	$-0.289503\pi$
$359$	23.2726	1.22828	0.614141	+	0.789196i	$0.289503\pi$
$360$	0	0
$361$	−17.0878	−0.899359
$362$	0	0
$363$	−13.5059	−0.708878
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−33.0153	−1.72338	−0.861692	−	0.507431i	$-0.830595\pi$
$367$	−33.0153	−1.72338	−0.861692	+	0.507431i	$0.830595\pi$
$368$	0	0
$369$	4.65254	0.242202
$370$	0	0
$371$	−1.98892	−0.103260
$372$	0	0
$373$	9.49778	0.491777	0.245888	−	0.969298i	$-0.420920\pi$
$373$	9.49778	0.491777	0.245888	+	0.969298i	$0.420920\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−11.5435	−0.594522
$378$	0	0
$379$	−16.3585	−0.840282	−0.420141	−	0.907459i	$-0.638019\pi$
$379$	−16.3585	−0.840282	−0.420141	+	0.907459i	$0.638019\pi$
$380$	0	0
$381$	0.648673	0.0332325
$382$	0	0
$383$	−8.95965	−0.457817	−0.228908	−	0.973448i	$-0.573516\pi$
$383$	−8.95965	−0.457817	−0.228908	+	0.973448i	$0.573516\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	14.3204	0.727946
$388$	0	0
$389$	15.3185	0.776679	0.388340	−	0.921516i	$-0.373049\pi$
$389$	15.3185	0.776679	0.388340	+	0.921516i	$0.373049\pi$
$390$	0	0
$391$	6.04457	0.305687
$392$	0	0
$393$	1.37878	0.0695502
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−36.8128	−1.84758	−0.923790	−	0.382901i	$-0.874925\pi$
$397$	−36.8128	−1.84758	−0.923790	+	0.382901i	$0.874925\pi$
$398$	0	0
$399$	0.687578	0.0344220
$400$	0	0
$401$	−33.1232	−1.65409	−0.827046	−	0.562134i	$-0.809981\pi$
$401$	−33.1232	−1.65409	−0.827046	+	0.562134i	$0.809981\pi$
$402$	0	0
$403$	2.97854	0.148372
$404$	0	0
$405$	0	0
$406$	0	0
$407$	20.5663	1.01943
$408$	0	0
$409$	5.63298	0.278533	0.139266	−	0.990255i	$-0.455526\pi$
$409$	5.63298	0.278533	0.139266	+	0.990255i	$0.455526\pi$
$410$	0	0
$411$	−4.00000	−0.197305
$412$	0	0
$413$	9.33693	0.459440
$414$	0	0
$415$	0	0
$416$	0	0
$417$	5.92699	0.290246
$418$	0	0
$419$	0.695239	0.0339647	0.0169823	−	0.999856i	$-0.494594\pi$
$419$	0.695239	0.0339647	0.0169823	+	0.999856i	$0.494594\pi$
$420$	0	0
$421$	33.3926	1.62745	0.813727	−	0.581247i	$-0.197435\pi$
$421$	33.3926	1.62745	0.813727	+	0.581247i	$0.197435\pi$
$422$	0	0
$423$	10.0143	0.486910
$424$	0	0
$425$	0	0
$426$	0	0
$427$	5.47163	0.264791
$428$	0	0
$429$	−13.9544	−0.673727
$430$	0	0
$431$	−10.2684	−0.494612	−0.247306	−	0.968937i	$-0.579545\pi$
$431$	−10.2684	−0.494612	−0.247306	+	0.968937i	$0.579545\pi$
$432$	0	0
$433$	8.79513	0.422667	0.211333	−	0.977414i	$-0.432219\pi$
$433$	8.79513	0.422667	0.211333	+	0.977414i	$0.432219\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1.38282	0.0661491
$438$	0	0
$439$	10.5226	0.502214	0.251107	−	0.967959i	$-0.419205\pi$
$439$	10.5226	0.502214	0.251107	+	0.967959i	$0.419205\pi$
$440$	0	0
$441$	15.9814	0.761021
$442$	0	0
$443$	31.0424	1.47487	0.737435	−	0.675419i	$-0.236037\pi$
$443$	31.0424	1.47487	0.737435	+	0.675419i	$0.236037\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−5.69312	−0.269275
$448$	0	0
$449$	19.5565	0.922930	0.461465	−	0.887158i	$-0.347324\pi$
$449$	19.5565	0.922930	0.461465	+	0.887158i	$0.347324\pi$
$450$	0	0
$451$	−10.2832	−0.484215
$452$	0	0
$453$	−6.98842	−0.328345
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3.29001	0.153900	0.0769502	−	0.997035i	$-0.475482\pi$
$457$	3.29001	0.153900	0.0769502	+	0.997035i	$0.475482\pi$
$458$	0	0
$459$	24.6866	1.15227
$460$	0	0
$461$	−12.6229	−0.587907	−0.293954	−	0.955820i	$-0.594971\pi$
$461$	−12.6229	−0.587907	−0.293954	+	0.955820i	$0.594971\pi$
$462$	0	0
$463$	−35.4008	−1.64521	−0.822607	−	0.568610i	$-0.807481\pi$
$463$	−35.4008	−1.64521	−0.822607	+	0.568610i	$0.807481\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	14.4624	0.669241	0.334620	−	0.942353i	$-0.391392\pi$
$467$	14.4624	0.669241	0.334620	+	0.942353i	$0.391392\pi$
$468$	0	0
$469$	1.65254	0.0763074
$470$	0	0
$471$	−9.35300	−0.430963
$472$	0	0
$473$	−31.6512	−1.45532
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−7.31975	−0.335149
$478$	0	0
$479$	32.0288	1.46343	0.731717	−	0.681608i	$-0.238719\pi$
$479$	32.0288	1.46343	0.731717	+	0.681608i	$0.238719\pi$
$480$	0	0
$481$	13.1822	0.601055
$482$	0	0
$483$	0.497230	0.0226247
$484$	0	0
$485$	0	0
$486$	0	0
$487$	9.76446	0.442470	0.221235	−	0.975221i	$-0.428991\pi$
$487$	9.76446	0.442470	0.221235	+	0.975221i	$0.428991\pi$
$488$	0	0
$489$	−8.45708	−0.382443
$490$	0	0
$491$	21.9722	0.991593	0.495796	−	0.868439i	$-0.334876\pi$
$491$	21.9722	0.991593	0.495796	+	0.868439i	$0.334876\pi$
$492$	0	0
$493$	−20.2238	−0.910836
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4.40310	−0.197506
$498$	0	0
$499$	−20.9002	−0.935620	−0.467810	−	0.883829i	$-0.654957\pi$
$499$	−20.9002	−0.935620	−0.467810	+	0.883829i	$0.654957\pi$
$500$	0	0
$501$	0.740305	0.0330744
$502$	0	0
$503$	44.1300	1.96766	0.983831	−	0.179101i	$-0.0573190\pi$
$503$	44.1300	1.96766	0.983831	+	0.179101i	$0.0573190\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	0.823778	0.0365853
$508$	0	0
$509$	7.07334	0.313520	0.156760	−	0.987637i	$-0.449895\pi$
$509$	7.07334	0.313520	0.156760	+	0.987637i	$0.449895\pi$
$510$	0	0
$511$	−8.43293	−0.373051
$512$	0	0
$513$	5.64756	0.249346
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−22.1337	−0.973439
$518$	0	0
$519$	−17.9020	−0.785809
$520$	0	0
$521$	−41.9344	−1.83718	−0.918589	−	0.395214i	$-0.870671\pi$
$521$	−41.9344	−1.83718	−0.918589	+	0.395214i	$0.870671\pi$
$522$	0	0
$523$	12.5676	0.549544	0.274772	−	0.961509i	$-0.411398\pi$
$523$	12.5676	0.549544	0.274772	+	0.961509i	$0.411398\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	5.21830	0.227313
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	34.3624	1.49120
$532$	0	0
$533$	−6.59108	−0.285491
$534$	0	0
$535$	0	0
$536$	0	0
$537$	7.38482	0.318678
$538$	0	0
$539$	−35.3225	−1.52145
$540$	0	0
$541$	11.9804	0.515079	0.257540	−	0.966268i	$-0.417088\pi$
$541$	11.9804	0.515079	0.257540	+	0.966268i	$0.417088\pi$
$542$	0	0
$543$	−10.8930	−0.467463
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−10.8045	−0.461966	−0.230983	−	0.972958i	$-0.574194\pi$
$547$	−10.8045	−0.461966	−0.230983	+	0.972958i	$0.574194\pi$
$548$	0	0
$549$	20.1370	0.859428
$550$	0	0
$551$	−4.62661	−0.197100
$552$	0	0
$553$	−5.72129	−0.243294
$554$	0	0
$555$	0	0
$556$	0	0
$557$	43.6053	1.84762	0.923809	−	0.382855i	$-0.125059\pi$
$557$	43.6053	1.84762	0.923809	+	0.382855i	$0.125059\pi$
$558$	0	0
$559$	−20.2871	−0.858055
$560$	0	0
$561$	−24.4477	−1.03218
$562$	0	0
$563$	18.3794	0.774598	0.387299	−	0.921954i	$-0.373408\pi$
$563$	18.3794	0.774598	0.387299	+	0.921954i	$0.373408\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	−2.80419	−0.117765
$568$	0	0
$569$	27.7863	1.16486	0.582430	−	0.812881i	$-0.302102\pi$
$569$	27.7863	1.16486	0.582430	+	0.812881i	$0.302102\pi$
$570$	0	0
$571$	8.35355	0.349585	0.174793	−	0.984605i	$-0.444075\pi$
$571$	8.35355	0.349585	0.174793	+	0.984605i	$0.444075\pi$
$572$	0	0
$573$	−19.5685	−0.817486
$574$	0	0
$575$	0	0
$576$	0	0
$577$	4.43610	0.184677	0.0923385	−	0.995728i	$-0.470566\pi$
$577$	4.43610	0.184677	0.0923385	+	0.995728i	$0.470566\pi$
$578$	0	0
$579$	16.8136	0.698748
$580$	0	0
$581$	−3.35167	−0.139051
$582$	0	0
$583$	16.1783	0.670036
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−32.4980	−1.34133	−0.670667	−	0.741758i	$-0.733992\pi$
$587$	−32.4980	−1.34133	−0.670667	+	0.741758i	$0.733992\pi$
$588$	0	0
$589$	1.19379	0.0491893
$590$	0	0
$591$	−15.6248	−0.642717
$592$	0	0
$593$	−26.0098	−1.06809	−0.534046	−	0.845455i	$-0.679329\pi$
$593$	−26.0098	−1.06809	−0.534046	+	0.845455i	$0.679329\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	12.9935	0.531787
$598$	0	0
$599$	20.9853	0.857434	0.428717	−	0.903439i	$-0.358966\pi$
$599$	20.9853	0.857434	0.428717	+	0.903439i	$0.358966\pi$
$600$	0	0
$601$	−4.35000	−0.177440	−0.0887202	−	0.996057i	$-0.528278\pi$
$601$	−4.35000	−0.177440	−0.0887202	+	0.996057i	$0.528278\pi$
$602$	0	0
$603$	6.08180	0.247670
$604$	0	0
$605$	0	0
$606$	0	0
$607$	11.8117	0.479424	0.239712	−	0.970844i	$-0.422947\pi$
$607$	11.8117	0.479424	0.239712	+	0.970844i	$0.422947\pi$
$608$	0	0
$609$	−1.66363	−0.0674135
$610$	0	0
$611$	−14.1868	−0.573937
$612$	0	0
$613$	−20.7767	−0.839164	−0.419582	−	0.907718i	$-0.637823\pi$
$613$	−20.7767	−0.839164	−0.419582	+	0.907718i	$0.637823\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	24.3059	0.978518	0.489259	−	0.872138i	$-0.337267\pi$
$617$	24.3059	0.978518	0.489259	+	0.872138i	$0.337267\pi$
$618$	0	0
$619$	1.83581	0.0737873	0.0368937	−	0.999319i	$-0.488254\pi$
$619$	1.83581	0.0737873	0.0368937	+	0.999319i	$0.488254\pi$
$620$	0	0
$621$	4.08409	0.163889
$622$	0	0
$623$	−1.47163	−0.0589595
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−5.59290	−0.223359
$628$	0	0
$629$	23.0947	0.920845
$630$	0	0
$631$	24.7003	0.983305	0.491652	−	0.870792i	$-0.336393\pi$
$631$	24.7003	0.983305	0.491652	+	0.870792i	$0.336393\pi$
$632$	0	0
$633$	17.9047	0.711648
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−22.6403	−0.897041
$638$	0	0
$639$	−16.2046	−0.641044
$640$	0	0
$641$	6.91141	0.272984	0.136492	−	0.990641i	$-0.456417\pi$
$641$	6.91141	0.272984	0.136492	+	0.990641i	$0.456417\pi$
$642$	0	0
$643$	−3.88404	−0.153172	−0.0765858	−	0.997063i	$-0.524402\pi$
$643$	−3.88404	−0.153172	−0.0765858	+	0.997063i	$0.524402\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	17.4296	0.685229	0.342614	−	0.939476i	$-0.388688\pi$
$647$	17.4296	0.685229	0.342614	+	0.939476i	$0.388688\pi$
$648$	0	0
$649$	−75.9485	−2.98124
$650$	0	0
$651$	0.429260	0.0168240
$652$	0	0
$653$	7.04624	0.275741	0.137870	−	0.990450i	$-0.455974\pi$
$653$	7.04624	0.275741	0.137870	+	0.990450i	$0.455974\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−31.0354	−1.21081
$658$	0	0
$659$	6.69124	0.260654	0.130327	−	0.991471i	$-0.458397\pi$
$659$	6.69124	0.260654	0.130327	+	0.991471i	$0.458397\pi$
$660$	0	0
$661$	23.8653	0.928253	0.464126	−	0.885769i	$-0.346368\pi$
$661$	23.8653	0.928253	0.464126	+	0.885769i	$0.346368\pi$
$662$	0	0
$663$	−15.6700	−0.608571
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3.34579	−0.129549
$668$	0	0
$669$	0.493067	0.0190631
$670$	0	0
$671$	−44.5073	−1.71819
$672$	0	0
$673$	15.5318	0.598706	0.299353	−	0.954142i	$-0.403229\pi$
$673$	15.5318	0.598706	0.299353	+	0.954142i	$0.403229\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	34.2339	1.31572	0.657858	−	0.753142i	$-0.271463\pi$
$677$	34.2339	1.31572	0.657858	+	0.753142i	$0.271463\pi$
$678$	0	0
$679$	−9.74168	−0.373851
$680$	0	0
$681$	4.32805	0.165851
$682$	0	0
$683$	16.5788	0.634371	0.317186	−	0.948363i	$-0.397262\pi$
$683$	16.5788	0.634371	0.317186	+	0.948363i	$0.397262\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−8.40410	−0.320636
$688$	0	0
$689$	10.3696	0.395051
$690$	0	0
$691$	−2.36774	−0.0900732	−0.0450366	−	0.998985i	$-0.514340\pi$
$691$	−2.36774	−0.0900732	−0.0450366	+	0.998985i	$0.514340\pi$
$692$	0	0
$693$	8.67518	0.329543
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−11.5473	−0.437387
$698$	0	0
$699$	5.47457	0.207067
$700$	0	0
$701$	40.8849	1.54420	0.772101	−	0.635500i	$-0.219206\pi$
$701$	40.8849	1.54420	0.772101	+	0.635500i	$0.219206\pi$
$702$	0	0
$703$	5.28338	0.199266
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1.90587	0.0716778
$708$	0	0
$709$	−11.9354	−0.448242	−0.224121	−	0.974561i	$-0.571951\pi$
$709$	−11.9354	−0.448242	−0.224121	+	0.974561i	$0.571951\pi$
$710$	0	0
$711$	−21.0559	−0.789657
$712$	0	0
$713$	0.863303	0.0323310
$714$	0	0
$715$	0	0
$716$	0	0
$717$	19.7262	0.736690
$718$	0	0
$719$	−7.21331	−0.269011	−0.134506	−	0.990913i	$-0.542945\pi$
$719$	−7.21331	−0.269011	−0.134506	+	0.990913i	$0.542945\pi$
$720$	0	0
$721$	−4.43791	−0.165277
$722$	0	0
$723$	−11.1737	−0.415555
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−49.3326	−1.82964	−0.914822	−	0.403856i	$-0.867670\pi$
$727$	−49.3326	−1.82964	−0.914822	+	0.403856i	$0.867670\pi$
$728$	0	0
$729$	−1.11400	−0.0412592
$730$	0	0
$731$	−35.5424	−1.31458
$732$	0	0
$733$	−27.0029	−0.997375	−0.498687	−	0.866782i	$-0.666184\pi$
$733$	−27.0029	−0.997375	−0.498687	+	0.866782i	$0.666184\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−13.4421	−0.495147
$738$	0	0
$739$	−15.2999	−0.562816	−0.281408	−	0.959588i	$-0.590801\pi$
$739$	−15.2999	−0.562816	−0.281408	+	0.959588i	$0.590801\pi$
$740$	0	0
$741$	−3.58482	−0.131692
$742$	0	0
$743$	11.6767	0.428377	0.214189	−	0.976792i	$-0.431289\pi$
$743$	11.6767	0.428377	0.214189	+	0.976792i	$0.431289\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	−12.3351	−0.451316
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−34.3589	−1.25377	−0.626886	−	0.779111i	$-0.715671\pi$
$751$	−34.3589	−1.25377	−0.626886	+	0.779111i	$0.715671\pi$
$752$	0	0
$753$	−20.6949	−0.754164
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−37.6164	−1.36719	−0.683596	−	0.729861i	$-0.739585\pi$
$757$	−37.6164	−1.36719	−0.683596	+	0.729861i	$0.739585\pi$
$758$	0	0
$759$	−4.04457	−0.146809
$760$	0	0
$761$	−15.2788	−0.553856	−0.276928	−	0.960891i	$-0.589316\pi$
$761$	−15.2788	−0.553856	−0.276928	+	0.960891i	$0.589316\pi$
$762$	0	0
$763$	−8.87583	−0.321327
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−48.6799	−1.75773
$768$	0	0
$769$	32.0346	1.15520	0.577598	−	0.816321i	$-0.303990\pi$
$769$	32.0346	1.15520	0.577598	+	0.816321i	$0.303990\pi$
$770$	0	0
$771$	−19.6107	−0.706262
$772$	0	0
$773$	32.3425	1.16328	0.581639	−	0.813447i	$-0.302412\pi$
$773$	32.3425	1.16328	0.581639	+	0.813447i	$0.302412\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	1.89978	0.0681543
$778$	0	0
$779$	−2.64169	−0.0946483
$780$	0	0
$781$	35.8157	1.28159
$782$	0	0
$783$	−13.6645	−0.488330
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−48.7784	−1.73876	−0.869381	−	0.494142i	$-0.835482\pi$
$787$	−48.7784	−1.73876	−0.869381	+	0.494142i	$0.835482\pi$
$788$	0	0
$789$	8.23516	0.293180
$790$	0	0
$791$	2.01108	0.0715058
$792$	0	0
$793$	−28.5274	−1.01304
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−25.9481	−0.919129	−0.459564	−	0.888144i	$-0.651994\pi$
$797$	−25.9481	−0.919129	−0.459564	+	0.888144i	$0.651994\pi$
$798$	0	0
$799$	−24.8548	−0.879299
$800$	0	0
$801$	−5.41598	−0.191364
$802$	0	0
$803$	68.5951	2.42067
$804$	0	0
$805$	0	0
$806$	0	0
$807$	12.4902	0.439676
$808$	0	0
$809$	−42.9009	−1.50831	−0.754157	−	0.656694i	$-0.771954\pi$
$809$	−42.9009	−1.50831	−0.754157	+	0.656694i	$0.771954\pi$
$810$	0	0
$811$	24.3947	0.856615	0.428308	−	0.903633i	$-0.359110\pi$
$811$	24.3947	0.856615	0.428308	+	0.903633i	$0.359110\pi$
$812$	0	0
$813$	21.6963	0.760922
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−8.13103	−0.284469
$818$	0	0
$819$	5.56044	0.194297
$820$	0	0
$821$	50.3621	1.75765	0.878825	−	0.477145i	$-0.158328\pi$
$821$	50.3621	1.75765	0.878825	+	0.477145i	$0.158328\pi$
$822$	0	0
$823$	−10.4334	−0.363686	−0.181843	−	0.983328i	$-0.558206\pi$
$823$	−10.4334	−0.363686	−0.181843	+	0.983328i	$0.558206\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−26.1756	−0.910215	−0.455107	−	0.890437i	$-0.650399\pi$
$827$	−26.1756	−0.910215	−0.455107	+	0.890437i	$0.650399\pi$
$828$	0	0
$829$	19.3776	0.673012	0.336506	−	0.941681i	$-0.390755\pi$
$829$	19.3776	0.673012	0.336506	+	0.941681i	$0.390755\pi$
$830$	0	0
$831$	−19.8720	−0.689353
$832$	0	0
$833$	−39.6650	−1.37431
$834$	0	0
$835$	0	0
$836$	0	0
$837$	3.52581	0.121870
$838$	0	0
$839$	15.5611	0.537229	0.268614	−	0.963248i	$-0.413434\pi$
$839$	15.5611	0.537229	0.268614	+	0.963248i	$0.413434\pi$
$840$	0	0
$841$	−17.8057	−0.613990
$842$	0	0
$843$	7.78625	0.268173
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−11.8948	−0.408710
$848$	0	0
$849$	−0.533473	−0.0183087
$850$	0	0
$851$	3.82073	0.130973
$852$	0	0
$853$	−19.7293	−0.675518	−0.337759	−	0.941233i	$-0.609669\pi$
$853$	−19.7293	−0.675518	−0.337759	+	0.941233i	$0.609669\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−56.9689	−1.94602	−0.973010	−	0.230764i	$-0.925877\pi$
$857$	−56.9689	−1.94602	−0.973010	+	0.230764i	$0.925877\pi$
$858$	0	0
$859$	34.9414	1.19218	0.596092	−	0.802916i	$-0.296719\pi$
$859$	34.9414	1.19218	0.596092	+	0.802916i	$0.296719\pi$
$860$	0	0
$861$	−0.949891	−0.0323722
$862$	0	0
$863$	−8.83953	−0.300901	−0.150451	−	0.988618i	$-0.548072\pi$
$863$	−8.83953	−0.300901	−0.150451	+	0.988618i	$0.548072\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−14.6797	−0.498548
$868$	0	0
$869$	46.5381	1.57870
$870$	0	0
$871$	−8.61586	−0.291937
$872$	0	0
$873$	−35.8520	−1.21341
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−12.1845	−0.411441	−0.205720	−	0.978611i	$-0.565954\pi$
$877$	−12.1845	−0.411441	−0.205720	+	0.978611i	$0.565954\pi$
$878$	0	0
$879$	4.87817	0.164537
$880$	0	0
$881$	−43.8764	−1.47823	−0.739116	−	0.673578i	$-0.764757\pi$
$881$	−43.8764	−1.47823	−0.739116	+	0.673578i	$0.764757\pi$
$882$	0	0
$883$	−41.3713	−1.39225	−0.696127	−	0.717918i	$-0.745095\pi$
$883$	−41.3713	−1.39225	−0.696127	+	0.717918i	$0.745095\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	7.03886	0.236342	0.118171	−	0.992993i	$-0.462297\pi$
$887$	7.03886	0.236342	0.118171	+	0.992993i	$0.462297\pi$
$888$	0	0
$889$	0.571292	0.0191605
$890$	0	0
$891$	22.8099	0.764160
$892$	0	0
$893$	−5.68603	−0.190276
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−2.59240	−0.0865579
$898$	0	0
$899$	−2.88843	−0.0963345
$900$	0	0
$901$	18.1672	0.605237
$902$	0	0
$903$	−2.92374	−0.0972958
$904$	0	0
$905$	0	0
$906$	0	0
$907$	−4.53105	−0.150451	−0.0752255	−	0.997167i	$-0.523968\pi$
$907$	−4.53105	−0.150451	−0.0752255	+	0.997167i	$0.523968\pi$
$908$	0	0
$909$	7.01413	0.232644
$910$	0	0
$911$	−23.0849	−0.764837	−0.382419	−	0.923989i	$-0.624909\pi$
$911$	−23.0849	−0.764837	−0.382419	+	0.923989i	$0.624909\pi$
$912$	0	0
$913$	27.2632	0.902280
$914$	0	0
$915$	0	0
$916$	0	0
$917$	1.21430	0.0400998
$918$	0	0
$919$	11.5758	0.381852	0.190926	−	0.981604i	$-0.438851\pi$
$919$	11.5758	0.381852	0.190926	+	0.981604i	$0.438851\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	22.9564	0.755620
$924$	0	0
$925$	0	0
$926$	0	0
$927$	−16.3327	−0.536437
$928$	0	0
$929$	30.8660	1.01268	0.506340	−	0.862334i	$-0.330998\pi$
$929$	30.8660	1.01268	0.506340	+	0.862334i	$0.330998\pi$
$930$	0	0
$931$	−9.07417	−0.297394
$932$	0	0
$933$	11.1711	0.365724
$934$	0	0
$935$	0	0
$936$	0	0
$937$	0.127614	0.00416896	0.00208448	−	0.999998i	$-0.499336\pi$
$937$	0.127614	0.00416896	0.00208448	+	0.999998i	$0.499336\pi$
$938$	0	0
$939$	−10.7090	−0.349475
$940$	0	0
$941$	−44.5703	−1.45295	−0.726475	−	0.687193i	$-0.758843\pi$
$941$	−44.5703	−1.45295	−0.726475	+	0.687193i	$0.758843\pi$
$942$	0	0
$943$	−1.91037	−0.0622101
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−36.5467	−1.18761	−0.593804	−	0.804610i	$-0.702375\pi$
$947$	−36.5467	−1.18761	−0.593804	+	0.804610i	$0.702375\pi$
$948$	0	0
$949$	43.9667	1.42722
$950$	0	0
$951$	10.7740	0.349370
$952$	0	0
$953$	14.9439	0.484081	0.242040	−	0.970266i	$-0.422183\pi$
$953$	14.9439	0.484081	0.242040	+	0.970266i	$0.422183\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	13.5323	0.437436
$958$	0	0
$959$	−3.52283	−0.113758
$960$	0	0
$961$	−30.2547	−0.975958
$962$	0	0
$963$	0	0
$964$	0	0
$965$	0	0
$966$	0	0
$967$	−32.3258	−1.03953	−0.519763	−	0.854310i	$-0.673980\pi$
$967$	−32.3258	−1.03953	−0.519763	+	0.854310i	$0.673980\pi$
$968$	0	0
$969$	−6.28048	−0.201758
$970$	0	0
$971$	−13.5125	−0.433638	−0.216819	−	0.976212i	$-0.569568\pi$
$971$	−13.5125	−0.433638	−0.216819	+	0.976212i	$0.569568\pi$
$972$	0	0
$973$	5.21995	0.167344
$974$	0	0
$975$	0	0
$976$	0	0
$977$	48.4226	1.54918	0.774588	−	0.632466i	$-0.217957\pi$
$977$	48.4226	1.54918	0.774588	+	0.632466i	$0.217957\pi$
$978$	0	0
$979$	11.9705	0.382579
$980$	0	0
$981$	−32.6654	−1.04293
$982$	0	0
$983$	−21.7142	−0.692576	−0.346288	−	0.938128i	$-0.612558\pi$
$983$	−21.7142	−0.692576	−0.346288	+	0.938128i	$0.612558\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−2.04457	−0.0650793
$988$	0	0
$989$	−5.88005	−0.186975
$990$	0	0
$991$	−25.6728	−0.815524	−0.407762	−	0.913088i	$-0.633691\pi$
$991$	−25.6728	−0.815524	−0.407762	+	0.913088i	$0.633691\pi$
$992$	0	0
$993$	−16.9010	−0.536336
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−32.4643	−1.02815	−0.514077	−	0.857744i	$-0.671866\pi$
$997$	−32.4643	−1.02815	−0.514077	+	0.857744i	$0.671866\pi$
$998$	0	0
$999$	15.6042	0.493696

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4600.2.a.ba.1.2	✓	4
4.3	odd	2		9200.2.a.cp.1.3		4
5.2	odd	4		4600.2.e.t.4049.6		8
5.3	odd	4		4600.2.e.t.4049.3		8
5.4	even	2		4600.2.a.bb.1.3	yes	4
20.19	odd	2		9200.2.a.cn.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4600.2.a.ba.1.2	✓	4	1.1	even	1	trivial
4600.2.a.bb.1.3	yes	4	5.4	even	2
4600.2.e.t.4049.3		8	5.3	odd	4
4600.2.e.t.4049.6		8	5.2	odd	4
9200.2.a.cn.1.2		4	20.19	odd	2
9200.2.a.cp.1.3		4	4.3	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:	\( N \)	\(=\)	\( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4600.a (trivial)

\(f(q)\)	\(=\)	\(q-0.751385 q^{3} -0.661751 q^{7} -2.43542 q^{9} +O(q^{10})\)	\(q-0.751385 q^{3} -0.661751 q^{7} -2.43542 q^{9} +5.38282 q^{11} +3.45017 q^{13} +6.04457 q^{17} +1.38282 q^{19} +0.497230 q^{21} +1.00000 q^{23} +4.08409 q^{27} -3.34579 q^{29} +0.863303 q^{31} -4.04457 q^{33} +3.82073 q^{37} -2.59240 q^{39} -1.91037 q^{41} -5.88005 q^{43} -4.11192 q^{47} -6.56209 q^{49} -4.54180 q^{51} +3.00554 q^{53} -1.03903 q^{57} -14.1094 q^{59} -8.26841 q^{61} +1.61164 q^{63} -2.49723 q^{67} -0.751385 q^{69} +6.65372 q^{71} +12.7434 q^{73} -3.56209 q^{77} +8.64568 q^{79} +4.23753 q^{81} +5.06486 q^{83} +2.51397 q^{87} +2.22384 q^{89} -2.28315 q^{91} -0.648673 q^{93} +14.7211 q^{97} -13.1094 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - q^{7} + 6 q^{9}+O(q^{10}) \) 4 * q - q^7 + 6 * q^9	\( 4 q - q^{7} + 6 q^{9} + q^{11} + 3 q^{13} + 2 q^{17} - 15 q^{19} + 8 q^{21} + 4 q^{23} + 27 q^{27} + q^{29} - 12 q^{31} + 6 q^{33} + 18 q^{37} - 3 q^{39} - 9 q^{41} - 9 q^{43} - 4 q^{47} - 3 q^{49} - 2 q^{51} + 6 q^{57} - 5 q^{59} + 14 q^{61} + 39 q^{63} - 16 q^{67} - 2 q^{71} + 21 q^{73} + 9 q^{77} - 21 q^{79} + 44 q^{81} - 9 q^{83} + 17 q^{87} - 16 q^{89} + 33 q^{91} - 29 q^{93} + 40 q^{97} - q^{99}+O(q^{100}) \) 4 * q - q^7 + 6 * q^9 + q^11 + 3 * q^13 + 2 * q^17 - 15 * q^19 + 8 * q^21 + 4 * q^23 + 27 * q^27 + q^29 - 12 * q^31 + 6 * q^33 + 18 * q^37 - 3 * q^39 - 9 * q^41 - 9 * q^43 - 4 * q^47 - 3 * q^49 - 2 * q^51 + 6 * q^57 - 5 * q^59 + 14 * q^61 + 39 * q^63 - 16 * q^67 - 2 * q^71 + 21 * q^73 + 9 * q^77 - 21 * q^79 + 44 * q^81 - 9 * q^83 + 17 * q^87 - 16 * q^89 + 33 * q^91 - 29 * q^93 + 40 * q^97 - q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more