LMFDB - Embedded newform 3800.2.a.w.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3800,2,Mod(1,3800)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3800, 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("3800.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3800 = 2^{3} \cdot 5^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3800.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:	$30.3431527681$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 760)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$2.34292$ of defining polynomial
Character	$\chi$	$=$	3800.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.34292 q^{3} -1.19656 q^{7} +2.48929 q^{9} +O(q^{10})$	$q+2.34292 q^{3} -1.19656 q^{7} +2.48929 q^{9} +4.97858 q^{11} +6.63565 q^{13} -1.48929 q^{17} +1.00000 q^{19} -2.80344 q^{21} +0.510711 q^{23} -1.19656 q^{27} -7.88240 q^{29} -2.97858 q^{31} +11.6644 q^{33} +7.14637 q^{37} +15.5468 q^{39} +1.66442 q^{41} +6.39312 q^{43} +9.95715 q^{47} -5.56825 q^{49} -3.48929 q^{51} +11.4219 q^{53} +2.34292 q^{57} -11.8396 q^{59} +3.66442 q^{61} -2.97858 q^{63} -7.61423 q^{67} +1.19656 q^{69} -13.8396 q^{73} -5.95715 q^{77} +12.6858 q^{79} -10.2713 q^{81} +8.68585 q^{83} -18.4679 q^{87} -4.87819 q^{89} -7.93994 q^{91} -6.97858 q^{93} +6.81079 q^{97} +12.3931 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} + q^{7}+O(q^{10}) $ 3 * q + q^3 + q^7	$ 3 q + q^{3} + q^{7} + 11 q^{13} + 3 q^{17} + 3 q^{19} - 13 q^{21} + 9 q^{23} + q^{27} - 7 q^{29} + 6 q^{31} + 8 q^{33} + 20 q^{37} + 3 q^{39} - 22 q^{41} + 10 q^{43} + 12 q^{49} - 3 q^{51} + 7 q^{53} + q^{57} + 11 q^{59} - 16 q^{61} + 6 q^{63} + q^{67} - q^{69} + 5 q^{73} + 12 q^{77} + 26 q^{79} - 13 q^{81} + 14 q^{83} - 33 q^{87} - 6 q^{89} + 29 q^{91} - 6 q^{93} - 8 q^{97} + 28 q^{99}+O(q^{100}) $ 3 * q + q^3 + q^7 + 11 * q^13 + 3 * q^17 + 3 * q^19 - 13 * q^21 + 9 * q^23 + q^27 - 7 * q^29 + 6 * q^31 + 8 * q^33 + 20 * q^37 + 3 * q^39 - 22 * q^41 + 10 * q^43 + 12 * q^49 - 3 * q^51 + 7 * q^53 + q^57 + 11 * q^59 - 16 * q^61 + 6 * q^63 + q^67 - q^69 + 5 * q^73 + 12 * q^77 + 26 * q^79 - 13 * q^81 + 14 * q^83 - 33 * q^87 - 6 * q^89 + 29 * q^91 - 6 * q^93 - 8 * q^97 + 28 * 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
$3$	2.34292	1.35269	0.676344	−	0.736586i	$-0.263563\pi$
$3$	2.34292	1.35269	0.676344	+	0.736586i	$0.263563\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−1.19656	−0.452256	−0.226128	−	0.974098i	$-0.572607\pi$
$7$	−1.19656	−0.452256	−0.226128	+	0.974098i	$0.572607\pi$
$8$	0	0
$9$	2.48929	0.829763
$10$	0	0
$11$	4.97858	1.50110	0.750549	−	0.660815i	$-0.229789\pi$
$11$	4.97858	1.50110	0.750549	+	0.660815i	$0.229789\pi$
$12$	0	0
$13$	6.63565	1.84040	0.920200	−	0.391449i	$-0.128026\pi$
$13$	6.63565	1.84040	0.920200	+	0.391449i	$0.128026\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−1.48929	−0.361206	−0.180603	−	0.983556i	$-0.557805\pi$
$17$	−1.48929	−0.361206	−0.180603	+	0.983556i	$0.557805\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	0	0
$21$	−2.80344	−0.611761
$22$	0	0
$23$	0.510711	0.106491	0.0532453	−	0.998581i	$-0.483043\pi$
$23$	0.510711	0.106491	0.0532453	+	0.998581i	$0.483043\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	−1.19656	−0.230278
$28$	0	0
$29$	−7.88240	−1.46373	−0.731863	−	0.681452i	$-0.761349\pi$
$29$	−7.88240	−1.46373	−0.731863	+	0.681452i	$0.761349\pi$
$30$	0	0
$31$	−2.97858	−0.534968	−0.267484	−	0.963562i	$-0.586192\pi$
$31$	−2.97858	−0.534968	−0.267484	+	0.963562i	$0.586192\pi$
$32$	0	0
$33$	11.6644	2.03052
$34$	0	0
$35$	0	0
$36$	0	0
$37$	7.14637	1.17486	0.587428	−	0.809277i	$-0.300141\pi$
$37$	7.14637	1.17486	0.587428	+	0.809277i	$0.300141\pi$
$38$	0	0
$39$	15.5468	2.48948
$40$	0	0
$41$	1.66442	0.259939	0.129970	−	0.991518i	$-0.458512\pi$
$41$	1.66442	0.259939	0.129970	+	0.991518i	$0.458512\pi$
$42$	0	0
$43$	6.39312	0.974941	0.487470	−	0.873139i	$-0.337920\pi$
$43$	6.39312	0.974941	0.487470	+	0.873139i	$0.337920\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	9.95715	1.45240	0.726200	−	0.687483i	$-0.241285\pi$
$47$	9.95715	1.45240	0.726200	+	0.687483i	$0.241285\pi$
$48$	0	0
$49$	−5.56825	−0.795464
$50$	0	0
$51$	−3.48929	−0.488598
$52$	0	0
$53$	11.4219	1.56892	0.784458	−	0.620182i	$-0.212941\pi$
$53$	11.4219	1.56892	0.784458	+	0.620182i	$0.212941\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	2.34292	0.310328
$58$	0	0
$59$	−11.8396	−1.54138	−0.770690	−	0.637211i	$-0.780088\pi$
$59$	−11.8396	−1.54138	−0.770690	+	0.637211i	$0.780088\pi$
$60$	0	0
$61$	3.66442	0.469181	0.234591	−	0.972094i	$-0.424625\pi$
$61$	3.66442	0.469181	0.234591	+	0.972094i	$0.424625\pi$
$62$	0	0
$63$	−2.97858	−0.375265
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−7.61423	−0.930226	−0.465113	−	0.885251i	$-0.653986\pi$
$67$	−7.61423	−0.930226	−0.465113	+	0.885251i	$0.653986\pi$
$68$	0	0
$69$	1.19656	0.144049
$70$	0	0
$71$	0	0		−	1.00000i	$-0.5\pi$
$71$	0	0			1.00000i	$0.5\pi$
$72$	0	0
$73$	−13.8396	−1.61980	−0.809899	−	0.586570i	$-0.800478\pi$
$73$	−13.8396	−1.61980	−0.809899	+	0.586570i	$0.800478\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−5.95715	−0.678881
$78$	0	0
$79$	12.6858	1.42727	0.713635	−	0.700518i	$-0.247048\pi$
$79$	12.6858	1.42727	0.713635	+	0.700518i	$0.247048\pi$
$80$	0	0
$81$	−10.2713	−1.14126
$82$	0	0
$83$	8.68585	0.953395	0.476698	−	0.879067i	$-0.341834\pi$
$83$	8.68585	0.953395	0.476698	+	0.879067i	$0.341834\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−18.4679	−1.97996
$88$	0	0
$89$	−4.87819	−0.517087	−0.258544	−	0.966000i	$-0.583243\pi$
$89$	−4.87819	−0.517087	−0.258544	+	0.966000i	$0.583243\pi$
$90$	0	0
$91$	−7.93994	−0.832332
$92$	0	0
$93$	−6.97858	−0.723645
$94$	0	0
$95$	0	0
$96$	0	0
$97$	6.81079	0.691531	0.345765	−	0.938321i	$-0.387619\pi$
$97$	6.81079	0.691531	0.345765	+	0.938321i	$0.387619\pi$
$98$	0	0
$99$	12.3931	1.24555
$100$	0	0
$101$	−2.29273	−0.228135	−0.114068	−	0.993473i	$-0.536388\pi$
$101$	−2.29273	−0.228135	−0.114068	+	0.993473i	$0.536388\pi$
$102$	0	0
$103$	6.51806	0.642243	0.321122	−	0.947038i	$-0.395940\pi$
$103$	6.51806	0.642243	0.321122	+	0.947038i	$0.395940\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−7.71462	−0.745800	−0.372900	−	0.927872i	$-0.621637\pi$
$107$	−7.71462	−0.745800	−0.372900	+	0.927872i	$0.621637\pi$
$108$	0	0
$109$	15.5468	1.48912	0.744558	−	0.667558i	$-0.232660\pi$
$109$	15.5468	1.48912	0.744558	+	0.667558i	$0.232660\pi$
$110$	0	0
$111$	16.7434	1.58921
$112$	0	0
$113$	0.753250	0.0708598	0.0354299	−	0.999372i	$-0.488720\pi$
$113$	0.753250	0.0708598	0.0354299	+	0.999372i	$0.488720\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	16.5181	1.52709
$118$	0	0
$119$	1.78202	0.163357
$120$	0	0
$121$	13.7862	1.25329
$122$	0	0
$123$	3.89962	0.351617
$124$	0	0
$125$	0	0
$126$	0	0
$127$	−10.4177	−0.924419	−0.462210	−	0.886771i	$-0.652943\pi$
$127$	−10.4177	−0.924419	−0.462210	+	0.886771i	$0.652943\pi$
$128$	0	0
$129$	14.9786	1.31879
$130$	0	0
$131$	−17.9572	−1.56892	−0.784462	−	0.620177i	$-0.787061\pi$
$131$	−17.9572	−1.56892	−0.784462	+	0.620177i	$0.787061\pi$
$132$	0	0
$133$	−1.19656	−0.103755
$134$	0	0
$135$	0	0
$136$	0	0
$137$	9.83956	0.840650	0.420325	−	0.907374i	$-0.361916\pi$
$137$	9.83956	0.840650	0.420325	+	0.907374i	$0.361916\pi$
$138$	0	0
$139$	−0.978577	−0.0830018	−0.0415009	−	0.999138i	$-0.513214\pi$
$139$	−0.978577	−0.0830018	−0.0415009	+	0.999138i	$0.513214\pi$
$140$	0	0
$141$	23.3288	1.96464
$142$	0	0
$143$	33.0361	2.76262
$144$	0	0
$145$	0	0
$146$	0	0
$147$	−13.0460	−1.07601
$148$	0	0
$149$	−8.24989	−0.675857	−0.337928	−	0.941172i	$-0.609726\pi$
$149$	−8.24989	−0.675857	−0.337928	+	0.941172i	$0.609726\pi$
$150$	0	0
$151$	13.8568	1.12765	0.563824	−	0.825895i	$-0.309330\pi$
$151$	13.8568	1.12765	0.563824	+	0.825895i	$0.309330\pi$
$152$	0	0
$153$	−3.70727	−0.299715
$154$	0	0
$155$	0	0
$156$	0	0
$157$	15.7073	1.25358	0.626788	−	0.779190i	$-0.284369\pi$
$157$	15.7073	1.25358	0.626788	+	0.779190i	$0.284369\pi$
$158$	0	0
$159$	26.7606	2.12225
$160$	0	0
$161$	−0.611096	−0.0481611
$162$	0	0
$163$	−5.80765	−0.454891	−0.227445	−	0.973791i	$-0.573037\pi$
$163$	−5.80765	−0.454891	−0.227445	+	0.973791i	$0.573037\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	13.8322	1.07037	0.535184	−	0.844735i	$-0.320242\pi$
$167$	13.8322	1.07037	0.535184	+	0.844735i	$0.320242\pi$
$168$	0	0
$169$	31.0319	2.38707
$170$	0	0
$171$	2.48929	0.190361
$172$	0	0
$173$	−14.8108	−1.12604	−0.563022	−	0.826442i	$-0.690361\pi$
$173$	−14.8108	−1.12604	−0.563022	+	0.826442i	$0.690361\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−27.7392	−2.08500
$178$	0	0
$179$	15.6644	1.17081	0.585407	−	0.810740i	$-0.300935\pi$
$179$	15.6644	1.17081	0.585407	+	0.810740i	$0.300935\pi$
$180$	0	0
$181$	−14.7862	−1.09905	−0.549526	−	0.835477i	$-0.685192\pi$
$181$	−14.7862	−1.09905	−0.549526	+	0.835477i	$0.685192\pi$
$182$	0	0
$183$	8.58546	0.634656
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−7.41454	−0.542205
$188$	0	0
$189$	1.43175	0.104144
$190$	0	0
$191$	−6.36748	−0.460735	−0.230367	−	0.973104i	$-0.573993\pi$
$191$	−6.36748	−0.460735	−0.230367	+	0.973104i	$0.573993\pi$
$192$	0	0
$193$	19.5970	1.41062	0.705312	−	0.708897i	$-0.250807\pi$
$193$	19.5970	1.41062	0.705312	+	0.708897i	$0.250807\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−10.7862	−0.768487	−0.384244	−	0.923232i	$-0.625538\pi$
$197$	−10.7862	−0.768487	−0.384244	+	0.923232i	$0.625538\pi$
$198$	0	0
$199$	2.80344	0.198731	0.0993654	−	0.995051i	$-0.468319\pi$
$199$	2.80344	0.198731	0.0993654	+	0.995051i	$0.468319\pi$
$200$	0	0
$201$	−17.8396	−1.25831
$202$	0	0
$203$	9.43175	0.661979
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1.27131	0.0883620
$208$	0	0
$209$	4.97858	0.344375
$210$	0	0
$211$	−11.2541	−0.774764	−0.387382	−	0.921919i	$-0.626621\pi$
$211$	−11.2541	−0.774764	−0.387382	+	0.921919i	$0.626621\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	3.56404	0.241943
$218$	0	0
$219$	−32.4250	−2.19108
$220$	0	0
$221$	−9.88240	−0.664762
$222$	0	0
$223$	−17.4966	−1.17166	−0.585831	−	0.810433i	$-0.699232\pi$
$223$	−17.4966	−1.17166	−0.585831	+	0.810433i	$0.699232\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	3.84942	0.255495	0.127748	−	0.991807i	$-0.459225\pi$
$227$	3.84942	0.255495	0.127748	+	0.991807i	$0.459225\pi$
$228$	0	0
$229$	−14.6430	−0.967637	−0.483818	−	0.875168i	$-0.660750\pi$
$229$	−14.6430	−0.967637	−0.483818	+	0.875168i	$0.660750\pi$
$230$	0	0
$231$	−13.9572	−0.918313
$232$	0	0
$233$	−11.9572	−0.783339	−0.391670	−	0.920106i	$-0.628102\pi$
$233$	−11.9572	−0.783339	−0.391670	+	0.920106i	$0.628102\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	29.7220	1.93065
$238$	0	0
$239$	15.5897	1.00841	0.504206	−	0.863583i	$-0.331785\pi$
$239$	15.5897	1.00841	0.504206	+	0.863583i	$0.331785\pi$
$240$	0	0
$241$	−16.0575	−1.03436	−0.517178	−	0.855878i	$-0.673018\pi$
$241$	−16.0575	−1.03436	−0.517178	+	0.855878i	$0.673018\pi$
$242$	0	0
$243$	−20.4752	−1.31349
$244$	0	0
$245$	0	0
$246$	0	0
$247$	6.63565	0.422217
$248$	0	0
$249$	20.3503	1.28965
$250$	0	0
$251$	23.9143	1.50946	0.754729	−	0.656037i	$-0.227768\pi$
$251$	23.9143	1.50946	0.754729	+	0.656037i	$0.227768\pi$
$252$	0	0
$253$	2.54262	0.159853
$254$	0	0
$255$	0	0
$256$	0	0
$257$	24.4324	1.52405	0.762025	−	0.647548i	$-0.224206\pi$
$257$	24.4324	1.52405	0.762025	+	0.647548i	$0.224206\pi$
$258$	0	0
$259$	−8.55104	−0.531336
$260$	0	0
$261$	−19.6216	−1.21455
$262$	0	0
$263$	14.4935	0.893707	0.446854	−	0.894607i	$-0.352544\pi$
$263$	14.4935	0.893707	0.446854	+	0.894607i	$0.352544\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	−11.4292	−0.699458
$268$	0	0
$269$	−14.0000	−0.853595	−0.426798	−	0.904347i	$-0.640358\pi$
$269$	−14.0000	−0.853595	−0.426798	+	0.904347i	$0.640358\pi$
$270$	0	0
$271$	−6.76060	−0.410677	−0.205339	−	0.978691i	$-0.565830\pi$
$271$	−6.76060	−0.410677	−0.205339	+	0.978691i	$0.565830\pi$
$272$	0	0
$273$	−18.6027	−1.12589
$274$	0	0
$275$	0	0
$276$	0	0
$277$	25.4292	1.52789	0.763947	−	0.645279i	$-0.223259\pi$
$277$	25.4292	1.52789	0.763947	+	0.645279i	$0.223259\pi$
$278$	0	0
$279$	−7.41454	−0.443897
$280$	0	0
$281$	9.07896	0.541605	0.270803	−	0.962635i	$-0.412711\pi$
$281$	9.07896	0.541605	0.270803	+	0.962635i	$0.412711\pi$
$282$	0	0
$283$	−12.0147	−0.714199	−0.357100	−	0.934066i	$-0.616234\pi$
$283$	−12.0147	−0.714199	−0.357100	+	0.934066i	$0.616234\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−1.99158	−0.117559
$288$	0	0
$289$	−14.7820	−0.869531
$290$	0	0
$291$	15.9572	0.935425
$292$	0	0
$293$	−23.6718	−1.38292	−0.691460	−	0.722415i	$-0.743032\pi$
$293$	−23.6718	−1.38292	−0.691460	+	0.722415i	$0.743032\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	−5.95715	−0.345669
$298$	0	0
$299$	3.38890	0.195985
$300$	0	0
$301$	−7.64973	−0.440923
$302$	0	0
$303$	−5.37169	−0.308596
$304$	0	0
$305$	0	0
$306$	0	0
$307$	16.0246	0.914570	0.457285	−	0.889320i	$-0.348822\pi$
$307$	16.0246	0.914570	0.457285	+	0.889320i	$0.348822\pi$
$308$	0	0
$309$	15.2713	0.868754
$310$	0	0
$311$	−27.5468	−1.56204	−0.781019	−	0.624508i	$-0.785300\pi$
$311$	−27.5468	−1.56204	−0.781019	+	0.624508i	$0.785300\pi$
$312$	0	0
$313$	−10.1751	−0.575133	−0.287566	−	0.957761i	$-0.592846\pi$
$313$	−10.1751	−0.575133	−0.287566	+	0.957761i	$0.592846\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−29.6289	−1.66413	−0.832063	−	0.554681i	$-0.812840\pi$
$317$	−29.6289	−1.66413	−0.832063	+	0.554681i	$0.812840\pi$
$318$	0	0
$319$	−39.2432	−2.19719
$320$	0	0
$321$	−18.0748	−1.00883
$322$	0	0
$323$	−1.48929	−0.0828662
$324$	0	0
$325$	0	0
$326$	0	0
$327$	36.4250	2.01431
$328$	0	0
$329$	−11.9143	−0.656857
$330$	0	0
$331$	30.4679	1.67467	0.837333	−	0.546694i	$-0.184114\pi$
$331$	30.4679	1.67467	0.837333	+	0.546694i	$0.184114\pi$
$332$	0	0
$333$	17.7894	0.974851
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−26.1396	−1.42392	−0.711958	−	0.702222i	$-0.752192\pi$
$337$	−26.1396	−1.42392	−0.711958	+	0.702222i	$0.752192\pi$
$338$	0	0
$339$	1.76481	0.0958512
$340$	0	0
$341$	−14.8291	−0.803039
$342$	0	0
$343$	15.0386	0.812010
$344$	0	0
$345$	0	0
$346$	0	0
$347$	7.31415	0.392644	0.196322	−	0.980539i	$-0.437100\pi$
$347$	7.31415	0.392644	0.196322	+	0.980539i	$0.437100\pi$
$348$	0	0
$349$	0.628308	0.0336325	0.0168163	−	0.999859i	$-0.494647\pi$
$349$	0.628308	0.0336325	0.0168163	+	0.999859i	$0.494647\pi$
$350$	0	0
$351$	−7.93994	−0.423803
$352$	0	0
$353$	8.23267	0.438181	0.219090	−	0.975705i	$-0.429691\pi$
$353$	8.23267	0.438181	0.219090	+	0.975705i	$0.429691\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	4.17513	0.220972
$358$	0	0
$359$	12.1323	0.640318	0.320159	−	0.947364i	$-0.396264\pi$
$359$	12.1323	0.640318	0.320159	+	0.947364i	$0.396264\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0	0
$363$	32.3001	1.69531
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−17.9572	−0.937356	−0.468678	−	0.883369i	$-0.655270\pi$
$367$	−17.9572	−0.937356	−0.468678	+	0.883369i	$0.655270\pi$
$368$	0	0
$369$	4.14323	0.215688
$370$	0	0
$371$	−13.6669	−0.709552
$372$	0	0
$373$	−29.6289	−1.53413	−0.767064	−	0.641571i	$-0.778283\pi$
$373$	−29.6289	−1.53413	−0.767064	+	0.641571i	$0.778283\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−52.3049	−2.69384
$378$	0	0
$379$	−6.80344	−0.349469	−0.174735	−	0.984616i	$-0.555907\pi$
$379$	−6.80344	−0.349469	−0.174735	+	0.984616i	$0.555907\pi$
$380$	0	0
$381$	−24.4078	−1.25045
$382$	0	0
$383$	20.2253	1.03347	0.516733	−	0.856147i	$-0.327148\pi$
$383$	20.2253	1.03347	0.516733	+	0.856147i	$0.327148\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	15.9143	0.808970
$388$	0	0
$389$	0.393115	0.0199317	0.00996587	−	0.999950i	$-0.496828\pi$
$389$	0.393115	0.0199317	0.00996587	+	0.999950i	$0.496828\pi$
$390$	0	0
$391$	−0.760597	−0.0384650
$392$	0	0
$393$	−42.0722	−2.12226
$394$	0	0
$395$	0	0
$396$	0	0
$397$	−17.4292	−0.874748	−0.437374	−	0.899280i	$-0.644091\pi$
$397$	−17.4292	−0.874748	−0.437374	+	0.899280i	$0.644091\pi$
$398$	0	0
$399$	−2.80344	−0.140348
$400$	0	0
$401$	13.1281	0.655585	0.327792	−	0.944750i	$-0.393695\pi$
$401$	13.1281	0.655585	0.327792	+	0.944750i	$0.393695\pi$
$402$	0	0
$403$	−19.7648	−0.984555
$404$	0	0
$405$	0	0
$406$	0	0
$407$	35.5787	1.76357
$408$	0	0
$409$	−18.7862	−0.928919	−0.464460	−	0.885594i	$-0.653751\pi$
$409$	−18.7862	−0.928919	−0.464460	+	0.885594i	$0.653751\pi$
$410$	0	0
$411$	23.0533	1.13714
$412$	0	0
$413$	14.1667	0.697098
$414$	0	0
$415$	0	0
$416$	0	0
$417$	−2.29273	−0.112276
$418$	0	0
$419$	−5.22219	−0.255121	−0.127560	−	0.991831i	$-0.540715\pi$
$419$	−5.22219	−0.255121	−0.127560	+	0.991831i	$0.540715\pi$
$420$	0	0
$421$	−7.68164	−0.374380	−0.187190	−	0.982324i	$-0.559938\pi$
$421$	−7.68164	−0.374380	−0.187190	+	0.982324i	$0.559938\pi$
$422$	0	0
$423$	24.7862	1.20515
$424$	0	0
$425$	0	0
$426$	0	0
$427$	−4.38469	−0.212190
$428$	0	0
$429$	77.4011	3.73696
$430$	0	0
$431$	−17.7073	−0.852929	−0.426465	−	0.904504i	$-0.640241\pi$
$431$	−17.7073	−0.852929	−0.426465	+	0.904504i	$0.640241\pi$
$432$	0	0
$433$	−17.9901	−0.864551	−0.432275	−	0.901742i	$-0.642289\pi$
$433$	−17.9901	−0.864551	−0.432275	+	0.901742i	$0.642289\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	0.510711	0.0244306
$438$	0	0
$439$	14.8438	0.708454	0.354227	−	0.935159i	$-0.384744\pi$
$439$	14.8438	0.708454	0.354227	+	0.935159i	$0.384744\pi$
$440$	0	0
$441$	−13.8610	−0.660047
$442$	0	0
$443$	−22.4078	−1.06463	−0.532314	−	0.846547i	$-0.678677\pi$
$443$	−22.4078	−1.06463	−0.532314	+	0.846547i	$0.678677\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	−19.3288	−0.914223
$448$	0	0
$449$	−40.7434	−1.92280	−0.961400	−	0.275156i	$-0.911271\pi$
$449$	−40.7434	−1.92280	−0.961400	+	0.275156i	$0.911271\pi$
$450$	0	0
$451$	8.28646	0.390194
$452$	0	0
$453$	32.4653	1.52536
$454$	0	0
$455$	0	0
$456$	0	0
$457$	−8.01721	−0.375029	−0.187515	−	0.982262i	$-0.560043\pi$
$457$	−8.01721	−0.375029	−0.187515	+	0.982262i	$0.560043\pi$
$458$	0	0
$459$	1.78202	0.0831775
$460$	0	0
$461$	−25.1281	−1.17033	−0.585166	−	0.810914i	$-0.698971\pi$
$461$	−25.1281	−1.17033	−0.585166	+	0.810914i	$0.698971\pi$
$462$	0	0
$463$	31.7795	1.47692	0.738459	−	0.674298i	$-0.235554\pi$
$463$	31.7795	1.47692	0.738459	+	0.674298i	$0.235554\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−20.9210	−0.968110	−0.484055	−	0.875038i	$-0.660837\pi$
$467$	−20.9210	−0.968110	−0.484055	+	0.875038i	$0.660837\pi$
$468$	0	0
$469$	9.11087	0.420701
$470$	0	0
$471$	36.8009	1.69570
$472$	0	0
$473$	31.8286	1.46348
$474$	0	0
$475$	0	0
$476$	0	0
$477$	28.4324	1.30183
$478$	0	0
$479$	27.7220	1.26665	0.633324	−	0.773886i	$-0.281690\pi$
$479$	27.7220	1.26665	0.633324	+	0.773886i	$0.281690\pi$
$480$	0	0
$481$	47.4208	2.16220
$482$	0	0
$483$	−1.43175	−0.0651469
$484$	0	0
$485$	0	0
$486$	0	0
$487$	−3.20390	−0.145183	−0.0725914	−	0.997362i	$-0.523127\pi$
$487$	−3.20390	−0.145183	−0.0725914	+	0.997362i	$0.523127\pi$
$488$	0	0
$489$	−13.6069	−0.615325
$490$	0	0
$491$	−21.5212	−0.971238	−0.485619	−	0.874171i	$-0.661406\pi$
$491$	−21.5212	−0.971238	−0.485619	+	0.874171i	$0.661406\pi$
$492$	0	0
$493$	11.7392	0.528706
$494$	0	0
$495$	0	0
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−23.1709	−1.03727	−0.518637	−	0.854995i	$-0.673560\pi$
$499$	−23.1709	−1.03727	−0.518637	+	0.854995i	$0.673560\pi$
$500$	0	0
$501$	32.4078	1.44787
$502$	0	0
$503$	1.88240	0.0839322	0.0419661	−	0.999119i	$-0.486638\pi$
$503$	1.88240	0.0839322	0.0419661	+	0.999119i	$0.486638\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	72.7054	3.22896
$508$	0	0
$509$	5.46365	0.242172	0.121086	−	0.992642i	$-0.461362\pi$
$509$	5.46365	0.242172	0.121086	+	0.992642i	$0.461362\pi$
$510$	0	0
$511$	16.5598	0.732564
$512$	0	0
$513$	−1.19656	−0.0528293
$514$	0	0
$515$	0	0
$516$	0	0
$517$	49.5725	2.18019
$518$	0	0
$519$	−34.7005	−1.52318
$520$	0	0
$521$	11.2222	0.491653	0.245827	−	0.969314i	$-0.420941\pi$
$521$	11.2222	0.491653	0.245827	+	0.969314i	$0.420941\pi$
$522$	0	0
$523$	29.1867	1.27624	0.638122	−	0.769935i	$-0.279711\pi$
$523$	29.1867	1.27624	0.638122	+	0.769935i	$0.279711\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	4.43596	0.193233
$528$	0	0
$529$	−22.7392	−0.988660
$530$	0	0
$531$	−29.4721	−1.27898
$532$	0	0
$533$	11.0445	0.478392
$534$	0	0
$535$	0	0
$536$	0	0
$537$	36.7005	1.58375
$538$	0	0
$539$	−27.7220	−1.19407
$540$	0	0
$541$	20.2499	0.870611	0.435305	−	0.900283i	$-0.356640\pi$
$541$	20.2499	0.870611	0.435305	+	0.900283i	$0.356640\pi$
$542$	0	0
$543$	−34.6430	−1.48667
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−42.8830	−1.83355	−0.916773	−	0.399409i	$-0.869215\pi$
$547$	−42.8830	−1.83355	−0.916773	+	0.399409i	$0.869215\pi$
$548$	0	0
$549$	9.12181	0.389309
$550$	0	0
$551$	−7.88240	−0.335802
$552$	0	0
$553$	−15.1793	−0.645491
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−28.2583	−1.19734	−0.598671	−	0.800995i	$-0.704304\pi$
$557$	−28.2583	−1.19734	−0.598671	+	0.800995i	$0.704304\pi$
$558$	0	0
$559$	42.4225	1.79428
$560$	0	0
$561$	−17.3717	−0.733433
$562$	0	0
$563$	20.2253	0.852396	0.426198	−	0.904630i	$-0.359853\pi$
$563$	20.2253	0.852396	0.426198	+	0.904630i	$0.359853\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	12.2902	0.516140
$568$	0	0
$569$	−31.1365	−1.30531	−0.652655	−	0.757655i	$-0.726345\pi$
$569$	−31.1365	−1.30531	−0.652655	+	0.757655i	$0.726345\pi$
$570$	0	0
$571$	27.4868	1.15029	0.575143	−	0.818053i	$-0.304947\pi$
$571$	27.4868	1.15029	0.575143	+	0.818053i	$0.304947\pi$
$572$	0	0
$573$	−14.9185	−0.623230
$574$	0	0
$575$	0	0
$576$	0	0
$577$	−23.7820	−0.990058	−0.495029	−	0.868876i	$-0.664843\pi$
$577$	−23.7820	−0.990058	−0.495029	+	0.868876i	$0.664843\pi$
$578$	0	0
$579$	45.9143	1.90813
$580$	0	0
$581$	−10.3931	−0.431179
$582$	0	0
$583$	56.8647	2.35510
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−24.0147	−0.991192	−0.495596	−	0.868553i	$-0.665050\pi$
$587$	−24.0147	−0.991192	−0.495596	+	0.868553i	$0.665050\pi$
$588$	0	0
$589$	−2.97858	−0.122730
$590$	0	0
$591$	−25.2713	−1.03952
$592$	0	0
$593$	−46.6148	−1.91424	−0.957121	−	0.289688i	$-0.906449\pi$
$593$	−46.6148	−1.91424	−0.957121	+	0.289688i	$0.906449\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	6.56825	0.268821
$598$	0	0
$599$	−44.6002	−1.82231	−0.911156	−	0.412061i	$-0.864809\pi$
$599$	−44.6002	−1.82231	−0.911156	+	0.412061i	$0.864809\pi$
$600$	0	0
$601$	−17.5787	−0.717051	−0.358526	−	0.933520i	$-0.616720\pi$
$601$	−17.5787	−0.717051	−0.358526	+	0.933520i	$0.616720\pi$
$602$	0	0
$603$	−18.9540	−0.771867
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−37.2467	−1.51180	−0.755899	−	0.654688i	$-0.772800\pi$
$607$	−37.2467	−1.51180	−0.755899	+	0.654688i	$0.772800\pi$
$608$	0	0
$609$	22.0979	0.895451
$610$	0	0
$611$	66.0722	2.67300
$612$	0	0
$613$	14.8866	0.601265	0.300632	−	0.953740i	$-0.402802\pi$
$613$	14.8866	0.601265	0.300632	+	0.953740i	$0.402802\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−3.70727	−0.149249	−0.0746245	−	0.997212i	$-0.523776\pi$
$617$	−3.70727	−0.149249	−0.0746245	+	0.997212i	$0.523776\pi$
$618$	0	0
$619$	17.7648	0.714028	0.357014	−	0.934099i	$-0.383795\pi$
$619$	17.7648	0.714028	0.357014	+	0.934099i	$0.383795\pi$
$620$	0	0
$621$	−0.611096	−0.0245224
$622$	0	0
$623$	5.83704	0.233856
$624$	0	0
$625$	0	0
$626$	0	0
$627$	11.6644	0.465832
$628$	0	0
$629$	−10.6430	−0.424364
$630$	0	0
$631$	40.3074	1.60461	0.802307	−	0.596912i	$-0.203606\pi$
$631$	40.3074	1.60461	0.802307	+	0.596912i	$0.203606\pi$
$632$	0	0
$633$	−26.3675	−1.04801
$634$	0	0
$635$	0	0
$636$	0	0
$637$	−36.9490	−1.46397
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−27.6728	−1.09301	−0.546506	−	0.837455i	$-0.684042\pi$
$641$	−27.6728	−1.09301	−0.546506	+	0.837455i	$0.684042\pi$
$642$	0	0
$643$	−19.0277	−0.750379	−0.375190	−	0.926948i	$-0.622422\pi$
$643$	−19.0277	−0.750379	−0.375190	+	0.926948i	$0.622422\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	32.0403	1.25964	0.629818	−	0.776743i	$-0.283130\pi$
$647$	32.0403	1.25964	0.629818	+	0.776743i	$0.283130\pi$
$648$	0	0
$649$	−58.9442	−2.31376
$650$	0	0
$651$	8.35027	0.327273
$652$	0	0
$653$	−25.3142	−0.990619	−0.495310	−	0.868716i	$-0.664945\pi$
$653$	−25.3142	−0.990619	−0.495310	+	0.868716i	$0.664945\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−34.4507	−1.34405
$658$	0	0
$659$	−0.860981	−0.0335391	−0.0167695	−	0.999859i	$-0.505338\pi$
$659$	−0.860981	−0.0335391	−0.0167695	+	0.999859i	$0.505338\pi$
$660$	0	0
$661$	4.55356	0.177113	0.0885564	−	0.996071i	$-0.471775\pi$
$661$	4.55356	0.177113	0.0885564	+	0.996071i	$0.471775\pi$
$662$	0	0
$663$	−23.1537	−0.899216
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−4.02563	−0.155873
$668$	0	0
$669$	−40.9933	−1.58489
$670$	0	0
$671$	18.2436	0.704287
$672$	0	0
$673$	11.7465	0.452795	0.226398	−	0.974035i	$-0.427305\pi$
$673$	11.7465	0.452795	0.226398	+	0.974035i	$0.427305\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	22.2854	0.856497	0.428248	−	0.903661i	$-0.359131\pi$
$677$	22.2854	0.856497	0.428248	+	0.903661i	$0.359131\pi$
$678$	0	0
$679$	−8.14950	−0.312749
$680$	0	0
$681$	9.01890	0.345605
$682$	0	0
$683$	7.23833	0.276967	0.138483	−	0.990365i	$-0.455777\pi$
$683$	7.23833	0.276967	0.138483	+	0.990365i	$0.455777\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	−34.3074	−1.30891
$688$	0	0
$689$	75.7917	2.88743
$690$	0	0
$691$	32.7434	1.24562	0.622809	−	0.782374i	$-0.285992\pi$
$691$	32.7434	1.24562	0.622809	+	0.782374i	$0.285992\pi$
$692$	0	0
$693$	−14.8291	−0.563310
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−2.47881	−0.0938915
$698$	0	0
$699$	−28.0147	−1.05961
$700$	0	0
$701$	−29.4208	−1.11121	−0.555604	−	0.831447i	$-0.687513\pi$
$701$	−29.4208	−1.11121	−0.555604	+	0.831447i	$0.687513\pi$
$702$	0	0
$703$	7.14637	0.269530
$704$	0	0
$705$	0	0
$706$	0	0
$707$	2.74338	0.103176
$708$	0	0
$709$	−5.32885	−0.200129	−0.100065	−	0.994981i	$-0.531905\pi$
$709$	−5.32885	−0.200129	−0.100065	+	0.994981i	$0.531905\pi$
$710$	0	0
$711$	31.5787	1.18429
$712$	0	0
$713$	−1.52119	−0.0569691
$714$	0	0
$715$	0	0
$716$	0	0
$717$	36.5254	1.36407
$718$	0	0
$719$	8.45317	0.315250	0.157625	−	0.987499i	$-0.449616\pi$
$719$	8.45317	0.315250	0.157625	+	0.987499i	$0.449616\pi$
$720$	0	0
$721$	−7.79923	−0.290459
$722$	0	0
$723$	−37.6216	−1.39916
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−2.56825	−0.0952511	−0.0476256	−	0.998865i	$-0.515165\pi$
$727$	−2.56825	−0.0952511	−0.0476256	+	0.998865i	$0.515165\pi$
$728$	0	0
$729$	−17.1579	−0.635479
$730$	0	0
$731$	−9.52119	−0.352154
$732$	0	0
$733$	−35.5212	−1.31201	−0.656003	−	0.754759i	$-0.727754\pi$
$733$	−35.5212	−1.31201	−0.656003	+	0.754759i	$0.727754\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−37.9080	−1.39636
$738$	0	0
$739$	−37.0508	−1.36294	−0.681468	−	0.731848i	$-0.738658\pi$
$739$	−37.0508	−1.36294	−0.681468	+	0.731848i	$0.738658\pi$
$740$	0	0
$741$	15.5468	0.571127
$742$	0	0
$743$	−23.2039	−0.851269	−0.425634	−	0.904895i	$-0.639949\pi$
$743$	−23.2039	−0.851269	−0.425634	+	0.904895i	$0.639949\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	21.6216	0.791092
$748$	0	0
$749$	9.23098	0.337293
$750$	0	0
$751$	51.2285	1.86935	0.934677	−	0.355499i	$-0.115689\pi$
$751$	51.2285	1.86935	0.934677	+	0.355499i	$0.115689\pi$
$752$	0	0
$753$	56.0294	2.04182
$754$	0	0
$755$	0	0
$756$	0	0
$757$	43.3717	1.57637	0.788185	−	0.615438i	$-0.211021\pi$
$757$	43.3717	1.57637	0.788185	+	0.615438i	$0.211021\pi$
$758$	0	0
$759$	5.95715	0.216231
$760$	0	0
$761$	24.7753	0.898104	0.449052	−	0.893506i	$-0.351762\pi$
$761$	24.7753	0.898104	0.449052	+	0.893506i	$0.351762\pi$
$762$	0	0
$763$	−18.6027	−0.673462
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−78.5632	−2.83675
$768$	0	0
$769$	−11.4893	−0.414314	−0.207157	−	0.978308i	$-0.566421\pi$
$769$	−11.4893	−0.414314	−0.207157	+	0.978308i	$0.566421\pi$
$770$	0	0
$771$	57.2432	2.06156
$772$	0	0
$773$	−27.0863	−0.974227	−0.487113	−	0.873339i	$-0.661950\pi$
$773$	−27.0863	−0.974227	−0.487113	+	0.873339i	$0.661950\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−20.0344	−0.718731
$778$	0	0
$779$	1.66442	0.0596342
$780$	0	0
$781$	0	0
$782$	0	0
$783$	9.43175	0.337063
$784$	0	0
$785$	0	0
$786$	0	0
$787$	−13.4563	−0.479666	−0.239833	−	0.970814i	$-0.577093\pi$
$787$	−13.4563	−0.479666	−0.239833	+	0.970814i	$0.577093\pi$
$788$	0	0
$789$	33.9572	1.20891
$790$	0	0
$791$	−0.901307	−0.0320468
$792$	0	0
$793$	24.3158	0.863481
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−30.4496	−1.07858	−0.539290	−	0.842120i	$-0.681307\pi$
$797$	−30.4496	−1.07858	−0.539290	+	0.842120i	$0.681307\pi$
$798$	0	0
$799$	−14.8291	−0.524615
$800$	0	0
$801$	−12.1432	−0.429060
$802$	0	0
$803$	−68.9013	−2.43147
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−32.8009	−1.15465
$808$	0	0
$809$	−29.7047	−1.04436	−0.522182	−	0.852834i	$-0.674882\pi$
$809$	−29.7047	−1.04436	−0.522182	+	0.852834i	$0.674882\pi$
$810$	0	0
$811$	32.2902	1.13386	0.566931	−	0.823765i	$-0.308130\pi$
$811$	32.2902	1.13386	0.566931	+	0.823765i	$0.308130\pi$
$812$	0	0
$813$	−15.8396	−0.555518
$814$	0	0
$815$	0	0
$816$	0	0
$817$	6.39312	0.223667
$818$	0	0
$819$	−19.7648	−0.690638
$820$	0	0
$821$	9.06427	0.316345	0.158173	−	0.987411i	$-0.449440\pi$
$821$	9.06427	0.316345	0.158173	+	0.987411i	$0.449440\pi$
$822$	0	0
$823$	7.35448	0.256361	0.128181	−	0.991751i	$-0.459086\pi$
$823$	7.35448	0.256361	0.128181	+	0.991751i	$0.459086\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−2.02204	−0.0703132	−0.0351566	−	0.999382i	$-0.511193\pi$
$827$	−2.02204	−0.0703132	−0.0351566	+	0.999382i	$0.511193\pi$
$828$	0	0
$829$	−36.3675	−1.26309	−0.631547	−	0.775337i	$-0.717580\pi$
$829$	−36.3675	−1.26309	−0.631547	+	0.775337i	$0.717580\pi$
$830$	0	0
$831$	59.5787	2.06676
$832$	0	0
$833$	8.29273	0.287326
$834$	0	0
$835$	0	0
$836$	0	0
$837$	3.56404	0.123191
$838$	0	0
$839$	25.8223	0.891486	0.445743	−	0.895161i	$-0.352939\pi$
$839$	25.8223	0.891486	0.445743	+	0.895161i	$0.352939\pi$
$840$	0	0
$841$	33.1323	1.14249
$842$	0	0
$843$	21.2713	0.732623
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−16.4960	−0.566810
$848$	0	0
$849$	−28.1495	−0.966088
$850$	0	0
$851$	3.64973	0.125111
$852$	0	0
$853$	−8.54262	−0.292494	−0.146247	−	0.989248i	$-0.546719\pi$
$853$	−8.54262	−0.292494	−0.146247	+	0.989248i	$0.546719\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−27.8322	−0.950730	−0.475365	−	0.879789i	$-0.657684\pi$
$857$	−27.8322	−0.950730	−0.475365	+	0.879789i	$0.657684\pi$
$858$	0	0
$859$	33.4868	1.14255	0.571277	−	0.820757i	$-0.306448\pi$
$859$	33.4868	1.14255	0.571277	+	0.820757i	$0.306448\pi$
$860$	0	0
$861$	−4.66611	−0.159021
$862$	0	0
$863$	29.2614	0.996071	0.498036	−	0.867157i	$-0.334055\pi$
$863$	29.2614	0.996071	0.498036	+	0.867157i	$0.334055\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	−34.6331	−1.17620
$868$	0	0
$869$	63.1575	2.14247
$870$	0	0
$871$	−50.5254	−1.71199
$872$	0	0
$873$	16.9540	0.573807
$874$	0	0
$875$	0	0
$876$	0	0
$877$	46.7852	1.57982	0.789911	−	0.613221i	$-0.210127\pi$
$877$	46.7852	1.57982	0.789911	+	0.613221i	$0.210127\pi$
$878$	0	0
$879$	−55.4611	−1.87066
$880$	0	0
$881$	−30.7581	−1.03627	−0.518133	−	0.855300i	$-0.673373\pi$
$881$	−30.7581	−1.03627	−0.518133	+	0.855300i	$0.673373\pi$
$882$	0	0
$883$	−0.435961	−0.0146713	−0.00733563	−	0.999973i	$-0.502335\pi$
$883$	−0.435961	−0.0146713	−0.00733563	+	0.999973i	$0.502335\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	2.15310	0.0722939	0.0361469	−	0.999346i	$-0.488492\pi$
$887$	2.15310	0.0722939	0.0361469	+	0.999346i	$0.488492\pi$
$888$	0	0
$889$	12.4653	0.418074
$890$	0	0
$891$	−51.1365	−1.71314
$892$	0	0
$893$	9.95715	0.333203
$894$	0	0
$895$	0	0
$896$	0	0
$897$	7.93994	0.265107
$898$	0	0
$899$	23.4783	0.783047
$900$	0	0
$901$	−17.0105	−0.566701
$902$	0	0
$903$	−17.9227	−0.596431
$904$	0	0
$905$	0	0
$906$	0	0
$907$	36.9002	1.22525	0.612626	−	0.790373i	$-0.290113\pi$
$907$	36.9002	1.22525	0.612626	+	0.790373i	$0.290113\pi$
$908$	0	0
$909$	−5.70727	−0.189298
$910$	0	0
$911$	−36.9504	−1.22422	−0.612111	−	0.790772i	$-0.709679\pi$
$911$	−36.9504	−1.22422	−0.612111	+	0.790772i	$0.709679\pi$
$912$	0	0
$913$	43.2432	1.43114
$914$	0	0
$915$	0	0
$916$	0	0
$917$	21.4868	0.709556
$918$	0	0
$919$	19.1537	0.631823	0.315911	−	0.948789i	$-0.397690\pi$
$919$	19.1537	0.631823	0.315911	+	0.948789i	$0.397690\pi$
$920$	0	0
$921$	37.5443	1.23713
$922$	0	0
$923$	0	0
$924$	0	0
$925$	0	0
$926$	0	0
$927$	16.2253	0.532910
$928$	0	0
$929$	−56.9185	−1.86744	−0.933718	−	0.358009i	$-0.883456\pi$
$929$	−56.9185	−1.86744	−0.933718	+	0.358009i	$0.883456\pi$
$930$	0	0
$931$	−5.56825	−0.182492
$932$	0	0
$933$	−64.5401	−2.11295
$934$	0	0
$935$	0	0
$936$	0	0
$937$	24.1176	0.787888	0.393944	−	0.919135i	$-0.371110\pi$
$937$	24.1176	0.787888	0.393944	+	0.919135i	$0.371110\pi$
$938$	0	0
$939$	−23.8396	−0.777975
$940$	0	0
$941$	3.19656	0.104205	0.0521024	−	0.998642i	$-0.483408\pi$
$941$	3.19656	0.104205	0.0521024	+	0.998642i	$0.483408\pi$
$942$	0	0
$943$	0.850040	0.0276811
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−3.63504	−0.118123	−0.0590614	−	0.998254i	$-0.518811\pi$
$947$	−3.63504	−0.118123	−0.0590614	+	0.998254i	$0.518811\pi$
$948$	0	0
$949$	−91.8345	−2.98107
$950$	0	0
$951$	−69.4183	−2.25104
$952$	0	0
$953$	48.0821	1.55753	0.778766	−	0.627315i	$-0.215846\pi$
$953$	48.0821	1.55753	0.778766	+	0.627315i	$0.215846\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−91.9437	−2.97212
$958$	0	0
$959$	−11.7736	−0.380189
$960$	0	0
$961$	−22.1281	−0.713809
$962$	0	0
$963$	−19.2039	−0.618837
$964$	0	0
$965$	0	0
$966$	0	0
$967$	23.4637	0.754540	0.377270	−	0.926103i	$-0.376863\pi$
$967$	23.4637	0.754540	0.377270	+	0.926103i	$0.376863\pi$
$968$	0	0
$969$	−3.48929	−0.112092
$970$	0	0
$971$	17.4868	0.561177	0.280589	−	0.959828i	$-0.409470\pi$
$971$	17.4868	0.561177	0.280589	+	0.959828i	$0.409470\pi$
$972$	0	0
$973$	1.17092	0.0375381
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−20.2316	−0.647266	−0.323633	−	0.946183i	$-0.604904\pi$
$977$	−20.2316	−0.647266	−0.323633	+	0.946183i	$0.604904\pi$
$978$	0	0
$979$	−24.2865	−0.776199
$980$	0	0
$981$	38.7005	1.23561
$982$	0	0
$983$	−28.1249	−0.897046	−0.448523	−	0.893771i	$-0.648050\pi$
$983$	−28.1249	−0.897046	−0.448523	+	0.893771i	$0.648050\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	−27.9143	−0.888522
$988$	0	0
$989$	3.26504	0.103822
$990$	0	0
$991$	−33.9718	−1.07915	−0.539576	−	0.841937i	$-0.681415\pi$
$991$	−33.9718	−1.07915	−0.539576	+	0.841937i	$0.681415\pi$
$992$	0	0
$993$	71.3839	2.26530
$994$	0	0
$995$	0	0
$996$	0	0
$997$	23.8223	0.754461	0.377231	−	0.926119i	$-0.376876\pi$
$997$	23.8223	0.754461	0.377231	+	0.926119i	$0.376876\pi$
$998$	0	0
$999$	−8.55104	−0.270543

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3800.2.a.w.1.3		3
4.3	odd	2		7600.2.a.bp.1.1		3
5.2	odd	4		3800.2.d.n.3649.1		6
5.3	odd	4		3800.2.d.n.3649.6		6
5.4	even	2		760.2.a.i.1.1	✓	3
15.14	odd	2		6840.2.a.bm.1.2		3
20.19	odd	2		1520.2.a.q.1.3		3
40.19	odd	2		6080.2.a.br.1.1		3
40.29	even	2		6080.2.a.bx.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
760.2.a.i.1.1	✓	3	5.4	even	2
1520.2.a.q.1.3		3	20.19	odd	2
3800.2.a.w.1.3		3	1.1	even	1	trivial
3800.2.d.n.3649.1		6	5.2	odd	4
3800.2.d.n.3649.6		6	5.3	odd	4
6080.2.a.br.1.1		3	40.19	odd	2
6080.2.a.bx.1.3		3	40.29	even	2
6840.2.a.bm.1.2		3	15.14	odd	2
7600.2.a.bp.1.1		3	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:	\( N \)	\(=\)	\( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3800.a (trivial)

\(f(q)\)	\(=\)	\(q+2.34292 q^{3} -1.19656 q^{7} +2.48929 q^{9} +O(q^{10})\)	\(q+2.34292 q^{3} -1.19656 q^{7} +2.48929 q^{9} +4.97858 q^{11} +6.63565 q^{13} -1.48929 q^{17} +1.00000 q^{19} -2.80344 q^{21} +0.510711 q^{23} -1.19656 q^{27} -7.88240 q^{29} -2.97858 q^{31} +11.6644 q^{33} +7.14637 q^{37} +15.5468 q^{39} +1.66442 q^{41} +6.39312 q^{43} +9.95715 q^{47} -5.56825 q^{49} -3.48929 q^{51} +11.4219 q^{53} +2.34292 q^{57} -11.8396 q^{59} +3.66442 q^{61} -2.97858 q^{63} -7.61423 q^{67} +1.19656 q^{69} -13.8396 q^{73} -5.95715 q^{77} +12.6858 q^{79} -10.2713 q^{81} +8.68585 q^{83} -18.4679 q^{87} -4.87819 q^{89} -7.93994 q^{91} -6.97858 q^{93} +6.81079 q^{97} +12.3931 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} + q^{7}+O(q^{10}) \) 3 * q + q^3 + q^7	\( 3 q + q^{3} + q^{7} + 11 q^{13} + 3 q^{17} + 3 q^{19} - 13 q^{21} + 9 q^{23} + q^{27} - 7 q^{29} + 6 q^{31} + 8 q^{33} + 20 q^{37} + 3 q^{39} - 22 q^{41} + 10 q^{43} + 12 q^{49} - 3 q^{51} + 7 q^{53} + q^{57} + 11 q^{59} - 16 q^{61} + 6 q^{63} + q^{67} - q^{69} + 5 q^{73} + 12 q^{77} + 26 q^{79} - 13 q^{81} + 14 q^{83} - 33 q^{87} - 6 q^{89} + 29 q^{91} - 6 q^{93} - 8 q^{97} + 28 q^{99}+O(q^{100}) \) 3 * q + q^3 + q^7 + 11 * q^13 + 3 * q^17 + 3 * q^19 - 13 * q^21 + 9 * q^23 + q^27 - 7 * q^29 + 6 * q^31 + 8 * q^33 + 20 * q^37 + 3 * q^39 - 22 * q^41 + 10 * q^43 + 12 * q^49 - 3 * q^51 + 7 * q^53 + q^57 + 11 * q^59 - 16 * q^61 + 6 * q^63 + q^67 - q^69 + 5 * q^73 + 12 * q^77 + 26 * q^79 - 13 * q^81 + 14 * q^83 - 33 * q^87 - 6 * q^89 + 29 * q^91 - 6 * q^93 - 8 * q^97 + 28 * q^99

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