LMFDB - Embedded newform 6028.2.a.e.1.11

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6028,2,Mod(1,6028)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6028, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("6028.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6028 = 2^{2} \cdot 11 \cdot 137 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6028.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:	$48.1338223384$
Analytic rank:	$0$
Dimension:	$27$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.11
Character	$\chi$	$=$	6028.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.700606 q^{3} +3.28744 q^{5} -1.66773 q^{7} -2.50915 q^{9} +O(q^{10})$	$q-0.700606 q^{3} +3.28744 q^{5} -1.66773 q^{7} -2.50915 q^{9} -1.00000 q^{11} -6.85014 q^{13} -2.30320 q^{15} +3.53862 q^{17} +0.358170 q^{19} +1.16842 q^{21} +5.56883 q^{23} +5.80724 q^{25} +3.85974 q^{27} -0.870870 q^{29} -4.81797 q^{31} +0.700606 q^{33} -5.48257 q^{35} -1.10810 q^{37} +4.79924 q^{39} +10.1016 q^{41} +1.82313 q^{43} -8.24868 q^{45} -0.204844 q^{47} -4.21867 q^{49} -2.47918 q^{51} -13.2389 q^{53} -3.28744 q^{55} -0.250936 q^{57} -2.13329 q^{59} -0.535298 q^{61} +4.18460 q^{63} -22.5194 q^{65} +2.91807 q^{67} -3.90155 q^{69} +9.21797 q^{71} +10.8949 q^{73} -4.06858 q^{75} +1.66773 q^{77} +15.0471 q^{79} +4.82330 q^{81} +9.57441 q^{83} +11.6330 q^{85} +0.610137 q^{87} +2.33036 q^{89} +11.4242 q^{91} +3.37549 q^{93} +1.17746 q^{95} +16.5704 q^{97} +2.50915 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 27 q + 5 q^{3} - 2 q^{5} + 7 q^{7} + 28 q^{9}+O(q^{10}) $ 27 * q + 5 * q^3 - 2 * q^5 + 7 * q^7 + 28 * q^9	$ 27 q + 5 q^{3} - 2 q^{5} + 7 q^{7} + 28 q^{9} - 27 q^{11} + 2 q^{13} + 10 q^{15} + 17 q^{17} + 4 q^{19} - 4 q^{21} + 25 q^{23} + 21 q^{25} + 32 q^{27} - q^{29} + 8 q^{31} - 5 q^{33} + 30 q^{35} - 24 q^{37} + 10 q^{39} + 31 q^{41} + 17 q^{43} - 15 q^{45} + 27 q^{47} + 40 q^{49} + 30 q^{51} + 2 q^{55} + 13 q^{57} + 31 q^{59} + 2 q^{61} + 61 q^{63} + 2 q^{67} - 15 q^{69} + 40 q^{71} + 17 q^{73} + 30 q^{75} - 7 q^{77} + 33 q^{79} + 31 q^{81} + 66 q^{83} + 26 q^{85} + 43 q^{87} + 22 q^{89} + 18 q^{91} - 9 q^{93} + 95 q^{95} - 21 q^{97} - 28 q^{99}+O(q^{100}) $ 27 * q + 5 * q^3 - 2 * q^5 + 7 * q^7 + 28 * q^9 - 27 * q^11 + 2 * q^13 + 10 * q^15 + 17 * q^17 + 4 * q^19 - 4 * q^21 + 25 * q^23 + 21 * q^25 + 32 * q^27 - q^29 + 8 * q^31 - 5 * q^33 + 30 * q^35 - 24 * q^37 + 10 * q^39 + 31 * q^41 + 17 * q^43 - 15 * q^45 + 27 * q^47 + 40 * q^49 + 30 * q^51 + 2 * q^55 + 13 * q^57 + 31 * q^59 + 2 * q^61 + 61 * q^63 + 2 * q^67 - 15 * q^69 + 40 * q^71 + 17 * q^73 + 30 * q^75 - 7 * q^77 + 33 * q^79 + 31 * q^81 + 66 * q^83 + 26 * q^85 + 43 * q^87 + 22 * q^89 + 18 * q^91 - 9 * q^93 + 95 * q^95 - 21 * 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
$3$	−0.700606	−0.404495	−0.202247	−	0.979334i	$-0.564825\pi$
$3$	−0.700606	−0.404495	−0.202247	+	0.979334i	$0.564825\pi$
$4$	0	0
$5$	3.28744	1.47019	0.735093	−	0.677966i	$-0.237138\pi$
$5$	3.28744	1.47019	0.735093	+	0.677966i	$0.237138\pi$
$6$	0	0
$7$	−1.66773	−0.630344	−0.315172	−	0.949035i	$-0.602062\pi$
$7$	−1.66773	−0.630344	−0.315172	+	0.949035i	$0.602062\pi$
$8$	0	0
$9$	−2.50915	−0.836384
$10$	0	0
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−6.85014	−1.89989	−0.949943	−	0.312424i	$-0.898859\pi$
$13$	−6.85014	−1.89989	−0.949943	+	0.312424i	$0.898859\pi$
$14$	0	0
$15$	−2.30320	−0.594683
$16$	0	0
$17$	3.53862	0.858241	0.429121	−	0.903247i	$-0.358824\pi$
$17$	3.53862	0.858241	0.429121	+	0.903247i	$0.358824\pi$
$18$	0	0
$19$	0.358170	0.0821699	0.0410849	−	0.999156i	$-0.486919\pi$
$19$	0.358170	0.0821699	0.0410849	+	0.999156i	$0.486919\pi$
$20$	0	0
$21$	1.16842	0.254971
$22$	0	0
$23$	5.56883	1.16118	0.580591	−	0.814196i	$-0.302822\pi$
$23$	5.56883	1.16118	0.580591	+	0.814196i	$0.302822\pi$
$24$	0	0
$25$	5.80724	1.16145
$26$	0	0
$27$	3.85974	0.742808
$28$	0	0
$29$	−0.870870	−0.161717	−0.0808583	−	0.996726i	$-0.525766\pi$
$29$	−0.870870	−0.161717	−0.0808583	+	0.996726i	$0.525766\pi$
$30$	0	0
$31$	−4.81797	−0.865332	−0.432666	−	0.901554i	$-0.642427\pi$
$31$	−4.81797	−0.865332	−0.432666	+	0.901554i	$0.642427\pi$
$32$	0	0
$33$	0.700606	0.121960
$34$	0	0
$35$	−5.48257	−0.926723
$36$	0	0
$37$	−1.10810	−0.182171	−0.0910855	−	0.995843i	$-0.529034\pi$
$37$	−1.10810	−0.182171	−0.0910855	+	0.995843i	$0.529034\pi$
$38$	0	0
$39$	4.79924	0.768494
$40$	0	0
$41$	10.1016	1.57761	0.788806	−	0.614642i	$-0.210700\pi$
$41$	10.1016	1.57761	0.788806	+	0.614642i	$0.210700\pi$
$42$	0	0
$43$	1.82313	0.278025	0.139013	−	0.990291i	$-0.455607\pi$
$43$	1.82313	0.278025	0.139013	+	0.990291i	$0.455607\pi$
$44$	0	0
$45$	−8.24868	−1.22964
$46$	0	0
$47$	−0.204844	−0.0298795	−0.0149397	−	0.999888i	$-0.504756\pi$
$47$	−0.204844	−0.0298795	−0.0149397	+	0.999888i	$0.504756\pi$
$48$	0	0
$49$	−4.21867	−0.602666
$50$	0	0
$51$	−2.47918	−0.347154
$52$	0	0
$53$	−13.2389	−1.81851	−0.909253	−	0.416245i	$-0.863346\pi$
$53$	−13.2389	−1.81851	−0.909253	+	0.416245i	$0.863346\pi$
$54$	0	0
$55$	−3.28744	−0.443278
$56$	0	0
$57$	−0.250936	−0.0332373
$58$	0	0
$59$	−2.13329	−0.277730	−0.138865	−	0.990311i	$-0.544345\pi$
$59$	−2.13329	−0.277730	−0.138865	+	0.990311i	$0.544345\pi$
$60$	0	0
$61$	−0.535298	−0.0685379	−0.0342689	−	0.999413i	$-0.510910\pi$
$61$	−0.535298	−0.0685379	−0.0342689	+	0.999413i	$0.510910\pi$
$62$	0	0
$63$	4.18460	0.527210
$64$	0	0
$65$	−22.5194	−2.79319
$66$	0	0
$67$	2.91807	0.356499	0.178249	−	0.983985i	$-0.442957\pi$
$67$	2.91807	0.356499	0.178249	+	0.983985i	$0.442957\pi$
$68$	0	0
$69$	−3.90155	−0.469692
$70$	0	0
$71$	9.21797	1.09397	0.546986	−	0.837142i	$-0.315775\pi$
$71$	9.21797	1.09397	0.546986	+	0.837142i	$0.315775\pi$
$72$	0	0
$73$	10.8949	1.27516	0.637578	−	0.770386i	$-0.279936\pi$
$73$	10.8949	1.27516	0.637578	+	0.770386i	$0.279936\pi$
$74$	0	0
$75$	−4.06858	−0.469799
$76$	0	0
$77$	1.66773	0.190056
$78$	0	0
$79$	15.0471	1.69293	0.846464	−	0.532446i	$-0.178727\pi$
$79$	15.0471	1.69293	0.846464	+	0.532446i	$0.178727\pi$
$80$	0	0
$81$	4.82330	0.535922
$82$	0	0
$83$	9.57441	1.05093	0.525464	−	0.850816i	$-0.323892\pi$
$83$	9.57441	1.05093	0.525464	+	0.850816i	$0.323892\pi$
$84$	0	0
$85$	11.6330	1.26177
$86$	0	0
$87$	0.610137	0.0654135
$88$	0	0
$89$	2.33036	0.247018	0.123509	−	0.992343i	$-0.460585\pi$
$89$	2.33036	0.247018	0.123509	+	0.992343i	$0.460585\pi$
$90$	0	0
$91$	11.4242	1.19758
$92$	0	0
$93$	3.37549	0.350022
$94$	0	0
$95$	1.17746	0.120805
$96$	0	0
$97$	16.5704	1.68247	0.841234	−	0.540671i	$-0.181830\pi$
$97$	16.5704	1.68247	0.841234	+	0.540671i	$0.181830\pi$
$98$	0	0
$99$	2.50915	0.252179
$100$	0	0
$101$	−13.3073	−1.32412	−0.662062	−	0.749449i	$-0.730319\pi$
$101$	−13.3073	−1.32412	−0.662062	+	0.749449i	$0.730319\pi$
$102$	0	0
$103$	5.67714	0.559385	0.279692	−	0.960090i	$-0.409768\pi$
$103$	5.67714	0.559385	0.279692	+	0.960090i	$0.409768\pi$
$104$	0	0
$105$	3.84112	0.374855
$106$	0	0
$107$	3.68730	0.356465	0.178233	−	0.983988i	$-0.442962\pi$
$107$	3.68730	0.356465	0.178233	+	0.983988i	$0.442962\pi$
$108$	0	0
$109$	2.18921	0.209689	0.104844	−	0.994489i	$-0.466566\pi$
$109$	2.18921	0.209689	0.104844	+	0.994489i	$0.466566\pi$
$110$	0	0
$111$	0.776343	0.0736872
$112$	0	0
$113$	5.18829	0.488074	0.244037	−	0.969766i	$-0.421528\pi$
$113$	5.18829	0.488074	0.244037	+	0.969766i	$0.421528\pi$
$114$	0	0
$115$	18.3072	1.70715
$116$	0	0
$117$	17.1880	1.58903
$118$	0	0
$119$	−5.90147	−0.540987
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−7.07727	−0.638136
$124$	0	0
$125$	2.65374	0.237357
$126$	0	0
$127$	4.15172	0.368406	0.184203	−	0.982888i	$-0.441030\pi$
$127$	4.15172	0.368406	0.184203	+	0.982888i	$0.441030\pi$
$128$	0	0
$129$	−1.27730	−0.112460
$130$	0	0
$131$	16.5639	1.44719	0.723596	−	0.690224i	$-0.242488\pi$
$131$	16.5639	1.44719	0.723596	+	0.690224i	$0.242488\pi$
$132$	0	0
$133$	−0.597332	−0.0517953
$134$	0	0
$135$	12.6887	1.09207
$136$	0	0
$137$	−1.00000	−0.0854358
$137$	−1.00000	−0.0854358
$138$	0	0
$139$	0.145613	0.0123507	0.00617537	−	0.999981i	$-0.498034\pi$
$139$	0.145613	0.0123507	0.00617537	+	0.999981i	$0.498034\pi$
$140$	0	0
$141$	0.143515	0.0120861
$142$	0	0
$143$	6.85014	0.572837
$144$	0	0
$145$	−2.86293	−0.237753
$146$	0	0
$147$	2.95562	0.243776
$148$	0	0
$149$	3.39155	0.277847	0.138923	−	0.990303i	$-0.455636\pi$
$149$	3.39155	0.277847	0.138923	+	0.990303i	$0.455636\pi$
$150$	0	0
$151$	8.05906	0.655837	0.327919	−	0.944706i	$-0.393653\pi$
$151$	8.05906	0.655837	0.327919	+	0.944706i	$0.393653\pi$
$152$	0	0
$153$	−8.87893	−0.717819
$154$	0	0
$155$	−15.8388	−1.27220
$156$	0	0
$157$	−5.27505	−0.420994	−0.210497	−	0.977594i	$-0.567508\pi$
$157$	−5.27505	−0.420994	−0.210497	+	0.977594i	$0.567508\pi$
$158$	0	0
$159$	9.27526	0.735576
$160$	0	0
$161$	−9.28732	−0.731944
$162$	0	0
$163$	−13.5515	−1.06143	−0.530717	−	0.847549i	$-0.678077\pi$
$163$	−13.5515	−1.06143	−0.530717	+	0.847549i	$0.678077\pi$
$164$	0	0
$165$	2.30320	0.179304
$166$	0	0
$167$	−9.59968	−0.742845	−0.371423	−	0.928464i	$-0.621130\pi$
$167$	−9.59968	−0.742845	−0.371423	+	0.928464i	$0.621130\pi$
$168$	0	0
$169$	33.9244	2.60957
$170$	0	0
$171$	−0.898703	−0.0687256
$172$	0	0
$173$	−21.0118	−1.59750	−0.798750	−	0.601663i	$-0.794505\pi$
$173$	−21.0118	−1.59750	−0.798750	+	0.601663i	$0.794505\pi$
$174$	0	0
$175$	−9.68492	−0.732111
$176$	0	0
$177$	1.49459	0.112340
$178$	0	0
$179$	10.0967	0.754666	0.377333	−	0.926078i	$-0.376841\pi$
$179$	10.0967	0.754666	0.377333	+	0.926078i	$0.376841\pi$
$180$	0	0
$181$	−10.8949	−0.809811	−0.404905	−	0.914359i	$-0.632696\pi$
$181$	−10.8949	−0.809811	−0.404905	+	0.914359i	$0.632696\pi$
$182$	0	0
$183$	0.375033	0.0277232
$184$	0	0
$185$	−3.64282	−0.267825
$186$	0	0
$187$	−3.53862	−0.258769
$188$	0	0
$189$	−6.43702	−0.468225
$190$	0	0
$191$	−18.8014	−1.36042	−0.680211	−	0.733017i	$-0.738112\pi$
$191$	−18.8014	−1.36042	−0.680211	+	0.733017i	$0.738112\pi$
$192$	0	0
$193$	8.52772	0.613839	0.306920	−	0.951735i	$-0.400702\pi$
$193$	8.52772	0.613839	0.306920	+	0.951735i	$0.400702\pi$
$194$	0	0
$195$	15.7772	1.12983
$196$	0	0
$197$	2.47256	0.176162	0.0880812	−	0.996113i	$-0.471926\pi$
$197$	2.47256	0.176162	0.0880812	+	0.996113i	$0.471926\pi$
$198$	0	0
$199$	14.7793	1.04768	0.523838	−	0.851818i	$-0.324500\pi$
$199$	14.7793	1.04768	0.523838	+	0.851818i	$0.324500\pi$
$200$	0	0
$201$	−2.04441	−0.144202
$202$	0	0
$203$	1.45238	0.101937
$204$	0	0
$205$	33.2085	2.31938
$206$	0	0
$207$	−13.9730	−0.971193
$208$	0	0
$209$	−0.358170	−0.0247752
$210$	0	0
$211$	3.36313	0.231527	0.115764	−	0.993277i	$-0.463069\pi$
$211$	3.36313	0.231527	0.115764	+	0.993277i	$0.463069\pi$
$212$	0	0
$213$	−6.45816	−0.442506
$214$	0	0
$215$	5.99343	0.408749
$216$	0	0
$217$	8.03508	0.545457
$218$	0	0
$219$	−7.63306	−0.515794
$220$	0	0
$221$	−24.2400	−1.63056
$222$	0	0
$223$	0.543869	0.0364201	0.0182101	−	0.999834i	$-0.494203\pi$
$223$	0.543869	0.0364201	0.0182101	+	0.999834i	$0.494203\pi$
$224$	0	0
$225$	−14.5712	−0.971416
$226$	0	0
$227$	28.3543	1.88194	0.940970	−	0.338490i	$-0.109916\pi$
$227$	28.3543	1.88194	0.940970	+	0.338490i	$0.109916\pi$
$228$	0	0
$229$	−2.77846	−0.183606	−0.0918030	−	0.995777i	$-0.529263\pi$
$229$	−2.77846	−0.183606	−0.0918030	+	0.995777i	$0.529263\pi$
$230$	0	0
$231$	−1.16842	−0.0768766
$232$	0	0
$233$	19.9581	1.30750	0.653749	−	0.756712i	$-0.273195\pi$
$233$	19.9581	1.30750	0.653749	+	0.756712i	$0.273195\pi$
$234$	0	0
$235$	−0.673410	−0.0439284
$236$	0	0
$237$	−10.5421	−0.684781
$238$	0	0
$239$	15.8371	1.02442	0.512209	−	0.858861i	$-0.328827\pi$
$239$	15.8371	1.02442	0.512209	+	0.858861i	$0.328827\pi$
$240$	0	0
$241$	20.3260	1.30931	0.654657	−	0.755926i	$-0.272813\pi$
$241$	20.3260	1.30931	0.654657	+	0.755926i	$0.272813\pi$
$242$	0	0
$243$	−14.9585	−0.959586
$244$	0	0
$245$	−13.8686	−0.886032
$246$	0	0
$247$	−2.45351	−0.156113
$248$	0	0
$249$	−6.70789	−0.425095
$250$	0	0
$251$	−15.6894	−0.990303	−0.495152	−	0.868807i	$-0.664887\pi$
$251$	−15.6894	−0.990303	−0.495152	+	0.868807i	$0.664887\pi$
$252$	0	0
$253$	−5.56883	−0.350109
$254$	0	0
$255$	−8.15014	−0.510381
$256$	0	0
$257$	−5.51792	−0.344198	−0.172099	−	0.985080i	$-0.555055\pi$
$257$	−5.51792	−0.344198	−0.172099	+	0.985080i	$0.555055\pi$
$258$	0	0
$259$	1.84802	0.114830
$260$	0	0
$261$	2.18515	0.135257
$262$	0	0
$263$	−15.7696	−0.972394	−0.486197	−	0.873849i	$-0.661616\pi$
$263$	−15.7696	−0.972394	−0.486197	+	0.873849i	$0.661616\pi$
$264$	0	0
$265$	−43.5221	−2.67354
$266$	0	0
$267$	−1.63267	−0.0999176
$268$	0	0
$269$	−1.75340	−0.106907	−0.0534534	−	0.998570i	$-0.517023\pi$
$269$	−1.75340	−0.106907	−0.0534534	+	0.998570i	$0.517023\pi$
$270$	0	0
$271$	23.6275	1.43527	0.717633	−	0.696421i	$-0.245225\pi$
$271$	23.6275	1.43527	0.717633	+	0.696421i	$0.245225\pi$
$272$	0	0
$273$	−8.00386	−0.484416
$274$	0	0
$275$	−5.80724	−0.350189
$276$	0	0
$277$	−15.5901	−0.936720	−0.468360	−	0.883538i	$-0.655155\pi$
$277$	−15.5901	−0.936720	−0.468360	+	0.883538i	$0.655155\pi$
$278$	0	0
$279$	12.0890	0.723750
$280$	0	0
$281$	14.1135	0.841940	0.420970	−	0.907075i	$-0.361690\pi$
$281$	14.1135	0.841940	0.420970	+	0.907075i	$0.361690\pi$
$282$	0	0
$283$	2.41050	0.143290	0.0716448	−	0.997430i	$-0.477175\pi$
$283$	2.41050	0.143290	0.0716448	+	0.997430i	$0.477175\pi$
$284$	0	0
$285$	−0.824936	−0.0488650
$286$	0	0
$287$	−16.8469	−0.994438
$288$	0	0
$289$	−4.47817	−0.263422
$290$	0	0
$291$	−11.6093	−0.680550
$292$	0	0
$293$	1.06977	0.0624968	0.0312484	−	0.999512i	$-0.490052\pi$
$293$	1.06977	0.0624968	0.0312484	+	0.999512i	$0.490052\pi$
$294$	0	0
$295$	−7.01304	−0.408315
$296$	0	0
$297$	−3.85974	−0.223965
$298$	0	0
$299$	−38.1472	−2.20611
$300$	0	0
$301$	−3.04050	−0.175251
$302$	0	0
$303$	9.32316	0.535602
$304$	0	0
$305$	−1.75976	−0.100763
$306$	0	0
$307$	23.1435	1.32087	0.660433	−	0.750885i	$-0.270373\pi$
$307$	23.1435	1.32087	0.660433	+	0.750885i	$0.270373\pi$
$308$	0	0
$309$	−3.97743	−0.226268
$310$	0	0
$311$	1.05967	0.0600882	0.0300441	−	0.999549i	$-0.490435\pi$
$311$	1.05967	0.0600882	0.0300441	+	0.999549i	$0.490435\pi$
$312$	0	0
$313$	−31.0591	−1.75556	−0.877782	−	0.479060i	$-0.840978\pi$
$313$	−31.0591	−1.75556	−0.877782	+	0.479060i	$0.840978\pi$
$314$	0	0
$315$	13.7566	0.775096
$316$	0	0
$317$	17.6361	0.990540	0.495270	−	0.868739i	$-0.335069\pi$
$317$	17.6361	0.990540	0.495270	+	0.868739i	$0.335069\pi$
$318$	0	0
$319$	0.870870	0.0487594
$320$	0	0
$321$	−2.58335	−0.144188
$322$	0	0
$323$	1.26743	0.0705216
$324$	0	0
$325$	−39.7804	−2.20662
$326$	0	0
$327$	−1.53378	−0.0848180
$328$	0	0
$329$	0.341624	0.0188344
$330$	0	0
$331$	18.8553	1.03638	0.518191	−	0.855265i	$-0.326606\pi$
$331$	18.8553	1.03638	0.518191	+	0.855265i	$0.326606\pi$
$332$	0	0
$333$	2.78040	0.152365
$334$	0	0
$335$	9.59296	0.524119
$336$	0	0
$337$	7.16705	0.390414	0.195207	−	0.980762i	$-0.437462\pi$
$337$	7.16705	0.390414	0.195207	+	0.980762i	$0.437462\pi$
$338$	0	0
$339$	−3.63495	−0.197423
$340$	0	0
$341$	4.81797	0.260907
$342$	0	0
$343$	18.7097	1.01023
$344$	0	0
$345$	−12.8261	−0.690535
$346$	0	0
$347$	28.1336	1.51029	0.755145	−	0.655558i	$-0.227567\pi$
$347$	28.1336	1.51029	0.755145	+	0.655558i	$0.227567\pi$
$348$	0	0
$349$	−2.34285	−0.125410	−0.0627050	−	0.998032i	$-0.519973\pi$
$349$	−2.34285	−0.125410	−0.0627050	+	0.998032i	$0.519973\pi$
$350$	0	0
$351$	−26.4398	−1.41125
$352$	0	0
$353$	29.9916	1.59629	0.798145	−	0.602466i	$-0.205815\pi$
$353$	29.9916	1.59629	0.798145	+	0.602466i	$0.205815\pi$
$354$	0	0
$355$	30.3035	1.60834
$356$	0	0
$357$	4.13461	0.218827
$358$	0	0
$359$	−12.8125	−0.676220	−0.338110	−	0.941107i	$-0.609788\pi$
$359$	−12.8125	−0.676220	−0.338110	+	0.941107i	$0.609788\pi$
$360$	0	0
$361$	−18.8717	−0.993248
$362$	0	0
$363$	−0.700606	−0.0367723
$364$	0	0
$365$	35.8164	1.87472
$366$	0	0
$367$	12.1896	0.636294	0.318147	−	0.948041i	$-0.396939\pi$
$367$	12.1896	0.636294	0.318147	+	0.948041i	$0.396939\pi$
$368$	0	0
$369$	−25.3466	−1.31949
$370$	0	0
$371$	22.0790	1.14628
$372$	0	0
$373$	−6.83562	−0.353935	−0.176968	−	0.984217i	$-0.556629\pi$
$373$	−6.83562	−0.353935	−0.176968	+	0.984217i	$0.556629\pi$
$374$	0	0
$375$	−1.85922	−0.0960098
$376$	0	0
$377$	5.96558	0.307243
$378$	0	0
$379$	8.90420	0.457378	0.228689	−	0.973500i	$-0.426556\pi$
$379$	8.90420	0.457378	0.228689	+	0.973500i	$0.426556\pi$
$380$	0	0
$381$	−2.90872	−0.149018
$382$	0	0
$383$	22.4087	1.14503	0.572514	−	0.819895i	$-0.305968\pi$
$383$	22.4087	1.14503	0.572514	+	0.819895i	$0.305968\pi$
$384$	0	0
$385$	5.48257	0.279417
$386$	0	0
$387$	−4.57452	−0.232536
$388$	0	0
$389$	−32.2001	−1.63261	−0.816306	−	0.577620i	$-0.803981\pi$
$389$	−32.2001	−1.63261	−0.816306	+	0.577620i	$0.803981\pi$
$390$	0	0
$391$	19.7060	0.996574
$392$	0	0
$393$	−11.6047	−0.585382
$394$	0	0
$395$	49.4663	2.48892
$396$	0	0
$397$	14.5518	0.730334	0.365167	−	0.930942i	$-0.381012\pi$
$397$	14.5518	0.730334	0.365167	+	0.930942i	$0.381012\pi$
$398$	0	0
$399$	0.418495	0.0209509
$400$	0	0
$401$	−15.6393	−0.780987	−0.390494	−	0.920606i	$-0.627696\pi$
$401$	−15.6393	−0.780987	−0.390494	+	0.920606i	$0.627696\pi$
$402$	0	0
$403$	33.0037	1.64403
$404$	0	0
$405$	15.8563	0.787905
$406$	0	0
$407$	1.10810	0.0549266
$408$	0	0
$409$	−9.89235	−0.489145	−0.244573	−	0.969631i	$-0.578648\pi$
$409$	−9.89235	−0.489145	−0.244573	+	0.969631i	$0.578648\pi$
$410$	0	0
$411$	0.700606	0.0345583
$412$	0	0
$413$	3.55775	0.175066
$414$	0	0
$415$	31.4753	1.54506
$416$	0	0
$417$	−0.102017	−0.00499581
$418$	0	0
$419$	27.2632	1.33189	0.665947	−	0.745999i	$-0.268028\pi$
$419$	27.2632	1.33189	0.665947	+	0.745999i	$0.268028\pi$
$420$	0	0
$421$	13.1817	0.642437	0.321219	−	0.947005i	$-0.395908\pi$
$421$	13.1817	0.642437	0.321219	+	0.947005i	$0.395908\pi$
$422$	0	0
$423$	0.513983	0.0249907
$424$	0	0
$425$	20.5496	0.996802
$426$	0	0
$427$	0.892734	0.0432024
$428$	0	0
$429$	−4.79924	−0.231710
$430$	0	0
$431$	−0.889964	−0.0428680	−0.0214340	−	0.999770i	$-0.506823\pi$
$431$	−0.889964	−0.0428680	−0.0214340	+	0.999770i	$0.506823\pi$
$432$	0	0
$433$	−18.8506	−0.905902	−0.452951	−	0.891535i	$-0.649629\pi$
$433$	−18.8506	−0.905902	−0.452951	+	0.891535i	$0.649629\pi$
$434$	0	0
$435$	2.00579	0.0961700
$436$	0	0
$437$	1.99459	0.0954141
$438$	0	0
$439$	−28.6796	−1.36880	−0.684402	−	0.729105i	$-0.739937\pi$
$439$	−28.6796	−1.36880	−0.684402	+	0.729105i	$0.739937\pi$
$440$	0	0
$441$	10.5853	0.504061
$442$	0	0
$443$	−30.8611	−1.46626	−0.733128	−	0.680091i	$-0.761940\pi$
$443$	−30.8611	−1.46626	−0.733128	+	0.680091i	$0.761940\pi$
$444$	0	0
$445$	7.66092	0.363163
$446$	0	0
$447$	−2.37614	−0.112388
$448$	0	0
$449$	0.716183	0.0337988	0.0168994	−	0.999857i	$-0.494621\pi$
$449$	0.716183	0.0337988	0.0168994	+	0.999857i	$0.494621\pi$
$450$	0	0
$451$	−10.1016	−0.475668
$452$	0	0
$453$	−5.64623	−0.265283
$454$	0	0
$455$	37.5563	1.76067
$456$	0	0
$457$	7.89489	0.369307	0.184654	−	0.982804i	$-0.440884\pi$
$457$	7.89489	0.369307	0.184654	+	0.982804i	$0.440884\pi$
$458$	0	0
$459$	13.6582	0.637508
$460$	0	0
$461$	−12.7002	−0.591508	−0.295754	−	0.955264i	$-0.595571\pi$
$461$	−12.7002	−0.591508	−0.295754	+	0.955264i	$0.595571\pi$
$462$	0	0
$463$	30.8214	1.43239	0.716196	−	0.697899i	$-0.245882\pi$
$463$	30.8214	1.43239	0.716196	+	0.697899i	$0.245882\pi$
$464$	0	0
$465$	11.0967	0.514598
$466$	0	0
$467$	−0.324568	−0.0150192	−0.00750960	−	0.999972i	$-0.502390\pi$
$467$	−0.324568	−0.0150192	−0.00750960	+	0.999972i	$0.502390\pi$
$468$	0	0
$469$	−4.86656	−0.224717
$470$	0	0
$471$	3.69573	0.170290
$472$	0	0
$473$	−1.82313	−0.0838277
$474$	0	0
$475$	2.07998	0.0954360
$476$	0	0
$477$	33.2185	1.52097
$478$	0	0
$479$	−22.7591	−1.03989	−0.519946	−	0.854199i	$-0.674048\pi$
$479$	−22.7591	−1.03989	−0.519946	+	0.854199i	$0.674048\pi$
$480$	0	0
$481$	7.59065	0.346104
$482$	0	0
$483$	6.50675	0.296067
$484$	0	0
$485$	54.4741	2.47354
$486$	0	0
$487$	−22.5613	−1.02235	−0.511175	−	0.859477i	$-0.670789\pi$
$487$	−22.5613	−1.02235	−0.511175	+	0.859477i	$0.670789\pi$
$488$	0	0
$489$	9.49425	0.429345
$490$	0	0
$491$	−16.1750	−0.729969	−0.364984	−	0.931014i	$-0.618926\pi$
$491$	−16.1750	−0.729969	−0.364984	+	0.931014i	$0.618926\pi$
$492$	0	0
$493$	−3.08168	−0.138792
$494$	0	0
$495$	8.24868	0.370750
$496$	0	0
$497$	−15.3731	−0.689579
$498$	0	0
$499$	−7.87799	−0.352667	−0.176334	−	0.984330i	$-0.556424\pi$
$499$	−7.87799	−0.352667	−0.176334	+	0.984330i	$0.556424\pi$
$500$	0	0
$501$	6.72559	0.300477
$502$	0	0
$503$	33.5883	1.49763	0.748814	−	0.662781i	$-0.230624\pi$
$503$	33.5883	1.49763	0.748814	+	0.662781i	$0.230624\pi$
$504$	0	0
$505$	−43.7469	−1.94671
$506$	0	0
$507$	−23.7676	−1.05556
$508$	0	0
$509$	5.02651	0.222796	0.111398	−	0.993776i	$-0.464467\pi$
$509$	5.02651	0.222796	0.111398	+	0.993776i	$0.464467\pi$
$510$	0	0
$511$	−18.1699	−0.803787
$512$	0	0
$513$	1.38245	0.0610364
$514$	0	0
$515$	18.6632	0.822400
$516$	0	0
$517$	0.204844	0.00900901
$518$	0	0
$519$	14.7210	0.646181
$520$	0	0
$521$	−13.3499	−0.584868	−0.292434	−	0.956286i	$-0.594465\pi$
$521$	−13.3499	−0.584868	−0.292434	+	0.956286i	$0.594465\pi$
$522$	0	0
$523$	11.9621	0.523067	0.261533	−	0.965194i	$-0.415772\pi$
$523$	11.9621	0.523067	0.261533	+	0.965194i	$0.415772\pi$
$524$	0	0
$525$	6.78531	0.296135
$526$	0	0
$527$	−17.0489	−0.742664
$528$	0	0
$529$	8.01187	0.348342
$530$	0	0
$531$	5.35274	0.232289
$532$	0	0
$533$	−69.1977	−2.99728
$534$	0	0
$535$	12.1218	0.524070
$536$	0	0
$537$	−7.07383	−0.305258
$538$	0	0
$539$	4.21867	0.181711
$540$	0	0
$541$	43.6958	1.87863	0.939316	−	0.343054i	$-0.111461\pi$
$541$	43.6958	1.87863	0.939316	+	0.343054i	$0.111461\pi$
$542$	0	0
$543$	7.63302	0.327564
$544$	0	0
$545$	7.19690	0.308282
$546$	0	0
$547$	5.09652	0.217911	0.108956	−	0.994047i	$-0.465249\pi$
$547$	5.09652	0.217911	0.108956	+	0.994047i	$0.465249\pi$
$548$	0	0
$549$	1.34314	0.0573240
$550$	0	0
$551$	−0.311920	−0.0132882
$552$	0	0
$553$	−25.0945	−1.06713
$554$	0	0
$555$	2.55218	0.108334
$556$	0	0
$557$	−11.7302	−0.497025	−0.248512	−	0.968629i	$-0.579942\pi$
$557$	−11.7302	−0.497025	−0.248512	+	0.968629i	$0.579942\pi$
$558$	0	0
$559$	−12.4887	−0.528216
$560$	0	0
$561$	2.47918	0.104671
$562$	0	0
$563$	−29.4516	−1.24124	−0.620619	−	0.784112i	$-0.713119\pi$
$563$	−29.4516	−1.24124	−0.620619	+	0.784112i	$0.713119\pi$
$564$	0	0
$565$	17.0562	0.717559
$566$	0	0
$567$	−8.04397	−0.337815
$568$	0	0
$569$	3.75591	0.157456	0.0787279	−	0.996896i	$-0.474914\pi$
$569$	3.75591	0.157456	0.0787279	+	0.996896i	$0.474914\pi$
$570$	0	0
$571$	20.4117	0.854205	0.427102	−	0.904203i	$-0.359534\pi$
$571$	20.4117	0.854205	0.427102	+	0.904203i	$0.359534\pi$
$572$	0	0
$573$	13.1724	0.550283
$574$	0	0
$575$	32.3395	1.34865
$576$	0	0
$577$	24.3579	1.01403	0.507017	−	0.861936i	$-0.330748\pi$
$577$	24.3579	1.01403	0.507017	+	0.861936i	$0.330748\pi$
$578$	0	0
$579$	−5.97457	−0.248295
$580$	0	0
$581$	−15.9676	−0.662446
$582$	0	0
$583$	13.2389	0.548300
$584$	0	0
$585$	56.5045	2.33618
$586$	0	0
$587$	−10.7566	−0.443973	−0.221987	−	0.975050i	$-0.571254\pi$
$587$	−10.7566	−0.443973	−0.221987	+	0.975050i	$0.571254\pi$
$588$	0	0
$589$	−1.72565	−0.0711042
$590$	0	0
$591$	−1.73229	−0.0712568
$592$	0	0
$593$	−20.3771	−0.836787	−0.418393	−	0.908266i	$-0.637407\pi$
$593$	−20.3771	−0.836787	−0.418393	+	0.908266i	$0.637407\pi$
$594$	0	0
$595$	−19.4007	−0.795352
$596$	0	0
$597$	−10.3545	−0.423780
$598$	0	0
$599$	−35.9658	−1.46952	−0.734761	−	0.678327i	$-0.762705\pi$
$599$	−35.9658	−1.46952	−0.734761	+	0.678327i	$0.762705\pi$
$600$	0	0
$601$	−32.0182	−1.30605	−0.653026	−	0.757336i	$-0.726501\pi$
$601$	−32.0182	−1.30605	−0.653026	+	0.757336i	$0.726501\pi$
$602$	0	0
$603$	−7.32187	−0.298170
$604$	0	0
$605$	3.28744	0.133653
$606$	0	0
$607$	1.15478	0.0468710	0.0234355	−	0.999725i	$-0.492540\pi$
$607$	1.15478	0.0468710	0.0234355	+	0.999725i	$0.492540\pi$
$608$	0	0
$609$	−1.01755	−0.0412330
$610$	0	0
$611$	1.40321	0.0567676
$612$	0	0
$613$	−6.46122	−0.260966	−0.130483	−	0.991451i	$-0.541653\pi$
$613$	−6.46122	−0.260966	−0.130483	+	0.991451i	$0.541653\pi$
$614$	0	0
$615$	−23.2661	−0.938179
$616$	0	0
$617$	7.35677	0.296173	0.148086	−	0.988974i	$-0.452689\pi$
$617$	7.35677	0.296173	0.148086	+	0.988974i	$0.452689\pi$
$618$	0	0
$619$	10.2526	0.412086	0.206043	−	0.978543i	$-0.433941\pi$
$619$	10.2526	0.412086	0.206043	+	0.978543i	$0.433941\pi$
$620$	0	0
$621$	21.4943	0.862535
$622$	0	0
$623$	−3.88643	−0.155706
$624$	0	0
$625$	−20.3122	−0.812488
$626$	0	0
$627$	0.250936	0.0100214
$628$	0	0
$629$	−3.92115	−0.156347
$630$	0	0
$631$	6.52981	0.259948	0.129974	−	0.991517i	$-0.458511\pi$
$631$	6.52981	0.259948	0.129974	+	0.991517i	$0.458511\pi$
$632$	0	0
$633$	−2.35622	−0.0936515
$634$	0	0
$635$	13.6485	0.541625
$636$	0	0
$637$	28.8984	1.14500
$638$	0	0
$639$	−23.1293	−0.914981
$640$	0	0
$641$	−15.6580	−0.618455	−0.309227	−	0.950988i	$-0.600070\pi$
$641$	−15.6580	−0.618455	−0.309227	+	0.950988i	$0.600070\pi$
$642$	0	0
$643$	−39.9357	−1.57491	−0.787456	−	0.616371i	$-0.788602\pi$
$643$	−39.9357	−1.57491	−0.787456	+	0.616371i	$0.788602\pi$
$644$	0	0
$645$	−4.19903	−0.165337
$646$	0	0
$647$	11.8528	0.465980	0.232990	−	0.972479i	$-0.425149\pi$
$647$	11.8528	0.465980	0.232990	+	0.972479i	$0.425149\pi$
$648$	0	0
$649$	2.13329	0.0837388
$650$	0	0
$651$	−5.62942	−0.220635
$652$	0	0
$653$	−45.4668	−1.77925	−0.889627	−	0.456689i	$-0.849035\pi$
$653$	−45.4668	−1.77925	−0.889627	+	0.456689i	$0.849035\pi$
$654$	0	0
$655$	54.4527	2.12764
$656$	0	0
$657$	−27.3371	−1.06652
$658$	0	0
$659$	28.1122	1.09510	0.547548	−	0.836774i	$-0.315561\pi$
$659$	28.1122	1.09510	0.547548	+	0.836774i	$0.315561\pi$
$660$	0	0
$661$	40.5350	1.57663	0.788315	−	0.615272i	$-0.210954\pi$
$661$	40.5350	1.57663	0.788315	+	0.615272i	$0.210954\pi$
$662$	0	0
$663$	16.9827	0.659553
$664$	0	0
$665$	−1.96369	−0.0761487
$666$	0	0
$667$	−4.84973	−0.187782
$668$	0	0
$669$	−0.381038	−0.0147318
$670$	0	0
$671$	0.535298	0.0206649
$672$	0	0
$673$	−34.2314	−1.31952	−0.659762	−	0.751475i	$-0.729343\pi$
$673$	−34.2314	−1.31952	−0.659762	+	0.751475i	$0.729343\pi$
$674$	0	0
$675$	22.4144	0.862732
$676$	0	0
$677$	39.1034	1.50286	0.751432	−	0.659811i	$-0.229363\pi$
$677$	39.1034	1.50286	0.751432	+	0.659811i	$0.229363\pi$
$678$	0	0
$679$	−27.6350	−1.06053
$680$	0	0
$681$	−19.8652	−0.761235
$682$	0	0
$683$	34.3118	1.31291	0.656453	−	0.754367i	$-0.272056\pi$
$683$	34.3118	1.31291	0.656453	+	0.754367i	$0.272056\pi$
$684$	0	0
$685$	−3.28744	−0.125606
$686$	0	0
$687$	1.94661	0.0742677
$688$	0	0
$689$	90.6884	3.45495
$690$	0	0
$691$	−19.8861	−0.756501	−0.378251	−	0.925703i	$-0.623474\pi$
$691$	−19.8861	−0.756501	−0.378251	+	0.925703i	$0.623474\pi$
$692$	0	0
$693$	−4.18460	−0.158960
$694$	0	0
$695$	0.478694	0.0181579
$696$	0	0
$697$	35.7459	1.35397
$698$	0	0
$699$	−13.9827	−0.528876
$700$	0	0
$701$	−6.11548	−0.230979	−0.115489	−	0.993309i	$-0.536844\pi$
$701$	−6.11548	−0.230979	−0.115489	+	0.993309i	$0.536844\pi$
$702$	0	0
$703$	−0.396889	−0.0149690
$704$	0	0
$705$	0.471795	0.0177688
$706$	0	0
$707$	22.1930	0.834654
$708$	0	0
$709$	−5.89901	−0.221542	−0.110771	−	0.993846i	$-0.535332\pi$
$709$	−5.89901	−0.221542	−0.110771	+	0.993846i	$0.535332\pi$
$710$	0	0
$711$	−37.7554	−1.41594
$712$	0	0
$713$	−26.8304	−1.00481
$714$	0	0
$715$	22.5194	0.842177
$716$	0	0
$717$	−11.0956	−0.414372
$718$	0	0
$719$	2.52512	0.0941712	0.0470856	−	0.998891i	$-0.485007\pi$
$719$	2.52512	0.0941712	0.0470856	+	0.998891i	$0.485007\pi$
$720$	0	0
$721$	−9.46795	−0.352605
$722$	0	0
$723$	−14.2405	−0.529611
$724$	0	0
$725$	−5.05735	−0.187825
$726$	0	0
$727$	22.0991	0.819611	0.409805	−	0.912173i	$-0.365597\pi$
$727$	22.0991	0.819611	0.409805	+	0.912173i	$0.365597\pi$
$728$	0	0
$729$	−3.98991	−0.147774
$730$	0	0
$731$	6.45137	0.238613
$732$	0	0
$733$	19.3792	0.715787	0.357893	−	0.933762i	$-0.383495\pi$
$733$	19.3792	0.715787	0.357893	+	0.933762i	$0.383495\pi$
$734$	0	0
$735$	9.71642	0.358395
$736$	0	0
$737$	−2.91807	−0.107488
$738$	0	0
$739$	30.7007	1.12934	0.564671	−	0.825316i	$-0.309003\pi$
$739$	30.7007	1.12934	0.564671	+	0.825316i	$0.309003\pi$
$740$	0	0
$741$	1.71895	0.0631471
$742$	0	0
$743$	−3.95948	−0.145259	−0.0726296	−	0.997359i	$-0.523139\pi$
$743$	−3.95948	−0.145259	−0.0726296	+	0.997359i	$0.523139\pi$
$744$	0	0
$745$	11.1495	0.408486
$746$	0	0
$747$	−24.0236	−0.878979
$748$	0	0
$749$	−6.14944	−0.224696
$750$	0	0
$751$	35.4664	1.29419	0.647094	−	0.762410i	$-0.275984\pi$
$751$	35.4664	1.29419	0.647094	+	0.762410i	$0.275984\pi$
$752$	0	0
$753$	10.9921	0.400573
$754$	0	0
$755$	26.4937	0.964203
$756$	0	0
$757$	−23.8780	−0.867861	−0.433931	−	0.900946i	$-0.642874\pi$
$757$	−23.8780	−0.867861	−0.433931	+	0.900946i	$0.642874\pi$
$758$	0	0
$759$	3.90155	0.141617
$760$	0	0
$761$	−22.1459	−0.802788	−0.401394	−	0.915905i	$-0.631474\pi$
$761$	−22.1459	−0.802788	−0.401394	+	0.915905i	$0.631474\pi$
$762$	0	0
$763$	−3.65103	−0.132176
$764$	0	0
$765$	−29.1889	−1.05533
$766$	0	0
$767$	14.6133	0.527656
$768$	0	0
$769$	42.6594	1.53834	0.769169	−	0.639045i	$-0.220670\pi$
$769$	42.6594	1.53834	0.769169	+	0.639045i	$0.220670\pi$
$770$	0	0
$771$	3.86589	0.139226
$772$	0	0
$773$	35.9213	1.29200	0.645999	−	0.763338i	$-0.276441\pi$
$773$	35.9213	1.29200	0.645999	+	0.763338i	$0.276441\pi$
$774$	0	0
$775$	−27.9791	−1.00504
$776$	0	0
$777$	−1.29473	−0.0464483
$778$	0	0
$779$	3.61811	0.129632
$780$	0	0
$781$	−9.21797	−0.329845
$782$	0	0
$783$	−3.36134	−0.120124
$784$	0	0
$785$	−17.3414	−0.618940
$786$	0	0
$787$	7.30010	0.260220	0.130110	−	0.991500i	$-0.458467\pi$
$787$	7.30010	0.260220	0.130110	+	0.991500i	$0.458467\pi$
$788$	0	0
$789$	11.0483	0.393328
$790$	0	0
$791$	−8.65269	−0.307654
$792$	0	0
$793$	3.66686	0.130214
$794$	0	0
$795$	30.4918	1.08143
$796$	0	0
$797$	42.4647	1.50418	0.752088	−	0.659063i	$-0.229047\pi$
$797$	42.4647	1.50418	0.752088	+	0.659063i	$0.229047\pi$
$798$	0	0
$799$	−0.724863	−0.0256438
$800$	0	0
$801$	−5.84724	−0.206602
$802$	0	0
$803$	−10.8949	−0.384474
$804$	0	0
$805$	−30.5315	−1.07609
$806$	0	0
$807$	1.22844	0.0432432
$808$	0	0
$809$	25.5627	0.898736	0.449368	−	0.893347i	$-0.351649\pi$
$809$	25.5627	0.898736	0.449368	+	0.893347i	$0.351649\pi$
$810$	0	0
$811$	−34.0961	−1.19727	−0.598637	−	0.801020i	$-0.704291\pi$
$811$	−34.0961	−1.19727	−0.598637	+	0.801020i	$0.704291\pi$
$812$	0	0
$813$	−16.5535	−0.580558
$814$	0	0
$815$	−44.5496	−1.56051
$816$	0	0
$817$	0.652992	0.0228453
$818$	0	0
$819$	−28.6650	−1.00164
$820$	0	0
$821$	18.8438	0.657652	0.328826	−	0.944391i	$-0.393347\pi$
$821$	18.8438	0.657652	0.328826	+	0.944391i	$0.393347\pi$
$822$	0	0
$823$	40.8867	1.42522	0.712611	−	0.701560i	$-0.247513\pi$
$823$	40.8867	1.42522	0.712611	+	0.701560i	$0.247513\pi$
$824$	0	0
$825$	4.06858	0.141650
$826$	0	0
$827$	−13.1177	−0.456149	−0.228074	−	0.973644i	$-0.573243\pi$
$827$	−13.1177	−0.456149	−0.228074	+	0.973644i	$0.573243\pi$
$828$	0	0
$829$	17.2274	0.598332	0.299166	−	0.954201i	$-0.403292\pi$
$829$	17.2274	0.598332	0.299166	+	0.954201i	$0.403292\pi$
$830$	0	0
$831$	10.9225	0.378899
$832$	0	0
$833$	−14.9283	−0.517233
$834$	0	0
$835$	−31.5583	−1.09212
$836$	0	0
$837$	−18.5961	−0.642776
$838$	0	0
$839$	−27.9098	−0.963554	−0.481777	−	0.876294i	$-0.660008\pi$
$839$	−27.9098	−0.963554	−0.481777	+	0.876294i	$0.660008\pi$
$840$	0	0
$841$	−28.2416	−0.973848
$842$	0	0
$843$	−9.88799	−0.340560
$844$	0	0
$845$	111.524	3.83655
$846$	0	0
$847$	−1.66773	−0.0573040
$848$	0	0
$849$	−1.68881	−0.0579599
$850$	0	0
$851$	−6.17083	−0.211533
$852$	0	0
$853$	−12.3777	−0.423804	−0.211902	−	0.977291i	$-0.567966\pi$
$853$	−12.3777	−0.423804	−0.211902	+	0.977291i	$0.567966\pi$
$854$	0	0
$855$	−2.95443	−0.101039
$856$	0	0
$857$	21.6035	0.737962	0.368981	−	0.929437i	$-0.379707\pi$
$857$	21.6035	0.737962	0.368981	+	0.929437i	$0.379707\pi$
$858$	0	0
$859$	−5.02714	−0.171524	−0.0857618	−	0.996316i	$-0.527332\pi$
$859$	−5.02714	−0.171524	−0.0857618	+	0.996316i	$0.527332\pi$
$860$	0	0
$861$	11.8030	0.402245
$862$	0	0
$863$	−3.30630	−0.112548	−0.0562739	−	0.998415i	$-0.517922\pi$
$863$	−3.30630	−0.112548	−0.0562739	+	0.998415i	$0.517922\pi$
$864$	0	0
$865$	−69.0751	−2.34862
$866$	0	0
$867$	3.13743	0.106553
$868$	0	0
$869$	−15.0471	−0.510437
$870$	0	0
$871$	−19.9891	−0.677307
$872$	0	0
$873$	−41.5776	−1.40719
$874$	0	0
$875$	−4.42572	−0.149617
$876$	0	0
$877$	−47.9894	−1.62049	−0.810243	−	0.586094i	$-0.800665\pi$
$877$	−47.9894	−1.62049	−0.810243	+	0.586094i	$0.800665\pi$
$878$	0	0
$879$	−0.749489	−0.0252797
$880$	0	0
$881$	−30.7430	−1.03576	−0.517878	−	0.855454i	$-0.673278\pi$
$881$	−30.7430	−1.03576	−0.517878	+	0.855454i	$0.673278\pi$
$882$	0	0
$883$	26.3133	0.885514	0.442757	−	0.896642i	$-0.354000\pi$
$883$	26.3133	0.885514	0.442757	+	0.896642i	$0.354000\pi$
$884$	0	0
$885$	4.91338	0.165161
$886$	0	0
$887$	−38.4081	−1.28962	−0.644809	−	0.764344i	$-0.723063\pi$
$887$	−38.4081	−1.28962	−0.644809	+	0.764344i	$0.723063\pi$
$888$	0	0
$889$	−6.92397	−0.232222
$890$	0	0
$891$	−4.82330	−0.161587
$892$	0	0
$893$	−0.0733688	−0.00245519
$894$	0	0
$895$	33.1924	1.10950
$896$	0	0
$897$	26.7262	0.892361
$898$	0	0
$899$	4.19582	0.139939
$900$	0	0
$901$	−46.8475	−1.56072
$902$	0	0
$903$	2.13019	0.0708883
$904$	0	0
$905$	−35.8162	−1.19057
$906$	0	0
$907$	−16.4044	−0.544700	−0.272350	−	0.962198i	$-0.587801\pi$
$907$	−16.4044	−0.544700	−0.272350	+	0.962198i	$0.587801\pi$
$908$	0	0
$909$	33.3900	1.10748
$910$	0	0
$911$	4.89641	0.162225	0.0811126	−	0.996705i	$-0.474153\pi$
$911$	4.89641	0.162225	0.0811126	+	0.996705i	$0.474153\pi$
$912$	0	0
$913$	−9.57441	−0.316867
$914$	0	0
$915$	1.23290	0.0407583
$916$	0	0
$917$	−27.6241	−0.912229
$918$	0	0
$919$	−54.0915	−1.78431	−0.892156	−	0.451727i	$-0.850808\pi$
$919$	−54.0915	−1.78431	−0.892156	+	0.451727i	$0.850808\pi$
$920$	0	0
$921$	−16.2144	−0.534284
$922$	0	0
$923$	−63.1444	−2.07842
$924$	0	0
$925$	−6.43501	−0.211582
$926$	0	0
$927$	−14.2448	−0.467860
$928$	0	0
$929$	−24.3653	−0.799401	−0.399700	−	0.916646i	$-0.630886\pi$
$929$	−24.3653	−0.799401	−0.399700	+	0.916646i	$0.630886\pi$
$930$	0	0
$931$	−1.51100	−0.0495210
$932$	0	0
$933$	−0.742409	−0.0243054
$934$	0	0
$935$	−11.6330	−0.380439
$936$	0	0
$937$	9.91688	0.323970	0.161985	−	0.986793i	$-0.448210\pi$
$937$	9.91688	0.323970	0.161985	+	0.986793i	$0.448210\pi$
$938$	0	0
$939$	21.7602	0.710117
$940$	0	0
$941$	35.5968	1.16042	0.580212	−	0.814465i	$-0.302970\pi$
$941$	35.5968	1.16042	0.580212	+	0.814465i	$0.302970\pi$
$942$	0	0
$943$	56.2544	1.83189
$944$	0	0
$945$	−21.1613	−0.688377
$946$	0	0
$947$	27.7981	0.903315	0.451658	−	0.892191i	$-0.350833\pi$
$947$	27.7981	0.903315	0.451658	+	0.892191i	$0.350833\pi$
$948$	0	0
$949$	−74.6318	−2.42265
$950$	0	0
$951$	−12.3559	−0.400668
$952$	0	0
$953$	30.6374	0.992444	0.496222	−	0.868196i	$-0.334720\pi$
$953$	30.6374	0.992444	0.496222	+	0.868196i	$0.334720\pi$
$954$	0	0
$955$	−61.8084	−2.00007
$956$	0	0
$957$	−0.610137	−0.0197229
$958$	0	0
$959$	1.66773	0.0538539
$960$	0	0
$961$	−7.78721	−0.251200
$962$	0	0
$963$	−9.25200	−0.298142
$964$	0	0
$965$	28.0343	0.902458
$966$	0	0
$967$	−10.0507	−0.323209	−0.161604	−	0.986856i	$-0.551667\pi$
$967$	−10.0507	−0.323209	−0.161604	+	0.986856i	$0.551667\pi$
$968$	0	0
$969$	−0.887967	−0.0285256
$970$	0	0
$971$	−29.4093	−0.943788	−0.471894	−	0.881655i	$-0.656429\pi$
$971$	−29.4093	−0.943788	−0.471894	+	0.881655i	$0.656429\pi$
$972$	0	0
$973$	−0.242844	−0.00778522
$974$	0	0
$975$	27.8703	0.892565
$976$	0	0
$977$	4.63731	0.148361	0.0741803	−	0.997245i	$-0.476366\pi$
$977$	4.63731	0.148361	0.0741803	+	0.997245i	$0.476366\pi$
$978$	0	0
$979$	−2.33036	−0.0744788
$980$	0	0
$981$	−5.49307	−0.175380
$982$	0	0
$983$	54.0770	1.72479	0.862395	−	0.506237i	$-0.168964\pi$
$983$	54.0770	1.72479	0.862395	+	0.506237i	$0.168964\pi$
$984$	0	0
$985$	8.12838	0.258992
$986$	0	0
$987$	−0.239344	−0.00761840
$988$	0	0
$989$	10.1527	0.322838
$990$	0	0
$991$	52.4156	1.66504	0.832518	−	0.553998i	$-0.186899\pi$
$991$	52.4156	1.66504	0.832518	+	0.553998i	$0.186899\pi$
$992$	0	0
$993$	−13.2101	−0.419211
$994$	0	0
$995$	48.5860	1.54028
$996$	0	0
$997$	40.7880	1.29177	0.645885	−	0.763435i	$-0.276489\pi$
$997$	40.7880	1.29177	0.645885	+	0.763435i	$0.276489\pi$
$998$	0	0
$999$	−4.27699	−0.135318

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6028.2.a.e.1.11	✓	27

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6028.2.a.e.1.11	✓	27	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 6028 = 2^{2} \cdot 11 \cdot 137 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6028.a (trivial)

\(f(q)\)	\(=\)	\(q-0.700606 q^{3} +3.28744 q^{5} -1.66773 q^{7} -2.50915 q^{9} +O(q^{10})\)	\(q-0.700606 q^{3} +3.28744 q^{5} -1.66773 q^{7} -2.50915 q^{9} -1.00000 q^{11} -6.85014 q^{13} -2.30320 q^{15} +3.53862 q^{17} +0.358170 q^{19} +1.16842 q^{21} +5.56883 q^{23} +5.80724 q^{25} +3.85974 q^{27} -0.870870 q^{29} -4.81797 q^{31} +0.700606 q^{33} -5.48257 q^{35} -1.10810 q^{37} +4.79924 q^{39} +10.1016 q^{41} +1.82313 q^{43} -8.24868 q^{45} -0.204844 q^{47} -4.21867 q^{49} -2.47918 q^{51} -13.2389 q^{53} -3.28744 q^{55} -0.250936 q^{57} -2.13329 q^{59} -0.535298 q^{61} +4.18460 q^{63} -22.5194 q^{65} +2.91807 q^{67} -3.90155 q^{69} +9.21797 q^{71} +10.8949 q^{73} -4.06858 q^{75} +1.66773 q^{77} +15.0471 q^{79} +4.82330 q^{81} +9.57441 q^{83} +11.6330 q^{85} +0.610137 q^{87} +2.33036 q^{89} +11.4242 q^{91} +3.37549 q^{93} +1.17746 q^{95} +16.5704 q^{97} +2.50915 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 27 q + 5 q^{3} - 2 q^{5} + 7 q^{7} + 28 q^{9}+O(q^{10}) \) 27 * q + 5 * q^3 - 2 * q^5 + 7 * q^7 + 28 * q^9	\( 27 q + 5 q^{3} - 2 q^{5} + 7 q^{7} + 28 q^{9} - 27 q^{11} + 2 q^{13} + 10 q^{15} + 17 q^{17} + 4 q^{19} - 4 q^{21} + 25 q^{23} + 21 q^{25} + 32 q^{27} - q^{29} + 8 q^{31} - 5 q^{33} + 30 q^{35} - 24 q^{37} + 10 q^{39} + 31 q^{41} + 17 q^{43} - 15 q^{45} + 27 q^{47} + 40 q^{49} + 30 q^{51} + 2 q^{55} + 13 q^{57} + 31 q^{59} + 2 q^{61} + 61 q^{63} + 2 q^{67} - 15 q^{69} + 40 q^{71} + 17 q^{73} + 30 q^{75} - 7 q^{77} + 33 q^{79} + 31 q^{81} + 66 q^{83} + 26 q^{85} + 43 q^{87} + 22 q^{89} + 18 q^{91} - 9 q^{93} + 95 q^{95} - 21 q^{97} - 28 q^{99}+O(q^{100}) \) 27 * q + 5 * q^3 - 2 * q^5 + 7 * q^7 + 28 * q^9 - 27 * q^11 + 2 * q^13 + 10 * q^15 + 17 * q^17 + 4 * q^19 - 4 * q^21 + 25 * q^23 + 21 * q^25 + 32 * q^27 - q^29 + 8 * q^31 - 5 * q^33 + 30 * q^35 - 24 * q^37 + 10 * q^39 + 31 * q^41 + 17 * q^43 - 15 * q^45 + 27 * q^47 + 40 * q^49 + 30 * q^51 + 2 * q^55 + 13 * q^57 + 31 * q^59 + 2 * q^61 + 61 * q^63 + 2 * q^67 - 15 * q^69 + 40 * q^71 + 17 * q^73 + 30 * q^75 - 7 * q^77 + 33 * q^79 + 31 * q^81 + 66 * q^83 + 26 * q^85 + 43 * q^87 + 22 * q^89 + 18 * q^91 - 9 * q^93 + 95 * q^95 - 21 * q^97 - 28 * q^99

Introduction

L-functions

Modular forms

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