LMFDB - Embedded newform 5376.2.a.bi.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5376,2,Mod(1,5376)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5376, 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("5376.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5376 = 2^{8} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5376.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:	$42.9275761266$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.1016.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x + 2 $ x^3 - x^2 - 6*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 2688)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.85577$ of defining polynomial
Character	$\chi$	$=$	5376.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} -1.29966 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} -1.29966 q^{5} -1.00000 q^{7} +1.00000 q^{9} +5.01121 q^{11} -3.71155 q^{13} -1.29966 q^{15} -4.41188 q^{17} +7.71155 q^{19} -1.00000 q^{21} +2.41188 q^{23} -3.31087 q^{25} +1.00000 q^{27} +7.71155 q^{29} -3.71155 q^{31} +5.01121 q^{33} +1.29966 q^{35} -10.3109 q^{37} -3.71155 q^{39} +11.0112 q^{41} -2.59933 q^{43} -1.29966 q^{45} +10.0224 q^{47} +1.00000 q^{49} -4.41188 q^{51} -10.3109 q^{53} -6.51289 q^{55} +7.71155 q^{57} -8.82376 q^{59} +9.11222 q^{61} -1.00000 q^{63} +4.82376 q^{65} +8.00000 q^{67} +2.41188 q^{69} -7.61054 q^{71} +12.5993 q^{73} -3.31087 q^{75} -5.01121 q^{77} +10.5993 q^{79} +1.00000 q^{81} -9.19866 q^{83} +5.73396 q^{85} +7.71155 q^{87} +4.98879 q^{89} +3.71155 q^{91} -3.71155 q^{93} -10.0224 q^{95} -12.0224 q^{97} +5.01121 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{3} - 3 q^{7} + 3 q^{9}+O(q^{10}) $ 3 * q + 3 * q^3 - 3 * q^7 + 3 * q^9	$ 3 q + 3 q^{3} - 3 q^{7} + 3 q^{9} - 4 q^{11} + 4 q^{13} - 2 q^{17} + 8 q^{19} - 3 q^{21} - 4 q^{23} + 13 q^{25} + 3 q^{27} + 8 q^{29} + 4 q^{31} - 4 q^{33} - 8 q^{37} + 4 q^{39} + 14 q^{41} - 8 q^{47} + 3 q^{49} - 2 q^{51} - 8 q^{53} - 20 q^{55} + 8 q^{57} - 4 q^{59} + 20 q^{61} - 3 q^{63} - 8 q^{65} + 24 q^{67} - 4 q^{69} + 4 q^{71} + 30 q^{73} + 13 q^{75} + 4 q^{77} + 24 q^{79} + 3 q^{81} - 12 q^{83} - 36 q^{85} + 8 q^{87} + 34 q^{89} - 4 q^{91} + 4 q^{93} + 8 q^{95} + 2 q^{97} - 4 q^{99}+O(q^{100}) $ 3 * q + 3 * q^3 - 3 * q^7 + 3 * q^9 - 4 * q^11 + 4 * q^13 - 2 * q^17 + 8 * q^19 - 3 * q^21 - 4 * q^23 + 13 * q^25 + 3 * q^27 + 8 * q^29 + 4 * q^31 - 4 * q^33 - 8 * q^37 + 4 * q^39 + 14 * q^41 - 8 * q^47 + 3 * q^49 - 2 * q^51 - 8 * q^53 - 20 * q^55 + 8 * q^57 - 4 * q^59 + 20 * q^61 - 3 * q^63 - 8 * q^65 + 24 * q^67 - 4 * q^69 + 4 * q^71 + 30 * q^73 + 13 * q^75 + 4 * q^77 + 24 * q^79 + 3 * q^81 - 12 * q^83 - 36 * q^85 + 8 * q^87 + 34 * q^89 - 4 * q^91 + 4 * q^93 + 8 * q^95 + 2 * q^97 - 4 * 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
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	0	0
$5$	−1.29966	−0.581227	−0.290614	−	0.956840i	$-0.593859\pi$
$5$	−1.29966	−0.581227	−0.290614	+	0.956840i	$0.593859\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	5.01121	1.51094	0.755468	−	0.655185i	$-0.227409\pi$
$11$	5.01121	1.51094	0.755468	+	0.655185i	$0.227409\pi$
$12$	0	0
$13$	−3.71155	−1.02940	−0.514699	−	0.857371i	$-0.672096\pi$
$13$	−3.71155	−1.02940	−0.514699	+	0.857371i	$0.672096\pi$
$14$	0	0
$15$	−1.29966	−0.335572
$16$	0	0
$17$	−4.41188	−1.07004	−0.535019	−	0.844840i	$-0.679696\pi$
$17$	−4.41188	−1.07004	−0.535019	+	0.844840i	$0.679696\pi$
$18$	0	0
$19$	7.71155	1.76915	0.884575	−	0.466398i	$-0.154449\pi$
$19$	7.71155	1.76915	0.884575	+	0.466398i	$0.154449\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	2.41188	0.502912	0.251456	−	0.967869i	$-0.419091\pi$
$23$	2.41188	0.502912	0.251456	+	0.967869i	$0.419091\pi$
$24$	0	0
$25$	−3.31087	−0.662175
$26$	0	0
$27$	1.00000	0.192450
$28$	0	0
$29$	7.71155	1.43200	0.715999	−	0.698101i	$-0.245972\pi$
$29$	7.71155	1.43200	0.715999	+	0.698101i	$0.245972\pi$
$30$	0	0
$31$	−3.71155	−0.666613	−0.333307	−	0.942818i	$-0.608164\pi$
$31$	−3.71155	−0.666613	−0.333307	+	0.942818i	$0.608164\pi$
$32$	0	0
$33$	5.01121	0.872340
$34$	0	0
$35$	1.29966	0.219683
$36$	0	0
$37$	−10.3109	−1.69510	−0.847549	−	0.530718i	$-0.821922\pi$
$37$	−10.3109	−1.69510	−0.847549	+	0.530718i	$0.821922\pi$
$38$	0	0
$39$	−3.71155	−0.594323
$40$	0	0
$41$	11.0112	1.71966	0.859831	−	0.510579i	$-0.170569\pi$
$41$	11.0112	1.71966	0.859831	+	0.510579i	$0.170569\pi$
$42$	0	0
$43$	−2.59933	−0.396394	−0.198197	−	0.980162i	$-0.563509\pi$
$43$	−2.59933	−0.396394	−0.198197	+	0.980162i	$0.563509\pi$
$44$	0	0
$45$	−1.29966	−0.193742
$46$	0	0
$47$	10.0224	1.46192	0.730960	−	0.682420i	$-0.239073\pi$
$47$	10.0224	1.46192	0.730960	+	0.682420i	$0.239073\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−4.41188	−0.617787
$52$	0	0
$53$	−10.3109	−1.41631	−0.708154	−	0.706058i	$-0.750472\pi$
$53$	−10.3109	−1.41631	−0.708154	+	0.706058i	$0.750472\pi$
$54$	0	0
$55$	−6.51289	−0.878198
$56$	0	0
$57$	7.71155	1.02142
$58$	0	0
$59$	−8.82376	−1.14876	−0.574378	−	0.818590i	$-0.694756\pi$
$59$	−8.82376	−1.14876	−0.574378	+	0.818590i	$0.694756\pi$
$60$	0	0
$61$	9.11222	1.16670	0.583350	−	0.812221i	$-0.301742\pi$
$61$	9.11222	1.16670	0.583350	+	0.812221i	$0.301742\pi$
$62$	0	0
$63$	−1.00000	−0.125988
$64$	0	0
$65$	4.82376	0.598314
$66$	0	0
$67$	8.00000	0.977356	0.488678	−	0.872464i	$-0.337479\pi$
$67$	8.00000	0.977356	0.488678	+	0.872464i	$0.337479\pi$
$68$	0	0
$69$	2.41188	0.290356
$70$	0	0
$71$	−7.61054	−0.903205	−0.451602	−	0.892219i	$-0.649147\pi$
$71$	−7.61054	−0.903205	−0.451602	+	0.892219i	$0.649147\pi$
$72$	0	0
$73$	12.5993	1.47464	0.737320	−	0.675544i	$-0.236091\pi$
$73$	12.5993	1.47464	0.737320	+	0.675544i	$0.236091\pi$
$74$	0	0
$75$	−3.31087	−0.382307
$76$	0	0
$77$	−5.01121	−0.571080
$78$	0	0
$79$	10.5993	1.19252	0.596259	−	0.802792i	$-0.296653\pi$
$79$	10.5993	1.19252	0.596259	+	0.802792i	$0.296653\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	−9.19866	−1.00968	−0.504842	−	0.863212i	$-0.668449\pi$
$83$	−9.19866	−1.00968	−0.504842	+	0.863212i	$0.668449\pi$
$84$	0	0
$85$	5.73396	0.621936
$86$	0	0
$87$	7.71155	0.826764
$88$	0	0
$89$	4.98879	0.528811	0.264405	−	0.964412i	$-0.414824\pi$
$89$	4.98879	0.528811	0.264405	+	0.964412i	$0.414824\pi$
$90$	0	0
$91$	3.71155	0.389076
$92$	0	0
$93$	−3.71155	−0.384869
$94$	0	0
$95$	−10.0224	−1.02828
$96$	0	0
$97$	−12.0224	−1.22069	−0.610346	−	0.792135i	$-0.708970\pi$
$97$	−12.0224	−1.22069	−0.610346	+	0.792135i	$0.708970\pi$
$98$	0	0
$99$	5.01121	0.503645
$100$	0	0
$101$	11.3221	1.12659	0.563295	−	0.826256i	$-0.309534\pi$
$101$	11.3221	1.12659	0.563295	+	0.826256i	$0.309534\pi$
$102$	0	0
$103$	3.71155	0.365709	0.182855	−	0.983140i	$-0.441466\pi$
$103$	3.71155	0.365709	0.182855	+	0.983140i	$0.441466\pi$
$104$	0	0
$105$	1.29966	0.126834
$106$	0	0
$107$	2.98879	0.288937	0.144469	−	0.989509i	$-0.453853\pi$
$107$	2.98879	0.288937	0.144469	+	0.989509i	$0.453853\pi$
$108$	0	0
$109$	11.4231	1.09413	0.547067	−	0.837089i	$-0.315744\pi$
$109$	11.4231	1.09413	0.547067	+	0.837089i	$0.315744\pi$
$110$	0	0
$111$	−10.3109	−0.978665
$112$	0	0
$113$	2.00000	0.188144	0.0940721	−	0.995565i	$-0.470012\pi$
$113$	2.00000	0.188144	0.0940721	+	0.995565i	$0.470012\pi$
$114$	0	0
$115$	−3.13464	−0.292306
$116$	0	0
$117$	−3.71155	−0.343132
$118$	0	0
$119$	4.41188	0.404436
$120$	0	0
$121$	14.1122	1.28293
$122$	0	0
$123$	11.0112	0.992847
$124$	0	0
$125$	10.8013	0.966102
$126$	0	0
$127$	18.0224	1.59923	0.799616	−	0.600512i	$-0.205037\pi$
$127$	18.0224	1.59923	0.799616	+	0.600512i	$0.205037\pi$
$128$	0	0
$129$	−2.59933	−0.228858
$130$	0	0
$131$	−16.6217	−1.45225	−0.726124	−	0.687563i	$-0.758680\pi$
$131$	−16.6217	−1.45225	−0.726124	+	0.687563i	$0.758680\pi$
$132$	0	0
$133$	−7.71155	−0.668676
$134$	0	0
$135$	−1.29966	−0.111857
$136$	0	0
$137$	8.59933	0.734690	0.367345	−	0.930085i	$-0.380267\pi$
$137$	8.59933	0.734690	0.367345	+	0.930085i	$0.380267\pi$
$138$	0	0
$139$	9.19866	0.780220	0.390110	−	0.920768i	$-0.372437\pi$
$139$	9.19866	0.780220	0.390110	+	0.920768i	$0.372437\pi$
$140$	0	0
$141$	10.0224	0.844040
$142$	0	0
$143$	−18.5993	−1.55535
$144$	0	0
$145$	−10.0224	−0.832317
$146$	0	0
$147$	1.00000	0.0824786
$148$	0	0
$149$	−5.11222	−0.418809	−0.209405	−	0.977829i	$-0.567153\pi$
$149$	−5.11222	−0.418809	−0.209405	+	0.977829i	$0.567153\pi$
$150$	0	0
$151$	20.6217	1.67817	0.839087	−	0.543997i	$-0.183090\pi$
$151$	20.6217	1.67817	0.839087	+	0.543997i	$0.183090\pi$
$152$	0	0
$153$	−4.41188	−0.356679
$154$	0	0
$155$	4.82376	0.387454
$156$	0	0
$157$	−1.11222	−0.0887646	−0.0443823	−	0.999015i	$-0.514132\pi$
$157$	−1.11222	−0.0887646	−0.0443823	+	0.999015i	$0.514132\pi$
$158$	0	0
$159$	−10.3109	−0.817705
$160$	0	0
$161$	−2.41188	−0.190083
$162$	0	0
$163$	5.19866	0.407190	0.203595	−	0.979055i	$-0.434737\pi$
$163$	5.19866	0.407190	0.203595	+	0.979055i	$0.434737\pi$
$164$	0	0
$165$	−6.51289	−0.507028
$166$	0	0
$167$	2.59933	0.201142	0.100571	−	0.994930i	$-0.467933\pi$
$167$	2.59933	0.201142	0.100571	+	0.994930i	$0.467933\pi$
$168$	0	0
$169$	0.775566	0.0596590
$170$	0	0
$171$	7.71155	0.589717
$172$	0	0
$173$	−11.5241	−0.876161	−0.438080	−	0.898936i	$-0.644342\pi$
$173$	−11.5241	−0.876161	−0.438080	+	0.898936i	$0.644342\pi$
$174$	0	0
$175$	3.31087	0.250278
$176$	0	0
$177$	−8.82376	−0.663235
$178$	0	0
$179$	7.03363	0.525718	0.262859	−	0.964834i	$-0.415335\pi$
$179$	7.03363	0.525718	0.262859	+	0.964834i	$0.415335\pi$
$180$	0	0
$181$	−11.1346	−0.827631	−0.413815	−	0.910361i	$-0.635804\pi$
$181$	−11.1346	−0.827631	−0.413815	+	0.910361i	$0.635804\pi$
$182$	0	0
$183$	9.11222	0.673594
$184$	0	0
$185$	13.4007	0.985237
$186$	0	0
$187$	−22.1089	−1.61676
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	18.4119	1.33224	0.666118	−	0.745846i	$-0.267955\pi$
$191$	18.4119	1.33224	0.666118	+	0.745846i	$0.267955\pi$
$192$	0	0
$193$	19.7340	1.42048	0.710241	−	0.703959i	$-0.248586\pi$
$193$	19.7340	1.42048	0.710241	+	0.703959i	$0.248586\pi$
$194$	0	0
$195$	4.82376	0.345437
$196$	0	0
$197$	2.31087	0.164643	0.0823214	−	0.996606i	$-0.473767\pi$
$197$	2.31087	0.164643	0.0823214	+	0.996606i	$0.473767\pi$
$198$	0	0
$199$	−22.8462	−1.61952	−0.809761	−	0.586759i	$-0.800403\pi$
$199$	−22.8462	−1.61952	−0.809761	+	0.586759i	$0.800403\pi$
$200$	0	0
$201$	8.00000	0.564276
$202$	0	0
$203$	−7.71155	−0.541244
$204$	0	0
$205$	−14.3109	−0.999515
$206$	0	0
$207$	2.41188	0.167637
$208$	0	0
$209$	38.6442	2.67307
$210$	0	0
$211$	12.8238	0.882824	0.441412	−	0.897305i	$-0.354478\pi$
$211$	12.8238	0.882824	0.441412	+	0.897305i	$0.354478\pi$
$212$	0	0
$213$	−7.61054	−0.521465
$214$	0	0
$215$	3.37825	0.230395
$216$	0	0
$217$	3.71155	0.251956
$218$	0	0
$219$	12.5993	0.851384
$220$	0	0
$221$	16.3749	1.10149
$222$	0	0
$223$	8.00000	0.535720	0.267860	−	0.963458i	$-0.413684\pi$
$223$	8.00000	0.535720	0.267860	+	0.963458i	$0.413684\pi$
$224$	0	0
$225$	−3.31087	−0.220725
$226$	0	0
$227$	−1.40067	−0.0929659	−0.0464829	−	0.998919i	$-0.514801\pi$
$227$	−1.40067	−0.0929659	−0.0464829	+	0.998919i	$0.514801\pi$
$228$	0	0
$229$	−3.71155	−0.245266	−0.122633	−	0.992452i	$-0.539134\pi$
$229$	−3.71155	−0.245266	−0.122633	+	0.992452i	$0.539134\pi$
$230$	0	0
$231$	−5.01121	−0.329713
$232$	0	0
$233$	12.5993	0.825409	0.412705	−	0.910865i	$-0.364584\pi$
$233$	12.5993	0.825409	0.412705	+	0.910865i	$0.364584\pi$
$234$	0	0
$235$	−13.0258	−0.849708
$236$	0	0
$237$	10.5993	0.688500
$238$	0	0
$239$	−23.2356	−1.50299	−0.751494	−	0.659739i	$-0.770667\pi$
$239$	−23.2356	−1.50299	−0.751494	+	0.659739i	$0.770667\pi$
$240$	0	0
$241$	15.4455	0.994933	0.497466	−	0.867483i	$-0.334264\pi$
$241$	15.4455	0.994933	0.497466	+	0.867483i	$0.334264\pi$
$242$	0	0
$243$	1.00000	0.0641500
$244$	0	0
$245$	−1.29966	−0.0830325
$246$	0	0
$247$	−28.6217	−1.82116
$248$	0	0
$249$	−9.19866	−0.582941
$250$	0	0
$251$	14.0224	0.885087	0.442544	−	0.896747i	$-0.354076\pi$
$251$	14.0224	0.885087	0.442544	+	0.896747i	$0.354076\pi$
$252$	0	0
$253$	12.0864	0.759868
$254$	0	0
$255$	5.73396	0.359075
$256$	0	0
$257$	−3.01121	−0.187834	−0.0939170	−	0.995580i	$-0.529939\pi$
$257$	−3.01121	−0.187834	−0.0939170	+	0.995580i	$0.529939\pi$
$258$	0	0
$259$	10.3109	0.640686
$260$	0	0
$261$	7.71155	0.477333
$262$	0	0
$263$	13.5881	0.837879	0.418940	−	0.908014i	$-0.362402\pi$
$263$	13.5881	0.837879	0.418940	+	0.908014i	$0.362402\pi$
$264$	0	0
$265$	13.4007	0.823197
$266$	0	0
$267$	4.98879	0.305309
$268$	0	0
$269$	10.7452	0.655145	0.327572	−	0.944826i	$-0.393769\pi$
$269$	10.7452	0.655145	0.327572	+	0.944826i	$0.393769\pi$
$270$	0	0
$271$	−18.5577	−1.12730	−0.563651	−	0.826013i	$-0.690604\pi$
$271$	−18.5577	−1.12730	−0.563651	+	0.826013i	$0.690604\pi$
$272$	0	0
$273$	3.71155	0.224633
$274$	0	0
$275$	−16.5915	−1.00050
$276$	0	0
$277$	25.1571	1.51154	0.755770	−	0.654837i	$-0.227263\pi$
$277$	25.1571	1.51154	0.755770	+	0.654837i	$0.227263\pi$
$278$	0	0
$279$	−3.71155	−0.222204
$280$	0	0
$281$	9.62511	0.574186	0.287093	−	0.957903i	$-0.407311\pi$
$281$	9.62511	0.574186	0.287093	+	0.957903i	$0.407311\pi$
$282$	0	0
$283$	−10.5129	−0.624926	−0.312463	−	0.949930i	$-0.601154\pi$
$283$	−10.5129	−0.624926	−0.312463	+	0.949930i	$0.601154\pi$
$284$	0	0
$285$	−10.0224	−0.593677
$286$	0	0
$287$	−11.0112	−0.649971
$288$	0	0
$289$	2.46469	0.144982
$290$	0	0
$291$	−12.0224	−0.704767
$292$	0	0
$293$	−6.12343	−0.357734	−0.178867	−	0.983873i	$-0.557243\pi$
$293$	−6.12343	−0.357734	−0.178867	+	0.983873i	$0.557243\pi$
$294$	0	0
$295$	11.4679	0.667688
$296$	0	0
$297$	5.01121	0.290780
$298$	0	0
$299$	−8.95180	−0.517696
$300$	0	0
$301$	2.59933	0.149823
$302$	0	0
$303$	11.3221	0.650437
$304$	0	0
$305$	−11.8428	−0.678118
$306$	0	0
$307$	−4.91020	−0.280240	−0.140120	−	0.990135i	$-0.544749\pi$
$307$	−4.91020	−0.280240	−0.140120	+	0.990135i	$0.544749\pi$
$308$	0	0
$309$	3.71155	0.211142
$310$	0	0
$311$	15.2211	0.863108	0.431554	−	0.902087i	$-0.357965\pi$
$311$	15.2211	0.863108	0.431554	+	0.902087i	$0.357965\pi$
$312$	0	0
$313$	8.22443	0.464872	0.232436	−	0.972612i	$-0.425330\pi$
$313$	8.22443	0.464872	0.232436	+	0.972612i	$0.425330\pi$
$314$	0	0
$315$	1.29966	0.0732278
$316$	0	0
$317$	2.31087	0.129792	0.0648958	−	0.997892i	$-0.479329\pi$
$317$	2.31087	0.129792	0.0648958	+	0.997892i	$0.479329\pi$
$318$	0	0
$319$	38.6442	2.16366
$320$	0	0
$321$	2.98879	0.166818
$322$	0	0
$323$	−34.0224	−1.89306
$324$	0	0
$325$	12.2885	0.681641
$326$	0	0
$327$	11.4231	0.631698
$328$	0	0
$329$	−10.0224	−0.552554
$330$	0	0
$331$	26.0224	1.43032	0.715161	−	0.698960i	$-0.246353\pi$
$331$	26.0224	1.43032	0.715161	+	0.698960i	$0.246353\pi$
$332$	0	0
$333$	−10.3109	−0.565032
$334$	0	0
$335$	−10.3973	−0.568066
$336$	0	0
$337$	35.8204	1.95126	0.975631	−	0.219418i	$-0.0704160\pi$
$337$	35.8204	1.95126	0.975631	+	0.219418i	$0.0704160\pi$
$338$	0	0
$339$	2.00000	0.108625
$340$	0	0
$341$	−18.5993	−1.00721
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	0	0
$345$	−3.13464	−0.168763
$346$	0	0
$347$	−25.8350	−1.38689	−0.693447	−	0.720508i	$-0.743909\pi$
$347$	−25.8350	−1.38689	−0.693447	+	0.720508i	$0.743909\pi$
$348$	0	0
$349$	11.5095	0.616091	0.308045	−	0.951372i	$-0.400325\pi$
$349$	11.5095	0.616091	0.308045	+	0.951372i	$0.400325\pi$
$350$	0	0
$351$	−3.71155	−0.198108
$352$	0	0
$353$	−7.83497	−0.417013	−0.208507	−	0.978021i	$-0.566860\pi$
$353$	−7.83497	−0.417013	−0.208507	+	0.978021i	$0.566860\pi$
$354$	0	0
$355$	9.89114	0.524967
$356$	0	0
$357$	4.41188	0.233501
$358$	0	0
$359$	7.61054	0.401669	0.200834	−	0.979625i	$-0.435635\pi$
$359$	7.61054	0.401669	0.200834	+	0.979625i	$0.435635\pi$
$360$	0	0
$361$	40.4679	2.12989
$362$	0	0
$363$	14.1122	0.740699
$364$	0	0
$365$	−16.3749	−0.857101
$366$	0	0
$367$	−13.1987	−0.688964	−0.344482	−	0.938793i	$-0.611945\pi$
$367$	−13.1987	−0.688964	−0.344482	+	0.938793i	$0.611945\pi$
$368$	0	0
$369$	11.0112	0.573221
$370$	0	0
$371$	10.3109	0.535314
$372$	0	0
$373$	8.62175	0.446417	0.223209	−	0.974771i	$-0.428347\pi$
$373$	8.62175	0.446417	0.223209	+	0.974771i	$0.428347\pi$
$374$	0	0
$375$	10.8013	0.557779
$376$	0	0
$377$	−28.6217	−1.47409
$378$	0	0
$379$	−0.374895	−0.0192570	−0.00962852	−	0.999954i	$-0.503065\pi$
$379$	−0.374895	−0.0192570	−0.00962852	+	0.999954i	$0.503065\pi$
$380$	0	0
$381$	18.0224	0.923316
$382$	0	0
$383$	7.42309	0.379302	0.189651	−	0.981852i	$-0.439264\pi$
$383$	7.42309	0.379302	0.189651	+	0.981852i	$0.439264\pi$
$384$	0	0
$385$	6.51289	0.331928
$386$	0	0
$387$	−2.59933	−0.132131
$388$	0	0
$389$	18.5129	0.938641	0.469320	−	0.883028i	$-0.344499\pi$
$389$	18.5129	0.938641	0.469320	+	0.883028i	$0.344499\pi$
$390$	0	0
$391$	−10.6409	−0.538135
$392$	0	0
$393$	−16.6217	−0.838456
$394$	0	0
$395$	−13.7756	−0.693124
$396$	0	0
$397$	−22.3109	−1.11975	−0.559875	−	0.828577i	$-0.689151\pi$
$397$	−22.3109	−1.11975	−0.559875	+	0.828577i	$0.689151\pi$
$398$	0	0
$399$	−7.71155	−0.386060
$400$	0	0
$401$	0.599328	0.0299290	0.0149645	−	0.999888i	$-0.495236\pi$
$401$	0.599328	0.0299290	0.0149645	+	0.999888i	$0.495236\pi$
$402$	0	0
$403$	13.7756	0.686210
$404$	0	0
$405$	−1.29966	−0.0645808
$406$	0	0
$407$	−51.6699	−2.56118
$408$	0	0
$409$	8.59933	0.425209	0.212605	−	0.977138i	$-0.431805\pi$
$409$	8.59933	0.425209	0.212605	+	0.977138i	$0.431805\pi$
$410$	0	0
$411$	8.59933	0.424174
$412$	0	0
$413$	8.82376	0.434189
$414$	0	0
$415$	11.9552	0.586856
$416$	0	0
$417$	9.19866	0.450460
$418$	0	0
$419$	−25.4007	−1.24090	−0.620452	−	0.784244i	$-0.713051\pi$
$419$	−25.4007	−1.24090	−0.620452	+	0.784244i	$0.713051\pi$
$420$	0	0
$421$	−33.1571	−1.61598	−0.807988	−	0.589199i	$-0.799443\pi$
$421$	−33.1571	−1.61598	−0.807988	+	0.589199i	$0.799443\pi$
$422$	0	0
$423$	10.0224	0.487307
$424$	0	0
$425$	14.6072	0.708552
$426$	0	0
$427$	−9.11222	−0.440971
$428$	0	0
$429$	−18.5993	−0.897984
$430$	0	0
$431$	−2.78678	−0.134234	−0.0671171	−	0.997745i	$-0.521380\pi$
$431$	−2.78678	−0.134234	−0.0671171	+	0.997745i	$0.521380\pi$
$432$	0	0
$433$	0.224434	0.0107856	0.00539280	−	0.999985i	$-0.498283\pi$
$433$	0.224434	0.0107856	0.00539280	+	0.999985i	$0.498283\pi$
$434$	0	0
$435$	−10.0224	−0.480538
$436$	0	0
$437$	18.5993	0.889727
$438$	0	0
$439$	−6.84618	−0.326750	−0.163375	−	0.986564i	$-0.552238\pi$
$439$	−6.84618	−0.326750	−0.163375	+	0.986564i	$0.552238\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	−12.6363	−0.600369	−0.300185	−	0.953881i	$-0.597048\pi$
$443$	−12.6363	−0.600369	−0.300185	+	0.953881i	$0.597048\pi$
$444$	0	0
$445$	−6.48375	−0.307359
$446$	0	0
$447$	−5.11222	−0.241800
$448$	0	0
$449$	8.80134	0.415361	0.207681	−	0.978197i	$-0.433409\pi$
$449$	8.80134	0.415361	0.207681	+	0.978197i	$0.433409\pi$
$450$	0	0
$451$	55.1795	2.59830
$452$	0	0
$453$	20.6217	0.968894
$454$	0	0
$455$	−4.82376	−0.226141
$456$	0	0
$457$	−40.8462	−1.91070	−0.955352	−	0.295471i	$-0.904524\pi$
$457$	−40.8462	−1.91070	−0.955352	+	0.295471i	$0.904524\pi$
$458$	0	0
$459$	−4.41188	−0.205929
$460$	0	0
$461$	−19.3221	−0.899919	−0.449960	−	0.893049i	$-0.648562\pi$
$461$	−19.3221	−0.899919	−0.449960	+	0.893049i	$0.648562\pi$
$462$	0	0
$463$	23.2211	1.07917	0.539587	−	0.841930i	$-0.318580\pi$
$463$	23.2211	1.07917	0.539587	+	0.841930i	$0.318580\pi$
$464$	0	0
$465$	4.82376	0.223697
$466$	0	0
$467$	−19.2211	−0.889445	−0.444723	−	0.895668i	$-0.646698\pi$
$467$	−19.2211	−0.889445	−0.444723	+	0.895668i	$0.646698\pi$
$468$	0	0
$469$	−8.00000	−0.369406
$470$	0	0
$471$	−1.11222	−0.0512482
$472$	0	0
$473$	−13.0258	−0.598926
$474$	0	0
$475$	−25.5319	−1.17149
$476$	0	0
$477$	−10.3109	−0.472102
$478$	0	0
$479$	−8.20202	−0.374760	−0.187380	−	0.982288i	$-0.560000\pi$
$479$	−8.20202	−0.374760	−0.187380	+	0.982288i	$0.560000\pi$
$480$	0	0
$481$	38.2693	1.74493
$482$	0	0
$483$	−2.41188	−0.109744
$484$	0	0
$485$	15.6251	0.709499
$486$	0	0
$487$	−4.62175	−0.209431	−0.104716	−	0.994502i	$-0.533393\pi$
$487$	−4.62175	−0.209431	−0.104716	+	0.994502i	$0.533393\pi$
$488$	0	0
$489$	5.19866	0.235091
$490$	0	0
$491$	−27.8574	−1.25719	−0.628593	−	0.777734i	$-0.716369\pi$
$491$	−27.8574	−1.25719	−0.628593	+	0.777734i	$0.716369\pi$
$492$	0	0
$493$	−34.0224	−1.53229
$494$	0	0
$495$	−6.51289	−0.292733
$496$	0	0
$497$	7.61054	0.341379
$498$	0	0
$499$	−15.7980	−0.707215	−0.353607	−	0.935394i	$-0.615045\pi$
$499$	−15.7980	−0.707215	−0.353607	+	0.935394i	$0.615045\pi$
$500$	0	0
$501$	2.59933	0.116129
$502$	0	0
$503$	−21.1987	−0.945201	−0.472601	−	0.881277i	$-0.656685\pi$
$503$	−21.1987	−0.945201	−0.472601	+	0.881277i	$0.656685\pi$
$504$	0	0
$505$	−14.7149	−0.654805
$506$	0	0
$507$	0.775566	0.0344441
$508$	0	0
$509$	−22.7003	−1.00617	−0.503087	−	0.864236i	$-0.667803\pi$
$509$	−22.7003	−1.00617	−0.503087	+	0.864236i	$0.667803\pi$
$510$	0	0
$511$	−12.5993	−0.557361
$512$	0	0
$513$	7.71155	0.340473
$514$	0	0
$515$	−4.82376	−0.212560
$516$	0	0
$517$	50.2244	2.20887
$518$	0	0
$519$	−11.5241	−0.505852
$520$	0	0
$521$	−42.8540	−1.87747	−0.938735	−	0.344641i	$-0.888001\pi$
$521$	−42.8540	−1.87747	−0.938735	+	0.344641i	$0.888001\pi$
$522$	0	0
$523$	−20.0000	−0.874539	−0.437269	−	0.899331i	$-0.644054\pi$
$523$	−20.0000	−0.874539	−0.437269	+	0.899331i	$0.644054\pi$
$524$	0	0
$525$	3.31087	0.144498
$526$	0	0
$527$	16.3749	0.713302
$528$	0	0
$529$	−17.1828	−0.747080
$530$	0	0
$531$	−8.82376	−0.382919
$532$	0	0
$533$	−40.8686	−1.77022
$534$	0	0
$535$	−3.88442	−0.167938
$536$	0	0
$537$	7.03363	0.303523
$538$	0	0
$539$	5.01121	0.215848
$540$	0	0
$541$	28.5353	1.22683	0.613414	−	0.789761i	$-0.289796\pi$
$541$	28.5353	1.22683	0.613414	+	0.789761i	$0.289796\pi$
$542$	0	0
$543$	−11.1346	−0.477833
$544$	0	0
$545$	−14.8462	−0.635940
$546$	0	0
$547$	−7.42309	−0.317388	−0.158694	−	0.987328i	$-0.550728\pi$
$547$	−7.42309	−0.317388	−0.158694	+	0.987328i	$0.550728\pi$
$548$	0	0
$549$	9.11222	0.388900
$550$	0	0
$551$	59.4679	2.53342
$552$	0	0
$553$	−10.5993	−0.450729
$554$	0	0
$555$	13.4007	0.568827
$556$	0	0
$557$	−5.11222	−0.216612	−0.108306	−	0.994118i	$-0.534543\pi$
$557$	−5.11222	−0.216612	−0.108306	+	0.994118i	$0.534543\pi$
$558$	0	0
$559$	9.64752	0.408047
$560$	0	0
$561$	−22.1089	−0.933437
$562$	0	0
$563$	14.0224	0.590974	0.295487	−	0.955347i	$-0.404518\pi$
$563$	14.0224	0.590974	0.295487	+	0.955347i	$0.404518\pi$
$564$	0	0
$565$	−2.59933	−0.109355
$566$	0	0
$567$	−1.00000	−0.0419961
$568$	0	0
$569$	43.4903	1.82321	0.911605	−	0.411067i	$-0.134844\pi$
$569$	43.4903	1.82321	0.911605	+	0.411067i	$0.134844\pi$
$570$	0	0
$571$	33.0706	1.38396	0.691981	−	0.721916i	$-0.256738\pi$
$571$	33.0706	1.38396	0.691981	+	0.721916i	$0.256738\pi$
$572$	0	0
$573$	18.4119	0.769167
$574$	0	0
$575$	−7.98543	−0.333016
$576$	0	0
$577$	−18.0448	−0.751216	−0.375608	−	0.926779i	$-0.622566\pi$
$577$	−18.0448	−0.751216	−0.375608	+	0.926779i	$0.622566\pi$
$578$	0	0
$579$	19.7340	0.820116
$580$	0	0
$581$	9.19866	0.381625
$582$	0	0
$583$	−51.6699	−2.13995
$584$	0	0
$585$	4.82376	0.199438
$586$	0	0
$587$	−34.6442	−1.42992	−0.714959	−	0.699167i	$-0.753555\pi$
$587$	−34.6442	−1.42992	−0.714959	+	0.699167i	$0.753555\pi$
$588$	0	0
$589$	−28.6217	−1.17934
$590$	0	0
$591$	2.31087	0.0950566
$592$	0	0
$593$	−10.4343	−0.428485	−0.214243	−	0.976780i	$-0.568728\pi$
$593$	−10.4343	−0.428485	−0.214243	+	0.976780i	$0.568728\pi$
$594$	0	0
$595$	−5.73396	−0.235070
$596$	0	0
$597$	−22.8462	−0.935032
$598$	0	0
$599$	44.4343	1.81554	0.907768	−	0.419472i	$-0.137785\pi$
$599$	44.4343	1.81554	0.907768	+	0.419472i	$0.137785\pi$
$600$	0	0
$601$	10.0000	0.407909	0.203954	−	0.978980i	$-0.434621\pi$
$601$	10.0000	0.407909	0.203954	+	0.978980i	$0.434621\pi$
$602$	0	0
$603$	8.00000	0.325785
$604$	0	0
$605$	−18.3411	−0.745673
$606$	0	0
$607$	17.6475	0.716291	0.358145	−	0.933666i	$-0.383409\pi$
$607$	17.6475	0.716291	0.358145	+	0.933666i	$0.383409\pi$
$608$	0	0
$609$	−7.71155	−0.312488
$610$	0	0
$611$	−37.1987	−1.50490
$612$	0	0
$613$	20.0000	0.807792	0.403896	−	0.914805i	$-0.367656\pi$
$613$	20.0000	0.807792	0.403896	+	0.914805i	$0.367656\pi$
$614$	0	0
$615$	−14.3109	−0.577070
$616$	0	0
$617$	−13.7980	−0.555486	−0.277743	−	0.960655i	$-0.589586\pi$
$617$	−13.7980	−0.555486	−0.277743	+	0.960655i	$0.589586\pi$
$618$	0	0
$619$	0.0448365	0.00180213	0.000901065	−	1.00000i	$-0.499713\pi$
$619$	0.0448365	0.00180213	0.000901065		1.00000i	$0.499713\pi$
$620$	0	0
$621$	2.41188	0.0967854
$622$	0	0
$623$	−4.98879	−0.199872
$624$	0	0
$625$	2.51625	0.100650
$626$	0	0
$627$	38.6442	1.54330
$628$	0	0
$629$	45.4903	1.81382
$630$	0	0
$631$	−18.0224	−0.717461	−0.358731	−	0.933441i	$-0.616790\pi$
$631$	−18.0224	−0.717461	−0.358731	+	0.933441i	$0.616790\pi$
$632$	0	0
$633$	12.8238	0.509699
$634$	0	0
$635$	−23.4231	−0.929517
$636$	0	0
$637$	−3.71155	−0.147057
$638$	0	0
$639$	−7.61054	−0.301068
$640$	0	0
$641$	22.8686	0.903255	0.451628	−	0.892207i	$-0.350844\pi$
$641$	22.8686	0.903255	0.451628	+	0.892207i	$0.350844\pi$
$642$	0	0
$643$	6.06402	0.239142	0.119571	−	0.992826i	$-0.461848\pi$
$643$	6.06402	0.239142	0.119571	+	0.992826i	$0.461848\pi$
$644$	0	0
$645$	3.37825	0.133019
$646$	0	0
$647$	23.0482	0.906118	0.453059	−	0.891481i	$-0.350333\pi$
$647$	23.0482	0.906118	0.453059	+	0.891481i	$0.350333\pi$
$648$	0	0
$649$	−44.2177	−1.73570
$650$	0	0
$651$	3.71155	0.145467
$652$	0	0
$653$	−26.5129	−1.03753	−0.518765	−	0.854917i	$-0.673608\pi$
$653$	−26.5129	−1.03753	−0.518765	+	0.854917i	$0.673608\pi$
$654$	0	0
$655$	21.6027	0.844087
$656$	0	0
$657$	12.5993	0.491547
$658$	0	0
$659$	−26.2099	−1.02099	−0.510496	−	0.859880i	$-0.670538\pi$
$659$	−26.2099	−1.02099	−0.510496	+	0.859880i	$0.670538\pi$
$660$	0	0
$661$	−19.5095	−0.758833	−0.379416	−	0.925226i	$-0.623875\pi$
$661$	−19.5095	−0.758833	−0.379416	+	0.925226i	$0.623875\pi$
$662$	0	0
$663$	16.3749	0.635948
$664$	0	0
$665$	10.0224	0.388653
$666$	0	0
$667$	18.5993	0.720169
$668$	0	0
$669$	8.00000	0.309298
$670$	0	0
$671$	45.6632	1.76281
$672$	0	0
$673$	2.66335	0.102665	0.0513323	−	0.998682i	$-0.483653\pi$
$673$	2.66335	0.102665	0.0513323	+	0.998682i	$0.483653\pi$
$674$	0	0
$675$	−3.31087	−0.127436
$676$	0	0
$677$	−11.5241	−0.442907	−0.221454	−	0.975171i	$-0.571080\pi$
$677$	−11.5241	−0.442907	−0.221454	+	0.975171i	$0.571080\pi$
$678$	0	0
$679$	12.0224	0.461378
$680$	0	0
$681$	−1.40067	−0.0536739
$682$	0	0
$683$	−2.98879	−0.114363	−0.0571815	−	0.998364i	$-0.518211\pi$
$683$	−2.98879	−0.114363	−0.0571815	+	0.998364i	$0.518211\pi$
$684$	0	0
$685$	−11.1762	−0.427022
$686$	0	0
$687$	−3.71155	−0.141604
$688$	0	0
$689$	38.2693	1.45794
$690$	0	0
$691$	26.8462	1.02128	0.510638	−	0.859796i	$-0.329409\pi$
$691$	26.8462	1.02128	0.510638	+	0.859796i	$0.329409\pi$
$692$	0	0
$693$	−5.01121	−0.190360
$694$	0	0
$695$	−11.9552	−0.453485
$696$	0	0
$697$	−48.5801	−1.84010
$698$	0	0
$699$	12.5993	0.476550
$700$	0	0
$701$	−45.4039	−1.71488	−0.857441	−	0.514582i	$-0.827947\pi$
$701$	−45.4039	−1.71488	−0.857441	+	0.514582i	$0.827947\pi$
$702$	0	0
$703$	−79.5128	−2.99888
$704$	0	0
$705$	−13.0258	−0.490579
$706$	0	0
$707$	−11.3221	−0.425811
$708$	0	0
$709$	−4.57691	−0.171889	−0.0859447	−	0.996300i	$-0.527391\pi$
$709$	−4.57691	−0.171889	−0.0859447	+	0.996300i	$0.527391\pi$
$710$	0	0
$711$	10.5993	0.397506
$712$	0	0
$713$	−8.95180	−0.335248
$714$	0	0
$715$	24.1729	0.904014
$716$	0	0
$717$	−23.2356	−0.867751
$718$	0	0
$719$	−23.4231	−0.873534	−0.436767	−	0.899575i	$-0.643877\pi$
$719$	−23.4231	−0.873534	−0.436767	+	0.899575i	$0.643877\pi$
$720$	0	0
$721$	−3.71155	−0.138225
$722$	0	0
$723$	15.4455	0.574425
$724$	0	0
$725$	−25.5319	−0.948233
$726$	0	0
$727$	−12.2885	−0.455754	−0.227877	−	0.973690i	$-0.573178\pi$
$727$	−12.2885	−0.455754	−0.227877	+	0.973690i	$0.573178\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	11.4679	0.424157
$732$	0	0
$733$	40.3333	1.48974	0.744872	−	0.667207i	$-0.232510\pi$
$733$	40.3333	1.48974	0.744872	+	0.667207i	$0.232510\pi$
$734$	0	0
$735$	−1.29966	−0.0479388
$736$	0	0
$737$	40.0897	1.47672
$738$	0	0
$739$	10.3973	0.382471	0.191236	−	0.981544i	$-0.438751\pi$
$739$	10.3973	0.382471	0.191236	+	0.981544i	$0.438751\pi$
$740$	0	0
$741$	−28.6217	−1.05145
$742$	0	0
$743$	23.6105	0.866187	0.433093	−	0.901349i	$-0.357422\pi$
$743$	23.6105	0.866187	0.433093	+	0.901349i	$0.357422\pi$
$744$	0	0
$745$	6.64416	0.243423
$746$	0	0
$747$	−9.19866	−0.336561
$748$	0	0
$749$	−2.98879	−0.109208
$750$	0	0
$751$	18.8013	0.686071	0.343035	−	0.939322i	$-0.388545\pi$
$751$	18.8013	0.686071	0.343035	+	0.939322i	$0.388545\pi$
$752$	0	0
$753$	14.0224	0.511005
$754$	0	0
$755$	−26.8013	−0.975401
$756$	0	0
$757$	−13.8204	−0.502311	−0.251156	−	0.967947i	$-0.580811\pi$
$757$	−13.8204	−0.502311	−0.251156	+	0.967947i	$0.580811\pi$
$758$	0	0
$759$	12.0864	0.438710
$760$	0	0
$761$	−21.2805	−0.771417	−0.385708	−	0.922621i	$-0.626043\pi$
$761$	−21.2805	−0.771417	−0.385708	+	0.922621i	$0.626043\pi$
$762$	0	0
$763$	−11.4231	−0.413544
$764$	0	0
$765$	5.73396	0.207312
$766$	0	0
$767$	32.7498	1.18253
$768$	0	0
$769$	−35.8204	−1.29172	−0.645858	−	0.763457i	$-0.723500\pi$
$769$	−35.8204	−1.29172	−0.645858	+	0.763457i	$0.723500\pi$
$770$	0	0
$771$	−3.01121	−0.108446
$772$	0	0
$773$	−7.56893	−0.272236	−0.136118	−	0.990693i	$-0.543463\pi$
$773$	−7.56893	−0.272236	−0.136118	+	0.990693i	$0.543463\pi$
$774$	0	0
$775$	12.2885	0.441414
$776$	0	0
$777$	10.3109	0.369901
$778$	0	0
$779$	84.9134	3.04234
$780$	0	0
$781$	−38.1380	−1.36468
$782$	0	0
$783$	7.71155	0.275588
$784$	0	0
$785$	1.44551	0.0515924
$786$	0	0
$787$	28.0000	0.998092	0.499046	−	0.866575i	$-0.333684\pi$
$787$	28.0000	0.998092	0.499046	+	0.866575i	$0.333684\pi$
$788$	0	0
$789$	13.5881	0.483750
$790$	0	0
$791$	−2.00000	−0.0711118
$792$	0	0
$793$	−33.8204	−1.20100
$794$	0	0
$795$	13.4007	0.475273
$796$	0	0
$797$	50.1683	1.77705	0.888526	−	0.458827i	$-0.151730\pi$
$797$	50.1683	1.77705	0.888526	+	0.458827i	$0.151730\pi$
$798$	0	0
$799$	−44.2177	−1.56431
$800$	0	0
$801$	4.98879	0.176270
$802$	0	0
$803$	63.1379	2.22809
$804$	0	0
$805$	3.13464	0.110481
$806$	0	0
$807$	10.7452	0.378248
$808$	0	0
$809$	−17.4231	−0.612563	−0.306282	−	0.951941i	$-0.599085\pi$
$809$	−17.4231	−0.612563	−0.306282	+	0.951941i	$0.599085\pi$
$810$	0	0
$811$	22.8013	0.800663	0.400332	−	0.916370i	$-0.368895\pi$
$811$	22.8013	0.800663	0.400332	+	0.916370i	$0.368895\pi$
$812$	0	0
$813$	−18.5577	−0.650848
$814$	0	0
$815$	−6.75651	−0.236670
$816$	0	0
$817$	−20.0448	−0.701280
$818$	0	0
$819$	3.71155	0.129692
$820$	0	0
$821$	20.5353	0.716687	0.358344	−	0.933590i	$-0.383342\pi$
$821$	20.5353	0.716687	0.358344	+	0.933590i	$0.383342\pi$
$822$	0	0
$823$	52.2469	1.82121	0.910605	−	0.413277i	$-0.135616\pi$
$823$	52.2469	1.82121	0.910605	+	0.413277i	$0.135616\pi$
$824$	0	0
$825$	−16.5915	−0.577641
$826$	0	0
$827$	18.2099	0.633219	0.316610	−	0.948556i	$-0.397456\pi$
$827$	18.2099	0.633219	0.316610	+	0.948556i	$0.397456\pi$
$828$	0	0
$829$	22.1089	0.767872	0.383936	−	0.923360i	$-0.374568\pi$
$829$	22.1089	0.767872	0.383936	+	0.923360i	$0.374568\pi$
$830$	0	0
$831$	25.1571	0.872689
$832$	0	0
$833$	−4.41188	−0.152863
$834$	0	0
$835$	−3.37825	−0.116909
$836$	0	0
$837$	−3.71155	−0.128290
$838$	0	0
$839$	−52.4197	−1.80973	−0.904865	−	0.425699i	$-0.860028\pi$
$839$	−52.4197	−1.80973	−0.904865	+	0.425699i	$0.860028\pi$
$840$	0	0
$841$	30.4679	1.05062
$842$	0	0
$843$	9.62511	0.331506
$844$	0	0
$845$	−1.00798	−0.0346754
$846$	0	0
$847$	−14.1122	−0.484902
$848$	0	0
$849$	−10.5129	−0.360801
$850$	0	0
$851$	−24.8686	−0.852485
$852$	0	0
$853$	27.5095	0.941908	0.470954	−	0.882158i	$-0.343910\pi$
$853$	27.5095	0.941908	0.470954	+	0.882158i	$0.343910\pi$
$854$	0	0
$855$	−10.0224	−0.342759
$856$	0	0
$857$	−13.1908	−0.450589	−0.225295	−	0.974291i	$-0.572334\pi$
$857$	−13.1908	−0.450589	−0.225295	+	0.974291i	$0.572334\pi$
$858$	0	0
$859$	36.9102	1.25936	0.629680	−	0.776855i	$-0.283186\pi$
$859$	36.9102	1.25936	0.629680	+	0.776855i	$0.283186\pi$
$860$	0	0
$861$	−11.0112	−0.375261
$862$	0	0
$863$	18.0370	0.613986	0.306993	−	0.951712i	$-0.400677\pi$
$863$	18.0370	0.613986	0.306993	+	0.951712i	$0.400677\pi$
$864$	0	0
$865$	14.9775	0.509249
$866$	0	0
$867$	2.46469	0.0837054
$868$	0	0
$869$	53.1155	1.80182
$870$	0	0
$871$	−29.6924	−1.00609
$872$	0	0
$873$	−12.0224	−0.406897
$874$	0	0
$875$	−10.8013	−0.365152
$876$	0	0
$877$	13.2435	0.447201	0.223600	−	0.974681i	$-0.428219\pi$
$877$	13.2435	0.447201	0.223600	+	0.974681i	$0.428219\pi$
$878$	0	0
$879$	−6.12343	−0.206538
$880$	0	0
$881$	−5.23564	−0.176393	−0.0881966	−	0.996103i	$-0.528110\pi$
$881$	−5.23564	−0.176393	−0.0881966	+	0.996103i	$0.528110\pi$
$882$	0	0
$883$	5.40067	0.181747	0.0908735	−	0.995862i	$-0.471034\pi$
$883$	5.40067	0.181747	0.0908735	+	0.995862i	$0.471034\pi$
$884$	0	0
$885$	11.4679	0.385490
$886$	0	0
$887$	−7.04820	−0.236655	−0.118328	−	0.992975i	$-0.537753\pi$
$887$	−7.04820	−0.236655	−0.118328	+	0.992975i	$0.537753\pi$
$888$	0	0
$889$	−18.0224	−0.604453
$890$	0	0
$891$	5.01121	0.167882
$892$	0	0
$893$	77.2883	2.58636
$894$	0	0
$895$	−9.14135	−0.305562
$896$	0	0
$897$	−8.95180	−0.298892
$898$	0	0
$899$	−28.6217	−0.954589
$900$	0	0
$901$	45.4903	1.51550
$902$	0	0
$903$	2.59933	0.0865002
$904$	0	0
$905$	14.4713	0.481042
$906$	0	0
$907$	−35.8428	−1.19014	−0.595071	−	0.803673i	$-0.702876\pi$
$907$	−35.8428	−1.19014	−0.595071	+	0.803673i	$0.702876\pi$
$908$	0	0
$909$	11.3221	0.375530
$910$	0	0
$911$	23.6105	0.782252	0.391126	−	0.920337i	$-0.372086\pi$
$911$	23.6105	0.782252	0.391126	+	0.920337i	$0.372086\pi$
$912$	0	0
$913$	−46.0964	−1.52557
$914$	0	0
$915$	−11.8428	−0.391512
$916$	0	0
$917$	16.6217	0.548898
$918$	0	0
$919$	45.1155	1.48822	0.744111	−	0.668056i	$-0.232873\pi$
$919$	45.1155	1.48822	0.744111	+	0.668056i	$0.232873\pi$
$920$	0	0
$921$	−4.91020	−0.161797
$922$	0	0
$923$	28.2469	0.929756
$924$	0	0
$925$	34.1380	1.12245
$926$	0	0
$927$	3.71155	0.121903
$928$	0	0
$929$	8.20987	0.269357	0.134678	−	0.990889i	$-0.457000\pi$
$929$	8.20987	0.269357	0.134678	+	0.990889i	$0.457000\pi$
$930$	0	0
$931$	7.71155	0.252736
$932$	0	0
$933$	15.2211	0.498316
$934$	0	0
$935$	28.7341	0.939705
$936$	0	0
$937$	−22.6666	−0.740485	−0.370242	−	0.928935i	$-0.620725\pi$
$937$	−22.6666	−0.740485	−0.370242	+	0.928935i	$0.620725\pi$
$938$	0	0
$939$	8.22443	0.268394
$940$	0	0
$941$	13.2548	0.432095	0.216048	−	0.976383i	$-0.430683\pi$
$941$	13.2548	0.432095	0.216048	+	0.976383i	$0.430683\pi$
$942$	0	0
$943$	26.5577	0.864839
$944$	0	0
$945$	1.29966	0.0422781
$946$	0	0
$947$	−41.4309	−1.34632	−0.673162	−	0.739495i	$-0.735064\pi$
$947$	−41.4309	−1.34632	−0.673162	+	0.739495i	$0.735064\pi$
$948$	0	0
$949$	−46.7630	−1.51799
$950$	0	0
$951$	2.31087	0.0749352
$952$	0	0
$953$	−17.4231	−0.564389	−0.282195	−	0.959357i	$-0.591062\pi$
$953$	−17.4231	−0.564389	−0.282195	+	0.959357i	$0.591062\pi$
$954$	0	0
$955$	−23.9293	−0.774333
$956$	0	0
$957$	38.6442	1.24919
$958$	0	0
$959$	−8.59933	−0.277687
$960$	0	0
$961$	−17.2244	−0.555627
$962$	0	0
$963$	2.98879	0.0963124
$964$	0	0
$965$	−25.6475	−0.825623
$966$	0	0
$967$	−28.8238	−0.926910	−0.463455	−	0.886121i	$-0.653390\pi$
$967$	−28.8238	−0.926910	−0.463455	+	0.886121i	$0.653390\pi$
$968$	0	0
$969$	−34.0224	−1.09296
$970$	0	0
$971$	−51.9168	−1.66609	−0.833045	−	0.553206i	$-0.813404\pi$
$971$	−51.9168	−1.66609	−0.833045	+	0.553206i	$0.813404\pi$
$972$	0	0
$973$	−9.19866	−0.294895
$974$	0	0
$975$	12.2885	0.393546
$976$	0	0
$977$	−22.4197	−0.717271	−0.358635	−	0.933478i	$-0.616758\pi$
$977$	−22.4197	−0.717271	−0.358635	+	0.933478i	$0.616758\pi$
$978$	0	0
$979$	24.9999	0.798999
$980$	0	0
$981$	11.4231	0.364711
$982$	0	0
$983$	−54.2693	−1.73092	−0.865460	−	0.500977i	$-0.832974\pi$
$983$	−54.2693	−1.73092	−0.865460	+	0.500977i	$0.832974\pi$
$984$	0	0
$985$	−3.00336	−0.0956950
$986$	0	0
$987$	−10.0224	−0.319017
$988$	0	0
$989$	−6.26927	−0.199351
$990$	0	0
$991$	20.6217	0.655071	0.327536	−	0.944839i	$-0.393782\pi$
$991$	20.6217	0.655071	0.327536	+	0.944839i	$0.393782\pi$
$992$	0	0
$993$	26.0224	0.825796
$994$	0	0
$995$	29.6924	0.941311
$996$	0	0
$997$	−3.91356	−0.123944	−0.0619719	−	0.998078i	$-0.519739\pi$
$997$	−3.91356	−0.123944	−0.0619719	+	0.998078i	$0.519739\pi$
$998$	0	0
$999$	−10.3109	−0.326222

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5376.2.a.bi.1.2		3
4.3	odd	2		5376.2.a.bh.1.2		3
8.3	odd	2		5376.2.a.bj.1.2		3
8.5	even	2		5376.2.a.bg.1.2		3
16.3	odd	4		2688.2.c.g.1345.2	✓	6
16.5	even	4		2688.2.c.h.1345.2	yes	6
16.11	odd	4		2688.2.c.g.1345.5	yes	6
16.13	even	4		2688.2.c.h.1345.5	yes	6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2688.2.c.g.1345.2	✓	6	16.3	odd	4
2688.2.c.g.1345.5	yes	6	16.11	odd	4
2688.2.c.h.1345.2	yes	6	16.5	even	4
2688.2.c.h.1345.5	yes	6	16.13	even	4
5376.2.a.bg.1.2		3	8.5	even	2
5376.2.a.bh.1.2		3	4.3	odd	2
5376.2.a.bi.1.2		3	1.1	even	1	trivial
5376.2.a.bj.1.2		3	8.3	odd	2

Overview	Random
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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:	\( N \)	\(=\)	\( 5376 = 2^{8} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5376.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} -1.29966 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} -1.29966 q^{5} -1.00000 q^{7} +1.00000 q^{9} +5.01121 q^{11} -3.71155 q^{13} -1.29966 q^{15} -4.41188 q^{17} +7.71155 q^{19} -1.00000 q^{21} +2.41188 q^{23} -3.31087 q^{25} +1.00000 q^{27} +7.71155 q^{29} -3.71155 q^{31} +5.01121 q^{33} +1.29966 q^{35} -10.3109 q^{37} -3.71155 q^{39} +11.0112 q^{41} -2.59933 q^{43} -1.29966 q^{45} +10.0224 q^{47} +1.00000 q^{49} -4.41188 q^{51} -10.3109 q^{53} -6.51289 q^{55} +7.71155 q^{57} -8.82376 q^{59} +9.11222 q^{61} -1.00000 q^{63} +4.82376 q^{65} +8.00000 q^{67} +2.41188 q^{69} -7.61054 q^{71} +12.5993 q^{73} -3.31087 q^{75} -5.01121 q^{77} +10.5993 q^{79} +1.00000 q^{81} -9.19866 q^{83} +5.73396 q^{85} +7.71155 q^{87} +4.98879 q^{89} +3.71155 q^{91} -3.71155 q^{93} -10.0224 q^{95} -12.0224 q^{97} +5.01121 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{3} - 3 q^{7} + 3 q^{9}+O(q^{10}) \) 3 * q + 3 * q^3 - 3 * q^7 + 3 * q^9	\( 3 q + 3 q^{3} - 3 q^{7} + 3 q^{9} - 4 q^{11} + 4 q^{13} - 2 q^{17} + 8 q^{19} - 3 q^{21} - 4 q^{23} + 13 q^{25} + 3 q^{27} + 8 q^{29} + 4 q^{31} - 4 q^{33} - 8 q^{37} + 4 q^{39} + 14 q^{41} - 8 q^{47} + 3 q^{49} - 2 q^{51} - 8 q^{53} - 20 q^{55} + 8 q^{57} - 4 q^{59} + 20 q^{61} - 3 q^{63} - 8 q^{65} + 24 q^{67} - 4 q^{69} + 4 q^{71} + 30 q^{73} + 13 q^{75} + 4 q^{77} + 24 q^{79} + 3 q^{81} - 12 q^{83} - 36 q^{85} + 8 q^{87} + 34 q^{89} - 4 q^{91} + 4 q^{93} + 8 q^{95} + 2 q^{97} - 4 q^{99}+O(q^{100}) \) 3 * q + 3 * q^3 - 3 * q^7 + 3 * q^9 - 4 * q^11 + 4 * q^13 - 2 * q^17 + 8 * q^19 - 3 * q^21 - 4 * q^23 + 13 * q^25 + 3 * q^27 + 8 * q^29 + 4 * q^31 - 4 * q^33 - 8 * q^37 + 4 * q^39 + 14 * q^41 - 8 * q^47 + 3 * q^49 - 2 * q^51 - 8 * q^53 - 20 * q^55 + 8 * q^57 - 4 * q^59 + 20 * q^61 - 3 * q^63 - 8 * q^65 + 24 * q^67 - 4 * q^69 + 4 * q^71 + 30 * q^73 + 13 * q^75 + 4 * q^77 + 24 * q^79 + 3 * q^81 - 12 * q^83 - 36 * q^85 + 8 * q^87 + 34 * q^89 - 4 * q^91 + 4 * q^93 + 8 * q^95 + 2 * q^97 - 4 * q^99

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