LMFDB - Embedded newform 5376.2.a.bl.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:	$1$
Dimension:	$4$
Coefficient field:	4.4.19664.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 5x^{2} + 2x + 2 $ x^4 - 2x^3 - 5x^2 + 2*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 168)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.771876$ 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)

$f(q)$	$=$	$q-1.00000 q^{3} -1.12875 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})$	$q-1.00000 q^{3} -1.12875 q^{5} +1.00000 q^{7} +1.00000 q^{9} +4.76717 q^{11} -0.456247 q^{13} +1.12875 q^{15} +0.415006 q^{17} -7.63843 q^{19} -1.00000 q^{21} +1.58499 q^{23} -3.72593 q^{25} -1.00000 q^{27} -6.72593 q^{29} +5.89592 q^{31} -4.76717 q^{33} -1.12875 q^{35} +5.89592 q^{37} +0.456247 q^{39} +0.415006 q^{41} -9.43967 q^{43} -1.12875 q^{45} -11.2769 q^{47} +1.00000 q^{49} -0.415006 q^{51} +7.63843 q^{53} -5.38093 q^{55} +7.63843 q^{57} -4.00000 q^{59} +1.80125 q^{61} +1.00000 q^{63} +0.514988 q^{65} -8.09467 q^{67} -1.58499 q^{69} +10.2068 q^{71} +3.34500 q^{73} +3.72593 q^{75} +4.76717 q^{77} +4.83001 q^{79} +1.00000 q^{81} +5.53434 q^{83} -0.468436 q^{85} +6.72593 q^{87} -4.92999 q^{89} -0.456247 q^{91} -5.89592 q^{93} +8.62185 q^{95} +16.4468 q^{97} +4.76717 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{3} - 2 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) $ 4 * q - 4 * q^3 - 2 * q^5 + 4 * q^7 + 4 * q^9	$ 4 q - 4 q^{3} - 2 q^{5} + 4 q^{7} + 4 q^{9} - 6 q^{11} - 4 q^{13} + 2 q^{15} + 2 q^{17} - 8 q^{19} - 4 q^{21} + 6 q^{23} + 12 q^{25} - 4 q^{27} - 4 q^{31} + 6 q^{33} - 2 q^{35} - 4 q^{37} + 4 q^{39} + 2 q^{41} - 8 q^{43} - 2 q^{45} + 4 q^{49} - 2 q^{51} + 8 q^{53} - 4 q^{55} + 8 q^{57} - 16 q^{59} + 4 q^{63} - 8 q^{65} - 12 q^{67} - 6 q^{69} - 14 q^{71} + 4 q^{73} - 12 q^{75} - 6 q^{77} + 20 q^{79} + 4 q^{81} - 28 q^{83} + 20 q^{85} - 10 q^{89} - 4 q^{91} + 4 q^{93} - 20 q^{95} + 20 q^{97} - 6 q^{99}+O(q^{100}) $ 4 * q - 4 * q^3 - 2 * q^5 + 4 * q^7 + 4 * q^9 - 6 * q^11 - 4 * q^13 + 2 * q^15 + 2 * q^17 - 8 * q^19 - 4 * q^21 + 6 * q^23 + 12 * q^25 - 4 * q^27 - 4 * q^31 + 6 * q^33 - 2 * q^35 - 4 * q^37 + 4 * q^39 + 2 * q^41 - 8 * q^43 - 2 * q^45 + 4 * q^49 - 2 * q^51 + 8 * q^53 - 4 * q^55 + 8 * q^57 - 16 * q^59 + 4 * q^63 - 8 * q^65 - 12 * q^67 - 6 * q^69 - 14 * q^71 + 4 * q^73 - 12 * q^75 - 6 * q^77 + 20 * q^79 + 4 * q^81 - 28 * q^83 + 20 * q^85 - 10 * q^89 - 4 * q^91 + 4 * q^93 - 20 * q^95 + 20 * q^97 - 6 * 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
$5$	−1.12875	−0.504791	−0.252395	−	0.967624i	$-0.581218\pi$
$5$	−1.12875	−0.504791	−0.252395	+	0.967624i	$0.581218\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	1.00000	0.333333
$10$	0	0
$11$	4.76717	1.43736	0.718678	−	0.695343i	$-0.244747\pi$
$11$	4.76717	1.43736	0.718678	+	0.695343i	$0.244747\pi$
$12$	0	0
$13$	−0.456247	−0.126540	−0.0632701	−	0.997996i	$-0.520153\pi$
$13$	−0.456247	−0.126540	−0.0632701	+	0.997996i	$0.520153\pi$
$14$	0	0
$15$	1.12875	0.291441
$16$	0	0
$17$	0.415006	0.100654	0.0503268	−	0.998733i	$-0.483974\pi$
$17$	0.415006	0.100654	0.0503268	+	0.998733i	$0.483974\pi$
$18$	0	0
$19$	−7.63843	−1.75237	−0.876187	−	0.481970i	$-0.839921\pi$
$19$	−7.63843	−1.75237	−0.876187	+	0.481970i	$0.839921\pi$
$20$	0	0
$21$	−1.00000	−0.218218
$22$	0	0
$23$	1.58499	0.330494	0.165247	−	0.986252i	$-0.447158\pi$
$23$	1.58499	0.330494	0.165247	+	0.986252i	$0.447158\pi$
$24$	0	0
$25$	−3.72593	−0.745186
$26$	0	0
$27$	−1.00000	−0.192450
$28$	0	0
$29$	−6.72593	−1.24897	−0.624487	−	0.781035i	$-0.714692\pi$
$29$	−6.72593	−1.24897	−0.624487	+	0.781035i	$0.714692\pi$
$30$	0	0
$31$	5.89592	1.05894	0.529469	−	0.848329i	$-0.322391\pi$
$31$	5.89592	1.05894	0.529469	+	0.848329i	$0.322391\pi$
$32$	0	0
$33$	−4.76717	−0.829858
$34$	0	0
$35$	−1.12875	−0.190793
$36$	0	0
$37$	5.89592	0.969283	0.484642	−	0.874713i	$-0.338950\pi$
$37$	5.89592	0.969283	0.484642	+	0.874713i	$0.338950\pi$
$38$	0	0
$39$	0.456247	0.0730581
$40$	0	0
$41$	0.415006	0.0648130	0.0324065	−	0.999475i	$-0.489683\pi$
$41$	0.415006	0.0648130	0.0324065	+	0.999475i	$0.489683\pi$
$42$	0	0
$43$	−9.43967	−1.43954	−0.719768	−	0.694214i	$-0.755752\pi$
$43$	−9.43967	−1.43954	−0.719768	+	0.694214i	$0.755752\pi$
$44$	0	0
$45$	−1.12875	−0.168264
$46$	0	0
$47$	−11.2769	−1.64490	−0.822449	−	0.568839i	$-0.807393\pi$
$47$	−11.2769	−1.64490	−0.822449	+	0.568839i	$0.807393\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−0.415006	−0.0581124
$52$	0	0
$53$	7.63843	1.04922	0.524609	−	0.851343i	$-0.324211\pi$
$53$	7.63843	1.04922	0.524609	+	0.851343i	$0.324211\pi$
$54$	0	0
$55$	−5.38093	−0.725565
$56$	0	0
$57$	7.63843	1.01173
$58$	0	0
$59$	−4.00000	−0.520756	−0.260378	−	0.965507i	$-0.583847\pi$
$59$	−4.00000	−0.520756	−0.260378	+	0.965507i	$0.583847\pi$
$60$	0	0
$61$	1.80125	0.230626	0.115313	−	0.993329i	$-0.463213\pi$
$61$	1.80125	0.230626	0.115313	+	0.993329i	$0.463213\pi$
$62$	0	0
$63$	1.00000	0.125988
$64$	0	0
$65$	0.514988	0.0638764
$66$	0	0
$67$	−8.09467	−0.988922	−0.494461	−	0.869200i	$-0.664634\pi$
$67$	−8.09467	−0.988922	−0.494461	+	0.869200i	$0.664634\pi$
$68$	0	0
$69$	−1.58499	−0.190811
$70$	0	0
$71$	10.2068	1.21133	0.605665	−	0.795720i	$-0.292907\pi$
$71$	10.2068	1.21133	0.605665	+	0.795720i	$0.292907\pi$
$72$	0	0
$73$	3.34500	0.391503	0.195751	−	0.980654i	$-0.437285\pi$
$73$	3.34500	0.391503	0.195751	+	0.980654i	$0.437285\pi$
$74$	0	0
$75$	3.72593	0.430233
$76$	0	0
$77$	4.76717	0.543270
$78$	0	0
$79$	4.83001	0.543419	0.271709	−	0.962379i	$-0.412411\pi$
$79$	4.83001	0.543419	0.271709	+	0.962379i	$0.412411\pi$
$80$	0	0
$81$	1.00000	0.111111
$82$	0	0
$83$	5.53434	0.607473	0.303737	−	0.952756i	$-0.401766\pi$
$83$	5.53434	0.607473	0.303737	+	0.952756i	$0.401766\pi$
$84$	0	0
$85$	−0.468436	−0.0508090
$86$	0	0
$87$	6.72593	0.721095
$88$	0	0
$89$	−4.92999	−0.522578	−0.261289	−	0.965261i	$-0.584148\pi$
$89$	−4.92999	−0.522578	−0.261289	+	0.965261i	$0.584148\pi$
$90$	0	0
$91$	−0.456247	−0.0478277
$92$	0	0
$93$	−5.89592	−0.611378
$94$	0	0
$95$	8.62185	0.884583
$96$	0	0
$97$	16.4468	1.66992	0.834962	−	0.550308i	$-0.185490\pi$
$97$	16.4468	1.66992	0.834962	+	0.550308i	$0.185490\pi$
$98$	0	0
$99$	4.76717	0.479119
$100$	0	0
$101$	−16.4056	−1.63242	−0.816209	−	0.577757i	$-0.803928\pi$
$101$	−16.4056	−1.63242	−0.816209	+	0.577757i	$0.803928\pi$
$102$	0	0
$103$	−17.1728	−1.69208	−0.846042	−	0.533117i	$-0.821021\pi$
$103$	−17.1728	−1.69208	−0.846042	+	0.533117i	$0.821021\pi$
$104$	0	0
$105$	1.12875	0.110154
$106$	0	0
$107$	−12.7672	−1.23425	−0.617125	−	0.786865i	$-0.711703\pi$
$107$	−12.7672	−1.23425	−0.617125	+	0.786865i	$0.711703\pi$
$108$	0	0
$109$	−7.27685	−0.696996	−0.348498	−	0.937310i	$-0.613308\pi$
$109$	−7.27685	−0.696996	−0.348498	+	0.937310i	$0.613308\pi$
$110$	0	0
$111$	−5.89592	−0.559616
$112$	0	0
$113$	−3.34500	−0.314671	−0.157336	−	0.987545i	$-0.550290\pi$
$113$	−3.34500	−0.314671	−0.157336	+	0.987545i	$0.550290\pi$
$114$	0	0
$115$	−1.78906	−0.166830
$116$	0	0
$117$	−0.456247	−0.0421801
$118$	0	0
$119$	0.415006	0.0380435
$120$	0	0
$121$	11.7259	1.06599
$122$	0	0
$123$	−0.415006	−0.0374198
$124$	0	0
$125$	9.84937	0.880954
$126$	0	0
$127$	10.4468	0.927007	0.463504	−	0.886095i	$-0.346592\pi$
$127$	10.4468	0.927007	0.463504	+	0.886095i	$0.346592\pi$
$128$	0	0
$129$	9.43967	0.831117
$130$	0	0
$131$	−6.25749	−0.546720	−0.273360	−	0.961912i	$-0.588135\pi$
$131$	−6.25749	−0.546720	−0.273360	+	0.961912i	$0.588135\pi$
$132$	0	0
$133$	−7.63843	−0.662335
$134$	0	0
$135$	1.12875	0.0971471
$136$	0	0
$137$	12.9618	1.10740	0.553702	−	0.832715i	$-0.313215\pi$
$137$	12.9618	1.10740	0.553702	+	0.832715i	$0.313215\pi$
$138$	0	0
$139$	−18.3644	−1.55764	−0.778822	−	0.627245i	$-0.784183\pi$
$139$	−18.3644	−1.55764	−0.778822	+	0.627245i	$0.784183\pi$
$140$	0	0
$141$	11.2769	0.949682
$142$	0	0
$143$	−2.17501	−0.181883
$144$	0	0
$145$	7.59187	0.630471
$146$	0	0
$147$	−1.00000	−0.0824786
$148$	0	0
$149$	15.6384	1.28115	0.640575	−	0.767896i	$-0.278696\pi$
$149$	15.6384	1.28115	0.640575	+	0.767896i	$0.278696\pi$
$150$	0	0
$151$	15.2769	1.24321	0.621606	−	0.783330i	$-0.286480\pi$
$151$	15.2769	1.24321	0.621606	+	0.783330i	$0.286480\pi$
$152$	0	0
$153$	0.415006	0.0335512
$154$	0	0
$155$	−6.65500	−0.534543
$156$	0	0
$157$	21.7824	1.73843	0.869214	−	0.494437i	$-0.164626\pi$
$157$	21.7824	1.73843	0.869214	+	0.494437i	$0.164626\pi$
$158$	0	0
$159$	−7.63843	−0.605767
$160$	0	0
$161$	1.58499	0.124915
$162$	0	0
$163$	4.60966	0.361056	0.180528	−	0.983570i	$-0.442219\pi$
$163$	4.60966	0.361056	0.180528	+	0.983570i	$0.442219\pi$
$164$	0	0
$165$	5.38093	0.418905
$166$	0	0
$167$	−22.9618	−1.77684	−0.888420	−	0.459032i	$-0.848196\pi$
$167$	−22.9618	−1.77684	−0.888420	+	0.459032i	$0.848196\pi$
$168$	0	0
$169$	−12.7918	−0.983988
$170$	0	0
$171$	−7.63843	−0.584125
$172$	0	0
$173$	−0.216252	−0.0164413	−0.00822067	−	0.999966i	$-0.502617\pi$
$173$	−0.216252	−0.0164413	−0.00822067	+	0.999966i	$0.502617\pi$
$174$	0	0
$175$	−3.72593	−0.281654
$176$	0	0
$177$	4.00000	0.300658
$178$	0	0
$179$	−22.6165	−1.69044	−0.845220	−	0.534419i	$-0.820530\pi$
$179$	−22.6165	−1.69044	−0.845220	+	0.534419i	$0.820530\pi$
$180$	0	0
$181$	−7.73310	−0.574797	−0.287398	−	0.957811i	$-0.592790\pi$
$181$	−7.73310	−0.574797	−0.287398	+	0.957811i	$0.592790\pi$
$182$	0	0
$183$	−1.80125	−0.133152
$184$	0	0
$185$	−6.65500	−0.489285
$186$	0	0
$187$	1.97840	0.144675
$188$	0	0
$189$	−1.00000	−0.0727393
$190$	0	0
$191$	−10.2068	−0.738541	−0.369271	−	0.929322i	$-0.620392\pi$
$191$	−10.2068	−0.738541	−0.369271	+	0.929322i	$0.620392\pi$
$192$	0	0
$193$	5.96407	0.429303	0.214651	−	0.976691i	$-0.431138\pi$
$193$	5.96407	0.429303	0.214651	+	0.976691i	$0.431138\pi$
$194$	0	0
$195$	−0.514988	−0.0368791
$196$	0	0
$197$	−18.8084	−1.34004	−0.670022	−	0.742341i	$-0.733715\pi$
$197$	−18.8084	−1.34004	−0.670022	+	0.742341i	$0.733715\pi$
$198$	0	0
$199$	3.27685	0.232290	0.116145	−	0.993232i	$-0.462946\pi$
$199$	3.27685	0.232290	0.116145	+	0.993232i	$0.462946\pi$
$200$	0	0
$201$	8.09467	0.570954
$202$	0	0
$203$	−6.72593	−0.472068
$204$	0	0
$205$	−0.468436	−0.0327170
$206$	0	0
$207$	1.58499	0.110165
$208$	0	0
$209$	−36.4137	−2.51879
$210$	0	0
$211$	−10.1628	−0.699637	−0.349819	−	0.936817i	$-0.613757\pi$
$211$	−10.1628	−0.699637	−0.349819	+	0.936817i	$0.613757\pi$
$212$	0	0
$213$	−10.2068	−0.699361
$214$	0	0
$215$	10.6550	0.726665
$216$	0	0
$217$	5.89592	0.400241
$218$	0	0
$219$	−3.34500	−0.226034
$220$	0	0
$221$	−0.189345	−0.0127367
$222$	0	0
$223$	−21.9668	−1.47101	−0.735504	−	0.677520i	$-0.763055\pi$
$223$	−21.9668	−1.47101	−0.735504	+	0.677520i	$0.763055\pi$
$224$	0	0
$225$	−3.72593	−0.248395
$226$	0	0
$227$	−7.91752	−0.525504	−0.262752	−	0.964863i	$-0.584630\pi$
$227$	−7.91752	−0.525504	−0.262752	+	0.964863i	$0.584630\pi$
$228$	0	0
$229$	13.1606	0.869676	0.434838	−	0.900509i	$-0.356806\pi$
$229$	13.1606	0.869676	0.434838	+	0.900509i	$0.356806\pi$
$230$	0	0
$231$	−4.76717	−0.313657
$232$	0	0
$233$	−17.2769	−1.13184	−0.565922	−	0.824459i	$-0.691480\pi$
$233$	−17.2769	−1.13184	−0.565922	+	0.824459i	$0.691480\pi$
$234$	0	0
$235$	12.7287	0.830330
$236$	0	0
$237$	−4.83001	−0.313743
$238$	0	0
$239$	10.5219	0.680603	0.340301	−	0.940316i	$-0.389471\pi$
$239$	10.5219	0.680603	0.340301	+	0.940316i	$0.389471\pi$
$240$	0	0
$241$	−6.10686	−0.393378	−0.196689	−	0.980466i	$-0.563019\pi$
$241$	−6.10686	−0.393378	−0.196689	+	0.980466i	$0.563019\pi$
$242$	0	0
$243$	−1.00000	−0.0641500
$244$	0	0
$245$	−1.12875	−0.0721130
$246$	0	0
$247$	3.48501	0.221746
$248$	0	0
$249$	−5.53434	−0.350725
$250$	0	0
$251$	−19.8743	−1.25446	−0.627228	−	0.778836i	$-0.715811\pi$
$251$	−19.8743	−1.25446	−0.627228	+	0.778836i	$0.715811\pi$
$252$	0	0
$253$	7.55594	0.475038
$254$	0	0
$255$	0.468436	0.0293346
$256$	0	0
$257$	−14.7550	−0.920391	−0.460195	−	0.887818i	$-0.652221\pi$
$257$	−14.7550	−0.920391	−0.460195	+	0.887818i	$0.652221\pi$
$258$	0	0
$259$	5.89592	0.366355
$260$	0	0
$261$	−6.72593	−0.416325
$262$	0	0
$263$	−15.7600	−0.971804	−0.485902	−	0.874013i	$-0.661509\pi$
$263$	−15.7600	−0.971804	−0.485902	+	0.874013i	$0.661509\pi$
$264$	0	0
$265$	−8.62185	−0.529636
$266$	0	0
$267$	4.92999	0.301711
$268$	0	0
$269$	−21.8331	−1.33119	−0.665593	−	0.746315i	$-0.731821\pi$
$269$	−21.8331	−1.33119	−0.665593	+	0.746315i	$0.731821\pi$
$270$	0	0
$271$	18.8328	1.14401	0.572005	−	0.820250i	$-0.306166\pi$
$271$	18.8328	1.14401	0.572005	+	0.820250i	$0.306166\pi$
$272$	0	0
$273$	0.456247	0.0276134
$274$	0	0
$275$	−17.7622	−1.07110
$276$	0	0
$277$	24.4684	1.47017	0.735083	−	0.677977i	$-0.237143\pi$
$277$	24.4684	1.47017	0.735083	+	0.677977i	$0.237143\pi$
$278$	0	0
$279$	5.89592	0.352979
$280$	0	0
$281$	−14.5150	−0.865892	−0.432946	−	0.901420i	$-0.642526\pi$
$281$	−14.5150	−0.865892	−0.432946	+	0.901420i	$0.642526\pi$
$282$	0	0
$283$	−16.5509	−0.983850	−0.491925	−	0.870638i	$-0.663707\pi$
$283$	−16.5509	−0.983850	−0.491925	+	0.870638i	$0.663707\pi$
$284$	0	0
$285$	−8.62185	−0.510714
$286$	0	0
$287$	0.415006	0.0244970
$288$	0	0
$289$	−16.8278	−0.989869
$290$	0	0
$291$	−16.4468	−0.964131
$292$	0	0
$293$	−9.44377	−0.551711	−0.275855	−	0.961199i	$-0.588961\pi$
$293$	−9.44377	−0.551711	−0.275855	+	0.961199i	$0.588961\pi$
$294$	0	0
$295$	4.51499	0.262873
$296$	0	0
$297$	−4.76717	−0.276619
$298$	0	0
$299$	−0.723150	−0.0418208
$300$	0	0
$301$	−9.43967	−0.544094
$302$	0	0
$303$	16.4056	0.942477
$304$	0	0
$305$	−2.03315	−0.116418
$306$	0	0
$307$	−0.361575	−0.0206362	−0.0103181	−	0.999947i	$-0.503284\pi$
$307$	−0.361575	−0.0206362	−0.0103181	+	0.999947i	$0.503284\pi$
$308$	0	0
$309$	17.1728	0.976925
$310$	0	0
$311$	−11.2769	−0.639452	−0.319726	−	0.947510i	$-0.603591\pi$
$311$	−11.2769	−0.639452	−0.319726	+	0.947510i	$0.603591\pi$
$312$	0	0
$313$	−25.5837	−1.44607	−0.723037	−	0.690809i	$-0.757255\pi$
$313$	−25.5837	−1.44607	−0.723037	+	0.690809i	$0.757255\pi$
$314$	0	0
$315$	−1.12875	−0.0635977
$316$	0	0
$317$	0.361575	0.0203081	0.0101540	−	0.999948i	$-0.496768\pi$
$317$	0.361575	0.0203081	0.0101540	+	0.999948i	$0.496768\pi$
$318$	0	0
$319$	−32.0637	−1.79522
$320$	0	0
$321$	12.7672	0.712594
$322$	0	0
$323$	−3.16999	−0.176383
$324$	0	0
$325$	1.69995	0.0942960
$326$	0	0
$327$	7.27685	0.402411
$328$	0	0
$329$	−11.2769	−0.621713
$330$	0	0
$331$	7.80403	0.428948	0.214474	−	0.976730i	$-0.431196\pi$
$331$	7.80403	0.428948	0.214474	+	0.976730i	$0.431196\pi$
$332$	0	0
$333$	5.89592	0.323094
$334$	0	0
$335$	9.13684	0.499199
$336$	0	0
$337$	8.76186	0.477289	0.238645	−	0.971107i	$-0.423297\pi$
$337$	8.76186	0.477289	0.238645	+	0.971107i	$0.423297\pi$
$338$	0	0
$339$	3.34500	0.181675
$340$	0	0
$341$	28.1069	1.52207
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	1.78906	0.0963196
$346$	0	0
$347$	2.07717	0.111509	0.0557543	−	0.998445i	$-0.482244\pi$
$347$	2.07717	0.111509	0.0557543	+	0.998445i	$0.482244\pi$
$348$	0	0
$349$	−18.7381	−1.00303	−0.501514	−	0.865149i	$-0.667224\pi$
$349$	−18.7381	−1.00303	−0.501514	+	0.865149i	$0.667224\pi$
$350$	0	0
$351$	0.456247	0.0243527
$352$	0	0
$353$	−13.2450	−0.704961	−0.352481	−	0.935819i	$-0.614662\pi$
$353$	−13.2450	−0.704961	−0.352481	+	0.935819i	$0.614662\pi$
$354$	0	0
$355$	−11.5209	−0.611468
$356$	0	0
$357$	−0.415006	−0.0219644
$358$	0	0
$359$	−5.69186	−0.300405	−0.150202	−	0.988655i	$-0.547993\pi$
$359$	−5.69186	−0.300405	−0.150202	+	0.988655i	$0.547993\pi$
$360$	0	0
$361$	39.3455	2.07082
$362$	0	0
$363$	−11.7259	−0.615452
$364$	0	0
$365$	−3.77566	−0.197627
$366$	0	0
$367$	−11.2769	−0.588647	−0.294323	−	0.955706i	$-0.595094\pi$
$367$	−11.2769	−0.588647	−0.294323	+	0.955706i	$0.595094\pi$
$368$	0	0
$369$	0.415006	0.0216043
$370$	0	0
$371$	7.63843	0.396567
$372$	0	0
$373$	7.08751	0.366977	0.183489	−	0.983022i	$-0.441261\pi$
$373$	7.08751	0.366977	0.183489	+	0.983022i	$0.441261\pi$
$374$	0	0
$375$	−9.84937	−0.508619
$376$	0	0
$377$	3.06869	0.158046
$378$	0	0
$379$	−6.67781	−0.343016	−0.171508	−	0.985183i	$-0.554864\pi$
$379$	−6.67781	−0.343016	−0.171508	+	0.985183i	$0.554864\pi$
$380$	0	0
$381$	−10.4468	−0.535208
$382$	0	0
$383$	18.6550	0.953226	0.476613	−	0.879113i	$-0.341864\pi$
$383$	18.6550	0.953226	0.476613	+	0.879113i	$0.341864\pi$
$384$	0	0
$385$	−5.38093	−0.274238
$386$	0	0
$387$	−9.43967	−0.479845
$388$	0	0
$389$	20.3428	1.03142	0.515709	−	0.856764i	$-0.327528\pi$
$389$	20.3428	1.03142	0.515709	+	0.856764i	$0.327528\pi$
$390$	0	0
$391$	0.657782	0.0332654
$392$	0	0
$393$	6.25749	0.315649
$394$	0	0
$395$	−5.45186	−0.274313
$396$	0	0
$397$	−10.9031	−0.547210	−0.273605	−	0.961842i	$-0.588216\pi$
$397$	−10.9031	−0.547210	−0.273605	+	0.961842i	$0.588216\pi$
$398$	0	0
$399$	7.63843	0.382400
$400$	0	0
$401$	29.1756	1.45696	0.728479	−	0.685068i	$-0.240228\pi$
$401$	29.1756	1.45696	0.728479	+	0.685068i	$0.240228\pi$
$402$	0	0
$403$	−2.69000	−0.133998
$404$	0	0
$405$	−1.12875	−0.0560879
$406$	0	0
$407$	28.1069	1.39321
$408$	0	0
$409$	31.7237	1.56864	0.784318	−	0.620359i	$-0.213013\pi$
$409$	31.7237	1.56864	0.784318	+	0.620359i	$0.213013\pi$
$410$	0	0
$411$	−12.9618	−0.639360
$412$	0	0
$413$	−4.00000	−0.196827
$414$	0	0
$415$	−6.24687	−0.306647
$416$	0	0
$417$	18.3644	0.899306
$418$	0	0
$419$	−19.2769	−0.941736	−0.470868	−	0.882204i	$-0.656059\pi$
$419$	−19.2769	−0.941736	−0.470868	+	0.882204i	$0.656059\pi$
$420$	0	0
$421$	−19.3234	−0.941765	−0.470882	−	0.882196i	$-0.656064\pi$
$421$	−19.3234	−0.941765	−0.470882	+	0.882196i	$0.656064\pi$
$422$	0	0
$423$	−11.2769	−0.548299
$424$	0	0
$425$	−1.54628	−0.0750057
$426$	0	0
$427$	1.80125	0.0871684
$428$	0	0
$429$	2.17501	0.105010
$430$	0	0
$431$	−14.4150	−0.694346	−0.347173	−	0.937801i	$-0.612858\pi$
$431$	−14.4150	−0.694346	−0.347173	+	0.937801i	$0.612858\pi$
$432$	0	0
$433$	−9.27685	−0.445817	−0.222908	−	0.974839i	$-0.571555\pi$
$433$	−9.27685	−0.445817	−0.222908	+	0.974839i	$0.571555\pi$
$434$	0	0
$435$	−7.59187	−0.364002
$436$	0	0
$437$	−12.1069	−0.579150
$438$	0	0
$439$	0	0		−	1.00000i	$-0.5\pi$
$439$	0	0			1.00000i	$0.5\pi$
$440$	0	0
$441$	1.00000	0.0476190
$442$	0	0
$443$	15.6465	0.743388	0.371694	−	0.928355i	$-0.378777\pi$
$443$	15.6465	0.743388	0.371694	+	0.928355i	$0.378777\pi$
$444$	0	0
$445$	5.56472	0.263793
$446$	0	0
$447$	−15.6384	−0.739672
$448$	0	0
$449$	−12.6550	−0.597226	−0.298613	−	0.954374i	$-0.596524\pi$
$449$	−12.6550	−0.597226	−0.298613	+	0.954374i	$0.596524\pi$
$450$	0	0
$451$	1.97840	0.0931594
$452$	0	0
$453$	−15.2769	−0.717769
$454$	0	0
$455$	0.514988	0.0241430
$456$	0	0
$457$	23.9237	1.11910	0.559551	−	0.828796i	$-0.310974\pi$
$457$	23.9237	1.11910	0.559551	+	0.828796i	$0.310974\pi$
$458$	0	0
$459$	−0.415006	−0.0193708
$460$	0	0
$461$	6.74558	0.314173	0.157086	−	0.987585i	$-0.449790\pi$
$461$	6.74558	0.314173	0.157086	+	0.987585i	$0.449790\pi$
$462$	0	0
$463$	11.1700	0.519113	0.259557	−	0.965728i	$-0.416424\pi$
$463$	11.1700	0.519113	0.259557	+	0.965728i	$0.416424\pi$
$464$	0	0
$465$	6.65500	0.308618
$466$	0	0
$467$	−3.70935	−0.171648	−0.0858242	−	0.996310i	$-0.527352\pi$
$467$	−3.70935	−0.171648	−0.0858242	+	0.996310i	$0.527352\pi$
$468$	0	0
$469$	−8.09467	−0.373777
$470$	0	0
$471$	−21.7824	−1.00368
$472$	0	0
$473$	−45.0005	−2.06913
$474$	0	0
$475$	28.4602	1.30585
$476$	0	0
$477$	7.63843	0.349739
$478$	0	0
$479$	2.17501	0.0993788	0.0496894	−	0.998765i	$-0.484177\pi$
$479$	2.17501	0.0993788	0.0496894	+	0.998765i	$0.484177\pi$
$480$	0	0
$481$	−2.69000	−0.122653
$482$	0	0
$483$	−1.58499	−0.0721197
$484$	0	0
$485$	−18.5643	−0.842962
$486$	0	0
$487$	−4.51499	−0.204594	−0.102297	−	0.994754i	$-0.532619\pi$
$487$	−4.51499	−0.204594	−0.102297	+	0.994754i	$0.532619\pi$
$488$	0	0
$489$	−4.60966	−0.208456
$490$	0	0
$491$	−1.91221	−0.0862967	−0.0431483	−	0.999069i	$-0.513739\pi$
$491$	−1.91221	−0.0862967	−0.0431483	+	0.999069i	$0.513739\pi$
$492$	0	0
$493$	−2.79130	−0.125714
$494$	0	0
$495$	−5.38093	−0.241855
$496$	0	0
$497$	10.2068	0.457840
$498$	0	0
$499$	−27.4065	−1.22688	−0.613442	−	0.789740i	$-0.710216\pi$
$499$	−27.4065	−1.22688	−0.613442	+	0.789740i	$0.710216\pi$
$500$	0	0
$501$	22.9618	1.02586
$502$	0	0
$503$	17.1368	0.764094	0.382047	−	0.924143i	$-0.375219\pi$
$503$	17.1368	0.764094	0.382047	+	0.924143i	$0.375219\pi$
$504$	0	0
$505$	18.5178	0.824030
$506$	0	0
$507$	12.7918	0.568105
$508$	0	0
$509$	21.6437	0.959342	0.479671	−	0.877449i	$-0.340756\pi$
$509$	21.6437	0.959342	0.479671	+	0.877449i	$0.340756\pi$
$510$	0	0
$511$	3.34500	0.147974
$512$	0	0
$513$	7.63843	0.337245
$514$	0	0
$515$	19.3837	0.854148
$516$	0	0
$517$	−53.7587	−2.36430
$518$	0	0
$519$	0.216252	0.00949241
$520$	0	0
$521$	−28.3137	−1.24045	−0.620223	−	0.784426i	$-0.712958\pi$
$521$	−28.3137	−1.24045	−0.620223	+	0.784426i	$0.712958\pi$
$522$	0	0
$523$	−15.0687	−0.658908	−0.329454	−	0.944172i	$-0.606865\pi$
$523$	−15.0687	−0.658908	−0.329454	+	0.944172i	$0.606865\pi$
$524$	0	0
$525$	3.72593	0.162613
$526$	0	0
$527$	2.44684	0.106586
$528$	0	0
$529$	−20.4878	−0.890774
$530$	0	0
$531$	−4.00000	−0.173585
$532$	0	0
$533$	−0.189345	−0.00820145
$534$	0	0
$535$	14.4109	0.623038
$536$	0	0
$537$	22.6165	0.975976
$538$	0	0
$539$	4.76717	0.205337
$540$	0	0
$541$	46.2990	1.99055	0.995274	−	0.0971017i	$-0.0309572\pi$
$541$	46.2990	1.99055	0.995274	+	0.0971017i	$0.0309572\pi$
$542$	0	0
$543$	7.73310	0.331859
$544$	0	0
$545$	8.21372	0.351837
$546$	0	0
$547$	−2.03714	−0.0871020	−0.0435510	−	0.999051i	$-0.513867\pi$
$547$	−2.03714	−0.0871020	−0.0435510	+	0.999051i	$0.513867\pi$
$548$	0	0
$549$	1.80125	0.0768753
$550$	0	0
$551$	51.3755	2.18867
$552$	0	0
$553$	4.83001	0.205393
$554$	0	0
$555$	6.65500	0.282489
$556$	0	0
$557$	11.1234	0.471315	0.235658	−	0.971836i	$-0.424276\pi$
$557$	11.1234	0.471315	0.235658	+	0.971836i	$0.424276\pi$
$558$	0	0
$559$	4.30683	0.182159
$560$	0	0
$561$	−1.97840	−0.0835282
$562$	0	0
$563$	37.2437	1.56963	0.784817	−	0.619727i	$-0.212757\pi$
$563$	37.2437	1.56963	0.784817	+	0.619727i	$0.212757\pi$
$564$	0	0
$565$	3.77566	0.158843
$566$	0	0
$567$	1.00000	0.0419961
$568$	0	0
$569$	−22.9369	−0.961564	−0.480782	−	0.876840i	$-0.659647\pi$
$569$	−22.9369	−0.961564	−0.480782	+	0.876840i	$0.659647\pi$
$570$	0	0
$571$	−35.1634	−1.47154	−0.735770	−	0.677231i	$-0.763180\pi$
$571$	−35.1634	−1.47154	−0.735770	+	0.677231i	$0.763180\pi$
$572$	0	0
$573$	10.2068	0.426397
$574$	0	0
$575$	−5.90558	−0.246280
$576$	0	0
$577$	13.7918	0.574162	0.287081	−	0.957906i	$-0.407315\pi$
$577$	13.7918	0.574162	0.287081	+	0.957906i	$0.407315\pi$
$578$	0	0
$579$	−5.96407	−0.247858
$580$	0	0
$581$	5.53434	0.229603
$582$	0	0
$583$	36.4137	1.50810
$584$	0	0
$585$	0.514988	0.0212921
$586$	0	0
$587$	26.9862	1.11384	0.556920	−	0.830566i	$-0.311983\pi$
$587$	26.9862	1.11384	0.556920	+	0.830566i	$0.311983\pi$
$588$	0	0
$589$	−45.0355	−1.85566
$590$	0	0
$591$	18.8084	0.773675
$592$	0	0
$593$	−31.8337	−1.30725	−0.653627	−	0.756817i	$-0.726753\pi$
$593$	−31.8337	−1.30725	−0.653627	+	0.756817i	$0.726753\pi$
$594$	0	0
$595$	−0.468436	−0.0192040
$596$	0	0
$597$	−3.27685	−0.134113
$598$	0	0
$599$	−34.9006	−1.42600	−0.712999	−	0.701165i	$-0.752664\pi$
$599$	−34.9006	−1.42600	−0.712999	+	0.701165i	$0.752664\pi$
$600$	0	0
$601$	0.175010	0.00713881	0.00356941	−	0.999994i	$-0.498864\pi$
$601$	0.175010	0.00713881	0.00356941	+	0.999994i	$0.498864\pi$
$602$	0	0
$603$	−8.09467	−0.329641
$604$	0	0
$605$	−13.2356	−0.538104
$606$	0	0
$607$	32.1705	1.30576	0.652881	−	0.757461i	$-0.273560\pi$
$607$	32.1705	1.30576	0.652881	+	0.757461i	$0.273560\pi$
$608$	0	0
$609$	6.72593	0.272548
$610$	0	0
$611$	5.14503	0.208146
$612$	0	0
$613$	−8.37869	−0.338412	−0.169206	−	0.985581i	$-0.554120\pi$
$613$	−8.37869	−0.338412	−0.169206	+	0.985581i	$0.554120\pi$
$614$	0	0
$615$	0.468436	0.0188892
$616$	0	0
$617$	−28.9618	−1.16596	−0.582980	−	0.812487i	$-0.698113\pi$
$617$	−28.9618	−1.16596	−0.582980	+	0.812487i	$0.698113\pi$
$618$	0	0
$619$	−6.85497	−0.275524	−0.137762	−	0.990465i	$-0.543991\pi$
$619$	−6.85497	−0.275524	−0.137762	+	0.990465i	$0.543991\pi$
$620$	0	0
$621$	−1.58499	−0.0636036
$622$	0	0
$623$	−4.92999	−0.197516
$624$	0	0
$625$	7.51221	0.300488
$626$	0	0
$627$	36.4137	1.45422
$628$	0	0
$629$	2.44684	0.0975619
$630$	0	0
$631$	−40.2055	−1.60056	−0.800278	−	0.599629i	$-0.795315\pi$
$631$	−40.2055	−1.60056	−0.800278	+	0.599629i	$0.795315\pi$
$632$	0	0
$633$	10.1628	0.403936
$634$	0	0
$635$	−11.7918	−0.467945
$636$	0	0
$637$	−0.456247	−0.0180772
$638$	0	0
$639$	10.2068	0.403777
$640$	0	0
$641$	−24.0737	−0.950854	−0.475427	−	0.879755i	$-0.657706\pi$
$641$	−24.0737	−0.950854	−0.475427	+	0.879755i	$0.657706\pi$
$642$	0	0
$643$	23.7559	0.936841	0.468421	−	0.883506i	$-0.344823\pi$
$643$	23.7559	0.936841	0.468421	+	0.883506i	$0.344823\pi$
$644$	0	0
$645$	−10.6550	−0.419540
$646$	0	0
$647$	−32.6218	−1.28250	−0.641249	−	0.767333i	$-0.721583\pi$
$647$	−32.6218	−1.28250	−0.641249	+	0.767333i	$0.721583\pi$
$648$	0	0
$649$	−19.0687	−0.748512
$650$	0	0
$651$	−5.89592	−0.231079
$652$	0	0
$653$	14.4109	0.563942	0.281971	−	0.959423i	$-0.409012\pi$
$653$	14.4109	0.563942	0.281971	+	0.959423i	$0.409012\pi$
$654$	0	0
$655$	7.06313	0.275979
$656$	0	0
$657$	3.34500	0.130501
$658$	0	0
$659$	23.3315	0.908866	0.454433	−	0.890781i	$-0.349842\pi$
$659$	23.3315	0.908866	0.454433	+	0.890781i	$0.349842\pi$
$660$	0	0
$661$	44.1468	1.71711	0.858555	−	0.512721i	$-0.171362\pi$
$661$	44.1468	1.71711	0.858555	+	0.512721i	$0.171362\pi$
$662$	0	0
$663$	0.189345	0.00735356
$664$	0	0
$665$	8.62185	0.334341
$666$	0	0
$667$	−10.6606	−0.412779
$668$	0	0
$669$	21.9668	0.849287
$670$	0	0
$671$	8.58685	0.331492
$672$	0	0
$673$	11.6960	0.450846	0.225423	−	0.974261i	$-0.427624\pi$
$673$	11.6960	0.450846	0.225423	+	0.974261i	$0.427624\pi$
$674$	0	0
$675$	3.72593	0.143411
$676$	0	0
$677$	0.648756	0.0249337	0.0124669	−	0.999922i	$-0.496032\pi$
$677$	0.648756	0.0249337	0.0124669	+	0.999922i	$0.496032\pi$
$678$	0	0
$679$	16.4468	0.631172
$680$	0	0
$681$	7.91752	0.303400
$682$	0	0
$683$	−22.4909	−0.860589	−0.430294	−	0.902689i	$-0.641590\pi$
$683$	−22.4909	−0.860589	−0.430294	+	0.902689i	$0.641590\pi$
$684$	0	0
$685$	−14.6306	−0.559007
$686$	0	0
$687$	−13.1606	−0.502107
$688$	0	0
$689$	−3.48501	−0.132768
$690$	0	0
$691$	45.2681	1.72208	0.861039	−	0.508538i	$-0.169814\pi$
$691$	45.2681	1.72208	0.861039	+	0.508538i	$0.169814\pi$
$692$	0	0
$693$	4.76717	0.181090
$694$	0	0
$695$	20.7287	0.786285
$696$	0	0
$697$	0.172230	0.00652366
$698$	0	0
$699$	17.2769	0.653470
$700$	0	0
$701$	1.03091	0.0389370	0.0194685	−	0.999810i	$-0.493803\pi$
$701$	1.03091	0.0389370	0.0194685	+	0.999810i	$0.493803\pi$
$702$	0	0
$703$	−45.0355	−1.69855
$704$	0	0
$705$	−12.7287	−0.479391
$706$	0	0
$707$	−16.4056	−0.616996
$708$	0	0
$709$	−10.5481	−0.396144	−0.198072	−	0.980188i	$-0.563468\pi$
$709$	−10.5481	−0.396144	−0.198072	+	0.980188i	$0.563468\pi$
$710$	0	0
$711$	4.83001	0.181140
$712$	0	0
$713$	9.34500	0.349973
$714$	0	0
$715$	2.45504	0.0918131
$716$	0	0
$717$	−10.5219	−0.392946
$718$	0	0
$719$	7.00502	0.261243	0.130622	−	0.991432i	$-0.458303\pi$
$719$	7.00502	0.261243	0.130622	+	0.991432i	$0.458303\pi$
$720$	0	0
$721$	−17.1728	−0.639547
$722$	0	0
$723$	6.10686	0.227117
$724$	0	0
$725$	25.0603	0.930718
$726$	0	0
$727$	−17.8028	−0.660270	−0.330135	−	0.943934i	$-0.607094\pi$
$727$	−17.8028	−0.660270	−0.330135	+	0.943934i	$0.607094\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	0	0
$731$	−3.91752	−0.144895
$732$	0	0
$733$	34.3300	1.26801	0.634004	−	0.773330i	$-0.281410\pi$
$733$	34.3300	1.26801	0.634004	+	0.773330i	$0.281410\pi$
$734$	0	0
$735$	1.12875	0.0416345
$736$	0	0
$737$	−38.5887	−1.42143
$738$	0	0
$739$	0.0946726	0.00348259	0.00174129	−	0.999998i	$-0.499446\pi$
$739$	0.0946726	0.00348259	0.00174129	+	0.999998i	$0.499446\pi$
$740$	0	0
$741$	−3.48501	−0.128025
$742$	0	0
$743$	−13.2755	−0.487032	−0.243516	−	0.969897i	$-0.578301\pi$
$743$	−13.2755	−0.487032	−0.243516	+	0.969897i	$0.578301\pi$
$744$	0	0
$745$	−17.6518	−0.646713
$746$	0	0
$747$	5.53434	0.202491
$748$	0	0
$749$	−12.7672	−0.466502
$750$	0	0
$751$	−15.4187	−0.562637	−0.281318	−	0.959615i	$-0.590772\pi$
$751$	−15.4187	−0.562637	−0.281318	+	0.959615i	$0.590772\pi$
$752$	0	0
$753$	19.8743	0.724261
$754$	0	0
$755$	−17.2437	−0.627562
$756$	0	0
$757$	−9.96685	−0.362251	−0.181126	−	0.983460i	$-0.557974\pi$
$757$	−9.96685	−0.362251	−0.181126	+	0.983460i	$0.557974\pi$
$758$	0	0
$759$	−7.55594	−0.274263
$760$	0	0
$761$	−23.4837	−0.851283	−0.425642	−	0.904892i	$-0.639952\pi$
$761$	−23.4837	−0.851283	−0.425642	+	0.904892i	$0.639952\pi$
$762$	0	0
$763$	−7.27685	−0.263440
$764$	0	0
$765$	−0.468436	−0.0169363
$766$	0	0
$767$	1.82499	0.0658966
$768$	0	0
$769$	22.9369	0.827125	0.413562	−	0.910476i	$-0.364284\pi$
$769$	22.9369	0.827125	0.413562	+	0.910476i	$0.364284\pi$
$770$	0	0
$771$	14.7550	0.531388
$772$	0	0
$773$	−35.0956	−1.26230	−0.631150	−	0.775660i	$-0.717417\pi$
$773$	−35.0956	−1.26230	−0.631150	+	0.775660i	$0.717417\pi$
$774$	0	0
$775$	−21.9678	−0.789106
$776$	0	0
$777$	−5.89592	−0.211515
$778$	0	0
$779$	−3.16999	−0.113577
$780$	0	0
$781$	48.6578	1.74111
$782$	0	0
$783$	6.72593	0.240365
$784$	0	0
$785$	−24.5869	−0.877542
$786$	0	0
$787$	−17.9480	−0.639778	−0.319889	−	0.947455i	$-0.603646\pi$
$787$	−17.9480	−0.639778	−0.319889	+	0.947455i	$0.603646\pi$
$788$	0	0
$789$	15.7600	0.561071
$790$	0	0
$791$	−3.34500	−0.118934
$792$	0	0
$793$	−0.821814	−0.0291835
$794$	0	0
$795$	8.62185	0.305785
$796$	0	0
$797$	−34.4199	−1.21922	−0.609608	−	0.792703i	$-0.708673\pi$
$797$	−34.4199	−1.21922	−0.609608	+	0.792703i	$0.708673\pi$
$798$	0	0
$799$	−4.67996	−0.165565
$800$	0	0
$801$	−4.92999	−0.174193
$802$	0	0
$803$	15.9462	0.562729
$804$	0	0
$805$	−1.78906	−0.0630560
$806$	0	0
$807$	21.8331	0.768561
$808$	0	0
$809$	−1.89314	−0.0665592	−0.0332796	−	0.999446i	$-0.510595\pi$
$809$	−1.89314	−0.0665592	−0.0332796	+	0.999446i	$0.510595\pi$
$810$	0	0
$811$	10.3400	0.363086	0.181543	−	0.983383i	$-0.441891\pi$
$811$	10.3400	0.363086	0.181543	+	0.983383i	$0.441891\pi$
$812$	0	0
$813$	−18.8328	−0.660495
$814$	0	0
$815$	−5.20314	−0.182258
$816$	0	0
$817$	72.1042	2.52261
$818$	0	0
$819$	−0.456247	−0.0159426
$820$	0	0
$821$	−2.40029	−0.0837706	−0.0418853	−	0.999122i	$-0.513336\pi$
$821$	−2.40029	−0.0837706	−0.0418853	+	0.999122i	$0.513336\pi$
$822$	0	0
$823$	25.8655	0.901616	0.450808	−	0.892621i	$-0.351136\pi$
$823$	25.8655	0.901616	0.450808	+	0.892621i	$0.351136\pi$
$824$	0	0
$825$	17.7622	0.618399
$826$	0	0
$827$	17.0928	0.594375	0.297188	−	0.954819i	$-0.403951\pi$
$827$	17.0928	0.594375	0.297188	+	0.954819i	$0.403951\pi$
$828$	0	0
$829$	−41.8893	−1.45488	−0.727438	−	0.686174i	$-0.759289\pi$
$829$	−41.8893	−1.45488	−0.727438	+	0.686174i	$0.759289\pi$
$830$	0	0
$831$	−24.4684	−0.848801
$832$	0	0
$833$	0.415006	0.0143791
$834$	0	0
$835$	25.9181	0.896933
$836$	0	0
$837$	−5.89592	−0.203793
$838$	0	0
$839$	−24.5500	−0.847560	−0.423780	−	0.905765i	$-0.639297\pi$
$839$	−24.5500	−0.847560	−0.423780	+	0.905765i	$0.639297\pi$
$840$	0	0
$841$	16.2381	0.559936
$842$	0	0
$843$	14.5150	0.499923
$844$	0	0
$845$	14.4387	0.496708
$846$	0	0
$847$	11.7259	0.402908
$848$	0	0
$849$	16.5509	0.568026
$850$	0	0
$851$	9.34500	0.320342
$852$	0	0
$853$	38.7193	1.32572	0.662862	−	0.748742i	$-0.269342\pi$
$853$	38.7193	1.32572	0.662862	+	0.748742i	$0.269342\pi$
$854$	0	0
$855$	8.62185	0.294861
$856$	0	0
$857$	45.2074	1.54425	0.772127	−	0.635468i	$-0.219193\pi$
$857$	45.2074	1.54425	0.772127	+	0.635468i	$0.219193\pi$
$858$	0	0
$859$	57.1890	1.95126	0.975631	−	0.219418i	$-0.0704159\pi$
$859$	57.1890	1.95126	0.975631	+	0.219418i	$0.0704159\pi$
$860$	0	0
$861$	−0.415006	−0.0141434
$862$	0	0
$863$	45.1106	1.53558	0.767791	−	0.640701i	$-0.221356\pi$
$863$	45.1106	1.53558	0.767791	+	0.640701i	$0.221356\pi$
$864$	0	0
$865$	0.244094	0.00829944
$866$	0	0
$867$	16.8278	0.571501
$868$	0	0
$869$	23.0255	0.781086
$870$	0	0
$871$	3.69317	0.125138
$872$	0	0
$873$	16.4468	0.556641
$874$	0	0
$875$	9.84937	0.332969
$876$	0	0
$877$	−17.6600	−0.596337	−0.298168	−	0.954513i	$-0.596376\pi$
$877$	−17.6600	−0.596337	−0.298168	+	0.954513i	$0.596376\pi$
$878$	0	0
$879$	9.44377	0.318530
$880$	0	0
$881$	6.78998	0.228760	0.114380	−	0.993437i	$-0.463512\pi$
$881$	6.78998	0.228760	0.114380	+	0.993437i	$0.463512\pi$
$882$	0	0
$883$	−28.6015	−0.962516	−0.481258	−	0.876579i	$-0.659820\pi$
$883$	−28.6015	−0.962516	−0.481258	+	0.876579i	$0.659820\pi$
$884$	0	0
$885$	−4.51499	−0.151770
$886$	0	0
$887$	23.5919	0.792138	0.396069	−	0.918221i	$-0.370374\pi$
$887$	23.5919	0.792138	0.396069	+	0.918221i	$0.370374\pi$
$888$	0	0
$889$	10.4468	0.350376
$890$	0	0
$891$	4.76717	0.159706
$892$	0	0
$893$	86.1374	2.88248
$894$	0	0
$895$	25.5284	0.853319
$896$	0	0
$897$	0.723150	0.0241453
$898$	0	0
$899$	−39.6555	−1.32259
$900$	0	0
$901$	3.16999	0.105608
$902$	0	0
$903$	9.43967	0.314133
$904$	0	0
$905$	8.72871	0.290152
$906$	0	0
$907$	2.02096	0.0671050	0.0335525	−	0.999437i	$-0.489318\pi$
$907$	2.02096	0.0671050	0.0335525	+	0.999437i	$0.489318\pi$
$908$	0	0
$909$	−16.4056	−0.544139
$910$	0	0
$911$	−52.5524	−1.74114	−0.870569	−	0.492046i	$-0.836249\pi$
$911$	−52.5524	−1.74114	−0.870569	+	0.492046i	$0.836249\pi$
$912$	0	0
$913$	26.3832	0.873156
$914$	0	0
$915$	2.03315	0.0672139
$916$	0	0
$917$	−6.25749	−0.206641
$918$	0	0
$919$	21.5237	0.710002	0.355001	−	0.934866i	$-0.384480\pi$
$919$	21.5237	0.710002	0.355001	+	0.934866i	$0.384480\pi$
$920$	0	0
$921$	0.361575	0.0119143
$922$	0	0
$923$	−4.65685	−0.153282
$924$	0	0
$925$	−21.9678	−0.722296
$926$	0	0
$927$	−17.1728	−0.564028
$928$	0	0
$929$	−15.7500	−0.516739	−0.258370	−	0.966046i	$-0.583185\pi$
$929$	−15.7500	−0.516739	−0.258370	+	0.966046i	$0.583185\pi$
$930$	0	0
$931$	−7.63843	−0.250339
$932$	0	0
$933$	11.2769	0.369188
$934$	0	0
$935$	−2.23312	−0.0730307
$936$	0	0
$937$	−17.0687	−0.557610	−0.278805	−	0.960348i	$-0.589938\pi$
$937$	−17.0687	−0.557610	−0.278805	+	0.960348i	$0.589938\pi$
$938$	0	0
$939$	25.5837	0.834892
$940$	0	0
$941$	−26.5456	−0.865362	−0.432681	−	0.901547i	$-0.642432\pi$
$941$	−26.5456	−0.865362	−0.432681	+	0.901547i	$0.642432\pi$
$942$	0	0
$943$	0.657782	0.0214203
$944$	0	0
$945$	1.12875	0.0367181
$946$	0	0
$947$	20.3265	0.660522	0.330261	−	0.943890i	$-0.392863\pi$
$947$	20.3265	0.660522	0.330261	+	0.943890i	$0.392863\pi$
$948$	0	0
$949$	−1.52615	−0.0495408
$950$	0	0
$951$	−0.361575	−0.0117249
$952$	0	0
$953$	38.6855	1.25315	0.626573	−	0.779362i	$-0.284457\pi$
$953$	38.6855	1.25315	0.626573	+	0.779362i	$0.284457\pi$
$954$	0	0
$955$	11.5209	0.372809
$956$	0	0
$957$	32.0637	1.03647
$958$	0	0
$959$	12.9618	0.418559
$960$	0	0
$961$	3.76186	0.121350
$962$	0	0
$963$	−12.7672	−0.411416
$964$	0	0
$965$	−6.73192	−0.216708
$966$	0	0
$967$	−26.6606	−0.857346	−0.428673	−	0.903460i	$-0.641019\pi$
$967$	−26.6606	−0.857346	−0.428673	+	0.903460i	$0.641019\pi$
$968$	0	0
$969$	3.16999	0.101835
$970$	0	0
$971$	−3.27685	−0.105159	−0.0525796	−	0.998617i	$-0.516744\pi$
$971$	−3.27685	−0.105159	−0.0525796	+	0.998617i	$0.516744\pi$
$972$	0	0
$973$	−18.3644	−0.588734
$974$	0	0
$975$	−1.69995	−0.0544418
$976$	0	0
$977$	12.0387	0.385153	0.192576	−	0.981282i	$-0.438316\pi$
$977$	12.0387	0.385153	0.192576	+	0.981282i	$0.438316\pi$
$978$	0	0
$979$	−23.5021	−0.751131
$980$	0	0
$981$	−7.27685	−0.232332
$982$	0	0
$983$	4.42188	0.141036	0.0705181	−	0.997510i	$-0.477535\pi$
$983$	4.42188	0.141036	0.0705181	+	0.997510i	$0.477535\pi$
$984$	0	0
$985$	21.2299	0.676442
$986$	0	0
$987$	11.2769	0.358946
$988$	0	0
$989$	−14.9618	−0.475758
$990$	0	0
$991$	9.10184	0.289129	0.144565	−	0.989495i	$-0.453822\pi$
$991$	9.10184	0.289129	0.144565	+	0.989495i	$0.453822\pi$
$992$	0	0
$993$	−7.80403	−0.247653
$994$	0	0
$995$	−3.69873	−0.117258
$996$	0	0
$997$	−42.8556	−1.35725	−0.678625	−	0.734485i	$-0.737424\pi$
$997$	−42.8556	−1.35725	−0.678625	+	0.734485i	$0.737424\pi$
$998$	0	0
$999$	−5.89592	−0.186539

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5376.2.a.bl.1.2		4
4.3	odd	2		5376.2.a.bp.1.2		4
8.3	odd	2		5376.2.a.bm.1.3		4
8.5	even	2		5376.2.a.bq.1.3		4
16.3	odd	4		168.2.c.b.85.1	✓	8
16.5	even	4		672.2.c.b.337.6		8
16.11	odd	4		168.2.c.b.85.2	yes	8
16.13	even	4		672.2.c.b.337.3		8
48.5	odd	4		2016.2.c.e.1009.6		8
48.11	even	4		504.2.c.f.253.7		8
48.29	odd	4		2016.2.c.e.1009.3		8
48.35	even	4		504.2.c.f.253.8		8
112.13	odd	4		4704.2.c.c.2353.6		8
112.27	even	4		1176.2.c.c.589.2		8
112.69	odd	4		4704.2.c.c.2353.3		8
112.83	even	4		1176.2.c.c.589.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.2.c.b.85.1	✓	8	16.3	odd	4
168.2.c.b.85.2	yes	8	16.11	odd	4
504.2.c.f.253.7		8	48.11	even	4
504.2.c.f.253.8		8	48.35	even	4
672.2.c.b.337.3		8	16.13	even	4
672.2.c.b.337.6		8	16.5	even	4
1176.2.c.c.589.1		8	112.83	even	4
1176.2.c.c.589.2		8	112.27	even	4
2016.2.c.e.1009.3		8	48.29	odd	4
2016.2.c.e.1009.6		8	48.5	odd	4
4704.2.c.c.2353.3		8	112.69	odd	4
4704.2.c.c.2353.6		8	112.13	odd	4
5376.2.a.bl.1.2		4	1.1	even	1	trivial
5376.2.a.bm.1.3		4	8.3	odd	2
5376.2.a.bp.1.2		4	4.3	odd	2
5376.2.a.bq.1.3		4	8.5	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-1.00000 q^{3} -1.12875 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{3} -1.12875 q^{5} +1.00000 q^{7} +1.00000 q^{9} +4.76717 q^{11} -0.456247 q^{13} +1.12875 q^{15} +0.415006 q^{17} -7.63843 q^{19} -1.00000 q^{21} +1.58499 q^{23} -3.72593 q^{25} -1.00000 q^{27} -6.72593 q^{29} +5.89592 q^{31} -4.76717 q^{33} -1.12875 q^{35} +5.89592 q^{37} +0.456247 q^{39} +0.415006 q^{41} -9.43967 q^{43} -1.12875 q^{45} -11.2769 q^{47} +1.00000 q^{49} -0.415006 q^{51} +7.63843 q^{53} -5.38093 q^{55} +7.63843 q^{57} -4.00000 q^{59} +1.80125 q^{61} +1.00000 q^{63} +0.514988 q^{65} -8.09467 q^{67} -1.58499 q^{69} +10.2068 q^{71} +3.34500 q^{73} +3.72593 q^{75} +4.76717 q^{77} +4.83001 q^{79} +1.00000 q^{81} +5.53434 q^{83} -0.468436 q^{85} +6.72593 q^{87} -4.92999 q^{89} -0.456247 q^{91} -5.89592 q^{93} +8.62185 q^{95} +16.4468 q^{97} +4.76717 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{3} - 2 q^{5} + 4 q^{7} + 4 q^{9}+O(q^{10}) \) 4 * q - 4 * q^3 - 2 * q^5 + 4 * q^7 + 4 * q^9	\( 4 q - 4 q^{3} - 2 q^{5} + 4 q^{7} + 4 q^{9} - 6 q^{11} - 4 q^{13} + 2 q^{15} + 2 q^{17} - 8 q^{19} - 4 q^{21} + 6 q^{23} + 12 q^{25} - 4 q^{27} - 4 q^{31} + 6 q^{33} - 2 q^{35} - 4 q^{37} + 4 q^{39} + 2 q^{41} - 8 q^{43} - 2 q^{45} + 4 q^{49} - 2 q^{51} + 8 q^{53} - 4 q^{55} + 8 q^{57} - 16 q^{59} + 4 q^{63} - 8 q^{65} - 12 q^{67} - 6 q^{69} - 14 q^{71} + 4 q^{73} - 12 q^{75} - 6 q^{77} + 20 q^{79} + 4 q^{81} - 28 q^{83} + 20 q^{85} - 10 q^{89} - 4 q^{91} + 4 q^{93} - 20 q^{95} + 20 q^{97} - 6 q^{99}+O(q^{100}) \) 4 * q - 4 * q^3 - 2 * q^5 + 4 * q^7 + 4 * q^9 - 6 * q^11 - 4 * q^13 + 2 * q^15 + 2 * q^17 - 8 * q^19 - 4 * q^21 + 6 * q^23 + 12 * q^25 - 4 * q^27 - 4 * q^31 + 6 * q^33 - 2 * q^35 - 4 * q^37 + 4 * q^39 + 2 * q^41 - 8 * q^43 - 2 * q^45 + 4 * q^49 - 2 * q^51 + 8 * q^53 - 4 * q^55 + 8 * q^57 - 16 * q^59 + 4 * q^63 - 8 * q^65 - 12 * q^67 - 6 * q^69 - 14 * q^71 + 4 * q^73 - 12 * q^75 - 6 * q^77 + 20 * q^79 + 4 * q^81 - 28 * q^83 + 20 * q^85 - 10 * q^89 - 4 * q^91 + 4 * q^93 - 20 * q^95 + 20 * q^97 - 6 * q^99

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